Computational Modelling of Concrete Structures

COMPUTATIONAL MODELLING OF CONCRETE STRUCTURES

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PROCEEDINGS OF EURO-C 2010, ROHRMOOS/SCHLADMING, AUSTRIA, 15–18 MARCH 2010

Computational Modelling of Concrete Structures Editors

Nenad Bi´cani´c Department of Civil Engineering, University of Glasgow, Scotland, UK

René de Borst Department of Mechanical Engineering, Eindhoven University of Technology, The Netherlands

Herbert Mang Institute for Mechanics of Materials and Structures, Vienna University of Technology, Vienna, Austria

Günther Meschke Institute for Structural Mechanics, Department of Civil Engineering, Ruhr University, Bochum, Germany

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CRC Press/Balkema is an imprint of the Taylor & Francis Group, an informa business © 2010 Taylor & Francis Group, London, UK Typeset by Vikatan Publishing Solutions (P) Ltd., Chennai, India Printed and bound in Great Britain by Antony Rowe (a CPI Group Company), Chippenham, Wiltshire All rights reserved. No part of this publication or the information contained herein may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, by photocopying, recording or otherwise, without written prior permission from the publisher. Although all care is taken to ensure integrity and the quality of this publication and the information herein, no responsibility is assumed by the publishers nor the author for any damage to the property or persons as a result of operation or use of this publication and/or the information contained herein. Published by: CRC Press/Balkema P.O. Box 447, 2300 AK Leiden, The Netherlands e-mail: [email protected] www.crcpress.com - www.taylorandfrancis.co.uk - www.balkema.nl ISBN: 978-0-415-58479-1 (Hbk) ISBN: 978-0-203-84833-3 (eBook)

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Table of contents

Foreword

XI

Keynote lectures Modeling of concrete creep and hygrothermal deformations, and computation of their structural effects Z.P. Bažant & Q. Yu

3

Alternate approaches to simulating the performance of ductile fiber-reinforced cement-based materials in structural applications S.L. Billington

15

Recent developments on computational modeling of material failure in plain and reinforced concrete structures A.E. Huespe, J. Oliver, G. Díaz & P.J. Sánchez

31

Concrete under various loadings, way to model in a same framework: Damage, fracture and compaction J. Mazars, F. Dufour, C. Giry, A. Rouquand & C. Pontiroli

45

Upscaling quasi-brittle strength of cement-based materials: A continuum micromechanics approach B. Pichler & C. Hellmich

59

C-Crete: From atoms to concrete structures F.-J. Ulm, R.J.-M. Pellenq & M. Vandamme

69

Constitutive and multi-scale modelling Pull-out behaviour of a glass multi-filaments yarn embedded in a cementitious matrix H. Aljewifi, B. Fiorio & J.L. Gallias

77

How to enforce non-negative energy dissipation in microplane and other constitutive models for softening damage, plasticity and friction Z.P. Bažant, J.-Y. Wu, F.C. Caner & G. Cusatis

87

A multiscale approach for nonlinear hysteretic damage behaviour of quasi-brittle disordered materials J. Carmeliet, S. Mertens & P. Moonen

93

Modeling of reinforced cementitious composites using the microplane damage model in combination with the stochastic cracking theory R. Chudoba, A. Scholzen, R. Rypl & J. Hegger

101

A statistical model for reinforced concrete bond prediction Z. Dahou, Z.M. Sbartaï, A. Castel & F. Ghomari

111

Introduction of an internal time in nonlocal integral theories R. Desmorat & F. Gatuingt

121

Elastoplastic constitutive model for concretes of arbitrary strength properties G. Etse & P. Folino

129

Properties of concrete: A two step homogenization approach E. Gal & R. Kryvoruk

137

V

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Stress state influence on nonlocal interactions in damage modelling C. Giry, F. Dufour, J. Mazars & P. Kotronis A model for the uniaxial tensile behaviour of Textile Reinforced Concrete with a stochastic description of the concrete material properties J. Hartig & U. Häußler-Combe Multi-axial modeling of plain concrete structures based on an anisotropic damage formulation M. Kitzig & U. Häußler-Combe Determination of cement paste mechanical properties: Comparison between micromechanical and ultrasound results S. Maalej, Z. Lafhaj & M. Bouassida The simulation of microcracking and micro-contact in a constitutive model for concrete I.C. Mihai & A.D. Jefferson Simulations of dynamic failure in plain and reinforced concrete with regularized plasticity and damage models J. Pamin, A. Winnicki & A. Wosatko

145

153 163

173 179

187

Micromechanical approach to viscoelastic properties of fiber reinforced concrete V.F. Pasa Dutra, S. Maghous & A.C. Filho

197

Boundary and evolving boundary effects in non local damage models G. Pijaudier-Cabot & F. Dufour

207

Homogenization-based model for reinforced concrete E. Rumanus & G. Meschke

217

Development of constitutive model of shear stress transfer on concrete crack surface considering shear stress softening Y. Takase, T. Ikeda & T. Wada

225

Microplane approach for modeling of concrete under low confinement N.V. Tue & J. Li

233

Meso- and macroscopic models for fiber-reinforced concrete S.M. Vrech, G. Etse, G. Meschke, A. Caggiano & E. Martinelli

241

Gradient damage model with volumetric-deviatoric split A. Wosatko & J. Pamin

251

Advances in numerical methods Continuous and discontinuous modeling of cracks in concrete elements J. Bobi´nski & J. Tejchman

263

Topological search of the crack path from a damage-type mechanical computation M. Bottoni & F. Dufour

271

On the uniqueness of numerical solutions of shear failure of deep concrete beams: Comparison of smeared and discrete crack approaches J. Cervenka & V. Cervenka

281

Lattice Discrete Particle Model for Fiber reinforced concrete (LDPM-F) with application to the numerical simulation of armoring systems G. Cusatis, E.A. Schauffert, D. Pelessone, J.L. O’Daniel, P. Marangi, M. Stacchini & M. Savoia

291

Nonlocal damage based failure models, extraction of crack opening and transition to fracture F. Dufour, G. Pijaudier-Cabot & G. Legrain Convergence aspects of the eXtended Finite Element Method applied to linear elastic fracture mechanics W. Fleming & D. Kuhl

VI

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301

309

Applicability of XFEM for the representation of crack bridge state in planar composite elements J. Jeˇrábek, R. Chudoba & J. Hegger

319

Localization properties of damage models M. Jirásek & M. Horák

327

Numerical multiscale solution strategy for fracturing of concrete Ł. Kaczmarczyk, C.J. Pearce & N. Bi´cani´c

337

A 3D lattice model to describe fracture process in fibrous concrete J. Kozicki & J. Tejchman

347

Limit analysis of 3D reinforced concrete frames K.P. Larsen, P.N. Poulsen & L.O. Nielsen

355

The role of domain decomposition techniques for the study of heterogeneous quasi-brittle materials O. Lloberas Valls, D.J. Rixen, A. Simone & L.J. Sluys Modelling cohesive crack growth applying XFEM with crack geometry parameters J.F. Mougaard, P.N. Poulsen & L.O. Nielsen

363 373

Strong discontinuities, mixed finite element formulations and localized strain injection, in fracture modeling of quasi-brittle materials J. Oliver, I.F. Dias & A.E. Huespe

381

Model for the analysis of structural concrete elements under plane stress conditions: Finite element implementation M. Pimentel & J. Figueiras

391

A partition of unity finite element method for fibre reinforced concrete F.K.F. Radtke, A. Simone & L.J. Sluys

401

A discrete cracking model for sequentially linear analysis A.V. van de Graaf, M.A.N. Hendriks & J.G. Rots

409

Relations between structure size, mesh density, and elemental strength of lattice models M. Voˇrechovský & J. Eliáš

419

Time-dependent and multi physics phenomena Prediction of the permeability of damaged concrete using a combined lattice beam-crack network approach M. Abreu, J. Carmeliet & J.V. Lemos

431

Modelling the CaO hydration in expansive concrete B. Chiaia, A.P. Fantilli, G. Ferro & G. Ventura

441

A two-scale approach for fluid flow in fracturing porous media R. de Borst, J. Réthoré & M.-A. Abellan

451

A probabilistic approach for modelling long-term behaviour and creep failure of a concrete structure subjected to calcium leaching T. de Larrard, F. Benboudjema, J.B. Colliat, J.M. Torrenti & F. Deleruyelle

461

A coupled transport-crystallization FE model for porous media H. Derluyn, R.M. Espinosa-Marzal, P. Moonen & J. Carmeliet

471

A numerical model for early age concrete behavior G. Di Luzio, L. Cedolin & G. Cusatis

481

Hygro-mechanical model for concrete specimens at the meso-level: Application to drying shrinkage A.E. Idiart, C.M. López & I. Carol

VII

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487

Comparison of approaches for simulating moisture content changes in concrete A.D. Jefferson & P. Lyons Modeling of Chloride and CO2 transport in intact and cracked concrete in the context of corrosion predictions of RC structures M. Kemper, J.J. Timothy, J. Kruschwitz & G. Meschke

497

503

Thermal activation of basic creep for HPC in the range 20°C–80°C W. Ladaoui, T. Vidal, A. Sellier & X. Bourbon

513

Modelling of the THM behaviour of concrete at the macroscopic and mesoscopic scale T.T.H. Le, H. Boussa & F. Meftah

519

Application of enhanced elasto-plastic damage models to concrete under quasi-static and dynamic cyclic loading I. Marzec & J. Tejchman

529

Propagation of cracks and damage in non ageing linear viscoelastic media Nguyen Sy Tuan, L. Dormieux, Y. Le Pape & J. Sanahuja

537

Micromechanical modelling of concrete V.P. Nguyen, M. Stroeven & L.J. Sluys

547

3D finite element analysis of concrete under impact load J. Ožbolt, V. Travaš & I. Kožar

553

Investigation into the form of the load-induced thermal strain model C.J. Robson, C.T. Davie & P.D. Gosling

563

Development of service life model CHLODIF++ I. Stipanovi´c Oslakovi´c, D. Bjegovi´c, D. Mikuli´c & V. Krsti´c

573

Modelling of concrete structures Some remarks concerning the shear failure in prestressed RC beams B. Belletti & C. Damoni

581

Verification of experimental tests on roller bearings by means of numerical simulations S. Blail & J. Kollegger

591

Numerical crack modelling of tied concrete columns C. Bosco, S. Invernizzi & G. Gagliardi

595

Numerical study of a massive reinforced concrete structure at early age: Prediction of the cracking risk of a massive wall L. Buffo-Lacarrière & A. Sellier

603

Numerical modelling of failure mechanisms and redistribution effects in steel fibre reinforced concrete slabs L. Gödde & P. Mark

611

Transverse rebar affecting crack behaviors of R.C. members subjected to bending D. Han, M. Keuser & L. Ruediger

623

A numerical method for RC-boxgirders under combined shear bending and torsion U. Häußler-Combe

629

Computation of optimal concrete reinforcement in three dimensions P.C.J. Hoogenboom & A. de Boer

639

Sequentially linear modelling of local snap-back in extremely brittle structures S. Invernizzi, D. Trovato, M.A.N. Hendriks & A.V. van de Graaf

647

Simulation of masonry beams retrofitted with engineered cementitious composites M.A. Kyriakides, M.A.N. Hendriks & S.L. Billington

655

VIII

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Setting and loading process simulation of Push-in anchor for concrete Y.-J. Li, N. Chilakunda & B. Winkler

665

Nonlinear FE modelling of shear behaviour in RC beam retrofitted with CFRP Y.T. Obaidat, O. Dahlblom & S. Heyden

671

Material optimization for textile reinforced concrete applying a damage formulation E. Ramm & J. Kato

679

A multifibre approach to describe the ultimate behaviour of corroded reinforced concrete structures B. Richard, F. Ragueneau & Ch. Crémona

689

Simulation of shear load behavior of fifty year old post-tensioned concrete bridge girders A. Schweighofer, M. Vill, H. Hengl & J. Kollegger

699

Multi-scale modelling of concrete beams subjected to three-point bending Ł. Skar˙zy´nski & J. Tejchman

703

Experimental and numerical analysis of reinforced concrete corbels strengthened with fiber reinforced polymers R.A. Souza

711

Computational modeling of the behaviour up to failure of innovative prebended steel-VHPC beams for railway bridges S. Staquet & F. Toutlemonde

719

Numerical investigations of size effects in notched and un-notched concrete beams under bending E. Syroka, J. Bobi´nski, J. Górski & J. Tejchman

729

FE modeling and fiber modeling for RC column failing in shear after flexural yielding K. Tajima, N. Shirai, E. Ozaki & K. Imai

737

Numerical simulation and experimental testing of a new bridge strengthening method W. Traeger, J. Berger & J. Kollegger

749

Failure studies on masonry infill walls: Experimental and computational observations K. Willam, B. Blackard & C. Citto

757

Numerical study on mixed-mode fracture in LRC beams R.C. Yu, L. Saucedo & G. Ruiz

767

Hazards, risk and safety (fire, blast, seismicity) Numerical investigation of damage and spalling in concrete exposed to fire C.T. Davie & H.L. Zhang

775

Numerical simulation of slender structures with integrated dampers P. Egger & J. Kollegger

785

Textile reinforced concrete sandwich panels: Bending tests and numerical analyses J. Finzel & U. Häußler-Combe

789

Structural behavior of tunnels under fire loading including spalling and load induced thermal strains T. Ring, M. Zeiml & R. Lackner

797

Numerical assessment of the failure mode of RC columns subjected to fire S. Sere˛ga

805

Ultimate load analysis of a reactor safety containment structure B. Valentini, H. Lehar & G. Hofstetter

815

Author index

821

IX

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Foreword

The long standing EURO-C conference series returned again this spring to the mountains of Austria, with the conference held in Schladming, Salzburgerland from the 15th to 18th March 2010. These proceedings comprise over 90 invited and contributed papers presented at the conference, which keeps its main focus and its traditional format in line with earlier conferences in the series (Innsbruck 1994, Badgastein 1998, St Johann im Pongau 2003, Mayrhofen 2006). The EURO-C series has emerged as a joint undertaking following an increase of research in computational mechanics of concrete generated by the ICC 1984 conference held in Split, Croatia, the SCI-C 1990 conference in Zell am See, Austria and the two Concrete Mechanics Colloquia, held in Delft, The Netherlands in 1981 and 1987. Once again we express our sincere gratitude to the members of the International Advisory Panel (Zdenek Bažant, Sarah Billington, Alberto Carpinteri, Tony Jefferson, Milan Jirásek, Karen Lundgren, Koichi Maekawa, Jacky Mazars, Javier Oliver, Jerzy Pamin, Gilles Pijaudier-Cabot, Marco di Prisco, Ekkehard Ramm, Jan Rots, Tadahiko Tanabe, Franz-Josef Ulm and Kaspar Willam) for their invaluable support and considerable effort in reviewing around 130 abstracts. EURO-C conference series keeps its tradition of a very rigorous reviewing process, thereby ensuring the high quality of presented papers. In order to reflect the current advances in computational modelling of concrete and concrete structures, conference papers are grouped into five distinct, yet strongly related, sections – (A) Constitutive and Multiscale Modelling of Concrete (B) Advances in Computational Modelling, (C) Time Dependent and Multiphysics Problems (D) Modelling of Concrete Structures and (E) Hazard, Risk and Safety. We hope that the EURO-C 2010 Conference proceedings will continue to provide a valuable referential source for an up-to-date information and scientific debate on the research advances in computational modelling of concrete and concrete structures, as well as on their application and relevance to the structural engineering practice. Nenad Bi´cani´c René de Borst Herbert Mang Günther Meschke Glasgow, Eindhoven, Vienna, Bochum January 2010

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Keynote lectures

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Modeling of concrete creep and hygrothermal deformations, and computation of their structural effects Zdenˇek P. Bažant & Qiang Yu Northwestern University, Evanston, IL, USA

ABSTRACT: This study discusses a physically based formulation of material model for concrete creep and shrinkage, and presents an effective computational approach for structural analysis of creep and shrinkage effects. As an instructive case study, excessive deflections of a prestressed box girder bridge of world-record span, which was built in Palau in 1977 and collapsed after remedial prestressing in 1996, is investigated. A new version of the step-by-step computational algorithm, based on the continuous retardation spectra of compliance curves for different ages at loading, is implemented as a driver program for repeated use of ABAQUS for three-dimensional analysis. The excessive creep deflections are studied by finite element analysis, and their causes are identified. They reveal a need to improve the current standard recommendations of engineering societies. A limited improvement can be achieved by statistical analysis of the existing database from worldwide testing. However, a major improvement will require adopting a model based on the theory of physical processes of creep and shrinkage in the nano-porous structure of cement gel. A model based to a large extent on such a theory is model B3. Its basic features are reviewed. 1

2

INTRODUCTION

Portland cement concrete is a rather unusual porous material, characterized by both capillary porosity and sub-capillary nano-porosity. The physical and chemical processes in the nano-pores (or gel pores) are believed to be the cause of complex creep properties (Bažant 2001), very different from those of other viscoelastic materials. The nano-porosity of cement gel has intriguing consequences whose mathematical formulation material model B3 will be reviewed. This model rests on the microprestress-solidification theory (Bažant & Prasannan 1989a,b, Bažant et al. 1997, Bažant et al. 1997), which is a theory that has achieved a unified description of the known creep properties of concrete, including the long-term aging after the hydration process has terminated, the Pickett effect, and the transitional thermal creep. The purpose of this paper is to summarize (based on a report by Bažant et al. 2007) the lessons from excessive long-time deflections of a record-setting segmentally erected prestressed concrete box girder, discuss the importance of selecting the correct material model for creep and shrinkage, critically comment on the statistical validation of the material model, and review the theoretical physical basis of a realistic material model.

EXCESSIVE DEFLECTIONS AND COLLAPSE OF KB BRIDGE IN PALAU

The Koror-Babeldaob (KB) Bridge (Yee 1979, Mc-Donald et al. 2003, Burgoyne & Scantlebury 2006) was built in 1977 in the Republic of Palau, situated in the tropical western Pacific. It connected the islands of Koror and Babeldaob, the former containing the airport and the latter the country capital. As shown in Figure 1a, the main span consisted of two symmetric simultaneously erected cantilevers connected at mid-span by a horizontally sliding hinge. Each cantilever consisted of 25 cast-in-place segments of depths varying from 14.17 m (46.5 ft.) to 3.66 m (12 ft.). The segmental erection of the box girder took about 6–7 months. At the time of completion, the main span of 241 m (790 ft.) set the world record for a prestressed concrete box girder. In design, the long-term deflection of the bridge was expected to remain in the tolerable range with the final mid-span sag ranging from 0.46 to 0.58 m (18.2 to 23 in.). In the early years, the deflections were benign but then accelerated unexpectedly. After 18 years, the deflection increase measured since the installation of the mid-span hinge that joined the opposite segmentally erected cantilevers reached 1.39 m (54.6 in.) and kept growing (Fig. 1b). If compared to the design camber, an additional creep deflection of 0.22 m (9 in.)

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analyses, which can be most effectively carried out by a commercial general purpose finite element program. The program ABAQUS was selected. Similar to other general purpose programs, ABAQUS is not designed to handle concrete creep. Therefore, a user subroutines that implements the creep integration in time and calls ABAQUS in each time step has been formulated. Another material subroutine was developed to describe the constitutive model for concrete creep and shrinkage. If all the equations of the constitutive model, converted to a rate form, are put in a proper incremental form, an incremental elastic problem with eigenstrains (or inelastic strains) is obtained for each subsequent time step. The incremental elastic moduli matrices are generally anisotropic and are different for each integration point of each finite element of each time step. So are the eigenstrains, which are non-isotropic. The incremental elastic relations can be obtained from the differential equation of Kelvin chain model according to the exponential algorithm which is unconditionally stable (Jirasék & Bažant 2002). When the non-aging Kelvin chain model is applied to model B3, one and the same relaxation spectrum, determined in advance, can be used for all the time steps (Bažant & Xi 1995). Since the cantilevers are symmetric, only one half of the bridge is modelled and simulated. A computational model with 5036 8-node hexahedral elements for concrete and 6764 bar elements for prestressed and unprestressed steel bars is built in ABAQUS; see Figure 2. 316 tendons (Dywidag alloy bars), which are densely packed in up to 4 layers within the top slab, run through the pier. The jacking force of each tendon is about 0.60 MN (135 kips), and the total jacking force at the pier is about 190 MN (42660 kips). The segmental construction sequence, the moving of the formwork traveller and the process of pre-stressing are all reproduced in the simulation by utilizing the functions provided by ABAQUS. There is no material model in the ABAQUS material library to capture the characteristics of the creep

Figure 1. (a) KB Bridge after construction; (b) sag at the mid-span; (c) collapse of KB Bridge.

was accumulated earlier during segmental erection, and so the total deflection was 1.61 m (63.6 in.). The serviceability of the bridge was severely impaired by the excessive deflection, and so a retrofit was carried out in 1996. The mid-span hinge was removed and external prestress within the box was added. Unfortunately, three months later the bridge collapsed (with 2 fatalities and many injuries); see Figure 1c. After legal litigation in which the deflections and collapse remained unexplained, all the data were sealed in perpetuity. However, on November 6, 2007, the 3rd Structural Engineers’ World Congress in Bangalore endorsed a resolution (proposed by the first writer, with the support of many experts) which called, on the grounds of engineering ethics, for the release of all the technical data necessary for analyzing major structural collapses, including the bridge in Palau. The resolution was circulated to major engineering societies. Two months later, the Attorney General of Palau gave his permission to release the technical data. This made it possible to analyze the deflections. 3

NUMERICAL MODELING OF KB BRIDGE

Since structural creep analysis can be broken down to a series of many incremental elastic finite element

Figure 2. 3-dimensional model of KB Bridge built in ABAQUS.

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compliance functions J (t, t  ) of the creep and shrinkage prediction models used in practice (t = duration of a unit uniaxial stress and t  = age when this sustained stress is applied). Therefore, the appropriate material model has been developed and supplied to ABAQUS. Given the fact that no severe cracking damage was found in the box girders and the observed cracks were sparsely distributed, the concrete of KB Bridge has been assumed as isotropic, characterized by a time-independent Poisson’s ratio ν = 0.21 (JICA 1990). As a compromise between simplicity and accuracy, the creep in concrete is generally considered to follow aging linear viscoelasticity, which implies the principle of superposition in time. The direct application of this principle gives the stress-strain relation in the form of a history integral. However, for the sake of efficiency in large-scale computer analysis, it is advantageous to avoid computation of history integrals. This is made possible by converting the compliance function to an equivalent rate-type form, which has here been based on the Kelvin chain model (Fig. 3). In model B3, which is based on the solidification theory, the creep is defined for a non-aging constituent of growing volume (≈ cement gel) (Bažant & Baweja 1995, Bažant & Baweja 2000), and in that case the conversion of the compliance function to a rate-type creep law is particularly easy. This can be done according to the solidification theory, in which the aging is taken into account by means of volume growth of the solidifying component, and by a gradual increase with age of the flow term viscosity. This makes it possible to use a non-aging compliance function for the solidifying component, for which one can uniquely determine a continuous retardation spectrum by a simple explicit formula (Widders’s formula) for inversion of Laplace transform (Bažant & Xi 1995). The parameters of the Kelvin chain model are in this case constant (i.e., non-aging) and are simply obtained as a discrete representation of the continuous spectrum. For example, if J (t, t  ) = φC(t − t  ) where φ is a non-aging factor, the continuous retardation spectrum can be

expressed as: Aμ = φ(t  )L(τμ ) ln 10(log τμ ) (−kτμ )k (k) C (kτμ ) k→∞ (k − 1)!

L(τμ ) = − lim

(1) (2)

where τμ is μth retardation time, k is a positive integer, and C (k) represents the kth order derivative of function C (K ≤ 3 is usually sufficient). In Figure 3, the non-aging spectrum of the basic creep of model B3 is plotted. Note the spectral value Aμ does not diminish as τμ increases. The reason is that the basic creep according to model B3 is unbounded. For empirical models, such as those of ACI (2008), CEB (1990) (or ‘fip’ 1999), JSCE (1991) and GL (Gardner 2000, Gardner & Lockman 2001), the creep analysis is slightly more complicated since the aging is not separated from the compliance function. Therefore, compliance curves that change with the age at loading must be used. Such a situation was handled in the 1970s by considering the retardation (or relaxation) spectrum to be age dependent, and the age dependence of Kelvin chain elastic moduli was identified from tests data by simultaneous fitting of creep data for various ages at loading. However, the identification problem turned out to be ill-conditioned and the resulting moduli as functions of age non-unique. As a new simpler approach, one exploits the fact that, during a short time step, compliance function may be considered as age-independent. The continuous retardation spectrum can thus be obtained easily from Eqs. (1) and (2) corresponding to the loading age at each time step, but the spectrum is different for each different age. This continuous retardation spectrum is then approximated by a set of discrete spectral values Aμ (μ = 1, 2, 3, . . . ), one set for each time step. These spectral values are then used in the individual time steps of the exponential algorithm based on Kelvin chain. No continuous function for Aμ (t) need be identified and used. A surface describing the compliance function of ACI for different ages is plotted in Figure 3, with the x-axis representing the retardation time, τμ , and the y-axis showing age t  . The disappearance of Aμ at large retardation time τμ is due to the fact that the ACI compliance function is (incorrectly) assumed to be bounded. After obtaining the Kelvin chain moduli, the exponential algorithm, which enables more data and more efficient creep computation, is implemented; see the flowchart in Figure 4. In this algorithm, the stress is assumed to vary linearly in each time step. The initial time steps since the bridge closing at mid-span were 0.1, 1, 10 and 100 days. After that, the time step was kept constant at 100 days up to 19 years.

Figure 3. Kelvin chain model and spectrums of non-aging and aging models.

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Figure 5. Brooks. Figure 4.

4

B3 model compared with 10-year creep data by

Flowchart of algorithm for creep calculation.

COMPARISON OF PREDICTIONS BASED ON DIFFERENT MODELS

For the concrete creep and shrinkage properties, models B3, GL, ACI, CEB (identical to ‘fib’ 1999) and JSCE are considered and predictions compared. Model B3, in contrast to the others, does not necessarily give a unique prediction because, in addition to concrete design strength, it involves several input parameters depending on the composition of concrete mix, on which there exists no information. These parameters can be set to their default values, but they can also be varied over their plausible range in order to ascertain the range of realistic predictions. The findings are as follows:

4.

1. For model B3, reasonable values of input parameters can be found to match all the measured deflections (as well as the few existing creep tests of duration longer than 10 years; Brooks 1984,2005), while for other models the maximum deflections cannot be approached and the recorded shape of laboratory creep curves cannot be reproduced; see Figure 5. 2. The 18-year deflections calculated by threedimensional finite elements according the ACI, CEB, JSCE and GL models (with consideration of differential shrinkage and drying creep) are about 66%, 62%, 46% and 53% less than the observed values. On the other hand, the observed deflections are closely matched by calculation on the basis of model B3, provided that model B3 is calibrated to match the 10-year creep tests of Brooks. If the default parameters are used in model B3, its predicted deflection is 43% less than the observed value; see Fig. 6. 3. The bridge in Palau is unique in that the pre-stress loss in grouted tendons was measured by stress

5.

6.

7.

relief tests. Three sections of each of three tendons were bared, gages installed, tendons cut and the stress determined from the shortening of the tendon. The result was an average prestress loss of 50%, much larger than the prestress loss of 22% assumed in design. Comparisons of the predictions of various models with the measured prestress loss are shown in Figure 7. The deflection is highly sensitive to prestress loss because it represents a small difference of two large numbers (deflections due to self-weight, and to prestress). According to the ACI, CEB, JSCE and GL models, the prestress loss obtained by the same finite element code is, respectively, about 56%, 52%, 42% and 46% smaller than the measured prestress loss, respectively. Contrarily, the prestress loss predicted by model B3 is very close to what measured in the strain relief tests. Model B3 is in agreement with the measurements if three-dimensional finite elements with step-bystep time integration are used to calculate both the deflections and the prestress losses, and if the differences in shrinkage and drying creep properties caused by differences in slab thickness, temperature and damage are taken into account; Figures 6, 7. The shear lag cannot be neglected. What makes its effect large is that it increases the downward deflection due to self-weight much more than the upward deflection due to prestress. The traditional beam-type analysis, in which the creep and shrinkage properties are assumed to be uniform throughout each cross section, gives grossly incorrect predictions for deflections and prestress loss. These differences can be captured closely by model B3, but poorly or not at all by other models. The compliance function must be determined separately for each slab in the cross section, depending on its thickness and humidity exposure (as well as temperature).

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Figure 6. Deflections obtained by B3, ACI, JSCE, CEB and GL models compared with measured. Left: linear scale; right: logarithmical scale.

Figure 7. Prestress loss obtained by B3, ACI, JSCE, CEB and GL models compared with the strain relief tests. Left: linear scale; right: logarithmical scale.

calibration by a larger database, with a rational statistical calibration procedure compensating for the database bias for short times and for small specimen sizes; c) Adjustment of the model to the shape of creep curves observed in individual longtime tests (5 to 30 years), which is obscured when the database is considered only as a whole; and d) rational physical basis of the model. 2. Three-dimensional analysis of deflections and prestress loss, which is much more realistic than the beam bending analysis, especially because it can capture different effects of shear lag on the downward deflection due to self-weight and on the upward deflection due to prestress. 3. Realistic representation of nonuniform shrinkage and drying creep properties in the cross sections, caused by the effect of different wall and slab thicknesses on the shrinkage and drying creep

8. The results agree with a recent statistical study of Bažant & Li (2009a), which shows that model B3 gives smaller errors in comparison to a comprehensive database than do the ACI, CEB and GL models (whose errors can be of the order of 100%). 9. Creep and shrinkage are notorious for their relatively high random scatter. Therefore, the design should be based on the 95% confidence limits. These limits can be calculated by Latin hypercube sampling of the input parameters analysis. The reasons why model B3 performs better than the other models can be summarized as follows: 1. A significantly higher long-time creep in model B3, compared to the ACI, CEB and GL models. There are four causes of that: a) theoretical advances on nano-porous materials during the last three decades, incorporated in model B3; b) Model

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rates and half-times, as well as by the differences in permeability due to temperature differences (Bažant & Kaplan 1996) and cracking (Bažant et al. 1987). 4. A larger number of input parameters in model B3, which include the water-cement ratio and aggregate-cement ratio. If these parameters are not specified, their variation allows exploring a greater range of responses, compared to the ACI, CEB and GL models. These models are inflexible because of missing these input parameters, and thus provide a unique response for a given design strength of concrete.

where q1 , q2 , q3 , q4 and n are primary material parameters to be determined from concrete composition, if known; t0 is the age at the start of drying; ζ = t − t  = stress duration; C(t, t  ) is the compliance of basic creep (independent of moisture loss); and Cd (t, t  , t0 ) is the compliance function of additional creep caused by the drying process (or moisture content change). In model B3, the drying creep and shrinkage strain sh can be written as:   Cd (t, t  , t0 ) = q5 (e−8H (t−t0 ) − e−8H (t −t0 ) ) sh (t, t0 ) = −sh∞ (1 −

Similar conclusions have been obtained by studying the observations in some other large-span prestressed box girders on which sufficient data could be accessed, particularly the Tsukiyono Bridge, Koshirazu Bridge, Konaru Bridge and Urado Bridge in Japan, and the Dˇecín Bridge in Czech Republic. Together with those bridges, the experience with the KB Bridge in Palau reveals that the current design practice based on empirical creep and shrinkage models and beam-type analysis may lead to dangerous underestimates of the longterm deflections of prestressed large box girders and prestress losses. Improvements in the currently used creep and shrinkage prediction models are, therefore, required. The improvement needs to based on better statistical evaluation of the existing data. However, the existing data are insufficient by far. Consequently, a physical justification based on the processes in the nano-porous microstructure is vital.

5

= −sh∞ (1 −

− t0 )  (t − t0 )/τsh

h3e ) tanh

(5)

(6)

where q5 is material constant; he is environmental relative humidity; sh∞ is ultimate shrinkage strain determined by the concrete property; τsh = shrinkage halftime = kt (ks D)2 , where Ks is the cross section shape factor, D is the effective cross section thickness, kt is a parameter determined by diffusivity (or permeability); and H (ˆt ) = 1 − (1 − he )S(ˆt ). There is a mathematical reason—the self similarity in time—for which the unbounded individual physical processes involved in creep and shrinkage should be described by power laws, and those that approach a finite bound by decaying exponentials. Eqs. (3) to (6) give a simplified model for practice which approximately describes the mean behavior of a cross section of a slab of thickness D when the bending moment is negligible, i.e., for essentially central in-plane loading. For a point-wise constitutive model, which can be applied only when the thickness of each slab is subdivided into many finite elements, the solidificationmicroprestress theory can be characterized by the nonlinear first-order differential equation for the microprestress in the nano-porous structure:

MODEL BASED ON PHYSICALLY BASED THEORY AND STATISTICS

Among the existing models, model B3 is the only one with some theoretical foundation—the solidification theory and the theory of microprestress buildup and relaxation in the nano-porous structure of cement gel. The theoretical basis include the following phenomena: 1) process without a characteristic time and asymptotic limit; 2) solidification process; 3) microprestress relaxation; 4) activation energies of creep and hydration; 5) diffusion of water; 6) arguments based on capillarity, surface tension, and free and hindered adsorbed water; 7) cracking or damage; and 8) rate of chemical processes causing autogeneous volume change. After certain idealizations and simplifying hypotheses, consideration of the phenomena led to a compliance function of the form: J (t, t  ) = q1 + C(t, t  ) + Cd (t, t  , t0 ) q  n−1 q4 ˙ t  ) = 2 + q3 nζ + C(t, tm 1 + ζn t

h3e )S(t

˙ s˙ + as2 = c|T lnh|

(7)

The microprestress s initially produced by the disjoining pressures in nano-pores and by volume mismatch of various constituents relaxes with time at a decaying rate over many years. If there is drying or wetting, or a change of temperature T , the fields of pore humidity h (and temperature) must be calculated for each time step from the nonlinear diffusion equation of drying, and the humidity and temperature change at each integration point of each finite element causes a change in the right-hand side, which produces a buildup of microprestress, and thus an acceleration of creep. ˙ at each material The free shrinkage rate ˙sh = ksh h, point, is proportional to the local pore humidity rate. This is much simpler than the approximate expression in Eqs. (5) and (6) for the overall average shrinkage of

(3) (4)

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the entire cross section, which includes the approximate effect of the self-equilibrated nonuniform shrinkage stresses and of the cracking that they produce. Although the form of the constitutive law is physically based, the model B3 parameters have to be obtained by statistical calibration from test data, same as the parameters of purely empirical models. Therefore, statistical study on a large database, is important. Can the best creep and shrinkage model be identified purely by a standard statistical regression of the existing database? The answer is—no (unless a statistically perfect database were available, which is not the case). In the last several decades, numerous tests have been conducted around the world to investigate the concrete creep and shrinkage. In 1978, the first comprehensive database, consisting of about 400 creep tests and approximately 300 shrinkage tests, was compiled at Northwestern University (Bažant & Panula 1978). A slight expansion based on this database led to what known as RILEM database (M¨uller & Hilsdorf 1990, M¨uller 1993, M¨uuller et al. 1999), which was widely used to calibrate various creep and shrinkage models. Now a significantly expanded database, named the NU-ITI Database, is assembled at Northwestern, comprising 621 creep tests and 490 shrinkage tests (Bažant & Li 2009b). Although there are thousands of test points in this expanded database, the statistical comparison is useless since the scatter is huge and very little difference can be seen among different models if the entire database is treated as a statistical population (ensemble statistics). The reasons are three: 1) The data points are not sampled uniformly in terms of the loading or drying duration, age of loading, start of drying, and cross section thickness; 2) The scatter caused by various concrete compositions is enormous; and 3) The trends of creep and shrinkage evident from individual tests are obfuscated. Most laboratory test data on creep and shrinkage have durations ≤5 years, while the data for 10 to 30 years are very scant (<1% of the total database) and incomplete (Bažant 2000, Bažant et al. 2008). Therefore, steps must be taken to eliminate or minimize the statistical bias before any meaningful statistical comparisons can be made. As argued in detail in a recent study (Bažant & Li 2009a), it is reasonable to assign the same weight to the total of all test data within each interval of time, size, humidity, and age at loading or start of drying. The subdividion into intervals is based on the regression trend of the corresponding variable. For example, the subdivision of the load duration t − t  is properly made by subdividing the logarithmical scale into equal intervals. Another plague of previous comparisons has been the invention and use of various nonstandard statistical indicators, which violate the principles of statistics. The standard statistical approach—the method of least squares–must be used because it maximizes the likelihood function and is consistent with the central

limit theorem. The statistical analysis of the database based proper statistics, with minimized bias, shows that model B3 to provide the best fit.

6

BASIC PHYSICAL PROPERTIES OF CREEP AND SHRINKAGE

As transpired from the numerical modeling of the KB bridge in Palau, the advantages of model B3 over other empirical models arise mainly from its theoretical basis. The nano-porous structure of concrete is the source of creep in hardened Portland cement (Bažant et al. 1997). Creep is most likely caused by separation, slip and restoration of bonds in the calcium silicate hydrates (C-S-H), which is a noncrystalline and strongly hydrophilic material. The weakest bonds are doubtless those crossing the nano-pores of thickness less than about 3 nm. Except for relative humidities h of water vapor in capillary pores less than about 12%, the adjacent nano-pores are filled with water (Fig. 8a), called the hindered adsorbed water, which gives rise to the so-called disjoining pressure pd , a concept due to Deryaguin (1955). At 100% humidity, this pressure can reach thousands of atm (or hundreds of MPa). It must be balanced by similarly large tensions (or microprestress) elsewhere in the nano-structure, which are large enough to cause nano-cracks. These tensions obviously promote creep. This explains why the creep is higher at a higher moisture content, provided that the material in thermodynamic moisture equilibrium (a uniform chemical potential in all phases of water). Concrete, however, hardly ever reaches such equilibrium (at normal temperatures) and, during drying, the opposite is observed—the creep is higher at a lower environmental humidity, which drives the drying. This long discussed and long misunderstood phenomenon is known as the drying creep effect (also called the Pickett effect, or the stress-induced shrinkage). Since a change of temperature alters the thermal equilibrium, it produces similar phenomena, particularly the transient thermal creep. A change of pore humidity causes a change in disjoining pressure pd , and thus changes the thickness of the hindered adsorbed water layers and the nano-pores. This is one source of shrinkage sh (deformation due to a change in humidity at no change in load). Diffusion (migration) of water molecules along the nano-pores, and in or out of the adjacent capillary pores (>micrometer in size), is one source in the delay of shrinkage and of the part of creep due to drying. But normal structures are so thick that the rate of shrinkage and drying creep of a structure is controlled by diffusion of moisture through structure. This diffusion is highly nonlinear, with the permeability and diffusivity

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a thermodynamically legitimate constitute law with aging is to take into account the fact that the newly produced hydrates are deposited on the surface of capillary voids in an unstressed state. Thus aging may be considered as a consequence of the growth of volume fraction v(t) of a certain solidifying constituent (approximately the solid cement gel), which itself is considered to have non-aging properties (Fig. 8e). This represents the essence of the solidification theory, which simplifies the analysis of aging creep in structures. The absence of aging for the solidifying constituent makes it possible to co-opt the theory of viscoelasticity of polymers, including the rate-type approximation of the non-aging compliance function by a Kelvin (or Maxwell) chain model (Fig. 8e) with an easily obtainable retardation spectrum. This spectral approach avoids the use of history integrals. Also, it is amenable to introducing the effects of variable pore humidity h, temperature T (Bažant et al. 2004) and the softening (Bažant 1995) due to distributed cracking (which cannot be introduced into a Volterra history integral). Although the Kelvin chain model corresponds to a discrete retardation spectrum, it is better to start with a continuous spectrum. Such a spectrum can be identified from the compliance curve by means of Widder’s formula, known from the viscoelasticity of polymers. Discretization of this formula at discrete times, which are best chosen to be spaced by decades in the log-time scale, yields then the Kelvin chain moduli. A direct least-square determination of the Kelvin chain moduli is possible and was used in the 1970s, but it suffers from ill-conditioning and near non-uniqueness. The continuous retardation spectrum is unique, and thus eliminates the problem with ill-conditioning. Besides, its use is computationally more efficient, as demonstrated by the analysis of the KB Bridge in Palau. The effect of temperature rise on creep and shrinkage is twofold: a) Acceleration of creep, which can be described as a decrease of viscosities of Kelvin units (Fig. 8e) according to the Arrhenius equation, characterized by an activation energy of creep; and b) at the same time an increase in the rate of aging, characterized again by another Arrhenius equation with a different activation energy that governs the rate of hydration. The latter reduces creep. Thus the temperature effect on creep is a competition of accelerating and decelerating influences. For shrinkage it is similar. Temperature increase accelerates diffusion but accelerated hardening hinders shrinkage. One consequence of ignoring the thermodynamic restrictions due to chemical aging is the condition of non-divergence of the compliance function J (t, t  ), defined as the strain at time t due to a unit stress applied at age t  ; see Fig. 8e. If this condition is ignored, the principle of superposition can produce, after unloading, a recovery reversal (Fig. 8f), and, the relaxation

decreasing about 20-times as the h decreases from 95% to 65%. Despite this nonlinearity, the drying (weight loss) as well as the shrinkage and drying creep at constant environmental humidity exhibit three basic properties of diffusion processes (Bažant & Xi 1994): a) The drying half-times scale as a square of structure size D (or cross section thickness) (Fig. 8b, c); b) They begin as a square-root function of time (Fig. 8b); and c) They approach the final equilibrium state as a decaying exponential of a power of time (Fig. 8c). Asymptotic matching of properties b) and c) for shrinkage leads to the tanh-function of the duration (Fig. 8c), normalized by half-time τsh which is proportional to D2 . A similar function should apply to drying creep compliance J . Since concretes have thousands of different compositions, it is important to conduct, before design, shorttime tests of creep and shrinkage, so as to update the parameters of creep prediction equations (Fig. 8c, d). For creep, which has no upper bound, the update is simple: It suffices to scale the entire creep curve up or down (Fig. 8d). Not so for shrinkage. There is a trap in that the half-time value and the final asymptotic shrinkage value cannot be predicted from short-time shrinkage measurements alone. An important idea in 1995, which still has not penetrated the practice, is that these values can be easily predicted (by a certain linear regression) if the loss of weight (due to loss of pore water) is simultaneously recorded, and if the total water loss upon heating of the shrinkage specimen at the end of the short-time tests (typically of 1 month duration) is determined. In this manner one can avoid wrong extrapolations shown at lower left and lower right of Fig. 8d. A convenient property of concrete creep is that, in absence of microcracking, it is linear (within the range of service stresses allowed in structure), i.e., follows the Volterra principle of superposition in time. However there is major complication—the aging. It causes that the creep properties vary significantly with time (at a decaying rate), because of two phenomena: a) the chemical hydration of cement, which lasts at normal temperature for about a year; and b) the gradual relaxation of microprestress in the nano-structure (possibly combined with polymerization of C-S-H), which continues for decades. The use of microprestress appears indispensable as a unifying concept that captures simultaneously the drying creep (Pickettt effect), the transitional thermal creep and the long-time aging. However, it is not yet clear what is its precise physical mechanism in the C-S-H. Researches at Northwestern University (by H. Jennings) and at M.I.T. (by F.-J. Ulm) may clarify this fundamental question. The aging causes some thermodynamic restrictions on the formulation of the constitutive equation, which have often been ignored. The easiest way to formulate

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Figure 8.

Basic physical phenomena of creep and shrinkage in concrete.

such compliance functions are still used in the design codes of ACI, JSCE and CEB (or ‘fib’). The effect of aging on the conventional elastic modulus E (which is really the inverse of compliance function for the duration of about 15 min.) can be greatly simplified by noting that the compliance

curves at constant strain can cross into the opposite sign. These phenomena violate the second law of thermodynamics if realistic Kelvin or Maxwell chain models with aging are considered (violation, though, need not occur for some contrived rheologic models that give unrealistic compliance curves). Nevertheless,

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Therefore, design measures that mitigate deflections should be sought. To have a better estimate of long-term deformation in bridge design, the following is necessary: a. Use a realistic creep and shrinkage model (such as B3 model built on solidification theory); b. update the model by tests of creep, shrinkage and water loss; c. analyze the structure three-dimensionally, in time steps; and d. calculate the 95% confidence limits instead of mean response.

Figure 9. Creep curves for different ages at loading and the asymptotic modulus E0 .

Practical measures can also be undertaken to mitigate the deflections, for example: a. no mid-span hinge; b. choice of a concrete with low creep, especially a low long-time creep; c. use of a higher level of prestress, even so high that an upward deflection is predicted; d. giving concrete more time to gain strength before prestressing; e. using stiffer girders of low slenderness; and f. installing some empty ducts into which additional tendons can be placed later.

curves for different ages approximately intersect at one point when plotted versus the 0.1-power of stress duration (Fig. 9). The intersection point gives the socalled asymptotic modulus E0 , which corresponds to a fictitious extrapolation of creep curve to the load duration of about 1 ns. With this concept, the compliance function automatically includes the dependence of the conventional elastic modulus E(t) on age t (Fig. 9).

7

Two more points deserve mention: In continuous girders, the deflections are particularly sensitive to tendon layout and can be reduced by the right layout. But a layout benefiting the stress state can at the same time be harmful from the deflection viewpoint. Second, although the absence of a mid-span hinge reduces deflections, excessive deflections may occur, as documented by the Dˇecˇ ín in Bridge.

CONCLUSIONS

If the advances in research of creep and shrinkage in last three decades were incorporated into the practice, and if the models and methods available today were used at the time of design, the observed deflections of the KB Bridge in Palau would have been expected. This would have forced a radical change of design of this bridge, precluded the fatal retrofit, and thus prevented the final catastrophe. Numerical simulation of the KB Bridge deflections shows that the main causes of error in design are as follows:

ACKNOWLEDGEMENT Financial support from the U.S. Department of Transportation through Grant 0740-357-A222 from the Infrastructure Technology Institute of Northwestern University is gratefully appreciated. Thanks are due to Dr. Khaled Shawwaf of DSI, Inc., Bolingbrook, Illinois, for providing valuable information on the analysis, design and investigations of the bridge in Palau.

1. incorrect compliance and shrinkage functions which are empirically based and rely on a database overwhelmingly dominated by short-term tests (≤3 years duration) and small size tests; 2. neglect of differential shrinkage and drying creep compliance due to a) different thicknesses of cross section walls; b) different temperatures; and c) cracking; 3. lack of three-dimensional analysis. The simplified one- or two-dimensional model leads to inaccurate calculation of shear lags in the slabs and webs; 4. absence of a statistical estimate of the 95% confidence limits; 5. inaccurate time integration and neglect of temperature and cracking; 6. lack of updating of the material parameters by short-time tests. 7. errors in top-bottom differences (a 5% error in prestress can cause a 50% error in the total deflection).

REFERENCES ACI Committee 209. 2008. Guide for Modeling and Calculating Shrinkage and Creep in Hardened Concrete ACI Guide 209.2R-08, Farmington Hills. Bažant, Z.P. 1995. Creep and damage in concrete Materials Science of Concrete IV, J. Shalny and S. Mindess, Eds., Am. Ceramic. Soc., Westerville, OH, 355–389. Bažant, Z.P. 2000. Criteria for rational prediction of creep and shrinkage of concrete. Adam Neville Symposium: Creep and Shrinkage—Structural Design Effects, ACI SP-194,

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A. Al-Manaseer, ed., Am. Concrete Institute, Farmington Hills, Michigan, 237–260. Bažant, Z.P. 2001. Creep of concrete. Encyclopedia of Materials: Science and Technology, K.H.J. Buschow et al., eds., Elsevier, Amsterdam, Vol. 2C, 1797–1800. Bažant, Z.P. and Baweja, S. 1995. Creep and shrinkage prediction model for analysis and design of concrete structures: Model B3. Materials and Structures 28, 357–367. Bažant, Z.P. and Baweja, S. 2000. Creep and shrinkage prediction model for analysis and design of concrete structures: Model B3. Adam Neville Symposium: Creep and Shrinkage—Structural Design Effects, ACI SP-194, A. Al-Manaseer, ed., 1–83. (update of RILEM Recommendation published in Materials and Structures Vol. 28, 1995, 357–365, 415–430, and 488–495). Bažant, Z.P., Cusatis, G. and Cedolin, L. 2004. Temperature effect on concrete creep modeled by microprestresssolidification theory. J. of Eng the Mechanics ASCE 130 (6) 691–699. Bažant, Z.P., Hauggaard, A.B. and Baweja, S. 1997. Microprestress-solidification theory for concrete creep. II. Algorithm and verification, J. of Engrg. Mech. ASCE 123(11), 1195–1201. Bažant, Z.P., Hauggaard, A.B., Baweja, S. and Ulm, F.-J. 1997. Microprestress-solidification theory for concrete creep. I. Aging and drying effects, J. of Engrg. Mech. ASCE 123(11), 1188–1194. Bažant, Z.P. and Kaplan, M.F. 1996. Concrete at High Temperatures: Material Properties and Mathematical Models, Long-man (Addison-Wesley), London (2nd printing Pearson Education, Edinburgh, 2002). Bažant, Z.P. and Li, G.-H. 2009a. Unbiased statistical comparison of creep and shrinkage prediction models. ACI Materials Journal, 106(6), 610–621. Bažant, Z.P. and Li, G.-H. 2009b. Comprehensive database on concrete creepa and shrinkage. ACI Materials Journal, 106(6), 635–638. Bažant, Z.P., Li, G.-H. and Yu, Q. 2008. Prediction of creep and shrinkage and their effects in concrete structures: critical appraisal. Proc., 8th International Conference on Concrete Creep and Shrinkage (CONCREEP-8), IseShima, Japan, T. Tanabe et al. eds., CRC Press/Balkema, 1275–1289. Bažant, Z.P. and Panula, L. 1978. Practical prediction of time-dependent deformations of concrete. Part I: shrinkage. Part II: creep. Materials and Structures, 11(65), 307–328. Bažant, Z.P. and Prasannan, S. 1989a. Solidification theory for concrete creep: I. Formulation. Journal of Engineering Mechanics ASCE 115(8), 1691–1703. Bažant, Z.P., and Prasannan, S. 1989b. Solidification theory for concrete creep: II. Verification and application Journal of Engineering Mechanics ASCE, 115(8), pp. 1704–1725. Bažant, Z.P., Sener, ¸ S. and Kim, J.-K. 1987. Effect of cracking on drying permeability and diffusivity of concrete. ACI Materials Journal, 84, 351–357. Bažant, Z.P. and Xi, Y. 1994. Drying creep of concrete: Constitutive model and new experiments separating its mechanisms. Materials and Structures, 27, 3–14. Bažant, Z.P. and Xi, Y. 1995. Continuous retardation spectrum for solidification theory of concrete creep J. of Engrg. Mech. ASCE 121(2), 281–288.

Bažant, Z.P., Yu, Q. and Li, G.-H., Klein, G., and Kˇrístek, V. (2007). ‘‘Explanation of excessive long-time deflections of collapsed record-span box girder bridge in Palau’’, Prelim. Structural Engrg. Report 08-09/A222e, Infrastructure Technology Institute (ITI), Northwestern University. Brooks, J.J. 1984. Accuracy of estimating long-term strains in concrete. Magazine of Concrete Research, 36(128), 131– 145. Brooks, J.J. 2005. 30-year creep and shrinkage of concrete. Magazine of Concrete Research, 57(9), 545–556. Burgoyne, C. and Scantlebury, R. 2006. Why did Palau bridge collapse? The Structural Engineer, 30–37. CEB-FIP Model Code 1990. Model Code for Concrete Structures. Thomas Telford Services Ltd., London, Great Britain; also published by Comité euro-international du béton (CEB), Bulletins d’Information No. 213 and 214, Lausanne, Switzerland. Deryagin, B.V. 1955. The definition and magnitude of disjoining pressure and its role in the statics and dynamics of thin fluid films, Kolloid Zh. 17, 205–214. FIB 1999. Structural Concrete: Textbook on Behaviour, Design and Performance, Updated Knowledge of the CEB/FIP Model Code 1990. Bulletin No. 2, Federation internationale du béton (FIB), Lausanne, Vol. 1, 35–52. Gardner, N.J. 2000. Design provisions of shrinkage and creep of concrete Adam Neville Symposium: Creep and Shrinkage- Structural Design Effect ACI SP-194, A. AlManaseer, eds., 101–104. Gardner, N.J. and Lockman, M.J. 2001. Design provisions for drying shrinkage and creep of normal strength ACI Materials Journal 98(2), Mar.–Apr., 159–167. Japan International Cooperation Agency (JICA). 1990. Present Condition Survey of the Koror-Babelthuap Bridge, Feburary, 42 pages. Jirásek, M. and Bažant, Z.P. 2002. Inelastic analysis of structures, John Wiley & Sons, London and New York. JSCE 1991. Standard Specification for Design and Construction of Concrete Structure, Japan Society of Civil Engineers (JSCE), in Japanese. McDonald, B., Saraf, V. and Ross, B. 2003. A spectacular collapse: The Koro-Babeldaob (Palau) balanced cantilever pre-stressed, post-tensioned bridge The Indian Constrete Journal Vol. 77, No.3, March 2003, 955–962. M¨uller, H.S. 1993. Considerations on the development of a database on creep and shrinkage tests. Creep and Shrinkage of Concrete, Proceedings of the 5th International RILEM Symposium, Barcelona, Spain, Bažant Z.P. and Carol I., eds., E&F Spon, London, UK, 859–872. M¨uller, H.S., Bažant, Z.P. and Kuttner, C.H. 1999. Database on creep and shrinkage tests. RILEM Subcommittee 5 Report, RILEM TC 107-CSP, 81 pages. M¨uller, H.S. and Hilsdorf, H.K. 1990. Evaluation of the TimeDependent Behaviour of Concrete: Summary Report on the Work of the General Task Force Group No. 199, Comité Euro-Internationale du Béton, Lausanne, Switzerland, 201 pages. Pickett, C. 1942. The effect of change in moisture-content on the creep of concrete under sustained load. ACI J. 38, 333–356. Yee, A.A. 1979. Record span box girder bridge connects Pacific Islands Concrete International 1 (June), 22–25.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Alternate approaches to simulating the performance of ductile fiber-reinforced cement-based materials in structural applications S.L. Billington Stanford University, CA, USA

ABSTRACT: Structural-scale applications of ductile, fiber-reinforced cement-based materials are being evaluated by researchers and implemented in practice more and more. These materials exhibit significant ductility in direct tension and generally do not spall in compression. Little guidance exists in codes or specifications for designing with these materials. Validated simulation approaches are called for to understand how these materials can best be applied and will perform in structural applications. This paper presents several approaches to simulating plain and steel-reinforced ductile, fiber-reinforced cement-based materials in both new design and retrofit applications, with an emphasis on the ability of simple approaches, using basic material properties, to predict performance. Specifically, the use of nonlinear finite element analysis, phenomenological spring modeling, and sequentially linear finite element analysis is presented. 1

Methods of simulating ductile cement-based composites in structures and structural components range from sequentially linear finite element analysis (Billington, 2009) to nonlinear finite element analysis for both two-dimensional (Lee & Billington, 2008, Suwada & Fukuyama, 2006, Hung & El-Tawil, 2010) and three-dimensional simulations (Sirijaroonchai & El-Tawil, 2007) to multi-phase and multi-scale methods (Chuang & Ulm, 2002, Kabele, 2007) to phenomenological spring models for cyclic loading (Lignos et al., 2009). In this paper, several modeling approaches for two ductile fiber-reinforced cement-based materials are evaluated by modeling a variety of experiments conducted on components and structural configurations

INTRODUCTION

Many researchers and practitioners are investigating structural-scale applications of ductile, fiberreinforced cement-based materials. These materials exhibit fine, multiple cracking in direct tension, reaching strains of 0.5 to 3 percent and generally exhibit little to no spalling in compression. The materials are often referred to as high performance fiber-reinforced concrete (HPFRC) or high performance, fiber-reinforced, cement-based composites (HPFRCC) if they contain no coarse aggregate. One class of HPFRCC being heavily researched is engineered cementitious composites (ECC). The term high performance is used when the fiber volume fraction is relatively low (2% or less) and the tensile properties are either high strength (e.g. increase in tensile strength after first cracking) or high ductility (e.g. >3% strain while maintaining full tensile carrying capacity) (Figure 1). Applications in service and under investigation range from bridge decks, to energy-dissipating hinge regions in seismicresistant frames, to repair and retrofit of structures. Little guidance exists in codes or specifications for designing with these materials. Therefore validated simulation approaches are called for to understand how these materials can best be applied and will perform in structural applications. This paper presents several example approaches to simulating ductile fiberreinforced cement-based materials in both new design and retrofit applications in particular for resistance to seismic loads, with an emphasis on the ability of simple approaches, using basic material properties, to predict performance.

Figure 1. Schematic of uniaxial tensile responses possible from ductile fiber-reinforced cement-based composites.

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using these materials. The two materials focused on here are ECC (Li, 2003) and a self-compacting HPFRC (SC-HPFRC) developed by Liao et al., (2006). Nonlinear finite element analysis, phenomenological spring modeling, and sequentially linear finite element analysis are evaluated. An additional example of modeling ECC as applied to masonry beams is given in Kyriakides et al. (2010).

2

2.1

NONLINEAR FINITE ELEMENT ANALYSIS TO IDENTIFY OPTIMAL ECC PROPERTIES FOR DUCTILE BRIDGE PIERS Figure 2.

Background and motivation

Ductile fiber-reinforced cement-based materials have been investigated for use in regions of high moment and shear (often referred to as hinge regions, i.e. where plastic hinges will likely form under overloads) in particular for bridge supports. Both experimental (Rouse, 2004, Saiidi and Wang, 2006) and analytical (Lee & Billington, 2008) research has been conducted, including the use of nonlinear finite element analysis to interpret an unexpected experimental failure mode (Lee & Billington, 2010). Here, an example application of nonlinear finite element analysis to evaluate and propose alternate material properties to improve structural performance of bridge columns is presented. In this case, ECC was used in the hinge regions of precast segmental bridge columns that were reinforced with vertical, unbonded post-tensioning. While the performance of the columns with ECC hinge regions was superior to that of a column using all traditional concrete segments, it was believed that more ductility should be possible in a column with ECC hinge regions than was observed experimentally. As its name suggests, ECC can be engineered to achieve a variety of tensile properties (Li and Leung, 1992). Therefore a study was carried out to evaluate the impact of alternate ECC properties on structural performance to identify what ECC properties lead to improved structural performance, which was defined as increased ductility under cyclic lateral loading. The original experiment modeled is briefly reviewed here and further detail on both the experiments and simulations are given in Rouse (2004) and Lee and Billington (2008). The experiment simulated was one of a pair of large-scale unbonded post-tensioned bridge columns tested together (Figure 2a). The column specimen had a height of 3.7 m and a 460-mm-square cross-section. The specimen consisted of four precast segments each 1.067 m in length, with the two end segments embedded into a cap and a foundation block. Reinforcement details of the column segments can be seen in Figure 2b. The precast segments, which had been match cast, were joined using a flowable epoxy mortar,

Test set-up and bridge column segment details.

and had no continuous, bonded reinforcing across the segmental joints. The segments were post-tensioned together with six 15.2-mm-diameter low-relaxation strands stressed to 690 MPa. Two columns were tested simultaneously to achieve double curvature in order to represent the behavior of a column in a multiple-column bent configuration subjected to lateral loading. Specimens were oriented horizontally (longitudinal axis parallel to the floor) with their foundation blocks connected to a steel reaction frame and cap blocks connected to one another to provide rotational restraint. The specimens were subjected to quasi-static cyclic lateral loads while under a constant axial load of 720 kN (applied with a hydraulic actuator), representing dead load from a bridge superstructure. The hinge segments were precast with ECC that demonstrated a compressive strength of roughly 40 MPa and a uniaxial tensile response as shown schematically in Figure 3 as ‘‘ECC’’ (based on experimental testing in Rouse, 2004). Lightweight concrete was used for the remaining segments (including cap and foundation blocks) with a nominal compressive strength of 55 MPa. The reinforcement in the column segments was detailed to meet only shear and shrinkage requirements.

2.2

Finite element modeling

A plane stress model was used in the analyses, as outof-plane stresses were assumed to be negligible for the testing configuration. Each column was modeled individually. The finite element model for a single column is shown in Figure 4. The concrete and ECC were modeled using nine-noded, quadrilateral isoparametric plane stress elements with a 3 by 3 Gauss integration scheme. All longitudinal and transverse bonded, mild steel reinforcing bars were modeled with threenoded embedded reinforcement elements, which were assumed to have perfect bond with the surrounding

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tendons, was applied to the truss elements. Cracking was not modeled explicitly between the segmental joints (e.g. with interface elements) because it was neither observed in this region experimentally nor expected (the epoxy mortar joining the segments a was stronger in tension than the segment materials). The foundation block was modeled as fixed by providing pin supports at all nodes along the bottom of the model. The top nodes of the specimen (at the top of the cap block) were modeled as being rotationally fixed to represent the fixity provided by the connection to the cap of the other specimen. Because rotational degrees of freedom do not exist for the quadrilateral elements, rigid three-noded beam elements were added along the top of the cap and their rotational degrees of freedom were constrained to provide the appropriate fixity. Point loads with a total magnitude equal to the magnitude of the applied axial load were applied to the top nodes of the cap block. Lateral loading was applied through applied displacements at two control nodes in the cap. Geometric nonlinearity was included in the analyses in addition to material nonlinearity as described next. The concrete elements in the footing and the cap were modeled as linear elastic, as non-linear behavior (manifested by cracking or crushing) in those regions was neither expected nor observed during testing. The elastic modulus for the concrete in these and all other concrete segments was assumed to be 24.8 GPa based on the American Concrete Institute (ACI) 318-05 (2005) equation for lightweight concrete using measured compressive strength data. A smeared, rotating crack model with a total strain formulation was used (Feenstra et al., 1998) for the concrete in tension. The tensile modulus was assumed equivalent to the compressive modulus (24.8 GPa). Post-cracking behavior was defined to be fracture energy-based with linear tensile softening. The tensile strength was assumed to be 3.45 MPa (ACI, 2005) and the tensile fracture energy was assumed to be 0.4 N-mm/mm2 (CEB-FIP, 1990). A bilinear model was used for the bonded, mild steel reinforcement with yield strength of 460 MPa, elastic modulus of 200 GPa, and post-yield (hardening) slope of 2 percent of the elastic modulus. The post-tensioning tendons were modeled as linear elastic, as designed and as observed during testing, with an elastic modulus of 186 GPa. The ECC was modeled with a total strain-based, rotating crack model developed by Han et al. (2003) using the loading and unloading parameters proposed therein. This model is based on the observed responses from a series of reversed cyclic tests on uniaxially loaded ECC specimens and was validated against cyclically loaded components. The envelope curves are shown in Figure 5. The unloading and reloading paths of the constitutive model were developed to capture the unique cyclic behavior of ECC’s. The tensile

Figure 3. Tensile properties of tested ECC (ECC) and three alternate ECC tensile property designs (Models 1–3).

Figure 4. Finite element model of bridge column with concrete and ECC segments joined with unbonded posttensioning.

plane stress elements. The bonded longitudinal reinforcing bars in the column segments did not have adequate development length to reach their yield strength in all areas. Where reinforcing bars did not have adequate development, the yield stress of the elements comprising the reinforcing bar were reduced in a stepwise fashion from 100 percent of the yield stress at the point of full development to zero at the end of the bar in order to approximate the linear reduction in bond stress (Lee & Billington, 2010). The unbonded post-tensioning tendons were modeled with two-noded truss elements that were constrained at their end nodes to the concrete element nodes at the anchorage locations. This connectivity allowed the strains to be distributed evenly along the length of the post-tensioned tendons. An initial stress of 690 MPa, equal to the experimental prestress in the

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Figure 5. Envelope curves for ECC constitutive model in tension (left) and compression (right).

stress and strain values for the envelope curve of the model were based on uniaxial tensile tests on the ECC (Kesner et al., 2003). The peak compressive stress was based on cylinder compression tests and the elastic modulus and peak compressive strain were estimated based on tests performed on ECCs with similar mix proportions and similar compressive strengths. To determine the effect of the ECC tensile parameters on the global response, the model was also analyzed for three additional tensile stress-strain responses (Models 1–3), shown in Figure 3. The three models represent behavior that can reasonably be expected from an ECC mix. 2.3

Simulation results

A cyclic analysis was performed and the loaddisplacement response is shown in Figure 6. The envelope of the simulated cyclic response generally follows the experiment well. During the lower drift cycles, low residual displacements (displacement at zero load) are seen in both the experiment and the simulation. Additionally, up to the 3 percent drift cycle, the simulation captures similar peak strength and displacement as the experiment and the observed gradual softening to 3 percent drift is also simulated. It is noted that the downward slope to increasing drifts (e.g. from 2 to 3 percent drift) is indicative of P-delta effects due to the gravity load on the column and is less prominent in the simulation, although geometric nonlinearity is included in the simulations. At the largest drift cycle (to 3 percent drift) the simulated response no longer models the experimental response well particularly in unloading. The simulated response displays the expected flag-shaped (origin-oriented) response while the experimental response follows a much less stiff unloading path. The less stiff experimental unloading is caused by the change in boundary conditions in the experiment relative to the simulation (i.e. the lower stiffness of deformation in single-curvature once one of the two specimens tested simultaneously begins to form a plastic hinge, versus double-curvature at the beginning of the test, Figure 2a). The wider hysteresis

Figure 6. Cyclic response of experiment and simulation using measured ECC material properties (top), and regions of crack and compressive strain localization after peak load (bottom).

loops are attributed to the spread of ECC crushing not captured in the simulation and additional effects that are not modeled such as possible slip of bonded steel reinforcement after cracks localize. The column reaches its peak lateral load soon after crack localization occurs in the column near the joint between the end segments and the adjacent concrete segments (as shown in Figure 6, bottom). Although there is longitudinal, bonded mild steel reinforcement at this section, it is unable to contribute significantly to the moment capacity because it cannot reach its yield strength due to its development length (recall that the mild steel reinforcement is not continuous across the segmental joints). The ECC is therefore relied on primarily to resist tensile stresses. After the ECC begins to soften in tension (crack localization), the crack progresses toward the compression region, reducing the size of the compressive zone and causing the ECC in compression to quickly reach its peak strain and begin to soften. At this point the column begins to lose its

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lateral load resistance. The fact that there is no steel yielding leads to a relatively brittle and therefore undesirable failure. The formation of the localized crack near the joint and the subsequent compressive failure of the ECC were both observed in the experiment. The column with ECC as simulated above had a low drift capacity and had a relatively brittle failure mode (both experimentally and in the simulation), both of which are unfavorable for structures in seismic regions. As ECC is an engineered material, its tensile response can be tailored to meet the requirements of the engineer. Therefore, a parameter study was performed to determine the effect of the ECC tensile parameters on the load-drift behavior of the columns. The goal of this study was to evaluate possible benefits of engineering new ECC mix designs, specifically with respect to increasing drift capacity and introducing greater ductility to the column prior to failure. Three alternate sets of tensile properties for ECC were evaluated as described above and shown in Figure 3. Particular attention is given here to the results of Model 3. The load-drift response of the column using Model 1 led to a 2 percent increase in the peak load and 36 percent increase in the drift at peak load. The localization of cracking in this case occurred in the concrete segments above the ECC segments rather than in the ECC segments themselves due to the cracking strength of the ECC exceeding the concrete cracking strength. The column fails when the compressive strength of the ECC is reached and begins to soften shortly after at the segmental joint between the ECC and concrete segments near where the crack localization occurs, just as observed in the original simulation. Model 2 led to a lightly higher increase in strength and ductility (details in Lee and Billington, 2008) but the critical section remained in the joint region where the mild steel reinforcement was unable to contribute significantly to the moment capacity due to its development length. Using ECC with properties of Models 1 and 2 would provide only minimal improvements to the overall behavior of the column relative to what was tested. Model 3 provided an increase in the strain hardening capacity of the ECC in tension while limiting the peak strength so as not to exceed the tensile strength of the concrete. The resulting load-drift response for Model 3 is compared in Figure 7 with the experimental response and the simulated response using the original ECC model. The response of the column with ECC Model 3 shows a distinct difference from the ECC used to model the experiment. By 3 percent drift, the column is continuing to carry additional load. In this case, the maximum tensile strains occur at the ends of the columns (column-base interface), and localization of cracking does not occur near the segment joints between the ECC and concrete. Cracking and strain hardening of the ECC occur throughout most

Figure 7. Cyclic response of experiment, original simulation (‘‘Baseline’’ using model ECC) and Model 3.

of the hinge segment, with the most tensile straining occurring at the base of the column. Therefore, the mild steel is able to contribute significantly to the moment capacity (and ductility) because in this region it has adequate development length to yield. 2.4

Further discussion

The finite element model was found to capture the response of the tested column well, with respect to both global behavior (lateral load versus displacement response) as well as local, damage (development of damage and mode of failure). The mode of failure of the column at a relatively low drift was captured well. The parameter study of the ECC tensile behavior revealed that the best method of improving the performance of the columns is to increase the ductility of the ECC without increasing the ultimate tensile strength. Increasing the ultimate tensile strength of the ECC to a value greater than that of the tensile strength of the concrete is predicted to lead to crack localization above the base segment (the plastic hinging region). High ECC tensile strength would lead to compressive failure near the segmental joint between the ECC and adjacent concrete segment. When increasing the tensile ductility while keeping the peak tensile stress below the tensile strength of the concrete, the tensile benefits of the ECC are better utilized. Cracking is predicted to occur throughout the ECC segment and a crack will eventually localize at the base of the column. In addition, the failure location is shifted away from sections where mild steel reinforcement is not able to fully yield (i.e. near the joints) to sections where it can contribute more significantly to moment capacity and improve ductility (i.e. at the ends of the column). Experimental validation of

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the proposed ECC material properties for improving column performance is needed and it is recommended that additional experimental projects on realistic structural applications be conducted to evaluate the ability of achieving a variety of structural performance characteristics through altering ECC material properties alone.

3

3.1

The panels to be modeled were single panels representing one half of the double-height panels shown in Figure 8, fixed at the base and laterally loaded at the top of a single panel (Figure 9a), representing the mid-point of the double height panels, or, the point of inflection for the panel system under lateral load. The reinforcing details of the two panels modeled are shown in Figure 9b–c. All reinforcement was modeled as embedded, which implies perfect bond to the concrete and/or SC-HPFRC material. Perfect bond was assumed to be representative of the deformed bars used but not the welded wire fabric (WWF) mesh reinforcement, which was made up of smooth wire. All modeling was 2-dimensional using 8-noded plane stress elements. The geometry for each panel was meshed with a user-defined mesh discretization size of ∼25 mm squares. The boundary condition at the base was fixed against vertical and horizontal deflection. The SC-HPFRC and concrete materials were considered to be isotropic. In compression, all materials were modeled as linear elastic as no compressive failure was expected (nor observed) during testing. The concrete panel was assumed to be brittle in tension (tensile strength of 4.1 MPa), while the SC-HPFRC was idealized as multi-linear with hardening and then softening in tension (similar to the ECC in Section 2). For both materials, a total strain fixed crack model was used in tension (Feenstra et al., 1998) with a constant shear retention factor of 0.2 and secant unloading and reloading. The multi-linear tension model for the SC-HPFRC was selected based on an inverse analysis of the performance of unreinforced SC-HPFRC modulus-of-rupture beams (Billington & Olsen, 2009). The steel for the connections was modeled as linear elastic, as designed (E = 200 GPa). The composite

NONLINEAR FINITE ELEMENT ANALYSIS TO SIMULATE HPFRC INFILL PANEL RESPONSE AND PREDICT STRUCTURAL PERFORMANCE 2D plane stress analysis of infill panels in cyclic flexure and shear

An infill panel system using a self-compacting HPFRC (SC-HPFRC) has been proposed to retrofit fracturecritical steel frames for seismic loading (Ol-sen & Billington, 2009) (Figure 8). The infill panels are precast, and intended to be both easily installed and rapidly replaced after an earthquake, if damaged. Panels were tested and proved to be robust under repeated cyclic load. With the SC-HPFRC a ductile, flexure failure was observed rather than a more brittle shear failure as seen in an equivalent panel cast with traditional concrete. In order to best design the infill panels for various retrofit configurations, two-dimensional nonlinear finite element analysis is proposed to evaluate strength, ductility and failure modes of various infill panel designs. Modeling strategies are under investigation and one example is presented here, in comparison to the modeling of a traditional concrete panel. Only measured material properties are used in the initial analyses as the objective of the investigation is to evaluate the ability of the simulation approach to be predictive of behavior.

Figure 8. Ductile fiber-reinforced concrete infill panel system proposed for seismic retrofitting of steel frame structures.

Figure 9. (a) Single panel test set-up, and details of the (b) SC-HPFRC Panel and (c) Concrete panel modeled here.

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connection area where the panel base was sandwiched with the U-channel with pre-stressed bolts and grout was also modeled as linear elastic (E = 69 GPa) since no damage was expected in this region during testing. The modulus of elasticity in the connection region was increased based upon a strain compatibility analysis of the different materials and thicknesses used in the connection. Finally, the reinforcing steel was modeled with the Von Mises elas-to-hardening yield plasticity assumption using measured material properties from uniaxial tensile tests (9.5 mm-dia. bars: E = 200 GPa, fy = 483 MPa, fu = 656 MPa at a strain of 0.2, and WWF: E = 200 GPa, fy = 690 MPa, fu = 828 MPa at a strain of 0.05). Tensile fracture was not modeled. The controlled cyclic displacement pattern that was used in the experiments was also used for the cyclic simulations, except drift cycles were not repeated in the simulations as they were in the experiments. The displacement step size was chosen to avoid numerical divergence of the solution while minimizing computation time. A Newton-Raphson iteration scheme was used with a force norm for the convergence criteria with a tolerance of 0.07%. If convergence could not be reached, the simulations were not allowed to continue. A line search algorithm was also implemented to aid in convergence. Figure 10 compares the hysteretic response of the SC-HPFRC panel experiment with the finite element analysis. Strength is well captured by the simulation. However the stiffness, strength degradation and hysteretic energy dissipation are not well captured with this simplified approach to modeling. The higher simulated stiffness is attributed primarily to connection flexibility and connection slip, which were not modeled using the fixed base assumption. The inability to capture post-peak strength degradation is attributed to the WWF reinforcement being modeled as having perfect bond to the SC-HPFRC. The contribution to shear strength provided by the

WWF, particularly the horizontal bars likely did not occur in the experiment because the WWF could not be fully developed as discussed above. Further evaluation of the WWF is given at the end of this sub-section. Finally, the level of pinching seen in the experiments is not well captured and is attributed to the idealized perfect bond assumption (by using embedded reinforcement) and the use of secant unloading rather than the more accurate ECC-type unloading/reloading proposed by Han et al. (2003). As large cracks form, some level of slip would be expected leading to a more pinched response. Figure 11a shows the principle tensile strain contours at +1% drift (∼9 mm, used as a reference point to compare failure locations and failure modes between the simulations and experiments) with the maximum contour representing 0.002 strain, which is where tensile softening is expected from the multi-linear tensile approximation for the SC-HPFRC. The damage is located near the base and matches the observed experimental results (Figure 11b). The average compressive strain in the compression region was observed to be around 0.001, which is in the elastic range of the material. Figure 12 shows the concrete panel simulation vs. experimental hysteretic response. The simulated panel capacity is accurate in the negative drift direction, and slightly underestimated in the positive direction. The simulated initial stiffness is very close to that seen in the experiments. The concrete panel simulation also uses a brittle tension model instead of the SC-HPFRC multi-linear tension model. The simulation was unable to converge after the peak was reached. Figure 13 shows the principal tensile strains at peak load and at 1% drift (∼9 mm). While there is evidence

Figure 10. Cyclic response of the SC-HPFRC panel experiment and initial simulation.

Figure 11. (a) Simulated principal tensile strains at 1% drift, and (b) final failure of SC-HPFRC panel experiment.

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ECC under cyclic loading with its unique loading and unloading response. Regarding the tensile response of the SC-HPFRC material, performing uniaxial tensile tests to characterize cast material is difficult to standardize and simpler methods are needed to estimate this necessary material response for finite element simulations of structural configurations. As an example of the impact such changes in assumptions can make, Figure 14 shows a monotonic pushover analysis for the SC-HPFRC panel where the full WWF is modeled and where only the vertical WWF is modeled, both compared to the cyclic experimental response. In these simulations, a stronger multi-linear tensile curve for the SC-HPFRC accounts for the difference in strength relative to the simulation shown in Figure 10. However the significant difference is seen in the post-peak behavior when no horizontal WWF steel is included. Finally, for a more rapid evaluation of strength and ductility for various designs, sequentially linear analysis (discussed in Section 4) is currently being investigated for these panel designs.

Figure 12. Cyclic response of the concrete panel experiment and simulation.

3.2

Seismic analysis of a frame structure with HPFRC infill panels modeled as rotational springs

Large-scale hybrid (pseudo-dynamic) testing of the infill panels described in Section 3.1 installed in a two-story steel frame is scheduled for early 2010. The testing involves a physical subassembly of a one bay sub-frame that includes the SC-HPFRC panels, and a numerical subassembly to represent the remainder of the moment resisting frame and modeled in the Open System for Earthquake Engineering Simulation (OpenSees, 2009) platform. Two horizontal translational degrees of freedom at each one of the floor levels of the experimental subassembly are to be controlled during the hybrid test.

Figure 13. (a) Simulated principal tensile strains at 1% drift, and (b) final failure of the concrete panel experiment.

of cracking damage throughout the panel, most damage is concentrated at mid-height and is at a 45-degree angle, indicative of shear cracking and failure, which is expected in the concrete material in this geometry. The location of damage at mid-height is likely influenced by the 9.5 mm-diameter dowel reinforcing bar cutoff at this height (Figure 9c). Both the location and orientation of the most severe damage corresponds to that seen in the experiment. Several improvements to this simplified modeling approach are currently being investigated such as implementing springs at the base to better represent the flexibility in that connection, reducing the bond (or area) for the WWF, which cannot fully develop, evaluating the sensitivity of the panel response to variations in the multi-linear tension model where tension hardening, yielding and tension softening show significant differences in component post-peak behavior, and adopting the Han et al. (2003) material model for

Figure 14. Evaluation of SC-HPFRC panel response with and without horizontal WWF reinforcement.

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Prior to performing the experiment, it is necessary to perform a ‘‘virtual test’’ wherein simulations represent both the portion of the structure to be modeled analytically during testing and the portion to be modeled experimentally. This modeling work is described here. The prototype structure that is used for seismic retrofitting with the SC-HPFRC panels is a two story 3-bay office building with perimeter steel moment resisting frames designed based on 1980s U.S. seismic provisions. The building does not meet the retrofit objectives based on current accepted guidelines in the U.S. A 2/3-scale model of the east west (EW) moment resisting frame (predominant period of 0.75 seconds) of the building is to be retrofit with five SC-HPFRC double panels per story installed in the first bay. The 2/3-scale frame of the prototype frame is designed with a W10 × 45 exterior and interior columns and W14 × 26 and W10 × 30 first and second floor beams, respectively. These sizes are based on similitude laws for strength and stiffness based on Mon-carz and Krawinkler (1981). The predominant period of the retrofitted scale frame is 0.39 sec. The bf /2tf and hf /tw ratios of the selected scaled sections are almost the same with the ones of the prototype steel momentresisting frame, i.e. the deterioration parameters of the components of the scaled frame represent reasonably well those of the prototype frame. The geometry of the physical subassembly is shown schematically in Figure 15 and details are given in Lignos & Billington (2010). Connection details of the panels are described in detail in Olsen and Billington, (2009) and Lignos et al. (2009). Although the prototype frame was designed based on pre-Northridge seismic provisions, the four steel moment connections in the physical subassembly are designed as standard welded, unreinforced flange-bolted web connections per FEMA-350 (2000) provisions to avoid any control instability of actuators during testing due to beam-column joint fracture. Fracture is simulated in the numerical portion of the hybrid model though recognizing the possibility of having fracture(s) at design level earthquake events.

During hybrid testing the two horizontal translational degrees of freedom are the control quantities and the physical subassembly is connected with a link that is designed to behave elastically. Two 975-kN dynamic actuators will impose the computed displacements and also measure the force and displacement quantities from the physical subassembly. The 2-dimensional test frame is modeled in OpenSees with elastic beam-column elements that have concentrated plasticity springs at their ends. The hysteretic response of these springs is bilinear. The springs simulate component deterioration based on the modified Ibarra-Krawinkler model (Lignos and Krawinkler, 2009). Deterioration parameters of the components are determined from relationships for deterioration modeling that were derived from a recently developed database of steel components for deterioration modeling (Lignos and Krawinkler, 2007, 2009). The modified deterioration model is able to simulate brittle fracture with an ultimate deformation parameter θ u that is set to be 2% for one end of the first floor exterior beam, recognizing the possibility of having a brittle fracture at an early inelastic cycle as reported in FEMA-351 (2000). The analytical model developed to capture the hysteretic response of the SC-HPFRC infill panels is shown in Figure 16. Two rigid links are connected together with a hinge connection in the middle that allows vertical movement of one panel with respect to the other. Each infill panel at its fixed end has a concentrated plasticity spring that utilizes the modified Ibarra-Krawinkler deterioration model with peakoriented hysteretic behavior. Experimental data provided by Hanson and Billington (2009) is used to calibrate the infill panel model. The calibrated moment vs. rotation diagram of the SC-HPFRC double panel is shown in Figure 16c. P-Delta effects are simulated numerically with a leaning column that does not contribute to the lateral stiffness of the building. Two percent Rayleigh damping is assigned to the bare steel moment frame. For the retrofitted steel moment resisting frame, 3% Rayleigh damping is assumed in order to consider the effect of the SC-HPFRC panels on viscous damping of the building. Two testing phases are scheduled wherein the same steel moment frame is used but the SC-HPFRC infill panels are replaced between the two phases. In both phases the ground motion records are scaled appropriately to represent levels of intensity that are of particular interest for the engineering profession. The two testing phases are summarized in Table 1. Phase I is concerned with seismic performance of the retrofitted 2-story frame during two subsequent design level earthquakes (DLE). Phase II involves experimental testing with a maximum considered earthquake (MCE) based on the scaled component of the Petrolia record from the Cape Mendocino earthquake

Figure 15. Coupled simulation model of two-story steel frame with SC-HPFRC infill panel retrofit in one bay.

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Figure 16. panels.

Table 1. Level of intensity Phase I Service level Design level Design level

in 1992. After the end of this event the frame is subjected to the unscaled component of the JR Takatori record from the 1995 Kobe earthquake in Japan. Simulation results for the two testing phases are presented here based on a new simulation method that couples two or more displacement-based structural finite element analysis programs together through a generic adapter element approach, which is implemented in the Open-source Framework for Experimental Setup and Control (OpenFresco) (Takahashi and Fenves, 2006; Schellenberg et al., 2007). The numerical subassembly of the 2-story steel moment-resisting frame shown in Figure 15 is analyzed in OpenSees and is connected with a generic Super-Element within OpenSees that represents the physical subassembly tested in the laboratory (referred to as the Master Program). The physical subassembly itself is modeled in OpenSees but as a separate input file (referred to as the Slave Program). The master program imposes boundary conditions on all of the subassemblies. The 2-node adapter element connects to the interface nodes of the physical subassembly in the slave program and is responsible for imposing trial displacements on such subassembly. Details of the couple simulation theory, adapter elements and implementation are given in Schellenberg et al. (2007, 2008). During DLE-I and DLE-II motions of Phase I (Table 1) the retrofitted frame does not exceed a maximum story drift ratio (SDR) of roughly 1.5% compared to 2.5% of the bare frame ground motions. Figure 17 illustrates the drift histories of the bare versus retrofitted frame for DLE-I level of intensity. This result indicates that during the DLE events no fracture occurs in any of the steel moment frame connections. At the end of both DLE-I and II levels of Phase I the residual drift ratios of the retrofitted frame are almost zero compared to a prediction of roughly 0.6% in the bare frame at both stories. After replacing the SC-HPFRC infill panels with new ones (Phase II) and subjecting the retrofitted frame to the MCE level ground motion (105% of the unscaled Petrolia record, Table 1) peak story drift ratios of the retrofitted frame stay below 2.5% at the first story. Due to fracture of the connection at the end location of the first floor beam of the bare frame, the story drift ratio increases in the bare frame are about 30% larger compared to the retrofitted ones indicating that the proposed retrofit system is effective for retrofitting existing steel moment frames with fracture-critical beam-column connections. Similarly, residual story drifts of the retrofitted frame are reduced by roughly 50% compared to those of the bare frame (Figure 18). Based on the ‘‘virtual’’ hybrid simulation for the two scheduled testing phases it was found that: (1) peak story drift ratios of the retrofitted frame are expected to be reduced by roughly 30% compared to the peak story drift ratios of the bare frame when they are both subjected to a design level and maximum considered

Modeling of double-height SC-HPFRC infill

Hybrid testing experimental program. Gr. motion Earthquake Notation intensity record SLE

30%

DLE-I

70%

DLE-II

100%

Phase II Service SLE level Maximum MCE considered level Collapse CLE level

30% 105% 100%

Petrolia (Cape Mendocino, 1992) Petrolia (Cape Mendocino, 1992) Canoga Park (Northridge, 1994) Petrolia (Cape Mendocino, 1992) Petrolia (Cape Mendocino, 1992) JR Takatori (Kobe, 1995)

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4

SEQUENTIALLY LINEAR ANALYSIS OF ECC IN FLEXURE

For large-scale structural analyses, sequentially linear finite element analysis has been proposed as an alternative to nonlinear finite element analysis (Rots & Invernizzi, 2004). To remain a simple modeling approach, sequential linear analysis is most useful for monotonic loading. Applied to structures using ductile cement-based materials, sequential linear analysis has the potential to simplify the evaluation of alternate designs of large-scale applications, such as the new designs proposed in Section 2 or the retrofit system presented in Section 3, where a monotonic pushover analysis could be employed for initial design evaluations in terms of panel design details and locations within a steel frame. An advantage of using sequential linear analysis would be that detailed behavior (e.g. local damage, failure modes) of the ductile cementbased materials could be obtained relatively easily (e.g. no convergence issues) using basic material properties as input. The evaluation of alternate tensile properties, as demonstrated in Section 2, or alternate reinforcing details as performed experimentally for the infill panels in Section 3, could be performed through simulation to identify the best material and component designs for desired structural-scale behavior. Sequentially linear finite element analysis has been evaluated for modeling traditional fiber-reinforced concrete, which shows tensile-softening behavior after cracking (Belletti et al., 2008) and more recently tensile hardening materials such as ECC (Billington, 2009). Here, a comparative analysis of small-scale slender beams made of ECC and tested in four-point bending is performed using nonlinear analysis with smeared cracking and using sequentially linear analysis. The beams to be modeled are shown in Figure 19 and displacement was measured at mid-span (details given in Billington, 2009). For the nonlinear analyses, two-dimensional, 8-noded, plane stress elements of constant thickness and a 3 × 3 Gaussian integration scheme were used to model the beam with 4 elements along the height and 48 along the length. The ECC was modeled as linear elastic in compression as no softening in compression was expected nor observed in the experiments. Young’s modulus was measured from an average of 4 cylinders tested in compression to be E = 11,032 MPa. A total strain-based fixed crack model (Feenstra et al., 1998)

Figure 17. Comparison of story drift ratio histories between the bare and retrofitted 2-story frame during design level earthquake I of testing Phase I.

Figure 18. Comparison of story drift ratio histories between bare and retrofitted 2-story frame during maximum considered level earthquake of testing Phase II.

event. The implication is that brittle fracture of beamto-column moment connections does not occur during design level events and is delayed during maximum considered earthquake events, and (2) residual deformations of the retrofitted frame are expected to be zero for a design level event and reduced by roughly 50% during a maximum considered event. Hybrid simulation testing of the retrofitted 2-story steel moment frame is under way at the Network for Earthquake Engineering Simulation (NEES) experimental facility at The University of California at Berkeley in order to confirm and improve pre-test analytical simulations and validate if the proposed SC-HPFRC infill panel retrofit system and modeling approach can be used to design retrofits of existing steel moment frames.

Figure 19.

ECC flexural beam test set-up.

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was used with a trilinear tensile stress-strain response assumed based on previous tensile tests of ECC dogbone specimens and defined by the following four stress-strain points: 1) (0,0), 2) (3.1 MPa, 0.000281), 3) (4.1 MPa, 0.025), and 4) (0, 0.05). A shear retention factor of 0.05 was used for the cracking and Poisson’s ratio was taken as 0.15. While the shear retention factor for concrete is often taken to be 0.2 to represent, for instance, aggregate interlock and dowel action of reinforcement across crack surfaces, a lower value of 0.05 is adopted here because the materials contain no steel reinforcement or coarse aggregate. This value 0.05 for shear retention did not cause ill-conditioning of the stiffness matrix. The analyses were run in displacement control with a step size of 0.05 mm for 100 steps followed by 100 steps of 0.025 mm, all applied at the loading point. The nonlinear analyses used a regular NewtonRaphson iteration scheme, including line estimation, with a maximum of 30 iterations. Both a displacement and force tolerance of 1% were used for the convergence criteria. For the sequential linear analyses, the same finite element model for the nonlinear analysis was used as described above, with the exception of using a 2 × 2 integration scheme rather than 3 × 3 in the elements. The ECC was modeled as linear elastic in compression as no softening in compression was expected nor observed in the experiments. Young’s modulus was measured from an average of 4 cylinders tested in compression to be E = 11,032 MPa. In tension, the same multi-linear tensile stress-strain curve from the nonlinear analysis was adopted for the baseline curve. To preserve fracture energy dissipation, a sawtooth model using an upper and lower bound curve, relative to the baseline curve, was adopted as shown in Figure 20. The upper and lower bound curves for the tension hardening and tension softening regions have the same stiffness as the baseline curve and represent a roughly 15% increase and decrease in stress at first cracking and peak cracking strength. Twenty stiffness reductions (similar to damage states) resulted to represent the full stress strain response. The grey triangles in Figure 20 highlight the preservation of fracture

Figure 20.

energy dissipation using the sawtooth model, whereby the triangle above the baseline curve is ‘‘extra’’ energy dissipated by loading along E2 up to the upperbound curve, and the triangle below the curve is the ‘‘missing’’ energy dissipation upon reloading along E3, again to the upperbound curve. These two triangles are roughly equal, thus allowing for the preservation of fracture energy dissipation with reasonable accuracy. Again, a shear retention factor of 0.05 was adopted and Poisson’s ratio was taken as 0.15. The linear analyses were run in load control as explained above and were repeated for 2000 steps. Figure 21 compares the load-displacement response of the nonlinear analysis, the sequentially linear analysis and the three experiments on the ECC beams. The nonlinear analysis failed to converge in the second step after the peak load was reached, which coincided with the first integration point to reach the softening branch of the ECC constitutive model. The sequential linear analysis captured the likely snap-back experienced during failure (although snapback was not attempted to be measured experimentally). In sequentially linear analysis, only one critical point is identified in each analysis and as a result, an asymmetric failure mode can easily be captured, which is representative of experimental results. However, the nonlinear analysis will typically predict symmetric failure modes if no prior imperfections are introduced (Figure 22). The colors (shading) in Figure 22 represent different regions in the ECC tensile strain response as labeled in the figure. Both analysis methods show that at this displacement, the ECC has experienced significant multiple cracking but that failure has localized in one or a pair of cracks. Upon localization of the failure cracks and unloading of the beam, the regions of the beam that had been tensile-hardening also unloaded in both analyses (top portion of beam, between loading points) as observed in experiments. In Figure 23, the mesh sensitivity of the sequentially linear analysis is observed when comparing the original analysis (mesh of Figure 20) with an analysis

Figure 21. Load-displacement response of experiments, and nonlinear and sequentially linear analyses.

Saw-tooth modeling of ECC in tension.

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The sequential linear analysis is able to capture likely snapback behavior after crack localization. In the nonlinear analysis, a more advanced solution procedure would be required to capture the snap back. Mesh dependence was observed in the sequential linear analysis for the ECC beams, as the adopted crack model was not regularized for mesh size. A regularization procedure for tensile hardening-softening materials may be required but further study is warranted. 5

CONCLUSIONS

Numerous simulation methods exist for modeling ductile fiber-reinforced cement-based materials. For structural applications, it is desirable to have simple modeling methods that can be used for rapid evaluation of alternate component designs and structural configurations using these ductile cement-based materials. Three different modeling approaches have been reviewed here including simulations of structural applications for new design as well as seismic retrofitting of existing structures. Using measured material properties and standard assumptions, nonlinear finite element analysis was demonstrated to capture well the cyclic response of precast segmental bridge piers using unbonded posttensioning and ECC in hinging regions, a complex combination of materials and components. It was demonstrated that this modeling approach can be used to identify optimal ECC properties to achieve a variety of structural responses, including a more robust, ductile response as would be desired for seismic design. A similar approach using nonlinear finite element analysis to capture the response of SC-HPFRC as well as traditional concrete infill panels to retrofit steel frames was observed to predict strength and failure modes well. However in the absence of detailed information on bond-slip characteristics of smooth and deformed steel reinforcement in SC-HPFRC materials, softening and hysteretic energy dissipation is not well predicted. Furthermore it was observed that panel simulation response post-peak is particularly sensitive to the presence of the smooth welded wire fabric reinforcement, whose contribution is not clear given the limited of bond and development possible. When experimental data is available, phenomenological rotational spring models can be adopted and an example application of such models to predict the seismic performance of a retrofitted frame was demonstrated here to limit beam-column joint fracture in a fracturecritical steel frame and reduce residual displacements after a seismic event. Finally, it was demonstrated that sequential linear analysis could be used to predict the tensile-hardeningsoftening performance of ECC in flexure. In comparison with a nonlinear analysis using a total-strain based

Figure 22. Principle strain contours of beams at failure (roughly zero load).

Figure 23. Load-displacement response of ECC beams from sequential linear analysis and two mesh sizes.

performed on a beam with twice as many elements in each direction. Adopting the same saw-tooth softening model as for the original mesh, one would expect to see a more brittle response using the fine mesh, which is observed in Figure 21. A regularization procedure for tensile hardening-softening materials may be needed but further study, particularly for ECC materials reinforced with conventional reinforcement is warranted. In summary, both the nonlinear finite element analysis with a smeared, fixed crack model based on total strain, and the sequential linear finite element analysis are able to capture the load-displacement response of deflection hardening ECC beams well using a tensile hardening-softening crack model. In both analyses, similar extents of multiple cracking were observed and softening of the ECC began at the same displacement (and load). A symmetrical pair of localized cracks formed in the nonlinear analysis due to the symmetry of the beam, loading and test set-up. With the sequential linear analysis, an asymmetrical failure mode, as expected experimentally is easily captured.

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fixed crack model, both analyses predict deflectionhardening response well. Unlike the nonlinear analysis, the sequential linear analysis is able to predict an asymmetric failure mode (as expected experimentally) as well as snap-back behavior with little computational effort. Applying sequential linear analysis to large-scale structural evaluations of ductile cementbased materials could prove to be an easy-to-adopt approach for designers and researchers to study both local and global behavior of a wide variety of material and structural designs.

Modelling of Concrete Structures, Proceedings of EURO-C 1998, de Borst, Bicanic, Mang & Meschke (eds), Balkema, Rotterdam, 13–22. Han, T.S., Feenstra, P.H. and Billington, S.L. (2003). Simulation of Highly Ductile Cement-Based Composites. ACI Structural Journal, 100(6): 749–757. Hanson, J.V. and Billington, S.L. (2009). Cyclic testing of a ductile fiber-reinforced concrete infill panel system for seismic retrofitting of steel frames, Report TR. 173, John Blume Earthquake Engineering Center, Stanford, CA. Hung, C-C. and El-Tawil, S. (2009). ‘‘Cyclic Model for High Performance Fiber Reinforced Cementitious Composite Structures,’’ Proc. ATC-SEI conference, Improving the Seismic Performance of Existing Buildings and Other Structures, December 9–11, 2009 San Francisco, CA. Kabele, P. (2007). Multi-scale framework for modeling of fracture in high performance fiber reinforced cementitious composites, Engineering Fracture Mechanics, 74: 194–209. Kesner, K.E., Billington, S.L. and Douglas, K.S. (2003). Cyclic Response of Highly Ductile Fiber-Reinforced Cement-Based Composites. ACI Materials J., 100(5): 381–390. Kyriakides, M.A., Hendriks, M.A.N. and Billington, S.L. (2010). Simulation of Masonry Beams Retrofitted with Engineered Cementitious Composites, Computational Modeling of Concrete Structures, Proc. EURO-C 2010, de Borst, Bicanic, Mang & Meschke (eds), March 15–18, Rohrmoos, Austria. Lee, W.K. and Billington, S.L. (2008). ‘‘Simulation of Self-Centering Fiber-Reinforced Concrete Columns,’’ Proceedings of ICE, Engineering and Computational Mechanics, 161(2): 77–84. Lee, W.K. and Billington, S.L. (2010). Simulation and Performance-Based Earthquake Engineering Assessment of Self-Centering Post-Tensioned Concrete Bridge Systems, Research Report, Pacific Earthquake Engineering Research (PEER) Center, to appear. Li, V.C. (2003). On Engineered Cementitious Comopsites (ECC)—A Review of the Material and Its Applications, J. Advanced Concrete Technology, 1(3): 215–230. Li, V.C. and Leung, C. (1992). Steady-State and Multiple Cracking of Short Random Fiber Composites, Journal of Engineering Mechanics, 118(11): 2246–2264. Liao, W.-C., Chao, S.-H., Park, S.-Y. and Naaman, A.E. (2006). ‘‘Self-Consolidating High Performance Fiber Reinforced Concrete (SCHPFRC)— Preliminary Investigation,’’ Report UMCEE 06-02, Dept. of Civil & Env. Engineering, Univ. of Michigan, Ann Arbor, MI, 76 pp. Lignos, D.G. and Billington, S.L. (2010). Hybrid Testing of a Retrofitted Steel Moment Resisting Frame with High Performance Fiber Reinforced Concrete Infill Panels, 9th US National & 10th Canadian Conference on Earthquake Engineering, Toronto, Canada, July. Lignos, D.G., Hunt, C.M., Krebs, A.D. and Billington, S.L. (2009). Comparison of retrofitting techniques for existing steel moment resisting frames, Proceedings ATC/SEI Conference, San Francisco, CA, December. Lignos, D.G. and Krawinkler, H. (2007). A database in support of modeling of component deterioration for collapse prediction of steel frame structures, ASCE Structures Congress, Long Beach CA, SEI.

ACKNOWLEDGEMENTS Financial support provided by the National Science Foundation (CMS-0530383), by the Pacific Earthquake Engineering Research (PEER) Center, and the Faculty of Civil Engineering at the Technical University of Delft, The Netherlands is gratefully acknowledged. The modeling research presented herein was carried out by the author as well as graduate research assistants Dr. Won Lee, Cole Olsen and post-doctoral scholar Dr. Dimitrios Lignos. The opinions expressed in this paper do not reflect those of the financial sponsors. REFERENCES American Concrete Institute. (2005). Building Code Requirements for Structural Concrete and Commentary. ACI 318-05 and ACI 318R-05, Farmington Hills, Michigan, 2005. Belletti, B., Hendriks, M.A.N. and Rots, J.G. (2008). Finite Element Modelling of FRC Structures —Pitfalls and How to Avoid Them, 7th Int’l RILEM Symp. on Fibre Reinforced Concrete: Design and Applications, BEFIB-2008, Chennai, India, September. Billington, S.L. (2009). ‘‘Evaluation of Sequentially Linear Finite Element Analysis to Simulate Nonlinear Behavior in Mortar and Engineered Cementitious Composites in Flexure,’’ ACI Special Publication 265-12, November. CEB-FIP. (1990). Bulletin d’information, CEB, Lausanne, Switzerland. Chuang, E. and Ulm, F.-J. (2002). Two-phase composite model for high performance cementitious composites, J. Engineering Mechanics, 128(12): 1314–1323. Federal Emergency Management Agency—FEMA 350 (2000). Recommended seismic design criteria for new steel moment frame buildings, Washington, DC, July. Federal Emergency Management Agency—FEMA 351 (2000). Recommended seismic evaluation and upgrade criteria for existing welded steel moment-frame buildings, Washington, DC: Federal Emergency Management Agency, July. Feenstra, P.H., Rots, J.G., Arnesen, A., Teigen, J.G. and Hoiseth, K.V. (1998). A 3D constitutive model for concrete based on a con-rotational concept. Computational

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Saiidi, M.S. and Wang, H. (2006). Exploratory Study of Seismic Response of Concrete Columns with Shape Memory Alloys Reinforcement, ACI Structural J., 103(3): 435–442. Schellenberg, A.H. (2008). ‘‘Advanced implementation of hybrid simulation’’, PhD Dissertation, Civil and Environmental Engineering, University of California, Berkeley. Schellenberg, A., Mahin, S. and Fenves, G. (2007). Software framework for hybrid simulation of large structural systems, Proceedings, Structures Congress, ASCE, Long Beach, CA, United States. Sirijaroonchai, K. and El-Tawil, S. (2007). Three Dimensional Plasticity Model for High Performance Fiber Reinforced Cement Composites, Proc. HPFRCC-5, H.W. Reinhardt and A.E. Naaman eds., Mainz, Germany, July,. Suwada, H. and Fukuyama, H. (2006). Nonlinear Finite Element Analysis on Shear Failure of Structural Elements using HPFRCC, J. Advanced Concrete Technology, 4(1): 45–57. Takahashi, Y. and Fenves, G. (2006). Software framework for distributed experimental computational simulation of structural systems, Earthquake Eng. and Structural Dynamics, 35(3): 267–291.

Lignos, D.G. and Krawinkler, H. (2009). Sidesway collapse of deteriorating structural systems under seismic excitations, Report TR 172, John A. Blume Earthquake Engineering Center, Stanford University, Stanford, CA. Moncarz, P.D. and Krawinkler, H. (1981). Theory and application of experimental model analysis in earthquake engineering, Report No. 50, John A. Blume Earthquake Engineering Center, Stanford University, Stanford, CA. Olsen, C. and Billington, S.L. (2009). Evaluation of precast, high-performance fiber-reinforced concrete infill panels for seismic retrofit of steel frame building: Phase 1-cyclic testing of single panel components, Technical Report No. TR 158, John A Blume Earthquake Engineering Center, Stan-ford University, Stanford CA. OpenSees. (2009). ‘‘Open System for Earthquake Engineering Simulation,’’ Pacific Earthquake Engineering Research Center (PEER), (http://opensees.berkeley.edu). Rots, J.G. and Invernizzi, S. (2004). Regularized sequentially linear saw-tooth softening model, International Journal for Numerical and Analytical Methods in Geomechanics, 28: 821–856. Rouse, J.M. (2004). Behavior of Bridge Piers with Ductile Fiber Reinforced Hinge Regions and Vertical Unbonded Post-Tensioning, PhD thesis, Cornell Univ., New York.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Recent developments on computational modeling of material failure in plain and reinforced concrete structures A.E. Huespe, J. Oliver & G. Díaz Technical University of Catalonia, Campus Nord UPC, Barcelona, Spain

P.J. Sánchez INTEC-UNL-CONICET, Santa Fe, Argentina

ABSTRACT: New developments for the computational simulation of plain and reinforced concrete structures are presented. Two models based on different length scales are proposed: one, at the macroscopic level, considers a homogenized material. Another, at the mesoscale level, describes specifically each mechanical component of the reinforced concrete members. Numerical applications of both approaches addressed to different problems are presented. Particular attention is given to the plain concrete simulation in confined regimes in order to capture the high stress triaxiality compressive states, which are typical of reinforced structures. The plain concrete is described via a new phenomenological model which can be characterized by a low number of parameters. 1

plane concrete constitutive model which is capable of capturing the wide range of failure modes observed in these structures. For example, the (distributed or concentrated) crack patterns displayed in tensile stress zones of RC structural members and the crushing effect produced in zones of high stress confinement. In order to take into account this characteristic response, in subsection 2.2 we present a new phenomenological constitutive relation for concrete that is well adapted for capturing this salient behavior. Its regularization is performed via the Continuum StrongDiscontinuity Approach (CSDA) Oliver, (2000) summarized in section 3. Numerical applications of this model are presented in section 4. The mesoscopic scale approach is adopted for simulating corroded reinforced concrete structures. In these structures, the effect of the slip fiber-concrete mechanism plays a crucial role in the structural strength. We show in section 5 that, although the corrosion problem is essentially a 3D phenomenon, it is amenable to be analyzed as a 2D problem via a two stages procedure. Finally, in section 6, some concluding remarks close this work.

INTRODUCTION

Reinforced concrete (RC), constituted by concrete with long fibers (reinforcements) oriented in different directions embedded in it, can be analyzed following two different conceptual models implying different length scales: i. a macroscopic scale model describing the response of the composite material via a homogenized constitutive model. In this case, the success of the model relies on the homogenization procedure, which becomes the key issue in this type of conceptual approach; ii. a mesoscopic scale model describing the response of every composite material constituent (matrix, interface and reinforcement) as an independent subsystem that is mechanically interacting with the neighbor constituents. The numerical simulation of mesoscopic models requires demanding computational costs. In this contribution, we present applications of both conceptual models addressed to analyze RC structures. Following the macroscopic scale approach, it is possible to develop a rather simple homogenization procedure based on the mixture theory, which provides a computational model that captures the most salient phenomena governing the failure of reinforced concrete structures. In this case, an important feature is that it only requires a reduced computational effort. This procedure is described in section 2. In any case, whatever is the adopted conceptual approach, it is necessary to include a regularized

2

MACROSCOPIC APPROACH BASED ON A HOMOGENEIZED RC MODEL

We summarize in this section a macroscopic model based on the mixture theory, which is adopted as a procedure for homogenization of the RC material response. This model was initially proposed by Linero, (2006), see also Oliver et al., (2008b), FOR additional details.

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2.1

Constitutive model for the composite

Reinforced concrete is assumed to be a composite material constituted of a matrix (concrete) and long fibers (steel rebars) arranged in different directions, as shown in Figure 1. According to the basic hypothesis of the mixture theory, the composite material is a continuum where each infinitesimal volume is occupied by all the constituents in terms of a volumetric fraction given by the factor k i ≤ 1 (for the i-th constituent). Assuming a parallel layout, all constituents are subjected to the same composite deformation ε. The composite stress, σ, is obtained by summing up the stress of each constituent, weighted according to its corresponding volumetric fraction k i , as shown in Figure 2. Thus, the matrix strain, εm , coincides with the composite strain, ε: εm = ε

(1) Figure 2. Constitutive model for the composite using a mixf f ture theory:  m , σ , τ are the constitutive material laws for the matrix, the fiber uniaxial effect and dowel action constituents, respectively.

Considering a fiber f in the direction r, a local orthogonal reference frame (r, s), can be associated with it, as shown in Figure 1. Then, the extensional strain, εf , of the fiber f is equal to the component εrr of the composite strain field in that direction, that is: εf = r · ε · r

rf (f = 1, 2, . . ., nf ) can be obtained using the following weighted sum of each contribution:

(2) σ = k m σm (εm ) +

γf =r·ε·s 2

k f σ f (εf )(r f ⊗ r f )

f =1

In order to take into account the dowel action, the fiber shear strains, γ f , are obtained as the shear components of the composite strain field. The fiber shear f component εrs is given by: εfrs =

nf 

+ 2τ f (γf )(rf ⊗ sf )sym

(4)

where k m and k f , are the volumetric fraction of the matrix and fiber f , respectively, σm is the matrix stress tensor, σf is the fiber axial stress, and τf is the fiber shear stress component. In eq. (4) it is assumed that the normal and tangential stress components of the fibers are related to the corresponding strains by means of specific constitutive equations in a completely decoupled behavior of the matrix response. The incremental form of the composite constitutive equation can be written as:

(3)

The stresses of a composite having nf fibers (or fiber bundles) oriented in different directions

σ˙ = Dtg : ε˙ Dtg = k m Dm tg +

nf 

f

k f [Etg (rf ⊗ rf ) ⊗ (rf ⊗ rf )

f =1

+

f 4Gtg (rf

⊗ sf )sym ⊗ (rf ⊗ sf )sym ]; f

Figure 1.

(5) f

m m f f where: Dm tg = ∂σ /∂ε ; Etg = ∂σ /∂ε ; Gtg = f f ∂σ /∂ε are the tangent moduli for the involved constitutive relations of concrete and fiber respectively.

Reinforced concrete as a composite material.

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2.2

Constitutive model for concrete

Table 1. Constitutive damage-plastic model for the concrete constituent.

Concrete shows very different responses for either tensile or compressive stress regimes. In each case, the most striking difference is observed in the failure mode. Under tensile stresses, the concrete displays a much lower strength than in compressive states. Also, it shows a higher brittleness in tensile stress conditions. In fact, formation of cracks is expected only if tensile states are observed. While in compressive states, the concrete behaves like a plastic material, sometimes displaying failure mechanisms like shear bands. Additionally, in reinforced concrete structures, concrete is generally subjected to high confinement stress states due to stirrups, a condition that plays a very important role in the structural strength. Therefore, it is advisable to use a concrete constitutive relation having the ability to capture the phenomenology observed under both tensile and compressive stress conditions. Previous concrete models addressed to capture this phenomenology where reported by Feenstra & de ˇ Borst, (1996) and Cervenka & Papanikolaou, (2008). The present constitutive model for the concrete matrix is described by a phenomenological damageelasto-plastic law, with the following main features:

Stress strain relationship: q σ = (1 − d)C : (ε − εp ) = σ¯ r Brittle-Ductile behavior: If (σm ≥ 0) or (σm = 0) & Tr( C : ε˙ ≥ 0) then Damage model, only possible evolution of variable d(˙εp = 0) ft ft r|t=0 = r0 = √ ; fc = ; η E √ −1 g(σ , q) = τσ − q; τσ = σ : C : σ;

r˙ = λ;

q˙ = H (r)˙r ; λ ≥ 0;

H < 0;

g ≤ 0;

qo = ro

λg = 0;

else if (σm < 0) or (σm = 0) & Tr(C: ε˙ < 0) then Elasto-Plastic model, only possible evolution of variable ε˙ p (d˙ = 0) rˆ (e, θ ) ψ(σ, ξ ) = Sβ + ξ(α)σm − km ; 2 ξ(α) ∈ [ξo , ξmax ]; ξ > 0; 1 S2 + χ ξ G σm − kmG ; 2 G ]; ξ G ∈ [ξoG , ξmax ∂G ∂G ; α˙ = −γ G ε˙ p = γ ∂σ ∂ξ γ ≥ 0; ψ ≤ 0; γ ψ = 0 end G(σ, ξ G ) =

i. the fracture phenomenon, typical of tensile stress states, are described by an isotropic damage evolution law, identical to the model presented in Oliver, (2000), which is regularized by the CSDA methodology. The damage variable evolution is only admissible in tensile states that are identified by the mean stress, σm , being positive (σm ≥ 0). In this case, the material softening, caused by the damage evolution, induces material instability and strain localization, this being the precursor mechanism for the crack formation; ii. in compressive stress regimes, (σm ≤ 0), the concrete behaves like a plastic material that follows closely the Willam’s elastoplastic model, see Willam & Warnke, (1974). There is no damage evolution and, furthermore, the plastic model is equipped with a hardening modulus in order to induce a stable material response.

ii. λ and γ are standard damage and plastic multipliers; iii. ft and fc are the uniaxial tensile and compressive ultimate stress, respectively, and η = ft /fc is a material parameter governing the ratio between them (fc  ft ). iv. (r, α) is the set of scalar, strain-like, internal variables, r being related with the damage model and α with the lastic model, respectively. v. (q, ξ ) is the set of scalar stress-like internal variables; from the thermodynamic point of view there is a duality relation between q and r. vi. The standard damage variable is defined by: d = 1 − q/r. Damage and plastic strain evolutions are mutually exclusive. Decision of which inelastic mechanism evolves depends only on the sign of σm . vii. In tensile regimes, while the material is not completely degraded, it is verified that: 0 < q/r ≤ 1. Thus, the damage criterion is given by the function: g(σ, q) = 0. Geometrically, the surface g = 0, as defined in Table 1, is an ellipsoid in the Haigh-Westergaard’s space. The intersection between this surface and the octahedral plane is

In Table 1, the governing basic equations of the damage-plastic model are presented. The notation and some important observations are the following: i. σ, ε and εp are, respectively, the Cauchy stress, the total strain tensor and the plastic component of the strain tensor; σ¯ is the effective stress, associated with the damage model, being defined by the relation: σ¯ = C : (ε − εp ). The mean stress is: σm = tr(σ)/3 and the deviatoric stress tensor: S = dev(σ). With the symbol C we refer to the fourth order isotropic elastic Hooke’s tensor. E is the Young’s modulus;

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√ a circle whose radius is: R = S = ( 2μ)q. During damage evolution, q → 0, and therefore the radius R tends to zero, or equivalently, the circle shrinks to a point. viii. The plastic yield surface is given by ψ(σ , ξ ) = 0. This surface intersects the octahedral plane as the circle of radius: R = S = (2km )1/β , since rˆ = 1 whenever σm = 0 (see next point). Thus, defining the parameter km through the simple restriction  2μq = (2km )1/β (6)

Note that rˆ depends on the eccentricity function e(σm ) which, in the present work, is empirically defined in terms of the mean stress value as follows: e(σm ) = eˆ + [1 − eˆ ] exp(3σm ); e(σm = 0) = 1;

  √ 1 3 3J3 θ = arccos ; 3/2 3 2J2 J3 = det(S);

rˆ (e = 1, θ) = 1

J2 =

1 (S : S) 2

(9)

x. The plastic-flow evolution is non-associative. In order to define this law, the potential function G(σ, ξ G ) is included. For determining the plastic potential surface G(σ, ξ G ) = 0 the parameter kmG (which is immaterial) could be computed from the equation:  2μq = (2kmG )1/2 (10)

θ + (2e ; 2(1 − e2 ) cos θ  +(2e − 1) 4(1 − e2 ) cos2 θ + 5e2 − 4e

rˆ = 

(8)

where eˆ is the eccentricity coefficient. Following to Kang, (1999), this parameter must satisfies: eˆ ∈ [0.5, 1]. In this contribution we have adopted eˆ = 0.54 Also, rˆ depends on the Lode’s angle θ defined by:

we force the continuity of both surfaces g(σ, q) = 0 and ψ(σ, ξ ) = 0 onto the octahedral plane for σm = 0, for any parameter q and, therefore, for any value of the damage variable d. Characterization of ξ is given in the next section. ix. The function rˆ (e, θ ), due to Willam & Warnke, (1974) (see also Kang, (1999)), defines the roundness of the plastic yield surface octahedral planes, see Figure 3-b, and it is given by: 4(1 − e2 ) cos2

e(σm → −∞) = eˆ

− 1)2

(7)

Characterization of ξ G is given in the next section. The scalar factor χ accounts for the degree of dilatancy introduced in the model. xi. From the discussed expressions it is clear that the evolution of damage modifies the plastic yield surface ψ = 0 as well as the potential surface G = 0, but it does not modify the set of plastic internal variables {ε p , α}. On the other hand, and since the hardening of the elasto-plastic model affects only the pressure dependent term, the evolution of plasticity have not any influence on the damage constitutive relation, neither in the value of their internal variables {q, r} nor in the evolution of its limit surface g = 0. 2.2.1 Parameter characterization for the concrete model A salient feature of this model is the relatively small number of parameters necessary to characterize the material. Similarly to the isotropic damage model presented in Oliver, (2000), the required damage model parameters, are the standard elastic parameters: Young’s modulus E and Poisson’s ratio ν; the tensile strength ft , and the fracture energy Gf which identifies the intrinsic softening parameter H¯ as will be explained in the next section. Furthermore, the plastic model is characterized by the (pseudo) cohesion km which is fully defined

Figure 3. Reinforced concrete model; a) damage and plastic yield surfaces in the meridian plane; b) cross-sections of the plastic yield surface with octahedral planes.

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through the variable q, as indicated in eq. (6). The parameter β adjusts the failure surface profile in the plane (σm S), see Figure 3-a. By using several experimental works of concrete failure in confinement stress regimes, see for example the literature cited in Kang, (1999), a good fitting is obtained with β = 1.48. In order to preserve the material stability, the hardening function ξ(α) is a monotonous increasing function with an asymptotic limit: ξmax = lim α→∞ ξ(α) as shown in Figure 4. ξmax can be characterized through a uniaxial compression test with an ultimate compressive strength given by fc , while ξo , which determines the initial yield condition, is identically characterized using a uniaxial compressive strength of value: fc /3. With this material model is possible to reproduce reasonably well a number of classical and well known uniaxial, biaxial and triaxial concrete tests. For example, Figure 5 displays the classical biaxial experimental results reported by Kupfer in 1973. We remark

the close description of the model, particularly, in the second and fourth quadrants. 2.3

Constitutive model for the steel fiber

Steel fibers are regarded as one-dimensional elements embedded in the matrix. They can contribute to the composite mechanical behavior introducing axial or shear strength and stiffness. The axial contribution of each fiber bundle depends on its mechanical properties and the matrix fiber bond/slip behavior. The combination of both mechanisms is modeled by the slipping-fiber model described below. 2.3.1 Bond-slip effect The fiber axial contribution can be modeled through one-dimensional constitutive relations, relating extensional strains with normal stresses. The assumed compatibility between matrix and fiber strains allows capturing the slip effect due to the bond degradation by means of a specific strain component associated with the slip. Thus, the fiber extensional strain, ε f , given by eq. (2), can be assumed as a composition of two parts: one due to fiber mechanical deformation, ε d , and the other related to the equivalent relaxation due to the bond-slip in the matrix-fiber interface, εi : εf = ε d + ε i

(11)

Assuming a serial composition between fiber and interface, as illustrated in Figure 6-a, the normal stress of the slipping-fiber model, σ f , is equal to each component of stress:

Figure 4. Reinforced concrete model; description of the function ξ(α) governing the plastic hardening.

σf = σd = σi

(12)

The stress associated to the fiber elongation, as well as the one associated with the matrix-fiber slip effect, can be related to the corresponding strain component by means of a uniaxial linearly elastic/perfectly plastic constitutive model as shown in Figure 6-b with material parameters defined from the composition of both models: Ef =

1/E d

1 ; + 1/E i

i σyf = min[σyd , σadh ]

(13)

in which E d and σyd are the Young’s modulus and yield stress of the steel, respectively, E i is the interd face elastic modulus and σadh is the interface bond limit stress, which is an upper bound for the fiber—matrix f interface adherence: σy .

Figure 5. Reinforced concrete model; profile of the failure criteria (ξ = ξmax ) under biaxial conditions. Results with the proposed model are plotted in solid line.

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f

τ˙ f = Gtg γ˙ f

(14) f

where the shear tangent modulus Gtg is given by: ⎧ f ⎪ ⎨G f Gtg = Gf H f τ ⎪ ⎩ f G + Hf τ

(elastic/unloading) (loading)

(15)

The constitutive elastic shear modulus, G f , is given f by: G f = E f /2 and the yield shear stress, τy , is given f f √ by τy = σy / 3 . The fiber shear hardening/softening modulus is commonly assumed as H f τ = 0. 3

CONSTITUTIVE REGULARIZATION VIA CONTINUUM-STRONG DISCONTINUITY APPROACH (CSDA)

Introducing strain softening (H (r) < 0) in the damage model, forces the consideration of some type of constitutive regularization technique in order to preserve the mathematical and physical consistency of the (local) Boundary Value Problem. For this purposes, the so called Continuum Strong Discontinuity Approach (CSDA) was adopted in the present work. The basic foundations and theoretical concepts about the CSDA can be found in many previous contributions, see for example Oliver et al., (2002) and references cited therein. Here, we only summarize its main features. As a consequence of material softening, strain localization modes and (probably) macro discontinuities (like fractures or cracks) can develop in the concrete under tensile regimes. This particular phenomenology is considered in the CSDA by means of an enhanced kinematical representation which introduces additional discontinuous modes in the displacement field. Such kinematics (commonly know as Strong Discontinuity Kinematics) can be formally described in terms of the nomenclature displayed in Figure 7.

Figure 6. Slipping-fiber model: the composition (+) of both elements (d) and (i) must be understood as a serial mechanical system, in the sense that, deformations are additive (ε. f = ε d + εi ) and the stresses are common (σ f = σ d = σ i ).

2.3.2 Parameter characterization of the bond— slip model: slipping-fiber model The parameters required for the bond/slip model characterization can be calibrated from the pull-out test, in which a bar embedded into a concrete core is subjected to a force applied at its free end. Additional details can be seen in Linero, (2006). 2.3.3 Constitutive model of rebars in shear mode (shear-resistant fiber). Dowel action model In a reinforced concrete member, when cracks open in mode II, the internal locking between particles (aggregate interlock) withstands some shear forces across the crack interfaces. The steel bars also introduce an important contribution to the shear strengthening effect, known as dowel action. This phenomenon has been widely studied and several authors have included this effect into their numerical simulations, see Belletti et al., (2001; Kollegger & Mehlhorn, (1990; Pietruszczak & Winnicki, (2003). Based on the previous considerations, the dowel action is modeled by means of a one-dimensional shear stress-strain elasto-plastic constitutive model, similar to the previously mentioned one for the axial stressstrain. In this case, the 1D constitutive model relates the fiber shear stress, τ f , associated to the local coordinate system (r, s) (see Figure 1), with the corresponding fiber shear stain, γ f , by means of:

3.1

Strong discontinuity kinematics and its regularization

Let us consider a body  displaying a discontinuous displacement field across the surface S with normal n and splitting the body  into two disjoints parts, + and − . An admissible displacement field, u(x), exhibiting discontinuities can be described by: u(x, t) = u(x, t) + HS (x) · [[u]] (x, t)

(16)

where u(x) is a smooth field, HS is the Heaviside/step function shifted to S. The second term captures the displacement jump, given by [[u]], at the discontinuity interface S.

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path. Additional details of this particular aspect can be seen in Oliver, (2000) and Oliver et al., (2002).

4 4.1

Figure 7.

The strain field, kinematicaly compatible with the discontinuous displacement field u(x), is given by: sym ε = ∇ sym u = ∇ sym S n ⊗ [[u]])

 u + (δ

  singular

μS = ∇ sym u + (n ⊗ [[u]])sym k

(17)

where the emerging surface Dirac’s delta function, δ s , has been regularized in (17) in terms of a, very small, regularization parameter k and a collocation function μs (x) on the discontinuity interface S. 3.2

Constitutive model regularization

A key point in the (CSDA) is the fact that bounded stress states σ S should appear at the discontinuity interface S, even when singular strains (εS ) are present, see equation (17). This is achieved via a regularization of the continuum strain softening modulus which is defined in a distributional sense: H = δS−1 H

Reinforced concrete beam test

In order to assess the macroscopic RC approach of the previous sections, we simulate one of the classical beam cases that were tested by Vecchio & Shim, (2004) and previously by Bresler & Scordelis, (1963) more than 40 years ago. Vecchio et al. observes that the failure mode of this series of beams is highly influenced by crushing of concrete below the loading plate. Also, they have observed that triaxial confinement effect in that zone is important and must be taken into account in the numerical approach. We simulate the case B3, in the reference work, using a 3D numerical FE model. Details of the beam dimensions, as well as the distribution and quantity of the reinforcement bars and stirrups, are depicted in Figure 8. In this case, these authors reported that the observed failure mode corresponds to flexurecompression induced by crushing of the concrete in the compressive zone. The material parameters for the numerical model are taken from Table 2, which closely correspond with that values reported in the reference work. In Figure 9 we depict the load vs. mid-point vertical displacement curve and compare the numerical solution with the experimental results. The simulation excellently matches the initial beam stiffness and reasonably captures the ultimate loading states.

Strong discontinuity kinematics.

regular

NUMERICAL EXAMPLES WITH THE MACROSCOPIC MODEL

(18)

where H¯ is an intrinsic softening modulus that can be computed from the classical parameters used in Fracture Mechanics of quasi-brittle materials. An adequate definition of the function: H¯ = f (fc ; Gf ; E); makes the dissipative effects, captured by the continuum constitutive relation, consistent with the specific fracture energy Gf of the material. The re-interpretation given in expression (18) assures the compatibility between a singular kinematics and the continuum damage material model depicted in Table 1. Besides, when the strong discontinuity kinematics, equations (16) and (17), is consistently introduced in this continuum setting, a classical cohesive-type constitutive law (traction vs. displacement jump) is reproduced at the fracture interface as a projection of the original continuum constitutive model onto the strong discontinuity

Figure 8. Beam case ‘‘B3’’ Vecchio et al. (2004) experimental test. a) main geometrical dimensions; b) cross section; c) reinforcement distribution; d) cross section FE discretization.

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Table 2.

Material parameters of the concrete beam test.

Concrete Poisson’s ratio Young’s modulus Compresive strength Tensile strength Fracture energy

ν = 0.18 E = 26,075 [MPa] fc = 34.3 [MPa] ft = 2.65 [MPa] Gf = 0.063 [N/mm]

Steel Young’s modulus Yield stress Ultimate stress Dowel action stiffness Yield shear stress No slip effect was assumed

E = 200 [GPa] σy = 436 [MPa] σu = 700 [MPa] G f = E/2 √ f τy = σy / 3 f (E f = E; σy= σy )

Figure 10. Beam case ‘‘B3’’ Vecchio et al. (2004): a) iso-displacement lines in the central zone; b) experimental result displaying the crack pattern distribution; c) numerical distribution of the damage variable.

Figure 11. Beam case ‘‘B3’’ Vecchio et al. (2004): Contour fill of the scalar variable (ltrack) displaying the crack pattern distribution captured by the numerical strategy.

the numerically tracked cracks. They include all those cracks that appeared and were tracked at some stage of the loading process but not necessarily active at the end of analysis. Figure 11 displays a very large number (more than 100) of these cracks. This shows the ability of the numerical model to handle a large number of evolving three-dimensional cracks, and, thus, its qualification for reproducing the crack distribution in heavily reinforced concrete structures (characterized by a large number of cracks).

Figure 9. Beam case ‘‘B3’’ ,Vecchio et al. (2004). Load vs. mid-span vertical displacement.

Description of the simulated crack pattern in concrete is shown in Figure 10-a. This figure displays, through the x-isodisplacement lines, four well developed vertical cracks (macro cracks) in the center zone of the beam. Figure 10-c presents the beam damage distribution obtained with the numerical model at the end of analysis. This result agrees with the experimental crack pattern displayed in Figure 10-b showing a rather distributed crack pattern. We have to remark that the crushing zone below the upper load, although considered in the model, is not represented in this damage variable distribution. Figure 11 confirms the results of Figure 10-c. In this figure we plot the scalar field (ltrack) defining

4.2

L-shaped panel

A series of RC L-shaped structural member was tested by Winkler et al., (2001). In the present subsection, we show the numerical solution obtained with the macroscopic model of the panel corresponding to the series D reported in that paper. We compare our results with the experimental test. Dimensions of the specimen, loading condition and the reinforcement distribution are displayed in

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Figure 12. L-shaped panel: a) geometric properties and boundary conditions (specimen width = 100 mm, the load P is located at 20 mm away from the free vertical face). Dimensions in mm; b) 3D FE model; c) FE grid accounting for the homogenized composite material model. Remaining elements are modeled as plane concrete.

Table 3.

Figure 13. L-shaped panel. Plot of the Force P vs. the vertical displacement of the load application point. Experimental and numerical solutions.

Material parameters.

Concrete Poisson’s ratio Young’s modulus Compresive strength Tensile strength Fracture energy

ν = 0.18 E = 26,075 [MPa] fc = 29.45 [MPa] ft = 2.65 [MPa] Gf = 0.063 [N/mm]

Steel Young’s modulus Yield stress Ultimate stress

E = 179.0 [GPa] σy = 526 [MPa] σu = 584 [MPa]

Figure 14. L-shaped panel. Comparison of the crack pattern distribution: experiments vs. numerical solution. Cracks in the numerical solutions are depicted through the contours of iso-displacement lines.

Figure 12-a. As can be seen there, a welded orthogonal reinforcement grid at 45◦ has been used in this specimen series. Steel bars of 6 mm diameter were used in the reinforcement grid having a grid spacing of 50 mm. The material parameters are specified in Table 3. We remark the large scattering of the concrete tensile strength, depending on the test technique, reported in the reference work. Figure 13 plots the load P vs. the vertical displacement of the load application point, computed with the numerical model and compared with the experimental results. It can be observed a rather good approximation in the initial stages, for vertical displacement less than 1.5 mm. After that point, the numerical solution shows a spurious drop not observed in the panel, which exhibits a ductile behavior. Figure 14 compares the crack pattern distribution of the numerical results with the experimental observation. As it can be seen in this case, results compare qualitatively well with the experiment.

5 5.1

MESOSCOPIC MODEL: AN APPLICATION TO CORRODED RC BEAMS Problem description

The corrosion phenomenon observed in reinforced concrete structures, largely limits the service life of these structures. In the present example we show an application of the proposed model to simulate the mechanical consequences of it: typically the loss of the structural load carrying capacity due to the reinforcement corrosion. The mechanism for the steel corrosion to cause a loss of the structural strength is the expansion of the

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corroded rebar, which induces cracking in the concrete cover, loss of steel-concrete bond, as well as the net area reduction of the steel fiber. The effects of the mentioned mechanisms on the structural load carrying capacity can be analyzed as a function of the reinforcement corrosion degree. Therefore, the model makes possible to determine the influence and sensitivity of this key variable, the reinforcement corrosion level, in the structural deterioration problem. The proposed numerical strategy can be applied to beams, columns, slabs, etc., through two successive and coupled 2-D meso and macroscopic mechanical analyzes, as follows: i. For a number of structural cross sections of the beam we simulate, at mesoscopic level, the reinforcement expansion due to the volume increase of the steel bars as a consequence of the corrosion products accumulation (see next subsection 5.1.1). Then, the damage distribution and crack patterns in the concrete bulk and cover are evaluated, and the corresponding concrete loss of stiffness in the structural member is calculated. ii. A second macroscopic longitudinal analysis, now considering the results of the previous analysis in terms of the initial damage distribution and the rebars net section, evaluates the mechanical response of the structural member subjected to an external loading system. This evaluation, presented in subsection 5.1.2, determines the global response and the macroscopic mechanisms of structural failure. iii. An intermediate evaluation is performed in order to couple both analyses. Results of the stage i) are projected onto the structural analysis ii) as it will be explained in subsection 5.1.3.

Figure 15. Corroded beam problem: cross section analysis of the structural member (expansion mode).

interface. Each of them is characterized by a different constitutive response, and FE technology, that takes into account the main mechanisms involved in the corrosion process. A steel-concrete interface model is considered in order to capture the possible friction and slipping between both constituents once the concrete fractures. A contact linear triangular element adopted in the present model has been taken from Oliver et al., (2008a), see Figure 16-a, where additional details about its formulation can be obtained. In Figure 17, we display two different numerical solutions related to the expansion mechanism of steel bars for a predefined corrosion attack depth, and the degradation induced in concrete cover at the cross section level. From a qualitative point of view, it can be observed that the proposed (mesoscopic plane strain) numerical model captures physically admissible failure mechanisms. The introduction at the interface of friction/contact finite elements becomes crucial to obtain consistent crack patterns that match very well the semi analytical predictions published in the literature.

5.1.1 Cross section analysis of the structural member (expansion mode) Let us consider the cross section of an arbitrary RC structural member, as displayed in Figure 15-(b), whose reinforcement bars are experiencing a corrosion process, not necessarily identical in all of them. The products derived from the steel bar corrosion, such as ferric oxide rust, reduce the net steel area and accumulate, causing volumetric expansion of the bars (see Figure 15-(a)), what induces a high hoop tensile stress state in the surrounding concrete. As a consequence the cover concrete undergoes damage and a degradation process which results in two typical fracture patterns: (i) inclined cracks and (ii) delamination cracks, as observed in Figure 15-(c). Clearly, these induced cracks can increase the rate of corrosion process in the structural member. The two-dimensional plane strain mesoscopic model, idealized in Figure 15-(b), considers three different domains of analysis: (i) the concrete matrix, (ii) the steel reinforcement bars and (iii) the steel-concrete

5.1.2 Macroscopic model to simulate the structural load carrying capacity (flexural mode) The model of subsection 5.1.1 provides qualitative information related to the concrete degradation mechanisms due to the steel expansion. Nevertheless, it does not give additional information about the mechanical

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behavior of a deteriorated RC structure subjected to external loads. Therefore, a 2-D macroscopic homogenized composite model, as the one described in section 2, is used to evaluate the residual load carrying capacity of the corroded RC member. The idealized scheme of the discrete model is shown in Figure 18. We have modeled the plain concrete by means of the model presented in section 2.2 which is regularized by means of the CSDA approach of section 3. 5.1.3 Coupling strategy Figure 19 shows an idealized scheme of the adopted strategy to couple the models presented above in sections 5.1.1 and 5.1.2. We transfer, from the domain of the generic geometric cross section, to the structural member domain, the average value of the damage variable ‘‘d’’ across horizontal slices in the cross section geometry. This projection is consistent because both analyses use the same continuum isotropic damage model for simulating the concrete domain. Thus, the final degradation state of concrete, induced by the rebar volumetric deformation process, is considered to be the initial damage condition for the subsequent longitudinal structural analysis. This means that we are assuming that the two models are coupled in only one direction, neglecting the structural load effects on the concrete damage evaluation determined in the cross section analysis.

Figure 16. Corroded beam problem. Contact finite element at the interface: a) representative scheme; b) typical contact element; c) scheme of the constitutive law for the contact normal stress σηη ; d) scheme of the constitutive law for the friction shear stress σηt .

5.1.4 Numerical results of the corroded structural member The structural analysis of the RC beam with the model of sections 5.1.1–5.1.3. provides structural responses taht change with the rebar corrosion attack depth. Sánchez et al., (2009) have used this approach to compare the numerical solution of several beam series tests performed by Rodriguez et al., (1995). From this study, in Figure 20 we reproduce the structural response of an identical beam subjected to different

Figure 17. Analysis of the rebar corrosion problem for two different reinforcement bar distribution. Plane strain expansion analysis: a) iso-displacement contour lines (pattern of cracks); b) scaled deformed configuration; c) Damage contour fill.

Figure 18. Corroded beam problem. Macroscopic FE model to simulate the structural load carrying capacity of a structural member (flexural mode).

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Figure 21. Corroded beam problem Numerical results of the macroscopic FE model: a) contour fills of the damage variable displaying the crack pattern distribution captured by the model; b) iso-displacement contour lines displaying the active crack pattern at the end of analysis.

corrosion levels. There, it can be observed the sensitivity of the model with the reinforcement corrosion attack. The same structural analysis provides the damage distribution across the structural member, as shown in Figure 21-(a) for one case of corrosion level. Also, Figure 21-(b) shows the iso-displacement contour lines, which display the active crack pattern at the end of analysis and, therefore, the final failure mechanism. Figure 19. Corroded beam problem. Coupling strategy between the cross section analysis and the structural member analysis.

6

CONCLUSIONS

Two different approaches for modeling the nonlinear mechanical behavior of RC structural members have been presented: i. a macroscopic material model that considers reinforced concrete as a composite material, with concrete and rebars as constituents, and ii. a mesoscopic model that takes into account the mechanical response of every constituent of the RC member in a separate way and includes, explicitly, their interaction effects. The specific ingredients of the constitutive models for plane concrete and rebars, and the framework for the material failure simulations have also been described. Within the resulting setting, a number of threedimensional simulations have been run, for assessment purposes, in heavily reinforced linear structural members (beams and panels). In most of cases they have displayed acceptable agreements between the physical reality, observed from experiments, and the simulation results. Ongoing research is done to assess if the proposed methodology can be extended to more complex reinforced concrete members as slabs and shells.

Figure 20. Corroded beam problem. Numerical results of the macroscopic FE model to simulate the structural load carrying capacity (load P vs. mid-span vertical displacement plots). The curves correspond to an identical RC beam subjected to a range of corrosion attack depths.

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ACKNOWLEDGMENT

Oliver, J., Huespe, A.E. & Cante, J.C. 2008a. An implicit/ explicit integration schemes to increase computability of non-linear material and contact/friction problems. Comput. Meth. App. Mech. Eng., 197, 1865–1889. Oliver, J., Huespe, A.E., Pulido, M.D.G. & Chaves, E. 2002. From continuum mechanics to fracture mechanics: the strong discontinuity approach. Engineering Fracture Mechanics, 69(2), 113–136. Oliver, J., Linero, D.L., Huespe, A.E.& Manzoli, O. 2008b. Two-dimensional modeling of mateiral failure in reinforced concrete by means of a continuum strong discontinuity approach. Comp. Meth. Appl. Mech. in Eng., 197, 332–348. Pietruszczak, S. & Winnicki, A. 2003. Constitutive model for concrete with embedded sets of reinforcement. Journal of Engineering Mechanics—ASCE, 129(7), 725–738. Rodriguez, J., Ortega, L. & Casal, J. (1995). Load carrying capacity of concrete structures with corroded reinforcement. Proc. of the 4th Int. Conf. on Structure Faults and Repair.: Engineering Tech. Press., Edinburgh, U.K. Sánchez, P.J., Huespe, A.E., Oliver, J. & Toro, S. 2009. Mesoscopic model to simulate the mechanical behavior of reinforced concrete members affected by corrosion. Int. J. Solids & Struct., published ‘‘on-line’’, doi: 10.1016/j.ijsolstr.2009.10.023. Vecchio, F.J. & Shim, W. 2004. Experimental and analytical reexamination of classical concrete beam tests. J. of Struct. Eng., ASCE, 130(3), 460–469. Willam, K. & Warnke, E. (1974). Constitutive model for triaxial behaviour of concrete. Proc. of the Concrete Struct. Subjected to triaxial stresses: Institution Assoc. for Bridges and Struct. Eng.,19, Section III., Zurich. Winkler, B., Hofstetter, G. & Niederwanger, G. 2001. Experimental verification of a constitutive model for cracking. Proc. Instn. Mech. Engrs., 215(Part L), 75–86.

Financial support from the Spanish Ministry of Science and Innovation through grant BIA2008-00411 is gratefully acknowledged.

REFERENCES Belletti, B., Cerioni, R. & Iori, I. 2001. Physical approach for reinforced-concrete (PARC) membrane elements. Journal of Structural Engineering. ASCE, 127(12), 1412–1426. Bresler, B. & Scordelis, A.C. 1963. Shear strength of reinforced concrete beams. J. Am. Concr. Inst., 60(1), 51–72. ˇ Cervenka, J. & Papanikolaou, V.K. 2008. Three dimensional combined fracture—plastic material model for concrete. Int. J. of Plast., 24(12), 2192–2220. Feenstra, P.H. & de Borst, R. 1996. A composite plasticity model for concrete. Int. J. os Solids & Struct., 33(5), 777–730. Kang, H.D. & Willam, K.J. 1999. Localization characteristics of triaxial concrete model. ASCE, JEM, 941–950. Kollegger, J. & Mehlhorn, G. 1990. Material model for the analysis of reinforced concrete surface structures. Computational Mechanics, 6, 341–357. Linero, D.L. (2006). A model of material failure for reinforced concrete via Continuum Strong Discontinuity Approach and mixing theory.PhD. Thesis, Technical University of Catalonia (UPC), Barcelona. Oliver, J. 2000. On the discrete constitutive models induced by strong discontinuity kinematics and continuum constitutive equations. International Journal of Solids and Structures, 37(48–50), 7207–7229.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Concrete under various loadings, way to model in a same framework: Damage, fracture and compaction J. Mazars, F. Dufour & C. Giry Lab. 3S-R, Institut Polytechnique de Grenoble, France

A. Rouquand & C. Pontiroli DGA/C.E.G. Gramat, France

ABSTRACT: The causes of the non linear behavior of concrete until failure are numerous and complex, particularly for non monotonic and rapid loadings. We present hereafter a strategy based on damage mechanics or a coupling of damage model and plasticity model including several effects: development and closure of cracks, damping, compaction, strain rate effects,...The idea being to describe with the same tools a wide variety of problems: static loading including the evaluation of crack opening, young concrete behaviour, dynamic loading at low and high velocity . . . The models are of explicit form, that makes possible their implementation into explicit numerical scheme well adapted to the treatment of fast dynamic problems. In this context the F.E. ‘‘Abaqus explicit’’ code is used for the PRM model which couples damage and plasticity. It has been successfully applied during the past few years to model the response of reinforced concrete structures subjected to severe loadings. In this paper the main model concepts are presented and some examples of numerical simulations are given (with some related to impact) and compared to experimental results in order to demonstrate the efficiency of the concepts used. 1

INTRODUCTION

It has been shown (Mazars, 1986) that basically three different damage modes have to be considered (see Figure 1): A/Situation dominated by mode I related to local extension (εi > 0), B/Situation dominated by mode II (or/and 3) without any local extension, C/Situation related to the application of a strong hydrostatic pressure which leads to consolidation (pore collapse in the cement matrix). To simulate the behavior of concrete, plasticity (Ottosen 1979), damage models (Mazars 1986, Mazars et al. 1989, La Borderie et al. 1994, Jirasek 2004, Gatuingt et al. 2008) or fracture based approabreak ches (Bazant et al. 1996) are used. They are adapted to simulate the situation ‘‘A’’, often present in classical reinforced concrete structures. For severe loadings, related to natural or technological hazards (accidental or intentional actions) two further aspects must be considered: the dynamic nature of the loading and locally, high confinement pressure. However very few models are able to simulate both phenomena. Figure 1 presents the different situations listed before related to the type of loading. To model the behaviors which arise, the strategy is the coupling

A

B

With extensions in at least one direction

Without any extension

Under hydrostatic pressure

C

Mode I

Mode II

Pore collapse

With instability

With ductility

With compaction

Damage Model local extension state pilots threshold and damage evolutions

Plasticity Model Von Mises stress is used for plastic threshold and plastic flow

Compaction is created after a threshold through the evolution of the bulk modulus

Figure 1. Local damage modes in relation with the type of loading (Mazars 1986) and the kind of models used.

of a damage model and a plasticity model including compaction effects. This paper presents how the models have been set up, the experiments requisite to identify the material

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parameters and series of applications at the material level and on reinforced concrete structures for which experimental results testify their effectiveness.

is expressed in terms of the principal extensions. An equivalent strain is defined as:  3 εi 2+ (4) ε eq =

2

where •+ is the McCauley bracket and εi are the principal strains. Accounting for isotropic hardening κ(D) the yield criterion of damage follows:

i=1

MODELLING CONCEPTS

Constitutive laws for concrete are based on the principles of damage mechanics following the usual approach (Lemaitre et al. 1990). After choosing the state variables and the expression of free energy, derivations give the state laws that lead to the constitutive equations. Two different models are presented hereafter, the first one adapted to monotonic loadings having one scalar damage variable, and the second one adapted to cyclic loadings having two scalar damage variables and including crack closure and permanent strain effects. 2.1

f (ε, D) = εeq − κ(D)

Two evolution laws for damage are considered for tension and compression (index i refers either to traction or compression): Di = 1 −

εd0 (1 − Ai ) − Ai exp(−Bi (εeq − εd0 )) (6) εeq

εd0 is the initial damage threshold; Ai and Bi are material parameters. The resulting damage to be introduced in the constitutive equation is a combination of those two scalar damage variables using the following weighting coefficients αt and αc [9]:

Mazars damage model

Concrete—like most of the geomaterials and ceramics—is perceived like brittle in tension and more ductile under compression loading. During experimental tests, a network of microscopic cracks nucleates perpendicularly to the direction of extension, which coalesces until complete rupture. Whereas under uniaxial tension a single crack propagates, under compression and due to the presence of heterogeneities in materials (aggregate surrounded by a cement matrix) tensile transverse strains generate a self-equilibrated stress field orthogonal to the loading direction. A pure mode I (extension) is thus considered to describe the behavior in compression. The influence of microcracking due to external loads is introduced via a single scalar damage variable D ranging from 0 (undamaged material) to 1 (completely damaged material). The free energy ψ for this model takes the following form: 1 ρψ = ε : (D) : ε (1) 2 (D) is the Hooke elasticity tensor depending on the actual value of D through the form (D) = 0 (1 − D), 0 being the elasticity tensor for the virgin material. From the state equations, σ = ρ∂ψ/∂ε, the constitutive state law for a scalar damage model coupled to elasticity leads to:    1 σ = (1 − D) KTr(ε)I + 2G ε − Tr(ε)I (2) 3 or, ε = 1/E(1 − D)[(1 + ν)σ − νTr(σ )I]

(5)

β

D = αt Dt + αcβ Dc

(7)

We denote σ + and σ − (σ = σ + +σ − ) the tensors in which appear only the positive and the negative principal stresses, respectively, and ε t , εc the strain tensors defined as: ε t = −1 : σ + and εc = −1 : σ −

(8)

(D) is a fourth-order symmetric tensor interpreted as the secant stiffness matrix and it is a function of damage. The weights αt and αc are defined by the following expressions: αt =

3  1

Hi

3  εti (εti + εci ) εci (εti + εci ) αc = Hi (9) 2 2 εeq εeq 1

αt and αc define the contribution of each type of damage. αt (respectively αc ) ranges from 0 (pure 3D compression state—respectively traction state) to 1 (pure 3D traction state—respectively compression state). Hi = 1 if εi = εci + εti ≥ 0, otherwise Hi = 0. From equation (9) it can be verified that for uniaxial tension αt = 1, αc = 0, D = Dt and vice versa for compression. β is a shear factor, generally equal to 1.06. Responses under compression and tension of this model are presented Figure 2. For the general case of a loaded structure, D is calculated from the local maximum value of εeq reached during the loading. Then, whatever is the following loading path (compression or traction) the local Young’s modulus is equal to E(1-D) which cannot allow to

(3)

K and G are the bulk modulus and the shear modulus respectively, E and ν are the Young’s modulus and the Poisson’s ratio respectively. I denotes the second order identity tensor. In order to introduce the nonsymmetric behavior of concrete, the failure criterion

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2.2

Applications

Such a model is useful for different kind of applications for monotonous loading. It allows to describe global and local behaviour including damage contours and can be completed in order to simulate thermomechanical effects and to evaluate crack opening. 2.2.1 Damage field and crack opening The sample presented Figure 3 is notched (2 notches at different levels) and loaded with a uniaxial traction until the propagation of cracks. Calculation is performed using the non local damage model presented before and a post treatment is used to deduce the crack opening from the strain field. The procedure used is from (Dufour et al. 2008), it consists: – to fit the line where ε¯ eq is maximum which gives the crack path (2 cracks are found in the present case, which is consistent with the experiment, cf. figure 3). – to consider short segments along the crack path and to get a perpendicular profile in the middle of each segment; the strain tensor is projected on these profiles in order to get the local evolution for εeq on each profile. If on the same path, a crack is considered instead of a damage field, displacement is locally discontinue and the corresponding strain tensor projected on the same profiles as before, is a Dirac function the intensity of which is directly related to the local crack opening; this Dirac function regularized using the non local expression for εeq (equation 10) gives a curve which can be compared to the one coming from the non local damage modelling. The crack opening is obtained assuming same maximum value for

Figure 2. Traction and compression response of the Mazars damage model.

simulate the well-known crack- closure effect (unilateral effect) which appears when the stress goes from traction to compression. This model is used in a non local version to avoid mesh dependency with the following concepts (Pijaudier-Cabot et al. 1987): – An average value for εeq is used at point M(x):  ε¯ eq (x) =

φ(x



− s)εeq (s)dv

φ(x − s)dv

(10)

– is the characteristic volume on which ε¯ eq is calculated and φ is the weighting function:  

x − s 2 φ(x − s) = exp − 2 lc

(11)

It has been shown that when is a circle (plane problem) with a constant size during the whole loading process some artifacts appear (close to an edge or close to a crack). A new proposition is done by (Giry et al. 2010) in which, during the loading,

becomes elliptic, the orientation and the size of which depending on the local stress field.

Figure 3. Notched specimen in traction—On the left ‘‘non local damage’’ - On the right the crack opening deduced from post treatment; bottom, zoom on the experimental crack (Shi et al. 2000).

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both non local ε¯ eq curves. This has been performed on the notched sample presented in Figure 3 and the results (crack path and opening) are consistent with the experimental data (Shi et al. 2000).

in the French design code giving the strength at day j: fcj = 0.685 fc28 log(j + 1). For the general case of concrete structures, some others phenomena must be taken into account such as dilation and shrinkage or creep and relaxation. This is done through the use of a partition of the strain tensor:

2.2.2 Prevision of the behaviour of young concrete For young concrete, hydration of the cement paste generates a progressive increase of mechanical properties. This can be taken into account through the use of a new state variable: maturity M , a function of temperature T and the actual time t: M=

K(T ) 1 + K(T )

ε = εe (E, D, T ) + ε d (T , M ) + εc (E, M )

2.3

(12)

K(T ) = k derives from the Arrhenius law, T is the temperature at time t, t0 corresponds to the beginning of the hardening phase, k and b are material parameters. M evolves between 0 at t = t0 and 1 at tf (infinite time), when hydration is over. The Young’s modulus and the Poisson’s ratio evolve with maturity. Simple laws have been proposed (Mazars et al. 1998): and

ν(M ) = 0.5(1 − M ) + νf M (13)

Ef and νf are respectively the Young’s modulus and the Poisson’s ratio at complete maturation (M = 1). Loading of young concrete generates damage. This can be forecast using the following coupling: E(M , D) = Ef M (1 − D)

d

PRM damage model for cyclic loading

Based on the previous model (Mazars, 1986) and from works of Pontiroli (1995) and Rouquand (1995 & 2005), this model, named ‘‘PRM model’’, simulates the cyclic behavior of concrete for low confinement (type A in figure 1). The main additionnal aspect is crack-closure effects and permanent strains. As for the previous model, the PRM model distinguishes the behavior under traction and the behavior under compression, but Dt and Dc have, each one, the status of a state variable individually instead of D (issue from the coupling of Dt and Dc –equation (7)) in the Mazars model. Between these two loading states a transition zone is defined by (σft , εft ), where σft and εft are the crack closure stress and the crack closure strain respectively. The main constitutive equations of the PRM model for a uniaxial loading are the following.

exp(− Tb )

E(M ) = Ef M

(15)

ε is the total strain, ε is the elastic strain, ε is the dilation strain and εc the creep strain. Details can be found in (Mazars et al. 1998). e

for traction:

(14)

(σ − σft ) = E(1 − Dt ) · (ε − εft )

Then, introducing this expression in equation (3), one can study the effects of time, temperature and loading on the material. In that way, the serie of curves Figure 4 gives the evolution of the compression behaviour from 0.5 to 28 days of a concrete cured at 20◦ C. These results are consistent with results reported

(16)

for compression: (σ − σft ) = E(1 − Dc ) · (ε − εft )

(17)

Before any damage in compression εft = εft0 and σft = σft0 (material parameters), afterwards σft is directly link to Dc as follows: σft = σft0 (1 − Dc )2

(18)

E is the initial Young’s modulus; Dt = 1 −

εd0t (1 − At ) − At exp(−Bt (εeq − εd0t )) εeq (19)

(εfc , σfc ) defines a focal point from which the damage compression Young’s modulus E(1 − Dc ) is deduced (see figure 5). Then: Dc = Figure 4. Mazars model, coupling of Damage and Maturity: prevision of the evolution of the compression behaviour with time (temperature of curing is T = 20◦ C; stress and strain are considered positive for compression).

DcM 1 − εfc /εm

(20)

with DcM = f (εeq , ε0c , Ac , Bc ) as in equation (6), εm is the maximum compression strain reached during the loading path.

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σ d = σ − σ ft ; ε d = ε − εft where σ ft and εft are the crack closure stress and strain tensors respectively, used to manage crack-closure effects. D remains a scalar and is issued from the combination law of the two damage modes Dt and Dc seen before (equation 22). αt evolves between 0 and 1 and the actual value depends on the sign of Tr(σ − σ ft ). This formulation is an explicit one. It has been implemented into ‘‘Abaqus explicit’’ and is used mainly for dynamic structural simulations. For more details see (Mazars 1986, Rouquand & Pontiroli 1995, Rouquand 2005).

Remarks: – for both equations (19), (20), the equivalent strain is used with an expression adapted to the presence of permanent  strains: 2 εeq = i xi + and, depending on the sign of the stress in the direction i, xi = ει in compression and xi = (ε − εft )i in traction. – the thresholds in traction ε0t and compression ε0c can be different Figure 5 illustrates the corresponding response for a uniaxial cyclic loading. One can observe that if we use the combination damage rule as in equation (7), the behaviour can be described by the classical equation: σd = E0 (1 − D)εd

2.3.1 Strain rate effects—Internal friction, damping It is well known that concrete is strain rate dependent, particularly under tensile loading. This effect is d d accounted for using dynamic thresholds (ε0t and ε0c ) instead of static one’s (ε0t and ε0c ) through the use of a dynamic increase factor R = ε0d /ε0 . Its value for a compressive dynamic loading takes the following form:

(21)

with σd = σ − σft , εd = ε − εft and D = αt Dt + (1 − αt )Dc

(22)

αt is the activation factor. In the uniaxial case: αt = 1 if σd > 0

and

αt = 0

if σd < 0

Rc = min(1.0 + ac ε˙ bc , 2.50) and for a dynamic traction loading:

In the framework of isotropic damage evolutions (Dt and Dc are scalars), the general 3D constitutive equations are set up on the scheme used for 1D situation, which leads to: 



1 σ d = (1 − D) KTr(εd )I + 2G ε d − Tr(ε d )I 3

Rt = min[max(1.0 + at ε˙ bt , 0.9˙ε0.46 ), 10.0]

(26)

ac , bc and at , bt are material coefficients identified from experimental results. For a high strain rate, the traction dynamic increase factor is supposed to follow an empirical formula (0.9˙ε0.46 ) that well agrees with the experimental data obtained by Brara & Klepaczko 1999 on Hopkinson bar tests on ordinary micro concrete (necessary due to the size of the tested specimen). Looking forward new results it is assumed that the same trend can be used for the concrete considered in the applications hereafter. Figure 6 illustrates the evolution of the compressive (dashed line) and tensile (continuous line) dynamic increase factors versus the strain rate.



(24)

Figure 5.

(25)

(23)

Uniaxial response of the PRM damage model.

Figure 6.

PRM damage model: strain rate effects.

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For cyclic loading, as the one encountered during an earthquake loading, friction stresses induce significant dissipated energy during unloading and reloading cycles. To account for this important phenomenon, an additional damping stress is introduced in the model: σ − σ ft = (σ − σ ft )damage + σ damping

confined cyclic loading (Rouquand 2005). For very high dynamic loads leading to a higher pressure level, an elasto-plastic model is more appropriate. For example, the impact of a projectile striking a concrete plate at 300m/s induces local pressures near the projectile nozzle of several hundred MPa, which induces a combination of type B and C modes seen Figure 1. The previous damage models can neither simulate the pore collapse phenomena nor the shear plastic strain occurring at this pressure level. To overcome these limitations, the elasto-plastic model proposed by (Krieg 1978), for geomaterials has been chosen to simulate this kind of problem. From this simple elasto-plastic model an improvement has been introduced in order to take into account the non linear elastic behavior encountered during an unloading and reloading cycle under a high pressure level. Another recent improvement has been done to take into account the water content effects, introducing an effective stress theory as described by C. Mariotti et al. (2003). This effect induces changes on the pressure volume curve and on the shear plastic stress limit. Finally, both models have been coupled to simulate the combination of the three types of situation (A, B, C) seen Figure 1.

(27)

The damping stress generates a hysteretic loop during the unloading and the reloading cycle. This stress is calculated from the damping ratio ξ , classically defined as the ratio between the area under the closed loop and the area under the linear elastic-damage stress curve, which gives for the uniaxial case: ξ=

Ah E(1 − D)(εmax − εft )2

(28)

Ah is the loop area under the stress strain curve, E(1 − D) is the current material stiffness. εmax is the maximum strain before unloading; εft is the closure strain that defines the transition point seen before. From (28) the damping stresses are computed in such a way that the damping ratio ξ is related to the damage D according to the relation: ξ = (β1 + β2 D)

2.4.1 The modified Krieg plastic model The Krieg model can be applied to describe the behavior of a dry material. It is based on a classical elastic purely plastic description using a parabolic deviatoric plastic threshold including a cut-off linked to the porosity of the material. We will see hereafter that this cut-off is also linked to the water content η figure 8 and § 2.4.2 for the relation between qmax and η):

(29)

β1 is a damping ratio for an undamaged and perfectly elastic material. β1 + β2 is the damping ratio for a fully damaged material. Usually β1 can be chosen equal to 0.02 and β2 can be chosen equal to 0.05. Figure 7 shows, for cyclic traction-compression loading, the strain-stress curve including damping stresses.

q = min (q0 = 2.4

Damage and plasticity. PRM coupled model

a0 + a1 P + a2 P 2 , qmax )

(30)

with q = (3/2σ d : σ d )1/2 (Von Mises stress), the stress tensor being σ = −P I + σ d , P is the confinement pressure and σ d the deviatoric stress tensor. The improvement made here concerns the spheric part of the behavior which is non linear and pressure

The previous damage model is very efficient to simulate the behavior of concrete for unconfined or low

Figure 8. Krieg modified model: shear yield threshold for dry and partially satured materials (qmax decreases when the water content η increases).

Figure 7. PRM damage model: cyclic loading including damping stresses.

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dependent. This non linearity is more pronounced when the pore collapse phenomenon has progressed a lot. Figure 9 shows a typical P − εv curve used in this model (εv = Tr [ε], is the volumic strain). For pressure values lower than P1 , the behavior is linear and elastic. For a pressure greater than P1 , the pore collapse mechanism becomes effective. During the loading process the pressure-volume response follows a curve identified from experiments. During the unloading the behavior is elastic and non linear. The bulk modulus becomes pressure dependent. It is equal to Kmax at the first unloading point and decreases to Kmin when the tensile pressure cut-off Pmin is reached (this value is generally negative, which means that traction is necessary to recover the initial volume). This pressure cut-off becomes smaller and smaller as the maximum pressure Pmax increases. When Pmax reaches Pcons , Kmax becomes equal to Kgrain and Kmin is equal to K0grain . So the non linearity becomes more and more important as the pore collapse phenomenon has taken place a lot. When Pmax reaches Pcons , all the voids are crashed in the material which means that the pore collapse phenomenon is achieved. At this pressure level the material is consolidated and the behavior is purely elastic and non linear.

Figure 10. tionship.

Water content effect on pressure volume rela-

partially saturated material, the relation between pressure and volume is given by the response of the dry material until all the voids (part of the pores without water) are removed from the medium (at pressure Pvps ). Thereafter, the dashed curve gives the response of the solid and water mixture. The intersection of the dotted curve with the horizontal axis gives the porosity of the dry material. The intersection of the dashed curve with the horizontal axis gives the ‘‘free porosity’’ εvps of the partially saturated material. Consequently, when the material becomes drier and drier, the dashed curve moves to the right. To define the behaviour of this solid and water mixture (dashed curve) the pressure is assumed to increase in the same way in both phases (solid and liquid). So an iterative procedure as to be run in order to find the relative volume changes of each phase (see Rouquand 2005). Schematically the behaviour remains similar to the behaviour of a dry material until all the voids are removed. Thereafter the pressure difference between the two phases remains constant. At this point, the effective stress concept can be introduced:

2.4.2 Improvement for partially saturated materials Concrete and geologic media contain an open porous network. Internal water can move through the porous media from one void to another. To understand more easily the water effect, the material structure can be studied as a mixture of a solid medium with a void partially filled with water. For high dynamic loads, the time scale is very low (few milliseconds or less) so the water has no time to move inside the material, then undrained conditions can be considered. Figure 10 shows the generic pressure—volumic strain response of a partially saturated material. For a dry material, the response follows the solid curve. When the pressure is sufficient to collapse all the voids, the response is given by the dotted curve. In case of a

Peff = Pdry (εv )

if Pmax < Pvps

Peff = Pdry (εv ) − Pw (εv − εvps )

if Pmax > Pvps (31)

Pw comes from the state equation of Mie-Gruneisen (Jonhson 1968) which describes the liquid behaviour under dynamic pressure. The solid phase behaviour is the non linear elastic model seen before (Figure 9). Water content η(ratio of the water volume on the pores volume)has also an effect on the shear behaviour. In the Krieg model, the plastic shear strength q (computed as the Von Mises stress) is pressure dependent (Figure 8). As the pressure increases, the shear yield stress increases too. This effect is the consequence of the porous structure of the material. During the pore collapse phenomenon the void ratio decreases, the contact area of the solid grains inside the material matrix increases so the pressure increases and

Figure 9. Krieg modified model: pressure—volumic strain behavior (P and εv are considered positive for compression).

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the shear forces inducing sliding motions between the solid grains also increases. When all the voids have collapsed, the shear strength remains constant and becomes pressure independent because of the contact area which cannot increase any more. The material becomes ‘‘homogeneous’’ and the shear strength reaches a limit that is material dependent. Then the proposition is to relate shear strength to the effective pressure. For a dry material, the effective pressure is always equal to the total pressure. But for a partially saturated material, the effective pressure is the total pressure like in dry material until all the voids are removed (equation (31)). After consolidation the interstitial (or the effective) pressure does not increase any more because the pressure in the solid phase and in the water increases together and consequently the shear strength remains constant. As the water content increases, the pressure level Pvps at the consolidation point decreases and then the shear strength q also decreases. Figure 8 illustrates the effect of the effective pressure concept. The solid line gives the shear strength versus the pressure for a dry material. For a partially saturated one, the shear strength follows the solid line until the pressure Pvps is reached. Afterwards the shear strength does not increase and it follows the dashed horizontal line. Then the shear yield limit can be described by:   2 q = min q0 = a0 + a1 Peff + a2 Peff ,

allows to perform triaxial tests on cylindrical samples ( = 7 cm, h = 14 cm) with a confining pressure possible up to 1 GPa. Figure 11 shows a serie of tests on dry concrete used to calibrate the model. The hydrostatic loading part of five triaxial tests is used to fit the P − εv curve of the model and, on the shear yield threshold is fitted from a series of triaxial tests performed at increasing confinement (Gabet et al. 2008—Vu, Malécot et al. 2009). It is assumed that the threshold is reached when the evolution of the volume variation is reversed (dεv /dσ > 0). 2.5.2 Ability of the model to simulate various loading situations Figures 12 (experiment) and 13 (model) show the static response obtained on a cylindrical specimen for triaxial tests with increasing lateral pressure. The response of the model is in good agreement. It exhibits the activation of the damage part for low confinement, the plasticity part for high confinement (visible when there is a plateau), a combination of

 qmax (Pvps) (32)

The PRM damage model has been coupled with the modified Krieg model. The coupling procedure ensures a perfect continuity between the two model responses. The damage model is activated if the maximum pressure is too low to start the pore collapse phenomenon or if the shear stress is too low to reach the shear yield stress. If not, the plastic model is the one activated and it drives the evolutions until the extensions sufficiently increase to lead to a damage failure. This model has been implemented in Abaqus explicit (Hibbitt et al. 2000) and to avoid mesh dependency the Hillerborg regularization method is used (Hillerborg 1976). This method insures constant fracture energy (Gf ) whatever is the size of finite elements. 2.5 Response of the model at the material level 2.5.1 Identification of the model In order to identify and to validate the model, specific tests have been performed at 3S-R Grenoble. As for the Mazars model the material parameters for the PRM damage model, are calibrated from uniaxial tests (traction and compression) and from existing data base for strain rate effects and damping. The identification of the modified Krieg model needs very specific tests. The GIGA machine at 3S-R

Figure 11. Tri-axial tests performed on the GIGA machine (3S-R)—Above, calibration of the P − εv response of the model—At the bottom, identification of the shear yield threshold (P, ε are considered >0 for compression, PRP means proportional test and TRX means triaxial test).

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Beam elements, 2D plane stress elements and 3D solid elements are used to model the reinforced concrete beam. A single element is used in the depth direction with the 3D model. Taking advantage of the symmetry, only one half of the beam is modelled. The reinforcement material model is the classical Johnson & Cook (1983) plasticity model. The concrete and the steel reinforcement are supposed to be perfectly bonded. Figure 15 shows, at the end of the dynamic test, the tensile damage contours on the 3D beam model (upper part of the figure). The lower part shows the corresponding observed crack pattern. The computed cracks are mainly concentrated in the central part of beam like in the experiment.

Figure 12. Experimental response of concrete subjected to triaxial compression for various confinements (Gabet et al. 2008).

450 120 MPa

400

Axial stress (MPa)

350

80 MPa

300 40 MPa

250

20 MPa

200 150

10 MPa

100

5 MPa

50 0

-1

0 MPa

0

1

2

3 4 5 6 Axial strain (%)

7

8

9

10

Figure 13. Response of the PRM coupled model for triaxial compression at various confinements.

both in between and the reactivation of damage when failure occurs.

3 3.1

APPLICATION TO REINFORCED CONCRETE STRUCTURES

Figure 14. devices.

Dynamic three point bending tests: experimental

Figure 15.

Computed and observed crack pattern.

Dynamic three points bending test on a reinforced concrete beam

Figure 14 shows the experimental device and the beam characteristics (in mm). These tests have been performed by (Agardh, Magnusson & Hanson 1999) in Sweden on a high strength reinforced concrete beam.

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Figure 16 shows the evolution in time of the measured force (cross points) compared to the three computed forces resulting from the three different meshes. The beam model gives the lower force. The 2D and 3D models give very similar results.

3.2

Impact of a soft projectile on a plate

The problem presented here concerns the crash of a soft missile impacting a reinforced concrete structure. This kind of impact is not a very highly dynamic event but it can generate severe structural damage. Numerical simulations of this problem are not easy since there are strong interactions between the target behaviour and the missile crash behaviour. Correct predictions suppose that the response of the target and of the missile is properly modelled. In order to evaluate the capabilities of the explicit finite element code Abaqus including the PRM coupled model, 3-D numerical simulations of Meppen tests (tests n◦ 12 and n◦ 20 presented below) have been done (Riech, 1984, Rouquand, 2006). A 3-D finite element model with solid brick elements is the most appropriate to reproduce the local and the global complex strain field generated during the impact on the target. The projectile is composed of a thin steel tube which can be efficiently modelled using 3-D shell elements. The target size is 6.5 m by 6.0 m the thickness is 70 cm in test n◦ 12 and only 50 cm in test n◦ 20 (see the scheme for test n◦ 12 and test n◦ 20 on Figure 19 and figure 20 respectively). Figure 17 shows a view of the meshes. About 30 000 3-D solid elements are used to model the target. The reinforcement is introduced using the Abaqus rebar option. This option allows to take into account the stiffness and the mass contribution of the reinforcement in the elementary stiffness matrix and in the elementary mass matrix associated to the 3-D solid elements. The rebar definition requires the definition of the reinforcement constitutive material.

Figure 16.

Figure 17. Soft shock: mesh of the projectile and of the target with the location of the points where displacements are measured (tests n◦ 12).

The projectile mesh is composed of about 6500 ‘‘S4R’’ shell elements used for thin or moderately thick structures. This projectile is a generic missile. Its length is 6 m and its diameter is 0.6 m. The thickness of the steel envelop is 7 mm in a first part and 10 mm in the second part. An additional mass is incorporated on the rear part of the projectile to model ballast. The behaviour of the metallic bars is modelled using a standard elastic and plastic model without any strain rate effect which remains low in the reinforcement. The behaviour of the metallic missile material is modelled using the Johnson Cook (1983) elasto-plastic model. Strain rate is now accounted for because during the projectile crash, it can reach about 1000/s (Souroushian et al., 1987, Karagiozova et al. 2002). In the physical tests, the reinforced concrete slab was put on a vertical position at the end of a rail. The projectile is accelerated along the rail and impacts the plate in its middle point. The slab is supported on a very stiff metallic frame of 5.4 m square centred on its rear face. To simulate this support, all the nodes on the rear face located along the frame, are fixed (zero displacement) in the plate normal direction. The missile initial velocity is 241.5 m/s in test n◦ 12 and 197.7 m/s in test n◦ 20. The missile axis and the velocity vector are perpendicular to the reinforced concrete plate. 3.2.1 Test n◦ 12 Figure 18 shows a comparison between measured and computed displacements on three points (w10, w6 and

Measured and computed dynamic load.

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Figure 18. Soft shock: comparison between measured (dashed line) and computed (solid line) displacements for test n◦ 12.

O

w8—see figure 17). The agreement is good with a difference less than 10%. 3.2.2 Test n◦ 20 Figure 19 shows the comparison between the observed crack pattern and damage areas. In fact for practical reasons the contours are related to maximum tensile strains which are directly linked to damage through the equivalent strain (see § 2.2). This comparison shows that the conical zone with open tensile cracks corresponds more or less to the computed cracked zone (light grey contours). Figure 20 shows the comparison between the computed maximum tensile strains and the observed crack pattern. In this experiment severe damage is obtained in the concrete ahead the projectile nozzle. Because of the reduction of the slab thickness in this test, a concrete plug is now clearly observed in the experiment. This damage mode is the result of large shear strains around the plug. This kind of damage is correctly predicted in the numerical simulation.

Figure 19. Soft shock (test n◦ 12): comparison between computed and observed damages (O: impact location).

3.3 Shock wave propagation in a partially saturated media (tuff specimen) R.P. Swift (1973) details a serie of experimental tests where a spherical LX—0401 explosive charge, embedded in a geologic tuff specimen, detonates (Figure 21). The charge diameter is 2.5 cm. The size of the geologic material is large enough to prevent the comeback of reflecting waves during the first microseconds. For each test, 3 electromagnetic velocity gages are incorporated into the tuff sample to measure the material velocity (numbered from 1 to 3 on Figure 21). Four physical tests have been performed with different water content. The Abaqus explicit finite element code simulates the detonation of the pyrotechnic device and the shock

Figure 20. Soft shock (test n◦ 20): comparison between computed and observed damages (O: impact location).

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Figure 21. Experimental shock wave tests in a geologic tuff specimen with various saturation ratios.

wave propagation through the tuff material. A single row of 2D axi-symmetric elements is used to model the problem. The total number of the finite elements is 550. The size of each element is 1 mm. The J.W.L. equation of state (Lee et al. 1973) simulates the behaviour of the explosive combustion products. The PRM coupled model with adapted material parameters is used for the tuff. As an example, the upper part of Figure 22 shows a comparison between experimental and numerical velocities at different points in the geologic specimen. Here, we consider a test with a water content of 58.8%. Three sets of curves are given. Each refers to a particular velocity transducer point (see Figure 21). The amplitude of the peak velocity and the arrival time of the shock wave are correctly predicted. The numerical simulation gives a stiffer material response so the arrival time is a little bit overestimated. The lower part of figure 22 shows the same kind of results for a water content of 81.4%. These results clearly show the great influence of the water contents on the shock wave characteristics (amplitude and arrival time) and confirm the ability of the PRM coupled model to simulate this effect. 3.4

Figure 22. Material velocity profiles for two water contents (58.8% and 81.4% respectively).

Impact on a T shape reinforced concrete structure (hard shock)

This study is related to the analysis of the vulnerability of concrete structures under intentional actions. More specifically, the effect of a projectile of about 80 kg striking at more than 300m/s a reinforced concrete structure is studied. Such an experiment has been done by E. Buzaud et al. (2003). The 35NCD16 steel projectile has an ogival nozzle. Its diameter is 160 mm and its length is 960 mm. An accelerometer system is placed inside the projectile to measure the axial and lateral accelerations during the tests. Figure 23 shows the test configuration with the T shape concrete target. The size of each reinforced concrete square plate composing the target is 3 m. The thickness of the front part of the concrete structure is 400 mm and the thickness of the rear part is 300 mm. Reinforcement is composed of two steel layers (one on each side of the concrete plate) with 16 mm diameter bars.

Figure 23. Hard shock: test configuration, impact on T structure and the projectile mesh.

Other 10 mm diameter bars link each reinforcement mesh node of the face to face layers. The distance that separates each bar is 100 mm. The distance between the reinforcement layer and the top (or the bottom) plate surface is 50 mm (concrete cover). 3D numerical simulations have been done using the ABAQUS explicit finite element code. The total number of the finite elements is about 530 000 for the entire

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model. The projectile material (Figure 23) is simulated using an elastic and perfectly plastic model with a plastic yield stress of 1300 Mpa. The reinforcement is also modelled with an isotropic hardening elastoplastic model. The initial yield stress is 600 Mpa and reaches 633 MPa for a failure strain ε = 0.13. The concrete behaviour is simulated with the PRM coupled model and an erosion method, applied to high twisted elements, is used for perforation zones. On Figure 24, the measured deceleration is compared to the computed value. Some differences can be seen but the overall deceleration shape is correctly predicted. Figure 25 shows the tensile damage contours at the end of the numerical simulation (T = 20 ms). The first part of the target is perforated and a rebound on the rear

310

Experiment

Velocity (m/s)

280

PRM coupled model

250 220 190 160 0

2

4

6

8 10 12 14 16 18 20

Time (ms)

Figure 26. velocity.

Hard shock: measured and computed projectile

part is noticed. This has been observed experimentally. The projectile velocity is also close to the measured one (Figure 26).

4

CONCLUSION

In the framework of damage and plasticity mechanics general constitutive models has been developed for concrete and concrete structures submitted to a large range of loadings (monotonic, cyclic, at high velocity and high confinement). Two models are presented. The simplest one (Mazars model) includes one scalar damage variable, is developed with non local concepts and is proposed with two recent improvements: the evaluation of crack opening and the simulation of young concrete behaviours. The second one (PRM coupled model) includes two damage variable and can simulate a lot of physical mechanisms like crack closure effects, strain rate effects, material damping induced by internal friction, shear plastic strains and compaction of porous media under high pressure. To validate this particular coupling of plasticity and damage, an extensive experimental programme has been performed at 3S-R Grenoble using the GIGA machine which permits high confinement up to 1 GPa (Gabet et al. 2008), and a new program is in progress on the large Hopkinson bar at JRC Ispra to complete the data base under high velocity loading. The new model implemented into the F.E. Abaqus explicit code, has been extensively used and can advantageously simulate a large range of problems going from quasi-static simulations on concrete structures to high dynamic problems related to the effect of low (Mazars et al., 2009) and high velocity impacts. The simulations presented here, compared to experimental results, show the relevance of the modelling used which allows to carry out true numerical

Figure 24. Hard shock: measured and computed projectile decelerations.

Figure 25. Hard shock: tensile damage contours at 20 ms. The projectile has perforated the upper part and penetrated the right part after a rebound.

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experiments very useful for complex structures and/or extreme loadings.

La Borderie C., Mazars, J., Pijaudier-Cabot G., 1994. ‘‘Damage mechanics model for reinforced concrete structures under cyclic loading’’, A.C.I, 134:147–172, edited by W. Gerstle and Z.P. Bazant. Lee E., Finger M., Collins W., 1973. JWL equation of state coefficients for high explosives, Technical report UCID16189, Laurence Livermore National Laboratory, Livermore CA, USA. Lemaitre, J. and Chaboche, J.L., 1990. Mechanics of solids material. Cambridge University Press. Mariotti, C., Perlat, J.P., Guerin, J.M., 2003. ‘‘A numerical approach for partially saturated geomaterials under shock’’, International Journal of Impact Engineering 28 717–741. Mazars, J., 1986. ‘‘A description of micro and macro scale damage of concrete structures,’’ Engineering Fracture Mechanics, 25(5/6), pp. 729–737. Mazars, J., Pijaudier-Cabot, G., 1989. ‘‘Continuum damage theory—application to concrete,’’ Journal of Engineering Mechanics. 115(2), pp. 345–365. Mazars J., Bournazel J.P., 1998, Modelling of Damage Processes due to Volumic Variations for Maturing and Matured Concrete in Concrete: from Material to Structure, Edit. JP Bournazel & Y. Malier, RILEM publications, Paris. Mazars J., Rouquand A., Pontiroli C., Berthet-Rambaud P., Malécot Y., 2009, ‘‘Damage tools to model severe loading effects on reinforced concrete structures’’, Proc. ACI fall convention, New Orleans, USA. Ottosen, N.S., 1979. ‘‘Constitutive model for short time loading of concrete’’, Journal of Engineering Mechanics, ASCE, 105, pp. 127–141. Pijaudier-Cabot, G. and Bažant, Z.P., 1987. ‘‘Nonlocaldamage theory’’, Journal of Engineering Mechanics, 113, 1512–1533. Pontiroli, C., 1995. ‘‘Comportement au souffle des structures en béton armé, analyse expérimentale et modélisation,’’ PhD thesis ENS Cachan—France. Riech, H., Rüdiger, E., 1984. ‘‘Results on MEPPEN TESTS II/11 to II/22)’’, Technischer Bericht 1500 408 (RS 467). Rouquand, A., Pontiroli, C., 1995. ‘‘Some considerations on explicit damage models,’’ Proc.FRAMCOS-2, Ed.F.H. Wittmann, AEDIFICATIO Publish., Freiburg. Rouquand, A., 2005. ‘‘Presentation d’un modèle de comportement des géomatériaux, applications au calcul de structures,’’ C.E.G., report T2005-00021/CEG/NC. Shi, C., van Dam, A.G., van Mier, J., Sluys, B., 2000. Crack Interaction in Concrete, in Materials for Buildings and Structures, Editor F.H. Wittmann, Publisher J. Wiley & Sons. Soroushian, P., Choi, K.B., 1987. ‘‘Steel mechanical properties at different strain rates,’’ Journal of structural engineering, 4, pp. 663–205. Swift, R.P., 1973. Dynamic response of earth media to spherical stress waves, Final report, Physics International Compagny. Vu, X.H., Malecot, Y., Daudeville, L., Buzaud, E., 2009. ‘‘Experimental analysis of concrete behavior under high confinement: Effect of the saturation ratio,’’ International Journal of Solids and Structures, 46, 1105–1120.

ACKNOWLEDGEMENT The authors thank the French research network VOR, the ANR programme Mefisto and the French ministry of defense (DGA) which participated in this research financing numbers of tests and calculations performed. REFERENCES Agardh, L., Magnusson, J., Hansson, H., 1999. ‘‘High strength concrete beams subjected to impact loading, an experimental study’’, FOA Defence Research Establishment report, FOA-R-99-01187-311—SE. Bazant, Z.P., 1994. ‘‘Nonlocal damage theory based on micromechanic of crack interaction’’, Journal of Engineering Mechanics ASCE 120, pp. 593–617. Brara, A., 1999. «Etude expérimentale de la traction dynamique du béton par écaillage», PhD thesis Metz university—France. Dufour, F., Pijaudier-Cabot, G., Choinska, M., Huerta, A., 2008. ‘‘Extraction of a crack opening from a continuous approach using regularized damage models’’, Computers and Concrete, 5(4), 375–388. Gabet, T., Malecot, Y., Daudeville, L., 2008. ‘‘Triaxial behavior of concrete under high stresses: Influence of the loading path on compaction and limit states,’’ Cement and Concrete Research, 38(3), 403–412. Gatuingt, F., Desmorat, R., Chambart, M., Combescure, D., Guilbaud, D., 2008. ‘‘Anisotropic 3D delay-damage model to simulate concrete structures’’. Revue Européenne de Mécanique Numérique. 17, pp. 740–760. Giry, C., Dufour, F., Mazars, J., Kotronis, P., 2010. ‘‘Stress state influence on nonlocal intractions in damage modelling, Euro-c 2010’’, Rohrmoos-Shladming, Austria. Hibbit, Karlssonn & Sorensen Inc., 2000. Abaqus manuals, version 6.4. Hillerborg, A. et al., 1976. ‘‘Analysis of crack formation and growth in concrete beams of fracture mechanics and finite elements’’, Cement and Concrete Research, 6, pp. 773–782. Jirásek, M., 2004. ‘‘Non-local damage mechanics with application to concrete’’, Revue française de génie civil, 8, pp. 683–707. Johnson, G.R., Cook, W.H., 1983. ‘‘A constitutive model and data for letals subjected to large strains, high strain rates and high temperatures’’. Proc. 7th International Symposium on Ballistics. pp. 541–547. Johnson, James N., 1968. ‘‘Single-particle Model of a Solid: the Mie-Grüneisen Equation’’, American Journal of Physics, 36(10), pp. 917–919. Karagiozova, D., Jones, N., 2002. ‘‘Stress wave effects on the dynamic axial buckling of cylindrical shells under impact,’’ in Structures under shock and impact VII, Jones N., edit. CA Brebia. Krieg, R.D., 1978. ‘‘A simple constitutive description for soils ans crushable foams,’’ Sandia National Laboratories report, SC-DR-72–0833, Albuquerque, New Mexico.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Upscaling quasi-brittle strength of cement-based materials: A continuum micromechanics approach Bernhard Pichler & Christian Hellmich Institute for Mechanics of Materials and Structures, Vienna University of Technology (TU Wien), Vienna, Austria

ABSTRACT: It is well known from experiments that the strength of cement-based materials depends linearly on the degree of hydration, once a critical hydration degree has been surpassed. It is less known about the microstructural material characteristics which drive this dependence, nor about the nature of the hydration degree-strength relationship before the aforementioned critical hydration degree is reached. In order to elucidate the latter issues, we here present a micromechanical explanation for the hydration degree-strength relationships of cement pastes covering a large range of water-to-cement ratios: Therefore, we extend the one step elastobrittle homogenization scheme of Pichler, Hellmich, and Eberhardsteiner (2009) to a more realistic two-step approach. Accordingly, we envision, at a scale of fifteen to twenty microns, a hydrate foam (comprising spherical water and air phases as well as needle-shaped hydrate phases oriented isotropically in all space directions), which, at a higher scale of several hundred microns, acts as a contiguous matrix in which cement grains are embedded as spherical clinker inclusions. It is the mixture- and hydration degree-dependent load transfer of overall, cement paste-related, uniaxial stress states down to deviatoric stress peaks within the hydrate phases triggering local brittle failure, which determines the first nonlinear, and then linear dependence of brittle paste strength on the degree of hydration. When extending this approach to the concrete level, it is expected to support, among others, the performance-based design of shotcrete in the framework of the New Austrian Tunneling Method. 1

Hellmich 2008). Following the related line of elastic limit-based (‘‘brittle’’) strength homogenization, we here extend and validate a micromechanics-based strength model for hydrating cement paste against experimental data by (Taplin 1959) who measured early-age strength evolutions of hydrating cement pastes with water-cement ratios ranging from 0.157 to 0.8. This contribution is structured as follows. After recalling fundamentals of continuum micromechanics and presenting cement paste as a hierarchical twolevel material (Section 2), we describe continuum micromechanics models for upscaling elasticity and compressive strength (Section 3). In Section 4, we identify, from dense hydrate foams with very low porosity, elastic properties which are, on average, representative for all hydrates. On this basis, our new micromechanics models predict elasticity and strength of cement pastes as functions of watercement ratio and degree of hydration (Section 5). Finally, the upscaling scheme for cement paste strength is quantitatively verified through the landmark experiments of Taplin (1959), in Section 6, before we conclude by comparing our brittle upscaling scheme with recently published ductile schemes related to the upscaling of confined hardness measurements on cement pastes and concretes (Section 7).

INTRODUCTION

Cement paste is the binder for cement-based materials, including cement mortar, concrete, shotcrete, and soilcrete. Therefore, a reliable prediction of mechanical properties of cement paste is paramount for subsequent modeling activities, be they related to material behavior of cement-based composites or to the structural behavior of engineering constructions built up from these materials. Challenging applications even require modeling of the evolution of mechanical properties of hydrating cement-based materials, e.g. drill and blast tunneling according to the principles of the New Austrian Tunneling Method (NATM), where shotcrete tunnel shells are loaded by the inward moving rock, while the material still exhibits rather small maturities and undergoes the chemical hydration process. This provides the motivation for the present contribution which focuses on upscaling elasticity and strength of hydrating cement paste by means of continuum micromechanics. Within cement paste, hydration products establish the links that constitute a network of connected particles. Their non-spherical phase shape as well as their brittle failure behavior are essential for reliable micromechanics-based prediction of the strength evolution of cement-based materials (Pichler, Hellmich, and Eberhardsteiner 2009; Pichler, Scheiner, and

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2 2.1

FUNDAMENTALS OF CONTINUUM MICROMECHANICS

equilibrium conditions (disregarding volume forces) divσ (x) = 0,

Representative volume elements and separation of scales principle

and linear strain-displacement relations 1 (3) (∇ξ + t ∇ξ ), 2 where x denotes the position vector, σ and ε, respectively, stand for the second-order tensors of stresses and strains, C for the fourth-order elastic stiffness tensor, and ξ for the displacement vector. The boundaries ∂ of the RVEs are subjected to linear displacements corresponding to a second-order strain tensor E, i.e. we prescribe so-called Hashin boundary conditions (Hashin 1983), also referred to as uniform strain boundary conditions ε(x) =

In continuum micromechanics (Hill 1963; Suquet 1997; Zaoui 1997; Zaoui 2002), a material is understood as a macro-homogeneous, but micro-heterogeneous body filling a representative volume element (RVE) with characteristic length . The separation of scales requirement implies (i)   d, where d is standing for the characteristic length of inhomogeneities within the RVE, and (ii)   D, where D stands for the characteristic lengths of dimensions or loading of a structure built up by the material defined on the RVE. Notably, ‘‘much smaller ()’’ does not necessarily imply more than a factor of 4 to 5 between the characteristic length of the heterogeneities and that of the RVE (Drugan and Willis 1996). In general, the microstructure within one RVE cannot be described in complete detail. Therefore, quasi-homogeneous subdomains with known physical quantities (such as volume fractions or elastic properties) are identified. They are called material phases. Once their mechanical behavior, their dosages within the RVE, their characteristic shapes, and the mode of their interactions are identified, the ‘‘homogenized’’ mechanical behavior of the overall material can be estimated, i.e. the relation between homogeneous deformations acting on the boundary of the RVE and resulting (average) stresses, or the ultimate stresses sustainable by the RVE, respectively. In the framework of multiscale homogenization theory, a material phase, identified at a specific scale of observation ‘‘A’’, exhibits a heterogeneous microstructure on a lower scale of observation ‘‘B’’. The mechanical behavior of this microheterogeneous phase can be estimated by that of an RVE with a characteristic length being smaller than or equal to the characteristic length of the aforementioned phase, i.e. that of inhomogeneities identified on observation scale ‘‘A’’, see, e.g. (Fritsch and Hellmich 2007). 2.2

ξ (x) = E · x. 2.3

(4)

Homogenization of elasticity

The geometric compatibility of the microscopic strain field ε(x) with boundary condition (4) implies the following strain average rule   1 ε(x)dV = fp ε p , (5) E=  p 

where p denotes an index running over all phases of the considered RVE, fp stands for the volume fraction of phase p, and ε p for the second-order tensor of average phase strains defined as  1 ε(x) dV . (6) εp = p p

In (6), p denotes the subvolume of the RVE occupied by phase p. Analogously to (5), macroscopic stresses  are set equal to the spatial average of the equilibrated local stresses σ (x) inside the RVE,   1 σ (x) dV = fp σ p , (7) =  p 

with σ p as the second-order tensor of average phase stresses, defined by analogy to (6). Linearity of the field equations (1)–(3) implies a linear strain concentration rule

Statement of the studied problem

For cement paste, we employ two RVEs: The first one relates to the a polycrystalline hydrate foam (with spherical phases representing water and air, and with needle-shaped phases of hydration products exhibiting isotropically distributed orientations), and the second one relates to cement paste (with a spherical phase representing clinker embedded in a continuous hydrate foam matrix), see Figure 1. We are left with presenting the field equations and the boundary conditions. Within the volume  of both RVEs, we consider field equations of linear elasticity, i.e. generalized Hooke’s law accounting for linear elastic material behavior σ (x) = C(x) : ε(x),

(2)

εp = Ap : E,

(8)

with Ap as the fourth-order strain concentration tensor of phase p. Specification of the elastic constitutive law of phase p σ p = C p : εp ,

(9)

for strain concentration rule (8) and insertion of the resulting expression for the phase stresses σ p into the stress average rule (7) delivers a relation between macrostress  and macrostrain E. Comparison of this relation with the macroscopic elastic law

(1)

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Figure 1. Micromechanical representation of cement paste microstructure through a two-step homogenization scheme: a) polycrystalline RVE of ‘‘hydrate foam’’ built up of water, hydrates, and air; (b) RVE of matrix-inclusion composite ‘‘cement paste’’ where clinker is embedded in hydrate foam matrix.

 = Chom : E allows for identification of the homogenized elasticity tensor as  Chom = fp C p : A p . (10)

Pichler, Scheiner, Hellmich 2008), on bone biomaterials (Fritsch, Dormieux, Hellmich, and Sanahuja 2009), and on gypsum (Sanahuja, Dormieux, Meille, Hellmich, and Fritsch 2009), we here model quasibrittle fracture in the framework of elastic limit analysis. Microscopic phase failure resulting in the failure of the entire RVE is governed by strain peaks rather than by average phase strains defined in (6). Strain levels corresponding to volumetric and deviatoric strain peaks can be estimated by corresponding quadratic strain averages (Dormieux, Molinari, and Kondo 2002). They can be easily derived from the elastic energy stored in the RVE, provided that the phases within the RVE exhibit isotropic elastic behavior

p

Eq. (10) highlights that knowledge of phase strain concentration tensors Ap allows for homogenization (upscaling) of phase stiffnesses to the homogenized elasticity tensor. As a rule, the concentration tensors Ap are not known up to analytical precision. Still, they can be estimated based on classical matrixinclusion problems of Eshelby (Eshelby 1957) and Laws (Laws 1977), for details see Zaoui (Zaoui 2002) and Benveniste (Benveniste 1987) Ap = [I + P0p : (Cp − C0 )]−1 −1  : fq [I + P0q : (Cq − C0 )]−1 .

Cp = 3kp J + 2μp K,

(11)

(13)

q

where Cp denotes the fourth-order isotropic elasticity tensor of phase p, kp and μp , respectively, denote the bulk modulus and the shear modulus of phase p, K stands for the deviatoric part of the fourth-order unity tensor, being defined as K = I−J, with I as the symmetric fourth-order unity tensor with components Iijrs = 1/2 (δir δjs + δis δjr ), and J = 1/3(1 ⊗ 1) stands for the volumetric part of the fourth-order unity tensor, where 1 denotes the second-order unity tensor with components δij (Kronecker delta), δij = 1 for i = j, and δij = 0 otherwise. In the context of modeling the uniaxial compressive strength evolution of cement-based materials, the quadratic phase average of the deviatoric strain field ε dev (x) = ε(x) − [tr ε(x)/3]1 over hydrates with specific orientations is of interest (Pichler, Hellmich, and Eberhardsteiner 2009; Pichler, Scheiner, and Hellmich 2008), see also Section 3.4. At this stage, we recall the expression for the quadratic average of the deviatoric strain field over a general phase p, which reads according to

Specification of (10) for (11) delivers the related estimate of the homogenized elastic stiffness tensor as Chom =



fp Cp : [I + P0p : (Cp − C0 )]−1

p

:



fq [I +

P0p

−1

: (Cq − C )] 0

−1

(12) .

q

In (11) and (12), P0p denotes the fourth-order Hill tensor, accounting for the characteristic shape of phase p embedded in a matrix with stiffness C0 . Choice of C0 describes the interactions between material phases, see Sections 3.2 and 3.3 for details. 2.4 Homogenization of strength In the line of earlier work on cementitious materials (Pichler, Hellmich, and Eberhardsteiner 2009;

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(Dormieux, Molinari, and Kondo 2002) as  εpdev =

 1 p

 =

products, and air:

p

1 dev (x) 2ε

: ε dev (x) dV (14)

1 4 fp E

:

∂Chom : E. ∂μp

Related deviatoric stress peaks follow simply as  σpdev

=

 1 p

p

1 2 s(x)

: s(x) dV = 2μp εpdev

fclin (ξ ) =

20(1 − ξ ) ≥ 0, 20 + 63(ω/c)

fH2 O (ξ ) =

63[(ω/c) − 0.42ξ ] ≥ 0, 20 + 63(ω/c)

fhyd (ξ ) =

43.15ξ , 20 + 63(ω/c)

fair (ξ ) =

3.31ξ . 20 + 63(ω/c)

In (16), ω/c denotes the water-to-cement mass ratio, and ξ stands for the hydration degree which is defined as the mass of currently formed hydrates over the mass of hydrates formed at completed hydration. Notably, air-filled pores are created since hydration products occupy a smaller volume than the reactants clinker and water.

(15)

where s(x) denotes the field of the deviatoric stress tensor, defined as s(x) = σ (x) − [tr σ (x)/3]1.

3.2 3

CONTINUUM MICROMECHANICS OF CEMENT PASTE

Homogenization step I: hydrate foam

The polycrystal-type mutual interaction of hydration products, water, and air suggests usage of the selfconsistent scheme (Hershey 1954; Kröner 1958; Hill 1965; Budiansky 1965) for determining the homogenized stiffness tensor of the hydrate foam: Chom hf . Accordingly, C0 in (12) is chosen to be equal to the homogenized stiffness of the hydrate foam itself, and the sum over p in (12) now extends over the air and water phases, as well as over an infinite amount of hydrate phases oriented in all space directions indicated by polar angles ϕ and ϑ. Consequently, we arrive at the following implicit tensorial expression for Chom hf :

3.1 Model inputs for cement paste Herein, we collect input data required for specifying the discussed elasticity and strength models for cement paste. Elasticity constants listed in Table 1 comprise intrinsic values for clinker, water, and air taken from the open literature, while properties of ‘‘hydration products’’ represent average elasticity constants of all types of hydrates, including Portlandite, Ettringite, and Calcium Silicate Hydrates (C-S-H) of all densities, see Section 4 for the related identification strategy. The evolution of cement paste-related phase volume fractions during hydration is accounted for by Powers’ hydration model (Powers and Brownyard 1948), providing the relative volumes of clinker, water, hydration

Chom hf =



 −1 hf f˜p Cp : I + Psph : (Cp − Chom hf )

p

+ f˜hyd Chyd : Table 1. Intrinsic mechanical properties of microstructural constituents of clinker, water, and air. Phase

Bulk modulus k [GPa]

Shear modulus μ [GPa]

Clinker

kclin = 116.7

μclin = 53.8

Water (drained RVE) Water (sealed RVE)

kH2 O = 0.0

μH2 O = 0.0

kH2 O = 2.3

μH2 O = 0.0

Hydration products Air

khyd = 18.7

μhyd = 11.8

kair = 0.0

μair = 0.0

(16)

2ππ 

hf

I + Pcyl (ϕ, ϑ)

−1 sin ϑdϑdϕ  ) : (Chyd − Chom hf 4π   −1 hf : f˜p I + Psph : (Cp − Chom hf ) 0 0

Source

p

(Acker 2001)

+ f˜hyd

2π π  0

(Bilaniuk and Wong 1993) Section 4

hf

I + Pcyl (ϕ, ϑ)

0

−1 sin ϑdϑdϕ −1 . 4π

: (Chyd − Chom hf )

(17) In (17), the hydrate foam-related volume fractions of hydration products, water, and air (f˜hyd , f˜H2 O , and

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f˜air ) follow from the cement paste-related volume fractions of clinker, water, hydration products, and air (fclin , fH2 O , fhyd , and fair , given in (16)) according to fj f˜j = 1 − fclin

⎧ ⎨hyd, j = H2 O, ⎩air.

almost no pre-peak nonlinearities (Pichler, Hellmich, and Eberhardsteiner 2009), strength of cement paste may be estimated through an elastic limit analysis. In this context, we essentially envision that each hydrate behaves linear elastically as long as microscopic deviatoric stress peaks remain below a specific critical value:

(18)

dev dev ≤ σhyd,crit max σhyd,ϕ,ϑ

For detailed explanations regarding the numerical evaluation of (17), as well as for computation of cp the Hill tensors for spherical material phases, Psph , cp and for cylindrical phases, Pcyl , respectively, see (Pichler, Hellmich, and Eberhardsteiner 2009; Pichler, Scheiner, and Hellmich 2008).

ϕ,ϑ

Eq. (22), together with (20) and (21) constitutes a micromechanics-based brittle failure criterion for cement pastes, considering through derivation of the homogenized stiffness tensor (19), the effect of their constituents’ elastic properties, as well as of their microstructures and compositions. In words, this criterion reads as follows: If, due to (compressive uniaxial) dev macroscopic load increase, this critical value σhyd,crit is reached in the most heavily stressed region of the hydrate phase, the elastic limit on the microscale is reached, which, in turn, corresponds to the macroscopic elastic limit of cement paste, associated with failure of the material (under macroscopic uniaxial compression). An algorithm for evaluation of (20), i.e. for computation of deviatoric stress peaks in needle-shaped hydrates within a two-step homogenization approach, is presented in (Pichler, Scheiner, Hellmich 2008).

3.3 Homogenization step II: cement paste The interaction of the spherical clinker phase embedded in a continuous matrix built up by the hydrate foam suggests usage of the Mori-Tanaka scheme (Benveniste 1987; Mori and Tanaka 1973; Zaoui 2002) for determining the homogenized stiffness tensor of 0 cement paste: Chom cp . Accordingly, C in 12 is chosen to be equal to the homogenized stiffness of the hydrate foam, resulting in an explicit expression for Chom cp : hom Chom cp = {(1 − fclin )Chf −1 + fclin Cclin : [I + Psph : (Cclin − Chom hf )] } hf

hf

: {(1 − fclin ) I + fclin [I + Psph −1 −1 : (Cclin − Chom hf )] } .

3.4

4 (19)

Deriving the homogenized stiffness of cement paste (19) with respect to the shear modulus of a hydrate phase with a specific orientation (ϕ, ϑ) provides access to the deviatoric stress peak within hydrates of that orientation. According to (14) and (15) the deviatoric stress peak is expressed as a function of the macrostrain Ecp imposed on a RVE of cement paste dev σhyd,ϕ,ϑ =

μ2hyd fhyd,ϕ,ϑ

Ecp :

∂Chom cp ∂μϕ,ϑ

: Ecp

(20)

the macrostrain being itself related to the macrostress cp through the inverse elasticity law according to −1  : cp , Ecp = Chom cp

ELASTICITY OF HYDRATION PRODUCTS

For the sake of simplicity, we introduce only one elasticity tensor representing the average behavior of all different types of hydration products including Portlandite, Ettringite, and Calcium Silicate Hydrates (C-S-H) of all densities. In this context, it would be desirable to produce a material sample consisting of hydration products only, and to perform elasticity tests on such a material. Deplorably, this is not possible. As a remedy, we consider the material coming closest to the aforementioned ‘‘pure hydrate agglomerate’’, i.e. to the situation where, in Figure 1, the volume fractions of water, air, and clinker are close to zero, while the volume fraction of hydrates approaches one. According to Powers’ hydration model, the aforementioned volume fractions are most closely attained in a fully cured (ξ = 1) cement paste with water-cement ratio amounting to 0.42. These fractions would be:

Strength modeling



(22)

fclin = fH2 O = 0,

(21)

fhyd = 92.9%,

fair = 7.1%. (23)

To get experimental access to the elastic properties of such a ‘‘close-to-a-pure-hydrate-agglomeration’’ cement paste, we consider data from (Helmuth and Turk 1966). These authors tested three different types of cement pastes: one was produced using almost

with Chom cp following from (19). Since macroscopic stress-strain diagrams obtained in uniaxial compression tests on cement pastes exhibit

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exclusively tricalcium silicate (C3 S), and the remaining two contained Portland cements (two different products). Water-cement ratios ranged from 0.3 to 0.6. Young’s modulus was deduced from measurements of the fundamental flexural and torsional resonance frequencies of specimens that were allowed to cure for six to 14 months (C3 S samples) and for six to 24 months (Portland cement samples), respectively. Interpolating between experimental data by (Helmuth and Turk 1966) implies that the almost pure hydrate agglomeration (fully-cured cement paste with ω/c = 0.42) can be expected to exhibit a macroscopic Young’s modulus of 25 GPa, see circle in Figure 2. When studying the almost pure hydrate agglomeration by means of the proposed micromechanics model, (23) implies that the two-step homogenization scheme illustrated in Figure 1 degenerates to a one-step scheme where the hydrate foam (consisting exclusively of hydration products and air) is equal to cement paste. Hydrate foam-related volume fractions follow from inserting (23) into (18) as f˜H2 O = 0 ,

f˜hyd = 92.9% ,

f˜air = 7.1% .

and Lemarchand 2003), as being representative for all hydration products. Specifying (17) for the volume fractions (24), for vanishing bulk and shear moduli of air, for νhyd = 0.24, and for Ehyd = 29.158 GPa (this is equivalent to the elasticity constants of hydration products listed in Table 1) results in the sought result Ehf (ξ = 1, ω/c = 0.42) = 25 GPa. A first relevance check of the hydrate elasticity in terms of Ehyd and νhyd can be made when predicting the experimental data of Figure 2, referring to cement pastes with a wide range of water-cement ratios: Therefore, the two-step homogenization scheme (17) and (19) needs to be specified for the identified average hydrate elasticity constants, for the intrinsic elastic properties of clinker, for the vanishing effective stiffnesses of the drained fluid phases air and water, see Table 1, as well as for the phase volume fractions (16) and (18) specified for the maximum attainable hydration degree ⎧ ⎨ ω/c for ω/c ≤ 0.42 max ξ = 0.42 (25) ⎩ 1 for ω/c ≥ 0.42

(24)

Model predictions and experiments agree very well (Figure 2).

In the following, we use the elasticity model (17) to identify the average elasticity constants of the hydration products which, together with the vanishing bulk and shear moduli of air, results in a prediction of macroscopic Young’s modulus of the almost pure hydrate agglomeration as Ehf (ξ = 1, ω/c = 0.42) = 25 GPa. Since experimental data by (Helmuth and Turk 1966) provide access to one constant of isotropic elasticity only, we have to make an assumption on the average Poisson’s ratio of the hydration products. Considering that Calcium Silicate Hydrates are commonly the dominating hydration products, we treat their Poisson’s ratio νCSH = 0.24, see (Bernard, Ulm,

5

PERFORMANCE OF ELASTICITY AND STRENGTH MODELS

Herein, we provide model-predicted evolutions of elasticity and strength of hydrating cement pastes with water-cement ratios ranging from 0.157 to 0.8. As expected, the hydration degree-induced stiffness increase is the larger the lower the water-cement ratio (Figures 3 and 4), and interestingly, this increase is of convex nature for low water-cement ratios, but of S-shaped nature for higher water-cement ratios, for both bulk and shear modulus evolutions (see Figures 3 and 4). In all model predictions, non-zero stiffnesses are reached already for very early hydration degrees,

Figure 2. Final Young’s modulus of cement paste: experimental results by (Helmuth and Turk 1966), taken from (Sanahuja, Dormieux, and Chanvillard 2007), and outputs of the proposed model.

Figure 3. Model-predicted evolution of bulk modulus of hydrating cement pastes.

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testing, the prisms were broken into two halves resulting in two specimens (2 by 0.5 by 0.5 in): On each of them, two (destructive) uniaxial compression tests in cross direction were carried out. This was possible, since the load plates were of rather small size, such that in each test an effective specimen length of less than three-quarters of an inch was destroyed. Because the crushed part did not always correspond exactly to the area of the load plate, (Taplin 1959) published the actually measured ultimate forces, rather than postprocessed compressive strength values, see the ordinate in Figure 6. After testing, (Taplin 1959) determined the bound water content of the crushed samples, which is a measure for the hydration degree, for a more elaborate theoretical discussion see, e.g. (Byfors 1980). In order to make our model predictions comparable to the experimental results by (Taplin 1959), we have to establish a link between the ordinates of dev Figures 5 and 6, whereby the unknown value of σhyd,crit should, preferably, not influence the result. This provides the motivation to divide the ordinate of Figure 5 by the dimensionless compressive strength predicted for a fully-cured cement paste with ω/c = 0.157, and the ordinate of Figure 6 by the maximum compressive strength measured on samples with ω/c = 0.157, see Figure 7 for the obtained result. This normalization makes sense since the largest compressive strength measured by (Taplin 1959) on a sample with ω/c = 0.157 exhibits ξ = 0.366 (see Figure 6), which is by only 2% smaller than the largest hydration degree possible at this ω/c-value, amounting to max ξ = 0.157/0.42 = 0.374, see (25). Model-predicted strength evolutions agree satisfactory with corresponding independent results by (Taplin 1959), see Figure 7. It appears worth emphasizing that the model predictions illustrated in Figure 7 do not depend on the unknown deviatoric strength of the hydration products, but only on the elasticity model proposed herein, i.e. the elasticity constants listed in Table 1, on Powers’ hydration model-related volume

Figure 4. Model-predicted evolution of shear modulus of hydrating cement pastes.

Figure 5. Model-predicted dimensionless compressive strength evolution of hydrating cement pastes.

and this is due to the prolate (‘‘infinitely long’’) hydrate shape adopted herein. It might be more realistic to introduce ‘‘cigars’’ with an aspect ratio somewhat smaller than infinity, if precise percolation modeling would be the key feature of interest. Herein, however, our focus is on strength, see Figure 5 for the modelpredicted uniaxial compressive strength of cement paste, normalized by the deviatoric hydrate strength. In contrast to the elastic predictions, strength turns out to be an always convex function of the hydration degree, whose slope obviously increases with decreasing water-cement ratio.

6

VALIDATION OF THE STRENGTH MODEL

Aiming at a strictly quantitative test of the predictive capabilities of the presented strength model, we now compare model predictions with independent experimental data. (Taplin 1959) produced cement pastes with water-cement ratios ranging from 0.157 to 0.8. Specimens were rectangular prisms measuring 4 by 0.5 by 0.5 in, which were cured at 25◦ C. At the time of

Figure 6. Compressive strength values of hydrating cement pastes: experimental results by (Taplin 1959).

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the deviatoric hydrate strength, which can be identified from classical macroscopic tests on cement pastes. Combining, for instance, the uniaxial compressive strength fcu = 54.1 MPa of a cement paste with a water-cement ratio amounting to 0.50 (Bernard, Ulm, and Germaine 2003), with related model predictions for a fully-cured paste from Figure 5, suggests a deviatoric hydrate strength of magnitude 50 MPa. As to our knowledge, except for our earlier approaches (Pichler, Hellmich, and Eberhardsteiner 2009; Pichler, Scheiner, and Hellmich 2008), no random homogenization tools for upscaling strength of cementbased materials have been proposed in the open literature so far. However, there has been intensive activity in strength upscaling in the framework of yield design approaches and nanoindentation testing, see, e.g. (Constantinides and Ulm 2007; Constantinides 2006). Interestingly, the latter approaches suggest hydrate hardness values of some hundreds of MPa, which could be regarded as somehow contradictory to our deviatoric hydrate strength in the tens of MPa range. On a second glance, however, this turns out to be not contradictory at all, but just the outcome of considering one problem (‘‘strength of cement-based materials’’) from two different viewpoints, which, in the end, might be integrated into a consistent overall understanding of the problem. In this context, two threads of argumentation are of interest:

Figure 7. Strength evolution of hydrating cement pastes: experimental results by (Taplin 1959) and predictions of the proposed model.

fractions (16) and (18), as well as on the morphological representation of cement paste including the hierarchical organization, the mode of phase interactions, and the phase shapes illustrated in Figure 1. Nonetheless, the comparison of model predictions with experimental results is of strict quantitative nature: the quadratic correlation coefficient is as high as r 2 = 93%, while the mean prediction error and the standard deviation amount to satisfactory −3% and to 7.3%, respectively. Given the simplicity of the proposed model and the vast range of considered water-cement ratios, we conclude that model validation is accomplished successfully.

7

1. Nanoindentation tests with indentation depths ranging from 170 ± 66 nm to 201 ± 90 nm (Constantinides and Ulm 2007) can be expected to probe the plastic material hardness at a scale of 700–800 nm (Ulm, Vandamme, Jennings, Vanzo, Bentivegna, Krakowiak, Constantinides, Bobko, and Van Vliet 2010). This is much smaller than 3 to 5 microns, which is the scale at which we introduce the hydrates as a material phase. Hence, the nanoindentation tests might give access to strength at a level well below that where we introduce the deviatoric strength value, and the latter might represent the strength of hydrate agglomerations with weak interfaces in between (which the nanoindentation tests do not ‘‘see’’), rather than the strength of ‘‘individual’’ hydrate phases probed by nanoindentation testing. 2. The hardness obtained from nanoindentation tests with indentation depths ranging from 170 ± 66 nm to 201 ± 90 nm (Constantinides and Ulm 2007) relates to a state of confined plastic (‘‘ductile’’) flowing of the hydrates, while the uniaxial tests considered in the present contribution refer to quasi-brittle failure of unconfined hydrates. In other words, our modeling scheme turns out to be well suited for representing brittle failure of unconfined hydrates, but it may have its limitations for more confined loading states. First computations of

DISCUSSION AND CONCLUSIONS

For the sake of simplicity, we do not distinguish between different types of hydrates, but identify, from dense hydrate foams with very low porosity, elastic properties which are, on average, representative for all hydration products. On their basis, we then relate overall stresses acting on the cement pastes to higherorder deviatoric stress averages in the hydrates. When the latter reach a critical strength value, the overall stresses refer to the (brittle) macroscopic uniaxial compressive strength of cement pastes. A single deviatoric hydrate strength can explain, via higher-order average upscaling within a two-step random homogenization scheme, the uniaxial compressive strengths of cement pastes, as functions of the hydration degree, over a wide range of water-cement ratios. Since the experimental data were all normalized to a maximum strength attained by a material with a very low water-cement ratio, our upscaling scheme did not even require a single strength parameter to be introduced—this may be regarded as an additional strong indication for the validity of our approach. When absolute, rather than normalized, values are of interest, however, we need to introduce a number for

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ours (to be documented in a later publication) have actually shown that a dependence of hydrate failure on hydrostatic stress states is beneficial to be introduced if non-uniaxial strengths at the macroscopic level are to be represented (and depending on the magnitude of this hydrostatic stress, the relevant deviatoric hydrate strength will increase, possibly even up to the hundreds of MPa regime). From the extension of our model towards a Drucker-Prager type hydrate failure (comprising the current formulation as the special case for zero hydrostatic stress, the relevance of which was shown herein), it is only a small conceptual steps towards a full elastoplastic upscaling of strength of cement-based materials; in particular, since a very first model of that type was already proposed for bone (Fritsch, Hellmich, and Dormieux 2009), and since the ‘‘transformation field analysis by (Dvorak and Benveniste 1992)’’ has been, very recently, extended to general microstructures (including those of Figure 1 representation) with eigenstrains (inclusive of plastic strains), (Pichler and Hellmich 2010). However, the aforementioned ‘‘small conceptual step’’ will trigger extensive algorithmic work to be done—and this will probably open a new chapter in the scientific community devoted to the ‘‘Computational Modeling of Concrete Structures’’.

Constantinides, G. and F.-J. Ulm (2007). The nanogranular nature of CSH. Journal of the Mechanics and Physics of Solids 55(1), 64–90. Dormieux, L., A. Molinari, and D. Kondo (2002). Micromechanical approach to the behavior of poroelastic materials. Journal of Mechanics and Physics of Solids 50(10), 2203–2231. Drugan, W. and J. Willis (1996). A micromechanics-based nonlocal constitutive equation and estimates of representative volume element size for elastic composites. Journal of the Mechanics and Physics of Solids 44(4), 497–524. Dvorak, G.J. and Y. Benveniste (1992). On transformation strains and uniform fields in multiphase elastic media. Proceedings of the Royal Society London 437, 291–310. Eshelby, J. (1957). The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proceedings of the Royal Society of London A 241, 376–396. Reprinted in (Markenscoff and Gupta 2006). Fritsch, A., L. Dormieux, C. Hellmich, and J. Sanahuja (2009). Mechanical behavior of hydroxyapatite biomaterials: An experimentally validated micromechanical model for elasticity and strength. Journal of Biomedical Materials Research—Part A 88(1), 149–161. Fritsch, A. and C. Hellmich (2007). Universal microstructural patterns in cortical and trabecular, extracellular and extravascular bone materials: Micromechanics-based prediction of anisotropic elasticity. Journal of Theoretical Biology 244(4), 597–620. Fritsch, A., C. Hellmich, and L. Dormieux (2009). Ductile sliding between mineral crystals followed by rupture of collagen crosslinks: Experimentally supported micromechanical explanation of bone strength. Journal of Theoretical Biology 260(2), 230–252. Hashin, Z. (1983). Analysis of composite materials—a survey. Journal of Applied Mechanics 50(3), 481–505. Helmuth, R. and D. Turk (1966). Elastic moduli of hardened portland cement and tricalcium silicate pastes: Effect of porosity. Symposium on structure of Portland cement paste and concrete 90, 135–144. Highway Research Board, Washington, DC. Hershey, A. (1954). The elasticity of an isotropic aggregate of anisotropic cubic crystals. Journal of Applied Mechanics (ASME) 21, 226–240. Hill, R. (1963). Elastic properties of reinforced solids. Journal of the Mechanics and Physics of Solids 11(5), 357–372. Hill, R. (1965). A self-consistent mechanics of composite materials. Journal of the Mechanics and Physics of Solids 13(4), 213–222. Kröner, E. (1958). Berechnung der elastischen Konstanten des Vielkristalls aus den Konstanten des Einkristalls [Computation of the elastic constants of a polycrystal based on the constants of the single crystal]. Zeitschrift für Physik A Hadrons and Nuclei 151(4), 504–518. In German. Laws, N. (1977). The determination of stress and strain concentrations at an ellipsoidal inclusion in an anisotropic material. Journal of Elasticity 7(1), 91–97. Markenscoff, X. and A. Gupta (Eds.) (2006). Collected Works of J.D. Eshelby—The Mechanics of Defects and Inhomogeneities, Volume 133 of Solid Mechanics and Its Applications. Springer.

REFERENCES Acker, P. (2001). Micromechanical analysis of creep and shrinkage mechanisms. In F.-J. Ulm, Z. Bažant, and F. Wittmann (Eds.), Creep, Shrinkage and Durability Mechanics of Concrete and other Quasi-brittle Materials, 6th International Conference CONCREEP@MIT, Amsterdam, pp. 15–26. Elsevier. Benveniste, Y. (1987). A new approach to the application of Mori-Tanaka’s theory in composite materials. Mechanics of Materials 6(2), 147–157. Bernard, O., F.-J. Ulm, and J. Germaine (2003). Volume and deviator creep of calcium-leached cement-based materials. Cement and Concrete Research 33(8), 1127–1136. Bernard, O., F.-J. Ulm, and E. Lemarchand (2003). A multiscale micromechanics-hydration model for the early-age elastic properties of cement-based materials. Cement and Concrete Research 33(9), 1293–1309. Bilaniuk, N. and G. Wong (1993). Speed of sound in pure water as a function of temperature. The Journal of Acoustical Society of America 93(3), 1609–1612. Budiansky, B. (1965). On the elastic moduli of some heterogeneous materials. Journal of the Mechanics and Physics of Solids 13(4), 223–227. Byfors, J. (1980). Plain concrete at early ages. Technical report, Swedish Cement and Concrete Research Institute, Stockholm, Sweden. Constantinides, G. (2006). Invariant properties of calcium silicat hydrates (C-S-H) in cement-based materials: instrumented nanoindentation and microporomechanical modeling. Ph.D. thesis, MIT.

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Mori, T. and K. Tanaka (1973). Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metallurgica 21(5), 571–574. Pichler, B. and C. Hellmich (2010, conditionally accepted for publication). Estimation of influence tensors for eigenstressed multiphase elastic media with non-aligned inclusion phases of arbitrary ellipsoidal shape. Journal of Engineering Mechanics. Pichler, B., C. Hellmich, and J. Eberhardsteiner (2009). Spherical and acicular representation of hydrates in a micromechanical model for cement paste—Prediction of early-age elasticity and strength. Acta Mechanica 203 (3–4), 137–162. Pichler, B., S. Scheiner, and C. Hellmich (2008). From micron-sized needle-shaped hydrates to meter-sized shotcrete tunnel shells: Micromechanical upscaling of stiffness and strength of hydrating shotcrete. Acta Geotechnica 3(4), 273–294. Powers, T. and T. Brownyard (1948). Studies of the physical properties of hardened portland cement paste. Research Laboratories of the Portland Cement Association Bulletin 22, 101–992. Sanahuja, J., L. Dormieux, and G. Chanvillard (2007). Modelling elasticity of a hydrating cement paste. Cement and Concrete Research 37(10), 1427–1439.

Sanahuja, J., L. Dormieux, S. Meille, C. Hellmich, and A. Fritsch (2009). Micromechanical explanation of elasticity and strength of gypsum: from elongated anisotropic crystals to isotropic porous polycrystals. Journal of Engineering Mechanics (ASCE). Online in advance of print, doi:10.1061/(ASCE)EM.1943.7889.0000072. Suquet, P. (1997). Continuum Micromechanics, Volume 377 of CISM Courses and Lectures. Springer Verlag, Wien New York. Taplin, J. (1959). A method of following the hydration reaction in portland cement paste. Australian Journal of Applied Science 10(3), 329–345. Ulm, F.-J., M. Vandamme, H. Jennings, J. Vanzo, M. Bentivegna, K. Krakowiak, G. Constantinides, C. Bobko, and K. Van Vliet (2010). Does microstructure matter for statistical nanoindentation techniques? Cement and Concrete Composites 32(1), 92–99. Zaoui, A. (1997). Structural morphology and constitutive behavior of microheterogeneous materials. In P. Suquet (Ed.), Continuum Micromechanics, Vienna, pp. 291–347. Springer. Zaoui, A. (2002). Continuum micromechanics: Survey. Journal of Engineering Mechanics (ASCE) 128(8), 808–816.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

C-Crete: From atoms to concrete structures Franz-Josef Ulm Massachusetts Institute of Technology, Cambridge, USA

Roland J.-M. Pellenq Massachusetts Institute of Technology, Cambridge, USA Centre Interdisciplinaire des Nanosciences de Marseille, CNRS and Marseille Universite, Marseille, France

Matthieu Vandamme Ecole des Ponts Paris Tech, Universite Paris-Est, France

ABSTRACT: We review recent developments of a multiscale bottom-up approach for concrete. The approach starts at the electron and atomic scale to nanoengineer the fundamental building block of concrete; to assess the properties by nanoindentation; and upscale strength, fracture and stiffness properties from nanoscales to macroscales of day-to-day concrete engineering applications. The key to all this is mechanics at the interface of physics and engineering. 1

empirical approaches, we have chosen a bottom-up approach that starts at the electron and atomic scale to nanoengineer the fundamental building block of concrete; to assess the properties by nanoindentation; and upscale strength, fracture and stiffness properties from nanoscales to macroscales of day-today concrete engineering applications. The key to all this is mechanics at the interface of physics and engineering. This paper reviews some recent developments of this bottom-up approach.

INTRODUCTION

More concrete is produced than any other synthetic material on Earth. The current worldwide cement production stands at 2.3 billion tons, enough to produce more than 20 billion tons or one cubic meter of concrete per capita per year. There is no other material that can replace concrete in the foreseeable future to meet our societies’ legitimate needs for housing, shelter, infrastructure, and so on. But concrete faces an uncertain future, due to a non-negligible ecological footprint that amounts to 5–10% of the worldwide CO2 production. It now appears that mechanics can be the discipline that enables the development of a sustainable green concrete future. We here adopt the perspective originating from Galileo’s Strength of Materials Theory, that weight, and thus CO2 -emission, increases with the volume of the produced material, while strength of structural members increases with the section. Hence, as one increases the strength of a material by a factor of x, one reduces the environmental footprint by 1/x for pure compressive members such as columns and perfect arches and shells, x−2/3 for beams in bending, and x−1/2 for slabs. Similarly, if one adopted a Linear Elastic Fracture model, an increase of the fracture toughness KIc = y KIc0 , would entail a reduction of the environmental footprint by 1/y for columns and y−4/5 for (notched) beams in bending or in torsion. All this hints towards a critical role of mechanics, and in particular Strength of Materials and Fracture Mechanics of concrete, in redesigning concrete materials and structures for the coming of age of global warming. In contrast to the classical top-down

2

A REALISTIC MOLECULAR MODEL FOR CALCIUM-SILICATE-HYDRATES

The first step in setting up a bottom-up approach is to address the fundamental unit of concrete material behavior at electron and atomic scale. But, despite decades of studies of calcium-silicate-hydrate (C-S-H), the structurally complex binder phase of concrete, the interplay between chemical composition and density remains essentially unexplored. Together these characteristics of C-S-H define and modulate the physical and mechanical properties of this ‘‘liquid stone’’ gel phase. 2.1

Background

Much of our knowledge of C-S-H has been obtained from structural comparisons with crystalline calcium silicate hydrates, based on HFW Taylor’s postulate that real C-S-H was a structurally imperfect layered hybrid of two natural mineral analogs: tobermorite of

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14-Å interlayer spacing and jennite. While this suggestion is plausible in morphological terms, this model is incompatible with two basic characteristics of real C-S-H; specifically the calcium-to-silicon ratio (C/S) and the density. Recently, small-angle neutron scattering measurements have fixed the C/S ratio at 1.7 and the density at 2.6 g/cm3 (Allen et al. 2007), values that clearly cannot be obtained from either tobermorite (C/S = 0.83, 2.18 g/cm3 ) or jennite (C/S = 1.5 and 2.27 g/cm3 ); see for instance Shahsavari et al. (2009). From the standpoint of constructing a molecular model of C-S-H, this means that these crystalline minerals are not strict structural analogs. This brought about the development of a realistic molecular of C-S-H, based on a bottom-up atomistic simulation approach that considers only the chemical specificity of the system as the overriding constraint (Pellenq et al. 2009). By allowing for short silica chains distributed as monomers, dimers, and pentamers, this C-S-H archetype of a molecular description of interacting CaO, SiO2 , and H2 O units provides not only realistic values of the C/S ratio and the density computed by grand canonical Monte Carlo simulation of water adsorption at 300 K. The model, displayed in Figure 1, with a chemical composition of (CaO)1.65 (SiO2 ) (H2 O)1.75 , also predicts other essential structural features and fundamental physical properties amenable to experimental validation. This model suggest that the C-S-H gel structure includes both glass-like shortrange order and crystalline features of the mineral tobermorite.

2.2

Molecular properties of C-S-H

One of the great advantages of having a realistic molecular model of C-S-H is that it is possible to probe the structure mechanically. The first quantity of interest is the elasticity content of the molecular structure, which is given in Table 1 in form of the components of the elasticity tensor. As one may expect from a glassy—layered hybrid, the elasticity exhibitsa high degree of anisotropy. For further applications it will be useful to consider the random polycrystal properties, in form of the Voigt-Reuss-Hill approximation classically used in mineralogy. As shown by Povolo & Bolmaro (1987), both Voigt and Reuss models are built using the invariance of the trace of the 9×9 matrix representing the stiffness and compliance tensors, respectively. This leads to the observation (made by Hill) that the Voigt and Reuss averages only use 9 of the 21 independent elastic constants. Denoting by I1 = Ciijj and I1∗ = Ciijj the traces (or linear invariants) of tensors Ciikl and Cijjl , respectively, the Voigt average is obtained from a comparison of those traces with their corresponding isotropic expressions, leading to: K Voigt =

1 1 I1 ; G Voigt = (3I ∗ − I1 ) 9 30 1

(1)

Applying a similar procedure to the compliance −1 , the Reuss average is obtained: tensor Sijkl = Cijkl K Reuss =

1 15 ; G Reuss = ∗ J1 6J1 − 2J1

(2)

where J1 = Siijj and J1∗ = Sijij are the corresponding traces of the compliance tensors Siikl and Sijjl , respectively. The Reuss-Voigt-Hill polycrystal properties are obtained as the arithmetic mean of Reuss and Voigt bounds, which yields for C-S-H: K C−S−H = 49GPa; G C−S−H = 23GPa

(3)

A second useful information the molecular model can provide is about the strength behavior, by simulating for instance the stress-strain behavior of the Table 1. Elastic tensor of C-S-H determined from molecular dynamic simulations (Voigt notation) [from Pellenq et al. 2009].

Figure 1. Computational molecular model of C-S-H: the blue and white spheres are oxygen and hydrogen atoms of water molecules, respectively; the green and gray spheres are inter and intra-layer calcium ions, respectively; yellow and red sticks are silicon and oxygen atoms in silica tetrahedra (adapted from Pellenq et al. 2009).

Cij/GPa

1

2

3

4

5

6

1 2 3 4 5 6

93.5

45.4 94.9

26.1 30.0 68.5

0.6 −4.6 −4.3 19.2

−0.1 1.8 −2.7 0.3 16.1

3.46 −3.0 −0.6 1.82 −0.4 31.2

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C-S-H model in affine shear deformation (strain controlled) after first relaxing the computational cell using MD at 300 K, under constant NVT ensemble conditions. A series of shear strains in increments of 0.005 is imposed; after each increment the atomic configuration is relaxed and the shear stress determined from the virial expression. Figure 2 displays the shear-stress-strain response of the C-S-H model. Two configurations are herein considered: the ‘‘wet’’ model, with water present particular in the inter-layer space, and the ‘‘dry’’ model, in which all water molecules have been removed. The stressstrain response shows that the present of water in the inter-layer space leads to a localization of deformation into a narrow band defined by a wet interlayer, akin to a fracture process. By contrast, the ‘‘dry’’ model shows a plastic deformation behavior, with irreversible (plastic) deformation upon unloading. This shows that there are some strength reserves at a molecular scale of C-S-H that could enhance strength and (ductile) deformation behavior of C-S-H. 2.3

the comparison with nanoindentation experiments. Nanoindentation measurements of C-S-H do not probe the particle properties, but rather the C-S-H gel properties. From a straightforward dimensional analysis, the link between particle properties and microstructure is provided by (Ulm et al. 2007): P = hs × H (μ, η, η0 ) Ac (dP/dh)hmax M =c √ = ms × M (v, η, η, η0 ) Ac

H=

(4)

where H the hardness, the indentation modulus, that are accessible from the applied force, Pand the indentation depth, h, and the projected contact area Ac . In return, hs and ms are particle hardness and plane stress modulus, the latter being defined for an isotropic material by: ms = 4G

3K + G = 65 GPa 3K + 4G

(5)

where we used the values for K and G in Eq. (3) determined from MD simulations. Furthermore, functions H and M in Eq. (4) are dimensionless functions of the friction coefficient μ, the Poisson’s ratio ν, and the packing density η. Linear and nonlinear microporomechanics (Dormieux et al. 2006) provides a convenient way to determine these functions analytically (Constantinides & Ulm, 2007; Cariou et al. 2008). Moreover, combining the elastic properties determined from the C-S-H model with these micromechanics models with no adjustable parameters, one can probe the texture and extent of anisotropic structures within cement paste at micrometer length scales of randomly oriented C-S-H particles. Figure 3 compares the prediction of two micromechanics models along with

From molecular properties to microtexture

The molecular model of C-S-H has been validated against several experimentally accessible properties, including density, extended X-ray absorption fine structure (EXAFS) spectroscopy signals measuring short-range order around Ca atoms, longer range correlations revealed in X-ray diffraction intensity, vibrational density of states measured by infrared spectroscopy, and nanoindentation measurements of elasticity and strength properties (for details, see Pellenq et al. 2009). Here we restrict ourselves to

Figure 3. Nanoindentation data (obtained by Vandamme & Ulm 2009) compared with two micromechanics model. The input for the model are the elastic C-S-H properties (indentation modulus = 65 GPa, Poisson’s ratio = 0.3). [adapted from Pellenq et al. 2009].

Figure 2. Relationship between the shear stress and the shear strain for the C-S-H model with (solid line) and without (dashed line) water molecules (results obtained by A. Kushima and S. Yip; adapted from Pellenq et al. 2009).

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nano-indentation results; one is a porous bicontinuous matrix approach captured by the so-called MoriTanaka scheme, and the other a granular approach captured by the self-consistent scheme. From this comparison, we observe first that the granular approach better describes the experimental data over the entire domain of C-S-H particle packing fractions. Second, both approaches give acceptable predictions at larger packing fractions. That is, at the micrometer-scale, Mori-Tanaka and self consistent micromechanics approaches, parameterized only with nanoscale derived elasticity constants, indicate that cement paste can be conceptualized as a cohesive granular material rather than a porous bi-continuous matrix. 3

3.2

Implementation for back-analysis of packing density distributions

Consider then a series of N indentation tests on a heterogeneous material. What is measured in these tests are N times (M , H ) values at each indentation point representative of a composite response. Assuming that the solid phase is the same, i.e. C-S-H with now well defined elasticity properties, and that all what changes between indentation points is the packing density, there are two particle unknowns (hs and μ) and N packing densities. Hence, for N  2, the system is highly over-determined, which makes it possible to assess microstructure and particle properties. One can then proceed with a statistical analysis of the packing density distributions, as shown in Figure 5. The interesting feature which emerges from this analysis is the presence of different C-S-H gel phases in the microstructure; namely a Low-Density (LD)

PROBING C-S-H MICROSTRUCTURE BY NANOINDENTATION

With the molecular properties of C-S-H in hand it becomes possible to assess the microstructure of cement-based materials at micrometer scale, and ultimately to make the link between microstructure, and meso- and macro-scale properties. The key herein are the dimensionless scaling relations H and M in Eq. (4). 3.1

Does particle shape matter?

The determination of functions H and M requires the choice of particle morphology. For perfectly disordered materials, the key quantity to be considered is the percolation threshold, that is the critical packing density below which the composite material has no strength, nor stiffness. This percolation threshold depends on the particle shape (Sanahuja et al. 2007). Clearly, as seen in TEM images of C-S-H, the elementary particle has an aspect ratio. However, as far as the mechanics response is concerned, it turns out that particle shape does not matter as soon as the packing density of the porous material is greater than 60%. This is obviously the case of cement-based materials (as their industrial success is due to their strength performance), and also of most other natural composite materials like bone, compacted clays (shale), etc. One can thus conclude that the effect of particle shape is negligible as far as the mechanical performance is concerned (Ulm & Jennings 2008); see Fig. 4. The negligible effect of the particle shape on the homogenized mechanical properties largely simplifies the micromechanical analysis. It thus suffices, to consider spherical particles and a distinct disordered morphology of the solid phase, similar to a polycrystal, characterized by a solid percolation threshold of η0 = 0.5. These relations can be found, for the indentation modulus in Constantinides & Ulm (2007), and for the indentation hardness in Cariou et al. (2008).

Figure 4. Normalized stiffness vs. packing density scaling relations. The percolation threshold of 0.5 corresponds to a perfectly disordered material composed of spherical particles, while lower percolation thresholds are representative of disordered materials with particle shape (here ellipsoids). [Adapted from Ulm & Jennings 2008].

Figure 5. Probability density plot (PDF) of packing density distribution determined from 400 nanoindentation tests on a w/c = 0.3 cement paste; together with a phase deconvolution obtained by fitting Gaussian distributions to the experimental PDF [adapted from Vandamme & Ulm, 2009].

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phase and High Density (HD) phase as originally proposed by Jennings and co-workers (for a recent review, see Jennings et al. 2007), with mean packing densities close to two limit packing densities of spherical objects, namely the 64% packing density of the random close-packed limit for the LD-phase, and 74% for the ordered face-centered cubic (fcc) or hexagonal close-packed (hcp) packing for the HD-phase. In addition, a third Ultra-High-Density (UHD) phase appears in the packing fraction distribution, particularly for cement pastes with low water-to-cement (w/c) ratios (Vandamme et al. 2010). Indeed, a comprehensive nanoindentation analysis of cement pastes prepared at different w/c ratio shows that LD dominates cement-based materials prepared at high w/c mass ratios; HD and UHD control the microstructure of low w/c ratio materials (Fig. 6). This shows that nanoindentation combined with micromechanicsbased indentation analysis provides a means to probe the microstructure of the C-S-H gel at micrometer scale of cement-based materials. 3.3

allow the determination of the packing density distribution. The difference between the two types of test is in a holding phase: following a fast loading (5s), in nanoindentation creep tests a constant load is applied for some 180s; in contrast to microstructure tests in which the dwelling time is 5s. During the 180s dwelling time, the change in indentation depth h(t) = h(t)−h0 is recorded as a function of time. In all tests on C-S-H phases, after correcting for thermal drift effects of the indenter equipment, a least square fitting of h(t) = h(t) − h0 demonstrates an indentation creep compliance of the C-S-H phases that is logarithmic with regard to time (Fig. 7). The indentation creep compliance rate is determined as (Vandamme & Ulm 2006, 2009): ˙ max = 1/(Ct) L˙ = 2aU h/P

(6)

√ where aU = Ac /π is the contact radius at the end of the dwelling phase; while C, which has the same dimension as an elastic modulus, is justly termed contact creep modulus (Vandamme & Ulm 2009). This creep modulus C depends on the packing density as shown in Figure 8. It is on the order of CLD = 120.4 ± 22.6 GPa, CHD = 183.6 ± 30.5 GPa and CUHD = 318.6 ± 32.2 GPa. The value for CLD = 120.4 ± 22.6 GPa comes remarkably close to macroscopic values of the long-term creep of normal strength concrete, while the value for CUHD = 318.6 ± 32.2 GPa is not far off from the long-term creep values of high-strength concrete -with one major difference: a nanoindentation creep test takes 3 minutes, a longterm macroscopic creep test requires several years of force, deformation control under highly controlled environmental conditions (Acker & Ulm 1999). This observation opens new perspectives for creep assessment of concrete materials and structures: By probing micrometer-sized volumes of materials, nanoindentation creep experiments provide quantitative results in a 6 orders of magnitude shorter time

Nanogranular origin of concrete creep

It is now possible to make the link between microstructure and mechanical performance. Besides elasticity and strength, a main concern for concrete is creep. Concrete creep occurs at a rate that deteriorates the durability and truncates the lifespan of concrete structures. One challenge in establishing this link between creep properties and microstructure is that creep of C-S-H needs to be experimentally measured at the scale of the microstructure of the C-S-H gel. We here consider nanoindentation creep tests, in addition to nanoindentation tests of stiffness and elasticity that

Figure 7. During the 180s dwelling time, the change in indentation depth is recorded as a function of time, and fit with a logarithmic function (for details see Vandamme & Ulm 2009).

Figure 6. Volume fraction distributions in the microstructure: (a) volume fractions of the cement paste composite; (b) volume fractions of the hydration phases [adapted from Van-damme etal. 2010].

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As engineers move into the fundamentals of the physics of materials, and physicist move into engineering applications, the boundary between physics and engineering is finally blurred. It is on this basis that we expect that progress on the concrete front will translate into societal benefits. Along this way, we believe that computational and experimental mechanics of concrete will be redefined.

REFERENCES Acker P. & Ulm F.-J. 2001. Creep and shrinkage of concrete: physical origins and practical measurements. Nuclear Engineering and Design 203(2–3): 143–158. Allen A.J., Thomas J.J. & Jennings, H.M. 2007. Composition and density of nanoscale calcium-silicate-hydrate in cement. Nature Materials 6(4): 311–316. Cariou S., Ulm F.J. & Dormieux L. (2008). Hardness-packing density scaling relations for cohesive-frictional porous materials. Journal of the Mechanics and Physics of Solids 56: 924–952. Constantinides G. & Ulm F.-J. (2007). The nanogranular nature of C-S-H. Journal of the Mechanics and Physics of Solids 55(1): 64–90. Dormieux L., Kondo D. & Ulm, F.-J. (2006). Microporomechanics. J. Wiley & Sons, Chichester, UK. Jennings H.M., Thomas J.J., Gevrenov J.S., Constantinides G. & Ulm, F-J. 2007. A multi-technique investigation of the nanoporosity of cement paste. Cem. Concr. Res. 37 (3): 329–336. Pellenq R.J-M., Kushima A., Shahsavari R., Van Vliet K.J., Buehler, M.J., Yip S. & Ulm F.-J. 2009. A realistic molecular model of cement hydrates. Proc. Natl. Acad. Science 106(38): 16102–16107. Povolo F. & Bolmaro R.E. 1987. Average elastic-constants and tensor invariants. Phys. Stat. Sol. 99(2) 423–436. Sanahuja J., Dormieux L. & Chanvillard G. 2007. Modelling elasticity of a hydrating cement paste. Cem. Concr. Res. 37: 1427–1439. Shahsavari R, Buehler M.J., Pellenq R.J.-M. & Ulm F.-J. 2009. First-Principles Study of Elastic Constants and Inter-layer Interactions of Complex Hydrated Oxides: Case Study of Tobermorite and Jennite. J. American Ceramic Society 92(10): 2323–2330. Ulm F.J. & Jennings H.M. 2008. Does C-S-H particle shape matter? A discussion of the paper ‘Modelling elasticity of a hydrating cement paste’, by Julien Sanahuja, Luc Dormieux and Gilles Chanvillard. CCR 37 (2007) 1427–1439. Cem. Concr. Res. 38(8–9): 1126–1129. Ulm F.-J., Vandamme M., Bobko C., Ortega J.A., Tai K. & Ortiz C. 2007. Statistical indentation techniques for hydrated nanocomposites: Concrete, bone, and shale. J. Am. Ceram. Soc. 90 (9): 2677–2692. Vandamme M. & Ulm F.-J. 2006. Viscoelastic solutions for conical indentation. Int. J. Solids and Structures 43(10): 3142–3165. Vandamme M. & Ulm F.-J. 2009. Nanogranular origin of concrete creep. Proc. Natl. Acad. Science 106(26): 10552–10557. Vandamme M., Ulm F.-J. & Fonollosa P. 2010. Nanogranular packing of C-S-H at substochiometric conditions. Cem. Concr. Res. 40(1): 14–26.

Figure 8. Creep modulus—packing density relation [adapted from Vandamme & Ulm 2009].

and on samples 6 orders of magnitude smaller in size than classical macroscopic creep tests. This ‘‘lengthtime equivalence’’ (large time scales can be accessed by looking at small length scales) may turn out invaluable for the implementation of sustainable concrete materials whose durability will meet the increasing worldwide demand of construction materials for housing, schools, hospitals, energy and transportation infrastructure, and so on.

4

CONCLUSIONS

The premise of the bottom-up approach is to identify new degrees of freedoms for concrete material and structural design, from electrons and atoms to structures. While still in its infancy, the first results obtained with this bottom-up approach reveal some interesting perspectives for enhancing the sustainability of this omnipresent material. With a focus on strength enhancement, we note: • On the atomic scale, strength enhancement may be achieved by fine tuning the chemistry of cementbased materials. Classical cement-based materials exhibit a calcium-to-silicon ratio (C/S) of 1.7, which determines mechanical properties of the molecular structure. A change in chemistry is expected to enhance those properties. • At the scale of the microtexture, a denser packing of elementary C-S-H particles has the premise to enhance both the strength behavior and the durability performance, for instance in terms of concrete creep. • From micro-to-macro of concrete structures, continuum micromechanics provides a powerful framework to translate progress in materials science into day-to-day engineering applications.

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Constitutive and multi-scale modelling

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Pull-out behaviour of a glass multi-filaments yarn embedded in a cementitious matrix H. Aljewifi, B. Fiorio & J.L. Gallias Université de Cergy-Pontoise, Laboratoire de Mécanique et Matériaux du Génie Civil, Cergy-Pontoise cedex, France

ABSTRACT: This paper describes the pull-out behavior of glass multi-filaments yarns embedded in a cementitious matrix. The pull-out behavior is characterized by direct pull-out tests performed on different glass rovings. For these tests, different pre-treatments of the roving are used, that induce different impregnation patterns. In this way, the effect of the impregnation on the pull-out behavior is characterized. The effect of the constitutive parameters of the yarn on the mechanical behavior is also studied through the use of three different glass yarns for the experimental study.

1 1.1

INTRODUCTION

yarn consists of several hundreds up to thousands of single filaments. Therefore the fineness of the yarn, in tex (g/km), depends on the number of filaments, filament diameter (range between 10 to 30 μm) and fibre density. Filaments can be gathered in two different ways to produce a yarn: direct roving and assembled roving. Direct roving consists in a mass agglomeration of thousands of single filaments, coated with a sizing (see below). Assembled roving consists in the agglomeration of strands, themselves constituted from hundreds of single filaments. Assembled roving are a three levels structure (filament-strand-yarn), when direct roving presents a two level structure (filament-yarn). In most of the applications, filaments are coated with a sizing which goal is to enhance the interaction with the matrix or to ease the building process of the yarn. The sizing material is a chemical mixture of polyhydroxyphenols, silane, polymer emulsion (PVAC) and additives. The detail of the chemical composition of the sizing is generally unknown, because of the industrial protection imposed by manufacturers. The type of sizing influences the interfacial properties of the cement matrix —filaments bond. Therefore, the mechanical properties of TRC may be influenced by the sizing, as the quality of the adhesion between filaments and matrix influences the composite performances. But the main element that influences the mechanical comportment of yarns reinforced concrete is the specific structure of the yarn itself. The tensile load-strain relationship of the yarn is significantly influenced by the specific organisation of the filaments in the yarn (Chudoba et al. 2006). If all filaments in the yarn were strictly straight and aligned, the stress in each filament of a yarn submitted to a given strain should

Textile reinforced concrete

Textile reinforced concrete (TRC) is a new cementitious composite that appeared in the early 1980’s, when the combination of new yarn types with 3D-production processes leads to an increasing number of new textile applications. Materials used for TRC application are mainly alkali-resistant glass fibres and carbon fibres. The yarn structures include cabled yarns and friction spun yarns which were developed to improve the bonding behaviour. They both have a structure that looks like the one of steel reinforcements with rods. The development of TRC is based on the fundamentals of shortcut glass fibre reinforced concrete. In order to increase the effectiveness of the fibres embedded in the concrete matrix, the fibres are aligned in the direction of the tensile stresses similar to ordinary steel reinforced concrete. Textile reinforced concrete offers many advantages compared to traditional concrete (Hanisch and et al., 2006). The most important reason for a reinforcement of concrete parts with textiles is that the concrete parts can be very thin as there is no risk of corrosion of the reinforcement materials. This allows to reduce the weight of the concrete parts, and therefore to contribute to the reduction of the environmental impact of construction. In addition, the reinforcement is more flexible and therefore the shape of the concrete elements can be varied in a wide range.

1.2

Multi-filaments reinforcement

1.2.1 Yarns characteristics Yarns are multi-filaments reinforcements made of a bundle of elementary fibres, so-called filaments. One

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be homogeneous in the yarn and the stress-strain relationship of the yarn would correspond to those of filaments, i.e. to the linear elastic brittle behaviour of the bulk material, in the case of glass or carbon filaments. In this case, the failure of the yarn should be associated to the simultaneous failure of all constitutive filaments. Real yarns lead to a different behaviour, as filaments are not initially aligned. As an example, Figure 1 shows the tensile behaviour of a glass yarn measured by the way of a direct tensile test. The first stage of the loading is characterized by the progressive stiffening of the mechanical behaviour, induced by the differed tension of the filaments due to their initial mis-alignement. When all filaments are tensed, the load-strain curve is linear until the first filaments begin to fail. This results in the progressive softening of the behaviour until the load reaches its maximum values Pmax . For increasing strain, filaments breaking continue until full failure of all filaments. Residual friction between failed filaments then results in a residual load that progressively reduces to zero for high strain. The main observation is that during yarn elongation, the filaments work almost independently one from the other, and that the yarn behaviour results from the coupling action of the individual behaviour of all yarns.

1mm

Penetration depth

Figure 2. Cross section (left); longitudinal section (right) of multi-filaments yarn embedded in micro concrete.

The inner filaments are not reached by the hydrated cement paste and thus are not directly anchored in the cementitious matrix. Only the friction between these filaments can generate a certain bond resistance when the yarn is pulled out, when differential slips appear between filaments. This phenomenon induces a non-uniform stress and strain distribution in the yarn. Two interactions between the yarn and the matrix can been formed. Ohno et al., 1994 & Langlois, 2004 assume two families of yarns: sleeve filaments with direct contacts with the matrix and core filaments without direct contacts. Sleeve filaments are mechanically anchored to the matrix. Core filaments are submitted to tension due to friction with sleeve filaments. As filaments are not straight and parallel into the yarn, a filament that is located in the center part of the yarn, in a given cross section of the yarn, can be located in the impregnated peripherical part of the yarn in another cross section. This means that the relative importance of the sleeve filaments and core filaments does not only depends on the impregnation process, but depends also on the embedded length of the yarn. The longer this length is, the more numerous the anchored sleeves filaments are. This phenomenon also depends on the fiber/matrix chemical bond strength (Kabele et al., 2006) that conditions the filament debonding and the filament/ filament and filament/matrix slip properties. From this point of view, the sizing may also play an important part in the pull-out behaviour. As explained, the impregnation process greatly influences the bond conditions of each individual filament of the yarn. The resulting bond conditions of the yarn are then also mainly guided by the impregnation process, as well as by the matrix and yarn properties. This specificity of multi-filaments reinforcements, compared to monolithic reinforcements, induces a very specific mechanical behaviour, which cannot be modelized with the same methods as monolithic reinforcements. The aim of this paper is to give some experimental explaination of the mechanisms involved in the yarn/matrix interaction, and to provide the modelization community with experimental data that describe the pull-out behaviour of multi-filaments reinforcement.

1.2.2 Influence of the yarn impregnation on the mechanical behaviour Due to the specific geometry of the yarn and to the penetration of the cementitious matrix inside the yarn during the casting of TRC, the behaviour of impregnated yarn in the cementitious matrix is very dissimilar from the behavior of ordinary steel rods. The penetration of the cementitious matrix is not homogeneous: the cross section of the impregnated yarn shows that the outer filaments are embedded in a hardened cement paste that provides them a good anchorage (figure 2). The penetration depth of the cement paste inside the yarn is not sufficient to ensure a full impregnation of the yarn.

Figure 1. Typical load vs. strain curve observed for multifilaments yarn in direct tension.

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2

EXPERIMENTAL PROCESS

2.1

Objectives of the experimentation

The hereafter-presented experimentation focuses on the influence of the fibre impregnation on the pull-out behaviour of a glass multi-filaments yarn embedded in a cementitious matrix, in relation with the constitutive parameters of the yarn. In this experiment, pull-out tests have been performed to characterize the micromechanical behaviour of the multi-filaments yarn/cementitious matrix interface. In these tests, a yarn was embedded into a matrix with various condition of impregnation and loaded in tension until the filaments slip, or break. Parameters of the study were the yarn properties, the embedded length and the matrix impregnation. Yarn properties were controlled through the use of three different types of yarns. The impregnation was varying trough the use of three preparation process applied to the yarn before the casting of the pull-out samples. 2.2

Materials

2.2.1 Glass yarn Three types of glass yarns (named in the following OC1, OC2 and SG2) have been used in this study. OC1 and OC2 yarns are made from E-glass filaments and came from a first manufacturer. SG2 yarn is made from AR-glass and came from a second manufacturer. OC1 and SG2 yarns are assembled rovings. OC2 is a direct roving (see 1.2.1). The main characteristics of these yarns are given in table 1. Figure 3 gives the results of tensile tests performed on the three yarns. These tests were performed on 10 cm long yarns of the three studied type. The five curves in each chart correspond to the same type of yarn and give an idea of the variability classically observed in the case of yarns.

Figure 3.

composition). The compressive and flexural strength were measured according to NF 196-1. Mean values of these properties were 55 MPa and 10.83 MPa respectively.

2.2.2 Cementitious matrix Fine grain (1.25 mm maximum size) CEM I 52.5 mortar was used as cementitious matrix. Mixing proportion was as follow: W/ C = 0.5, S/C = 1.4 and SP/C = 0.0035 (W, C, S and SP are respectively the mass of water, cement, sand and superplasticizer used for the

2.3

Yarn OC1 OC2 SG2

12 17 14

n

Fineness tex (g/km)

Young’s modulus GPa

Glass bulk density kg/m3

39 / 30

2400 2400 2450

57.73 59.38 49.11

2530 2530 2680

Samples

2.3.1 Samples preparation Samples were 3.4 cm diameter cylinder made from the mortar described in the previous part. A straight yarn was positioned along the axis of the cylinder. The height of the cylinder varied from 1 to 25 cm, which results in a variation of the embedded length of the yarn. One of the three hereafter described pre-treatment was applied to each yarn before casting the sample:

Table 1. Characterization of the glass multi-filaments yarns (n: number of strands in assembled rovings). Filament diameter μm

Tensile test on yarns.

• Pre-wetting (W): yarn was saturated with water so that the inter-filaments voids are filled with water, which prevent the cement paste to enter the yarn

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during casting. It should be noticed that in this case, capillarity forces induce the agglomeration of the glass filaments which became more parallel than without this treatment. • Drying (D): yarn was air dried at room temperature before casting. Capillarity leads to penetration of water and cement particles in the yarn at the time of casting. As the filaments act as a filter, penetration of cement particles into the yarn was limited. • Cement pre-impregnation (PI): the yarn was manually saturated with a cement slurry before casting. Saturation was obtained by manual action on the yarn placed in a slurry batch. The cement slurry composition corresponds to the micro-concrete matrix cement paste composition.

filaments partially interlocked due to the mechanical action during pre-treatment. Dried yarn (D) shows an intermediate facies, with the main part of the filaments embedded in the matrix, which signifies that the free length of most of the filaments (the distance between two embedded points) is smaller than the length of the specimen (4 cm). Similar observations were made for all other types of yarns. The penetration depths of the cementitious matrix in the yarns were estimated from the SEM observations. An impregnation index iy was calculated from these values as the ratio of he impregnated area of the yarn to the apparent area of its cross section. Values of iy for the different combinations of yarns and pretreatments are given in table 2. Flow tests realized by Aljewifi et al., 2009 give complementary information about the porosity of the impregnated yarn. This test allows to measure the water flow rate along an embedded yarn under a constant pressure gradient of 107.5 kPa/cm. For a given yarn, the measured flow rate decreases when the impregnation becomes more important (see figure 5). Figure 5 also shows that the lowest value of the flow rate corresponds to the PI pre-treated yarns and that this value is roughly independent from the type of yarn. This is the signature of a small waterflow inside the tortuous cement paste porosity of the impregnated yarn.

After 24 h hardening at room conditions (about 50% relative humidity and 20◦ C), samples were removed from mould and placed in 20◦ C water during 24 days. They were then removed from water for one day to allow the air drying of the yarn. The free end of the yarn was glued in between two epoxy plates for fastening in the pull-out device. 2.3.2 Characterization of the yarn impregnation Aljewifi et al., 2010 have studied the impregnation of the yarns obtained from the three pre-treatments applied to the three types of yarns used in this study. Different methods of characterization were used, among them scanning electron microscopy (SEM) observation of longitudinal sections of yarns and flow tests. The main result of these SEM observations is the strong influence of the pre-treatment on the impregnation. As an example, figure 4 gives the three SEM photographs corresponding to each pre-treatment applied to OC1 yarn. The pre-wetting of the yarn (W) leads to straight parallel mainly un-impregnated filaments. Conversely, the pre-impregnation with the cement slurry (PI) induces a full impregnation of the yarn, with

W

D

PI

Table 2. Diameter of the impregnated yarns (mm) and values of the impregnation index iy (%).

OC1 OC2 SG2

W

D

PI

3.52 mm 38% 2.69 mm 24% 2.84 mm 29%

3.26 mm 67% 3.39 mm 19% 3.14 mm 70%

4.19 mm 100% 4.26 mm 100% 3.42 mm 100%

1 mm

1 mm

1

Figure 4. OC1 yarns; longitudinal sections corresponding to W, D and PI yarn pre-treatment.

Figure 5.

Flow rate vs. impregnation index iy .

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In the case of the lowest impregnation index, the flow rate is higher and influenced by the yarn type. This is explained by a flow of water that takes place inside the inter-filaments porosity of the un-impregnated center part of the yarn. As the water is in direct contact with the filaments in this case, the chemical composition of the filament surface (sizing) influences the flow, which explains the differences observed for the yarns in this situation. As a conclusion of this part, it is shown that there are strong evidences that the yarn impregnation is only partial in most of the case. As a consequence, the pullout behaviour of the yarn will be strongly influenced by this fact and the overall behaviour will diverge from the pull-out behaviour of monolithic reinforcements.

2.4

along the fibre/matrix interface. Once the debonding process has reached the end of the embedded fibre length, a dynamic mechanism of pull-out is observed. Moreover, a displacement at the end of the free length is also accompanied by a displacement at the embedded end (Naaman et al., 1991). In the case of yarn, the above described behaviour is observed for individual filaments but cannot be directly applied to the whole yarn.

3 3.1

Pull-out tests are performed on yarns embedded in a cylinder of micro-concrete. As explain in 2.3.1, the free end of the yarn (i.e. the end which is not embedded in the micro-concrete) is glued between two epoxy plates. A universal mechanical device is used to perform the tests (see also figure 6). The micro-concrete cylinder is positioned in a specific basket clamped on the upper grip of the test device. The free end of the yarn is clamped on the lower grip of the tension device. The mortar cylinder is precisely centred so as to ensure that the tension load is applied parallel to the embedded yarn. For all test, the free length LL of the yarn (i.e. the length of the yarn from the micro-concrete cylinder to the epoxy plates, figure 6) has a constant values of 10 cm. Tests are realized at constant speed (1 mm.min−1 ). In this test, a load P is applied to the tip of the yarn embedded in a cementitious matrix over an embedded length Le which was a parameter of the study. In the case of monolithic fibre, a monotonic increase in the load P is accompanied by a displacement from the tip of the fibre and leads to progressive debonding

Fixed member: Gl: lower grip D: displacement transducer

Gu

D

S

E P Gl

Figure 6.

3.1.2 Three stages behaviour Generally, pull-out curves are divided into three stages (Hegger and et al., 2004). In our study, all of the obtained results obey this statement, except may be for the third stage, as explain later. The first stage is determined by the elasticity of the adhesion bond and corresponds to the progressive tension of the filaments that constitute the yarn. Stiffening is generally observed at the beginning of the loading, associated to the delayed tension of the filaments. The slope of the linear ascending portion is not similar for all curves and seems to be an increasing function of the embedded length.

Le Mobile member: Gu: upper grip LL B: sample basket L: load cell

SS

B

Load/displacement curves

3.1.1 Measurements As the stress is not homogeneous in all filaments of the tested yarns, it was chosen to present the results of the pull-out tests as load/displacement curves instead of stress/strain curves. Load and displacement are measured through the press sensor. In particular, the displacement is measured through the displacement transducer of the test device: it is the relative displacement of the upper grip supporting beam regardless to the lower grip support (figure 6). Therefore, the measured displacement is not directly the extracted length of the yarn but also take into account the strain of the free length of the yarn. As this length stay constant over all the tests, it is considered that the comparison between the behaviours observed in all different cases that have been tested remains pertinent. This assumption is clearly true when working with a given type of yarn. It remains roughly true for comparison between two different types of yarns, as yarns mechanical properties are in the same range. Figures 7 to 9 give the load/displacement curves for each of the three tested yarns. Each figures presents three charts, one for each type of yarn’s pre-treatment. In each chart, the results corresponding to the different measurements made for a given embedded length are presented as a single average behaviour curve. This curve is obtained from three tests performed in the same conditions.

Pull-out test

L

RESULTS

Sample: S: mortar cylinder with embedded yarn E: epoxy plates P: applied load

Experimental setup.

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W pre-treated OC1 yarn

W pre-treated OC2 yarn

D pre-treated OC1 yarn

D pre-treated OC2 yarn

PI pre-treated OC1 yarn

Figure 7. Load vs. displacement curves for OC1 yarn and different embedded lengths in fine concrete.

PI pre-treated OC2 yarn

Figure 8. Load vs. displacement curves for OC2 yarn and different embedded lengths in fine concrete.

In the second stage of the behaviour, the chemical bond between the strands and the matrix break (the smooth surface of the filaments do not ensure a good mechanical anchorage of the strands on short length. The breaking of the strand/matrix bond is accompanied by filaments breaking. This phenomenon induces a smoothing of the mechanical behaviour. When broken filaments are in sufficient number, failure progressively appears as a slow decrease of the load. After the breaking of all the filaments and strands (tensile or adhesion failure), broken filaments are extracted from the matrix. During this stage, frictional stresses appear at the filament/filament and filament/matrix contacts. This results in a residual load which progressively reduces to almost zero, when all filaments become extracted from the matrix. This observation corresponds to the third stage of the behaviour mentioned by Hegger and et al., 2004.

3.1.3 Complementary elements on the residual load The values of the residual frictional load for all types of yarn and embedded lengths show a tendency of the residual load level to increase with the embedded length, which is consistent with the increase of contact points along the filaments. In some case, this tendency is not respected and some lower embedded length gives highest frictional level. This can be attributed to the appearance of local disorganization in the yarn, due to differential slip of the filaments. Evidences of this phenomenon have been seen on longitudinal sections of samples made after the pull-out test (figure 10). By comparison with reference longitudinal sections made before pull-out test, the structure of the loaded yarns has been disorganized. In some location, node looking

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structures (see detail on figure 10) appear and seem to be the consequence of the blocking of a group of filaments in a small cement cluster pulled by another groups of filaments. It should be noted that this third stage of the behaviour is in some cases reduced to almost nothing. This is systematically the case for PI pre-treated yarns, whatever the embedded length was. The explanation is, that in this case, the anchorage length necessary to anchor the whole yarn (i.e. all its filaments) is very short. Most of the filaments’ failures then take place in the vicinity of the surface of the concrete, producing failed filaments with short embedded length. As the embedded length of the failed filaments is short, the remaining frictional load is very low and comes back to zero for small extraction displacement. 3.2

Behaviour law parameters

3.2.1 Maximum pull-out load Pmax Raw data described in 3.1 were used to determine the maximum value Pmax of the pull-out load during the test. Values of Pmax are given in figure 11. As a general trend, it is seen that Pmax is an increasing function

Figure 9. Load vs. displacement curves for SG2 yarn and different embedded lengths in fine concrete.

Figure 10. Longitudinal section of OC1 D yarn after pullout test. Arrow: direction of extraction.

Figure 11.

Values of Pmax vs. embedded length.

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of the embedded length for the low values of the embedded length. For higher values of the embedded length, Pmax remains roughly constant but do not reach the tensile strength of the yarn, which indicates that a portion of the filaments remains inactive during the pull-out test. SG2 yarn gives the best efficiency (about 90% of the tensile strength), when OC1 and OC2 give lower efficiencies. Concerning the effect of the pre-treatment, D and PI pre-treatment does not show major differences for the evolution of Pmax : PI pre-treatment gives higher values of Pmax than D pre-treatment, which itself gives higher values than W pre-treatment. In the case of OC1 yarn, values of Pmax are rather independent of the pre-treatment. 3.2.2 Optimum embedded length and efficiency of the yarn. The minimum embedded length Lmin necessary to reach the maximum value of Pmax was determined for each pre-treated yarn. The maximum value Pmax lim of Pmax was determined as the average values of Pmax for embedded length greater than Lmin . The pull-out efficiency epo was calculated as the ratio of Pmax lim to the tensile strength of the yarn. Values of theses parameters are given in table 3. The efficiency is all the more high that the pre-treatment favours the penetration of cement paste in the yarn. The best efficiency is obtained for SG2 yarn. 3.2.3 Stiffnesses κbond and κdebond are respectively the ascending and descending stiffnesses measured for the ascending and descending part of the behaviour curve. Calculation is made for load values in the range 0.5 to 0.8 Pmax . Figure 12 gives the values of these two parameters for the different configurations used in the study. Values lower than those measured in direct tension (right bars on figure 12) reveal a lake of adhesion of

Table 3. yarns.

Figure 12. length.

Values of κbond and κdebond vs. the embedded

Optimum embedded length and efficiency of Lmin cm

Pmax lim N

epo %

OC1

W D PI

10 5 3

333 334 354

29 30 34

OC2

W D PI

10 10 5

206 367 484

21 36 48

SG2

W D PI

15 15 3

395 452 748

49 55 81

the sleeve filaments. This is specially the case for OC2 W pre-treated yarn. OC2 and SG1 yarns give κbond values for pull-out tests comparable to those measured in direct tension. κdebond is an increasing function of the embedded length. The observed values for κdebond are dependent of the pre-treatment. In particular, PI pre-treated yarns give high values of κdebond , allowing κdebond to reach the values measured in direct tension. This signifies that in this case the anchorage of the sleeve filaments is very efficient and that this family of filaments behaves as if they were in direct tension.

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3.3

Core and sleeve filaments

the value of the ratio is in some case an over estimated value of the real fraction of core filaments (this fact is particularly true for PI yarns, which explains why in some case the ratio exceeds 100%). Figure 14 gives the values of the ratio mLe /mLL for the different yarns, pre-treatments and embedded lengths. The main trends that should be noted are the decrease of the ratio when the embedded length increases (for OC1, the ratio falls to zero when the embedded length is high enough), the similarity of W and D pre-treated yarns (except for OC2, PI pretreatment leads to much better anchoring of the filaments, which appears as a marked reduction of the mLe /mLL ratio) and the specificity of OC2 yarn, which kept a large amount of core filaments, even for high embedded length.

To evaluate the ratio between sleeve and core filaments, the ratio of the linear mass mLe of the extracted part of the yarn (damaged due to filaments failure) to the linear mass mLL of the undamaged yarn was determined after each pull-out test (figure 13). This ratio gives an idea of the amount of un-anchored core filaments in the loaded yarn. As these filaments could include some small cement particles in between them,

Free length LL: Embedded length Le : Linear mass mLL Linear mass mLe

4

RELATIONSHIP TO THE IMPREGNATION

Figure 13. Determination of the yarn’s linear masses after the pull-out test.

4.1

Figure 14.

Figure 15 gives the relationship between the optimal embedded length Lmin , the efficiency and the impregnation index Iy determined from SEM observation (see table 2). The optimal embedded length is a decreasing function of the impregnation index. This is directly connected to the increase of cement paste presence inside the yarn, which reduces the distance between the anchorage points of the filaments. The efficiency tends to increase when the impregnation index increases, which is linked to the increase of the sleeve filaments due to the penetration of the matrix in the yarn. It should be noted that the efficiency never reach 100%. OC1 and OC2 data are very similar, despite of the difference of structure of these two yarns. This may be related to the type of sizing, not design in this case for a use with concrete. Sizing for SG2 yarn is dedicated to concrete applications and favours the load transfer between yarn and matrix. Concerning the effect of the impregnation index on the stiffness, the three yarns, despite of their differences, roughly present the same increasing of κbond lim with the impregnation index (figure 16). This is consistent with the evolution of Lmin : when Lmin becomes lower, the working length of the filaments (i.e. length from the grip to the first anchorage point in concrete) decreases. As a consequence, the displacement is reduced for a given load, which traduces an increase of stiffness. Values of κdebond lim also increase with the impregnation index. They are about ten times highest than κbond lim one for full impregnation. κbond lim and κdebond lim are the values of κbond and κdebond that correspond to Lmin .

Values of mLe /mLL vs. embedded length.

Optimal embedded length and efficiency

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However, despite of this complexity, the experimental approach presented above allow the main parameters that determined the pull-out behaviour of the yarn to be highlighted. These parameters are the following: the structure of the yarn itself, its sizing, the embedded length, the state of the yarn before the casting of concrete. Most of these parameters influence the way the concrete penetrate the yarn at the time of casting and the resulting impregnation appears as playing a major part in the control of the pull-out behaviour. Impregnation, by influencing the partition between sleeve and core filaments, influences the maximum pull-out load and the efficiency of the yarn. It also determined the anchorage length necessary to reach the maximum efficiency of the yarn. What is now needed is to take into account this complexity in models, so as to evaluate the effect of the different scenarios that can explain the pull-out behaviour.

Figure 15.

Figure 16. index iy .

5

REFERENCES

Lmin and epo vs. impregnation index iy .

κbond lim and κdebond lim vs.

Aljewifi, H., Fiorio, B., Gallias, J.L., 2010. Caracterization of the impregnation by a cementitious matrix of five glass multi-filaments yarns. European Journal of Environmental and Civil Engineering, EJECE (in press). Aljewifi, H., Fiorio, B., Gallias, J.L., 2009. Quantitative methods used to characterize the impregnation of a glass multifilament yarn by a cementitious matrix. 4th Colloquium on Textile Reinforced Structures (CTRS4). 3–5 June Dresden: Germany. Chudoba, R., Voˇrechovsky, M., Konrad, M., 2006. Stochas tic modeling of multi-filament yarns. I. Random properties within the cross-section and size effect. International Journal of Solids and Structures 43: 413–434. Hanisch, V., Kolkmann, A., Roye, A., Gries, T., 2006. Yarn and textile structures for concrete reinforcements. FERRO8, Bangkok, Febuary 6th. Hegger, J., Bruckermann, O., Chudoba, R., 2004. A smeared bond-slip relation for multi-filament yarns embedded in fine concrete. 6th International RILEM Symposium on Fibre Reinforced Concretes. Kabele, P., Novák, L., Nemecek, J., Kopecký, L., 2006. effects of chimical exposure on bond between synthetic fiber and cementitious matrix. ICTRC—1st International RILEM Conference on Textile Reinforced Concrete 10: 91–99. Langlois, V., 2004. Etude du comportement mécanique des matériaux cimentaires à renforts synthétiques longs ou continues, PhD, Université de Cergy-Pontoise, France. Naaman, A.E., Namure, G.G., Alwan, J.M., Najm, H.S., 1991. Fiber pullout and bond slip. I: Analytical study. ASCE, Journal of Structural Engineering. 1117(9): 2769–2790. Naaman, A.E., Namure, G.G., Alwan, J.M., Najm, H.S., 1991. Fiber pullout and bond slip. II: Experimental validation. ASCE, Journal of Structural Engineering. 1117(9): 2791–2800. Ohno S., Hannant, D.J., 1994. Modelling the stress-strain response of continuous fiber reinforced cement composites. ACI Materials Journal, vol. 91, pp. 306–312.

impregnation

CONCLUSION

The results presented in this paper give detailed information on the influence of the pull-out behaviour of multi-filaments yarns. They show all the complexity induced by the non-monolithic structure of the yarn. This complexity is also a consequence of the variability generated by the specific constitution of yarns.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

How to enforce non-negative energy dissipation in microplane and other constitutive models for softening damage, plasticity and friction Zdenˇek P. Bažant Northwestern University, Evanston, Illinois, USA

Jian-Ying Wu State Key Laboratory of Subtropical Building Science, South China University of Technology, China

Ferhun C. Caner UPC, Barcelona, Spain

Gianluca Cusatis RPI, Troy, New York, USA

ABSTRACT: Material constitutive models must be formulated in such a way that the energy dissipation can never become negative during deformation increments within the range of intended applications. However, checking this obvious thermodynamic condition for complex models such as the microplane model (e.g. Bažant 1984, Bažant and Caner 2000) is not a trivial task and is often complicated by incomplete, ambiguous or unrealistic definition of unloading or reloading. Ignoring such incompleteness may result in a misleading appraisal of the performance of the model for monotonic loading. Here an attempt is made to clarify this problem and suggest a simple way of ensuring non-negativity of dissipation. The condition of non-negative increment of energy dissipation density at each continuum point of each loading step in an incremental computation of structural response is formulated in the context of the microplane model. If a negative dissipation is detected, the trial constitutive law is adjusted by a change in the unloading compliance and, if necessary, also by a change of the final stresses in the loading step. This adjustment represents an integral part of the constitutive law and must be considered in calibrating the model by test data. A similar correction is then formulated for tensorial constitutive models. Further it is pointed out that without specifying the unloading behavior, the dissipation inequality cannot be checked, and that by modifying the hypothesis about unloading, negative dissipation increments can be changed to positive. Thus the dissipation inequality is not too important for constitutive models intended only for monotonically applied loads, provided that unloading for the individual microplane strain components either does not occur or occurs only rarely. The dissipation check is very sensitive to the assumption about unloading, and so it makes no sense to get alarmed by a check of the dissipation inequality for constitutive models whose characterization of unloading is known to be simplistic and unrealistic. But for models intended for cyclic loading, this inequality is, of course, an essential criterion of soundness. 1

INTRODUCTION AND DEFINITIONS

where the superior dots denote the derivatives with respect to time t. Two expressions for U may be considered:

In constitutive models intended to describe damage such as distributed microcracking, the elastic stiffness tensor Eijkl as well as is inverse, the compliance tensor Cijkl , is variable (the subscripts refer to Cartesian coordinates xi , i = 1, 2, 3). Under isothermal con˙ is the ditions, the rate of energy dissipation density, D, rate of work of stress tensor σij on the rate of strain tensor, ˙ij , minus the rate of change of the stored strain energy U (e.g., Jirásek and Bažant 2002). Thus we have: ˙ = σij ˙ij − U˙ ≥ 0 D

U =

1 e  Eijkl kle 2 ij

(2)

or, equivalently, U =

1 σij Cijkl σkl 2

(3)

where  e = elastic part of strain tensor. Under isothermal conditions, the former represents the Helmholtz free energy density (or isothermal potential energy),

(1)

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and the latter the Gibbs free energy density (or the isothermal complementary energy). In some works (e.g., Lubliner 2006), Eq. (1) is enhanced by the term k Qk θk where θk are internal variables and Qk are the thermodynamically associated internal forces. However, such an enhancement is appropriate only if some physically different types of work, e.g., the work of a muscle driven by chemical energy (Lubliner 2006), are present, which is not the case considered here. Alternatively, the internal forces may be considered as a partial or full replacement of the term σij ˙ij ; but this is appropriate only if the plastic work itself is described by a set of internal variables (Rice 1970). Otherwise the internal variables do not belong into Eq. (1).

2

U

O (a)

DISSIPATION IN DAMAGE MODELS

The contribution to U from the plane of stress σi j versus strain ij is represented by the cross-hatched triangular areas in Fig. 1(a). Generally, the unloading from a damaged state cannot terminate at the origin. An exception is the special case of an isotropic damage model, for which unloading terminates at the initial stress-free state for which the residual stresses σi0j vanish (which corresponds to perfect closing of all microcracks). In this special case Eqs. (2) and (3) may be written as U =

4

2

5

C dC 1 0

1−ω e 0 e ij Eijkl kl 2

D d = 1 2dC 2 Dp = d

1

3

6

7

(4)

d

p

(b)

or U =

1 0 σkl σij Cijkl 2(1 − ω)

Figure 1. Areas representing various parts of work or energy dissipation in the one-dimensional case.

(5) where σ ,  and C are the stress, strain and compliance. This equation may be rewritten as

0 0 where Eijkl and Cijkl are the fourth-order tensors of the initial elastic moduli and compliances. In real materials, though, plastic frictional deformations always accompany microcracking. The consequence is that the material cannot unload to its initial stress-free state and, after unloading of the material, nonzero residual stresses σij0 or strains ij0 are always locked in. Thus U , as given by Eq. (2) or (3), represents the triangular area cross-hatched in Fig. 1(a). Consider now the one-dimensional case, representing the uniaxial loading, or one pair of tensorial components, or one component of the microplane stress or strain vector. The increment of energy dissipation density is, according to Eq. (1)

˙ = σ ˙ − d D dt



1 Cσ 2 2



˙ = 1 σ Cσ ˙ + σ (˙ − C σ˙ − Cσ ˙ ) = 1 σ Cσ ˙ + σ ˙ p D  2 2    ˙d D

˙p D

(7) ˙ . Eq. (7) has a simwhere we have set ˙ p = ˙ −C σ˙ −Cσ ple geometrical interpretation in the one-dimensional stress-strain diagram of Fig. 1(b) depicting an infinitesimal loading increment from point 1 to point 2 for the case of post-peak softening damage: The total ˙ is represented by the energy dissipation increment D area 42634 which is first-order small in terms of strain increment  (cf. Bažant 1996), by subtracting the area of the triangular 2672 (i.e., the change

1 ˙ (6) = σ (˙ − C σ˙ ) − σ Cσ 2

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of elastic strain energy, U˙ ) from the area of parallelogram 42734 (i.e., the rate of work of stress, σ ˙ ). The triangular area 1241 is second-order small and thus negligible in comparison. The first term ˙ d , is equal to triangular area 4534 and of Eq. (7), D represents the energy dissipation by damage alone. ˙ p , corresponds to parallelogram The second term, D area 52635 and represents the frictional-plastic energy dissipation. The special case in which the second term vanishes for all load increments (i.e., area 52635 = 0, or length 36 = 0) represents the unloading to the origin, for which there is no plastic-frictional energy dissipation. In concrete, however, the plastic-frictional deformation in the fracture process zone generally dissipates more energy than the microcracking (Bažant 1996).

3

unknown parameters α and β as follows: D =

(σN δN + σ T · δT ) d

D = 0



˙ N + σ T · ˙T )d − U˙ ≥ 0 (σN D

(11)

still assuming that β = 1. This means that the unloading compliances are changed from CN and CT are changed to αCN and αCT . However, if the new αCN or new αCT is greater than the initial elastic compliance for one or more microplanes (which is inadmissible), a revised α and a new β must be obtained from the condition that both αCN − CN0 ≥ 0 and αCT − CT0 ≥ 0 for all the microplane while D > 0. The minimum value of β satisfying these inequality conditions should be used, which is achieved by decreasing β in small steps until all the aforementioned inequalities are satsified. The α and β corrections are implemented at each integration point of each finite element at the end of calculation of each loading step. These corrections, which adjust the unloading moduli and the final microplane stresses, must be regarded as part of the microplane constitutive law. Thus the initially assumed constitutive law of the microplane model represents only a trial constitutive law, and the α and β corrections based on Eq. (10) complete the definition of the constitute law. These corrections must, of course, be considered in data fitting and calibration of the microplane constitutive model.

(8)

and the energy dissipation density is ˙ = 3 D 2π

(10)

This equation represents a summation of the contributions defined by Eq. (7) over all the microplane stress components and all the microplanes. Subscripts i and j label the beginning and end of the loading step in which the strain increments are prescribed; subscripts N and T label the microplane normal and shear components; CN and CT are the normal and shear compliances specified by the microplane constitutive law; and σN and σT represent the normal component and the shear stress vector on each microplane. The use of averages such as 12 (βσN , j + σN ,i ) makes Eq. (6) a central difference approximation. In computations, the integral over the unit hemisphere surface is approximated by a summation based on an optimal Gaussian integration formula. At the end of computation of each small loading step (ti , tj ), one evaluates D from Eq. (8) assuming that α = β = 1. If D, no change is made. But if D is detected, one solves a new value of α from the condition





1 (βσN , j + σN ,i )(N ,j − N ,i ) 2

− (αCN , j · βσN2 , j − αCN ,i · βσ 2T ,i )

− (αCT , j · βσ 2T , j − αCT ,i · βσ 2T ,i ) d

The microplane model was conceived as a counterpart of the classical Taylor model (Taylor 1938, Batdorf and Budianski 1949), permitting the softening to be modeled. In this model, the total energy density dissipated, D, is the sum of the energies dissipated on all the microplanes. The contribution to D from each microplane can be positive or negative but the sum (or integral) of all these contributions must be nonnegative. In the fitting of complex multiaxial data for complex loading histories, such as those for concrete, it is often not easy to ensure a priori that the dissipation inequality be always satisfied. In microplane model M1 (Bažant and Oh 1986), in which only the normal and shear components of the stress and strain vectors on the microplanes are considered, the virtual work equation is given by 3 2π



+ (βσ T , j + σ T ,i )(T ,j − T ,i )

ENFORCING NON-NEGATIVE DISSIPATION IN MICROPLANE MODEL

δW =

3 2π

(9)

Therefore, it is proposed to make in each small loading step from time ti to time tj the following correction to the a priori assumed constitutive law: If a negative increment of the total D is detected, the compliance increment or the stress increment, or both, are reset so as to be make D non-negative. To this end, one may introduce, for an explicit finite element program,

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4

ADAPTATION TO MICROPLANE MODEL WITH VOLUMETRIC-DEVIATORIC SPLIT

are also an equilibrium system of forces, but such a postulate is not consistent with the calculation of the dissipated work. ˙ in microplane Therefore, the energy dissipation D models M2, M3 and M4 should be expressed as

In microplane models M2 (Bažant and Prat 1988), M3 and M4 (Bažant et al. 2000), the normal strain and stress on the microplanes are split into the volumetric and deviatoric components. Upon substitution of the relations N = V + D and σN = σV + σD Eq. (10), based on the principle of virtual work, becomes δW =

3 2π 

˙ VD ˙ =W ˙ − U˙ = W ˙ − U˙ +W D  m  ˙m D



˙m = W ˙ m − U˙ ≥ 0. It is easy to ensure that D Therefore, it is only necessary to enforce the condition ˙ VD ≥ 0, i.e., W  ˙ VD = 3 W (σD ˙V ) d = 3σ¯ D ˙V ≥ 0 (18) 2π

(σV δV + σD δD + σT · δT ) d

  δWm  3 + σD δV d

(12) 2π

   δWVD

where σ¯ D is the average deviatoric stress over all microplanes  1 σ¯ D = σD d

(19) 2π

The microplane volumetric and deviatoric stress components are expressed separately in terms of the corresponding microplane strains V and D ; σV = fV (V ),

σD = fD (D )

Condition (18) requires that the sign of the average deviatoric stress σ¯ D be the same as that of V . If V ≥ 0, the average deviatoric stress σ¯ D should be non-negative, i.e., the positive deviatoric stress on the microplanes under deviatoric tension should overall be greater in magnitude than the negative deviatoric stress on those under deviatoric compression. If V < 0, the average deviatoric stress σ¯ D should be negative (this property is necessary to describe the di-latancy exhibited under uniaxial and biaxial compression loadings).

(13)

In microplane model M2, functions fV and fD are formulated as a microplane damage model, whereas in microplane models M3 and M4, these functions are implied by the strain and stress boundaries. In model M4, they are further constrained by the condition:  σV = min

1 2π





σN d , fV (V )

(14)

In M4 (Bažant et al. 2000), only the first term, δWm (Eq. 12), is considered in the virtual work equation, i.e.,  3 (σV δV + σD δD + σ T · δT ) d (15) δW = 2π

5



σV V d

ENFORCING NON-NEGATIVE DISSIPATION IN TENSORIAL FORM OF CONSTITUTIVE MODEL

For a constitutive law in the classical tensorial form, the increment of energy dissipation density loading step (tr , ts ) is given by

The microplane volumetric stress σV is defined by the virtual work equation 3 σkk δmm = 3 2π

(17)

1 1 (σ r + σ s ) :  − βσ s : αCs : βσ s 2 2 1 + σ r : Cr : σ r ≥ 0 2

D =

(16)

which leads to σV = σkk /3. This is an equilibrium definition of microplane volumetric stress σV . As one can see from Eqs. (8), (12), (15) and (16), the stress tensor σij , microplane normal stress σN and shear stress vector σ T are an equilibrium system of forces, and can thus be used to calculate the first-order work and energy dissipation, which underlies Eq. (10). One can, of course, introduce a postulate that the stress tensor σij , the microplane volumetric stress σN , the deviatoric stress σD and the shear stress vector σ T ,

(20)

(now σ ,  and C are all tensors). At the end of the computation of each load step, the procedure is as follows: • Set first α = β = 1. • Check if D ≥ 0. • If satisfied, go to the next integration point. If not, find α from the condition D = 0, which amounts to adjusting the constitutive law for damage.

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• But if 12 σ s : αCs : σ s < 12 σ s : C0 : σ s , then reset also β so that, in this condition, ‘<’ be changed to ‘=’ while D = 0 would remain valid. 6

booster grant from the Initiative for Sustainability and Energy (ISEN) at Northwestern University. The application to fiber reinforced concrete has been supported by grant W911NF-09-1-0043 from the Army Research Office, Durham, to Northwestern university, monitored by ERDC, Vicksburg. The application to concrete infrastructure problems has been supported by the U.S. Department of Transportation through Grant 23120 from the Infrastructure Technology Institute of Northwestern University.

DISCUSSION

The material damage is in constitutive models often described by a reduction of secant compliance, with an unloading path that points to the origin. But often the unloading behavior might be considered beyond the scope of the model, or the unloading might be expected to follow some simple rule different from secant compliance, for example, the initial compliance or some nonlinear unloading rule. This point needs to be realized in interpreting the example in Table 1 of Carol et al. (2001), which examined microplane model M2 (Bažant and Prat 1988). In this model, all the inelastic behavior is described by a variation of the secant compliance. Under the hypothesis that unloading follows the secant compliance, Carol et al. (2001) demonstrated for M2 the existence of a loading cycle with negative energy dissipation. However, if the unloading is assumed to follow the initial compliance, the dissipation during this cycle would be positive, and so it would be for a certain range of intermediate unloading rules. The hypothesis of secant unloading was not followed in subsequent extension of model M2 for unloading and cyclic loading, and was replaced by a curved unloading path (Ožbolt and Bažant 1992). It may thus be observed that, for constitutive models not intended to describe unloading, the dissipation inequality might not necessarily be an important consideration. If the loads are applied monotonically, the strains on the microplanes, of course, might not necessarily evolve monotonically, but it many application they do so, at least approximately. On the other hand, for constitutive models intended to cover significant unloading and hysteretic loops, the dissipation inequality is an essential check. Finally note that the checks for stability and bifurcation of the equilibrium path are a different matter. They deal with the positivity or vanishing of the secondorder-small triangular area 1421 in Fig. 1(b), which is irrelevant for the dissipation inequality.

REFERENCES Batdorf, S.B., and Budianski, B., 1949. ‘‘A mathematical theory of plasticity based on the concept of slip.’’ Technical Note No. 1871, Nat. Advisory Committee for Aeronautics, Washington, D.C. Bažant, Z.P., 1996. ‘‘Analysis of work-of-fracture method for measuring fracture energy of concrete.’’ J. of Engrg. Mechanics ASCE, 122 (2), 138–144. Bažant, Z.P., Caner, E.C., Carol, I., Adley, M.D., and Akers, S.A., 2000. ‘‘Microplane model M4 for concrete: I. Formulation with work-conjugate deviatoric stress.’’ J. of Engrg. Mechanics ASCE, 126 (9), 944–953. Bažant, Z.P., and Oh, B.-H., 1985. ‘‘Microplane model for progressive fracture of concrete and rock.’’ J. of Engrg. Mechanics ASCE, 111, 559– 582. Bažant, Z.P., and Planas, J., 1998. ‘‘Fracture and Size Effect in Concrete and Other Quasibrittle Materials.’’ CRC Press, Boca Raton. Bažant, Z.P., and Prat, P.C., 1988. ‘‘Microplane model for brittle plastic material: I. Theory,’’ J. of Engrg. Mech. ASCE, 114, 1672–1688. Carol, I., Jirásek, M., and Bažant, Z.P., 2001. ‘‘A thermodynamically consistent approach to microplane theory. Part I: Free energy and consistent microplane stresses.’’ Int. J. of Solids and Structures, 38 (17), 2921–2931. Jirásek, M., and Bažant, Z.P., 2002. ‘‘Inelastic Analysis of Structures.’’ J. Wiley & Sons, London and New York. Lubliner, J., 2006. Plasticity Theory (revised edition), Pearson Education, Edinburgh. Ožbolt, J., and Bažant, Z.P., 1992. ‘‘Microplane model for cyclic triaxial behavior of concrete.’’ J. of Engineering Mechanics ASCE, 118 (7), 1365–1386. Rice, J.R., 1970. ‘‘On the structure of stress-strain relations for time-dependent plastic deformation of metals.’’ J. of Applied Mechanics, ASME, 37 (Sept.), 728–737. Taylor, G.I., 1938. ‘‘Plastic strain in metals.’’ J. Inst. Metals, 62, 307–324.

ACKNOWLEDGMENT The theoretical work of J.-Y. Wu, motivated by applications to braided composites, was supported by a

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

A multiscale approach for nonlinear hysteretic damage behaviour of quasi-brittle disordered materials J. Carmeliet Chair of Building Physics, Swiss Federal Institue of Technology Zürich (ETH), Zürich, Switzerland Laboratory of Building Science and Technologies, Swiss Federal Laboratories for Materials Testing and Research (EMPA), Dübendorf, Switzerland

S. Mertens & P. Moonen Chair of Building Physics, Swiss Federal Institue of Technology Zürich (ETH), Zürich, Switzerland

ABSTRACT: A description of the complete failure process in quasi-brittle materials is obtained by embedding a continuous and discontinuous theory into a common framework. The framework employs the partition of unity (PU) concept and introduces a cohesive zone model, capturing the entire failure process starting from the growth and coalescence of micro-defects until the formation of macrocracks. A gradual transition is considered between a constitutive model considering both intact material bonds at the process zone and damaged material bonds, which are considered traction-free. As damage grows, material bonds are broken and a macro-crack is formed. The continuous-discontinuous constitutive model is further extended to include nonlinear hysteretic behaviour during cyclic loading. The tangent stiffness of the undamaged continuous material and the tangent acoustic tensor of the discontinuous model, describing the relation between tractions and damage-induced elongations of the material bonds, are considered to depend on loading history. The latter also depends on damage, in order to properly model the change of nonlinear hysteretic behaviour due to damage development. The nonlinear and hysteretic behaviour is due to the presence of nonlinear contacts in the disordered material, which open and close at different stress levels. The nonlinear contacts are modelled by a phenomenological approach at lower levels, also known as the Preisach-Mayergoyz (PM) model. The PM model shows that with growing damage more nonlinear contacts are generated, but that these nonlinear contacts behave more and more less hysteretic. Upon implementation in a X-FEM framework, mesh independent results are obtained which correspond well with the experiments. 1

INTRODUCTION

models, which are enriched by higher order formulations (nonlocal or gradient, Pijaudier-Cabot and Bazant (1987), Borst et al. (1995)) in order to adequately capture the localization phenomenon. A trend

The cyclic behaviour of concrete in tensile loading shows a rich palette of observations: nonlinear elasticity in pre-peak loading, strain softening in the post-peak region, a stiffness reduction and permanent deformations in unloading, and hysteresis in reloading (Figure 1). This nonlinear elastic hysteretic damage behaviour is generally attributed to the evolution of disorder in the mesoscopic structure of concrete. Nonlinear elasticity is attributed to opening of existing micro-cracks, hysteresis attributed to the presence of nonlinear contacts and permanent deformations in unloading to the non-perfect closing of microcracks, while softening is related to the growth, coalescence of microcracks leading finally to a localisation into a macrocrack. Different models capturing parts of this behaviour have been proposed. Nowadays, efforts are being made to incorporate effects of crack initiation and propagation into the continuum theory. In a continuum description, strain-softening and stiffness reduction are often modelled by means of damage plasticity

Figure 1. Experimental behaviour of plain concrete in monotonic and cyclic tensile loading (Reinhardt et al. 1986).

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(Preisach 1935; Mayergoyz 1986), showed that the nonlinear hysteretic behaviour finds its origin in multiple hysteretic contacts in disordered materials and that the strain can be adequately modelled by the opening or closing of a distribution of these hysterectic contacts. However, the models as described above are generally not capable to capture all observed phenomena as observed. A reason is that the origin of all observed phenomena is not captured with models. In this paper, we present a two-scale model, which takes benefit of a macroscopic formulation, but which is enriched by information fed from lower scales incorporating the multiple contacts at meso and lower scales. On the macroscale, we model the gradual transition from continuous damage to localised cracks using a continuous-discontinuous framework as proposed by Moonen et al. (2008).

towards coupled continuous-discontinuous modelling techniques can be observed. Regarding the modelling of failure, this idea has been pursued by many authors. Ren and Bi´cani´c (1997) employ an element removal technique to obtain discrete cracks at the final stages of failure. Jirásek and Zimmermann (2001) propose a method in which smeared cracks are combined with embedded discontinuities. Simone et al. (2003) introduce a traction-free discontinuity at the final stages of failure. Similar to Ren and Bi´cani´c (1997), the continuum model governs the softening behaviour. Wells et al. (2001) make use of the cohesive zone assumption (Dugdale (1960), (Barenblatt, 1962)) and the fracture energy concept (Hillerborg et al. 1976) to describe the gradual formation of cracks in quasi-brittle materials. Here a traction-separation model governs the nonlinear behaviour in the fracture process zone and the continuum remains elastic at all times. All mentioned approaches have in common that a single moment exists at which the continuous model is replaced by a discontinuous model. The distinct feature of the model proposed by Moonen et al. (2008) is that this transition takes place gradually. On the other hand, the modelling of hysteretic behaviour of quasi-brittle materials has attracted a great deal of attention. Gylltoft (1984), Reinhardt et al. (1986), Yankelevsky et al. (1989), Hordijk (1991) proposed semi-empirical relations for describing the cyclic behaviour. Duda (1991) proposed a rheological model for the simulation of damage-dependent cyclic behaviour based on springs and sliders. Oliveira (2002) proposed an interface cyclic model based on plasticity theory, while Rageneau et al. (2000) introduced a damage-plasticity model, where the stiffness reduction is modelled by damage and the hysteresis phenomenon is modelled by a plasticity model. The origin of nonlinear elasticity and hysteresis of disordered materials has been subject of intensive studies in the past. The nonlinear response of mesoscopic materials is found to be sensitive to damage and used as signature for non destructive testing of materials and structures. The nonlinear and hysteretic behaviour finds its origin in the bond system between the aggregates, which is rich of of defects, cracks, pores, microflaws, intergrain and crack contacts, asperities . . . randomly distributed in the material and of different size. When these micro- and mesoscopic features are loaded by compressive or shear forces, a complex network of contacting forces originates, which are nonlinear and hysteretic in nature. A physical understanding of these hysterectic contacts has been explored by Aleshin and Van Den Abeele (2005, 2007a,b ). This network of contact forces results in macroscopic nonlinear elastic and hysteretic behaviour. Phenomenological models, such as formulated by and Mc. Call and Guyer (1996) for rock, originating from the original Preisach-Mayergoyz (PM) approach

2

MACROSCOPIC FORMULATION

A damage zone is assumed to consist of a discrete crack, where no stresses are transferred and a cohesive zone rich of microdamage, where still a part of the stresses can be transferred. Assume an infinitesimal part of a cohesive zone with damage d exposed to a continuum (average) stress σ . Traction forces can only be transferred through the undamaged part of the material. When assuming that micro-damage influences the traction field only locally, we can define effective tractions, which follow from a force equilibrium: σ ndA = teff (1 − d)dA

(1)

with n the normal to the surface dA in the cohesive zone. As damage grows, the active area decreases and higher effective tractions are needed to maintain equilibrium with continuum stresses. The redistribution of the tractions causes additional deformations in the undamaged material bonds. Therefore, the effective traction can be additively decomposed into two terms: (i) the initial traction prior to damage growth and (ii) the traction related to the damage-induced elongation of the material bond, or: teff = σ n + γ −1 Qu

(2)

where Q = nEn is the acoustic tensor, E is the fourthorder constitutive tensor describing the constitutive behavior of the bulk material, u is the displacement jump corresponding to the elongation of the material bond and γ is one unit length follows from dimensional analysis (omitted from now on). Combining equations 1 and 2, and introducing Hooks law for the undamaged material, we get the following constitutive

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3

law for the cohesive zone: t = (1 − d)(Eεn + Qu)

(3)

In this section we describe first the PM-model for the nonlinear hysteretic behaviour of the undamaged material. Then, the PM model is extended to damaged materials. We limit for simplicity to a one-dimensional formulation. The total strain for the bulk material is decomposed in elastic and inelastic part. Applying Hooks law, where E0 is the elastic modulus at zero stress and zero damage, and introducing the inelastic compliance h, which is history dependent, we may write:

Damage evolution law is governed by the KuhnTucker conditions, supplemented with the consistency condition: d˙ ≥ 0,

f ≤ 0,

d˙ f = 0

and d˙ f˙ = 0

PHENOMONOLOGICAL HYSTERESIS MODEL

(4)

where f is a damage criterion. A traction-based Rankine-type criterion is used to describe damage evolution for mode I-dominated failure:

ε = εe + εin =

f (teff, d) = t eff,eq − κ(d)

(5)

d = α1 exp(−α2 u) + (1 − α1 ) exp(−α3 u)

(6)

with ft the tensile strength. Note that damage evolution is not explicitly given in terms of e.g. the equivalent traction, but in function of the displacement jump. We now formulate in the same framework the constitutive law for the nonlinear elastic hysteretic behaviour. We assume that during unloading and reloading in a hysteretic loop no damage occurs. The nonlinear elastic hysteretic behaviour arises from nonlinear contacts in the bulk material and from new nonlinear contacts developing in the damage zone. The hysteretic nonlinear behaviour of the initial material is modelled by a tangent stiffness modulus [E], where the operator [ ] denotes the dependence on loading history to include hysteresis. The hysteretic nonlinear behaviour of the damaged material is modelled by a tangent acoustic tensor [Q] dependent on loading history and on damage. The rate formulation of the constitutive law for nonlinear hysteretic behaviour (no damage evolution, or d˙ = 0) then reads: t˙ = (1 − d)[E]˙ε − [Q(d)]u ˙

(8)

The inelastic compliance [h] is determined using a PM model, where the nonlinear hysteretic behaviour is modelled by an ensemble of bistable elements also called non-classical units (NCU) or hysterons (Figure 2). A NCU can only be in one of the two states, i.e. in open or closed state (open = subscript o, closed = subscript c). The behaviour of a single NCU is such that it is closed with length Lc . When the tensile stress σ increases to σ0 , it abruptly opens to length L0 , and remains open as the stress continues to increase. When the applied stress σ decreases, the element closes to its original length Lc at stress σc , which can be different to σ0 , creating hysteresis and energy dissipation at NCU-scale. The corresponding characteristic strain of a non-classical unit, can be written as γ = (L0 − Lc )/L0 . NCU’s behave either in a reversible or hysteretic manner. A NCU is reversible when σ0 = σc . A large number of these elements with different σ0 , σc and γ describe the nonlinear contacts in the bond system, and the outcome of the behaviour of all NCU’s during loading describe the global nonlinear strain response of the material. In order to determine the response of all NCU’s as a function of the applied stress, the NCU’s are mapped onto a stress space, where σ0 and σc are used as coordinates for a NCU (see Figure 3a). This space is half-filled, since σ0 ≥ σc always applies. This space is called the PM-space. Reversible NCU’s reside on the diagonal of the PMspace. Hysteretic NCU’s are mapped away from the diagonal. The NCU’s are distributed over the PM space

For the tensile failure, the equivalent effective traction is taken equal to the effective traction and κ is the residual strength of the damaged material. The following expression is used for κ(d): κ(d) = (1 − d)ft

σ σ + [h] σ = E0 [E]

(7)

This equation shows that two origins of nonlinear hysteretic behaviour in the damaged material are considered: the nonlinear contacts in the still intact material and the increase of nonlinear contacts due to damage development in the damaged part of the material. The history dependent stiffness [E] and history and damage dependent acoustic tensor [Q] are described using a phenomenological approach, which will be described in the following section.

Figure 2. Bistable units or NCU’s, a) hysteretic NCU, b) non-hysteretic NCU.

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Figure 3. a) Mapping of the NCU’s on a half-space, b) resulting distribution for the 1st quadrant.

according to a PM density distribution ρ (σ0 , σc ) (see Figure 3b). The integral over the domain gives the nonlinear elastic strain:  εin = ρ(σ0 , σc )dσ0 dσc (9)

Figure 4. Staircase line in the PM space due to a history of un- and reloading.

1 = [Q]

o

where o represents the domain of the open NCU’s. The NCU element strain γ is following Carmeliet and Van Den Abeele (2002) incorporated into the functional description of the PM density function and taken equal to 1. Figure 4 gives as an example the PM space with the domain o of the open NCU’s and the domain c of closed NCU’s after a given loading history. These domains are generally separated by a staircase line, characterized by vertexes which correspond to extremes in the stresses of the loading history. The inelastic compliance h can then be determined using the PM model taking the line integral over respectively the horizontal or vertical line segments for unloading or reloading. [h] =

∂εin = ∂σ

3.1

θ(t0 , tc ; d)dt

(12)

t

Determination of the PM space distributions

We note that due to damage development, the residual strength of the material reduces, which is the upper bound of the PM space. This means that during damage development, the PM space will become populated by new nonlinear contacts or NCU’s and the upper boundary will shrink to lower stress levels. The determination of the PM-density distributions for the continuous bulk material ρ(σ0 , σc ) and for the damaged material θ(t0 , tc ; d) is an undetermined problem, i.e. the number of constraints that can be obtained from a hysteretic loop is much smaller than the number of points defining the PM-density, causing an illposed problem. To avoid the ill-posed problem, following Carmeliet and Van Den Abeele (2002), the PMdistributions are modelled as a product of exponential functions:

 ρ(σ0 , σc )dσ



(10)

σ

For the damaged material, we also assume that nonlinear contacts exists, which open and close at a different tractions respectively t0 and tc and are described by a distribution function θ(t0 , tc ). The displacement jump u is the outcome from the opening (or closing) of a number of NCU’s. We assume that no damage occurs during a hysteretic loop, which means that the PM distribution does not change during an hysteretic cycle. However, damage influences the shape of the hysteretic loop and will therefore influence the number of nonlinear contacts and as such the PM distribution. The displacement jump u and the acoustic tensor are then defined as:  u = θ(t0 , tc ; d)dt0 dtc (11)

ρ(σ0 , σc ) = (ac exp(bc σm )) exp(−cc σd )

(13)

θ(t0 , tc ; ) = (ad exp(bd tm )) exp(−cd td )

(14)

where a, b, c are functional parameters (subscript c and d refer to the continuous and discontinuous distributions). The mean stress σm is defined as (σ0 + σc )/2, the distance in the PM space from the diagonal σd is defined as (σ0 −σc )/2. For the discontinuous formulation the PM distribution depends on the mean traction tm = σm n and td = σd n the distance to the diagonal. The first exponential describes the PM density on the diagonal. The second exponential describes the decay from the diagonal with c a decay coefficient.

o

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The PM-distribution parameters for each hysteretic loop are determined by indirect parameter estimation. The parameters of the discontinuous part, ad , bd , cd are function of the damage and are determined for each different hysteretic loop of un the cyclic tensile test.

4

allows to determine the nonlinear hysteretic elastic modulus [E], at the Gauss points (see Figure 6). For integration points in damaged zones, we keep track of two PM spaces, one for the undamaged bonds and one for the discrete crack, which is also dependent on the damage.

STRONG DISCONTINUITY APPROACH AND X-FEM IMPLEMENTATION

5

Figure 7 compares the simulated results using the proposed multiscale model implemented in a XFEM framework with the experimental results of the tensiletensile test by Reinhardt, 1986. We observe a good agreement between measurements and simulation. To show mesh objectivity the behavior was simulated with a fine and coarse mesh (Figure 8). The coarse mesh consists of 3 two node bar elements and the fine mesh of 21 two node bar elements. Mesh-objectivity is observed. The simulation parameters are: E0 = 36,707 MPa; for the damage evolution (equation 6): α1 = 0.482, α2 = 173.704 and α3 = 38.131; for the continuum PM distribution (equation 13): ac = 3.4E-6, bc = 0.4 and cc = 1.46. The parameters describing the discrete PM distribution as a function of damage are plotted in Figure 9. We observe that parameters ad and cd

The total displacement field u of a body  crossed by a strong displacement discontinuity is given by: u = uˆ + H u˜

(15)

where uˆ and u˜ are continuous functions and H is the Heaviside function. In the model, uˆ describes the nonlinear hysteretic displacement at a given loading history without the presence of damage and u˜ gives the displacement due to crack opening. The Heaviside function is zero for all points x ∈ − and one for all points x ∈ + (see Figure 5). The strain corresponding with the displacement is obtained by taking the symmetric gradient ∇ s of the displacement: ε = ∇ s u = ∇ s uˆ + H ∇ s u˜ + δ(u˜ ⊗ n)s

RESULTS AND DISCUSSION

(16)

where n is the normal to the discontinuity, δ is the Dirac impulse distribution centered at discontinuity. The Dirac impulse distribution is the derivative of the Heaviside function. The constitutive model including the continuous-discontinuous damage formulation and nonlinear hysteretic behaviour are implemented in X-FEM (Mertens 2009). The implementation is based on a two-scale approach. A cohesive zone is introduced or grows at the macroscale according to the damage criterion (equation 5) and the damage evolution law (equation 6). At the meso-scale, we keep track of the loading history for every point by updating the staircase line defining the boarder between open and closed NCU’s. For the undamaged part, we keep track of the PM space, which

Figure 6. Two-scale model for the undamaged material. For the undamaged part, we keep track of the PM space, which allows to determine the nonlinear hysteretic elastic modulus [E], at the Gauss points.

Figure 5. a) Displacement u composed of 2 continuous functions and the Heaviside function. b) body  crossed by a strong discontinuity with normal n dividing the body in + and − .

Figure 7. Comparison of stress-displacement response of experimental data (Reinhardt et al. 1986) and simulation of plain concrete loaded in cyclic tensile loading.

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Figure 8. meshes.

into account the gradual transition of an undamaged continuum to a damaged material, combining a continuous-discontinuous approach, capturing the entire failure process starting from the growth and coalescence of micro-defects until the formation of macrocracks. The framework employs the partition of unity (PU) concept and introduces a cohesive zone model, capturing the entire failure process starting from the growth and coalescence of micro-defects until the formation of macrocracks. The model is enriched to a multiscale model to take into account nonlinear hysteretic behaviour. The nonlinear and hysteretic behaviour is due to the presence of nonlinear contacts in the disordered material, which open and close at different stress levels. The nonlinear contacts are modelled by a phenomenological approach, also known as the Preisach-Mayergoyz (PM) model. Upon implementation in a X-FEM framework, mesh independent results are obtained which correspond well with experiments. It is concluded that with increasing damage more nonlinear contacts are generated in the disordered material, especially in crack contacts, and that these contacts behave less hysteretic when the cracks develop. However, the global response of all nonlinear contacts results in a more hysteretic response of the material.

Study of the mesh objectivity for two different

Figure 9. Evolution of the parameters describing the discrete PM-distribution with damage.

ACKNOWLEDGEMENTS increase, while the parameter bd can be considered (taking into account the uncertainty in determining these parameters) to remain constant. This means that with developing damage the distribution on the diagonal becomes more populated with nonlinear contact units (increasing ad ), but that the shape of the distribution function does not change considerably (constant bd ). The parameter cd describing the decay of the PM distribution normal to the diagonal, increases with damage, which indicates that the nonlinear contacts become more densely located near the diagonal, which means they behave less hysteretic. We may conclude that with increasing damage more nonlinear contacts are generated in the cracks and that these contacts behave less hysteretic. However, the global response of all nonlinear contacts results in a more hysteretic response.

6

The help of Miguel Abreu in processing the paper in time is highly appreciated. REFERENCES Aleshin, V. & Van Den Abeele, K. 2005. Micropotential model for stress-strain hysteresis of microcracked materials. J. Mech. and Phys. Solids, Vol. 53(4), pp. 795–824. Aleshin, V. & Van Den Abeele, K. 2007a. Microcontact based theory for acoustics in microdamaged materials. J. Mech. and Phys. Solids, Vol. 55, pp. 366–390. Aleshin, V. & Van Den Abeele, K. 2007b. Friction in unconforming grain contacts as a mechanism for tensorial stress-strain hysteresis. J. Mech. and Phys. Solids, Vol. 55, pp. 765–787. Barenblatt, G.I. 1962. The mathematical theory of equilibrium cracks in brittle fracture. Advances in Applied Mechanics, Vol. 7, pp. 55–129. Carmeliet, J. & Van Den Abeele, K.E.A. 2002. Appliaction of the Preisach-Mayergoyz space model to analyze moisture effects on the nonlinear elastic response of rock, Geophysical research letters, Vol. 29(7). de Borst, R., Pamin, J., Peerlings, R.H. & Sluys, L.J. 1995. On gradient-enhanced damage and plasticity models for failure in quasi-brittle and frictional materials. Computational Mechanics, 17, 1–2, 130–141. Duda, H. 1991. Bruchmechanisches verhalten von Beton unter monotoner und zyklischer zugbeantspruchung,

CONCLUSIONS

The cyclic behaviour of concrete in tensile loading shows a rich palette of observations: nonlinear elasticity in pre-peak loading, strain softening in the post-peak region, a stiffness reduction and permanent deformations in unloading, and hysteresis in reloading. A proper constitutive model is proposed to take

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Deutscher aussschutz für stahlbeton, heft 419, Beuth verlag GmbH. Dugdale, D.S. 1960. Yielding of steel sheets containing slits, Journal of the Mechanics and Physics of Solids, Vol. 8(2), pp. 100–104. Gylltoft, K. 1984. Fracture mechanics model for fatigue, Concrete, Materials and Structures, Vol. 17(97), 55–58. Hillerborg, A., Modeer, M. & Pettersson, P.E. 1976. Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements, Cement and Concrete Research, Vol. 6(6), pp. 773–782. (doi:10.1016/0008-8846(76)90007-7). Hordijk, D.A. 1991. Local approach to fatigue of concrete, PhD-report, Technische Universiteit Delft, the Netherlands. Jirásek, M. & Zimmermann, T. 2001. Embedded crack model. Part II: Combination with smeared cracks, International Journal for Numerical Methods in Engineering, Vol. 50, pp. 1291–1305. Mayergoyz, I.D. 1986. Mathematical models of hysteresis, Physical Review Letters, Vol. 56, 15, 1518–1521. McCall, K.R. & Guyer, R.A. 1996. A new theoretical paradigm to describe hysteresis, discrete memory and nonlinear elastic wave propagation, Rock, Nonlinear Processes in Geophysics, 3, 89–101, 3113–3124. Mertens, S. 2009. Hysteresis, Damage and Moisture Effects in Quasi-Brittle Porous Materials, PhD-report, Katholieke Universiteit Leuven, Belgium. Moonen, P., Carmeliet, J. & Sluys, L.J. 2008. A continuousdiscontinuous approach to simulate fracture Processes in quasi-brittle materials, Philosophical Magazine, Vol. 88(28–29), pp. 3281–3298. (doi:10.1080/1478643 0802566398).

Oliveira, D.V. 2002. Experimental and Numerical analysis of blocky masonry structures under cyclic loading, PhD-report, Universidade do Minho, Portugal. Pijaudier-Cabot, G. & Bazant, Z.P. 1987. Nonlocal damage theory, Journal of Engineering Mechanics, 113, 1512–1533. Preisach, F. 1935. über die magnetische nachwirkung, Zeitschrift für Physik, 94, 277–302. Rageneau, F., La Borderie, C. & Mazars, J. 2000. Damage model for concrete-like materials coupling cracking and friction, contribution to structural damping: first uniaxial applications, Mechanics of Cohesive Materials, 5, 607–625. Reinhardt, H.W., Cornelissen, H.W. & Hordijk, D.A. 1986. Tensile tests and failure analysis of concrete, Journal of structural engineering, Vol. 112(11), 2462–2477. Ren, Z. & Bi´cani´c, N. 1997. Simulation of progressive fracturing under dynamic loading conditions, Communications in Numerical Methods in Engineering, Vol. 13, pp. 127–138. Simone, A., Wells, G.N. & Sluys, L.J. 2003.From continuous to discontinuous failure in a gradi-ent-enhanced continuum damage model, Computer Methods in Applied Mechanics and Engineering, Vol. 192(41–42), pp. 4581– 4607. (doi:10.1016/S0045-7825(03)00428-6). Wells, G.N. & Sluys, L.J. 2001. A new method for modelling cohesive cracks using finite elements, International Journal of Numerical Methods in Engineering, Vol. 50(12), pp. 2667–2682. (doi:10.1002/nme.143). Yankelevsky, D.Z. & Reinhardt, H.W. 1989. Uniaxial behavior of concrete in cyclic tension, ASCE Journal of structural Engineering, Vol. 115(1), 166–182.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Modeling of reinforced cementitious composites using the microplane damage model in combination with the stochastic cracking theory R. Chudoba, A. Scholzen, R. Rypl & J. Hegger Institute of Structural Concrete, RWTH Aachen University, Germany

ABSTRACT: Short fiber reinforced concrete and textile-reinforced concrete are composites exhibiting initial anisotropy introduced by the distribution of the fibers or by the orientation of the textile fabrics. The cracking of the matrix is also directionally dependent so that an anisotropic damage model is required to capture the complex failure patterns at the macroscopic scale. The present paper applies the microplane damage model as a phenomenological framework for the directionally dependent damage response of reinforced cementitious composites. The microplanes oriented in a reinforced direction are associated with the damage function that reflects the strain-hardening effect due to the multiple cracking of the matrix. The specification of the directionally dependent damage function is linked to the mesoscopic model for multiple cracking of reinforced composite. 1

INTRODUCTION

The highly heterogeneous material structure of textile reinforced concrete (TRC) leads to complex damage patterns including several mechanisms such as crack bridge initiation, debonding and filament rupture. It is important to include these micro-structural effects in the modeling strategy of TRC. On the other hand it is inevitable to use a smeared representation of the material behavior in order to allow for an efficient calculation. The material model presented here uses a meso-macroscopic representation of the damage process applying a smeared cracking approach as well as an implicit representation of the textile reinforcement. The approach is applicable for damage patterns exhibiting a sufficiently fine degree of regularity (fine crack patterns). Such patterns occur for example in tensile zones of plates and shells. The present contribution starts with a short review of the applied microplane model in Sec. 2. The ability to capture the initial anisotropy is then exemplified for a reinforced specimen with different orientations of the continuous reinforcement with respect to the loading direction. For the preliminary studies shown in Sec. 3 the direction-dependent damage functions were specified intuitively. The quantitative identification of the damage functions is described in Sec. 4. In Sec. 5 a mesoscopic model for multiple cracking is described that is used to establish the link between the strain-hardening response of the composite at the macroscopic level and the elementary failure mechanisms at the mesoscopic level of the material structure, i.e. matrix cracking and debonding. Finally, in Sec. 6 the possibility to use the projection

of the reinforcement ratio onto an oriented crack plane as an additional criterion for constructing the direction-dependent damage function is presented. 2

MICROPLANE DAMAGE MODEL

The decomposition of the macroscopic behavior into different spatial directions is achieved by a spherical discretization of the material point into a set of oriented microplanes. It shall be emphasized that the microplanes do not have the purpose to reflect the microscopic material structure. They can rather be interpreted as a polar/spherical discretization of the stress-strain space. The basic steps in the formulation of a microplane model are schematically depicted in Figure 1 (a) for an early formulations of the microplane model including softening (Bazant and Gambarova 1984): • geometric projection of the macroscopic strain tensor yielding a set of microplane strain vectors • evaluation of constitutive laws linking the microplane strains and stresses • an energetic homogenization of the microplane stress vectors to obtain the macroscopic stress tensor. The energetic homogenization is based on the principle of virtual work stating that the work at the macroscopic level corresponds to the work at the microplane level (Carol and Bazant 1997). Alternative formulations base their derivation on the equivalence of the Helmholtz free energy. It has been shown that for the basic microplane model as depicted in Figure 1 (a)

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(a)

(b)

energetic homogenisation

projection

constitutive laws

(c)

(d)

Figure 1. Classification of different types of microplane models: (a) basic principle of the microplane approach; (b) microplane formulation with split on the microplane level; (c) microplane formulation with split on the macroscopic level; (d) microplane formulation with explicit representation of the elasticity tensor.

both approaches are equivalent and lead to the same formulations (Carol, Jirasek, and Bazant 2001). The involved integration at the microplane level can be evaluated numerically (Stroud 1971) as the weighted sum of all microplane contributions. Today a variety of microplane formulations exists. They differ in their focus and fields of application. The first microplane model as described in (Bazant and Oh 1985) was limited in its ability to reproduce the linear-elastic case and allowed only a certain range of values for the Poisson’s ratio. This restriction was eliminated in the succeeding models (Bazant and Prat 1987) by splitting the microplane normal strains and stresses into a volumetric (eV , sV ) and deviatoric part (eD , sD ). The basic scheme of this class of microplane models is depicted in Figure 1 (b). The shear component of the microplane strain/stress vectors (ei , si ) is denoted in the Figure by (eT , sT ). The introduction of the split leads to the correct reproduction of arbitrary values of Poisson’s ratio as well as a better behavior under triaxial compression. Despite of these improvements and convincing practical applicability of the model in terms of good fits and prediction of experimental behavior (Bazant, Caner, Carol, Adley, and Ankers 2000) for a wide range of loading cases, this version of the model did not fulfill the requirement on thermodynamic consistency (Carol, Jirasek, and Bazant 2001; Kuhl, Steinmann, and Carol 2001). On the other hand, if the split is introduced directly at

the macroscopic level instead of the microplane level it leads to a thermodynamically consistent formulation. Further improvement of the microplane formulation within a thermodynamically sound framework has been achieved by combining the kinematic and static constraint. This extension was primarily motivated by the need to improve the model behavior in the softening regime for uniaxial tension (Bazant and Caner 2005). In (Kuhl and Ramm 2000; Leukart and Ramm 2003) another thermodynamically consistent microplane formulation has been derived. In this particular formulation, the macroscopic strain tensor is split into a volumetric and a deviatoric part as depicted schematically in Figure 1 (c). An alternative split based on spectral decomposition of the strain tensor was proposed in (Cusatis, Beghini, and Bazant 2008) for laminate composites. In both models the motivation behind the introduction of the split is to assign different constitutive laws to the decomposed parts in order to better capture the underlying failure mechanisms of the material in different loading situations. An alternative formulation that does not need the split was provided in (Jirasek 1999). The derived model is thermodynamically consistent and falls into the class of microplane models depicted in Figure 1 (d). It is based on theoretical considerations devised in (Carol and Bazant 1997). In contrast to the original microplane scheme in Figure 1 (a) the connection between the effective macroscopic strain

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Evolution of damage for a polar discretization with 14 microplanes Evolution of damage for a polar discretization with 100 microplanes Macroscopic stress-strain response

Figure 2. tension.

Anisotropic damage development for two different microplane resolutions of a single 2D material point loaded in

tensor and stress tensor is established explicitly using the elasticity tensor. This places this microplane model into the context of classical damage mechanics. For undamaged material the model can exactly reproduce the linear-elastic response. Furthermore, the Poison’s effect is correctly reproduced for arbitrary values of the Poison’s ratio. 2.1

Damage induced anisotropy

In order to visualize the way how the microplane model represents the damage, Figure 2 shows the response of a single 2D material point with initially isotropic, quasi-brittle behavior loaded in uniaxial tension. The longitudinal axis of the cylindric plot corresponds to the imposed macroscopic strain. The damage evolution obtained from the two-dimensional microplane formulation is depicted for two polar discretizations with 14 and 100 microplanes (cf. Fig. 2 (a) and (b)). The blue color corresponds to an undamaged material, i.e. φ = 1, while the red color reflects complete material degradation in the direction of the corresponding microplane, i.e. φ = 0. At the beginning of the loading process none of the microplanes exhibits damage as the material is still in the linear-elastic regime. With increasing macroscopic strain the damage initiates at the microplanes with positive (tensile) microplane strain. Due to the Poisson’s effect the microplanes orthogonal to the loading direction exhibit negative (compressive) strains and remain undamaged even at large macroscopic strains. The macroscopic damage is captured by a fourth order damage tensor accumulating the contributions of the damage variables at each microplane. The

Figure 3. Damage function with different residual integrity for varying inclination angle.

stress-strain curve depicted in Figure 2 (c) has been obtained using 14 microplanes which documents already a sufficient accuracy. 2.2

Initial anisotropy

The microplane damage model introduced by (Jirasek 1999) has been utilized to incorporate a directiondependent damage function. The explicit notion of microplane orientation makes it possible to associate different parameters of the damage function of a microplane depending on the orientation. Further, the geometrical and mechanical interpretation of the parameters can be easily established. In order to reflect the effect of reinforcement, the damage law corresponding to the reinforced direction must show a residual integrity. This has been implemented by introducing the variable φr into the damage law representing the horizontal asymptote at the desired level of residual integrity (cf. Fig. 3).

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Figure 4. Simulated response of the composite for varied inclinations of the reinforcement; First column: reinforcement inclination; Second column: stress-strain response; Third column: development of the lateral strain versus control strain in the loading direction.

⎧ 1 if emax ≤ ep ⎪  ⎨   ep emax − ep φ = f (emax ) = (1 − φr ) e exp − + φr e − e max p f ⎪ ⎩ if emax ≥ ep (1)

In the example shown in Figure 4 a value of φr = 0.40 for the reinforced direction and a linear transition from the reinforced to the non-reinforced direction has been chosen to exemplify the computational procedure. The value of φr stays constant for the microplanes with an orientation that differs ±π/4 from the reinforced direction. From that point on, φr decreases linearly to zero until the direction orthogonal to the reinforced direction is reached. 3

ELEMENTARY STUDIES

In order to capture the discussed effects in a smeared approach on the macroscopic level the numerical model must be capable to reproduce the different behavior of the material in different directions.

Figure 4 exemplifies this for a TRC disk reinforced in one direction for different orientations of the reinforcement direction (indicated by the thick gray lines) with respect to the loading direction (indicated by the arrows). For the case of loading the disk perpendicularly to the reinforcement, the damage of the material in this direction can be described by a strain softening law. The same disk loaded in the reinforced direction exhibits multiple-cracking resulting in a pronounced tension stiffening effect in this direction. The obtained direction-dependent response is first reviewed using the stress-strain diagrams. Then, the lateral response is discussed using the diagrams of the principle strains. For the case of loading the disk in the reinforced direction (α = 0, first row in Figure 4) the composite exhibits multiple cracking in terms of a pronounced strain hardening in the response curve. The same disk loaded perpendicularly to the reinforcement (α = π2 , last row in Figure 4) exhibits the strain softening response of plain concrete. The transition between these two extreme cases is illustrated in the second and π third row for angles α = 16 ≈ 11◦ , α = π8 ≈ 22◦ . In the former case of small inclination, the stress-strain diagram is almost identical to the case α = 0, only

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a slight decrease of the stiffness can be recognized after the formation of cracks has finished. For α = π8 the tension stiffening effect is less pronounced and the stiffness is further reduced as compared to the cases with aligned reinforcement. The sudden drop of stress at the strain level of 0.15% corresponds to the failure of microplanes in the unreinforced direction. The neighboring ‘‘reinforced’’ microplanes are still able to transfer the tensile stresses and prevent the ultimate failure. In order to examine the material response in the lateral direction the third column in Figure 4 shows also the ε1 −ε2 diagrams. The initial slope of the ε1 −ε2 curve in all shown cases corresponds to a Poison’s ration of 0.2. As the damage in the ε1 -direction develops, the slope of the lateral strain ε2 starts to decrease. This reflects the fact that only the undamaged material, i.e. the effective strains, contributes to a lateral contraction of the material in 2-direction. For the aligned loading (α = 0) the slope of the lateral strain (ε2 ) is reduced only slightly and remains constant after the onset of damage in 1-direction. An interesting lateral response is observed for α = π8 with a sudden drop of stress at the strain level of approximately 0.15%. In the corresponding ε1 −ε2 diagram this drop in stress coincides with a sign change of the lateral strain ε2 and its subsequent rapid growth. As a consequence, further loading in the 1-direction results in a lateral expansion of the disintegrated specimen in the 2-direction. This can be interpreted as a crack aligned with the reinforcement direction. Nevertherless, the pickup of stresses after the damage localization is unnatural and can be ascribed to the oversimplified shape of the damage function (Eq. 3) with constant residual integrity φr = 0.40. As a consequence, the material is still able to transfer stress even at extremly large microplane strains. More realistic damage function with a drop to zero corresponding to ultimate failre of the composite is shown later in Figure 6. In order to provide a qualitative comparison of the individual stress-strain diagrams the calculated curves have been assembled in Figure 5 (left). Although the directional dependency has been only introduced very

roughly, the qualitative trends observed in the experiment could be reproduced by the model as shown in Figure 5.

4

CALIBRATION

The elementary study shown in Sec. 3 demonstrates the capability of the model to capture the anisotropic characteristics of the material. The tests have been evaluated at the material point level yielding the described stress-strain curves. The residual integrity has been intuitively chosen as a control parameter in order to obtain a response similar the the behavior of the material in experiments with inclined reinforcement. For a practical application of the model, a systematic calibration procedure is required to deliver a unique set of parameters derived from experimental data. In order to cover a possibly wide range of reinforcement configurations a numerical calibration procedure has been devised for obtaining the appropriate damage function automatically. For demonstration purpose, an isotropic damage function (identical for all microplane orientations) is to be fitted such that the stress-strain curve of a tensile test is reproduced. The top diagram in Figure 6 shows the experimental curve together with the iteration steps of the numerical calibration. In each load step, the simulation starts with a trial ω = 0 and calculates the response of the undamaged material. In the second trial step ω = 1 is set to get the response of a fully damaged material. After that the line search procedure is used to minimize the lackof-fit in the current load step. Once the value of the damage parameter ω within a given tolerance has been found, the state update is performed and the next load increment is imposed. The resulting damage function is depicted in the bottom diagram of Figure 6. It is represented by a piecewise linear function with a resolution corresponding to the resolution of the time stepping procedure. The described fitting example is limited by the fact that all microplane directions must have an identical

Figure 5. Qualitative comparison of the model response with the measured test data; Left: simulation with varied inclination of the reinforcement; Right: stress-strain response measured in tensile tests with reinforced TRC specimens.

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order to reflect localization within the characteristic length correctly.

5

As mentioned above, a severe limitation of the phenomenological damage model described is the fact that the identified damage function is only valid for a prescirbed reinforcement ratio. Obviously, the strainhardening response in the tensile tests changes for varying reinforcement ratios. The phenomenologically formulated anisotropic damage model without any explicit notion of the multiple cracking and debonding process cannot reflect the changes in the strainhardening response. A simple way how to extend the validity range of the macroscopic damage model is to include a mesoscopic, one-dimensional idealization of the multiple cracking and debonding of the tensile specimen. By choosing these elementary failure mechanisms an explicit relation between strain and stress can be established for a uniaxial loading condition as described e.g. by (Aveston, Cooper, and Kelly 1971). The model applied here uses the stochastic description of the matrix strength that has been incorporated by (Cuypers and Wastiels 2002). In general, the strainhardening response is decomposed into three stages: I–undamaged, II–multiple cracking, III–saturated cracking.

Figure 6. Fitting of uniaxial stress-strain diagram. Top: illustration of the incremental fitting procedure; Bottom: damage function associated to each microplane.

damage function. Such a damage model would only be appropriate for materials with uniform distribution of reinforcement in all directions. This might be the case for short-fiber reinforced specimens. However, for an aligned reinforcement, the calibration procedure must account for several directions of loading with respect to the orientation of the fabric simultaneously. For such an anisotropic response, the calibration procedure must be extended as follows: Given the set of curves with direction dependent response (cf. Fig. 5) the fitting is performed for a simultaneously running set of simulations, each of which is associated with one experimental curve. In each load step, the set of damage parameters ωα is sought such that the lack-of-fit in all response curves gets minimized.

Initial stage: The linear elastic response in the stage I is characterized by the initial effective elastic modulus that is obtained using the mixture rule E¯ = ρEf + (1 − ρ)Em ,

(2)

where ρ denotes the reinforcement ratio of a composite specimen given as ρ=

Remarks: • The obtained damage function is only valid for structural elements with the same reinforcement ratio as the source test specimen. Further mesoscopic detailing of the model is required in order to extend the validity range of the model. • Even though the model provides sufficient flexibility to cover the anisotropic damage behavior in different directions, the calibration requires a whole range of tests for α ∈ 0, π/2. • For under-reinforced directions that exhibit strain softening usual regularization techniques (non-local averaging, crack band model) must be employed in

MESOSCOPIC MODEL FOR MULTIPLE CRACKING

Af . Am + A f

(3)

Here, Af and Am denote the cross sectional area of the matrix and of the fibers, respectively. The matrix stress is related to the loading stress of the composite in regions without debonding by the relation σm =

Em σ¯ . E¯

(4)

Stress transfer in the crack vicinity: Upon a crack intiation at the level of the matrix stress σm the length of the debonded interface with a constant frictional stress

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τ can be expressed as δ=

(1 − ρ) rEm (1 − ρ) rσm σ¯ , · = · ρ 2τ ρ 2τ E¯

(5)

where r denotes the fiber radius. Final crack spacing: At both sides of an existing crack no new crack can occur within the stress transfer length δ. Due to the random positioning of the cracks, the average crack distance in the final cracking stage is within the range < δ, 2δ >. Moreover, regarding the positioning of cracks as a problem of random placement of objects with the length δ along a line the average crack spacing in the final stage III (no more cracks fit in) can be quantified as (Widom 1966) csF = 1.337δ.

Figure 7. Strains between two cracks with nonoverlapping debonding zones.

(6)

Evolution of crack spacing: During the cracking stage II, multiple cracks in the matrix occur. The matrix strength is assumed random with the probability of failure prescribed by the cummulative Weibull distribution

σm m . (7) P(σm ) = 1 − exp − σmu Here, m denotes the Weibull modulus and σmu the scale parameter of the distribution. As the loading level σ¯ increases, new cracks occur along the specimen at random locations. Their average spacing cs is getting finer until the final average crack spacing csF (see Eq. 6) is reached. The hypothesis behind the stochastic cracking model is that in the stage II the crack spacing is inverse proportional to the cummulative probability distribution of the matrix strength: cs =

1 csF . P(σm )

Using Eqs. (7) and (4) the development of crack spacing from the start to the saturated stage is given as 

−1 σ¯ m cs = csF 1 − exp − . σ¯ R

(8)

¯ mu /Em is the referential comHere, the σ¯ R = Eσ posite stress, i.e. the composite stress at average level of the matrix strength.

Figure 8. Strains between two cracks with overlapping debonding zones.

debonding zones the strain profile becomes triangular (Fig. 8). The macroscopic (effective) strain can be obtained using the standard averaging technique: the effective composite strain ε¯ can be obtained as the average strain in the reinforcement εf  within a sufficiently long segment of the specimen. More precisely, due to the periodicity of the strain field the necessary and sufficient averaging volume is equal to the half of the crack spacing at the instantaneous stress level σ¯ .

Averaging of strains: With the above aligns at hand it is possible to specify the strain profile between two neighbouring cracks at any level of the loading stress σ¯ . For low stress level, the debonding zones do not ovelap so that the strain profile has a trapezoidal shape with strain equality between the debondinz zones (Fig. 7). For higher stress level with overlapping

ε¯ =

2 cs

cs 2

εf (x) dx.

0

For low stress level, with 2δ < cs, the trapezoidal shape of the εf stress profile (Fig. 7) the averaging delivers the effective strain in the form 1 ε¯ = E¯

αδ σ¯ , 1+ cs



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(9)

where (1 − ρ) Em . α= ρ Ef Once the debonding zones start to overlap 2δ > cs (Fig. 8) the averaging of fiber strains renders

1 α cs ε¯ = − σ¯ . (10) ρEf 4δ E¯ Remarks: • The strain hardening curve obtained using Eqs. (9) and (10) does not depend on the the frictional stress level τ and radius of the fiber r. • The stochastic cracking model does not reflect the effect of the reinforcement ratio on the matrix strength. • The described mesoscopic resolution of the damage process into cracking of the matrix and debonding is limited to a simple frictional behavior. In spite of the mentioned limitations, the model can be used in its present form as a meso-scopic representation of the failure process that can be integrated within the macroscopic damage model. Moreover, the limitations regarding the effect of the reinforcement ratio on the level of matrix cracking stress and the microscopic effects in the crack bridge can be included in the model by further enhancements. An extension of the stochastic cracking model with the microscopic representation of a crack bridge has been presented in (Konrad, Jerabek, Vorechovsky, and Chudoba 2006). 6

DIRECTION DEPENDENT REINFORCEMENT RATIO

The application of the described mesoscopic model has been primarily motivated by the need to reflect the effect of the reinforcement ratio on the strainhardening behavior of the composite. The same model shall be employed for reflecting the directionally dependent response of the composite due to the orientation of the reinforcement. Indeed, by introducing an assumption that the effective reinforcement ratio changes with the angle between the crack plane and the reinforcement orientation the closed form of the strain-hardening curve given by Eqs. (9) and (10) could be applied to extrapolate the response measured in a tensile test with aligned reinforcement to a response with inclined reinforcement. For the present paper a simple projection of the reinforcement ratio on the crack plane is used (see Fig. 9). Full alignement of the yarn with the loading direction is assumed. Further, the effective cross sectional area of the yarn is assumed to remain constant in the crack bridge. For a composite reinforced with uni-axial fiber reinforment (no textile fabrics)

Figure 9. plane.

Projection of the reinforcement ratio on the crack

these assumptions lead to the following relationship between the reinforcement ratio and the inclination α of the reinforcement: ρ(α) = ρ0 cos(α),

(11)

with ρ0 representing the reinforcement ratio for α = 0. This mapping accounts for the two extreme cases of full reinforcement ratio ρ(α = 0) = ρ0 and no reinforcement ration ρ(α = π2 ) = 0. The transition between those two extremes is described by cos(α) reflecting an increasing distance between the yarns in the plane parallel to the crack. The assumptions applied for the simple directional mapping of the reinforcement ratio using Eq. (11) do not reflect the local microscopic effects occurring in a crack bridge. In particular, the snubbing, debonding and lateral pressure at the crack edges are not included. As dicussed in (Bruckermann 2007) the complete alignement of the filament with the loading direction holds primarily for the outer filaments. The link can be improved by a more detailed micromechanical resolution of the crack bridge distinguishing individual filament groups. The micromechanical models provide a more realistic description of the crack bridge behavior. They can be linked with the mesoscopic model in terms of the fiber crack bridging stress and the corresponding stress transfer length δ (Konrad, Jerabek, Vorechovsky, and Chudoba 2006). However, even for the simplified projection of the reinforcement ratio to the crack plane the validity of the macroscopic model is significantly extended by

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establishing a link to the mesoscopic failure mechanisms, namely the matrix cracking and debonding.

7

CONCLUSIONS

Figure 10 gives an overview of the entire modeling approach. The tensile test specimen (1) is used to calibrate the mesoscopic stochastic cracking model (SCM). The mesoscale model together with the direction dependent reinforcement ratio introduced in Sec. 6 can then be used to obtain the response for varying inclinations of the reinforcement with respect to the loading direction (2). This information is the basis for the calibration procedure of the directiondependent damage functions (3) of the microplane damage model (MDM). After the model has been calibrated the multi-axial behavior of a TRC-structure can be simulated as exemplified by a depicted quarter of a thin plate with hinge supports at the corners (4). The red/dark color shows the amount of dissipated energy. Summarizing, the modeling framework combines the mesoscopic model for uniaxial tension reflecting multiple cracking and debonding with the macroscopic damage model. The flexibility of the microplane model is exploited in order to construct a direction dependent damage functions. Further adjustments of the introduced directional dependency of the damage specification for TRC are necessary. In particular the kinematic behavior of the model during the localization process must be adjusted in order to reflect the meso-level damage mechanisms occurring in the tested material. The microscopic aspects of the crack bridge behavior, as well as effects connected with unloading and cyclic loading (Konrad, Chudoba, and Kang 2006) have not been considered in the present paper.

ACKNOWLEDGEMENT The work has been supported by Deutsche Forschungsgemeinschaft (DFG) in the framework of the collaborative research center SFB 532 Textilereinforced concrete, development of a new technology. The support is gratefully acknowledged.

REFERENCES

Figure 10. Steps included in the modeling of the cementitious composite.

Aveston, J., G. Cooper, and A. Kelly (1971). Single and multiple fracture, the properties of fibre composites. Proc. Conf. National Physical Laboratories, IPC Science and Technology Press Ltd. London, 15–24. Bazant, Z. and F. Caner (2005). Microplane model M5 with kinematic constraint for concrete fracture and inelasticity. I: Theory. Journal of Engineering Mechanics, ASCE 131(1), 31–40.

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Bazant, Z., F. Caner, I. Carol, M. Adley, and S. Ankers (2000). Microplane model M4 for concrete: I. Formulation with work conjugate deviatoric stress. Journal of Engineering Mechanics, ASCE 126(9), 944–953. Bazant, Z. and B.H. Oh (1985). Microplane model for progressive fracture of concrete and rock. Journal of Engineering Mechanics, ASCE 111, 559–582. Bazant, Z. and P. Prat (1987). Microplane model for brittle material. I: Theory. Journal of Engineering Mechanics, ASCE 113(7), 1050–1064. Bazant, Z.P. and P.G. Gambarova (1984). Crack shear in concrete: crack band microplane model. Journal of Structural Engineering, ASCE 110, 2015–2036. Bruckermann, O. (2007). Zur Modellierung des Zugtragverhaltens von textilbewehrtem Beton. Ph.D. thesis, RWTH Aachen University, Germany. Carol, I. and Z.P. Bazant (1997). Damage and plasticity in microplane theory. International Journal of Solids and Structures 34, 3807–3835. Carol, I., M. Jirasek, and Z. Bazant (2001). A thermodynamically consistent approach to microplane theory. part ii. free energy and consistent microplane stresses. Int. Journal of Solids and Structures 38, 2921–2931. Cusatis, G., A. Beghini, and Z. Bazant (2008). Spectral stiffness microplane model for quasib-rittle composite laminates I: Theory. Journal of Applied Mechanics 75(2). Cuypers, H. and J. Wastiels (2002). Application of a stochastic matrix cracking theory on a E-GFRCC. In Tenth Eourpean Conference on Composite Materials, Brugge, Belgium.

Jirasek, M. (1999). Comments on microplane theory. In Mechanics of Quasibrittle Materials and Structures, pp. 55–77. Hermes Science Publications. Konrad, M., R. Chudoba, and B. Kang (2006). Numerical and experimental evaluation of damage parameters for textile reinforced concrete under cyclic loading. In ECCM 2006 III European Conference on Computational Mechanics. Konrad, M., J. Jerabek, M. Vorechovsky, and R. Chudoba (2006). Evaluation of mean performance of cracks bridged by multifilament yarns. In M.B. Meschke, de Borst (Ed.), EURO-C 2006: Computational Modelling of Concrete Structures, pp. 873–880. Kuhl, E. and E. Ramm (2000). Microplane modelling of cohesive frictional materials. European Journal Mechanics and Solids 19, 121–134. Kuhl, E., P. Steinmann, and I. Carol (2001). A thermodynamically consistent approach to microplane theory. Part II. Dissipation and inelastic constituive modelling. Int. Journal of Solids and Structures 38, 2933–2952. Leukart, M. and E. Ramm (2003). A comparison of damage models formulated on different material scales. Computational Materials Science (28), 749–762. Stroud, A.H. (1971). Approximate calculation of multiple integrals. Prentice-Hall, Englewood Cliffs, N.J. Widom, B. (1966). Random sequential addition of hard spheres to a volume. J. Chem. Phys. 44, 3888–3894.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

A statistical model for reinforced concrete bond prediction Z. Dahou University of Béchar, Department of Civil Engineering, Béchar, Algeria

Z.M. Sbartaï Université Bordeaux, Talence, France

A. Castel Université de Toulouse UPS, INSA, LMDC, Toulouse cedex, France

F. Ghomari University of Tlemcen, Department of Civil Engineering, Tlemcen, Algeria

ABSTRACT: This study proposes a statistical approach, based on artificial neural network model, for modelling the bond between conventional ribbed steel bars and concrete. Then and according to RILEM test configuration (Rilem 1970), the ultimate pull-out load is predicting from the concrete mix constituents and from the steel bar diameter. More than hundred pullout tests were carried out in order to investigate experimentally the bond behaviour between three concrete mixes, with different constituent proportions, and two different diameters of ribbed bars. These experimental results show that the pull-out load is as affected by concrete mixes as strength of concrete and the bar diameter. On the whole, six input data were used for the ANN model and the ultimate pull-out load was the output data. The network was Multi-Layer-Perceptron trained according to a back-propagation algorithm. The agreement between experimental results of other authors and simulation of the proposed model proves a satisfactory accuracy. 1

INTRODUCTION

A fundamental property of reinforced concrete is the bond between the reinforcement and the concrete. In anchorage zones, the two important aspects that control the bond are the force transfer mechanism between the reinforcement and the surrounding concrete and the capacity of the concrete to resist reinforcement pull-out. Chemical adhesion, friction and mechanical actions are the mechanisms that assure bond efficiency. The contribution of each of these mechanisms depends on the surface and geometry of the reinforcement. In the case of ribbed bars, the force transfer is mainly governed by the blocking of the ribs in the concrete (Eligehausen et al. 1999). Thus, the bond depends on numerous and various factors, which basically concern the reinforcing bars, the concrete and the stress state in both the reinforcing and the surrounding concrete. Generally, bond strength can be assessed by different test in laboratories. Also in codes of practice, equations are proposed for the bond evaluation. This relations, referred usually to a correlation between the strength of concrete and the bond stress.

The main goal of this study was to develop a bond model allowing the pull out load prediction from concrete mix constituents, concrete maturity and the ribbed steel bar diameter. So, an experimental program was developed by conducting pull-out tests on ribbed bars with a nominal diameter of 10 mm and 12 mm. Three concretes (two grades 40 and one grade 30) were studied. The specimens were tested at various ages ranging from 1 to 28 days. Based on these experimental data, an Artificial Neural Network (ANN) was used in order to predict the ultimate pull-out load. The basic strategy for implementing a neural based model of material behaviour is to train a neural network on the results of a series of experiments on a material. If the experimental results contain the relevant information about the material behaviour, then the trained neural network will contain sufficient information to qualify as a material model. Such a trained neural network would not only be able to reproduce the experimental results on which it was trained but should also, through its generalization capability, be able to approximate the results of other experiments (Ghaboussi et al. 1991).

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2

ARTIFICIAL NEURAL NETWORKS

An artificial neural network (ANN) can be represented as a simplified model of the nervous system consisting of a large number of information processing elements. ANN has been successfully applied to a wide range of engineering and scientific applications such as classification, pattern recognition, process control and prediction based on historical data. The fundamental principle of ANN has been reported by several researchers (Dreyfus et al. 2002), (Jodouin 1994), (Bishop, 1995), (Rafiq et al. 2001). The basic element of the method is the artificial neuron called the Perceptron, which is a mathematical model of a biological neuron as presented in (Fig. 1). Artificial neurons connected together form a network. Depending on the way in which the neurons are connected, many types of artificial neural networks are distinguished (Dreyfus et al. 2002), (Jodouin 1994). The Multy-Layer Perceptron (MLP) is the most popular (Yeh 1998), (Dias et al. 2001), (Nehdi et al. 2001), and is adopted in this work. The MLP is composed of an input layer, one or more hidden layers and an output layer. The relationship between the inputs (xi ) and the output (y) can be written as follows:   n  (1) y =F b+ w i xi i=1

where F is an activation or transfer function. Several functions are available but the sigmoid is the must common (Fig. 2). Upper and lower limits of output from this function are generally 1 and 0 respectively. It is therefore recommended to normalise the input and output data before presenting them to the network. The vectors wi and b design weights and bias vectors. They governed the rate of information transmitted between input and output. These network parameters are optimised by the use of a training process. Several training algorithms exist but back-propagation (BP) commonly provides satisfactory results (Rumelhart et al. 1987). BP is a gradient descent algorithm which consists of changing the weights and bias according to the negative of the error function. They are expressed

Figure 2.

by equations 2 and 3. This process must be repeated until the network error is minimized. To enhance the generalization capacity of the network, a crossvalidation method is generally used which consists of testing the network on new data at each iteration (Weigend et al. 1991), (Morgan et al. 1990). These new data are not used in the training processes and are composed of approximately 20% of the database. Evaluation of the network prediction must therefore be based on the testing data prediction. wi+1 = wi − η∇Ei/w

(2)

bi+1 = bi − η∇Ei/b

(3)

wi+1 is the corrected weight value, wi is the weight value at iteration i, η refers to the learning rate, ∇i is the error gradient computed at iteration i: ∂Ei (4) ∂w Ei designates the Root Mean Square Error (RMSE) and it is defined by:   N 1  |en |2 (5) Ei =  N ∇Ei/w =

n=1

3

EXPERIMENTAL PROCEDURES

The standard RILEM pull-out test was used to study the anchoring capacity of the rebar in the concrete. The influence of the diameter of the ribbed bars, the concrete mixes and maturity were studied experimentally. Concrete mix variables were the type of cement (R for fast hydration and N for normal hydration), the water to cement ratio (W/C), the gravel to sand ratio (G/S) and the crushed to rolled gravel ratio (Gc /Gr ). Only standard 52.5 R (MPa) and 52.5 N (MPa) grade Portland cement was used. 3.1

Figure 1.

Sigmoid function.

Concrete mixtures

Three concretes (two grade 40 MPa and one grade 30 MPa) were studied. Table 1 summarizes the mix proportions. The ratio of crushed gravel to rolled

Artificial neuron (Perceptron).

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Composition of the vibrated concretes.

Table 2.

425

375

710 532.5 532.5 185 0.435 1.5 1

755 336 790 187.5 0.5 1.5 2.35

Digital VC30 (kg/m3 )

Silico

Bond length L

VC40b (kg/m3 )

325 811 382 731 195 0.6 1.37 1.91

Total length: 2 x

Cement 52.5 R Cement 52.5 R Sand 0/4 Rolled Gravel 4/10 Crushed Gravel 10/14 Total water W/C G/S Gc/Gr

VC40a (kg/m3 )

5 (no

Table 1.

Fixed Rebar

Plastic

Mechanical properties of the concretes.

Concretes

Compressive strength MPa

Elastic modulus GPa

VC40a VC40b VC30

43.9 43.8 36.8

32.0 34.2 25.1

F Figure 3. tions.

gravel, ranging from 1 to 2.35, was the main difference between the two 40 MPa grade concretes. Also, the water to cement ratio was significantly higher for concrete VC40b. The specimens were removed from their moulds 24 hours after casting and stored for 28 days in a controlled environment (T◦ = 20◦ C, RH% = 60%). The compressive strength and the instantaneous elastic modulus were measured at 28 days on concrete cylinders (diameter = 110 mm, height = 220 mm). Table 2 shows the mechanical properties of the tested concrete.

Pull-out test according to RILEM recommenda-

The bond strength was calculated assuming a uniform distribution of bond stresses along the bond length. It was calculated from the ultimate pull-out load using the relation (6): τu =

Fu π ·φ·L

(6)

where τu is the ultimate bond stress (MPa), φ is the rebar diameter (mm), L is the bond length (mm) and Fu is the ultimate pull-out load (N).

4

TESTS RESULTS AND PARAMETERS OF THE BOND MODEL

3.2 Pull-out tests

4.1

3.2.1 Description of the tests specimens The pull-out test was carried out according to Figure 3. The experimental set up was similar to the pull-out test described in the RILEM recommendations (Rilem 1970). The specimens had a concrete cross-section of 10 × 10 cm. There were two possible diameters for the ribbed bars used: 10 mm and 12 mm. All bars had a yield strength of 500 MPa.

In each pull-out test, the slip of the free end of the reinforcement was recorded on a plotting table versus the load applied. For the three concrete, pull-out tests were performed for each ribbed bar diameter (10 mm and 12 mm) at different concrete maturities. The VC40a specimens were tested at 5, 7, 14 and 28 days. For the VC40b, the tests were conducted at 1, 3 and 11 days. Experiments on VC30 concrete were carried out both at early age and at 7, 14 and 28 days. For each configuration (concrete mix and diameter of the bar), four specimens were tested. For all specimens, failure occurred at the interface between the reinforcing bar and the surrounding concrete and was due to the steel pull-out. For example, the curve of the Figure 4 shows the typical load-slip behaviour obtained. It correspond to

3.2.2 Testing The pull-out bond specimens were tested up to bond failure by applying progressively a tensile load to the end of the bar. It was pulled from the concrete cube by controlling the applied force, at rate of 0.1 kN/s. To plot the bond stress—slip relationship, the slip was measured at the free end of the bar by a digital transducer with an accuracy of 0.001 mm (Fig. 3).

Load-slip curves

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a VC30 concrete specimen, with a 10 mm or a 12 mm ribbed bar, carried out after 14 days of curing. The curves show a different behaviour in relation to the diameter. Due to the simplified uniform bond stress distribution assumed, this difference do not appear clearly for the ultimate bond stress (Fig. 5) given by relation (6). For the whole set of specimens tested, more than 95% showed a standard deviation of less than 3.5 KN (Fig. 6).

4.2

Effect of the parameters of materials

In turn for the concrete and the bars, the impact of the individual parameters was studied to establish their influence on the ultimate pull-out load. For both bars tested the Figures 7–9, show the variation of the ultimate pull-out load with respect to the age, for the three concretes mixtures respectively. The results reported correspond to the average ultimate pull-out load obtained from four tests. For all the concrete mixes, the ultimate pull-out load increased from 10 kN to 30 kN for the specimens with the 10 mm

35 HA10

HA12

60

25

HA12

HA10

22

25

50 Ultimate Pull-out load (kN)

Pull out load (kN)

30

20 15 10 5 0

40 30 20 10

0

1

2

3

4

5

6

7

8

9

10

Slip (mm)

0 1

Figure 4. Pull-out-load versus slip curves of the two deformed bars (VC30 concrete used and specimens tested at 14 days).

4

7

10

13

16

Age (days)

19

28

Figure 7. Average ultimate pull-out load versus age for the VC40a concrete. 60

16 HA10

HA12

HA12

HA10

22

25

50

12

Ultimate Pull-out load (kN)

Bond stress (MPa)

14

10 8 6 4

40 30 20 10

2 0 0

1

2

3

4

5

6

7

8

9

0

10

1

Slip (mm)

Figure 5. Bond stress versus slip curves of the two deformed bars (VC30 concrete used and specimens tested at 14 days).

4

7

10

13

16

Age (days)

19

28

Figure 8. Average ultimate pull-out load versus age for the VC40b concrete. 60 HA12

30%

HA10

50 Ultimate Pull-out load (kN)

25%

Frequency

20% 15% 10%

40 30 20 10

5%

0

0% 0,5

1

1,5

2

2,5

3

3,5

4

4,5

5

1

5,5

4

7

10

13

16

Age (days)

19

22

25

28

Standard Deviation [KN]

Figure 6.

Frequency distribution of the standard deviation.

Figure 9. Average ultimate pull-out load versus age for the VC30 concrete.

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diameter bars and from 15 kN to 50 kN for the specimens with the 12 mm diameter bars. The ultimate pullout load measured on the VC30 concrete remained lower than those obtained on the VC40 specimens. So, the influence of the ribbed bar diameter and the concrete compressive strength (VC40 and VC30) was clearly apparent. Also, the ultimate pull-out load increased significantly with the concrete maturity. After 1 day and 3 days, grade 40 MPa concretes, cast with fast hydration cement type R, reached about 50% and 75% respectively (average value obtained on the 10 mm and 12 mm diameter steel bars) of the bond strength measured after 28 days. For the grade 30 MPa concrete, cast with normal hydration cements type N, only about 35% and 55% (average value obtained on the 10 mm or 12 mm diameter steel bars) of the bond strength measured after 28 days was reached after 1 day and 3 days respectively. It is well known that concrete constituents, such as the type of cement, the water-cement ratio and the granularity, influence the strength of concrete, which is a key factor for enhancement of the bond with ribbed steel bars. The water to cement ratio is obviously a key parameter influencing the bond strength but results obtained on both VC40 show that the ratio of crushed gravels to rolled gravels is also significant. VC40a and VC40b have water to cement ratios equal to 0.435 and 0.5 respectively (Table 1) and, according to Figures 7 and 8, show almost the same bond strength. This is due to the crushed to rolled gravel ratio, which is significantly higher for the VC40b. Therefore, both grade 40 MPa concretes had the same compressive strength after 28 days (Table 2) in spite of their different watercement ratios. Only grade 52.5 MPa cement was used in this experimental program, but two types were studied: N (Normal hydration) and R (Fast hydration). According to the experimental results, this factor influences the compressive strength of the concrete at early age and, thus, the bond strength. Finally, as the gravel interaction with the steel ribs significantly increases the bond strength, the characteristics of the gravels (sizes and types) affect both the compressive strength (Guinea et al. 2002) and the bond strength and will therefore be taken into account in the development for the statistical ANN bond model.

4.3

Instead of considering the concrete compressive strength as input data to the ANN model, the water to cement ratio (W/C), gravel to sand ratio (G/S), and crushed to rolled gravel ratio (Gc/Gr) were used. Moreover, the type of cement (R: fast hydration or N: normal hydration) was added because it influences the concrete strength at early age. The age of the concrete and the ribbed steel bar diameter were directly used as ANN input factors. Thus, a total of six factors were considered as input data of the ANN bond model.

5

DEVELOPMENT OF THE ANN PULL-OUT MODEL

The experimental set of data was divided into two sets: a training set and a testing set. The first one represented 80% of the whole database and the reset was reserved for the testing process. MATLAB software was used for the numerical implementation of the ANN and the training algorithm. The network structure: six inputs, one hidden layer of 10 neurons and one output were adopted. The structure of the network is shown in Figure 10. The variations of training RMSE and testing RMSE are plotted with respect to the number of epochs in Figure 11.The training process in this figure was limited when the optimum error was reached. It’s corresponding to a minimum error of the testing data of about 1.2 × 10−3 achieved after 2000 epoch. The performance of the model was assessed mainly by analyzing the prediction ability related to data set. bH

[W]

W/C

1

[Z]

G/S

2 b0

Gc /Gr

3 Fu

Cement

…

9

Summary and ANN input parameters

Experimental results show that the concrete compressive strength, the concrete maturity and the diameter of the bar tested have a significant effect on the ultimate pull-out load. Thus, the input data of the proposed ANN-bond model must be selected to take into account the influence of these factors on the pull-out load, chosen as the output of the ANN model.

10 age INPUT

Figure 10.

HIDDEN LAYER

OUTPUT LAYER

Structure of the network bond model.

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The correlation coefficient (R) is equal to about 0.98 for the training set and to about 0.97 for the testing set. Moreover, 90% of the training patterns were located near the perfect prediction line with an absolute accuracy range of ±3 kN. For the testing set, 81% of the patterns showed the same accuracy. Only 5% of the tested patterns presented absolute errors exceeding ±5 kN. 6

Experimental results obtained by other researchers (Castel et al. 2006), (Daoud et al. 2002), (Koning et al. 2001), (Zhu et al. 2004), (Xiao et al. 2007) were considered in this section in order to test the validity and the accuracy of the proposed ANN bond model. Table 3 gives the specifications of concrete and reinforced for each research. As some experimental factors did not fit with our experimental range used for the development of the ANN bond model, the following strategy was applied:

Figure 11. Model network training and testing RMSE variation versus the number of epochs. 52 R² = 0.97

ANN Pull-out load (kN)

48 44 40 36 32 28 24 20

– In the case of concrete mixes incorporating only crushed or only rolled gravels, the crushed to rolled gravel ratio could not be taken to be equal to infinity or zero. So the bond model was used with this ratio fixed at the smallest value (Gc/Gr = 1) for mixes with rolled gravel and the largest value (Gc /Gr = 2.35) for concrete with crushed gravel. – For the application of the proposed model to concrete mixes with lower grades of cement, we assumed that this factor had the same effect on the bond strength as on the compressive strength of concrete.

16 12 8 4 0 0

4

8

12

16

20

24

28

32

36

40

44

48

52

Experimental Pull-out load (kN)

ANN Pull-out load (kN)

Figure 12. ANN-predicted versus measured ultimate pullout load—Training data set. 52 48 44 40 36 32 28 24 20 16 12 8 4 0

VALIDATION OF THE ANN MODEL ON EXPERIMENTAL RESULTS FROM THE LITERATURE

R² = 0.94

The Bolomey law (7) (Dreux et al. 1998), assumes a linear relationship between the concrete compressive strength (fc ) and the cement grade (FCE ) by taking the cement to water ratio (C/W) and the coefficient (G), Table 3. A detail of Concretes mixes and ribbed bars for referenced research.

0

4

8

12

16

20

24

28

32

36

40

44

48

Authors

52

Cement W/C G/S

φ Bond Gc /Gr [mm] length

Experimental Pull-out load (kN)

Castel et al. Figure 13. ANN-predicted versus measured ultimate pullout load—Testing data set.

Predicted and measured values are presented and compared in Figure 12 for the training set and Figure 13 for the testing set. They show that predicted and measured ultimate pull-out loads are well correlated.

32.5 R 52.5 R Daoud et al. 52.5 R Konig et al. 42.5 R Zhu et al. 42.5 N 42.5 N Xiao et al. 32.5 R

0.5 0.43 0.57 0.55 0.68 0.68 0.43

1.51 1.49 1.93 2.17 1.15 1.15 2.24

2.58 1 C* R* C* C* C*

12 12 16 10 12 20 10

5φ 5φ 5φ 5φ 10φ 6φ 5φ

* The letters C or R specifies that only crushed or rolled gravel was used.

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Castel et al. Daoud et al. Konig et al. Zhu et al. Xiao et al.

Experimental τu [MPa]

(*)

(**)

Variation [MPa]

12.70 22.47 15.11 20.17 10.24 7.89 17.38

20.07 22.11 13.34 26.05 10.67 7.97 28.31

12.04 / / 20.84 8.54 6.38 16.99

0.66 0.36 1.77 0.67 1.70 1.51 0.39

* Corresponding to calculated values. ** Corresponding to equivalent values.

which characterizes the quality of gravel, as constant. In our case, G was equal to 0.45 and, for a fixed cement to water ratio, grade 42.5 MPa and grade 32.5 MPa cement led to a reduction in compressive strength of about 20% and 40% respectively in comparison to a concrete cast with grade 52.5 MPa cement. Therefore, a 20% or 40% reduction was applied to the calculated ANN pull-out load when the cement grade used was 42.5 or 32.5 MPa respectively. The pull-out load obtained is labelled equivalent pull-out load.  C − 0.5 (7) fc = G · FCE W

7

PARAMETRIC STUDY OF THE ANN BOND MODEL

45 40 35 30 25 20 15 10 5 0 60

Barre diameter 10 mm Barre diameter 12 mm 0.435 0.45 0.5 0.55 0.6 (a)

Barre diameter 10 mm Barre diameter 12 mm 0.435 0.45 0.5 0.55 0.6 (b)

50 40 30 20 10 0

Barre diameter 10 mm Barre diameter 12 mm 0.435 0.45 0.5 0.55 0.6 (c)

Figure 14. Comparison of the effect of water-to-cement ratio on value of the pull-out load for two ribbed bars. (a) Concrete maturity 3 days (b) Concrete maturity 7 days (c) Concrete maturity 28 days.

45 N cement

Ultimate Pull-out Load (kN)

Table 4 shows ultimate bond stress measured experimentally by different researchers and those estimated by the proposed model. The comparison shows that the model is able to predict the bond strength with acceptable accuracy. Although the coefficients characterizing the granularity were, in the majority of the concrete mixes reported in table 3, outside our experimental margin, the model reproduced the experimental ultimate bond stress with a maximal variation of 2 MPa.

Ultimate Pull-out Load (kN)

Authors

Ultimate Pull-out Load (kN)

Estimated τu [MPa]

40 35 30 25 20 15 10 5 0

Ultimate Pull-out Load (kN)

Table 4. Evaluation of the bond model for predicting ultimate bond stress.

R cement

40

35

30

25

20 0

The ANN bond model can be used to simulate the effects of concrete mix on the ultimate pull out load. The simulation results are shown in Figures 14 to 17. Firstly, figures 14a, b, and c compare the ultimate pull-out load for both diameters for different value of the water to cement ratio. The ultimate pull-out load, and so the bond strength, increase significantly with the decrease of the water to cement ratio. At 28 days, for a water-cement ratio increasing from 0.435 to 0.6, the decrease in ultimate pull-out load is 30% for the 10 mm diameter bar and about 35% for the 12 mm diameter bar. So, the water to cement ratio has the

2

4

6

Age (days)

8

10

12

14

Figure 15. Effect of type of cement (N or R) on ultimate pull-out load.

same effect on both the bond strength and compressive strength. Figure 15 shows the variation in ultimate pullout load versus the type of 52.5 MPa cement. Fast hydration cement (R) leads to a significant increase in the bond at early age compared to normal hydration

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The concrete mixes used did not contain any chemical or mineral admixtures. The experimental results provided a database for implementing a Neural Network model for the ultimate pull-out prediction. The following conclusions can be drawn from the present study:

60

Ultimate Pull out Load [KN]

3 days

7 days

14 days

28 days

50 40 30 20

– The results show that artificial neural networks can be implemented to model the experimental relationship between the ultimate pull-out load and parameters such as water-cement ratio, gravel-sand ratio, crushed-rolled gravel ratio, type of cement, diameter of bar and concrete maturity. The bond strength can then be deduced according to the RILEM method. – The statistical model, trained, tested and validated according to the large laboratory database, shows good accuracy in the ultimate pull-out load prediction and a high capacity for generalization. – The ANN model was based only on experimental results with bar diameters of 10 mm or 12 mm, the prediction of bond strengths measured by different researchers on specimens with larger diameter bars was also accurate.

10 0 1,36

1,38

1,4

1,42

1,44

1,46

1,48

1,5

1,52

Gravel-sand ratio

Figure 16. ratio.

Ultimate pull-out load versus gravel to sand

60 1

1,5

2

2,35

Ultimate pull out load (kN)

50 40 30 20 10

Further experiments including other input concrete parameters could extend the field of application of the proposed ANN bond model. The influence of the cement grade could be investigated experimentally, which may enhance the accuracy of the prediction. Concrete mixes including water reducers or plasticizers could be studied. A wider variation range of the experimental input parameters, such as water to cement ratio or gravel to sand ratio, could be also useful.

0 3

7

14

28

Age (days)

Figure 17. Effect of crushed to rolled gravel ratio on ultimate pull-out load.

cement (N). This accelerated hydration effect on bond strength disappears with increasing concrete maturity. For concrete mixtures with gravel to sand ratios increasing from 1.37 to 1.5, the ultimate pull-out load calculated increases of about 17% for any concrete maturity (Fig. 16). An increase in crushed gravel proportion also leads to a significant improvement of the ultimate pull-out load (Fig. 17). After 28 days, an increase in the Gc /Gr ratio from 1 to 2.35 leads to an increase in the ultimate pull-out load of about 18%. This result is in accordance with other studies (Appa et al. 2002) which have shown that concrete shear strength, and thus bond strength with ribbed steel bars, increases as the roughness of the aggregate increases. 8

CONCLUSIONS

The bond between concrete and steel bars in reinforced concrete structures is governed by numerous factors. Experimental work has been performed to study how the ultimate pull-out load, according to the RILEM recommendation, is affected by the concrete mix proportions, the concrete maturity and the diameters of the ribbed bars. Two grades of concrete strength, of about 30 and 40 MPa, were considered.

REFERENCES Appa Rao G, Raghu Prasad BK. 2002. Influence of the roughness of aggregate surface on the interface bond strength. Cement and Concrete Research 32(2):253–257. Bishop CM. 1995. Neural Networks for Pattern Recognition. Oxford University Press. Castel A, Vidal T, Viriyametanont K, Raoul F. 2006. Effect of Reinforcing Bar Orientation and Location on Bond with Self-Consolidating Concrete. ACI structural journal 103(4):559–567. Daoud A, Lorrain M, Elgonnouni M. 2002. Résistance à l’arrachement d’armatures ancrées dans du béton autoplaçant. Materials and Structures 35(7):395–401. Dias WPS, Pooliyadda SP. 2001. Neural networks for predicting properties of concretes with admixtures. Construction and Building Materials 15(7):371–379. Dreux G, Festa J. 1998. Nouveau guide du béton et de ses constituants. Eyrolles, 8e edition. Dreyfus G et al. 2002. Réseaux de neurones-Méthodologie et applications-. Eyrolles. Eligehausen R, Bigaj van Vliet A. 1999. Bond Behaviour and models. In: Fib Bulletin No.1, vol.1: Introduction- Design Process-Materials; 161–187.

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Ghaboussi J, Garett JH, Wu X. 1991. Knowledge-based modelling of materials behaviour with neural networks. Journal of Engineering Mechanics 117(1):132–153. Guinea GV, El-Sayed K, Rocco CG, Elices M, Planas J. 2002. The effect of the bond between the matrix and the aggregates on the cracking mechanism and fracture parameters of concrete. Cement and Concrete Research 32(12):1961–1970. Jodouin JF. 1994. Les réseaux de neurones- principe et définitions-. Paris : Hermes. Koning G, Holschemacher K, Dehn F, Weisse B. 2001. Selfcompacting concrete – time development of material properties and bond behaviour. In: Ozawa K, Ouchi M, editors. Proceedings of the second RILEM international symposium on self-compacting concrete. Kochi University of Technology, Japan: COMS Engineering Corporation; 507–516. Morgan N, Bourlard H. 1990. Generalization and Parameter Estimation in Feedforward Nets: Some Experiments. In: Conference on Advances in Neural Information Processing Systems (NIPS), vol. 2., p. 630–637. Nehdi M, El Chabib H, El Naggae H. 2001. Predicting Performance of Self-Compacting Concrete Mixtures Using Artificial Neural Networks. ACI Materials Journal 98(5): 394–401. Ni HG, Wang JZ. 2000. Prediction of compressive strength of concrete by neural networks. Cement and Concrete Research 30(8):1245–1250.

Rafiq MY, Bugmann G, Esterbrook DJ. 2001. Neural network design for engineering applications. Computers & Structures 79(17):1541–1552. RILEM. 1970. Essai portant sur l’adhérence des armatures du béton : essai par traction. Materials and Structures 3(3):175–178. Rumelhart DE, Hinton GE, Williams RJ. 1987. Learning internal representations by error propagation. In: Parallel Distributed Processing-Vol.1, Eds: Cambridge, MA: MIT Press, Chapitre 8. Weigend A, Rummelhart D, Huberman B. 1991. Generalization by Weight Elimination with Application to Forecasting. In: Conference on Advances in Neural Information Processing Systems (NIPS), vol. 3., p. 875–882. Xiao J, Falkner H. 2007. Bond behaviour between recycled aggregate concrete and steel rebars. Construction and Building materials 21(2):395–401. Yeh IC. 1998. Modeling of strength of high-performance concrete using artificial neural networks. Cement and concrete research 28(12):1797–1808. Zhu W, Sonebi M, Bartos PJM. 2004. Bond and interfacial properties of reinforcement in self-compacting concrete. Materials and Structures 37(7):442–448.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Introduction of an internal time in nonlocal integral theories Rodrigue Desmorat & Fabrice Gatuingt LMT-Cachan, ENS Cachan/CNRS/Université Paris 6/PRES Universud Paris, Cachan, France

ABSTRACT: Nonlocal damage models are now commonly used. Their ability to make finite element computations with softening laws robust and mesh independent is well established. There are nevertheless still a few open questions as the identification of the so-called internal length lc , as its loading or its boundary independency. One focuses in the present note on the boundary conditions problem in nonlocal integral approaches and on the feature that points separated by a crack or a hole should not interact as they do in Pijaudier-Cabot and Bazant initial nonlocal theory. Instead of defining an internal length one proposes to make the nonlocal weight function as a function of the information time propagation of an elastic wave normalized by an internal time τc . 1

INTRODUCTION

The effect and the formulation of boundary conditions— such as free edges, notches and initial cracks— remain an open question for nonlocal models. The main drawback of the classical nonlocal integral theory (Pijaudier-Cabot & Bazant (1987)) consists in the nonphysical interaction, through the nonlocal averaging process, of points across a crack or a hole. The definition of natural boundary conditions of vanishing strain normal derivative at a free edge is still under discussion for gradient formulations (Aifantis (1987); Peerlings et al (1996)). The continuous nucleation of a crack of zero thickness is not so simple as the thickness of a localization band is more or less proportional to the internal length introduced. Local behavior along free edges—i.e. with a vanishing internal length—has been obtained by some authors (Pijaudier-Cabot et al (2007); Krayani et al (2009)). The consideration of an internal length evolving with damage (Geers et al (1998); Pijaudier-Cabot et al (2004)) seems a way to properly bridge Damage Mechanics and Fracture Mechanics as the internal length may then vanish for large values of damage. One attempts here to propose a solution—bringing also questions—to these main difficulties. The idea is to keep the nonlocal averaging process but to quantify the distance between points as an effective distance, i.e. as a distance function for instance of the geometry and the matter encountered between interacting points. One proposes to define such an effective distance with respect to a dynamic process: how information or wave propagates between interacting points. This can be made through the introduction of an internal time τc , constant, instead of a internal length c , measured as evolving. Dynamics is important to define a link between a characteristic time and a characteristic length, either

when viscosity is introduced (Needleman (1988); Allix & Deü (1997)) or when the physical defects obscuration phenomenon encountered in high speed dynamics and multi-fragmentation is taken into account (Denoual C. & Hild (2000)). Some authors even introduce the classical nonlocal theory by comparing the characteristic wavelength of the deformation field to an intrinsic length of the material (Jirasek (2003)), still a dynamics vocabulary. Wave propagation will give us information on interacting/non interacting points for the definition of the nonlocal averaging. One does not intend here to solve the problem in the general case and focuses only on the nonlocal integral theories. 2

NONLOCAL THEORY WITH INTERNAL TIME

Softening constitutive equations classically lead to spurious dissipation modes and to mesh dependency. The need of the definition—and the introduction—of an internal length in the models is now established. But in which form? In a gradient form? In an integral form? From an internal viscosity (delay-damage) combined then with dynamics? The main idea of such regularizations is to average the variable—i.e. the thermodynamics force, denoted next V in the general case or Y or ˆ for damage—responsible for the strain localization. The procedure to define a nonlocal variable V nl from its local expression V introduces a characteristic length lc considered as a material parameter. 2.1

Nonlocal integral theories—Boundary effect

The classical nonlocal theory (Pijaudier-Cabot & Bazant (1987)) uses the integral

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   1 x − s V(s)dVs V (x) = ψ Vr  lc    x − s dVs Vr (x) = ψ lc  nl

This illustrates the need to deal with such a boudary condition effect but also to make equivalent a real crack and a highly damaged zone.

(1) (2)

2.2 over the whole domain  in order to define a nonlocal quantity V nl —built from the local variable V—to be used in thermodynamics state or evolution laws. In Eq. (1) ψ is the nonlocal weight function, positive and decreasing with respect to the distance x − s between the considered point x and all the structure points s. The normalizing factor is denoted Vr (x) and lc is an internal or characteristic length. Two classical weight functions are (Bazant & Jirasek (2002)). ψ(ξ ) = e



− 12 ξ 2

or ψ(ξ ) = 1 − ξ

 2 2

In order to solve the problem, one proposes in this work to keep the nonlocal integral framework and to consider the information time propagation τxs between points x and s instead of the classical distance x − s. A nonlocal theory with internal time is then simply defined by replacing Eq. (1) by:    1 τxs V(s)dVs ψ Vr  τc    τxs Vr (x) = dVs ψ τc 

V nl (x) =

(3)

The expressions (1) do then define the same contribution to V nl of points x and s – across a crack than any points x and s separated by the same distance x − s (for instance path x1 –x6 versus path x2 –x5 of figure 1); this flaw has been pointed out and studied recently by Jirasek & Marfia (2006) and Pijaudier-Cabot et al (2007). A numerical averaging adaptation neglecting the communication between some integration points can be found in (de Vree et al (1993)); – across a damaged zone: for example the interaction between points x3 –x4 is not affected by the zone at a damage level D0 , i.e. at a Young’s E(1 − D0 ) much lower than the Young’s modulus of the virgin material; much less studies address this point.

x2

x3

x1

x2

x3

x6

x5

x4

x6

x5

x4

Specimen (c) x1

x2

x3

D=D0 x6 Figure 1.

x5

(5)

Example: nonlocal damage models

For example, local isotropic damage theories for quasibrittle materials define

Specimen (b)

x1

(4)

with τxs the information propagation time (Desmorat et al (2009)) taken next as the time for a wave to propagate from point x to point s and τc a material parameter. As the wave time propagation τsx from point s to point x is identical to τxs the nonlocal weight function thus built is symmetric. Such a nonlocal averaging process may apply to physical laws of different nature. Concerning continuum mechanics and elasticity, plasticity and/or damage, the variables V are often equivalent strains, accumulated plastic strains or strain energy densities. 2.3

Specimen (a)

Time instead of distance

˜ with – the damage as a loss of stiffness D = 1 − E/E ˜E (resp. E) the damaged (resp. initial) Young modulus, equation also rewritten in 3D as the elasticity law coupled with isotropic damage σ = E (1 − D) : 

(6)

with σ ,  and E respectively the stress, the strain and the Hooke tensors; – the damage evolution as a function of a local variable V either equal to the thermodynamics force Y = 12  : E :  associated with damage (Marigo model) √ or to an equivalent strain as Mazars strain ˆ = + : + , D = g(V)

x4

(7)

with g a nonlinear function and ·+ the positive part of a tensor (in terms of principal values).

Notched and damaged specimens.

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The non local damage law is simply written D = g(V nl )

– Induced damage anisotropy governed by the positive extensions, (8)

D = λ˙ ˜ 2+

instead of Eq. (7) with the nonlocal averaging process (4). For concrete, the microcracks due to tension are mainly orthogonal to the loading direction, when the microcracks due to compression are mainly parallel to the loading direction. The damage state has then to be represented by a tensorial variable D either a fourth rank tensor or a second rank tensor (Murakami & Ohno (1978); Cordebois & Sidoroff (1982); Murakami (1988); Lemaitre & Desmorat (2005); Lemaitre et al (2009)). The use of a second order damage tensor is more convenient for practical applications (as well as for the material parameters identification) and this is the choice which has been made. The damage anisotropy induced by either tension or compression is simply modeled by the consideration of damage evolution laws ensuring a damage rate proportional to the positive part of the strain tensor, i.e. a damage governed by the principal extensions (Mazars et al (1990)). The full set of constitutive equations for the local anisotropic damage model reads (Desmorat (2004); Desmorat et al (2007); Souid et al (2009))

There are 5 material parameters introduced: E, ν for elasticity, κ0 as damage threshold and A and a as damage parameters. The model is simply made nonlocal, either from the classical integral theory or from the new integral nonlocal with internal time theory, by replacing Mazars equivalent strain ˆ by its nonlocal form in the damage criterion function, becoming f = ˆ nl − κ(tr D )

 D σ˜ = (11 − D )−1/2σ D (11 − D )−1/2

1 tr σ  + − −tr σ  1 3 1 − tr D

x − s = c τxs

(16)

with c the information celerity taken as the wave speed. Then, if the internal time is related to the internal length as lc = c τc the weight functions are equal,     x − s τxs ψ =ψ (17) lc τc

(10)

where(.)D stands for the deviatoric part and . for the positive part of a scalar. – Damage criterion, f = ˆ − κ(tr D )

Effective or ‘‘dynamic’’ distance—Vanishing internal length

In a plain and uncracked medium the internal length and internal time concepts are equivalent as

(9)

with E the Young modulus and ν the Poisson ratio. – Effective stress,

(14)

instead of Eq. (11) with the nonlocal averaging process (4),    1 τxs ˆ nl (x) = ψ (15) ˆ (s)dVs Vr (x)  τc 2.4

ν 1+ν σ˜ − tr σ˜ 1 E E

(13)

In the rate independent formulation, the damage multiplier λ˙ is determined from the consistency condition f = 0, f˙ = 0.

– Elasticity, =

˜ = E −1 : σ

(11)

so that the condition f < 0 → elastic loading or unloading, f = 0, f˙ = 0 → damage growth, where √ ˆ =  + :  + is Mazars equivalent strain built from the positive part  + of the strain tensor and where  κ

tr D 0 + arctan (12) κ(tr D ) = a · tan aA a

In a homogeneously damaged medium at D = D0 , the wave speed is proportional to the square root of the damaged Young’s modulus and depends on the damage level as

c˜ = c 1 − D0 (18) One has in this case τxs =

x − s c˜

(19)

and x − s x − s x − s τxs = = √ > τc c˜ τc lc lc 1 − D0

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(20)

which shows that the effective or ‘‘dynamic’’ distance x − seff

x − s = √ 1 − D0

and on upper damaged zone sides. When the damage D0 becomes large (0.99 in the figure), notched and damaged specimens are found equivalent.

(21)

3.1

between two points increases, as expected, with damage. Eq. (20) defines in an equivalent manner an effective evolving internal length

˜lc = lc 1 − D0 (22) which tends to zero when D0 tends to unity in accordance with Pijaudier-Cabot et al. results (PijaudierCabot et al (2007)) of a material behavior becoming local on free edges (at least in the direction normal to the edge). In 1D and in the non uniform case, the effective distance is gained as the integral over the path [s, x] as x | s (1 − D(x ))−1/2 dx |, in 3D as c τxs . 3

EQUIVALENCE BETWEEN A CRACK AND A DAMAGED ZONE

X2

X3

0.1 m

X4

X6

X5

X4

y z x

1.414

0

1

1.414

1

1

0

1

1.414

1.414

1

0

1

1

1.414

1

0

1.414

2.236

2

1

1

⎤

⎥ 1.414⎥ ⎥ 2.236⎥ ⎥ ⎥ 2 ⎥ ⎥ ⎥ 1 ⎦ 0 (23)

The classical connectivity matrix C is the same (and is symmetric) for the 3 structures (a), (b) and (c). It does not take into account the presence of notches nor the occurence of damage as C16 = C25 = C34 = l/lc . In order to determine the connectivity matrix with the new internal time formulation, one proceeds as follows:

X3

⎡

X6

2.236

X2

X1

0.5 m

X1

2

For the concrete square plate (E = 35000 MPa, ν = 0.2, ρ = 2400 kg/m3 ) and notched specimen one ends up to:

X3

X5 0.5 m

1

Specimen (c)

Specimen (b)

0.5 m

X6

0 ⎢ ⎢ 1 ⎢ l ⎢ ⎢ 2 C= ·⎢ lc ⎢2.236 ⎢ ⎢ ⎣1.414

– an impulse force according to the out of plane z-axis (close to a Dirac) is applied at each point x = xp in a dynamic finite element analysis of a 3D model of a plate (thickness of 0.1 m, 720000 nodes and 360000 elements, free boundary conditions), – the times τxs at which the first pic of the z-acceleration information arrives at point s = xq are recorded. Divided by τc they are put in the form of connectivity matrices C τ , – the expressions for τxs /τc are synthesized by use of the relation lc = c τc in the form C τ = llc · A and compared to table 1 results for the different specimens.

Specimen (a)

X2

⎡

1

In order to illustrate the formulation ability, consider the specimens of figure 2: (a) a square plate, (b) a notched specimen and (c) a specimen with a damaged zone at D = D0 . Vertical z-acceleration fields are also plotted at the same instant for the 3 specimens. It is clearly noticed that the information initiated as an impulse at x = x1 takes longer to reach point x6 for both notched and damaged specimens (the wave generated in the example has to turn around the notch). Note the waves reflexion on upper notch

X1

Connectivity matrices

To quantify the approach, consider six points xp in these specimens. The distances between these points, used in the classical nonlocal theory, are calculated in table 1 for the plate (a) (with l = 0.5 m). The connectivity table√ is rewritten √ in a matrix form as a connectivity matrix ( 2 and 5 replaced by their numerical values to make easier further comparisons),

X5

X4

C (a) τ

Damage zone at D = D0

Figure 2. Square plate, notched and damaged specimens. Geometry and pictures of wave propagation at the same instant.

0

⎢ 1 ⎢ ⎢ l ⎢ ⎢2.008 = ·⎢ lc ⎢2.233 ⎢ ⎢ ⎣1.420 1

1

2.008

2.233

1.420

0

1

1.420

1 1.420

1

0

1

1.420

1

0

1

1

1.420

1

0

1.420

2.233

2.008

1

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1

⎤

1.420⎥ ⎥ ⎥ 2.233⎥ ⎥ ⎥ 2.008⎥ ⎥ ⎥ 1 ⎦ 0 (24)

3.2

Table 1. Connectivity table built from the classical distances x − s = xp − xq . x1

x2

x3

x1

0

l lc

x2

l lc 2l l √c

0

2l lc l lc

x3

5l lc √ 2l lc l lc

x4 x5 x6

⎡ ⎢ ⎢ l ⎢ ⎢ (b) Cτ = ·⎢ lc ⎢ ⎢ ⎣

l lc √ 2l lc l lc √ 2l lc

0 l lc √ 2l lc √ 5l lc

x4 √

x5 √ 2l lc l lc √ 2l lc l lc

5l

l √c 2l lc l lc 0 l lc 2l lc

0 l lc

x6 l lc √ 2l lc √ 5l lc 2l lc l lc

1

2.008

2.233

1.420

1.494

1

0

1

1.420

1

1.420

2.008

1

0

1

1.420

2.233

2.233

1.420

1

0

1

2.008

1.420

1

1.420

1

0

1

1.494

1.420

2.233

2.008

1

0

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(25) where the notch presence is taken into account (boxed terms), leading for the notched specimen to Cτ(b)16 = (b) (a) /τc = 1.494 l/lc intead of Cτ(a)16 = τ16 /τc = l/lc τ16 for the plate with no notch. For the specimen with the damaged zone at D = D0 = 0.99, the connectivity matrix corresponding to the nonlocal internal time analysis reads ⎡ (c) Cτ

⎢ ⎢ l ⎢ = ·⎢ lc ⎢ ⎢ ⎣

0

1

2.008

2.233

1.420

1.500

1

0

1

1.420

1

1.420

2.008

1

0

1

1.420

2.233

2.233

1.420

1

0

1

2.008

1.420

1

1.420

1

0

1

1.500

1.420

2.233

2.008

1

0

(27)

√

e 1 − 1 − D0 l τ16 1+ · √ = τc lc l 1 − D0

(28)

or

0

0

√ e 1 − 1 − D0 l + · √ c c 1 − D0

τ16 =

where e is the thickness of the damaged zone at D = D0 . The figure 3 shows the normalized increase of time (c) with respect to the damage value D0 and τ34 = τ34 for different ratios e/l. The ratio (τ34 /τc )/(l/lc ) in case (c) is equal (resp. close) to unity for a zero (resp. small) damage, the nonlocal theory with internal time recovering then the classical nonlocal theory. The very large increase obtained for large values of the damage, enhanced by a large damaged zone thickness, proves that both a real crack and a highly damaged zone are equivalent in the proposed nonlocal framework. This property is emphasized when the ratio of the weight functions ψ(τ34 /τc )/ψ(l/lc ) is drawn (figures 4 to 6) for l/lc = 5, 1 and 0.2 and for the Gaussian weight 1 2 function ψ(ξ ) = e− 2 ξ : the nonlocal spatial interaction between points across a damaged zone strongly diminishes with damage increase, the larger the points

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(26)

10 e/l=0,1 e/l=0,5 e/l=1

8 (tau/tau_c)/(l/l_c)

xp − xq  lc

Straight 1D wave propagation

For a better understanding, focus on times τ16 , τ25 , τ34 wave propagation in specimen (c) from x1 to x6 , from x2 to x5 , from x3 to x4 . The corresponding distances are equal in the 3 cases (equal to l). Due to the presence of the notch the time τ16 is of course larger than the time τ25 correspnding to a straight path with no notch (and τ25 /τc = l/lc ). A wave propagating along the path x1 − x6 in specimen (c) slows down to the celerity c˜ when meeting the damaged zone. If one only consider the straight path x1 − x6 for the wave propagation, one has

6

4

2

Again ony the terms 16 and 61 of the connectivity matrices are changed and found close to the value 1.494 l/lc obtained with reals notches: the damaged zone behaves as a notch, damage and notch being both taken into account by the proposed nonlocal with internal time analysis.

0 0

0,2

0,4

0,6 Damage

Figure 3.

Ratio

τ34 l τc / lc

vs damage D0 .

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0,8

1

1,2

distance l with respect to the internal length lc , the stronger the effect.

1,2 e/l=0,1

e/l=0,5

e/l=1

Ratio of weigth functions

1

0,8

4

0,6

As an example emphasizing how a damaged zone is taken into account, consider a bar in tension at a stress level σ . The length of the bar is 2L, a damaged zone at large D = D0 corresponds to x ∈ [−l, l]. The non local analyses perform the integrals (1) over the whole bar,

0,4

0,2

BAR WITH A DAMAGED ZONE

0 0

0,2

0,4

0,6

0,8

1

 nl (x) =

1,2

Damage



Figure 4.

Ratio

2

τ exp − 34 2

exp

τc 2 − l2 lc







L

−L

 ψ · (s)ds

Vr (x) =

L

ψ ds (29)

−L

with as strain (x) = σ/E if |x| > l, (x) = σ/ E(1 − D0 ) else and where ψ = ψ( |x−s| lc ) for the

 vs damage D0 for l/lc = 5.

30

1,2 e/l=0,1

e/l=0,5

e/l=1

a)

1

25

0,8 20

0,6

Vr

Ratio of weigth functions

1 Vr

15

0,4

Vr0 Vrnew 10

0,2

5

0 0

0,2

0,4

0,6

0,8

1

1,2

Damage 0

exp

Figure 5.

Ratio







 vs damage D0 for l/lc = 1.

exp

τ2 − 34 τc2 2 − l2 lc

-60

-20

0

20

40

60

x 100

b) 80

1,2 e/l=0,1

e/l=0,5

e/l=1

1 60

nl0 nlnew local

nl /

0,8

.

Ratio of weigth functions

-40

40

0,6

0,4

20

0,2 0 -60

0 0

0,2

0,4

0,6

0,8

1

-40

-20

0

20

40

60

x

1,2

Damage



Figure 6.

Ratio

τ2

exp − 34 τ2 exp



c 2 − l2 lc

Figure 7. a) Comparison of normalizing factors Vr (x); b) Comparison of normalized nonlocal strains (lengths in cm, black: nonlocal with internal time, grey: classical nonlocal, dot: local strain).

  vs D0 for l/lc = 0.2.

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classical nonlocal analysis, ψ = ψ( ττxsc ) for the new nonlocal analysis with internal time. The Gaussian 1 2 weight function ψ(ξ ) = e− 2 ξ is considered. The normalizing factors Vr (x) are compared for both analyses in figure 7a where D0 = 0.99, l = 2.5 cm, L = 50 cm and where the characteristic length is taken as lc = 10 cm (twice the size 2l of the damaged zone). The classical normalization does not ‘‘see’’ the damaged zone and averages across it when the new nonlocal with internal time approach behaves for the undamaged domains almost as for two independent bars, as expected. The nonlocal strains obtained with both approaches are compared in figure 7b. In this particular piecewise constant strain field, the formulation with internal time gives as expected a nonlocal strain field closer to the local strain field than the classical nonlocal approach for which too much importance is given to points over the damaged zone when performing the integral (29). Note that in case of structural failure, strain localization leads to non homogeneous fields. The proposed nonlocal averaging then fully acts and makes the solution regular.

5

between a localized zone and a crack will not be obtained. To gain this last feature, two possibilities are: – to consider an elasticity coupled with damage even if the plasticity part of the model remain not affected by damage, – to define τxs from plastic waves propagation (no need of damage then).

CONCLUSION

A new nonlocal integral formulation is proposed. An internal time is introduced leading to the equivalent definition of an effective or ‘‘dynamic’’ distance and of an evolving internal length. The important feature is that the distance bewteen points is not the argument of the averaging weight function anymore. It is replaced by the information or wave time propagation between these points. The nonlocal weight function build is then symmetric, even in non homogeneous bodies. Pre-computations of wave propagation in the considered structure allow to build the corresponding nonlocal connectivity matrix, with of course the open question of the wave type. The cracks and notches presence—and if necessary their closure—are naturally taken into account within the wave propagation study and PijaudierCabot and Bazant nonlocal theory is recovered far from the boundaries. The proposed approach makes equivalent a crack and a highly damaged zone, as points across a notch have a small contribution in the nonlocal averaging. Important point, no assumption on the medium isotropy is made. The proposed nonlocal framework includes anisotropy, either initial or induced. Last, when no damage is considered as in plasticity with negative hardening models, the internal time concept may still be used, for example by making nonlocal—through Eq. (4)—the accumulated plastic strain. The presence of existing notches and cracks will be naturally taken into account if propagation of elastic waves defines the time τxs but the equivalence

REFERENCES Aifantis E. The physics of plastic deformation. Int. J. Plasticity, 3: 211–47, 1987. Allix O. and De¨u J.F., Delay-damage modelling for fracture prediction of laminated composites under dynamic loading, Engineering Transactions, 45: 29–46, 1997. Bazant Z.P. and Jirasek M., Nonlocal integral formulations of plasticity and damage: Survey of progress, Journal of Engineering Mechanics ASCE, 128: 1119–1149, 2002. Cordebois J.P. and Sidoroff J.P., Endommagement anisotrope en élasticité et plasticité, J.M.T.A., Numéro spécial, 45–60, 1982. Denoual C. and Hild F., A Damage Model for the Dynamic Fragmentation of Brittle Solids. Comp. Meth. Appl. Mech. Eng., 183 (3–4): 247–258, 2000. Desmorat R., Modèle d’endommagement anisotrope avec forte dissymétrie traction/compression, 5è journèees du Regroupement Francophone pour la Recherche et la Formation sur le Béton (RF2B), Liège, Belgium, 5–6 july, 2004. Desmorat R., Gatuingt F. and Ragueneau F., Nonlocal anisotropic damage model and related computational aspects for quasi-brittle materials, Engineering Fracture mechanics, 74: 1539–1560, 2007. Desmorat R., Gatuingt F. and Ragueneau F., Non standard thermodynamics framework for robust computations with induced anisotropic damage, International Journal of Damage Mechanics, 2009, doi:10.1177/105678 9509104839. de Vree J.H.P., Brekelmans W.A.M. and van Gils M.A.J., Comparison of nonlocal approaches in Continuum Damage Mechanics, Computers and Structures, 55(4): 581–588, 1995. Geers M., de Borst R., Brekelmans W. and Peerlings R., Strain-based transient-gradient damage model for failure analyses, Comput. Methods. Appl. Mech. Engng, 160: 133–153, 1998. Jirasek M., Int. J. Engng Science, 4: 1553–1602, 2003. Jirasek M. and Marfia S., Nonlocal damage models: displacement-based formulations, Euro-C conference Computational modelling of Concrete Structures, Ed. G. Meschke, R. de Borst, H. Mang & N. Bicanic, Mayrhofen, Austria, 2006. Krayani A., Pijaudier-Cabot G. and Dufour F., Boundary effect on weight function in nonlocal damage model, Engineering Fracture Mechanics, 76(14): 2217–2231, 2009. Lemaitre J. and Desmorat R., Engineering Damage Mechanics: Ductile, Creep, Fatigue and Brittle Failures, Springer, 2005. Lemaitre J., Chaboche J.L., Benallal A. and Desmorat R., Mécanique des matériaux solides, Dunod Paris, 2009.

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Mazars J., Berthaud Y. and Ramtani, S., The unilateral behavior of damage concrete, Eng. Fract. Mech., 35: 629–635, 1990. Murakami S. and Ohno N., A constitutive equation of creep damage in pollicristalline metals, IUTAM Colloquium Euromech 111, Marienbad, 1978. Murakami S., Mechanical modeling of material damage, J. App. Mech., 55: 280–286, 1988. Needleman A., Material rate dependence and mesh sensitivity in localization problems, Comp. Meth. Mech. Engng, 67: 69–85, 1988. Peerlings R., de Borst R., Brekelmans W. and de Vree J., Gradient-enhanced damage model for quasi-brittle materials. Int. J. Numer. Methods Engng, 39: 391–403, 1996.

Pijaudier-Cabot G. and Bazant Z., Nonlocal damage theory. J. Engng Mech., ASCE, 113: 1512–33, 1987. Pijaudier-Cabot G., Haidar K. and Dubé J.-F., Nonlocal Damage Model with Evolving Internal Length, Int. J. Num. Anal. Meth. Geomech., 28: 633–652, 2004. Pijaudier-Cabot G., Krayani A. and Dufour F., Comments on boundary effects in non local damage based models, chapter in Nonlocal Modeling of Materials Failure, H. Yuan and F.H. Wittmann Ed., Aedificio Pubs, 2007. Souid A., Ragueneau F., Delaplace A. and Desmorat R., Pseudodynamic testing and nonlinear substructuring of damaging structures under earthquake loading, Engineering Structures, 31: 1102–1110, 2009.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Elastoplastic constitutive model for concretes of arbitrary strength properties Guillermo Etse & Paula Folino Faculty of Engineering, University of Buenos Aires, Argentina

ABSTRACT: The increasing use of high strength concretes in civil constructions and structures is demanding the development of reliable constitutive laws for accurate structural analysis by means of computational methods. One of the main difficulties in this regard is the strong variation of the relevant mechanical features of concretes with the intrinsic quality of this material. This variation is not only relevant when comparing normal (NSC) with high strength concretes (HSC), but also within the wide spectrum of high strength concretes. Experimental evidence demonstrates that besides maximum strength, one of the mechanical properties of concrete most affected by the variation of material quality is the ductility and, therefore, the transition point from ductile to brittle failure modes. Most of the constitutive laws for concrete in the literature are only valid for NSC. Very few of them can be also used for HSC with the same level of accuracy as in case of NSC. But the most significant limitation is that up to date very few proposals can be found, related to concrete models covering the wide spectrum from NSC to very HSC. In this work, a performance dependent constitutive model for concretes of arbitrary strengths is presented. The model is based on the flow theory of plasticity and takes into account fracture energy concepts for the formulation of the softening law. The maximum strength surface of the model as well as hardening/softening evolution laws are described in terms of the so-called concrete performance parameter. This parameter is introduced to evaluate concrete quality in terms of uniaxial strength and of the water/binder ratio, as proposed by the authors [Folino, Etse & Will 2009]. The pre and post-peak behavior of concrete are described by means of non-uniform hardening and isotropic softening formulations. Volumetric dilatancy of concretes of arbitrary strengths is described by means of a volumetric non-associated flow rule that is also expressed in terms of the performance parameter. The proposed model is being implemented in finite element tool for failure analysis of concrete elements of different qualities. The most relevant features of model formulation as well as of the considerations made to take into account the influence of concrete quality in the response behavior of this quasi-brittle material in pre and post-peak regimes are described in this work. 1

INTRODUCTION

Several authors have extensively discussed the fundamental differences between mechanical response behavior of NSC and HSC. Among others, we refer here to the works by Imran & Pantazopoulou 1996 and Xie et al. 1995. A very important fact that follows from the experimental evidence is that the improvement of concrete mechanical properties, particularly the ductility in pre and post regimes, but also the reduction in of the volumetric dilatancy during monotonic compressive loading does not linearly vary with the increase of the uniaxial compressive strength ( fc ). Actually, this variation is highly complex and nonlinear. The same can be said regarding the variation of the mechanical response sensitivity with respect to the acting confinement. By far, this sensitivity is higher in case of NSC. Contrarily, there are other mechanical features that decrease (in a non-linear form as well) for increasing fc or concrete strength properties, such as the fracture energy release in Mode I and II

type of failure. Finally, another relevant aspect to be considered is that the variation of the uniaxial tensile strength ( ft ) does not linearly depend on the variation of fc . This means that in fact, maximum strength surface of concrete cannot isotropicaly vary when going from NSC to HSC. This relevant property was taken into account by the authors, see Folino et al. 2009, for the formulation of the performance dependent failure criterion for concrete. In this work a model for concrete of arbitrary strength and performance is presented. The model is based on the flow theory of plasticity. A fundamental feature of the proposed formulation is that it encompasses in one single equation the yield surfaces in pre and post-peak regimes for all stress stages and all types of considered concrete qualities. In hardening zone the yield surfaces have cap-cone form with C1 -continuity. This allows a simpler numerical implementation of the elostoplastic model in finite element codes. Cone portion of yield condition agrees with the maximum strength surface, while the cap portion is based on an

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The evolution of the state variables t κ˙ is defined by the hardening/softening laws. In the next sections, the failure criterion, hardening and softening formulations of the proposed material model for concretes of arbitrary performance are described.

elliptic formulation that perpendicularly intersects the hydrostatic axis. Thus, concrete non-linear response under pure hydrostatic loading can also be realistically simulated with this model. Under increasing plastic loading in the pre-peak regime the cone-cap intersection moves to the confinement direction. In the proposed formulation the ductility of the hardening mechanical response depends on both concrete quality and current confining pressure. To account for mesh regularization during post peak regime softening formulation of the model is based on fracture energy concepts following original proposals by Etse & Willam 1994. The presented model incorporates the influence of the performance dependent parameter in the evaluation of the fracture energy-based characteristic length both in mode I and II type of fracture. In this way the ductility in post-peak regime depends on the material quality, on current confining pressure and on the fracture mode. To reduce excessive volumetric dilatancy of concrete when loading in the low confining region the proposed formulation includes a non-associative formulation of the plastic flow based on volumetric variation of the yield condition. The volumetric nonassociativity depends on both the confining pressure and the concrete quality by means of the performance parameter.

3

PERFORMANCE DEPENDENT FAILURE CRITERION FOR CONCRETE

The performance dependent failure criterion (PDFC) for concretes of arbitrary strength considered in this model was proposed by the authors (see Folino, Etse & Will 2009) on the basis of the so-called performance parameter. It covers the entire spectrum of concrete qualities from NSC to HSC. The PDFC is defined in terms of the Haigh Westergaard stress coordinates ξ and ρ which are functions of the first and second invariant to the stress and deviatoric tensor, respectively, and of the Lode angle θ . The last one is function of the second and third invariants of the deviatoric stress tensor. According to this criterion, concrete failure occurs when actual 2nd Haigh Westergaard stress coordinate ρ reaches the shear strength ρ ∗ F=

2

THEORETICAL FRAMEWORK

The proposed constitutive formulation is based on smeared crack approach and the elasto-plastic incremental flow theory. Only infinitesimal strains are considered. Elasticplastic coupling is neglected, accepting the additive Prandtl-Reuss decomposition of the infinitesimal strain rate tensor into its elastic and plastic parts described as p

ε˙ ij = ε˙ ije + ε˙ ij

ρ −1=0 ρ∗

(4)

In the deviatoric plane, the C1 -continuity elliptic interpolation of Willam & Warnke (1974) between second stress coordinate in compressive ρc∗ and tensile ρt∗ meridians is followed ρ = ρc∗ /r

(5)

being r the ellipticity factor r=

4(1 − e2 ) cos2 θ + (2e − 1)2  − 1) 4(1 − e2 ) cos2 θ + 5e2 − 4e

2(1 − e2 ) cos θ + (2e

(1)

( 6)

The elastic constitutive response is defined by the generalized Hooke law

and e = ρt∗ /ρc∗ the so-called eccentricity. The failure surface is represented by (7) F = A r 2 ρ ∗ 2 + Bc rρ ∗ + C ξ¯ − 1 = 0

σ˙ ij = Eijkl ε˙ kl

(2)

In the above equation σ˙ ij is the Cauchy stress rate tensor, Eijkl the fourth order elasticity tensor depending on the material Young’s modulus Ec and Poisson’s ratio υ. Inelastic material response is governed, in general, by the following non associated flow rule p ε˙ ij = λ˙ mij

where mij =

∂Q ∂σij

(3)

Being Q (σ ; κ) = 0 the plastic potential. Plastic parameter follows from the well-known consistency condition.

Whereby ρ ∗ = ρ ∗ /fc and ξ¯ = ξ/fc are the normalized first and second stress coordinates. For the compressive and tensile meridians the failure surface reduces, respectively, to Fπ/3 = Aρc∗ 2 + Bc ρc∗ + C ξ¯ − 1 = 0 F0 =

Aρt∗ 2

+

Bt ρt∗

+ C ξ¯ − 1 = 0

(8) (9)

Coefficients A, Bc , Bt and C in Eqs. (7) to (9) are defined by means of explicit expressions, see Folino, Etse & Will (2009), of four material parameters, as follow:

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– uniaxial compressive strength fc , – uniaxial tensile strength ft (through uniaxial strength ratio αt = ft /fc ), – biaxial compressive strength fb (through the biaxial compressive strength ratio αb = fb /fc ), – parameter m: tangent to the compressive meridian on the peak stress’s shear component corresponding to the uniaxial compression test. Considering the difficulties involved in determining material parameters αt , αc and m, internal calibration functions were proposed by the authors in terms of the performance parameter βP βp = 0.001 fc /(W /B)

(10)

being (W /B) the water/binder ratio, considered as a fundamental property of concrete mix controlling the material performance. Consequently, βp is directly related to the porosity of the mortar. When W /B is unknown, a range of possible βp can be defined, limited by upper and lower bounds proposed by Folino et al. (2009). Thus, once fc and βp are defined, the whole set of remaining material parameters αt , αb and m can be evaluated using calibration functions and, subsequently, coefficients A, Bc , Bt and C from the explicit expressions by Folino, Etse & Will (2009). As a result, the performance dependent maximum strength criterion is fully defined. Figure 1 illustrate maximum strength meridians predicted by the PEDFC for different concrete qualities. Figures 2 and 3 depicts comparisons of experimental results in terms of peak stresses with the proposed failure criterion. In Figures 4 and 5 validations of triaxial and biaxial tests results are presented. As can be observed good agreements between experimental results and the proposed failure criterion are obtained in all cases.

Figure 1. Compressive and tensile meridians predicted by the PDFC for different concrete qualities—Normalized plot.

Figure 2. Triaxial tests validation, for two concretes of different qualities: fc = 65 and 120 MPa—Normalized plots.

Figure 3. Biaxial tests validation, for two concretes of different quality: fc = 43 and 97 MPa—Normalized plots.

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of C 1 -continuity. Cone portion of yield surfaces in hardening agrees with the failure criterion F defined by Equation (7) while cap sector in meridian plane is defined by ellipse portions that perpendicularly intersect the hydrostatic axis. fh can be mathematically expressed as  F =0 if 2 fh =  ξ¯ − ξ¯cen /a2 + r 2 ρ¯ 2 /b2 − 1 = 0 if

ξ¯ ≥ ξ¯1 (k) ξ¯ < ξ¯1 (k) (11)

Figure 4. Loading surfaces in hardening fc MPa—Normalized plot.

=

20

whereby the hardening parameter k controls the evolution of coefficients a, b and ξ¯cen defining the ellipse’s size and location in the meridian plane. Haigh Westergaard second stress coordinate of the cap-cone intersection point P1 in current position of hardening yield locus is defined in terms of k as  (12) ρ¯p1 = k 2/3 Haigh Westergaard first stress coordinate of P1 depends on concrete quality as well. During hardening process under increasing confining pressure, P1 evolutes along the maximum strength surface defining starting point of current cap. The evolution of the hardening parameter k from its initial lower bound ko has no defined limit. Figures 4 and 5 illustrate normalized plots of loading surfaces in hardening regime corresponding to two different concrete qualities. 4.2

Initial loading surface in cap regime

The initial cap requires definition of the lower bound ko of the hardening parameter and of k = kel corresponding to the initiation of plastic process in uniaxial compression test. Initial cap in compressive meridian is an ellipse centered in the ξ axis. Coordinates of this ellipse centre are ξ¯ = ξ¯cen o and ρ = 0. For evaluation of ξ¯cen o and the two ellipse radiuses ao , bo the following three boundary conditions are introduced, see Fig. 6: Figure 5. Loading surfaces in hardening fc MPa—Normalized plot.

4

= 120

HARDENING FORMULATION FOR PRE-PEAK REGIME

In this section the hardening formulation of the performance dependent constitutive model for concrete is described. 4.1

Yield surfaces in hardening

Meridian traces of the yield criterion in hardening regime fh are defined by cap-cone loading surfaces

1. Initial ellipse passes through P1 on the compressive meridian of the failure surface with coordinates ξ¯1o and ρ¯1o . 2. C 1 -continuity at P1 . 3. Initial ellipse also passes through P2 corresponding to plastic process initiation of uniaxial compression test. P2 with coordinates ξ¯2o and ρ¯2o lies in the line between the uniaxial compression maximum strength and the stress coordinates origin. From these conditions result explicit expressions for ξ¯cen o , ao and bo , see Folino & Etse (2009). Additional geometrical constrains of ellipse radiuses a2 > 0, b2 > 0 and a2 /b2 > 0 imply, as expected, that ξ¯1 > ξ¯2 . On the other hand, it also implies that

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Figure 7. Hardening level as a function of the level of confinement and of the developed work hardening. Figure 6.

Initial cap ellipse—Normalized plot.

ρ¯1 o min < ρ¯1 o < ρ¯1 o max , defining an admissible range for the second stress coordinate of P1 at initial ellipse. 4.3

Hardening surface evolution

Subsequent cap yield surfaces after reaching elastic limit are obtained through the evolution of P1 along the maximum strength surface that, in turn is controlled by the evolution of k. New cap ellipses keep the ratio between major and minor axes invariant with respect to the initial cap. From this condition follow explicit equations for the ellipse parameters ξ¯cen , a and b in current position during hardening regime. 4.4 Hardening evolution law In this proposal, hardening parameter k is defined in terms of the normalized work hardening measure κh by means of an elliptic variation as follow (See Fig. 9)   k = ko + k max − ko κh (2 − κh ) (13) being k max the maximum possible hardening parameter corresponding to actual work hardening measure κh that, in turn, is compatible with current confinement level. k max is defined as    −Bc + Bc2 − 4A 1 − C ξ¯ 1 max = √ (14) k 2A 2/3 while the evolution law of the work hardening measure is p

κ˙ h = ω˙ ap /Wt p σij ε˙ ji

(16)

p

εmax =

ISOTROPIC SOFTENING FORMULATION FOR POST-PEAK REGIME

In this section, the softening law of the proposed model is summarized. It is based on fracture energy concepts for Mode I and II type of failure. 5.1

Yield surface evolution in softening regime

Once the cone is reached, a progressive softening process starts, controlled by the softening parameter cs that varies between 1 and 0. Yield surfaces in softening regime are defined by the expression fs = A r 2 ρ¯ 2 + B r ρ¯ + C ξ¯ − cs = 0

(18)

Figures 8 and 9 show compressive and tensile meridian views of yield surfaces in softening regime for two different concrete qualities. 5.2

and p Wt

5

(15)

Thereby are ω˙ ap =

p

where εmax is the maximum plastic strain at peak stress under current confinement pressure. A bilinear interpolation formula is proposed for numerical evaluation p p of Wt , see Folino and Etse (2009). Once Wt is known, the evolutions of plastic work and of work hardening measure can be obtained from previous Eqs. (16) and (15), respectively. Then, the hardening parameter can be updated with Eq. (14).

p

σij ε˙ ji

(17)

0

Fracture energy-based softening model

Similarly to the formulation by Etse & Willam (1994), softening parameter cs defines the degradation of the tensile strength during post-peak regime in terms of fracture parameters and variables  σt −κs ≥0 (19) 1 ≥ cs =  = exp ft ur

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6

NON-ASSOCIATED FLOW RULE

To limit volumetric dilatancy during plastic range in low confinement regime, a restricted form of nonassociativity is considered that only involved the volumetric plastic flow. In other words, plastic potential in hardening/softening regime is based on a volumetric modification of the yield condition in cone and/or cap zone, as follow Plastic potentials in hardening: (22) Qcone = A r 2 ρ ∗ 2 + B rρ ∗ + Cηcone ξ¯ − 1 = 0 2 2 2 2 2 ¯ ¯ Qcap = ηcap [ξ −ξcen ] /[a ] + r ρ¯ /[b ] − 1 = 0 (23) being Figure 8. Meridian views of yield softening—fc = 20 MPa—Normalized plot.

surfaces

  ηcone = ηcone βp ,

in

  ηcap = ηcap t κ; βp

(24)

Plastic potential in softening: Qs, cone = A r 2 ρ ∗ 2 + B rρ ∗ + Cηcone ξ¯ − cs = 0 (25) with

  ηs, cone = ηs, cone cs ; βp

(26)

Thereby are ηcone , ηcap and ηs, cone the grade of volumetric non-associativity in terms of the performance parameter and the state parameter corresponding to hardening/softening regime. Explicit expressions of the grades of volumetric non-associativity are given in Folino & Etse (2009).

7 Figure 9. Meridian views of yield surfaces softening—fc = 120 MPa—Normalized plot.

in

Thereby κs is the fracture energy-based softening measure and ur the maximum crack opening displacement. In this formulation it is considered that the evolution lay of κs is defined in terms of

 (20) κ˙ s = κ˙ s δ; ε˙˜ f ; lcmII ; βp whereby, δ defines the shape of the decay function, βp is the performance parameter and incorporate the influence of concrete quality in the post-peak evolution law, ε˙˜ f is the rate of fracture strain that extracts only tensile components of the rate of plastic strain tensor in principal components. Finally, lcmII is the characteristic length for general mode II type of fracture lcmII = [GfII /GfI ]ht

(21)

with GfII and GfI the fracture energy releases in mode II and mode I type of rupture, respectively, and ht the characteristic length in uniaxial tensile failure mode.

CONCLUSIONS

In this work a constitutive model is proposed to evaluate failure behavior of concrete of arbitrary performance and strength capacity. The material model is formulated within the mathematical framework of the flow theory of plasticity. Maximum strength criterion of the model is sensitive not only to the variation of basic material parameters of concrete such as friction, cohesion and uniaxial compressive/tensile strengths, but, moreover, of the particular concrete quality that is considered. Concrete quality is defined by means of the so-called performance parameter together with uniaxial compressive strength. Constitutive model is completed with the formulation of hardening/softening laws as well as non-associated flow rule. All of them include the performance parameter in their respective equations to incorporate the influence of the material quality in pre and post-peak ductility as well as in the volumetric dilatancy of concrete when loading in the low confinement regime. One particularity of the model is the formulation of yield condition in hardening regime that represents a cap-cone surface with C 1 -continuity. The cone portion coincides with the maximum strength surface while the cap one is describe by an elliptic trace that evolves during work hardening process. Softening

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formulation includes fracture mechanics concepts and, consequently, incorporates a characteristic length allowing regularization of post-peak behavior. Both hardening and softening laws are sensitive to the level of confinement in order to reproduce the increase of ductility in pre and post-peak regime with the acting confining pressure.

ACKNOWLEDGEMENTS The authors acknowledge partial financial supports for this work by FONCYT (Argentine agency for research & technology) through Grant PICT1232/6, by CONICET (Argentine council for science & technology) through Grant PIP6201/05 and by University of Buenos Aires through Grant UBACYT2006-2009 Project I813.

REFERENCES Chen, W.F. & Han, D.J. 1988. Plasticity for structural engineers, Springer Verlag. Dvorkin, E., Cuitiño, A. & Gioia, G. 1989. A concrete material model based on non-associated plasticity and fracture. Journal Engineering Computations, Vol. 6, No. 4, 281–294. Etse, G. & Willam, K. 1994. Fracture energy formulation for inelastic behavior of plain concrete. ASCE Journal of Engineering Mechanics, Vol. 120, No. 9, 1983–2011. Folino, P., Etse, G. & Will, A. 2006. Modelación inelástica de hormigones de distintas resistencias basada en el índice de prestación. Mecánica Computacional Vol. XXV, 1915–1925. Folino, P. & Etse, G. 2008. Endurecimiento a través de superficies con capa dependientes del grado de prestación del hormigón, Mecánica Computacional Vol. XXVII, 909–926. Folino, P., Etse, G. & Will, A. 2009. A performance dependent failure criterion for normal and high strength concretes, ASCE Journal of Engineering Mechanics, in press. Folino, P. and Etse, G. 2009. Performance dependent constitutive model for concretes of arbitrary strength. Submitted to Int J. Comp. Methods in Applied Mech & Eng.

Fossum, A.F. & Fredrich, J.T. 2000. Cap plasticity models and compactive and dilatant pre-failure deformation. Proc. 4th North American Rock Mechanics Symp., A.A. Balkema, Rotterdam, 1169–1176. Grassl, P., Lundgren, K. & Gylltoft, K. 2002. Concrete in compression: a plasticity theory with a novel hardening law. International Journal of Solids and Structures, Vol. 39, 5205–5223. Han, D.J. & Chen, W.F. 1987. Constitutive modeling in analysis of concrete structures. ASCE Journal of Engineering Mechanics, Vol. 113, No. 4, 577-593. Hussein, A. & Marzouk, H. 2000. Behavior of High-Strength Concrete under biaxial stresses. ACI Materials Journal, Vol. 97, No. 1, 27–36. Imran, I. & Pantazopoulou, S.J. 1996. Experimental study of plain concrete under triaxial stress. ACI Materials Journal, Vol. 93, No. 6, 589–601. Kang, H. & Willam, K. 1999. Localization characteristics of triaxial concrete model. ASCE Journal of Engineering Mechanics, Vol. 125, No. 8, 941–950. Lade P. 1977. Elasto-plastic stress-strain theory for cohesionless soil with curved yield surfaces. International Journal of Solids and Structures, Vol. 13, 1019–1035. Lu, X. 2005. Uniaxial and triaxial behavior of high strength concrete with and without steel fibers. Phd Thesis, New Jersey Institute of Technology. Ohtani, Y. & Chen, W.F. 1988. Multiple hardening plasticity for concrete materials. ASCE Journal of Engineering Mechanics, Vol. 114, No. 11, 1890–1910. Oller, S. 1988. Un modelo de daño continuo para materials friccionales. Tesis Doctoral UPC, Barcelona, España. Rossi, P., Ulm, F.J. & Hachi, F. 1996. Compressive behavior of concrete: physical mechanisms and modeling. ASCE Journal of Engineering Mechanics, Vol. 122, No. 11, 1038–1043. Sfer, D., Carol, I., Gettu, R. & Etse, G. 2002. Experimental study of the triaxial behavior of Concrete. ASCE Journal of Engineering Mechanics, Vol. 128, No. 2, 156–163. van Geel, E. 1998. Concrete Behavior in multiaxial compression. Doctoral Thesis, Technische Universiteit Eindhoven. van Mier, J.G. 1997. Fracture Processes of Concrete. CRC Press. Willam, K.J. & Warnke, E.P. 1974. Constitutive model for the triaxial behavior of concrete. Proc. Intl. Assoc. Bridge Struct. Engrg., Report 19, Section III, Zurich: 1–30. Xie, J., Elwi, A. & Mac Gregor, J. 1995. ‘‘Mechanical properties of three high-strength concretes containing slica fume’’, ACI Materials Journal, Vol. 92, No. 2, 135–145.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Properties of concrete: A two step homogenization approach E. Gal & R. Kryvoruk Department of Structural Engineering, Ben-Gurion University of The Negev, Beer-Sheva, Israel

ABSTRACT: This paper describes the development of a two step homogenization approach for evaluating the elastic properties of concrete. For that purpose a finite element model of the concrete unit cell is generated. Prior to the generation of the unit cell finite element model the Interface Transition Zones (ITZ) and the aggregates are homogenized using an analytical approach. This approach makes it possible to exclude the ITZs from taking part in the finite element model of the unit cell. This is essential for achieving a practical calculating effort, since the ITZ typical dimensions are three orders of magnitude smaller than the typical aggregate dimensions; it thus requires a much more detailed model to represent explicitly the ITZ shells. For verification and validation purposes three types of concrete were tested, in two of them the aggregate distribution was obtained using sieve analysis while in the third the aggregate distribution was obtained according to the Fuller curve. 1

INTRODUCTION

Evaluating the elastic properties of concrete becomes complicated due to the fact that concrete has a variety of microstructures and eventually has no one distinctive microstructure. The variety of micro-structures include: addition of fibers made up from different materials; variation of aggregate size, shape and type; water to cement ratio etc.. The authors suggest that the use of multi-scale analysis evidently is the appropriate way to model the behavior of concrete structures by coupling between the concrete micro-structures and its macroscopic properties needed to analyze the concrete structure (e.g. Markovic & Ibrahimbegovic 2004, Ibrahimbegovic & Markovic 2003, Gitman et al. 2006, 2007, 2008, He et al. 2009, Kouznetsova et al. 2001, 2002, Feyel 2003, Gutierrez 2004, Ghosh et al. 2001, Nadeau 2003, Lee et al. 2009, Pichler et al. 2007, Mang et al. 2003, de Borst et al. 1999, Füssl et al. 2008, Asferg et al. 2007, Oliver 1996, Jirasek 2000, Wells & Sluys 2001, Moes & Belytschko 2002, Dumstorffz & Meschke 2007, de Borst 2002, Meschke & Dumstorff 2007, Simone & Sluys 2004, Wriggers & Moftah 2006, Haffner et al. 2006, Wang et al. 1999, Cusatis & Cedolin 2007, Gal et al. 2008). The method for obtaining the macroscopic behavior of the concrete, based on its microstructure is referred to as the theory of homogenization, by which the heterogeneous material is replaced by an equivalent homogeneous continuum. The method is performed on a statistically representative sample of material, referred to as a material unit cell. Numerous theories have been developed to predict the behavior of composite materials. Starting from the various effective properties obtained by the models of Eshelby (1957),

Hashin (1962), Mori & Tanaka (1973), self-consistent approaches of Hill (1965) and various mathematical homogenization methods (e.g. Christensen 1979) pioneered by Bensoussan (1978) and Sanchez-Palencia (1980). Unfortunately, most of these analytical models can only give estimates or boundaries for the macroscopic properties, and the simplifying assumptions used, result in, the major differences obtained. Computational procedures for implementing homogenization have been an active area of research starting with the contribution by Guedes & Kikuchi (1990) for linear elasticity problems. Over the past decade major contributions have been made to extend the theory of computational homogenization to nonlinear domains Terada & Kikuchi (1995), Fish et al. (1997), Fish & Shek (1999), Fish & Yu (2001) and improving fidelity and computational efficiency of numerical simulations (Terada & Kikuchi 2001, Matsui et al. 2004, Aboudi 1991, Aboudi et al. 2003, Aboudi 2003, Smit et al. 1998, Miehe & Koch 2002, Kouznetsova et al. 2001, Feyel & Chaboche 2000, Ghosh et al. 1995, 1996, Geers et al. 2001). These developments established the Finite Element Method (FEM) as one of the most efficient numerical methods, whereby the macroscopic responses can be obtained by volumetrically averaging numerical solutions of unit cells (e.g. Zohdi & Wriggers 2001, 2005). It has been established that the presence of the aggregates in the mortar causes a thin layer of the mortar material surrounding each inclusion to be more porous than the other surrounding matrix. This thin layer is named the interfacial transition zone (ITZ) Bentz et al. (1992), (1993), Scrivener & Nemati (1996), Sun et al. (2007). The ITZ between the paste matrix (mortar) and aggregate phases played an important role in the properties of a concrete composite

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of the following quadratic equation where the positive square root is the physical answer.    2 G G +C =0 (1) + 2B A G2 G2

He et al. (2009), Scrivener & Gariner (1988), Prokopski & Halbiniak (2000). It has been found by experiment that the elastic moduli of concrete are intimately related to the elastic modulus and volume fraction of the ITZ regions Simeonov & Ahmad (1995), Nilsen & Monteiro (1993). Therefore the concrete microstructure is to include three phases consisting of aggregates, paste matrix and ITZ. The variety of homogenization techniques suggest modeling the concrete microstructure using a spherical aggregate to be surrounded by a concentric layer of ITZ, all embedded in the cement paste matrix. This paper adopts this assumption which enables performing a two step homogenization for evaluating the concrete module of elasticity. As the first step the properties of a spherical aggregate and its concentric ITZ layer were homogenized via the procedure suggested by Gaboczi & Berryman (2001). Then the homogenization procedure suggested by Gal et al. (2008) is applied to obtain the macroscopic properties of the concrete. Finally verification and validation of three types of concrete were performed, in two of them the aggregate distribution was obtained using sieve analysis while in the third the aggregate distribution was obtained according to the Fuller curve.

where G2 and G3 are the shear modulus of the ITZ and aggregate respectively and A = 8z (4 − 5ν2 ) η1 p10/3 − 2 (63zη2 + 2η1 η3 ) p7/3 + 252zη2 p5/3 − 50z(7 − 12ν2 + 8ν22 )η2 p + 4 (7 − 10ν2 ) η2 η3

B = − 2z (1 − 5ν2 ) η1 p10/3 + 2 (63zη2 + 2η1 η3 ) p7/3 − 252zη2 p5/3 + 75z (3 − ν2 ) η2 ν2 p +

3 (15ν2 − 7) η2 η3 2

2.1

(2b)

C = 4z (5ν2 − 7) η1 p10/3 − 2 (63zη2 + 2η1 η3 ) p7/3 + 252zη2 p5/3 + 25z(ν22 − 7)η2 p − (7 − 5ν2 ) η2 η3

(2c)

η1 = z (7 − 10ν2 ) (7 + 5ν3 ) + 105 (ν3 − ν2 ) η2 = z (7 + 5ν3 ) + 35 (1 − ν3 )

2

(2a)

MACROSCOPIC MATERIAL PROPERTIES OF CONCRETE

(2d)

η3 = z (8 − 10ν2 ) + 15 (1 − ν2 ) G3 −1 G2 3  r p= r+t

z=

Homogenization of the aggregate and the ITZ-Step 1

Here, developing the homogenization of the concrete starts from homogenizing the aggregates and the ITZs. This is essential since in the following stage is using the finite element method (FEM) to evaluate the homogenized property of the concrete. Using the FEM requires a detailed model so that each constituent of the concrete has to be represented through using many finite elements. As the differences in typical dimensions of the aggregate (∼2000–19000 μm) and the ITZ (∼10–50 μm) is about three orders of magnitude, creating a 3D finite element model which links, the finite elements representing the ITZs and the finite elements representing the aggregates, probably will be unpractical or/and inaccurate. Therefore we suggest excluding the ITZ from our finite element model by homogenizing it with the aggregate. The homogenization of a spherical particle surrounded by a spherical shell of different elastic properties can be performed as suggested by Christensen (1979), (1990). In this paper we follow the formulation suggested by Christensen (1990) as it has been applied by Gaboczi & Berryman (2001) to homogenize the concrete aggregate with its ITZ. Following Gaboczi & Berryman (2001), the effective shear modulus is obtained from the solution

(2e)

where ν2 and ν3 are the Poisson’s ratio of the ITZ and aggregate respectively, r is the aggregate radius and t is the ITZ thickness. The effective bulk modulus, K, is given by: K = K2 +

p (K3 − K2 ) 1 + (1 − p)

2 ) (K3 −K K2 + 43 G2

(3)

where K2 and K3 are the bulk modulus of the ITZ and aggregate respectively. 2.2

Asymptotic theory of homogenization-Step 2

The asymptotic theory of homogenization is based on the following asymptotic expansion of the displacement field: u (x, y) = u0 (x, y) + ξ u1 (x, y) + ξ 2 u2 (x, y)

(4)

where x is the macroscopic scale position vector; y = x/ξ are the micro scale position vectors as 0 < ζ << 1. Assuming that u0 (x, y) = u (x) and

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that u1 (x, y) = ε0 (x) χimn (y) where ε 0 (x) is the macroscopic strain vector we obtain the following two (macroscopic and microscopic) coupled problems. The macroscopic problem is formulated using the following boundary volume problem: find u(x) on  such that Lijlm ε lm,xj + bi = 0 ui = ui

on u ;

on 

(5)

σ ij nj = t i

on t

where  is the macroscopic domain, u is the macro scale displacement; b the average unit cell body force, L the macroscopic effective material properties, ui on u are the essential boundary conditions, σ ij nj = t i on t are the natural boundary conditions with normal nj , σ ij are the macro scale stress components and the macro scale strain components are   1 ∂um ∂un (6) εmn = u(m,xm ) = + 2 ∂xn ∂xm summation convention is employed for repeated indices. The microscopic problem with periodic boundary conditions is formulated using the following boundary volume problem: find χimn (y) on such that   Lijlm χ(k,yi )mn + Iklmn ,yj = 0 on χimn (y) = χimn (y + Y ) on ∂ χimn (y) = 0 on ∂ ver

(7)

where χ are the micro scale influence functions; is the unit cell domain of size Y, ∂ represents the unit cell boundaries, ∂ ver represents the unit cell vertices, Iklmn is Iklmn = (δmk δnl + δnk δml )/2

(8)

where δmk is the Kronecker delta. The homogenized constitutive tensor components which represents the macroscopic material properties is given by:  1 Lijlm = σ mn d (9) | | ij where σijmn are stress influence functions induced by applying an overall unit strain εmn defined as σijmn = Lijkl (χ(k,yj )mn + Iklmn )

(10)

Solution of the unit cell problem using the finite element method is obtained by resolving the unit cell problem to multiple RHS vectors (six in a 3D case due to symmetry of indices mn). In the matrix implementation, Lijlm is a 6 × 6 matrix where ij represents six rows and mn six columns. Each column in Lijlm can be extracted by multiplying Lijlm with a unit overall strain, εmn = 1. For implementation in a commercial

package, it is convenient to select εmn in the form of a unit thermal strain. However, due to the idealization of the homogenized behavior of concrete as being statistically isotropic (see also Wrigges & Moftah 2006) one loading state is sufficient for describing its overall linear elastic behavior. Thus, the effective bulk and shear modulus of the concrete are given by σD : σD tr (σ ÷ 3) (11) and 2μ = 3κ = tr (ε ÷ 3) εD : ε D where κ is the effective bulk modulus, μ is the effective shear modulus of the concrete and σD = σ − tr (σ ÷ 3) I

and

εD = ε − tr (ε ÷ 3) I (12)

are the deviatory parts of σ and ε respectively.

3

UNIT CELL GENERATION

The suggested framework executes the multi-scale analysis of concrete structures by incorporating an original concrete unit cell generator into a commercial finite element software package intended for simulating nonlinear solid mechanical problems, for more details see Gal et al. (2008). Essential information needed for creating a concrete unit cell is the aggregate distribution. This information is used to monitor the size and amount of the different aggregates within the concrete. In the suggested framework the fracture volume of the aggregate is adjusted to include the volume fracture of the ITZ. The aggregates can be distributed in three different ways according to a sieve analysis, the Fuller curve or direct access as suggested by Gal et al. (2008).

4

VERIFICATION AND VALIDATION

The concrete macroscopic properties are determined for the unit cells generated according to the presented development. Three types of concrete were tested, in two of them the aggregate distribution was obtained using sieve analysis while in the third the aggregate distribution was obtained according to the Fuller curve. The following uniform strains ε0 = { 1

0

0

0

0

0 }T

(13)

were applied to all the unit cell elements and according to the theory of homogenization the macroscopic properties were calculated. The results obtained from the presented finite element model were compared with experimental data found in the literature see Tables 1 to 5.

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Table 1. Aggregate sieve analysis—results. Aggregate Sieve size Total percentage Total percentage type (mm) retained (%) passed (%) I

12.70 9.50 4.75 2.36

II

19.00 12.70 9.50 4.75 2.36

Table 2.

0 23 74 100

100 77 26 0

0 3 39 90 98.6

100 97 61 10 1.4

Sieve analysis—material data of constituents.

Material property

Mortar material

Aggregate material I

Aggregate material II

E(GPa) ν

19.0 0.20

72.0 0.20

62.0 0.20

4.1

Sieve analysis

Sieve analysis results of experiments presented by Hirsch (1962) are given in Table 1 while Table 2 contains the microscopic material properties for two types of concrete. For concrete type I the maximum aggregate size is 12.7 mm while for type II the maximum aggregate size is 19 mm. The experimental investigation performed by Hirsch (1962) included several types of aggregates for use in concrete mixes. Various proportions of these aggregates were mixed with a selected cement paste in order to observe the effect of their batch proportions on the elastic modulus of the concrete. The secant moduli of the concrete specimens were determined from stress–strain curves as given by the slope of a straight line drawn from the point of zero stress to a stress of 6.9 GPa. The ITZ properties were selected as: thickness 50 μm, E = 12 GPa and ν = 0.35. The macroscopic modulus of elasticity is computed using the suggested development see Table 3. The finite element model generated using the presented algorithms for the case of 30% aggregate is shown in Figure 1. Each sphere in Figure 1 represents an aggregate surrounded by 50 μm thickness of ITZ. The stress influence functions used to obtain the macroscopic properties are given in Figure 2. Table 3 clearly shows that the numerical simulation is in excellent agreement with the experimental results. In addition, the finite element analysis results compare well with the classical boundaries of Hashin & Shtrikman (1963), Reuss (1929) and Voigt (1889) which give theoretical ranges for the effective elastic modulus of concrete with respect to a certain volume fraction of the constituents.

Figure 1. The unit cell finite element model and the aggregate distribution for the type I 20% aggregate concrete.

4.2

The Fuller curve

The same procedure is followed while, computing the macroscopic modulus of elasticity for the case of material properties and experimental data presented by Stock et al. (1979) as given in Table 4. In the experimental tests the aggregate size of 0.15 ÷ 19 mm were used while in the numerical model the smallest aggregate size is 2.36 mm. Stock et al. (1979) tested different aggregate volume fractions of values were the Elastic modulus of the concrete was measured on cylindrical specimens of 100mm in diameter and 300 mm in length. The modulus of elasticity was determined by the secant to 33% of the ultimate stress in compression. The ITZ properties were selected as: thickness 50 μm, E = 12 GPa and ν = 0.35.

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Figure 2. The Von-Misses stress influence functions for type I 20% aggregate concrete.

Table 3.

Elastic modulus of concrete-sieve analysis.

Aggregate Volume Current Experimental type fraction (%) research E (GPa) results E (GPa) I

20 30

23.65 26.35

26.10 30.75

II

20 30

23.36 25.83

22.96 27.44

Table 4.

Material data of constituents-Fuller curve.

Material property

Mortar material

Aggregate material

E (GPa) ν

11.6 0.20

74.5 0.20

Figure 3. The unit cell finite element model and the aggregate distribution for the Fuller curve 30% aggregate concrete.

The finite element generated using the presented algorithm for the case of 30% aggregate volume is shown in Figure 3 while stress influence function used to obtain the macroscopic properties is given in Figure 4. Each sphere in Figure 3 represents an aggregate surrounded by 50 μm thickness of ITZ. The finite element analyses of unit cells based on the Fuller curve aggregate distribution (see Table 5) gives good results in comparison with the experimental data of Stock et al. (1979). In summary, the results obtained via the finite element analysis are in good agreement with the experimental data of Hirsch (1962) and results obtained by Stock et al. (1979). In addition, all the computed results are completely within the classical boundaries of Hashin and Shtrikman (1963), Reuss (1929) and Voigt (1889) in spite of the fact that these boundaries

Figure 4. The Von-Misses stress influence function for the Fuller curve 30% aggregate concrete.

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Table 5.

Elastic modulus of concrete-Fuller curve.

Volume fraction (%)

Current research E (GPa)

Experimental results E (GPa)

20 40

15.95 22.33

15.64 23.22

are usually applicable to samples of infinitely large sizes. 5

SUMMARY AND FUTURE WORK

This paper outlines the development of a concrete unit cell used for multi scale analysis of concrete structures based on a two step homogenization approach. The creation of the concrete unit cell finite element model follows an analytical homogenization of the aggregates and the ITZs. By that we allow the ITZs to be excluded from the finite element model of the unit cell. This is essential to make the calculation practical, since the ITZ typical dimension are three orders of magnitude smaller than the aggregate typical dimension. The results, obtained using the presented algorithm, are in very good agreement with experimental results obtained by the sieve analysis case and the Fuller curve case. In future works the unit cell generator will be extended to include fibers for the analysis of fiber reinforced concrete structures in addition a non linear reduced order multi-scale analysis will be developed based on the generated unit cell to enable damage analysis of concrete/fiber reinforced concrete. REFERENCES Aboudi, J. 1991. Mechanics of Composite Materials—A Unified Micromechanical Appqroach. Amsterdam: Elsevier. Aboudi, J. 2003. Micromechanical Analysis of the Finite Elastic-Viscoplastic Response of Multiphase Composites. Int. J. Solids and Structures 40: 2793–2817. Aboudi, J., Pindera, M.-J. & Arnold, S.M. 2003. HigherOrder Theory for Periodic Multiphase Materials With Inelastic Phases. International Journal of Plasticity 19(6): 805–847. Asferg J.L., Poulsen, P.N. & Nielsen, L.O. 2007. A consistent partly cracked XFEM element for cohesive crack growth. Int. J. Numer. Meth. Engng 72: 464–485. Benssousan, A., Lions, J.L. & Papanicoulau, G. 1978. Asymptotic Analysis for Periodic Structures. Amsterdam: North-Holland. Bentz, D.P., Garboczi, E.J. & Stutzman, P.E. 1993. Computer modeling of the interfacial transition zone in concrete. In Maso, J.C. (ed.), Interfaces in cementitious composites, London: E. & F.N. Spon.

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Mang, H.A., Lackner, R., Meschke, G. & Mosler, J. 2003. Computational modeling of concrete structures. Comprehensive structural integrity. Numerical and computational methods 3: 541–606. Markovic, D. & Ibrahimbegovic, A. 2004. On micro–macro interface conditions for micro scale based FEM for inelastic behavior of heterogeneous materials. Comput. Methods Appl. Mech. Engrg. 193: 5503–5523. Matsui, K., Terada, K. & Yuge, K. 2004. Two-scale finite element analysis of heterogeneous solids with periodic microstructures. Computers and Structures 82: 593–606. Meschke, G. & Dumstorff, P. 2007. Energy-based modeling of cohesive and cohesionless cracks via X-FEM. Comput. Methods Appl. Mech. Engrg. 196: 2338–2357. Miehe, C. & Koch, A. 2002. Computational Micro-To-Macro Transition of Discretized Microstructures Undergoing Small Strain. Arch. Appl. Mech. 72: 300–317. Moes, N. & Belytschko, T. 2002. Extended finite element method for cohesive crack growth. Engineering Fracture Mechanics 69: 813–833. Mori, T. & Tanaka, K. 1973. Average Stress in the Matrix and Average Elastic Energy of Materials With Misfitting Inclusions. Acta Metall. 21: 571–574. Nadeau, J.C. 2003. A multiscale model for effective moduli of concrete incorporating ITZ water–cement ratio gradients, aggregate size distributions, and entrapped voids. Cement and Concrete Research 33: 103–113. Nilsen, A.U. & Monteiro, P.J.M. 1993. Concrete: a threephase material. Cem Conc Res 23(1): 147–51. Oliver, J. 1996. Modeling strong discontinuities in solid mechanics via strain softening constitutive equations. Part 1: Fundamentals. Part 2: Numerical simulation. International Journal for Numerical Methods in Engineering 39: 3575–3623. Pichler, C., Lackner, R. & Mang, H.A. 2007. A multiscale micromechanics model for the autogenous-shrinkage deformation of early-age cement-based materials. Engineering Fracture Mechanics 74: 34–58. Pichler, B., Hellmichz, C. & Mang, H.A. 2007. A combined fracture-micromechanics model for tensile strainsoftening in brittle materials, based on propagation of interacting microcracks. Int. J. Numer. Anal. Meth. Geomech. 31: 111–132. Prokopski, G. & Halbiniak, J. 2000. Interfacial transition zone in cementitious materials. Cem Conc Res 30(4): 579–83. Reuss, A. 1929. Berechnung der Fliessgrenz von Mischkristallen auf Grund der Plastizitatsbedingung fur Einkristalle. Z. Angew Math. Mech. 9: 49–58. Sanchez-Palencia, E. 1980. Non-Homogeneous Media and Vibration Theory: Lecture notes in physics. NJ: Springer. Scrivener, K.L. & Nemati, K.M. 1996. The percolation of pore space in the cement paste/aggregate interfacial zone of concrete. Cem Conc Res 26(1): 35–40. Scrivener, K.L. & Gariner, E.M. 1988. Microstructural gradients in cement paste around aggregate particles. In Mindess, S., & Shah, S.P. (eds.) Bonding in cementitious composites; Mater Res Soc Symp Proc. Simeonov, P. & Ahmad, S. 1995. Effect of transition zone on the elastic behavior of cement-based composites. Cem. Conc. Res. 25(1): 165–76. Simone, A. & Sluys, L.J. 2004. The use of displacement discontinuities in a rate dependent medium. Comput. Methods Appl. Mech. Engrg. 193: 3015–3033.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Stress state influence on nonlocal interactions in damage modelling Cédric Giry, Frédéric Dufour & Jacky Mazars Grenoble Institute of Technology/UJF/CNRS Lab 3S-R, Grenoble, France

Panagiotis Kotronis Ecole Centrale de Nantes, Institut GeM, UMR 6183 CNRS, Nantes, France

ABSTRACT: This paper presents a modification of an integral nonlocal damage model used to describe concrete behaviour. It aims at providing a better treatment of areas close to a boundary and a fracture process zone where the interactions between points should vanish. Modifications on the original integral nonlocal model are introduced by considering the stress state of points in the weight function used to compute the nonlocal variables. Computations show that local information such as strain or damage profiles are significantly different, leading to a narrower region where damage equal to 1 in the case of the modified nonlocal model. It allows to better approach a discontinuity of the displacement field upon failure and thus, improves the estimation of the crack opening that has been developed in post-processing for this type of calculation. 1

INTRODUCTION

Damage models have been widely developed and used these last decades to describe the behaviour of concrete structures. Due to strain softening, these models sometimes involve media with microstructure (‘‘local’’ second gradient models) (Chambon et al. 2001), (Kotronis et al. 2005) or more often nonlocality that is introduced through integral (Pijaudier-Cabot and Bažant 1987) or gradient (Peerlings et al. 1996) regularisation techniques. Allowing retrieving the objectivity of the results, the nonlocal approach can find its origins in the micromechanics of cracks interaction (Bažant 1994). However, there are still some pending issues. The nonlocal interactions are set constant all along calculation leading to unacceptable damage profile at the end. A damage equal to 1 corresponds actually to a crack and should be concentrated on a line, whereas a band is obtained. Moreover, in the vicinity of a boundary, the weight function is chopped off and normalised. This results to a damage field preferably developed near the boundaries. It has already been pointed out that the regularisation technique should be altered close to boundaries (Krayani et al. 2009). Besides and in a more general sense, the apparition of a crack can be seen as the creation of a new boundary and thus, the interactions between points on both sides of this crack should vanish. A crack separates two areas that can no more interact with each other. As the nonlocality is the influence of a point on an other point, it is clear that the nonlocal model should evolve when cracks initiate. For the original nonlocal model, these two points still interact. As a consequence, a

new parameter should be introduced in the modelling reflecting the state of each point regarding the nonlocality. Several workers have proposed evolution law for the internal length, parameter governing the area of interaction, or equivalent parameter regarding the method of regularisation used. Geers and coworkers proposed (Geers et al. 1998) a direct coupling between the material length parameter and the local strain state of the material leading to a transient behaviour of the nonlocal effect and a localisation of the strain at high state of damage. Voyiadjis and Abu Al-Rub gave (Voyiadjis and Abu Al-Rub 2005) for gradient plasticity theory, an evolution equation for the length scale parameter depending on several parameters among which the plastic strain. This equation, based on experimental observations, leads to a decrease of the length scale parameter with the plastic strain. Desmorat and Gatuingt proposed (Desmorat and Gatuingt 2007) also an interesting aspect to improve the original integral nonlocal model. Instead of using an internal length to treat the interaction between two points, they introduced an internal time corresponding to the time needed for a wave to propagate from point x to point s. More particularly, it allows to reduce the interactions between two points across a crack as the wave has to turn around the crack to go from one point to the other. In spite of its direct physical considerations, this approach seems to be prohibitive regarding the computational time. In the same way, Pijaudier-Cabot and Dufour replace (Pijaudier-Cabot and Dufour 2010) the geometrical distance between two Gauss points used in the weight

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function by a distance that take into account the state of damage between these two points. This method, showing good results for 1D problem, can however present some difficulties in terms of computation when 2D or 3D specimen are studied. Indeed, the integration of damage along a path is not trivial to calculate. Bažant has developed (Bažant 1994) also an evolution of the shape of the domain of interaction depending on the state of stress and the location of the two points considered. These developments, based on micromechanics of cracks interaction, give an anisotropic additional nonlocal term to the original nonlocal parameter. In this development, the stress state is introduced in the regularisation however, a nonlocal term remains during all the calculation avoiding the possibility for a point to become local when damage evolves. For local second gradient models, the width of the localized zone can evolve freely (Kotronis et al. 2008): for a quasi-brittle (bi-linear) constitutive law the localization zone stays constant, whereas for a ductile (softening) constitutive law it decreases. A simple 1D damage model able to control a priori the evolution of the localisation zone has been proposed recently (Kotronis 2008). In the present paper, we propose an evolution of the regularisation that yields a decrease of interaction during the nonlinear process, as proposed by the different authors quoted previously, and depending on the state of stress in an approach similar as the one developed by Krayani and coworkers (Krayani et al. 2009). First, we present the original non local damage model. Then, we introduce the modifications according to the the previous remarks. The performance of the model is finally illustrated studying the case of a bar under tension.

2

σij and εkl are the components of the Cauchy stress and strain tensors, respectively (i, j, k, l ∈ [1, 3]) and Cijkl are the components of the fourth-order elastic stiffness tensor. D is a damage scalar variable quantifying material degradation that grows from zero (virgin material) to one (completely degraded material). Damage is determined as a linear combination of two damage variables (equation 2): Dt and Dc which are damage due to tension and compression respectively: D = αt Dt + αc Dc

The parameters αt and αc depend on the stress state. The damage evolution is characterised by the following law. Dc,t = 1 −

MODELS FOR DAMAGE AND POST-PROCESSING FOR CRACK OPENING

We recall hereafter the main equations involved in the nonlocal damage model considered in this study (Mazars 1986; Mazars and Pijaudier-Cabot 1996). Then, a modification of the weight function is proposed in order to deal with the problems mentioned previously. 2.1

Damage model

The scalar isotropic damage model developed by Mazars for describing the non linear behaviour of concrete under monotonic loading is used (Mazars 1986). The general stress-strain relationship is: σij = (1 − D)Cijkl : εkl

(2)

(1)

εD0 (1 − Ac,t ) Ac,t − Y exp[Bc,t (Y −εD0 )]

(3)

At , Bt , Ac and Bc are the parameters governing the shape of the evolution law. εD0 is the strain threshold for the initiation of damage. Classical values can be found in (Mazars 1986). The state variable Y definition is: Y = max(Y , εeq )

(4)

with Y = εD0 initially, and the equivalent strain εeq defined by Mazars as:   3  εi 2+ εeq =  (5) i=1

εi + denotes the positive parts of the principal strains. This constitutive relation exhibits strain softening. As a consequence, a regularisation technique should be used in order to avoid mesh dependency and illposedness of the governing equations of equilibrium. Among the existing techniques, we can quote the integral nonlocal or gradient enhanced models (see e.g. (Bažant and Jirasek 2002)). An internal length is added with these models allowing to dissipate energy in a non zero area. The developments made in the present work deal with the integral nonlocal model. 2.2

Classical integral nonlocal model

In the nonlocal damage model, the equivalent strain given by the equation 5 is replaced by an average equivalent strain εeq over a volume  in the equation governing the growth of damage as defined by Pijaudier-Cabot (Pijaudier-Cabot and Bažant 1987).  φ(x − s)εeq (s)ds ε eq (x) =   (6)  φ(x − s)ds φ(x − s) is the weight function defining the interaction between the point located at x considered and

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the neighbouring points located at s inside the volume of the structure . This formulation fulfils the requirement of non alteration of an uniform field. The basic nonlocal weight function is often taken as the polynomial bell-shaped function or here as the Gauss distribution function:  

2 x − s 2 (7) φ(x − s) = exp − lc

made by the authors to avoid this phenomenon by considering an evolution of the domain of regularisation in the vicinity of free boundaries. Another problem that has been pointed out, is the treatment of the fracture process zone at high damage level. On figure 2b), quarter of a plate with a central crack is represented. It is submitted to tension of equal magnitude in both directions. The circles given on the figure represent isovalues of the Gaussian function, corresponding to the influence of the point located at the center of these circles. In the classical nonlocal model, this point can influence a point on the other side of the crack as only the distance between them is considered in the model. However, the interaction should vanish as soon as the fracture process zone passes between the two points considered. A solution to this problem can be to define an evolution of the interaction that should decrease in the vicinity of a cracked area.

where lc is the internal length of the model and x − s the distance between points located at x and s. The point located at s (distributing point) influences its neighbourhood equivalently in all directions and during all the calculation (figures 1 and 2, b)) except close to boundaries. The shape of the Gaussian function depends only on the initial internal length. As a consequence, the area close to boundaries are not correctly taken into account. It can be observed an attraction of the damage by the boundaries due to the truncation of the volume of interaction (Krayani et al. 2009). Considering a point at the level of the boundary, the number of distributing points with an influence non negligible on it is divided by two. As a consequence, the term at numerator in equation 7 is also divided by two, leading thus to a double amplitude of the interactions close to the boundary. A first proposal was

2.3

Modified integral nonlocal model

In this paper, we propose to modify the integral nonlocal regularisation in order to deal with the problems quoted previously. We restrict the domain of application of our modification for problems with mode I crack opening under tensile stress. During the calculation, an evolution of the interaction between points is considered through a coefficient. This coefficient depends on the stress state of the distributed points and it is introduced in the Gaussian function. At each step, the stress state of the point located at s, expressed in its principal space, is known.It is denoted hereafter σI (s), σII (s) and σIII (s). We define an ellipsoid of center point the point located at s, corresponding to an homothety of the original Gaussian function with the following ratio along principal stress directions: | 1st axis : | σIf(s) t

Figure 1. Influence of a distributing point. Gaussian function. (a) modified nonlocal model, b) original nonlocal model).

2nd axis : | σIIft(s) | 3rd axis : | σIIIft(s) |

Figure 2. Isovalues of the influence of different points in the specimen (a) modified nonlocal model, b) original nonlocal model).

with ft the tensile strength of concrete. The choice of ft leads to no modification at peak. Furthermore, the length of the axes can not overcome 1 in order to prevent high values for principal stress in compression. In the original integral nonlocal model, the influence of point located at s is isotropic. We define now an anisotropic influence of point located at s depending on its stress state and the position of the point located at x. The weight function now reads:   2

2 x − s (8) φ(x − s) = exp − lc β(x, σ prin (s))

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with β(x, σ prin (s)) equal to the ratio of the homothety defined previously in the direction (s − x). In 3D, we obtain for it: β(x, σ prin (s))2 =

1 sin2 ϕ cos2 θ a2

+

sin2 ϕ sin 2 θ b2

+

cos2 ϕ c2

(9) By noting u and v, the vectors associated respectively to the highest and the lowest pincipal eigenvalues of the Cauchy stress tensor at the point located at s, we have θ = (u, s − x) and ϕ = (v, s − x). A point under high stress influences widely its neighbourhood whereas a point that encounters very low stress does not affect the surrounding points. Figure 1 (a)) gives an example of the modified Gaussian function. The coefficient β is also calculated for point in elastic domain in order to avoid the interaction of two non damage points across a crack that encounter unloading due to the creation of the crack. In order to avoid mathematical problems with zero stress for some principal directions and thus a zero volum ellipsoid, a minimal value for β is fixed. Furthermore, the stress state used numerically to modify the nonlocal regularisation is the one of the last converged step thus, the model remains explicit. The general algorithm of the original nonlocal is unchanged, only the calculation of the weight function is found affected. When we considered a point close to a free boundary, we have the following condition for its stress state: σ prin (s) · n = 0 with n normal vector to the boundary. As a consequence, for the modified nonlocal model, this point interacts only with other points in the direction parallel to the boundary. We retrieve here the result developed in (Krayani et al. 2009) with a vanishing length of the diameter of the ellipsoid perpendicular to the boundary. The advantage of this method compared with the modification of the domain of interaction as proposed in (Krayani et al. 2009), is its adaptability to various structures. Indeed, there is no need to give the area for which the connectivity needs to be modified as it is directly taken into account through the stress state; the modification is based on mechanics rather than on geometry. For the case of study on figure 2 (a), we observe an evolution of the isovalues of the Gaussian function corresponding to the influence of different points in the specimen. The upper point is in a non disturbed area, as a consequence, with the prescribed loading, the isovalues are close to circles. For the middle point, in the vicinity of the crack tip, its stress state is highly disturbed and oriented, leading to ellipses for the isovalues. The lower point is shielded by the crack, as a consequence, it encounters very low stress state. In other words, it has a local behaviour and no influence on the surrounding points.

2.4

Crack opening estimation

A post-processing method has been developed to estimate crack opening in mode I from a continuous modelling calculation. First, the finite element calculation is performed. Then, the location of the crack is determined by using the maximum state variable. The profile of the crack obtained is discretized in several segments. For each one, a perpendicular profile segment is defined in order to calculate the crack opening (Bottoni and Dufour 2010). For each profile, projection of the equivalent strain by interpolation is made. Then, the method developed by (Dufour et al. 2008) to estimate the crack opening is used. Analytically, the crack is considered as a displacement jump [u] on a 1D profile at the location of the crack (figure 3). It corresponds to a Dirac when we consider the strain profile (figure 4). The concept of the method is to compare the strain profile obtained numerically with the one obtained analytically by considering a strong discontinuous displacement. However, the strain profile obtained analytically corresponds to a Dirac which can be hard to use directly for comparison. As a consequence, this profile as well as the one obtained numerically are regularised by means of a convolution product (figure 5). Finally, hypothesis of equal magnitude of the strain at the location of the crack leads to an estimation of the crack opening (equation 10): [U ] =

ε¯ eq (x0 ) φ(0)

(10)

An advantage of this method is to give an estimation of the error made between the analytical profile and the numerical one.

Figure 3.

Displacement profile of the strong discontinuity.

Figure 4.

Strain profile of the strong discontinuity.

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Figure 6.

Figure 5.

Mesh of the bar.

Nonlocal strain profile of the strong discontinuity.

An aim of the modification made on the nonlocal regularisation is to better treat the post-peak behaviour of concrete structure. By reducing the highly damage area, we get closer to a strong discontinuity and a Dirac for the strain profile. We can expect to reduce the error as defined previously.

3

Figure 7. Evolution of the force versus displacement for the modified (31 elements: dot-and-dash line) (61 elements: continuous line) and original nonlocal models (61 elements: dash line).

EXAMPLE

The test used here to illustrate the functioning of the modified nonlocal formulation is a bar under tension with a 2D geometry and a 1D loading (no Poisson’s effect). In spite of its simplicity this test allows to show the change in the behaviour obtained with this improvement more particularly when dealing with high damaged state. All the calculations presented in this paper are made with the finite element code Cast3M developed by the C.E.A. (French Nuclear Agency)(wwwcast3m.cea.fr/). The bar is 1 m long and 0.01 m high (figure 6). Only one element is put on the height. We have used square elements with linear interpolation. The following parameters have been taken for the damage model describing concrete. The central element is weakened with a lower Young’s modulus in order to initiate the damage in the bar. Eb = 33.7 GPa

At = 1

Ebendo = 31 GPa

υ = 0.

Bt = 20000

An arc-length technique is used to follow such a global response which exhibits snap-back. 3.1 Global behaviour The objectivity of the results is studied. Two sizes of mesh are used (31 and 61 elements in the length of the bar). Figure 7 gives the evolution of the force versus displacement for both mesh size and nonlocal versions. We can observe the objectivity of the results for both. The small difference at the end for the modified nonlocal model is due to the fact that the smaller size mesh can more localised the damage at the end than the coarse one. This tendency is also observed in the case of gradient-enhanced damage model (Simone 2007).

We observe also that the modified nonlocal approach dissipate less energy than the original one for the same initial internal length since the area of interaction (and in some way of dissipation of the energy) reduces with the decreasing stress state. The peak force is identical as we recover the value of the internal length lc for the modified nonlocal model. 3.2

Local response

Locally, the figures 8, 9, 10 and 11 give the evolution of the strain and damage profiles along the bar for both nonlocal approaches and with 61 element in the mesh. The chosen state for the curves are from Fmax up to max 0.01Fmax with a step of F10 . The classical integral nonlocal method leads to a completely damage area spread over several elements when the force vanishes. At each step, the increase of damage is observed over a wider surface. This is due to the fact that the interactions between Gauss points for the original nonlocal model do not evolve during the test. In reality, we can suppose only one element should encountered a damage equal to 1 whereas the surrounding elements should be partially damaged. The intensity of damage should decrease as we go further from the weakened points. Experimentally, it could be seen by measuring the evolution of density of microcracks from the main crack up to the virgin area. The modified nonlocal model leads to more localised damage and strain profiles. The values given are projected on the nodes of the mesh and due to the linear interpolation in the elements, maximum strain and damage are reached for two nodes. Points with damage equal to one are concentrated in one element. As the Gauss points along the bar have encountered their

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Figure 8. model.

Evolution of the strain for the original nonlocal

Figure 11. model.

Figure 9. model.

Evolution of the strain for the modified nonlocal

Figure 12. Regularised effective strain profiles from the strong discontinuity approach (continuous line) and from FE computations using the original nonlocal model (continuous line with points).

Figure 10. model.

Evolution of the damage for the original nonlocal

peak stress, we observe a decrease of the interactions. Each point tends to be more local. As a consequence, the area with highest damage decreases at each step to finally be reduced to the weakened element introduce at the beginning of the calculation. For the strain field along the bar, we observe that with the modified nonlocal model, it approaches a Dirac better than with original damage model. Regarding crack opening the figures 12 and 13 compare the regularised strain profile obtained from FE calculation and from strong discontinuity at F =

Evolution of damage for the modified nonlocal

Figure 13. Regularised effective strain profiles from the strong discontinuity approach (continuous line) and from FE computations using the modified nonlocal model (continuous line with points).

0.01Fmax . One can see that at this state, the modified nonlocal model gives a strain profile closer to a strong discontinuity than the original one. The figures 14 and 15 give the evolution of the crack opening in function of the displacement and the error between the regularised FE strain profile and the regularised strong discontinuity strain profile in function of crack opening.

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Figure 14. Crack opening as the function of the displacement (modified non local: continuous line) (original non local: dash line).

given in a previous article by the 2nd author (Krayani et al. 2009). The modification has been illustrated through the example of a bar of concrete under tension. The objectivity of results is conserved and our proposal allows to get closer to a strong discontinuity at high damage state as the results on crack opening shows. This last result is an important point as we can see nowadays that the information on crack opening is a hot topic. It has been shown that continuous modelling can provide this kind of information. By improving the FE calculation and more particularly by taking into account the effect of a damage zone on the neighbourhood in the calculation, we although improve the results we can provide for the crack opening.

ACKNOWLEDGEMENTS Financial support from the Agence Nationale de la Recherche through the sustainable cities programme (contract VD08_323065) is gratefully acknowledged.

REFERENCES

Figure 15. Relative error as the function of the crack opening (modified non local: continuous line) (original non local: dash line).

The first part of the curves correspond to the error just after the peak. In both cases, the strain field is far from a Dirac as a consequence, a relatively high error is observed. However, when the crack initiates, the error with the strong discontinuity decreases faster in the case of the modified nonlocal model validating the observations we could have made previously on the strain and damage profiles. It confirms the capability of the modified nonlocal model to approach the discontinuity without loosing the advantages of the original nonlocal model.

4

CONCLUSIONS

We have proposed in this paper a modification of integral nonlocal model in order to adapt the regularisation close to boundary and fracture process zone in the case of mode I failure. The stress state of each point is used during the calculation in order to create an evolution of the interaction between points. Each point interacts with its neighbourhood in function of the intensity of its principal stress values. This modification has been presented in 3D case. It has allowed to retrieve results

Bažant, Z.P. (1994). Nonlocal damage theory based onmicromechanics of crack interactions. Journal of Engineering Mechanics 120, 593–617. Bažant, Z.P. and M. Jirasek (2002). Nonlocal integral formulations for plasticity and damage: Survey of progress. Journal of Engineering Mechanics 128, 1119–1149. Bottoni, M. and F. Dufour (2010). Topological search of the crack path from a damage-type mechanical computation. In EURO-C 2010 Computational Modelling of Concrete Structures. Chambon, R., D. Caillerie, and T. Matsushima (2001). Plastic continuum with microstructure, local second gradient theories for geomaterials, localization studies. International Journal of Solids and Structures (38), 8503–8527. Desmorat, R. and F. Gatuingt (2007). Introduction of an internal time in nonlocal integral theories. Internal report LMTCachan, number 268, year 2007, ENS Cachan/CNRS/ Université Paris 6/PRES Universud Paris. Dufour, F., G. Pijaudier-Cabot, M. Choinska, and A. Huerta (2008). Extraction of a crack opening from a continuous approach using regularized damage models. Computers and Concrete 5(4), 375–388. Geers, M., R. de Borst, W. Brekelmans, and R. Peerlings (1998). Strain-based transientgradient damage model for failure analyses. Computer Methods in Applied Mechanics and Engineering 160, 133–153. Kotronis, P. (2008). Stratégies de Modélisation de Structures en Béton Soumises à des Chargements Sévères. Habilitation à Diriger des Recherches, Université Joseph Fourier, http://tel.archives-ouvertes.fr/tel-00350461/fr/. Kotronis, P., S. Al Holo, P. Bésuelle, and R. Chambon (2008). Shear softening and localization: Modelling the evolution of the width of the shear zone. Acta Geotechnica 3(2), 85–97.

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Kotronis, P., R. Chambon, J. Mazars, and F. Collin (2005). Local second gradient models and damage mechanics: Application to concrete, cd paper 5712. In ICF (Ed.), 11th International Conference on Fracture, 20–25 March. Krayani, A., G. Pijaudier-Cabot, and F. Dufour (2009). Boundary effect on weight function in non-local damage model. Engineering Fracture Mechanics 76, 2217–2231. Mazars, J. (1986). A description of micro- and macroscale damage of concrete structures. Engineering Fracture Mechanics 25(5–6), 729–737. Mazars, J. and G. Pijaudier-Cabot (1996). From damage to fracture mechanics and conversely: a combined approach. International Journal of Solids and Structures 33, 3327–3342. Peerlings, R.H.J., R. de Borst, W.A.M. Brekelmans, and J.H.P. de Vree (1996). Gradient enhanced damage for quasi-brittle materials. International Journal for Numerical Methods in Engineering 39, 937–953.

Pijaudier-Cabot, G. and Z. Bažant (1987). Nonlocal damage theory. Journal of Engineering Mechanics 113, 1512–1533. Pijaudier-Cabot, G. and F. Dufour (2010). Nonlocal damage model: boundary and evolving boundary effects. European Journal of Environmental and Civil Engineering (Submitted). Simone, A. (2007). Explicit and implicit gradientenhanced damage models. European Journal of Environmental and Civil Engineering 11/7- 8, pp.1023–1044. Voyiadjis, G. and R. Abu Al-Rub (2005). Gradient plasticity theory with a variable length scale parameter. International Journal of Solids and Structures 42(14), 3998–4029.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

A model for the uniaxial tensile behaviour of Textile Reinforced Concrete with a stochastic description of the concrete material properties J. Hartig & U. Häußler-Combe Institute of Concrete Structures, Technische Universität Dresden, Dresden, Germany

ABSTRACT: In this contribution, a model for the uniaxial tensile behaviour of Textile Reinforced Concrete, which is a composite of a cementitious matrix and a reinforcement of multi-filament yarns of high-performance fibres processed to textiles, is enhanced with a stochastic formulation of the constitutive law for the cementitious matrix and the bond strength between matrix and reinforcement. The model, which is implemented in the finite element method, consists of one-dimensional bar elements and bond-link elements arranged in a regular lattice scheme. While the bar elements represent either the matrix or parts of the reinforcement, the bond elements are used to model the load transfer between those. The fluctuations of the material properties of the matrix are modelled by means of random fields. With this model parametric studies are performed to investigate the influence of the scatter in the material properties on the material behaviour of Textile Reinforced Concrete. 1

INTRODUCTION

Textile Reinforced Concrete (TRC) is a composite consisting of a matrix of fine-grained concrete with a maximum aggregate size of usually 1 mm and a reinforcement of multi-filament yarns of high-performance fibres processed to textiles. Usually alkali-resistant glass or carbon fibres are applied, which do not suffer from corrosion in the matrix. This facilitates the construction of thin structural elements with a thickness of a few centimetres because no additional concrete layer for corrosion protection is necessary as in the case of ordinary steel reinforced concrete. There are multiple fields of application as for instance thin shell structures or the strengthening and retrofitting of existing reinforced concrete structures. In both fields, first applications were realised, see for instance (Curbach et al. 2007) and (Weiland et al. 2008). A special characteristic of TRC compared to other composites as for instance fibre-reinforced polymers is that the yarns are usually not completely penetrated with matrix due to the high viscosity of the cement slurry and filtrating effects due to the small distances between the filaments. Besides other reasons, this leads to a complex material behaviour of the composite even in the case of uniaxial tensile loading, which to model is the topic of this contribution. Although the comprehension of the material behaviour of TRC and its controlling mechanisms has been improved in the recent past, still open issues exist. For instance, the question of how scattering material parameters of the constituents of the composite influence its material behaviour needs more elaboration. However, there already exist a number

of works concerning scattering material parameters in TRC. In (Chudoba et al. 2006) and (Voˇrechovský & Chudoba 2006), the influence of scattering material parameters of plain multi-filament yarns were studied. Furthermore, (Cuypers & Wastiels 2006) developed an analytical model for the uniaxial tensile behaviour of TRC incorporating a stochastic description of the tensile strength of the matrix and (Konrad et al. 2006) studied the influence of scattering reinforcement and bond properties on the performance of cracks bridged with multi-filament yarns. In this contribution, a reduced two-dimensional finite element model for the simulation of the uniaxial tensile behaviour of TRC previously published in (Hartig et al. 2008) is enhanced concerning the description of the material properties. In the model, the matrix and reinforcement are represented by serial connections of one-dimensional bar elements, which are connected at corresponding nodes with zero thickness bond elements having nonlinear bond laws. The reinforcement is subdivided into a number of so-called segments to consider variations of the bond conditions over the cross section. The model incorporates further material nonlinearities as for instance limited tensile strength for matrix and reinforcement. Hitherto, the material properties were modelled deterministically. This approach seems to be not very realistic as the material properties show usually more or less scatter. Therefore, random fields are applied as they offer a good means to model smooth fluctuations of the material properties. Here, an approach by (Voˇrechovský 2008) for the simulation of crosscorrelated random fields incorporating KarhunenLoève expansion is used. The stochastic modelling

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of the material parameters is limited to selected properties of the matrix in a first approach. The contribution is structured as follows. In the subsequent section, the experimentally observable material behaviour of TRC exposed to uniaxial tensile loading is briefly described. Afterwards in Section 3, the formulation of the model is summarized regarding its geometrical properties and the deterministic as well as stochastic description of the material behaviour. In Section 4, results of parametric studies with stochastically modelled material properties of the matrix are presented. Finally, a short summary and some conclusions are given. 2

EXPERIMENTAL MATERIAL BEHAVIOUR

To set a stage, the material behaviour of TRC as it is observable under monotonic uniaxial tensile loading is described briefly in the following. The investigations will be limited here to the case of alkali-resistant glass as reinforcement. For the characterization of the material behaviour, tensile specimens are used by (Jesse 2004) with an overcritical amount of reinforcement, which means that the reinforcement can bear the forces redistributed from the matrix in case of matrix cracks, see the inset in Figure 1. In these tests, forces F and deformations u are measured, which are transferred to mean stresses by relating F to the cross-sectional area of the concrete and mean strains by relating u to the measurement length of 200 mm. The resulting stress-strain relations, see Figure 1, can be subdivided into three main states: the uncracked state, the state of multiple cracking of the matrix and the post-cracking state until failure. The stiffness of the uncracked state is mostly influenced by the Young’s modulus of the matrix. When the tensile strength of the matrix is reached for the first time, the state of multiple cracking starts associated with a reduction of the mean stiffness, which results 30 ultimate state

post-cracking state

mean stress [N/mm2]

multiple 25 cracking

from further matrix cracks and sudden stress drops. During the stress drops the reinforcement is activated along the so-called load transmission length starting at both crack faces. With increased loading more cracks develop on increasing stress levels, which can be explained at least partially with scatter of the matrix tensile strength. When the crack spacing becomes too small to retransmit enough forces back from the reinforcement to the matrix to reach the matrix tensile strength again, cracking stops and the post-cracking state starts. In this state, the stiffness increases again, which is mostly influenced by the Young’s modulus of the reinforcement. However, the stiffness in the post-cracking state is lower compared to the stiffness of the plain reinforcement. A final explanation for this effect is missing so far. Furthermore, also the matrix participates in load-carrying between the cracks in the post-cracking state leading to the well-known tension stiffening effect. The specimen fails usually in a quasi-brittle manner when the tensile strength of the reinforcement is reached as well. The latter statement appears quite simple. However, the effects leading to the failure of the reinforcement are rather complex because of the vast number of filaments, all with fluctuating tensile strengths in longitudinal and transverse direction. It is beyond the scope of this paper to analyse these effects. Nonetheless, there exists the class of fibre bundle models based on the initial work by (Daniels 1944), which is well suited for such problems, see (Chudoba et al. 2006; Voˇrechovský & Chudoba 2006). In this paper, the tensile strength of the reinforcement is assumed deterministic. Besides the characteristics described previously, TRC exhibits further special properties. As already mentioned in the introduction, in TRC multi-filament yarns are usually not completely penetrated with matrix. Thus, there exist essentially two zones with different bond conditions. In the so-called fill-in zone where the filaments are embedded in the matrix relatively strong adhesional bond exists. In contrast, in the core zone of the yarns only frictional load transfer at the contact points of the filaments is possible. This leads to inhomogeneous loading of the filaments, which also reduces ultimate loads of the composite.

20

3

F

15

specimen [mm]:

3.1

50 0

10

u

0

8

5

reinforcement: AR glass yarns (310 tex); 1.8 % fibre volume content

0

0.5

F

100

1.0 mean strain [%]

Figure 1. Experimental schematic test setup.

stress-strain

1.5

relations

2.0

and

MODEL Geometry

From the geometrical point of view, the model for the description of the uniaxial behaviour of TRC was already presented in detail in previous publications, see e. g. (Hartig et al. 2008). Thus, the following presentation is limited to the essential characteristics. The model can be classified as a regular lattice model consisting of one-dimensional bar elements and bond-link elements. The bar elements are used to

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dimensions of the specimens with 800 mm2 , see also Figure 1. The total cross-sectional area of the reinforcement is assumed with about 14.6 mm2 , which results from the cross-sectional area of one yarn of about 0.11 mm2 and a number of yarns of 133. This corresponds to a reinforcement ratio of about 1.8 %. The reduction of the cross-sectional area of the matrix due to the embedded reinforcement is neglected. For the bar elements, an element length of 0.2 mm is chosen, which leads with the specimen length of 0.5 m to 2500 elements per bar element strand.

model the axial rigidity and the failure of either the matrix or parts of the reinforcement while the bond elements reflect the force transmission between both components. In the model, the bar elements are arranged as serial connections, which are called here strands. While the matrix is always modelled as one of such strands, the reinforcement is usually subdivided into so-called segments. In one segment, a number of filaments with similar material and bond properties are summarised, because it is hitherto impossible to model each filament separately. A schematic visualisation can be seen in Figure 2. In this figure also the regular connection scheme of the bar element strands with bond elements is observable. Two different bond laws hcr and hrr , see section 3.2, are used, which reflect the different load transfer mechanisms inside the yarns already mentioned. The determination of the geometrical properties of the bar elements and the bond elements is described in (Hartig et al. 2008). For the computations, a model with one matrix strand and five yarn segment strands is used. The filaments in the fill-in zone are modelled with one socalled sleeve segment corresponding to the assumption of global load sharing in this zone and the assumption that in circumferential direction of the yarns no essential differences in the material and bond properties exist. For the core of the yarns, local load sharing has to be assumed, which necessitates a finer resolution chosen in this case with four so-called core segments. The core segments have also the shape of annuli except for the innermost segment, which is a circle. The boundary conditions are applied at the first node of the concrete strand, which is fixed, and at the last concrete node where displacements are prescribed. The cross-sectional area of the matrix is given by the

Segmentation approach segment

3.2

Deterministic constitutive relations

The material behaviours of the matrix and the reinforcement are modelled linear-elastic up to failure, which is assumed to occur in a brittle manner. Thus, Hooke’s law for the relation between stresses σ and strains ε is applied σ = Eε

with the Young’s modulus E of the respective material equal to  E for σ ≤ ft E= (2) 0 for σ > ft where ft is the tensile strength of the respective material. While this is a priori an appropriate approach for the glass fibres, it is known that cementitious matrices like concretes and mortars show certain post-cracking resistance also known as tension softening. However, previous investigations, see (Hartig et al. 2009), showed that considering realistic values of the fracture energy, the stress-strain response of the composite is

Discretization Transverse direction:

matrix

Longitudinal direction: no matrix cracking

no matrix cracking

yarn sleeve direction (j )

(1)

u core direction (i)

i

j

i

h cr

x

h rr

longitudinal ng dinal inal direction (x ) r

h rr node

matrix element

reinforcement element

bond element

Deterministic constitutive relations

stress

bond stress

E 1

t

strain

Figure 2.

8

Bond law h rr : 10

max)

bond degradation

6 4 2 0

friction (sres, res) unloading 0.2 0.4 0.6 0.8 1.0 1.2 slip s [10-5 m]

[N/mm²]

[N/mm²]

ƒt

Bond law h cr : 10 (smax,

bond stress

Tensile material law for matrix and reinforcement:

8 6 4 2 0

(smax,

max)

= (sres,

res)

friction unloading

0.2 0.4 0.6 0.8 1.0 1.2 slip s [10-5 m]

Segment model (schematic) and deterministic constitutive laws.

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only less affected. Since it also further complicates the computations, tension softening of the matrix is neglected here. The bond behaviour between the matrix and the reinforcement as well as between parts of the reinforcement is modelled with bond laws formulated as bond stress-slip (τ -s) relations. As the bond law was also presented in some detail in (Hartig et al. 2008) only important properties will be described here. Corresponding to the two bond zones inside the yarns, two bond laws are used, which only differ concerning the parametrization. In the case of matrix-filament interaction in the fill-in zone, the bond law hcr starts with a steep increase corresponding to the assumption of adhesional load transfer until the bond strength τmax at the slip smax , see also Figure 2. Subsequently, bond degradation with a transition from adhesion to friction is assumed, which is finished at the residual bond stress value τres at the slip sres . Afterwards, purely frictional load transfer is assumed, which is modelled with a constant bond stress equal to τres . The bond law also includes a description of unloading based on the concept of plasticity. The interpolation between the supporting points of the bond law, e. g. (smax , τmax ) is performed with the PCHIP method, see (Fritsch & Carlson 1980). The load transfer in the core of the yarns is assumed to occur by friction. Thus, in the respective bond law hrr the values of τmax and τres are equal, see also Figure 2. The limited tensile strength of matrix and reinforcement as well as the nonlinear bond laws lead to a nonlinear system of equations, which needs to be solved with an incremental iterative solution method. Therefore, the BFGS approach in combination with line search, see for instance (Matthies & Strang 1979), is used. In the case of reaching the tensile strength of a bar element, the Young’s modulus is set to zero. In every load step, only one bar element is allowed to crack and subsequently the system is solved again at the same load level. The load steps are chosen such small that after a crack at least one load step without element failure follows. However, the resulting cracking stresses are always somewhat larger than the tensile strength depending on the load step length, which might influence a statistic evaluation of results of the model.

properties of both the aggregates and the hardened cement paste. Thus, the material properties of a certain material region, which is assumed to be a representative section of the whole body, are homogenized properties of the material points in this region. As the body is at least initially assumed to be a continuum, it might be assumed that also the material properties show some continuous distribution due to the influence of the closer neighbourhood. This influencing is called correlation in statistics and smooths the spatial distribution of the material parameters. Also between different material properties correlation might exist, which is called cross-correlation. A method to model smooth fluctuations of the material properties is given with random fields. A convenient approach for the simulation of cross-correlated random fields was presented by (Voˇrechovský 2008) and is applied here. As the concrete and the reinforcement are modelled each essentially one-dimensional according to Section 3.1, the approach by (Voˇrechovský 2008) can be substantially simplified. Because the bar and bond elements used in the model have only one integration point, the material properties are constant in one element and, thus, the number nRV of the random variables X of a random field describing one material property is equal to the number of considered elements nel , e. g. the number of bar elements representing the concrete. It is assumed that all random variables have the same underlying distribution function. Depending on the position, two random variables X and X  at the coordinates x and x are assumed to influence one another more or less, which can be modelled with the autocorrelation coefficient ρ(X, X  ) where the approach     |x − x | 2  ρ(X, X ) = exp − (3) lcor

3.3 Stochastic constitutive relations All described material parameters show more or less scatter in reality. While on the nanoscopic scale and eventually also on the microscopic scale the material parameters might fluctuate purely randomly, on the mesoscopic and the macroscopic scale some kind of homogenization occurs. For instance Young’s modulus and tensile strength of concrete on the macroscale, which is also the scale that is considered for the concrete in the model, depend on the respective

is used and where lcor is the correlation length, which determines the range of influence. Small values of lcor lead to very local influence while large values extend the influence to larger regions. The values of the mutual correlation of all random variables can be assembled in a so-called auto-correlation matrix Cauto , which is real, symmetric, dense and of order nRV . The Karhunen-Loève expansion is applied to characterize the random fields in a domain , which incorporates the solution of the Fredholm integral of the second kind defining an eigenvalue problem  Cauto (X , X  )ψ i (X  )ddim = λi ψ i (X ) (4) 

with eigenvectors ψ i and eigenvalues λi of Cauto . Equation (4) needs not to be solved explicitly. Cauto rather can be decomposed into independent matrices of eigenvectors  auto and eigenvalues auto via Cauto =  auto auto  Tauto .

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(5)

An advantageous property of this decomposition is that the only accumulation point of the eigenvalues is zero, see (Sudret & Der Kiureghian 2000), which can be used to reduce the number of random variables at a given accuracy by means of truncating the series after the nred -th term, see (Voˇrechovský 2008). The eigenvectors ψ i and the eigenvalues λi resulting from Equation (5) are used to expand the random field H H (X ) =

nred √ 

λi ξ i ψ i (X )

(6)

i=1

where ξ is a vector of length i of uncorrelated standard Gaussian random coefficients. In the one-dimensional case, H is a vector containing the realisations of the random field. The previously described procedure is only valid if the underlying distribution function is Gaussian. In the case of non-Gaussian distributions, which are often needed to model e. g. the tensile strength, the procedure is more complicated. At first, the correlation coefficients determined with Equation (3) have to be transformed from the non-Gaussian domain to the Gaussian domain. Therefore, usually the Nataf transformation is applied 

∞



∞



F −1 ( (X )) − E(X ) ρNG (X , X ) = D(X ) −∞ −∞  −1   F ( (X )) − E(X  ) × D(X  ) 





× φ2 (X , X , ρG (X , X ))dXdX





(7)

where ρNG (X , X  ) and ρG (X , X  ) are the correlation coefficients in the non-Gaussian and the Gaussian domain, E(.) and D(.) are expected value and standard deviation of the non-Gaussian variables, (·) is the standard Gaussian cumulative distribution function (CDF), F −1 (·) is the percent point function (PPF) of the non-Gaussian distribution and φ2 is the bivariate standard Gaussian probability density function (PDF). The solution of Equation (7) with respect to ρG (X , X  ) is quite complicated as it has to be performed iteratively and it is not always guaranteed to have a solution, see (Vio et al. 2001; Lebrun & Dutfoy 2009). Having solved for ρG (X , X  ) the next steps are identical to the case of a Gaussian distribution. Only at the end, the values of H corresponding to Equation (6) have to be transformed back to the (original) non-Gaussian domain by means of (X ) = F −1 [ (H (X ))], H

(8)

which (Grigoriu 1998) calls translation process. As mentioned previously, also between different material properties some correlation, called crosscorrelation in this case, might exist depending on

the underlying physical process, as e. g. for the tensile strength of the concrete and the bond strength between matrix and reinforcement. In the approach by (Voˇrechovský 2008), the cross-correlations between nprop different material properties, which are each modelled as random fields, are introduced with constant factors assembled in the so-called crosscorrelation matrix Ccross of order nprop . Ccross has entries in the range −1 to 1 with entries equal to 1 at the main diagonal and it is symmetric and dense. Similar to Cauto , the correlation coefficients of Ccross have to be transformed from the, in general, underlying non-Gaussian domain to the Gaussian domain using Equation (7). In contrast to the case of Cauto , the underlying distribution functions are not identical. Again, Ccross in the Gaussian domain has to be decomposed into eigenvectors  cross and eigenvalues cross similar to Equation (5). Again, an eigenvalue truncation is possible, but because nprop is usually small the reduction is often negligible. With the entries of Ccross a so-called block crosscorrelation matrix D of order nprop · nRV is virtually established, where only the main diagonal and the main sub-diagonals of the nRV × nRV partial blocks have non-zero entries. The eigenvectors  D and eigenvalues D of D are nRV -multiples of  cross and cross as (Voˇrechovský 2008) showed. This is used to calculate a cross-correlated random vector

χ D =  D D ξ (9) where ξ is a vector of independent standard Gaussian random variables of order nprop · nRV . χ D can be j splitted into subvectors χ D of order nRV where j = 1, 2, . . . , nprop . Cauto is identical for all nprop fields. Thus, the realisations of each random field H j corresponding to a certain material property can be calculated using Equation (6) with the eigenvectors ψ i , the eigenvalues λi of the (Nataf-corrected) auto-correlation matrix Cauto as well as the respective cross-correlated random j subvector χ D . Finally, the realisations of the random fields have to be transformed back into the (original) non-Gaussian domain via Equation (8) leading j . to random fields H In the following simulations, only the material properties of the concrete are modelled stochastically. Although, this would be also possible for the reinforcement in principle, this exceeds the scope of this paper and has to be devoted to further investigations. For the simulation of the random fields, a correlation length lcor of 2 mm is assumed, which corresponds to two times the maximum grain size of the matrix. This has to be seen as a first approach. A better approximation might be possible e. g. with the approach by (Baxter & Graham 2000). The expected value of the Young’s modulus of the matrix Ec , which is modelled with a logarithmic normal distribution,

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Between these material properties a relatively strong correlation is assumed while the correlation between fct and τmax is expected to be slightly higher than with Ec . This has to be seen also as a first approach. The realisations of the random fields are computed at the positions of the integration points of the bar elements. As the integration point positions of the bond elements and the respective bar elements do not coincide, an interpolation of the bond strength values is necessary, which is realised by means of the calculation of the mean value of the bond strength at the integration points of the two neighbouring bar elements of the bond element under consideration. Furthermore, it is assumed that the concrete only cracks in a range of 0.3 m in the centre of the model while at the ends in ranges of 0.1 m each corresponding to the clamping zones in the experiments the concrete is not allowed to crack. In contrast, Ec and τmax are modelled stochastically along the whole length of the model. In order to reduce disturbances of the random fields at the boundaries the discretization range is artificially increased for 0.02 m at both ends of the model.

4

RESULTS AND DISCUSSION

relations is carried out first. The material properties are assumed to be the mean values given in the previous section while the geometrical properties of the model were specified in Section 3.1. In Figure 3, the simulated stress-strain relation is observable where the stress is the calculated reaction force at the concrete’s end nodes divided by the cross-sectional area of the concrete while the strain is the mean strain of the concrete elements at a length of 0.2 m in the centre of the specimen. Despite the deterministic modelling of the material parameters, a relatively good agreement between the simulated and the experimental results is observable, cp. Figure 1. While the linear-elastic part corresponding to the uncracked state essentially coincides in the simulation with the experiments, most differences occur in the state of multiple cracking. The first difference is the stress where the first concrete crack occurs, which is considerably lower in the experiments. Furthermore, the experiments show a relatively smooth increase of the stress-strain relation during the crack development, which is not reproduced by this model. In the simulation, the cracks occur at several discrete stress levels. In the presented case, the last cracks develop between existing cracks at a mean strain of about 0.5%. Concerning the post-cracking state, the simulation overestimates the stiffness compared to the experiments. The calculated stiffness corresponds to the stiffness of the reinforcement, which is not the case in the experiments as mentioned in Section 2. As the ultimate strain in the simulation corresponds to those of the experiments, the ultimate stress is overestimated in the simulation. 4.2

Scatter of matrix tensile strength

In the following, the influence of the scatter of the matrix tensile strength on the material behaviour of TRC is investigated. Therefore, only the tensile 30 25 mean stress [N/mm2]

is assumed with 28,500 N/mm2 , according to (Jesse 2004), with a relative standard deviation of 2% based on tests on a similar matrix by (Brockmann 2005). The tensile strength of the matrix fct , which is modelled with a two-parametric Weibull distribution, is assumed with an expected value of 6.7 N/mm2 and a relative standard deviation of 10% corresponding to results of flexural tensile strength tests by (Jesse 2004). The Young’s modulus and the tensile strength of the reinforcement is modelled deterministically with values of 79,950 N/mm2 for the Young’s modulus and 1357 N/mm2 for the tensile strength, see (Abdkader 2004). The parametrisation of the bond laws is chosen corresponding to Figure 2. For the bond strength τmax of the bond law hcr , which has an expected value of 9 N/mm2 , a relative standard deviation of 10% corresponding to the tensile strength is assumed, because experimental results are missing. Furthermore, τmax is also modelled with two-parametric Weibull distribution. The cross-correlation between fct , Ec and τmax is assumed as ⎤ ⎡ (f ) ct 1.0 0.8 0.9 (E ) c 1.0 ⎦ Ccross = ⎣ 0.8 (10) 0.8 (τmax ) 1.0 0.9 0.8

20 15 10 5

4.1 Deterministic case

0

In order to determine the influence of the stochastic modelling of the material parameters on the simulated behaviour of the tensile specimen corresponding to Section 2, a simulation with deterministic constitutive

0

0.5

1.0 mean strain [%]

1.5

2.0

Figure 3. Simulated stress-strain relation with deterministic material laws.

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strength is modelled stochastically with expected value and standard deviation according to Section 3.3. Ten simulations with different realisations of the random field for the tensile strength are carried out, see Figure 4. As previously mentioned, limited tensile strength is only applied to the concrete elements in a range of 0.3 m in the centre of the specimen. The simulated stress-strain relations, see Figure 5, show a more realistic course compared to the deterministic calculation. As to be expected, the results of the simulations show scatter because of the stochastic modelling of the tensile strength. The state of multiple cracking is strongly influenced by the varying tensile strengths. At least the first crack can be forecast from the simulated strength distribution. For instance, for the simulation with the black strength distribution curve in Figure 4, the first crack will occur at the minimum at x = 0.1542 m. Subsequently, cracking depends also on the force transmission length between matrix and reinforcement while the local strength minima are only preference locations for 9 8 7 fct [N/mm2]

6 5 4 3 2 1 0

0.1

Figure 4.

0.15

0.2

0.25 x [m]

0.3

0.35

0.4

Realizations of random fields for fct .

cracks. Compared to the deterministic case, the stress at the first matrix crack decreases in all cases. Furthermore, contrary to the deterministic simulation, the stress levels of subsequent cracks increase successively similar to the experiments. However, at least the standard deviation seems to be chosen somewhat too high as the experimental results show a flatter mean slope at the begining of the multiple cracking state. For the post-cracking state, the statements given for the deterministic simulation apply. Additionally, the courses of the stress-strain relations in the post-cracking state show scatter. While this can be explained to some extent with different tension stiffening of the matrix due to different numbers of cracks also the slope slightly varies between the computations. This might be explained with different activations of the reinforcement via bond. 4.3

Scatter of matrix Young’s modulus

Further simulations are carried out to investigate the influence of variations of Ec along x. Therefore, again ten simulations were carried out with the stochastic properties according to Section 3.3. The respective realisations of the random fields are shown in Figure 6. Because of the smaller standard deviation of Ec compared to random fields for fct , the fluctuations are considerably lower. The simulated stress-strain relations are shown in Figure 7. In principle, similar effects as for fct should be observable as the matrix stress σc is coupled to Ec via Equation (1). As the standard variation is considerably smaller for Ec and the matrix is also connected to the reinforcement, the impact on the stress-strain relations should be, however, smaller. In Figure 8, the qualitative courses of σc before and after the first crack as well as Ec along x are shown for one of the simulations. The course of σc before the crack is smoother compared to Ec as the stiffness differences

30 35 30

20

25 Ec [103 N/mm2]

mean stress [N/mm2]

25

15

20 15

10

10

5 0

5

0

0.5

1.0 mean strain [%]

1.5

2.0

0

Figure 5. Simulated stress-strain relations with stochastic modelling of fct .

0

Figure 6.

0.1

0.2

x [m]

0.3

0.4

Realizations of random fields for Ec .

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0.5

are balanced by the activation of the reinforcement via bond. Furthermore, it is observable that the matrix cracks in one of the stiffest elements first. Thus, the stress at the first crack is always smaller in the stressstrain relations compared to deterministic simulation although the effect is subordinated in this parameter combination. The same applies for the subsequent matrix cracking where at the beginning only small differences between the simulations with fluctuating Ec but also to the deterministic case exist. Scatter in the stress-strain relation starts to appear only at the crack development between existing cracks. Furthermore, the stresses where cracks occur successively increase, see Figure 7, in contrast to the deterministic simulation. As in the case of fluctuating fct , variations of the stress-strain courses between the simulations appear in the post-cracking state, which can be explained similar to the case of varying fct . 30

mean stress [N/mm2]

25 20 15

4.4

Scatter of bond strength

Another parametric study with stochastically modelled bond strength τmax of bond law hcr between matrix and reinforcement was performed. Again all other material properties were assumed to be deterministic and ten simulations were carried out. The courses of τmax are similar to those of fct , see Figure 4, as only the expected value is assumed differently between both properties. At least in this parameter combination, the stochastic modelling of τmax has only minor influence on the stress-strain relations, which are shown in Figure 9. As the bond laws influence the load transmission length between matrix and reinforcement, only the crack plateaus of the matrix cracks, which develop between existing cracks, show scatter in the stress-strain relations. Furthermore, it is observable that the cracking stresses do not successively increase as in the cases of stochastically modelled fct and Ec but several crack plateaus with almost constant stress develop. This can be explained with the constant value of fct , which does not allow for such stress fluctuations. The scatter in the post-cracking state appears somewhat lower compared to the previous cases and seems to be essentially a result of different extents of tension stiffening due to different numbers of cracks.

10

4.5

5

In a final parametric study, the three parameters fct , Ec and τmax , which were studied in the previous sections independently of each other, are modelled now simultaneously with cross-correlated random fields with the stochastic properties given in Section 3.3. In this study twenty simulations were carried out, where an exemplary realisation of the cross-correlated random fields for the three stochastically modelled properties

0

0

0.5

1.0 mean strain [%]

1.5

2.0

Figure 7. Simulated stress-strain relations with stochastic modelling of Ec .

matrix cracking allowed

Simultaneous scatter of matrix tensile strength, matrix Young’s modulus and bond strength

30 25 mean stress [N/mm2]

Ec σc (right before first crack) σc (right after first crack)

20 15 10 5

0

0.1

0.2

x [m]

0.3

0.4

0

0.5

Figure 8. Qualitative courses of Ec and σc along x before and after cracking.

0

0.5

1.0 mean strain [%]

1.5

2.0

Figure 9. Simulated stress-strain relations with stochastic modelling of τmax of bond law hcr .

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is observable in Figure 10. The cross-correlations between the different properties are observable. In Figure 11, the stress-strain relations resulting from the simulations are shown. The courses are similar to those of the simulations with varying fct , which seems reasonable because this property appears to have the strongest impact on the stress-strain behaviour with the chosen parameter combination as the previous parametric studies showed. However, the strong cross-correlation between the properties should lead to a smaller variation of the matrix cracking stresses because now two types of preference locations for cracking exist. As shown in the Sections 4.2 and 4.3, both locations with low fct and high Ec are preferred for matrix cracking while in this parameter combination the locations with low fct have more impact. Regarding the post-cracking state, the variation of the slopes increases compared to the previous parametric studies, which needs further investigations to be explained completely. However, the explanations given in Section 4.2 on this issue certainly apply. The influence of the variation of τmax , which was already Ec [103 N/mm 2]

35 30

fct [N/mm 2]

max

[N/mm 2]

25 12 10 8 6 4 9 8 7 6 5 4 0

0.1

0.2

0.3

0.4

0.5

x [m]

Figure 10. Realizations of cross-correlated random fields for fct , Ec and τmax . 30

mean stress [N/mm2]

25

relatively small when it was varied exclusively, seems to be completely masked by the effects of fct and Ec . 5

SUMMARY AND CONCLUSIONS

In this contribution, a model for the uniaxial tensile behaviour of TRC with a stochastic constitutive law for the concrete and the bond between concrete and reinforcement was presented. Based on parametric studies on tensile specimens the applicability of the model was shown and first conclusions can be drawn. For instance, it can be concluded that although the variations of the matrix tensile strength has the largest influence on matrix cracking also the other stochastically modelled parameters influence the cracking behaviour of the matrix. This complicates a clear identification of the respective impact if all material parameters are modelled simultaneously in a stochastic manner. Furthermore, the estimation of stochastic material properties is difficult, especially for the tensile strength where only minimum values are accessible by means of experiments but the whole range of occurring values is needed to establish a distribution function. Although is was not studied in this work, the results of the simulations strongly depend on the chosen correlation length where a consistent theory for its estimation seems to be missing. Furthermore, the enhanced model allows for investigations concerning size effects. While the description of the energetic size effect was already possible with the deterministic model, the stochastic modelling of the material properties allows also to incorporate the statistical size effect. However, to complete the model also the material properties of the reinforcement need to be modelled stochastically. While the application of the random fields to the reinforcement is unproblematic, the considerable increase of the system size, which is necessary for a sufficiently fine discretization of the reinforcement up to single filaments, is still challenging. ACKNOWLEDGEMENTS

20

The authors gratefully acknowledge the financial support of this research from Deutsche Forschungsgemeinschaft DFG (German Research Foundation) within the Sonderforschungsbereich 528 (Collaborative Research Center) ‘‘Textile Reinforcement for Structural Strengthening and Retrofitting’’ at Technische Universität Dresden.

15 10 5 0

0

0.5

1.0 mean strain [%]

1.5

2.0

REFERENCES Figure 11. Simulated stress-strain relations with simultaneous stochastic modelling of fct , Ec and τmax of bond law hcr .

Abdkader, A. (2004). Charakterisierung und Modellierung der Eigenschaften von AR-Glasfilamentgarnen für die

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Betonbewehrung. Ph. D. thesis, Technische Universität Dresden, Dresden. Baxter, S. & Graham, L. (2000). Characterization of random composites using moving-window technique. Journal of Engineering Mechanics 126(4), 389–397. Brockmann, T. (2005). Mechanical and fracture mechanical properties of fine grained concrete for textile reinforced composites. Ph. D. thesis, RWTH Aachen, Aachen. Chudoba, R.; Voˇrechovský, M. & Konrad, M. (2006). Stochastic modeling of multi-filament yarns. I.Random properties within the cross-section and size effect. International Journal of Solids and Structures 43, 413–434. Curbach, M.; Graf, W.; Jesse, D.; Sickert, J. & Weiland, S. (2007). Segmentbrücke aus textilbewehrtem Beton: Konstruktion, Fertigung, numerische Berechnung. Beton- und Stahlbetonbau 102(6), 342–352. Cuypers, H. & Wastiels, J. (2006). Stochastic matrix-cracking model for textile reinforced cementitious composites under tensile loading. Materials and Structures 39, 777–786. Daniels, H. (1944). The statistical theory of the strength of bundles of threads. i. Proceedings of the Royal Society of London A183, 405–435. Fritsch, F. & Carlson, R. (1980). Monotone piecewise cubic interpolation. SIAM Journal on Numerical Analysis 17(2), 238–246. Grigoriu, M. (1998). Simulation of stationary non-gaussian translation processes. Journal of Engineering Mechanics 124(2), 121–126. Hartig, J.; Häußler-Combe, U. & Kitzig, M. (2009). Effect of matrix tension softening on the uniaxial tensile behaviour of textile reinforced concrete. In Smojver, I. & Sori´c, J. (eds.), Proceedings of the 6th International Congress of Croatian Society of Mechanics (ICCSM), Dubrovnik, 30 September–2 October 2009, p. 46 (Book of Abstracts and CD of full papers). Zagreb: Croatian Society of Mechanics. Hartig, J.; Häußler-Combe, U. & Schicktanz, K. (2008). Influence of bond properties on the tensile behaviour of textile reinforced concrete. Cement & Concrete Composites 30(10), 898–906.

Jesse, F. (2004). Load Bearing Behaviour of Filament Yarns in a Cementitious Matrix (in German). Ph. D. thesis, Technische Universität Dresden, Dresden. Konrad, M.; Jerabek, J.; Voˇrechovský, M. & Chudoba, R. (2006). Evaluation of mean performance of cracks bridged by multi-filament yarns. In Meschke, G.; de Borst, R.; Mang, H. & Bicanic, N. (eds.), Computational Modelling of Concrete Structures—Proceedings of the EURO-C 2006, Mayrhofen, 27–30 March 2006, pp. 873–880. London: Taylor & Francis. Lebrun, R. & Dutfoy, A. (2009). An innovating analysis of the nataf transformation from the copula viewpoint. Probabilistic Engineering Mechanics 24(3), 312–320. Matthies, H. & Strang, G. (1979). The solution of nonlinear finite element equations. International Journal for Numerical Methods in Engineering 14, 1613–1626. Sudret, B. & Der Kiureghian, A. (2000). Stochastic finite element methods and reliability: A state-of-the-art report. Technical Report UCB/SEMM-2000/08, Berkeley: University of California. Vio, R.; Andreani, P. & Wamsteker, W. (2001). Numerical simulation of non-gaussian random fields with prescribed correlation structure. Publications of the Astronomical Society of the Pacific 113, 1009–1020. Voˇrechovský, M. (2008). Simulation of simply cross correlated random fields by series expansion methods. Structural Safety 30(4), 337–363. Voˇrechovský, M. & Chudoba, R. (2006). Stochastic modeling of multi-filament yarns. II. Random properties over the length and size effect. International Journal of Solids and Structures 43, 435–458. Weiland, S.; Ortlepp, R.; Hauptenbuchner, B. & Curbach, M. (2008). Textile Reinforced Concrete for Flexural Strengthening of RC-Structures—Part 2: Application on a Concrete Shell. In A. Dubey (Ed.), ACI SP-250: Textile Reinforced Concrete, pp. 41–58. Farmington Hills: ACI.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Multi-axial modeling of plain concrete structures based on an anisotropic damage formulation M. Kitzig & U. Häußler-Combe Institute of Concrete Structures, Technische Universität Dresden, Dresden, Germany

ABSTRACT: In this contribution, a strain-based constitutive law for concrete within the framework of continuum damage mechanics is proposed. The model allows for the multi-axial simulation of predominantly tensile loaded plain concrete. A second-order integrity tensor is chosen as the internal damage variable to consider the phenomenon of load-induced anisotropy. The model is implemented in the finite element method. Hence, a fracture-energy based regularization approach is included to overcome the mesh-dependence of the computational results. This local regularization method is contrasted with a non-local model version, in which the damage variable is obtained based on weighted strain averages over a spatial neighborhood of the point under consideration. Consistent tangent stiffness formulations are derived for both model versions to achieve fast convergence within the incremental-iterative solution procedures. The applicability of the model and the influence of the used regularization technique on the numerically obtained results are demonstrated by means of the simulation of a well documented experiment with plain concrete specimens. 1

INTRODUCTION

Concrete is one of the dominating construction materials in various fields of civil engineering. The demand for slender supporting structures in combination with an increase of load-carrying capacity reserves requires the appropriate specification of the complex multiaxial material behavior in constitutive laws. Up to now, such universally valid descriptions only succeed with restriction, especially with regard to practical applications. The material characteristics of concrete are wellknown from a large number of comprehensive experiments. Investigations on concrete for biaxial and triaxial stress states were performed e.g. by (Kupfer & Hilsdorf 1969), (Imran & Pantazopoulou 1996), (Hussein & Marzouk 2000) and (Lee et al. 2004). Major characteristics of the quasi-brittle material are the strength increase under multiaxial compressive loading compared to the uniaxial compressive strength, the pronounced non-linear stress-strain response for tensile and compressive loading due to propagation and growth of micro-defects and a softening behavior after exceeding the limit states of the material. A further important material property is the loadinduced anisotropy, i.e. the phenomenon that the initially isotropic elastic stiffness becomes cumulatively anisotropic for increased loading. Continuum damage mechanics provides a constitutive theory for the macro-scale description of the progressive degradation of material stiffness due to micro-cracking. For a theoretical framework refer

e.g. to (Carol et al. 1994). The appropriate choice of the internal damage variable is one of the kernel issues of a damage formulation within this framework. This state variable, introduced to quantify the orientation and density of defects on the micro-scale, can be of scalar-valued, second-order or fourth-order tensor-valued type. The use of scalar internal variables enables the description of isotropic damage, wheras tensor-valued state variables are needed for anisotropic damage formulations. Detailed information can be found e.g. in (Lemaitre 1992), (Krajcinovic 1996) and (Skrzypek & Ganczarski 1999). In this contribution, a strain-based formulation is proposed, which allows for the multi-axial simulation of predominantly tensile loaded plain concrete structures. Based on the concept of effective and nominal stresses and strains as well as the energy equivalence requirement in terms of both quantities, the secondorder integrity tensor is chosen as the internal damage variable, see (Carol et al. 2001a). This choice enables the specification of an orthotropic material behavior. The eigenvalues of the state variable 1 can be understood as damage measures in the directions of the associated eigenvectors. These directions are established under the assumption that the principal axes of the strain tensor coincide with those of the incremental integrity tensor. Hence, load-induced anisotropy can be described. The proposed constitutive law is implemented in the finite element method. The use of material models with softening in numerical calculations leads to localization phenomena, i.e. the concentration of large

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strains in narrow bands, and to highly meshsensitive results. Hence, regularization methods are required to overcome the dependence of the results on the used discretization. A popular approach is based on a rescaling of the post-peak branch of the stress-strain curve as a function of the element size, see e.g. (Pietruszczak & Mróz 1981), (Bažant & Oh 1983), (Willam et al. 1986). Advanced techniques to ensure mesh insensitivity of numerical results exist in form of localization limiters. This group includes the Cosserat continuum theory, see e.g. (de Borst 1991), nonlocal models of the integral or gradient type, see e.g. (Pijaudier-Cabot & Bažant 1987) and (M¨uhlhaus & Aifantis 1991), and higher-order gradient approaches, see e.g. (Aifantis 1984). A computationally convenient localization limiting technique is provided by the concept of nonlocal weighted averaging of a proper quantity over a spatial neighborhood of the point under consideration. For the motivation and a survey of nonlocal integral formulations refer e.g. to (Bažant & Jirásek 2002). In this contribution, the constitutive law proposed in Section 2 includes a fracture-energy based regularization approach, which is described in Section 2.3, and a nonlocal version of the integral type presented in Section 3. Consistent tangent stiffness formulations are derived for both the local and the nonlocal model version to achieve fast convergence in incrementaliterative procedures like the Newton-Raphson method in the finite element analysis. The applicability of the constitutive law is demonstrated with the numerical simulation of a well documented experimental setup in Section 4. In this context, the independence of the computationally obtained structural response on the chosen discretization as well as the influence of the chosen regularization method on the determination of the localization zone geometry in the calculation is presented.

2

that the initially isotropic stiffness becomes cumulatively anisotropic for increased loading. This internal variable is obtained based on the concept of effective and nominal stresses and strains. Within this concept, effective quantities are defined as stresses and strains to which the material between the distributed microdefects is subjected, whereas nominal quantities are the stresses and strains that can be measured externally. The requirement of energy equivalence, i.e. the same elastic energy shall be stored in terms of the effective and in terms of the nominal quantities, leads to the choice of the second-order integrity tensor φ¯ as the internal damage variable, see e.g. (Carol et al. 2001a). According to this choice, E = λφ¯ ⊗ φ¯ + 2μφ¯ ⊗ φ¯

can be written for the secant stiffness tensor. In Equation (2), ⊗ and ⊗ symbolize the dyadic products (A ⊗ B)ijkl = Aij Bkl and (A ⊗ B)ijkl = 12 (Aik Bjl + Ail Bjk ) of second-order tensors A and B. Furthermore, the Lamé constants λ = ν 0 E 0 /[(1 + ν 0 )(1 − 2ν 0 )] and μ = E 0 /[2(1+ν 0 )] are defined in terms of the Young’s modulus E 0 and the Poisson’s ratio ν 0 of the initially isotropic material. 2.2

In the following, an anisotropic formulation for concrete within the framework of continuum damage mechanics is proposed. A characteristic equation of elastic degradation is the secant stress-strain relation σ =E:

(1)

Damage evolution and limit condition

The formulation of the damage evolution starts with the spectral decomposition of the incremental integrity ˙¯ The principal axes of the incremental damtensor φ. age variable are assumed to coincide with those of the (i) strain tensor  and the principal values φ˙¯ are related to damage measures D(i) according to  (i)  −D ˙ (i) d(i) ⊗ d(i) . φ˙¯ = φ˙¯ d(i) ⊗ d(i) = √ (i) 2 1 − D i=1 i=1 (3) 3

ANISOTROPIC DAMAGE CONSTITUTIVE LAW

2.1 Secant relations and damage variable

(2)

3

The vectors d(i) in Equation (3) are the eigenvectors of the strain tensor . The damage measures D(i) are functions of equivalent strain measures κd(i) ⎧ 0 κd(i) ≤ ed0 ⎪ ⎪ ⎞gd ⎛ ⎨ (i) ⎜ κd − ed0 ⎟ D(i) = (4) −⎝ ⎠ ⎪ ⎪ ed ⎩ (i) 1−e κd > ed0

which couples Cauchy stresses σ and elastic strains  via the fourth-order secant stiffness tensor E. Plastic strains are assumed to be negligible since predominant tension is considered. An internal damage variable must be chosen to describe the density and orientation of distributed micro-defects in the material. This variable must be a tensor of order two or higher to consider the load-induced anisotropy, i.e. the phenomenon

with (i) κd

(i) (i)

=β 

,

β

(i)

=

αt  (i) ≥ 0 . −1  (i) < 0

(5)

This coupling enables the description of damage in three mutually orthogonal material directions. Values for the damage measures D(i) lie within the interval

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[0, 1] with D(i) = 1 representing the completely damaged state in direction i. In Equation (5),  (i) are the eigenvalues of the strain tensor, while the damages parameters αt , ed0 , ed and gd in Equations (4) and (5) are material constants, which can be obtained from uniaxial stress-strain relations. Their determination is described in detail in (Häußler-Combe & Hartig 2008). Values for the damage parameters are listed in Table 1 for the particular concrete of the L-shaped panel test described in Section 4. The stress-strain relation for tensile loading in the uniaxial special case is shown in Figure 1. ˙ (i) remain to be defined. The damage multipliers D For this purpose, loading functions Fd(i) (i)

(i)

Fd = β (i)  (i) − κd

(i)

˙ (i) ≥ 0, D

˙ (i) = D

β (i) h(i) d

(d(i) ⊗ d(i) ) : ˙

(8)

(i) (i) with 1/h(i) d = ∂D /∂κd . Substitution of Equation (8) in Equation (3) results in the formulation for the incremental damage variable φ˙¯ (i)

 −β (i) /h d φ˙¯ = (d(i) ⊗ d(i) ⊗ d(i) ⊗ d(i) ) : ˙ . √ (i) 2 1 − D i=1 3

(9)

(6)

˙ (i) F (i) = 0. D d

(7)

The corresponding consistency conditions F˙ d(i) = 0 lead to the definition of the damage multipliers Table 1.

(i)

˙ (i) = 0 β (i) (d(i) ⊗ d(i) ) : ˙ − h(i) d D

are introduced which control the evolution of damage and differentiate between damage in tension and compression. Damage can increase only if the current material state reaches the boundary of the elastic domain Fd(i) = 0. This is expressed by the loadingunloading or Kuhn-Tucker conditions Fd(i) ≤ 0,

(i)

∂Fd ∂Fd ∂κd (i) ˙ =0 D : ˙ + (i) ∂ ∂κd ∂D(i)

Material parameters for the L-shaped panel test.

2.3

Regularization based on fracture energy

The application of the proposed damage law in the tension regime leads to a limited tensile strength followed by a descending branch in the stress-strain relation, see Figure 1. It is well-known that the use of material models with softening for numerical calculations leads to localization phenomena, i.e. the concentration of large strains in narrow bands or localization zones, respectively, and to highly mesh-sensitive results. A fractureenergy based calibration of the softening part of the uniaxial stress-strain curve provides an opportunity to overcome this mesh dependence. For this purpose, the equivalent strains κd(i) are replaced by modified values (i) κd,mod

L-shaped panel Damage parameter ed0 Damage parameter ed Damage parameter gd Damage parameter αt Young’s modulus E 0 Poisson’s ratio ν 0 Strengths fct / fc Fracture energy GF

[-] [-] [-] [-] [MN/m2 ] [-] [MN/m2 ] [Nm/m2 ]

(i) κd,mod = κd,fct + (1 − γ )κd,fct ln

−1.99 · 10−4 3.07 · 10−3 2.00 11.49 25, 850 0.18 2.7 / − 31.0 90

κd(i) − γ κd,fct (1 − γ )κd,fct

(10)

if and only if the equivalent strain κd,fct corresponding to the concrete tensile strength is exceeded. Otherwise, (i) (i) the relation κd,mod = κd holds. This is demonstrated in Figure 2. The modified equivalent strains shall

Kd,mod 0.005

without regularization = 0.2 = 0.3 = 0.4

0.004 0.003 0.002 0.001

Kd Kd,fct

Figure 1.

Uniaxial tensile stress-strain curves.

Figure 2.

0.004

0.006

Modified equivalent strains.

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0.008

ensure a continuously differentiable stress function at the position of the maximum stress. This requirement is fulfilled by the approach presented in Equation (10). In this equation, γ is a further model parameter depending on the characteristic length lel,ch of the elements used in the finite element computation. Values for γ can be obtained by means of involvement of the fracture energy GF , i.e. the energy required to induce a tensile crack of unit area. For the determination of γ it is assumed that the limit strain κd,fct is exceeded within the whole width bcal = f (lel,ch ) of the localization zone in the calculation. Then the regularization parameter γ can be derived from the relations for the specific fracture energy gf ,cal , which is the area below the uniaxial stress-strain curve   GF (i) gf ,cal = = σ (κd,mod , ) d . (11) bcal fct

the accurate formulation of the tangent stiffness tensor Etan to achieve fast convergence. The incremental constitutive relation, which relates increments of the stresses with increments of the strains, is obtained by means of differentiation of Equation (1) with respect to time: ˙ :  = Etan : ˙ . σ˙ = E : ˙ + E (12)

With a presumed width of the localization zone, which varies between one and two times the characteristic element length, Equation (11) can be solved for γ , see Figure 3. For the numerical simulation of the L-shaped √ panel test described in Section 4 a value bcal = 2lel,ch is chosen. Modified uniaxial stressstrain relations are shown in Figure 1. The advantages of the fracture-energy based regularization are the numerical robustness and relatively small effort with respect to the implementation in the finite element method. On the other hand, an essential drawback is the fact that the width of the localization zone must be a priori predefined in dependence on the element length by the choice of bcal . This disadvantage can be eliminated by means of a refined regularization technique based on weighted averaging of a local quantity over a certain neighborhood of the point under consideration, which is presented in Section 3.

˙ :  in Equation (12) needs further elabThe term E oration. The utilization of compact tensor formalism, see e.g. (Rizzi & Carol 2001), and of the symmetries of both the strain and the integrity tensor, leads to the expression ¯˙ E˙ :  = [λ(φ¯˙ ⊗ φ¯ + φ¯ ⊗ φ) ˙¯ :  + 2μ(φ˙¯ ⊗ φ¯ + φ¯ ⊗ φ)] = [λ((φ¯ : )Is + φ¯ ⊗ ) ¯ + (φ) ¯ ⊗ I )] : φ˙¯ + 2μ(I ⊗ (φ) ˙¯ ˆ : φ, =E

which in association with Equation (9) can be substituted in Equation (12) to obtain the nonsymmetric fourth-order tangent stiffness tensor Etan ˆ : Etan = E + E

The proposed constitutive equations are implemented in a finite element code. Equilibrium iterations are performed by means of the Newton-Raphson method. The use of this incremental-iterative procedure requires

Figure 3.

Mesh-dependent parameter γ .

3  −β (i) /h(i) d d(i) ⊗ d(i) √ (i) i=1 2 1 − D

⊗ d(i) ⊗ d(i) , Etan = E + E . 3 3.1

2.4 Tangent stiffness

(13)

(14)

NONLOCAL DAMAGE MODEL Nonlocal averaging

The constitutive law proposed in Section 2 is extended to a nonlocal model version. This advanced regularization method introduces an additional material parameter, the characteristic length, which can be understood as a measure of material inhomogeneities and controls the width of the localization zone in numerical calculations. The introduction of nonlocal approaches to model the multiaxial material behavior of concrete is motivated by a number of reasons. First of all, in continuum models the heterogeneous microstructure of concrete is homogenized on a small scale compared to the overall dimensions of the structure. The externally measurable stress, which is averaged over an representative volume element (RVE), depends not only on the strain in one material point, but on the average strain values within this RVE. A further cause is the mutual interaction of microcracks, which in dependence on their orientation and size can shield or amplify the stress intensity of one another. Then, microcrack growth is controlled not only by the stresses and strains

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in the particular point of the continuum representing the center of the crack, but by the energy release from a spatial domain around the crack. Finally, a different and not physically motivated reason for the introduction of nonlocal constitutive laws is the fact that damage models with strain-softening behavior lead to localization of damage in a zero thickness or volume zone, which is associated with a mesh-sensitivity in the finite element method. In this regard, nonlocal strain-softening models may serve as localization limiters. For more details regarding the motivations of nonlocality refer e.g. to (Bažant & Jirásek 2002). In this contribution, attention is focused on a nonlocal continuum model of the integral type. In this approach, the local elastic strains  are replaced by their nonlocal counterparts ˜ , which are obtained by weighted averaging over a spatial neighborhood of each point x under consideration  g(x, ξ ) (ξ ) dξ . (15) ˜ (x) = V

Based on the nonlocal quantity, the internal dam¯  ) is determined as a function of age variable φ˜¯ = φ(˜ the nonlocal strain tensor, and the secant stress-strain relation Equation (1) can be rewritten as ˜ : σ˜ = E(˜ ) :  = E

(16)

with ˜ = λφ¯˜ ⊗ φ¯˜ + 2μφ¯˜ ⊗ φ¯˜ . E

(17)

The weight function g in Equation (15) remains to be specified. Without an influence of a boundary of the considered body, g is only a function of the distance r = ||x − ξ || between the point x under consideration and the affecting point ξ in the neighborhood of x. Near a boundary, the nonlocal quantity is obtained by averaging only over the domain part that lies within the body. Hence, the weight function must be scaled to satisfy the normalizing condition  g(x, ξ ) dξ = 1 ∀x ∈ V . (18) V

This can be done by reformulating the weight function in the form 

g(x, ξ ) =  V

g (r) . g (||x − ζ ||) dζ 

(19)

The function g  can be chosen e.g. as the Gaussian distribution function or as a bell-shaped polynomial approach. In this distribution, a formulation ⎧ ⎪ 0 |r| > R ⎪ ⎨    2 g (r) = (20) r2 ⎪ ⎪ |r| ≤ R ⎩ 1− 2 R

Figure 4. Nonlocal damage law: (a) interaction length R, (b) weight function g.

is chosen, see Figure 4(b). The interaction length R determines how far away a point ξ can be located from the point x under consideration to have an influence on the nonlocal quantity. Thus, Equation (20) represents a bounded weight function, which is advantageous with respect to the bandwidth of the stiffness matrix if the nonlocal model is implemented in the finite element method. 3.2

Tangent stiffness

In analogy to Section 2, a consistent tangent stiffness formulation shall be derived also for the nonlocal model to achieve fast convergence in incrementaliterative procedures like the Newton-Raphson method. For this purpose, the incremental constitutive relations are derived in tensorial notation similarly to Section 2. Subsequently, these equations are rewritten in a form suited for the implementation in the finite element method. A respective derivation was shown for an isotropic damage law e.g. in (Jirásek & Patzak 2002). Firstly, the incremental integrity tensor based on the nonlocal strain tensor ˜ is obtained in analogy to (i) Section 2.2 if the local quantities D(i) , h(i) and d , β (i) (i) (i) ˜ , h˜ , β˜ (i) d are replaced by their counterparts D d and d˜ (i) determined from ˜ ˙ φ˜¯ =

3  −β˜ (i) /h˜ (i) d  (d˜ (i) ⊗ d˜ (i) ⊗ d˜ (i) ⊗ d˜ (i) ) : ˙˜ . ˜ (i) i=1 2 1 − D (21)

Differentiation of Equation (16) with respect to time leads to the incremental relation ˜ˆ ˙˜¯ ˜ : ˙ + E σ˙˜ = E˜ : ˙ + E˙˜ :  = E :φ =

˜ : ˙˜ . ˜ : ˙ + E E

(22)

˜ˆ similar to Equations (12), (13) and (14). E is obtained ˜ ˆ ¯ ¯ if φ is replaced by φ in the expression for E. In the following, the tangent stiffness is presented for the proposed anisotropic approach in the form

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required for the implementation in the finite element method. For this purpose, the notation for stresses, strains and elastic stiffness is changed from tensor to matrix and vector notation. Starting point is the vector f int of internal nodal forces  BT (x) σ˜ (x) dx (23) f int =

which in association with Equation (22) can be substituted in Equation (27): K=

f int =



wi BTi · σ˜ i

(24)

i

with the weight factors wi for the numerical integration. In a next step, σ˜ i is replaced by the expression ˜ i ·  i as the matrix notation of Equation (16) σ˜ i = E ˜ i and the (6 × 1) with the (6 × 6) elasticity matrix E local strain vector  i at the integration point under consideration. Substitution in the previous equation leads to  ˜ i · i. f int = wi BTi · E (25) i

The tangent stiffness matrix K is obtained as the derivative of the vector f int of internal nodal forces with respect to the vector u of the nodal displacements K=

∂f int . ∂u

= K sec +

=



∂

i



˜ ·B wi wj gij BTi · E i j

(29)

j

with the secant stiffness matrix K sec and the matrix ˜ of the fourth-order tensors E˜ and ˜ i and E notations E i ˜ respectively, from Equation (22). The obtained E, expression for K can immediately be utilized for the implementation in a finite element code. From Equation (29) it can be seen that the tangent stiffness is equal to the secant stiffness if for all integration points the stress-strain relations are either still in the linear elastic range or if unloading occurs according to the Kuhn-Tucker conditions Equation (7). Moreover, it becomes evident that the nonlocal tangent stiffness matrix turns out to be nonsymmetric. One of the reasons is the fact that the relation gij = gji for the weight function g is valid only if the mesh of the finite element model is regular and extends over an infinite domain. Since irregular discretizations with existing boundaries are used in practice, g is not symmetric with respect to the considered integration points i and j. The second reason for the nonsymmetry of K results from the expressions of the fourth-order ˜ˆ tensor E which possesses no major symmetry, but this holds for the respective quantity in the local model version, too. Hence, the nonsymmetry of the nonlocal tangent stiffness matrix means no important drawback compared to the constitutive relations presented in Section 2.

(26)

˜ i · i wi BTi · E ∂u

˜ tan wi BTi · E i · Bi .

j

 i

Substitution of Equation (25) in Equation (26) results in K=

   ˜ · ˜ i · Bi + E wi BTi · E w g B i j ij j

i

V

with the strain-displacement matrix B and the (6 × 1) vector σ˜ of the stress components. Within the finite element method, the integral in Equation (15) is replaced by the sum over the finite number of integration points i of the elements used for the discretization. For reasons of convenience, a quantity with subscript i denotes a scalar, vector or matrix at the integration point i with the coordinates xi in the following, e.g. Bi = B(xi ). Thus, Equation (23) can be rewritten in the form



(27)

i

Now we replace the integral form Equation (15) by the sum over all integration points j in the considered neighborhood of the integration point i, see Figure 4(a), to obtain  wj gij  j (28) ˜ i = j

4

APPLICATION

Experimental data are commonly used for the calibration of material models and for the demonstration of their applicability. The L-shaped panel (LSP) test performed by (Winkler 2001) provides appropriate data for the validation of constitutive models under mode-I conditions. The geometry and the boundary conditions of the test specimens are shown in Figure 5, values for the elastic constants, strengths and fracture energy, which are indicated in Table 1, are taken from (Feist et al. 2004). Two discretizations of the concrete specimen are investigated to demonstrate the meshobjectivity of the numerical results and the dependence of the computed crack paths on the mesh alignment. The first mesh consists of 2161 nodes and 2079 fournode quadrilaterals with regularly arranged elements in the region of the expected process zone, see Figure 6. In contrast, 1829 three-node triangular elements and

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500 250

220

30

uh

500

250

t = 100

250

F, uv

[mm]

Figure 5. L-shaped panel (LSP): geometry and boundary conditions. Figure 7.

LSP mesh 2: triangular elements.

mesh 1 nonlocal

mesh 1 local

mesh 2 local

exp. scatter

Figure 8.

Figure 6.

LSP mesh 1: quadrilaterals.

LSP: load-displacement uv relation.

mesh 1 nonlocal

968 nodes are used for the second discretization, see Figure 7. Plain stress states are considered in both cases. All nodes along the lower horizontal edge of the vertical leg are fixed. The vertical tensile load F is applied under displacement control. The displacements of the point of load application and of the upper left corner have been measured. First of all, computations were performed for both discretizations using the local fracture-energy based regularization technique presented in Section 2.3. The corresponding force-displacement relations for the point of load application are contrasted with the experimental data in Figures 8 and 9. The maximum test load is well approximated for both meshes, whereas the value for the triangular element mesh exceeds the one obtained with the quadrilateral mesh by 6%. This

mesh 2 local mesh 1 local

experiment

Figure 9.

LSP: load-displacement uH relation.

minor difference leads to variances in the descending branches, which nevertheless exhibit similar courses and qualitatively agree with the test data. Generally, the slope of the pre-peak load-displacement curve

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5

Figure 10.

(a)

(b)

(c)

(d)

Crack paths: (a) test; (b) mesh 2; (c), (d) mesh 1.

observed in the tests is considerably lower compared to the numerical results. This fact can be explained by the disregard of the elasticity of the support construction in the experiment, see (Kitzig et al. 2009). The involvement of a rigid body rotation of the specimen leads to an improved agreement of the numerical and the test data, which is not demonstrated in the present contribution. The computational crack paths are plotted for mesh 2 in Figure 10(b) and for mesh 1 in Figure 10(c) and are compared to the observed scatter in the experiment in Figure 10(a). For this purpose, the maximum eigenvalues of the tensorial damage variable φ¯ in the last calculated step are presented. It can be clearly seen, that the light inclination of the cracks in the test is adequately reproduced only by the discretization using triangular elements. In contrast, a straight horizontal crack arises for the regular quadrilateral mesh, which reflects the dependence of the path on the element alignment. However, the widths of the damaged bands are in accordance with the a priori predefined value bcal described in Section 2.3. A further computation was performed based on the nonlocal version of the constitutive law presented in Section 3 using the quadrilateral mesh. An interaction length R = 12 mm, see Equation (20), was chosen, which corresponds to 1.5 times the maximum aggregate size of the concrete used in the test. Again, the corresponding force-displacement relation for the point of load application is shown in Figures 8 and 9. The experimental maximum load is well approximated before numerical problems occur. The crack path for the last computed load step represented by the maximum eigenvalue of φ˜¯ is plotted in Figure 10(d). In contrast to the result of the local model version, this path exhibits a clearly observable inclination.

SUMMARY AND OUTLOOK

An anisotropic damage formulation was proposed, which allows for the simulation of predominantly tensile loaded plain concrete. The choice of the second-order integrity tensor as the internal damage variable allows for the consideration of the loadinduced anisotropy. The material directions were assumed to coincide with the eigenvectors of the strain tensor, while the decoupled damage measures were associated with equivalent damage strains in the respective directions. The proposed model was implemented in the finite element method. Hence, regularization techniques are necessary to overcome the mesh-sensitivity of the numerically obtained results. For this purpose, firstly a local approach was described in Section 2 based on a fracture-energy dependent rescaling of the softening parts of the uniaxial stressstrain relations in the considered material point. Furthermore, a more refined regularization technique was formulated in Section 3 in association with a nonlocal version of the constitutive law including the replacement of the elastic strains by their nonlocal counterparts obtained by means of averaging over a certain neighborhood of the considered point. The major advantage of the latter method is the fact, that the width of the localization zone does not need to be a priori predefined as in the firstly mentioned approach. Indeed, the local regularization technique is more robust and less time consuming with respect to its implementation in numerical calculation procedures. Finally, consistent tangent stiffness formulations were derived for the local as well as for the nonlocal model version to achieve fast convergence in incremental-iterative procedures. The applicability of the proposed damage law was demonstrated in Section 4 by means of the simulation of a well-documented experiment with plain concrete specimens. Moreover, this section also served for the presentation of the characteristics of the implemented regularization techniques. The experimental loads were well approximated by both the local and the nonlocal model version. The width of the numerically obtained localization zone in the case of the energy-based regularization was in good agreement with the value bcal predefined in Section 2.3, whereat the slightly inclined crack paths observed in the tests could only adequately be reproduced using the nonregular discretization. Furthermore, the maximum test load as well as the inclined band of damage were proven to be numerically obtained if the nonlocal regularization technique is applied and the interaction length R is properly chosen in dependence on the aggregate size, which can be understood as a measure of the material inhomogeneity for concrete. Further investigations are related to the extension of the proposed constitutive law by a plastic model part

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to enable the simulation of typical concrete properties like dilatancy under compressive loading. Moreover, the mutual interaction of the presently decoupled damage directions shall be considered. While a tensile damage in one direction has only a marginal influence on the damage measures in the remaining directions, this effect is stronger pronounced for compressive damage. In this case and in the case of simultaneously appearing damage in multiple directions in one material point, a mutual dependence is obvious. ACKNOWLEDGEMENTS The authors gratefully acknowledge the financial support of this research from Deutsche Forschungsgemeinschaft DFG (German Research Foundation) within the project ‘‘Multiaxial Concrete Constitutive Laws Based on Anisotropic Damage and Plasticity’’. REFERENCES Aifantis, E.C. (1984). On the microstructural origin of certain inelastic models. Journal of Engineering Materials and Technology 106, 326–330. Bažant, Z. & Jirásek, M. (2002). Nonlocal integral formulations of plasticity and damage: Survey of progress. Journal of Engineering Mechanics 128, 1119–1149. Bažant, Z. & Oh, B. (1983). Crack band theory for fracture of concrete. Materials and Structures 16, 155–177. Carol, I., Rizzi, E. & Willam, K. (1994). A unified theory of elastic degradation and damage based on a loading surface. International Journal of Solids and Structures 31, 2835–2865. Carol, I., Rizzi, E. & Willam, K. (2001). On the formulation of anisotropic elastic degradation. I. Theory based on a pseudo logarithmic damage tensor rate. International Journal of Solids and Structures 38, 491–518. de Borst, R. (1991). Simulation of strain localization: a reappraisal of the cosserat continuum. Engineering Computations 8, 317–332. Feist, C., Kerber, W., Lehar, H. & Hofstetter, G. (2004). A comparative study of numerical models for concrete cracking. In Neittaanmäki, P., Rossi, T., Korotov, S., Oñate, E., Périaux, J. & Knörzer, D. (Eds.), Proceedings of ECCOMAS 2004, Jyväskylä, Finland, 1–19. Häußler-Combe, U. & Hartig, J. (2008). Formulation and numerical implementation of a constitutive law for concrete with strain-based damage and plasticity. International Journal of Non-Linear Mechanics 43, 399–415.

Hussein, A. & Marzouk, H. (2000). Behavior of high-strength concrete under biaxial stresses. ACI Structural Journal 97, 27–36. Imran, I. & Pantazopoulou, J. (1996). Experimental study of plain concrete under triaxial stress. ACI Materials Journal 93, 589–601. Jirásek, M. (1998). Nonlocal models for damage and fracture: Comparison of approaches. International Journal of Solids and Structures 35, 4133–4145. Jirásek, M. & Patzak, B. (2002). Consistent tangent stiffness for nonlocal damage models. Computers & Structures 80, 1279–1293. Kitzig, M. & Häußler-Combe, U. (2009). Damage modeling of plain concrete based on an anisotropic constitutive law. In Smojver, I. & Sori´c, J. (Eds.), Proceedings of 6th ICCSM 2009, Dubrovnik, Croatia. Krajcinovic, D. (1996). Damage Mechanics. North-Holland, Elsevier, 1996. Kupfer, H. & Hilsdorf, H.K. (1969). Behavior of concrete under biaxial stresses. ACI Journal 66, 656–666. Lee, S.-K., Song, Y.-C. & Han, S.-H. (2004). Biaxial behavior of plain concrete of nuclear containment building. Nuclear Engineering and Design 227, 143–153. Lemaitre, J. (1992). A Course on Damage Mechanics. Springer, 1992. Mühlhaus, H.B. & Aifantis, E.C. (1991). A variational principle for gradient plasticity. International Journal of Solids and Structures 28, 845–857. Pietruszczak, S. & Mróz, Z. (1981). Finite element analysis of deformation of strain-softening materials. International Journal for Numerical Methods in Engineering 17, 327–334. Pijaudier-Cabot, G. & Bažant, Z.P. (1987). Nonlocal damage theory. Journal of Engineering Mechanics 113, 1512–1533. Rizzi, E. & Carol, I. (2001). A formulation of anisotropic elastic damage using compact tensor formalism. Journal of Elasticity 64, 85–109. Skrzypek, J. & Ganczarski, A. (1999). Modeling of Material Damage and Failure of Structures. Springer, 1999. Willam, K., Bi´cani´c, N. & Sture, S. (1986). Composite fracture model for strain-softening and localised failure of concrete. In Hinton, E. and Owen, D.R.J. (Eds.), Computational Modelling of Reinforced Concrete Structures, 122–153. Pineridge Press, Swansea. Winkler, B. (2001). Traglastuntersuchungen von unbewehrten und bewehrten Betonstrukturen auf der Grundlage eines objektiven Werkstoffgesetzes für Beton. Ph.D. thesis, Universität Innsbruck.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Determination of cement paste mechanical properties: Comparison between micromechanical and ultrasound results S. Maalej & Z. Lafhaj LML UMR CNRS 8107, Ecole Centrale de Lille, Villeneuve d’Ascq, France

M. Bouassida Unité de Recherche Ingénierie Géotechnique, Ecole Nationale d’Ingénieur de Tunis, Tunisia

ABSTRACT: This work presents a comparison between micromechanical and experimental elastic properties of a cement paste material. First, principal micromechanical models found in the literature are described. Second, the ultrasonic and porosity methods used to constitute the experimental database are detailed. The investigated material in this study is a cement paste prepared with three water/cement ratios. Finally, a comparison between micromechanical and experimental obtained results is discussed and analyzed. 1

INTRODUCTION

A composite is a combination of two materials or more, which are not soluble in each other. Generally, one component is designed as the matrix (the continuous component) and the other constituents are designed as inclusions. Thus, a cementitious material (cement paste, mortar or concrete) is considered to be a composite material made up of components of different sizes (inclusions) held together by nonporous cement paste (matrix). The determination of effective elastic properties of composite materials is of great interest for many studies mainly geophysical and mechanical fields since they are considered as indicator of stability and durability. Various computational and experimental methods have been investigated to estimate mechanical, hydraulic and acoustic properties of a composite material. The first theoretical development, focused on determining overall macroscopic transport phenomena, was proposed by (Maxwell, 1873). Voigt determined the effective mechanical properties of composite material with a contribution by Reuss (Hill, 1963, Watt et al. 1976). Later, several researches (Berryman & Berge, 1996, Kaczmarek & Goueygou, 2006, NolenHoeksema, 2000, Watt et al. 1976) were interested in developing micromechanical models to estimate the effective properties of composite materials. But, few ones were interested in cementitious materials. To estimate experimentally the effective elastic properties of cementitious porous material, ultrasonic techniques were adopted. These methods are considered as non-destructive promising tools. They are characterized by the elastic property of ultrasonic

waves and the well-known relationships between pulse velocities and elastic moduli. The composite and its constituents are considered to be isotropic and homogenous. The overall bulk, K hom , and shear, Ghom , moduli are related to the longitudinal and transverse wave velocities (VL and VT ) and overall density, ρ hom , by (Brown, 1997):   4 K hom = ρ hom VL2 − VT2 ; G hom = ρ hom VT2 (1) 3 Micromechanical formulas for effective elastic moduli are used to evaluate some properties which are considered as indicators of the durability of material mainly porosity, permeability or diffusivity. In this paper, micromechanical models relating macroscopic and microscopic properties are described. These models are applied to both dry and fully saturated cement paste material. The latter is modelled as a two-phase porous material. In the first part, four micromechanical models were detailed. In the second part, ultrasonic measurements were carried out to determine the longitudinal and transverse velocities. Finally a comparison between the micromechanical models’ estimates and the experimental results obtained on cement paste will be presented and analyzed. 2

TWO PHASE MICROMECHAICAL MODELS

The micromechanical models are based on the knowledge of mechanical properties of the considered composite’s constituents. In this study, micromechanical models are formulated for cement paste material. The latter is perceived as two-phase composite

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with multiple inclusions of the same material (dry or saturated pores) within a matrix. The composite is considered to be isotropic. Four micromechanical models are presented below; (K1 , G1 ) and (K0 , G0 ) denote respectively the bulk and shear moduli, of inclusions and cement matrix (at zero porosity).

and

2.1 Dilute inclusion model

It can be observed that for dry material, the lower Hashin bound are equal to zero. This is due to the fact that elastic properties of air, which is considered as the material of the pore phase, are null. The range between Hashin-Shtrikman bounds becomes too wide to be of any practical interest when the difference between the phases’ moduli is very large. Several authors showed that the Hashin-Shtrikman bounds are considerably tighter than the Voigt and Reuss bounds (Kaczmarek & Goueygou, 2006, Watt et al. 1976).

The dilute inclusion model was introduced by Eshelby. He gives (Eshelby, 1957) the overall elastic moduli of a material containing a dilute dispersion of ellipsoidal inclusions within a uniform strain. This was accomplished by using a set of imaginary cutting, straining and welding operations. The pores are assumed to be spherical and diluted at small volumetric concentration, in the matrix. The expressions of effective shear and bulk moduli are as follows: K hom = K0 (1 + Ap)−1 ;

G hom = G0 (1 + Bp)−1 (2)

where p is the porosity, A and B are constants depending on phases’ elastic properties such that: A=

K1 − K0 ; (K0 − K1 )α − K0

α=

3K0 ; 3K0 + 4G0

β=

B=

G 1 − G0 (G0 − G1 )β − G0

2 3K0 + 6G0 5 3K0 + 4G0

It can be noted that the interaction between the inclusions is neglected, thus expressions above are valid only for low porosities, which represent the inclusion’s volume fraction. 2.2

Hashin-Shtrikman bounds

Hashin and Shtrikman (Hashin, 1960, Hashin & Shtrikman, 1963) gave bounds for the effective shear and bulk moduli of multiphase composite using variational principles in elasticity. The inclusions are supposed to be spherical. The effective bulk K hom and shear G hom moduli are comprised between lower hom and upper bounds values denoted respectively (Klow , hom hom hom Kup ) and (Glow , Gup ) such that: hom hom < K hom < Kup ; Klow

hom hom Glow < G hom < Gup (3)

Where: hom Klow = K1 + hom = K0 + Kup

1−p 1 K0 − K1

+

3p 3K1 + 4G1

;

hom = G1 + Glow hom = G0 + Gup

2.3

+

3(1 − p) 3K0 + 4G0

;

+

6(K1 + 2G1 )p 5G1 (3K1 + 4G1 )

+

6(K0 + 2G0 )(1 − p) 5G0 (3K0 + 4G0 )

p 1 G1 − G0

Kuster-Toksöz model

Kuster and Toksöz derived (Kuster & Toksöz 1974a,b) a multiphase model for porous material by considering a process of wave propagation in an inhomogeneous elastic material. In the Kuster-Toksöz model, the effective bulk and shear modulus of the composite material are given by:   K0 K0 + 43 G0 + 43 G0 (K1 − K0 ) α 12 p K hom = (4) K0 + 43 G0 − (K1 − K0 )α 12 p G hom = ζ0 =

G0 (G0 + ζ0 ) + (G1 − G0 ) ζ0 β 12 p ; G0 + ζ0 − (G1 − G0 ) β 12 p G0 9K0 + 8G0 6 K0 + 2G0

(5)

Coefficients α 12 and β 12 depends on the aspect ratio of pores and are function of the elastic moduli of both matrix and inclusion phases (Berryman & Berge, 1996). It can be noted that for porous cement paste with spherical pores (for all saturation states), the KusterToksöz approximations coincide with the upper Hashin bounds. 2.4

Mori-Tanaka model

The non-interaction assumption (between the composite phases) considered in Eshelby model is unrealistic. Thus, involving slight modifications in the dilute model, Mori and Tanaka (Mori & Tanaka, 1973) carried out a generalized formulation for the effective moduli of a composite. Mori-Tanaka suggested the following expressions (Benveniste, 1987, Berryman & Berge, 1996):

p 1 K1 − K0

1−p 1 G0 − G1

K hom = K0 +

(K1 − K0 ) α 12 p (1 − p) + α 12 p

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(6)

G hom = G0 +

(G1 − G0 ) β 12 p (1 − p) + β 12 p

(7)

where α 12 and β 12 are expressions that depends on the aspect ratio of pores and the elastic moduli of phases (Berryman & Berge, 1996). These expressions are the same as those presented in Kuster-Toksöz model. It is to notice that for spherical inclusion, and for all saturation degree, the Mori-Tanaka approximations coincide with the upper Hashin bounds. In the particular case of two-phase porous material with spherical pores, the effectives moduli given by Kuster-Toksöz, Mori-Tanaka and upper Hashin bound coincide.

3

EXPERIMENTAL SETUP

In order to obtain various porosity values, cement paste samples were prepared with three different water/ cement (w/c) ratios, namely 0.3, 0.4 and 0.5. The cement paste used is made up of cement CPA CEM I 52.5. The samples were either dry or fully saturated. Porosity was measured by the gravity method (AFPC-AFREM 1977), using vacuum saturation. Porosity, p, is then determined using the following formula: p=

Msat − Mdry ρw V

(8)

where ρ w is the unit mass of water, V is the volume of the sample, Mdry and Msat denote, respectively, the weight of the dried and fully saturated sample. Broadband ultrasound spectroscopy (Eggers & Kaatze 1996) was used to obtain ultrasonic parameters of the materials (longitudinal and transverse velocities). Here, this technique is applied in pulsed mode, using a conventional through-transmission setup with a pair of compression and shear wave contact transducers (Lafhaj et al. 2006, Ould Naffa et al. 2002). The initially emitted ultrasonic pulse is wideband, with a central frequency of 500 kHz. The pulse velocity is obtained from: VL,T =

2e t

(9)

where e is the thickness of the sample and t is the time delay between the first and the second received signals.

4

RESULTS AND DISCUSSION

Each sample of cement paste was subjected to the ultrasonic tests described previously. For each w/c ratio and water saturation state, the presented result is an average of measurements performed on three samples taken from the same core.

Table 1. Dynamic cement paste elastic properties at zero porosity. State

Bulk (MPa)

Shear (MPa)

Dry Saturated

28857 28393

13878 15635

In order to evaluate the micromechanical models, elastic moduli of the cement paste with diminishing pore volume is needed. The latter are determined by a linear regression of the experimental data describing relationships between dry and fully saturated modulus and porosity. The zero porosity parameters were computed and obtained data are given in Table 1. Dry pores are considered as voids and thus both bulk and shear moduli are null. On the other hand, the fully saturated samples presented full water pores with null shear modulus and with 2.2 GPa bulk modulus. The bulk and shear moduli of dry and fully saturated cement paste were evaluated using the above micromechanical models based on two-phase formulations and assuming spherical inclusions (dry and fully saturated pores). The obtained results are presented in Figures 1, 2. Figures 1, 2 show the variation of normalized bulk and shear moduli for the two saturation states. The micromechanical and experimental elastic properties decrease when porosity increases. Bulk and shear moduli estimated by the dilute inclusion model, fit the data with a good agreement for low porosities, up to 20% (Fig. 1b). But, in general, dilute inclusion model overestimates the moduli, especially for higher porosity samples. Such result was predicted as this model assumes low inclusion fraction. It can be noted that Hashin-Shtrikman upper bound curve coincides with the Kuster-Toksöz and MoriTanaka models, which assume spherical inclusions. Such result was predicted from the formulation of the three models as noted previously. According to Figures 1, 2, the Hashin-Shtrikman upper bound, the Kuster-Toksöz and Mori-Tanaka models present a general good agreement with the experimental determined bulk and shear moduli. This agreement is observed for all the porosity ranges, even relatively high porosity (around 35%). This is due to the fact that these models take into account the interaction between inclusions. The discrepancy between micromechanical and experiment bulk moduli, remarked in figure 1b, would be caused by the effect of capillary pressure present in the saturated state and not considered when modeling the saturated cement paste. Comparison between Figure 1a and 1b shows that the measured and theoretical dynamic bulk modulus varies significantly between the dry and the fully saturated states. For instance, the measured bulk modulus

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Normalized saturated bulk modulus (Knom/K0)

Normalized dry bulk modulus (Knom/K0)

0.8 0.7 0.6 0.5 0.4 0.3 15

20

25 30 Porosity (%)

35

40

(a)

0.8 0.7 0.6 0.5 0.4 0.3 15

20

25 30 Porosity (%)

35

40

(b)

Normalized saturated shear modulus (Ghom/G0)

Normalized dry shear modulus (Ghom/G0)

Figure 1. Normalized experimental and computed dynamic bulk modulus vs porosity (a) dry (b) saturated. • Experiments Eshelby _ _ _ Hashin upper bound, Mori-Tanaka, Kuster-Tuksöz. 0.8 0.7 0.6 0.5 0.4 0.3 15

20

25 30 Porosity (%)

35

40

(a)

0.8 0.7 0.6 0.5 0.4 0.3 15

20

25 30 Porosity (%)

35

40

(b)

Figure 2. Normalized experimental and computed dynamic shear modulus vs porosity (a) dry (b) saturated. • Experiments Eshelby _ _ _ Hashin upper bound, Mori-Tanaka, Kuster-Tuksöz.

increases by 13.3% to 36.82% between the dry and the saturated samples. However, the dynamic shear modulus (Fig. 2) presents a little increase between the dry and saturated states for both measured and theoretical values. These results were predicted by theoretical micromechanical formulations. In fact, for the case of shear modulus, the pores’ shear modulus was considered, for the both saturation cases, to be null. So, the shear modulus was expected to be independent from the saturation state. While, the pores’ bulk modulus was considered to be non-null for the case of saturated cement paste and null for the dry one.

5

CONCLUSION

In this paper, the relationship between homogenized elastic parameters (bulk and shear) of a dry and fully

saturated cement paste and porosity was investigated. Transverse and longitudinal pulse velocities were used to determine the experimental dynamic elastic parameters. Four models were used to express theoretically the bulk and shear parameters as a function of porosity. The adopted micromechanical models considered cement paste as two phase material: the matrix (cement paste) and spherical inclusions (pores). Comparisons showed that the relationship between elastic moduli and porosity is correctly represented by dilute inclusion model for low porosity and by the three other models for higher porosities. The large scatter observed for the case of dilute inclusion model in high porosity is due to the fact that this model assumes no interaction between inclusions and that is defined for low porosities. The presented work, dealt with the particular case of two-phase composite with spherical inclusion. Considering spheroidal inclusions with different

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aspect ratio can influence the elastic moduli predictions. Considering micromechanical models that depend on inclusions’ aspect ratio is the subject of ongoing works. REFERENCES AFPC-AFREM 1977. Concrete durability: determination of apparent density and water accessible porosity: 121–125. Benveniste, Y. 1987. A new approach to the application of Mori-Tanaka’s theory in composite materials. Mechanics of Materials 6: 147–157. Berryman, J.G. & Berge, P.A. 1996. Critique of two explicit schemes for estimating elastic properties of multiphase composites. Mechanics of Materials 22, pp. 149–164. Brown, A.E. 1997. Rational and summary of methods for determining ultrasonic properties of materials. Report at Lawrence Livermore national laboratory, California. Eggers, F. & Kaatze, U. 1996. Broad-band ultrasonic measurement techniques for liquids. Measurement Science & Technology 7: 1–19. Eshelby, J.D. 1957. The Determination of the Elastic Field of an Ellipsoidal Inclusion, and Related Problems. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 241: 376–396. Hashin, Z. 1960. The elastic moduli of heterogeneous materials. Technical Report 9 Office of Naval Research. Hashin, Z. & Shtrikman, S. 1963. A variational approach to the theory of the elastic behaviour of multiphase materials. J. Mech. and Phys. of Solids 11: 127–140.

Hill, R. 1963. Elastic properties of reinforced solids: some theoretical principles. J. Mech. Phys. Solids 11: 357–372. Kaczmarek, M. & Goueygou, M. 2006. Dependence of elastic properties of materials on their porosity: Review of models. Journal of Porous Media 4: 335–355. Kuster, G.T. & Toksöz, M.N. 1974a. Velocity and attenuation of seismc waves in two-phase media: Part I. Theoretical formulations. Geophysics 39: 587–606. Kuster, G.T. & Toksöz M.N. 1974b. Velocity and attenuation of seismic waves in two-phase media: Part II. Experimental results. Geophysics 39: 607–618. Lafhaj, Z., Goueygou, M., Djerbi, A. & Kaczmarek, M. 2006. Correlation between porosity, permeability and ultrasonic parameters of mortar with variable water/cement ratio and water content. Cement and Concrete Research 36(4): 625–633. Maxwell, J.C. 1873. A treatise on electricity and magnetism. Clarendon Press, Oxford. Mori, T. & Tanaka, K. 1973. Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metall. 21: 571–574. Nolen-Hoeksema, R.C. 2000. Modulus–porosity relations, Gassmann’s equations, and the low-frequency elasticwave response to fluids. Geophysics 65: 1355–1363. Ould Naffa, S., Goueygou, M., Piwakowski, B. & BuyleBodin, F. 2002. Detection of chemical damage in concrete using ultrasound. Ultrasonics 40: 247–251. Watt, J.P., Davies, G.E.R. & O’Connell, J. 1976. The elastic properties of composite materials. Reviews of geophysics and space physics 14(4): 541–563.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

The simulation of microcracking and micro-contact in a constitutive model for concrete I.C. Mihai & A.D. Jefferson Cardiff University School of Engineering, UK

ABSTRACT: The paper describes a recently developed model for concrete which is based on micromechanical solutions. The model employs solutions of a material with a matrix phase, spherical inclusions, penny-shaped cracks and combines these with a rough surface contact sub-model. The components of the two-phase composite are modelled using the Eshelby inclusion solution and its properties are derived using the Mori-Tanaka scheme. This approach allows the model to predict the development of tensile stresses within the mortar phase under uniaxial compression and thus the model is able to simulate compressive splitting cracks in a natural way. A novel aspect of the model is the inclusion of a multi-component rough contact sub-model which aims to represent the surface of a crack, and associated post-cracking contact behaviour, at three scales; namely those appropriate to hardened cement paste, fine aggregate and coarse aggregate. An attempt to derive the parameters for the three components from micrographs of cracked concrete is also described. The inclusion of the embedded contact sub-model allows the model to simulate the dilatant behaviour of concrete subject to compression with reasonable accuracy. Finally, the paper presents a consistent formulation for the constitutive tangent matrix and a number of examples which describe predictions for uniaxial and biaxial compressive and tensile loading paths. 1

INTRODUCTION

The attainment of a robust and accurate constitutive model for concrete remains a goal for researchers in the field. A number of macroscopic phenomenological models based on damage, plasticity and plasticdamage theories have been developed over the years (Comi & Perego 2001, Prisco & Mazars 1996, Este & Willam 1994, Feenstra & de Borst 1995, Grassl et al. 2002). Many of these models have been implemented in FE codes and have been used effectively. However, some such models use parameters that are difficult to establish and no one model is able to fully describe and predict the complete range of the complex behaviour of concrete. Considerable work has been undertaken in recent years to develop micromechanical models as an alternative approach in order to overcome the drawbacks experienced with the phenomenological models. Pensée et al. (2002) formulated an anisotropic damage model based on a micromechanical solution for an elastic solid containing penny-shaped micro-cracks and an energy release rate-based damage criterion that includes crack closure effects. Gam-barotta (2004) proposed an anisotropic friction-damage model using the solution of an elastic body containing plane cracks. More recently, Zhu et al. (2008, 2009) developed an anisotropic model based on the classic Eshelby inclusion solution and Ponte-Castaneda and Willis homogenization scheme that takes into account the

interaction and spatial distribution of microcracks and unilateral effects. A thermodynamics framework is used to derive the damage evolution law which is then coupled with frictional sliding on closed cracks based on a Coulomb friction criterion. The model presented in this paper is based on the work of Jefferson & Bennett (2007, 2009) and employs micromechanical solutions in an attempt to capture the macroscopic behaviour of concrete by modelling the physical phenomena that take place at microscale. 2

GENERAL PRESENTATION OF THE MODEL

The model utilizes an Eshelby-based framework to simulate a two-phase composite comprising a matrix (m) that represents the mortar and spherical inclusions () that represent the coarse aggregate particles. The composite incorporates randomly distributed pennyshaped microcracks with rough surfaces on which stress can be recovered. It is assumed that the microcrack initiation and propagation takes place in the matrix phase. Figure 1 illustrates the basic concepts of the model. The theoretical aspects of the two-phase composite as well as the solutions for microcracking and stress recovery on crack surfaces that regain contact are described in detail in Jefferson & Bennett (2007, 2009). However, for completeness a summary is given below.

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Figure 1.

Microcracking and rough contact concepts. Figure 2.

2.1

Reference system for a penny-shape microcrack.

Two-phase composite solution

Eshelby based micromechanical solutions and the Mori-Tanaka averaging method for a non-dilute distribution of inclusions (Nemat-Nasser & Hori 1999) are employed in order to derive the equivalent homogeneous elastic tensor Dm . Similarly, a fourth-order tensor Wm  that links the matrix and average stresses is obtained.

all directions (integrating over a hemisphere) and considering that the crack density parameter is a function of orientation f = f (θ , ψ): ⎛

⎜1 εa = ⎝ 2π

  2π

⎞

⎟ NT : Cα : N f (θ , ψ) sin(ψ)dψdθ⎠ : σ¯

π 2

σ¯ = Dm : (¯ε − ε a )

(1)

(4)

σ m = Wm : σ¯

(2)

Cα contains the elastic compliance terms in Equation 3 and N is the transformation matrix (Jefferson 2003). 29 sample directions corresponding to McClauren integration rule are used in numerical simulations in order to evaluate the integral over a hemisphere.

where Dm = (f D : T + fm Dm ) : (f T + fm I4s )−1 , Wm = Dm : (f D : T + fm Dm )−1 · fm and f = volumetric proportions of the matrix material and of the inclusions, noting that fm + f = 1. T = (I4s + S : A ), A = [(D − Dm ) : S = Dm ]−1 ; (Dm − D ), S = Eshelby tensor for spherical inclusions (Nemat-Nasser & Hori 1999), D and Dm = elasticity tensors for the inclusion and matrix phase respectively and I4s = fourth order identity tensor. σ¯ and ε¯ = average or far-field stress and strain tensors respectively and εa = total additional strain due to penny-shaped microcracks with random orientation.

2.3

Using the Mori-Tanaka averaging scheme for a nondilute distribution of inclusions throughout all stages of damage leads to an over brittle post peak response in compression because if the matrix strength is reduced in a direction lateral to a compressive principal axis, the formulation leads to a reduction in strength in the compressive direction, as explained more fully in Jefferson and Bennett (2009). Hence, a cracking stressed is introduced which ensures that initial cracking depends on the matrix stress whereas the latter stages are governed by the average stress as follows:

2.2 Additional strain due to penny-shaped microcracks First, the additional strains generated by a dilute set of microcracks that have the same orientation, given by the normal vector r = [r s t]T (Fig. 2), are written in terms of a crack density parameter f employing the classical solution of Budiansky and O’Connell (1976): ⎡ ⎤ ⎤ σ¯ rr εαrr 2) 16(1 − ν ⎢ 4 σ¯ ⎥ ε α = ⎣γαrs ⎦ = f ⎣ 2−ν rs ⎦ = f Cα : s¯ 3E γαrt 4 σ¯

Cracking stress and damage evolution

sα = N :  : σ¯

(5)

where  is a tensor that provides the transition from the matrix to the average stress.  = rWm + 1(−r)I4s ,

⎡

ξ −εtm

2−ν rt

(3) Next, the total additional strain is calculated by transforming and summing the contributions from

where r = e− 7εtm is the transition function. ζ = effective local strain parameter and εtm = strain in the matrix at uniaxial damage. Expressing damage in terms of the crack density parameter of Budiansky and O’Connell, f proved to be rather inconvenient and it was found to be more

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advantageously characterized by a one-dimensional damage parameter ω. For the undamaged state ω = 0 whereas for the fully damaged state ω = 1. The equivalence between f and ω is described by:  f Cα =

 ω CL 1−ω 

where CL =

1 E

(6)

1 0

0

4 2−ν

0 0

0

0

4 2−ν

 . This leads to the relation-

ship between the local stress tensor and the equivalent local strain tensor given by: s = (1 − ω)DL εL

s∗m (x) = N : Dm : [I4s − SE (x) : (A + S )−1 ] : (f T + fm I4s )−1 : ε¯

(7)

where DL = C−1 L · εL incorporates the elastic strain and the additional strain due to microcracking εL = εLe + εα . The damage evolution law ω = ω(ζ ) has the same expression in all directions and is an exponential function in terms of the effective local strain parameter: ω =1−

The consequence of the use of an effective or average matrix stress is that microcracks may initiate anywhere in the matrix phase. The authors are currently examining the possibility of obtaining a better estimate for the cracking stress by assuming that microcracks first appear in the interfacial transition zone between aggregate particles and the hardened cement paste. A framework that employs the so called exterior point Eshelby solution is proposed in order to achieve this and the effective matrix stress s∗m in equation (10) can be replaced accordingly by:

εm e ζ

tm −5 εζ −ε 0 −εtm

(8)

where ε0 = uniaxial strain at full damage. The damage function that governs the effective local strain parameter ζ includes both normal and shear components of the effective local strain vector εL : 2 =0 F = γL2 − q 2 ζ 2 − (q 2 + μ2 )ζ εLrr + μ2 εLrr

(9)

The effective local strain εL can be written in a similar fashion as the cracking stress of Equation 5, considering the transition function as well as the stress and strain transformation matrices: εL = rCL : s∗m + (1 − r)Nε : ε¯

(12)

where SE (x) = the exterior point Eshelby tensor and A is a fourth order phase-mismatch tensor. This work is currently under study and will be addressed in detail in a forthcoming journal paper. 2.4

Rough crack contact model—Stress recovery

The rough crack contact model implemented at this stage is based on the one used in the Craft concrete model (Jefferson 2003). The motivation behind crack closure model is that contact can be regained with normal and shear movement and consequently stress can be recovered across rough crack surfaces, as indicated by experimental evidence. The main principles are presented in Figures 4 and 5. The asperities of the crack surface are represented as right circular cones. The degree of roughness of the crack surface is characterized by parameter mg which also defines the slope of the interlock contact surface. Mathematically, rough crack contact is introduced by splitting the local stress tensor into two components: an average stress on undamaged material and a

(10)

∗ where sm = effective matrix stress obtained when applying the Mori-Tanaka averaging scheme:

s∗m = N : Dm : (f T + fm I4s )−1 : ε¯

Figure 3.

(11)

Figure 4. Rough crack with elastic region either side showing the angle of contact surface.

Damage surface.

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relationship is obtained σ¯ = Ds : ε¯ , where    1 NT : Ccα : Ds = I4s + Dm 2π 2π

π 2

−1

N :  sin(ψ)dψdθ 3 Figure 5.

Contact regions in relative displacement space.

recovered stress on microcracks in contact: s − su + sr = [(1 − ω)DL : ε L ] + [Hf (εL )ωDL : g] (13) The recovered stress depends on the state of damage—through the damage parameter ω—and on the state of contact—through the embedment strain g. (14)

where  is a matrix that depends on the state of contact, as follows:  = 0 if crack surfaces are not in contact (open region).  = g if crack surfaces are in contact (interlock region).  = I if are in fully closed region. The expression of the contact matrix in the interlock region was demonstrated to be: 1 g = 1 + mg2



∂φint ∂ε L



∂φint ∂ε L

T

∂ 2 φint + φint ∂ε 2L



(15)  where φint (εL ) = mg εLrr − εL2 rs εL2 rt is the mathematical expression for the interlock contact surface. Hf is a shear contact reduction function that accounts for the fact that the potential for shear transfer reduces with increasing crack opening: Hf = e−ηe , where −εtm ηe = εLrr εrange · εrange defines the strain range on which function stress recovery mechanism is active. Making use of Equations 13 and 14 and removing the elastic compliance, the added compliance with incorporated contact is obtained:   Ccα = [(1 − ω)I2s + Hf ω]−1 − I2s · CL

(16)

Rearranging Equation 1 and making use of Equations 4, 5, 6 and 16 the average stress—average strain

(17)

PRESENT DEVELOPMENTS

Present investigations include exploration of potential improvements of the contact model, and the transition function. Also, the use of the so called exterior point Eshelby solution for improving the initial cracking criterion as well as the derivation of a consistent formulation for the constitutive tangent matrix has been examined. The present paper focuses on the developments of the contact model and on the constitutive tangent matrix formulation. 3.1

g =  : εL

Dm

Multilevel contact

The use of non-destructive and non-invasive techniques has led to progress in understanding the process of microcrack formation, growth and propagation. Several assumptions regarding crack evolution, supported by experimental evidence provided by the aforementioned techniques, are exerted in this model. The interfacial transition zone between hardened cement paste and aggregate particles, fine and coarse, is generally considered to be the weakest link in normal concrete (van Mier 1997) and therefore microcracks are presumed to form initially at the cement—aggregate interface. With the increase of the macrostrain they link with neighbouring cracks through the hcp following paths that go around, rather than through, fine and coarse aggregate particles (Avram et al. 1981). The process eventually results in macrocrack formation, frequently referred to as damage localization. Following the investigation of micrographs (van Mier 1997, Avram et al. 1981, Mouret et al. 1999, Nichols & Lange 2006, Elaqra et al. 2007) it has been concluded that the roughness of the crack surface varies at different stages of damage. Stress recovery on crack surfaces that regain contact depends upon surface tortuosity through parameter mg . Thus, the contact model has been enlarged in order to capture the variation of the crack surface unevenness. For this, three contact components have been introduced at different scales each of them being linked to material constituents as follows: the first component characterizes microcracks in hcp, the second component is related to the crack surface in the vicinity of fine aggregate particles and the third component characterizes the crack surface around coarse aggregate particles.

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A set of three parameters, namely, proportion p, slope of the interlock surface mg and exponent ηe of the shear transfer reduction function defines each contact component. Following a parametric study and careful examination of micrographs the parameters for the three components were proposed (Table 1): A rigorous study of micrographs indicated that the surfaces of fine micrographs in hcp are, at a microscale, locally approximately smooth (van Mier 1997, Avram et al. 1981). This observation explains a value of mg that tends to infinity for the first contact component. As cracks propagate and link with neighbouring cracks they form sinuous paths that gently bridgeover fine aggregate particles increasing the roughness of the crack surface accordingly (i.e. mg decreases). Around coarse aggregate particles the tortuosity is further increased due to particle debonding. The theoretical development of the contact model is presented by Jefferson (2002) in which the author derives the expression of the interlock surface in the form of a linear function: int (u, v) = mg u − |v|

Fine aggregate mg=0.35

shear strain

0.00016

Hcp Fine aggregate particles Coarse aggregate particles

∞ 0.35 0.22

12εtm 25εtm ε0

0.2 0.5 0.3

0.0008

Contact model

Coarse aggregate mg=0.22 0.0004

shear strain

0.0003 0.0002 0.0001 0 0

0.001

0.002

0.003

normal strain

Walraven&Reinhardt

Contact model

Figure 7. Contact surface for coarse aggregate component.

3.2

Constitutive tangent matrix

The consistent formulation for the constitutive tangent matrix is derived below for the case of a two-phase model incorporating micromechanics based damage evolution and crack closure. The tangent constitutive matrix is subsequently used in a New-ton-Raphson method based algorithm which significantly improves the convergence properties in comparison with those of a direct iteration method. For the aforementioned case the constitutive relationship can be written as: σ¯ = Ds : ε¯

(20)

Proportion p

0.0006

Figure 6. Contact surface for fine aggregate component.

k=1

εrange

0.0004

Walraven&Reinhardt

where τ = shear stress, fcu = compressive strength. In the multi-component contact model each component is active on a certain relative-displacement range, given in the exponent of the shear contact reduction function. The values of mg from Table 1 generate a sensible match for Walraven and Reinhardt’s function in those specific ranges (Figs. 6, 7). Replacing the single-component with the multicomponent contact model the added compliance given in equation (16) becomes: ⎤ ⎡ −1 3  pk Hfk k − I2s⎦ · CL Ccα = ⎣ (1 − ω)I2s + ω

mg

0.0002

normal strain

fcu + [1.8u−0.80 + (0.234u−0.707 − 0.20) · fcu ] · v 30 (19)

Component

0.00004

0

(18)

Table 1.

0.00008

0

where u = crack opening and v = shear displacement at which contact is regained. The linear function of Jefferson reasonably matches, in a particular relativedisplacement range, the regression analysis-based relation of Walraven and Reinhardt (1981) that in turn fits experimental results with considerable accuracy: τ =−

0.00012

(21)

 −1 d T where Ds = I4s +Dm ni=1 Ni Ccαi Ni i wi · Dm is the secant constitutive matrix and Ccαi is given by Equation 20. Differentiating Equation 21 gives: d σ¯ = Ds : d ε¯ +

∂Ds d ε¯ : ε¯ ∂ ε¯

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(22)

For expediency, the parameter r was kept constant within each strain increment. the following  d Using NiT Ccαi Ni i wi , B = notations, A = I4s + Dm ni=1  (1 − ω)I2s + ω 3j=1 pj Hfj j and the expression of the derivative of an inverse

∂Y−1 ∂x

−1 = −Y−1 ∂Y ∂x Y :

∂Ds ∂A d ε¯ = −A−1 d ε¯ A−1 Dm ∂ ε¯ ∂ ε¯

(23)

and:    nd   ∂A ∂Bi NiT −B−1 d ε¯ = Dm d ε¯ · B−1 i · i CL Ni i wi ∂ ε¯ ∂ ε¯ i=1

(24)

For each sample direction i = 1..nd , mined using the chain rule:

∂B ∂ ε¯

is deter-

4

NUMERICAL RESULTS

A number of single point strain path simulations were undertaken with the proposed model. Numerical predictions for uniaxial tension, uniaxial compression as well as a biaxial failure envelope are shown below (Figs. 8–12). The values of the material parameters used for the simulations are (Case 1): Em = 32000 N/mm2 , E = 49000 N/mm2 , νm = 0.1, ν = 0.28, f = fm = 0.5, q = 3.67, μ = 1.76, ε0 = 0.003, ftm = 1.5 N/mm2 . Figure 9a indicates that the three component contact model significantly improves the predicted ductility relative to the previous version in which an average degree of the crack surface roughness was assumed at all stages of damage (single component contact model). The model is able to capture the post-peak dilatancy observed in experiments (Figs 10a, b, Case 2).

∂B ∂ω ∂ζ ∂ε L ∂B · d ε¯ = · · d ε¯ ∂ ε¯ ∂ω ∂ζ ∂ε L ∂ ε¯ 4

3  ∂B ∂Hf j ∂ε L d ε¯ + · · ∂Hf j ∂ε L ∂ ε¯ 3  ∂B ∂j ∂ε L · · d ε¯ ∂ ∂ε L ∂ ε¯ j j=1

xx-stress (MPa)

j=1

+

(25)

It can be proven that the differential of the embedded strain in Equation 14 is: dg =  : dε L

2 1.5 1 0.5 0 0

∂A−1 ∂A ∂A d ε¯ k = −A−1 · d ε¯ k · y = −A−1 · · y · d ε¯ k ∂ ε¯ k ∂ ε¯ k ∂ ε¯ k (27)

Column k (k = 1, 2, . . . 6) of matrix Dadt is hence obtained: (28)

0.001

0.002

0.003

xx-strain

Figure 8.

Predicted uniaxial tensile response, Case 1.

single component contact Case 1 3-component contact Case 1 40 xx-stress (compression +ve) MPa

∂ Hence, ∂ε = 0 and the second summation in L Equation 24 vanishes. The partial derivatives can be straightforwardly derived. The consistent tangent matrix is then formed in a ‘‘column by column’’ manner so that dε¯ is extracted element by element as shown below. Denoting y = A−1 · Dm : ε¯ one can obtain:

D(k) adt

3 2.5

(26)

∂A = −A−1 · ·y ∂ ε¯ k

uniaxial tension

3.5

35 30 25 20 15 10 5 0 0

Equation 22 becomes:

0.0005 0.001

0.0015

0.002 0.0025

0.003

xx-strain (compression +ve)

d σ¯ = (Ds + Dadt ) : d ε¯

(29) Figure 9a. Predicted response for uniaxial compressive paths, Case 1.

Consequently, Dt = Ds + Dadt .

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3-component contact Case1

xx-stress (compression +ve) MPa

single component contact Case 1

40 35 30 25 20 15 10 5 0 -0.0008

-0.0004

0

0.0004

volumetric strain

xx-stress (compression +ve) MPa

Figure 9b. Predicted dilatancy for uniaxial compressive paths, Case 1.

Predicted biaxial failure envelope, Case 1.

the model. Therefore, the authors acknowledge that the current model is not complete and believe that the incorporation of microplasticity will address its shortcomings. This work is underway and the complete model will be the subject of a forthcoming publication. The introduction of the multi-component contact sub-model generates a biaxial failure envelope (Fig. 11) that favourably matches the experimental data of Kupfer (1969).

40

sx/sy=1/0 Case 2

35

Figure 11.

30 25 20 15 10 5 0 0

0.001

0.002

0.003

0.004

5

xx-strain (compression +ve)

xx-stress (compression +ve) MPa

Figure 10a. Case 2.

-0.0005

Predicted uniaxial compressive response,

40 35 30 25

sx/sy=1/0 Case2

20 15 10 5 0

CONCLUSION

The constitutive model presented in this paper is able to predict many of the key characteristics of the complex behaviour of concrete under uniaxial and biaxial tension and compression. The expanded rough crack contact model enables the model to improve the predicted ductility. On-going work on the model involves the development of a crack criterion based on an exterior point Eshelby solution as well as the introduction of microplasticity on the crack planes.

REFERENCES 0

0.0005

0.001

volumetric strain

Figure 10b. Predicted dilatancy for uniaxial compressive path, Case 2.

However, it proved necessary to modify some of the material parameters in order to obtain such a response. Also, following a parametric study it was observed that a reasonable ductile and dilatant response could not be predicted simultaneously with this version of

Avram, C. Facaroiu, I. Filimon, I. Mirsu, O. & Tertea, I. 1981. Concrete strength and strains. Developments in civil engineering. Elsevier. Budiansky, B. & O’Connell, R.J. 1976. Elastic moduli of a cracked solid. International journal of solids and structures. 42: 81–97. Comi, C. & Perego, U. 2001. Fracture energy based bidissipative damage model for concrete. International Journal of Solids and Structures. 38: 6427–6454. di Prisco, & M. Mazars, J. 1996. Crush-crack: a non-local damage model for concrete. Mechanics of Cohesive Frictional Materials.; 1: 321–347.

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Elaqra, H. Godin, N. Peix, G. R’Mili, M. & Fantozzi, G. 2007. Damage evolution analysis in mortar during compressive loading using acoustic emission and X-ray tomography: effects of the sand/cement ratio. Cement and concrete research. 37: 703–713. Este, G. & Willam, K. 1994. Fracture energy formulation for inelastic behaviour of plain concrete. Journal of Engineering Mechanics ASCE. 120(9): 1983–2011. Feenstra, P.H. & de Borst, R. 1995. A plasticity model and algorithm for mode-I cracking in concrete. International Journal for Numerical Methods in Engineering. 38: 2509–2529. Grassl, P. Lundgren, K.G. & Gylltoft, K. 2002. Concrete in compression: a plasticity theory with a novel hardening law. International Journal of Solids and Structures; 39: 5205–5223. Gambarotta, L. 2004. Friction-damage coupled model for brittle materials. Engineering fracture mechanics. 71: 824–836. Jefferson, A.D. 2002. Constitutive modelling of aggregate interlock in concrete. International journal for numerical and analytical methods in geomechanics. 26: 515–535. Jefferson, A.D. 2003. Craft – a plastic–damage–contact model for concrete. I. Model theory and thermodynamic Considerations. International journal of solids and structures. 40: 5973–5999. Jefferson, A.D. & Bennett, T. 2007. Micro-mechanical damage and rough crack closure in cementitious composite materials. International journal for numerical and analytical methods in geomechanics. 31(2): 133–146. Jefferson, A.D. & Bennett, T. 2009. A model for cementitious composite materials based on micro-mechanical solutions and damage-contact theory. Computers and structures. In press.

Kupfer, H.B. Hilsdorf, H.K. & Ruch, H. 1969. Behaviour of concrete under biaxial stresses. Journal of ACI. 66(8): 656–666. Mouret, M. Bascoul, A. & Escadeillas, G. 1999. Microstructural features of concrete in relation to initial temperature—SEM and ESEM characterization. Cement and concrete research. 29: 369–375. Nemat-Nasser, S. & Hori, M. 1999. Micromechanics: Overall properties of heterogeneous materials. North-Holland. Nichols, A.B. & Lange, D.A. 2006. 3D surface image analysis for fracture modelling of cement-based materials. Cement and concrete research. 36: 1098–1107. Pensee, V. Kondo, D. & Dormieux, L. 2002. Micromechanical analysis of anisotropic damage in brittle materials. Journal of engineering mechanics ASCE. 128(8): 889–897. van Mier, J.G.M. 1997. Fracture processes of concrete. CRC Press. Walraven, J.C. & Reinhardt, H.W. 1981. Theory and Experiments on the Mechanical Behaviour of Cracks in Plain and Reinforced Concrete Subjected to Shear Loading. Heron. 26(1A). Zhu, Q.Z. Kondo, D. & Shao, J.F. 2008. Micromechanical analysis of coupling between anisotropic damage and friction in quasi brittle materials: Role of homogenization scheme. International journal of solids and structures. 45: 1358–1405. Zhu, Q.Z. Kondo, D. & Shao, J.F. 2009. Homogenizationbased analysis of anisotropic damage in brittle materials with unilateral effect and interactions between microcracks. International journal for numerical and analytical methods in geomechanics. 33: 749–772.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Simulations of dynamic failure in plain and reinforced concrete with regularized plasticity and damage models Jerzy Pamin, Andrzej Winnicki & Adam Wosatko Faculty of Civil Engineering, Cracow University of Technology, Poland

ABSTRACT: The paper presents an overview of two enhanced constitutive models: Burzy´nski-Hoffman viscoplastic consistency model and scalar gradient damage model. They are used for the simulation of wave propagation in two-dimensional benchmark configurations: Brazilian split test and direct tension test. The influence of loading rate on the results is examined. 1

INTRODUCTION

The paper is focused on a comparison of the performance of two regularized continuum models in the simulations of selected benchmarks of dynamic response. These models are: Burzy´nski-Hoffman viscoplastic consistency model (Winnicki et al. 2001) and scalar damage model enhanced with higher order strain gradients (Peerlings et al. 1996). For the latter model a coupling to plasticity can also be considered (de Borst et al. 1999). The numerical modelling of dynamic processes using continuum models equipped with localization limiters was for the first time fully covered in (Sluys 1992). Only for nonlocal or rate-dependent models a proper formation of standing localization wave can be reproduced and pathological mesh sensitivity of discrete solutions can be avoided. In the paper the employed models are briefly reviewed. The results of two standard wave propagation tests on plain and reinforced concrete are presented: Brazilian split of a cylinder and direct tension of a bar. Two- and three-dimensional finite element models are built, but the paper is limited to 2D configurations. Linear kinematic relations are assumed. The constitutive models have been implemented in the FEAP package (Taylor 2001). The standard Newmark scheme combined with the consistently linearized Newton procedure is used. The issue of failure prediction is discussed whereby the influence of loading rate is considered. The physical significance and regularization effect of the gradient-enhancement and viscosity are assessed. 2 2.1

idea proposed by (Wang 1997) for metals, named ‘‘viscoplastic consistency model’’. A similar approach was proposed by (Kleiber 1998) and later developed by (Heeres et al. 2002). In the considered model the yield function can expand or shrink depending on the current viscoplastic strain rate. The stress state remains on the yield surface and the consistency condition is invoked. There is no additional equation defining a viscoplastic multiplier and this approach is very close to the classical rate independent plasticity. Therefore, the established numerical algorithms like the closest point projection or the cutting plane algorithm (Simo and Hughes 1997) can easily be adapted. The Burzy´nski-Hoffman yield surface in its isotropic form is selected, since it was successfully employed in the analysis of concrete structures (Bi´cani´c et al. 1994; Pearce 1993). This yield surface has the form: f vp = 3J2σ + I1σ (fc − ft ) − fc ft = 0

(1)

where J2σ and I1σ are the usual stress invariants, fc and ft are the uniaxial compressive and tensile strengths, respectively. Two internal variables κc and κt are postulated which are both functions of the equivalent viscoplastic strain. They separately describe the material hardening/ softening in compression and tension, respectively. Moreover, two more internal variables ηc and ηt are assumed to determine the increase/decrease of compressive and tensile strengths due to the actual equivalent viscoplastic strain rate. Thus, the respective strengths are:

ENHANCED MATERIAL MODELS fc = fc (κc , ηc )

Overview of Burzy´nski-Hoffman viscoplasticity

The first employed model for concrete has been developed within the viscoplasticity theory and follows the

and

ft = ft (κt , ηt )

(2)

The rates of the internal variables depend on the current stress and the rates of internal variables

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κ and η:

The dependence of fc on κc and ηc is formulated in a general way as:

κ˙ c = gc (σ )κ˙

and

κ˙ t = gt (σ )κ˙

(3)

fc = fc Hc (κc )Sc (ηc )

η˙ c = gc (σ )η˙

and

η˙ t = gt (σ )η˙

(4)

where fc is the initial compressive strength. Functions Hc and Sc can for instance be specified as piecewise linear. Similarly, the actual tensile strength is computed as:

In the above equations gc and gt are scalar functions accounting for independent processes of damage in compression and tension. In turn, κ˙ is defined as an equivalent viscoplastic strain rate assuming work hardening, while η˙ depends on the time derivative of the viscoplastic strain rate: 1 κ˙ = − σ : ˙ vp σ˜

and

η˙ =

1 σ : ¨ vp σ˜

ft = ft Ht (κt )St (ηt )

1

˙ = ˙ e + ˙ vp

n : σ˙ +

and Hooke’s law is valid for the elastic part: σ˙ = E : ˙

(7)

The viscoplastic flow is defined similarly to the classical associated plasticity: ˙ ˙ vp = λn,

n=

∂f vp ∂σ

∂f vp ∂f vp ∂f vp ∂f vp κ˙ c + η˙ c κ˙ t + η˙ t = 0 ∂κc ∂ηc ∂κt ∂ηt (11)

Using Equations (3), (4), (5), (8) derivatives κ˙ c , κ˙ t , η˙ c , η˙ t can be expressed as:

(6)

e

(10)

Example functions, based on experimental data from (Kupfer 1973) and (Suaris and Shah 1985), are presented in (Winnicki 2007). In order to establish the viscoplastic multiplier λ˙ the consistency equation is used:

(5)

where σ˜ = (3J2σ ) 2 . It should be noted that in the course of loading the yield surface can change its shape due to the separate hardening/softening processes of the compressive and tensile strengths, but the surface remains convex at all times. It depends not only on the internal parameter κ, but also on the additional one η. Since the total value of η depends on the first derivatives of viscoplastic strains, the yield surface is rate dependent, i.e. it expands for higher and shrinks for lower viscoplastic strain rates. As a result, this model can correctly predict basic viscoplastic phenomena like creep and relaxation. The strain rate is decomposed into its elastic and viscoplastic parts:

(9)

(8)

˙ κ˙ t = gt g λ, ˙ η˙ c = gc g λ, ¨ η˙ t = gt g λ¨ κ˙ c = gc g λ, where g is equal to: g=

1 σ : n σ˜

(13)

Using the above relationships the consistency equa¨ tion can be written in a form depending on λ˙ and λ: n : σ˙ − hλ˙ − sλ¨ = 0

(14)

where h is the classical generalized plastic modulus and s is the generalized viscoplastic modulus. Due to the last term the consistency equation is no longer an algebraic equation for the viscoplastic multiplier, but a differential equation of the first order, to be solved for an appropriate initial condition. The generalized plastic and viscoplastic moduli are computed as: h = ac Sc hc + at St ht ,

The functions gc and gt from Equations (3)–(4) are selected according to the experimental evidence taking into account the influence of the damage process in compression on the concrete strength in tension and, conversely, the influence of the damage process in tension on the concrete strength in compression. In (Winnicki 2007) two options are considered: (1) gc = gt = 1 (damage is assumed to be an isotropic phenomenon) and (2) gc + gt = 1 (with extreme cases of dominant compressive stress gc = 1, gt = 0 and dominant tensile stress gc = 0, gt = 1).

(12)

s = ac sc Hc + at st Ht

(15)

where: hc =

dHc dHt dSc dSt , ht = , sc = , st = dκc dκt dηc dηt

(16)

In turn, coefficients ac and at are: ac = fc (ft − tr σ )gc g,

at = ft (fc + tr σ )gt g

(17)

When functions Sc and St are constant, their derivatives vanish and Equation (14) reduces to the form known from the classical rate independent plasticity.

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It is discussed in (Winnicki 2007) how the material model parameters can be linked with fracture energies Gf t and Gfc which are the actual material properties (fib 1999; van Mier 1984; Vonk 1992) via the width of the localization zone. For the case St = Sc = 1, i.e. when the presented model is not a localisation limiter, the width of the localisation zone usually coincides with one row of finite elements. However, when viscous effects are active the width of localisation zone becomes larger and can only be estimated a posteriori.

The equivalent strain measure ˜ can be defined in different ways. In this paper we employ the modified von Mises definition involving the first and second strain invariants, I1 and J2 , respectively, (de Vree et al. 1995) and depending on the ratio of compressive and f tensile strengths k = fc : t    (k − 1)I1 1 (k − 1)I1 2 12kJ2 + ˜ = + 2k(1 − 2ν) 2k 1 − 2ν (1 + ν)2 (21)

2.2 Summary of gradient damage model The simplest model of continuum damage mechanics which can describe elastic stiffness degradation in quasi-brittle materials is scalar damage. This means that one damage measure ω, which grows monotonically from 0 to 1 (Kachanov 1958), is a function of damage history parameter κ d and depends on the deformation of a body. For a material without any defect (no damage) the parameter ω equals to 0. For a material with a complete loss of stiffness (complete damage) we have ω = 1. Considering the damage evolution we distinguish the actual body with strains  and stresses σ and its fictitious undamaged counterpart with stresses σˆ and strains ˆ . The fictitious counterpart represents the undamaged ‘‘skeleton’’ of the body, and the stresses σˆ acting on it are called effective. We adopt the postulate that the strains  observed in the actual body and in its undamaged representation ˆ are equal (Lemaitre 1971; Simo and Ju 1987). Then, the real stress tensor σ and its effective counterpart σˆ are related by the above-mentioned parameter ω: σ = (1 − ω)σˆ

(18)

where: σˆ = E :  e

(19)

here E is the elastic stiffness operator. The elastic strain tensor  e is equal to the strain tensor  when the standard elasto-damage model is considered, but the model can easily be coupled with a plastic behaviour of the undamaged ‘‘skeleton’’ (de Borst et al. 1999), and then e =  − p. The strain equivalence is related with a loading function f d , also called damage activation function, defined in the strain space: f d (, κ d ) = ˜ () − κ d = 0

(20)

Here ˜ is an equivalent strain measure. During the damage evolution the history parameter κ d is equal to the largest value of ˜ reached in the loading history and obeys the standard loading/unloading conditions.

Although this definition introduces the sensitivity of the model to the sign of strains and allows for damage not only under tension but also under (biaxial) compression, the interaction between tensile and compres-sive action in concrete is not as well represented as in the Hoffman model. In order to improve the description, one would need to use an isotropic version of the damage model with two damage parameters degrading the tensile and compressive stiffness separately, see for instance (Mazars and Pijaudier-Cabot 1989; Comi 2001). It is assumed that κ d grows from damage threshold κo and damage ω asymptotically increases but never reaches 1. We define the damage growth function according to (Mazars and Pijaudier-Cabot 1989):  κo  d ω(κ d ) = 1 − d 1 − α + αe−η(κ −κo ) (22) κ The respective parameters η and α are responsible for material ductility and residual stress, respectively. The former parameter is thus related with concrete fracture energy Gf . The latter one prevents the complete loss of material stiffness and leads to a more stable numerical response. If unloading is considered, irreversible strains are usually observed in concrete. In this case the motivation for coupling the damage model with plasticity is substantial. A combination of a plasticity theory formulated in the effective stress space with the above damage theory formulated in the strain space is described for instance in (de Borst et al. 1999). In addition, the constitutive relations can incorporate a crack-closing projection operator which is important for cyclic loading and extensive stress redistributions, see e.g. (Pamin et al. 2003). Following (Peerlings et al. 1996), the damage evolution in the gradient-enhanced model is governed by the following damage loading function: f d (, κ d ) = ¯ (˜ ()) − κ d = 0

(23)

where the averaged (nonlocal) strain measure ¯ satisfies the Helmholtz equation: ¯ − c∇ 2 ¯ = ˜ .

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(24)

The parameter c > 0 has a unit of length squared and is related to an internal length scale. It is assumed here to be constant, although, with some modifications in the formulation, it can be made a function of ˜ or ¯ (Geers 1997), which might be physically relevant.

Analogically, at integration points we decompose secondary fields, for example the stress:

2.3 FE implementation

σ (i+1) = σ (i) + σ

In this subsection Voigt’s notation is used. We consider a certain domain B, occupied by the material body, with boundary ∂B. The weak form of motion equations is the virtual work equation:   δuT ρ udV ¨ + δ T σ dV B

B

 =

B

δuT bdV +

 ∂B

δuT tdS

(25)

where the superscript T is the transpose symbol, ρ is the material density, b is the body force vector and t is the traction vector. The finite element algorithm for the Burzy´nskiHoffman viscoplastic model is similar to the classical rate-independent plasticity, i.e. linearization is performed and the weak form of incremental motion equations is discretized (Winnicki 2007). For the gradient damage model a two-field FE model is required. The weak form of Equation (24) is derived as follows. The variation of the averaged strain measure δ ¯ , Green’s formula and the natural boundary condition (∇ ¯ )T ν = 0 are introduced to obtain:    δ ¯ dV ¯ + (∇δ ¯ )T c∇ ¯ dV = δ ¯ dV ˜ (26) B

B

B

In the ensuing two-field formulation averaged strain measure ¯ is discretized in addition to displacements u. These primary fields are interpolated in the following way: u=Na

and ¯ = hT e

(27)

where N and h contain suitable shape functions. From the above interpolations the secondary fields  and ∇ ¯ are computed:  =Ba

and

∇ ¯ = gT e

(28)

where B = L N, gT = ∇hT , L is a differential operator matrix. After discretization, applied also for variations δu and δ , ¯ Equations (25) and (26) must hold for any admissible δa and δe. The boundary value problem is linearized, hence at nodal points the increments of the primary fields from iteration i to iteration i + 1 are introduced according

to the following decomposition: ai+1 = ai + a

and

ei+1 = ei + e

(29)

(30)

Hence the constitutive relation is written in its incremental version:

σ = (1 − ωi )E  − ω σˆ i

(31)

The increment of damage ω depends on the increment of averaged strain ¯: 

ω =

∂ω ∂κ d

i 

∂κ d ∂ ¯

i

¯ = G i hT e

(32)

In the Helmholtz equation the increment of equivalent strain measure ˜ is computed from the interpolated displacement increment a: 

˜ =

d˜ d

i

 = [sT ]i B a

(33)

Finally, we rewrite Equations (25) and (26) in a matrix form, so that the gradient damage formulation can be written as the coupled matrix problem (Peerlings et al. 1996):   Maa 0 a¨ 0 0 e¨    K Kae a f − fint + aa = ext (34) Kea Kee e f − f e In the absence of damage growth Ka¯ = 0 and the motion equations are uncoupled from the averaging equation. It is noted that no structural damping is included in the analysis, however both employed models involve energy dissipation, so they are capable of representing material damping. 3

SIMULATIONS FOR PLAIN CONCRETE—BRAZILIAN SPLIT TEST

This section presents the results of simulations of the Brazilian split test on concrete specimen using the described regularized models. A comparison of simulations of the same configuration under static loading is presented in (Wosatko et al. 2009). The geometry of the Brazilian tests is based on papers (Feenstra 1993; Winnicki et al. 2001; Ruiz et al. 2000). Plane strain

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conditions are analyzed. Due to double symmetry only a quarter of the domain is considered. If not stated otherwise, the load is applied through a stiff elastic platen. The platen is perfectly connected with the specimen. The fundamental material data for concrete are: Young’s modulus Ec = 37700 MPa, Poisson’s ratio νc = 0.15, density ρ = 2405 kg/m3 . For the platen Es = 10 · Ec and the other parameters are the same as for concrete.

3.1 Viscoplasticity simulations

2

2

1.5

1.5

1

St

Sc

In this section the cylinder radius is equal to 40 mm according to (Feenstra 1993; Winnicki et al. 2001). Standard eight-noded finite elements are used. The adopted material parameters are: initial tensile strength ft = 3 MPa, initial compressive strength fc = 30 MPa. Isotropic version of the model κt = κc is assumed. Linearly decreasing softening moduli are adopted with Hc decreasing from 1 to 0 for κc = 0.015 and Ht decreasing from 1 to 0 for κt = 0.001. In Figure 1 the employed piecewise linear functions Sc and St are shown. The load is assumed to grow linearly with time. Figure 2 presents the structural response for three different loading rates: (1) ‘‘ slow’’ dynamic process (vertical displacement of the platen equal to 0.2 mm is reached in 300 s), (2) ‘‘moderate’’ (300e-4 s) and (3) ‘‘fast’’ dynamic process (300e-5 s). The higher strain rates result in the larger maximum loads and

0.5 0

corresponding displacements. Also the post critical behaviour is less steep for the higher strain rate. Figure 3 presents contour plots for the total strain values (horizontal component 11 , vertical component 22 and internal variables κt = κc at the end of the loading process for the smallest strain rate (slow process) and the highest strain rate (fast process). The horizontal strain component is highly localised whereas the vertical component exhibits a rather distributed pattern. Damage is localised in the narrow vertical zone in the middle of the specimen forming a wedge at the top under the platen. For the higher strain rates the width of the localisation zone becomes larger which proves the viscoplastic regularization is active. A comparison for the fast loading between the dynamic test computed in the quasi static manner (i.e. without inertia effects using the standard Newton-Raphson procedure) and the full dynamic procedure using the Newmark algorithm for integration in time domain shows that inertia effects for the considered range of strain rates are negligible (Winnicki 2007).

(a) Slow, ∈11

(b) Slow, ∈22

(c) Fast, ∈11

(d) Fast, ∈22

1 0.5

0

0.2

0.4

0.6

0.8

1

0

0

(a) Function Sc

Figure 1.

0.2

0.4

0.6

0.8

1

(b) Function St

Material functions for viscoplasticity.

0.8 0.7

P/2 [kN]

0.6 0.5 0.4 0.3 fast rate medium rate slow rate

0.2 0.1 0

0

0.02

0.04

0.06

0.08 [mm]

0.1

0.12

0.14

Figure 2. Load-displacement plots for three different load rates (8-noded elements).

Figure 3. Contour plots of strains and internal variables for different loading rates.

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0

ti Figure 4.

td

vertical displacement

[mm]

p pi

t

Coarse Medium Fine

0

Loading history for gradient damage simulations.

0.00002 0.00004 0.00006 0.00008 0.0001 0.00012 t [s]

(a) Displacement in time. 1000 0

Coarse Medium Fine

0

0.00002 0.00004 0.00006 0.00008 0.0001 0.00012 t [s]

(b) Velocity in time. 1e+09 Coarse acceleration a[ mm ] s2

The input data adopted in this section are similar to (Ruiz et al. 2000). The radius of the cylinder is 25.4 mm. The material data adopted for the scalar gradient damage model are as follows: internal length l = 4 mm, exponential softening with threshold κo = 0.0001195 (tensile strength ft = 4.53 MPa), ductility parameter η = 600 and to prevent the complete loss of material stiffness α = 0.98. The modified von Mises definition of damage loading function with parameter κ = 14.8 (compressive strength fc = 67 MPa) is selected. A linear-constant type of the pulse loading is adopted as shown in Figure 4, where for instant ti = 48 μs the traction intensity pi = 175 MPa is reached, then remains constant till time td = 100 μs and drops to 0. The elements with quadratic interpolation of the displacements, linear interpolation of the averaged strain measure and 2 × 2 Gauss integration are employed for three mesh refinements. In Figure 5 the displacement, velocity and acceleration in time are plotted in history diagrams for a point at the top of the platen. We notice differences for the final stage of deformation history, i.e. when damage is advanced and the specimen is close to failure. If we compare results for specimens with or without platen (Wosatko 2008) the character of diagrams and crack patterns is quite similar, so next aspect of computations is performed for the specimen without platen. Displacement history diagrams in Figure 6 are obtained for cases with different times when the load changes from linear to constant. The shorter time ti is the more rapidly failure follows. Hence, the largest loading rate when ti = 40 μs induces the most sudden response and displacements go to infinity very fast. In Figure 7 quite similar damage patterns are observed for each mesh. The plots are depicted for time t = 110 μs, so 10 μs after the load is removed. The splitting process in the central vertical zone in each case is present, while the damage pattern is a continuum representation of crack distribution. The split zones are observed in each contour plot of Figure 8, however for the case with ti = 40 μs damage is visible also at the top where the load is apllied.

velocity V [

mm ] s

Gradient damage simulations

5e+08 0

Medium Fine

0

0.00002 0.00004 0.00006 0.00008 0.0001 0.00012 t [s]

(c) Acceleration in time.

Figure 5. History diagrams for gradient simulations—three different mesh densities.

damage

0 vertical displacement ν [mm]

3.2

ti = 56 μs ti = 48 μs ti = 40 μs

0

0.00005

0.0001 t [s]

0.00015

0.0002

Figure 6. Different loading rates—displacement history diagrams for gradient damage simulations.

4

SIMULATIONS FOR REINFORCED CONCRETE–DIRECT TENSION TEST

The aim of this test is a two-dimensional analysis of tensile wave propagation in a bar not only made from plain concrete, like in (Pamin 2005), but also

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(a) Coarse mesh.

Figure 7.

(b) Medium mesh.

Damage patterns for three different meshes in instant t = 110 μs.

(a) Case ti = 40 μs, t = 65 μs. Figure 8.

(b) Case ti = 48 μs, t = 90 μs.

(c) Case ti = 56 μs, t = 210 μs.

Damage patterns for different loading rates.

A

Figure 9.

(c) Fine mesh.

Dynamic direct tension test.

strengthened with reinforcement (Wosatko et al. 2006; Wosatko 2008). As shown in Figure 9, we consider a bar supported at the symmetry axes and loaded with a time-dependent normal traction at both ends. The reinforcement is located along the axis of the bar. The length of the bar is L = 250 mm, the width B = 60 mm. Figure 9 also shows the employed discretization, where the central zone is refined because of the expected localization. Eight-noded two-field gradient damage elements for concrete and elastic truss finite elements for the reinforcement are introduced. Plane stress with thickness t = 50 mm and full bond between steel and concrete are assumed. For the gradient damage model of concrete the material data are as follows: E = 18000 MPa, ν = 0.0,

ρ = 2320 kg/m3 . Exponential damage growth function is used with κo = 0.00188 (tensile strength ft = 3.4 MPa), α = 0.99 and η = 500. The modified von Mises definition of the equivalent strain is employed. For the internal length scale parameter 3 values, namely l = 2/4/8 mm, are considered. The steel reinforcement is modelled with E = 200000 MPa, ν = 0.0 and ρ = 7800 kg/m3 and assumed cross section A = 30 mm2 (the reinforcement ratio equals 1%). The time step is 2 × 10−6 s, ti = 3 × 10−5 s and the traction intensity is pi = 2.4 MPa. The diagrams depicted in Figure 10 confront the behaviour of the bar with and without the reinforcement. For plain concrete the elongation tends to infinity due to fracture, while for reinforced concrete the horizontal displacement oscillates around a certain state. Nevertheless, for both responses the standing wave and localization are observed in the centre of the bar, cf. Figures 10(e) and 10(f). Figure 10 shows the internal length study of this test for plain and reinforced concrete. The elongationtime diagrams for plain concrete, which are presented in Figure 10(a), reveal that the smaller the value of

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0.15

0.025

0.1

0.05

0

Figure 10.

l = 2 mm l = 4 mm l = 8 mm

Elongation at point A [mm]

Elongation at point A [mm]

0.2

0

0.02 0.015 0.01 l = 2 mm l = 4 mm l = 8 mm

0.005 0

5e-05 0.0001 0.00015 0.0002 0.00025 0.0003 t [s] (a) Diagrams for plain concrete.

0

0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 t [s] (b) Diagrams for reinforced concrete.

(c) Averaged strain, plain, l = 2 mm.

(d) Averaged strain, reinforced, l = 2 mm.

(e) Averaged strain, plain, l = 4 mm.

(f) Averaged strain, reinforced, l = 4 mm.

(g) Averaged strain, plain, l = 8 mm.

(h) Averaged strain, reinforced, l = 8 mm.

Internal length study for plain concrete and reinforced bar.

parameter l is, the faster the elongation grows. Figure 10(b) points out that for reinforced concrete together with the decrease of l the amplitude of vibrations diminishes. For l = 2 mm a much smaller period of the vibrations occurs. We observe a similar upper limit of the elongation for each l, which is related to the reinforcement action. All contour plots in Figure 10, which illustrate the averaged strain distribution, are performed for the time instant t = 0.0006 s (after 300 time steps). Considering the results, for plain concrete and l = 2 mm (Figure 10(c)) two separated zones of localization appear, which can be a numerical effect. For the other cases one central zone emerges, which is consistent with the analytical solution for the strain-softening bar presented in (Baˇzant and Belytschko 1985). The standing decohesion wave is located exactly in the centre. The general rule for gradient-enhanced models that the width of zone expands together with larger l is valid. For plain concrete the localization zones are parallel to the width of the domain and averaged strain contours form even bands, which is expected due to the loading direction. On the right-hand side of Figure 10 the averaged strain distributions for reinforced concrete are shown. Here again with the growth of l the zone of localization enlarges, but a new visible effect is that the reinforcement delays the progress of cracking along

the whole width. Along the bar axis localization does not seem to occur. The most active zones are the farthest from the reinforcement, but similarly to plain concrete the standing wave and damage zone are placed in the central part of the tensile bar. The full bond between concrete and the rebar influences the shape of the damage zones, which do not look very realistic. In reality more than one localized cracks, which gradually develop in the vertical direction next to the reinforcement, are observed. To simulate this behaviour a proper representation of bond slip and, from the numerical point of view, the implementation of interface elements are required as shown in (Sluys and de Borst 1996). 5

FINAL REMARKS

Two constitutive models for concrete have been summarized and applied in the simulation of the Brazilian split test. The Burzy´nski-Hoffman viscoplastic consistency model is quite general, since it incorporates separate representations of damage in tension and compression as well as intrinsic loading rate dependence. Since it involves many parameters, an extensive case study is needed and has been to a certain extent provided in (Winnicki 2007). The scalar gradient damage model can be extended to its isotropic version with

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two independent damage variables, thus enabling a more accurate representation of damage in tension and compression. Nevertheless, the scalar model is capable of representing the splitting effect in the dynamic Brazilian test. It has also been used to predict a standing localization wave in the direct tension test, whereby a comparison of the response of the plain versus reinforced bar has been performed. The gradient damage simulations of the dynamic four-point bending test have been presented in (Wosatko et al. 2006; Wosatko 2008). REFERENCES Bažant, Z.P. and T. Belytschko (1985). Wave propagation in a strain-softening bar: exact solution. ASCE J. Eng. Mech. 111(3), 381–389. Bi´cani´c, N., C.J. Pearce, and D.R.J. Owen (1994). Failure predictions of concrete like materials using softening Hoffman plasticity model. In H. Mang, N. Bi´cani´c and R. de Borst (Eds.), Computational modelling of concrete structures, Euro-C 1994, Volume 1, Innsbruck, Austria, pp. 185–198. Pineridge Press. Comi, C. (2001). A non-local model with tension and compression damage mechanisms. Eur. J. Mech. A/Solids 20(1), 1–22. de Borst, R., J. Pamin, and M.G.D. Geers (1999). On coupled gradient-dependent plasticity and damage theories with a view to localization analysis. Eur. J. Mech A/Solids 18(6), 939–962. de Vree, J.H.P., W.A.M. Brekelmans, and M.A.J. van Gils (1995). Comparison of nonlocal approaches in continuum damage mechanics. Comput. & Struct. 55(4), 581–588. Feenstra, P.H. (1993). Computational aspects of biaxial stress in plain and reinforced concrete. Ph.D. dissertation, Delft University of Technology, Delft. fib (1999). Structural Concrete. The Textbook on Behaviour, Design and Performance. Updated knowledge of the CEB/FIP Model Code 1990, Volume 1, Bulletin No. 1. fib. Geers, M.G.D. (1997). Experimental analysis and computational modelling of damage and fracture. Ph.D. dissertation, Eindhoven University of Technology, Eindhoven. Heeres, O., A. Suiker, and R. de Borst (2002). A comparison between the Perzyna viscoplastic model and the Consistency viscoplastic model. Eur. J. Mech. A/Solids 21, 1–12. Kachanov, L.M. (1958). Time of rupture process under creep conditions. Izd. Akad. Nauk SSSR, Otd. Tekh. Nauk 8, 26–31. (in Russian). Kleiber, M. (1998). Plasticity problems. In M. Kleiber (Ed.), Handbook of Computational Mechanics. Survey and Comparison of Contemporary Methods, Berlin, Germany, pp. 201–252. Springer Verlag. Kupfer, H. (1973). Das Verhalten des Betons unter mehrachsiger Kurzzeitbelastung under besonderer Berücksichtigung der zweiachsigen Beanspruchung. Number 229. Berlin: Deutscher Ausschuss fur Stahlbeton. Lemaitre, J. (1971). Evaluation of dissipation and damage in metals. In Proc. I.CM., Volume 1, Kyoto, Japan. Mazars, J. and G. Pijaudier-Cabot (1989). Continuum damage theory—application to concrete. ASCEJ. Eng. Mech. 115, 345–365.

Pamin, J. (2005). Gradient plasticity and damage models: a short comparison. Computational Materials Science 32, 472–479. Pamin, J., A. Wosatko, and A. Winnicki (2003). Two- and three-dimensional gradient damage-plasticity simulations of cracking in concrete. In N. Bi´cani´c et al. (Eds.), Proc. EURO-C 2003 Int. Conf. Computational Modelling of Concrete Structures, Rotterdam/Brookfield, pp. 325–334. A.A. Balkema. Pearce, C.J. (1993). Computational aspects of the softening Hoffman plasticity model for quasi-brittle solids. M.Sc., University Of Wales, College Swansea. Peerlings, R.H.J., R. de Borst, W.A.M. Brekelmans, and J.H.P. de Vree (1996). Gradient-enhanced damage for quasi-brittle materials. Int. J. Numer. Meth. Engng 39, 3391–3403. Ruiz, G., M. Ortiz, and A. Pandolfi (2000). Threedimensional finite-element simulation of the dynamic Brazilian tests on concrete cylinders. Int. J. Numer. Meth. Engng 48, 963–994. Simo, J. and T. Hughes (1997). Computational Inelasticity. New York, Berlin: Springer Verlag. Simo, J.C. and J.W. Ju (1987). Strain- and stress-based continuum damage models–I. Formulation, II. Computational aspects. Int. J. Solids Struct. 23(7), 821–869. Sluys, L.J. (1992). Wave propagation, localization and dispersion in softening solids. Ph.D. dissertation, Delft University of Technology, Delft. Sluys, L.J. and R. de Borst (1996). Failure in plain and reinforced concrete–an analysis of crack width and crack spacing. Int. J. Solids Struct. 33, 3257–3276. Suaris, W. and S.P. Shah (1985). Constitutive model for dynamic loading of concrete. J. of Struct Engng ASCE 111(3), 563–576. Taylor, R.L. (2001). FEAP—A Finite Element Analysis Program, Version 7.4, User manual. Technical report, University of California at Berkeley, Berkeley. van Mier, J.G.M. (1984). Strain-softening of concrete under multiaxial loading conditions. Ph.D. dissertation, Eindhoven University of Technology, Eindhoven. Vonk, R.A. (1992). Softening of concrete loaded in compression. Ph.D. dissertation, Eindhoven University of Technology, Eindhoven. Wang, W.M. (1997). Stationary andpropagative instabilities in metals—a computational point of view. Ph.D. dissertation, Delft University of Technology, Delft. Winnicki, A. (2007). Viscoplastic and internal discontinuity models in analysis of structural concrete. Series Civil Engineering, Monograph 349, Cracow University of Technology, Cracow. Winnicki, A., C.J. Pearce, and N. Bi´cani´c (2001). Viscoplastic Hoffman consistency model for concrete. Comput. & Struct. 79, 7–19. Wosatko, A. (2008). Finite-element analysis of cracking in concrete using gradient damage-plasticity. Ph.D. dissertation, Cracow University of Technology, Cracow. Wosatko, A., J. Pamin, and A. Winnicki (2006). Gradient damage in simulations of behaviour of RC bars and beams under static and impact loading. Archives of Civil Engineering 52(1), 455–477. Wosatko, A., A. Winnicki, and J. Pamin (2009). Numerical analysis of brazilian split test on concrete cylinder. Computers & Concrete. Submitted for publication.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Micromechanical approach to viscoelastic properties of fiber reinforced concrete V.F. Pasa Dutra, S. Maghous & A. Campos Filho Universidade Federal do Rio Grande do Sul, Porto Alegre, Brasil

ABSTRACT: This paper studies some aspects of fiber reinforced concrete (FRC) within a micromechanical framework. As a starting point, the overall elastic properties are determined by modeling the fibers as flat prolate spheroids and implementing a Mori-Tanaka homogenization scheme. In a second step, the formulation of the non-aging viscoelastic behavior is carried out by making use of results previously obtained in elastic homogenization and the Elastic-Viscoelastic Correspondence Principle. Adopting a Zener model for the behavior of the concrete matrix the homogenized creep compliance functions are derived analytically. The validity of the model is established by means of comparison with available experiment measurements of creep strain in steel fiber reinforced concrete under compressive load and analytical models formulated within a one-dimensional setting. 1

INTRODUCTION

Fiber reinforced concrete (FRC) is a composite material formed by the association of a cement matrix and embedded fibers. Several studies have shown that the addition of steel fibers causes improvement of the mechanical properties of concrete such as ductility, post-cracking behavior, and resistance to static and dynamic actions. In the perspective of a wide use in structural engineering, a comprehensive formulation of the corresponding behavior is necessary. The micromechanics-based approach aims at establishing a connection between the macroscopic behavior of FRC and the characteristics of its constituents. A considerable advantage of the micromechanical approach lies in the fact that the closed-form expression for the homogenized properties makes it possible to easily analyze the impact of the reinforcement characteristics at the macroscopic level. This study focuses on elastic and viscoelastic behavior of the steel FRC by means of a micromechanical approach. Reasoning on the representative elementary volume (REV) of such a composite, the overall elastic characteristics are determined from the knowledge of the elastic properties of its individual constituents. Modeling the fibers as prolate inclusions embedded within the concrete matrix, the Mori-Tanaka scheme, which is based on the results established by Eshelby (1957), is used to derive estimates of the elastic coefficients. It was found that the micromechanical predictions for the overall stiffness proved to be considerably close to the experimental data, as well as to the finite element solutions obtained from numerical analysis of a REV

of steel FRC (modeled as a randomly heterogeneous medium). The formulation of the non-aging viscoelastic behavior is then carried out by making use of results previously obtained in elastic homogenization and the Elastic-Viscoelastic Correspondence Principle.

2 2.1

MACROSCOPIC ELASTIC BEHAVIOR OF FIBER REINFORCED CONCRETE Estimates of the elastic properties of FRC

This section deals with the evaluation of the elastic behavior of FRC through homogenization approach. The main interest of this approach lies on the possibility to use the obtained effective behavior to perform computations at the scale of the structure, considering the homogenized structure instead of the initial heterogeneous one. A central concept of the homogenization procedure is the existence of a representative elementary volume, which must comply with the scale separation condition (Zaoui 2002). The effective behavior of the composite is obtained from the response of its REV to a mechanical loading analyzed in the framework of a boundary value problem (Suquet 1987). In the framework of linear elastic homogenization of random heterogeneous material, considering the imposition of homogeneous strain boundary conditions, the macroscopic constitutive equation reads  = C hom : E =

∼

=

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(1)

where E represents in fact the macroscopic strain and

with

it is the volume average of the microscopic strain field over the whole REV,  is the macroscopic stress ten-

[1 + P m,α : (c f − c m )]−1 fib

=

=

∼

sor and it is taken by definition as the volume average of the microscopic stress field over the whole REV and C hom denotes the overall homogenized (macroscopic)

2π π

∼

∼

n  r=1

ϕ=0 θ =0

fr c r : Ar ∼

×

(2)

∼

∼

∼

conditions E 0 at infinity. =

The Mori-Tanaka scheme (Mori & Tanaka 1973), which is based on the results established by Eshelby, considers a composite with a matrix phase m clearly identified surrounding all the others. This homogenization scheme adopts this matrix as the reference medium in the Eshelby inhomogeneity problem: c 0 = ∼

c m and takes into account the interaction between the

∼

∼

∼

∼

∼

sin θ dθdϕ 4π

(4)

∼

∼

∼

[1 + P m,α (ϕ, θ) : (c f − c m )]−1

stiffness tensor of fibers, respectively. The fibers are represented by prolate spheroids which differ in orientation (θ, ϕ). The shape and orientation of the fibers are taken into account by the Hill tensor P m,α for pro-

the stiffness tensor of phase r. The fourth-order tensor A is the so-called strain concentration tensor and Ar is the corresponding volume average over the domain occupied by phase r. As emphasized by the Equation 2, the determination of the overall elasticity tensor requires being able to compute estimates of the average of strain concentration tensor over each phase. This is usually achieved through an appropriate homogenization scheme integrating some information on the microstructure morphology. From the viewpoint of Eshelby-based theory (Eshelby 1957), the average strain concentration tensor is estimated from the elastic solution of an ellipsoidal inhomogeneity embedded within an infinite medium (reference medium) with stiffness c 0 subjected to uniform strain boundary

∼

where f and c f denotes the volume fraction and the

in which n is the number of different phases of the REV, fr = the volume fraction of phase r and c =

∼

late spheroid with aspect ratio α (Mura 1987). fib is the volume average over the domain occupied by the fibers and is obtained through integration over all the orientation of space, as it can be seen in Equation 4. The assumption of elastic isotropy for both concrete matrix and fibers, and the random distribution of the fibers imply the elastic isotropy of FRC at the macroscopic level. The homogeneity assumed for the concrete matrix can result of a preliminary homogenization process, which accounts for possible microcracks. The homogenized elasticity tensor C hom,MT ∼

can therefore be completely defined by means of the homogenized bulk modulus K hom and shear modulus G hom , respectively Equations 5 and 6. These coefficients are function of the elastic modulus of concrete matrix (k m , g m ) and fibers (k f , g f ), as well on the volume fraction f and aspect ratio α of fibers. For simplicity, we shall restrict the subsequent analysis to the situation of fibers with infinite value for the aspect ratio (α → ∞), that is the case where the diameter parallel to the axis of revolution is considerably larger than the diameter perpendicular to the axis of revolution.

phases in an indirect way. More precisely, this interaction is expressed by means of the value of the reference strain E 0 which is equal to the average strain in the =

K hom =

matrix: E 0 = ε m . As a result of the latter consider=

∼

=

elasticity tensor: C hom =

∼

((3g m + g f )[(1 − f )k m + fk f ] + 3k m k f ) ((3g m + g f ) + 3(fk m + (1 − f )k f )) (5)

=

ations, the homogenized stiffness tensor of the MoriTanaka scheme for a uniform orientation distribution of fibers is ∼

[1 + P ∼

m,α

∼

: (c f − c m )]−1 fib } : ∼

∼

ni

5 

di

i=1

n1 = 9fk m k m (g f )3 ,

∼

∼

(c f − c m )]−1 fib ∼



∼

(1 − f )1 + f : [1 + P ∼

=

6  i=1

C hom,MT = {(1 − f )c m + f c f : ∼

G

hom

∼

m,α

:

n2 = 3g m (g f )2 (7fg f (k m + k f ) + 15k m k f (1 + f ) (3)

+ 5g f k m ),

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(6)

n3 = (g m )2 g f (35(g f )2 (35 + 37f )

1.5%}. Figures 1 and 2 respectively display the MoriTanaka estimates of E hom and ν hom , together with the experimental results. The Ashour et al. experiments (2000) correspond to steel fibers with steel fibers with an aspect ratio of α = 75 and f ∈ {0.5%, 1%}. The comparison of the micromechanical estimates of the homogenized Young modulus with the experimental results are shown in Figure 3. As it appears from the above figures, the MoriTanaka estimates prove to be slightly lower than the measured values provided in the mentioned experimental works (Thomas & Ramaswamy 2007, Ashour et al. 2000). A good agreement is obtained and the consistency with experimental data can be termed as reasonable. The discrepancy observed between the micromechanical predictions and the experimental results remains lower than 10%.

+ 15k g (5 + 3f ) + 3k g (35 + 17f ) m f

f

f

+ 10k m k f (9 − f )), n4 = (g m )3 ((g f )2 (145 + 39f ) + 21k m g f (5 − f ) + 3k f g f (40 − 19f ) + 45k m k f (1 − f )), n5 = (g m )4 (45k m (1 − f ) + 15k f (1 − f ) + g f (125 − 61f )), n6 = 15(g m )5 (1 − f ), d1 = 3k m (g f )2 (5g f (1 − f ) + 3k f (5 − 4f )), d2 = g m g f (35(g f )2 (1 − f ) + 3k m g f (25 − 13f ) + 21k f g f (5 − 4f ) + 90k m k f ) d3 = (g m )2 ((g f )2 (145 − 73f ) + 15k m g f (7 + f ) + 12k f g f (10 + 3f ) + 9k m k f (5 + 4f ), d4 = (g m )3 (9k m (5 + 4f ) + 3k f (5 + 16f ) + g f (125 + 59f )), d5 = (g m )4 (15 + 49f ).

2.2

Comparison with experimental results

The micromechanical predictions of FRC elastic properties using a results Mori-Tanaka scheme are compared herein to available experimental results performed on steel fiber reinforced concrete. The values of Young modulus and Poisson ratio of the composite components as provided in the experimental studies are listed in Table 1. In Thomas & Ramaswamy’s (2007) experiments, the aspect ratio of the used steel fibers was α = 55 and several volume fractions were tested f ∈ {0.5%, 1%, Table 1.

Figure 1. Young modulus: Mori-Tanaka estimate and experimentally measured value (Thomas & Ramaswamy 2007).

Material properties of steel fiber and concrete.

Material

Young modulus Poisson E (MPa) ratio ν

Thomas & Ramaswamy (2007) Steel Fiber Concrete 35 MPa (C35) Concrete 65 MPa (C65) Concrete 85 MPa (C85)

210,000 28,700 37,500 41,700

0.300 0.182 0.201 0.210

Ashour et al. (2000) Steel Fiber 210,000 Normal Strength Concrete NSC 24,612 Medium Strength Concrete MSC 35,443 High Strength Concrete HSC 38,423

0.300 0.200 0.200 0.200

Figure 2. Poisson ratio: Mori-Tanaka estimate and experimentally measured value (Thomas & Ramaswamy 2007).

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2.3

Finite element numerical solutions

In order to asses to accuracy of the Mori-Tanaka estimates, the concentration problem (Eq. 1) is now solved numerically through the implementation of a finite element (FE) method. For sake of simplicity, a cubic REV of side l is considered. Owing to the macroscopic isotropy of FRC, the restriction of the analysis to a loading mode defined by uniaxial compression is sufficient to obtain the overall elastic properties. Hence, a macroscopic stress  = −l e1 ⊗ e1 with l > 0 is applied to the REV. It is recalled that the scale separation condition ensures the equivalence between homogeneous boundary strain and stress conditions on the REV (Hill 1967). Twenty nodded quadratic hexahedral elements were used for the concrete discretization of the concrete matrix geometry. As regards the reinforcement components, the fibers are randomly generated and embedded within the concrete matrix finite elements.

Figure 3. Young modulus: Mori-Tanaka estimate and experimentally measured value (Ashour et al. 2000).

Table 2.

It is recalled that perfect bonding is assumed between fibers and concrete matrix. We herein refer to the socalled ‘‘embedded model’’ (Elwi & Hrudey 1989) in which each fiber has the same kinematics than the coincident points of the embedding concrete matrix finite element. The spatial distribution and orientation of fibers are randomly generated by means of a specific procedure using the intrinsic function RAN of Fortran programming language. Once the finite element displacement solution is obtained, the macroscopic strain could also be computed. It is expected that E = −El e1 ⊗ e 1 + Et (e 2 ⊗ e 2 + e 3 ⊗ e 3 ), =

with El , Et ≥ 0

(7)

The homogenized elastic parameters are then obtained by E hom = l /El and ν hom = Et /El . In the objective to qualitatively address the scale effect, two values for the REV side l = 20, 40 cm as well as two values for the fiber length d = 1, 3 cm have been considered in the numerical study. The values adopted for the aspect ratio and volume fraction of fibers range between 10–100 and 0.5%–5%, respectively. The numerical results for each of the above configuration defined by the set {l, d, α, f } are given in Tables 2 and 3, where the Mori-Tanaka predictions are also reported. It is noted that E hom and ν hom stand for the numerical estimates of the effective Young modulus and Poisson ratio obtained from the FE simulations. As it clearly appears from the above results, the micromechanical predictions are very close to the FE numerical solutions. This emphasizes the consistency of the Mori-Tanaka scheme to estimate the elastic properties of FRC. In addition even though the condition d << l necessary for scale separation (Zaoui 2002) is roughly satisfied for the adopted values of d and l, the obtained results seem to be low sensitive

FE element calculations and Mori-Tanaka prediction of the homogenized Young modulus of FRC. Young modulus E hom (MPa) d = 3 cm

Fiber REV α

f(%)

10 10 10 10 40 40 80 80 100 100

0.5 1 3 5 3 5 3 5 3 5

l = 20 cm

3259.9 3366.9 3250.4 3347.7 3242.5 3353.3 3248.3 3355.1

d = 1 cm l = 40 cm

3252.3 3363.6 3245.5 3353.9

l = 20 cm

l = 40 cm

Mori-Tanaka predict

3106.5 3132.7 3248.4 3362.8

3106.6 3132.6

3129.4 3159.0 3279.5 3403.3 3293.7 3427.1 3295.2 3429.7 3295.5 3430.1

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Table 3.

FE element calculations and Mori-Tanaka prediction of the homogenized Poisson ratio of FRC. Poisson ratio ν hom (MPa) d = 3 cm

Fiber REV α

f(%)

10 10 10 10 40 40 80 80 100 100

0.5 1 3 5 3 5 3 5 3 5

l = 20 cm

0.209 0.214 0.209 0.214 0.209 0.213 0.209 0.213

d = 1 cm l = 40 cm

0.209 0.214 0.209 0.214

l = 20 cm

l = 40 cm

Mori-Tanaka predict

0.203 0.204 0.209 0.214

0.204 0.205

0.200 0.201 0.203 0.205 0.203 0.205 0.203 0.205 0.203 0.205

to the value of ratio d/l, at least for the considered data. This justifies that the volume of FRC tested in the numerical simulation has the status of REV. Clearly enough, addition of steel fibers in amounts as small as those commonly used for FRC has a negligible effect on the effective elastic properties. From a practical point of view, computing the effective elasticity of FRC is not in itself a relevant task. However, this is a necessary step for further upscaling towards the formulation of a consistent micromechanical model describing the overall elastoplastic behavior of FRC.

makes use of the homogenization result in elasticity in the Laplace (or Laplace-Carson) transformed space and the Elastic-Viscoelastic Correspondence Principle (Salençon 1993). Formally, the concentration problem to be solved for non-ageing viscoelastic constitutive materials is identical to the elastic case defined by Equation 1. The constitutive equations at the macroscopic scale are therefore obtained by inverse Laplace (or Laplace-Carson) transform. In terms of Carson-Laplace transform variables, the concentration problem governing the stress and strain in the REV reads

3

 ∗ = C hom ∗ : E ∗

NON-AGING VISCOELASTIC BEHAVIOR OF FIBER REINFORCED CONCRETE

=

Under constant load, a concrete specimen first exhibits instantaneous elastic deformation then followed by a slow further increase of deformation in time, corresponding to concrete creep phenomenon. Incorporation of fibers such as steel fibers in concrete reduces the magnitude of creep deformation and associated damage, as has been well experimentally evidenced by Mangat & Azari (1985). The mechanism at the basis of the effect of fiber on matrix creep strain which is frequently invoked relies on the assumption that shear stress is induced at the fiber/matrix interface. Fibers do provide restraint to the sliding action of the matrix relative to the fiber due to the flow component of creep. However, up to now, few works have been dedicated to linking macroscopic FRC creep behavior to composite micromechanics. Furthermore, the models developed in this context are essentially restricted to uniaxial creep of FRC (Mangat & Azari 1985, Zhang 2003). This section deals with the investigation of nonaging viscoelastic properties of steel FRC by means of a micromechanics approach. Basically, the study

∼

(8)

=

which formally expresses an elastic type relationship in Carson-Laplace space. The elastic homogenization procedure carried out in Section 2 as well as related results hold in terms of Carson-Laplace transform fields. C hom ∗ represents the Carson-Laplace trans∼

form of the macroscopic elasticity tensor. In particular, the Carson-Laplace transform of the Mori-Tanaka estimate C hom,MT ∗ can be used as approximation of C hom ∗ ∼

(i.e. C hom ∗ ≈ C hom,MT ∗ ). ∼

∼

∼

To go further, we shall first specify the viscoelastic behavior of each constituent of FRC. 3.1

Vicoelastic behavior of concrete matrix and steel fibers

As a first approximation, the uniaxial non-ageing viscoelastic behavior of concrete matrix is described by the Zener rheological model. This simple and commonly used model for linear viscoelastic solid consists in the association in series of a spring (stiffness E1 ) and a Kelvin element with spring stiffness E2 and dashpot

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viscosity η. Even though this model is conceptually very simple, it reasonably describes the main expected characteristics of the response under creep and relaxation observed experimentally for concrete (Salençon 1993). The relaxation compliance E Zener (t) associated to the Zener model reads E Zener (t) =

E1 (E2 + E1 e−t/τr ) E1 + E 2

Equation 5 and 6, we obtain:

(9)

where τr = η/(E1 + E2 ) is the relaxation characteristic time. It is observed that E Zener (0) = E1 and E Zener (+∞) = E1 E2 /(E1 + E2 ) respectively refer to the instantaneous and asymptotic relaxation moduli. To describe the three-dimensional non-aging viscoelastic behavior of isotropic concrete matrix, the relaxation modulus under tensile stress is first approximated by the Zener relaxation compliance (Eq. 9), while a constant Poisson ratio ν m is adopted. The stiffness parameters of the model are identified by observing that E1 represents the elastic instantaneous stiffness, while the asymptotic relaxation modulus E(+∞) ranges between 3 and 5 times the instantaneous stiffness E1 , depending on certain parameters such as the quality of concrete, humidity and temperature (Mehta & Monteiro 1994, Neville et al. 1983). As regards the viscosity coefficient η, its value may be evaluated from the value of the characteristic time in relaxation τr , which usually ranges between 100 and 300 days (Mehta & Monteiro 1994, Neville et al. 1983). Finally, the analysis is restricted herein to steel fibers, for which the delayed effects can be neglected at usual temperatures. Accordingly, a linear elastic behavior is adopted for fibers, E f and ν f being the Young modulus and Poisson ratio of steel, respectively. 3.2 Homogenized viscoelastic properties of FRC The calculation of the homogenized viscoelastic properties needs first the acquisition of elastic homogenization result in the Carson-Laplace transformed space. The first step is to calculate the transforms of the bulk and shear moduli of concrete and steel. The Correspondence Principle indicates that the Carson-Laplace transforms of the homogenized bulk and shear moduli of FRC are formally obtained by replacing in the expressions of K hom and G hom , respectively the Equation 5 and 6, the bulk and shear moduli of FRC components by their Carson-Laplace transforms. For simplicity, we shall restrict the subsequent analysis to the situation of fibers with infinite value for the aspect ratio (α → ∞). This consideration is justifiable, since this parameter showed little influence on the elastic behavior of the FRC. Thus, considering the

K hom ∗ =

NK (k m∗ , g m∗ , k f ∗ , g f ∗ , f ) DK (k m∗ , g m∗ , k f ∗ , g f ∗ , f )

(10)

G hom ∗ =

NG (k m∗ , g m∗ , k f ∗ , g f ∗ , f ) DG (k m∗ , g m∗ , k f ∗ , g f ∗ , f )

(11)

The crucial step of such a viscoelastic homogenization process lies in the ability to compute the CarsonLaplace inverse transform of K hom∗ and G hom∗ . Generally, such an inverse transform is achieved by means of numerical procedures such as those given in Abate & Whitt (2006). In the present situation, the particular structure of K hom∗ and G hom∗ makes it possible to derive analytically the expression of the relaxation moduli K hom (t) and G hom (t). The latter moduli can formally be written as ⎡

⎤ 2  Q j K hom (t) = ⎣Q0 + (1 − e−qj t )⎦ H (t) qj

(12)

j=1

⎡

⎤ 4  R j (1 − e−rj t )⎦ H (t) G hom (t) = ⎣R0 + r j j=1

(13)

where H (t) is the Heaviside step function. The above viscoelastic behavior of (FRC) is illustrated below. The select data corresponding to the usual concrete and steel fibers are given in Table 4. The parameters of the Zener model describing the viscoelastic behavior of concrete are E1 = 31 Pa, E2 = 15.43 GPa, η = 4650 GPa × days. The fact that these model parameters are considered as constant is consistent with the approximation of non-aging concrete behavior. It is observed that the latter could reasonably be justified for instance if concrete is loaded at advanced age (Berthollet 2003). The variations in time of the creep compliance function in bulk JKhom (t) and creep compliance in shear JGhom (t) are displayed in Figures 4 and 5, respectively. It should be recalled that these functions are related to K hom (t) and G hom (t) in the Carson-Laplace Table 4.

Viscoelastic properties of concrete and steel fiber.

Concrete Instantaneous relaxation modulus Asymptotic relaxation modulus Relaxation time Poisson ratio

Em (0) Em (+∞) τr νm

31,000 MPa 10,300 MPa 100 days 0.2

Steel Young modulus Poisson ratio

Ef νf

210,000 MPa 0.3

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Figure 4. (FRC).

Model-predicted creep compliance bulk for steel

significant in the present case as it can be illustrated trough a comparison between the relaxation characteristic times of concrete and homogenized material. Indeed, the characteristic time for the variations of G hom is about 102 days whereas it is equal to 100 days for the considered concrete. This small difference is attributed to the particular viscoelastic model considered for concrete as well as to the selected data. It is likely that the memory effects would be more important if the amount of fiber increases. The next sections are intended to validate and verify the proposed theory and model for non-ageing viscoelastic behavior of FRC. The model predictions will be compared to available experimental data as well as to some theoretical models already developed for the compressive creep of concrete matrix reinforced by steel fibers. It should be kept in mind that only non-aging basic creep of concrete is addressed in the proposed model. All the other components and mechanisms of concrete creep (thermal, autogeneous and drying shrinkage) are disregarded in the modeling. Moreover, the model calibration disregards the dependence of non-aging properties of concrete on effective load-bearing structure of calcium silicate hydrates (C-S-H). This phenomenon was explained and verified by Bažant & Prasannan (1989a, b) in the context of solidification theory. Consequently, the results as well as the related comments of comparison of model predictions with experimental data should be considered with caution.

3.3

Figure 5. Model-predicted creep compliance in shear for steel (FRC).

domain through JKhom∗ = 1/K hom∗ and JGhom∗ = 1/G hom∗ . Two values commonly used for volume fractions of steel fiber have been considered: f = 1% and f = 5%. As shown by Figures 4 and 5, the addition of steel fibers • does not significantly affect the elastic instantaneous properties; • do provide restraint to the deformation of the concrete matrix along the fiber matrix interfacial bond. These observations are consistent with previous works (Mangat & Azari 1985, Chern & Young 1989). Besides, these figures clearly show that the reinforcing effect due to fibers increases with time. Finally, it also appears from these figures that the so-called ‘memory effects’, generally induced by homogenization (see for example Maghous & Creus 2003), are not

Comparison with experimental data

An experimental investigation on creep of steel fiber reinforced concrete has been performed by Mangat & Azari (1985). The tests were carried out on specimens 100 mm × 100 mm × 500 mm of concrete reinforced by means of steel fibers with aspect ratio ranging between α = 56 and α = 61. The volume fraction of fibers ranged between f = 0 and f = 3%. The specimens were cured in a temperature and humidity controlled room for 28 days and then loaded under uniaxial compression up to a stress/strength ratio of 0.3. A temperature of 20◦ C and a relative humidity of 55% were maintained in the controlled curing room. Table 5 summarizes the properties of the materials used for the experimental analysis. For comparison purpose, the procedure used to identify the model parameters (Zener parameters) is the following. The instantaneous relaxation modulus is determined by condition E Zener (0) = E1 = σc /εm (0), whereas the value of E2 is fixed by the asymptotic condition E Zener (+∞) = E1 E2 /(E1 + E2 ) = σc /εm (+∞). These conditions yield E1 = 20.4 GPa and E2 = 7.7 GPa. The determination of the dashpot viscosity η, or equivalently the characteristic time in relaxation, is achieved by curve fitting from

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Table 5. Material properties for creep tests by Mangat & Azari (1985). Concrete Instantaneous elastic strain Asymptotic creep strain Poisson ratio

ε m (0) ε m (+∞) νm

365 10−6 1335 10−6 0.2

Steel Young modulus Poisson ratio

Ef νf

210,000 MPa 0.3

observed at early stages of loading is clearly attributed to the fact that the proposed model does not account for concrete aging. It should be emphasized that the experimental results analyzed in this paper clearly indicate no slip on the matrix-fiber interface, which is compatible with the micromechanical model assumptions.

3.4

Figure 6. Creep strain of steel fiber reinforced concrete versus loading time: comparison between model predictions and experiments (Mangat & Azari 1985).

the experimental creep curve of unreinforced concrete. This leads to the value η = 184.1 GPa × days. Figure 6 shows the relationship between uniaxial creep strain of steel FRC and time under load at 0.3 stress/strength ratio. The continuous curves represent to model predictions whereas the symbols refer to the experimental points obtained by Mangat & Azari (1985). It is first recalled that all the theoretical model predictions correspond to the asymptotic value α → +∞ for the fiber aspect ratio. The relevancy of such a model assumption seems to be justified by the experimental measures. Indeed, the experimental results shown in Figure 6 suggest that the value of the fiber aspect ratio α has a slight effect on the creep properties of (FRC). Actually, this result confirms that established in elasticity (small influence of α on the overall elastic properties) still hold in viscoelasticity. It can be seen from this figure that the theoretical predictions fairly fit the experimental results, particularly for advanced loading time. The discrepancy

Comparison with theoretical models

Additionally to experiments described earlier, two theoretical models for the assessment of creep of steel reinforced concrete under uniaxial compression were found in the literature: the Mangat & Azari (1985) and Zhang (2003) models. Mangat & Azari (1985) and later Zhang (2003) developed analytical models for creep of steel reinforced concrete under uniaxial compression. These two models were formulated within quite similar frameworks in terms of basic assumptions. First, both models are based on the assumption that creep of the reinforced material in the loading direction is restrained by an equivalent distribution of aligned fibers parallel to the direction of sustained applied stress. In addition, the two models assume that the fibers provide restraint to creep of the concrete matrix through the fiber/matrix interfacial shear stress τ . The formulation of the analytical expression for creep strain of steel reinforced concrete matrix is achieved in the approach developed by Mangat & Azari (1985) as well in that proposed by Zhang (2003) by introducing heuristic considerations. Comparison of the proposed model predictions with those derived from the model of Mangat & Azari (1985) and the model of Zhang (2003) are shown in Figures 7 and 8. The experimental results obtained by Mangat & Azari (1985) are also reported in these figures for two values of the fiber aspect ratio α = 56 and α = 59. It is observed from these figures that the models exhibit a good correlation with the experiments. The model of Mangat & Azari (1985) and of Zhang (2003) better fit the experiment results for early stages of loading, whereas the proposed model presents better predictions with respect to experiments for higher values of loading time. Therefore, it can be emphasized from the above analysis that, although its relative simplicity, the proposed viscoelastic model showed a good correlation with a series of experimental results available from two independents studies. Moreover the proposed model presented similar performance when compared to two analytical models formulated for uniaxial creep of (FRC). The main advantage of the proposed model lies in its applicability to multiaxial solicitations. Clearly enough, the proposed model still needs to be improved in order to address important aspects of (FRC) behavior, such as ageing or micro-cracking of

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the matrix. The model can however be used in its current form, as a preliminary approach, in both material optimization and performance prediction with regard to creep of (FRC). 4

Figure 7. Creep strain as a function of loading time for fiber aspect ratio α = 56. Predictions of proposed model versus predictions of Mangat & Azari (1985) and Zhang (2003) models.

Figure 8. Creep strain as a function of loading time for fiber aspect ratio α = 59. Predictions of proposed model versus predictions of Mangat & Azari (1985) and Zhang (2003) models.

CONCLUSION

In this paper the effective elastic properties of (FRC) is estimated by a Mori-Tanaka linear scheme. This model successfully predicts the values of the Young’s modulus and Poisson’s coefficient of such a composite, as emphasized by comparison with either finite element solutions developed in the present work or available experimental data. A considerable advantage of the micromechanical approach lies in the fact that the closed-form expression for the homogenized elastic properties makes it possible to easily analyze the impact at the macroscopic level of the reinforcement characteristics (fiber stiffness, fiber volume fraction, fiber aspect ratio). In the situation, covering in particular the case of a loading applied to concrete at advanced age, where ageing could be neglected for the viscoelastic behavior of concrete, the homogenized behavior of (FRC) has been determined. Reasoning within the framework of Carson-Laplace transform, the conjunction of the results previously obtained in elastic homogenization with the correspondence principle allows deriving closed-form expressions for the homogenized bulk and shear relaxation moduli. The validity of the model has been assessed by mean of comparison with available experiment results of creep strain of (FRC) specimens under compressive load. In addition, the model predictions prove to be similar to those derived from two analytical models formulated in one-dimensional setting. The main advantage of the present model lies in its potential applicability to (FRC) under multiaxial solicitation. In view of a more comprehensive modeling of (FRC) behavior, a number of improvements and developments are still necessary. In this respect, the validation of the homogenized viscoelastic model formulated in Section 3 through comparison with experimental data and finite element solutions is a fundamental issue to be addressed. As regards the theoretical aspect, a key challenge for next future extensions is to account for ageing. Indeed, the viscoelastic behavior concrete exhibits ageing and a specific homogenization procedure is required (Maghous & Creus 2003), since the reasoning based on Carson-Laplace transform does not apply. Moreover, a coupled viscoelasticity-damage modeling would be a relevant framework to address the question of slip at the matrix/fiber interface. Finally, the real task which remains to be done consists in dealing with nonlinear phenomena which are necessary for addressing strength. Actually, the formulation of the macroscopic strength properties of (FRC)

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by means of limit analysis homogenization techniques is the matter of on-going research. REFERENCES Abate, J. & Whitt, W. 2006. A Unified Framework for Numerically Inverting Laplace Transforms. INFORMS Journal on Computing 18: 408–421. Ashour, S.A., Wafa, F.F. & Kamal, M.I. 2000. Effect of the concrete compressive strength and tensile reinforcement ratio on the flexural behavior of fibrous concrete beams. Engineering Structures 22: 1145–1158. Bažant, Z.P. & Prasannan, S. 1989a. Solidification theory for concrete creep: I. Formulation. Journal Engineering Mechanics 115(8): 1691–1703. Bažant, Z.P. & Prasannan, S. 1989b. Solidification theory for concrete creep: II. Verification and application. Journal Engineering Mechanics 115(8): 1691–1703. Berthollet, A. 2003. Contribution à la modélisation du béton vis-à-vis du vieillissement et de la durabilité: interaction des déformations de fluage et du comportement non-linéaire du matériau. PhD Thesis, Ecole doctorale de Mécanique, Energétique, Génie civil et Acoustique, Lyon. Chern, J.C. & Young, C.H. 1989. Compressive creep and shrinkage of steel fibre reinforced concrete. The International Journal of cement Composites and Lightweight Concrete 11(34): 205–214. Elwi, A.E. & Hrudey, T.M. 1989. Finite element model for curved embedded reinforcement. Journal of Engineering Mechanics Division 115: 740–754. Eshelby, J.D. 1957. The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proceedings of the Royal Society of London A 241(1226): 376–396. Hill, R. 1967. The essential structure of constitutive laws for metal composites and polycrystals. Journal of the Mechanics and Physics of Solids 15: 79–95.

Maghous, S. & Creus, G.J. 2003. Periodic homogenization in thermoviscoelasticity: case of multilayered media with ageing. International Journal of Solids and Structures 40: 851–870. Mangat, P.S. & Azari, M.M. 1985. A theory for the creep of steel fibre reinforced cement matrices under compression. Journal of Materials Science 20: 1119–1133. Mehta, P.K. & Monteiro, P.J.M. 1994. Concrete: Structure, Properties, and Materials. New York: Prentice Hall. Mori, T. & Tanaka, K. 1973. Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metallurgica 21: 571–574. Mura, T. 1987. Micromechanics of Defects in Solids. The Netherlands: Nijhoff. Neville, A.M., Dilger, W.H. & Brooks, J.J. 1983. Creep of Plain and Structural Concrete. London: Construction Press. Salençon, J. 1993. Viscoélasticité. Paris: Presse de l’Ecole Nationales des Ponts et Chaussées. Suquet, P.M. 1987. Elements of homogenization for inelastic solid mechanics, Homogenization Techniques for Composite Media. In E. Sanchez-Palencia & A. Zaoui (eds), Lecture notes in physics No. 272: 193–278. New York: Springer. Thomas, J. & Ramaswamy, A. 2007. Mechanical properties of steel fiber-reinforced concrete. Journal of Materials in Civil Engineering 19: 385–392. Zaoui, A. 2002. Continuum Micromechanics: Survey. Journal of Engineering Mechanics 128: 808–816. Zhang, J. 2003. Modeling of the influence of fibers on creep of fiber reinforced cementitious composite. Composites Science and Technology 63: 1877–1884.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Boundary and evolving boundary effects in non local damage models Gilles Pijaudier-Cabot Laboratoire des Fluides Complexes, Université de Pau et des Pays de l’Adour, Anglet, France

Frédéric Dufour Laboratoire 3S-R, Institut National Polytechnique de Grenoble, Grenoble, France

ABSTRACT: The present contribution discusses boundary effects in non local damage modelling. We recall first that on a boundary, interaction stress components normal to the surface should vanish. These interaction stresses are at the origin of non locality and therefore the material response of points located on the boundary should be local. Then we discuss a tentative modification of the classical non local damage model aimed at accounting for this effect due to existing boundaries and also boundaries that arise from crack propagation. One-dimensional computations show that the profiles of damage are quite different compared to those obtained with the original formulation. The region in which damage is equal to one is small, with a triangular profile of damage according to the modified formulation instead of a rectangular one. The modified model performs better at complete failure, with a consistent description of discontinuity of the displacement field after failure and with a consistent simulation of a spalling failure. 1

INTRODUCTION

Most failure models for strain softening materials involve non locality. Such constitutive relations provide consistent continuum models for progressive cracking in quasi-brittle materials (see e.g. (Bažant and Jirasek 2002)). Non locality finds its origin in the interaction between growing defects in the course of failure, see e.g. Refs. (Bažant 1994; Pijaudier-Cabot and Berthaud 1990). Nearby the boundary of the solid, interactions between defects are expected to be different compared to those observed in the bulk material. On the boundary of a solid the material response should be local according to (Krayani et al. 2009). Cracks should also modify the non local interactions and should induce some shielding effect: the interaction between two points located apart from a crack should not exist. This means also that non locality, to some extend, should depend on the state of damage in the material and, for instance, that the internal length entering in non local models should evolve in the course of failure. Capturing existing and evolving boundaries in non local model is a pending issue with various approaches: • In integral models, the weight function involved in the non local average is chopped off and normalized (Pijaudier-Cabot and Bažant 1987). • In gradient enhanced models, the free boundary condition on the non local variable is the same as the condition that would be induced by an axis of symmetry (no normal flux of the regularised variable). Therefore, the non local interactions

nearby a boundary of the solid are the same as the non local interactions that would be observed nearby an axis of symmetry. • In damage models with non local displacements, the local displacements are taken equal to the non local displacements at the boundary (Rodriguez-Ferran et al. 2005). This condition is rather arbitrary but it is the only case where it is consistent with the assertion of locality of the material response on the boundary. Krayani and co-workers (Krayani et al. 2009) proposed some modifications of the integral non local damage model aimed at accounting for the locality of the material response on the boundary. In order to evaluate the pertinence of the modification of the non local model numerical simulations of size effect were considered. The issue was to analyse size effect in specimen geometries where the crack initiates from a notch or from a flat surface (un-notched specimens). It was demonstrated that the original non local model was not capable to describe size effect in both cases with a single set of material parameters. On the contrary, the modified model which accounts for a local behaviour at the surface of the specimen (where the crack initiates in un-notched specimens) provided a more consistent description of size effect in both geometries. This result indicated that a better approach to the boundary effect provided more consistent numerical simulations (although it remains to be checked against experiments). The purpose of this paper is twofold: first we discuss some justification based on micro-mechanics to

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the fact that the constitutive model should be local at the boundary of the solid. We also discuss the pertinence of using non local quantities for governing damage growth within the format of local stress-strain relation. Second we consider how boundary effects can be folded into a new non local damage formulation. A crack is composed of two free surfaces facing each other. In a continuum damage setting, a crack is represented by a band of intense damage. When damage is almost maximum, close to 1 at material failure reproducing macro-crack formation, a local response should be recovered and non local interactions should vanish. Modification of the non local damage model are proposed accordingly. Finally, one dimensional examples illustrate the capabilities of this new model to describe progressive failure and also complete failure with a view to the estimate of crack opening and to spalling failure. 2

Infinite elastic body containing two voids.

Figure 2.

Superposition scheme in sub-problem II.

INTERACTIONS IN AN ELASTIC SOLID WITH MICRO-CRACKS

In this section, we are going to look at the material at a scale where the microstructure can be described in details. We will discuss the case of a damageable material, viewed as an elastic material which contains a distribution of defects. The purpose is to exhibit the various interactions which develops at this level, between cracks and between cracks and the boundary of the solid. We are going to discuss the case where the distance between the micro-cracks is not very small compared to their length (of the same order of magnitude). Under this assumption, interaction is the same as if the cracks were two voids. The spatial distribution of the interaction forces is the same, except that the prefactor may differ between cracks and voids (Bažant 1994; Pijaudier-Cabot and Bažant 1991). Thus we will speak about cracks or voids indifferently. 2.1

Figure 1.

Infinite body

Let us consider for the sake of simplicity the case of an elastic material containing two voids and subjected to a remote stress field σ∞ (Figure 1). The two voids denoted as S1 and S2 are of diameter a1 and a2 respectively and the distance between their center is denoted as r. We may compute the state of stress in the solid, accounting for the presence of the voids and their interaction, according to a superposition scheme that is similar to the technique devised by Kachanov for crack interaction (Kachanov 1987; Fond and Berthaud 1995). We decompose this problem into two sub-problems: • Sub-problem I: the solid is considered without any void and loaded by the remote boundary condition corresponding to σ∞ .

• Sub-problem II: the remote traction σ∞ is trans1 = P  2 = −σ∞ · n formed into distributed forces P acting inside each void whose inner surface is defined by the normal vector n. Consider now sub-problem II. Again, we can apply the principle of superposition (Figure 2), in order to compute the interaction stress field due to the presence of the voids. In sub-problem II-1, the void S1 is considered alone loaded by a force distribution p1 on its inner surface. These surface forces generate on the imaginary contour of void S2 stress vectors denoted as p21 . Similar considerations are performed for void S2 in subproblem II-2. Superposition of these sub-problems II yields:  1 = p1 + p12 P  2 = p21 + p1 P

(1)

The difficulty is now that the distributions of the forces pi acting inside void Si and of the influence

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pressure pji on the imaginary contours of void Sj are not known. Usually, these distributions are approximated. A possibility is to consider the distribution of each component to be piecewise linear on the contour of the voids (Pijaudier-Cabot and Bažant 1991). The distribution of each component can be also expanded into Fourier series with constant and periodic terms (Kachanov 1987). The level of approximation required for a proper representation depends on the complexity of the interactions. The resulting state of stress (stress tensor) in the body at point x in the infinite solid is the sum of the remote stress, and the interaction stresses due to the force distributions pi inside void Si denoted as σ (x)xi : σ (x) = σ∞ +

N 

σ (x)xi

Figure 3.

(2)

Semi-infinite body with a single inclusion.

i=1

where we have extended the results to the case of N voids, each void i being located at point xi . The sum in the right handside term of this equation is typically a non local term. It depends on the states of stress at the location of voids i = 1, . . . , N . In the very simplistic case of an elastic medium subjected to a remote isotropic stress containing weakly interacting voids, these terms are proportional to the remote stress field at the location of each void and depend on the ratio of the diameter of the void to the distance to the void ai /r(x, xi) (Pijaudier-Cabot et al. 2004).

inner contour of S1 . Superposition yields:  1 = p1 + p1, P 0 = p ,1 + p

2.2 Semi-infinite body The same infinite elastic body is now cut into two pieces (Figure 3). The two voids have the same radius and are symmetrically located with respect to this new boundary. We shall be interested in the body containing void S1 only. The objective is the same as in the previous problem—we want to compute the elastic strain in the body—but now we have to deal with a free boundary . In sub-problem I, the solid without void S1 is subjected to the remote stress field. The corresponding distribution of stress satisfies the boundary conditions on the free surface : σ∞ · n = 0

(3)

In the new sub-problem II, the interaction between the void and the free surface is computed. A formulation that is similar to the previous one in section 2.1 may be followed. The surface forces p1 acting on the inner surface of void S1 generate on  surface forces denoted as p,1 . Surface forces acting on  are introduced and they generate surface forces p1, on the

(4)

Compared to Eq. (1-b), the left handside term in Eq. (4-b) is zero because of the condition of free surface on . The resulting interaction stress vectors on this surface should vanish. The same type of reasoning can be extended without difficulties to many interacting cracks nearby the boundary of a solid and to any type of boundary condition. The resulting, total interaction stress vectors should vanish on the free boundary of a solid. If non locality of the constitutive response is directly related to the presence of these interaction terms, non locality vanishes on the free surface, in the normal direction to this surface. 3

UPSCALING AND CONSEQUENCES IN CONTINUUM MODELLING

The equivalent homogeneous material which is the macroscopic counterpart of the above solid with microcracks is a continuum with an overall stiffness which is a function of the density, size and shape of the microcracks (see e.g. (Budiansky and O’Connel 1976)). The present issue is not to define accurately what should be the function relating the distribution of micro-cracks to the overall stiffness but rather how the variable which defines the growth of damage in the continuum model can be obtained from upscaling techniques. As suggested in the continuum based constitutive relationship proposed by Bažant, (Bažant 1994), the interaction stresses computed in the previous section are at the origin of non locality in a

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locality at the macro scale in the presence of strong gradients which does not vanish upon upscaling. Non locality vanishes only in three cases: (i) when the distribution of micro-craking is uniform and the solid is subjected to a homogeneous remote state of stress, local quantities are equal to non local ones; (ii) at the boundary of a solid, non locality should vanish in the direction normal to the boundary; (iii) upon a macrocrack formation, non locality should also vanish on the macro-crack faces (in their normal direction).

continuum description. Another framework based on a generalisation of Hashin-Shtrikman variational principles (Hashin and Shtrikman 1962) was discussed by Drugan and co-workers (Drugan and Willis 1996; Monetto and Drugan 1962) and it is quite instructive as it discusses non local effects. In the case of a two phase composite with isotropic and statistically uniform distribution of phases, Drugan and Willis arrived to a non local constitutive relation in the elastic regime relating the average stress < σ > to the average strain <  >: <σ >= L :<  > −  T : ∇ 2 (<  >) where the average strain <  > is defined as:  (x, ξ )dξ <  >= 0 +

(5)

4

We are going to examine now how such results can be incorporated in a continuum formulation and extended to the case of evolving boundaries, macro-crack surfaces described by a zone in which damage is equal to 1 according to the continuum model (Mazars and Pijaudier-Cabot 1996). Although the constitutive relations are discussed in a full 3D format, we will restrict applications of the modified non local model to the one-dimensional case which is simple but not entirely representative of full boundary effects (no tangential effects on the boundary). 2D and 3D generalisation are left for further developments.

(6)



Function (x, ξ ) is a measure of the local interactions derived with the help of Green’s functions and 0 is the strain in the homogeneous comparison solid (equal to the strain in the matrix if inclusions are very dilute). The above equation is very similar to Eq. (2), except that it is expressed in term of strain rather than stress. Since the material is elastic, the difference is not really fundamental. In the approach by Drugan (Drugan and Willis 1996), there is an evaluation of the size of the representative volume element (RVE) in the homogenised material. The size of the RVE is such that the second order gradient appearing in Eq. (5) becomes negligible. In other words, the constitutive equations relating the average stress to the average strain become local. This definition of the RVE was also used by Delaplace and co-workers (Delaplace et al. 1996) in order to compute a correlation length in a lattice model. According to Drugan and Willis, the size of the RVE for a material containing voids with a volume fraction less than 0.33 is about two times the diameter of the void. As pointed out in the same paper, this size of RVE is much less than what is commonly expected in heterogeneous materials. A ratio of five to ten times the size of the voids or cracks is usually used for defining the size of the RVE needed in order to average in a statistical sense the fluctuations of the stress or strain. Hence, the operator relating the stress to the elastic strain may be defined locally as non local second order gradient terms are negligible but at the same size of RVE, averages of strains and stresses may not be statistically representative. A non local strain may be required in order to capture damage growth, although the constitutive relations may remain defined locally since higher order terms are negligible. Pijaudier-Cabot and Dufour (Pijaudier-Cabot and Dufour 2010) have discussed this issue and arrived to the conclusion that crack interaction at the microstructural scale results into non

NON LOCAL DAMAGE MODEL

4.1

Isotropic (scalar) damage model

The classical stress-strain relation for this type of model reads: σij = (1 − D)Cijkl εkl

(7)

where σij and εkl are the components of the stress and strain tensors, respectively (i, j, k, l ∈ [1, 3]) and Cijkl are the components of the fourth-order elastic stiffness tensor. The damage variable D represents a measure of material degradation which grows from zero (undamage material with the virgin stiffness) to one (at complete loss of integrity). The material is isotropic, with E and ν the initial Young’s modulus and Poisson’s ratio respectively. For the purpose of defining damage growth, a scalar equivalent strain εeq is introduced, which quantifies the local deformation state in the material in terms of its effect on damage. In this contribution, Mazars’ definition of the equivalent strain is used (Mazars 1984):

εeq

  3  =  (εi + )2

(8)

i=1

where εi + are the positive principal strains. Damage growth is governed by the loading function: g(ε, k) = εeq (ε) − k

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(9)

k equals the damage threshold εD0 initially, and during the damage process it is the largest ever reached value of εeq . The evolution of damage is governed by the Kuhn-Tucker loading-unloading condition: g(ε, k) ≤ 0,

k˙ ≥ 0,

˙ kg(ε, k) = 0

(10)

The damage variable D is determined as a linear combination of two damage variables Dt and Dc , that represent tensile damage and compressive damage respectively, by the help of two coefficients αt and αc which depend on the type of stress state (Mazars 1984): D = αt Dt + αc Dc Dt,c = 1 −

(11)

1 − At,c At,c + εeq exp(Bt,c (εeq − εD0 ))

(12)

Standard values of the model parameters in the damage have been given in Ref. (Mazars 1984). 4.2

with respect to its arguments x and ξ . This lack of symmetry leads to the non-symmetry of the tangent operator (Bažant and Pijaudier-Cabot 1988; PijaudierCabot 1995; Jirasek and Patzák 2002). A symmetric non local formulation exists also (Borino et al. 2003).

Original non local formulation

In the integral-type non local damage models, the local equivalent strain is replaced by its weighted average: 

(x, ξ )εeq (ξ )dξ (13) ε¯ eq (x) = 

with  the volume of the structure and (x, ξ ) the weight function. It is required that the non local operator does not alter the uniform field, which means that the weight function must satisfy the condition: 

(x, ξ )dξ = 1 ∀x ∈  (14)

4.3

The goals of the modified non local formulation are twofold: (1) at the boundary of the solid, non locality should vanish; (2) when a crack has formed—i.e. when a band of damage close to 1 has been created, the constitutive relations should again be local in this band. By comparison to the modifications discussed in (Krayani et al. 2009) which dealt with a boundary effect only, here we try to account also for boundary effects occurring when a new free surface has formed in the process of cracking. We are going to rely on a simple argument: in the process of cracking, the quantity of information that can be transmitted between two neighbors depends on the state of damage. The argument is rather similar to that of Desmorat and Gatuingt (Desmorat and Gatuingt 2007). Some internal time is introduced in the constitutive relation which induces a decrease of the internal length upon damage growth. Here, the analogy with the propagation of wave is used but the weight function is regarded as an attenuation function in the transmission of information from one point to its neighbor. Let us denote as s the distance appearing in Eq. (16), defined as x − ξ . This distance is now modified according to the following incremental relation: du =



For this reason, the weight function is recast in the following form (Pijaudier-Cabot and Bažant 1987):

0 (x − ξ ) r (x) 

0 (x − ξ )dξ r (x) =

(x, ξ ) = with

Modified non local formulation

(15)



where r (x) is a representative volume and 0 (x − ξ ) is the basic non local weight function which may be taken as a polynomial bell-shaped function (Bažant and Jirasek 2002), or here as a Gauss distribution function:  4x − ξ 2

0 (x − ξ ) = exp − (16) lc2 lc is the internal length of the non local continuum. Preserving the uniform field in the vicinity of the boundary makes the averaging in Eq. (15) not symmetric

x − dξ  ds = γ (ξ ) γ (ξ )

(17)

where γ (ξ ) is a function of the state of damage at point of coordinate ξ . In the computation of the average centered at point x, we use now the distance u instead of the distance s:  ds u= (18)  γ (ξ ) In this remapping of the neighborhood around point of coordinate x, each point of coordinate ξ is now defined by the new coordinate u. In a spherical coordinate system centered at point x, the distance is defined by the above equation and the two angles are kept the same, they are invariant through the mapping function. The weighted average in Eq. (13) becomes:  ε¯ eq (x) =



(u)εeq (x + u)du

(19)

γ (ξ ) should be equal to 1 when the material is not damaged at point ξ , and it decreases when damage

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grows. As a consequence, the distance between two close neighbors will be increased as damage grows and in the weighted average process, the non local effect of point ξ will decrease. Function γ (ξ ) remains to be defined completely. We use for this an analogy with the attenuation during the propagation of waves. In the non local average defined in Eq. (13) the weight function (x−ξ ) is seen as the attenuation applied to a wave of amplitude εeq generated at point ξ and propagating toward point x.

(x−ξ )·εeq is the interaction of point ξ on point x. It is attenuated as the information is propagated in the solid. We may now consider that the wave speed depends on the state of damage: the time needed for the interaction effect to cover distance s is going to increase if the material on the path of propagation of the wave is getting damaged. the wave √ √In a one-dimensional setting, speed c = E/ρ becomes cd = (1 − D) · E/ρ where cd is the velocity of the damaged material. This increase of time for the information to be transmitted in the damaged medium between the two points ξ and x is converted into a fictitious increase of distance between these two points. The time needed for the interaction to propagate between the two points is kept the same, but since we consider now that the interaction is propagated in an undamaged medium the distance between the points is increased. When this distance is increased, the attenuation is increased accordingly. Let us consider Eq. (17) and divide both terms by the wave speed c: du ds ds = = c γ (ξ ) ∗ c cd

(20)

This equation states that the time needed for an information to propagate over a distance du in an undamaged medium is the same as the time needed for an information to propagate over the length ds in the damaged medium. Upon damage, the distance between these two points is increased, which is equivalent to a slower wave propagation and consequently, it is more attenuated than in the case where damage is not observed. From this equation we have: γ (ξ ) =

cd c

(21)

This qualitative reasoning implies that damage enters in the function γ . The difficulty is that the formulation becomes implicit. In order to keep the simplicity of the approach, we substitute to the non local value of damage the local expression and we take: γ (ξ ) = (1 − F(εeq (ξ )))

1 2

arbitrary. In fact, it fulfills the requirements stated above: when the material is not damaged, this function is equal to 1 and it becomes equal to zero if the material is totally damaged locally. At this stage, the incremental value of the modified distance between two points becomes infinite and the non local interaction in between them vanishes. The exponent 1/2 has been chosen by analogy to wave propagation. This new definition of the non local averaging needs also to fulfill the conditions on the free boundary demonstrated in the previous section. This can be performed by considering that on a free boundary local damage is set equal to 1. According to Eq. (17), a close neighbor to a point located on the boundary of the solid is located at a distance du which becomes infinite. Consequently, the non local effect of this neighbor to the non local average centered at the boundary is equal to zero.

5

COMPARISON BETWEEN THE ORIGINAL AND MODIFIED MODELS

In order to compare the original and the modified non local formulation, we are going to consider two simple one-dimensional tests. 5.1

Dynamic failure of a bar

The first test is quite classical (see e.g. (PijaudierCabot and Bažant 1987)). A bar of length 2L, in which two constant strain waves converge toward its center is considered. The amplitude of the wave is 0.7 times the deformation at the peak load in tension. When the two waves meet at the center, the strain amplitude is doubled, the material enters the softening regime suddenly and failure occurs in the middle of the bar. The bar length is taken equal to 30 cm. The parameters used in this example are: the mass per unit volume ρ = 1 kg/cm3 , the Young’s modulus E = 1 N/cm2 and the velocity boundary condition v = 0.7 cm/s applied at the two bar ends. The other model parameters are At = 1, Bt = 2, εD0 = 1 and the internal length lc is 4 cm (there is no damage in compression). A fixed mesh of 99 constant strain elements is used. Time integration is performed according to an explicit, central difference scheme. The time step is t = 0.3s. Figures 5 present the evolution with time of the profiles of damage and non local strain. The two waves meet at the center of the bar at time t = 15s.

(22)

where F(εeq ) is the function defined in Eq. (12). F(εeq ) is the local value of damage computed at the considered point. This definition of function γ is rather

X U(-L, t) = -vt

U(L, t) = vt L

Figure 4.

Principle of the one dimensional computation.

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Figure 6. Evolution of the damage profile with time over the bar according to the original damage model.

Figure 5. Evolution of the damage (top) and non local strain (bottom) profiles with time over the bar.

Note that damage develops over a band of finite—non zero–width. Compared to computations with the original damage model shown in Fig. 6, the profile of damage is almost triangular instead of being almost rectangular, forming a band of damage equal to 1 according to the original formulation. This difference is due to the modification of the weight function as damage develops. This is illustrated in Figs. 7. The weight function centered in the middle element where complete failure is expected to occur (e.g. where damage is equal to 1) shrinks progressively as damage develops. When damage is equal to 1 in this element, it is a Dirac delta function and the material response becomes local. When damage is equal to 1, it is as if the bar would be cut into two pieces. At this point, the material response is local and this is in agreement with micro-mechanics of crack interaction as demonstrated in this paper. In neighboring elements, however, the weight function evolves differently. We have computed the weight function nearby the element in which failure occurs, two elements farther on the left. Figure (7) shows that the weight is cut at the center of the bar. Information coming from material points located behind the point at which failure occurs is screened by damage. This is a difference with the original non local damage formulation in which non local interactions (weights) is transmitted even across a macro-crack. We have also

Figure 7. Evolution of weight function with damage: weight centered in the middle element (top), weight centered near the middle element (bottom).

checked in Fig. (8) that the distribution of damage is not subject to spurious mesh dependency. For the finite element meshes used which are already quite refined, the largest element size is smaller than the internal length used in the computations, the profiles of damage are almost the same. Convergence of the damage profiles with respect to mesh refinement means also that the energy dissipated at failure is a constant. It is the sum of the energy dissipated due to damage at each material point in the damage band.

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Figure 8. meshes.

Evolution of the damage profiles for different

Figure 9. Evolution of the maximum strain il the middle element with the finite element size. l0 = l/2

The damage profiles are not dependent on the finite element size. The issue is now whether the strain distribution should be also independent of the finite element size or not. Before complete failure, this result is expected. Without this property, the damage band which is a function of the maximum non local strain recorded in each finite element over the history of loading would not be expected to be mesh independent. When damage is equal to one in the middle element, convergence of the strain with respect to the size of this element may be questioned. Indeed, in the finite element located in the middle of the mesh, where complete failure occurs, the stress is vanishing and the relative displacement at the extremity of the finite element should be equal to the crack opening. According to the finite element discretisation however, there is no displacement and strain discontinuity and the crack opening is smeared over the element. Let us denote as h the length of this finite element, the crack opening [u] is: [u] =  × h

(23)

If the finite element calculation has converged with respect to mesh refinement, upon complete failure the crack opening should be independent of the element size and therefore that the strain in the element located in the middle of the mesh evolves as a function of h−1 . We have checked this in Fig. (9). According to the modified non local model, the maximum strain is indeed a power law of the finite element size. Exponent −1 is recovered. At the same time, the crack opening displacement is measured. We obtain [u] = 4,81 cm. Interestingly, the same crack opening displacement, computed according to the technique proposed by Dufour and co-workers (Dufour et al. 2008) is rather close [u] = 5,07 cm. For this calculation, we have chosen a constant weight function for the computation of the estimate of the crack opening. It is important to remark that the original non local formulation is far from exhibiting the same property. The

tension t < t0 u = ct t > = t0, u = 0 l = t0 E r

Figure 10.

compression

Principle of the dynamic tension test.

maximum strain is constant and therefore, in this simple application it may not provide the displacement discontinuity properly. 5.2

Spalling test

The second example is the spalling test proposed by (Krayani et al. 2009). This example has been designed so that the location of localised failure may occur very close to one extremity of the bar, involving modified non local interactions due to the extremity of the bar and to the occurrence of failure at the same time. The geometry and applied load is shown in Fig. 10. A square compression signal is generated in the bar. Upon reflection at the end of the bar, the compression signal turns into a tensile one. This signal is added to the incoming compression. If the absolute amplitude of the compression signal is greater than the tensile strength, failure is initiated at a distance from the boundary equal to half the signal length. Depending on the duration of the compression signal, it is possible to initiate failure in the material at any location, near the boundary or far from it. The bar length is 20 cm. The parameters used in this example are: the mass per unit volume ρ = 1 kg/cm3 , the Young’s modulus E = 1 N/cm2 and the velocity boundary condition c = −1.5 cm/s applied at the left bar end. The other model parameters are the same as in the previous computations. Time integration is performed according to an explicit, central difference scheme. The signal length is calculated as l = t0 υ and

its amplitude is c/υ where υ is

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E ρ.

The time step is

t = 0.2s and the signal length is 4s so that failure is initiated at a distance from the boundary equal to half the internal length. Figure (11) shows the evolution of the damage profiles according to the original non local model. We observe that maximum damage is reached exactly at the bar end. This result is in agreement with those obtained in (Krayani et al. 2009). On the contrary, the location of maximum damage is approximately located at 2 cm from the bar end according to the modified formulation (Fig. 12). The modified computation of the non local average devised in this paper provides results that are close to those proposed by Krayani and co-workers who discussed boundary effects, but not the case of cracks viewed as emerging boundaries. The modified formulation is capable of describing spalling failure with a spall of finite thickness, same as the modified formulation in (Krayani et al. 2009). There is, however, a difference between the two models that needs to be further investigated: the distribution of damage in Figure (12) decreases slowly in between the location of the maximum of damage and the finite element located at the bar end, and then it jumps to zero in the last finite element. In the formulation by

Figure 11. Evolution of the damage profiles during spalling failure according to the original formulation.

Krayani and co-workers, the non local strain becomes local as it reaches the boundary and the neighborhood over which the average is computed is decreased gradually as the point at which the average is computed is getting close to this boundary. This produces a smooth decreasing damage profile going towards the boundary after the peak damage located inside the bar.

6

CONCLUSIONS

Boundary effects are difficult issues in non local models. In either integral or gradient formulations, boundary conditions are rather arbitrary. We have recalled first that on a free boundary non local interactions should vanish. In the course of failure, when a macrocrack is formed, new boundary surfaces are appearing. This should also be taken into account in non local formulation, with the same requirements as for an initial boundary in the solid. We have presented a prototype damage model that accounts for the progressive shielding effect induced by a crack appearing in the material. This is achieved by a remapping of the non local averaging, in the same spirit as in (Krayani et al. 2009), but with a different mathematical formulation. One dimensional finite element calculations show that the modified non local model describes failure with a finite non zero fracture energy, that the damage profiles are triangular and not rectangular as observed in the original formulation, that the model is capable of approaching a discontinuous formulation at complete failure, whereas the original formulation is not capable of reaching this limit case, and that the model is capable of capturing spalling failure more properly that the original one. Grassl and Jirasek (Grassl and Jirasek 2006) showed that boundary effects had an influence on the energy of fracture. The variation of fracture energy, measured experimentally and derived from finite element computations, may help at the validation of the present model, e.g. with the help of structural size effect tests.

ACKNOWLEDGEMENTS Financial support from ERC advanced grant Failflow (27769) to the first author is gratefully acknowledged.

REFERENCES

Figure 12. Evolution of the damage profiles during spalling failure according to the modified formulation.

Bažant, Z.P. (1994). Nonlocal damage theory based on micromechanics of crack interactions. Journal of Engineering Mechanics 120, 593–617. Bažant, Z.P. and M. Jirasek (2002). Nonlocal integral formulations for plasticity and damage: Survey of progress. Journal of Engineering Mechanics 128, 1119–1149.

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Bažant, Z.P. and G. Pijaudier-Cabot (1988). Non-local continuum damage, localization instability and convergence. Journal of Applied Mechanics ASME 55, 287–294. Borino, G., B. Failla, and F. Parinello (2003). A symetric nonlocal damage theory. International Journal of Solids and Structures 40, 3621–3645. Budiansky, B. and R. O’Connel (1976). Elastic moduli of cracked solids. International Journal of Solids and Structures 12, 81–97. Delaplace, A., G. Pijaudier-Cabot, and S. Roux (1996). Progressive damage in discrete models and consequences on continuum modelling. Journal of the Mechanics and Physics of Solids 44(1), 99–136. Desmorat, R. and F. Gatuingt (2007). Introduction of an internal time in nonlocal integral theories. In Internal report n 268, Cachan, France, pp. 31. LMT-Cachan, Ecole normale suprieure de Cachan. Drugan, W. and J. Willis (1996). A micromechanics-based nonlocal constitutive equation and estimates of representative volume element size for elastic composites. Journal of the Mechanics and Physics of Solids 44(4), 497–524. Dufour, F., G. Pijaudier-Cabot, M. Choinska, and A. Huerta (2008). Extraction of a crack opening from a continuous approach using regularized damage models. Computers and Concrete 5, 375–388. Fond, C. and Y. Berthaud (1995). Extensions of the pseudotractions technique for friction in cracks, circular cavities and external boundaries; effect of the interactions on the homogenised stiffness. International Journal of Fracture 74, 1–28. Grassl, P. and M. Jirasek (2006). Damage plastic model for concrete failure. Computer Methods in Applied Mechanics and Engineering 43, 7166–7196. Hashin, Z. and S. Shtrikman (1962). On some variational principles in anisotropic and nonhomogeneous elasticity. Journal of the Mechanics and Physics of Solids 10, 335–342. Jirasek, M. and B. Patzàk (2002). Consistent tangent stiffness for nonlocal damage models. Computers and Structures 80, 1279–1293.

Kachanov, M. (1987). Elastic solids with many cracks— a simple method of analysis. International Journal of Solids and Structures 23, 23–43. Krayani, A., G. Pijaudier-Cabot, and F. Dufour (2009). Boundary effect on weight function in nonlocal damage model. Engineering fracture Mechanics 76, 2217–2231. Mazars, J. (1984). Application de la mécanique de l’endommagement au comportement non linéaire et à la rupture de béton de structure. Ph.D. thesis, Université Pierre et Marie Curie. Mazars, J. and G. Pijaudier-Cabot (1996). From damage to fracture mechanics and conversely: a combined approach. International Journal of Solids and Structures 33, 3327–3342. Monetto, I. and W. Drugan (1962). On some variational principles in anisotropic and nonhomogeneous elasticity. Journal of the Mechanics and Physics of Solids 10, 335–342. Pijaudier-Cabot, G. (1995, Chapt. 5, 105–144). Non local damage. In Continuum Models for Materials with Microstructure, H.B. Muhlhaus Ed., John Wiley Pubs. Pijaudier-Cabot, G. and Z. Bažant (1987). Nonlocal damage theory. Journal of Engineering Mechanics 113, 1512–1533. Pijaudier-Cabot, G. and Z. Bažant (1991). Cracks interacting with particles or fibers in composite materials. Journal of Engineering Mechanics 117, 1611–1630. Pijaudier-Cabot, G. and Y. Berthaud (1990). Effets des interactions dans l’endommagement d’un milieu fragile. formulation non locale. Comptes rendus de l’Acadmie des sciences 310, 1577–1582. Pijaudier-Cabot, G. and F. Dufour (2010). Non local damage model: boundary and evolving boundary effect. European Journal of Environ-mental and Civil Engineering, in press. Pijaudier-Cabot, G., K. Haidar, and J.-F. Dubé (2004). Nonlocal damage model with evolving internal length. International Journal for Numerical and Analytical Methods in Geomechanics 28, 633–652. Rodriguez-Ferran, A., I. Morata, and A. Huerta (2005). A new damage model based on non local displacements. International Journal for Numerical and Analytical Methods in Geomechanics 29, 473–493.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Homogenization-based model for reinforced concrete Erkan Rumanus & Günther Meschke Institute for Structural Mechanics, Ruhr University Bochum, Germany

ABSTRACT: In the paper, a constitutive model for reinforced concrete considering cracking and the relevant interactions between steel and concrete, such as bond slip and dowel action, is presented. Without explicit discretization of the reinforcing bars, reinforced concrete is modeled as a composite material by adopting concepts of continuum micromechanics to describe the macroscopic mechanical response. The formulation is based on the Mori-Tanaka homogenization scheme assuming a three-phase composite material formed by a continuous matrix (concrete) and two sets of straight rebars (reinforcement). For each phase, the nonlinear preand post-peak behavior is described separately, while considering the interactions between the reinforcement bars and the concrete. Concrete cracking is modeled by means of a fracture-energy-based elasto-plastic damage model. A classical J2 -plasticity model describes the elasto-plastic response of the steel reinforcement. Interfacial debonding is taken into account by adopting a two-phase serial system consisting of the steel bar and of the bond surface. The redistributed stresses due to crack formation at the steel-concrete interface are governed by slip strains. The related macroscopic bond-slip vs. bond-stress relationship is determined from classical pull-out tests. The residual shear stiffness of cracked reinforced concrete (dowel action) is derived from homogenization of the shear force vs. displacement relation obtained from the ‘‘beam on elastic foundation’’ theory considering each set of steel bar as a beam embedded in the surrounding concrete. The proposed model is implemented in the finite element code Msc.Marc and validated by means of selected experimental investigations on reinforced concrete panels and beams.

1

INTRODUCTORY REMARKS

The efficiency of the load-carrying characteristics of reinforced concrete (RC) is attributed to the compensation of the brittle behavior of concrete in tension by the ductile material properties of steel. The combination of these two materials generates a more ductile composite material and a more distributed mode of cracking with reduced crack widths. Both factors play a predominant role with regards to the serviceability and the ultimate load-carrying capacity of reinforced concrete structures. Since the material behavior of the composite differs completely from the material properties of the constituents, besides the constitutive laws for steel and concrete, also the corresponding interactions have to be considered adequately. Even though the interactions between steel and concrete are acting at the scale of the steel bars, their influence on the structural level is relevant and must be considered in a computational model for reinforced concrete structures. In the first part of the paper, the proposed nonlinear constitutive laws for concrete and steel are presented separately. Subsequently, the main steelconcrete interactions—bond slip and the dowel action—and the respective models for their numerical representation are summarized. Adopting concepts of continuum micromechanics, the composite material consisting of concrete and steel is formulated and

the homogenized constitutive relations for reinforced concrete considering the steel-concrete interactions in cracked concrete are established in the following sections. The performance of the proposed reinforced concrete model is finally demonstrated in the last section of this paper by means of re-analyzes of selected experimental investigations performed on RC panels and beams.

2 2.1

CONSTITUTIVE MODELS Concrete matrix material

The constitutive model for (plain) cracked concrete is originally formulated within the framework of poromechanics according to the Biot-Coussy theory (Coussy 2004). Using an elasto-plastic damage model, this multiphase concept allows consideration of shrinkage induced damage and its evolution of damage and long-term creep deformations resulting from combined external and thermo-hygral loading conditions (Meschke and Grasberger 2003; Grasberger and Meschke 2004). Since the focus of the present study lies on the mechanical behavior of concrete, the time-dependent state variables such as temperature, moisture and long-term creep are not included in the formulation of the model in this paper.

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The linearized strain tensor εm of the concrete is assumed to be small and can therefore be additively decomposed into elastic strains εem and irrecoverable p strains εm ε m = εem + εpm .

(1)

In the following, the index m refers to the matrix (concrete). To characterize the behavior of concrete in tension and in compression, a multi-surface fracture energy-based damage-plasticity theory is adopted Meschke et al. 1998. Degradation mechanisms and inelastic deformations are controlled by four threshold functions fk defining a region of admissible stress states in the space of the matrix stresses σ m E = {(σ m , qk )| fk (σ m , qk (αk )) ≤ 0,

In the following, the subscript s refers to the steel reinforcement. The secant stress-strain relationship is obtained as σ s = Cs0 : ε es = Cs0 : (ε s − εps ),

where Cs0 denotes the isotropic tensor of elasticity. The admissible stress field σ s within the steel reinforcement is controlled by a J2 –plasticity model (Simo and Hughes 1998)  fs (S, α) = S  − 2/3[σy + Kαs ] ≤ 0.

k = 1, . . ., 4}. (2)

Cracking of concrete is accounted for by means of a fracture energy based Rankine criterion, employing three failure surfaces perpendicular to the axes of principal stresses fR,A (σ m , qR ) = σA − qR (αR ) ≤ 0,

A = 1, 2, 3

(4)

with Cm0 denoting the undamaged elasticity tensor and ψ the isotropic integrity parameter capturing the damage state of the concrete. The evolution of the irrecovp erable strains εm associated with the mismatch of the crack faces is defined in an associative format. The evolution of the integrity ψ is governed by (ψ˙−1 ) =

3 

γ˙k

k=1

ε˙ dm C0 ε˙ dm ε˙ dm : σ

,

(5)

where εdm are the damage strains representing the effect of the stiffness degradation. Details of the formulation and the numerical implementation within a finite element code are contained in (Meschke et al. 1998). 2.2 Steel reinforcement The tensor of total strains ε s of the steel rebar is decomp posed into an elastic ε es and a plastic part ε s ε s = ε es + εps .

(8)

In 8 s denotes the deviatoric stress tensor, K a constant isotropic hardening plastic modulus and αs the isotropic hardening parameter governed by α˙ s =

 2/3γs ,

(9)

with γs as the consistency parameter. The evolution of p the irreversible plastic strains εs follows an associative flow rule.

(3)

with qR (αR ) = −∂U /∂αR denoting the softening parameter and the index A refers to the principal direction. The ductile behavior of concrete subjected to compressive loading is described by a hardening/ softening Drucker-Prager plasticity model. Based on a secant formulation, the stress field within the matrix σ m can be calculated by the elastic strains εem given in equation (1) σ m = ψCm0 : εem = Cm : (ε m − εpm )

(7)

(6)

3

CONCRETE-STEEL INTERACTIONS

In this section the two relevant mechanical interactions between concrete and steel, bond slip and dowel action, are briefly discussed and the adopted models are presented. While bond slip is concerned with the transmission of forces in the longitudinal direction of the steel bar, dowel mechanisms become relevant when shear forces across a crack are transmitted by the reinforcing bar. 3.1

Bond slip

As long as the interface between the concrete and the embedded rebars is undamaged, a full (perfect) bond can be assumed between the constituents which is manifested by strain compatibility. By exceeding the tensile strength of the concrete, however, relative displacements between the steel bar and the surrounding concrete occur, inducing local bond stresses in the vicinity of the cracks. Since the focus of the proposed model lies on structural analyses, a macroscopic approach is adopted for the modeling of bond behavior. Accordingly, the assumption of strain compatibility between steel and concrete in the bar direction is maintained and therefore no additional degree of freedoms need to be introduced. Furthermore, a macroscopic description of the bond behavior can easily be formulated within the framework of homogenization concepts. Hence, without focusing on local bond mechanisms, the proposed bond model is based on redistribution of steel stresses due to bond damage (Luccioni et al. 2005;

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Linero 2006; Manzoli et al. 2008). Adopting a twophase serial system consisting of the steel bar and of the bond surface, the strain tensor of the steel given in equation (6) is extended by slip strains εi εsi = εs + εi = εse + εsp + εi .

(10)

Note that in the local coordinate system of the steel bar, the strain tensor εi consists only of the component in the bar direction. The modified steel law including slip strains has the format σ s = Cs0 : ε es = Cs0 : (ε si − ε ps − ε i ).

(11)

Adopting suggestions made in (Linero 2006; Manzoli et al. 2008) a relatively simple elasto-plastic bond slip model is adopted (Figure 1). Note, that consequently the decreasing branch of the shear stress— slip law observed in experiments is neglected. The parameters (Ei , τmax ) may be obtained from classical pull-out tests where bond damage origins from sliding of the rebar along the steel-concrete interface as Ei =

P l , u As

τmax =

Pmax ≤ σy As

(12)

with As denoting the cross section of the embedded bar. 3.2 Dowel action The transmission of global shear stresses across a crack is governed—in absence of aggregate interlock—by dowel mechanisms of the reinforcing bar which undergoes additional shear and bending deformations in the vicinity of the crack. The dowel action can have a considerable influence on the structural stiffness, in particular when shearing or bending loading conditions are taken into account. Besides the global loading conditions the direction of the crack αc has a significant influence on the residual shear stiffness of a cracked reinforced element. In Figure 2a the dowel

Figure 2. a) Dowel action mechanisms. b) ‘‘Beam on elastic foundation’’—theory (He and Kwan 2001).

action is illustrated schematically. The cracked reinforced concrete element loaded by shear strains γ ∗ , has a residual (dowel) stiffness G D which considerably depends on the crack direction αc . In the proposed model the residual shear stiffness of cracked reinforced concrete is derived from the relation between shear stresses and strains obtained from the ‘‘Beam on elastic foundation’’—theory (He and Kwan 2001; El-Ariss 2006) in which the foundation modulus kf depends on the surrounding concrete (see Figure 2b). Without considering the influence of the crack direction αc , the related differential equation based on the Timoshenko beam—theory is first solved analytically. Afterwards, by means of a cracked reinforced element with an arbitrary crack direction αc the equilibrium state is formulated, leading to the effective dowel stiffness G D (αc ) (He and Kwan 2001; El-Ariss 2006). In this work, the dowel action is incorporated in the constitutive law of the reinforcement. Consequently, the shear modulus of steel Gs is replaced by the effective dowel modulus G D (αc ). Since the proposed dowel action model is formulated within a continuum mechanics model concept, it is consistent with the adopted continuum approaches for concrete cracking and the reinforcement.

4 4.1

Figure 1. Adopted macroscopic bond slip law derived from pull-out experiments.

CONTINUUM MICROMECHANICS Reinforced concrete as a homogenized composite

Reinforced concrete is represented as a three-phase composite material consisting of a continuous matrix formed by the concrete and by two sets of straight rebars representing the steel reinforcement and forming the reinforcement layer. In this Section, all parameters associated with concrete are subscripted with ‘‘m’’ (matrix) and the steel sets with ‘‘1’’ for the first set and ‘‘2’’ for the second set, respectively. The direction of the rebars and the geometry of the

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cross section may be arbitrary within the (2–3)—plane. Figure 3 contains an illustration of the composite material ‘‘reinforced concrete’’. Such a configuration is typical for reinforced shell-like and plate structures as well as beam structures. Besides steel bars also fibers made of steel or of other materials (glass, polymers) and textile reinforced concrete may be covered by the proposed homogenization-type approach. In (Richter 2005), a similar micromechanical framework has been employed for the modeling of concrete reinforced by textile materials. In order to cover a broad class of reinforced composite materials the proposed micromechanically oriented material model for RC presented in the following section is formulated in a rather general format. 4.2

The representative volume element (RVE)

Figure 4.

A widely used approach in continuum micromechanics is based on the consideration of a representative volume element (RVE) representing an arbitrary material point of a structure. Thereby, the complex morphology of the microstructure is captured in a simplified manner by the RVE in order to estimate the related effective (macroscopic) response by means of an averaging procedure. The RVE can therefore be regarded as a homogeneous counterpart to a heterogeneous material with effective material properties. To confirm the representative character of the RVE, its size l has to be large enough in order to ensure a statistical distribution of the constituents with a characteristic size d and at the same time it has to be essentially smaller than a length of the structure L d  l  L.

(13)

In Figure 4 the assumed RVE is depicted schematically. The considered microstructure is governed by two straight steel rebars with the tensors C1 , C2 which are embedded in a continuous matrix (concrete) with the tensor Cm . It should be emphasized that in the present work the material tensors C1 and C2 consist

Representative volume element (RVE).

only of the stiffnesses in the longitudinal direction of the rebar provided by the Young’s modulus of steel Es and by the shear stiffnesses given either by the shear modulus of steel Gs or by the effective dowel stiffness G D (αc ). C1 and C2 have to transformed to the global coordinate system. Depending on the related angles α1 , α2 and volume fractions of each rebar, the expected effective mechanical response of the RVE is therefore in general anisotropic or transversal isotropic. Hill’s condition requires the equality of the energy on the micro and macro level independently of the constitutive law. This condition is a priori fulfilled by homogeneous strain boundary conditions applied by prescribing linear displacements at the boundary of the RVE (Zohdi and Wriggers 2005) u(x) = ε ∗ · x,

x ∂V ,

(14)

where ε∗ defines the macroscopic (constant) strain tensor. 4.3

Micro-macro mapping

In order to evaluate the homogenized values of the strains and stresses, the local strain and stress fields within the RVE are averaged over the total volume V   1 <ε>V = ε(x)dV = ci <ε>i (15) V v i   1 <σ>V = σ (x)dV = ci <σ>i . (16) V v i

Figure 3. Illustration of reinforced concrete as a three-phase composite material.

Since the local averaged field values are assumed to be constant within each phase (σ i =<σ (x)>i and εi =<ε(x)>i ), they can be summed up according to the volume fraction ci = Vi /V , whereby Vi is the total volume of the phase i within the RVE. Hence,

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the volume of the RVE of the considered three-phase composite is assumed to be filled completely by all phases i.e. c1 + c2 + cm = 1. According to the average strain theorem, for any perfectly bonded heterogeneous body the averaged strains <ε>V given by equation (15) can be identified as the macroscopic strain tensor ε ∗ applied on the RVE i.e. <ε>V = ε∗ , which is independent of the considered constitutive laws (Zohdi and Wriggers 2005). For a three-phase composite, a reformulation of equation (15) and (16) leads to ∗

ε =<ε>v = c1 ε1 + c2 ε2 + cm ε m

(17)

describing the homogeneous macroscopic strains ε∗ applied on the composite material and to σ ∗ = <σ>v = c1 σ 1 + c2 σ 2 + cm σ m

(18)

for the macroscopic stresses σ ∗ representing the composite stress field. The related local stress tensors σ i (εi ) are calculated according to the constitutive laws which are already given in equations (4) and (7). The unknown local strain fields ε i have to estimated by means of the forth-order localization (concentration) tensor Ai which relates the homogenized macroscopic strains ε∗ to the local strains within each phase εi = Ai : ε ∗ ,

i = 1, 2, m.

(19)

The tensor Ai of each phase accounts for the morphology of the microstructure by considering the elasticity, the volume fraction, the aspect ratio, the orientation and the shape of each constituent. Since the distribution of the stiffness within a reinforced concrete structure is discontinuous, the strains of the matrix may differ from the strains of the reinforcement even when full bonding between reinforcement and matrix is assumed. Therefore the concentration tensor of the matrix Am usually differs from that of the reinforcement. It should be emphasized that Ai relates micro and macro quantities and depends on the chosen micromechanical model. If a three-phase composite is considered only two concentration tensors have to be known. The third one can be determined from the average value V = c1 A1 + c2 A2 + cm Am = 11,

(20)

with 11 denoting the forth-order unit tensor. Due to the different orientation and shape of the inhomogeneities, which is captured by Ai , the mechanical response of the related homogenized stiffness tensor C ∗ is in general anisotropic even if all constituents are isotropic. As long as all constituents are in the elastic regime, the mechanical constitutive relation for a composite

material is defined by σ ∗ = C ∗ : ε∗ ,

(21)

where C ∗ can be derived from the localization tensors of each phase  C ∗ =V = ci Ci : Ai , i = 1, 2, m. (22) i

4.4

Three-phase Mori-Tanaka approach

An appropriate homogenization scheme to derive the effective mechanical response of a RVE is provided by the widely used Mori-Tanaka approach (Mori and Tanaka 1973). This micromechanical model ensures the continuity of the matrix phase and accounts for mechanical interactions between the inclusions in an average manner. According to this homogenization scheme, the reference material playing the predominant morphological role of the composite is the continuous matrix. The inclusions and their states of strain and stress are directly affected by the matrix material. Within this approach, which is also denoted as effective field theory, for a two-phase composite (c2 = 0) the limit cases are covered. If, for example, no inclusions are considered (c1 = 0), the macroscopic stiffness is given by the matrix phase C ∗ = Cm and if no matrix phase is considered (cm = 0) the macroscopic stiffness is solely manifested by the inclusion material C ∗ = C1 . In the following, the Mori-Tanaka equations for a non-linear three-phase composite are summarized. The relation between the strains of the phases ε i and the applied macroscopic strains ε∗ on the boundary of the RVE is formulated in the general format : ε∗ , ε i = AMT i

i = 1, 2, m.

(23)

In order to identify the fourth-order concentration of each phase, the related assumptions tensor AMT i of the Mori-Tanaka approach have to be taken into account. As mentioned before, the average strains of the inclusions (ε 1 , ε2 ) are defined by the average strains of the matrix εm ε 1 = T1 : ε m

and

ε 2 = T2 : ε m .

(24)

Based on Eshelby’s equivalent inclusion approach, the fourth-order tensors T1 and T2 of each inclusion can be estimated by reformulating the inclusion inhomogeneity problem as a homogeneous problem with eigenstrains (Eshelby 1957). The solution for a single elastic inhomogeneity with an ellipsoidal shape perfectly bonded to a surrounding homogeneous matrix is given by Tj = [11 + S j : (Cm−1 : Cj − 11)]−1

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j = 1, 2

(25)

For an ellipsoidal geometry of the inclusions the related forth-order Eshelby tensor Sj is solely dependent on the aspect ratio of the inclusion and on Poisson’s ratio νm of the surrounding isotropic matrix Cm . Since a cylindrical shape can be regarded as an ellipsoidal geometry with a particular aspect ratio, the solution of the Eshelby tensor for the considered straight rebars can be computed explicitly (Eshelby 1957). In Figure 5, the cylindrical inhomogeneity representing a single rebar is illustrated. Depending on the aspect ratio s = a2 /a1 the cross section of the rebar may have an elliptical or circular shape. In this Figure, over-bars are used to characterize the local coordinate system. Note that for each set of rebars with the orientation αj ( j = 1, 2) within the RVE, the related Eshelby tensor Sj has to be transformed from the local to the global coordinate system. The tensor T given by equation (25) correlates exactly to the localization tensor resulting from the dilute approach, where no interactions between inclusions are considered (Benveniste 1987, Gross and Seelig 2006). The application of the dilute approach is, however, limited to composites with very small volume fractions of the inclusions. For the present three-phase composite the concenintroduced in equation (25) related tration tensors AMT i to each phase can be identified from combining equation (17) for the homogenized macroscopic strains ε∗ with the microscopic strains ε i given by equation (24) as −1 AMT + cm T1−1 ]−1 1 = [c1 11 + c2 T2 : T1

(26)

−1 + c2 11 + cm T2−1 ]−1 AMT 2 = [c1 T1 : T2

(27)

−1 AMT m = [c1 T1 + c2 T2 + cm 11] .

(28)

is a function of T1 and T2 , it is obviSince AMT i ous that the strains in each phase are affected by the

other constituents, which allows for the consideration of micromechanical interactions within the MoriTanaka strategy. 4.5

Influence of damage and plasticity

As soon as the mechanical response of the matrix becomes inelastic, the fourth-order tensor Tj given in equation (25) and the related concentration tensor AMT have to be defined according to the actual damage i state. Hence, in the post-cracking regime the tensors Tj are calculated with the degenerated secant stiffness of the matrix phase Cm = ψCm0 where ψ is the remaining integrity (see equation (4)). The secant stiffnesses of the steel reinforcement C1 and C2 , however, remain unchanged even in the post-yielding regime. Since the Mori-Tanaka homogenization approach is based on the existence and dominance of the matrix phase, for ψ → 0 the calculated composite shear stiffness tends to zero. Hence, the dowel stiffness would be reduced to zero. In the longitudinal direction of the rebar however, for ψ → 0 the composite stiffness is represented correctly by the axial steel stiffness. In order to reproduce the correct residual shear stiffness (dowel action) for a completely cracked matrix material (ψ = 0), the shear components of the Eshelby tensor S are scaled with ψ 2 . This modification allows to reproduce the correct limits regarding the macroscopic shear behavior of a cracked reinforced concrete element. In the post-cracking range of the matrix or in the yielding regime of the rebars, the homogenized tangent operator of the composite C ∗,tan needs to be computed in the context of the solution of the linearized equilibrium equations. Hence, depending on the damage or yielding state, the tangent stiffness Citan for each phase is first derived from the adopted constitutive laws given in Sections 2.1 and 2.2, respectively, Citan = dσ i /dε i

i = 1, 2, m.

(29)

Finally, the homogenized non-linear tangent operator C ∗,tan which is used for formulating the stiffness matrix of the composite can be calculated dσ ∗ dσ 1 dσ 2 dσ m = c1 ∗ + c2 ∗ + cm ∗ dε∗ dε dε dε dσ 1 dε1 dσ 2 dε2 dσ m dε m = c1 : + c2 : + cm : dε1 dε ∗ dε 2 dε∗ dε m dε∗

C ∗,tan =

tan MT tan MT = c1 C1tan : AMT 1 + c2 C2 : A2 + cm Cm : Am

= Cmtan + c1 (C1tan − Cmtan ) : AMT 1 Figure 5.

+ c2 (C2tan − Cmtan ) : AMT 2 .

Representation of an inhomogeneity.

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(30)

As mentioned before, in the non-linear regime also is affected by the damthe concentration tensor AMT i age or yielding state, which is manifested by Ti given in equation (25). As long as all constituents are within the elastic range, equation (30) coincides with equation (22) and the classical micromechanical laws of elastic composites are restored.

5

VALIDATION OF THE MODEL

5.1 3-point-bending tests The performance of the proposed reinforced concrete model is illustrated by reanalyzes of 3-point-bending tests by means of the Finite-Element-Method using 1116 volume-elements. Two identical concrete beams of the dimensions 1350 · 300 · 50 mm3 (length · height · depth) with different steel ratios of ρs = 0.065% and ρs = 0.13% were loaded by prescribed displacements. The structures were reinforced with ripped steel bars of 2φ2.5 mm and 4φ2.5 mm, respectively. The experimental procedure, boundary conditions and the related material parameters are given in (Ruiz et al. (1998)). For the concrete and for the steel reinforcement the proposed elasto-plastic damage model described in Section 2.1 and the elasto-plastic material model (section 2.2), respectively, are used. The bond properties are estimated by the performed pull-out experiments documented in (Ruiz et al. (1998)). Figure 6 shows the numerical results vs. the measured data. For each beam three simulations are performed. The load-displacement curve obtained for the same beam analyzed as a plain concrete structure is included in dark grey. The two numerical results depicted in black correspond to the reinforced structure, whereby the solid curve is obtained by assuming a small notch in the center of the beam (with predamage). A comparison of the experimental data (light grey) with the numerical simulations shows good agreement for both reinforced concrete beams.

Figure 6. Re-analyzes of Three-Point-Bending tests on RC beams (Ruiz et al. 1998). Steel ratio: ρs = 0.065% (left) and ρs = 0.13% (right).

5.2

Shear RC panel

In order to investigate the influence of the bond quality and dowel action on the structural load carrying capacity, a shear loaded reinforced panel with the dimensions 890 · 890 · 70 mm3 (length · height · depth) is reanalyzed numerically. The panel is reinforced biaxially with the steel ratio ρy = ρz = 1.79% with two layers of reinforcement in thickness direction. The experimental set-up, the boundary conditions, the measuring technique and the governing material parameters are documented in (Collins et al. 1985; Vecchio and Collins 1986). In the documentation of the experimental investigation, the considered panel is denoted as PV27. For the numerical simulation, some missing material parameter are estimated by the measured compressive strength according to (Vecchio and Collins 1986; CEB-FIP 1990). Since no information concerning the bond quality and dowel action are documented in the experimental study, these properties are assumed according to lower and upper limits in the numerical analyses performed with the proposed reinforced concrete model including the presented bond slip and dowel action mechanisms. For the finiteelement simulation, 1024 volume-elements are used whereby only one element layer is chosen in the thickness direction. Since the reinforcement is distributed homogeneously within the panel, the material properties of reinforced concrete have been assigned to all finite elements. Elements containing plain concrete elements are not used in this analysis. The shear-stresses τ versus the equivalent shear-strains γ are shown in Figure 7. The structural shear-strains are computed from the prescribed displacements at the edges of the panel, and the corresponding shear-stresses are obtained by averaging the local shear-stresses also calculated at the edges of the panel. The solid lines in black and dark gray color are obtained by assuming the same (low) bond quality. The dashed line corresponds to the dark gray curve (with dowel action), assuming, however, an improved

Figure 7. Re-analysis of a shear test on RC panels (Collins et al. 1985; Vecchio and Collins 1986).

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bond quality. The numerical result in which the dowel mechanisms are completely disregarded (black curve) underestimates the structural response considerably. If the residual shear stiffness of the cracked reinforced concrete is taken into account (solid dark gray curve), a relatively good agreement between the experimental and numerical results is obtained with regards to the structural stiffness. To capture the structural load carrying capacity accurately, however, an improved bond behavior has to be adopted. In this analysis the maximum load carrying capacity of the panel is caused by exhausting the ultimate bond stresses and by concrete crushing. Yielding of the reinforcement was not observed. 6

CONCLUDING REMARKS

Based on a continuum micromechanics oriented concept, a constitutive model for reinforced concrete including bond slip and dowel action mechanisms is presented in this paper. For each constituent, the nonlinear pre- and post-peak behavior is described separately. The proposed homogenization approach allows considering steel-concrete interactions without an explicit discretization of the reinforcing bars. As was shown by selected validation examples on a structural level, the main mechanisms governing the structural behavior of RC structures are well captured by the proposed model. The concrete model presented in Section 2.1 is originally formulated within a poromechanics framework. This more general model framework allows to employ the proposed model for durability-oriented numerical analyses of reinforced concrete structures considering also moisture and heat transport. This becomes relevant for the numerical analysis of corrosion induced damage in RC structures which is the topic of future research. REFERENCES Benveniste, Y. (1987). A new approach to the application of Mori-Tanaka’s theory in composite materials. Mechanics of Materials 6, 147–157. CEB-FIP (1990). Model Code 1990, Bulletin d’Information. Lausanne, Switzerland: CEB. Collins, M., Vecchio, F. and Mehlhorn, G. (1985). An international competition to predict the response of reinforced concrete panels. Canadian Journal of Civil Engineering 12, 624–644.

Coussy, O. (2004). Poromechanics. Chichester, England: Wiley. El-Ariss, B. (2006). Shear mechanism in cracked concrete. International Journal of Applied Mathematics and Mechanics 2(3), 24–31. Eshelby, J.D. (1957). The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc. Roy. Soc. London Series A, 241, 376–396. Grasberger, S. and Meschke, G. (2004). Thermohygromechanical degradation of concrete: From coupled 3D material modelling to durability-oriented multifield structural analyses. Materials and Structures 37, 244–256. Gross, D. and Seelig, T. (2006). Bruchmechanik—Mit einer Einführung in die Mikromechanik. Springer. He, X. and Kwan, A. (2001). Modeling dowel action of reinforcement bars for finite element analysis of concrete structures. Computers and Structures 79, 595–604. Linero, D. (2006). A model of material failure for reinforced concrete via continuum strong discontinuity approach and mixing theory. Ph. D. thesis, Universitat Politècnica de Catalunya, Barcelona. Luccioni, B., López, D. and Danesi, R. (2005). Bond-slip in reinforced concrete elements. Journal of Structural Engineering 131(11), 1690–1698. Manzoli, O., Oliver, J., Huespe, A. and Diaz, G. (2008). A mixture theory based method for three dimensional modeling of reinforced concrete members with embedded crack finite elements. Computers and Concrete 5(4), 401–416. Meschke, G. and Grasberger, S. (2003). Numerical modeling of coupled hygromechanical degradation of cementitious materials. Journal of Engineering Mechanics (ASCE) 129(4), 383–392. Meschke, G., Lackner, R. and Mang, H. (1998). An anisotropic elastoplastic-damage model for plain concrete. International Journal for Numerical Methods in Engineering 42, 703–727. Mori, T. and Tanaka, K. (1973). Average stress in the matrix and average elastic energy of materials with misfitting inclusions. Acta Metall. 21(5), 571–574. Richter, M. (2005). Entwicklung Mechanischer Modelle zur analytischen Beschreibung der Materialeigenschaften von textilbewährtem Feinbeton. Ph.D. thesis, TU Dresden, Germany. Ruiz, G., Elices, M. and Planas, J. (1998). Experimental study of fracture of lightly reinforced concrete beams. Materials and Structures 31, 683–691. Simo, J. and Hughes, T. (1998). Computational inelasticity. Berlin: Springer. Vecchio, F. and Collins, M. (1986). The modified compression-field theory for reinforced concrete elements subjected to shear. ACI Material Journal, March– April, 219–231. Zohdi, T.I. and Wriggers, P. (2005). Introduction to Computational Micromechanics. Springer.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Development of constitutive model of shear stress transfer on concrete crack surface considering shear stress softening Y. Takase & T. Ikeda Research Institute of Technology, Tobishima Corporation, Chiba, Japan

T. Wada Hokkaido Polytechnic College, Hokkaido, Japan

ABSTRACT: In a previous paper by the authors, local stress softening phenomenon found on surface of crack subjected to under shear loading was observed by constant contact ratio tests for the first time. Therefore, the authors constructed a nouveau constitutive model that enables to estimate for shear stress transfer on concrete crack surface. The proposed model was based on certain physical theory. Details of the proposed model, which consides stress softening based on the experimental results are presented in this paper. Moreover, the relidity of this model is comfirmed by comparing with experimental results. 1

2

INTRODUCTION

Studies for shear stress transfer mechanism on concrete crack surface has begun since the 1960s, and until now a multitude of shear transfer models with various interpretive concepts has been proposed. Current studies indicate that ‘‘shear stress transfer models of physical contact type’’ possess higher reliability and possibilities than other types of models. In order to construct a model of this type, experimental data with a high degree of accuracy is indispensable. Accurate verification of constitutive model can only be possible with the following requirements: a shape measurement method, which is able to measure concavo-convex shape on local crack surface accurately; an analytical method and also a shear loading experiment equipment which is accurately controllable on the amount of physical local contact between crack surfaces. It is seemed that there are no studies that fulfill the above requirements. Under the previously mentioned study background, the authors conducted study by taking following steps: 1) Shape measurement and analysis by light-projection method; 2) Development of highly accurate shear loading equipment based on the PID (Proportional, Integral, Deferential) control theory; 3) Experimental verification of shear stress transfer mechanism based on the contact theory. With the findings in the previous studies by the authors, completed with the work carred out in the study fulfill the requirements mentioned above, a nouveau constitutive model was developed. The validity of the model was then verified by comparing the experimental results with those previously obtained.

MODELING OF CONTACT STRESS-CONTACT DISPLACEMENT CURVE WITH STRESS SOFTENING

Figure 1 shows mechanism of shear stress transfer and, Figure 2 shows the conceptual depiction of contact stress operating on local crack surface hypothecating that effect of friction stress is ignorable. In the shear stress transfer model based on the contact theory, shear stress transfer is acquired by integration of contact stress occurring on a local surface with regards to the whole crack surface. As a for re-result, the contact stress-contact displacement curve for a local surface is found to be vital and inegrectable in the shear stress transfer model. That is the reason that the authors conducted constant contact ratio tests in a previous study to get to know the behavior of contact stress of local surface. The results will be discussed in the later section.

Vertical Stress

Shear Disp.

Shear Stress

Crack Width

Figure 1.

Mechanism of shear transfer on crack surface.

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Shear displacement Crack width

cn

Contact stress (MPa)

cn

Contact stress Incline angle

Contact displacement

Figure 2. Conceptual depiction of contact stress operating on local crack surface hypothecating that friction stress is ignorable.

With the results, it became clear that the contact stress curve in a post-peak area is in a nonlinear trend accompanied by stress softening. Based on these experimental results, the authors try formulated a mathematical model for σcn –ωθ curve which considers stress softening. In the final part of this paper, discussion are on implement this model using the contact theory to construct the proposed model for shear stress transfer on concrete crack surface. 2.1 Envelope model of contact stress-displacement curve

σcn =

√

e·

  ω2 σcnc · EXP − θ2 · ωθ ωθ c 2ωθ c

(1)

Unloading

Reloading K0

K0

Contact displacement (mm)

Figure 3. Conceptual behavior shape of curve model proposed in this paper.

Contact displacement is given by the equation (2) which is used in the same previous models of physical contact type by the shear displacement and the crack width. ωθ = δ · sin θ − ω · cos θ(ωθ ≥ 0) 2.2

Figure 3 shows conceptual behavior shape of curve model proposed in this study adopted in. In the contact ratio constant tests by the authors, it is difficult to evaluate σcn –ωθ curve on arbitrary incline angle on local surface for the lack of data at this point, because incline angle of contacting local surface is confined to a certain degree during the shear loading. However it is possible to know approximate behavior of σcn –ωθ curve from ‘‘the average contact stress—average contact displacement’’ curve analyzed by the crack shape model of experimental results. For the above reason, applying the experimental results to develop σcn –ωθ model needs next two steps. First, using examples from ‘‘average contact stress—average contact displacement’’ curve for modeling it. Next introduction the model in contact theory, simulating experimental curve of constant contact ratio, if confirmed by the possibility of simulated the experimental curves, the formulation model is judged as contact stress—contact displacement curve on local surface. In addition, previous contact stress models are mostly elastic—plastic models, so it is clearly different from the experimental results of contact ratio constant tests by the authors. And so the actual nonlinearity and the softening phenomenon of post-peak areas are duplicated by exponent function in this model. Then the coordinate of peak value has shown by the peak contact stress and the contact displacement.

K0: Initial contact

(2)

Impact evaluation of cyclic history

Behavior of the model with cyclic shear history is preferable to con figure modeling based on the experimental results. However, it was difficult to figure out qualitative nature on cyclic behavior using highprecision controlled loading system of the authors. In the proposal model, as shown in Figure 2, incremental contact stress is written in the form of the equation (3) supposing stress process under unloading and reloading is able to duplicate by the initial contact stiffness.  √ dσ cn = e · σ cnc ωθ c · dωθ (ωθ q ≤ ωθ ≤ ωθ p ) (3) In the equation above ωθ p represent maximum contact displacement history on the local surface. ωθ q is the contact displacement when the contact stress zero, drawing a tilted straight line of initial contact stiffness from stress value of contact displacement. 2.3

Maximum contact displacement on local surface occurs in adjacent spaces and the displacement

Figure 4 shows stress field by usual contact theory. The Figure 4 enables to understand that only detached local surfaces have contact with each other. However Figure 5 and Figure 6 show contact concept on local surface and the model noticing ‘‘steep-sided surface’’ and ‘‘mildly-sided surface’’ in the crack surface. From the results of contact ratio constant tests, as the contact ratio increases the peak value of contact stress decreases, stress softening curve of post-peak becomes shallow curve, and the displacement on peak

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ωθ c = 2.5 · cos3 θ

Contact area cn

Figure 4.

3

Stress field by usual contact theory.

Steep-sided surface Aggregate

Contact Area

Figure 5.

Adequacy of the contact model will be verified by comparison with the experimental results later.

Crack width Shear displacement Contact displacement Contact stress

cn

Contact stress circumferentially exerted in actual phenomenon

Due to modeling shear stress transfer stress on concrete crack surface, the process applying the contact behavior model, which was developed in previous chapter with the shape model on crack surface, to contact theory will be described in this chapter.

Contact stress evaluated in previous models

3.1

Equilibrium equation of stress based on contact theory

In the previous contact theory, it is assumed that the sum of contact stress balances with shear stress transfer stress occurring throughout crack surfaces. The proposal model adopts the same assumption. In order to set up shear stress and normal stress based on the assumption, formulated contact stress into shear element and vertical element, then each component are multiplied by the crack shape function and integrated. The formulation is shown in equation (6) and (7).

contact stress exerted in Contact stress evaluated actual phenomenon Mildly-sided in previous models surface Aggregate

Figure 6.

ADOPTATION OF CONTACT STRESS-DISPLACEMENT MODEL TO CONTACT THEORY

Contact concept on steep-sided surface.

Contact Area

Contact area evaluated in contiguous steep-sided surface

π/2

Contact concept on mildly-sided surface.

τ= tends to increase. These results show that contact phenomenon occurs around local areas and the breakdown progression quickens on a steep-sided local surface with larger contact quantity. On the other hand, the breakdown progression slows on a mildlysided local surface with small contact displacement. Above-mentioned two types of contact condition were reflected into σcn –ωθ curve model. In order to apply contact condition on crack surface to modeling, formulation of model that maximum contact stress vary with incline angle was attempted. However critical contact angle of the contact ratio constant tests were based on only three experimental data in the 8π /36 to 10π /36 range. For this reason modeling beyond the range just had to be estimated from the experimental results. In the proposal model, the equation (4) shows the maximum contact stress using incline angle to estimate the experimental results of above mentioned three data. σcnc = 480 sin3 |θ|

(5)

(4)

Also the equation (5) shows the contact displacement under the maximum stress with fracture mechanism changing, as the incline angle on local surface is getting moderate.

kcn · σcn · sin θ · (θ)dθ

(6)

kcn · σcn · cos θ · (θ)dθ

(7)

−π/2

π/2 σ = −π/2

The kcn in the equation above is the experimental coefficient, which is the physical quantity to compute stress, imponderable only with the simple 2D contact, on local crack surface in the proposal model. And this kcn shown in the equation (8) is compatible with previous experimental results, however, further analyses will be a future agenda. kcn = 15/8 3.2

(8)

Simulation of crack surface shape

Figure 7 is a chart that compares 2D incline density function model with the process of measurement result. There are many possible elements in shape property on concrete crack surface. Constructing high accuracy model needs formulation with various physical parameters. Nevertheless the proposal model is a simplified

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Table 1.

1.5

List of specimens.

(a) Constant contact ratio series.

Average incline density 1.0

Proposal equation

0.5

Specimen

Cont. ratio

Critical cont. angle (rad.)

Cont. area (mm2 )

RCN0157 RCN0271 RCN0439

0.0157 0.0271 0.0439

10/36π 9/36π 8/36π

118 203 309

Cont. = contact. (b) Constant crack width series.

0.0 0 θ (rad.)

δ( mm)

Figure 7. Chart that compares 2D incline density function model with the process of measurement results.

one because the development is the initial step. In the proposal model, 2D incline density function model, which is regarded as significant influence for shear stress transfer in published papers, is used. Approximation of 2D incline density function model is shown in the equation (9) and (10). (θ ) =

 3 a0 + a1 · |θ|n · cos3 θ 2

n + 1  π −1 , a0 = 2n 2

a1 = −

Specimen

ω0 (mm) +1 −1

+2

−2

+3 −3

W03δ06O3 W05δ08O3 W10δ09O3 W01δ05R1 W03δ05R1 W05δ08R1 W10δ08R1 W03δ06R2 W05δ09R2 W10δ09R2

0.3 0.5 1.0 0.1 0.3 0.5 1.0 0.3 0.5 1.0

0.45 0.6 0.7 – – – – 0.6 0.9 0.9

– – – – – – – −0.6 −0.9 −0.9

0.6 0.8 0.9 – – – – – – –

– – – −0.5 −0.5 −0.8 −0.8 −0.4 −0.6 −0.6

– – – – – – – – – –

(9) Table 2.

Concrete mix.

n + 1  π −(n+1) 2n

0.3 0.4 0.5 0.5 0.5 0.8 0.8 0.4 0.6 0.6

2

Mix (kg/m3 )

(10)

Size of agg.

W/C

Water

Cem.

Agg.

Adm.

In the above equation, set n to 0.6. Undoubtedly the approximation has high adequacy of the measurement result of shapes on crack surface in Figure 7.

20 mm

42%

173

412

1, 718

4.491

4

Table 3.

ADEQUACY VALIDATION OF EXPERIMENTAL VALUE OF SHEAR STRESS TRANSFER MODEL

In this chapter, the adequacy will be verified by comparing the experimental results of the proposal model with the previous model. 4.1 Active parameter of specimen in comparison Table 1 shows active parameter list of specimen, Table 2 shows concrete mix, and each material property are listed in Table 3. Adequacy of σcn –ω model on a local surface will be verified among specimens of constant contact ratio series shown in Table 1(a). As their specimen named, RCN code and contact ratio values are used. And Table 1(b) shows specimen of three types of constant crack width series with different loading history. Using

Cem. = cement, Agg. = aggregate, Adm. = admixture.

Material properties of concrete.

Comp. strength

Tensile strength

Young’s modulus

40.1 N/mm2

2.61 N/mm2

23.69 N/mm2

those specimens, the comparison between the proposal model and previous models will be noted. Name of specimen is composed with a combination of decided crack width, maximum shear displacement, and loading cycle. 4.2

Verifying of contact stress σcn -contact displacement ωθ in the proposal model

There is no previous study verifying adequacy or compatibility of contact stress—contact displacement on

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a local surface. As far as the authors know, above mentioned data of constant contact ratio test conducted by the authors is the only experimental data of the behavior of contact stress σcn –ωθ displacement on local surface. In this chapter, implementing σcn –ωθ model previously formulated on local crack surface in contact theory, and the adequacy of the model will be verified by comparing experimental value with the answer of numerical calculation under the contact ratio constant condition. Focus on the shear stress-shear displacement curves shown in each center figures in three different contact ratio specimens in Figure 8 (a), (b) and (c), maximum shear stress value and the tendency of experimental curve, like softening behavior on a post-peak, have been reproduced roughly by the proposal model. About shear stress—normal stress curve in figures on the right side, experimental value of stress gradient and peak value in the proposal model shows overall tendency. Seeing the comparison result, the σcn –ωθ model in Chapter Three can be identified as a predictable model for contact stress-contact displacement behavior on an actual local surface. 4.3

Compatibility validation of the proposal model and LI model on cyclic loading

Using three types of experimental results in Table 1(b), the compatibility of the proposal model and Li model, (mm) 1.6

RCN0157 fc = 40MPa 1.2 0.8 0.4 0

(MPa) 4

4.3.1 In case of the cyclic shear history of one way three cycle Figure 9(a), (b) and (c) show the experimental results of specimen W03d06O3, W05d08O3, and W10d09O3, setting each crack width to 0.3 mm, 0.5 mm, and 1.0 mm, with one way cyclic shear pressing and analysis values of each models. In those figures, figures on the left show crack width–shear displacement curves, center figures show shear stress—shear displacement curves, and figures on the right show shear stress normal stress curves. What follows is comparison between experimental values, the proposal model, and Li model in all of those figures. At first the result of W03d06O3 specimen, target crack width to 0.3 mm, is shown in Figure 9(a). In the center, shear stress τ –δ shear displacement curve, the proposal model is almost correspond with the experimental value on the maximum value of shear stress and shear stiffness, but somewhat higher curve. However, in Li model, the proposal model is almost coincident with the experimental value on the maximum stress value, but both loading and unloading rigidity were neither reproducible nor describable. Looking at the shear stress-normal stress curve on the right side, the proposal model estimates normal stress rather higher, but the tendency is almost coincident with

(MPa) 4

Starting point

Model

lim

con =

Target line 2 ch1 ch2 ch3 ch4 tan lim=1.192 (mm) 1 2 0

which is most frequently used among published papers, most reliable and expansible among previously analyzed models, will be verified in this chapter.

2 Model

0.3

Exp.

Exp. (mm) 2 0

1

(mm) RCN0271 fc = 40MPa 1.2

(MPa) 4

2

-0.1 0

0.8 0.4 0

(MPa)

3

0.5

Model 0.0271 1

Model (mm) 2 0

0.8 0.4

3

(MPa) 6

0.1

0

6

0

1.4

Exp. Model Exp.

con =

(MPa)

12

(mm) ch1 ch2 ch3 fc = 40MPa ch4 1

(mm) 1

0

(MPa)

12

Proposal Li-Maekawa Exp.

0.0439 1

Model 3

(mm) 2 0

(mm)

12

W10 09O3 0 = 1.0mm 1.0

Starting point lim

(MPa) -1 2

0

Proposal Li-Maekawa Exp.

(MPa) -1 2

0

(MPa)

6

Target line 3 ch1 ch2 ch3 ch4 tan lim=0.839 (mm) 1 2 0

(MPa) Proposal Li-Maekawa Exp.

(b) W05d08O3 specimen

(MPa)

(mm) RCN0439 1.2 fc = 40MPa

12

(mm) 1

0

W05 08O3 0 = 0.5mm

(b) RCN0271 specimen 1.6

(mm)

0.9

Exp.

Exp.

lim con =

(MPa) Proposal Li-Maekawa Exp.

(a) W03d06O3 specimen

6 Starting point

Target line 3 ch1 ch2 ch3 ch4 tan lim=1.000 (mm) 1 2 0

(mm) ch1 ch2 ch3 fc = 40MPa ch4 1

(MPa)

6

12

W03 06O3 0 = 0.3mm

(a) RCN0157 specimen 1.6

(mm)

0.7

0.0157

3

(MPa) 6

0.60

(mm) ch1 ch2 ch3 fc = 40MPa ch4 1

(MPa) Proposal Li-Maekawa Exp.

0

12

(mm) 1 0

(MPa) Proposal Li-Maekawa Exp.

(MPa) -1 2

(c) W10d08O3 specimen

(c) RCN0439 specimen

Figure 8. Comparisons between experimental results and models of contact ratio constant.

Figure 9. Comparison between experimental results and models of one way three cycles.

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the experimental value. Previous models are almost coincident with the experimental value. Specimen W05d08O3 target crack width to 0.5 mm in Figure 9(b) is the next topic. In the shear stress-shear displacement curve in the middle, the same with the specimen W05d08O3 setting crack width to 0.3 mm in the same figure, magnitude of maximum shear stress is nearly described. However, it is impossible to duplicate a concave upward carve, which is shown in the experiment, neither in the proposal model nor in Li model. Nonetheless, the curve of unloading time in the proposal model is mostly pursuing experimental values and it is more compatible than Li model. In the shear stress-normal stress curve on the right side of the same figure, the proposal model is almost coincident with experimental results but in Li model the normal stress is computed rather low. And finally, specimen W10d09O3 target crack width to 1.0 mm in the same figure will be verified. It is visible in shear stress—shear displacement curve in the middle of the same figure that the proposal model is estimated smaller than the experimental value. On the other hand, the stress value of Li model is near the experimental value. The adaptation of differences, as described in Section 3 Paragraph 3, including the experimental results of other loading type, target crack width to 1.0 mm, will be conciderd again at the end of this paper. Aside from the stress discussion, large and small, the proposal model describes stress gradient well as the comparison between both models about shear stress-normal stress curve on the right side of the same figure. Adaptation of the proposal model has been cleared. Compatibility comparison of the proposal model and previous Li model have been mentioned above. It will be later described that the results of verification of specimen series, other RV-1 and RV-2, differ little from the consideration in this paragraph. In the validation, starting from the next paragraph, Li model will be mentioned for reference, but not thorough detailed consideration. 4.3.2 In case of shear history of reversed one cycle Figure 10(a), (b), (c) and (d) show that comparison results of specimen W01d05R1, W03d05R1, W05d08R1, and W10d08R1 target crack width to 0.1 mm, 0.3 mm, 0.5 mm, and 1.0 mm under the shear pressing of reversed one cycle. In those figures, analysis curves of the experimental results, the proposal model, and Li model were shown together. First of all, the result of specimen setting and kept the crack width to 0.1 mm in Figure 9(a) is observed. The specimen target crack width to 0.1 mm is the only one among three loading types, therefore Li model will be noted here. The proposal model almost coincide the maximum stress value of the experimental value rather than shear stress-shear displacement curve in

(mm)

0.5

P roposal Li-Maekawa Exp.

0 =0.1mm

0.1

(mm)

(MPa) 15 Proposal Exp.

(MPa)

15

W01 05R1 fc = 40MPa

0

(mm)

0

(MPa)

ch1 ch2 ch3 ch4

LiMaekawa

-0.3 -1

1 -15-1

0

0

1

-15

0

-1 5

(a) W01d05R1 specimen (mm)

0.7

(MPa)

12

(MPa)

12

Proposal

W03 05R1 fc = 40MPa

Proposal

Li-Maekawa Exp.

0 =0.3mm

0.3

(mm)

0

Exp. (mm)

0

(MPa)

ch1 ch2 ch3 ch4 -0.1 -1

LiMaekawa 0

1

-12 -1

0

1

-12

0

-1 2

(b) W03d05R1 specimen 0.9

(mm)

12

W05 08R1 fc = 40MPa 0 =0.5mm

(MPa)

(MPa)

12

Proposal Li-Maekawa

Proposal

Exp. 0.5

(mm)

0

(mm)

Exp.

0

(MPa)

ch1 ch2 ch3 ch4 0.1 -1

LiMaekawa 0

1

-12 -1

0

1

-12

0

-1 2

(c) W05d08R1 specimen 1.4

(mm)

12

W10 08R1 fc = 40MPa 0 =1.0mm

1.0

(mm)

(MPa)

12

Proposal Li-Maekawa Exp.

0

Exp. Proposal (mm)

0

ch1 ch2 ch3 ch4 0.6 -1

(MPa)

(MPa) LiMaekawa

0

1

-12 -1

0

1 -12 0

-1 2

(d) W10d08R1 specimen

Figure 10. Comparison results of reversed one cycle between models and experimental results.

the middle of the same figure, but whole hysteresis behavior has a bigger tendency than the experimental value. However, shear displacement of Li model yields with shear displacement as 0.2 mm and it has completely different behavior from the experimental curve. Also looking at the τ –σ curve on the right side of Figure 9(a), the proposal model estimates normal stress rather higher but basically compatible with the experimental values. Next, focus on the central figure of Figure 10(b) target crack width to 0.3 mm, the proposal model is able to pursue the τ –δ curve with high accuracy. About the τ –σ relation on the right side of the same figure, it is clear that the proposal model almost correspond with the experimental value. Furthermore, the specimen setting crack width to 0.5 mm in Figure 10(c) will be verified. Looking at the maximum value of the shear stress in the middle

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figure and the normal stress in the right figure, it is distinctive in the experimental value that the stress in the negative loading is bigger than the one in the positive loading. It is impossible to evaluate the phenomenon from the theoretical assumption of the proposal model, it is clear that the experimental values are basically traceable in both figures; shear stress-shear displacement curve in the middle of same figure and shear stress—normal stress curve on the right side of the same figure. Finally, the maximum stress value of the proposal model is smaller than the experimental value in middle figure in Figure 10(d), target crack width to 1.0 mm, which has same tendency with the specimen in Figure 9(c). And about the shear stress-normal stress curve in the same figure on the right side, stress value of the proposal model is underestimated but stress ratio of normal and shear is properly described.

4.3.3 In case of cyclic shear history of reversed two cycles Figure 11(a), (b) and (c) show comparison between experimental results of specimen W03d06R2, W05d09R2, and W10d09R2, target crack width to 0.3 mm, 0.5 mm, and 1.0 mm under the shear loading of reversed two cycle, and two models.

0.7

ω (mm)

12

W03δ 06R2 fc = 40MPa

τ (MPa)

12

Proposal

ω 0 =0.3mm

τ (MPa) Exp.

Li-Maekawa Exp.

0.3

(mm)

δ 0

(mm)

Proposal

δ 0

(MPa)

ch1 ch2 ch3 ch4 -0.1 -1

0

1

-12 -1

0

1

-12

5

σ

0

ω (mm)

12

W05δ 09R2 fc = 40MPa

τ (MPa)

12

Proposal

ω 0 =0.5mm

-1 2

τ (MPa) Proposal

Li-Maekawa Exp.

0.5

(mm)

δ 0

(mm)

Exp.

δ 0

(MPa)

ch1 ch2 ch3 ch4 0.1 -1

0

1

-12 -1

0

1

-12

σ

LiMaekawa 0

-1 2

(b) W05d08R2 specimen 1.4

ω (mm)

12

W10δ 09R2 fc = 40MPa ω0 =1.0mm

1.0

τ (MPa)

12

Proposal

τ (MPa) Exp.

Li-Maekawa Exp.

(mm)

δ 0

(mm)

δ 0

ch1 ch2 ch3 ch4 0.6-1

Proposal σ (MPa) LiMaekawa

0

1

-12 -1

0

1 -12 0

CONCLUSION

In this study, a unique contact stress model, which adapts the softening phenomenon to the contact stress of local concrete crack surface, was developed. Adapting this model to the previous theory create a 2D shear stress transfer model. In addition, by comparing the proposed model with the representative previous model, in terms of comparison of some experimental data; the compatibility of the model has verified. Summaries of this study are shown as follows:

LiMaekawa

(a) W03d06R2 specimen 0.9

First, focus on the specimen in Figure 11(a) setting crack width to 0.3 mm, it appears that the experimental results are totally accurate by the proposal model. Though there is a difference of stress values between the proposal model and the experimental curve during the negative loading in the shear stress-shear displacement curve. The proposal model is able to pursue the experimental value more than the shear stressnormal stress curve on the right side of the same figure. Figure 11(b) target crack width to 0.5 mm is the next topic. In the first cycle, positive and negative, the proposal model has lower initial stiffness than the shear stress—shear displacement curve in the middle of the same figure, and the transition of stress is smaller than the experimental value. In the second cycle, stiffness, maximum stress value, and the experimental value of unloading curve are described with a high degree of accuracy. In the shear stress-normal stress curve on the right side of Fig. 11, stress value of the proposal model is slightly smaller but it seems that the compatibility is higher than Li model. Then looking into the specimen in Figure 11(c) setting crack width to 1.0 mm, the maximum stress value of positive loading is slightly smaller in the shear stress-shear displacement curve in the middle, but it is clear that the experimental curve is captured by the proposal model in negative loading. In the shear stress-normal stress curve on the right side of Figure 11(c), it shows that the proposal model is able to pursue experimental values with high accuracy.

-1 2

(c) W10d08R2 specimen

Figure 11. Comparison results of reversed two cycles between given models and experimental results.

1. The proposal analysis curve, which adapts σcn –ωθ model of local surface to contact theory, is compatible with the curve of contact ratio constant tests. The proposal model has sufficient validity as contact stress σcn –ωθ displacement model on local concrete crack surface. 2. In τ –δ curve of specimen with each crack width, the proposal model is not only compatible with stiffness under loading and peak value, but also enable unloading stiffness and reloading curve to reproduce with high accuracy. 3. In τ –σ curve of each specimen, the proposal model, which almost coincides with stress gradient of the experimental curve, show rather high compatibility.

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The authors will continue the experiment to study shear stress transfer mechanism for different concrete compressive strengths to develop extended models based on this proposed model.

ACKNOWLEDGEMENTS The authors would like to express our gratitude to the graduates of 2005 and 2006 in Wada laboratory of Hokkaido Polytechnic College, for their generous effort in tests conducted for this study.

REFERENCES Bazant, Z.P. & Gambarova, P. 1980. Rough Cracks in Reinforced Concrete, Journal of Structure Division, ASCE, Vol. 106, No. 4: 819–842. Bujadham, B. 1991. The Universal Model for Transfer across Crack in Concrete, Department of Civil Engineering, The Graduate School of The University of Tokyo. Fenwick, R.C., & Paulay, T. 1968. Mechanism of Shear Resistance of Concrete Beams, Journal of Structure Division, ASCE, Vol. 94, No. 10: 2325–2350. Feenstra, P.H., Borst, R., & Rots, J.G. 1991. Numerical Study on Crack Dilatancy Part 1 Models and Stability Analysis, Journal of Engineering Mechanics, ASCE, Vol. 117, No. 4: 733–753. Li, B., Maekawa, K., & Okamura, H. 1989. Contact Density Model for Stress Transfer across Cracks in Concrete. Journal of the Faculty of Eng. The University of Tokyo(B), Vol. 40, No. 1: 9–52. Michael N. Fardis, & Oral, B. 1979. Shear Transfer Model for Reinforced Concrete, ASCE, Vol. 105, No. EM 2: 255–275. Millard, S.G. & Johnson, R.P. 1984. Shear Transfer across Cracks in Reinforced Concrete due to Aggregate Interlock and Dowel Action, M. of Concrete Research, Vol. 36, No. 126: 123–137. Sato, R., Wada, T., & Ueda, M. 2001. Fast Fourier One-dimensional Analysisi of Concrete Crack Surface, Proceedings of Fracture Mechanism of Concrete and Concrete Structures, Framcos-4: 423–430. Sato, R., Wada, T., & Ueda, M. 2003. Study on Shape Properties of Concrete Crack Surface in Freaquency Domain, Proceedings, Computation modelling of concrete structures, 315–324. Shinohara, Y., Kawamichi, K., & Ishitobi, S. 2001. Shear Behavior in Precracked Concrete under Cyclic Loading at Constant Crack Width, Journal of Struct. Constr. Eng., AIJ, No. 548: 101–106. (in Japanese) Takase, Y., Ueda, M., & Wada, T. 2007. Proposal of Optimal Experimental Method for Evaluation of the Concrete Shear Transfer Mechanism, Proceedings of Fracture Mechanism of Concrete and Concrete Structures, Framcos6: 333–339. Catania, Italy. Takase, Y., Wada, T., & Ueda M. 2007. Verification of Shear Transfer Mechanism Based on Contact Theory—Study on shear transfer mechanism on concrete crack surface Part 1-, Journal of Struct. Constr. Eng., AIJ, No. 622: 155–162. (in Japanese)

Van Mier, J.G.M., Nooru-Mohamed, M.B., & Timmers, G. 1991. An experimental study of shear fracture and aggregate interlock in cement based composites, Heron, Vol. 36, No. 4. Wada, T., Sato, R., Ishikawa, C., & Ueda, M. 1996. Development of Mesurement of the Concrete Crack Surface by Laser Beam and Proposal of 2-Demensional Analytical Method of the Measured Image A basic study on shape properties of the concrete crack surface Part. 1-, Journal of Struct. Constr. Eng., AIJ, No. 490: 179–188. (in Japanese) Wada, T., Sato, R., Ishikawa, C., & Ueda, M. 1998. Dimensional Shape Analysis of the Concrete Crack Surfaces Introduced by Various Kinds of Stress—A basic study on shape properties of the concrete crack surface Part. 2-, Journal of Struct. Constr. Eng., AIJ, No. 504: 81–86. (in Japanese) Wada, T., Sato, R., Ishikawa, C., & Ueda, M. 1999. Proposal of 3-dimensional analytical method for the concrete crack surface image measured by laser beam—A basic study on shape properties of the concrete crack surface Part. 3-, Journal of Struct. Constr. Eng., AIJ, No. 524: 111–118. (in Japanese) Wada, T., Sato, R., Ishikawa, C., & Ueda, M. 2000. Dimensional shape analyses of the concrete crack surfaces introduced by various kinds of stress—A basic study on shape properties of the concrete crack surface Part. 4-, Journal of Struct. Constr. Eng., AIJ, No. 534: 103–111. (in Japanese) Walraven, J.C. 1981. Fundamental Analysis of Aggregate Interlock, ASCE, Vol. 107, No. ST11: 2245–2270.

APPENDIX The following symbols are used in this paper: a0 = a1 = δ = dσcn = dωθ = n = θ = θlim = σ = σcn = σcnc = τ = (θ) = ω = ωθ = ωθ c = ωθ p ωθ q

Modulus of incline density function Modulus of incline density function Shear displacement Increment of contact stress Increment of contact displacement Multiplier of incline density function Contact angle Critical contact angle Normal stress Contact stress Maximum contact stress Shear stress Incline density Crack width Contact displacement Contact displacement of maximum contact stress = Maximum contact displacement = Contact displacement when the contact stress zero, drawing a tilted straight line of initial contact stiffness from stress value of maximum contact displacement

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Microplane approach for modeling of concrete under low confinement Nguyen Viet Tue & Jiabin Li Leipzig University, Leipzig, Germany

ABSTRACT: This paper mainly deals with the computational modeling of concrete subjected to low confinement. To this end, a new concrete material model is developed on the basis of the well known microplane theory. The new model, called model M4L, follows the same approach as a previous microplane model M4 but with enhanced constitutive formulation (stress-strain boundaries). The main contributions include a novel concept of confinement-adjusted effective microplane moduli, two confining pressure dependent microplane deviatoric stress boundaries and a new shear stress boundary. Numerical experiments indicate that model M4L is able to fit a wide range of test data in a realistic way. Finally, the model is implemented into the finite element software (SOFiSTiK) and used to model the behavior of tied high-strength concrete (HSC) and normal-strength concrete (NSC) column as well as a sandwich joint between HSC column and NSC slab subjected to axial compression. Numerical results indicate that the simulated response agree well with the experimental observations.

1

for concrete under low confinement, both from the material and structural level.

INTRODUCTION

The failure of concrete under uniaxial compression is often characterized by volume dilitancy. However, the tendency for internal cracking and volume increase is reduced when lateral confining pressure (σL ) exists. As a result, the ultimate strength, deformation capacity and the post peak ductility of concrete are greatly enhanced. In conventional reinforced concrete structures, the confining pressure level is relatively low (smaller than 30% of the uniaxial compressive strength fc ). Reliable numerical modeling of confined concrete structures and members requires a concrete material model that can accurately represent the responses of concrete under low confinement. Various theories have been used to describe the behavior of concrete subjected to low confining pressure, such as the elasticity, plasticity, damage, coupled damage-plasticity as well as endochronic and microplane theory. Each theory has its own advantages and disadvantages. Amongst of them, the microplane theory is a promising approach. The appealing aspects of microplane theory are the conceptual simplicity and the three dimensional formulation. In this paper, a new material model that can realistically capture the behavior of concrete under low confinement is developed within the microplane framework. The new model is called microplane model M4L and can be regarded as an enhanced version of the model M4 developed by Prof. Bažant and his coworkers (Bažant et al. 2000, Caner and Bažant 2000). The new model follows the same approach as model M4 but with improved constitutive formulation. Numerical experiments reveal that the model M4L can accurately reproduce a wide range of experimental observations

2 2.1

MICROPLANE THEORY Theoretical framework

The essential idea of the microplane modeling approach is to characterize the constitutive law in terms of stress and strains vectors on various planes in the material, called microplanes. A microplane is an arbitrary plane which cuts through the material at a point and is defined by the orientation of its normal unit vector n = (n1 , n2 , n3 ), which represents one point in the spatial unit sphere. In microplane models for concrete, the microplanes might be imagined as the damage planes or weak planes in the micro-structure, such as the inter particle contact planes, interfaces between mortar and aggregates or planes of microcracks, as shown in Figure 1. In order to generalize a stable post peak softening damage, a kinematic constraint is usually used in modeling the nonlinear behavior of quasi-brittle materials. That is, the strain vectors on microplanes are assumed to be the projection of the macroscopic strain tensor. Given the strain vectors on various microplanes, the stress vectors of work conjugacy for various microplanes can be determined through the predefined stress-strain relationships for the microplanes. Once the microscopic stress components for each microplane are known, the macroscopic stress tensor can be obtained though the principle of virtual work (Bažant et al. 1996) or the principle of energy equivalence (Carol et al. 2001). The flow path of the microplane model is given in Figure 2.

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strain is imposed, ensuring energy dissipation on the microplane. After the stress components on each microplane are known, the macroscopic stress tensor σij is derived by accumulating the contributions of each microplane as:   δij σD Nij − 3   + σM Mij + σL Lij d

σij = σv δij +

Figure 1.

Flow path of microplane modeling approach.

Microplane model M4

The microplane model M4 was developed by Prof. Bažant and his coworkers in 2000 (Bažant et al. 2000; Caner and Bažant 2000). This model adopted a volumetric-deviatoric-tangential (V-D-T) split for describing the strain components on each microplane. A concept of stress-strain boundary is used to simulate the inelastic behavior of concrete. There are altogether 6 stress-strain boundaries in microplane model M4. They include: – A normal stress boundary (σNb ); – Two deviatoric stress boundaries for compression and tension (σDb− , σDb+ ); – A shear stress boundary (σTb ); and – Two volumetric stress boundaries for compression and tension (σVb− , σVb+ ). Within the boundaries, the behavior of the microplane is incrementally elastic, i.e., σv = Ev εv ,  σM = ET εM ,

 σD = ED  εD , and σL = ET εL

(2)

Numerical accuracy of microplane model M4

Numerical experiments indicate that the microplane model M4 is very powerful and versatile in capturing many complex inelastic behavior of concrete under various stress states (Caner and Bažant 2000), such as tension and compression softening, cracking and damage, volume dilitancy and strength dependence on high confining pressure. The model has been adopted by some popular finite element packages such as ATENA and OOFEM, as a standard concrete model. However, it was found that the microplane model M4 still exhibits some undesired responses (Tue et al. 2008). For instance, the model predictions for concrete under low confinement are quite inadequate. This can be seen from Figure 3, which shows the model M4 reproductions and the test data of Imran (Imran 1994) for concrete subjected to low confinement. A comparison between the model response and the test data indicates the following noticeable aspects: – The model leads to an increase of the peak stress with the increase of the confining pressure, which is consistent with the experimental observations;

90 Model M4 Test

75 in MPa

2.2

 

A full explicit computational algorithm is adopted in the microplane model M4, which is very helpful for achieving numerical efficiency.

Imagined microplanes in concrete.

2.3

Figure 2.

3 2π

60 45 8.4

30

(1)

15

where, EV , ED and ET are the microplane volumetric, deviatoric and tangential moduli, respectively. The exceeding of the stress-strain boundary value is never allowed. If the stress exceeds the boundary, a vertical drop to the boundary at the current

0

0.0 0

5

1.05 10

2.1

15 in ‰

4.2

20

25

Figure 3. A comparison of model M4 simulations and test data for concrete under low confinement.

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0.8

Model M4 Ottosen model Test

0.6 0.4 0.2 0.0 0.0

0.2

0.4

0.6 /f c

0.8

1.0

1.2

Figure 4. A comparison of model M4 simulations and test data for the Poisson’s ratio of concrete in uniaxial compression.

– The magnitudes for the peak stress and strain at peak stress from model M4 are considerably lower than that in the test; – The model reproductions for the softening branch are also less satisfactory. As can be seen, the softening branches for different confining pressures are almost parallel to each other indicating that the lateral confinement has nearly no influence on the post peak ductility. Furthermore, model M4 gives also insufficient prediction for the lateral response of concrete under uniaxial compression, as shown in Figure 4. It can be seen that model M4 gives too much lateral expansion at an early stage. This will lead to an overestimation of the stiffness in case that the concrete expansion is an important issue, e.g. passive confinement. The above two weaknesses must be eliminated before the model can be used to accurately predict the behavior of confined concrete members in nonlinear analysis. The aim of this paper is to remedy these insufficiencies through formulating a new microplane model, called model M4L.

3.1

θ=

−σ1  ≥0 fc

x =

MICROPLANE MODEL M4L: FORMULATION AND VALIDATION

(3)

where, σ1 is the first principle stress; fc is the uniaxial compressive strength of concrete, and x + |x| 2 1.2

Confinement-adjusted effective microplane elastic moduli

1.0

In model M4, the initial values for the microplane elastic moduli EV , ED and ET are always used in representing the elastic behavior of the microplanes. Through the use of the pre-defined stress-strain boundaries, the decrease of the toughness of the stress-strain curve, i.e., the damage to the elastic modulus with increased of stress (or strain) level, can be correctly predicted which can be seen from the uniaxial compression or tension simulations in Caner and Bažant (2000). This is because most of the stress-strain boundaries are

0.8 E*/E

3

strain dependent (except for the shear stress boundary), the damage induced by strain increasing can be captured. However, for concrete subjected to multiaxial compression, experimental observations indicate that the slope of the stress-strain curve is influence by both the strain and the stress states. Although it is usually reasonable to assume that the initial tangential elastic modulus under multiaxial compression is independent on the confining pressure (Attard and Setunge 1996). However, if the initial hydrostatic pressure is taken into account, namely, if the stress-strain curve is plotted to begin from the confining pressure, a reduction of the tangential elastic modulus here denoted as E∗ with the increase of the confining pressure can be observed (Dahl 1992, Sfer et al. 2002). Figure 5 presents the test data of Sfer et al. (2002). It can be clearly seen from Fig. 5 that the tangential elastic modulus E ∗ decreases with the increase of the confinement. This reduction can be approximately expressed in a bilinear manner. However, the combination of constant EV , ED and ET and the stress-strain boundaries is insufficient in reproducing this behavior since the stress-strain boundaries are insufficient for reflecting the influence of the stress states. In order to improve the model performance for concrete under low confinement, the elastic description of the microplane is improved in the new model through re-determining the microplane moduli EV , ED and ET according to the stress states. A simple way is to make the microplane elastic moduli dependent on the confining pressure. The following index is chosen for representing the confining pressure in the model M4L.

0.6 0.4 0.2 0.0 0.0

0.4

0.8

1.2 L

1.6

2.0

/f c

Figure 5. Influence of confining pressure on the elastic modulus of concrete (Sfer et al. 2002).

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1.0

ED( )/ED

0.8 0.6 0.4 0.2 0.0 0.0

0.2 =

0.4 ( 1, f c )

0.6

0.8

0.6

0.8

(a) ED 1.0

ET( )/ET

0.8 0.6 0.4 0.2 0.0 0.0

0.2 =

0.4 ( 1, f c )

(b) E T Figure 6. Variation of microplane elastic moduli with lateral confining pressure.

In model M4L, the following microplane elastic moduli are used in describing the elastic behavior of the microplane: EV is kept constant while ED and ET are made to be varied with θ , as shown in Figure 6.This idea is on the basis of experimental observations and the work of Phillips and Zienkiewicz (1976). The detailed equation for the ED and ET can be found in Li (2009). Finally, the elastic behavior for the microplane in model M4L is calculated as: σv = Ev εv ,

 σD = ED (θ ) εD ,

 σM = ET (θ )εM ,

and σL = ET (θ )εL (4)

3.2 Stress-strain boundaries Following the same approach as in model M4, the inelastic behavior of concrete is also characterized through the concept of stress-strain boundary in the new model M4L. The normal stress boundary σNb simulates the tensile fracture of concrete while the volumetric boundaries

σVb− , σVb+ simulate the behaviour of concrete under hydrostatic loadings. They have only slightly influence on the model responses for concrete under low confinement. These boundaries remain in model M4L. However, the deviatoric stress boundaries σDb− , σDb+ and the shear stress boundary σTb must be enhanced for better simulating the responses of concrete subjected to low confining pressure and the lateral responses under uniaxial compression. The failure of concrete under unconfined compression or low pressure is a result of the formulation and propagation of splitting tension cracks parallel to the loading direction. This means that the cohesion plays a dominant role. Beyond the peak, with increasing deformation, the cohesion continues to decrease until it vanishes, resulting in strain softening. The influence of the confining pressure can be well reflected in the deviator space. As found by Li (2009), the microplane deviatoric strain εD comes from the macroscopic deviatoric strain tensor and is oriented in the normal direction of the microplane, which represents the interparticle cohesion in the material. In model M4, two deviatoric boundaries including a compressive and a tensile one are used to simulate the compressive softening behavior of concrete. The compressive deviatoric boundary controls the axial strain of concrete under unconfined and low confinement while the tensile deviatoric boundary controls the lateral behavior, the volume change as well as the transverse opening of the axial distributed cracks under compression. However, numerical simulations show that the two boundaries in model M4 fail to capture the behavior of lightly confined concrete in a realistic way. With the new procedure for determining the elastic behavior of the microplane with confinement-adjusted effective microplane elastic moduli, the increased deformation capacity in multiaxial compression can be obtained, however, the enhancement in the ultimate strength and especially the post peak ductility is still less satisfactory. This means that the deviatoric stress boundaries in model M4 has to be improved. In model M4L, two new deviatoric stress boundaries are developed, shown in Figure 7. The mathematic expressions are given in Li (2009). The distinct characteristics of the new boundaries are both of the boundaries are scaled according to θ. The two boundaries, together with the new procedure for describing the microplane elastic behavior, is very efficient in capturing the improvement of the ultimate strength, the deformation capacity and the post peak ductility for concrete under low confinement. This can be clearly seen in the examples given later. The shear stress boundary σTb in model M4 given in Eq. (5) has a nonlinear form and describes the frictional interaction between the normal stress σN and shear stress σT . When the normal stress σN is not

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Decrease of

( 1, fc)

3.0 in MPa

-10.0

4.0

-20.0 -30.0

1.0

-40.0 -12.0 -9.0 -6.0 -3.0 0.0 3.0 6.0 in ‰ D (a) Compressive deviatoric stress boundary

0.0 -1.2

in MPa b+ D

6.0

Decrease of ( 1, fc) 0.0 -6.0 -3.0 0.0 3.0 6.0 9.0 in ‰ D (b) Tensile deviatoric stress boundary

0.3

0.6

12.0

3.3

Deviatoric stress boundaries in model M4L.

very large, the shear boundary is an approximate linear function. This boundary simulates the friction and slip. ET k1 k2 c10 −σN + σN0  ET k1 k2 + c10 −σN + σN0  (5a) ET k1 c11 1 + c12 εv 

-0.3 0.0 in ‰

is given in Li (2009). The newly formulated σN0 makes it possible for a realistic prediction for the Poisson’s ratio of concrete subjected to uniaxial compression, as shown later. A similar numerical algorithm as model M4 is adopted in model M4L. For more details, see Li (2009).

3.0

σN0 =

-0.6

Figure 8. σN0 in model M4L.

9.0

σTb = FT (−σN , εv ) =

-0.9

V

12.0

Figure 7.

2.0

0 N

bD

in MPa

0.0

(5b)

However, the shear stress boundary in model M4 is found to be too low in the elastic regime. As a result, the microplane shear stresses σM and σL are determined from the boundary value. This shifts the shear stress components into inelastic regime too early. Hence, the Poisson’s ratio under uniaxial compression begins to increase when the stress level is still very low, as shown in Fig. 4. A very direct remedy is to make the shear boundary higher through enlarging σN0 . As found by Di Luzio (2007) an increase of the parameter from its reference value 0.2 to 0.95 is very beneficial for producing realistic Poisson’s ratio of concrete under uniaxial compression. However, such a way makes the shear stress boundary rather high in the inelastic regime and shifts the peak strain under uniaxial compression too far away. In model M4L, a new formulation of σN0 is developed, as illustrated in Figure 8. The detailed expression

Numerical validation

To validate the accuracy of the model M4L, a comparison of the model responses and the test data in the literature is carried out. The used model parameters can be found in Li (2009). Figure 9 (a) shows the simulation of model M4L and the test data of Imran (1994) for concrete under low confinement. It is obvious that the simulated response is very close to the experimental observations. The increase of ultimate strength, deformation capacity as well as the post peak ductility with increasing the confining pressure can be successfully represented by model M4L. This mainly attributes to the newly developed procedure for evaluating the microplane elastic behavior and the novel deviatoric stress boundaries. Figure 9 (b) illustrates the model simulations for the low confined compression data obtained by Richart et al. (1928). One can see that the model reproduces the test data fairy well. Figures 9 (c) and 9 (d) show a comparison between the simulated responses and the test data of Sfer et al. (2002) and Candappa et al. (2001), respectively. It is evident that the model M4L reproduces the both of the test data quite accurate. A comparison of the model predicted Poisson’s ratio under uniaxial compression and the measured test data by Ma (2009) is presented in Figure 9 (e). It is clear that model M4L realistically captures the variation of the Poisson’s ratio with the relative stress. The mode predicted Poisson’s ratio under different lateral pressures is shown in Figure 9 (f). It can be seen from the figure that the simulated response is quite consistent with the experimental observations of Dahl (1992).

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75

50

75 9.0

4.2 2.1

30

1.05 0.0

15

15 in ‰ (a) Stress-strain response (Imran 1994) 120 0

5

10

20

25

90 in MPa

3.79

30 20

1.24

10

Model M4L Test

0

60

5.38 in MPa

45

0

40

8.4 in MPa

in MPa

60

10

20

30 40 in ‰ (b) Stress-strain response (Richart et al. 1928) 1.0 Model M4L Test 0.8

50

0.6 60

1.5 0.0

0

0

Model M4L Test

5

10 in ‰ (c) Stress-strain response (Sfer et al. 2002) 0.8 0.0 1.05 0.6 2.1 4.2 8.4 0.4

15

20

0.4 0.2

0.2

30 0 -20 -15 -10 -5

Model M4L Test

0 5 10 15 20 25 in ‰ (d) Stress-strain response (Candappa et al. 2001)

Figure 9.

30 15

Model M4L Test 0

4.5

45

0.0 0.0

0.2

0.4

0.6 /f c

0.8

1.0

1.2

(e) Poisson’s ratio in uniaxial compression (Ma 2009)

0.0 0.0

0.2

0.4

0.6 /f c

0.8

1.0

1.2

(f) Poisson’s ratio under low confinement

Comparisons of model M4L simulations and test data for concrete under low confinement.

The above numerical simulations imply that model M4L is adequate for representing the behavior of concrete under low confinement.

4

STRUCTURAL APPLICATION EXAMPLE

To further evaluate the performance of model M4L, the model is implemented in the finite element software SOFiSTiK and used to model a common problem in high rise buildings—the transmission of high-strength concrete (HSC) column loads through normal-strength concrete (NSC) slabs. Due to the different lateral deformation properties of HSC and NSC, the slab is subjected to the confinement from the column. As a result, the bearing capacity of the joint is higher than that of column with the same concrete. A corner joint is considered in this paper as this kind of joints is more critical owing to the absence of a surrounding slab. The test specimen by Lee et al. (2007) is simulated with model M4L. The test consists of a sandwich specimen and two reference columns, shown in Figure 10. 4.1

Specimen details

All the specimens have a cross section of 250 × 250 mm2 and a height of 750 mm. In the sandwich specimen, the NSC joint layer has a thickness of 150 mm. The layout of the longitudinal and the transverse reinforcements as well as the measured properties of concrete and reinforcement for each specimen are given in (Lee et al. 2007).

Figure 10.

4.2

Specimen details (Lee et al. 2007).

Finite element model

The finite element model for each specimen is illustrated in Figure 11. The concrete was modeled by Hexahedral elements with 8 integration points. Truss elements were used for modeling the reinforcement. Each model consists of a sum of 4608 concrete elements and 384 reinforcement elements. The total number of the nodes is 5577. The model parameters are calibrated according to the uniaxial compression behavior and the knowledge about the variation of the Poisson’s ratio for different concretes in uniaxial compression. The parameters used in the calculation are given in Li (2009). The predicted Poisson’s ratio under uniaxial compression for both concretes of the sandwich specimen is illustrated in Figure 12.

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(a) HSC column (a) HSC column

Figure 13.

(c) Sandwich column

Finite element mesh.

NSC HSC

1.0 0.8 0.6 0.4

Figure 12.

HSC

4000 NSC

3000 2000

Predicted Test

1000

0.2 0.0 0.0

Deformed mesh at failure.

5000

1.2

0 0

0.2

0.4

0.6 /f c

(c) Sandwich column

6000

P in kN

Figure 11.

(b) NSC column

(b) NSC column

0.8

1.0

1

2

3

4

5

in ‰

1.2

Figure 14. Predicted and observed load-deformation relation for HSC and HSC columns.

Model predicted Poisson’s ratio for concrete.

6000

Numerical results

5000

The simulated responses for each specimen are illustrated in the following. Figure 13 shows the deformed FE-mesh at failure. The model is proven to be able to correctly reproduce the failure mode. For the sandwich column, the predicted failure by the numerical model is due to the crushing of the joint under compression, as observed in the test. Figure 14 illustrates a comparison of the predicted load-deformation behavior of each specimen. One can see from the figure that for both the HSC and the NSC columns, the calculated results are in good agreement with the test data. This implies that model M4L is adequate for simulating the behavior of confined concrete columns under axial compression. The predicted response for the sandwich column is also shown in Figure 15. It can be seen that the numerical model successfully simulates the strength and deformation enhancement of the joint due to the confinement of the top and bottom HSC column. However, the predicted peak load and strain at failure are a little higher than that of the test. A better simulation might be achieved through optimizing the parameters for the concretes both in the top and bottom columns and slabs. In this calculation, the model parameters for the concretes are calibrated only from

P in kN

4.3

4000 3000 2000 Predicted Test

1000 0 0

1

2

3

4

5

in ‰

Figure 15. Predicted and observed load-deformation relation for sandwich columns.

uniaxial compression test data in the axial direction, this might be insufficient.

5

CONCLUSIONS

In this paper, a new material model, called model M4L for representing the behavior of concrete under low confinement is developed. The model is formulated on the basis of microplane theory and follows the same approach as a previous microplane model M4. Numerical results indicate that the model M4L can accurately

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describe the enhanced ultimate strength, deformation capacity as well as the post peak ductility of concrete subjected to low confinement. REFERENCES Attard, M.M. & Setunge, J.A.S.D. 1996. Stress strain relationship of confined and unconfined concrete. ACI Materials Journal, 93(5), 432–442. Bažant, Z.P., Caner, F.C., Carol, I., Akers, S.A. & Adley, M.D. 2000. Microplane mode M4 for concrete. Part I: Formulation with work-conjugate deviatoric. Journal of Engineering Mechanics, ASCE, 126(9), 944–953. Bažant, Z.P., Xiang, Y. & Prat, C. 1996. Microplane mode M4 for concrete. I: Stress-strain boundaries and finite strain. Journal of Engineering Mechanics, ASCE, 126(9), 944–953. Candappa, D.C., Sanjayan, J.G. & Setunge, S. 2001. Complete triaxial stress-strain curves of high-strength concrete. Journal of Materials in Civil Engineering, ASCE, 13(3), 209–215. Caner, F.C. & Bažant, Z.P. 2000. Microplane mode M4 for concrete. Part II: Algorithm and calibration. Journal of Engineering Mechanics, ASCE, 126(9), 954–961. Carol, I., Jirásek, M. & Bažant, Z.P. 2001. A thermodynamically consistent approach to microplane theory. Part I. Free energy and consistent microplane stresses. International Journal of Solids and Structures, 38(17), 2921–2931.

Dahl, K. 1992. A Failure Criterion for Normal- and Highstrength Concrete. Technical University of Denamrk. Di Luzio, G. 2007. A symmetric over nonlocal microplane model M4 for fracture in concrete. International Journal of Solids and Structures 44(13), 4418–4441. Imran, I. 1994. Application of Nonassociated Plasticity in Modelling the Mechanical Response Concrete. PhD thesis, University of Toronto. Lee, J.-K., Yoon, Y.-S., Cook, W. & Mitchell, D. 2007. Benefits of using puddle HSC with fibers in slabs to transmit HSC column loads. Journal of Structural Engineering, ASCE, 133(12), 1843–1847. Li, J. 2009. Development of a New Material Model for Concrete on the Basis of Microplane Theory. PhD Thesis, Leipzig University. Ma, J. 2009. Production and Performance of Ultra-highperformance Concrete. PhD Thesis, Leipzig University. Richart, F.E., Brandtzaeg, A. & Brown, R.L. 1928. A Study of the Failure of Concrete under Combined Compressive Stresses. Bulletin No. 185, University of Illinois. Sfer, D., Carol, I., Gettu, R. & Este, G. 2002. Study of concrete under triaxial compression. Journal of Engineering Mechanic, ASCE, 128(2), 156–163. Tue, N.V., Li, J., Caner, F.C. & Püschel, T. 2008. A new Microplane constitutive model for concrete. The Eleventh East Asia-Pacific Conference on Structural Engineering and Construction (EASEC-11), pp. 548–549.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Meso- and macroscopic models for fiber-reinforced concrete Sonia M. Vrech & Guillermo Etse CONICET, Universities of Tucuman and Buenos Aires, Argentina

Günther Meschke Institute for Structural Mechanics, Ruhr University Bochum, Bochum, Germany

Antonio Caggiano & Enzo Martinelli Department of Civil Engineering, University of Salerno, Fisciano (SA), Italy

ABSTRACT: Fiber reinforced concrete is analyzed and modeled at two different levels of observation. On the one hand, a macroscopic formulation based on the non-linear microplane theory is presented. Following approaches recently proposed in Pietruszczak & Winnicki (2003) and Manzoli et al. (2008), the mixture theory is used to describe the coupled action between concrete and the fiber reinforcement. The parabolic DruckerPrager maximum strength criterion is considered at the microplane level. Post-peak behavior is formulated in terms of the fracture energy release under mode I and/or II failure modes. On the other hand, a mesoscopic model of fiber reinforced concrete (FRC) is also presented which is based on three constituents: aggregate, mortar and aggregate-mortar interfaces. Aggregates are considered to be elastic while cracks are represented in a discrete format by means of interface elements. The presence of steel fibers is considered within the framework of the mixture theory. Consequently, mortar-mortar interfaces account for both fiber-mortar debonding and dowel effects according to the fiber volume content. After describing the constitutive models the paper focuses on numerical analysis of FRC failure behavior including re-analyzes of the experimental tests of Hassanzadeh (1990). The capabilities and shortcomings of both approaches for FRC failure analyses are evaluated. 1

INTRODUCTION

Fundamental deficiencies of cement-based materials like concrete and mortar such as low tensile strength and brittleness can be mitigated by adding steel fibers into the matrix. Fibers play a major role in the postcracking behavior of fiber reinforced mortar composites (FRMC) by bridging the cracks and providing resistance to the crack opening process. Actually, FRMC may achieve quasi-ductile response exhibiting strain-hardening response with multiple cracks and relatively large energy absorption prior to fracture localization. In this case the composite takes the name of high performance steel fiber reinforced mortar composite (HPFRMC). Regarding structural behavior of concrete members, the addition of fibers leads also to significant improvements in the ductility in pre- and post-peak regimes as well as in the tensile peak stress. This is a consequence of the increase of dissipation attributed to the action of the fibers bridging opening microcracks and the reduction of volumetric expansion in the low confinement regime. Other relevant advantages attributed to FRMC is the reduced water permeability.

Different approaches on different scales have been proposed for the modeling of FRMCs. They can be broadly categorized as follows: • Micro-scale models (more correctly denoted as meso-scale models): models which describe the interaction among the phases of the composite material, i.e. fiber, matrix, fine and coarse aggregates, and interfacial zones between them on the scale of the individual fibers. • Macro-scale models: the fibers and the matrix inside the FRMC at this scale of observation are indistinguishable. In this context FRMC is considered as a homogeneous material. Among others, we refer to the macro-scale models for FRMC by Hu et al. (2003) who proposed a single smooth biaxial failure surface for steel fiber reinforced concrete (SFRC), the proposal by Seow & Swaddiwudhipong (2005) based on a five parameter failure criterion for FRC with straight and hooked fibers and that of Minelli & Vecchio (2006) based on a modification to the compression field theory. Other relevant works are those by Guttema (2003), Pfyl (2003), etc. • Structural-scale models: these models capture the essence of the material behavior at the structural

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level, for example, cross-sectional moment versus curvature behavior or panel shear force versus lateral displacements. Semi-analytical models, for the flexure behavior of fiber-reinforced concrete (FRC) materials, based on the equilibrium of forces in the critical cracked section, have been proposed by Zhang & Stang (1997). Stang & Olesen (1998) present a design approach for fiber reinforced concrete structures. Lee & Barr (2003) characterize the complete load/stress-deformation curve, under various loading conditions, using one continuous four-exponential function. • Multi-scale models: the performance of these models are based on coupling ingredients of different scale models: micro, meso, macro and structural models (e.g. Kabele (2002)). The present work deals with failure analysis of FRC material at the macro and mesoscopic level of observation. Both approaches use the composite theory as a basis for the simulation of the interaction between matrix and steel fibers. The main aim of this research is the evaluation of fundamental properties at the mesoscopic level controlling the mechanical response behavior of FRMC during monotonic loading beyond the elastic range. To this end, macroscopic constitutive formulation which incorporates information on the fibre-mortar-interaction at the meso-level is used. For the 2D mesoscopic analysis in this work a new methodology is followed based on considering FRMC as a three phase material: aggregate, mortar and the interfaces between the other two constituents. The non-linear behavior of steel fiber reinforced mortar is captured by means of zero-thickness joint elements. To this end the original interface model by Carol et al. (1997) is reformulated to include the interaction between steel fibers and mortar based on the composite theory by Manzoli et al. (2008). The macroscopic approach is based on the microplane theory combined with the flow theory of plasticity and the parabolic Drucker-Prager maximum strength criterion. In this case, the composite theory accounts for the interaction between steel fibers and concrete (including the effect of aggregates at the macroscopic level). The numerical analysis presented in this work include predictions of both approaches of FRC failure behavior in direct shear and uniaxial traction. Also the stress paths of experimental tests are considered. The results demonstrate the potentials of the mesoscopic approach in this work to evaluate relevant aspects of FRC such as the influence of the aggregate maximum size, of the ratio between this size and the fiber length, and of the failure mechanism of this complex material. On the other hand, the microplane based theory for macroscopic evaluations of FRC failure behavior allows the consideration of arbitrary and

multiple directions for steel fibers and, eventually, of non-homogeneous materials. The next steps of this research consider the inclusion of strain gradients in the constitutive formulation of the microplanes to capture the size effect of FRC and, particularly, the influence of the ratio between maximum aggregate size and the fiber length.

2

CONSTITUTIVE MESO-MECHANICAL MODEL

Fracture analysis of FRMC on the meso-level follows the approach proposed by Vonk (1992) and then also used by Lopez et al. (2008a,b). While continuum mortar elements are assumed as linear elastic, the non-linear dissipative behavior of plain and fiber reinforced mortar is fully localized in the interface elements. According to the basic hypothesis of the mixture theory, the composite is considered as a continuum in which each infinitesimal volume is ideally occupied simultaneously by all constituents. In the plane of the interface, all constituents are subjected to the same strain fields and the corresponding composite stresses are given by the weighted (in terms of the volume fraction) sum of the constituent stresses. Being u the interface relative displacement vector, axial displacement of the fiber (in n direction) is given by uN = u · n, while in the transversal direction, uT = u · nT , being the unit vector perpendicular to n, see Figure (1). Consequently, the axial and angular fiber strains are given by εN = uN /lf and γT = uT /lf respectively, being lf the length of the fiber. Similarly to the flow theory of plasticity, the interface constitutive model is formulated in rate form. According to the composite theory, the rate of the stress vector ˙t = [σ˙ , τ˙ ]t in the interface plane is calculated as the sum of each constituent weighted by the volumetric fraction, k

Figure 1. fiber.

Schematic configuration of joint crossed by one

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m t˙ = k m t˙ + k f σ˙ f (˙εN ) nt + k f τ˙f (γ˙T ) (nT )t

(1)

The superscripts m and f refer to mortar matrix and fibers, respectively, while t means the transposition operation for tensors. The incremental stress-displacement relation of the proposed composite joint model can be expressed in compact form as t˙ = E ep · u˙

rate form is defined by u˙ = u˙ el + u˙ cr

(4)

u˙ el = C −1 t˙

(5)

t˙ = C (u˙ − u˙ cr )

(6)

(2)

where the constitutive tangent matrix E ep is ep

E ep = k m C ep + k f

Ef

lf

ep

nt n + k f ep

Gf lf

(nT )t nT

(3) ep

With C ep = ∂t/∂u, Ef = ∂σf /∂εN and Gf = ∂τf /∂γT as tangent operators defined in the following sections. General aspects of the constitutive formulation, which models the discontinuity behavior at the interface in presence of steel fibers, are the following: • Fracture energy-based joint constitutive law: the constitutive model is formulated in terms of normal and shear stresses on the interface plane, and corresponding relative displacements. The failure criterion (F = 0) of the interface model is described by the three-parameter hyperbola by Carol et al. (1997). Main features of the elastoplastic interface model formulation are summarized in the Section 2.1. • Fiber bond-slip effects are considered as a combination of the uniaxial elasto-plastic model, for fibers, and a uniaxial debonding dissipative model for the interface mortar-fibers, resulting in a global constitutive model for the constituent slipping fiber. • Dowel action of reinforcement short fibers crossing cracks in mortar is modeled in a smeared manner. ‘‘Beam on elastic foundation theory’’ is utilized to derive the dowel force-displacement law, which is expressed in terms of dowel stress and strain in order to be compatible with the previously indicated constitutive laws. Both the bond-slip axial model of the fiber and the dowel action of short fiber reinforcements crossing cracks in mortar/concrete are similarly considered in both meso- and macroscopic approaches. These models are detailed in Section 4. 2.1 Fracture-based interface constitutive model In this section, the interface model, originally proposed by Gens et al. (1988), is summarized. The elasto-plastic formulation of the interface model in

where u˙ t = [˙u, v˙ ] is the rate vector of relative displacements, decomposed into the elastic and crack opening components u˙ el and u˙ cr , respectively. C defines a fully uncoupled normal/tangential elastic stiffness at the interface  C=

kn 0 0 kt

 (7)

The yield-loading condition of the interface constitutive model is defined as F = τ 2 − (c − σ tanφ)2 + (c + χ tanφ)2

(8)

withτ and σ as the interface stress components. The tensile strength χ (vertex of the hyperbola), the shear strength c (cohesion strength) and the internal friction angle φ are model parameters. In Eq. (8), two limit situations can be distinguished: (a) cracking under pure tension, with zero shear stress (Mode I), when the yield surface is reached along the horizontal axis, and (b) cracking under shear and very high compression, when the yield surface is reached in its asymptotic region, where the hyperbola approaches a MohrCoulomb criterion. The last one is called ’’asymptotic Mode II (or Mode IIa)’’. The evolution of fracture process is driven by the cracking parameters χ and c, which depends on the energy release during interface degradation Wcr . Details are given in Carol et al. (1997). 3

MACROSCOPIC MODEL BASED ON MICROPLANE THEORY

For the mesoscopic analysis of FRC failure behavior a fracture energy-based elastoplastic microplane model was developed. Instead of the existing spherical microplane models, see a. o. Beghini et al. (2007), Carol et al. (2001), Kuhl et al. (2001), the proposed constitutive theory considers a 2D stress and strain fields using disk microplanes according to the proposal by Park & Kim (2003). As a result, a reduced number of microplanes is required. According to this approach, the fiber-reinforced concrete is idealized as a disk of unit radius and constant

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thickness b, which agrees with that of the analyzed material patch. The following assumptions are considered:

stress K in terms on the strain-like internal variable κ, and the friction and cohesion parameters α and β, respectively, that are defined as functions of the uniaxial compression and tensile strengths, ft and fc , respectively, according to

• Macroscopic stresses are uniform in the disk and are equilibrated by the surface tractions on the microplanes. • Microscopic strains in normal (ε) and tangential (γr ) directions to each microplane with normal direction n, are obtained from macroscopic strains ε (kinematic constrains) ε = ni nj εij γr =

1 [ni δjr + nj δir − 2ni nj nr ]εij 2

∂[ρ0 ψ0mic ] , ∂ε

τr =

∂[ρ0 ψ0mic ] ∂γr

with ρ0 denoting the material density. The macroscopic free-energy potential per unit mass of material in isothermal conditions, ψ0mac (ε, κ), with κ a set of thermodynamically consistent internal variables, result ψ0mac

1 = bπ

κ˙ =

(10)

(11)

β=

3 fc

∂F = λ˙ ∂K



nσ m=  1−

(12)

V

with V as the disk volume. 3.1 Microplane constitutive laws

(15)

p pdil

→ if p > 0

nσ

→ if 0 < p < pdil

(13)

in terms of the pressure p, the second invariant of the deviatoric stress tensor J2 , the dissipative plastic

(16)

where pdil is a model parameter representing the value at which the dilatancy vanishes, see Figure (2). In the post-peak regime the evolution of the dissipative stress, due to micro-fracture process at the microplane level, is defined through the homogenization process of the fracture energy released during crack formation with the plastic dissipation of an equivalent continuum, similarly to the fracture energybased plasticity model by Willam et al. (1985) and

The microplane constitutive law is based on the mixture theory by Truesdell & Toupin (1960) and, similarly to the interface model formulation used in the analysis at the mesoscopic level of observation, on the hypothesis of FRC to be represented by a composite model (Oliver et al. (2008)). Then, the rate of stress vector t˙ at each microplane is obtained by Eq. (1) where superscript m refers now to the concrete matrix variables. The constitutive model of the concrete matrix is based on the parabolic Drucker-Prager strength criterion. The yield condition in hardening/softening regime is defined by the unified equation F(t, k) = βJ2 + αp − (ft − K(κ))

(14)

A non-associated flow is adopted in order to avoid the excessive inelastic dilatancy. The plastic potential is based on a volumetric modification of the yield condition in the compressive regime. Then, the gradient tensor m of to the plastic potential can be obtained by a modification of the gradient tensor nσ to the yield surface as

 ψ0mic (tε , κ)dV

 fc − ft , fc

The evolution of the internal variable is defined in terms of the plastic parameter rate λ˙ as

(9)

• Normal and tangential microscopic stresses t = [σ , τr ]t are obtained from the microscopic free energy potential σ =

 α=3

Figure 2.

Macroscopic constitutive model.

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Etse & Willam (1994), as



K˙ = ft 1 − exp −5

I ht Gf ε˙ fr ur GfIIa

The yield criterion, in tension as well as in compression, is represented by the following expression



Ff = |σf | − (σy, f + Qf )

(17)

where σy, f is the elastic limit. The evolution, in the post-elastic regime of the 1D surface is driven by the stress-like internal variable Qf , given in incremental form as

with the equivalent fracture strain ε˙ fr = mI  κ˙

(18)

˙ f = λ˙ f H f Q

where ht represents the characteristic length associated with the active fracture process and, more specifically, the distance or separation between microcracks. Moreover, ur represents the maximum crack opening displacement in mode I type of failure. GfI and GfIIa are the fracture energies in modes I and II of failure, respectively. The McAuley brackets in Eq. (18) indicate that only tensile principal plastic strains contribute to the fracture strain during fracture process. In the special case of uniaxial tension state, the evolution of the dissipative stress can be obtained with the simplified expression    ht K˙ = ft 1 − exp −5 ε˙ fr ur

4

(23)

(24)

with ε˙ p// = λ˙ f ∂Ff /∂σf = λ˙ f sign[σf ] representing the plastic flow law, λ˙ f is the non-negative plastic multiplier and H f is the hardening/softening modulus. The incremental stress-strain relationship is ep

σ˙ f = Ef ε˙

(25) ep

where the elasto-plastic tangent modulus Ef takes the two following distinct following values, see Figure (3)  ep Ef = E f → Elastic response (26) ep Ef = Ef E /H1 f +1 → Elasto-plastic regime f

(19)

FIBER-MORTAR/CONCRETE INTERACTION

It is assumed that the fiber strain ε is decomposed in two additive parts, one due to the intrinsic fiber uniaxial deformation εs and another one associated with the interface debonding εd

In this section the models for the interaction between steel fibers and mortar (mesoscopic model), as well as steel fibers and concrete (macroscopic model), considering the bond-slip and dowel effects, are presented.

ε = εs + ε d

(27)

Assuming a serial model constituted by the fiber and the fiber-mortar joint the corresponding total deformability 1/Ef is given by

4.1 Bond-slip axial model of the fiber

1/Ef = 1/Es + 1/Ed

The uniaxial behavior of the steel fiber is approcimated by means of a simple 1D elasto-plastic model. The following set of equations is considered

where Es and Ed are the steel Young’s modulus and an equivalent elastic modulus of matrix-fiber interface, respectively. Two limit situations can be recognized:

ε˙ = ε˙ el + ε˙ p ε˙ el =

σ˙ f Ef

σ˙ f = Ef (˙ε − ε˙ p )

(20)

(21)

(22)

where the rate of the axial fiber strain ε˙ is decomposed into a elastic part and a plastic component, ε˙ el and ε˙ p , respectively. Ef represents an equivalent uniaxial elastic modulus including the uniaxial response of the steel and the bond-slip effect of the short reinforcement. σ˙ f is the rate of bond-slip axial stress of the steel fiber.

Figure 3.

Uniaxial bond-slip model for the fiber.

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(28)

• Ed → 0: the stiffness of the complete structure becomes null and the effect of the fibers vanishes. • Ed → ∞: represents the case of a perfect adherence between matrix-fibers.

following values ⎧ ⎨Gfep = Gf ⎩Gfep = Gf

In this context, and to complete the bond-slip axial constitutive models presented by the Eqs. (20) to (26), the following material parameter are defined σy, f = min[σy,s , σy,d ]  Hs H = Hd f

(30)

in which σy,s and σy,d are the material yield stress and the equivalent interface elastic limit, respectively; while the super-indices s and d refer to steel and debonding, respectively. The parameters Ed , σy,d and H d required for the bond/slip model characterization can be calibrated from a simple pull-out test (Oliver et al. 2008).

γ˙ = γ˙ el + γ˙ pl γ˙ el =

(31)

τ˙f Gf

(32)

τ˙f = Gf (γ˙ − γ˙ pl )

(33)

being γ˙ the rate of the shear fiber strain, which is decomposed into a elastic and a plastic part, γ˙ el and γ˙ pl , respectively. Gf represents the shear modulus, while τ˙f is the rate of dowel shear stress of the interaction between fiber-matrix. The model is completed with:

→ plastic regime

Vd = Es Is λ3 

(35)

(36)

in which Is = πdf4 /64 is the moment of inertia of the fiber (df diameter of the fiber) and λ parameter representing the relative stiffness between the fiber and the foundation, defined as  kc df λ= 4 (37) 4Es Is

4.2 Dowel action of reinforcement short fibers crossing cracks in mortar matrix. The dowel effect of fibers crossing cracks in mortar, is taken account in the joint model by means of a 1D shear stress-strain elasto-plastic constitutive model, similar to the previously mentioned one for the axial stressstrain. In this case, the following equations are utilized

Gf /H dow + 1

H dow is the hardening/softening modulus of the uniaxial dowel model, commonly assumed as H dow = 0. Dowel effect can be analyzed treating each reinforcement fiber as beam on elastic foundation (Winkler theory) to deal with the interaction between the fibers and the surrounding mortar (He and Kwan, 2001, Rumanus and Meschke, 2010). The fiber can be treated as a semi-infinite beam on the elastic foundation, loaded with a concentrated load at one extreme representing the dowel resultant Vd . The analytical solutions of the beam on elastic foundation in Figure (4) results in the following forcedisplacement relationship Vd − 

(29)

If σy,s < σy,d otherwise

→ elastic response 1

Thereby is kc the foundation modulus of the surrounding mortar that governs the dowel stiffness. Experimental data, available for RC specimens (Dei Poli et al. 1992), show that the same coefficient take values ranging from 75 to 450 N/mm3 . Other tests (Soroushian et al. 1987) show that the coefficient kc increases as the strength of surrounding mortar increases and when the volume fraction of the reinforcement increases. An equivalent shear elastic modulus can be calculated as Vd = Es Is λ3  = Gf

 Af Lf

⇒

Gf = Es Is λ3

Lf Af (38)

where As = π df2 /4 is the cross area section of the bar.

• yielding criterion, similar to Eq. (23); • hardening/softening law, similar to Eq. (24). The incremental shear stress-strain relation, can be written as ep

τ˙f = Gf γ˙

(34)

where, in a similar manner as in the bond model, ep the tangent shear modulus Gf takes the two distinct

Figure 4.

Dowel effect based on Winkler beam theory.

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At the limit stage, local crushing of the surrounding mortar and/or yielding of the dowel bar occurs. Based on experimental results for RC specimens, the following equation has been proposed by Dulacska (1972) for the dowel force at the limit stage    (39) Vdu = kdow df2  fc σy,s  In (39) kdow is a non-dimensional coefficient (kdow = 1.27, for RC-structures), while fc is the compressive strength of the concrete or the surrounding mortar. Finally, the equivalent shear yield stress, τy, f results as Vdu τy, f = (40) Af 5

NUMERICAL ANALYSIS

In this section numerical analyses are performed with both the mesoscopic and macroscopic models and considering failure behavior of FRMC and FRC, respectively. 5.1 Mesoscopic evaluation of FRMC failure Behavior

Firstly, a uniaxial tensile test is performed by imposing homogeneous vertical displacements to all four nodes of the upper quadrilateral element. The results in terms of vertical nominal stress vs. vertical displacements are shown in Figure (6) when the test is performed with different amount of fibers crossing the interface (df = constant = 0.8 mm), and in Figure (7) where the specific case of 5 fibers are considered but with six different values of df . The results in Figures (6) and (7) indicate that the proposed model is able to reproduce the sensitivity of FRMC regarding peak load and post-peak ductility to both the amount of fibers and the diameter of the fibers (when equal number of fibers are considered). Moreover, the sensitivity of the post-peak ductility is significantly more important than that of the peak load with respect to fiber amount and diameter. Post-peak responses in both figures show increasing reloading effects with the increment in the amount and diameter of the fibers. This is due to the increasing composite (inhomogeneity) effect that results by enlarging the amount or diameter of the fibers. Figures (8) and (9) show the model performance when compared against the experimental results by Hassanzadeh (1990) (included with dotted lines). These tests are characterized by imposing combined normal and shear relative displacements to

To evaluate the predictive capabilities of the proposed non-linear dissipative interface model for FRMC the element patches shown in Figure (5) are considered. Thereby, one interface element is placed between 2D plane stress isoparametric four node elements subjected to the indicated boundary conditions in terms of impeded displacements. As indicated in the same Figure (5) six different cases are considered with one, two, three, five, seven and nine steel fibers crossing the interface element (all of them having the same diameter df ). Figure 6. Normal stress vs. vertical displacement performed with different amount of fibers with df = constant = 0.8 mm.

Figure 5. Interface configuration with: (a) one, (b) two, (c) three, (d) five, (e) seven and (f) nine steel fibers (df = 0:8 mm).

Figure 7. Normal stress vs. vertical displacement performed with six different values of df (nf = constant = 5).

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Figure 8. Single crack crossed by fibers: numerical tests with θ = 30◦ , df = constant = 0.8 mm and different number of fibers: (a) normal stress vs. relative displacement and (b) shear stress vs. relative displacement.

the peak strength is reached. From that point, normal and shear relative displacements are applied simultaneously in a fixed proportion characterized by a constant value of the relation tanθ = u/v, with u and v the normal and tangential relative interface displacements, respectively. These tests were re-analyzed with θ = 30◦ and θ = 60◦ , both for df = 0.8 mm. Model parameters value used in these numerical analysis are: kN = 200 MPa/m , kT = 200 MPa/m, tanφ = 0.9, χ0 = 2.8 MPa, c0 = 7.0 MPa, GfI = 0.1 N/mm, σdil = 30 MPa, αχ = 0 and αc = 0, for the interface, while ν = 0.2 and Em = 25000 MPa, were used for the elastic modulus and the Poisson’s ratio, respectively, for the mortar. Results in Figure (8a) show, as expected, that the tangential and normal displacement control in the second part of the Hassanzadeh test for θ = 30◦ is responsible for a stronger softening of the normal tensile stress than in case of the pure tension test. The fiber content affects mainly the last portion of the curves in the reloading zone. With other words, the strong post-peak decrease (connected with severe cracking) in the first portion of the softening regime of this test practically suppresses the fiber contribution to the ductility (to compare the results with that corresponding to plain mortar). In contrast, results in terms of shear stress vs. tangential relative displacement in Figure (8b) show relevant fiber contribution in the preand post-peak regimes as well as in the maximum strength. When comparing these curves with results in Figures (9a) and (9b) we observe, as expected, less severe softening both in normal and shear stress components. Moreover, only in case of plain mortar or low content of fibers the softening branch leads to zero in the compressive regime of the normal stress. With three fibers (or more then three) the softening regime fully remains in the tensile regime indicating a significant increase of the ductility as compared to plain mortar case. In conclusion, the proposed interface model for mesoscopic analyses of FRMC failure behavior seems to provide realistic predictions of peak stresses, ductility and post-peak behavior of this material when different fiber directions and fiber contents are considered crossing a single crack.

Figure 9. Single crack crossed by fibers: numerical tests with θ = 60◦ , df = constant = 0.8 mm and different number of fibers: (a) normal stress vs. relative displacement and (b) shear stress vs. relative displacement.

a developing crack in a prismatic concrete specimen of 0.07 × 0.07 m2 square cross section with a 0.015 m deep notch. During first part of the numerical test, a pure tension stress state is imposed until

5.2

Macroscopic evaluation of failure behavior of FRC

To evaluate the capabilities of the proposed macroscopic model based on microplane theory to simulate failure behavior of FRC, preliminary numerical studies considering a unidirectional and an isotropic distribution of fibers, are performed. A single element problem in plane strain conditions and subjected to a homogeneous stress/strain state is considered. The concrete matrix is characterized by

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Figure 10. Uniaxial tensile test for microplane model. Fibers oriented in loading direction.

Figure 12. Uniaxial compression test for microplane plasticity. Fibers oriented in loading direction.

Figure 13. Uniaxial compression test for microplane plasticity. Isotropic distribution of fibers. Figure 11. Uniaxial tensile test for microplane model. Isotropic distribution of fibers.

the following material parameters E = 19000 MPa, ν = 0.2, fc = 22 MPa, ft = 3.0 MPa, ht = 108 mm, ur = 0.127 mm, GfIIa /GfI = 10, pil /fc = 1.2. Fiber material parameters are those of Subsection 5.1. Firstly, a uniaxial tensile test is performed. Results for fibers oriented in the loading direction and with isotropic distribution of fibers are illustrated in Figures (10) and (11). Different volume contents Vf of the fibers are considered including the extreme case of plain concrete (Vf = 0).The elastic stiffness increases in case of bias fiber as compared to the plain concrete case and with the increment of Vf . In both cases of fiber orientations a slight increase of the peak tensile strength and a re-stiffening effect in post-peak regime are observed. Figures (12) and (13) show the results obtained in the uniaxial compression test when both uniaxially oriented fibers and an isotropic distribution of

fiber directions are considered. In case of bias fiber, as expected, the stiffness in the pre-peak regime and the overall dissipated energy during post-peak regime increase with Vf . 6

CONCLUSIONS

Mesoscopic and macroscopic models for fiber reinforced cement composite materials were presented. The mesoscopic model takes into account a three phase mesostructure composed by elastic aggregates, mortar and mortar-aggregate interfaces. This model also includes mortar-mortar interfaces to simulate the dissipative response behavior of this constituent. The macroscopic model is formulated within the theoretical framework of microplane theory. Both the interface model and the microplane model for meso- and macroscopic analyses, respectively, are based on flow rule of plasticity, mixture theory by

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Truesdell & Toupin (1960) and composite model by Oliver et al. (2008). The interactions between steel fibers and mortar/concrete associated with debonding and dowel effects are considered in both models. Preliminary numerical studies presented in this paper demonstrate (partially on a conceptual level) their capabilities to reproduce the most relevant aspects of failure behavior of steel fiber reinforced concretes under tensile, shear and compressive stresses.

ACKNOWLEDGEMENTS The first two authors acknowledge financial support for this work by FONCYT (Argentine agency for research & technology) through Grant PICT1232/6, and by CONICET (Argentine council for science & technology) through Grant PIP6201/05.

REFERENCES Beghini, A., Bazant, Z.P., Zhou, Y. Gouirand, O. and Caner, F.C. 2007. Microplane Model M5f for Multiaxial Behavior and Fracture of Fiber-Reinforced Concrete. Journal of Engineering Mechanics. 133: 66–75. Carol, I., Prat, P.C. and Lopez, C.M. 1997. Normal/shear cracking model: Applications to discrete crack analysis. Journal of Engineering Mechanics. 123(8): 765–773. Carol, I., Jirasek, M. and Bazant, Z. 2001. A thermodynamically consistent approach to microplane theory. Part I. Free energy and consistent microplane stresses. International Journal of Solids and Structures. 38: 2933– 2952. Dei Poli, S., Di Prisco, M. and Gambarova, P.G. 1992. Shear response, deformations, and subgrade stiffness of a dowel bar embedded in concrete. ACI Struct. J. 89(6): 665–75. Dulacska, H. 1972. Dowel action of reinforcement crossing cracks in concrete. ACI J. 69(12): 754–7. Etse, G. and Willam, K. 1994. A fracture energy-based constitutive theory for inelastic behavior of plain concrete. J. Engineering Mechanics, ASCE. 120: 1983–2011. Gens, A., Carol, I. and Alonso, E. 1988. An interface element formulation for the analysis of soil-reinforcement interaction. Comput. Geotechnics. 7: 133–151. Guttema, T.B. 2003 Ein Beitrag zur realitatsnahen Modellierung und Analyse von stahlfaserverstarkten Stahlbeton und Stahlbetonflachentragwerken. PhD Thesis. Univ. Kassel. Hassanzadeh, M. 1990. Determination of fracture zone properties in mixed mode I and II. Engineering Fracture Mechanics. 35 (4/5): 845–853. He, X. and A. Kwan (2001). Modeling dowel action of reinforcement bars for finite element analysis of concrete structures. Computers and Structures 79: 595–604. Hu, X.D., Daz, R. and Dux, P. 2003. Biaxial failure model for fiber reinforced concrete. Journal of materials in civil engineering. 15(6): 609–615. Kabele P. 2002. Equivalent continuum model of multiple cracking. Engineering Mechanics. 9(1/2): 75–90. Kuhl, E., Steinmann, P. and Carol, I. 2001 A thermodynamically consistent approach to microplane theory. Part II.

Disipation and inelastic constitutive modeling. International Journal of Solids and Structures. 38: 2921–2931. Lee, M.K. and Barr, B.I.G. 2003. A fourexponential model to describe the behavior of fibre reinforced concrete. Materials and Structures. 37(7): 464–471. Lopez, C.M., Carol, I. and Aguado, A. 2008a. Mesostructural study of concrete fracture using interface elements. I: numerical model and tensile behavior. Materials and Structures. 41: 583–599. Lopez, C.M., Carol, I. and Aguado, A. 2008b. Mesostructural study of concrete fracture using interface elements. II: compression, biaxial and Brazilian test. Materials and Structures. 41: 601–620. Manzoli, O.L., Oliver, J., Huespe, A.E. and Diaz, G. 2008. A mixture theory based method for three-dimensional modeling of reinforced concrete members with embedded crack finite elements. Computers and Concrete. 5(4): 401–416. Minelli, F. and Vecchio, F.J. 2006. Compression Field modeling of fiber-reinforced concrete members under shear loading. ACI Structural Journal. 106(2): 244–252. Oliver, J., Linero, D.L., Huespe, A.E. and Manzoli, O.L. 2008. Two-dimensional modeling of material failure in reinforced concrete by means of a continuum strong discontinuity approach. Comput. Methods Appl. Mech. Engrg. 197: 332–348. Park, H. and Kim, H. 2003. Microplane model for reinforcedconcrete planar members in tension-compression. Journal of Structural Engineering. 129: 337–345. Pfyl, T. 2003 Tragverhalten von Stahlfaserbeton. PhD. Thesis. Eidgenssischen Technischen Hochshule Zrich. Switzerland. Pietruszczak, S. and Winnicki, A. 2003. Constitutive Model for Concrete with Embedded Sets of Reinforcement. Journal of Engineering Mechanics. 129(7): 725–738. Rumanus, E. and Meschke, G. 2010. Homogenization-based model for reinforced concrete. Computational Modeling of Concrete Structures (EURO-C 2010), in print. Seow, P.E.C. and Swaddiwudhipong, S. 2005. Failure Surface for Concrete under Multiaxial Load—A Unified Approach. Journal of Material in Civil Engineering. 17(2): 219– 228. Soroushian, P., Obaseki, K., Rojas, M.C. 1987. Bearing strength and stiffness of concrete under reinforcing bars. ACI Mater. J. 84(3): 179–84. Stang, H. and Olesen, J.F. 1998. On the interpretation of bending tests on FRC materials. Fracture Mechanics of Concrete Structures. 1: 511–520. Truesdell, C. and Toupin, R. 1960. The classical field theories. Handbuch der Physik, Springer Verlag, III/I, Belin. Vonk, R. 1992. Softening of concrete loaded in compression. Ph.D. thesis, Technische Universiteit Eindhoven, Postbus 513, 5600 MB Eindhoven, the Netherlands. Willam, K., Hurbult, B. and Sture, S. 1985. Experimental and constitutive aspects of concrete failure. In US-Japan Seminar on F.E. Anal. of R.C. Struct. ASCE-Special Public. 226–254. Zhang, J. and Stang, H. 1998. Applications of Stress Crack Width Relationship in Predicting the Flexural Behavior of Fibre—Reinforced Concrete. Cement and Concrete Research. 28(3): 439–452.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Gradient damage model with volumetric-deviatoric split Adam Wosatko & Jerzy Pamin Faculty of Civil Engineering, Cracow University of Technology, Poland

ABSTRACT: The decomposition of the isotropic damage model according to a volumetric-deviatoric split involves two damage parameters and in the case of gradient enhancement can produce a two- or three-field formulation depending on the adopted strain measure. The paper discusses the theory of isotropic gradient damage starting from the constitutive equations and finishing with proper matrix systems. The theory is verified by means of one-element test, however a more advanced simulation of the splitting effect in the Brazilian test is also performed. 1

INTRODUCTION

The constitutive equation becomes:

Departing from scalar damage, but remaining within the isotropic description, it is possible to generalize the damage model with two strain measures and/or two damage parameters (Ju 1990). One way of such modification can be the decomposition of damage into two parts related to tensile and compressive actions, cf. for instance (Mazars and Pijaudier-Cabot 1989; Comi 2001). However, the proposal discussed in this paper results from a volumetric-deviatoric split given for example in (Lubliner et al. 1989; Comi and Perego 2001). The first damage parameter influences the bulk modulus and the second one reduces the shear modulus. In the simplest approach one damage history parameter and one damage loading function are assumed, while two different damage growth functions are distinguished. Then, in a gradient-enhanced continuum model, one Helmholtz averaging equation is still the basis of a formulation as for the scalar gradient damage (Peerlings et al. 1996). Total separation leads to a model with two history parameters, corresponding to the volumetric and deviatoric strain measures. In this proposal two damage loading functions are introduced and two different averaging equations are applied. The theory is described in Sections 2 and 3. Computational verification of both approaches is performed in Sections 4 and 5.

σ = E KG 

2

and the variables on the right-hand side are: θ = IIT 

where: E KG = (1 − ωK )KT + 2(1 − ωG )GQ

(2)

σ is the stress tensor and  is the strain tensor, both in a vector form. In the above relation the damage parameter ωK degrades bulk modulus K and the parameter ωG reduces the deviatoric stiffness represented by shear modulus G. The following relations are introduced: 1 Q = Q0 − T 3 1 Q dev = I − T 3

(3) (4)

T where in  three dimensions:  = [1, 1, 1, 0, 0, 0] and 1 1 1 Q0 = diag 1, 1, 1, 2 , 2 , 2 . Note that the strain and stress vectors are split into volumetric and deviatoric parts:

1 θ +  dev 3 σ = p + ξ =

CONSTITUTIVE MODEL

The continuum damage formulation satisfies the isotropy condition if two damage parameters ωK and ωG for the volumetric and deviatoric parts respectively are considered, cf. (Lubliner et al. 1989; Ju 1990; Comi and Perego 2001; Carol et al. 2002).

(1)

(5) (6)

– dilatation, dev = Q –deviatoricstrain, p = 13 T σ – pressure and ξ = Q dev σ – deviatoric stress. The stress rate is obtained differentiating Equation (1): σ˙ = E KG ˙ − ω˙ K KT  − 2ω˙ G GQ

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(7)

The basic issue is how the damage and the equivalent strain which governs it should be measured. Two approaches can be proposed. 2.1

One strain measure

The simplest case is to assume κ d = κkd = κGd so that one damage loading function: f d (, κ d ) = ˜ () − κ d = 0

ω˙ G =

(8)

ωK = ωK (κ d )

(9)

ωG = ωG (κ d )

(10)

3

ω˙ K =

dωK dκ d d ˜ ˙ dκ d d ˜ d

(11)

ω˙ G =

dωG dκ d d ˜ ˙ dκ d d ˜ d

(12)

3.1

¯ − c∇ 2 ¯ = ˜

A more general concept involves a total separation of damage multipliers applied to the bulk and shear modulus, hence it is called ‘‘two strain measures’’. Now two damage loading functions are employed— separately for the volumetric strains (dilatation):

f d (, κ d ) = ¯ (˜ ()) − κ d = 0

(13)

(14)

ω˙ K =

∂ωK ∂κ d ˙ ¯ = GK1 ˙¯ ∂κ d ∂ ¯

(19)

ω˙ K =

∂ωG ∂κ d ˙ ¯ = GG1 ˙¯ ∂κ d ∂ ¯

(20)

The aim of distinguishing two history parameters κKd and κGd is that damage then increases independently for the volumetric and deviatoric strains: ω˙ K =

dωK dκKd d θ˜ ˙ dκKd d θ˜ d

(18)

Hence the damage rates are computed as:

and for the deviatoric (shear) strains: =0

(17)

is still the basis of the two-field formulation like for scalar gradient damage (Peerlings et al. 1996). The above relation involves the second gradients of the averaged strain ¯ . The parameter c > 0 has a unit of length squared and it is connected with the internal length scale l of a material. The relation c = 12 l 2 is derived for instance in (Askes et al. 2000). Instead of the local equivalent strain ¯ the averaged strain ¯ now governs the damage progress:

2.2 Two strain measures

κGd

Gradient enhancement for one strain measure

If ‘‘one strain measure’’ approach is assumed, the Helmholtz equation:

and during unloading both ω˙ K and ω˙ G are equal to 0.

fKd (θ, κKd ) = θ˜ (θ ) − κKd = 0

ISOTROPIC GRADIENT DAMAGE

This section concerns the derivation of isotropic damage with a gradient enhancement. The gradientenhanced model according to (Peerlings et al. 1996; Geers 1997; Pamin 2004) is nonlocal and guarantees producing mesh-independent results. During a failure process, from the onset of localization and until the total loss of the stiffness, the governing system of equations remains elliptic and regularization allows one to avoid a spurious mesh sensitivity. The equations of the boundary value problem (BVP) are almost the same as for scalar damage, but the tangent stress-strain relation is defined in Equation (7).

This option is further called ‘‘one strain measure’’. If one history parameter κ d governs the damage evolution, the rates of damage parameters ωK and ωG during loading are respectively:

= ˜dev ( dev ) −

(16)

During unloading it can be assumed that either ω˙ K or ω˙ G are equal to 0. A similar approach with total separation of damage parameters is described in (Carol et al. 2002), where two damage variables influence two loading functions and that model is called ‘‘bi-dissipative isotropic model’’.

satisfying the Kuhn-Tucker conditions is applied with the equivalent strain measure ˜ and the damage history parameter κ d which grows from the initial value κ o . This means that function () ˜ can be defined like in the scalar damage model. However, two different damage growth functions are distinguished:

fGd ( dev , κGd )

dωG dκGd d ˜dev ˙ dκGd d ˜dev d

(15)

The weak form of equilibrium equations can be written as follows:    δ T σ dV = δuT bdV + δuT tdS (21) B

B

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∂B

where u is the displacement field, b is the body force vector, t is the traction vector. A weak form of Equation (17) is derived using Green’s formula and the non-standard boundary condition (∇ ¯ )T ν = 0:    T δ ¯ ¯ dV + (∇δ ¯ ) c∇ ¯ dV = δ ¯ ˜ dV (22) B

B

B

Independent interpolations of displacements u and averaged strain measure ¯ are employed in the semidiscrete linear system and a two-field formulation ensues. The primary fields are interpolated in this way: u=Na

¯ = hT e

and

(23)

where N and h contain suitable shape functions. From the above interpolations the secondary fields can be computed in the following way:  = Ba

and

∇ ¯ = g e T

(24)

where B = LN and g T = ∇hT . The discretized equations must hold for any admissible δa and δe, therefore:    BT σ dV = N T bdV + N T tdS (25) 

B

B

B



(hhT + cgg T )edV =

B

∂B

h˜ dV

(26)

The BVP is linearized, hence at nodal points the increments of the primary fields from time instant t to time instant t + t are introduced in such a way: at+ t = at + a

et+ t = et + e

and

(27)

Analogically, at the integration points we employ decomposition of , σ and ˜ . The equilibrium equations then become:  BT (σ t + σ )dV b

=

T t+ t

N b B

dV +

t hT e

ωK = GK1

(31)

t

ωG = GG1 hT e

(32)

and the derivatives are determined at instant t. We rewrite Equation (28) in a matrix form: KG t+ t t K KG aa a + K ae e = f ext − f int

T t+ t

∂B

N t

dS

 K KG aa =

K KG ae = −

f t+ t = ext  f tint =

t t BT [GK1 KT + 2GG1 GQ] t hT dV

(35)

N T bt+ t dV +

N T t t+ t dS

BT σ t dV

B

(36)

∂B

(37)

In Equation (29) the increment of equivalent strain measure ˜ is computed from the interpolated displacement increment a: t



d ˜

˜ = d

 = [sT ]B a

(38)

and Equation (29) can be formulated as follows: K ea a + K ae e = f t − f te

(39)

The matrices and vectors in Equation (39) are similar to the scalar damage formulation, cf. (Peerlings et al. 1996):  h[sT ]t BdV

(40)

(hh + cgg T )dV

(41)

B

  f t =

B

B

h˜ t dV

f te = K ee et

(29)

t t

σ = E tKG B a − [GK1 KT + 2GG1 GQ] t hT e (30)

(34)



B

(28)

The incremental constitutive relation is derived starting from Equation (7):

B



K ee = −

B

BT E tKG Bd V

B



and averaging equation is derived as:   T T t (hh + cgg )(e + δe)dV = h(˜ t + ˜)dV B

(33)

where:

K ea = −





where the damage increments have been calculated as:

(42) (43)

Eventually, the following system of equations is used: 

K KG aa K ea

K KG ae K ee



  t+ t 

a f ext − f tint =

e f t − f te

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(44)



 3.2

Gradient enhancement for two strain measures

In this option two damage loading functions for two averaged strain measures are introduced: fKd (θ, κKd )

= θ¯ (θ˜ (θ )) −

κKd

=0

(45)

fGd ( dev , κGd ) = ¯dev (˜dev ( dev )) − κGd = 0

(46)

The damage rates depend on two different fields: ∂ωK ∂κKd ˙ θ¯ = GK2 θ˙¯ ∂κKd ∂ θ¯

(47)

∂ωG ∂κGd ω˙ G = ˙¯ dev = GG2 ˙¯ dev ∂κGd ∂ ¯dev

(48)

ω˙ K =

B

(hhT + cG gg T )d dV =

B

h˜dev dV

which are valid for any δo and δd. The BVP is linearized and corresponding increments of the primary fields at nodal points and the secondary fields at integration points from time instant t to time instant t + t are written. The equilibrium equation (28) holds, but instead of Equation (29) we obtain:   ˜ (hhT + cK gg T )(ot + o)dV = h(θ˜ t + θ)dV B

Therefore, two different averaging equations are introduced: θ¯ − cK ∇ 2 θ¯ = θ¯

(49)

¯dev − cG ∇ ¯dev = ˜dev

(50)

2

(55)

B

 B

(56)

(hhT + cG gg T )(d t + d)dV  =

t h(˜dev + ˜dev )dV

B

(57)

As previously, the incremental form of constitutive relation is derived from Equation (7), but now the interpolation of three fields is considered:

and in result a three-field formulation is obtained. Note that in this approach we can adopt two different values of internal lengths, namely cK for the volumetric part and cG for the deviatoric part. The weak form of equilibrium equations is identical to the previous approach. Using Green’s formula and the non-standard boundary conditions (∇ θ¯ )T v = 0 and (∇ ¯dev )T v = 0 for Equations (49) and (50) the weak forms of the above averaging equations are obtained:    ˜ δ θ¯ θ¯ dV + (∇δ θ¯ )T cK ∇ θ¯ dV = δ θ¯ θdV (51)

Here the damage increments are functions of different discretized fields in such a way:



We can formulate Equation (28) using matrix notation:

B

B

B

δ ¯dev ¯dev dV +  =

B

B



(∇δ ¯dev ) cG ∇ ¯dev dV T

B

δ ¯dev ¯dev dV

(52)

θ¯ = hT o

and ¯dev = h d T

(53)

It is noted that the same shape functions are used here for the volumetric and deviatoric averaged strain, but in general they can be different. The discretized equilibrium is governed by Equation (25). In addition we obtain two discretized averaging equations:   ˜ (hhT + cK gg T )o dV = hθdV (54) B

t GQ t hT d + 2GG2

(58)

t

ωK = GK2 hT o

(59)

t GG2 hT d

(60)

ωG =

t+ t t K KG aa a + K ao o + K ad d = f ext − f int

(61)

where:

Three primary fields are distinguished, so now we employ interpolations of displacements u, volumetric averaged strain measure θ¯ and deviatoric averaged strain measure ¯dev : u = N a,

t

σ = E tKG B a − GK2 KT  t hT o

B

 K ao = −

B

t Bt GK2 Kt  t ht dV

(62)

t Bt GG2 KGQεt ht dV

(63)



K ad = −2

B

The increments of equivalent strain measures θ˜ and ˜dev , which occur in Equations (56) and (57), can be calculated on the basis of primary fields as follows: 

θ˜ =

d θ˜ d

t

 = [sKT ]t B a

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(64)



d ˜dev d

˜dev =

t T t

 = [sG ] B a

(65)

Hence, Equations (56) and (57) take the matrix form: K oa a + K oo o = f tvol − f to K da a + K dd d =

f tdev

−

f td

The matrices and vectors denote:  K oa = − h[sKT ]t B dV  K ee = − f tvol =

B

(67) ω(κ d ) =

(hhT + cK gg T ) dV

hθ˜ t dV

f to = K oo ot  T t K da = − h[sG ] B dV

(69)

ω(κ d ) = 1 −

(70)

are employed. Using Equation (77) for the description of damage evolution firstly κ d exceeds threshold κo and then grows to ultimate value κu which corresponds to total damage. The parameters η and α in Equation (78) are responsible for the rate of softening and residual stress which in one dimension tends to (1 − α)Eκo . In our computations in the case of linear softening, we assume equal or different ultimate values κu for volumetric and deviatoric degradation. For exponential softening ductility parameter η is varied. Hence lower index i = K, G is applied in order to distinguish the analyzed tests. The cases with identical data for both damage growth functions can be treated as reference, because these in fact are computations using scalar damage. The same parameter α = 0.98

(71) (72)

  f tdev =

B

B

(hhT + CG gg T ) dV

t h˜dev dV

(73) (74)

f td = K dd d t

(75)

Finally, the matrix system of equations for the threefield formulation has the form: ⎤⎡ ⎤ ⎡ ⎡ KG t ⎤ K aa K ao K ad f t+ t

a ext − f int ⎥⎣ ⎦ ⎣ t ⎢ t 0 ⎦ o = f vol − f o ⎦ (76) ⎣ K oa K oo

d f tdev − f td K da 0 K dd 4

(77)

and exponential softening law (Mazars and PijaudierCabot 1989; Peerlings et al. 1998):

B

K dd = −

κu κ d − κo κ d κu − κo

(68)

B

B



(66)

the degradation of bulk modulus K and shear modulus G. This permits one to control damage evolution governed by dominating volumetric or shear failure. All calculated cases are set in Table 1. There are presented symbols used to distinguish a given case, and corresponding damage growth data. During the damage evolution κ d can grow in different ways. In this paper linear softening law (Peerlings et al. 1996):

 κo  d 1 − α + αe−η(κ −κo ) d κ

(78)

Table 1. Tension in one direction—all cases computed using one strain measure. Linear softening Ultimate value of history parameter

ONE-ELEMENT BENCHMARK TEST

One three-dimensional finite element (FE) with eight nodes is subjected to static tension in one direction. Such a simple test permits one to observe the numerical response of the isotropic damage model at the point level. Better understanding is necessary to perform more advanced analyses. The following material data are introduced: Young’s modulus E = 20000 MPa, Poisson’s ratio ν = 0.20.

Symbol of case

Volumetric

Deviatoric

lin lin, K lin, G lin, K&G

κuK κuK κuK κuK

κuG κuG κuG κuG

••• ..... –––

One damage threshold κo (equal here to 0.0001) is used in the loading function (18). The simple modification of the scalar theory makes it possible to adopt two different damage growth functions, separately for

= 0.002 = 0.002 = 0.003 = 0.003

Exponential softening Ductility parameter Symbol of case

4.1 Computations for one strain measure

= 0.002 = 0.003 = 0.002 = 0.003

exp exp, K exp, G exp, K&G

---–––

Volumetric

Deviatoric

ηK ηK ηG ηK

ηK ηG ηG ηG

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= 1000 = 750 = 1000 = 750

= 1000 = 1000 = 750 = 750

in all cases with exponential softening is assigned. It is also possible to adopt two different damage laws during the failure process, cf. (Wosatko 2008). The normalized elastic energy release rate (Ju 1989) is applied in computations as the loading function, however Mazars definition (Mazars 1984) and modified von Mises definition (de Vree et al. 1995) for one three-dimensional FE subjected to tension in one direction are all equivalent. The diagrams of strain 11 versus ωK and ωG in the final phase of damage are shown in Figure 1 for the chosen cases. These diagrams are helpful in understanding how different damage growth functions can influence the results. Firstly, in Figure 2 cases with linear softening are depicted. As it was expected, after the peak, diagrams for lin, K and lin, G start to descend between diagrams

1 0.95 K

0.9 G

0.85 0.0005

0.001

11

0.0015

for lin and lin, K&G. However, the nonlinear character of softening is observed although the damage growth function related to linear softening is assumed for cases lin, K and lin, G. Moreover, in case lin, G, where the deviatoric part has larger κuG arc-length control must be applied during computations because a snapback occurs. It suffices to change the ultimate value κu for either the volumetric or deviatoric part in order to obtain nonlinear response. In Figure 3 we can see stress-strain diagrams for damage growth functions related to exponential softening. The interpretation is easy since each softening branch has an exponential character. For cases exp, K and exp, G with different ductility parameters the diagrams run between the extreme cases exp and exp, K&G. Analogically to the cases with linear functions a certain tendency can be noted. If the fracture energy for the deviatoric part is increased (compare cases exp and exp, G), a larger difference in response is noticed than when fracture energy grows for the volumetric part. It is doubtful to consider fracture energy only for a chosen part of the stiffness, however in order to simplify the explanations this concept is used. Before the next figure is described the following definition of Poisson’s ratio depending on the stiffness

0.002

(a)Case lin,K.

2

1

lin,K&G

σ11 [MPa]

K

0.9 G

0.85 0.0005

lin,G

1.5

0.95

0.001

11

0.0015

0.002

lin,K

1

lin

0.5

(b)Case lin,G. 1

0

0

0.0005

0.001

0.0015

0.002

0.0025

0.003

11

0.95 K

Figure 2. Influence of ultimate κui (i = K, G) in linear softening for one strain measure.

0.9 G

0.85 0.0005

0.001

11

0.0015

0.002 2 exp,K &G

(c)Case exp,K. 1 σ11 [MPa]

0.95 K

0.9 0.001

11

0.0015

exp,K exp

1 0.5

G

0.85 0.0005

exp,G

1.5

0.002 0

(d)Case exp,G.

0

0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 11

Figure 1. Final variation of damage parameters in damagestrain diagrams for chosen cases—one strain measure.

Figure 3. Influence of ductility parameter ηi (i = K, G) in exponential softening for one strain measure.

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For linear softening (77) ultimate value κu = 0.002. If exponential softening (78) is considered, then η = 750 and α = 0.98. However, for two history parameters (κKd for volumetric part and κGd for deviatoric part) two damage thresholds are employed. For two averaged strain measures we introduce two equivalent strain definitions. The simplest proposal is:

degradation is introduced: νω =

3(1 − ωK )K − 2(1 − ωG )G 2[3(1 − ωK )K + (1 − ωG )G]

(79)

To distinguish the Poisson’s ratio which is given as an elastic material parameter from the one defined in Equation (79) the subscript ω is additionally applied. Hence, parameter νω is computed during the damage process. As it is shown in Figure 4, if only damage growth functions are predefined for a particular case then this parameter changes. Furthermore, in the case of linear softening the value of νω drastically tends to a lower or upper limit. These limits can be perceived as controversial results and they are not physically motivated. Such extreme behaviour in simulation and as a consequence of nonlinear relation between 11 and σ11 seems to be undiserable. A complete degradation for the volumetric part in case lin, G gives finally νω equal to –1. On the other hand the zeroed shear stiffness in case lin, K leads to νω = 0.5 like for incompressible materials, cf. (Carol et al. 2002). Concrete is rather a material where microcracks under tension decrease the Poisson’s effect during the failure process (Ju 1990). The problem is that non-physical values below zero appear. If exponential softening is used (see Figure 4, case exp, G) a smooth drop to zero is observed, but this is not always so. Naturally the starting value of νω and also the configuration of the considered test decide on whether νω becomes negative. For quasibrittle materials like concrete generally it is expected that Poisson’s ratio tends to 0 during the damage evolution (Carol et al. 2002). Summarizing, case exp, G seems to be physically correct. 4.2

θ˜ = 0.5(I1 + |I1 |)  ˜dev = 3J2

(80) (81)

where I1 and J2 are strain invariants. Analogically to the previous computations, the considered cases with symbols and corresponding data are given in Table 2. The thresholds for case t are chosen in such manner that they are equivalent to threshold κo = 0.0001 for scalar damage. For case t, K&G the choice of values is connected with a higher threshold κo = 0.00015 for scalar damage. Figure 5 shows stress-strain diagrams for the cases with linear softening. It is visible that the lower value of damage threshold decides about the onset of softening. Moreover, for case t, G only the deviatoric part of stiffness undergoes degradation, while the volumetric part remains elastic. For the opposite case t, K the deviatoric stiffness remains without changes and the

Computations for two strain measures

In this subsection we continue the analysis for one three-dimensional FE, but the proposal of two strain measures in the formulation is verified. The same damage growth data is assumed to focus our attention.

Table 2. Tension in one direction—cases computed using two strain measures. Damage threshold Symbol of case

Volumetric

Deviatoric

t t, G t, K t, K&G

κoK κoK κoK κoK

κoG κoG κoG κoG

= 0.00006 = 0.00009 = 0.00006 = 0.00009

= 0.00012 = 0.00012 = 0.00018 = 0.00018

3 0.5 0.4

2.5

0.2

2

σ11 [MPa]

t

0

t,G

1.5

lin,K exp,K

t,

1

exp,K &G

0.5

exp,G lin,G

0

0.0005 0.001

0

0.0015 0.002 0.0025 0.003 0.0035

0

0.0005

0.001

0.0015

0.002

11

11

Figure 4. Sensitivity of Poisson’s ratio νω to adopted damage growth functions for one strain measure.

Figure 5. Influence of damage threshold κoi (i = K, G) in linear softening for two strain measures.

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3 t, &G

[MPa]

2.5

t

2

t,G t,

11

1.5 1 0.5 0

0

0.0005

0.001 0.0015 0.002 0.0025 0.003

0.0035

11

Figure 6. Influence of damage threshold κoi (i = K, G) in exponential softening for two strain measures.

Figure 7. Table 3.

volumetric part is reduced. In spite of this behaviour a linear character of softening is preserved. Analogical diagrams for exponential softening are presented in Figure 6. Like previously, the lower value of damage threshold activates the failure process. It is observed that damage of either volumetric or deviatoric part causes a change of Poisson’s ratio νω , so for the volumetric degradation (here case t, K) νω still decreases below zero.

5

SIMULATION OF BRAZILIAN TEST

The splitting test is used to establish the concrete tensile strength, because the compression between the loading platens induces the perpendicular tensile force action in the centre of the specimen. This phenomenon and the snapback response in the Brazilian test is not simple to reproduce in numerical computations, because damage can localize directly under the platen. In fact, different mechanical models have been verified numerically using this test, starting from plasticity theories like in (Chen and Chang 1978). Due to a double symmetry and in order to reduce the computation time only a quarter of the domain (with radius equal to 40 mm) is considered. The general geometry data are based on (Winnicki et al. 2001), but plane strain conditions are assumed. The load is applied to the specimen indirectly via a stiff platen (width – 5 mm, height – 2.5 mm). The platen is perfectly connected with the specimen. In these computations only one mesh shown in Figure 7 is employed and attention is focused on the details of isotropic version of the gradient damage model. However, mesh insensitivity for gradient damage is widely discussed in (Wosatko 2008). The load acts downwards at the top of the platen. The material data are presented in Tables 3 and 4. It is obvious that in the gradient model not only the internal length parameter decides about the results of the test, but also other parameters, for instance the choice of the damage growth function. Here four

Applied mesh for Brazilian test. Brazilian test—material model data.

Specimen

damaging

Young’s modulus: Poisson’s ratio: Equivalent strain measure: Fracture energy: Internal length scale: Threshold:

Ec = 37700 MPa ν = 0.15 modified von Mises, k = 10 Gf = 0.075 N/mm l = 6 mm, i.e. c = 18.0 κo = 7.9576 × 10−5

Platen

elastic

Young’s modulus: Poisson’s ratio:

Es = 10 · Ec ν = 0.15

Table 4. Brazilian test—cases computed using one strain measure. Symbol of case

ηK

ηG

Damage growth

exp exp, K exp, G exp, K&G

1200 600 1200 600

1200 1200 600 600

more intensive ωK < ωG ωK > ωG less intensive

---–––

options of exponential softening are analyzed, where different combinations of values of ductility parameter η decide whether the damage process is more or less brittle. The parameter α equals 0.99 for each case. Because of a possible snapback response the test is computed using the arc length method. Load-displacement diagrams in Figure 8 are plotted for the four considered cases. It is noticed that for cases exp,K and exp the softening paths are mono-tonically decreasing and without snapback. The same value of parameter ηG = 1200 governs the solution. On the other hand, for cases exp,G and exp,K&G the snapback response is retrieved. This means that deviatoric damage is more important in the stiffness degradation

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P/2[kN]

0.45 0.4

exp, &G

0.35 0.3 0.25 0.2

exp,G

A

exp, exp

0.15 0.1 0.05 0

B

0

0.05

0.1

0.15

0.2

0.25

0.3

(a)

K,

exp,G, point A.

(b)

K,

exp,G, point B.

(c)

K,

exp,K, point A.

(d)

K,

exp,K, point B.

(e)

G,

exp,G, point A.

(f )

G,

exp,G, point B.

(g)

G,

exp,K, point A.

(f )

G,

exp,K, point B.

[mm]

Figure 8. Brazilian test—influence of ductility parameter ηi (i = K, G), load-displacement diagrams.

(a) Case exp,G, point A.

(c) Case exp,K, point A.

Figure 9.

(b) Case exp,G, point B.

(d) Case exp,K, point B.

Brazilian test—contour plots of averaged strain ¯ .

and decides about the proper behaviour in the Brazilian test. Contour plots in Figures 9–10 are presented for the peak and the final state (in Figure 8 the respective points A and B are marked). The splitting is observed only for the cases which correspond to the larger value of fracture energy Gf for the deviatoric part, i.e. the ductility ηG = 600. It is confirmed by means of Figure 9, where the distributions of averaged strain are plotted for the two stages—points A and B. Therefore the interaction between the compressive loading and the tensile response seems to be transferred via the deviatoric characteristics in the model. Figure 10 shows the damage patterns for cases exp,G and exp, K. According

Figure 10.

Damage patterns in Brazilian test.

to the assumptions included in Table 4, the domination of damage paramter ωK in case exp, G and inversely κG —in case exp, K is visible. The splitting effect observed for case exp, G is expected in the Brazilian test and the coincidence with the decrease of νω during the process (not shown in this paper) is understandable for concrete. However, the problem with negative values is still present, but the aim in our computations is to observe how sensitive the model is

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to the change of parameters, even if one strain measure approach is used. 6

CONCLUSIONS

Two proposals of gradient enhancement for isotropic damage with the volumetric-deviatoric split are formulated and examined. The option with one strain measure assumes one damage loading function, one averaging equation and two different damage growth functions. Then two-field finite element formulation as for the scalar gradient damage is obtained. A more general approach introduces two damage loading functions, two different averaging equations and a threefield formulation is derived. One of the features of the isotropic models is that evolving Poisson’s ratio is simulated, which is characteristic for degrading quasibrittle materials. It is questionable whether negative values of this ratio should be admitted in the simulated process. In the proposed models this is not excluded, hence only by means of appropriate values of input parameters negative Poisson’s ratio can be avoided. However, this problem can be removed if the model is implemented with restrictions similar to the ones presented in (Ganczarski and Barwacz 2004). REFERENCES Askes, H., J. Pamin, and R. de Borst (2000). Dispersion analysis and element-free Galerkin solutions of secondand fourth-order gradient-enhanced damage models. Int. J. Numer. Meth. Engng 49, 811–832. Carol, I., E. Rizzi, and K. Willam (2002). An ‘extended’ volumetric/deviatoric formulation of anisotropic damage based on a pseudo-log rate. Eur. J. Mech. A/Solids 21(5), 747–772. Chen, W.F. and T.Y.P. Chang (1978). Plasticity solutions for concrete splitting tests. ASCE J. Eng. M Div. 104(EM3), 691–704. Comi, C. (2001). A non-local model with tension and compression damage mechanisms. Eur. J. Meek A/Solids 20(1), 1–22.

Comi, C. and U. Perego (2001). Numerical aspects of nonlocal damage analyses. Revue europenne des lmentsfinis 10(2-3-4), 227–242. de Vree, J.H.P., W.A.M. Brekelmans, and M.A.J. van Gils (1995). Comparison of nonlocal approaches in continuum damage mechanics. Comput. & Struct. 55(4), 581–588. Ganczarski, A. and L. Barwacz (2004). Notes on damage effect tensors of two-scalar variables. Int. J. Damage Mechanics 13, 287–295. Geers, M.G.D. (1997). Experimental analysis and computational modelling of damage and fracture. Ph.D. dissertation, Eindhoven University of Technology, Eindhoven. Ju, J.W. (1989). On energy-based coupled elasto-plastic damage theories: constitutive modeling and computational aspects. Int. J. Solids Struct. 25(1), 803–833. Ju, J.W. (1990). Isotropic and anisotropic damage variables in continuum damage mechanics. ASCE J. Eng. M 116(12), 2764–2770. Lubliner, J., J. Oliver, S. Oller, and E. Ofiate (1989). A plastic-damage model for concrete. Int. J. Solids Struct. 25(3), 299–326. Mazars, J. (1984). Application de la mecanique de l’edommagement au comportement non lineaire et a la rupture du beton de structure. Ph.D. dissertation, Universite Paris 6, Paris. Mazars, J. and G. Pijaudier-Cabot (1989). Continuum damage theory—application to concrete. ASCE J. Eng. Mech. 115, 345–365. Pamin, J. (2004). Gradient-enhanced continuum models: formulation, discretization and applications. Series Civil Engineering, Monograph 301, Cracow University of Technology, Cracow. Peerlings, R.H.J., R. de Borst, W.A.M. Brekelmans, and J.H.P. de Vree (1996). Gradient-enhanced damage for quasi-brittle materials. Int. J. Numer. Meth. Engng 39, 3391–3403. Peerlings, R.H.J., R. de Borst, W.A.M. Brekelmans, and M.G.D. Geers (1998). Gradient-enhanced damage modelling of concrete fracture. Mech. Cohes.-frict. Mater. 3, 323–342. Winnicki, A., C.J. Pearce, and N. Bicanic (2001). Viscoplastic Hoffman consistency model for concrete. Comput. & Struct. 79, 7–19. Wosatko, A. (2008). Finite-element analysis of cracking in concrete using gradient damage-plasticity. Ph.D. dissertation, Cracow University of Technology, Cracow.

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Advances in numerical methods

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Continuous and discontinuous modeling of cracks in concrete elements J. Bobi´nski & J. Tejchman Gdansk University of Technology, Gdansk, Poland

ABSTRACT: The paper presents results of numerical simulations of concrete elements using two different approaches to model cracks. First, the cracks were modeled in a smeared way by defining an elasto-plastic, a continuum damage or a smeared crack model. In elasto-plasticity, a Rankine criterion was used. The degradation of the stiffness in the second model was described as a scalar variable of a equivalent strain measure. A rotating crack model and a multi-fixed orthogonal crack model were also used. To ensure mesh-independent results, all these models were enriched by a characteristic length of a micro-structure using a non-local theory. Second, the cracks were simulated as discontinuities with the aid of cohesive elements. Two benchmark problems of concrete elements under mixed mode conditions (simultaneous occurrence of the failure mode I and II) were examined: a Nooru-Mohamed test (1992) and Schlangen (1993) test. The obtained numerical results were compared with the corresponding experimental ones.

1

INTRODUCTION

Concrete belongs to quasi-brittle materials which exhibit a gradual decrease of strength with increasing strain after the peak. It is a highly heterogeneous and non-linear material. A choice of the constitutive model, the knowledge about its limitations and drawbacks is essential to obtain results that agree with experimental outcomes. This problem is especially important when the formation and evolution of cracks is considered. There are two main approaches to simulate cracks in concrete within continuum mechanics. The first one simulates cracks in a smeared sense, i.e. it assumes a finite width of cracks by using elasto-plasticity, damage mechanics or a smeared crack approach. These continuum models have to be enriched in a softening regime by a characteristic length of micro-structure to properly capture strain localization, to restore the well-posedness of a boundary value problem and to obtain objective numerical solutions. The alternative options using e.g. cohesive (interface) elements analyze cracks as zero width discontinuities while the material remains continuous in the remaining region. The more advanced numerical techniques utilize a strong discontinuity approach or XFEM. The concrete can be also modeled as a set of discrete bonded grains by DEM. The behaviour of concrete always follows a continuous-discontinuous failure description. At the beginning of failure, a localized fracture zone of diffused micro-cracks with a defined thickness is created and next a discrete crack is formed with increasing strain. Thus, a realistic constitutive concrete model should reflect a finite width of the fracture zone at the

beginning and its discrete nature at the end of the cracking process. The aim of our paper is to present FE results of concrete specimens under quasi-static conditions obtained with different continuum constitutive laws (Marzec et al. 2007, Majewski et al. 2008, Bobi´nski & Tejchman 2008). The continuum models (elastoplastic, damage mechanics, smeared crack) were enriched by a integral non-local theory. In the case of an elasto-plastic model, a Rankine criterion in a tensile regime was adopted. Within damage mechanics, different definitions of the strain measure were used (by Rankine and modified von Mises). In the case of smeared cracks, a rotational and fixed crack model were applied. Alternatively, cohesive elements were used to simulate a crack formation. In interface elements, a damage constitutive relationship between the traction vector and separation vector (both quantities with normal and shear terms) was assumed. All models were implemented into the commercial FE-code Abaqus/Standard. Two benchmark problems with curved cracks in concrete were carefully analyzed: a Nooru-Mohamed (1992) test and a Schlangen (1993) test. The FE results were compared with corresponding laboratory outcomes. Advantages and disadvantages of constitutive models were outlined. 2 2.1

CONSTITUTIVE MODELS Elasto-plasticity

A Rankine criterion was used with a yield function f defined as: f = max {σ1 , σ2 , σ3 } − σt (κ),

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(1)

where σ1 , σ2 and σ3 = the principal stresses, σt = the tensile yield stress and κ = the hardening/softening parameter (equal to the maximum principal plastic p strain ε1 ). An associated flow rule was assumed. To define the softening under tension, a curve proposed by Hordijk (1991) was chosen: σt (κ) = ft ((1 + A1 κ 3 ) exp(−A2 κ) − A3 κ),

way in a tension-compression regime. The coefficient c reflects the influence of the principal compression stress. With c = 0, Equation (6) is recovered. In the text, Equation (8) will be called a ‘modified Rankine’ definition. Alternatively, a modified von Mises definition in terms of strains was used:   k −1 ε 2 1 12k k −1 Jε I1 + I1 + ε˜ = 2k(1 − 2ν) 2k 1 − 2ν (1 + ν)2 2 (9)

(2)

where A1 , A2 and A3 are defined as A1 =

c1 , κu

A2 =

c2 , κu

A3 =

 1  1 + c13 exp (−c2 ) κu (3)

where I1ε and J2ε = the first invariant of the strain tensor and second invariant of the deviatoric strain tensor, respectively:

with κu = the ultimate value of the softening parameter κ (material constants c1 = 3.0 and c2 = 6.93).

I1ε = ε11 + ε22 + ε33 , 2.2

Damage mechanics

Damage of the material was described by a scalar parameter D growing from zero (undamaged) to one (total damage). The damage variable D acts as a stiffness reduction factor: e εkl , σij = (1 − D) Cijkl

κ(t) = max{˜ε(τ )}.

(5)

τ ≤t

During loading, the parameter κ grew, and during unloading and reloading, it remained constant. To define the equivalent strain measure ε˜ , a so-called Rankine definition was used:

D =1−

2.3

ε˜ =

eff σ1

e εkle . σij = Cijkl

−

E eff

eff



(8)

with σ1 > σ2 and a non-negative coefficient c. This formulation is equivalent to Equation (6) in a tension—tension regime, but it behaves in a different

(13)

Between stresses and cracked strains, the following relation held (in a local coordinate system):

(7)

eff c−σ2

(12)

The stresses were related to the elastic strains by the following traditional relationship:

eff

Due to some numerical problems, Eq. (6) was modified as:

(11)

Smeared crack approach

εij = εije + εijcr .

where E = the modulus of elasticity and σi = the principal values of the effective stress tensor defined as: eff

κ0 (1 − α + αe−β(κ−κ0 ) ), κ

The total strains εij were decomposed into elasticεije and inelastic strains εijcr (coupled with the existence of cracks):

(6)

e σij = Cijkl εkl .

(10)

wherein κ0 = the initial value of the damage threshold parameter κ, and α and β = the material parameters.

eff

max{σi } E

1 1 εij εij − (I1ε )2 . 2 6

The parameter k in Eq. (9) denotes the ratio between the compressive and tensile strength of the material. To describe the evolution of the damage variable D, an exponential softening law was used for all definitions of ε˜ :

(4)

e where Cijkl = the elastic stiffness matrix and εkl = the total strain tensor. The growth of the variable D was controlled by a damage threshold parameter κ which was defined as the maximum of the equivalent strain measure ε˜ reached during the entire load history:

ε˜ =

ε J21 =

cr cr σij = Cijkl εkl

(14)

cr with the secant cracked matrix Cijkl (defined only for open cracks). To simplify the calculations, the matrix cr was defined as a diagonal one. A crack was creCijkl ated when the maximum tensile stress σ1 exceeded the tensile strength ft . To define material softening in a normal direction under tension, a Hordijk (1992) curve was adopted again. In this model, the parameter

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κu in Eqs. (2)–(3) was replaced by an ultimate value of the cracked strain in tension εnu and the parameter κ by εmax , respectively. The quantity εmax was the largest cracked strain εicr in a i-direction obtained during the entire loading history. By combining Eqs. (12)–(14), the following relationship between stresses and total strains was derived (in a local coordinate system): s σij = Cijkl εkl

separation δi vector (both quantities with normal and shear terms) was assumed: ti = (1 − D) E0 Iij δj

with the penalty (dummy) stiffness E0 and unit tensor Iij . To take into account both the normal and shear terms in the separation vector, an effective displacement was introduced (Camacho & Ortiz 1996):

(15)

s with the secant stiffness matrix Cijkl defined as: s e e e cr −1 e = Cijkl − Cijrs (Crstu + Crstu ) Ctukl . Cijkl

δeff = (16)

s Cijij

(20)

   ft σt (δmax ) = ft exp −β δmax − , E0

(17)

δmax = max δ(τ ),

(22)

where β = the model parameter. 3

NON-LOCAL THEORY

As a regularization technique, a non-local theory was used. It is based on a spatial averaging of tensor or scalar state variables in a certain neighborhood of a given point. In plasticity, softening parameters κ were treated non-locally according to the Brinkgreve’s (1994) proposal:  κ(x) ¯ = (1 − m)κ(x) + m

(18)

α0 (x − ξ )dκ(ξ )dξ  , α0 (x − ξ )dξ (23)

where κ¯ = the non-local softening parameter, m = the coefficient greater than one, x = the coordinates of the considered (actual) point, ξ = the coordinates of the surrounding points and α0 = the weighting function. As the weighting function α0 , the Gauss distribution was assumed:

Cohesive elements

Cracks were simulated as the discontinuities in a bulk continuum using cohesive elements. These elements were defined at the edges between standard finite elements to nucleate cracks and propagate them following the deformation process. They governed the separation of crack flanks in accordance with irreversible cohesive laws. The bulk elements were modeled as linear elastic ones. In the interface elements, a simple damage constitutive relationship between the traction vector ti and

(21)

with:

where εsu = the ultimate value of the cracked strain in shear (usually εsu = εnu ) and p = the material parameter. 2.4

δn 2 + ηδs2

τ ≤t

The second formulation allowed one a creation of three mutually orthogonal cracks in 3D-problems (2 cracks in plane simulations, respectively). The orientation of the crack was described by its primary inclination at the onset, i.e. the crack could not rotate during loading. A shear modulus G was reduced by a shear retention factor β which was constant (usually β = 0.2) or it decreased with increasing strains across the crack. Here, the power law proposed by Rots and Blaauwendraad (1989) was chosen:   εcr p , β = 1− i εsu



with a coefficient η. To describe softening after the cracking, an exponential law was assumed (Camacho & Ortiz 1996):

After cracking, the isotropic elastic stiffness matrix was replaced by the orthotropic one. Two formulations were investigated: a rotating crack model and a multi-fixed orthogonal crack model. In the first approach, only can crack was created which could rotate during deformation. To keep the principal axis of total strains and stresses aligned, the secant stiffness coefficient was calculated according to: σii − σjj = . 2(εii − εjj )

(19)

r 2 1 α0 (r) = √ e−( l ) , l π

(24)

where r = x − ξ  = the distance between points x and ξ , and l = the characteristic length of microstructure. It should be noted that the averaging was restricted only to a small area around each material point; the influence of points at the distance of r = 3 × l is only of 0.01%.

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In a damage model, a local definition of the equivalent strain measure ε˜ from Eqs.(6) or (9) was replaced in Eq. (5) by its non-local counterpart ε¯ :  α0 (x − ξ )˜ε(ξ )dξ ε¯ (x) =  . (25) α0 (x − ξ )dξ In a smeared crack approach, a constitutive law in the form: s (¯εkl )εkl σij = Cijkl

(26)

was adopted following the proposal by Jirásek and Zimmerman (1998). The symbol ε¯ kl denotes the nonlocal total strain tensor defined as (independently for all tensor components):  α0 (x − ξ ) εkl (ξ ) dξ  ε¯ kl (x) = . (27) α0 (x − ξ ) dξ s was calculated from Thus, the secant matrix Cijkk the non-local strain tensor.

4

NOORU-MOHAMED TEST

4.1 Problem A double-edge notched (DEN) specimen under various different loading paths of combined shear and tension was analyzed (Nooru-Mohamed (1992)). The dimensions of the largest specimen and boundary conditions are presented in Fig. 1. The length and height of the element was 200 mm. The thickness was 50 mm. Two notches with dimensions of 25 × 5 mm2 were placed in the middle of the vertical edges. The loading

was prescribed by rigid steel frames glued to concrete. During one of the loading paths (number 4), a shear force Ps was applied until it reached a specified value, while the horizontal edges were free. At the second stage, the shear force remained constant and vertical tensile displacement was prescribed. In the experiment, two curved cracks with an inclination depending on the shear force (for a small value of Ps —almost horizontal, for a large value of Ps —highly curved) were obtained. The following elastic material parameters were chosen in the FE-analyses: E = 32.8 GPa and v = 0.2. A FE-mesh was composed of 12600 3-node triangle finite elements. 4.2

FE-results within elasto-plasticity

The tensile strength was assumed as ft = 2.6 MPa and parameter κu = 0.033. The characteristic length was equal to l = 1 mm and the non-local parameter was m = 2. Figure 2 presents the obtained FE results with the shear force Ps = 5 kN (path 4a). A very good agreement was achieved with respect to both the forcedisplacement curve and strain localization, although the calculated maximum force P was too large. The FE-results for the path 4b (Ps = 10 kN) are shown in Fig. 3. The force-displacement curve is satisfactorily reproduced. Two curved localized zones

Figure 1. Geometry of DEN specimen (Nooru-Mohamed (1992)) (dimensions in mm).

Figure 2. The force-displacement curves and the contour map of non-local parameter κ¯ for shear force Ps = 5 kN within elasto-plasticity (l = 1 mm, κu = 0.033).

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Figure 3. The force-displacement curves and contour map of non-local parameter κ¯ for shear force Ps = 10 kN within elasto-plasticity (l = 1 mm, κu = 0.033).

Figure 4. The force-displacement curves and contour map of parameter D for shear force Ps = 10 kN within damage mechanics (Rankine definition, Eq. 6).

were numerically obtained again, but they were too flat as compared to the experiment (wherein they were more curved and the distance between them was larger). 4.3

FE-results within damage mechanics

First, the Rankine definition of the equivalent strain measure ε˜ was used, Eq. (6). The following material parameters were assumed: κ0 = 7 × 10−5 , α = 0.92, β = 100 and l = 0.5 mm. Figure 4 presents results at the shear force Ps = 10 kN. The force-displacement curve overestimated the maximum value, and its slope after the peak was too high. One curved localized zone was numerically obtained instead of two experimental ones. To improve the behaviour of the damage model, the modified Rankine definition (Eq. (8)) was also used. The coefficient c was taken as 0.15. The pattern of localized zones and force-displacement diagram reflect the experimental results much better than those with a standard Rankine definition (Fig. 5). Afterwards, the modified von Mises definition was used with the same set of parameters as for the Rankine definition. The coefficient k in Eq. (9) was equal to 10. Both the force-displacement curve and pattern of localized zones are in good agreement with the experiment (Fig. 6). 4.4

FE-results with smeared crack model

The characteristic length was equal to l = 2 mm. The tensile strength was taken as ft = 2.2 MPa and the

Figure 5. The force-displacement curves and contour map of parameter D for shear force Ps = 10 kN within damage mechanics (modified Rankine definition, Eq. 8).

ultimate normal crack strain ε nu = 0.02. For fixed crack models, the shear retention parameters were assumed as: εsu = 0.02 and p = 8. Figure 7 shows results obtained with a multi-orthogonal fixed crack model. The force-displacement curve was reproduced quite well. One obtained two straight localized zones similarly as in the experiment.

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Figure 6. The force-displacement curves and the contour map of the parameter D for shear force Ps = 10 kN for damage model with modified von Mises definition.

Figure 7. The force-displacement curves and the contour map of the εmax for shear force Ps = 5 kN for multi-fixed orthogonal crack model.

The calculations with a rotating crack model were also carried out, but they were not successful. Serious numerical problems with convergence took place shortly after the peak. The obtained localization pattern

Figure 8. The force-displacement curves and deformed specimen for shear force Ps = 5 kN using cohesive elements.

Figure 9. The force-displacement curves and deformed specimen for shear force Ps = 10 kN using cohesive elements.

was similar to that obtained with a damage model using a Rankine definition of the equivalent strain measure (one horizontal localized zone or two almost horizontal localized zones were created).

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4.5

FE-results with cohesive elements

The following parameters were assumed: ft = 2.2 MPa, η = 0.0 and β = 30000. Figures 8 and 9 show the results obtained for a shear force Ps equal to 5 kN and 10 kN, respectively. In both cases, a very good agreement was obtained between experimental and numerical crack patterns and force-displacement curves.

5

SCHLANGEN TEST

5.1 Problem A single-edge notched (SEN) concrete beam under four-point shear loading (anti-symmetric loading) was analyzed (Schlangen, 1993), Fig. 10. The length and height of the beam were equal to 440 mm and 100 mm, respectively. The depth of the notch was equal to 20 mm. The thickness was 5 mm. In the experiments, a curved crack starting from the lower-right part of the notch towards a point to the right of the lower right support was obtained. A FE-mesh consisted of 6556 3-node triangle finite elements. The Young modulus was taken as E = 35 GPa and the Poisson ratio as ν = 0.2. The deformation was induced by linearly increasing the distance δ2 (due to the snap-back behaviour of vertical displacements at the points where the forces were applied) (Fig. 10). Therefore, a so-called indirect displacement control method was implemented into the program Abaqus Standard. It required the definition of the second independent mesh (the same as the basic one) to collect the needed data and to calculate the loading factor (Bobi´nski et al. 2009). 5.2

Figure 11. The force-displacement curves and contour map of non-local parameter κ¯ in central part of beam within elastoplasticity (l = 1 mm, κu = 0.04).

FE-results within elasto-plasticity

Figure 11 demonstrates results obtained with the ela-sto-plastic model (ft = 3 MPa, κu = 0.040 and l = 1 mm). The localized zone was curved and its shape matched well the experiment. A satisfactory agreement between a numerical and experimental force-displacement diagram was also achieved.

Figure 12. The force-displacement curves and contour map of damage parameter D in central part of beam within damage mechanics (with Rankine equivalent strain definition, Eq. 6).

5.3

Figure 10. Geometry of SEN specimen (Schlangen, 1993) (dimensions in mm).

FE-results within damage mechanics

The numerical calculations were performed with the Rankine definition (Eq. (6)) of the equivalent strain measure. The following material parameters were assumed: l = 1 mm, κ0 = 8.5 × 10−5 , α = 0.92 and

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shown that the influence of the material description in the tensile-compression regime has to be taken into account. A fixed crack model was not able to reproduce curved cracks. The worst results were obtained with a rotating crack model. In turn, the approach with cohesive elements provided the best approximation of experiments. ACKNOWLEDGEMENTS The FE-calculations were performed at the Academic Computer Centre in Gdansk TASK. REFERENCES

Figure 13. The force-displacement curves and contour map of damage parameter D in central part of beam within damage mechanics (with modified von Mises equivalent strain definition).

β = 150. Both the force displacement-diagram and strain localization differ significantly from the experimental outcome (Fig. 12). To obtain a better agreement with the experiment, again the modified von Mises definition (Eq. (9)) of the equivalent strain measure was used with the following parameters: l = 1 mm, κ0 = 8.0 × 10−5 , α = 0.92 and β = 150. Although, the slope of the calculated force-displacement curve was too sharp, the shape of the localized zone was properly reproduced (Fig. 13). 6

CONCLUSIONS

The results of our FE simulations within continuum mechanics of the concrete behaviour under mixed mode conditions have shown that a proper choice of a constitutive law is a very important issue. The models show a different capability to capture the crack phenomenon. In general, the elasto-plastic model with the Rankine failure criterion was the most effective among continuous models. The use-fulness of an isotropic damage model depended on the definition of the equivalent strain measure. Our simulations have

Bobi´nski, J. & Tejchman, J. 2008. Quasi-static crack propagation in concrete with cohesive elements under mixedmode conditions. In B.A. Schrefler, U. Perego (eds.), Proc. 8th World Congress on Computational Mechanics WCCM 2008, Venice. Bobi´nski, J., Tejchman, J. & Gorski, J. 2009. Notched concrete beams under bending–calculations of size effects within stochastic elasto-plasticity with non-local softening. Archives of Mechanics, 61(3–4): 1–25. Brinkgreve, R.B.J. 1994. Geomaterial Models and Numerical Analysis of Softening. PhD Thesis, Delft University of Technology. Camacho, G.T. & Ortiz, M. 1996. Computational modelling of impact damage in brittle materials. Int J Sol Struct, 33(20–22): 2899–2938. Hordijk, D.A. 1991. Local approach to fatigue in concrete. PhD Thesis, Delft University of Technology. Jirásek, M. & Zimmermann, T. 1998. Rotating crack model with transition to scalar damage. J Eng Mech-ASCE, 124(3): 277–284. Majewski, T., Bobi´nski, J. & Tejchman, J. 2008. FE-analysis of failure behaviour of reinforced concrete columns under eccentric compression. Engineering Structures, 30(2): 300–317. Marzec, I., Bobi´nski, J. & Tejchman, J. 2007. Simulations of crack spacing in reinforced concrete beams using elasticplasticity and damage with non-local softening. Computers and Concrete, 4(5): 377–403. Nooru-Mohamed, M.B. 1992. Mixed mode fracture of concrete: an experimental approach. PhD Thesis, Delft University of Technology. Rots, J.G. & Blaauwendraad, J. 1989. Crack models for concrete, discrete or smeared? Fixed, Multi-directional or rotating?. Heron, 34(1): 1–59. Schlangen, H.E.J. 1993. Experimental and numerical analysis of fracture processes in concrete. PhD Thesis, Delft University of Technology, 1993.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Topological search of the crack path from a damage-type mechanical computation Marina Bottoni Électricité de France, Research and Developement, Clamart, France

Frédéric Dufour Grenoble Institute of Technology, Grenoble, France

ABSTRACT: In this paper, we present a method to track the crack path from a continuum finite element simulation using non linear material models. The work belongs to a larger project, aiming at extracting the crack opening from the results of a computation performed by means of a regularized damage model, where a damaged zone represents the crack in a diffuse manner. In the developed algorithm, we look at the distribution of the history variable rather than the internal variable, since it is not limited to 1 and thus its profile is sharper. Then, we find the crack path by means of a step-by-step procedure. At each step, a new point is found as the location where the history variable is maximum at a given direction and distance. The algorithm makes use of some parameters, whose range of use is discussed and calibrated. Finally, we applied the method to two examples and the crack path is automatically found along the ridge of the studied variable. 1

INTRODUCTION

A correct evaluation of crack opening is important in many concrete structural applications. For some type of structure, crack opening estimation is required in relation with permeability issues, in order to limit gas or liquids releases. Examples are confinement vessels and cooling towers of nuclear power plants, dams and silos. In other cases, where concrete is exposed to environmental agents, such as for bridges, the notion of crack opening is necessary to assess durability in the sense of penetration of external chemical aggressive agents or simply for aesthetic reasons. Another approach to assess structural durability is the use of a matching law which relates material permeability with damage variable (Pijaudier-Cabot et al. 2009). Even in this method a simple analytical relation between crack opening and damage has been used and can be improved. Finite elements and continuum damage models are an useful and widespread method to compute concrete structures in the nonlinear range. They are often employed together with nonlocal (Pijaudier-Cabot and Bažant 1987) or gradient enhanced models (Peerlings et al. 1998), since conventional continuum damage descriptions suffer from ill-posedness beyond a certain level of damage. Some other computational methods seem to be more suited to compute crack opening, such as finite elements with re-meshing of the crack (Askes and Sluys 2000) and methods incorporating discontinuities, such as the X-FEM (Belytschko

and Black 1999; Moës et al. 1999). However, the application of the first one is limited by excessive computational time, whereas the use of the second one is limited to crack propagation only. Another method consists in combining the continuum formulation for the bulk response and a discrete cohesive model to represent the crack formation, see for example (Simo and Oliver 1994; Oliver 1996). However, this approach has intrinsic difficulties by establishing the transition point between the two kind of modelisations, in terms of stresses, damage or strain equivalence (Comi et al. 2007; Simone et al. 2003). Our approach consists in the extraction of the opening directly from a damage computation, as detailed by Dufour and co-workers (Dufour et al. 2008). This procedure is able to predict crack opening, if the crack path is a priori known. The aim of this paper is to obtain the line representing the crack path in automatic way. We restrict ourself to 2D domains with a single fracture process zone FPZ. Coherently with damage mechanics, a crack is represented by a damage distribution. The challenge is to develop a procedure which extract a 1D geometry, i.e. the crack path, from a 2D space, i.e. the numerical FPZ. However, damage as internal variable is limited to the value of 1, resulting in an undifferentiated zone where it takes the same value. Hence, we assumed the history variable (maximum value taken by the equivalent strain during loading) as representative of the crack, since this quantity has no upper limit. We propose here a procedure to extract this discrete

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information from the distribution of the history variable. The main idea is to identify the crack path with the ‘‘ridge’’ of this distribution on the process zone. One of the advantages of our algorithm, is that it is applied as a post-processing and not incrementally at each time-step of the simulation, since informations about the loading history are incorporated in the history variable. In this paper, we remind firstly basic notions of regularized damage models. Then, we explain the algorithm in details. In the third part, we apply it to an artificial result in order to study the sensitivity to parameters. Finally, we test it to a more complex artificial result and to one mechanical simulation.

loses mesh-objectivity. In literature, there are different regularization techniques to overcome this problem. In finite elements with C 0 continuity, regularization can be performed, for example, by nonlocal models or implicit gradient models. In nonlocal models (Pijaudier-Cabot and Bažant 1987; Jirásek and Bažant 2002), a non local variable g(ε) is computed by averaging the local g(ε) with a weighting function  defined on a neighbourhood of the point of coordinates x, so that:   (ξ − x) g(ε(ξ )) dξ (4) g(ε(x)) = V  V  (ξ − x) dξ

2

For , a number of functions can be employed; often, a gaussian function is used:     2ξ − x 2 (ξ − x) = exp − (5) lc

2.1

NON LOCAL DAMAGE APPROACH Damage models

In damage models, damage growth is controlled by a loading function f , which can be expressed in terms of stresses or strains. In the following, this function is given in terms of strains, the most common choice to be implemented in finite element codes (strain driven problem). Supposing for simplicity to have a single scalar variable for damage (isotropic damage), the loading function can be written as: f = f (g(ε) − Y )

where lc is a characteristic length. In implicit gradient models (Peerlings et al. 1998; Geers et al. 2000), the regularized variable is related to the local quantity by the following differential relationship and associated boundary conditions: g(ε(x)) − c2 ∇ 2 g(ε(x)) = g(ε(x)) ∇g(ε) · n = 0

(1)

˙ ≥ 0, D

˙ =0 fD

(2)

The choice of function g(ε) is related to the material characteristics. For concrete, usually damage is related to positive principal strains, see among others (Mazars and Pijaudier-Cabot 1989; Desmorat et al. 2007; Papa and Taliercio 1996). The constitutive law can be written as: σ = (1 − D) Eε

(6b)

where n is the vector normal to the boundary and c is the characteristic length of the gradient model. For implementation in a finite element code, Equation 6a is then restated in a weak form. In the nonlocal/gradient enhanced formulation of damage models, the constitutive relationship given by Equation 3 remains unaltered, whereas the evolution of damage depend on regularized quantities: D = D(Y ).

where g(ε) is a function of strains and Y is the history variable, which starts from an initial value Y0 and is, for each time instant, equal to the maximum value taken by g(ε) during the loading history. An evolution law for damage, D = D(Y ), must also be defined. The damage growth is controlled by the Kuhn-Tucker conditions: f ≤ 0,

(6a)

(3)

being E the elastic stiffness tensor. D varies from 0 for a sound material to 1 upon the total loss of integrity. 2.2 Regularization techniques for damage models For many materials, such as concrete or rock, damage models introduce strain softening and consequently the ill-posedness of the continuum. After the domain discretization due to finite elements, the solution

3 3.1

THE NUMERICAL PROCEDURE Damage and history variable: an example

We present in this section an example of the damage and history variable fields to show the difference in their distribution. Fields are issued from a 2D-mechanical simulation of a brazilian test using Mazars’ damage model (Mazars and Pijaudier-Cabot 1989), performed with the finite element code Code_Aster (www.codeaster.org). In Figure 1 it is shown the obtained distribution of damage. The field is very flat in the FPZ since damage is saturated to 1 by definition. For this reason, the extraction of the crack path from this quantity is not possible. On the contrary, the history variable has a much more pronounced ridge, as shown in Figure 2, easier to follow during our procedure.

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interpolation functions are used for displacements and linear interpolation functions for regularized strains. Hence, regularized strains are defined on the apex of elements; all quantities derived from regularized strains, such as the equivalent strain and the history variable Y are then also known on the same nodes and have the same degree of approximation on the finite element. Nonlocal integral models can be applied to finite elements with all possible approximations, but are usually applied to linear elements due to the large size of non-zero components in the connectivity matrix. In the damaged zone, the search consists in a stepby-step procedure, according to which a new point is found at each step. Except for the first two points (initialisation), a starting point and a search direction must always be fixed. At each step, the following actions must be performed (see also the scheme of Figure 3):

1

D

0

0 40

15

30

x Figure 1.

20 10

30

y

0 45

Damage map for a brazilian test.

Figure 2. Y map for a brazilian test.

3.2

The algorithm

A precise definition of the ridge cannot be given; however, this notion lies on some intuitive assumptions. To begin with, there is somehow the idea that the ridge contains the maxima of the field; still, it cannot be easily defined as the geometric one-dimensional locus containing the field maxima, because the field varies strongly along the crack path, from the crack notch to the point where the crack is first activated. The second implicit assumption is that the damaged zone from which the crack is extracted, is mainly distributed in one direction; to that is related the idea of a directional search. Thirdly, it can be stated that the tangent to the crack is continuous; this hypothesis is not verified in true concrete due to heterogeneities, but acceptable in the frame of continuum mechanics. An important clarification is about the discretization of the history variable, its level of approximation depending on the degree of the interpolation polynoms of the adopted finite elements. The examples shown in this contribution (see Section 5) are based on a linear interpolation of the field Y , which is the most common for both regularisation techniques. In fact, equations describing the gradient enhanced method are usually implemented in finite element codes by means of a mixed formulation; quadratic

1. the starting point is the point found in the previous step; 2. the search direction is determined by the last two points found; 3. the next point is first estimated by moving from the starting point in the search direction, at a distance a (‘‘step length’’); actually, this choice for the search direction is related to the assumption of continuous tangent; 4. a line orthogonal to the search direction is traced; 5. the field Y is projected onto this line, by means of the element shape functions; 6. the profile Y (s) obtained by projecting the field is regularized in order to find a curve without discontinuities in the first derivative; in fact, since the history variable field is stepwise linear due to the discretization, the projected field on the orthogonal line also shares the same property. Then, the new point of the ridge is the point where Y (s) is maximum. In this regard, it is easy to understand why the

Figure 3.

Scheme of the topological search procedure.

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regularisation on the orthogonal profile is important. In fact, since Y (s) is stepwise linear, the maxima of the field would be necessarily at an intersection between the orthogonal line and an element edge; the precision in finding the good ridge would be then strongly dependent on mesh refinement. We will see in Section 4 how the precision of the procedure can be improved with respect to mesh refinement. To obtain a smooth curve, a convolution product is used, analogously to the regularization operated by the nonlocal integral model, so to obtain the quantity Y(s) as a function of the curvilinear abscissa s on the orthogonal profile :  Y (s) =

l

(|ς − s|) Y (ς ) dς  l (|ς − s|) dς

Search direction

a 3 (P1 ) 1

Orthogonal profile

r=a Circular profile

(7)

where l is the orthogonal line and a Gaussian function is employed as weight function :     2|ς − s| 2 (ς − s) = exp − lreg

2 (P2)

Figure 5.

Initialisation of the topological search.

(8)

An example of the projected field Y (s) and its regularization Y (s) is shown in Figure 4. As far as the initialisation procedure is concerned, the scheme is shown in Figure 5. The procedure starts from the point where the field Y is maximum. Due to the discretization, this point is located on a node. Hence, an iteration is required in order to set its position again. Moreover, if this node does not belong to the geometry boundaries, a search must take place in two opposite directions. In the following list, we detail the actions to find the first two points: 1. the point with the maximum value of Y is found on the damaged zone; due to the linear interpolation of the field, the maximum is always on a node;

Figure 4. Projected fields Y (s) and Y (s) on the orthogonal profile, obtained by regularization through the convolution product.

Figure 6. Projected fields Y (s) and Y (s) on the circle obtained by regularization through the convolution product.

2. a circle is drawn, with radius equal to the step length a; 3. the field Y is projected onto this circle, by means of the element shape functions; 4. the projected Y (s) field is regularized in order to find a curve without discontinuities in the first derivative. A convolution product is used, analogously to the typical step (see Equations 7 and 8), by integration on the circle. An example of the projected field and its regularization are shown in Figure 6; 5. the second ridge point (P2 ) is the point where Y (s) is maximum; 6. two possible search directions are determined by −−→ −−→ the points P1 and P2 , i.e. P1 P2 and P2 P1 . After the two first ridge points are found, the search is managed as detailed for the typical step.

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The procedure is then stopped in one direction when the identified ridge value is smaller than a given value defined a priori, which is another parameter of the algorithm. However, damage models for brittle or quasi brittle materials already contain a damage threshold in terms of an equivalent strain, below which damage cannot take place. In the present work, the same value is employed as a threshold for the search algorithm, beyond which the crack is not propagated, i.e. no damage has occured. A higher value can eventually be set, in case it is decided that for the corresponding value of damage there is no real crack, but only diffuse microcracks. Furthermore, if the crack path is looked for computing the crack opening then the limit value will be the one below which the opening is computed with an associated error larger than a given limit according to Dufour and co-workers (Dufour et al. 2008). Moreover, there are other situations requiring the search to stop in one direction, that is, when the crack path reaches the material boundary. Obviously, the procedure is arrested when all points on the orthogonal profile lay outside the material, but this condition is very strict and seldom verified; hence, in the current implementation, the procedure is simply stopped if the prediction point is located outside the material.

4

size Lx = 52 and Ly = 32 (the units are not important). Three squared meshes with different refinements have been used. The three meshes are here denoted as coarse, medium and fine; corresponding element sizes are  = 2,  = 1.333 and  = 0.8. The crack path is given by the following equation:

CALIBRATION OF PARAMETERS

y = 10 sin ([0.045 (x + 10)]2 ) + 15

whereas on the orthogonal plane, the field is determined by an inferiorly limited parabola, with the minimum value equal to Y min = 5 · 10−7 : 

10−3 (−t 2 + 10) Y min

Y =

if Y > Y min if Y ≤ Y min

(10)

t is the distance between any point belonging to the line orthogonal and the crack path in the xy plane. The intersections of the parabola with the plane z = Y min determine the numerical fracture process zone width, in this case equal to about 6.32. Hence, there are about 3 elements in the process zone in the case of the coarse mesh, 7 for the fine mesh. The function representing the theoretical crack path (Equation 9) is represented in Figure 7. The curve has remarkable curvatures at points A and B, that is 0.127 in A and 0.382 in B; these values correspond to radii of the osculating circles equal to 7.86 and 2.62 in A and B

On the whole, the procedure requires the calibration of three parameters:

30 A

25

• the search step a; • the length of the orthogonal profile lort ; • the regularisation length lreg ;

(9)

y

20 15 10 5

4.1 Definition of the testing field

0

In order to test the numerical procedure, we have built on a given mesh a numerical field representing the history variable. The aim is to compare the crack path predicted by the algorithm with a theoretical curve, while controlling the parameters defining the field. In order to build the field, first a mesh must be determined on the geometric domain xy. Then, two functions are needed, the first one defining the crack path in the xy−plane, and the second one the values of the field on all nodes of the geometric domain. The second function is defined on all points of the crack path, on the straight line orthogonal to the crack path itself; it has a maximum on the crack path and is always positive, since it represents the history variable. For the calibration of the procedure parameters, we set a constant width for the process zone, and a constant value for the history variable on the ridge. The field is defined on a rectangle in the plane xy, having

B 0

10

20

30 x

40

50

60

Figure 7. Theoretical ridge for the testing of the procedure and parameters calibration.

Figure 8. Numerical field used for parameters calibration; the color of each element facet is associated to average value of the field on the element nodes.

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respectively. We tested the algorithm also for high curvatures, even for values which are hardly found in a real crack. 4.2

Calibration

Search step. At first, the influence of the search step a is studied. In the following simulations, the values of all other parameters are constant: lort = 8, lreg = 0.18lort . In Figure 9, the obtained ridge points are shown together with the theoretical curve for the coarse mesh (element size equal to 2) and for three values of the search step. The algorithm provides points with good correspondence with the initial curve for a large range of a. For big values of the search step, the algorithm find it difficult to follow the real path near point B. Looking closely to the curves of Figure 9, an error can be observed near extremities; the numerical crack path ridge deviates from the theoretical one. This loss of accuracy is due to the fact that there the field Y is projected only onto the part of the orthogonal profile belonging to the geometric domain; the resulting orthogonal profile can be much smaller and asymetrical with respect to the prediction point, thus boundary problems can arise, in analogy of what happens with nonlocal formulations (Peerlings et al. 2001). However, this boundary error remains relatively small and confined in a very small region. Furthermore, looking at a recent work (Krayani et al. 2009), one can conclude that for nonlocal damage simulation cracks are always othogonal to free boundaries. Thus, the symmetry of the orthogonal profile will be maintained. To compare quantitatively the results, the average residue defined in Equation 11 is used as an error indicator: Np 1  · |yi,theo − yi,num | Q= Np i

with Np being the number of points identified by the numerical procedure, yi,num their ordinates on the xy plane, and yi,theo the theoretical ordinates at the same abscissa. In Figure 10, the error is depicted as a function of the search step a and for the three meshes. The points near the crack path extremities are not taken into account, in order to exclude the boundary effect. The error remains nearly constant for steps going from 0.4 to 2.6; then, from 2.6 the error increases rapidly. However, if the geometric domain of the field is reduced to x ≤ 24, so excluding the area with the strong curvature near point B, the error remains of the same order of magnitude until 4, the last tested value of the search step. We can conclude that the step length must be smaller than the radius of the osculating circle (equal to 2.62 in B) in order to well approximate the curve. A final remark is about the behaviour of the algorithm for different meshes. In fact, the error depends strongly on the mesh refinement, as pointed out again on Figure 10. On the contrary, the value of the step from which the algorithm cannot follow the theoretical path is the same for the three meshes since it depends on the geometrical properties of the path (curvature).

(11)

Figure 9. Crack paths identified by the developed algorithm compared with the theoretical path: results for different values of the search step, coarse mesh.

Length of the orthogonal profile. Figure 11 shows the error for different values of the orthogonal profile lort . The other parameters are constant and have the following values: a = 1 and r = 0.28. As seen for the calibration of the step length, the error depends on the refinement. For lort bigger than a certain value, which depends on mesh refinement, the error is minimum for that refinement and more or less constant; this value is about 4−5 times the finite element size. It is anyway recommended to adopt a value larger than the FPZ width. Regularization length. Figure 12 shows the error defined by Equation 11 for different values of the

Figure 10. Error for different values of the search step: results for the three meshes.

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near extremities are excluded. As remarked by the calibration of the search step and of the length of the orthogonal profile, the error depends on mesh refinement. A loss of accuracy can be observed for all three meshes when r is bigger than 0.52; the lower limit of the optimal range depends slightly on the refinement. This fact is easily explained by noticing that for lreg → 0, the regularized curve Y (s) coincides with the original, stepwise linear curve Y (s), so that the points of the crack path are on the element edges due to the linear discretization of the field. In Figure 13, the obtained crack paths for three values of r are shown. For larger values of r, the curvature at point B is caught less precisely. Figure 11. Error for different values of the length of the orthogonal profile: results for the three meshes.

5 5.1

EXAMPLES Artificial result

We first tried our procedure onto an artificial result, obtained as described in Section 4.1. The crack path is the fourth order polynom running through points P1 = [10, 0], P2 = [35, 20] P3 = [60, 0], P4 = [85, 20], P5 = [110, 0]. We modulated the field along the crack path assuming for the parabola y = a1 x2 + a2 on the orthogonal profile two coefficients a1 ,a2 , varying with x with equation: a1 · 10−6 = −60 cos (0.1x) + 100 a2 · 10 Figure 12. Error for different values of the regularisation length: results for the three meshes.

−6

= 500 cos (0.3x) + 1000

(12a) (12b)

The obtained field is shown in Figure 14. The identified crack path is shown in Figure 15; for the search procedure, we used the following parameters: a = 2, lort = 20, lreg = 0.18lort . The procedure follows the ridge with accuracy also when the fracture process zone width and the value of the field on the ridge have strong variations along the crack path. 5.2

Double-notched specimen in tension

In this section, the procedure is applied to the results of a F.E. mechanical simulation, performed with the

Figure 13. Crack paths identified by the developed algorithm compared with the theoretical path: results for different values of the regularization length, medium mesh.

regularization length lreg and for the three meshes. The error is represented as a function of the ratio between the regularization length and the length of the orthogonal profile, r = lreg /lort . The other parameters are constant and are: a = 1 and lort = 8. The points

Figure 14.

Artificial results for testing of the procedure.

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Figure 15.

means of a non local integral model. For the presented FE-simulation, we made use of the following parameters: Young modulus E = 31000 MPa, Poisson ratio ν = 0.2, damage threshold εd0 = 0.000097, At = 0.95, Bt = 9000, Ac = 1.25, Bc = 1000; the internal lenght of the nonlocal model is lc = 0.0216 m. The distribution of damage on the specimen at the end of the computation is shown in Figure 17. A single damaged zone is observable in the center of the specimen. Nevertheless, the simulation predicts two distinct cracks developing from the notches. These two cracks are still recognizable from two distinct zones in the history variable map (see Figure 17). Since the algorithm applies to one crack at a time, the specimen has been divided into two separated zones by tracing an

Crack path for the manifactured results.

thickness = 10 mm

2 mm 120 mm

5 mm

40 mm 10 mm

10 mm

60 mm

(a)

(b)

Figure 16. (a) geometry and (b) boundary conditions of the mechanical test.

software Cast3M (www.cast3m.cea.fr). Geometry and boundary conditions are shown in Figures 16a, b. For the analysis of the double-notched specimen, the Mazars damage law has been employed. According to this law, the damage (scalar) variable D is a combination of two components: Dt which is damage due to tension and Dc which is damage due to compression: D = αt Dt + αc Dc

(13)

where αt and αc depend on the strain tensor. Damage in tension is: Dt = 1 −

εd0 (1 − At ) At − Y exp (Bt (Y − εd0 ))

Figure 17. Scalar map of damage on the double-notched specimen in tension.

(14)

where εd0 is the damage threshold (Y0 in Section 2.1), At , Bt are other two material parameters. Damage in compression Dt is calculated in the same way by using material parameters Ac , Bc instead of At , Bt . For this law, function g(ε) defined in Section 2.1 is an equivalent strain εeq , equal to: εeq



3

 =  ε 2

i +

(15)

i=1

εi + denotes the positive part of the principal strain εi . Strain regularization has been performed by

Figure 18. Scalar map of the Y field and crack path for the double-notched specimen in tension: zoom on the central part of the specimen.

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horizontal line at the center of the specimen. Then, the procedure has been run onto the two zones in order to find the two cracks. In Figure 17, the points of the cracks path identified by our algorithm are plotted. For the search, we used the following parameters: a = 0.001 m, lort = 0.015 m, lreg = 0.1lort . 6

CONCLUSIONS

In this paper, we describe a method to track the crack path from a finite element computation making use of a regularized damage model. The main advantage is to use continuum models and to apply the tracking algorithm as a post-processing, so without making the simulation computationally more expensive. The point is to perform a topological search, so to find the ‘‘ridge’’ of the history variable field, which is less flat than the damage field. The notion of crack direction is essential for the search; moreover, the regularization of the history variable field decreases appreciably the error on the crack path. The method makes use of some parameters, such as the search step, the regularization length and the length of the orthogonal profile. The method performs well, working for a wide range of the parameters. More into details, the search step must be smaller of the minimum radius of the osculating circle; the length of the orthogonal profile must be bigger than about 5 times the finite element size and of the fracture process zone width; the regularization length has an optimal range depending slightly on mesh refinement and ranged between about 0.2 and 0.52. The total error on the crack path depends on mesh refinement too. We have also tested the procedure on an artificial result, which shows a strong curvature of the crack path and a modulation on the ridge value and of the numerical FPZ width. The other application is on the results of a mechanical simulation, performed with an integral nonlocal model on a double-notched specimen loaded in tension. In both applications, the algorithm has managed well to find the crack path, even with a rather coarse mesh. REFERENCES Askes, H. and L.J. Sluys (2000). Remeshing strategies for adaptative ale analysis of strain localisation. European Journal of Mechanics A/Solids 19(3), 447–467. Belytschko, T. and T. Black (1999). Elastic crack growth in finite elements with minimal remeshing. Int. J. Num. Methods Eng. 45, 601–620.

Comi, C., S. Mariani, and U. Perego (2007). An extended fe strategy for transition from continuum damage to mode I cohesive crack propagation. International Journal for Numerical And Analytical Methods in Geomechanics 31, 213–238. Desmorat, R., F. Gatuingt, and F. Ragueneau (2007). Nonlocal anisotropic damage model and related computational aspects for quasi-brittle materials. Engineering Fracture Mechanics 74(10), 1539–1560. Dufour, F., G. Pijaudier-Cabot, M. Choinska, and A. Huerta (2008). Extraction of a crack opening from a continuous approach using regularized damage models. Computers and Concrete 5(4), 375–388. Geers, M., R. Peerlings, W. Brekelmans, and R. de Borst (2000). Phenomenological nonlocal approaches based on implicit gradient-enhanced damage. Acta Mecanica 144, 1–15. Jir´asek, M. and Z.P. Bažant (2002). Non local integral formulations of plasticity and damage: survey of progress. J. Eng. Mech. 128(11), 1119–1149. Krayani, A., G. Pijaudier-Cabot, and F. Dufour (2009). Boundary effect on weight function in non-local damage model. 76, 2217–2231. Mazars, J. and G. Pijaudier-Cabot (1989). Continuum damage theory—Application to concrete. J. Eng. Mech. Div. 115(2), 345–365. Moës, N., J. Dolbow, and T. Belitschko (1999). A finite element method for crack growth without remeshing. Int. J. Num. Methods Eng. 46, 131–150. Oliver, J. (1996). Modelling strong discontinuities in solids mechanics via strain softening discontinuities equations. Part 1: Fundamentals. Int. J. Numeric. Methods Eng. 39(21), 3575–3600. Papa, E. and A. Taliercio (1996). Anisotropic damage model for the multiaxial static and fatigue behaviour of plain concrete. Engineering Fracture Mechanics 55(2), 163–179. Peerlings, R., R. de Borst, W. Brekelmans, and M. Geers (1998). Gradient-enhanced damage modelling of concrete fracture. Mechanics of Cohesive-Frictional Materials 3, 323–342. Peerlings, R., M. Geers, R. de Borst, and W. Brekelmans (2001). A critical comparison of nonlocal and gradientenhanced softening continua. IJSS 38, 7723–7746. Pijaudier-Cabot, G. and Z. Bažant (1987). Nonlocal damage theory. ASCE Journal of Engineering Mechanics 113(2), 1512–1533. Pijaudier-Cabot, G., F. Dufour, and M. Choinska (2009). Permeability due to the increase of damage in concrete: from diffuse to localised damage distributions. 135(9), 1022–1028. http://dx.doi.org/10.1061/(ASCE)EM.1943 7889.0000016. Simo, J. and J. Oliver (1994). A new approach to the analysis and simulation of strong discontinuities, pp. 25–39. E&FN Spon. Simone, A., H. Askes, R. Peerlings, and L. Sluys (2003). Interpolation requirements for implicit gradientenhanced continuum damage models. Commun. Numer. Meth. Engng 19, 563–572.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

On the uniqueness of numerical solutions of shear failure of deep concrete beams: Comparison of smeared and discrete crack approaches J. Cervenka & V. Cervenka Cervenka Consulting Ltd., Prague, Czech Republic

ABSTRACT: Numerical modeling of shear failure of reinforced concrete beams with or without shear reinforcement still remains a challenging task even after several decades of active research. The paper concentrates on the shear failure analysis of large beams without and with shear reinforcement. As an example, it uses the experiments performed by Collins and Yoshida (2006). The paper discusses the main issues affecting the reliability of shear strength predictions and will evaluate the effectiveness of the discrete and smeared crack approaches in resolving the numerical problems in shear failure modeling of strain softening materials. 1

INTRODUCTION

Numerical models for brittle materials such as concrete were for the first time introduced already in the 70’s by landmark works of Ngo & Scordelis (1967), Rashid (1968) and Cervenka V. & Gerstle (1971). Many material models for concrete and reinforced concrete were developed in 70’s, 80’s and 90’s such as for instance the models by Suidan & Schnobrich (1973), Lin & Scordelis (1975), De Borst (1986), Rots (1989), Pramono & Willam (1989), Etse (1992) or Lee & Fenves (1998). These models were based on the finite element method. A concrete material model was formulated as a special constitutive model that is used at each integration point for the evaluation of internal forces. It was soon realized that material models with strain softening, if not formulated properly, exhibit severe mesh dependency (De Borst & Rots 1989), and tend to zero energy dissipation if the element size is reduced (Bažant 1976). This was attributed to the local nature of the constitutive material description, which results in the loss of hyperbolicity of the governing differential equation in the softening region (Belytschko et al. 1986). This deficiency means that mathematically a solution can be found, but its uniqueness cannot be guaranteed. In numerical analysis this results in mesh sensitivity and/or numerical instabilities such as convergence problems. The crack band approach was proposed by Bažant and Oh (1983) to remedy the convergence towards ˇ zero energy dissipation. It was shown by Cervenka V. (1995) that proper formulation of the crack band size can severely reduce also the mesh bias of these smeared crack approaches. A more rigorous solution of the ill-posed nature of the strain softening problem is the introduction of higher-order continuum models: such as non-local

damage model by Bažant & Pijaudier-Cabot (1987), gradient plasticity model by de Borst & Muhlhaus (1992) or gradient damage model by de Borst et al. (1996). The non-local models introduce additional material parameters related to an internal material length scale, which is however difficult to derive from existing material tests. Currently these models are mathematically rigorous, but appear to be too fundamental for practical applications. Another solution for the strain softening problem is the discrete crack model, where the discontinuities arising from strain localization are directly included into the numerical model. This model was first introduced with automatic remeshing and crack propagation by Saouma & Ingraffea (1981). In the classical form of the discrete crack approach a crack is simulated as a cohesive interface, which is inserted into the finite element model whenever a certain criterion for crack initiation or propagation is satisfied. This means that whenever a crack is initiated or an existing crack propagates remeshing is necessary. In recent years, a whole new class of methods has emerged based on enhanced finite element formulations or various mesh free methods. A comprehensive treatise on these approaches is for instance provided in de Borst et al. (2003) or Jirásek (2003). The mesh free method was proposed by Belytschko et al. (1994). This method has a strong potential for solving crack propagation problems, but it contains still many open issues such as large computational demand, difficulties in 3D implementation and the need to use a background mesh for the numerical integration. The enhanced finite element formulations such as EED (elements with embedded discontinuities) or X-FEM (extended finite elements) are nicely classified by Jirásek (2003). These models attempt to bridge the gap between the discrete and smeared models by enhancing the finite element formulation to

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better capture the discontinuity arising from the strain localization. Their basic idea is however very close to the classical discrete approach, where the ‘‘traditional’’ remeshing is replaced by enhancing the finite element formulation in the elements where cracking occurs. Similarly to the classical discrete crack approach it is often necessary to trace the crack propagation through the model, guarantee its continuity across element boundaries, solve crack branching, crack intersection etc. The paper discusses the main issues affecting the reliability of shear strength predictions. Second objective is to evaluate the effectiveness of the discrete approach in resolving the ill-posed nature of the underlying mathematical problem; since it is often believed that introduction of discontinuities into the numerical formulation can resolve this issue. The reliable prediction of shear strength in real structures requires a rather complex material formulation, which should capture at least the most important features of concrete behavior, such as: compressive crushing, compressive softening, shear response of cracked concrete and reinforcement yielding. The paper evaluates the importance of these factors on the reliability of shear strength predictions.

2

FRACTURE-PLASTIC MATERIAL MODEL

The smeared crack analyses presented in this paper ˇ were performed with program ATENA (Cervenka et al. 2009) using the combined fracture-plastic model ˇ of Cervenka & Pappanikolaou (2008). The material model formulation assumes small strains, and is based on the strain decomposition into p f elastic (εije ), plastic (εij ) and fracture (εij ) components. The stress development can be then described by the following rate equations describing the progressive degradation (concrete cracking) and plastic yielding (concrete crushing): p

f

σ˙ ij = Dijkl · (˙εkl − ε˙ kl − ε˙ kl )

to define analogous quantities for the fracturing model, i.e. λ˙ f is the inelastic fracturing multiplier respectively and g f is the potential defining the direction of inelastic fracturing strains in the fracturing model. The consistency conditions can be than used to evaluate the change of the plastic and fracturing multipliers.

The constitutive equations of the both models can be summarized as follows: Flow rule governs the evolution of plastic and fracturing strains: p

p

Fracture model: ε˙ ij = λ˙ f · mij , f

f

p

mij = f

∂g p ∂σij

mij =

∂g f ∂σij

nij =

f f˙ f = nij · σ˙ ij + H f · λ˙ f = 0,

nij =

p

f

∂f p ∂σij

(4)

∂f f ∂σij

(5)

H p and H f is hardening modulus for plastic model and fracturing model respectively. This represents a system of two equations for the two unknown multiplier rates λ˙ p and λ˙ f , and is analogous to the problem of multisurface plasticity (Simo et al. 1988). The details of ˇ the model implementation can be found in Cervenka ˇ et al. 1998 and Cervenka & Pappanikolaou (2008). The model is using Rankine criterion for tensile fracture with exponential softening of Hordijk (1991) (see Figure 1). The compressive behavior is modeled by the plasticity model, which is using the three parameter surface of Menentrey & Willam (1995) (see Figure 2) and hardening softening is defined according to the laws p described in Figure 3 where εeq is the equivalent plastic strain. The softening in tension and compression is adjusted using a crack band approach of Bažant & Oh (1983). The crack band Lt as well as crush band size Lc are adjusted with regard to the crack orientaˇ tion approach proposed by Cervenka V. et al. (1995). This method is described in Figure 4 and in (6).

(1)

Plastic model: ε˙ ij = λ˙ p · mij ,

p f˙ p = nij · σ˙ ij + H p · λ˙ p = 0,

Lt = γ Lt

and

Lc = γ Lc

γ = 1 + (γmax − 1)

θ , 45

θ ∈ 0; 45

The basic idea is to adjust the crack band size depending on the crack orientation with respect to the element edges. This reflects the fact that a crack cannot localize into a single element if the crack direction is not aligned with the element edges.

(2) (3)

where λ˙ p is the plastic multiplier rate and g p is the plastic potential function. Following the unified theory of elastic degradation of Carol et al. (1994) it is possible

(6)

Figure 1.

Tensile softening (Hordijk 1991).

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2.1

Special features of reinforced concrete

When it comes to nonlinear analysis of reinforced concrete, i.e. when reinforcement is to be considered, it becomes important to consider additional special issues related to the reinforcement and the composite reinforced concrete material. The most important phenomena are:

Figure 2. Three paramater (Menetrey & Willam 1995).

criterion

for

concrete

a. shear strength and stiffness of cracked concrete, i.e. aggregate interlock b. compressive strength reduction due to crack opening in perpendicular direction c. reinforcement yielding d. tension stiffening e. dowel action and bending stiffness of the reinforcement e. bond failure between concrete and reinforcement. In the used constitutive model, the items (a) and (b) are considered according to the modified compression field theory of Collins (Bentz et al. 2006). In this theory, the compressive strength is reduced using the formula σc = rc fc rc =

(7)

1 , 0.8 + 170 ε1

rclim ≤ rc ≤ 1.0

(8)

where ε1 is the tensile strain in the crack. In ATENA the largest maximal fracturing strain is used for ε1 and the compressive strength reduction is limited by rclim . In this work rclim = 0. The shear strength of crack concrete is also assumed according to the modified compression field theory MCFT (Bentz et al. 2006) as:

Figure 3.

σij ≤

Hardening and softening in compression.

 0.18 fc 0.31 +

24 w ag +16

;

i = j

(9)

where fc is the compressive strength in MPa, ag is the maximum aggregate size in mm and w is the maximum crack width in mm at the given location. The modified compression field theory does not give any formula for the shear stiffness, but it is an important parameter, which significantly affects the reinforced concrete response. In the present formulation the crack shear stiffness Ktcr is calculated directly from crack normal stiffness using a scaling factor sF . Ktcr = sF Kncr

(10)

where Kncr comes directly from the tensile softening law in Figure 1 as: Figure 4. Crack band size adjustment based on crack direction orientation.

 Kncr = ft (wt ) wt

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(11)

This appears to be a very natural assumption as this makes the shear stiffness dependent on the crack opening displacement. Reinforcement is modeled using the embedded approach with truss elements, and a multi-linear stressstrain law is used to capture reinforcement yielding. Tension stiffening can be activated in the present ˇ model, but was not used. It was shown by Cervenka & Margoldová (1995) that if sufficiently fine mesh is used the tension stiffening effect can be very well captured by an appropriate cracking model. The dowel action and reinforcement bending stiffness is not considered in the present model. The analyzed beams are only slightly reinforced; therefore these effects cannot play a major role. Reinforcement bond failure can play an important role in the analyzed problems. A bond modeling was discussed by authors in a separate paper Jendele & ˇ Cervenka (2006). It was shown that a bond model can strongly improve the results if large finite elements are used in heavily reinforced structures. The problems presented in this paper are only lightly reinforced and the largest element size is 200 mm. It was therefore decided not to use the bond model to limit the number of investigated parameters. 3

DISCRETE CRACK MODEL

The analyses calculated with the discrete crack model in this paper are using a simple approach, where a crack is modeled as a zero thickness interface with Mohr-Coulomb type of criterion with tension cut-off (see Figure 5). τ ≤ c − σ φ;

σ ≤ ft

(12)

where c is cohesion and φ is frictional coefficient. The Hordijk’s (1991) law is used for tensile softening. The cohesion softening is also modeled by the same law but the displacement values in the softening diagram for cohesion are 10 times increased. This approximately corresponds to the assumption that the shear response should be more ductile then the tensile one, and that

Figure 5. elements.

Mohr-Coulomb criterion for discrete crack

the mode II fracture energy GIIF is about 10 times larger then the fracture energy for mode I. GFII ∼ 10 GFI 4

(13)

SHEAR FAILURE IN PLAIN CONCRETE

In the first example, the discrete and smeared crack models are compared on a typical shear problem without reinforcement (see Figure 6). This is the well known Iosipescu’s shear beam. The geometry corresponds to the SEN beams tested by Schlangen (1993). The tested beams have dimensions 440 × 100 × 100 mm. They were cast from concrete with modulus of elasticity E = 35 GPa, Poisson’s ratio ν = 0.15, tensile strength ft = 2.8 MPa and the specific fracture energy GF = 70 N/m. This test setup was originally proposed by Iosipescu (1967) for shear tests of metals. This test was later used by Bažant & Pfeifer (1986) for shear testing of concrete. It was discovered by Ingraffea & Panthaki (1985) that the crack propagation in this kind of test is mainly dominated by mode I, i.e. tensile cracking. Since then it has become a typical test problem for crack propagation analysis, because it is a common believe that smeared crack models cannot predict the behavior correctly and some kind of enhanced formulation is necessary. The load displacement diagrams are compared in Figure 7. The figure shows a single discrete crack analysis and several smeared crack results. The discrete crack analysis has been performed previously ˇ by Cervenka J. (1994). The peak load in the discrete crack analysis is captured very well as well as the overall shape of the response. In post-peak the response is slightly lower. This could be probably improved by increasing the shear properties of the cohesive interface model. The crack path was determined (see Figure 8) by the direction of maximal principal stresses at the crack tip. The discrete crack

Figure 6. Geometry of the modified Iosipescu’s beam (Schlangen 1993), the dimensions are in mm.

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bias, and an incorrect vertical crack is reported, which ends to left of the bottom plate. This improved behavior of the current model can be attributed to the crack band size formulation (6). The smeared crack analyses labeled with ‘‘Std’’ indicate analyses using the model described in Section 2, but the special features for reinforced concrete are not activated. The shear factor sF is set to the low value of 20, which means that the shear stiffness on the crack surface is almost identical to the normal one. The main findings from this study can be summarized as follows: Figure 7. Comparison of load-displacement diagrams for the Iosipescu’s beam.

Figure 8.

Crack pattern by discrete analysis.

a. The results confirm that the crack propagation is mainly in mode one. b. The shear properties of the crack do not influence the results significantly. c. The cracked area is quite localized so no numerical problems occur in the smeared crack analysis d. The peak load is predicted correctly by all models. e. The coarse finite element models show lower peak values, which is quite common situation in the crack band model. f. The discrete model predicts more accurately the curved shape of the crack path. g. The crack path predicted by the smeared model is acceptable for practical applications.

5

Figure 9. Crack pattern by smeared analysis. (a) coarse mesh 10 mm, (b) fine mesh 2 mm, (c) experiment (Schlangen 1993).

model can nicely capture the curved shape of the crack path. On the contrary, the crack path curvature is even slightly overestimated. This is caused by extending the crack by a certain non-infinitesimal length a at each propagation. Because of that the crack extension is overestimated, and the crack needs to curve strongly to return to the correct path. Figure 1 shows also the results from several smeared crack analyses. All smeared crack analyses showed the crack path depicted in Figure 9, i.e. a more or less straight crack path towards the right side of the bottom loading plate. So the curved crack path is not obtained, but the crack ends at the right side of the loading plate. For instance the smeared crack results reported by Schlangen (1993) are strongly affected by the mesh

SHEAR FAILURE IN LARGE RC BEAMS

In the next example large beams tested at the University of Toronto by Collins and Yoshida (2000) were investigated numerically. Two beams from the experimental program of Yoshida are considered: Beam YB2000/0 with bending reinforcement and no shear reinforcement and beam YB2000/4 with vertical reinforcement by 8 T-headed bars. The beams are schematically depicted in Figures 10 and 11. The longitudinal reinforcement in both beams is identical. The reinforcing ratio of bottom reinforcement of 6×M30 bars is 0.0074. The ratio of vertical reinforcement of T-headed bars T#4, spacing 0.59 m is 0.00071. The beams are only lightly reinforced. The shear span ratio a/d = 2.86 indicates a shear critical geometry.

Figure 10.

Beam YB2000/0 dimensions and reinforcement.

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is considered in the analysis but not included in the monitored load P. 5.1

Figure 11.

Beam YB2000/4 dimensions and reinforcement.

Table 1. Concrete material properties of RC beams, this material set is denoted as ‘‘Std’’, i.e. ‘‘Standard’’. Concrete property

Value

Elastic modulus Ec [MPa] Compressive strength fc [MPa] Tensile strength ft [MPa] Specific fracture energy Gf [N/m] Poisson ratio μ [−] Plastic strain at fc (peak) εcp [−] Plastic end displacement wd [mm] Shear factor sF MCFT fc reduction MCFT aggregate interlock

34 000 37 2.8 80 0.2 0.001 0.5 20 none none

Table 2.

Value

Elastic modulus Es MPa Yield stress fy MPa Max. stress fs,max MPa Limit strain εlim

200 000 470 680 0.11

fctm = fck + 8 = 30 + 8 = 38∼37 MPa

(14)

This was the initial set used for the analyses. This set of parameters is very similar to the one used in Section 4. This set of parameters does not include any of the special provisions for reinforced concrete analysis from Section 2.1. It provided very good results for the Iosipescu’s beam (see Figure 7) and also for the case of beam YB2000/0 (see Figure 10). The loaddisplacement diagrams for this beam are compared in Figure 22. However for the beam with stirrups (Figure 11), the peak load was greatly underestimated. These results are reported in Figure 22 under the label ‘‘FP-Std’’. The peak load is underestimated by almost 50%. The input parameters had to be modified in order to obtain a good agreement. This best-fit response is shown in Figure 13, and the adjusted parameters are listed in Table 3.

Reinforcement properties of RC beams.

Steel property

Discussion on best-fit results

The material properties denoted as ‘‘Std’’ and listed in Table 1 correspond to a standard material setup. It is approximately identical with the standard EC2 concrete class C30/37. The used set of material parameters can be recognized as mean properties of this concrete class.

The experimental study Yoshida (2000) offered for concrete property only a compressive strength at the date of testing, which was obtained from cylinder tests. In the tests, slightly different properties were found in two specimens. However, in this study it was decided to use identical concrete properties in both specimens in order to keep the effect of different shear reinforcing not influenced by other parameters. The assumed set of parameters for concrete and reinforcement is shown in Tables 1 and 2. The parameters reported in this table are referred to a ‘‘Standard’’ or ‘‘Std’’. In some analysis certain parameters are modified to evaluate their influence on the results. The finite element analysis was done for a symmetrical half of the beam in plane stress representation. Quadrilateral 4-node isoparametric elements, sizes 50–200 mm, were used for concrete and embedded truss elements for bars. The total load P = 2 V acting in the top centre of the beam is considered as the global resistance. Like in experiment, self weight

Figure 12. Beam YB200/0 L-D diagram comparison. Analysis is based on ‘‘Std’’ properties and mesh 200 mm.

Figure 13. Beam YB2000/4 L-D diagram comparison. Analysis is based on modified ‘‘Std’’ properties, sF = 300, εcp = 0.002, wd = 50 mm.

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Table 3. Adjusted parameters for best fit for beam YB2000/4. Concrete property

Value

Plastic strain at fc (peak): εcp [−] Plastic end displacement: wd [mm] Shear factor: sF

0.002 50 300

Figure 15.

Figure 14.

Beam YB2000/0 crack pattern comparison.

Beam YB2000/4 crack pattern comparison.

Figure 16. Beam YB2000/4 concrete crushing and yielding of stirrups in numerical analysis.

From the adjusted parameters it is clear that the deficiencies of the initial parameter set were: – brittle response in compression – low shear stiffness of the cracked material. The beam YB2000/0 is failing due to a diagonal cracking. The diagonal cracks can fully open and therefore no significant shear stress can be transferred across the cracks. This failure pattern is nicely documented in Figure 14, which also shows a good agreement between the calculated and observed crack patterns. This should be contrasted by the behavior of beam YB2000/4. This is a beam with shear reinforcement. The reinforcement limits the crack opening so the crack cannot open so much, and significant shear is transferred across each crack. If the shear stiffness is underestimated, a premature failure is calculated. Figure 16 shows the calculated failure mode for this beam. The final failure is due to concrete crushing near the top loading plate and stirrups yielding. Also the bottom bending reinforcement is yielding at this point. However, to obtain a ductile response as in the experiment, it is necessary to increase the ductility of the concrete in compression; otherwise the concrete near the top loading plate fails by a brittle compression failure. In both examples, it is rather difficult to obtain a stable solution in the post-peak. This can be attributed to the following facts: – It is a large beam with lot of elastic energy, which needs to be released.

– Large areas of the model are cracked, and there exist multiple similar solutions, which of these cracks should close and which to open. The second point exactly corresponds to the deficiency of the smeared crack models reported in the introduction to this paper. The mathematical problem of the strain-softening material becomes ill-posed and the uniqueness of the solution is not guaranteed. 5.2

Discrete crack analyses

Now, it will be interesting to explore if an application of discrete crack model can help to resolve this issue of non-uniqueness and numerical stability. Both beams are analyzed using a discrete crack model with cohesive zero thickness elements as described in Section 3. These elements are placed along the expected crack paths. It should be noted that in this study no automatic remeshing and crack propagation is used. It is not necessary since the objective is to verify if the addition of discrete discontinuities into the model can help to resolve the localization problem of the strain softening material. During the localization process some of the initially created cracks need to close while some should open. The results from the discrete crack analyses are summarized in Figure 17 for the beam YB2000/0, i.e. the beam with no shear reinforcement. In this figure several discrete analysis are shown with different number of inserted cohesive cracks. The number of assumed cracks ranges from 1 to 11. The first crack,

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Figure 17. analyses.

Figure 18. analyses.

Beam YB2000/0 results including discrete

Beam YB2000/4 results including discrete

Figure 19. Discrete models for beam YB2000/0 with 1 and 11 discrete cracks.

which quite naturally comes to our mind, is the diagonal shear crack (Figure 19 top), which corresponds to the final failure mode of this beam as it was shown in the previous Section 5.1 in Figure 14. The diagonal crack is not the first crack that appears in reality. The previous analyses showed that the cracking is first initiated in the middle of the beam as bending cracks that later on spread through the whole bottom part of the beam. When these bending cracks are not included

Figure 20. Discrete model for beam YB2000/4 with 22 discrete cracks.

in the discrete model an extremely stiff response is obtained as shown in Figure 17. In order to correct the pre-peak stiffness, new models were created with multiple bending cracks in the middle of the beam. One such model is shown in Figure 19 (bottom) with altogether 11 discrete cracks. It is interesting to note that as the number of discrete cracks in the model is increasing the stiffness of the pre-peak is improving as well. Figure 17 also shows that once the number of discrete cracks increases to 11 it becomes very difficult to obtain a stable post-peak solution. Analogical results were obtained for the beam YB2000/4. In this case, a model with cca 22 discrete cracks was used (Figure 20). The load displacement diagram is shown in Figure 18. It is clear that even 22 discrete cracks are not enough to capture the reduction of stiffness due to the diffused crack pattern in the pre-peak regime. The results show that similarly to the smeared model in Figure 13, it is necessary to modify the shear properties of the crack model to obtain at least a correct peak load. Standard discrete parameters underestimated the peak load by more then 50%. It was necessary to increase the cohesion to 4.2 MPa and friction coefficient to 0.35 to obtain good match of the peak load (Standard values were c = 2.8 MPa and φ = 0.3). It was also very difficult to obtain a stable solution once the peak load was reached. Many of the discrete cracks are opened, many similar solutions exist. For the numerical solver it is difficult to determine which of them should close and which should continue to open and localize the failure. 5.3

Effect of special reinforced concrete features

Various special issues related to the constitutive modeling of reinforced concrete were introduced in Section 2.1. It will be interesting to examine their effect on the numerical solution. Some effects were already discussed in Section 5.1 and 5.2 and additional results are shown in Figures 21 and 22.

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Figure 21.

Beam YB200/0 effect MCFT and ductility.

Figure 22.

Beam YB2000/4 effect of MCFT and ductility.

Figure 21 clearly shows that these special features play only a minor role when no shear reinforcement is present. This is also confirmed by the results of the Iosipescu’s beam in Section 4. Totally different situation is in the case of beam YB2000/4, i.e. the one with shear reinforcement. Although the beam is only lightly reinforced, its strength is determined by reinforcement yielding as shown in Figure 16. The results also show that the shear properties of the crack concrete, i.e. both the shear strength as well as the shear stiffness should be considered properly. In this case the shear stiffness of the cracked concrete was a major factor. The MCFT (Bentz et al. 2006) parameters such as the aggregate interlock and the reduction of compressive strength due to cracking did not play a major role. As already pointed out in Table 3 it was the concrete compressive ductility and the shear stiffness of cracked concrete that proved to be critical for good prediction of the beam behavior.

6

CONCLUSIONS

Paper discusses various aspects of numerical predictions of shear strength of plain and reinforced concrete structures. One of the objectives is to verify whether

the introduction of strong displacement discontinuities into the numerical solution can be used as a remedy for the known problem of softening materials, i.e. the ill-posed nature of the mathematical solution, which results in a non-unique solution. In plain concrete the discrete crack model definitely improves the crack path predictions; however a good smeared crack model can provide almost identical results. This is especially true if the randomness and heterogeneity of the concrete material is taken into account. In reality, the crack path will always differ in all tests, so minor deviations from the exact path should be tolerated. In reinforced concrete, the discrete crack model is applicable only if large number of discontinuities is introduced into the model. This may be difficult to accomplish with the classical form of the model with remeshing, but can be handled by its modern variants such as X-FEM. With increasing number of discontinuities, i.e. cracks, it is apparent that the same problem of solution non-uniqueness will appear. This shows that the enhanced finite element method cannot be used as a remedy to this problem of softening materials. The only proper solution would be a nonlocal approach or a full dynamic analysis with rate dependent formulation. The reinforced concrete beam shows that shear properties of the crack concrete are critical for good predictions, although the current level of knowledge is quite limited in this area. The aggregate interlock as well as the fc reduction proposed by Modified Compression Field theory of Bentz et al. 2006 did not play an important role for the shear strength of the analyzed beams.

ACKNOWLEDGEMENTS This research was partially supported by research grants from Czech Grant Agency no. 103/07/1660. and 103/08/1527. The financial support is greatly appreciated.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Lattice Discrete Particle Model for Fiber reinforced concrete (LDPM-F) with application to the numerical simulation of armoring systems G. Cusatis & E.A. Schauffert Rensselaer Polytechnic Institute, Troy, NY, USA

D. Pelessone ES3, Solana Beach, CA, USA

J.L. O’Daniel Engineer Research and Development Center, Vicksburg, MS, USA

P. Marangi, M. Stacchini & M. Savoia University of Bologna, Bologna, Italy

ABSTRACT: In this paper, the Lattice Discrete Particle Model (LDPM) is extended to include the effect of dispersed fibers with the objective of simulating the behavior of fiber reinforced concrete for armoring system applications. Within the LDPM framework, the effect of dispersed fibers is taken into account through the following procedure. 1) Fibers are randomly placed in the volume of interest according to the given fiber volume ratio and fiber geometry. 2) The number and orientation of fibers crossing each facet are computed along with the fiber embedment length on each side of the facet. 3) At the facet level, fibers and plain concrete are assumed to be coupled in parallel. 4) The contribution of each fiber to the facet response is formulated on the basis of a micromechanical model of fiber-matrix interaction. The developed model, named LDPM-F, is validated by carrying out numerical simulations of direct tension and three-point bending tests on fiber reinforced concrete mixes characterized by various fiber volume fractions. Finally, LDPM-F is applied to the analysis of the penetration resistance of fiber reinforced slabs. 1

THE LATTICE DISCRETE PARTICLE MODEL

Since the mid-eighties, many mesoscale models for concrete have appeared in the literature. The main advantage of these models over classical constitutive models for concrete is their ability to simulate material heterogeneity and its effect on damage evolution and fracture. Noteworthy examples of mesoscale models include: Roelfstra et al. (1985); Wittmann et al. (1988); Bažant et al. (1990); Schlangen & VanMier (1992); Bolander & Saito (1998); Bolander et al. (1999); Bolander et al. (2000); Carol et al. (2001); Lilliu & Van Mier (2003); Cusatis et al. (2003a,b); Cusatis et al. (2006); and Cusatis & Cedolin (2006). In this paper, recent results obtained at Rensselaer Polytechnic Institute with the Lattice Discrete Particle Model (LDPM) are presented and discussed. LDPM simulates concrete mesostructure by taking into account only the coarse aggregate pieces, typically with characteristic size greater than 5 mm. The mesostructure is constructed through the following steps. 1) The coarse aggregate pieces, whose shapes are assumed to be spherical, are introduced into the

concrete volume by a try-and-reject random procedure. 2) Zero-radius aggregate pieces (nodes) are randomly distributed over the external surfaces. 3) A three-dimensional domain tessellation, based on the Delaunay tetrahedralization of the generated aggregate centers, creates a system of cells interacting through triangular facets, which can be represented in a two-dimensional sketch by straight line segments (Fig. 1). A vectorial constitutive law governing the behavior of the model is imposed at the centroid of the projection of each single facet (contact point) onto a plane orthogonal to the straight line connecting the particle centers (edges of the tetrahedralization). The projections are used instead of the facets themselves to ensure that the shear interaction between adjacent particles does not depend on the shear orientation. The straight lines connecting the particle centers define the lattice system. Rigid body kinematics describes the displacement field along the lattice struts and the displacement jump, [uC ], at the contact point. The strain vector is defined as the displacement jump at the contact point divided

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way similar to simple damage models (see derivation in Cusatis et al. 2003a): σN = σ

εN ; ε

σM = σ

αεM ; ε

σL = σ

αεL ε

(4)

The effective stress, σ , is incrementally elastic (σ˙ = E0 ε˙ ), and must satisfy the inequality 0 ≤ σ ≤ σbt (ε, ω). The strain dependent boundary, σbt (ε, ω), can be expressed as:   εmax − ε0 (ω) (5) σbt = σ0 (ω) exp −H0 (ω) σ0 (ω) in which the brackets • are used in Macaulay sense: x = max{x, 0}. The internal variable ω is defined as follows (Cusatis et al. 2003a): √ εN σN α tan ω = √ = σT αεT Figure 1. a) Mesostructure tessellation. b) Threedimensional discrete particle. c) Definition of nodal degrees of freedom and contact facets in two-dimensions.

by the inter-particle distance, L. The components of the strain vector in a local system of reference, characterized by the unit vectors n, l, and m, are the normal and shear strains: εN =

nT [uC ] ; L

εL =

lT [uC ] ; L

εM =

mT [uC ] (1) L

The unit vector n is orthogonal to the projected facet and the unit vectors l and m are mutually orthogonal and lie in the projected facet. The elastic behavior is described by assuming that the normal and shear stresses are proportional to the corresponding strains:

(6)

and it characterizes the coupling between normal and shear strains (or stresses). The σbt boundary evolves exponentially as a function of the maximum effective strain, which is a history-dependent vari-

able defined as εmax = εN2 ,max + αεT2 ,max , where εN ,max (t) = max[εN (τ )], and εT ,max (t) = max[εT (τ )] τ
τ
are the maximum normal and total shear strains, respectively, attained during the loading history. In the absence of unloading εmax ≡ ε. The function σ0 (ω) is the strength limit for the effective stress and is defined as follows:  − sin(ω) + sin2 (ω) + 4α cos2 (ω)/rst2 σ0 (ω) = σt 2α cos2 (ω)/rst2 (7)

where EN = E0 , ET = αE0 , E0 = effective normal modulus, and α = shear-normal coupling parameter. E0 and α are assumed to be material properties. For tensile loading (εN > 0), fracturing behavior is formulated through a relationship between the effective strain and the effective stress (Cusatis et al. 2003a), defined as:   (σM + σL )2 2 2 2 ε = εN + α(εM + εL ); σ = σN2 + α (3)

in which the ratio between the shear strength (cohesion), σs , and the tensile strength, σt , is represented by rst = σs /σt . In the stress space σN − σT , Eq. 7 represents a parabola with its axis coincident with the σN -axis. The exponential decay of the σbt boundary starts when the maximum effective strain reaches its elastic limit, ε0 (ω) = σ0 (ω)/E0 , and the decay rate is governed by the post-peak slope (softening modulus), which is assumed to be a power function of the internal variable ω:  nt 2ω H0 (ω) = Ht (8) π

By using the effective strain and the effective stress, the relationship between normal and shear stresses versus normal and shear strains can be calculated in a

Equation 8 provides a smooth transition from softening behavior under pure tensile stress (ω = π/2, H0 (ω) = Ht ) to perfectly plastic behavior under pure

σN = EN εN ;

σM = ET εM ;

σL = ET εL

(2)

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shear (ω = 0, H0 (ω) = 0). In order to preserve the correct energy dissipation during mesoscale damage localization (Bažant & Oh 1983), the softening modulus in pure tension is expressed as Ht = 2E0 / (Lcr /L − 1), where Gt is the mesoscale fracture energy, Lcr = 2E0 Gt /σt2 , and L is the length of the tetrahedron edge associated with the particular facet under consideration. For compressive loading (εN < 0), the normal stress is computed by imposing the inequality −σbc (εD , εV ) ≤ σN ≤ 0, where σbc is a straindependent boundary depending on the volumetric strain, εV , and the deviatoric strain, εD . The volumetric strain is computed by considering the interaction of the four particles located at the vertices of each Delaunay tetrahedron. For a constant deviatoric strain to volumetric strain ratio, rDV = εD /εV , the compressive boundary, σbc (rDV , εV ), is assumed to have an initial linear evolution (modeling pore collapse) followed by an exponential evolution (modeling compaction and rehardening). One can write the following: σbc = σc0 + −εV − εc0 Hc

(9)

for −εV ≤ εc1 , and σbc = σc1 exp [(−εV − εc1 )Hc /σc1 ]

(10)

otherwise, where σc0 = yielding compressive stress, εc0 = σc0 /E0 = volumetric strain at the onset of pore collapse, Hc = initial hardening modulus, εc1 = λc0 εc0 = volumetric strain at which rehardening begins, λc0 = material parameter governing the onset of rehardening, and σc1 = σc0 + (εc1 − εc0 )Hc . For increasing rDV , the slope of the initial hardening modulus must tend to zero in order to simulate the observed horizontal plateau featured by typical experimental data. This can be achieved by setting: Hc (rDV ) =

Hc0 1 + κc2 rDV − κc1 

(11)

where Hc0 = κc0 EN , and κc0 , κc1 , and κc2 are material parameters. In the presence of compressive stresses, the shear strength increases due to frictional effects. This effect can be simulated effectively through classical incremental plasticity. Incremental shear stresses can be p calculated as σ˙ M = ET (˙εM − ε˙ M ) and σ˙ L = ET (˙εL − p ε˙ L ), where the plastic strain increments are assumed p ˙ to obey the normality rule: ε˙ M = λ∂ϕ/∂σ M , and p ˙ ε˙ L = λ∂ϕ/∂σL , where λ is the plastic multiplier.  The plastic potential can be expressed as ϕ = σM2 + σL2 − σbs (σN ), in which the shear strength,

σbs , is formulated with a nonlinear frictional law: σbs = σs + (μ0 − μ∞ )σN 0 − μ∞ σN − (μ0 − μ∞ )σN 0 exp(σN /σN 0 )

(12)

where σs = cohesion, μ0 and μ∞ are the initial and final internal friction coefficients, respectively, and σN 0 = the normal stress at which the internal friction coefficient transitions from μ0 to μ∞ . Finally, equations governing the shear stress evolution must be completed by the loading-unloading conditions ϕ λ˙ ≤ 0 and λ˙ ≥ 0. Current LDPM formulation has been implemented into MARS, a multi-purpose computational code for the explicit dynamic simulation of structural performance (Pelessone 2009). 2

SIMULATION OF FIBER EFFECTS

Herein, LDPM is extended to include the effects of randomly dispersed fibers in order to simulate the behavior of fiber reinforced concrete (FRC). During the preprocessing phase, each individual fiber is inserted into the specimen volume. Fiber positions and orientations are randomly generated, and the intersections between fibers and LDPM facets are detected. Description of this procedure is provided in Sec. 2.1. By assuming a parallel coupling between the fibers and the concrete matrix, stresses on each LDPM facet can be computed as: σ = σc +

1  Pf (w, nf , df , Lsf , Llf ) Ac

(13)

f ∈Ac

where σ = [σN σM σL ]T , σ c = [σNc σMc σLc ]T , Pf = [PNf PMf PLf ]T , Ac = facet area, wf = facet crack opening, df = fiber diameter, Lsf = short embedment length, Llf = long embedment length, and nf = fiber orientation with respect to the crack (facet) plane. The concrete stress components, σNc , σMc , and σLc , are computed according to the LDPM constitutive law presented in Sec. 1. The fiber contribution to the crack bridging force, Pf , is computed according to the micro-structural fiber-matrix interaction model developed by Lin et al. (1999), and summarized in Sec. 2.2. 2.1

Fiber generation

At the global level, the distribution of fiber-facet intersections is accomplished by generating a uniform distribution of fibers throughout a simulated specimen. From a geometric point of view, physical fibers can be described using a few parameters: fiber density, length, diameter, and tortuosity. The latter parameter

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is used to characterize the fiber shape: a straight fiber has no tortuosity while a fiber with many ‘‘kinks’’ is very tortuous. These geometric parameters are used in MARS for generating random fibers inside a control volume, which can either be the volume of the entire specimen or a larger volume that contains the specimen. Each fiber is modeled using a sequence of one or more segments linked together. Single segments are sufficient for generating straight fibers. Multiple segments are necessary for generating ‘‘tortuous’’ fibers. Fiber location and orientation are assumed to be randomly distributed. All fibers are completely contained in the control volume. For cast FRC parts, the control volume should be equal to the volume of the simulated specimen. For machined parts, the control volume should be larger than the simulated specimen. The fibers that intersect the external surface of the part are treated as cut. The portion of a cut fiber which lays inside the part is shorter than the length of original fiber and this affects the mechanical characteristics of the fiber-facet interaction near the external surfaces. In the spirit of the discrete, multi-scale physical character of LDPM, the occurrences of fiber-facet intersection are determined by actually computing the locations where fibers cross inter-cell facets. This computation can require significant resources for large models in terms of both time and memory. For this reason, an efficient bin-sorting algorithm was developed for computing these intersections. For each intersection, the fiber length on each side of the facet, and the orientation at which the fiber intersects the facet (crack plane) are computed. These parameters are saved in the facet data structure and used during the simulation for calculating the fiber contribution to the LDPM facet behavior. It is worth noting that the uniform distribution of fibers throughout a volume is sometimes not realistic due to the intrinsic technical difficulties of dispersing the fibers evenly in an actual concrete specimen. This issue needs to be taken into account carefully when calibrating and validating the model (see Sec. 3.2). 2.2

Figure 2.

Crack bridging effect of a single fiber.

the crack-bridging segment and expressed as: Pf = Pf nf

(15)



2

2

2 + wMf + wLf . where nf = wf /wf and wf = wNf As shown in Figs. 2d and e, at the exit point, i.e. where the fiber begins to protrude beyond the crack face, there is a kink in the fiber. At this point, the matrix acts as a ‘‘frictional pulley’’. In general, therefore, the force Pf in the bridging segment will be higher than the forces Psf and Plf acting at the ends of the short and long embedded segments, respectively. This phenomenon is typically referred to in the literature as ‘‘snubbing effect’’, and it can be modeled as follows:



Pf = eksn ϕf Psf (vsf ) = eksn ϕf Plf (vlf )

Fiber-matrix interaction law

(16)

where ϕf = arccos(nfT nf ) and ksn is the snubbing parameter. Equation 16 also implies that: Psf (vsf ) = Plf (vlf )

(17)

Consider a crack crossed by a straight fiber (Fig. 2a) characterized by the embedment lengths Lsf and Llf . If one neglects the fiber bending stiffness and the elastic deformation of the crack-bridging segment, the length of such a segment (distance between points A and B in Fig. 2a) can be computed as:

which expresses the equilibrium of the bridging segment. In addition, compatibility between the length of the bridging segment and the short and long side slippages (vsf and vlf , respectively), can be enforced by writing:

wf = wf + 2sf nf

wf = wf + 2sf + vsf + vlf

(14)

where wf = crack opening vector and sf = reduction of embedment lengths due to micro-spalling. In addition, the fiber force can be assumed to be coaxial with

(18)

 2 2 2 where wf = wNf + wMf + wLf . For a given spalling length, sf , Eqs. 17 and 18 can be used to compute

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vsf and vlf once the load versus slippage law, P(v), is defined. According to Lin et al. (1999), for the debonding phase (Fig. 2b), characterized by v < vd , one can write: P(v) =

π 2 Ef df3 2

1/2 (τ0 v + Gd )

(19)

and for the frictional pull-out phase (Fig. 2c), characterized by v > vd :    β(v − vd ) v − vd 1+ P(v) = P0 1 − L df

(20)

The critical slippage, vd , represents the extent of slippage leading to full debonding, and is defined as: vd =

2τ0 L2 + Ef d f



8Gd L2 Ef df

1/2 (21)

In Eqs. 19 to 21: P0 = πLdf τ0 ; Ef is the fiber Young’s Modulus; τ0 is the frictional bond strength; Gd is the debonding fracture energy associated with ‘‘tunnel-type’’ cracking over the embedment length as the fiber debonds from the matrix; β is a dimensionless material parameter characterizing the extent to which the frictional pullout of the fiber will be slip-hardening (low values, near zero or negative, will cause the frictional pullout to be a softening process); and L is the current embedment length, defined as L = Lsf − sf and L = Llf − sf for the short and long embedment lengths, respectively. The previous equations are valid only if the fiber force does not cause fiber rupture. This can be checked by making sure that the fiber force magnitude is less than the rupture force, Puf : Pf ≤ Puf =

0.25πdf2 σuf e−krup ϕf , where krup is a material parameter and σuf is the ultimate tensile strength of the fiber associated with the theoretical situation where the fiber’s bridging segment and its embedded segments are collinear. For the definition of the spalling length, sf , various models have been proposed in the literature, including Cailleux et al. (2005) and Leung & Li (1992). Herein, the formula proposed by Yang et al. (2008) is adopted under the assumption that the shear components of the fiber force do not cause spalling: sf =

Since the spalling length, sf , depends on the fiber force, and the force versus slippage relationship is nonlinear, the problem of computing the fiber force for a given crack opening is highly nonlinear and needs to be solved iteratively.

PfN sin(ϕ/2) ksp σt df cos2 (ϕ/2)

3

SIMULATION OF EXPERIMENTAL DATA

In this section, the Lattice Discrete Particle Model for Fiber Reinforced Concrete (LDPM-F), formulated above, is validated by comparison with experimental data relevant to direct tension tests and three-point bending tests. 3.1

Direct tension tests

Simulation of the fiber effect on the tensile fracturing behavior of concrete is shown in Fig. 3. The experimental data are relevant to experiments reported by Li et al. (1998). In this experimental investigation, rectangular specimens were subjected to direct tension. The tests were controlled through displacement measurements over a measured length of 120 mm to ensure stability in the post-peak softening regime. The simulated fibers are Dramix steel fibers with hooked ends characterized by a diameter of 0.5 mm and a length of 30.0 mm. Figure 3a shows experimental and numerical stress versus displacement curves for four different fiber volume fractions (Vf ): 0% (plain concrete), 2%, 3%, and 6%. LDPM-F is able to predict the increased strength

(22)

where ϕ = arccos(nfT wf )/wf is the orientation of the crack-bridging segment in the absence of spalling (Fig. 2d), and ksp is a dimensionless material parameter.

Figure 3.

Direct tension tests of FRC specimens.

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and ductility due to the effect of fibers. The behavior gradually transitions from softening for plain concrete and low Vf , to hardening for high Vf . LDPM-F numerical results are further investigated in Fig. 3b, where contours of the mesoscale crack opening at the end of the simulations are reported for three fiber volume fractions. For plain concrete, the crack pattern is characterized by one localized crack that propagates from one side of the specimen towards the other. As fracture propagates, material outside the crack unloads as the overall load applied to the specimen tends to zero. For the 2% Vf , there is still one main crack propagation, but the entire specimen features diffuse cracking and no unloading occurs. Absence of unloading outside the main crack is due to the fact that even though the overall behavior is softening, the stress versus displacement curve shows a non-zero residual stress associated with the fiber crack bridging effect. Finally, for the 6% Vf , the crack pattern is characterized by several branched cracks whose propagation is arrested by the effect of the fibers. No unloading occurs outside the main cracks since the overall behavior is strain-hardening and, up to a displacement of 0.5 mm (average nominal strain of 0.5 mm/120 mm ≈ 0.42%), no reduction of the load carrying capacity can be observed. 3.2 Three-point bending tests In this section, LDPM-F is validated by simulating three-point bending tests (Buratti et al. 2008) carried out on prismatic notched specimens of plain concrete and FRC with two different Vf values: 0.26% and 0.45%. Figure 4 shows the specimen geometry and

Figure 4. Specimen geometry and discretization for the three-point bending tests.

the adopted numerical model, in which only the central part (where damage is expected to occur due to the presence of the notch) is modeled through LDPM-F, while the two lateral parts are modeled with standard elastic finite elements. All the specimens have an out-of-plane thickness of 150 mm. The LDPM-F model parameters were calibrated by fitting the load versus CMOD (crack mouth opening displacement) curves relevant to plain concrete and 0.26% Vf as reported in Figs. 5a and b. In Fig. 5a, both the numerical and experimental curves represent the average of the response of two specimens. In Fig. 5b, the numerical and experimental response of seven specimens is reported along with the average curves (solid curve and circles, respectively). The seven different numerical simulations were obtained by seven different random fiber distributions. The agreement between the numerical results and the experimental data is very good. Figure 5c shows the same comparison for the 0.45% Vf . Again the response of seven different specimens and the average curves are reported. In this case the numerical results produce a post-peak load carrying capacity higher than the experiments. For a CMOD of 0.8 mm the difference between the average numerical and experimental curves is about 45%. In order to clarify this apparent discrepancy, it is useful to compare the actual number of fibers bridging the crack in the experiments and the in numerical simulations. During the experimental campaign, postmortem evaluation was performed and the number of fibers on the crack surface was counted for all specimens of both FRC mixes. In Buratti et al. (2008), the authors report the number of fibers in each third of the ligament length (i.e. the reduced cross-section width associated with the notch, as shown in Fig. 6a). The same information was extracted from the numerical simulations. In Fig. 6b, for CMOD = 0.8, the load increment due to the effect of fibers, calculated as 100(1 − P/P0 ), where P is the load for the FRC specimens and P0 is the average load for the plain concrete specimens, is plotted versus the number of fibers in the lower two-thirds of the ligament length, i.e. the two-thirds closest to the notch. The choice of plotting the number of fibers in the lower two-thirds of the ligament length was motivated by analyzing the crack pattern in the numerical simulations. Figure 6a shows gray-scale isosurfaces from a typical mesoscale crack opening. Each isosurface corresponds to a different crack opening value. The minimum value that was plotted (very light gray) corresponds to a crack opening magnitude of 2vd (see Sec. 2.2). This corresponds to a situation at which a straight fiber, orthogonal to the crack surface, would experience full debonding (onset of the frictional pullout phase). At this level of crack opening, fibers can be considered fully active. As one can see from Fig. 6a, the zone characterized by fully active fibers extends to

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Figure 6. a) Crack opening isosurfaces; b) load increment due to fiber versus number of active fibers.

Figure 5. Load versus CMOD curves: a) Vf b) Vf = 0.26%; c) Vf = 0.45%.

= 0%;

approximately two-thirds of the ligament length above the notch. The plot shows clearly a linear relationship between the load increment and the number of fibers detected, and that both the experimental data and the numerical results have a similar trend. There is, however, an

inconsistency in the number of detected fibers for the experimental specimens. For the numerical specimens, the range of detected fibers increases from (22–39) to (47–70) as the reported fiber volume fraction nearly doubles from Vf = 0.26% to Vf = 0.45%. For the experimental specimens, the increase in number of detected fibers, from (17–39) to (17–50), is not consistent with the near doubling of Vf . For the higher Vf , the numerically simulated specimens feature more active fibers than the experimental ones. This explains the discrepancy in the curves shown in Fig. 5c, while confirming the validity of the fiber-matrix interaction strategy adopted in this study. The difference between the number of active fibers within the experimental and numerical crack ligaments is due to the fact that in the numerical model a uniform fiber distribution was obtained, while such uniformity apparently was not obtained in the experimental specimens with higher Vf . Based on the fact that fiber dispersion can be significant in terms of load carrying capacity, LDPM-F should be extended to include non-unform fiber distribution, and, in addition,

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experimental data should provide as much information as possible on the actual fiber distribution obtained during specimen casting. 4 4.1

PROJECTILE PENETRATION OF FRC SLABS Effect of fibers on penetration resistance

In this section, simulations of steel projectile impacts into FRC slabs at various impact velocities are presented. The numerical simulations were performed by using the same material parameters used in Sec. 3.2. The simulated slabs were 508 mm squares with two different thicknesses, t = 50.8 mm and 101.6 mm. Impact velocities up to 450 m/sec and three different fiber volume fractions (Vf = 0%, 2%, and 3%) were considered. Figure 7 reports the projectile residual velocity, Vr , as a function of the striking velocity, Vs , for the different simulations. As one can see, the ballistic limit (the highest strike velocity associated with zero residual velocity) increases for increasing Vf . The FRC slabs with 2% Vf have ballistic limits that are approximately 175% and 80% higher than the plain concrete, for t = 50.8 mm and 101.6 mm, respectively. The 3% Vf simulations show, however, that an additional increase in fiber content does not necessarily correspond to a significant additional increase of the ballistic limit. Finally, the effect of the fibers becomes less and less significant for high striking velocities. This is due to the fact that at very high striking velocity, the penetration phenomenon is governed more by the mass of the system and the confined compressive resistance, rather than by the tensile fracturing behavior, which is significantly influenced by the presence of the fibers. 4.2

design of lightweight and low-cost armoring systems. Within this effort, experimental characterization of a very high strength (fc = 157 MPa) FRC mix was performed both quasi-statically (Akers, et al. 1998) and dynamically. The dynamic experiments consisted of impacts from a Fragment Simulating Penetrator (FSP) into FRC panels. Parameters of the tests, including FSP velocity and target panel thickness, were chosen to generate extreme deformation and failure of the

Design of armoring systems

The Engineer Research Development Center (ERDC) is currently performing research towards the innovative

Figure 7.

Figure 8. Numerical simulations of penetration into FRC slabs performed at ERDC.

Projectile penetration of concrete slabs.

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material, including generation of surface craters and perforation through the target. Of concern were the damage generated by the FSP to the target plates and the residual velocity when the projectile exited the targets. The adopted fibers were steel of 25.4 mm length and bent ends, and were randomly distributed during the casting phase. Penetration experiments were conducted to measure the FRC’s resistance to ballistic projectile penetration. Panels tested were 304.8 mm squares of 25.4, 50.8, and 76.2 mm thickness. Some preliminary results relevant to the LDPM-F simulations of these tests are reported herein. LDPM-F was used to model the target panels. The FSP was modeled with hexahedral finite elements. A typical LDPM-F mesh for a 50.8 mm thick panel contained approximately 20,000 fibers, 115,000 LDPM particles, and 670,000 LDPM tetrahedral elements. Figures 8a, b, and c show snapshots of the penetration event characterized by the formation of entrance and exit craters. Figure 8d shows a comparison between the post-penetration damage observed during the experiments (left) and that predicted by the simulation (right).

5

CLOSING REMARKS

In this paper the Lattice Discrete Particle Model was extended to include the effect of randomly dispersed fibers. Fiber-matrix interaction was modeled by using an earlier formulated micromechanical theory providing the fiber crack bridging force as a function of the crack opening. This law was coupled to the LDPM constitutive behavior at the facet level. The contribution of each individual fiber is taken into account by detecting the intersection between the fibers and the LDPM facets. The formulated model, named LDPM-F, is able to reproduce the fiber toughening mechanisms and, once calibrated, is able to predict the macroscopic fracturing properties as a function of different fiber volume fractions. Finally, LDPM-F was used to investigate the effect of fibers on the penetration resistance of FRC slabs. Preliminary numerical results show that fibers influence significantly the penetration response for striking velocities close to the ballistic limit, whereas their effect tends to become less significant as striking velocity increases.

ACKNOWLEDGMENT This effort was sponsored in part by the US Army Engineer Research and Development Center (ERDC). Permission to publish was granted by the Director, Geotechnical and Structures Laboratory, ERDC. The work of the first two authors was also supported under

DTRA grant HDTRA1-09-1-0029 to Rensselaer Polytechnic Institute. This financial support is gratefully acknowledged. REFERENCES Akers, S.A., Green, M.L. and Reed, P.A., 1998. Laboratory characterization of very high-strength fiber-reinforced concrete. US Army Corps of Engineers, Waterways Experiment Station, TR, SL-98-10. Bažant, Z.P. and Oh, B.H., 1983. Crack band theory for fracture of concrete. Mater. Structures, RILEM, 16(93): 155–177. Bažant, Z.P., Tabarra, M.R., Kazemi, T. and PijaudierCabot, G., 1990. Random particle model for fracture of aggregate or fiber composites. J. Eng. Mech., 116(8): 1686–1705. Bolander, J.E. and Saito, S., 1998. Fracture analysis using spring network with random geometry. Engng. Fracture Mech., 61(5–6): 569–591. Bolander, J.E., Yoshitake, K. and Thomure, J., 1999. Stress analysis using elastically uniform rigid-body-spring networks. J. Struct. Mech. Earthquake Engng., (JSCE), 633 (I-49): 25–32. Bolander, J.E., Hong, G.S. and Yoshitake, K., 2000. Structural concrete analysis using rigid-body-spring networks. J. Comp. Aided Civil and Infrastructure Engng., 15: 120–133. Buratti, N., Mazzotti, C., Savoia, M. and Thooft, H., 2008. Study of the behavior of concrete reinforced through steel and polymeric fibers (in Italian). XVII Congress C.T.E., 5–8 November, 2008. Roma, Italia. Cailleux, E., Cutard, T. and Bernhart, G., 2005. Pullout of steel fibers from a refractory castable: experiment and modeling. Mechanics of Materials, 37: 427–445. Carol, I., López, C.M. and Roa, O., 2001. Micromechanical analysis of quasi-brittle materials using fracture-based interface elements. Internat. J. Numer. Methods Engrg., 52: 193–215. Cusatis, G., Bažant, Z.P. and Cedolin, L., 2006. Confinementshear lattice model for fracture propagation in concrete. Comput. Methods Appl. Mech. Engrg., 195: 7154–7171. Cusatis, G. and Cedolin, L., 2006. Two-scale analysis of concrete fracturing behavior. Engng. Fracture Mech., 74: 3–17. Cusatis, G., Bažant, Z.P. and Cedolin, L., 2003a. Confinement-shear lattice model for concrete damage in tension and compression: I. Theory. J. of Engrg. Mech. (ASCE), 129(12): 1439–1448. Cusatis, G., Bažant, Z.P. and Cedolin, L., 2003b. Confinement-shear lattice model for concrete damage in tension and compression: II. Computation and validation. J. of Engrg. Mech. (ASCE), 129(12): 1449–1458. Leung, C.K.Y. and Li, V.C., 1992. Effect of fiber inclination on crack bridging stress in brittle fiber reinforced brittle fiber matrix composites. J. Mech. Phys. Solids, 40(6): 1333–1362. Li, Z., Li, F., Chang, T.-Y.P. and Mai, Y.-W., 1998. Uniaxial tensile behavior of concrete reinforced with randomly distributed short fibers. ACI Material Journal, 95(5): 564–574. Lilliu, G. and VanMier, J.G.M., 2003. 3D lattice type fracture model for concrete. Engng. Fracture Mech., 70: 927–941.

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Lin, Z., Kanda, T. and Li, V.C., 1999. On interface characterization and performance of fiber-reinforced cementitious composites. Concr Sci Engng., 1: 173–184. Pelessone, D., 2009. MARS: Modeling and Analysis of the Response of Structures—User’s Manual, ES3, Solana Beach (CA), USA. http://www.es3inc.com/mechanics/ MARS/Online/MarsManual.htm. Roelfstra, P.E., Sadouki, H. and Wittmann F.H., 1985. Le béton numérique. Mater. Struct., 18: 327–335. Schlangen, E. and VanMier, J.G.M., 1992. Shear fracture in cementitious composites, Part II: Numerical simulations. Fracture mechanics of concrete structures, Proc. FraMCoS-1, Bažant Z.P., ed., Elsevier, London, 671–676.

Wittmann, F.H., Roelfstra, P.E. and Kamp, C.L., 1988. Drying of concrete: an application of the 3L-approach. Nucl. Eng. Des., 105: 185–198. Yang, E.H., Wang, S., Yang, Y. and Li, V.C., 2008. FiberBridging Constitutive Law of Engineered Cementitious Composites. Journal of Advanced Concrete Technology, 6(1): 181–193.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Nonlocal damage based failure models, extraction of crack opening and transition to fracture Frédéric Dufour 3S-R Lab. (Soils, Solids, Structures—Risks), Grenoble Institute of Technology, France

Gilles Pijaudier-Cabot Laboratoire des Fluides Complexes, Univ. de Pau et des Pays de l’Adour, France

Grégory Legrain GeM Research Institute, Ecole Centrale de Nantes, France

ABSTRACT: Damage models are capable to represent initiation and somehow crack propagation in a continuum framework. Thus crack openings are not explicitly described. However for concrete structures durability analysis, crack opening through transfer properties is a key issue. Therefore, in this contribution we present a new approach that is able from a continuum modelling to locate a crack from internal variable field and then to estimate crack opening along its path. Results compared to experimental measures for a three point bending test are in a good agreement with an error lower than 10% for a widely opened crack (40 μm). 1

INTRODUCTION

For many concrete structures, crack opening is a key parameter needed in order to estimate durability. Cracks are preferential paths along which fluids or corrosive chemical species may penetrate inside concrete structural elements. For structures such as confinement vessels, reservoirs or nuclear waste disposals for instance, tightness to gas or liquids is a major serviceability criterion that is governed by Darcy’s relation in which permeability of the material is involved. Hence, the prediction of the durability of structural components requires models that describe failure, crack locations and crack openings in the present example too when damage has localised. Enhanced continuum and integral damage models are capable of representing diffuse damage, crack initiation and possibly crack propagation (Pijaudier-Cabot and Bažant 1987; Peerlings et al. 1996). They regard cracking as an ultimate consequence of a gradual loss of material integrity. These models, however, do not predict crack opening as they rely on a continuum approach to fracture. Fictitious crack models are based on an explicit description of the discontinuity within the material (e.g. cohesive crack model model (Hillerborg et al. 1976)). They relate the crack opening to the stress level and they are based on the linear elastic (or plastic) fracture mechanics. Cohesive crack models needs proper algorithms for crack propagation, and more importantly they are not capable of describing crack initiation.

Ideally, the prediction of durability that involves inception of failure, crack location, propagation and crack opening would require to merge the continuum damage approach and the discrete crack approach into a single, consistent, computational model bridging the continuous and discrete approaches. Bridges between damage and fracture have been devised in the literature (see e.g. (Mazars and Pijaudier-Cabot 1996; Planas et al. 1993)). They rely on the equivalence between the dissipation of energy due to damage and the energy dissipated in order to propagate a crack. Given the energy dissipated in the damage process, the equivalent crack length is computed, knowing the fracture energy. Generally, the entire energy that is dissipated in the fracture process zone is ‘‘converted’’ into a crack length (Mazars and Pijaudier-Cabot 1996). Some part of this energy may be dissipated in the process zone outside from the crack and it follows that the crack length and opening are probably overestimated. The strong discontinuity approach initiated by (Simo et al. 1993) and widely used over the last decade (e.g. (Oliver et al. 2002; Larsson et al. 1998)) offers the possibility of merging in the same formulation a continuous damage model for the bulk response and a cohesive model for the discontinuous part of the kinematics. It is certainly a combination of continuum—discrete modelling that is sound from a theoretical point of view and appealing from the point of view of the physics of fracture. The issue in combining the continuum based model for crack initiation and then a discrete crack model for propagation is, however, the threshold upon which one switches from one

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analysis to the other. Usually, it is considered that the discontinuity appears when damage, stresses or strain energy reach a certain threshold fixed beforehand, which remains arbitrary (Comi et al. 2007; Simone et al. 2003). Besides as damage and fracture models do not rely on the same material description and thus on the same internal variables, jumps in time are observed on variables of interest (strain and stress) at the switch time. As we will see further, one of the outcome of the present paper is to provide an indicator on the basis of which the appearance of a discontinuity during a damage process can be defined, with an indication of accuracy. Instead of trying to combine continuum and discrete models in computational analyses, it would be attractive to derive from the continuum approach an estimate of crack opening, without considering the explicit description of a discontinuous displacement field in the computational model. This derivation could be based on some post-processing of the distribution of strain and damage in the considered structure. The main purpose of this paper is to present such an estimate of crack opening derived from a continuum model description. First, we recall the continuum approach that will be considered: the (integral) nonlocal damage model. Nonlocal models are known to possess shortcomings such as spurious boundary effects on fracture propagation (Jirasek et al. 2004; Pijaudier-Cabot et al. 2009) or incorrect initiation of damage at a crack tip (Simone et al. 2004). Still these defects in the model formulation do not alter their ability to capture a fully localised (mode I) crack. The location of the crack in the computational domain and the estimate of its opening are discussed in the second part in which we propose an improvment of an existing approach (Dufour et al. 2008). Finally, we compare our numerical procedure with experimental results obtained on a 3 point bending test on plain concrete beam in terms of crack location and opening.

2 2.1

NUMERICAL MODELLING Nonlocal damage approach

The scalar isotropic damage model (Mazars and Pijaudier-Cabot 1989) will be used in the finite element computations for representing the progressive failure. This constitutive relation exhibits strain softening. Thus a regularization technique shall be considered in order to avoid mesh dependency and ill-posedness of the governing equations of equilibrium. In this model the tensorial stress σ —strain ε relationship is expressed as follows: σ = (1 − D)C : ε

(1)

where D is the damage scalar variable and C is the elastic stiffness tensor of the sound material. Damage is a combination of two components: Dt and Dc which are damages due to tension and compression based loads respectively: D = αt Dt + αc Dc

(2)

αt and αc depend on both strain and stress tensors. Damage evolution laws for both traction and compression components read:   YD0 1 − At,c At,c Dt,c = 1 − (3) − B Y¯ −YD0 )] ¯ [ ( t,c Y e where At , Ac , Bt , Bc and YD0 are model parameters and Y¯ is defined by:   (4) Y¯ = max Y¯ , ε¯ eq with Y¯ = YD0 initially. The nonlocal equivalent strain ε¯ eq (Pijaudier-Cabot and Bažant 1987) is defined as a weighted average of the local equivalent strain εeq :  φ(x − s)εeq (s)ds (5) ε¯ eq (x) =    φ(x − s)ds Several weight functions exist in the literature, we choose the most used, i.e. the Gaussian function:     2x − s 2 (6) φ (x − s) = exp − lc where lc is the internal length of the model. Finally the local equivalent strain is defined according to Mazars criterion:

3

ε =  ε 2 (7) i +

eq

i=1

+ denotes the positive part of the principal strain εi . 2.2

Location of a crack

In (Dufour et al. 2008), an approach was proposed to extract an equivalent crack opening from a nonlocal damage computation. However, the crack position was supposed a-priori known and the computational domain was reduced to 1D. In order to apply this approach in a more general context (2D and 3D with unknown crack position), it is necessary to be able to locate an idealized crack from the nonlocal computation. Some approaches have already been proposed in the field of damage/fracture transition in order to update the crack position during the propagation. In (Comi et al. 2007), the authors proposed to fit a fourthorder polynomial on the damage field, then to propagate the crack in a direction that is perpendicular to the maximum curvature of the polynomial at the crack-tip. The main drawback is that when the damage profile does not exhibit a clear peak but a region with a small

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curvature, the fitting may be obtained with a degraded accuracy. Moreover, the accuracy of the fitting may not be sufficient at the crack-tip, which should lead to extra difficulties for the estimation of the crack direction. In (Mariani and Perego 2003), the authors proposed a similar procedure, but working on the stress-field in a half disc centered at the crack-tip. Finally, the crack is introduced perpendicular to the fitted maximal principal stress. This approach allows to work on a ‘‘sharper’’ mechanical field, but the influence of the degree of the polynomial fitting was not discussed. However, the authors reported that a third order polynomial fit was not sufficient, and that in the proposed examples a fourth order one provided consistent results. Alternatively, averaging approaches have been proposed in (Mediavilla et al. 2006) and (Wells et al. 2002). In the later, the directions pointing from the tip to the most damaged points at various radius are averaged on a V-shaped window. A last approach was proposed by (Oliver and Huespe 2004a), called ‘‘Global tracking algorithm’’. This approach was first used within the strong discontinuity approach (SDA) (Oliver and Huespe 2004b) to evaluate the crack propagation direction. The resolution of a heat-conduction like problem leads to a scalar function whose iso-values represent all the possible directions of propagation. The selection of the iso-value emanating from the crack-tip makes possible its propagation. The approach has been modified (Feist and Hofstetter 2007) in order to restrict the heat-conduction problem on a subset of elements already or potentially crossed by the crack. The approach was also applied in the context of the extended Finite Element Method (Moes et al. 1999) by (Dumstorff and Meschke 2007), and compared with various crack branching criteria. Here, we propose to use this approach in order not to propagate the crack since we use a continuum modeling but to locate it from mechanical variables at hand.

T(x)

Crack

Figure 1. Global tracking algorithm: envelopes of the vector field T(x), possible crack path and real crack.

This condition can be formulated as the following linear boundary value problem (Oliver and Huespe 2004a): div(K Grad θ) = 0

in ()

(−K Grad θ) · n = 0 θ = θd

on ∂q 

(9)

on ∂θ 

where () is the domain occupied by the solid, n is the unit vector normal to ∂q , θd is a prescribed value for the Dirichlet boundary condition and K is a second order tensor defined as: K (x) = T (x) ⊗ T (x)

2.2.1 Global tracking algorithm According to this approach, the evaluation of the propagation direction is obtained as a separate problem (linked to the mechanical one). The crack is assumed to be located along a surface (or a line in 2D) which is tangent to a vector field T (x) (with unit norm). The construction of the envelopes of T (x) supplies all the possible discontinuity lines at time t (see Figure 1). The envelopes of T (x) are described by a function θ(x) whose level contours (θ (x) = constant) define all the possible discontinuity lines, as described in Figure 1. The gradient of this function must be normal to T (x) in each point: ∂θ T (x) · Grad θ = =0 ∂ T

Possible crack path

(8)

(10)

The θ field can be assimilated as a temperature field, −K Grad θ as a heat flux, and K as an anisotropic conductivity tensor. If the Dirichlet boundary conditions are compatible with Equation (8), then a solution satisfying: ∂θ =0 (11) ∂ T is solution of the boundary value problempresented in Equation 10. In order to overcome the singularity of the problem (K is rank one), the conductivity tensor is modified as (Oliver and Huespe 2004a):

θ(x) = constant ;

K (x) = T (x) ⊗ T (x) + I

(12)

where  is a small isotropic algorithmic conductivity, and I is the second order identity tensor. Once the problem is solved, the crack can be propagated along the path defined by the iso-value of θ that passes at the crack-tip. 2.2.2 Location of the crack using the global tracking algorithm To apply this approach to the problem at hand, two main ingredients have to be adapted: (1) the definition of the T field, and (2) the location of one point of the crack. We make here the hypothesis that the idealized crack is perpendicular to the principal direction

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associated to the maximum principal strain εmax which represents the opening direction in mode I dominated loading. The T field is thus taken perpendicular to the principal direction associated to εmax . The knowledge of this field in the body makes possible to solve the boundary value problem and obtain the θ field. The last operation consists in selecting the right isovalue. We make here a last hypothesis by considering that the crack passes at the Gauss point where εmax is maximal on the body. One could have chosen the most damage Gauss point, but this choice proved to be a bad one since the damage profile is very flat, leading to crack paths that depends on the loading level. The algorithm that summarizes the process is presented in Figure 2. In practice, the thermal-like problem is not solved on the full structure, but only on the damaged zone. This allows to speed-up the process and decrease the computer requirement of the procedure. 2.3

Reduction to a 1D problem

At the end of the tracking process, a mesh of the crack is built using the iso-temperature defining the crack. The second step consists now in evaluating the opening across this idealized crack. In this contribution, it is proposed to re-use the approach that was presented in (Dufour et al. 2008). Once the crack is meshed, it is possible to apply the 1D approach on lines perpendicular to the elements (segments in 2D, triangles in 3D) defining the crack surface. In 2D, for example, a set of lines is generated from the middle of each segment of the crack (see Figure 3). In 3D this set of lines would emanate from the center of the elements that are used to discretize the crack surface.

Once these profiles are defined, the component of the local strain field along the 1D profile εN = N ·ε· N is first computed (see Figure 4). Then, this axial strain field is projected on the 1D profile as an input for the 1D crack opening procedure.

2.4

Estimation of a crack opening

We summarize in this part the key idea developped by (Dufour et al. 2008) to estimate the crack opening in a 1D structure that we use along perpendicular profiles to the idealized crack. If we assume a bar upon failure, the displacement field is a step with a jump {U } at the crack location x0 . The derivation of the displacement field gives a Dirac function for the local strain and a nonlocal strain with an amplitude of {U } and the same shape than the averaging function ψ used in the convolution product. Remark: For any regularized damage models ψ can be defined independently of the mechanical model. However, since with the nonlocal model used in the present work we already have defined a weighting function φ, we keep it, thus ψ = φ (see Equation (6)). With this procedure a nonlocal measure of strain is analytically obtained assuming a strong discontinuity kinematical field upon failure. The key point of (Dufour et al. 2008) is to compare this function with the nonlocal strain obtained by the FE mechanical computations. Several possibilities do exist in order to compare two profiles to each other. In the original paper, only the strong link were developped, i.e. the crack opening is computed so that both profiles are equal at their maximum x = x0 : ε¯ sd (x0 ) = ε¯ eq (x0 ) ⇒ ⇒

Figure 2.

Algorithm for the location of the idealized crack.

Figure 3. (red).

1D profiles (blue) generated from the crack mesh

{U }ψ(0) = ε¯ eq (x0 ) ψ(x 0 − s)ds

 ε¯ eq (x0 ) ψ(x0 − s)ds {U } = ψ(0) 

(13)

Figure 4. 1D profiles (blue) generated from the crack mesh (red).

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Furthermore, the distance between both profiles gives an indicator of the quality of the solution obtained by the FE computation using nonlocal damage model with respect to an analytical strong discontinuity approach:

  ε¯ sd − ε¯ eq  d

(14) εI (x) = ε¯ eq d

(c)

(b)

(a)



Thus, it is not an error on the crack opening itself but on the capacity of nonlocal damage to reproduce local kinematic field across the crack as in the strong discontinuity approach. A new profile comparison technique, named weak form, is proposed in the present work by equating the integral of both profiles, i.e.:

ε¯ eq d = ε¯ sd d

(15)



(d)

Thus it gives a different value for the crack opening. Since this error measure gives only a quality estimation of the model, we have performed experimental test in order to estimate by comparison the error on the crack opening itself.

Figure 5. (a) Initial picture in undeformed state, (b) picture during the crack propagation, (c) horizontal displacement field and its 3D view (d). 3

(1) 2,5

Force (kN)

2.5 Comparison vs experimental results 200

(5)

(9) (6)

1

(7) (8)

0,5

-25

Opening [micron]

160

Top of the beam

In order to quantify our approach against experimental measurements, three point bending test were performed on a notched beam. The beam dimensions are 400 mm span, 100 mm high, 50 mm thick and the notch is 20 mm high. The test is driven by the CMOD measure at the notch mouth. In order to measure the crack length and opening, we use a Digital Image Correlation technique (see Figure 5). For practical reasons, the picture frame is limited to 55 mm high from the notch tip. The crack is assumed to be vertical and thus the crack opening is estimated as the horizontal displacement jump in the horizontal direction. In order to get the crack opening evolution along the crack, 30 horizontal profiles are drawn and the displacement jumps are estimated along those profiles. For particular values of CMOD 20 (corresponding to peak load), 30, 40, 50, 60, 80, 100, 150 and 200 microns, a linear curve is fitted through the 30 measurements (see Figure 6). The fitted solid lines are prolongated in dashed line up to the CMOD measure at the notch mouth. For the numerical simulation of this test, we use the nonlocal version of Mazars’ damage model described in 2.1. Model parameters are fitted (before any crack opening estimation) on the experimental global response, i.e. force vs CMOD. Comparison between experimental and numerical curves is shown in Figure 7.

(2) (3) (4)

2 1,5

0 0

50

100

150

200

(8)

COD (micron)

120 (7)

80

(6) (5) (4) (3) (2)

40

(1)

0

0

25

Y [mm] Picture frame

Figure 6. process.

(9)

50

75

Notch

Crack opening at different stage of the loading

A good fit is obtained for material parameters summarized in Table 1. For a given CMOD, the crack shape is compared between experimental measurements and numerical estimation both using the strong and the weak link approaches. A relative error is computed between experimental crack opening and its numerical counterpart. Just after the peak (CMOD = 50 microns), the two numerical approaches are quite similar (see Figure 8.a) and slightly underestimate the measured crack opening. However for large CMOD (200 microns) the strong approach yield a large error (see Figure 8.b and d) and the weak approach always

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properties for a structure are naturally dominated by large crack openings. The numerical approach systematically underestimates the experimental crack opening, at least for a 3 point bending test. Although it is not on the safe side for an engineering use, it can be clearly explain from crack propagation considerations and by recalling that experimental crack opening are measured on the surface whereas the numerical one is performed on a 2D plane stress simulation:

Figure 7. Experimental and numerical Force vs CMOD responses.

Table 1.

Parameter fitting using nonlocal damage model. ν

E [GPa] 30

At

0.2

0.9

Bt

YD0

lc [mm]

4 000

410−5

8

50

200

Opening [micron]

Opening [micron]

Expe Num strong Num weak

40 30 20

120 80

10

40

0 -25

Expe Num strong Num weak

160

0

0

25

50

75

-25

0

25

Y [mm]

75

(b)

0.6

0.4

Strong approach Weak approach

0.5

0.3 0.4

Error [-]

Error [-]

50

Y [mm]

(a)

0.3

Strong approach Weak approach 0.2

0.2 0.1 0.1 0

0 0

10

20

30

Y [mm]

(c)

40

50

60

0

10

20

30

40

50

60

Y [mm]

(d)

Figure 8. Comparison betweem strong and weak approaches vs experimental crack opening for COD = 50 microns (a) and 200 microns (b). Corresponding errors between numerical and experimental crack openings for COD = 50 microns (c) and 200 microns (d).

provides a better estimation of the measured crack opening. The strong approach relies only on the regularized equivalent strain at one given point that may be affected by boundary effect for instance (Pijaudier-Cabot et al. 2009) and is thus more sensitive to numerical perturbations. The larger the crack opening, the better the estimation. This is a rather important result since the transfer

• The stress state is close to a plane stress condition at the beam free surfaces whereas it is close to a plane strain condition in the bulk of the beam that reduces the crack propagation velocity due to confinement. • Due to casting process the material contains less aggregate close to the boundaries and is thus weaker in the sense that aggregates are obstacles for cracking. For these two reasons, on the surface the crack is more developped in length and opening than in the core of the beam. It is clearly proved if one looks carefully at the experimental measurements of the crack opening for CMOD of 200 microns (see Figure 8-b). The extension of the plot gives a zero opening above the top of the beam, i.e. the neutral axis is out of the beam. For a bending test it means that the applied load is null. However in Fig 7 one can see that for CMOD of 200 microns the bearing capacity is not yet zero and thus outside the surface the crack has not yet propagated to the beam limits. Besides due to 2D assumption the numerical modelling gives an average crack geometry between the surface and the core of the beam. In contrary, the experimental measurements are made on the surface where the crack length and opening are the largest. Furthermore, part of the inaccuracy in the crack opening estimation is due to the spreading of strain profile that occurs in nonlocal damage models. At the end the strain profile width is related to the internal length of the model (Giry et al. 2010).

3

CONCLUSIONS

In this contribution we have presented a complete procedure used in a post-treatment analysis to get crack location, crack opening and an estimation of the error done for tensile failure. The tracking is perform solving a conduction-like FE problem based on mechanical variables. The crack opening is estimated by nonlocal strain profile comparisons with those analytically obtained from the strong discontinuity approach. Results are in good agreement with crack opening measured on a 3 point bending test by DIC technique.

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Moes, N., J. Dolbow, and T. Belytschko (1999). A finite element method for crack growth without remeshing. International Journal for Numerical Methods in Engineering 46, 131–150. Oliver, J. and A.E. Huespe (2004a). Continuum approach to material failure in strong discontinuity settings. Computer Methods in Applied Mechanics and Engineering 193(30–32), 3195–3220. Oliver, J. and A.E. Huespe (2004b). Theoretical and computational issues in modelling material failure in strong discontinuity scenarios. Computer Methods in Applied Mechanics and Engineering 193(27–29), 2987–3014. Oliver, J., A.E. Huespe, M.D.G. Pulido, and E.W.V. Chaves (2002). From continuum mechanics to fracture mechanics: the strong discontinuity approach. Engineering Fracture Mechanics 69, 113–136. Peerlings, R.H.J., R. de Borst, W.A.M. Brekelmans, and J.H.P. de Vree (1996). Gradient enhanced damage for quasi-brittle materials. International Journal for Numerical Methods in Engineering 39, 937–953. Pijaudier-Cabot, G. and Z. Bažant (1987). Nonlocal damage theory. Journal of Engineering Mechanics 113, 1512–1533. Pijaudier-Cabot, G., A. Krayani, and F. Dufour (2009). Boundary effect on weight function in nonlocal damage model. Engineering Fracture Mechanics 76(14), 2217–2231. Planas, J., M. Elices, and G.V. Guinea (1993). Cohesive cracks versus nonlocal models: Closing the gap. International Journal of Fracture 63, 173–187. Simo, J.C., J. Oliver, and F. Armero (1993). An analysis of strong discontinuities induced by strain-softening in rateindependent inelastic solids. Computational Mechanics 12, 277–296. Simone, A., H. Askes, and L.J. Sluys (2004). Incorrect initiation and propagation of failure in non-local and gradient-enhanced media. International Journal of Solids and Structures 41, 351–363. Simone, A., G.N. Wells, and L.J. Sluys (2003). From continuous to discontinuous failure in a gradient-enhanced continuum damage model. Computer Methods in Applied Mechanics and Engineering 192(41–42), 4581–4607. Wells, G., L.J. Sluys, and R. de Borst (2002). Simulating the propagation of displacement discontinuities in a regularized strain-softening medium. International Journal for Numerical Methods in Engineering 53(5), 1235–1256.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Convergence aspects of the eXtended Finite Element Method applied to linear elastic fracture mechanics Wagner Fleming Department of Civil Engineering, Universidad Católica del Norte, Chile

Detlef Kuhl Institute of Mechanics and Dynamics, Universität Kassel, Germany

ABSTRACT: In the framework of linear elastic fracture mechanics, the non-smoothness of the solution influences the accuracy of the classical p-finite element solution and consequently the optimal convergence rate, associated with smooth solutions, is not achieved. For example, in the two-dimensional case, the convergence rate for the energy norm is limited to be of order O(h1/2 ), where h is the mesh size, being independent of the order of the finite element basis (4; 12). Considering that the determination of the crack propagation direction is a crucial step in fracture modelling and that an inaccurate calculation of this parameter can lead to locking in the solution, other techniques are needed in order to improve the accuracy of the p-finite element solution. In this paper, the extended finite element method (10) is exploited in order to improve the convergence rate of the classical finite element method for problems involving cracks in brittle materials. The extended finite element method allows to model a discontinuity in the displacement field by means of the enrichment of classical finite element basis with a generalized Heaviside function and specific near-tip functions that consider the singularity of the stresses at the crack tip within the framework of the partition of unity (1). Different near-tip enrichment techniques like the classical approach enriching only the nodes of the element containing the crack tip, the enrichment of a domain of fixed size around the crack tip, the use of a cut-off function and the gathering of the degrees of freedom are considered. The influence of all these approaches on the convergence rates of the following mechanical quantities of interest is studied: • • • •

L2 -norm over the entire domain and over a subdomain not containing the crack tip. Energy norm over the entire domain and over a subdomain not containing the crack tip. Stress intensity factors. T -stress.

This paper complements the works (3; 5; 7) in order to give a better understanding of the performance of the extended finite element in fracture modelling. 1

cific near-tip functions, which take into account the discontinuity in the displacement field and the local behaviour of the solution near the crack tip. The properties of the XFEM have been analyzed in (3; 5; 7), and some variations in order to improve the method have been proposed:

INTRODUCTION

The modelling of crack propagation using the finite element method has several difficulties, the mesh needs to match the crack and due to this, remeshing is needed as the crack propagates. Furthermore, in linear elasticity, the finite element solution over regular meshes can not capture the singularity of the stress field properly and special elements are needed at the crack tip, see for example the quarter-point elements (2). In recent years, several techniques have been proposed in order to modell the fracture process (6). One if this techniques is the extended finite element method XFEM, introduced by Moës and co-workers (10), which offers the possibility of modelling crack growth without remeshing. The idea behind this method is to enrich the classical finite element approximation with a generalized Heaviside step function and spe-

• The enrichment with near-tip functions of a domain of fixed size around the crack tip. • The gathering of the degrees of freedom associated to the near-tip functions. • The use of a cut-off function. In this paper, the effect of these variations on the errors in the L2 -norm, energy norm, stress intensity factors (SIFs) and T -stress is studied for different Lagrange finite elements of polynomial order p = 1, 2 and 3.

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The paper is structured as follows: The strong and weak form of the model problem are formulated in Section 2. In Section 3, the finite element method is briefly described, together with the classical extended finite element method and its above-mentioned variations. The convergence properties of the different methods are analyzed in Section 4. Finally, concluding remarks are given in Section 5.

2

FORMULATION OF THE PROBLEM

In this section, classical results of linear elasticity are summarized in order to describe the problem in its strong form and weak form. 2.1

where σ is the Cauchy stress tensor, ε is the linear strain tensor, b the vector force per unit volume and C the elasticity tensor. Equations (1) and (2) define the strong form of the problem. 2.2

This section is devoted to obtain the weak form of the problem. For this purpose a test function δu is considered, which is zero on the boundary u and has at least square-integrable derivatives (δu ∈ H 1() ), that is δu = 0

∀X ∈ u .





Considering now that δu · Div(σ ) = Div (δu · σ ) − ∇(δu) : σ ,



 ∇(δu) : σ dV = 

 δu · b dV .

(6)



Defining now δε = ∇ s (δu) and considering the symmetry of the stress tensor, it is easy to see that ∇(δu) : σ = ∇ s (δu) : σ = δε : σ .

(7)

Replacing this expression in (6) follows (2)



 δε : σ dV =



δu · (σ · n) dS 

 δu · b dV .

+ Ω

Separating the boundary integral in the domains u , σ , d+ and d− , and using the boundary conditions (2) together with (3) leads to   δu · (σ · n) dS = δu · t ∗ dS. (9)

t

+ d d

e1



t* Cracked body and boundary conditions.

(8)



n

u

Figure 1.

δu · (σ · n) dS 

(1)

u(X ) = u∗ (X ) ∀ X ∈ u , σ (X ) · n = t ∗ (X ) ∀ X ∈ σ , σ (X ) · n = 0 ∀ X ∈ d+ , σ (X ) · n = 0 ∀ X ∈ d− ,

e3

(5)

replacing this in (4) and using the Gauss’s theorem, the previous equation can be re-written as

+

Div (σ ) + b = 0, σ = C : ε, in   1 ∇u + ∇ T u , ε= 2 together with the boundary conditions

e2

(3)

Multiplying (1) by the test function δu and integrating over the domain  leads to   δu · Div (σ ) dV + δu · b dV = 0. (4)

Governing equations

A cracked body with open domain  ∈ R3 and piecewise smooth boundary  is considered. The boundary of the domain is the junction of the disjointed parts u , σ , d+ and d− , being n its outward unit normal vector. Prescribed displacements u∗ are imposed on Dirichlet boundary u , while prescribed stresses t ∗ are imposed on Neumann boundary σ . d+ and d− represent the crack surfaces, which are assumed to be traction-free, see Figure 1. The problem is to find the displacement u, satisfying the governing equations

Weak form

 

σ

Finally, substituting (9) into (8) yields   δε : σ dV = δu · t ∗ dS + δu · b dV . σ

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(10)

Here K is the stiffness matrix, F the vector of external forces and u, abusing of the notation, the vector grouping the nodal degrees of freedom in the form

T (17) u = u11 u21 u12 u22 . . . u1NN u2NN .

This is the weak form of the equilibrium equation. No assumption related to the material behaviour is implicit. The weak form of the problem can now be formulated as follows: Given b, C, u∗ and t ∗ . Find u(X ) ∈ U such that ∀ δu(X ) ∈ U0 , equation (10) is satisfied. Here, U and U0 are the spaces defined by U = {u(X ) ∈ H 1() | u(X ) = u∗ (X ) on u }, U0 = {δu(X ) ∈ H 1() | δu(X ) = 0 on u }. 3

(11) (12)

CRACK MODELLING WITH XFEM

This Section describes the different numerical methods analyzed in this paper. The analysis is restricted to two-dimensional problems. 3.1 FEM In order to fix the notation, the classical FEM approach is resumed as follows. First, the domain  is partitioned into NE only at their boundaries overlapping elements e with 1 ≤ e ≤ NE. The element vertices are called nodes and the total number of nodes in the domain is NN . Associated to every node X i , i ∈ I = {1, 2, . . . , NN }, there is a shape function φ i , which takes the value 1 at this node and 0 at all the other nodes. Formally  1 if i = j, i j φ (X ) = δij = i, j ∈ I . (13) 0 if i = j.

3.2

i∈I

+

Thus, the support wi of the shape function φ i is given by the union of the element sub-domains connected to the node X i . In this paper, only Lagrange shape functions are considered. Now, the basic idea of the finite element method is to approximate the displacement  uand the test function δu using the same basis B = φ i i∈I , that is, u ≈ uh (X ) =



ui φ i (X ),

δu ≈ δuh (X ) =

ψ i (X ) ⎝

i∈IH p 4  

⎞ j i,j Fk (X )ck ⎠,

(18)

j=1 k=1

where IH is the set of nodes enriched with the Heaviside function and IT the set of nodes enriched with the near-tip functions   (19) IT = i ∈ IPr | X t ∈ wi ,   / wi . (20) IH = i ∈ I | wi ∩ d = ∅ ∧ X t ∈

(14)

δui φ i (X ),



⎛

i∈IT

The Heaviside function takes the value 1 on one side of the crack, and the value 0 on the other side. j The near tip functions Fk are given by

i∈I



XFEM approach

The principal idea behind XFEM is to enrich the classical finite element approximations (14) and (15), with local information about the solution u in order to capture its local features. Thus, in the context of Linear Elastic Fracture Mechanics considered in this work, the facts that the displacement field is discontinuous on the crack and that the asymptotic behaviour of the solution in a neighborhood of the crack tip is known (see Appendix A), can be incorporated into the finite element approximation. The discontinuity across the crack is treated using a generalized Heaviside step function while the asymptotic behaviour is incorporated by means of near-tip functions. With this new formulation, the method allows to model crack growth without remeshing. Consider a Lagrange finite element basis of order p, B = {φ i (X )}i∈I , where φ i is the shape function associated to the node X i . On the same mesh a Lagrange basis of order 1, B1 = {ψ i (X )}i∈IPr can be constructed, where IPr is the set of primary nodes of the mesh. With this definitions, a first version of the XFEM approximation for a boundary crack (see Figure 2) can be written as   uh (X ) = φ i (X )ui + φ i (X )H (X )bi

(15)

i∈I j

2j−1 2

j

2j−1 2

j

2j−1 2

j

2j−1 2

F1 (r, θ) = r

ui

where are the unknown nodal degrees of freedom and δui are arbitrary vectors. Inserting this approximations in the weak form (10), a system of linear equations, called the structural equation, is obtained

F3 (r, θ) = r

Ku = F.

F4 (r, θ) = r

F2 (r, θ) = r

(16)

(2j − 1)θ , 2 (2j − 1)θ cos , 2 (2j − 5)θ sin , 2 (2j − 5)θ cos . 2 sin

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(21)

At this point some remarks about the XFEM approximation (18) are worth pointing out in order to achieve an optimal convergence rate (the convergence rate of the FEM for smooth solutions): First, the shape functions multiplying the discontinuous function H (X ) need to be of the same order as the finite element basis B. This had been shown in (7) comparing the p-FEM and the XFEM approximations for the displacement jump. If p-FEM is considered, the displacement jump is approximated by a p-order polynomial. On the other hand, in the XFEM approach, if an element completely cut by the crack is considered, the displacement jump is given only by the discontinuous part of the displacement approximation, which is associated with the Heaviside function. Thus, if for example only a linear basis is considered for the Heaviside enrichment, the displacement jump would be approximated linearly, which is not compatible with the p-FEM approximations and will lead to a nonoptimal convergency rate. Another way to see this is comparing the DOFs of both approximations. Second, p sets of 4 near-tip functions (F-functions) need to be added to the displacement approximation. This functions span the singular part of the displacement field (see the asymptotic expansion in Appendix 5), related to the fractional potencies of r, which is responsible for the well known lack of convergence in the classical FEM approach. Furthermore, it is neither neccesary nor desirable to multiply the F-functions by the high order functions φ i and the order-1 ψ functions are the right choice. This is due to the fact that it is only neccesary to be able to reproduce the F-functions, which can be achieved simply combining the products ψF. This reduces the number of extra degrees of freedom and minimizes the convergence problems in the partially enriched elements (blending elements).

Figure 2. Modified near-tip enrichment: All nodes within a radius amax are enriched by the near-tip functions.

  IH = i | wi ∩ d = ∅ ∧ X t ∈ / wi ,

where X t is the position of the crack tip, X i the position of the node i and wi its support. It will be shown in Section 4, at least numerically, that if amax is independent of the mesh size h, this enrichment strategy leads to the optimal convergence rate. Nevertheless, a consequence of this strategy is that many extra degrees of freedom are now added to the classical FEM approach, and due to this, the conditioning of the stiffness matrix deteriorates with respect to classical FEM stiffness matrix. In order to reduce this effect, a specialized preconditioner is proposed in (3), while in (7) the gathering of the extra degrees of freedom related to the near-tip enrichment is considered. The second approach is studied in the next section. 3.4

3.3 Near-tip enrichment on a fixed area As the mesh size goes to zero, the influence zone of the near-tip enrichment vanishes and the optimal convergence rate is not achieved. Due to this, a modification of the enrichment strategy is considered, see also (3; 7). The idea is to enrich all the nodes within a radius amax by the near-tip functions. In this zone, the singular behaviour of the stress field is assumed to dominate the solution. Thus, a node with coordinates X i is enriched by the near-tip functions if its distance to the crack tip X t is less than amax . Figure 2 shows this enrichment strategy for a boundary crack. Within this framework, there are nodes with both enrichments, Heaviside and near-tip. The sets IH and IT can be formally defined as follows 



IT = i | X t ∈ w i ∨ X t − X i ≤ amax ,

(23)

DOF gathering

In order to reduce the number of extra degrees of freedom and, consequently, to improve the conditioning of the stiffness matrix, the gathering of the degrees of freedom associated with the near-tip enrichment is considered. Inserting the gathering condition i,j

j

ck = d k

(22)

and

i ∈ IT , k = 1, 2, 3, 4 j = 1, 2, . . . , p,

(24)

in the XFEM approximation (18), leads to uh (X ) =

 i∈I

φ i (X )ui +



φ i (X )H (X )bi

i∈IH

p 4   j j + χ h (X ) Fk (X )d k , j=1 k=1

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(25)

with χ h (X ) =



ψ i (X ).

(26)

i∈IT

where r is the distance of a point X to the crack tip, see Figure 3. This approach leads to the optimal convergence rate, provided that χ is smooth enough such p  j j that u − χ j=1 4k=1 d k Fk ∈ H k+1 for some conj

Compared with the XFEM approach without gathering, which adds 8p degrees of freedom for each node enriched with near-tip functions, the DOF gathering technique reduces the number of extra degrees of freedom to only 8p for each crack tip. Note that χ h (X ) is identical to one on those elements with all their nodes enriched with the near-tip functions, vanishes in the standard elements and varies continuously from zero to one in the blending elements (elements with only some enriched nodes). The fact that the zone where χ h falls from 1 to 0 vanishes as h → 0, leads to a lost of 1/2 in the convergence rate in the energy norm (over the entire domain), independently of the order of the polynomial approximation, see (7).

stants dk , see (12). Functions of class C k , with k = 1, 2, 3, lying in H k+1 can be defined as a function of x = (r − r1 )/(r2 − r1 ) as follows χ |[0 1] = 1 − 3x2 + 2x3

⇒

χ|[0 1] = 1 − 10x3 + 15x 4 − 6x5

χ ∈ C1 ⇒

χ ∈ C2

χ|[0 1] = 1 − 35x4 + 84x 5 − 70x6 + 20x7 ⇒ χ ∈ C3 (28)

4

ANALYSIS

In this section, five methods are compared: • The finite element method (FEM), Section 3.1. • The classical extended finite element method, where only the nodes whose support contains the crack tip are enriched with near-tip functions (XFEM), Section 3.2. • XFEM with near-tip enrichment on a fixed area (XFEM-FA), Section 3.3. • XFEM considering the gathering of the degrees of freedom associated to the near-tip enrichment (XFEM-FA-Gath), Section 3.4. • XFEM using a cut-off function (XFEM-Cutoff), Section 3.5.

3.5 Cut-off function In this section the use of a cut-off function varying continuously from 1 to 0 in a zone independent of the mesh size h is studied. This idea is actually very old and was considered in the 70’s by Strang & Fix in the framework of the classical FEM (12), and more recently by Chahine and co-workers (5) in the framework of the extended finite element method. Within this approach, the XFEM approximation is similar to (25) but with χ h replaced by a mesh independent cut-off fuction χ given by ⎧ 1 if r < r1 ⎪ ⎨ χ (r) = smooth if r1 < r < r2 ⎪ ⎩ 0 if r2 < r

r2

2

r1 1

d

r 1 , r 2 constant! Figure 3.

Enrichment with a cut-off function.

For each method, the convergence rate r, in terms of the mesh size h, is studied for the following parameters: (27)

• L2 -norm over the entire domain and over a subdomain not containing the crack tip. • Energy norm over the entire domain and over a subdomain not containing the crack tip. • Stress intensity factors. • T -stress. The tests are carried out considering Lagrange finite elements of order p = 1, 2 and 3. The domain  is the unit square [0, 1] × [0, 1] (in meters) and the crack is defined by the segment going from (0, 0.5) to (0.5, 0.5). Different regular meshes are obtained by dividing the unit square. Divisions going from 5 to 75 (in both directions) are considered, see Figure 4 for an example mesh. The mesh size h is choosen as the side length of the elements. In the case of the methods XFEM-FA and XFEMFA-Gath a value of amax = 0.1 m is choosen, while for the XFEM-Cutoff the values r1 = 0.1 m and r2 = 0.4 m are considered.

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The L2 -norm, energy norm and stress intensity factors are calculated for the solution obtained by imposing the asymptotic displacement field (30) with KI = KII = 1 N/mm3/2 as Dirichlet boundary condition, while for the T -stress the values KI = KII = 10 N/mm3/2 and T = 1 N/mm2 are used. Considering that the exact solution of the problem does not involve terms of order r 3/2 or higher, only one set of near-tip functions is added in the XFEM approximation.

Since the Heaviside enrichment is active at the Dirichlet boundary u , the known nodal degrees of freedom at the boundary are calculated by minimizing 

u∗ − uh L2 (u ) = u∗ − uh 2 d. (29) u

The subdomain considered for the L2 -norm and energy norm is the square [0.6, 0.8] × [0.6, 0.8] in the case of the XFEM methods, while for the FEM the subdomain is given by [0.625, 0.875]×[0.625, 0.875]. In the case of the stress intensity factors and T -stress, they are extracted from the numerical solution using the domain interaction integral J (1,2) and appropiate auxiliary solutions, as described in Appendix B. The domain over which J (1,2) is evaluated, is choosen as composed of all the elements with a node within a circle of radius 0.2 m around the crack tip. Thus, this domain depends only weakly on the mesh size. In all cases, the stiffnes matrix of the element containing the crack tip was obtained using the integration technique proposed in (7), where the integration points of the sub-triangles with vertex on the crack tip are obtained from the unit square via mapping. The results of the convergence test are summarized in Table 1, while Figure 5 shows the convergence curves for the stress intensity factor KII . From this results some remarks are worth mentioning:

u = u * (KI = 1, KII = 1)

L

• The finite element method shows a convergence rate equal to 1 for all the studied parameters, with exception of the energy norm in the entire domain, where a poor convergence rate r = 1/2 is achieved. This

L Figure 4.

Table 1.

Example mesh used for the analysis.

Convergence rate r in terms of the mesh size h (side length of the elements). L2 -norm

L2 -norm-sub

Energy norm

Energy norm-sub

KII

T -stress

FEM p=1 p=2 p=3

1.0 1.0 1.0

1.0 1.0 1.0

0.5 0.5 0.5

1.0 1.0 1.0

1.0 1.0 1.0

1.0 1.0 1.0

XFEM p=1 p=2 p=3

0.9 1.4 1.5

0.9 1.0 1.0

0.5 0.5 0.5

1.0 2.0 2.9

1.0 1.0 1.0

0.9 1.0 1.0

XFEM-FA p=1 p=2 p=3

2.4 3.3 4.2

2.4 3.1 3.9

1.2 2.2 3.1

1.0 2.0 3.0

2.5 4.2 −

2.5 4.2 −

XFEM-FA-Gath p=1 p=2 p=3

0.9 2.6 3.6

0.9 3.0 3.8

0.5 1.5 2.6

1.0 2.0 3.0

1.0 3.0 4.8

0.9 3.0 4.6

XFEM-Cutoff p=1 p=2 p=3

1.7 3.2 3.9

1.7 3.0 4.0

0.9 1.9 2.9

0.8 2.0 3.0

1.8 3.8 5.7

1.8 3.9 5.6

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10

02

10

04

10

06

FEM p=1 (r=1.0) FEM p=2 (r=1.0)

K II 10

FEM p=3 (r=1.0)

1

XFEM p=1 (r=1.0) XFEM p=2 (r=1.0)

08

XFEM p=3 (r=1.0) XFEM FA p=1 (r=2.5) XFEM FA p=2 (r=4.2)

10

10

10

12

10 Figure 5.

2

h/L

10

1

Convergence analysis for the stress intensity factor KII and the different enrichment strategies.

results are independent of the order of the finite element basis p. The classical XFEM presents the same behaviour, the only surprising result was a convergence rate r ≈ p for the energy norm in a subdomain not containing the crack tip. This result needs further investigation. • The methods XFEM-FA and XFEM-Cutoff show a similar behaviour, with convergence rates r ≈ p + 1 in the L2 -norm and r ≈ p in the energy norm. In the case of the stress intensity factors and T -stress, a convergence rate r ≈ 2p is achieved. For the XFEM-FA with p = 3 it is difficult to recognize a convergency rate in the case of the stress intensity factors and T -stress, probably due to the fact that the accuracy of the method is comparable with the machine precision and that the stiffnes matrix becomes very sensitive due to its poor conditioning. Actually, the condition number for p = 3 is of order O(h−8.7 ), while the XFEM-Cutoff shows an order O(h−3.0 ). • The method XFEM-FA-Gath shows a lost of convergence with respect the optimal convergence rates, but the convergence rates increase with the order of

the finite element basis. In the energy norm over the entire domain, the method shows a lost of half an order (independent of p). The same occurs for the L2 -norm over the entire domain with exception of the case p = 1, where one order was lost. In the case of the SIFs and T -stress, one order was lost in the convergence rate (again independent of p), with respect to the XFEM-FA and XFEM-Cutoff methods. This lost of convergence is attributed to the blending elements, see (7).

5

CONCLUDING REMARKS

In this paper, a numerical analysis of the extended finite element method and some of its variations was carried out. The considered variations were: • The enrichment of a domain of fixed size around the crack tip. • The gathering of the extra degrees of freedom associated to the near-tip enrichment. • The use of a cut-off function.

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The main point of analysis was the convergence rate for quantities of special interest as the L2 -norm, energy norm, SIFs and T -stress. Even though the enrichment of a fixed domain presents the best results from the convergence point of view, the fact that many extra degrees of freedom need to be added, presents an important disadvantage for this method. In comparison, the extended finite element method with a cut-off function achieves similar convergence rates without adding many extra degrees of freedom. The performance of the XFEM with the gathering technique is also worth mentioning, the method shows an improvement over the classical XFEM and since the dof gathering is nothing more than a static condensation, this technique can be easily included in a commercial software. Finally, in the context of quasi-brittle materials, when a polynomial cohesive law is considered, the asymptotic fields have been given in ?, showing that the first term of the displacement is proportional to r 3/2 . Thus, the solution u is smoother than in the case of brittle materials and better convergence rates are expected in the case of the FEM and the classical XFEM.

APPENDICES

f21n (r, θ) =

    nθ r n/2 n κ − − (−1)n sin √ 2 2 2G 2π +

f22n (r, θ) =

r n/2 √ 2G 2π



 n   n sin −2 θ 2 2



    n nθ − κ − + (−1)n cos 2 2  n   n − cos −2 θ 2 2

 (31)

and κ is the Kolosov’s constant defined as ⎧ ⎨3 − 4ν κ = 3−ν ⎩ 1+ν

Plane strain, (32) Plane stress.

The displacements corresponding to n = 0 are rigid body translations, representing the displacement of the crack tip. In (30), KI1 and KII1 are the stress intensity factors, denoted normally by KI and KII , respectively. Furthermore KI2 is related to the T-Stress by T = √ 4KI2 / 2π .

A asymptotic displacement field The asymptotic solution for the displacement field is given in polar coordinates relatively to the crack tip by (see for example (9))

B SIFs AND T-STRESS EXTRACTION



Consider a structure containing a crack and suppose that by some means, for example a numerical method, the displacements, strains and stresses are known, at least approximately. Then, the question is how to extract from this solution the stress intensity factors. In this appendix, the attention is concentrated on the extraction of the SIFs using the domain interaction integral following (8; 11). The main idea is to consider two admissible states (u(1) , ε(1) , σ (1) ) and (u(2) , ε (2) , σ (2) ), and their associated stress intensity factors (KI(1) , KII(1) ) and (KI(2) , KII(2) ), respectively. Consider the well known expresion for the J -integral for linear elastic fracture



 ∞   f11n (r, θ ) f12n (r, θ ) = f21n (r, θ ) f22n (r, θ ) u2 (r, θ) n=0 u1 (r, θ)

 ×

 KIn , KIIn

(30)

where f11n , f12n , f21n & f22n are given by f11n (r, θ) =

    r n/2 n nθ κ + + (−1)n cos √ 2 2 2G 2π  n   n − cos −2 θ 2 2

f12n (r, θ) =



B.1 Extraction of the SIFs

J =

    nθ r n/2 n κ + − (−1)n sin √ 2 2 2G 2π

KI2 KII2 + , E∗ E∗

(33)

where

∗

 n   n −2 θ − sin 2 2

E =

⎧ ⎨ ⎩

E 1 − ν2

Plane strain,

E

Plane stress.

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(34)

(2)

J (1+2) =

(KI(1) + KI(2) )2 + (KII(1) + KII(2) )2 E∗

= J (1) + J (2) +

(1)

KI

2 (K (1) K (2) + KII(1) KII(2) ). E∗ I I (35)

 P : ∇q d,

=

E ∗ (1,Mode I) J . 2

(42)

In a similar way, the stress intensity factor KII(1) can be extracted from the numerical solution, considering the auxiliary state as the Mode II of fracture (KII(2) = 1 and KI(2) = 0), which leads to

On the other hand, for a traction-free crack, the J -integral can be written in a domain form as J =−

(2)

asymptotic solution for Mode I (KI = 1 and KII = (1) 0), the stress intensity factor KI can be obtained from

For the superposition of these two states, the J integral calculated from (33) for in-plane loading, takes the form

KII(1) =

(36)

E ∗ (1,ModeII) . J 2

(43)



B.2 Extraction of the T -stress

where P is the Eshelby tensor P = ω1 − ∇ T u · σ ,

The extraction of the T -stress is very similar to that of the SIFs. In this case, the auxiliary solution can be taken as the leading term of the solution associated to a concentrated load F per unit thickness, acting at the crack tip in the e1 direction, see Figure 6. The stress field of this auxiliary solution, satisfying Div(σ ) = 0, is given by

(37)

and q(X ) is an auxiliary vector field, which vanishes on a prescribed contour 1 and takes the value e1 on the crack tip, formally  q=

0

∀ X ∈ 1 ,

e1

if X = 0.

(38)

Here, the coordinate system is defined locally, with the basis vector e1 tangent to the crack at its tip, as shown in Figure 6. Furthermore, in order for (36) to be valid, q needs to be tangent to the crack (see (11)), that is q·n=0

on

d .

σrr = −

F cos θ , πr

(44)

or equivalently,

σ11 = −

F cos3 θ , πr

σ22 = −

F cos θ sin2 θ πr

(39)

Considering again two states and evaluating (36) for the sum of these states gives J (1+2) = J (1) + J (2) + J (1,2) ,

σrθ = σθ θ = 0,

σ12 = −

F cos2 θ sin θ , πr (45)

(40)

where J (1,2) is the domain interaction integral given by J (1,2) = −



(σ (1) : ε(2) 1 − ∇ T u(2) · σ (1)



− ∇ T u(1) · σ (2) ) : ∇q d.

(41)

Equalling now (40) and (35), and considering the state 1 as the numeric solution and state 2 as an auxiliary solution, for example, the leading term of the

Figure 6. Concentrated load per unit thickness acting at the crack tip.

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and the displacement field derivatives are −F (κ cos θ + cos 3θ ), 8π Gr F ((2 − κ) cos θ + cos 3θ ), u2,2 = 8π Gr −F ((2 + κ) sin θ + sin 3θ ), u1,2 = 8π Gr F (κ sin θ − sin 3θ ). u2,1 = 8π Gr u1,1 =

(46)

The J -integral vanishes for this auxiliary solution. Now, if the J -integral is calculated for the superposition of the auxiliary solution and the general asymptotic solution (30), including the r −1/2 singularity, this yields J (1+2) = J (1) + J (2) +

FT , E∗

(47)

where the last term can be identified as the domain interaction integral and then, after choosing F = 1, the T -stress can be calculated by T = E ∗ J (1,Aux) ,

(48)

where J (1,Aux) is the domain interaction integral defined in (41). REFERENCES [1] I. Babuška and J.M. Melenk. The partition of unity method. International Journal for Numerical Methods in Engineering, 40, 725–758, 1997. [2] R.S. Barsoum. Application of quadratic isoparametric finite elements in linear fracture mechanics. International Journal of Fracture, 10, 603–605, 1974. [3] E. Bechet, H. Minnebo, N. Moës and B. Burgardt. Improved implementation and robustness study of the X-FEM for stress analysis around cracks. International Journal for Numerical Methods in Engineering, 64, 1033–1056, 2005.

[4] S. Brenner and C. Carstensen. Finite Element Methods. In Encyclopedia of Computational Mechanics, Vol. 1, E. Stein, R. de Borst and T.J.R. Hughes, editors, 2004, John Wiley & Sons. [5] E. Chahine, P. Laborde and Y. Renard. Crack tip enrichment in the XFEM method using a cut-off function. International Journal for Numerical Methods in Engineering, 00, 1–15, 2006. [6] A. Combescure, R. de Borst and T. Belytschko. Proceedings of the IUTAM Symposium on Discretization Methods for Evolving Discontinuities. Lyon, France, 4–7 September, 2006 [7] P. Laborde, J. Pommier, Y. Renard and M. Salaün. High-order extended finite element method for cracked domains. International Journal for Numerical Methods in Engineering, 64, 354–381, 2005. [8] F.Z. Li, C.F. Shih and A. Needleman. A comparison of methods for calculating energy release rates. Engineering Fracture Mechanics, 21, 405–421, 1985. [9] X.Y. Liu, Q.Z. Xiao and B.L. Karihaloo. XFEM for direct evaluation of mixed mode SIFs in homogeneous and bi-materials. International Journal for Numerical Methods in Engineering, 59, 1103–1118, 2004. [10] N. Moës, J. Dolbow and T. Belytschko. A finite element method for crack growth without remeshing. International Journal for Numerical Methods in Engineering, 46, 131–150, 1999. [11] N. Moës, A. Gravouil and T. Belytschko. Non-planar 3D crack growth by the extended finite element and level sets—Part I: Mechanical model. International Journal for Numerical Methods in Engineering, 53, 2549–2568, 2002. [12] G. Strang and G.J. Fix. An Analysis of the Finite Element Method, Prentice-Hall, 1973, Englewood Cliffs, NJ. [13] Q.Z. Xiao and B.L. Karihaloo. Asymptotic fields at frictionless and frictional cohesive crack tips in quasibrittle materials. Journal of Mechanics of Materials and Structures, 5, 881–910, 2006. [14] J.F. Yau, S.S. Wang and H.T. Corten. A mixed-mode crack analysis of isotropic solids using conservation laws of elasticity. Journal of Applied Mechanics, 47, 335–341, 1980.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Applicability of XFEM for the representation of crack bridge state in planar composite elements J. Jeˇrábek RWTH Aachen University, Aachen, Germany

R. Chudoba & J. Hegger Institute of Structural Concrete, RWTH Aachen University, Aachen, Germany

ABSTRACT: This paper reviews an enriched finite element representation of crack bridges suitable for simulating the complex damage processes in textile reinforced concrete. The heterogeneity of both the matrix and the reinforcement occurs at similar length scales of the material structure. Consequently, an improved accuracy of approximation at the hot-spots of damage is required in order to capture the relevant damage mechanisms. The explicit representation of the matrix crack is thus inevitable. Besides that, the quality of the strain and stress fields along the discontinuity is of special importance. This paper presents numerical studies focused on the reflection of the peak values in the crack bridge for selected elementary configurations. The studied approach combines the XFEM approximation of matrix displacement field with higher-order approximation of the reinforcement field. After showing the difficiencies of this approach, the possibilities of a consistent enrichment of both fields are briefly discussed. 1

INTRODUCTION

Textile reinforced concrete (TRC) is a composite material combining the advantages of fiber reinforced concrete and steel reinforced concrete. Textile reinforcement made of glass, carbon or aramid is embedded as fabrics in a cementitious matrix. The heterogeneity of both the matrix and the reinforcement occurring on similar length scales of the material structure demands for an improved accuracy to capture the relevant damage mechanisms. In the present paper, we focus on the strategy applicable for the simulation of coarse crack pattern emerging in structural details and in shear zones. In these zones a complex stress state develops leading to localization of matrix damage in a few dominant cracks (Fig. 1). With further loading the bond and yarn damage develops particularly at the crack bridges. Therefore, an accurate assessment of the ultimate failure is only possible with an improved kinematics reflecting the discontinuities in the displacement fields. We remark that in zones with fine and regular crack pattern developing under uniaxial loading (Fig. 2) the smeared approach to the damage modeling is appropriate as documented e.g. in (Scholzen, Chudoba, and Hegger, 2008) presenting a smeared model with initial and damage-induced anisotropy. The possibility to introduce discrete crack in order to better reflect the stress state in the crack bridge is studied using 2D implementation. The constructed displacement approximation is intended to serve as a

basis for improved coupled meso-micro scale simulation of planar boundary value problems with coarse crack pattern. The explicitly obtained values of crack opening and crack sliding provide the necessary input for the micro-scale representation of critical crack bridges. For the related work on this issue we refer to (Konrad, Chudoba, and Jeˇrábek 2007; Konrad, Jeˇrábek, Voˇrechovský, and Chudoba 2006). We emphasize that this work is primarily focused on the comparison and verification of the formulated enrichment approaches. An improved local kinematics in the vicinity of the crack bridge can be achieved with the help of the

Figure 1.

Shear zone with few dominant cracks.

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Figure 2.

TRC tensile specimens with fine crack patterns.

extended finite element method (XFEM) providing an efficient and elegant tool for introducing discontinuities and material interfaces into an originally smooth discretization. The discontinuities are included in a standard finite element framework by augmenting the set of displacement approximation functions with additional discontinuous fields through the partition of unity method (Melenk and Babuška, 1996). The major advantage of this method is that the enrichment is independent of the mesh and, thus, no or only minimum remeshing is required. XFEM has its roots in the work of (Belystchko and Black, 1999) who used the local partition of unity enrichment of finite elements. They enriched the nodes around a crack tip with the near-tip, linear elastic fracture mechanics (LEFM) solution. An enrichment for the nodes the support of which is fully cut by the crack introduced by (Moës, Dolbow, and Belytschko 1999). By enriching these nodes with a Heaviside function the treatment of cracks in the interior of elements became possible. In the following years the method was further enhanced and applied to numerous problem dealing with discontinuities. XFEM for three-dimensional crack modeling was first used by (Sukumar, Moës, Moran, and Belystchko 2000). An algorithm that couples the XFEM with the level set method (Osher and Sethian 1988) by applying a discontinuous function in modeling two-dimensional linear elastic crack-tip displacement fields was presented in (Stolarska, Chopp, Moës, and Belytschko 2001). Above mentioned methods were unified and further extended in (Belytschko, Moës, Usui, and Parimi 2001). A new crack tip elements and their application to cohesive cracks introduced in (Belytschko, Zi, Xu, and Chessa 2003; Zi and Belytschko 2003; Dumstorff and Meschke 2007).

2

VARIATIONAL FORMULATION OF TWO-FIELD BOUNDARY VALUE PROBLEM

 with the boundary . The domain  is decomposed according to the layout of the material components: m for the cementitious matrix, f for the fibers or reinforcement and mf for the intersection of these subdomains (see Fig. 3). Further, we introduce the zones of potential debonding e ⊆ mf where the displacement fields are explicitly resolved in the numerical representation. The boundaries of the subdomains are denoted by m and f and are further distinguished into essential (um , uf ) and natural boundary conditions (tm , tf ). The stress and strain components are aggregated in vectors σ (·) = {σ(·)xx , σ(·)yy , σ(·)xy }T and ε (·) = {ε(·)xx , ε(·)yy , ε(·)xy }T , respectively. The index (·) stands for the two material phases [m, f ]. The surface friction is denoted as τ mf = {τx , τy }T For the sake of simplicity the derivations shall be demonstrated on one-dimensional debonding problem. The local equilibrium of both layers with an implicitly assumed unit thickness is given as (see Fig. 3 right demonstrating the uni-axial stress state) ∂ T σ m − τ mf = 0,

∂ T σ f + τ mf = 0

where ∂ T denotes the differential operator with the following distribution of partial derivatives along x and z   · (·), y (·), x ∂T = . · (·), y (·), x The corresponding essential and natural boundary conditions are specified as um = u¯ m uf = u¯ f

on um on uf

nσ m = t¯m

and and

nσ f = t¯f

on tm

on tf

(2)

where n represents the boundary operator to project the stress tensor into surface tractions (n1 , n2 are the components of the normal surface vector)   · n2 n . n= 1 · n2 n1 The weak form of the boundary value problem given in Eqs. (1) and (2) can be constructed using the variation fields vm and vf 

 v m , ∂ T σ m − τ m  + (vm , um − u¯ m )um     + vm , nσ m + t¯m t + vf , ∂ T σ f + τ m  m   + (uf , uf − u¯ m )u + uf , nσ f + t¯f t = 0 f

The variational framework is established for a twolayered, two-dimensional body occupying the domain

(1)

(3)

f

where (u, v)V denotes the integration of the product of the terms u, v over V . The constitutive laws for matrix and fibers and the corresponding kinematic relations

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Figure 3.

Left: Two-field boundary value problem; Right: Infinitesimal segment of two-phase composite.

have the form σ (·) = D(·) ε(·) ,

ε(·) = ∂ u(·) · · ·

(4)

Substituting Eqs. (4) into Eq. (3) and applying the integration by parts and assuming that essential boundary conditions are implicitly satisfied by the chosen approximation we obtain the reduced weak formulation as 

 ∂ T vm , Dm ∂um  + (v m , τ mf )     − vm , t¯m  + ∂ T v f , Df ∂uf  t   − (v f , τ mf ) − v f , t¯f  = 0.

(5)

t

In the next two sections the approximation for the sought displacement fields um , uf leading to a discrete form of Eq. (5) shall be formulated and tested.

3

GLOBAL TWO-FIELD APPROXIMATION

The reinforcement displacement field is approximated using standard shape functions uf (x) = N d f =



Ni (x)df , i

(6)

i∈I

where I denotes the set of all nodes, Ni is the standard finite element shape function at node i and df , i the corresponding degree of freedom. The displacement field of the matrix is additionally enriched to introduce a crack at xξ applying the XFEM method (see Fig. 4) um (x) = N d m =

 i∈I

Ni (x)d¯ m, i +



Figure 4. XFEM enrichments N s (xξ ) by example of four node element bilinear shape functions and a single crack ξ at position xξ = 0.5.

with J denoting the set of nodes with a crack cutting its support, d¯ m and dˆ m representing the standard and additional degrees of freedom, respectively and with j introducing the XFEM jump function for the enrichment of node j: j (x) = sign( (x)) − sign( j ),

j ∈ J.

(8)

Here, represents the level set function implicitly specifying the position xξ of the crack as (xξ ) = 0. By substituting the displacement approximations (6) and (7) into the variational formulation given in Eq. (5) and applying the kinematics to both the trial and the test functions (u(·),x = N,x d (·) = Bd (·) ) the discrete weak form of the boundary value problem is obtained as     δd Tm BT , Em Am B  d m − N T , τmf 

Nj (x)j (x)dˆ m, j

     − N T , ¯tm  + δd Tf BT , Ef Af B  d f t



j∈J

(7)

+ N T , τmf

 

   − N T , ¯tf t = 0.

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(9)

Here, also the fact was exploited that the variations δd f and δd m are constant within the spatial integrals. The above equation must be fulfilled for arbitrary variations of δd f and δd m resulting in two sets of possibly nonlinear equilibrium equations  R(d) =

   Rm (d m , d f ) 0 = Rf (d m , d f ) 0

Inserting Eqs. (14) and (15) into Eq. (13) yields the algorithmic stiffness matrix at the iteration step (k)   T = B E A B dx + N T Dτ(k−1) N dx K (k−1) m m mm 

=− K (k−1) mf

(10)

K (k−1) =− fm with the residuals Rm and Rf defined as 

= K (k−1) ff



    Rm = BT , Em Am B  d m − N T ,τmf  − N T, ¯tm t 



    Rf = BT , Ef Af B  d f + N T , τmf  − N T , ¯tf  . t (11) In case of nonlinear material behavior assumed either for the matrix, reinforcement or bond, Eq. (10) must be prepared for iterative solution strategies by means of linearization i.e. by Taylor expansion neglecting quadratic and higher order terms. The expansion up to the linear term reads

R(d

(k)

) ≈ R(d

(k−1)

∂RT (d) )+ d (k) . ∂d d (k−1)

(12)

By substituting for R(d) in Eq. (10) and denoting the derivatives of residuals K (ij) = ∂R(i) /∂d (j) with i, j ∈ [m, f ] we can write down the standard iterative expression for achieving equilibrium in the form

K (k−1) mm (k−1)

K fm

K (k−1) mf



(k−1)

K ff

d (k) m

d f (k)

=−



Rm(k−1) R(k−1) f

. (13)

For the sake of simplicity, throughout this paper the material behavior of matrix and reinforcement is assumed linear elastic. A generally nonlinear bond law can be introduced for τmf (s) with the slip field defined as s = um − uf . The derivatives of the constitutive law with respect to the degrees of freedom then read ∂τmf ∂τmf ∂s ∂(−um + uf ) = = Dτ (s) ∂d m ∂s ∂d m ∂d m

(14)



 



 



N T Dτ(k−1) N dx N T Dτ(k−1) N dx 

BT Ef Af B dx +



N T Dτ(k−1) N dx. (16)

The applicability of this approach is shown on the example of a two layer 2D composite rectangle (3 × 2 meters) with uniform displacement loading of the reinforcement layer uf in x-direction (ur = 0.3 m) at the right end. The bond law is assumed linear in the form τmf = Dτ s with constant stiffness Dτ . The matrix layer um is cut by a single crack with varied orientation. The values of the material parameters are provided in Tab. 1. The example was computed using of n = 30 × 20 elements for both the matrix and the reinforcement fields with the total of 2688 degrees of freedom (including degrees of freedom for the XFEM contribution in the cracked matrix element). The obtained stress/strain state fields (Fig. 5) shall now be examined in a more detail with respect to their • convergence to the exact solution for perpedicular crack with macroscopically uniform stress field, and • ability to reproduce the kink in the reinforcement at the crack bridge. The convergence to an exact solution has been studied for the case of the matrix crack perpendicular (90◦ ) to the uniform tensile loading (σxx = 1 MPa). At the left hand side, both fields were fixed (um = 0, uf = 0). The strain/stress in y-direction is constant and the stress level of the composite in x direction is equal to one (Fig. 6). At the crack bridge σmexact = 0 and σfexact = 1. Evidently, the linear element with constant strain in the x-direction cannot capture the peak stress/strain in the reinforcement. Such a pure reflection of strains in the crack bridge, the hot spot of fiber damage, calls for further improvements of the approximation. The

= −Dτ (s)N Table 1.

and, analogically ∂τmf = Dτ (s)N . ∂d f

Parameters of the running example.

Em [Pa]

νm [−]

Ef [Pa]

νf [−]

Dτ [Pa/m]

α [◦ ]

2.0

0.0

1.0

0.0

10.0

30, 45, 60, 90

(15)

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Figure 5. Tensile specimen with inclined matrix crack discretized by 30 × 20 bilinear elements with varying crack angle (measured relatively to the loading direction); a) 90◦ ; b) 60◦ ; c) 45◦ ; d) 30◦ .

cracked

Figure 7. Maximal value of the stress in the reiforcement in the crack bridge to number of degrees of freedom in the direction of loading.

simplest option is the uniform h- and p-refinement. Figure 7 shows the convergence to an exact value of fiber stress for an increasing number of DOFs. Even for higher-order elements very slow convergence can be observed. The fact, that the kink in the reinforcement strain field is ignored leads to an underestimation of the peak stress. The fundamental problem lies in the inability of the shape function to reflect the shape of exact strain profile in the cracked element. The situation is qualitatively sketched in Figure 8. The profile of the reinforcement strain is compared

with the shape function derivatives of the linear, quadratic and cubic approximation. For linear elastic bond, the profile of the exact solution consists of two convex functions centered at the crack. Obviously, the linear and quadratic approximation can only reflect the average constant strain within the element. For a cubic element, the continuous approximation leads to concave reinforcement strain in the crack element. This contradiction to the convexity of the exact solution results in the underestimation of the peak strain. As the applied error criterion is local, it would be more appropriate to compare the number of DOFs for

Figure 6. Displacement and strain profile of specimen (a).

90◦

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non-uniform refinement with higher-order elements applied only in the in the crack-bridging elements. However, this would require kinematic constraints between elements of different order and even then the

Figure 8. Qualitative comparison between the exact solution and linear, quadratic and cubic approximation of σf .

Figure 9. Comparison of the quality of the approximation of the stress peak in the reinforcement field.

mismatch between the convexity of the exact solution and concavity of the shape functions would remain. This mismatch has the consequence, that isolated inclusion of the displacement jump into the matrix without corresponding kink in the strain field cannot reproduce a stress free crack in the matrix. For a generally inclined crack shown in Figure 5a, b and c, additional characteristics of the strain field at the crack bridge can be recognized. For linear elements, the low order of strains results in saw-tooth profile of the fiber stress along the crack bridge depicted in detail in Figure 9 (left). The consequence is the same as for the perpendicular crack: underestimated peak stress in the reinforcement and spurious stress transfer across the matrix crack. The depicted ‘‘hangovers’’ in the stress field can be reduced using the element with higher polynomial approximation. In the profile shown in Figure 9 (right) the serendipity bicubic elemets were used. Their polynomial basis is able to form the peak and visibly decreeses the strain incompatibilities at the element edges. Still, as for the perpendicular crack, the shape of the strain profile with a kink cannot be accurately reproduced using higherorder elements. The advantage of the described approach is that it can be implemented into finite element software with XFEM support without the need to modify the core components. Figure 10 shows an example of a rectangle with multiple matrix crack subjected to tensile loading. The cubic serendipity elements have been used showing an acceptable quality of the peak

Figure 10. Tensile specimen with inclined matrix cracks discretized by 30 × 20 serendipity bicubic elements; a) ideaqlization and discretization; b) strain field in the reinforcement; c) slip filed; d) strain field in the matrix.

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reflection. The example shows also the associated slip field and the matrix stresses. However, for the purpose of local improvement of the field approximation in the crack bridge, this approach seems to be a numerical overkill. Moreover, the present study has shown that an isolated inclusion of a crack into the matrix without a kink reinforcement strain field leads to an inconsistency in the strain fields and very slow convergence to an exact local value. More focused improvements with a consistent enrichment of the fiber displacement field is desirable. 3.1

CONSISTENT ENRICHMENT OF THE REINFORCEMENT

The displacement jump in the matrix um induces a kink in the gradient of the reinforcement displacement field uf . This holds for any type of the bond law between the matrix and reinforcement. Therefore, a consistent enrichment could be achieved by introducing a weak discontinuity in uf . For this purpose, the abs-enrichment and its extensions could be applied (Moës, Choirec, Cartaud, and Remacle 2003; Fries 2007). In these methods, the weak discontinuity is introduced only approximately. Even though the sufficient convergence rate in the global sense (L2 -norm) is achievable, the required quality of the strain field along the discontinuity is not generally guaranteed. An alternative approach to achieving the consistent fields has been suggested by the authors (Chudoba, Jeˇrábek, and Peiffer 2009). The performed construction of the enrichment can be summarized in two steps: • use the same XFEM approximation for the common part of uf and um , introducing a crack in both fields, • add an additional jump term with oposite signs to both the matrix and the reinforcement fields, • constrain the reinforcement field to recover the displacement continuity.       +N˜ d˜ m um 1  = N d + ξ N d + uf 1 −N˜ d˜ f

The major advantage of such an approach is the possibility to center the enrichment in the crack and to keep the original discretization in the regions without cracks. Thus, the matrix and reinforcement fields are only distinguished in the zones with debonding.

4

CONCLUSIONS

The purpose of the present study was to assess the feasibility of the numerical approximation for twofield problems with XFEM discontinuity introduced in one field. The reproduction of the field values in the discontinuity has been used as a criterion for qualitative evaluation of the approximation. The isolated inclusion of the jump in one field leads to an underestimated value of the bridging stresses in the reinforcement phase. This issue is critical, since the failure of the reinforcement initiates the ultimate failure of the structure. The work motivates further improvements of the crack bridge enrichments. One possible approach has been presented recetly by the authors in (Chudoba, Jeˇrábek, and Peiffer 2009).

ACKNOWLEDGEMENT

The approximation has the following structure: 

Even though the formulation of the kinematic constraint for 2D problems is possible, it leads to a rather complex data structure required for the implementation. The complexity arises from the need to distinguish the DOFs at the edges cut by the crack. Further, for higher order elements the number and positioing of the kinematic constraints must reflect the order of the shape functions. • In the latter case, the variational formulation of the equilibrium condition is augmented with terms corresponding to the crack bridging forces. The most robust approach of this class seems to be the Nitsche’s method (Hansbo 2005).

(17)

The shared part of the displacement (first term) introduces a jump in both fields. The same local approximation term containing a jump N˜ is added to both fields with an oposite sign. As a consequence, the approximation of both fields remains of the same order. In order to recover the continuity of uf in Eq. (17), either the kinematic or the static constraint can be applied: • The recovery of the reinforcement continuity using a kinematic constraint for uniaxial case has been presented in (Chudoba, Jeˇrábek, and Peiffer 2009).

The work has been supported by Deutsche Forschungsgemeinschaft (DFG) in the framework of the collaborative research center SFB 532 Textile-reinforced concrete, development of a new technology. The support is gratefully acknowledged.

REFERENCES Belytschko, T. and T. Black (1999). Elastic crack growth in finite elements with minimal remeshing. International Journal for Numerical Methods in Engineering 45, 601–620. Belytschko, T., N. Möes, S. Usui, and C. Parimi (2001). Arbitrary discontinuities in finite elements. International Journal for Numerical Methods in Engineering 50, 993–1013.

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Belytschko, T., G. Zi, J. Xu, and J. Chessa (2003). The extended finite element method for arbitrary discontinuities. In Proccedings of Computational Mechanics— Theory and Practice, Barcelona, Spain. CIMNE. Chudoba, R., J. Jeˇrábek, and F. Peiffer (2009). Crack-centered enrichment for debonding in two-phase composite applied to textile reinforced concrete. International Journal for Multiscale Computational Engineering 7(4), 309–328. Dumstorff, P. and G. Meschke (2007). Crack propagation criteria in the framework of x-fembased structural analyses. Int. J. Numer. Anal. Meth. Geomech. 31, 239–259. Fries, T.P. (2007). A corrected xfem approximation without problems in blending elements. Int. J. Numer. Meth. Engng. 75, 503–532. Hansbo, P. (2005). Nitsches method for interface problems in computational mechanics. GAMM-Mitt. 28, 183–206. Konrad, M., R. Chudoba, and J. Jeˇrábek (2007). Influence of the heterogeneity at the microlevel on the multi-cracking performance of textile reinforced concrete. In SEMC 2007: The Third International Conference on Structural Engineering, Mechanics and Computation. Konrad, M., J. Jeˇrábek, M. Voˇrechovský, and R. Chudoba (2006). Evaluation of mean performance of cracks bridged by multifilament yarns. In EURO-C 2006: Computational Modelling of Concrete Structures, pp. 873–880. Taylor and Francis Group, London. Melenk, J. and I. Babuška (1996). The partition of unity finite element method basic theory and applications. Moës, N., M. Choirec, P. Cartaud, and J.F. Remacle (2003). A computational approach to handle complex microstructure geometries. Comp. Meth. Appl. Mech. Engng 192, 3163–3177.

Moës, N., J. Dolbow, and T. Belytschko (1999). A finite element method for crack growth without remeshing. International Journal for Numerical Methods in Engineering 46. Osher, S. and J. Sethian (1988). Front propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations. Journal of Computational Physics 79, 12–49. Scholzen, A., R. Chudoba, and J. Hegger (2008). Numerical simulation of textile reinforced concrete using a microplane-type model with initial anisotropy. In B. Topping and M. Papadrakakis (Eds.), Ninth International Conference on Computational Structures Technology. Stolarska, M., D. Chopp, N. Moës, and T. Belytschko (2001). Modelling crack growth by level sets in the extended finite element method. International Journal for Numerical Methods in Engineering 51(8), 943–960. Sukumar, N., N. Moës, B. Moran, and T. Belytschko (2000). Extended finite element method for three-dimensional crack modelling. International Journal for Numerical Methods in Engineering 48(11), 1549–1570. Zi, G. and T. Belytschko (2003). New crack-tip elements for XFEM and applications to cohesive cracks. International Journal for Numerical Methods in Engineering 57, 2221–2240.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Localization properties of damage models Milan Jirásek & Martin Horák Department of Mechanics, Faculty of Civil Engineering, Czech Technical University in Prague, Czech Republic

ABSTRACT: The classical localization condition based on vanishing determinant of the acoustic tensor is explored in detail for several types of damage models. It is shown that the localization properties of isotropic models are not in full agreement with intuitive expectations. More satisfactory results are found for the rotating crack model, considered as a special type of an anisotropic damage model. 1

a discontinuity in certain components of the strain rate. Combination of the traction continuity conditions, rate form of the constitutive equations and Maxwell compatibility conditions leads to the classical criterion for the loss of ellipticity based on singularity of the acoustic tensor. The fundamental question is under which conditions the inelastic strain increments can localize in one or more narrow bands separated from the remaining part of the body by weak discontinuity surfaces. Across such surfaces, the displacement field remains continuous but the strain field can have a jump. At the onset of localization, the current strains are still continuous and the jump appears only in the strain rates. Classical localization analysis was inspired by the early works of Hadamard (1903) and Hill (1958) and developed, among others, for plasticity by Rice (1975) and Runesson (1991) and for damage by Rizzi et al. (1996). Under certain assumptions, the necessary condition for the formation of a weak discontinuity is the existence of a unit vector n for which the tensor

INTRODUCTION

Realistic description of the mechanical behavior of quasibrittle materials such as concrete requires constitutive laws with softening. From the physical point of view, softening can be attributed to the propagation and coalescence of defects, e.g. voids and cracks. It is well known that softening may lead to localization of inelastic strain into narrow process zones. For traditional models formulated within the classical framework of continuum mechanics, such zones have an arbitrarily small thickness, and failure can occur at extremely low energy dissipation, which is not realistic. The mathematical model becomes ill-posed and the numerical solutions suffer by pathological sensitivity to the discretization parameter, e.g. to the size of finite elements. It is therefore important to clearly understand the conditions under which localization may occur, and to limit the application of traditional continuum damage mechanics to the range of material states that do not allow for localization of damage into arbitrarily thin bands. Beyond this range, special enhancements acting as localization limiters are necessary. This paper analyzes the localization properties of

Q =n·D·n

• a family of simple isotropic damage models with one damage variable and with different definitions of the damage-driving equivalent strain (Mazars, Rankine, modified Mises, energy release rate), • an isotropic damage model for concrete with two damage variables characterizing separately tensile and compressive damage, • a rotating crack model, which can be interpreted as an anisotropic damage model.

2

INCIPIENT WEAK DISCONTINUITIES

From the mathematical point of view, the onset of localization can be characterized as the appearance of

(1)

becomes singular. Here, D denotes the tangent material stiffness tensor. Vector n for which the localization condition det Q = 0

(2)

is satisfied represents the normal to a potential discontinuity surface. The normalized nontrivial solution m of the homogeneous equation Q·m=0

(3)

is the so-called polarization vector, which serves as an indicator of the discontinuity mode. The angle between vectors m and n characterizes the failure mode, ranging from tensile splitting with m = n (Fig. 1b) to shear sliding with m perpendicular to

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(a)

(b)

n

nm

(c) n m

Figure 1. (a) Body with a localization band, (b) tensile splitting (mode I), (c) shear sliding (mode II).

n (Fig. 1c) and further to compaction with m = −n. From the mathematical point of view, singularity of the localization tensor1 Q indicates the loss of ellipticity. The localization tensor defined in (1) depends on the tangent stiffness tensor D and on the unit normal to the discontinuity surface, n. With certain exceptions (e.g. models with multiple loading conditions or incrementally nonlinear models), the tangent stiffness can be considered as dependent on the current state only and thus known. The vector n, however, is not given in advance. Therefore, localization analysis consists in searching for a unit vector n for which the localization tensor becomes singular. If such a vector does not exist, the strain field must remain continuous. Singularity of the localization tensor for a certain vector n indicates that a strain jump can develop across a surface with normal n. One should bear in mind that this condition for the appearance of a weak discontinuity is only necessary but not sufficient because the localization analysis presented here is purely local, restricted to the level of a material point and its infinitely small neighborhood. Whether the discontinuity surface indeed develops in a finite body depends on the state of the surrounding material and on the boundary conditions. Nevertheless, analysis of the localization tensor is widely used as a powerful indicator of potential discontinuous failure modes. In the artificial but illustrative case of a body under uniform stress, the localization condition (2) can be satisfied at all points simultaneously. The local discontinuities potentially appearing at individual points can then easily merge into global discontinuity planes. For instance, one can imagine solutions with a band (layer) under uniform strain rate, enclosed by two parallel discontinuity planes that separate the band from the remaining part of the body, in which the strain

1 The localization tensor Q is sometimes referred to as the acoustic tensor, because if the tangent stiffness tensor D is taken as the elastic stiffness tensor De , the eigenvalues of the corresponding acoustic tensor Qe = n · De · n divided by the mass density are squares of the speeds of elastic waves propagating in the direction of n. The corresponding eigenvalues are polarization vectors that determine the type of waves (longitudinal, transversal, mixed).

rate is also uniform but different from the strain rate inside the band; see Fig. 1a. If the polarization vector m is aligned with the normal vector n, the difference between the strain rates inside and outside the band corresponds to stretching of the band in the normal direction; see Fig. 1b. This discontinuous mode is a precursor to splitting failure and is denoted as mode I. On the other hand, if the polarization vector m is perpendicular to vector n, i.e., parallel with the discontinuity planes, the failure occurs by shear slip and this is referred to as mode II; see Fig. 1c. For general vectors m, failure is of a mixed type, and the angle between m and n characterizes the failure mode. Under compressive loading, compaction failure with a negative scalar product m · n can arise. 3 3.1

SIMPLE ISOTROPIC DAMAGE MODELS Basic Equations

In this section, we consider a family of simple isotropic damage models with one scalar damage variable ω, driven by the equivalent strain. The basic equations consist of the stress-strain law σ = (1 − ω)De : ε

(4)

damage law ω = g(κ)

(5)

and loading-unloading conditions f (ε, κ) ≤ 0,

κ˙ ≥ 0,

f (ε, κ) κ˙ = 0

(6)

in which g is the damage evolution function, f (ε, κ) = εeq (ε) − κ

(7)

is the damage loading function, εeq is a scalar measure of the strain level called the equivalent strain, and κ is an internal variable that corresponds to the maximum level of equivalent strain ever reached in the previous history of the material. The choice of the specific expression for the equivalent strain directly affects the shape of the elastic domain in the strain space and, as will be shown in Section 3.4, also the localization properties of the model. From the rate form of the basic equations it is easy to derive the (elastic-damaged) tangent stiffness tensor Ded = (1 − ω)De − g  σ¯ ⊗ η

(8)

Here, (1 − ω)De is the unloading (secant) stiffness, g  = dg/dκ is the derivative of the damage function, σ¯ = De : ε is the effective stress, and η = ∂εeq /∂ε is a second-order tensor obtained by differentiation of

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the expression for equivalent strain with respect to the strain tensor. 3.2

Localization condition

The specific form of the necessary condition for the onset of localization (incipient weak discontinuity) depends on the particular choice of the equivalent strain definition, but its general form can be elaborated for the entire family of simple isotropic damage models (4)–(7). As shown in detail e.g. by Jirásek (2007), it is fully sufficient to restrict attention to the case when the tangent stiffness is the same on both sides of the discontinuity, which leads to the localization condition det Qed = 0

(9)

where Qed = n · Ded · n is the elasto-damage localization tensor. Substituting from (8), we obtain Q ed = (1 − ω)Qe − g  (n · σ¯ ) ⊗ (η · n)

(10)

the simplest case of a one-dimensional damage model. All tensors become scalars, the elastic stiffness tensor De is replaced by Young’s modulus E, the equivalent strain εeq is the strain εeq itself (we consider monotonic tensile loading), and tensor η is replaced by the scalar η = dεeq /dε = 1. The unit normal vector n is also replaced by the scalar n = 1, and so there is no difference between the localization tensor and the tangent stiffness. Realizing that the effective stress under uniaxial loading is σ¯ = Eε and substituting into (8) we get the tangent modulus Eed = (1 − ω)E − g  Eε = (1 − ω − g  ε)E

The localization condition det Qed = 0 is now written as Eed = 0, which means that the loss of ellipticity occurs when the peak of the stress-strain curve is reached. This is of course the result that would be expected intuitively. The value of g  corresponding to vanishing tangent stiffness Eed is

where Qe = n · De · n is the elastic acoustic tensor. Due to the special structure of the localization tensor (10), it is easy to show that condition (9) is satisfied if and only if 

g n·η·

Q −1 e

· σ¯ · n = 1 − ω

 gcrit =

 gcrit =

1−ω ¯ · n) maxn=1 (n · η · Q −1 e ·σ

1−ω ε

(14)

This is exactly what we obtain from the general formula (12) by substituting 1 for η and n, E for Qe and Eε for σ¯ .

(11)

¯ · n depends on the elasThe product n · η · Q −1 e ·σ tic constants, on the current state of the material and on the assumed direction of discontinuity plane. For given η, De and σ¯ , it is a fourth-order polynomial of the components of the unit vector n. Variable g  (the derivative of the damage function with respect to the equivalent strain) indicates how ‘‘dramatically’’ damage evolves. If g  is sufficiently small, the left-hand side of (11) is less than the right-hand side for all unit vectors n and the strain must remain continuous. The minimum value of g  for which (11) can hold is

(13)

3.4

Multi-dimensional case

Formula (13) has been derived under uniaxial loading but it turns out that the tangent modulus Eed is directly related to the the slope of the stress-strain curve even under multiaxial stress, provided that the loading is proportional. Suppose that the strain evolution is described by

(12)

 , the localization tensor Q ed is regular If g  < gcrit for all possible directions n, and strain discontinuities  , the localization tensor Qed are excluded. If g  < gcrit is singular for that particular direction n which maximizes n · η · Q −1 ¯ · n, and a weak discontinuity e · σ across a plane perpendicular to that direction can start  , there exist infinitely evolving. Finally, if g  < gcrit many directions n for which Qed is singular, and the discontinuity can evolve even ‘‘more easily’’.

3.3 One-dimensional case To get more insight into the meaning of the critical value of g  , let us reduce the results derived so far to

ε = μεref

(15)

where μ is a scalar multiplier that parameterizes the loading process and εref is the reference value of the strain tensor, which specifies the type of loading. Assuming that the expression for equivalent strain is positively homogeneous of degree 1 (which is true for all the equivalent strain definitions presented in Section 3.5), the corresponding evolution of stress under proportional loading with monotonically increasing parameter μ is expressed from (4)–(7) as σ = [1 − g(μκref )]μσ¯ ref

(16)

where κref = εeq (εref ) is the equivalent strain at the reference state and σ¯ ref = De : εref is the effective stress at the reference state. Denoting Dν = De /E = elastic stiffness tensor constructed for a unit value of Young’s

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modulus, we can rewrite (16) and its rate form as

terms of the principal components gives the fourthorder polynomial

σ = Eu Dν : ε

(17)

σ˙ = Eed Dν : ε˙

(18)

⎡ ⎤ 3 3 3 ηI n2I σ¯ I n2I ⎥ 1 ⎢ 2 I =1 I =1 ϕ(n1 , n2 , n3 ) = ⎣ ηI σ¯ I nI − ⎦ G 2(1 − ν) I =1

where

(25)

Eu = [1 − g(μκref )]E

(19)

is the unloading modulus and Eed = [1 − g(μκref ) − μκref g  (μκref )]E

(20)

is the tangent modulus. Obviously, (20) is a generalized form of (13). At the onset of localization, g  is  equal to gcrit evaluated from (12) with ω = g(μκref ). The corresponding ratio between the tangent and unloading moduli is 

Eed Eu

 =1− crit

κref maxn=1 ϕ(n)

(21)

where ¯ ref · n ϕ(n) = n · η · Q−1 e (n) · σ

(22)

in which ηI and σ¯ I , I = 1, 2, 3, are the principal values of η and σ¯ , and NI , I = 1, 2, 3, are the components of n with respect to the principal coordinate system. This polynomial has to be maximized under the constraint n21 + n22 + n23 = 1. Renaming n21 as N1 , n22 as N2 and n23 as N3 , we convert the objective function f into a quadratic function of arguments N1 , N2 and N3 , and the constraint into a linear one, but additional constraints N1 ≥ 0, N2 ≥ 0 and N3 ≥ 0 must be imposed. The resulting optimization problem has a strictly concave objective function and a convex admissible domain, and so the maximum exists and is unique. It can be obtained by methods of quadratic programming. Two-dimensional localization analysis under plane stress conditions can be based on the general formulae (12) or (21), but all tensors must be interpreted as two-dimensional ones, with indices running from 1 to 2. Furthermore, the elastic acoustic tensor must be derived from the elastic stiffness tensor for plane stress, and its inverse is given by

is an auxiliary function of the unit vector n. The critical stiffness ratio (21) is much easier to interpret than the critical value of g  . For a specific version of simple damage model (with a given expression for the equivalent strain) and given value of Poisson’s ratio, the critical stiffness ratio depends only on the type of loading, characterized by the reference effective stress σ¯ ref . Evaluation of εref = D−1 : σ¯ ref , κref = εeq (εref ) and η(εref ) e is straightforward. The only demanding step is the maximization of function ϕ. In the general three-dimensional setting, the inverse of the elastic acoustic tensor is given by Q−1 e =

1 G

  n⊗n δ− 2(1 − ν)

(23)

where G is the elastic shear modulus and δ is the unit second-order tensor (Kronecker delta). The function ϕ to be maximized with respect to n is thus

Q−1 e

1 = G





1+ν δ− n⊗n 2

(26)

It is also important to realize that even if the problem is analyzed under plane stress conditions, the evaluation of equivalent strain must take into account the outof-plane normal strain ε3 , which is in general nonzero and can be expressed from the condition of zero outof-plane normal stress as ε3 (ε1 , ε2 ) = −

ν (ε1 + ε2 ) 1−ν

(27)

where ε1 and ε2 are the in-plane principal strains. If εeq (ε1 , ε2 , ε3 ) is the original expression for equivalent strain, in plane-stress calculations it is replaced by ∗ εeq (ε1 , ε2 ) = εeq (ε1 , ε2 , ε3 (ε1 , ε2 ))

(28)

and its derivatives are evaluated as

  1 (n · η · n)(n · σ¯ ref · n) n · η · σ¯ ref · n − ϕ(n) = G 2(1 − ν)

η1∗ =

(24) The principal directions of tensors σ¯ ref = De : εref and η = ∂εeq /∂ε are the same, and (24) rewritten in

η2∗ =

∗ ∂εeq

∂ε1 ∗ ∂εeq

∂ε2

=

∂εeq ∂εeq ∂ε3 η3 ν + = η1 − ∂ε1 ∂ε3 ∂ε1 1−ν

(29)

=

∂εeq ∂εeq ∂ε3 η3 ν + = η2 − ∂ε2 ∂ε3 ∂ε2 1−ν

(30)

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Principal values η1∗ and η2∗ are then substituted into (25) instead of η1 and η2 , and the sums are taken over I = 1, 2. After all the described modifications, we obtain a polynomial  1 ∗ ϕ (n1 , n2 ) = η∗ σ¯ 1 n21 + η2∗ σ¯ 2 n22 (1 − ω)G 1  1+ν ∗ 2 (η1 n1 + η2∗ n22 )(σ¯ 1 n21 + σ¯ 2 n22 ) − 2 (31) which has to be maximized under the normalizing constraint n21 + n22 = 1. Renaming n21 as N1 and n22 as N2 = 1−N1 , we convert the objective function ϕ ∗ into a quadratic function of one single argument N1 and we automatically satisfy the normalizing constraint, but additional inequality constraints 0 ≤ N1 ≤ 1 must be imposed. The coefficient multiplying the quadratic term N12 is a positive constant times −(η1∗ − η2∗ ) (σ¯ 1 − σ¯ 2 ). Without any loss of generality, we can order the principal stresses such that σ¯ 1 ≥ σ¯ 2 . The equality sign corresponds to the special cases of equibiaxial tension or equibiaxial compression. In all other cases, the function to be maximized is strictly concave, provided that η1 > η2 (which is indeed verified for all the expressions for equivalent strain presented in the next section). The value of N1 at which the first derivative of ϕ ∗ with respect to N1 vanishes is N1∗

η=

∂εeq ε = ∂ε εeq

(34)

2. For the Rankine-type definition of equivalent strain σ¯ 1 E

εeq =

(35)

we get η=

1 ∂ σ¯ 1 ∂ σ¯ 1 : = (p1 ⊗ p1 ) E ∂ σ¯ ∂ε E   ν 1 : De = δ + p 1 ⊗ p1 1 + ν 1 − 2ν

(36)

where σ¯ 1 is the maximum principal effective stress and p1 is the unit vector in the corresponding principal direction. 3. de Vree, Brekelmans, and van Gils (1995) introduced the modified von Mises expression εeq =

1 (k − 1)I1ε + 2k(1 − 2ν) 2k



12kJ2ε (k − 1)2 2 I + (1 − 2ν)2 1ε (1 + ν)2

(37)

η∗ σ¯ 1 + νη2∗ σ¯ 2 − (1 + ν)(η1∗ σ¯ 2 + η2∗ σ¯ 1 )/2 = 1 (1 + ν)(η1∗ − η2∗ )(σ¯ 1 − σ¯ 2 ) (32)

If N1∗ is between 0 and 1, the unit normal to the potential discontinuity curve has coordinates n1,crit =

± N1∗ and n2,crit = ± 1 − N1∗ . Formally, four solutions are obtained, but only two of them represent physically different directions, which are symmetrically placed with respect to the principal axes. If formula (32) gives N1∗ ≥ 1, then the unit normal has components n1,crit = 1 and n2,crit = 0 and the discontinuity is perpendicular to the major principal axis. If formula (32) gives N1∗ ≤ 0, then the unit normal has components n1,crit = 0 and n2,crit = 1 and the discontinuity is perpendicular to the minor principal axis. 3.5

where εI , I = 1, 2, 3, are the principal strains, and the brackets . . . denote the positive part. The corresponding tensor η is obtained as

Specific expressions for equivalent strain

Localization properties will be scrutinized for models incorporating the following formulae for equivalent strain: 1. Mazars (1984) defined the equivalent strain as  3 εeq = ε  = εI 2 (33) I =1

in which I1ε = δ : ε

(38)

is the first strain invariant (trace of the strain tensor), J2ε =

1 e:e 2

(39)

is the second deviatoric strain invariant, e = ε − (I1ε /3)δ is the deviatoric part of strain, and k is a model parameter controling the ratio between the uniaxial compressive strength fc and uniaxial tensile strength ft . The corresponding tensor η is given by (k−1)2

I δ + (1+ν)2 e k −1 (1−2ν)2 1ε δ+  η= 2k(1 − 2ν) (k−1)2 2 12kJ2ε 2k (1−2ν)2 I1ε + (1+ν) 2 6k

(40) 4. Formulations based on the energy release rate can be presented in the format considered here if the

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equivalent strain is defined as the scaled energy norm, ε : De : ε E

(41)

which leads to η=

1 De : ε = Eεeq εeq



I1ε δ e + 3(1 − 2ν) 1 + ν

critical stiffness ratio

εeq =

0

-0.5

-1 -135

(42)

3.6 Localization characteristics To get an overall idea about the influence of the definition of equivalent strain on localization properties, the localization condition has been evaluated for proportional loading under plane stress conditions with different ratios of the in-plane principal stresses σ1 and σ2 . The type of stress is characterized by the ratio σ2 /σ1 or, more conveniently, by the stress angle ζ chosen such that tan ζ = σ2 /σ1 . This can be achieved by setting the principal values of the reference effective stress to σ¯ ref ,1 = E cos ζ and σ¯ ref ,2 = E sin ζ . To get σ1 ≥ σ2 , values of ζ are considered in the range from −135◦ (equibiaxial compression) to 45◦ (equibiaxial tension). Values ζ = −90◦ , −45◦ and 0◦ correspond respectively to uniaxial compresion, shear and uniaxial tension. The principal strains at the reference state are given by

simple Mazars Rankine modified Mises Mises energy

0.5



-90

-45

90

simple Mazars Rankine modified Mises Mises energy

75 60 45 30 15 0 -135

-90

-45

εref ,2 = sin ζ − ν cos ζ

(44)

εref ,3 = −ν(cos ζ + sin ζ )

(45)

Depending on the specific version of the model, the equivalent strain at the reference state κref = εeq (εref ) an the tensor η(εref ) are evaluated according to the appropriate formulae (33)–(42). For each value of stress angle ζ , the critical stiffness ratio (21) is then calculated. The direction of the potential discontinuity plane is characterized by the critical angle, defined as the angle between the normal vector n and the direction p1 of maximum (in-plane) principal stress. The mode of discontinuity is characterized by the polarization angle, defined as the angle between the normal n and the polarization vector m. The results are graphically presented in Figs. 2–4 for Poisson’s ratio ν = 0.18. For the modified Mises definition of equivalent strain, the ratio between the compressive and tensile strength is set to k = 10. For comparison, the (non-modified) Mises definition with k = 1 is also considered. Fig. 2 shows the dependence of the critical ratio Eed /Eu on the stress angle ζ . It is interesting to note that the critical stiffness remains nonpositive only for the energy-based equivalent strain, which is the only

0

45

stress angle [deg]

Figure 3. Simple isotropic damage models, plane stress analysis: dependence of the critical angle on the stress angle ζ .

polarization angle [deg]

(43)

45

Figure 2. Simple isotropic damage models, plane stress analysis: dependence of the critical stiffness ratio Eed /Eu on the stress angle ζ .

180

εref ,1 = cos ζ − ν sin ζ

0

stress angle [deg]

critical angle [deg]



1

simple Mazars Rankine modified Mises Mises energy

135

90

45

0 -135

-90

-45 stress angle [deg]

0

45

Figure 4. Simple isotropic damage models, plane stress analysis: dependence of the polarization angle on the stress angle ζ .

approach that leads to a symmetric tangent stiffness. For the classical Mises definition based on the second deviatoric invariant (i.e., on the part of elastic energy stored in the change of shape), the critical modulus can become slightly positive. The other definitions of equivalent strain lead for certain stress combinations to relatively large positive values of critical modulus. This means that if the damage law is formulated such that the stress-strain curve has a hardening part, localization can occur already before the peak. This is the case for stress states close to shear or uniaxial compression, but for the modified Mises model even for uniaxial tension. On the other hand, for stress states

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introduced two damage parameters, ωt and ωc , that are computed from the same equivalent strain (33) using two different damage functions, gt and gc . Function gt is identified from the uniaxial tensile test and gc from the compression test. The damage parameter entering the constitutive equation (4) is ω = ωt under tension and ω = ωc under compression. Under general stress, the value of ω is obtained as a linear combination

(a)

(b)

Figure 5. Results of finite element simulations on skewed meshes showing numerically computed localized strain for simple damage model with (a) Mazars and (b) Rankine definition of equivalent strain.

close to biaxial tension or biaxial compression the critical modulus is negative and localization can occur only if the softening is sufficiently ‘‘steep’’. Fig. 3 shows the dependence of the critical angle on the stress angle ζ . Here it is perhaps surprizing that, under uniaxial tension, all formulations (with the exception of the modified Mises model) predict a discontinuity plane that is not perpendicular to the loading direction. The deviation is considerable not only for the energy-based and classical Mises formulations, which are not really appropriate for quasibrittle materials, but also for the Rankine and Mazars formulations. This is confirmed by the results of a finite element simulation of the uniaxial tensile test in Fig. 5. Skewed meshes containing layers of elements of different inclinations are used, in order to allow for the formation of a localized damage band in an ‘‘arbitrary’’ direction. It appears that, on such meshes, no imperfection is needed to trigger the bifurcation from a uniform state. The numerical solution tends to spontaneously localize into one layer of elements with the most favorable orientation. This orientation is very close to the theoretically predicted one (indicated in the figure by the inclined line). Fig. 4 shows the dependence of the polarization angle on the stress angle ζ . Again, under uniaxial tension all formulations with the exception of the modified Mises model predict a nonzero polarization angle, i.e., a mixed-mode discontinuity rather than pure tensile splitting.

(46)

where the coefficients αt and αc take into account the nature of the stress state. In the original model, Mazars (1984) used αt = α and αc = 1 − α where α=

3 εtI εI 2 εeq I =1

(47)

is a dimensionless factor that depends on the principal values εtI , I = 1, 2, 3, of the part of strain εt = D−1 e : De : ε

(48)

that corresponds to the positive part of stress. In more recent implementations of Mazars model, an additional improvement is achieved by setting αt = α β ,

αc = (1 − α)β

(49)

where the exponent β = 1.06 slows down the evolution of damage under shear loading (i.e., when principal stresses do not have the same sign). Note that if all principal stresses are nonnegative we have αt = 1, αc = 0, and ω = ωt , and if all principal stresses are nonpositive we have αt = 0, αc = 1, and ω = ωc . These are the ‘‘pure tensile’’ and ‘‘pure compressive’’ stress states. For intermediate stress states, the value of ω is between ωt and ωc , depending on the relative magnitudes of tensile and compressive stresses.

MAZARS MODEL (a)

(b)

3

0

2.5

-5 stress [MPa]

For concrete and other materials with a high ratio of compressive to tensile strength, the simple models presented in the preceding section can provide realistic results only if the failure has a predominantly tensile character. If the model should be used in rather general loading situations, certain modifications must be introduced.

stress [MPa]

4

ω = αt ωt + αc ωc

2 1.5 1 0.5 0

A popular damage model specifically designed for concrete was proposed by Mazars (1984, 1986). He

-15 -20 -25

0

0.1

0.2

0.3

0.4

strain [1/1000]

4.1 Basic equations

-10

-30

-3 -2.5 -2 -1.5 -1 -0.5

0

strain [1/1000]

Figure 6. Stress-strain curves for Mazars damage model constructed for (a) uniaxial tension, (b) uniaxial compression.

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Functions characterizing the evolution of damage were proposed by Mazars (1984) in the form ε0 − At e−Bt (κ−ε0 ) κ ε0 gc (κ) = 1 − (1 − Ac ) − Ac e−Bc (κ−ε0 ) κ

gt (κ) = 1 − (1 − At )

(50)

Switch factors HIε or HIσ = H (σI ) turn on and off certain terms depending on the signs of principal strains or stresses. Substituting (55)–(56) into (54) and using (47) and the relation ε˙ eq = η : ε˙ leads to

(51)

where ε0 is the equivalent strain at the elastic limit and At , Bt , Ac , and Bc are material parameters related to the shape of the uniaxial stress-strain diagrams. Formulae (50)–(51) are valid above the initial damage threshold, i.e., for κ > ε0 , while for κ ≤ ε0 both gc and gt vanish and the response is purely elastic. To ensure a continuous variation of slope of the compressive stress-strain curve, it is necessary to satisfy the condition Ac Bc ε0 = Ac − 1, which reduces the number of independent parameters to four. A sample set of parameters used by Saouridis (1988) is ε0 = 10−4 , At = 0.81, Bt = 10450, Ac = 1.34, and Bc = 2537. 4.2

ξ=

ω˙ = k  ξ : ε˙ + g  η : ε˙

k =

for β > 1 for β = 1

  3 ε˙ tI εI + εtI ε˙I 2εtI εI ˙εeq α˙ = − 2 3 εeq εeq I =1

ε˙I = HIε PI : ε˙

(53)

(54)

(55)

where PI = pI ⊗pI and HIε = H (εI ), with H denoting the Heaviside function, and also that ε˙ tI = ξ I : ε˙

ν ξI = 1 − 2ν

g  = (αt gt + αc gc − 2αk  )

(61)

(62)

is again a rank-1 correction of the elastic stiffness tensor, but the term g  η from (8) is replaced by k  ξ + g  η, and g  is not the derivative of a given damage function g but depends on the derivatives of functions gt and gc and on the current state according to (61). The corresponding localization tensor Q ed = n · Ded · n is singular if and only if

Due to material isotropy, the strain, stress and effective stress tensors as well as the tensor εt have the same principal directions pI , I = 1, 2, 3. Using the spectral representation, it is possible to show that

where

kα , εeq

Ded = (1 − ω)De − σ¯ ⊗ (k  ξ + g  η)

(52)

where η is given by (34), and ωt − ωc ωt − ωc

(60)

The resulting tangent stiffness tensor

= kα α˙ + (αt gt + αc gc )η : ε˙

kα =

(59)

where

ω˙ = α˙ t ωt + αt ω˙ t + α˙ c ωc + αc ω˙ c

β(1 − α)β−1

3 1 (εI ξ I + HIε εtI P I ) εeq I =1

Finally, using (58), expression (52) for the damage rate can be rewritten as

Differentiating (46)–(49) with respect to time we obtain

βα β−1

(58)

where

Localization condition



ξ − 2αη : ε˙ εeq

α˙ =

(56)

n · (k  ξ + g  η) · Q−1 ¯ ·n=1−ω e ·σ

(63)

For a material with no damage, the left-hand side vanishes (for all directions n) and the right-hand side is equal to 1. Due to damage, the left-hand side increases (at least for some directions n) and the right-hand side decreases. Therefore, the onset of localization can be characterized by the condition min L(n) = 0

n=1

(64)

where  HIσ

+ HIσ P I −

3 ν σ − H 1 + ν J =1 J

ν 1+ν

3

HJσ P J



¯ ·n L(n) = 1 − ω − n · (k  ξ + g  η) · Q −1 e ·σ

I

(57)

J =1

(65)

is the localization indicator. Of course, instead of L one could use the determinant of the localization tensor, but the expression for L is simpler and, in some cases, equation (64) can be solved analytically.

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4.3

Localization characteristics

Under monotonic proportional loading, scalars α, αt , αc and tensors ξ and η remain fixed, while σ¯ grows proportionally to the load multiplier μ and scalars ω, g  and k  evolve in a nonlinear fashion, depending on the selected form of damage functions gt and gc . The minimum value of the localization indicator becomes a function of μ, and the critical value of μ for which the localization condition (64) is satisfied can be found numerically. This has been done for plane stress conditions, with the stress type characterized by the stress angle ζ related to the ratio of the in-plane principal stresses, as explained in Section 3.6. The dependence of the critical stiffness ratio, critical angle and polarization angle on the stress angle ζ is plotted in Figs. 7–8. The results are only slightly affected by parameter β and are similar to the results obtained for the simple model with Mazars definition of equivalent strain. A peculiar feature is that the dependence of the localization properties on the stress angle exhibits a discontinuity at ζ = −90◦ , i.e., at uniaxial compression. This is caused by the jump change of the switch factor H2σ , which depends on the sign of principal stress σ2 . 1

5

0

-0.5

-1 -135

-90

-45

0

45

σ = De : (ε − εc )

stress angle [deg]

Figure 7. Mazars model, plane stress analysis: dependence of the initial and critical stiffness ratio Eed /Eu on the stress angle ζ .

critical and polarization angle [deg]

180

135

ROTATING CRACK MODEL

Many different anisotropic damage formulations have been proposed in the literature, and it is not possible to cover all of them or to derive generally valid statements. For illustration, we will look at a simple rotating crack formulation, which can be interpreted as a special type of anisotropic damage. Rotating or fixed crack models are usually presented in the engineering notation. The physical motivation and detailed derivation can be found e.g. in Jirásek and Zimmermann (1998). For the purpose of localization analysis it is useful to rewrite the basic equations in the tensorial notation. Based on the additive split of strain into the elastic part and the inelastic part due to cracking, the stress-strain law can be written as

initial, beta=1.06 beta=1.00 critical, beta=1.06 beta=1.00

0.5 stiffness ratio

Fig. 7 shows not only the critical stiffness ratio at the onset of localization but also the initial stiffness ratio at the onset of damage propagation. Under uniaxial or biaxial compression, the stress-strain curve is smooth and the tangent stiffness at the onset of damage is equal to the elastic stiffness, which gives initial stiffness ratio equal to 1, while for uniaxial and biaxial tension softening starts right at the elastic limit and the initial stiffness ratio is negative (and in a certain range is below the critical stiffness ratio). For stress angles ζ between −100◦ and 19◦ , the critical modulus is positive and localization can occur already on the hardening branch of the stress-strain diagram. The largest critical stiffness ratio Eed /Eu = 0.468 is found for stress angle ζ = −86.5◦ , i.e., for principal stress ratio σ2 : σ1 = −1 : 0.061.

(66)

For the rotating crack model, the inelastic (cracking) strain εc is fully described by a single variable c , which is the only nonzero component of crackεnn ing strain in the local coordinate system aligned with the crack (it is the normal strain in the direction perpendicular to the crack). The cracking strain tensor is expressed as

polarization angle, beta=1.06 beta=1.00 critical angle, beta=1.06 beta=1.00

90

c c εc = εnn p1 ⊗ p1 = εnn P1

(67)

45

0 -135

-90

-45 stress angle [deg]

0

where p1 is the unit vector in the direction of maximum principal strain (assumed to be normal to the crack), and P 1 = p1 ⊗ p1 . The cracking law relates the cracking strain ε c to the normal component of the traction transmitted by the (cohesive) crack,

45

Figure 8. Mazars model, plane stress analysis: dependence of the critical angle and polarization angle on the stress angle ζ .

σnn = p1 · σ · p1 = P 1 : σ

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(68)

For our purpose, it is sufficient to consider the cracking law in the rate form c σ˙ nn = Dc ε˙ nn

(69)

where Dc is the tangent crack stiffness. By eliminating the cracking strain from the rate form of (66)–(67), it is possible to derive the tangent stiffness tensor c ˜ N− Dec = De − εnn

D e : P1 ⊗ P 1 : D e e Dc + D1111

(70)

where N˜ is a certain fourth-order tensor. The specific expression for N˜ is not important because in our localization analysis we will focus attention on the state c just after crack initiation, when the cracking strain εnn c ˜ is zero and the term εnn N can be deleted from (70). The stiffness tensor then becomes a rank-1 modification of the elastic stiffness tensor, and the conditions for a vanishing determinant of the localization tensor can be established using the same techniques as in the localization analysis of isotropic damage models. The resulting necessary condition for an incipient weak discontinuity is e (n · De : P 1 ) · Q−1 e · (n · De : P 1 ) = Dc + D1111

(71)

where as usual, n is a unit vector normal to the discontinuity and Qe = n · De · n is the elastic acoustic tensor. The critical value of the softening modulus is e Dc,crit = max ϕ(n) − D1111 n=1

(72)

where ϕ(n) = (n · De : P 1 ) · Q −1 e · (n · De : P 1 )

(73)

e and D1111 = P 1 : De : P 1 is an elastic constant. For isotropic elasticity in a three-dimensional setting, we obtain e = D1111

E(1 − ν) (1 + ν)(1 − 2ν)

Eν ϕ(n) = 1 − ν2



(74)

ν2 + 2(p1 · n)2 + (p1 · n)4 1 − 2ν



The scalar product p1 · n represents the cosine of the angle between the discontinuity plane and the crack, and it can vary between 0 and 1. It is easy to show that the maximum of ϕ(n) is attained for n = p1 , e and the maximum turns out to be equal to D1111 , which means that the critical modulus is Dc,crit = 0.

The actual softening modulus Dc used by the rotating crack model is always negative and thus smaller than the critical one. The weak discontinuity can appear right at the onset of cracking and is perfectly aligned with the crack, independently of the Poisson ratio and of the stress state. It is also possible to show that the discontinuity type corresponds to pure mode I, which is consistent with the idea of an opening crack. This confirms that the model is appropriate for quasibrittle materials such as concrete. Of course, the crack is initiated only if the maximum principal stress reaches the tensile strength, and so compressive failure cannot be captured. ACKNOWLEDGMENT Financial support of the Czech Science Foundation ˇ under project GACR 106/08/1508 is gratefully acknowledged. REFERENCES de Vree, J.H.P., W.A.M. Brekelmans, and M.A.J. van Gils (1995). Comparison of nonlocal approaches in continuum damage mechanics. Computers and Structures 55, 581–588. Hadamard, J. (1903). Leçons sur la propagation des ondes. Paris: Librairie Scientifique A. Hermann et Fils. Hill, R. (1958). A general theory of uniqueness and stability in elastic-plastic solids. Journal of the Mechanics and Physics of Solids 6, 236–249. Jirásek, M. (2007). Mathematical analysis of strain localization. Revue Européenne de Génie Civil 11, 977–991. Jirásek, M. and T. Zimmermann (1998). Analysis of rotating crack model. Journal of Engineering Mechanics, ASCE 124, 842–851. Mazars, J. (1984). Application de la mécanique de l’endommagement au comportement non linéaire et à la rupture du béton de structure. Thèse de Doctorat d’Etat, Université Paris VI., France. Mazars, J. (1986). A description of micro and macroscale damage of concrete structures. International Journal of Fracture 25, 729–737. Ottosen, N. and K. Runesson (1991). Properties of discontinuous bifurcation solutions in elastoplasticity. International Journal of Solids and Structures 27, 401–421. Rizzi, E., I. Carol, and K.Willam (1996). Localization analysis of elastic degradation with application to scalar damage. Journal of Engineering Mechanics, ASCE 121, 541–554. Rudnicky, J.W. and J.R. Rice (1975). Conditions for the localization of deformation in pressuresensitive dilatant materials. Journal of the Mechanics and Physics of Solids 23, 371–394. Saouridis, C. (1988). Identification et numérisation objectives des comportements adoucissants: Une approche multiéchelle de l’endommagement du béton. Ph. D. thesis, Université Paris VI.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Numerical multiscale solution strategy for fracturing of concrete Łukasz Kaczmarczyk, Chris J. Pearce & Nenad Bi´cani´c Department of Civil Engineering, University of Glasgow, Glasgow, UK

ABSTRACT: This paper presents a numerical multiscale modelling strategy for simulating fracturing of concrete where the fine-scale heterogeneities are fully resolved. The fine-scale is modelled using a hybrid-Trefftz stress formulation for modelling propagating cohesive cracks. The very large system of algebraic equations that emerges from detailed resolution of the fine-scale structure requires an efficient iterative solver with a preconditioner that is appropriate for fracturing heterogeneous materials. This paper presents a two-grid strategy for construction of the preconditioner that utilizes scale transition techniques derived for computational homogenization and represents an adaptation and extension of the work of Miehe and Bayreuther (IJNME, 2007). For the coarse scale, this paper investigates both classical C 0 -continuous displacement-based finite elements as well as C 1 -continuous elements. The preconditioned GMRES Krylov iterative solver with dynamic convergence tolerance is integrated with a constrained Newton method with local arc-length control and line searches. The convergence properties and performance of the parallel implementation of the proposed solution strategy is illustrated on a numerical examples. 1

INTRODUCTION

Multiscale analysis aims to predict the macroscopic constitutive behaviour of materials with heterogeneous microstructures. Such techniques not only determine macroscopic ‘‘effective’’ continuum material properties but also provide understanding of the relationship between microstructural phenomena and the overall macroscopic behaviour. Computational approaches (Moulinec and Suquet 1998, Miehe and Koch 2002, Kouznetsoca et al. 2002) typically utilize nested multilevel finite element analyses with discretisation at both the microscale and macroscale—so-called computational homogenization. A fundamental restriction of these techniques is a clear separation of scales, such that the characteristic length of a representative volume element (RVE) is sufficiently small compared to the macrostructural characteristic length. Clear separation of scales permits the assumption of uniformity of the macroscopic strain field across the microstructure, as adopted in first-order homogenization schemes. In cases where the existence of an RVE necessitates a less well defined separation of scales, the assumption of uniform strains may be inappropriate in some situations, e.g. strain localization, boundary layers, etc. Second-order schemes have been proposed to overcome such short-comings (Kouznetsova et al. 2002, Feyel 2003, Kaczmarczyk et al. 2008), whereby the macroscopic material behaviour is described using a higher-order continuum theory (e.g. strain gradient, Cosserat, micropolar). In such cases the material response at a macroscopic point also depends on the response in the neighbourhood of

that point, thereby introducing a material length scale into the macroscopic constitutive model, and enables geometrical size effects to be captured. Fracturing leads to an evolving microstructure which makes it impossible to define a priori the size of the RVE. Once strains start to localise or fractures coalesce, material instability occurs, scale separation is no longer possible, the RVE becomes undefined and it is not possible to use scale-transition homogenization techniques. Various strategies for such situations have been presented in the literature, including Belytschko et al. (Belytschko et al. 2008), Gitman (Gitman et al. 2007, Gitman et al. 2008) and Markovic and Ibrahimbegovic (Markovic and Ibrahimbegovic 2006). Miehe and Bayreuther (Miehe and Bayreuther 2007) presented unifying computational procedures for the analysis of heterogeneous materials in the extremes of scale separation and in particular a multi-grid solution strategy (referred to as numerical multiscale) for situations without scale separation. This approach was inspired by the formulations, and in particular the scale transition techniques, of computational homogenization. Here Miehe and Bayreuther’s numerical multiscale solution strategy (Miehe and Bayreuther 2007) is extended for the case of fracturing in concrete at the level of observation below the macroscale (so-called meso-level (1–10 cm)), identifying individual aggregates embedded in a matrix, with a weak interfacial transition zone. An efficient two-grid (fine and coarse mesh) preconditioner for Krylov iterative solvers is constructed and derivation of the homogenizationbased scale transition operator is fully described.

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stiffness matrix is very small and computationally efficient to solve. The bulk response of the material is described by stress and displacement fields which are approximated by means of a HTS approximation. The stress field within an element is approximated directly as:

The analysis of fracturing heterogeneous materials requires a robust solution strategy for tracing the unstable equilibrium path. Thus the preconditioned iterative solver is embedded in a constrained Newton method with local arc-length control and line searches. In this paper, the cohesive crack methodology is utilized together with Hybrid-Treffz stress elements (Kaczmarczyk and Pearce 2009) for the fine mesh that fully resolves the heterogeneous mesostructure. Next, the overall solution strategy for fracturing heterogeneous materials is discussed, before describing in detail the proposed two-grid preconditioner and in particular the construction of the homogenization-based scale transition operators. Finally performance of the proposed model is demonstrated with a numerical example and investigates the use of both C 0 -continuous and C 1 -continuous elements for the coarse mesh. 2

σ = Sv v

where Sv is a matrix of field approximation functions and v is the unknown vector of generalised stress degrees of freedom. In Equation (1), the stress approximation field is chosen so as to automatically satisfy equilibrium: LT Sv = 0.

(2)

An additional and independent approximation of displacements u on the traction boundary σ of the element is introduced:

HYBRID-TREFFTZ STRESS ELEMENTS

The analysis of fracturing heterogeneous materials necessitates full resolution of the fine-scale structure, that evolves during mechanical loading. This requires a robust model for cohesive cracking, where multiple cracks, crack branching and crack coalescence are the norm and where different constitutive models are required for the various phases of matrix, inclusions and interface. A hybrid-Trefftz stress (HTS) formulation is adopted, the detailed description of which can be found in (Kaczmarczyk and Pearce 2009) for the extension to heterogeneous quasi-brittle materials. The current formulation is restricted to 2D but is fully extendable to 3D. Displacement discontinuities are restricted to element interfaces and the material response within each element is assumed to be hyperelastic. Such an approach is deemed realistic (Tijssens 2001), and can be justified by the observation of fracturing phenomena, for which localization occurs and material unloads in the vicinity of the crack. Unstructured fine-scale meshes are adopted in order to reduce the influence of the mesh on fracture propagation. Furthermore, for heterogeneous materials, the finite element mesh size is significantly constrained by the size and spacing of the inclusions. It has been shown that for the type of problem considered here, the results are mesh objective (Kaczmarczyk and Pearce 2009). The HTS finite element formulation is characterised by the approximation of stresses within the domain of the element and by the fact that the stiffness can be expressed via a boundary, rather than domain, integral. Thus, compared to their classical FEM counterpart, HTS elements exhibit faster convergence of the stress fields. Furthermore, the displacements are approximated on element boundaries and the displacement basis is defined independently on each element interface. Consequently, the overall bandwidth of the

(1)

u = U q.

(3)

The primary unknowns are the stress degrees of freedom v within the element and the displacement degrees of freedom on the element boundary q. However, the stress degrees of freedom can be eliminated conveniently from the global system of equations by application of static condensation, leaving only the displacement degrees of freedom to be determined. Following solution of the displacement degrees of freedom, the stress degrees of freedom can be recovered on an element by element basis, a process which lends itself to parallelization. The cohesive crack model on the element interfaces adopted here assumes that all inelastic deformation in the vicinity of a crack is concentrated onto a line and expressed in terms of tractions and displacements (Hillerborg et al. 1976). In order not to over-complicate the formulation, a straightforward material model for cohesive failure in two-dimensions is adopted. The model response depends on three material parameters: tensile strength ft , fracture energy Gf and α which assigns different weights to the shear and normal opening displacements. As such, a discontinuity (i.e. displacement jump) κ is introduced as a history variable when the effective traction ˜t on an element face exceeds the tensile strength of the material; the subsequent traction transferred across the interface is dependent on the magnitude of the displacement jump. A bilinear softening law is adopted here. Full details of the HTS element formulation for modelling cohesive cracking in heterogeneous materials are described in (Kaczmarczyk and Pearce 2009). The current paper focuses on the solution strategy for this class of problem.

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3

3.2

SOLUTION STRATEGY

Local arc-length control

The concept of local arc-length control with line searches was presented in (Alfano and Crisfield 2003) for delamination analysis and is adapted here to the analysis of fracturing heterogeneous materials. The basic idea of local arc-length control is to restrict attention to those DOF’s associated with the active process zone and thereby control the dissipative evolution of the structure. The arc-length vector b in (7) is a function of the displacement jumps along element interfaces. The magnitude of the displacement jumps in the direction normal to the interface at the integration points are collected together in δ as:

The analysis of fracturing heterogeneous materials represents a significant computational challenge. First, tracing the unstable equilibrium path, including overcoming critical stability points, requires a robust solution scheme. Second, the full resolution of the heterogeneous fine-scale structure results in a large system of algebraic equations that needs to be solved efficiently. This section will initially discuss Newton’s method with local arc-length control and line searches for tracing the nonlinear response before going on to present a numerical multi-scale preconditioner for Krylov solvers that is specifically designed for this application.

δ T = [g1 , g2 , . . . , gN ] 3.1

Constrained Newton method

The equilibrium equation can be expressed as r(q, λ) = f ext (λ) − f int (q) = 0

(4)

where q are the displacement degrees of freedom, λ is the load parameter, f ext and f int are the external and internal forces, and r is the vector out-of-balance forces. In order to trace the dissipative load-displacement path, an arc-length scheme is adopted. The corresponding constraint equation is written as: r λ = b(q) − l = 0

where gi is the displacement jump in the normal direction to the face, at integration point i. Only active cracks are included, i.e. gi > 0 in the previous increment. A critical value, gic , is associated with each component of δ, and is chosen to be the displacement jump associated with the change in slope of the adopted interface bilinear softening curve. The magnitude of this value will vary for different phases of the composite material.

where l represents the increment length for the current load step, the magnitude of which is computed via an automatic incremental procedure described later. The equilibrium and constraint equations together form a nonlinear system of equations that is solved using Newton’s method in combination with an efficient iterative solver for the associated linear system of equations. Linearization of (4) results in: (6)

where K = −∂q r is the tangent stiffness matrix and p = ∂λ f ext is the vector of externally applied reference loads. Since it is assumed that function b(q) remains unchanged during each load step (Alfano and Crisfield 2003), b in Equation (5) can be expressed as b = bT q

with

(δ c )T = [g1c , g2c , . . . , gNc ].

bT δq = 0

(7)

where q is the displacement increment and δq is the iterative change in displacement.

(9)

The constraint equation is now written as a function of displacement jumps:

(5)

Kδq − pδλ = r

(8)

bT q =

N 

vi sgn(δis−1 )(δi − δis−1 ) = l

(10)

i=1

with 1 δc

vi = N i

1 i=1 δic

.

(11)

Although local arc-length control has proven to be generally effective, the procedure is still found to be unstable in some specific situations, typically related to overcoming sharp limit points and non-smooth nonlinearities. Line searches have been shown to provide an effective remedy (Alfano and Crisfield 2003) and is adopted here. 3.3

Global system of equations

The linearized problem (6) and Eq. (5) can be expressed together in matrix form as: 

K bT

−p 0



q λ



 =

r rλ

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 (12)

where δ has been omitted from δq and δλ for simplicity. In condensed form, the above equation can be rewritten as: ˆ qˆ = fˆ K

4

(13)

PRECONDITIONER

Solution to this system of equations is found using a Generalized Minimum Residual Method (GMRES) solver. Although the GMRES method is well founded theoretically, it can suffer from slow convergence for problems of the type discussed in this paper. In general, preconditioning can be applied to the system described by Eq. (13) (Balay et al. 2009, Saad 1996) as follows: ˆ R−1 )(MR q) (ML−1 KM ˆ = ML−1 fˆ

(14)

where ML and MR indicate ‘‘left’’ and ‘‘right’’ preconditioning matrices respectively. In this paper, attention is restricted to left preconditioning, i.e. MR = I & ML = M. An important feature of this solution strategy is that the convergence tolerance is dynamic and ensures that the GMRES algorithm does not converge to an unnecessarily small tolerance while the Newton iterations are still far from equilibrium, thereby representing an important speed-up with respect to direct solvers. 4.1

A multi-grid preconditioning strategy utilises coarser meshes for fast smoothing of the long-wave modes of the error. Typically, a hierarchy of coarse meshes would be used, although here we restrict ourselves to just two meshes—fine and coarse. To construct the two-grid preconditioner, it is convenient to consider the variables in Eq. (12) separately. The first equation in (12) can be written as: (15)

where the superscript ‘‘f ’’ has been added to indicate the fine mesh. Rearranging this equation, we obtain an additive decomposition of qf as: qf

f −1

f −1

= (K ) r + λ(K ) p = qrf + λqλf

and

qλc = (K c )−1 Rp

(17)

where the superscript ‘‘c’’ has been added to indicate the coarse mesh. The construction of the stiffness matrix Kc on the course mesh will be discussed in the next section. The grid transfer operator R, and its counterpart P (introduced shortly), are at the heart of the proposed solution strategy since they account for the fine-scale heterogeneities, including cracks. Detailed derivation of these are also presented in the next Section. Typically, since the size of the coarse mesh system of equations is significantly smaller than that of the fine mesh, the coarse mesh solution can be computed using a direct solver. The solutions computed on the coarse mesh (Eq. (17)) are prolongated back onto the fine mesh to give: qrf = Pqrc = P(K c )−1 Rr

(18)

and qλf = Pqλc = P(K c )−1 Rp

(19)

Thus, the improved solution for qf can be determined from Eq. (16) once λ has been calculated. Substituting (16) into the second equation of (12) gives:   bT qrf + λqλf = r λ

(20)

which can be rearranged to give λ:

Two-grid preconditioner

r = K f qf − λp

qrc = (K c )−1 Rr

(16)

An approximation to qrf and qλf can be obtained by first restricting r and p onto the coarse mesh using a restriction operator R as:

  λ = r λ − bT qrf /bT qλf

(21)

The algorithm also includes a relaxation process to remove high frequency errors, which exhibit local variations in the solution, with a straightforward relaxation method such as Gauss-Seidel (Xu 1997). In subsequent sections we focus on the key issue of deriving the coarse-to-fine prolongation operator P, fine-to-coarse restriction operator R and coarse stiffness matrix K c .

5

GRID TRANSFER OPERATORS

The two-grid preconditioner utilises a coarse mesh in addition to the fine mesh. A patch of fine mesh elements, which fully resolves the fine-scale heterogeneities, is associated with a single coarse mesh element, see Figure 1. The construction of the homogenization-based coarse mesh stiffness matrix K c is described later. Attention is currently focused on derivation of the prolongation operator P based on a decomposition of the fine-scale displacements

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at hand, the fine-to-coarse mesh restriction operator R, is defined as

Nodal displacements & element shape functions qf = Pqc , uf = uc

R ≡ PT .

qf

Geometric centre for Taylor expansion

=

B.C. s & Solve BVP uf + rf = uc

Pave qc ,

5.1

Elements of different shades depict heterogeneities

qf = Phom qc uf = x

   P = A N c xif

Figure 1. Long-wave and short-wave prolongation operators. Top: long-wave operator P. Middle: average component of the short-wave operator  Pave . Bottom: homogeneous component of short-wave operator  Phom .

(22)

The long-wave contribution of the fine-scale displacements are associated with the homogeneous contribution of the coarse mesh approximation and determined from interpolation of the coarse mesh displacements. The short-wave displacements represent fluctuations due to the fine-scale heterogeneous structure of the patch, e.g. inclusions and deformation induced cracks. Long- and short-wave contributions to the fine mesh approximation of displacements are obtained by prolongation of the coarse mesh displacements qc qf = Pqc

and  qf =  Pqc .

5.2

(23)

(24)

This decomposition is also described in Figure 1. With the coarse-to-fine mesh prolongation operator P

Short-wave prolongation operator  P

The short-wave component of the prolongation operator,  P, reflects the influence of the heterogeneous nature of the fine mesh patch of elements corresponding to a single coarse-scale element. Following (Miehe and Bayreuther 2007), this contribution can be formulated using the scale transition techniques for computational homogenization. A truncated Taylor series expansion of the displacement vector about the geometric centre of the coarse mesh element, using Voigt notation, yields:

As a consequence, the prolongation operator P in (??) is decomposed into long-wave P and short-wave fluctuation  P components as: P P = P +

(26)

where matrix function A(. . .) depends on the fine mesh face approximation functions adopted and relates the interpolated coarse mesh nodal displacements with the fine mesh face displacements of the fine mesh patch. In a classical geometric multi-grid strategy, this long-wave prolongation operator associated with the homogeneous part of the deformation represents the only component of the total prolongation operator, i.e. P = P. However, following the approach in (Miehe and Bayreuther 2007), the long-wave prolongation operator is augmented by a short-wave component reflecting the fine-scale heterogeneities.

into long-wave and short-wave contributions by the additive split:  + qf qf   long-wave short-wave

Long-wave prolongation operator P

The long-wave coarse-to-fine mesh prolongation operator P relates the displacement field in a coarse mesh element to the displacement field in the corresponding patch of fine mesh elements and is computed from interpolation of the displacement field on the coarse mesh. Here two different discretization methods are used, i.e. displacement type finite elements and HTS elements for the coarse and fine mesh problems, respectively. For such problems, the prolongation operator relating coarse mesh nodal degrees of freedom with those on the faces of the corresponding fine mesh patch is given by

X

qf =

(25)

u(x) = u0 + Xε + w(x)

(27)

where u0 is the displacement vector at the geometric centre of the coarse mesh element, ε is the strain state at element centre, w(x) is the displacement fluctuation

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over the fine mesh patch and X is a matrix of relative position coordinates:   1 2x 0 y (28) X= 0 2y x 2

boundary conditions represent a minimal condition that fulfill the Hill-Mandel theorem, e.g. see (Kaczmarczyk et al. 2008, Kouznetsova et al 2002, Miehe and Bayreuther 2007). Given the HTS discretization of displacements (3) and the expression for the displacement fluctuation (29), the constraint equation (35) is expressed as:

Rearranging (27) yields an expression for the displacement fluctuation field within the fine mesh patch: w(x) = u(x) − Xεε − u0

Cqf − DBc qc = 0

(29)

where C is a constraint matrix given by

Relating the displacement fluctuation to the coarse element degrees of freedom qc yields, after HTS discretization (3), the short-wave component of the prolongation operator is introduced as:

C=

and D as

and

D=

Phom qc Xεε = U

(38)

The terms of matrix H reflects the specific nature of the boundary conditions used, i.e. linear deformation or uniform traction. The influence of the type of boundary condition will be investigated with numerical examples in the next section. Each term in matrix H can be interpreted as an admissible distribution of nodal traction forces on the boundary of the elements patch. Details on the construction of H can be found in (Kaczmarczyk et al. 2008). The solution of the discretized BVP for the finescale patch of elements can be expressed as a constrained quadratic problem. A common method to solve such a problem is to introduce Lagrange multipliers. However, such an approach increases the number of unknowns and alters the character of the system matrix (to an indefinite saddle point problem). Moreover, the numerical solution of Euler’s conditions for the stationary point of the Lagrangian is rather inefficient and therefore not suitable for solving computationally complex multiscale problems where the constrained quadratic problem has to be solved for every integration point. Alternative numerical techniques include the penalty method or Uzawa method (Zienkiewicz and Taylor). However, in order to express the short wave prolongation operator in closed-form, the approach taken in (Kaczmarczyk et al. 2008) is briefly presented, where the work of Ainsworth (Ainsworth 2001) is applied. The fine-scale BVP can be expressed as:

Substitution of (30) and (31) into (29) yields Pqc = U ( U Pave −  Phom )qc

HUT Xd ∂c

(31)

(32)

where the short-wave component of the prolongation operator is: (33)

The homogeneous component of this prolongation operator,  Phom , associated with coarse mesh deformations, simply takes the form:  Phom = A(Xi Bc )

(37)

(30)

where U is the matrix of displacement approximation functions on the element faces in the fine mesh patch. Similarly,

 Phom P = Pave − 

HUT U d, ∂c

w(x) = U Pqc

Pave qc + u0 u(x) = U

(36)

(34)

where Xi is evaluated at the fine mesh nodes and the approximation prolongation function A is the same function as used in (26). Bc is the coarse mesh straindisplacement matrix. The fluctuation component of the short-wave prolongation operator,  Pave , is determined by the solution of a boundary value problem (BVP), whereby deformation of the fine mesh patch is enforced, via appropriate boundary conditions, according to a given coarse element average strain. The boundary conditions for the fine mesh patch must satisfy:

w ⊗ n d = 0 (35) ∂c

K f qf + C T λ = 0

(39)

where n is the normal vector to the coarse element boundary domain ∂c . It can be shown, that such

Cqf = DBc qc

(40)

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where λ are the Lagrange multipliers. Following (Ainsworth 2001), the following matrices are defined: Q = I − CT (CCT )−1 C,

F = CT (CCT )−1

element volume. Inserting (47) into the above, the coarse element stress-strain relationship is obtained:

(41)

σ = Cεε

Matrix Q projects the fine mesh stiffness matrix K f onto a sub-space that is orthogonal to that of the constraints expressed by CT C. Solution for qf is then given as: ˘ q =K f

−1 ˘

f

5.4

and (43)

where ˘ = CT C + QT K f Q K

(44)

and f˘ , expressed in terms of the coarse element displacement degrees of freedom, is: f˘ = g˘ Bc qc ,

  g˘ = CT D − QT K f FD

(45)

Thus, Equations (42) & (45) provide an expression for  Pave as: Pave qc qf = 

˘ −1 g˘ Bc with  Pave = K

The coarse scale element stiffness is computed from averaging of the evolving fine-scale stiffness and represents a coupled-volume approach (Gitman et al. 2007, Gitman et al. 2008). For computation of the coarse element stiffness Kc , it is convenient to rewrite (46) in terms of the coarse element strain ε¯ , rather than degrees of freedom qc , using an alternative prolongation operator as: ˘ −1 ˘ with  Pave ε =K g

1 T 1 D λ = − DT FT K f qf V V

(50)

(51)

where

(47)

(48)

Generalization to second-order continuum

w(x) = u(x) − XBc qc − ZGc qc − u0

1 Z= 4

To determine the coarse element stress-strain relationship, the average stress can be expressed in terms of the Lagrange multipliers (Kaczmarczyk et al. 2008) as σ =

(49)

where η = ∂xx u is a second-order strain measure. Detailed description of second-order computational homogenization can be found in Kaczmarczy et al. 2008 Here we limit ourselves to the salient equations required for extension of the first-order scheme developed in the previous sections. The displacement fluctuation in (50) expressed in terms of the coarse mesh degrees of freedom have the form

5.3 Construction of coarse element stiffness

Pave q = ε ε

1 T T f ave D F K Pε V

1 u(x) = u0 + x · ε + x ⊗ x : η + w(x) 2

(46)

Pave to give  P, Recall that  Phom is subtracted from  which is then added to P to yield the prolongation operator P = P + ( Pave −  Phom ).

f

C=−

For problems involving softening and fracturing of materials, there is further potential for improvement of the preconditioner. The two-grid preconditioner described above can be described as a first-order approach, adopting only first-order deformation modes in the computation of Pave . A natural extension of this can be achieved by taking into account secondorder strain measures, in the spirit of the second-order homogenization method (Kouznetsova et al. 2002, Kaczmarczyk et al. 2008). In the computation of Pave , such an approach enhances the coarse mesh displacement field approximation space and improves numerical efficiency of the preconditioner, which is manifested by faster convergence of the Newton method with Krylov iterative solver. This will be demonstrated with numerical examples in the next section. A second-order scheme is realized by expanding the fine mesh displacements with a Taylor series, truncated after the second-order term

(42)

λ = −FT K f qf

where



2x2 0

0 2y2

2y2 0

0 2x2

xy 0

0 xy

 (52)

and Gc is the coarse element second-order equivalent of Bc . The homogeneous part of the prolongation operator has the form  Phom = A(Xi Bc + Zi Gc )

where the final expression has been arrived at by substitution of λ from Equation (43) and V is the coarse

(53)

where Xi and Zi are evaluated at the fine mesh nodes.

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The boundary conditions for the fine mesh patch, in addition to Equation (35), are given by

n ⊗ x ⊗ w d = 0. (54) ∂c

This equation enforces deformation of the fine mesh patch according to the average coarse mesh second-order strain η. Following discretization of the fine mesh displacements, the boundary conditions (35) and (54) are expressed in a modified constraint equation (compare with (36)): Cqf − DBc qc + EGc qc = 0

(55)

where

HUT Zd E=

(56)

∂c

With this enhanced constraint equation (55) the second-order fluctuation part of the short-wave contribution to the coarse-to-fine mesh operator  Pave can be constructed in a similar manner to the first-order scheme. In order to keep the coarse mesh problem as simple and efficient as possible for practical implementation, the formulation is restricted to the antisymmetric part of η . Thus, in 2D, there are two bending modes in addition to the simple first-order deformation modes. 6

NUMERICAL EXAMPLE

The fracture of dog-bone concrete specimens subject to tensile loading is analysed in order to demonstrate the performance of the proposed solution strategy for relatively large-scale problems and fracturing heterogeneous materials. Details about geometry, boundary conditions, aggregate size distribution, material model and parameters can be found in (Vliet 2000). In Figure 2, a simplified geometry for the specimens is Mesostructure 0.25D

50mm

D

D

100mm

presented. Three sizes of dog-bone samples are considered, D = 50 mm, D = 100 mm and D = 200 mm. Each sample size has a meso-structure generated with the same aggregates size distribution. Specific details of the aggregate structure can be found in Appendix B of (Vliet 2000) and in (Walraven 1980). The mesostructure for all problems is fully resolved, whereby aggregates, matrix and the interfacial transition zone are discretely modelled. The characteristic size of the fine mesh for all sample sizes are the same and controlled by the aggregate size and spacing. Such an approach has been shown to lead to mesh independent results (Kaczmarczyk and Pearce 2009). The circular shape of aggregates is discretized by a piecewise linear approximation, with the length of the segment equal to 1/10 of the radius of the smallest aggregate. The characteristic size of the coarse mesh is chosen arbitrarily as the distance between aggregate centres. Table 1 shows the number of degrees of freedom for both the fine and coarse meshes. For the aggregate, the elastic modulus is EA = 98 MPa and for the matrix EM = 35 MPa. Poisson’s ratio ν = 0.2 for all phases. The steel loading plates are also modelled as an elastic material with ES = 420 MPa and ν = 0.3. The cohesive crack material parameters for matrix, aggregate and interfacial transition zone have the following values: ftM = 12 MPa, GfM = 190 N/m, ftA = 6 MPa, GfA = 190 N/m, ftITZ = 2.0 MPa and GfITZ = 30 N/m, respectively. For all interfaces α = 1.0 is assumed. For the smallest sample D = 50 mm attention is focused on convergence for four types of grid transfer operators, as in the previous example, i.e. CST/DISP, CST/TRAC, GRADIENT/DISP and GRADIENT/ TRAC, see Figure 3 and Table 1. In all cases the preconditioned GMRES solver was used in combination with local arc-length control and line searches. The analysis using uniform traction boundary conditions to compute the short-wave prolongation operator together with elements (i.e. CST/TRAC) does not converge. The analyses with linear and quadratic displacement boundary conditions (i.e. CST/DISP and GRADIENT/DISP) exhibit good convergence. The C 1 -continuous elements with linear distribution of tractions (GRADIENT/TRAC) shows the fastest convergence. The relative number of iterations and relative cumulative computational time are shown in Table 2. The GRADIENT/TRAC analysis, with 8 processors, is 10% faster than GRADIENT/DISP, whereas the cumulative number of iterations is 40%

Ls = 1.2D

Table 1.

0.2D

200mm

Figure 2. Dog-bone geometry (left), mesostructure for different specimen sizes (right).

Number DOF’s for each sample size.

Size

Coarse DOF

Fine DOF

50 mm 100 mm 200 mm

3342/1114 9084 28296

71496 (+114324) 248080 (+390204) 868936 (+1364160)

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Nominal Stress [MPa] Number of iterations GRADIENT/DISP GRADIENT/TRAC 2.5 CST/DISP 5k CST/TRAC 2.0

Nominal Stress [MPa]

mag. x103

not converged

6k

4k 3k 2k

0.1

mag. x102

2.0

1.0 0.5

0

50mm 100mm 200mm

2.5 1.5

1k 0

3.0

10

20

30

40

Elongation of Ls [ m] Relative Error

1.5

0.0 60

50

1.0 0.5

0.01 0.001

0.0

0

10

20

30

40

50

60

Elongation of Ls [μm] resNewton 1000

1020

1040

1060

1080

resGMRES 1100

1120

Number of iterations

Figure 3. Model performance for 50 mm specimen. Nominal stress-displacement response and cumulative number of iterations (top left). Deformed fine mesh at peak and post-peak (top right). Illustration of convergence of GRADIENT/TRAC analysis, demonstrating the influence of the dynamic tolerance adopted for the GMRES algorithm (bottom).

Figure 4. Nominal stress vs displacement plots for D = 50 mm, 100 mm and 200 mm dog bone specimens. Experimental peak stress and standard deviation also show.

50mm

100mm Table 2. Relative number of iterations and relative cumulative computational time. Analysis

Rel. num. iterations

Rel. time

GRADIENT/TRAC CST/TRAC GRADIENT/DISP CST/DISP

1.0 not converged 1.3 1.4

1.0 – 1.2 1.1

200mm

smaller. However, since the current implementation of the second-order scheme is not optimized, the authors believe that there is potential to significantly improve efficiency. All analyses with displacement boundary conditions (DISP) converged more slowly than for the second-order scheme with traction boundary conditions (GRADIENT/TRAC). The reasons for this is that where a crack crosses the fine mesh patch boundary, displacement fluctuation continuity is not enforced by the traction boundary conditions. Furthermore, the coarse mesh problem gives too stiff a response as a result of the residuals restricted from the fine mesh in the case of displacement boundary conditions. In summary, the type of preconditioner adopted (CST/DISP, CST/TRAC, GRADIENT/DISP or GRADIENT/TRAC) does not affect the response of the structure, it only affects the rate of convergence and consequently the solution time. GRADIENT/TRAC was found to be the most efficient. Finally, for each of the D = 50 mm, D = 100 mm and D = 200 mm specimens, four random mesostructures were generated, resulting in 12 separate analyses. The GRADIENT/TRAC preconditioner was used

Figure 5. Fracture patterns for D = 50 mm, 100 mm and 200 mm dog bone specimens.

for all analyses. Figure 4 shows the nominal stressdisplacement response for all 12 of these analyses and it can be seen that there is a degree of scatter in the response for all specimen sizes. This scatter can be directly attributed to the randomness of the fine-scale structure only and is not the result of the preconditioner used. Moreover, these numerical results support the experimental observation of van Vliet (Vliet 2000), that for a range of small sample sizes (i.e. 50–200 mm) no clear size effect for nominal strength is observed (i.e. reducing with size). It can also be seen that the response becomes more brittle with increasing size. Figure 5 shows the crack patterns that developed, illustrating that the shape of the developing crack is clearly influenced by the spatial arrangement of aggregates.

7

CONCLUSIONS

This paper presents a modelling strategy for simulating the behaviour of fracturing heterogeneous materials

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such as concrete where the fine-scale heterogeneities are fully resolved. The fine-scale is modelled using HTS elements, where cracks are restricted to element interfaces. This represents an efficient framework for modelling propagating cohesive cracking. The very large nonlinear system of algebraic equations that emerges from this fine scale resolution requires an efficient iterative solver with a preconditioner that is appropriate for fracturing heterogeneous materials. A significant extension to the work of Miehe and Bayreuther (Miehe and Bayreuther 2007) has been proposed that constructs a preconditioner using a two-grid strategy that utilizes the scale transition techniques derived for computational homogenization. Both displacement and traction boundary conditions are considered in construction of the grid transfer operators for both first and second-order schemes. The proposed two-grid strategy has been demonstrated for both elastic and fracturing heterogeneous materials, illustrating both the convergence properties of the proposed scheme and efficiency of the parallel implementation. The use of the preconditioned GMRES Krylov iterative method with dynamic tolerance in combination with a constrained Newton method with local arc-length control and line searches represents a robust and efficient simulation framework.

ACKNOWLEDGEMENTS The authors gratefully acknowledge the support of the UK Engineering and Physical Sciences Research Council (Grant Ref: EP/D500273). The authors also wish to thank Dr. Lee Margetts of The University of Manchester for access to HECToR (UK’s national supercomputing service) through the UK Engineering and Physical Sciences Research Council HECToR capability challenge grant (Grant Ref: EP/F055595/1).

REFERENCES Ainsworth, M. (2001). Essential boundary conditions and multi-point constraints in finite element analysis. Computer Methods in Applied Mechanics and Engineering 190(48), 6323–6339. Alfano, G. and M.A. Crisfield (2003). Solution strategies for the delamination analysis based on a combination of localcontrol arc-length and line searches. International Journal for Numerical Methods in Engineering 58(7), 999–1048. Balay, S., K. Buschelman, W. Gropp, D. Kaushik, M. Knepley, L.C. McInnes, B. Smith, and H. Zhang (2009). PETSc Web page. http://www.mcs.anl.gov/petsc. Belytschko, T., S. Loehnert, and J.H. Song (2008). Multiscale aggregating discontinuities: A method for circumventing loss of material stability. International Journal for Numerical Methods in Engineering 73(6), 869–894.

Feyel, F. (2003). A multilevel finite element method (fe2) to describe the response of highly non-linear structures using generalized continua. Comput. Methods. Appl. Mech. Engrg. 192(-), 3233–3244. Gitman, I.M., H. Askes, and L.J. Sluys (2007). Representative volume: Existence and size determination. Engineering Fracture Mechanics 74(16), 2518–2534. Gitman, I.M., H. Askes, and L.J. Sluys (2008). Coupledvolume multi-scale modelling of quasi-brittle material. European Journal of Mechanics A-Solids 27(3), 302–327. Hillerborg, A., M. Moder, and P.E. Petersson (1976). Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cement and Concrete Research 6(6), 773–781. Kaczmarczyk, L. and C.J. Pearce (2009). A corotational hybrid-trefftz stress formulation for modelling cohesive cracks. Computer Methods in Applied Mechanics and Engineering 198(15–16), 1298–1310. Kaczmarczyk, L., C.J. Pearce, and N. Bicanic (2008). Scale transition and enforcement of rve boundary conditions in second-order computational homogenization. International Journal for Numerical Methods in Engineering 74(3), 506–522. Kouznetsova, V., M.G.D. Geers, and W.A.M. Brekelmans (2002). Multi-scale constitutive modelling of heterogeneous materials with a gradient-enhanced computational homogenization scheme. International Journal for Numerical Methods in Engineering 54(8), 1235–1260. Markovic, D. and A. Ibrahimbegovic (2006). Complementary energy based fe modelling of coupled elasto-plastic and damage behavior for continuum microstructure computations. Computer Methods in Applied Mechanics and Engineering 195(37–40), 5077–5093. Miehe, C. and C.G. Bayreuther (2007). On multiscale fe analyses of heterogeneous structures: From homogenization to multigrid solvers. International Journal for Numerical Methods in Engineering 71(10), 1135–1180. Miehe, C. and A. Koch (2002). Computational micro-tomacro transitions of discretized microstructures undergoing small strains. Archive of Applied Mechanics 72(4–5), 300–317. Moulinec, H. and P. Suquet (1998). A numerical method for computing the overall response of non-linear composites with complex microstructure. Computer Methods in Applied Mechanics and Engineering 157, 69–94. Saad, Y. (1996). Iterative Methods for Sparse Linear Systems. PWS. Tijssens, M.G.A. (2001). On the cohesive surface methodology for fracture of brittle heterogeneous solids. Ph.D. thesis, Technische Universiteit Delft. Vliet, M.V. (2000). Size effect in tensile fracture of concrete and rock. Ph.D. thesis, Technische Universiteit Delft. Walraven, J. (1980). Aggregate interlock: A theoretical and experimental analysis. Ph.D. thesis, Technische Universiteit Delft. Xu, J. (1997). An introduction to multigrid convergence theory. In Iterative methods in scientific computing, pp. 169–241. Springer. Zienkiewicz, O. and R. Taylor.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

A 3D lattice model to describe fracture process in fibrous concrete J. Kozicki & J. Tejchman Gda´nsk University of Technology, Gda´nsk-Wrzeszcz, Poland

ABSTRACT: Paper deals with simulations of fracture process in concrete including steel fibres. A discrete geometric linear lattice 3D model was used. Concrete was described at a meso-scale as a five-phase material composed of aggregates, cement matrix, steel fibres and interfacial zones between matrix and aggregate and between matrix and fibres. The lattice elements were stochastically distributed in the form of a mesh using a Delaunay’s construction scheme. The calculations were carried out with notched concrete specimens subjected to uniaxial extension. 1

INTRODUCTION

Concrete is still the most widely used construction material since it has the lowest ratio between cost and strength as compared to other available materials. However, it has two undesirable properties, namely: low tensile strength and large brittleness that cause the collapse to occur shortly after the formation of the first crack. Therefore, the application of concrete subjected to impact, earth-quaking and fatigue loading is strongly limited. To improve these two negative properties and to achieve a partial substitute of conventional reinforcement, the addition of short discontinuous randomly oriented fibres (steel, glass, synthetic and natural) can be practiced among others. Steel fibres are the most used in concrete applications due to economy, manufacture facilities, reinforcing effects and resistance to the environment aggressiveness. By addition of steel fibres, the following properties of plain concrete: tensile splitting strength, flexural strength, first cracking strength, toughness (energy absorption capacity), stiffness, durability, impact resistance, fatigue and wear strength increase, and deflection, crack width, shrinkage and creep are reduced (Shah 1971, Bentur & Mindess 1990, Balaguru & Shah 1992, Zollo 1997). In turn, compressive strength can slightly increase (Mohammadi et al. 2008) or slightly decrease (Altun et al. 2007). The addition of steel fibres aids in converting the brittle characteristics to a ductile one. Fibres limit the formation and growth of cracks by providing pinching forces at crack tips. They bear some stress that occurs in cement matrix themselves and transfer the other portion of stress at stable cement matrix portions. Real effects of fibre addition can be observed as a result of the bridging stress offered by the fibres after the peak load. The fibre reinforced concrete specimens develop first a pattern of fine distributed cracks instead of directly failing in one localized crack. Fibre-reinforced concrete has found many applications in tunnel linings, wall

cladding, bridge desks, pavements, slabs on grounds, factory (industrial) floors and slabs, dams, pipes, fire protection coatings, spray concretes (Balaguru and Shah 1992, Krstulovic-Opara et al. 1995, Falkner & Henke 2000, Schnüttgen & Teusch 2001, Walraven & Grünewald 2002). It can be also used as a method for repair, rehabilitation, strengthening and retrofitting of existing concrete structures (Li et al. 2000). The degree of improvement of fibrous concrete depends upon many different factors such as: size, shape, aspect ratio, volume fraction, orientation and surface characteristics of fibres, ratio between the fibre length and maximum aggregate size, and volume ratio between long and short fibres and concrete class. The fibre orientation depends on the specimen size and flow direction of the fresh concrete. On the other hand, fibres hinder the flowability of fresh concrete decreases workability by increasing the pore volume, resulting in strength decrease (what is visible during compression). The most suitable volume values for concrete mixes are between 0.5% and 2.5% by volume of concrete. The aspect ratios of steel fibers used in concrete mix are varied between 50 and 100. A probability of heterogeneous distribution and flocculation of fibers in concrete mix is increased by increasing aspect ratios. Homogeneous distribution of fibers at mixing and placing is required regardless of the type of fibers. A better workability of concrete mix is obtained as the percentage of shorter fibres increased in the mix in comparison to percentage of longer fibres (Mohammadi et al. 2008). A strengthening effect in concrete increases with the ratio of the fibre length to the size of the coarse aggregate (optimum ratio: 1.5–2.0, Chenkui & Guofan 1995) and by combining two different fibre types: short and long fibers (optimum content: 65% long and 35% short, Mohammadi et al. 2008). A small addition of non-metallic fibres results in good fresh concrete properties and reduced early age cracking (Sivakumar & Santhanam 2007) and an increased impact energy, toughness and

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ductility (Komlos et al. 1995). The toughness indices are higher for lightweight concrete than for normal weight concrete and for normal strength concrete than for high strength concrete (Balendran et al. 2002). In spite of positive properties, fibrous concrete did not find such acknowledgment and application as usual concrete. There do not still exit consistent dimensioning rules due to the lack sufficient large-scale static and dynamic experiments taking into account the effect of the fibre orientation. There is a general lack of confidence in the design particularly under bending in spite of existing tools on different scale. An analytical micro-scale approach has been proposed in (Lim et al. 1987). On the meso scale, a truss model has been used to study the material behavior (Li et al. 2006, Bolander & Saito 1997, van Hauvert & van Mier 1998). In turn, on the macro scale, constitutive models have been developed that can be used in finite element computations (Al-Taan & Ezzadeen 1995, Kooiman et al. 2000). Recently, a numerical continuum approach was proposed by Radtke et al. (2008) wherein the existing continuum approach to model concrete failure was combined with a discrete representation of fibres by adding extra nodal forces at fibre ends measured during the pull-out of a fibre from a matrix specimen. In turn, a semi-analytical method was proposed by Jones et al. (2008) to predict the flexural behaviour of steel fibers using a stress-block approach. Kabele (2007) has used a multi-scale framework for modeling of fracture in high performance fiber reinforced cementitious composites. The intention of this paper is to describe the fracture process at the meso-scale in fibrous concrete using a geometrical linear lattice type model (Kozicki & Tejchman 2007, Kozicki & Tejchman 2008). In the model, individual steel fibres were explicitly modeled. Three dimensional calculations were performed with concrete considered as five-phase material (aggregate, cement matrix, steel fibres and interfacial transition zones between both cement matrix and aggregate and cement matrix and steel fibres). Attention was paid to the effect of the amount of steel fibres and their orientation on the material behavior during uniaxial tension and uniaxial compression. Our lattice model was successfully used to model the fracture process in two-dimensional and three-dimensional concrete specimens subject to uniaxial tension, uniaxial compression, three-point bending and shear-extension test (Kozicki & Tejchman 2007, 2008). The effect of aggregate density, mean aggregate size and specimen size was realistically captured. 2

(Schlangen et al. 1997, van Mier et al. 1995, van Mier & van Vliet 2003) in that it consists of rods with flexible nodes and longitudinal deformability, rotating in the form of a rigid body rotation (Fig. 1). Thus, shearing, bending and torsion are represented by a change of the angle between rod elements connected by angular springs. This quasi-static model is of a kinematic type. The calculations of element displacements are carried out on the basis of the consideration of successive geometrical changes of rods due to translation, rotation and normal and bending deformation. Thus, the global stiffness matrix is not built and the calculation method has a purely explicit character. In spite of necessity of the application of small displacement increments (what is the inherent property of explicit numerical procedures), the computation time is significantly reduced as compared to implicit solutions. In our model, the quasi-brittle material was discretized in the form of a 3D tetrahedral grid including lines. The distribution of elements was assumed to be completely random using a Delaunay’s construction scheme. First, a tetrahedral grid of nodes was created in the material with the side dimensions equal to g. Then each node was randomly displaced by a vector of the random magnitude smaller than s. The nodes randomized in this way were connected with each other. Thus, each edge in the Delaunay mesh formed a lattice. The model needs 2 parameters to randomly distribute elements in the lattice. In the calculations, we assumed mainly the parameters g and s as g = 2 and s = 0.6 g. The elements possessed longitudinal stiffness described by the parameter kl (controlling the changes of the element length), bending stiffness described by the parameter kb (controlling the changes of the angle between elements) and torsional stiffness described by the parameter kt (controlling torsion between two elements in 3D calculations). The parameter kr was assumed kr = kt . The displacement of the center of each rod element (Fig. 2) was calculated as the average displacement of its two end nodes from the previous iteration step  i X

=

A  i X

+ BiX 2

(1)

wherein AX and BX are the displacement of the end nodes A and B in the rod element i, respectively.

LATTICE MODEL

Our linear lattice model (Kozicki & Tejchman 2007, 2008) differs from a classical lattice beam model composed of beams connected by non-flexible nodes

Figure 1. Rods connected by angular springs (rods do not bend) (Kozicki & Tejchman 2007).

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Figure 2.

General scheme to calculate displacements of elements in the 2D lattice (Kozicki & Tejchman 2007).

The displacement vector of each element node was obtained by averaging the displacements of the end of elements belonging to this node caused by translation, rotation, normal and bending deformation (Fig. 1):

X =

j

 iW  + iR  jn sum i  +

1    i i d init (i D i kl + iB i kb + iT i kt ) ,  1 i i dinit (i kl + i kb + i kt )

Figure 3. Lattice composed of rods to model five phases of concrete: fibres, aggregate, cement matrix and interfaces between aggregate and cement matrix and between fibres and cement matrix.

(2)

 = wherein: i X = resultant node displacement, i W  = node displacement due to the rod translation, i R  = node node displacement due to the rod rotation, i D displacement due to a change of the rod length  = node displacement due to (induced by kl ), i B a change of the angle between rods (induced by kb ),  = node displacement due to torsional spring i T between two neighboring rods (induced by kt ), i = successive rod number connected with node j, j = node number, j nsum = number of rods belonging to node j and i dinit = initial length of rod i. The node displacements were calculated successively during each calculation step beginning first from elements along boundaries subject to prescribed displacements. By applying Eq. 2, the equilibrium of strains was obtained in each node (what required always about 10 iterations). The resultant force F in a selected specimen’s cross-area A was determined with the aid of corresponding normal strain ε, shear strain γ, stiffness parameters kl , kl , kt , modulus of elasticity E, shear modulus G and cross-section area A:

F =A



(kl εE + kb γ G),

(3)

where the sum is extended over all elements that cross a selected specimen’s cross-section. For the bending stiffness parameter kb = 0 (Eq. 2), the elements behave as simple bars. The element was removed from the lattice if the local critical tensile strain εmin was exceeded. All presented numerical calculations were strain-controlled. The strain increment in a single calculation step expressed as scrit = gεmin /l (where l—displacement of specimen edge, εmin —local critical tensile strain, g—average rod length) should be larger than 500. It means that minimum 500 iterations were required to remove a single rod. In the calculations, different properties were prescribed to lattice elements to simulate the behaviour of aggregate, cement matrix, bond between aggregate and cement matrix, fibres and bond between fibres and cement matrix (Fig. 3, Tab. 1). The aggregate volume percentage was 60% in 3D, and the mean aggregate diameter was d50 = 12 mm (Fig. 4). The aggregate distribution was generated following the method given

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Table 1.

Material parameters used in calculations (five-phase material, p = kb /kl ). Young’s modulus E [GPa]

p = kb /kl tension

p = kb /kl compression

kl

Tensile strain εmin [%]

Cement matrix

20

0.6

0.2

0.01

0.2

Aggregate Aggregate interface

60 14

0.6 0.6

0.2 0.2

0.03 0.007

0.133 0.05

20 14 14 14 14 14 14 14 14 160

0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6

0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

0.01 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.007 0.08

0.2 0.025 0.05 0.1 0.2 0.5 1 10 90 90

Fibrous interface

Steel fibres

Figure 5. Approximation of the grading curve with discrete number of aggregate sizes (Cusatis et al. 2003). Figure 4. Aggregate sieve curve (for two different specimens), d50 = 12 (Kozicki & Tejchman 2007).

by Cusatis et al. (2003) and Eckardt anf Könke (2006). First, a grading curve was chosen (Fig. 4). Next, the certain amounts of particles with defined diameters were generated according to curve in Fig. 5. Finally, the spheres describing aggregates were randomly placed in the specimen preserving the particle density and a certain mutual minimum distance (van Mier et al. 1995)

Dp > 1.1

D1 + D 2 , 2

(4)

where D is the distance between two neighboring particle centers and D1 and D2 are the diameters of these two particles. In 3D calculations, the minimum element length was 0.6 mm, and the maximum one was

about 4 mm (2 mm on average). The size of 3D concrete specimen was 5 × 5 × 5 cm3 . The fibre length was 20 mm. The amount of elements was 200 000. Tab. 1 includes the material parameters assumed for analyses. The material stiffness parameters kl and kb for aggregate, cement matrix and bond between matrix and aggregate were determined empirically (Kozicki & Tejchman 2007, 2008) to match experimental results at the macro-scale with numerical ones on the basis of a uniaxial tension and compression test for concrete (van Mier et al. 1995). The calculations resulted in the following values of the material parameters kl and εmin : kl = 0.01 and εmin = 0.2% in the cement matrix, kl = 0.03 and εmin = 0.133% in the aggregate and kl = 0.007 and εmin = 0.05% in the bond zones (Kozicki et al. 2007, Kozicki et al. 2008). The ratios between the parameters kl and εmin for different phases were assumed in a similar way as the ratios between the moduli of elasticity and tensile strengths for each concrete phase (van Mier et al.

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1995). The weakest phase was bond between aggregate and cement matrix. A different value of the parameter kb was used in compression and in tension (Kozicki & Tejchman 2008). For the sake of simplicity, one assumed that same ratio of p = kb /kl for all phases (0.6 in tension and 0.2 in compression) (Tab. 1). The chosen ratio of p resulted in the following Poisson’s ratios: 0.22 (uniaxial compression) and 0.07 (uniaxial tension) (Kozicki & Tejchman 2008). In the case of the fibrous interface, different cases were taken into account (Tab. 1) by changing the parameter εmin (the values of E and p were similar as for the aggregate interface): A. no interface between fibre and cement matrix (properties the same as for cement matrix), B. strength of fibrous interface smaller than strength of aggregate interface (local critical tensile strain εmin = 0.025%), C. strength of fibrous interface equal to strength of aggregate interface (εmin = 0.05%), D. strength of fibrous interface stronger than strength of aggregate interface (εmin = 0.1%, 0.2%, 0.5%, 1.0%, 10% and 90%).

Figure 7. Calculated stress-strain curves for 5 × 5 × 5 cm3 notched concrete specimen with 0.35% fibre volume subjected to uniaxial extension: a) pure concrete, b) fibrous interface with εmin = 0.1%, c) fibrous interface with εmin = 0.2%, d) fibrous interface with εmin = 0.5% (σ22 —vertical normal stress, ε22 —vertical normal strain).

The material parameters for steel fibres were: E = 160 GPa, p = 0.6 (tension), p = 0.2 (compression), kl = 0.08 and εmin = 90% (Tab. 1).

3

SIMULATIONS OF UNIAXIAL TENSION

The results of calculated stress-strain curves during uniaxial tension with a notched cubical 5 × 5 × 5 cm3 concrete specimen with smooth horizontal edges are shown in Figs. 6–11 for different properties of the

Figure 8. Calculated stress-strain curves for 5 × 5 × 5 cm3 notched concrete specimen with 0.35% fibre volume subjected to uniaxial extension: a) pure concrete, b) fibrous interface with εmin = 1%, c) fibrous interface with εmin = 10%, d) fibrous interface with εmin = 90% (σ22 —vertical normal stress, ε22 —vertical normal strain).

Figure 6. Calculated stress–strain curves for 5 × 5 × 5 cm3 notched concrete specimen with 0.35% fibre volume subjected to uniaxial extension: a) pure concrete, b) without fibrous interface, c) fibrous interface with εmin = 0.025%, d) fibrous interface with εmin = 0.05% (σ22 —vertical normal stress, ε22 —vertical normal strain).

Figure 9. Calculated stress-strain curves for 5 × 5 × 5 cm3 notched concrete specimen with 2.25% fibre volume subjected to uniaxial extension: a) pure concrete, b) without fibrous interface, c) fibrous interface with εmin = 0.025%, d) fibrous interface with εmin = 0.05% (σ22 —vertical normal stress, ε22 —vertical normal strain).

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Figure 10. Calculated stress-strain curves for 5 × 5 × 5 cm3 notched concrete specimen with 2.25% fibre volume subjected to uniaxial extension: a) pure concrete, b) fibrous interface with εmin = 0.1%, c) fibrous interface with εmin = 0.2%, d) fibrous interface with εmin = 0.5% (σ22 —vertical normal stress, ε22 —vertical normal strain). Figure 12. Distribution of aggregate in concrete specimens 5 × 5 × 5 cm3 subjected to uniaxial extension (with 2.25% fibre volume).

Figure 11. Calculated stress-strain curves for 5 × 5 × 5 cm3 notched concrete specimen with 2.25% fibre volume subjected to uniaxial extension: a) pure concrete, b) fibrous interface with εmin = 1%, c) fibrous interface with εmin = 10%, d) fibrous interface with εmin = 90% (σ22 —vertical normal stress, ε22 —vertical normal strain).

fibrous interface (Tab. 1). The distribution of aggregate and steel fibres in the concrete specimen is shown in Figs. 12 and 13. In turn, Fig. 14 demonstrates the deformed concrete specimens at failure. The results show that both concrete strength and ductility increase with increasing amount of steel fibres if the fibrous interface is significantly stronger than the aggregate interface, i.e. εmin ≥ 0.2% (both strength and ductility increase with increasing εmin ≥ 0.2%). Due to a high particle density of 60%, percolation of bond zones occurs early in the loading history. Since the interface between cement matrix and aggregate is the weakest component of the system, the material becomes initially weak there, cracks

Figure 13. Distribution of steel fibres in concrete specimens 5 × 5 × 5 cm3 subjected to uniaxial extension (with 2.25% fibre volume).

are created along aggregate and the pre-peak nonlinearity does not appear. The cracks propagate from the notch in the mid-region of the specimen. Since the amount of aggregates is relatively large, the cracks cannot propagate in long lines. Instead of this, several discontinuous macro-cracks propagate in a tortuous manner. Between fibres, the cracks overlap and form

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branches. The crack propagation is clearly disturbed by the presence of steel fibres which delay their development. If the fibrous interface is weaker than the aggregate interface, both material strength and ductility decrease due to the fact that several straight cracks are only created along fibres which act as imperfections promoting cracks (Kozicki & Tejchman 2009).

4

CONCLUSIONS

The following conclusions can be drawn from numerical results for fibrous concrete described as five-phase material at meso-scale using a linear lattice model where individual steel fibres were explicitly modeled. The lattice model in spite of its simplicity is capable to simulate fracture process. The obtained results of stress-strain curves for fibrous concrete during uniaxial tension qualitatively compare with experimental results. By using an elastic-purely brittle local fracture law at the particle level of the material, global softening behavior is obtained. The advantage of our quasi-static lattice model lied is its explicit character. Thus, a large amount of elements could be taken into account when using parallelized computers. Both strength and ductility of concrete specimens increase with increasing amount of fibres during uniaxial extension, if strength of the fibrous interface is significantly higher than strength of interfacial zones between aggregate and cement matrix. In this case, the crack propagation is delayed by the presence of steel fibres. The calculations with a lattice model will be continued. Further calibration studies will be performed by taking into account the real micro-structure of fibrous concrete specimens. The possibility of a crack closure will be considered. In addition, inertial forces will be taken into account during dynamic calculations.

REFERENCES

Figure 14. Crack distribution at failure in concrete specimens 5 × 5 × 5 cm3 with 2.25% fibre volume subjected to uniaxial extension including fibrous interface with: a) εmin = 0.1%, b) εmin = 10%, c) εmin = 90%.

Al-Taan, S.A. & Ezzadeen, N.A. 1995. Flexural analysis of reinforced fibrous concrete members using the finite element method. Computers and Structures 56(6), 1065–1072. Altun, F., Haktanir, T. & Ari, K. 2007. Effects of steel fiber addition on mechanical properties of concrete and RC beams. Construction and Building Materials 21(3), 654–662. Balaguru, P. & Shah, S.P. 1992. Fiber reinforced cement composites. McGraw Hill, USA. Balendran, R.V., Zhou, F.P., Nadeem, A. & Leung, A.Y. 2002. T. Influence of steel fibres on strength and ductility of normal and lightweight high strength concrete. Building and Environment 37, 1361–1367. Bentur, A. & mindess, S. 1990. Fiber reinforced cementitious composites. Elsevier Applied Science, New York.

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Bolander, J.E. & Saito, S. 1997. Discrete modeling of shortfiber reinforcement in cementitious composites. Adv. Cem. Based Mater. 6, 76–86. Chenkui, H. & Guofan, Z. 1995. Properties of steel fibre reinforced concrete containing larger coarse aggregate. Cement and Concrete Composites 17, 199–206. Cusatis, G., Bazant, Z.P. & Cedolin, L. 2003. Confinement-shear lattice model for concrete damage in tension and compression: I. theory. ASCE Journal of Engineering Mechanics 129(12), 1439–1448. Eckardt, S. & Könke, C. 2006. Simulation of damage in concrete structures using multiscale models. Computational Modelling of Concrete Structures, EURO-C (eds.: G. Meschke, R. de Borst, H. Mang and N. Bicanic), Taylor and Francis, 77–83. Falkner, H. & Henke, V. 2000. Stahlfaserbeton-konstruktive Anwendungen. Beton- und Stahlbetonbau 95(10), 597–606. Jones, P.A., Austin, S.A. & Robins, P.J. 2008. Predicting the flexural load-deflection response of steel fibre reinforced concrete from strain, crack-width, fibre pull-out and distribution data. Materials and Structures 41, 449–463. Kabele, P. 2007. Multiscale framework for modeling of fracture in high performance fiber reinforced cementitious composites. Engineering Fracture Mechanics 74, 194–209. Kooiman, A.G., van der Veen, C. & Walraven, J.C. 2000. Design relation for steel fibre reinforced concrete. Heron, 45, 275–307. Komlos, K., Babal, B. & Nürnbergerova, T. 1995. Hybrid fibre-reinforced concrete under repeated loading. Nuclear Enginering and Design 156, 195–200. Kozicki, J. & Tejchman, J. 2007. Effect of aggregate structure on fracture process in concrete using 2D lattice model. Archives of Mechanics 59(4–5), 1–20. Kozicki, J. & Tejchman, J. 2008. Modelling of fracture processes in concrete using a novel lattice model. Granular Matter 10(5), 399–405. Kozicki, J. & Tejchman, J. Modeling of fracture process in fibrous concrete using a lattice model. X International Conference on Computational Plasticity—COMPLAS X.-E. Oñate and D.R.J. Owen, eds), CIMNE, Barcelona, 1–4, 2009. Krstulovic-Opara, N., Haghayeghi, A.R., Haidar M. & Krauss, P.D. 1995. Use of conventional and highperformance steel-fiber reinforced concrete for bridge deck overlays. ACI Mater. J. 92(6), 669–677. Li, V.C., Horii, H., Kabele, P., Kanda, T. & Lim, Y.M. 2000. Repair and retrofit with engineered cementitious composites. Engineering Fracture Mechanics 65, 317–334.

Li, Z., Perez Lara, M.A. & Bolander, J.E. 2006. Restraining effects of fibers during non-uniform drying of cement composites. Cement and Concrete Research, 36, 1643–1652. Lim, T.Y., Paramaivam, P. & Lee, S.L. 1987. Analytical model for tensile behaviour of steel-fibre concrete. ACI Materials Journal. Mohammadi, Y., Singh, S.P. & Kaushik, S.K. 2008. Properties of steel fibrous in fresh and hardened state. Construction and Building Materials 22, 956–965. Radtke, F., Simone, A. & Sluys, L.J. 2008. An Efficient computational model for fibre reinforced concrete incorporating information from multiple scales. Proc. 8th Worls Congress on Computational Mechanics WCCM 2008 (eds.: B.A. Schrefler, U. Perego), Venice 30.06–4.06. Schlangen, E. & Garboczi, E.J. 1997. Fracture simulations of concrete using lattice models: computational aspects. Engineering Fracture Mechanics 57, 319–332. Schnütgen, B. & Teutsch, M. 2001. Beonbauwerke aus Stahlfaserbeton beim Umgang mit umweltgefährdenden Stoffen. Beton- und Stahlbetonbau 96(7), 451–457. Shah, S.P. & Rangan, B.V. 1971. Fiber reinforced concrete properties. ACI Journal 68(2), 126–135. Sivakumar, A. & Santhanam, M. 2007. Mechanical properties of high strength concrete reinforced with metallic and non-metallic fibres. Cement and Concrete Composites 29, 603–608. van Hauwaert, A. & van Mier, J.G.M. 1998. Computational modeling of the fibre-matrix bond in steel fibre reinforced concrete. Fracture Mechanics of Concrete Structures (eds.: H. Mihashi, K. Rokugo), Aedificatio Publishers, Freiburg, Germany, 561–571. van Mier, J.G.M., Schlangen, E. & Vervuurt, A. 1995. Lattice type fracture models for concrete. Continuum Models for Materials with Microstructure, H.-B. Mühlhaus (Ed.), John Wiley & Sons, 341–377. van Mier, J.G.M. & van Vliet, M.R.A. 2003. Influence of microstructure of concrete on size/scale effects in tensile fracture. Engineering Fracture Mechanics 70(16), 2281–2306. Walraven, J.C. 7 Grünewald, S. 2002. Regelung und Anwendung des Stahlfaserbetons in den Niederlanden. Stahlfaserbeton—ein unberechenbares Material. Proc. Bauseminar 2002, Braunschweig, 164, 47–63. Zollo, R.F. 1997. Fiber-reinforced concrete: an overview after 30 years of development. Cement Concrete Composite 19(2), 107–122.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Limit analysis of 3D reinforced concrete frames Kasper P. Larsen Ramboll Denmark DTU Byg, Technical University of Denmark

Peter N. Poulsen & Leif O. Nielsen DTU Byg, Technical University of Denmark

ABSTRACT: In this paper we present a new finite element framework for lower bound limit analysis of reinforced concrete beams subjected to loadings in three dimensions. The method circumvents the need for a complex section force based yield criteria by creating a discrete representation of the internal stress state in the beam, and employ the yield criteria on the stress state level. The stress state is decomposed into concrete and reinforcement stresses and separate yield criteria are applied to each stress component. Simple upper- and lower bounds are applied to the reinforcement stresses and the classical Mohr-Coulomb criteria are applied to the concrete stresses. The exact Mohr-Coulomb criteria are implemented using Semidefinite Programming and an approximation using Second-Order Cone Programming is developed for improved performance. The element is verified by comparing the numerical results with analytical solutions. 1

INTRODUCTION

Structural analysis of reinforced concrete structures applying limit state analysis always results in an optimization problem. In order to solve the optimization problem the yield criteria needs to be established for the given structural component. Originally, yield criteria have often been sought on a sectional level, e.g. for beams (Damkilde and Hoyer 1993), plates (Krabbenhoft and Damkilde 2002), stringers (Damkilde, olsen, and Poulsen 1994) and disks (Poulsen and Damkilde 2000). In most of these yield criteria approximations have been applied in order to simplify the formulation. Another way of establishing the load bearing capacity of a structural component would be to divide the section into smaller parts/zones for which the yield criteria is well-known and then get the capacity by summing up the different contributions. In a beam the yield criteria could be established on a sectional level as the N-M-diagram can be found by analyzing typical situations for a beam. For a 2D beam this approach is applicable but for a 3D beam this turns out to be very troublesome. Instead the capacity for a 3D beam can be formulated by dividing the crosssection into zones. This approach has been applied in (Niebling, Vinther, and Larsen 2007) where a reinforced concrete 3D-beam was established using the zone model, though the cross section modeling capabilities was limited to rectangular beams with simple reinforcement. Recent advances in optimization algorithms and computational power has made it possible to solve medium to large scale problems with complex yield criteria within reasonable time. The Finite

Element Method (FEM) is the most common method for numerical analysis of structures. In this paper we present an element which expands on the work done by (Niebling, Vinther, and Larsen 2007) by allowing engineers to model a wider range of cross section types. Similar to the element presented in (Niebling, Vinther, and Larsen 2007), a discrete formulation of the internal stress state is used, and the yield criteria are applied at the stress level instead of at the section force level. The classic Mohr-Coulomb failure criteria are applied to the concrete while simple upper- and lower bounds are applied to the reinforcement. We will here implement the exact Mohr-Coulomb criteria using Semidefinite Programming (SDP) as described in (Larsen, Poulsen and Nielsen 2009). We will also develop an approximate method based on second-order cones (SOCP) which solves faster and more efficiently. The lower bound limit analysis problem is formulated as minimize subject to

−λ −Rλ + H σ = R0 σ T ∈ Ct

(1.1)

where λ is the scalar load factor sought optimized (by minimizing the negative load factor we will find the maximum load capacity). The vector σ contains the n stress parameters used to define the stress state in within the structure. The matrix H defines the equilibrium equations and the continuity conditions, the vectors R and R0 are the variable and static load vectors respectively, and the Ct is a convex cone.

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2

ELEMENT MODEL

Here we consider a geometric linear beam element with local coordinate axes as shown on Figure 1. The element length is measured along the local x-axis and the width and height is measured along the y- and z-axis respectively. Here, the x direction will also be referred to as the longitudinal direction and the y- and z-direction will be referred to as the transverse directions. The element is capable of carrying loads in the longitudinal and transverse directions as well as torsional moments as illustrated on Figure 1. The element is intended to be compatible with a shell element, such that full 3D structures can be modeled. The external load intensities are coupled to the internal section forces through the following equilibrium equations. py + Vy,x = 0

(2.1)

pz + Vz,x = 0

(2.2)

tx + Mx,x = 0

(2.3)

The bending moments are coupled to the external loads through the shear forces by two additional equilibrium equations Mz,x + Vy = 0

(2.4)

My,x − Vz = 0

(2.5)

Finally, a set of static boundary conditions are required to get at complete description of the internal section forces. These are applied at the beam ends, see Figure 1, and defined as QA = {−NxA , −VyA , −VzA , −MxA , −MyA , −MzA }

(2.6)

QB = {NxB , VyB , VzB , MxB , MyB , MzB }

(2.7)

2.1 Zone model To obtain a lower bound solution, the element must be in a safe state i.e. the yield criterion must be obeyed at all points within the element. To circumvent the need for a yield criterion based on section forces, we

Figure 1.

3D Beam element model.

apply the material constraints on a stress state level. In (Niebling, Vinther, and Larsen 2007), a zone model was used to discretize the internal stress state in the beam element. Here, we will adopt this zone model and extend it to work with an arbitrary number of zones, hereby enable engineers to model and analyze more complex cross sections. The zones must be aligned with the local coordinate axis and the reinforcement is smoothed over the zone area. We will here assume that the resulting transverse normal stresses, i.e. the sum of transverse reinforcement and concrete stresses, are equal to zero. The transverse normal stresses in the concrete and reinforcement does not have to be zero, as long as the total transverse stresses are. Additionally, only the transverse shear stresses, τxy and τxz , are considered in the cross section. The stress state at a point within the beam is then defined by ⎤ ⎡ σx τxy τxz 0⎦ (2.8) σ = ⎣τxy 0 0 τxz 0 Besides the section force equilibrium equations described in the previous section, the stress state within a zone must also fulfil the following equilibrium equation σx,x + τxy,y + τxz,z + fx = 0

(2.9)

where fx is the volume load in the longitudinal beam direction. The normal stresses, σx , are chosen constant in the yz-plane and the shear stresses, τxy and τxz , varies linearly in the y- and z-direction respectively. A fourth degree variation of the normal stresses and a third degree variation of the shear stresses in the longitudinal direction requires 5 normal stress variables and 4 shear stress variables along each side of the zone, see Figure 2. The stress state in a zone can be parameterized by a total of 21 stress variables. To obtain a statically admissible stress field within the element, traction continuity between zones must be ensured. Because the zones are axis-aligned and transverse normal stresses are disregarded, this is simply done by ensuring shear stress continuity along shared zone faces. Because the internal stress state is discretized by the zones, the 6 section forces required by the equilibrium

Figure 2.

Sketch of zone model.

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equations and the static boundary conditions can be computed from a set of summations e.g. Mx =

n  1 (2) (1) (2) (2) Ai ((τxz + τxz )yi + (τxy + τxy )zi ) (2.10) 2 i=1

where n is the number of zones, Ai is the cross section area of zone i and yi and zi are the geometric center point coordinates of the zone cross section. 3

YIELD CRITERION

The equilibrium equations defined in the previous section ensures that the obtained solution is statically admissible. Because a lower bound solution requires the structure to be in a safe state, a set of material constraints must be applied as well. The stress state given in Eq. (2.8) is decomposed into a set of concrete- and reinforcement stresses as σ = σc + As σs where ⎡

σcx σc = ⎣τcxy τcxz

τcxy σcy 0

σc + kαI  0 σc − β I  0 β ≤ fA

(3.1) ⎤ τcxz 0 ⎦ σcz

are the concrete stresses, ⎡ ⎤ σsx 0 0 σs = ⎣ 0 σsy 0 ⎦ 0 0 σsz

β≤ (3.2)

(3.3)

Since only normal stresses are considered in the reinforcement, simple upper- and lower bounds are applied to these (3.5)

where fYc and fYt are the compression and tensile strength of the rebars respectively. The Mohr-Coulomb failure criterion, which consists of a sliding- and a separation criteria as shown in Eq. (3.6) and Eq. (3.7), will be applied to the concrete stresses. kσ1 − σ3 ≤ fc

(3.6)

σ1 ≤ fA

(3.7)

In Eq. (3.7), fA is the separation strength and fc is the uni-axial compression strength of the concrete.

(3.8)

fc −α k

where α and β are scalar variables. 3.1

are the reinforcement stresses and As is then a diagonal matrix containing reinforcement area per unit area perpendicular to the local element axes ⎡ ⎤ Asx 0 0 As = ⎣ 0 Asy 0 ⎦ (3.4) 0 0 Asz

−fYc ≤ σs ≤ fYt

The separation strength, fA , will often be equal to the uni-axial tensile strength, ft . k is a parameter determined from the frictional angle, φ. Here, k = 4 are used which is equivalent to a frictional angle of approximately 37◦ . The principal stresses are sorted as σ1 ≥ σ2 ≥ σ3 . The yield criteria on the reinforcement and the concrete will be applied in a number of control points along the length of the beam element. The SDP formulation of the Mohr-Coulomb criteria is, see (Larsen, Poulsen, and Nielsen 2009) for further details: For a given stress state σc in the concrete, the yield criteria can be written as two linear matrix inequalities and two linear inequalities

Approximation to Mohr-Coulomb using SOCP

Despite the attractive properties of the SDP formulation of the Mohr-Coulomb criteria given in the previous section, performance considerations in practical applications calls for a yield criterion based on second-order cones. The major challenge here is to model the tri-axial effects which occurs in areas with reinforcement in all three directions. Initially, we will make the assumption that the separation strength of the concrete material is equal to zero. Because of the low separation strength found in concrete, this assumption is often made in practical engineering. All tensile stresses must therefore be carried by the reinforcement and transverse reinforcement must be present in zones that carry shear stresses. Figure 3 shows a rectangular beam reinforced with longitudinal reinforcement bars and stirrups. The beam has been discretized into 9 zones as marked with the dashed lines. The figure also illustrates the three possible zone types used when modeling cross sections. The zone types differ by the transverse reinforcement present in the zone. The first type has no transverse reinforcement and because the concrete is not able to carry any tensile stresses, only a uni-axial stress state is possible in these zones. The second zone type has transverse reinforcement in one of the transverse directions making it similar to a reinforced concrete disk with a biaxial stress state. In the third zone type, reinforcement is present in both of the transverse directions making a tri-axial stress state possible. The zone types

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t 2

3

h

where

t

1

2

3

i =

t

Asi fYt Aci fc

for i = {x, y, z}

(3.12)

This type of constraint can be handled by some solvers (these are known as Quadratic Constrained Programming (QCP) problems) but it is often recommended to use cone constraints because they solve more efficiently, (MOSEK). Eq. (3.11) can be translated into a rotated quadratic cone as shown below

2

2x1 x2 ≥ x32 + x42 2

3

3

1 x1 = − σcx y z 2

t

w

x2 = 1 x3 = y τcxy x4 = z τcxz

Figure 3. Rectangular beam example with longitudinal rebars and stirrups.

are marked by the numbers 1–3 on Figure 3. Separate yield criteria are applied in each of the three zone types. 3.1.1 Triaxial stress state Here, we will propose an approximation to the full Mohr-Coulomb using second order cone constraints only. The approximation will, to some extent, be able to utilize the tri-axial stress state in the concrete. The approximation is based on the observation that zones subjected to tri-axial stress states often will have the same amount of reinforcement in the transverse directions. Since zones with reinforcement in both direction often will be corner zones, see Figure 3, in which a stirrup wraps around the longitudinal reinforcement, this assumption seems reasonable. The principal stresses are determined by the eigenvalues of the concrete stress tensor σc ⎡ ⎤ σcx − λ τcxy τcxz ⎣ τcxy σcy − λ 0 ⎦=0⇒ (3.9) τcxz 0 σcz − λ (σcx − λ)(σcy − λ)(σcz − λ) 2 (σ − λ) − τ 2 (σ − λ) = 0 − τcxy cz cxz cy

For the sliding criterion, Eq. (3.6), we will assume that the transverse normal stresses are of equal magnitude i.e., σcy = σcz . If the transverse normal stresses are not equal, we apply the largest (compression is negative) of the transverse normal stresses in both directions. It should be noted that if the difference between the reinforcement in each direction becomes large, the criterion will be too restrictive. Therefore, a simplified criterion must be applied in the biaxial stress state. First, we will assume that σcz is the largest of the transverse normal stresses. If we substitute σcy with σcz in Eq. (3.10) we get (σcx − λ)(σcz − λ)2 2 2 − τcxy (σcz − λ) − τcxz (σcz − λ) = 0

It is evident that one eigenvalue must be given by λ1 = σcz

(3.10)

In (Andreasen 1985) the separation criterion, Eq. (3.7), for a reinforced solid is utilizing Eq. (3.1) and Eq. (3.10) approximated on quadratic form as   σcx − x − y z fc  + z

τcxy fc

2

 + y

τcxz fc

2 ≤0

(3.13)

(3.11)

(3.14)

The remaining two eigenvalues are computed from

σcx + σcz λ2 = λ3 2   ±

2

1 (σcx − σcz ) 2

2 + τ2 + τcxy cxz

(3.15)

Since we do not know the order of the three eigenvalues we must check every combination hereof.

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From Eq. (3.15) we see that λ2 ≥ λ3 which means that it is sufficient to check three combinations of the eigenvalues. In order to formulate the yield criterion on a second-order cone form, we introduce the auxiliary variable 1 1 =

1 (σcx − σcz ) 2

    σcx + σcz 1 fc − (k − 1) k +1 2



σcx + σcz 2 2 + τ2 + 1 + τcxy cxz 2

 ≤ fc ⇒ (3.17)

σ + σcy 2 + τ 2 ≤ f − k · σ + cx 12 + τcxy (3.18) c cz cxz 2



2 + τ2 ≤ α 12 + τcxy 1 cxz

(3.24)

2 + τ2 ≤ α 22 + τcxy 2 cxz

(3.25)

1 =

1 (σcx − σcz ) 2

(3.26)

2 =

 1 σcx − σcy 2

(3.27)



The above inequality is a quadratic cone constraint. Case II: k · λ2 − λ1 ≤ fc Here, λ1 is assumed to be the smallest eigenvalue making λ2 the largest. Inserting this into Eq. (3.6) we get  k·

 σcx + σcz 2 2 2 + 1 + τcxy + τcxz − σcz ≤ fc ⇒ 2 (3.19)

f σ σ + σcz 2 + τ 2 ≤ c + cz − cx 12 + τcxy cxz k k 2

α1 ≤ fc − k · σcz +

Case III: k · λ2 − λ3 ≤ fc The last combination is when λ1 lies between λ2 and λ3 which gives the following constraints when inserted into Eq. (3.6) k·

σcx + σcz + 2 

−



σcx + σcz − 2



+

2 τcxy

+

fc σcz σcx + σcz + − k k 2

(3.29)

α1 ≤

    1 σcx + σcz fc − (k − 1) k +1 2

(3.30)

α2 ≤ fc − k · σcy +

2 τcxz



σcx + σcz 2 +



(3.32)

α2 ≤

    σcx + σcy 1 fc − (k − 1) k +1 2

(3.33)

 (k − 1)

4

2 + τ 2 (k + 1) ≤ f ⇒ 12 + τcxy c cxz

(3.22)

(3.31)

σcy σcx + σcy fc + − k k 2

≤ fc ⇒ (3.21)

σcx + σcy 2

α2 ≤

 2 + τ2 12 + τcxy cxz

(3.28)

α1 ≤

 12

σcx + σcz 2

(3.20)

which yields another quadratic cone constraint.



(3.23)

The above conditions are used when σcz is the largest of the transverse normal stresses. The case when σcy is the largest normal stress is handled by interchanging the z-indicies with y in the above constraints adding an additional three cones and another auxiluary variable 2 to the set of inequality constraints. We can reduce the number of cones by introducing two more auxiluary variables, α1 and α2 . The complete yield criterion can be written as a combination of two cones, two linear equality constraints and six linear inequality constraints as shown in Eq. (3.24)–(3.33).

(3.16)

Case I: k · λ1 − λ3 ≤ fc Here we assume that λ1 is the largest of the three eigenvalues. λ3 must therefore be the smallest and when inserted into Eq. (3.6) we get k · σcz −

2 + τ2 ≤ 12 + τcxy cxz

NUMERICAL TESTS

This section shows some numerical examples of the beam element presented in this paper. In the first

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example a single element is subjected to some basic load cases and the results are compared to analytical solutions for verification. The second examples utilizes the flexibility of the zone model to analyze a beam with a more complex cross section. 4.1

reinforcement varies along the length as described in the figure. Because a single element must have the same shear reinforcement throughout, 5 elements are required to model the beam. The cross section is modelled using 37 zones as illustrated on Figure 4 and the material parameters are noted on the figure. Initially, the uni-axial bending capacity of the cross section with respect to the y-axis is found to be My = −2073 kN/m. When a constant line load is applied along the geometric center line, the load bearing capacity is found to be pz = 119 kN/m. It should be noted that the maximum bending moment is only 689 kN/m The load bearing capacity is therefore governed by the shear strength of the beam. As part of a renovation project in Copenhagen, Ramboll conducted a series of tests on beams similar to the one shown on Figure 4. Five tests were done and the capacity was here found to be pz = 136 kN/m. The material strength given on Figure 4 are design strength provided by the

Basic load cases

For the basic element tests, a simple rectangular beam modeled using the 9 zones shown on Figure 3 is used. The width and height of the beam is 250 mm and 600 mm respectively and the thickness, t, of the outer zones is 76 mm. The reinforcement diameters are 20 mm and 6 mm for the longitudinal and the stirrups respectively. The material parameters for concrete and reinforcement are as shown below fYt = 550 MPa fYc = 0 fc = 25 MPa ft = 0

k=4

All examples are analyzed using both the exact and the approximate Mohr-Coulomb model. The tests results are presented in Table 1. 4.2 Analysis of inverse T-Beam The rectangular beam analyzed in the previous section is similar to the one presented in (Niebling, Vinther, and Larsen 2007). The major advantage of the element presented here is its ability to model more complex cross sections. This example demonstrates how the flexible zone model can be used to determine the load bearing capacity of the inverse T-beam shown on Figure 4. The beam is 6.8 m long and the shear

Figure 4. Cross section of inverse T-beam with zone division model.

Table 1. Results obtained from the exact, and approximate method compared to an analytical solution. Forces are in kN, moments in kNm and line loads in kN/m. Static model

SDP SOCP

Analytical

Comments

Fx

691

691

691

Fx

4626

4626

4500

272

272

273

The lower capacity found by the numerical method is caused by the zone meshing.

22

22

21

The maximum moment is identical to the numerical moment capacity found above.

47

47

51

The lower capacity found by the numerical method is caused by the combined shear stresses in the corner zones.

My

The numerical and analytical solutions are identical because no approximations are made to the tensile strength of the reinforcement. The higher capacity found by the numerical method is caused by the tri-axial effects found in the corner zones.

pz

Mx

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manufacturer which are often lower than the actual strength. It is therefore expected that the capacities found from tests are higher than the one found by the numerical model. 5

and analyzed and the numerical results were compared to those obtained from physical tests. The load bearing capacity found by the numerical model was only 13% less than those found from the tests which is considered very good when taking the the uncertainties of the material properties into account.

CONCLUSION

We have presented a method for determining the load bearing capacity of 3D beams subjected to both bending- and torsional moments as well as normal and shear forces. The method circumvent the need for a complex yield criteria based on section forces by representing the internal stress state by a discrete set of zones. The material constraints are then applied directly at the stress-state level. The Mohr-Coulomb failure criteria is used to constrain the stress state in the concrete. Two different methods are employed to solve the non-linear problem posed by this criteria. The first method is Semidefinite Programming which makes it possible to implement the exact Mohr-Coulomb criteria. It does suffer from some performance issues on the current solvers. A method using only second-order cones was therefore developed to reduce computation times. This method makes good approximations to the exact Mohr-Coulomb and is sufficient for most analysis cases. The numerical model was tested using a range of basic load cases. The results were then compared with analytical results and it was found that the correlation was very good. An inverse T-beam was also modelled

REFERENCES Andreasen, Bent Steen Nielsen, M.P. (1985). Armering af beton i det tredimensionale tilfælde. Bygningsstatiske Meddelelser 56(2–3). Damkilde, L. and O. Hoyer (1993). An efficient implementation of limit state calculations based on lower-bound solutions. Computers and Structures 49(6), 953–962. Damkilde, L., J.F. Olsen, and P.N. Poulsen (1994). A program for limit state analysis of plane, reinforced concrete plates by the stringer method. Bygningsstatiske Meddelelser 65(1), 1–26. Krabbenhoft, K. and L. Damkilde (2002). Lower bound limit analysis of slabs with nonlinear yield criteria. Computers and Structures 80(27–30), 2043–2057. Larsen, K.P., P.N. Poulsen, and L.O. Nielsen (2009). Limit analysis of solid reinforced concrete structures. In The First International Conference on Computational Technologies in Concrete Structures. MOSEK. The mosek optimization toolbox for matlab manual. Niebling, J., A. Vinther, and K.P. Larsen (2007). Numerisk modellering af plastiske betonkonstruktioner. Poulsen, P.N. and L. Damkilde (2000). Limit state analysis of reinforced concrete plates subjected to in-plane forces.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

The role of domain decomposition techniques for the study of heterogeneous quasi-brittle materials O. Lloberas Valls, D.J. Rixen, A. Simone & L.J. Sluys Delft University of Technology, Delft, The Netherlands

ABSTRACT: We focus on the analysis of fracture in quasi-brittle materials by exploiting the potential of domain decomposition techniques. More specifically, we restrict our attention to the FETI (Finite Element Tearing and Interconnecting) method which is used as a solver in our non-linear solution scheme. We develop criteria to anticipate the linear/non-linear character of different regions in the structure according to a damage constitutive model. A first application of our scheme focuses on the efficiency increase of a monoscale analysis by simplifying the computations in those areas that remain linear. The second application treats the problem in a multiscale fashion where the resolution of the non-linear domains is increased in order to describe failure phenomena with a higher degree of accuracy. Both applications represent a significant improvement from a computational standpoint when the main non-linear regions are small compared to the size of the whole specimen. This is often the case for brittle and quasi-brittle materials where strain localization is expected to occur upon failure. 1

INTRODUCTION

The study of cracking and failure phenomena is of utmost importance in the design of many engineering materials. Damage nucleation and growth can eventually cause the collapse of an existing structure. For this reason, the accurate modeling of these phenomena has been a topic of ongoing research in the last decades. It is therefore crucial to model, with great care, those regions in the material that can potentially show failure and strain localization. In this study we develop strategies to focus the computational effort preferentially at those regions of the structure that undergo the main non-linear processes. A domain decomposition method is adopted to split the discretized structure in several domains and, based on certain criteria, allow for a different treatment between linear and non-linear regions. Some preliminaries of the selected domain decomposition technique are introduced in Section 2. The criteria to predict the linear and non-linear constitutive behaviour of the material is described in Section 3. Two applications of the present framework are elaborated in Sections 4 and 5 including an illustrative example for each of them. They basically consist on: (i) an improvement of the solver performance by simplifying a number of operations at the elastic domains; (ii) a zoom-in at the areas where damage growth is taking place. 2

equations. Consequently, different strategies based on a divide and conquer approach are well suited for these complex systems. One of the most popular choices is the use of domain decomposition techniques. These methods can be used as parallel efficient solvers for the partitioned systems of equations arising from the mechanical analysis of a decomposed structure. In the following we introduce the basic principles of the FETI (Finite Element Tearing and Interconnecting) technique introduced by Farhat and Roux (1991) which constitutes the starting point of the present framework. Consider a body which is divided in two domains (Fig. 1). Continuity of the solution field between domains is enforced by the interface constraint u(1) = u(2)

at I .

(1)

The variational forms of the equilibrium problem arising from each of the domains in combination with the continuity condition (Eq. 1) results in a hybrid variational principle. Finite element discretization using a standard Galerkin procedure transforms the hybrid

(2)

(1)

THE FETI METHOD I

The computation of structures with a high degree of resolution leads to the solution of large systems of

Figure 1.

Decomposition in two domains.

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The operator F I represents the flexibility of the interface. In absence of the rbm the matrix F I sets the relation between the forces λ and the displacements d at the interface. The operator G I is built of the rigid body modes of each domain restricted onto the interface and e is the part of the applied force that is out of balance with respect to the subdomain rigid body modes. The FETI method can be regarded as a solver for a decomposed structure which is formed by a blend of direct solvers for the independent local problems (Eq. 3) and iterative solvers for the coupled interface problem (Eq. 5). A short overview of primal and dual domain decomposition methods is presented by Rixen (2001). The FETI method, and in particular its dual-primal version has become one of the most used parallel solution techniques.

variational principle in the set of equations ⎡

K (1) ⎣ 0 B(1)

0 K (2) B(2)

⎤ ⎡ ⎤ T ⎤⎡ B(1) f (1) u(1) T (2) (1) ⎦, B(2) ⎦ ⎣ u ⎦ = ⎣ f λ 0 0 (2)

where K (s) and B(s) represent the stiffness and signed Boolean matrices for domain (s) . The solution field is composed by the domain displacements u(s) and the connecting forces λ. The solution of Equation 2 for a given set of λ forces and a particular configuration of the rigid body modes (rbm) reads +

T

u(s) = K (s) (f (s) − B(s) λ) − R(s) α (s) .

(3)

(s) +

is introduced in The generalized inverse K order to handle local singularities induced by floating domains (i.e. domains which are not fixed by any boundary condition once they are disconnected from its neighboring domains). Conversely, the matrix + K (s) is the inverse of K (s) if it is non-singular (i.e. the domain is isostatically or hyper-statically constrained). All floating domains exhibit rbm R(s) spanning the null space of K (s) . The amplitudes α (s) of the rbm are determined by imposing self-equilibrium between the prescribed external forces f (s) and the tying forces T B(s) λ between domains: R(s) (f (s) − B(s) λ) = 0. T

T

3

The formulation of the domain strain indicators is obviously linked to the considered constitutive relation. In the following we restrict to a particular damage model for the modeling of softening materials. All inertia effects are neglected in this study (i.e. quasistatic loading) and we consider small deformations and rotations. However, it should be stressed that similar predictors can also be formulated on the basis of different inelastic laws.

(4)

The tying forces λ and the amplitudes α of the rbm are determined by the solution of the interface problem by substituting Equation 3 in the compatibility condition (Eq. 1) and taking into account the orthogonality condition in Equation 4:      FI GI λ d = , (5) T α e GI 0

3.1

FI =

σ = (1 − ω)De ε, B(s) K

(s)+

B

(s)T

Ns 

+

B(s) K (s) f (s) ,

f (˜ε, κ) = ε˜ − κ,

s=1

GI =



B(1) R(1) . . . B(s) R(s)

⎤ α (1) ⎢ . ⎥ α = ⎣ .. ⎦ ⎡

α (s)

ω ∈ [0, 1].

(7)

The range of the scalar variable ω represents the transition from a virgin material (ω = 0) with intact elastic moduli De into a fully damaged one (ω = 1). Damage growth is controlled by the damage loading function

,

s=1

d=

Constitutive model and implementation in the FETI framework

Failure mechanisms are simulated in this study considering a continuous degradation of the elastic material moduli via the introduction of a damage variable as described by Lemaître and Chaboche (1994). The total stress σ and strain  tensors are related through the isotropic elasticity-based damage constitutive law

with Ns 

LINEAR/NON-LINEAR STRAIN PREDICTORS



,

⎡

⎤ T R(1) f (1) .. ⎢ ⎥ and e = ⎣ . ⎦.

(6)

R(s) f (s) T

(8)

where κ is a history dependent parameter which reflects the loading history. The equivalent strain ε˜ corresponds to an invariant of the strain tensor. In the following we adopt a specific definition of the equivalent strain introduced by Mazars and Pijaudier-Cabot (1989) in which only the tensile strains are considered relevant. The initial strain κ0 dictates the initiation of damage in a particular point. The evolution of the

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deformation history parameter κ is governed by the Kuhn-Tucker relations f ≤ 0,

κ˙ ≥ 0,

f κ˙ = 0.

(9)

An exponential damage evolution law ω(κ) is considered in this study where critical damage (ω = 1) can only be reached asymptotically at infinite strain values. Without a regularization strategy strains would localize into a narrow band of infinitesimal width while its value would approach infinity. This would cause the problem to become ill-posed. A way to circumvent this drawback is the introduction of a non-local strain quantity as argued by Bazant and Pijaudier-Cabot (1988). In this study we consider a differential version of the non-local damage model introduced by Peerlings et al. (1996) as the Gradient Enhanced Damage model. The governing equations result in a coupled system between the equilibrium equation and a diffusion equation. In the absence of body forces the governing equations read ∇ ·σ =0 ε˜ (x) = ε˜ nl (x) − c∇ 2 ε˜ nl (x),

(10)

preconditioners can improve the convergence as indicated by Rixen and Farhat (1999). 3.2

Domain strain indicators

The goal of an appropriate indicator is to predict sufficiently early (i.e. at least one step ahead of the most recent computed information) the linear or non-linear behaviour of a domain (or part of it). Moreover, the information on which the predictor is based should not be expensive. A convenient strategy seems to be the use of solution field data so it is not needed to descend to the Gauss point level. 1In the current constitutive model the non-local equivalent strain ε˜ nl , available at the solution field, and its history can be used to estimate a strain prediction for the coming steps. This procedure can be performed domain-wise (i.e for each domain (s) ) with the evaluation of the corresponding domain loading function f (s) :

p,(s) 

c,(s)  p,(s) f (s) ˜eq − κ (s) ˜eq = ˜eq , (12) with p,(s)

c,(s) c,(s) = ˜eq + δ ˜eq , c,(s) c ˜eq = max(˜eq,i ), ∈ (s) ,

c,(s) 

c,(s) (s)  ,κ . = max ˜eq κ (s) ˜eq

˜eq

ε˜ nl (x) being the non-local equivalent strain and ε˜ (x) 2 the local equivalent strain. Moreover c = l2 and l is the internal length scale of the gradient enhancement. This parameter represents the internal length scale needed to regularize the localization of deformation and is related to the width of the localization band. The variational formulation of the governing equations results in the hybrid variational form, in which the inter-domain continuity condition is accounted for. After discretization using FE and linearization, a set of equations analogous to the one shown in Equation 2 (partition in two domains) is recovered. The field of connecting forces λ now reads   λx , (11) λ= λε˜nl being λx the field of spatial connecting forces and λε˜nl the connector that glues the non-local equivalent strain dof arising from the diffusion equation. The FE stiffness resulting from the implementation of the coupled governing equations turns out to be non-symmetric and, consequently, the flexibility operator F I of the interface problem (Eq. 5) becomes non-symmetric as well. If an iterative solver is chosen for the solution of the interface problem, the Bi-Conjugate Stabilized Gradient (Bi-CGSTAB) or Generalized Minimal Residual (GMRES) can be appropriate candidates as explained by Barret et al. (1993). For the case of a highly heterogeneous interface problem (e.g. when considering heterogeneous materials or growth of damage in a structure) special

(13)

The superscripts c and p denote current and predicted values, respectively. The subscript i indicates the ith nodal point of the corresponding domain (s). The growth of f (s) is controlled by the Kuhn-Tuker loading-unloading conditions f (s) ≤ 0,

κ˙ (s) ≥ 0,

f (s) κ˙ (s) = 0.

(14)

The construction of a safe strain increment for each c,(s) is of crucial importance to build a cordomain δ ˜eq rect strain prediction. In Equation 15, two different c,(s) are described choices for the strain increment δ ˜eq considering a constant load stepping. If the external load increments are not constant the strain increment should be scaled to the future load step increment. The first option (∗) is a more optimal choice whilst the latter (∗∗) provides always a safer guess. This is because the first choice computes the maximum of all nodal increments. The second choice computes the difference between minimum and maximum strains at step t and t − 1 of different nodal points. The proposed strain increments read    c,t c,(s) c,t−1  (∗)δ ˜eq = max ˜eq,i − ˜eq,i ,

(s) (s)  c,(s) (∗∗)δ ˜eq = max δ ˜1 , δ ˜2 , 

c,t 

c,t−1  (15)  δ ˜1(s) = max ˜eq,i − min ˜eq,i  , ∈ (s) , 

c,t−1 

c,t   δ ˜2(s) = min ˜eq,i − max ˜eq,i  , ∈ (s) .

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The non-linear behaviour of domain (s) is deterp,(s) mined when the predicted strains ˜eq reach the initial (s) domain strain κ0 . In the absence of non-linearities over the whole specimen the transition to a non-linear regime can be predicted at (s) using strain increments which are based on past information. It is stressed that deformations and rotations are small and, for this reason, no geometrical non-linearities can occur. Nevertheless, when damage growth is initiated in a certain region of the specimen the strain history at a particular linear domain (s) might not be sufficient to construct a safe strain increment. In this scenario, it is expected p,(s) that the predicted strains ˜eq for step step t are found c,(s) to be smaller than the current strains at time t ˜eq . In this situation, it might be dangerous to assume a linear behaviour at domain (s) at time t (see Section 4) and a correction should be done in the previous step (i.e. the linear enhancements described in Section 4 can not be applied) in order to avoid deviating from the true solution path. Considering the previous domain strain indicators it is possible to predict the linear character of the forthcoming steps using the criteria shown in the following flowchart (Fig. 2). Three different situations are possible: linear loading (switch = 0), non-linear loading (switch = 1) and linear unloading (switch = −1). The value assigned to the switch is exploited in the next sections in order to trigger the enhancements to the FETI framework.

(s)

c (s) eq

+

<

(s) 0

Yes Switch = 0 Elastic

No Evaluate p (s) f (s) ( eq )

f

(s)

(

p (s) eq )

<0

Yes

4

Structural softening behaviour is caused by the progressive growth and coalesce of damage in the material. At ultimate loading stages, the growth of deformations typically localizes in a small area and the rest of the material tends to unload. Strain localization is a well known phenomenon in quasi-brittle materials such as concrete and rock and is related to failure mechanisms. 4.1

Assembly and factorization enhancements

In this application we focus the computational effort of the FETI framework at the non-linear areas by simplifying the operations in those regions which remain linear (i.e they remain elastic or unload linearly). In linear domains, the stiffness matrix K (s) i and internal force vector δf (s) int,i do not need to be explicitly assembled for each global iteration i. Hence, it is possible to bypass the computation of these quantities by assum(s) (s) (s) (s) (s) (s) ing K i = K l and δf int,i = K l δui , where K l is (s) the stiffness matrix of the linear domain and δui is the displacement increment in (s). The linear stiff(s) ness K (s) l is equal to the elastic stiffness K 0 when the (s) initial domain strain κ0 is not exceeded. If domain (s) experiences linear unloading the linear stiffness (s) K (s) l is equal to the secant stiffness K s computed at the beginning of the unloading stage. In the case of plasticity the initial stiffness K (s) 0 can be used during the unloading stages. After the assembly of the FE quantities, the gen+ eralized inverse K (s) and the rbm R(s) are needed in order to form the components of the interface problem (Eq. 5) and solve the local independent problems + shown in Equation 3. The computations with K (s) are performed considering its decomposition in a lower (s) (s) and upper triangular matrix K (s) i = Li U i . Hence, the lower and upper factors L(s) and U (s) together i i (s) with the rbm Ri do not need to be recomputed for each linear domain. The assembly and factorization enhancements summarized in Figure 3 are triggered by the linear/non-linear switch described in Section 3.

U nloading 4.2

No

Example: Tension test of a concrete specimen

In this section we present an illustrative example of strain localization in a concrete sample. Two concrete specimens of different sizes are loaded in tension (Fig. 4). Concrete is simulated in this study as a three phase material containing aggregates, matrix and an interfacial transition zone (ITZ) between the

Switch = 1 Loading Figure 2.

APPLICATION I: EFFICIENCY IMPROVEMENT FOR THE FINE SCALE ANALYSIS OF QUASI-BRITTLE MATERIALS

Switch criteria.

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Linear Domains (s) (s) Li = Ll (s) (s) U i = Ul (s) (s) Ri = Rl

Factorization Stage

Linear Domains (s) (s) Ki = Kl (s) (s) (s) f int, 1 = K l u i

Assembly Stage

(a)

Switch On / Off user criterion

(1) (2)

Figure 3. Assembly and factorization enhancements of a linear domain.

ux

(1) Aggregates (2) ITZ (3) Matrix

ux

1.5 × h h

(3)

(b)

Figure 5. FE meshes (a) and mixed decompositions of the concrete samples (b).

w 1.5 × w Figure 4. Table 1.

Tension test for the concrete specimens. General test data.

w

h

particle

ux

66.67 mm

30 mm

2–12 mm

0.03 mm

Table 2.

Material data.

Phase data

Aggregates

Matrix

ITZ

E (N/mm2 ) ν ˜nl κo c (mm2 ) ω(κ) α β

35.0e+3 0.2 Mazars dummy 0.75 Exp. 0.999 500

30.0e+3 0.2 Mazars 0.124e−4 0.75 Exp. 0.999 500

20.0e+3 0.2 Mazars 0.1e−4 0.75 Exp. 0.999 500

aggregates and the matrix (Fig. 5a). The ITZ is considered a weak porous region of the specimen were damage is expected to grow and, consequently, propagate through the matrix until complete failure of the material.

The objective of this example is to show that the simulation of strain localization and failure in quasi brittle materials can be seen as providing an advantage, from a computational point of view, since the main non-linearities tend to concentrate in a small area compared to the whole sample. For this reason, one can take advantage of the assembly and factorization enhancements in order to improve the efficiency of the solver. Furthermore, the level of efficiency of our scheme depends on the type of decomposition that we chose. For this example, we chose a structural decomposition in which aggregates and ITZs constitute individual domains while the matrix is partitioned according to a certain criteria (Fig. 5b). The decomposed samples are equivalent in terms of the size of the domains and its discretization using finite elements. At a certain loading stage the tensile strength is exceeded and structural softening occurs (Fig. 7 (top)). Strain localization is observed in Figure 6 for both small and large specimens. The amount of mesh that enters the factorization and assembly routines (here denoted as active mesh) is plotted in Figure 7 (bottom). The active mesh is proportional to the work load of the solver and is significantly reduced during the elastic and unloading stages. It is also observed that the amount of active mesh at ultimate loading stages is proportionally lower for the large sample (Fig. 7 (middle)). This effect is due to the fact that the non-linear area is proportionally lower for larger samples upon strain localization.

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5

Figure 6. Non-local equivalent strain contours at ultimate loading stages.

Mechanical response 0.4

Stress [N/mm^2]

0.3 Small specimen Large specimen

0.2 0.1 0

0

0.0001

0.0002 0.0003 Strain

0.0004

0.0005

Active mesh 1

APPLICATION II: MULTISCALE EXTENSION OF THE FETI METHOD

The goal of this application is to improve the resolution of the material at those areas where the main non-linearities are taking place. The resolution increment allows for the description of the material at a smaller scale including features that can not be treated at the larger scale. We assume that the heterogeneous components of the material at the lower scale play an important role during fracture processes. However, their averaged effective properties at the larger scale are sufficient for an accurate analysis during the elastic stage. The multiscale analysis is sketched in Figure 8 for an idealized L-shaped structure with a regular distribution of voids at the lower scale. Computations start with a coarse description of the structure which is partitioned into several domains. Effective properties are considered at coarse domains during the elastic regime. The linear/non-linear strain indicators introduced in Section 3 are used to predict the initiation of non-linearities at the domains. The domain resolution is changed right before entering in the non-linear regime. The history and solution field is exported to the finer resolution and computations are continued. The increment of resolution can be combined with the earlier proposed enhancements as indicated in Table 3. In the following, two main ingredients for the multiscale extension are described. The first one indicates how to connect meshes with a different resolution and the second one addresses the transition from a low resolution to a high resolution domain.

0.8

5.1

Small specimen Large specimen

Active mesh

0.6

Gluing different domain resolutions

During a multiscale analysis it is expected to encounter different domain resolutions which are based on a

0.4 0.2 0

0

200

400 Step

600

800

High resolution domains

Active mesh (Small specimen) 1 Low resolution domains with effective properties

0.8 Safe predictor Optimal predictor

Active mesh

0.6

0.4

Figure 8.

Sketch of a multiscale analysis.

0.2

Table 3. 0

0

200

400 Step

600

Resolutions for the different loading stages.

800

Low resolution

Figure 7. Top to bottom: Average stress-strain curves, active mesh curves for both specimens and active mesh curves for both predictors.

Linear elastic Loading Linear unloading

× Initial step

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High resolution × ×

successive refinement. We glue them using simple collocation with master-slave interfaces assuming that some nodes are matching (independent nodes) at the common boundaries between coarse and fine domains. Another option, which is not considered in this text, would be to use Mortar techniques. In order to restrain the dof of a non-conforming interface we apply linear multipoint constraints (LMPC) at the nodes of the fine mesh which do not have a corresponding pair at the adjacent coarse domain (dependent nodes). The collocation of the dependent nodes on to the interface is illustrated in Fig. 9. The set of homogeneous LMPC can be cast in a matrix form for the case of two domains as  

u(1) = 0, (16) Cu = C (1) C (2) (2) u u(s) , being the solution field for domain s . The matrix C (s) , which contains the multipoint constraints, sets the link between dependent and independent dof at domain s . The multipoint constraints can be implemented by adding extra equations via the use of Lagrange multipliers. These extra equations are (s) contained in the modified Boolean matrices B¯ by simply concatenating rowwise the constraint matrices C (s) and the original Boolean matrices B(s) as    (1)  C C (2) (1) (2) = . (17) B¯ B¯ (1) (2) B B The extended field of Lagrange multipliers  contains the forces arising from the multipoint constraints μ and the ones that glue the adjacent domains λ:   μ = . (18) λ The continuity condition (Eq. 1) written in terms of (s) the modified Boolean matrices B¯ reads (1) (2) B¯ u(1) + B¯ u(2) = 0

Coarse resolution

(19)

λ

Fine resolution

μ

and the system in Equation 20 can be rewritten in the equivalent form ⎤⎡ ⎡ ⎤ ⎡ ⎤ (1)T 0 B¯ K (1) u(1) f (1) T ⎥ ⎢ (2) ⎦ ⎣ u(2) ⎦ = ⎣ f (2) ⎦. (20) ⎣ 0 K (2) B¯ (1) (2)  0 0 B¯ B¯ The domain stiffness matrices K (s) remain unal(s) tered and the modified Boolean matrices B¯ loose their original Boolean character because of the addition of the new LMPC. For this reason, if an iterative solver is used for the solution of the interface problem, an appropriate preconditioner needs to be set up (s) using the modified Boolean matrices B¯ as indicated by Rixen (2002). The incorporation of extra rows in (s) the B¯ matrices representing the LMPC is no longer needed when the resolutions are equivalent at both sides of the interface. For this reason the constraints are released when the resolution at both sides of an interface match. During a multiscale analysis it is expected that the number of high resolution domains will vary. Hence, the interface problem needs to be re-computed every time a domain resolution is changed. 5.2

The zoom-in strategy is employed for the transition of a low resolution domain into a high resolution domain. It consists of (i) the solution of a boundary value problem (BVP) in order to update the solution field of the fine domain to the deformed configuration of the coarse domain (the shape functions of the coarse domain are used to interpolate the displacements to all boundary nodes of the fine domain) and (ii) a global iteration in which the internal forces of the newly inserted domain are in equilibrium with the external forces acting on the boundary of the fine scale domain. At the end of the zoom-in event, both displacement compatibility and equilibrium are reached at the whole structure. The zoom-in strategy is summarized in Figure 10. During the global relaxation (Fig. 10b) it is possible to monitor the energy imbalance E12 after inserting a refined (s) domain as |E1 − E2 |, with Ei = f (s)T int,i ui being the energy at stage i of the zoom-in process. Low values of E12 indicate that the deformation energies of the coarse and fine domains do not differ significantly. 5.3

Domain Connector LMPC

Figure 9.

Zoom-in strategy

Example: L-shape specimen

The following example demonstrates the performance of the multiscale FETI for an heterogeneous material. An L-shaped structure is loaded according to the boundary conditions sketched in Figure 11. The decomposed coarse discretization is shown in Figure 8. When entering the non-linear regime, the coarse

Independent d.o.f. Dependent d.o.f.

LMPC at non-conforming adjacent domains.

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Coarse resolution

Table 4.

Fine resolution

Test data.

Test data E (N/mm2 ) ν ˜nl κo c (mm2 ) ω(κ) α β ux

BVP (update process)

+

=

BVP

(0)

(1)

35.0e+3 0.2 Mazars 5.0e−4 0.2 Exp. 0.999 50 0.8

(2)

r (s)

(s) r (s) = fext

(s) fint

r (s) = 0

r (s) = 0 (s)T (s) E 1 = fint, 1u 1

(s)T (s) E 2 = fint, 2u 2

(b) Global relaxation

Figure 10.

Zoom-in technique.

100 mm

50 mm

ux

100 mm Figure 11.

50 mm

Boundary conditions for the L-shaped specimen.

domains are refined taking into account the underlying heterogeneous structure which is represented here as a regular distribution of voids. The computations during the linear regime are performed using the coarse description of the material. The cell containing one void is assumed to be representative of the regular void distribution at the fine scale. For this reason, its

Figure 12. Damage contours at an initial (top), medium (middle) and ultimate (bottom) loading stages.

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displacement curves are plotted for different choices of the effective Young’s modulus (Fig. 13 (middle)). It is observed that when the effective properties are not accurate enough the load displacement curve drifts away from the true fine scale analysis in which all domains are refined. Moreover a higher jump is observed during the zoom-in episodes due to the fact that the deformation energies of the coarse and fine domains do not fully coincide. The difference between both energies is shown in Figure 13 (bottom) for the different choices of the Young’s modulus. There is a good agreement between |E1 − E2 | and the displacement jumps observed during the zoom-in episodes. By monitoring the energy change in the system one can assess the choice of the effective properties used during the elastic regime. If these properties are not computed accurately the solution will differ from the fine scale computations.

Active mesh 1

Active mesh

0.8

Safe predictor

0.6

0.4 0.2 0

0

100

50

Step

100

150

E=27.85 MPa E=32 MPa E=40 MPa Fine scale analysis

6

0

0.2

0.4

0.8

0.6

1

Strain Energy imbalance after "Zoom-in"

|E1-E2| [Nmm]

1.2 1

E=27 MPa E=32 MPa E=40 MPa

0.8 0.6 0.4 0.2 0 0

10

20

30

40

50

CONCLUSIONS

A FETI framework is proposed for the efficient simulation of failure in quasi-brittle materials. Two main applications are explored. In the first application the efficiency of the solver is improved by simplifying the computations in the linear domains. This strategy proves to significantly reduce the size of the problem to be solved specially when strain localization occurs during the loading process. The second application constitutes a multiscale extension for the FETI framework in which the non-linear areas are simulated with a higher degree of accuracy. The multiscale simulations show a good agreement with the fine scale solution when the correct effective properties are assigned to the coarse discretization. The multiscale strategy presents a significant reduction of the complete fine scale problem and is compatible with the simplification of operations proposed in the first application of this framework.

50 0

Stress [N/mm^2]

200

Mechanical response (Different effective Youngs modulus)

60

Step Figure 13. Top to bottom: active mesh, load deflection curves and energy imbalance during zoom-in events.

effective elastic properties are homogenized at each macroscopic element. In this example the safe predictor is used in order to trigger the zoom-in events early enough since they are now predicted using a coarser description of the underlying material. For this reason we observe in Figure 12 that more domains than necessary are computed in a high resolution. The amount of active mesh is now relative to the highest number of elements (i.e. when using the high resolution for all domains) (Fig. 13 (top)). The load

ACKNOWLEDGMENTS Funding for this work was provided by the Flemish Institute for Science and Technology (SBO project IWT 03175).

REFERENCES Barrett, R., M. Berry, T.F. Chan, J. Demmel, J.M. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H.V.D. Vorst (1993). Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods. SIAM Press, Philadelphia, PA.

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Bazant, Z. and G. Pijaudier-Cabot (1988). Nonlocal continuum damage, localization, instability and convergence. Journal of Applied Mechanics 55, 287–293. Farhat, C. and F. Roux (1991). A method of finite element tearing and interconnecting and its parallel solution algorithm. International Journal for Numerical Methods in Engineering 32(6), 1205–1227. Lemaître, J. and J. Chaboche (1994). Mechanics of Solid Materials. Cambridge University Press. Mazars, J. and G. Pijaudier-Cabot (1989). Continuum damage theory—application to concrete, Journal of Engineering Mechanics 115(2), 345–365. Peerlings, R., R. de Borst, W.A.M. Mrekelmans, and J. de Vree (1996). Gradient enhanced damage for quasibrittle materials. International Journal for Numerical Methods in Engineering 39(19), 3391–3403.

Rixen, D.J (2001). Parallel processing. In Encyclopedia of Vibration, pp. 990–1001. Oxford: Elsevier. Rixen, D.J (2002). Extended preconditioners for the FETI method applied to constrained problems. International Journal for Numerical Methods in Engineering 54(1), 1–26. Rixen, D.J. and C. Farhat (1999). A simple and efficient extension of a class of substrcture based preconditioners to heterogeneous structural mechanics problems. International Journal for Numerical Methods in Engineering 44(4), 489–516.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Modelling cohesive crack growth applying XFEM with crack geometry parameters J.F. Mougaard, P.N. Poulsen & L.O. Nielsen Department of Civil Engineering, Technical University of Denmark, Lyngby, Denmark

ABSTRACT: When modelling discrete cracks using standard eXtended Finite Element Method (XFEM) crack geometry parameters are not included in the degrees of freedom, which may lead to cumbersome iterations and ill-conditioned equations. A formulation introducing crack parameters as variables in the XFEM equations is presented. The crack parameters are introduced in a general manner, but in the present work exemplified in 2D plane stress for cohesive crack growth. The crack parameters are introduced through the principal of virtual work, and the crack-tip conditions for crack growth are established and included in the FE-equations in order to determine the crack parameters. As an example a partly cracked Constant Strain Triangle (CST) element is analyzed, where the extended FE-equations including crack parameters are set up. 1

INTRODUCTION

The eXtended Finite Element Method (XFEM) has shown to be an efficient tool when modelling cohesive crack growth (Belytschko and Black 2003). Lately an effort has been put into formulations of partly cracked elements. These works show good results and allow us to model cohesive crack growth in rather coarse standard finite element meshes see e.g. (Mougaard et al. 2009b). At the stage where a crack is propagated to the point where the crack mouth becomes traction free and is about to start steady state crack growth, convergence problems are often seen. It seems that the crack length is an important parameter for the problem. The crack parameters have in most cases not been included directly in the governing equations. This means that the tangent stiffness is not consistent and therefore not a correct predictor for an incremental solution. Traditionally the condition that the crack tip stress equals the tensile strength has been achieved using some special iteration procedure within nonlinear finite element solutions. When using a tangential stiffness which is not consistent, these iterations may result in convergence problems. Introducing the crack length as a direct parameter in the equations makes the tangent stiffness consistent, whereby the crack tip criterion can be easier fulfilled within the iterations of the traditional nonlinear solution strategy. Introducing this more numerically stable solution strategy for modelling cohesive crack growth is believed to stabilize solutions with simultaneous growth of several cracks. Other methods like smeared cracks do not have a well defined crack parameters i.e. the stiffness of

the system changes rapidly each time a new element cracks without any prediction. These methods do therefore not offer any possibility of introducing the important crack length parameter directly in the FEM equations. With the above mentioned partly cracked elements for XFEM a direct crack length parameter exist for the element. The purpose of the present work is to include the crack geometry parameters in the variational formulation in order to achieve a more robust and efficient solution strategy. In the present work extended FE-equations are set up for a partly cracked CST element. A general survey is given by (Meschke and Dumstorff 2007), where crack parameters have been included in the governing equations by a potential energy formulation. Contrarily the present work use a virtual work formulation because of the material nonlinearities, and analyzes the possibility of using the crack initiation criterion by (Hillerborg et al. 1976). 2

FEM WITH CRACK GROWTH VARIABLES

We consider a structure discretised in finie elements see Figure 1. The structure contains a crack with a geometry described by a set of parameters a1 , a2 . . . = aT , e.g. the crack length l and the crack orientation described by the angle α. In order to solve the FE-equations efficiently we want to include these crack parameters as variables in the equations. In order to do so a must be included in the description of the displacement field, from where the effect of a can be determined. The formulation of the displacement field is based on the principles of XFEM, where displacements can be decomposed in a continuous and a

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2.1

Displacement increments

An incremental formulation is used, i.e. from a given state (u, , σ , f , a) of the finite element meshed structure, the incremented state (u + du,  + d, σ + dσ , f + df , a + da) is wanted. Here  is the generalized strains with the corresponding generalized stresses σ and f is the external load both containing boundary and domain load. The change da of the crack geometry parameters can only influence the element containing the crack-tip, when we assume that the crack path is fixed in the elements which are fully cracked. The virtual displacements Equation 5 now gives with increments on the crack parameters a δu(x, y) = Nc (x, y)δvc + Nd (x, y, a + da)δvd

(6)

= Nc (x, y)δvc + Nd (x, y, a)δvd  +

∂Nd (x, y, a) da1 ∂a1

+

∂Nd (x, y, a) da2 + · · · ∂a2

 δvd

(7)

On matrix form Figure 1. The given and the incremented state of a FE-discretised structure.

δu(x, y) = Nc (x, y)δvc + Nd (x, y, a)δvd  d  ∂N (x, y, a) ∂Nd (x, y, a) + ··· ∂a1 ∂a2 ⎡ ⎤ da1 ⎢da2 ⎥ d × ⎣ ⎦ δv .. .

discontinuous part: u(x, y) = Nc (x, y)vc + Nd (x, y)vd

(1)

where u is the interpolated displacement vector, Nc is the continuous interpolation matrix, with the corresponding continuous Degree Of Freedom (DOF) vector vc and Nd is the discontinuous interpolation matrix, with the corresponding discontinuous DOF vector vd . For the uncracked elements the displacement formulation is only of continuous type i.e. standard interpolation, and if we choose the virtual displacements of the same shape, we get for the uncracked elements u(x, y) = Nc (x, y)vc

(2)

δu(x, y) = Nc (x, y)δvc

(3)

(8)

= Nc (x, y)δvc + Nd (x, y, a)δvd +

∂Nd (x, y, a) daδvd ∂a

(9)

(x,y,a) . In the followwhere last equal sign defines ∂N ∂a ing we use a compact notation, and introduce therefore d

d(δud ) =

∂Nd (x, y, a) daδvd ∂a

(10)

This simplifies Equation 9 to For the cracked elements there is discontinuous contributions. Only the discontinuous displacement contribution can depend on the crack parameters a. u(x, y) = Nc (x, y)vc + Nd (x, y, a)vd

(4)

δu(x, y) = Nc (x, y)δvc + Nd (x, y, a)δvd

(5)

δu(x, y) = Nc (x, y)δvc + Nd (x, y, a)δvd + d(δud ) (11) In the following the transposed of Equation 10 is needed. Equation 12 defines this. All operations are not shown, but the reader will find the same using matrix algebra on Equation 8.

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 (d(δud ))T = δvdT

∂NdT (x, y, a) ∂NdT (x, y, a) ··· ∂a1 ∂a2



⎤ da1 ∂NdT (x, y, a) ⎥ ⎢ × ⎣ da2 ⎦ = δvdT da .. ∂a . (12) ⎡

crT ∂BdT (x,y,a) and ∂B ∂a(x,y,a) are defined ∂a d (x,y,a) matrices as ∂N ∂a in Equation 9.

where

∂N (x, y, a) . ∂a



=

An incremental virtual work is made based on Figure 1. In an arbitrary given state (u, , σ , f , a), where the elements are grouped in two parts, one containing only the crack-tip element (tip) and one containing all other elements (nontip), the virtual work is written

=

allel

=

tipel



δu f +

δuT f

(13)

tipel

In the incremented state the same virtual DOFs are used as in the given state. This gives the incremented virtual work

δuT df +

δuT df + d(δudT )f (17) tipel

el

tipel



δu df +

d(δudT )f

T

el



dδ dT σ = δvdT tipel

tipel−cr

δ T (σ + dσ )

+

+

cr

(δ T + d(δ dT ))(σ + dσ ) tipel

δu (f + df ) T



nontip el

dδudT f = δvdT

tipel

+

(δuT + d(δudT )(f + df )

tipel−cr

≡ δvdT ra da

where d(δ dT ) are the strains from d(δudT ). In the crack-tip element this is written out using Equation 10. In the continuum part of the crack-tip element the generalized strains are ∂BdT (x, y, a) da ∂a

(19)  ∂NdT (x, y, a) f da ∂a

(14)

tipel

d(δ dT ) = δvdT

∂BdT (x, y, a) σ ∂a

 ∂BcrT (x, y, a) σ da ∂a

≡ δvdT qa da



(18)

tipel

In the following only the crack-tip contributions from Equation 18 will be studied in detail. Applying the Equations 15, 16 and 10 the crack-tip contributions from Equation 18 can be rewritten

nontip el

=



allel

T

nontip el



δ T dσ + dδ dT σ tipel

which is rearranged using the new group allel which include all elements both uncracked and cracked.

δ T dσ + dδ dT σ

δ T σ

nontip el

δ dσ +

nontip el

δ T σ +

in similar

T

nontip el

2.2 Virtual work



(16)

Subtracting Equation 14 from 13, neglecting 2. order terms as dδ T dσ , gives

dT

where last equal sign defines

∂BcrT (x, y, a) da ∂a

d(δ dT ) = d(δ crT ) = δvdT

(15)

In the crack it self the generalized strains (opening,sliding) are

(20)

where the last identity in Equation 19 and 20 defines qa (internal element nodal force matrix from crack growth) respectively ra (element load matrix from crack growth). 2.3

FEM equations

The virtual work gives equations corresponding to the virtual continuity DOFs δvc and the virtual discontinuity DOFs δvd on the system level.

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Considering the crack-tip contributions from Equation 19 and 20 the standard FEM equations are extended to 

KTcc KTdc

KTcd KTdd

0 Qa − R a



⎡

⎤   dVc c ⎣ dVd ⎦ = dRd dR da

(21)

where KT is the system tangent stiffness matrix which is partitioned after continuous and the discontinuous DOFs (capital letters are used to indicate system level). In Equation 21 there are more unknowns than equations, i.e. they cannot be solved. The missing equations comes from the conditions connected to the state immediately in front of the growing crack-tip. Considering plane stress, two crack parameters must define the crack growth (e.g. crack length and crack direction). For the growing crack the maximum stress σmax (a + da) just in front of the incremented crack-tip must be equal to the tensile strength ft of the applied material. Moreover the direction of this stress must be perpendicular to the crack direction with normal n(a + da) at the growing crack-tip, i.e. with an ordering of the principal stresses in the plane σI > σII and the corresponding principal directions defined by unit vectors nI , nII σ I (a + da) = ft

(22)

nI (a + da) = n(a + da)

(23)

If the crack path is assumed fixed, the crack parameters only influence the crack-tip element. I.e. the crack parameter variables can be eliminated locally within the crack-tip element leaving the equations on system level unchanged. Crack growth stops when the computed crack length increment becomes non-positive, and the equations reduces to the original form. For closing/reopening of an existing crack no special precautions are needed at the crack-tip, the crack parameter extension of the equations is excluded/included for closing/reopening of the crack. The constitutive modelling (typical a plasticity/damage modelling) is used for an existing crack, see e.g.(Nielsen et al. 2009) and (Mougaard et al. 2009a).

3

can vary solutions can be found. In the present example a CST element has been analyzed, here stresses are constant within regions of the element, and in order to have varying stresses a weighting between parts in the element has been used. The CST element allow us to write a closed form solution showing how the crack parameters influence the original equations. The FEM-equations with crack growth are set up for a partially cracked (CST) element 123 as shown in Figure 2 for plane stress. The element and the load are symmetric about the x-axis. The element is supported in the x-direction along 23 and supported in 1, i.e. only displacements in the y-direction is possible. The material is homogenous. Outside the crack the material is linear elastic with the material stiffness DT ⎡

E DT = ⎣ 0 0

0 E 0

⎤ 0 0 ⎦ G

(24)

where E is Young’s modulus, with Poisson ratio’s ν = 0. In the crack a linear traction separation law is applied, with the crack opening stiffness Dcr T  Dcr T =

Dn 0

0 Ds

 (25)

where Dn and Ds represent opening and sliding stiffness respectively. 3.1

Displacements

The continuous displacements uyc are expressed using the area coordinates ζ1 , ζ2 , ζ3 . The displacements are

EXAMPLE

In the following section the FEM-equations extended with crack parameter variables are set up for a partly cracked triangular element. It is essential to note that stresses need to vary within the crack-tip element in order to find solutions satisfying the conditions in Equation 23. For higher order elements where stresses

Figure 2. Geometry, load and support conditions for the partly cracked CST element.

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applied symmetrically along the x-axis, this means that v3c = −v2c and v3d = −v2d , as shown in Figure 2. uyc (ζi ) = ζ2 v2c + ζ3 v3c = (ζ2 − ζ3 )v2c

(26)

The discontinuous displacements uyd are defined in the sub triangle s1 s2 s3 containing the crack OT with the crack-tip T in s1. uyd are expressed by the sub triangle area coordinates ζ1 , ζ2 , ζ3 

uyd (ζi )

=

H2 (ζi )ζ2 v2d

+

H3 (ζi )ζ3 v3d

−ζ3 v2d + side = ζ2 v2d − side (27)

where Hi is the 2D Heaviside step function defined for each enrichment node in the triangle. These Heaviside step functions are defined as zero on the same side of the discontinuity as the enrichment nodes and 1 on the opposite side of the discontinuity. For further details see e.g. (Mougaard, Poulsen, and Nielsen 2009b). The crack opening un is un = uyd+ − uyd− = (ζ3 v2d − ζ2 v2d )

for ζ2 = ζ3

= −2ζ3 v2d

(28)

To make the crack parameters (here only the crack length l) visible in the displacement expression, the sub triangle coordinates are transformed to the element triangle area coordinates. ⎡

⎤

⎡

l h

ζ1 ⎣ ζ2 ⎦ = ⎢ ⎣ 12 − ζ3 1 2 −

0 l 2h l 2h

⎤⎡

0

1

⎥⎢ 0⎦ ⎣

0

1

ζ1 ζ2 ζ3

⎤ ⎥ ⎦

(29)

i.e. ζ1

Writing Equation 26 on the matrix form [uyc (ζi )]

= [ζ2 − ζ3 ][v2c ]

(33)

Defines uc = [uyc (ζi )]

Nc = [ζ2 − ζ3 ]

vc = [v2c ]

(34)

Writing Equation 35 on the matrix form      d − ζ + 1ζ 1 − h [v ] for y > 0 [uyd (ζi )] =   3 1 2  1 h l  d 2 for y < 0 [v2 ] ζ2 + 2 ζ1 1 − l (35) defines outside the crack ud = [uyd ]      d − ζ3 + 12 ζ1 1 − hl [v2 ] for y > 0 d      N = for y < 0 [v2d ] ζ2 + 12 ζ1 1 − hl vd = [v2d ] 3.2

(36)

Strains

Outside the crack the strains are defined as usual. Here x = 0, so the strain vector reduces to     uy,y y = (37) = γxy uy,x From the continuous displacements in Equation 26 are obtained

h = ζ1 l 



c yc = uy,y =





=

h 1 ζ2 = ζ2 + ζ1 1 − 2 l ζ3

1 In the crack ζ2 = ζ3 = 1−ζ 2 . Then Equation 28 gives      h d h 1 d v2 = − 1 − ζ1 v2 un = −2 ζ3 + ζ1 1 − 2 l l (32)

h 1 = ζ3 + ζ1 1 − 2 l

∂ (ζ2 − ζ3 )v2c ∂y 

(30)

With Equation 30 and 27 the discontinuous displacements can be expressed in area coordinates from triangle 123     − ζ + 1 ζ 1 − hl v2d for y > 0  uyd (ζi ) =  3 1 2 1 ζ2 + 2 ζ1 1 − hl v2d for y < 0 (31)

= c = γxyc = uy,x

−x3 + x1 −x1 + x2 − 2A 2A

v2c

v2c 2h c v = 2 b 2 12 h2b

(38)

∂ (ζ2 − ζ3 )v2c ∂x 

=



y 1 − y2 y3 − y1 − 2A 2A

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 v2c = 0

(39)

Writing Equation 38 and 39 on the form 

yc γxyc





 =



1 b

=

 d

v2c

0

determines  c  y c = γxyc

(40)

 Bc =



1 b

(41)

0

From the discontinuous displacements in Equation 35 is obtained in the upper half sub triangle d = yd = uy,y

∂ ∂y



    h 1 − ζ3 + ζ1 1 − v2d 2 l

d = γxyd = uy,x

h v2d 2 12 h2b

=

v2d 2b

(42)

    ∂ 1 h − ζ3 + ζ1 1 − v2d ∂x 2 l 

 = −

1 h y1 − y2 − 1− 2A 2 l



vd hb v2d = 2 = 1 2l l2 2 h2b

y 2 − y3 2A



 cr = [un ]

3.3

(47) for

y<0

(48)

The crack parameter l is found in Equations 36, 47 and 49. The non-zero crack growth derivatives occur in Equations 9, 15 and 16, and they give   ∂NdT (x, y, l) 1 h = − ζ1 2 ∂l 2 l

 v2d

⎧  ⎪ 0 − 2l12 ⎨ ∂B (x, y, l) =   ⎪ ∂l ⎩ 0 12 2l dT

(43)

(44)

(49)

Crack growth derivatives

3.4

(50)

for y > 0 (51) for y < 0

(52)

Crack growth matrices

The element load matrix from crack growth is defined in Equation 20, inserting Equation 50 gives

i.e. v2d 2b

γxyd = −

v2d 2l

(45)

ra = tipel−cr

Writing Equation 42 and 43 on the form

yd γxyd

y>0



   h Bcr = − 1 − ζ1 l

yd (−y) = yd (y)



for

   h [un ] = − 1 − ζ1 [v2d ] l

  ∂BcrT (x, y, l) h = −ζ1 2 ∂l l

yd =

1 2b −1 2l



In the crack the strains are given by Equation 16, in this case only un is non-zero. Writing the equation on the form

Symmetry about the x-axis gives in the lower half sub triangle γxyd (−y) = γxyd (y)

1 2b 1 2l

defines

    −x1 + x2 1 h −x2 + x3 d = − − 1− v2 2A 2 l 2A =

yd γxyd

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ d B =  ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

⎧ ⎪ ⎪ ⎪ ⎪  ⎪ ⎨ =  ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

1 2b 1 2l 1 2b −1 2l

 v2d

=

for y > 0



(46) v2d

∂NdT (x, y, a) f ∂a

for y < 0

1 h − ζ1 2 fy = 2 l subtri

=−

1 1 − ζ1 fy 2 l subtri

bfy ≡ ra 6

(53)

The internal element nodal force matrix from crack growth is defined in Equation 19, and gives inserting

which determines

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the results from Equation 51 and 52, when symmetry along the x-axis is utilized

qa = tipel−cr

+ cr

0

uhs

−ζ1

+ cr

− 2l12





BcT DT Bc triangle



∂BdT (x, y, a) σ ∂a

= triangle

(54)

k cd = k dc =

σy τxy



triangle

(55)

b

0

 0

E 0

=

Eh b

(60)

0 G



1 2b ± 2l1

 =

El 2b

BdT DT Bd +

subtriangle

subtriangle

  E c bh bl 1 + σyd = v2 h + v2d l = ft (57) = σyc A A A 2

The incremental form of Equation 57 is 1 1 hdv2c + ldv2d + v2d dl = 0 2 2

(58)

FEM equations

Extending the FEM equations 21 with 58 gives ⎤⎡ c ⎤ ⎡ c ⎤ 0 dv2 dr qa − r a ⎦ ⎣ dv2d ⎦ = ⎣ dr d ⎦ 1 d dl 0 2 v2

(59)

where the matrix elements k cc , k cd , k dc , k dd are obtained as follows

+

cr



=

1 h−l 1 bl bh bl = σyc 2b + σyc+d 2 = σyc + σyd 2 A 2 A A A (56)

k cd k dd 1 2l

1 b

k dd =

where the stress superscript indicates the contributing field. Using the constitutive condition σy = E y , where E is Young’s modulus outside the crack. Equation 56 gives together with Equation 38,42 and 23

k cc ⎣ k dc h

 1 

0 G

(61)

h σy ≡ q a l2

The nominal stresses in the area immediately in front of the crack-tip is determined with an area weighting of the triangle and sub triangle stresses at T. Because of symmetry the nominal shear stress τxy vanish, while

⎡



=

3.6

E 0

BcT DT Bd

3.5 Crack tip conditions

σynom

0

triangle

where uhs means the upper half sub triangle.

σynom



1 b

∂BcrT (x, y, a) σ ∂a 

=2

k cc =

1 2b

±

1 2l



E 0

cr BcrT Dcr T B

0 G



1 2b ± 2l1



   

  h h [Dn ] − 1 − ζ1 − 1 − ζ1 l l cr

  l b 1 1 E +G + Dn l = 4 b l 3

(62)

Equation 59 represent the extended system of equilibrium including the crack length as a variable. 4

CONCLUSION

In the present work crack geometry parameters has been included as direct variables in a XFEM formulation. By including these crack geometry parameters directly in the formulation and introducing the incremental relations the propagation of the crack can be handled more directly and with fewer iterations. The crack parameters are introduced through the principle of virtual work, and appear as direct variables in the equations. A set of conditions concerning crack growth has been set up in order to determine the crack parameters. An example with a partly cracked CST element has been considered. In this example a weighting of the stresses has been applied. The example shows the setup the crack growth matrices. In the present work the scope has been to clarify the principles of introducing crack parameters as variables, and the example with the partly cracked CST element allowed for calculations by hand. A natural extension of this work is to implement the LST element, where the performance can be tested.

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REFERENCES Belytschko, T. and T. Black (2003). Elastic crack growth in finite elements with minimal remeshing. International Journal for Numerical Methods in Engineering (45), 601–620. Hillerborg, A., M. Moder, and P.-E. Peterson (1976). Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cem. Concr. Res (6), 773–782. Meschke, G. and P. Dumstorff (2007). Energy-based modeling of cohesive and cohesionless cracks via x-fem. Computer Methods in Applied Mechanics and Engineering 196(21–24), 2338–2357.

Mougaard, J.F., P.N. Poulsen and L.O. Nielsen (2009a). Modelling concrete structures applying xfem with a mixed mode constitutive model. Submitted to FraMCoS7. Mougaard, J.F., P.N. Poulsen and L.O. Nielsen (2009b). A partly and fully cracked xfem element based on higher order polonomial shape functions for modeling cohesive fracture. Submitted to: International Journal for Numerical Methods in Engineering. Nielsen, L.O., J.F. Mougaard, J.S. Jacobsen and P.N. Poulsen (2009). A mixed mode model for fracture in concrete. Submitted to FraMCoS7.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Strong discontinuities, mixed finite element formulations and localized strain injection, in fracture modeling of quasi-brittle materials J. Oliver & Ivo F. Dias Technical University of Catalonia, Campus Nord UPC, Edifici C-1, Barcelona, Spain

A.E. Huespe INTEC-UNL-CONICET, Santa Fe, Argentina

ABSTRACT: The presented work explores a combination of strain localization methods and strong discontinuity approaches to remove the flaws (stress-locking and mesh bias dependence) of the classical strain localization methods, and reduce the sophistication of the strong-discontinuity based methods. The concept of strong discontinuity is initially substituted by that of weak discontinuity capturing the displacement jump through its distribution (smearing) in a finite length: the oriented with of a finite element through which the fracture passes in a rather diffuse manner. In other words, the classical strain localization concept is brought to the approach, no strong discontinuity enrichment is made and standard constitutive models are used in a classical strain localization setting. The standard mesh refinement recovers the strong discontinuity concept by taking the element width to zero. However, in order to remove the spurious mesh orientation dependence, in a second stage specific localization modes are injected, via mixed finite elements formulations, to the path of elements that are going to capture the cracks. These modes, as well as their injection time, are characterized from the information provided by the SDA. Then, the obtained approach enjoys the benefits of the strong discontinuities one, at a complexity similar to the classical, and simpler, localization methods. 1

MOTIVATION

A review of the available approaches used for numerical modeling of material failure in quasi-brittle materials, allows roughly classifying them as: a. Strain localization methods: They use standard finite elements and classical continuum stressstrain constitutive models equipped with strain softening (Borst, Sluys et al. 1993). It is well known that they suffer from classical spurious mesh size and mesh-bias dependences. Mesh size dependence can be easily overcome by using appropriate regularizations of the softening modulus values and resorting to the fracture energy concept. However, the spurious mesh-bias dependence cannot be easily removed. Only non-trivial, and sometimes unphysical, modifications of the constitutive model by introducing an internal characteristic length (viscous, non-local or gradient regularizations) and requiring very fine meshes to capture fracture line/surfaces have proved to remove it. However, their use for engineering purposes, in modeling fracture of real 3D concrete and reinforced concrete structures, seems limited due to their inherent high computational cost (Oliver and Huespe 2004). b. A subsequent sophistication, stemming from the 90’s of the past century, brought the so-called

strong discontinuity approach (SDA) (Simo, Oliver et al. 1993). There, a real discontinuity in the displacement field (strong discontinuity) was aimed at being modeled and captured via an appropriate mechanical theory (the strong discontinuity theory) and different families of standard finite elements enriched by discontinuous deformation modes, namely: finite elements with embeddeddiscontinuities (elemental-based enrichment) or X-fem methods (Belytschko, Moes et al. 2001) (nodal-based enrichment) (Oliver, Huespe et al. 2006). At the cost of resorting to some additional sophisticate tools, like multiple crack tracking algorithms, specific integration algorithms etc., the method proves to be efficient in providing results free from the mesh-size and orientation dependences, and showed able to account for the propagation of multiple cracks in two and even three-dimensional settings at a relative low computational cost (Oliver and Huespe 2004). However, the sophistication of the required tools seems to place a limitation on their incorporation to commercial simulation codes and to their use for real-life industrial purposes. The presented work explores a combination of both approaches to remove the flaws of the classical strain localization methods and reduce the sophistication of

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the strong-discontinuity based methods. The concept of strong discontinuity is initially substituted by that of weak discontinuity capturing the displacement jump through its distribution (smearing) in a finite length: the oriented with of a finite element through which the fracture passes in a rather diffuse manner. In other words, the classical strain localization concept is brought to the approach, no strong discontinuity enrichment is made and standard constitutive models are used in a classical strain localization setting. The standard mesh refinement recovers the strong discontinuity concept by taking the element width to zero. However, in order to remove the spurious mesh orientation dependence, in a second stage specific localization modes are injected, via mixed finite elements formulations, to the path of elements that are going to capture the cracks. These modes, as well as their injection time, are characterized from the information provided by the SDA. Then, the obtained approach enjoys the benefits of the strong discontinuities one, at a complexity similar to the classical, and simpler, localization methods. 2

Let us consider a strong discontinuity in the displacement field, u(x, t), occurring at a discontinuity line, S, of unit normal n, in a body B (see Figure 1): (1)

where u¯ stands for the regular (continuous) displacement field, [[u]] is the displacement jump and HS is the step (Heaviside) function on S. The regularization of the corresponding infinitesimal strain, ε, (weak discontinuity kinematics) reads: ε ≡ ∇ s u = ε¯ +

μS ([[u]] ⊗ n)S h

(2)

where ε¯ is the regular strain, h is the (very small, thus tending to zero) width of the regularization band,

Figure 1.

 ε¯ (x, t) ε(x, t) ≈ 1 S h ([[u]] ⊗ n) (x, t)

x ∈ B/Bh x ∈ Bh

(3)

On the other hand, strain localization can be understood as a concentration and intensification of the strain in a geometrical band, Bloc ⊂ B, of width , containing the path of a strong discontinuity. The corresponding strain field reads:  ε¯ (x, t) ε(x, t) ≡ εloc (x, t)

x ∈ B/Bloc x ∈ Bloc

(4)

Comparison of equations (3) and (4), and the fact that they aim at capturing the same physical phenomenon (a propagating discontinuity) suggest that a physically meaningful localized strain field should take the format:  ε¯ (x, t) ε(x, t) ≡ εloc (x, t) = (β ⊗ n)S

STRAIN LOCALIZATION AND WEAK DISCONTINUITIES

u = u¯ + HS [[u]]

Bh , containing S, and μS stands for the h-regularized Dirac’s delta function whose support is Bh . In this sense, the strain field can be approximated as:

x ∈ B/Bloc x ∈ Bloc

(5)

β = (1/) [[u]] (x, t) where β is a field proportional to the jump, obtained by comparison of equations (3) and (4), in terms of the inverse of the localization bandwidth . Equation (5) is the basis of the strain injection concept in the proposed finite element approach to strain localization and the term εloc (x, t) = (β ⊗ n)S will be termed, from now on, the strain localization mode. 2.1

Mesh size and mesh bias dependency and the strain localization mode

On the light of equation (5) one could think that the ability of a given finite element to appropriately capture strain localization depends on the measure that the localization mode can be reproduced and propagated trough the finite element mesh. Regarding this ability there are two issues to be considered here: a. The dependence of the strain on the localization bandwidth . In fact, this gives raise to the classical mesh-size dependence issue. Different kinds of remedies provide a solution for this problem: e.g. by resorting to concepts of the Strong Discontinuity Approach (SDA) the regularization of the softening modulus, H = Hint , in terms of the intrinsic softening modulus Hint which is considered a material

Strong discontinuity in a body

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property, computed, (for isotropic continuum damage models) as: Hint =

σy2

(6)

2EGf 

in terms of the uniaxial peak stress, σy , the Young modulus E and the Fracture Energy Gf , overcomes such a dependence also for strain localization approaches. b. The dependence of the resulting strain localization on the orientation of the finite element mesh (mesh-bias dependence): i.e. the obtained localization path tends to propagate spuriously parallel to the sides of the elements, thus dramatically changing with the finite element mesh orientation in structured meshes. In the context of strain localization methods, using standard finite elements, this flaw can only be overcome by changing the assumed physical properties of the material: i.e.: the constitutive model. Introduction of rate dependence (viscous models), non-locality or gradient dependency, partially solves the problem at the cost of using tinny elements at the localization zone, thus substantially affecting the computational cost. In this work, we consider the injection, in the finite element mesh, of specific strain localization modes as a possible remedy for the spurious mesh-bias dependency.

3

solution, n = 0 and, in general, two solutions (n ≡ n, n ≡ m) for the normal to the weak discontinuity. Let us also consider the constitutive model equipped with strain softening, and the strain-like internal variable, α (where, as usual, α˙ = 0 for elastic/unloading cases and α˙ > 0 for loading cases) and the corresponding stress-like internal variable, q(α) defining the hardening/softening law (see Figure 2). In terms of this, the injection zone is determined as the geometrical locus of the in-loading bifurcated points (see Figure 3), i.e.: ˙ t) > 0} Binj (t) := {x ∈ B | t ≥ tB (x); α(x, 3.2

(8)

Injection of an elemental constant strain mode

As soon as a given finite element (whose bifurcation time is characterized, for numerical implementation purposes, by that of its central point) belongs to the injection domain Binj , it is equipped with a strain field whose ability to capture and propagate localization modes is superior to that of the standard underlying element. Thinking of two-dimensional problems, we consider here the four-noded bilinear quadrilateral as the underlying element. Then, a first injection of an elemental constant strain mode is performed via a mixed u − ε formulation restricted to Binj . The

THE INJECTION PROCEDURE

The proposed method lies on the following steps: a) determination of the injection domain, b) determination of the localization domain and c) the injection of specific deformation modes. They will be described in next sections. 3.1

Figure 2.

Hardening softening law and injection threshold.

Figure 3.

Injection and localization zones.

Determination of the injection domain

Again, we borrow from the SDA the concept of the inception of a weak discontinuity indicated by the discontinuous bifurcation analysis. According to this, a weak discontinuity onsets as the following condition takes place, for the first time, at the bifurcation time tB , at a given material point x: det Q(x, tB ) = 0;

Q(n(x), tB ) = n · C(x, tB ) · n (7)

where Q is the so called localization tensor, computed in terms of the tangent constitutive operator C (fulfilling σ˙ = C : ε˙ ), which stems from the chosen constitutive model. Equation (7) provides the bifurcation time tB (the first time that it produces a nontrivial

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corresponding variational equations, in rate form, read as follows:  ˙ s u)dB ∇ s η : (∇ ˙ B\Binj

+

  B(e)

e∈Binj



(e) ∇ s η : (ε˙¯ )dB − Gext = 0

(e)

B(e)

μ(e) : {∇ s u˙ − ε˙¯ }dB = 0

∀e ∈ Binj

(9a)

(9b)

where η stands for the virtual admissible displacements, (•) for the function returning the stresses in the constitutive model, ε¯ (e) the element-wise constant injected strain field and μ(e) the corresponding element-wise constant weighting function. Solution of equation (9)-(b) is then trivially found as: (e) (e) ε¯˙ = ∇ s u˙

(10)

(e)

where (•) stands for the mean value of (•) in the element (e). Substitution of equation (10) into (9)-(a) yields:  ˙ s u)dB ∇ s η : (∇ ˙

where Fint and Fext stand, respectively, for the internal and external forces, A is the assembling operator, B(e) is the standard deformation matrix, σ is the stress vector, K is the tangent stiffness matrix and D is the tangent constitutive matrix (Voigt’s notation is considered everywhere in the equations). As it will be shown in the examples, the injection of that constant strain mode at the injection domain exhibits remarkable properties as for the correct propagation of the strain localization in initial stages. However, as deep localization takes place, in some cases it still exhibits a limited capability to replicate the strain localization mode, εloc , of equation (5), and some degree of stress-locking can appear. Therefore, an additional injection mode is required.

3.3

Recalling the bifurcation time, tB , already defined as the first time that equation (7) is fulfilled, the corresponding value of the stress-like internal variable is defined as: qbif (x) = q(x, tbif ) : ∀x ∈ Binj )

+

e∈Binj

(e)

B(e)

˙ s u˙ )dB − Gext = 0 ∇ s η : (∇ qloc (x) = γ qbif (x) : γ ∈ [0, 1]

(11)

Fint − Fext = 0  Fint = A e∈B\Binj

B(e)

˙ s u˙ (e) )dB B(e)T · σ(∇

 + A

e∈Binj

(e)

B(e)

 A

e∈B\Binj

B(e)

B(e)T · D · B(e) dB

 + A

e∈Binj

B(e)

B

(e)T

·D·B

(e)

˙ t) > 0} Bloc (t) := {x ∈ Binj |q(x, t) ≤ qloc (x) : α(x, (15) where parameter γ in equation 14 defines the degree of softening of points in Bloc ⊂ Binj . Then Bloc is the locus of bifurcated in-loading points, which have experienced a sufficient degree of softening determined by γ . For practically uses it is taken γ ∈ [0.95 − 0.99].



˙ s u˙ )dB B(e)T · σ(∇

 3.4



Injection of an elemental localization mode

Similar to what has been done in section 3.2, a specific localization mode is injected in all the elements whose central point belongs to Bloc .

 dB

(14)

Consequently, the localization zone is defined as (see Figure 3):

which justifies the use if the term injection for the used procedure. In fact, equation (11) corresponds to the virtual work principle where the strain field, ∇ s u˙ (e) , is injected in the constitutive equation in those elements belonging to the injection zone Binj . After the corresponding discretization procedure, equation 3.3 yields the following equations in matrix form:

K=

(13)

Then, a new variable, controlling the softening depth, is defined at each point as:

B\Binj

 

Determination of the localization domain

(12)

(e) (x, t) = (β (e) ⊗ n(e) )S εloc

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(16)

such that:

The corresponding variational equations read:  B\Binj

  e∈Bloc

(e)

B(e)

e∈Binj \Bloc

+

(•)tt = (t ⊗ t) : (•) = t · (•) · t





+

Ploc : (•) = (•) − (•)tt (t ⊗ t)

˙ s u)dB ∇ s η : (∇ ˙

∇ s η : (ε˙¯ )dB

(17a)

and Ploc : (•) ∈ Eloc . After the corresponding discretization procedure, equation (18) yields the following equations in matrix form:

(e)

B(e)

(21)

∇ s η : (˙εloc )dB − Gext = 0

Fint − Fext = 0  B(e)

(e) μ(e) : {∇ s u˙ − ε˙¯ }dB = 0



∀e ∈ Binj \Bloc

Fint =

(17b)  B(e)

(e) δβ (e) ⊗ n(e) : {∇ s u˙ − ε˙ loc }dB = 0

A

e∈B\Binj

 ∀e ∈ Bloc

+

A

e∈Binj \Bloc

(17c)

 + A

Solving equations (17)-(b)-(c) and inserting them into equation (17)-(a) leads to: 

B(e)

˙ ε(e) )dB B(e)T · σ(˙

e∈Bloc

B(e)

B(e)



T (e) ˙ ε˙¯ )dB B(e) · σ(

˙ ε(e) B(e)T · σ(˙ loc )dB



 (22)

where, in matrix notation, ˙ s u)dB ∇ η : (∇ ˙ s

B\Binj

+



 e∈Binj \Bloc

+

  e∈Bloc

B(e)

∇ s u˙ (e) ≡ ε˙ (e) → B(e) · u˙ (e) (e)

B(e)

˙ s u˙ )dB ∇ s η : (∇

˙ (e) : ∇ η : (P loc s

∇ s u˙

(e) ¯ (e) · u˙ (e) ≡ ε˙¯ → B

Ploc : ∇ s u˙

(e) ∇ s u˙ )dB

(e)

(e) ¯ (e) (e) (e) ≡ ε˙ (e) ·u˙ = B¯ loc · u˙ (e) loc → πloc · B  

(e) B¯ loc

(23)

− Gext = 0

(18)

¯ (e) is the B(e) being the standard elemental B-matrix, B (constant) B-matrix evaluated at the center of the element and B(e) loc is the localization B-matrix computed (e) in terms of the matrix version, πloc , of the localization (e) projector Ploc in equation (19):

(e) is the fourth order localization projector where Ploc defined as: (e) Ploc = I − t(e) ⊗ t(e) ⊗ t(e) ⊗ t(e)

(19)

where I stands for the symmetric fourth order unit tensor and t(e) is the tangent unit vector orthogonal to n(e) (see Figure 1) computed at the center of the element. It is trivially shown that Ploc : (•) is the orthogonal projection of a symmetric second order tensor (•) onto the strain localization sub-space εloc defined as: εloc := {εloc = (β ⊗ n)s }

(e)

(e) ¯ (e) B(e) loc = πloc · B ;

(20)

a(e)

(e) πloc = 1 − a(e) ⊗ a(e)

⎧ ⎫ ⎨(t1 )2 ⎬ = (t2 )2 ⎩2t t ⎭ 1 2

(24)

where 1 is the second order unit tensor and t1 and t2 are the components of the tangent vector t ≡ [t1 , t2 ]T .

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As for the stiffness matrix, stemming from equations (22) and (24), it reads:  K=

A

e∈B\Binj

B(e)T · D · B(e) dB

B(e)

 +

A

e∈Binj \Bloc

B(e)

 + A

e∈Bloc

B(e)

B

(e)T

·D·B

4.2



(e)

B(e)T · D · B(e) loc dB

Figure 4 shows the geometric description and the spatial and temporal loading conditions of the problem (Bazant and Cedolin 1980). The diagonal compression forces, F2 , are initially introduced together with the wedge loads, F1 , increasing along the time, until reaching 3.78 [KN]. Then, the diagonal loads remain constant while the wedge loads increase. The material parameters are, Young’s modulus: E = 30500 [MPa], Poisson’s ratio: ν = 0.2, fracture energy: Gf = 100 [N/m], ultimate tensile strength: σu = 3 [MPa] and specimen thickness: 50.8 [mm]. The reported experimental crack path follows a straight line (inclined α = 19◦ with the vertical axis) Therefore, it is rather simple to construct structured meshes such that the discontinuity path intersects the elements in arbitrary directions. Those meshes will

 dB

 (25)

where the non-symmetric character of the stiffness matrix in the last term of equation (25) can be noticed.

4

Injection of an element wise constant strain mode in an isotropic continuum damage model: double cantilever beam (DCB) with diagonal loads

REPRESENTATIVE NUMERICAL SIMULATIONS

In order to assess the effectiveness of the proposed methodology in capturing strain localization processes, a set of two-dimensional examples have been considered.

Table 2.

Elasto-plastic model.

Free energy:

φ(εe , α) = 12 ε e : Ce : εe + φ p (α) ε = εe + εp

Internal variables

α˙ = λ˙ ; ε˙ p = λ˙ ξ ;

4.1 Constitutive models As representative of a broad set of models, two families of constitutive models for computational material failure have been tackled: isotropic damage models and elastoplastic models (summarized in Table 1 and 2). The reader is referred to (Oliver, Huespe et al. 2006) for an specific description of both models.

Table 1.

Constitutive equation

σ = Ce : εe = Ce : (ε − ε p )

Damage/ yield function

g(σ , q) = (σ ) − q

Loading-unloading condition

λ˙ ≥ 0;

Free energy: Internal variables

ψ(ε, α) = (1 − d)ψo ; ψo = : εd(α) = 1 − q(α)/α √ α˙ = λ˙ ; αt=0 = αo = σu / E q α

C   : ε =

Constitutive equation

σ = (1 − d)C : ε =

Damage/yield function

g(ε, α) ≡τε (ε) − α τε (ε) ≡ σ¯ : (Ce )−1 · σ¯

Loadingunloading condition Stress-like internal variable evolution Constitutive tangent operator

λ˙ ≥ 0;

g ≤ 0;

q α

Ce

:

g ≤ 0;

λ˙ g = 0

Stress-like internal variable evolution

q˙ = −H α; ˙ q|t=0 = σu ;

Constitutive tangent operator

σ˙ = Ctan  : ε˙u C ≡ Ce tan C = Cl ≡ Ce −

Isotropic continuum damage model. 1 2ε

αt=0 = 0 ξ = ∂σ (σ )

q≥0 q|t=∞ = 0

Ce :ξ ⊗Ce :ξ ξ :Ce :ξ +H

σ¯ ;

σ¯

(a)

(b)

˙ =0 λg

q˙ = H α; ˙ q ≥ 0√ q|t=0 = αo = σu / E ;

q|t=∞ = 0

σ˙ =  C : ε˙ ; Cu ≡ (1 − d)Ce = αq Ce C= α Cl ≡ αq Ce − q−H σ¯ ⊗ σ¯ α3

Figure 4. Double cantilever beam with diagonal loads: (a) geometrical data and (b) loading data.

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strongly challenge the standard localization formulations based on quadrilateral elements, which perform particularly well when the propagation direction is parallel to the element sides but exhibit mesh-bias dependence otherwise. Figure 5 shows the results (in plane stress), in terms of the localization pattern, obtained with one of those meshes for two cases: (a) using the standard bi-linear quadrilateral element and, (b) injecting in the domain Binj an element-wise constant strain mode, following the procedure indicated in section 3.2. There, it

is displayed the good match of the obtained localization pattern in case (b) with the reported experimental crack. In Figure 6 the evolution of the injection domain, Binj is displayed in four representative time steps. There, it can be observed a bulb-shaped domain, at the tip of the advancing localization band, where the material initially bifurcates and remains in inloading state, so that the elemental-wise constant deformation mode is injected. Soon later, most of the bifurcated elements behind the bulb unload (i.e.: α˙ = 0) so they leave the Binj domain, according to equation (8), excepting for an inclined band, behind the bulb and encompassing one element size, which remains in inelastic loading and defines the localization band. It is remarkable the sharp definition of this localization band, characterizing a good resolution of the weak discontinuity in a one-element-width band, propagating independently of the mesh bias (unlike for the standard quad element) and following the reported experimental crack path. No further type of injection was required in this example.

(a) 19º

4.3

Injection of constant strain and localization modes in a J2 plasticity model: strip in homogeneous uniaxial tensile stress state

Figure 7 sketches the uniaxial tensile test in a plane strain state. The material properties are the following: Young’s modulus: E = 120 [Kpa], Poisson’s ratio: ν = 0.49, fracture energy: Gf = 22.72 [N/m], and ultimate tensile strength: σu = 1 [Kpa]. In the context of the strong discontinuity approach the solution of the problem consists of a straight shear band inclined 45◦ (Oliver and Huespe 2004).

(b) 19º

Figure 5. Double cantilever beam with diagonal loads: localization patterns (iso-displacement contours) with different approaches a) standard quadrilateral element b) Injection of an elemental constant strain mode.

t=1

t=2

t=3

t=4

Figure 6. Double cantilever beam with diagonal loads: evolution of the injection domain (shaded zone) , Binj , along different times of the analysis.

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(a)

P

(b)

0.5m

P

Figure 7. Strip under uniaxial tensile stress state. Oriented mesh of quadrilaterals and theoretical discontinuity path.

In order to trigger the discontinuity, the initially homogeneous problem is slightly perturbed in terms of the tensile strength, reduced in a 10%, at two elements placed at the upper edge of the strip. The theoretical solution in terms of the P − δ curve is also displayed in Figure 10 for the linear softening case. In order to check the ability to circumvent the mesh bias dependence, a structured mesh of quadrilaterals, with an orientation of 65◦ is considered (see Figure 7) and checked in front of a number of formulations. Three of alternatives are then considered: I. The standard (irreducible) formulation II. Injection of an element wise constant strain mode in Binj ≡ Bloc (as explained in section 3.2) III. Injection of an element wise constant strain mode in Binj and, then, a localization mode in Bloc ⊂ Binj (as explained in section 3,4).

(c)

Figure 8. Strip under uniaxial tensile stress state: localization patterns with the different approaches: (a) I-standard element, (b) II-injection of a constant strain mode and (c) III-injection of a localization mode.

(a)

In Figure 8 the obtained localization patterns, in terms of the contours of the total displacement field, are shown. In addition, in Figure 9 the corresponding localization domains, for the alternatives II and III, are shown. Finally in Figure 10 the load-displacement curves are presented. There, the following facts can be noticed: • The very poor resolution, both in terms of the propagation and the sharpness of the localization pattern for the standard formulation in alternative I • The good resolution of the constant strain injected deformation modes (alternative II above) as for the propagated localization pattern. However, it exhibits still some stress-locking, manifested in terms of a little diffuse localization pattern, which encompasses more than one element. • The clear improvement, in terms of the localization pattern (now much sharper) introduced by the injection of the localization mode (alternative III above). • As for the quantitative results, although alternative II translates into a significant improvement with

(b)

Figure 9. Strip under uniaxial tensile stress state: Localization domain with the different approaches: (a) II-injection of a constant strain mode and (b) III-injection of a localization mode.

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Research on the extension of these techniques to other target elements and formulation is currently undertaken.

Theoretical Option III Option II Option I

0.7 0.6

P(kN)

0.5 0.4

ACKNOWLEDGEMENT

0.3

Financial support from the Spanish Ministry of Science and Innovation through grant BIA2008-00411 is gratefully acknowledged.

0.2 0.1 0 0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

REFERENCES

(m)

Figure 10. Strip under uniaxial tensile stress state: Load deflection curves for the different approaches.

respect to standard formulations (alternative I), the injection of a localization mode in alternative III, results in a softer action-response curve, this stating the practical cancelling of the stress-locking effects for this case. 5

FINAL REMARKS

In the preceding sections an exploratory work, about the effects of some new techniques in the ability of standard quadrilateral elements to capture mesh size and mesh-bias indifferent strain localization, has been presented. The presented preliminary results show that the combination of some concepts borrowed from the Strong Discontinuity Approach (SDA) and the injection of specific deformation modes, via mixed formulations, translate into a large increase of the strain localization properties of that target element. The result is a new technique that enjoys the benefits of both the SDA, in terms of mesh-size and mesh bias indifference, and the simplicity of the classical strain localization methods. These benefits have been obtained with two representative constitutive models (continuum damage and elasto-plasticity), which is encouraging in terms of the broadness of the application of the new technique.

Bazant, Z. and L. Cedolin 1980. ‘‘Fracture mechanics of reinforced concrete.’’ Journal of the Engineering Mechanics Division ASCE: 1287–1305. Belytschko, T., N. Moes, et al. 2001. ‘‘Arbitrary discontinuities in finite elements.’’ International Journal for Numerical Methods in Engineering 50(4): 993–1013. Borst, R.d., L.J. Sluys, et al. 1993. ‘‘Fundamental issues in finite element analyses of localization of deformation.’’ Engineering Computations 10: 99–121. Oliver, J. and A. Huespe 2004. ‘‘Continuum approach to material failure in strong discontinuity settings.’’ Computer Methods in Applied Mechanics and Engineering 193: 3195–3220. Oliver, J. and A.E. Huespe 2004. ‘‘Continuum approach to material failure in strong discontinuity settings.’’ Computer Methods in Applied Mechanics and Engineering 193(30–32): 3195–3220. Oliver, J. and A.E. Huespe 2004. ‘‘Theoretical and computational issues in modelling material failure in strong discontinuity scenarios.’’ Computer Methods in Applied Mechanics and Engineering 193(27–29): 2987–3014. Oliver, J., A.E. Huespe, et al. 2006. ‘‘Stability and robustness issues in numerical modeling of material failure in the strong discontinuity approach.’’ Comput. Methods Appl. Mech. Engng. 195: 7093–7114. Oliver, J., A.E. Huespe, et al. 2006. ‘‘A comparative study on finite elements with embedded discontinuities: E-FEM vs. X-FEM.’’ Computer Methods in Applied Mechanics and Engineering 195: 4732–4752. Simo, J., J. Oliver, et al. 1993. ‘‘An analysis of strong discontinuities induced by strain-softening in rate-independent inelastic solids.’’ Computational Mechanics 12: 277–296.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Model for the analysis of structural concrete elements under plane stress conditions: Finite element implementation M. Pimentel & J. Figueiras Laboratory for the Concrete Technology and Structural Behaviour (LABEST), Faculty of Engineering of the University of Porto, Portugal

ABSTRACT: The implementation in a finite element code of a recently developed material model for the analysis of reinforced concrete (RC) cracked membranes is presented. The model aims for the analysis of large scale structural elements that can be considered an assembly of membrane elements, such as bridge girders, shear walls, transfer beams or containment structures. The code was implemented as a user-supplied sub-routine in the general purpose finite element software DIANA 9.3. The equilibrium equations of the cracked membrane element are established directly at the cracks eliminating the need to resort to averaged stress-strain relations. The average stress fields are obtained as a by-product of the local behaviour at the cracks and of the bond stress transfer mechanisms between the reinforcement and concrete. Special attention is devoted to the issues related with the implementation of the material model in a finite element code. A total strain based formulation was developed allowing the direct calculation of the stresses given a trial average strain tensor, without having to resort to inner loops at the constitutive level. The material model, previously calibrated at the element level with the results of monotonic loading tests on RC panels under in-plane shear and axial stresses, is now generalized to accommodate all possible stress trajectories, such as loading reversals, biaxial compression and the existence of two orthogonal cracks. The path dependent behaviour of the model is illustrated and some validation examples with experimental results from full scale tests with common shear critical structural elements are presented. 1

INTRODUCTION

In this work a macroscopic scale treatment of the reinforced concrete (RC) behaviour is adopted. The accurate reproduction of RC behaviour at the membrane level is taken as a departure point for the analysis of large scale RC structures via the finite element method. In the past 30 years a strong investment has been made on the research of suitable constitutive laws for reinforced concrete membranes under in-plane shear and axial stresses. Given the present state of knowledge, a phenomenological approach is adopted instead of the existing theoretical frameworks that can also be used for modelling concrete behaviour, such as the plasticity theory, continuum damage theory, microplane theory, etc. The direct implementation of such a phenomenological approach usually leads to less elegant theoretical formulations than the ones based on the above mentioned theories. Nonetheless, it has the advantage of being rooted on strong experimental evidence and on mature physical models specially developed for RC. The behaviour of the basic cracked membrane element is discussed in references (Pimentel 2010; Pimentel et al. 2010), where the detailed formulation and extensive validation of the post-cracking concrete relations is presented. One of the most striking aspects

of structural concrete behaviour is that, after cracking, the slip between concrete and reinforcing bars leads to a highly irregular stress field. This aspect is commonly dealt with by treating reinforced concrete as a new material with its own stress-strain characteristics, which are valid only in spatially average terms (Hsu & Zhu 2002; Maekawa et al. 2003; Vecchio 2000; Vecchio & Collins 1986). The relation between these averaged constitutive laws and the mechanical phenomena taking place at the cracks—which are known to govern the behaviour of cracked concrete elements—is not always straightforward. Therefore, it is not possible to establish a link between the usual averaged constitutive relations and the significant research effort that has been made throughout the last decades on RC mechanics topics like bond, aggregate interlock, concrete tensile fracture, etc. The main advantage of the proposed model is that the equilibrium equations of the cracked membrane element are established directly at the cracks eliminating the need to resort to averaged stress-strain relations. Henceforth, the response the cracked membrane element is computed considering the individual contributions of the involved mechanical phenomena in a transparent manner. The basic membrane element is assumed to contain several cracks. Therefore, the strain field calculated

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from the nodal displacements must be understood as a spatially-averaged strain field. The strain profile along the reinforcing bars is recovered from the averaged strain field assuming and stepped rigid bond shear stress-slip law between the rebars and the surrounding concrete according to the Tension Chord Model (TCM) (Marti et al. 1998). With the TCM the steel stresses at the cracks can be calculated allowing the establishment of the equilibrium in terms of stresses at the cracks. Moreover, the bond stress transfer mechanisms can be accounted for in a rational manner both in the pre- and post- yielding regimes, and the crack spacing and crack widths can be calculated from first principles instead of relying in empirical based expressions. In the following the generalization of the proposed model allowing its implementation in a finite element code is described. This comprised the formulation of the uncracked concrete nonlinear behaviour, the establishment of suitable loading/unloading conditions, the existence of two orthogonal cracks in the same integration point, and the treatment of the strain localization issues. 2 2.1

TOTAL STRAIN BASED MODEL Uncracked concrete

For uncracked concrete, an efficient elasticity based isotropic formulation is adopted, requiring only one state variable for tracing the damage evolution and establishing the loading/unloading/reloading conditions. 2.1.1 Equivalent stress The equivalent stress S is here defined as a scalar measure of the applied stress level under biaxial plane stress conditions. For S = 0 concrete is totally unloaded while for S = 1 the biaxial failure envelope has been reached. As proposed by Maekawa & Okamura (1983), the equivalent stress can be defined as a function of the mean and deviatoric invariant components of the stress tensor, σm and τd :  2  2 a b σ + τ (1) S= m d fc fc √ √ 2 2 |σ1 − σ2 | (2) σm = (σ1 + σ2 ) ; τd = 2 2 In the above expressions, fc is the standard cylinder uniaxial compressive strength and the two parameters a = 0.61 and b = 1.27 were determined from experimental data for the uniaxial compression and equal biaxial compression stress trajectories. In Figure 1 the resulting failure envelope (S = 1) is presented in the σ1 − σ2 plane and it is compared to the Kupfer &

1.5

1.0 2/

Kupfer and Gerstle

.52:

f c'

1

Present criteria: S = 1

0.5 1

Crack detection surface

0.0 0.0

0.5

1.0 1/

1.5

f c'

Figure 1. Graphical representation of the proposed failure criteria in the σ1 − σ2 stress plane.

Gerstle (1973) failure criteria, which was adopted by the CEB in the Model Code 1990 (CEB 1993). The crack detection envelope represented in the figure is used for detecting if cracking has occurred. When this envelope is violated, the cracked concrete subroutines are invoked, see section 2.2. 2.1.2 Equivalent total strain The equivalent total strain E is a scalar measure of the current biaxial total strain level. Similar to the equivalent stress, for E = 0 concrete is unloaded while for E = 1 the failure envelope is reached. The equivalent total strain is also defined as a function of the two invariants of the total strain tensor (Maekawa & Okamura 1983), εm and γd :  E= √ εm =

c εm ε0

2

 +

2 (ε1 + ε2 ); 2

d γd ε0

2

γd =

(3) √ 2 |ε1 − ε2 | 2

(4)

The two constants c = 0.62 and d = 1.10 were derived such as to fit the peak strains obtained in the uniaxial compression and equal biaxial compression tests and ε0 is peak strain in a standard uniaxial compression test. 2.1.3 Stress-strain relations and loading/unloading conditions In a monotonic loading process it is assumed that the equivalent stress S can be univocally obtained from the equivalent total strain E through the following nonlinear relationship:

S=

⎧ k E − E2 ⎪ ⎪ ⎪ ⎨ 1 + (k − 2) E ,

E≤1

⎪ ⎪ ⎪ ⎩

E>1

1+

η 2

1

, 1 − 2E + E 2



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(5)

For the ascending branch, E ≤ 1, the Sargin’s law was adopted (fib 1999), while for the descending (post-peak) branch a modified version of the function proposed by Krätzig & Pölling (2004) was defined, ensuring a smooth transition between the two branches. The parameter k determines the shape of the curve in the ascending branch. With k = 2, a parabolic relationship is obtained. In general k = Eci · ε0 /fc , with Eci being the initial tangential modulus of elasticity in a uniaxial compressive test. The parameter η controls the shape of the curve in the descending branch. In order to guarantee objectivity of the results with respect to the mesh size, the post-peak response is regularized according to the dimension of the strain localization band, hC , which is related to the finite element size. It is proposed to use 1 η= 2



π f  ε c 0  GC hC − fc ε0 2

a region in the total strain space where no damage evolution occurs—the reversible process domain. This region expands as the maximum total strain increases and can never contract. The resulting algorithm is very simple as can be summarized as follows: 1. calculate the current equivalent total strain E t+t from current the total strain tensor ε t+t according to expression (3); 2. calculate the current maximum equivalent total t+t strain and the equivalent strain ratio: Emax = t , E t+t ), r = E t+t/E t+t , respectively; max(Emax max t+t 3. calculate the equivalent stress Smax corresponding t+t to Emax according to expression (5); 4. calculate the current equivalent stress S t+t = t+t r Smax .

2 (6)

where GC is the compressive fracture energy. Unless otherwise stated, GC is estimated using the Compressive Damage Zone (CDZ) model proposed by Markeset & Hillerborg (1995) considering k = 3, r = 1.25 mm, wc = 0.6 mm and Ld = 2.5 · e, being e the element thickness (k, r, wc and Ld are the CDZ model parameters): GC = 1500 GF e + 0.0003fc

(7)

In Figure 2, the resulting uniaxial stress-strain curves for an element with length/thickness of 0.25 × 0.10 m2 are presented for 5 different uniaxial concrete strengths. The fracture energy GF , the Young modulus Eci and the peak uniaxial strain ε0 were calculated according to the expressions given in (fib 1999). In the present model, a loading step always leads to damage evolution while in an unloading/reloading step damage remains constant. The variable that is used to monitor damage evolution is the maximum equivalent total strain, Emax , which by definition is a non-decreasing variable. A given Emax value bounds 1

100 f ' c = 20 40 60 (Mpa) 80 100

S

-

0

2

0

0

E

3

0

-

0.006 2

Figure 2. Graphical representation of the proposed stressstrain curve in the uniaxial compression case for 5 different concrete strengths: (a) Equivalent stress vs. equivalent strain; (b) Resulting uniaxial stress-strain curves.

2.1.4 Constitutive and iteration stiffness matrices So far, only a scalar relationship between the equivalent stress and equivalent total strain was defined. To complete the formulation, a directional relationship between the total strain and stress tensors is required. In the present model, before cracking, concrete is treated as an isotropic material. At higher loading levels this is a simplifying assumption, as shown by Maekawa & Okamura (1983). Nonetheless, it greatly simplifies the model while the error that is introduced is small and has little significance on the overall response of a RC member under biaxial stress conditions. Taking into account the definition of equivalent stress given by (1), the secant constitutive matrix relating concrete stresses with the total strains is derived in the form of a nonlinear elasticity matrix: ⎡ ⎧ ⎫ 1 ⎨ σcx ⎬ Ec,sec ⎢ν σcy = ⎣ ⎩τ ⎭ 1 − ν 2 0 cxy Ec,sec =

fc  ε0  a

fc

ν 1 0

⎤⎧ ⎫ 0 ⎨ε ⎬ 0 ⎥ εx ⎦ 1−ν ⎩ y⎭ γxy 2

S 2  εm + fb 1−ν c

γd 1+ν

2

(8)

(9)

where Ec,sec is the secant Young modulus and ν is the Poisson ratio for which a constant value ν = 0.2 may be taken. The secant stiffness matrix of (8) is used to calculate stresses for the given strain state. The tangent stiffness matrix Dc is used for construction of an element stiffness matrix for the iterative solution at the structural level. This matrix can readily be obtained in the global coordinate system replacing in (8) Ec,sec by the tangential modulus Ec,tan , which is the slope of the equivalent stress-strain curve at a given total strain: Ec,tan = dS/dE. In the present implementation, whenever the slope of the curve is less than the minimum value (Ec,tan )min the value of the tangent modulus is set equal to the minimum. This occurs in the softening range and near the compressive peak.

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2.1.5 Examples In Figure 3 the results from the analytical model are compared to a classical set of experimental data (Kupfer & Gerstle 1973) with three compressive biaxial stress trajectories (see also Figure 1). The agreement is quite good in view of the simplifications considered in the model, such as the isotropy assumption and the constant value for the Poisson coefficient. In Figure 4 the model response for the case of non proportional loading under biaxial compression stress states is presented. In a first stage, a uniaxial compression loading was applied till the equivalent stress reached S = 0.7, after which followed an unloading stage. Then a biaxial compression loading path was applied following the stress trajectory σ1 :σ2 = −.52:−1 till the equivalent stress reached S = 0.85. After unloading, a biaxial compression loading path was finally applied following the stress trajectory σ1 :σ2 = −1:−1. The stress trajectories are

1.5 0.52/ 1

1.0

1: 1 1:

' 2 / fc

2=

0: 1

0.5

0.0 1.0

0.0 1

1.0

/ '0

2

2.0

/ '0

Figure 3. Comparison of analytical model (k = 2.375, ν = 0.2) with experimental data from Kupfer & Gerstle (1973). 1.5

represented in Figure 4a and the corresponding strain trajectories in Figure 4b.

2.2

Cracked concrete

After cracking, concrete is treated as an orthotropic material and a local coordinate system is introduced (n-t coordinates) where the constitutive laws are established. The total strains in the global coordinate system are transformed into the local coordinate system through the usual strain transformation relationship. Once cracking occurs, two uniaxial stress fields are assumed along the two orthogonal n-t directions and the Poisson effect in the cracked concrete is neglected. The crack direction is determined by the principal tensile direction of the concrete stress tensor at impending cracking, θr . The crack angle θr remains fixed and is kept in memory as a state variable. 2.2.1 Compression model parallel to the cracks direction The compressive stress parallel to the crack direction is determined from the total strain along the corresponding local using the base curve (5) already adopted in the uncracked stage. However, it is well known (Belarbi 1991; Vecchio & Collins 1986) that the compressive strength of cracked concrete is reduced (or softened, as usually termed in RC literature) with increasing tensile strains in the orthogonal direction. Therefore, the compressive strength must be replaced by the effective compressive strength fc,ef = ζf · ζe · fc . Also the peak uniaxial strain is reduced to ε0,ef = ζf · ζe · ε0 . The softening coefficients are formulated as (ε⊥ is the strain in the direction normal to the compressive stress being evaluated):

1.5

ςf = (fc0 /fc ) 3; 1

1.0 2/

1.0

S=1.0

f c'

E=1.0 2/

0.52: 1 0.5

0: 1

1:

2=

ςe =

'0

1: 1 0.85

1 Cm (1.08 + 81ε⊥ )

(10)

0.5 0.568

0.7

0.411

0.0

with fc0 = 30 MPa. For details refer to (Pimentel 2010; Pimentel et al. 2010). For each local axis (n- or t-), the equivalent stress and the equivalent strain are given by

0.0

0.0

0.5

1.0 1/

1.5

0.0

0.5

f c'

1.0 1/

(a)

1.5

'0

(b) 1.5

1.0

E = 0.411 S = 0.7

2: 1

S=−

1.0 E = 0.568 S = 0.85

S

' 2 / fc

0.5

2: 2

0.5

0.0

σi fc,ef

;

E=−

εi ε0,ef

;

i = n or t

(11)

0.0 0.0

0.5

1.0

E

(c)

1.5

0.0

1.0

0.5 1

/ '0 ;

2

1.5

/ '0

(d)

Figure 4. Model behaviour under non-proportional biaxial compression loading paths: (a) Load paths in the principal stress space and (b) corresponding principal strain trajectories; (c) equivalent stress-equivalent strain relationship; (d) minimum compressive stress ratio versus the two principal strain ratios.

2.2.2 Tension model normal to the cracks direction At the cracks, the normal stresses are due to crack bridging stresses and to crack dilatancy stresses arising from the shear transfer mechanics. For very small crack widths the former govern while the later are important for fully developed cracks. The dilatancy stresses are dealt by the crack shear transfer model. The crack bridging stresses are calculated with the

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expression proposed by Hordijk (1992):   3  wr wr wr σi e−6.93 wc − 0.027384 = 1 + 27 fct wc wc (12) GF wc = 5.14 , fct

wr = hεi ,

i = n or t

(13)

where h is the crack band width, which is related to the finite elements size. 2.2.3 Coupled tension-compression model The direct combination of the compression and tension models results in a uniaxial path-dependent model. The unloading/reloading is performed secant to the origin as already illustrated for the uncracked state. For reproducing this type of behaviour two state variables are required for each local coordinate system direction: the maximum and minimum total strain experienced in the past history. In the present implementation, instead of the minimum compressive strain, the state variable governing the compressive behaviour is the maximum equivalent total strain in the past history, given by expression (11)2 . The loading/unloading determination is made independently for the n- and t-direction. Once concrete is cracked, the biaxial confinement effects cannot be recovered. 2.2.4 Shear transfer model in the local coordinate system 2.2.4.1 One active crack in the coordinate system The last stage of the tensile fracture process corresponds to the formation of a macroscopic crack that cannot transmit normal tensile stresses. Shear transfer across these cracks cannot be simply formulated as a relation between shear stress and shear displacement, but is a more complex mechanism, in which shear stress, shear displacement, normal stress and crack width are involved. The formulation of the equilibrium conditions directly at the cracks, allows the adoption of previously developed theoretical models for reproducing this complex behaviour, instead of adopting some oversimplified formulations, such as the ones based on shear retention factors. A closed form solution based on the crack density model (Li et al. 1989), which was posteriorly adapted by Pimentel (Pimentel 2010; Pimentel et al. 2010), is adopted for calculating the crack shear and crack dilatancy stresses from the normalized shear displacement β = δr,t /δr,n . If a regular array of cracks is formed then it is possible to show that βi = γnt /εi,max , where γnt is the current shear strain in the local n-t coordinate system and εi,max is the maximum normal strain along the local axis perpendicular to the crack. The state variable εi,max (see 2.2.2) is used

instead of the current normal strain εi in the determination of the normalized shear strain, βi . This was required for ensuring: (1) the stability in the algorithm in the unloading/reloading regimes; and (2) a smooth transition from loading to unloading states. For loading/unloading states in the positive side of the normalized shear strain, β > 0, the shear and dilatancy stresses are given by: βi2 1 + βi2   |βi | π σdil,i = −τLIM ,i − cot−1 |βi | − 2 1 + βi2

τLIM ,i = 3.83fc1/3 g εi,max , srmθ τagg,i = τLIM

g εi,max , srmθ =

(14)

0.5 ≤1 0.31 + 200 srmθ εi,max

In the expressions above srmθ is an estimate of the crack spacing according to the TCM (Pimentel 2010; Pimentel et al. 2010). In the negative side of the normalized shear strain, β < 0, a minus sign must be placed in the τagg expression. For simplicity of the resulting algorithm, especially when two cracks arise in the same integration point, loading and unloading are performed following the same curve. This equivalent to disregarding the energy dissipation occurring at the crack lips during shear loading reversals. 2.2.4.2

Equilibrium and compatibility for two orthogonal cracks Material models for reinforced concrete must deal with situations were at least two cracks arise in the same integration point. In the present formulation, postcracking stresses are evaluated in the local orthotropy axes, which are, by definition, orthogonal. Therefore, only two orthogonal cracks are allowed. In an RC element containing two cracks the total shear strain γnt must be distributed by each crack (Figure 5) so that the corresponding shear and dilatancy stresses can be calculated. Equilibrium requires that the shear stress transmitted along both cracks is the same. Therefore,

(a) Compatibility

(b) Equilibrium

(c) Problem to be solved

Figure 5. Equilibrium and compatibility for two-way orthogonal cracks.

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the problem is governed by the two following conditions: τagg = τagg (β1 ) = τagg (β2 );

2.3

γnt = γnt,1 + γnt,2 (15)

 . with β1 = γnt,1 /εn,max and β2 = γnt,2 /εt,max It can be shown that the determination of the shear strains fulfilling the equilibrium and compatibility conditions (15) can be reduced to the problem of finding the roots of a polynomial of fourth degree. Since only one real root exists in the interval, γnt,1 ∈ [0, γnt ], the bisection method, which can be shown to be unconditionally convergent, was implemented for finding the solution. The algorithm starts by checking how many active cracks exist. An active crack is defined as having tensile normal strains, otherwise is considered to be closed or inactive. If two active cracks are detected, a subroutine is called for calculating the corresponding crack shear strains γnt,1 and γnt,2 from the total shear strain γnt . The crack shear stress τagg and the crack dilatancy stresses σdil,1 and σdil,2 are then calculated according to (14).

2.2.5 Assembly of the concrete stress vector In the local n-t coordinate system, the concrete stress vector at the cracks is simply obtained from: σcr,n = σn + σdil,1 σcr,t = σt + σdil,2 τcr,nt = τagg

(16)

2.2.6 Iteration stiffness matrix In the local coordinate system, the stiffness matrix used for the iterative process has the form:

Dc,nt

⎡ D11 =⎣0 0

0 D22 0

⎤ 0 0⎦ D33

(17)



σsr,i = f εi , Es,i , fsy,i , fsu,i , εsu,i , srm,i , τb0,i , τb1,i

1 D33,1

1 +

1 D33,2

(19)

with the subscript (·)i indicating the x- or y- reinforcement direction, εi the average strain along the reinforcement i-direction, Es,i the steel Young modulus, fsy,i , fsu,i and εsu,i the steel yielding strength, tensile strength and corresponding rupture strain, respectively, srm,i the component of the crack spacing along the reinforcement i-direction, and τb0,i and τb1,i the plastic bond stresses prior to and after the i-reinforcement yielding. For implementation in a finite element procedure, the model needs to be complemented with suitable loading/unloading/reloading conditions. These can be derived considering rigid-plastic behaviour of the bond shear stress-slip law as depicted in Figure 7. In the unloading stage, it is here assumed that bond shear stress drops to zero. In reality the bond shear stress may reverse sign and, although a rapid degradation of bond is usually seen, negative bond shear stresses are expected in the unloading stage. This may be important for an accurate modelling of the cracks widths in service conditions and should be subject of future generalization of the present model. If reversed bond action is neglected, when the unloaded length is equal to the crack spacing, the

The non-zero members of (17) are determined in a closed form by differentiation of the corresponding stress-strain laws. In the case of two active cracks, the shear stiffness is obtained from: 1 = D33

Reinforcing steel

The reinforcement stresses at the cracks are calculated from the average strains, as described elsewhere (Pimentel 2010; Pimentel et al. 2010). The algorithm starts by calculating the maximum diagonal crack spacing srm0 given the crack angle θr , the reinforcement ratios, the assumed rigid plastic bond shear-stress slip law, the concrete stresses at the cracks—given by (16)—, and the fact that maximum concrete tensile stresses at the centre between the cracks cannot exceed the concrete tensile strength fct . Figure 6 provides polar representations of the solution. The steel stresses at the cracks are calculated in the form (Pimentel 2010; Pimentel et al. 2010):

x= xy /f ct =

s rmy0

3 2 1

3

x=

y

Øx = 2 Øy cr,nt

3

y

Øx = 2 Øy

/f ct = 0

cr,nt

/f ct = 0.5

(18)

In theory, the adoption of a tangent stiffness matrix neglecting the effect of the cross terms weakens the convergence characteristics of the algorithm if a fully Newton-Raphson procedure is adopted for the incremental-iterative solution procedure. Nonetheless, this effect was found not to be severe and good convergence rates can still be achieved.

s rm0

s rm0 r

0 0

xy /f ct =

3 2 1

r

s rmx0

0

s rmx0

Figure 6. Polar representations of the maximum diagonal crack spacing srm0 .

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(a)

(b)

Figure 8. Reinforcement stresses under non-proportional loading (Note: εsm = εi ).

(c)

(d)

Figure 7. Formulation of the reinforcement model under non-proportional loading: (a) Bond stress-slip law; (b) Stress and strain distributions between the cracks (pre-yielding stage); (c) Resulting stress-strain laws for the steel; (d) Corresponding average stresses in the concrete (Note: εsm = εi ).

stresses in the reinforcement are equal to σs,min , see Figure 7b. From that point on, the naked bar stressstrain relation is adopted. For simplicity, a linear relationship is assumed until the unloaded length is equal to the crack spacing. It can be shown that the unloading/reloading stiffness in this region is twice the steel young modulus Es if the steel stresses along the tension chord are all above or below the tensile strength fsy . In these cases the following equalities hold: σsr = σsr − σs,min = 2(σsr − σsm )

(20)

σsr = 2 Es ε

(21)

In Figure 7c and d the resulting loading/unloading/ reloading behaviour is illustrated. The determination of the notable points of the unloading/reloading σsr (ε) curve, i.e. the points where the unloading/reloading slope changes 2·Es to Es , is made using the equilibrium conditions of the differential element as discussed in (Pimentel 2010; Pimentel et al. 2010) and the constitutive laws for the reinforcement steel and bond shear stress. For the cases where partial yielding has occurred along the tension chord, i.e. σs,min < fsy < σsr , the equalities (20) and (21) are only good approximations of the real solution. Nonetheless, these were adopted for simplicity of the resulting algorithm.

The naked bar stress-strain law is adopted also when the stresses go onto the compressive side. The plastic strains are used as offsets of the steel response, as depicted in Figure 8. 2.3.1 Assembly of the stress vector The composite stress vector in the reinforcement directions is finally assembled as:  σ xy = Tσ (θr )σ nt + ρx σsr,x

ρy σsr,y

T

0

(22)

with Tσ being the stress transformation matrix and ρx and ρy the reinforcement ratios in the x- and ydirections.

3

VALIDATION

Following a previously performed validation at element level, which included an extensive set of results obtained from tests on RC panels (Pimentel et al. 2010), here the validation at the structural level of the proposed constitutive model is presented. The validation is made using the results from a set of experimental tests on shear critical structural elements. Four validation examples are presented here. 3.1

Beams VN2 and VN4

The first two beams, VN2 and VN4, belong to an experimental campaign performed by Kaufmann and Marti at the ETH Zürich. The goal of the VN test series was to investigate the behaviour of webs of structural concrete girders with no plastic deformation in the chords and having low shear reinforcement ratios (ρw = 0.335%). The state of stress is similar to

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that observed outside the support regions of a continuous girder. For that purpose, a specially developed testing facility was developed allowing the investigation of elements of beams rather than entire girders. The beams were subjected to a constant shear force. The rotation of the beams ends was restricted and the forces at the element ends were controlled such that the axial force remained constant. In beam VN2 the axial force was null while in beam VN4 a compressive force of 1 MN was applied. In the case of beam VN2 collapse was triggered by stirrup rupture after some minor spalling of the web cover concrete. Beam VN4 failed after crushing of the web concrete. For further details refer to test report (Kaufmann & Marti 1996) and to (Pimentel 2010). The beam elements are 5.84 long and 0.78 m high, see Figure 9. The model predicts the same failure modes as observed in the tests. In the case of cracked concrete, the state variable Emax = −εt /ε0,ef is an indicator of whether concrete crushing has or has not occurred. If Emax > 1 the post-peak branch of the stress strain curve has been achieved. In this stage of the curve, signs of concrete cover spalling or concrete delamination are to be expected in real specimens. The contour levels with the values of Emax at failure are presented in Figure 10. Only minor signs of web cover spalling are predicted near the flanges for beam VN2. Similarly to the test, collapse was triggered by rupture of the vertical reinforcement. This can be confirmed from inspection of Figure 11, where the contour levels with

the vertical reinforcement stresses at the cracks are presented. The red colour corresponds to regions where the tensile strength of the stirrups ( fsu = 604 MPa) was exceeded. The cracking patterns at failure are depicted in Figure 12 and the web crack widths, which constitute a direct output of the analysis, are presented in Figure 13. In Figure 14 the calculated and the measured average vertical deformations in a 1.60 m long web segment at the centre of the beams are compared. In the case of beam VN2 the curve is presented till the load step where the steel stresses reached fsu .

Figure 11. Beam VN2: deformed shape (x10) with the σsr,y contour levels at failure.

(a)

(b)

Figure 12. Experimental and calculated cracking patterns at failure: (a) VN2; (b) VN4. 2.5

LS 4 wr [mm]

Exp.

Calculated 0 1.6

LS 3 wr [mm]

Figure 9. Beams of the VN series: internal forces, cross section and adopted finite element mesh.

0 -2

0

2

x [m]

Figure 10. Deformed shape (x10) with the Emax contour levels at failure.

Figure 13. Experimental (Kaufmann & Marti 1996) and calculated web crack widths of beam VN2. The load steps LS3 and LS4 are identified in Figure 14.

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Peak load

Peak load

F [kN]

LS3

LS5

LS4

LS2

LS3 LS2

Exp.

LS1

Exp.

VN4

Calculated

LS5

Exp.

LS1

Exp.

VN2

LS4

Calculated

0 0

3

y

Figure 14.

3.2

(x10 )

30 0

3

y

(x10 )

30

Force vs. average vertical strain curves.

Beams MVN2 and MVN4

The four beams of the series MVN have the same geometry and the same shear reinforcement content as the ones from the series VN (Kaufmann & Marti 1996). However, in contrast to series VN, an additional vertical jack was provided at midspan. The loading procedure simulated the behaviour of an (inverted) intermediate support of a continuous girder, see Figure 15. In this case, bending moments have a significant influence on the behaviour of the girder and usually chords are plastically deformed. This bending-shear interaction can be explained by the decreasing softened compressive strength of the web concrete with increasing longitudinal strains. Here only the analyses to specimens MVN2 and MVN4 are presented. In the first phase of the test, horizontal forces were controlled such that moments at the element ends were zero. In the second phase, after reaching the yield moment at midspan, rotations of both element ends were prevented. This second phase of the test simulated a redistribution of moments from the support region (midspan) into the span (element ends) of a continuous girder. The forces applied at the element ends were controlled such that normal forces remained constant throughout the test and shear forces on either side of the concentrated load were of equal magnitude. Specimens MVN2 and MVN4 were subjected to axial compression of 1.30 MN before being loaded in shear and the axial force was held constant during the test. The specimen MVN4 was designed not to yield at midspan during the entire test and was post-tensioned with a bonded VSL 6-7 cable (Pu = 1.85 MN) tensioned to an initial force of 1.30 MN. All the girders contained the same amount of web reinforcement (ρw = 0.335%, equal to the one adopted in the series VN). When specimen MVN2 failed, half of the girder was shifted over the remaining part of the element along an inclined failure surface made up of existing web shear cracks (see Figure 18a). Also some spalling of the concrete cover in the web was observed. This failure mode is here classified as a web sliding failure. At peak load the model predicts that some regions of the web

are crushing, as can be seen in Figure 16a. Nonetheless, collapse only occurred with concrete crushing in the top flange, near the loading platen. This can be confirmed by the incremental deformed shape in the post peak branch of the load-deformation curves, see Figure 16b, which is an indication of the occurrence of a sliding shear failure. The specimen MVN4 exhibited a typical web crushing failure (Figure 18b). Failure was more ductile and the web concrete crushed progressively on both sides of the girder. Altough this failure mode was correctly predicted the failure load is slightly underestimated (Fu,exp /Fu,calc = 1.09). In Figure 17 the calculated and the measured average vertical deformations are compared. The deformations were averaged over two symmetrically located and 0.80 long web segments. In general, the

Figure 15.

Beams of the MVN series.

Figure 16. (a) Deformed shape (x10) with the Emax contour levels at failure; (b) Beam MVN2: incremental deformed shape (x1000) after the peak load with the contour levels of the crack shear strains γnt . 2000 Peak load Peak load

F [kN]

su 700

MVN2

Exp. Exp. Calculated

MVN4

0 0

Figure 17.

ε y (x103)

20 0

ε y (x103)

Force vs. average vertical strain curves.

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20

REFERENCES

(a)

(b) Figure 18. Experimental (Kaufmann & Marti 1996) and calculated cracking patterns at failure: (a) MVN2; (b) MVN4.

agreement between the measurements and the calculations is good. It must be remarked that these are average web strains which are directly related with the shear behaviour. Usually, a good fit between numerical and experimental results is much easier to obtain if flexure related parameters, (midspan displacements, per example) are compared. The observed and calculated cracking patterns at failure are depicted in Figure 18. 4

CONCLUSIONS

A new numerical model is presented for the analysis of large scale structural concrete elements. A macroscopic scale treatment of the RC behaviour is adopted and the accurate reproduction of structural concrete behaviour at the membrane level is taken as a departure point for the analysis of large scale RC structures via the finite element method. The model uses a classical kinematic description of the displacement field based on the concept of ‘‘weak discontinuities’’. The developed total strain formulation revealed to be robust and showed good convergence characteristics. The results from the preliminary validation examples showed that the model can successfully reproduce some of typical shear failure models usually observed on RC beams. ACKNOWLEDGMENTS The support by the Portuguese Foundation for Science and Technology (FCT) through the PhD grant SFRH/BD/24540/2005 attributed to the first author is gratefully acknowledged.

Belarbi, A. (1991). ‘‘Stress-Strain Relationships of Reinforced Concrete in Biaxial Tension-Compression,’’ Doctoral Thesis, University of Houston, Houston. CEB. (1993). CEB-FIP Model Code 1990, Thomas Telford, London. fib. (1999). ‘‘Bulletin n◦ 1: Structural Concrete. Text book on Behaviour, Design and Performance. Vol. 1.’’ fib, Lausanne. Hordijk, D.A. (1992). ‘‘Tensile and tensile fatigue behaviour of concrete: experiments, modelling and analyses.’’ Heron, 37(1), 1–79. Hsu, T.T.C. & Zhu, R.R.H. (2002). ‘‘Softened Membrane Model for Reinforced Concrete Elements in Shear.’’ ACI Structural Journal, 99(4), 460–469. Kaufmann, W. & Marti, P. (1996). ‘‘Versuche an Stahlbetonträgern unter Normal- und Querkraft.’’ Swiss Federal Institute of Technology Zürich, Zürich. Krätzig, W.B. & Pölling, R. (2004). ‘‘An elasto-plastic damage model for reinforced concrete with minimum number of material parameters.’’ Computer & Structures, 82, 1201–1215. Kupfer, H.B. & Gerstle, K.H. (1973). ‘‘Behaviour of Concrete Under Biaxial Stresses.’’ Journal of Engineering Mechanics Division, 99(EM4), 853–866. Li, B., Maekawa, K. & Okamura, H. (1989). ‘‘Contact Density Model for Stress Transfer across Cracks in Concrete.’’ Journal of the Faculty of Engineering of Tokyo, XL(1), 9–52. Maekawa, K. & Okamura, H. (1983). ‘‘The Deformational Behavior and Constitutive Equation of Concrete Using the Elasto-Plastic and Fracture Model.’’ Journal of the Faculty of Engineering of Tokyo, XXXVII(2). Maekawa, K., Pimanmas, A. & Okamura, H. (2003). Nonlinear Mechanics of Reinforced Concrete, Spoon Press, London. Markeset, G. & Hillerborg, A. (1995). ‘‘Softening of concrete in compression—Localization and size effects.’’ Cement and Concrete Research, 25(4), 702–708. Marti, P., Alvarez, M., Kaufmann, W. & Sigrist, V. (1998). ‘‘Tension Chord Model for Structural Concrete.’’ Structural Engineering International, 98(4), 287–298. Pimentel, M. (2010). ‘‘Advanced numerical modelling applied to the safety examination of existing concrete bridges,’’ Doctoral Thesis, Faculty of Engineering of the University of Porto, Porto. Pimentel, M., Brühwiler, E. & Figueiras, J. (2010). ‘‘Extended cracked membrane model for the analysis of RC panels.’’ Engineering Structures, (Submitted for publication). Vecchio, F.J. (2000). ‘‘Disturbed Stress Field Model for Reinforced Concrete: Formulation.’’ Journal of Structural Engineering, 126(9), 1070–1077. Vecchio, F.J. & Collins, M.P. (1986). ‘‘The Modified Compression-Field Theory for Reinforced Concrete Elements Subjected to Shear.’’ ACI Journal, 83(2), 219–231.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

A partition of unity finite element method for fibre reinforced concrete F.K.F. Radtke, A. Simone & L.J. Sluys Faculty of Civil Engineering and Geosciences, Delft University of Technology, Delft, The Netherlands

ABSTRACT: In this contribution we present a partition of unity based approach to model discrete fibres embedded in a matrix material without discretization of the fibres. For that purpose, the fibres are superimposed on a background mesh. Fibres are incorporated into the model by enriching the displacement field. We develop an element that incorporates fibre material characteristics, geometrical fibre position, bonding between fibre and matrix and matrix material behaviour. For the constitutive behaviour of the matrix simple fracture energy regularised isotropic damage with an exponential softening law is used. The behaviour of the fibre-matrix bond follows a non-linear relation. The performance of the approach and the influence of different numerical and material parameters are investigated. 1

INTRODUCTION

Fibres are employed in a number of materials to improve their mechanical behaviour. Notable examples are fibre reinforced concrete and fibre reinforced polymers. With the extensive use of these materials, there is an increasing need for understanding the influence of fibres and for predicting the interaction between the micro-structure of the fibre reinforced material and the mechanical properties of the composite. Finite elements provide a very flexible and easy to use tool for this purpose. Unfortunately, accurate analysis of fibre reinforced materials are commonly unfeasible due to the prohibitive costs of the meshing process. Here, we present an approach based on the partition of unity property of finite elements that allows the inclusion of an arbitrary number of discrete fibres in an element without meshing them.

2

APPROACH

Fibres are incorporated into finite elements by employing the partition of unity properties of the finite element method (Duarte et al. 2000) (pufem approach). By means of a suitable enrichment function it is possible to superimpose discrete fibres on a background mesh as depicted in Figure 1. The action of the fibres is represented by the enrichment function χ as depicted in Figure 2. The enrichment function is equal to one at the fibre and zero elsewhere. We make the following main assumptions: 1) the fibre thickness is small; 2) the interface between fibre and matrix is imperfect, thus a relative deformation between matrix and fibre may occur already from the beginning of the simulation; 3) all fibres are fully embedded in the matrix material. Since we treat fibre, matrix and

fibre matrix interface independently we have to specify a constitutive relation for each of them. The matrix is modelled using simple fracture energy regularised isotropic damage with an exponential softening law (Jirásek and Patzák 2002). The tangential component of the fibre matrix bond is described using a relation given by (Häußler-Combe and Hartig 2007), shown in Figure 3, which has been modified including secant unloading. The normal component of the bond law is chosen such that a normal opening of the gap between matrix and fibre is prevented. No coupling between normal and tangential component is assumed. The fibre is considered linear elastic.

3

VALIDATION OF THE PUFEM APPROACH IN THE LINEAR ELASTIC RANGE

We compare the proposed pufem approach to the shear lag theory as described in Bentur and Mindess (2006) for fibre reinforced concrete to validate the results in the linear elastic range. For that purpose the shear stress along the fibre and the normal stress in the fibre are computed following both approaches. Contrary to the pufem approach the shear lag theory assumes perfect bond between fibre and matrix. It does not account for an interface allowing for slip between fibre and matrix, while this is one of the basic assumptions of the pufem approach (see Section 2). Thus, to compare both approaches it is necessary to consider an almost rigid interface in the pufem approach to minimise slip along the fibre. The influence of the interface stiffness is studied in Section 3.2 in more detail. Furthermore, the analytical solution neglects the influence of the fibre on the overall strain field of the matrix. But as long as this influence is limited, the results of both approaches can be compared.

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Figure 1.

matrix

fibres

fibre reinforced concrete

background mesh

discrete fibres

fibres do not coincide with mesh

Discrete fibres distributed in a continuum matrix discretised with a background mesh.

thick fibre

thin fibre mechanical system

(a)

matrix

F

F

enrichment function

(b) dfib Figure 2.

dfib

Enrichment function χ for thick and thin fibres being pulled out from a matrix material.

elastic

debonding

at its left end and pulled at its right end. The test is performed using displacement control with the right end of the sample being pulled by 0.1 mm. This leads to a theoretical strain in the sample of

frictional

(stmax , ttmax )

tt

ε11 = εm =

(stres , ttres ) unloading branch st Figure 3. Relation between the tractions tt and the relative deformation between fibre and matrix st describing the tangential component of the fibre-matrix bond.

Test setup

For the comparison between the pufem approach and the analytical solution given by the shear lag theory a test setup as shown in Figure 4 is used. The fibre is embedded in the centre of a sample which is fixed

0.1 mm = 0.01, 10 mm

which is used as input for the analytical solution. The fibre has a length of 6 mm while the specimen has a length of 10 mm. The distances between fibre and sample end have been chosen such that the influence of boundary effects on the stresses in the fibre are minimised (Goh et al. 2004). The fibre has a diameter of 0.025 mm. Poisson’s ratio of matrix and fibre are chosen to be 0.2. Matrix, fibre and interface are linear elastic. 3.2

3.1

0

Influence of the stiffness of the fibre matrix interface in comparison to the analytical solution

In this example the influence of the stiffness of the fibre matrix interface in the pufem approach is studied in comparison to the shear lag theory assuming a rigid

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1

fibre

2

Figure 4.

6

2

[mm]

Test setup for the comparison between analytical solution and pufem approach. 8

analytical

6 4

/

0

2 0

pufem

Dbt = 10

3

N/mm

2

pufem

Dbt = 10

4

N/mm

2

pufem

Dbt = 10

5

N/mm

2

pufem

Dbt = 10

6

N/mm

2

Dbt = 10

7

N/mm

2

pufem

0

0.1

0.2

0.3

0.4

0.5 x / lf

0.6

0.7

0.8

0.9

1

/

0

0

0.95

1

x / lf

Figure 5. Influence of the stiffness of the fibre matrix interface on the shear stress at the fibre matrix interface in comparison to the analytical solution; x-axis is normalised with respect to the fibre length; y-axis is normalised with respect to the maximum shear stress in the interface following the shear lag theory.

interface. In the pufem approach the stiffness of the fibre matrix interface is varied from Dbt = 103 N/mm2 to Dbt = 107 N/mm2 . The test setup has been described in the previous section. Young’s modulus of matrix and fibre are chosen to be equal (Ef = Emat = 20000 N/mm2 ). The shear stresses along the fibre are plotted in Figure 5. The x-axis is normalised with respect to the fibre length while the y-axis is normalised with respect to the maximum shear stress from the analytical solution. Regarding the shear stress along the fibre the influence of the interface stiffness and the difference between the different approaches is barely visible. In the detail shown in the lower part of Figure 5 it can be seen that the increase of the stiffness of the fibre matrix interface leads to higher shear stresses and higher gradients at the fibre ends, which seems to agree with experimental findings as described in (Bentur and Mindess 2006).

The normal stresses in the fibre are plotted along the fibre axis in Figure 6. The x-axis is normalised as before. The y-axis is normalised with respect to the maximum normal stress computed with the shear lag theory. The weak interface allows for rather large amounts of slip that lead to a less steep increase of the normal stresses in the fibre. The higher the interface stiffness the more the pufem approach approximates the analytical solution. But naturally the pufem approach always considers some small amount of slip in the interface, while the analytical solution does not. An interesting aspect is that the pufem approach predicts higher stresses in the fibre than the analytical solution does. This is due to the fact that the local strain field around the fibre endpoints is influenced by the fibre. This is not taken into account by the analytical solution but occurs in measurements of real systems (Bentur and Mindess 2006).

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3.3

maximum shear stress following from the analytical solution for the case of Ef /Em = 1. For all cases the differences between the pufem approach and the analytical solution are small apart from the endpoints where the same phenomenon can be observed as has been described already in Section 3.2: the pufem approach yields higher values than the shear lag theory, which agrees with experimental results according to Bentur and Mindess (2006). The plots of the normal stresses in the fibre as given in Figure 8 show more clearly the influence of the stiffness variation. The x-axis is normalised with

Comparison of different ratios between fibre and matrix Young’s modulus

We compare the response of the sample using different ratios between fibre and matrix stiffness computed with the pufem approach to the response following from the analytical solution. The interface stiffness is set to Dbt = 107 N/mm2 . We vary the ratio between fibre and matrix Young’s modulus Ef /Em from 1 to 8. The shear stress distributions are plotted in Figure 7. The x-axis is normalised with respect to the fibre length while the y-axis is normalised with respect to the

1.2

analytical

1 0.8

pufem

Dbt = 10

3

N/mm

2

pufem

Dbt = 10

4

N/mm

2

Dbt = 10

N/mm

2

0.6

pufem

Dbt = 10

6

N/mm

2

pufem

Dbt = 10

7

N/mm

2

0.4

f

/

0

pufem

5

0.2 0

0

0.2

0.4

0.6

0.8

1

x / lf

Figure 6. Influence of the stiffness of the fibre matrix interface on the normal stress in the fibre in comparison to the analytical solution; x-axis is normalised with respect to the fibre length; y-axis is normalised with respect to the maximum normal stress in the fibre following the shear lag theory. 15 10

analytical pufem

Ef /Em =2

analytical pufem

Ef /Em =4

analytical pufem

Ef /Em =8

analytical pufem

/

0

5

Ef /Em =1

0

0

0.2

0.4

0.6

0.8

1

x / lf 0

/

0

analytical solution

pufem approach

0.975

1

x / lf

Figure 7. Comparison of the shear stress at the fibre interface computed following the shear lag theory and the pufem approach for different ratios between fibre and matrix Young’s modulus; x-axis is normalised with respect to the fibre length; y-axis is normalised with respect to the maximum shear stress following the shear lag theory for the case of an equal Young’s modulus in fibre and matrix.

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9

analytical pufem

Ef / Em = 8

8 7

0

5

f

6

4

Ef / Em = 4

3

Ef / Em = 2

2 Ef / Em = 1

1 0

0

0.2

0.4

0.6

0.8

1

1.2

x / lf

Figure 8. Comparison of the normal stress in the fibre computed following the shear lag theory and the pufem approach for different ratios between fibre and matrix Young’s modulus; x-axis is normalised with respect to the fibre length; y-axis is normalised with respect to the maximum normal stress in the fibre following the shear lag theory for the case of an equal Young’s modulus in fibre and matrix.

respect to the fibre length. The y-axis is normalised using the maximum normal stress in the fibre for a ratio between fibre and matrix stiffness of 1. The results show good agreement between the analytical and the pufem solution. Only for a ratio Ef /Em = 8 the results differ considerably. This is due to the fact that in this case the fibre influences the strain field of the sample not only locally around the fibre endpoints, but also globally by stiffening the sample in the centre part. This basically means that the fibre is pulled less than assumed in the analytical solution and thus the normal stress in the fibre computed with the pufem approach is lower than the one following from the analytical solution. In general, the pufem approach and the shear lag theory compare well if the fibre matrix interface in the pufem approach is chosen sufficiently stiff. Differences are mainly due to the fact that the analytical solution does not take into account the influence of the fibre on the strain field of the sample—neither locally around the fibre endpoints nor globally on the strain field further away from the fibre. The higher shear stresses produced by the pufem approach at the fibre endpoints seem to be in better agreement with reality than the ones coming from the shear lag theory (Bentur and Mindess 2006).

4

EXAMPLES USING NON-LINEAR CONSTITUTIVE EQUATIONS

We present a comparison of three different fibre distributions as shown in Figure 9, each sample containing 9 fibres. One specimen contains only parallel, horizontal

fibres while the fibres in distribution 1 and 2 are arbitrarily placed in the sample. The samples are squares of 4 mm length. The system is discretised with 625 bilinear quadrilateral elements. The sample is fixed at its left boundary and pulled at its right boundary. The results are given in Figures 9 and 10. To begin with, we study the linear elastic case. The matrix has a Young’s modulus of 20000 N/mm2 and Poisson’s ratio of 0.2. The fibre is 0.2 mm thick, it has a Young’s modulus of 500000 N/mm2 and a Poisson’s ratio of 0.2. The interface stiffness is set to Dbt = 50000 N/mm2 . Naturally, the sample containing 9 horizontal fibres is stiffer than the other samples. The stiffer response of the second distribution compared to the first distribution can be explained by examining the main orientation of the fibres in the samples with respect to the horizontal loading direction: in distribution 2 more fibres are orientated horizontally than in distribution 1 leading to a higher influence of the fibres. Next, the influence of the nonlinear bond law is examined while the matrix material is kept linear elastic. As shown in Figure 3 the following parameter are chosen: stmax = 10−6 mm, ttmax = 50 N/mm2 , stres = 10−3 mm and ttmax = 2 N/mm2 . The debonding process and the frictional pull-out stage can be observed for all samples. Again, the horizontal fibre sample shows the stiffest reaction. Furthermore, distribution 1 is slightly less ductile than distribution 2. This shows the influence of the orientation of the fibres with respect to the loading direction. Compared to the linear elastic example the influence of the fibres diminishes with the use of the nonlinear bond law. This is due to the shape of the bond law as shown in Figure 3. While

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linear elastic case 160 no fibres horizontal fibres distribution 1 distribution 2

F [N]

120

80 mechanical system

u

40

0

0

0.004

0.008 u [mm]

0.012

0.016

horizontal fibres

F [N]

120

80 no fibres horizontal fibres distribution 1 distribution 2

40

0

0

0.004

0.008 u [mm]

0.012

distribution 1

0.016

distribution 2

no fibres horizontal fibres distribution 1 distribution 2

F [N]

20

10

0

Figure 9. bond.

0

0.004

0.008 u [mm]

0.012

0.016

Comparison of three fibre distributions employing nonlinear constitutive behaviour for matrix and fibre-matrix

the interface is rather stiff in the beginning, its stiffness decreases rapidly after the fibre has been pulled out of the matrix for a distance larger than a specified peak value stmax . But as soon as the interface weakens, the load carrying capacity of the fibre as part of the composite material reduces. Finally, we allow damage growth in the matrix material. The matrix strength is set to 5 N/mm2 . In the middle of the sample a vertical zone of weak material is assumed to have a strength of 2 N/mm2 . The pure matrix sample responds in a very brittle fashion as can bee seen in the force-displacement plot shown in the bottom of Figure 9. The addition of fibres not only leads to a strong increase in peak strength but also to a more ductile post-peak behaviour. Furthermore, the results clearly show the influence of the discrete fibre distributions. All fibre samples develop

the same strength which is mainly determined by the matrix strength outside of the weak zone in the middle of the sample. While the horizontal fibre sample shows the stiffest behaviour in the pre-peak part, fibre distribution 2 behaves clearly more ductile in the post peak part. Distribution 1 is only slightly more ductile than the horizontal fibre sample. Apart from the fact that in the horizontal fibre sample all fibres undergo the debonding process simultaneously while in the arbitrarily distributed fibre samples some fibres are already debonding while others are still in the elastic branch the described behaviour can be explained by considering the damage patterns as depicted in Figure 10. Without fibres damage localises in the weak part in the centre of the sample. Adding horizontal fibres leads to failure at the supports of the sample. Damage localises at the support which is

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no fibres

horizontal fibres

distribution 1

distribution 2

ω Figure 10.

0

1

Comparison of damage patterns of three different fibre distributions (damage plots are smoothed).

pulled while only a small amount of damage occurs at the clamped support. Regarding distribution 1 some damage occurs at the weak zone in the middle of the sample although localisation takes place at the right support of the sample. The amount of damage is only slightly higher than in the sample with horizontal fibres. Thus only a minor increase in ductility is visible in Figure 9 between the horizontal fibre sample and distribution 1. In distribution 2 damage occurs at both supports of the sample and parts of the weak zone in the middle of the specimen. This leads to a higher energy distribution and thus to an increased ductility compared to the other samples. For all cases, the purely linear elastic setting, the nonlinear fibre bond, and the nonlinear fibre bond in combination with damage in the concrete matrix,

the influence of the discrete fibre distribution on the mechanical response of fibre reinforced concrete and thus the importance to include discrete fibres in the study of fibre reinforced materials has clearly been shown in this example.

5

CONCLUSIONS

We have presented an approach based on the partition of unity finite element method in connection with damage mechanics. This approach enables us to study the influence of discrete fibre distributions in a sample on the mechanical properties of the material. The computationally expensive mesh generation process

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of each fibre is avoided. Since we treat fibre reinforced concrete as a composite material consisting of concrete matrix, fibres and interfaces between fibres and matrix, the constitutive behaviour of each constituent can be directly taken into account. This enables a detailed study of the mechanical behaviour of fibre reinforced concrete or other fibre reinforced materials. REFERENCES Bentur, A. and S. Mindess (2006). Fibre reinforced cementitious composites (2. ed.). Taylor & Francis.

Duarte, C., I. Babuška, and J. Oden (2000). Generalized finite element methods for three-dimensional structural mechanics problems. Compt. Struct. 77, 215–232. Goh, K., R. Asden, K. Mathias, and D. Hukins (2004). Finite–element analysis of the effect of material properties and fibre shape on stresses in an elastic fibre embedded in an elastic matrix in a fibre–composite material. Proc. R. Soc. Lond. A. 460, 2339–2352. Häußler-Combe, U. and J. Hartig (2007). Bond failure mechanisms of textile reinforced concrete (TRC) under uniaxial tensile loading. Cem. Concr. Compos. 29, 279–289. Jirásek, M. and B. Patzák (2002). Consistent tangent stiffness for nonlocal damage models. Compt. Struct. 80, 1279–1293.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

A discrete cracking model for sequentially linear analysis A.V. van de Graaf, M.A.N. Hendriks & J.G. Rots Delft University of Technology, Delft, The Netherlands

ABSTRACT: Over the past few years it has been shown that the sequentially linear analysis scheme is able to model fracture in quasi-brittle materials in an effective manner. So far, in nearly all publications a smeared cracking model was adopted. However, some phenomena can be modeled in a better way by using discontinuum interface elements. Consider for instance the case of a pull-out test in which the bond-slip relation plays an important role. The aim of this paper is to start the implementation of this family of finite elements by considering the simplest case, viz. a discrete cracking model. It is demonstrated that results obtained with the new implementation match the results obtained with finite element analyses based on an incremental-iterative scheme. It is also shown that a saw-tooth softening law constructed with the so-called ripple model gives smoother results in terms of load-displacement diagrams than a saw-tooth softening law constructed with a model based on equal energy dissipation per damage increment. 1

INTRODUCTION

Numerical simulation of fracture in quasi-brittle materials by means of a finite element analysis based on an incremental-iterative scheme can be a real hassle, in particular for large-scale structures. In these cases it may be hard to obtain a properly converged solution. To address the problem of poor convergence, Rots (2001) introduced the sequentially linear analysis (SLA) technique which was later elaborated by Rots & Invernizzi (2004), DeJong et al. (2008) and DeJong et al. (2009). In nearly all the work that followed, a smeared cracking model was adopted. Rots et al. (2006b) have initiated the development of a sawtooth softening model for interface elements, but was never continued nor extended to fields other than discrete cracking. The aim of this paper is to resume this work by reproducing the discrete cracking model for sequentially linear analysis and to present a new model to set up saw-tooth softening laws based on equal energy dissipation per damage increment. 1.1

Brief review of the sequentially linear analysis procedure

integration point the maximum stress remains smaller than the respective strength. Next, a damage increment is applied to the critical integration point which comes down to an instantaneous reduction in stiffness and strength of that point in the next linear analysis. This implies that the constitutive law needs to be discretized, resulting in a so-called saw-tooth softening law. Summarizing, the outlined procedure amounts to continuously executing the following steps: 1. Perform a linear-elastic analysis with a representative (unit) load. 2. Determine which integration point has the largest maximum stress over tensile strength ratio (i.e. identify the critical integration point). 3. Multiply the representative (unit) load by the inverse of the largest ratio of step 2. In this way a critical stress state is obtained. 4. Update the stiffness and strength properties of the critical integration point in accordance with the adopted saw-tooth softening law.

2

Traditionally physically nonlinear finite element analysis is carried out by applying an incremental-iterative scheme. The basic concept of SLA is to replace this scheme by a series of scaled linear analyses. After each linear analysis, the applied loads are to be multiplied by a scalar λ which has to be chosen such that a critical stress state is obtained. This means that in one integration point—the so-called critical integration point—the maximum stress equals the current strength of that point whereas in every other

2.1

CRACK SIMULATION WITH INTERFACE ELEMENTS: A DISCRETE CRACKING MODEL Basic formulation

Interface elements can be used in a finite element model to anticipate geometrical discontinuities due to fracture or debonding processes. For these structural interface elements the relative displacements across the interface u are related to the tractions t acting at both sides of the interface. In a two-dimensional

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model the relative displacement vector u consists of two components   un , (1) u = ut

tn 3

ft

2.5

mother curve 2 1.5

where un represents a crack opening displacement and ut a crack sliding displacement. Similarly, the traction vector t consists of the components   t t= n , (2) tt where tn and tt are the normal and tangential tractions respectively. The relation between these two vectors can be written compactly as t = Du,

(3)

where D is the constitutive matrix which—in case of discrete cracking without dilatancy—reads   0 k , (4) D= n 0 kt

p1 ×fft t

1 0.5

p2 ×fftt

kn( i )

0 0

0.02

0.04

0.06

un 0.08

0.1

un;ult

Figure 1. Saw-tooth model 1 which is based on a fixed band around the mother curve.

and p2 (see Fig. 1) are determined from an iterative procedure such that the following two conditions are fulfilled. First, the obtained saw-tooth softening law should be invariant with respect to the fracture energy GfI , i.e. GfI − GfI ∗ GfI

= 0,

(7)

with kn the normal stiffness and kt the shear stiffness. 2.2

Crack initiation and growth criterion

In the context of SLA a crack arises or grows if for one and only one integration point the following equation holds ∗

tn (λ) = ft ,

where GfI ∗ is the area enclosed by the saw-tooth softening law. Simultaneously, the obtained saw-tooth softening law should be invariant with respect to the ultimate relative displacement un;ult , i.e. ∗ un;ult − un;ult

un;ult

(5)

where ft∗ equals the current tensile strength of the integration point under consideration. For all other integration points the following inequality should hold tn (λ) < ft∗ .

(6)

SAW-TOOTH DISCRETISATION OF A LINEAR STRAIN-SOFTENING LAW

In this section two ways to set up a saw-tooth softening law are presented. The first saw-tooth softening model is a slightly modified version of the ripple model which is based on a fixed band around the mother curve (Rots et al. 2006a). The second saw-tooth model is based on equal energy dissipation per damage increment. 3.1

Model 1: Saw-tooth model based on a fixed band around the mother curve (ripple model)

The input parameters for this model are the initial dummy stiffness kn(1) and the desired number of sawteeth N . Then the unknown band width parameters p1

(8)

∗ where un;ult is the ultimate relative displacement of the saw-tooth softening law. Note that both conditions (7) and (8) have been normalized to ease the iterative procedure for finding p1 and p2 .

3.2 3

= 0,

Model 2: Saw-tooth model based on equal energy dissipation per damage increment

Starting point of this new saw-tooth softening model is that the area under the softening curve—which equals the mode I fracture energy GfI —is split in N triangles of equal area (see Fig. 2). In other words, after each damage increment an equal amount of fracture energy GfI is dissipated, assuming secant unloading. Then for every triangle the relative displacement of its rightmost vertex un(i) is calculated. These values are used as input for the saw-tooth softening model. Starting from the ultimate relative displacement a new series of triangles is constructed. The height of every triangle—which effectively comes down to a traction drop—is calculated from the required triangle area and the corresponding relative displacement. Every triangle is

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225

tn 3

225 F

2.5

100

2

G = I f

1.5

G If N

1

t = 100 mm

0.5

E = 30,000 N/mm2

un

0 0

0.02

0.04

0.06

(a)

0.08

50

= 0.2

0.1

5

un(i )

tn

Figure 3. Model setup of the analyzed single-edge-notched beam. All dimensions are in mm.

3 2.5 2

G If * = G If

1.5 1

(a) 0.5

un

0 0

0.02

0.04

0.06

0.08

0.1

(b) (b)

Figure 2. Saw-tooth model 2 which is based on equal energy dissipation per damage increment. Divide the area under the softening curve in N equal parts of GfI and calculate un(i) (i)

for every triangle (a). Then use un as a basis to construct a new series of triangles with an individual area of GfI ∗ (b).

(c)

positioned such that it fits exactly on top of the previous triangle (see Fig. 2b).

4

4.1

(d)

Figure 4. Adopted meshes: medium meshes with linear (a) and quadratic (b) elements and fine meshes with linear (c) and quadratic (d) elements.

NUMERICAL SIMULATION OF A THREE-POINT-BEND TEST ON A NOTCHED BEAM Beam model and meshes

The effectiveness of the presented discrete cracking model is demonstrated by numerical simulation of a three-point-bend test on a single-edge-notched beam (Grassl & Jirásek, 2005). The simply supported beam has a span of 450 mm, a depth of 100 mm and a thickness of 100 mm (see Fig. 3). At midspan a notch of 50 mm deep and 5 mm wide is present. The unreinforced concrete has a Young’s modulus of 30,000 N/mm2 , a Poisson’s ratio of 0.2 and has been modeled with triangular plane stress elements. The anticipated crack has been indicated in Figure 3 with a dashed line and has been modeled with line interface elements. To investigate the objectiveness of the obtained solution with respect to the chosen mesh, four meshes have been applied (see Fig. 4). Variations regard the mesh size

(10 interface elements for the medium mesh vs. 20 interface elements for the fine mesh) and the element shape functions (linear versus quadratic). Table 1 shows the characteristics of each mesh. 4.2

Adopted saw-tooth softening laws

For the adopted saw-tooth softening laws linear softening was assumed with a tensile strength ft of 2.4 N/mm2 and a mode I fracture energy GfI of 113 J/m2 (see Fig. 5). The ultimate relative displacement un;ult was obtained as (2*GfI )/ft ≈ 0.094 mm. To investigate the effect of the adopted saw-tooth softening law six laws have been set up. Variations regard the adopted saw-tooth model (the models of Sections 3.1 and 3.2) and the number of saw-teeth (25, 50 and 100). See Table 2 for an overview

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Table 1.

Characteristics of the adopted meshes.

Label

Mesh refinement

Mesh 1 Mesh 2 Mesh 3 Mesh 4

tn

Element shape functions

Medium Medium Fine Fine

25 teeth kn(1) = 1.0 104 N/mm3

2.5

Linear* Quadratic** Linear* Quadratic**

p1 = 12.3%

2

p2 = 12.0% 1.5

* Numerical integration of the line interface elements was carried out with a two-point Newton-Cotes scheme. ** Numerical integration of the line interface elements was carried out with a three-point Newton-Cotes scheme.

p1 ft

1 0.5

tn

p2 ft

kn(i )

0 0

0.02

un 0.04

0.06

0.08

0.1

(a)

3

ft = 2.4 N/mm 2

ft

Saw-tooth law A

3

2.5

tn

G If = 0.113 Nmm/mm 2 un ;ult

2

Saw-tooth law B

3

50 teeth kn(1) = 1.0 104 N/mm3

0.094 mm 2.5

1.5

p1 = 6.16%

2

p2 = 5.91%

1

G If

1.5

0.5

1

un

0 0

0.02

0.04

0.06

0.08

0.1

0.5

un;ult

un

0

Figure 5. For the line interface elements a linear softening relation relation was assumed.

0

0.02

Characteristics of adopted saw-tooth softening Number of saw-teeth

Saw-tooth law A Saw-tooth law B Saw-tooth law C Saw-tooth law D Saw-tooth law E Saw-tooth law F

Model 1* Model 1 Model 1 Model 2** Model 2 Model 2

25 50 100 25 50 100

0.1

100 teeth kn(1) = 1.0 104 N/mm3

2.5

Saw-tooth model

0.08

Saw-tooth law C

3

Label

0.06

(b)

tn Table 2. laws.

0.04

p1 = 3.13%

2

p2 = 2.88% 1.5 1 0.5

* Based on the ripple model. ** Based on equal energy dissipation per damage increment.

of the characteristics of each saw-tooth softening law. Figures 6 and 7 show the adopted saw-tooth softening laws. Note that the initial dummy stiffness cannot be chosen freely for saw-tooth model 2. Figure 8 shows the cumulative energy dissipation as a function of the relative displacement un for both models. Note that for model 1 the dissipated energy per damage increment increases with un whereas for model 2 it is an equal amount per damage increment. 4.3 Analysis results An overview of all performed analyses is given in Table 3. For comparison reasons, also nonlinear finite

un

0 0

0.02

0.04

0.06

0.08

0.1

(c)

Figure 6. Saw-tooth softening laws which have been constructed with the saw-tooth model of Section 3.1 with 25 (a), 50 (b) and 100 (c) saw-teeth. For all saw-tooth laws an initial dummy stiffness of 1.0 × 104 N/mm3 was used.

element analyses (NLFEA) have been carried out using an incremental-iterative scheme. Figure 9 shows the load-displacement curves obtained with a medium mesh consisting of linear elements. The applied saw-tooth softening laws have been based on the ripple model. Obviously, all models are able to capture the course of the nonlinear finite element analysis sufficiently accurate. It is also clear that if more saw-teeth are adopted a smoother structural response

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tn

(

Saw-tooth law D

4 3.5 3

G If ) [Nmm/mm 2 ]

0.12

25 teeth kn(1) = 1.023 ×103 N/mm3

0.08

G If = 4.52 × 10 3 Nmm/mm 2

0.06

Saw-tooth law A

0.1

2.5 0.04

2 0.02

G If

1.5

un [mm]

0 0

1

G

I f

(

0.5

0.02

0.05

0.075

0.1

0.04

0.06

0.08

0.05

0.075

0.1

G If ) [Nmm/mm 2 ]

0.12

un

0 0

0.025

(a)

0.1

Saw-tooth law D

0.1

(a) 0.08

tn

0.06

Saw-tooth law E

4 3.5

k 3

(1) n

0.04

50 teeth = 2.071× 103 N/mm3

G = 2.26 × 10 Nmm/mm I f

3

0.02

un [mm]

0

2

0

2.5

0.025

(b)

2

Figure 8. Cumulative energy dissipation (GfI ) as a function of the relative displacement un for saw-tooth model 1 (a) and saw-tooth model 2 (b).

1.5 1 0.5

un

0 0

0.02

0.04

0.06

0.08

Table 3.

(b)

tn

Saw-tooth law F

4 3.5

kn(1)

3

100 teeth = 4.167 ×103 N/mm3

G If = 1.13 × 10 3 Nmm/mm 2

2.5 2 1.5 1 0.5

un

0 0

0.02

0.04

0.06

0.08

Overview of all performed analyses.

0.1

0.1

(c)

Figure 7. Saw-tooth softening laws which have been constructed with the saw-tooth model of Section 3.2 with 25 (a), 50 (b) and 100 (c) saw-teeth. The initial dummy stiffness depends on the number of saw-teeth applied.

is obtained. Figure 10 shows the results for the same mesh, but for these analyses the applied saw-tooth softening laws have been based on equal energy dissipation per damage increment. From Figure 10a it is clear that the initial dummy stiffness of saw-tooth law D is not large enough compared to the stiffness of the surrounding continuum elements. Also note that the obtained curves are less smooth compared to those of Figure 9 and that a few

Label

Adopted mesh

Adopted saw-tooth law

SLA01 SLA02 SLA03 SLA04 SLA05 SLA06 SLA07 SLA08 SLA09 SLA10 SLA11 SLA12 SLA13 SLA14 SLA15 SLA16 SLA17 SLA18 SLA19 SLA20 SLA21 SLA22 SLA23 SLA24

Mesh 1 Mesh 2 Mesh 1 Mesh 2 Mesh 1 Mesh 2 Mesh 3 Mesh 4 Mesh 3 Mesh 4 Mesh 3 Mesh 4 Mesh 1 Mesh 2 Mesh 1 Mesh 2 Mesh 1 Mesh 2 Mesh 3 Mesh 4 Mesh 3 Mesh 4 Mesh 3 Mesh 4

Saw-tooth law A Saw-tooth law A Saw-tooth law B Saw-tooth law B Saw-tooth law C Saw-tooth law C Saw-tooth law A Saw-tooth law A Saw-tooth law B Saw-tooth law B Saw-tooth law C Saw-tooth law C Saw-tooth law D Saw-tooth law D Saw-tooth law E Saw-tooth law E Saw-tooth law F Saw-tooth law F Saw-tooth law D Saw-tooth law D Saw-tooth law E Saw-tooth law E Saw-tooth law F Saw-tooth law F

minor snap-backs are observed as well as a small bump around a displacement of 0.35 mm. In Figures 11 and 12 the load-displacement curves obtained with a medium mesh and quadratic elements are shown for different numbers of saw-teeth. For the

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2

2

nonlin NLFEA

nonlin NLFEA sla0 SLA01

sla0 SLA13

1.5

Load [kN]

Load [kN]

1.5

1

1

0.5

0.5

0

0 0

0.25

(a)

0.5

0.75

0

1

0.25

0.5

1

2

2

nonlin NLFEA

nonlin NLFEA sla0 SLA03

sla0 SLA15

1.5

Load [kN]

1.5

Load [kN]

0.75

Displacement [mm]

(a)

Displacement [mm]

1

1

0.5

0.5

0

0

0

0

0.25

0.5

0.75

0.25

(b)

(b)

0.5

1

Displacement [mm]

Displacement [mm] 2

2

nonlin NLFEA

nonlin NLFEA

sla0 SLA17

1.5

sla0 SLA05

Load [kN]

1.5

Load [kN]

0.75

1

1

1

0.5

0.5 0 0

0 0

0.25

0.5

0.75

1

(c)

0.25

0.5

0.75

1

Displacement [mm]

(c)

Figure 9. Load-displacement diagrams for a medium mesh with linear elements and saw-tooth softening laws produced with model 1 using 25 (a), 50 (b) and 100 (c) saw-teeth.

Figure 10. Load-displacement diagrams for a medium mesh with linear elements and saw-tooth softening laws produced with model 2 using 25 (a), 50 (b) and 100 (c) saw-teeth.

curves in Figure 11 the saw-tooth model of Section 3.1 was applied. Just like the curves in Figure 9, the curve obtained with the nonlinear finite element analysis is resembled well. Also here it holds true that when more saw-teeth are applied, a smoother response is obtained. Compared to linear elements, quadratic elements do not yield significantly better results, except for a slight reduction in the band width of the curve. Compared to Figure 10 the load-displacement curves of Figure 12 look better, particularly because the

strange little bump has now disappeared. Nevertheless, the minor snap-backs remain. So for saw-tooth model 2 switching to quadratic elements yields an improvement in response. Figure 13 reveals the effect of a mesh refinement. In that case the band of small jumps and drops—which is characteristic for results obtained with SLA— becomes even smaller. The almost perfect resemblance of the load-displacement curve obtained with nonlinear finite element analysis is remarkable. Note that the the curve in Figure 13a is slightly rougher than the curve in Figure 9b, despite that on average

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2

2 nonlin NLFEA

sla0 SLA02

sla0 SLA14

1.5

Load [kN]

Load [kN]

1.5

nonlin NLFEA

1

0.5

1

0.5

0

0 0

0.25

0.5

0.75

1

0

Displacement [mm]

(a)

0.25

(a)

2

0.5

sla0 SLA16

1.5

Load [kN]

Load [kN]

nonlin NLFEA

sla0 SLA04

1.5

1

0.5

1

0.5

0

0 0

0.25

0.5

0.75

1

0

Displacement [mm]

(b)

0.25

0.5

0.75

1

Displacement [mm]

(b)

2

2 nonlin NLFEA

nonlin NLFEA

sla0 SLA06

sla0 SLA18

1.5

Load [kN]

1.5

Load [kN]

1

2 nonlin NLFEA

1

0.5

1

0.5

0

0

0

(c))

0.75

Displacement [mm]

0.25

0.5

0.75

1

0

Displacement [mm]

(c)

0.25

0.5

0.75

1

Displacement [mm]

Figure 11. Load-displacement diagrams for a medium mesh with quadratic elements and saw-tooth softening laws produced with model 1 using 25 (a), 50 (b) and 100 (c) saw-teeth.

Figure 12. Load-displacement diagrams for a medium mesh with quadratic elements and saw-tooth softening laws produced with model 2 using 25 (a), 50 (b) and 100 (c) saw-teeth.

the same amount of energy—fracture energy times area per integration point—is dissipated per damage increment. The load-displacement curves in Figure 14 also show a smoothening of the structural response upon mesh refinement. However note that the peak value of the load is still slightly underestimated. Also note that a mesh refinement does not help to get rid of the minor snap-backs that we have seen before.

Figure 15 shows the obtained load-displacement curves for a fine mesh consisting of quadratic elements. As expected this combination gives the best results. In particular for Figure 15c it yields that there is no visible deviation between the curve obtained with SLA and the one obtained with NLFEA. Figure 16 shows that if saw-tooth laws based on the saw-tooth model of Section 3.2 are used the results are improving upon mesh refinement and an increasing number of saw-teeth. Nevertheless, it

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2

2 nonlin NLFEA

sla0 SLA07

sla0 SLA19

1.5

Load [kN]

Load [kN]

1.5

nonlin NLFEA

1

0.5

1

0.5

0

0 0

0.25

(a)

0.5

0.75

1

0

Displacement [mm] 2

nonlin NLFEA

sla0 SLA09

sla0 SLA21

1.5

Load [kN]

Load [kN]

0.75

nonlin NLFEA

1

0.5

1

1

0.5

0

0 0

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Figure 13. Load-displacement diagrams for a fine mesh with linear elements and saw-tooth softening laws produced with model 1 using 25 (a), 50 (b) and 100 (c) saw-teeth.

should be concluded that in all cases saw-tooth softening laws set up with the ripple model give the best results.

5

0.25

(a)

CONCLUSIONS

This paper has confirmed that sequentially linear analysis can be applied successfully to finite element models that contain interface elements with discrete cracking models. A new saw-tooth model has been presented based on equal energy dissipation per damage

0.25

0.5

0.75

1

Displacement [mm]

Figure 14. Load-displacement diagrams for a fine mesh with linear elements and saw-tooth softening laws produced with model 2 using 25 (a), 50 (b) and 100 (c) saw-teeth.

increment. Although the quality of the obtained structural response with this new saw-tooth model can be called reasonable, it was shown that the best results (in terms of smooth curves and resemblance of the curve obtained by an incremental-iterative approach) are obtained with the ripple model. Apparently, it is more important to resemble the original softening curve as closely as possible by adopting a constant stress overshoot line—like in the ripple model—than having equal energy dissipations per damage increment. Vice versa, the snap-backs in the global loaddisplacement curves, which are so typical for SLA

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2

2 nonlin NLFEA

sla0 SLA08

sla0 SLA20

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Figure 15. Load-displacement diagrams for a fine mesh with quadratic elements and saw-tooth softening laws produced with model 1 using 25 (a), 50 (b) and 100 (c) saw-teeth.

0.25

0.5

0.75

1

Displacement [mm]

Figure 16. Load-displacement diagrams for a fine mesh with quadratic elements and saw-tooth softening laws produced with model 2 using 25 (a), 50 (b) and 100 (c) saw-teeth.

REFERENCES results with relatively rough meshes and/or rough sawtooth diagrams, are more related to the variations of tensile strengths than to variations in dissipated energy per damage increment. Furthermore, for both saw-tooth models it was observed that a mesh refinement and/or an increase in the number of saw-teeth improved the quality of the obtained results.

DeJong, M.J, Belletti, B., Hendriks, M.A.N. & Rots, J.G. 2009. Shell elements for sequentially linear analysis: lateral failure of masonry structures. Engineering Structures 31(7): 1382–1392. DeJong, M.J., Hendriks, M.A.N. & Rots, J.G. 2008. Sequentially linear analysis of fracture under non-proportional loading. Engineering Fracture Mechanics 75(18): 5042–5056.

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Grassl, P. & Jirásek, M. 2005. Nonlocal damage-plasticity model for failure of plain concrete. Proc. 11th Int. Conf. Frac., Turin, Italy. Rots, J.G. 2001. Sequentially linear continuum model for concrete fracture. In R. de Borst, J. Mazars, G. PijaudierCabot & J.G.M. van Mier (eds), Fracture mechanics of concrete structures: 831–839. Rotterdam: Balkema. Rots, J.G., Belletti, B. & Invernizzi, S. 2006a. On the shape of saw-tooth softening curves for sequentially linear analysis. In G. Meschke, R. de Borst, H. Mang & N. Bi´cani´c (eds), Computational modelling of concrete structures: 431–442. Leiden: Taylor & Francis/Balkema.

Rots, J.G. & Invernizzi, S. 2004. Regularized sequentially linear saw-tooth softening model. Int. J. Numer. Anal. Meth. Geomech. 28: 821–856. Rots, J.G., Invernizzi, S. & Belletti, B. 2006b. A sequentially linear saw-tooth model for interface elements. In G. Meschke, R. de Borst, H. Mang & N. Bi´cani´c (eds), Computational modelling of concrete structures: 203–212. Leiden: Taylor & Francis/Balkema.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Relations between structure size, mesh density, and elemental strength of lattice models M. Voˇrechovský & J. Eliáš Institute of Structural Mechanics, Faculty of Civil Engineering, Brno University of Technology, Brno, Czech Republic

ABSTRACT: We study the effect of discretization of lattice models. Two basic cases are examined: (i) homogeneous lattices, where all elements share the same strength and (ii) lattices in which the properties are assigned to the elements according to their correspondence to three phases of concrete, namely matrix, aggregates, and the interfacial transitional zone. These dependencies are studied with both, notched and unnotched beams loaded in three point bending. We report the results for regular discretization and irregular networks obtained via Voronoi tessellation. This is done for two types of models: with and without rotational springs (normal and shear springs are always present). All the springs are ideally brittle, i.e. after reaching the strength criterion; they are irreversibly removed from the structure. The dependence of strength is compared to various size effect formulas, and we show that in the case of homogeneous lattices, the fineness of discretization of the specimens of the same size can mimic variations in the size of lattice models with the same discretization density. In the case of heterogeneity (ii), we report how both the peak force and fracture energy depend on the mesh resolution both for notched and unnotched structure. 1

INTRODUCTION

Lattice models are well established tool for fracture modeling and they appear to be very helpful especially thanks to the increasing power of modern computers. In classical lattice models, the material is represented by a set of discrete elements interconnected by springs. The combination of simple constitutive models with a material structure incorporated from the meso-level (Lilliu & van Mier 2003, Bolander et al. 1998) or by randomness of material parameters which somehow mimics this structure (Grassl & Bažant 2009, Alava et al 2008) makes it a powerful tool able to model quasibrittle structural response. It is an alternative to relatively complex constitutive laws applied in classical continuum models. The simplest models are those involving only elasto-brittle springs. This type of model is studied in this contribution. The weak point of using purely brittle springs is strong dependency of the results on a network density. Since the network does not represent any real underlying structure, this dependency is understood as a bias which should be removed. If one insists on keeping the brittleness of elements as we do (no softening of elements is incorporated), the mesh size dependency can be overcome e.g. by scaling the strength of elements according to their lengths and a chosen internal length parameter (Jagota & Bennison 1995) or, as is believed, by incorporating the material inhomogeneities (voids, grains, microcracks) that introduces an internal length as well. In this paper, we extend our recent results (Voˇrechovský & Eliáš 2009) by the effect of rotational

springs acting on facets of Voronoi cells. Correction functions for previously derived scaling formulas are found and compared with the results. We study both homogeneous and heterogeneous lattice models. By homogeneous models, we mean a lattice in which all elements share the same deterministic material strength criterion (and elastic modulus E). Otherwise, there are several ways to represent disorder or heterogeneity of material. This can be achieved e.g. (a) by spatial randomization of the properties of elements or, (b) by attributing element properties depending on their phase which is obtained by projecting a granular structure on the mesh. From here on, heterogeneous models are those obtained by alternative (b), i.e. by projecting the simulated meso-level material structure on the network and changing the properties based on the phase classification. Both types of models can be used with either regular (structured, REN) or irregular (IRN) geometry of the network (or mesh). In this paper, we use both types of discretization. If we speak of structured network (REN), we use unstructured meshes with a regular mesh in a certain small region of interest. In this work, we focus on the effect of varying the network density (or mesh density) on the overall structural response. We consider that varying the network density in homogeneous models corresponds to changes in structural size of the structure modeled. In other words, by changing the network density, we might model different sizes of the specimen. Several papers concerning the effect of network density have been published but, according to authors’ knowledge, two issues have not been studied yet:

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Figure 1. Specimens with a central notch: nTPBT (relative notch depth α = 1/3). Two types of meshes around a notch are presented: REN and IRN. The Delaunay triangulation corresponding to the dual graph of the Voronoi tessellation is illustrated for a given mesh density. The configuration of uTBPT is identical except for the missing notch.

(i) the effect of size on the strength of the homogeneous model with a random network geometry (IRN) and, (ii) the effect of grain microstructure projected on the specimen with varying network densities (the grain layout properties to be used to classify elements is kept, but the network density—mesh—is varied). These mesh-density effects are studied for both: specimens that fail by crack initiated from a smooth surface and notched specimens. In particular, we have performed numerous simulations with either notched or unnotched three-point-bent specimens (denoted either nTPBT or uTPBT). The geometry of the specimens is illustrated in Figure 1 top. The span S = 3D is equal to three times the specimen depth D. We have found this topic interesting because the irregularity of the network influences the strength unexpectedly. Not all the sources of the observed behavior were identified and analytically analyzed, thus, the contribution predominantly present our (mostly numerical) observations. The first part of the contribution describes briefly the model adopted. The following sections (3–6) are devoted to the observed size effects in homogeneous lattice models with both REN and IRN. The last part presents a short study describing how (whether) the meso-level concrete structure projected onto the model reduces these size (or mesh density) effects.

(1998). The fracture criteria are taken from the same article, i.e. Mohr-Coulomb surface with tension cutoff is adopted. More detailed description can be found in Eliáš (2009). In this paper, we study two different mechanical models that differ in how internal forces (between rigid bodies) are transmitted at the connections of adjacent facets. In the first model type (denoted NS), only normal and shear springs act. In the NSR model type, also rotational springs transferring local bending moments are added. However, only stresses in normal and shear spring contribute to the fracture criteria in both model types. As mentioned above, in the case of homogeneous models, the strength criterion defined by the breaking stress is identical for all springs. Tensile strength of all elements is set to 5 MPa. Also the E-modulus is the same for all springs. Forces carried by springs are influenced by the corresponding cross-sectional area A, spring length l and Poisson ratio ν. The cross-sectional area is calculated from the contact area between the rigid bodies (Figure 1). The springs representing the contact areas operate on the actual eccentricity coming from the discretization. 2.2

It has been proven by several authors (e.g. Schlangen & Garboczi 1997, Jirásek & Bažant 1995) that irregular geometry of the network helps to avoid directional preference of crack propagation. Thus, it has been chosen for the present model. The meshing algorithm is based on Voronoi tessellation, which is performed on the set of pseudorandomly placed triangulation nodes within the domain. The only restriction is that their minimal mutual distance equals to a predefined parameter l min . When a notch is to be modeled, it is included by mirroring nodes by the notch line in the notch vicinity, see Figures 1b and d. Voronoi tessellations then creates a straight line and all springs on that line are subsequently removed to model the notch. In order to place the notch tip exactly at the desired coordinate, three points are placed with a prescribed distance from the tip. This procedure guarantees an exact location of the shared vertex—the interface of the three corresponding rigid bodies at the notch tip. 2.3

2

Meshing algorithm

BRIEF DESCRIPTION OF THE MODEL

2.1 Mechanics of discrete model Several lattice-type models can be found in literature. Here, the rigid-body-spring network developed by Kawai (1978) is used. In basics, the model is very similar to the one published by Bolander & Saito

Deviations from the theory of elasticity

The stiffness of springs is derived to represent an underlying imaginary isotropic, linearly elastic homogeneous continuum (Kawai 1978). However, the elastic behavior differs from the assumed theory. The effect is clearly described by Schlangen & Garboczi (1996). Simply, the isotropic elastic material should exhibit uniform stress under uniform strain. Voronoi tessellation can satisfy this

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criterion for zero Poisson’s ratio ν. However, as showed by Bolander et al. (1999), for nonzero Poisson’s ratios, the stress distribution of a body under remote uniform uniaxial strain is not uniform any more. The greater the deviation from zero ratio ν, the more fluctuation in stress occurs (see error bars in Fig. 2). We also observed that Poison’s ratio severely influences the stress in the surface layer of elements. The average values of stresses in the lowermost elements of uTPBT diverge from the linear stress profile approximately obtained for zero ratio ν (Fig. 2a). Elsewhere, the average stresses roughly correspond to values given by ν = 0. The higher the Poisson’s ratio is and the finer discretization is used, the higher mean value of the stress is received in the boundary layer. This was observed also in a simulation of uniaxial tensile test without friction (i.e. uniform applied stress). Due to

Figure 4. Dependency of peak load on the REN network density (structural size D). Comparison with the size effect formulas (Equations 8 and 15).

Figure 5. Crack patterns at the peak load for various sizes of the unnotched beam with irregular network geometry. Left horizontal lines indicate the average height cf reached by the crack and its standard deviation.

Figure 2. Effect of poisons ratio on stresses σxx . a) the lowermost part of stress profile of uTPBT and b) stress profile close to the notch tip of nTPBT with regular net geometry loaded by force 10 N. Error bars show averages and sample standard deviations computed from 50 realizations.

this surface effect, the nominal strength is decreased, see Fig. 4. There is no significant effect of Poisson’s ratio on nominal strength observed in the case of notched specimens (nTPBT). The stresses in the elements around the notch tip are shown in Figure 2b. The greater is the deviation of Poisson’s ratio from 0, the greater is the variation in the stress. But the averages seem to be independent of value ν selected.

3

Figure 3. On derivation of the peak moment in a bent specimen.

SIZE EFFECT SIMULATIONS

The size of a concrete specimen typically affects the observed nominal strength. Several sources of this phenomenon are documented (Bažant & Planas 1998),

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we name the statistical and deterministic effects. Two main types of the deterministic size effects are distinguished. Structures with preexisting notches (positive geometry exhibiting type II size effect) and structures without any notch or with a small notch with respect to material internal length (negative geometry exhibiting type I size effect). Notched (type II, nTPBT) and unnotched (type I, uTPBT) are used to study this size effect in homogeneous brittle-spring networks. The density of the network is denoted as l min . Since there is no internal length in our constitutive law/model, we can represent varying size by varying network density. The characteristic size (depth) D is kept constant at a reference size D0 = 0.1 m, whereas the network density l min is varied; and we can mimic varying of the intrinsic size D by writing:

D = D0

l0min l min

Let us now deliver a closed-form expression for the observed size effect. Consider the midspan rectangular cross-section BD. The depth is discretized into 2N rigid bodies’ contacts of the same size, see Figure 3. Therefore the stress profile is a piecewise constant function along the depth D and approximates the actual (almost perfectly) linear profile. When the outermost spring reaches the extreme tensile stress f ∞ , the crosssection reaches its maximum bending moment M. Due to the symmetry along the neutral axis we can consider only the lower bottom of the depth (N elements) and calculate the bending moment as a doubled sum of force contributions times the corresponding arm. Each force contribution can be written as (Figure 6): Ti = B ·

(1)

where l0min = 0.02 m is the selected reference mesh density. Since we deal, in fact, with models of the same size, it is not necessary to report the size dependence on nominal strength (nominal stress at peak load). It suffices to report the loading forces F (D). On the other hand, however, the lengths (e.g. crack length) must be recalculated in a similar fashion as we did for D (see Equation 1). Removal of one element of the same size is interpreted as a crack of different lengths in models of various mesh densities. In order to evaluate the effect of network irregularity, all the results are computed for REN and IRN. Since the network in REN models is only regular in the vicinity of notch or midspan, the rest of the specimen (meshed by a lattice of irregular geometry) causes fluctuations of forces acting on the ‘‘crack faces’’. Subsequently, the obtained nominal forces are scattered. This effect is emphasized in unnotched structures, see e.g. Figure 4. In the regular networks, the rupture of the first element (beam or spring) causes the collapse of the whole structure. This holds both in the nTPBT and uTPBT. Therefore, the measured peak loads F p equal the elastic limits F e in the case of REN.

4

D i − 1/2 ∞ · f , 2N N − 1/2

i = 1, . . . , N

(2)

where B is the bar thickness [m], the second factor is the bin width l min = D/(2N ) [m] and the third factor is the corresponding constant stress in that bin [N/m2 ]. Each such a force has the following arm from the neutral axis:   1 D i− , i = 1, . . . , N (3) ri = 2N 2 where again, the first factor is the bin width. The resisting moment is a double of the sum (i = 1, . . . , N ): M (N ) = 2

N  i=1

Ti ri =

 N  1 2 BD2 f ∞  i − 4N 2 N − 1/2 2 i=1 (4)

SIZE EFFECT OF UNNOTCHED STRUCTURE

In the case of regular mesh geometry (REN), the crack can only propagate along the axis of symmetry through regularly placed squared elements of exact size l min . Figure 4 shows the maximal load (that is the elastic limit at the same time) depending on the density of the REN net (or, the size of the structure D).

Figure 6. Plot of elastic limits and peak loads of beams with irregular network and smooth bottom surface. Average values and standard deviations are computed from 50 realizations for every size.

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Calculating the sum yields:   BD2 ∞ 2N + 1 f M (N ) = 6 2N

where f ∞ is the strength limit for infinitely large structures and Db = 2 cf , i.e. double of the thickness of the boundary layer of cracking. If we take r = 1 which is a special case derived by Bažant and Li (1995), and rewrite Equation 9 in forces:

(5)

As N grows to infinity, the bending moment converges to the well-known value: M

∞

=f

∞

BD /6 2

(6)

The external moment equals the support reaction times the half span: M = F/2 · 3D/2. Putting this equal to Equation (5) yields:   2BD ∞ 2N + 1 F= f (7) 9 2N Equation (7) can be transformed into the dependence of peak force on bin width l min = D/(2N ): c

Fmax

c

       2BD ∞ l min l min 1+ = f = F∞ 1 + (8) 9 D D

This equation is plotted in Figure 4 and compared to the computed data. We can introduce a new length constant Db = l min = 20 mm to make it identical with Equation 10 introduced later. What remains to be clarified is the choice of the extreme stress f ∞ . An obvious choice would be the direct tensile strength (fl∞ = 5 MPa) of the model. This is because very large specimens fail at initiation of crack right at the midspan bottom face, which must equal the tensile strength. It would yield the asymptotic force Fl∞ = 11.11 kN. Unfortunately, the stress profile in not perfectly linear in reality. The real stress profile is affected by wall effects (the span of the beam is only 3D) and by the local compressive stress concentration around the point load. The nonzero Poisson’s ratio causes additional deviation from linear stress profile, see Sec. 2.3. As an approximation, we used nonlinear square fitting procedure to determine the two free parameters Db and F ∞ in Equation 8. One can calculate form fitted constant F ∞ the theoretical stress at the bottom layer for infinitely small mesh caused by load 10 N, see Sec. 2.3. This stress σ ∞ is added into Fig. 2 to show consistency with our fits. Is it worth pointing that another way exists to nicely fit the data—to consider the bent specimen being made of a quasibrittle material. One can assume a linear stress profile along the depth except for the damaged zone in the bottom tensile part. If we consider that the boundary layer of cracking has a constant size cf irrespective the specimen size, the following size effect formula can be derived (see pages 41–43 of Bažant 2005) for scaling of nominal strength (modulus of rupture):   rDb 1/r σN = fr (D; f ∞ , Db , r) = f ∞ 1 + (9) D

Fmax

       2BD ∞ Db Db 1+ = f = F∞ 1 + 9 D D    F∞

(10) when Db = l min , Equations 8 and 10 match. Now, a question appears: What if we consider rotational springs? In each element (or contact area), the spring add a new additional moment Mi , see the last strip in Figure 6 right. These contributions are equal (for i = 1, . . . , 2N ). In each bin there is a pair of forces Ti that represent two triangles (below and above the constant stress σi . Each of these triangles are half of the strip long (= D/(4N)) and the maximum stress difference is σ . The stress σ is one half of the difference between the current strip and the adjacent strip:   f∞ i − 0.5 − i + 1.5 (σi − σi−1 ) = σ = 2 2 N − 1/2 =

f∞ 2N − 1

(11)

The pair of forces Ti representing the two triangles are (for i = 1, . . . , N ): Ti =

1 D BD f∞ BD σ · B · = σ = · 2 4N 8N 2N − 1 8N (12)

Each of these two forces act over the distance of D/(6N ) from the ‘‘neutral’’ state and form an additional moment increase to the total bending moment. The magnitude of each such moment contribution (in each strip-bin) is twice the arm times force Ti : Mi = 2 · Ti ·

BD2 D = f∞ 2 6N 24N (2N − 1)

(13)

In total, there are 2N such partial moments over the whole cross-section and therefore the total moment increment is 2N times the contribution Mi : M = 2N · Mi = f∞

1 BD2 6 2N (2N − 1)

(14)

This moment increment is not reflected in the failure condition. Transforming it into the increment of

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at the notch tip leads to collapse of the whole beam. The peak forces of REN (that are the elastic limits at the same time) are plotted against the net density lmin (or size D) in loglog plot. They fall exactly on a line of slope –1/2 (see Figure 7). This result is not new and corresponds to the remedy of size dependency of homogeneous regular lattice models proposed by Jagota & Bennison (1995). Again, rotational springs cause deviation from the theoretical size effect (LEFM power law). The deviation diminishes with increasing size of the structure (refining the mesh). We use the same procedure as in Section 4 to modify the power law. In large structures (or fine meshes) the stress profile behind the notch tip follows the well known formula

maximal force gives F =

1 4 M = F∞ 3D 2N (2N − 1)

= F∞

l min l min D D − l min

(15)

Adding this increment to the total force from Equarot = tion 8 yields the upgraded size dependence Fmax Fmax + F. The first term uses Db , while the increment F uses l min . In other words F increases the maximal load, especially for small sizes. If the structure is represented by just one element (N = 1/2), the increment is infinite (an arbitrarily high load is supported by the rotational spring, no tensile forces appear). As the size becomes larger (or discretization finer), the total moment tend to that in Equation 6. The irregularity of the network geometry (IRN) allows the model to choose the ‘‘weakest’’ area to initiate and propagate the crack. That is why the elastic limits are, on average, lower in IRN compared to REN, see Figure 6. The load applied to break the first spring F e in IRN model is, on average, also much lower than the peak forces. The peak forces in IRN models are greater than those of REN. The first crack appears at the weakest spring loaded by high forces: the crack prefers short facets. Qualitatively, however, both force dependencies of IRN are similar to REN and follow the tendency proposed by Equation (8), respectively (15). The deviations for larger specimens (finer mesh densities) are caused by local stress deviation described in Section 2.3. Namely, we mean the stress fluctuations in the lowermost layer caused by Poisson’s ratio. Both the elastic forces and peak forces can drop below this horizontal asymptote; see Figure 9. Instead of one crack, many small cracks are created inside the bottom area of the specimen (Figure 5) and the model allows for redistribution of forces after many such local ruptures. These cracks do not form a continuous line. On average, the thickness of the boundary zone of distributed cracking is >l min . The fact that the zone has approximately the same height for all sizes (Figure 5), supports our claim that the data can be approximated reasonably well by Equations 8 and 10.

5

σxx = √

KI 2π r

,

(16)

One could compute the F increment (correction) assuming this function over the whole stress profile. This would have to be done numerically because expression for Mi is too complicated to be summed analytically. However, this nonlinear stress function is valid only in a short region close to the notch tip. The correction for rotational spring is only necessary when the mesh is very coarse (small structure). There however, the stress profile is almost linear, not singular as in Equation 16. Therefore we finally assume linear stress profile and derive the correction term F for notched specimens caused by additional moments the same way as we did in Equations 11–15. The only difference is that instead of the total beam depth D, we substitute 2/3 D, because 1/3 Dis occupied by the notch. The formula for pair of forces Ti in strip i using σ from Equation 11 reads: Ti =

1 2 DB DBf ∞ σ = . 2 3 4N 12N (2N − 1)

(17)

SIZE EFFECT OF NOTCHED STRUCTURE

A somewhat different situation appears when the beam fails by crack propagation from sharp notch (such as our nTPBT). In our numerical simulations, the notched specimens have similar features to the uTPBT. In the case of regular geometry (REN), the first rupture of the beam

Figure 7. Simulations of nTPBT beams with regular mesh REN. Each circle is an average of 50 simulations, error bars are not included as the standard deviation is extremely small. Fitted by LEFM straight line of slope –1/2.

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Contribution to the moment is given by multiplying by arm of the pair of forces (similar to Eq. (13)). Mi = 2Ti

2 D D2 Bf ∞ = 3 6N 54N 2 (2N − 1)

(18)

Summing all moment contributions, we receive a moment correction M that can be transformed into the load correction F: M = 2

N 

Mi = 2N · Mi =

i=1

D2 Bf ∞ 27N (2N − 1) (19)

F =

4DBf ∞ 4 M = 3D 81N (2N − 1)

Figure 8. Simulations of nTPBT beams with irregular mesh (IRN). Each circle is an average value of 50 simulations.

(20)

Finally, we rewrite the function in terms of l min using N = D/(3 l min ): F =

2(l min )2 4Bf ∞ (l min )2 = F∞ min 9(2D − 3l ) D(2D − 3l min )

(21)

Adding this correction function to the power law obtained for large sizes, we can explain the trend of the data also for small sizes (see Fig. 7). The effect of springs is underestimated because the basic stress profile was assumed to be linear. Network irregularity (IRN), brings a new effect. Since the element placed right above the notch tip is angled and has varying size, the external load F e necessary to break it is affected. One has to break usually more than one element to reach peak force F p , thus F e < F p. The elastic F e limit obeys the LEFM slope of –1/2 and lies very close to the previous fit with regular nete e works (FIRN = 1.92D−0.5 , FREN = 1.94D−0.5 ). This is surprising because two effects working one against the other appear here. (i) The angle of the first element (deviation from the horizontal direction) increases the elastic limit force, i.e. shifts the line upwards. (ii) The average area of the first broken element is lower in IRN than in REN where all broken elements have the area l min × thickness. This leads to downward shift of the size effect line. Apparently, effects of those the upward and downward shifts cancel each other. Looking at the peak force data, a good fit in loglog plot is a straight line of slope –0.424, see Figure 8. The source of the observed deviation from the LEFM slope of –1/2 was found in the crack behavior. Figure 9 shows crack patterns at the peak load for all considered sizes. These crack length are recalculated into the intrinsic magnitudes (Equation 1). Apparently, the larger the specimen, the longer the length at the peak load: the crack initiation from the notch is followed by an increase in peak crack length with size D. This increase can be fitted by a power law with exponent

Figure 9. Crack patterns at the peak load for various sizes of the notched IRN beam (with irregular mesh geometry).

1/2 (see Fig. 10) both for NS and NSR models. The slower declination of the fitted power law in Figure 8 (–0.424) can be attributed to the described growth in crack length with specimen size D.

6

DISCUSSION

Some interesting points have been shown in the two preceding section and we now discuss some of the features in a more detail. It was mentioned previously that the length of a crack at the peak load seems to be increasing in the nTPBT IRN simulations while the average crack length at the peak load is about constant in uTPBT, see Figures 5 and 9.

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Let us also mention recent results of Alava et al (2008) who show, using random fuse model, that strength of notched beams of various sizes and notch depths is influenced by the amount of disorder. The strength dependence on notch depth deviates from LEFM power law with an increasing disorder. Specimen strength made of highly disordered material, with a small notch, is driven mainly by the disorder and not only by the stress concentration. 7

REDUCTION OF SPURIOUS SIZE EFFECT

The influence of network density has to be understood as a spurious phenomenon, because the mesh is arbitrary, artificial and does not arise from any real material structure. Some authors believe that the network density dependency might be removed by projecting the material inhomogeneities onto the lattice. This introduces the internal length, which decrease this dependency (van Mier & van Vliet 2003). In the following, this expectation is subjected to critical study, which shows limits of such a procedure. 7.1 Figure 10. Length of the crack at the peak for notched and unnotched TPBT: a) NS model, b) NSR model with rot. springs.

Surprisingly, the increase in the crack length of IRN nTPBT specimens obeys a power law with exponent 1/2 (see the thin angled line in Figure 10ab). In the case of unnotched specimens, the constant peak crack length is dependent on the selected reference network density l0min . This was checked numerically by additional simulations with various lengths l0min (otherwise kept constant here); see the thin horizontal lines in Figure 10a which shows that the average peak crack length roughly lies in the range of 1.1–1.3 l0min . In conclusion, scaling both the network size and the specimen size by the same positive scaling factor yields an identical result as for the original sizes. The peak force depends only on the quality of the stress profile approximation. We have performed a sensitivity analysis to identify the influence of various parameters on the elastic limit load and the peak load. In particular, Spearman nonparametric correlation coefficient was used. The greatest absolute correlation to the peak load was found with the maximum vertical coordinate of the crack—i.e. the thickness of the zone with distributed cracking cf . In the case of notched specimens, the elastic limit force F e is sensitive to the initial crack length as well as to the inclination of the initial crack from vertical direction (corr. coeff. approx. 0.8). Unfortunately, no dominant variable affecting the peak load was identified.

Incorporating of grain structure

The grain structure that is used here is generated by computer algorithm using the Fuller curve (see e.g. Cusatis et al. 2006). Typically, maximal grain diameter dmax is chosen according to the real batch contents, and the minimal dmin according to the network density. The length of network elements should be at least three times smaller than dmin (van Mier 1997), otherwise the particles coalesce in the mesh. Grains smaller than dmin are ignored in the procedure. The larger dmin , the coarser mesoscopic structure is incorporated; yet, the generated coarse grains still correspond to the requested content of coarse grains, i.e. reducing dmin has no effect on coarse grains—it only adds finer aggregates. Grains were projected onto the lattice to attribute springs with the three material phases—aggregate, matrix and ITZ. These are distinguished according to the positions of nodes with respect to the mesostructure (see van Mier et al. 1997). Each phase has a different strength and Young’s modulus. Values from the article by Prado & van Mier (2003) were used. In the following part, we will test the hypotheses that: when finer the mesoscopic structure is considered a lower mesh sensitivity of the model is observed. For this reasons, six different grain contents were generated. The first one corresponds to the homogeneous case without any grains (studied above), the others differ by dmin (Figure 11); whereas maximum grain diameter dmax is kept equal to 32 mm. Mesh varied from density 2.5 mm up to density 0.625 mm. Note that not all the densities can be used for all the grain contents. The finer the grains, the finer

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mesh is required. For all possible combinations, 50 realizations of notched and unnotched TPBT were simulated. In order to have an idea about the model behavior, Figure 12 shows average load-deflection diagrams for some of considered densities and mesostructures. Notice the drastic strength decrease when one incorporates even only few grains in the unnotched structure. The crack position can be sampled from many possible positions and thus fracture always propagates along the weak ITZ. Such a steep drop in strength is not observed with notched structures: because the crack propagates from the deep notch independently of positions of grains. The results for the peak loads are shown in Figure 13. The homogeneous model of the notched TPBT follows a straight line of slope 0.424 in the notched case (previously described in Figure 8). Models with a grain contents seem to reduce this dependency. The

best results (almost a horizontal line) are achieved by the most detailed grain contents. In the unnotched case, the presence of grains apparently increases the dependence on network density. The more detailed structure, the less dependency is visible. But still, the dependence of the peak force on the mesh resolution is stronger than in the homogeneous models. The second monitored parameter is the area under load-deflection curves, which has meaning of energy. No substantial reduction in the dependence of this energy on the mesh density (by incorporation of the grain structure) was observed for the notched structure—Figure 14a documents that all the lines

Figure 11. An example of crack patterns observed in a simulation of (a) nTPBT and (b) uTPBT with the model including concrete mesoscopic grain structure of varying fineness.

Figure 13. Dependence of maximum load for (a) notched TPBT and (b) uTBPT on grain structure incorporated.

Figure 12. Average load-deflection diagrams of 50 realizations of (a) nTPBT and (b) uTPBT for some of the considered densities and mesostructures.

Figure 14. Dependence of the area under load-deflection curve for (a) notched TPBT and (b) unnotched TPBT on grain structure incorporated.

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share approximately the same rising slope. The energy dependency is only slightly reduced for the unnotched structure, see Figure 14b. Yet, the line plotted for the finest resolution is far from being horizontal. 8

CONCLUSIONS

The effect of discretization of lattice models was studied. The basic cases are examined: (a) homogeneous lattices, where all elements share the same strength and (b) lattices in which the properties are assigned to the elements according to their correspondence to three phases of concrete, namely matrix, aggregates, and the interfacial transitional zone (ITZ). These dependencies are studied with both, notched and un-notched beams loaded in three point bending. We report the results for regular discretization and irregular networks obtained via Voronoi tessellation. The dependence of strength is compared to various size effect formulas and we show that in the case of homogeneous lattices, the fineness of discretization of the specimens of the same size can mimic variations in the size of lattice models with the same discretization. In the case of heterogeneity (b), we report that even though the peak force dependence is influenced by the mesh resolution, and almost disappears in notched structures, a strong dependence of the fracture energy remains. This is important for fracture studies with lattice models. ACKNOWLEDGEMENTS This research was conducted with the financial support of the Ministry of Education, Youth and Sports of the Czech Republic under project No. 1M06005 within activities of the CIVAK research centre, and with the financial support of the Czech Science Foundation, project GACR 103/09/H085. The second author acknowledges the support from the Fulbright commission within the framework of Fulbright-Masaryk fellowship. REFERENCES Alava, M.J., Nukala, P.K.V.V & Zapperi, S. 2008. Fracture size effects from disordered lattice models. International Journal of Fracture 154: 51–59. Bažant, Z.P. (2005). Scaling of Structural Strength; 2nd updated ed., Elsevier Butterworth-Heinemann. Bažant, Z.P. & Li, Z. (1995). Modulus of rupture: size effect due to fracture initiation in boundary layer. J. of Struct. Engrg—ASCE, 121(4): 739–746. Bažant, Z.P. & Planas, J. 1998. Fracture and Size Effect in Concrete and Other Quasibrittle Materials. CRC Press, Boca Raton and London.

Bolander, J.E., Hikosaka, H. & He, W.-J. 1998. Fracture in concrete specimens of different scale. Engineering Computations 15: 1094–1116. Bolander, J.E. & Saito, S. 1998. Fracture analyses using spring networks with random geometry. Engineering Fracture Mechanics 61: 569–591. Bolander, J.E., Yoshitake, K. & Thomure, J. 1999. Stress analysis using elastically homogeneous rigid-body-spring networks. Journal of Structural Mechanics and Earthquake Engineering 633: 25–32. Cusatis, G., Bažant, Z.P. & Cedolin, L. 2006. Confinementshear lattice CSL model for fracture propagation in concrete. Computer Methods in Applied Mechanics and Engineering 195: 7154–7171. Eliáš, J. 2009. Discrete simulation of fracture processes of disordered materials. Ph.D. thesis, Brno University of Technology, Faculty of Civil Engineering, Brno, Czech Rep. Grassl, P. & Bažant, Z.P. 2009. Random lattice-particle simulation of statistical size effect in quasi-brittle structures failing at crack initiation. Journal of Eng. Mechanics 135: 85–92. Jagota, A. & Bennison, S.J. 1995. Element breaking rules in computational models for brittle materials. Modelling and Simulation in Materials Science and Engrg 3: 485–501. Jirásek, M. & Bažant, Z.P. 1995. Particle model for quasibrittle fracture and application to sea ice. Journal of Engineering Mechanics 121: 1016–1025. Kawai, T. 1978. New discrete models and their application to seismic response analysis of structures. Nuclear Engineering and Design 48: 207–229. Lilliu, G. & van Mier, J.G.M. 2003. 3D lattice type fracture model for concrete, Eng. Fracture Mechanics 70: 927–941. Schlangen, E. & Garboczi, E.J. 1997. Fracture simulations of concrete using lattice models: computational aspects. Engineering Fracture Mechanics 57: 319–332. Schlangen, E. & Garboczi, E.J. 1996. New method for simulating fracture using an elastically uniform random geometry lattice. Int. Journal of Engineering Science 34: 1131–1144. Prado, E.P. & van Mier, J.G.M. 2003. Effect of particle structure on mode I fracture process in concrete. Engineering Fracture Mechanics 70: 1793–1807. van Mier, J.G.M. 1997. Fracture Processes of Concrete: Assessment of Material Parameters for Fracture Models. CRC Press, Boca Raton, Florida. van Mier, J.G.M. & van Vliet, M.R.A. 2003. Influence of microstructure of concrete on size/scale effects in tensile fracture. Engineering Fracture Mechanics 70: 2281–2306. van Mier, J.G.M., Chiaia, B.M. & Vervuurt, A. 1997. Numerical simulation of chaotic and self-organizing damage in brittle disordered materials. Computer Methods in Applied Mechanics and Engineering 142: 189–201. Voˇrechovský, M. & Eliáš, J. 2009. Mesh dependency and related aspects of lattice models. FRaMCoS 7, Proc. of 7th Int. Conf. on Fracture Mechanics of Concrete and Concrete Structures, Jeju, South Korea, May 23–28, 2010, in press.

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Time-dependent and multi physics phenomena

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Prediction of the permeability of damaged concrete using a combined lattice beam-crack network approach M. Abreu Swiss Federal Institute of Technology ETHZ Zürich, ETH-Hönggerberg; Materials Department, Portuguese National Laboratory for Civil Engineering, LNEC-Lisbon

J. Carmeliet Swiss Federal Institute of Technology ETHZ Zürich, ETH-Hönggerberg; Swiss Federal Laboratories for Materials Testing and Research, EMPA-Dübendorf, Building Technologies

J.V. Lemos Concrete Dams Department, Portuguese National Laboratory for Civil Engineering LNEC-Lisbon

ABSTRACT: The article describes a combined damage and permeability 3D model for concrete. The objective of the model is to predict the permeability of damaged concrete. For the mechanical model a lattice of beam elements is used where the damage is modelled by a step-by-step removal of beams. For the calculation of the permeability the removed beams are connected by transport elements with a aperture proportional to the relative displacement between the nodes previously connected by the beans. The numerical simulations are then compared with experimental results from a diffuse tensile cracking and permeability test. Two lattice models are compared with the experimental results: in the first one the concrete is considered as a homogenous material and in the second the aggregates are explicitly modelled. It is shown that when the aggregates are included, the simulation agrees better with the test results for lower strains, while for higher strains the homogeneous model agrees better. 1

INTRODUCTION

The accurate knowledge of the concrete permeability is crucial for the assessment of durability of concrete, since liquid transport into the material is a determining key factor. The penetration of liquids may initiate several damage processes, such as frost damage, corrosion, chemical degradation and salt damage, etc. Damage in the form of cracks significantly modifies the permeability of the material and allows a preferential transport along the crack network into the material several orders of magnitude faster compared to the transport in sound concrete (Roels et al. 2006). Therefore, when designing durable concrete constructions, not only the prediction of cracking is needed, but also the estimation of the permeability of cracked concrete is necessary. The permeability of cracks highly depends on the average crack width, the crack tortuosity and connectivity (Carmeliet et al. 2004). Therefore, we present in this paper a model taking into account these important crack aspects. In order to predict the permeability of damaged concrete a combined crack and transport model is presented. The crack model is based on a three dimensional lattice of beam elements where damage is

modelled by the consecutive removal one-by-one of beams. Therefore the model can be classified as of a repetitive linear elastic type. The transport elements are formed by connecting the centre of the removed beams and their individual permeability depends on the crack width, taken as the relative displacement between the nodes previously connected by the beans. The global permeability of the cracked concrete is then determined from the obtained crack network considering saturated transport. The model accuracy depends on the properties of the beam elements and the criterion used for their removal, which are obtained by calibration against experiments. 2

LATTICE MODEL

The lattice crack model is based on developments at the Delft University and therefore often known as Delft Lattice (Schlangen & Garboczi 1996, 1997) (Lilliu. & van Mier 2000). In this type of model the lattice of beams is composed of one dimensional finite elements with axial, bending and shear stiffness. In a 3D model there are 6 degrees of freedom per node, these are the three displacements and the three rotations (figure 1).

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A beam element will connect two nodes, therefore the elemental stiffness matrix has 12×12 dimensions (figure 1) (CALFEM, 1999) were Ke is the stiffness matrix of a beam element, u1...12 are the displacements and rotations at the beam nodes, E is the beam material modulus of elasticity, Iy,z are the beam inertia in each direction, L is the beam length, k1 is the beam axial stiffness defined as EA/L where A is the beam area and k2 is the beam shear stiffness defined as GJ/L where G is the beam shear modulus and J is the polar inertia. To generate the model first a pack of spheres is created. If a regular lattice is used the model nodes will be located on the centre of the spheres, if a irregular lattice is used the nodes are randomly located inside a cube with side dimensions equal to the spheres diameter. The distortion factor z controls how irregular the lattice will be (Lilliu & van Mier, 2000). If z = 0 the lattice is regular and the node is in the centre of the sphere, if z = 1 the distortion is maximal as the node may be located in any position inside the cube. In all the models presented in this article z = 0.9, this means that the nodes may be located anywhere inside the cube up to 0.05x the distance from its borders. After all the nodes are placed, a type of 3D Delaunay tessellation is used to connect them. The objective of the tessellation is to create regular tetrahedrons as close as possible and that adjacent tetrahedrons have adjoining faces. The beam elements are the edges of the tetrahedrons. Figure 2 shows three small examples of one regular and two irregular meshes, where z = 0.5 and 1.0. After the boundary conditions are applied the fundamental equation Ku = F is solved, where K is the lattice global stiffness matrix obtained from assembling all the individual elemental matrices accordingly to

Figure 1. Beam element degrees of freedom and elemental stiffness matrix (CALFEM, 1999).

their connectivity, u are the displacements and rotations of the nodes and F the forces and moments. At each calculation step, a linear elastic analysis is performed and the forces and displacement are calculated. A beam will be removed when it fails according to a defined failure condition. The beam to be removed is the one closest to failure. The failure condition used in the presented lattice models is a maximum value for the beam positive (tensile) axial force. Negative axial forces (compression), bending moments, torsion and transversal forces are not considered for the failure condition. This failure condition was selected since it is result in a less brittle behaviour for a lattice under a tensile load (Schlangen, 1993) and agrees better with the experimental results. On the next calculation step the stiffness of the beam is removed from the global stiffness matrix. The procedure is then repeated until the lattice breaks completely and therefore in no longer able to transfer the load. Figures 3a and 3b show an example of a uni-axial tensile simulation. The forcedisplacements required to break each beam are plotted sequentially in figure 3b. As it can be seen in this image, frequently the strain necessary to break a element is smaller than the previous one, which creates ‘‘backward steps’’.

Figure 2. Lattice mesh examples for z = 0 (left), z = 0.5 (center) and z = 1.0 (right).

Figure 3a. Example of a lattice simulation of a tensile test on a concrete cylinder with a thinner middle section, the removed beams are highlighted.

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Figure 3b. Force-displacement graphic for the lattice shown in figure 3a.

3

Figure 4. Possible locations of the transport element when three beams are removed.

TRANSPORT NETWORK MODEL

For the transport network model a finite element mesh of triangles with three nodes is used. There is only one degree of freedom per node: the pressure. The pressure varies linearly within the finite element and the flow is constant. The transport triangles are introduced in the lattice of tetrahedrons when the beams on the edge of a tetrahedron are removed. They are created when at least one of the four nodes of the tetrahedron is no longer directly connected to another node belonging to the same tetrahedron. When this happens the tetrahedron is considered split and a transport triangle is created at the splitting surface. The triangle will connect the centre points of the removed beams, which become nodes of the transport model. At least three beams belonging to the same tetrahedron must be removed in order for a transport element (triangle) to be created. Depending on the position of the three removed beams, the triangle location on the tetrahedron will change as shown in figure 4, where the removed beams are highlighted (bold lines). If four beams are removed two cases are considered: 1) if the two remaining beams are adjacent which means that three of the lattice nodes are still connected, then only one transport triangle is created, as shown in figure 4; 2) if the two reaming beams are not adjacent to each other this means that the tetrahedron is now split by a square. As shown in figure 5 there are three possible positions for this square surface. Note that in order to use only triangular finite elements, the square surface is divided in two triangles. If five beams are removed from the tetrahedron several of the situations presented in figure 4 and 5 are combined. There will be two triangles and one square inside the same tetrahedron. The possible combinations are: (using the numbers and letters

Figure 5. Possible locations of the transport element when four beams are removed.

shown on figures 4 and 5) A+1+2, A+3+4, B+1+4, B+2+3, C+1+3 and C+2+4. The presence of several splitting surfaces inside a tetrahedron means that several crack surfaces are connected on this tetrahedron showing crack splitting or branching. Finally if all the six beams are removed, four triangles and three squares will be present inside the tetrahedron showing a complex pattern of crack connectivity. The aperture of the transport triangle is calculated at each of the three nodes (figure 6). First the relative displacement between the nodes previously connected

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Figure 6.

The elemental permeability matrices are assembled in the global permeability matrix and the fundamental equation for the saturated (steady) case K1 P = Q can then be solved; where K1 is the global permeability P is the pressure at the nodes and Q the flow. The pezrmeability for the entire lattice is generally obtained by imposing a unitary pressure gradient between two of the lattices faces, the sum of the flows at one of those borders is then equal to the lattice permeability. In figure 7 we present an example of the calculated flow across a simulated crack surface. The crack in this example is almost planar because a regular and very brittle lattice is used. Nevertheless some irregularities do exist and they divert the flow increasing its velocity around them, as it can be seen in the image.

Aperture of the transport triangle at each node.

4

Figure 7. Example of calculated flow inside a simulated crack surface: a) lattice at failure with the removed beams highlighted; b) stress-strain curve; d) flow in the direction of the arrow, perspective and top views.

SIMULATION OF A DIFFUSE CRACKING TEST (P.I.E.D.)

In this section, the model is used to simulate a diffuse cracking test on concrete and the results are compared to the ones presented in Gérard et al, (1994). In a P.I.E.D. (Pour le Identification de le Endommagement Diffuse) test, the test apparatus allows for a permeability test to be performed on the specimen during different stages of the damage process. As shown in figure 8, in this type of test a concrete cylinder specimen is attached between two steel plates with a cylindrical hole. The diameter of these holes is half the concrete cylinder diameter (55 mm). The holes in the steel plates allow water to cross the sample during the permeability test. The steel plates are submitted to tensile loading, so that the deformation is applied indirectly to the concrete. As the steel plates are loaded the concrete is subjected to quasi uni-axial tensile stresses.

with a beam is calculated. The aperture is assumed to be the relative displacement normal to the triangle surface. The aperture at each of the three triangle nodes is then averaged so that an element with constant aperture is obtained. Knowing the geometry of the transport model defined by the transport triangles and their corresponding aperture and applying a pressure difference as boundary condition the transport problem can be solved. The element permeability for each transport triangle is calculated using the equations for flow thorough parallel plates as presented in equation (1) (Vandersteen, 2002): ⎡ ⎤ k1 0 0 ρ2 (1) K1e = ⎣ 0 k1 0 ⎦ k1 = t 12μ 0 0 k 1

where: Ke1 is the elemental permeability matrix, ρ is the density of the fluid, μ is the dynamic viscosity of the fluid and t is the aperture of the transport triangle.

Figure 8.

P.I.E.D. test scheme (B. Gérard et al, 1994).

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Because the concrete is attached to the steel plates the damage will not be localized in one simple crack, but a diffuse crack pattern is obtained. Figure 9 shows two cracking patterns obtained with the P.I.E.D. test for a normal strength concrete (left) and a high strength concrete (right). In both cases two main cracks are present, but for the normal concrete a diffuse secondary cracking is clearly visible. The secondary cracks probably do not connect with each other and with the main crack, they probably also do not cross the entire thickness of the cylinder therefore not contributing much to the permeability. In figure 10 the lattice model of the P.I.E.D. test is shown. The lattice consists of about 7000 nodes and 50,000 beams. With this number of elements it is possible to simulate the geometry of the P.I.E.D. test, although as shown in the figure the model geometry is still quite rough. The thickness of the steel plates is increased to 8 mm in order to facilitate its modelling. To generate the model a pack of spheres with 5 mm

Figure 9. Diffuse cracking patterns obtained with a P.I.E.D. test for a normal strength concrete (left) and a high strength concrete (right) (B. Gérard et al, 1994).

Figure 10.

Lattice model 1 (reference).

diameter is used. This means that the average length of the beams will be close to this value. The beams are cylindrical with a 2.25 mm diameter; this value is used so that the entire lattice behaves as a material with a Poisson coefficient of about 0.2. The beams at the interface between the concrete and the steel plates have the same properties as the beams modelling the concrete. The elastic modulus of the beams modelling the concrete is 60 GPa; this value is used so that the elastic modulus of the entire lattice will be about 20 GPa. The elastic modulus of the beams modelling the steel is 1000x higher than the concrete ones. The limit stress for the beams modelling the concrete is 6 MPa. Note that this stress only limits the beam positive axial force. The 6 MPa value is chosen so that the lattice breaks for tensile stress of about 2 MPa and at a strain of 1.0E-4 m/m. For the steel an infinite value was used for this limit stress in order to guarantee that there will be no damage in the steel plates. Figure 11 compares the damage in the concrete cylinder obtained by the lattice simulation and by a characteristic P.I.E.D. test (we have no information if the results are for a normal or high strength concrete). Damage is defined according to the loss of stiffness of the concrete: damage is zero when the concrete stiffness equals its elastic stiffness, and the damage has a value of one when the concrete stiffness is zero, therefore D = 1 − Ei /Ee where D is the damage, Ei is the concrete stiffness for calculation step i and Ee is the concrete initial or elastic stiffness. As shown in figure 27 the point when damage starts is accurately simulated by the lattice but then the lattice behavior is more brittle, showing a too fast increase of damage in the simulation. In figure 12 the results for the permeability are presented. The permeability is presented in a logarithmical scale as it increases very fast and ranges over several orders of magnitude. The permeability is calculated every 100 calculation steps by solving equation (1) given a pressure gradient of 1.0 Pa between the back and the front of the cylinder. Because of the characteristic lattice ‘‘backward steps’’ (figure 3) the

Figure 11.

Damage versus strain for model 1 (reference).

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Figure 12. ence).

Permeability versus strain for model 1 (refer-

simulated permeability is not a perfect curve. The simulated results lower limited is 5.0e-13 which is the value obtained for the permeability of the sound concrete. In the simulations the permeability remains low for too long and starts to increase only for a strain of 6.3E-04 m/m, then it quickly rises and for values above 8.5E-4 m/m the simulation and test results agree. Although the permeation tests for the P.I.E.D. experiments are performed in unloaded specimens, the results are plotted for the maximum strain achieved during the loading. Nevertheless the obtained permeability values seem to be comparable to the ones simulated with the lattice model, possibly because: 1) in the P.I.E.D. test the steel plates are yield up to plastic load levels, so they will not be able to recover all their strain; 2) as stated by the authors; small pieces of concrete seem to detach inside the cracks preventing their closing after unloading (B. Gérard et al, 1994). In figure 13 the simulated crack distribution and flow across the cylinder for step calculation 4800 and a strain of 6.8E-4 m/m is plotted. Comparing with the experimental results shown in figure 9 we may observe also two main cracks, one slightly bellow and the other above the centre of the cylinder. As it can be seen in the image in perspective the maximum flow is located in the crack bellow the centre of the cylinder. In figure 14 the simulated crack distribution and flow is again plotted but for calculation step 7300 and for a strain of 1.0E-3 m/m. It is now possible to observe considerable more damage but the main flow is still located in the same crack. In the next lattice larger aggregates are included in the model. For the aggregate positioning a set of 9 images is used. These images are shown in figure 15. The aggregates are idealized as spherical or ellipsoidal particles. The images are shown along the x axis of the model, and the steel plates can be seen on the left and right side of the images. The model length in the x direction is 110 mm (this is also the diameter of the cylinder so the 9 images are positioned at intervals of 12.2 mm. During the material assignment

Figure 13. Simulated crack distribution and flow across the cylinder for step 4800 and a strain of 6.8E-4 m/m for model 1 (reference).

step the model scan the images and assign properties according to the material present in the image pixel closest to each node. If both nodes of a beam element belong to material A, material properties A are assigned to that beam. If one node of a beam element is associated with material A while the other node is associated with material B, the beam will be considered as belonging to an interface and the properties will be assigned accordingly. For the beams modelling the mortar the same properties were considered them in the previous model, the beams modelling the aggregates are two times more resistant, and the interface beams connecting the aggregate and the mortar are two times less resistant. In figure 16 the resulting lattice model is shown. Some of the aggregates at the edges of the concrete cylinder are highlighted.

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Figure 15. Set of 9 images used for the positioning of the aggregates on model 2.

Figure 14. Simulated crack distribution and flow across the cylinder for step 7300 and a strain of 1.0E-3 m/m for model 1 (reference).

The damage versus strain for the lattice model including the aggregates is shown in figure 17. Comparing with figure 11 we can conclude that adding aggregates has almost no influence on the damagestrain behaviour. In figure 18 the permeability is plotted for the lattice model with aggregates. The permeability is calculated every 50 simulation steps. Compared with figure 12, the permeability now starts to increase earlier and agrees with the tests results, but as the strain increases the increase of the permeability is slower. For large strains the permeability is lower than the one estimated by the lattice without aggregates and does not agree so well with the test results. The simulated crack distribution and flow across the cylinder for step 2000 and a strain of 2.7E-4 m/m is presented in figure 19. Comparing with the experimental results shown in figure 9 we note that in this simulation there is only one main crack crossing the specimen. This crack is not planar, it shows a wave

Figure 16. Lattice model 2, some of the aggregates are highlighted.

Figure 17. gates).

Damage versus strain for model 2 (with aggre-

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Figure 18. Permeability versus strain for model 2 (with aggregates).

Figure 20. Simulated crack distribution and flow across the cylinder for step 4000 and a strain of 5.2E-4 m/m and for model 2 (with aggregates).

pattern, probably formed by the presence of the aggregates around which the crack develops. Most of the flow (dark colour) is located in the centre of the crack, which coincide with the middle of the cylinder. In figure 20 the simulated crack distribution and flow for calculation step 4000 and a strain of 5.2E-4 m/m is plotted. At this level of strain there is a lot more damage, but most of this damage consists of secondary isolated cracks that don’t contribute to the transport. The main flow is still on the main crack near the centre of the cylinder as it was for step 2000.

5 Figure 19. Simulated crack distribution and flow across the cylinder for step 2000 and a strain of 2.7E-4 m/m and for model 2 (with aggregates).

CONCLUSIONS AND FUTURE DEVELOPMENTS

The article describes a combined damage and permeability 3D model for concrete where for the damage model a lattice of beam elements is used and for the

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calculation of the saturated permeability a transport network model is used. The numerical simulations are then compared with experimental results from a diffuse tensile cracking and permeability test (P.I.E.D.). Two lattice models are compared with the experimental results: in the first one the concrete is considered as a homogenous material and in the second the aggregates are explicitly modelled. We may conclude that adding aggregates has almost no influence on the damage-strain (mechanical) behaviour of the lattice, but it influences the simulated permeability. When the aggregates are added the permeability agrees better with the experimental results for small strains, but for the large strains the results from the lattice without aggregates agree better with the experimental results. The early increase of the model permeability probably is a direct result of the lower resistance of the beams modelling the mortar-aggregate interface. The too low permeability of the model with aggregates for the high strains is probably a result of the fact that now the crack is less planar, developing around the aggregates, which means that there will be a longer path for the water to cross the model. The simulation results may depend significantly on the aggregates geometry and position, therefore it is advisable to test different aggregates distributions. An accurate prediction of the cracks geometry improves the prediction of its permeability. The lattice of beams crack model is capable of simulating quite realistic crack geometries and therefore it seems a good candidate for a model when coupling between damage and permeability is required. The main drawback of the lattice model is that, because of its simplicity, there are too few control parameters that may be changed to fine-tune the model behaviour. Currently an experimental study is planned where the main aim is to develop a test capable of simultaneous loading to produce damage, measuring the permeability and imaging the cracks in 3D. This test would be used to further calibrate the lattice-transport model.

ACKNOWLEDGMENTS Founding for this work was provided by the Portuguese Foundation for Science and Technology (SFRH/BD/ 30435/2006). REFERENCES Schlangen, E. 1993. Experimental and numerical analysis of fracture processes in concrete. Dissertation. Delft University of Technology. Gérard, B. et al. 1996, Cracking and permeability of concrete under tension. Materials and Structures/Materiaux et Constructions, Vol. 29, April 1996, pp. 141–151. Schlangen, E. & Garboczi, E.J. 1996. New method for simulating fracture using an elastically uniform random geometry lattice. Int. J. Engineering Sci. Vol. 34, No. 10, pp.1131–1144. UK: Elsevier. Schlangen, E. & Garboczi, E.J. 1997. Fracture simulation of concrete using lattice models: Computational aspects. Engineering Fracture Mechanics Vol. 57, No. 2/3, pp.319–332. UK: Elsevier. CALFEM, A finite element toolbox to MATLAB, Version 3.3 Reference book. Division of Structural Mechanics and the Department of Solid Mechanics, Lund University, Sweden, 1999. Lilliu, G. & van Mier, J.G.M. 2000. Simulation of 3D crack propagation with the lattice model. Proceedings Materials Week 2000. Available in 2005-07-08 at: http://www.proceedings.materialsweek.org/proceed/ mw2000_634.pdf Vandersteen, K. 2002. Unsaturated water flow in fractured porous media. Dissertation. Catholic University of Leuven. Carmeliet, J. Delerue, J.-F. Vandersteen, K. Roels, S. 2004. Three-dimensional liquid transport in concrete cracks. Int. J. Num. Anal. Meth. Geomech. 28: p. 671–-687. Roels, A. Moonen, P. de Proft, K. Carmeliet, J. 2006, A coupled discrete-continuum approach to simulate moisture effects on damage processes in porous materials. International Journal for Computational Methods in Applied Mathematics 195(52): 7139–7153.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Modelling the CaO hydration in expansive concrete B. Chiaia, A.P. Fantilli, G. Ferro & G. Ventura Politecnico di Torino, Torino, Italy

ABSTRACT: In order to investigate the average expansion of concrete members having the same percentage of free lime and impurities, but not the same grain diameter, a size-effect theory is introduced. More specifically, by drawing an analogy with the structural size-effect, the expansion of lime hydration must show a characteristic length, whose dimension remains constant at all scales. Thus, if a grain of CaO is assumed to be a sphere, only a thin layer on the external surface hydrates and doubles the volume. As the thickness of such layer is constant, independently of lime manufacture, the expansion of CaO is proportionally lower in grains of larger diameter. This phenomenon, confirmed by the results of a 2D finite element model, agrees with some experimental observations. For practical application, a simplified chemo-mechanical model of lime expansion is eventually proposed, to predict the consequences of uncombined lime in concrete structures. 1

INTRODUCTION

Most of the burnt lime (CaO) contained in the Portland cement is generally hydrated at the moment of concrete mixing, in accordance with the well known ‘‘slaking’’ reaction: CaO + H2 O ⇒ Ca(OH)2 + 15.5 kcal

(1)

Eq. (1) is a strongly exothermic reaction, in which the formation of the calcium hydroxide has a larger volume (expansive reaction). The ratio of volume change from CaO particle to Ca(OH)2 is 33/17 ∼ =2 (Nagataki & Gomi, 1998). Sometimes, the presence of residual grains of burnt lime, not yet hydrated, can be detected within the concrete members. In such particles, a very slow slaking reaction takes place, and continues for months after concrete casting. Generally, this lime consists of high-density CaO grains, originated by a deadburn process, which hydrate very slowly because of a reduced porosity. If during the hydration of cement the formation of the calcium hydroxide from such CaO grains is incomplete, the slaking process [Eq. (1)] continues also during the concrete hardening. In some cases the expansion by hydration of CaO, similarly to the alkali aggregate reaction, can cause concrete cracking or the so-called pop-out phenomenon (Lee & Lee, 2009). The latter consists of the rising, and the subsequent expulsion, of concrete around the grains closer to the concrete surface. The presence of magnesium oxide (MgO), which derives from dolomitic limestone (MgCO3 ), can intensify the deleterious effects of expansion by hydration, as it shows a slower hydration.

Obviously, if expansion takes place in an unrestrained concrete, it can cause cracking. Conversely, if the expansion is properly restrained, its magnitude reduces and a prestress develops. When the compressive stress is between 0.1 and 0.7 MPa, which is adequate to compensate the tensile stress from restrained drying shrinkage, a sort of shrinkage compensation can be achieved (Metha & Monteiro, 2006). This compensation has been largely used in making crack-free concrete structures, such as pavements and slabs. It must be remarked that the mechanism of shrinkage compensation generated by expansive agents like CaO, differs from that produced by the Surface Reducing Admixture (SRA). In fact, SRA, not considered in the presents work, reduces drying and/or autogenous shrinkage due to the decrease of the surface tension of water within the cement paste (Collepardi et al., 2005). Only the use of CaO-based expansive agents is here investigated. In accordance with Collepardi (2003), in concretes having the same hardening conditions and the same amount of reinforcement, the type of CaO plays a fundamental rule in reducing the deleterious effect of shrinkage. Precisely, although the slaking reaction [Eq. (1)] is always the same, the physical and chemical properties of calcium oxide can vary, as well as its expansion.

2

PROPERTIES OF CALCIUM OXIDE

X-ray diffraction reveals that a pure calcium oxide crystallizes in the cubic system depicted in Figure 1. The edges of the cube are about 4.8 Å in length, with calcium atoms located in between (Boynton, 1966). The physical and chemical properties of quicklime, which is the name of CaO when it is obtained by the

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Figure 1. Crystal structure (unit cell) of calcium oxide (Boynton, 1966).

calcination process (i.e., cooking of limestone at a temperature higher than 575◦ C), depend, respectively, on the cooking temperature (the so-called calcination temperature) and on the purity of limestone (CaCO3 ). 2.1 Physical properties of CaO The physical properties of CaO, such as porosity, bulk density, the dimensions of grains, and hydration rate are strongly related to the calcination temperature. Depending on the temperature reached in the kiln where limestone is cooked, hard-burned lime, averageburned lime and soft-burned lime can be obtained (Schiele & Berens, 1976). With respect to the hard-burned lime (resulting from high calcination temperature), soft-burned lime (resulting from low calcination temperature) shows the following properties: – – – – –

are generally made by CaO aggregation, and show cracked and corrugated surfaces. In this quicklime, large contact surfaces between grains, caused by a sintering process at high temperature, can be detected. In hard-burned limes, the diameter of grains can be larger than 10 μm, whereas the diameter of pores reaches 20 μm (Schiele & Berens, 1976). Table 1 shows the values of the main physical properties of the soft-burned, average-burned and hardburned limes, having the same unit weight (Schiele & Berens, 1976). The calcination temperature, the bulk density, and the specific surface as well, are strictly connected. In the case of soft-burned limes the specific surface can be three times larger than that measured in hard-burned limes. As a consequence, also the average diameter of grains increases in hard-burned limes. All these observations are extremely important in evaluating the chemical reactivity of lime (and the consequent expansion). It is unanimously accepted that the surface area is the most reliable parameter which can be used to evaluate the property of quicklime hydration: the higher this value, the higher the chemical reactivity. This has been confirmed by various chemical tests (Boynton, 1966). Also fineness, as determined by particle size or grain diameter, is indicative of the degree of reactivity. However, both the specific surface and the mean diameter of grains are related to the calcination temperature. As reported in Figure 2, if calcination temperature increases, but is maintained for the same duration, porosity (Fig. 2a) and specific surface area Ss (Fig. 2b) decreases, while grain diameter size (CaO ) increases (Fig. 2c). Thus, hard-burned limes are generally characterized by moderate to low chemical reactivity (Boynton, 1966; Schiele & Berens, 1976). 2.2

Chemical properties of CaO

The hydration or slaking reaction of quicklime [Eq. (1)] is strongly related to the temperature. Up to 100◦ C, the rate of reaction increases. In such situations, as supplementary heat is developed, a further increment of temperature can be generated and, consequently, the chemical reaction is accelerated.

Lower grain dimensions. Larger specific surface. Higher porosity (but a lower dimension of pores). Lower bulk density. Higher chemical reactivity.

Generally, soft-burned lime is made of little grains, whose maximum diameter varies between 1 μm and 2 μm, and shows a bulk density of about 1.51 g/cm3 . Most of the pores have a diameter comprised between 0.1 μm and 1 μm. If the calcination temperature increases, an agglomeration of the single grains can be observed, although pores increase in diameter (all the observed cavities are in the range 1 ÷10 μm). These effects are even more pronounced in the case of hard-burned limes, which can show a bulk density of 2.44 g/cm3 . Moreover, due to the high calcination temperature, grains

Table 1. Some of the physical properties measured in the most common limes (Schiele & Berens, 1976). Type of cooking

Unit weight (g/cm3 ) Bulk density (g/cm3 ) Porosity (%) Specific surface (m2 /g)

soft

average

hard

3.35 1.5÷1.8 46÷55 >1.0

3.35 1.8÷2.2 34÷46 0.3÷1.0

3.35 >2.2 <34 <0.3

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Figure 2. Some of the physical properties of CaO as a function of calcination temperature (Schiele & Berens, 1976): a) Porosity vs. calcination temperature; b) Specific surface vs. calcination temperature; c) Diameter of grains vs. calcination temperature.

When the temperature is higher than 100◦ C, the rate of hydration progressively reduces. In particular, at 540◦ C, Eq. (1) becomes a reversible reaction, as dehydration, affected by subsequent heating, recreates quicklime and volatilize the water of hydration as vapour (Schiele & Berens, 1976). Dry carbon dioxide (CO2 ) does not react with quicklime at ordinary temperature, although slight traces of recarbonation have been observed according to the following reaction (Boynton, 1966): 42.5 kcal + CaCO3 ⇔ CaO + CO2

(2)

If temperatures are increased, recarbonation will start. However, no adsorption of CO2 occurs until about 400◦ C, whereas at about 600◦ C the recarbonation is very pronounced and rapid. But the complete process, even at higher temperatures, does not occur. This is due to the fact that adsorption is a surface phenomenon, when a shell of calcium carbonate forms around the CaO particle. Thus, it is more difficult for the CO2 molecule to penetrate through the narrowing pores of the previously formed CaCO3 and unite with the interior CaO molecules. Finally, recarbonation varies depending upon the size of particle. The smaller the CaO particles, the greater the CO2 adsorption. The above observations are really important not only for the storage and transport of CaO, but also for the hardening stage of cement-based concrete. During this stage, only the slaking reaction of quicklime occurs, whereas no traces of recarbonation are generally observed. Usually, the quicklime is not exclusively made of pure calcium oxide; neither has derived from pure limestone. For these reasons, in the field of concrete construction, the so called hydraulic lime is generally used. It consists of a chemically impure lime

with hydraulic properties of varying extent, which possesses appreciable amounts of silica, alumina, and iron. The degree of hydraulicity of these limes depends on the type and the amount of the above-mentioned impurities. Precisely, hydraulic limes are classified into three different grades as follows (Boynton, 1966): feebly hydraulic lime, moderately hydraulic lime, and eminently hydraulic lime (so-called Roman lime). Traditionally, hydraulic limes have been defined by the Cementation Index (CI), which is evaluated on the base of the following scientific and empirical assumptions: 1. The hydraulic properties are imparted by the formation of compounds of calcium and magnesium with silica, alumina, and iron. 2. The silica combines with the lime molecularly to form tricalcium silicate (3CaO · SiO2 ). 3. Alumina combines with the lime to form dicalcium aluminate (2CaO · Al2 O3 ). 4. Magnesia reacts molecularly the same as lime, except at a slower rate. 5. Iron oxide has the same equivalent molecular reaction as alumina. According to these statements, the following cementation index formula can be derived (Boynton, 1966): CI =

2.8 × %SiO2 + 1.1 × %Al2 O3 + 0.7 × %Fe2 O3 %CaO + 1.4 × %MgO

(3)

Based on Eq. (3) the three groups of hydraulic limes are classified by the Cementation Index as follows: – Feebly hydraulic lime, when 0.3 ≤ CI ≤ 0.5. – Moderately hydraulic lime, when 0.5 < CI ≤ 0.7. – Eminently hydraulic lime, when 0.7 < CI ≤ 1.1.

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3

EXPERIMENTAL OBSERVATIONS ON CaO HYDRATION

The presence of high quantity of CaO within the concrete mixture can be due to: – Lime-based expansive agents introduced with the aim of preventing, or reducing, the cracking produced by restrained drying shrinkage (expansive concretes). – High quantity of lime contained in the Portland cement, resulting, for instance, from a wrong cement manufacture (expansive cements). In order to introduce a chemo-mechanical model capable of predicting the CaO expansion in a cementbased concrete, both expansive concretes and expansive cements are analyzed in the following sections. 3.1

Expansive concretes

As is well known, the final volume of hydrated concrete is less than the sum of the volumes of the components (i.e., cement, aggregates, additives, and water). Such chemical shrinkage is also called Le Chatelier contraction, in honour of the French scientist who first observed the phenomenon. The chemical shrinkage is ascribed to the loss of the so-called evaporable water, which does not react with the cement. However, before hardening, the chemical shrinkage is not constrained and, therefore, it coincides with the whole shrinkage of concrete mixture. As the rigid network of the hydrated products starts to develop, the chemical shrinkage is lower than that measured (and generally called autogenous shrinkage), because the stiffness of the concrete skeleton restrains the volume change (Metha & Monteiro, 2006). Deleterious effects produced by autogenous shrinkage are widely described in the literature (Metha & Monteiro, 2006; Collepardi, 2003), mainly in high strength concrete. Nevertheless, several mitigation strategies for reducing the autogenous shrinkage cracking have been introduced in the last years (Bentz & Jensen, 2004). This is the case of the expansive concretes, which can show a large expansion, able to balance or outweigh the shrinkage contraction (Taylor, 1997). From a practical point of view, expansive agents are mixed to the classical components of concrete. The additives that increase the formation of ettringite are generally called type-K expansive agents. Developed originally by Alexander Klein in the 1960s, they are generally made of calcium aluminate cements (Metha & Monteiro, 2006). Those based on the hydration of CaO [Eq. (1)] are called type-O expansive agents, named after the Onada Cement Co., Ltd. (Taiheiyo Cement Corp. at present) that in the early 1970s introduced for the first time such additives in Japan. The use of type-O

expansive agents is more advantageous with respect to ettringite-based additives, for the shorter period of wet curing time (1–2 days vs. 5–7 days) needed to develop the potential expansion (Collepardi et al., 2005). In what follows, Expan (2005) and HyperExpan (2005), nowadays merchandised by Taiheiyo Cement Corp. as type-O expansive agents, are taken into consideration. The percent compositions by mass of both the additives are reported in Table 2. If 30 kg of Expan (2005) or 20 kg of HyperExpan (2005) are added to a cubic meter of normal concrete, the autogenous shrinkage is drastically reduced (Fig. 3), without compromising the mechanical properties, such as the compressive strength (Nagataki & Gomi, 1998). According to Table 1, with respect to the lime usually contained in the Portland cement, percentages of 0.84% (Expan, 2005) and 0.62% (HyperExpan, 2005) in weight of CaO are added to the concrete, respectively. As Figure 3 shows, the shrinkage observed when Expan (2005) and HyperExpan (2005) are mixed as additives, is about three times lower than that observed in normal concrete. This difference, quantified in terms of strain (εCaO ), is due to the surplus of CaO, and is generally constant after 60 days. 3.2

Expansive cements

The CaO contained in a Portland cement is generally hard-burned by the high temperature reached in the kiln where the clinker is cooked. Thus, according to Figure 2, this lime is characterized by a low chemical reactivity. However, if the content of CaO is higher than 1% in weight, expansion by hydration can Table 2. Percent compositions by mass of Expan (2005) and HyperExpan (2005).

Expan HyperExpan

CaO (%)

MgO (%)

SiO2 (%)

Al2 O3 (%)

Fe2 O3 (%)

67.3 74.0

0.4 0.5

9.6 4.2

2.5 1.1

1.3 1.3

Figure 3. The shrinkage of concrete with Expan (2005) or HyperExpan (2005) are used as additives.

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cause pop-out phenomenon (Lee & Lee, 2009). Conversely, a lower content of calcium oxide does not lead to deleterious expansions (Metha & Monteiro, 2006). Sometimes, elevate CaO contents (higher than 1% by mass) do not generate remarkable expansions. This is the case of the four cements reported in Table 3 and named S1, S2, S3 and S4, respectively. As the lime is hard-burned, the grain diameter CaO = 10 μm is relatively high (see Fig. 2c). In accordance with the mix design indicated in Table 4, a concrete specimen is cast with each cement. The geometrical properties of the four specimens are reported in Figure 4a. As they are used to measure the free shrinkage strain in hardened concrete (UNI 11307, 2008), all the dimensions were chosen on the base of the indications given by UNI EN 12390-1 (2002). After casting, the specimen were kept in the steel formworks for 24 ± 1 hours at a temperature of 20 ± 2◦ C, covered with waterproof sheets. As depicted in Figure 4a, steel basements were located on both the ends of the specimens, in order to measure the variation of the length L due to shrinkage. In particular, Table 3. Lime content and diameter of CaO grains in the cements considered in the present paper. Specimen

Type of cement

CaO (%)

CaO (μm)

S1 S2 S3 S4

CEM V/A (S-P) 32.5 R CEM V/A (S-P) 32.5 R CEM V/A (S-P) 32.5 R CEM III/A 32,5 N LH

2.46 3.09 3.39 4.26

10 10 10 10

Table 4. and S4.

The constituents of the specimens S1, S2, S3

Water content (kg/m3 ) Cement content (kg/m3 ) Diameter of coarse aggregates (mm) Unit weight (kg/m3 )

the length decrements L (and therefore the strains L/L) of each specimen have been evaluated at 2, 3, 7, 28, 60 and 90 days from casting (UNI 11307, 2008). The measurements, as reported in the strain-time diagram of Figure 4b, do not show any relevant shrinkage compensation. Only the specimen made with the cement S4 shows a shrinkage contraction lower (of about 20%) than those of the other specimens. In other words, when the content of CaO is about 3% in weigth, ordinary shrinkage takes place (i.e., 0.5 mm/m after 90 days). If the chemical reactivity of the quicklime contained in the expansive cements (Table 3) has to be defined, the difference (εCaO in Fig. 4b) between the average shrinkage strain of the specimens S1, S2 and S3 and that of the specimen S4 needs to be analyzed. In fact, εCaO is due to the surplus of CaO (1.28%) contained by the specimen S4 (4.26%) with respect to the average content of lime detected in the other specimens (2.98%). Although the extra amount of CaO (i.e., about 1.28%), only εCaO = 0.1 mm/m was observed in the considered expansive cements after 90 days (see Fig. 4b). Conversely, when Expan (2005) or HyperExpan (2005) are added to normal concrete, εCaO = 0.2 mm/m is obtained at 90 days (see Fig. 3) just with a minimum surplus of CaO (0.7%). As shrinkage compensation, the considered expansive cements are definitely less effective than the expansive concretes (if made with the additive agents described in the previous section). The lower expansion of these cements can be ascribed to the reduced chemical reactivity of the CaO, confirmed by the presence of large grain diameter (10 μm, Table 3). 4

175 350 20 2300

MODELLING THE EXPANSION OF A SINGLE CaO GRAIN

From the observations reported in the previous Sections, if a constant CaO weight content is assumed

Figure 4. Shrinkage contraction in concrete made with expansive cements: a) geometrical properties of the specimens (UNI EN 12390-1, 2002); b) strain vs. time diagrams measured for the specimens S1, S2, S3, and S4 (UNI 11307, 2008).

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a)

b)

Figure 5. A CaO particle in a concrete cell. a) the hydration of a CaO grain; b) Finite Element Model for the evaluation of the particle dimensional variation (Lusas FEM model).

and the typology of the chemical reaction does not change, it follows that grains with larger diameter give a lower mean expansion and a dimensional scale effect is therefore evident. Similarly to structural size effects (see e.g. Rilem TC QFS, 2004), the expansion phenomenon must show, at different scales, a dimension (or characteristic length) that remains always unchanged. In the present case, from the results of the previous Sections, it can be reasonably assumed that in a spherical CaO particle only a small external layer of thickness s is hydrated and doubles its volume (Fig. 5a). The particle expansion phenomenon is examined (similarly to approached introduced by Ulm et al., 2000) with a plane stress model. The CaO grain can be represented by a circular particle in a square cell with size B representing the concrete matrix. Given the CaO particle size, the concrete cell size B is determined by assuming a CaO constant weight content equal to 1.5%. Consider two limes originated by two production processes, one at a calcination temperature of 1000◦ and the other at a temperature of 1200◦ . From the data illustrated in Figure 2, the CaO particles will have different physical characteristics (the diameter in particular), and consequently the concrete cells will have different sizes. Table 5 summarizes the simulation data and the examined models. Assuming a constant hydration thickness s = 0.02 μm, the simulation cells will have the geometry reported in the first two columns of Table 5 (Models I and II). As a second hypothesis, it is assumed that the hydration thickness s is a given percentage (5%) of the cell diameter. The third and fourth columns of Table 5 report the relevant data (Models III and IV). Table 5 reports all the data used in the numerical finite elements models as well as the computed mean strain in the cell.

Table 5. Data for the considered finite element models and computed mean strain in the cell. Model

Calcination temperature (◦ ) CaO diameter - CaO - (μm) CaO unit weight (g/cm3 ) Concrete unit weight (g/cm3 ) CaO weight percentage (%) Young moduli ratio Cell side - B- (μm) Hydration thickness s- (μm) Mean strain (%)

4.1

I

II

III

IV

1000 1.55 2.1 2.4 1.5 0.62 12 0.02 0.08

1200 10 2.65 2.4 1.5 0.62 68.8 0.02 0.02

1000 1.55 2.1 2.4 1.5 0.62 12 0.078 0.27

1200 10 2.65 2.4 1.5 0.62 68.8 0.5 0.34

2D-FEM analyses

The numerical simulations of the CaO particle expansion in the concrete matrix have been run by the finite element program LUSAS® , version 14.1 (Lusas, 2008). The analysis has been preformed in plane stress condition differentiating the assumed finite elements for the CaO particle and the concrete, to better capture the stress/strain fields in the concrete cell. The CaO particle has been discretized by triangular constant strain elements (Fig. 5b). This allowed to accurately describing the round geometry of the particle with a polar symmetric mesh. A high accuracy is not required, being the particle much stiffer than the surrounding concrete and subjected to uniform expansion, as better specified in the following. The concrete cell has been discretized by 4 nodes mixed (assumed strain) elements, reproducing linear displacement and strain fields. This allows for a better solution approximation in the cell, so that the concrete strain field, which is the target of the analysis, is accurately determined.

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Figure 6. strains.

Model I: Process temperature 1000◦ C, s = 0.02 μm. From left: model undeformed and deformed mesh, principal

Figure 7. strains.

Model II: Process temperature 1200◦ C, s = 0.02 μm. From left: model undeformed and deformed mesh, principal

Figure 8. strains.

Model III: Process temperature 1000◦ C, s = 5% CaO . From left: model undeformed and deformed mesh, principal

Figure 9. strains.

Model IV: Process temperature 1200◦ C, s = 5% CaO . From left: model undeformed and deformed mesh, principal

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The simulation cell has been constrained to exclude rigid body motions without any effect on the stress/ strain fields. The CaO particle expansion has been modelled through a uniform thermal expansion giving, in the free (unconstrained) particle, the expansion listed in Table 5, and illustrated in Figure 5b. Then, the computed thermal load has been applied to the complete particle/cell model (Fig. 5a). The expansion of the particle generates a compression field in the particle and a tension field in the cell. Both fields are of course in a self equilibrium state. Figures 6–9 report the results of the four models described in Table 5, showing the discretization mesh, the deformed geometry and the principal strains. It can be observed in the first two models (I and II, constant hydration thickness), that the higher process temperature implies a significantly lower mean expansion (see the last row of Table 5). This can be explained as due to the higher particle diameter. Therefore, a larger expansion is associated with the smaller particle diameter, in accordance with the experimental results described in the previous Sections. Analyzing the results of models III and IV, where the hydration thickness is assumed equal to 5% of the particle diameter, the opposite mean cell expansion trend is obtained. Comparing the last row of Table 5, a greater expansion is obtained for the larger particle diameter, i.e. at higher process temperature. Therefore, this result is in contrast to the experimental evidences described by Boynton (1966). For complying with experimental results it must be concluded that the CaO hydration process is characterized by a constant hydration depth, regardless of the grain diameter, and therefore a size effect on the expansion phenomenon clearly arises.

5

A SIMPLYFIED CHEMO-MECHANICAL MODEL OF CAO EXPANSION IN CEMENT-BASED MATERIALS

According to the diagrams of Figure 2b, Ss depends on the calcination temperature, as well as on the porosity (Fig. 2a) and the diameter CaO of the grains (Fig. 2c). The evolution in time of the CaO hydration, and its subsequent expansion, is difficult to evaluate, during the hardening stage of concrete. For practical applications, it is therefore more interesting to quantify the difference, εCaO , between the average shrinkage strain of expansive concrete and that of ordinary concrete, after a long period of time from casting (in the present case 90 days). In fact, as observed in the Figures 3–4, εCaO stabilizes and assumes a constant value after about 60 days. Under the above condition, Eq. (4) can be simplified as follows: εCaO = α PCaO Ss (1 − CI )

The coefficient α = const. of Eq. (5) includes all the physical variables of Eq. (4) which are always constant, such as the hydration thickness s (which remains unknown), and those that becomes constant for long time observations. To calibrate the value of α, the experimental data measured by Taiheiyo Cement Corp. for Expan (2005) and HyperExpan (2005) are here taken into consideration. Despite the different amount of CaO contained by these two expansive agents, the same εCaO has been measured after 90 days from casting (see Fig. 3). Such values, together with the relative weight content, the specific surface and the cementation index of CaO, are reported in Table 6. In the same Table, the values of α computed with Eq. (5) are also reported in the last column. Both for Expan (2005) and HyperExpan (2005), the same coefficient α = 1.2 · 10−5 g/cm2 has been obtained. Therefore, the hypothesis that such a coefficient must be a constant is definitely proved. 5.1

A chemo-mechanical model of the CaO expansion within a cement-based concrete can be summarized by the following formula: εCaO = ε(PCaO , CI , s, Ss , t)

(5)

(4)

where, εCaO = expansion produced by CaO in hydrated concrete; PCaO = weight percentage of CaO content with respect to the global weight of concrete; CI = degree of hydraulicity of CaO [i.e., the Cementation Index computed with Eq. (3)]; s = hydration thickness in a single CaO grain (assumed to be constant); t = time passed from the concrete cast; Ss = specific surface of CaO (in cm2 /g).

Experimental validation on expansive cements

Eq. (5) can be also used to justify the behaviour of the specimens S1, S2, S3, and S4 made with the expansive cements described in the Section 3.1. Specifically, the specimens S4, despite the surplus of CaO (PCaO = 1.28 %) with respect to the other specimens (see Table 3), shows a relatively low expansion Table 6. Chemical properties of concretes made with Expan (2005) and HyperExpan (2005).

Expan Hyper Expan

Ss (cm2 /g)

PCaO (%)

εCaO (%)

CI

α (g/cm2 )

3500 3450

0.84 0.62

0.019 0.020

0.45 0.18

1.2 · 10−5 1.2 · 10−5

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be better analyzed. In particular, the hydration thickness, assumed to be a characteristic length of the CaO hydration, has to be experimentally measured. ACKNOWLEDGEMENT The authors wish to thank ARCOS Engineering Company for the technical support given in the experimental analyses reported in the Section 3.2. REFERENCES Figure 10. Comparison between the theoretical value of CaO , obtained with the proposed chemo-mechanical model, and that measured by Schiele & Berens (1976).

(εCaO = 1.2 · 10−4 at 90 days). For the specimen S4, if the values α1.2·10−5 g/cm2 [previously obtained for Expan (2005) and HyperExpan (2005)], CI = 0.315 [in absence of direct measurements, CI is assumed as the average of the values computed with Eq. (3) for Expan (2005) and HyperExpan (2005)], Eq. (5) gives a specific surface Ss = 1200 cm2 /g. As reported in Figure 10, where for the sake of clarity the diagrams of Figures 2b-c are depicted, such specific surface is obtained when the calcination temperature is 1200 ◦ C. At this temperature, a grain diameter CaO = 10μm was measured by Schiele & Berens (1976), which coincides with that of the expansive cements S1, S2, S3, and S4 reported in Table 3.

6

CONCLUSIONS

In the present paper, with the aim of investigating the behaviour of expansive cements, and predicting some experimental results, a chemo-mechanical model has been introduced. On the base of the numerical and experimental observation, the following conclusions can be drawn: 1. The cements S1, S2, and S3 show an ordinary shrinkage strain, despite the high content of uncombined CaO (i.e., 2.98%). 2. The reduced shrinkage can be ascribed to the low chemical reactivity of the CaO, characterized by a low value of the specific surface and large grains (CaO = 10 μm). 3. Only the specimen S4, having 4.26% in weight of CaO, shows a low shrinkage strain, which is correctly predicted by the proposed chemo-mechanical model [Eq. (5)].

Bentz, D.P. & Jensen, O.M. 2004. Mitigation strategies for autogenous shrinkage cracking. Cement and Concrete Composites, 26(6), 677–685. Boynton, R.S. 1966. Chemistry and technology of lime and limestone. New York: John Wiley & Sons. Collepardi, M. 2003. New concrete. Villorba: Tintoretto [in Italian]. Collepardi, M., Borsoi, A., Collepardi, S., Ogoumah Olagot, J.J. & Troli, R. 2005. Effects of shrinkage reducing admixture in shrinkage compensating concrete under nonwet curing conditions. Cement and Concrete Composites, 27 (6), 704–708. Expan. 2005. Expan, expansive additive for concrete. Taiheiyo Materials Corporation, http://www.taiheiyo-m.co. jp/en/images/802/2.pdf. HyperExpan. 2005. HyperExpan, low dosage type expansive additive for concrete. Taiheiyo Materials Corporation, http://www.taiheiyo-m.co.jp/en/images/204/2.pdf. Lee, M.-H. & Lee, J.-C. 2009. Study on the cause of pop-out defects on the concrete wall and repair method. Construction and Building Materials, 23(1), 482–490. Lusas. 2008. Modeller Reference Manual. Surrey, Lusas, www.lusas.com. Mehta, P.K. & Monteiro, P.J.M. 2006. Concrete— Microstructure, Properties, and Materials. New York: McGraw-Hill. Nagataki, S. & Gomi, H. 1998. Expansive admixtures (mainly ettringite). Cement and Concrete Composites, 20(2–3), 163–170. Schiele, E. & Berens, L.W. 1976. La calce: calcare, calce viva, idrato di calcio, fabbricazione, caratteristiche, impieghi. Milano: Edizioni Tecniche ET [in Italian]. Taylor, H.F.W. 1997. Cement Chemistry. London: Thomas Telford Ltd. Rilem, T.C. QFS. 2004. Quasibrittle fracture scaling and size effect—Final report. Materials and Structures, 37, 547–568. Ulm, F.-J., Coussy, O., Kefei, L. & Larive, C. 2000. ThermoChemo-Mechanics of ASR Expansion in Concrete Structures. ASCE Journal of Engineering Mechanics, 126(3), 233–242. UNI 11307. 2008. Testing for hardened concrete—Shrinkage determination. UNI EN 12390-1.2002. Testing hardened concrete—Shape, dimensions and other requirements for specimens and moulds.

Finally, the coefficient α of Eq. (5), here defined on the basis of previous experimental data, needs to

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

A two-scale approach for fluid flow in fracturing porous media René de Borst Department of Mechanical Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands

Julien Réthoré Université de Lyon, CNRS INSA-Lyon, LaMCoS UMR, France

Marie-Angèle Abellan LTDS-ENISE - UMR CNRS, Saint-Etienne, France

ABSTRACT: A derivation is given of two-scale models that are able to describe deformation and fluid flow in a progressively fracturing porous medium. From the micromechanics of the flow in the cavity, identities are derived that couple the local momentum and the mass balances to the governing equations for a porous medium, which are assumed to hold on the macroscopic scale. By exploiting the partition-of-unity property of the finite element shape functions, the position and direction of the fractures are independent from the underlying discretization. The finite element equations are derived for this two-scale approach and integrated over time. The resulting discrete equations are nonlinear due and the nonlinearity of the coupling terms and the possible use of a cohesive crack model. Examples are given to show the versatility of the approach. 1

INTRODUCTION

Since the pioneering work of Terzaghi (1943) and Biot (1965) the flow of fluids in deforming porous media has received considerable attention. A general framework for accommodating multi-field problems has been presented by Jouanna and Abellan (1995), and more recently, Lewis and Schrefler (1998) have published a text book on this topic which is crucial for understanding and predicting the physical behaviour of many systems of interest, including durability of concrete structures. In spite of the importance of the subject, flow in damaging porous media has received much less attention. Yet, the presence of damage, such as cracks, faults, and shear bands, can markedly change the physical behaviour. Furthermore, the fluid can transport contaminants which can dramatically reduce the strength of the solid skeleton. To account for such phenomena, the fluid flow must be studied also in the presence of discontinuities in the solid phase. The physics of the flow within such discontinuities can be very different from that of the interstitial fluid in the deforming bulk material. These differences affect the flow pattern and therefore also the deformations in the vicinity of the discontinuity. As we will show at the end of the paper, the local differences in flow characteristics can even influence the flow and deformations in the entire body of interest. In this contribution, we will describe a general numerical methodology to capture deformation and

flow in progressively fracturing porous media, summarizing and unifying the recent work reported in (de Borst et al. 2006, Réthoré et al. 2007a, b, 2008). The model allows for flow inside an evolving crack to be in the tangential direction. This is achieved by a priori adopting a two-scale approach. At the fine scale the flow in the cavity created by the (possibly cohesive) crack can be modelled in various ways, e.g. as a Stokes flow in an open cavity, or using a Darcy relation for a damaged porous material. Since the cross-sectional dimension of the cavity is small compared to its length, the flow equations can be averaged over the width of the cavity. The resulting equations provide the momentum and mass couplings to the standard equations for a porous medium, which are assumed to hold on the macroscopic scale. Numerically, the two-scale model which ensues, imposes some requirements on the interpolation of the displacement and pressure fields near the discontinuity. The displacement field must be discontinuous across the cavity. Furthermore, the micromechanics of the flow within the cavity require that the flow normal to the cavity is discontinuous, and in conformity with Darcy’s relation which, at the macroscopic scale, is assumed to hold for the surrounding porous medium, the normal derivative of the fluid pressure field must also be discontinuous from one face of the cavity to the other. For arbitrary discretizations, these requirements can be satisfied by exploiting the partition-of-unity property of finite element shape functions (Babuska & Melenk 1997), as has been done

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successfully in applications to cracking in single-phase media (Belytschko & Black 1999, Moës et al. 1999, Wells & Sluys 2001, Wells et al. 2002a, b, Remmers et al. 2003, 2008, Réthoré et al. 2005a, b). To provide a proper setting, we will first briefly recapitulate the governing equations for a deforming porous medium under quasi-static loading conditions. The strong as well as the weak formulations will be considered, since the latter formulation is crucial for incorporating the micromechanical flow model properly. This micromechanical flow model is discussed in the next section, where it will be demonstrated how the momentum and mass couplings of the micromechanical flow model to the surrounding porous medium can be accomplished in the weak formulation. Example calculations are given of stationary and propagating cracks. 2

BALANCE EQUATIONS

∂(ρπ vπ ) + ∇ · (ρπ vπ ⊗ vπ ) ∂t (1)

with σ π the stress tensor, ρπ the apparent mass density, and vπ the absolute velocity of constituent π. As in the remainder of this paper, π = s, f , with s and f denoting the solid and fluid phases, respectively. Further, g is the gravity acceleration and pˆ π is the source of momentum for constituent π from the other constituent, which takes into account the possible local drag interaction between the solid and the fluid. Evidently, the latter source terms must satisfy the momentum production constraint:  pˆ π = 0 (2) π =s, f

We now neglect convective, gravity and acceleration terms, so that the momentum balances reduce to: ∇ · σ π + pˆ π = 0

(3)

Adding both momentum balances, and taking into account eq. (2), one obtains the momentum balance for the mixture: ∇ ·σ = 0

∂ρπ + ∇ · (ρπ vπ ) = 0 ∂t

(4)

(5)

Again neglecting convective terms, the mass balances can be simplified to give: ∂ρπ + ρπ ∇ · vπ = 0 ∂t

(6)

We multiply the mass balance for each constituent π by its volumetric ratio nπ , add them and utilize the  constraint π =s, f nπ = 1 to give: ∇ · vs + nf ∇ · (vf − vs ) +

We consider a two-phase medium subject to the restriction of small displacement gradients and small variations in the concentrations (Jouanna & Abellan 1995). Furthermore, the assumptions are made that there is no mass transfer between the constituents and that the processes which we consider, occur isothermally. With these assumptions, the balances of linear momentum for the solid and the fluid phases read: ∇ · σ π + pˆ π + ρπ g =

where the stress is, as usual, composed of a solid and a fluid part, σ = σ s + σ f . In a similar fashion as for the balances of momentum, one can write the balance of mass for each phase as:

nf ∂ρf ns ∂ρs + = 0 (7) ρs ∂t ρf ∂t

The change in the mass density of the solid material is related to its volume change by: ∇ · vs = −

Ks ns ∂ρs Kt ρs ∂t

(8)

with Ks the bulk modulus of the solid material and Kt the overall bulk modulus of the porous medium. Using the Biot coefficient, α = 1 − Kt /Ks (Lewis & Schrefler 1998), this equation can be rewritten as (α − 1)∇ · vs =

ns ∂ρs ρs ∂t

(9)

For the fluid phase, a phenomenological relation is assumed between the incremental changes of the apparent fluid mass density and the fluid pressure p (Lewis & Schrefler 1998): nf 1 dρf dp = Q ρf

(10)

with the overall compressibility, or Biot modulus α − nf nf 1 = + Q Ks Kf

(11)

where Kf is the bulk modulus of the fluid. Inserting relations (9) and (10) into the balance of mass of the total medium, eq. (7), gives: α∇ · vs + nf ∇ · (vf − vs ) +

1 ∂p =0 Q ∂t

(12)

The field equations, i.e. the balance of momentum of the saturated medium, eq. (4), and the balance of

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mass, eq. (12), are complemented by the boundary conditions

with T the material tangent stiffness matrix of the discrete traction-separation law:

n · σ = tp ,

T=

vs = vp

(13)

which hold on complementary parts of the boundary ∂t and ∂v , with  = ∂ = ∂t ∪ ∂v , ∂t ∩ ∂v = ∅, tp being the prescribed external traction and vp the prescribed velocity, and nf (vf − vs ) · n = qp ,

p = pp

(14)

which hold on complementary parts of the boundary ∂q and ∂p , with  = ∂ = ∂q ∪ ∂p and ∂q ∩ ∂p = ∅, qp and pp being the prescribed outflow of pore fluid and the prescribed pressure, respectively. The initial conditions specify the displacements uπ , the velocities vπ , and the pressure field at t = 0, uπ (x, 0) = uπ0 , vπ (x, 0) = vπ0 , p(x, 0) = p0 3

(15)

CONSTITUTIVE EQUATIONS

δn =0

with σ the stress across the fracture process zone. Evidently, cohesive-zone models as defined above are equipped with an internal length scale, since the quotient Gc /E, with E a stiffness modulus for the surrounding continuum, has the dimension of length. In a standard manner we adopt Darcy’s relation for the flow in the bulk of the porous medium, nf (vf − vs ) = −kf ∇p

(17)

(18)

with κ a history parameter. For linear elastic fracture mechanics this contribution vanishes. After linearization, necessary to use a tangential stiffness matrix in an incremental-iterative solution procedure, one obtains: t˙d = Tδ˙

with kf the permeability coefficient. The equations for the flow in the cavity, which close the initial value problem, will be detailed in a subsequent section. 4

¯ tan has been used. In where the notation Dtan = ns D the examples, a linear-elastic behaviour of the bulk material has been assumed, and we have set Dtan = D, the linear-elastic stiffness tensor. At the discontinuity d a discrete relation holds between the interface tractions td and the relative displacements δ : td = td (δδ , κ)

(22)

(16)

¯ tan is the fourth—order tangent stiffness tenwhere D sor of the solid material and the d - symbol denotes a small increment. Since the effective stress in the solid skeleton is related to the partial stress by σ s = σ s /ns , the above relation can be replaced by σ s = Dtan : d s dσ

(20)

A key element in cohesive zone models is the presence of a work of separation or fracture energy, Gc , which governs crack growth and enters the interface constitutive relation (18) in addition to the tensile strength ft . It is defined as the work needed to create a unit area of fully developed crack:  ∞ Gc = σ dδn (21)

The effective stress increment in the solid skeleσ s is related to the strain increment d s by an ton, dσ incrementally linear stress-strain relation for the solid skeleton, ¯ tan : d s σ s = D dσ

∂td ∂td ∂κ + ∂δδ ∂κ ∂δδ

(19)

WEAK FORM

To arrive at the weak form of the balance equations, we multiply the momentum balance (4) and the mass balance (12) by kinetically admissible test functions for the displacements of the skeleton, η , and for the pressure, ζ . After substitution of Darcy’s relation for the flow in the porous medium, eq. (22), integrating over the domain  and using the divergence theorem then leads to the corresponding weak forms:    (∇ · η ) · σ d + [[ηη · σ ]] · nd d = η · tp d 

d



(23) and   − αζ ∇ · vs d + kf ∇ζ · ∇p d 

 −



ζ Q−1 p˙ d +

 =







d

nd · [[ζ nf (vf − vs )]]d

ζ n · qp d

(24)

Because of the presence of a discontinuity inside the domain , the power of the external tractions

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on d and the normal flux through the faces of the discontinuity are essential features of the weak formulation. Indeed, these terms enable the momentum and mass couplings between the discontinuity—the microscopic scale—and the surrounding porous medium—the macroscopic scale. The momentum coupling stems from the tractions across the faces of the discontinuity and the pressure applied by the fluid in the discontinuity onto the faces of the discontinuity. We assume stress continuity from the cavity to the bulk, so that we have: σ · nd = td − pnd

of mass and momentum at the microscale must be considered. Different assumptions can be made for the cavity. First, we will assume an open cavity which is totally filled with a Newtonian fluid. Then, the balance of mass for the ‘micro’-flow in the cavity reads: ρ˙f + ρf ∇ · v = 0 subject to the assumptions of small changes in the concentrations and that convective terms can be neglected, cf. eq. (6). We assume that the first term can be neglected because the problem is monophasic in the cavity and the velocities are therefore much higher than in the porous medium. With this assumption, and focusing on a two-dimensional configuration, the mass balance inside the cavity simplifies to:

(25)

with td the cohesive tractions, which are given by eq. (18). Therefore, the weak form of the balance of momentum becomes:   (∇ · η ) · σ d + [[ηη ]] · (td − pnd )d 

d

 =



η · tp d

(26)

Since the tractions have a unique value across the discontinuity, the pressure p must have the same value at both faces of the discontinuity, and, consequently, this must also hold for the test function for the pressure, ζ . Accordingly, the mass transfer coupling term for the water can be rewritten as follows:   − αζ ∇ · vs d + kf ∇ζ · ∇p d 

 −



=

∂v ∂w + =0 ∂x ∂y

−1



ζ Q p˙ d +  

with ν = v · td and w = v · nd the tangential and normal components of the fluid velocity in the discontinuity, respectively, see Figure 1. Accordingly, the difference in the fluid velocity components that are normal to both crack faces is given by:  [[wf ]] = −

ζ nd · nf [[vf − vs ]]d

ζ n · qp d

v(y) =

(27)

where qd = nf [[vf − vs ]] represents the difference in the fluid fluxes through both faces of the discontinuity. The above identity for the coupling of the mass transfer can be interpreted as follows. Part of the fluid that enters the cavity through one of its faces flows away tangentially, that is in the cavity. Therefore, the fluid flow normal to the cavity is discontinuous. Because the fluid flow between the cavity and the surrounding porous medium has to be continuous at each of the faces of the discontinuity, and because the fluid velocity is related to the pressure gradient via Darcy’s law, the gradient of the pressure normal to the discontinuity must be discontinuous across the crack.

h

−h

∂v dy ∂x

(29)

To proceed, the velocity profile of the fluid flow inside the discontinuity must be known. From the balance of momentum for the fluid in the cavity and the assumption of a Newtonian fluid, the following velocity profile results:



d

(28)

1 ∂p 2 (y − h2 ) + vf 2μ ∂x

where an integration has been carried out from y = −h to y = h and μ the viscosity of the fluid. The essential boundary v = vf has been applied at both faces of the cavity, and stems from the relative fluid velocity in the porous medium at y = ±h: vf = (vs − nf −1 kf ∇p) · td

y

2h

n Γd t Γd

5

(30)

x

MICRO-MACRO COUPLING

To quantify the influence of the ‘micro’-flow inside the discontinuity on the ‘macro’-scale, the balances

Figure 1.

Geometry and local coordinates in cavity.

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Substitution of eq. (30) into eq. (29) and again integrating with respect to y then leads to: 2 ∂ [[wf ]] = 3μ ∂x



 ∂vf ∂p 3 h − 2h ∂x ∂x

h

−h

∂ ∂x

+

(31)

nf (vf − vs ) = −kd



h −h

−h

 α∇ · vs dy = 

=

h

−h

α

h −h

 α

∂vs ∂ws + ∂x ∂y

∂vs dy − α[[ws ]] ∂x

∂p ∂x

(36)

nf ∇ · (vf − vs )dy

∂ = nf [[wf − ws ]] − ∂x

Because the width of the cavity 2h is negligible compared to its length, the mass balance is again enforced in an average sense over the cross section. For the first term, we obtain: h

(35)

with kd the permeability of the damaged, porous material inside the cavity. The dependence of the permeability inside the cavity, kd , on y will be negligible compared to that on x and accordingly, kd can be assumed not to depend on y. However, the decohesion inside the cavity will affect the permeability, and therefore kd = kd (h). Upon substitution of eq. (36) into eq. (35), the following relation is obtained:

1 ∂p α∇ · vs + nf ∇ · (vf − vs ) + =0 Q ∂t



∂h  nf (vf − vs )dy − nf vf (h) − vs (h) ∂x −h h

We now introduce Darcy’s relation projected onto the axis tangential to the crack,

nd · qd = nf [[wf − ws ]]

Another possibility for closure of the initial value problem is to assume that the cavity is partially filled with solid material. The mass balance for the fluid inside the cavity then reads:



∂(−h)  − nf vf (−h) − vs (−h) ∂x

This equation gives the amount of fluid attracted in the tangential fluid flow. It can be included in the weak form of the mass balance of the ‘macro’-flow to ensure the coupling between the ‘micro’-flow and the ‘macro’-flow. Since the difference in the normal velocity of both crack faces is given by [[ws ]] = 2 ∂h ∂t , the mass coupling term becomes:

  3 2 ∂vf ∂h 2h ∂ p 2h2 ∂p ∂h + − 2h − 2 = nf 3μ ∂x2 μ ∂x ∂x ∂x ∂t (32)

nf ∇ · (vf − vs )dy = nf [[wf − ws ]]



∂p kd dy ∂x −h h



  ∂p(h) ∂h ∂p(−h) ∂(−h) − kd + kd ∂x ∂x ∂x ∂x



(37)

At the faces of the cavity the permeability equals that of the bulk, and for continuity reasons ∂p(h) = ∂x ∂p(−h) , so that the second term becomes: ∂x

dy



(33)

h

−h

nf ∇ · (vf − vs )dy

where vs and ws are the component of the solid velocity tangential and normal to the crack, respectively. Because α depends on the capillarity pressure only, it can be assumed as constant over the cross section. Assuming furthermore that vs varies linearly with y, and defining vs = 12 (vs (h) + vs (−h)), the integral can be solved analytically:    h ∂vs ∂vs dy = 2αh (34) α ∂x ∂x −h

For the third term, neglecting variations of the pressure over the height of the cavity, one obtains:

Applying the same operations to the second term of eq. (29), the following expression ensues:

Recalling that the term nf [[wf − ws ]] can be identified as the coupling term nd · qd of the weak form of

= nf [[wf − ws ]] − 2h



h −h

∂kd (h) ∂p ∂ 2p − 2kd h 2 (38) ∂x ∂x ∂x

1 ∂p 2h ∂p dy = Q ∂t Q ∂t

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(39)

the mass balance, we obtain: 

nd · qd = −

2h ∂p ∂h ∂vs + 2α − 2αh Q ∂t ∂t ∂x

+ 2h

6

bands, in finite element models such that the discontinuous character of cracks and shear bands is preserved. With the standard small-strain assumption that the strain-rate field of the solid, ˙ s , is derived from the symmetric part of the gradient of the velocity field, we obtain:



∂kd (h) ∂p ∂ 2p + 2kd h 2 ∂x ∂x ∂x

(40)

˙ s = ∇ s v¯ s + Hd ∇ s vˆ s

away from the discontinuity d , with the superscript s denoting the symmetric part of the gradient operator. With respect to the interpolation of the pressure p we note that the fluid flow normal to the discontinuity can be discontinuous. Since the fluid velocity is related to the pressure gradient via Darcy’s relation, the gradient of the pressure normal to the discontinuity can therefore also be discontinuous across the discontinuity. Accordingly, the enrichment of the interpolation of the pressure must be such that the pressure itself is continuous, but has a discontinuous normal first spatial derivative. The distance function Dd defined as

DISCRETIZATION

A finite element method that can accommodate the propagation of discontinuities through elements was proposed by Belytschko and co-workers (Belytschko & Black 1999, Moës et al. 1999), exploiting the partitionof-unity property of finite element shape functions (Babuska & Melenk 1997). Since finite  element shape functions ϕj form partitions of unity, nj=1 ϕj = 1 with n the number of nodal points, the components vi of a velocity field v can be interpolated as

n m   (41) ϕj a¯˙ j + ψk aˆ˙ jk vi = j=1

(43)

nd · ∇Dd = Hd

k=1

(44)

with a¯ j the ‘regular’ nodal degrees-of-freedom for the displacements, ψk the enhanced basis terms, and aˆ jk the additional displacement degrees-of-freedom at node j which represent the amplitude of the kth enhanced basis term ψk . Next, we consider a domain  that is crossed by a single discontinuity at d (see Figure 2). The velocity field vs can then be written as the sum of two continuous velocity fields v¯ s and vˆ s :

We now discretize the fields vs and p and the test functions η and ζ for the velocity and the pressure, respectively, in a Bubnov-Galerkin sense:

vs = v¯ s + Hd vˆ s

(42)

vs = N(a˙¯ + Hd a˙ˆ ),

η = N(w ¯ + Hd w) ˆ

where Hd is the Heaviside step function centred at the discontinuity. The decomposition in eq. (42) has a structure similar to the interpolation in eq. (41). Accordingly, the partition-of-unity property of finite element shape functions enables the direct incorporation of discontinuities, including cracks and shear

p = H(p¯ + Dd p), ˆ

ζ = H(¯z + Dd zˆ )

satisfies this requirement, and accordingly, the interpolation of p will be such that: p = p¯ + Dd pˆ

Γ= ∂ Ω Ω+

up

nΓd tp

Γu

Γt Ω− Γd

Figure 2. Body composed of continuous displacement fields at each side of the discontinuity d .

(45)

(46)

where the matrix N contains the shape functions Ni used as partition of unity for the interpolation of the velocity field vs , and H contains the shape functions Hi used as partition of unity for the interpolation of the pressure field p. a˙¯ and a˙ˆ are the nodal arrays assembling the amplitudes that correspond to the standard and enhanced interpolations of the velocity field, and p¯ and pˆ assemble the amplitudes that correspond to the standard and enhanced interpolations of the pressure field. The choice for the interpolants in N and H is driven by modelling requirements. Indeed, the modelling of the fluid flow inside the cavity requires the computation of second derivatives of the pressure, see eqs (32) or (40). Hence, the order of the finite element shape functions Hi has to be sufficiently high, otherwise the coupling between the fluid flow in the cavity and the bulk will not be achieved. Further, the order of the finite element shape functions Ni must be greater than or equal

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to that of Hi for consistency in the discrete momentum balance. Inserting eqs (46) into eqs (26) and (27) and requiring that the result holds for all admissible w, ¯ z¯ , w ˆ and zˆ gives:

the mass and momentum couplings can be removed from the tangent stiffness matrix, and the iterations in the example calculations in the next section have been carried out with such a symmetrized stiffness matrix.

fa¯int = fa¯ext faˆint = faˆext fp¯int fpˆint

=

fp¯ext

=

fpˆext

7

with the external force vectors:  NT tp d fa¯ext = 

 faˆext

=



Hd NT tp d

 fp¯ext =



(48) HT nT qp d

 fpˆext

=



Dd HT nT qp d

and the internal force vectors:  fa¯int = ∇NTσ d 

 faˆint =



 Hd ∇NTσ d +



fp¯int = −



 −



d

NT (td − pnd )d



αHT mT vs d +



kf ∇HT ∇pd

Q−1 HT p˙ d

 fpˆint = −  +  +

EXAMPLES

(47)





d

Dd αHT mT vs d  kf ∇(Dd H)T ∇pd − Dd Q−1 HT p˙ d 

HT nT qd d

(49)

where, for two dimensions, mT = [1, 1, 0]. The time integration has been carried out using a backward finite difference scheme, see Réthoré et al. (2008) for details. Therein, the linearization is also discussed, necessary for the use of a NewtonRaphson method. The resulting expressions depend on the subscale model that has been chosen for the microflow, and are generally very complicated. In any case, the coupling terms cause the tangent stiffness matrix to become unsymmetric. To restore symmetry, the non-symmetric contributions that arise from

First, an open, stationary crack is considered in a linear-elastic medium and no tractions are transferred between both sides of the crack. Hence, the first subscale model for the fluid flow in the cavity is used and singularity functions that stem from linear-elastic fracture mechanics are added as enrichment functions at the crack tips (Réthoré et al. 2007b). The specimen is a square-shaped fractured block under plane-strain conditions. A normal fluid flux qo = 10−4 ms−1 starting at t = 0 s is imposed at the bottom, while the top face is assigned a drained condition with zero pressure. Both left and right faces have undrained boundary conditions. No mechanical load is applied, but essential boundary conditions have been applied in order to remove rigid body motions. The block is 10 m × 10 m and consists of a porous material with a fluid volume fraction nf = 0.3. The absolute mass densities are ρs = ρs /ns = 2000 kg/m3 for the solid phase and ρf = ρf /nf = 1000 kg/m3 for the fluid phase. The solid constituent is assumed to behave in a linear elastic manner with a Young’s modulus E = 9 GPa and a Poisson’s ratio ν = 0.4. The Biot coefficient α has been set equal to 1, and the Biot modulus has been assigned a value Q = 1018 GPa so as to simulate a quasi-incompressible fluid. This is not a limitation of the model, but more clearly brings out the influence of a fault. The bulk material has a permeability kf = 10−9 m3 /Ns. Initially, one fault is considered. This fault, at the centre of the specimen, is 2 m long, and is inclined with the horizontal axis. Simulations have been run for a period of 10 s using 75 time steps. A coarse mesh has been used composed of 16 × 16 quadrilateral elements equipped with quadratic shape functions, and a fine mesh with 32 × 32 elements. The results obtained for an inclination angle of 30◦ are shown in Figures 3–5. Because of the imposed fluid flux at the bottom, the pressure increases in the specimen and inside the fault, which subsequently opens, and fluid can flow inside, see Figure 3. The discontinuity in the normal derivative of the pressure is illustrated in Figure 4. Its jump has an opposite sign at both tips of the fault. The pore fluid flows into the cavity at the left tip where the gradient of the fault opening is positive. At the right tip, the gradient of the fault opening is negative and the fluid in the cavity flows back into the bulk. Figure 5 gives the absolute velocity field of the fluid. The ‘macro’-flow is oriented from the bottom to

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Fluid velocity (m/s) 0.0002

Figure 3. Deformed mesh for a crack angle of 30◦ (magnified ×1000).

Figure 5.

0.09

0.18

Fluid velocity field for a crack angle of 30◦ .

Figure 6. L2 —Norm of the pressure gradient.

Figure 4. Normal derivative of the pressure nd · ∇p for a crack angle of 30◦ . The top picture, where the data have been plotted as a piecewise-constant field, shows the results for the coarse mesh. In the bottom picture, the results for the fine mesh have been smoothed and are depicted with filled iso-values.

the top of the specimen. The ‘micro’-flow in the cavity is oriented by the direction of the fault. In the cavity, the velocity of the fluid is very high compared with the velocity in the bulk because there is no resisting solid skeleton.

Next, a set of ten stationary faults is randomly generated. The length of the faults is between 1 and 3 m, while the fault angles vary from −10◦ to +30◦ . Figure 6 shows the influence of the faults on the norm of the pressure gradient. The global fluid flow is strongly affected by the ‘micro’-flow inside the faults. From Figure 6 it is observed that the main effect is due to the two longest faults. Evidently, the effect of a fault on the ‘macro’-flow increases with its length because more fluid can flow inside the cavity. Finally, a cohesive crack is simulated which propagates in a plate under plane-strain conditions. Now, the second model for fluid flow inside the cavity has been used, in which progressive decohesion within the crack is assumed (Réthoré et al. 2008). The plate has sides of 0.25 m and an initial notch which is located at the symmetry axis and is 0.05 m deep. A fixed vertical velocity v = 2.35 × 10−2 μm/s is prescribed in

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100.0

80.0

Only solid skeleton With interfaces terms No interface term

60.0 F (kN)

a opposite direction at the bottom and at the top of the plate (tensile loading). All other boundaries of the plate are assumed to be impervious. The mesh consists of 20 × 20 quadrilateral elements with bilinear shape functions for the pressure and the displacement fields. The time step size is 1 s and the analysis is continued until the crack reaches the right side of the plate. Obviously, the amount of fluid that flows into the cavity is closely related to the crack opening displacement and vanishes at the crack tip. As a consequence the water pressure decreases in the vicinity of the crack. Figure 7 gives the pressure fields for a simulation with a full coupling at the interface derived in the previous section and for a simulation without a mass transfer coupling term (and also without pressure enrichment). In the latter case, the crack is not seen as a discontinuity in the pressure field and the fluid phase flows through the crack as it does in the

40.0

20.0

0.0 0.0

Figure 8.

5.0

10.0

15.0

20.0

25.0

30.0

Load-displacement curves.

bulk. Accordingly, the pressure and pressure gradient are continuous at the interface. Indeed, the results presented in the two graphs of Figure 7 are very different. Because of the mass transfer coupling, the water is sucked into the crack and high negative pressures occur around the crack. As a consequence, the water saturation decreases in this zone and intense cavitation occurs. Because of the negative values of the water pressure, sucking tractions modify the global response of the plate. As shown in Figure 8, the loaddisplacement curve obtained with the coupling term results in a higher load-carrying capacity. Indeed, the effect of the mass transfer coupling is strong and changes the fluid flow in the entire domain. Moreover, the fluid velocity is increased by an order of magnitude.

8

Figure 7. Pressure field in Pa. Top: The case with full coupling. Bottom: The case without coupling.

CONCLUDING REMARKS

A methodology has been proposed to insert discontinuities such as cracks, faults, or shear bands, in a porous medium. The discontinuities can be located arbitrarily, not related to the underlying discretization. For the fluid flow in the fractured porous medium a two-scale approach has been chosen, where the flow of the fluid inside the discontinuity (the ‘micro’ -scale) is modelled independently from the flow of the pore fluid in the surrounding porous medium (the ‘macro’-scale). The mechanical and the mass transfer couplings between the two scales are obtained by inserting the homogenized ‘constitutive’ relations of the ‘micro’-flow into the weak form of the balance equations of the bulk. The assumptions made for the fluid flow in and near the discontinuity require the addition of special enrichment functions for the displacement and the pressure fields. These conditions are satisfied by exploiting the partition of unity property of the finite element polynomial shape functions.

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REFERENCES Babuska, I. & Melenk, J.M. 1997. The partition of unity method. International Journal for Numerical Methods in Engineering 40: 727–758. Belytschko, T. & Black, T. 1999. Elastic crack growth in finite elements with minimal remeshing. International Journal for Numerical Methods in Engineering 45: 601–620. Biot, M.A. 1965. Mechanics of Incremental Deformations. Chichester: John Wiley & Sons. de Borst, R., Réthoré, J. & Abellan, M.A. 2006. A numerical approach for arbitrary cracks in a fluid-saturated medium. Archive of Applied Mechanics 75: 595–606. Jouanna, P. & Abellan, M.A. 1995. Generalized approach to heterogeneous media. In Modern Issues in Non-Saturated Soils: 1–128. Wien—New York: Springer-Verlag. Lewis, R.W. & Schrefler, B.A. 1998. The Finite Element Method in the Static and Dynamic Deformation and Consolidation of Porous Media, Second Edition. Chichester: John Wiley & Sons. Moës, N., Dolbow, J. & Belytschko, T. 1999. A finite element method for crack growth without remeshing. International Journal for Numerical Methods in Engineering 46: 131–150. Remmers, J.J.C., de Borst, R. & Needleman, A. 2003. A cohesive segments method for the simulation of crack growth. Computational Mechanics 31: 69–77. Remmers, J.J.C., de Borst, R. & Needleman, A. 2008. The simulation of dynamic crack propagation using the cohesive segments method. Journal of the Mechanics and Physics of Solids 56: 70–92. Réthoré, J., Gravouil, A. & Combescure, A. 2005a. An energy-conserving scheme for dynamic crack growth using the extended finite element method. International Journal for Numerical Methods in Engineering 63: 631–659.

Réthoré, J., Gravouil, A. & Combescure, A. 2005b. A combined space—time extended finite element method. International Journal for Numerical Methods in Engineering 64: 260–284. Réthoré, J., de Borst, R. & Abellan, M.A. 2007a. A discrete model for the dynamic propagation of shear bands in a fluid-saturated medium. International Journal for Numerical and Analytical Methods in Geomechanics 31: 347–370. Réthoré, J., de Borst, R. & Abellan, M.A. 2007b. A two-scale approach for fluid flow in fractured porous media. International Journal for Numerical Methods in Engineering 71: 780–800. Réthoré, J., de Borst, R. & Abellan, M.A. 2008. A two-scale model for fluid flow in an unsaturated porous medium with cohesive cracks. Computational Mechanics 42: 227–238. Terzaghi, K. 1943. Theoretical Soil Mechanics. Chichester: John Wiley & Sons. Wells, G.N. & Sluys, L.J. 2001. Discontinuous analysis of softening solids under impact loading. International Journal for Numerical and Analytical Methods in Geomechanics 25: 691–709. Wells, G.N., de Borst, R. & Sluys, L.J. 2002a. A consistent geometrically non—linear approach for delamination. International Journal for Numerical Methods in Engineering 54: 1333–1355. Wells, G.N., Sluys, L.J. & de Borst, R. 2002b. Simulating the propagation of displacement discontinuities in a regularized strain-softening medium. International Journal for Numerical Methods in Engineering 53: 1235–1256.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

A probabilistic approach for modelling long-term behaviour and creep failure of a concrete structure subjected to calcium leaching T. de Larrard, F. Benboudjema & J.B. Colliat LMT-ENS Cachan, CNRS/UPMC/PRES UniverSud Paris, France

J.M. Torrenti LCPC, Université Paris-Est, France

F. Deleruyelle IRSN/DSU/SSIAD/BERIS, Fontenay-aux-Roses, France

ABSTRACT: This work aims at developing a predictive model based on a probabilistic approach for the mechanical durability of concrete structures under calcium leaching attack. For this purpose, an important experimental campaign is being carried out to investigate the statistical variability of some material properties (such as porosity, compressive strength or resistance to an ammonium-nitrate leaching attack). A finite volume computer code for leaching modelling is developed to run probabilistic calculations taking into account the variability of calcium diffusion parameters. This code is used for an inverse analysis approach aiming at identifying the parameters of non-linear diffusion. Then numerical simulations are run to investigate the influence of the material variability on the effects of leaching and creep failure in a concrete structure such as a post. The calculations take into account the variability of porosity and tortuosity accounting for the leaching process on the one hand, and the variability of the compressive strength accounting for creep on the other hand. 1

in a concrete post. The lifespan of the concrete structure under a mechanical load and a calcium leaching attack is evaluated, taking into account the variability of porosity, tortuosity and compressive strength of the material. It was assumed that only the sound part of concrete would carry the load and no stress could be transmitted to the degraded area. Thus, the effective area of concrete would reduce as leaching progresses, as proposed by et al. (2008).

INTRODUCTION

The work presented here is part of a project named APPLET and funded by the National Research Agency (France). This project aims at investigating the variability of the characteristics of concrete, such as its mechanical properties or durability indicators. It has two main purposes: the first one being to acquire statistical data on concrete properties through a large experimental campaign, and the second one being to propose a modelling approach to assess the long-term behaviour of concrete structures taking into account this variability. This paper presents the application of the project with regards to leaching and creep failure. In a first time, the bases of the numerical model developed for leaching of concrete will be discussed. The model is based on the finite volume theory and is expected as light as possible so as to be suitable for running Monte-Carlo simulations. Then the experimental campaign will be briefly presented and related to the developed and formerly described model thanks to an inverse analysis approach aiming at identifying diffusion parameters that cannot be measured directly. Eventually, numerical simulations will be presented, aiming at investigating the influence of the material variability on the effects of leaching and creep failure

2 2.1

MODELLING LEACHING WITH A FINITE VOLUME SCHEME The finite volume method applied to the non-linear equations of leaching

The base of the model for leaching in cement pastes is the mass balance equation for calcium, as it appears in (1) and was initially proposed by Buil et al. (1992). In this equation, SCa is the solid calcium concentration, CCa is the liquid calcium concentration, D is the calcium diffusivity in porous material and φ is the porosity. One can recognise the two main phenomena involved in the leaching process. On the one hand, the solid and liquid phases of calcium in the cement paste and in the porous solution are in chemical equilibrium.

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On the other hand, the ionic calcium species diffuses through the porosity of the material.

Adenot, 1992 Richet, 1997 Model [Tognazzi, 1998] 2

Diffusivity [m /s]

∂SCa ∂(CCa φ) = −div(−D(φ)grad(CCa )) − ∂t ∂t

2e-11

(1)

3

Liquid calcium concentration [mol/m ]

Among the several variables appearing in this problem, it has been decided to chose only one as a reference variable, the solid calcium concentration SCa . The values of the other variables derives from this reference one. Figure 1 illustrates the empirical laws used to derive the values of the liquid calcium concentration (Figure 1(a), after Berner (1990); cf. also (Gérard et al. 1998; Ulm et al. 1999, Gérard et al. 2002) about the use of such empirical laws for the chemical equilibrium of calcium) and of the porosity (Figure 1(b), after Revertégat et al. (1992)) from the solid calcium concentration. The non-linearity of the equations for leaching is also due to the dependancy of the calcium diffusivity through the cement paste on the porosity. The law used to model this dependancy (2) was proposed by Mainguy et al. (2000) and fits accurately experimental data (Adenot 1992; Richet et al. 1997) as it appears on Figure 2. The values for the diffusion parameters proposed by Mainguy et al. (2000) are D0 = 2.355 · 10−13 m2 /s and k = 9.95. It is important to notice that these results have been validated on sound materials, their porosity changing with the water/cement ratio. It is a fundamental hypothesis that the diffusivity

0

1 1,5 Ratio CaO/SiO2 (solid phase)

2

2,5

(a) Liquid calcium concentration 60 Revertegat, 1991 Model

Porosity [%]

50

40

40

50

(2)

The next step is to propose an accurate modelling approach for calcium leaching in concrete from the formerly expressed laws, which are adapted for leaching in cement pastes. This requires a few more assessments, the first one being that the leaching phenomenon concerns only the calcium solid phases of the cement paste (meaning that the aggregates do not interfere with the dissolution process). Another hypothesis is that diffusion occurs only through the cement paste porosity. The last hypothesis stands for the opposite effects of aggregates on the diffusion process, because on the one hand aggregates are an area where no diffusion occurs, but on the other hand they could introduce around them an Interfacial Transition Zone where the diffusivity could be increased. This leads to the introduction of a tortuosity parameter standing for these opposite effects of aggregates; this parameter was proposed by Nguyen et al. (2006) and appears in (3) in order to express the diffusivity of calcium species through concrete depending on the porosity of the cement paste. A new parameter is introduced in (3): fp/m is the volumetric proportion of cement paste in mortar. One can also recognise fm/c the volumetric proportion of mortar in concrete in (4) so as to derive the porosity of concrete from the porosity of cement paste. These two new parameters accounts for the concrete mix design.

Flint, 1934 Taylor, 1950 Kalousek, 1952 Greenberg, 1965 Chen, 2004 Model

0,5

20 30 Porosity [%]

D(φ) = D0 e k φ

5

0

10

follows the same evolution when the porosity increases through the leaching process.

10

0

0

Figure 2. Empirical law to model the calcium diffusivity in cement paste with respect to the porosity.

20

15

1e-11

  = τ fp/m D 0 ek φ D(φ)

(3)

 = fp/m · fm/c · φ φ

(4)

30

20

0

4

8 Mass of CaO [g]

12

2.2

16

(b) Porosity

Figure 1.

Solid calcium concentration as reference variable.

Validation of the model

The model was validated by comparing experimental results found in the literature to numerical simulations (the input parameters correspond to the experimental

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Pearson product-moment correlation coefficient

10

Degradation depth [mm]

8

6

4 Cement paste [Tognazzi] Cement paste [FVM] Mortar [Tognazzi] Mortar [FVM]

2

0

0

1

2 3 1/2 Square root of time [days ]

4

5

(a) Leaching of cement paste and mortar

1 0,8 0,6 -13

D0 : [ 1,8.10 ;2,8.10 k : [ 8,5;10,5 ] Tau : [ 0,15;0,45 ] fp/m : [ 0,45;0,60 ]

0,4 0,2 0 -0,2

fm/c : [ 0,55;0,70 ] XS : [ 0,85;1,0 ]

-0,4

Xphi : [ 1,0;1,38 ]

-13

]

Ca

-0,6 -0,8 -1 5

10 15 1/2 Square root of time [days ]

20

30

Figure 4.

Degradation depth [mm]

25 20 15 10 Nguyen FVM implicit FVM semi-implicit

5 0

0

10 5 1/2 Square root of time [days ]

15

(b) Leaching of concrete

Figure 3. Validation of the model through comparison with experimental data.

set up described in the articles). On Figure 3, two validation tests are illustrated. On Figure (3a) stand the experimental results by Tognazzi (1998) for leaching of cement paste and mortar in a 6 mol/L ammonium nitrate solution. On Figure 3(b) stand the experimental results by Nguyen et al. (2007) for leaching of concrete under the same conditions. In both case the results of the numerical simulations are highly satisfactory. One can notice on Figure 3(b) that two numerical schemes were tested. A completely implicit scheme was tested first and the results were very satisfactory. A semiimplicit scheme was tested then, in which only the diffusivity was explicitly expressed (meaning that at the time step tn , the diffusivity D(tn ) was used instead of D(tn + 1 ) in the totally implicit scheme). The gain in computing time was not significant. Therefore, the fully implicit scheme (which is a little more accurate) has been kept. 2.3

Parametric survey.

Parametric survey

A parametric survey was achieved so as to determine which parameters of the model should focus the attention and be the object of the identification process. Among these parameters, three of them are devoted only to the expression of diffusivity (cf. (2) and (3)): D0 , k and τ (named ‘‘Tau’’ in Figure 4). Two other parameters are related to the concrete formulation: the

volumetric proportion of paste in mortar fp/m and the volumetric proportion of mortar in concrete fm/c . The two last parameters are multiplier factors applied to the models adopted for the equilibrium of the phases of calcium in the porous medium (cf. Figure 1(a)) and for the porosity (cf. Figure 1(b)) depending on solid calcium concentration: XSCa and Xphi , as they appear in Figure 4. These two parameters are to be related to the quality of the cement paste and the nature of the cement used in the concrete mix. Figure 4 represents the Pearson product-moment correlation coefficient calculated for all 7 parameters mentioned above, with regard to the degradation depth at the experimental terms for the accelerated leaching test (28, 56, 98 and 210 days). The Pearson correlation coefficient is a well-known indicator of the linear correlation between two parameters, x and y, and is obtained by dividing the covariance of the two variables by the product of their standard deviations (5).  (xi − x¯ )(yi − y¯ ) (5) cor =   (xi − x¯ )2 (yi − y¯ )2 To achieve this study, more than 200 parameters sets were generated thanks to a simulated annealing algorithm to complete a Latin Hypercube sampling. The range for each parameter was determined after the values found in literature for materials equivalent to the concrete mixes tested in the APPLET project. Figure 4 shows that the most important parameter is the porosity deduced from the solid calcium concentration. Then come the parameters related to diffusivity: k and the tortuosity τ . One should keep in mind that porosity appears not only in the mass balance equation of calcium but also in the expression of the calcium diffusivity (3), which explains why the value of the Pearson coefficient for Xphi is so important. The large influence of the parameters of diffusion on the degradation depth is related to the hypothesis of the instantaneous local chemical equilibrium and the fact that the kinetics of the leaching process is

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mainly conditioned by the diffusion phenomenon. It can also be noticed that the influence of the parameters remains approximatively constant for the duration of the leaching test. This parametric survey leads to focus on only two main parameters for the model proposed for leaching: the initial porosity φ of the material, which can be measured, and the coefficient of tortuosity τ , which cannot be directly measured.

o

5 C o 15 C o 25 C o 35 C

Degradation depth [mm]

8

6

4

2

2.4 Taking into account a variable temperature

0

Two types of ambient conditions, in terms of temperature, were tested. The leaching process is tested at different constant temperatures; these tests were performed at the CEA (Pierre et al. 2009). The leaching process was also tested in realistic outside conditions, meaning that the concrete specimens undergo a variable temperature through the test. Thus, the influence of a variable temperature had to be implemented in the computer code. The leaching process is mainly the combination of two main phenomena: the dissolution of the solid calcium phases of the cement paste and the diffusion of the ionic calcium species through the porous solution. Both phenomena are affected by temperature. Portlandite, which is the first of all cement hydrates to dissolve and whose sharp dissolution front stands for the degradation depth, has a lower solubility when temperature increases, which means that the dissolution process is less important under high temperature conditions. It may be assumed that the effect of temperature on the equilibrium between the cement components and the pore solution could be described by an Arrhenius equation (Atkins and De Paula 2002) as for any other thermally activated process. It was proposed by Gawin et al. (2009) to use a global equation (6) using an apparent activation energy EAC equal to −3222 J/mol, accounting for the whole calcium leaching process. This value of EAC was determined by interpolation of available experimental data (Yokozekia et al. 2004). The reference temperature T0 is 298.15 K. The negative value of the activation energy stands for the decreasing solubility of portlandite with an increasing temperature. 

EC C = C0 exp − A R



1 1 − T T0

0

1

2

3 1/2 Time [days ]

4

5

6

Figure 5. Simulations vs. experimental data for leaching at different constant temperatures.

law (7), as it was proposed and experimentally validated by Peycelon et al. (2006), using a value of energy of activation EAD of 44 kJ/mol.    1 ED 1 − D = D0 exp − A R T T0

 (6)

The thermoactivation of the diffusion process is mainly related to the evolution of the properties of water contained in the pores, especially the reduction of water viscosity with an increasing temperature, according to the well-known law of Stock-Einstein. This thermal dependancy of the ionic species diffusivity can be described thanks to a thermoactivation

(7)

This model for leaching taking into account the temperature was validated through comparison between experimental results for leaching tests under different constant values of temperature (Pierre et al. 2009) and corresponding simulations. A concrete sample was separated into four parts, each of them being immersed in a 6 mol/L ammonium nitrate solution. Each part of the sample was degraded at a different temperature: 5, 15, 25 and 35◦ C (these values range from the minimum to the maximum temperatures encountered during the experimental campaign). Figure 5 shows both experimental and numerical simulation results for each temperature tested, which are in good accordance. 3 3.1

PARAMETRIC IDENTIFICATION THROUGH INVERSE ANALYSIS Experimental data: porosity and degradation depth

The experimental campaign is led in partnership with the Vinci Company, and its principle is to follow two real building operations, corresponding to two different kinds of concrete mix design, the first one being a high-performance concrete with fly ash and the second one being an ordinary concrete. For each building site and so for each concrete formulation, 40 batches are characterised through several tests performed in laboratory (such as compressive and tensile strength, static and dynamic Young modulus, electrical resistivity, etc.). All the concrete samples necessary for the tests are directly provided by the building site

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Table 1.

Water porosity.

φ [%]

Nbr

Mean value

Std Dev.

COV [%]

A1 A2

40 40

12.87 14.22

1.0 1.0

7.9 7.0

to the involved laboratories. This campaign aims at acquiring statistical data on the material characteristics and investigating the variability of these characteristics within several batches of a same formulation and also between different concrete mix design. Another goal is to investigate correlations between all measures and indicators considered. As mentioned above, one of the main parameters involved in the leaching process is the porosity measured on sound concrete. Each batch of concrete studied in the project is characterised by the measurement of its water porosity at 1 year, which is calculated on the basis of three mass measurements: two performed on a saturated slice of concrete (full atmospheric and hydrostatic weight) and one performed on the dried slice (105◦ C in a furnace till constant mass). The data collected on porosity are summarised in Table 1: number of samples tested, mean values, standard deviation and coefficient of variation for each concrete mix design. As expected, the high performance concrete with fly ash (referenced as A1) has a lower porosity than the ordinary concrete (A2). For each batch, a concrete sample is devoted to a leaching test. Leaching usually presents a very slow kinetic and one of the most common ways to accelerate it is to use ammonium nitrate solution (Lea 1965; Goncalves and Rodrigues 1991; Carde et al. 2006; Le Bellégo 2001; Perlot 2005). Solubility of portlandite increases significantly (from 21 mmol/L to 2.7 mol/L for a concentration of 6 mol/L of ammonium nitrate), which accelerates the degradation kinetics at least by a factor 100. At fixed dates (28, 56, 98 and 210 days), the degradation depth is revealed with phenolphthalein (cf. Fig. 6). The specimens are degraded at the age of one year. Phenolphthalein is a pHindicator which becomes pink on the sound area of the sample, where the ammonium nitrate did not turn the initial basic pH of the concrete into values lower than 9. The corresponding data appears in Table 2. Once again, as expected, the degradation depths are more important for the ordinary concrete than for the high performance concrete. 3.2

Tortuosity through Artificial Neural Network

The Artificial Neural Networks (ANN) have the ability to be used as an approximation mechanism for an unknown function (Hertz et al. 1991; Haykin 1998), and this mechanism can be trained from observed data, so as to improve its performance. In our case, the

(a) 28 days

(b) 56 days

(c) 98 days

(d) 210 days

Figure 6. Degradation depths in leaching test, revealed with phenolphtalein. Table 2. Summary of the degradation depth collected for a leaching test in ammonium nitrate solution. d [mm]

Nbr

Mean value

Std Dev.

COV [%]

A128 A228 A156 A256 A198 A298 A1210 A2210

32 40 24 32 40 24 24 8

3.8 5.3 5.5 8.0 8.8 10.1 13.8 16.3

0.6 0.9 0.6 1.6 1.5 1.1 1.0 1.6

15.0 17.8 10.5 20.6 16.8 11.1 7.6 9.8

unknown function shall be the identification process for the coefficient of tortuosity from the porosity and degradation depths of a concrete sample. The finite volume model described in §2.1 uses the porosity and the tortuosity as input data. It produces a simulation for the degradation depths in time and especially at the dates for which experimental data have been collected. It also needs the history of temperature during the degradation, which is monitored hour per hour during the test. It would not have been possible to use an Artificial Neural Network with 24 values of temperature for each of the 210 days of the leaching test. Then it was investigated if equivalent results could be obtained for the degradation depths evaluated at 28, 56, 98 and 210 days with numerical simulations using hourly registered temperature on the one hand, or with only one value of temperature standing for

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each period of degradation (0 to 28 days, 28 to 56 days, 56 to 98 days and 98 to 210 days) on the other hand. This single value for a period of degradation could be the mean value of temperature on this period, as well as an equivalent value of temperature, given in (8), where ti stands for the time-steps and Ti for the corresponding value of temperature. This equivalent temperature Teq results from the quasi-linearity of the degradation depth with regard to the square root of time. (8)

One can see on Figure 7 the results of the simulations with a hourly-variable temperature, with a mean-per-period temperature and with an equivalentper-period temperature. All three can be compared with the experimentally observed degradation depths for a concrete sample, at the fixed dates of 28, 56, 98 and 210 days. These satisfactory results allowed us to use a mean value for the temperature for each degradation period as an input parameter for the Artificial Neural Network. The Artificial Neural Network is used as illustrated on Figure 8: the input data are the values measured for the porosity and the degradation depths for each sample, and a mean value of temperature for each period of degradation. The output data is an identified value of tortuosity. 500 parameter sets of initial porosity (0.11 ≤ φ ≤ 0.18) and tortuosity (0.13 ≤ τ ≤ 0.25) were generated thanks to a simulated annealing algorithm, as well as representative data-set of temperature, within ranges accurate for the material tested here. 500 corresponding degradation depths were calculated with the finite volume model. Finally, the 500 complete data-set were used for the training of the Artificial Neural Network. 15 Hourly variable temperature Equivalent temperature Mean temperature

Degradation depth [mm]

12

9

Temperature: 0 – 28 days 28 – 56 days 56 – 98 days 98 – 210 days

Tortuosity

Input data Initial porosity

Figure 8.

Artificial Neural Network.

(A1-33) (A1-34) (A1-35) (A1-36) (A1-37) (A1-38) (A1-39) (A1-40)

12

Degradation depth [mm]

N √ √  ti+1 − ti Ti Teq = i=0 √  N √ ti+1 − ti i=0

Output data

Input data Degradation depths: 28 days 56 days 98 days 210 days

9

6

3 6

9

12

15

1/2

Time [days ]

Figure 9.

Validation of the identification approach.

To verify this inverse analysis framework, concrete samples have been considered, for which the initial porosity and the degradation depths after 28, 56, 98 and 210 days of leaching in ammonium nitrate solution have been measured. These experimental data were submitted to the Artificial Neural Network to identify a value of tortuosity for each of the samples. Then, numerical simulations have been performed, using the measured value of porosity and the identified value of tortuosity as input parameters for the finite volume model; and corresponding degradation depths have been calculated. On Figure 9, one can compare the experimental degradation depths with the simulated ones (the number in the legend of the graph is the reference of the sample in the experimental campaign) for 8 of these concrete samples.

6

4

3

0

0

3

6

9 Time [days1/2]

12

4.1

15

Figure 7. Equivalence for the estimation of degradation depth at fixed dates between different ways to consider the history of temperature.

PREDICTION OF THE LIFESPAN OF A STRUCTURE Modelling creep failure in a post under calcium leaching attack

To investigate the influence of the variability of the material properties on the long-term behaviour of concrete structures, a very simple structure such as

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a post only made of concrete is considered, without any reinforcement, submitted to a constant mechanical load, and thus subjected to creep, and exposed to a calcium leaching attack by pure water at the same time. Monte-Carlo simulations are run to evaluate the variability of the lifespan of this structure, accounting on the input variability on the material characteristics. The coupling between creep and leaching was modelled as proposed by Torrenti et al. (2008). The post is loaded in pure compression, and the load is equivalent to a stress of 30% of the characteristic strength fc of the material. It is assumed that only the sound material can sustain loads and the stresses in the degraded area of the concrete are neglected. This means that, as long as the degradation depth increases, the surface carrying the mechanical load decreases as well. A modelling for the behaviour of concrete under creep was proposed by Benboudjema et al. (2001) and Reviron et al. (2007). It is considered from the experimental results by Li (1994) that failure is reached in compression when the stress in the material overcomes 85% of the real compressive strength σc of the concrete. It is to be noticed that this level of stress basically corresponds to the end of secondary creep (which is not yet exactly the failure of the structure). Anyway, the time necessary to reach tertiary creep is so much more important than the duration of this phase of creep itself (between the 85% of real compressive strength fc and the failure of the structure). For instance, creep failure occurs only after 30 minutes when concrete is loaded at 85% of the real compressive strength fc (Li 1994). Therefore, the duration of tertiary creep can be neglected. Under these hypothesis, one can express the critical degradation depth d(Tlife ) for which the sound material corresponds to the section where the stress reaches 85% of the real compressive strength of the concrete σc . The load is equivalent to 30% of the characteristic strength fc applied on the whole initial section (r0 is the initial radius of the post, 20 cm). This expression is presented in (9). The numerical modelling for leaching presented in §2.1 can then be used to evaluate the time Tlife required for the degradation depth to reach this critical value, basically being the lifespan of the structure. ⎛ d(Tlife ) = r0 ⎝1 −



⎞ 0.3 fc ⎠ 0.85 σc

(9)

4.2 Input variability There are three main input parameters for this modelling of leaching and creep failure in a concrete post: on the one hand compressive strength σc for creep, and

on the other hand porosity φ and tortuosity τ for leaching. It was decided to consider, for these parameters, the variability experimentally observed for the first concrete mix design presented in §3.1. The concrete mix is presented in Table 3. The distribution of the parameters was chosen to follow normal laws, mean values and standard deviation being deduced from the experimental data (through direct measurements for compressive strength and porosity, and through inverse ana-lysis from ammonium-nitrate degradation for tortuosity) and presented in Table 4. In order to run Monte-Carlo numerical simulations, 1000 parameter sets were randomly generated, and the distributions obtained for each of them are presented on Figure 10. It was supposed that all parameters were independent from one another. The value of the characteristic strength fc of the material is also the value observed from the experimental data-set, equal to 70 MPa.

4.3

Output of interest

For each simulation, the critical degradation depth is calculated from the real compressive strength σc of the post, and a numerical simulation of leaching with the corresponding values of porosity and tortuosity is led to determine the time necessary to reach this degradation depth. The distribution of the estimated lifespans of the posts is presented on Figure 11, and follows a normal law as well as the input parameters. Mean values, standard deviations and coefficients of variation for input parameters as well as for lifespans appear in Table 4.3. One can observe that the order of magnitude of the coefficient of variation is basically the same for the input parameters and for the output of interest of our problem, around 9%.

Table 3. Concrete mix design (quantities for 1 m3 of concrete). Cement Fly ash Sand Coarse aggregate Water Plasticizer

Table 4.

σc φ τ Tlife

350 kg 80 kg 900 kg 950 kg 170 kg 3 kg

Variability of input parameters and lifespan.

[MPa] [%] [-] [year]

Mean value

Std Dev.

COV [%]

85.9 12.9 0.13 133.0

8.1 1.0 0.01 11.8

9.5 7.9 9.5 8.8

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150

140

120

100 100

80

60 50

40

20

0 50

60

70

80

90

100

110

0 90

120

Figure 11.

(a) Compressive strength

150

5

50

0 10

11

12

13

14

15

16

17

0.15

0.16

0.17

Porosity [%]

(b) Porosity 140

120

100

80

60

40

20

0 0.09

0.1

0.11

0.12

0.13

0.14

(c) Tortuosity

Figure 10.

110

120

130

140

150

160

170

180

Distribution of lifespan.

CONCLUSIONS AND PERSPECTIVES

A model based on the finite volume theory was developed to predict the degradation of concrete under leaching, taking into account the history of temperature during the degradation. An experimental campaign is led to acquire statistical data on porosity and the kinetics of leaching for several batches of a same concrete mix design, so as to quantify the variability of the material characteristics, such as its compressive strength. An inverse analysis method was developed, based on the Artificial Neural Network theory, to identify, from the measures of porosity and degradation depths during a leaching test in ammonium nitrate solution under variable temperature, the coefficient of tortuosity of the material. Once the complete data-set about the variability of tortuosity and porosity between different batches for a same concrete formulation would have been collected, the next step will be to investigate the influence of this variability on the long-term behaviour of concrete structures by generating random fields of porosity and tortuosity in the geometry of the structure with accurate statistical characteristics. Monte-Carlo simulations based on the observed variability of the material characteristics have been run to highlight the influence of this variability on the long-time behaviour of concrete structures, and evaluate the propagation of uncertainty for the problem of a concrete post under simultaneous creep and leaching. This simple application led to the conclusion that the variability of the lifespan of concrete structures was not to be neglected. The computer code for modelling leaching has to be adapted for 3-dimensional simulations, so that the long-term behaviour of real structures (such as nuclear waste containers, for which leaching is a major issue) could be simulated, taking into account the variability of the material characteristics.

100

9

100

Lifespan [years]

Compressive strength [MPa]

Distribution of input parameters.

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ACKNOWLEDGEMENT The investigations and results reported herein are supported by the National Research Agency (France) under the APPLET research program (grant ANR06-RGCU-001-01). REFERENCES Adenot, F. (1992). Durabilité du béton: caract érisation et modélisation des processus physiques et chimiques de dégradation du ciment. Ph. D. thesis, Université d’Orléans. Atkins, P. and J. De Paula (2002).Physical Chemistry. New York: Oxford University Press. Benboudjema, F., F. Meftah, J.M. Torrenti, G. Heinfling, and A. Sellier (2001, May). A basic creep model for concrete subjected to multiaxial loads. In 4th International Conference on Fracture Mechanics of Concrete and Concrete Structures, Cachan, France, pp. 161–168. Berner, U. (1990). A thermodynamic description of the evolution of the pore water chemistry and uranium speciation during the degradation of cement. NAGRA TR 90–12 & PSI Ber. n◦ 62. Buil, M., E. Revertégat, and J. Oliver (1992). A Model of the Attack of Pure Water or Undersaturated Lime Solutions on Cement, Volume 2nd, pp. 227–241. Philadelphia: American Society for Testing and Materials. Carde, C., R. Francois, and J.M. Torrenti (2006). Leaching of both calcium hydroxyde and CSH from cement paste. modeling the mechanical behavior. Cement and Concrete Research 26(8), 1257–1268. Gawin, D., F. Pesavento, and B. Schrefler (2009). Modeling deterioration of cementitious materials exposed to calcium leaching in nonisothermal conditions. Computer Methods in Applied Mechanics and Engineering. Gérard, B., C. Le Bellégo, and O. Bernard (2002). Simplified modelling of calcium leaching of concrete in various environment. Materials and Structures 35, 632–640. Gérard, B., G. Pijaudier-Cabot, and C. Laborderie (1998). Coupled diffusion-damage modelling and the implications on failure due to strain localization. Int. J. Solids Struct. 35(31–32), 4107–4120. Goncalves, A. and X. Rodrigues (1991). The resistance of cements to ammonium nitrate attack. In Durability of concrete, 2nd International Conference. Haykin, S. (1998). Neural Networks: A Comprehensive Foundation (2nd ed.). Prentice Hall. Hertz, J., A. Krogh, and R. Palmer (1991). An Introduction to the Theory of Neural Computation. Addison Wesley. Le Bellégo, C. (2001). Couplage chimiem écanique dans les structures en béton attaquées par l’eau: étude expérimentale et analyse numérique. Ph. D. thesis, ENS Cachan.

Lea, F.M. (1965). The action of ammonium salts on concrete. Magazine of concrete research 52, 115–116. Li, Z. (1994). Effective creep poisson’s ratio for damages concrete. International Journal of Fracture 66, 189–196. Mainguy, M., C. Tognazzi, J.M. Torrenti, and F. Adenot (2000). Modelling of leaching in pure cement paste and mortar. Cement and Concrete Research 30, 83–90. Nguyen, V.H., H. Colina, J.M. Torrenti, C. Boulay, and B. Nedjar (2007). Chemomechanical coupling behaviour of leached concrete. part 1: experimental results. Nuclear Engeneering and Design 237, 2083–2089. Nguyen, V.H., B. Nedjar, H. Colina, and J.M. Torrenti (2006). A separation of scales homogenisation analysis for the modelling of calcium leaching in concrete. Computer Methods in Applied Mechanics and Engineering (195), 7196–7210. Perlot, C. (2005). Influence de la décalcification de matériaux cimentaires sur les propriétés de transferts: application au stockage profond des déchets radioactifs. Ph. D. thesis, Universités de Toulouse et de Sherbrooke (Canada). Peycelon, H., C. Blanc, and C. Mazoin (2006). Longterm behaviour of concrete: Influence of temperature and cement binders on the degradation (decalcification/ hydrolysis) in saturated conditions. Revue Européenne de Génie Civil 109), 1107–1125. Pierre, M., P. Le Bescop, and S. Poyet (2009). Projet ANR APPLET (GT1) caractérisation de la variabilité des bétons : perméabilité au gaz et dégradation accélérée. Rapport DRT 09MMHQ000142, CEA/DEN/DANS/DPC/SCCME/ LECBA. Revertégat, E., E. Richet, and P. Gegout (1992). Effect of pH on the durability of cement pastes. Cement and Concrete Research 22, 259–272. Reviron, N., F. Benboudjema, J.M. Torrenti, G. Nahas, and A. Millard (2007, June). Coupling between creep and cracking in tension. In 6th International Conference on Fracture Mechanics of Concrete and Concrete Structures, Italie. Richet, C., C. Le Callonec, C. Mazoin, M. Pin, and F. Adenot (1997). Amélioration du modèle de dégradation de la fiabilité du modèle DIFFUZON. Technical Report RT SESD/97.60, CEA. Tognazzi, C. (1998). Couplage fissurationdégradation chimique dans des matériaux cimentaires: caractérisation et modélisation. Ph. D. thesis, INSA Toulouse. Torrenti, J.M., V.H. Nguyen, H. Colina, F. Le Maou, F. Benboudjema, and F. Deleruyelle (2008, June). Coupling between leaching and creep of concrete. Cement and Concrete Research 38(6), 816–821. Ulm, F.J., J.M. Torrenti, and F. Adenot (1999). Chemoporoplasticity of calcium leaching in concrete. Journal of Engeneering Mechanics 125(10), 1200–1211. Yokozekia, K., K.Watanabe, N. Sakata, and N. Otsuki (2004). Modeling cementitious materials used in underground environment. Applied clay science 26, 293–308.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

A coupled transport-crystallization FE model for porous media H. Derluyn ETH Zürich, Chair of Building Physics, Zürich, Switzerland

R.M. Espinosa-Marzal EMPA, Laboratory for Building Science and Technology, Dübendorf, Switzerland

P. Moonen ETH Zürich, Chair of Building Physics, Zürich, Switzerland

J. Carmeliet ETH Zürich, Chair of Building Physics, Switzerland EMPA, Laboratory for Building Science and Technology, Dübendorf, Switzerland

ABSTRACT: In this paper, a finite element model is presented, describing moisture, heat and salt transport combined with salt crystallization of single salts in porous materials. The model can be applied on different types of building materials, e.g. masonry and concrete. Salt crystallization is described by the kinetics of phase changes. The salt influence on the transport variables is taken into account by using parameter functions including non-ideality of solutions based on the Pitzer ion interaction approach. Numerically, all crystallization processes are covered by one equation, which improves stability and convergence of the system within the finite element framework. The model is illustrated with three academic examples on two different materials: mortar and ceramic brick. In the first two examples crystallization and deliquescence due to drying/wetting of brick are simulated. In the third example cooling of a mortar saturated with a salt solution is studied. The moisture, heat and salt transport as well as the crystallization induced by cooling and drying are covered by the developed model. 1

INTRODUCTION

Durability and sustainability are two main issues in the development of new building materials and renovation techniques. During their lifetime, building materials are subjected to different types of loading, a.o. mechanical stresses, repeated hygro-thermal loading and chemical attack. Reducing the life cycle cost and extending the service life time of materials and structures require therefore the knowledge of the underlying physical degradation mechanisms. Water and dissolved salt ions can penetrate into building materials due to both diffusive and convective fluxes within the porous matrix and through existing cracks. Upon drying salt may crystallize at the surface (efflorescence) or beneath the surface (subflorescence or cryptoflorescence). Efflorescence leads to esthetical discomfort. However, if subflorescence occurs, salt crystallizes in the pore space causing high tensile stresses in the solid matrix and leading eventually to spalling and cracking. To understand and avoid these salt related issues a thorough knowledge of the coupled heat and mass transfer in building materials is needed, including salt transport and crystallization. Eventually

this has to be coupled with the mechanical behaviour of the material. In this paper we describe the development of a transport and crystallization model. The crystallization model incorporates the kinetics of the salt crystallization (Espinosa et al. 2008). Up to now, the model can describe the crystallization of single salts (no mixtures). The properties of salt crystals and solution are mainly based on Steiger et al. (2008). The transport and crystallization equations are embedded in a finite element model which has recently been developed by Moonen (2009). This model couples moisture and heat transport with the mechanical behaviour to predict damage and failure. It has been extended with a salt library for sodium sulfate (Na2 SO4 ) and sodium chloride (NaCl), two of the most damaging salts mentioned in literature. Sulfates are present in contaminated soil water (by a.o. factories), causing damage in building materials due to capillary rise. NaCl is especially damaging in coastal areas (sea spray) and in cold climates where it is used as deicing salt. E.g. Petkovic (2005) and Lubelli (2006) studied the damaging character of NaCl in layered specimens of plaster/mortar on sandstone/brick.

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Both salts are also interesting because of their significant different behaviour. Sodium sulfate has two stable phases—mirabilite (Na2 SO4 .10H2 0) and thenardite (Na2 SO4 (V)) and two metastable phases— heptahydrate (Na2 SO4 . 7H2 0) and thenardite (Na2 SO4 (III)). Each of these phases are stable (or metastable) in a determined range of temperature and relative humidity. Sodium chloride however only has one stable crystal phase above the freezing point: halite (NaCl). Just by cooling down a sodium sulfate solution mirabilite or heptahydrate may form at the required critical supersaturation (Espinosa-Marzal & Scherer 2008), while by drying thenardite may also precipitate. In contrast, crystallization of halite can only be induced by drying. The paper consists of three parts. In the first part, we derive the governing equations. Next, the numerical implementation is discussed. In the third part, the performance of the model is illustrated by means of some academic examples.

2

BALANCE EQUATIONS

2.1 Conservation of mass The theories used to develop the moisture and salt mass balance equations are described by Hassanizadeh & Gray (1979), Hassanizadeh (1986) and Lewis & Schrefler (2000). In every representative elementary volume (REV) four main phases are considered: (1) the solid material matrix, (2) the gas phase, (3) the liquid phase and (4) the crystal phase. In these phases components π can be considered, for the gas phase we consider dry air and vapour, for the liquid phase water and salt ions. Each phase α has a volume fraction ηα . The volume fraction of the solid material matrix equals ns = 1 − n where n = (δυ g + δυ l + δυ cr )/δυ is the total porosity; δυ α is the volume of phase α in the REV and δυ the volume of the REV. For each phase the volume fraction is written as: ηα = nSα with Sα = δυ α /(δυ g + δυ l + δυ cr ) the degree of saturation of phase α. Consequently Sg + Sl + Scr = 1. The general formulation of the mass balance equation for a component π in phase α is: ∂(ηα ρ α cα,π ) + ∇(ηα ρ α cα,π vα ) + ∇J α,π = eαβ,π (1) ∂t with ρ α the density of phase α, cα,π the concentration of component π in phase α, vα the velocity of phase α, Jα,π the diffusive mass flow of component π in phase α and eαβ,π expressing the mass exchange of component π between phase α and all other phases β = α. Summation over all components π gives the general mass balance equation for phase α:

∂(ηα ρ α ) + ∇(ηα ρ α vα ) = eαβ ∂t Using the total or material derivate, defined as: ∂f Dα f = + (∇f ) · vα Dt ∂t equations 1 and 2 can be written as:

(2)

(3)

Dα (ηα ρ α cα,π ) + ηα ρ α cα,π ∇ · vα + ∇Jα,π = eαβ,π Dt (4) Dα (ηα ρ α ) + ηα ρ α ∇ · vα = eαβ Dt  with eαβ = eαβ,π .

(5)

π

For the developed mass balances, the following assumptions are made: – The mean velocity of the solid phase equals zero, thus no deformations of the solid phase are taken into account and the density of the solid material matrix ρ s and the total porosity n is constant. By this, the mass balance for the solid phase is fulfilled. – The total gas pressure is constant and equals the atmospheric pressure. Consequently resolving a mass balance for the air in the gaseous phase is not necessary. – In the absence of an electric field, the dissolved salt (i.e. the ions) does not separate macroscopically and the ions are always transported together. – Mass exchange occurs for water between the liquid and the gas phase and between the liquid and the crystal phase. For salt ions, mass exchange happens between the liquid and the crystal phase. Under these assumptions, the transport in the multiphase medium can be described by the following four equations: 1. vapour component in gas phase: Dg (6) (nSg ρ v ) + nSg ρ v ∇ · vg + ∇Jg,v = egl,w Dt where ρ v = ρ g cg,v with cg,v the concentration in kg vapour/kg gas and vg equal to zero based on the assumption of a constant gas pressure. 2. liquid phase: Dl (7) (nSl ρ l ) + nSl ρ l ∇(vl ) = elg,w + elcr,w+i Dt where the superscript w represents water and i represents ions. 3. salt ions in liquid phase: Dl elcr,w+i (8) (nSl ρ l ci ) + nSl ρ l ci ∇(vl ) + ∇Jl,i = cr Dt ρ · Vmol_cr

with ci the concentration in mol/kg liquid solution and Vmol_cr the molar volume of the crystal.

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Lv is the latent heat of evaporation and K v the vapour permeability defined by: pv (13) Kv = δv ρw R v T

4. crystal phase: Dcr (nScr ρ cr ) + nScr ρ cr ∇(vcr ) = ecrl,w+i Dt

(9)

where the mean velocity of the salt crystal vcr is assumed to be zero.

The functions γ and β describe the dependence of capillary pressure to respectively the temperature and the concentration. This is done by using the YoungLaplace equation for capillary pressure:

Equation 6 is incorporated in equation 7 in order to eliminate the exchange term elg,w . This leads eventually to three balances: the liquid, salt ion and salt crystal balance. 2.1.1 Mass balance of the liquid phase By combining equations 6, 7 and 9 and using the law of Darcy and Fick, the mass balance of the liquid phase becomes: n

∂(Sg ρ v + Sl ρ l + Scr ρ cr ) = ∇(K l ∇(pc ) + δ v ∇(pv )) (10) ∂t

where the vapour density and the change of vapour density with time is assumed to be negligible and so the derivative ∂(Sg ρ v )/∂t is neglected. K l is the liquid permeability and δ v the vapour permeability. Thus, the first term on the right hand side represents the liquid flow, the second term the vapour flow. pv is the vapour pressure and is function of the capillary pressure pc , temperature T and the water activity aw as given by the modified Kelvin relation (e.g. Coussy 2006, Nguyen et al. 2007):   pc (11) pv = aw pv,sat exp ρw Rv T

2σ cos θ (14) r where σ is the surface tension between liquid and vapour and θ the contact angle. γ and β are then defined by: pc =

1 σ 1 β= σ γ =

∂σ ∂T ∂σ ∂ci

(15)

with Rv the gas constant of water vapour. The water activity is dependent on the salt concentration and the temperature and is calculated here with the parameterizations described by Steiger et al. (2008) based on the Pitzer ion interaction approach. The higher the salt concentration, the lower the water activity will be and equation 11 indicates that this corresponds to lower vapour pressures. Equation 11 allows expressing the vapour flow in function of the variables capillary pressure, temperature and concentration and the liquid mass balance becomes:

Experimental data for the surface tension of NaCl as a function of temperature and concentration are applied in the model (Horibe 1996). For Na2 SO4 data can be found in the CRC Handbook of Chemistry and Physics (online 90th edition). The dependence of the contact angle on temperature and capillary pressure is not considered, as no clear data were available. Sghaier et al. (2006) found for certain substrates that the product σ cos θ does not change appreciably with the NaCl concentration at room temperature. However they discuss that when taking into account liquid wetting films (thick films) the contact angle variations can have a large effect on salt transport. More experimental research should be performed on this topic to be able to formulate a conclusive answer for different salts and different substrates. The liquid permeability K l changes when salt ions are present in the liquid phase, due to a change in density ρ and viscosity η. This can be taken into account by considering an intrinsic constant permeability (independent of the fluid) equal to K l η/ρg. (Hall 1997). The liquid permeability then becomes: ηw ρi K l = Kwl (16) ρw ηi

∂(Sl ρ l + Scr ρ cr ) n ∂t = ∇((K l + K v )∇pc ) ⎞ ⎛⎛ v  ⎞ pc T ∂ρw K ρ L + p (γ T − 1) − w v c ⎜⎜ T ⎟ ⎟ ⎜  ρw ∂T ⎟ ∇T ⎟  +∇ ⎜ ⎝⎝ ⎠ ⎠ ∂aw pc +δ v psat exp ρw Rv T ∂T      ∂aw pc ∇ci +∇ K v pc β + δ v psat exp ρw Rv T ∂ci (12)

The dependence of the viscosity on temperature and NaCl concentration can be found in Kestin et al. (1981). Data for Na2 SO4 can be found in Korosi and Fabuss (1968). The liquid saturation degree is determined by expressing the moisture content of the material in function of the capillary pressure. Due to crystallization the liquid and gas porosity decrease, causing a reduction of the moisture content and the liquid and vapour permeability. This reduction is calculated in a simplified way by using the approach of Nicolai and Scheffler (2009). They propose to multiply the

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moisture retention curve and the permeabilities with the factor (1 − Scr ). This factor expresses that without crystals everything stays unchanged; when the pores are fully filled with crystals (Scr = 1), no moisture is present anymore and the permeabilities reduce to zero. However when describing crystallization in the unsaturated material, we should consider a more accurate pore filling by the salts by using pore network models as developed by Carmeliet et al. (1997), or by assuming a certain pore filling as e.g. starting in the large pores and continuing into the smaller ones (Poupeleer 2007). 2.1.2 Mass balance of the salt ions By combining equations 8 and 9 and applying again the laws of Fick and Darcy, the mass balance of the salt ions becomes: n

∂(Sl ∂t

ρl c

i)

+n

∂Scr /Mvol_cr ∂t

= ∇(ρ l Dli ∇ci ) + ∇(ci K l ∇pc )

(17)

with Dli the salt diffusion coefficient. The first term on the right hand side represents the diffusive ion flow, the second term the convective ion flow. The diffusion coefficient is given by Buchwald (2000): Dli = τ −1 D[ci ] · n · (Sl )ns

(18)

where D[ci ] is the diffusion coefficient in function of concentration in a non dilute solution, τ the direction dependent tortuosity of the porous material and ns the saturation exponent. The reduction of the diffusion coefficient due to salt crystals is also taken into account by multiplying equation 18 with the factor (1 − Scr ) as described in 2.1.1. 2.1.3 Mass balance of the salt crystal ∂(Scr ρ cr ) = ecrl,w+i ∂t

crystallization can start (Espinosa-Marzal & Scherer 2008). Once crystals are present, the threshold value Ustart rapidly reduces to 1. Dissolution is described by a similar type of equation, only the kinetic parameters and the threshold value differ: ecrl,w+i = Kdiss (1 − U )gdiss

for

U <1

(21)

The kinetic parameters K, g and Ustart have to be determined experimentally. The supersaturation U is calculated taking into account the non-ideality of the solutions by using the parameterizations of Steiger et al. (2008). 2.3

Conservation of energy

Heat transport is described by the energy balance: ∂h (22) + div(qh ) ±  = 0 ∂t with h the enthalpy, qh the heat fluxes and  a source or sink term. In the developed model, no source or sink terms are included. Equation 22, with the variation over time of the water vapour and dry air enthalpy considered negligible, becomes: ∂((cs ρ s + ncl Sl ρ l + nccr Scr ρ cr )(T − T0 ) + nLcr Scr ρ cr ) ∂t ⎤ ⎡ λ∇T + ⎥ ⎢ (cl (T − T0 ) · K l + (cv (T − T0 ) + Lv )K v )∇pc ⎥ ⎢ ⎛ ⎛ ⎞⎞ ⎥ ⎢ ρ L + p (γ T − 1) w v c ⎥ ⎢ v ⎥ ⎢ ⎜K ⎜ ⎟⎟ ⎥ ⎢ ⎟ ⎜ ⎝ ⎠ T ∂ρ p ⎥ ⎢ c w ⎟ ⎜ T ⎥ − ⎢ +(cv (T − T0 ) + Lv ) ⎜ ∇T ⎟ ⎥ ⎢ ρw  ∂T ⎟ ⎜  ⎥ ⎠ ⎝ =∇⎢ ∂a p ⎥ ⎢ c w v +δ psat exp ⎥ ⎢ ⎥ ⎢ ρw Rv T ∂T ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎞ ⎛ v ⎥ ⎢ K pc β+  ⎥ ⎢  ⎣ +(cv (T − T0 ) + Lv ) ⎝ ⎠ ∂aw ∇Cs ⎦ pc v δ psat exp ρw Rv T ∂Cs

(19)

(23)

where we have to define the mass exchange ecrl,w+i between the liquid and the crystal phase.

with cs the heat capacity of the solid matrix, ccr the heat capacity of the crystals, cl and cv the heat capacity of the liquid and the vapour, λ the heat conduction coefficient of the solid matrix and Lcr the heat of crystallization. The thermal properties of the crystals can be calculated using the method described by Steiger and Asmussen (2008). Properties for NaCl crystals are also available in the CRC Handbook of Chemistry and Physics (online 90th edition).

n

2.2

Salt crystallization

The salt crystallization term ecrl,w+i is described by a model for the kinetics of phase changes of salts in pores, as introduced by Espinosa et al. (2008). This model considers that the supersaturation is the driving force for crystallization, and consequently the crystallization rate is given by: ecrl,w+i = Kcr (U − 1)gcr

for U > Ustart

(20)

where Kcr and gcr are the kinetic parameters and U is the supersaturation ratio. For primary crystallization (i.e. if no salts are present in the solution) the supersaturation has to reach a certain limit value Ustart before

3

NUMERICAL IMPLEMENTATION

The transport equations 12, 17, 19 and 23 are solved using the finite element method. The equations are embedded in the finite element code developed by Moonen (2009). A salt library has been introduced in

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order to include the properties of crystals and solution. The primary variables are the capillary pressure pc , the salt concentration ci , the crystal saturation degree Scr and the temperature T . In order to obtain a mass and energy conservative system of equations, a mixed form of the capacitive terms is used, as described in Janssen et al. (2007). First, the weak formulation of the transport equations is described. The equations are further developed using the Green-Gauss theorem and by discretizing the primary variables in space and time. A staggered solution scheme is selected. Each equation is solved implicitly using the Newton-Raphson method. Numerical integration is performed by means of a Gauss-Lobatto scheme. This has a similar accuracy as the more commonly used Gauss-Legendre scheme, but suppresses oscillations in the solution field for our set of highly non linear PDE’s (Moonen). Every equation is expressed in matrix form as:

and the connectivity of the pore system. At last, it also has to be taken into account that dissolution can only occur when there are still crystals present. In order to get a stable numerical system incorporating all salt crystallization effects, the crystallization term ecrl,w+i is implemented as:



– crystallization: available pore volume for the salt crystals, when the volume is full, no crystallization occurs anymore; – dissolution: only occurs when there are still crystals present;

 C + K x = Fext − Fint t

(24)

where x represents the change of the primary variable during the iteration step and Fext and Fint are the external and internal load vector, respectively. For example for the liquid phase the element matrices Ce , K e and Fint,e are given by:  Ceij = e

Kije =

  ∂Sl t+t,ni Ni nρ l Nj d ∂pc ∇ T Ni (K l + K v )∇Nj d

⎞ ρlt+t,ni Slt+t,ni − ρlt Slt + n ⎟ ⎜ ⎟d t Fint,e = Ni ⎜ t+t,ni i ⎠ ⎝ t Scr − Scr cr nρ e t ⎛ l ⎞ (K + K v )∇pt+t,ni c ⎜ ⎛ v ⎛ ρ L + p (γ T − 1) ⎞ ⎞ ⎟ w v c ⎜ ⎟ ⎜ ⎜ K ⎝ pc T ∂ρw ⎟ ⎠⎟ ⎜ ⎜ T ⎟ ⎟ − t+t,ni ⎜+ ⎜ ⎟ ⎟ ∇T  ρ ∂T w ⎜ ⎜ ⎟ ⎟  T ⎟d pc ∂aw ⎠ + ∇ Ni ⎜ ⎜ ⎝ +δ v psat exp ⎟ ⎜ ⎟ ρw Rv T ∂T⎞ e ⎜ ⎛ ⎟ v ⎜ ⎟ K pc β+  ⎜ ⎟  t+t,ni ⎝+ ⎝ ⎠ ⎠ pc ∂aw ∇ci v δ psat exp ρw Rv T ∂ci (25) e



⎛

where Ni is the shape function of node i. The global matrices C, K and Fint are composed by assembling the element matrices given by equation 24. For the modeling of the crystallization, next to the kinetics, also the confined pore volume space needs to be considered. If no space is available for crystals to grow, the crystallization will stop and consequently the supersaturation will become higher than when the crystals could grow freely. The presence of crystals in certain pores can also influence the crystallization in the neighboring pores, as the crystals can grow through the pore space, depending on the diffusion of the ions

ecrl,w+i = f (Scr , 1 − Sl ) · Kcr (max(U , Uthr ) − Uthr )gcr + f (Scr , 0) · Kdiss (1 − min(U , 1))gdiss

(26)

with

    x−r  f (x, r) = sign(r − x) · 1 − exp −  HBW  Uthr = 1 + (Ustart − 1) · exp(−κScr )

(27)

The function f represents two checks:

and it also contains a smoothing part through the exponential function. In this way poor numerical performance due to oscillations between crystallization and dissolution stages are prevented. The function Uthr represents the drop of the crystallization threshold from Ustart to 1. This drop goes as fast as numerically possible, but the function is necessary to avoid divergence of the system. A value of 0.01 is taken for the half band width HBW and a value of 4000 for κ. 4 4.1

ACADEMIC EXAMPLES Material properties

To illustrate the transport-crystallization model, material properties of masonry samples described in Gonçalves (2007) are used. The masonry samples consist of a Dutch red ceramic brick and the mortar MEP-SP® by Strasservil (France). This mortar is described as an industrial trass-(air)lime-based plastering/rendering mortar which includes mass water-repellent additives and is aimed to be a salt accumulating plaster. The material properties were measured using different techniques: capillary absorption test, drying test, vapour diffusion test and mercury intrusion porosimetry. The moisture transport properties of the brick and mortar material are summarized in Table 1. We note that the capillary absorption coefficient of the brick is much larger than the absorption coefficient of the mortar and that there is a large difference between saturated and capillary moisture content of the mortar. This indicates that only 26% of the available 53% porosity is

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used for capillary moisture transport, which is possibly caused by the mass water repellent additives present in the mortar. The pore volume distribution was measured using mercury intrusion porosimetry, the data and curve fits are given in Figure 1. One additional curve is added representing the pore space accessed during capillary transport in the mortar. We observe that the large pores of the mortar (0.4–10 μm) do not take part in the capillary moisture transport. The wetting capillary pressure curves in the model are described by a multimodal function of the van Genuchten type (Carmeliet and Roels 2002): w = wsat

k   m li 1 + (ai pc )ni i

(28)

i=1

with a, n, m shape parameters and l a weight parameter. The parameters for brick and mortar are given in Table 2. The liquid permeability of the brick is modeled using equation 29 (based on Van Genuchten 1980) with the parameters of Table 2; τ is the tortuosity, for Table 1.

Moisture transport properties of brick and mortar.

Property

Brick

Mortar

Density (kg/m3 ) Porosity (%) Saturated moisture content (kg/m3 ) Capillary moisture content (kg/m3 ) Capillary absorption coefficient (kg/m2 s1/2 ) Dry vapour resistance factor (−)

1840 26 260 260

1780 53 530 260

0.29 12*

0.015 6

*This value was assumed, as no measurements on the brick were available. The value corresponds to a tortuosity of 3, which is in the range of expected tortuosities for rocky and ceramic materials (Hall & Hoff, 2002).

PVD (m3/m3log(m))

0.30

⎡

⎢ ×⎣

mi  ⎤2   (ai ·pc )ni li · ai · 1 − 1+(a n i i ·pc ) ⎥  ⎦ li · ai

(29)

Table 2. Parameters of capillary pressure curve for brick and mortar (equation 1). i=1

i=2

i=3 −4.1 10−6 11.91 −1.827 0.063

Brick

ai ni mi li

−2.6 10−6 4.87 −0.782 0.172

−1.58 10−5 15.63 −0.064 0.765

Mortar

ai ni mi li

−6.76 10−8 1.66 −2.858 0.169

−2.03 10−6 29.78 −0.034 0.321

1E-07

curve fit PVD brick

1E-08

data PVD mortar

1E-09

curve fit PVD mortar curve PVD mortar capillary uptake

1E-10

0.10

 −mi τ li · 1 + (a · pc )ni

For mortar, the methodology as described by Carmeliet et al. (2007) is followed. The saturated liquid permeability for this material has a value of 1.45 10−14 s. The permeability curves for both materials are given in Figure 2. The modeled material properties are validated by simulating a drying experiment described by Gonçalves (2007). The experiment was conducted on a cylindrical specimen of 20 mm diameter. A mortar layer of ±20 mm was applied on a substrate of 20 mm brick. The test started with saturating the substrate (the brick layer) by water. This was done by maintaining the bottom 5 mm of the brick immersed in water for some minutes, until the wet fringe attained the brick/plaster interface. Drying of the sample was

data PVD brick

0.20



Kl = K s ·

Liquid permeability (s)

0.40

which a value of 3 is assumed and Ks is the saturated liquid permeability, having a value of 10−7 s.

brick mortar

1E-11 1E-12 1E-13 1E-14 1E-15 1E-16 1E-17

0.00

1E-18 -9

-8

-7

-6

-5

-4

2

Figure 1.

3

4

5

6

7

8

log pc (log Pa)

log radius (log(m))

Pore volume distribution of brick and mortar.

Figure 2.

Liquid permeability of brick and mortar.

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9

measured by use of the 1D MRI technique (Pel 1995) for 20 hours. The sample was bottom and laterally sealed to assure a one dimensional moisture flow and test conditions were 0% RH at 18◦ C. The measured data points and the simulation results are given in Figure 3, showing a good agreement between both. This indicates that the modeled material properties are physically correct and can be applied in the following crystallization simulations. 4.2 Crystallization simulations The presented 1D simulations are performed on 1 cm thick specimens. The kinetic parameters used in the simulations are given in Table 3. The values for Na2 SO4 .10H2 O (mirabilite) are taken from Espinosa et al. (2008). It must be remarked that these values are determined on other brick materials. In reality the kinetics in the Dutch red ceramic brick and the MEP-SP® mortar may differ, as the amount of nucleation sites on the pore walls is different for every material. 4.2.1 Drying experiment As a first test case, we take a brick specimen, 50% filled with a saturated (6.13 molal) NaCl solution at 20◦ C (Sl = 0.5). Over the 1 cm length, a uniform evaporation flow is applied on all nodes, defined by a surface vapour coefficient of 2 10−8 s/m and a relative humidity of 10%. Due to the evaporation, the 0.25

0h 4h 8h 12 h 16 h 20 h 2h 6h 10 h 14 h 18 h

Water content (m³/m³)

0.2

0.15

2h 6h 10 h 14 h 18 h 0h 4h 8h 12 h 16 h 20 h

BRICK MORTAR

0.1

supersaturation ratio increases above 1 and induces crystallization. The decrease of the liquid mass (water and ions in solution) and the increase of the crystal mass with time are given in Figure 4. At the end of the drying process (around 3240 s), the last ions precipitate from the thin water layer which is left, and the supersaturation goes back to 1. It has to be remarked that the decrease of the moisture content would be expected to go slower at the end of the drying process. That we don’t see this clearly here has to do with the dimensionality (1D) of the simulation and the fact that not all physical processes are modeled precisely yet. 4.2.2 Deliquescence experiment A second simulation illustrates deliquescence. We start from a dry brick sample with an initial NaCl crystallization degree Scr of 0.25 at 20◦ C. The 1 cm long brick specimen is exposed on one side to an environment of 95% RH, so moisture uptake takes place due to sorption, which dissolves the present NaCl salts. A surface vapour transfer coefficient of 2 10−6 s/m is used. The other boundary is sealed (no flux condition). The kinetic parameters for dissolution are used for the deliquescence process. It must be remarked that the real K value for deliquescence can be lower than the one for dissolution, as for deliquescence we have to go through the process of water adsorption before dissolution can start. However the aim of this example is to show the calculation possibilities of the model. Figure 5a shows the change of the liquid saturation degree. This changes the fastest at the boundary, where the 95% RH is imposed. Consequently, the crystals are dissolved the fastest at this position also, as given in Figure 5b. Deeper in the material the liquid saturation degree evolves slower and less crystals are dissolved as the salt concentration is higher there. After 4.2 hours the sample is completely filled.

0.05 160

0 10

15

20

25

30

35

40

NaCl Na2 SO4 .10H2 O

gcr

1 8

0.41 2.52

1 1.91

1.020

100 1.015 80 1.010 60 1.005

40

1.000

20 0

Kdiss (kg/m3 s)

gdiss

0.013

1

Supersaturation (-)

Kinetic parameters for NaCl and Na2 SO4 .10H2 O. Kcr (kg/m3 s)

1.030 1.025

120

Figure 3. Measured (markers) and simulated (lines) water content profiles in a masonry sample with an initially saturated brick part at different points in time.

Ustart

Crystal content

45

Position (mm)

Mass content (kg/m3)

5

Table 3.

Liquid content Supersaturation

140

0.995 0

600

1200

1800

2400

3000

Time (s)

Figure 4. Uniform drying of a brick containing a saturated NaCl solution.

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0.30

9

1.00 4.2 hours

Supersaturation

time ↑: plot after every 15 min

8

Crystal saturation degree

Supersaturation (-)

0.60

0.40

0.20

6 5

0.15 4 3

0.10

2 0.05 1

0.20

0

Crystal saturation degree (-)

Liquid saturation degree (-)

0.25 7

0.80

0.00 600 1200 1800 2400 3000 3600 4200 4800 5400 6000

0

Time (s)

0.00 0

2

4

6

8

10

Position (mm)

Figure 5a. Liquid saturation degree of an initially dry brick containing NaCl crystals, on one side exposed to 95% RH.

Figure 6a. Supersaturation and crystal saturation degree of a mortar sample containing a 3 molal Na2 SO4 solution cooled to 4◦ C. 294 292 290

Temperature (K)

Crystal saturation degree (-)

0.30

0.25

0.20 time ↑: plot after every 15 min

0.15

288 286 284 282 280 278

0.10 276 0

1000

2000

4.2 hours

0.05

2

4

4000

5000

6000

Figure 6b. Temperature change in a mortar sample containing a 3 molal Na2 SO4 solution cooled to 4◦ C.

0.00 0

3000

Time (s)

6

8

10

Position (mm)

Figure 5b. Crystal saturation degree of an initially dry brick containing NaCl crystals, on one side exposed to 95% RH.

4.2.3 Cooling experiment In this simulation we start from a mortar which is capillary saturated with a 3 molal Na2 SO4 solution at 20◦ C. A cooling rate is applied on the sample defined by a heat transfer coefficient of 23 W/m2 K and an ambient temperature of 4◦ C. The conductivity λ is taken to be 1.85 W/mK. The supersaturation first has to go up to a value of 8 (see Table 3) before crystallization starts. This is represented in Figure 6a. A steep drop of the supersaturation is observed once crystals start to form, as the crystallization of mirabilite is a fast process. The corresponding temperature change is given in Figure 6b. We observe an increase in temperature when crystallization starts, due to the release of heat of crystallization.

5

CONCLUSIONS

In this paper, we presented a finite element model for heat, moisture, salt transport and salt crystallization in porous materials. Crystallization is well described by the kinetics of the process. All crystallization processes are modeled by one equation, which improves stability and convergence of the system within the finite element framework. The model is illustrated with three academic examples, using real physical material properties. We note from these simulations that it is very important to know the correct material properties and the parameters for the salt kinetics if we want to simulate crystallization processes in real materials. Consequently, further research will focus on the combination of experimental and modeling research to refine the model based on physical crystallization phenomena. Furthermore, the model will be numerically improved for large simulations and to couple the crystallization with the mechanical model of Moonen (2009) to assess damage risks due to salt presence.

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ACKNOWLEDGEMENTS We thank Teresa Diaz-Gonçalves for providing the experimental data of her PhD measurements (Gonçalves 2007).

REFERENCES Buchwald, A. 2000. Determination of the ion diffusion coefficient in moisture and salt loaded masonry materials by impedance spectroscopy, 3rd Int. PhD Symposium 11.-13.10.2000 Vienna, Vol. 2, 475–482. Carmeliet, J., Descamps, F. & Houvenaghel, G. 1999. A multiscale network for simulating moisture transfer properties of porous media, Transport in porous media, 35: 67–88. Carmeliet, J. & Roels, S. 2002. Determination of the moisture capacity of porous building materials, Journal of Thermal Envelope and Building Science, 25(3): 209–237. Carmeliet, J., Janssen, H. & Derluyn, H. 2007. An improved moisture diffusivity model for porous building materials, Proceedings of the 12th Symposium for Building Physics, Vol. 1, pp.228–235, Dresden, Germany, March 29–31 2007. Coussy, O. 2006. Deformation and stress from in-pore drying-induced crystallization of salt, Journal of the Mechanics and Physics of Solids 54: 1517–1547. CRC Handbook of Chemistry and Physics. 2009–2010. Online 90th edition: http://www.hbcpnetbase.com/. Espinosa, R.M., Franke, L. & Deckelmann, G. 2008. Phase changes of salts in porous materials: Crystallization, hydration and deliquescence, Construction and Building Materials, 22: 1758–1773. Espinosa-Marzal R.M. & G. Scherer. 2008. Study of sodium sulfate salts crystallization in limestone, Env. Geo, 56: 605–621, DOI 10.1007/s00254-008-1441-7. Gonçalves, T.D. 2007. Salt crystallization in plastered or rendered walls, PhD Thesis, Technical University of Lisbon, Portugal. Hall, C. 1997. Barrier performance of concrete: a review of fluid transport theory, in: Penetration and permeability of concrete, RILEM report 16, Edited by H.W. Reinhardt, E & FN Spon, London. Hall, C. & Hoff, W.D. 2002. Water transport in brick, stone and concrete, Spon Press, New York. Hassanizadeh, S.M. & Gray, W. 1979. General conservation equations for multi-phase systems: 1. Averaging procedure, Advances in Water Resources, Vol. 2, September: 131–144. Hassanizadeh, S.M. 1986. Derivation of basic equations of mass transport in porous media, Part 1. Macroscopic balance laws, Advances in Water Resources, Vol. 9, December: 196–206. Hassanizadeh, S.M. 1986. Derivation of basic equations of mass transport in porous media, Part 2. Generalized Darcy’s and Fick’s laws, Advances in Water Resources, Vol. 9, December: 207–222. Horibe, A., Fukusako, S. & Yamada, M. 1996. Surface tension of low-temperature aqueous solutions, International Journal of Thermophysics, Vol. 17, 2: 483–493. Janssen, H., Blocken, B. & Carmeliet, J. 2007. Conservative modelling of the moisture and heat transfer in building

components under atmospheric excitation, International Journal of Heat and Mass Transfer 50: 1128–1140. Kestin, J., Ezzat Khalifa, H. & Correia, R.J. 1981. Tables of the dynamic and kinematic viscosity of aqueous NaCl solutions in the temperature range 20–150◦ C and the pressure range 0.1–35 MPa, J. Phys. Chem. Ref. Data, Vol. 10, 1: 71–87. Korosi, A. and Fabuss, B.M. 1968. Viscosities of binary aqueous solutions of NaCl, KCl, Na2 SO4 , and MgSO4 at concentrations and temperatures of interest in desalination processes, Journal of chemical and engineering data, 13 (4): 548–552. Lewis, R.W. & Schrefler, B.A. 2000. The Finite Element Method in the Static and Dynamic Deformation and Consolidation of Porous Media. Chichester: Wiley. Lubelli, B. 2006. Sodium chloride damage to porous building materials, PhD Thesis, T.U. Delft, The Netherlands. Moonen, P. 2009. Continuous-discontinuous modeling of hygrothermal damage processes in porous media, PhD Thesis, T.U. Delft, The Netherlands & K.U. Leuven, Belgium. Moonen, P., Alfaiate, J., Carmeliet, J. & Sluys, L.J. Comparative study on integration rules for hygric simulations, in preparation. Nicolai, A. 2007. Modeling and numerical simulation of salt transport and phase transitions in unsaturated porous building materials, PhD Thesis, Syracuse University, USA. Nicolai, A. & Scheffler, G.A. 2009. Moisture storage and transport properties influenced by salt crystallization inside porous materials, Proceedings of 4th International Building Physics Conference, 15–18 June 2009, Istanbul, Turkey. Nguyen, T.Q., Petkovic, J., Dangla, P. & Baroghel-Bouny, V. 2007. Modelling of coupled ion and moisture transport in porous building materials, Construction and Building Materials, DOI:10.1016/j.conbuildmat.2007.08.013. Pel, L. 1995. Moisture transport in porous building materials, PhD Thesis, T.U. Eindhoven, The Netherlands. Petkovic, J. 2005. Moisture and ion transport in layered porous building materials: a Nuclear Magnetic Resonance study, PhD Thesis, T.U.Eindhoven, The Netherlands. Poupeleer, A.S. 2007. Transport and crystallization of dissolved salts in cracked porous building materials, PhD Thesis, K.U.Leuven, Belgium. Sghaier, N., Prat, M. & Ben Nasrallah, S. 2006. On the influence of sodium chloride concentration on equilibrium contact angle, Chemical Engineering Journal. 122: 47–53. Steiger, M., Kiekbusch, J. & Nicolai, A. 2008. An improved moisture diffusivity model incorporating Pitzer’s equations for calculation of thermodynamic properties of pore solutions implemented into an efficient program code, Construction and Building Materials, 22: 1841-1850. Steiger, M. & Asmussen, S. 2008. Crystallization of sodium sulfate phases in porous materials: the phase diagram Na2 SO4 -H2 O and the generation of stress, Geochimica and Cosmochimica Acta 72: 4291–4306. Van Genuchten, M. Th. 1980. A closed-form equation for predicting the hydraulic conductivity of unsaturated soils, Soil Science Society of America Journal, Vol. 44: 892–897.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

A numerical model for early age concrete behavior Giovanni Di Luzio & Luigi Cedolin Department of Structural Engineering, Politecnico di Milano, Milano, Italy

Gianluca Cusatis Department of Civil and Environmental Engineering, Rensselaer Polytechnic Institute, Troy, NY, USA

ABSTRACT: In this paper, a new constitutive model for early age concrete mechanical behavior is presented. This model, which amalgamates the microplane model and the solidification-microprestress theory, takes into account all the major aspects of concrete behavior, such as creep, shrinkage, thermal deformation, and cracking starting from the initial stages of its maturing up to the age of several years. The aging viscoelastic behavior is described according to the solidification-microprestress theory. Cracking behavior is modeled through an agedependent microplane model, in which the free parameters are assumed to be functions of an aging variable whose evolution depends on hydration degree and temperature. Calibration and validation of the model is performed by simulating numerically the age-dependent response of sealed and unsealed specimens subject to sustained loads and/or shrinkage. The numerical simulations show that the model can reproduce the interplay between shrinkage, creep, and cracking phenomena during maturing and drying processes. 1

INTRODUCTION

A reliable constitutive model for young concrete must be able to reproduce creep, shrinkage, thermal phenomena, and linear and non-linear deformations. In addition, it must take into account complex phenomena such as (1) creep under variable hygro-thermal conditions during concrete aging; (2) autogenous and drying shrinkage; (3) evolution with age of mechanical properties (i.e. compressive and tensile strength, fracture energy, etc.). In this paper this is achieved by amalgamating the Microprestress-Solidification theory (Bažant and Prasannan 1989a; Bažant and Prasannan 1989b; Bažant et al. 1997a; Bažant et al. 1997b; Bažant et al. 2004) and the Microplane Model (Bažant et al. 2000; Bažant and Di Luzio 2004; Di Luzio 2007). The solidification theory, in which the solidifying constituent (the hardened cement gel) is assumed to be age-independent and the chemical aging is interpreted as a volume growth of the solidifying constituent, greatly reduces the number of material parameters in the modeling of aging creep and it allows a unique identification from test data of the age-dependent moduli of Kelvin (or Maxwell) chains typically used to model concrete creep. This is a consequence of the fact that, according to the solidification theory, these chains are characterized by a spectrum of spring moduli and relaxation times that vary in proportion to a function of the solidified volume growth. The microprestress theory Bažant et al. (1997a) assumes that drying creep and long-term creep can be

explained by the same physical theory, resting on the idea of relaxation of self-equilibrated stresses (microprestresses) created in the solid nanostructure of the cement gel by microscopic volume changes of various chemical species during hydration and by the imbalance of chemical potentials among the four phases of pore water (vapor, capillary, adsorbed, and hinderedadsorbed phases). The microplane model, initially formulated by Bažant and coworkers (Bažant et al. 2000; Bažant and Di Luzio 2004; Di Luzio 2007), is a general three-dimensional constitutive law able to reproduce accurately strain softening in tension and compression as well as the reverse from softening to hardening upon confinement. The microplane model has been extensive calibrated and validated with comparison to experimental data. The M4 version of the microplane model as improved by Di Luzio (Di Luzio 2007) is adopted in this study. 2

AGING MODEL

The phenomenon known as aging, which is due to the change in the relative proportions and physical properties of the basic constituents of concrete during the chemical reaction of hydration, at the macroscopic level is observed as the change in the mechanical properties of concrete correlated with the hydration degree (Neville 1997). In a previous study by the authors, a hygro-thermo-chemical model of concrete at early ages has been formulated (Di Luzio and

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Cusatis 2009a) and validated (Di Luzio and Cusatis 2009b). This model is capable of predicting the physical quantities (hydration degree, relative humidity, and temperature) that strongly influences the mechanical performance of concrete at early ages and beyond, and in particular, the evolution of strength with age and deformations (shrinkage and creep strains). Creep In the Solidification theory of Bažant and Prasannan (1989a) the aging law v(t), which represents the volume fraction of solidified matter, is expressed as function of the equivalent time. In view of its physical meaning, a new expression (function of the degree of reaction, α(t), is here proposed   1 α∞ n (1) = v(t) α(t) where α∞ is the degree of reaction at infinite time and n is an empirical parameter. The degree of reaction, α(t), is an average measure of the total binder reaction which in the case of portland cement coincides with the degree of hydration. Strength There is experimental evidence that the evolution of concrete strength depends not only on the degree of hydration, but also on the kinetics of the hydration reaction; see among others, (Carino 1981; Kjellsen and Detwiler 1998; Kim et al. 1998). For instance, the lower the curing temperature the higher the strength at a given degree of hydration (Verbeck and Helmuth 1968). In view of this, it is concluded that aging cannot be related directly to the hydration degree, and therefore, the mechanical properties cannot be obtained without consideration of the hydration kinetics. A realistic aging model must be established, in which the evolution law of mechanical properties must be formulated in terms of at least hydration degree and temperature. The aging model adopted here is the one proposed by Cervera et al. (1999), which considers explicitly the effect of the curing temperature. An aging internal variable λ, called aging degree, is introduced, so that the compressive strength of concrete, fc , can be express as fc (λ) = λfc∞

(2)

where λfc∞

is the compressive strength at infinite time and λ varies from zero to unity. The evolution of the aging degree is related to the degree of reaction and to the temperature, as λ˙ = λT (T )λ˙ α (α)α˙

In Eq. 4 Tref is the reference temperature for the determination of fc∞ , T0 represents the maximum temperature at which hardening of concrete may occur, and nT is a material property. The term λ˙ α (α) represents the relation between the hydration rate and the hardening rate. In this work it is assumed a parabolic function of λα (α), and therefore, the derivative λ˙ α (α) can be written as λ˙ α (α) = Af α + Bf

α ≥ αset

(5)

where αset is a value defining the end of the setting phase, just when the concrete may begin to be considered a solid. Values αset = 0.1 ÷ 0.4 have been proposed in the literature, depending on the type of cement and the water/cement ratio (de Schutter and Taerwe 1999). Imposing the initial (λα (α0 ) = 0) and final (λα (α = 1) = 1) conditions, one obtains that only one parameter in Eq. 5 must be calibrated.

3

MECHANICAL MODEL

In a one-dimensional setting, concrete strain rate ˙ can be decomposed as follows ˙ = ˙ i + ˙ ve + ˙ f + ˙ cr + ˙ sh + ˙ T

(6)

where ˙ i is the instantaneous strain; ˙ ve is the viscoelastic strain; ˙ f is the purely viscous strain; ˙ cr is the inelastic strain due to cracking; and ˙ sh and ˙ T are the shrinkage and thermal strains caused by variations of humidity and temperature, respectively. The triaxial generalization of all the strain components can be based on the restrictions of material isotropy (including the cracking strain, Bažant and Planas (1997). Microprestress-solidification theory The instantaneous strain rate ˙ i , i.e., the strain appearing immediately after applying the stress rate σ˙ , may be written as ˙ i = q1 G σ˙

(7)

At room temperature (about 23◦ C) and for the saturation condition (relative humidity of 100%), the coefficient q1 (expressed in MPa−1 ) is age independent as demonstrated by Bažant and Baweja (1995). The viscoelastic strain rate ˙ ve , originating in the solid gel of calcium silicate hydrates, is described according to the solidification theory (Bažant and Prasannan 1989a),

(3)

where the term λT (T ) takes into account the influence of the curing temperature   T0 − T nT (4) λT (T ) = T0 − Tref

for

γ˙ (t) v(α)

˙ ve (t) =  γ (t) =

t

(t − t0 )G σ˙ dτ

0

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(8)

(9)

where γ is the visco-elastic micro-strain rate of cement gel, v(α) represents the volume fraction of solidified material (see Eq. 1), and the non-aging microcompliance function (t − t0 ) is defined as follows:     t − t0 n (t − t0 ) = q2 ln 1 + (10) λ0

Shrinkage, thermal deformation and cracking Humidity changes cause free hygrometric strains  sh (free shrinkage or swelling) and temperature variations cause thermal strains  T . Both these strains can expressed as

where t −t0 is the loading time duration, q2 (in MPa−1 ) is a free parameter of the model while the other parameters are fixed, with the following values: n = 0.1 and λ0 = 1 day. For numerical calculation, it is convenient to expand the function (t −t0 ) into a Dirichlet series, which is equivalent to modeling the viscoelastic strain of cement gel in terms of a non-aging Kelvin chain. After some mathematical manipulations (Bažant et al. 2004), Eqs. 8 and 9 may be rewritten in incremental form as

where the coefficients ksh and kT can be assumed to be constant for each concrete. The cracking strain  cr is defined through the Microplane Model (Di Luzio 2007), extended in order to take into account the effect of aging by making the free model parameters proportional to the aging degree previously defined (Eq. 3). The microplane model constitutive law gives an explicit formulation of the tensor stress corresponding to a given tensor strain  σ n+1 = fMPL  el−cr +  el−cr (16) n n

ve ve  ve n = An Gσ n +  n

∗

(11)

where the subscript n indicates the time step tn (from ve∗ the time tn to tn+1 = tn + tn ), Ave n and  n are calculated from the series expansion of function (t − t0 ) (Bažant et al. 2004). The purely viscous strain rate ˙ f is the completely irrecoverable part of the creep strain, which is explained with the relaxation of the microprestresses S in the micro-pores containing the hindered adsorbed water. The breakage of these bridges due to the overstress and their restoration with adjacent atoms, leads to a slip of planes of hindered water and consequently to the flow creep (Bažant et al. 1997a; Bažant et al. 1997b). Therefore, the macroscopic viscosity is related to the microprestress and is formulated as ˙ f (t) =

Gσ (t) η(S)

˙ ˙ sh = ksh h(t)I

1 = cbS b−1 η(S)

(12)

where c0 (in MPa−1 day−1 ) and k1 (in MPa K−1 ) are free parameters of the model. Note that while the general formulation depends separately on c and c0 , only their ratio c/c0 governs the basic creep. The purely viscous strain increment at the time step tn can be written as

f An

∗

(15)

where  el−cr is the elastic and cracking deforman tion of the microplane model and fMPL is a function which defines the microplane model formulation. Since the inverse formulation of Eq. 16 can not be directly obtained, one may proceed as follow. According to Eq. 16, the stress increment for a given strain increment  el−cr can be written as n  el−cr +  el−cr (17) − σn σ n = fMPL  n n

where η(S) is a viscosity parameter defined as function of the microprestress S (Bažant et al. 1997a), b = 2 is a constant and c (in MPa−2 day−1 ) is a free parameter of the model. According to Bažant et al. (2004), under variation of temperature and humidity the evolution of microprestress can be assumed as    ˙  h  2 ˙ + c0 S(t) = k1 T˙ lnh + T  S(t) (13)  h

 fn = Afn Gσ n +  fn

and ˙ T = kT T˙ (t)I

(14)

f∗

where and  n are constant in each time step tn . Details can be found in Bažant et al. (2004)

where σ n is the initial stress at the beginning of the time step increment. Assuming that the unloading happens with the initial stiffness E, the stress increment in the time step can be written as  el−cr σ n = E el −  cr n = E  n n  +  el−cr − σn = fMPL  el−cr n n

(18)

Solving the previous Eq. 18 for the the cracking strain increment one obtains el−cr  cr − E−1 n = n

  × fMPL  el−cr +  el−cr − σn n n

4

(19)

COMPUTATIONAL MODEL FOR EARLY-AGE BEHAVIOR OF CONCRETE

The total strain incremental at the time step n, , can be expressed as cr sh t  n =  ve n +  n +  n +  n

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(20)

With the substitution of Eqs. 7, 11, 14, 15, and 19 it becomes  n = q1 Gσ n +

Ave n Gσ n

+

∗  ve n

+

Afn Gσ n

∗

+  fn + ksh hn I + kT Tn I +  cr n Eq. 21 can be rewritten concisely as  " cr σ n = Dve n  n −  n −  n

(21)

1 f

q1 + Ave n + An ∗

 σ n = fMPL  el−cr +  el−cr − σn n n

(25)

el−cr  cr − E−1 σ n n =  n

(26)

5 (22)

" where the quantities Dve n and  n are known at the beginning of each time step

Dve n =

unknown  cr n and σ n

G −1

(23)

∗

f  "n =  ve n +  n + ksh hn I + kT Tn I

(24)

Since the inverse formulation of Eq. 16 can not be obtained in closed form, the constitutive equation (Eq. 21) must be solved iteratively. For this purpose the general algorithm proposed by Di Luzio (2009) is adopted. According to this approach the nonlinear system of equations 22 can be solved in the only unknown  el−cr . One can then calculate the other n

NUMERICAL EXAMPLES

The first numerical example deals with the experimental investigation of Laplante (1993), in which two different concrete mixes were tested: (1) an ordinary portland concrete (OPC) with a water to cement ratio (w/c) of 0.5 without additives; (2) a high-performance concrete (HPC) with silica fume and superplasticizers with w/c = 0.3. For these tests cylindrical specimens of diameter 160 mm and length 100 mm were adopted. Fig. 1 shows the comparison between the experiments and the numerical simulation for the evolution of the compressive strength for both mixes. The specimens were subjected to an axial compressive stress of 30% of the compressive strength at the age of loading, so the applied load is increased with the age at loading. Fig. 2 shows strain versus time curves for OPC and HPC loaded at different ages, t = 24 h, 1, 3, 7, and 28

Basic Creep - OPC 8

Compression strength (fc)-OPC

Numerical simulations Experimental data

50

Strain x 10 -4

6

fc (MPa)

40

30

20

4

2

Numerical simulation Experimental data

10

0

0

10 1

10

100

1000

100

1000

10000

time [hours]

10000

time [hours]

Basic Creep - HPC Compression strength (fc)-HPC

10

Numerical results Experimental data

100

8

Strain x 10 -4

fc (MPa)

80

60

40

20

6

4

2

Numerical simulation Experimental data

0

0 1

10

100

1000

10

10000

100

1000

10000

time [hours]

time [hours]

Figure 1. Evolution of the compressive strength for (a) ordinary portland concrete (OPC) and (b) a high-performance concrete (HPC).

Figure 2. Strain versus time under constant load at different ages for (a) ordinary portland concrete (OPC) and (b) a highperformance concrete (HPC).

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7

7

Horizontal load [kN]

5 4 3 2

5 4 3 2 1

1

20

Experimental data Numerical simulations

6

Horizontal load [kN]

Numerical simulations Experimental data

6

30

fc (MPa)

WST - NS

WST - NS

Compression strength (fc) - NS 40

0

0 0

0.4

0.8

0

1.2

c)

Numerical simulation Experimental data

Horizontal load [kN]

1

10

100

1000

10000

time [hours]

5 4 3 2

4 3 2

0 0

0.4

0.8

1.2

CMOD [mm]

0

0.4

0.8

1.2

CMOD [mm]

Figure 4. Strain versus time under constant load at different ages for (a) ordinary portland concrete (OPC) and (b) a highperformance concrete (HPC).

60

fc (MPa)

Experimental data Numerical simulations

5

1

0

80

1.2

WST - HS 6

1

Compression strength (fc) - HS

0.8

7 Numerical simulations Experimental data

6

0 0.1

d)

WST - HS 7

Horizontal load [kN]

10

0.4

CMOD [mm]

CMOD [mm]

40

20

concept of crack band model (Bažant and Oh 1983) was adopted in the numerical analysis. It is worth nothing that the data of the evolution of compressive strength and the WST results for the age of 14.75 days have been used to calibrated the model parameters. The other curves in Fig. 4 can be used for evaluating the prediction capabilities of the proposed model which appear satisfactory.

Numerical simulation Experimental data

0 0.1

1

10

100

1000

10000

time [hours] Figure 3. Evolution of the compressive strength for (a) normal strength concrete (NC) and (b) high strength concrete (SC).

days. The agreement between the experiments and the model simulation is notably good. In the second example, the experimental investigations of Kim et al. (2004) is considered. They experimentally investigated the fracture characteristics of concrete at early ages using the Wedge-Splitting Test (WST) performed on cubical specimens with an initial notch at the edge. By taking different concrete mixes at various ages the load-crack mouth opening displacement (CMOD) curves were obtained. Fig. 3 show the comparison between the experimental data and the model simulation for the evolution of the compressive strength for two mixes, normal- (NS) and high-strength (HS) concrete. Fig. 4 shows the comparison between the experimental and numerical results of the horizontal load components versus CMOD for NS (Figs. 4a and b) and HS (Figs. 4c and d) concretes loaded at different ages (about 3, 7, 14 and 28 days). To prevent pathological spurious mesh sensitivity, the

6

CONCLUSIONS

This paper describes a mechanical model that accounts for many of the features observed in the behavior of concrete at early ages. A novel aging model for creep and microplane model is introduced. The model, implemented in a finite-element program, shows its capabilities in the presented numerical simulations. The qualitative and quantitative agreement between the model results and the available experimental data is good.

REFERENCES Bažant, Z. and J. Planas (1997). Fracture and size effect in concrete and other quasibrittle materials (W.F. CHEN ed.). New directions in civil engineering. CRC Press. Bažant, Z.P. and S. Baweja (1995). Justification and refinement of Model B3 for concrete creep and shrinkage. 1.Statistics and sensitivity. Materials and Structures 28, 415–430.

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Bažant, Z.P., F.C. Caner, I. Carol, M.D. Adley, and S.A. Akers (2000). Microplane model M4 for concrete. I: Formulation with workconjugate deviatoric stress. Journal of Engineering Mechanics, ASCE 126(9), 944–953. Bažant, Z.P., G. Cusatis, and L. Cedolin (2004). Temperature effect on concrete creep modeled by microprestresssolidification theory. Journal of Engineering Mechanics 130(6), 691–699. Bažant, Z.P. and G. Di Luzio (2004). Nonlocal microplane model with strain-softening yield limits. International Journal of Solids and Structure 41(24–25), 7209–7240. Bažant, Z.P., A.B. Hauggaard, S. Baweja, and F.J. Ulm (1997a). Microprestress- Solidification Theory for Concrete Creep. I: Aging and Drying Effects. Journal of Engineering Mechanics, ASCE 123, 1188–1194. Bažant, Z.P., A.B. Hauggaard, S. Baweja, and F.J. Ulm (1997b). Microprestress- Solidification Theory for Concrete Creep. II: Algorithm and Verification. Journal of Engineering Mechanics, ASCE 123, 1195–1201. Bažant, Z.P. and B.H. Oh (1983). Crack Band Theory for Fracture of Concrete. Matériaux et Constructions 16, 155–177. Bažant, Z.P. and S. Prasannan (1989a). Solidification theory for concrete creep. I: Formulation. Journal of Engineering Mechanics, ASCE 115, 1691–1703. Bažant, Z.P. and S. Prasannan (1989b). Solifification theory for concrete creep. II: Verification and application. Journal of Engineering Mechanics, ASCE 115, 1704–1725. Carino, N.J. (1981). Temperature effects on the strengthmaturity relation of mortar. Struc. Res. Lab. Rep. NBSSIR 81-2244, National Bureau of Standards, Washington, D.C. Cervera, M., J. Oliver, and T. Prato (1999). Thermo-chemomechanical model for concrete. II: Damage and creep. Journal of Engineering Mechanics, ASCE 125(9), 1028–1039.

de Schutter, G. and L. Taerwe (1999). Degree of hydration based description of mechanical properties of early age concrete. Materials and Structures 29, 335–344. Di Luzio, G. (2007). A symmetric over-nonlocal microplane model M4 for fracture in concrete. International Journal of Solids and Structure 44(13), 4418–4441. Di Luzio, G. (2009). Numerical model for timedependent fracturing of concrete. Journal of Engineering Mechanics, ASCE 135(7), 632–640. Di Luzio, G. and G. Cusatis (2009a). Hygrothermo- chemical modeling of high performance concrete. I: Theory. Cement and Concrete Composites 31(5), 301–308. Di Luzio, G. and G. Cusatis (2009b). Hygrothermo- chemical modeling of high performance concrete. II: Numerical implementation, calibration, and validation. Cement and Concrete Composites 31(5), 309–324. Kim, J.-K., Y. Lee, and S.-T. Yi (2004). Fracture characteristics of concrete at early ages. Cement and Concrete Research 34(3), 507–519. Kim, J.-K., Y.-H. Moon, and S.-H. Eo (1998). Compressive strength development of concrete with different curing time and temperature. Cement and concrete Research 28(12), 1761–1773. Kjellsen, K.O. and R.J. Detwiler (1998). Laterage strength prediction by a modified maturity model. Material Journal, ACI 90(3), 220–227. Laplante, P. (1993). Mechanical properties of hardening concrete: A comparative analysis of classical and high strength concretes. Ph. D. thesis, Ecole Nationale des Ponts et Chaussées, Paris (in French). Neville, A. (1997). Properties of concrete. New York: John Wiley and Sons. Verbeck, G.J. and R.H. Helmuth (1968). Bonded anchors in high performance fibre reinforced concrete. In 5th Int. Symp. on the Chemistry of Cement, pp. 1–32.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Hygro-mechanical model for concrete specimens at the meso-level: Application to drying shrinkage A.E. Idiart OXAND S.A., Avon, France (formerly at ETSECCPB, UPC, Barcelona, Spain)

C.M. López & I. Carol ETSECCPB (School of Civil Engineering), UPC (Technical University of Catalunya), Barcelona, Spain

ABSTRACT: A previously developed FE-based mesomechanical model for concrete, based on the systematical use of zero-thickness interface elements to represent cracks, is extended to the domain of diffusion-driven phenomena. In particular, the model is applied to the hygro-mechanical coupled analysis of drying shrinkage in concrete specimens, in which the influence of microcracks on the diffusion of moisture is explicitly accounted for and evaluated. First, the model predictions are quantitatively compared with classical drying shrinkage experiments on concrete specimens, and the coupled behavior of the hygro-mechanical model is studied. In addition, the effect of the main heterogeneities of concrete on the drying process and drying-induced microcracking is addressed.

1

INTRODUCTION

Drying shrinkage of concrete is governed by the drying through the pore system of the hardened cement paste. It is currently accepted that moisture migration is driven by a non-linear diffusion process in which, from the initial uniformly saturated state, moisture moves towards the concrete surface in contact with the lower relative humidity environment. Shrinkage strain is then related to the water loss, different at each point of the material. Deformation compatibility leads in general to cross-sectional or structural stresses and, in some cases, cracking. Strains measured at the drying surface depend not only on the external (and internal) restraints to deformation, but also (and especially) on the non-uniform moisture distribution through a cross-section of a specimen or concrete member. Traditionally, studies on drying shrinkage of concrete have been addressed at the macroscopic level, considering the material as continuous and homogeneous. Although this corresponds to the classical engineering approach to concrete, it does not seem optimal from the material modeling viewpoint, since concrete is a composite material formed by a shrinking matrix with aggregates which typically do not shrink. Models explicitly considering the two phases may be simpler and more general and accurate, as the overall response is the outcome of the combination of simpler ingredients in each of the phases. Additional advantages of considering concrete as a two-phase composite may be found for instance in the study of creep (the matrix experiences visco-elastic deformations while

the aggregates do not), and in general all other physical or chemical processes in which there is a well differentiated behavior between aggregates and cement paste or mortar (e.g. high temperatures or sulfate attack). These and related considerations have led to some previous attempts to model the main features of drying shrinkage of concrete with the help of mesostructural models (Tsubaki et al. 1992, Sadouki & van Mier 1997, Sadouki & Wittmann 2001, Schlangen et al. 2007). Most of that previous work, however, considers a reduced number of particles, focuses only on either the moisture diffusion or the mechanical analysis of shrinkage, or exhibits some limited representation of the mechanical behavior and cracking. A sound mechanical representation is an essential basis for an overall prediction of the serviceability, which is precisely the main advantage of the approach presented in this paper, which is an extension of a well-demonstrated meso-mechanical model for concrete in both 2D and 3D. Among other aspects, the model encompasses fracture mechanics-based crack propagation, describes well the localization process (from multiple distributed microcracks to single or few macrocracks), and yields accurate predictions of practically all known aspects of concrete behavior under mechanical loading (Carol et al. 2001, López et al. 2001, 2008, Caballero et al. 2006, 2007). The objective of the present paper is to evaluate the effect of drying-induced microcracks and the material heterogeneities on the overall response of concrete specimens in terms of strains, weight losses, moisture distribution and crack patterns. To this end,

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a mesostructural representation with zero-thickness interface elements is adopted, allowing for explicitly considering the effect of the aggregates and cracks on the overall behavior of the material. Also a quantitative comparison with existent drying shrinkage experiments is pursued.

2 2.1

MESO-MECHANICAL MODEL FOR CONCRETE Geometry and mesh generation

The stochastic procedure used in this work for the generation of 2D geometries is based on the Delaunay/ Voronoï tessellation theory (López et al. 2008). The input data for such representation consist of fundamental parameters related to the mix design (aggregate size, volume fraction, and shape), as well as some other parameters for controlling the randomness of the generation process. Only the largest aggregate particles are explicitly represented. In turn, the surrounding matrix represents the homogenized behavior of mortar plus smaller aggregates. This is motivated by the fact that the fracture process and failure in concrete are generally governed by the main heterogeneities in the material. Additionally, in order to represent the potential crack propagation paths, zero-thickness interface elements are introduced a priori in all the aggregatematrix contacts and also within the matrix, in some predetermined positions. The advantages and drawbacks of this type of mesostructural representation of concrete as compared to other kind of meso-models for mechanical and diffusion-driven phenomena in 2-D have been previously discussed elsewhere (Carol et al. 2001, Idiart, 2009). Extension to 3D in the case of purely mechanical loading has also been addressed in Caballero et al. 2006, 2007). 2.2

Interface constitutive law with aging

The constitutive law for the interface elements is based on the formulation used previously (Carol et al. 1997, López et al. 2008). It also incorporates time as a variable to simulate the aging effect. The model is formulated in terms of the normal and tangential stress components in the mid-plane of the joint element σ = [σN , σT ]t and the corresponding relative displacements u = [uN , uT ]t (t = transposed). The constitutive formulation conforms to work softening elasto-plasticity, in which plastic relative displacements can be identified with crack openings. The initial loading (failure) surface F = 0 is given as three-parameter hyperbola (tensile strength χ , asymptotic ‘‘cohesion’’ c, and asymptotic friction angle tan φ, (curve 0 in Fig. 1). When cracking starts, the loading surface begins to shrink, (curve 2 in Fig. 1).

Figure 1. Cracking surface considering the aging effect: evolution with time and degradation due to energy spent in fracture.

This is achieved by means of softening laws in which the surface parameters are functions of the work spent in fracture processes, Wcr . In order to control the process of evolution of F, the model has two parameters that represent the classical fracture energy in Mode I, GIF (pure tension) and a second energy under ‘‘Mode IIa’’ defined under shear and high compression, GIIa F , with values generally higher than its mode I counterpart. Under pure tension the loading surface shrinks and moves to become another hyperbola with vertex at the coordinate origin. Under mixed-mode, it degenerates further, asymptotically becoming a pair of straight lines (curve 3 in Fig. 1), representing the residual pure friction after all roughness of the crack surface has been eliminated. The aging effect is considered through the evolution of the main parameters of the fracture surface (χ, c) with time, as well as the fracture energies GIF and GIIa F , through a monotonic increasing function of the exponential asymptotic type (Idiart 2009). As a result, the initial fracture surface will expand in time (from curve 0 to curve 1 in Fig. 1). The consideration of the aging effect causes the model to exhibit two counteracting effects: on one side the contraction of the fracture surface is determined by the energy spent in the fracture process, leading to softening behavior; on the other hand, the evolution of the main parameters in time results in an expansion of the fracture surface. This leads to a much richer behavior of the joint element, since the updated fracture surface will depend on the resulting combination of the loading state and the time interval considered. A constitutive verification of the model response may be found in (Idiart et al. 2010, Idiart 2009).

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2.3

Deff (H ) = D0 + (D1 − D0 ) f (β, H )

Aging viscoelastic model for the matrix-phase

In order to simulate the time-dependent deformations in concrete at the meso-level, a basic creep model for the matrix phase needs to be introduced. Aggregates are assumed to remain linear elastic and timeindependent. The implemented rheological model consists of an aging Maxwell-chain, which is equivalent to a Dirichlet series expansion of the relaxation function R(t,t’), dual to the usual compliance function J(t,t’) (t’ = age at loading). It is based on previous work by Bazant & Panula (1978). Because the matrix exhibits a time-dependent mechanism while the aggregates do not, the parameters of the Maxwell Chain for the matrix have to be set in order to produce the desired overall viscoelastic behavior corresponding to the concrete (López et al. 2001). In the present case, this adjustment is made with respect to the compliance function J(t,t’) given in the Spanish code (EHE 1998).

3

in which, D0 and D1 are constants determining the value of Deff at H = 0 and at H = 1, respectively, and the dependence on H is introduced by a hyperbolic function given by (β representing a shape factor)  −1 f (β, H ) = e−β · H · 1 + (e−β − 1) · H 3.2

(4)

K1 (u) = η · u3

(5)

and

K0 (u) = αK1 (u)

where u[cm] is the crack opening, η[1/(cm.s)] is a parameter relating the crack width with the diffusivity, α and βK are two model parameters and f is the function given by Eq. 3 (with β replaced by βK ).

Diffusion through the uncracked porous media

Since the early work of Bazant & Najjar (1972), it is generally accepted that moisture movement in concrete can be described by a non-linear diffusion equation which may be advantageously written in terms of the relative humidity at the point (denoted as H ), which may be written as   dH dwe = C(H ) · = −div Deff (H )grad(H ) dt dt

Moisture diffusion through the cracks

KL (H , u) = K0 + [K1 − K0 ] f (βK , H )

3.3 3.1

(3)

Cracks may affect diffusivity since they represent potential preferential channels for moisture migration out of the material. In the case that a crack forms and/ or propagates, the diffusion process will be affected. To analyze this problem, the same FE mesh is used for the diffusion and the mechanical calculations, allowing the use of a staggered strategy as coupling scheme. This has required the formulation and implementation of interface elements with double nodes also for the diffusion analysis (Segura & Carol 2004). The longitudinal diffusion KL of the element presents two situations: before cracking, KL takes a zero value, and after the crack has opened an expression similar to Eq. 2 is used, in which the diffusivity for saturated flow K1 is in this case given by the so-called ‘‘cubic law’’:

MODEL DESCRIPTION FOR DRYING SHRINKAGE

In order to extend the applicability of the model described in the previous section to the case of hygromechanical (H-M) coupled problems, a connection between the purely mechanical analysis and the drying process is needed. On one hand, the moisture loss due to drying induces a volumetric shrinkage which, if restrained, generates a self-equilibrated state of stresses within a specimen cross-section. On the other hand, an acceleration of the drying process is expected in the case of micro/macro cracking. The last consideration is usually neglected in most of the models proposed in the literature, mainly due to the complexity of a correct implementation and the difficulties of validating a theoretical model.

(2)

(1)

Desorption isotherm model

An important ingredient for the modeling of the drying process is the so-called desorption isotherm, relating H to the evaporable water content in the pores we [g/cm3 ], the latter needed for calculating the overall specimen weight loss and for predicting the shrinkage at a point. The desorption isotherm will also have an effect on the H field and will enter the nonlinear problem through the capacity matrix. In this study, the desorption isotherm proposed by Norling (1997) has been adopted, which is written as we (H )/c = a1 · (1 − e−a3 ·H ) + a2 · (ea3 ·H − 1)

[g/cm3 ]

is the evaporable water content, where we C(H ) is the capacity matrix [g/cm3 ], calculated as the derivative of the desorption isotherm (with respect to H ), the latter relating evaporable water losses and relative humidity. Deff [cm2 /s] is the effective diffusion coefficient, which strongly depends on H itself (Roncero, 1999):

a1 =

0.15αhydr , a2 = 1 − e−a3

w0/c

(6)

− αhydr/3 , −1

ea3

(7) a3 = −w0/c · f1 + f2

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where c[g/cm3 ] is the cement content, αhydr is the degree of hydration, w0 /c the initial water-cement ratio, and f1 , f2 represent two shape factors. The moisture capacity matrix C(H ) is calculated as the derivative of the desorption isotherm. Recent studies by Baroghel-Bouny (2007) have indicated that the aggregate content does not have an effect on the resulting desorption isotherms shape, thus allowing to employ these curves determined on concrete or cement paste for the behavior of the matrix in a meso-scale simulation (assuming impervious aggregates).

3.5

3.4 Shrinkage strains due to drying The difficulties encountered for experimentally determining the shrinkage coefficient αshr , relating shrinkage to variation in weight losses at a local level, have traditionally lead most researchers to assume a constant value of this coefficient: εshr = αshr · we

4 4.1

(8)

 ε˙ shr dt =

αshr (we )w˙ e dt + cst

(9)

in which the shrinkage coefficient can be expressed in a general way as αshr (we ) =

∞ nεshr w¯ n−1 − (we0 )n e

(weenv )n

(10)

∞ εshr represents the final volumetric shrinkage strain corresponding to the environmental H env considered, weenv and we0 are the moisture contents corresponding to H env and the initial internal H , respectively (calculated from the desorption isotherm), w¯ e represents the average moisture content within the considered time interval, and n is an integer. These expressions have been derived by forcing the same (fixed) final dryinginduced shrinkage value for all cases. In this way, only the drying rate is modified when considering different n values. More details of this formulation, and a comparison with experimental data on different concretes, may be found in (Idiart et al. 2010, Idiart 2009).

NUMERICAL RESULTS Simulation of tests by Granger et al. (1997)

This section presents the main results of the analysis of drying shrinkage experiments on cylindrical specimens of different concretes Granger et al. (1997). Specimens were subjected to H = 50% (±5%), allowing a radial drying (top and bottom faces sealed). A normal strength concrete has been chosen for the simulations. Details of the test and material properties are summarized in Table 1. These experiments are particularly relevant due to the fact that the overall relation between longitudinal strains and total weight losses was measured, which is a basic feature for evaluating a coupled H-M model. The cylindrical shape of the specimen is a disadvantage for the present 2D mesostructural model, since an axisymmetric analysis is not suitable for this kind of representation (aggregates would represent bodies of revolution, i.e. rigid ‘donuts’ within a flexible matrix). Nonetheless, the lack of similar experimental results using prismatic samples has forced the selection of these tests. In order to eliminate to some extent these undesired effects, a ‘semi-axisymmetric’ analysis has been performed for the moisture diffusion simulation (axis of revolution at center of the mesh, D/2, D being

However, this linear relationship may sometimes yield only rough approximations. In the present work, shrinkage has been assumed to be related linearly to the local water loss per unit volume in a first stage (López et al. 2005). Further calibration of model parameters with experimental results has put in evidence the advantages of considering a nonlinear relation between strains and weight losses, as will be shown in Section 4.1 (Idiart et al. 2010). The dependency adopted in this paper is based on the work of van Zijl (1999). The local volumetric strains can then be integrated as follows 

Staggered coupling scheme

The H-M coupling is achieved through a staggered approach: one code (DRACFLOW) performs the nonlinear moisture diffusion analysis, and the results in terms of volumetric strains at the local level serve as input to the second code (DRAC), solving the mechanical problem. The updated displacement field (nodal variables) obtained in the latter, from which new crack openings are derived, will in turn alter the diffusion analysis. This loop must be successively repeated within each time step until a certain tolerance is satisfied (in terms of water losses), before passing to the next time interval.

Table 1. Composition & mechanical properties of the concrete used, and summary of test setup (Granger et al. 1997). Penly concrete mixture details & summary of test setup Specimens size (φ, high in cm) G:12.5/25 (kg/m3 ) g:5/12.5 (kg/m3 ) s: 0/5 (kg/m3 ) Filler (kg/m3 ) Cement (kg/m3 ) Water (kg/m3 )

16, 100 682 330 702 50 350 202

Density (kg/m3 ) E28d (GPa) fc28d (MPa) ft28d (MPa) Humid., Temp. ◦ C Measurement base Curing period

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2270 36,2 34,3 3,4 50%, 20◦ 50 cm 28 days

the diameter of cylinder; spin limited to a π value), coupled with a 2D plane stress analysis for the mechanical problem, over the same FE mesh (which is a necessary condition of the model). In this way, an undesired behavior due to the ‘donut shape’ of the aggregates in the mechanical simulations is avoided, and the cylindrical shape for the moisture diffusion problem, which is an essential feature, is kept in the simulation. The FE mesh used in the simulations (see Fig. 2) has dimensions 16 × 50 cm2 (height of the mesh equal to strain measurement basis used in the experiments). The aggregate volume fraction is 26% and the max. & min. sizes are 25.9 & 10 mm (corresponding approx. to G fraction, Table 1), respectively. In order to simulate the central part of a larger specimen, the end faces were forced to remain planar in the simulation, with free horizontal displacements. Material parameters finally adopted are summarized in Table 2. The results in terms of weight losses (as % of the specimen’s weight) vs. time are compared to the experimental values in Fig. 3, for the coupled and uncoupled cases. It can be observed that the effect of coupling is small and could be neglected in this case, and that the numerical results agree well with the experiments. When the same weight loss is represented against the longitudinal strains (average of left and right faces), as in Fig. 4, the numerical results (coupled and uncoupled cases) exhibit some departure from the experimental ones. Moreover, the effect of coupling yields slightly smaller strains, as compared to the uncoupled case, for the same state of drying. The latter is to be expected, since the coupling effect manifests through a higher degree of microcracking, thus decreasing axial shrinkage strains.

Table 2. Adopted parameters for diffusion and mechanical models yielding the best fit to experimental results. Material parameters adopted Diffusion analysis: matrix Initial humidity 100% D0 (cm2 /day) 5 × 10−5 2 2 × 10−1 D1 (cm /day)

Mechanical anal.: continuum Maxwell chain Ematrix Eaggr (MPa) 70,000 νmatrix ; νaggr 0.2; 0.2

β(−) 3.0 C(gcem /cm3 mat.) 0.473 αhydration 0.90 0.50 w0 /c f 1; f 2 5.0; 8.0 αshr 0.01

Mechanical anal.: joint elem. χ (MPa) 2.0; 4.0 c (MPa) 7.0; 14.0 tan φ 0.6; 0.6 tan φresidual 0.2; 0.2 GFI 0.03; 0.06

Diffusion anal.: joint elem. η(1/(cm.day)) 100 × 106 0.01 K0 /K1 βk (−) 0.0

GFIIa σdil (MPa) pχ , pc , pGF Kχ , Kc , KGF

10.GFI 40 0.4, 0.5, 0.8 1.0, 1.0, 1.0

Figure 3. Comparison of simulations (coupled & uncoupled analyses) and experiments in terms of weight loss vs. time.

0.6

(mm/m) 0.5

Experimental: Penly concrete Uncoupled simulation

0.4

Coupled simulation 0.3 0.2 0.1

w e (%)

0 0

Figure 2. tions.

Mesh & boundary conditions used in the simula-

0.5

1

1.5

2

2.5

3

3.5

Figure 4. Comparison of numerical and experimental results: weight loss vs. longitudinal strains [mm/m] for coupled and uncoupled simulations (constant shrinkage coeff. = 0.01[cm3/g]).

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Although the skin microcracking effect is well captured, reflected by small longitudinal strains for large weight losses at the beginning of drying, two experimental features seem not well represented by the simulations: the numerical curve seems to be shifted horizontally with respect to the experimental one, and the second curved part of the experimental curve, in which strains grow more slowly than weight losses, is not well captured by the model. A deeper study of the results has led to two main potential reasons for these differences. On one side, the consideration of a constant shrinkage coefficient could be the cause of the lack of a second curved part in the diagrams. For this reason, simulations have been repeated considering a linear (n = 2 in Eq. 10) and a parabolic (n = 3) dependence of the shrinkage coeff. on the weight loss (the final local shrinkage corresponds to that of a constant shrinkage coeff. of 9 × 10−3 [cm3 /g]). Fig. 5 presents the results for the uncoupled cases. A considerable improvement of the global behavior can be observed, especially for advanced drying states (second curved part), which is now satisfactorily captured. However, a small gap still exist between experimental and simulation results. The second factor that might have an influence on the results is the elasto-plastic nature of the model used for the interface elements, which implies that the crack closure effect is not taken into account. Indeed, relative displacements are irreversible on unloading with the initial stiffness (which is assigned a very high value). This fact may be of importance, since skin microcracks eventually unload and are subjected to compression stresses when the drying front advances towards the interior of the sample. Although the introduction of this effect could probably improve the approximation, further investigation would require a new constitutive interface model with secant unloading, which is at present the focus of on-going work.

0.5

(mm/m) 0.4

4.2

Aggregate effects on drying-induced cracking

Recent experimental observations by Bisschop & van Mier (2002) highlighted the influence of the size and quantity of mono-sized aggregates in the microcrackpatterns of cementitious materials subjected to drying. Due to a restraining effect of the aggregates immersed in a shrinking matrix, microcracks radiating from the aggregates and in the aggregate-matrix interface, may develop. The occurrence and extent of cracking depend on the size and number of inclusions, the rheological behavior of the matrix, the aggregate-matrix bond, the mismatch between the stiffness of inclusions and matrix, and the interaction between inclusions. This is an important aspect for determining the effect of drying on mechanical properties. In order to evaluate the ability of the present H-M mesostructural model to capture the effect of the aggregate distribution, number and size on the crack patterns of samples subjected to drying, a significant number of mesostructural meshes of 120 × 40 mm2 has been generated and numerically tested, in the spirit of these experimental tests. Polygonal aggregates inscribed in circumferences of constant diameter in each mesh have been adopted (experiments were performed with mono-sized glass spheres as aggregates, varying the size for different specimens), with three different volume fractions. Aggregates of 2, 4 & 6 mm have been used, with volume fractions of 20, 30 & 40%, yielding a total of nine cases. Four different meshes have been generated and simulated for most of these cases in order to assess the effect of random aggregate distribution (except in the 2 mm series, with higher computational cost). Geometry and boundary conditions for diffusion and mechanical analyses are shown in Fig. 6. Samples are assumed to be initially saturated. At the age of 28 days H = 0.3 is imposed on the top face of the specimen (the remaining ones are sealed). All simulations were coupled. A set of model parameters with realistic values for a conventional concrete (ordinary Portland cement with w/c of 0.45) has been used. Aging viscoelasticity is assumed for the matrix (and aging interface elements). Only the central part of the specimen

PENLY CONCRETE SIMULATION ( Parabolic shr. coeff.) SIMULATION( Linear shr. coeff.)

0.3

SIMULATION( Constant shr. coeff.)

H = 0.3

0.2

0.1

w e(%) 0 0

0.5

1

1.5

2

2.5

3

Figure 5. Comparison of (uncoupled) numerical and experimental results: weight losses vs. long. strains[mm/m]. Influence of different shrinkage coeff. functions on the drying behavior.

Figure 6. Geometry and boundary conditions for mechanical and diffusion analyses (2 mm aggregates, 30% volume fraction).

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(40 × 40 mm2 ) is used for quantification of the crack patterns, to eliminate end effects. In order to compare simulations made with different matrix volume fractions, and thus different initial moisture contents, the degree of drying is defined as the weight loss at time t with respect to the initial water content (before drying starts). More details on the simulations setup and results on different cases may be found in (Idiart 2009, Idiart et al. 2010). Rosette diagrams have been generated as a postprocess for all the simulated cases in order to quantify the crack patterns and determine the orientation of microcracks, as shown e.g. in Fig. 8. These diagrams are constructed by adding up the length of all crack-segments of the matrix with the same orientation. Aggregate-matrix cracked interfaces are not considered in the quantification, similarly to the experiments. The rosettes for the cases of 4 and 6 mm aggregates are computed as the average of 4 different meshes. Rosettes help us determining the mechanisms acting on drying-induced microcracking. A homogeneous material will mostly produce cracks perpendicular to the drying surface, resulting in rosettes with vertically elongated shapes. In contrast, when cracking due to the effect of aggregate restraining is predominant, cracks will randomly radiate from the inclusions, producing cracks along all possible directions. In this case, it is expected that the rosettes have a rounded shape. The intermediate case, with both mechanisms present, will show rosettes with a shape varying between these two limiting cases (Bisschop & van Mier 2002). Fig. 7 shows the experimental results obtained by Bisschop & van Mier (2002). The influence of the aggregate volume fraction at constant aggregate size and of the aggregate size at constant volume fraction was evaluated for cementitious composites with glass spheres. These two cases show that an increase in size or volume fraction yields an increase of the degree of microcracking and of the aggregate restraining effect (trends shown here were not as marked as in Fig. 7 for all the experimental series). Fig. 8 depicts the microcrack patterns of the simulations with 4 mm aggregates and 3 different volume fractions (20, 30 & 40%). Fig. 8a shows the microcrack patterns at 30% drying, in terms of the fracture energy spent. It can be observed that the depth of penetration of microcracks does not increase with increasing volume fraction, which is in agreement with experimental observations (Fig. 7). In Fig. 8b, the averaged polar diagrams at the same drying state for each case are presented. It can be clearly seen that an increase in volume fraction results in much less elongated polar diagrams (note also the increase of the aspect ratio, calculated as the relation between horizontal and vertical axis of the rosettes). This is due to the aggregate restraining effect, which becomes more important when increasing the number of inclusions. For the lowest volume

(a)

90

90

135

45

20

30 45

20

135

0

0

225

315

10

0

180

0

225

0

180

0

225

315

315

270

270

45

20

10

10 180

90

30

30 135

270

(b)

90

90

26

135

45

45

135

45

13

13 180

90

26

26 135

0 180

0

225

315

13 0

0

225

315

180

225

315 270

270

270

0

0

Figure 7. Experimental results of microcracking: crackmaps (samples central part, 4 superposed cross-sections) and rosettes. (a) Effect of aggregate size (2, 4 & 6 mm left-to-right; vol. fraction = 35%; 30% drying); (b) effect of aggregate vol. fraction (10, 21 & 35%, left-to-right; size =4 mm; 10% drying). Results by Bisschop & van Mier 2002, with permission of the author.

4mm - 20% vol

4mm - 30% vol

4mm - 40% vol

(a) (b)

5 cm

90

90

120

150

30

2,5 0

180

120

60

0

210

330 240

300 270

0.18

150

5 cm

90

60

2,5 0

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120

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0

210

330 240

300 270

0.22

150

5

cm

60 30

2,5 0

180

0

210

330 240

300

0.45

270

Figure 8. Effect of aggregate volume fraction (20, 30 & 40%) on drying-induced microcracking (4 mm aggregates, 30% drying): (a) microcrack-maps of 40 × 40 mm2 midspecimens; (b) averaged (of 4 different geometries) rosettes and aspect ratios.

fraction, an elongated diagram is found, suggesting that aggregate restraining is not very important in this case. The results are in qualitative agreement with the experiments (Fig. 7).

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2mm - 20%

4mm – 20%

6mm – 20%

(a) (b) 135

90 2 1.5

cm

90 2 45

135

1 0 180

315 270

90 2 45

135

0.363

1.5

cm 45

1

0.5

0

225

cm

1

0.5 180

1.5

0.5

0

0

225

315 270

0.735

0

180

225

0

315 270

0.693

Figure 9. Effect of aggregate size (2, 4 and 6 mm aggregates) on drying-induced microcracking (20% vol. fraction, 30% drying): (a) microcrack-maps; (b) rosettes & aspect ratios for each case.

The experimental results depicted in Fig. 7 also showed that with increasing aggregate size, at constant volume fraction, the rosettes turn out more round-shaped, and also the total crack length measured turns out considerably higher. The first batch of results obtained with the present model seemed not to reproduce too clearly these experimental trends. After careful examination, that was attributed to the need of filtering (i.e. eliminating from the plotted results) microcracks with crack openings smaller than a certain threshold (microcracks with widths much smaller than 1 micron are not captured by the experimental technique, even though the exact threshold is not clear). Thus, it seems reasonable to consider a threshold of 0.1 microns in the simulations. Results finally obtained by filtering microcracks with the above criterion as a post-process are shown in Fig. 9, in terms of crack patterns and polar diagrams. It can be seen that the trends of absolute crack lengths, cracks penetration depth and polar diagrams are in qualitative agreement with the experiments (Bisschop & van Mier 2002). Although the rosettes seem to be sensitive to the crack filtering with very small crack widths, the model is capable of capturing the effect of aggregate size on drying-induced microcracking.

5

CONCLUDING REMARKS

be captured by only representing the largest aggregate pieces while the rest (smaller aggregates and sand) are considered to be represented by the matrix phase. Cracking is introduced via zero-thickness interface elements equipped with an elasto-plastic fracturebased constitutive law, and the additional moisture diffusion through open cracks is explicitly accounted for in the simulation. The present model is able to satisfactorily represent the essential features of drying shrinkage in concrete, such as the non-uniform moisture distribution due to the presence of aggregates and cracks, strains vs. weight loss relation, and crack patterns. Simulations have shown that the effect of coupling may only mean a slight increase in the drying state, especially at the beginning of the drying process, when drying through surface microcracks is most important. This seems to suggest that in many practical cases uncoupled analyses can be performed without major loss of consistency of the results and with a significant reduction of computational cost. The model parameters have been adjusted to existing experimental results of concrete specimens (Granger et al. 1997), and the resulting numerical predictions have been found to agree well with experimental measurements. In addition, it has been shown that the consideration of a nonlinear local relationship between shrinkage strains and weight losses can be more accurate for simulating drying shrinkage experiments than the commonly employed linear relationship. Finally, the effect of aggregates on the dryinginduced microcracking has been studied and qualitatively compared with experimental findings in (Bisschop & van Mier 2002). The performance of the model in this regard has proven satisfactory in most cases, showing larger degrees of internal microcracking for higher drying states, and increasing aggregate volume fraction and size. ACKNOWLEDGEMENTS Research supported by grants BIA2006-12717 and BIA2009-10491, funded by MEC (Madrid, Spain). Special thanks to Dr. Jan Bisschop for kindly providing the experimental data in Fig. 7. REFERENCES

The applicability of the FE mesomechanical model previously developed for concrete specimens under purely mechanical loading has been extended to the analysis of hygro-mechanical coupled analysis of drying shrinkage in concrete. A basic underlying assumption of the meso-level analysis is that the main aspects of mechanical behavior as well as cracking and degradation due to diffusion-driven phenomena may

Baroghel-Bouny, V. (2007). Water vapour sorption experiments on hardened cementitious materials. Part I and Part II. Cem. Conc. Res., 37:414–454. Bazant, Z.P. & Najjar, L.J. (1972). Nonlinear water diffusion in nonsaturated concrete. Mat. Struct., 1:461–473. Bazant, Z.P. & Panula, L. (1978). Practical prediction of timedependent deformations of concrete. Part II: basic creep. Mat. Struct., 11:317–328.

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Bisschop, J. & van Mier, J.G.M. (2002). Effect of aggregates on drying shrinkage micro-cracking in cement-based composites. Mat. Struct., 35:453–461. Caballero, A., López, C.M. & Carol, I. (2006). 3D mesostructural analysis of concrete specimens under uniaxial tension. Comp. Meth. Appl. Mech. Eng., 195:7182–195. Caballero, A., Carol, I. & López, C.M. (2007). 3D mesomechanical analysis of concrete specimens under biaxial loading. Fatigue & Fracture of Engng. Mat. & Struct., 30:877–886. Carol, I., Prat, P. & López, C.M. (1997). Normal/shear cracking model. Application to discrete crack analysis. ASCE J. Engng. Mech., 123(8):765–773. Carol, I., López, C.M. & Roa, O. (2001). Micromechanical analysis of quasi-brittle materials using fracturebased interface elements. Int. J. Num. Meth. Engng., 52:193–215. EHE, (1998). Instrucción de Hormigón Estructural EHE. Min. de Fomento, Madrid (in Spanish). Granger, L., Torrenti, J.M. & Acker, P. (1997). Thoughts about drying shrinkage: scale effects and modelling. Mat. Struct., 30:96–105. Idiart, A.E. (2009). Coupled analysis of degradation processes in concrete specimens at the meso-level. Doctoral Thesis, UPC, Barcelona, Spain. Idiart, A.E., López, C.M. & Carol, I. (2010). Modeling of drying shrinkage of concrete specimens at the meso-level. Submitted for publication. López, C.M., Carol, I. & Murcia, J. (2001). Mesostructural modeling of basic creep at various stress levels. In Ulm, Bazant & Wittmann (eds) Concreep6, Cambridge (US), pp. 101–106. López, C.M., Segura, J.M., Idiart, A.E. & Carol, I. (2005). Mesomechanical modeling of drying shrinkage using interface elements. In Pijaudier-Cabot, Gérard & Acker (eds), Concreep7, Nantes, France, pp. 107–112.

López, C.M., Carol, I. & Aguado, A. (2008). Meso-structural study of concrete fracture using inter-face elements. I: numerical model and tensile behavior. II: compression, biaxial and Brazilian test. Mat. Struct., 41:583–620. Norling, K. (1997). Moisture Conditions in High Performance Concrete. PhD Thesis, Chalmers Univ. Tech., Goteborg, Sweden. Roncero, J. (1999). Effect of superplasticizers on the behavior of concrete in the fresh and hardened states: implications for high performance concretes. PhD thesis, UPC, Barcelona, Spain. Sadouki, H. & van Mier, J.G.M. (1997). Meso-level analysis of moisture flow in cement compo-sites using a latticetype approach. Mat. Struct., 30:579–587. Sadouki, H. & Wittmann, F.H. (2001). Damage in a composite material under combined mechanical and hygral load. Lecture Notes in Physics, Cont. & Discont. Mod. of Cohesive-frictional Mat., Springer-Verlag Berlin Heidelberg, 568/2001, pp. 293–307. Schlangen, E., Koenders, E.A. & van Breugel, K. (2007). Influence of internal dilation on the fracture behaviour of multi-phase materials. Engng. Fract. Mech., 74:18–33. Segura, J.M. & Carol, I. (2004). On zero-thickness interface elements for diffusion problems. Int. J. Num. Anal. Meth. Geomech., 28(9):947–962. Tsubaki, T., Das, M. & Shitaba, K. (1992). Cracking and damage in concrete due to nonuniform shrinkage. In ed. Z.P. Bazant (ed.) Fracture Mechanics of Concrete Structures (FraMCoS1), Colorado, USA, pp. 971–976. van Zijl, G.P. (1999). Computational modelling of masonry creep and shrinkage. PhD thesis, Delft Univ. Tech., The Netherlands.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Comparison of approaches for simulating moisture content changes in concrete A.D. Jefferson Cardiff University School of Engineering, Cardiff, UK

P. Lyons LUSAS, Kingston Upon Thames, Surrey, UK

ABSTRACT: A coupled thermo-hygro formulation is presented for the prediction of concrete behaviour due to hydration, drying and general changes in moisture content. The approach is essentially similar to that of Gawin et al (2006) but neglects the gas phase on the basis of the assumption that the gas phase is maintained at atmospheric pressure. Whilst there are certain minor aspects of the formulation which are unique, the primary aim of the work is not to present a new formulation but rather to find a reliable approach to predicting the parameters of the model with a view to its implementation in the commercial FE code LUSAS. As part of this work, comparisons were made between simulations of the drying experiments of Kim and Lee (1999) using the present formulation and the CEB-FIP method. The range of certain parameters found in the literature is explored and issues associated with establishing reliable values discussed. In addition, a comparison is made between the effective diffusion coefficient from the developed model and the diffusion coefficient from the CEB-FIP model.

1

INTRODUCTION

The established approach for simulating drying shrinkage, first developed by Bazant & Najjir (1972), involves using an empirical equation for internal humidity which is linked to a shrinkage potential (CEB-FIP, 1993). The relative humidity is computed from a diffusion analysis with moisture content or relative humidity as the primary variable. This approach has been included in codes of practice (CEB_FIP, 1993) and has been validated using large amounts of experimental data. The numerical advantage of approaches based upon simulating relatively humidity or moisture content directly is that they are relatively simple and result in equations that are wellconditioned. However, as argued by Gawin et al. (2006) such approaches have certain inherent disadvantages which are a consequence of their phenomenological basis.The fact that certain physical processes are not considered explicitly makes extending the approaches to more general cases more difficult. As further pointed out by Gawin et al. (2006), such approaches do not account for phase changes in the moisture, which are important during early age drying when temperature gradients are significant and are not easily adjusted to account for changes in porosity, permeability,density and stiffness, all of which are very significant in early age concrete behaviour.

Alternative, more mechanistic approaches, based on solving the mass balance equations of gas, liquid and temperature flow and coupling them to mechanical behaviour, have been developed in recent years by Coussy et al. (2004), Gawin et al (2006) and Bary et al. (2008). Gawin et al. (2006) modelled the field equations using the mass balance equations for dry air, water vapour and liquid water as well as the enthalpy balance equation (i.e. heat flow). This reduces to three sets of equations with gas pressure (pg ) water capillary pressure (pc ) and temperature (T) as the independent variables. Bary et al. (2008) used an essentially similar approach but argued that a simplified approach could be used for moderate temperatures (up to 200◦ C) with only two primary flow variables pc and T. The simplification was achieved by assuming that the water vapour pressure is constant at atmospheric (patm ). The reduction in the problem size is significant and generally leads more well-conditioned equations. The present authors followed the route of using only two equations but believe that it is better to assume that the combined dry-air vapour mix is constant at atmospheric pressure and that vapour diffuses through the gas phase. Studies on gas flow in concrete show that the time to reach steady state is relatively short (Gardner et al., 2008) and would be negligible in comparison with the time scale of drying shrinkage. Hence, provided that there are not rapidly changing temperature

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gradients, as in a fire or in a pressure vessel, the current assumption is considered valid. The overall aim of the work described in this paper is to establish a reliable and robust implementation of the model in the commercial program LUSAS and to give users a reliable means of defining and establishing the model parameters. As part of this work, the model has been compared with the ‘traditional’ model, as described in the CEB-FIP model code 1990 and the possibility of allowing users to derive parameters from those of the CEB-FIP model explored. For completeness the equations will include moisture and heat flow, although the paper will concentrate on moisture flow aspects.

2 2.1

FLOW EQUATIONS

pg = pv + pda = patm

(5)

Darcy’s flow equation for phase π is taken in the form used by Lewis and Schrefler (1998) as follows 1 ki krπ ρπ vπ = − (∇pπ − ρπ g) ρπ μπ

nSπ vπ =

(6)

= −kπ (∇pπ − ρπ g) in which ki = ‘intrinsic’ permeability, krπ = relative permeability of phase π , μ = viscosity, g = gravity vector. Fourier’s equation for heat flow is given by q = −kT ∇T

(7)

Coupled thermo-hygro balance equations

The mass balance equations for the water (w) and water vapour (v) phases and the enthalpy balance equation involving the temperature (T) may be written as follows; ˙v = 0 ρ˙¯ w + ∇ · (ρ¯w vw ) + m

(1)

˙v = 0 ρ˙¯ v + ∇ · Jv − m

(2)

ρC T˙ + ∇ · q − Q = 0

(3)

In which, n = porosity, Sπ = Degree of saturation of phase π and ρπ = density of phase π. Subscript π denotes the phase, i.e. water (w) and gas (g). vπ is the velocity of the phase π , Jv mass flux of vapour = ˙ v = rate of vapour mass transJv = ρv nSg vvg , m fer from liquid water (positive is liquid reduces and vapour increases).





Sw = 1 +

pc ac patm

−1 b  b  b−1

(8)

The formulation allows for vapour diffusion within the air phase. Diffusion is governed by Fick’s equation which, with the present assumptions, becomes;

ρC = (1 − n) ρs Cs + nSg ρg Cg + nSw ρw Cw , Cπ = Specific heat of phase π, Hv = Specific heat of vaporisation, q = heat flux and Qh = A term to account for heat generated (W/m3 ) The primary variables will be capillary pressure (pc ) and temperature (T). The capillary pressure is related to the water and gas pressures (pw and pg respectively) as follows; pc = pg − pw

where kT = is the thermal conductivity and is a function of degree of saturation and void ratio. A key element of this type of formulation, and probably the most contentious, is the isotherm equation which relates moisture content to capillary pressure. Although a single function is often used, it is well known that a pronounced capillary hysteresis is evident from experimental Sw v pc curves (Coussy, 2004), which is ignored in the present, and many other, such formulations. The importance of this hysteresis is an issue to be addressed as part of the present work. For the present, the relationship used by Gawin et al. (2006), originally developed for soils by van Genuchten (1980), will be adopted

(4)

The gas is assumed to comprise dry air (da) and water vapour (v) and, by Dalton’s law of partial pressures, the gas pressure is the sum of the two partial pressures. However the assumption is used that the gas pressure is maintained at atmospheric patm

Jv = −Dmv ∇pv in which

Dmv = Dv

Mv RT

(9)

The vapour pressure within pores is that which is in equilibrium with the external vapour pressure. The saturated vapour pressure (pvs ), which is a function of temperature alone, is that appropriate to external environment, i.e. that pertaining to pure liquid over a flat surface. With these definitions, the relationships between pc and pv may be obtained using Kelvin’s equation as follows;

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  −ρw RT pv pc = ln Mw pvs

Using the Galerkin weight procedure in the discretisation of the equations and making use of the constitutive relations given in equations (5) to (13) leads to a set of discretised equations.

(10)

in which R = gas constant and Mπ = molar mass of phase π . The ideal gas law relates gas density to pressure as follows; ρπ RT pπ = Mπ

(11)

The intrinsic permeability and relative permeability of gas and water depend upon the degree of saturation and are assumed to be governed by the following equations (Gawin et al. 1999, 2006)   ki = ki0 10AK (T −T0 )+A (1−h )

(m 2 ) (12)

krw = SwAw In which AK = 0.005, A = 2.25, Aw = 3. On the basis of work by Perre et al. (1980), Gawin et al (2006) suggested the following expression for the diffusion coefficient  Dmv = n (1 − Sw )Av fs Dv0

T T0

Bv

patm pg

h˙ − ∇ · (D · ∇h) − h˙ s = 0

(15) The formulation allows for Dirichlet and the following mixed boundary conditions

(17)

In which hs = change in relative humidity due to self-desiccation and D is the moisture diffusion coefficient, which is given by  D = D1 α +

1−α 1 + [(1 − h) / (1 − hc )]n

 (18)

In which, using CEB-FIP values, α = 0.05, hc = 0.8 n = 15, D1 = value at h = 1 and D0 = D at h = 0 and may, in the absence of more detailed information, be taken as 1 × 10−9 m2 /s. Here, it is assumed that self-desiccation may be reasonably simulated using the expression  hs = h∞ e−t/τs − 1

(14)

 ρC T˙ + ∇ · q − Q − Hv ρ˙¯ w + ∇ · (ρ¯w vw ) = 0

Moisture content diffusion equations

Kim and Li (1999) consider their own experimental data using the CEB-FIP 1990 model, developed by Bazant and Najjar (1972), but added a term which allows for self-desiccation. It should be noted that Bazant and co-workers later developed an updated version of the 1972 model (Bazant an Xi, 1994). The governing equation assumes that the moisture content W, which in terms of previous variables = n · (Sg · ρv + Sw · ρw ), is a function of the relative humidity h. Thus the equation may expressed in terms of h, as follows

(13)

In which, Av is given as in the range 1 to 3 (here taken as 1), Bv =1.667 and Dv0 =2.58*10−5 m2 /s. fs in the range 0.001−0.01. The aim of this section is to derive two main governing equations with pc and T as the primary variables. Adding equations (1) and (2) eliminates m ˙ v from these equations and then m ˙ v is eliminated from (3) using (1). This gives the following governing equations ρ˙¯ w + ρ˙¯ v + ∇ · (ρ¯w vw ) + ∇ · Jv = 0

2.2

(19)

In which h∞ is the total change in final change in h due to self-drying and τ is the ultimate hydration time parameter. The boundary condition for this diffusion type equation is as follows (D · ∇h) · n = −βh (h − henv )

(ρ¯w vw + Jv ) · n − qwv − βc (ρv − ρvenv ) = 0 on qc

(20)

In which βh = boundary transfer coefficient in terms of h.

(q − Hv ρ¯w vw ) · n − qT − αc (T − Tenv ) = 0 on qT (16) where qwv = the water species boundary flux, qT = boundary heat flux, n is the normal to the boundary and βc and αc are the flux boundary transfer coefficients for moisture and temperature respectively.

3

PARAMETER COMPARISONS

This section will firstly examine the differences between the effective overall moisture flow coefficients employed by the two models.

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Arranging the terms of the first mechanistic approach to obtain the equivalent overall diffusion constant Deqv gives 

Deqv

∂pv = n −Sw ρw kw + Dmv ∂pc



Table 1.

ki

∂pc ∂h

(21)

The following graph show comparisons between • Deqv , using parameters appropriate for simulating the examples of Kim and Lee (1999) (Deqv ), • Deqv but with Dmv set to 0, to show relative contribution of vapour diffusion within the gas phase (Deqv_0 ) • D using CEB-FIP parameters (DCEB ) • D using parameters to give best fit to Deqv (Dbf ) The latter curve is included to test whether the CEBFIP function can in fact match the present model’s effective diffusion function (or perhaps, vice versa). The graphs shown in Figure 1a and 1b give the D values for h from 0 to 1 and 0 to 0.4 respectively. 6.E-09 Deqv 5.E-09

kef

DCEB

2.E-09 1.E-09 0.E+00 1

h

1.E-10 Deqv Deqv0

8.E-11

8.5 × 10−17 to 19.3 × 10−17 (G NSC) 2.7 × 10−17 to 7.3 × 10−17 (G HSC) 3 × 10−21 NS(∞) 2 × 10−22 to 5 × 10−22 HS(∞)

Gardner et al. (2008)

LP (Gas) HP (Gas) LP (Water) HP (Water)

1 × 10−23 (Sw = 0.1) to 2 × 10−18 (Sw = 1.0) (W) 1 × 10−16 (Sw = 0.0) to 1 × 10−22 (Sw = 0.9) (G)

BaroghelBouny et al. (1999) Chung & Consolazio. (2005)* Bary et al. (2008)*

The differences between the standard CEB-FIP values and those which provide a close fit to the data of Kim et al. are considerable. However, the comparisons also show that the CEB-FIP and Deqv can be made effectively the same and also that, for normal environmental relative humidity values, the second term within the brackets in (21) is effectively negligible. To further explore the issue of parameters for the mechanistic model, a range of permeability values found in the literature were gathered and these are displayed in Table 1. Particular care has to be taken with how the permeability is defined in each case and under what conditions it was measured. The table below certainly illustrates the challenge a user may be faced with in selecting an appropriate value for the intrinsic permeability.

D 3.E-09

0.5

Ref

NSC = Normal strength concrete; HSC = High strength concrete. LP = Low permeability concrete; HP = High permeability concrete. ∞ = long term; W = Water; G = Gas. * = High temp study; ** Moderate temps, includes microcracking.

Dbf

0

Values m2

2.24 × 10−17 2.17 × 10−17 8.49 × 10−19 9.82 × 10−17

Deqv0

4.E-09

Comparison of permeability values.

Dbf

4

DCEB

6.E-11 D

One example is presented to illustrate the sensitivity of solutions to the effective diffusion parameter. The example considers a test from a series by Kim and Lee (1999) who undertook a number of experiments on drying shrinkage using prismatic specimens, as illustrated in Figure 2, which were cast and then subject to moist curing for 3 days. After this, one surface was exposed to a relative humidity of 50% and a temperature of 20◦ C, whilst all other surfaces were sealed. The specimens designated medium (M) are considered for analysis. These were cast using a typical concrete

4.E-11

2.E-11

0.E+00 0

0.1

0.2

0.3

0.4

h

Figure 1.

EXAMPLE

1a (above) and 1b (below). D vs h.

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All others surfaces sealed except exposed face

Exposed surface

1.000 0.950 0.900

200 30

70

0.850 30 days Exp 60 days Exp 30 Days Num TH 60 Days Num TH 30 days Num H 60 days Num H 30 days Num H_std 60 days Num H_std

0.800

120

0.750 0.700 0.650 0.600 0.550 0.500 28

48

68

88

108

Distance from exposed face (mm)

Long section 100

Figure 3.

Numerical—Experimental comparisons.

(Dims. in mm) 100

range of simulations is needed to enable such guidance to be supplied reliably. Cross-section

Figure 2.

Testing arrangement of Kim & Lee (1999).

5

made with w/c 0.4 and maximum coarse aggregate size of 20 mm. The specimen was modelled with a two-dimensional mesh of 4-noded elements. In Figure 3 TH denotes the first formulation with pc and T as primary variables, H denotes the CEB-FIP method, but with the diffusion constant chosen for best fit, and H_Std uses the CEB-FIP model with default values for all parameters. The differences in relative humidity between the CEB-FIP analysis with standard parameters and the other analyses are approximately 10% over much of the specimen. To get a first indication of what this means in terms of free shrinkage a relationship is used which provides a reasonable match to the data of Baroghel-Bouny et al. (1999) for normal strength concrete in the range 0.55 < h < 0.9 εsh = 2150 · h · 10−6

(22)

Although grossly simplified, this suggests a potential free shrinkage strain error of approximately 2 × 10−4 , which is very significant. At present it is thought advantageous to include an option for supplying parameters in the form of the CEB-FIP model since users are more likely to be familiar with these parameters and guidance is available based on a wide range of experimental data. It is planned to also provide guidance on key parameters of the model presented and to allow the user to input these directly. It is believed, however, that a good

CONCLUDING REMARKS

A coupled thermo-hygro formulation was presented for the prediction of concrete behaviour due to hydration, drying and general changes in moisture content. The issue of how to supply users with reliable means of identifying parameters was discussed. To explore this issue some comparisons were made with the CEBFIP 1990 diffusion model which is expressed in terms of relative humidity. It was shown that for a simple case, the effective diffusion coefficient from the coupled model, developed in terms of capillary potential and temperature, can match the D vs h curve from the CEB-FIP model. This suggests that it would possible to use the more advanced mechanistic formulation but to provide an option for the users to supply limited parameters in terms of those of the CEB-FIP model in cases where more detailed information was not available. The work presented in this paper gives only a first indication of issues associated with the implementation and parameter selection of the hygro-thermal model for the prediction of drying behaviour. The full study will be the subject of a forthcoming journal publication.

REFERENCES Bazant Z.P. & Najjar L.J. 1972. Nonlinear water diffusion in non-saturated concrete. Materials and Structures, Vol. 5, pp. 3–20. Bazant Z.P. & Xi Y. 1994. Drying creep of concrete: Constitutive model and new experiments separating its mechanisms. Materials and Structures, 27, 3–14.

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Bary B., Ranc G., Durand S. & Carpentier O. 2008. A coupled thermo-hydro-mechanical-damage model for concrete subjected to moderate temperatures. International Journal of Heat and Mass Transfer 51, 2847–2862. Baroghel-Bouny V., Mainguy, M., Lassabatere, T. & Coussy, O. 1999. Characterisation and identification of equilibrium and transfer moisture properties for ordinary and high-performance cementitious materials. Cement and concrete research, 29, 1225–1238. CEB-FIP 1993. Model code 1990. Comité Euro-International du Béton. Coussy O. 2004. Poromechanics, Wiley. Coussy O., Dangla P., Lassabatère & Baroghel-Bouny V. 2004. The equivalent pore pressure and the swelling and shrinkage of cement-based materials. Concrete Science and Engineering, 37, 15–20. Chung J.H. & Consolazio G.R. 2005. Numerical modelling of transport phenomena in reinforced concrete exposed to elevated temperatures. Cement and Concrete Research. 35, 597–608. Gawin D., Majorana C.E. & Schrefler B.A. 1999. Numerical analysis of hygro-thermal behaviour and damage of concrete at high temperatures. Mechanics of CohesiveFrictional Materials, 4, 47–74. Gawin D., Pesavento F. & Schrefler B.A. 2006. Hygrothermo-chemo-mechanical modelling of concrete at early ages and beyond. Part I: Hydration and hygro-thermal phenomena. International Journal for Numerical Methods in Engineering, 67, 299–331.

Gawin D., Pesavento F. & Schrefler B.A. 2006. Hygrothermo-chemo-mechanical modelling of concrete at early ages and beyond. Part II: Shrinkage and creep of concrete. International Journal for Numerical Methods in Engineering, Vol. 67, pp. 331–363. Gardner D.R., Jefferson A.D. & Lark R.J. 2008. An experimental, numerical and analytical investigation of gas flow characteristics in concrete. Cement and Concrete Research, 38, 360–367. Kim J-K. & Lee C-S. 1999. Moisture Diffusion of concrete considering self-desiccation at early ages. Cement and Concrete Research, 29, 1921–1927. Lewis R.W. & Schrefler B.A. 1998. The finite element method in the static and dynamic deformation and consolidation of porous media. Wiley. Perre P. 1980. Measurements of softwoods’ permeability to air: importance upon the drying model. Int. Comm. Heat Mass Transfer, 14, pp. 519–529. van Genuchten M.T. 1980. Closed-form equation for predicting the hydraulic conductivity of unsaturated soils., Vol. 44, pp. 892–898.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Modeling of Chloride and CO2 transport in intact and cracked concrete in the context of corrosion predictions of RC structures M. Kemper, J.J. Timothy, J. Kruschwitz & G. Meschke Institute for Structural Mechanics, Ruhr-University Bochum, Germany

ABSTRACT: Besides mechanical loads environmental conditions may have a great impact on the long term deterioration and, consequently, on the lifetime of reinforced concrete structures. In addition to hygric and thermal effects the ingress of chloride ions and carbon dioxide is often responsible for a premature loss of serviceability and of structural integrity. Corrosion not only causes the loss of bond and load carrying capacity due to the reduced steel cross section, but also local cracking and spalling of the concrete cover in consequence of the pressure exerted by the products of the corrosion reaction. In this paper, a numerical method to predict the progress of corrosion and corrosion induced deterioration is presented. The initiation time of corrosion is determined by a numerical model based on the finite element method. It considers the transport of chloride ions and CO2 through the concrete cover considering the state of moisture and temperature. An electro-chemical microcell approach based on the laws of NERNST and FARADAY provides the rate of corrosion and the amount of corrosion products. Finally the propagation of corrosion induced cracks is demonstrated by 2D analyses of a cross section of a RC beam.

1

INTRODUCTION

In concrete structures reinforcement corrosion is a major cause of a premature loss of serviceability and often also of structural integrity. Because of the porous character of concrete, concrete structures are permanently interacting with the environment. Corrosion of the reinforcement is a complex process which depends on various environmental influences and simultaneous physico-chemical processes. These interactions are still an issue of current research. In this paper, a numerical method for prognoses of the initiation time of steel corrosion and the rust production is presented. The ingress of moisture, carbon dioxide and chloride ions into the concrete cover is considered as well as their effects on the corrosion reaction. De-icing salts or marine environments are major sources of chloride ions transported into concrete structures by aqueous solution. Chloride ions dissolved in the pores surrounding the reinforcement initiate the corrosion process which leads to the reduction of the steel cross-section accompanied by the loss of bond and load carrying capacity. The volume of corrosion products is larger than the volume of uncorroded steel. This expansion causes severe pressures in the vicinity of the rebars resulting in cracking and cover spalling. The second corrosive substance considered in this paper is carbon dioxide which, ingressing from the atmosphere, influences the corrosion process indirectly. Within the cement matrix CO2 reacts

with the basic constituents of the pore solution. This reaction produces a corrosion-promoting environment at the concrete-reinforcement interface by lowering the pH value of the pore solution. In this neutralized milieu the passive layer, the natural corrosion protection of the steel, is destroyed and becomes permeable for ingressing chloride ions. For the presented concept for durability prognoses the complex corrosion process is structured into three phases (Liu and Weyers 1998). First the diffusion or initiation phase is defined by the time of the corrosive substances (e.g. chloride ions) to penetrate the concrete cover. The second phase, denoted as corrosion, considers the period of time from the initiation of corrosion to the appearence of first cracks in concrete cover. The deterioration phase describes the development of cracks until the loss of serviceability. In Section 2 a one dimensional transport model according to (Steffens 2000) is presented to predict the initiation time by considering the transport mechanisms of moisture, carbon dioxide and chlorides. The determination of moisture diffusivities based on the microstructure of intact and cracked concrete is presented in Section 3. In Section 4 the corrosion process is explained in more detail and a numerical concept based on electrochemistry (Maekawa and Ishida 2002) is presented. This approach provides an estimation of the corrosion rate and the amount of corrosion products, respectively, depending on the concentration of corrosive substances, which is predicted by the transport model.

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In Section 5 the propagation of corrosion induced cracks as a result of rust expansion is illustrated by numerical studies.

2

TRANSPORT MODEL

2.2

The nature of steel corrosion is physico-chemical and its evolution is strongly moisture dependent. The consideration of transport mechanisms of moisture and dissolved corrosive substances is a prerequisite for life-time oriented structural analyses. The presented numerical model considers temperature ϑ, relative humidity ϕ, the concentration of gaseous CO2 κ and of Chloride ions c dissolved in pore solution as the primary variables. In the formulation of the energy and mass balance equations Ficks law is considered. The model provides the distribution of the field variables along the cover depth. A one dimensional structure of the model is considered as sufficient for analyses of beam, slab and plate structures exposed to environmental conditions. The initiation time of corrosion is defined by exceeding a critical chloride concentration in the cement matrix at the reinforcement-concrete interface. From the CO2 transport model the carbonation depth is obtained, which allows a prediction of the time required for the neutralization of the basicity in the pore solution at the location of the reinforcement. For the numerical solution of the system of differential equations the finite element method with a problem-oriented spatial discretization with a fine solution near the surface is used. The time discretization (see also Subsection 2.6) is based on a Newmark iteration scheme enhanced by an adaptive time stepping procedure according to (Kuhl, Bangert, Becker, Krimpmann, Rumanus, and Meschke 2006). The balance equation in the generalized 1D form is given in C · v˙ − div[D · ∇v] = Q

(1)

using v as a substitute for the considered field variable in (1). C is a term of storage properties and D is the diffusion coefficient specifying the transport properties. Q is the sum of internal sources or drains. In the following the balance and transport equations are specified for the transport of heart, moisture, carbon dioxide and chlorides. 2.1

of dry concrete. The term of storage properties Cϑ is specified by the product of density ρc and heat capacity cc . Internal thermal sources or drains do not exist (Q = 0). Moisture transport

The relative humidity ϕ is the primary variable for describing the balance of moisture   Cϕ · ϕ˙ − div Dϕ · ∇ϕ = 0.

(3)

Both liquid and gaseous phase are taken into account ϕ = wl (ϕ) + wv (ϕ).

(4)

Analogously Dϕ includes the diffusion coefficients of vapour and liquid water and Cϕ includes the storage properties of both phases. The content of vapour wv is described by the partial vapour pressure p, which is equal to the product of relative humidity ϕ multiplied by the saturation vapour pressure ps wv (ϕ, ϑ) =

φv · ps (ϑ) · ϕ. RT

(5)

Therein, φv is the vapour-filled pore volume, T is the absolute temperature and R denotes the universal gas constant. The connection of the relative humidity ϕ with the water content is defined by the moisture storage function wl (ϕ) which is illustrated in Fig. 1 according to (Steffens 2000). 2.3

Carbonation

The transport of carbon dioxide is considered in the vapour phase only, since according to (Steffens 2000) convective transport mechanisms of CO2 dissolved 120

content of liquid water wl [kg/m³] 100

80

60

Heat transport

40

Considering the heat balance Cϑ · ϑ˙ − div [Dϑ · ∇ϑ] = 0

(2)

with Cϑ = ρc · cc and Dϑ = λˆ dry , the diffusion coefficient Dϑ is given as the thermal conductivity λˆ dry

20

0 0

Figure 1.

relative humidity ϕ 0.2

0.4

0.6

Moisture storage function.

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0.8

1

in pore water are negligible. The balance of carbon dioxide concentration is described by κ˙ − div[Dκ · ∇κ] = −κ˙ r .

2000). The resulting balance equations are written in matrix form ⎞ ⎛˙⎞ ⎛ Cϑ Cϑϕ 0 Cϑc ϑ ⎜Cϕϑ Cϕ 0 Cϕc ⎟ ⎜ ϕ˙ ⎟ · ⎝ 0 0 1 0 ⎠ ⎝ κ˙ ⎠ 0 0 0 1 c˙ ⎡⎛ ⎞ ⎛ ⎞⎤ 0 Dϑ Dϑϕ 0 ϑ 0 Dϕc ⎟ ⎢⎜Dϕϑ Dϕ ⎜ ϕ ⎟⎥ − div ⎣⎝ · ∇ ⎝ ⎠⎦ (11) κ 0 0 Dκ 0 ⎠ 0 Dc c 0 Dcϕ ⎞ ⎛ 0 ⎜0, 4091 · κ˙ r ⎟ =⎝ ⎠ −κ˙ r −˙cb

(6)

The concentration κ of gaseous CO2 is the primary variable. Taking into account the development of the carbonation process the content of CO2 which reacts with the alkaline constituents dissolved in the pore solution is relevant. This concentration of reacted carbone dioxide is included in (6) by κr as an internal drain. The reaction equation of carbonation is given as (g)

(l) Ca(OH)(dis) + CO2 ⇒ CaCO(s) 2 3 + H2 O .

(7)

Calcium hydroxide (Ca(OH)2 ) is a constituent of the cement matrix which is dissolved in the pore solution. In combination with gaseous CO2 it reacts to insoluble calcium carbonate CaCO3 and water. This reaction affects the pH value, since the dissolved hydroxide ions (OH− ) are neutralized by producing water. Consequently, the pH value decreases. For the determination of the stage of carbonation the concentration of reacted CO2 κr is determined in a post calculation process. The velocity of carbonation reaction, given by the rate of reacted CO2 κ˙ r , is described using the ARRHENIUS equation. 2.4

Chloride transport

The balance of chloride concentration has the form c˙ − div[Dc · ∇c] = −˙cb .

(8)

In the cement matrix, a part of the ingressing chloride ions is physically adsorbed at the pore surface. The amount of bound chloride ions cb can be determined by sorption isotherms. Frequently the LANGMUIR isotherm (Steffens 2000),Iqbal and Ishida 2009; Glasser Marchand, and Samson 2008) cb =

α·c 1+β ·c

(9)

The temperature controls the moisture balance directly via the saturation vapour pressure ps (ϑ). With the description of the uncoupled heat balance in Equation (2) only the heat capacity of dry concrete is considered. However, thermal energy is stored in both phases of water. Therefore, in Equation (11), this storage term is replaced with the heat capacities of vapour and liquid water. In analogy, the diffusion coefficient of vapour Dϕg is depending on the saturation vapour pressure ps (ϑ). An additional convective component of heat transport is provided by the moisture flow. An internal water source, considered by the term Qϕ = 0, 4091 · κ˙ r , results from the carbonation reaction in Equation (7) and the stoichiometry according to which 0, 4091g water are produced by the reaction of 1g CO2 . With respect to the transport of chloride ions, in addition to diffusion also advective transport is considered in the term Dcϕ = wcl · Dϕl . Dϕl is the diffusion coefficient of the liquid water phase. Due to the hygroscopic characteristics of salt, the moisture balance is affected. Beyond a relative humidity of 75% chloride ions attract water molecules from the surrounding pore solution. According to RAOULT’S law the saturation vapour pressure of an ionic solution changes. Fig. 2 illustrates the moisture storage function wl (ϕ, c) for different chloride concentrations according to (Steffens 2000).

or the FREUNDLICH isotherm cb = k · c n

(10)

2.6

Temporal discretization

are used. The parameters α, β, k and n have to be determined for different concrete compositions.

For the numerical solution of the presented initial value problem of first order a Newmark type time integration scheme is used.

2.5

V˙ n+1 =

Interaction of primary variables

In this subsection the complete system of differential equations including the coupled effects of the 4 field variables is summarized. This summary follows closely the work of (Oberbeck 1995) and (Steffens

1 1−γ · (Vn+1 − Vn ) − V˙ n · γ t γ

(12)

In addition, an adaptive time stepping procedure offering an optimization of the computation time is implemented (Kuhl, Bangert, Becker, Krimpmann,

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120

content of liquid water wl [kg/m³] 2

c = 0 kg/m³ c = 10 kg/m³ c = 5 kg/m³

80

free chloride concentration in point A [kg/m³]

ϕ = 0,55 ϕ = 0,90

1

40

relative humidity ϕ 0 0.5

0.75

1 0

Figure 2. Dependence of the moisture storage function on chloride content.

Rumanus, and Meschke 2006). This procedure is characterized by an error indicator γ t · (V˙ i,n+1 − V˙ i,n ). By using a reference value Vn+1 the error measure e (Kuhl, Bangert, Becker, Krimpmann, Rumanus, and Meschke 2006) is defined as: e = γ t ·

 i=ϑ,ϕ,κ,γ

||V˙ i,n+1 − V˙ i,n || ||Vi,n+1 ||

(13)

The determination of the norm is performed separately for each component i = ϑ, ϕ, κ, γ to take care of different dimensions. For example, typical values for the temperature ϑ are in the range of 20C while the CO2 concentration is in the range of 10−4 mkg3 . By comparing the error measure e with user defined error bounds νe · ν1 and νe · ν2 the required precision is obtained and an unnecessary large number of small time steps is avoided. νe · ν1 ≤ e ≤ νe · ν2

(14)

If Equation (14) is fulfilled, the time step remains unchanged. Otherwise, t will be adapted according to  tnew = told ·

νe ν

(15)

0

Numerical example

The behavior of the presented transport model is illustrated by means of a numerical study. A part of a reinforced concrete structure with a cover thickness of 3 cm is simulated (Fig. 3). At the surface of the structure a chloride concentration of 3 kg/m3 is assumed. To demonstrate the influence of moisture on the chloride balance two (constant) relative humidities of ϕ = 55%

10

15

20

Figure 3. Temporal evolution of chloride concentration in point A for two different levels of relative humidity.

and 90% are compared. For both cases the concentration of unbound chloride ions is determined over a period of 20 years. Fig. 3 contains the geometry and the boundary conditions of the considered structure. The two graphs indicate the chloride concentration at the surface of the reinforcement bars (point A) versus time. After a period of 20 years in the wet concrete with a humidity of 90% a chloride concentration c of nearly 2 kg/m3 is reached at the steel surface. In the dry concrete (ϕ = 55%) c only reaches a level of 1,4 kg/m3 . The comparison of both results shows a considerably faster ingress of chloride ions in wet concrete than in dry concrete. This is due to the fact that in the wet cement matrix more pores are water saturated and therefore less chloride ions are bound at the pore surfaces. Consequently, more unbound chloride ions penetrate into the structure. These free ions are relevant for the corrosion process. Therefore, the differentiation between free and bound chloride ions is essential.

3

for the next time step. For e ≤ νe · ν2 the last time step is repeated with tnew . 2.7

5

DETERMINATION OF MOISTURE DIFFUSIVITY BASED ON MICROMECHANICS

The moisture diffusivity Dϕ in Equation (3) depends on the porosity and, consequently, the water-cement ratio of the concrete. Furthermore, it is restricted to uncracked concrete. To allow for an identification of moisture diffusivities based on the microstructure of intact and cracked concrete, respectively, a micromechanics oriented model is presented in this section. In the following, a representative volume element RVEuc consisting of interconnected pores in-between solid inclusions is considered to represent uncracked

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concrete (Fig. 4). The solid inclusions are considered to be spherical inclusions. Hence a homogenization of the heterogeneous RVE consisting of the pore space and the solid inclusions is required. The homogenized diffusivity in uncracked concrete is obtained by using the Mori-Tanaka scheme (Mori and Tanaka 1973; Dormieux and Ulm 2005) as Duc = D0 ·

(16)

where D0 is the diffusivity in free space and

Figure 5.

2φ 3−φ

=

is used for describing the relationship between the microscopic and the macroscopic concentration gradient

(17)

denotes the effective porosity of the uncracked structure depending on the porosity φ. Similarly, the RVE for cracked concrete RVEc , illustrated in Fig. 4, is characterized by distributed microcracks embedded in a homogenized uncracked porous matrix (RVEuc ). The cracks are represented as prolate spheroids visualized in Fig. 5. Hence the domain of the RVE  is composed of c and uc .  = c ∪ uc

∇x c = A(x) · H .

(18)

4 3 c πa N 3 a

(21)

The homogenized diffusion tensor is represented in terms of the concentration tensors on the individual domains c and uc

The porosity of the cracked domain is defined as ϕc =

Representation of cracks as prolate spheroids.

Dhom = Dc ϕc A¯ c + Duc (1 − ϕc )A¯ uc .

The average concentration gradient in the microscopic scale is equal to the macroscopic concentration gradient hence the relationship I − ϕc A¯ c A¯ uc = 1 − ϕc

(19)

(22)

(23)

with c denoting the half length and a the symmetry radius. N denotes the crack density. The concentration tensor

is obtained. According to the homogenized diffusion tensor defined in Equation (22) the homogenized flux on the macroscopic scale is given as

A(x) = ϕc A¯ c + (1 − ϕc )A¯ uc

J = −Dhom · ∇c.

RVE uc

(20)

Hence A¯ c is required to obtain the homogenized diffusion tensor which is a function of the HillPolarization tensor P which takes into consideration the geometry and the interaction of the inclusion and the matrix.

RVE c

A¯ c = (I + (Dc − Duc )P)−1

Ωinc Figure 4. (right).

Ωφ

Ωc

(24)

(25)

Using the Hill-Polarization tensor for prolate spheroids (Torquato 2002), the homogenized diffusion tensor is obtained as follows

Ωuc

RVE of uncracked (left) and cracked concrete

Dhom = Duc ϕc Y + Duc I

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(26)

with ⎡

1− ⎢ + (1 − )(1 − 2QI ) Y=⎢ 0 ⎣ 0

0

0

1−

+ (1 − )QI 0

0 1−

+ (1 − )QI

⎤ ⎥ ⎥ ⎥ ⎦

To use this 3D theory in the presented 1D transport model Equation (26) is reduced to one dimension Dhom = ϕc Duc

1− + Duc .

+ (1 − )(1 − 2Q I )

(27)

Duc replaces the diffusion coefficient of uncracked materials which is equal to Dϕ described in Section 2.2. Therefore Equation (27) can be re-written in the form   1− Dϕc = Dϕ 1 + ϕc . (28)

+ (1 − )(1 − 2QI ) which is written in the notation according to the transport model. 4 4.1

CORROSION Corrosion mechanisms

This section contains a short overview over the chemical reactions associated with corrosion and an electrochemical approch proposed for determining the amount of corrosion products. The corrosion process consists of two partial reactions. In the anodic reaction Fe ⇒ Fe2+ + 2e−

(29)

ferreous ions are oxidized by the separation of two or three electrons and dissolved in the pore solution. For simplicity, the dissolution of two electrons is considered in the following. In the cathodic reaction 1 O2 + H2 O + 2e− ⇒ 2OH− . 2

(30)

the atmospheric oxygen is reduced by accepting the electrons released during the anodic reaction. In the presence of water hydroxid ions are produced. If both reactions products are available in the pore solution iron hydroxid (Fe(OH)2 ), a chemical precusor of rust (FeO) is generated. In uncracked concrete the anodes and cathodes are disributed more or less uniformly along the surface of the steel bars. In a cracked cement matrix with an accelerated ingress of environmental substances along the cracks leads to a local arrangement of anodes and cathodes resulting in a more localized pitting corrosion (Fig. 6). In the presented

Figure 6. Uniform and pitting corrosion of steel reinforcement.

model, however, only uniformly distributed corrosion, referred as microcell corrosion, is considered. Local pitting (macrocell) corrosion is not taken into account. For prognoses of corrosion induced damage the amount of produced rust is decisive. FARADAYS law Q =n·z·F

(31)

with  Q=

icorr (t) · Adt

(32)

provides the corrosion rate depending on the corrosion current density icorr induced by the electron exchange between the andodic (Eq 29) and cathodic (Eq 30) reaction. F is the Faraday constant and z replaces the number of charge. Q denotes the electrical charge, A is the steel surface where corrosion takes place and t

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is the time by which the amount of ferreous n is transformed into rust. Considering the definitions of molar mass M = m/n and density ρ = m/V = m/(d · A) the reduction of the steel diameter  M (33) icorr (t)dt dsteel = zFρsteel can be determined. The volume of corrosion induced expansion ranges between the two- and sixfold of the transformed steel volume. The volume expansion varies with the chemical composition of rust depending on the conditions of the cement matrix surrounding the steel. Fig. 7 visualizes the definition of the effective cross-sectional expansion of the reinforcement bars d. With d = drust − dsteel = (fv − 1)dsteel

Figure 8. icorr .

In Equation (35) R is the ideal gas constant and F is the Faraday constant. T is the absolute temperature and E 0 denotes the standard electrode potential. The ferreous concentration c[Fe] = 1 is constant, since a total disintegration of the steel rebars is not considered. c[Fe] describes the concentration of Fe2+ ions dissolved in pore solution. According to (Bažant 1979) a value of 10−12,2 mol/l is assumed corresponding to a pH value of 12, 6. The general equation for determining the cathode potential is given as follows

(34)

an estimation of the volume expansion is provided by using the reduced steel diameter resulting from Equation (33). The factor fv is defined by the ratio of the produced rust volume with respect to the reduced steel volume. 4.2

Determination of icorr

In this section a method for determinig the corrosion current density icorr , which is required for the estimation of expansion, similar to the approach proposed by (Maekawa, Ishida, and Kishi 2009), is presented. Between anodic and cathodic reaction an exchange current is generated. The graphs in Fig. 8 visualize the anodic and the cathodic potentials and their point of intersection which denotes the state equilibrium associated with icorr . In the constant ranges up to the exchange current densities ia0 and ic0 the potentials EFe and EO2 are described based on NERNST equation Accordingly, the anodic potential is determined as 0 + EFe = EFe

RT · ln zF



 c[Fe2+ ] . c[Fe]

(35)

Determination of the corrosion current density

EO2 = EO0 2 +

RT · ln zF



p O2 p∞

 − 0, 05916 · pH . (36)

Provided that the partial pressure of oxygen pO2 inside the pores is equal to the atmospheric pressure p∞ , Equation (36) can be simplified to EO2 = EO0 2 − 0, 05916 · pH .

(37)

pH denotes the pH value. With an increasing cathodic potential EO2 in consequence of an decreasing pH value the effects of carboation reaction are taken into account with an incresing icorr . In the range of overvoltage, above the exchange current densities ia0 and ic0 , the potentials follow a linear gradient. The gradients are defined by the TAFEL slopes  ηa = η · lg

ia ia0



 and ηc = −η · lg

ic ic0

 (38)

with η = 2, 303 ·

Figure 7.

RT 0, 5 · zF

resulting from the BUTTLER-VOLMER equation. A more detailed description of the presented method for

Rust layer formed in reinforcement bars.

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estimating the corrosion current density is given in (Maekawa, Ishida, and Kishi 2009). In (Hussain and Ishida 2007) a semi-empirical approach for considering the concentration of chloride ions by adapting the anodic potential in Equation (35) is presented. In consequence of an increasing chloride concentration in pore solution the anodic potential decreases and therefore icorr increases.

5

DETERIORATION

In this section the prediction of the third phase of corrosion processes, the deterioration phase, is analyzed. This phase describes the development of cracks until the loss of serviceability. Only if the amount of corrosion products is beyond a critical volume, radial micro cracks in the vicinity of the reinforcement are induced. The products of corrosion are compressible like a gel. Therefore, the rust grouts the pores in the cement matrix in the vicinity of the rebars, before a pressure is exerted at the surrounding concrete eventually creating cracking. In the literature different models and approaches for predicting the material properties of corrosion products are discussed. The difficulties for developing a reliable material model are similar to those of the definition of the exact volume expansion, since the chemical composition of corrosion products is affected by many factors. Some of them are, for example, the type of cement or aggregates used in the concrete mix and the environmental conditions such as the pH level or the moisture content. In (Toongoenthong and Maekawa 2005) an average stiffness of the corroded system, which is defined by the remaining steel bar and the corrosion products, is determined. Depending on the level of corrosion the stiffness is computed based upon the stiffness of steel and corroded substance. The equivalent stiffness of the steel and rust conglomerate is defined by Es,eq =

1 + γ (fv − 1) 1−γ Es

+

γ fv G

.

than νr = 0.5 and a bulk modulus of Kr = 2.0 GPa are used. The Young’s modulus is assumed by Er = 3(1 − 2νr ) · Kr . In (Liu and Weyers 1998) an approach for determining the pressure exerted at the concrete/rust interface is presented. Considering the stage of corrosion when the pressure is initiated, the steel bar is surrounded by a rust layer d0 exerting no stresses. Considering the radius of the conglomerate of steel and strainless rust layer R = Rsteel + d0 , the pressure p can be expressed as p=

(1 + ϕcr )2R

2Ec ds 

(C+R)2 +R2 (C+R)2 −R2

+ νc

.

(40)

Ec is the elastic modulus of concrete and νc the Poisson’s ratio. C denotes the thickness of concrete cover and ds is the tickness of corrosion products generating tensile stresses. In the following a structural analyses of corrosion induced cracks is presented using the model predictions for the volume expansion of the rust layer. As a result of corrosion radial micro cracks around the reinforcement bars start to open and finally lead to cover cracking. The arrangement of these cracks is demonstrated by the following numerical example. Fig. 9 illustrates a detail of a RC structure containing five rebars, which is used for numerical analyses of the damage process. The diameter of the reinforcing bars are 10 mm and their spacing is assumed as 50 mm. The thickness of the concrete cover is 30 mm. The corrosion expansion is assumed equal for all rebars. Both the steel and the concrete are discretized by plane straine finite elements. According to the thickness of the considered rust layer, expansive volumetric strains are

(39)

γ is a factor which describes the level of corrosion ranging between 0 and 1. fv denotes the volume expansion with fv = Vrust /Vsteel . The parameters Es and G replace the stiffness of steel and corrosive substances (rust), respectively. Another approach according to (Molina, Alonso, and Andrade 1993) describes the modeling of the two superposed effects of decreasing the stiffness of steel and increasing its volume. For predicting the material properties of corrosion products a linear variation of the steel material properties from steel to rust is assumed. The properties of rust are assumed to be similar to those of liquid water. Representing the material parameters of rust a Poisson ratio slightly lower

Figure 9.

Detail of a RC structure (dimensions in cm).

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applied onto the finite elements representing the steel bars. In this example, the initial stage is the level of corrosion corresponding to the initiation of corrosion induced pressure. According to the material properties of corrosion products defined by (Molina, Alonso, and Andrade 1993) no bond between concrete and corroded steel is assumed. For the analyses, different stages of corrosion induced expansion d are considered. Fig. 10 illustrates the cracking strains caused by the expansion with 200-fold magnification. In the first stage of corrosion shown in Fig. 10 radial cracks develop around the reinforcement bars. The horizontal cracks shown in the second picture are opening simultaneously and propagate from one rebar to the other. In a further stage also vertical cracks between the rebars are starting to open. In the fourth picture a more advanced stage of damage is illustrated. At this stage, the cracks opens additional means or transport for corrosive substances which accelerates the corrosion process. The crack width in this stage is approximately 0,1 mm. Therefore, the transport mechanisms differ considerably from those of an intact structure. However, this coupling effect is not considered in the present numerical study.

6

CONCLUSIONS

One focus of the paper was laid on the modelling of the transport mechanisms for chlorids and carbon dioxide. To this end, 1D models which consider the major characteristics of the transport mechanisms. These models are supplemented by a micromechanically oriented approach to determine the permeabilities of intact and cracked concrete. This approach allows to correlate the degree of damage and the porosity with the permeabilitiy of concrete. An electro-chemical model has been suggested to predict the rust production. Finally, preliminary numerical studies have illustrated the crack evolution in the vicinity of the steel rebars leading to spalling of the concrete cover.

REFERENCES

Figure 10.

Bažant, Z.P. (1979). Physical model for steel corrosion in concrete sea structures—theory. Journal of the Structural Division 105(6), 1137–1153. Dormieux, L. and F.-J. Ulm (2005). The determination of the elastic field of an ellipsoidal inclusion, and related problems. CISM courses and lectures, Springer, Wien 480. Glasser, F.P., J. Marchand, and E. Samson (2008). Durability of concrete—degradation phenomena involving detrimental chemical reactions. Cement and Concrete Research 38, 226–246. Hussain, R. and T. Ishida (2007). Modeling of corrosion in rc structures under variable chloride environment based

Structural analyses.

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on thermodynamic electro-chemical approach. Journal of SSMS, JAPAN 3, 104–113. Iqbal, P. and T. Ishida (2009). Modeling of chloride transport coupled with enhanced moisture conductivity in concrete exposed to marine environment. Cement and Concrete Research 39, 329–339. Kuhl, D., F. Bangert, C. Becker, S. Krimpmann, E. Rumanus and G. Meschke (2006). Numerical methods for multifield durability analyses. In R. de Borst, N. Biˆcani´c, H. Mang and G. Meschke (Eds.), Computational Modelling of Concrete Structures, Rotterdam. Balkema. Liu, Y. and R.E. Weyers (1998). Modeling the time-tocorrosion cracking in chloride contaminated reinforced concrete structures. ACI Materials Journal 95(6), 675–681. Maekawa, K. and T. Ishida (2002). Modeling of structural performanaces under coupled environmental and weather actions. Materials and Structures 35, 591–602. Maekawa, K., T. Ishida and T. Kishi (2009). Multiscale Modeling of Structural Concrete, Volume 1. Taylor and Francis Group. Molina, F.J., C. Alonso and C. Andrade (1993). Cover cracking as a function of bar corrosion: Part 2—numerical model. Materials and Structures 26, 532–548.

Mori, T. and K. Tanaka (1973). Average stress in the matrix and average elastic energy of materials with misfitting inclusions. Acta Metallica 21(5), 571–574. Oberbeck, N. (1995). Instation¨arer Wärme- FeuchteSchadstoff-Transport in Beton. Ph.D. thesis, Institut für Statik, Technische Universität Braunschweig. Steffens, A. (2000). Modellierung von Karbonatisierung und Chloridbindung zur numerischen Analyse der Korrosionsgefährdung der Betonbewehrung. Ph.D. thesis, Institut für Statik, Technische Universität Braunschweig. Toongoenthong, K. and K. Maekawa (2005). Simulation of coupled corrosive product formation, migration into crack and propagation in reinforced concrete sections. Journal of Advanced Concrete Technology 3, 253–265. Torquato, S. (2002). Random Heterogeneous Materials: Microstructure and Macroscopic Properties, Volume 1. Springer-Verlag, New York.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Thermal activation of basic creep for HPC in the range 20◦ C–80◦ C W. Ladaoui, T. Vidal & A. Sellier Université de Toulouse, UPS, INSA, LMDC (Laboratoire Matériaux et Durabilité des Constructions), Toulouse Cedex 04, France

X. Bourbon Agence Nationale pour la gestion des Déchets Radioactifs; Direction Scientifique/Services Colis—Matériaux, Châtenay—Malabry Cedex, France

ABSTRACT: The work presented in this paper is part of an extensive research program conducted by Andra (French National Agency for Management of Nuclear Waste). Its objective is to provide a high level of understanding of the long-term behaviour of concrete disposal cells. The creep of high performance concrete (HPC) envisioned for these applications must be characterized in conditions as similar as possible to real ones. In particular, in the field, thermal and moisture conditions have to be considered (temperature ranging from 20◦ C to 80◦ C during more than a century, high water content). The two objectives of our study are to experimentally assess the long-term strain capabilities of concrete so as to improve the understanding of creep phenomena in such an environment, and to propose a realistic model taking these results into account for the prediction of HPC long term behaviour. 1

INTRODUCTION

The paper deals more precisely with the thermal activation of basic creep for the High Performance Concretes (HPC) chosen as reference materials by ANDRA. The experiments were performed at 3 temperatures: 20, 50 and 80◦ C (covering the predicted exploitation temperature range). The experimental programme concerned four HPC (two types of cement, CEM I and CEM V with or without silica fume and steel fibres). To avoid interaction between shrinkage and creep, specimens were previously water saturated. Tests were carried out with sealed samples to prevent desiccation. The tests were performed on concrete after its compressive strength had stabilized, i.e. when hydration was considered as completed. Specimens were first heated and the temperature was maintained while the specimens were loaded at 30% of their compressive strength Volumetric changes and axial strains were measured through LVDT sensors and gauges over more than six months. The results presented in this paper concern the four HPC at 20◦ C and 50◦ C (tests at 80◦ C are in progress). They show that the temperature increase considerably influences the creep kinetics. The results are discussed in the framework of a recent basic creep model based on a non-linear rheological scheme in which only viscosity is affected by the temperature. The temperature effect on viscosities is modelled using an Arrhenius formula initially proposed by Bazant & al (2004) for OPC. To illustrate the temperature effect on basic creep, a theoretical

quantification of the role of natural temperature fluctuation on creep phenomena is proposed. Two calculations are compared: one with a specimen subjected to an average temperature that remains constant over 50 years and the other assuming a periodic variation. Results give an idea of the error that occurs with the constant temperature assumption classically used in civil engineering design rules. 2

STATE OF THE ART

Maréchal (1969, 1970) has shown that creep of an old concrete where hydration is stabilized and the evaporable water has been removed by desiccation follows an Arrhenius law. This observation is limited to creep tests performed with a sustained load of less than 50% of the compressive strength during a relatively short period of two months. According to the author, this behaviour is rather similar to that of a crystalline material characterized by weak creep. He showed that the energy of activation was constant for several ordinary concretes. However, for a non-dried, although old, concrete, the variation of the creep rate at thermo-hydraulic equilibrium for different temperatures is more difficult to understand. The rate shows a maximum for 50–60◦ C but decreases when the temperature exceeds 105◦ C. Bazant et al. (2004) proposed to model the influence of temperature by using an equivalent time calculated from an Arrhenius law. From experimental data

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obtained on ordinary concretes, the activation energy relative to the gas constant, Q/R, was found to be around 5000◦ K. On the basis of Bazant & al’s model, Benboudjema & Torrenti (2008) recently modelled the temperature effect on basic creep by affecting the viscosities and stiffness of a rheological model incorporating several Kelvin Voigt elements. This model keeps the characteristic times but modifies the amplitude, which is quite different from the approach of Bazant & al (2004). The ratio of activation energy to gas constant, Q/R, of ordinary concretes was found to be around 2000◦ K. 3

EXPERIMENTAL PROGRAM

Four different types of HPC with CEM I and CEM V cement were studied: two with fibres and silica fume, called CEM IF and CEM VF, and two others without fibres and without silica fume, called CEM I and CEM V. The basic creep tests were performed in a creep system located in a heated room, following RILEM TC 129—MTH (January-February 2000; May 2000) and ASTM (2005) recommendations. Figure 1 presents the experimental set up. All the mixtures incorporated limestone aggregates (sand 0/4 and gravel 4/12.5). The Water/Binder ratio was 0.44 for CEM I, 0.40 for CEM V, and 0.35 for concretes with fibres and silica fume (CEM IF and CEM VF). After demoulding, the specimens were cured under water (20◦ C) until the age of creep loading, i.e. 14 months (stabilized hydration and saturated porosity). During the creep tests, the specimens were sealed to avoid desiccation. The various mixture compositions are given in Table 1 and the mechanical characteristics in Table 2. 3.1 Experimental results Figures 2 to 5 show the development of basic creep for the four HPC (plots). The continuous lines correspond to the fitted model explained further in the article. The two CEM V concretes developed less creep strain than CEM I did. It can be noted that silica fume and fibres did not significantly modify creep amplitude. These two parameters probably had opposite effects on creep rate and counteracted each other. The comparative analysis of creep results showed that the strain at 50◦ C was twice that at 20◦ C after 250 days. 4

PRESENTATION OF THE RHEOLOGICAL MODEL

The rheological model used was developed by Sellier & Buffo-Lacarrière (2009). In its full version, this model is written in a poro-mechanical framework to

Figure 1.

Experimental set up.

consider drying creep. This is the reason why the total stress is divided into a spherical part on the solid skeleton, a deviatoric part, and water stress due to capillary depression πw . This is a similar technique to that used in the models proposed by Bernard et al. (2003) and Ulm et al. (1999) and also adopted by Benboudjema

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Compositions of concrete in kg/m3 .

Components

CEM I

CEM V

CEM IF

CEM VF

CEMI 52.5R CE PM-ES-CP2 (Lafarge,Val d’Azergues)

400

−

454

−

CEMV/A 42.5 N PM-ES-CP1 (Calcia, Airvault) Limestone sand 0/4 mm, (Boulonnais)

−

Limestone aggregate 4/12.5 mm (Boulonnais)

945

Superplasticizer Glenium 27, MBT Silica fume Condensil S95 DM Stainless fibre IFT (L = 30 mm, Ø = 0.6 mm) Superplasticizer SIKA Viscocrete 5400F Total water

10

Table 2.

−

450

800 700 600 500 400 300 200 100

454

0 0

50

100

150

exp 50 C exp 20 C

858

800

984

250

300

Figure 2. 984

672

50 C model 20 C model

984 Basic creep for CEM I at 20◦ C and 50◦ C.

672

−

−

11.25

−

45

Basic creep strain ( m/m)

900

−

45

800 700 600 500 400 300 200 100 0

−

−

85

0

85

50

100

150

exp 50 C exp 20 C

−

−

13.70

17.25

178

183

172

178

Figure 3.

CEM V

CEM IF

CEM VF

86 45450 0.27

75 44752 0.27

113 46216 0.29

99 45073 0.28

μe =

Ee 2(1 + ν)

300

50 C model 20 C model

Basic creep for CEM IF at 20◦ C and 50◦ C.

800 700 600 500 400 300 200 100 0 0

50

100

150

200

250

300

Time (days)

et al. (2001). Figure 6 below presents the rheological model. In our work, the samples were saturated in order to have no capillary stress and avoid any interaction with the drying creep. In Figure 6, k e corresponds to the compressive modulus, μe is the shear modulus, the stage called e corresponds to elasticity: and

250

900

CEM I

Ee 3(1 − 2ν)

200

Time (days)

exp 50 C exp 20 C

ke =

200

Time (days)

Mechanical characteristics.

fcm (MPa) Ecm (MPa) ν

Basic creep strain ( m/m)

900

Basic creep strain ( m/m)

Table 1.

(1)

The creep is simulated by the KV (Kelvin Voigt) and M (Maxwell) stages. The Kelvin Voigt stage

Figure 4.

50 C model 20 C model

Basic creep for CEM V at 20◦ C and 50◦ C.

models the reversible part of creep and the Maxwell stage models the viscous part. The specificity of the model lies in the fact that the viscosity of the Maxwell stage evolves as a non-linear function of strain. The choice of this non-linearity is explained in detail by Sellier & Buffo Lacarriere (2009). In this model, the temperature is first assumed to affect only the viscosity, uniformly on all viscous elements of Figure 6. This assumption finally proved to be sufficient to model our experimental results.

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Kelvin-Voigt stage associated with reversible creep; ηMs = the Maxwell fluid viscosity. For deviatoric deformations, in the same way, we obtain:

Basic creep strain ( m/m)

900 800 700 600 500



400 300 200 100 0 0

50

100

150

200

250

50 C model

exp20 C

20 C model

where μe = the shear elastic modulus; μKV and ηKVd the shear modulus and viscosity associated with reversible deviatoric creep; ηMd = non-linear viscosity associated with the permanent deviatoric creep. To model this non-linearity, the coefficient of consolidation Cc is introduced as:

Basic creep for CEM VF at 20◦ C and 50◦ C.

Figure 5.

(4)

300

Time (days) exp50 C

2μe εijed = 2μKV εijKVd + ηKVd ε˙ ijKVd 2μe εijed = ηMs ε˙ ijMs



ηMs = Cc ηMs0 ηMd = Cc ηMd0

(5)

ηM (s or d)0 is the initial viscosity of the material. With: Cc = exp

Figure 6.

The basic creep deformation mentioned in this article concerns only the two bottom stages of the scheme of Figure 6: εC = ε KV + εM

(2)

where ε = the basic creep strain; ε = the reversible creep strain, modelled by the Kelvin Voigt linear solid; εM = the permanent creep strain modelled by a Maxwell solid with non-linear viscosity. 4.1

KV

4.2

(6)

Temperature effect

As indicated above, we proposed to take the effect of temperature into account simply by changing viscosity, which physically means that, for the temperature range considered (below 80◦ C), the microstructure is not affected in terms of stiffness or potential creep. Only the rate of the phenomenon is activated by heat. By adopting an Arrhenius law similar to that proposed by Bazant, the following relationship can be written:

Rheological laws

The rheological laws are first presented for a given temperature. The effect of temperature will be discussed later. The classical laws of viscoelasticity are used: 

k e εes = k KVs εKVs + ηKVs ε˙ KVs k e εes = ηMs ε˙ Ms

εMsk

where ε Msk = a parameter called ‘‘characteristic strain of consolidation’’ controlling the rate of consolidation of concrete, and εMs = the strain creep in the Maxwell element.

Rheological model.

C

  ε Ms 

ηMs0 (T ) ηMd0 (T ) ηKVs0 (T )   =   =   Ms0 Md0 η η ηKVs0 Tref Tref Tref =

  ηKVd0 (T ) Q 1 1   = exp − − R T Tref ηKVd0 Tref (7)

(3)

where k e = the elastic modulus of compressibility; k KVs and ηKVs the modulus and viscosity of the

In this expression, Tref is the reference temperature (in ◦ K), T the current temperature (◦ K), Q is the activation energy of the viscous processes (J/mol), R = 8.31 is the gas constant (J/mol.◦ K).

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4.3

Fitting of the model parameters

Table 4. HPC.

The fitting of the basic creep model is facilitated by the possibility of analytically integrating the constitutive equations in the case of a constant stress test. The expression of the axial viscous basic creep strain is then given by formula (8) (Sellier & Buffo Lacarriere, 2009).   σ 1 M + 2β ln 1 − Ms0 Msk t (8) (t) = −ε Msk εaxial 3 3η ε

Qv /R (◦ K)

η 2(1 + υ) = Ms0 = KVs0 3(1 − 2υ) η η

CEM V

CEM IF

CEM VF

5992

4093

10631

5326

T Ouargla Tref

40

T Bordeaux T QØbec

30

Temperature ( C)

β≈

ηKVd0

CEM I

50

With: Md0

Activation energies calculated on the 4 types of

(9)

20

10

0 0

50

100

150

200

250

300

350

400

-10

The curve fitting (Figs 2–5) is performed by considering that only the viscosity is affected by temperature. The results are given in Table 3. Formula (7) allows the activation energy to be calculated from the ratio of viscosities. It is interesting to note that, although the results were obtained for HPC, the values of activation energy found (Table 4) are of the same intensity as those proposed by Bazant et al. (2004) for OPC (except for CEM IF).

-30

Annual temperature record.

1600

1400

Prediction of long-term strains depending on constant or variable temperature

To quantify the effect of temperature for a concrete in its utilization situation, the simulations of the basic creep of HPC CEM I subjected to a constant stress of 25.8 MPa were calculated. These simulations were performed for three temperature histories corresponding to meteorological records in different regions: Bordeaux (France), Quebec (Canada) and Ouargla (Algerian Sahara). The corresponding annual mean temperatures are 13.4◦ C in Bordeaux, 4.6◦ C in Quebec and 23.7◦ C in Ouargla. The simulation is also provided for a constant reference temperature of 20◦ C. A first calculation was made incrementally using an explicit scheme in time, constant stress and variable temperature. A second calculation was performed by taking a constant temperature equal to the average annual value of the temperature of the region. The simulations were made over a period of 50 years (Fig. 8). Table 3.

Time (days)

Figure 7.

Basic creep strain ( m/m)

4.4

-20

1200

1000

800

600

400

Tvar Ouargla Tvar Bordeaux Tvar QuØbec Tref

200

Tcst Ouargla (23.7 C) Tcst Bordeaux (13.4 C) Tcst QuØbec (4.6 C)

0 0

Figure 8.

3650

7300

Time (days)

10950

18250

Simulation for 50 years.

As expected, the annual thermal cycles led to creep amplitudes that differed significantly according to the region. In Quebec and Ouargla, creep strains had amplitudes of −20% and +20% respectively relative to those of Bordeaux. It was also observed that the amplitudes were different from those obtained with calibration at 20◦ C (usually used for standards testing). This difference could reach 25% for Quebec. This is significant and should be taken into account in standards of structural design.

Rheological characteristics obtained by curve fitting.

εMsk ηMs0 (20◦ C) (MPa.j) ηMs0 (50◦ C) (MPa.j) ηKV (20◦ C) /k(j) EKV /Ee

14600

CEM I

CEM V

CEM IF

CEM VF

7.32×10−5 31.44×107 4.71×106 7 3.10

5.34×10−5 24×107 6.56×106 3 6.33

4.04×10−5 44.57×107 1.53×106 7 3.87

5.44×10−5 42.57×107 7.87×106 4.84 5.17

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experimental study to desiccation creep in temperature. These results as a whole will enable predictions to be made of the long-term behaviour of a structure for the deep storage of radioactive wastes subjected to temperature variations in the range 23◦ C to 70◦ C, over a period of 100 years.

600

Basic creep strain ( m/m)

500

400

300

200

Tvar Ouargla

Tcst Ouargla (23.7 C)

Tvar Bordeaux

Tcst Bordeaux (13.4 C)

Tvar QuØbec

Tcst QuØbec (4.6 C)

100

ACKNOWLEDGEMENTS

Tref 0 0

Figure 9.

50

100

150

200

Time (days)

250

300

350

400

We thank Andra for their collaboration in the study and the Algerian Meteorological Center for providing us the meteorological data.

Simulation for 1 year.

The comparison between the amplitudes estimated with the cycles of temperature and those calculated with a constant temperature equal to the average temperature of the region shows that the calculation of creep considering a constant temperature can be satisfactory in most cases. The maximum difference between the two calculations is only 3.2% at 50 years for Ouargla. Concerning short-term creep, Figure 9 shows a significant effect of the annual thermal cycle on the first year of creep. It would be relevant to analyse the deformation of structures during their first year bearing the phenomenon in mind. 5

CONCLUSION

This article provides results of compressive basic creep for four HPC (of which two with fibres and silica fume) subjected to 20◦ C and 50◦ C. The creep strains at 50◦ C are approximately twice those obtained at 20◦ C. The phenomenon can be easily modelled using a rheological model of basic creep where viscosities are affected by temperature through an Arrhenius law with an activation energy of 42000 J/mol. The numerical simulation of an HPC with CEM I under sustained loading in various geographical locations tends to prove that the temperature parameter has to be taken into account in order to avoid significant errors, which can reach 25%. The effect of temperature can be integrated in a simplified way by considering a constant value (equal to the mean value during a year at the geographical location) in the creep calculation. The error induced by the simplification does not exceed 3% compared to a calculation based on a variable temperature. The continuation of this work will verify the validity of the law for temperatures up to 80◦ C, in particular through data from experimental tests in progress at the LMDC. A further step will be to extend the

REFERENCES ASTM C 512–02, Standard test method for creep of concrete in compression, 2005. Bazant, Z.P. Gianluca, C. & Luigi C. 2004. Temperature Effect on Concrete Creep Modeled by MicroprestressSolidification Theory, JOURNAL OF ENGINEERING MECHANICS © ASCE / JUNE 2004, p. 691–699. Benboudjema, F. Meftah, F. Sellier, A. Torrenti J.-M. & Heinfling, G. 2001. A basic creep model for concrete subject to multiaxials loads, FRAMCOS IV, Cachan June 2001, Balkema, p. 161–168. Benboudjemaa, F. & Torrenti, J.-M. 2008. Early age behavior of concrete nuclear containments, Nuclear Engineering and Design, 2008, doi:10.1016/j.nucengdes.2008.04.009. Bernard, O. Ulm, F.-J. & Germaine, J. 2003. Volume and deviator creep of calcium-leached cement-based materials, Cement and Concrete Research, vol. 33, 2003, p. 1127–1136. Maréchal, J. -C. 1969. Le fluage du béton en fonction de la température, Rilem Conference, Munich 1968, Matériaux et Constructions N◦ 8, March-April 1969, p. 111–115. Maréchal, J. -C. 1970. Fluage du béton en fonction de la température Compléments expérimentaux, Matériaux et Constructions, Vol. 3 N◦ 18, March-April 1970, p. 395–406. RILEM TC 129-MTH, Test methods for mechanical properties of concrete at high temperatures, Part 8 : Steady-state creep and creep recovery for service and accident conditions, Materials and Structures, Vol. 33, January-February 2000, p. 6–13. RILEM TC 129-MTH, Test methods for mechanical properties of concrete at high temperatures, Part 9: Shrinkage for service and accident conditions, Materials and Structures, Vol. 33, May 2000, p. 224–228. Sellier, A. & Buffo-Lacarriere, L. 2009. Toward a simple and unified modelling of basic creep shrinkage and drying creep for concrete, European Journal of Environment and Civil Engineering, N◦ 10/2009, Dec 2009. Ulm, F.-J. Le Maou, F. & Boulay, C. 1999. Creep and shrinkage coupling: new review of some evidence, Revue Française de Génie Civil, vol. 3, 1999, p. 21–37.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Modelling of the THM behaviour of concrete at the macroscopic and mesoscopic scale T.T.H. Le & H. Boussa Université Paris-Est, Centre Scientifique et Technique du Bâtiment (CSTB), Département TIDS, Division MOD-EVE, Champs sur Marne, Marne la Vallée Cedex 2, France

F. Meftah Université Cergy-Pontoise, Laboratoire Mécanique et Matériaux du Génie Civil, Neuville sur Oise, Cergy-Pontoise Cedex, France

ABSTRACT: The aim of this contribution is to proceed to the numerical modelling of the thermo-hygromechanical (THM) behaviour of concrete by starting at the mesoscale level: the scale at which coarse aggregates are embedded in a mortar matrix (hardened cement paste and fine aggregates). For this purpose, a finite element THM modelling approach of heterogeneous cementitious material is presented. The approach is developed within the framework of partially saturated porous media. A bi-dimensional finite element configuration of representative elementary volume (REV) of concrete is generated with a random location of the aggregates. The objective is then to illustrate the model capabilities in simulating THM behaviour of concrete at both macroscopic and mesoscopic scale and to identify the influence of the microstructure on the concrete behaviour at high temperatures. Finally, some numerical simulations of the experimental test realised by Mindeguia (2009) are performed at both macroscopic and mesoscopic scale and results are discussed. 1

INTRODUCTION

The description of the concrete behaviour under thermal loading is important in the field of building civil engineering. In fact, concrete is a partially saturated porous medium the microstructure of which presents a random and highly heterogeneous character at different scales of observation. When concrete is subjected to a high temperature, the material is the seat of numerous degradation processes. Among them, the aggregate-paste incompatibility may have a direct influence on the material stability: a deterioration of the mechanical properties of the material and its spalling in summary. Indeed, the thermal expansion of aggregates contrasts with the drying and dehydration induced shrinkage that arises in the cement paste due to physical and chemical transformations when temperature increases. Therefore, self equilibrated tensile stresses arise in the material which generate micro-cracking in the bulk of the cement matrix and at the matrix-granular inclusion interface. The assessment of this degradation process requires developing full coupled thermo-hygro-mechanical modelling approaches at the scale of occurrence of these degradation processes. However, the THM behaviour of concrete was mainly investigated at the macroscopic scale, at which concrete is regarded as a homogeneous

material (Alnajim 2004), (Gawin et al. 2003), (Obeid et al. 2001), which partially precludes the effects of the microstructure on the overall THM behaviour. On the other hand, the thermo-mechanical mesoscopic modelling of behaviour of damaged concrete has been studied on two dimensional configurations and where aggregates morphology has been idealized (Grondin et al. 2007). In particular, the effect of the microstructures on the propagation and percolation of the rising micro-cracks induced by the incompatibility between the cement matrix and of the granular inclusions needs further investigations. This issue is of a prime importance in order to assess the role that the microstructure may have in the material spalling: granular presence may lead to preferential percolation paths of microcracking which may in turn control the flak geometry. In this contribution, a finite element THM modelling of concrete at the meso-scale level is presented. Bi-dimensional finite element configurations of representative elementary volumes (REV) are generated by distinguishing the cement paste form the aggregates with different considered size distributions. The THM approach is developed within the framework of partially saturated media. Numerical simulations are then performed in order to carry out the influence of the microstructure on the behaviour of concrete at high temperatures.

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2

MULTI-SCALE MODELLING

given by: ˙ dehyd ; m ˙ rs = m

The purpose is to proceed to the simulation of the THM behaviour of a concrete by starting at the mesoscale: the scale at which coarse aggregates are embedded in a mortar matrix (hardened cement paste and fine aggregates). The inclusions (coarse aggregates) may be seen either as hygrally inert or behaving as a partially saturated medium. In this analysis, studied aggregates are considered to have a weak connect porous network and a low initial saturation. Their hygral behaviour is then marginal and a thermal model is sufficient to describe their behaviour within the concrete mix. Therefore, only the mortar matrix is considered to have full thermo-hygral behaviour with an ad hoc model. Concerning the mechanical behaviour, a damage model (Ung Quoc 2003) is adopted for mortar and aggregates (as a first approach). Two mechanical damage modes have been considered: one damage according to a volumetric strain, and one damage according to a deviatoric strain. 2.1

m ˙ rv = m ˙ vap ;

∂mπ ˙ rπ + ∇ · (mπ νπ−s ) = m ∂t

ml νl−s = −K

mv = ρv (1 − Sl )φ;

= −K

ρl krl ∇pl μl

(4)

ρv krg Mv Ma ∇pg − Dρg ∇ μg Mg2

ma νa−s = ma νg−s + ma νa−g = −K

ρa krg Mv Ma ∇pg − Dρg ∇ μg Mg2





pv pg

pa pg

 (5)  (6)

where νg−s = velocity of the gas mixture with respect to the solid phase; νv−g and νa−g = velocity of the vapour and dry air with respect to the gas mixture; pg = pv + pa = pressure of gas phase; pl = pressure of liquid water; K = intrinsic permeability; krπ = relative permeability of π phase (π = g for gas and π = l for liquid); μπ = dynamic viscosity; D = effective diffusivity; and Mv , Ma and Mg = molar mass of vapour, dry air and gas phase, considered as ideal gases. Furthermore, the energy balance equation of the whole medium reads:

(1)



(2)

where ρπ = corresponding density; φ = porosity; and ˙ rπ is Sl = liquid saturation degree. The source term m

(mπ Cπ )

π =s,l,v,a

 ∂T (mπ Cπ νπ −s ) · ∇T + ∂t π =g,l

˙ vap + Hdehyd m ˙ dehyd + ∇ · q = −Hvap m

(7)

where Cπ = heat capacity; q = heat flux given by Fourier’s law; Hvap = enthalpy of vaporization; and Hdehy = enthalpy of dehydration. The finite element model is derived starting from the weak forms of the set of the previous balance equations, with p∗ and T ∗ weighting functions nil on p and T . The weak forms are then given by: 

ml = ρl Sl φ; ma = ρa (1 − Sl )φ

(3)

mv νv−s = mv νg−s + mv νv−g

where t = stands for time; mπ = mass per unit volume of each constituent (π = s for solid, l for liquid, v for vapour and a for dry air); m ˙ rπ = source term; and νπ −s = velocity of each constituent with respect to the solid phase. The mass mπ of each constituent is given by: ms = (1 − φ)ρs ;

m ˙ ra = 0

where m ˙ vap = rate of mass due evaporation/ condensation phase change; and m ˙ dehyd = rate of mass due to dehydration. The mass transport of each constituent within the porous network is given by Darcy’s law (for permeation) and Flick’s law (for diffusion). Therefore, the mass fluxes write:

Thermo-hygral model

2.1.1 Mortar The thermo-hygral model for mortar starts from a set of microscopic balance equations of mass and energy of each constituent of the medium: solid skeleton, liquid water, vapour water and dry gas. The approach is based on a space averaging theory proposed by Schrefler and co-workers (Gawin et al. 2003), (Lewis & Schrefler 1998). It allows describing the major mechanisms of mass transport, heat transfer, phase change (vaporisation and dehydration). The governing equations of the model are given in terms of the chosen state of variables. In this approach, they are the dry air pressure pa , the water vapour pressure pv and the absolute temperature T . The mass balance equations of the constituent can put in the following generic form:

m ˙ rl = −m ˙ vap − m ˙ dehyd ;



p∗

∂mπ d − ∂t 

=−



∇p∗ · (mπ vπ −s )d



p∗ (mπ vπ −s ) · nd +

¯p 

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p∗ m ˙ rπ d

(8)



 T

∗



 −

  ∂T  + (mπ Cπ ) (mπ Cπ vπ ) · ∇T d ∂t π π ∇T ∗ · qd +





−



point at the previous converged step (at time stage tn ) and by cumulating the residues during the iterations. This TH model has been implemented in the finite element code Cast3M developed by the French research centre for nuclear energy (CEA).

T ∗ Hvap m ˙ vap d

 ∗



T Hdehyd m ˙ dehyd d = − 

∗

T q · nd

2.1.2 Aggregates A pure thermal model is used for the simulation of aggregates behaviour within the concrete mix. This model starts from the standard linearized balance equation of heat:

(9)

T

where p and T are parts of the boundary, of the domain , concerned with Dirichlet’s type conditions, ¯ p and  ¯ T are the complementary parts where while  Neumann’s type conditions are introduced, and n is the outward unit normal. By introducing the conventional space discretisation and fields interpolation:

ρCp

∂pπ = Np p˙ π ; p∗ = Np p∗ ; ∂t ∂T ˙ T ∗ = NT T∗ ; T = NT T; = NT T; ∂t ∇pπ = Bp pπ ; ∇p∗ = Bp p∗ ; (10)

together with theta-method time discretisation scheme for the time interval [tn , tn+1 = tn + t], then the weak forms can recast in the following compact final form:   Cαα + θ Kαα xαn+1 + Kαα xαn t   Cαβ + θKαβ xβn+1 − Kαβ xβn = fαn+θ − t   Cαγ + θ Kαγ xγn+1 − Kαγ xγn − (11) t where α = v, a, T , β, γ = α β = γ are superscripts referring to the adopted primary variables x• with(xv = pv , xa = pa , xv = T), xαn+1 = xαn+1 − xαn and t are the variable and time increments, K•• and C•• are matrices and fαn+θ are second member vectors that are obtained from special discretisation. The set of non linear equations (11) is solved using a staggered iterative scheme with two nested levels of iterations. The first level (local iterations) concerns the convergence process for each of the three equations when solved for one variable xα while the two other variables (xβ , xγ ) are kept constant. The second level (global iteration) concerns the convergence of the interaction between the three equations when considering simultaneously the three updated variables (pv , pa , T). Moreover, a quasi-Newton solving algorithm is adopted; each equation is solved for the total increment xαn+1 by keeping fixed the linearization

(12)

where ρ = density of the medium; Cp = specific heat; and q = heat flux given by Fourier’s law. Furthermore, an ideal interface is assumed between the aggregates and the mortar matrix. Therefore, no jump in the temperature field can be captured when crossing the interface. This aspect needs to be improved since water transport in the mortar phase can lead to strong temperature gradients close to the aggregates. The finite element model is derived starting from the weak form of this balance equation, with T ∗ weighting function nil on T . The weak form is then given by:     ∂T T ∗ ρCp − r d − ∇T ∗ · qd ∂t

pπ = Np pπ ;

∇T = BT T ; ∇T ∗ = BT T∗ ;

∂T +∇ ·q=0 ∂t



 =−



T ∗ q · nd

(13)

T

where  is a part of the boundary, of the domain , concerned with Dirichlet’s type conditions, while ¯ T is the complementary part where Neumann’s type  conditions are introduced, and n is the outward unit normal. By introducing the conventional space discretisation and fields interpolation together with thetamethod time discretisation scheme for the time interval [tn , tn+1 = tn + t], then the weak form can recast in the following compact final form:   CTT (14) + θKTT Tn+1 + KTT Tn = fTn+θ t where Tn+1 = Tn+1 − Tn and t are the variable and time increments, KTT and CTT are matrices and fTn+θ are second member vectors that are obtained from special discretisation. The equation (14) is solved using a staggered iterative scheme. A quasi-Newton solving algorithm is adopted; the equation is solved for the total increment Tn+1 by keeping fixed the linearization point at the previous converged step (at time stage tn ) and by cumulating the residues during the iterations.

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2.2

√

Mechanical model

ε˜ 0d

Because the presence of aggregate, cement paste, micro-cavities and micro-cracks in the material, the mechanical behaviour of concrete cannot be modelled by only one strain mechanism. According to experimental observations, Ung (2003) have suggested a macroscopic damage model, MODEV, with two mechanical damage mechanisms, a damage associated to the deviatoric strain and a damage associated to the volumetric strain. In order to model the unilateral behaviour of concrete, only the deviatoric damage is taken into account for the compressive behaviour and both deviatoric and volumetric damages are introduced for the tensile behaviour. The damage mechanism depends on three strain energies, the total elastic strain energy, the distortion energy to take into account the slip mechanism between the cracks lips, and the volumetric strain energy to take into account the influence of the cracks growth according to the hydrostatical stress. The stress-strain relation is written as follow: σ = (1 − d)E : εe

2fcis (1 + ν) 3E

ds = 1 −

ε˜ 0s exp[−Bt (˜ε s − ε˜ 0s )] ε˜ s

d d = 1 − exp[−Bc (˜ε d − ε˜ 0d )]

1 trε 3

(16)

3

(22)

3.1

NUMERICAL SIMULATIONS Studied configuration

The model has been used at the macroscopic and mesoscopic scale in order to simulate an experimental test performed by Mindeguia (2009). This test aimed to measure simultaneously the temperature and the gas pressure at various positions within a concrete specimen (30 × 30 × 12 cm3 ) that is heated on one face up to 600◦ C while the surrounding far field temperature at the opposite side is kept equal to the ambient one. The detailed experimental setup can be found in (Mindeguia 2009). Figure 1 gives the scheme of the experimental setup.

(17)

 (ε1d )2 + (ε2d )2 + (ε3d )2 + αc εH

(18)

where εid = main strains of the deviatoric strain tensor and αc = coupling coefficient reflecting the material consolidation under a compressive load. If ε˜ s and ε˜ d reach a threshold, the associated damage variables d s and d d evolve. The initial damage threshold is defined by experimental tests, uniaxial tensile tests and shear tests, with: ft (1 − 2ν) 3E

(21)

where Bt = damage parameter related to the fracture energy to limit the mesh sensitivity and Bc = material parameter associated to its compressive strength. This macroscopic damage model has been implemented in the finite element code Cast3M. We have adapted it on the scale of the REV for simulations of the behaviour of concrete at high temperatures.

where d s and d d are respectively the volumetric and the deviatoric damage variables. According to the equivalent strain introduced by Mazars (1984), two equivalent strains are retained in this model, ε˜ s for the volumetric mechanism and ε˜ d for the deviatoric mechanism. These equivalent strains correspond to the local slip in micro-cracks and the hydrostatic state. They are obtained from the deviatoric strain part ε d and the volumetric strain part εs of the strain tensor by:

ε˜ 0s =

(20)

(15)

(1 − d) = (1 − d s )(1 − d d )

ε˜ d =

√

where E = Young modulus, G = shear modulus, ν = Poisson’s ratio of the material, ft = its tensile strength and fcis = its shear strength. Ung (2003) has suggested these following evolutions of the volumetric damage and the deviatoric damage, according to the Mazars’s model (Mazar 1984):

where εe is the elastic strain tensor, E is the elastic stiffness tensor, and d the total damage variable is defined by:

ε˜ s = ε H =

=

2fcis = 3G

(19)

Figure 1.

Scheme of the experimental set-up.

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The concrete specimen has a thickness of 12 cm, and a height and a width of 30 cm. The composition of the concrete, also its mixture process, is presented in detail in the work of Mindeguia (2009). It is an ordinary concrete mix (OC M40) which contains a cement paste with a water cement ratio of W/C = 0.54. The cement used is of type Ciment CEM II 42.5 R (Airvault). The OC M40 contains calcareous aggregates (Chalonnes sur Loire) and sand (Chazé). The principal thermal, hygral and mechanical properties of the M40 concrete are obtained experimentally by (Mindeguia 2009). Concerning its components (mortar and calcareous aggregates), only some properties are available in the literature for the associated concrete mix (Menou 2004), (Gaweska Hager 2004), (Bazant et al. 1996), (Dreux et al. 1995) and (Tourenq et al. 1997). Therefore, the missing parameters are estimated, when possible, by an inverse analysis using Mori-Tanaka’s homogenization scheme (Benveniste 1986) for thermal and hygral parameters and by a numerical homogenization for mechanical parameters. All phase’s properties are presented in Table 1 and Table 2. Thermal, hygral and mechanical properties of concrete and mortar (Parameters linked to mass transfer and fluid properties, Young Modulus, coefficient of thermal expansion. . .) are supposed to depend upon temperature. The aggregate size distribution retained is presented in Table 3 for each volume fraction (Vf) according to aggregates diameter (D) from experimental data (Mindeguia 2009). In order to perform twodimensional simulations at the mesoscale, we have generated finite element meshes of the two phases: coarse aggregates and mortar matrix. By assuming that only the medium and large aggregates have a significant effect on the thermo-mechanical behaviour and on the degradation of the concrete at high temperature, only aggregates greater than 5 mm are explicitly meshed. The complementary part is regarded as homogeneous mortar matrix. The volume fraction of mortar Table 1. Principal thermal and hygral parameters of the M40 phases. Concrete Intrinsic permeability [m2 ] at 80◦ C Density [kg/m3 ] at 80◦ C Porosity [%] at 80◦ C Thermal conductivity [W/m K] at 20◦ C Heat capacity of the solid skeleton [J/kg K] at 20◦ C

Mortar

Aggregate

5.53 10−16 1.15 10−15 − 2285

1948

2750

13.85

23.88

−

2.55

1.7

4.32

910

934

877

Table 2. Principal mechanical parameters of the M40 phases at 20◦ C.

E [MPa] ν α fc [MPa] ft [MPa] fcis [MPa] Gf [N/mm]

Concrete

Mortar

Aggregate

30 000 0.2 5.00 10−6 40 5 9 0.75

25 000 0.2 5.79 10−6 30 4 5 0.05

60 000 0.2 4.16 10−6 75 7 14 0.1

Table 3. Aggregate size distribution of the OC M40 (Mindeguia 2009). D (mm)

Vf (%)

5 6.3 8 10 12.5 16 20

0.15 0.32 1 4.5 7.3 15 13.7

is about 58% and therefore is 42% for coarse aggregates. A bi-dimensional finite element configuration of a concrete representative elementary volume (REV) is generated with a random location of the inclusions (Fig. 2). The inclusions are assumed to have a circular shape and several sizes (inclusion radii) are considered in order to fit with the aggregate gradation curve: the graded aggregates are divided into 7 grades, a representative average radius is fixed for each grade, then the surface fractions of the idealised aggregate are computed by basing on Table 3 and summed in order to retrieve the overall surface concentration of inclusions. The REV has the same thickness than the specimen (120 mm); on the other hand its height is equal to three times the largest aggregates (60 mm). For the simulation at the macroscopic scale, we have kept the same mesh, however the material is considered as a homogeneous concrete. In the numerical simulations, the REV is heated on one superior face up to 600◦ C while the temperature at the opposite face is equal to 20◦ C. The surrounding external relative humidity is hext r = 50% and convective mass boundary conditions are considered between these two surfaces of the specimens and the surrounding medium. Because the specimen is thermally and hygrally insulated on all lateral faces; the heat, liquid and gas fluxes are equal to zero on both lateral faces of REV (Fig. 3). Concerning the initial condition, the specimen is stored in air-conditioned room at 20◦ C (± 2◦ C) and 50% (± 5%) of relative humidity.

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T [°C] 400 20 300

10

200

40 30

100

Simul-Macro Test

0 0

Macroscopic Figure 2.

100

150

200

250 300 Time [min]

Figure 4. Macroscopic simulation results of temperature time evolutions at different depths compared to the experimental results given by Mindeguia 2009 (the number after arrow is the distance in mm from the heated surface).

Mesoscopic

Meshes of REV of the specimen.

T = 600 C, HR = 50%

50

T = 600 C, HR = 50%

T [°C] 400 ql = 0, qv = 0, qa = 0, qT = 0

ql = 0, qv = 0, qa = 0, qT = 0

ql = 0, qv = 0, qa = 0, qT = 0

ql = 0, qv = 0, qa = 0, qT = 0

20 10

300

200

40 30

100

Simul-Macro Simul-Meso

T = 20 C, HR = 50%

T = 20 C, HR = 50%

Macroscopic

Figure 3.

0 0

Mesoscopic

Boundary conditions.

The mass water content and the degree of saturation of the specimen after 154 days of conservation are respectively equal to 2.59% and 43%. We have used these values for our simulation. 3.2

Thermo-hygral results

Some first results are presented here in terms of thermo-hygral fields within the studied specimen. 3.2.1 Temperature Figure 4 shows test and macroscopic simulation results of the temperature time evolutions at different depths of the concrete specimen. The obtained results show a good agreement between experiment and macroscopic computation. Figure 5 presents the evolution of temperature versus time for both macroscopic and mesoscopic simulations. The obtained results are almost

50

100

150

200

250 300 Time [min]

Figure 5. Comparison between the macroscopic and mesoscopic simulation results of temperature time evolutions at different depths (10, 20, 30 and 40 mm) from the heated face.

similar for these two modes of simulations, especially at 10 and 20 mm from the heated surface. However, from 30 mm the difference becomes more important; it may be due to the process of transport of mass which is different for two modes of simulation. Figure 6 presents in detail the comparison between the macroscopic simulation, the mesoscopic simulation and the experimental results of the temperature evolutions at 20 mm from the heated face. These three results quasi coincide. This shows that the both modes of simulations are suitable for the prediction of the temperature vs. time evolution. Now we discuss the temperature distribution within the specimen. Figure 7 presents the iso-values of temperature inside the specimen for both macroscopic

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Figure 8 shows test and macroscopic simulation results of the gas pressure evolution versus time at different depths from the heated face. In the macroscopic simulation the gas pressure is homogeneous for a fixed depth; (Figure 11—left). The curve in continuous line presents the simulated (macroscopic) gas pressure while, the points with fluctuation present the average gas pressure measured for 2 specimens at different depths. The maximum deviation between the two measured values of gas pressure is about 0.4 MPa. It takes place at 10 mm from the heated face. Figure 9 compares macroscopic and mesoscopic simulation results of the gas pressure evolution versus time at different depths of the concrete specimen. In the mesoscopic simulation the gas pressure is heterogeneous; for a fixed depth and time, gas pressure

T [°C] 400

300

200

Simul-Macro

100

Simul-Meso Test

0 0

50

100

150

200

250 300 Time [min]

Figure 6. Comparison between the macroscopic simulation, mesoscopic simulation and experimental results of temperature time evolutions at 20 mm of distance from the heated surface.

Pg [MPa] 1.6

30

40

20 20

1.2

40

10

60 80 100

0.8

120 140 160 180

0.4

200

Simul-Macro

220

Test

240 260

0

280

0

300

50

100

150

200

320

250 300 Time [min]

340 360 380

Simul-Macro

Simul-Meso

Figure 7. Iso-values of the temperature at 300 min of heating of the macroscopic and mesoscopic simulation.

Figure 8. Macroscopic simulation results of gas pressure time evolutions at different depths compared to the experimental results given by Mindeguia 2009 (the number at peak is the distance in mm from the heated surface).

1.6

and mesoscopic simulations after 300 min of heating. The temperature distribution is almost not affected by the material heterogeneity even for the mesoscopic simulation. We note a slight difference (limited to 5%) between the macroscopic and mesoscopic results. Thus, a macroscopic simulation seems to be acceptable for the prediction of temperature. Nevertheless, this result has to be further analyzed in terms of either thermal conductivity contrast between constituents (aggregates and mortar matrix) or imperfect thermal interfaces between them.

Pg [MPa]

1.2 10 0.8

0.4 Simul-Macro Simul-Meso 0 0

3.2.2 Gas pressure Here we present the gas pressure in the homogeneous concrete specimen (for macroscopic simulation) and in the mortar matrix only, since the aggregates are hygrally inert (for mesoscopic simulation).

40

30 20

50

100

150

200

250 300 Time [min]

Figure 9. Comparison between the macroscopic and mesoscopic simulation results of gas pressure time evolutions at different depths (the number at peak is the distance in mm from the heated surface).

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fluctuates around an average value (see iso-value of gas pressure on Figure 11—right). The curves in discontinuous lines present the average of the mesoscopic results of gas pressure at each depth. These lines present the fluctuation. The maximum deviation between values of gas pressure at the same depth and time in the mesoscopic simulation is about 0.58 MPa. It takes place at 10 mm from the heated face. This deviation agrees with the experimental one. That is may be the way to explain the variability of the gas pressure noticed during experimental tests. Figure 10 compares in detail macroscopic, mesoscopic and experimental results of gas pressure evolutions at 20 mm from the heated face. The average of gas pressure in the mesoscopic simulation quasi coincides with the macroscopic results. Moreover, these obtained results show a good agreement between experiment and both modes of computation. The picks are correctly predicted, either in terms of time stages or amplitudes. Figure 11 presents the iso-values of the gas pressure within the studied specimen with macroscopic and mesoscopic simulation at 120 min and 300 min of heating. The gas pressure distribution is homogeneous for the macroscopic simulation while the gas pressure distributions within specimen show strong local effects for the mesoscopic simulation due the heterogeneity. The gas pressure values locally depend on aggregates presences. Gas pressure induces tensile stresses which may lead to a microstructural effect (micro-cracking that arises when tensile stresses attain mortar matrix tensile strength which decreases with temperature); amplifying thus the structural effect generated by tensile stresses due to thermal expansion gradients at the macroscopic scale. Aggregates will then control crack patterns, namely percolation paths as shown by Pg [MPa] 1.6

20

0.8 Simul-Meso Simul-Macro Test 0 0

50

100

150

200

Simul-Macro - 120 min

Simul-Meso - 120 min 0.00E+6 0.08E+6 0.16E+6 0.24E+6 0.32E+6 0.40E+6 0.48E+6 0.56E+6 0.64E+6 0.72E+6 0.80E+6 0.88E+6 0.96E+6 1.04E+6 1.12E+6 1.20E+6 1.28E+6 1.36E+6 1.44E+6 1.52E+6 1.60E+6

Simul-Macro - 300 min

Simul-Meso - 300 min

Figure 11. Iso-values of the gas pressure at 120 min (top) and 300 min (bottom) of heating of the macroscopic and mesoscopic simulation.

damage profiles hereafter. Furthermore, in the mesoscopic simulation, gas pressure distribution also shows strong variations around large aggregates which may be expected to amplify local mechanical degradations at inclusion-matrix interfaces. 3.2.3 Mass loss Figure 12 shows results of mass loss of the studied concrete specimens either with a macroscopic and mesoscopic simulation. There is no significant difference between simulated mass losses of these two modes of simulations. The mass loss after 5h of heating is equal to 1.6%. The initial water content is equal to 2.59%. So, the specimen is not yet fully dried after 5h of heating. The mode of simulation seems to have no significant effect on the global mass-loss of this concrete specimen. Thus, a macroscopic simulation appears to be acceptable for this purpose.

1.2

0.4

0.00E+6 0.08E+6 0.16E+6 0.24E+6 0.32E+6 0.40E+6 0.48E+6 0.56E+6 0.64E+6 0.72E+6 0.80E+6 0.88E+6 0.96E+6 1.04E+6 1.12E+6 1.20E+6 1.28E+6 1.36E+6 1.44E+6 1.52E+6 1.60E+6

250 300 Time [min]

Figure 10. Comparison between the macroscopic simulation, mesoscopic simulation and experimental results of gas pressure time evolutions at 20 mm of distance from the heated surface.

3.3

Mechanical results

Some first results are presented here in terms of damage profiles within the studied specimen and

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Mass loss [%] 2 Simul-Macro Simul-Meso

1.5

1

0.5

0 0

50

100

150

200

250 300 Time [min]

Figure 12. Comparison between the macroscopic and mesoscopic simulation results of mass loss time evolutions.

deformation of specimens sketched in Figures 13 at 240 minutes. Figure 13 presents the comparison between the macroscopic and mesoscopic simulation results of damage and deformation. The result of the macroscopic simulation shows that there is damage at the superior heated part. The distribution of damage is rather homogeneous. On the other hand, in the mesoscopic simulation, the damage is also at the superior part but appears heterogeneous, damage is more important at the aggregates—mortar interface which is normally the place with the weakest resistance in the concrete. The effect of aggregates can be assessed by comparing damage profiles and crack patterns given by figures 13 which concern mesoscopic simulation. In the macroscopic simulation, cracks are almost inclined and cross aggregates while, in the mesoscopic simulation, cracks arise at the mortar—coarse aggregates interfacial zone and then cross the mortar. 4

CONCLUSION

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Simul-Macro

Simul-Meso

Simul-Macro

Simul-Meso

Figure 13. Comparison between the macroscopic and mesoscopic simulation results of damage profile (top) and deformed shape (bottom).

An original thermo-hydro-mechanical (THM) modelling approach of concrete at the meso-scale level is presented and used to investigate the effects of microstructures on the concrete behaviour during heating. The obtained results show that the heterogeneity does not affect neither the overall thermo-hygral behaviour of the material nor the local temperature evolution. By contrast, the local hygral behaviour and the mechanical degradation are strongly affected by the microstructure of the material. The obtained results of temperature show a good agreement between experiment, macroscopic and mesoscopic computation. Indeed, the mesoscopic simulation shows that the hygral fields (gas pressure in this contribution, but also saturation) are strongly heterogeneous in the specimen. The distribution of the gas pressure in the mortar depends on the presence of aggregates. The obtained results of both modes of simulation are similar with the experimental results of gas pressure. The picks are correctly predicted, either in terms of time stages or amplitudes. The mesoscopic results of gas pressure present a big fluctuation. That is maybe the way to explain the gas pressure variability noticed during experimental tests. With regard to the mechanical behaviour, the first results show that the damage occurs more important at the interface mortar / aggregates in the mesoscopic simulation and then develops into the mortar. The damage distribution seems heterogeneous in the mesoscopic simulation and homogeneous in the macroscopic simulation. These interesting new results are to be completed by extending the mechanical model in order to account for fluid pressure (gas and liquid when over-saturation

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occurs) effect at this scale, shrinkage and transient creep in the cement matrix. Furthermore, a 3D THM analysis is also under investigation. These are some perspectives to this work. ACKNOWLEDGEMENTS Numerical simulations are performed at Laboratoire de Modélisation et Simulation Multi Echelle (MSME) of Université Paris-Est Marne-la-Vallée, France. The authors are very grateful. REFERENCES Alnajim, A. 2004. Modélisation et simulation du comportement du béton sous hautes températures par une approche thermo-hygro-mécanique couplée. Application à des situations accidentelles. PhD thesis, Université de Marnela-Vallée, France. Bazant, Z. & Kaplan, M. 1996. Concrete at high temperatures: material properties and mathematical models, Concrete Design and Construction Series, Longman Group Limited. Benveniste, Y. 1986. On the effective thermal conductivity of multiphase composites. Journal of Applied Mathematics and Physics 37: 696–713. Dreux, G. & Festa, J. 1995. Nouveau guide du béton, Eyrolles. Gaweska Hager, I. 2004. Comportement à haute température des bétons à haute performance—évolution des principales propriétés mécaniques. PhD thesis, Ecole Nationale des Ponts et Chaussées, France. Gawin, D., Pesavento, F. & Schrefler, B.A. 2003. Modelling of hygro-thermal behaviour of concrete at high temperature with thermo-chemical and mechanical material degradation. Comput. Methods. Appl. Mech. Engrg 192: 1731–1771.

Gawin, D., Pesavento, F. & Schrefler, B.A. 2005. Towards prediction of the thermal spalling risk through a multiphase porous media model of concrete. Comput. Methods Appl. Mech. Engrg. Grondin, F., Dumontet, H., Ben Hamida, A., Mounajed, G. & Boussa, H. 2007. Multi-scales modelling for the behaviour of damaged concrete. Cement and Concrete Research 37: 1453–1462. Kanema, M., De Morais, M.V.G., Noumowe, A., Gallias, J.-L. & Cabraillac R. 2007. Experimental and numerical studies of thermo-hydrous transfers in concrete exposo high temperature. Heat and Mass Transfer 44 (2). Lewis, R.W. & Schrefler, B.A. 1998. The Finite Element Method in the Static and Dynamic Deformation and Consolidation of Porous Media. John Wiley & Sons. Mazars, J. 1984. Application de la mécanique de l’endommagement au comportement non linéaire et à la rupture du béton de structure. PhD thesis, Université Paris 6, France. Menou, A., Mounajed, G., Boussa, H., Pineaud, A. & Carre, H. 2006. Residual fracture energy of cement paste, mortar and concrete subject to high temperature. Theoretical and Applied Fracture Mechanics 45. Mindeguia, J.-C. 2009. Contribution expérimentale à la compréhension des risques d’instabilité thermique des bétons. PhD thesis, Université de Pau et des Pays de l’Adour, France. Obeid, W., Mounajed, G. & Alliche, A. 2001. Mathematical formulation of thermo-hygro-mechanical coupling problem in non saturated porous media. Comput. Methods Appl. Mech. Engrg. 190: 5105–5122. Tourenq, C., Durville, J.-L. & Massieu, E. 1997. Les propriétés des roches: guide pratique, Mémento, Mines et Carrières, vol. 3, LCPC. Ung Quoc, H. 2003. Théorie de dégradation du béton et développement d’un nouveau modèle d’endommagement en formulation incrémentale tangente. Calcul à la rupture appliqué au cas des chevilles de fixation ancrées dans le béton. PhD thesis, Ecole Nationale des Ponts et Chaussées, France.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Application of enhanced elasto-plastic damage models to concrete under quasi-static and dynamic cyclic loading I. Marzec & J. Tejchman Gdansk University of Technology, Gdansk, Poland

ABSTRACT: The paper presents results of FE simulations of the cyclic concrete behavior under using three different coupled elasto-plastic-damage continuum models. To ensure the mesh-independence, to properly reproduce strain localization and to capture size effects, the constitutive models were enhanced by a characteristic length of micro-structure by means of a non-local theory. Numerical results were compared with corresponding laboratory tests on concrete beams. 1

INTRODUCTION

An analysis of concrete elements under cyclic loading is complex mainly due to a stiffness degradation caused by strain localization in the form of cracks and shear zones. The main problem is to formulate a constitutive continuum model which is able to describe both stiffness degradation and irreversible strain. The aim of the paper is to show the capability of different coupled elasto-plastic damage continuum models to describe strain localization and stiffness degradation in a concrete elements subjected to cyclic loading. First, a coupled elasto-plastic damage model proposed by Pamin & de Borst (1999) was used. Second, a coupled model using a concept proposed by Carol et al. (2001) and Hansen & Willam (2001) was applied. Third, the model following Meschke et al. (1998) was taken into account. The first model combines non-local damage with hardening plasticity and assumes that total strains are equal to strains in an undamaged skeleton. Plastic flow can occur only in undamaged specimens, thus an elasto-plastic model is defined in terms of effective stresses. In the second model, plasticity and damage are connected by two loading functions describing the behaviour of concrete in compression and tension. A damage approach simulates the behaviour of concrete under tension while plasticity describes the concrete behaviour under compression. A failure envelope is created by combining a linear Drucker-Prager formulation in compression with a damage formulation based on a conjugate force tensor and a pseudo-log damage rate in tension. In turn, in the third formulation, an elasto-plastic criterion was enriched by new components including stiffness degradation. Following the partitioning concept of strain rates, an additional scalar parameter was introduced into a constitutive formulation. Thus, the splitting of irreversible strains into components associated with plasticity and damage was obtained.

To capture properly strain localization in the frame of continuum models, a characteristic length of microstructure was included. It prevented the pathological discretization sensitivity and ill-posedness of the underlying incremental boundary value problem caused by strain softening and localization formation (differential equations of motion did not change a elliptic type during quasi-static calculations and a hyperbolic type during dynamic calculations). Thus, objective and properly convergent numerical solutions for localized deformation were achieved (meshinsensitive load-displacement diagram and meshinsensitive deformation pattern). A characteristic length was included by means of a non-local theory. It allowed us also to take into account a deterministic size effect, i.e. dependence of strength and other mechanical properties on the size of the specimen. The FE results were compared with corresponding laboratory tests on concrete notched beams under cyclic loading performed by Hordijk (1991) and Perdikaris & Romeo (1995). Finally, a formulation providing a full coupling between plasticity and damage by Lubliner et al. (1989) was discussed.

2

CONSTITUTIVE MODELS FOR CONCRETE

The first formulation (called model ‘1’’) according to Pamin & de Borst (1999) combines elasto-plasticity with scalar damage enhanced by non-locality. The idea of such coupling assumes the strain equivalence hypothesis; total strains εij are equal to strains in an eff undamaged skeleton εij (effective strains). Degradation is caused by microcracks. When considering material subjected to loading, one may observed that the area of the material cross-section that remains intact decreases. A decrease of this area leads to the

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idea of effective stress and strain which are experienced by the material skeleton between microcracks. Since plastic flow occurs only in the undamaged specimen, an elasto-plastic model was defined in terms of effective stresses eff

e σij = Cijkl εkl

(1)

The following failure criterion to describe the material response in an elasto-plastic regime was used fp = σ (σ eff ) − σy (κp ),

(2)

wherein σy = uniaxial tension yield stress and κp = hardening parameter equal to plastic strain in uniaxial tension. In Equation 2, an elasto-plastic criterion may be used e.g. by Drucker-Prager, by Rankine (Bobinski & Tejchman 2004, 2006) defined with effective stresses. As a result of elasto-plastic calculations in the effective stress space, a strain rates decomposition was obtained. In the second step, the material degradation was calculated with the aid of an isotropic damage model, described with a single scalar damage parameter D growing monotonically from zero (undamaged material) to one (completely damaged material). As a consequence, the stress-strain function was represented by the following relationship eff

σij = (1 − D)σij

(3)

The damage parameter D acts as a stiffness reduction factor (the Poisson ratio ν is not affected by damage). The growth of the parameter D was controlled by a threshold parameter κ which was defined as a maximum of the equivalent strain measure ε˜ reached during the load history up to time t κ = max ε˜ (τ )

(4)

τ ≤t

(5)

where κ 0 is the initial value of κ when damage starts. If the loading function f is negative, damage does not develop. During monotonic loading, the parameter κ grows (it coincides with ε˜ ) and during unloading it remains constant. To describe the equivalent strain measure ε˜ , a modified definition of the failure criterion by von Mises in terms of strains was used (Peerlings, 1999) k −1 1 ε˜ = I1 + 2k (1 − υ) 2k



1 1 1 (7) εkk and J2 = εij εij − I12 3 2 6 To describe the evolution of the damage parameter D, an exponential softening law proposed by Peerlings (1999) was chosen  κ  D =1− 1 − α + αe−β(κ−κ0 ) , (8) κ0

I1 =

wherein α and β are the material parameters. The equivalent strain measure ε˜ can be defined in terms of total strains εij or elastic strains εije . In the second model (called model ‘2’), a twosurface isotropic damage/plasticity model combining damage mechanics and plasticity in a single formulation was used (Carol, Rizzi & Willam 2001 and Hansen & Willam 2001). The plastic region was described with aid of a linear Drucker-Prager criterion. The material experienced permanent deformation under sustained loading with no loss of the material stiffness. In turn, damage was formulated in the spirit of plasticity by adopting the concepts of a strain rate decomposition and failure condition. Thus, damage was formulated by a decomposition of the total strain rate into the sum of the elastic strain rate dε e and degraded strain rate dεd dε = dε e + dε d

(9)

The boundary between an elastic and a damage region was governed by the failure criterion fd = f (σ , qd ),

(10)

where qd is the damage history variable describing the evolution of the damage surface. The stress rate was defined as dσ = C s (dε e − dεd ).

The loading function of damage was equal to f (˜ε, κ) = ε˜ − max{κ, κ0 },

tensile strength of the material. The invariants I1 and J2 are

12k (k − 1)2 2 I + J2 , 2 1 (1 − 2ν) (1 − ν)2 (6)

where I1 = first invariant of the strain tensor, J2 = second invariant of the deviatoric strain tensor, ν = Poisson ratio, and k = ratio between compressive and

(11)

The degraded strains dε d were a result of the decreasing stiffness. The reduction of the elastic stiffness to the secant stiffness was due to the material damage parameter D C s = (1 − D)C e .

(12)

The effective stress and effective strain were again experienced by the material between micro-cracks. Assuming the energy equivalence, the relationship between the nominal and effective stress and strain was √ √ σ = 1 − Dσ eff , ε eff = 1 − Dε (13) and σ eff ε eff = σ ε,

(14)

with σ eff = C e εeff

and

σ = (1 − D)C e ε

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(15)

The loading function (Eq. 10) was defined in terms of the principal tensile conjugate forces fd =

3 

(−ˆy(i) ) − r(L),

Meschke et al. (1998). The strain decomposition was written as dε pd = dεp + dε d .

(16)

i

where −ˆy(i) = principal components of the tensile conjugate force tensor and r(L) = resistance function. The components of −ˆy(i) were written with aid of the effective stresses/strains 1  eff   eff  −ˆy(i) = σ εi . (17) 2 i

The plastic damage strains dε was calculated analogously to classical plasticity. The component associated with degradation and plasticity was obtained by introducing a scalar parameter β varying from zero to one (0 ≤ β ≤ 1) dε p = (1 − β)dε pd

−

Gf r0

L

,

(18)

with the fracture energy Gf , elastic part r0 and pseudolog damage variable L. However, this function caused mathematical problems in a softening regime. Therefore, a new resistance function was proposed  1 L (2 − α) , (19) r(L) = Eε02 exp 2 α

κ 1+ κ0

3

(22)

where dεer is the elastic-reversible strain rate. The third model (called model ‘3’) is related to another concept of coupling. The elasto-plastic criteria by Drucker-Prager and Rankine (Bobinski & Tejchman 2004) were enhanced by new components to describe the stiffness degradation according to

(27)

NON-LOCAL THEORY

To capture properly strain localization, to preserve the well-posedness of the boundary value problem and to obtain mesh-independent FE-results, a characteristic length of micro-structure was included by means of a non-local theory (Pijaudier-Cabot & Bazant 1987). When using a first coupled elasto-plastic damage model (model ‘1’), non-locality can be applied in elasto-plasticity or damage. In the paper, the second option was chosen (i.e. softening was not allowed in elasto-plasticity). The equivalent strain measure in Equation 6 was replaced by its nonlocal counterpart

ε¯ (xka ) = ω(r)˜ε(xk )dV , (28)

(21)

(23)

2 ,

where ft = tensile strength and κ0 = parameter adjusted to fracture energy.

The plastic strains are permanent while elastic and damage strains are reversible. Therefore the total strain rate is dε = dεer + dε p ,

ft

σt (κ) = 

When simultaneously considering both damage and plasticity, the total strain rate becomes the sum of the elastic, damage and plastic rate dε = dε e + dε d + dε p .

(25)

n+1

with the tensile strength ft , the ‘‘damaged’’ stiffness modulus Ept and the rate of change of the stiffness modulus nt . The variable L was 1 . 1−D

dεd = βdεpd

The stresses were updated analogously to the standard plasticity theory. To simulate concrete softening in tension, a hyperbolic softening law was chosen

wherein E = Young modulus, ε0 = threshold value of strain and α = parameter describing softening. The resistance function adopted by Nguyen (2005) was also used 2  E + Ept e−L·nt 1f2 r(L) = t , (20) 2 E Ee−L + Ept e−L·nt

L = ln

and

The parameter β enabled one a simple splitting of effects connected with an inelastic slip process (which caused an increase of inelastic strains) and a deterioration of microstructure (which contributed to an increase of the compliance modulus). The evolution law for the compliance tensor was

T ∂f ∂f ∂σ ∂σ Dn+1 = Dn + β T . (26) ∂f σ ∂σ

In the original formulation by Carol et al. 2001 and by Hansen & Willam 2001, the resistance functions had the following form r(L) = r0 e

(24) pd

where xka = coordinates of the considered (actual) point, xk = coordinates of the surrounding points, and ω = weighting function. In the second coupled model, non-locality was introduced by adopting the concept of non-local plasticity proposed by Rolshoven (2003); non-locality was prescribed in tension to the damage energy release Y (Marzec 2009). The non-local damage energy was

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composed of a local and non-local term calculated in the current (i) and previous iteration (i − 1) ∗ = (1 − m+mAkl )Y(i) + m(Y¯ (i−1) − Y(i−1) Akl ), Y¯ (i)

(29) wherein m = nonlocal parameter and Akl = component of nonlocal coefficient matrix. Finally, in the third model, the rates of the softening parameter were averaged according to the Brinkreve’s formula independently of both tension and compression  ω(r)dκi (xk )dV − dκi dκ¯ i (xka ) = dκi + m (30)

Figure 1. Geometry and boundary conditions of a notched beam under three-point bending (Perdikaris & Romeo 1995). Table 1.

As a weighting function ω, the Gauss distribution was used

ω(r) =

1 − √ e lc π

r lc

2

4.1

Beam size

Depth d [mm]

Width b [mm]

Span S [mm]

Notch a [mm]

Small Medium Large

64 128 254

127 127 127

254 508 1016

20 39 78

(31)

The models were implemented into the Abaqus Standard program with the aid of the subroutine UMAT (user constitutive law definition) and UEL (user element definition). For the solution of the nonlinear equation of motion governing the response of a system of finite elements, the calculations were performed with a symmetric elastic global stiffness matrix. The calculations with a full Newton-Raphson method resulted in a poor convergence in the softening regime due to the fact that the determination of a tangent stiffness matrix within a non-local theory is virtually impossible. The calculations were carried out using a large-displacement analysis In this case, the actual configuration of the body was taken into account. The Cauchy stress was taken as the stress measure. The conjugate strain rate was the rate of deformation. The rotation of the stress and strain tensor was calculated with the Hughes-Winget method. The non-local averaging was performed in the current configuration. The FE-simulations were performed under plane strain conditions. The triangle finite elements with linear shape functions (one integration point) were used.

4

Beams dimensions (Perdikaris & Romeo 1995).

FE-CALCULATIONS Three-point cyclic bending

The FE-calculations were performed for different concrete notched beams under cycling loading (Fig. 1, Table. 1) (Perdikaris & Romeo 1995). In the experiments under three-point cycling bending, a deterministic size effect in 3 different concrete beams was investigated.. The deformation was induced by imposing a vertical displacement at the node at the top, in the central part of the beam. The modulus of elasticity was E = 45.6 GPa and the Poisson ratio was ν = 0.2. The

number of finite elements was equal to 2292, 5213, 9211 for a small, medium and large beam, respectively. The characteristic length was lc = 5 mm. In the model ‘1’, the von Mises yield criterion with linear hardening (with the modulus hp = E/2) was assumed in a plastic region. The crucial point was to obtain the onset of damage and yielding almost at the same time. In a damage regime, the following parameters were assumed: κ0 = 1.55 × 10−4 , α = 0.99, β = 700 in the formulation with elastic strains ε˜ (εije ) and κ0 = 1.65 × 10−4 , α = 0.99, β = 550 in the formulation with total strains ε˜ (εij ). The yield stress was σy = 4.8 MPa and σy = 5.5 MPa in the formulation with elastic and total strains, respectively. Figures 2a and 2b show load-displacement diagrams for a small- and large-size beam with a first coupled elasto-plastic-damage model using the equivalent strain measure defined as total strains ε˜ (εij ) or elastic strains ε˜ (εije ) compared with the experimental data. The results with a model using total strains ε˜ (εij ) match well the experimental results. The calculated ultimate vertical forces differ by 3.3–12.1% only. In the case of a solution with elastic strains ε˜ (εije ), the differences in maximum forces increase up to 5.3–16.2% (thus, the elastic strains lead to an overestimation of the ultimate vertical force). In second coupled model (model ‘2’), the resistance function adopted by Nguyen (2005) was assumed with the following parameters: Ept = 38 GPa, nt = 0.1 and ft = 4.6 MPa. The results for a small and large size beam are depicted on Figures 3a and 3b. The loaddisplacement curves show satisfactory agreement with the experiments in the case of the load-bearing capacity only. The calculated maximum vertical forces differ by 4.6–13.6% as compared with the experimental ones. However, the calculated stiffness degradation is overestimated as compared with the experiments.

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Figure 2. Experimental (Perdikaris & Romeo 1995) and calculated load-displacement curves with the first coupled elasto-plastic-damage model with non-local softening using elastic and total strains: a) small size beam, b) large size beam.

The results of a deterministic size effect were confronted with the size effect law by Bažant (Bažant and Planas 1998) for notched beams σN = √

Bft , 1 + D/D0

Figure 3. Experimental (Perdikaris & Romeo 1995) and calculated load-displacement curves with the second coupled elasto-plastic-damage model: a) small-size beam, b) large-size beam.

(32)

wherein B = geometry-related parameter D = specimen size and D = specimen characteristic size. The FE results show good agreement with the analytical formula by Eq. 33. (Fig. 4). 4.2

Four-point cyclic bending

The numerical simulations were also performed with a concrete notched beam under four-point cycling bending (Hordijk 1991), Fig. 5. The deformation was induced by imposing a vertical displacement at two nodes at the top and central part of the beam. The modulus of elasticity was E = 40 GPa, Poisson ratio ν = 0.2 and characteristic length lc = 5 mm. The load-displacement diagrams are shown in Fig. 6. In turn, Figure 7 presents the calculated contours of localized zones near the notch for 3 coupled models. The evolution of non-local parameters (nonlocal equivalent strain measure (model ‘1’), non-local pseudo-log damage variable (model ‘2’), non-local

Figure 4. Calculated deterministic size effect for concrete notched beams as compared with size effect law by Bazant (Bazant and Planas 1998).

softening parameter) (model ‘3’) is demonstrated in Fig. 8. For a first coupled model (model ‘1’) with one surface in hardening plasticity, the von Mises yield criterion with the yield stress σy = 6.5 and σy = 5.9 MPa was assumed. A linear hardening parameter (with the modulus hp = E/2) was chosen. The following

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Figure 5. Geometry and boundary conditions of a notched beam under four point bending (Hordijk 1991). Figure 7. Calculated contours of localized zones near the notch in a beam under four-point bending with different enhanced coupled models: a) model ‘1’, b) model ‘2’ and c) model ‘3’.

Figure 6. Calculated load-displacement curves with coupled elasto-plastic-damage model with non-local softening: model ‘1’ (a), model ‘2’ (b) and model ‘3’ (c) (four point bending, Hordijk 1991).

parameters were used: κ0 = 9.5 × 10−5 and β = 140 for a model with total strains ε˜ (εij ) and κ0 = 8.6×10−5 and β = 170 for a model with elastic strains ε˜ (εije ). The parameter α as α = 0.92. Figure 6a shows the calculated load-displacement curves with a coupled elasto-plastic damage model using total strains ε˜ (εij ).

Figure 8. Evolution of non-local parameter above the notch in a beam under four-point bending with different enhanced coupled models: a) model ‘1’, b) model ‘2’ and c) model ‘3’.

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The load reversals exhibit a proper gradual decrease of the elastic stiffness. The calculated ultimate vertical force with elastic strain differs by only 1.3% as compared with the experiment. In the case of the formulation with total strains, the difference increases up to 5.5%. The width of the localized zone above the notch in the beam is about 2.4 cm (4 ÷ 5 × lc ), Fig. 7a. Using the second coupled model, the resistance function by Nguyen (2005) was assumed with the following parameters: Ept = 37 GPa, nt = 0.175 and ft = 2.85 MPa. The numerical results agree well with the experimental data in the case of the load bearing capacity of beams (Fig. 6b). The calculated ultimate vertical force differs by 4.6% as compared with the experimental values. However, the calculated stiffness degradation is higher than in the experiment. As a consequence, the evolution of a damage zone leads to an increase of the width of the localized zone as compared to the first coupled model. The width of the localized zone grows up to 3.2 cm (6 ÷ 7 × lc ) (Fig. 7b). In the third model, the following parameters were assumed: ft = 2.45 MPa, κ0 = 1.85 × 10–3 (Rankine criterion with a hyperbolic softening law) and β = 0.2. The calculated ultimate vertical force differs by 3.4% as compared with the experimental value (Fig. 6c). The calculated stiffness degradation is properly described. The width of the localized zone above the notch is 1.4 cm (2 ÷ 3 × lc ), Fig. 7c. The shape of a localized zone above the notch depends on the formulation (Fig. 7). 5

CONCLUSIONS

The presented FE calculations show that coupled elasto-plastic-damage models enhanced by a characteristic length of micro-structure in a softening regime can properly reproduce the experimental loaddisplacement diagrams and strain localization in concrete notched beams under quasi-static cyclic bending. The models are also able to properly capture a deterministic size effect. The models ‘1’ and ‘3’ are also able to describe the stiffness degradation. The shape and thickness of a localized zone depends on the formulation. Further research on a more consistent formulation of damage and plasticity will be conducted. One of the possibilities is to use a yield condition (Lubliner et al. 1989) which takes into account a different strength evolution under tension and compression. The formulation ensures a complete material description in both tension with softening and compression with initial hardening followed by softening, a different degradation of the elastic stiffness in tension and compression and stiffness recovery effects during cyclic loading. In the next step, the calculations of concrete cyclic behavior will be carried out with inertial forces and

rate effects (Ožbolt et al. 2006). A rate-dependent elasto-plastic damage model will be developed. First, the rate effect on the stress-strain relationship during slow loading will be taken into account according to Bažant et al. (2000). Later, the rate dependency will be incorporated through viscosity terms in an elastic and inelastic regime (Pedersen et al. 2008). ACKNOWLEDGMENTS The numerical calculations were performed on supercomputers of Academic Computer Centre in Gdansk TASK. REFERENCES Abaqus Theory Manual, Version 5.8. 1998. Hibbit, Karlsson & Sorensen Inc. Bažant, Z.P., Caner, F.C., Adley, M.D. & Akers, S.A. 2000. Fracturing rate effect and creep in microplane model for dynamics. Journal of Engineering Mechanics, ASCE 1126(9): 962–970. Bobinski, J. & Tejchman, J. 2004. Numerical simulations of localization of deformation in quasi-brittle materials within non-local softening plasticity. Computers and Concrete 4: 433–455. Bobinski, J. & Tejchman, J. 2006. Modeling of strain localization in quasi-brittle materials with coupled elastoplastic-damage model. Journal of Theoretical and Applied Mechanics 44(4): 767–782. Brinkgreve, R.B.J. 1994. Geomaterial models and numerical analysis of softening. PhD Thesis, Delft University of Technology. Carol, I., Rizzi, E. & Willam, K. 2001. On the formulation of anisotropic elastic degradation. Int. J. of Solids and Structures 38: 491–518. Hansen, E. & Willam, K. 2001. A two-surface anisotropic damage-plasticity model for plane concrete. Proceedings Int. Conf. Fracture Mechanics of Concrete Materials (R. de Borst, ed.): 549–556, Paris: Balkema. Hordijk, D.A. 1991. Local approach to fatigue of concrete. PhD Thesis, Delft University of Technology. Lubliner, J., Oliver, J., Oller, S. & Oñate, E. 1989. A plasticdamage model for concrete. International Journal of Solids and Structures 25(3): 229–326. Marzec, I., Bobinski, J. & Tejchman, J. 2007. Simulations of crack spacing in reinforced concrete beams using elasticplasticity and damage with non-local softening. Computers and Concrete 4(5): 377–403. Marzec, I. 2009. Application of coupled elasto-plasticdamage models with non-local softening to concrete cyclic behaviour, PhD Thesis, Gda´nsk University of Technology. Nguyen, G.D. 2005. A thermodynamic approach to constitutive modelling of concrete using damage mechanisc and plasticity theory, PhD Thesis, Trinity College, University of Oxford. Meschke, G., Lackner, R. & Mang, H.A. 1998. An anisotropic elastoplastic-damage model for plain concrete. International Journal for Numerical Methods in Engineering 42(4): 702–727.

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Ožbolt, J., Rah, K.K. & Meštrovi´c, D. 2006. Influence of loading rate on concrete cone failure, International Journal of Fracture: 139: 239–252. Pamin, J. & de Borst, R. 1999. Stiffness degradation in gradient-dependent coupled damage-plasticity. Arch. Mech. 51(3–4): 419–446. Pedersen, R.R., Simone, A. & Sluys, L.J. 2008. An analysis of dynamic fracture in concrete with a continuum viscoelastic visco-plastic damage model. Engineering Fracture Mechanics 75: 3782–3805. Peerlings, R.H.J. 1999. Enhanced damage modeling for fracture and fatigue. PhD Thesis, TU Eindhoven.

Perdikaris, P.C. & Romeo, A. 1995. Size effect on fracture energy of concrete and stability issues in three-point bending fracture toughness testing. ACI Mater. J. 92(5): 483–496. Pijaudier-Cabot, G. & Bazant, Z.P.1987. Nonlocal damage theory. ASCE J. Eng. Mech., 113, 1512–1533. Rolshoven, S. 2003. Nonlocal plasticity models for localized failure. PhD Thesis, École Polytechnique Fédérale de Lausanne.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Propagation of cracks and damage in non ageing linear viscoelastic media Nguyen Sy Tuan Laboratoire Navier, ENPC, Champs-sur-Marne, France EDF E&D, Moret-sur-Loing, France

L. Dormieux Laboratoire Navier, ENPC, Champs-sur-Marne, France

Y. Le Pape & J. Sanahuja EDF E&D, Moret-sur-Loing, France

ABSTRACT: The proposed paper is devoted to the modeling of damage induced by microcracks in a viscoelastic medium. First, the case of a single crack propagating in a non aging linear viscoelastic (n.a.l.v.) material is considered within the framework of the thermodynamics of irreversible processes. The second part of the paper aims at the determination of the effective behavior of a microcracked linear viscoelastic solid. An approximation of the effective behavior in the framework of a Burger model is derived analytically. 1

INTRODUCTION

This question occurs in the simulation of the response of the concrete wall which constitutes the containment system surrounding the reactor in a nuclear plant. This concrete wall is prestressed in order to close the microcracks and to prevent their propagation in case of an increase of the internal pressure associated with an accident. However, the creep of the material is expected to reduce the prestress. It is therefore important to analyze the behavior of a microcracked structure in the case of viscous strains. The rate dependent mechanical behavior of concrete is often approximated by a linear viscoelastic model, the simplest formulation of which is a non ageing one. The practical interest lies in the well known correspondence principle which transforms a time dependent boundary value problem into a linear elastic one. The case of a single crack is illustrated on the 3 points bending experiment in which the effect of the rate of loading is investigated. The determination of the effective behavior in the framework of homogenization amounts to solving a boundary value problem defined on a representative elementary volume (r.e.v.). As already reported in the literature (see e.g. [1]), homogenization of n.a.l.v. heterogeneous materials can be expected from a combination of the correspondence principle and the Eshelby-based homogenization schemes for random elastic heterogeneous media. However, this paper shows that the non linearity of the strain concentration in the cracks prevents a straightforward application of

this reasoning. Based on an alternative approach, the contribution of the present paper is to provide a quantitative assessment of damage in viscoelastic media and to derive a simple damage-dependent Burger model. Notations: Let us introduce the second order identity tensor 1, the fourth order identity tensor I, the fourth order projectors of the spherical and deviatoric parts respectively denoted by J = 1 ⊗ 1/3 and K = I − J. The average of the field a(z) in the r.e.v.  (resp. in the subset α ⊂ ) is denoted by a (resp. aα ):   1 1 a= a(z)dV ; aα = α a(z)dV (1) ||  | | α 2

3D NON AGEING LINEAR VISCOELASTIC BEHAVIOR

A class of three-dimensional isotropic n.a.l.v. constitutive behaviors is obtained in transposing classical unidimensional rheological models to the tensorial context. Burger’s model, namely a Maxwell system connected in series with a Kelvin-Voigt one (figure 1) turns out to model the linear viscoelastic behavior of concrete quite well. In the following, the subscripts

Figure 1.

Rheological model for concrete.

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M and K respectively refer to the ‘‘Maxwell’’ and the ‘‘Kelvin’’ part. In order to derive the state equation, the total strain is decomposed into the respective contributions of the Maxwell and Kelvin parts:

the Laplace-Carson transform:  +∞  ∗  −pt a (t)e dt = p a (p) =

ε = εM + ε K

When applied to (6), the Laplace-Carson transform yields the state equation of the solid in the form σ ∗ = C∗ (p) : ε∗ with:

−∞

(2)

Each element of the model (spring or dashpot) is characterized by an isotropic fourth order tensor, related to its elasticity or viscosity: CeK = 3kK J + 2μK K; CvK = ηKs J + ηKd K s d CeM = 3kM J + 2μM K; CvM = ηM J + ηM K

(3)

−∞

a(t)e−pt dt (9)

C∗ (p) = (X + pY + p2 Z)−1 : (pCeK + p2 CvK )

(10)

Owing to (3) and (8), C∗ (p) appears as an isotropic tensor which can be put in the form 3k ∗ J + 2μ∗ K with

Accordingly, kα and μα (α = K or M ) denote the bulk and shear moduli, whereas ηαs and ηαd represent a bulk and shear viscosity, respectively. The Maxwell strain εM is related to the total stress by: ε˙ M = SeM : σ˙ + SvM : σ

+∞

(4)

3 1 1 1 + s + = k ∗ (p) kM pηM kK + pηKs /3

(11)

and 1 1 2 1 = + d + μ∗ (p) μM pηM μK + pηKd /2

(12)

where SeM is the tensor of elastic compliance (inverse of CeM ) and SvM is the inverse of the viscosity tensor CvM . Similarly, in the state equation of the Kelvin part, the total stress σ is related to the Kelvin contribution εK to the total strain:

Note that a Poisson coefficient ν ∗ (p) can be defined as usual:

σ = CeK : εK + CvK : ε˙ K

In the resolution of a time-dependent boundary value problem, it should be recalled that the implementation of the correspondence theorem requires that the boundary conditions keep the same nature (stress or displacement boundary conditions) throughout the time interval of the considered loading. In contrast, in the case of propagating cracks, the propagation can be viewed as such a change in the nature of boundary conditions. This makes the correspondence theorem inappropriate and motivates to look for an alternative numerical approach. The starting point is the differential equation (6) for which an incremental formulation is sought. To do so, the first and second time derivatives at time t are replaced by the following first (resp. second) order difference quotients:

(5) CeK

in which appear the stiffness and viscosity tensors and CvK . Combining (2), (4), (5) and their time derivatives, a second-order differential equation w.r.t. time is obtained: X : σ + Y : σ˙ + Z : σ¨ = CeK : ε˙ + CvK : ε¨

(6)

in which: X = CeK : SvM ; Y = I + CeK : SeM + CvK : SvM Z = CvK : SeM

(7)

According to (7), it is readily seen that: 3kK 2μK K s J+ d ηM ηM     ηs ηd kK μK + sK J + 1 + + dK K Y= 1+ kM ηM μM ηM

X=

ηd ηs Z= K J+ K K 3kM 2μM

(8)

ν∗ =

3k ∗ − 2μ∗ 2(3k ∗ + μ∗ )

a˙ 

a(t + dt) − a(t) dt

a¨ 

a(t + dt) + a(t − dt) − 2a(t) dt 2

(13)

(14)

where dt denote the time increment. Introduction of (14) into (6) yields: 3

σ (t + dt) = C : (ε(t + dt) − ε(t)) + σ0

INCREMENTAL FORM OF THE STATE EQUATION

(15)

where For determining the response of a structure made up of a n.a.l.v. material, the classical approach is to implement the correspondence theorem is based on

C =

k1 k2 J+ K c1 c2

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(16)

 σ0 =

   c5 c3 c4 c6 J + K : σ (t) − J + K : σ (t − dt) c1 c2 c1 c2     k4 k3 (17) J + K : ε(t) − ε(t − dt) − c1 c2

The coefficients ci (1 ≤ i ≤ 6) and kj (1 ≤ j ≤ 4) are defined at appendix 7. Hence, the incremental formulation of state equation (6) is analogous to that of a linear elastic behavior with pre-stress. The latter, denoted by σ0 includes information concerning the stresses and strains at times t and t − dt. 4

If the current fields of viscous strains {ε vα }(t) (α = M or K) are regarded as given, the total strain field ε appears in the light of (22) as the solution of a problem of linear elasticity with a pre-stress field being equal to σ p . The total strain field ε(t) at time t therefore linearly depends on E(t) and on {ε v }(t), as well as on the geometry of the system which is characterized by (t). Accordingly, the elastic energy stored in the viscous system can be viewed as a function of the current values E(t) and (t) of the loading parameter and of the crack length, but also of the current fields of viscous strains {εvα } : W (t) = W (E(t), (t), {ε vα }(t)). Hence, the dissipation takes the form:

ENERGY ANALYSIS OF CRACK PROPAGATION

Let us now consider a domain () in which a plane crack with length  is propagating. The elastic energy, dissipation, stiffness and force considered in this section refer to the unit length in the normal-to-plane direction. For the sake of simplicity, the loading is defined by the evolution in time of a scalar kinematic parameter denoted by E(t). Let Q denote the external force associated with E. The mechanical rate of ˙ In the energy provided to the system then reads Q · E. linear elastic case (see e.g. [8]), the current value of the elastic energy of the system is a function of the loading parameter E(t) and of the current crack length (t): W = W EL (E, ). The dissipation associated to the crack propagation reads:

∂W ∂W D˙ = Q · E˙ − {ε˙v } E˙ − ∂E |,{εvα } ∂{εvα } |E , α −

EL

C  () 2 1 Q (20) F = − C  ()E 2 = − 2 2C()2 In the viscoelastic case, the elastic energy stored in the system is obtained by integration of a volume density which incorporates the contribution of the Maxwell part and that of the Kelvin one:  1 el 1 W = ε : CeM : ε el + ε vK : CeK : ε vK dV (21) 2 2 () To go further, it is useful to start from an alternative formulation of the state equation (6), which reads: with σ p = −CeM : εv

(23)

Considering first a reversible evolution, that is {ε˙vα } = 0 and ˙ = 0, one obtains:

∂W D˙ = − (E, )˙ (18) ∂ |E The energy release rate F = −∂W /∂ thus appears to be the driving force of propagation, so that the criterion for propagation takes the form of a threshold on the latter. Since W EL (E, ) is a quadratic function of E, it takes the form: 1 W EL (E, ) = C()E 2 (19) 2 which introduces the overall stiffness C() of the system (). Accordingly, the driving force of crack propagation is:

σ = CeM : ε + σ p

∂W ˙ ∂ |E ,{εvα }

(22)

Q=

  ∂W = C() E − E vis (, {ε vα }) ∂E |,{εvα }

(24)

C() represents the system instantaneous stiffness. It is identical to the stiffness encountered in (19), for an elastic system in which the local elastic stiffness tensor is CeM . E vis can be interpreted as the value of the kinematic parameter after an instantaneous total unloading (Q = 0). The term involving the viscous strain rates {ε˙vα } in (23) represents the viscous dissipation D˙ vis . Note that the derivative is taken with respect to the fields of viscous strain: ∂W D˙ vis = − (25) {ε˙v } ∂{εvα } |E , α The energy dissipated in the propagation appears to be the complementary of the viscous term in the total dissipation: ∂W D˙ − D˙ vis = − ˙ ∂ |E ,{εvα }

(26)

The driving force of propagation therefore reads: F =−

∂W (E, , {εvα }) ∂ |E ,{εvα }

(27)

The new feature lies in the fact that its value depends on the viscous strain field which in turn depends on the loading history E(0 → t) and on the evolution of the crack length (0 → t) on the considered time interval [0, t]. With a similar reasoning to the energy analysis performed in microelastoplasticity (see e.g. [3]), the elastic energy can be put in the form: 2  1 W = C() E − E vis + Wres (, {ε vα }) (28) 2

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The first term represents the energy that is recoverable in an instantaneous unloading (Q = 0). The second term can be interpreted as the residual elastic energy which is stored in the system immediately after this unloading. Since E vis and Wres both depend on , the determination of the driving force F does not reduce to that of C  (). This is the essential difference with the linear elastic case. The propagation criterion is based on the assumption that the non viscous dissipation is proportional to ˙ D˙ − D˙ vis = Gc , ˙ in which Gc is the propagation rate : termed the critical energy. The comparison with (26) yields a propagation criterion to be written on F : F < Gc ⇒ ˙ = 0;

F = Gc ⇒ ˙ ≥ 0

For h = L/4, g(X ) can be approximated by: g(X ) =

1.99 − X (1 − X )(2.15 − 3.93X + 2.7X 2 ) √ π (1 + 2X )(1 − X )3/2 (31)

Considering again (20) (with F/T replacing Q), it appears that the overall stiffness C() of the structure defined by: F = C()U T

(32)

satisfies (29)

The equality F (E, , {ε vα }) = Gc allows to derive the crack propagation induced by the loading history E(0 → t). Even if it assumed that Gc is a material constant, it is emphasized that the viscous strain fields explicitly appears in this equality. The loading rate is therefore expected to affect the crack propagation. This is illustrated on an analytical example in the next section.

−

1 9π L2 C  () = g 2 C() E 2h4

 2  h

(33)

An integration w.r.t.  yields the structure compliance: 1 1 1 9πL2 − = C() C(0) E 2h2



 h

Xg(X )2 dX

(34)

0

where C(0) is the classical stiffness of the notch-free beam (unit thickness): 5

THE NOTCHED BEAM

Let us now consider the three-points bending test on a notched beam (see figure 2). U denotes the displacement of the point where the force F is applied. The purpose of this section is to implement the numerical approach introduced at section 3 together with the theoretical framework presented at section 4 for dealing with crack propagation.

C(0) =

E c(0)

(35)

where c(0) is determined numerically. It is eventually found that: c(/h) 1 = C() E

(36)

with 5.1 Compliance of the structure in the linear elastic case

c(X ) = c(0) +

In the linear elastic isotropic case, the literature (see e.g. [9]) provides an expression of the energy release rate as defined in (20): F=

1 9π F 2 L2 g E 4h4 T 2

Figure 2.

 2  h

5.2

(30)

9π L 2 ( ) 2 h



X

xg(x)2 dx

(37)

0

Response of the structure in the n.a.l.v. case

The material behavior is now that described at section 2. The loading is defined by a constant loading rate U˙ . We successively examine the response of the structure without crack propagation (section 5.2.1) and with propagation (section 5.2.2). 5.2.1 Three-points bending without crack propagation We expect that there is a critical rate of loading U˙ cr such that crack propagation takes place only if U˙ > U˙ cr . The determination of U˙ cr will be discussed later. First, the response of the structure in the absence of crack propagation is considered.

3-points bending test on a notched beam.

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In the absence of crack propagation ( = o ), the response of the structure can be determined analytically from the correspondence theorem. With the notation introduced at (9), it is found that:

concepts. The displacement is prescribed in the form U (t) = U˙ tH(t), where H(t) is the Heaviside function. The assumption about the existence of ∞ and F∞ suggests that the local stress state anywhere in the structure also has a limit σ∞ as t → ∞. In view of the rheological model of section 2, this in turn implies that the behavior of the material tends to become purely viscous. In other words, the local state equation reduces to

F ∗ (p) = C ∗ (o , p)U ∗ (p) (38) T In the above equation, the structural stiffness C ∗ (o , p) is derived from (36): 1 C ∗ (o , p)

=

c(o /h) E ∗ (p)

σ∞ = CvM : ε˙ ∞

(39)

It therefore turns out that the asymptotic response at the scale of the viscous structure is formally that of a linear elastic one, that is (32), provided that U is replaced by U˙ and that a pseudo Young modulus is derived from the viscous tensor CvM :

The ‘‘Young modulus’’ E ∗ is derived from the ‘‘stiffness tensor’’ C∗ (p) of (10). Let us further discuss the asymptotic response (t → ∞). It is assumed that the velocity U˙ is constant after some time to . The leading term in U ∗ (p) in the vicinity of p = 0 is U ∗ (p) ≈ U˙ /p. Similarly, it is found from (10) that E ∗ (p) ≈ pE∞ with   1 2 −1 (40) E∞ = 3 s + d ηM ηM

F∞ = C(∞ , CvM )U˙ T

(43)

The structure stiffness C(∞ , CvM ) is derived from (36) with  = ∞ and E = E∞ given in (40):

Combining (38) and (39), it is found that F ∗ (p) tends toward a limit as p → 0. This means that F(t) tends toward the same limit as t → ∞: F(t) F ∗ (p) E∞ ˙ lim = lim = U (41) t→∞ T p→0 T c(o /h)

F∞ E∞ ˙ = U T c(∞ /h)

(44)

This provides a first equation between F∞ and ∞ . The second one is derived from the condition of propagation (29). In the case of an homogeneous structure, it can be shown numerically that the contribution of the residual energy to F is negligible, so that the latter can be approximated as:  2 1 F∞ /T F =− C  (∞ , CeM ) (45) 2 C(∞ , CeM )

Figure 3 compares the numerical response and the Laplace-Carson analytical one. Two different loadings are considered. In loading 1, there is no initial jump of displacement: U1 (t) = t U˙ H(t) (H(t) is the Heaviside function). In loading 2, the effect of an initial jump Uo of displacement is investigated: U2 (t) = (Uo + t U˙ )H(t). 5.2.2 Three-points bending with crack propagation At higher velocity U˙ , we expect that the crack propagation will occur to an asymptotic force F∞ and an asymptotic crack length ∞ . Based on this assumption, the purpose of this section is to determine ∞ and F∞ and to check numerically the validity of these

(42)

It is emphasized that C(∞ , CeM ) in the above equation refers to the instantaneous elastic stiffness of the structure, which is derived from the elastic component of the Maxwell part (stiffness tensor CeM ): C(∞ , CeM ) =

EM c(∞ /h)

(46)

Combining (44) and (45), the condition of propagation yields an equation which can be used to determine the asymptotic crack length ∞ : c (∞ /h) 2Gc EM =h 2 2 c(∞ /h)2 U˙ E∞

Figure 3. Variation of the force F in the absence of crack propagation: L = 1 m, h = 25 cm, o = 10 cm, T = 12.5 cm, U˙ = 5/6 10−12 m/s, Uo = 5 10−5 m, the viscoelastic moduli of the solid phase are given by the table 1.

(47)

Let us introduce the maximum ρmax of the function c /c2 which appears in the lefthand side, which is reached at /h = Xcr , as well as the critical length cr = hXcr : ρmax = max

c (X ) c (Xcr ) = 2 2 c (X ) c (Xcr )

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(48)

The equation (47) has no solution ∞ if the velocity is smaller than a critical threshold: U˙ < U˙ cr (see 4), where:

Table 1.

2Gc EM U˙ cr2 = h 2 ρmax E∞

‘‘M’’ ‘‘K’’

(49)

In other words, irrespective of the initial crack length o , no propagation occurs if U˙ < U˙ cr . (2) In contrast, two mathematical solutions (1) ∞ < ∞ are found if U˙ > U˙ cr . However, the first solution (1) ∞ lies on the increasing branch of the c /c2 plot ((1) ∞ < cr ) (see 4). This branch is physically meaningless because the solution ∞ is expected to be an increasing function of the loading rate U˙ . The physically acceptable solution is therefore the largest one, that is (2) ∞, which lies on the decreasing branch ((2) >  ). cr ∞ If the initial crack length o is less than the critical length cr (at which c /c2 reaches its maximum), the asymptotic crack length is (2) ∞ , which is indeed greater than o . In contrast, if o > cr , two cases can be encountered. If (2) ∞ < o , no crack propagation occurs for the considered rate U˙ . In turn, if (2) ∞ > o , crack propagation takes place and the asymptotic length is

c h 2G U2

c c2

U < Ucr

EM E2

U > Ucr

(1) o

cr

ηs (GPa.s)

ηd (GPa.s)

24.42 39.27

13.27 14.07

22.108 15.2.107

7.75.108 2.54.107

(50)

If U˙ < U˙ ∗ (o ), no crack propagation takes place. In turn, if U˙ > U˙ ∗ (o ), crack propagation takes place and the asymptotic crack length is the solution (2) ∞ of (47). Note that U˙ ∗ (cr ) = U˙ cr and that U˙ ∗ (o ) is an increasing function of o . Figure 5 illustrates different case of a loading rate and initial crack lengths for the beam of dimensions L×h×T = 1×0.25×0.125 (m3 ). The viscoelastic moduli of the solid phase are given by the table 1.

HOMOGENIZATION OF VISCOELASTIC MICROCRACKED SOLID

Let us now investigate the damage induced by microcracks in a n.a.l.v. material of the type described in section 2. The most natural way to address this question consists in taking advantage of the results concerning the effective behavior of microcracked linear elastic materials (see e.g. [2],[6],[3]) and to apply the correspondence principle. However, some precaution has to be taken in the transposition of mathematical results from elasticity to viscoelasticity. 6.1

Figure 5. Crack propagation for U˙ cr = 4.10−12 (m/s); cr = 0, 1(m): (a) U˙ = 0.75U˙ cr ; (b) U˙ = 1.25U˙ cr and o = 0.05(m); (c) U˙ = 1.25U˙ cr and o = 0.13(m) < (2) ∞ = (2) 0.17(m); (d) U˙ = 1.25U˙ cr and o = 0.18(m) > ∞ .

μ

2Gc EM c (o /h) =h 2 2 2 c(o /h) U˙ ∗ (o ) E∞

(2) (2) o

Figure 4. Critical length and loading rates. Crack propagation takes place if U˙ > U˙ cr and o < (2) ∞.

k (GPa)

equal to (2) ∞ . It is emphasized that the asymptotic crack length (2) ∞ as defined by (47) is not affected by the initial crack length o provided that o < (2) ∞ . The same property holds for the asymptotic force F∞ which also only depends on U˙ , irrespective of the value of o (see (44)). For o > cr , the above discussion suggests to introduce another critical loading rate U˙ ∗ (o ) > U˙ cr depending on the initial crack length and defined as:

6 (1)

Vicoelastic moduli of concrete.

A review of the linear elastic case

More precisely, it is well known that the usual homogenization schemes in heterogeneous elasticity, such as the dilute scheme or the Mori-Tanaka scheme, are based on the concept of strain localization which, in turn, takes root in the linearity of the homogenization problem. In short, given a r.e.v.  and some macroscopic strain E, linear displacement boundary conditions are prescribed on the boundary of : (∀z ∈ ∂) : ξ (z) = E · z

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(51)

If the response is linear, the local strain ε(z) is linearly related to E by a fourth-order so-called ‘‘strain localization’’ tensor: ε(z) = A(z) : E

p

Chom = Cs : (I − ϕA )

(53)

p

where A is the average of A(z) over the pore space p . p Classical estimates of A are derived from the solution of the Eshelby inhomogeneity problem. The latter considers a single ellipsoidal inhomogeneity I embedded in an infinite homogeneous medium made up of an elastic material. Given some macroscopic strain E, linear displacement boundary conditions of the form: ξ (z) → E · z

(56)

Similarly, the rate-type formulation of the Eshelby problem for a spheroidal cavity now yields:  −1 ˙ :E (57) ε˙ I = I − S(ω) in which ω now refers to the aspect ratio in the current configuration of the spheroidal cavity. Eventually, the p p use of Adil or AMT in (53) leads to an estimate of the tangent effective stiffness. For mathematical reasons related to the fact that the crack porosity ϕ is proportional to ω (see e.g. [4]), it turns out that this tangent effective stiffness is in fact independent of ω. This renders the effective behavior linear elastic. However, from a rigorous point of view, the rate-type reasoning is indispensable in order to avoid the troubles associated with large strain in the direction of the normal to the cracks.

(54)

are adopted at infinity. Then, the strain εI in the inhomogeneity proves to be uniform. For instance, in the case of a cavity embedded in an infinite solid medium with stiffness tensor Cs , εI is given by: εI = (I − S)−1 : E

ε˙ (z) = A(z) : E˙

(52)

The effective stiffness tensor Chom of the composite is then shown to be the average C(z) : A(z), where C(z) denotes the local stiffness tensor. In the case of a porous medium with a homogeneous solid phase of stiffness tensor Cs , one obtains (see e.g. [3]):

|z| → ∞ :

should be replaced by a strain rate concentration tensor so that (52) becomes

(55)

where S is the Eshelby tensor of the cavity which depends on the geometry of the cavity and on the bulk elasticity tensor Cs [5]. Hence, recalling (52), p the dilute scheme simply amounts to taking Adil = p (I − S)−1 as an estimate for A . The Mori-Tanaka scheme can be viewed as a refined analysis aiming at capturing mechanical interaction between elementary pores (resp. cracks). The corresponding estimate comprises a correcting term w.r.t. the dilute one and reads p AMT = (I − S)−1 : ((1 − ϕ)I + ϕ(I − S)−1 )−1 . The usual 3D crack model is a flat spheroid characterized by its aspect ratio ω  1. In this case, the Eshelby tensor S is a function of ω. Let the subscript n be associated with the direction of the normal to the crack. The mathematical problem encountered specifically in the case of cracks lies in the fact that the coefficients nnαβ (as well as nαnα) of the tensor (I − S)−1 are of the order of 1/ω. This implies that the ratio of normal strain εnn to the macroscopic strain E is of the order of 1/ω. This in turn induces non negligible variations of the aspect ratio ω and is in contradiction with the assumption of linearity on which the concept of strain concentration tensor is based. This question has been discussed into details in [4]. In order to overcome this difficulty, the idea consists in considering the rate-type formulation of the problem. In other words, the strain concentration tensor

6.2

The viscoelastic case

The standard extension to n.a.l. viscoelasticity of the homogenization schemes in linear elasticity is based on the Laplace-Carson transform. it amounts to replacing the elastic homogenization rule Chom = C : A by ∗

Chom = C∗ : Av

(58)

In (58), the strain concentration tensor Av has now to be estimated for a fictitious r.e.v. having the same geometry as the real one and elastic properties characterized by the elastic stiffness C∗ . Clearly enough, since the laplace-Carson transform is a linear operator, it can only be applied to a linear set of equations. Now, in the case of microcracks, the previous section has emphasized the existence of a non linearity at the local scale in the relationship between the crack strain and the macroscopic strain. This implies that the homogenization of a viscoelastic cracked medium is not as straightforward as (58) could be since it cannot be based on the strain concentration concept. The purpose of this section is to present an alternative approach. We therefore consider a r.e.v.  made up the n.a.l.v. solid described at section 2 and of a network of plane penny-shaped cracks. As opposed to the 3D ellipsoidal crack model considered in the Eshelby-type approach, the mathematical penny-shaped crack is a 2D concept in nature. In particular, it does not refer to an aspect ratio. On the time interval of the loading, the displacement ξ (z, t) is prescribed on the boundary ∂ . The latter is related to the history of the macroscopic strain E(t) by ξ (z, t) = E(t) · z. Accordingly, the macroscopic strain is related to the microscopic strain field in the solid s by an average rule which has to be

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corrected by a term accounting for the strain in the cracks:    s 1 ε dV + [ξ ]i ⊗ ni dS (59) E= || s Ci i where Ci denotes crack no i, ni the unit normal to the crack plane and [ξ ]i the displacement discontinuity between the two lips of the crack defined according to the orientation of ni . Let (t) and σ (z, t) respectively denote the macroscopic stress tensor and the microscopic stress field, which are related by the average rule:  = σ . In the case of empty cracks, it takes the form:  1 σ dV (60) = || s

In turn, with shear stresses at infinity, say  ∗ =  ⊗ t, with t parallel to the crack plane, the discontinuity is in the crack plane (mode II) and reads: [ξt ]∗ =

The idea consists in anticipating that both the microscopic strain field in the solid and the displacement discontinuity vectors [ξ ]i linearly depend on the macroscopic strain, which will have to be confirmed a posteriori. Thus, it is justified to apply Laplace-Carson transform to (59), which yields: E∗ =

1 ||

 s

ε∗ dV +

   s 1 [ξ ]∗i ⊗ ni dS || Ci i (61)

We now seek the relationship between the macroscopic strain E∗ and stress  ∗ . To begin with, combining the stress average rule (60) with the state equation of the solid in the form σ ∗ = C∗ (p) : ε ∗ , the strain average rule (61) now reads: E∗ = C∗ (p)−1 :  ∗ +

   s 1 [ξ ]∗i ⊗ ni dS || Ci i (62)

We now look for an estimate of [ξ ]∗i as a function of  ∗ . To do so, we resort to the so-called stress-base dilute scheme [7]. This amounts to consider a single penny-shaped crack in an infinite elastic medium of stiffness C∗ (p) with a remote stress state  ∗ . The displacement discontinuity [ξ ]∗i is then estimated by the solution of this classical problem of linear elastic fracture mechanics. More precisely, consider the cylindrical coordinates system defined w.r.t. the axis of symmetry of the crack. The radius of the penny-shaped crack is denoted by a. If the remote stress is isotropic, say  ∗ =  ∗ 1, the discontinuity is normal to the crack plane (mode I) and reads: [ξn ]∗ =

4(1 − ν ∗ )  ∗ 2 a − ρ2 π μ∗

s

∗n

(63)

4 ∗ 1 − ν∗ 2 a − ρ2 π μ∗ 2 − ν ∗

(64)

(63) or (64) linearly relate the elementary displacement discontinuity and the macroscopic stress. In turn, the use of the latter in (62) clearly yields a linear relationship between  ∗ and E∗ , which can be put ∗ in the form  ∗ = Chom (p) : E∗ . This equation characterizes the effective behavior, up to an inverse Laplace-Carson transform. Combined with (64) and (63), it also a posteriori confirms the assumption of a linear relationship between the local displacement discontinuity and the macroscopic strain. For further investigation, it is necessary to specify the distribution of crack orientations. In the sequel, the method is illustrated on the case of an isotropic distribution of crack orientations. 6.3

Isotropic distribution of crack orientations

In this section, an isotropic distribution of crack ori∗ entations is considered. This implies that Chom (p) is an isotropic tensor which sought in the form: ∗

∗

∗

Chom (p) = 3k hom (p)J + 2μhom (p)K ∗

(65) ∗

In order to determine k hom (p) and μhom (p), we successively consider an isotropic and a deviatoric loading. 6.3.1 Effective behavior under isotropic loading Under an isotropic loading in which the macroscopic stress reads  ∗ 1, the elementary contribution of a crack to the macroscopic strain in (62) is derived from (63): 

s

[ξ ]∗ ⊗ n dS = C

8a3  ∗ (1 − ν ∗ ) n⊗n 3 μ∗

(66)

Assuming that all cracks have the same radius a, an integration over all orientations on the unit sphere yields the total crack contribution:    s 1 8Na3  ∗ (1 − ν ∗ ) [ξ ]∗i ⊗ ni dS = 1 || 9 μ∗ Ci i (67)

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in which N denotes the number of cracks per unit volume. Introducing (67) into (62) leads to the homogenized state equation under isotropic loading: ∗

 ∗ (p) = k hom (p)tr E∗ (p)

(69) and (70). This yields:

(68) 1 s ηM ()

with

k

hom ∗

1 1 + κM  ; = kM () kM

k∗ 16 = ; Q∗ = 1 + Q∗ 9

1 − ν∗2 1 − 2ν ∗

(69)

where  = Na3 , often referred to as crack density parameter [2], appears to characterize the damage level. For practical implementation, it is appealing to seek an effective Burger model for the cracked medium; This amounts to seeking the appropriate s () and ηKs (). set of parameters kM (), kK (), ηM Recalling (11), the latter should meet the following condition: 1 1 3 1 = + s + ∗ kM () pηM () kK () + pηKs ()/3 k hom (p) (70) ∗

Recalling the expression (69) of k hom (p) and that of ν ∗ (p) (see (13)), it is readily seen that (70) cannot be satisfied rigorously. Our purpose is rather to identify the best approximation of the effective behavior in the class of Burger models. The idea is to satisfy the series expansion of (70) to the first order at p = 0 and p = ∞. Recalling that

=

s  1 + vM ; s ηM

1 1 + κK  = kK () kK 1 ηKs ()

=

1 + vKs  ηKs

(73)

s where the set of constants κM , κK , vM and vKs are determined from the following system (see also figure 6):

κM = Qo∞ ;

Qo κM − Qoo κK − Qoo + = 3 s1 kM kK ηM

s Q∞ vs − Q∞ vM − Qo∞ + K s o = −1 ; s ηM ηK 3kM

s vM = Qoo (74)

6.3.2 Effective behavior under deviatoric loading Given a cartesian orthonormal frame (e1 , e2 , e3 ), consider for example a deviatoric loading in which the macroscopic stress reads:  ∗ =  ∗ (e1 ⊗ e1 − e3 ⊗ e3 )

(75)

The elementary contribution of an elementary crack to the macroscopic strain in (62) is derived from (64) and (63), according to its orientation. Integration over all crack orientations on the unit sphere yields:    s M ∗ ∗ 1 [ξ ]∗i ⊗ ni dS =  (p) || 2μ∗ Ci i

(76)

with lim a(t) = lim a∗ (p);

t→∞

p→0

lim a(t) = lim a∗ (p) (71)

t→0

p→∞

M∗ =

32 (1 − ν ∗ )(5 − ν ∗ ) 45 2 − ν∗

(77)

the sought Burger model is expected to be an excellent approximation in the short and long term. For forthcoming reference, we note the series expansion of Q∗ in the vicinity of p = 0 and p = ∞: Q∞ Q∗ = Qoo + Q1o p + O(p2 ); Q∗ = Qo∞ + −1 p   1 +O p2

(72)

∞ are explicitly determined where Qoo , Q1o , Qo∞ and Q−1 by the viscoelastic moduli of the solid phase. The damaged stiffness and viscosity parameters, that is kM (), s kK (), ηM () and ηKs (), must ensure the compatibility between the series expansions in the vicinity of p = 0 and p = ∞ of the bulk ‘‘compliance’’ derived from

Figure 6. Approximation of the effective behavior of a microcracked medium by a Burger model, the viscoelastic moduli of the solid phase are given in the table 6.

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Introducing (77) into (62) leads to the homogenized state equation under deviatoric loading:

7

ANNEXE

c1 =

  kK ηKs 1 ηs 3kK 1 1 + + 2 K + + s s ηM dt kM ηM dt 3kM

c2 =

  ηKd 1 ηKd 2μK 1 μK + + + (83) 1 + d d dt μM dt 2 2μM ηM ηM

c3 =

  ηs 2 ηs 1 kK + sK + 2 K 1+ dt kM ηM dt 3kM

1 1 1 2 = + + d ∗ μhom (p) μM () pηM () μK () + pηKd ()/2 (79)

c4 =

  ηd 2 ηd 1 μK + dK + 2 K 1+ dt μM dt 2μM ηM

(84)

This equation is to be satisfied in the vicinity of p = 0 and p = ∞. The series expansion of M ∗ in the vicinity of p = 0 reads:

c5 =

1 ηKs ; dt 2 3kM

(85)

k1 =

1 1 3kK + 2 ηKs ; dt dt

k3 =

1 s η ; dt 2 K

∗

∗

 ∗ (p) = 2μhom (p) E∗ (p)

with μhom =

μ∗ 1 + M ∗ (78)

As done previously for an isotropic loading, we seek an effective Burger model for the cracked medium under shear. The latter is characterized by the approprid () and ηKd () ate set of parameters μM (), μK (), ηM defined by the following condition (see (12)):

M ∗ = Moo + M1o p + O(p2 ); M ∗ = Mo∞ +

∞ M−1 + O(1/p2 ) p

by the viscoelastic moduli of the solid phase. The damaged stiffness and viscosity parameters, that is μM (), d μK (), ηM () and ηKd (), must ensure the compatibility between the series expansions in the vicinity of p = 0 and p = ∞ of the shear ‘‘compliance’’ derived from (78) and (79). This yields:

1 d ηM ()

=

d  1 + vM ; d ηM

1 1 + mK  = μK () μK 1 ηKd ()

=

1 + vKd  ηKd

(81)

d and vKd are where the set of constants mM , mK , vM determined from the following system (see also figure 6):

mM − Moo Mo mK − Moo + = 2 d1 μM μK ηM mM = Mo∞

;

1 ηKd dt 2 2μM k2 =

1 1 2μK + 2 ηKd dt dt

(80)

∞ where Moo , M1o , Mo∞ and M−1 are explicitly determined

1 1 + mM  ; = μM () μM

c6 =

;

d vM = Moo

∞ d M−1 vM vKd − Mo∞ − Mo∞ + = d 2μM ηM ηKd (82)

k4 =

1 d η dt 2 K

(86)

REFERENCES [1] Beurthey, S. and Zaoui, A. Structural morphology and relaxation spectra of viscoelastic heterogeneous materials. Eur. J. Mech. A/Solids, 2000. [2] Budiansky, B. and O’Connell, R. Elastic moduli of a cracked solid. Int. J. Solids Struct., 12:81–97, 1976. [3] Luc Dormieux, Djimédo Kondo, and Franz-Josef Ulm. Microporomechanics. Wiley, 2006. [4] Deudé V., Dormieux, L., Kondo, D. and Maghous S. Micromechanical approach to non linear poroelasticity: application to cracked rocks. J. Eng. Mech., 128:848–855, 2002. [5] Eshelby, J.D. The determination of the elastic field of an ellipsoidal inclusion, and related problems. Proc. R. Soc. Lond., A 241:376–396, 1957. [6] Horii, H. and Nemat-Nasser, S. Overall modulii of solids with microcracks: load-induced anisotropy. J. Mech. Phys. Solids, 31:155–171, 1983. [7] Dormieux, L. and Kondo, D. Stress-based estimates and bounds of effective elastic properties: The case of cracked media with unilateral effects. Computational Mater. Sci., 2009. [8] Leblond, J.B. Mécanique de la rupture fragile et ductile. Germes, 2003. [9] Shah, S.P., Swartz, S.E. and Ouyang, C. Fracture mechanic of concrete. John Wiley & Sons, 1995.

Despite the fact that the damaged Burger model is identified from the short and long term regimes, it appears to be an excellent approximation in between these limit cases.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Micromechanical modelling of concrete V.P. Nguyen, M. Stroeven & L.J. Sluys Faculty of Civil Engineering and Geosciences, Delft University of Technology, Delft, The Netherlands

ABSTRACT: In this contribution, we present the mechanical behaviour of the hardening cement paste explicitly taking into account its complex microstructure. The proposed model is a combination of a hydration model and a mechanical model. The microstructure of the cement paste is obtained via numerical cement hydration simulation. A simple nonlocal isotropic damage model is adopted for different constituents of the cement paste in finite element computations. This micromechanical model for concrete is believed useful in the process of building a multiscale model for concrete. 1

INTRODUCTION

In this paper, we present preliminary results of the mechanical behavior of the hardening cement paste (CP). The cement paste, at micro level, is composed of unhydrated cement particles, hydration products and capillary pores. It is the topology of this microstructure of the CP that controls the behaviour of concrete at mesoscale and macroscale. We have used a cement hydration model by (Navi and Pignat 1999; Stroeven 1999; Bishnoi and Scrivener 2009) to create the microstructure of the cement paste. This microstructure is then discretized into finite elements wherein a simple nonlocal damage constitutive law is applied to each microstructural constituent. The microscale result will be used in a mesoscale model thereby

forming a micro-meso two-scale damage model for concrete. Since our work employs purely computational techniques, it complements experimental mechanics particularly at the microscale since it is difficult or even impossible to perform experiments on concrete samples at this very small scale (typically about 100 μm). The analysis presented in this paper focusses on the microscale of concrete specimens. The procedure is such that a three-dimensional representation of the CP is obtained through cement hydration simulations of which a slice is input to a mesh generation code that produces a mesh which will be used for mechanical tests, see Fig.1. The paper is organized as follows. The utilized cement hydration model is given in Section 2. The next section presents the finite element model whereas Section 4 gives some numerical examples. Section 5 closes this contribution with some conclusions and remarks. 2

Figure 1. Procedure of numerical modelling of cement pastes: (a) cement hydration model (3D) and (b) nonlocal FE damage model (2D).

NUMERICAL CEMENT HYDRATION MODEL

To simulate the microstructure of the cement paste, we have used the cement hydration model of (Navi and Pignat 1999) and implemented it in a flexible manner following the work described in (Bishnoi and Scrivener 2009). It is emphasized that the cement particles are modelled as spheres of arbitrary size. Moreover, if a sufficiently large number of particles, say one million particles, is adopted the assumption on spherical shape for the particles is good enough to model the realistic microstructure of CP. Although the implemented model is, in principle, able to model any particulate chemical reactions, in this presentation only the reaction of tricalcium silicate (C3 S) with water producing silicate hydrates (CSH) and Portlandite (CH) is considered.

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Figure 3. Different meshing strategies for multi-phase materials: (a) Compatible mesh, (b) Gauss point method and (c) Hanging node method. Figure 2. The simulated three dimensional microstructure of a hardening cement paste.

Figure 2 shows a three dimensional microstructure of a hardening cement paste for which periodic boundary conditions have been taken into account.

3

FINITE ELEMENT MODEL

3.1

Finite element discretization

A common issue in micromechanical finite element computations is the discretization of microstructures that are typically complex multi-phase materials. The obvious solution is to utilize a compatible mesh1 , see Figure 3(a). In case that structured meshes are preferable, say when combined with multigrid solvers, the Gauss point method proposed by (Zohdi, Feucht, Gross, and Wriggers 1998), Figure 3(b) or the hanging node method developed in (Löhnert 2004) Figure 3(c) might be suitable alternatives to the compatible meshes. Partition of unity methods like XFEM (Moës N. 1999) have also been adopted to model microstructure independently of the mesh e.g., (Sukumar, Chopp, Moës, and Belytschko 2001). When dealing with the complicated 3D microstructure of cement pastes, none of the Gauss point method, the hanging node method and XFEM is applicable, at least practically2 . That might explain the reason in (Hain and Wriggers 2008) why the 3D CP specimen is meshed by replacing each voxel by one eight-node hexahedron element. Although this meshing technique is simple, it creates, however, meshes containing too many elements which necessitate the use of parallel computing. Since meshing the whole 3D CP specimen will result in a too large FE problem, we had to confine to two dimensional FE computations on slices of a 3D cement paste. To mesh the microstructure of the cement paste we have used the meshing module of

Figure 4.

Meshing the microstructure of cement paste.

the OOF program3 . The input of OOF’s mesher is a digital image which can easily be obtained from the cement hydration program. Figure 4 shows an example of using OOF’s mesher to obtain the unstructured mesh of a slice of a three-dimensional CP. Starting from a 2D sliced image in PPM4 format, the materials are assigned to different phases based on the color, or more precisely the gray scales, of different phases of the image, the meshing process starts with a regular triangle mesh superimposed on the whole image. This regular mesh is then refined adaptively by minimizing a cost function which is the sum of energy value E of all elements. The element’s energy E is defined as a linear combination of a shape parameter Eshape and a homogeneity parameter Ehom : E = αEhom + (1 − α)Eshape ;

0≤α≤1

where α is a weighting factor. The shape parameter Eshape indicates how much a triangle element differs from being an equilaterial, and is given by 36 A Eshape = 1 − √ 2 3L

1 Also

known as aligned mesh. can be explained by the fact that utilizing those methods leads to non-standard finite element formulations that unnecessarily complicate the problem. 2 This

(1)

3 For

details, see http://www.ctcms.nist.gov/oof for Portable Pixel Map.

4 stands

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(2)

where A and L are the area and perimeter of a triangle element, respectively. The homogeneity parameter, which Ehom measures the homogeneity of the element, is minimized with a value of zero when the element contains only one material. It is defined as Ehom =

N  1 − ai (T ) 1 − N1 i

de Borst, Brekelmans, and de Vree 1996), given by ¯eq − c∇ 2 ¯eq = eq

where c denotes an internal length scale and the local equivalent strain eq defined as (Mazars and PijaudierCabot 1989)

(3) eq =

where N denotes the number of pixel categories i.e., the number of micro-phases, ai (T ) represents the area fraction of phase i in the element T . These two parameters are easily understood by the illustration given in Fig. (5).

(7)

 1 2 + 2 2 + 3 2

(8)

In order to simulate the tensile damage of the CP, we have employed the simple isotropic damage model regularized by the gradient enhanced method (Peerlings, de Borst, Brekelmans, and de Vree 1996). For sake of completeness, this model is summarized here. The stress-strain relation is given by (Lemaitre 1996)

wherein i represents the principal strain. In summary, the utilized constitutive model is fully characterized by six material parameters: the Young’s modulus E, the Poisson ratio ν, the three inelastic parameters α, β, κI and the length scale c. The values of the various material parameters are tabulated in Table 1. The Young’s moduli and Poisson ratios, which are normally obtained through nanoindentation methods, are taken from (Constantinides and Ulm 2004; Hain and Wriggers 2008) whereas the inelastic parameters have been chosen without any experimental base for at the time of this writing no experimental data was available.

σ = (1 − ω)D : 

3.3

3.2 Constitutive models

(4)

where the scalar variable ω describes the damage and the second order tensor D contains the elastic moduli. Damage is governed by the following exponential law ω =1−

κ [1 − α + α exp−β(κ−κI ) ] κ ≥ κI κI

(5)

where α (residual stress), β (softening slope) and κI (damage threshold) denote the inelastic parameters. The variable κ is a scalar measure of the largest strain ever reached which is defined through the following loading function f f = ¯eq − κ

(6)

with ¯eq being the nonlocal equivalent strain which is, according to the gradient enhanced model (Peerlings,

Pore modelling

In (Hain and Wriggers 2008) the authors have written that ‘‘a stable finite element calculation must include pores since preliminarily removing unconnected voxels is computationally costly’’. They have chosen, based on a series of numerical computations in linear elastic regime, the values of the Young’s modulus and the Poisson ratio for the pores to be of unity and zero, respectively. Here we are interested in the effect of the pore in the post peak regime. To this end, we are going to compare the response of the cement paste with and without pores being modelled. To mesh the microstructure without considering the pores, we proceed as follow: (i) meshing the microstructure using oof as before however, the pore elements are marked with a special tag; (ii) those elements having this special tag are omitted from the mesh; (iii) the algorithm given in the Box 1, a simple application of the fast marching method (Sethian 1999), is then applied to this mesh to remove the isolated elements. Table 1.

Figure 5. Shape (a) and homogeneity (b) parameters used in adaptive meshing, taken from (N. Chawla Patel, M. Kooopman, Chawla, R. Saha, B.R. Patterson, Fuller, and S.A. Langer 2003).

Material parameters of different phases.

E[N/mm2 ] ν κI α β

Unhydrated

Hydrates

Pore

132700 0.3 – – –

24000 0.24 5e-06 0.95 1500

1.0 0.0 – – –

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Box 1.

We have performed uniaxial tension tests on two meshes, one with the pores of which the elastic parameters are the ones given in Table 1 and another one where the pores are not meshed. Figure 8 confirms the conclusion given in (Hain and Wriggers 2008) that in the linear elastic regime, if the pore is to be modelled their elastic constants should be E = 1 N/mm2 and ν = 0. However there is a substantial difference between modelling the pore (with E = 1 N/mm2 and ν = 0) and not modelling the pore. In order to determine the proper values for the elastic constants of the pore in case it is to be modelled, the uniaxial tension test previously presented has been conducted again with varying Young’s moduli from 0.001 N/mm2 to

Isolated elements removal.

1. Two element sets: accepted (A) and front (F) 2. Choose one element as seed element, put it in A and F 3. while F is not empty, do (a) pop the first element E of F (b) get neighbors of E, loop on the neighbors I (c) if I is not in A, put I in A and F Neighbors = element sharing one common edge.

4 4.1

NUMERICAL RESULTS Pore effect

In this section, we present some results of a hardening CP specimen of a 200 × 200 × 200 μm3 unit cell containing 886383 cement particles whose diameters range from 0.5 μm to 40 μm according to the Seive curve shown in Figure 6. The water to cement ratio is 0.4. This CP was hydrated for a period of 8 simulated years. Figure 7a shows a partial mesh of a slice through the center of the specimen. The red color denotes unhydrated material, the blue color for the hydration products and the yellow color represents the pores. Figure 7b is the mesh as in 7a but now the pores are not meshed.

Figure 6. Cumulative volume size distribution of the utilized cement.

Figure 7. FE discretizations of a slice of a CP specimen: with and without meshing pores.

Figure 8. 200 × 200 × 200 μm3 specimen-Meshing and no meshing pores: load-displacement comparison.

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E=1 E=0.1 E=0.01 E=0.001 without pore meshed

load [N]

0.015

0.01

0.005

0 0

1e-05 2e-05 3e-05 4e-05 5e-05 6e-05 displacement [mm]

(a) 28 days, αd=0.86, fp=11.54%

Figure 9. Comparison between pore and meshed pores with varying Young’s moduli. E=1 E=0.1 without pore meshed

load [N]

0.015

0.01

0.005

0 0

1e-05 2e-05 3e-05 4e-05 5e-05 6e-05 displacement [mm]

(b) 243 days, αd=0.97, fp=5.47%

E=1 E=0.1 without pore meshed

load [N]

0.015

0.01

0.005

0 0

1e-05 2e-05 3e-05 4e-05 5e-05 6e-05 displacement [mm]

(c) 730 days, αd=0.99, fp=4.38%

Figure 11. Comparison between pore and meshed pores with varying Young’s moduli.

Figure 10. Localized damage pattern in an uniaxially loaded CP specimen obtained with pores (a) being meshed and (b) without for X slice.

1 N/mm2 . Figure 9 indicates that the Young’s modulus of the pore must be chosen to be of 0.001 N/mm2 . Localized damage distribution in the uniaxially loaded CP is depicted in Figure 10. In Figure 11, the result of uniaxial tensition test is shown where αd and fp are the degree of hydration

(being zero for unhydrated cement paste and unity for fully hydrated CP) and the pore fraction, respectively. In reducing the Young’s modulus of the pores, the load-displacement curves converge to the one obtained without meshing the pores. For specimens which are more hydrated, see Figure 11b-c, even a small reduction in Young’s modulus led to the response obtained without meshing the pores. This is expected since more hydrated material means less pores. 5

CONCLUSIONS

We have presented a numerical framework for damage analysis of concrete specimens at microscale. The

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proposed model, which is a combination of a discrete model (a cement hydration model) and a continuum model (nonlocal damage model) provides a promising means to understand the behaviour of concrete at microscopic scale. Furthermore it might complement experimental work. The discretization of the microstructure of the cement paste and the effect of meshing the pores in contrast to not incorporating them at all in the mesh have been given. The numerical tests showed that the pores should not be discretized. In an other case, the Young’s modulus of the pores must be chosen carefully. The next logical step is to devise a method to upscale this microscale model of the cement paste to the matrix phase of the three phase matrix-aggregate-ITZ mesoscale model of concrete. Due to the big jump in length scales between micro and meso scales, application of coupled volume methods such as domain decomposition methods, variational multiscale methods would result in a mesoscopic finite element model of impractically solvable size. We are, therefore, working on a homogenizationbased scheme which is able to bridge the scales and is objective to the size of the microscopic model. ACKNOWLEDGEMENTS The financial support from the Delft Center for Computational Science and Engineering (DCSE) is gratefully acknowledged. REFERENCES Bishnoi, S. and K. Scrivener (2009). Mic: a new platform for modelling the hydration of cements. Cement and Concrete Research 39, 266–274. Constantinides, G. and F. Ulm (2004). The effect of two types of csh on the elasticity of cementbased materials:

Results from nanoindentation and micromechanical modeling. Cement and Concrete Research 34, 67–80. Hain, M. and P. Wriggers (2008). Numerical homogenization of hardened cement pastes. Computational Mechanics 42, 197–212. Lemaitre, J. (1996). A course on damage mechanics. Springer Verlag. Löhnert, S. (2004). Computational homogenization of Microheterogeneous materials at finite strain including damage. Ph. D. thesis, Universit sität Hannover, Deutschland. Mazars, J. and G. Pijaudier-Cabot (1989). Continuum damage theory—application to concrete. Journal of Engineering Mechanics Division ASCE 115, 345–365. Moës N.J. and Dolbow, T. B. (1999). A finite element method for crack growth without remeshing. International Journal for Numerical Methods in Engineering 46(1), 131–150. Navi, P. and C. Pignat (1999). Simulation of cement hydration and the connectivity of the capillary pore space. Advances in Cement Based Materials 4, 58–67. Chawla, N. Patel, B. Koopman, M. Chawla, K.R. Saha, B.R. Patterson, E. Fuller, and S.A. Langer (2003). Microstructure-based simulation of thermomechanical behavior of composite materials by object-oriented finite element analysis. Materials Characterization 49, 395–407. Peerlings, R.R. de Borst, W. Brekelmans, and J. de Vree (1996). Gradient enhanced damage for quasi-brittle materials. International Journal for Numerical Methods in Engineering 39, 3391–3403. Sethian, J.A. (1999). Fast marching methods. SIAM Rev 41(2), 199–235. Stroeven, M. (1999). Discrete numerical modelling of composite materials. Ph. D. thesis, Delft University of Technology. Sukumar, N. D. Chopp, N. Moës, and T. Belytschko (2001). Modeling holes and inclusions by level sets in the extended finite element method. International Journal for Numerical Methods in Engineering 190(47), 6183–6200. Zohdi, T.M. Feucht, D. Gross, and P. Wriggers (1998). A description of macroscopic damage through microstructural relaxation. International Journal for NumericalMethods in Engineering 43, 493–506.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

3D finite element analysis of concrete under impact load J. Ožbolt Institute of Construction Materials, University of Stuttgart, Stuttgart, Germany University of Rijeka, Rijeka, Croatia

V. Travaš & I. Kožar University of Rijeka, Rijeka, Croatia

ABSTRACT: In the present paper the results of numerical 3D FE analysis of concrete compact tension specimen and plain concrete beam loaded by impact load are presented and discussed. The spatial discretization is performed by the finite element method using Update Lagrange formulation. To account for cracking and damage of concrete, specimens are modeled by the rate sensitive microplane model for concrete. Green-Lagrange stain tensor is used as a strain measure together with the co-rotational stress tensor. Damage and cracking phenomena are modeled within the concept of smeared cracks. In the case of plain concrete beam, the contact-impact analysis is based on the mechanical interaction between two bodies—concrete beam and dropping hammer falling on the mid span of the beam. To investigate the influence of loading rate on the failure mode, parametric studies are carried out. The numerical results are discussed and compared with test results known from the literature. 1

INTRODUCTION

Various theoretical and experimental studies were in the past conducted in order to investigate the effect of loading rate on the response of concrete structures (Dilger et al, 1978; Reinhardt, 1982; Curbach, 1987; CEB, 1988; Bažant & Gettu, 1992; Weerheijm, 1992; Ožbolt & Reinhardt, 2001, 2005a, 2005b; Ožbolt, et al., 2005; Saatci & Vecchio, 2009; Travaš, et al., 2009). In most of the studies different constitutive relations, similar to the spring-dashpot models of visco-elasticity, were employed. Some of the models cover only a limited range of loading rates whereas the other are more general and applicable over many orders of magnitude of loading rate. However, apart from the influence of the loading rate on the structural response, an additional problem at high impact mechanics is rather complex energy transfer mechanism over the contact surfaces between the bodies under collision. Therefore, impact loading cannot be simply viewed only as a problem of strain rate. As discussed by Bentur et al. (1987) and Banthia et al. (1987), in an experiment it is difficult to satisfy the energy balance through the measuring of mechanical energies. In contrary to the experimental investigations, assuming isothermal conditions, in numerical analysis the transformation of impact kinetic energy into other mechanical energies can be easily calculated. Due to the fact that the failure process in concrete occurs in a very short period of time, numerical study

is useful and necessary for better understanding of damage phenomena at impact loading. Currently there are only a limited number of numerical and experimental studies in which the failure mode is investigated as a function of loading rate (Sukontasukku & Mindess, 2003; Ožbolt & Reinhardt, 2005a, 2005b; Saatci & Vecchio, 2009; Travaš, et al., 2009). In the here presented numerical studies rate dependent microplane material model for concrete is used. The model was originally proposed by Bažant et al. (2000b). It is based on the rate process theory (Krausz & Krausz, 1988) of bond ruptures, which is coupled with the microplane model for concrete (Ožbolt et al., 2001; Ožbolt et al., 2006). The aim of the study is to check whether the numerical model is able to realistically predict the rate dependent failure mechanism of plain concrete specimens and to see the influence of impact velocity on the response their response. The numerical experiments are conducted by 3D finite element simulation of the compact tensile specimen and of the free fall of hammer on the plain concrete beam, where the dropped hammer falls from different heights. In the case of the compact tensile specimen the load was applied by the displacement control. However, in the case of the beam the contact force between the dropped hammer and the concrete beam is unknown, therefore, the mechanical interaction is simulated. The investigated range of impact velocities is such that the strain rates are high but still smaller than the strain rates at which the dropping

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Under the assumption that the deformation of bodies does not cause mechanical interaction (contact), the weak form of dynamic equilibrium equation for body Bκ (1) can be obtained by applying the Hamilton’s variational principle of least action:

hammer would cause extreme local concrete damage of the impact zone.

2

STRONG AND WEAK FORM OF THE GOVERNING DIFFERENTIAL EQUATIONS

Let’s consider a system of bodies Bκ , with κ denoting each of them. Adopting index notation, for the entire deformation history the balance of linear momentum reads: κ σij,j

+

ρ κ fiκ

=ρ

κ

u¨ iκ

for i, j = 1, 2, 3

on uκ

(2)

κ is known displacement field. Denoting in which ui,0 the boundary surface unit outward normal vector with nκi , the additional Neumann traction boundary condiκ tions ti,0 on the boundary tκ reads: κ σijκ nκi = ti,0

on tκ

(3)

To solve the problem of contact between two or more bodies in the framework of continuum mechanics, there are two basic additional restrictions on the displacement and traction field over the contact surfaces Cκ . From the fact that two material points cannot occupy the same space at the same time, the penetration of one into another body is not possible. Let’s assume that the negative gap value giκ indicates the magnitude of the non-physical penetration, then the first condition, condition of impenetrability, states that no penetration is allowed: giκ ≥ 0

κ

=0

κ σij,j δ ui d κ



ρ κ fiκ δ ui d κ −

κ

tiκ δ ui d  κ = 0

(7)

κ

Since in general there are more than two bodies, they can mechanically interact between each other. Consequently, the contact surfaces are time dependent. To account for this scenario, the displacement field has to be restricted by excluding those displacements that cause non-physical penetration. From the various strategies available for the enforcement of contact condition (Wriggers, 2002), in the present formulation the Lagrange multiplier method is used (Belytschko et al., 2001; Wriggers, 2002). From the mechanical point of view, Lagrange multiplier λ represents force that is needed for the separation of bodies after a non-physical penetration is detected (4). The contact situation can be energetically described by introducing the artificial ‘‘parasite’’ contact energy κC stored on the contact interface Cκ . The contact energy is defined by the work of Lagrange multipliers on not-allowable gaps gnκ . Assuming that the frictional energy on the contact surface is fully dissipative, only normal contact forces λκn contribute to contact energy. The entire contact energy is calculated as: κ 

 =

C

λκn gnκ dCκ

(8)

Cκ

The variation and minimization of (8), that is a weak form of (4), and its combination with (7) yields to the weak form of dynamic equilibrium equation for bodies Bκ in contact:

(4)



ρ κ u¨ iκ δui dκ +

κ

(6)

σijκ δui,j dκ −

tiκ δui d κ + ∂

−

∂



ρ κ fiκ δ ui dκ

κ κ 

=0

(9)

C

κ

with

 κ



(5)

Finally, combination of impenetrability condition (4) with the intensility condition (5) yields to the so-called complementarity condition: giκ tiκ

 κ

 −

The second condition assures the compressive character of the normal traction acting on the contact surface Cκ . This is so called intensility condition: tiκ ≤ 0 on Cκ

ρ κ u¨ iκ δ ui d κ +

(1)

where σij,j is divergence of the true Cauchy stress tensor, ui is displacement with dot indicating its time derivatives. If the boundary conditions between the starting time t0 (initial time) and current time t are known, the trajectory of each point in the deformable continuum is uniquely defined. The Dirichlet displacement boundary conditions on surface uκ are: κ uiκ = ui,0



κ  C

 =

λκn δgnκ dCκ +

Cκ

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 Cκ

δλκn gnκ dCκ

3

where the nodal displacement increments are calculated using standard explicit update:   t −1 D . un+1 = M + C 2   (13) t 2 t {Kun −Rn }+ Cun−1 +M{2un − un−1 } 2

FE DISCRETIZATION

The numerical approximation of (9) is performed by finite elements. Following the standard finite element formulation and accounting for constrains on contact surfaces of finite elements, the discrete form of (9) becomes (Voight notation): Mu¨ + Cu˙ + Ku + GT λ = R

(10a)

G{u + X} = 0

(10b)

where M is the global diagonal mass matrix, K the global secant stiffness matrix and R the known load vector. The influence of vicious damping is introduced through damping matrix C obtained by linear combination of matrix M and K. Note, that the product Ku represents internal nodal forces due to deformation of bodies. Matrix G is the element contact displacement constraint matrix and λ is the vector of unknown Lagrange multipliers. The last term on the left hand side of (10a) is the discrete form of contact forces and (10b) is the discrete form of the contact displacement constrains, where X is the material coordinate vector. The system of differential algebraic equations (10) is solved using the explicit time integration scheme and Update Lagrange formulation. In the solution strategy the multi-step central difference method is adopted. The equation of motion (10a) for the time step tn and displacement constrains (10b), which are calculated at the time step tn+1 and related to contact forces obtained at time tn , can be written as: Mu¨ n + Cu˙ n + Kun +

T Gn+1 λn

= Rn

Gn+1 {un+1 + X} = 0

The computation proceeds with activating the socalled global contact detection algorithm, where the search over the possible non-physical penetrations of the boundary mesh nodes is tested (Hutter & Fuhrmann, 2007). For those contractor nodes that violate the contact constraint equation (11b) and penetrate into a mesh domain, an additional local detection strategy is activated. As a result, data needed for calculation of constraint matrix Gn+1 are obtained. The contact forces, represented by the Lagrange multiplier λn , are than calculated for each contact element as:  −1

D  T λn = t 2 Gn+1 M−1 Gn+1 Gn+1 un+1 + X − un (14) To remove the penetrated nodes from contact, in the contact correction phase, incremental contact disC are calculated by distributing the conplacements un+1 tact forces from equation (14) to each contact element node using matrix Gn+1 : C T = −t 2 M−1 Gn+1 λn un+1

(11a) (11b)

1 {un+1 − un−1 } 2t 1 {un−1 − 2un + un+1 } u¨ n = t 2

(15)

Equations (13) to (15) are solved using GaussSeidel iterative algorithm (Carpenter et al., 1991). To avoid numerical instability of the explicit algorithm, the time increment t must be smaller than the critical time increment tcr (Wriggers, 2002).

with: u˙ n =

4

Note that in the Update Lagrange formulation the equilibrium is required on the current configuration xn that is continuously updated as: xn+1 = X + un+1

(12)

in which xn+1 is the coordinate vector of the current mesh configuration calculated by adding displacement vector un+1 to the vector of material coordinates X. The unknown displacement un+1 at the end of the time interval t can be calculated in four steps. Neglecting for a moment the influence of the contact in (11a), the computation starts with the contact predictor phase

RATE DEPENDENT MICROPLANE MODEL FOR CONCRETE

In the microplane model the material is characterized by the relation between stress and strain components on planes of various orientations. These planes may be imagined to represent damage planes or weak planes in the microstructure, such as those that exist at the contact between aggregate and cement matrix. In contrast to phenomenological models for concrete, which are based on tensor invariants, in the microplane model the tensorial invariance restrictions need not be directly enforced. Superimposing in a suitable manner the responses from all the microplanes automatically satisfies them. The used microplane model (Ožbolt et al., 2001) is based on the so-called relaxed kinematic constraint concept. It is a modification of the M2 microplane

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model proposed by Bažant and Prat (1988). Each microplane is defined by its unit normal vector components ni (Fig. 1). Microplane strains are assumed to be the projections of macroscopic strain tensor εij (kinematic constraint). On each microplane considered are normal and shear stress-strain components (σN , σTr , εN , εTr ). To realistically model concrete, the normal microplane stress and strain components have to be decomposed into volumetric and deviatoric parts (σN = σV + σD , εN = εV + εD ). Based on the micromacro work conjugancy of volumetric-deviatoric split and using in advance defined microplane stress-strain constitutive laws, the macroscopic stress tensor is calculated as an integral over all possible, in advance defined, microplane orientations: σij = σV δij + 3 + 2π



3 2π





δij σD ni nj − dS 3

S

(16)

σTr (ni δrj + nj δrj )dS 2

S

where S denotes the surface of the unit radius sphere and δij is Kronecker delta. For more detail refer to Bažant et al. (2000a) and Ožbolt et al. (2001). At the constitutive level the rate dependency consists of two parts: (i) the rate dependency related to the formation of the microcracks and (ii) the rate dependency due to the creep of concrete between the microcracks. The influence of inertia forces on the rate effect is not a part of the constitutive law. However, this effect is automatically accounted for in dynamic analysis in which the constitutive law interacts with inertia forces. Based on the activation energy theory (Krausz & Krausz, 1988), the influence of strain rate on the rate insensitive microplane stress σm0 (εm ), where m indicates microplane stress-strain components V , D and Tr , can be written as (Bažant et al., 2000a; Ožbolt et al., 2006):

with: γ˙ =

1 ε˙ ij ε˙ ij 2

c1 =

c0 scr

(17)

where c0 and c2 are material constants, which have to be calibrated by fit of test data. The calibration of the constitutive law was carried out for moderate loading rates for which inertia forces have not much influence on the rate dependent response of concrete, i.e., only the rate of the crack growth controls the response (Ožbolt et al., 2006).

5

NUMERICAL EXAMPLES—COMPACT TENSION SPECIMEN AND PLAIN CONCRETE BEAM

Compact tension specimen is frequently used in the material science to measure fracture properties of concrete and other materials. It is well known that the cracking response of the compact tension specimen depends on the rate of loading. Higher loading rates lead to higher resistance. Moreover, with increase of loading rate failure mode also changes (Brandon, 1987). To demonstrate these effects of loading rate, 3D transient finite element analysis of compact tension specimen is carried out. The geometry, boundary conditions and finite element mesh of the specimen are shown in Fig. 2. The dimensions of the specimen are: width-heightthickness = 200 × 250 × 50 mm and the notch length is 100 mm. The mechanical properties of concrete

  2γ˙ σm (εm ) = σm (εm ) 1 + c2 ln c1 z microplane

z microplane integration point T

n

D

K

y

V M

x

y microplane

FE integration point

x

Figure 1. Decomposition of the macroscopic strain vector into microplane strain components—normal (volumetric and deviatoric) and shear components.

Figure 2. Geometry of the compact tension specimen (all in mm), finite element mesh, loading (displacement control) and boundary conditions.

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are: Young’s modulus Ec = 30000 N/mm2 , Poisson’s ratio ν = 0.18, uni-axial compressive strength fc = 40.0 N/mm2 , tensile strength ft = 3.50 N/mm2 , fracture energy GF = 0.09 N/mm and concrete mass density ρc = 2400 kg/m3 . The load is applied by controlling horizontal displacement of the notch 50 mm from the top of the specimen. To prevent local failure at the application of load, elements around the applied load are assumed to be linear elastic. The specimen is first analyzed assuming quasi-static loading condition. Subsequently dynamic analysis is carried out, with three different loading rates: /dt = 100, 500, 1000, 2500, 5000 and 10000 mm/s. The typical calculated load displacement curves are shown in Fig. 3. It can be seen that with increase of loading rate peak load increases and for high loading rates it is much larger than the static resistance. The peak load measured relative to the quasi-static resistance is plotted as a function of loading rate in Fig. 4. It can be seen that for loading rates up to 100 mm/s there is only a slight increase of resistance. In this region of loading rate resistance can be predicted only by the use of rate dependent constitutive law. However, for higher loading rates structural inertia forces have a major influence on resistance. In Fig. 5 are shown crack patterns in terms of maximal principal strains, assuming critical crack width wcr = 0.2 mm. As can be seen, for quasi-static load and relatively low loading rates (up to 100 mm/s) there is only one mode-I crack, which propagates starting from the notch tip. However, for /dt = 1000 mm/s there is a branching of the crack. The crack starts as a single one from the notch tip and subsequently it branches into two inclined cracks. With increasing loading rate there is a multiple branching of cracks (see Fig. 5). For the highest loading rate there is a cluster of multiple cracks with rather complex mixed failure mode. Responsible for the change of the failure mode are structural inertia forces.

Figure 3. Predicted load-displacement curves for loading rates up to 1000 mm/s.

Figure 4. Relative dynamic resistance as a function of loading rate.

Figure 5. Predicted failure modes: (a) quasi static analysis (b) /dt = 100 mm/s (c) /dt = 500 mm/s and (d) /dt = 1000 mm/s (e) /dt = 2500 mm/s and (f) /dt = 5000 mm/s.

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The evaluation of the numerical results for the case shown in Fig. 5d show that crack branching takes place at the crack velocity of approximately 0.5 Rayleigh wave speed vR , which is for the used concrete vR = CR (Gc /ρc )0.5 = 2140 m/s. Note, that the constant CR depends on Poisson’s ratio. For here used concrete it is equal to CR = 910 (for crack velocity in m/s) and Gc is shear modulus of concrete. Similar to the example of compact tensile specimen, available numerical and experimental results of investigations on plain and reinforced concrete beams (CEB, 1988; Sukontasukku and Mindess, 2003; Travaš et al., 2009) loaded by three-point bending show that the resistance and brittleness of beams increases with increase of loading rate. Furthermore, it was shown that for relatively low loading rate the failure is due to bending (mode I fracture), however, with increase of loading rate there is a transition of the failure mode from bending to shear. The numerical analysis (Travaš et al., 2009) was carried out in order to: (1) investigate whether the proposed numerical model is able to reproduce the test results qualitatively correct and (2) to investigate in more detail the response of concrete beam, e.g. cracking rate, distribution of energies, transition of failure mode etc., for different loading rates. The analysis is performed using the above discussed numerical approach. Cracking and damage phenomena in concrete are modeled using the concept of smeared cracks. To obtain mesh objective results crack band method is employed (Bažant and Oh, 1983; Ožbolt et al., 1996). The plain concrete beam is loaded by impact hammer at the mid-span (see Fig. 6). The dimensions of the beam are: length-height-width = 3000 × 600 × 300 mm. The mechanical properties of concrete are: Young’s modulus Ec = 30000 N/mm2 , Poisson’s ratio ν = 0.18, uni-axial compressive strength fc = 45.0 N/mm2 , tensile strength ft = 2.70 N/mm2 , fracture energy GF = 0.10 N/mm and concrete mass density ρc = 2300 kg/m3 . The load is applied through the kinetic energy of dropping hammer. The length of the hammer is 600 mm and the cross-section area is 300 × 300 mm. The behavior of hammer is assumed to be linear elastic with Young’s modulus of 200000 N/mm2 and mass density of 8000 kg/m3 . On the beam-hammer contact surface, the frictional coefficient μ = 0.5 is assumed. The numerical analysis is performed for quasi-static loading and for impact loading velocities of 2, 4, 6 and 8 m/s. The results of the quasi-static analysis show typical mode I type of failure (bending). Fig. 7 shows predicted failure modes for impact analysis in terms of maximal principal strains. The crack (red zone) is plotted in terms of maximal principal strains. It is assumed the critical crack opening wcr = 0.2 mm what corresponds to the critical strain of εcr = wcr /he = 0.0033, with he equal to average element size (60 mm). For

Figure 6. Geometry of the investigated plane concrete beam.

Figure 7. Failure modes for: (a) impact velocity of 2 m/s and (b) impact velocity of 8 m/s.

impact velocity of 2 m/s dominates mode-I fracture (bending failure). However, for impact velocity higher than 4 m/s dominates shear failure mode. Between impact velocities of 2 m/s and 4 m/s there is a transition from bending to shear failure. Similar results were obtained by experimental investigations (Sukontasukku and Mindess, 2003). Note, that these ‘‘limit’’ impact velocities are valid only for the here investigated beam-hammer geometry and their mechanical properties. For other geometrical and mechanical properties these limit velocities would change, however, the observed failure modes would principally be the same.

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(a)

(a) 5

350 250

hammer kinec energy

200

beam strain energy

Impact load [MN]

beam kinec energy

150 hammer strain energy

50

3 2 impact vel.4m/s 1 imapct vel.2m/s 0,44

0,39

0,34

0,29

0,24

0,20

0,15

0,10

0,00

0,42

0,37

0,33

0,28

0,23

0,19

0,14

0,09

0,05

0,00

0,05

0

0

Time [ms]

Time [ms]

(b)

(b) total energy

beam strain energy

2 hammer strain energy

impact vel.6m/s

-150

impact vel.8m/s

0,55

0,50

0,44

0,39

0,00

0,42

0,37

0,33

0,28

0,23

0,19

0,14

0,09

0,05

impact vel.4m/s

-200

0 0,00

impact vel.2m/s

-100

0,33

1

0 -50

0,28

3

beam kinec energy

0,22

hammer kinec energy

0,11

4

50

0,17

5

Reacon force [kN]

100

0,06

Energy [J]

4

100

Energy [kJ]

impact vel.8m/s impact vel.6m/s

total energy

300

Time [ms]

Time [ms]

Figure 9. Dynamic analysis: (a) impact load at the mid span versus time and (b) support reaction versus time.

Figure 8. Energy balance for: (a) impact velocity of 2 m/s and (b) impact velocity of 8 m/s.

In Fig. 8 are plotted computed distributions of energies in time, for impact velocities of 2 m/s and 8 m/s. Considered is the time interval up to the moment when the beam-hammer contact force drops approximately to zero. Within this time period the relevant damage of the beam took place. It can be seen that, because of relatively high stiffness of the hammer, its deformation energy is negligible compared to the deformation energy of the beam. After approximately 0.30 ms the total kinetic energy of the hammer is transformed into deformational and kinetic energy of the beam. Fig. 5 shows that the total energy slightly decreases with time. For all loading rates the decrease is obvious only up to the point of transition of the total kinetic energy of the hammer into the beam (approximately up to t = 0.30 ms). The reason for this slight drop is the frictional energy between the beam and hammer, which is not included in the total energy plotted in Fig. 8. The smaller part of the energy loss is caused by numerical error. The energy curves show that the sum of deformational and kinetic energy is equal to the total energy (isothermal conditions), what confirms that dynamic equilibrium is fulfilled. The predicted mid-span impact loads and reactions are plotted in Fig. 6. The impact load is much higher than the quasi-static peak (failure) load and with increase of impact velocity the impact load increases. Compared to the impact load, the reaction forces are relatively small. They are activated

after the beam is already significantly damaged and impact load already reduces almost to zero. This indicates that the load transfer takes place in a relatively small zone of the beam, close to impact zone, and that the impact load is almost entirely in equilibrium with inertia forces. With increase of loading velocity, the zone of the load transfer tends to be smaller, i.e. more localized (see Fig. 4). It can also be seen that just after the impact of the hammer, the reactions start to act in direction opposite to the impact load (positive reactions), however, once the beam in the zone of impact is damaged, the left and the right part of the beam tend to be lifted up (negative reactions). In Fig. 10 are plotted velocities of bending-crack tip as a function of time. They are obtained from the evaluation of the results of numerical analysis. Shown are absolute (Fig. 10a) and relative (Fig. 10b) velocities. The relative velocities are related to Rayleigh wave speed, which is for the used concrete vR = CR (Gc /ρc )0.5 = 2140 m/s. It is interesting to observe that after crack initiation there is almost linear increase of velocity of the crack tip up to the maximal velocity of approximately 0.55 vR . This is in good agreement with theoretical prediction (Freund, 1972a,b). From Fig. 7 can be seen that maximal crack speed only slightly increases with increase of impact velocity. According to theoretical solution for dynamic propagation of a single crack (Freund, 1972a,b), for relative velocities greater than 0.5 crack branching of mode-I crack is possible.

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have no or little effect on the failure mode that is of shear type.

(a)

2

maximal velocity of crack propagaon

1,5

imapct vel.2m/s imapct vel.4m/s imapct vel.6m/s imapct vel.8m/s

1 0,5

6

0,39

0,44

0,50

0,55

0,44

0,50

0,55

0,33

0,39

0,28

0,22

0,17

0,11

0,06

0 0,00

Apsolute crack velocity [km/s]

2,5

Time [ms]

(b) 0,8 0,6

imapct vel.2m/s imapct vel.4m/s imapct vel.6m/s imapct vel.8m/s

0,4 0,2

0,33

0,28

0,22

0,17

0,11

0,06

0 0,00

Relavy crack velocity [v/vR]

1

Time [ms]

Figure 10. Propagation of the bending crack as a function of time: (a) absolute velocity of the crack tip and (b) relative velocity of the crack tip.

This is also confirmed by the here presented numerical results. Namely, Fig. 7 shows that after reaching relative velocity of 0.5 the bending crack tends to branch into two inclined cracks. Note, that because of relatively coarse FE mesh, the results related to the dynamic crack propagation can be considered only as a qualitative. To get more detailed view into the dynamic crack propagation, numerical analysis should be carried out using finer discretization of the cracking zone. The comparison between numerical and experimental results (Sukontasukku and Mindess, 2003), shows that the used numerical model is able to correctly capture the rate dependant failure mechanism of plain concrete beams. Recent numerical studies (Ožbolt and Reinhardt, 2005; Ožbolt et al., 2006) show that for lower loading rates (bending failure) the rate sensitive response is controlled by local inertia forces at the micro-crack tip and by the viscosity. In the numerical analysis for these effects is accounted for by the rate dependent constitutive law of concrete. Furthermore, it is shown that for high and extremely high strain rates (shear failure mode), the structural inertia forces govern structural response and that the rate dependency at the constitutive level is much less important. The consequence is that the structural response is strongly dependent on the geometry of the structure. The same turns out to be the case in the present numerical results, what implies that, in contrary to lower loading rates, for higher loading rates bending reinforcement would

SUMMARY AND CONCLUSIONS

In the present paper the theoretical background, which is used in the numerical study of failure of compact tensile specimen and plain concrete beam loaded by impact 3-point loading is briefly discussed. The numerical model is formulated in the framework of continuum mechanics, following basic principles of irreversible thermodynamics. Based on the experimental results from the literature and presented numerical results of rate dependent failure of plain concrete specimens under impact load, the following can be concluded. (i) Loading rate has significant influence on the resistance and failure mode. The comparison between numerical and experimental results shows that the used numerical model is able to correctly predict the rate dependent failure. (ii) For quasistatic load and relatively low impact load velocity, the specimens fail in bending (mode-I fracture). With increase of loading rate (impact velocity) there is a transition of failure mode from dominant bending (single crack) to dominant shear with multiple cracks. In the case of plain concrete beam the failure tends to be localized closer to impact zone. (iii) For relatively low strain rates (mode-I fracture) local inertia forces at the micro-crack tip control the response. The rate dependent constitutive law for concrete can account for the rate dependent response. However, for higher loading rates the influence of loading rate on the response is mainly controlled by structural inertia forces. (iv) Because of the change of the failure mode with increase in loading rate, from bending to shear, in reinforced concrete beams loaded by higher loading rates, bending reinforcement would be ineffective. Instead, the shear reinforcement is required to prevent failure of the beam. (v) Velocity of the bending crack tip increases almost linearly up to the peak value of approximately 0.55 Rayleigh wave speed. The maximal velocity only slightly increases with the increase of the impact velocity. (vi) The analysis shows the branching of bending (mode-I) crack that occurs at the crack speed of approximately 0.55 Rayleigh wave speed.

REFERENCES Banthia, N.P., Mindess, S. and Bentur, A. 1987. Impact behaviour of concrete beams. Materials and Structures, 20: 293–302. Bažant, Z.P., Adley, M.D., Carol, I., Jirasek, M., Akers, S.A., Rohani, B., Cargile, J.D. and Caner, F.C. 2000a. LargeStrain Generalization of Microplane Model for Concrete

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and Application. Journal of Engineering Mechanics, ASCE, 126(9): 971–980. Bažant, Z.P., Caner, F.C., Adley, M.D. and Akers, S.A. 2000b. Fracturing rate effect and creep in microplane model for dynamics. Journal of Engineering Mechanics, ASCE, 126(9): 962–970. Bažant, Z.P. and Gettu, R. 1992. Rate effect and load relaxation in static fracture of concrete. ACI Material Journal, 89: 456–468. Bažant, Z.P. and Oh, B.H. 1983. Crack band theory for fracture of concrete. RILEM, 93(16): 155–177. Bažant, Z.P. and Prat, P.C. 1988. Microplane model for brittle-plastic material—parts I and II. Journal of Engineering. Mechanics, ASCE, (114): 1672–1702. Belytschko, T., Liu, W.K. & Moran, B. 2001. Nonlinear Finite Elements for Continua and Structures. New York: John Wiley & Sons Ltd. Bentur, A., Mindess, S. and Banthia, N. 1987. The behaviour of concrete under impact loading: Experimental procedures and method of analysis, Materials and Structures, (19): 113. Brandon, D.G. 1987. Dynamic loading and fracture, In T.Z. Blazynski (Ed.) Materials at high strain rates, Elsevier, London, 187–218. Carpenter, N.J., Taylor, J.R. and Katona, M.G. 1991. Lagrange constraints for transient finite element surface contact. International Journal for numerical methods in engineering, (32): 103–128. Comite Euro-International Du Beton (CEB), 1988. Concrete Structures under Impact and Impulsive Loading. Syntesis Report, Bulletin D’Information No 187. Curbach, M. 1987. Festigkeitssteigerung von Beton bei hohen Belastungs-geschwindigkeiten. PhD. Thesis, Karlsruhe University, Germany. Dilger, W.H., Koch, R. and Kowalczyk, R. 1978. Ductility of plained and confined concrete under different strain rates. American Concrete Institute, Special publication, Detroit, Michigen, USA. Freund, L.B. 1972a. Crack Propagation in an Elastic Solid Subjected to General Loading-I. Constant Rate of Extension. J. Mech. Phys. Solids, (20): 129–140. Freund, L.B. 1972b. Crack Propagation in an Elastic Solid Subjected to General Loading-II. Non-uniform Rate of Extension. J. Mech. Phys. Solids, (20): 141–152. Hutter, M. and Fuhrmann, A. 2007. Optimized continuous collision detection for deformable triangle meshes, research grant KF0157401SS5 in the PRO INNO II program, Germany.

Krausz, A.S. and Krausz, K. 1988. Fracture kinetics of crack growth. Kluwer, Dordrecht, The Netherlands. Ožbolt, J. and Bažant, Z.P. 1996. Numerical smeared fracture analysis: nonlocal microcrack interaction approach. International Journal for numerical methods in engineering, (39): 635–661. Ožbolt, J., Li, Y. and Kožar, I. 2001. Microplane model for concrete with relaxed kinematic constraint. International Journal of Solid and Structures, (38): 2683–2711. Ožbolt, J., Rah, K.K. and Mestroviæ, D. 2006. Influence of loading rate on concrete cone failure. International Journal of Fracture, (139):239–252. Ožbolt, J. and Reinhardt, H.W. 2001. Three-dimensional finite element model for creep-cracking interaction of concrete. Proceedings of the sixth international conference CONCREEP-6, Ed. By Ulm, Bažant & Wittmann, 221–228. Ožbolt, J. and Reinhardt, H.W. 2005a. Dehnungsgeschwindig-keitsabhängiger Bruch eines Kragträgers aus Beton. Bauingenieur, Springer VDI, (80): 283–290. Ožbolt, J. and Reinhardt, H.W. 2005b. Rate dependent fracture of notched plain concrete beams. Proceedings of the 7th international conference CONCREEP-7, Ed. By Pijaudier-Cabot, Gerard & Acker, 57–62. Reinhardt, H.W. 1982. Concrete under impact loading, Tensile strength and bond. Heron, 27(3). Saatci, S. and Vecchio, J.V. 2009. Effect of Shear Mechanisms on Impact Behavior of Reinforced Concrete Beams. ACI Structural Journal, 106(1): 78–86. Sukontasukkul, P. and Mindess, S. 2003. The shear fracture of concrete under impact loading using end confined beams. Materials and Structures, (36): 372–378. Travaš, V., Ožbolt, J. and Kožar, I. 2009. Failure of plain concrete beam at impact load: 3D finite element analysis. Int. J. Fracture, (160): 31–41. Weerheijm, J. 1992. Concrete under impact tensile loading and lateral compression. Dissertation, TU Delft, The Netherlands. Wriggers, P. 2002. Computational Contact Mechanics, John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Investigation into the form of the load-induced thermal strain model C.J. Robson, C.T. Davie & P.D. Gosling Newcastle University, Newcastle Upon Tyne, UK

ABSTRACT: Investigation into the temperature function and the normalised load level function, the product of which is postulated to form the equation which governs the load-induced thermal strain of cementitious materials. Using data from the literature, statistical methods and assumed forms of the equations involved, the normalised load level function is shown to be well modelled simply by the stress:cold compressive strength ratio and as a result, it is indicated that the temperature function cannot be universally plotted above a temperature of approximately 500◦ C. 1

INTRODUCTION

When a cementitious body is heated for the first time whilst under load, the largest strain component tends to be the load-induced thermal strain (LITS) for even relatively low levels of temperature increase and loading. The underlying mechanism which results in the occurrence of LITS is an ongoing research topic but it is known that the causes reside within the hardened cement paste phase itself. The role of the aggregate inclusions is simply one of restraint. It has also been widely accepted since the discovery of LITS in 1966 (Hansen and Eriksson 1966) that LITS is present only when the previous maximum temperature is exceeded, and that, consistent with this, LITS does not occur when the temperature is decreased (though, some studie have previously shown that creep increases in cooling which suggests that there may be a LITS-like effect at work, i.e. (Wallo, Yuan et al. 1965; Fahmi, Polivka et al. 1972) as referenced in (Bažant, Cusatis et al. 2004)). It is known that the excess deformation resulting from the coupling between loading and heating is far in excess of the additional strain which can be expected from the sum of the high temperature basic creep and the drying creep that is observed when concrete is simultaneously loaded and allowed to dry, though these two physical mechanisms surely contribute to LITS. LITS can be defined as (Khoury, Grainger et al. 1985) εLITS (T , σ ) = εtot (T , σ ) − εts (T ) − εel (T0 , σ )

(1)

where εLITS (T , σ ) is the LITS when a cementitious body is heated to temperature T whilst under load σ , εtot (T , σ ) is the total strain when the body is heated and loaded in the same conditions, εts (T ) is the thermal strain when the body is heated to the same temperature under no applied load and εel (T 0, σ ) is the elastic

strain produced by an identical load on the body with no change in temperature. LITS is thought to consist of several separate deformation mechanisms, transitional thermal creep, drying creep, temperature-dependent basic creep and changes to the elastic strain. Of these components, TTC is dominant (Khoury 2006) and is also the least understood. Having said that, decomposing the LITS into all separate components is currently an inexact process likely to introduce errors unnecessarily, therefore LITS is generally modelled as a single component (e.g. (Khoury, Grainger et al. 1985; Thelandersson 1987; Terro 1998; Pearce, Davie et al. 2003)). One of the most influential experimental studies to have guided attempts to model LITS was performed in 1985 at Imperial College in London (Khoury, Grainger et al. 1985). Results from this investigation suggested that the LITS behaviour of concrete is governed by a ‘master’ curve which is common to all cementitious materials of similar curing regimes and preheating conditions. Heating rate was shown to play only a minor role in defining the magnitude of LITS and the age was shown to be relatively unimportant, particularly for the temperature range below about 450◦ C. The study suggested that a linear relationship between LITS and the stress: cold compressive strength ratio1 was an accurate approximation for applied compressive loads between about 10% and 60% of the compressive strength at the initial temperature. These results and others have influenced researchers attempting to produce realistic models for LITS. Some researchers have chosen to represent LITS as the product of a temperature function and the normalized load level (Thelandersson 1987; Terro 1998; Nielsen, Pearce et al. 2004). Other models for LITS

1 Referred to in the remainder as ‘normalised load level’.

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have been proposed which are explicitly based on physical models (Sabeur and Meftah 2008), whilst others prefer to model LITS in the framework of creep models as an increase in magnitude of creep due to the changing temperature (Schneider 1988; Bažant, Cusatis et al. 2004).

2

AIMS & OBJECTIVES

The aims of this investigation are to investigate the universality of the form of the LITS temperature function and, by postulating that Equation 2 is the general form of the expression for LITS, εLITS (T , σ ) = f (T )g(σ/fc )

(2)

to find the most appropriate forms of the consitituent temperature function f and normalised load level function g. To achieve these aims, a material point-style model will be used to simulate experimental conditions to produce total strain results that can be compared to reported total strain values. The performance of a chosen LITS model compared with the performance of of the same model optimised over several data files will provide evidence for the universality of LITS. The same model will be used to perform a similar investigation for three LITS models, each of which describes the temperature function of LITS in a different numerical method, to compare differing forms of the temperature function and the performance of these in comparison to experimental data. Another model shall be used to determine the most appropriate form of the normalised load level function in a model for LITS. To do this, five possible forms of the function shall be assumed, each of which has one free parameter. Using strain ratios at corresponding data points, the free parameter can be fixed for every paired data point. Comparison of the values found via this method should indicate which model is the most accurate.

3 3.1

METHODOLOGY Details of numerical models used

The material point model used here produces total strain results for any given heating and loading conditions using the strain components of Equation 1. Therefore, an initial elastic strain component, a thermal strain component and a LITS component must be calculated. The initial elastic strain can be calculated using the reported stiffness parameters. Data for the thermal strain behaviour in the same heating conditions are available in the literature. Therefore, using interpolation methods, the free thermal strain can be

approximated at any temperature point. This method was assumed to be more accurate than fitting a polynomial to the data. Models from the literature were used to produce the LITS component in the model (see section 3.2 for details). No previous information was assumed with respect to the parameters of the LITS models, other than taking note of prior values that have been used. This allowed a set of parameter maxima to be chosen that were far in excess of typical values. It was decided to choose parameters which resulted in no LITS as the parameter minima. Once chosen, the parameters were systematically varied between these extremes using a large number of permutations. The program the compared the total strain vs temperature data produced with the experimental data file in order to calculate an error measure which could be recorded in a data file adjacent to the particular permutation of parameters used. The error was defined as the mean of the square of the strain differences between the simulated results and the experimental data. To remove bias towards temperature intervals which were more densely populated with data points, a weighting method was used in the error calculations. Potential error due to comparison of data points with misaligned temperature values was not present as the models used outputs results at the same temperatures as the experimentat data files. Once the data files containing the error measures were produced, the permutation of model parameters which produced the strain history most closely matching the reported data could be determined. Using multiple files, the model parameters could be optimised for more than one set of conditions. An error normalisation procedure was used in order to prevent bias towards files containing error minima of the largest magnitude. The material point program was tested using numerically exact data to show how accurate the model is in terms of the forms of error inherent in the methodology (e.g. that due to the linear interpolation method). Thermal strain data files were produced by fitting fifth order polynomials to the original thermal strain files and replacing the strain values at each temperature point. Total strain data files were produced using the now fully characterised thermal strain, the loading and stiffness parameters from the reported data sets and Nielsen’s LITS model (see section 3.2.1 for details) with known values of the parameters. The program was then applied to these data files. Results demonstrated that the model predicted that the values of optimal values of the LITS parameters were consistently very close to the known parameters. When the analysis was performed over all of the data sets, the accuracy increased and the parameters predicted were exactly correct. Another model, to be referred to as the SRAP (strain ratio analysis program), was used to directly investigate the nature of the relationship between LITS and

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normalised load level. The total strain data of the reported experimental data sets (see section 3.3 for details) were converted into LITS data by removing the thermal strain (calculated by linear interpolation of the reported thermal strain data) and the initial elastic strain. Assuming that LITS is the product of a temperature function f and a normalised load level function g, as in Equation 2, the ratios of the LITS values at the same temperatures are also the ratios of the normalised load level functions. Therefore, analysis of the experimental data can provide evidence as to which form of g is the most appropriate. To do this, the form of the function g must be assumed. Five different forms shall be investigated in this manner, g(σ/fc ) = (σ/fc ) + k1 ,

(3.1)

g(σ/fc ) = −(σ/fc ) + k2 ,

(3.2)

g(σ/fc ) = −(σ/fc + k3 )2

(3.3)

g(σ/fc ) = −|σ/fc | , and

(3.4)

g(σ/fc ) = −(σ/fc ) + k5 (σ/fc ),

(3.5)

2

k4 2

where fc is the initial compressive strength. By equating the values of the LITS and the ratios of the normalised load level function, a value of the free parameter k can be calculated for every pair of LITS data points which correspond to different load levels but identical temperatures, i.e. k1 =

(σ/fc )1 − x(σ/fc )2 , (x − 1)

(4.1)

(σ/fc )21 + x(σ/fc )22 , (x − 1) √ (σ/fc )1 − x(σ/fc )2 , k3 = √ ( x − 1) √ (σ/fc )1 + x(σ/fc )2 , k3 = √ (− x − 1)

k2 = −

(4.2) (4.3a) (4.3b)

k4 =

ln(x) , and [ln(σ/fc )1 − ln(σ/fc )2 ]

(4.4)

k5 =

−(σ/fc )21 + x(σ/fc )21 , −(σ/fc )1 + x(σ/fc )2

(4.5)

where X =

εLITS (T , σ1 ) . εLITS (T , σ2 )

(4.6)

If the experimental data is produced by a product of two functions of the form of Equation 2 and the load function is given by one of Equations 3.1 to 3.5, then it would be expected that the value of ki calculated using the corresponding equation from Equations 4.1 to 4.5 would be consistently calculated to be a very

similar magnitude to that which best describes the LITS behaviour. Therefore, the model calculates the standard deviation of the calculated values of the free parameter ki for each model for every data point available in order to evaluate the most consistent model. To ensure that the analysis is unbiased towards certain models of low ki values, the standard deviation will be normalised to the magnitude of the calculated mean value of the free parameter for each model. Note that there are two solutions for the free parameter of the third form of the function, Equation 3.3 as a result of the square root operator. Both of these solutions shall be found, Equations 4.3a and 4.3b, and the method which produces the best results, as previously defined, shall be taken as representative of this form of the loading function. Two of the experimental data sets (see section 4.3) were obtained under identical normalised loading conditions. Hence, a model which conforms to the form of the LITS equation of Equation 2 would predict that the LITS curves of each should be identical. Therefore, the ratio of all of the respective LITS data points should clearly be 1.0. Hence, a value for the intrinsic error of the method can be found by statistical analysis of the actual ratio. If it is assumed that the error is entirely due to innaccuracy in the value of the free parameter of the normalised load level function and not due to inaccuracy in the loading conditions, temperature measurement, strain measurement (and interpolation) or the temperature function, an upper bound measure of the possible error in the free parameter is obtained. Using the error measure term, a minimum and a maximum of the free parameter for each model can be calculated for each of the LITS ratios. This means that a set of error-adjusted values of the parameter can be found and the standard deviation can be found in the usual way. Clearly, there are more forms that g could take than simply the Equations 3.1–3.5, particularly if more than one free parameter is allowed. These forms can be possibly investigated in the future if necessary. The SRAP methodology was tested in a similar manner to that in which the material point model was tested. As with the material point model test, fifth order polynomials were used to produce thermal strain data files, the known stiffness parameters were used to produce the initial elastic strain component and the LITS behaviour was produced using an equation with chosen parameters. The LITS equation was based on Equation 2 with a quadratic temperature function, chosen for simplicity. In order that each normalised load level model was directly tested, a set of data files was produced for each of the five assumed forms of the load level function of Equations 3.1–3.5. The test was very useful as it demonstrated several issues with the methodology which were then able to be addressed. It became clear that the data in the low temperature range could not be included in the analysis

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as the strain ratios in this region tend to be much larger than in the rest of the temperature range due to numerical instability caused by the low values of the strain. The resultant predictions of the free parameter of the load function were wildly different from those of the remaining temperature region. Even comparisons of the G45 and the T45 data used to define the intrinsic error calculated relatively large error in this lower temperature region. Therefore, the temperature range used in comparisons of the LITS ratios was narrowed such that the minimum temperature allowable was set to at least 100◦ C which was roughly the temperature at which the strain ratios became more consistent. This replaced the original setting which was based on the minimum temperatures in the data files. When this issue was resolved, the intrinsic error was reduced from about 10% to about 1%. Another issue highlighted by the tests is that comparisons between some of the data sets should be prevented as they tend to lead to erroneous values of the free parameter k (i.e. for comparisons of the results of the P44 test and the G45 or T45 test and of the T675 test and the G60 test). The predicted values of k can be very much larger than for other comparisons and, as a result, the mean value can be vastly different from the correct value. This is, again, a numerical issue which is due to the value of the LITS ratio being so close to 1.0. Therefore, results produced from the ratios of the aforementioned tests shall not be used in the process of finding the values of the free parameter k for each model. When both of these issues were resolved, each calculated mean value of k was far closer to the correct value than they had been. One final issue emerged from the testing of the model. This was that the relative standard deviation was not an indicator of the superior form of the normalised load level function. In fact, in one case, the model which was used to predict the results was, in fact, the model with the largest associated measure of relative standard deviation. This suggests that, whilst the methodology is valid, the principle that consistency of the calculated value of the free parameter is a reliable indicator of the most realistic model is highly questionable. Therefore, to quantify the quality of the data fit, a temperature function will be produced by inversion of Equation 2 to give the temperature function of the LITS equation. The function shall be calculated between a minimum temperature and a maximum temperature that is within the range of all of the data sets such that all of the data sets can be used to produce a mean temperature function. With reference to this mean, a standard deviation can be calculated which is taken as the error measure of the temperature function. The standard deviation can be made into a relative value using the magnitude of the mean temperature function for each temperature at which the quantity is calculated.

Some of the proposed models of Equations 5.1–5.5 can predict a non-zero value of LITS when no stress is applied, which seems objectionable, given that the definition of LITS used in this investigation is given by Equation 1. However, it should be noted that, according to experimental results (Khoury, Grainger et al. 1985), the relationship between normalised load level and LITS is approximately linear in the normalised load level range of 10% to 60% with the relationship tending towards a decreasing function of normalised load level for the lower stress range and an increasing function of normalised load level in the higher stress range. Hence, to fit experimental observations, the model for the mid-stress range would not be expected to pass through the origin when extrapolated to the state of zero applied stress. This possibility applies even if a loading function is approximated over the whole stress range. Thus, despite this conceptual problem, a model which is accurate in the mid-stress range may predict that LITS is present even when there is no agent present which can produce LITS. To avoid this problem, should a normalised load level function be selected with a value of k which produces non-zero LITS in situations of no applied stress, the range of applicability of the LITS model should explicitly remain in the normalised load level in which the investigation is carried out. 3.2

Details of LITS models used

As the material point model has no means of calculating the evolving hygral state of the cementitious material or explicit structural effects which may occur over the course of a test, only LITS models which do not explicitly require hygral parameters can be investigated with this method in its present form. Three models from the research field which fulfill this condition shall be investigated here using the material point-type model. The models shall be referred to as Nielsen’s model (Nielsen, Pearce et al. 2004), Thelandersson’s model (Thelandersson 1987) and Terro’s model (Terro 1998). A brief description of each of these models is provided below. Each of the models conforms to the general form of the LITS model of Equation 2 with the normalised load level given by Equation 3.1 with k1 set to zero. 3.2.1 Nielsen’s model The temperature function of the LITS model of Nielsen and colleagues is biparabolic with a transition temperature of 470◦ C. According to the model, LITS is given by εLITS (T , σ ) = (σ/fc )y(θ),

(5.1)

where, for 0 ≤ θ ≤ θ∗ = 4.5, y(θ) = Aθ 2 + Bθ,

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(5.2a)

and, for θ > θ ∗ ,

Table 1. Details of the parameters of the test data used in the investigation.

y(θ ) = c(θ − θ ∗ )2 + Aθ ∗ (2θ − θ ∗ ) + Bθ ∗ ,

(5.2b)

where A, B and C are the parameters of the model, and θ is a normalised measure of temperature given by (T − T0 ) θ= , 100◦ C def

(5.2c)

where T is the elevated temperature of interest and T0 is the initial temperature. 3.2.2 Thelandersson’s model Thelandersson’s LITS model is given by εLITS (T , σ ) = (σ/fc )βεts (T , 0),

3.2.3 Terro’s model The temperature function of Terro’s LITS model is a fourth order polynomial in temperature (fifth order if Thames river gravel is used as aggregate) which was originally fitted to the ‘master’ curve of Khoury for a normalised load level of 0.3. The form of the normalised load level function of Equation 3.1 is used to model LITS in different loading conditions. An additional factor was also initially included to account for materials of different aggregate contents but this is ignored here as this effect is assumed to be intrinsically accounted for in the value of the initial compressive strength, an assumption presumably inherent in the previous two models. With this in mind, according to Terro’s model, LITS is given by σ  ) (An T n ), fc

Normalised load level

Temperature range (◦ C)

Ratio of elastic modulus to compressive strength

G15 G30 G45 G60 T225 T45 T675 P44

−0.150 −0.300 −0.450 −0.600 −0.225 −0.450 −0.675 −0.440

20–700 20–700 20–500 20–450 20–800 20–600 20–500 20–250

455.7 455.7 455.7 455.7 540.0 540.0 540.0 558.4

(5.3)

where β is the only parameter of the model. The temperature function of the model is directly proportional to the unloaded thermal strain, which is generally expressible as a nonlinear function of temperature.

n=4

εLITS (T , σ ) = (k +

Test code

(5.4)

n=0

As a result of the experimental observation that LITS is only present when the temperature exceeds the previous maximum temperature, the curing regime of the concrete can be very important for attempts to model the LITS of concrete. For instance, the central regions of large concrete structures can develop temperatures as high as 80◦ C during hydration (Neville 1995). Therefore, a prediction of LITS in future heating could be absent in this low temperature region (assuming the subsequent heating does not allow sufficient time to pass that LITS recovery takes place). Therefore, experimental conditions which do not allow the hydration process to cause significant temperature increase are necessary for this investigation. Eight sets of experimental data, from the literature, were chosen to perform the analyses of this investigation [four were found in (Gawin, Pesavento et al. 2004), three were found in (Thelandersson 1987), and one found in (Petkovski and Crouch 2008)]. The experimental details are shown in Table 1.

4 4.1

where Ai and k are the parameters of the model. Terro’s model should be investigated by finding a value for the parameter k first of all using the SRAP methodology, then the coefficients can be found for the polynomial temperature function. 3.3 Details of experimental data sets used In order to conduct this investigation effectively, data for a broad range of load levels is required, as well as over a large temperature range. Experimental conditions which minimise gradients within the cementitious specimen would be advantageous as the material point-style model does not include structural effects, which should ideally, therefore, be kept to a minimum.

RESULTS & DISCUSSION Investigation into the form of the LITS temperature function

4.1.1 Results of using temperature function of Nielsen’s model The parameters found by the material point model to produce the total strain behaviour most closely matching each set of experimental results using Nielsen’s LITS model are shown in Table 2. The optimal parameters for each data set vary to a fairly high degree as is shown in the table. However, the actual LITS produced by each set of parameters is similar as is demonstrated in Fig. 1, which shows the LITS produced by applying each set of parameters to a representative stress/strength of 45%. Note that the optimal parameters produced for data sets G60 and P44 are not plotted beyond 470◦ C because no information

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When more than one data set is analysed at once, the best LITS model parameters that are found should result in the most accurate version of the LITS model for the data sets analysed. If there is a universal LITS function, then the parameters produced when multiple data sets are analysed like this should converge to the same or a very similar set of parameters. To investigate if there is any evidence in these data sets for a universal LITS function, multiple data sets were analysed simultaneously in order to find evidence for a convergence of parameters. The data in Table 3 suggests that there may be a universal LITS function. The optimised parameters barely change at all when one any one of the data sets is removed from the analysis which suggests the parameter set produces good results for all the data files.

Table 2. The LITS parameters found to reproduce the strain with the least error for each individual set of test conditions. Test code G15 G30 G45 G60 T225 T45 T675 P44

Parameter A (10−4 )

B (10−3 )

C (10−2 )

Error measure (10−6 )

10.0 9.4 1.7 4.1 7.5 2.5 1.0 7.3

0.0 0.0 2.8 1.4 0.3 2.1 2.8 0.0

0.9 0.7 7.7 * 0.2 1.6 4.4 *

0.605 1.490 0.113 0.224 0.116 0.139 0.149 0.004

*The highest temperature for the test used does not reach the region in which the parameter C is active.

Figure 1. Graph shows the LITS produced by the model parameters optimised for each different data set with an applied normalised load level of −0.45.

4.1.2 Results of using temperature function of Thelandersson’s model Thelandersson’s model was the most simple to fit to data as there was only one free parameter. Similarly to the results of section 4.1.1, the optimised parameter for one data set is not the same for any of the others. However, problems were found with this model. The model can match the experimental data fairly well up to a temperature of approximately 500◦ C but, after this point, the LITS curves appear to qualitatively change in a manner that the curves produced using the Thelandersson model are unable to capture. It was found that, initially, in order to minimise values of the error measure (i.e. the sum of the squares of the difference in strain between the predicted curves and the experimental curves), the model was producing values of the free parameter β which would cause the total strain curve to pass through a mid-point of the total strain curve. This would cause the error measure to be lower overall than it would if the parameter were chosen such that the region of the strain curve in the lower temperature region were well simulated.

is available regarding the value of the model parameter C in this region. Clearly, the results of the graph suggest that each of the optimised sets of parameters produce results which are in fairly close correlation for the lower region of the temperature range. For the higher range of temperatures, there is a relatively large degree of scatter which should be expected because the quantity of experimental data available in this range is much more sparse. In two cases (G45 and T675), there is only one data point in this range. Therefore, the model will favour a value of the parameter C which results in a curve which passes very close to this point, regardless of the subsequent projection of the curve. This can clearly be seen for the data sets G45 and T675 as the optimal value of the final parameter for both is substantially higher than those of the other data sets.

Table 3. The optimised LITS parameters obtained when seven data sets are analysed at once. Test code missed out G15 G30 G45 G60 T225 T45 T675 P44 None*

Parameter A (10−4 )

B (10−3 )

C (10−2 )

Normalised error

8.8 8.8 8.6 8.9 8.5 8.6 8.8 7.3 8.8

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.8 0.0

0.2 0.2 0.2 0.2 1.0 0.2 0.2 0.2 0.2

7.083 7.209 6.574 7.537 4.525 5.890 6.768 6.317 6.985

*‘‘None’’ means that all eight data sets were analysed at once.

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To avoid this unsatisfactory situation, the model had to be run only in the lower temperature region. This produced a better model in this region of the temperature field but the total strain in the large temperature region was very inaccurate. As can be seen as an example in Fig. 2, the Thelandersson model cannot produce the downturn in strain that is observed in practice. 4.1.3 Results of using temperature function of Terro’s model Details of the calculation of the optimal parameter k are presented in section 4.2. The best value of the parameter which was found was 0.0. Therefore, the temperature function of the LITS model of Equation 2 can be expressed in terms of the LITS data divided by the normalised load level. Collating all results from all of the data sets of section 3.3, a group of data is produced. Using the plotting software ‘gnuplot’, a fourth order polynomial was fitted to the group of data. The parameters, of the Terro model, are therefore shown in Table 4. In general, application of this model was reasonably successful. However, the error measures are consistently higher than those of the Nielsen model. Also, the fourth order temperature function appears to be possibly inadequate. The LITS function actually decreases in magnitude slightly between 150◦ C and 350◦ C. This behaviour is not observed in experimental data and may be a result of the temperature range over which the parameters were set (20–250◦ C).

Figure 2. Graph compares the total strain produced using the Thelandersson LITS model optimised to the data plotted on the same graph (data set is T225). Table 4.

k

The optimised parameters of the Terro equation.

A0 (10−3 )

A1 (10−4 )

A2 (10−6 )

A3 (10−9 )

A4 (10−12 )

0.0 −6.25157 1.99317 −1.23681 2.85134 −1.85289

4.1.4 Discussion of the temperature function investigation The results of this investigation demonstrated that there is a qualitative change in the behaviour of LITS at high temperatures. The one parameter model of Thelandersson does not have the versatility to capture this high temperature behaviour. Fig. 2 is a typical result from the investigation which shows that, while the model seems ideally suited to capturing the behaviour of the LITS in the lower temperature range and seems to have a plausible physical footing, the strain behaviour in the higher temperature conditions implies that the model must be developed with perhaps another parameter included in the model in order that it can simulate the strain behaviour over a wider range of temperature. The basis for Thelandersson’s model seems to be that LITS is a coupling between the temperature increase and the loading. Hence, it is a natural progression to attempt to model the LITS behaviour as being proportional to the product of the strain produced by loading the cementitious body without heating and the strain produced by the heating of the cementitious body without loading. If this is the cause of the LITS in a direct way then perhaps the improvements to be made to this model are in analysing the conditions that are perhaps incorrectly taken as the same in the low temperature conditions as in the high temperature conditions, e.g. the compressive strength of the material is perhaps reduced significantly at high temperatures. This would result in an increased normalised loading level in the higher temperature region and perhaps could partially explain the increasing difference between the model predictions and the observed strain data in the high temperature regions. Nielsen’s model already has the ability to track the increased LITS at high temperatures with an extra parameter. This is a welcome addition to the model as it allows for steep gradients in the higher temperature regions without the requirement of high powers of temperature as the Terro model has. However, just as with the Thelandersson model, more research is required in order to ensure that the model is able to find a reliable way to model the high temperature LITS behaviour accurately without the requirement of changing the parameter C to fit the data available.

4.2

Investigation into the form of the LITS normalised load level function

4.2.1 Results of the strain ratio analysis program In order to assess the intrinsic accuracy of the method, the LITS ratios of the G45 and T45 tests were compared between a temperature minimum and maximum (100◦ C and the lower of the two temperature maxima in the results files) at a set number of equally spaced temperature values. The largest strain in magnitude

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was assigned as the numerator term and the other was assigned as the denominator term. This meant that the ratio was taken such that it was always larger than 1.0; this was in order to reflect the subsequent investigation of the LITS ratios whereby the larger strain and the smaller strain are always taken to be the numerator term and the denominator term of the strain ratio respectively. The intrinsic error in the strain ratios was then taken to be the mean of the difference between this value and 1.0. The result is shown in Fig. 3 along with the mean error, which was calculated to be about 0.21. The subsequent program determined the ratios of the LITS values found from the reported experimental data. The previously calculated mean error was then applied to the ratios such that a maximum and minimum possible value of the parameter were found for each model according to the maximum and minimum possible values of the LITS ratio. Table 5 shows the calculated mean values of the free parameter k for each normalised load level model. Analysis of the results show that, for each model, values calculated for k for a small number of the pairs

of tests are anomalously large. These values are so different from the remainder of the calculated values of k that the mean is significantly altered. As shown in table 5, the mean values of k after the removal of the anomalous results leads for all models to far more moderate quantities when compared with the load levels of use in the functions. This method of selecting which results to use is somewhat arbitrary and far less objective than is desirable. Nonetheless, the mean values obtained after removing the few values of k considered to be anomalies are noted along with another statistical quantity which can be objectively calculated from the data sets of k values without being significantly skewed by the presence of the anomalous results, the median. The median values of k found for each form of the normalised load level function are shown in Table 6. By inverting the LITS equation and calculating the temperature function which results from using these forms of the normalised load level functions, a standardised measure of error can be calculated for each of the values of k applied for the respective models as shown in Table 7. According to the results, the third load level model produces the most similar temperature functions over the experimental data are used in Table 6. Median values of the free parameter k found for each form of the normalised load level function.

Figure 3. Graph showing the erratic nature of the LITS ratio of the two sets of test data produced using the same normalised load level. Table 5. Mean values of free parameter k calculated both with and without the anomalous results for each form of the normalised load level function.

Model number

Initial mean value of k

Mean value of k after removal of anomalous results*

1 2 3 4 5

1.007 0.480 1.644 1.135 1.029

0.072(1) −0.074(1) −0.230(1) 0.888(2) −0.076(1)

*Figure in parentheses represents number of anomalous results removed to produce new mean value of k.

Model number

Median value of k

1 2 3 4 5

0.051 −0.077 −0.245 1.133 0.039

Table 7. Error measure found for the mean and median values of k by using relative standard deviation methodology. Model number

k

Relative standard deviation*

1 1 2 2 3 3 4 4 5 5

0.072 0.051 0.123 0.121 0.076 0.071 0.077 0.092 1.252 0.434

37.639 34.465 32.771 32.740 32.598 32.503 35.341 33.127 89.669 56.966

*Relative standard deviation calculated at 100 evenly spaced temperature intervals. The values quoted are the sum of these values over the entire temperature range. For the mean value, simply divide by 101.

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Figure 4. The temperature function produced by inversion of the LITS data using Equation 3.3 and k = 0.071.

this investigation and a lower level of error is achieved by using the median value of k rather than the mean value. The resultant temperature function is shown in Fig. 4. The relative standard deviation produced for the first form of the load function with k set equal to zero is 33.667. This is the form of the function that is assumed by several models (e.g. Nielsen’s model and Thelandersson’s model). When compared with values of the relative standard deviation calculated for various forms of the load function in Table 7, it can be seen that it is among the best observed. The scatter graph of temperature functions produced by each model also looks very similar in terms of variance from a mean of the functions produced. In actual fact, the normalised load level functions produce very similar results with most of the divergence occurring at the higher load levels. 4.2.2 Discussion of the normalised load level investigation The relatively modest increase in accuracy (as defined in the methodology) gained by using the normalised load level function Equation 3.3 in the general LITS equation given by Equation 2 suggests that the assumption of a simple form of the normalised load level function, such as that of Equation 3.1 with k set equal to zero, is a relatively good one in the mid-stress range considered here. The investigation provides no strong evidence statistically that the other simple forms which the function could take provide any kind of significant improvement. Fig. 4 is evidence that a quite clear temperature function is produced by the methods employed in this investigation but only in the temperature range below approximately 500◦ C. Above this temperature, as already noted earlier when discussing the Thelandersson model, the LITS behaviour seems to change

qualitatively in these high temperature conditions. As a result, it is fully expected that no normalised load level function could significantly improve on the temperature function produced as the LITS of the high temperature region is perhaps governed by a different equation than Equation 2 or possibly by a modification of this equation. Fig. 3 shows the relative error that is produced using the methodology of this investigation from two data sets which should have identical LITS behaviour according to Equation 2. The clearly erratic behaviour that is identified here and, at times, in the subsequent investigation suggests that this method, though plausible in theory, may be too heavily reliant on the accuracy of the data to produce reliable results. However, the methodology has produced some results that can guide future research efforts. The results produced consistently shows that the temperature function shows far more scatter in the temperature region greater than about 500◦ C than in the lower temperature region. This is in agreement with the previous observations (Khoury, Grainger et al. 1985) that the common behaviour of cementitious materials is present only in temperatures below about 450◦ C. Above this temperature, the LITS behaviour seems very much dependent on the specific concrete used. In the higher temperature region of Fig. 4, the divergent data can clearly be seen to be separated into two distinct curves, the upper set of data points belongs to the experimental data sets G15 and G30 while the lower set of data points belong to the experimental data set T225. This clear distinction in LITS behaviour between different types of concrete may be due to dissimilar damage-type behaviour occurring in the two respective concretes. Also possible is that the reaction of the aggregate material to the increasing load placed on it over the course of a LITS test due to the stress relaxation in the cement paste phase is responsible for a proportion of the deformation and, hence, a disparity between the LITS observed for concretes made with different aggregate materials may tend to grow with time as a result of this. However, this hypothesis would not explain why the difference in LITS only seems to become noticeable in the temperature range above 500◦ C rather than a steady increase with time and it is questionable whether the increased load on most typical aggregate materials would cause such a difference in observed strain. The question of the role of the stress state of the cement paste phase however is of high interest as regards LITS analysis. As the cement paste phase is known to be the phase which undergoes LITS, the role of the variation of the average stress in the cement paste phase is an area to be explored in the future. Further to this, future work in this area shall investigate the relationship between damage evolution and the LITS behaviour in the higher temperature region. A greater understanding on the effects of damage should improve the modelling potential of all of the

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LITS models used in this investigation so far. Also, the current investigation will be continued such that further LITS models are examined, such as those of Bažant, Schneider, and Sabeur, certainly with changes in the methodology employed in this investigation, such as finite element modelling to allow structural effects and explicit hygral phenomena. 5

CONCLUSIONS

This investigation has provided evidence that the concept of a universal temperature function for LITS is limited to the temperature region below approximately 500◦ C, at which a qualitative change occurs in the behaviour of LITS. The investigation has determined that the three LITS models used in this paper are illequipped to model LITS in these high temperature conditions. It has been shown that whilst minor improvements can be made to the normalised load level function, use of the normalised load level in itself as the loading function in Equation 2 is probably sufficient. The next area of reseach to be carried out in the field of LITS modelling is in reliably modelling the high temperature strain behaviour without necessity of changing parameters for different tests. ACKNOWLEDGEMENTS This work was carried out with the financial support of Halcrow Group Ltd. This contribution is gratefully acknowledged. REFERENCES Bažant, Z.P., G. Cusatis, et al. (2004). ‘‘Temperature effect on concrete creep modeled using microsprestress-solidification theory.’’ Journal of Engineering Mechanics 130(6): 691–699. Fahmi, H.M., M. Polivka, et al. (1972). ‘‘Effect of sustained and cyclic temperature on creep of concrete.’’ Cement and Concrete Research 2: 591–606.

Gawin, D., F. Pesavento, et al. (2004). ‘‘Modelling deformations of high-strength concrete at elevated temperatures.’’ Materials and Structures 37: 218–236. Hansen, T.C. and L. Eriksson (1966). ‘‘Temperature change effect on behavior of cement paste, mortar, and concrete under load.’’ Journal of the American Concrete Institute 63(4): 489–504. Khoury, G.A. (2006). ‘‘Strain of heated concrete during two thermal cycles. Part 1: strain over two cycles, during first heating and at subsequent constant temperature.’’ Magazine of Concrete Research 58(6): 367–385. Khoury, G.A., B.N. Grainger, et al. (1985). ‘‘Strain of concrete during first heating to 600◦ C under load.’’ Magazine of Concrete Research 37(133): 195–215. Khoury, G.A., B.N. Grainger, et al. (1985). ‘‘Transient thermal strain of concrete: literature review, conditions within specimen and behaviour of individual constituents.’’ Magazine of Concrete Research 37(132): 131–144. Neville, A.M. (1995). Properties of Concrete. Harlow, Essex, Pearson Education Limited. Nielsen, C.V., C.J. Pearce, et al. (2004). ‘‘Improved phenomenological modelling of transient thermal strains for concrete at high temperatures.’’ Computers and Concrete 1(2): 189–204. Pearce, C.J., C.T. Davie, et al. (2003). A transient thermal creep model for the hygral-thermal-mechanical analysis of concrete VII International Conference on Computational Plasticity, Barcelona, COMPLAS VII. Petkovski, M. and R.S. Crouch (2008). ‘‘Strains under transient hygro-thermal states in concrete loaded in multiaxial compression and heated to 250◦ C.’’ Cement and Concrete Research 38. Sabeur, H. and F. Meftah (2008). ‘‘Dehydration creep of concrete at high temperatures.’’ Materials and Structures 41(1): 17–30. Schneider, U. (1988). ‘‘Concrete at high temperatures— A general review.’’ Fire safety journal 13(1): 55–68. Terro, M.J. (1998). ‘‘Numerical modeling of the behaviour of concrete structures in fire.’’ ACI Structural Journal 95(2): 183–193. Thelandersson, S. (1987). ‘‘Modeling of combined thermal andmechanical action in concrete.’’ Journal of Engineering Mechanics 113(6): 893–906. Wallo, E.M., R.L. Yuan, et al. (1965). Sixth progress report: Prediction of creep in structural concrete from short time tests. T&AM. Urbana, Illinois, University of Illinois.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Development of service life model CHLODIF++ I. Stipanovi´c Oslakovi´c Institut IGH d.d. (Civil Engineering Institute of Croatia), Zagreb, Croatia

D. Bjegovi´c Institut IGH d.d. (Civil Engineering Institute of Croatia), Zagreb, Croatia Faculty of Civil Engineering, University of Zagreb, Croatia

D. Mikuli´c Faculty of Civil Engineering, University of Zagreb, Croatia

V. Krsti´c The College of New Jersey, School of Engineering, USA

ABSTRACT: The concept of reinforced concrete durability started to develop only 25 years ago, when it became obvious that reinforced concrete could have serious durability problems, especially when exposed to the actions of aggressive environment. In marine environment chlorides are always the most decisive environmental action for the corrosion initiation. Most engineering models for the calculation of service life are based on the mechanism of chloride ingress into concrete due to diffusion process. More than 10 years ago started a development of the numerical model CHLODIF for the calculation of service life of reinforced concrete structures exposed to chloride action, based on Fick’s second law and deterministic approach. In this paper basis of the model are presented, together with the improvements based on the experimental researches, laboratory and field results and computational analysis. The aim of this research is to improve the existing numerical model to become fully operational for the structural and materials’ engineers. 1

INTRODUCTION

Durability of a structure is defined as its ability to preserve functionality, stability and aesthetic properties under expected environmental influences without larger maintenance and repair costs during designed service life. With the requirements of 100-year design life for major bridge structures and enormous rehabilitation and repair costs associated with inability to satisfy these requirements, durability of civil engineering structures is today one of the key problems of structures worldwide. Numerous examples of early deterioration of reinforced concrete structures in marine environment have shown that designing, constructing and maintaining durable concrete structure is rather difficult with today’s approach towards durability. Procedures for bearing capacity design are well defined, mathematically precise, based on the principles of mechanics, standardized and commonly used in practice. On the other hand, durability design procedures for reinforced concrete structures in aggressive environment are for a large extent still empirical. They are based on

deem-to-satisfy rules (e.g. minimum cover, maximum water/cement ratio) and the assumption that if these rules are met, the structure will achieve an acceptably long but unspecified service life. [HRN EN 206-1:2000] Analogously, the same as load-bearing capacity design, durability design can be performed, under condition that environmental loadings and performance of materials are known. Four different levels of durability design are shown in Figure 1 [fib Bul. 34, 2006]. Moving from right hand side, levels of durability design are upgrading, being more complex but at the same time more precise in predicting service life of concrete structures. The majority of service life models are based on the deterioration mechanisms due to corrosion, since the reinforcement corrosion is one of the most frequent deterioration processes in concrete structures. Corrosion of the reinforcement and subsequent damage of the concrete cover are highly complex phenomena and still not completely understood. However it is necessary to simplify many of the complex deterioration processes in the concrete and make assumptions where insufficient knowledge is available.

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(reinforced concrete slab) for a constant value of the initial concentration of chloride ions and timedependent diffusion coefficient is described by Fick’s second law. The period of initiation of corrosion process (t0 ) initiated by chlorides can be determined by diffusion analysis of chloride ions as the time necessary for the concentration of chloride ions to reach the critical value Ccr on the level of reinforcement. Chloride threshold value is usually taken from the actual regulations [HRN EN 206-1:2000], but it can be also experimentally determined and defined in the model. For a continuous diffusion process of chloride ions in a reinforced concrete structure for a time-varying diffusion coefficient and chloride surface concentration, according to [Krsti´c 1994] the process is described by following experimental equation:

DURABILITY DESIGN

Full probabilistic design probabilistic models - resistance - loads - geometry

limit states

Figure 1.

2 2.1

Deemed to satisfy design

Avoidance of deterioration

design values - characteristic values - partial safety factors - combination factors

exposure classes

exposure classes

design equations

design provisions

design provisions

Partial factor design

Different levels of durability design [2].

NUMERICAL MODEL CHLODIF++ Performance-based approach in service life design

Performance-based approach to the durability design of reinforced concrete structures means that it is based on durability indicators like materials parameters measured on built-in materials and real environmental conditions. Concrete permeability is a major factor that determines durability of concrete structures. Aggressive fluids that penetrate through concrete and cause severe degradation can penetrate by four basic transport mechanisms: capillary absorption, permeability, diffusion and migration. Chloride diffusivity is found as reliable indicator for corrosion process in the reinforced concrete induced by chlorides. This parameter is quantifiable by laboratory tests in a reproducible manner and with clearly defined test procedures, and therefore it is in most cases basis for mathematical service life models for chloride ingress into concrete. [Chlortest -1, 2005] 2.2

for

  x C(x, t) = [Co + k (t − 1)] 1 − erf √ 2 τ     2 2 x x x − x 1 − erf √ −√ +k 1+ e 4τ 1 2τ 1 2 τ1 πτ1 (2) until reaching the maximum surface concentration C0 = Cmax when the following solution is valid:   x , C(x, t) = C0 1 − erf √ 2 τ

Development of CHLODIF

C0 = Cmax

(3)

where:

More than 10 years ago at the Faculty of Civil Engineering University of Zagreb started a development of the numerical model CHLODIF for the calculation of service life of reinforced concrete structures exposed to chloride action, based on Fick’s second law and deterministic approach. Krsti´c, et al. [Krsti´c 1994, Bjegovi´c et al., 1995] proposed the criterion of corrosion limit state, according to which is necessary to prove that the service life calculation tc is higher or at least equal to the designed tp : to + t1 = tc ≥ tp

0 ≤ C0 < Cmax

• C0 —initial concentration of chloride ions, • Cmax —maximum surface concentration of chloride ions, • τ —the substitution by which variation of DCl with time is taken into account, and

(1)



t

dτ = D(t)dt thus τ =

D(s)ds;

τ1 = τ (t = 1)

0

(4) • k—coefficient of linear increase of initial concentration.

where: • to —initiation period of reinforcement corrosion in concrete • t1 —propagation period of reinforcement corrosion in concrete. 2.3

Predicting the initiation period

The solution of the diffusion problem in a semiinfinite, anisotropic and non-homogeneous medium

2.4

Predicting the propagation period

The period of propagation can be computed using the algorithm for evaluation of the corrosion rate given by [Andrade et al. 1989], which is in this model modified by presentation of coefficient of corrosion current increase and safety factors [Krstic 1994]. The introduced coefficient of corrosion current density is taking into account the influence of carbonation, sulphates and concrete cracking under basic loads.

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There are many other factors that do not directly affect the corrosion process, but contribute to an increased failure probability of the bearing capacity of the design cross-section where the corrosion process is in progress. These factors cannot be quantified, but they can be included in a factor of safety which gives the following modified equation: φ(t) = φi − γ (0.023 × p × icorr × t)

(5)

internal influencing factors related to materials parameters such as cement type, mineral additions, duration of curing etc., and second group of parameters related to external influencing factors such as age of concrete, temperature, humidity, wind influence etc. [Stipanovic Oslakovic 2009]. Following simplified equation represents this method of assuming actual diffusion coefficient: DCl = D0 × fint (cem, SG, FA, cur, crack)

where: • φ(t)—the reinforcement diameter at time t (mm), • φi —the initial reinforcement diameter at time t = 0 (mm) • 0.023—conversion coefficient (μA/cm2 ) in (mm/year), • icorr —corrosion current density (μA/cm2 ); • t—time passed since the corrosion process initiated (years) • p—correction coefficient of corrosion current density, • γ —corrosion limit state safety coefficient, calculated using fuzzy set logic [Bjegovic et al 2006], γ ∈ {1.2, 1.4, 1.6, 1.8, 2}.

× fext (t, T , RH , ws , chl)

(7)

where D0 is reference or nominal diffusion coefficient, and fint internal factors and fext external factors describing different influences on the change of chloride diffusivity.

By equalization of reinforcement cross section during propagation period and limit value of 10% percent decrease φu = 0.9 × φi , it is possible to calculate the propagation period from the following equation:

3.1.1 Nominal chloride diffusion coefficient Input of reference or nominal diffusion coefficient D0 can be given either as an assumed—calculated value (on the basis of concrete mixture data) or as a ‘‘real’’—experimental one. On the basis of laboratory research and literature data following empirical formula was accepted for the theoretical calculation of nominal chloride diffusion coefficient based on water-cement ratio [Stipanovic Oslakovic 2009]:

φ(t1 ) = φi − γ (0.023 · p · icorr · t1 ) = φu = 0.9 · φi

D0 = 5 × 10−13 × e4,8708×v/c

⇒ t1 =

3

0.1 · φi γ (0.023 · p · icorr )

(6)

INPUT PARAMETERS

3.1 Chloride diffusion coefficient In the current study chloride diffusion coefficient is considered using a multifactor law, which includes internal factors such as mineral composition of cement, mixture proportion, curing conditions, cracking, and external factors such as age of the structure, temperature, relative humidity, wind, chloride content. For the value of diffusion coefficient there are following possibilities: 1. theoretically calculated value on the basis of concrete mixture proportions and exposure conditions; 2. apparent or effective diffusion coefficient from laboratory tests, where internal factors are set to the value 1, and the external factors are defined depending on the exposure conditions; 3. apparent diffusion coefficient achieved from the real structure after certain exposure time, when all correcting factors are set to the value 1. The actual chloride diffusion coefficient is defined with multifactor method, where first group represents

(8)

where D0 is nominal diffusion coefficient (cm2 /s), and w/c is water-cement ratio (%). 3.1.2 Internal factors The addition of silica fume, slag, fly ash, superplasticizer, type of cement, type of the formwork and cracking are taken into account in coefficient fint for the internal influences (Table 1). For the combined effect of different factors it is used the product of chosen coefficients. That means for example, for the cement with the addition of slag and superplasticizer that coefficient DCl would be: DCl = D0 × 0.3 × 0.8 = 0.24 × D0 Table 1.

(9)

Coefficient fint .

fint

Influence

1–1.35 0.5–1 0.4–0.9 0.08–0.12 0.3 0.8 1–1.3 0.04

sulphates increase of C3 A quantity addition of flying ash addition of silica fume addition of slag addition of superplasticizer cracking of concrete under basic loads embedding in fabric formwork

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3.1.3 External factors 3.1.3.1 Age of concrete Coefficient fext (t) describes the influence of the age of concrete, which reflects the cement hydration-induced reduction in the concrete porosity with time and consequently reduction of chloride diffusion coeffcient. A number of researches have shown that the relationship between diffusivity and time is best described by a power law [Stipanovic Oslakovic 2009, Chlortest-2 2005, Nokken et al. 2006, Luping & Gulikers, 2007], where the exponent is potentially a function of both the materials (e.g. mix proportions) and the environment (e.g. temperature and humidity). The following equation for coefficient fext (t) is accepted:  fext (t) =

tref t

m (10)

where tref is the reference age from which D0 is derived, t is the actual age of concrete (year) and m is an age factor (depending on mix proportions). Factor m can be chosen from the existing data in the literature [Thomas & Bentz 2001, DuraCrete 2000] or can be given as specific value, experimentally determined for a specific type of concrete. Within the framework of this research chloride diffusion coefficients were determined on different types of concrete, same as concrete mixture used during construction of three bridges in Croatia at the Adriatic coast. Chloride diffusion coefficients were determined from chloride profiles on specimens taken from different locations on these three bridges, exposed to direct influence of marine chlorides for the period of 5, 10 and 25 years and compared with chloride penetration testing in laboratory at the age of 1, 3, 6 and 12 months. Based on this research following relation is suggested to evaluate change of chloride diffusion in time depending to the % of mineral admixtures in the concrete [Stipanovic et al. 2007, Stipanovic Oslakovic et al. 2008b, Stipanovic Oslakovic 2009]: m = 0, 0075 × MA(%) + 0, 30

3.1.3.3 Relative humidity The effect of the environment humidity on the diffusion coefficient is considered through the model proposed by Bazant and Najar, 1972: 



RH fext (RH ) = 1 + 256 · 1 − 100

4 −1 (13)

where RH is the relative humidity of the environment [%]. 3.1.3.4 Wind influence In order to evaluate the effect of wind on the acceleration of sea water penetration into the concrete cover, laboratory and on-site experiments are under development and the results will be taken into account in the model. Currently wind velocity and pressure are calculated according to the actual standards and on the basis of wind map of Croatia, and the influence is taken into account as the increase of water penetration into the concrete [Bjegovic et al. 2008, Stipanovic Oslakovic et al. 2008a] At the northern part of the eastern Adriatic coast blows a gusty downslope windstorm, called bora, form NE. Related hourly mean wind speeds surpassing 20 m/s, with gusts reaching up to 50 or even 70 m/s, in the coastal mountain lee areas are common (hurricane speeds). This wind brings sea water to all parts of the structure, much higher then tidal and splashing zone, as visible in Figure 2. In Figure 3 the analysis of surface chloride concentration for columns of big arch at the Krk bridge is presented in relation to height above the sea level and to the side of structural element. High chloride surface concentration at the level of 40 to 60 metres above the sea level on north and east side of columns are explained by the influence of bora wind. In Figure 4 numerical simulation of chloride surface concentration is presented for this case study of Krk bridge columns.

(11) 3.2

where m is the age factor and MA percentage of mineral addition such as slag, fly ash and silica fume.

Surface chloride concentration

For the assumption of a linear increase of the surface concentration of chloride ions with time, this change

3.1.3.2 Temperature influence Following relationship is accepted to account for temperature-dependent changes in diffusion:    U 1 1 − fext (T ) = exp · (12) R Tref T where U is the activation energy of the diffusion process [J/mol], R is the universal gas constant [J/mol.K], Tref is the reference temperature [K] and T is the average temperature of each month. [Thomas & Bentz 2001, Martin Perez, 1999]

Figure 2.

Bora wind blowing at the location of Krk bridge.

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Table 2. Coefficient k for the initial chloride ion concentration.

Figure 3. Surface chloride concentration depending on height above sea level and on the orientation of the structural element.

Environmental condition

k

Wetting zone Tidal zone Atmospheric zone—for structures < 10 m away from the sea Atmospheric zone—for structures at 10–50 m away from the sea Atmospheric zone—for structures at 50–250 m away from the sea Atmospheric zone—for structures > 250 m away from the sea Continental area—application of de-icing salt

10 1 0.8 0.5 0.2 0.1 0.1

Table 3. Chloride surface concentration depending on the exposure zone. east side

Relation of C smax to the height above sea level

north side

70,0

west side 65,0

south side approximation curve

60,0

Zone Description

CS max (% by weight of Correction concrete) factor (-)

55,0 Zone under influence of wind (from 30 to 70 m above sea level), function of wind speed f (v 2ref ), (~XS3)

50,0

height (m)

45,0

40,0

35,0

30,0 "Reference zone" (from 10 to 30 m above sea level), function of reference wind speed f (vref)

25,0

20,0

15,0

XS 1 Up to 10 m distance from the sea Between 10 and 100 m distance from the sea Between 100 and 250 m distance from the sea More than 250 m from the sea XS 2 Submerged

0,80

0,85

XS 3 Tidal and splashing

1,00

0,50 0,60 0,60

10,0 Splashing zone (up to 10 m above sea level), exponential law (XS3)

5,0

0,0 0,00

0,10

0,20

0,30

0,40

0,50

0,60

Wind influence fw1 Wind influence fw3 Wind influence fw3 Wind influence fw4 Influence of geometrical shape of element fgeom −

chloride surface concentration (% by weight of concrete)

Figure 4. Analysis of chloride surface concentration depending on orientation of the element and height above sea level.

can be expressed as: C(0, t) = k · t,

0 ≤ C0 < Cmax ,

practical approach to the actual regulations, meaning to define quantitatively exposure classes with chloride surface concentration, humidity, temperature, wind etc. [Stipanovic et al. 2004, Stipanovic et al. 2007, Stipanovic Oslakovic et al. 2008a, 2008b, Stipanovic Oslakovic 2009].

(14) 4

where k is a coefficient of a linear increase of the initial chloride ion concentration. Values of coefficient k are given in Table 2 [Krstic 1994, Roy et al. 1993, Stipanovic Oslakovic 2009]. Based on this research numerical characterization for marine environment classes XS1, XS2 and XS3 defined in EN 206-1 is suggested, as presented in Table 3, on the basis of experimental results performed on structures at the Adriatic coast. The aim is to improve service life design procedures with the

CONCLUSION

The majority of models are based on the deterioration mechanisms due to steel corrosion, since the reinforcement corrosion is one of the most frequent deterioration processes in concrete structures. Corrosion of the reinforcement and subsequent damage of the concrete cover are highly complex phenomena and still not completely understood. However it is necessary to simplify many of the complex deterioration processes in the concrete and make assumptions where

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insufficient knowledge is available. This accounts with the responsibility of the users to be aware of any assumptions and potential limitations. Performance-based approach to the durability design of reinforced concrete structures, included in the new CHLODIF++ model, means that it is based on durability indicators like materials properties measured on laboratory and on site specimens and geometrical characteristic of cross section of reinforced concrete element. With these values, as input parameters, service life modelling should be performed, in order to determine the reliability level or safety level of structure. The aim of this research is to develop fully operational mathematical model, which should become a part of standard design procedures, for the structural and materials’ engineers.

ACKNOWLEDGEMENTS This research was performed within scientific projects ‘‘The Development of New Materials and Concrete Structure Protection Systems’’, 082-0822161-2159, and ‘‘From Nano- to Macro-structure of Concrete’’, 082-0822161-2990, funded by Croatian Ministry of education, science and sport.

REFERENCES Andrade, C., Alonso, C., Gonzales, J.A. & Rodriguez, J. 1989. Remaining Service Life of Corroding Structures, IABSE Symposium on Durability of Structures, Lisbon (Portugal), 359–364. Bazant, Z.P. & Najjar, L.J. 1972. Nonlinear Water Diffusion of Nonsaturated Concrete, Materials and Structures Journal, 5(1), 3–20. Bjegovi´c, D., Krsti´c, V., Mikuli´c, D. & Ukrainczyk, V. 1995. C-D-c-t Diagrams for Practical Design of Concrete Durability Parameters, Cement and concrete research 25, 1; 187–196. Bjegovi´c, D., Mikuli´c, D., Serdar, M., Milovanovi´c, B. & Gabrijel, I. 2008. Service life estimation of concrete structure exposed to agressive environmental loads, Concrete Engineering in Urban Development, 4th CCC Scientific Symposium, Zagreb, 561–568. Bjegovi´c, Dubravka, Krsti´c, Vedrana, Mikuli´c, Dunja. 2006. Design for durability including initiation and propagation period based on the fuzzy set theory. Materials and Corrosion-Werkstoffe und korrosion. Materials and Corrosion-Werkstoffe und Korrosion (0947-5117) 57, 8; 642–647 8. CHLORTEST -1 project GRD1-2002-71808, 2005. Resistance of concrete to chloride ingress—from laboratory tests to in-field performance: WP 5 Report—Final evaluation of test methods. CHLORTEST -2 project GRD1-2002-71808, 2005. Resistance of concrete to chloride ingress—from laboratory tests to in-field performance: WP 4 Report—Modeling of chloride ingress.

EU—Brite EuRam III 2000. DuraCrete Final Technical Report: Probabilistic Performance Based Durability Design of Concrete Structures, Document BE951347/R17. fib Bulletin 34, 2006. Model Code for Service Life Design. HRN EN 206-1: 2000: Concrete—Part 1: Specification, performance, production and conformity. Krsti´c, V. 1994. Numerical model for durability design of reinforced concrete structures, Master Thesis, Faculty of Civil Engineering, University of Zagreb, 1994. (in Croatian). Luping, T. & Gulikers, J. 2007. On the mathematics of timedependent apparent chloride diffusion coefficient in concrete, Cement and Concrete Research 37, 589–595. Martín Pérez, B.: Service Life Modelling of R.C. Highway Structures Exposed to Chlorides. 1999. Ph.D. thesis (168 pp.). Department of Civil Engineering, University of Toronto. Nokken, M., Boddy, A., Hooton, R.D. & Thomas, M.D.A. 2006. Time Dependent Diffusion in Concrete—Three Laboratory Studies, Cement and Concrete Research, 36(1), 200–207. Roy, S.K., Chye, L.K. & Northwood, D.O. 1993. Chloride Ingress in Concrete as Measured by Field Exposure Tests in the Atmospheric, Tidal and Submerged Zones of a Tropical Marine Environment, Cement and Concrete Research, Vol. 23, 1289–1306. Stipanovi´c Oslakovi´c, I. 2009. Prediction and measuring transport of chlorides in concrete, Ph.D. Thesis, Faculty of Civil Engineering, University of Zagreb, p. 209. Stipanovi´c Oslakovi´c, I., Bjegovi´c, D. & Mikuli´c, D. 2008a. Chlorides ingress as an environmental load on Krk bridge, Tailor made concrete structures. New solutions for our society, International fib Symposium, Amsterdam. Stipanovi´c Oslakovi´c, I., Serdar, M., Bjegovi´c, D. & Mikuli´c, D. 2008b. Modeling of time depended changes of chloride diffusion coefficient, Durability of Building Materials and Components 11: Globality and Locality in Durability, Volume 1: Durability of Materials, Istanbul Technical University, 203–212. Stipanovi´c, I., Bjegovi´c, D. & Rako, D. 2004. Chlorides Impact on Reinforced Concrete Structures, Proceedings of the International Symposium ‘‘Durability and Maintenance of Concrete Structures’’, SECON HDGK, Zagreb, 583–590. Stipanovi´c, I., Bjegovi´c, D., Mikuli´c, D. & Serdar, M. 2007. Time dependent changes of durability properties of concrete from Maslenica Bridge at the Adriatic coast, Integral Service Life Modelling of Concrete Structures, RILEM Publications S.A.R.L., Guimaraes, Portugal, 87–95. Takewaka, K. & Mastumoto, S. 1988. Quality and Cover Thickness of Concrete Based on the Estimation of Chloride Penetration in Marine Environments, ACI-SP 109–117, Concrete in Marine Environment, Detroit (USA), 381–400. Thomas, M.D.A. & Bentz, E.C. 2001. Life-365, Service Life Prediction Model, Computer Program for Predicting the Service Life and Life-Cycle Costs of Reinforced Concrete Exposed to Chlorides, University of Toronto.

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Modelling of concrete structures

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Some remarks concerning the shear failure in prestressed RC beams B. Belletti & C. Damoni DICATeA, University of Parma, Parma, Italy

ABSTRACT: In this paper the behavior of prestressed reinforced concrete beams and in particular the shear resistance is analyzed. Even if several different theoretical approaches and numerical models have been proposed over the last years to calculate the shear capacity of prestressed reinforced beams, the problem is not fully solved. A general agreement among researchers is not achieved and also formulations implemented in codes are mostly based on experimental test data. In this paper it is demonstrated that the last formulations contained in Eurocode 2 (EN 1992-1-1: 2004) bettered the prediction of the shear capacity of prestressed beams by a new formulation of the coefficient αc . NLFE analyses have been carried out to simulate the behavior of the prestressed beams tested in laboratory by Levi and Marro (Levi and Marro, 1992) and to check the effectiveness of αc values recommended by Eurocode 2 (EN 1992-1-1: 2004). The PARC constitutive model (Belletti, 2001), theoretically formulated at the University of Parma and implemented in the user’s subroutine UMAT.for is adopted to carry out NLFE analyses with ABAQUS Code. 1

INTRODUCTION

In this paper the shear failure mechanism and the main parameters that influence it are investigated for prestressed reinforced concrete beams with shear reinforcement. For prestressed members there are two primary modes of cracking in shear: web shear and flexure shear. For moments less than the cracking moment, the section is not cracked and the shear strength is controlled by web shear. Web shear cracking initiates in location of high shear that are also subjected to low flexural stress. This cracking mode is typically observed in the end regions of thin-webbed prestressed members and occurs when the principal tensile stresses in the web exceed the tensile strength of the concrete. Several parameters have been identified as having a significant influence on the contribution of the shear resistance mechanism and thus the ultimate shear capacity even if until now no unified theory exists that is capable of fully describing the complex behavior of reinforced concrete elements subjected to shear. According to the previous version of Eurocode 2 (ENV 1992-1-1: 1991) two alternative methods to calculate the ultimate shear capacity were initially proposed: the standard method and the variable inclination method (Walraven, 2002). According to the standard method the shear capacity is the sum of two terms: one is the shear reinforcement term, based on a truss mechanism with concrete struts inclined of 45◦ to the member axis and the other one is the concrete term regarding the effects of the uncracked compression area, the dowel action of the longitudinal reinforcement and the aggregate interlock. In

reality macro-cracks that develop after increasing loads don’t necessarily coincide with the direction of the initial micro-cracks formed at the beginning of the loading and non linear stress distribution takes place also before the formation of visible cracks. This phenomenon induces a transfer of shear force across the cracks that consequently influence the cracking angle which becomes important in the evaluation of the shear capacity (Vecchio and Collins, 1988, Dei Poli and Gambarova, 1987). According to these considerations only the variable inclination method is proposed in the current version of Eurocode 2 (EN 1992-1-1: 2004) in which struts could have an inclination θ to longitudinal beam axis ranging from limit values (1≤ cot θ 2.5), Fig. 1. The new Eurocode 2 formulation, based on variable inclination method, is a simple model suitable for practicing engineers and for everyday work. Some researchers affirm (Cladera and Marì, 2007) that the variable inclination method is a gross oversimplification of the complex problem of shear resistance of RC members that neglects important key variable. For this reason other international Codes have been taken

Figure 1. Variable inclination truss model and notation for shear reinforced members.

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into account in this paper for the evaluation of shear capacity, such as Canadian Standards Associations (CSA Commettee, 2004) which consider as well as EC2 a variable angle truss model, and the ACI formulation (ACI Committee, 2005) which considers a 45◦ truss angle model. The formulation proposed are better investigated in the following paragraphs and compared with analytical and experimental results. 2

CODE FORMULATIONS FOR SHEAR CAPACITY

The variable inclination method proposed in Eurocode 2 (EN 1992-1-1: 2004) is a simple equilibrium method which gives a transparent view of the flow of force in the structure. If the shear reinforcement yields the truss can, by rotation of the compression struts to a lower inclination, activate more reinforcement, Fig. 2; the shear resistance provided by stirrups is given in Eq. (1): Asw · z · fyw · cot θ (1) s where Asw and s are the cross sectional area and the spacing of the shear reinforcement, z is lever arm, fyw is the steel yield strength and bw is the width of the web. Due to strut rotation the stress in the concrete struts increases, consequently rotation can only continue until the crushing of the concrete occurs, Fig. 3; the VR,s =

corresponding shear force is given in Eq. (2): VR,c =

bw · z · fc1 · αc cot θ + tan θ

where fc1 = νfc and αc is a coefficient that takes into account the prestressing effects; it increases the shear resistance and depends on the compressive stress level and it is used as a multiplier of the compressive concrete strength. If the beam is not prestressed αc can be taken equal to 1. According to ACI 318-08 model (ACI Committee, 2005) the shear capacity VR is the sum of the concrete contribute, the shear reinforcement contribute and a term taking into account the effect of prestressing force N which gives a benefit in the ultimate shear capacity calculation, Eq. (3) (Joint ACI-ASCE, 1962):      1 N VR = fc + ρsw fyw bw d (3) 1+ 6 14 · A where A is the gross area of the section, ρsw is the shear reinforcement ratio and d is the depth of the beam. The CSA formulation, as mentioned above, is based on a variable inclination truss model and refers to the MCFT theory (Vecchio, Collins, 1986). According to this formulation the ultimate shear capacity is, as in ACI Code, the sum of the concrete contribute and the steel contribute and it is given as:    Asw fyw dv cot θ 0.4 fc b w d v + (4) VR = 1 + 1500εx s where dv is taken as max{0.72h; 0.9d} and the strut inclination, given by Eq. (5), depends on the longitudinal strain εx at mid-depth of the member, Eq. (6):

z (cotgθ + cotgα) C

θ = 29◦ + 7000εx z M1

V

εx = α

θ T

z (cotgθ + cotgα)

α

Fs

Figure 2. Variable inclination truss model: shear resistance provided by reinforcement.

z (cotgθ + cotgα) C

σcw z Fc θ

θ T

α

(2)

V M 2

z (cotgθ + cotgα) sinθ

Figure 3. Variable inclination truss model: shear resistance provided by concrete struts.

V (1 + M /Vdv ) + 0.5N − Ap fp0 2(Es As + Ep Ap )

(5) (6)

where Ap , Ep , fp0 are the area, Young modulus and yielding stress of prestressing reinforcement; while Es and As are the Young modulus and area of longitudinal reinforcing bars on the flexural tension side of the member. Several authors during the years (Haddadin and Hong, 1971, Gupta, 1995, Yoshida, 2000) have studied the shear failure mechanism and compared the experimental results obtained with the international codes mentioned above. Because of the difficulties of the physical reality in many cases the semi-empirical formulations adopted in the international codes are too conservative in the calculation of the shear capacity. In this paper the effects of the coefficients that influence the shear capacity, especially those factors regarding the crushing of concrete and the effect of the prestress, will be better analyzed. The coefficient ν takes into account that after cracking the compressive

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(a)

(b)

Figure 4. Experimental results of shear tests on prestressed beams compared with the shear capacity calculated with Eurocode 2 (a) without considering and (b) by considering compressive stress effects by αc parameter (Walraven, 2002).

strength of the struts decreases because of the occurrence of transversal tension. In the years ν has become into several changes according to experimental results. This coefficient has been studied and modified by several authors, such as Watanube and Kabeyasawa (Watanube and Kabeyasawa, 1998) whom theories have been included in the design guidelines of the Architectural Institute of Japan (AIJ, 1988). In Table 1 different relationship for ν calculation are shown according to different codes. Table 1.

Relationship proposed for ν coefficient.

MODEL CODE 1978 MODEL CODE 1990 EC2 Part1 1992 EC2 Part 1-3 EC2 2005 Watanube, Kabeyasawa CSA Code

0.6 fck 250  fck 0.5 fck < 50 MPa 0.7 − ≥ 0.4 fck > 50 MPa 200 1−

fck 0.7 − 200 ⎧ ⎨ 0.6 fck < 60 MPa f ⎩ 0.9 − ck fck > 60 MPa 200 0.85 − 0.004 fc ≥ 0.5 1 0.8 + 170ε1∗

If the same rules described above are applied to prestressed members with shear reinforcement it can be seen that there is apparently an increase of both safety and scatter. In Fig. 4(a) and 4(b) respectively the experimental results obtained for prestressed beams are compared with the shear resistance calculated according to the variable inclination method proposed by Eurocode 2 without considering and by considering the effects of compressive stresses, respectively (Walraven, 2002). It can be noted that by neglecting compressive stresses effects (which means by considering αc = 1), an high scatter of results occurs in terms of ψ versus nominal shear strength curves (being ψ = ρsw fyw /fc1 ), while a better fitting is observed if the values of αc coefficient are evaluated according to the new formulation proposed by Eurocode 2 (EN 1992-1-1: 2004), Eq. (7): ⎧ ⎪ for 0 < σcp < 0.25 fc ⎨αc = (1 + σcp /fc ) αc = 1.25 for 0.25 fc < σcp < 0.5 fc (7) ⎪ ⎩α = 2.5(1 − σ /f ) for 0.5 f < σ < f c cp c c cp c In Fig. 5 the formulation proposed for αc in the previous version of Eurocode 2, calculated according with Eq. (8) is also reported: 

∗ principal

tensile strain in the cracked concrete calculated from Mohr’s circle

σcp αc = 1.67 · 1 − fcd



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(8)

Figure 6.

Kinematic quantities for the model PARC.

Figure 5. Comparison of different formulation of αc (Walraven, 2002).

0

0

Un-cracked concrete Cracked concrete

Several authors have taken into account the influence of the prestressing by the αc coefficient that multiply the coefficient ν described above. Nielsen (Nielsen 1990) states that prestressing has a positive influence on the shear capacity of beams with shear reinforcement. He proposed to evaluate αc with Eq. (9) : αc = 1 + 2σcp /fc

with σcp /fc < 0.5

fc

(9)

A comparison with 93 tests results shows, however, that this expression is not sufficiently conservative to serve as a safe lower bond over the whole region of test results. Another proposal is given by Fourè (Fourè 2000), Eq. (10):

fc

Figure 7.

Constitutive model for concrete in compression.

– for low compressive levels (0 < σcp < 0.4fcd ): αc = (1 − 0.67σcp /fctm )

(10)

– for high compression levels (0.4fcd < σcp < fcd ):

0.5 αc = 1.2(1 − σcp /fcd )(1 + σcp /fcd ) (11) In Fig. 5 the different formulation proposed for αc are compared to each other.

Belarbi and Hsu (Belarbi and Hsu, 1991) suggested to reduce both the peak strength and strain by means of a softening coefficient for strain, νε , and a softening coefficient for stresses, νσ , Fig. 7. These softening coefficients depend on the orientation of the cracks to the reinforcement (angle β): νε = √

3

PARC MODEL FOR NON LINEAR FINITE ELEMENT ANALYSES

1 1 + Kε · ε1

(12)

0.9 1 + Kσ · ε1

(13)

νσ = √

Non linear finite element analyses have been carried out with ABAQUS Code. The PARC constitutive matrix (Belletti, 2001), implemented in the users’s subroutine UMAT for describes the material inelasticity. Total strains are evaluated by considering as kinematic variables the crack opening w, the crack slip ν and the concrete strain εc2 , Fig. 6. PARC model has been modified in this study by adopting Belarbi and Hsu (Belarbi and Hsu, 1991) relation for concrete struts in compression. The Saenz’s stress-strain relation is reduced to consider the deterioration in compression resistance due to cracking.

Kε is a value linearly varying from 0 to 160 for β varying from 90◦ to 45◦ , respectively, while Kσ is value linearly varying from 250 to 400 for β varying from 90◦ to 45◦ , respectively. 4

CASE STUDY

Experimental tests (Levi, Marro, 1992) have been carried out on twelve reinforced and prestressed concrete beams (RC and PC beams) differing to each other by

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Figure 8.

Figure 9. 1992).

Geometry of Levi and Marro’s beams (Levi and Marro, 1992).

Crack pattern for PC60B2 beam (Levi and Marro,

stirrups (Vy ) followed by the reaching of the ultimate shear strength Vmax . The geometrical and mechanical features of the beams analyzed in this paper are summarized in Table 2. In Fig. 10 the comparison between the experimental tests and the variable inclination method curve are represented considering the different contributions of shear capacity of the beam tested. The shear capacity has been plotted considering the steel contribution and the concrete contribution taking into account both αc = 1 and αc = 1.

spacing of stirrups, compressive concrete strength, diameter of longitudinal reinforcement and amount of prestressing. All the beams are supported at the two extremities and loaded with two concentrated loads. In Fig. 8 the geometry of the PC beams is shown. The beams are prestressed by means of 0.6 strands, 6 to 12 per beam, to an average prestressing value at the centre of gravity of approximately fc /12. The tendons are tensioned before casting and, following the transfer, are provided with anchorages to prevent early slip induced failure. The evolution of the test carried out on the PC beams reveals the appearance of cracks at an inclination of about 30◦ , which crosses web and which are close to one other, with a distance of about 15 cm, as shown in Fig. 9. These cracks remains virtually unchanged up to failure. The failure is due to complete yielding of the

5 5.1

DISCUSSION OF RESULTS Modeling

The behavior of the beams tested by Levi and Marro (Levi and Marro, 1992) has been studied by NLFE analyses carried out with ABAQUS Code. Since the structure is symmetric along y axis in term of geometry, boundary and loading conditions, half structure has been analyzed in order to reduce the time for the implementation. A mesh of four node membrane elements has been adopted. The average element size is 50 mm; non linear analyses have been carried out in displacement control. Different materials have been used to distinguish elements having different thicknesses and reinforcement ratios (smeared reinforcement is assumed in

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Table 2.

Geometrical and mechanical features of Levi and Marro’s beams.

BEAMS

BARS

7 wire tendons 0.6

s (mm)

ρsw

P (kN)

σsp (MPa)

σcpm (MPa)

fc (MPa)

Vcr (kN)

Vy (kN)

Vmax (kN)

PC30A1 PC60A2 PC60B1

26 26 30

N◦ 6 N◦ 12 N◦ 12

225 225 150

0.0084 0.0084 0.1205

798 1318 1324

957 790 793

2.58 3.93 3.94

30 50 50

250 400 400

850 1150 1327

880 1207 1400

s is the stirrup spacing; P is the prestressing force; Vcr is the shear cracking; Vy is the shear at yielding of the stirrups; Vmax is the ultimate shear capacity; σsp is the stress in the reinforcement; σcpm is the average compressive stress; θexp is the strut inclination.

PC30A1 VR,s V

R,s

1800

VR,c c≠1 VR,c c≠1

1600

Vu [kN]

cut-off

2000

Figure 11.

1400

1000

Vu,exp

PARC model). In Figure 11 the mesh and the boundary conditions are shown; each grey shade represents a different material. A Gauss integration technique in the element plane has been adopted. Rebar layers, embedded in the ‘‘host’’ memebrane elements, have been used in order to model strands. By specifying the cross-sectional area, the material and the isoparametric direction of each rebar, the strands can be modeled in a quite appropriate way. The prestressing action is defined in the rebars as a given initial condition which can change during an equilibrate static analysis step. Finally a selfequilibrating stress state established itself in the structure. In this way the prestressing force does not remain constant during all the analysis steps but changes following the development of the crack pattern up to failure.

800 600 400 200 0 45

40

35

30

25

20

15

[˚]

PC60A2 cut-off

2000

VRd,s VR,s

1800

VR,C alfa=1 VRd,c c≠ 1 V

R,c VRd,c

1600

c=1

1400 1200

Vu [kN]

Mesh adopted for the modeling of the beams.

1200

Vu,exp

1000 800 600 400 200

(b) 0 45

40

35

30

25

20

5.2

15

[˚]

PC60B1 cut-off

2000

VRd,s VR,s

1800

VRd,c VR,C alfa=1 c≠ 1

1600

VR,c c=1 VRd,c Vu,exp

1400

Vu [kN]

1200 1000 800 600 400 200

(c) 0 45

40

35

30

25

20

15

[˚]

Figure 10. (a) Shear capacity for PC30A1, (b) Shear capacity for PC60A2, (c) Shear capacity for PC60B1.

NLFEA results

In Table 3 the mechanical characteristics of material adopted for the NLFEA of the PC30A1, PC60A2 and PC60B1 beams are shown. In Fig. 12 the shear- deflection curves obtained with NLFE analyses are shown. It can be noted that the shear resistances obtained with NLFE analyses are very close to the experimental ones. As the prestressing increases, the ultimate shear capacity of the beam increases and as the shear reinforcement increases the stiffness of the structure in the cracked phase increases. The cracks develop most of all in the zones between the load and the support, where the shear force is constant and maximum, while in the middle zone the cracks are limited and mostly due to bending moment. In Fig. 13(a), (b) and (c) are reported, for PC60B1 beam, the values of crack opening, strain of stirrups and strain of concrete, respectively.

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Table 3.

Mechanical properties of materials.

Concrete fck fctm Ec (MPa) (MPa) (MPa) εc2 PC30A1 25 PC60A2 50 PC60B1 50

2.6 4.1 4.1

εc2u

31000 0.2% 0.35% 37000 0.2% 0.35% 37000 0.2% 0.35%

Reinforcement steel Es (MPa)

Ehard (MPa)

εy

εu

196000

31388

0.2%

1.6%

Es (MPa)

Ehard (MPa)

f0.2 (MPa)

fu (MPa)

196000

31388

1620

1800

Prestressing steel

Figure 13. Contours at failure point of (a) crack opening, (b) strain of stirrups, (c) strain of concrete values for PC60B1 beam.

Figure 12.

Shear-deflection curve of the beams tested.

Fig. 14 demonstrates that for low values of prestressing force (PC30A1 beam) the stirrups start to work earlier than for high values of prestressing force (PC60 beams). Furthermore, for PC60 beams the activation of stirrups is quite contemporary but the slope of shear versus stirrups stresses curves and the yielding points are different since the stirrup spacing is different. In Table 4 the experimental shear capacity of the beams tested by Levi and Marro (Levi and Marro, 1992) is compared with the shear resistance obtained with different codes and with the shear resistance obtained with NLFE analyses. 5.3

Some remarks regarding the evaluation of the internal lever arm

Initially the internal lever arm was fixed, as a common simplified assumption, equal to z = 0.9d. This

Figure 14. Table 4.

Shear versus stirrups stresses curves. Shear capacity of the beams tested.

VR,EC2 BEAMS (KN)

VR,ACI (KN)I

VR,CSA (KN)I

PC30A1 646.88 643.32 862.24 PC60A2 943.32 942.52 876.81 PC60B1 1116.19 942.75 948.79

Vmax,exp (KN)

Vmax,fem (KN)

880 1207 1327

855.37 1231.7 1393

approximation doesn’t take into account the effects of prestressing forces and that the occurrence of cracking changes the application point of the resultant of the internal forces, so afterward the lever arm (zfem ) value has been calculated in a rigorous way by considering

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the arm of the couple produced by the internal forces coming from NLFE analyses. In Fig. 15 the results obtained from the analyses carried out for PC60B1 beam are shown and compared with the shear capacity calculated with Eurocode 2, by adopting z = 0.9d = 897 mm and z = zfem = 873 mm. It can be noted that considering zfem the shear capacity calculated with Eurocode 2 (EN 1992-1-1: 2004) are closer to NLFEA results. In Table 5 the values of the internal lever arm evaluated with NLFEA results, zfem , and by assuming z = 0.9d are reported. It can be noted that the difference between the z values increases as the prestressing force increases.

axes which remain fixed after cracking, Fig. 16(a). Stresses in concrete struts are due to axial compressive stresses along 2 direction, axial tensile stresses along 1 direction and tangential stresses due to aggregate interlock phenomena. In the perspective of the variable inclination angle, the inclination θ of concrete struts coming from Eurocode 2 formulations, has been compared with the inclination of principal direction of stresses (σI and σII ) of cracked concrete, Fig. 16(b). In Fig. 17 the elements used to calculate θ value are represented. In Table 6 the values of strut inclination obtained with experimental tests (Levi and Marro, 1992), θexp , with Eurocode 2 formulations (EN 19921-1: 2004), θEC2 , and with NLFEA results, θfem are reported.

5.4 Some remarks concerning the evaluation of the θ angle

5.5

Because of PARC constitutive model is a fixed crack model, the stresses can be referred to the orthotropic

Some remarks concerning the evaluation of the coefficient αc

For PC60B1 the coefficient αc,fem has been calculated from the NLFEA analyses as: αc,fem = Vconcrete /VR,c

PC60B1-z =0.9 d

(14)

cut-off

2000

VRd,c VR,C alfa=1 c≠ 1

1800

VRd,c VR,c

1600

c=1

VR,s VRd,s

1400 1200

VRd [kN]

(a)

2

Vu,exp 1

( 1,

2

)

12

12

1000

1

2

II

12

800

I 2

1

600

VR,C

NLFEA

VR,s

NLFEA

( 2,

400

)

12

200

(a) 0 45

40

35

30

25

20

(b)

15

[˚]

I II

PC60B1-zfem 2000

II

1800

VR,c VRdc 1600

c=1

VR,s VRd,s

1400

Vu [kN]

I

cut-off

VRd,c VR,C alfa=1 c≠1

Figure 16.

Vu,exp

Evaluation of θ angle from NLFEA results.

1200 1000 800 600

VR,C

NLFEA

VR,s

NLFEA

400 200

(b) 0 45

40

35

30

25

20

15

[˚]

Figure 15. Shear capacity-inclination of concrete strut curve considering (a) z equal to 0.9d and (b) zfem . Table 5.

Comparison among z values.

Figure 17. Element considered in order to simulate the inclined variable method. Table 6.

Comparison among θ values.

BEAMS

σpc /fc (-)

z = 0.9d (mm)

zfem (mm)

BEAMS

θexp

θEC2

θfem

PC30A1 PC60A2 PC60B1

0.11 0.09 0.09

875.27 878.46 897.35

867.27 865.19 873

PC30A1 PC60A2 PC60B1

33 28 33

31 23 29

35 30 30

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being

2.5

Vconcrete =

σc,fem · bw · z cot θ + tan θ

2

(15)

Nielsen (1990) EC2 (2005) EC2 (1992) NLFE analyses

1.5

In Table 7 the values of this coefficient evaluated with Eurocode 2 (EN 1992-1-1: 2004) prescriptions, αc , and NLFEA results, αfem , for PC30A1, PC60A2 and PC60B1 beams are reported. The value of compressive stresses due to prestressing are quite low for these beams, so a parametric study on PC60B1 beam is carried out by varying the applied prestressing force. This parametric study helps to evaluate the effects of high compressive stress values on shear capacity of beams. In Table 8 (Boiardi, 2008) the values of prestressign force considered are reported; it should be noted that in order to distinguish the applied prestressing force, the name of each beams contains a digit which represents the ratio of the applied prestressing force to the prestressing force of PC60B1. For example beam PC60B1(8/4) is a beam with a prestressing force which is the double of the prestressing force applied to PC60B1. The αfem values obtained with NLFE analyses by applying different values of prestressing force are reported in Table 8 and compared in Fig. 18 with the different formulations proposed for αc coefficient. It should be noted that the assumed prestressing forces values and consequently the assumed compressive stresses and σcp /fc ratios are those usual for reinforced concrete beams and correspond to the first branch of the three linear relation proposed by EC2 (EN 1992-1-1: 2004).

Table 7.

Comparison among αc values.

BEAMS

Vconcrete (KN)

VR,c (KN)

αc

αfem

PC30A1 PC60A2 PC60B1

615.8 1099 1130.4

636.56 1087.79 1116.92

1.113 1.092 1.092

0.97 1.010 1.012

Table 8. forces.

αc values obtained with different prestressing

BEAMS

P (kN)

ϑr (◦ )

zfem(mm)

PC60B1 PC60B1(5/4) PC60B1(6/4) PC60B1(7/4) PC60B1(8/4) PC60B1(9/4) PC60B1(11/4)

1324 1655 1975 2317 2610 2963 3704

30.0 28.7 27.9 27.9 27.8 27.1 26.3

873.3 867.7 863.2 796.0 770.8 749.7 710.1

c

1

0.5

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

cp/fc

Figure 18. Comparison among NLFEA results and different formulation proposed for αc coefficient.

6

CONCLUSIONS

In this paper the shear capacity of reinforced concrete prestressed beams is analyzed. To this aim the prestressed beams tested by Levi and Marro are analyzed and experimental measurements are compared with NLFEA results and Code prescriptions. Particular attention has been paid on the values of αc and ν parameters proposed by Eurocode 2 to evaluate the shear resistance provided by inclined concrete struts forming an angle θ with respect longitudinal beam axis. The comparison between experimental measurements and NLFEA results demonstrated that the effect of prestressing forces on shear capacity is properly considered by Eurocode 2 prescriptions for low values of compressive stresses. The case of high values of compressive stresses are not analyzed in this paper and will be deeply investigated in further works. The case of high value of compressive stresses is important because is typical of reinforced concrete columns. Consequently, the right evaluation of the dependency of the shear capacity on the compressive stress level is really important also for the design of frames subjected to seismic actions. Indeed, in the perspective of the capacity design approach the evaluation of the shear capacity becomes an essential task for the prediction of the earthquake resistance of the entire building.

REFERENCES ACI Committee 318, 2005, ‘‘Building Code Requirements for Structural Concrete (ACI 318-05) and Commentary (318R-05)’’, American Concrete Institute, Farmington Hills, Michigan. Belarbi A., Hsu T.T.C, 1991, ‘‘Constitutive laws of reinforced concrete in biaxial tension-compression’’, Research Report UHCEE 91-2, University of Houston, Houston, Texas.

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Belletti B., Cerioni, R., Iori, I., 2001, ‘‘A Physical Approach for Reinforced Concrete (PARC) membrane elements’’, Journal of Structural Engineering, ASCE, 127(12), pp. 1412–1426. Boiardi A., 2008, ‘‘Studio sulla resistenza a taglio di travi in cement armato precompresso’’, MASc thesis, Department of Civil Engineering, University of Parma, Italy. Cladera, A., Marì, A., 2007, ‘‘Shear strength in the new Eurocode 2. A step forward?’’ Print ISSN: 1464–4177, Volume: 8, Issue: 2, Thomas Telford and fib, pp. 57–66 CSA Commettee A23.3, 2004, ‘‘Design of concrete structures’’, Canadian Standards Association, Mississauga, Ontario, Canada. Dei Poli S.D., Gambarova P.G., Karakoc C., 1987, ‘‘Aggregate interlock role in R.C. thin- webbed beams in shear’’, Journal of the Structural Division, ASCE V. 113, No. 1, pp. 1–19. Eurocode 2,’’Design of concrete structures - Part 1-1: General rules and rules for buildings’’, ENV 1992-1-1: 1991. Eurocode 2, 2005. Eurocode 2,’’Design of concrete structures—Part 1-1: General rules and rules for buildings’’, EN 1992-1-1: 2004. Fourè B., 2000, ‘‘Proposal for rewriting item 6.2.4 in section 6.2 (Ultimate Limit State-Shear’’, Note to Project Team for EC-2, May. Gupta P.R., 1995, ‘‘Shear behavior of reinforced concrete beams subjected to high axial compression’’, PhD thesis, University of Toronto, Canada. Joint ACI-ASCE Committee 326, 1962, ‘‘Shear and diagonal tension’’, ACI Journal, Proceedings V. 59, No. 1-3, pp. 130, 277–344, 352–396.

Haddadin M., Hong S.T., Mattok A.H., 1971, ‘‘ Stirrup effectiveness in reinforced concrete beams with axial force’’, Journal of the structural division, ASCE, V. 97, No. ST9, pp. 2277–2297. Levi F., Marro P., 1993, ‘‘Shear tests on HSC prestressed beams - proposals of new interpretative models’’, Conference on High Strength Concrete, Lillehammer 20–24 June, Proceedings, pp. 293–305. Nielsen M.P., 1990, ‘‘Commentaries on Shear and Torsion’’, Eurocode 2 editorial group, 1st draft, October. Vecchio F.J., Collins M.P.,1988, ‘‘Predicting the response of reinforced concrete beams subjected to shear using modified compression field theory’’, ACI Structural Journal, V. 85, No. 3, pp. 258–268. Vecchio F., Collins M.P., 1986, ‘‘ The modified compression field theory for reinforced concrete elements subjected to shear’’, ACI Journal, Proceedings, V. 83 No. 2, pp. 219–231. Watanabe F., Kabeyasawa T., 1998, ‘‘Shear Strength of RC Members with High-Strength Concrete’’, High Strength Concrete in Seismic Regions, SP-176, C.W. W. French and M.E. Kreder, eds., American Concrete Institute, Farmington Hills, Mich., pp. 379–396. Walraven J.C., 2002, ‘‘Background document for prENV 1992-1-1:2001, 6.2 SHEAR’’, Delft University of Technology, The Netherlands. Yoshida Y., 2000, ‘‘Shear reinforcement for large, lightly, reinforced members’’, MASc thesis, Department of Civil Engineering, University of Toronto, Canada.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Verification of experimental tests on roller bearings by means of numerical simulations S. Blail & J. Kollegger Institute for Structural Engineering, Vienna University of Technology, Vienna, Austria

ABSTRACT: Roller bearings are an important part of a new construction procedure for large concrete bridges which has been developed at Vienna University of Technology. This new method will considerably speed up construction without relying on erection girders. In the ‘‘balanced lift method’’ the proposal is made to build the bridge girders in vertical position and to rotate the bridge girders into the final horizontal position. The bridge girders can be built in combination with the pier using self climbing formworks which will allow for lower production costs and shorter construction time. Two different versions of this construction method can be applied, either construction with compression struts or construction with tension ties. An innovative feature of the new bridge construction method is the design of the joints, where large rotations have to be accommodated. Curved steel plates serve as formwork for the bridge girders or the compression struts along the contact surfaces. In case of the tension tie bridge, the hinges are the ends of the bridge girders, whereas for the compression strut bridge they are the ends of the compression struts. Therefore, a preliminary test on roller bearings was carried out in July 2008. The hinges had a radius of 2 m, a steel thickness of 10 mm and a concrete strength of 42 N/mm2 . Further tests were carried out in July 2009 in which the parameters radius (R = 0.5 m, 1.0 m, 2.0 m), steel plate thickness (t = 10 mm, 30 mm) and quality of concrete (C20/25, C50/60) were varied. The test series with a total of 20 hinges was accompanied by numerical simulations using nonlinear finite element programs. In the nonlinear analyses the nonlinear material behaviour as well as the contact problem at the line of contact between two hinges has to be taken into account. 1 1.1

NEW CONSTRUCTION METHOD Idea of the balanced lift method

The balanced lift method is a new bridge construction method which was invented in 2006 and was patented worldwide in 2009 [1]. This method proposes to build the bridge girders in a vertical position and to rotate them into the final horizontal position with the aid of compression struts (see Figure 1) or tension ties (see Figure 2). Pier, bridge girders and struts form a statically determinate mechanism whose geometry can be changed by lifting the base points. Due to the tension ties, or alternatively the compression struts, considerable mass savings can be achieved, because the ties or struts reduce the span length. Furthermore time and cost savings can be achieved. 1.2

was build with a span of 15 m [2]. An important part of the new bridge construction method is the design of the nodes. Curved steel plates serve as a formwork for the bridge girders along the contact surfaces and transfer the force at the line of contact through Hertz pressure [3]. The maximum Hertz pressure at the line of contact between the two cylinders occurs when the bridge girder has reached the final horizontal position.

Nodes

Field tests for the balanced lift method were carried out at Vienna University of Technology in January 2008 in a scale of 1:10. With the aid of these tests the operability of the hinges could also be demonstrated for large rotations. At first the tension tie bridge was build with a span of 17 m and then the compression strut bridge

Figure 1.

Lifting process of the compression strut bridge.

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The Hertz pressure can be calculated according to the formula    1 F ·E 1 · (1) + pmax = π · l · (1 − v 2 ) d1 d2 where pmax = maximum Hertz pressure between two cylinders; F = compressive force; E = modulus of elasticity (210 000 N/mm2 for steel); l = length of contact line; v = Poisson’s ratio (0.3 for steel) and d1 , d2 = diameter of first and second cylinder. The calculation of the Hertz pressure was done by using the modulus of elasticity of steel. This approach of course is not correct because two different materials are involved (steel at the contact surface and concrete in the node regions). The occurrence of large rotations at the connection points of the structural members during the lifting process is an inherent feature of the balanced lift method. It is proposed to shape the joints, which transfer compressive forces, as cylinders. The lifting points of the compression strut bridge were the end points of the

Figure 2.

Lifting process of the tension tie bridge.

compression struts, whereas for the tension tie bridge (see Figure 3) they are the bridge girders. The joints of the field test, shown in Figure 3 were scaled versions of joints designed for real applications (scaling factor 1:10). The joints were fitted with curved steel plates (thickness 3 mm) in order to obtain rolling surfaces with accurate geometry and to enable the joints to carry high compressive stresses at the line of contact. 2 2.1

FULL SCALE TESTS ON ROLLER BEARINGS Previous tests on roller bearings

Already in the late 1930’s the application of roller bearings turned out to have many advantages. At this time the ‘‘Neckarbau’’—Department in Stuttgart developed a concept of a bridge near Obereßlingen, where joints made of concrete and fitted with steel plates serving as formwork were applied [4]. This bridge consisted of a three-hinged arch with a span of 43.3 m and a camber of 4 m. Therefore roller bearings could be used at the abutments and at the centre of the arch. The costs of the concrete roller bearings fitted with steel were even at that time lower than those of steel hinges [4]. Further practical advantages, such as the suitable transportation of the relatively lightweight steel plates, the simple assembly, the precise adjustability and the unproblematic casting of the roller bearings became evident after the first application. During the lowering of the bridge, a close contact of the steel plates was visible all along the contact line. However it was unpredictable how the behaviour of the roller bearings in the contact line affects the calculation of the Hertz pressure. Therefore, a series of comparative tests (roller bearings with and without reinforcement, reinforced concrete with and without steel plates) (see Figure 4) were carried out at the Universitiy of Technology in Stuttgart [4]. 2.2

Test program

Full scale tests on roller bearings started in July 2008 at Vienna University of Technology. The tests were planned for a variation of the most important

Figure 3. test.

Lifting points of the tension tie bridge in the field

Figure 4. plates.

Test on roller bearings with concrete without steel

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parameters radius (R = 0.5 m, 1.0 m, 2.0 m), steel plate thickness (t = 10 mm, 30 mm) and quality of concrete (C20/25, C50/60). The objective of this experimental program is to determine the influence of the design parameters on the behaviour of the roller bearings regarding the serviceability and ultimate limit states. Table 1 contains information on the complete test series with altogether 20 roller bearings. 2.3 Preliminary test The preliminary test on two roller bearings (see Figure 5) was carried out under monotonically increasing compressive loading in July 2008. The roller bearings had a radius of 2.0 m, a steel thickness of 10 mm and a concrete strength of 42.1 N/mm2 . The maximum compressive force in the test was larger than 18 MN. The roller bearings could not be destroyed completely, because the contact area between the two cylinders increased during loading through flattening of the steel plate. This test indicated that the failure of roller bearings designed to be used for the balanced lift method will occur in a very ductile manner. A clear distinction can be made between the almost linear behaviour and presumably very low rolling friction in the serviceability limit state and the ultimate limit state with high bearing capacity and also high rolling friction, due to the deformed contact surface. Table 1.

Test series of the roller bearings.

Specimens

R [mm]

t [mm]

fcm Loading [N/mm2 ] [kN]

W_2000_10_25 W_2000_30_25 W_1000_10_25 W_1000_30_25 W_500_10_25 W_500_30_25 W_1000_10_60 W_1000_30_60 W_500_10_60 W_500_30_60

2000 2000 1000 1000 500 500 1000 1000 500 500

10 30 10 30 10 30 10 30 10 30

42.1 39.2 31.4 39.2 32.1 31.4 81.9 81.9 81.9 81.9

Figure 5.

monotonic incl. rotation incl. rotation incl. rotation incl. rotation incl. rotation incl. rotation incl. rotation incl. rotation incl. rotation

2.3.1 Numerical simulation of the preliminary test In order to predict the course of the preliminary test under monotonically increasing static load, the nonlinear finite element program ATENA [5] was used for the numerical simulation. ATENA contains very advanced models for simulating concrete behaviour, but unfortunately does currently not include the possibility to solve contact problems. A 2D—model in the program ATENA was prepared. The geometric properties of the preliminary test are radius R = 2.0 m, thickness t of steel plate = 10 mm and width equal to 1.0 m. The mean uniaxial concrete strength was chosen to be equal to 42.1 N/mm2 . The flattening of the steel plate at the line of contact, which was also observed in the field tests, was modelled in the nonlinear calculation by using several analyses and by predefining the width of the contact zone. The loading was applied in several load levels with a deformation in each case of −5 ∗ 10−5 m. On the bottom side the roller bearing itself is fixed in vertical direction and moveable in horizontal direction (see Figure 6). The ram was shaped by a linear elastic material with a calculative slightly higher elasticity modulus. Thus, a more comfortable distribution of the punctual loading is achieved. The steel plate was as well shaped by a linear elastic material with a steel quality of S235. The diagram with the calculated results (Figure 7) shows the deformation u directly under the steel plate as a function of the applied force F. Up to a compressive force of 12 MN an almost linear behaviour of the roller bearing can be observed. The displacement u directly under the steel plate amounts to 0.6 mm at this load stage. Upon further loading of the roller bearing the nonlinear material behaviour of concrete causes a disproportionate increase of the deformations. In the numerical analysis the maximum force F was equal to 15 MN. 2.4

Further tests on rotating roller bearings

The preliminary test with monotonically increasing compression force showed that the tests can be carried

Figure 6. Model of the preliminary test on roller bearings with bearings conditions.

Preliminary test on roller bearings.

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Figure 9. Roller bearings in a rotation position with a compressive force of 10000 kN.

Figure 7. Force—deformation curve and distribution of deformations at F = 12 MN.

Furthermore 2D models for the rotating roller bearings were prepared in the program ABAQUS. This program is more specialized in modeling contact problems. Concrete is shaped by a linear elastic material and the steel plate with elastic plastic behaviour. The results will be the stresses at the contact line, the deformations during the loading levels and the length of the contact line of the steel plates in the last load step.

3

Figure 8.

System with the rotating roller bearings.

out in full scale and yielded also valuable information on the behaviour of roller bearings in the limit states. However, a direct determination of the size of roller friction as a function of the Hertz pressure at the line of contact is not possible using the test set—up of the preliminary test. In order to model the actual situation of the roller bearings during the erection of a bridge according to the balanced lift method more closely the test set—up was changed in order to permit rotations of the roller bearings during compression loading (see Figure 8). With the aid of the four hydraulic jacks the bearings can be rotated to a top position with a relative rotation of 8 degrees. Then the compressive force is applied and the roller bearings are rotated to a lower position with 16 degrees relative rotation. Figure 9 shows the roller bearings with radius 0.5 m, steel thickness of 30 mm and concrete strength of 81.9 N/mm2 with a compression load of 10000 kN.

CONCLUSION

In the course of the development of a new bridge construction method it turned out that roller bearings made of concrete with a steel plate on the outside surface would be an economic solution to transfer high compressive forces and at the same time to permit large rotations. Since no design equations are available for such bearings numerical simulations and full scale tests were carried out on 20 specimens. It can be concluded that the roller friction of the bearings is very low in the foreseen service load condition during construction, that the ultimate strength of such bearings is high and that the bearings fail is a ductile mode.

REFERENCES [1] Kollegger, J., Tilt—Lift Method for Erecting Bridge, World Intellectual Property Organisation, WO 2008/022359, 2008. [2] Kollegger, J. & Blail, S. 2008. Structural Engineering International, 3/2008, Balanced Lift Method for Bridge Construction: 283–289. [3] Blail, S. & Kollegger, J. 2009. Feldversuche mit dem Brückenklappverfahren, Beton- und Stahlbetonbau 104, Heft 2 (2009): 97–104. [4] Burkhardt, E. 1933. Die Bautechnik—11. Jahrgang, Heft 48, Betongelenke mit gepanzerter Wälzfläche: 651–658. Berlin. [5] Cervenka Consulting, 2001. Computer Program for Nonlinear Finite Element Analysis of Reinforced Concrete Structures ATENA—Revison 2001/5.

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Numerical crack modelling of tied concrete columns C. Bosco, S. Invernizzi & G. Gagliardi Politecnico di Torino, Torino, Italy

ABSTRACT: In the present paper the problem of monotonically compressed concrete columns is studied numerically, accounting for transverse steel reinforcement and concrete cracking. The positive confinement effect of the ties on the core concrete is modeled explicitly and studied in the case of distributed or concentrated vertical load. The main aim is to investigate the influence of transverse reinforcement steel characteristics on the column load carrying capacity and ductility, in order to provide an evaluation about some standards requirements about the class and ductility of steel to be used for ties. 1

INTRODUCTION

It is well known that, when particular collapse mechanisms are taking place in the structure, the concrete columns must provide not only a given strength but also the necessary ductility in order to allow for sufficient energy dissipation without a sensible reduction of the bearing capacity. This becomes crucial in the case of seismic repeated loading. The ductility of concrete columns is basically a function of the concrete class of resistance, of the amount and mechanical characteristics of the longitudinal reinforcements, as well as of the amount, characteristics and position of the transverse reinforcement, i.e. the ties. The ties provide a level of confinement of the column concrete core that depends also on the shape of the column cross-section (e.g. circular or rectangular). In recent years, several authors carried out experimental tests, e.g. (Ki-Nam et al. 2006, SunKyoung & Hyun-Do 2004), and some prescriptions have also been incorporated in National Standards (e.g. EC8, ACI 318 and NZS 3101). In the present paper the results of a set of finite element numerical simulations are presented. The numerical analyses have been carried out with the commercial finite element code DIANA9.3. The presence of transverse reinforcement has been explicitly taken into account, allowing for the assessment of the influence of the ties arrangement. In the present case, perfect-bonding special embedded reinforcement elements have been used. In addition, no-bonding or a given shear-slip law can be considered for the reinforcement. Thanks to a smeared crack approach based on total strain, it has been possible to obtain the correct crack pattern, included the ability to model the spalling of concrete parts outside the confined core. Some of the experimental results available in the literature concern columns loaded on a limited area of the cross section. This aspect has been considered

in details, since many convergence difficulties arise in this case. In fact, at the border of the loaded area the classical discontinuities of contact mechanics are detected in the stress and displacement fields. Moreover, in the transition zone the finite element reach a critical condition both for compression and tension (and cracking), with the consequent local instabilities. This problem has been overcome adopting a spring bedding in correspondence of the loaded area. Furthermore, it has been investigated the influence of tie class of ductility on the overall ductility of the column. This aspect is particularly relevant to some European country where the National Standards ask for some questionable ductility requirements for the transverse reinforcement. In general, the proposed numerical approach provides a way for the interpretation of the experimental results and a valuable strategy for the study and design of non-standard cases. 2

EXPERIMENTAL EVIDENCES

When the load is applied to the structure acting on a very limited area (concentrated load) a transition zone (discontinuity) can be recognized, where a complex three-dimensional stress state takes place. In this case (Figure 1), three main critical areas can be considered. The region right below the loaded area, characterized by high compressive stresses; the region aligned with the load axis, characterized by tensile tractions (bursting stresses) that tend to split the element, and the surface region around the loaded area, also subjected to tensile stresses which are responsible for surface cracking and possible expulsion of material (spalling). The phenomenon was firstly analyzed by Morsch (1924) who introduced the strut and tie model. The approach was based on the equilibrium of internal forces (compression or traction) with linearized

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Figure 1.

Fotoelasticity results obtained by Tesar (1932).

trajectories. In this way, it is possible to calculate the resultant of the tractions normal to the external load, i.e. the bursting force. In the fifties, Guyon (1953) provided a theory based on the analogy of the symmetric prism, and on the fotoelasticity experimental results of Tesar (1932). In later time, those results were also confirmed numerically by Yettram (1969). Given the bursting force Z, which is the integral of tensile stresses, it is possible to design the tie reinforcement that has to be put in place to avoid the splitting of the element region close to the concentrated load. The influence of tie reinforcement distribution was studied by Adeghe and Collins (1986) by means of a nonlinear finite element calculation. They emphasized that if the reinforcement is not adequately smeared over the splitting region, the behavior can be quite fragile. In addition, they showed that due to the limited diffusion of stresses allowed in the nonlinear analysis, with respect to the linear one, the position of the bursting force is closer to the loaded area and, therefore, the reinforcement area has to be increased. In more recent times it has been recognized that the effect of the transversal reinforcement is not limited to sustain the tensile forces that originate below the loaded area, but pertain also a favorable threedimensional compressive stress state of the concrete below the loaded area (Figure 2). This effect has been accounted for both in the CEN (2004) Eurocodes and in the CEB-FIB (1993) Model Code, by means of empirical expressions for the increased strength and increased ultimate strain. The increased ultimate strain, i.e. ductility, plays a crucial role especially in the case of the seismic resistance of columns. This effect has been investigated experimentally by several Authors. Madner et al. (1988) considered circular and rectangular cross section columns, varying the amount of transverse reinforcement. They emphasized that in addition to the increase of the compressive strength and ultimate deformation, the tie reinforcement provides a great increase of the dissipated energy. Sheikh & Uzumeri (1980), considering concrete with a cylindrical strength comprised between 30 MPa and 40 MPa, noted that the positive effect

Figure 2. Scheme of the concrete confinement with cicular (a), and rectangular ties (b).

of confinement takes place starting from a mean axial deformation of 0.0015. If the deformation is increased, the inner concrete core carries increasing stresses, while the outer part progressively unload. At a deformation of approximately 0.004, the outer concrete is spalled off from the column. The use of smooth rebars does not influence sensibly the behavior of the columns, since the spalling of the outer concrete vanishes the exchange of shear stresses between steel and concrete. On the other hand, a key role is played by the tie arrangement, which can sensibly improve the confinement in rectangular cross section columns. In recent years, the effect of confinement in selfcompacting and or high strength concrete has also been investigated (Khayat et al. 2001). In this case, the positive confinement effect is less sensible due to the limited Poisson’s coefficient. Finally, experimental results have been made available about the confinement of concrete with fiberreinforced composites (Karantzikis et al. 2005).

3

MONOTONIC COMPRESSION OF COLUMNS

The numerical simulations have been carried on considering a square concrete column 600 mm by 600 mm, with 1200 mm height. A regular quadratic brick element mesh was adopted. Four concentrated load

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configurations have been considered (Figure 3) in addition to distributed load. Embedded reinforcement has been added to the mesh (Figure 4). Two different arrangements were considered: square ties, and square ties with additional reinforcement ligaments between the tie midpoints.

3.1

Compressive behavior of concrete

Concrete subjected to compressive stresses shows a pressure-dependent behavior, i.e., the strength and ductility increase with increasing isotropic stress. Due to the lateral confinement, the compressive stressstrain relationship is modified to incorporate the effects of the increased isotropic stress. Furthermore, it is assumed that lateral cracking influences the compressive behavior. To model the lateral confinement effect, the parameters of the compressive stress-strain function, fc and εp , are determined with a failure function which gives the compressive strength as a function of the confining stresses in the lateral directions. The base function in compression can be modeled with a number of different curves, included the Thorenfeldt and the parabolic curve. The class of the concrete used in the analyses was C30. 3.1.1 Lateral confinement The increase of the strength with increasing isotropic stress is modeled with the four-parameter Hsieh-TingChen failure surface, which is defined as: f = 2.0108

√ J2 J2 + 0.9714 fcc2 fcc

+ 9.1412

fc1 I1 + 0.2312 − 1 = 0 fcc fcc

(1)

with the invariants J2 and I1 defined in terms of the stress in the concrete, and fc1 the maximum principal stress (DIANA, 2008). The parameters in (1) can be determined by fitting of experimental data on concrete specimens.

Figure 3. Geometry of the column (a); different loaded areas: a/d = 0.2, 0.3, 0.4, 0.5 (b).

3.1.2 Lateral cracking In cracked concrete, large tensile strains perpendicular to the principal compressive direction reduce the concrete compressive strength. The compressive strength is consequently not only a function of the internal variable in a certain direction, but also a function of the internal variables governing the tensile damage in the lateral directions. If the material is cracked in the lateral direction, the strength parameters are reduced with the factor βεcr for the peak strain, and with the factor βσcr for the peak stress, which are functions of the average lateral damage variable. The relationship for reduction due to lateral cracking is the model according to Vecchio & Collins (1993), shown in Figure 5. 3.2

Figure 4. Discretization mesh obtained exploiting symmetry and position of the ties.

Cracking of concrete

The behavior of concrete subjected to tension has been modeled with the smeared cracking approach. A formulation based on the so-called total strain crack model was adopted (DIANA 2008), which combine

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10

Stress [MPa]

0 -4

-3

-2

-1

0

1

-10 -20 -30 -40

Strain [‰]

Figure 5. Thorenfeldt and parabolic constitutive laws (DIANA, 2008). Figure 8.

Stress-strain constitutive law for concrete.

600

Stress [MPa]

500 400 300 Type A (H=1.2)

200

Type C (H=1.05) 100

Ideal plasticity

0 0

Figure 6. Influence of lateral confinement on compressive stress-strain curve (DIANA, 2008).

3.3

60

80

100

120

Bilinear constitutive law for reinforcement steel.

Reinforcement

As far as the reinforcement is concerned, a bilinear constitutive law was adopted. The initial linear part of the diagram is characterized by a Young’s modulus Es = 200 GPa, up to the yielding stress fy = 435 MPa. Three different case have been considered for the inelastic part, namely: perfect plasticity, 1.2 linear hardening up to the limit deformation of 10% (representative of type C steels) and 1.05 linear hardening up to the limit deformation of 5% (representative of type A steels). The constitutive laws for steel are shown in Figure 9.

4 the tensile and compressive behavior in a unique relationship. This constitutive model is based on the Modified Compression Field Theory, originally proposed by Vecchio & Collins (1986). Due to the amount and disposition of the reinforcement in the columns under consideration, the coaxial concept was adopted, which allows for a rotation of the crack set following the principal stress redistribution. The behavior of concrete in tension is linear up to the reaching of the tensile strength (in the present case ft = 2.9 MPa), while the post peak behavior is linear (Figure 8).

40

Strain [‰]

Figure 9.

Figure 7. Peak stress reduction factor due to lateral cracking (DIANA, 2008).

20

NUMERICAL RESULTS

When the load is applied on a limited loading area, many convergence difficulties can arise. In fact, at the border of the loaded area the classical discontinuities of contact mechanics are detected in the stress and displacement fields. In addition, in the transition zone the finite element reach a critical condition both for compression and tension (and cracking), with the consequent local instabilities. This problem has been overcome adopting a spring bedding in correspondence of the loaded area.

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Plain concrete

z/d

The first set of simulations has been carried on taking into consideration a plain concrete column. Six different loading area ratios are studied, namely a/d = 0.2; a/d = 0.3; a/d = 0.4; a/d = 0.5; a/d = 0.6 and a/d = 0.7. Three main reference longitudinal deformations are chosen to represent the stress state in the column, 0.2%, 2% and 5% respectively. In particular, 0.2% is chosen as a reference to the initial loading range. Figure 10 shows the evolution of tensile stress, in the y direction, along the z-axis for a/d = 0.5. The curves are calculated starting from the symmetry axes (x/d = 0), and approaching the surface of the column (x/d → 0.5). On the other hand, Figure 11 shows how the tensile stress varies along the x-axis with reference to different relative depth. The two diagrams agree well with the experimental results from photoelasticity (Tesar 1932) and with the numerical results obtained by Yettram (1969). 2 1,8 1,6 1,4 1,2 1 0,8 0,6 0,4 0,2 0 0,00

The integration of the tensile stress field provides the bursting force Z. Figure 12 compares the magnitude of the numerical bursting force normalized with respect to the applied axial force P, with the empirical results from Morsch (1924) and Guyon (1953). In Figure 13 the comparison is shown between the maximum tensile stresses. Figure 14 shows the comparison considering the position of the tensile 0,3

Morsch (1924) Numerical results Guyon (1953)

0,2

Z/P

4.1

0,1

0 0

0,2

0,4

0,6

0,8

a/d

Figure 12. Comparison between the normalized bursting force (Z/P) obtained numerically and the results from Morsch (1924) and Guyon (1924) for different load ratios (a/d). 0,8 x/d=0

Numerical results

x/d=0,1

Yettram & Robbins (1971)

0,6

x/d=0,2

Guyon (1953)

x/d=0,3

0,4

x/d=0,4

0,05

0,10

0,15 0,20 σyy / σ0

0,25

0,2

0,30

0,0

Figure 10. Normalized tensile stresses (in the y direction) versus normalized height (z/d). Diagram along the symmetry axis (x/d = 0) and approaching the column surface (x/d → 0.5).

yy

0,00 0,0 0,1

0,05

0,10

0,15

/

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

a/d

Figure 13. Normalized maximum bursting tensile stress as a function of the loaded area (a/d).

0

0,20

0,25

0,30

0,5

0,35

0,4

z/d=0,5

0,3

z/d=0,4

x/d

0,2 0,3

z/d=0,3 z/d=0,2

0,2

Yettram & Robbins (1971)

0,1

Guyon (1953) Numerical Results

0

0,4

0

0,2

0,4

0,6

0,8

a/d

0,5

Figure 11. Normalized tensile stresses (in the y direction) versus normalized abscissa (x/d). Diagrams for increasing normalized depth (z/d).

Figure 14. Comparison between the normalized positions of the bursting force, and results from Yettram & Robbins (1971) and Guyon (1953).

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stress resultant. It can be noticed that the numerical calculation generally provides higher values of the maximum tensile stress, while the resultant is often located closer to the loaded area. As the mean deformation of the column increase, it is possible to observe the migration of tensile stresses due to the cracking and damage below the loaded area. This phenomenon can be clearly detected in the diagram of Figure 15. In addition, the magnitude of the bursting force first increases for increasing deformations, and then decreases (Figure 16). With this respect, it is also interesting to note that the Morsch (1924) prevision is more accurate for low levels of deformation.

2,0 1,8 1,6 1,4 1,2 1,0 0,8 0,6 0,4 0,2 0,0

Figure 17. First principal tensile stress (on the symmetry axis): comparison between the plain and reinforced case (a/d = 0.5).

2‰ 20‰

1200

50‰

1000

Height [mm]

z/d

4.1.1 Square ties When ties are added to the column, the damage below the loaded area is greatly limited. Due to the presence of transverse reinforcement an almost constant region bearing the tensile stresses takes place. In addition, in the lower part of the column the horizontal stresses become negative, providing a favorable three-dimensional stress state.

This phenomenon can be confirmed looking at the confinement stresses in Figure 18 that shows the second principal component of stress, calculated on the symmetry axis. The curves are oscillating due to the tie spacing. The minimum stresses (i.e. maximum confinement effect) are reached in correspondence of the positions of the ties. The axial stress in the transverse reinforcement is shown in Figure 19, where the abscissa equal to zero refers to the tie midpoint, while 270 mm is the tie

800 600 2‰

400

20‰

200

0,0

0,1

0,2

0,3

0,4

0,5 yy

/

0,6

0,7

0,8

0,9

50‰

1,0

0

0

-14

-12

-10

-8

-6

-4

-2

0

2

4

Stress [MPa]

Figure 15. Migration of the normalized principal tensile stresses (calculated on the symmetry axis) away from the loaded area (a/d = 0.5).

0,20

Figure 18. Confinement stresses in correspondence of increasing mean column deformation. The oscillations correspond to the tie spacing.

2‰ 20‰

0,18

50‰

Z/P

0,16

Morsch (1924)

0,14 0,12 0,10 0,1

0,2

0,3

0,4

0,5

0,6

0,7

a/d

Figure 16. Bursting force for increasing mean deformations of the column, for different loaded areas compared with Morsch (1924).

Figure 19. C and A).

Evolution of the stress state in the ties (steel type

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corner. It is not possible to appreciate any sensible difference when different types of steel are used. The same behavior appears in term of the load bearing capacity of the column (Figure 20). Therefore, it can be said that an improved ductility class of steel does not provide a valuable additional strength and ductility, nor dissipated energy. Figure 21 shows the evolution of the crack pattern. It is well recognizable the spalling mechanism which affects the outer region of the concrete column.

Figure 23. Tie strain decrease due to additional ligaments.

Figure 20. Load displacement curve: plain concrete, reinforcement type A, and reinforcement type C.

4.1.2 Square tie with additional ligaments When the ties arrangement is changed, e.g. adding two additional ligaments between the tie midpoints, the confinement effect is sensibly improved. This can be appreciated in term of load bearing capacity and dissipated energy (Figure 22). A quantitative assessment can be obtained also considering the magnitude of tensile stresses in the tie in the two cases (Figure 23). On the other hand, the effect of the tie steel class of ductility is negligible.

5

Figure 21. Evolution of the crack pattern in one quarter of the column.

CONCLUSIONS

A numerical study of the problem of monotonically compressed concrete columns has been presented, accounting for transverse steel reinforcement and concrete cracking. The positive confinement effect of the ties on the core concrete has been modeled explicitly and studied in the case of distributed or concentrated vertical load. The obtained results show that the influence of transverse reinforcement steel class of ductility is negligible both on the column load carrying capacity and on its ductility. Also the dissipated energy is basically unchanged. In view of these evidences, some standards requirements about the steel class of ductility to be used for ties appear to be rather questionable.

REFERENCES

Figure 22. Load displacement curve in the case of ties with additional ligaments.

Adeghe L.N. & Collins M.P., 1986, A finite element model for studying reinforced concrete detailing problems, Department of civil engineering, University of Toronto, Publ. No. 86–12. CEB-FIP, 1993, Model Code 1990, CEB Bulletin No. 213–214, Lausanne. CEN, 2004, Eurocode 2: Design of concrete structures, Brussels, Ref. No. EN 1992-1-1:2004: E. Guyon Y., 1953, Prestressed Concrete, John Wiley and Sons, New York.

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Karantzikis M., Papanicolaou C.G., Antonopoulos C.P., & Triantafillou T.C., 2005, Experimental Investigation of Nonconventional Confinement for Concrete using FRP, Journal of Composites for Construction, 9(6): 480–487. Khayat K.H., Paultre P., & Tremblay S., 2001, Structural Performance and In-Place Properties of Self-Consolidating Concrete Used for Casting Highly Reinforced Columns. ACI Materials Journal 102(4): 560–568. Ki-Nam H, Sang-Hoon H, & Seong-Tae Y, 2006, Highstrength concrete columns confined by low-volumetricratio lateral ties, Engineering Structures 28: 1346–1353. Mander, J.B., Priestley, M.J.N., & Park. R., 1988, Observed stress-strain behavior of confined concrete. J. Struct. Engrg., ASCE 114(8): 1827–1849. Sheikh, S.A., & Uzumeri, S.M., 1980, Strength and ductility of tied concrete columns. J. Struct. Div., ASCE, 106(5), 1079–1102.

Sun-Kyoung H, & Hyun-Do Y, 2004, Effects of transverse reinforcement on flexural behaviour of high-strength concrete columns, Engineering Structures 26:1–12. Tesar, M., 1932, Determination expérimentale des tensions dans les extrémités des pièces prismatiques munies d’une semi-articulation, Abh IVBHI. Vecchio, F.J., & Collins, M.P. 1986, The modified compression field theory for reinforced concrete elements subjected to shear. ACI Journal 83(22): 219–231. Vecchio, F.J., & Collins, M.P. 1993, Compression response of cracked reinforced concrete. J. Str. Eng. ASCE 119(12): 3590–3610. Yettram A.L., 1969, Anchorage Zone Stresses in Axially Post-tensioned Members of Uniform Rectangular Section, Magazine of Concrete Research 21(67): 103–112.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Numerical study of a massive reinforced concrete structure at early age: Prediction of the cracking risk of a massive wall L. Buffo-Lacarrière & A. Sellier Université de Toulouse, UPS, INSA, LMDC (Laboratoire Matériaux et Durabilité des Constructions), Toulouse Cedex 04, France

ABSTRACT: The present paper deals with the prediction of the cracking risk of massive structures at early age. The proposed model is presented through the case study of a massive wall instrumented by the French company EDF. First, the hydration of binder is predicted using a multiphasic model that takes account of temperature and moisture effects on the kinetic reaction of several phases (cement, pozzolanic additions). Then, the results of this physico-chemical model are used in a non linear creep model associated to anisotropic damage in order to evaluate the cracking risk of the structure studied. 1

INTRODUCTION

The work deals with the assessment of early age cracking in massive concrete structures. Firstly an original hydration model is presented; it has the particularity of taking into account, at the structural scale, a coupling not only with temperature but also with drying. Next, the coupling with a mechanical model is clarified to predict crack localization and their opening. It considers the stress relaxation due to the creep at early age together with anisotropic damage. The application to an instrumented thick wall was carried out in the framework of the French research program CEOS.FR (ref). It points out the benefits and deficiencies of the approach, which will lead to a complementary research program being carry out in CEOS.FR in the coming years. 2

nuclear plant. For the early age behavior of the structure, the concrete may be subject to damage because of the heat released by the cement hydration. Therefore, EDF chose to build an experimental wall, representative of the nuclear plant walls, in order to evaluate the risk of early age cracking for the two concretes envisioned: one cast with an ordinary Portland cement (C11 in Table 1) and the other cast with cement and silica fume (HPC in Table 1). z y x

Tubes for prestressing

PRESENTATION OF THE STRUCTURE STUDIED: EDF CIVAUX MASSIVE WALL

10 m The field structure chosen for this study was a massive experimental wall built by EDF on the site of the Civaux nuclear power plant in France. The wall is 1.2 m wide, 1.9 m high and 20 m long and has a plane of symmetry at mid-length that justifies modeling only one half of the structure (Figure 1). The wall is reinforced by two reinforcement planes located at 5 cm from the lateral surface, and contains vertical, regularly spaced holes reserved for the prestressing cable guide tubes as presented in Figure 1. 2.1

Material

The concrete used for this structure was designed to withstand the high stresses that can be observed in a

Figure 1. Table 1.

Mesh of the half structure. Concrete formulations.

(kg/m3 )

Cement Silica fume (kg/m3 ) Water (kg/m3 ) Adjuvant (kg/m3 ) Sand (kg/m3 ) Aggregates (kg/m3 )

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HPC

C11

266 40.3 161 9.08 873 1099

350 0 195 1.22 769 1072

3

NUMERICAL PREDICTION OF THE PHYSICOCHEMICAL CHANGES

The wall behavior was assessed with the following model implemented in the Castem finite element code (Cast3M). This model has the specificity of being able to take account of coupling, not only with the temperature but also with the moisture. It is described below. 3.1

Multiphasic hydration model

The variation of degrees of hydration (Eq. 1) is thus coupled with the water mass balance equation (Eq. 2) and heat balance equation (Eq. 3). ⎧  α, W , T ) ⎪ ⎨α˙ = F( −−→  W · α˙ ˙ W = div(DW · grad W ) + Q th ⎪ −→ ⎩ρc · T˙ = div(λ · −  T · α˙ grad W ) + Q th

(1)

– Thermal activation: Temperature effects on the hydration kinetics are reproduced by means of the conventionally used Arrhenius law (Arrhenius, 1915). This modeling approach can, at any instant, combine the effects of temperature and water content on the hydration of several solid phases. As recommended binders are increasingly composed of various products having different heat activation and different hydration kinetics, it is necessary to model the hydration development of these binders using separate phases (clinker, silica fume, . . . ). This splitting of the reaction kinetics is one of the interesting aspects of this model, especially in the case of concrete cast in hot weather, for which high initial and external temperature will activate the hydration development of clinker and pozzolanic components differently (different activation energy). 3.2

where: − → – X = (X1 · ·Xi · ·Xn ) and (i = 1→clinker, i = 2, .., n → pozzolans), – αi is the degree of hydration of phase ‘‘i’’ – W is the total water content of concrete – Dw is the water transfer coefficient (modeled by the law proposed by Mensi et al. 1988 with a variation of the porosity according to hydration) W – Qth i is the water needed for total hydration of phase ‘‘i’’ – T is temperature – λ is thermal conductivity – ρ and c are concrete density and specific heat T – Qth i is the heat produced by complete hydration of phase ‘‘i’’ The hydration kinetics law presented in equation (1) is composed of several functions modeling each particular aspect of hydration development. This multiphasic model is described in (Buffo-Lacarrière et al. 2007) and only the general chemophysical effects modeled are presented below: – Chemical activation: In the first stages of hydration, when the concrete can be considered as a diluted medium, reactions are accelerated by supersaturation of the interstitial solution. – Water accessibility to anhydrous phase: During hydration, reaction kinetics are decreased due to the difficulty of contact between water and anhydrous components caused by the progressive formation of hydrated products. – Secondary pozzolanic reaction delaying parameter: In the case of pozzolanic component hydration, the reaction kinetics depends on the progression of the primary reaction (depending on the portlandite content).

Parameter fitting for the two binders

The parameters of the hydration kinetics law for each anhydrous phase were determined using a quasi adiabatic test performed with a Langavant calorimeter. For calibration, a least squares method was used to determine the material parameters that allowed the experimental hydration heat curve to be reproduced. Thus, a calibrated law was obtained using only accumulated heat results of the quasi adiabatic test on the chosen binder, results which are usually provided by cement manufacturers. Moreover, as the phenomenological law contains only three calibration parameters for each kinetics law, the calibration by means of a standard least squares method is easy. For binder with a binary composition (such as HPC), the parameters of the cement used as the basis for the binder are first determined in a calorimetric test performed on the pure cement. Then, the parameters of mineral additions (silica fume in HPC) are deduced from calorimetric tests performed on the composed binder. 3.3

Thermal boundary conditions

At early age, the wall is encased with a plywood formwork. The thermal exchanges between the concrete surfaces and the surroundings are thus influenced by this formwork and the convective exchanges at the concrete surface are modeled by the following equation: −−→ [Heq (Text − Tface )] · n − λcon · grad T = 0

(2)

where: – Text and Tface are the ambient and block surface temperatures respectively – Heq is the equivalent heat transfer defined by Eq. (3).

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ef 1 1 = + Heq Hair λf

and Hair = a · Swb

(3)

where: – λf and ef are the thermal conductivity and the thickness, respectively, of the formwork – Hair is the convective exchange coefficient in air determined according to the wind speed Sw (expressed in m/s), in which a and b are model parameters determined in (McAdams, 1954). The external temperature was measured from concrete casting onwards and the variation observed is illustrated in Figure 2. 3.4

are presented in Figures 5 and 6. The model correctly reproduces the temperature in the field. It can be seen that, because of its large dimensions, the structure undergoes a large increase in temperature, mainly in the core. The surfaces experience thermal transfer by convection with the environment and have a lower temperature than the core.

1.0 0.8 0.6

C

0.4

SF

Results and discussion

After the fitting of hydration parameters on a Langavant test (quasi adiabatic test), the multiphasic hydration model was applied to the structures with the mesh presented in Figure 1 and in situ conditions (see thermal boundary conditions in previous section). Note on Figure 1 that the soil is partly modeled in order to reproduce the heat transfers in the bottom slab and in the base of the wall. The variations obtained for the hydration degrees of cement (and silica fume for HPC) in the structures are given in Figure 3 and Figure 4. The temperature variations obtained for HPC- and C11-based structures at the core and near the surface

Value

λf (J/(h.m.K)) ef (m) Sw (m/s) a b

2100 0.02 1.44 7.3 0.78

0.0 0

40

80

120

Figure 3. Variation of hydration degree of clinker (C) and silica fume (SF) for HPC structure.

1.0 0.8 0.6 0.4

Table 2. Thermal boundary condition parameters. Parameter

0.2

0.2 0.0 0

40

80

120

Figure 4. Variation of hydration degree of cement for C11 structure.

80 T (˚C)

25 Text (˚C) C11 HPC

20

60 at core

15

40 10

20 near the surface

5 Time (h) 0

20

40

60

80

100

Time (h)

0

0

0

120

Figure 2. Variation of external temperature for the 2 structures.

20

40

60

80

100

120

Figure 5. Variation of temperature for HPC in the core and near the surface (line: model; rectangles: experiments).

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where:

80 T (˚C)

– ε is the strain, – T is the temperature, – β is the thermal dilatation coefficient (parameter Pi ) (=10−5◦ C−1 ), – Cc is the consolidation variable, – k and μ are the bulk and shear moduli respectively (parameter P i ), – η is the viscosity (parameter P i ). – X˙ − = X˙ < 0, 0 otherwise.

at core

60

40

20

near the surface

Time (h)

0 0

20

40

60

80

100

120

The consolidation variable Cc expresses the fact that the viscosity is increased when the C-S-H layers come closer together during settling. The consolidation variable is thus defined in the proposed model by Eq. (5) (Sellier & Buffo-Lacarrière 2009).

Figure 6. Variation of temperature for C11 in the core and near the surface (line: model; rectangles: experiments).

4 4.1

NUMERICAL PREDICTION OF THE MECHANICAL BEHAVIOR AT EARLY AGE



(s)

|εM | Cc = exp k εM

Mechanical model for hardening concrete

The cracking risk associated with these physicochemical variations was assessed using a non-linear mechanical model combining a non-linear creep model and an anisotropic damage model. It has been shown that, at early age, it is necessary to take account of creep phenomena in order to avoid an overestimation of the stress (Benboudjema & Torrenti 2008). The proposed creep model was inspired by the physical observations of Acker & Ulm (2001). The stress is split into spherical (superscript (s) in equations (4)) and deviatoric (superscript (d)) parts in order to reproduce the different creep behavior under these two kinds of load. Each part of the stress is evaluated using a system including an elastic level (0 in Eq. (4)) for instantaneous behavior, and two creep levels. The first creep module is a Kelvin-Voigt viscoelastic module, which reproduces the reversible creep (KV in Eq. (4)), and the second is a Maxwell module with a non-linear viscosity, which models the irreversible creep (M in Eq. (4)). The model is mathematically expressed by the equation set (4) for the spherical part of the stress (for the deviatoric part, the bulk modulus (k) must be replaced by the shear modulus (μ) and superscript (s) by (d)). In this system, the elastic parts of the behavior laws are written in the adaptive incremental form presented in (Buffo-Lacarrière & Sellier 2009) in order to adapt to the mechanical behavior of concrete subjected to hydration.

 (5)

where: (s) – εM is the strain of the Maxwell module associated with the medium part of the stress, k is a characteristic strain that controls the consol– εM idation rate.

⎧˙ 0 0 σ˜ (s) = k 0 ε˙ (s) + k˙ 0 − ε(s) ⎪ ⎪ ⎪ ⎪ ⎨σ˜˙ (s) = k KV ε˙ KV + k˙ KV εKV + η˙ KV ε˙ KV + ηKV ε¨ KV (s) (s) (s) (s) (s) (s) ˙ (s) = (η˙ M · Cc + ηM · Cc) ˙ ε˙ M + (ηM · Cc) ε¨ M ⎪ σ ˜ ⎪ (s) (s) (s) (s) (s) ⎪ ⎪ ⎩ H KV M ε˙ (s) = ε˙ (s) + ε˙ (s) + ε˙ (s) + β · T˙ (4)

The effective stress given by this rheological model is used in an anisotropic damage model to assess the cracking risk through a damage variable as introduced in (Kachanov 1986). The proposed damage variable is inspired by (Sellier & Bary 2002) and is described in (Sellier 2006, Buffo-Lacarrière 2007). It is based on a tensorial writing of damage associated with a Rankine criterion in tension and on a scalar damage variable associated with a Drucker-Prager criterion in compression. In the framework of hardening concrete, the damage variables depend not only on the effective stress state but have also to be updated at each time step according to the variation of hydration degree. This updating leads to a decrease in mechanical damage under the effect of hydration (which leads to self healing of the cement paste) (Buffo-Lacarrière & Sellier 2009). 4.2

Variation laws for mechanical properties

The variation of the elastic characteristics according to hydration development is fitted by equation (6). These laws were inspired by the Young’s modulus variation law proposed by (De Schutter 2002, Laube 1990). 

α − αs 2/3 1 − αs  α − αs 2/3 0 0 μ (α) = μ∞ 1 − αs 0 k 0 (α) = k∞

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(6)

where: – kth and μth are the theoretical elastic characteristics of the completely hydrated concrete; they are the fitting parameters. – αs is the hydration degree that characterizes the percolation threshold (which can be considered equal to 0.15 for concrete (Byfors 1980; Torrenti et al. 2005)) – X  is the positive part of X (X if X > 0, 0 elsewhere). The variation of the creep parameters is similar to that of the instantaneous elastic parameter (Eq. (6)). The parameters related to the spherical part of the creep model are thus proportional to k0 while the deviatoric creep parameters are proportional to μ0 . This assumption has been previously proposed in De Shutter’s works (De Schutter 2002; Benboudjema & Torrenti 2008), and implies that the characteristic time of creep is independent of the hydration degree. 4.3

this leads to stresses too small to produce superficial cracks just after the formwork removal. The cracking observed on the structure 96 hours after casting (see Figure 7) was due to the global cooling of the wall that led to internal thermal contraction. As these strains were restrained by the bottom slab on the wall base, they produced tensile stresses in the entire wall, with higher values at the core (where the temperature reached its maximal value). The crack was initiated near the prestressing cable guide tubes, which constitute weak points in the structure. Figure 7 highlights the good concordance between the model results and the observations made in situ. In the proposed model, the crack opening can be evaluated by the unsteadiness of the horizontal displacement of the node around the crack (see Figure 8). The values of crack opening for different distances to the surface are shown in Table 4. It can be noted that the crack opening is partially restrained by the reinforcement near the surface and is maximal in the core.

Prediction of the risk of wall cracking

The mechanical model for hardening concrete was applied using the physicochemical model’s results as inputs (hydration degree and temperature variations). The averaged hydration degree (evaluated as a weighted average of the hydration degree of clinker and of silica fume for HPC) was used to calculate the variation of mechanical properties (Eq. (6) with the characteristics of totally hydrated concrete presented in Table 3). 4.3.1 Results for HPC For an HPC structure, the difference between surface and core temperatures observed in Figure 5 generates a system of self balancing stresses with tension at the surface and compression at the core. On this structure,

Figure 7. Longitudinal damage field (dxx ) 96 hours after casting on half structure for HPC. 20 ( m) Ux x (m) 0

Table 3. Parameters variations.

0 (GPa) k∞ μ0∞ (GPa) kKV ∞ (GPa) μKV ∞ (GPa)

(s)∞ (GPa.s) ηKV (d)∞ ηKV (GPa.s) (s) ηM ∞ (GPa.s) (d) ηM ∞ (GPa.s) k εM (m/m)

Rc∞ (MPa) Rt∞ (MPa)

for

mechanical

0

characteristic

0.2

0.4

0.6

0.8

1

-20

HPC

C11

-40

27.2 16.7 104.2 38.9

28.8 17.4 110.3 40.5

-60

3.56.107

3.77.107

2.01.107

2.10.107

9.00.107

1.72.108

1.75.109 2.8.10−5 80.1 4.13

3.06.109 10.5.10−5 61.8 4.6

-80

Figure 8. Crack opening calculation using the unsteadiness of the displacement. Table 4.

Crack opening.

Measurement location

Crack opening

At core (y = 63 cm) At y = 35 cm Near the steel (y = 7 cm)

27.2 16.7 4.13

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4.3.2 Results for C11 For the C11 wall, the behavior was slightly different and the first crack observed was a horizontal one at the base of the wall (see Figure 9) 44 hours after casting. This crack was the conjugated consequence of the restrained thermal shrinkage and the thermal gradient between core and surface. Figure 9 shows that this first crack is not prejudicial because it occurs only on the first element and it does not progress to the core due to the effect of reinforcement placed near the lateral faces. This figure also shows the initiation of vertical cracking near the the prestressing tubes (weak points). As for the HPC structure, this vertical cracking occurs by the limiting of strains due to the bottom slab. The first localized vertical crack occurs 68 hours after casting and is presented in Figure 10. The crack opening was obtained by discontinuity of the displacement. Numerical results are given in Table 5. 78 hours after casting, a second vertical crack is numerically predicted as illustrated in Figure 11. The crack opening was evaluated at this second cracking date and the results are given in Table 6. By comparing the result for crack n◦ 1 with the value obtained 68 hours after casting, we can observe an increase of this opening with time.

Table 5.

Crack opening 68 h after casting.

Measurement location

Crack opening

At core (y = 63 cm) At y = 35 cm Near the steel (y = 7 cm)

156.7 141.7 90.6

Figure 11. Longitudinal damage field (dxx ) 78 hours after casting on half structure for C11. Table 6.

Crack opening 78 h after casting. Crack opening

Measurement location

Crack n◦ 1

Crack n◦ 2

At core (y = 63 cm) At y = 35 cm Near the steel (y = 7 cm)

189.5 176.8 105.6

321 310 239

(a): C11

Figure 9. Longitudinal damage field (dxx ) 52 hours after casting on half structure for C11.

(b): HPC

Figure 12.

4.4

Figure 10. Longitudinal damage field (dxx ) 68 hours after casting on half structure for C11.

Experimental crack observation on the walls.

Experimental observations and discussion

The experimental observations of the cracking development led to the cracking map represented in Figure 12 for HPC and C11 structures. For the C11 wall, 8 cracks were observed while only one was found on the HPC structure. Some measurements of the crack opening were performed and led to a value between 40 and 500 μm for C11 and a value of 100 μm for HPC. The crack openings obtained with the proposed model globally reproduce the order of magnitude but underestimate the opening values for both structures. This underestimation may be explained by the fact that

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the modeling of adherence between steel and concrete that was not considered to be affected by hydration or damage (except indirectly by the variation of concrete tensile strength in the finite element surrounding the reinforcement mesh). The model also underestimates, for the C11 structure, the number of cracks (and thus their spacing). An explanation of this underestimation may be that the thermal dilatation coefficient was modeled as constant and equal to 10−5 whereas it is higher at very early age (just before the percolation threshold) and consequently leads to higher thermal expansion. Another way of improving the model could be a better understanding and modeling of the phenomena occurring near the percolation threshold (fluid-solid transition of the cement paste). 5

CONCLUSION

The FEM used for this application has several original features: among others, a coupling between hydration and drying at early age and an anistropic poromechanical damage modeling coupled with a creep model. Although the model is able to realistically capture crack localization and opening, the cracking assessment at early age remains difficult and, despite all the physical phenomena considered in the modeling, the crack frequency and opening remain slightly under estimated. The causes could reside in misrecognition of, on the one hand, the concretereinforcement bond at early age and, on the other hand, the thermo-mechanical properties around the percolation threshold. A complementary research program including these aspects has started as part of the follow up of the French research program CEOS.FR. ACKNOWLEDGMENTS This work was performed in the framework of the French national project, CEOS.fr (see, in French, http://www.ceosfr.org). The authors are also grateful to CEA for providing the finite element code, Castem.

Benboudjema, F. & Torrenti, J.M. 2008. Early age behaviour of concrete nuclear containments. Nuclear Engineering & Design 238 (10): 2495–2506. Buffo-Lacarrière, L. 2007. Prévision et évaluation de la fissuration précoce des ouvrages en béton. PhD Thesis, Université de Toulouse. Buffo-Lacarrière, L., Sellier, A., Escadeillas, G. & Turatsinze, A. 2007. Multiphasic finite element modeling of concrete hydration. Cement and Concrete Research 37(2): 131–138. Buffo-Lacarriere, L. & Sellier, A. 2009. Numerical modelling of effects of chemical evolution on mechanical behaviour of concrete. In Alexender-Bertron, (ed.), RILEM TC 211-PAE Final Conference. Concrete in Agressive Aqueous Environments; Toulouse, 3–5 June 2009. Byfors, J. 1980. Plain concrete at early ages. PhD Thesis, Swedish, Cement and Concrete Institute, Sweden. Cast3M, 2005. CEA—DEN/DM2S/SEMT, web page: http:// www-cast3m.cea.fr/cast3m/index.jsp. De Schutter, G. 2002. Finite element simulation of thermal cracking in massive hardening concrete elements using degree of hydration based material laws. Computers and Structures 80: 2035–2042. Kachanov, L.M. 1986. Introduction to continuum damage mechanics. Ed. Martinus Nijhoff, ISBN 90-247-3319-7, 135p. Laube, M. 1990. Werkstoffmodell zur Berechnung von Temperaturspannungen in massigen Betonbauteilen im jungen Alter. Dissertation TU Braunschweig. McAdams, W.H. 1954. Heat Transmission. McGraw Hill Series in Chemical Engineering, McGraw Book Company, New York. Sellier, A. 2006. Modélisations numériques pour la durabilité des ouvrages de Génie Civil. Mémoire d’habilitation, Université Paul Sabatier, Toulouse III, 156p. Sellier, A. & Bary, B. 2002. Coupled damage tensors and weakest link theory for the description of crack induced anisotropy in concrete. Engineering Fracture Mechanics 69: 1925–1939. Sellier, A. & Buffo-Lacarrière, L. 2009. Toward a simple and unified modelling of basic creep, shrinkage and drying creep for concrete, European Journal of Environment and Civil Engineering 10. Torrenti, J-M. & Bendoudjema F. 2005. Mechanical threshold of cementitious materials at early age. Materials and Structures 38(277): 299–304.

REFERENCES Acker, P. & Ulm, F.-J. 2001. Creep and shrinkage of concrete: physical origins and practical measurements. Nuclear Engineering and Design 203: 143–158. Arrhenius, S. 1915. Quantitative Laws in Biological Chemistry. G. Bell and Sons, London

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Numerical modelling of failure mechanisms and redistribution effects in steel fibre reinforced concrete slabs L. Gödde & P. Mark Institute of Concrete Structures, Ruhr-University Bochum, Germany

ABSTRACT: A layered finite element shell model is developed to investigate steel fibre reinforced concrete slab structures. It considers effects of anisotropic orientation and accumulation of fibres caused by manufacturing processes. The material model starts from a plasticity based damage model developed by Lubliner et al. for modelling of plain concrete. Its range of application is expanded to steel fibre reinforced concretes by elaborating material functions and material parameters. This leads to a consistent description of the nonlinear material behaviour of plain and steel fibre reinforced concrete. Comparisons of numerical results, e.g. load-deflection curves, crack pattern or cross sectional stress distributions are in good accordance to analytical and experimental data. 1

Steel fibre reinforced concrete (SFRC) is increasingly used in many fields of application in structural engineering (Falkner & Teutsch 2006, Brite Euram 2002). One field are redundant slab systems like e.g. floor constructions. Due to pronounced redistribution effects those systems can provide an overall ductile behaviour under bending stress in spite of a softening material behaviour of the SFRC. However, the system’s bearing behaviour is considerably affected by anisotropic orientations and accumulation of fibres caused by manufacturing processes. In addition to experimental investigations numerical simulations are useful to analyse the bearing behaviour of such slab systems in more detail. Therefore, a layered finite element shell model is developed to investigate steel fibre reinforced concrete structures mainly subjected to bending. The main focus is on large-sized slab structures. The model includes variability in important parameters like geometry, discretisation, loading and material properties. Additionally it accounts for effects of anisotropic orientations and accumulation of fibres.

2 2.1

Lee & Fenves (1998) for modelling of plain concrete. Based on an elasto-plastic damage theory a nonassociated flow rule is used. Starting from an additive split of strain rate (decomposition in elastic and plastic parts)

INTRODUCTION

MATERIAL MODEL Description and basic equations

SFRC is specified as a layer wise homogeneous material. The nonlinear material behaviour of plain and SFRC is modelled consistently using a continuum model (Abaqus 2008). Originally, the model was developed by Lubliner et al. (1989) and modified by

ε˙ = ε˙ el + ε˙ pl

(1)

summarised basic equations of the model read:   σ¯ = D0 (ε − εpl ) ∈ σ¯ |F(σ¯ , ε˜ pl ) ≤ 0 σ = (1 − d(σ¯ , ε˜ pl ))σ¯ ε˙˜ = h(σ¯ˆ , ε˜ pl )ε˙ˆ , pl

pl

(2) ε˙ pl = λ˙

∂G(σ¯ ) . ∂ σ¯

The stress-strain relation is governed by the elastic stiffness matrix of the material D. Assuming isotropic degradation of the initial elastic stiffness D0 due to cracking, a scalar damage variable d is used with 0 ≤ d < 1 : D = (1 − d)D0 . Consequently, effective values of stresses are defined σ¯ = σ /(1 − d). While no cyclic loads are investigated, and no closing and reopening of cracks occurs, a degradation of elastic stiffness is not considered in the simulations (d = 0). The yield function F separates states with linearelastic material response from those causing plastic strains and damage in effective stress space.

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F(σ¯ , ε˜ pl ) =

1 [¯q − 3α p¯ +β(˜ε pl ) σ¯ max  1−α − γ −σ¯ max  ] − σ¯ c (εcpl ) ≤ 0

α=

with

Fcc − 1 , 2Fcc − 1

γ =

Using effective values of p and q, the flow potential G in the non-associated flow rule shown in Equation 2, is defined  G = −¯p tan ψ + (αe σt0 tan ψ)2 + q¯ 2 . (7)

(3)

3(1 − Kc ) , 2Kc − 1

For ae = 0 the flow potential is reduced to the well know Drucker-Prager function, where the dilatation angle ψ defines the inclination to the hydrostatic axis in p-q-plane. σt0 is the initial yield stress in tension. Parameter αe controls the rate of exponential deviation of G from linear Drucker-Prager function in p-q-plane for small confining pressures. This provides a smooth form of G and an uniquely definition of the flow direction. The application range of this material model can be extended to SFRC by elaborating material functions and material parameters as shown in the remainder.

(4) pl

β=

σ¯ c (εc ) pl

σ¯ t (εt )

(1 − α) − (1 + α)

Thereby, p = −I1 /3 is the hydrostatic pressure and √ q = (3J2 ) expresses the von Mises equivalent stress (Chen & Han 1988). The yield function is adjusted to experimental data using the ratio of equibiaxial to uniaxial compressive yield stress Fcc = σcc /σc and the ratio of stress invariants on tensile and compressive meridians Kc = qTM /qCM . Evolution of the yield function through isotropic hardening or softening is controlled by the uniaxial pl material response in tension σ¯ t (εt ) and compression pl σ¯ c (εc ) depending on two corresponding plastic strains pl pl εt and εc . These two strain variables are related to the multiaxial conditions using a stress weight factor r(σˆ¯ ). Their evolution is described by h(σˆ¯ ,ε˜ pl ) shown in Equation 2. ⎡ ⎤ ˆ¯ ) 0 r( σ 0 ⎦ h(σˆ¯ , ε˜ pl ) = ⎣ 0 0 r(σˆ¯ ) − 1 ⎧ 3 ⎪ ⎪ ⎪ σ¯ i  ⎪ ⎪ ⎪ ⎨ i=1 , 3 r(σˆ¯ ) = σˆ¯ = 0 | | σ ¯ ⎪ i ⎪ ⎪ i=1 ⎪ ⎪ ⎪ ⎩ 0, σˆ¯ = 0

ε˜ pl = ⎝

⎞

⎛

pl

ε˙ 1

⎞

⎛

σ¯ 1

⎞

⎜ ⎟ ⎜ ⎟ ⎜ ⎟ pl ⎟ ˆ ⎠ ,ε˙ˆ pl = ⎜ ¯ , σ = ⎟ ⎜ σ¯ 2 ⎟ , ⎜ ε ˙ 2 pl ⎠ ⎝ ⎠ ⎝ εc pl ε˙ 3 σ¯ 3 pl

εt

x =

1 (x + |x|) 2

(6)

Thereby the eigenvalues of the plastic strain rate tensor pl pl pl are ordered to ε˙ 1 ≥ ε˙ 2 ≥ ε˙ 3 The yield function and the influences of its parameters are presented in Figure 1.

Uniaxial material response in compression

Comparing with plain concrete, practical fibre dosages do not lead to significant changes until the compressive strength fc is reached (Falkner & Teutsch 2006, Brite Euram 2002). Therefore, analytical functions which describe the stress-strain behaviour of plain concrete can be used for SFRC as well. Linear-elastic behaviour is limited to a stress level of σc0 = 0, 4fc (Fig. 1). The following hardening domain up to compressive strength is described using a material function acc. to (DIN 2008) and (Mark 2006). Furthermore, Young’s Modulus Ec , compressive strength fc , and the according strain εc1 need to be determined. A softening behaviour is not considered as it causes no appreciable effect (see paragraph 4.2). 2.3

(5)

and ⎛

2.2

Uniaxial material response in tension

Before reaching the tensile stress fct it is satisfactory to assume linear-elastic behaviour both for plain and SFRC (Fig. 1). Following with the beginning of micro-cracking in SFRC, forces can be transferred by ‘‘aggregate interlock’’ and steel fibres. The bearing contingent of concrete can be described e.g. using a stress-crack opening relation σct (w) depending on the fracture energy Gf . An approach developed by Strack (2007) is used to model bearing behaviour of steel fibres σf (w). An initial fibre induced stress σf 0 is multiplied by a coefficient ηpullout (w) to consider the reduction of the bond length and the decreasing number of fibres crossing a crack with increasing crack width w. Another factor ηbond (w) integrates fibre activation and shape-bond (Fig. 2). For simplification two efficiency factors are introduced. κI relates the initial stress σf 0 to the concrete tensile strength fct regarding fibre orientation and fibre

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influence parameter Kc

-

deviatoric plane ( 1-3 < 0)

1

Kc = 0.5 Kc = 2/3 Kc = 1.0

influence parameter Fcc

evolution by uniaxial material response

plane stress

compression

Fcc =

c domain

B

fc (-

c /0/0)

A

c0

Ec

Kc = 0.5

Kc = qTM / qCM (- c /0/0)

(-

Kc = 2/3 Kc = 1.0

q

Figure 1.

meridian plane

(- c /0/0)

(-

cc /-

cc /0)

inel

t( t

)

1

A

Fcc1 Fcc2 Fcc3

Fcc1 > Fcc2 > Fcc3

c

el c

t

ft p

2

elastic domain

B

A

e.g. plain stress

t0

B

Ec

Ec

t

inel el t t

q without stiffness degradation (d = 0):

=

, p = p , q = q and ~ pl = ~ inel

Parameters of yield function: definitions and influences.

a)

Nf

content. κII quantifies the influences of shape-bond as well as effects due to degradation of the concrete matrix or fibre failure. The total σ -w-curve of SFRC is determined by the sum of concrete and steel fibre load bearing behaviour (Fig. 2).

shapebond on cti i r f

on cti re rce i d re fo

σfct (w) = σct + σf

II

I

tension

q

p

cc /- cc /0)

Ec

inel c

TM

q

cc /- cc /0)

CM

meridian plane

(-

3

Tensile Meridian Compressive Meridian = 0˚ = 60˚

-

2

C A

e.g. plain stress

C

60˚

2

B

elastic

2

0˚

-

1

1

(0/- c/0) Fcc1 Fcc2 Fcc3

cc c

inel c( c )

w

= fct e−w·fct

shape-bond fibre activation lpeak

fibre pullout with friction bond



Gf

+ σf 0 · ηbond ·ηpullout    with κII

(8)

w

(for details see (Strack 2007)) with

lH

σf 0 (w = 0) = ηϕ Vf τb lf /df or σf 0 (w = 0) = κI fct

b)

(9) lH = lPeak

fct

In Equation 9 Vf , lf , df , ηϕ denote fibre content, length, diameter and orientation factor respectively. τb characterises the mean bond stress. Beside the fibre geometry, parameters Gf , κI , κII and lpeak have to be determined (Fig. 2).

lH lH = lf/2

2.4

Gf

w lH

Figure 2. Material response of SFRC in uniaxial tension: a) fibres only; b) fibres and concrete matrix.

Localisation and mesh dependency

Cracking of plain and SFRC as a kind of local damage is modelled by means of inelastic strains in terms of ‘‘smeared’’ cracks. Here, the typical strain softening behaviour of SFRC induces strain localisations in ‘‘cracked’’ zones which may lead to severe mesh

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dependencies. To minimize mesh dependency, the material functions for uniaxial tension are formulated as stress crack-opening relations, based on the ‘‘Fictitious Crack Model’’ (FCM) of Hillerborg (1980 & 1983) as shown in Figure 3a. The numerical implementation requires the introduction of a parameter often called characteristic length lch (Oliver 1989). It relates the fracture energy Gf (per unit of area) to the specific energy gf (per unit of volume) at the integration points of the ‘‘cracked’’ elements. Consequently lch can be interpreted as the length over which the crack width is ‘‘smeared’’ at the element’s integration points. lch =

Gf gf

with

εcrack = εCB − εel = εCB −

lch =

or

w εcrack

(10) σt Ec

(11)

According to that definition, lch is not a material property but a pure numerical parameter. It transfers the discrete crack width w of the analytical FCM into a ‘‘crack-strain’’ εcrack in the numerical continuum model. lch depends on form, type, and integration rule of the used elements as well as on crack orientation (Oliver 1989). It can be determined by relating the a)

FCM acc. to Hillerborg l/2 fct t

analytical energy dissipation in the cracked zone to the numerical solution (cp. Eq. 10). Especially for irregular meshes and arbitrary crack directions this may lead to complex determinations. In Figure 3a the determination of lch is presented for the case of localisation in a rectangular crack band parallel to the crack and with a consistent strain εCB inside the band. Such a foramtion arises in all simulations presented in paragraph 5. Here, it results from Equation 10 that lch equals the length of the crack band lCB independent of the discretisation inside the band. For verification the load-deformation behaviour of a tensile bar with constant stress field is determined analytically as well as numerically using the FCM (Figs 3a, 3b). Localisation is induced artificially by a small reduction of tensile strength. Thereby, the crack band is modelled with different discretisations and dimensions. The results are presented in Figure 3b and show accordance of the analytical and numerical solutions. It can be conducted the material model provides objective results independent from the used mesh. Further, the characteristic length not only depends on the element properties and the crack direction, but also on the modelling of the problem and the resulting form of the crack band. Hence, lch needs to be determined with respect to this, and should be checked carefully with regard to the developed localisation.

numerical implementation with smeared cracks l/2 geometry: t

• l = 120 mm

f*ct t

l

fct

t

0.5fct Gf 0.01fct w 5 0.2 1

w

t

material: • Ec = 30000 N/mm2 • v = 0.2 •fct = 5 N/mm2; f*ct > fct

el

gf

lCB fct

elongation: a) l = w + t/Ec l b) l = crack lCB + t/Ec l

el crack CB

f*ct

fct

t

Gf lch = g = f l/2

l/2

b) discretisation with continuum-shell elements 1 3 4 2

= lCB

el

stress-elongation curve 5

t

5 4

- analytical solution - numerical solutions 1

3

5

2 1 lCB = 120 mm

Figure 3.

24 mm

12 mm

6 mm

24 mm = lch

l

0 0

1

2

Smeared crack modelling using the ‘‘Fictitious Crack Model’’ by Hillerborg.

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3

4

5

3

MATERIAL PARAMETERS

To determine the required material parameters with adequate accuracy two types of experimental tests are presumed acc. to the related design codes (DIN 2008, DAfStb 2009): uniaxial compression tests and beam bending tests. While the parameters for the uniaxial material response in compression could directly be identified from tests, the parameters for uniaxial tensile response need to be deviated from bending tests. Therefore, an inverse analysis applying a ‘‘plastic-hinge model’’ developed by Strack (2007) is used. Compared to an inverse analysis using the FE-model this is less time consuming and more fail-safe. Modelling the multiaxial strength of plain concrete the accordant yield function parameters Fcc and Kc can be taken as constant values with sufficient accuracy (Table 1). For estimation and verification of those

Table 1.

Material parameters.

Elastic Ec = Ecm = (7600 + 1900

acc. (DIN 2008)

1/3 fcm /88)fcm

or from UCT*/BBT**

[N/mm2 ]

ν = 0.2 [−]

values a database was used which contains more than 900 experimental strength results in uniaxial, biaxial and triaxial stress states (Gödde et al. 2009). The addition of steel fibres leads to appreciable increase of compressive strength in multiaxial stress states, depending on effect of fibres. Comparison to numerous experimental strength results has shown that this can be modelled with the yield surface as well, by adjusting its parameters (Gödde et al. 2009). Parameter Fcc may be determined directly by biaxial compression tests for particular SFRC. To avoid such extensive tests an analytical prediction was derived (Gödde et al. 2009), based on an approach by Murugappan et al. (1993) shown in Figure 4. Hence, the equibiaxial compressive state of stress at failure in a SFRC specimen is analogous to stress state in a plain concrete specimen with additional external pressure in third direction. This pressure substitutes internal confining effects provided by the steel fibres bridging the arising cracks. It can be approximated by the initial stress σf 0 (Eq. 9). Thus an evaluation of the yield function (Eq. 3), for the case shown in Figure 4b, leads to a formulation of Fcc depending on the effect of fibres considered by the efficiency factor κI (Eq. 9).   κI fct 1 + α 0 + γ 0 0 1+ Fcc = Fcc fc 1 − α0

(12)

Poisson’s ratio

Here the index 0 identifies parameters for plain concrete. Applying these parameters (Table 1) to Equation 12 incorporating Equation 4 leads to:

Uniaxial response in compression fc = fcm [N/mm2 ]

from UCT*

εc1 = 2.0 [%◦]

acc. (DIN 2008)

Fcc = 1.16 +

or from UCT Uniaxial response in tension fct = fctm = 0.3(fcm − 8)2/3

from fibre geometry

κI [−], κII [−], lpeak [mm],

from BBT** with

Gf [N/mm]

(13)

Admittedly this is a simplification in some respects, e.g. the fibre pullout effect is not considered. Although, a comparison (Gödde et al. 2009) of analytical values to several experimental results has shown a satisfactory accordance.

acc. (DIN 2008)

[N/mm2 ] lf , lh [mm]

κI fct 5.438. fc

inverse analysis a) steel fibre reinforced concrete

Yield function for multiaxial stress states Kc = 2/3 [−] for p/fcm ≤ 1

[Gödde 2009]

-f cc

-f cc

= 0.69 [−] for 1 < p/fcm ≤∼ 4 Fcc = 1.16 + 5.438κI fct /fc

b) plain concrete

-f cc

-f cc

(→ 1.16 for plain

~ = -f ccf

concrete) -f cc

Flow potential ψ = 15 [◦ ]

[Gödde 2009]

αe = 0.1 [−]

[Gödde 2009]

-f cc

-f cc

biaxial compressive analogue stress state stress state note: crack patterns simplified for clarity

* UCT = uniaxial compression test ** BBT = beam bending test

Figure 4. Modelling of biaxial compressive strength of SFRC acc. to (Murugappan et al. 1993).

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Adjustment of the yield function (Eq. 3 & Fig. 1) to experimental strength results of SFRC in triaxial compressive stress states has shown that the same values of parameter Kc can be used for both plain and SFRC (Gödde et al. 2009). This implies that the addition of fibres leads to an increase of strength which is approximately equal along tension and compression meridians. An absence of fibres (κI = 0) results in plain concrete properties. The material parameters are summarised in Table 1. 4

DISCRETISATION AND MODELLING

4.1

Elements

With respect to the intended application reduced integrated 8-node 3-dimensional continuum-based shell elements with linear shape functions are used (Fig. 5 & Abaqus 2008). In few cases 6-node elements with triangular in-plane shape are used. These continuum based shell elements are similar to 3-dimensional solid elements, but their kinematic and constitutive behaviour is based on a shell-theory. In general, elements can be stacked and subdivided internally into concrete and reinforced layers. Thus, strain distributions with pronounced nonlinearities along the section height are captured. Moreover, the typically layered concentration of fibres can be accounted for. Fibre content and orientation can be varied along the thickness to realistically model the effects of compaction and pre-orientation of fibres. Further, arrangement of integration points and integration-scheme can be adjusted flexible in thickness direction. In comparison to solid-elements the numerical performance can be improved significantly, especially in case of large-sized slab structures. For the analysed bending problems with strain localisation in a ‘‘cracked’’ area, the incorporation of a change in shell thickness through in-plane deformation may lead to severe artificial stiffening. 7

8

5

4 6

3

1

2

2

3

It occurs if material parameter of Poisson’s ration is used (Fig. 6). The effect vanishes for Poisson’s ratio ν = 0. The element formulation allows controlling thickness changes independently from material definition by a parameter νeff . Because thickness changes are not important for the analysed structures they are neglected for all simulations setting νeff = 0 (Fig. 6). 4.2

Discretisation in thickness direction

To obtain sufficient accuracy and minimisation of computational costs at the same time, suitable mesh geometries and integration point distances along the section height were determined from extensive numerical studies. For reference results of experimental beam bending tests are used as well as results from according numerical simulations. The latter one means using a reference-configuration with a huge number of elements and integration points in thickness direction. This is shown exemplary for one test series in Figure 7. Six elements along cross-section height are found to be sufficient to capture nonlinear strain devolution in ‘‘cracked’’ cross-sections. This devolution is due to the modelling of discrete crack width by means of inelastic strains depending on the mesh size (cp. paragraph 2.4). The number of integration points is adaptively increased to the boundaries of the cross-sections to take account into a complex stress distribution with steep gradients within the highly localised compressive zones. To obtain results at the cross-section boundaries, integration in thickness direction is done using Simpson’s rule in the outer elements. The other elements are integrated using more effective Gauss quadrature (Fig. 7b).

beam test

F [kN] F/2

F/2

plastic strains

IP

simulation,

3

1 2

1

=

= 0.2

4

2 1

simulation,

eff

=0

layer 1 (material 1) layer 2 (material 2)

3 layerwise variable IP - number thickness direction

Figure 5.

eff

experiment [mm]

layer i (material i)

Figure 6. Artificial stiffening effect due to thickness chance of shell elements in crack bands.

Continuum-based shell element.

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b = 150

b)

material

F/2

-6.0

concrete: C35/30 fibre: Trefilarbed HE 1/50 35 kg/m3 1

for details see (Strack 2007)

l = 600

reference-discretisation fct

6 elements - 22IP

adjusted-discretisation fct

24, 36, 48 elements

48 elements

F [kN]

25.0

0

2.0

integration points: using Simpson using Gauss -35.0

-0.8 -0.4

-25.0

-15.0

[‰] 0 0.4 0.8 1.2 reference nEl,l = 36 nEl,l = 24 nEl,l = 48 -5.0 0

2

h 10.0 1

lch =

5.0

0

Figure 7.

0.5

1.0

reference nEl,l = 24 nEl,l = 36 nEl,l = 48

l nEl,l/2

1.5

2.0

2.5

5.0 [N/mm2]

experiments mean value

15.0

4.0 [N/mm2]

fct

0.05 0.025 w [mm] w

results of simulations and experiments

20.0

0

-2.0

= 0.1 mm

h

[mm]

-4.0

cracked uncracked

beam-test

30 elements - 90IP

h = 150

a)

2

= 3.0 mm 3.0 2.4 1.8 1.2 0.6 w [mm ] w

[‰]

[mm] -120 0 120

3.5

3.0

Discretisation and integration in thickness direction.

Figure 7a shows a comparison of load-deflection curves of the beam test series determinated experimentally and numerically, using both the reference and the adjusted configuration. Additionally for the adjusted configuration simulations are carried out with varying number of elements over the length nEl,l . The results from the different numerical simulations are nearly identical, and show good accordance to the experimental results. This demonstrates that the numerical solution is almost mesh-independent. Further, the ability of the present material model and the suitability of the adjusted configuration in thickness direction are shown. Additionally, an evaluation of the numerical stress and strain distribution in the ‘‘cracked’’ cross-section is presented in Figure 7b for two different deflections (0.1 and 3.0 mm). By regarding the tensile strains and the related crack widths (Eqs 10, 11) the numerical realisation of Hillerborg’s FCM can be observed. It is noticeable, that at a greater deflection and a corresponding crack width a linear strain distribution occurs for all used discretisations. Although one expects a non-linear strain distribution the load deflection behaviour still corresponds to the test results. This is due to the fact that for larger crack widths the compressive zone is highly localised. Thus deviations of compressive strain have only a negligible influence on the deformation and bearing capacity of the cross-section. Therefore, a softening behaviour in

compression can be disregarded for the analysed structures (cp. paragraph 2.2). This can be proved in numerous numerical studies. Certainly for structures with additional rebars or external compressive forces the influence of the compression zone increases. In these cases it might be necessary to take into account softening behaviour in compression. 4.3

Modelling of anisotropic fibre orientation and accumulation

Combination of material formulation with the employed elements allows to model characteristic effects of fibre orientation and accumulation caused by the manufacturing processes. In slab structures a fibre alignment perpendicular to the compacting direction and parallel to the formwork leads to a predominantly 2-dimensional in-plane orientation of fibres (Fig. 8). For bending loads perpendicular to the slab plane and isotropic material definition this can be considered by using two different fibre orientations. In-plane fibre orientation (ηplane ) is used for the definition of uniaxial tensile material response (Eqs 8, 9). Fibre orientation in thickness direction (ηthickness ) defines yield function parameter for multiaxial states of stress (Eq. 13). Fibre sedimentation in direction of casting and compaction is modelled by an isotropic material definition, varying layer wise in thickness direction depending

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simulation. In some cases localisation due to cracking must be induced artificially in the numerical models by definition of slightly different tensile strengths. The importance of a correct definition of the characteristic length lch is emphasised by simulating series 2. Here two unfounded definitions of lch are used. The conducted modelling leads to a localisation in two element rows (Fig. 9). Thus according to Figure 3 lch equals two times element length in beam direction (lch = 2nEl,l ). ) or to Setting lch equal to element length (lch = nEl,l√ the square root of element in-plane area (lch = AEl ) leads to severe deviances of the structural response as shown in Figure 9.

direction of casting and compacting

direction wise anisotropic fibre orientation

layer wise anisotropic fibre content yield function parameters ←

plane thickness

uniaxial mat- ← erial response

Figure 8. Modelling of anisotropic fibre orientation and accumulation caused by the manufacturing processes.

on the particular fibre content Vf ,layer (Fig. 8). An interaction of effects can be considered as well. 5

NUMERICAL SIMULATION OF BENDING PROBLEMS

The capability of the developed model is verified by numerical simulations of several experimental flexural tests, carried out on beam and slab members. The iterative solution is conducted using NewtonRaphson method incorporating a Line-Search algorithm (Wriggers 2001). 5.1 Beams Three different experimental test series are used for verification of the model. While series 1 and 2 (Fig. 9) where carried out at the Ruhr-University in Bochum in the context of material testing, series 3 is taken from an international research project Brite Euram (2002). Dimensions, load setups and basic material parameters are summarised in Figure 9. More detailed information can be found in (Strack 2007). Test series 3 was performed with different fibre contents of 25 and 75 kg/m3 . Following the test performances crack bridging fibres were count to evaluate fibre orientation factors. This evaluation can be used to prove the capability of the developed model to consider changes of fibre orientation and fibre content. Hence, the test series with a fibre content of 25 kg/m3 is simulated twice. Once by adjusting material parameters directly to the tests, and again deviating fibre effect (with κI ) from test series with a fibre content of 75 kg/m3 taking into account different fibre content and orientation (cp. Eq. 9). Comparing experimental and numerical results in terms of load-deflection curves, good accordance is found (Fig. 9). The mesh-independency of the numerical solution is verified employing different element numbers nEl,l over the length for every

5.2

Slabs

Numerical simulations of selected small circular SFRC slabs are carried out, as part of an experimental test series performed by Bernard (1998). Although these slabs do not exactly meet the aspired application of model (large-sized roof constructions) this test series is particularly suitable for the verification of the presented model. For example additional flexural tests of beam elements sawn from several failed slabs were gained. Dimensions, load setups and basic material parameters are summarised in Figure 10. Three slab geometries are simulated with constant height, but varying diameter and support span. Due to the support conditions all experiments and simulations result in the same crack pattern as shown in Figure 10. Hence, to relieve a consistent definition of the characteristic length, discretisation is done that a regular mesh with rectangular crack-parallel elements exists in the ‘‘crack area’’. The remaining parts are discretised using an optimisation method to provide a preferably regular mesh (Fig. 10). To prove the meshindependency of the numerical solution, four meshes with different refinement in the ‘‘crack area’’ are used. The conducted discretisation leads to a localisation in two element rows which is considered in definition of the characteristic length as described before (Fig. 10). The material parameters, determined using the beam tests, are assigned to the beam and slab models thereafter. Although beam tests are only documented up to a crack width of about 2 mm the material model allows an extrapolation to greater crack widths for the recalculation of slab tests. The simulations of the slabs show almost mesh-independency. Crack pattern of the numerical simulation agree well with the experimental results (Fig. 11). A comparison of the load-deflection curves shows a good accordance of the numerical results with the experimental ones for the slab series with the largest diameter. For the simulations of the two slab series with smaller diameters, an underestimation of the experimental load bearing capacity is observed. This underestimation is about 50% for the slab series with the smallest diameter and 25% for the

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Figure 9.

Figure 10.

Experimental test setups & numerical and experimental results of flexural test series on beams.

Experimental test setups & finite element meshes used for numerical simulation of slabs.

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Figure 11.

Numerical and experimental results of flexural test series on beams and round slabs.

middle one respectively (Fig. 11). Here, a fictitious increase of fibre effect (with κI ) with corresponding magnitude leads to a good accordance of numerical and experimental results again. The reason for this underestimation might be an additional bearing contingent in the experiments which is not reproduced by the numerical model. For example membrane actions activated by support constraints are supposable. However, for the intended application considering of these effects is not necessary.

6

CONCLUSIONS

SFRC can effectually be used in redundant slab systems as they can provide an overall ductile behaviour under bending stress in spite of a softening material behaviour of the SFRC. However, anisotropic orientation and accumulation of fibres, caused by manufacturing processes, can considerably affect the system’s bearing capacity. Numerical simulations are very useful to analyse and evaluate the bearing behaviour of those structures more detailed. Due to both, the demand on time and costs of experiments, numerical simulation allows an extended variation of basic parameters. Moreover, a selective analysis of effects of particular fibre orientation and accumulation of interest can be determined. By contrast, this can

hardly be adjusted in experiments. Finally, simulation helps to understand inner bearing mechanisms, e.g. by providing spatial distribution and development of stresses and strains. Such simulations can be performed extending a material model for plain concrete by elaborating material functions and defining material parameters. Furthermore, an appropriate element choice as well as suitable mesh geometries and integration point distances along the section height have to be elaborated. Here, characteristics of the load bearing behaviour of SFRC and the capability to model large-sized slab structures have to be taken into account.

REFERENCES Abaqus Theory Manual, Version 6.8. U.S.A.: Dassault Systemes Simulia. Bernard, E.S. 1998. Behaviour of Round Steel Fibre Reinforced Concrete Panels under Point Loads. Australia: CE8. Brite Euram BRPR-CT98-0813 2002. Test and Design Methods for Steel Fibre Reinforced Concrete. Chen, W.F. & Han, D.J. 1988. Plasticity for Structural Engineers. New York: Springer-Verlag. DAfStb 2009. DAfStb-Richtlinie Stahlfaserbeton (Schlußentwurf)—Ergänzungen zur DIN 1045, Teile 1 bis 3. Berlin: Beuth Verlag GmbH.

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DIN 2008. DIN 1045 Teile 1 und 2: Tragwerke aus Beton, Stahlbeton und Spannbeton. Berlin: Beuth Verlag GmbH. Falkner, H. & Teutsch, M. 2006. Stahlfaserbeton—Anwendungen und Richtlinie. In: Betonkalender 2006 Part 1: 667–703. Berlin: Ernst & Sohn. Gödde, L., Strack, M. & Mark, P. 2009. Numerische Simulation von Stahlfaserbetonbauteilen. In: Proc. Deutsche Simulia-Konferenz 2009. Aachen: Dassault Systemes Simulia GmbH. Hillerborg, A. 1980. Analysis of fracture by means of the fictitious crack model, particularly for fibre reinforced concrete. In: Int. J. of Cement Composite 2 (4): 177–184. Construction Press. Hillerborg, A. 1983. Analysis of one single crack. In: Fracture mechanics of concrete: 223–249. Amsterdam: Elsevier Science Publishers. Lee, J. & Fenves, G.L. 1998. Plastic-damage Model for Cyclic Loading of Concrete Structures. In: J. Eng. Mechanics 124 (8): 892–900. U.S.A.: ASCE.

Lubliner, J., Oliver, J., Oller, S. & Onate, E. 1989. A plasticdamage model for concrete. In: Int. J. Solids Structures 25 (3): 299–326. England: Pergamon Press plc. Mark, P. 2006. Zweiachsig durch Biegung und Querkräfte beanspruchte Stahlbetonträger. Habilitation, RuhrUniversity-Bochum. Murugappan, K., Paramasivam, P. & Tan, K.H. 1993. Failure Envelope for Steel-Fiber Concrete Under Biaxial Compression. In: J. Mat. in civil Eng. 5 (4): S.436–446. U.S.A.: ASCE. Oliver, J. 1989. A Consistent Characteristic Length for Smeared Cracking Models. In: Int. J. for Numerical Methods in Eng. 28: 461–474. U.S.A.: John Wiley & Sons, Ltd. Strack, M. 2007. Modellbildung zum rissbreitenabhängigen Tragverhalten von Stahlfaserbeton unter Biegebeanspruchung. Ph.D. thesis, Ruhr-University-Bochum. Wriggers, P. 2001. Nichtlineare Finite-Element-Methoden. Berlin: Springer Verlag.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Transverse rebar affecting crack behaviors of R.C. members subjected to bending D. Han & M. Keuser University of the German Armed Forces Munich, Neubiberg, Germany

L. Ruediger German Non Comissioned Officer Airforce Academy, Appen, Germany

ABSTRACT: Crack behaviour of reinforced concrete (hereinafter RC) member, includes RC-beam and RC-slab, is the one of major parts in design for the serviceability limit state. It has a significant influence on durability and serviceability. To study on transverse reinforcement effect on crack behaviours, twelve concrete slab-strips have been conducted to study the cracking behaviour of R.C. members subjected to uniaxial bending moment in the presence of transverse reinforcement. Specimens involved the testing of three groups with concrete grade (Normal, High-strength and Light-weight concrete) were mainly designed to qualitatively study the extent of concrete grade transverse reinforcement effect on crack behaviours. In this paper, the measured average values of the final crack were compared to values calculated by the expression of DIN1045-1. These test results indicate transverse reinforcement primarily affect crack pattern and crack spacing. Furthermore this effect was different with different concrete grade.

1

INTRODUCTION

Crack behaviour of RC member subjected to bending is the one of major parts in design for the serviceability limit state. It has a significant influence on durability and serviceability. All bending R.C. concrete members are generally reinforced in two directions along flexure direction and mainly subjected to uniform load. There are many research reports on crack behaviour in reinforced concrete members. Bond between steel and concrete is regarded as the main influence factor in recent researches Nilson (2004). The second important factor of crack behaviour is the stress in the reinforcement, which has been confirmed by studies hold by Gergely & Lutz (1968) and others (Kaar & Mattock 1963) that crack width is proportional to the steel stress. Furthermore, experiments by Broms 1965 and others have shown that crack spacing and crack width are related to the concrete cover distance, which is measured from the centre of the bar to the face of the concrete. Reinforcement stress and concrete cover are considered in recent codes (DIN1045-1 2008 & ACI318 2005) sufficiently. However, the influence of transverse reinforcement (here is vertical to the bending direction) which affects crack behaviour as a single variable was considered rarely; although, it was studied in the R.C. members subjected to pure

tension by Rizkalla et al. (1987) and others (Keuser & Purainer 2004, 2005 & 2006) and Keuser et al. (2008) in two decades. In order to quantitatively and qualitatively analysis crack behaviours, which considers the influence of transverse reinforcements on R.C. slabs subjected to tension and flexure load, a large number of experimental and theoretical investigations of RC panel members subjected to biaxial tension and bending loading were performed at the Institute for Construction Engineering of the University of the German Armed Forces Munich in last eight years. To study on specific parameters of transverse reinforcement effect on crack behaviours of R.C. panel members subjected to bending moments. Four R.C. slab-strip series, sponsored by German Research Foundation (DFG), have been continuously being conducted at University of the German Armed Forces Munich. The first series, which were studied in this paper, involved the testing of twelve reinforced concrete slabstrips and was mainly designed to qualitatively study transverse reinforcement effect on crack behaviour, and examine the applicability of the existing equation for predicting crack spacing. The segments were reinforced in unidirectional bending moments. The main parameters varied were the amount of concrete

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grade (Normal concrete, high strength concrete and light-weight concrete) and the position of transverse reinforcement lay, as given in Table 1 & Figure 1. The measured values of the final crack spacing were compared to the values based on equations of DIN 1045-1 (2008). In the following three ongoing experimental series involve the testing of 36 reinforces concrete slabstrips. They were divided with concrete cover, transverse bar spacing and direction, and concrete grade, within each series, all segments were identical in all parameters; expect the varied parameter which include series (a) Transverse rebar lay; (b) Transverse direction; (c) Concrete cover and (d) transverse rebar spacing. The series are designed to study the influence of transverse reinforcement factors on crack behaviour. These testing results and expression will be regarded as a foundation of methodology for predicting the crack behaviours of R.C. panel member subjected to biaxial bending moments with variable transverse reinforcement.

Table 1.

Variables considered in testing.

Concrete

Type

No.

c*

fctm

eff (%)**

C30

A A B B

PS01 PS02 PS03 PS04

– – – –

2.25 2.25 2.25 2.25

0.90 0.89 1.84 1.83

C100

A A B B

PS05 PS06 PS07 PS08

41(3) 35(2) 18(2) 9(3)

4.59 4.59 4.59 4.59

0.76 0.88 1.71 3.41

LC30

A A B B

PS09 PS10 PS11 PS12

38(5) 35(2) 17(2) 17(2)

2.02 2.02 2.02 2.02

0.81 0.87 1.79 1.78

2 2.1

EXPERIMENT OF UNIAXIAL BENDING TEST OF R.C. SLAB-STRIPS Test specimens

The first series (a) has been planned to crack behavior and deflection behavior of RC slab-strip with a view to (1) define the variable position of transverse rebar, (2) hold the constant value of, that is the distance between the longitudinal rebar and concrete surface, (3) study the cracking behavior. The experimental series consisted of casting and testing 12 RC slab-strips in which the main variables are (1) position of transverse rebar and (2) concrete grade. The RC slab-strip were designated PS01 to PS12. The slabstrips have overall dimensions of 3300 mm 300 mm 200 mm (length width height). Casting and testing 12 slab-strips were divided into three test groups based on different concrete grades (Normal concrete C30, High strength concrete C10 and Light weight concrete LC30). The slab-strips in each group were differentiated into Type A and Type B from different transverse bar position to longitudinal bar (Fig. 1 & Fig. 2), which was fixed with longitudinal bar by assembling reinforcement. 2.2

Testing apparatus and procedure

A diagram of the test equipment and the specimen is shown in Figure. 3. The figure shows that the loading method adopted with reverse loading for easy to observe cracking. This kind of reverse loading method means that two synchronic hydraulic cylinders were fixed on position of both ends of the specimen. The central area of the specimen was the observation area of cracks with 1800 mm length (Han 2009). While testing the RC slab-strip, the load and the deflection were measured. Also the propagation and expansion of cracks were marked at each stage of

Notes: * Concrete cover were measured by PS200 HILTI, and coefficient of variation. ** (%) is according to DIN 1045-1. *** Concrete cover is 30 mm, Transverse Bar spacing is 10 mm. Type B Longitudinal Reinforcement

Transverse reinforcement

Type A Figure 1. Transverse longitudinal bar.

Type B bar

position

related

Type A

with

Figure 2. Reinforcement details of the RC slab-strip of Type B and Type A.

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A

Camera

L+S Imager 5006i

PS200s Ferroscan

Figure 3.

Hydraulic cylinders

B

Slab-strip specimens

Test setup and instrumentation.

loading; moreover crack was marked with different colours with different loading ranges. Crack patterns were recorded by three approaches: (1) handwork measuring with ruler, (2) Digital photos, which were shot by digital camera with 10.2 million effective pixels (Nikon D60). (3) 3D Point Cloud, which were scanned by the Laser scanner (L+S Imager 5006i, Zoller+Froehlich GmbH, 2008). In Post processing, Digital photos were processed by Photoshop and Cracks spacing were measuring by used of Digital Photogrammetric Technique Lin (2002). 3D Point Cloud were processed by Processing Software (Leica Cyclone) and MATLAB [15]. The results from these three approaches were compared and analyzed. Finally, the second approach was commended in this paper. The loading was done in increments of 4 kN per level up to the formation of the first crack and later the increment was about 7 kN per level. Each level has a 7 min holding time. Loading was continued until the ultimate load was reached which indicated by a decrease in load with increase in deflection.

3

EXPERIMENTAL RESULTS AND COMPARISON

Type A are similar to normal RC beams which we are familiar with, and are different from Type B. These differences include (a) crack pattern: Crack pattern of Type A concentrate in several crack areas, which followed the transverse rebar pattern areas along the width direction. Corresponsively, Crack pattern of Type B has no correlativity with the transverse rebar pattern. And cracking spread to various directions even along the long direction. (b) cracking timeliness. As loads are of Type A gradually increased above the cracking load, both the number and the width of cracks increase. After the load up to 50% ultimate load, crack widths increase further, although the number of cracks is more or less stable. Cracking process completed in 30 seconds in hold load. However, for Type B, the number and width increasing in each load level (Fig. 4), moreover cracking time lasted minimum 5 min to hold load. (c) The crack spacing of Type A specimens were long and sparse. Corresponsively, crack spacing of Type B specimens were short and dense (Fig. 5). 3.2

It was observed that most of the surface cracks for all 12 tested specimens coincided approximately with the location of the mid transverse longitudinal reinforcing bars. The final crack pattern was reached at 60 kN, which was approximately 90% of the ultimate load. 3.1

Figure 4. Influence of Transverse bar position for crack pattern (A) Type A-PS10A, (B) Type B-PS12B.

Influence on final crack pattern

The crack pattern under the load 60 kN, which was approximately 90% of the ultimate load, was shown in Figure 4. The characteristic of crack patterns include crack position and direction, cracking timeliness, crack distance and crack width. Characteristics of

Influence on final crack spacing

3.2.1 Governing equations According to the general crack theory (Zilch & Zehetmaier 2005), the transfer length between le cracks is le =

ds · eff 7.2

(1)

The minimum is Srmin = le , maximum and average crack spacing are (DIN1045-1 2008): Sr,max =

ds σ s · ds ≤ 3.6eff 3.6fct,eff

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(2)

Table 2.

reinforcement; ds is the reinforcing bar diameter and σs is the stress in the reinforcing steel in state II and is the effective tensile strength of the concrete as specified.

Measured and calculated crack spacing.

No.

Concrete grade

SC∗

Sr,m

α ∗∗

PS01A PS02A PS03B PS04B

C30

112,6 96,2 96,9 111,6

208,9 209,6 102,1 102,5

0,54 0,46 0,95 1,09

PS05A PS06A PS07B PS08B

C100

137,4 116,4 45,1 55,7

248,0 212,4 109,6 55,0

0,55 0,55 0,41 1,01

PS09A PS10A PS11B PS12B

LC30

57,5 52,1 30,1 24,3

232,8 215,2 104,8 105,2

0,25 0,24 0,29 0,23

3.2.2 Comparison of computed and measured crack spacing In this study, the maximum and minimum crack spacing were compared to the average crack spacing for each specimen at the final crack pattern. Using the crack spacing expression of DIN1045-1 (2008), the average crack spacing were computed for all segments (Table 2) with the measured concrete cover (Table 1) and then compared with measured values from the test. Fig. 5 illustrates comparison between measured and computed crack spacings based on DIN 1045-1 (2008). The figure shows different properties, which separated among of normal concrete, high strength concrete and light-weight concrete.

Notes: ∗ is the average spcing of measured cracks; ∗∗ α is the ratio of S to S ; in unit of mm. C r,m Crack spacing (mm)

4

Figure 5. Comparison between measured and computed crack spacings based on DIN 1045-1 (2008). ∗ The color strip are the calculated possible arrange between Sr,min and Sr,max . ∗∗ The percentage is the ratio of cracks which appeared in a range of calculated crack spacing.

Sr,m = 1.35 · le =

ds 4.8·eff

ANALYSIS OF THE EXPERIMENTS

In test of type A, crack positions are affected obviously by the transverse rebar, that’s because (a) transverse rebar formed cavities outside of the longitudinal rebar and, cavities became the potential crack with 8 mm width; (b) transverse rebar occupied large bond area between concrete and longitudinal rebar, which decreased the effect of bond force. In this case, with increased load, rebar force and concrete force continuous increased. Due to the bond force of concrete, the rebar force increased suddenly near the field of transverse and the concentrated rebar force happened on the top of the transverse rebar, from where the Crack expanded. Hence the final crack pattern concentrates in a certain field along width direction (Fig. 4 & Fig. 5). In contrast, the negative effects of transverse rebar are reduced in test of type B. Meanwhile, the active effects of longitudinal rebar are reacted. Crack pattern is more likely a random expansion process, which represents (a) crack uniform distribution, (b) continuous cracking to various direction (Fig. 4 & Fig. 5). Furthermore, due to the increased the bond force of the longitudinal bar, the effect of the longitudinal bar controlling crack expansion is enhanced significantly. Thereby, concrete grade plays an important role in affecting crack behaviours of R.C. member subjected to bending moments.

5

CONCLUSIONS AND ACKNOWLEDGEMENTS

(3)

The following observation and conclusions are based on the test series discussed in this study.

Where eff is the effective reinforcement ratio as follows: eff = As /Ac,eff and; As is the area of the

1. The spacing of cracks in reinforced concrete members subjected to bending moments are affected

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2.

3.

4.

5.

primarily by the transverse reinforcement steel parallel to the direction of the cracks. The cracks developing process, as the bending increasing, are affected by transverse reinforcement. The parameter of transverse reinforcement should be considered as a independent factor in expressions of predicting crack spacing of R.C. members subjected to bending moments. The expression of DIN1045-1(2008) for average crack spacing can estimate crack spacing of Type B specimens made of Normal concrete, as Type B specimens reduces the influence of transverse reinforcement structurally. However, the expression is highly overestimate the spacing of specimens made of High-strength concrete and Light-weight concrete. The expression of DIN1045-1(2008) is absolute overestimates the spacing in all kinds of Type A specimens, as it ignore the influence of transverse reinforcement. The effect of Concrete grade should be considered as an independent factor.

In the following two years, more variables of transverse reinforcing bar (spacing and direction) and concrete cover will be studied with different concrete grade. Base on obtained results, new expressions will be studied and presented for predicting average crack spacing and width in reinforced concrete members subjected to bending moments with known transverse bar spacing and concrete grade. Finally the authors would like to give their acknowledgements to the German Research Foundation for financial support of the ongoing research work reported about in this paper. REFERENCES Broms, B.B. 1965. Crack Width and Crack Spacing in Reinforced Concrete Member, ACI Structure; vol. 62(10): 1237–1256. ACI 318–05. 2005. Building Code Requirements for Structural Concrete, American Concrete Institute, Farmington Hills, MI, 2005. DIN 1045–1. 2008. Reinforced and Prestressed Concrete structures: 126–134. Berlin: Design and Constrution, Beuth Publisher. Frosch, R.J. 1999. Another Look at Cracking and Crack Control in Teinforced Concrete. ACI Structure; vol. 96(3): 437–442.

Gergely, P. & Lutz, L.A. 1968. Maximum Crack Width in Reinforced Concrete Flexural Members. in Causes, Mechanisms, and Control of Cracking in Concrete, SP-20, American Concrete Insitute, Deroit, MI : 1–17. Han, D. 2009. Factor of transverse rebar affecting crack behaviors of concrete element subjected to bending. IABSE International Association for Bridge and Structural Engineering, Bangkok, 9–11 September 2009. Han, D., Heunecke, O., Keuser, M., Liebl, W., Neumann, I. & Nichelmann, K. 2009. Anwendung des TLS zur Untersuchung des Last-Verformungsverhaltens von Flächentragwerken aus Stahlbeton. In: Wunderlich, T. (Hrsg.): Beiträge des 16. Internationalen Kurses für Ingenieurvermessung, München, Wichmann Verlag. Kaar, P.H. & Mattock, A.H. 1963. High-Strength Bars as Concrete Reinforcement-Part 4: Control of Cracking. Journal, PCA Research and Development Laboratories, vol. 5(1): 15–38. Keuser, M. & Purainer, R. 2004. Cracking and load-carrying behaviour of reinforced concrete panels subjected to biaxial tension, 5th International PhD-Symposium in Civil Engineerin: Delft. Keuser, M. & Purainer, R. 2005. A new concept for realistic modelling of cracks in biaxial stressed reinforced concrete. Fib Symposium ‘‘Keep Concrete Attractive’’, Budapest: Hungary. Keuser, M., Ruediger, L. & Hallermann, N. 2007. Platten aus Stahlbeton unter zweiaxialer Biegebeanspruchung. Beton-und Stahlbetonbau, Verlag Ernst & Sohn, Berlin, Vol. 102(5): 296–309. Lin, L., Ye, L. & Cheng, J. 2002. Application of Digital Photogrammetric Technique for Deformation Measuring in Structural Experiment. Proceeding of the 7th International Symposium on Structural Engineering for Young Experts. Beijing: Science Press, vol. 1: 263–267 Nilson, A.H., Darwin, D. & Dolan, W.D. 2004. Design of Concrete Structures (13th edition): 203–204. Singapore: Mc Graw Hill Higher Education. Purainer, R. & Keuser, M. 2006. Versuche an Stahlbetonscheiben und -platten unter Zugbeanspruchung. Berichte aus dem Konstruktiven Ingenieurbau. vol. 06(3). Rizkalla, S.H. & Hwang, L.S. 1984. Transverse reinforcement effect on cracking behavior of R.C. members. Canadian journal of civil engineering ISSN 0315–1468 vol. 10(4): 566–581. Ruediger, L., Keuser, M. & Berger, J. 2008. Numerische Berechnungen zur Untersuchung des Einflusses der Bewehrungsrichtung auf das Last-Verformungsverhalten von Stahlbetonplatten. Beton- und Stahlbetonbau, Verlag Ernst & Sohn, Berlin, vol. 103(6): 388–397. Zilch, K. & Zehetmaier, G. 2005. Bemessung im konstruktiven Betonbau: 302–305. Berlin Heidelberg New York, Berlin: Springer. Zoller+ Fröhlich GmbH. 2008. Specification of the product, www.zf-laser.com.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

A numerical method for RC-boxgirders under combined shear bending and torsion U. Häußler-Combe Institute of Concrete Structures, Technische Universität Dresden, Dresden, Germany

ABSTRACT: Computational mechanics provides a number of methods for the description of nonlinear behavior of structures with complex geometries. Nevertheless, numerical methods for the description of the spatial behavior of reinforced concrete structures with a unified approach for serviceability states up to ultimate limit states are rarely available for practical applications. In particular, the modeling of crack formation in concrete is a major task, which has a large influence on the model behavior. A smeared crack approach and simple biaxial constitutive law for concrete are described for the simulation of plane RC plates. This serves as a base for the modeling of box girders, which are composed of RC plates arranged in space. Major aspects of nonlinear behavior of RC box girders are described by this relatively simple approach. 1

INTRODUCTION

Simulation of the behavior of complex structures is enabled by computational mechanics. A realistic description of force-displacement behavior in the range of serviceability states up to ultimate limit states should be reached with a uniform, consistent model. Regarding structural concrete, major aspects of this task are, besides others, discussed in (Mehlhorn 1995), (Mehlhorn 2001), (CEB-FIP 2008). A central point is the correct description of the material behavior. The materials of structural concrete are given with the concrete itself, which is considered on the macroscopic level and assumed as a homogeneous material in the following, and the reinforcement. Both yield a compound material connected by bond, which can be considered as a further distinct structural element. The compound material is characterized by cracking or discontinuous displacement fields, respectively. This may be demonstrated with experimental observations, see Figure 1 with the cracking pattern of a reinforced deep beam (Leonhardt 1966). Typically, a large number of cracks arise, whose exact course obviously is influenced by stochastic effects. The exact simulation of the crack pattern, if possible at all, needs numerical methods with the capability to model discontinuities like XFEM (Areias 2005) or EFG (Belytschko 1996), is very costly for a large number of cracks and does not necessarily predict the correct global load-displacement behavior. An alternative approach is given by the smeared crack method, which does not consider discrete, single cracks, but covers regions of cracking. Corresponding to smearing of cracks also a smearing of reinforcement is appropriate, i.e. reinforcement bars of a single direction are treated as a directional layer

with a thickness according to the reinforcement ratio. Several directed reinforcement layers can be used for different reinforcement directions and be superposed with a concrete layer. A rigid bond is assumed in the following, i.e. concrete layer and reinforcement layers in the same geometric point are assumed to have the same displacement. This is not necessarily required, but simplifies the computational approach. Altogether, a homogenization is performed including concrete, cracks and reinforcement. This idealization basically allows simulating the behavior of larger reinforced concrete structures with more complex geometries. A major question remains with the appropriate description of the material behavior of concrete. With the restriction to plane structures a biaxial material law is sufficient. This simplifies

Figure 1.

Crack development of a plate.

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the formulation of material laws to a large extent. The basic ingredients, homogenization and biaxial material laws, have been used by several theories. The Modified Compression Field Theory (MCFT) is cited as a well-known example (Vecchio 1986), (Vecchio 1990), (Vecchio 2000). First of all, these theories are applied to simple geometries like rectangular plates, but may describe some complex mechanisms of the load bearing behavior of reinforced concrete (CEB-FIP 2008, 7.2.3). Another problem arises with the spatial structural behavior of reinforced concrete structures. Generally, beam theories are assumed as a base. A plane of cross section is considered and the resulting internal forces—two bending moment components, two shear force components, a normal force and a torsional moment—are assumed to be known from a superordinated calculation. With known internal forces, the extended MCFT assumes a constant distribution of shear stresses while ensuring cross sectional equilibrium (Rahal 1995), (Rahal 2003). From a local point of view, shear stresses are combined with normal stresses to ensure local equilibrium. Additional normal stress contributions result from bending moments and normal forces under the assumption of the Bernoullihypothesis, whereby the MCFT is incorporated for the calculation of plane stress states. An extension with additional kinematic degrees of freedom and the adaptation to the specific properties of reinforced concrete is performed with an extended beam theory (Hartung 2004), (Krebs 2004). A system of coupled differential and integral equations arises regarding the cross sectional plane, which has to be solved by numerical methods. Both mentioned approaches allow including biaxial nonlinear material laws for concrete combined with smeared cracks. As nonlinear numerical methods are necessary in both approaches, both for the global system to determine resulting internal forces for indeterminate beams and for the cross sectional plane to determine the distribution of strains and stresses, the question arises whether these extended beam theories are still an useful simplification. Restricting to thin walled cross sections—an assumption used by all current models for spatial behavior of reinforced concrete beams— alternatives are offered by folded plates theories. Such an approach will be discussed in this contribution. A spatial structure will be composed from plates with different orientations. These plates can sustain only load directions within their plane. A spatial load carrying behavior results from an appropriate arrangement of plates, whereby external loads are applied in the edges of plates. This approach allows using the material laws of the homogenized reinforced concrete plate with smeared cracks. No assumptions will be used regarding the biaxial distribution of strains and stresses within plates. In particular, a differentiation with e.g. St. Venant torsion and warping torsion is

not necessary. The corresponding model uses finite element methods and ensures integral equilibrium, kinematic compatibility of deformations and nonlinear material laws for reinforced concrete including cracking . The model can be applied likewise upon serviceability states and ultimate limit states. An important application is given for box girders under combined bending moment, torsion and shear. This contribution is organized as follows: Section 2 describes simple material models for concrete, smeared cracking and reinforcing steel. In Section 3 the basics of discretization, implementation of material models, special aspects of solution methods and the application to plane problems are discussed. Section 4 describes the arrangement of plates into spatial structures. The application on reinforced concrete box girders is demonstrated in Section 5. Finally, Section 6 gives some conclusions. 2

MATERIAL BEHAVIOR

The material behavior of concrete is characterized by its limited tensile strength. Upon reaching the tensile limit localization arises, i.e. a narrow band with large strains. Within this crack band increasing strains are connected with decreasing stresses. A fusion of micro cracks into a macro crack can be observed on the mesoscopic scale. On the macroscopic scale this may be modeled with the cohesive crack (Hillerborg 1976), where a fictitious crack width w1 serves as kinematic variable. The corresponding course of stresses normal to the crack surface is schematically shown in Figure 2. As the formation of a macro crack is an irreversible process the marked area indicates the dissipated energy or crack energy, respectively. The crack energy causes size effects and may contribute to the ductile behavior of concrete. It is assumed as a material constant. The implementation of the cohesive crack model into finite element methods may be accomplished with the smeared crack concept. Thus, a strain normal to the crack surface is determined by 1 = 1el + ξ 1cr ,

Figure 2.

1cr =

w1 , h

Cohesive crack model.

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ξ=

h L

(1)

with the strain 1el outside the crack band, the width h of the crack band, the fictitious crack width w1 , the nominal strain 1cr within the crack band and a characteristic element length L. To simplify the argumentation a linear elastic law up to reaching the tensile strength is assumed for concrete in the following. Thus, the lateral stress σ2 is determined by 1el =

1 (σ1 − ν σ2 ) Ec

(2)

considering a plane stress state with Young’s modulus Ec and Poissons’s ratio ν of concrete. Furthermore, the lateral strain is determined with 1 2 = (3) (σ2 − ν σ1 ) Ec Finally, a simple bilinear approach is used for the cohesive crack model    0 < w1 ≤ wcr fct 1 − wwcr1 σ1 = (4) 0 w1 > wcr where the critical crack width wcr = 2Gf /fct ensures that a prescribed crack energy value Gf is reached with an increasing crack band deformation. With fixed given parameters E, ν, h, L, fct , Gf and variable but given values 1 , 2 there remain five unknown values σ1 , σ2 , w1 , 1el , 1cr and five equations. With the tensile limit strain ct = fct /Ec und α = L/wcr the stress solution for 1 > ct , w1 < wcr is given by σ1 = fct

1 − α1 − αν2 1 − αct (1 − ν 2 )

(1 − αct )2 − αct ν 1 + ct ν σ2 = Ec 1 − αct (1 − ν 2 )

(5)

or in a general formulation σ = Csec (Ec , ν, fct , Gf , L, ) ·  σ˙ = Ctan (Ec , ν, fct , Gf , L) · ˙

(6)

with a biaxial stress σ , a biaxial strain , their increments or time derivatives σ˙ , ˙ , a material secant stiffness Csec and a material tangent stiffness Ctan . Taking into account the crack energy Gf and a characteristic element length L in the material law, a regularization is ensured to gain mesh objectivity of results in the softening range (Häußler-Combe 2007). Initially, Eqns. (6) are valid in the local system of principal strains 1 , 2 , which in turn are determined from a given strain state x , y , γxy measured in the global system. To begin with, 1 > ct und 2 < ct is assumed. The principal strain system may be rotated with an angle ϕ compared to the global system. Upon exceeding the tensile limit strain by the larger principal strain and the initiation of the cohesive crack a crack plane arises with an inclination angle ϕ + π/2. The concept of a rotating crack is used in the following, i.e. the inclination of the cohesive

crack follows the principal strain direction. This may be motivated as macro cracking is not distinctive with achieving the tensile limit strain, but the orientation of growing micro cracks may change with principal strain directions. A rotating crack concept is advantageous from a methodical point of view, as principal strain and stress directions coincide and a shear force transmission across crack planes does not arise. An inhomogeneous loading of reinforced concrete structures is associated with internal redistributions of stresses. Considering a cracked point it cannot be assumed that the crack width w1 will monotonously increase with a monotonous increase of global loads. In contrast, un- and reloading processes will occur in local points. The un- and reloading procedure in the cohesive crack model used here is shown in Figure 3. Similar relations like Eqns. (1)–(4) can be formulated for that and solved for σ1 , σ2 , w1 , 1el , 1cr . Furthermore, with exceeding the critical crack width w1 > wcr and disappearing cohesive σ1 -stresses often a state 2 > ct occurs and a further crack orthogonal to the initial crack. A further unknown arises with the fictitious crack width w2 and a further relation σ2 = f (w2 ) has to be formulated, e.g. in analogy to Eq. (4). This case also has closed solutions, if necessary with consideration of un-and reloading states. The procedure for a cracked point may be summarized as follows: with a given principal strain state a feasible assumption has to be made concerning the cracking state (loading with increasing w, unloading, reloading, consideration of a second crack). As has been discussed before, a value can be determined for w1 and for w2 if necessary. Then it has to be proven whether the assumptions for their determination are fulfilled. If not, the procedure has to be repeated. After all, stress σ and tangential material stiffness have to be transformed into the global system with a rotation angle ϕ. Tensile behavior of concrete has been discussed up to now. The combination of the cohesive crack with a linear elastic law for the bulk material is not mandatory. Nonlinear material laws and nonlinear relations for the crack-traction depending on crack-width may

Figure 3.

Bilinear cohesive crack with un- and reloading.

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with the matrix B of the derivatives of the element shape functions, the global stresses σ α and the global tangential material stiffnesses Cαtan . Four-node quadrilateral isoparametric elements are used as it is common for plates. The numerical integration is performed with 2 × 2 Gauss-quadrature. With the assembly of element contributions a vector f of global nodal forces results, which depends on the global nodal displacements u. Furthermore, a load vector p(t) is given with 0 ≤ t ≤ 1. The time parameter t controls the load intensity. A loading may also include prescribed displacements. At a time ti global equilibrium has the condition

be regarded, but this leads to a system of nonlinear equations on the local point level. As far as solutions can be determined for a given strain state the incorporation of nonlinear relations does not alter the overall context. To emphasize this overall context with simplified details the linear approaches with a limited concrete tensile strength are used here as discussed before. As a consequence, the concrete compressive stresses are not limited. Insofar the following considerations are at first valid for under-reinforced structures, which fail by yielding of reinforcement. Basically, a limited compressive strength may be regarded by corresponding material laws. Besides concrete, the behavior of the reinforcing steel has to be considered. A uniaxial ideal elastoplastic approach is sufficient in this context. The stress-strain relation is given with  Es ( − p ) for − p <  < p (7) σ = sign() fsy otherwise with a Young’s modulus Es , a yield stress fsy , the sign function and a current plastic strain p . This considers loading, unloading and reloading both for compression and tension. The evolution law for the current plastic strain p is given with ˙p = ˙

if  ˙ > 0 and |σ | = fsy

(8)

The underlying strain  is determined by the transformation of the given global strain x , y , γxy into the reinforcement orientation. Accordingly, local reinforcement stress and stiffness have to be transformed to the global system. Several reinforcement layers of different orientation may be incorporated in the same way. 3

A PLATE MODEL FOR RC

The material laws will be u sed for a reinforced concrete plate with a plane stress state. The corresponding model is composed of a concrete part and superposed reinforcement parts for every reinforcement orientation. A reinforcement part is given by an anisotropic layer in the global system, whereby the anisotropy is ruled by the particular orientation and the layer thickness by the reinforcement ratio. Every part is discretized with finite elements with a rigid bond between concrete and reinforcement parts, which share a node at the same point. Each part is identified with an index α. Internal nodal forces and tangential element stiffness matrix of an element I are given by α   fI = BT · σ α dV VI (9) α   BT · Cαtan · B dV kI =

r = fi − pi = 0

(10)

with fi = f (ui ), ui = u(ti ), pi = p(ti ). Because of the nonlinear material behavior f has a nonlinear dependence on u. The nodal displacements are determined with an incremental iterative method in the course of t. With the prescription of a loading pi+1 the iteration is controlled by   (j) δu(j+1) = [K (j) ]−1 · pi+1 − fi+1 (11) (j+1) (j) ui+1 = ui+1 + δu(j+1) (j)

(j)

with the iteration index (j) and fi+1 = f (ui+1 ). Iter(0) = ui . It stops with sufficiently ation starts with ui+1 small δu and r. Well known methods like the Newton-Raphson method can be derived from Eq. (11) using the global tangential stiffness matrix for K (j) , which is assembled by the element tangential stiffness matrices kI from Eq. (9). But the underlying material law for concrete with a cohesive crack often yields discontinuous tangential stiffnesses. Regarding Figure 3, a continuous course of stresses is given, but the derivatives have jumps in the stress kinks. Thus, the Newton-Raphson method often will not lead to a convergence of the equilibrium iteration. As an alternative, a quasi-Newton method is used with an iteration matrix K (j) which fulfills a secant condition

VI

K (j) · δu(j) = δr(j) (j) (j−1) δr(j) = r(j) − r(j−1) = fi+1 − fi+1

(12)

This condition does not uniquely determine K (j) . The BFGS-specification (Luenberger 1984) has proven to be adequate in this context. The inverse of the secant matrix is given by

T −1   (j) −1 δu(j) · δr(j) = I− · K (j−1) K T (j) (j) δr · δu

T T δu(j) · δu(j) δr(j) · δu(j) + × I− T T δr(j) · δu(j) δr(j) · δu(j) (13)

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with the unit matrix I. Iteration starts with K (0) as the tangential stiffness of the final equilibrium state of the preceding time step. The inversion is not performed explicitly, a triangular decomposition suffices. As this decomposition is performed once for the whole iteration, Eq. (13) can be efficiently implemented with a recursive approach. The convergence behavior is often improved by adding a line search. Within an iteration step (j) the scalar-product g(s) = r(s)T · δu(j+1) (j)

r(s) = f (ui+1 + s δu(j+1) ) − pi+1

(14)

is minimized by the variation of s along a search direction δu(j+1) . This should make the residual r orthogonal to the search direction. A rough estimation s of the minimizing value is sufficient for practical applications. The improved total displacement is determined with (j+1)

(j)

ui+1 = ui+1 + s δu(j+1)

Figure 4.

Example reinforced concrete plate.

Table 1.

Model parameters for plate problem.

(15)

The equilibrium iteration is embedded in a stepwise increasing loading pi = p(ti ), ti = i t. Nodal displacement ui as well as internal nodal forces fi are determined as a result of every loading step. Within this scheme initiation of cracks is regarded with a semi-explicit approach: after an equilibrium iteration the crack criterion—principal strain exceeding of tensile limit strain—is controlled in all integration points of all elements. That integration point with the largest principal strain exceeding tensile limit strain is chosen to change into a cohesive crack. From a formal point of view a system change takes place. A new equilibrium state is iteratively determined for the modified system without increasing the load. This procedure might be repeated for more crack initiations. The next load step is applied not before a complete crack and equilibrium control is performed. Altogether, a load-deformation behavior is computed for a loading history. A displacement control is appropriate due to decreasing system stiffness and limited ultimate loads. A number of experimental investigations were performed about the nonlinear behavior of reinforced concrete plates, recent results are collected in (Fehling 2008). A major issue concerns the nonlinear behavior of concrete struts with lateral tension, in particular with shear panels and orthogonal reinforcement meshes. As has been discussed before, nonlinear compression behavior of concrete will not be treated in this contribution. First of all, the application is restricted to under-reinforced structures failing by reinforcement failure. Standard investigations about this case are still given with (Leonhardt 1966), where single-span slabs are studied. An example is schematically shown in Figure 4. This case will be investigated by a simulation in the following. The symmetry of the system is utilized

Concrete Young’s modulus Ec Poisson’s ratio ν Tensile strength fct Crack energy Gf Characteristic element length L Slab thickness t

MN/m2 – MN/m2 Nm/m2 m m

32000 0.2 4.7 150 0.08 0.10

Reinforcement Young’s modulus Es Yield limit lower reinforcement fsy Cross section lower reinforcement As Yield limit mesh reinforcement fsy Ratio mesh reinforcement ρs

MN/m2 MN/m2 cm2 MN/m2 %

210000 420 2.0 220 0.15

for the discretization, which was done with 50 elements and 66 nodes as an overlay of one concrete and two reinforcement layers. A major component is given with the lower longitudinal reinforcement, which was modeled with 2-node bar elements and a material law according to Eq. (7). Due to node sharing between bar elements and quad-elements five bar elements are placed in the model. The model parameters are listed in Table 1. Apart from the crack energy the same values are chosen as given in (Leonhardt 1966). The chosen crack energy value conforms to the tensile strength value. Furthermore, a characteristic element length has to be determined, see Eq. (5) or (6) respectively. An approach L=

1 AFE 2

(16)

is chosen for plane-stress quad-elements with the area AFE of an element. As these elements are integrated

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with a 2 × 2 Gauss quadrature and a smeared crack may √ arise in every sampling point the length measure AFE was divided by two. Separate boundary and loading elements were used in the model to consider the particular conditions of the experiment. Overall, the chosen discretization is coarse. But the homogenization of real behavior is a major item in the model under discussion. Insofar a finer discretization does not necessarily lead to a better prediction of the global behavior. Figure 5 shows the computed load depending on the mid-span deflection of the upper edge. A characteristic property is a saw-tooth effect in the transition phase from uncracked state into the state of stabilized cracking. This effect is caused as single elements or groups of elements exceed tensile limit strain and loose their stiffness. The simulated load-displacement curve is compared to the experimental curve. The considerable difference in the initial stiffness cannot be comprehended, however experimental results of different specimen with same nominal stiffness show a high scatter in the uncracked state. The experimental behavior is observed up to a deflection of 3.5 mm with a total load of 1.12 MN. The simulation leads to a total load of 1.16 MN for this deflection, whereas a strain value of 3.6% is calculated for the lower reinforcement. Regarding the relatively simple approach for the material behavior, which has been used for the model, the agreement seems to be satisfactory. Due to homogenization and widespread cracked areas the tensile contribution of concrete between cracks or tension stiffening is not yet regarded. This effect may be considered by a modification of the stress-strain law of the reinforcement. Furthermore, the computed concrete compressive stresses exceed the compressive strength in smaller areas near the support. A consideration of local load redistributions due to nonlinear concrete behavior will presumably not lead to a larger difference in the global load-displacement behavior.

4

Figure 5.

Figure 6.

Results of reinforced concrete plate.

SPATIAL STRUCTURES OF PLATES

As it has been shown with the truss model for torsion (Leonhardt 1974), several plane structural elements carrying loads only in their plane may provide spatial load carrying capacity through an appropriate arrangement in space. Thus, spatial truss models will be extended such that plane plates will be used instead of plane trusses as basic elements. The basic plates themselves are modeled by finite elements as described before and may include a variety of nonlinear material laws for concrete, reinforcement and bond. In the following the material as discussed in Section 2 will be considered. As the dicretization of a particular plate is performed with plane finite elements every node will get two degrees of freedom for the displacements. If these elements are arranged in space, a node will get an additional degree of freedom. But this additional degree of freedom cannot be related to a stiffness and such a system cannot be solved as a spatial system. Hence, an edge of a particular plate is considered and another plate is arranged along this edge with some inclination. The nodes on the edge, which are shared by both plates, will have a stiffness for all degrees of freedom in space. A stiffness component remains to be related to all nodes, which do not belong to an edge between two plates with inclination. A first possibility is to introduce a constraint, which allows only displacements within a prescribed plane. This yields another equation for every interior node, which makes the system principally solvable in space. As an alternative a so-called numerical support will be used in the following. A volume element will be attached to a plate element on one side, i.e. an isoparametric 8-node brick-element to a 4-node quad-element with four nodes shared, see Figure 6. Hence, nodes can be distinguished as edge-nodes, interior-nodes and auxiliary-nodes. A linear elastic behavior is assumed for the volume elements with such a low stiffness, that the influence of these elements on the system behavior

Numerical support of plate elements.

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can be neglected. This support of plate elements may also be regarded as a membrane spanned by the edges of a plate. Such a membrane is flat in the undeformed configuration and gets warped due to the deformation of the underlying plate during loading. This also determines the displacement of interior nodes normal to the plane they originate from. This approach seems to have two drawbacks. Additional degrees of freedom arise at the auxiliary nodes, which require additional costs to solve but do not give immediately usable results. As this approach uses a relative coarse discretization due to homogenization the overall costs are relatively small. Hence, additional costs are not critical considering the performance of current hardware. Furthermore, the influence of the stiffness of the additional elements has to be investigated in further parametric studies. The stiffness has to be small to be of negligible influence on the system behavior. On the other hand it has to be large enough that the system’s numerical conditioning does not deteriorate. This stiffness parameter may be regarded as another generalized discretization parameter and its influence on computation results may be handled like other discretization parameters. In further developments the additional volume elements might be utilized for the consideration of local lateral bending or a simplified modeling of solid sections. With the extension into the spatial dimension and the inclusion of volume elements no essential changes in the procedures occur regarding discretization and nonlinear solution method, as have been described in Section 3. Detail changes concern the transformation of element nodal forces and stiffness matrices according to Eq. (9) from a plane system into a spatial system according to a plates spatial orientation. This will lead to zero entries or linear dependencies in the global tangential stiffness matrix, which will be corrected by adding the volume elements. Some application limits remain to be discussed. Regarding a thin walled structure, originally flat plates will obtain a warping to ensure the kinematic compatibility during spatial displacements. With the given approach the geometry of this warping is determined like the warping of a membrane spanned between deforming edges. The projection of this warping onto longitudinal and circumferential directions lead to corresponding curvatures. Hence, variable strain- and stress states will occur in the thickness direction of a plate. Regarding concrete struts, this might lead to a major reduction of their load carrying capacity. This is not relevant here, as concrete compressive strength has not been limited. But another effect is given with secondary bending moments, in e.g. the circumferential direction, proportional to t 3 with the thickness t, which do not establish an equilibrated system. Insofar thick walled systems are not yet covered. Finally, a loading has to be applied along edges if its directions do not coincide with a plate direction.

5

APPLICATION ON BOX GIRDERS

A simple case of a spatial loading is given with pure torsion. Experimental investigations were performed, besides others, by (Leonhardt 1974), (Rasmussen 1995), (Zedler 2006), but thereof only a small number with hollow box girders. A typical box girder case is investigated in (Leonhardt 1974). This case will be considered for a simulation, see Figure 7. Corresponding to the experimental reinforcement configuration width and height of the model are chosen with 0.29 m. The effective wall thickness t is assumed with 6 cm, which is 2 cm less than the geometric thickness of the specimen. Some aspects of effective thickness will be discussed at the end of this section. The value of t influences the stiffness of uncracked areas or in the initial state of cracking. The behavior in the cracked state and in the ultimate limit state is not influenced as concrete compressive strength is not limited here. A web reinforcement ratio is chosen in circumferential and longitudinal direction with the same reinforcement. The parameters are chosen according to (Leonhardt 1974), if available there, and listed in Table 2. A common value is taken for the crack energy. The characteristic element length is determined with Eq. (16). The end cross sections are stiffened by linear elastic plate elements, which act as diaphragms and have some influence on the

Figure 7. Table 2.

System of box girder with torsion Model parameters for torsion box girder.

Concrete Young’s modulus Ec Poisson’s ratio ν Tensile strength fct Crack energy Gf Characteristic element length L Wall thickness t

MN/m 2 − MN/m 2 Nm/m2 m m

25 000 0.2 2.6 100 0.07 0.06

Reinforcement Young’s modulus Es Yield limit web reinforcement fsy Ratio web reinforcement ρs

MN/m 2 MN/m 2 %

210 000 460 0.47

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structural behavior. The shape deformation of the end cross sections was only partially prevented in the experiment. Thus, some margin remains for the determination of the stiffness of the diaphragms. Again a rough discretization with 15 element rows in the longitudinal direction and 4 × 3 plate elements per row in the circumferential direction is chosen. Eight linear volume elements are placed within the box girder for every row of plate elements as numerical support, as has been discussed in Section 4. Their Young’s modulus is chosen with E = 0.1MN/m2 and their Poisson’s ratio with ν = 0. A variation of Young’s modulus below 1MN/m2 does not have a significant influence on the computed results. Figure 8 shows the computed torsion moments MT depending on the twist θ compared to the experimental results. Regarding initial cracking, an overshoot of the computed torsion is obvious. This can be related to the multi-linear characteristic of stresses with sharp bends, see Figure 3, and furthermore to the small difference between initial crack moment and ultimate limit moment in this case. A characteristic property of the simulation results again arises with a saw-tooth effect during formation of cracks before the final cracking state, which is connected to a larger redistribution of internal stresses in this example. This also leads to some snap-back effects, as the twist was determined in a central section. Furthermore, compared to the experiment the model shows a larger stiffness in the final cracking state before reinforcement yielding. This can be explained with differences in the reinforcement stress-strain behavior, which is ideal elasto-plastic in the model but has a smooth transition into yielding in the experiment. Principal compressive stresses and reinforcement stresses in integration points of a central section are shown in Figure 9. The stress state corresponds to the classical truss-model for torsion, as a wall of concrete struts develop with an inclination of 45◦ while the reinforcement reaches the yield limit. Insofar, nothing is new. But beyond that, the simulation yields

Figure 8.

Torsion of box girder depending on twist.

Figure 9. Middle section stresses of torsion box girder in ultimate limit state.

Figure 10.

Box girder under shear and torsion.

displacements and the load-displacement behavior from serviceability states up to ultimate limit states and complements the classical theory for this case. A box girder with combined bending, shear and torsion is examined as another example. The classial approach faces the problem of bending shear and torsion shear superposition. They are regarded in different models, both amounts of reinforcement are added and empirical interaction rules for compression of concrete struts have to be obeyed. A more consistent approach is discussed in (Rahal 2007). As experimental investigations of hollow box girders with combined loading are rare, only a simulation example will be presented in the following. A single-span girder is examined with an eccentric load in midspan, see Figure 10. The girder is stiffened by diaphragms in the end and midspan cross sections. To simplify the computation, the diaphragms itself are assumed as linear elastic. The discretization is performed utilizing symmetry, otherwise similar like the preceding example. RC-plate elements are used for flange plates and web plates, linear elastic plate elements for the diaphragms and linear elastic volume elements for numerical support. This results in

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160 RC-plate elements, 2 × 12 elastic plate elements for the diaphragms, 120 volume elements with totally 263 nodes, each with three degrees of freedom. The model parameters are given in Table 3, the characteristic element length is determined with Eq. (16). The parameters of the numerical support are chosen with E = 0.1 MN/m2 , ν = 0. The Young’s modulus EQT of the diaphragms is varied to vary their stiffness as a whole. Figure 11 shows the computed load for two extreme diaphragm stiffnesses depending on the vertical displacement of the loaded point. The importance of stiff diaphragms becomes obvious, as the ultimate limit loads differs by a factor of two. As in the other examples, a saw-tooth effect is given during increasing crack formation. Its shape is to some extent influenced by the value of the crack energy, larger values of Gf reduce this effect. Figure 12 shows the deformation of both cases at the last computation step with the same vertical displacement of the loading point. Warping of flange plates and and web plates is obvious, as well as the cross section shape deformation for the case with low diaphragm stiffness. The deformation state of RC-plates has to be discussed with some details. First of all, a biaxial strain state of a reference plane is given. As has been indicated before, a strain gradient in the thickness Table 3. torsion.

Model parameters for box girder under shear and

Concrete Young’s modulus Ec Poissons’s ratio ν Tensile strength fct Crack energy Gf Characteristic element length L Wall thickness t

MN/m2 – MN/m2 Nm/m2 m m

30 000 0.2 3.5 100 0.17 0.15

Reinforcement Young’s modulus Es Yield limit web reinforcement fsy Ratio web reinforcement ρs

MN/m2 MN/m2 %

200 000 500 2.0

Figure 11. Load-displacement curve for girder under shear and torsion.

Figure 12. Deformed box girder under shear and torsion (scaling factor 10)

Figure 13. Ultimate load vs. diaphragm stiffness for girder under shear and torsion

direction has to be superposed due to the warping of a plate. In case of compression of the reference plane an elongation of a surface may occur for high strain gradients. This is connected with the reduction of the compressed wall thickness, which can be calculated with the model used here. A wall thickness reduction has to be regarded in case of a limited tensile and compressive strength of concrete, e.g. with means of an effective thickness (Rahal 1995). The same problem arises with the elongation of the reference plane and a compression of a surface, but a modified concrete contribution in the tensile regime is of minor importance in case of reinforced concrete. Another local effect is given with spalling of concrete corners along edges due to the direction change of concrete struts (Zedler 2006). Modeling of this effect requires a triaxial approach. The influence of the diaphragms will be investigated in a small parametric study. It examines the influence of the stiffness of the diaphragm stiffness on the ultimate limit load. The latter is defined as the load with a vertical displacement of 5 cm, i.e. 1/200 of the span. The result is shown in Figure 13. A decreasing ultimate limit load is given with decreasing diaphragm stiffness, as those parts of the girder,

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which are not directly loaded, are drawn to load bearing to less extent. Roughly estimated, only one half of the girder is utilized in case of vanishing diaphragm stiffness. The small study is exemplary. Basically, a variety of other cases may be investigated, i.e. other cross section shapes, varying cross section shape dimensions in the longitudinal direction, variable wall thicknesses and reinforcement ratios, continuous girders and prestressing. As nonlinear problems are given with nonlinear material laws, the feasibility has to be proven in every single case. 6

CONCLUSIONS

Consistent computation models for spatial concrete structures are rare regarding practical applicability. This contribution proposes a simplified model for thin walled structures by using relatively simple plates as a basic element and arranging them in space. This model considers material nonlinearities for concrete and reinforcement, biaxial behavior and cracking of concrete and—without any further assumptions—fulfills equilibrium conditions and kinematic boundary conditions in an integral sense. It covers serviceability states and ultimate limit states in the same way. It is shown by application examples that basic mechanisms of reinforced concrete behavior are correctly described. Potential application fields are at first given with the simulation of complex stress distributions and redistributions within thin walled reinforced concrete structures and of their deformations, in cases both of which are strongly influenced by concrete cracking. Further extensions concern the consideration of the nonlinear biaxial concrete behavior in the compressive range. If such a material law is given, it may be incorporated in this method. Furthermore, the rotating smeared crack approach may be principally replaced by a fixed smeared crack or a combination of both. Within that context the effect of transmission of shear forces across cracks has to be covered. Both items—nonlinear compressive behavior and shear force transmission—each open a field of own complexity which have not been touched in this contribution. REFERENCES Areias, M.A. & Belytschko, T. (2005). Analysis of threedimensional crack initiation and propagation using the extended finite element method. Int. J. Numer. Meth. Engng. 63: 760–788. Belytschko, T., Krongauz, Y.,Organ, D., Fleming, M. & Krysl, P. (1996). Meshless Methods: An overview and recent developments. Comput. Methods Appl. Mech. Engrg. 139: 3–46. CEB-FIP (2008). Practitioners’ guide to finite element modelling of reinforced concrete structures. Bulletin Nr. 45.

Lausanne : International Federation for Structural Concrete fip. Fehling, E., Leutbecher, T. & Röder, F.K. (2008). Zur DruckQuerzug-Festigkeit von Stahlbeton und stahlfaserverstärktem Stahlbeton in scheibenförmigen Bauteilen. Schriftenreihe Baustoffe und Massivbau Heft 11. Kassel : kassel university press. Hartung, B. & Krebs, A. (2004). Erweiterung der Technischen Biegelehre. Teil 1: Aufstellung der Bedingungsgleichungen mit Überprüfung der Lösbarkeit. Beton- und Stahlbetonbau 99: 378–387. Häußler-Combe, U. (2007). Zur Verwendung von Stoffgesetzen mit Entfestigung in numerischen Rechenverfahren. Bauingenieur 82: 286–298. Hillerborg, M., Modeer, M. & Petersson, P.E. (1976). Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements. Cement and Concrete Research 6: 773–782. Krebs, A., Schnell, J. & Hartung, B. (2004). Erweiterung der Technischen Biegelehre. Teil 2: Iterative Lösung der Bedingungsgleichungen, Bestätigung der Theorie durch Versuche sowie Anwendung bei einer Brückensanierung. Beton- und Stahlbetonbau 99: 536–551. Leonhardt, F. & Schelling, G. (1974). Torsionsversuche an Stahlbetonbalken. DAfStb Heft 239. Berlin : Verlag Ernst & Sohn. Leonhardt, F. & Walther, R. (1966). Wandartige Träger. DAfStb Heft 178. Berlin : Verlag Ernst & Sohn. Luenberger, David G. (1984). Linear and Nonlinear Programming (2nd ed.). Reading, Massachusetts : AddisonWesley. Mehlhorn, G. (ed.) (1995). Der Ingenieurbau—Rechnerorientierte Baumechanik. Berlin : Ernst & Sohn, 1995. Mehlhorn, G. (2001). Beiträge zu rechnerorientierten physikalisch nichtlinearen Berechnungen im Betonbau mit der FEM. Beton- und Stahlbetonbau 96: 412–426. Rahal, K.N. (2007). Combined Torsion and Bending in Reinforced and Prestressed Concrete Beams Using Simplified Method for Combined Stress-Resultants. ACI Structural Journal 104: 402–411. Rahal, K.N. & Collins, M.P. (1995). Analysis of Sections Subjected to Combined Shear and Torsion—A Theoretical Model. ACI Structural Journal 92: 459–469. Rahal, K.N. & Collins, M.P. (2003). Combined Torsion and Bending in Reinforced Concrete Beams. ACI Structural Journal 100: 157–165. Rasmussen, L.J. & Baker, G. (1995). Torsion in Reinforced Normal and High-Strength Concrete Beams—Part 1: Experimental Test Series. ACI Structural Journal 92: 56–62. Vecchio, F.J. (1990). Reinforced Concrete Membrane Element Formulations. Journal of Structural Engineering 116 (3): 730–750. Vecchio, F.J. (2000). Disturbed Stress Field Model for Reinforced Concrete: Formulation. Journal of Structural Engineering 126: 1070–1077. Vecchio, F.J. & Collins, M.P. (1986). The Modified Compression-Field Theory for Reinforced Concrete Elements Subjected to Shear. ACI Journal 83: 219–231. Zedler, Th. & Mark, P. (2006). Druckstrebentragfähigkeit torsionsbeanspruchter Stahlbetonbalken mit üblicher Betondeckung. Beton und Stahlbetonbau 101: 681–694.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Computation of optimal concrete reinforcement in three dimensions P.C.J. Hoogenboom Delft University of Technology, Faculty of Civil Engineering and Geosciences, Delft, The Netherlands

A. de Boer Ministry of Transport, Public Works and Water Management, Centre for Public Works, Utrecht, The Netherlands

ABSTRACT: A method is proposed for determining the required reinforcement based on stresses that have been computed by the finite element method using volume elements. Included are, multiple load combinations, compression reinforcement, confinement reinforcement and crack control. The method is illustrated by several stress examples and a structural example. 1

INTRODUCTION

Many computer programs for structural analysis have post processing functionality for designing reinforcement and performing code compliance checks. For example the moments and normal forces computed with shell elements can be used to determine the required reinforcement based on the Eurocode design rules [1, 2, 3]. However, for finite element models containing volume elements the codes do not provide design rules. Software companies that are developing structural analysis programs are in the process of extending their program capabilities with volume elements. Consequently, also the algorithms for computing reinforcement requirements need to be extended for use with volume elements. In 1983, Smirnov pointed out the importance of this problem for design of reinforced concrete in hydroelectric power plants [4]. In 1985, Andreasen and Nielsen derived formulas for the optimal reinforcement for three-dimensional stress states [5]. They also designed a flow chart for determining which formula to use. In 1994, Kamezawa et al. proposed and tested several formulas for three-dimensional reinforcement design [6]. In 2002 and 2003, Foster, Marti and Mojsilovi´c published two thorough studies on the subject [7, 8]. In 2008, Hoogenboom and de Boer used analytical and numerical methods for computing three-dimensional reinforcement requirements [9]. This paper continues on this path. An algorithm is proposed for computing the optimal reinforcement for multiple load combinations. The load combinations are related to the ultimate limit state or the serviceability limit state. Not only tension reinforcement is considered but also compression reinforcement and confinement reinforcement. A maximum crack width is imposed for the serviceability limit state. Numerical

results are compared to analytical results of elementary stress states. The algorithm has been implemented in a finite element program. A structural example is included. 2

PROBLEM FORMULATION

It is assumed that reinforcing bars are present in the x, y and z direction only (Fig. 1, App. 2). The reinforcement ratios are ρx , ρy and ρz , respectively. The smallest amount of reinforcement is obtained when the volume reinforcement ratio is minimised. Minimise ρx + ρy + ρz

(1)

Clearly, the reinforcement ratios need to be positive which gives a constraint on the solution.

crack face xx

reinforcing bar sz

sy

xy

xz

yy

sx

yz zz

Figure 1. Elementary part of reinforced concrete. Shown are a crack and the reinforcing bars that bridge this crack.

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ρx ≥ 0

3

ρy ≥ 0 ρz ≥ 0

For load combinations related to the ultimate limit state the stresses σsx , σsy , σsz in the reinforcing steel need to be no larger than the yield value fy . −fy ≤ σsx ≤ fy −fy ≤ σsy ≤ fy

(3)

−fy ≤ σsz ≤ fy The concrete principal stresses σc1 , σc2 , σc3 need to be in compression or zero. This is because the concrete might not have tensile strength locally due to shrinkage cracks that can occur during hardening. In this paper the principal stresses are ordered from small to large σc3 ≤ σc2 ≤ σc1

(4)

The stresses in a structural part can be computed in a linear or non-linear finite element analysis. In this paper the stress tensor in a point is written as ⎡ ⎤ σxx σxy σxz ⎣ σxy σyy σyz ⎦ . (8) σxz σyz σzz From the material stress tensor the concrete stress tensor can be derived [9]. ⎡ ⎤ σxy σxz σxx − ρx σsx ⎣ ⎦ σxy σyy − ρy σsy σyz (9) σxz σyz σzz − ρz σsz The concrete principal stresses are the eigenvalues of this tensor. The invariants of the concrete stress tensor are Ic1 = σcx + σcy + σcz

(10)

Ic2 = σcx σcy + σcy σcz + σcz σcx 2 − σxy − σxz2 − σyz2

Therefore, it is sufficient to require that σc1 ≤ 0.

CONCRETE STRESSES

(2)

(11)

Ic3 = σcx σcy σcz + 2σxy σxz σyz 2 − σcx σyz2 − σcy σxz2 − σcz σxy

(5)

In this paper the Mohr-Coulomb yield condition is used for preventing concrete compressive failure.

where the concrete stresses are

σc3 σc1 −  ≤1 ft fc

σcy = σyy − ρy σsy

(6)

(12)

σcx = σxx − ρx σsx (13)

σcz = σzz − ρz σsz .

where, fc is the uniaxial concrete compressive strength and ft is the concrete mean tensile strength. Here, the tensile strength is larger than zero because it is an average value instead of a local value. For load combinations related to the serviceability limit state, the crack width w needs to be limited.

It can be proved that the condition σc1 ≤ 0 is fulfilled if and only if Ic1 ≤ 0, Ic2 ≥ 0 and Ic3 ≤ 0 [9]. In this paper it is assumed that reinforced concrete is a ductile material. Prager’s second law (lower bound theorem of plasticity theory) is applied to load combinations for the ultimate limit state [10, 11].

w ≤ wmax

4

(7)

This condition is imposed for aesthetics and to prevent corrosion of the reinforcing steel. In reinforced concrete beam design it is customary to include at least a minimum reinforcement. This is to ensure ductile failure and distributed cracking. However, in many situations the minimum reinforcement requirements result in much more reinforcement than reasonable. Therefore, in this paper it is not considered. Of course, a design engineer can decide to apply at least minimum reinforcement according to the governing code of practice.

CRACK WIDTH COMPUTATION

The linear elastic strains computed by a finite element analysis could be used for determining the crack width. However, these strains would not be very accurate because they strongly depend on Young’s modulus of cracked reinforced concrete which can only be estimated. On the other hand, the stresses do not depend on Young’s modulus.1 Therefore, the computation of 1 Except

for temperature loading and foundation settlements in statically indetermined structures. For these

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crack widths starts from the stresses. In essence, the adopted equations are part of the Modified Compression Field Theory [12] simplified for the serviceability limit state and extended for three dimensional analysis. Eqs (9) and (13) can be rewritten to ⎤ ⎡ ⎤ ⎡ σc1 0 0 σxx σxy σxz ⎣ σxy σyy σyz ⎦ = P ⎣ 0 σc2 0 ⎦ P T 0 0 σc3 σxz σyz σzz ⎡ ⎤ ρx σsx 0 0 ρy σsy 0 ⎦ +⎣ 0 0 0 ρz σsz (14) where σc1 , σc2 , σc3 are the concrete principal stresses and ⎡ ⎤ cos α1 cos α2 cos α3 P = ⎣ cos β1 cos β2 cos β3 ⎦ . (15) cos γ1 cos γ2 cos γ3 The columns in P are the vectors of the concrete principal directions. Note that in general these principal directions are not the same as the linear elastic principal directions. Since yielding is supposed not to occur in the serviceability limit state, the constitutive relations for the reinforcing bars are linear elastic. The constitutive relation for compressed concrete is also approximated as linear elastic in the principal directions. Poisson’s ratio is set to zero. The constitutive relation for tensioned concrete is σci =

ft √ 1 + 500εi

i = 1, 2, 3

The CEB-fib Model Code 90 is applied for computing crack widths [13]. The mean crack spacings s for uniaxial tension in the reinforcement directions are sx =

2 dx 2 dy 2 dz , sy = , sz = 3 3.6 ρx 3 3.6 ρy 3 3.6 ρz

(18)

where dx , dy , dz are the diameters of the reinforcing bars in the x, y, z direction. The crack spacing s in principal direction i is computed from |cos αi | |cos βi | |cos γi | 1 = + + i = 1, 2, 3. (19) si sx sy sz The mean crack width in the principal direction i is wi = si (εi − εc − εs )

i = 1, 2, 3.

(20)

where εc is the concrete strain and εs is the concrete shrinkage. The value of εc is positive and the value of εs is negative. For simplicity, in this paper is assumed that they cancel each other out.

(16)

where ft is the concrete mean tensile strength [12]. For the crack width computation it is assumed that aggregate interlock can carry any shear stress in the crack. It is assumed that the concrete principal stresses and the principal strains have the same direction. The principal strains ε1 , ε2 , ε3 are the eigenvalues of the strain tensor. ⎡ ⎤ 1 1 ⎡ ⎤ εxx 2 γxy 2 γxz ε1 0 0 ⎢1 ⎥ 1 = P ⎣ 0 ε2 0 ⎦ P T (17) ⎣ 2 γxy εyy 2 γyz ⎦ 0 0 ε3 1 1 εzz 2 γxz 2 γyz

5

REINFORCMEMENT OPTIMISATION

The optimisation problem can be visualised in a graph (Fig. 2). The axis of this graph represent ρx , ρy and ρz . The condition Ic3 = 0 is shown as a surface. The objective is to find the smallest possible value of ρx +ρy +ρz . The shape of the surface depends on the linear elastic stress tensor and on the steel stresses. Not only interior solutions but also corner solutions and boundary solutions are possible. For each load combination related to the ultimate limit state four of such surfaces occur as a result of possible interior solution possible boundary solution possible corner solution z

Ic 3 = 0 y

x

From Eqs (14) to (17) the strain tensor can be solved numerically by the Newton-Raphson method. cases an accurate estimate of Young’s modulus of cracked reinforced concrete needs be used in the linear elastic analysis. Alternatively, a physical nonlinear analysis can be used.

x

Figure 2. problem.

+ y + z = constant

Conceptual presentation of the optimisation

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the equations Ic1 = 0, Ic2 = 0, Ic3 = 0, σc1 /ft − σc3 /fc  = 1. For each load combination related to the serviceability limit state one surface occurs as a result of the equation w = wmax . Each surface gives a lower bound to the amount of reinforcement. The minimum can be on a crossing line of two surfaces or on a crossing point of three surfaces. At first sight, efficient use of the reinforcement requires that the steel yields in tension or in compression. However, sometimes it is necessary to consider less steel stress. This can be explained in a simple example. Suppose that one load combination requires a large amount of reinforcing steel. Suppose that another load combination results in hardly any stresses. If we insist on applying the yield stresses in both load combinations than in the second load combination the concrete is compressed considerably by the reinforcement and the Mohr-Coulomb constraint might not be fulfilled. In reality, the steel stresses and concrete stresses will be small too during the second load combination. Consequently, if the yield stress is used in the optimization, the Mohr-Coulomb condition can work as an artificial upper bound in the optimization problem instead of as a lower bound. This upper bound is removed when the steel stresses are properly reduced in the second load combination. Consequently, not only the reinforcement ratios but also the steel stresses σsx , σsy , σsz need to be varied in the optimisation problem. 6

ρx =

ρz =

σyz2 σzz − ± fy fy σyy

ρx =

σxx σ2 − xz ± fy fy σzz

ρy =

σyz2 σyy − ± fy fy σzz

The optimization problem is reduced considerably if only one load combination is present. It simplifies even further when the Mohr-Coulomb condition and crack control are ignored. In this case eleven sets of solutions can be derived for the optimal reinforcement [9]. Each of these sets fulfil the condition Ic3 = 0 which means that one of the concrete principal stresses is zero. ρy = 0,

ρx = 0,

ρy =

ρx =

I3 2) fy (σxx σyy − σxy

I3 , fy (σxx σzz − σxz2 )

I3 , fy (σyy σzz − σyz2 )

ρx = 0 ρy =

ρz =

2 σxy

σyy − ± fy fy σxx

σzz σ2 − xz ± ρz = fy fy σxx

 

ρy = 0,

σxz σxy σyz − fy σxx fy σxz σxy σyz − fy σxx fy

(21-1)

ρz = 0

(21-2)

ρz = 0

(21-3)

σyz σxy σxz − fy σyy fy



ρy = 0

REINFORCEMENT FORMULAS

ρx = 0,



2 σxy σxx − ± fy fy σyy

(21-4)

(21-5)  



σyz σxy σxz − fy σyy fy σxz σyz σxy − fy σzz fy σxz σyz σxy − fy σzz fy





(21-6)

ρz = 0 ρx =

σxx + σxy + σxz fy

ρy =

σyy + σxy + σyz fy

ρz =

σzz + σxz + σyz fy

ρx =

σxx + σxy − σxz fy

ρy =

σyy + σxy − σyz fy

ρz =

σzz − σxz − σyz fy

ρx =

σxx − σxy − σxz fy

ρy =

σyy − σxy + σyz fy

ρz =

σzz − σxz + σyz fy

ρx =

σxx − σxy + σxz fy

ρy =

σyy − σxy − σyz fy

ρz =

σzz + σxz − σyz fy

ρx =

σxy σxz σxx − fy fy σyz

σyy σxy σyz − fy fy σxz σxz σyz σzz ρz = − fy fy σxy ρy =

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(21-7)

(21-8)

(21-9)

(21-10)

(21-11)

where I3 is the determinant of the linear elastic stress tensor.

The functions involved are σc1 = σc1 (ρx σsx , ρy σsy , ρz σsz )

I3 = σxx σyy σzz + 2σxy σxz σyz

σc3 = σc3 (ρx σsx , ρy σsy , ρz σsz )

2 − σxx σyz2 − σyy σxz2 − σzz σxy

w = w(ρx , ρy , ρz )

(22)

Which of these eleven sets of formulas (21-xx) to use can be determined in four steps. First, calculate I1 , I2 and I3 . If I1 ≤ 0, I2 ≥ 0 and I3 ≤ 0 than tension reinforcement is not needed. Second, ignore the sets of formulas that give negative reinforcement ratios. Third, calculate Ic1 and Ic2 by Eqs (13), (10), (11) and ignore the sets for which Ic1 > 0 or Ic2 < 0. Fourth, select the set of formulas for which ρx + ρy + ρz is smallest. With the results of step three the concrete compressive stress can be calculated and checked. σc3

1 = Ic1 − 2



1 Ic1 2

2

− Ic2 ≥ −fc

Note that B is positive for feasible solutions. It goes to infinity if any of the constraints is almost violated. An advantage of the barrier method compared to other methods of computational optimisation is that only interior points are evaluated. Interior points are ‘‘sufficient reinforcement’’ for which the computation of the crack width w converges quickly. The minimisation can be performed by any unconstrained optimisation algorithm such as the down-hill simplex method or Newton’s method. A good starting point for the Barrier method is the envelope of the requirements for the individual load combinations. The required reinforcement for a load combination related to the serviceability limit state can be quickly approximated by assuming that the reinforcement ratios are proportional to the steel stresses.

(23)

The ± signs in Eq. (15, 4), (15, 5) and (15, 6) can be replaced by the absolute value. The proof for this is presented in Appendix 1.

8 7

BARRIER METHOD

The barrier method is a method for computational optimization [14]. In this method, a large cost is imposed on points that lie close to the boundary of the feasible region. This cost is called the barrier because it makes sure that a new point is not picked outside the feasible region. minimise ρx + ρy + ρz + rB

(24)

where r is a factor that is reduced in subsequent steps and B is the barrier. A suitable barrier function for the problem of this paper is B=

0.01 0.01 0.01 + + ρx ρy ρz nu  ft + + − σc1,i 1− i=1

+

ns  i=1

wmax wmax − wi

1 σc1,i ft

+

σc3,i fc

(25)

where nu is the number of load combinations related to the ultimate limit state and ns is the number of load combinations related to the serviceability limit state.

STRESS EXAMPLES

Table 1 shows 13 results of the proposed algorithm. The rows contain computation examples. Columns σxx , σyy , σzz , σxy , σxz , σyz contain the input stresses in N/mm2 . The reinforcement yield stress is fy = 500 N/mm2 for each example. Column ρx , ρy , ρz contain the output reinforcement ratios in %. Column σc1 , σc2 , σc3 contain the output principal concrete stresses. Column Eq. shows the formula number that gives the same result. Except for the last example all examples involve just one load combination. In the last computation example two load combinations are included. Example 1 to 7 have also been used by Andreasen et al. [5]. Their results and the results in this paper are the same. Example 8 and 9 have also been studied by Foster et al. [7]. In example 8 the same results have been found. In example 9, Foster selected ρx = 0.75%, ρy = 0, ρz = 0.75%. Table 1 shows that the optimal reinforcement differs considerably. However, the total reinforcement is almost the same (Foster; 0.75 + 0.00 + 0.75 = 1.50%, Table 1; 0.89 + 0.00 + 0.57 = 1.46%). It is noted that Forster et al. selected this reinforcement without trying to find the optimum. In fact, the optimum is an edge solution which was not considered in their publications [7, 8]. Example 10 shows that in a plane stress state several formula sets provide the optimum reinforcement. Example 11 and 12 are included for comparison with Example 13. Example 13 includes two load

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Table 1.

Stress computation examples.

Case

σxx

σyy

σzz

σxy

σxz

σyz

ρx

ρy

ρz

σc3

σc2

σc1

Eq.

1 2 3 4 5 6 7 8 9 10 11 12 13

1 −5 −5 −5 1 1 1 2 −3 3 15 . 15 .

2 2 −6 −6 2 −2 2 −2 −7 . . . . .

3 3 3 −6 3 3 3 5 . 10 . . . .

−1 1 1 1 −1 2 −1 6 6 . . 5 . 5

3 3 3 3 −3 3 2 −4 −4 5 . . . .

−4 4 4 4 −4 −4 4 2 2 . . . . .

1.00 . . . 0.60 0.50 0.40 2.40 0.89 1.60 3.00 1.00 3.00 3.00

1.40 1.36 . . 1.00 0.13 1.00 0.40 . . . 1.00 0.33 0.33

2.00 1.88 1.69 . 2.00 1.80 1.80 1.40 0.57 3.00 . . . .

−10.65 −10.31 −10.15 −10.44 −10.58 −10.17 −9.36 −15.21 −14.76 −10.00 . −10.00 −1.67 −16.67

−5.35 −5.89 −6.30 −6.31 −1.42 . −0.64 −0.79 −2.52 . . . . .

. . . −0.24 . . . . . . . . . .

21-10 21-4− 21-1 21-8 21-11 21-7 21-8 21-5+ 21-5−, 7, 10 21-3, 7, 9 21-6−, 7, 8

The dots (.) represent zeros (0) in order to improve readability of the table.

combinations. The volume reinforcement ratio is ρx + ρy + ρz = 3.00 + 0.33 + 0.00 = 3.33%. Alternatively, we could have selected the envelope of the reinforcement requirements for the individual load combinations, which are 11 and 12. The volume reinforcement ratio applying the envelope method is max(3.00, 1.00) + max(0.00, 1.00) = 4.00%. Consequently, the envelope method gives a safe approximation but it overestimates the required reinforcement substantially.

1000

z 200

9

STRUCTURAL EXAMPLE

x

Figure 3 shows a square concrete block that is fixed at one face of the block. The block is loaded by a vertical force of 1000 kN over an area of 0.20 × 0.20 m (25 N/mm2 ). Just one load is considered. Dead load (24 kN) is neglected. Young’s modulus is 30000 N/mm2 and Poisson’s ratio is 0.15. The concrete compressive strength is 35 N/mm2 . Its tensile strength is 4 N/mm2 . The steel yield strength is 550 N/mm2 . One load case related to the ultimate limit state is considered. Load and resistance factors are not included. The proposed algorithm has been implemented in a finite element program. An eight node brick element was used. The element dimensions are 0.10 × 0.10 × 0.10 m. A linear elastic analysis is performed. The normal stress σxx is shown in Fig. 4. The required reinforcement ratios for the ultimate limit state are computed by the proposed algorithm (Figs. 5, 6 and 7). The results provide sufficient information for a structural engineer to select bar diameters and bar spacing. Subsequently, the reinforcing cage can be designed by applying reinforcing principles (hoops, hooks, hairpins, development length). Note that not

Top view

50

200

50

25 N/mm 2 x

y

1000

Side view

1000 mm Figure 3. Concrete block loaded by a vertical force (dimensions in mm).

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fixed

front z

x y

y

Figure 4. The normal stress σxx on the surface of the concrete block. The largest value is 6.17 N/mm2 . The smallest value is −8.52 N/mm2 .

z x

Figure 7. Optimal reinforcement ratio ρz The largest value is 1.02%.

only reinforcement for bending and shear are needed but also splitting reinforcement is needed for introducing the load into the concrete. The authors recommend that the reinforcement detailing is checked by mentally visualising the force flow with a strut-andtie model. This does not mean that the reinforcement needs to be quantified with a strut-and-tie model. There is no need for this time consuming task because the required amounts are already determined by the proposed algorithm.

fixed

z

x

10

CONCLUSIONS

y

Figure 5. Optimal reinforcement ratio ρx The largest value is 1.48%.

A numerical algorithm is proposed for computing the required reinforcement in solid concrete. It starts from the stresses in the integration points of a finite element model. In subsequent improvements the algorithm finds the reinforcement ratios ρx , ρy , ρz for which the sum is smallest. Constraints are imposed on the steel stresses and the concrete stresses for load combinations related to the ultimate limit state. A constraint is imposed on the crack widths for load combinations related to the serviceability limit state. The algorithm shows to be robust, fast and accurate.

LITERATURE f ixed

z

x y

Figure 6. Optimal reinforcement ratio ρy The largest value is 0.56%.

[1] Eurocode 2, Design of concrete structures, EN 19921-1, 2004. [2] Marti P., Design of concrete slabs for transverse shear, ACI Structural Journal, Vol. 87 (1990) No. 2, pp. 180–190. [3] Lourenço P.B., Figueiras J.A., Solution for the design of reinforced concrete plates and shells, Journal of Structural Engineering, May 1995, pp. 815–823.

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[4] Smirnov, S.B., Problems of calculating the strength of massive concrete and reinforced-concrete elements of complex hydraulic structures, Power Technology and Engineering (formerly Hydrotechnical Construction), Springer New York, Volume 17, Number 9/September, 1983, pp. 471–476. [5] Andreasen B.S., Nielsen M.P., Armiering af beton I det tredimesionale tilfælde, Bygningsstatiske meddelelser, Vol. 56 (1985), No. 2–3, pp. 25–79 (in Danish). [6] Kamezawa Y., Hayashi N., Iwasaki I., Tada M., Study on design methods of RC structures based on FEM analysis, Proceedings of the Japan Society of Civil Engineers, Issue 502 pt 5–25, November 1994, pp. 103–112 (In Japanese). [7] Foster S.J., Marti P., Mojsilovi´c N., Design of reinforced concrete solids using stress analysis, ACI Structural Journal, Nov.-Dec. 2003, pp. 758–764. [8] Marti P., Mojsilovi´c N., Foster S.J., Dimensioning of orthogonally reinforced concrete solids, structural concrete in Switzerland, The first fib-Congress, Swiss Group of fib, Osaka, Japan, Oct. 13–19, 2002, pp. 18–23. [9] Hoogenboom P.C.J., de Boer A., Computation of reinforcement for solid concrete, Heron, Vol. 53 (2008), No. 4, pp. 247–271. [10] Prager W., Hodge P.G., Theory of Perfectly Plastic Solids, New York, Wiley, 1951. [11] Nielsen M.P., Limit analysis and concrete plasticity, second edition, CRC Press, 1999. [12] Vecchio F.J., Collins M.P., The modified compression-field theory for reinforced concrete elements subjected to shear, ACI Journal, Vol. 83, No. 2, March-April 1986, pp. 219–231. [13] CEB-fib Model Code 1990, Design Code, Thomas Telford, London, 1993, ISBN 0 7277 1696 4. [14] Nocedal J., Wright S., Numerical Optimization, Springer, New York, 1999, ISBN 0-387-98793-2

≥ 0 it is concluded that the values of σyy From  σ Ic2 σ and ± yzσyyxy − σxz need to have the same sign. Form Ic1 ≤ 0 it is concluded that this sign needs to be negative. σ σ  Consequently, σyy < 0 and ± yzσyyxy − σxz ≥ 0. Q.E.D. APPENDIX 2 Reinforcing bars do not need to be in the x, y and z directions. For reinforcement in any direction the concrete stress tensor is ⎡ ⎤ σcx σxy σxz ⎣ σxy σcy σyz ⎦ σxz σyz σcz n  vxi ρi σsi σcx = σxx − i=1

σcy = σyy − σcz = σzz −

n  i=1 n 

vyi ρi σsi vzi ρi σsi

i=1

where n is the number of bars, ρi is the reinforcement ratio of bar i, σsi is the normal stress in bar i and vxi , vyi , vzi are the components of the unit length direction vector of bar i.  The volume reinforcement ratio is ρ = ni=1 ρi , which can be minimised with the proposed algorithm.

APPENDIX 1 The ± sign in Eqs (21, 4), (21, 5) and (21, 6) can be replaced by the absolute value. Here this is proven for Eqs (21, 5). Substitution of Eqs (21, 5) in Eqs (10), (11) and (12) gives Ic1 = σyy +

2 + σ2 σxy yz

 σyz σxy

Ic2 =

±

σyy

σyy

 ±



σyz σxy − σxz 2 σyy

2 + (σ ± σ )2 ) − σxz (2σyy yz xy

σyy

Ic3 = 0

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Sequentially linear modelling of local snap-back in extremely brittle structures S. Invernizzi & D. Trovato Politecnico di Torino, Torino, Italy

M.A.N. Hendriks & A.V. van de Graaf Delft University of Technology, Delft, The Netherlands

ABSTRACT: In the present paper a sequentially linear approach is proposed for the study of extremely brittle structures. A peculiar definition of the saw-tooth approximation of the softening branch is presented, which allows for a straightforward discretization of the softening branch in the case of snap-back on constitutive level. In this way the well-known limit of the smeared crack approach is overcome, and extremely brittle structures, i.e. with a very low ratio between the fracture energy and the finite element size, can be modeled without the need of an extreme mesh refinement. The model is applied to some experimental results obtained on reinforced glass beams. 1

INTRODUCTION

The sequentially linear approach in modelling strain softening materials and structures has recently been demonstrated to be effective especially when the ratio between the elastic energy stored and the energy that can be dissipated by fracture is very large, i.e. in the case of real scale structures (Rots & Invernizzi, 2004). The model, introduced by Rots, replaces the downward stress-strain curve by a saw-tooth curve, either saw-tooth tension-softening for unreinforced material or saw-tooth tension stiffening for reinforced material. A linear analysis is performed, the most critical element is traced, the stiffness and strength of that element are reduced according to the saw-tooth curve, and the process is repeated. The sequence of scaled critical steps provides the global load displacement response. In the present paper we consider some experimental results obtained from bending of a reinforced threelayer glass beam. The mechanical response of the beam is extremely brittle, although the presence of reinforcement at the beam intrados, since the glass fracture energy is very small and the span of the beam quite large. Therefore, this structure can be considered representative of a class of extremely brittle structures. Due to the very limited ductility of the structure, the phenomenon of snap-back take place several times during the loading procedure (Carpinteri, 1984). From a numerical point of view, one possibility could be to model the fracture occurrence in glass by means of discrete interface elements, where the

cohesive law is given in terms of crack aperture. Unfortunately, this approach is not well suited in the case of reinforced structures, since many cracks arise during the tests and the position of cracks cannot be easily determined in advance. On the other hand, smeared cracking can be adopted, which provides automatic detection of the crack pattern. The dilemma is the role of the negative slope of the softening branch. Incorporating the negative slope in the tangent stiffness of a Newton Raphson scheme to solve the systems of equations, might turn out to be crucial for finding localizations of cracks along with a structural snapback behavior. At the same time the use of the negative slope in the tangent stiffness might make the analysis less stable, if not unstable. When the fracture energy divided by the crack bandwidth is very small, snapback behavior even shows up on constitutive level. A rigorous numerical treatment of the softening becomes impossible. From a practical point of view, apart from adapting the material parameters, the only way out with extremely brittle structures is to perform the analysis with extremely refined mesh, which is often unaffordable due to the required computational effort. In the paper it is shown how adopting the sequentially linear approach can effectively solve the problem. In fact, a mesh-independent saw-tooth discretization of the softening branch can be used even in the case of snapback behavior on constitutive level, allowing for affordable coarser mesh discretizations. The formulation of the methods, some examples and comparison with experimental results are provided and commented.

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2

THE OVERALL ‘‘EVENT-BY-EVENT’’ PROCEDURE

The locally brittle snap-type response of many brittle and quasi-brittle structures inspired the idea to capture these brittle events directly rather than trying to iterate around them in a Newton–Raphson scheme (Rots, 2001). A critical ‘‘event’’ is traced and subsequently a secant restart is made from the origin for tracing the next critical event. Hence, the procedure is sequential rather than incremental. The sequence of critical events governs the load—displacement response. To this aim, the softening diagram is replaced by a saw-tooth curve and linear analyses are carried out sequentially (Rots & Invernizzi, 2004). The global procedure is as follows. The structure is discretized using standard elastic continuum elements. Young’s modulus, Poisson’s ratio and initial strength are assigned to the elements. Subsequently, the following steps are sequentially carried out: • Add the external load as a unit load. • Perform a linear-elastic analysis. • Extract the ‘critical element’ from the results. The ‘critical element’ is the element for which the stress level divided by its current strength is the highest in the whole structure. • Calculate the ratio between the strength and the stress level in the critical element: this ratio provides the ‘global load factor’. The present solution step is obtained rescaling the ‘unit load elastic solution’ times the ‘global load factor’. • Increase the damage in the critical element by reducing its stiffness and strength, i.e. Young’s modulus E and tensile strength ft , according to a saw-tooth constitutive law as described in the next section. This corresponds to a local damage ‘‘event’’. • Repeat the previous steps for the new configuration, i.e. re-run a linear analysis for the structure in which E and ft of the previous critical element have been reduced. Trace the next critical saw-tooth in some element, repeat this process till the damage has spread into the structure to the desired level. In the case of non-proportional loading scheme, the determination of the global load factor is not as straightforward (DeJong et al., 2008), but can be approached basically with the same approach. 2.1 The saw-tooth approximation The way in which the stiffness and strength of the critical elements are progressively reduced at each ‘‘event’’ constitutes the essence of the model. In other words, it is necessary to provide a saw-tooth approximation of the constitutive stress–strain relation. Different approaches have been investigated in previous works (Rots et al., 2004), namely the linear softening with stepwise reduction of the Young’s modulus (Figure 1a)

Figure 1. Analytical reduction of Young’s modulus by a fixed factor (a), or by division of softening line into equidistant portions (b). Bandwidth model for linear tension softening (c), and for nonlinear tension softening (d).

and the linear softening with stepwise uniform reduction of the strength (Figure 1b). More recently, a new generalized tooth size approach has been presented (Rots et al., 2008), which allows for a straightforward unification of saw-tooth constitutive laws, not only linear softening (Figure 1c), but also nonlinear softening (Figure 1d) and generic constitutive law based on the definition of a strength bandwidth around the softening tail. At each tooth the stiffness reduction can be expressed in the following general way: Ei =

Ei−1 , ai

(1)

where ai depends on the adopted saw-tooth. It is worth noting that the underlying idea of each those saw-tooth approximations, is actually the same, i.e. to provide that the area below the curve is equal to the fracture energy divided by the element size. In this way, in fact, the proposed approach has been proved to be mesh-size independent (Rots & Invernizzi, 2004). 2.2

Damage anisotropy due to cracking

In the simplest hypothesis, the Young’s modulus can be (saw-tooth wise) reduced in all directions. Although this isotropy assumption allows for the simulation of cracking in case of direct tension or bending (i.e. when the phenomenon is basically driven by a localized crack in a one-dimensional stress field), a substantial improvement is necessary when dealing with reinforced structures. In fact the isotropic reduction of stiffness is a rather rough approximation, and does not represent the compressive struts that develop parallel to the cracks.

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Therefore, in analogy to the pioneering approach of Rashid (1968), the initial isotropic stress-strain law can be replaced by an orthotropic law upon crack formation (Rots et al., 2006), with the axes of orthotropy being determined according to a condition of crack initiation. Referring to the plane stress situation, and to a local coordinate system oriented parallel to the crack plane, the following constitutive relation is assumed (e.g. Rots et al., 1985): ⎡ En ⎧ ⎫ ⎢ ⎨σnn ⎬ ⎢ 1 − νnt νtn ⎢ σtt = ⎢ νtn Et ⎩σ ⎭ ⎢ nt ⎣ 1 − νnt νtn 0

νnt En 1 − νnt νtn Et 1 − νnt νtn 0

⎤ ⎥⎧ ⎫ ⎥ ⎨εnn ⎬ ⎥ , ⎥ ε 0 ⎥ ⎩ tt ⎭ ⎦ εnt Gred (2) 0

where n is the normal to the crack, t the crack plane. En and Et represent Young’s moduli normal and tangential to the crack face respectively. It is assumed that νtn relates the lateral contraction (εtt ) to the longitudinal extension (εnn ) through εtt = −νtn εnn . Due to symmetry of the stress-strain matrix, νnt En = νtn Et , therefore νnt is not an independent material parameter. When the first crack arises (primary crack initiation) En is reduced while Et is kept equal to the initial stiffness E0 . Simultaneously, the tensile strength in normal direction is reduced, whereas the tensile strength in tangential direction is kept equal to the initial tensile strength. In addition, νnt is also reduced according to νtn = ν0 (En /E0 ). The reduced shear modulus Gred depends on the chosen shear retention relation, being possible to adopt full shear retention, constant shear retention, or also a stepwise decreasing shear stiffness. This formulation is based on the work of Bazant & Oh (1983) extended with shear retention. Extension towards axi-symmetry, plane strain and 3D is straightforward. The equation can be rewritten in compact form as follows: σnt = Dnt εnt .

(3)

In addition to Young’s modulus, also the shear retention factor and Poisson’s ratio are reduced with increasing crack opening. In the present implementation a stepwise reduction is assumed: ⎧ N −i ⎪ ⎪ ⎪ ⎨βi = N , (4) 0 ≤ i ≤ N, ⎪ ⎪ N − i ⎪ ⎩νi = N where i is the current tooth, and N the number of teeth adopted in the discretization. Given the following transformations for the strain and stress vectors:  εnt = Tε (φ)εxy , (5) σnt = Tσ (φ)σxy

(a)

(b)

Figure 2. Bandwidth saw-tooth diagrams for elastic-plastic behavior in compression (a), and for symmetric behavior in tension and in compression (b).

eq. 3 can be easily transposed in terms of global stress and strain components by pre- and post-multiplication with the transformation matrices: σxy = T−1 σ (φ)Dnt Tε (φ)εxy . 2.3

(6)

Saw-tooth approximation for ductile material

It is straightforward to obtain a saw-tooth approximation even in the case of ductile material behavior (Figure 2). In this way, it is possible to model also reinforced structures or the crushing phenomenon in compressed quasi-brittle and brittle materials.

3

MODELING OF EXTREMELY BRITTLE MATERIALS

When the smeared crack approach is adopted to model cracking, the cohesive crack law (tractions vs. crack

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opening displacement) is substituted by an equivalent stress-strain constitutive law, provided that the crack is ‘‘smeared’’ over the finite element size h (crack bandwidth). In this case, it is well known (Bazant & Oh, 1983) that it is necessary to regularize the stress-strain constitutive law to avoid the mesh dependence of the model. In other words, the slope of the softening tail is modified, such that the area below the curve equals the fracture energy divided by the element size. In this way the constitutive law is adapted for every finite element based on the size, such that smaller elements are given a more ductile curve while steeper softening tails characterizes greater elements. When the softening slope become positive the numerical algorithm become unstable and no solution can be found. This behavior mimics, numerically, the physical behavior of snap-back (Carpinteri, 1896, Carpinteri et al., 1999) of a displacementcontrolled test. Limiting the element size can often circumvent this situation, but is not an affordable possibility in the case of extremely brittle materials. In fact, when dealing with materials like glass, because of the very limited fracture energy the maximum element size become usually too small, and the degree of freedom of the model too large, with respect to the available computational power. 3.1

Figure 3. Scheme of the saw-tooth approximation in case of positive slope softening.

for the following ones:   2Gf h 2Gf h ft − , −p ft E ft

εu+ =

Saw-tooth for positive slope softening

While classical nonlinear analysis can not be performed in the case of positive slope softening tail (i.e. with snap-backs), the sequentially linear analysis can work, provided that an adequate saw-tooth approximation is given. Actually, the philosophy does not differ that much from the more common negative slope case, and can be summarized as follow. The ultimate deformation, in the case of linear softening, is calculated in the classical way, but note that its value will be lower than the deformation at the peak load. 2Gf h . (7) εu = ft

ft1− = ft (1 − p) , ε1 =

Dεu+ , D−E

+ fti+1 =

Ei Dεu+ , D − Ei

− + = fti+1 − 2pft , fti+1

εi+1 =

Dεu+ , D − Ei

(9)

(10)

(11)

(12) (13)

(14)

The softening modulus, which is positive, can be obtained as follows: ft ft D= = f > 0. (8) 2G h t εe − ε u − f

Ei+1 =

Then, it is necessary to define two curves parallel to the softening tail, uplifting and downscaling of a constant amount equal to p times the tensile strength. The quantities pertaining the curve above will be referenced with the apex +, while the other with the apex −. The following equations provide the strength, strain and stiffness in correspondence of the first tooth and

It is worth noting that the percentage p can be optimized to obtain the desired number of teeth and to provide that the area under the curve equals the fracture energy divided by the element size up to the desire tolerance.

E

ft

− fti+1

εi+1

.

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(15)

3.2

Model validation

In order to validate the proposed saw-tooth definition, it is necessary to make a comparison with a simple benchmark problem. Note that a direct comparison with the nonlinear analysis calculation is not possible, for the reasons outlined above. Therefore the reference curve obtained with the nonlinear calculation will be obtained only for the finer mesh. We considered a typical notched beam subjected to three-point bending. The structural scheme and dimension is shown in Figure 4, while Figures 4a–c show the three adopted meshes. Table 1 shows the mechanical parameters. Note that the fracture energy was chosen small enough that for the coarse mesh (Figure 4c) the linear softening slope become positive. 3.2.1 Nonlinear analysis results The nonlinear results are shown in Figure 5. The loaddisplacement curves are well overlapping in the case

Figure 5. Reference load-displacement curves obtained with non-linear analyses (fine, medium and coarse mesh).

Figure 6. Sequentially linear analysis and saw-tooth approximation for the fine mesh.

of fine and medium meshes, while no results can be obtained in the coarse mesh case. Figure 4. Geometry of the notched beam: h = 5 mm fine (a), h = 10 mm medium (b) and h = 20 mm coarse (c) mesh discretization. Table 1. beam.

Material parameters of the validation

Young Modulus Poisson’s ratio Tensile strength Fracture energy Shear ret. Factor

E v ft GF β

30 GPa 0.2 5 MPa 4.15 J/m2 0.001–0.2

3.2.2 Sequentially linear analysis results The corresponding load displacement curves, obtained with the sequentially linear analysis are shown in Figures 6–8. In the case of the fine mesh (Figure 6) the classical saw-tooth approximation was used (Rots et al., 2008), since the softening modulus is negative. In the case of the medium mesh (Figure 7) the sequentially linear analysis has been performed adopting a quite peculiar saw-tooth approximation. In fact, the dimension of elements and Fracture energy were chosen on purpose such that the softening tail becomes

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4

Figure 7. Sequentially linear analysis and saw-tooth approximation for the medium mesh.

REINFORCED GLASS BEAMS

The proposed approach as been applied to simulate numerically some experimental results obtained from four-point bending tests of reinforced glass beams (Louter, 2007). The beams were obtained with an unconventional construction method for large span structural glass beam. The procedure is based on two fundamental concepts that are the Reinforced glass (SGP) and the Adhesively bonded glass segments (Acrylate). Briefly, the first one consists in a small stainless steel section that is adhesively bonded at the tensile zone of the glass beam acts as reinforcement. The second concept consist in an overlapping operation of glass segments adhesively each other bonded, to create beams with a span exceeding the standard maximum size of glass panel. The structural scheme and the cross sections are shown in Figure 9. The mechanical properties adopted for the glass composite and for the reinforcement are shown in Table 2. It is worth noting that different values of the shear retention factor can be assigned, in order to better represent the composite in the two cases. Four main phases of crack propagation are distinguished during the application of the load: a. Small cracks (initial failure) The beam specimen shows linear elastic behavior until the global tensile bending stress at the lower edge exceeds the local tensile strength of the glass. One or several small cracks occur, which originate

(a)

(b)

Figure 8. Sequentially linear analysis and saw-tooth approximation for the coarse mesh.

vertical. In this very peculiar case, each tooth of the curve exactly match the mother curve, and no correction is necessary regardless the number of teeth. In the third case (Figure 8), we have a positive slope softening tail, and the proposed procedure is adopted. All the three cases matches egregiously with the reference curve obtained with the nonlinear analysis. It is worth noting that the increased scatter obtained in the third case can be reasonably ascribed to the coarseness of the last mesh.

Figure 9. Glass beams: structural scheme (a); sections and numerical idealization (b). Table 2. Material parameters of the reinforced glass beams.

Young Modulus Poisson’s ratio Tensile strength Fracture energy Density Shear ret. Factor

E v ft GF ρ β

Glass

Steel

70 GPa 0.2 50 MPa 3 J/m2 2650 kg/m3 0.01–0.2

190 GPa 0.27 500 MPa – 7850 kg/m3 –

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at the lower edge of the glass beam and run over 2/3 of the total beam height before being stopped in the upper compression zone. The cracks generally only run in one glass layer and do not affect the other glass layers in the beam laminate. Depending on the applied adhesive, the cracks are V-shaped or consist of a single line of fracture. b. Large cracks at the seams As loading is continued, bending stiffness is slightly decreased. The load increases again until, at a certain point, large vertical cracks appear. These cracks are generally located at the seams and run in all glass layers present at this seam. These cracks might occur at once or in segments at small time intervals (total crack growth within a second). Depending on the applied adhesive the reinforcement might be torn from the glass locally on either side of the crack origin due to the shock load upon glass failure. A sudden increase in vertical displacement of the specimen causes a drop in load (displacement controlled tests). As loading is continued the load starts to rise again until, at a second peak-load, similar cracks occur at a different position along the edge of the beam and existing small cracks start to propagate. Depending on the applied reinforcement geometry and applied adhesive, progressive detachment of reinforcement might occur. The stress displacement diagram clearly shows a decrease in bending stiffness after each peak-load. Four different meshes were used in the analysis to investigate the eventual influence of the mesh size and mesh alignment.

c. Horizontal crack propagation The existing large (vertical) cracks start to propagate horizontally and start to grow towards each other. Bending stiffness gradually decreases until final failure occurs. d. Collapse (ultimate failure) At the final failure stage the beam has largely lost its bending stiffness and collapses. Two different failure mechanisms have been observed: – Detachment of reinforcement; the reinforcement has been torn from the glass. Tensile forces cannot be transferred anymore and the beam collapses. – Buckling; lateral instability due to decreased cohesion of the glass by excessive cracking causes the beam to buckle. Which final failure mechanism occurs is dependent on the beam cross-section geometry, the geometry of reinforcement and the type of glass-reinforcement adhesive bond. Figure 12 shows the horizontal stress distribution plots and the principal total strain plots for different stages of the sequentially linear analysis. The four main stages in fracture propagation can be recognized quite well.

Figure 11. Two different meshes used in the simulations, varying the mesh size and the meshing algorithm. All the structural schemes exploit the symmetry.

Figure 10. Extremely sharp saw-tooth diagram for refined mesh ‘a’ with h = 6,25 mm (a); and saw-tooth curve for steel.

Figure 12. Representation of contour plot of the horizontal stress and of the principal maximum strain for different steps of SLA for the Mesh ‘a’.

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On the other hand, Figure 13 shows the evolution of the corresponding tensile stress in the reinforcement. Again the four stages are recognizable, as well as the stress migration when the yielding of the reinforcement takes place. Figure 14 shows some details of the crack pattern at different load stages. It is worth noting that the numerical simulations are able to catch correctly the different phases of the crack propagation.

Figure 15 compare the load-displacement curves obtained from the sequentially linear analysis with the experimental results. A quite good agreement is found, provided that the optimum shear retention factor is adopted.

5

CONCLUSIONS

A sequentially linear approach has been proposed for the study of extremely brittle (e.g. glass) structures. A specific definition of the saw-tooth approximation of the softening branch is proposed and validated, which allows for a straightforward discretization of the softening branch in the case of snap-back on constitutive level. In this way the well-known limit of the smeared crack approach is overcome, and extremely brittle structures, i.e. with a very low ratio between the fracture energy and the finite element size, can be modeled without the need of an extreme mesh refinement. The model has proved efficient in simulating some experimental results obtained on reinforced glass beams, being able to represent correctly the loaddisplacement curve as well as the main feature of the crack pattern.

Figure 13. Evolution of the tensile stress along the reinforcement for increasing load.

REFERENCES

Figure 14. Details of the numerical crack pattern compared to the experimental results. Diffused V-shaped at the initial stage of cracking (a); sub-horizontal cracking and reinforcement debonding (b).

Figure 15. Load displacement diagrams: comparison with the experimental results.

Bažant, Z.P. and Oh, B.H. 1983, Crack band theory for fracture of concrete, Materials and Structures, 16(93): 155–177. Carpinteri, A. 1986, Mechanical Damage and Crack Growth in Concrete: Plastic Collapse to Brittle Fracture. Martinus Nijhoff Publishers, Dordrecht. Carpinteri, A. and Monetto, I. 1999, Snap-Back analysis of fracture evolution in multi-cracked solids using boundary element method, International Journal of Fracture, 98:225–241. DeJong, M.J., Hendriks, M.A.N. and Rot, J.G. 2008, Sequentially linear analysis of fracture under non-proportional loading, Engineering Fracture Mechanics 75: 5042–5056. Louter, and Christian, 2007, Adhesively bonded reinforced glass beams, HERON Volume 52(1/2), special issue: Structural glass. Rashid, Y.R. 1968, Analysis of prestressed concrete pressure vessels, Nuclear Engng. And Design, 7(4):334–344. Rots, J.G. 2001, Sequentially linear continuum model for concrete fracture. In: de Borst R, Mazars J, PijaudierCabot G, van Mier JGM, (eds) Fracture mechanics of concrete structures: 831–9. Lisse: Balkema. Rots, J.G. and Invernizzi, S. 2004, Regularized sequentially linear saw-tooth softening model, Int. Journal for Numerical and Analytical Methods in Geomechanics, 28:821–856. Rots, J.G., Belletti, B. and Invernizzi, S. 2007, Robust modeling of RC structures with an ‘‘event-by-event’’ strategy. Engineering Fracture Mechanics, 75(3–4):590–614. Rots, J.G., Invernizzi, S. and Belletti, B. 2006, Saw-tooth softening/stiffening—a stable computational procedure for RC structures. Computers and Concrete, 3(4):213–233.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Simulation of masonry beams retrofitted with engineered cementitious composites M.A. Kyriakides Stanford University, California, USA

M.A.N. Hendriks Delft University of Technology, The Netherlands

S.L. Billington Stanford University, California, USA

ABSTRACT: Two-dimensional nonlinear finite element analyses using two different micro-modeling approaches are used to simulate unreinforced masonry beams retrofitted with a thin layer of ductile fiberreinforced cement based material referred to as Engineered Cementitious Composites under four-point bending. In a detailed approach each material is modeled independently and in a simplified approach, expanded brick units with zero thickness mortar elements are used. The adequacy of these models to capture the experimental response is examined. It was found that both modeling approaches can be used to capture the response of the retrofitted masonry beams well, with the exception of the initial stiffness. Attention is required in modeling the ECC-masonry adhesion areas, as extensive cracking of the masonry may occur, which was not observed experimentally. 1

INTRODUCTION

A thin layer of ductile fiber-reinforced mortar material referred to as Engineered Cementitious Composites, or ECC has been experimentally investigated as a seismic retrofit for unreinforced masonry infill walls in non-ductile reinforced concrete frames. A series of small scale tests including compression and flexural tests of retrofitted masonry, as well as in-plane cyclic tests of 1/5th scale non-ductile concrete frames with ECC retrofitted masonry infills were performed for the retrofit validation (Kyriakides & Billington 2008). Non-linear finite element analysis of these experiments is underway. In this paper, analyses of the ECC retrofitted masonry beams subjected to fourpoint bending are presented. This study supports on-going research to develop reliable methodologies for researchers and practitioners to assess the performance of unreinforced masonry infilled nonductile concrete structures retrofitted with ECC using nonlinear finite-element analysis. Flexural tests have indicated that the strength and ductility of a masonry beam under four-point bending are increased tremendously with a thin ECC layer applied to the tension face. Plain brick beams demonstrated very brittle failure with approximately 20–25 times lower strength (Kyriakides & Billington 2008).

Two different two-dimensional micro-modeling approaches of masonry samples are most prevalent in the literature (e.g. Lotfi & Shing 1994, Lourenço et al. 1998, Douglas et al. 2003, ElGawady et al. 2006, Binda et al. 2006, Chaimoon & Attard 2009). The first modeling approach, referred to here and by others as the detailed micro-modeling approach, represents all parts of the masonry sample independently, i.e. the brick units, the mortar joints and the brick/mortar adhesion areas. The second approach referred to here and also by others as the simplified approach, uses zero-thickness interface elements to connect the brick units together and expanded brick units such that the actual geometry of the specimen remains unchanged (Rots, 1997). The expanded brick units have modified material properties to account for the material properties of the brick units and the mortar joints with the adhesion areas. Although this method is not as accurate as the detailed model due to the fact that the Poisson’s effect of the mortar is not included, it has been adopted to analyze relatively large structures in a detailed manner, with less computational cost compared to the detailed model but more than a macro-modeling approach would require. Both micro-modeling approaches are examined in this paper using a commercial finite-element program (DIANA 2009) to predict the performance of the ECC

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retrofitted masonry beams under four-point bending. These experiments were performed to give insight to the potential response of a retrofitted masonry wall in out-of-plane bending as well as in direct tension from in-plane lateral loading. Alternate modeling approaches of the bond between the ECC layer and the masonry surface are examined. 2 2.1

EXPERIMENTS Fabrication of specimens

The masonry specimens were built by a professional mason in order to ensure real practice conditions. The materials used were 94 mm × 196 mm × 58 mm clay bricks (ASTM C216, Grade MW, Type FBS) and mortar with mix ratio 1:1:5 (cement: lime: sand), a typical type of mortar used for building masonry walls in the 1920’s in western United States. All brick beams had nine brick units with eight mortar joints, each 10 mm thick, and the specimens’ dimensions were measured approximately 602 mm × 196 mm in plan and 94 mm deep. All specimens were retrofitted with 13 mm of sprayable ECC. ECC is a micromechanically designed fiber-reinforced cement based material that exhibits high tensile strain-hardening behavior with less than 2% fibers by volume. ECC has been used to repair and retrofit structures (Li et al. 2000). An ECC suitable for wet-mixture shotcreting (sprayable ECC) has recently been developed using a micromechanics- and rheology-based design and was demonstrated to spray on to a vertical surface achieving a thickness of 45 mm (Kim et al. 2003).

Figure 1. Flexural test setup according to ASTM E518-03 with a quarter-point bending configuration. All dimensions are in mm.

2.2 Test setup and instrumentation Flexural tests were performed according to the international standard ASTM E518-03 but with quarter-point bending instead of third-point bending. The test setup was accommodated by an MTS loading table fixed on the ground, and the load was applied with a displacement controlled procedure at a rate of 0.006 mm/sec by a 245 kN MTS hydraulic actuator. The deflection was measured by two string potentiometers attached on either side of the brick beam at midspan and at the brick midheight (Fig. 1). The specimen was placed into the setup such that its retrofitted side was always facing its tensile side. 2.3 Experimental response and failure mode Three out of the five ECC retrofitted specimens gave a similar and more representative response of this group and their load-displacement responses are given in Figure 7 when compared with the simulations. In particular, the mortar-brick interface at the 3rd joint initially started to open, followed by multiple cracking of

Figure 2. a) Typical failure mode of ECC retrofitted masonry beams under quarter-point bending. b) ECC multiple cracking at joints (view under beam, after testing).

the ECC layer below the mortar joints at the constant moment region and finally failure of the ECC below the 3rd joint (Fig. 2). Despite the similar failure mode of the three retrofitted specimens mentioned above, variability in the load-displacement responses in terms of initial stiffness, load-carrying capacity and ductility were observed experimentally in all specimens. This variability is attributed to the different mechanical properties of the mortar developed when placed and cured within the brick units, by the variation found in the fabrication of the mortar joints by the mason in terms of actual brick-mortar adhesion area size and by possible imperfections and thickness inconsistency in the ECC layer (13 mm ± 5 mm). However, it is

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clear that by troweling a thin layer of ECC onto the masonry beam surface, the load-carrying capacity of the specimens is increased by 20–25 times compared to the plain masonry beams. More importantly, a very ductile response is achieved rather than a brittle one that occurs when plain brick beams are tested under four-point bending.

3

MODELING APPROACHES FOR ECC RETROFITTED BEAMS

Finite element analyses are conducted for understanding the behavior of the different materials in the retrofitted masonry beams as well as to predict the performance of alternate retrofit designs. From the experimental responses it appears that the mortar joints and in particular the adhesion areas between the mortar joints and the brick units contribute significantly to the deformation behavior and failure mode of the masonry beams. Furthermore, the bond between the masonry surface and the ECC layer is a parameter that plays a major role in the performance of the retrofitted masonry beams given that the load and ductility increases depend on ECC’s adhesion to the masonry. A detailed and simplified micro-modeling approaches are used to examine i) the impact of each basic unit to the response of the retrofitted beam, and ii) the adequacy of each model to capture the experimental response. A schematic of the basic units of the two approaches used with the different parts are shown in Figure 3. 3.1

Detailed micro-modeling approach

3.1.1 Geometric model and loading In the detailed micro-modeling approach, all parts of the retrofitted masonry beam are modeled independently i.e. the brick units, the mortar joints, the

brick-mortar adhesion areas, the ECC and the ECCmasonry adhesion areas. The brick units, the mortar and the ECC are modeled with 8-noded plane stress elements with 3 × 3 Gaussian integration points. ‘‘Zero-thickness’’ interface elements with 3 Gaussian integration points are introduced between the brick units and the mortar to simulate the brick-mortar adhesion areas. For mesh compatibility, ‘‘zero-thickness’’ interface elements with 3 Gaussian integration points are introduced between the ECC continuum elements located directly below the edges of the brick units and the mortar joints. Due to symmetry, half of the beam is modeled for simplicity as the response of a whole and half beam model gave nearly identical results. Figure 4a shows the finite element mesh of the ECC retrofitted masonry beam. The general constitutive and geometric properties of the models are given in the following sections. For the nonlinear finite element analyses, the regular Newton-Raphson iterative scheme is adopted with a 0.0025 mm step size and 100 iterations per step. When the force norm or the displacement norm is less than 10−2 or 10−3 times the corresponding norm at the start of each step, respectively, then convergence occurs.

3.1.2 Constitutive models and material properties The material properties used in the analyses were obtained from material characterization tests performed and are listed in Table 1, with the exceptions of (1) the clay brick units are assumed to have a tensile strength of 10% of its measured compressive strength, (2) the Poisson’s ratio of all materials is assumed to be 0.15, and (3) the Young’s and shear moduli of the brick-mortar joints used to calculate the brick-mortar interface stiffnesses are assumed to be equal to those of the mortar. The brick units are modeled with a total strain, fixed crack model in tension (Feenstra et al. 1998). A linear elastic model in compression is adopted as no failure of the compression zone was observed during the tests. A tension cut-off criterion is adopted in order to capture brick cracking.

Figure 3. Schematic representation of the two micromodeling approaches: a) Detailed model, and b) Simplified model.

Figure 4. Finite element mesh of half of the ECC retrofitted brick beam analyzed using a) the detailed approach, and b) the simplified approach. The numbers correspond to the mortar joints.

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Table 1.

Material properties and model parameters.

Parameter

Symbol

Value

Brick Unit (continuum) Young’s Modulus Poisson’s ratio Compressive strength Tensile strength Shear retention factor

Ebrick vbrick fcbrick ft brick βbrick

16,547 N/mm2 0.15 49.3 N/mm2 4.93 N/mm2 0.2

Mortar Joint (continuum) Young’s Modulus Poisson’s ratio Compressive strength Tensile strength Fracture Energy Shear retention factor

Emortar vmortar fc mortar ft mortar Gf βmortar

3385 N/mm2 0.15 11.3 N/mm2 0.96 N/mm2 1.75 N/mm 0.2

ECC (continuum) Young’s Modulus Poisson’s ratio Compressive strength Shear retention factor

EECC vECC fcECC βECC

11,245 N/mm2 0.15 53.2 N/mm2 0.2

Brick-Mortar Interface Linear normal stiffness Linear tangential stiffness Bond tensile strength

kn kt tn

355,600 N/mm3 154,500 N/mm3 0.145 N/mm2

ECC-Brick Interface Linear normal stiffness Linear tangential stiffness

kn kt

88,500 N/mm3 38,500 N/mm3

ECC-ECC Interface Linear normal stiffness Linear tangential stiffness

kn kt

4425 kN/mm3 1925 kN/mm3

For the mortar joints a total strain, fixed crack model in tension is also used. A parabolic fracture energybased diagram for mortar in compression is used with a brittle failure in tension. For the interface elements introduced between the brick units and the mortar to simulate the brick-mortar adhesion areas, linear elastic models in both tension and shear are adopted. A discrete crack-brittle failure mode with a low tensile stress cut-off representing the brick-mortar bond strength is assigned to these elements. This tensile strength corresponds to the stress of the tension side of the unretrofitted brick beam at its peak load-carrying capacity obtained experimentally under four-point bending. For the ECC elements a total strain, fixed crack model in tension is used as well. The Thorenfeldt model is adopted in compression (Thorenfeldt et al. 1987), and a multi-linear crack model (Fig. 5) derived from an inverse analysis of the simulation of four-point bending tests of 25 mm × 25 mm × 305 mm ECC beams fabricated during the retrofit of the brick beams, is used in tension. ‘‘Infinite’’ linear elastic stiffness is used for the interface elements introduced between the ECC continuum elements.

For the bond between the ECC layer and the masonry, linear elastic and elastic-perfectly plastic models in both normal and transverse directions are examined for the interface elements (discussed in section 4.1). 3.2

Simplified micro-modeling approach

3.2.1 Geometric model and loading For the simplified modeling approach, expanded brick units are used with zero thickness interface elements between them, keeping the dimensions of half of the beam that is modeled, unchanged. The finite element mesh is shown in Figure 4b. Similar to the nonlinear finite element analyses of the detailed micro-models, the regular NewtonRaphson iterative scheme is adopted with a 0.0025 mm step size and 100 iterations per step. When the force norm or the displacement norm is less than 10−2 or 10−3 times the corresponding norm at the start of each step, respectively, then convergence occurs. 3.2.2 Constitutive models and material properties For the simplified micro-modeling approach the same types of elements, material properties and constitutive

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stiff elements for simplicity. A multi-linear model in the normal direction for the ECC-ECC interface elements following the ECC uniaxial tensile response given in Figure 5b is also examined.

4 4.1

Figure 5. a) Load-displacement response of an ECC beam (experiment & simulation), b) ECC uniaxial tensile stress-strain response derived from inverse analysis of the ECC beam.

models are used as with the detailed micro-modeling approach described in section 3.1.2 except for the mortar-brick interfaces and the ECC interface elements directly below. ‘‘Zero-thickness’’ brick-mortar interface elements are used with normal stiffness kn and shear stiffness kt that incorporate the properties of both the brick units and the mortar according to (1) and (2) as follows: kn =

Ebrick Emortar hmortar (Ebrick − Emortar )

(1)

kt =

Gbrick Gmortar hmortar (Gbrick − Gmortar )

(2)

where Ebrick is the Young’s modulus of the brick units, Emortar is the Young’s modulus of the mortar, Gbrick and Gmortar are the corresponding shear moduli and hmortar is the actual thickness of the mortar joint. The derivation of the expressions in (1) and (2) for the interface stiffnesses can be found in Rots, 1997. For these elements, a discrete crack-brittle failure model in tension is adopted. For the case of the ECC-ECC interface elements directly below the mortar interface elements, linear elastic interface elements with very large values for their stiffnesses are used assuming ‘‘infinitely’’

NUMERICAL SIMULATION OF ECC RETROFITTED MASONRY BEAMS ECC-masonry interface model

A study of the ECC-masonry bond was conducted given that no experimental data were recorded nor found in the literature to describe this bond performance in both the normal and transverse directions. For simplicity, linear elastic (LE) and elastic-perfectly plastic (EP) models were evaluated (Fig. 6). No ECCmasonry interface sliding was observed visually during the experiments. It is noted that the amount of sliding simulated was so small that could not have been observed visually. For the case of the elastic-perfectly plastic models, a sensitivity study of the normal and shear bond strengths as variables was performed. The purpose of this study was to determine the range of values of the bond strengths in the tensile and shear directions for which the retrofitted beam failure modes were representative of those observed experimentally. The compressive response of the EP interface model was following that of the ECC. However, in the analyses the interface elements deformed within the elastic range in compression. Figure 7 shows the analytical and experimental load-displacement responses obtained with both the detailed and simplified modeling approaches assuming linear elastic behavior for the ECC-masonry interfaces (LE Models). The overall response was

Figure 6. ECC-masonry interface models evaluated: a) Linear elastic, and b) Elastic-perfectly plastic.

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Experiments Detailed LE Model Simplified LE Model

25

Load (kN)

20 15 10 5 A

0 0.0

0.5

1.0

1.5

B

2.0

2.5

Displacement at midspan (mm)

Figure 7. Load-displacement responses of ECC retrofitted masonry beams obtained from both experiments and analyses. The dots indicate the displacement levels where the analyses did not converge.

adequately captured, with the exception of the initial stiffness (discussed further in section 4.2). The principal tensile strain contour plots at displacements A and B shown in Figure 8 indicate brick cracking close to the ECC-masonry adhesion area, especially at the constant moment region (i.e. to the right of the loading point). The colors used in the contour plots of the masonry beam and the ECC retrofit correspond to those used in the material tensile strain diagrams shown in Figures 9a & b respectively. At several displacement levels indicated in Figure 7, the finite element analyses did not converge (nor did it diverge) due to the extensive cracking developed in the brick units. When elastic-perfectly plastic models were adopted, local in-plane bending of the less stiff ECC layer below the brick units occurred. It was found that for values of bond strength less than 0.14 MPa in tension and less than 0.69 MPa in shear, the analysis diverges due to early ECC-masonry bond failure close to the 1st mortar joint. For models with bond strength values more than 1.7 MPa in tension and 1.4 MPa in shear, the overall failure mode of the retrofitted beam is captured adequately, but again extensive cracking in the brick units close to the ECC-masonry adhesion area is obtained. For tensile bond strength values within the range of 0.14 MPa to 1.7 MPa and shear bond strength values from 0.69 MPa to 1.4 MPa no brick cracking is observed and the analysis response and the failure mode are in good agreement with the experimental results. Figure 10 presents the load-displacement responses obtained with the detailed micro-modeling approach using the elastic-perfectly plastic ECC-masonry interface models (EP models, Fig. 6) for two different tensile (σn ) and shear (σt ) bond strength values.

Figure 8. Principal tensile strain contour plots of (a) the simplified, and (b) the detailed modeling approach. The shading indicates the strain state of each material according to Figure 9. The deformed shapes are enlarged 20 times.

Figure 9. Tensile stress-strain diagrams of a) brick and mortar, and b) ECC model.

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Experiments Detailed EP Model Simplified EP Model

25

Load (kN)

20 15 10 5 A B

C

D

0 0.0

0.5

1.0

1.5

2.0

2.5

Displacement at midspan (mm)

Figure 10. Load-displacement responses of retrofitted beams obtained from detailed micro-modeling analyses using the EP ECC-masonry interface models from Fig. 6.

Towards higher tensile bond strength values, the local in-plane bending of the ECC layer below the brick units that occurs analytically is prevented, together with an early failure of the ECC layer. Consequently, the response becomes more ductile. On the other hand, towards higher shear bond strength values, sliding of the ECC layer below the masonry beam is restricted. As a result, ECC cracking mainly occurs below the mortar joints leading to a less ductile response. For the comparison of the simplified and detailed modeling approaches presented herein, the tensile bond strength was selected to be 0.69 MPa and the shear bond strength 1.2 MPa; values with which the ductility and load-carrying capacities obtained experimentally are well captured. 4.2

Figure 11. Load-displacement responses of retrofitted masonry beams obtained from both experiments and analyses using elastic-perfectly plastic ECC-masonry interface models.

Simplified EP Model

Displacement at A (Fig. 11): Δ =0.22mm, L =13.79kN

Comparison of the detailed and simplified modeling approaches

The analytical and the experimental load-displacement responses of the ECC retrofitted masonry beams are given in Figure 11. Both modeling approaches capture adequately the overall behavior of the specimens with the exception of the initial stiffness. For the different displacement levels A, B, C and D indicated in Figure 11, the principal tensile strain contour plots of the masonry beam and the ECC layer are obtained and presented in Figures 12 & 13. The colors used in the contour plots of the masonry beam and the ECC retrofit, correspond to those used in the material tensile strain diagrams shown in Figures 9a & b respectively, and indicate the principal tensile strain state of each material at the corresponding displacement levels. First cracking in the ECC occurs once the ECC exhibits a tensile strain over 0.04% as indicated by the uniaxial tensile stress-strain diagram of the material (Fig. 9b). At displacement level A, the simplified model indicates full cracking in the ECC under

Displacement at C (Fig. 11): Δ =1.10mm, L =18.28kN Figure 12. Principal tensile strain contour plots of the simplified EP modeling approach. The shading indicates the strain state of each material according to Figure 9. The deformed shapes are enlarged 20 times.

the mortar joints located in the constant moment region. Degradation in the initial stiffness of the loaddisplacement response is observed as the ECC finite elements located below the 3rd and 4th joints (constant moment region) fully enter the strain hardening branch of the ECC tensile diagram. At almost the same displacement level (displacement level B) the same response is obtained by the detailed model. At displacement level C of the simplified model, ECC

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Detailed EP Model

Table 2. Comparison of the initial stiffness K, and the ductility 2 − 1 , obtained by the specimens experimentally and analytically. The ratio of the analytical values over the experimental ones is indicated in brackets.

Displacement at B (Fig. 11): Δ =0.25mm, L =13.59kN

K F1 (MPa) (kN)

F2 (kN)

2 − 1 (mm)

Experiments*

998

1.27

1.68

1.17

Detailed EP Model

2368 [2.37]

1.36 1.76 1.15 [1.07] [1.05] [0.98]

Simplified EP Model 2510 [2.52]

1.40 1.73 0.93 [1.10] [1.04] [0.79]

* Average of 3 specimens.

Displacement at D (Fig. 11): Δ =1.37mm, L =18.00kN Figure 13. Principal tensile strain contour plots of the detailed EP modeling approach. The shading indicates the strain state of each material according to Figure 9. The deformed shapes are enlarged 20 times.

below the 3rd and 4th mortar joints starts to fail forcing the retrofitted beam to failure. At a later stage (displacement level D), the detailed model captures the failure of the retrofitted beam through failure of the ECC below the 3rd mortar joint. A direct comparison of the two modeling approaches with the experimental responses in terms of initial stiffness, K, load-carrying capacities F1 and F2, and ductility (2 − 1 ) is given in Table 2. 1 corresponds to the midspan displacement of the beam when the stiffness of the specimen reduces to 90% of its elastic stiffness. The midspan displacement 2 corresponds to the displacement at which the applied load is less than 5% of the peak load. F1 and F2 correspond to the load-carrying capacities at midspan displacements 1 and 2 respectively. As expected, there is an agreement between the two models with respect to the initial stiffness, K; however the initial stiffness obtained analytically by the detailed and the simplified model is approximately 2.4 and 2.5 times higher than the experimental one, respectively. This discrepancy in stiffness is attributed to the material properties used, the brick-mortar adhesion area and the ECC-masonry interface stiffnesses adopted. The material properties used in the models that were obtained from characterization tests do not perfectly describe their actual properties as developed during curing of the masonry beams. In order to capture the experimental stiffness, the mortar stiffness alone would need to be reduced approximately 10 times. The

brick-mortar adhesion area is assumed to cover the entire brick surface, which was not always the case in the fabrication of the masonry specimens. Assuming that the adhesion area is half of the brick surface area (by visual inspection this was likely), the analytical stiffness is reduced by approximately 25%. Finally, the large stiffnesses adopted for the ECC-masonry interface elements (kn = 88,500 N/mm3 and kt = 38,500 N/mm3 ) also contribute to the higher stiffness obtained analytically. The experimental stiffness can be captured by reducing the interface stiffnesses to kn = 8.9 N/mm3 and kt = 3.9 N/mm3 . However, this reduction will lead to a shear crack in the masonry beam prior yielding of the ECC layer, different crack pattern compared to the experiments. It is therefore most likely that a combination of the above parameters would better represent the actual behavior of the retrofitted beams in terms of initial stiffness. The load carrying capacities F1 and F2 of the retrofitted beam are mainly controlled by the tensile capacity of the ECC layer. Both models capture adequately the corresponding experimental load-carrying capacities within 10%. In terms of ductility, the simplified model is slightly underestimating the ductile response of the retrofitted beam compared to the detailed model. The higher stiffness obtained by the simplified model after yielding of the ECC below the mortar joints and the early failure of the retrofitted beam compared to the detailed model is attributed to the modeling approach of the ECC below the mortar joints. For the case of the simplified model, infinitely stiff linear elastic interface elements are introduced for simplicity between the ECC continuum elements directly below the brick-mortar interfaces, and therefore the crack propagates into the adjacent ECC continuum elements. For the case of the detailed model, ECC cracking and failure is initially developed in the ECC continuum elements located directly below the mortar elements where failure of the ECC also occurs. A modified simplified approach is examined in section 4.3 where the infinitely stiff linear elastic ECC-ECC interface elements are replaced with

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Modified Simplified EP Model

ECC-ECC interface elements that incorporate ECC material properties.

4.3

Modified simplified approach

One of the main differences in the response of the simplified and detailed micro-modeling approaches is found in the way the failure of the ECC is captured by the two models. For the case of the detailed approach, the ECC failure is mainly captured by the continuum elements located directly below the mortar joints. When the simplified approach is used, the ECC failure is captured by the continuum elements adjacent to the ECC-ECC interfaces and located below the brick units. While good agreement in the analytical response was obtained by the two approaches, an alternate model is examined for the ECC-ECC interfaces in order to better understand its impact on the global response of the beam. In the ‘‘modified’’ simplified approach, the ‘‘infinitely’’ stiff linear elastic ECCECC interface elements of the simplified approach are replaced by interface elements that account for the ECC uniaxial tensile response (Fig. 5b) in the normal direction. In this way, the brick-mortar crack can also propagate to the ECC-ECC interfaces directly below the brick-mortar interfaces and not just the adjacent ECC continuum elements. The comparison of the load-displacement responses obtained by the detailed, the simplified and the modified simplified EP approaches shown in Figure 14, shows an increased predicted ductility of the modified simplified approach relative to the simplified approach. This change in ductility is attributed to the ability of the ECC-ECC interface of the modified simplified approach to deform in the normal direction. Good agreement is also found in the crack propagation and failure mode as shown in Figure 15.

25

Load (kN)

15 10 5 A

B

0 0.0

0.5

1.0

1.5

2.0

Displacement at B (Fig. 14): Δ =1.54mm, L =18.28kN Figure 15. Principal tensile strain contour plots of the modified simplified EP approach. The shading indicates the strain state of each material according to Figure 9. The deformed shapes are enlarged 20 times.

At displacement level A in Figure 14, both the interface elements and the adjacent continuum elements of the ECC layer below the mortar joints at the constant moment region are fully cracked causing the degradation of the initial slope of the beam. Failure of the retrofitted beam is attributed to the failure of the ECC-ECC interface elements at the constant moment region, at displacement level B.

5

Detailed EP Model Simplified EP Model Modified Simplified EP Model

20

Displacement at A (Fig. 14): Δ =0.30mm, L =13.82kN

2.5

Displacement at midspan (mm)

Figure 14. Analytical load-displacement responses of ECC retrofitted masonry beams using the detailed, the simplified and the modified simplified EP approaches.

CONCLUSIONS

Nonlinear finite element analyses using two different micro-modeling approaches were performed to predict the response of ECC retrofitted masonry beams under four-point bending. It was found that both the simplified and detailed micro-modeling approaches were able to adequately capture the overall performance of the specimens with the exception of the initial stiffness. Therefore, the simplified model is preferred over the detailed model as a less complicated finite element mesh is required. When linear elastic models are assigned to the ECC-masonry interface elements, extensive cracking in the masonry close to the ECC-masonry adhesion areas occurs. Masonry cracking is avoided when elastic-perfectly plastic models are used and the ECCmasonry tensile and shear bond strengths assumed are

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within the range of 0.14 MPa to 1.7 MPa and 0.69 MPa to 1.4 MPa, respectively. An experimental investigation of the response of the ECC-masonry bond in both its normal and transverse directions is warranted for a better representation of its behavior in simulations. Finally, analysis of the ECC retrofitted specimen using the simplified approach with ECC-ECC interface elements following the ECC uniaxial tensile stress-strain response, led to a very similar response to the one obtained with the detailed model in terms of post-yield stiffness and in that more ductility was achieved than the simplified model with the ‘‘infinitely’’ stiff linear elastic ECC-ECC interface elements. ACKNOWLEDGEMENTS Financial support provided by the National Science Foundation (NSF-NEESR Grant No. 0530737) and the TU Delft research fellowship program is gratefully acknowledged. The opinions expressed in this paper do not necessarily reflect those of the sponsors. REFERENCES Binda L., Pina-Henriques J., Anzani A., Fonatana A. and Lourenço P.B. 2006. A contribution for understanding of load-transfer mechanisms in multi-leaf masonry walls: Testing and modeling. Eng Struct, 28(8): 1132–48. Chaimoon K. and Attard M.M. 2009. Experimental and numerical investigation of masonry under three-point bending (in-plane). Eng Struct, 31(1):103–112. DIANA. 2009. Finite element analysis. Release 9.3. TNO Diana. Delft, The Netherlands. Douglas, K.S., Rots, J.G., Netzel, H.D. and Billington, S.L. 2003. Predicting tunneling-induced settlement damage for concrete frame structure with masonary façade. In N. Bicanic, R. de Borst, H. Mang & G. Meschke (Eds.), Computational modelling of concrete structures (pp. 695–705). Lisse: Balkema publishers.

ElGawady, M.A., Lestuzzi, P. and Badoux, M. 2006. Shear strength of URM walls upgraded with FRP. Eng Struct, 28(12): 1658–1670. Feenstra, P.H., Rots, J.G., Arnesen A., Teigen, J.G. and Hoiseth, K.V. 1998. A 3D constitutive model for concrete based on a con-rotational concept. Computational Modelling of Concrete Structures, Proceedings of EURO-C 1998, de Borst, Bicanic, Mang&Meschke (eds), Balkema, Rotterdam, 13–22. Kim, Y.Y., Kong, H.J. and Li, V.C. 2003. Design of engineered cementitious composite suitable for wet-mixture shotcreting. ACI Materials Journal, 100(6): 511–518. Kyriakides, M.A. and Billington, S.L. 2008. Seismic retrofit of masonry-infilled non-ductile reinforced concrete frames using sprayable ECC, Proceedings of the 14th World Conference on Earthquake Engineering (14WCEE), Beijing, China, October. Li, V.C., Horii, H., Kabele, P., Kanda, T. and Lim, Y.M. 2000. Repair and Retrofit with engineered cementitious composites. Int’l J. of Engineering Fracture Mechanics, 65(2–3): 317–334. Lotfi, H.R. and Shing, P.B. 1994. Interface model applied for fracture of masonry structures. J Struct Eng, 120(1): 63–80. Lourenço, P.B. 1998. Experimental and numerical issues in the modeling of the mechanical behavior of masonry. In: Roca P et al., editor. Structural analysis of historical constructions II. CIMNE: Barcelona, pp. 57–91. Lourenço, P.B. and Rots, J.G. 1997. A multi-surface interface model for analysis of masonry structures. J Eng Mech. 123(7): 660–668. Lourenço, P.B., Rots J.G. and J. Blaauwendraad 1998. Continuum model for masonry: Parameter estimation and validation, ASCE J Struct Eng. 124(6): 642–652. Rots, J.G. 1997. Structural masonry. An experimental/ numerical basis for practical design rules. TNO Building and Construction Research, Rijswijk, Netherlands. Thorenfeldt, E., Tomaszewicz, A. and Jensen, J.J. 1987. Mechanical properties of high-strength concrete and applications in design. Proc. Symp. Utilization of High-Strength Concrete (Stavanger, Norway) Trondheim, Tapir.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Setting and loading process simulation of Push-in anchor for concrete Y.-J. Li, N. Chilakunda & B. Winkler Corporate Research & Technology, Hilti Corporation, Schaan, Liechtenstein

ABSTRACT: In order to develop innovative fastening products for concrete, a deep knowledge on failure mechanism of concrete during setting and loading process is necessary. Numerical simulation is a suitable method to investigate the process. With the aid of simulation a more thorough understanding of fastening system and the failure mechanism of anchor during setting and loading process can be significantly improved. In this paper a special anchor, namely Push-in anchor, is presented as an example to show how the simulation is applied to support Hilti innovative anchor development. 1

INTRODUCTION

Push-in anchor is a metallic anchor which requires a setting tool to drive the internal cone into the anchor, causing the anchor sleeve to expand into the concrete. This kind of anchor belongs to the group of expansion anchors and is normally used in un-cracked concrete (Eligehausen et al. 2006). The anchor consists of two separate parts: a steel tubular expansion shield with sleeves and a solid expander cone as shown in Figure 1. In order to set the anchor into concrete a borehole is first drilled into the concrete up to a specified depth. Then the anchor with the internal cone is inserted to the concrete borehole. The cone is driven into the anchor with a special device. When the cone moves into the anchor, the expansion sleeves are pressed into concrete. The setting process is pictured in Figure 2. To develop an efficient Push-in anchor, the challenge is to optimize both cone and sleeve geometries to achieve as high as possible load carrying capacity with a given input energy defined by EOTA (European Organization for Technical Approvals) approval test (EOTA 2006). The EOTA approval test is a standard test to certify the construction anchors, including Push-in anchor into European market. In the approval test certain conditions both for setting and for loading process are constrained. The setting process is defined to be performed by a special device as shown in Figure 3. The weight of the hammer and the height of fall vary according to the dimension of anchor. The number of impacts is defined based on different anchor sizes as shown in Figure 4. With the defined number of impact blows, the cone is pushed into anchor to expand the sleeves into concrete. When the cone base and the anchor base are aligned, the anchor is installed as full expansion. An efficient design of Push-in anchor should be when the cone is installed and produces the full expansion with the defined input energy (number of blows in

Fig. 4). Meanwhile, the sleeves are strongly expanded into concrete to ensure the load carrying capacity of pull-out test. Sometime, the cone inserted with the defined blows does not reach full expansion, which causes the sleeves do not properly expand into the concrete. Consequently, the load carrying capacity by pullout test will be negatively influenced. Evidently, a

Figure 1.

Push-in anchor system (Salim et al. 2005).

Figure 2.

Setting process of Push-in anchor.

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Figure 3.

pull-out process under static loading are developed. For the setting process simulation, the commercial code Ls-Dyna is used. Based on the evaluation of existing material models for concrete in Ls-Dyna, a concrete model, namely continuous surface cap model (CSCM) (Murry et al. 2007, Schwer et al. 2006), is selected for simulating the setting process of Pushin anchor. According to the requirement of Push-in anchor design, the main task for setting process simulation is the parameter study to optimize the geometry parameters such as the overlap area between sleeve and cone, the slope angles of the sleeve and cone, the internal diameter and the length of sleeve as well as their interaction. For the loading process simulation, a special code developed by Hilti Corporation is used, in which the concrete is modeled by a fracture based model for tension and a plasticity based model for compression. The main requirement for this static analysis is to evaluate the load carrying capacity for the design with different combination of sleeve and cone geometries. During product development, a lot of information obtained from simulation is considered for the new product design. The development period and the amount of required tests is therefore reduced.

Push-in anchor setting device.

2

Figure 4.

Approval test specifications (EOTA 2006).

deep understanding of the working process of Push-in anchor during the setting and loading process, which is defined by the interaction among the cone, sleeves and concrete appears to be important for an efficient Push-in anchor development. Traditionally, the investigation tasks are performed by experiments. Considering time-consuming experiments and the limitation for getting information on failure mechanism during both setting and loading process, the numerical simulation is introduced to support the Push-in anchor development. Further studies show that the damage in concrete due to setting and the failure mode in concrete due to pull-out strongly influence the load carrying capacity in approval test, which can suitably be investigated by simulation. Due to the complexity of concrete material, the simulation of concrete behavior under complex loading condition is still a challenging research topic. During the last two decades Corporate Research & Technology of Hilti Corporation has been focused on the research in this field and has tried to improve the simulation capability for supporting our fastening product development. Concerning the simulation support for Push-in anchors, the simulation tools both for the setting process under dynamic loading and the

SETTING PROCESS SIMULATION OF PUSH-IN ANCHOR

The setting process simulation is performed by using the commercial code Ls-Dyna. 3D simulation model is established and only a quarter of specimen is simulated due to symmetry. The finite element discretization is shown in Figure 5. Since the loading device (see Fig. 2) is very large compared to the size of Push-in anchor, for simplification, only a part of the striker is modeled in the simulation system as seen in Figure 5. A comparison between the simplified striker and the complete loading device was performed and the result

Figure 5.

FE model for setting process simulation.

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Figure 6.

Velocity profile of the striker.

Experiments Displacement of plug

shows that the influence for the position of cone is very limited. Penalty contact model is used to the surfaces between striker and cone, cone and sleeves as well as sleeves and concrete. The friction coefficient between steel and steel is assigned as μ1 = 0.15 and between steel and concrete as μ2 = 0.35. The elastic material model is used for the striker with standard material property for steel as Young‘s modulus E = 210,000 MPa and Poisson‘s ratio υ = 0.3. The elasto-plastic material model is used for cone and anchor with yield strengths between 500 to 700 MPa. The rate-dependent CSCM is selected to model the behavior of concrete. The material property for concrete is defined based on experimental data. Typically, for normal strength concrete the material properties are assigned as Young’s modulus E = 30,300 MPa, Poisson’s ratio υ = 0.2, compressive strength fc = 26 MPa, tensile strength ft = 2.9 MPa and fracture energy Gf = 65 N/m. The initial condition for concrete is assumed as undamaged. Appropriate hourglass method is used to control zero energy modes. The components are discretized by using one point hexahedral elements. Care must be taken in providing suitable mesh for the concrete block as the material model used for concrete is sensitive to mesh distortions. Two different boundary conditions are applied for the simulation, namely displacement and dynamic boundaries. The displacement boundaries are applied during the complete simulation time. These conditions include the symmetry for all components, the outer surface and bottom of concrete block. The dynamic boundaries are applied to the striker with a velocity profile and variable during the simulation time. The input of kinetic energy (KE) is defined by EOTA for each impact, which is governed by the velocity of the striker. According to the volume of modeled striker the initial velocity (Vs ) can be calculated as Vs = (2KE/Ms )1/2 , where Ms is the mass of striker. To model the multiple impacts the striker is forced to move with defined velocity of Vs at each blow. The velocity profile of the striker for six impacts as an example is shown in Figure 6. The simulation with above setup was carried out and the simulated displacement of the internal cone during the setting process is shown in figure 7. Experimental results for the setting process resulting in six blows are given in the same figure. A group of specimens were tested therefore and the area of experimental band in Figure 7 corresponds to the scatter in the experimental data. From this figure we can see that the simulation results fit well within the experimental band. The flat part of the simulation curve signifies the end of each blow. It is worth to point out that for the first blow the cone displacement is larger than subsequent blows. This is due to the fact that during the first blow the resistance to the cone movement is small as the segments have a limited deformation and

Simulation

Time

Figure 7.

Cone displacement.

a fine gap maybe exist between anchor and concrete. For the subsequent blows the gap closes and the cone faces resistance from both concrete and anchor, consequently the resistance becomes larger and larger as the cone moves deeper in sleeves. Figure 8 shows the energy balance of the simulation system. The total energy represents the summary of energy supplied per blow, which corresponds to the input energy defined by EOTA. The sliding energy is the energy consumed for friction, which is an important value to evaluate the design suitability for the setting process. The internal energy is the energy consumed for stored elasticity, plasticity and damage both in steel parts and in concrete. The hourglass energy is the energy consumed for spurious deformation, which occurs in reduced integration elements and must be in a low level. The damage in concrete during the setting process is studied and shown in Figure 9. The number in the pictures represents the order of each blow. The damage level is presented based on the distribution of maximum principal strain in concrete at the end of each

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Figure 8.

Output of energy balance.

Figure 9.

Damage of concrete during setting process.

blow. The area and level of the damage in concrete after each blow could be clearly observed. Evidently, the damage area in concrete increase with increasing the number of impact blows. The final damage form after setting has a significant influence for the load carrying capacity of pullout test. Local damage near the edges of segment looks stronger, which represents the penetration of segment to concrete. Additionally some damage at the top of anchor due to pressure in the area could be observed.

3

parameters to the ultimate load. 2D axi-symmetry condition is used for the simulation. Considering the gaps between segments a special treatment is conducted for modeling the segments with releasing the circumferential stress. The finite element mesh and the simulation setup are shown in Figure 10. The dimension of the block depends on the anchor embedment depth. Normally, the length of the specimen should be larger than 3.5 times embedment depth and the height should be 2.5 times. Penalty contact model is used to the surfaces between striker and cone, cone and sleeves as well as sleeves and concrete. The friction coefficients and the other material properties both for concrete and for anchor keep the same as used in the dynamic setting process simulation as described in section 2. Concrete is simulated by a fracture based model for tension and plasticity based model for compression. The crack band method is used for the objectivity with respect to different meshes. The initial condition for concrete is assumed as undamaged. The loading process simulation is divided into two steps. In the first step the push-in process is simulated based on the final position obtained from dynamic simulation as described in last section. The internal cone is pushed downward until the desired position. Anchor is fixed at top in the vertical direction. With the cone is forced to move down in the anchor the segments are penetrated into concrete and caused the concrete damage in the area of expanded segments. The friction coefficient is an important parameter for the simulation, which depends on the coating of segment and cone, and has to be identified by experiments. Some typical load displacement curves simulated by different coefficients for the push-in process are shown in Figure 11, which illustrates that the static input energy is sensitively influenced by the friction coefficient. The geometry parameter study could be performed in a similar way and the simulated push-in curves could be compared to evaluate the suitability of different prototypes on setting. At the end of push-in process

LOADING PROCESS SIMULATION OF PUSH-IN ANCHOR

The static loading process is simulated by a Hilti developed code. The purpose of the simulation is to identify the load carrying capacity of the designed anchor and to evaluate the influence of different geometry

Figure 10.

FE model for loading process simulation.

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Load

Load

Friction coefficient=0.20

Simulatiom

Friction coefficient=0.15 Friction coefficient=0.10

Displacement

Displacement

Figure 11.

Push-in process simulation.

Figure 12. setting.

Damage and penetration in concrete due to

the deformed segment and the damaged area of concrete due to setting could be investigated as shown in Figure 12. The vertical line in the deformed segment meshes represents the borehole boundary of concrete. The part in the right side of vertical line is the part of sleeves which have been penetrated into concrete. From this figure the penetration of sleeves and the damaged area in concrete near the segments could be easily observed. For different prototypes the loaddisplacement curve of push-in process, the deformation of segment, the penetration in concrete and the concrete damage near the segments could be compared, which offers the basis for the optimized design. In the second step of loading process simulation the pull-out process is simulated with displacement control at the top of the anchor. The load carrying capacity could be evaluated by this simulation. The failure mode in general could be concrete cone failure, pull-through failure or their combination. The failure mode as concrete cone failure is the desired failure mode as the loading capacity in this mode is larger than others. From simulation the load-displacement curve can be obtained as shown in Figure 13. The final failure mode for pullout loading could be visible as seen in Figure 14.

Figure 13.

Load-displacement curve.

Figure 14.

Failure mode under pull-out loading.

Theoretically, the function of anchor is to transfer the carrying load to concrete. Therefore, the load carrying capacity is strongly dependent on the failure process in concrete, especially the failure mode at the ultimate load point. In order to identify the simulation model the initial simulated results have to be compared to a group of experiments and all parameters used in the simulation both for material and for contact models have to be evaluated. The maximum load and failure mode have to be carefully verified by experimental data. After the simulation system is evaluated, the virtual design of the prototypes could be performed. Normally the experimental study for a new prototype requires a certain period of time including manufacture, transportation and experiment in laboratory. For simulation much less time is required compared to experiment and the amount of experiments is subsequently reduced.

4

CONCLUSIONS

The simulation support for Push-in anchor development is presented in this paper. The dynamic simulation is conducted to simulate the setting process of

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Push-in anchor and the static simulation is applied to simulate the loading process. From dynamic simulation, the geometry influence on the setting position of internal cone with defined input energy could be investigated and the damage in concrete due to setting could be studied. The energy consumed for friction, anchor deformation and concrete damage are clearly visible and the influence to the desired expansion can be evaluated. From static simulation, the load carrying capacity of the simulated anchor could be predicted and the influence of the different geometric parameters to the ultimate load could be estimated. Based on the simulation results, suggestions for improving the design of Push-in anchor could be proposed. With the aid of simulation support, the virtual prototypes can be quickly evaluated and the mechanism of the failure process can be investigated in detail. During the period of product development a lot of information obtained from simulation has been considered for improving product design. The development period is subsequently reduced.

REFERENCES Eligehausen, R., Mallée, R. & Silva, J.F. 2006. Anchorage in Concrete Construction. Ernst & Sohn. EOTA. 2006. Guideline for European technical approval of metal anchors for use in concrete, Part 4 deformationcontrolled expansion anchors. European Organization for Technical Approvals. Brussels. Murray, Y.D., Abu-Odeh, A. & Bligh, R. 2007. Evaluation of LS-DYNA concrete material model 159. FHWA Report. FHWA-HRT-05-063. Salim, H., Dinan, R., Shull, J. & Townsend, P.T. 2005. Shock loading capacity of concrete expansion anchoring systems in un-cracked concrete. Journal of Structural Engineering. 131(8):1206–1215. Schwer, L.E. & Murray, Y.D. 2006. Continuous surface cap model for geomaterial modeling: A new LS-DYNA material type. In 7th International LD-DYNA Users Conference, Material Technology. (2), 16:35–50.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Nonlinear FE modelling of shear behaviour in RC beam retrofitted with CFRP Yasmeen Taleb Obaidat, Ola Dahlblom & Susanne Heyden Division of Structural Mechanics, Lund University, Lund, Sweden

ABSTRACT: A nonlinear 3-D numerical model has been developed using the ABAQUS finite element program, and it was used to examine the shear behaviour of beams retrofitted by CFRP. Two models were used to represent the interface between CFRP and concrete, a perfect bond model and a cohesive model. Validation of the model was performed using data obtained from an experimental study. The results showed that the cohesive model is able to simulate the composite behaviour of reinforced concrete beams retrofitted by CFRP in shear correctly. The model is then used to examine the influence of length and orientation of CFRP. It is shown that the length of CFRP and the orientation strongly influence on the behaviour of the retrofitted beams. INTRODUCTION

Reinforced concrete (RC) structural elements such as beams are subjected to significant flexure and shear. Strengthening or upgrading becomes necessary when these structural elements are not able to provide satisfactory strength and serviceability. Shear failure of RC beams could occur without any warning. Many existing RC members are found to be deficient in shear strength and need to be repaired. Shear deficiencies in reinforced concrete beams may occur due to many factors such as inadequate shear reinforcement, reduction in steel area due to corrosion, use of outdated design codes, increased service load and design faults. The application of carbon fibre reinforced polymer (CFRP) as an external reinforcement has become widely used recently. It is found to be important for improving the structural performance of reinforced concrete structures. A beam can be bonded with CFRP plates on either the soffit or the web. Generally, the soffit bonding is preferred for flexural retrofitting of beams, while web bonding is performed for shear retrofitting. For shear retrofitting of beams, different schemes can be employed, such as bonding vertical or inclined strips, or bonding continuous plates on the web. Most of the research done in the past on strengthening of existing RC beams focused on flexural strengthening (Ashour et al. 2004), (Esfahani et al. 2007), (Wang & Zhang 2008) (Wenwei & Guo 2006) and (Obaidat et al. 2009) and very few studies have specifically addressed the topic of shear strengthening (Sales & Melo 2001), (Santhakumar & Chandrasekaran 2004) and (Sundarraja & Rajamohan 2009). While experimental methods of investigation are extremely useful in obtaining information about the

composite behaviour of FRP and reinforced concrete, the use of numerical models helps in developing a good understanding of the behaviour at lower costs. In this paper, the efficiency of applying CFRP as external reinforcement to enhance the shear capacity of RC beams was investigated by the finite element method. ABAQUS (Hibbitt, Karlsson, & Sorensen Inc. 2000) is used to model the behaviour of a retrofitted beam; in the first part of the paper validation of the model is done using four beams tested by Obaidat (Obaidat 2007). The second part is to investigate the effect of different parameters on shear retrofitting. The test parameters included a variable length and orientation of CFRP.

2

EXPERIMENTAL DATA

The experimental data was obtained from (Obaidat 2007). This work consisted of four beams subjected to four point bending. All beams were identical in geometry and reinforcement. The geometry of the beams is shown in Figure 1 and the material properties are given in Table 1. Two beams were used as control beams and P/2

P/2

300 mm

1

520mm

520mm 1960 mm

520mm

2 10 8/400 mm 2 18 150 mm

Figure 1. Geometry, arrangement of reinforcement and local of the tested beams.

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Concrete CFRP

100 mm

520 mm

CFRP

507 MPa 210 GPa 0.3 30 MPa 165 GPa 2 GPa

100 mm

(a) RB 90/300. 520 mm 50 mm CFRP (b) RB 90/200.

50 mm

CFRP

50 mm

200 mm

fy Es ν  fc Ef ff

520 mm

100 mm

520 mm

50 mm

CFRP

100 mm

Steel

100 mm

300 mm

Mechanical properties of materials

300 mm

Table 1. used.

(c) RB 90/100.

Figure 2. The arrangement of the CFRP laminate in retrofitted beams.

CFRP

420 mm (d) RB 45/300.

CFRP

200 mm

100 mm

300 mm

100 mm

420 mm

Installation CFRP in experimental work.

(e) RB 45/200. 100 mm

the rest were retrofitted on both sides of the beams with CFRP. The CFRP had 50 mm width and 300 mm length and the spacing was 100 mm, see Figures 2 and 3. 3 3.1

NUMERICAL SIMULATIONS

420 mm (f) RB 45/100.

Figure 4.

Studied bodies

In order to study how the length and orientation of CFRP affect the shear behaviour of retrofitted beams, numerical simulations were conducted for the cases shown in Figure 4. Three different lengths of CFRP were used in the simulations. The orientation of the CFRP was also varied keeping the amount of CFRP as in 90◦ by using 35 mm width of CFRP. Two different models for the concrete-CFRP interface were evaluated in the validation. 3.2

CFRP

Material models

3.2.1 Concrete A plastic damage model was used to represent the behaviour of concrete. The model assumes that the two main failure mechanisms are tensile cracking and compressive crushing of the concrete material.

100 mm

Figure 3.

Studied CFRP configurations.

The softening curve of concrete under tension is shown in Figure 5, where fct is the tensile strength, and Gf is the fracture energy of concrete, (Hillerborg 1985). The tensile strength of concrete can be obtained from Equation 1 (ACI Comitte 318. 1999), and the fracture energy was assumed equal to 90 J/m2 .  (1) fct = 0.35 fc = 1.81 MPa The stress-strain curve under uni-axial compression is shown in Figure 6, (Saenz 1964). The modulus of elasticity was obtained using (ACI Comitte 318. 1999).  Ec = 4700 fc = 26000 MPa (2) 3.2.2 Steel reinforcement The constitutive behaviour of steel was modelled using an elastic perfectly plastic model, see Figure 7. The

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fct

fy

Es

Gf 0.3fct y

0.2

0

Figure 7. Figure 5. tension.

Stress-strain curve for the reinforcement steel.

Softening curve of concrete under uni-axial max

c,

MPa

35

Ko

30

Gcr

25 o

20

f

15

Figure 8.

10

E22 = E33 = 9.65 GPa and G12 = G13 = 5.2 GPa. υ23 and G23 were set to 0.45 and 3.4 GPa, respectively.

5 0 0 0.0025 0.005 0.01

0.015

0.02

0.025 0.03

Figure 6. Stress-strain behaviour of concrete under uniaxial compression.

parameters used to define this model are elastic modulus Es , yield stress, fy , and Poisson’s ratio, υ. The parameters from the experimental study were used; see Table 1. Perfect bond was assumed between the steel and the concrete. 3.2.3 CFRP The CFRP was assumed to be a linear elastic orthotropic material. The elastic modulus in the fibre direction of the unidirectional CFRP material used in the experimental study was specified by the manufacturer as 165 GPa. For the orthotropic material model E11 was set to 165 GPa. Using Rule of Mixture (Piggott 2002), Eepoxy = 2.5 GPa and the fibre volume fraction 75%, Efibre was found to be 219 GPa and υ12 = υ13 = 0.3. By use of Inverse Rule of Mixture (Piggott 2002),

Bilinear traction–separation constitutive law.

3.2.4 CFRP-concrete interface Two different models for the interface between CFRP and concrete were used in this study. In the first model, the interface was considered as a perfect bond. In the second model, the interface was modelled using a cohesive zone model. Cohesive elements were used together with a traction separation law which defines the traction as a function of the separation distance between the interface elements, see Figure 8. The material has an initial linear elastic behaviour. The elastic response is followed by damage initiation and evolution until total degradation of the elements. The nominal traction stress vector consists of three components: σn , τt , τs , which represents the normal and shear tractions, respectively. The initial stiffness matrix is directly related to the thickness of the cohesive layer and to the material stiffness G. A general expression of this relation is: K0 =

ti Gi

1 +

tc Gc

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(3)

where ti is the adhesive thickness, tc is the concrete thickness, and Gi and Gc are the shear modulus of adhesive and concrete respectively. An upper limit for the maximum shear stress, τmax , is provided by the expression (Ye, Lu & Chen 2005): τmax = 1.2 βw fct

(4)

where  βw = (2.25 − wf /sf )/(1.25 − wf /sf )

(5)

and wf is CFRP plate width, sf is spacing of CFRP strips and fct is concrete tensile strength. This equation gives τmax = 2.17 MPa. Numerical simulations showed that this value is too high; since CFRP rupture or concrete crushing induced the failure, instead of the CFRP debonding that occurred in the experimental study. Hence, τmax was reduced to 1.5 MPa. For fracture energy, Gcr , previous studies have indicated values from 300 J/m2 up to 1500 J/m2 (JCI 1998) and (JCI 2003). For this study the value 900 J/m2 , in the middle of the interval proposed by previous studies, was used. During separation of the cohesive element surfaces, the thickness increases and the stiffness degrades. The quadratic nominal stress criterion was used as damage initiation criterion:       σn  2 τs  2 τt  2 + + =1 (6) σn0 τ0s τ0t where σn and τs , τt are the cohesive tensile and shear strengths of the interface. The values used for this study were σn0 = fct = 1.81 MPa, and τ0s = τ0t = 1.5 MPa. 3.3

Abaqus/standard (Hibbitt, Karlsson, & Sorensen Inc. 2000) was used for these simulations. The total deflection applied was divided into a series of deflection increments. In addition automatic stabilization and small time increments were used to avoid a diverged solution. Since the geometry of the beams, loading and boundary conditions were symmetrical, only one quarter of a beam was modelled with typical finite element mesh as shown in Figure 9. Boundary conditions are shown in Figure 10.

980 mm 75 mm (a) By use of symmetry, one quarter of the beam was modelled.

(b) Finite element mesh of a quarter of a control beam.

Finite element analysis

The concrete and the steel were modelled using a linear tetrahedral element. This element has four nodes with three degrees of freedom at each node–translation in the x, y, and z directions. The element used is capable of plastic deformation and cracking in three orthogonal directions. An eight node reduced-integration element was used to model the CFRP composite and the steel plates under the load and at the support. This element also has three degrees of freedom at each node. Eight-node 3-D cohesive elements were used to model the interface layer. The cohesive interface elements are composed of two surfaces separated by a thickness. The relative motion of the bottom and top parts of the cohesive element measured along the thickness direction represents opening or closing of the interface. The relative motion of these parts represents the transverse shear behaviour of the cohesive element.

(c) Finite element mesh of a quarter of a strengthened beam.

Figure 9. analysis.

Geometry and elements used in the numerical

Figure 10.

Boundary conditions used in numerical work.

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4

VERIFICATION OF FINITE ELEMENT MODEL

To verify the finite element model of the reinforced concrete retrofitted with CFRP, four beams from an experimental study (Obaidat 2007) were simulated. The results from the FEM analysis were then compared with the experimental results. The load-deflection curves for the control beams are shown in Figure 11. In the linear part the FEM results are slightly stiffer than the experimental results. One explanation for this may be the assumption of perfect bond between concrete and steel. As shown in Figure 12 there is good agreement between the cohesive model and the perfect bond model in the first part of the curve, but when the crack starts to propagate the cohesive bond model gives a stronger softening effect in the beam. It is also clear from the figure that the cohesive model shows a very satisfactory agreement with the experimental response. The perfect bond model overestimates the ultimate load and deflection. This can be attributed to the fact that the perfect bond model fails to capture the

Debonding

Figure 13. Comparison of failure mode from FEM analysis and experiment.

softening of the beam and it is not able to represent the debonding failure that occurred in the experimental work. The cohesive model also shows good agreement with the experimental work in the debonding failure mode as shown in Figure 13. The following results have been obtained using the cohesive model.

300 250

Load (kN)

200 150 100

Control Beam (FEM) Control Beam (Exp)

50

5 5.1

0 0

5

10 Deflection (mm)

15

20

Figure 11. Load-deflection curves of control beams obtained by experiments and FEM.

300 250 Diagonal Shear Crack

200 Load (kN)

Debonding

150 Experimental FEM / Perfect Bond FEM / Cohesive Model

100 50 0 0

5

10 Deflection (mm)

15

20

Figure 12. Load-deflection curves of retrofitted beams obtained by experiments and FEM.

RESULTS AND DISCUSSION Effect of length of CFRP

To study the effect of CFRP length, three lengths were investigated, 100 mm (lower third of web of beam, RB 90/100), 200 mm (lower two thirds of web of beam, RB 90/200) and 300 mm (entire web of beam, RB 90/300). Figure 14 shows the load versus the mid-span deflection of a reinforced concrete beam retrofitted with CFRP. In all beams the failure mode was debonding due to concentration of shear stress resulting from the diagonal crack. Generally the stiffness and load capacity of the beam increases when the length of CFRP is equal to the web of the beam as shown in Figure 14. In Figure 14, it can be seen that for RB 90/200 and RB 90/100 the reinforcement seems not to strengthen the beam; this is due to the fact that the crack crossed the strips close to their end. This result supports what (Monti and Liotta 2005) found. For these cases the formation of such a crack was accompanied by yielding of internal shear reinforcement. The steel yielding caused the drop in the beam stiffness compared to the control beam as shown in Figure 14. Before the

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300

250

250

200

200

150 100 Opening Diagonal

50

Crack

0 0

5

Load (kN)

Load (kN)

300

RB 90/300 RB 90/200 RB 90/100 Control Beam

10 Deflection (mm)

15

150 RB 45/300 RB 90/300

100 50 0 0

20

5

10 Deflection (mm)

15

20

15

20

15

20

(a) 300

Figure 14. Load-deflection curves for different lengths of CFRP, obtained by FEM. Load (kN)

150 100

0 0

RB 45/200 RB 90/200

5

(b) 250 200 150 100

Effect of orientation of CFRP

RB 45/100 RB 90/100

50

Since a shear crack propagates in a diagonal manner the orientation of the CFRP may affect the behaviour of a retrofitted beam. Two orientations, 90◦ and 45◦ were studied. For the longest CFRP the failure mode was a shear crack while for the other beams the failure mode was debonding. It is clear from Figure 15 that when the angle of orientation was 45◦ the retrofitted beam carries more load for all lengths. It is interesting to note that the load capacity for RB 45/300, RB 45/200 and RB 45/100 was increased by 20.1%, 2% and 5% respectively, compared to the control beam while the change was 11.9% for RB 90/300, −1.8% for RB 45/200 and −1.1% for RB 90/100. This indicates that the performance of a beam retrofitted with CFRP is influenced by the orientation of the CFRP. When the CFRP did not cover the full depth of the beam the load capacity was actually decreased for 90◦ , while a small increase was obtained for 45◦ . From Figure 16, it can be concluded that the orientation of CFRP has a strong effect on the behaviour of the retrofitted beam for the same total amount of

10 Deflection (mm)

300

0 0

5

10 Deflection (mm)

(c)

Figure 15. Load-deflection curves of 90◦ and 45◦ of CFRP obtained by FEM.

300 250 200 Load (kN)

5.2

200

50

Load (KN)

yielding any increase in loading is shared by the reinforcing steel and CFRP. After the yielding most of the increased loading has to be carried by the plate until debonding occurs. It can be seen also that RB 90/200 failed at a lower load than RB 90/100. This is attributed to the fact that the area in the middle of the web is critical in the failure process and that the end of plate in RB 90/200 is in that area. Hence the stress concentration is more pronounced in this beam and debonding occurs earlier. This analysis, regarding the length of CFRP, verifies that the strip, when the orientation is 90 degrees, should cover the whole web of the beam to provide an improvement of beam behaviour.

250

Shear crack Debonding

150

RB 45/300 RB 45/200 RB 45/100 Control Beam

100 50 0 0

5

10 Deflection (mm)

15

20

Figure 16. Load-deflection curves of Control and retrofitted beams with 45◦ of CFRP and different lengths, obtained FEM.

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CFRP. A 45◦ orientation gives a better effect in terms of maximum load and deflection. 6

CONCLUSION

An FE model was used to investigate the behaviour of RC beams retrofitted in shear with CFRP by using the commercial program ABAQUS. The results from FE model were compared with experimental work by Obaidat (Obaidat 2007) to assess the accuracy of the proposed FEM model. The present numerical study has shown that the proposed FEM model is suitable for predicting the behaviour of RC beams retrofitted with CFRP plates in shear. It should also be noted that the perfect bond model cannot account for debonding failure of CFRP since this model does not take in to account the behaviour of the interface between the CFRP and the concrete. On the other hand, the cohesive model is capable to predict the debonding failure. This study also presents results of an investigation of the effects of length and orientation of CFRP. The following can be concluded: 1. Change in length of CFRP reinforcement may result in different behaviours of retrofitted beams. The longest CFRP presents a high stiffness and high load while when the CFRP strip do not cover the full depth of beam the load capacity decrease for 90◦ . Therefore it is not advisable to use a CFRP strip not covering the entire beam depth when retrofitting for shear. 2. The peak load and deflection is slightly affected by the orientation of CFRP. For 45◦ CFRP orientation, a larger increase in load capacity is obtained compared to 90◦ .

REFERENCES ACI Comitte 318. 1999. Building Code Requirements for Structural Concrete and Commentary (ACI 318-99). American Concrete Institute Detroit, MI. Ashour, A.F., El-Refaie, S.A. & Garrity, SW. 2004. Flexural strengthening of RC continuous beams using CFRP laminates. Cement & Concrete Composites, 26: 765–775. Esfahani, M., Kianoush, M., & Tajari. 2007. A. Flexural behaviour of reinforced concrete beams strengthened by CFRP sheets. Engineering Structures, 29: 2428–2444.

Hibbitt, Karlsson, & Sorensen, Inc. 2000. ABAQUS Theory manual, User manual Example Manual, Version 6.7. Providence, RI. Hillerborg, A. 1985. The theoretical basis of a method to determine the fracture energy Gf of concrete. Materials and Structures, RILEM 50-FMC, 108, pp 291–296. JCI. Technical report on continous fibre reinforced concrete. JCI TC952 on Continous Reinforced Concrete 1998; 116–124. JCI. Technical report on retrofit technology for concrete structures. Technical Committee on Retrofitting Technology for Concrete Structures 2003; 79–97. Monti, G., & Liotta, M.A. 2005. FRP strengthening in shear: Test and design equations. Proceeding of 7th International Symposium on Fiber Reinforce Polymer for Reinforced Concrete Structures (FRPRCS-7); 543–562. Obaidat, Y. 2007. Retrofitting of reinforced concrete beams using composite laminates, Master Thesis, Jordan University of Science and Technology. Obaidat, Y., Heyden, S., & Dahlblom, O. 2009. The Effect of CFRP and CFRP/Concrete Interface Models when Modelling Retrofitted RC Beams with FEM. Accepted for publication in Composite Structures. Piggott, M. Load bearing fibre composites, 2nd Edition. Kluwer Academic Publishers, Boston/Dordrecht/London, 2002. Saenz, L. 1964. Equation for the stress-strain curve of concrete. Desayi P, Krishnan S. ACI Journal, 61:1229–1235. Salles Neto, M., Melo, G.S. & Nagato, Y. 2001. T Beams Strengthened in shear with carbon sheet laminates, Fiberreinforced Plastic for Reinforced Concrete Structure, 1: 239–248. Santhakumar, R., & Chandrasekaran, E. 2004. Analysis of retrofitted reinforced concrete shear beams using carbon fibre composite. Electronic Journal of Structural Engineering, 4: 66–74. Sundarraja, M., & Rajamohan, S. 2009. Strengthening of RC beams in shear using GFRP inclined strips—An experimental study. Construction and Building Materials, 23: 856–864. Wang, J., & Zhang, C. 2008. Nonlinear fracture mechanics of flexural-shear crack induced debonding of FRP strengthened concrete beams. International Journal of Solid and Structures, 45: 2916–2936. Wenwei W., Guo L. 2006. Experimental study and analysis of RC beams strengthened with CFRP laminates under sustaining load. International Journal of Solids and Structures, 43: 1372–1387. Ye, L.P., Lu, X.Z, and Chen, J.F. Design proposals for the debonding strengths of FRP strengthened RC beams in the chinese design code. Proceeding of International Symposium on Bond Behaviour of FRP in Structures (BBFS 2005); 45–54.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Material optimization for textile reinforced concrete applying a damage formulation Ekkehard Ramm & Junji Kato Institute of Structural Mechanics, University of Stuttgart, Stuttgart, Germany

ABSTRACT: The present study discusses optimization strategies for maximizing the structural ductility of Fiber Reinforced Concrete (FRC) with long textile fibers. Due to material brittleness of both concrete and fibers in addition to complex interfacial behavior between above constituents the structural response of FRC is highly nonlinear. This material nonlinearity including the interface response has to be taken into account for an optimal fiber layout in the structural domain. In the present contribution three kinds of optimization strategies based on a damage formulation are described. The performance of the proposed method is demonstrated by a series of numerical examples; it is verified that the ductility can be substantially improved. 1

INTRODUCTION

Unlike conventional steel reinforcement, textile fibers used for FRC sometimes also called Textile Reinforced Concrete (TRC) are corrosion free due to their high alkali-proof; this property allows for the manufacturing of light-weight thin-walled composite structures. However FRC structures show very complex failure mechanisms resulting on the one hand from the material brittleness of both constituents, fibers and matrix, and on the other hand from their interface behavior introducing the necessary ductility. The specific characteristic of FRC is an ideal target for optimization applying the overall structural ductility as objective which ought to be maximized for a prescribed fiber volume. In this context the ‘structural ductility’ means ‘energy absorption capacity’ which is measured by the internal energy summed along the entire structure up to a prescribed displacement of a dominant control point, see Eq. 12. The influential parameters on the entire structural response of these composites are (i) material parameters involved in the interface, (ii) the material layout at the small scale level, and (iii) the fiber geometry on the macroscopic structural level. Some of these parameters are taken as design variables for the optimization problem. For this task we describe three kinds of optimization strategies. The first one is a multiphase material optimization introduced in Kato et al. (2009) where the ductility of FRC is maximized with respect to fiber size, length and the combination of different

fiber materials. This approach is considered as a material distribution problem derived from conventional topology optimization. However fiber materials are defined only in so-called prescribed ‘design element layers’, only straight fibers are allowed in the mentioned study. In order to cure this restriction Kato and Ramm (2009) apply so-called embedded reinforcement elements utilizing shape optimization where the fiber geometry is globally defined and is the target of the optimization procedure. These embedded reinforcement elements have been originally introduced by Phillips and Zienkiewicz (1976); thereafter the model has been refined by many researchers. The present contribution describes briefly these two optimization strategies, i.e. multiphase material optimization and shape optimization of fiber geometry. Furthermore the combination of the two above strategies, denoted multiphase layout optimization, is investigated. A detailed explanation of this approach is given in Kato (2010). The present method is not restricted to FRC but may as well be applied to other fiber reinforced composites, for example fiber reinforced glass (FRG). The materials for both concrete and fibers are modeled by a gradient-enhanced damage formulation, see (Peerlings et al. 1996, 1998 and Peerlings 1999). For the interface between concrete and fiber a discrete bond model is applied, see (Krüger et al. 2002a,b, 2003 and Xu et al. 2004). The optimization problem is solved by a gradient-based optimization scheme.

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2 2.1

governing interfacial response are the bond strength and the debonding behavior. The influence of material properties at a small scale level and the stresses perpendicular to the fiber direction are included in the material formulation as important parameters. The bond stress–slip (σLi − uiL ) relation is expressed as

APPLIED MATERIAL MODELS Material model for constituents concrete and fiber

First an equivalent strain measure is defined. For concrete matrix and fibers de Vree’s definition of equivalent strains εv is adopted, Vree et al. (1995), as follows εv (I1 , J2 ) =

 σLi

k−1 I1 2k(1 − 2ν)  (k − 1)2 2 1 12k + I1 − J2 , 2k (1 − 2ν)2 (1 + ν)2

=w ˜ · b + (1 − b) ·

1 1+w ˜R for

(1) where I1 denotes the first invariant of the strain tensor and J2 the second invariant of the deviatoric strain tensor. k indicates the ratio of compression relative to the tension strength and ν is Poisson’s ratio. For the damage evolution of both concrete and fibers we use an exponential damage law introduced by Mazars and Pijaudier-Cabot (1998) as κ0 D(κ) = 1 − (1 − α + αe−β(κ−κ0 ) ), if κ ≥ κ0 κ (2)

σm = σm, 0 ψ,

σf = σf , 0 ψ

· σ0 ,

uiL ≤ w1

(4)

 ψ = 1 + tanh αr  × 1−

σR − αf νεs 0.1fc −1 

r2

(r + h)2

(6)

where ψ denotes an additional parameter (1 < ψ < 2) which considers the influence of the kind of fiber material, the loading condition and the stresses perpendicular to a fiber direction. σm,0 and σf ,0 denote the initial adhesion strength and sliding friction strength, respectively. r describes a fiber (roving) radius, ν is Poisson’s ratio of a fiber and h is the surface roughness of a fiber. αr and αf are constants assuming the

(3)

Material model for interface

This model was obtained by experiments for different textile fiber materials and leads to a realistic interface response of FRC. The significant factors

(5)

with

where ∇ 2 denotes the Laplacean operator and c is a positive parameter of the dimension length squared regularizing the localization of the deformation. Thus in the Kuhn-Tucker equations εv is replaced by the non-local equivalent strain ε˜ v which is discretized in the finite element sense. Elastic unloading is included in the traditional way. 2.2

 R1 

where w ˜ = uiL /w0 denotes the normalized slip. uiL is the slip length. w0 is a factor defined by the initial tangent k1 , see Figure 1. k2 is the tangent at slip w1 where the bond stress achieves the maximum bond strength. b = k2 /k1 and σ0 = k1 · w0 are parameters to calculate the stresses and R defines the radius of curvature at slip w1 . The stress-slip relation for the range uiL > w1 is simply described by the adhesion strength σm and the friction bond strength σf ,

where D stands for the damage parameter (0 ≤ D ≤ 1), α defines the final softening stage and β governs the rate of damage growth. κ0 is a threshold variable which determines damage initiation and κ represents the most severe deformation the material has experienced during loading. In a conventional local damage model, κ is related to the local equivalent strain εv and the history variable κ is defined by the Kuhn-Tucker ˙ v − κ) = 0. relations, i.e. κ˙ ≥ 0, εv − κ ≤ 0, κ(ε For non-local damage, κ is related to a weighted volume average of the local equivalent strain εv , denoted as non-local equivalent strain ε˜ v . In the gradientenhanced damage model ε˜ v is approximated implicitly as follows ε˜ v − c∇ 2 ε˜ v = εv ,



Figure 1.

Discrete bond model.

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lateral deformation of a fiber. These properties are given depending on the kind of fiber material used. fc is the uniaxial compressive strength of concrete, εs the uniaxial strain and σR defines the stress perpendicular to a fiber. For a detailed description of this model it is referred to (Krüger et al. 2002a,b, 2003). In this model loading and unloading conditions are also considered. This one-dimensional interface model has originally been formulated for a fiber in a three-dimensional setting. If this model is utilized in a two-dimensional space, the interface has to be approximated to meet the original total interface area.

3

MULTIPHASE MATERIAL OPTIMIZATION

3.1 Overview The present methodology is strongly related to topology optimization, in particular to the Solid Isotropic Microstructure with Penalization of intermediate densities for a one-phase material, the so-called SIMP approach, see Bendsoøe et al. (2000) and Zhou and Rozvany (1991), and to its generalization to multiphase topology optimization, see Sigmund and Torquato (1997), for example used for composite structures. The development of these methods is briefly described in the sequel. It is well known that the ‘0-1’ integer topology optimization being a highly non-convex variational problem is ill-posed. For this many material models providing a regularization have been developed. The SIMP method is one of them and may be the most popular model due to its numerical robustness. In order to classify the present formulation within the multiphase optimization let us summarize the different concepts of material distribution problems (Fig. 2). The SIMP approach, Figure 2 (a1) uses the intermediate densities as mathematical vehicle to relax the ill-posed problem during optimization. The exponent η plays the role of a penalization factor without a physical meaning eventually leading to a pure or at least an almost pure layout for a single material structure. The concept of topology optimization may also be applied to a single material for which intermediate densities physically exist, for example polymer or metal foams. Here the porosity, limited by upper and lower bounds, can be used as design parameter which varies in different regions of the structure, Figure 2 (a2); the effective modulus Ceff is often defined by a power-law formula, see for example Gibson and Ashby (1999), where the similarity to the SIMP approach can be recognized; for an application see Lipka (2007), Lipka and Ramm (2005), in which the method is used to avoid local buckling in sensitive regions or to increase the overall ductility of the entire structure.

For a two-phase material the principle of multiphase topology optimization is sketched in Figure 2 (b1), in analogy to the SIMP approach; in other words the void phase is replaced by a second solid material. Again intermediate stage are allowed during optimization applying a penalized relaxation. Figure 2 (b1) also shows a slight variation where parts of the structure, for example most of the matrix, are not elements of optimization; the optimization is restricted to the design element s which may consist of several layers or even individual finite elements. Penalized functions based on the volume fraction r1 /r0 interpolate the material stiffness between the two phases C1 and C2 , see the added interpolation formula rendering the effective stiffness Ceff of the composite material. The same concept can be utilized if a material consists of two (or more) phases on a small scale, for example a material with a certain heterogeneous microstructure composed of several phases or a mixture of two sintered powders, this time allowing intermediate stages of a physically existing smeared material, Figure 2 (b2). In this case the interpolation represents the material behavior of a real mixture, macroscopically describing the constitutive behavior of a material point. Here ηˆ is a fitting variable rather than a penalization parameter, which guarantees the physically admissible intermediate stages and can be obtained by experiments or homogenization. The present approach applies this concept, as described in Figure 2 (b2), to the fiber layout of a FRC-structure; the two phases are the concrete matrix and the fiber material. For simplicity a power-law interpolation is used for the constitutive behavior of the smeared fiber/matrix material. After optimization a clear fiber layout is obtained controlled by the volume fractions. The present study extends this multiphase material optimization to materially nonlinear problems applying the above described damage formulation with strain softening in order to consider a more realistic physical behavior of FRC. Linear kinematics is assumed in this study for simplicity. 3.2

Two-phase model

This section introduces a two-phase material optimization applying the described damage formulation. Since this formulation includes three extra material parameters, i.e. initial equivalent strain κ0 and exponential softening parameters α and β shown in Eq. (2) in addition to Young’s modulus E for each material, the interpolation of the mixture according to Figure 2 (b2) is also applied to these additional parameters, namely ζ = (1 − sηˆ )ζ1 + sηˆ ζ2 ,

(7)

where ζ represents the effective material parameters of the four parameters described above. ζ1 and ζ2 stand

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Gibson and Ashby Lipka and Ramm (2005)

Figure 2.

Classification of material distribution problems in optimization, (a) single phase material, (b) multiphase material.

for the material properties of phase-1 (e.g. concrete matrix) and phase-2 (e.g. fiber), respectively, and are fixed values. The design variable s(= r1 /r0 ) indicates a ‘volume fraction’ of the constituent materials and is a function of the geometrical parameters r0 and r1 of a design element (see Fig. 2). A closer look shows that Eq. (7) is not always sufficient to express the interpolation for all damage parameters since they have their own characteristics. In order to understand the features of the individual material parameters its relation to the present objective f , the structural ductility, is considered. f is the area below the stress-strain curve and increases if either Young’s modulus E or initial equivalent strain κ0 increases under the condition that all other material parameters are kept constant, see Figures 3 (a), (b). On the other hand the ductility decreases if either one of the softening parameters α or β increases, see

Figures 3 (c), (d). Keeping this behavior in mind one can define related interpolation rules. It is obvious that Eq. (7) is a reasonable interpolation for the stiffness, namely the effective Young’s modulus if E1 ≤ E2 , see for example Bendsøe and Sigmund (2003). It is apparent that the stiffer phase-2 has a dominant influence on the mixture expressed by a larger gradient at s = 1 compared to that at s = 0, confer Figure 2 (b2). Since κ0 has essentially the same tendency it makes sense to use the same interpolation Eq. (7) also for this parameter, provided κ01 ≤ κ02 . The situation is reverse for both softening parameters α and β. Assuming again ζ1 ≤ ζ2 phase-1 is the ‘‘leading’’ constituent requiring a larger gradient of the interpolation function at s = 0; therefore the power law has to be concave and is expressed by ζ = (1 − s)ηˆ ζ1 + [1 − (1 − s)ηˆ ]ζ2 .

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(8)

Figure 3. Change of stress-strain relation of damage model with respect to one parameter increased under the condition that all other parameters are kept constant. (a) Young’s modulus E, (b) initial equivalent strain κ0 , (c) (d) softening parameters α and β.

It may happen that for either of the four parameter ζ1 > ζ2 . In this case the interpolation functions (7) and (8) respectively have to be interchanged. Let us summarize the interpolation rules,  ⎧ 1 − sηˆ ζ1 + sηˆ ζ2 ⎪ ⎪ ⎪ ⎪ ⎧ ⎪ ⎪ ⎪ ⎨ E, κ0 (ζ1 ≤ ζ2 ) ⎪ ⎪ ⎪ ⎪ for ζ : or ⎪ ⎪ ⎩ α, β (ζ > ζ ) ⎪ ⎪ 1 2 ⎪ ⎨ (9) ζ =   ⎪ ⎪ ηˆ ηˆ ζ + 1 − ⎪ (1 − s) ζ − s) (1 ⎪ 1 2 ⎪ ⎪ ⎪ ⎪ ⎧ ⎪ ⎪ ⎪ ⎨ E, κ0 (ζ1 > ζ2 ) ⎪ ⎪ ⎪ ⎪ or for ζ : ⎪ ⎩ ⎩ α, β (ζ ≤ ζ ) 1

2

Note that it is not necessarily required that the same value of the fitting parameter ηˆ is used for all four parameters. For a detailed description including the extension to a three-phase composite introducing two kinds of fibers it is referred to Kato et al. (2009) and Kato (2010).

spline. pj indicates the j-th control point. Of course we could apply another parameterization allowing more general geometries such as level set functions. The fiber approximated by Bézier-splines is embedded in the structure and the control points are moved in order to obtain the optimal fiber layout. The entire domain of structure is defined in a parametric space s (0 ≤ s ≤ 1), see Figure 4. Thus the normalized coordinates s of control points turn out to be the design variables defining the global fiber layout in the physical space. According to this the j-th position vector of the control point pj can be expressed as follows y

y

rj (sxj , sj ) = O(ˆx, yˆ ) + (sxj Lx , sj Ly )

where O stands for the coordinate origin of the structure; xˆ , yˆ are the corresponding global coordinates of O. L denotes the contour lengths of the structure and superscripts x, y on L and also on s indicate the direction. Inserting Eq. (10) into the general mathematical formulation of Bézier-splines leads to the geometric formulation of a fiber including the design variables s as follows r(ϑ, sx , sy ) =

4

(10)

SHAPE OPTIMIZATION OF FIBER GEOMETRY

nb 

y

j (ϑ)rj (sxj , sj )

j=0

For the layout of textile fibers in FRC often parallel fibers or a mesh of straight fibers are used. Slightly curved fibers are advantageous if an optimal structural response is looked for. The geometry of a continuous long fiber is defined in global coordinates. We approximate the fiber geometry by Bézier-splines, defined as parametric curves by control points. A quadratic Bézier-spline and its mathematical formulation is introduced in Figure 4, where r stands for a position vector of the spline; ϑ (0 ≤ ϑ ≤ 1) is the local coordinate system of the

with

j =

nb ! ϑ j (1 − ϑ)nb −j (nb − j)!j!

(11)

where nb is the order of the Bézier-spline. Note that the coefficients  are independent of the design variables s. Once the fiber geometry is defined by Eq. (11) the global coordinates for the intersections between the fiber and the fixed finite element mesh can be calculated in order to establish the stiffness matrix and afterwards the internal forces of embedded fiber elements. This procedure is detailed in Kato and Ramm (2009).

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Figure 4.

5

(a) Quadratic Bézier-spline and (b) concept of global layout of fiber geometry.

OPTIMIZATION PROBLEM

In general an optimization problem is defined by the objective f (ˆs), equality constraints h(ˆs) and inequality constraints g(ˆs). Here sˆ denotes a design variable vector consisting of all design variables sˆi in the design domain. sˆi may be either volume fractions (section 3) or control parameters for shape (section 4); for simplicity sˆ i is often replaced by s. In this study the objective is to maximize the structural ductility for a prescribed fiber volume. Again we restrict the following derivation to a two-phase material optimization. As the ductility is defined by the internal energy summed along entire structure for a prescribed nodal ˆ see Maute (1998) and Maute et al. displacement d, (1998), the mathematical formulation of the optimization problem can be written as follows

Equilibrium is satisfied in the weak sense utilizing the matrix formulation of the principle of virtual work (PVW) in the parameter space ξ at time t + 1,   T ˜ δε σ t+1 |J|d ξ = λt+1 δuT t0 |J|d (15) ξ,

ξ

 subject to

σ dεd

f (ˆs) = −

h(ˆs) =



∇s f = ∇sex f + ∇d f T ∇s d,

(16)

(12)

(13)

6

εˆ

ˆ =0 sˆi d s − V

s

sˆL ≤ sˆi ≤ sˆU ,

where the body forces are not considered without loss of generality. λ denotes the load factor with respect to a reference traction load t0 . The structural response in turn depends on the design variables. Thus the gradient of design functions, for example of the objective function f = f (ˆs, d(ˆs)), can be formulated as follows

where ∇sex (•) describes the explicit derivative with respect to the design variables. An optimality criteria method (Patnaik et al. 1995), and the method of moving asymptotes (Svanberg 1987) are applied solving the optimization problems.

  minimize

ξ

i = 1, . . . , ns

(14)

ˆ denotes the prescribed fiber volume, sˆL and where V sˆ U the lower and upper bounds of the design variables, and ns the number of design variables. εˆ stands for the strain tensor after convergence at each time step in the structural analysis.

SENSITIVITY ANALYSIS

Starting from Eq. (15) and applying the variational direct method leads to   ˜ δεT ∇s σ t+1 |J|d ξ = ∇s λt+1 δuT t0 |J|d ξ.

ξ

ξ

(17) The determinant |J| of the Jacobian matrix, metric ˜ virtual displacements δu, and virtual strains δε |J|,

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do not depend on the design variables therefore their derivatives with respect to those parameters vanish. In Eq. (17) the loads are assumed to be independent of the optimization variables. In order to determine the stress derivative ∇s σ t+1 , Eq. (17) is expressed in terms of the derivative of the nodal displacements ∇s d. In this section the influence of the non-local equivalent strains ε˜ v on the sensitivity is neglected for simplicity. First the main variables stress σ, damage parameter D, history variable κ, and local equivalent strain εv are described in their general relation to the design parameters sˆ σ

= σ (D, C(E(ˆs), ν(ˆs)), ε(ˆs)),

(18)

D

= D(κ, κ0 (ˆs), α(ˆs), β(ˆs)),

(19)

κ

=

(20)

εv

κ(εv , κu (ε u )),

= εv (I1 (ε(ˆs)), J2 (ε(ˆs)), k(ˆs), ν(ˆs)),

by the loading condition, i.e. ∂κ = ∂εv ∂κ = ∂κu





1

if

loading

0

if

un−/reloading

0

if

loading

1

if

un−/reloading

(25)

The actual εv is replaced by κ at time t + 1 when loading occurs, so that ∇εv κ becomes unity. However the threshold history variable κ at time t + 1 does not change when unloading starts, i.e. κ = κu . This results in ∇εv κ = 0 and ∇κu κ = 1. The derivative of κu with respect to the design variable sˆi in Eq. (24) is given by   ∂κu ∂κ ∂εv ∂I1 ∂εv ∂J2  ∂ε u = + ∂s ∂εv ∂I1 ∂ε ∂J2 ∂ε t=tu ∂s 

(21)

(26)

where κu and εu , respectively, denote the history variable and the strain tensor at the time tu when a potential unloading starts. The stress derivative ∇s σ t+1 is determined according to the chain rule

The strain derivative ∇s εu in Eq. (26) is expressed by the usual kinematic B-operator matrix as

∇s σ t+1 = CT ∇s ε t+1 + ∇sex σ t+1

∇s ε u = B∇s du ,

=1

(22)

where du indicates the total nodal displacement vector at time tu . Note that du and the term in the parentheses of Eq. (26) need to be stored and updated whenever ‘loading’ occurs. Inserting Eqs. (23) and (24) into Eq. (22) and afterwards Eq. (22) into Eq. (17) yields

with CT =

∂σ ∂σ ∂D ∂κ + ∂ε ∂D ∂κ ∂εv



∂εv ∂J2 ∂εv ∂I1 + ∂I1 ∂ε ∂J2 ∂ε

 , (23)

∇sex σ t+1 =

∂σ ∂D +



∂D ∂κ0 ∂D ∂α ∂D ∂β + + ∂κ0 ∂s ∂α ∂s ∂β ∂s

∂σ ∂D ∂κ ∂D ∂κ ∂εv 



∂εv ∂ν ∂εv ∂k + ∂k ∂s ∂ν ∂s

∂C ∂E ∂C ∂ν + ∂E ∂s ∂ν ∂s

+

∂σ ∂C

+

∂σ ∂D ∂κ ∂κu , ∂D ∂κ ∂κu ∂s

(27)





 BT CT B|J|d ξ ∇s d = ∇s λt+1

ξ



 −

˜ NT t0 |J|d ξ

ξ

BT ∇sex σ |J|d ξ . (28)

ξ

 It can be recognized that Eq. (28) has the format of the typical stiffness expression adding up all terms on the right hand side to a new pseudo load vector Ppse : (24)

where CT denotes the tangent modulus at the time step t + 1. Here ∂/∂s is the partial derivative with respect to an optimization variable sˆi , regarding other variables sˆk for k = i as constants. Each derivative in Eqs. (23) and (24) can be explicitly obtained, see Kato et al. (2009) and Kato (2010). The derivatives of the history variable κ are determined

KT ∇s d = Ppse = ∇s λt+1 P + P˜ pse .

(29)

KT denotes the tangent stiffness matrix at the time step t + 1. From Eq. (29) the derivative of the total nodal displacement vector ∇s d is calculated, see Kato et al. (2009) and Kato (2010). Finally, the total sensitivity of the objective function can be obtained by inserting ∇s d into Eq. (16) and accumulating all sensitivities of the load increments

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up to step number nstep as ∇s f =

nstep  t=1

∇s ft =

nstep 

(∇sex ft + ∇d ftT ∇s dt ),

(30)

ν κ0

t=1

where ft indicates the ductility increment in the t-th load increment.

7

NUMERICAL EXAMPLES

7.1 Multiphase material optimization For the following numerical investigations a bending beam with straight fiber reinforcement is chosen as displayed in Figure 5. The FRC structure is composed of a concrete matrix and four unidirectional fibers at the fixed positions. These fibers consist of a mixture with fibers and a concrete matrix. The properties of concrete matrix in the fixed domain are not part of the optimization. In this study phase-1, phase-2 and phase-3 are set to be concrete, AR-glass and carbon, respectively. Eight-node quadratic plane stress elements are applied for both concrete and mixture, and six-node numerically integrated elements are used for the interface. The material properties for the concrete and the fibers are given in Figure 5 and those for the interface are referred to Kato (2010). λ denotes the load factor as introduced in Eq. (15). The analysis is carried out by a displacement control; the prescribed nodal displacement uˆ (in −y direction) at the control point c is 5 mm. In this section two optimization problems are introduced; one is for a two-phase composite in which the ‘fiber size’ is the design variable and the other one for a three-phase composite in which the ‘fiber size’ and ‘fiber material combination’ are the design variables. For the two-phase composite the design element consists of concrete (phase-1) and AR-glass (phase-2) in the element. The total fiber volume is set to 4.5% for the entire structure and is kept constant throughout the optimization process. In the initial stage each design element layer consists of concrete (50%) and AR-glass (50%). For the three-phase composite, each initial design element layer consists of concrete (50%), AR-glass (25%) and carbon (25%). The total fiber volume is again set to 4.5% and is kept constant. However the kind of fiber material is a free variable which is determined through the optimization process. Figure 5 shows the results of the optimization procedure applying the damage formulation. The upper two figures (a) and (a’) indicate the original and optimized fiber layouts and the final damage distributions for the two-phase composite, respectively, and the third one (b) shows the result of the three-phase model.

3000

3000

1500

1500

0

0 0

2

4

0

2

4

Figure 5. Comparison of optimization results using a damage formulation, (a), (a’) two-phase, (b) three-phase composite.

Carbon fibers have in general high strength and stiffness but show brittle behavior in tension. On the other hand AR-glass fibers have less strength but exhibit more ductile interface response than carbon. Therefore it is possible to increase the ductility by considering the respective advantages of each material. In Figure 5 (b) the carbon fiber of the lowest layer leads to a stiff structure and the AR-glass fiber in the next layer contributes the ductile response even after the carbon fiber is damaged. The optimized three-phase composite considerably increases the structural ductility. If only two carbon fibers are embedded at the

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lower two layers, this structure does not lead to an optimal ductility; see dotted line depicted in graph of Figure 5 (b).

7.2

Shape optimization of fiber geometry vs. multiphase layout optimization

In this numerical simulation a FRC beam reinforced with four AR-glass fibers is investigated, see Figure 6 where also material properties are given. The material properties for the interface are again referred to Kato (2010). For the discretization applying an embedded reinforcement formulation a two-dimensional eight-node quadratic plane stress element is used for

10

0

5

the concrete. The interface slip is discretized by a three-node quadratic one-dimensional element using the kinematic assumption by Balakrishnan and Murray (1986). The fiber geometry is approximated by a symmetric biquadratic Bézier-spline. The fiber volume is kept constant (1.4%) during the optimization leading to an initial fiber thickness of 0.4 mm. For the detailed description it is referred to Kato (2010). Figure 6 shows the optimization results again using the damage model; (a) is the result obtained by pure material shape optimization with a constant fiber thickness 0.4 mm and (b) by two-phase layout optimization as a combination of shape and multiphase optimization in which the fiber thickness may vary within 0 mm < r1 < 0.8 mm. The ductility could be increased by 37% applying shape optimization only, however it is also found that upper fiber in Figure 6 (a) is not exploited in an optimal way. In comparison the results of multiphase layout optimization show that the fiber material can move between fibers in space. As a result the ductility was further improved to 145%.

0 0

0.2

0.4

8

CONCLUSIONS

Multiphase material optimization, shape optimization of fiber geometry and the combination of both methodologies, namely ‘multiphase layout optimization’ were applied for maximizing the structural ductility of textile fiber reinforced composites. The two brittle materials, namely concrete matrix and fibers, get the necessary ductility from the interface behavior of the two constituents. For this objective, it is of course not sufficient to base the optimization process on a linear material model so that it is mandatory to consider material nonlinearities in the optimization process. It was verified in the numerical example of multiphase material optimization that the three-phase composite using an AR-glass and carbon could optimally improve the ductility of FRC. As an additional strategy shape optimization of fiber geometry was developed. It could be shown that the ductility of FRC could be substantially increased with respect to the geometrical layout of continuous fibers. In order to obtain even more efficient designs multiphase layout optimization was developed as the combination of both schemes, multiphase and shape optimization. This methodology is suited also to the optimization of other fiber reinforced composites.

REFERENCES Figure 6. Comparison of optimization results for deep beam, (a) material shape optimization, (b) 2-phase layout optimization.

Balakrishnan, S. and D. Murray (1986). Finite element prediction of reinforced concrete behavior: Structural Engineering Report, No. 138. University of Alberta.

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Bendsøe, M.P., M.M. Neves, and O. Sigmund (2000). Some recent results on topology optimization of periodic composites. In Proc. of the NATO advanced research workshop on topology optimization of structures and composite continua, Budapest, Hungary, pp. 3–17. Bendsøe, M.P. and O. Sigmund (2003). Topology optimization, theory, method and applications. Springer-Verlag. Gibson, L.J. and M.F. Ashby (1999). Cellular solids: Structure and properties. Cambridge University Press, Cambridge, UK. Kato, J. (2010). Material optimization for fiber reinforced composites applying a damage formulation. Ph.D. thesis, Institut für Baustatik und Baudynamik, Universität Stuttgart, Germany. Kato, J., A. Lipka, and E. Ramm (2009). Multiphase material optimization for fiber reinforced composites with strain softening. Struct. Multidisc. Optim. 39, 68–81. Kato, J. and E. Ramm (2009). Optimization of fiber geometry for fiber reinforced composites considering damage. Finite Elem. Anal. Des., submitted. Krüger, M., J. Ozbolt, and H.-W. Reinhardt (2002). A discrete bond model for 3D analysis of textile reinforced and prestressed concrete elements. Otto-Graf-Journal 13, 111–128. Krüger, M., J. Ozbolt, and H.-W. Reinhardt (2003). A new 3D discrete bond model to study the influence of bond on the structural performance of thin reinforced and prestressed concrete plates. In Proc. of High Performance Fiber Reinforced Cement Composites (HPFRCC4) RILEM, Ann Arbor, USA, pp. 49–63. Lipka, A. (2007). Verbesserter Materialeinsatz innovativer Werkstoffe durch die Topologieoptimierung. Ph.D. thesis, Institut für Baustatik und Baudynamik, Universität Stuttgart, Germany. Lipka, A. and E. Ramm (2005). Optimization of foam filled structures using gradient algorithms. In Proc. of the IUTAM SYMPOSIUM: topoptSymp2005, Topological design optimization of structures, machines and materials status and perspectives, Copenhagen, Denmark, pp. 537–548. Maute, K. (1998). Topologie- und Formoptimierung von dünnwandigen Tragwerken. Ph.D. thesis, Institut für Baustatik, Universität Stuttgart, Germany.

Maute, K., S. Schwarz, and E. Ramm (1998). Adaptive topology optimization of elastoplastic structures. Struct. Optim. 15, 81–91. Mazars, J. and G. Pijaudier-Cabot (1998). Continuum damage theory-application to concrete. J. Eng. Mech. 115, 345–365. Patnaik, S., J. Guptill, and L. Berke (1995). Merits and limitations of optimality criteria method for structural optimization. Int. J. Num. Meth. Eng. 38, 3087–3120. Peerlings, R. (1999). Enhanced damage modelling for fracture and fatigue. Ph.D. thesis, Technische Universiteit Eindhoven, The Netherlands. Peerlings, R., R.d. Borst, W. Brekelmans, and M. Geers (1998). Gradient- enhanced damage modelling of concrete fracture. Mech. Cohes. Frict. Mater. 3, 323–342. Peerlings, R., R.d. Borst, W. Brekelmans, and J.d. Vree (1996). Gradient enhanced damage for quasi-brittle materials. Int. J. Num. Meth. Eng. 39, 3391–3403. Phillips, D. and O. Zienkiewicz (1976). Finite element nonlinear analysis of concrete structures. Proc. Inst. Civ. Engrs. Part 2, 61 (3), 59–88. Sigmund, O. and S. Torquato (1997). Design of materials with extreme thermal expansion using a three-phase topology optimization method. J. Mech. Phys. Solids 45, 6, 1037–1067. Svanberg, K. (1987). The method of moving asymptotes—a new method for structural optimization. Int. J. Num. Meth. Eng. 24, 359–373. Vree, J.d., W. Brekelmans, and M. Gils (1995). Comparison of nonlocal approaches in continuum damage mechanics. Comput. & Struct. 55, 581–588. Xu, S., M. Krüger, H.-W. Reinhardt, and J. Ozbolt (2004). Bond characteristics of carbon, alkali resistance glass, and aramid textiles in mortar. J. Mater. Civil Eng. ASCE 16, 4, 356–364. Zhou, M. and G. Rozvany (1991). The coc algorithm, part ii: Topological, geometrical and generalized shape optimization. Comp. Meth. Appl. Mech. Eng. 89, 309–336.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

A multifibre approach to describe the ultimate behaviour of corroded reinforced concrete structures B. Richard University Paris-Est, French Public Works Laboratory (LCPC), Paris, France

F. Ragueneau LMT-Cachan, ENS-Cachan/CNRS/Univ Paris 6/PRES UNIVERSUD, Paris, France

Ch. Crémona Directorate for research and innovation, Ministry of Ecology, Energy, Sustainable Development and Sea, La Défense, France

ABSTRACT: This contribution aims to present a multifibre approach allowing to deal with complex case-studies of reinforced concrete structures subject to coupled mechanical and corrosion actions. Constitutive equations for steel, concrete and steel-concrete interface are presented. Cyclic behaviour for concrete and coupled mechanics-corrosion mechanisms for steel and interface are taken into account in the proposed formulation. An implicit iterative procedure is used to introduce a relative slip between steel and concrete within the framework of beam theory. 1

INTRODUCTION

Identifying and quantifying the mechanical behaviour of reinforced concrete structures degraded by coupled mechanical and corrosion processes is an important problem in civil engineering fields: although a lot of work has been done during last decades, this problem remains opened (Almusallam et al. 2996, Almusallam 2001, Cabrera 1996, Castel et al. 2000, Rodriguez et al. 1997, Coronelli & Gambarova 2004, Spacone & Limkatanvu 2000, Wang & Liu 2004). The corrosion of reinforcement bars is the major cause of damages for reinforced concrete structures. It manifests in: – cross section reduction of the steel reinforcements and brittleness increase of steel, – cracking of the concrete cover, – evolution of the steel-concrete bond properties. This paper presents a simplified finite element model for assessing the load carrying capacity of reinforced concrete structures degraded by corrosion and mechanical loadings. It is focused on the use of specific multifibre elements accounting for relative slip between steel and concrete (Combescure & Wang 2007). The steel/concrete interface model is based on irreversible processes thermodynamics; this framework ensures coherence with respect to conservation and evolution physics principles. The interface degradation is caused by both mechanical and chemical effects. Concrete is described within the

continuum damage mechanics framework coupled with a frictional sliding model to ensure a satisfying response in the case of cyclic loadings. Steel bar properties are coupled to the level of corrosion thanks to the use of a classical ductile damage model coupled to plasticity. In order to treat the case of large scale structures, the multifibre approach with a non-perfect interface (whatever the beam kinematics assumption) is used and has been implemented in finite element software. Local responses at the Gauss point level are exposed in the cases of monotonic and cyclic stress path. The results help to point out the main features of the proposed constitutive equations. The efficiency and the reliability of the approach are assessed by presenting structural case studies which highlight a good agreement between numerical and experimental results. Structural case-studies emphasizing the model responses subject to cyclic loadings (earthquake) and coupled corrosion-mechanical loadings are also presented.

2 2.1

MULTIFIBER FINITE ELEMENTS General framework

In the field of seismic engineering, the multifiber numerical framework is often used in order to decrease the global computational cost needed to perform complex analyses (Spacone et al. 1996, Mazars et al. 2006).

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The multifiber approach authorizes to include non linear constitutive laws in a finite element model built from Timoshenko’s or Euler-Bernoulli’s beam elements. A relationship between the axial strain, the curvature, the rotations and the generalized stresses (normal load, shear load and bending moment) represents the constitutive behaviour operator. Each element cross section is described using classical twodimensional elements. At the cross section scale, each material is characterized by a one-dimensional behaviour constitutive equation linking the normal stress and the shear stresses respectively to the axial strain and to the distortions. The major point of the multifibre approach is to add kinematics assumptions to link the global nodal displacement (at the beam element scale) to the local strains (at the cross section scale). These hypotheses are generally provided by well known beam theories such as Timoshenko’s or Euler-Bernoulli’s theories. In the present study, the kinematics conditions provided by the Timoshenko’s beam theory are considered; the cross section remains plane but not necessarily perpendicular to the neutral axis. Therefore, considering the general case of a bending problem, for a given cross section denoted S(x) at the point x and for any point at a vertical location y, local strains εxx and εxy are linked to the axial displacement u(x), to the vertical displacement ν(x) and to the rotation θ (x) according to: εxx

du(x) dθ (x) = −y , dx dx

dν(x) dθ (x) 2εxy = − dx dx

Figure 1. Local stress state at the steel (in red)/concrete (in grey) interface due to the application of a tension force to the steel rebar.

The equilibrium condition can then be formulated according to the following equation: σ (s)ds − S(x)

(1)

2.2 Non perfect steel/concrete interface The specific mechanical behaviour of the steel/ concrete interface can be taken into account in an implicit way. Let us consider a set of fibers combining concrete and steel. The fundamental idea consists in assuming that the total strain in the steel fiber can be split into two parts: a first one is associated to the proper strain of the steel and a second one is related to the sliding strain occurring at the steel/concrete interface (Combescure & Wang 2007). This assumption can be expressed by the following equation: εxx = λεxx + (1 − λ)εxx =

S εxx

+

I εxx





(2)

where εxx is the axial strain and λ the so-called partition factor. The first term is associated to the steel and the second one to the steel/concrete interface. The partition factor can only vary from 0 to 1 by definition. Its computation is made by imposing a local equilibrium condition between the tension force in the steel and the shear force acting at the steel/concrete interface. Figure 1 depicts the local state of stress at the steel/concrete interface when the steel rebar is in tension.

τ (s)ds = 0

(3)

S(x)

where σ and τ are respectively the normal stress in the steel and the shear stress at the steel/concrete interface. The previous equation can be rewritten considering two constitutive operators for the steel and for the steel/concrete interface denoted S (.) and I (.) respectively. Integrating equation (3) and combining it to the constitutive operators, it comes: S (εxx , λ) S(x) − I (εxx , 1 − λ) laI P(x) = 0

(4)

where laI and P(x) denote respectively the anchorage length and the perimeter of the current cross section. Equation (4) introduces a unique variable which is the partition factor. This equation is generally nonlinear due to the nonlinearity of the behaviour operators and therefore requires invoking a classical Newtonbased numerical scheme for solving it. This numerical framework can be used with any constitutive operators for the steel and for the steel/concrete interface. In the next section, the proposed steel/concrete interface constitutive law is exposed. 2.3

Numerical implementation

A Newton based scheme is used for solving the equilibrium equation. Let us consider a total axial strain increment denoted εi . The total axial strain, at the current iterate, is εi+1 = εi + εi . The partition factor λk is initialized to λ0 , which is the value reached at the previous loading sep. The strain in the steel and in the steel/concrete interface can then be computed according to equation (5): S εi+1,k = λk εi+1 ,

I εi+1,k = (1 − λk )εi+1

(5)

Using the constitutive operators defined from the knowledge of the constitutive laws associated to the

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aims to account for the loss of ductility due to stress concentration so as to predict the brittleness increase with the level of corrosion. Plasticity models coupled to damage can be used for the description of ductile rupture of steel-like materials. Such model introduces material parameters like the plastic strain at rupture. The modification of the constitutive equations allows for the determination of such material parameters function of the level of corrosion. Thermodynamics variables define the state of a representative element volume. The classical choice for steel plasticity is: the strain tensor (ε), the plastic strain tensor (εp ), the isotropic damage (D) and the isotropic hardening (r). Through the strain equivalence principle and the effective stress concept, one can express the constitutive equations allowing defining the damaged Young’s modulus E˜ = (1 − D)E (Lemaître 1992, Ouglova 2004).

steel and to the steel/concrete interface, the corresponding stresses are: S = S (εi+1 , λk ), σi+1,k

I σi+1,k = I (εi+1 , 1 − λk )

(6) The forces in the steel rebar and at the steel/concrete interface can then be calculated: S S Fi+1,k = σi+1,k S(x),

I I Fi+1,k = σi+1,k P(x)laI

(7)

The total disequilibrium can be expressed as S I Ri+1,k = Fi+1,k − Fi+1,k

(8)

From the computation of the total disequilibrium, the next partition factor can easily be estimated according to a modified Newton scheme λk+1 = −

Ri+1,k 2π RlaI μ + πR2 Es

σ =ρ

(9)

where R is the rebar radius and Es is the steel Young modulus. As it can be noticed, equation (9) clearly shows that the stiffness is not updated during the iterations. The iterative process is stopped as soon as the disequilibrium is lower than a given tolerance. 3

CONSTITUTIVE EQUATIONS

3.1.1 Steel subject to coupled corrosion and mechanical loading The rust formation on the steel surface is the consequence of chemo-electro mechanisms due to the concrete carbonation and chloride intrusion inducing a pH decrease. This results in a phase transformation from the steel to oxides. At the structural level of a reinforced concrete structure, only the mechanical consequences of such phase transformation are taken into account. The constitutive equations of the steel are based on irreversible processes thermodynamics, being known the level of corrosion of the material. The experimental observations show two main consequences of steel oxidation. First of all, the loss of cross section (Almusallam et al. 1995) has to be taken into account in order to be able to predict the mechanical behaviour of corroded bars. Different experiments on steel bars show an increase of brittleness with the level of corrosion, the stress concentration due to notch effects is the major source of capacity bearing loss (Castel et al. 2000). This feature leads to an important decrease of peak displacement at rupture. Being known the level of corrosion by the reduction of the steel cross section, the present model

(10)

In order to ensure a coupling between damage and plasticity, the effective stress is introduced in the plastic criterion. For isotropic damage, the von Mises criterion f , defining the elasticity domain, takes the simple form: f =

In this section, the constitutive equations for concrete, steel and steel-concrete interface are presented.

∂ψ = E(1 − D)(ε − εp ) ∂ε

σeq − R − σy 1−D

(11)

σy is the initial elasticity threshold for uniaxial tension. So as to describe damage accumulation under monotonic and fatigue loading, using the same law and material parameters, a unified damage law is expressed using non-associated plasticity rules defining a plastic potential. The damage potential F D is chosen as a power law of its associated variable −Y . FD =

  S Y s+1 − (s + 1)(1 − D) S

(12)

The normality rules apply defining the evolution laws of internal variables. Ductile damage is initiated through a threshold function, introducing the coupling between plasticity and damage, allowing dealing with failure. Moreover, if we consider that close to rupture, the hardening is saturated, the damage evolution law may be simply integrated and allows introducing a linear dependence of the damage to the cumulated plastic strain. The damage threshold function is expressed by: fD = p − pD

(13)

with pD , the initial threshold at damage activation (corresponding to the plastic strain εD ). One may introduce a critical damage Dc defining the limit between a critical micro-crack density before macro-crack

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R(p) = σ − σy = Kp1/m

(14)

The corrosion modelling in the framework of mechanical load bearing prediction has been identified using the experimental data of Almusallam (1995). Considering 6 mm diameter steel bar sustaining different levels of corrosion under tensile loading, the important decrease of load bearing capacity and ductility is observed in figure 1a. The nonlinear hardening model parameters have been fitted on the virgin sample corresponding to a 0% level of corrosion. Results shown in figure 1b have been performed using the following values: E = 200.0 GPa, v = 0.3, DC = 0.2, σy = 500 Mpa, K = 500 Mpa and m = 2.78. The results are shown in terms of load/displacement curves allowing easier comparisons to experimental results. 3.1.2 Concrete subject to cyclic loadings Most of quasi-brittle materials, such as concrete, are subject to unilateral effects. They classically appear when the material is subject to a tension—compression loading path. It can be observed that crack closure is traduced by a gain of stiffness during the compression phase. Under robustness considerations, a scalar damage variable has been chosen. A major drawback lies

25

20

Load (kN)

initiation, corresponding to the cumulated plastic strain at rupture (the plastic strain εR ). For monotonic uniaxial loading, the plastic strain ε p , εD , εR corresponds to the cumulated plastic strain p, pD , pR . Analysing several experimental results on steel, the ratio εD /εR is often constant and equal to εD = 0.8εR . If the cross section reduction of the steel bars are simply accounted for in the finite element mesh, what ever the kind of elements (volumetric or uniaxial beams or rods), the dependency of the plastic strain at rupture to the level of corrosion has to be introduced in the constitutive equations. Using classical results from the literature (Almusallam et al. 1995) and from experimental tests performed in LMTCachan (Ouglova 2004), such a law can be directly identified by observing the plastic strain at rupture in function of the corrosion ratio for 4 steel bars of different diameters (6, 13, 20 and 25 mm). It appears that two regimes can clearly be identified. The phenomenological description of these mechanisms can be fitted by the following expression: For Tc ≤ 15% then εR = −0, 0111Tc + 0, 2345 else εR = −0, 0006Tc + 0, 077 The remaining parameters to be identified are the elastic constants (Young’s modulus and Poisson’s ratio, obtained on simple uniaxial tension tests) and the plasticity parameters determined by a uniaxial tension test performed up to the hardening saturation. Let us make σy stand for the original elasticity limit, the isotropic hardening function R(p) is classically introduced through a power law with two parameters K and m:

15

10

0% 2% 13 % 22.4 % 35 % 50 %

5

0

0

5

10 15 Elongation (mm)

20

25

Figure 2. Tensile tests on 6 mm diameter steel bars for different levels of corrosion: numerical response of the proposed model (Ouglova 2004).

in the difficulty to take into account total unilateral effects (if a single damage variable is considered, for thermodynamic adequacy, and not two different ones for tension and compression). Nevertheless, it can be partially considered. That means during a tension— compression loading path, the elastic modulus is not fully recovered but only partially. Sliding may occur between the cracks lips and therefore, friction has to be created. This effect is exhibited when the material is subject to a cyclic loading. In order to take into account this mechanism, the approaches proposed by (Ragueneau et al. 2000) have been considered. Nevertheless, in the present study (Richard et al. 2009), it is considered that the energy rate (related to sliding) only affects the deviatoric part of the free energy. This consideration can be justified by the fact that sliding and friction are mainly in relation with shear stresses. This hypothesis is consistent with the fact that in mode I (pure opening), no friction and no sliding appear. The thermodynamic potential takes the following form: ρ =

  1 κ (1 − d) < εkk >2+ − < −εkk >2+ 2 3   + 2(1 − d)μεijD εijD + 2dμ εijD − εijπ    D εij − εijπ + γ αij αij + H (z) (15)

where ρ is the material density, κ and μ are the bulk and shear coefficients respectively. εij is the second order total strain tensor, δij is the second order Kronecker’s tensor and d is the scalar damage variable (0 for virgin material and 1 for failed material). εijπ is the second

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order sliding tensor, γ is a material parameter, αij is the second order tensor associated to the kinematics hardening, z is the internal variable corresponding to the isotropic hardening and H its consolidation function. < Aij >+ denotes the positive part of the tensor Aij εijD = εij − 13 εkk δij is the second order deviatoric total strain tensor. From the state potential, the state laws are obtained by simple derivation regarding the internal variables: For the Cauchy stress tensor ∂ρ κ = ((1 − d) < εkk >+ − < −εkk >+ )δij ∂εij 3

(16) + 2(1 − d)μεijD + 2dμ εijD − εijπ

σij =

direct extension mechanisms (tension) and induced extensions mechanisms (compression): Y¯ = βYDir + YInd with YDir =

Y =−

(17)

The thermodynamic forces associated to kinematics and isotropic hardenings can be expressed by equations: ∂ρ = γ αij ∂αij

and Z =

∂ρ dH (z) = ∂z dz (18)

where Xij denotes the second order back stresses tensor associated to kinematic hardening and Z the thermodynamic force corresponding to the isotropic hardening. A damage variable is associated to the isotropic hardening variable in order to ensure that damage mechanism is not activated during an unloading step. On the contrary, the second order sliding tensor is associated with kinematics hardening variable. It allows an efficient control of the energy released during cyclic loadings. A threshold surface, denoted fd , is introduced according to equation: fd = Y¯ − (Y0 + Z)

εijDir =< εij >+ H(< εij >+ < σij >+ ),

εijInd

= εij − εijDir

(21)

A surface without any threshold is introduced in order to manage sliding mechanism associated with kinematics hardening. It takes the form of a Von Mises’s criterion (without hydrostatic effects) expressed as:

3 fπ = (sij − Xij )(sij − Xij ) (22) 2

And the damage energy release rate:

Xij =

1 Dir 1 ε Eijkl εklDir and YInd = εijInd Eijkl εklInd 2 ij 2

where Eijkl is the fourth order Hook’s tensor which can be computed from the elastic parameters κ and μ, β is a parameter driving the dissymmetry of the threshold surface between tension and compression. The direct and induced extensions tensors are obtained through the following decompositions:

The frictional stress tensor:

∂ρ σijπ = − π = 2dμ εijD − εijπ ∂εij ∂ρ 1 κ = < εkk >2+ + 2μεijD εijD ∂d 2 3



− 2μ εijD − εijπ εijD − εijπ

(20)

(19)

where Y¯ denotes energy-type variable driving damage and Y0 , an initial threshold. To define a different behaviour in tension and in compression, this variable is defined through decomposition between

where sij is the deviatoric part of the Cauchy’s stress tensor. The evolution of damage and isotropic hardening variables are postulated as associated ones. From the maximum dissipation principle, a unique Lagrange’s multiplier, denoted λ˙ d , needs to be introduced as followed: ∂fd = λ˙ d d˙ = λ˙ d ∂ Y¯

and

z˙ = λ˙ d

∂fd = −λ˙ d ∂Z

(23)

In order to keep a unique damage variable in tension and in compression, representing the state of damage independently from the state of stresses, the dissymmetry between tension and compression responses can be obtained by choosing an appropriate consolidation function which has been postulated in the following form: dH (z) −1 = dz 1+z   H(<εij >+ <σij >+) 1 − H(<εij>+ <σij >+) + × ADir AInd where ADir and AInd are material parameters which can be interpreted as brittleness terms respectively in tension and in compression. Let us note that the function dH dz is clearly continuously differentiable with respect to the variable z. Nevertheless, it is indexed by the parameters ADir and AInd .

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Concerning frictional-sliding, differently from the previous case, the flow rules are supposed not to be associated ones. From the maximum dissipation principle, they can be expressed as: ∂ϕπ ∂σijπ

ε˙ ijπ = λ˙ π

and α˙ ij = −λ˙ π

∂ϕπ ∂Xij

(24)

where ϕπ is a pseudo potential of dissipation and λ˙ π the associated Lagrange’s multiplier. It has been chosen according to the proposal from (Armstrong & Frederick 1966) and can be expressed as:

ϕπ =

3 a (sij − Xij )(sij − Xij ) + Xij Xij 2 2

(25)

It allows overcoming the main drawback of the Drucker’s criterion which is the linearity of the associated hardening. Let us note that this strategy ensures that the sliding tensor, the kinematics hardening tensor and the back stress tensor are pure deviatoric second order tensors. The local responses under a cyclic loading path are presented in the figure 3. The material parameters used are described in the table 1. 3.1.3 Steel/concrete interface Three different degradation steps can be observed during the failure process of a steel concrete interface. The 7

0.5

x 10

0

-0.5

Stress

11

(Pa)

-1

-1.5

-2

-2.5

-3

-3.5 -16

-14

-12

-10

-8

-6 Strain

-4

-2

0

2 x 10

11

Material parameters used for the concrete model.

Material parameters Young modulus Poisson’s ratio Initial threshold Brittleness in tension Brittleness in compression Kinematics hardening Non linear hardening

where ρ is the material density and μ the second I ,π is the inelastic sliding strain, Lamé’s coefficient. εxx α the kinematic hardening variable, z the isotropic kinematic variable, H the consolidation function and γ is a material parameter which needs to be identified. The scalar nature of the damage variable is physically motivated by the fact that cracks have a single and fixed orientation at the steel/concrete interface. The amount of friction along cracked surfaces has to be linked to the level of damage d; in order to describe suitable cyclic behaviours, stored energies due to both kinematic and isotropic hardenings are introduced in the free energy expression. Let us note that the proposed state potential is clearly thermodynamically admissible (differentiable with respect to each variable, convex, positive and null at the origin). The state potential expressed by the previous equation leads to the following state equations. The Cauchy’s shear stress is given by τ=

∂ρ I I I ,π = 2μ(1 − d)εxx + 2μd(εxx − εxx ) I ∂εxx

(27)

As for (Ragueneau et al. 2006), the friction shear stress τ π is obtained by

Values E ν Y0 ADir AInd γ0 a0

I I I I ,π I I ,π εxx + dμ(εxx − εxx )(εxx − εxx ) ρψ = (1 − d) μεxx γ + αα + H (z) (26) 2

-3

Figure 3. Damage, permanent strain and hysteretic loops for the concrete model subject to uniaxial tensioncompression loading. Table 1.

first step occurs when the chemical adhesion ends. It must be noted that this adhesion is very weak and is often analysed by elastic models. The second step allows observing the apparition of the first shear cracks between the rebar and the concrete. This third level concerns the post-peak domain. Sliding becomes high and the steel-concrete interface is entirely degraded. Degradation mode II (shearing) is entirely characterized by these three stages. Due to the Timoshenko’s kinematics hypotheses, the degradation mode I (opening) cannot be considered. From these observations, cracking effects are taken into account by introducing a continuous damage variable d, varying from 0 (virgin material) to 1 (ruined material). The Helmholtz free energy, denoted , is given by (Richard et al. 2009):

36000 MPa 0,2 200 Jm−3 1,6 10−3 Pa−1 1,6 10−5 Pa−1 7,0 109 Pa 5,0 10−7 Pa−1

τπ = −

∂ (ρ ) I ,π ∂εxx

I I ,π = 2μd(εxx − εxx )

(28)

It can be noticed that the damage variable and the inelastic sliding strain act in a coupled way on the friction shear stress. The energy released rate due to

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Maximum shear stress (MPa)

damage Y is ∂(ρ ) I I I I ,π I I ,π εxx − μ(εxx − εxx )(εxx − εxx ) = μεxx Y =− ∂d (29) The first term represents the energy rate released due to mode II (shearing) and the second term the energy rate released due to inelastic sliding and friction. The back stress X can also be defined and the thermodynamic force Z associated to the isotropic hardening are X =

∂(ρ ) = γα ∂α

and Z =

∂(ρ ) = H  (z) ∂z (30)

A unique yield surface, denoted fd , is considered to handle the joint evolution of the damage and of the isotropic hardening

15 10 5 0 0

Experimental data Model response after identification 2 4 6 Macroscopic corrosion degree (%)

8

for W allows fitting experimental results coupling the shear bearing capacity of an interface with the level of corrosion Tc. STRUCTURAL CASE-STUDIES

(31) 4.1

where Y0 in an initial threshold. Therefore, a single Lagrange’s multiplier λ˙ d needs to be considered in the normality rules ∂fd d˙ = λ˙ d = λ˙ d ∂Y

20

Figure 4. Model response for a steel/concrete interface constitutive equations coupling mechanical loading and corrosion.

4 fd = Y − (Z + Y0 )

25

and

z˙ = λ˙ d

∂fd = −λ˙ d ∂Z

(32)

Due to the Timoshenko’s hypotheses, swelling cannot be considered. Nevertheless, the bond strength variation can be included in the constitutive law. In the present work, the bond strength variation due to the corrosion phenomenon is considered as an additional energy rate initially released YR . This energy rate is function of a macroscopic corrosion degree Tc defined in terms of steel mass loss or in terms of steel cross section loss for instance. This energy rate can be included in the damage yield surface to modify the initial threshold, charactering the non corroded behaviour of the steel/concrete interface. Therefore, equation (31) becomes fd = Y − (Z + Y0 + W (YR (Tc )))

(33)

The inelastic strain variable and the kinematic hardening variable are handled as for the concrete model (see equation 24). The set of material parameters associated to the steel/concrete interface can be separated as followed: one elastic parameter (μ), two parameters for accounting the damage and the isotropic hardening mechanisms (Ad and Y0 respectively), one function for the corrosion phenomenon (W ), two parameters for the kinematic hardening mechanism (γ and a respectively) and one geometrical parameter which is the anchorage length (laI ). The use of a particular function

Seismic loadings

In the context of the French National Project CEOS.FR, a numerical benchmark has been performed. It aimed to quantify the ability of new numerical methods and refine constitutive models to deal with reinforced concrete structures subject to cyclic loadings. Especially, reinforced concrete beams subject to a cyclic four point bending test has been proposed. The geometry of the specimens is 1.5 × 0.20 × 0.15 m3 . There are four steel reinforcement bars with a diameter equal to 10 mm (top part) and 12 mm (bottom part). Stirrups with a diameter equal to 6 mm and a spacing equal 15 cm have also been considered. The displacement at the middle of the specimen has been prescribed and recorded. Hence, the loading history is available and has been used as an input data in mechanical models. Mechanical properties of constitutive materials (concrete and steel) have been estimated according to usual values used. For concrete, a compressive strength of 21 MPa and a tensile stress of 1.5 MPa have been considered and for steel, a yield stress equal to 525 MPa has been used. The loading (in terms of displacement vs time diagram) is shown in the next figure. The numerical results are depicted in figure 6. One can notice that the accordance between experimental data and numerical results is good. Nevertheless, it can be noticed that mechanical properties for concrete are very weak. These results allowed detecting defaults when the specimen has been cast. 4.2

Coupled corrosion-mechanical loading example

The efficiency and the reliability of the proposed approach for practical applications are illustrated on

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x 10-3

7

14

6

12

5 Load (N)

Midspan displacement (m)

16

10 8

3 2

4

1

2

0 0

-2 0

Figure 5.

500

1000

1500

2000

2500 3000 Pseudo time

3500

4000

4500

0.002

0.004 0.006 0.008 0.01 Middle span displacement (m)

0.012

0.014

Figure 7. Load-displacement curves obtained for various corrosion degrees (expressed in terms of cross section losses).

5000

Earthquake loading: mid-span displacement.

70 60

Exp. Num.

50 40 30 20 10 0 -10 -20 -30 0

Experimental data (Tc = 0.00 %) Experimental data (Tc = 1.25 %) Experimental data (Tc = 2.50 %) Experimental data (Tc = 5.00 %) Experimental data (Tc = 7.50 %) Numerical results (Tc = 0.00 %) Numerical results (Tc = 1.25 %) Numerical results (Tc = 2.50 %) Numerical results (Tc = 5.00 %) Numerical results (Tc = 7.50 %)

4

6

0

Load (kN)

x 104

1

2

3 Pseudo time

4

5

6 x 106

Figure 6. Model response: comparisons with the registered experimental load at the centre of the beam.

an experimental study performed by (Mangat et al. 1999). This test quantifies the effects to the corrosion phenomenon on the global behaviour of structural elements up to failure. The authors realized several beam specimens with dimensions 910 mm × 100 mm × 150 mm. The reinforcement is made of two longitudinal rebars with 10 mm diameter. According to the authors, these beams were under-reinforced in order to get a global beam failure driven by the local failure of the steel/concrete interface. The concrete had a Young modulus equal to 28000 MPa, a Poisson ratio equal to 0.2, a tensile stress limit equal to 3,45 MPa, a limit compressive stress equal to 40 MPa. These properties were measured using classical mechanical tests. The steel has a Young modulus equal to 206000 MPa, a Poisson ratio equal to 0.3, a yield stress equal to 510 MPa to avoid any yielding. Each specimen was subjected to corrosion by imposing a corrosion rate equal 2 mA · cm−2 . Different corrosion degrees were performed and the beams were tests with four-points bending tests up to failure. The experimental properties have been used to calibrate the model. The W function has not been modified because the peak shear stress values at

the steel/concrete interface have been supposed to be the same. This assumption can be justified by noting that the concrete and the steel used in the experience performed by (Mangat et al. 1999) are not very different from those used by (Almusallam et al. 1996). No particular difficulties were reported to calibrate the steel model because all the parameters could be identified from the experimental measurements (Ouglova 2004). The material parameters can be easily identified using the experimental measurements. Several four-points bending tests have been simulated at different corrosion degrees. The steel cross sections reduction has been taken into account by specifying their loss in the multifiber model. The numerical results are given in Figure 7 and a good agreement can be noticed. The three main degradation stages experimentally observed are well captured: the linear response, the concrete cracking and the failure stage characterized by the total degradation of the steel/concrete interface. The effects of the corrosion phenomenon are also satisfyingly captured.

5

CONCLUSION

Through this contribution, a simplified steel/concrete model has been proposed. This model helps to analyze the behaviour of corroded reinforced concrete structures subject to strong mechanical loadings. Based on a multifibre numerical framework, a set of constitutive equations capable of modelling the nonperfect behaviour of the steel/concrete interface has been introduced and discussed. Additional constitutive models for corroded steel and concrete subject to cyclic loadings are also presented. It is written within the rigorous framework of irreversible processes thermodynamics. The numerical implementation of the model has been realized using a coupling explicit/implicit integration scheme improving the robustness versus numerical instabilities. The main features of the proposed model have been presented at the Gauss point level for cyclic responses. From this analysis, it can be stated that the proposed model

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is clearly able to reproduce a homogenised behaviour based on three elementary behaviours (steel/concrete interface, steel reinforcement bar and concrete). The efficiency and the ability of the approach for practical cases have been also illustrated on two case studies: a beam subject to seismic action and the mechanical analysis up to failure of corroded reinforced concrete elements. A good accordance between simulations and experimental results has been obtained and is encouraging the authors to keep working on this simplified model. Nevertheless, due to the simplified nature of the multifibre numerical framework, no information related to the concrete cracking pattern or to crack openings could be obtained. It appears that the proposed approach constitutes an efficient and relevant tool for preliminary analyses of reinforced concrete structures. REFERENCES Almusallam, A.A. 2001. Effect of degree of corrosion on the properties of reinforcing steel bars. Construction and Building Materials, 15:361–368. Almusallam, A.A., Al-Gahtani, A.S., Aziz, A.R. & Rasheeduzzafar, M. 1996. Effect of reinforcement corrosion on bond strength. Construction and building materials, 10(2): 123–129. Armstrong, P.J. & Frederick, C.O. 1966. A mathematical representation of the multiaxial Bauschinger effect, G.E.G.B., Report RD/B/N. p. 731. Cabrera, J.G. 1996. Deterioration of concrete due to reinforcement steel corrosion. Cement and Concrete Composites, 18(1):47–59. Castel, A., François, R. & Arliguie, G. (2000). Mechanical behaviour of corroded reinforced concrete beamsPart 2: Bond and notch effects. Materials and Structures, 33:545–551. Castel, A., François, R. & Arliguie, G. 2000. Mechanical behaviour of corroded reinforced concrete beams-Part 1: Experimental study of corroded beams. Materials and Structures, 33:539–544. Combescure, D. & Wang, F. 2007. Assesments of existing RC structures under severe dynamic loading using non linear modelling, CONSEC 07, Tours, France. Coronelli, D. & Gambarova, P. 2004. Structural assessment of corroded reinforced concrete beams: Modeling guidelines. Journal of Structural engineering, 130(8): 1214–1224.

Lemaître, J. 1992. A Course on Damage Mechanics, SpringerVerlag. Mangat, P.S. & Elgarf, M.S. 1999a. Flexural strength of concrete beams with corroding reinforcement, ACI Structural Journal, 96(1):149–158. Mazars J., Kotronis, P., Ragueneau, F. & Casaux G. 2006. Using multifiber beams to account for shear and torsion: Applications to concrete structural elements, Computer Methods in Applied Mechanics in Engineering, 195(52), pp. 7264–7281. Ouglova, A. 2004. Mechanical behaviour of reinforced concrete strcutures subjected to corrosion (in French). PhD thesis. ENS-Cachan., Cachan, France. Ragueneau, F., Dominguez, N. & Ibrahimbegovic, A. 2006. Thermodynamic-based interface model for cohesive brittle materials: Application to bond slip in RC structures. Computer Methods in Applied Mechanics and Engineering, 195:7249–7263. Ragueneau, F., La Borderie, C. & Mazars, J. 2000. Damage Model for Concrete Like Materials Coupling Cracking and Friction, Contribution towards Structural Damping: First Uniaxial Application. Mechanics Cohesive Frictional Materials, 5:607–625. Richard, B., Ragueneau, F., Crémona, C., Adélaide, L. & Tailhan, J.L. 2009. Isotropic damage model coupled to frictional sliding to model the cyclic behaviour of reinforced concrete elements, COMPLAS 2009. Barcelona, Spain. Richard, B., Ragueneau, F., Crémona, C., Adélaide, L. & Tailhan, J.L. 2009. Thermodynamical assessments to model steel concrete interface behaviour including corrosion effects, COMPLAS 2009, Barlecona, Spain. Rodriguez, J., Ortega, L.M. & Casal. J. 1997. Load carrying capacity of concrete structures with corroded reinforcement. Construction and Building Materials, 11(4): 239–248. Spacone, E., Filippou, F.C. & Taucer, F.F. 1996. Fibber beam—column model for nonlinear analysis of R/C frames. I: Formulation. Earthquake Engineering and Structural Dynamics, 25(7):711–725. Spacone, E. & Limkatanyu, S. 2000. Responses of reinforced concrete members including bond-slip effects. ACI Structural Journal, 97(6):831–839. Wang, X. & Liu, X. 2004. Modelling bond strength of corroded reinforcement without stirrups. Cement and Concrete Research, 34:1331–1339. Wang, X. & Liu, X. 2006. Bond strength modelling for corroded reinforcements. Construction and Building Materials, 20:177–186.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Simulation of shear load behavior of fifty year old post-tensioned concrete bridge girders A. Schweighofer Institute for Structural Engineering, Vienna University of Technology, Austria

M. Vill ÖBB Infastruktur Bau AG

H. Hengl & J. Kollegger Institute for Structural Engineering, Vienna University of Technology, Austria

ABSTRACT: At the beginning of the 1950s, the use of prestressed concrete for bridge structures rapidly increased. Many of these structures, which are still in use, have already reached their maximum service life. Therefore, the question arises whether sufficient load-carrying-capacity and serviceability (according to the state of the art) can still be ensured or not. In the course of the destruction of an underground railway station (Fig. 1), the ÖBB Bridge Department and the Institute for Structural Engineering at the Vienna University of Technology was given the opportunity to carry out tests concerning the shear load carrying capacity on four bridge girders, which were built in 1960. 1

STRUCTURE

The structure was designed as a single-span girder with a supporting distance of 17.5 m. The girders (T-beams) were removed during the demolition work of the railway station. All T-beams had the original length of 17.5 m, a plate width of 3.3 m and a height of 1.15 m–1.40 m. The plate was prestressed in the transverse direction at a distance of 75 cm with tendons St 75/105 Rg Ø 26 mm, the webs contained with 14 tendons St 75/105 Rg Ø 26 mm along the bridge axis. In the center part of the girders transverse reinforcement (stirrups in web of T-beam) consisted of Ø 10 mm with a distance of 25 cm and close o the supports bars Ø 12 mm with a distance of 25 cm were placed (Fig. 2). A recalculation of the structure according to the current standard showed that the design load (dead and life load) is approximately 40% higher than the shear resistance according to Eurocode 2 [EN 1992-1-1, 2009]. VEd = 1, 40 · VRds

Furthermore, to obtain information about the durability of the structure, the properties of the materials used—concrete, reinforcing steel, pre-stressing reinforcements—were analysed by testing samples of bridge material in the laboratory.

Figure 1.

Removing of the girders.

(1)

whereas is: VEd Shear force—Load model bridge class 1—ULS VRds Shear resistance of transverse reinforcement

Figure 2. Cross section and reinforcement of web close to the supports.

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2 2.1

LOAD TESTS Test Setup ‘‘Load Tests’’

In the load tests the bridge girders were supported on two wood bearings at intervals of 15 meters. On each side of the girders a bearing block with a height of 1.05 meters was concreted. Two steel girders were placed on these bearing blocks. The bridge and the steel girders were tensioned together with four traverses and 16 post tensioned threaded bars with a diameter of 32 mm, and thus a closed system without additional foundations was created. The schematic test setup is shown in Figure 3. The bridge structure was subjected to a four point bending by applying two loads by hydraulic jacks, which were gradually increasing to maximum load. Three Tests were carried out where the load distances were varied from 2.75 m up to 3.5 m distance to the bearings. 2.2

Figure 5.

Removed tendon with duct and grouting.

Figure 6.

Test setup for actual pretensioning force.

Test Setup ‘‘Post-Tensioning Force’’

The actual post-tensioning force was not sufficiently known due to shrinkage, friction losses, creep and relaxation after 50 years of service life. To get more information about these time dependent behavior a simple test setup was developed. On the one hand the results are essential for the numerical simulation and on the other hand the results could be compared to the calculated pretension losses according to EC2 [EN 1992-1-1, 2009]. Furthermore, we wanted to obtain information about the bond and the quality of the

Figure 3.

Schematic test setup.

grouting. Figure 5 shows a removed tendon including the duct and the grouting. To determine the actual pretension force the ducts were uncovered in five areas on two bridge girders. Afterwards the ducts were sliced and the tendons were laid open. In the areas 3, 4 and 5 (see Fig. 6) three strain gauges (SG) were applied on three different tendons. The dismantling of the anchors was carried out by a concrete crusher and afterwards the tension was removed by cutting through the tendons with a cutting torch in the order of area 1, 2, 3, 4 and 5 (see Fig. 5). The determined post-tensioning force was about 200 kN per tendon. In regard to the bond different results appeared as result of the different grout quality due to the small distance between the tendons and the duct.

3

NUMMERICAL SIMULATION

The calculations were made with the program ATENA [ATENA, 2007]. 3.1

Figure 4.

Materials

3.1.1 Concrete For the concrete, the already implemented sBeta material model [ATENA, 2007] was used (see Fig. 7). Drill

Test setup.

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Figure 7.

sBeta model of ATENA (Picture by ATENA).

bits were taken and tested by the Technische Versuchsund Forschungsanstalt GmbH (TVFA; Institute for Testing and Research) for the calculation of the cube compressive strength. The result was a cube compressive strength of fc = 70 N/mm2 . The tensile strength ft was calculated by the determined pretensioning force and the load during the onset of cracking and averaged about 2,7 N/mm2 . The elastic module was 38.000 N/mm2 . The type of tension softening model was exponential (see Fig. 8). The specific fracture energy Gf was calculated with 102 N/mm. The compressive strain at the compressive strength in uniaxial compressive test εc amounted to 0,289%. The reduction of the compressive strength due to cracks was 0,8. 3.1.2 Prestressing steel Tendon patterns were tested by the TVFA. The 0,2% yield stress was 780 N/mm2 and the elastic modulus was 210000 N/mm2 . The material property of the prestressing steel was modeled with the stress–strain diagram. The 0,2% yield stress was 780 N/mm2 . 3.1.3 Reinforcement steel The reinforcement steel was also tested by the TVFA. For the modeling of the material behaviors of the reinforcement steel a bilinear material model with hardening was used. A yield stress of fy = 450 N/mm2 and a elastic modulus of Es = 209500 N/mm2 was chosen. The input of the reinforcement was made with individual bars and not with the option of a smeared reinforcement. 3.2 Boundary conditions and loads The support for the load cases ‘‘post-tensioning’’ and ‘‘dead load’’ was chosen in the middle of the wooden sleepers. For the load case ‘‘load by hydraulic jacks’’ the bearings were displaced in the area of the traverses (see Fig 3 and 4). The loading of the bridge girders by the hydraulic jacks was modeled on the one hand by a prescribed deformation of 0,5 mm per load step and on the other hand by an applied load with 50 kN per load step. The area of the load introduction and the bearing areas were modeled with steel plates. A size of 0,05 m for each macro element was chosen.

Figure 8. ATENA).

Type of tension softening model (Picture by

Figure 9.

2D model of ATENA.

Fig. 9 shows the half bridge girder with the transverse reinforcement, the longitudinal bars and the tendons including the prestressing force of 400 kN for 2 tendons.

4

RESULTS OF THE NUMERICAL ANALYSIS AND COMPARISON WITH THE FIELD TESTS

Fig. 10 shows the load—deflection—diagram of load test 3 and of the ATENA simulation. The ATENA simulation and the field tests correspond very well. The field tests as well as the numerical simulations show a bending failure and no shear force failure. Although the yield stress in the transverse reinforcement was achieved, the girder did not fail in shear due to the dowel effects of the prestressing steel and the shear transfer in the concrete. That agreed with the measured strain in the transverse reinforcement. The failure occurred when the yield stress of the longitudinal bars and the tendons was achieved. During the load tests maximum loads of two times 2.64 MN were measured before failure occurred. The load tests showed big deflections of 120 mm in the middle of the span, significant cracks widths of 4 mm, and a ductile behaviour of the T-beams. Table 1 shows a comparison of the calculated loads to the measured loads. During the tests maximum loads of 2,64 MN were measured wheras the calculated resistance of the shear reinforcement according to Eurocode 2 is only 0,863 MN. Therefore, a global safety factor of 2,8 for this test was calculated.

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5

CONCLUSION

The numerical simulation with ATENA is in accordance with the load test. During the load tests maximum loads of two times 2,64 MN were measured before failure occurred. In the ATENA simulation the maximum load averages to 2,55 MN. The load tests as well as the simulation showed big deflections, significant crack widths and a ductile behaviour of the T-beams. Finally, additional load bearing mechanism beside the part of the shear reinforcement like tensile strength of the concrete or dowel effects of the prestressing steel must be accountable for the high maximum loads of the tests. The results of the test-series are an important basis for a goal-oriented maintenance planning for the ÖBB Bridge Stock, which includes many similar structures. Furthermore, similar structures, which are currently in use, showing good maintenance conditions must not be replaced due to calculated insufficient shear load resistance.

Figure 10.

Deflection under load.

Figure 11.

Crack pattern by ATENA.

ACKNOWLEDGEMENT We thank ÖBB Infastruktur Bau AG for providing funds for the accomplishment of these tests. Also we thank Ing. Dr. Radomir Pukl, from Cervenka Consulting, for the excellent support. REFERENCES

Figure 12.

Table 1. Standard

Crack pattern of test 3.

Relation of results. [kN]

γ [−]

Vk

950

∼1,0

VEd

1.332 ∼1,4

VRd,s

863

Description

ÖN B4002 Shear force – Load model bridge class 1 – SLS ÖN B4002 Shear force – Load bridge class model 1 – ULS EC2 Shear resistance of transverse reinforcement EC2 Shear resistance of the concrete TEST Shear force from load test 3 ATENA Shear force from ATENA ‘‘TEST 3’’

ATENA. 2007. ATENA Program Documentation. Cervenka Consulting, Praha 2007. Hengl, H. 2009. Großversuche an Spannbetonträgern. Master’s thesis, TU - Wien. Maurer, R. und Bäätjer, G. 2006. Entwicklung von Konstruktions- und Bemessungsgrundsätze in Deutschland im Hinblick auf die Tragsicherheit von Spannbetonbrücken, München Massivbau Seminar 2006. ÖNORM EN 1992-1-1:2009. Bemessung und Konstruktion von Stahlbeton- und Spannbetontragwerken, Österreichisches Normungsinstitut, Wien, 2009. Vill, M., Brunner, H., Torghele, H. and Kollegger, J. 2007. Schubversuche an ausgebauten Trägern einer Straßenbrücke aus Spannbeton, Heft 65, Schriftenreihe der österreichischen Vereinigung für Beton- und Bautechnik, Wien, 2007.

∼0,9

VRd,max 2.300 ∼2,4 Vtest−3

2.640 ∼2,8

VATENA 2.580 ∼2,7

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Multi-scale modelling of concrete beams subjected to three-point bending Ł. Skar˙zy´nski & J. Tejchman Gda´nsk University of Technology, Gda´nsk, Poland

ABSTRACT: The paper deals with formation of strain localization in notched concrete beams subjected to quasi-static three-point bending. Concrete was described as heterogeneous three-phase material. The calculations were carried out with the FEM using isotropic damage constitutive model enhanced by a characteristic length of micro-structure by means of a non-local theory. The numerical results were compared with own laboratory test results using a non-invasive optical technique called Digital Image Correlation (Skar˙zy´nski et al. 2009), the tests by Le Bellˇego et al. (2003) and the size effect law by Bažant (2004).

1

directly compared with corresponding laboratory test results with notched concrete beams (Skar˙zynski et al. 2009) where the width and shape of localized zones on the surface of notched concrete beams was determined with a Digital Image Correlation (DIC) method. In addition, our results were also compared with corresponding laboratory size effect tests by Le Bellˇego et al. (2003) with the same beam geometry and the size effect law by Bažant (Bažant & Planas 1998, Bažant 2004) for notched concrete elements.

INTRODUCTION

Fracture process is a fundamental phenomenon in cementitious materials (Bažant & Planas 1998, Lilliu & van Mier 2003). It is very complex since it consists of main cracks with various branches, secondary cracks and micro-cracks. It is a major reason of damage in concrete material under mechanical loading contributing to a significant degradation of material strength. Fracture is always preceded by the formation of localized zones of a certain width. To describe strain localization and other global properties in a plastic region, a material micro-structure has to be taken into account (Nielsen et al. 1995, Bažant & Planas 1998, Sengul et al. 2002, Lilliu & van Mier 2003, Kozicki & Tejchman 2008, He et al. 2009). In particular, the presence of aggregate is important since its volume fraction can be as high as 0.70–0.75 in concrete. The intention of our paper is to present the results of FE investigations of the width and shape of localized zones in geometrically similar notched concrete beams subjected to quasi-static three-point bending using a meso-scale model. The simulations were carried out with a simple isotropic continuum constitutive damage model (Marzec et al. 2007, Skar˙zy´nski et al. 2009) enhanced by a characteristic length of micro-structure by means of a non-local theory (Pijauder-Cabot & Bažant 1987, Bažant & Jirasek 2002, Bobi´nski et al. 2009). The model can realistically describe tensile failure at macro-level (Marzec et al. 2007, Skar˙zy´nski et al. 2009). Concrete beam was assumed at mesoscale as a random heterogeneous material composed of three phases: aggregate, cement matrix and bond (interface) surrounding each aggregate. The main intention of our simulations was to determine a characteristic length of micro-structure lc used in a continuum model and to identify lc with aggregate size and aggregate density. The numerical results were

2

EXPERIMENTS

The three-point bending laboratory tests were carried out with concrete beams of a different size D ×L (D = beam height, L = 4 × D = beam length) with free ends (Skar˙zy´nski et al. 2009). The beams were geometrically similar (Fig. 1): a) small-size beams had the size 80 × 320 mm2 , b) medium-size beams 160 × 640 mm2 and c) large-size beam 320 × 1280 mm2 . The thickness of beams was always the same b = 40 mm, and the beams’ span was equal to 3 × D. A notch with a height of D/10 mm was located at the mid-span of the beam bottom. The quasi-static deformation in beams was induced by a vertical displacement prescribed at the beam mid-span at a very slow rate. Two different concrete mixes were composed of ordinary Portland cement, water and sand (with a mean aggregate diameter d50 = 0.5 mm and maximum aggregate diameter dmax = 3.0 mm) or gravel (d50 = 2.0 mm, dmax = 8.0 mm). The width and shape of the localized zone above the notch on the surface of beams was determined with a Digital Image Correlation (DIC) method, which is an optical way to visualize two dimensional deformations by successive post-processing of digital photos taken at a constant time increment using a professional digital

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6 5.5 Fracture zone width [mm]

5

c b

4.5 a

4 3.5 3 2.5 2 1.5

(a) (b) (c)

1 0.5

A)

Figure 1. Geometry of experimental concrete beams subjected to three-point bending (Le Bellˇego et al. 2003).

0 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

4

4.5

5

u/D [-] 6 c

5.5 Fracture zone width [mm]

5 4.5 4

a

b

3.5 3 2.5 2 1.5

(a) (b) (c)

1 0.5

B) 0 0

0.5

1

1.5

2

2.5

3

3.5

u/D [-]

Figure 3. Evolution of width of localized zone in experiments using DIC versus normalized beam deflection u/D (u = beam deflection and D = beam height): A) gravel concrete, B) sand concrete, a) small-size beam 80 × 320 mm2 , b) medium-size beam 160 × 640 mm2 and c) large-size beam 320 × 1280 mm2 (Skar˙zynski et al. 2009).

3

Figure 2. Formation of localized zone directly above notch in experiments using DIC: gravel concrete medium size beam 160 × 640 mm2 (top) sand concrete medium size beam 160×640 mm2 (bottom); vertical and horizontal axes denote coordinates in [mm], colour scales denotes strain intensity (Skar˙zynski et al. 2009).

camera (White et al. 2003). The experimental set-up and results were described in detail by Skar˙zy´nski et al. (2009). The beams of the same size were also used by Le Bellˇego et al. (2003). Figure 2 shows the formation of the localized zone on the surface of concrete beams above the notch using different concrete mixes. The zone occurred before the peak on the force-deflection diagram and was strongly curved. In some cases, it branched. The width of the localized zone increased during deformation due to material dilatancy (Fig. 3). The maximum width of the localized zone was equal to 3.5 mm−5.5 mm. It did not depend upon the mix type and beam size.

CONSTITUTIVE MODEL FOR CONCRETE

A simple isotropic damage continuum model was used (Marzec et al. 2007) which describes the material degradation with the aid of only a single scalar damage parameter D growing monotonically from zero (undamaged material) to one (completely damaged material). A damage variable D is associated with a degradation of the material due to the propagation and coalescence of micro-cracks and micro-voids. It is defined as the ratio between the damage area and the overall material area (Katchanov 1986). The stress-strain relationship is represented by: e σij = (1 − D)Cijkl εkl ,

(1)

e = linear elastic material stiffness matrix where Cijkl (including modulus of elasticity E and Poisson’s ratio υ) and εkl = strain tensor.

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The damage parameter D acts as a stiffness reduction factor (the Poisson’s ratio is not affected by damage) that changes from 0 to 1. The growth of damage is controlled by a threshold parameter κ which is defined as the maximum equivalent strain measure ε reached during the load history up to time t. The loading function of damage is

was calculated also on the basis of Equation 6 (which satisfies the normalizing condition). As a weighting function ω, a Gauss distribution function was used

f (˜ε, κ) = ε˜ − max{κ, κ0 },

where lc = characteristic length of micro-structure and the parameter r = distance between two material points. The averaging in Equation 7 is restricted to a small representative area around each material point (the influence of points at the distance of r = 3 × lc is only 0.01%). A characteristic length is usually related to material micro-structure and is determined with an inverse identification process of experimental data (Le Bellˇego et al. 2003). However, the determination of a representative characteristic length of material micro-structure is very complex in concrete since strain localization can include a mixed failure mode (cracks and shear zones) and the characteristic length (which is a scalar value) is related to the fracture region with a certain volume increasing during deformation (Bažant & Jirasek 2002). It depends also on the choice of the weighting function. In turn, other researchers concluded that the characteristic length depended upon the boundary value problem only (Ferrara & di Prisco 2001). The FE calculations were carried out using a large-displacement analysis. The non-local averaging was performed in the current configuration. The deformation was induced by prescribing a vertical displacement at the mid-point of the beam top. Concrete on meso-scale was considered as a three-phase material encompassing cement matrix, aggregate and interfacial transition zones (ITZ) between cement matrix and aggregate (Fig. 4). Aggregate was randomly distributed in cement matrix according to a sieve curve. Similarly as in experiments, two different concrete mixes were analyzed: sand and gravel concrete (the size of aggregate inclusions varied from amin = 0.5 mm up to amax = 8 mm).

(2)

where κ = initial value of κ when damage begins. If the loading function f is negative, damage does not develop. During monotonic loading, the parameter κ grows (it coincides with ε˜ ) and during unloading and reloading it remains constant. To define the equivalent strain measure ε˜ , a Rankine failure type criterion was assumed (Jirasek & Marfia 2005) ε˜ =

  1 eff − max σi , E

(3)

i = principal where E = modulus of elasticity and σeff values of the effective stress eff

σi

e = σijkl εkl .

(4)

To describe the evolution of the damage parameter, the exponential softening law was used (Peerlings et al. 1998) D =1−

 κ0  1 − α + αe−β(κ−κ0 ) , κ

(5)

where α and β are the material constants. The constitutive isotropic damage model for concrete requires the following 5 constants: E, υ, κ0 , α and β. The model is suitable for tensile failure (Marzec et al. 2007). However, it cannot realistically describe irreversible deformations, volume changes and shear failure at macro-level (Simone & Sluys 2004). To properly describe strain localization, to preserve the well-possedness of the boundary value problem and to obtain mesh-independent results, a non-local theory was used as a regularization technique (Bažant & Jirasek 2002, Bobi´nski & Tejchman 2004). In this theory, the principle of a local action does not take place any more. In the calculations, the equivalent strain measure ε˜ was replaced by its non-local value (Pijauder-Cabot & Bažant 1987)  ε¯ =

Vω (x V

− ξ ) ε˜ (ξ )dξ , ω (x − ξ ) dξ



1 − ω(r) = √ e lc π

r lc

2

,

(7)

(6)

where V = the body volume, x = the coordinates of the considered material point, ξ = the coordinates of surrounding points and ω = the weighting function. The equivalent strain measure ε˜ near boundaries

Figure 4. Three-phase concrete in the neighbourhood of the notch: aggregate of round shape, cement matrix and interfacial transition zones (ITZ).

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Aggregate had mainly a circular shape. The width of ITZs was assumed to be 50 μm (He et al. 2009). Aggregate was generated according to the method given by Eckardt & Konke (2006). The circles simulating aggregate were randomly placed starting with the largest ones and preserving a certain mutual distance (van Mier et al. 1995)

D >1·1

D1 + D2 , 2

(8)

where D = distance between two neighboring particle centers and D1 , and D2 = diameters of two particles, respectively. The aggregate density was ρ = 30%, ρ = 45% or ρ = 60%. The FE-meshes included 12000–1600000 triangular elements. The calculations were carried out with one set of material parameters for usual concrete only which was prescribed to finite elements corresponding to a specified concrete phase (Tab. 1). The interface was assumed to be the weakest component (Lilliu and van Mier 2003). The size of finite elements was: 0.5 mm (aggregate), 0.5 mm (cement matrix) and 0.1 mm (bond). The following numerical calculation program was assumed. First, three beams of a different size were modeled to be totally homogeneous (as one-phase material). Afterwards, a small-size beam (80 × 320 mm2 ) of gravel concrete was modeled: as a partially homogeneous and partially heterogeneous with a meso-section in the notch neighbourhood and as an entirely heterogeneous beam at meso-scale. The width of a heterogeneous meso-scale section bms varied between D/2 (40 mm) and D (80 mm) (D = beam height). These analyses allowed us to determine a representative width of a required heterogeneous region close to the notch (it was assumed to be always equal to the beam height bms = D). Next, the effect of different parameters was studied in a small-size beam. Finally, calculations were carried out with partially heterogeneous beams of a different size to determine a deterministic size effect. Three-five different stochastic realizations were usually performed for the same case.

4

NUMERICAL RESULTS (MACRO-SCALE APPROACH)

Figure 5 presents the numerical results of the nominal strength σn = 1.5Pl/bD2 of three different concrete beams versus the normalized deflection u/D, where P = vertical force, u = beam deflection, D = beam height, b = beam width, l = 3 × D = beam span) as compared to laboratory tests by Le Bellˇego et al. (2003). Concrete was treated as an entirely homogeneous one phase-material with the following material constants: E = 38500 MPa, υ = 0.2, κ0 = 1.3×10−4 , α = 0.95, β = 400 and lc = 2 mm. Totally, 12000–92000 triangular elements were assumed. In turn, Figure 6 shows the distribution of a non-local softening strain measure in beams. The numerical results of strength are in a satisfactory agreement with tests by Le Bellˇego et al. (2003). The deterministic size effect is realistically described (nominal strength and material ductility increase with decreasing beam size). The width of the localized zone above the notch is about wc = 6.0 mm (at u/D = 0.5) and approximately corresponds to the measured maximum value

Figure 5. Calculated and experimental nominal strength 1.5Pl/(bD2 ) versus normalised beam deflection u/D: A) FE results, B) experiments by Le Bellˇego et al. (2003), 1) smallsize beam, 2) medium-size beam, 3) large-size beam (homogeneous one-phase material, lc = 2 mm).

Table 1. Material properties assumed for FE calculations of concrete beams on meso-scale. Parameters

Aggregate

Cement

ITZ

E [GPa] υ [−] κ0 [−] α [−] β [−]

40 0.2 0.5 0.95 200

35 0.2 1 × 10−4 0.95 200

40 0.2 7 × 10−4 0.95 200

Figure 6. Distribution of non-local strain measure above notch from numerical calculations (at u/D = 0.5) in small, medium and large beam (homogeneous one-phase material, lc = 2 mm).

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of wc (5.5 mm) by DIC (Fig. 3B). However, in contrast to experiments, the calculated localized zone above the notch is always straight.

(a) (b) (c)

3 Force [kN]

5

4 3.5

NUMERICAL RESULTS (MESO-SCALE APPROACH)

2.5 2 1.5

(c)

1

5.1

(b)

Effect of aggregate density

0.5

The effect of the aggregate distribution on the loaddeflection diagram and strain localization is shown in Figures 7–10 (beam size 80 × 320 mm2 ) with material constants from Table 1 (lc = 1.5 mm). The aggregate density was ρ = 30% or ρ = 45% for sand concrete (d50 = 0.5 mm and dmax = 3 mm) and for gravel concrete (d50 = 2 mm, dmax = 8 mm), respectively. The load-displacement curves are the same in an almost entire elastic regime independently of the distribution of inclusions. However, they can be significantly different after the peak is reached (Figs 7 and 9) due to a localized zone propagating between aggregate distributed at random. The localized zone is always non-symmetric and curved. The width of the calculated localized zone is approximately wc = (4 × lc ) = 6 mm (ρ = 30%) and wc = (3 × lc ) = 4.5 mm (ρ = 45%) independently of d50 . The calculated localized

0

0.1

0.15

Figure 9. Calculated load-deflection curves for sand concrete (d50 = 0.5 mm, dmax = 3 mm) and three random distributions of aggregate (curves ‘a’, ‘b’, ‘c’) in a small-size beam 80 × 320 mm2 (lc = 1.5 mm, ρ = 45%).

a)

b)

c)

Figure 10. Calculated distribution of non-local strain measure for sand concrete (d50 = 0.5 mm, dmax = 3 mm) and three random distributions of aggregate (curves ‘a’, ‘b’, ‘c’) in a small-size beam 80×320 mm2 (lc = 1.5 mm, ρ = 45%).

(b) (a) (b) (c)

(a)

3 Force [kN]

0.05 Deflection [mm]

4 3.5

(a)

0

zone is created at u/D = 0.15 and its width linearly increases during deformation (as in the experiment).

2.5 (c)

2 1.5

5.2

1 0.5 0 0

0.05

0.1

0.15

Deflection [mm]

Figure 7. Calculated load-deflection curves for gravel concrete (d50 = 2 mm, dmax = 8 mm) and three random distributions of aggregate (curves ‘a’, ‘b’, ‘c’) in a small-size beam 80 × 320 mm2 (lc = 1.5 mm, ρ = 30%).

a)

b)

c)

Figure 8. Calculated distribution of non-local strain measure for gravel concrete (d50 = 2 mm, dmax = 8 mm) and three random distributions of aggregate (curves ‘a’, ‘b’, ‘c’) in a small-size beam 80×320 mm2 (lc = 1.5 mm, ρ = 30%).

Effect of aggregate size and aggregate density

Figures 11–12 demonstrate the effect of the aggregate size and aggregate density in sand concrete (d50 = 0.5 mm and dmax = 3 mm) for a small-size beam (80 × 320 mm2 ) using aggregate density ρ = 30%, ρ = 45% and ρ = 60% (lc = 1.5 mm). With increasing maximum aggregate and aggregate density, beam strength slightly increases (by 10–25%, Fig. 11). The width of the localized zone does not depend on the maximum aggregate size dmax (Figs. 11 and 12). This outcome is in contrast to statements by Pijauder-Cabot and Bažant (1987), and Bažant and Oh (1983) wherein the width of the localized zone was estimated to be about 3 × dmax . It is also in contrast to experimental results by Mihashi and Nomura (1996) which have shown that the width of the localized zone in the case of normal concrete increases with increasing aggregate size. The width of the localized zone is influenced by the aggregate density; the localized zone becomes narrower with increasing aggregate density: wc = 6 mm at ρ = 30%, wc = 4.5 mm at ρ = 45%,

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4.5

6

4

5.5

(a) (b) (c) (d) (e)

5 3.5

4.5 (a) (b) (c)

4 Force [kN]

Force [kN]

3 2.5 2 (b)

1.5

(c)

3.5 3 2.5 (c)

2

1

(a)

0.5 0

0.05

0.1

(e)

(a)

0.5

0

(d)

(b)

1.5

1

0

0.15

0

0.05

Deflection [mm]

Figure 11. Calculated load-deflection curves (small-size beam 80 × 320 mm2 ) for sand concrete (d50 = 0.5 mm, dmax = 3 mm), a) ρ = 30%, b) ρ = 45%, c) ρ = 60%.

0.1 Deflection [mm]

0.15

0.2

Figure 13. Calculated load-deflection curves for different characteristic lengths: a) lc = 0.1 mm, b) lc = 0.5 mm, c) lc = 1.5 mm, d) lc = 2.5 mm, e) lc = 5.0 mm (smallsize beam 80 × 320 mm2 , sand concrete d50 = 0.5 mm, dmax = 3 mm, aggregate density ρ = 30%).

A) a)

b)

c)

Figure 12. Calculated distribution of non-local strain measure for different aggregate densities (small-size beam 80 × 320 mm2 ): a) ρ = 30%, b) ρ = 45%, c) ρ = 60%, A) sand concrete (d50 = 0.5 mm, dmax = 3 mm), B) gravel concrete (d50 = 2 mm, dmax = 8 mm).

wc = 3 mm at ρ = 60%). The calculated results of wc compare will with the experimental ones (Fig. 3). The shape of the localized zone is affected by d50 (dmax ) and ρ (in particular for ρ = 60%). The width of the localized zone is influenced by the aggregate density: the localized zone becomes narrower with increasing aggregate density: wc = 6 mm at ρ = 30%, wc = 4.5 mm at ρ = 45%, wc = 3 mm at ρ = 60%). The calculated results of wc compare quite well with the experimental ones (Fig. 3). The shape of the localized zone is affected by both d50 (dmax ) and ρ (in particular for ρ = 60%). 5.3

c)

d)

e)

Figure 14. Calculated distribution of non-local strain measure for different characteristic lengths: a) lc = 0.1 mm, b) lc = 0.5 mm, c) lc = 1.5 mm, d) lc = 2.5 mm, e) lc = 5.0 mm (small beam 80 × 320 mm2 , sand concrete d50 = 0.5 mm, dmax = 3 mm, ρ = 30%).

B) a)

b)

Effect of characteristic length of micro-structure

of load-deflection curves for sand concrete (d50 = 0.5 mm and dmax = 3 mm) with aggregate density ρ = 30%. In turn, Figure 14 presents the distribution of a non-local softening strain measure above the notch for various lc changing between 0.1 mm and 5.0 mm. With increasing characteristic length, both beam strength and width of the localized zone obviously increase. The material softening decreases and material becomes more ductile. A pronounced deterministic size effect occurs. The localized zone propagating in a cement matrix between aggregate grains is strongly curved at lc = 0.1 − 2.5 mm, whereas it becomes more straight at lc > 2.5 mm. The width of the localized zone, wc = 4.5 − 6.0 mm = (3 − 4) × lc with lc = 1.5 mm and ρ = 30% (lc = 0.5 × dmax for sand concrete) is in good accordance with the experimental width of the fracture process zone on the surface of notched beams (3.5–5.5 mm, Fig. 3). 5.4

The effect of the characteristic length of microstructure on the force-deflection diagram and strain localization is shown in Figures 13 and 14 using the same stochastic distribution of aggregate. Figure 13 demonstrates the influence of lc on the evolution

Effect of beam size

The effect of the beam size is presented in Figures 15 and 16. Figure 15 shows the numerical results of the nominal strength σn = 1.5 Pl/bD2 versus the normalized deflection u/D for three different concrete beams compared to tests by Le Bellˇego et al. (2003). In turn,

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4.5

5 (a) (b) (c) (d) (e)

4

4.5 1

3

(A) (B)

1

1.5Pl/bD2 [MPa]

2

1.5Pl/(bD ) [MPa]

3.5

2.5 2

2 3

1.5

2

3

1

4

3.5

3

0.5

2.5

0 0

0.2

0.4

0.6 0.8 u/D [-]

1

1.2

0

1.4

0.1

0.2

0.3

0.4

D [m]

Figure 15. Calculated nominal strength 1.5Pl/(bD2 ) versus normalised beam deflection u/D (u = beam deflection, D = beam height): A) FE—results, B) experiments by Le Bellˇego et al. (2003): 1) small-size beam, (2) medium-size beam, (3) large-size beam (three-phase random heterogeneous material close to notch, bms = D).

Figure 17. Calculated and measured size effect in nominal strength 1.5Pl/(bD2 ) versus beam height D for concrete beams of a similar geometry (small-, medium—and large-size beam): a) our laboratory experiments, b) our FEcalculations (homogeneous one-phase material), c) our FEcalculations (heterogeneous material close to notch, bms = D), d) size effect law by Bažant (2004), e) experiments by Le Bellˇego et al. (2003).

where ft = tensile strength, B = dimensionless parameter depending upon the structure geometry and crack, D = beam height and D0 = size dependent parameter (called transitional size). The experimental and numerical results match quite well the size effect law by Bažant (Bažant and Planas 1998).

Figure 16. Calculated distribution of non-local strain measure above notch from numerical calculations (at u/D = 0.5) in small-, medium—and large-size beam (random heterogeneous three-phase material close to notch, bms = D).

Figure 16 presents the distribution of a non-local softening strain measure in beams. The calculations were carried out with gravel concrete of dmax = 8 mm, aggregate density of ρ = 30% and characteristic length of lc = 1.5 mm. The numerical results are in a satisfactory agreement with tests by Le Bellˇego et al. (2003). The width of the localized zone above the notch at u/D = 0.5 is 6 mm (ρ = 30%) for all beam sizes. The localized zone propagating between aggregate is always strongly curved, what satisfactorily reflects the experimental results (Fig. 2). Figure 17 shows a comparison between the measured and calculated size effect for concrete beams. In addition, the results of a deterministic size effect law by Bažant (Bažant and Planas 1998, Bažant 2004) are enclosed (which is valid for structures with preexisting notches) wherein the nominal strength is calculated as: Bft σn = √ , 1 − (D/D0 )

(9)

6

CONCLUSIONS

A meso-scale numerical model was used in this study to analyze strain localization in concrete. The FEcalculations reveal that an isotropic damage continuum model enhanced by a characteristic length of micro-structure is able to capture the evolution of strain localization in notched concrete beams under quasi-static three-point bending where concrete was treated as a heterogeneous three-phase material. The following conclusions can be drawn: – material micro-structure on meso-scale has to be taken into account in calculations of strain localization to obtain a proper shape of the localized zone, – the calculated strength, width and geometry of the localized zone are in a satisfactory agreement with experimental measurements, when the characteristic length is about 1.5 mm, – the width of the localized zone above the notch is about (2−4)×lc . It increases with decreasing aggregate density from 2 × lc (ρ = 60%) up to 4 × lc (ρ = 30%). It increases also with increasing characteristic length. It is not affected by the aggregate size and beam height, – beams strength increases with increasing characteristic length, aggregate density and decreasing beam height. It depends also on the aggregate distribution,

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– material softening is strongly influenced by the characteristic length, aggregate density and beam height, – the localized zone above the notch is strongly curved with lc = 1.0–2.5 mm, – the characteristic length of micro-structure may be also related to the grain size of cement matrix. Our calculations at meso-scale will be continued. First, 3D studies will be performed to capture more realistically micro-structure of concrete. The effect of the grain size in cement matrix will be also analyzed to identify it with a characteristic length of microstructure. Next, a multi-scale approach will be implemented to reduce computation time. Both scales will be linked by means of a Coupled Volume multi-scale approach (Gitman et al. 2007, 2008), where the size of a macro-element equals the size of a meso-cell (to avoid the assumption of any size of RVE). REFERENCES Bažant, Z.P. & Oh, B.H. 1983. Crack band theory for fracture of concrete. Materials and Structures, RILEM 16, 155–177. Bažant, Z. & Planas, J. 1998. Fracture and size effect in concrete and other quasi-brittle materials. CRC Press LLC, Boca Raton. Bažant, Z.P. & Jirasek, M. 2002. Non-local integral formulations of plasticity and damage: survey of progress. Journal of Engineering Mechanics, 128(11), 1119–1149. Bažant, Z. 2004. Probability distribution of energeticstatistical size effect in quasibrittle fracture. Probabilistic Engineering Mechanics, 19, 307–319. Bobi´nski, J. & Tejchman, J. 2004. Numerical simulations of localization of deformation in quasi-brittle materials with non-local softening plasticity. Computers and Concrete 4, 433–455. Bobi´nski, J., Tejchman, J. & Górski, J. 2009. Notched concrete beams under bending—calculations of size effects within stochastic elasto-plasticity with non-local softening. Archives of Mechanics, 61(3–4), 1–25. Eckardt, S. & Konke, C. 2006. Simulation of damage in concrete structures using multi-scale models. Computational Modelling of Concrete Structures, EURO-C (G. Meschke, R. de Borst, H. Mang and N. Bicanic, eds.), Taylor and Francis, 77–89. Ferrara, I. & di Prisco, M. 2001. Mode I fracture behaviour in concrete: non-local damage modeling. ASCE Journal of Engineering Mechanics, 127(7), 678–692. Gitman, I.M., Askes, H. & Sluys, L.J. 2007. Representative volume: Existence and size determination. Engineering Fracture Mechanics, 74, 2518–2534. Gitman, I.M., Askes, H. & Sluys, L.J. 2008. Coupled-volume multi-scale modelling of quasi-brittle material. European Journal of Mechanics A/Solids, 27, 302–327.

He, H., Guo, Z., Stroeven, P., Stroeven, M. & Sluys, L.J. 2009. Influence of particle packing on elastic properties of concrete. Proc. First International Conference on Computational Technologies in Concrete Structures (CTCS’ 09), Jeju, Korea, 1177–1197. Jirasek, M. & Marfia, S. 2005. Non-local damage model based on displacement averaging. International Journal for Numerical Methods in Engineering, 63, 77–102. Katchanov, L.M. 1986. Introduction to continuum damage mechanics, Dordrecht: Martimus Nijhoff. Kozicki, J. & Tejchman, J. 2008. Modeling of fracture processes in concrete using a novel lattice model. Granular Matter, 10(5) 377–288. Le Bellˇego, C., Dube, J.F., Pijaudier-Cabot, G. & Gerard, B. 2003. Calibration of nonlocal damage model from size effect tests. European Journal of Mechanics A/Solids, 22, 33–46. Lilliu, G. & van Mier, J.G.M. 2003. 3D lattice type fracture model for concrete. Engineering Fracture Mechanics, 70, 927–941. Marzec, I., Bobi´nski, J. & Tejchman, J. 2007. Simulations of crack spacing in reinforced concrete beams using elasticplastic and damage with non-local softening. Computers and Concrete, 4, 377–403. Mihashi, H. & Nomura, N. 1996. Correlation between characteristics of fracture process zone and tension softening properties of concrete. Nuclear Engineering and Design, 165, 359–376. Nielsen, A.U., Montiero, P.J.M. & Gjorv, O.E. 1995. Estimation of the elastic moduli of lightweight aggregate. Cement and Concrete Research, 25(2), 276–280. Peerlings, R.H.J., de Borst, R., Brekelmans, W.A.M. & Geers, M.G.D. 1998. Gradient enhanced damage modelling of concrete fracture. Mechanics of Cohesive— Frictional Materials, 3, 323–342. Pijauder-Cabot, G. & Bažant, Z.P. 1987. Non-local damage theory. ASCE Journal of Engineering Mechanics, 113, 1512–1533. Sengul, O., Tasdemir, C. & Tasdemir, M.A. 2002. Influence of aggregate type on mechanical behaviour of normal– and high-strength concretes. ACI Master Journal, 99(6), 528–533. Simone, A. & Sluys, L. 2004. The use of displacement discontinuities in a rate—dependent medium. Computational Methods in Applied Mechanics Engineering, 193, 3015–3033. Skar˙zy´nski, £. Syroka, E. & Tejchman, J. 2009. Measurements and calculations of the width of fracture process zones on the surface of notched concrete beams. Strain, DOI: 10.1111/j.1475-1305.2008.00605.x. van Mier, J.G.M., Schlangen, E. & Vervuurt, A. 1995. Lattice type fracture models for concrete. Continuum Models for Material and Microstructure (H.B. Muhlhaus, ed.), John Wiley & Sons, 341–377. White, D.J., Take, W.A. & Bolton, M.D. 2003. Soil deformation measurement using particle image velocimetry (PIV) and photogrammetry. Geotechnique, 53, 619–631.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Experimental and numerical analysis of reinforced concrete corbels strengthened with fiber reinforced polymers R.A. Souza Universidade Estadual de Maringá, Brazil

ABSTRACT: Nonlinear analysis based on the finite element method can be seen as a powerful virtual laboratory, capable of predicting structural behavior of many complex reinforced concrete structures. In the present paper, nonlinear analysis have been applied in order to predict the behavior of a simple reinforced concrete double corbel as well as a reinforced concrete double corbel strengthened with fiber reinforced polymer (FRP) and designed using a strut-tie-model. The reinforced concrete corbels have been tested experimentally and the obtained results were compared with those results obtained using two-dimensional and three-dimensional finite element models. Based on different constitutive models it was realized that nonlinear analysis is highly dependent on the user’s experience as well as in the potentiality of the selected constitutive models for treating cracks in concrete. 1

INTRODUCTION

At the beginning of the XX Century, Ritter and Mörsch introduced the ‘‘Truss Analogy’’, one of the most brilliant ideas developed concerning the design of structural concrete. The ‘‘Truss Analogy’’ was refined in the 1960s and many researchers have enhanced the available design models, contributing for the creation of solid background based on the ‘‘Theory of Plasticity’’. Since that, ‘‘Truss Analogy’’ has been facing a great evolution and its generalization has become known as ‘‘Strut-and-Tie Models’’. Despite the fact that ‘‘Strutand-Tie Models’’ was introduced many years ago by CEB-FIP Model Code (1978) and CSA (1984), this theory only obtained appropriated attention after the publication of the seminal papers by Marti (1985a, 1985b) and Schlaich et al. (1987). ‘‘Strut-and-Tie Model’’ has as principal idea the substitution of the real structure by a truss form resistant structure, which simplifies the original problem in a systematical way, as shown in Figure 1. In these hypothetical trusses, the compressive concrete elements are denominated struts, while the tensile steel elements are referred as ties. The points of intersection between struts and ties, i.e., the points where there is a distribution of forces, are referred as ‘‘nodal regions’’. The stress level established in the nodal regions, as well as in the struts, should be limited to a certain value of the compressive concrete strength, in a way of avoiding cracks or premature failure. However, there is a great difficult for establishing this level of stress, taking into account the diversity of possibilities regarding the geometry of nodal zones, as well as struts.

Nowadays, many papers have been demonstrating the potentiality of ‘‘Strut-and-Tie Models’’, as for example Schlaich (1991), MacGregor (1991), ASCEACI (1998) and Schäfer (1999). Besides that, many structural codes worldwide have included important procedures about this method, as for example: CEBFIP Model Code 1990 (1993), CSA (1994), EHE (1999) and ACI-318 (2005). Schlaich et al. (1987) have proposed the idea of subdividing a structure in ‘‘B-Regions’’ and ‘‘D-Regions’’, in order to introduce rational procedures for design reinforced/prestressed concrete structures. ‘‘B-Regions’’ follow the ‘‘Bernoulli Hypothesis’’, i.e, the hypothesis of linear deformations can be assumed through the whole cross section, since the beginning of the loading until the failure of the section. ‘‘B-Regions’’ correspond to linear members, where the stress state can be considered continuous and uniform. Also, the deformations generated by bending and axial forces are predominant when compared to the deformations caused by shear forces. For ‘‘B-Regions’’ the usual design procedures based on the ‘‘Beam Theory’’ may be applied.

Figure 1. Examples of ‘‘Strut-and-Tie Models’’: (a) Double corbel, (b) Beam and (c) Deep-Beam.

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For ‘‘B-Regions’’, the tensile force in the longitudinal reinforcement varies throughout the structural element, in order to balance the applied bending moment, keeping the internal level arm relatively constant. By another hand, the tensile force in the longitudinal reinforcement for ‘‘D-Regions’’ is likely to be constant, the internal level arm experiences some variation and the structure presents an ‘‘arch action’’ behavior. ‘‘D-Regions’’ presents non-linear deformations throughout the cross section and the usual design procedures based on ‘‘Beam Theory’’ become inadequate and even unsafe whether they are applied. In these regions, usually some details of a structure, there is a complex stress state mainly generated by shear deformations. As examples of ‘‘D-Regions’’ the following situations can be mentioned: pile caps, footings, deep beams, corbels, dapped end beams and prestressed anchorages. Generally, ‘‘D-Regions’’ are produced by static (loadings) and/or geometric perturbations, and the length of these discontinuity regions may be found using the Saint Venant Principle. The zones of dissipation of perturbations are usually defined based on the height of the member, as shown in the examples of Figure 2. For ‘‘D-Regions’’, usually designed in the past using rule-of-thumb rules, ‘‘Strut-and-Tie Models’’ are desirable to apply, once this method can provide a rational, safe and systematic procedures for design. However, for some structures, defining a strut-and-tie model is not so straightforward and for these situations the ‘‘Finite Element Method’’ may be an excellent tool. ‘‘Finite Element Method’’ may be used for defining the resistant truss based on the obtained flow of stresses in the interior of a structure. Also, non-linear analysis based on the cracking of concrete and yielding of steel reinforcement may be applied in order to verify the proposed detailing. In that way, ‘‘Finite Element Method’’ may act as an efficient virtual laboratory, providing interesting gain regarding time, money and safety. Into this context, the present paper aims at presenting the design of reinforced concrete double corbels using the ‘‘Strut-and-Tie Model’’. The experimental results of these reinforced corbels and their strengthening model with near surface mounted Fiber Reinforced Polymers (FRP) strips are compared with numerical

Figure 2. Examples of ‘‘D-Regions’’ in concrete structures (Source: ACI-318 (2005)).

results, in order to investigate the performance of ‘‘Strut-and-Tie Models’’ and nonlinear analysis based on the ‘‘Finite Element Method’’.

2

CORBEL’S DESIGN USING A STRUT-AND-TIE MODEL

In order to evaluate the near surface mounted FRP strengthening technique, two double corbels were designed and tested. The first specimen, denominated D-Corbel A, has principal reinforcement constituted by 3 (three) bars of 8.0 mm of diameter. The second corbel, denominated D-Corbel B, has principal reinforcement constituted by 3 (three) bars of 5.0 mm of diameter plus 2 (two) FRP rectangular strips of 2 mm × 8 mm (0.32 cm2 ). These two corbels were designed to support the same ultimate loading, intending to simulate a real situation of strengthening for the main tie. As one can observe in Figure 3(a), there are many alternatives of strut-and-tie models available for double corbels, as suggested by Singh et al. (2005) However, a very simple model was selected for the double corbels with dimensions shown in Figure 3(b). The selected strut-and-tie model is very similar to that one used for design two-pile caps. Basically, the main tie was designed considering a characteristic point load (P) of 100 kN for both situations tested. Also, the strength of the diagonal struts, nodal regions and bearing areas were verified using the recommendations of Schäfer & Schlaich (1991). The FRP trips used had ultimate characteristic tensile strength of 3900 MPa, elastic modulus of 240 GPa. However, for design purposes it was assumed a maximum deformation of 0.006 which conducts to a maximum design strength of 1440 MPa. The steel bars used in the specimens had ultimate characteristic yielding strength of 500 MPa and concrete was specified to have a characteristic compressive strength of 20 MPa. Additionally, strength factors (1.4 for concrete and 1.15 for steel) as well as load factors (factor of 1.54 for the point loading) were applied through the design process.

Figure 3. (a) Alternatives of strut-and-tie models for double corbels suggested by Singh (2005) and (b) Dimensions of the double-corbels tested experimentally.

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The double corbels were reinforced as shown in Figure 4. As can be seen, it was not detailed vertical reinforcements (stirrups) neither skin reinforcements, once there was a great interest about the effective necessity of these reinforcements, usually prescribed in the structural codes for corbels. The vertical column, necessary to apply the vertical point load, was reinforced with 4 longitudinal bars of 10.0 mm of diameter and closed stirrups of 5.0 mm of diameter spaced each 10 cm.

3

EXPERIMENTAL RESULTS

The specimens were tested after 28 days of casting, using the equipments available at the Structural Laboratory of the University of Maringá. The concrete deformations in the direction of the diagonal struts were registered, as well as the steel deformations for the main ties. The point load was applied at the top of the column by a load cell with capacity of 300 kN and the experimental results were stored using a data system connected to a desktop computer. D-Corbel A was monotonically loaded and for a point load of 69.85 kN the first cracks appeared in the lateral bottom/top face of the specimen, in the link region of the column and its sides. The first cracks developed slowly from the top until reach the bearing areas following the same inclination of the concrete struts. The yielding of the main reinforcement occurred for a point load of approximately 187.5 kN, while the failure load was reached for a point load of about 217.08 kN. The failure was caused by the rupture of a diagonal concrete strut and some localized damage effects acting in the region of the bearings were realized. D-Corbel B, strengthened with FRP strips, was also loaded monotonically and for a point load of 95.58 kN the initial cracking of the structure was registered. The first cracks also appeared in the lateral bottom/top face of the double corbel and they did not developed following the diagonal struts. The yielding of the main reinforcement occurred for a point load of approximately 125.00 kN, while the failure load was reached for a point load of about 257.97 kN. The failure was caused by a direct shear in the interface between the column and the sides of the double corbel. Table 1 shows the main properties obtained for the materials and Table 2 shows the main experimental results obtained for the double corbels. Figure 5 shows

Table 1.

Material properties for the tested corbels.

Table 2. corbels.

Cracking, yielding and failure loads for the tested

Figure 4. Details of reinforcements used for (a) D-Corbel A and (b) D-Corbel B. Dimensions in cm.

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Table 3. Numerical results obtained for D-Corbel A using a two-dimensional approach.

(a)

(b)

Figure 5. Crack propagation for (a) D-Corbel A and for (b) D-Corbel B.

the crack propagation and the failure configurations for the double corbels tested. As one can be seen, for the D-Corbel A there was a tendency of failure following the direction of the diagonal struts. By another hand, the D-Corbel B presented a direct shear failure, occurred in the interface between the column and one corbel extension. 4

NUMERICAL RESULTS

Nonlinear analysis can be applicable even to predict the ultimate limit state as well as to forecast the service limit state, contributing for the production of more economical and safer structures. Another great advantage of the nonlinear analysis is to provide an estimative of strength for damaged structures or structures which need to be strengthened for some reason. For modeling the concrete behavior of the tested double corbels, a fracture-plastic model based on the classical orthotropic smeared crack formulation implemented by Cervenka & Cervenka (2003, 2005) in the package software ATENA 2D and 3D was applied. The choice for this package software was made based on the previous good results obtained by Souza et al (2007) regarding the simulation of complex reinforced concrete structures constituted by ‘‘D-Regions’’. Reinforcements were modeled using an embedded formulation and different solution methods based on the ‘‘Newton-Raphson Method’’ and ‘‘Arc Length Method’’ were applied for the solution scheme. Boundary conditions and material properties were defined in order to accurately represent the experimental setup and the overall response was recorded using monitoring points. 4.1

Two-dimensional analysis using ATENA 2D

Many possibilities available in the selected package software were used for the bi-dimensional and three-dimensional simulations conducted. The results obtained for D-Corbel A, using bi-dimensional resources are presented in more details in Table 3.

As one can observe in Table 3, two smeared crack models were used in the two-dimensional simulations: ‘‘Rotating Crack Model’’ and ‘‘Fixed Crack Model’’. As mentioned by Feenstra & Borst (1993), Rots et al. (1985) and Rots & Blaauwendraad (1989), a ‘‘Rotating Crack Model’’ usually conducts to lower failure loads when compared to a solution using a ‘‘Fixed Crack Model’’. As can be seen in Table 3, this information is really true, but the shear retention factor (beta parameter) needs to be adequately defined for the ‘‘Fixed Crack Model’’. In the two-dimensional simulations, ‘‘NewtonRaphson Method’’ and ‘‘Arc Length Method’’ were applied as iterative schemes for elimination of unbalanced forces and restoring equilibrium state. ‘‘Line Search Technique’’ can be used in combination with both of them to accelerate a convergence rate. More information about these methods can be obtained in Cervenka & Cervenka (2003, 2005) and Crisfield (1983). Many researchers have been recommending the use of ‘‘Fixed Crack Models’’ for situations where shear force is the major problem. However, for the conducted simulations, the ‘‘Rotating Crack Model’’ using the ‘‘Newton-Raphson Method’’ improved by ‘‘Line Search Technique’’ conducted to the best response, even quantitative (cracking, yielding and failure loads) as well as qualitative (crack pattern), as shown in Figure 6(a). In this figure one should observe that cracks with different widths are presented. Kotsovos & Pavlovic (1995) concluded that reinforced concrete structures seem to be insensible to the shear retention factor when this parameter is defined between 0.1 and 0.5. Also, the mentioned authors

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Figure 6. Cracking pattern at failure for (a) D-Corbel A and (b) D-Corbel B. Table 4. Numerical results obtained for D-Corbel B using a two-dimensional approach.

found that some numerical problems may arise for shear retention factors defined in the extreme range, i.e, 0.01 and 0.9. For the conducted simulations there were no major problems defining these extreme values. However, the failure loads were cleared affected by the shear retention factor, as one can see in Table 3. Rots et al. (1985) concluded that high values defined for the shear retention factor, simulating rough crack faces, may lead to more distinctive cracks acting in narrow strips. By another hand, for low values of the shear retention factor, simulating crack faces without friction, it is not possible to obtain diagonal cracking patterns. In fact, when the shear retention factor is defined by a small number in the ‘‘Fixed Crack Model’’, the obtained results will tend to be similar to those results obtained using a ‘‘Rotating Crack Model’’, as one can verify in Table 3. For the D-Corbel B, the same characteristics used for D-Corbel A were used, with the exception of the FRP strips which were included and the main steel reinforced reduced. Table 4 presents the obtained numerical results using a two-dimensional approach.

Again, a great variation among the chosen possibilities was registered, strongly indicating the necessity of introduce a probabilistic component for numerical simulations. As can be realized, the numerical results were not very close to those results obtained experimentally, but the cracking pattern is very close to the experimental situation. Again, the ‘‘Rotating Crack Model’’ using the ‘‘Newton-Raphson Method’’ improved by ‘‘Line Search Technique’’ lead to the best response, even quantitative (cracking, yielding and failure loads) as well as qualitative (crack pattern), as shown in Figure 6(b). 4.2

Three-dimensional analysis using ATENA 3D

For the three-dimensional simulations, the material properties presented in Table 1 were maintained and brick elements were selected in order to construct the finite element model. A three-dimensional nonlinear rotating crack model (‘‘CC3DNonLinCementitious’’) was selected to work together with a solution scheme based on the ‘‘Newton-Rapshon Method’’ enhanced by the ‘‘Line Search Technique’’. Cervenka & Cervenka (2003, 2005) describes the ‘‘CC3DNonLinCementitious Model’’ as fractureplastic model that combines constitutive models for tensile (fracturing) and compressive (plastic) behavior. The fracture model is based on the classical orthotropic smeared crack formulation and crack band model. It employs ‘‘Rankine Failure Criterion’’, ‘‘Exponential Softening’’, and it can be used as rotated or fixed crack model. Loads were applied at the top of the column and the boundary conditions were defined in order to respect the experimental procedures. For the point load of 68.40 kN the cracking of the D-Corbel A in the bottom face was observed, as shown in Figure 7(a). The yielding of the main tie was registered for a point load of 267.10 kN while the failure load was 278 kN. The failure of the double corbel A occurred by excessive concrete deformations, however the cracking pattern was very complex as shown in Figure 7(b). In this figure only crack widths higher than 0.1 mm are presented. For the D-Corbel B, the first cracks also initiated at the bottom of the structure, for a point load of

Figure 7. (a) Initial cracking pattern and (b) failure pattern for D-Corbel A using ATENA 3D.

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58.40 kN, as show in Figure 8(a). The yielding of the main tie was registered for a load of 188.60 kN and for this moment the stress level in the FRP strips was 214.20 MPa. The failure load was registered for a point load of 309.30 kN with a stress level about 2153 MPa acting at the FRP strips. The failure pattern of double corbel is presented in Figure 8(b). This failure, characterized by direct shear in the interface between the column and one corbel was very similar to that one obtained experimentally. Table 5 presents a comparison among the results obtained using experimental procedures, ATENA 2D and ATENA 3D. One should realize that only the best approximations in the two-dimensional analysis were considered. Despite the fact that numerical results presented a wide range of variation regarding the experimental results, the qualitative results (cracking patterns) are very close to the observed experimental behavior. The double-corbels A and B presented different cracking patterns at failure in the experimental investigation and this behavior was obtained by the non-linear analysis conducted. Figure 9 shows the response load versus displacements at the top of the column obtained numerically using two and three-dimensional models. As can be seen, there is little difference regarding the performance of this models. Until a point load of 100 kN, a kind of limit load for linear behavior, the response of

Figure 8. (a) Cracking pattern and (b) failure pattern for D-Corbel B using ATENA 3D. Table 5. results.

Comparison between experimental and numerical

Figure 9. Load versus displacement behavior for D-Corbels A and B using ATENA 2D and 3D.

double corbels A and B are identical. However, after this load, there was a very significant different behavior between double corbel A and B. Unfortunately, it was not possible to register the displacements of the column in order to compare numerical results with experimental results.

5

CONCLUSIONS

Various constitutive models available for reinforced concrete in the selected commercial package software have been used and some recommendations based on these investigations are made. It can be observed that nonlinear analysis is highly dependent on the user’s experience as well as in the potentiality of the selected constitutive models for treating cracking in concrete. Also, the great numbers of parameters that usually control the nonlinear analysis are another point that users should be aware in order to obtain confident answers. Taking into account the numerical and experimental results obtained for the double corbels, some relevant conclusions can be draw regarding the application of non-linear analysis for behavior prediction: • The numerical prediction of the behavior of the double corbels (or two pile caps) tested was very effective, despite the fact that the main loads (cracking loads, yielding loads and failure loads) had a significant range of variation. In some cases the obtained error was about 3% (cracking of the corbel A using ATENA 3D), while in other situations the obtained error was about 50% (yielding of the main tie for Corbel B using ATENA 3D). In general, the mean error using ATENA 2D was 14% while the error using ATENA 3D was 16%; • The patterns of cracking, i.e, the cracking propagation produced by the package software applied are very close to the behavior obtained experimentally. Nonlinear analysis were able to even obtain the

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different patterns at failure realized for D-Corbel A and B. Unfortunately, it is difficult to conclude how the specimens failed using finite element method. The author strongly believe that the failures was due to concrete splitting of the diagonal struts as the main tie still had some possibility of deformation; • For D-Corbel B, taking into account the experimental test, there was a certain damage of the concrete in the bearing locations, probably indicating the lack of anchorage for the main tie at the failure. Once the applied numerical models had a hypothesis of perfect adherence between concrete and steel, higher loads of failure using nonlinear analysis were really expected. It can be easily seen in Table 5; • The differences between experimental and numerical results may explained by the fact that an experimental test is not always an absolute reality of the problem. Some eccentricities may arise during the test and some problems regarding the reading of data also may occur. Also, the great number of parameters that need to be informed in non-linear analysis may lead to significant differences. It can be easily seen in Tables 3 and 4 and users should be aware about the importance of some parameters. As suggestion, the shear retention factor (beta parameters) should always investigate when using ‘‘Fixed Cracking Models’’. In numerical analysis is usually possible to realize a tendency of symmetrical cracking. However, cracking patterns obtained from experimental investigation is not supposed to present this behavior, as there is some eccentricities in action and concrete can not be considered homogeneous as in a numerical analysis. Non-symmetrical cracking may give rise to transversal stresses, which by turn cause the weakening of the concrete struts. The presence of stirrups in corbels may be sufficient to control the mentioned transversal stresses introduced by non-forecasted eccentricities. The vertical and horizontal stirrups may also produce a symmetrical cracking, leading to higher failure loads. In this situation, one should expect a experimental behavior similar to that one obtained by a numerical behavior. However, the effort in order to construct a finite element model with a great number of reinforcement is not a easy task. The application of finite element models demands a great experience and sensitivity of their users. For problems where bending is predominant the available models can produce good results. However, for problems where shear force is predominant, the constitutive models need to be carefully selected. It is due to the fact that is difficult to produce the typical diagonal cracking produced by shear. Users should be aware about the difference between ‘‘numerical collapse’’ and ‘‘structural collapse’’. Sometimes, numerical collapse may be overcome using

‘‘Line Search Technique’’ or ‘‘Arc Length Method’’ options. Analyzing the diagram load versus displacement is always a good tool in order to conclude about structural collapse. Finally, a statistical component should be included for numerical simulations. In fact, a range constituted by maximum and minimum values should be defined when using non-linear analysis. In that way, the searched loads (cracking, yielding and failure) should to be selected within the defined range. It is impossible to obtain numerical values identical to the obtained experimental values taking into account the great number of parameters in both approaches.

REFERENCES ACI Committee 318. 2005. ‘‘Building Code Requirements for Structural Concrete (ACI 318-2005) and Commentary (ACI 318R-2002), APPENDIX A: Strut-And-Tie Models’’. American Concrete Institute, Detroit. ASCE-ACI Comittee 445 on Shear and Torsion. 1998. ‘‘Recent Approaches to Shear Design of Structural Concrete’’. Journal of Structural Engineering, v. 124, n. 12, dec. CANADIAN STANDARDS ASSOCIATION. 1984. ‘‘CSA Standard-A23.3-M84—Design of Concrete Structures’’. Ontario: Rexdale. CANADIAN STANDARDS ASSOCIATION. 1994. ‘‘CSA Standard-A23.3-94-Design of Concrete Structures’’. Rexdale. Cervenka, V. & Cervenka, J. 2003. ‘‘ATENA Program Documentation—Part 2–1: User’s Manual for ATENA2D’’, Prague. Cervenka, V. & Cervenka, J. 2005. ‘‘ATENA Program Documentation—Part 2–2: User’s Manual for ATENA3D’’, Prague. COMITÉ EURO-INTERNATIONAL DU BÉTON. 1993. ‘‘CEB-FIP Model Code 1990’’. Thomas Telford Services, Ltd., London. COMITÉ EURO-INTERNATIONAL DU BÉTON. 1978. ‘‘CEB-FIP Model Code for Concrete Structures’’. Thomas Telford Services, Ltd., London. Crisfield, M.A. 1983. ‘‘An Arc-Length Method Including Line Search and Accelerations’’, International Journal for Numerical Methods in Engineering, v.19, pp.1269–1289. EHE. 1999. ‘‘Instrucción de Hormigón Estructural’’. Spanish Code, Second Edition, Madrid. Feenstra, P.H. & Borst, R. 1993. ‘‘Aspects of Robust Computational Modeling for Plain and Reinforced Concrete. Heron, Delft, Netherlands, v. 38, n. 04, pp. 3–76. Kotsovos, M.D. & Pavlovic, M.N.. 1995. ‘‘Structural Concrete: Finite Element Analysis For Limit-State Design’’. Thomas Telford Publications, 1995. MacGregor, J.G. 1991. ‘‘Dimensioning and Detailing’’. In: IABSE COLLOQUIUM STRUCTURAL CONCRETE, v. 62, pp. 391–409, Stuttgart. Marti, P. 1985a. ‘‘Basic Tools of Reinforced Concrete Beam Design’’. ACI Journal, Proceedings, v. 82, n. 01, pp. 45–56. Marti, P. 1985b. ‘‘Truss Models in Detailing’’. Concrete International, v. 82, n. 1, p. 66–73.

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Rots, J.G. & Blaauwendraad, J. 1989. ‘‘Crack Models for Concrete: Discrete or Smeared? Fixed, Multi-directional or Rotating?’’. Heron, v. 34, n. 01, Delft, Netherlands. Rots, J.G.; Nauta, P.; Kusters, G.M.A. & Blaauwendraad, J. 1985. ‘‘Smeared Crack Approach and Fracture Localization in Concrete’’. Heron, v. 30, n. 1, Delft, Netherlands, pp. 1–48. Schäfer, K. 1999. ‘‘Deep Beams and Discontinuity Regions. Structural Concrete—Textbook on Behaviour, Design and Performance’’, v. 03, fib CEB-FIP, 1999. Schäfer, K. & Schlaich, J. 1991. ‘‘Design and Detailing of Structural Concrete Using Strut-and-Tie Models’’. The Structural Engineer, v. 69, n. 06. Schlaich, J. 1991. ‘‘The Need for Consistent and Translucent Models’’. IABSE Colloquium Structural Concrete, v. 62, pp. 169–184, Stuttgart. Schlaich, J.; Schafer, K. & Jennewein, M. 1987. ‘‘Toward a Consistent Design of Reinforced Concrete Structures’’. Journal of Prestressed Concrete Structures, v. 32, n. 03, pp. 74–150. Singh, Bhupinder; Mohammadi, Yaghoub & Kaushik, S.K. 2005. ‘‘Design of a Double Corbel Using the Strut-andTie Model’’. Asian Journal of Civil Engineering (Building and Housing), v. 06, n. 1–2, pp. 21–33, 2005.

Souza, R.A.; Kuchma, D.A.; Park, J. & Bittencourt, T.N. 2007. ‘‘Non-Linear Finite Element Analysis of Four-Pile Caps Supporting Columns Subjected to Generic Loading’’. Computers and Concrete, v. 04, pp. 363–376.

NOTATION fcm fyy fyu As ffrp ffrp,u Afrp Pcracking Pyielding Pfailure

Concrete compressive strength; Steel yielding strength; Steel ultimate strength; Reinforcement steel area for the main tie; FRP Stress level considered for design the strengthening; FRP ultimate strength; FRP reinforcement area used for strengthening the main tie; Cracking load registered for the double corbels; Yielding load registered for the double corbels; Failure load registered for the double corbels.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Computational modeling of the behaviour up to failure of innovative prebended steel-VHPC beams for railway bridges S. Staquet BATir, Université Libre de Bruxelles, Brussels, Belgium

F. Toutlemonde Université Paris-Est, LCPC, Paris, France

ABSTRACT: An extensive research has been carried out in the framework of the French National Project MIKTI in order to extend the Belgian technique of steel-concrete prebended beams to Very High Performance Concrete (VHPC: self-compacting C80/95 with silica fume). In comparison to the present realizations coming from Belgium, the main advantage of using VHPC instead of C50 concrete is to decrease the prestressing losses of the system thanks to a significant decrease of the creep deformations, together with the possibility to optimize the beam weight and its serviceability domain. Numerous measurements such as mechanical characterization creep and shrinkage tests were performed. A major issue of the programme consisted in validating a design method for the serviceability limit state (SLS) linked to the cracking of the concrete and for the ultimate limit state (ULS) corresponding to an instability (warping) or to yielding of the steel girder. Detailed results of the global response of two 13m-long beams as well as the validation of the scientific method for taking concrete creep into account are analyzed in conjunction with deflection measurements. The ultimate limit state computation based on the strength and elastic stability is always controlled by the serviceability limit state for this kind of structures. 1

RESEARCH SIGNIFICANCE

The aim of the research program that has been carried out at LCPC in the framework of the French National Project MIKTI was to extend the system of the Preflex beam to Very High Performance Concrete (VHPC) and to give background for further updating of the Eurocode 4 (EN 1994-2) to the design of this kind of structure. Actually, the concrete grade which has been used until now is lower or equal to C50/60 whereas the average compressive strength at 28 days of the concrete in this research is 110 MPa. In the comparison to the present realizations coming from the Belgian precast industry, the main advantage of using VHPC with silica fume is to reduce the prestressing losses of the system thanks to a significant decrease of the creep deformations, as predicted by Eurocode (EN 1992-2:2005) together with the possibility to optimize the beam weight and its serviceability domain. Applications might concern French Railways Bridges, where stiffness has to be maintained or increased, due to high speed train requirements, with maintained clearance and reduced beam inertia (accounting for ballast between the rails and the bridge structure). A first theoretical step developed the updated background of these beams design, relatively to delayed strains of concrete and their effects (Morano et al.,

2006). This development was confirmed by research work conducted in Belgium for prebended U-shaped Bridges (Staquet & Espion, 2005). Then, the interest of VHPC in optimizing the 1st phase concrete deadweight and the global stiffness was confirmed. Consequently, it was decided to carry out an experimental investigation of long-term performance under dead and fatigue loads and behaviour up to ultimate limit state and failure of VHPC-prebended beams, mainly aiming to validate the scientific method for taking concrete creep into account. The design principles for the service loads are partly based on the class II type of BPEL (BPEL, 1999) implying no decompression in concrete under frequent service load combinations and a maximum concrete tensile stress limited to the actual concrete tensile strength under rare service load combinations. The design criteria for the ultimate limit state (ULS) correspond to an instability (warping) of the steel profile or to yielding of the steel girder upper flange. The Preflex system has been particularly successful in Belgium because it allows long spans with a minimal construction depth and offers excellent fire resistance (Staquet et al., 2004). It is presently mainly used in Belgium for railway bridges due to its high fatigue performance (first verified experimentally by Verwilst, 1953).

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2

STEPS OF THE EXPERIMENTAL PROGRAM

The construction stages a to d (Fig. 1) were carried out at LCPC on two factory-made steel girders HEB 360, 13 m-long, using a specially designed self levelling VHPC (Roussel, 2007). The cross section of the beams illustrated on Fig. 2 shows that the thickness of the concrete situated below the bottom flange of the steel girder was only 55 mm. Moreover, in addition to the longitudinal passive reinforcements (diameter 12 mm), ribbed stirrups (diameter 8 mm) were disposed every 15 cm along the beam and steel square ribs (25 × 25 × 200 mm) were welded to the bottom flange of the steel girder every 45 cm (Fig. 2). Dimensions of the ribs were defined as usual for taking all the design shear stresses at steel lower flange—concrete interface. As usually realized, stirrups are fixed across the steel profile web, which also favours regular stress transfer between concrete and steel, ensuring the desired composite behaviour.

a

Two days after concrete casting, the prestressing was transferred by releasing the prebending loads (145.2 kN for beam P1 and 137.5 kN for beam P2) initially applied by two jacks at one quarter and at three quarters of the span (Fig. 3). Two months after (Fig. 4), the beams were submitted to permanent loads (40 kN applied by lead masses, representing the load of the upper concrete deck, superstructures and ballast). After 4 months, live loads representative of railway traffic were applied: 1000 cycles representing the effect of trains possibly transiting once a year (half UIC conveys), then one or two million cycles corresponding to more frequent heavy trains. The deflection was monitored for more than 8 months as well as numerous complementary strain measurements. Mechanical characterization (Young’s modulus), creep and shrinkage tests were also performed so that a correct analysis of the structural behaviour of the prebended beams can be done, focusing on the steel-concrete behaviour, and taking advantage of low delayed deformations of VHPC. Finally the beams were tested up to failure for quantifying their safety margin under service and ultimate loads, and checking the accuracy of the design predictions.

b

c

d

e Figure 1.

Construction steps of a Precambered beam. Figure 3. Transfer of prestressing to the composite beam at 48 hours of concrete age.

Figure 2. Cross section of the prebent beams tested (dimensions in mm).

Figure 4. Permanent loads (40 kN) applied on the composite beam at 57 days of age.

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3

VHPC MIXTURE PROPORTIONS

In the context of this research project, the mixture proportions of the VHPC were determined on basis of simulations made with the BetonLabPro® software developed by LCPC and are given hereafter: Coarse aggregates 4/12.5 (Limestone from Boulonnais, 920 kg/m3 ); Sand 0/4 (Limestone from Boulonnais, 379 kg/m3 ); Sand 0/4 (Silico-calcareous rounded sand, 369 kg/m3 ); Cement CEM I 52.5 N CE CP2 NF (500 kg/m3 ); Silica fume (50 kg/m3 ); Batch water (165 l/m3 ). The rheological criteria implied the choice (polyphosphonate superplasticizer) and the quantity (10 l/m3 ) of the admixture corresponding (Roussel, 2007) to a slump flow of 70 cm.

4

on cylinders with a diameter of 16 cm and a height of 100 cm, so that a correct analysis of the structural long-term behaviour of the beams could be done. The conditions of these tests are in direct link with the phases of construction and the history of loading of the instrumented beams. The two beams were exposed to variable ambient conditions (from 16 to 25◦ C and from 35 to 60% RH) from one day of concrete age and the prebending forces applied on the beams before the casting phase were released at two days of concrete age. The recommendations (Rilem, 1998) issued by RILEM TC 107 to perform the creep and shrinkage tests have been followed. The specimens have been exposed to drying conditions at one day. Creep specimens have been loaded at 2 days under a constant compressive stress of 23 MPa and one of them has been completely unloaded at 56 days.

LABORATORY INVESTIGATION OF VHPC BEHAVIOUR 5

MODELING THE CREEP AND SHRINKAGE

4.1 VHPC instantaneous mechanical properties The evolution of the mechanical properties of the VHPC has been determined at 1, 2, 7, 28, 56, 100 and 168 days, which corresponds to specific events of loading or bearings change of the instrumented beams. Standard size of cylindrical specimen has been used: a radius of 5.5 cm and a height of 22 cm, what corresponds to the thickness of VHPC around the bottom flange of the steel girder (from 5 to 5.5 cm). Table 1 shows the evolution of the VHPC mechanical properties: fc , average compressive strength; ft , average splitting tensile strength; E, average modulus of elasticity, determined on cylinders 11/22 cm exposed to 40% of relative humidity and 20◦ C at one day of age. The E value at 168 days is slightly lower than the one at 100 days, explained by an effect of rather severe drying conditions. 4.2

Creep and shrinkage of the VHPC

To take full advantage of using VHPC instead of HPC (C50 used in Belgium), shrinkage and creep tests have been carried out in standard: sealed and 20◦ C, 50% RH and variable ambient conditions (same as the beams) Table 1. VHPC mechanical properties of the cylinders 11/22 exposed to 40% RH and 20◦ C after 1 day of concrete age. Age (d)

fc (MPa)

ft (MPa)

E (GPa)

2 7 28 56 100 168

71.5 87.8 99.3 103.1 103.3 106.1

− 4.4 4.5 − − 5.6

38.2 40.6 42.8 43.4 43.5 43.3

At the beginning of this research devoted to prebent and prestressed composite bridge decks, a 26 m-span bridge deck belonging to a multi-span viaduct constructed in June 2000 at the entrance of Brussels South Station was instrumented to monitor the timedependent evolution of the strains of the two concrete phases and the steel girders. Besides, a large series of creep tests was performed under constant stress, followed by a period of recording of the creep recovery after complete unloading on C50 concrete used in situ cylindrical samples. These tests should provide input for detailed models, e.g. the two-function method (Yue, 1993). These data have been compared with different computation methods (Staquet, 2004; Staquet, 2005) and it was found that: – the modular ratio method gives a rather poor prediction of the strains of the instrumented bridge; – the creep and shrinkage of the concrete C50 cast in the instrumented bridge, assessed through series of laboratory tests, is well estimated by the CEB-FIP Model Code 1990 (fib,1999); – a far better, although not perfect, agreement with the measured strains is obtained by applying the step-by-step method of time-dependent analysis; this method, detailed in (Ghali, 2002), takes into account creep and shrinkage and strands relaxation; – after unloading, experimental data show that the creep recovery deviates strongly from the numerical predictions obtained by the application of the principle of superposition but seems to conform rather well with the recovery model proposed by Yue and Taerwe (Yue, 1993) in which the nonlinear behaviour caused by unloading is divided into parts: a classical linear creep law for loading according to the model CEB-FIP Code 1990 (fib,1999) and a creep recovery law for unloading;

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– the application of the recovery model with the two-function method proposed by Yue and Taerwe (Yue, 1993) yielded computed strains which are in rather good agreement with in situ measured strains, and in better agreement than the strains computed by the application of the principle of superposition. The applicability of this method is favoured by the fact that dead loads typically increase during the structure service life leading to progressively decreasing compressive stresses in concrete. Consequently and in accordance to the results presented in this paper, the computation method which was used to predict the evolution of the stress and strain of the beams is the two-function method by using the values predicted by the 1999 version of the CEB-MC90 model (fib, 1999) for creep and shrinkage and the expression proposed by Yue and Taerwe (Yue, 1993) for the creep recovery law. The computation method adopts a general numerical solver for the step-by-step solution by trapezoidal rule of the Volterra hereditary integral equations (Ghali, 2002). At each step of the calculations, the previous steps are stored so that the entire history of stress and strain is stored in computer memory. The time axis is divided in a series of time steps between each loading event at time ti according to the equation: tp − ti 1 = 10 4 tp−1 − ti where

(1)

t p = tp − tp−1

(2)

The trapezoidal rule is applied to the creep compliance and to the creep recovery law as it is detailed in the third equation hereafter:   εp = εs (t p ) − εs (t p−1 ) +

p−1  

In the equation (3): 

r (t p , t k ) + r (t p , t k−1 ) − r (t p−1 , t k )  −r (t p−1 , t k−1 ) = 0

when σk > 0 (corresponding to a reloading in compression) J (tp , tk ) =

1 + ϕ(tp , tk ) E(tk )

p−1  

J (tp , tk ): creep compliance; ϕ (tp , tk ): creep coefficient The stress variation σp at each time-step tp is computed as follows:

6

r (t p , t k ) + r (t p , t k−1 )

(8)

where

(ε(tp ) = ε(tp ) − ε(tp−1 )

and Ep =

 σ k −J (tp−1 , t k ) − J (tp−1 , t k−1 ) · 2

(6)

when σk < 0 (corresponding to unloading or partial decompression)  r (t p , t k ) + r (t p , t k−1 ) − r (t p−1 , t k )  −r (t p−1 , t k−1 ) = 0 (7)

where

J (tp , t k ) + J (tp , t k−1 )

(5)

where εs (tp ): shrinkage strain at time tp ; r (tp , tk ): creep recovery function; tpress : event of transfer of prestressing; trec : event of the 1st partial unloading (concrete decompression after the transfer of prestressing); E(tk ): Young’s modulus of concrete at tk ; E28 : Young’s modulus of concrete at 28d.

σp = Ep · (ε(tp ) − εp )

κ=1

+

 and αk = 1 − exp −0.1 · (tpress + 0.05  × (tk − tpress ))

2 J (tp , tp ) + J (tp , tp−1 )

(9) (10) (11)

DATA AND PREDICTED VALUES OF VHPC CREEP AND SHRINKAGE

k=trec

 σ k −r (t p−1 , t k ) − r (t p−1 , t κ−1 ) · 2

with

r (t p , t k ) =

(3)

tp − tk 1 1 + · E(t k ) E28 tp − tk + 300 · α k (4)

We report hereafter results from creep tests compared with the values predicted by the 1999 version of the CEB-MC90 model (fib, 1999) for creep and shrinkage under constant stress and by Eq. 2 (Yue, 1993) for the creep recovery law. Near the beams, the average temperature was 21.6◦ C and the average relative humidity was 43%. Fig. 5 shows the creep function measured on a specimen

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0.00007

Strain ( m/m)

Creep function J (1/MPa)

100 (t-2) in days

experimental total J, t1 = 56d, 43% average RH 0.00006

0 0.01 -100

predicted total J, t1 = 56d, with recovery function

0.1

1

10

100

1000

predicted total J, t1 = 56d, superposition principle -200

0.00005 -300 -400

0.00004

-500

0.00003

-600 -700

0.00002 -800

-1000

(t-2) in days 0 0.00001

JB1SL 4m (data)

JB1SL 4m (predicted)

JB3ML 4m (data)

JB3ML 4m (predicted)

JB5IL 4m (data)

JB5IL 4m (predicted)

-900

0.00001

-1100

0.0001

0.001

0.01

0.1

1

10

100

1000

Figure 5. Creep function of the specimen loaded at 2d, unloaded at 56d, exposed to variable conditions at 1d.

loaded at 2 days under a constant stress of 23 MPa, completely unloaded at 56 days, exposed to variable ambient air conditions at 1 day and the associated predicted values. The difference between measurements and predicted values after unloading is significant when the superposition principle is applied whereas the trend of the measurements is well reproduced when applying the creep recovery law.

Figure 6. Measured and computed strains in the VHPC at the cross-section X = 4 m.

Displacement (mm) at mid-span 10 (t-2) in days 0 0.01

0.1

1

10

100

1000

-10

-20

Deflection, X=6.5m (data) Deflection (predicted)

-30

-40

-50

-60

-70

7

MODELING THE CREEP AND SHRINKAGE

In the numerical simulations of the beam with the twofunction method, a value of relative humidity rounded 40% was used as drying condition. Fig. 6 shows the evolutions with time of the measured and computed strains by the two-function method. Very good agreement between these measurements and predicted values is found, particularly, at early age. Fig. 7 shows the evolution of the measured deflection of the beam at mid-span (X = 6.5 m) and the computed deflection by the two-function method. The evolution of the deflection is very limited when concrete age is beyond 2 months—corresponding to the period where a significant part of creep and shrinkage has already taken place, what confirms VHPC interest for this kind of prebent structural element. Moreover, a good agreement is found between the measurements and the computed deflection by the twofunction method. Fig. 8 shows the evolution of the stresses computed by the two-function method in three locations of the concrete part at X = 4 m cross-section. A positive stress value indicates a compression. At the bottom fiber of the concrete part (JB5IL), the maximum stress which was reached after transfer of prestressing was 19.5 MPa and the stress variation in tension due to the permanent loads applied two months after was 2.6 MPa. Eight months after the construction of the beam, the stress value at the VHPC bottom fiber reached 3 MPa in compression. At the upper fiber of the concrete part (JB1SL), the stress value reached 1.8 MPa in tension. In regard to the stress state in

Figure 7. Measured and computed deflection of the beam at mid-span (X = 6.5 m).

Stress (MPa) in X = 4m

Ambient drying condition: 40% RH

20 18

JB5IL, X = 4m JB3ML, X = 4m JB1SL, X = 4m

16 14 12 10 8 6 4 2 0 0.01 -2

0.1

1

10

100

(t-2) in days 1000

-4

Figure 8. Stresses computed with recovery function in the VHPC at the cross-section X = 4 m.

the concrete part, the criterion of keeping compression in concrete under permanent loads has probably not been met in the instrumented beam. In fact, the prebending loads would have to be higher than the effectively applied loads on the instrumented beam to prestress the concrete at a maximum stress level equal to 28 MPa at the bottom fiber instead of 19.5 MPa. Namely, before the step b (Fig. 1), the steel girder is submitted to several cycles of loading and unloading to remove the residual stresses due to its production process. As a consequence of this phase, the camber of the steel girder becomes smaller than its initial camber. In the case of this research project, the loss

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cracking pattern) and the theoretical value of 336 kN corresponds to the ultimate limit state (ULS) loading value. Table 2 shows the extreme first phase concrete stress values under permanent loads computed by using the CEB MC 90 model code in its version 99 (fib, 1999) for the prediction of creep and shrinkage of the VHPC. The stress values for the beam P1 are put in italic in Table 2 because, due to an extra-severe loading leading to an irrecoverable cracking that was applied on Beam P1 before the fatigue test and the test up to failure, the stress state of concrete is different from these computed values in some parts of this beam.

of camber after this phase was significantly higher than the predicted one. Since during VHPC casting, the formwork had to be maintained horizontally, the maximum possible prebending loads were smaller than the initially expected values.

8

LOADING UP TO FAILURE

The estimation of the cracking load (average stress equal to about 1.5 ft in the whole 1st phase concrete: 8.4 MPa), the warping load for a free span of 6 m (between application points of the loads by the jacks, Fig. 9) and the yield load of the upper flange of the steel girder (fy : 420 N/mm2 ) by neglecting the concrete participation, provided the following theoretical values of the load at each jack (EN 1993): 93 kN, 336 kN and 410 kN respectively. The tests were carried out in 3 phases (Figs. 10 and 11): up to 115 kN then unloading; up to 250 kN (3/4 of the warping theoretical load or 60% of the theoretical yield load) then unloading; and loading up to failure. The value of 115 kN is considered as an excess value of the design service load which is related to the damage occurrence in the concrete flange (stabilized

Figure 9.

RESULTS AND DISCUSSION

According to the loading configuration shown in the figure 12, the apparent bending stiffness can be determined from the deflection f at mid-span and the global applied load 2P, using the following equation: EI =

a(3l 2 − 4a2 ) 2P = αk 48 f

where k =

Static loading up to failure for the two beams.

400 375 350 325 300 275 250 225 200 175 150 125 100 75 50 25 0

Load P (kN)

Test up to failure

Beam P2 P2 Max: 386 kN Beam P1 P1 Max: 394 kN

time (h) 0

Figure 10.

9

1

2

3

4

5

6

7

8

9

10

Loading sequence up to failure of Beams P1 and P2.

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11

12

2P f

Figure 11.

View of loading of Beam P2.

Table 2. Extreme stress computed values in concrete before the test up to failure (negative value in compression).

P1

k (kN/mm)

EI (kN · m2 )

Variation

−0.55 −0.6

L 0-115 kN U 115 kN-0 L 0-250 kN U 250 kN-0

7.12 6.04 6.96 5.56

125,000 106,000 123,000 98,000

– −15.2 % −2.2 % −21.9 %

a

P2 L 0-115 kN U 115 kN-0 L 0-250 kN U 250 kN-0

k (kN/mm) 7.58 6.28 7.30 5.48

EI (kN · m2 ) 133,000 111,000 129,000 96,000

Variation – −17.2 % −3.7 % −27.7 %

Stress value in the 1st phase concrete (MPa)

Bottom fiber

Upper fiber

Average

Beam P1 Beam P2

−2.9 −2.8

+1.8 +1.7

P

Table 3. Static bending stiffness of the tested beams during the test up to failure (L:loading; U: unloading).

P

a

L Figure 12.

Loading configuration during test up to failure.

The static bending stiffness EI of the steel girder alone is 90,700 kN · m2 whereas the static bending stiffness of the composite beam corresponding to a full concrete contribution (with instantaneous modular ratio) is 152,700 kN · m2 , the concrete corresponding to one third of the global stiffness. The values of the stiffness k reported in Table 3 were obtained from a linear regression of the experimental relation loaddeflection at mid-span for each loading and unloading phase in the range of 0 to 20 kN. As the P1 beam had

been submitted to an extra-severe loading leading to irrecoverable cracking, its static bending stiffness at the beginning of the test was lower than the one of the P2 beam. After having reached the SLS load (115 kN), the bending stiffness loss which was obtained from the experimental relation force-deflection in the loading phase up to 250 kN, was only 2% for Beam P1 and 4% for Beam P2. The reduction of the static bending stiffness after having loaded the beam up to 250 kN and determined in the unloading phase was 22% for Beam P1 and 28% for Beam P2 and is less than 10% higher than stiffness of the steel alone. A significant reduction of the bending stiffness in the range of 15% to 17% was also obtained in the unloading phase from 115 kN. The bending stiffness was subsequently fully

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recovered after having completely unloaded the beam because the level of compressive stresses was high enough to close the cracks in the concrete flange of the two beams. The observation of the first cracking visible on the bottom surface of the concrete flange around mid-span appeared at a load of 40 kN for Beam 1 and 85 kN for Beam 2 or at a bending moment of 100 kN · m and 212 kN · m respectively. As the load increased up to the SLS value, the crack network extended until it reached a pattern in close relation with the steel mesh of reinforcement: ribbed stirrups every 15 cm, visible on lateral and upper surfaces along the beam and steel square ribs welded to the bottom flange of the steel girder, visible on the bottom surface every 45 cm along the beam (Fig. 13). The maximal crack opening reaches 0.15 mm under 115 kN and the crack pattern was stabilised, uniformly distributed between the two concentrated loads at this loading value. At a loading value of 50 kN for Beam P1 and 115 kN for Beam P2, a longitudinal crack was opened in the

Figure 13.

central part, corresponding to the lateral boundary of the steel square ribs. During the second loading, the maximal crack opening reached 0.3 mm under 140 kN and 0.4 mm under 180 kN for Beam P1 whereas 0.2 mm under 180 kN and 0.3 mm under 250 kN for Beam P2. The non-linearity of the load-deflection curve was observed since 40 kN for the two beams (Fig. 14). After the first loading phase up to 115 kN with a maximal deflection at mid-span of 38.3 mm for Beam P1 and 38.2 mm for Beam P2, the residual deflection was 0.6 mm and 1.4 mm respectively. After the second loading phase up to 250 kN with a maximal deflection at mid-span of 89.5 mm for Beam P1 and 91.2 mm for Beam P2, the residual deflection was 1.7 mm and 2.6 mm respectively. The global behaviour of the beam, relatively linear until 250 kN (Fig. 14) became very non linear from 310 kN for the two beams (Fig. 15) due to the development of cracks and slidings between cross-sections. During the third loading, the maximal crack opening under 350 kN reached 0.7 mm at the bottom surface of the concrete flange. The warping appeared at 394 kN for Beam P1 and 386 kN for Beam P2 with a deflection at mid-span equal to 24 cm at this stage and a typical diagonal cracking in concrete flange around mid-span due to the rotation of the sections (Fig. 16). The failure corresponds firstly to the instability of the steel upper flange as predicted because the maximum experimental loading value of about 390 kN is a little bit lower than the theoretical value of 410 kN, corresponding to the yield load of the upper flange of the steel girder (taking into account a safety factor according to EN-1993). The presence of VHPC does not modify the failure mode linked to the steel girder but

View of the cracking pattern.

Load (kN)120 260 100 240

80

220

60

200

40

180

20

160

0

1st and 2nd loading phases

up to 115 kN (beyond SLS)

0

4

8

12

16

20

24

28

32

36

Beam P2 Beam P1

40

140 120

up to 250 kN (cracked concrete)

100 80 60 40 20 0

Figure 14.

Deflection at mid-span (mm) 0

4

8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92

Loading of Beams P1 and P2 up to 250 kN.

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400 380 360 340 320 300 280 260 240 220 200 180 160 140 120 100 80 60 40 20 0 Figure 15.

Third loading up to failure

Load (kN) Beam P2 P2 Max: 386 kN Beam P1 P1 Max: 394 kN

0

15

30

45

60

75

Deflection at mid-span (mm) 90 105 120 135 150 165 180 195 210 225 240 255

Loading of Beams P1 and P2 up to failure.

10

Lateral displacement (cm)

Post-failure behavior of Beam P2

9 8 7 6 5 4 3 2

10 RESULTS AND DISCUSSION

1 0

Longitudinal abscissa (m) 0

1

Figure 18. Figure 16.

2

3

4

5

6

7

8

9

10

11

12

13

Lateral displacement of Beam P2 after failure.

View of the cracking pattern after failure.

provides important reserve with respect to an ultimate limit state of warping instability type by delaying its onset: about 390 kN instead of 336 kN. Moreover, its decreases the suddenness of the warping thanks to the energy which is dissipated by diagonal cracking of the concrete flange in torsion. After the final unloading, an elastic recovery still existed. The figure 17 displays the VHPC-steel beam P2 after loading up to failure. Figure 15 shows the lateral displacement between the position of the upper flange of the Beam P2 after failure due to the warping and its initial position. The maximal value of the lateral displacement occurred around mid-span and was equal to 9 cm (Fig. 18). 10 Figure 17.

CONCLUSIONS

This research has focused on the feasibility to extend the Belgian technique of steel-concrete prebended

View of Beam P2 after failure.

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beams to Very High Performance Concrete (VHPC: self-compacting C80/95 with silica fume) and on the validation of a design method for the prediction of stress redistribution due to concrete creep within this optimized cross-section with VHPC. A laboratory investigation was set up to study experimentally the creep and shrinkage of a VHPC. It was shown that, for the VHPC considered, the trend of the experimental data seems to be well reproduced in the different situations by the 1999 version of the CEB MC90 prediction model. After unloading, experimental data show that the creep recovery deviates strongly from the numerical predictions obtained by the application of the principle of superposition but seems to conform rather well to the recovery model proposed by Yue and Taerwe. The comparison between the detailed results of the experimental response of the beams in term of strains and deflection and the computed values by the twofunction method has shown that a rather good agreement is obtained by using the values predicted by the 1999 version of the CEB-MC90 model for creep and shrinkage and the expression proposed by Yue and Taerwe for the creep recovery law. In (Staquet S, 2005), it was shown that this computation method provided also a rather good agreement with in situ measurements of an instrumented prebent prestressed composite bridge deck with 26 m-span situated near Brussels South Station. These results confirm the robustness of this computation method for the prediction of stress redistribution due to creep and shrinkage in prebent prestressed steel-concrete structural elements. Moreover, the experiments realised on the two prebended steel-VHPC beams allow concluding that the presence of VHPC does not modify the failure mode linked to the steel girder but provides important reserve with respect to an ultimate limit state of warping instability type by delaying its onset and reducing its suddenness. The remarkable repeatitivity of the behaviour of the two beams brings a great credibility to the results. The ultimate limit state computation based on the strength and the stability of the single girder appearsas safe. This is even safer in case of 2nd phase concrete for a multi-beams structure. Practically, the design of prebended beams is always controlled by the serviceability limit state verifications due to the following criteria: – control of the delayed effects of concrete by limiting the compressive stress during the prestressing, which implies to have accurate data on creep and mechanical properties of the concrete; – control of cracking implying to keep compression under permanent and even frequent loads and to limit the tension under rare service loads.

ultimate limit state, and corresponding design methods are valid. ACKNOWLEDGEMENTS The authors are pleased to thank number of colleagues at the French Public Works Research Laboratory (LCPC) who participated to this study: BCC, MI and FDOA. The support of the National Project MIKTI (sponsored by the Ministry for Public Works—Civil Engineering R&D funds) is acknowledged. REFERENCES BPEL 1999. Règles BPEL 91 modifiées 99 (Règles techniques de conception et de calcul des ouvrages et constructions en béton précontraint suivant la méthode des états limites) fascicule n◦ 62 titre 1er section II du CCTG applicable aux marchés publics de travaux, JO du 16 février 1999. EN 1992-2 : 2005 Eurocode 2—Calcul des structures en béton—Partie 2 : Ponts en béton—calcul et dispositions constructives, CEN. EN 1993 Eurocode 3—Design of steel structures, CEN. EN 1994-2 Eurocode 4—Design of composite steel and concrete structures—Part 2: Bridges, CEN. fib-CEB-FIP, ‘‘Structural Concrete Volume 1’’, Bulletin fib, N◦ 1, 1999. Ghali A., Favre R. and Elbadry M., ‘‘Concrete structures: stresses and deformations’’, 3rd ed., E&FN Spon, 2002. Morano G.S. and Mannini C., ‘‘Preflex Beams: a method of calculation of creep and shrinkage effects’’, Journal of Bridge Engineering, 21, (1), ASCE, January 2006, pp. 48–58. Rilem TC 107-CSP, ‘‘Measurement of time-dependent strains of concrete’’, Materials and Structures, 31, 1998, pp. 507–512. Roussel N., Staquet S., D’Aloia L., Le Roy R. and Toutlemonde F., ‘SCC casting prediction for the realization of prototype VHPC-precambered composite beams’, Materials and Structures, 40, n◦ 9, 2007, 877–887. Staquet S., Rigot G., Detandt H. and Espion B. 2004. Innovative Composite Precast Precambered U-shaped Concrete Deck for Belgium’s High Speed Railway Trains. PCI Journal 49 (6): 94–113. Staquet S. and Espion B. 2005. Deviations from the principle of superposition and their consequences on structural behaviour. Shrinkage and Creep of Concrete ACI SP 227: 67–83. Verwilst Y. 1953. Essais pulsatoires et essai statique à la rupture d’une importante poutre de 14,50 m sur l’installation G.I.M.E.D. de l’Association des Industriels de Belgique (A.I.B.), revue l’Ossature Métallique (O.M.) 3: 165–169. Yue L.L. and Taerwe L., ‘‘Two-function method for the prediction of concrete creep under decreasing stress’’, Materials and Structures, 26, 1993, pp. 268–273.

Provided these criteria are met, composite behaviour is normally ensured under service loadings and up to

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Numerical investigations of size effects in notched and un-notched concrete beams under bending E. Syroka, J. Bobi´nski, J. Górski & J. Tejchman Faculty of Civil and Environmental Engineering, Gdansk University of Technology, Poland

ABSTRACT: Numerical FE investigations of a deterministic and statistical size effect in notched and un-notched concrete beams of a similar geometry under three point bending were performed. The FE analyses were carried out with different beam sizes. Deterministic calculations were performed assuming constant values of tensile strength. In turn, in statistical calculations, tensile strength took the form of random spatial fields described by a truncated Gaussian random distribution. In order to reduce the number of stochastic realizations without loosing the accuracy of the calculations, Latin hypercube sampling was applied. 1

INTRODUCTION

The size effect phenomenon (nominal strength varies with the size of structure) is an inherent property of the behavior of many engineering materials, i.e. the strength and brittleness increase with increasing element size under. Thus, concrete becomes perfectly brittle on a sufficiently large scale. The results from laboratory tests which are scaled versions of the actual structures cannot be directly transferred to them. Two sources of size effects are of a major importance: deterministic and statistical one. The first one is caused by strain localization in the form of a fracture process zone FPZ, whose size cannot be appropriately scaled in laboratory tests. Strain localization is not negligible to the cross-section dimensions and is large enough to cause a significant stress redistribution in the structure. The size of FPZ is related to a characteristic length of micro-structure lc (influencing both the thickness and spacing of localized zones). The specimen strength increases with increasing ratio lc /L (L—specimen size). In turn, a statistical (or stochastic) effect is caused by the spatial variability/randomness of local material strength. An increase of the specimen size causes an increase of weak spots what induces a decrease of the material strength. The aim of our research is to investigate experimentally and numerically a deterministic and stochastic size effect in concrete and reinforced concrete beams under quasi-static bending. In order to simulate the concrete behavior, an elasto-plastic model was used. To describe properly strain localization in concrete elements, a characteristic length of micro-structure was included by means of a non-local theory. The reinforcement was modeled as an elastic-ideally plastic material.

In this paper, the results of two-dimensional calculations with notched (Bobi´nski et al. 2009) and unnotched concrete beams under bending are presented. The deterministic calculations were carried out with a uniform distribution of tensile strength. In turn, the stochastic analyses were performed with a spatially correlated homogeneous distribution of tensile strength which was assumed to be random. Truncated Gaussian random tensile strength fields were generated using a conditional rejection method for correlated random fields. In order to reduce the number of stochastic realizations without losing the accuracy of the calculations, Latin hypercube sampling was applied.

2 2.1

CONSTITUTIVE MODEL Elasto-plastic model with non-local softening

An elasto-plastic model with isotropic hardening and softening was assumed (Majewski et al. 2008). In compression, a linear Drucker-Prager criterion was used   1 (1) f1 = q + p tan ϕ − 1 − tan ϕ σc (κ1 ) , 3 where q = the Mises equivalent stress, p = the mean stress, ϕ the internal friction angle, σc = the uniaxial compression yield stress and κ1 = the hardening/softening parameter (equal to plastic strain in p uniaxial compression ε11 ). The invariants p and q were defined as  1 3 sij sji , p = σkk , q = (2) 3 2

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where σij is the stress tensor and sij denotes deviatoric stress tensor. A non-associated flow rule was assumed. In tensile regime, a Rankine criterion was used with a yield function f2 (Marzec et al. 2007) defined as f2 = max{σ1 , σ2 , σ3 } − σt (κ2 ),

(3)

where σ1 , σ2 and σ3 = principal stresses, σt = tensile yield stress and κ2 = hardening (softening) parameter p (equal to the maximum principal plastic strain ε1 ). The associated flow rule was assumed. To model softening, the exponential curve by Hordijk (1991) was chosen   σt (κ) = ft (1 + A1 κ 3 ) exp(−A2 κ) − A3 κ , (4) where ft stands for the tensile strength of concrete. The constants A1 , A2 and A3 were A1 =

c1 c2 1 , A2 = , A3 = (1 + c13 ) exp(−c2 ), (5) κu κu κu

wherein κu = 0.005 denotes the ultimate value of the softening parameter, and constants ci : c1 = 3 and c2 = 6.93. The modulus of elasticity was assumed to be E = 38.5 GPa and the Poisson ratio was υ = 0.24. The edge and vertex in Rankine yield function were taken into account by the interpolation of 2–3 plastic multipliers according to the Koiter’s rule. As a regularisation technique, a non-local theory was used (Pijaudier-Cabot & Bažant 1987, Bažant & Jirasek 2002). It is based on a spatial averaging of tensor or scalar state variables in a certain neighbourhood of a given point. In plasticity, softening parameters κ were treated non-locally according to the Brinkgreve’s proposal (1994)  α0 (x − ξ ) dκ (ξ ) dξ κ¯ (x) = (1 − m) κ (x) + m  , α0 ( x − ξ ) dξ (6) where κ¯ is the non-local softening parameter, m denotes a non-local coefficient larger than one, x are the coordinates of the considered point, ξ denotes the coordinates of the surrounding points and α0 is the weighting function. As the weighting function α0 , the Gauss distribution was assumed r 2 1 α0 (r) = √ e−( l ) , l π

(7)

where r is the distance between points x and ξ , and l denotes a characteristic length of micro-structure. The averaging is restricted only to a small area around each material point (the influence of points at the distance of r = 3 × l is only of 0.01%). The internal length was chosen as l = 5 mm. The 2D and 3D non-local model was implemented in the commercial finite element code Abaqus (1998)

with the aid of subroutine UMAT (user constitutive law definition) and UEL (user element definition) for efficient computations (Bobi´nski and Tejchman 2004). For the solution of the non-linear equation of motion governing the response of a system of finite elements, the initial stiffness method was used with a symmetric elastic global stiffness matrix. The calculations were carried out using a large-displacement analysis available in the Abaqus finite element code (1998) (although the influence of such analysis was negligible). In this method, the current configuration of the body was taken into account. The Cauchy stress was taken as the stress measure. The conjugate strain rate was the rate of deformation. The rotation of the stress and strain tensor was calculated with the Hughes-Winget method. The non-local averaging was performed in the current configuration. To capture a snap-back behaviour in a very largesize beam, the so-called arc-length technique was used (Bobi´nski et al. 2009). The actual load vector P was defined as λP max where λ—multiplier and P max — maximum constant load vector. In general, the determination of the length of the arc the P–u space (u— displacement vector) involves the displacements of all nodes. However, for problems involving strain localization, it is more suitable to use an indirect displacement control method, where only selected nodal displacements are considered to formulate an additional condition in the P–u space. The horizontal distance between two nodes lying on the opposite sides of the notch was chosen as a control variable CMOD (crack mouth open displacement). The indirect displacement algorithm was implemented with the aid of two identical and independent FE-meshes and some additional node elements to exchange the information about the displacements between these meshes.

3 3.1

FE INPUT DATA Deterministic calculations

The behaviour of concrete beams with free ends without notch and with notch at the bottom at mid-span under three-point bending (Fig. 1) was simulated. The following three beams (which were experimentally investigated by Le Bellˇego et al. (2003)) were considered: small (h = 8 cm), medium (h = 16 cm) and large-size beam (h = 32 cm). Additional calculations were made with a very large beam (h = 192 cm). The beam span was L = 3h. The quadrilateral elements divided into triangular elements were used to avoid volumetric locking. In total, 7628 (small size beam), 14476 (medium size beam), 28092 (large size beam) and 104310 (very large size beam) triangular elements were used, respectively. The mesh was particularly very fine in the region of a notch to properly capture strain localization in concrete (where the element size

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Figure 1. Three-point bending test: geometry of notched beam and boundary conditions (Le Bellˇego et al. 2003).

was equal to 1/3 × lc ). The ratio between the width of this region and beam length was always the same. A quasi-static deformation of a small, medium and large beam was imposed through a constant vertical displacement increment u prescribed at the upper mid-point of the beam top. In the case of a very large beam (to capture the snap-back behavior), a procedure described in Section 2 was used. 3.2

Statistical calculations

In the paper the Monte Carlo method was used. Application of the method in stochastic problems of mechanics requires the following steps: simulation of random variables or fields describing the problem under consideration (variability of material parameters, initial imperfections in structure geometrics and others), solution of the problem for each simulated realization, creation of a set of results and its statistical description. Contrary to stochastic finite element codes, the Monte Carlo method does not impose any restriction to the solved random problems. Its only limitation is the time of calculations. For example, to reproduce exactly the input random data of initial geometric imperfection of a shell structure problem, at least 2000 random samples should be used (Bielewicz and Górski 2002). Any nonlinear calculations for such number of initial data are, however, impossible due to excessive computation times. To determine a minimal, but sufficient number of samples (which allows one to estimate the results with a specified accuracy), a convergence analysis of the outcomes was proposed. It was estimated that in case of various engineering problems only ca. 50 realizations had to be considered. A further decrease of sample numbers can be obtained using Monte Carlo variance reduction methods. The statistical calculations according to the proposed Latin sampling method were performed in two steps (Tejchman and Górski 2007, 2008). First, an initial set of random samples was generated in the same way as in the case of a direct Monte Carlo method. Next, the generated samples were classified

and arranged in increasing order according to the chosen parameters (i.e. their mean values and the gap between the lowest and the highest values of the fields). From each subset defined in this way, only one sample was chosen for the analysis. The selection was performed in agreement with the theoretical background of the Latin sampling method. The numerical calculations were performed only for these samples. It was proved that using the Latin sampling variance reduction method the results can be properly estimated by several realizations only (e.g. 12–15). To generate the random field, the conditional-rejection method described by Walukiewicz et al. (1997) was used. The method makes it possible to simulate any homogeneous or non-homogeneous truncated Gaussian random field described on regular or irregular spatial meshes. The simulation process was based on the original conditional, rejection method of generation. An important role in the calculations was played by the propagation base scheme covering sequentially the mesh points and the random field envelope which allowed one to fulfill the geometric and boundary conditions of the structure of the model. Random fields of practically unlimited sizes could be generated. Various properties of concrete may be considered as randomly distributed. In the present work, only fluctuations of tensile strength were taken into account. Two parameters described the random field should be chosen, i.e. the distribution of the random variable in a single point of the field and a function defining the correlation between these points. In the work, the distribution of a single random variable takes the form of truncated Gaussian function with the mean concrete tensile strength of 3.6 MPa (Fig. 2). Additionally it was assumed that the concrete tensile strength values changed between ft = 1.6 MPa and ft = 5.6 MPa (ft = 3.6 ± 2.0 MPa). To fulfil this condition, the standard deviation sft = 0.424 MPa was used in the calculations. Then, the coefficient of variations describing the field scattering was cov = sft /f¯t = 0.118 (f¯t —mean tensile strength). It is easily to notice that 5sft = 5 × 0.424 = 2.12 MPa and the cut of variables does not change the theoretical Gauss distribution distinctly. Randomness of tensile strength ft must be described by a correlation function. Here, the following, second order, homogeneous correlation function was adopted according to Bielewicz and Gorski (2002) K(x1 , x2 ) = sf2t × e−λx1 x1 (1 + λx1 x1 )e−λx2 x2 × (1 + λx2 x2 ),

(8)

where x1 and x2 is are the distances between two field points along the horizontal axis x1 and vertical axis x2 , λx1 and λx2 are the decay coefficients (damping

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Frequency

0.16 theoretical (Gauss) dustribution

0.12 0.08

mean value ft=3.6 MPa

ft = 3.6 MPa sft = 0.424 MPa

0.04 2.0 MPa

2.0 MPa

0 1 2 3 4 5 6 Concrete tensile strength [MPa] Figure 2. Distribution of the concrete strength values for a single point of the mesh.

parameters) characterizing a spatial variability of the specimen properties In finite element methods, continuous correlation function has to be represented by the appropriate covariance matrix. For this purpose, the procedure of local averages of the random fields proposed by Vanmarcke (1983) was adopted. After an appropriate integration of the function (Eq.9), the following expression describing the variances Dw and covariances Kw were obtained (Knabe et al. 1998): Dw (x1 , x2 ) =

correlation was approximately 80 mm in the horizontal direction and 30 mm in the vertical direction The smaller the lambda parameter, the shorter is the correlation range). The dimension of the random field was identical as the finite element mesh. The same random values were assumed in 4 neighboring triangular elements. Figure 3 shows a stochastic distribution of tensile strength in one concrete beam in the area close to the notch. Using the conditional-rejection method 2000 field realizations of the initial void ratio tensile strength were generated. Next, the generated fields were classified according to two parameters: the mean value of the tensile strength and the gap between the lowest and the highest value of the tensile strength. The joint probability distribution (so-called ‘‘ant hill’’) is presented in Fig. 4. One dot represents one random vector described by its mean value and the difference between its extreme values. The two variable domains

 3 2 sf2t 2 + e−λx1 x1 − λx1 x1 λx1 x1



× 1 − e−λx1 x1 ×

 2 sf2t 2 + e−λx2 x2 λx2 x2 

−

3 1 − e−λx2 x2 λx2 x2

Figure 3. The random distribution of tensile strength in medium-size beams with notch.



eλx1 x1 2 

2 sft cos λx1 x1 λx1 x1 e

− sin ( λx1 x1 )] + 2λx1 x1 − 1} ×

λx2 x2

λx2 x2

Random vector max-min gaps [MPa]

Kw (x1 , x2 ) =

2

2 sft



 

× cos λx2 x2 − sin λx2 x2 + 2λx2 x2 − 1

We took mainly into account a strong correlation of the tensile strength ft in horizontal direction λx1 = 1 1/m and a weaker in the vertical directions λx2 = 3 1/m in Eq. 9 (due to the way of specimen’s preparation). In this way, the layers forming during the concrete placing were modeled. The range of significant

4 3.6 3.2 2.8 2.4 2 3.5

3.55

3.6

3.65

3.7

Random vector mean values [MPa]

Figure 4. Selection of 12 pairs of random samples using Latin hypercube sampling: 1 − 4, 2 − 7, 3 − 3, 4 − 11, 5 − 5, 6 − 8, 7 − 1, 8 − 6, 9 − 2, 10 − 9, 11 − 10 and 12 − 12 (Bobi´nski et al. 2009).

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were divided in 12 intervals of equal probabilities (see vertical and horizontal lines in Fig. 4). Next, according to the Latin hypercube sampling assumptions, 12 random numbers in the range 1–12 were generated (one number appeared only once) using the uniform distribution. The generated numbers formed the following 12 pairs: 1−4, 2−7, 3−3, 4−11, 5−5, 6 − 8, 7 − 1, 8 − 6, 9 − 2, 10 − 9, 11 − 10 and 12 − 12. According to these pairs, the appropriate areas (subfields) were selected (they are presented as rectangles in Fig. 4). From each subfield only one realization was chosen and used as the input data to the FEM calculations. In this way the results of 12 realizations were analyzed.

4

above the notch is in Fig. 7. Moreover, the numerical results of a deterministic size effect compared to the size effect law by Bažant for concrete specimens (Bažant and Planas 1998) are shown in Fig. 8. The beam strength and beam brittleness obviously increased with increasing beam size. This pronounced

FE RESULTS

4.1 Deterministic results

a)

b)

c)

d)

Figure 5 and 6 show the evolution of the calculated normalized vertical force PL/tf t (0.9h)2 versus the normalized vertical beam displacement u/h for four different beam heights h: 8 cm, 16 cm, 32 cm and 192 cm (notched beams) and three different beam heights h: 8 cm, 16 cm and 32 cm (un-notched beams) with constant values of tensile strength of ft = 3.6 MPa. A distribution of the non-local softening parameter

Figure 7. Distribution of non-local softening parameter for notched beams in deterministic calculations; a) small beam (h = 8 cm), b) medium beam (h = 16 cm), c) large beam (h = 32 cm), d) very large beam (h = 192 cm) (Bobi´nski et al. 2009). a)

Figure 5. Normalized load-displacement curves for four notched concrete beams (Bobi´nski et al. 2009).

b)

Figure 6. Normalized load-displacement curves for three un-notched concrete beams.

Figure 8. FE-results for notched and un-notched beams as compared with the size effect law by Bažant (Bažant and Planas 1998) a) notched beams, b) un-notched beams.

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deterministic size effect is in agreement with the size effect model by Bažant (Bažant and Planas 1998). For a very large size beam, a so-called snap-back behaviour occurred (decrease of strength with decreasing deformation). The mean width of a localized zone above the notch was 15.08 mm (h = 8 cm), 15.10 mm (h = 16 cm), 18.02 mm (h = 32 cm) and 18.05 mm (h = 192 cm) at u/h = 1‰. The calculated vertical forces for a small, medium and large beam are in good accordance with the experiments by Le Bellˇego et al. (2003). The calculated width of the localized zone was similar as in experiments, i.e. about 20 mm (on the basis of acoustic emission, Pijaudier-Cabot et al. 2004). 4.2

Stochastic calculations

shown in Fig. 9 for 3 different beam heights h: 8 cm (small beam), 32 cm (large beam) and 192 cm (very large beam), respectively. Figure 10 demonstrates the calculated width of a localized zone above the notch. The normalized maximum vertical force decreases with decreasing beam height h. For h = 8 cm, it changes between 2.92–3.38 kN. The mean stochastic Pmax = 3.08 kN (with the standard deviation of 0.126 kN) is practically the same as the deterministic value Pmax = 3.13 kN (it is smaller by only 2%). If the beam height is h = 32 cm, the maximum vertical force varies between 7.73–8.85 kN and the mean stochastic force Pmax = 8.30 kN (with the standard deviation of 0.334 kN) is smaller by only 0.6% than the deterministic value (Pmax = 8.35 kN). For the beam height of h = 192 cm, the maximum vertical force varies between 26.05–28.72 kN and the mean

The 12 different evolutions of the vertical force P versus the vertical displacementu in notched beams are

a)

a)

b) b)

c) c)

Figure 9. Normalized force-displacement curves in the case of deterministic (dashed lines) and random calculation (solid lines) for 3 notched beams under three-point bending; a) small size beam (h = 8 cm), b) large size beam (h = 32 cm), c) very large size beam (h = 192 cm) (λx1 = 1 1/m, λx2 = 3 1/m, sft = 0.424 MPa) (Bobi´nski et al. 2009).

Figure 10. Distribution of non-local softening parameter above the notch in the case of deterministic (dashed lines) and random calculation (solid lines) for 3 notched beams under three-point bending: a) small size beam (h = 8 cm), b) large size beam (h = 32 cm), c) very large size beam (h = 192 cm) (λx1 = 1 1/m, λx2 = 3 1/m, sft = 0.424 MPa) (Bobi´nski et al. 2009).

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stochastic Pmax = 27.56 kN is again smaller by only 0.6% than the deterministic value of Pmax = 27.72 kN (the standard deviation equals 0.692 kN). The loaddisplacement curves for a very large beam are not smooth in softening regime when tensile strength is distributed stochastically. The scatter of the maximum vertical force around its mean value is similar for all beam sizes. The deformation field above the notch is strongly non-symmetric. The mean width of the localized zone above the notch is slightly higher than the deterministic value, namely: w = 16.56 mm (h = 8 cm), w = 18.88 mm (h = 32 cm) and w = 19.67 mm (h = 192 cm). In un-notched beams, the randomness of tensile strength more strongly affected the maximum vertical force (Fig. 11). The ratio between deterministic and stochastic calculations for a small, medium and large beam were respectively 3%, 8% and 14% (Fig. 11).

In Fig. 13, the summary results of deterministic and stochastic calculations of nominal beam strength are presented as compared to the size effect law SEL by Bažant (Bažant and Planas 1998). Both, the deterministic and mean stochastic results are in agreement with SEL. The effect of a random tensile distribution upon nominal strength clearly shows a decrease of nominal strength for larger structures. Figure 12 demonstrates the calculated width of a localized zone in un-notched beams. The localized zone is strongly non-symmetric and curved for all

a)

b)

a)

c) b)

Figure 12. Distribution of non-local softening parameter in tension in un-notched beams: a) small beam (h = 8 cm), b) medium beam (h = 16 cm), c) large beam (h = 32 cm).

c)

Figure 11. Normalized force-displacement curves in the case of deterministic (dashed lines) and random calculations (solid lines) for 3 un-notched beams under three-point bending: a) small beam (h = 8 cm), b) medium beam (h = 16 cm), c) large beam (h = 32 cm) (λx1 = 1 1/m, λx2 = 3 1/m, sft = 0.424 MPa).

Figure 13. FE-results for concrete beams from deterministic and stochastic calculations as compared with the size effect law SEL by Bažant (Bažant and Planas 1998).

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beams and also curved. The mean width of localized zones slightly increases with increasing specimen size from 15.04 mm (h = 8 cm) up to 16.58 mm (h = 32 cm). It is slightly smaller as compared to notched beams. The difference between the deterministic and mean stochastic width was negligible. 5

CONCLUSIONS

The following conclusions can be drawn from our nonlinear FE-investigations of a deterministic and statistical size effect in notched and un-notched concrete beams of a similar geometry: A deterministic size effect (nominal strength decreases with increasing specimen size) is very pronounced. It is caused by occurrence of tensile localized zone above the notch with a certain width. The material ductility increases with decreasing specimen size. A pronounced snap-back behaviour occurs for very large size beams (h/lc ≈ 400). The width of the localized zone above the notch slightly increases with increasing beam size. The solution of random non-linear problems on the basis of several samples is possible. A negligible stochastic size effect occurs for notched beams. A more pronounced stochastic size effect occurs for un-notched beams. The strength of several specimens with random sampling can be higher than a deterministic value. Localized zones in beams can be non-symmetric and curved. The correlated distribution causes larger scatter of nominal strength and lower mean stochastic value compared to distribution without correlation. Our research will be continued for reinforced concrete beams. Own laboratory experiments with different geometrically similar reinforced concrete beams with and without stirrups to investigate a deterministic size effect are now under way. In the case of laboratory tests to investigate a stochastic size effect (Koide et al. 1998), three reinforced concrete beams have a different length with the same cross-section.

Bažant, Z.P. & Jirasek, M. 2002. Non-local integral formulations of plasticity and damage: survey of progress. Journal of Engineering Mechanics, 128(11), 1119–1149. Bielewicz, E. & Górski, J. 2002. Shell with random geometric imperfections-simulation-based approach. Int. Journal of Non-linear Mechanics, 37, 4–5, 777–784. Bobi´nski, J. & Tejchman, J. 2004. Numerical simulations of localization of deformation in quasi-brittle materials within non-local softening plasticity. Computers and Concrete, 4, 433–455. Bobi´nski, J., Tejchman, J. & Górski, J. 2009. Notched concrete beams under bending-calculations of size effects within stochastic elasto-plasticity with non-local softening. Achives of Mechanics, 61, 3–4, 283–307. Koide, H., Akita, H. & Tomon, M. 1998. Size effect on flexural resistance due to bending span of concrete beams. In Fracture Mechanics of Concrete Structures (eds. Mihashi, H. Rokugp, K), Aedifactio Publishers: Freiburg, Germany, 2121–2130. Le Bellˇego, C., Dube, J.F., Pijaudier-Cabot, G. & Gerard, B. 2003. Calibration of nonlocal damage model from size effect tests. European Journal of Mechanics A/Solids 22, 33–46. Majewski, T., Bobi´nski, J. & Tejchman, J. 2008. FE-analysis of failure behaviour of reinforced concrete columns under eccentric compression. Engineering Structures, 30, 2, 300–317. Marzec, I., Bobi´nski, J. & Tejchman, J. 2007. Simulations of crack spacing in reinforced concrete beams using elasticplasticity and damage with non-local softening. Computers and Concrete, 4, 5, 377–403. Pijaudier-Cabot, G. & Bazant, Z.P. 1987. Nonlocal damage theory. ASCE J. Eng. Mech., 113, 1512–1533. Pijaudier-Cabot, G., Haidar, K. & Dube, J.F. 2004. Non-local damage model with evolving internal length. Int. J. Num. and Anal. Meths. in Geomech. 28, 633–652. Tejchman, J. & Górski, J. 2007. Computations of size effects in granular bodies within micro-polar hypoplasticity during plane strain compression. Int. J. for Solids and Structures, 45, 6, 1546–1569 . Tejchman, J. & Górski, J. 2008. Deterministic and statistical size effect during shearing of granular layer within a micro-polar hypoplasticity. Intern. J. for Numerical and Analytical Methods in Geomechanics. 32, 1, 81–107. Walukiewicz, H., Bielewicz, E. & Górski, J. 1997. Simulation of nonhomogeneous random fields for structural applications. Computers and Structures, 64, 1–4, 491–498.

REFERENCES Bažant, Z. & Planas, J. 1998. Fracture and size effect in concrete and other quasi-brittle materials. CRC Press LLC, Boca Raton.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

FE modeling and fiber modeling for RC column failing in shear after flexural yielding Kazuki Tajima, Nobuaki Shirai & Eisuke Ozaki Department of Architecture, College of Science and Technology, Nihon University, Tokyo, Japan

Kiwamu Imai KozoSoft Co., LTD, Tokyo, Japan

ABSTRACT: Objective of this study is to establish a fiber element-based analytical model capable of simulating inelastic behavior of the shear-critical RC columns including post-peak response. Experiment on the shearcritical column specimens was conducted under alternative cyclic loading, and image data were acquired to investigate failure and deformation mechanisms by means of the digital scanner and camera. In addition, the failure mechanism was investigated by the FE analysis and the total deformation was decomposed into flexural, shear and other components. FS specimen to be an object of the analysis failed in shear after flexural yielding, and the shear component in the hinge zone and the rotational one at the end section of the column were noticeable. The fiber model was extended to include shear behavior in the hinge zone and rotational behavior at the critical section by introducing sub-elements and limit state curves. Finally, validity of the modified fiber model was investigated through static monotonic and cyclic analyses on RC column and dynamic analysis on frame system. 1

INTRODUCTION

Various methods dealing with the damage evaluation have been presented in the previous studies. Williams et al. (1995) and Mehanny et al. (2000) reviewed the damage evaluation methodologies presented so far, and pointed out several issues. Among them, the following two items are important: 1. Researches on the shear-critical component, which may lead to overall collapse of the structure, are extremely limited, and 2. Reliable method that integrates local damage indices into global damage indices of the structure, is still not established yet. In Japan, the seismic performance of RC structures has been evaluated according to the Seismic Evaluation Standards (JBDPA, 2001). In other words, the evaluation method of residual seismic performance based on the Seismic Evaluation Standards has been applied to newly-designed (AIJ, 2004) or damaged buildings (JPDBA, 2001). According to this method, first the reduction factor of seismic performance is obtained for each component, and then the retained performance for each component is decreased by this factor. However, two issues stated in the above may be applicable to these methods as well. Mehanny and Deierlein (2000) presented the methodology to quantify system stability limit state in terms of the local damage indices of components in the

composite frames with RC columns and steel beams. They investigated destabilizing effects represented by the local damage indices, and integrated the destabilizing effects into the system stability limit states. However, they excluded shear-critical components from a viewpoint of the structural system adopted, and thus the damage evaluation of shear-critical members was out of the scope. Under these circumstances, the authors began with the research program to develop an analytical model for shear-critical RC columns subjected to static or dynamic loading, which is able to simulate complete inelastic behavior including post-peak response. The goal of this program aims at establishing a new numerical analysis-based evaluation method of the residual seismic performance for RC structures. For this reason, it is essential that the analytical model has capability to simulate overall behavior of structures. For extensive applications to research and design practices in the future, it is also important to enhance general-purpose capability. Under these requirements, the open-source code called ‘‘Opensees’’ shall be utilized as a computational code. In addition, analytical models to be developed are the extension of the socalled ‘‘Fiber model.’’ However, there remains a big challenge that the fiber model has to be extended to include shear failure of the component for achieving the objective of this study. This paper presents the test program, improvement processes of the fiber model and model verification.

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First, the failure test under gravity load on the RC column specimens is described. In the test, image data characterizing fracture behavior such as cracks and deformation were measured using the digital camera and scanner. Next, the finite element (FE) analysis was conducted to understand failure mechanism of the column, and then additional models to be included were investigated. In the present study, concept of the shear and joint sub-elements propsed by Filippou (1992) was introduced into the fiber model. Since restoring force characteristics to be provided for each sub-element is critical, how to evaluate them is investigated. The characteristics of the shear sub-element are determined by applying the modified compression field theory(MCFT) proposed by Vecchio and Collins (1986) and the concept of limit state curves by Elwood et al. (2003). Finally, the extended fiber model was verified.

2 2.1

GRAVITY LOAD FAILURE TEST

Figure 1. Table 1.

Geometry of RC column specimens. Material properties of concrete and reinforcement. Reinforcing bar (upper: D10, lower: D4)

Concrete

Compressive Young’s Yielding Young’s Year strength σB modulus Ec strength σB modulus Es 2007 26.0

2.99 × 104

2008 21.8

2.51 × 104

401 365 382 405

2.03 × 105 1.72 × 105 1.86 × 105 1.97 × 105

Test Program

The objective of the test program is to obtain restoring force characteristics of the RC column specimens failing in shear including post-peak response. The experiment was conducted for two years from 2007 to 2008. Two column specimens were made for the test in 2007; one is the specimen failing in shear after flexural yielding, and the other is the specimen failing in shear. The former is referred to as FS specimen and the latter S specimen. The ratios of the shear strength to the flexural strength of FS and S specimens were 1.05 and 0.79, respectively. The test in 2008 was limited to FS specimen and the image data regarding damages in the column was acquired in detail. Hereafter, the specimens shall be referred to as FS07, S07 and FS08 specimen. The geometry and the material properties are listed in Fig. 1 and Table 1, respectively. These specimens are representative of the interior columns at the lowest story in the low or medium story RC buildings. The size of specimen is about 1/3 scale of the actual size. Figure 2 shows the loading setup and measuring devices. Three actuators with capacity of 400 kN were used for loading. The actuator for lateral loading fixed to the reaction wall will produce an anti-symmetric distribution of the bending moment along the height of the column through the L-shaped loading beam. The pantograph was equipped to avoid rotation of the L-shaped beam during loading. Two actuators for axial loading fixed to the L-shaped beam have the sliding mechanism allowing to move horizontally. After applying constant axial load of 250 kN to the specimen, the lateral displacement of δH = ±1 mm was incrementally applied by the actuator. Consequently, alternative cyclic lateral load was applied to

Figure 2.

Loading setup and measuring devices.

the specimen. The load was determined with the load cell installed in the actuator. The lateral displacement was measured as the relative displacement between the top and bottom stubs with laser displacement transducer. The amount of axial expansion and contraction was measured with the displacement transducer of reel type. 2.2

Measurement of image data

2.2.1 Acquirement of image data When the scanner is used, damages are acquired by scanning over the surface of the specimen. In this experiment, the resolution of 600 dpi was selected by taking scanning time and accuracy of the image into consideration. Where, dpi is an abbreviation of ‘‘dot per inch’’, and it represents the number of pixels per one inch. Thus, the size of pixel for 600 dpi turns out to be about 0.04 mm. Crack width is measured by multiplying the size of single pixel by the number

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of pixels. The number of pixels at the cracking part can be determined by counting the number of pixels spanning over the crack width. This is a basic idea to evaluate the image data. Next, when the digital camera is used, accuracy of the image data depends on the number of pixels and the range of acquired image data. For example, if you will get the image data for the area of 120 mm × 160 mm by the camera with 12 million pixels, the size of pixel will be equivalent to one for the scanner with 600 dpi. However, deteriorating effect of the camera angle and the lightening condition on quality of the image is significant. For this reason, the fixed point observation was adopted in this test (Fig. 3). The image data over the surface of specimen were acquired by the digital camera with 10 million pixels, and the data were mainly used to decompose deformation of the column.

Figure 5 illustrates how to calculate flexural and shear components of the total deformation for a single measuring segment. The flexural component i δ flex can be calculated with the curvature φi obtained from the difference between the axial extensions hiL and hiR , the height of segment hi and the distance from the midheight of the specimen to the center of segment under consideration xi as follows:

2.2.2 Decomposition method of deformation with image data Total lateral deformation of the specimen δH can be decomposed into the flexural component δflex , the shear component δshear and the rotational component δslip from pullout displacement of the longitudinal bar and compressive deformation at both sides of the column. Figure 4 shows an example of the measuring points for decomposing deformation into components. For FS08, six measuring segments were arranged to measure shear deformation caused in the hinge zones. Three segments for FS07 and two segments for S07 were arranged.

Camera No.1

Camera No.2

Figure 3. camera.

Camera No.3

Measurement of deformation and damages by

for

Hinge Zone Hinge Zone

i δflex

for

flex

(1)

= φi · hi · xi

(2)

Now, the shear component i δshear can be calculated from the diagonal extensions lD13 and lD24 as follows: i δshear

shear

=

lD24 − lD13 2 cos θ

(3)

Finally, δ flex , δ shear and δ slip can be calculated as follows:  δflex = (4) i δflex δshrear =



i δshear

(5)

  δslip = δH − δflex + δshare

2.3

75 250 75

FS

hiL − hiR D · hi

φi =

(6)

Test results and failure mechanism

2.3.1 Lateral load—lateral displacement relationship and failure progression Figure 6 shows the lateral load (P)—lateral displacement (δH ) relationship. For FS07, tensile yielding of the longitudinal bars at the ends of column and flexure-shear cracking at the top of column (δH = 3.9 mm) were observed. After that, yielding of hoops (δH = 6.4 mm) occurred. The maximum load Pmax was attained at δH = 10 mm for both the positive and negative loading. After the peak load, the load degrades slowly. FS08 indicated the similar failure process to the present one.

xi

lD13 = lD13' lD13 lD24 = lD2'4' lD24 2 2'

xi

3 lD2'4' lD13

D hiL

hiR

lD24

hi

hi (a) i

Figure 4. tion.

1 flex

(b) i

3' lD13'

4 4' shear

Measuring points for decomposition of deformaFigure 5.

Flexural and shear deformation i δflex and i δshear .

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shear_other

shear

80

shear_hinge

Ratio Componet to

H

(%)

100

shear

60 slip

40 20

flex

(a) FS07 0

Figure 6.

5

10 15 H (mm)

slip

slip

flex

20 0

(b) S07 5 H (mm)

flex

10 0

Figure 7.

Ratios of δflex , δslip and δshear to δH .

Figure 8.

Final failure patterns of specimens.

(c) FS08

5 H (mm)

10

P – δH relationship.

For S07, shear cracks occurred near the top and bottom regions of column at δH = 2 mm. The maximum load Pmax was attained at δH = 3 mm in the positive loading and δH = −2.5 mm in the negative loading. Hoops yielded at δH = 2.5 mm, and then shear cracks over the height of column were observed. Finally, the axial limit load was attained at the same time of the abrupt strength reduction on the way of the cycle to δH = 11 mm. 2.3.2 Decomposed Deformations Figure 7 shows the ratio of decomposed component of each deformation to δH . Note that the shear components for the hinge zone and the other zone are indicated separately in FS08 specimen. For FS, the ratio of δflex + δslip to δH is about 80 % and the ratio of δshear to δH is about 20%. For FS08, the ratio of δslip to δH amounts to over 50%. Thus, this may lead to the difference in the post-peak behavior of P-δH response. Also, it is seen from the ratio of δshear to δH that the shear component concentrates on the hinge zones. For S07, the ratio of δshear to δH increases gradually and finally amounts to over 80%. 2.3.3 Final failure mode and mechanism Figure 8 shows the final failure patterns of the specimens. For FS, shear cracks caused in the hinge zones at the top and bottom of column are noticeable. For FS08, cracks along the boundary between stub and column have been opened, and thus pullout displacement of the longitudinal bars can be observed. For S specimen, shear cracks have been observed all along the height of column, and the shear component of deformation is dominant. Although the final failure mode for FS specimens was the shear failure after flexural yielding, big difference may be identified in their failure mechanisms. Behavior of FS08 is extremely complicated such that significant rotational deformation due to the pullout of bars from the stub has been measured in the hinge zones as well as the shear deformation. In this study, the authors will focus on analytical modeling for the FS08 specimen.

3 3.1

3-D FE ANALYSIS OF RC COLUMN FAILING IN SHEAR AFTER FLEXURAL YIELDING Objective of analysis

FS08 specimen was selected as the object and 3-D FE analysis was conducted to identify well-suited FE models for simulating the observed failure mechanisms. Four case studies listed in Table 2 are investigated. Case-1 is the reference case with the simplest models. Case-2 includes modeling of the bond slip between concrete and steel bar. Case-3 includes modeling discontinuous cracks along the boundary between stub and column. Further, Case-4 integrates the modeling of Case-2 and Case-3 to include the pullout displacement of steel bars causing opening displacement between stub and column. The computational code DIANA-9.3 was utilized. 3.2

Finite element models and procedure

Figure 9 shows the mesh division and the loading and boundary conditions. In Case-3 and Case-4, the plane interface elements were inserted into the boundary between stub and column to represent discrete cracks. Normal stiffness of the interface element was evaluated according to the tension softening curve of concrete. Shear or tangential stiffness of the interface element was assumed to have a shear retention factor of 5% after cracking in concrete to take aggregate interlocking and dowel action into account. The longitudinal bar spanning discrete crack was modeled by the point interface element. Stiffness of the normal spring

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was determined on the basis of the stress—strain relation of steel and stiffness of the tangential spring was assumed to be one hundred times of stiffness of the normal spring. After imposing the dead weight and the constant axial force of 250 kN on the top of specimen, the monotonic lateral load was applied to the top stub in an incremental manner by the displacement control as shown in Fig. 9. Figure 10 shows the assumed uniaxial stress-strain relationships for concrete. The compressive branches were represented by two parabolic curves: one for ascending branch and other for descending branch after the peak. The area under the descending curve after the peak stress is defined as the regularized compressive fracture energy Gfc by the characteristic element length Lc . In this study, Lc was set to be side length of the cube with volume equivalent to one of the element to be used. Gfc [N/mm] was determined according to the following empirical equation by Nakamura Table 2. Analtyical cases investigated and typical models used.

et al.(1999):  Gfc = 8.8 fc

(7)

Where, fc [N/mm2 ] is the compressive strength of concrete. The tensile stress—strain relationship for the ascending branch was assumed to be linear elastic until the tensile strength of concrete ft , and the descending curve was expressed by the tri-linear softening model. The area under the softening curve after ft is defined as the regularized fracture energy Gf by Lc . Gf [N/mm] was determined according to the following empirical equation by Oh-oka et al. (2000): Gf =

0.23fc + 136 1000

(8)

Figure 11 shows the assumed stress—strain relationship for steel. The stiffness after yielding was set to be 1/100 of the initial stiffness. Figure 12 shows the assumed bond stress-slip relationship for the longitudinal bar in Case-2 and Case-4. This follows the CEB model (Confined concrete Bond conditions:Good) (1993). The maximum bond stress τmax [N/mm2 ] was determined according to the following formula (AIJ, 2003):   σ0 23 τu = 0.7 1 + σB / σB

(9)

where, σ 0 [N/mm2 ] is the applied compressive stress in column and σ B [N/mm2 ] is the compressive strength of concrete. 3.3

(MPa)

Mesh Division, Loading and Boundary Condi-

Figure 13 shows the lateral load (P)—displacement (δH ) relationship. The predicted response up to the maximum load by Case-1 overestimates both the stiffness and strength. Although the predicted responses by Case-2 with the bond-slip and Case-3 with the interface elements between stub and column simulate the stiffness degradation from P = 60 kN, agreement with the test result is not sufficient. On the other hand, Case-4 well simulates the test result.

ft

Ec

0

P

Tensile Stress t (MPa)

fc Gfc / Lc u

Compressive Strain

Figure 10.

0

Ec G / L f c P

y

Stress

Compressive Stress c (MPa)

Figure 9. tions.

Results by FEM and discussion

wu

Es 0

Tensile Strain

Uniaxial stress-strain relationships for concrete.

Figure 11.

Es/100

Strain

Stress-strain relationship for steel.

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Bond Stress

(MPa) b

max

f

Figure 12.

2. In modeling of the fiber method, the shear behavior in the hinge zone and the rotational behavior at the column/stub interface due to the pullout of bars have to be included.

4 4.1

0

S1

S2 Slip S (mm)

Shear sub-element and joint sub-element

Based on the findings by the FEM, the shear subelement and the joint sub-element proposed by Filippou (1992) will be included in the fiber model. The concept of these sub-elements is schematically shown in Fig. 15. Note that these sub-elements have been implemented in the OpenSees.

S3

Bond stress-strain relationship.

4.2

Figure 13.

IMPROVEMENT OF FIBER MODEL

Verification of sub-elements

The shear sub-element may be characterized by such mechanical properties as the shear force—shear deformation relation in the hinge zone, and the joint subelement by the bending moment—rotational angle relation at the section along the stub/column interface. For verification of the sub-elements, first the mechanical properties to be provided for the sub-elements will be directly obtained from the test results. Then, the fiber analysis with the sub-elements will be conducted and the predicted results will be compared with the test results. Figure 16 shows the fiber mesh division and the assumed stress—strain relationship for concrete. The column was discretized with 11 segments and 12 elements along the height. The section of column was divided into the fibers of 20 × 20. The cyclic stress—strain relation for concrete was represented by the model called ‘‘Fedeas material’’ implemented in the OpenSees as shown in Fig. 16. Tensile resistance of concrete was disregarded. The bi-linear stress— strain relationship was assumed for reinforcing bars. The stub concrete blocks were assumed to be the rigid body. The bottom of the lower stub was set to be the fixed end, and only the rotational degree of freedom at

P − δH relationship.

Figure 14. Observed cracking pattern and predicted principal tensile strain distribution. Joint Sub-Element

Figure 14 shows the observed cracking pattern at the maximum load (δH = 10 mm) and the predicted principal tensile strain distribution by Case-4. This strain distribution simulates shear cracks observed in the test relatively well. In addition, the opening due to the pullout of bars from stub can be seen visually. Consequently, the following conclusions can be obtained through the FE analysis: 1. The modeling of Case-4 is effective for simulating the test result, and

Shear Sub-Element Fiber Model

Figure 15. element.

Concept of shear sub-element and joint sub-

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250 kN

150

Element Segment Ec = 2 x Fc / Fc

p

P (kN)

Fedeas Material

Fiber Discretization of RC Column Section Fcu p

u

100

50

Figure 16. Fiber mesh division and stress—strain relationship for concrete.

Test Analysis 0

5 H (mm)

10

Figure 18. P − δH response. Figure 17. Identified mechanical properties for shear and joint sub-elements.

4.3

Derivation of mechanical properties for shear sub-element

4.3.1 Application of modified compression field theory MCFT proposed by Vecchio and Collins (1986) shall be applied to evaluate the shear force-shear deformation relation for the shear sub-element; that is, the hinge zone of column. The theory was applied to the RC panel representing the hinge zone in the column. Figure 19 shows the comparison between the calculated and measured shear force—shear deformation relationships for the hinge zone. The shear force was calculated by multiplying the predicted shear stress

Q (kN)

the top of the upper stub was fixed. After imposing the constant axial force of 250 kN on the top of column, the monotonic lateral load was applied incrementally by the displacement control. Figure 17 shows the identified mechanical properties for the shear sub-element and the joint subelement obtained from the test results. The shear forces and bending moments inducing in the column can be directly obtained from the applied loads. The components of shear and rotational deformation decomposed with the image data were utilized. For verification, the curve up to δH = 10 mm for each sub-element was simplified by the tri-liner model as shown in Fig. 17. Figure 18 compares the calculated P − δH response by the modified fiber model with the test result. The prediction compares with the observed one fairly well. Thus, it can be said that an introduction of the subelements into the fiber model is effective to simulate the response of column with this type of failure mode.

150 100 50 0 0

Test MCFT 0.5

1.0 (mm) shear

1.5

Figure 19. Q − δshear relationships.

by the cross sectional area of column, and the shear deformation by multiplying the predicted shear strain by the length of hinge zone. The prediction simulates the measured response relatively well, and thus this approach is simple and useful for deriving shear behavior for the shear sub-element. 4.3.2 Limit state curves Unfortunately, MCFT can not predict the post-peak softening response. In other words, the theory provides no information about the limit states. Elwood et al. (2003) proposed two kinds of the limit state curves; that is, the shear limit state curve and the axial limit state curve, as shown in Fig. 20. The shear failure point or the axial failure point is defined as an intersection

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Figure 20.

Axial Load N

Shear Force Q

Shear Limit Shear State Curve Failure Axial Axial Failure Failure Axial Limit State Curve

Shear limit state curve and axial limit state curve.

of either the shear or axial limit state curve and the response curve of column. The shear limit state curve is defined by the following equation: 3 1 v 1 P 1 s = + 4ρ  − ≥  − L 100 500 fc 40 Ag fc 100

(10)

where, L [in] is the height of column, ρ  the steel ratio of hoop, ν [ksi] theshear stress, fc [ksi] the compressive strength of concrete, P [kips] the axial force, and Ag [in2 ] the cross sectional area. This equation is applicable until the chord angle of 1/100 and the unit system used is psi. On the other hand, the axial limit state curve is defined by the following equation: 4 1 + (tan θ )2 a  = s L 100 tan θ + P Ast fyt dc tan θ

(11)

where, θ is the critical crack inclination (assumed to be 65◦ ), s [in] the interval of hoop, Ast [in2 ] the area of hoop, fyt [ksi] the yield strength of hoop, and dc [in] the distance between the centers of consecutive hoops. s and a in Eqs. (10) and (11) are the lateral displacements when the shear and axial failures occur, respectively. The shear components have to be extracted from s and a to include the concept of the limit state curves in the shear sub-element. The fiber element and the sub-elements shall be connected in series in this modified fiber method. If the shear sub-element reaches its maximum strength and the softening begins, the fiber element and joint subelement will follow the unloading path. Consequently, the shear sub-element will control the overall behavior of column. In other words, the deformation induced in the shear sub-element becomes almost equivalent to the total deformation of column. To avoid this problem, the shear component to be used in the shear limit state curve to determine a shear failure point has to be determined by proportioning s according to the decomposed component. In this case, δshear was assumed to equal to 0.2s (see Fig. 7). On the other hand, a shall be used in the axial limit state curve to determine an axial failure point without extracting the shear component.

Figure 21. Assumed restoring force characteristics for shear sub-element.

Figure 21 shows the final restoring force characteristics for the shear sub-element. The ascending curve was predicted by applying MCFT to the hinge zone of column, and then the curve was represented by the bilinear model. The descending branch after the peak was modeled by the line connecting the shear failure point to the axial failure point. 4.4

Derivation of mechanical properties for joint sub-element

Mechanical properties of the joint sub-element may be characterized by flexural cracks along the stub/column interface, yielding of the longitudinal bars and pullout of the longitudinal bars from the stub. The bending moment at cracking Mcr was calculated according to the AIJ formula (AIJ, 1999): √ Mcr = 0.56 σB · Z

(12)

where, σB [N/mm2 ] is the compressive strength of concrete and Z [mm3 ] the section modulus. The yield bending moment My was determined according to the following formula (AIJ, 1999) as:  My = 0.8at σy D + 0.5ND 1 −

N bDFc

 (13)

where, at [mm2 ] is the area of tensile reinforcement, σy [N/mm2 ] the yield stress of bar, D [mm] the depth of column, b [mm] the width of column, N [N] the applied axial force and Fc [N/mm2 ] the specified strength of concrete. The compressive strength σB was used instead of Fc . The pullout displacement of the longitudinal bars Sy [mm] was calculated according to the AIJ formula (AIJ, 2003) as:

   23 Sy = (0.321ub σB / + 0.463) · εy · D 2

(14)

  ub = (1 + γ ) · σy · db 4D

(15)

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where, εy is the yield strain of longitudinal bar, γ the ratio of the area of compressive reinforcement to one of tensile reinforcement and db the diameter of bar. The rotational angle at the yielding was obtained from Sy . Figure 22 shows the bending moment—rotational angle relationship determined on the basis of the above procedure. This relation well simulates the test result obtained from the image data. The stiffness after the yielding was assumed to 1/10 of the Kjoint 4.5

Hysteresis model

Since the shear-critical column is the object of this analysis, it is essential to establish hysteresis model for the shear sub-element. The sub-element implemented in the OpenSees allows to input the unloading stiffness and the pinching factors px and py as shown in Fig. 23. The unloading stiffness is defined in terms of the ductility μ indicating the ratio of the maximum displacement to the elastic limit displacement and the unloading stiffness coefficient β. The data base on the column tests conducted in Japan was established to determine β. Figure 24 shows the relationship between β and the shear margin. The following fitting equation for β was identified from the test data as:  β = 0.77(Qsu Qmu )−0.9

(16)

On the other hand, the proper pinching factors have not been found, and thus the best fit vales to the test data was assumed in this study.

Figure 24. Relationship between unloading stiffness coefficient β and shear margin Qsu/ Qmu .

5 5.1

Bending moment—rotational angle relation-

Figure 23. factor.

Definition of unloading stiffness and pinching

Verification by pushover analysis for FS08

FS08 specimen was analyzed under the pushover lateral loading by the modified fiber model including the sub-elements. The mesh division and the loading and boundary conditions were the same as those stated in Section 4.2. The mechanical properties for the sub-elements were also given in Section 4.3. Figure 25 shows the comparison between the predicted and observed P − δH relationships. The result by the ordinary fiber model without sub-element is also shown in the figure. The modified fiber model simulates the test response including the post-peak softening behavior fairly well. 5.2

Figure 22. ship.

VERIFICATION OF MODIFIED FIBER MODEL

Verification by cyclic analysis for FS08

Now, FS08 specimen was analyzed under alternative cyclic lateral loading by the modified fiber model including the sub-elements. The hysteresis behavior for the shear sub-element was represented by the model implemented in the OpenSees. The unloading stiffness was determined by using β from Eq. (16). The pinching factors px and py were identified from the measured shear force—shear deformation relationship obtained by the image measurement. Consequently, β = 0.74, px = 0.78 and py = 0.54 were used for the computation. For the joint sub-element, the unloading stiffness was assumed to be equal to the initial stiffness and the reloading stiffness was assumed to be the peak-oriented type. Figure 26 shows the comparison between the predicted and observed cyclic lateral load—lateral displacement relationships. The modified fiber model compares the observed cyclic response as a whole. However, some difference in the unloading behavior may be observed in the post-peak region. This produces a little bit fatter hysteresis loop than the test. This issue is the remaining research work to be solved in the future.

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5.3

Verification by monotonic, cyclic and dynamic analyses for RC frame

The goal of this research is to evaluate the seismic performance of overall RC structure. As a first step, the shaking table test on the RC frame consisting of three columns tested by Elwood et al. (2003) was analyzed by the modified fiber method. 5.3.1 Outline of test program Figure 27 shows the geometry of test frame and the reinforcement detail of the columns. The interior column was so designed to fail in shear. The additional mass of about 23 ton was rested on the beams. The input acceleration time history is the scaled one of the acceleration recorded during Chile earthquake in 1985. In the test, all columns first experienced the flexural yielding, and then the interior column failed in shear.

Figure 25.

Predicted and observed P − δH relationships.

Figure 26. ships.

Predicted and observed cyclic P − δH relation-

Figure 27. Geometry of test frame and reinforcement detail of columns.

5.3.2 Analytical models and results Figure 28 shows the modeling of test specimen. The columns were modeled by the fiber elements, and the beam and footing were assumed to be rigid body. The joint sub-element was inserted into each end of member, and the shear sub-element was positioned at the top and bottom of the interior column. The section of column was discretized into 10 × 10 fiber elements. The stress-strain relationship was modeled by the Fedeas material, and the bi-linear stress-strain relation was provided to the reinforcing bars. Figure 29 compares the predicted base shear or shear force—lateral displacement relationship with the envelope curve of the test. The predicted response for the frame shows gradual degradation of the capacity and is similar to the test result. Also, the predicted response for the interior column shows degradation of the shear capacity due to shear failure after flexural yielding alike the test. Figure 30 shows the predicted and measured base shear—lateral displacement relationships. The frame was analyzed by the cyclic static procedure instead of the dynamic one as a preliminary study. After imposing the dead weight on the top of each column, the lateral displacement measured during the shaking table test was applied to the top of interior column. The analysis simulates the maximum strength and hysteresis loops very well. This suggests that the modified fiber model with the sub-elements is applicable to RC framed structures. Figure 31 shows the predicted base shear or shear force—lateral displacement relationship under seismic excitation. The acceleration record of N-S component (JMA Kobe NS) recorded during Kobe earthquake in 1995 was used for the dynamic analysis. Damping

Figure 28.

Modeling of test specimen.

Figure 29.

Base shear or Q − δH relationship.

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Figure 30. tionships.

Predicted and measured base shear – δH rela-

2. The failure mode of the RC column failing in shear after flexural yielding can be simulated by the FE analysis including the bond-slip model between concrete and steel and the discrete crack model along the stub/column interface. 3. The restoring force characteristics for the shear sub-element can be constructed by applying the modified compression field theory. The bending moment-rotation relation for the joint sub-element can be formulated in terms of the pullout displacement. 4. The strength degradation after shear failure can be modeled by introducing the limit state curve to the shear sub-element. The inclination of softening after the peak can be represented by a line connecting the shear failure point to the axial failure point. 5. According to the response results by the dynamic analysis on the frame system consisting of three RC columns, the hysteretic response in the positive side differed from one in the negative side. Thus, it is needed to evaluate the damage indices under seismic excitations; that is, dynamic loads

Figure 31. Predicted dynamic responses of frame and interior column under seismic excitation.

proportional to the tangential stiffness of the system was adopted, and the damping factor h was assumed to be h = 2.0%. The natural period of the system was about 0.2 sec. in both the test and analysis. As far as the response for the frame system is concerned, behaviors in the positive side are different from ones in the negative side, and an abrupt reduction of the capacity occurs in the negative side. At this moment, the interior column carries no shearing force in the negative side. It seems that this results shows a matter of great importance for the application of dynamic analysis.

6

ACKNOWLEDGEMENT This research was supported by Japan Society for the Promotion of Science under Grant-in-Aid for Scientific Research (C) (Head Investigator: Prof. Shirai).

REFERENCES

CONCLUSIONS

The advanced analytical model capable of simulating inelastic behavior of the RC column failing in shear after flexural yielding under seismic excitations was constructed by introducing the sub-elements to the ordinary fiber model. The following results were obtained through the investigation: 1. The image measurement on the RC column failing in shear after flexural yielding was conducted to acquire failure and deformation modes, and the total deformation was decomposed into the shear, flexural and other components. It was found that the shear deformation concentrates in the hinge zone after shear cracking and the rotational deformation occurs at the stub/column interface due to the pullout of longitudinal bars.

AIJ(Architectural Institute of Japan). 1999. AIJ Standard for Structural Calculation of Reinforced Concrete Structures. Japan. (in Japanese). AIJ. 2003. Design Guidlines for Earthquake Resistant Reinforced Concrete Buildings Based on Inelastic Displacement Concept. Japan. (in Japanese). AIJ. 2004. Guidlines for performance evaluation of earthquake resistant reinforced concrete buildings (Draft). Japan. (in Japanese). CEB. 1993. CEB—FIP MODEL CODE1990 DESIGN CODE. Thomas Telford Service Ltd. Elwood, K.J. and Moehle, J.P. 2003. Shake Table Tests and Analytical Studies on the Gravity Load Collapse of Reinforced Concrete Frames. PEER-2003/01. Filippou, F.C. 1992. Nonlinear Static and Dynamic Analysis of Reinforced Concrete Subassemblages, UBC/EERC92/-8. JBDPA(The Japan Building Disaster Prevention Association).2001. Standard for Seismic Evaluation of Existing RC Buildings. Japan (in Japanese). JBDPA. 2001. Guideline for Post-earthquake Damage Evaluation and Rehabilitation. Japan (in Japanese). Mehanny, S.S.F. and Deierlein, G.G. 2000. Modeling of assessment of seismic performance of composite frames with reinforced concrete columns and steel beams.

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Rep. No. 135. The John A. Blume Earthquake Engineering Center, Stanford Univ. Nakamura, H. and Higai, T. 1999. Compressive fracture energy and fracture zone length of concrete. Modeling of Inelastic Behavior of RC Structures under Seismic Loads. ASCE.: 471–487. Oh-oka, T., Kitsutaka, Y. and Watanabe, K. 2000. Influence of short cut fiber mixing and curing time on the fracture parameter of concrete. Journal of Structural and Construction Engineering, Transactions of AIJ. (529): 1–6. (in Japanese).

Vecchio, F.J. and Collins, M.P. 1986. The Modified Compression-Field Theory for Reinforced Concrete Elements Subjected to Shear, ACI JOURNAL: Williams, M.S. and Sexsmith, R.G. 1995. Seismic Damage Indices for Concrete Structures: A State-of-the-ArtReview, Earthquake Spectra, 11(2): 319–349.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Numerical simulation and experimental testing of a new bridge strengthening method W. Traeger, J. Berger & J. Kollegger Institute for Structural Engineering, Vienna University of Technology, Austria

ABSTRACT: Since bridges are constantly subjected to cyclic loading due to the high traffic load, deterioration in stiffness is often the result. A new strengthening method was developed and tested for its behaviour under fatigue. A thin bonded unreinforced overlay made of concrete for the upper region with exposed aggregate surface was added to the existing bridge deck to increase its load carrying capacity. As the overlay also serves as concrete carriageway, it reduces the weights on the bridge since no asphalt surface is needed anymore. In order to achieve a high efficiency of the fatigue testing facility, analytical and numerical analyses were carried out. Because of the stiffness, mass and damping features, the test specimen itself becomes the major element of the testing facility. In order to obtain values of the stiffness of the reinforced concrete, specimen calculations are made by using a finite element program, taking into account the nonlinear material behaviour and concrete cracking. With the previously determined stiffness, the resonance frequency was determined analytically and numerically by using the finite element method in a 3D- model. 1

2

INTRODUCTION

In recent years strengthening and restoration of existent bridges, particularly with concrete or cementbased overlays, have been of a great interest for the possibility of increasing loads and traffic intensities. It is therefore essential for better understanding and predicting the fatigue behavior of the composite section. The Association of the Austrian Cement Industry (VÖZ) assigned the Institute for Structural Engineering of the Vienna University of Technology to perform seven fatigue loading tests on seven specimens and two static loading tests until failure after the fatigue loading test. The set-up for dynamic fatigue testing using the resonance effect, developed at the Institute for Structural Engineering at Vienna University of Technology, saves time and energy in contrast to servo-hydraulic testing machines [Kollegger, 2005]. Since conventional servo-hydraulic machines achieve frequencies for large structural components lower than one cycle per second this new approach increases the frequency up to at least ten times and therefore has a strong impact on time and energy consumption. Depending on the specimen, the testing of large components could be carried out 20 to 40 times faster compared to conventional servo-hydraulic testing methods while the energy consumption dropped down three orders of magnitude.

EXPERIMENT DESCRIPTION

In the new restoration method the plain concrete overlay will not only be a part of the existing bridge, it should also present a concrete carriage-way in bond with the existing structure. A crucial factor in regard to the new method of constructing the composite section was the installation of the overlay and how to increase the overlay-substrate bond. At the beginning the substrate or the main panel was cast in site and the surface roughness of the substrate was increased by sandblasting. By increasing the surface roughness the substrate-overlay bond became stronger. In some of the slabs dowels were placed to enhance the bond behavior and to draw a comparison between overlays with and without dowels. After roughening the surface and placing the dowels in some of the specimens, a 2 mm thick highly elastic polyurethane layer was placed on the roughened surface of the substrate. Since the polyurethane layer had a smooth surface, a thin layer of primer and grains of sand were applied to roughen the surface of the polyurethane layer. At the end the plain concrete overlay was cast on the roughened surface of the polyurethane layer. The bond layer had a shear strength of 1 N/mm2 . The main task of this layer was to avoid the debonding between the two concrete sections and make the overlay influence effectively to the load bearing capacity of the section.

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Figure 1. The conventional remediation system and the newly developed one.

In conventional methods there are at least 5 to 6 cm intermediate layer between the existing structure and the new overlay which doesn’t support the composite action and requires an overlay with larger thickness. Figure 1 shows the difference between the conventional restoration system and the newly developed one. To accurately model the behavior of a box girder bridge in the field region and the flange region two different specimens were used. Since in the field range the moments are always positive compression zone supplements (CZS) were used. To model the flange region where the moments are negative, specimens with a tension zone supplement (TZS) were used. In the specimens with compression zone supplements the overlay works as a part of the structure and contributes to the load bearing capacity of the section, which is why the goal of using these specimens was to check the durability of the bond and its shear carrying capacity. It was also important to investigate the composite action between overlay and substrate to ascertain the effectiveness of the bonding layer. However in the tension zone supplements it was assumed that the plain concrete overlay wouldn’t increase the load carrying capacity since plain concrete doesn’t contribute to tension after cracking.

3 3.1

Figure 2. Alternative spring model of the fatigue tests on bending beams.

Figure 3.

ANALYTICAL CALCULATIONS Analytical calculation of the expected testing frequency

For reasons of comparison, analytical and numerical analyses were carried out [Berger, 2008]. The spring model of the test set-up results from two parallel connected springs, with the stiffness of the specimen and the auxiliary structure, as pictured in figure. 2. The stiffness of the auxiliary structure consists of the device structure like spring packs, threaded rods and channel sections. The stiffness calculation of the specimen is uncertain because of the transition of reinforced concrete from the uncracked to the cracked condition. 3.1.1 Mass and stiffness calculation of the auxiliary structure For the determination of the equivalent mass and stiffness of a specimen under flexural stress a monomial

Fundamental wave shape for the cantilever.

Ritz approach for dynamic deformations was used. For the crosshead the distribution function ϕ(x) (1) was chosen, which is affine to the bending line of a cantilever beam (Fig. 3).     x 2  x 3 1 3 (1) ϕ(x) = − 2 l l The generalized mass and stiffness of the crosshead were calculated with the following equations: l m ˜ =

2 ρA(x) ϕ(x) dx

(2)

33lρA 140

(3)

0

m ˜ stell section =

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k˜ =

l 2 EI ϕxx(x) dx

(4)

3EI k˜Steel section = 3 l

(5)

EI π 4 k˜Specimen = 2 · l3

(8)

0

The stiffness and mass of the auxiliary structure is shown in table 1. Because of large difference between the stiffness of the spring packs compared to the threaded rods and the crosshead the overall stiffness is close to the stiffness of the spring packs. The mass of the spring packs can be neglected because this mass is hardly swinging with the overall system. At the Uchannels a rigid body motion is taking place because of the deflection of the cantilever beam generated through bending. 3.1.2 Mass and stiffness calculation of the testing specimen As approach for the distribution function ϕ(x) (6) of the testing specimen a sinusoidal half- cycle, which is affine to the bending line of a single- span beam under a point load, was used (Fig. 4). ϕ(x) = sin

πx l

(6)

The generalized mass and stiffness of the testing specimen were calculated with the equations (2) and (4) with the distribution function ϕ(x) (6). m ˜ Specimen = ρA

l 2

(7)

Table 1. Overview of the masses and stiffnesses of the auxiliary structure.

Steel section Threaded rod Spring pack Stiffness-auxiliary struct. Mass—auxiliary struct.

The determination of the stiffness of reinforced concrete sections depends very much on the load. In the state of transition into the cracked condition, a sharp decrease of the stiffness can be observed. As limits for the determination of the stiffness the pure uncracked and the pure cracked conditions were analyzed. In the following table the respective stiffnesses for the two types of specimen are given. The calculation of the stiffness in the uncracked condition was carried out for a ratio of the Young’s modulus of reinforcing steel and concrete of n = 6.06 and a Young’s modulus for the concrete of Es = 33, 000 N/mm2 . The values given in table 2 show the strong stiffness decrement, especially in the tension zone supplements with a decrease factor around 16. 3.1.3 Calculation of the testing frequency on equivalent system The total mass m for the single degree of freedom model results from the sum of the individual vibrating elements of the test equipment. In addition to the already considered mass of the auxiliary structure and the generalized mass of the specimen, the mass of the vibration generator needs to be determined. The stiffness and the mass of the box sections for the support of the specimen can be neglected due to the large difference in the stiffness of the auxiliary structure and the specimen. In contrast to the crosshead, the box sections are conveniently loaded and should not resonate in the experimental setup. The overall stiffness follows from a summation of the stiffness of the auxiliary structure and the test specimen. Disregarding the damping, the expected frequencies are shown in table 3. The table shows the sharp drop of the frequency at the transition to the uncracked condition.

Stiffness [kN/cm]

Mass [kg]

1102, 5 × 2 562, 0 × 2 20, 9 × 2 19, 79 × 2 = 39, 6

154, 5 × 2 61 × 2 −

Table 2. Comparison of the stiffness of the specimens in the uncracked and cracked condition.

431

CZS TZS

EI-Uncracked [kN cm2 ]

EI-Cracked [kN cm2 ]

1,760 e9 1,760 e9

4,003 e8 1,0998 e8

Table 3. Comparison of the first natural frequencies of the specimens in the uncracked and cracked condition.

Figure 4.

Fundamental wave shape for a single-span beam.

CZS TZS

f0 -Uncracked [Hz]

f0 -Cracked [Hz]

26,21 26,21

13,51 8,65

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4

NUMERICAL SIMULATION

4.1

Nonlinear FE computation of the specimen stiffness

The uncracked and the cracked conditions are the two limits for the expected test frequency. In reality, it should, if the tensile strength of concrete is exceeded, lead to a lower reduction of the stiffness than a perfectly cracked cross- section. This drop largely depends on the actual tensile strength of the concrete. To obtain an estimation of the stiffness subjected to the tensile strength, two-dimensional FE- calculations were carried out, taking the non-linear material behavior of concrete into account. The determination of the stress- strain- diagram of concrete (C30/37) is based on EN 1992-1-1 [EN 1992-1-1, 2004] for non-linear calculations of concrete structures. For a short- acting, uniaxial compression, the relationship between σc and εc is described by equation (11). k · η − η2 σc = fcm 1 + (k − 2) · η

(9)

whereas is: η=

only transmit tensile forces. For the modeling of concrete a 4 node 2-D element for a state of plane stress was used. The assembly of the two elements took place by means of a complete bond condition. The release of the composite behavior is described by the deflection of the concrete tensile strength after crossing the tensile strength.

εc εc1

εc1 = 0, 2% (Compression of concrete at peak compressive stress) k = 1, 1 · Ecm ·

|εc1 | fcm

(fcm = 37, 9 N/mm2 ,

Ecm = 33.000 N/mm2 )

The calculations were carried out with the program ABAQUS. For the concrete, the already implemented plasticity model by Lee and Fenves [Lee & Fenves, 1998] was used on the basis of Drucker Prager failure surface [ABAQUS, 2006]. The tensile strength of the concrete varies in the range between 0.5 N/mm2 and 2.8 N/mm2 . The ‘‘tension stiffening’’ behavior of concrete can be described by the defection of the stress-strain-diagram. In the present case, a constant softening module Esoftening = 1030 N/mm2 was scheduled. With a tensile strength of 2 N/mm2 the elongation of each finite element is εs = 0.2% when the tensile strength drops to zero. This corresponds to the selected mesh refinement of a fracture energy of GIF = 120 N/m and thus is in an area that is common for nonlinear calculations. For the modeling of the material behavior of reinforcement steel a linear elastic material model was used. A yield stress of fy = 550 N/mm2 and a Young’s modulus of Es = 200, 000 N/mm2 were chosen. The reinforcement was modeled by bar elements which can

4.1.1 Compression zone supplement The steel reinforcement at the bottom of the base plate As = 22 cm2 (7Ø20). The concrete cover is 3 cm. For technical reasons an additional structural reinforcement of As = 5.65 m2 (5Ø12) was placed in the middle of the specimen. The single load at midspan Fo,CZS = 132.6 kN, which is also the upper load of the compression zone supplement (Fig. 5). For a specimen with a tensile strength of ft = 1 N/mm2 , the plastic distortions of concrete under the maximum upper load are shown in figure 6. With the program ABAQUS no cracks can be represented but the plastic distortion in the tensile zone of the concrete can be interpreted as ‘‘cracks’’. The plastic strain in the tension zone remains zero until it reaches the predetermined tensile strength of concrete. If the tensile strength is exceeded, cracks occur which is shown in the FE- model as plastic strain in the concrete. This means that there is a direct correlation between these two variables and the size of the plastic strain provides information on the size of the crack opening. Due to the reinforcement, a uniform crack pattern develops over the cross section. With the resulting deformation and with equation (9) the stiffness EI of the specimen can be calculated. Table 4 summarizes the results of the compression zone supplements. Convergence problems in the calculation with ft = 0.5 N/mm2 caused an increase in the fracture energy, therefore GIF was set to 25 Nm, and it shows that the stiffness at low tensile strength very rapidly approaches the analytically determined cracked condition. With a directed preload F > Fo,CZS = 132.6 kN it should be possible to influence the requested test frequency.

Figure 5.

FE- model of CZS.

Figure 6.

Plastic strains in concrete, CZS ft = 1 N/mm2 .

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4.1.2 Tension zone supplement The steel reinforcement at the top of the base plate As = 22.0 cm2 (7Ø20) with a concrete cover of 3 cm. For technical reasons an additional structural reinforcement of As = 5.65 cm2 (5Ø12) was placed in the middle of the specimen. The single load at midspan Fo,TZS = 132.6 kN, which is also the upper load of the tension zone supplement (Fig. 7). The plastic distortions in the concrete for a specimen with a tensile strength of ft = 1 N/mm2 are shown in figure 8 for Fo,TZS = 49.4 kN. Due to the large concrete cover, the entire topping layer acts as such and a dominant crack in the middle of the field occurs. A uniform rack distribution over the cross section is no longer given. Difficulties in regard to the convergence conditions of the calculations resulted in an adjustment of the fracture energy GIF for the specimen with a maxi2 mum tensile strength of 0.5 N/mm and 1.0 N/mm2 . For these two cases a fracture energy GIF = 25 Nm was selected. The resulting deformation leads to the stiffness EI of the specimen, and the expected test frequency can be determined, see table 5. It turns out that the stiffness approaches the analytically determined cracked conditions less rapidly and the topping layer shows a stiffening effect on the test specimen. For the tension zone supplement it can be observed that with a tensile strength for concrete of ft = 2.0 N/mm2 the specimen will not crack because of the Table 4. Estimated natural frequencies depending on the concrete tensile strength for CZS.

Cracked ft = 0, 5 N/mm2 ft = 1, 0 N/mm2 ft = 2, 0 N/mm2 ft = 2, 8 N/mm2 Uncracked

k[kN/cm]

f0 [Hz]

176,30 194,25 203,40 463,16 671,59 775,10

13,51 14,06 14,33 20,66 24,60 26,21

Figure 7.

FE- Modell for TZS.

Figure 8.

Plastic strains in concrete, TZS ft = 1 N/mm2 .

low load. This low level of loading results from the observance of the fatigue detection, which is carried out according to the code on fully cracked sections. Due to the not existing reinforcement in the concrete overlay, the static effective height is low compared to the cross section, and the result is a low flexural stiffness. According to the stiffness in the cracked condition, the forces were designed for the required stress analysis. For the experiments the cross section should be transferred with a directed preload into the cracked condition to simulate realistic circumstances for the composite joint. 4.2

3D-FE computation of the expected testing frequency

In addition to an analytical calculation the experimental setup was analyzed with a three-dimensional FE model, see figure 9. This approach proved to be in accordance with the analytical calculations. The stiffness of the specimen was thereby varied similar to the Table 5. Estimated natural frequencies depending on the concrete tensile strength for TZS.

Cracked ft = 0, 5 N/mm2 ft = 1, 0 N/mm2 ft = 2, 0 N/mm2 ft = 2, 8 N/mm2 Uncracked

Figure 9.

k[kN/cm]

f0 [Hz]

48,4 132,96 453,75 761,01 761,01 775,10

8,65 12,08 20,40 25,99 25,99 26,21

3D FE- model of the experimental set-up.

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Table 6. Comparison of analytically and numerically determined natural frequencies.

CZS Uncracked CZS Cracked TZS Uncracked TZS Cracked

Numerically [Hz]

Analytically [Hz]

26,19 13,80 26,19 8,78

26,21 13,51 26,21 8,65

previous calculations. The variation was done by using a fictitious Young’s modulus on parts or on the whole area of the specimen. Table 6 shows the analytical and numerical results. The calculations showed that there are two mode shapes of the rods in the range between 8 Hz and 25 Hz. Therefore these frequencies should not be used because of unfavorable resonance effects.

5

Figure 10. Experimental set-up, compression zone specimen during fatigue test.

OBSERVATION AND RESULTS

In the fatigue loading tests the bond and the plain concrete overlay are subjected to tension and compression depending on the configuration of the specimens (CZS or TZS). For the fatigue tests 4 million cycles were prescribed. To achieve the required stress range the specimens were primarily subjected to a mean load. The applied loads and fatigue stress ranges were chosen according to EN 1991-1-1 annex C. After applying the mean load to the specimen, the vibration generator initiated a force amplitude to reach the required stress range for 4 million cycles. The experimental set-up is shown in figure 10. For the specimens with a compression zone supplement which were not doweled, the fatigue loading test affected the stiffness by 5% comparing the stiffness at the beginning of the test and at the end as the overlay fully contributed to the section strength as you can see in figure 11. No serious cracks in the specimens were observed during the fatigue test. The bond between the overlay and the substrate was efficient during the fatigue loading test since the horizontal displacements were too small in the range 0.01 mm to 0.02 mm and also no deterioration in the bonding layer could be observed. More information on the bond layer could be gained from the static bending test. The static loading test after the fatigue loading also showed a very positive result since the loads exceeded the load capacity of the section. This can be contributed to the higher tensile strength of the tension reinforcement. But even more important was the bond behavior which showed great efficiency, although the section reached its loading capacity and the maximum corresponding horizontal displacements between the layers were nearly 1 mm. The shear strength of the bond reached only 60% of

Figure 11. Frequency/stiffness variation during the load history (CZS 1).

Figure 12. Amplitude/deflection at mid-span to the number of cycles, n (CZS 1).

its shear strength capacity of the bonding layer. The 2 mm thick polyurethane layer transferred the loads effectively and made the section behave as a homogenous section which is crucial for the new restoration method, but the elasticity of the layer led to bigger deflection in comparison to a homogenous section. In figure 12 the amplitude and the mid-span deflection are plotted with the number of cycles. The first specimen with a tension zone supplement and without dowels withstood the fatigue loading test

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after the overlay cracked at the early stages of the of the test. The existing tension reinforcement in the substrate withstood the tension forces applied to the specimen and the test could be continued. After reaching a certain height the vertical crack in the substrate crack started to propagate horizontally. The deterioration in stiffness in the specimen amounted to nearly 60% which was much higher than in the subsequent specimens with dowels but as cracks were propagating and the crack widths were increasing the deformation kept increasing till the end of the fatigue loading test. Also the bond behavior showed no overall deterioration as the horizontal displacements were very low. The second specimen with a tension zone supplement and without dowels showed in contrast to the first specimen without dowels a more efficient behavior as the stiffness deterioration was only 15%. Overall the specimen showed a higher strength than the rest of the specimens as the overlay cracked at a higher load in the preloading phase. The crack pattern consisted of a main crack in the overlay and branching from it, two cracks in the substrate. The doweled specimens with a tension zone supplement cracked in the preloading stage. Both specimens showed great similarities in behavior and also in stiffness degradation during the fatigue loading test. As both specimens withstood 4 million fatigue cycles the stiffness deterioration from the experimental data amounted to 23% and 22% respectively. Even more important is the fact that more than 80% of the stiffness deterioration occurred within the first million cycles which shows that the deflection and crack growth increased slowly or even stagnated in the last part of the fatigue test. No bond deterioration was observed as the horizontal movement between overlay and substrate was very low [Träger & Novoszel, 2008]. 6

CONCLUSION

The new bridge restoration method has shown great efficiency in strengthening the bridge section. Because of the thin bonding layer the overlay becomes an effective part of the section and contributes to its overall capacity. The bonding layer, which consists of a 0.002 m thick polyurethane layer, sustained all tests without any deterioration. Even when the section reached its loading capacity the ultimate shear strength capacity of the bond was not reached and the factor of safety was equal to 1.7. In the field region,

where the overlay is in compression, there is no concern considering the composite behavior and efficiency of the section, as the test results were indeed extremely positive. In the flange region, where the overlay is in tension, problems may occur due to the cracking of the plane concrete overlay, which leads to a much lower efficiency of the new restoration method. On average, specimens with dowels showed a higher stiffness, a better composite action and a lower stiffness degradation. Considering the crack propagation in section with tension zone supplement, only a tendency could be predicted as doweled specimens showed higher crack propagation ability than the other specimens. ACKNOWLEDGEMENT We thank Vienna University of Technology for providing funds for building the fatigue testing facility. We also thank ‘‘Österreichische Forschungsförderungsgesellschaft mbH’’ and ‘‘Vereinigung der Österreichischen Zementindustrie’’ for their support of the project and, last but not least, ‘‘AIT’’ for providing the unbalanced vibration generator. REFERENCES ABAQUS. 2006. ABAQUS Standard-Users Manual. Hibbitt, Karlsson & Sorensen, INC, Pawtucket, 133. Berger, J. 2008. Hochfrequente Dauerschwingversuche an Aufbetonschichten für Brückentragwerke-Analytische und Numerische FE Berechnungen. Master’s thesis, TUWien. 122. EN 1992-1-1, 2004. Eurocode 2: Bemessung und Konstruktion von Stahlbeton und Spannbetontragwerken, Teil 1-1: Allgemeine Bemessungsregeln und Regeln für den Hochbau. Kollegger, J., Koeberl, B., Pardatscher, H. & Vill, M. 2005. Verfahren zur Durchführung von Dauerschwingversuchen an einem Prüfkörper sowie Prüfvorrichtung. Patent: Österreich, No. AT 501 168 B1; submitted: 05-02-2005, granted: 07-15-2006, 2005. 33, 36, 119. Lee, J. & Fenves, G.L. 1998. Plastic Damage Model for Cyclic Loading of Concrete Structures. Journal for Engeneering Mechanics, 124(8):892–900, 1998. 133. Träger, W. & Novoszel, J. 2008. Hochfrequente Dauerschwing versuche für biege- und zugbeanspruchte Bauteile. Master’s thesis, TU-Wien. 122.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Failure studies on masonry infill walls: Experimental and computational observations Kaspar Willam & Ben Blackard University of Colorado Boulder, Boulder, USA

Carlo Citto Atkinson-Noland & Associates Inc., Boulder, USA

ABSTRACT: At the core of model-base simulation is the upscaling of the constitutive properties from ‘proper’ material tests to the structural level of observation. This issue raises fundamental questions when ‘softening’ is introduced to model the degradation of stiffness and strength of material properties. The paper addresses softening from the perspective of snap-back instabilities which lead to control problems of the localized energy release in numerical solutions and experimental investigations. 1

INTRODUCTION

The objective of this study is to quantify the origin of brittle failure in masonry construction and to assess the consequences of ‘Snap-Back’ instabilities: (a) due to material mismatch of brick- and mortar construction in compression and shear, and (b) due to brittle shear failure in non-ductile columns bounding masonry infill walls. After a brief review of recent experimental observations on two-wythe unreinforced brick masonry infill walls with and without openings, the paper addresses the underlying modeling issues of periodic brick and mortar construction. The masonry wall experiments and the concomitant prism tests illustrates the composite action of clay units and mortar layers which leads to intriguing failure mechanisms because of the inherent mismatch of the constituent properties.

2

MASONRY INFILL WALL TESTS

In a related effort, six 2/3 scale, single-story, singlebay, RC frames infilled with two-wythe unreinforced masonry walls were tested at the University of Colorado Boulder as part of the NSF NEESR-SG project, entitled ‘Seismic Performance Assessment and Retrofit of Non-Ductile RC Frames with Infill Walls’. The experimental observations [Blackard et al., 2009a] were used to develop backbone envelope curves for simplified analysis of masonry infill walls with and without openings within the format of elementary lattice models, [Blackard et al., 2009b]. The NEESR-Small Group project was directed by Prof. Benson Shing (PI) at the University of California,

San Diego (UCSD). The research at UCSD focused on shake table experiments and finite element modeling of masonry infilled RC frames [Shing et al., 2009]. The group project also included Prof. Sarah Billington at Stanford University (Co-PI) who developed retrofit techniques based on engineered cementitious composite materials (ECC) to strengthen and to enhance the ductility of infill wall panels [Billington et al., 2009]. In this context two of the six pushover tests at CU Boulder were performed on ECC retrofitted wall specimens. 2.1

Infill wall test with eccentric window

The cyclic pushover tests on single panels involved two-wythe masonry walls of solid clay bricks with and without window and door openings. The infilled frames were pre-loaded in the vertical direction by two jacks at 35 kip (156 KN) each to emulate the confinement of a multistory building during the pushover experiments. Cyclic lateral shear loads of increasing amplitude were applied by the high performance 220 kip (979 KN) MTS actuator of the Colorado NEES equipment site. Figure 1 and 2 illustrate the basic test setup of the masonry infill walls and the reinforcement of the RC bounding frame with non-ductile columns. As a representative example Figure 3 reproduces the response behavior of the first 20 lateral load cycles of the infill wall with an eccentric small window opening. The actual test was carried out beyond drift levels of ±2% without significant deterioration of the residual shear capacity. Note the large drop of the shear capacity at load stage N14 which occurred due to formation of large diagonal shear crack in the left pier of the infill wall accompanied by a large acoustic bang. Similarly,

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Figure 1. Infill Panel with Small Eccentric Window [Blackard et al., 2009a].

Figure 2. Layout of Geometry and Reinforcement [Blackard et al., 2009a].

Figure 3. Cyclic Pushover Response of Infill Panel with Small Window [Blackard et al., 2009a].

another sudden drop occurred at load stage P18 when the left column developed a very shallow shear crack at midheight at 1.2% drift. The photo in Figure 4 shows the rectangular window opening ovalizing at a drift level of 1.5% together with the formation of a shallow diagonal shear crack in the left column. The main failure modes are further illustrated in Figure 5 which shows the shallow shear crack in more details and the large opening of

Figure 4. Cracking of Infill Panel with Window (1.5% Drift) [Blackard et al., 2009].

Figure 5. Diagonal Shear Cracking of Columns (1.2% and 4% Drift) [Blackard et al., 2009].

the localized shear crack at the right top column joint at 4.0% drift. The remarkable aspect of this pushover experiment was the modest decrease of the peak and residual load capacities of the window opening when compared to the unperforated wall test [Blackard et al., 2009a]. Moreover, the infill wall did exhibit surprising resilience under significant drift levels in spite of the severe damage in the reinforce frame columns. Figure 6 summarizes the overall envelope behavior of the pushover experiments on four masonry infill walls with eccentric openings, (a) in form of a small window, SW, (b) a large window, LW and (c) a door opening, D, for comparison with (d) the solid infill wall without opening. In case of the large window, LW, we observe a much more ductile behavior than in the other three cases. In fact, little degradation was observed of the maximum shear capacity to the residual load level, while the other three cases did exhibit significant reductions by 50–100% accompanied by sudden energy release mechanisms. The surprising and reassuring aspects of the test series were the consistent residual shear strength of all test cases to a capacity level of 1/3 of the peak strength of the solid wall which remained constant beyond the ±2% drift levels shown in Figure 6.

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Figure 6. Summary of Response Envelopes [Blackard et al., 2009]. Figure 7.

3

Failure of Five Brick Masonry Prism.

MASONRY PRISM TESTS Material Compression Test 6.0 40.0 brick

5.0

35.0 30.0

4.0 25.0 3.0

20.0

prism

(MPa)

Axial Stress (ksi)

In order to understand the underlying failure mechanisms of the masonry wall, we briefly review the performance of the interaction of the brick and mortar constituents. To this end we examine the traditional masonry prism test when the brick units are mortared by layers of bed-joints (without furrows) and are loaded in axial compression. Figure 7 illustrates the prevalent failure mode of Prism Specimen #2 made of five fired clay brick units and four mortar bed joints. The dimensions of the brick units are hb = 2.25 in (57.2 mm) high, bb = 3.75 in (95.3 mm) wide, and l b = 7.75 in (197 mm) long. All mortar joints are made of 1:1:5 cement:lime:sand mix proportions and are placed without furrows hm = 3/8 in (9.53 mm) high. The prism failure included axial splitting in-plane as well as out-of plane. The overall load-deformation behavior is depicted in Figure 8 which compares the pre- and post-peak response behavior of the masonry prism with the compression test results of the brick units and the mortar specimens when cubes and cylinders were tested on their own. We recognize that the axial load capacity of the masonry prisms fall halfway between the stronger brick units and the weaker mortar specimens. Aside from this puzzling observation in view of the serial arrangement of the brick and mortar composite, we note the sudden drop of the compressive resistance of the masonry prism when compared to the gradual softening of the brick and mortar constituents. The two fundamental questions are, (1) why is the prism strength higher than that of the weakest mortar link, and (2) why is the degradation mechanism of the composite masonry fundamentally different from the softening behavior of the brick units and the mortar specimens when tested on their own? The answer to question (a) dates back to the fundamental observation of Prof. em. Hubert Hilsdorf [Hilsdorf, 1965]: the mismatch of the elastic material properties leads to triaxial states of stresses in the constituents introducing lateral tension in the brick units

15.0

2.0

10.0 mortar 1.0 5.0 0.0

0.0

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

Axial Strain (in/in)

Figure 8.

Axial Response Behavior of Prism Test.

and lateral compression in the mortar. The level of lateral brick tension and lateral mortar compression must form a self equilibrating load system at the mid planes such that hm σxm = −hb σxb . Assuming linear isotropic elastic behavior the magnitude of the lateral stresses generate triaxial compression in the mortar at the mid-planes of symmetry [Berto et al., 2005], and [Blackard et al., 2007], m = σ˙ lat

˙ b E m mlat (ν m E b − ν b E m )σ˙ axial − λE E b (1 − ν m ) +

hm m E (1 hb

− νb)

(1)

Vice versa the axial compression generates lateral tension in the brick units, b σ˙ lat =

˙ b E m mlat (ν b E m − ν m E b )σ˙ axial + λE E m (1 − ν b ) +

hb b hm E (1

− ν m)

(2)

Thereby the first term on the rhs in the numerator refers to the elastic mismatch among the isotropic brick and mortar properties, while the second term refers to

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the additional lateral effect in case of plastic mortar behavior. Here λ˙ denotes the plastic multiplier and mlat the lateral component of the plastic strain (i.e. the lateral gradient of the plastic potential). In short, the lateral stress distribution over the height of the masonry prism exhibits the zig-zag behavior between brick tension and mortar compression as shown by Hilsdorf [1965] and reproduced schematically in Figure 9. It remains to verify the triaxial mismatch among the constituents and to explain the embrittlement of the masonry prism when compared to the postpeak response behavior of brick and mortar on their own. For experimental validation several prism specimens with rectilinear and slanted mortar joints were tested and studied with the digital image correlation system Vic2D. Typical contour maps of in-plane axial and lateral displacements are shown in Figure 10, and the corresponding principal strain contours are shown in Figure 11 right after the peak load resistance of the axial prism test was reached. The axial displacement and strain contours show fairly uniform compaction in each brick unit and each mortar bed-joint. In

Figure 11.

Strain Contour Plots, Vic2D.

contrast, the lateral displacement and strain contours show formation of a displacement discontinuity and a concomitant jump of lateral strains in form of axial cracks which emanate from brick two and three (from top) and propagate in the axial direction. Please note that the strain contour plots also show the principal strain orientations in terms of short white lines which rotate in close proximity to the vertical crack formation. This axial splitting mode precedes the overall prism failure mechanism depicted in Figure 7. It remains to explore the source of enhanced prism strength values and the apparent embrittlement in form of the sudden drop of axial load resistance as opposed to the gradual softening behavior of the two constituents. 4

Figure 9. Triaxial Stress Distribution in Brick and Mortar Joints, Hilsdorf [1965].

SNAP-BACK ANALYSIS

The serial arrangement of elastic brick units and plastic softening joints leads to the classical argument of snap-back instabilities when displacement control fails [Koeberl & Willam, 2008]. Under uniaxial conditions it can be readily shown that perfectly brittle response behavior of the masonry prism takes place when the tangential softening stiffness of the mortar joints reach a critical value when the composite change of height h = hb + hm = 0. With hb = N b hb /Ab E b and hm = N m hm /Am E m , this renders the critical softening modulus of the mortar joint as a function of the relative thickness of the brick unit and the mortar joint. m Esoftg |crit = −

Axial Displacement Uy

Figure 10.

Lateral Strains x and Minor Principal Directions

Axial Strains y and Major Principal Directions

hm Ab b E , hb Am

or

hm |crit = −

m m A b Ecr h E b Ab (3)

Lateral Displacement Ux

Note the size effect of the mortar and brick geometry which enters the critical softening modulus for the

Displacement Contour Plots, Vic2D.

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m

composite structure in form of the thicknesses ratio hhb . Equilibrium requires that N b = N m for the serial system, whereby the surface areas of mortar joint and brick units, Ab = Am , in case of uniform contact of mortar joints without furrows. At the critical softening modulus the mortar-brick assembly results in perfectly brittle response with h = 0. This infers that the elastic strain energy release of the elastic brick is entirely dissipated by softening in the mortar joints. Consequently the brittle post-peak appearance of the prism experiments in Figure 8 is really the result of elastic rebounding in the brick units when softening takes place in the mortar joints. In conclusion, the post-peak response behavior of the prism composite in Figure 8 exhibits a pronounced size effect which differs fundamentally from the postpeak response of the brick and mortar constituents. Under stroke control the snap-back instability takes the form of perfectly brittle post-peak behavior in the laboratory, and introduces severe difficulties in postpeak response computations of localized softening domains. These fundamental arguments of controlling the energy release in softening problems hold also for softening in simple shear. The development of shear bands in mortar bed joints leads to the same critical softening argument among the tangential shear moduli of the two constituents, m Gsoftg |crit = −

h m Ab b G , hb Am

or hm |crit = −

Composite 1D Test of Weak Mortar Element.

Figure 13. Element.

Material Softening Model of Embedded Weak

m m A b Gcr h G b Ab (4)

On a final note, the snap-back arguments of localized softening hold for continuum-based models as well as for zero-thickness interface models. The main difference is the direct control of the energy release rate in terms of relative displacements rather than strain softening, which however does not retrofit the snapback problem. 4.1

Figure 12.

Escrit = −

hm b E = −357.5 [ksi] = −2465 [MPa] hb

Snap-back studies of brick-mortar composite

Let us review the basic features of snap-back instabilities from the view point of the masonry prism experiment. In this case the basic geometric properties of the masonry are the brick height hb = 2.25 in (57.2 mm) and the thickness of the mortar layer, hm = 3/8 in (9.53 mm). Let us first consider the simple 1D representation of the prism along the line of Figure 12, where one brick and one mortar layer have been isolated. The mechanical properties of the brick and mortar units have been determined experimentally as E b = 2145 ksi (14790 MPa), ν b = 0.13, and E m = 1425 ksi (9830 MPa), ν m = 0.22. Consequently the critical strain softening modulus of the mortar joint is according to Equation 1 for uniaxial condition,

(5)

Implementation within the simple bilinear relationship for the mortar behavior shown in Figure 13 requires the additional strength in compression of the mortar which was determined as σym = fcm = 1160 psi (8.00 MPa), while that of the brick was σyb = fcb = 4840 psi (33.37 MPa). Splitting tensile strength properties were measured as ftm = 141 psi (0.97 MPa) and ftb = 372 psi (2.56 MPa). For completeness, the interface bond property was in direct tension fb = 38 psi (0.26 MPa). Considering only elastic mismatch of brick and mortar in Equations 1, 2, this leads to triaxial comm pression in the mortar σ˙ lat = 0.152σ˙ axial and the lateral b m = −0.167σ˙ lat . In other tension on the brick unit σ˙ lat

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terms, a linear elastic solution will predict axial splitting of the brick unit when the lateral tensile stress reaches ftb = 372 psi (2.56 MPa) at an axial load level of σaxial = −14, 600 psi (−100.67 MPa). This is a factor four above the axial splitting capacity of the prism experiments shown in Figure 8. In short, linear elastic analysis significantly over-predicts the load bearing capacity of the prism and is not the predominant factor for prism failure. As an alternate failure scenario let us examine the prism strength in terms of the fundamental mismatch conditions in compression and the highly nonlinear snap-back instabilities when the plastic mortar joints initiate softening. The results of a 1D simulation with the Finite Element code LUSAS [2009] are shown in Figure 14 which depicts a strength value of the two element composite which under-predicts the prism strength by a factor of three. However it is of little surprise that the 1D strength corresponds to the axial load capacity of the mortar according to the weakest link concept. It also illustrates the effect of different mortar softening moduli Es -values which affect the structural force-displacement relationship in the post-peak regime. It clearly shows the influence of the degree of material softening: in the case of subcritical softening the structural softening response is readily obtained under displacement control, while in the case of critical and supercritical softening the perfectly brittle and snap-back behavior of the composite had to be stabilized by arc-length control using Rheinboldt’s continuation method in LUSAS [Rheinboldt, 1998]. Let us investigate the same two-element simulation in 3D rather than 1D assuming full (uniform) contact among the mortar bed joint and the brick unit, Am = Ab . The layout of the idealized composite model is shown in Figure 15 which illustrates the placement of the weak (softening) mortar element in series with the elastic hexahedral brick element. In this case it is important to recall the mismatch of the elastic properties. Assuming full bond introduces a triaxial state of

stress under axial loading when no interfacial slip is permitted among the two constituents. Implementing strain softening within the framework of von Mises plasticity leads to the 3D snap-back results shown in Figure 16. Note the same peak behavior of the composite as in the 1D case which demonstrates the effect of mismatch and triaxiality takes place in the postpeak response. Comparison with Figure 14 illustrates more severe snap-back tendencies when the composite mismatch problem is resolved in 3D rather than in 1D. For comparison the results of the 1D and 3D composite studies are superimposed for the softening modulus Es = −357.5 ksi (−2465 MPa) which is the critical softening modulus for 1D conditions. Figure 17 illustrates the perfectly brittle response of the 1D simulation in comparison to the 3D snap-back instability. It is important to recognize that the 3D snap-back solution degenerates to the brittle response behavior of the 1D case when a zero-thickness interface model is sandwiched between the mortar and the brick elements with near zero shear capacity.

Figure 15.

Composite 3D Test of Weak Mortar Element.

3D Model - Two Hexahedral Elements

1D Model - Two Bar Elements 10

1.4

10

1.4

Es=-250 ksi

Es=-250 ksi Es=-357.5 ksi

Es=-357.5 ksi

1.2

4

0.4

8

1.0 6

(MPa)

6 0.8

0.6

Es=-500 ksi

8

Normal Axial Stress (ksi)

Es=-500 ksi

1.0

(MPa)

Normal Axial Stress (ksi)

1.2

0.8

0.6

4

0.4 2

2 0.2

0.2

0.0

0.0

0 0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0 0

0.0014

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0.0014

Normal Axial Strain (in/in)

Normal Axial Strain (in/in)

Figure 14. Composite 1D Snap-Back Results of Weak Mortar Element.

Figure 16. Composite 3D Snap-Back Results of Weak Mortar Element.

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3D Model w/ Interface 10

1.4

1D Model 3D Model 3D Model w/ interface

8

Esoft = -357.5 ksi interf = 0.1 psi

1.0

6 0.8

0.6

(MPa)

Normal Axial Stress (ksi)

1.2

4

0.4 2 0.2

0.0

0 0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0.0014

Normal Axial Strain (in/in)

Figure 17. All Composite Snap-Back Results Including Weak Shear Interface Elements.

3D Model - Full Prism w/ Interface

Snap-back studies of masonry prism

3.0

F u = 2 kb(1 + b/4h)

σY where k = √ 3

(6)

20 1D Model 3D Model full bond

2.5

3D Model w/ interface Esoft = -286 ksi interf = 0.1 psi

2.0

1.5

15

10

(MPa)

Further on the brick-masonry composite let us examine snap-back instabilities of the entire masonry prism made of five brick units bonded by the four mortar layers. The model simulation starts with a serial arrangement of periodic mortar joints as shown in Figure 18. In this case there are multiple 1D mortar joints interspersed between the elastic brick units of equal yield properties σy = 1,160 psi (8.00 MPa) using the critical softening modulus of mortar Es = −4/5 357.5 = 286 ksi (1972 MPa) according to Equation 5. Figure 19 compares the results of 1D and 3D prism simulations using a 2 × 2 × 9 mesh of quadratic hexahedral elements for the brick units and mortar joints. The results show perfectly brittle response behavior of the 1D simulations when all mortar joints reach critical softening as expected. In contrast, the 3D simulation exhibits increasing compression capacity with little stiffness degradation when the 3D brick and mortar elements are fully bonded. In fact, the 3D simulation exhibits nearly threefold increase of the prism capacity above the uniaxial compressive strength of mortar because of the triaxial confinement when full bond is enforced with the elastic brick elements. Considering the brck-mortar mismatch the plastic mortar layers are constrained from developing plastic flow in the spirit of lower and upper bound plastic solutions of thin rigid plastic layers in compression under smooth and rough contact. Smooth contact or loss of bond reverts the response behavior of the mortar layer to the perfectly brittle case of the 1D simulation. In this case the triaxial confinement effect of the mismatch is eliminated because of negligible interface shear strength. In contrast, rough contact leads to an increase of the limit load capacity which is according to the upper bound estimate [Calladine, 1985]:

Normal Axial Stress (ksi)

4.2

Figure 18. Full Prism Layout of Five Brick Units and Four Mortar Joints.

1.0 5 0.5

0.0

0 0

0.0005

0.001

0.0015

0.002

0.0025

Normal Axial Strain (in/in)

Figure 19.

Full Prism Response of 1D and 3D Models.

Note b = bb is the width of the brick unit and h = hm the thickness of the mortar layers. In terms of the present geometry b/h ∼ 10 the limit load capacity of a perfectly plastic thin layer increases by a factor four above the uniaxial yield capacity of the mortar. In other terms, rough contact leads to plastic flow of mortar at a nominal axial prism stress which is close to the prism strength measured in Figure 8. Whereas all mortar elements exhibit initially the same softening behavior leading to perfectly brittle behavior of the prism because of loss of lateral confinement. However in the 3D case the failure mode localizes in two mortar layers. In fact, Figure 20 shows the plastic strain contours which reach a maximum at the front and back faces of the prism walls transversely to the in-plane direction. Considering this is the dominant direction of plastic flow, we recognize that axial splitting of the brick units occurs primarily out-ofplane which agrees with the dominant experimental mode of out-of-plane spalling shown in Figure 7. As a note of caution, one should be aware that thin layer plastic flow requires special provisions of the finite element mesh to accommodate incompressible

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plastic flow in case of von Mises plasticity. In our prism problem shear enhanced tri-linear elements did not capture the softening branch at all. In fact, tri-quadratic 3D elements had to be used with 2×2×2 point GaussLegendre integration to avoid incompressible locking, [Zienkiewicz & Taylor, 2005]. In a complementing study of mortar softening Figure 21 shows the snap-back results of the prism response when the softening slope varies below and above the critical value of the two element composite. In case of a weak interface this leads in one case to gradual softening, but in the other case it results in increasing snap-back instabilities because of no interface bond. In summary, the simulations exhibit enormous imperfection sensitivity with regard to interfacial

boundary conditions, spatial variations of the mortar strength, and in particular with regard to the overall ductility. In case of full bond we observe a large increase of axial load capacity, in case of degrading bond properties we see a large decrease of the prism capacity down to the mortar strength, and in the limit, violent snap-back instabilities. However, for a full picture further refinements such as gradual loss of adhesion in form of partial debonding among brick and mortar should be considered in addition to progressive plastic flow of the mortar interphase in concert with tensile cracking of the brick units see e.g. [Giambanco & Mroz, 2001], all this in light of the large uncertainties of bond under field conditions. 4.3

Masonry prism issues

In conclusion, the simple prism test in Figure 8 poses several challenging problems beyond the Hilsdorf [1965] tenet that the prism test is an indirect tension test: 1. Elastic and Inelastic Mismatch of Constituent Properties: mortar and brick units introduce triaxial stress states in the form of triaxial mortar compression and biaxial tension-compression in the brick units. 2. Interface Bond Properties: degrading bond leads to loss of triaxial confinement counteracting the triaxial confinement effects of elastic and inelastic mismatch. 3. Size Effect and Critical Softening Properties: snapback instabilities introduce geometric effects in the mortar-brick construction. 4. Lack of Uniform Contact: mortar furrows in on-site masonry construction introduces large uncertainties regarding the initial interface contact and hence large variations of bond properties. Figure 20. Prism.

Plastic Strain Contours in the Mortar Layers of

3D Model - Full Prism w/ Interface 10

1.4

Es=-150 ksi

1.2

Es=-450 ksi

1.0

Esoft =-286 ksi interf = 0.1 psi

8

6 0.8

0.6

4

(MPa)

Normal Axial Stress (ksi)

Es=-286 ksi

The main question remains what comes first: whether compression softening and snap-back initiates axial splitting of the masonry prism, or whether lateral tension starts the axial splitting process together with the loss of interface bond and the ensuing loss of triaxial confinement? All this comes with the understanding that there are large variations in the brick and mortar contact in real life masonry construction which introduce great uncertainties in the interface contact in addition to the environmental effects of drying shrinkage and thermal expansion.

0.4 2

5

0

The CU Boulder pushover test program did exhibit surprisingly moderate reductions of stiffness and strength properties of masonry infill walls with window and door openings. In fact, the initial elastic stiffness properties did not exhibit significant stiffness degradation

0.2

0.0 0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0.0014

Normal Axial Strain (in/in)

Figure 21. Interface.

Full Prism Response of 3D Models w/Weak

CONCLUSIONS

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when window and door openings were included in the masonry infill wall compared to the wall without perforations. Moreover the shear capacity at the residual level did not exhibit significant deterioration when window or door openings were included. Moreover, the eccentricity of the window and door openings had little influence on the cyclic performance of the pushover tests when compared to the performance of the masonry wall without opening. In conclusion, the infill masonry walls did exhibit surprising resilience up to drift levels of 2% in spite of the brittle nature of unreinforced masonry and nonductile bounding RC columns. This apparent ductility of the brittle composite components may be attributed to the overall confinement of the RC bounding frame and the confinement due to vertical prestressing. ACKNOWLEDGEMENTS The authors wish to dedicate the paper to Prof.em. Hubert Hilsdorf and his inspiring insight into the performance behavior of masonry. The authors acknowledge the support of this research effort by the US National Science Foundation under NSF NEESR-SG award #0530709. The close collaboration and exchange with Prof. Benson Shing at UC San Diego and Prof. Sarah Billington at Stanford University are highly appreciated in the context of the joint project ‘‘Seismic Performance Assessment and Retrofit of Non-Ductile RC Frames with Infill Walls’’. Opinions expressed in this paper are those of the authors and do not necessarily reflect those of the sponsor. REFERENCES Berto, L., Saetta, A., Scotta, R., & Vitaliani, R., 2005, Failure mechanism of masonry prism loaded in axial compression: computational aspects, Materials and Structures, 38: 249–256. Billington, S.L., Kyriakides, M.A., Blackard, B., Willam, K., Stavridis, A., & Shing, B., 2009, Evaluation of a sprayable, ductile cement-based composite for the seismic retrofit of

unreinforced masonry infills, Proc., Conf. on Improving the Seismic Performance of Existing Buildings and Other Structures, ATC and SEI, Dec. 9–11, 2009, San Francisco. Blackard, B., Kim, B., Citto, C., Willam, K., & Mettupalayam, S., 2007, Failure issues of brick masonry, Proc. of the Sixth International Conference on Fracture Mechanics of Concrete and Concrete Structures, FramCos6, Carpinteri, A., Gambarova, P., Ferro, G. and G. Plizarri, eds, Vol. 2, 1587–1594, Catania: Balkema. Blackard, B., Willam, K. & Mettupalayam, S., 2009a, Experimental observations of masonry infilled rc frames with openings, ACI SP-265-9, Thomas T.C. Hsu Symposium on ‘Shear and Torsion in Concrete Structures’, A. Belarbi, YL. Mo, and A. S. Ayoub, eds.: 199–222, Farmington Hills: American Concrete Institute. Blackard, B., Willam, K., & Mettupalayam, S., 2009b, Simplified Strut Model for Masonry Infilled RC Frames, SESM Report No. 09-04, University of Colorado: Boulder. Calladine, C.R., 1985, Plasticity for Engineers, Chichester: Ellis Horwood. Giambanco, G. & Mroz, Z., 2001, The Interphase Model for the Analysis of Joints in Rock Masses and Masonry Structures, Meccanica, 36: 111–130. Hilsdorf, H., 1965, Investigation into the Failure Mechanism of Brick Masonry Loaded in Axial Compression, A3A-39, Proc. of Designing, Engineering, and Constructing with Masonry Products: 34–41, Houston: Gulf Publ. Co. Koeberl, B. & Willam, K., 2008, The question of tension softening vs tension stiffening in plain and reinforced concrete, J. of Engrg. Mechanics, ASCE, 134(9): 804–808. LUSAS Bridge Plus, 2009, Version 14.3-3, LUSAS, Forge House, 66 High Street, Kingston upon Thames, Surrey, UK. Rheinboldt, W.C., 1998, Methods for Solving Systems of Nonlinear Equations, 2nd ed., Philadelphia: SIAM. Shing, P.B., Stavridis, A., Koutromanos, I., Willam, K., Blackard, B., Kyrikides, M.A., Billington, S.I., & Arnold, S., 2009, Seismic performance of non-ductile rc frames with brick infill, Proc., Conf. on Improving the Seismic Performance of Existing Buildings and Other Structures, ATC and SEI, Dec. 9–11, 2009, San Francisco. Zienkiewicz, O.C. & Taylor, R.L., 2005, Finite Element Method for Solid and Structural Mechanics, Sixth Ed., Linacre House, Jordan Hill, Oxford: Elsevier ButterworthHeinemann.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Numerical study on mixed-mode fracture in LRC beams R.C. Yu, L. Saucedo & G. Ruiz University of Castilla-La Mancha, Ciudad Real, Spain

ABSTRACT: The objective of this work is to model the static-dynamic propagation of fracture in mixed-mode in LRC beams. When a beam does not have enough shear reinforcement, fracture can initiate and propagate unstably and lead to failure through diagonal tension. In order to study this phenomenon numerically, a model capable of dealing with both static and dynamic crack propagation as well as the natural transition of these two regimes is necessary. A cohesive model for concrete fracture and an interface model for the deterioration between concrete and steel re-bar, both combined with an insertion algorithm, are chosen for this task. The static process is solved by dynamic relaxation (DR) method together with a modified technique to enhance the convergence rate. This same DR method is used to detect a dynamic process and switch to a dynamic calculation. The numerically obtained load-displacement curves, load-CMOD curves and crack patterns fit very well with their experimental counterparts, having in mind that we fed the calculations only with parameters measured experimentally. 1

INTRODUCTION

In this paper we investigate the evolution of 3D mixed-mode fracture in notched reinforced concrete beams subjected to static loading. As observed in the experimental results of Carmona, Ruiz & Del Viso 2007, under static loading conditions, mix-mode fracture can propagate in static or dynamic regime. The conditions of transition between these two regimes vary with beam geometry, reinforcement ratio, location and inclination of the rebars. Knowing those conditions beforehand is essential to beam design and safety of a whole structure, since without a rebar to slow down and eventually stop the dynamic fracture, the beam would collapse. The idea of adding reinforcements to transform a brittle behavior—beam collapsing—into a ductile one—rebar yielding—is many times more important than sole increasing of peak loads. In other words, mere high peak load may present false high load capacity if ductility is lacking. When a beam is not sufficiently reinforced to resist shear, the crack initiated from the notch can run unstably and leads to diagonal tension failure, see Figure 1. To numerically reproduce this entire progress of the crack initiation, propagation and ‘‘hinge’’ forming, is undoubtedly a challenging task. A computational model that is able to not only handle both static and dynamic fracture but also detect the transition in between is required. To tackle this problem, we adopt an explicit discrete cohesive model, with a three-dimensional discretization, incorporated with a modified dynamic relaxation (DR) method. The modified DR, see Yu and Ruiz, 2004, is served both as a solver and a detector for dynamic processes.

Figure 1. A lightly reinforced concrete beam failed through diagonal tension.

2

FINITE ELEMENT METHODOLOGY

We model the concrete bulk using the Neo-Hookean material extended to the compressible range. The fracture in concrete is modeled using a 3D cohesive model developed by Ortiz & Pandolfi 1999, with a lineardecreasing cohesive law. The steel rebar is explicitly represented using 10-node tetrahedrons and follows an elastic perfectly-plastic constitutive law. The steelconcrete interface is simulated through an interface element endorsed with a perfectly plastic bond-slip law. Both the cohesive elements and the interface elements are inserted upon crack initiation or bond deterioration, the geometry is updated through an effective fragmentation algorithm, see Pandolfi & Ortiz, 2002. The whole process is solved using a modified dynamic relaxation method (Yu & Ruiz 2004). All the material properties were measured through independent experiments and their values are listed in Table 1, where E represents the material elastic modulus, ft ,GF are

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Table 1. Mechanical properties of concrete, steel and the interface in between given in Carmona, Ruiz & Del Viso 2007. Material Concrete Steel Steel-concrete interface

E (GPa)

ft (MPa)

GF (N/m)

Sc0.2 (MPa)

Tc (MPa)

36.3 174 −

3.8 − −

43.4 0.42 −

− 563 −

− − 6.4

the tensile strength and the specific fracture energy of the concrete, Sc0.2 stands for the 0.2% yield strength of the steel rebar, Tc is the bond strength of the steelconcrete interface. Note that the value 174 GPa is the nominal modulus measured for a ribbed rebar. Figure 2. Geometry for the beams tested by Carmona, Ruiz & Del Viso (2007), where D is of 75, 150 and 300 mm for small (S), middle (M) and large (L) specimens.

3 3.1

NUMERICAL RESULTS Validation

In Figure 2, we show the geometry of the notched beams tested by Carmona, Ruiz and Del Viso (2007) and follow their nomenclature for a beam of different size (S, M or L), reinforced with longitudinal or shear rebars. For example, S01 is a small size beam, with zero longitudinal, one transversal reinforcement rebar. This was designed to provoke a single propagating crack in beams of different sizes and to facilitate the study of crack trajectories, peak loads and their relation with the amount and location of the reinforcements. Beams M00 and M20 are chosen as validation examples. The complete load-displacement and loadCMOD curves are plotted against their experimental counterparts in Figure 3, where letters A to E mark different stages of the crack propagation. Taking into consideration that all fed material parameters are measured ones, the fit is remarkable. It needs to be pointed out that, the calculations shown in Figure 3 had gone through a static-dynamicstatic transition. In particular, at the last stage of M20, the crack closes due to the formation of the ‘‘hinge’’ after the diagonal-tension failure. This last part cannot be captured by any sophisticated solvers if considering it static, since the crack propagation is intrinsically dynamic. The lower numerical load at post-peak compared to the experimental one is attributed to the material hardening of the rebar is not yet included in the numerical model.

3.2

Dynamic fracture under static loading

When a structural element fails due to surpassing of its loading limit, the collapsing process is usually

Figure 3. Numerical-experimental comparison for (from top to bottom) load-displacement, load-CMOD curves and crack patterns.

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unstable, and is accompanied with dynamic fracture. Such a dynamic fracture propagation under static loading is demonstrated in Figure 4 for the case of a small beam without any reinforcements S00. Figure 4(top) shows the numerical crack trajectory superimposed upon the experimental one. A perfect match is obtained. In Figure 4 (bottom), the discontinuous lines with empty symbols represent the experimentally recorded load-displacement curve. Note that, at post-peak stage, from peak load to zero load, no intermediate points were recorded due to the rapid failure of the beam. In other words, we cannot directly compare the loaddisplacement data after peak load, nevertheless, one can observe that the numerical softening curve tends to the experimental one taken at the end of the experiment. The matching of both the load-displacement curves and crack trajectories validates both the amount and the location of the energy spent. It needs to be pointed out that since the test was done with displacement control, it was the self-weight that provoked the dynamic fracture when the crack was advanced towards the loading plane. For a beam with reinforcements, the static-dynamic transition could occur (a) when the interface is broken

Figure 4. Experimental and numerical comparison for (top) crack patterns and (bottom) load-displacement curve for the case of S00, a small beam with neither longitudinal nor transversal reinforcements.

and (b) when the rebar is yielded or broken, see Figure 3 B-C and D-E, and Figure 4 B-C. Figure 3 shows that the longitudinal reinforcement of M20 actually stopped the dynamic crack propagation, consequently there was a dynamic-static transition from point C to D, apart from the static-dynamic ones BC and D-E. Such transitions are determined by the reinforcement ratio and the beam geometry. 3.3

Effect of longitudinal and transversal reinforcements

From Figure 3, comparing the case of M00 and M20, we observe that longitudinal rebars help to augment both the peak and the ultimate loads. Consequently, M20 is more ductile compared to the non-reinforced one M00. From Figures 4 and 5, for beam S01, however, the peak load is lower than the beam S00. By looking at the crack trajectories, we observe that the transversal rebar served to change the main crack trajectory at point D. This is clearly seen in Figure 6, where the experimental trajectories are put together for beams S00, S01, S10 and S11. The fractures paths remained the same at the first stage of the crack propagation, this shows neither the longitudinal nor the transversal rebar was activated. Then then the

Figure 5. Numerical and experimental crack patterns and load-displacement curve for S01, a small beam reinforced with one transversal re-bar.

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rebar. Note that the phenomenon of dowel action as shear transfer mechanism across cracks is reproduced naturally as a result of the explicit representation of rebar and the bond-slip interaction between concrete bulk and the rebar. Additionally observed is the sewing effect of both rebars and secondary cracks in concrete bulk. Correctly model each physical phenomenon individually and the interactions between reinforcement and concrete is fundamental for a right design of

Figure 6. Main crack trajectories observed in experiments for beams S00, S01, S10 and S11.

crack path curved towards the loading plane for S10, whereas it went the opposite direction for S01. The crack followed a straight line in the case of S00. S11 is a mixture of S01 and S10, but closer to S10 than to S01. This reveals the stronger influence of the longitudinal reinforcement than the transversal on fracture path. It bears emphasis that such complicated trajectory would be an impossible task for those methods that work with only pre-embedded cracks. 3.4

Size effect in reinforced beams

According to the size effect law, two geometrically similar beams, the smaller one resists proportionally more than the larger one. However, Figure 7 shows the well-distinguished smaller-is-stronger rule in plain concrete beams do not equally reproduce for reinforced ones. In Figure 7(top), the load-displacement values for S10 are doubled to compare better with M20. Surprisingly, S10 resists less peak load than M20. By looking at the movements of the two crack surfaces individually, see Figure 7(bottom), we observe that S10 and M20 show different failure mechanisms. Larger absolute amount of reinforcement in M20 has resulted that the left part of the beam is being dragged towards the loading plane. In other words, even though S10 and M20 share the same crack patterns, but the loading capacity do no follow the size effect law as observed in plain concrete beams. This phenomenon would not be captured by a non-explicit representation of the rebar and cracks. 3.5

Figure 7. Load-displacement comparison for S10 and M20 (top) and crack surface displacements versus loading-point displacement for S10 and M20 (bottom).

Dowel action and sewing effect of the rebar

Figure 8 is a snapshot of the fractured beam S11, reinforced with one longitudinal and one transversal

Figure 8. The dowel action, sewing effect of the rebar and secondary cracks modelled in beam S11.

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an reinforced concrete structure, since all those aforementioned factors contribute to the entire energy consumption therefore the resistance and global behavior of the beam. 4

ACKNOWLEDGEMENTS The authors acknowledge the financial support from Ministerio de Ciencia e Innovación, Spain, under grant MAT 2006-09105 and the Consejería de Educación y Ciencia, JCCM, Spain, under grant PAI08-0196.

CONCLUSIONS

We have modeled the static-dynamic mixed-mode fracture in reinforced concrete. The concrete bulk, steel rebar and the interface in between are all explicitly represented. A modified relaxation method is employed to solve the static process and detect a dynamic one. The methodology was validated against experimental results of Carmona, Ruiz and Del Viso. The salient features, such as micro cracking, changing of crack trajectory, pull out and the dowel action of the rebar are all naturally reproduced through the discrete fracture and explicit representation of the rebar. As a by product, the fundamental role of fracture mechanics in reinforced concrete structure design is thoroughly demonstrated.

REFERENCES Carmona, J.R., Ruiz, G. and Del Viso, J.R. Mixed-mode crack propagation through reinforced concrete. Engineering Fracture Mechanics. Engineering Fracture Mechanics, 74, 2788–2809. Ortiz, M. and Pandolfi, A. 1999 Finite-deformation irreversible cohesive elements for three-dimensional crackpropagation analysis. International Journal for Numerical Methods in Engineering, 44:1267–1282. Pandolfi, A. and Ortiz, M. 2002 An efficient adaptive procedure for three-dimensional fragmentation simulations. Engineering with Computers, 18:148–159. Yu, R.C. and Ruiz, G. 2004 Multi-cracking modeling in concrete solved by a modified DR method. Computers and Concrete, 1(4), 371–388.

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Hazards, risk and safety (fire, blast, seismicity)

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Numerical investigation of damage and spalling in concrete exposed to fire C.T. Davie & H.L. Zhang School of Civil Engineering and Geosciences, Newcastle University, Newcastle upon Tyne, UK

ABSTRACT: There exists continued controversy over the underlying causes of thermally induced spalling in concrete, with thermally induced stress and internal pore pressures being variously cited as the critical factors. The results of a series of numerical experiments designed to investigate these phenomena are presented here. The experiments were carried out using a fully coupled hygro-thermal-mechanical finite element model, with the development of mechanical damage considered indicative of certain spalling behaviours. The results indicate that thermally induced stresses are the dominant factor in the cases considered with pore pressures playing, at most, a secondary role. Initial investigations concerning the damage model indicate that the choice of equivalent strain measure is critical in producing observed spalling behaviours, but that other factors, while influencing the development of damage do not affect the major finding of this work. 1

INTRODUCTION

It is well known that when concrete structures are exposed to elevated temperatures, for example during a fire, they can experience significant damage that can ultimately threaten structural integrity. The effects of this damage are often observable as a change of colour and measureable as, amongst other things, a loss of strength and stiffness. A further and potentially more significant effect is the development of spalling (i.e. the fracturing and loss of material from the surface of concrete elements), which has been observed to occur in various explosive or non-explosive forms, and which can ultimately threaten structural integrity if sufficient cross-section is lost from structural members or if reinforcing steel is exposed directly to heat as cover material is lost. Over at least the last two decades, extensive experimental studies have been conducted in order to gain a better understanding of the driving mechanisms underlying spalling behaviour, e.g. (Phan 1996; Anderberg 1997; Kalifa, Menneteau et al. 2000; Hertz 2003), and at the same time, various numerical models have been presented for simulation of these complex phenomena at different levels of simplification, e.g. (Gawin, Majorana et al. 1999; Tenchev, Li et al. 2001; Khoury, Majorana et al. 2002; Davie, Pearce et al. 2006; Dwaikat and Kodur 2009). These investigations, have shown that spalling is not only dependent on the material properties of the concrete such as strength, moisture content and permeability, but also on the combination of heating rate, section shape and size, and constraint configuration. Despite this extensive work the fundamental mechanisms underlying the development of spalling are not well understood and there exists continued

controversy with some researchers stating that internal pore pressure is the most important factor e.g. (Phan, Lawson et al. 2001; Hertz 2003; Ichikawa and England 2004) while some emphasise the significance of thermally induced stresses e.g. (Bažant 1997; Ulm, Coussy et al. 1999; Gawin, Pesavento et al. 2006; Msaad and Bonnet 2006), and others believe spalling is caused jointly by these two factors e.g. (Anderberg 1997; Consolazio, McVay et al. 1997; Kalifa, Menneteau et al. 2000). In this work a numerical investigation is conducted in an attempt to clarify understanding of these behaviours. In order to achieve this and capture the complex behaviour of concrete under heating, the concrete is considered as a multi-phase, porous material and a hygro-thermal-mechanical (HTM) material description is adopted with strong coupling between the solid skeleton, liquid water and gas (water vapour and dry air) phases, in a detailed and fully generalised (3D) finite element implementation. Heat and mass transport of the fluid phases are modelled in a coupled manner such that an accurate description of the fluid transport processes in concrete is possible, illustrating in particular the redistribution of liquid and the increases in vapour content and pore pressure associated with the application of elevated temperatures. The mechanical behaviour of the solid skeleton is modelled by way of an isotropic thermo-mechanical damage model in which the degradation of the material due to both mechanical and thermal loading is accounted for. Coupling with the hygro-thermal components of the model allows for the internal pore pressures to be considered within the effective stress term and for the effects of material degradation on mass transport to be captured.

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The mathematical formulation, together with its finite element solution procedure is briefly described before the results of various numerical experiments are presented. Taking the examples of a concrete wall, a square column and an I-beam exposed to fire, the damage development for concretes with different combinations of initial permeabilities and moisture contents was analysed. These numerical experiments allowed the effects of pore pressures and thermally induced stresses to be isolated and the relative significance of each to be determined.

gas mixture and dehydrated water phases, respectively), ρθ is the density of a phase θ, ρ˜θ the mass of a phase θ per unit volume of gaseous material, Jθ the mass flux of a phase θ, ρC the heat capacity of concrete, k the effective thermal conductivity of concrete, λE and λD are the specific heats of evaporation and dehydration, σ  is the Bishop’s stress (also known as the effective stress in geomechanics), I the identity matrix, α is the Biot coefficient, PPore the pore pressure and b the body force. 2.2

2

MATHEMATICAL MODEL

In the mathematical formulation, concrete is treated as a multi-phase system consisting of solid, liquid and gas phases. The solid skeleton is assumed to develop isotropic elastic-damage deformations under mechanical and thermal loadings. The liquid phase consists of free liquid water in pores and adsorbed water physically bound to the surface of solid skeleton. The dehydrated water is considered as a part of free liquid water since chemically bound water is assumed to be initially released as liquid water. The gas phase is a mixture of dry air and water vapour, both of which are assumed to behave as ideal gases. Most of the material properties are variable (typically, either directly or indirectly, as a function of temperature). The complete description of the material properties can be found in (Pearce, Nielsen et al. 2004; Davie, Pearce et al. 2006). Herein, only a shortened description is given. 2.1

Governing equations

The model consists of four governing equations (Equations 1 to 4) defining the mass conservations of dry air and moisture (i.e., vapour and liquid water), conservation of energy and linear momentum balance, respectively. ∂(εG ρ˜A ) = −∇ · JA ∂t

(1)

∂(εG ρ˜V ) ∂(εL ρL ) ∂(εD ρL ) + − = −∇ · (JV + JL ) ∂t ∂t ∂t (2) ∂T ∂(εD ρL ) ∂(εL ρL ) − λE + (λD + λE ) ∂t ∂t ∂t = ∇ · (k∇T ) + λE ∇ · JL

(ρC)

  ∇ · σ  − αPPore I + b = 0

Fluid transport and energy equations

Liquid water flow is assumed to be driven by the pressure gradient, according to Darcy’s law, while gas flow is assumed to be driven by both the pressure gradient and the concentration gradient according to Fick’s law. The mass fluxes of dry air (JA ), water vapour (JV ) and liquid water (JL ), per unit area of concrete, are thus given by Equations 5 to 7.     kg KKG ρ˜A ∇PG − εG ρ˜G DAV ∇ JA = εG ρ˜A − μG ρ˜G (5)  JV = εG ρ˜V

(4)

where, εθ is the volume fraction of a phase θ (θ = L, V , A, G, D refer to liquid water, water vapour, dry air,

kg KKG μG



 ∇PG − εG ρ˜G DAV ∇

ρ˜V ρ˜G

 (6)

  KKL ∇PL JL = εL ρL − μL

(7)

where, K is the intrinsic permeability of the dry concrete, Kθ , μθ and Pθ are the relative permeability, dynamic viscosity and pressure of the phase θ, kg is the gas-slip modification factor and DAV is the coefficient of diffusion for the dry air/water vapour mixture within the porous concrete. Capillary pressure effects are accounted for within the liquid pressure term (Equation 8), with the capillary pressure, PC , calculated according to the Kelvin Equation (Equation 9). PL = PG − PC  PC =

(3)

−

−RV T ρL ln h for S > SSSP 0 for S ≤ SSSP

(8)

(9)

where, RV is the ideal gas constant for water vapour, T is the temperature, h is the relative humidity and S is the degree of saturation with liquid water. SSSP is the solid saturation point, which is the degree of saturation below which all ‘liquid’ water exists as adsorbed water, physically bound to the concrete skeleton (Gawin, Majorana et al. 1999).

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For the calculation of the total stress (Equation 4) the combined pore pressure, PPore , is derived from a pro rata weighting of the gas and liquid pressures, according to their volume fractions (Equation 10) (Davie, Pearce et al. 2006).

PPore

⎧ PG − PG,∞ ⎪ ⎪ ⎪ ⎨ S − SSSP = 1 − SSSP PL ⎪  ⎪ ⎪ S ⎩ PG − PG,∞ + 1 1−−SSSP

according to Equation 15, following the work of Peerlings et al. (Peerlings, de Borst et al. 1998) and Pearce et al. (Pearce, Nielsen et al. 2004). ω =1−

for S ≤ SSSP

for S > SSSP

κ0md =

where, PG,∞ is the pressure of the atmosphere external to the concrete.

The total strain rate decomposition for the mechanical component of the model, ε˙ , consists of elastic strain, ε˙ e , free thermal strain, ε˙ ft , and load induced thermal strain, ε˙ lits , (Equation 11).

σ  = (1 − ) D0 : ε e

γ =η

(12)

(13)

where, the thermal damage parameter, χ , is defined by the reduction in the elastic stiffness as a function of temperature (Equation 14), χ =1−

E(T ) E(T0 )

ft (T ) Gf (T )

κ md = max{κ0md , max(˜ε)}

where, is the scalar damage parameter and D0 is the initial elasticity matrix. The classical damage formulation, with a single scalar mechanical damage parameter, ω, is here modified to include a second thermal damage parameter, χ . This parameter, which accounts for the reduction of the elastic stiffness that results from thermally induced degradation of the cement paste, is introduced in a multiplicative manner following the work of Stabler (Stabler 2000) and Gawin et al. (Gawin, Pesavento et al. 2003). The scalar damage parameter, , can thus be further expressed as shown in Equation (13). = 1 − (1 − ω)(1 − χ )

(16)

(17)

where, η is a characteristic length dependent on the size of the localization zone, and κ md is the mechanical damage history parameter, which takes the maximum value of the damage threshold, κ0md , and the equivalent strain measure, ε˜ (Equation 18).

(11)

The development of (micro) fracturing in the concrete and the associated reduction in stiffness are accounted for by a straightforward isotropic damage model as described by Equation 12.

ft (T ) E(T )

γ (T ) is a ductility parameter related to the fracture energy, Gf , (Equation 17)

Mechanical constitutive equations

ε˙ = ε˙ m (σ  , T ) + ε˙ ft (T ) + ε˙ lits (σ  , T )

(15)

where, κ0md , which defines the threshold for the onset of damage, is a function of the tensile strength, ft (T ), and elastic modulus, E(T ), both of which are temperature dependent (Equation (16)),

(10)

2.3

κ0md (T ) −γ (T )(κ md −κ md (T )) 0 e κ md

(14)

where T0 is a reference temperature (typically 20◦ C), and the mechanical damage parameter, ω, is defined

(18)

where, the equivalent strain measure, ε˜ , describes the mapping of the tensorial strain state to a scalar variable. This may take numerous forms but, unless otherwise stated, a modified von Mises description (Pearce, Nielsen et al. 2004) is used in this work. 2.4

Numerical formulation

The governing equations, 1–4, are discretised in space using the standard finite element approximation, with the chosen primary variables of displacements, u, temperature, T , gas pressure, PG , and vapour content. The discrete set of equations is also discretised in time using a generalised mid-point finite difference scheme. Boundary conditions for the primary variables may be defined as Dirichlet, Neumann or Cauchy types, depending on the specific problem under consideration. The numerical solution procedure is implemented in the general purpose finite element programme FEAP (Zienkiewicz and Taylor 2005). 3

NUMERICAL ANALYSES

Three separate numerical problems were developed to investigate the relative behaviours of thermally

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induced stresses and pore pressures during the development of damage and spalling in concrete members exposed to fire. These problems were representative of a wall, a column and an I-beam exposed to fire on all sides. With each of these an extensive range of numerical experiments designed to separate and identify the contributions of thermally induced stresses and pore pressures were conducted. The philosophy for the series of experiments was based on the principle that the magnitude of the pore pressures built-up in the concrete during heating is significantly influenced by the permeability and moisture content of the concrete and that, through the effective stress concept, these pressures could theoretically result in the development of damage in the concrete. Thus, to investigate the significance of this behaviour a large range of combinations of initial permeabilities and moisture contents (controlled by the relative humidity) were considered with the aim of producing situations where different magnitudes of pore pressure would develop. In each of these analyses it was expected that thermally induced stresses would also develop due to the restraint conditions present in the various problem configurations and that their effects would be separated from those of the pore pressures in the analyses where only low pressures developed. However, in addition to these analyses, it was also possible to numerically isolate the contributions of thermally induced stresses from those of the pore pressures by ‘switching off ’ the PPore term in the momentum equation (Equation 4) and therefore a parallel set of analyses were conducted in which this was done. It may be noted that this could not be achieved physically and thus represents an advantage of the numerical modelling approach over a laboratory based experimental study. Although a thorough set of analyses was conducted, exploring the full range of parametric combinations, only the key sets of results are reported here. Table 1 summarises the initial permeabilities, relative Table 1. Summary of analyses indicating initial permeability (K0 ), relative permeability (h) and condition of PPore term.

Case

K0 (m2 )

h (%)

PPore (–)

1 – Wall 2 – Wall 3 – Wall 4 – Wall 5 – Column 6 – Column 7 – Column 8 – I-beam

1 × 10−17 1 × 10−17 1 × 10−21 1 × 10−21 1 × 10−19 1 × 10−19 1 × 10−21 1 × 10−19

1 1 90 90 65 65 80 65

On Off On Off On Off On On

humidities and on/off options of the PPore term as used in these simulation cases. Typical initial values for other key parameters were chosen from the literature, and set as shown in Table 2 for all of the simulations. Furthermore, in all analyses, the concrete elements were assumed to be exposed to the ISO834 standard fire curve, the far-field temperatures of which can be expressed as a function of time, t, as in Equation 19. T∞ = 20 + 345 log10 (8t + 1) + 273.15 3.1

(19)

Wall problem

Analyses 1 to 4 (Table 1) considered a concrete wall exposed to fire on both sides. The problem was modelled as an effectively one-dimensional analysis in central-plane symmetry, as illustrated in Figure 1. As can be seen, one (left) side was constraint-free and heated by the standard ISO834 fire curve, while the other (right) side was a symmetry plane and was therefore fixed and isolated, and had no heat or mass transfer across it. Figure 2 shows the predicted distributions of mechanical damages and pore pressures at different times for the concrete with an initial permeability of 1 × 10−17 m2 and an initial relative humidity of 1% (Case 1). As can be seen, the mechanical damage initially occurs after about 1200 s, increasing with time and developing towards the inside of the wall. It is noted that this pattern matches very well with the location and depth to which spalling is typically observed to occur (e.g. (Hertz 2003; Gawin, Pesavento et al. 2006)) and hence the mechanical damage is seen as a very good analogue for spalling behaviour. In Figure 2, the highest peak of the pore pressures is less than ∼0.2 MPa, which is much lower than the Table 2. Summary of key parametric values employed in all analyses. Parameter

Symbol

Initial value

Internal temperature (K) Internal gas pressure (MPa) Porosity (%) Young’s modulus (GPa) Poisson’s ratio (-) Tensile strength (MPa)

T0 PG0 φ0 E0 ν ft0

293 0.1 12.2 30 0.2 3.0

Figure 1.

Schematic representation of 1D wall model.

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Figure 2. Distributions of mechanical damage and pore pressure (Case 1—K0 = 1 × 10−17 m2 ; R.H. = 1%).

Figure 4. Distributions of mechanical damage and pore pressure (Case 5—K0 = 5 × 10−21 m2 ; R.H. = 90%).

Figure 3. Distributions of mechanical damage and pore pressure without pore pressure effects in elastic deformation (Case 2—K0 = 1 × 10−17 m2 ; R.H. = 1%).

Figure 5. Distributions of mechanical damage and pore pressure without pore pressure effects in elastic deformation (Case 6—K0 = 5 × 10−21 m2 ; R.H. = 90%).

tensile strength of the concrete (initially 3 MPa). Figure 3 shows the behaviour predicted for Case 2, which is the same as Case 1 but for the isolation of the pore pressure effects from the thermal induced stresses. As can be seen, the damage pattern is almost identical to that of Case 1 and hence it seems that, in this case, it is the thermally induced stresses that were the predominant cause of the damage zone. In contrast, Figure 4 and Figure 5 show the predicted results for concrete with an initial permeability of 1 × 10−21 m2 and an initial relative humidity of 90% (Cases 3 & 4). As can be seen, the mechanical damage again initiates after about 1200 s, increases with time and generally developing towards the inside of the wall, and again it may be noted that this pattern matches very well with the location of observed spalling behaviour. However, comparing Figures 4 & 5 (respectively with and without pore pressure effects), with Figures 2 & 3, clear differences can be seen.

In both cases the predicted pore pressures are much higher than before, in excess of 3 MPa, and exceeding the tensile strength of the concrete. This is to be expected given the increased moisture content that, upon evaporation, produces large quantities of water vapour, and the low permeability, which restricts fluid flow and allows vapour to build up. Furthermore, the damage pattern in Figure 4 is significantly different to that observed in the previous analyses with a flat zone that can be seen developing deeper into the wall, ahead of the main damage zone. It may be noted that, firstly this zone coincides with the peak pore pressures and that, secondly it is not present in Figure 5 where the effects of the pore pressures have been ‘switched off’. It therefore seems clear that the pore pressures have contributed directly to the development of the damage zone, accelerating its development towards the inside of the concrete and slightly increasing its magnitude throughout.

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Figure 6. Mechanical damage separately caused by pore pressures and by thermally induced stresses.

The distinct effects of the pore pressures and thermally induced stresses may be more clearly seen in Figure 6 where the damage attributable to each has been plotted separately. Despite the clear effect of the pore pressures it may still be considered that they play a secondary role in the development of spalling in terms of magnitude of damage.

3.2

Column problem

Analyses 5 to 7 (Table 1) considered a concrete column exposed to fire on all sides. This problem was modelled as a two-dimensional problem in quarter-plane symmetry, as shown in Figure 7. Similarly to the wall model (Section 3.1), two (top and right) sides were constraint-free and heated by the standard ISO834 fire curve, while the other (bottom and left) sides were symmetry planes and were therefore fixed and isolated and had no heat or mass transfer across them. A typical set of results are presented in Figure 8 showing the development of mechanical damage, first principal stresses and pore pressures, in this example for a concrete column with an initial permeability of 1 × 10−19 m2 and initial relative humidity of 65% (Case 5). It can be observed that the mechanical damage starts just after 160 s, developing quickly and extending to the heated boundaries after 200 s. The damage pattern can clearly be interpreted as defining a fracture across the corner of the column which is highly reminiscent of the often observed phenomenon of corner spalling, where a triangular corner piece is lost (e.g. (Anderberg 1997)). Anderberg, suggests that corner spalling is mainly caused by thermally induced stresses and this is again supported by the numerical results of these analyses. Firstly, it can be seen in Figure 8 that during heating tensile stresses are quickly built-up and concentrated at the corner, causing mechanical damage once they exceed the tensile strength of the concrete.

Figure 7.

Schematic representation of 2D column model.

Secondly, it can be seen that the pore pressure has a maximum value of just over 1 MPa at the time of 200 s. This is much lower than the tensile strength of the concrete (initially 3 MPa) and it may also be seen that this peak value occurs at some distance from the area of damage. In fact it can be seen that the pore pressure build-up is concentrated in a very small area at the edge and corner of the column and lags significantly behind the build-up of stresses. These results indicate that the thermally induced stresses play a more significant role than the pore pressures in the development of the mechanical damage. Further support for this conclusion can be found in the results of Case 6, which was the same as Case 5 but for the numerical isolation of the pore pressure effects from the thermal induced stresses. As can be seen in Figure 9, the developed pattern of damage is almost identical to that in Case 5 (Figure 8), indicating that the pore pressures had very little effect on the development of damage and that thermally induced stresses were the primary factor. The results of an extreme example of this problem, for a concrete with low initial permeability (5 × 10−21 m2 ) and high relative humidity (80%) (Case 7), are shown in Figure 10. In this case pore pressures higher than those in Cases 5 & 6 were expected. As can be seen, at the time of 200 s, a damage pattern similar to those seen in Cases 5 & 6 is again present. However, it may also be seen that there is a small patch of damage in the very corner of the column and that this corresponds to the location of the peak gas pressures (∼2.5 MPa). It may therefore be concluded that although under certain conditions pore pressures of magnitudes similar to that of the concrete strength can be developed and that these can be shown to contribute to damage of the concrete, they do not significantly affect the major damage zone that can be associated with observed spalling behaviour and,

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Figure 8. Evolution of mechanical damage, stress and pore pressure at: 160 s, 1801 s and 200 s (Case 5—K0 = 1×10−19 m2 , R.H. = 65%.)

3.3

Figure 9. Mechanical damage at 200 s without pore pressure effects in elastic deformation (Case 6—K0 = 1× 10−19 m2 , R.H. = 65%).

again, play only a secondary role. Thus, the effects of thermally induced stresses seem to be more significant in the development of damage, and hence spalling, of the concrete.

I-beam problem

Analyses 8 (Table 1) considered a concrete I-beam exposed to fire on both sides and its underside. The problem was modelled as a two-dimensional analysis in central-plane symmetry, as illustrated in Figure 11. As can be seen, two (right and bottom) sides were constraint-free and heated by the standard ISO834 fire curve, while the top surface was considered to be fixed and sealed by its attachment to a floor slab and the left side was a symmetry plane. Both these sides were therefore fixed and isolated, and had no heat or mass transfer across them. Again, a typical set of results are presented in Figure 12 showing mechanical damage, first principal stresses and pore pressures, in this example for a concrete column with an initial permeability of 1 × 10−19 m2 and initial relative humidity of 65% (Case 8) after 350 s of heating. Once again it can be observed that the mechanical damage has developed in zones associated with high levels of stress and that while areas of increased pore pressures have developed they again lag considerably

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Figure 10. Mechanical damage and pore pressures at 200 s (Case 7—K0 = 1 × 10−21 m2 , R.H. = 80%).

Figure 12. Mechanical damage, stress and pore pressures at 350 s (Case 8—K0 = 1 × 10−19 m2 , R.H. = 65%).

Figure 11.

Schematic representation of 2D I-beam model.

behind the stresses in moving towards the centre of the concrete beam. Once again this corroborates the findings of the previous analyses that the development of damage and hence spalling is principally the result of thermally induced stresses and that pore pressures are a secondary factor.

It is recognised that the pattern of damage developed in the I-beam is not as easily comparable to observed spalling behavior as in the previous two examples. One reason for this may be that the 2D analysis does not accurately capture the stress state of what is truly a 3D problem. Although the computational expense of the model is an issue at present, it is hoped to investigate this in the future. Another reason may be parametric behaviour of the damage model. Initial investigations into this aspect of the problem are summarised in the follow section.

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3.4

Damage development

It is recognised that the development of the damage zone may be highly dependent on the formulation of the model and on various chosen parametric values. Of particular interest are the formulation of the equivalent strain measure, ε˜ , the damage initiation threshold, κ0md , and the ductility, γ . In relation to the equivalent strain measure initial investigations, following the work of Peerlings et al. (Peerlings, de Borst et al. 1998), have shown that significantly different damage patterns are developed when formulations based on the energy release rate, a Mazars type description and a modified von Mises description are employed (Figures 8 & 13). In agreement with Peerlings et al. this work has found the von Mises description to give the most representative results. Since the von Mises description is weighted towards tensile effects this finding suggests that spalling is derived mainly through tensile behavior. However it is noted that this formulation employs an assumed ratio of tensile to compressive strength and that the sensitivity of the results to this ratio also warrants further investigation. In relation to the damage initiation threshold, κ0md (Equation 16), it may be noted that the choice of temperature dependent function for the tensile strength,

Figure 14. Normalised variation of damage initiation threshold, κ0md , (Equation 16) with temperature for various functions of tensile strength (ft (T )).

ft (T ), can have a significant effect on the variation of κ0md with temperature. Three examples employing only subtly different functions for ft (T ) taken from the literature are shown in Figure 14 and as can be seen significantly different damage thresholds are produced. The significance of this choice for the development of damage will be investigated further. Also shown to be significant is the choice of the ductility coefficient, γ . Higher values represent more brittle materials and although a value representative of the fracture energy should be selected (Equation 17) initial investigations have shown that the results are quite insensitive if a constant value is employed, even over a range of 3 orders of magnitude. However, using a more realistic, temperature dependent value for ductility, γ (T ), as proposed by Pearce et al. (Pearce, Nielsen et al. 2004) has been shown to produce results apparently less representative of observed spalling behavior; specifically, the narrow damage zone seen in these results (Figure 8) stops short of the heated edges of the concrete suggesting that the fracture remains internal to the column and that the corner piece could not fall away. The implications of this are not yet clear and again, further investigation is required. However, initial thoughts are that dynamic behavior, not captured by the model in its current form, may be important in rapid fracture development related to spalling or simply that all of the damage parameters discussed above are heavily inter-related. 4

Figure 13. Mechanical damage at 200 s using Energy Release rate and Mazars type equivalent strain descriptions.

CONCLUSIONS

In summary, the results of these investigations show that, in the cases considered, thermally induced stresses were primarily responsible for the development of damage, and hence spalling, with pore pressures playing only a secondary role. For concrete with high initial permeability and/or low relative humidity, the maximum pore pressures that developed were too low

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to directly affect the integrity of the concrete, and therefore, had negligible contribution to the development of damage. For concrete with low initial permeability and/or high relative humidity, high pore pressure can be built-up and these can directly cause mechanical damage. However, even in such cases, the effects of pore pressures were still secondary to those of thermally induced stresses, as, firstly, they tended not to occur in the locations associated with observed spalling phenomena and, secondly, the magnitude of damage caused by thermal induced stresses was significantly higher than that caused by the pore pressures. It was also shown that a modified von Mises equivalent strain measure produced damage patterns most representative of observed spalling behavior and that, while initial investigations showed that various components of the damage model are important in controlling the damage development they do not affect the first conclusion. Although it is clear that further investigation is required into certain aspects of the model it is noted that the results shown here match well with observed spalling behaviour both qualitatively in geometry and quantitatively in time.

ACKNOWLEDGEMENTS The support of the Engineering and Physical Sciences Research Council, UK (EP/E048935/1) is gratefully acknowledged.

REFERENCES Anderberg, Y. (1997). Spalling Phenomena of HPC and OC. International Workshop on Fire Performance of HighStrength Concrete Proceeding, NIST Special Publication 919. L.T. Phan, N.J. Carino, D. Duthinh and E. Garboczi. Gaithersburg, MD, NIST. Bažant, Z.P. (1997). Analysis of Pore Pressure, Thermal Stresses and Fracture in Rapidly Heated Concrete. International Workshop on Fire Performance of High-Strength Concrete Proceeding, NIST Special Publication 919. L.T. Phan, N.J. Carino, D. Duthinh and E. Garboczi. Gaithersburg, MD, NIST. Consolazio, G.R., M.C. McVay, et al. (1997). Measurement and Prediction of Pore Pressures in Cement Mortar Subjected to Elevated Temperature. International Workshop on Fire Performance of High-Strength Concrete Proceeding, NIST Special Publication 919. L.T. Phan, N.J. Carino, D. Duthinh and E. Garboczi. Gaithersburg, MD, NIST. Davie, C.T., C.J. Pearce, et al. (2006). ‘‘Coupled Heat and Moisture Transport in Concrete at Elevated Temperatures—Effects of Capillary Pressure and Adsorbed Water.’’ Numerical Heat Transfer, Part A 49(8): 733–763. Dwaikat, M.B. and V.K.R. Kodur (2009). ‘‘Hydrothermal model for predicting fire-induced spalling in concrete structural systems.’’ Fire Safety Journal 44(3): 425–434.

Gawin, D., C.E. Majorana, et al. (1999). ‘‘Numerical Analysis of Hygro-Thermal Behaviour and Damage of Concrete at High Temperature.’’ Mechanics of Cohesive-Frictional Materials 4: 37–74. Gawin, D., F. Pesavento, et al. (2003). ‘‘Modelling of HygroThermal Behaviour of Concrete at High Temperature with Thermo-Chemical and Mechanical Material Degradation.’’ Computer Methods in Applied Mechanics and Engineering 192: 1731–1771. Gawin, D., F. Pesavento, et al. (2006). ‘‘Towards prediction of the thermal spalling risk through a multi-phase porous media model of concrete.’’ Computer Methods in Applied Mechanics and Engineering 195(41–43): 5707–5729. Hertz, K.D. (2003). ‘‘Limits of Spalling of Fire-Exposed Concrete.’’ Fire Safety Journal 38: 103–116. Ichikawa, Y. and G.L. England (2004). ‘‘Prediction of moisture migration and pore pressure build-up in concrete at high temperatures.’’ Nuclear Engineering and Design 228(1–3): 245–259. Kalifa, P., F.-D. Menneteau, et al. (2000). ‘‘Spalling and Pore Pressure in HPC at High Temperatures.’’ Cement and Concrete Research 30: 1915–1927. Khoury, G.A., C.E. Majorana, et al. (2002). ‘‘Modelling of Heated Concrete.’’ Magazine of Concrete Research 54(2): 77–101. Msaad, Y. and G. Bonnet (2006). ‘‘Analyses of Heated Concrete Spalling due to Restrained Thermal Dilation: Application to the ‘‘Chunnel’’ Fire.’’ Journal of Engineering Mechanics 132(10): 1024–1132. Pearce, C.J., C.V. Nielsen, et al. (2004). ‘‘Gradient enhanced thermo-mechanical damage model for concrete at high temperatures including transient thermal creep.’’ International Journal for Numerical and Analytical Methods in Geomechanics 28(7–8): 715–735. Peerlings, R.H.J., R. de Borst, et al. (1998). ‘‘Gradientenhanced damage modelling of concrete fracture.’’ Mechanics of Cohesive-Frictional Materials 3(4): 323–342. Phan, L.T. (1996). Fire Performance of High-Strength Concrete: A Report of the State-of-the-Art, NISTIR 5934. Gaithersburg, MD, USA., Building and Fire Research Laboratory, National Institute of Standards and Technology. Phan, L.T., J.R. Lawson, et al. (2001). ‘‘Effects of Elevated Temperature Exposure on Heating Characteristics, Spalling and Residual Properties of High Performance Concrete.’’ Materials and Structures 34: 83–91. Stabler, J. (2000). Computational Modelling of Thermomechanical Damage and Plasticity in Concrete, PhD Thesis, The University of Queensland, Australia. Tenchev, R.T., L.Y. Li, et al. (2001). ‘‘Finite Element Analysis of Coupled Heat and Moisture Transfer in Concrete Subjected to Fire.’’ Numerical Heat Transfer, Part A 39: 685–710. Ulm, F.-J., O. Coussy, et al. (1999). ‘‘The ‘‘Chunnel’’ Fire. I: Chemoplastic Softening in Rapidly Heated Concrete.’’ Journal of Engineering Mechanics 125(3): 272–282. Zienkiewicz, O.C. and R.L. Taylor (2005). The Finite Element Method: Its Basis and Fundamentals, ButterworthHeinemann.

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Numerical simulation of slender structures with integrated dampers P. Egger & J. Kollegger Institute for Structural Engineering, Vienna University of Technology, Vienna, Austria

ABSTRACT: Several trends indicate that the design of structures is becoming more slender and filigree for esthetical and financial reasons. The innovative damping system offers new solutions for the contemporary construction with slender light-weight structures, higher strength materials and an increased awareness of earthquake risk. This paper describes how slender structures could be made more resistant against dynamical forces caused by wind and earthquake. On the structures dampers are mounted which are located parallel to the center line of the structure. The damping system can be implemented in different structure types like high rise buildings and bridges and in several ways along the structure. The kinematical relationships are described to explain the deformations and how to calculate them. Experiments were carried out in the laboratory in order to show the efficiency of the new damping system. In order to predict the structural damping ratio of future structures several numerical models were elaborated and the results were compared to the laboratory experiments. 1

INTRODUCTION

Humankind always tended to build higher and more slender structures. Many details have been solved to increase technical progress and adapt the strength of the construction material to new construction methods. Nowadays the precise analysis of the structures is one of the fundamental reasons that such slender construction can be built. Contemporary architecture and modern bridge design show an increased sensitivity for dynamic excitation. Due to these trends the innovative damping system could make a useful contribution to the future design of slender light-weight structures. There are many different types of damping systems for the reduction of the dynamic hazards of earthquakes or strong winds. The damping system depends on the one hand on the choice of the type of the damper e.g. metallic dampers, friction dampers, viscoelastic dampers, viscous fluid-dampers, tuned mass dampers and tuned liquid dampers, and on the other hand the method of implementation.

2

is activated as soon as the structure is subjected to bending and longitudinal elongation (Fig. 1). Under the dynamical loading the structure oscillates and the relative displacement causes the damper to dissipate energy. The innovation consists in the application and the shape of the dampers. For high rise buildings or chimneys the viscous dampers are running along the entire height of the building und are part of the supporting structure. The dampers can be integrated in columns or in structural walls by leaving just a void formed by a steel or plastic tube. In the tube a bar with ribbed or profiled surface connected to the structure at one point only is installed. After installation of the bar the tube is filled with a viscous liquid. The bar moves relatively to the tube when the structure oscillates [Kollegger, 2009]. For arch bridges the dampers could be placed along the hanger between the arch and the bridge deck or in a stay cable bridge, in the stays. It would also be possible to integrate the tube and the bar in the bridge beams

SYSTEM DESCRIPTION

The new method protects structures like e.g. high rise buildings, towers, chimneys and bridges from vibrations, due to wind or earthquake loading. The structure supports dead, variable and dynamical loads in horizontal and vertical direction. Waves produced by an earthquake or wind affect the structure less if the damper is integrated. A structure equipped with the damper can be seen as a compound beam which improves the bending resistance. The damper is situated parallel to the centre line of the structure and

Figure 1. Cantilever beam with an integrated damper [Kollegger, 2009].

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so that the damper would not only reduce the ambient vibrations but also the ones caused by cars and trains. One of the main objectives of the structural damping method is to simplify the installation of the dampers without any more expenditure for corrosion prevention.

results in ds ≈ dx. For a beam without longitudinal forces subjected to uniaxial bending the curvature can be determined by κζ =

1 dϕζ Mη =− . = ρζ dx EIη

l is calculated by the following relationship 3

l = ϕζ ∗ z

KINEMATICAL RELATIONSHIP FOR THE DETERMINATION OF THE AXIAL DEFORMATION L

The strain- deformation equation for a bending beam is determined by the curvature κ. κζ =

4

1 dϕζ =− ρζ ds

4.1

The curvature is depending on the radius of curvature of the beam axis through the centroids of the cross-section (Fig. 3). Considering the Bernoulli hypothesis, the linearization of the bending curvature

Figure 2.

For a more precise calculation of the axial deformation l the nonlinear mathematical relationship of the third order theory could be considered [Mang, 2008].

DESCRIPTION OF THE EXPERIMENTS Experiment to evaluate the damping ratio of different types of dampers

In the laboratory of the Institute for Structural Engineering at Vienna University of Technology experiments were carried out to prove the theoretical thesis of the topic. The first experiment consisted in testing the different types of dampers in order to evaluate their damping constants. The shells of the dampers were round fiberglass tubes with different diameters. The tubes were filled with water or silicon oil. Different shaped bars were plunged into the tubes. The bars were either threaded bars or threaded bars with screwed nuts and washers or threaded bars surrounded with perforated steel plates. In order to excite vibrations a shaker or an impulse hammer was used. The evaluated results have shown that the bars with the screwed nuts and washers had the highest damping effect but for the second experiment we chose the bar surrounded with perforated steel plates because the damping effect was just slightly lower and more perforated steel plates could be placed around the bar thus the damper gave the expected damping-values.

Determination of the curvature of a bending beam.

4.2

Figure 3. Experimental construction to test the new damping system [Neubauer, 2009].

Laboratory test of the damping system

The second experiment shows a structure with the implemented new damping system. The cantilever beam with the eccentric connected damper is shown in Figure 5. Steel plates jointed on both sides serve as spacers between the damper and the beam. The damper has the same shape as the one used in the first experiment where it was analyzed and the damping constant was ascertained. The damper is 2 m high and has an eccentricity of 0,2 m from the aluminum beam. In order to determine the stiffness of the structure a static experiment was carried out. A calibrated weight was placed on the aluminum beam in order to displace the structure, and with the laser—displacement transducer the precise distance was measured. Due to these results the stiffness of the structure could be calculated. After this the structure was excited

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by a sinusoidal force or an impulse hammer and the dynamic damping values could be determined. The plastic tube was gradually filled once with water and once with silicon oil with 5000 mm2 /s kinematic viscosity. The diagram (Fig. 4b) shows the result of the experiment using the profiled bar with perforated steel sheets (Fig 4a). By using silicon oil the damping ratio increases from 3 to 10% by gradually filling the tube from the zero to 1,8 m. The diagram shows that the damper filled with water is less efficient than the one filled with silicon oil. 4.3 Field experiment on a concrete cantilever beam A third experiment on a larger scale was carried out employing the bridge pillar which had been used for the balanced lift method experiment [Kollegger, 2008]. The pillar has a height of 8,3 m and on top a base area of 1,10 m by 0,40 m. The cantilever is

Figure 4. (a) Bar surrounded with perforated steel plates; (b) Diagram of the damping ratio [Neubauer, 2009].

Figure 5. (a) and (b) bridge pillar with implemented damping system; (c) electro dynamic force generator.

made of reinforced concrete with the concrete quality C40/50 and steel bars quality BSt 550. On top of the pillar four electro-dynamic force generators were placed to produce dynamic forces in the direction of the two principal axes and around its centre line in order to generate a torsion moment. The damper was placed in four different places eccentric to the centre to enable the analysis of several cases. For one experiment (Fig. 5a) the damper was applied in a distance of 0,80 m from the centre line of the cantilever beam. The damper was built once again using a fiberglass tube filled with silicon oil 5000 mm2 /s kinematic viscosity. The profiled bar with three rings of perforated steel sheets was moved along the tube every time the pillar moved. The damping ratio ζ of the structure increased by 0,5% in accordance with the numerical analysis. 5

NUMERICAL SIMULATION

To predict the structural damping ratio, computational models were developed by means of the commercial software SOFiSTiK. By using the dynamic tools DYNA and ASE (SOFiSTiK AG, 2009) the finite element model (Fig. 6a and 6b) was excited by the sinusoidal force. For the numerical calculation the modal analysis and the time integration method were chosen respectively. The results of the natural frequencies, the results of the displacements and accelerations of the numerical models are very much in accordance with the experimental laboratory tests. Figure 6c and 6d show a comparison of the dynamic displacement amplitudes of the numeric analysis and the laboratory tests after the transient oscillation. For the field tests the numerical simulation proved to be much more difficult. First of all the stiffness properties of the pillar could not be ascertained by a precise static experiment which had been performed in the second laboratory experiment. In the laboratory test, it could be assumed that the cantilever beam had a nearly rigid restraint, which was not the case in the field test, even if the threaded bars were prestressed in order to achieve a higher restraint. To develop the numerical model the eigenfrequencies of the field experiments and the simulation were compared. The bearing properties of the computational model were adjusted until the natural frequencies matched the ones evaluated from the experimental tests. By exposing the numerical model to the same dynamical forces as the pillar in field test, the results of the displacements and accelerations concurred. Unfortunately, due to the low force power of the four electro dynamic force generators and the high stiffness of the cantilever, the concrete pillar never reached the cracked state. In the state of resonance the pillar, applied with maximum sinusoidal force, the maximum oscillation displacement on top of the pillar amounted to ±30 mm at the natural frequency

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Figure 8. Damping system for shear deformation [Kollegger, 2009].

of 1,7 Hz. Consequently, there are no differences between linear or nonlinear material model simulations. Even when calculating with linearized geometrical relationships or a more precise third order theory, the approach made no significant alteration. 6

Figure 6. (a) (b) Numerical model; (c) time- displacement results from FEM; (d) time- displacement results from laboratory tests.

OUTLOOK

In the future we will analyze this easy way of placing dampers along structural walls for common buildings in order to reduce the shear deformations under earthquake loadings (Fig. 9) and to find out by using numerical models when the concrete structure changes from the uncracked state to the cracked state and to analyze the loss of the structural safety. 7

CONCLUSION

The innovative damping system can be integrated in many types of structures and also in many ways. Dynamical excitations are becoming more and more important and the newly developed system could offer a solution. Important issues are the simple handling during the construction process and the low cost maintenance over the lifespan of the structure. REFERENCES

Figure 7. (a) (b) Numerical model for the experiment shown in figure 5a; (c) Numerical model for the experiment shown in figure 5b.

Chopra, A.K. 2007. Dynamics of Structures: theory and applications to earthquake engineering. Upper Saddle River, NJ : Prentice Hall, 2007. Kollegger, J. & Blail, S. 2008. Balanced Lift Method for Bridge Construction. Structural Engineering International. Volume 18, 2008, 3/2008. Kollegger, J. & Egger, P. 2009. Tragkonstruktion. Patent application T12456 Austria, March 18, 2009. Mang, H. & Hofstetter, G. 2008. Festigkeitslehre. Wien: Springer, 2008. Neubauer, C. 2009. Untersuchungen zur Entwicklung einer Tragkonstruktion mit hoher Strukturdämpfung. Vienna: Vienna University of Technology, 2009. SOFiSTiK A.G. 2009. ASE. Oberschleißheim: SOFiSTiK, 2009. SOFiSTiK A.G. 2009. DYNA. Oberschleißheim: SOFiSTiK A.G, 2009.

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Textile reinforced concrete sandwich panels: Bending tests and numerical analyses J. Finzel & U. Häußler-Combe Technische Universität Dresden—Institut für Massivbau, Dresden, Germany

ABSTRACT: In an ongoing study the mechanical behavior of Textile Reinforced Concrete (TRC) sandwich panels under quasi-static and dynamic loading is being investigated. Experimental data from bending tests are compared with different numerical models. In a parametric study the influence of span, foam density and percentage of reinforcement in the TRC-layers are investigated. Results of the static tests indicate that the stiffness of the samples depends mainly on the core density and thus the shear stiffness, while an increase of TRC reinforcement enhances deflection capacity. To numerically model the behavior of sandwich elements, ‘classical’ sandwich theory calculation models were used. Furthermore two software products (ANSYS and ‘nmk’, an FEA-system developed by Häussler-Combe et al.) were employed. The test results are being compared with calculation models based on beam-theory, with FEA models based on linear-elastic, anisotropic and multilinear elastic material models. 1

INTRODUCTION

Sandwich constructions consisting of a relatively soft core material, covered by harder face sheets, have been in use in naval and aerospace applications for a long time. Their main advantages are light-weight construction and impact absorption capacity. Using foam cores also provides good thermal insulation. These advantages can be transferred to structural applications, such as prefabricated facade-elements. Yet the use of ‘traditional’ reinforced concrete as face layers induces thick, heavy profiles due to the necessary concrete cover. Substituting the steel rebars by high-strength carbon or glass fiber textiles that are not susceptible to crorrosion, and using fine-grained concrete with a maximum aggregate size of 1–2 mm, facilitates concrete layers of less than 1 cm thickness. These textile reinforced concrete (TRC) sandwich elements can even be used as load-bearing walls. First experimental buildings have been tested successfully by Schneider (2009). Possible applications are also in structures exposed to dynamic loads, like impacts of projectiles in a military context or meteorites in extraterrestrial structures, such as the projected ‘LunarBase’ on the moon, cf. HäusslerCombe (2009). TRC faces pose a good alternative to metal faces, because they can be produced at low costs and are less vulnerable to local indentations than thin metal layers. To assess the load bearing mechanism, load capacity and especially impact resistance, an experimental program has been planned. So far a number of static bending tests with different materials and at

varying spans have been conducted. ‘Classical’ sandwich theories have been tested in order to reproduce the test results. Furthermore FEA calculations have been used for his purpose. In the future, dynamic impact tests using a falling-weight tower, similar to the test setup proposed by Bentur (1987), will be carried out.

2 2.1

EXPERIMENTAL PROGRAM Test setup

The experimental program comprises a number of different test setups. The tested sandwich elements (Figure 1) (l = 600 mm/1200 mm; b = 100 mm) consist of an 80 mm polyurethane foam core covered by two 8 mm face sheets made of textile reinforced concrete (TRC), bonded by 1 mm thin layers of epoxyresin cement (Sikadur-30 DUE). The parameters variegated were span (30 cm; 50 cm and 100 cm), core density (80 kg/m3 and 200 kg/m3 ) and percentage of reinforcement (1.61%/1.79% alkali resistant glass and 0.78%/1.56% Carbon fibers) in the face layers. For each setup, four to six samples were used. The tests were conducted deflection-controlled at a speed of 2 mm/min using a Zwick hydraulic material testing machine (Type Z250) with the measuring data processing software Zwick-Roell testXpert V11.02. HBM WA50 inductive displacement transducers were used to measure deflections. Measuring data were being processed by HBM Catman software and later

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on imported to Microsoft Excel 2003 spreadsheet application. 2.2

Test results

For all test setups the structure reacted linear-elastic until the first crack in the bottom TRC-layer appeared. In the next stage, more cracks open and also propagate into the foam core. Finally continuous shear cracks in the foam layer lead to large deflections. In both the four-point- and three-point-bending tests the results in the linear elastic range displayed a low coefficient of variation (0.04 . . . 0.19). The slope of the second branch of the load-deflection curve can also be

Figure 1. 50 cm.

Test setup for three-point-bending at a span of

Figure 3.

Load-deflection-curves for the different test setups.

idealized linear with a relatively low coefficient of variation (0,10 . . . 0,31). This corresponds with the tensile behavior of TRC. Until crack initiation, TRC reacts linear elastic with the elastic modulus of the matrix. In the following stage, multiple cracks form, reducing the tensile stiffness. In the post-cracking state, an increasing amount of force is transferred to the reinforcement, resulting in an increase in stiffness until the tensile failure of the textile strands. A typical stress-strain diagram for TRC is shown in Figure 2. The final failure of all sandwich test samples was initiated by a core shear crack. In most cases, the shear crack abruptly propagated horizontally along the foam-concrete interface (delamination, see Figure 4). This type of failure is also described as typical in the

Figure 2. Typical stress-strain diagram for a TRC tensile test (reinforcement: alkali resistant glass, 1200 tex, 2 layers).

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Figure 4. tion.

Failure in four-point-bending due to delamina-

Figure 5. Failure in four-point-bending due to tensile failure of bottom TRC-layer.

corresponding code for ‘Self-supporting double skin metal faced insulating panels’ EN14509:2006. Only in the conducted static four-point-bending tests different final failure modes (tensile failure of the bottom TRC face (Figure 5) and delamination in the bottom layer (Figure 7) or between bottom layer and core as stated above) causing large scatter in the ultimate failure load values could be observed. However these effects only occured way beyond the serviceability state when the core shear cracks already have widened. Due to the relatively thick faces, a noteworthy indentation (which is a common failure mode for sandwiches with thin metal faces) only occurred in one single sample with a PU80 core (Figure 6). Mean test results for the different setups are displayed in Figure 3 and in Table 1, using the standardized nomenclature for the experiments: 1P_PU222_333_44_55 with: • • • •

1 = 3 or 4 (-point-bending test) 2 = 80 oder 200 (density of PUR-foam) 3 = 30, 50 or 100 (cm of span) 4 = A or C (reinforcement: Alkali resistent glass, 1200 tex or Carbon fibers, 800 tex) • 5 = 1, 2 or 3 (number of reinforcement layers)

Figure 6. Indentation of top TRC layer in a three-point-test with PU80 foam core.

Figure 7.

Delamination within the TRC layer.

Table 1. Test results for first cracking in TRC and for ultimate load state (mean values).

Test setup

Fcr (kN)

wcr (mm)

Fu (kN)

wu (mm)

3P_PU200_50_C_2 3P_PU80_50_C_1 3P_PU80_100_C_2 3P_PU80_50_C_2 4P_PU200_50_A_3 3P_PU200_50_A_2 3P_PU200_30_A_2

5.31 2.27 2.06 2.41 6.39 5.92 4.00

1.97 2.45 5.87 2.87 1.95 1.88 0.84

10.92 4.63 4.63 5.82 18.99 9.82 −

8.49 9.43 21.45 14.24 11.28 4.81 −

The tests at 30 cm span were only conducted in the linear-elastic range, since the same specimens were also to be used in 50 cm span tests afterwards. A comparison of the setups shows that the bending stiffness is mainly governed by the foam density, which is almost proportional to the modulus of elasticity and the shear modulus. The different types of textile reinforcement in the TRC face layers had no measurable effect on the bending stiffness in the cracked state, although the tensional stiffness of the face layers differs by a ratio of EAC,2layers,cr /EAC,1layer,cr = 1.73.

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Yet, substituting one layer of reinforcement by two layers, resulted in a higher ultimate deflection by factor 1.53.

The so calculated mid-span deflection show good agreement with the the measured deflections in the linear elastic case (cf. Table 3). Normal forces in the face layers can be estimated by:

3

Nf ,b,l =

NUMERICAL SIMULATIONS

3.1

‘Classical’ sandwich theories

The estimation of the load bearing behavior of sandwich structures has been researched since the 1930s. A synopsis of these researches is given by Duda (1965). For the classical theories, bending stiffness D and shear stiffness S are being considered separately. The simplest model for sandwich load bearing properties is based on the assumptions that the core has no bending rigidity and the faces have no Eigen area moment of inertia and are not deformed due to shear. So S only depends on the core properties, while D is determined by the tensile/compressive stiffness of the faces. For a soft core (Ef · t · d 2 /Ec · c3 ≤ 16.7) and thin faces (d/t ≤ 5.77), these assumptions cause less than 1% error in calculation. Thus the bending stiffness is determined by (indices: s = Sandwich; c = core; f = face − t = top; b = bottom) (E · A)t + (E · A)b 2 ·a 4 with a being the distance between the centroids of the face layers. The shear stiffness can be written with the core shear modulus and core dimensions as: D=

Yet the prediction of cracking load by this method did not give satisfactory results for the test setups. Hence the results are omitted here. 3.1.2 Modified beam theory A simpler and thus more common approach, as described by Zenkert (1995) and in the DIAB (2009) sandwich handbook, is based on beam theory and superposes bending and shear deformations. In this model, for a three-point-bending test, the deflection at midspan equals w=

wI =

P· D

l3

 ·

1 ·ψ α · λ2 

e ε= ; l β=

x ξ= ; l

λ=



1+α ; α·β

wb =

P · l3 P · l3 = E ·A 48 · D 48 · f 2 f · a2

ws =

P·l P·l = 4·S 4 · Gc · hc · b

In four-point-bending, the deflection equals wb = 2 ·

1 · (1 − ε) · ξ · (2ε − ε2 − ξ 2 ) 6 +

PL PL3 + 48D 4S

with the bending stiffness D and the shear stiffness S. Zenkert proposes testing equal sections at different spans to determine D and S. Yet the tests at different spans did not bear results which could be used for this determination, since this calculation is very susceptible to test scatter, a phenomenon also described by Fukuda (2001). In three-point-bending with equal top and bottom face layers the model leads to the following bending deflection and shear deflection at midspan:

S = Gc · h c · b 3.1.1 Differential equations Differential equations provide a solution for the displacements u(x), w(x) for a given load. Stamm (1974) derived simplified equations valid with the assumptions, that deformation can be divided into shear and bending deformation, materials are isotropic, shear stresses are constant in the core and no core compression occurs. In a setup with a single load, this yields the deflection

P·l 1 · ·ψ a 1+α

(3α − 4α 3 ) · P/2 · l 3 48 ·

Ef ·Af 2

· a2

Table 2. Mean values of tensile stiffness for the different types of TRC in uncracked and cracked state and failure stress (related to entire TRC-section).

Dt + Db α= ; Ds

Ds S · l2

TRC

EAuncr (GPa)

EAcr (GPa)

σu (MPa)

C_1 C_2 A_2 A_3

22.17 20.04 18.67 19.51

1.63 0.94 0.90 0.99

10.59 20.46 16.69 11.65

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ws = 2 ·

P/2 · α · l P·α·l = 2·S 2·V

3.2

which yields for the test geometry (α = a/l = 0.15/0.50 = 0.3):

wb = 0, 792 ·

P · l3 48 ·

Ef ·Af 2

· a2

P·l ws = 0.3 · 4·S According to these calculations, pure bending deformation amounts to only 6 . . . 8% of the total deflection in the given setups. To consider concrete in the cracked state, the bottom face can be assigned a lower Young’s modulus. Thus the bending stiffness is modified: Dcracked = D ·

FE-Analyses

Additional to the above stated hand-calculations, FEA analyses were done with ANSYS Version 11.0 and nmk (Numerische Methode Karlsruhe) Version 2009– 07, an FEA-program developed by Häussler-Combe et al. All Layers were represented by 2-dimensional rectangular elements. The models represent one half of the beam, making use of symmetry conditions.

2 · (Ec,cracked · Ec ) Ec,cracked + Ec

The so calculated deflection values for the linear elastic case are compared with the measured deflections in Table 3. Evidently all values for the softer foam core (PU80) display good agreement. The values for the PU200core are also within a reasonable range, except for the four-point-bending test setup. Here the real test sample reacts by far softer than the calculation model. Considering the equations above, the load-deflection ratios would have to be in the range F/w(3 − point)/ F/w(4 − point) = (0.3 . . . 0, 792) at midspan. For the real tests, the ratio is F/w(3 − point)/F/w(4 − point) = 0,875. Even with the assumption of a lower core density of Ec = 56 MPa instead of 76 MPa, a variation within the manufacturers tolerance specifications, the calculated result is 67% higher than the test result. This indicates that this analysis is not adequate for this kind of setup. FEM analyses (see below) showed much better agreement with the same input values.

3.2.1 ANSYS For an ANSYS FEA-model of similar structure, an inorganic phosphate cement (IPC)-sandwich structure reinforced with glass fibers, Cuypers (2001) proposes the use of PLANE82 elements or a combination of PLANE82 elements for the core and BEAM188 elements for the face layers. These models have been tested and for the linear elastic model, all layers of the sandwich beam could be modeled by PLANE82 elements. For modeling the Sikadur mortar layer, different models were tried. By using short BEAM3 elements in vertical direction plus inclined BEAM3 elements in between, the compressive properties and shear properties could be considered separately. Yet modeling the mortar layer by PLANE82 rectangular elements yielded better results and avoided the stress peaks at the beam-plane connecting points. To account for the nonlinearity of cracked concrete, the TRC layers were assigned the element type PLANE183. Thus a multilinear isotropic material model (MISO) could be applied. The resulting VonMises-stresses and corresponding Von-Mises-strains in a three-point-bending test are displayed in Figure 8. Typically, the calculations using data from tensile tests for the TRC multilinear stress-strain relation, yielded smaller forces than the real tests although damages in the foam core were not accounted for. An example is shown in Figure 9. Local stress peaks at the supports

Table 3. Comparison: Test (t) results (mean values) and calculations (MBT = modified beam theory; DE = differential equations) (kN/mm; (relative error in %)). Test setup

F 

3P_PU200_50 3P_PU80_100 3P_PU80_50 4P_PU200_50 3P_PU200_30

2.81 0.36 0.87 3.72 4.79

w t

F 

F 

w MBT

2.46 (−12%) 0.39 (−8%) 0.85 (−2%) 7.32 (122%) 4.74 (−1%)

w DE

2.49 (−12%) 0.41 (13%) 0.98 (11%) 3.68 (−1%) 4.75 (−1%)

Figure 8. Von-Mises-stresses and Von-Mises strains in the ANSYS analysis of the three-point-bending setup.

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Figure 9. Load-deflection: comparison of test results with a two-step ANSYS (continuous line) simulation and a multilinear isotropic ANSYS calculation (grey).

cause the premature yielding of the concrete. When the support area was assigned a linear elastic material, the stress peaks occurred at the border between linear elastic material and multilinear isotropic material in the bottom layer, which returned similar results. Still, the F/ w ratio in the cracked state matches the experimental results. If the load-deflection curve is considered as a bilinear relation, the linear elastic branch and the smoother branch due to cracked bottom layer could be calculated in two steps. The slope of the second branch is determined by calculating on a linear elastic basis, with a reduced modulus of elasticity for the bottom layer. Yet it was not possible to determine exactly the bending point, i.e. the initial cracking of the bottom TRC layer. 3.2.2 Nmk nmk (Numerische Methode Karlsruhe) is a C++− based FEA-program developed by Häussler-Combe et al. The program uses input files with an ABAQUSsyntax. It provides a number of material models, such as plasticity, anisotropy and different damage models proposed by Pröchtel (2008). Using this program also facilitates the development and implementation of other material models. Parametric studies are possible with an OptiSlang-interface or by modifying the input files. In this project, all layers were represented by 2-dimensional rectangular 4-node elements with plane stress distribution (CPS4). For the parametric study, a python-script which variegates the parameters was developed. By comparing the different setups, the elastic material parameters which yield correct results for all setups were determined. A sensitivity analysis proved the model to be insensitive to most parameters, except for the shear modulus of the mortar layer and by far predominantly the shear modulus of the core. Variations of the other parameters within a realistic range only altered the results by less than 1 percent. This still applies for an anisotropic material model with a

shear modulus completely independent of the elastic modulus. Figure 10 shows the dependence of calculation results for the PU200 test setups on core shear modulus. Values above the x-axis show that the calculation model reacts ‘softer’ than the real test setup. The ‘scatter’ of the values shows the small influence of the other parameters. It can be observed, that the calculations for three-point-bending with a span of 30 cm and 50 cm respectively, converge to the test results at a similar core shear modulus, while only a ≈30% lower core shear modulus can explain the test results in four-point-bending. A possible explanation would be that the foam cores from this lot had a lower density than the other cores. The manufacturer’s specifications indicate up to 30% of variation in elastic modulus and thus shear modulus. The material parameters which yielded correct results are listed and compared to the manufacturers specifications in Table 4. Considering cracked concrete by a lower elastic modulus of the TRC and and the cracked foam by an anisotropic foam model with a very low shear modulus (Ganiso ≈ 0.12 · Ec ), gave fairly good results for all test setups at 50 cm span, cf. Table 5. For the 100 cm span setup this numerical model is 20% softer than the real setup. This is probably because in reality cracks only appear close to mid span while in the model they are spread over the whole beam. In general, this approach with the cracks represented by a smeared damage of the whole core can

Figure 10. Relative error (calculations—test results) for different test setups, calculated with identical material parameters at different core shear modulus values. Table 4. Material parameters for the nmk-FEA calculations reproducing real tests–compared to manufacturers specifications (m.s.). Material

E (MPa)

E(m.s.) (MPa)

ν −

(uncracked) Concrete Sikadur PU80 PU200 PU200 (4-point-test)

27,000 12,800 25 76 56

(27,000) (12,800) (20–31.5) (52–75) (52–75)

0.20 0.00 0.10 0.05 0.05

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Table 5. Comparison: F/w-ratio of test (t) results (mean values) and calculations (cal) in post-cracking state (kN/mm; (relative error in %)). Test setup

F 

3P_PU200_50_C2 3P_PU200_50_A2 3P_PU80_100_C2 3P_PU80_50_C1 3P_PU80_50_C2 4P_PU200_50_A3

0.901 0.901 0.164 0.336 0.385 1.47

Figure 11.

w t

F  w cal

0.904 0.857 0.131 0.358 0.358 1.39

(error) (0%) (−5%) (−20%) (7%) (−7%) (−5%)

Drop-weight frame for future impact tests.

serve as a rough estimation of post-cracking deflection behavior.

4

CONCLUSIONS AND OUTLOOK

A number of TRC sandwich plates have been tested in different bending test setups. Due to the complex load bearing mechanism relatively large scatter in the failure loads and maximum deflections occurred. Yet, ‘hand-calculations’ can reproduce most test results in the serviceability range with reasonable accuracy. Especially at short spans, an FEA calculation will yield more precise results. For exact results in the analysis of cracked TRC sandwiches, an FE analysis with discrete cracks (as e. g. proposed by Carlsson) is required and will be used in future simulations. For future tests, the most important objective is to test samples on impact loading, using the drop-weight tower shown in Figure 11. Furthermore static tests will be monitored by a 3-dimensional digital image correlation (Aramis 5 M by GOM), as proposed by Jin (2007) and Hegger (2007) to validate the numerical models and to improve the understanding of the load bearing mechanisms.

REFERENCES Bentur, A.; Mindess, S.; Banthia, N.: Impact behaviour of concrete beams. In: Materials and Structures 20 (1987), Nr. 5, 293–302. ISSN 13595997. Carlsson, L.A.; Matteson, R.C.; Aviles, F.; Loup, D.C.: Crack path in foam cored DCB sandwich fracture specimens. In: Composites Science and Technology 65 (2005), Nr. 15–16, 2612–2621. ISSN 02663538. Cuypers, H.: Analysis and Design of Sandwich Panels with Brittle Matrix Composite Faces for Building Applications. Brussel, Vrije Universiteit Brussel, thesis, 2001. http:// wwwtw.vub.ac.be/memc/website/webpages/general_info/ pdf/heidi_phd.pdf DIAB Sandwich Concept—Sandwich Handbook http://www. diabgroup.com/europe/literature/e_pdf_files/man_pdf/ sandwich_hb.pdf (accessed October 20, 2009). DIN EN 14509—Selbsttragende Sandwich-Elemente mit beidseitigen Metalldeckschichten—werkmässig hergestellte Produkte—Spezifikationen Norm November 2006. Duda, A.: Berechnung dreischichtiger Platten und Schalen— Literaturübersicht. 1965. Fukuda, H.; Kawasaki, T.; Kataoka, A.; Tashiro, S.: Bending test for CFRP skin/foamed core sandwich plates. In: Advanced Composite Materials 10 (2001), Nr. 2/3, p199–208. ISSN 09243046. Häussler-Combe, U.; Hartig, J.; Finzel, J.: TextilbetonSandwichbauteile—Tragverhalten unter Anpralllasten. In: Breit, W. (Hrsg.); Schnell, J. (Hrsg.); Grümann, R. (ed.); TU Kaiserslautern and DLR (Veranst.): LunarbaseBauen für ein Leben auf dem Mond. Kaiserslautern, 2009, p. 61. Hegger, J.; Horstmann, M.; Scholzen, A.: Sandwich panels with thin-walled textile reinforced concrete facings. Oral presentation: ACI Fall Convention Puerto Rico 2007, published in ACI SP-251: Design and Applications of Textile Reinforced Concrete. CD-ROM (pdf-file SP-251-7), pp. 109–123. Jin, H.; Lu, W.Y.; Scheffel, S.; Hinnerichs, T.D.; Neilsen, M.K.: Full-field characterization of mechanical behavior of polyurethane foams. In: International Journal of Solids & Structures 44 (2007), Nr. 21, 6930–6944. ISSN 0020–7683. Pröchtel, P.: Anisotrope Schädigungsmodellierung von Beton mit Adaptiver Bruchenergetischer Regularisierung, TU Dresden, thesis, 2008. Schneider, H.N.; Schätzke, C.; Feger, C.; Horstmann, M.; Pak, D.: Modulare Bausysteme aus Textilbeton-Sandwichelementen. In: CTRS4-4. Kolloquium zu textilbewehrten Tragwerken, 2009, p. 565–576. Stamm, K.; Witte, H.; Sandwichkonstruktionen: Berechnung, Fertigung, Ausführung. Springer Wien, 1974. Zenkert, D.: An Introduction to Sandwich Construction. Engineering materials advisory services Ltd, 1995.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Structural behavior of tunnels under fire loading including spalling and load induced thermal strains Thomas Ring & Matthias Zeiml Institute for Mechanics of Materials and Structures(IMWS), Vienna University of Technology, Vienna, Austria

Roman Lackner Material-Technology Innsbruck (MTI), University of Innsbruck, Innsbruck, Austria

ABSTRACT: Fire accidents in tunnels considerably influence their structural safety. During fire exposure, the concrete lining is damaged due to various processes (i.e., thermal degradation, spalling etc.), eventually leading to failure of the tunnel structure. Simulation of the structural behavior of tunnels under fire loading can only capture the real structural behavior when all governing processes are taken into account with sufficient accuracy. This work concentrates on the influence of various parameters on the quality of prediction of the structural behavior of tunnels under fire loading (temperature distribution, material model, spalling). 1

INTRODUCTION

Fire design of tunnel structures can be performed on different levels of sophistication, introducing a certain degree of simplification. Starting with definition of the fire load itself (e.g., by prescribing temperature-time curves in the cavity such as RWS, HC, RABT etc.), non-linear temperature distributions within the cross-section can be obtained. Since it is not yet standard to use these non-linear temperature distributions in commercial design programs, they are transformed into so-called equivalent temperature distributions (Wageneder, 2002; Kusterle et al., 2004). This widely-used simplification of the temperature load together with a linear-elastic analysis considerably deviates from reality since essential physical phenomena such as plasticity, stress redistribution, spalling etc. are not taken into account. In this work, different approaches to simulate the structural behavior of concrete tunnel linings under fire loading are presented. In Section 2, the investigated model parameters and assumptions are described. The effect of the considered model parameters on the numerical results is investigated by simulation of a single finite element (Section 3) as well as a real circular tunnel structure (Section 4). 2 2.1

MODEL PARAMETERS Finite element

The finite-element analyses are performed using thick (layered) shell elements (see Fig. 1 and (Savov et al.,

Figure 1. Illustration of employed layer concept (Savov et al., 2005; Zeiml et al., 2008; Ring, 2008): (a) real cross-section, (b) layered finite element.

2005; Zeiml et al., 2008; Ring, 2008)). The layer concept enables for (i) assignment of different temperatures and, hence, of temperature-dependent material parameters to the respective layers and (ii) consideration of spalling by de-activation of the respective near-surface layers. Concrete and steel are considered by separate layers, the reinforcement bars are transformed into a homogeneous steel layer of equivalent thickness. In case of the non-linear analysis, the steel reinforcement is simulated by a 1D plasticity model formulated in the direction of the reinforcement bars, whereas a plane-stress plasticity model is used for concrete (see (Savov et al., 2005) for details). 2.2

Consideration of temperature loading

In engineering practice, determination of the structural safety of tunnels subjected to fire loading is based on the so-called equivalent-temperature concept, assuming linear-elastic material behavior. The equivalent temperature load is calculated from the stress resultants Nequ and Mequ within a clamped beam loaded by the real (nonlinear) temperature distribution

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Figure 2. Model for determination of equivalent temperature loading (Tm and T ) (Kusterle et al., 2004), giving the same stress resultants Nequ and Mequ as the corresponding nonlinear temperature distribution.

(see Fig. 2 and (Kusterle et al., 2004)): Tm =

Nequ αEequ A

and T =

Mequ , αequ I

(1)

• Linear-elastic analysis Linear-elastic material behavior together with the equivalent temperature (Tequ ) is considered. This combination is commonly employed in engineering practice, using standard beam-spring models. • Stress-strain curves given in standards (ÖNORM EN1992-1-2, 2007) The stress-strain curves suggested in EC2-1-2 (see Fig. 4) are average values obtained from numerous tests, including various non-linear material effects (such as plasticity and load-induced thermal strains, LITS) in a combined manner. • Separate consideration of physical phenomena In contrast to the curves given in EC2-1-2, the mentioned nonlinearities in the material behavior are considered separately. Hereby, stress-strain curves (see Fig. 5(b)) are used which are obtained from temperature-dependent relations for strength and

In Eq. (1), A [m2 ] and I [m4 ] are the cross-sectional area and the moment of inertia, respectively, whereas α [K−1 ] is the thermal expansion coefficient of concrete. Eequ [MPa] is the equivalent Young’s modulus, given by Eequ =

N  Ec,i (Ti )Ai i=1

A

,

(2)

where Ec,i (Ti ) [MPa] and Ai [m2 ] are Young’s modulus and cross-sectional area of the i-th layer (with N [–] as the number of layers). The parameters Tm [◦ C], T [◦ C/m], and Eequ serve as input for the linear-elastic structural analysis using standard beam elements. On the other hand, employing layered finite elements allows direct consideration of nonlinear temperature distributions, which are generally obtained from either uncoupled thermal or coupled analyses (solving energy and mass transport in a coupled manner). Fig. 3 shows the temperature distributions considered within this work.

Figure 3. Nonlinear temperature distributions over the cross-section for selected time instants.

2.3 Material model for concrete Within the simulations, the following parameters are used: • Compressive strength: fc,0 = 30 MPa • Tensile strength: ft = 0 MPa (according to (ÖNORM EN1992-1-2, 2007)) or ft = 3 MPa is considered in order to demonstrate the influence of variation of ft on the numerical results. • Thermal strains: εth (T ) for concrete with siliceous aggregates according to (ÖNORM EN1992-1-2, 2007) is used. In this work, material models of different level of sophistication are investigated:

ε

Figure 4. Stress-strain curves for concrete under compressive loading according to EC2-1-2 (ÖNORM EN1992-1-2, 2007).

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c

c

c

c

c

Figure 6. Investigated spalling scenario (ds∞ : final spalling depth).

2.4

Consideration of spalling

Spalling has a considerable effect on the structural safety of tunnels since it reduces the cross-sectional area and, hence, the load-carrying capacity of the structure. In order to demonstrate the effect of spalling on the structural behavior of a real tunnel, a simple spalling scenario is considered (see Fig. 6). Figure 5. (a) Temperature-dependent reduction of compressive strength fc (T ) and Young’s modulus Ec (T ) of concrete according to (CEB, 1991); (b) resulting stress-strain curves obtained with the above temperature-dependent material functions.

3

To show the influence of the different material models on the numerical results, the simple case of a single layered element is analyzed (see Fig. 7). The boundary conditions are: (1) clamped in one direction and (2) allowing free deformation/rotation in the second direction.

stiffness given in (CEB,1991)(see Fig. 5(a)). Since these stress-strain curves only account for plasticity, the effect of combined thermal and mechanical loading on the strain behavior of concrete (also referred to as LITS) needs to be considered separately, leading to strain decomposition of the form ε = εel (T , σ ) + εpl (T , σ ) + ε th (T ) + ε lits (T , σ ),

3.1

σ th ε , fc,0

(3)

(4)

(5)

where k [–] is an empirical factor with 1.7 ≤ k ≤ 2.35 (see(Thelandersson, 1987)), linking ε lits with the thermal strain ε th . The stress level s = σ/fc,0 [–] accounts for the stress dependence of εlits .

Linear-elastic analysis (Tequ )

In engineering practice, the equivalent-temperature concept (Tequ , containing Tm and T ) together with linear-elastic material behavior is used to determine the stress resultants required for the design of the reinforcement. Fig. 8 shows the obtained linear stressdistribution for different time steps. It can be observed that the stresses exceed the tensile strength of concrete and are therefore unrealistic. However, since the equivalent temperature is determined on the same clamped beam based on Tnonl , application of Tequ will produce the same stress resultants as if Tnonl would be applied. Nevertheless, this is only the case for this simple example, application of Tequ to more complex geometries leads to results deviating from a non-linear analysis, as will be shown in Section 4.

where εel [–] and εpl [–] are the elastic and plastic strain component, respectively. In Equation (4), the load-induced thermal strain ε lits [–] is implemented by using the empirical relationship proposed in (Thelandersson, 1987), reading

εlits = k

SINGLE ELEMENT

3.2

Nonlinear analysis (Tnonl )

As opposed to the equivalent-temperature concept, application of layered finite elements enables to introduce the real non-linear temperature distribution, leading to non-linear stress distributions (see Figs. 9 to 11). In this subsection, the influence of different material

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Figure 7. Single layered element with prescribed boundary conditions.

Figure 9. Nonlinear analysis with Tnonl : distribution of (a) σ1 (clamped direction) and (b) σ2 (free direction) at 30 and 180 min of fire loading. Figure 8. Linear-elastic analysis with Tequ : distribution of σ1 at 30 and 180 min of fire loading.

models as well as different boundary conditions is investigated: • Influence of Eigenstresses in consequence of thermal restraint Fig. 9 shows the stress distributions obtained using the stress-strain curves from national standards (ÖNORM EN 1992-1-2, 2007). Hereby, the conservative approach prescribed within (ÖNORM EN 1992-1-2, 2007) of setting the tensile strength ft to zero is compared with setting ft = 3 MPa. It is observed that restraint stresses in consequence of the nonlinear temperature distributions develop in the free direction (σ2 , see Fig. 9(b)) for the case of ft = 3 MPa. This stress development influences the stresses in the clamped direction (σ1 , see Fig. 9(a)) since a plane-stress element is employed (Savov et al., 2005). The case ft =0 on the other hand, can be interpreted as the 1D case since no stresses occur in the free direction and therefore no influence exists for the σ1 -direction. • Influence of boundary conditions

Figure 10. Nonlinear analysis with Tnonl : distribution of σ1 at 30 and 180 min of fire loading for different boundary conditions.

The influence of different boundary conditions in the σ2 -direction is illustrated in Fig. 10, comparing the distributions of σ1 under consideration of the boundary conditions shown in Fig. 7 with the

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these fixations are deactivated in order to investigate the role of longitudinal restraint. The described boundary conditions simulate the two limiting cases with the real situation (tunnel sections of 15–25 m) lying somewhere between these conditions, depending on the location and extent of fire loading. 4.2

Figure 11. Nonlinear analysis with Tnonl : distribution of σ1 at 30 and 180 min of fire loading considering different material models.

case of equal boundary conditions in σ1 - and σ2 direction. As already mentioned, the former case can be considered as 1D case (since σ2 = 0, see Fig. 9(b)), whereas the latter case is characterized by a equal stress state (σ1 = σ2 ). This influence in stresses (and, hence, stress resultants) is lateron also investigated for a circular tunnel cross-section. • Influence of considered material model Fig. 11 shows stress distributions (σ1 ) under consideration of different material models. In case the stress-strain curves shown in Fig. 5(b) are applied, large stresses due to thermal restraint are observed, almost meeting the uniaxial compressive strength. Using this curves together with the empirical LITSrelationship presented in Subsection 2.3 leads to considerable reduction of the thermal restraint and, hence, smaller compressive stresses. When the stress-strain curves given in EC2-1-2 are applied (implicitly taking LITS into account), the results lie in between the two former cases. This difference in numerical results illustrates the necessity to develop a suitable model to consider the path dependence in the strain behavior of concrete, requiring further experimental and modeling effort.

4

Linear-elastic analysis (Tequ )

Following the state-of-the-art in engineering practice, the circular cross-section is first analyzed considering linear-elastic material behavior together with the equivalent temperature. Fig. 13 shows stress distributions in circumferential direction in the tunnel shoulder for different time instants. The peaks in the stress distribution correspond to the stress in the reinforcement (considered in circumferential as well as longitudinal direction). The stresses after 30 min are higher than after 180 min of fire loading since the temperature gradient is larger at the beginning of fire loading and is reducing as temperatures penetrate the crosssection. As observed for the single finite element,

Figure 12. Circular tunnel cross-section: (a) situation at the considered cross-section and (b) numerical model.

CIRCULAR TUNNEL CROSS-SECTION

4.1 Basic information Fig. 12 shows the situation at the considered crosssection of the tunnel as well as the numerical model. The thickness of the tunnel lining is 40 cm. The temperature loading is only applied to the top and the bench since the rail bedding protects the tunnel bottom from fire loading (see Fig. 12(b)). Fig. 12(b) shows the boundary conditions, additionally, deformations (in z-direction) and rotations (in x- and y-direction) are hindered in longitudinal direction. For comparison,

Figure 13. Linear-elastic analysis with Tequ : distribution of circumferential stress in the tunnel shoulder (element 23, see Fig. 12(b)) at 0, 30, and 180 min of fire loading.

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tensile stresses exceeding the tensile strength of concrete (with values up to 17 MPa) are observed which are highly unrealistic. Fig. 14 shows the distribution of the bending moment over the whole circular cross-section for selected time instants. The maximum is observed at the tunnel shoulder (element 23). As for the stress distribution, the bending moments are larger after 30 min than after 180 min of fire loading. The vertical convergence of the tunnel (i.e., the relative displacement between the top and the bottom of the tunnel) is shown in Fig. 15 for the linear-elastic analysis together with Tequ . Small deformations are observed, the thermal restraint leads to an expansion of the whole structure.

Figure 16. Linear-elastic vs. nonlinear analysis: distribution of circumferential stress in the tunnel shoulder (element 23, see Fig. 12(b)) at 0, 30, and 180 min of fire loading.

4.3 Nonlinear analysis (Tnonl ) • Influence of material behavior Comparing the results obtained from the state-ofthe-art (linear-elastic) design with results obtained from the nonlinear analysis applying the stressstrain curves given in EC2-1-2 (see Figs. 16 to 18) shows the large discrepancy between these two approaches. As a result of the significantly different stress distributions (see Fig. 16), the bending

Figure 17. Linear-elastic vs. nonlinear analysis: distribution of bending moment at 0, 30, and 180 min of fire loading.

Figure 14. Linear-elastic analysis with Tequ : distribution of bending moment at 0, 30, and 180 min of fire loading.

Figure 18. Linear-elastic vs. nonlinear analysis: vertical convergence.

Figure 15. vergence.

Linear-elastic analysis with Tequ : vertical con-

moment is largely reduced applying the nonlinear instead of the linear-elastic approach (see Fig. 17, the maximum bending moment at tf = 30 min decreases from 505 kNm to 260 kNm). This illustrates the limitations of the equivalent-temperature

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concept which gave the same stress resultants for the simple case of a single finite element (see Section 3) but leads to considerable deviations for more complex geometries. The deformations of the tunnel show a completely different behavior for the two cases (see Fig. 18). Already before fire loading the deformations are different since plasticity occurs also for the thermally not loaded case. When fire loading is applied, the linear-elastic analysis predicts expansion of the tunnel whereas an increase in vertical convergence is observed for the (more realistic) nonlinear analysis. Only towards the end of fire, thermal restraint leads to expansion of the tunnel. • Influence of boundary conditions The influence of the boundary conditions in longitudinal direction is illustrated in Figs. 19 and 20 where longitudinal displacements and rotations of the nodes are either fixed or free. Similar conclusions can be drawn as for the single element, i.e., fixing longitudinal displacement results in an increase in stresses and, hence, stress resultants (see Fig. 19). At the tunnel shoulder (element 23) the bending moment is hardly affected by the different boundary conditions (BC). This is explained by the fact that in this part of the tunnel a plastic hinge occurs during fire loading, hence, no increase in bending moment is possible. Therefore, moment re-distributions occur, increasing the bending moment in other regions of the tunnel. Allowing longitudinal displacements leads to larger deformations due to a reduction in the effect of thermal restraint (see Fig. 20). Compared to the differences between linear-elastic and nonlinear analysis, the influence of boundary conditions can be considered as small, reality can be found in between the two displayed BC-scenarios.

• Influence of spalling The effect of a reduction in cross-section due to spalling is illustrated in Figs. 21 to 23. Besides reduction of the cross-section, spalling also leads to loss of the inner reinforcement layer. When spalling is considered, the stresses are higher in the remaining cross-section (see Fig. 21). The distribution of bending moment presented in Fig. 22 shows a significant reduction when spalling is taken into account (due to the reduced cross-section and the missing inner reinforcement layer). The sudden change in the distribution of bending moment is a result of the jump in thickness from 40 to 30 cm (from the lower not fire-loaded part, to the upper fire-loaded part of the tunnel, see Fig. 12(b)). Concerning the displacements of the tunnel, spalling leads to a significant increase in the compliance of the tunnel. As shown in Fig. 23, reduction of the cross-section by 25% leads to doubling of the deformations. This large difference develops during the period when

Figure 20. Nonlinear analysis with Tnonl : vertical convergence for different BC-scenarios.

Figure 19. Nonlinear analysis with Tnonl : distribution of bending moment at 0, 30, and 180 min of fire loading for different BC-scenarios.

Figure 21. Nonlinear analysis with Tnonl : distribution of circumferential stress in the tunnel shoulder (element 23, see Fig. 12(b)) at 0, 30, and 180 min of fire loading.

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Figure 22. Nonlinear analysis with Tnonl : distribution of bending moment at 0, 30, and 180 min of fire loading.

especially in terms of a realistic prediction of the deformations of the structure. Additionally, the great influence of a reduction of the cross-section due to spalling shows the need to further investigate this field (e.g., by development of a suitable criterion to predict spalling). The need of considering non-linear temperature distributions and nonlinear material effects (e.g., plasticity, LITS or spalling) was illustrated. Moreover, the path dependence of combined thermo-mechanical loading needs to be investigated both from the experimentally as well as from a modeling point of view. Therefore, future work focuses on the cooperation with software developers to implement more sophisticated models into the state-of-the-art in engineering practice.

ACKNOWLEDGMENTS The authors thank Artur Galek from Vienna University of Technology for the help during the simulations.

REFERENCES

Figure 23. Nonlinear analysis with Tnonl : comparison of vertical convergence with and without spalling.

spalling takes place. Subsequently, the deformations follow a similar trend.

5

CONCLUSIONS AND ONGOING WORK

Realistic analysis of the structural behavior of tunnels under fire loading requires numerical models which realistically reflect reality. In this work, the stateof-the-art in tunnel-fire design (linear-elastic material behavior together with equivalent temperature distributions) is compared to more complex numerical models considering nonlinear material behavior, nonlinear temperature distributions and spalling. In a first step, the simple case of a single finite element is analyzed, showing the influence of boundary conditions and different material models. Stress distributions are presented for different cases showing a reduction of the thermal restraint when load-induced thermal strains are considered. In a second step, a real tunnel (with circular crosssection) is investigated. The results show a great influence of the method to consider temperature loading for as well as the description of material behavior,

CEB (1991). Fire Design of Concrete Structures, Bulletin d’Information 208. CEB, Lausanne, Switzerland. Kusterle, W., Lindlbauer, W., Hampejs, G., Heel, A., Donauer, P.-F., Zeiml, M., Brunnsteiner, W., Dietze, R., Hermann, W., Viechtbauer, H., Schreiner, M., Vierthaler, R., Stadlober, H., Winter, H., Lemmerer, J., and Kammeringer, E. (2004). Brandbeständigkeit von Faser-, Stahl- und Spannbeton [Fire resistance of fiberreinforced, reinforced, and prestressed concrete]. Technical Report 544, Bundesministerium für Verkehr, Innovation und Technologie, Vienna. In German. ÖNORM EN1992-1-2 (2007). Eurocode 2 –Bemessung und Konstruktion von Stahlbeton- und Spannbetontragwerken—Teil 1–2: Allgemeine Regeln—Tragwerksbemessung für den Brandfall [Eurocode 2—Design of concrete structures—Part 1–2: General rules-Structural fire design]. European Committee for Standardization (CEN). In German. Ring, T. (2008). Finite element analysis of concrete structures subjected to fire load considering different element types and material models. Master’s thesis, Vienna University of Technology, Vienna, Austria. Savov, K., Lackner, R. and Mang, H.A. (2005). Stability assessment of shallow tunnels subjected to fire load. Fire Safety Journal, 40:745–763. Thelandersson, S. (1987). Modeling of combined thermal and mechanical action in concrete. Journal of Engineering Mechanics (ASCE), 113(6):893–906. Wageneder, J. (2002). Traglastuntersuchungen unter Brandeinwirkungen [Ultimate load investigations considering fire load]. Bauingenieur, 77:184–192. In German. Zeiml, M., Lackner, R., Pesavento, F. and Schrefler, B.A. (2008). Thermo-hydro-chemical couplings considered in safety assessment of shallow tunnels subjected to fire load. Fire Safety Journal, 43(2):83–95.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Numerical assessment of the failure mode of RC columns subjected to fire S. Sere˛ga Cracow University of Technology, Cracow, Poland

ABSTRACT: The paper presents the experimental and numerical study of the influence of a tie spacing on the fire resistance of axially loaded, high strength concrete columns. It was indicated in earlier research, that tie spacings according to recommendations in the building codes could be insufficient and may lead to a premature failure. However, this statement was not always confirmed in experiments. Due to this fact the research into the fire resistance of columns with different spacings of ties is carried out. The experiments revealed the positive influence of the reduced spacing of lateral reinforcement on the fire resistance of axially loaded columns. The numerical calculations confirmed the failure mechanism of columns observed in the tests. Moreover, it is shown, that due to a thermal gradient, the confinement effect in the columns with the reduced tie spacing does not appear and the decrease of the fire resistance of the columns with the basic spacing of transverse reinforcement can be attributed only to the effect of inelastic buckling of main reinforcing bars between adjacent ties. 1

INTRODUCTION

In recent years a great interest has been shown in the fire safety of structures. This results in development of many methods for prevention of fire and in case of its occurrence ensuring the required fire resistance. There are no doubts that high temperature is one of the most severe condition that may act on a structure in the whole period of the structure live. This is the main reason why the fire resistance is important in analysing of structural safety issues. From among of all materials used to built engineering structures concrete has the best reaction on high temperature. Concrete is a non-flammable material with relatively low thermal conductivity of 1.4 to 3.6 W/(m·K) at the room temperature which usually decreases when temperature increases. Due to this fact outer layers of concrete element have an insulation character which prevents from fast heating of both reinforcement and deeper located parts of the element cross-section. Moreover, up to 400◦ C mechanical properties of concrete are not essentially different from these at the room temperature. Usually concrete structures do not need an additional protection (e.g. insulation) to satisfy the required fire resistance criteria. From the fire resistance point of view a concrete structure is the best solution. Columns are the most important part of reinforced concrete frames. When a failure of a beam or a plate occurs destruction of a structure is usually limited to one or two storeys. The failure of a column (for example as a result of fire), which transfers loads from upper floors, leads to more serious destruction. Therefore, recognition of behaviour of reinforced concrete columns, improving their design procedures is still

an important subject of theoretical and experimental research. One of many aspects of research is the influence of transverse reinforcement (made of ties or spiral) on the load bearing capacity and ductility of reinforced concrete columns. The most important parameters considered in the experimental research are: spacing and detailing of transverse reinforcement, column slenderness ratio, load eccentricity, longitudinal reinforcement ratio, cross-section shape, compressive strength of concrete. More details can be found for example in the papers by Assa et al. (2001), Cusson & Paultre (1994), Korzeniowski (2000), Mander et al. (1988), Moehle & Cavanagh (1985), Nishiyama et al. (1993), Scott et al. (1982), Sheikh & Uzumeri (1980). Tests confirmed that due to the confinement effect the reduced tie spacing has a positive influence on the load bearing capacity of axially loaded columns. Moreover, the higher transverse reinforcement ratio is applied the more ductile behaviour of a column is observed, which is especially important for high strength concrete (HSC) structures and for structures located in seismic active regions. Experiments carried out on axially and eccentrically loaded columns reveal that during the failure inelastic buckling of reinforcing bars is likely to occur. This phenomenon is not taken into account in a typical design procedure of columns. Except for seismic regions, design codes usually assume that inelastic buckling of reinforcement does not take place when the transverse reinforcement spacing is limited to the following values: 20ø (EN 1992-1-1), 16ø (ACI 318-08), 10ø or 15ø (PN-B-03264:2002), where ø is a diameter of main reinforcing bars. These recommendations are a sufficient protection against buckling of

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reinforcing bars to fire. In this case inelastic buckling of reinforcing bars is very likely to occur. The reduced slenderness ratio sw /ø could limit this phenomenon and increase the fire resistance of compressed concrete elements. Up to now little experimental research into the influence of a tie spacing on the fire resistance of concrete columns has been performed. This existing, reported by Kodur et al. (2000) & Kodur et al. (2005), was carried out on the square cross-section columns made of HSC. On the other hand, in the majority of tests performed by Dotreppe et al. (1996), Franssen & Dotreppe (2003), the tie spacing was constant but different diameters of the main reinforcing bars were used. In such type of tests only the effect of a bar slenderness ratio can be recognized. In Figures 1 and 2 the experimental values of the fire resistance tfi against the slenderness ratio sw /ø of longitudinal reinforcing bars according to Kodur et al. (2000) & Kodur et al. (2005)—Figure 1 400

HS2-3 n=0.28

HS2-2 n=0.28

Fire resistance tfi [min]

350

300

HS8 n=0.20

HS2-1 n=0.25 HS7 n=0.17

250

HS9 n=0.25

n=Nc/NRc Nc - test load NRc = fcAc+fyAs

HS2-7 n=0.17

200 HS2-10 n=0.20

150 HS2-9 n=0.20

HS2-8 n=0.20

100 4

8

12

16

20

sw/ø [-] bxh = 30.5cm x 30.5cm, carbonate aggr. bxh = 40.6cm x 40.6cm, carb./silic. aggr. bxh = 40.6cm x 40.6cm, carbonate aggr.

Figure 1. Effect of reinforcement slenderness ratio sw /ø on fire resistance of square cross-section columns—test results.

200

n'=Nc/NRcd

C3 n'=0.57

Fire resistance tfi [min]

bars for service loads but may be insufficient if the load reaches the ultimate level. The experiments presented by Korzeniowski (2000) show that even for the transverse reinforcement spacing of 7.5ø the inelastic buckling of main reinforcing bars occurs and stresses reached in steel at the moment of the failure are less then the yield strength. The effect of inelastic buckling of reinforcing bars was studied for example by Bae et al. (2005), Bayrak & Sheikh (2001) in the context of behaviour of column sections in plastic hinge regions. The experimental research, carried out on the isolated bars, shows that there are noticeable differences between the stressstrain relationship for steel in tension and the average stress-strain curves for steel in compression. The average stress in compression is defined as the axial load divided by the cross-sectional area of the bar and the average strain is equal to the total axial shortening of the bar divided by its initial length. It can be concluded that the influence of the second order effects should be taken into account for the bar slenderness ratio sw /ø higher than 7, where sw is a distance between adjacent ties. It is worth noting that the experiments described by Bae et al. (2005), Bayrak & Sheikh (2001) do not consider the positive influence of concrete cover and negative effect of lateral volumetric increase of the concrete core on the second order effects of reinforcing bars. The positive influence of transverse reinforcement on the load capacity and ductility of reinforced concrete columns at the room temperature indicates that the gain in the fire load capacity NRfi , and indirectly, in the fire resistance tfi may also be observed. However, in the case of fire this presumption is not obvious. As it was mentioned at the beginning of the paper concrete has high thermal inertia. During fire high temperature on a surface of a concrete element generates heterogeneous temperature fields with high thermal gradients inside the element. This produces tensile stresses perpendicular to the axis of the element. The experiments carried out by Kristensen & Hansen (1994) on hardened cement paste cylinders show that during rapid heating tensile stresses generated by thermal gradients are high enough for the possibility of an appearance of the radial and tangential cracks. It means that in the case of axially loaded columns with tightly arranged ties rapid heating may eliminate the positive effect of confinement in the core of the concrete column. Hence, increasing of the transverse reinforcement ratio which increases the load capacity of axially loaded columns at the room temperature may not lead to the increase of the fire load capacity and fire resistance. In fire, the reduced tie spacing may be also important with reference to inelastic buckling of main reinforcing bars. Structures made of high strength concrete usually suffer from explosive thermal spalling. Besides weakening of concrete, thermal spalling exposes

Nc - test load NRcd - design strength

180

160

C4 n'=0.75

C1 n'=0.59

140

Circular column diameter 30cm , siliceous aggregate

C2 n'=0.83

120 8

10

12

14

sw/ł

16

18

[-]

Figure 2. Effect of reinforcement slenderness ratio sw /ø on fire resistance of circular cross-section columns—test results.

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and Franssen & Dotreppe (2003)—Figure 2 are plotted. The limited data from various studies presented in these figures do not explain if the positive effect of the reduced tie spacing and/or reinforcing bar slenderness ratio sw /ø on the fire resistance of concrete columns exists. Additionally, the experimental results strongly vary between the series of the comparable columns. The tests on columns HS7 to HS9, HS2-7 to HS2-10 and C1 to C4 indicate the positive influence of the reduced tie spacing or bar slenderness ratio on tfi values. However, if the specimens HS2-1 to HS2-3 or these presented in Dotreppe et al. (1996) are considered the positive effect of reduced value sw /ø disappears or the fire resistance decreases with simultaneous decrease of the tie spacing. The fact of variations in the test results and sometimes contradictory observations can be partly explained by different extent of concrete thermal spalling noticed during the tests as well as different load levels applied in the experiments on the individual elements in the comparable groups of columns. Due to the different load levels and different geometry (concrete cover, presence of cross ties, detailing of ties) as well as various extend of concrete thermal spalling it is very difficult or even impossible to draw the final conclusions. Moreover, on the basis of theoretical calculations Dotreppe et al. (2005) indicated that inappropriate detailing or insufficient spacing of ties can lead to a premature failure and pointed out that additional research should be done in order to create design recommendations on this subject. This is the main motivation for undertaking research into the influence of confining reinforcement on the fire resistance of axially loaded concrete columns.

2 2.1

EXPERIMENTAL STUDIES Description of column specimens

The experiments were carried out on five circular cross-section columns S-0 to S-4. The elements S-1 to S-4 were tested in a furnace and S-0 specimen was used as a witness column to determinate the load capacity at the room temperature. The specimens S-0 to S-4 had the same geometrical dimensions shown in Figure 3. Four different spacings of the ties were used: basic 15ø (S-0 and S-1), 10ø (S-2) and reduced 5ø (S-3), 2.5ø (S-4), where ø is a diameter of longitudinal reinforcing bars. The same basaltic aggregate concrete was used for fabrication of the columns. In order to prevent explosive thermal spalling 2.0 kg per cubic meter of polypropylene fibres was added to the concrete mix. The compressive strength of concrete after 28 days and 365 days (concrete age during tests) was equal to 65.4 MPa and 73.7 MPa, respectively.

Figure 3. View of column in furnace, details of reinforcement and thermocouples locations.

Reinforcement of the specimens S-0 to S-4 consisted of eight ø16 mm deformed bars and deformed ø8 mm circular stirrups with 29 mm concrete cover. Ends of the stirrups were bent and anchored in the column core. The yield stress was equal to 446 MPa for ø16 mm bars and 750 MPa for the stirrups. The full description of fabrication of the specimens is given by Sere˛ga (2009). During the experiments temperature along the diameter of the specimens was measured using the sets of

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eight chromel-alumel 1.5 mm thick thermocouples TS1 to TS8. Locations of the thermocouples in the column cross-section are shown in Figure 3. 2.2

Table 1.

sw tfi

Test equipment

Fire resistance tfi of tested columns. S-1

S-2

S-3

S-4

15ø 167 min

10ø 179 min

5ø 196 min

2.5ø 194 min

The elements S-1 to S-4 were tested in the Fire Testing Laboratory of the Building Research Institute in Warsaw, Poland. The test equipment consisted of the furnace chamber with dimensions of 5.00 m × 2.85 m in plane and 3.00 m in height, the steel frame and the loading system. Temperature in the furnace chamber was measured using six plate thermometers (TP1 to TP6) with shielded K type thermocouple located approximately 150 mm from the element surface—see Figure 3. During each test displacements of the steel beam located on the column top plate were measured using two LVDT transducers. 2.3 Test procedure S-1 to S-4 specimens were tested after one year since their prefabrication, according to the following procedure. The column was installed in the furnace and aligned on circular plates in order to minimize a load eccentricity. Next the load of 1530 kN was applied. This load was equal to 33% of the witness column (S-0) load capacity tested at the room temperature. Heating, according to ISO834 time-temperature curve, started approximately 15 minutes after stabilization of axial displacements of the top of the column. This was assumed as the initial condition of the axial displacements. The applied force of 1530 kN was maintained constant up to the failure of element. The measured loading variations during each test were negligible. The experiment was terminated if the set of hydraulic jacks could not maintain the load. 2.4 Test Results The experimental values of the fire resistance tfi for the tested columns are presented in Table 1. It is clearly seen that for the analyzed concrete columns the reduced tie spacing has the positive influence on tfi . However, this effect is relatively limited, the maximum gain in the fire resistance between S-1 and S-3 elements equals 29 minutes which is 17% of the experimental value tfi for column S-1. The compression failure in the middle of the heating zone of the tested elements was observed—see the photographs in Figure 4a,b for column S-1 and S-4, respectively. For the columns with the basic tie spacing (S-1 and S-2) in the damaged zone crushing of concrete and buckling of the main reinforcing bars occurred. For the elements S-3 and S-4 the reduced distance between ties to 5ø and 2.5ø prevented buckling of the bars and only concrete in the core was crushed.

(a)

Figure 4. test.

(b)

Photographs of S-1 (a) and S-4 (b) columns after

The vertical displacements of the top of the columns were very similar in all four cases. In the first stage of heating the columns expanded up to approximately 1.0 mm. After reaching the peak, deterioration of mechanical parameters occurred as well as creep effects became more pronounced and the continuous contraction of the columns was noticed, what is illustrated in Figure 9 (continuous lines) for elements S-1 and S-3. The addition of polypropylene fibers turned out to be a sufficient protection against spalling. Eliminating this phenomenon ensured axially symmetrical temperature fields in the columns cross-section and similar thermal conditions for the tested elements. The example of temperature-time curves measured during the test in the various points of cross-section is given in Figure 6. Generally, for the all specimens temperature measured for the symmetrically located pairs of thermocouples was almost identical. Only for TS3 and TS4 thermocouples temperature values differed which was probably caused by imperfectly stabilized ends of these thermocouples during concreting of the columns. The experimental values of tfi for S-1 to S-4 columns are plotted in Figure 5, where the theoretical fire load capacity against time of heating is given. The calculations were performed for the load eccentricities ea equal 5 mm and 10 mm on the basis of modified moving pivot method developed by Sere˛ga (2008).

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in perfect accordance with the theoretical predictions. This suggests that the effect of confinement, which enhances the load capacity of an axially loaded column at the room temperature, did not appear in the experiments. In the next section of the paper this issue will also be analyzed.

1900

Simplified theoretical calculations

Fire load capacity NRfi [kN]

1800

ea=5mm 1700

ea=10mm 1600

S-3 1530 1500

S-1

S-4

S-2

3

1400

1300 160

170

180

193 190

201 200

210

Time [min]

Figure 5. Comparison of test results and simplified theoretical calculations.

For the constant compressive force of 1530 kN the theoretical fire resistance is equal to 193 and 201 minutes for the load eccentricities ea = 10 mm and ea = 5 mm, respectively—see Figure 5. This indicates that for the elements with the basic tie spacing of 15ø and 10ø the premature failure occurs, what is probably caused by inelastic buckling of reinforcing bars. The following failure mechanism can be stipulated for the columns with the basic tie spacing (S-1 and S-2). An imperfect main reinforcing bar is protected against buckling by ties and concrete cover. As a result of temperature rise, the mechanical parameters of concrete in cover decrease significantly. For example, temperature in the concrete cover after 160 minutes of heating was between 650◦ C and 1060◦ C for the column S-1. At such temperature the maximum tensile strength of concrete is less than 10% of its original value as well as the elastic modulus of concrete is significantly reduced. In these conditions the concrete cover was pushed out by the slender reinforcing bar for S-1 and S-2 specimens. Lost of the additional horizontal support caused inelastic buckling of the bars between the adjacent ties, the increase of compressive force in the column core and, in consequence, the failure of the column. However, it is not obvious if the increase of stresses in the S-1 and S-2 column core due to buckling of the reinforcement is significant enough for causing crushing of concrete. This will be discussed in the next section of the paper. In the case of columns with the reduced lateral reinforcement spacing such a failure mechanism did not occur. Hence, almost identical experimental tfi values were obtained which are in good accordance with the theoretical calculations. The additional conclusion can be drawn from Figure 5. The theoretical dependence of the fire load capacity—heating time for the columns in question was calculated on the basis of unconfined concrete stress-strain relationships. The experimental values of the fire load capacities for element S-3 and S-4 are

NUMERICAL STUDY

In the previous section, on the basis of the experimental observations and simple theoretical calculations, the failure mechanism of elements with the basic tie spacing (15ø, 10ø) and these with ties tightly arranged (5ø, 2.5ø) was presumed. The premature failure of columns with ties located at the distances of 15ø and 10ø is attributed to inelastic buckling of main reinforcing bars. The main doubt, which needs to be explained, is if the part of compressive force carried out by the reinforcing steel transferred into the column core at the moment of buckling is high enough for crushing of concrete and a failure of element. Moreover, it seems to be also essential whether it is possible that the three dimensional compressive stress state in the column S-3 and S-4 core appears during any stage of heating which could increase the fire load capacity and the fire resistance. The main goal of the numerical analysis is to answer these issues. The numerical computations were performed by using concrete and steel models that are available in the finite elements system DIANA. The calculations consisted of following steps: – solution of the transient thermal problem in order to obtain time dependent temperature fields, – the temperature fields from the thermal analysis were used as a state variable in mechanical calculations to find the response of the S-1 to S-4 columns on both mechanical and thermal loadings. In the experiments the compressive failure mode without any visible influence of the global second order effects for the tested elements was observed. Therefore, these effects were neglected in the numerical calculations. Since the columns geometry, reinforcement and temperature fields were axisymmetrical the elements S-1 to S-4 were modelled in the axisymmetry. It should be pointed out that an imperfection eccentricity introduce small deviation of stresses and strains from the axisymmetry. This fact was consciously omitted in the analysis for the sake of simplicity. In the numerical calculations the same finite element mesh topology was used for both thermal and mechanical steps—see Figure 7a. Two different types of finite elements were used: the four-node isoparametric, axisymmetric solid ring element with linear interpolation of temperature and the eight-node isoparametric, axisymmetric solid ring element with

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quadratic interpolation of displacement fields for thermal and mechanical analyses respectively. For the mechanical calculations, temperature at mid-nodes was interpolated from the adjacent nodes. Because of the horizontal symmetry only the half column was modelled. The reinforcement was modelled by using embedded reinforcement elements available in DIANA (the so-called GRID elements—see Manie & Kikstra (2008)). Two independent grids were used with the equivalent thickness of 2.44 mm for the main bars and 0.21 mm, 0.31 mm, 0.63 mm, 1.26 mm for the transverse reinforcement of S-1, S-2, S-3 and S-4 column, respectively. 3.1

800

TS1, TS2 - calculations TS1,TS2 - experiment TS3, TS4 - calculations TS3, TS4 - experiment TS5, TS6 - calculations TS5, TS6 - experiment

o

Temperature [ C]

600

400

200

0 0

40

80

120

160

200

Time [min]

Figure 6. Comparison of the experimental and calculated temperature for S-3 specimen.

Thermal analysis

In order to obtain the thermal response of the analysed columns on the fire loading the Fourier-Kirchhoff equation was solved: ∂θ ρc cp = div (λc · grad(θ )) ∂t

3.2

It is the well-known fact that high temperature has usually the negative influence on mechanical properties of concrete and steel. When temperature increases the reduction of the elastic modulus, compressive and tensile strength are observed. Besides the changes in mechanical properties rheological strains are intensified. In the present study the thermo-elastic-plastic material models for concrete and steel which are available in DIANA are used.

(1)

where ρc , cp , λc are density, specific heat and thermal conductivity of concrete, respectively, θ is temperature and t is time. The mean temperature from the thermocouples TS7 and TS8 was taken as the boundary condition. Temperature of 15◦ C was assumed as the initial condition. The relative thermal conductivity λc (θ)/λc (20◦ C) as well as the specific heat of concrete were assumed according to EN 1992-1-2. The temperature dependent cp values were taken as for 3% moisture of concrete. The initial value of thermal conductivity coefficient, i.e. thermal conductivity at 20◦ C, was equal to 1.47 W/(m·K). The reduction of concrete density was modelled as for basaltic aggregate concrete according to the experimental results reported by Schneider (1982). The measured density of concrete at 20◦ C was equal to 2477 kg/m3 . The calculated temperature histories for S-3 column are compared with the experimental measurements in Figure 6. In the simplified approach described by equation (1) only the heat transfer by conduction is considered and simultaneous moisture transfer into the deeper layers of the column cross-section which also influences temperature fields is neglected. This approach generates the differences between calculated and measured temperature which are mostly pronounced at the first stage of heating for the thermocouples located near the centre of the section—see Figure 6. However, these differences are not significant at the subsequent stages of heating, which are more important from the point of view of carried out analysis, and where the satisfactory agreement between the experiment and theoretical prediction is obtained.

Mechanical analysis

3.2.1 Concrete model For concrete without the prior heating history the total strain vector ε can be expressed as a sum of the free thermal strain ε th , the instantaneous mechanical strain εmech , the transient creep strain εtr and the basic creep strain ε creep —see e.g. Andenberg & Thelandersson (1976), Borst & Peeters (1989), Khennane & Baker (1992): ε = εth + εmech + εtr + ε creep

(2)

It should be pointed out that the strain components εth , ε mech , εtr are apparent functions of time and only ε creep is directly time dependent. The test results reported by Andenberg & Thelandersson (1976) show that the basic creep component of the total strain is relatively insignificant comparing to the transient creep if high heating rates and short temperature duration of a few hours are considered, so ε creep was neglected in the analysis. The free thermal strain increment can be written in the following form: ε˙ th = α(θ) · θ˙ · π

(3)

where α(θ) is the temperature dependent coefficient of thermal expansion, and π = [1, 1, 1, 0, 0, 0]T . The values of α(θ) were calculated on the basis of the free thermal strain tests carried out on basaltic aggregate concrete and reported by Schneider (1982).

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The instantaneous mechanical strain is written as the sum of the elastic εel and the plastic εpl strain: εmech = ε el + εpl

(4)

The elastic strain ε el is given by the Hook law. The elastic modulus Ecm (θ) was calculated according to the following stress-strain relationships adopted to the analysis. The pre-peak part of σc – εc curve is described by Kodur et al. (2004):   H   σc (εc , θ) = fc (θ ) 1 − 1 − εc εc1 (θ ) (5) where H = 2.28 – 0.012 · fc (20◦ C)/MPa, and εc1 (θ ) is the strain at the peak stress fc (θ ). The elastic modulus Ecm (θ) is defined by the line between the origin of the coordinate system and the point on σc – εc curve for which the stress is equal 0.4fc (θ). The post-peak behaviour of σc – εc relationships is described by the softening branch: σc (εc , θ) = fc (θ ) · 10−2.5·(1−εc (θ )/εc )

2

(6)

The relative values of compressive strength were taken from Kodur et al. (2004) and εc1 (θ ) was assumed on the basis of experimental research reported by Cheng et al. (2004). In the literature a wide scatter of experimental results of Poisson’s ratio ν for concrete at elevated temperature is reported—see Bonnard & Gardell (1980), Marechal (1972). Due to this fact the constant, temperature independent ν value of 0.2 was adopted in the computations. The Drucker-Prager yield surface was assumed for concrete in compression. The plastic strain was given by the flow rule: ε˙ = λ˙ pl



∂g ∂σ

 (7)

where g is a plastic potential and λ is a plastic multiplayer. In the calculations the associated flow rule was assumed. The transient creep strain component from eq. (2) is expressed in the form (Borst & Peeters (1989)): ε˙ tr = Aσ θ˙

(8)

where: A= ⎡

αkc1 fc (20◦ C)

1 −νc ⎢−νc 1 ⎢ ⎢−ν −νc ×⎢ c 0 ⎢ 0 ⎣ 0 0 0 0

−νc −νc 1 0 0 0

⎤ 0 0 0 0 0 0 ⎥ ⎥ 0 0 0 ⎥ ⎥ 2(1 + νc ) 0 0 ⎥ ⎦ 0 2(1 + νc ) 0 0 0 2(1 + νc )

and kc1 , νc are material constants equal 2.33 and 0.285, respectively. Concrete in tension was described using the classical smeared crack model with a fixed crack direction. In the softening region the relation between crack stress and crack strain in the direction normal to the crack was modelled according to Hordijk (1991). The values of fracture energy Gf between 20◦ C and 450◦ C were taken from Zhang & Biˇcaniˇc (2006). Due to lack of data the constant value of fracture energy Gf (450◦ C) was assumed for temperatures higher than 450◦ C. The temperature dependent concrete tensile strength ft (θ) was adopted according to Model Code 90 (1991). The initial value ft (20◦ C) = 4.5 MPa was calculated on the basis of the mean compressive strength according to EN 1992-1-1.

3.2.2 Steel model For steel the total strain ε is expressed as the sum of the free thermal strain εth and the instantaneous mechanical strain ε mech . In the calculations the temperature dependent elastic properties of steel as well as the thermal expansion coefficient were taken from EN 1992-1-2. The associated von Mises plasticity model was assumed for steel. The effect of creep on the total strain was neglected in the computations. In order to study the influence of the slenderness ratio of main reinforcing bars on the fire load capacity and the fire resistance of the tested columns the average stress-strain relationships for steel in compression were numerically worked out using the method presented for example by Bae et al. (2005), Bayrak & Sheikh (2001). The calculations were performed for four applied in the experiments spacings of the ties sw : 15ø, 10ø, 5ø, 2.5ø and for the initial bar imperfection eimp = sw /300. The numerical model of the isolated bar with the initial imperfection eimp is presented in Figure 7b. Due to the horizontal symmetry only the half bar with proper boundary conditions was modelled. The classII beam element available in DIANA, numerically integrated over the cross-section was used. Loading was realized by the vertical translation ux of the upper bound. The calculations were performed for the stressstrain relationships of steel according to EN 1992-1-2. The Total Lagrange Approach was applied to model geometrically nonlinear behaviour of reinforcing bars. The average stress was equal to the compressive force divided by the area of the bar. The averaged strain was calculated by dividing the translation ux by the initial length of the bar. The example of numerical results are presented in Figure 8 where the average stress-strain curves for steel in compression at 500◦ C are plotted. It should be noted that such an approach for describing behaviour of reinforcing steel in compression does

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not take into account the positive influence of concrete cover and the negative effect of lateral volumetric increase of a concrete core on stability of reinforcing bars, what is the main drawback of the model. The following computational cases were considered: – Case 1: the models of columns S-1 to S-4 with ideal reinforcing bars where the stress-strain relationships

for steel according to EN 1992-1-2 were applied. These cases describe the columns where the main reinforcing bars are protected against buckling by ties and concrete cover during the whole period of heating, – Case 2: the models of columns S-1 to S-4 with imperfect main reinforcing bars. In this case the second order effects of the reinforcement was described by the average stress-strain relationships in compression σs = σs (εs ,θ , eimp , sw /ø). These models describe behaviour of columns where influence of concrete cover on the second order effects of reinforcing bars is neglected during the whole heating period. 3.3

(a)

(b)

Figure 7. (a) FEM model for S-1 to S-4 columns, (b) FEM model of isolated main reinforcing bar. 400

eimp = sw/300;

o

= 500 C

350

250 200

s

[MPa]

300

150

sw=15ø

100

sw=10ø sw=5ø

50

sw=2.5ø

0 0

0.01

0.02

0.03

0.04

[-]

Figure 8. Example of average stress—strain relationships for steel in compression.

Numerical results

The results of the numerical calculations are presented in Figures 9 to 11. In Figure 9 the measured and calculated axial deformations for S-1 (15ø) and S-3 (5ø) columns are compared. The calculations were made for Case 2 models. Generally, the numerical and experimental results fit well, though there are small differences. During the first stage of heating (i.e. up to the first 45 minutes) an immediate expanding of the elements was observed which is not properly modelled in the numerical simulation. This could be attributed to the fact that the influence of moisture transport on temperature fields was neglected in the thermal analysis which generates the differences between the measured and calculated temperatures at the initial stages of heating. As it was mentioned earlier and presented in Figure 6 for the initial heating stage higher temperature was obtained in the tests than in the simulations. Due to this fact the actual thermal expansion of the columns was greater than that obtained from the computations. In the subsequent stages of heating the differences between the tests and numerical predictions result from the simple transient creep model adopted to the calculations. In this model the proportionality between ε tr and ε th was assumed for the whole temperature range. The experiments carried out by Andenberg & Thelandersson (1976) indicate that this assumption is valid only for the temperature up to 550◦ C. At temperature above 550◦ C transient creep strains become more intensive than arise from equation (8). This explains the differences between the tests and predictions in the stage where the shortening of the column was observed. In Figure 10 the stresses in the main reinforcing bars against heating time are presented. In this figure the results of calculations for the ideal elastic-plastic material model of steel (Case 1) are also plotted—the line with crosses. In the first period of heating up to 30 minutes the stresses are almost the same for the all considered columns. After reaching the maximum σs becomes gradually smaller and the most significant

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2 0

Axial deformation [mm]

-2

0

30

60

90

120

150

180

210

Time [min]

-4 -6 -8

S-1- experiment S-1- calculations (Case 2) S-3- experiment S-3- calculations (Case 2)

-10 -12 -14 -16 -18

Figure 9. Axial deformations of S-1 and S-3 columns, calculations and experiment.

-400 -350

-250 -200

S-1; Case 2 S-2; Case 2 S-3; Case 2 S-4; Case 2 Case 1

s

[MPa]

-300

-150 -100 -50

S-1 and S-2 (Case 2) and the model with the ideal bars (Case 1) are equal to 113 MPa and 76 MPa, respectively. It can be estimated that at the moment of concrete cover falling-off the reductions of the fire load capacity are 182 kN for the element S-1 and 122 kN for the column S-2. Assuming the load eccentricity of the tested elements ea = 10 mm the fire load capacities NRfi at the failure moment of the elements S-1 and S-2 are equal to 1590 kN and 1540 kN, respectively. These values are very close to the applied load during the tests (1530 kN)—see Figure 5. Thus, the compressive force transferred into the concrete core during inelastic buckling of the reinforcing bars could lead to crushing of concrete in the column core and, in consequence, cause the failure of the column. In Figure 11 radial σrr and vertical σzz stresses in the centre of columns S-1 to S-4 are plotted. Because of high thermal gradients there are tensile stresses in radial direction σrr (and also in circumferential direction σθ θ ) in the core of the analysed columns during the whole period of heating. This fact explains why the expected confinement effect for the elements with the reduced tie spacing S-3 (5ø) and S-4 (2.5ø) did not appear in the experiments. 4

0 167 179 194 196 0

30

60

90

120

150

180

210

Time [min]

Figure 10. Predicted stress history in reinforcing bars for columns S-1 to S-4. -60 -50

S-1; Case 2 S-2; Case 2 S-3; Case 2 S-4; Case 2

c

[MPa]

-40 -30

zz

-20 -10

rr

167 179 194 196 0 10

Figure 11. to S-4.

0

30

60

90

120

150

180

210

Time [min]

Predicted stress history in centre of columns S-1

drop is observed for the element with the basic tie spacing of 15ø (model where the lateral influence of concrete cover on stability and the second order effects of main reinforcing bars are neglected). The differences in stresses between the models for elements

SUMMARY AND CONCLUSIONS

The experiments carried out on the columns with the circular cross-section prove that the positive effect of a tie spacing on the fire resistance exists. However, it should be noticed that the gain in the fire resistance tfi is relatively limited. The increase of the fire resistance was equal 7% to 17% of tfi obtained for S-1 (15ø) column. The observations made during the tests and the further numerical study indicate that for the elements with the basic tie spacing (15ø and 10ø) the premature failure occurs which is caused by inelastic buckling of main reinforcing bars. It also turned out that during the whole period of heating in the core of the tested elements tensile stresses in radial and circumferential direction appear and, in consequence, the expected effect of confinement in the core of the elements with the reduced tie spacing to 5ø and 2.5ø does not take place. Thus, the increase of the fire resistance of S-3 and S-4 elements can be attributed only to the fact that the tightly arranged transverse reinforcement prevents the longitudinal bars from inelastic buckling. Moreover, the simplified calculations presented in Figure 5 indicate that in the case of the analyzed columns requirements for the fire resistance class R180 are fulfilled. The experimentally obtained fire resistances for the elements with the basic tie spacing classify these columns to R120 class. This means that details like a tie arrangement could decrease the fire resistance class of a structure and this should be considerated during the design process.

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ACKNOWLEDGMENTS Author would like to thank prof. Zbigniew Janowski for his contribution to the research. This work has been done as a part of the research project N506 069 31/3256 financed by the Polish Ministry of Science and Higher Education in years 2006–2008.

REFERENCES Andenberg, Y. & Thelandersson, S. 1976. Stress and deformation characteristics of concrete at elevated temperatures. Experimental investigation and material behaviour model. Technical report, Lund Institute of Technology Lundt. Assa, B., Nishiyama, M. & Watanabe, F. 2001. New approach for modeling confined concrete: Circular columns. Journal of Structural Engineering, ASCE, 127(7):743–750. Bae, S., Mieses, A.M. & Bayrak, O. 2005. Inelastic buckling of reinforcing bars. Journal of Structural Engineering, ASCE, 131(2):314–321. Bayrak, O. & Sheikh, S. 2001. Plastic hinge analysis. Journal of Structural Engineering, ASCE, 127(9):1092–1100. Bonnard, M. & Gardel, M. 1980. Verhalten des warmen betons. Institut de Statique des Constructions, Final Report, Lausanne, (in German). Borst, R. & Peeters, P.J.M. 1989. Analysis of concrete structures under thermal loading. Computer Methods in Applied Mechanics and Engineering, 77(3):293–310. Building code requirements for structural concrete (ACI 31808) and commentary. American Concrete Institute, 2008. Cheng, F., Kodur, V. & Wang, T. 2004. Stress-strain curves for high strength concrete at elevated temperatures. Journal of Materials in Civil Engineering, ASCE, 16(1):84–90. Cusson, D. & Paultre, P. 1994. High-strength concrete columns confined by rectangular ties. Journal of Structural Engineering, ASCE, 120(3):783–804. Dotreppe, J.-C., Franssen, J., Braus, R., Vandevelde, P., Minne, R., Nieuwenburg, D.V. & Lambotte, H. 1996. Experimental research on the determination of the main parameters affecting the behaviour of reinforced concrete columns under fire conditions. Magazine of Concrete Research, 49(179):117–127. Dotreppe, J.-C., Ludwig, R., Kodur, V.K.R. & Fransen, J.-M. High strength concrete columns under fire conditions: experimental tests, numerical simulations and provisions of prEN1992-1-2. In FIB Symposium ‘‘Keep concrete Attractive’’, Budapest, 2005. EN 1992-1-1: Eurocode2: Design of concrete structures— Part 1: General rules and recommendations for buildings. EN 1992-1-2:2004. Eurocode 2: Design of concrete structures—Part 1.2: General rules—Structural fire design. Fire Design of Concrete Structures in Accordance with CEB/FIP Model Code 90. CEB Bulletin D’Information No. 208. Lausanne, Switzerland. Franssen, J.-M. & Dotreppe, J.-C. 2003. Fire tests and calculation methods for circular concrete columns. Fire Technology, 39(1):89–97. Hordijk, D.A. 1991. Local Approach to Fatigue of Concrete. PhD thesis, Delft University of Technology.

Khennane, A. & Baker, G. 1992. Thermoplasticity model for concrete under transient temperature and biaxial stress. Proceedings of The Royal Society A, 439:59–80. Kodur, V.K.R., McGrath, R., Latour, J. & MacLaurin, J. 2000. Experimental studies on the fire endurance of highstrength concrete columns. Technical Report No. 919, National Research Council, Institute for Research in Construction, Ottawa. Kodur, V.K.R., McGrath, R., Leroux, P. & Latour, J. 2005. Experimental studies for evaluating the fire endurance of high-strength concrete columns. Technical Report No. 197, National Research Council, Institute for Research in Construction, Ottawa. Kodur, V.K.R., Wang, T. & Cheng, F. 2004. Predicting the fire resistance behaviour of high strength concrete columns. Cement & Concrete Composites, 26(2):141–153. Korzeniowski, P. 2000. Spirally reinforced concrete columns. Tests and theory. Gdansk Univ. of Technology, (in Polish). Kristensen, L. & Hansen, T.C. 1994. Cracks in concrete core due to fire or thermal heating shock. ACI Materials Journal, 91(5):453–459. Mander, J., Priestley, M. & Park, R. 1988. Observed stressstrain behavior of confined concrete. Journal of Structural Engineering, ASCE, 114(8):1827–1849. Manie, J. & Kikstra, W.P. editors 2008. DIANA—Finite Element Analysis. User’s Manual Release 9.3. Marechal, J.C. 1972. Variations in the modulus of elasticity and Poisson’s ratio with temperature. ACI Special Publication 34, 1:495–503. Moehle, J. & Cavanagh, T. 1985. Confinement effectiveness of crossties in RC. Journal of Structural Engineering, ASCE, 111(10):2105–2120. Nishiyama, M., Fukushima, I., Watanabe, F. & Muguruma, H. 1993. Axial loading tests on high-strength concrete prisms confined by ordinary and high-strength steel. In Third International Symposium on the Utilization of High-Strength Concrete, volume V, pages 322–329, Lillehammer. PN-B-03264:2002. Concrete, reinforced concrete and prestressed concrete structures. Static calculations and design, (in Polish). Schneider, U. 1982. Behaviour of concrete at high temperatures. Deutscher Ausschuss fur Stahlbeton, Heft 337. Scott, B., Park, R. & Priestley, M. 1982. Stress-strain behavior of concrete confined by overlapping hoops at low and high strain rates. ACI Journal Proceedings, 79(1):13–27. Sere˛ga, S. 2008. A new simplified method for determining fire resistance of reinforced concrete sections. In Proceedings of 6th International Conference Analytical Models and New Concepts in Concrete and Masonry Structures, pages 383–384 and CD. Sere˛ga, S. 2009. Behaviour of high performance concrete columns at fire temperatures. PhD thesis, Cracow University of Technology, (in Polish). Sheikh, S. & Uzumeri, S. 1980. Strength and ductility of tied concrete columns. Journal of Structural Division, ASCE, 106(5):1078–1102. Zhang, B. & Biˇcaniˇc, N. 2006. Fracture energy of highperformance concrete at high temperatures up to 450C: the effects of heating temperatures and testing conditions. Magazine of Concrete Research, 58(5):277–288.

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Ultimate load analysis of a reactor safety containment structure Bernhard Valentini, Hermann Lehar & Günter Hofstetter Institute for Basic Sciences in Civil Engineering, University of Innsbruck, Austria

ABSTRACT: The paper deals with the ultimate load analysis of a prestressed reactor containment structure in the context of a Round Robin analysis program. The containment consists of a cylindrical wall and a spherical dome, which are connected by a ring beam. The complexity of the structure is increased by several openings and stiffeners. Both the development of the numerical model and the results of the ultimate load analysis are described in detail. 1

INTRODUCTION

The Bhabha Atomic Research Centre (BARC) in Trombay, India, and the Nuclear Power Corporation of India are conducting a Round Robin analysis program for predicting the structural response of a model of a 540 MWe pressurized heavy water reactor primary containment (Singh 2007). Thirteen teams from Austria (1 participant), the Czech Republic (1), Finland (1), France (2), India (5), the United Kingdom (1) and South Korea (2) are taking part in this competition, among them are the authors of this contribution. In contrast to the results of most of the numerical simulations presented in the literature, the predictions of the load carrying behaviour of a structure within the framework of a Round Robin analysis program are predictions in its original sense without knowing the test results in advance. The safety containment structure of the investigated Indian nuclear reactor type consists of two safety containments. The primary (inner) containment is a prestressed concrete shell structure and the secondary (outer) containment is a reinforced concrete shell structure. Within the framework of the Round Robin analysis only the model of the primary containment structure on a scale of 1:4 is investigated. It is composed of a cylindrical wall and a spherical dome. The complexity of the containment model results from several openings in the shell structure and the assembly of the different layers of the reinforcement and tendons across the shell thickness. In contrast to other reactor safety containments, the here examined containment has no additional gasket liner at the inner surface of the concrete shell. In the case of malfunctions the containment should prevent the release of radioactive materials into the atmosphere. Such a malfunction could be e.g. the loss of liquid or steam in one of the cooling circuits of the nuclear reactor or rupture of a high pressure pipe of the main cooling circuit which would lead to an

overpressure inside the reactor containment structure. A design specification of these containment structures requires that under specified design accidents (rupture of a high pressure pipe, missile impact, etc.) and earthquakes the structural and functional behaviour (leak tightness) is guaranteed. However, in case of rupture of a high pressure pipe, the internal pressure can increase beyond the design pressure of the containment. Then the concrete will be subjected to tensile stresses followed by the initiation of cracks in the concrete, which will propagate from the outer to the inner surface. Upon further increase of the internal pressure plastic strains will develop in the reinforcement and the prestressing tendons. Finally, the containment structure will collapse due to rupture of tendons. In this paper the numerical model of the primary reactor safety containment and the results of the ultimate load analysis due to an internal pressure increasing up to failure will be described in detail. Since the structural behaviour of the containment will be monitored during the tests by about 1100 sensors, the predicted numerical results can be compared with extensive experimental data in the near future, allowing a profound evaluation of the capabilities of the employed numerical model. 2

REACTOR SAFETY CONTAINMENT MODEL

The model of the 540 MWe pressurized heavy water reactor primary safety containment is on a scale of 1:4. It consists of a cylindrical wall and a spherical dome. The diameter of the containment model is 12.752 m and the height is 15.750 m without the foundation. The thickness of the cylindrical wall is 188 mm and the thickness of the spherical dome measures 164 mm. At the intersection of the cylindrical wall and the spherical dome the structure is stiffened by a ring beam with a thickness of 0.350 m and a height

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of 1.231 m. The cylindrical wall is also stiffened at the base, where the thickness amounts to 0.350 m over a height of 1.200 m. In the wall there are four vertical buttresses arranged at the azimuths of 41.5◦ , 131.5◦ , 221.5◦ and 311.5◦ for anchoring and prestressing the Mono-Strand steel tendons in hoop direction of the wall. The Mono-Strand steel tendons in the dome and the vertical tendons in the wall are anchored at one end in the foundation and at the other end in the ring beam. The tendons are grouted after application of the prestressing forces. The prestressing force per tendon is about 210 kN. In the containment model the reinforcement bars have a diameter of 12 mm and 16 mm, respectively, and the tendons are characterized by a diameter of 24 mm. Cross sectional views of the containment model are shown in Figure 1. Beside several small culverts (diameter of about 100 mm), there are three openings the cylindrical wall: main air lock (MAL, 1.500 m × 1.800 m), fuelling machine air lock (FMAL, 0.782 m × 0.782 m) and emergency air lock (EAL, diameter of 0.618 m). They are arranged at the azimuths of 270◦ , 180◦ and 90◦ , each approximately half way between two buttresses. In the spherical dome there are two circular steam generator openings (SG, diameter of 1.350 m). All five openings are surrounded by a thickened wall/dome section and additional reinforcement. For the test, the openings of the containment model are sealed with steel gates. The construction of the reactor containment model in India started in December 2006. The foundation was finished in May 2007. In June 2007 BARC started with the construction of the first of nine pours of the

C SG OPENING L 3241

EL 15,750

C SG OPENING L 3241

2044

EL 13,706 EL 12,475

R9794

188

13706

EL 5,875 INNER CONTAINMENT (I.C.) WALL

INNER CONTAINMENT (I.C.) WALL 12376

150

188 350

EL 0,000

94

3000

1800

3000

1800

1500

1500 15550

S

BU T

ES

TR

T BU

TR

EAL

ES

S

3241

1350 1850 2600

F /MAL

3241

1274 1754 2454

cylindrical wall. Thereafter, the stiffening ring and the dome, respectively, were made in one pour. The pouring of the containment model was finished in April 2008. To protect the model from thermal and other environmental influences, an auxiliary construction consisting of a roof which is supported by columns was built around the containment model (Singh 2009). The containment model will be tested by BARC with regard to its structural (load carrying behaviour) and functional (leakage rate) safety. For the test a pneumatic air pressure, with a design value of pd = 140 kN/m2 , will be applied at the interior of the containment model. During the test, the internal air pressure will be increased up to failure of the structure. The structural behaviour of the containment model will be monitored by about 300 surface sensors and 800 embedded sensors (strain gauges, earth pressure cells, displacement transducers, tilt meters and camera systems). 3 3.1

FINITE ELEMENT ANALYSIS MODEL Finite element mesh

The containment is modeled as a three dimensional shell structure. A shell model is preferred to a continuum model, because it requires a smaller number of degrees of freedom and less time and effort for generating the finite element model. An axisymmetric model does not allow a good representation of the actual structure because of the openings in the wall and the dome. Hence, the most efficient model for this type of structure seems to be a model consisting of shell elements, combined with beam and truss elements for the ring beam. The inner and outer surface of the shell is assumed to be parallel to the reference surface of the shell. Hence, thickness changes of the shell are modeled by stepped thicknesses resulting in a discontinuous shell surface. The foundation of the containment model is taken into account by appropriate boundary conditions at the bottom of the cylindrical part of the shell. The geometry and the finite element mesh are generated by means of the software program I-Deas (I-DEAS 2009). At first, the geometry is created as a three-dimensional solid model. Then the reference surface for the shell model is generated from the solid model. Afterwards the model is meshed and the layers of the elements are defined. The complexity of the numerical model for the BARC containment model results from

R =

76 63 82 = 62 R = 6188 R

BU T

MAL

TR

S

ES

ES

S

TR

T BU

Figure 1. Cross sectional view of the containment model on a scale of 1:4 (taken from (Singh 2007)): Vertical section (top) and plan view (bottom).

• different section thicknesses, • different section layouts (composite sections with different materials), • different offsets of stiffeners from the shell midsurface, • different directions of the reinforcement and tendons,

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• different diameters of the reinforcing bars and • different distances between reinforcing bars and tendons. To manage all these different geometric and material parameters within one model, 82 different regions have to be generated. Each region has its own specific properties. It’s assumed, that the section thickness, the material parameters, the amount and spacing of the reinforcement bars and tendons do not change within one region. Figure 2 shows an overview of the different regions of the shell model (without the steel parts of the openings in the wall), using different gray scale shadings for the specific regions. Each region is meshed with finite elements. Isoparametric, eight node quadrilateral shell elements with bi-quadratic shape functions are used for the discretisation of the cylindrical and spherical shell. The shell elements are based on a shell theory for thin shells and have three displacement and two rotational degrees of freedom at each node. A reduced integration scheme is used in the shell reference surface. In the direction normal to the reference surface, nine integration points are chosen. The ring beam is modeled by beam elements representing the concrete and by truss elements representing the tendons which are placed in the ring beam. Both, beam and truss elements are three node elements. The beam and truss elements are connected to the shell structure with two separate rows of shell elements. One row of shell elements extends from the beam/truss elements to the intersection of the cylindrical shell and the dome and the other one extends from the beam/truss elements to the dome.

Figure 2.

Containment model with different shaded regions.

Figure 3 shows the finite element mesh of the whole containment model. The finite element mesh consists of about 37000 layered shell elements and about 875000 degrees of freedom in total. The reinforcement and tendons are modeled with rebar layers embedded into the shell elements. These rebar layers are assumed to be parallel to the reference surface of the shell elements. Hence, stirrups are not considered in the model. 3.2

Boundary conditions

At the bottom of the containment, between the stiffened base of the cylindrical wall and the foundation, the displacements and rotations are assumed to be fully constrained. This assumption is based on the layout of the reinforcement extending from the foundation into the cylindrical wall. 3.3

Loading history

The loading history of the finite element analysis is spit into two load steps. In the first load step the permanent loads are applied. They consist of the dead load of the containment model and the prestressing forces. The latter are modeled by applying init = initial stresses to the tendons, prescribed as σtendon 1478.4 N/mm2 . In the second load step the air pressure in the interior of the containment is applied. The pressure at the inner surface is represented by distributed surface loads. These distributed surface loads are increased until failure occurs.

Figure 3.

Finite element mesh of the containment model.

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4 4.1

CONSTITUTIVE MODELS

Table 1.

Constitutive model for concrete

To model the cracking behaviour of the containment model, constitutive relations for concrete, based on plane stress conditions, were implemented into the finite element program Abaqus (ABAQUS 2009) by means of a material user subroutine. Since the nonlinear material behaviour of concrete in the containment is mainly due to tensile stresses, linear-elastic material behaviour of concrete is assumed in compression, whereas cracking is modeled by a smeared crack approach, formulated within the framework of plasticity theory. To this end, the tensile stresses are limited by a Rankine yield surface using a rounded corner region to obtain a smooth yield surface in biaxial tension (Winkler 2001). The yield function f , depending on the stress tensor σ and the softening variable α, is written as   σ11 − σ22 2 2 + σ12 + μ2E f (σ ; α) = 2 +

σ11 + σ22 − μ2E − q(α) = 0 2

(1)

with the round off parameter μE . The material response in tension is modeled as linear-elastic up to the tensile strength of concrete fctm and the post-peak region is described by the exponential isotropic softening law q(α) = fctm e

− αhfGctm f

.

(2)

To avoid mesh size dependent results, the softening response in tension is regularized by means of the specific fracture energy of concrete for mode I fracture Gf and the characteristic element lengths of the finite elements h. The evolution law of the softening variable is defined as α˙ = λ˙

(3)

˙ An associated with the consistency parameter λ. flow rule ε˙ p = λ˙

∂f ∂σ

(4)

is used for determining the plastic strain rate. The material parameters for concrete are summarized in Table 1. Following (Lackner and Mang 2003), the tension-stiffening (TS) effect is taken into account by increasing the specific fracture energy of plain concrete. In the present case a factor of six was determined. Hence the chosen fracture energy for the numerical simulation is GfTS = 0.4536 Nmm/mm2 .

Material parameters for concrete.

Elastic modulus E Poisson ratio ν Tensile strength fctm Fracture energy Gf

Table 2.

N/mm2 Nmm/mm2

200000 0.3 415 550

N/mm2 N/mm2 N/mm2

Material parameters for the tendons.

Elastic modulus E Poisson ratio ν Yield strength f0.2 Tensile strength fu

4.2

N/mm2

Material parameters for the reinforcement.

Elastic modulus E Poisson ratio ν Yield strength fy Tensile strength fu

Table 3.

33540 0.2 2.78 0.0756

189600 0.3 1683 1848

N/mm2 N/mm2 N/mm2

Constitutive model for the reinforcement and tendons

The reinforcement and tendons are modeled by an elastic-plastic material model using a von Mises type yield surface, an associated flow rule and a nonlinear isotropic hardening law. The hardening law is deduced from stress-strain curves, obtained from uniaxial tensile tests by BARC. Identical material behaviour of steel in tension and compression is assumed. The material parameters for the reinforcement and for the tendons are shown in Tables 2 and 3, respectively. 5

ANALYSIS RESULTS

In the numerical simulation cracks are already predicted due to application of dead-load and prestressing forces. They are located on the outer surface of the cylindrical wall in the vicinity of the base and the ring beam (Figure 4 (top)). For the containment subjected to the design pressure, only small additional cracked regions around the openings are computed (Figure 4 (middle)). If the pressure is increased to twice the design pressure, then the cracked regions around the openings increase (Figure 4 (bottom)) and new cracks are predicted at the inner surface of the dome next to the ring beam. At the pressure of 2.50 pd , the cracked regions are propagating from the main air lock towards the vertical buttresses. The cracked regions around the other opening are also increasing (Figure 5 (top)). In addition, cracks are formed at the inner surface of the ring beam and the adjacent regions in the dome. Further increase of the internal pressure results in

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Figure 4. Regions with cracks at three different pressure states (deformation 100-fold magnified). Top: 0.00 pd ; Middle: 1.00 pd ; Bottom: 2.00 pd . Light gray: small crack widths; Dark gray: medium crack widths; Black: large crack widths.

cracking of the vertical buttresses at each side of the main air lock. At 2.75 pd , the cylindrical wall is almost completely cracked through the thickness of the wall (Figure 5 (middle)). Upon small increase of the internal pressure the predicted fictitious crack strains are increasing considerably (Figure 5 (bottom)). Henceforward, very small load increments cause a large increase of the deformations and, thus, the load carrying capacity of the reactor safety containment model is reached at an internal pressure of about three times the design pressure. Figure 5 (bottom) shows the cracked regions in the containment at 2.91 pd , shortly before the load carrying capacity is reached.

Figure 5. Regions with cracks at three different pressure states (deformation 100-fold magnified). Top: 2.50 pd ; Middle: 2.75 pd ; Bottom: 2.91 pd . Light gray: small crack widths; Dark gray: medium crack widths; Black: large crack widths.

The radial displacements for the whole loading history are shown in Figure 7 for three different nodes of the containment model. Node 1 is located at the center of the quarter of the cylindrical wall without openings, about half way between the buttresses in horizontal direction and about half way between the foundation and the ring beam in vertical direction. Node 2 is next to the main air lock opening and node 3 is at the apex of the dome (Figure 6). During the first load step the numerical model is prestressed, hence the internal pressure is still zero but the displacements are not. Up to an inner pressure reaching the design pressure, the displacements are increasing linearly. Up

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Figure 6.

Location of nodes 1, 2 and 3.

12.5

2.5

Funding of this work by the Tyrolean Science Fund and the Austrian Ministry of Science BMWF as part of the UniInfrastrukturprogramm of the Forschungsplattform Scientific Computing at LFU Innsbruck is gratefully acknowledged.

0.00 p d (prestressing)

REFERENCES

-5.0 0.00

0.10

0.30 0.20 pressure [N/mm2]

2.75 p d 2.91 p d

2.50 p d

-2.5

2.00 p d

0.0 1.00 p d

radial displacement [mm]

5.0

ACKNOWLEDGEMENT

node 1 node 2 node 3

10.0 7.5

ring beam. Since for the present structure an axisymmetric model would only allow a very crude representation of the containment, a finite element model, consisting of shell elements for the cylindrical wall and the spherical dome combined with beam and truss elements for the ring beam, was developed. It is characterized by 875000 degrees of freedom. For values of the internal pressure, exceeding the design pressure, nonlinear structural behavior is mainly caused by cracking of concrete and plastic deformations of the reinforcement and tendons. The finite element analysis clearly shows the evolution of cracked regions of the containment and of plastic deformations of the reinforcement and tendons with increasing internal pressure until failure occurs. Since a model of the containment structure will be tested in the context of a Round Robin analysis program in the near future, the predicted numerical results will be compared with extensive experimental data, allowing a profound evaluation of the capabilities of the employed numerical model.

0.40

Figure 7. Radial displacements for three nodes of the reactor safety containment model.

to a pressure of 2.00 pd , in general the displacements are still increasing linearly, except for node 2. For larger pressures the overall structural behaviour is nonlinear. In particular, the radial displacement at node 1 is increasing rapidly (Figure 7).

6

ABAQUS (2009). ABAQUS Analysis User’s Manual. Providence, RI, USA: Dassault Syst‘emes Simulia Corp. I-DEAS (2009). I-DEAS 12. Köln, Germany: Siemens PLM Software. Lackner, R. and A.H. Mang (2003). Scale transition in steelconcrete interaction i: Model, ii: Applications. Journal of Engineering Mechanics, ASCE April 2003, 393. Singh, R. (2007). Barc containment model round robin analysis - release of model documents. Technical report, Bhabha Atomic Research Centre, Trombay, Mumbai, India. Singh, R. (2009). Pre-test report on international round robin analysis of barc containment (barcom) test model. Technical report, Bhabha Atomic Research Centre, Trombay, Mumbai, India. Winkler, B. (2001). Traglastuntersuchungen von unbewehrten und bewehrten Betonstrukturen auf der Grundlage eines objektiven Werkstoffgesetzes für Beton. Ph.D. thesis, Universitäat Innsbruck.

CONCLUSIONS

In this paper the development of a numerical model for a prestressed reactor containment structure was described. The containment consists of a cylindrical wall and a spherical dome, which are connected by a

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Computational Modelling of Concrete Structures – Bi´cani´c et al. (eds) © 2010 Taylor & Francis Group, London, ISBN 978-0-415-58479-1

Author index

Dormieux, L. 537 Dufour, F. 45, 145, 207, 271, 301

Abellan, M.-A. 451 Abreu, M. 431 Aljewifi, H. 77

Egger, P. 785 Eliáš, J. 419 Espinosa-Marzal, R.M. 471 Etse, G. 129, 241

Bažant, Z.P. 3, 87 Belletti, B. 581 Benboudjema, F. 461 Berger, J. 749 Bi´cani´c, N. 337 Billington, S.L. 15, 655 Bjegovi´c, D. 573 Blackard, B. 757 Blail, S. 591 Bobínski, J. 263, 729 Bosco, C. 595 Bottoni, M. 271 Bouassida, M. 173 Bourbon, X. 513 Boussa, H. 519 Buffo-Lacarriére, L. 603

Fantilli, A.P. 441 Ferro, G. 441 Figueiras, J. 391 Filho, A.C. 197 Finzel, J. 789 Fiorio, B. 77 Fleming, W. 309 Folino, P. 129

Caggiano, A. 241 Caner, F.C. 87 Carmeliet, J. 93, 431, 471 Carol, I. 487 Castel, A. 111 Cedolin, L. 481 Cervenka, J. 281 Cervenka, V. 281 Chiaia, B. 441 Chilakunda, N. 665 Chudoba, R. 101, 319 Citto, C. 757 Colliat, J.B. 461 Crémona, Ch 689 Cusatis, G. 87, 291, 481 Dahlblom, O. 671 Dahou, Z. 111 Damoni, C. 581 Davie, C.T. 563, 775 de Boer, A. 639 de Borst, R. 451 de Larrard, T. 461 Deleruyelle, F. 461 Derluyn, H. 471 Desmorat, R. 121 Di Luzio, G. 481 Dias, I.F. 381 Díaz, G. 31

Gagliardi, G. 595 Gal, E. 137 Gallias, J.L. 77 Gatuingt, F. 121 Ghomari, F. 111 Giry, C. 45, 145 Gödde, L. 611 Górski, J. 729 Gosling, P.D. 563 Han, D. 623 Hartig, J. 153 Häußler-Combe, U. 153, 163, 629, 789 Hegger, J. 101, 319 Hellmich, C. 59 Hendriks, M.A.N. 409, 647, 655 Hengl, H. 699 Heyden, S. 671 Hofstetter, G. 815 Hoogenboom, P.C.J. 639 Horák, M. 327 Huespe, A.E. 31, 381

Kaczmarczyk, L. 337 Kato, J. 679 Kemper, M. 503 Keuser, M. 623 Kitzig, M. 163 Kožar, I. 553 Kollegger, J. 591, 699, 749, 785 Kotronis, P. 145 Kozicki, J. 347 Krsti´c, V. 573 Kruschwitz, J. 503 Kryvoruk, R. 137 Kuhl, D. 309 Kyriakides, M.A. 655 Lackner, R. 797 Ladaoui, W. 513 Lafhaj, Z. 173 Larsen, K.P. 355 Le Pape, Y. 537 Le, T.T.H. 519 Legrain, G. 301 Lehar, H. 815 Lemos, J.V. 431 Li, J. 233 Li, Y.-J. 665 Lloberas Valls, O. 363 López, C.M. 487 Lyons, P. 497

Idiart, A.E. 487 Ikeda, T. 225 Imai, K. 737 Invernizzi, S. 595, 647

Maalej, S. 173 Maghous, S 197 Marangi, P. 291 Mark, P. 611 Martinelli, E. 241 Marzec, I. 529 Mazars, J. 45, 145 Meftah, F. 519 Mertens, S. 93 Meschke, G. 217, 241, 503 Mihai, I.C. 179 Mikuli´c, D. 573 Moonen, P. 93, 471 Mougaard, J.F. 373

Jeˇrábek, J. 319 Jefferson, A.D. 179, 497 Jirásek, M. 327

Nguyen Sy Tuan, 537 Nguyen, V.P. 547 Nielsen, L.O. 355, 373

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Rumanus, E. 217 Rypl, R. 101

O’Daniel, J.L. 291 Ožbolt, J. 553 Obaidat, Y.T. 671 Oliver, J. 31, 381 Ozaki, E. 737 Pamin, J. 187, 251 Pasa Dutra, V.F. 197 Pearce, C.J. 337 Pelessone, D. 291 Pellenq, R.J.-M. 69 Pichler, B. 59 Pijaudier-Cabot, G. 207, 301 Pimentel, M. 391 Pontiroli, C. 45 Poulsen, P.N. 355, 373 Radtke, F.K.F. 401 Ragueneau, F. 689 Ramm, E. 679 Réthoré, J. 451 Richard, B. 689 Ring, T. 797 Rixen, D.J. 363 Robson, C.J. 563 Rots, J.G. 409 Rouquand, A. 45 Ruediger, L. 623 Ruiz, G. 767

Sanahuja, J. 537 Sánchez, P.J. 31 Saucedo, L. 767 Savoia, M. 291 Sbartaï, Z.M. 111 Schauffert, E.A. 291 Scholzen, A. 101 Schweighofer, A. 699 Sellier, A. 513, 603 Sere˛ga, S. 805 Shirai, N. 737 Simone, A. 363, 401 Skar˙zy´nski, Ł. 703 Sluys, L.J. 363, 401, 547 Souza, R.A. 711 Stacchini, M. 291 Staquet, S. 719 Stipanovi´c Oslakovi´c, I. 573 Stroeven, M. 547 Syroka, E. 729 Tajima, K. 737 Takase, Y. 225 Tejchman, J. 263, 347, 529, 703, 729 Timothy, J.J. 503 Torrenti, J.M. 461

Toutlemonde, F. 719 Traeger, W. 749 Travaš, V. 553 Trovato, D. 647 Tue, N.V. 233 Ulm, F.-J. 69 Valentini, B. 815 van de Graaf, A.V. 409, 647 Vandamme, M. 69 Ventura, G. 441 Vidal, T. 513 Vill, M. 699 Voˇrechovský, M. 419 Vrech, S.M. 241 Wada, T. 225 Willam, K. 757 Winkler, B. 665 Winnicki, A. 187 Wosatko, A. 187 Wosatko, A. 251 Wu, J.-Y. 87 Yu, Q. 3 Yu, R.C. 767 Zeiml, M. 797 Zhang, H.L. 775

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