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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“ch-05” — 2010/2/1 — 20:07 — page 67 — #9

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 be