LifetimeOriented Structural Design Concepts
Friedhelm Stangenberg · Rolf Breitenbücher Otto T. Bruhns · Dietrich Har...
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LifetimeOriented Structural Design Concepts
Friedhelm Stangenberg · Rolf Breitenbücher Otto T. Bruhns · Dietrich Hartmann Rüdiger Höffer · Detlef Kuhl Günther Meschke (Eds.)
LifetimeOriented Structural Design Concepts
ABC
Prof. Dr.Ing. Friedhelm Stangenberg RuhrUniversity Bochum Institute for Reinforced and Prestressed Concrete Structures Universitätsstr. 150 44780 Bochum, Germany Email: sandra.krimpmann@ ruhrunibochum.de, friedhelm.stangenberg@ ruhrunibochum.de
Prof. Dr.Ing. Dietrich Hartmann RuhrUniversity Bochum Institute for Computational Engineering Universitätsstr. 150 44780 Bochum, Germany
Prof. Dr.Ing. Rolf Breitenbücher RuhrUniversity Bochum Institute for Building Materials Universitätsstr. 150 44780 Bochum, Germany
Prof. Dr.Ing. Detlef Kuhl University of Kassel Institute of Mechanics and Dynamics Mönchebergstr. 7 34109 Kassel, Germany
Prof. Dr.Ing. Otto T. Bruhns RuhrUniversity Bochum Institute of Mechanics Universitätsstr. 150 44780 Bochum, Germany
Prof. Dr.Ing. Günther Meschke RuhrUniversity Bochum Institute for Structural Mechanics Universitätsstr. 150 44780 Bochum, Germany
ISBN 9783642014611
eISBN 9783642014628
Prof. Dr.Ing. Rüdiger Höffer RuhrUniversity Bochum Building Aerodynamics Laboratory Universitätsstr. 150 44780 Bochum, Germany
DOI 10.1007/9783642014628 Library of Congress Control Number: Applied for c 2009 SpringerVerlag Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, speciﬁcally the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microﬁlm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a speciﬁc statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting by the Author. Production: Scientiﬁc Publishing Services Pvt. Ltd., Chennai, India. Cover Design: WMX Design GmbH, Heidelberg. Printed in acidfree paper 30/3100/as 5 4 3 2 1 0 springer.com
For Our Students, Colleagues and Engineers in Industry and Academia
The Team of SFB 398 Mark Alexander Ahrens • Hussein Alawieh • Matthias Baitsch • Falko Bangert • Yavuz Ba¸sar • Christian Becker • Ivanka Bevanda • J¨ org Bockhold • Ndzi Christian Bongmba • Dietrich Braess • Rolf Breitenb¨ ucher • Otto T. Bruhns • Christian Duckheim • Andreas Eckstein • Frank Ensslen • Olaf Faber • M´ ozes G´alﬀy • Volkmar G¨ ornandt • Jaroslaw Gorski • Stefan Grasberger • Klaus Hackl • Ulrike Hansk¨ otter • Gerhard Hanswille • Dietrich Hartmann • Anne Hartmann • Gunnar Heibrock • Martin Heiderich • Jan Helm • Christa Hermichen • Erich Heymer • R¨ udiger H¨ oﬀer • Norbert H¨olscher • JanHendrik Hommel • Wolfgang Hubert • Hur¸sit Ibuk • Mikhail Itskov • HansLudwig Jessberger • Daniel Jun • Dirk Kamarys • Michael Kasperski • Christoph Kemblowski •Olaf Kintzel • Andreas S. Kompalka • Diethard K¨ onig • Karsten K¨ onke • Stefan Kopp • Wilfried B. Kr¨ atzig • Sandra Krimpmann • Jens Kruschwitz • Detlef Kuhl • Jan Laue • Armin Lenzen • Roland Littwin • Ludger Lohaus • Dimitar Mancevski • G¨ unther Meschke • Kianoush MollaAbbassi • J¨ orn Mosler • Stephan M¨ uller • Thomas Nerzak • HansJ¨ urgen Niemann • Andrzej Niemunis • SamYoung Noh • Markus Peters • Lasse Petersen • Yuri Petryna • Daniel Pfanner • Tobias Pﬁster • Gero Pﬂanz • Igor Plazibat • Rainer P¨ olling • Markus Porsch • Thorsten Quent • Stefanie Reese • Christian Rickelt • Matthias Roik • Jan Saczuk • J¨ org Sahlmen • E. Scholz • Henning Sch¨ utte • Robert Schwetzke • Max J. Setzer • Bj¨ orn Siebert • Anne Spr¨ unken • Friedhelm Stangenberg • Zoran Stankovic • Sascha Stiehler • Mathias Strack • Helmut Stumpf • Theodoros ¨ undag • Heinz Waller • Claudia Walter • Heiner Triantafyllidis • Cenk Ust¨ Weber • Gisela Wegener • Andr´es Wellmann Jelic • Torsten Wichtmann • Xuejin Xu • Natalia Yalovenko
Preface
At the beginning of 1996, the Cooperative Research Center SFB 398 ﬁnancially supported by the German Science Foundation (DFG) was started at RuhrUniversity Bochum (RUB). A scientists group representing the ﬁelds of structural engineering, structural mechanics, soil mechanics, material science, and numerical mathematics introduced a research program on “lifetimeoriented design concepts on the basis of damage and deterioration aspects”. Two scientists from neighbourhood universities, one from Wuppertal and the other one from Essen, joined the Bochum Research Center, after a few years. The SFB 398 was sponsored for 12 years, until the beginning of 2008 – this is the maximum possible duration of DFG ﬁnancial support for an SFB. Safety and reliability are important for the whole expected service duration of an engineering structure. Therefore, prognostical solutions are needed and uncertainties have to be handled. A diﬀerentiation according to building types with diﬀerent service life requirements is necessary. Lifecycle strategies to control future structural degradations by concepts of appropriate design have to be developed, in case including means of inspection, maintenance, and repair. Aspects of costs and sustainability also matter. The importance of structural lifecycle management is well recognized in the international science community. Therefore, parallel corresponding activities are proceeding in many countries. In Germany, two other related SFBs were established: SFB 524 “Materials and Structures in Revitalisation of Buildings” at Weimar University and the still running SFB 477 “LifeCycle Assessment of Structures via Innovative Monitoring” at Braunschweig University of Technology. With these two SFBs, a fruitful cooperation was developed. The Cooperative Research Center for LifetimeOriented Design Concepts (SFB 398) at RuhrUniversity has carried out substantial work in many ﬁelds of structural lifetime management. Lifetimerelated fundamentals are provided with respect to structural engineering, structural and soil mechanics, material science as well as computational methods and simulation techniques. Stochastic aspects and interactions between various inﬂuences are included.
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Thus, a solid basis is provided for future practical use and, e.g. also for standardization. The wide range of scientiﬁc topics among the speciﬁcation and determination of external loading and the simulation based lifetimeoriented structural design concepts is presented in an extraordinary format. All scientists of the SFB 398, professors and Ph.D. students, have contributed with a great eﬀort in matchless team work to the present book. As a result of this, the present work is not only a collection of project reports, in fact it is almost written in the style of a monograph, whereby several authors fruitfully interact in all sections from the highest to the deepest level. Within this philosophy of joint authorship, authors are denoted in chapters and sections down to the third level. In special cases, where authors have contributed to a selected deeper section level, they are denoted beside the standard procedure in the regarding text episode. All members of SFB 398, with sincere thanks, acknowledge the ﬁnancial support of DFG over more than 12 years. The dedicated research work of all participating colleagues and of many guest scientists from diverse countries also is gratefully mentioned. Finally, the great eﬀorts of SpringerVerlag, Heidelberg, to produce this attractive volume is appreciated very much.
Bochum, March 26th, 2009
Friedhelm Stangenberg, Chairman of SFB 398 Otto T. Bruhns, Vicechairman of SFB 398
Contents
1
LifetimeOriented Design Concepts . . . . . . . . . . . . . . . . . . . . . . 1.1 LifetimeRelated Structural Damage Evolution . . . . . . . . . . . . 1.2 TimeDependent Reliability of Ageing Structures . . . . . . . . . . 1.3 Idea of WorkingLife Related Building Classes . . . . . . . . . . . . . 1.4 Economic and Further Aspects of ServiceLife Control . . . . . . 1.5 Fundamentals of LifetimeOriented Design . . . . . . . . . . . . . . . .
1 1 3 4 5 7
2
DamageOriented Actions and Environmental Impact . . . . 2.1 Wind Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Wind Buﬀeting with Relation to Fatigue . . . . . . . . . . . 2.1.1.1 Gust Response Factor . . . . . . . . . . . . . . . . . . . . 2.1.1.2 Number of Gust Eﬀects . . . . . . . . . . . . . . . . . . . 2.1.2 Inﬂuence of Wind Direction on Cycles of Gust Responses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2.1 Wind Data in the Sectors of the Wind Rosette . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2.2 Structural Safety Considering the Occurrence Probability of the Wind Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2.3 Advanced Directional Factors . . . . . . . . . . . . . 2.1.3 Vortex Excitation Including LockIn . . . . . . . . . . . . . . . 2.1.3.1 Relevant Wind Load Models . . . . . . . . . . . . . . 2.1.3.2 Wind Load Model for the Fatigue Analysis of Bridge Hangers . . . . . . . . . . . . . . . . . . . . . . . . 2.1.4 Micro and Macro Time Domain . . . . . . . . . . . . . . . . . . . 2.1.4.1 Renewal Processes and Pulse Processes . . . . . 2.2 Thermal Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 General Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Thermal Impacts on Structures . . . . . . . . . . . . . . . . . . .
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22 23 25 27 29 33 34 35 35 35
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2.2.3 Test Stand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Modelling of Short Term Thermal Impacts and Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.5 Application: Thermal Actions on a Cooling Tower Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Transport and Mobility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Traﬃc Loads on Road Bridges . . . . . . . . . . . . . . . . . . . . 2.3.1.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1.2 Basic European Traﬃc Data . . . . . . . . . . . . . . 2.3.1.3 Basic Assumptions of the Load Models for Ultimate and Serviceability Limit States in Eurocode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1.4 Principles for the Development of Fatigue Load Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1.5 Actual Traﬃc Trends and Required Future Investigations . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Aerodynamic Loads along HighSpeed Railway Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2.1 Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2.2 Dynamic Load Parameters . . . . . . . . . . . . . . . . 2.3.2.3 Load Pattern for Static and Dynamic Design Calculations . . . . . . . . . . . . . . . . . . . . . . 2.3.2.4 Dynamic Response . . . . . . . . . . . . . . . . . . . . . . . 2.4 LoadIndependent Environmental Impact . . . . . . . . . . . . . . . . . 2.4.1 Interactions of External Factors Inﬂuencing Durability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Frost Attack (with and without Deicing Agents) . . . . . 2.4.2.1 The ”Frost Environment”: External Factors and Frost Attack . . . . . . . . . . . . . . . . . 2.4.2.2 Damage Due to Frost Attack . . . . . . . . . . . . . . 2.4.3 External Chemical Attack . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3.1 Sulfate Attack . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3.2 Calcium Leaching . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Geotechnical Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Settlement Due to Cyclic Loading . . . . . . . . . . . . . . . . . 2.5.2 Multidimensional Amplitude for Soils under Cyclic Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Deterioration of Materials and Structures . . . . . . . . . . . . . . . 3.1 Phenomena of Material Degradation on Various Scales . . . . . 3.1.1 Load Induced Degradation . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1.1 Quasi Static Loading in Cementitious Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39 40 43 46 46 46 47
52 62 73 79 80 82 87 90 92 93 95 96 103 106 107 107 109 109 114 123 124 124 124
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3.1.1.1.1 Fracture Mechanism of Concrete Subjected to Uniaxial Compression Loading . . . . . . . . . . . 3.1.1.1.2 Fracture Mechanism of Concrete Subjected to Uniaxial Tension Loadings . . . . . . . . . . . . . . . . . . . . . . 3.1.1.1.3 Concrete under Multiaxial Loadings . . . . . . . . . . . . . . . . . . . . . . 3.1.1.2 Cyclic Loading . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1.2.1 Ductile Mode of Degradation in Metals . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1.2.2 QuasiBrittle Damage . . . . . . . . . . . 3.1.1.2.2.1 Cementitious Materials . . . . . . . . . . . . 3.1.1.2.2.2 Metallic Materials . . . . 3.1.2 Nonmechanical Loading . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2.1 Thermal Loading . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2.1.1 Degradation of Concrete Due to Thermal Incompatibility of Its Components . . . . . . . . . . . . . . . . . . . 3.1.2.1.2 Stresses Due to Thermal Loading . . . . . . . . . . . . . . . . . . . . . . . 3.1.2.1.3 Temperature and Stress Development in Concrete at the Early Age Due to Heat of Hydration . . . . . . . . . . . . . . . . . . . . . 3.1.2.2 ThermoHygral Loading . . . . . . . . . . . . . . . . . . 3.1.2.2.1 Hygral Behaviour of Hardened Cement Paste . . . . . . . . . . . . . . . . . . 3.1.2.2.2 Inﬂuence of Cracks on the Moisture Transport . . . . . . . . . . . . . 3.1.2.2.3 Freeze Thaw . . . . . . . . . . . . . . . . . . . 3.1.2.3 Chemical Loading . . . . . . . . . . . . . . . . . . . . . . . 3.1.2.3.1 Microstructure of Cementitious Materials . . . . . . . . . . . . . . . . . . . . . . 3.1.2.3.2 Dissolution . . . . . . . . . . . . . . . . . . . . . 3.1.2.3.3 Expansion . . . . . . . . . . . . . . . . . . . . . 3.1.2.3.3.1 Sulphate Attack on Concrete and Mortar . . . . . . . . . . . . . . 3.1.2.3.3.2 AlkaliAggregate Reaction in Concrete . . . . . . . . . . . . 3.1.3 Accumulation in Soils Due to Cyclic Loading: A Deterioration Phenomenon? . . . . . . . . . . . . . . . . . . . . . .
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125 126 129 129 131 131 137 140 140
140 141
142 143 143 147 148 150 150 152 157
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3.2 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Laboratory Testing of Structural Materials . . . . . . . . . 3.2.1.1 Micromacrocrack Detection in Metals . . . . . . 3.2.1.1.1 Electric Resistance Measurements . . . . . . . . . . . . . . . . . . 3.2.1.1.1.1 Introduction . . . . . . . . . 3.2.1.1.1.2 Measurement of the Electrical Resistance . . . . . . . . . . . 3.2.1.1.1.3 Calculation of the Electrical Resistance . . 3.2.1.1.1.4 Experiments . . . . . . . . . 3.2.1.1.1.5 Experimental Results . . . . . . . . . . . . . 3.2.1.1.2 Acoustic Emission . . . . . . . . . . . . . . 3.2.1.1.2.1 Location of Acoustic Emission Sources . . . . . . . . . . . . . 3.2.1.1.2.2 Linear Location of Acoustic Emission Sources . . . . . . . . . . . . . 3.2.1.1.2.3 Location of Sources in Two Dimensions . . . 3.2.1.1.2.4 Kaiser Eﬀect . . . . . . . . 3.2.1.1.2.5 Experimental Procedures . . . . . . . . . . 3.2.1.1.2.6 Experimental Results . . . . . . . . . . . . . 3.2.1.2 Degradation of Concrete Subjected to Cyclic Compressive Loading . . . . . . . . . . . . . . . 3.2.1.2.1 Test Series and Experimental Strategy . . . . . . . . . . . . . . . . . . . . . . . 3.2.1.2.2 Degradation Determined by Decrease of Stiﬀness . . . . . . . . . . . . . 3.2.1.2.3 Degradation Determined by Changes in StressStrain Relation . . . . . . . . . . . . . . . . . . . . . . . 3.2.1.2.4 Adequate Description of Degradation by Fatigue Strain . . . . 3.2.1.2.5 Behaviour of High Strength Concrete and AirEntrained Concrete . . . . . . . . . . . . . . . . . . . . . . . 3.2.1.2.6 Inﬂuence of Various Coarse Aggregates and Diﬀerent Grading Curves . . . . . . . . . . . . . . . .
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165 166 166 167 169
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3.2.1.2.7 Cracking in the Microstructure Due to Cyclic Loading . . . . . . . . . . . 3.2.1.2.8 Inﬂuence of Single Rest Periods . . . 3.2.1.2.9 Sequence Eﬀect Determined by TwoStage Tests . . . . . . . . . . . . . . . . 3.2.1.3 Degradation of Concrete Subjected to Freeze Thaw . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 HighCycle Laboratory Tests on Soils . . . . . . . . . . . . . . 3.2.3 Structural Testing of Composite Structures of Steel and Concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3.2 Basic Tests for the Fatigue Resistance of Shear Connectors . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3.2.1 Test Program . . . . . . . . . . . . . . . . . . 3.2.3.2.2 Test Specimens . . . . . . . . . . . . . . . . . 3.2.3.2.3 Test Setup and Loading Procedure . . . . . . . . . . . . . . . . . . . . . 3.2.3.2.4 Material Properties . . . . . . . . . . . . . 3.2.3.2.5 Results of the PushOut Tests . . . . 3.2.3.2.5.1 General . . . . . . . . . . . . . 3.2.3.2.5.2 Results of the Constant Amplitude Tests . . . . . . . . . . . . . . . 3.2.3.2.6 Results of the Tests with Multiple Blocks of Loading . . . . . . . 3.2.3.2.7 Results of the Tests Regarding the Mode Control and the Eﬀect of Low Temperature . . . . . . . . . . . . 3.2.3.2.8 Results of the Tests Regarding Crack Initiation and Crack Propagation . . . . . . . . . . . . . . . . . . . . 3.2.3.3 Fatigue Tests of FullScale Composite Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3.3.1 General . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3.3.2 Test Program . . . . . . . . . . . . . . . . . . 3.2.3.4 Test Specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3.5 Test Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3.6 Material Properties . . . . . . . . . . . . . . . . . . . . . . 3.2.3.7 Main Results of the Beam Tests . . . . . . . . . . . 3.3 Modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Load Induced Damage . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1.1 Damage in Cementitious Materials Subjected to Quasi Static Loading . . . . . . . . . 3.3.1.1.1 ContinuumBased Models . . . . . . . .
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3.3.1.1.1.1 Damage MechanicsBased Models . . . . . . . . 3.3.1.1.1.2 Elastoplastic Models . . 3.3.1.1.1.3 Coupled ElastoplasticDamage Models . . . . . . 3.3.1.1.1.4 Multisurface ElastoplasticDamage Model for Concrete . . . . . . . . . . . . 3.3.1.1.2 Embedded Crack Models . . . . . . . . 3.3.1.2 Cyclic Loading . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1.2.1 MechanismOriented Simulation of Low Cycle Fatigue of Metallic Structures . . . . . . . . . . . . . . . . . . . . . 3.3.1.2.1.1 Macroscopic ElastoPlastic Damage Model for Cyclic Loading . . . . . . . 3.3.1.2.1.2 Model Validation . . . . . 3.3.1.2.2 QuasiBrittle Damage in Materials . . . . . . . . . . . . . . . . . . . . . . 3.3.1.2.2.1 Cementitious Materials . . . . . . . . . . . . 3.3.1.2.2.2 Metallic Materials . . . . 3.3.2 Nonmechanical Loading and Interactions . . . . . . . . . . 3.3.2.1 ThermoHygroMechanical Modelling of Cementitious Materials  Shrinkage and Creep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2.1.1 Introductory Remarks . . . . . . . . . . . 3.3.2.1.2 State Equations . . . . . . . . . . . . . . . . 3.3.2.1.3 Identiﬁcation of Coupling Coeﬃcients . . . . . . . . . . . . . . . . . . . . 3.3.2.1.4 Eﬀective Stresses . . . . . . . . . . . . . . . 3.3.2.1.5 Multisurface DamagePlasticity Model for Partially Saturated Concrete . . . . . . . . . . . . . . . . . . . . . . . 3.3.2.1.6 LongTerm Creep . . . . . . . . . . . . . . . 3.3.2.1.7 Moisture and Heat Transport . . . . 3.3.2.1.7.1 Freeze Thaw . . . . . . . . . 3.3.2.2 ChemoMechanical Modelling of Cementitious Materials . . . . . . . . . . . . . . . . . . . 3.3.2.2.1 Models for Ion Transport and Dissolution Processes . . . . . . . . . . . .
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3.3.2.2.1.1 Introductory Remarks . . . . . . . . . . . . 3.3.2.2.1.2 Initial Boundary Value Problem . . . . . . . 3.3.2.2.1.3 Constitutive Laws . . . . 3.3.2.2.1.4 Migration of Calcium Ions in Water and Electrolyte Solutions . . . . . . . . . . . . 3.3.2.2.1.5 Evolution Laws . . . . . . 3.3.2.2.2 Models for Expansive Processes . . . 3.3.2.2.2.1 Introductory Remarks . . . . . . . . . . . . 3.3.2.2.2.2 Balance Equations . . . 3.3.2.2.2.3 Constitutive Laws . . . . 3.3.2.2.2.4 Model Calibration . . . . 3.3.3 A HighCycle Model for Soils . . . . . . . . . . . . . . . . . . . . . 3.3.4 Models for the Fatigue Resistance of Composite Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4.1 General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4.2 Modelling of the Local Behaviour of Shear Connectors in the Case of Cyclic Loading . . . 3.3.4.2.1 Static Strength of Headed Shear Studs without Any Predamage . . . 3.3.4.2.2 Failure Modes of Headed Shear Studs Subjected to HighCycle Loading . . . . . . . . . . . . . . . . . . . . . . . 3.3.4.2.3 Correlation between the Reduced Static Strength and the Geometrical Property of the Fatigue Fracture Area . . . . . . . . . . . 3.3.4.2.4 Lifetime  Number of Cycles to Failure Based on Force Controlled Fatigue Tests . . . . . . . . . 3.3.4.2.5 Reduced Static Strength over Lifetime . . . . . . . . . . . . . . . . . . . . . . . 3.3.4.2.6 LoadSlip Behaviour . . . . . . . . . . . . 3.3.4.2.7 Crack Initiation and Crack Development . . . . . . . . . . . . . . . . . . . 3.3.4.2.8 Improved Damage Accumulation Model . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4.2.9 Ductility and Crack Formation . . .
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3.3.4.2.10 Finite Element Calculations of the (Reduced) Static Strength of Headed Shear Studs in PushOut Specimens . . . . . . . . . . . . 3.3.4.2.11 Eﬀect of the Control Mode Eﬀect of Low Temperatures . . . . . . 3.3.4.3 Modelling of the Global Behaviour of Composite Beams Subjected to Cyclic Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4.3.1 Material Model for the Concrete Behaviour . . . . . . . . . . . . . . . . . . . . . 3.3.4.3.2 Eﬀect of HighCycle Loading on Load Bearing Capacity of Composite Beams . . . . . . . . . . . . . . . 3.3.4.3.3 Cyclic Behaviour of Composite Beams  Development of Slip . . . . . 3.3.4.3.4 Eﬀect of Cyclic Loading on Beams with Tension Flanges . . . . . 3.4 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Durability Analysis of a Concrete Tunnel Shell . . . . . . 3.4.2 Durability Analysis of a Cementitious Beam Exposed to Calcium Leaching and External Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Durability Analysis of a Sealed Panel with a Leakage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.4 Numerical Simulation of a Concrete Beam Aﬀected by AlkaliSilica Reaction . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.5 Lifetime Assessment of a Spherical Metallic Container . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Methodological Implementation . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Classiﬁcation of Deterioration Problems . . . . . . . . . . . . 4.1.2 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.4 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Generalization of Single and Multiﬁeld Models . . . . . 4.2.1.1 Integral Format of Balance Equations . . . . . . 4.2.1.2 Strong Form of Individual Balance Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Strategy of Numerical Solution . . . . . . . . . . . . . . . . . . . . 4.2.3 Weak Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3.1 Weak Form of Coupled Balance Equations . .
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4.2.3.2 Linearized Weak Form of Coupled Balance Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Spatial Discretization Methods . . . . . . . . . . . . . . . . . . . . 4.2.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4.2 Generalized Finite Element Discretization of Multiﬁeld Problems . . . . . . . . . . . . . . . . . . . . 4.2.4.2.1 Approximations . . . . . . . . . . . . . . . . 4.2.4.2.2 NonLinear Semidiscrete Balance . . . . . . . . . . . . . . . . . . . . . . . 4.2.4.2.3 Linearized Semidiscrete Balance . . 4.2.4.2.4 Generation of Element and Structural Quantities . . . . . . . . . . . . 4.2.4.3 pFinite Element Method . . . . . . . . . . . . . . . . . 4.2.4.3.1 Onedimensional HigherOrder Shape Function Concepts . . . . . . . . 4.2.4.3.1.1 Shape Functions of the LegendreType . . . 4.2.4.3.1.2 Comparison of Both Shape Function Concepts . . . . . . . . . . . . 4.2.4.3.2 3DpFinite Element Method Based on Hierarchical Legendre Polynomials . . . . . . . . . . . . . . . . . . . . 4.2.4.3.2.1 Generation of 3DpShape Functions . . . . . . . . . . . 4.2.4.3.2.2 Spatially Anisotropic Approximation Orders . . . . . . . . . . . . . . 4.2.4.3.2.3 Fieldwise Choice of the Approximation Order . . . . . . . . . . . . . . . 4.2.4.3.2.4 Geometry Approximation . . . . . . . 4.2.5 Solution of Stationary Problems . . . . . . . . . . . . . . . . . . . 4.2.5.1 Numerical Solution Technique . . . . . . . . . . . . . 4.2.5.2 Iteration Methods . . . . . . . . . . . . . . . . . . . . . . . 4.2.5.3 ArcLength Controlled Analysis . . . . . . . . . . . 4.2.6 Temporal Discretization Methods . . . . . . . . . . . . . . . . . . 4.2.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.6.1.1 Motivation . . . . . . . . . . . . . . . . . . . . . 4.2.6.1.2 Newmarkα Time Integration Schemes . . . . . . . . . . . . . . . . . . . . . . .
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4.2.6.1.3 Galerkin Time Integration Schemes . . . . . . . . . . . . . . . . . . . . . . . 4.2.6.2 Newmarkα Time Integration Schemes . . . . . 4.2.6.2.1 Nonlinear Semidiscrete Initial Value Problem . . . . . . . . . . . . . . . . . 4.2.6.2.2 Numerical Concept of Newmarkα Time Integration Schemes . . . . . . . . . . . . . . . . . . . . . . . 4.2.6.2.3 Time Discretization . . . . . . . . . . . . . 4.2.6.2.4 Approximation of State Variables . . . . . . . . . . . . . . . . . . . . . . 4.2.6.2.5 Algorithmic Semidiscrete Balance Equation . . . . . . . . . . . . . . . 4.2.6.2.6 Eﬀective Balance Equation . . . . . . . 4.2.6.2.7 Newmarkα Algorithm . . . . . . . . . . 4.2.6.3 Discontinuous and Continuous Galerkin Time Integration Schemes . . . . . . . . . . . . . . . . 4.2.6.3.1 Time Discretization . . . . . . . . . . . . . 4.2.6.3.2 Continuity Condition . . . . . . . . . . . . 4.2.6.3.3 Temporal Weak Form . . . . . . . . . . . 4.2.6.3.4 Linearization . . . . . . . . . . . . . . . . . . . 4.2.6.3.5 Temporal Galerkin Approximation . . . . . . . . . . . . . . . . . 4.2.6.3.6 Discontinuous BubnovGalerkin Schemes dG(p) . . . . . . . . . . . . . . . . . 4.2.6.3.7 Continuous PetrovGalerkin Schemes cG(p) . . . . . . . . . . . . . . . . . 4.2.6.3.8 NewtonRaphson Iteration . . . . . . . 4.2.6.3.9 Algorithmic SetUp of Galerkin Schemes . . . . . . . . . . . . . . . . . . . . . . . 4.2.7 Generalized Computational Durabilty Mechanics . . . . 4.2.8 Adaptivity in Space and Time . . . . . . . . . . . . . . . . . . . . 4.2.8.1 ErrorControlled Spatial Adaptivity . . . . . . . . 4.2.8.1.1 Variational Functional . . . . . . . . . . . 4.2.8.1.2 Interpolation . . . . . . . . . . . . . . . . . . . 4.2.8.1.3 Stress Computation . . . . . . . . . . . . . 4.2.8.1.4 Discretized Weak Form . . . . . . . . . . 4.2.8.1.5 Summary . . . . . . . . . . . . . . . . . . . . . . 4.2.8.1.6 Hanging Node Concept . . . . . . . . . . 4.2.8.1.7 Error Criteria . . . . . . . . . . . . . . . . . . 4.2.8.1.7.1 WarpingBased Error Criterion . . . . . . 4.2.8.1.7.2 ResidualBased Error Criterion . . . . . . 4.2.8.1.8 Program Flow . . . . . . . . . . . . . . . . . .
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4.2.8.1.9 Transfer of History Variables . . . . . 4.2.8.1.10 Examples . . . . . . . . . . . . . . . . . . . . . . 4.2.8.1.10.1 Uniaxial Bending (Beam of Uniform Thickness) . . . . . . . . . . 4.2.8.1.10.2 Uniaxial Bending (Beam of Variable Thickness) . . . . . . . . . . 4.2.8.1.10.3 Biaxial Bending (Thick Plate of Uniform Thickness) . . 4.2.8.2 ErrorControlled Temporal Adaptivity . . . . . . 4.2.8.2.1 Local a Posteriori h and pMethod Error Estimates . . . . . . . 4.2.8.2.2 Local a Posteriori h and pMethod Error Indicators . . . . . . . 4.2.8.2.3 Local Zienkiewicz a Posteriori Error Indicators . . . . . . . . . . . . . . . . 4.2.8.2.4 Adaptive Time Stepping Procedure . . . . . . . . . . . . . . . . . . . . . 4.2.8.2.5 Algorithmic SetUp . . . . . . . . . . . . . 4.2.9 Discontinuous Finite Elements . . . . . . . . . . . . . . . . . . . . 4.2.9.1 Overview and Motivation . . . . . . . . . . . . . . . . . 4.2.9.2 Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.9.2.1 Extended Finite Element Method (XFEM) . . . . . . . . . . . . . . . 4.2.9.2.1.1 Partition of Unity . . . . 4.2.9.2.1.2 XFEM Displacement Field . . . . . . . . . . . . . . . 4.2.9.2.1.3 Integrating Discontinuous Functions . . . . . . . . . . . 4.2.9.2.1.4 pVersion of the XFEM . . . . . . . . . . . . . . 4.2.9.2.1.5 3D XFEM . . . . . . . . . . 4.2.9.2.1.6 XFEM for Cohesive Cracks . . . . . . . . . . . . . . 4.2.9.2.2 Strong Discontinuity Approach and Enhanced Assumed Strain . . . 4.2.9.2.2.1 Kinematics: Modeling Embedded Strong Discontinuities . . . . . . .
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4.2.9.2.2.2 Numerical Implementation . . . . . . 4.2.9.2.2.3 Numerical Example: 3Point Bending Problem . . . . . . . . . . . . 4.2.9.3 Crackgrowth Criteria . . . . . . . . . . . . . . . . . . . . . 4.2.9.3.1 Hoop Stresses . . . . . . . . . . . . . . . . . . 4.2.9.3.2 ModeICrack Extension . . . . . . . . . 4.2.9.3.3 Minimum Energy . . . . . . . . . . . . . . . 4.2.9.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.9.4.1 Double Notched Slab . . . . . . . . . . . . 4.2.9.4.2 Anchor PullOut . . . . . . . . . . . . . . . . 4.2.10 Substructuring and Model Reduction of Partially Damaged Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.10.1 Motivation and Overview . . . . . . . . . . . . . . . . . 4.2.10.2 Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.10.3 Derivation of a Substructure Technique for Nonlinear Dynamics . . . . . . . . . . . . . . . . . . . . . . 4.2.10.3.1 CraigBampton Method . . . . . . . . . 4.2.10.3.2 Model Reduction of Linear Dynamic Structures . . . . . . . . . . . . . 4.2.10.3.2.1 Modal Reduction . . . . . 4.2.10.3.2.2 Proper Orthogonal Decomposition . . . . . . . 4.2.10.3.2.3 Pad´eViaLanczos Algorithm . . . . . . . . . . . 4.2.10.3.2.4 LoadDependent Ritz Vectors . . . . . . . . . 4.2.10.3.3 Substructuring in the Framework of Nonlinear Dynamics . . . . . . . . . . . . . . . . . . . . . . 4.2.10.3.3.1 Discretisation and Linearisation . . . . . . . . 4.2.10.3.3.2 Primal Assembly . . . . . 4.2.10.3.3.3 Solution of the Decomposed Structure . . . . . . . . . . . 4.2.10.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.11 Strategy for Polycyclic Loading of Soil . . . . . . . . . . . . . 4.3 System Identiﬁcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Covariance Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Subspace Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2.1 State Space Model . . . . . . . . . . . . . . . . . . . . . . . 4.3.2.2 Subspace Identiﬁcation . . . . . . . . . . . . . . . . . . . 4.3.2.3 Modal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . .
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4.4 Reliability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 General Problem Deﬁnition . . . . . . . . . . . . . . . . . . . . . . 4.4.2 TimeInvariant Problems . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2.1 Approximation Methods . . . . . . . . . . . . . . . . . . 4.4.2.2 Simulation Methods . . . . . . . . . . . . . . . . . . . . . . 4.4.2.2.1 Importance Sampling . . . . . . . . . . . . 4.4.2.2.2 Latin Hypercube Sampling . . . . . . . 4.4.2.2.3 Subset Methods . . . . . . . . . . . . . . . . 4.4.2.3 Response Surface Methods . . . . . . . . . . . . . . . . 4.4.2.4 Evaluation of Uncertainties and Choice of Random Variables . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 TimeVariant Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3.1 TimeIntegrated Approach . . . . . . . . . . . . . . . . 4.4.3.2 Time Discretization Approach . . . . . . . . . . . . . 4.4.3.3 Outcrossing Methods . . . . . . . . . . . . . . . . . . . . . 4.4.4 Parallelization of Reliability Analyses . . . . . . . . . . . . . . 4.4.4.1 Reliability Analysis of Fatigue Processes . . . . 4.4.4.2 Parallelization Example . . . . . . . . . . . . . . . . . . 4.5 Optimization and Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1 Classiﬁcation of Optimization Problems . . . . . . . . . . . . 4.5.2 Design as an Optimization Problem . . . . . . . . . . . . . . . . 4.5.3 Numerical Optimization Methods . . . . . . . . . . . . . . . . . 4.5.3.1 DerivativeBased Methods . . . . . . . . . . . . . . . . 4.5.3.2 DerivativeFree Strategies . . . . . . . . . . . . . . . . . 4.5.4 Parallelization of Optimization Strategies . . . . . . . . . . . 4.5.4.1 Parallelization with GradientBased Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.4.2 Parallelization Using Evolution Strategies . . . 4.5.4.3 Distributed and Parallel Software Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Application of LifetimeOriented Analysis and Design . . . . . . 4.6.1 Testing of BeamLike Structures . . . . . . . . . . . . . . . . . . . 4.6.1.1 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . 4.6.1.2 Identiﬁcation of Modal Data . . . . . . . . . . . . . . 4.6.1.3 Updating of the Finite Element Model . . . . . . 4.6.2 Lifetime Analysis for Dynamically Loaded Structures at BMW AG . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2.1 Works for the New 3Series Convertible . . . . . 4.6.2.2 The Shaker Test . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2.3 Approach 1: Time History Calculation and Amplitude Counting . . . . . . . . . . . . . . . . . . . . . 4.6.2.3.1 Structural Analysis Using Time Integration . . . . . . . . . . . . . . . . . . . . 4.6.2.3.2 Cycle Counting Using the Rainﬂow Method . . . . . . . . . . . . . . .
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4.6.2.3.3 Damage Calculation . . . . . . . . . . . . . 4.6.2.4 Approach 2: Power Spectral Density Functions and Calculation of Spectral Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2.4.1 Structural Analysis Using Power Spectral Density (PSD) Functions . . . . . . . . . . . . . . . . . . . . . 4.6.2.4.2 Analytical Counting Method . . . . . 4.6.2.4.3 Damage Accumulation for the Analytical Case . . . . . . . . . . . . . . . . . 4.6.2.5 Comparison of the Results . . . . . . . . . . . . . . . . 4.6.2.6 Summary and Outlook . . . . . . . . . . . . . . . . . . . 4.6.3 LifetimeOriented Analysis of Concrete Structures Subjected to Environmental Attack . . . . . . . . . . . . . . . . 4.6.3.1 HygroMechanical Analysis of a Concrete Shell Structure . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.3.1.1 Conclusive Remarks on the HygroMechanical Analysis . . . . . . 4.6.3.2 Calcium Leaching of Cementitious Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.3.2.1 Calcium Leaching of a Cementitious Bar . . . . . . . . . . . . . . . 4.6.3.2.1.1 Analysis of the Numerical Results . . . . 4.6.3.2.1.2 Adaptive Newmark Solution . . . . . . . . . . . . 4.6.3.2.1.3 Robustness of Galerkin Solutions . . . . 4.6.3.2.1.4 Error Estimates for Newmark Solutions . . . 4.6.3.2.1.5 Error Estimates for Galerkin Solutions . . . . 4.6.3.2.1.6 Order of Accuracy of Galerkin Schemes . . 4.6.3.2.2 Calcium Leaching of a Cementitious Beam . . . . . . . . . . . . . 4.6.3.2.2.1 Analysis of the Numerical Results . . . . 4.6.3.2.2.2 Robustness of Continuous Galerkin Solutions . . . . 4.6.4 Arched Steel Bridge Under Wind Loading . . . . . . . . . . 4.6.4.1 Deﬁnition of Structural Problem . . . . . . . . . . . 4.6.4.2 Probabilistic Lifetime Assessment . . . . . . . . . . 4.6.4.2.1 Micro Time Scale . . . . . . . . . . . . . . .
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4.6.4.2.2 Macro Time Scale . . . . . . . . . . . . . . Results of Structural Optimization . . . . . . . . . Parallelization of Analyses . . . . . . . . . . . . . . . . Final Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . Reinforced Concrete Bridge . . . . . . . . . . . . . . . . Numerical Simulation . . . . . . . . . . . . . . . . . . . . 4.6.5.1.1 Experimental Investigation on Mechanical Concrete Properties . . 4.6.5.1.1.1 Nondestructive Tests . . . . . . . . . . . . . . . 4.6.5.1.1.2 Destructive Tests . . . . . 4.6.5.1.1.3 Microscopic Analysis . . . . . . . . . . . . 4.6.5.1.1.4 Cyclic Tests . . . . . . . . . 4.6.5.1.2 Finite Element Model . . . . . . . . . . . 4.6.5.1.3 Material Model . . . . . . . . . . . . . . . . . 4.6.5.1.4 Damage Mechanisms . . . . . . . . . . . . 4.6.5.1.4.1 Corrosion of the Reinforcement Steel Bars . . . . . . . . . . . . . . . . 4.6.5.1.4.2 Fatigue of the Prestressing Tendons . . . . . . . . . . . . 4.6.5.1.5 Modelling of Uncertainties . . . . . . . 4.6.5.1.5.1 LongTerm Developement of Concrete Strength . . . . 4.6.5.1.5.2 Determination of Material Properties . . . 4.6.5.1.5.3 Modelling of Spatial Scatter by Random Fields . . . . . . . . . . . . . . 4.6.5.1.6 Lifetime Simulation . . . . . . . . . . . . . 4.6.5.1.7 Conclusions . . . . . . . . . . . . . . . . . . . . 4.6.5.2 Experimental Veriﬁcation . . . . . . . . . . . . . . . . . 4.6.5.2.1 State Space Model for Mechanical Structures . . . . . . . . . . . 4.6.5.2.2 White Box Model  Physical Interpretable Parameters . . . . . . . . 4.6.5.2.3 Identiﬁcation of Measured Mechanical Structures . . . . . . . . . . . 4.6.5.2.3.1 Black Box Model Deterministic System Identiﬁcation . . . . . . . .
4.6.4.3 4.6.4.4 4.6.4.5 4.6.5 Arched 4.6.5.1
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Contents
4.6.5.2.3.2 Diﬀerences between Theory and Experiment . . . . . . . . . 4.6.5.2.4 Experiments . . . . . . . . . . . . . . . . . . . 4.6.5.2.4.1 Cantilever Bending Beam . . . . . . . . . . . . . . . 4.6.5.2.4.2 TiedArch Bridge near H¨ unxe  Germany . . . . 4.6.5.2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . 4.6.6 Examples for the Prediction of Settlement Due to Polycyclic Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Future Life Time Oriented Design Concepts . . . . . . . . . . . . . 5.1 Exemplary Realization of Lifetime Control Using Concepts as Presented Here . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Reinforced Concrete Column under Fatigue Load . . . . 5.1.2 Connection Plates of an Arched Steel Bridge . . . . . . . . 5.1.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 LifetimeControl Provisions in Current Standardization . . . . . 5.3 Incorporation into Structural Engineering Standards . . . . . . .
638 641 641
642 645 646 653 653 653 655 658 658 659
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661 Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 711
List of Figures
1.1 1.2 1.3 1.4 1.5 1.6 1.7 2.1 2.2
2.3 2.4 2.5
2.6 2.7 2.8 2.9 2.10
Lifetimerelated aspects of structural concrete . . . . . . . . . . . . . . . Evolution of degradation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Timedependent reliability of structures . . . . . . . . . . . . . . . . . . . . . Timedependent reliability of structures with upgrading by repairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Workinglife related building classes . . . . . . . . . . . . . . . . . . . . . . . . Service Life control and economic aspects . . . . . . . . . . . . . . . . . . . Related Collaborative Research Centers . . . . . . . . . . . . . . . . . . . . . Typical wind load process (a), and related low frequency (b) and high frequency (c) response of a structure [572] . . . . . . . . . . Curve of the total variance of the base bending moment of a cantilever due to buﬀeting excitation plotted over frequencies [572] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of the occurence of repeated wind eﬀects at diﬀerent locations in Germany and a codiﬁed representation . . . Distribution of absolute frequencies of normalized gust responses into subsequent classes of diﬀerent levels of eﬀect . . . . Comparison of the distribution of cyclic stress amplitudes with the SN curve (W¨ohler curve) of stress concentration category 36* after [30] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rosettes of wind quantities at Hannover (12 sectors, 50 years return period) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Roughness lengths of the terrain in the farther vicinity of the building location [771] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sketch of a building contour and fa¸cade element exposed to a pressure coeﬃcient cp = −1.4 [32] . . . . . . . . . . . . . . . . . . . . . . . . . . ´rma ´n vortex trail formed by vortex shedding . . . . . . . . Von Ka Dependence of the vortex shedding frequency fv on the wind velocity u ¯. fn is the natural frequency of the structure . . . . . . . .
2 2 3 4 5 6 6
10
12 15 17
17 18 21 21 26 27
XXVI
2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.19 2.20 2.21 2.22 2.23
2.24 2.25 2.26 2.27 2.28 2.29 2.30 2.31 2.32 2.33 2.34 2.35
List of Figures
Wind velocity, measured and simulated deﬂection vs. time for the bridge hanger 1 (left) and 2 (right) . . . . . . . . . . . . . . . . . . Width of the lockin range for bridge tie rods . . . . . . . . . . . . . . . . Measured and simulated amplitude of the displacement within and outside of the lockin range . . . . . . . . . . . . . . . . . . . . . Sample realizations of a renewal process (left) and of a pulseprocess (right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wavelength of the visible light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Climatic load on a structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Test stand for the analysis of thermal actions on concrete specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measured temperature proﬁle during a summer day . . . . . . . . . . Rainﬂow analysis of the macroscopic temperature behaviour . . . Temperature behaviour due to a sudden change in solar radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Temperature distributions determined at 16 layers within a cooling tower shell under constant external load actions . . . . . . . Eﬀect of the mean wind speed on the development of the temperature diﬀerence of a cooling tower shell . . . . . . . . . . . . . . . Frequency distribution of the total weight G of the representative lorries per 24 hours based on traﬃc data of Auxerre in France (1986) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gross vehicle and axle weight distribution of recorded traﬃc data from England, France and Germany . . . . . . . . . . . . . . . . . . . Histogram of vehicle Type 3 and approximation by two separate distribution functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of measured and theoretical values for the density function of intervehicle distances . . . . . . . . . . . . . . . . . . . . Model for the vehicles and local irregularities and power spectral density of the pavement . . . . . . . . . . . . . . . . . . . . . . . . . . . Eigenvalues of the ﬁrst mode of steel and concrete Bridges [169] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cumulative frequency of the action eﬀects for diﬀerent vehicle speeds [530] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inﬂuence of the quality of the pavement on the dynamic ampliﬁcation factor ϕ [530] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inﬂuence of the span length and the number of loaded lanes on the dynamic ampliﬁcation factor ϕ . . . . . . . . . . . . . . . . . . . . . . Additional dynamic factor Δϕ taking into account irregularities of the pavement [9] . . . . . . . . . . . . . . . . . . . . . . . . . . . Determination of the characteristic values of the action eﬀects from the random generations of loads . . . . . . . . . . . . . . . . . Load Model 1 according to Eurocode 12 . . . . . . . . . . . . . . . . . . . . Comparison of the Load Model 1 in Eurocode 2 with the characteristic values obtained from real traﬃc simulations . . . . .
30 32 33 34 38 38 40 41 42 43 44 45
48 48 49 51 53 54 55 56 56 57 58 59 59
List of Figures
2.36 2.37 2.38 2.39 2.40 2.41 2.42 2.43 2.44 2.45 2.46 2.47 2.48 2.49 2.50 2.51 2.52 2.53 2.54
2.55 2.56 2.57 2.58 2.59 2.60 2.61
XXVII
Determination of the representative values and the corresponding dynamic factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Factors ψT R for frequent design situations acc. to [37] for average pavement quality with Φ(Ωh ) = 16 . . . . . . . . . . . . . . . . . . Inﬂuence of the pavement quality on the factor ΨT R for frequent design situations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Determination of stress spectra and damage accumulation due to fatigue loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fatigue strength curves for structural steel and reinforcement . . Typical examples for fatigue strength categories . . . . . . . . . . . . . . Set of lorries of Fatigue Load Model 4 in Eurocode 2 and contact surfaces of the wheels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Distribution of transverse location of centre line of vehicles and dynamic load ampliﬁcation factor near expansion joints . . . Linear damage accumulation and damage equivalent dynamic ampliﬁcation factor ϕf at . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inﬂuence of the pavement quality on the damage equivalent dynamic ampliﬁcation factor [530] . . . . . . . . . . . . . . . . . . . . . . . . . . Fatigue Load model 3 in Eurocode 12 and fatigue veriﬁcation for steel structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example for the damage equivalent factor λe [530] . . . . . . . . . . . Determination of the damage equivalent factor λ1 . . . . . . . . . . . . Factors λ1 for steel bridges given in Eurocode 32 . . . . . . . . . . . . Assumptions for the factor λ4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Damage equivalent factor λmax . . . . . . . . . . . . . . . . . . . . . . . . . . . . Development of the freight traﬃc on roads, railways and ships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Development of the number of heavy vehicles per day and relative frequency of the gross weight for articulated vehicles . . Development of the number of permits of heavy transports in Bavaria and NorthRhine Westphalia and examples for vehicles for heavy transports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Traﬃc records from the Netherlands recorded in 2006 . . . . . . . . Heavy vehicles on the basis of the modular concept (GigaLiners) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Axle spacing and allowable axle weights of ”GigaLiners” . . . . . Pressure time history at the trackside face of a 8 m high wall; at a ﬁxed position; V = 234.3 km/h, [573] . . . . . . . . . . . . . . Pressure distribution along the trackside face of a wall at two diﬀerent train speeds [573] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Full scale tests performed along the high speed line CologneRhine/Main . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Eﬀect of train speed stagnation pressure on the head pulse acting at the trackside face of a wall . . . . . . . . . . . . . . . . . . . . . . .
61 61 62 64 64 65 67 67 68 68 69 70 70 71 73 73 74 75
76 77 78 78 80 81 81 83
XXVIII
2.62 2.63 2.64 2.65 2.66 2.67 2.68 2.69 2.70 2.71 2.72 2.73 2.74 2.75 2.76 2.77 2.78 2.79 2.80
2.81 2.82 2.83 2.84
2.85 2.86 2.87 2.88
List of Figures
Pressure coeﬃcients of the head pulse from 34 passages (at the trackside wall face) at 1.65 m above track level . . . . . . . Distance between the pulse peaks and the zero crossing (ΔL1 = pressure maximum, ΔL2 = pressure minimum) . . . . . . . . . . . . Head pulse in a free ﬂow at various distances from the track axis [98] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Head pulse in the presence of a wall . . . . . . . . . . . . . . . . . . . . . . . . Load pattern over the height of the wall . . . . . . . . . . . . . . . . . . . . Variation of the time lag between maxima and minima of the head pulse over the wall height transformed to V = 300 m/s . . . Load factor for the load distribution over the height of the wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pattern of pressure coeﬃcients cp for the ICE3 train . . . . . . . . . Noise protection wall and mode shape of the 1st mode . . . . . . . . Time history of post top displacement calculated for a post in the middle of the wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Resonant ampliﬁcation of the displacement maximum vs. the natural frequency at train speeds between 200 and 300 km/h . . Resonant ampliﬁcation of the displacement minimum vs. the natural frequency at train speeds between 200 and 300 km/h . . Schematic diagram  Interaction of climate, environmental attack and damage process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Examples of reinforcement corrosion and concrete corrosion . . . Attacks on concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Surface of frost damaged concrete in situ . . . . . . . . . . . . . . . . . . . . Microcracking of cement paste(left); ESEM image of frost damaged concrete (right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Field exposure (left); Modiﬁed multiring electrode (right) . . . . Eﬀects at speciﬁc depths of water penetration(left); Dependence of Arrhenius factor b on moisture content (right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Air temperature and rainfall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Freezethaw cycle illustrated by example (left); Temperature curve during thaw phase on November 26 (right) . . . . . . . . . . . . . Exemplary illustration of the change in resistance at depth level 6.6 cm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Frequency of freezethaw cycles depending on minimum temperature (left) and maximum cooling and thawing rates (right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . External damage of concrete specimens after one winter . . . . . . Correlation between surface scaling and degree of visual damage on ﬁeld exposed specimens . . . . . . . . . . . . . . . . . . . . . . . . . Development of external damage . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of the surface scaling obtained in laboratory and in ﬁeld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
84 84 85 85 87 88 88 89 90 91 92 92 93 94 95 96 97 97
98 99 100 101
103 103 104 105 106
List of Figures
2.89 2.90 2.91 2.92 2.93 2.94 2.95 2.96 2.97 2.98 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17 3.18 3.19
XXIX
Concrete damage caused by thaumasite . . . . . . . . . . . . . . . . . . . . . Corrosion on mortar coatings in two drinking water reservoirs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sources of cyclic loading of soils . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cyclic stresses in a soil element a) due to a passing wheel load and b) due to an earthquake loading . . . . . . . . . . . . . . . . . . . Accumulation of settlement due to cyclic loading . . . . . . . . . . . . . Decomposition of a signal with varying amplitudes into packages of cycles with constant amplitude . . . . . . . . . . . . . . . . . . Distinction between uniaxial IP, multiaxial IP and OOPcycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hodograph for detrending of a strain path . . . . . . . . . . . . . . . . . . Multiaxial amplitude deﬁnition . . . . . . . . . . . . . . . . . . . . . . . . . . . . Complex strain loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Schematic stressstrain diagram of cementitous materials subjected to uniaxial compression [867] . . . . . . . . . . . . . . . . . . . . . Schematic stressstrain diagram of cementitous respectively geological materials due to tension [538] . . . . . . . . . . . . . . . . . . . . Stressdisplacement diagram of a concrete specimen subjected to cyclic tensile loading [381] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Biaxial failure envelope for concrete [467, 567] . . . . . . . . . . . . . . . Stressdisplacement diagrams obtained from triaxial compression tests for three levels of conﬁning pressure σ2 . . . . . Failure surface of concrete in principal stress space and crack patterns corresponding to diﬀerent triaxial loading conditions . . Ductile fracture surfaces of a round notched bar after 30 cycles with notch radius 2mm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (a) Void nucleation due to fracture of inclusions, (b) partition of inclusionmatrixarea, (c) void coalescence . . . . . . . . . . . . . . . . Schematic SN curves for concrete (W¨ ohler curves) . . . . . . . . . . . Fatigue fracture of concrete specimens due to cyclic compression load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Number of cycles to failure Nf for diﬀerent load levels and their variation [627] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stressstrain relation of concrete measured after diﬀerent number of cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Development of total longitudinal strain with the cycle ratio (N/Nf ) [383] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Change of secant modulus of elasticity [383] . . . . . . . . . . . . . . . . . Development of the value of the residual strength [70] . . . . . . . . W¨ ohler curves for tensile loads [207] . . . . . . . . . . . . . . . . . . . . . . . . W¨ ohler curves for ﬂexural loads [865] . . . . . . . . . . . . . . . . . . . . . . . Development of strains in tensile loading [207] . . . . . . . . . . . . . . . Development of strains in bending [662] . . . . . . . . . . . . . . . . . . . . .
108 108 110 111 111 113 115 116 117 119
125 127 127 128 128 129 130 130 131 132 132 133 134 135 135 136 137 137 138
XXX
3.20 3.21 3.22 3.23 3.24 3.25
3.26 3.27 3.28 3.29 3.30 3.31 3.32 3.33 3.34 3.35 3.36
3.37 3.38 3.39 3.40 3.41 3.42 3.43 3.44 3.45 3.46
List of Figures
Degradation process of relevant concrete properties due to tensile loadings [429] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Degradation process of relevant concrete properties due to ﬂexural loadings [866] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stiﬀness reduction by high cycle fatigue . . . . . . . . . . . . . . . . . . . . . Model for brittle damage by microcrack growth . . . . . . . . . . . . . . Stresses in a concrete slab at onesided, nonlinear cooling from the top [145] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Temperature and stress development during the ﬁrst hydration phase in restrained concrete elements [763, 145, 466] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hygric strains vs. relative humidity . . . . . . . . . . . . . . . . . . . . . . . . . Hygric strains vs. relative humidity & vs. water content . . . . . . . Hygric strains vs. surface free energy change . . . . . . . . . . . . . . . . . Hygric strains vs. surface free energy change & comparison between measured and calculated hygric strains . . . . . . . . . . . . . . Sorption isotherms vs. relative humidity . . . . . . . . . . . . . . . . . . . . Solid density vs. relative humidity . . . . . . . . . . . . . . . . . . . . . . . . . . Schematic diagram of hygric mechanisms and properties of hardened cement paste . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of macroscopic and microscopic situation of the microicelens model during the heating and cooling phase . . . . Volume fractions of constituents of hardened cement paste as a function of the water cement ratio [448] . . . . . . . . . . . . . . . . . . . Schematic illustration of the dissolution and loading induced longterm deterioration of concrete . . . . . . . . . . . . . . . . . . . . . . . . . Equilibrium states between the calcium concentration s and the ratio c/s: experimental [114, 115] and analytical [307, 308] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Decrease of compressive strength as a function of the increase in porosity resulting from calcium leaching [172] . . . . . . . . . . . . . Expansion behaviour of ﬂat mortar prisms with Portland cement during storage in sodium sulfate solution [502] . . . . . . . . Alkalisilica reaction damage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Accumulation of stress or strain, illustrated for the twodimensional case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Evolution of the electrical resistance vs. number of cycles during fatigue  plain and circular specimen . . . . . . . . . . . . . . . . . Evolution of the electrical resistance during fatigue  plain specimen blocktest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electrical potential  plane specimen . . . . . . . . . . . . . . . . . . . . . . . . Electrical potential  circular specimen . . . . . . . . . . . . . . . . . . . . . . Evolution of electrical resistance vs. crack length during fatigue  plain and circular specimen . . . . . . . . . . . . . . . . . . . . . . . Waveform parameters for a burstsignal . . . . . . . . . . . . . . . . . . . . .
138 139 139 140 141
142 144 144 145 146 146 147 148 149 151 153
154 155 158 159 161 167 167 168 168 169 170
List of Figures
3.47 3.48 3.49 3.50 3.51 3.52 3.53 3.54 3.55 3.56 3.57 3.58 3.59 3.60 3.61 3.62 3.63 3.64 3.65 3.66 3.67 3.68 3.69 3.70 3.71 3.72 3.73 3.74 3.75 3.76 3.77
XXXI
Location of the source in two dimensions . . . . . . . . . . . . . . . . . . . . Geometry of the plain specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geometry of the circular specimen . . . . . . . . . . . . . . . . . . . . . . . . . The position of AEtransducers on the plain specimen . . . . . . . . The position of AEtransducers on the circular specimen . . . . . . Rate of event counts during fatigue  plain specimen . . . . . . . . . . Rate of event counts during fatigue  circular specimen . . . . . . . Total event counts during fatigue  plain specimen . . . . . . . . . . . Total event counts during fatigue  circular specimen . . . . . . . . . Origin of acoustic emission  plain specimen . . . . . . . . . . . . . . . . . Origin of acoustic emission  circular specimen . . . . . . . . . . . . . . . Acoustic emission event counts vs. amplitude  plain specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acoustic emission event counts vs. amplitude  circular specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acoustic emission event counts vs. frequency  plain specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acoustic emission event counts vs. frequency  circular specimen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Execution of the singlestage and twostage test . . . . . . . . . . . . . Decrease and scatter of Estat at Smax /Smin = 0.675/0.10 (singlestatetests) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Decrease and scatter of Edyn at Smax /Smin = 0.675/0.10 (singlestatetests) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variation of maximal bearable number of load cycles to failure Nf [627] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measured longitudinal strain at Smax (Smax /Smin = 0.60/0.10) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stressstrain curves at diﬀerent number of cycles (Smax /Smin = 0.60/0.10) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Total strain at Smax /Smin = 0.675/0.10 . . . . . . . . . . . . . . . . . . . . Calculation of fatigue strain at Smax . . . . . . . . . . . . . . . . . . . . . . . Formation of fatigue strain (schematically) . . . . . . . . . . . . . . . . . . Fatigue strain at Smax /Smin = 0.675/0.10 . . . . . . . . . . . . . . . . . . . Correlation between the fatigue strain and the residual stiﬀness for diﬀerent load levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . Correlation between the fatigue strain and the residual stiﬀness of normal and high strength concrete . . . . . . . . . . . . . . . Correlation between the fatigue strain and the residual stiﬀness of normal and airentrained concrete . . . . . . . . . . . . . . . . Correlation between the fatigue strain and the residual stiﬀness subjected to diﬀerent aggregates in concrete . . . . . . . . . Correlation between the fatigue strain and the residual stiﬀness subjected to diﬀerent grading curves in concrete . . . . . . Light microscopy micrographs . . . . . . . . . . . . . . . . . . . . . . . . . . . .
172 173 173 174 174 175 175 176 176 177 177 178 178 179 179 181 182 183 183 184 184 186 186 187 187 188 188 189 189 190 191
XXXII
3.78 3.79 3.80 3.81 3.82 3.83 3.84
3.85 3.86 3.87 3.88 3.89 3.90 3.91 3.92 3.93 3.94 3.95 3.96 3.97 3.98 3.99 3.100 3.101 3.102 3.103 3.104 3.105 3.106 3.107
List of Figures
Load history with various rest periods [150] . . . . . . . . . . . . . . . . . Behaviour of the longitudinal strain at Smax /Smin = 0.675/0.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Related longitudinal strain at Smax /Smin = 0.675/0.10 . . . . . . . Correlation between the fatigue strain and the residual stiﬀness subjected to diﬀerent sequences of cyclic loading . . . . . Steps of exposure and measuring during CDF/CIF test [731] . . Example relationship between RDM and relative moisture uptake  concrete type 2 [610] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Internal damage due to freezethaw cycles at several depths of the specimen (left), Moisture uptake vs. number of freezethaw cycles (right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Test devices and deﬁnitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cyclic ﬂow rule (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cyclic ﬂow rule (II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Intensity of accumulation in drained cyclic element tests on soils (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Intensity of accumulation in drained cyclic element tests on soils (II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inﬂuence of the grain size distribution curve on D acc . . . . . . . . . . Undrained cyclic tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Eﬀect of cycles at σ = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Application of headed shear studs in composite bridges . . . . . . . Loaddeﬂection behaviour of headed shear studs embedded in solid concrete slabs under static loading . . . . . . . . . . . . . . . . . . Fatigue strength curve for cyclic loaded headed shear studs according [685] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Safety concept to determine the lifetime of composite structures subjected to high cycle loading . . . . . . . . . . . . . . . . . . . Tests with multiple blocks of loading . . . . . . . . . . . . . . . . . . . . . . . Tests to compare the eﬀect of the mode control  force control vs. displacement control  and the eﬀect of low temperature . . . Duration of the crack initiation phase and crack growth velocity due to very low cyclic loads [685] . . . . . . . . . . . . . . . . . . . Details of the pushout test specimen . . . . . . . . . . . . . . . . . . . . . . . Servo hydraulic actuators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Position of transducers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Development of plastic slip over the fatigue life in series S1  S4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Decrease of static strength vs. lifetime due to high cycle loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Test programme and loading parameters of the composite beam tests VT1 and VT2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Details of test beam VT1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Details of test beam VT2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
192 192 193 194 195 195
197 199 200 201 202 204 205 206 207 208 209 210 211 213 215 216 216 217 218 220 221 226 228 229
List of Figures
XXXIII
3.108 Test setup of test beams VT1 and VT2 . . . . . . . . . . . . . . . . . . . . . 3.109 Electric circuit to detect complete shear failure of headed studs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.110 Change of initial deﬂections due to cyclic loading . . . . . . . . . . . . 3.111 Loaddeﬂection behaviour of test beams VT1 and VT2 in the static tests after cyclic loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.112 Experimental determination of the reduced static strength of the steel section near midspan after high cycle preloading . . . . 3.113 Slip along the interfaces of steel and concrete after ﬁrst loading and after cyclic loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.114 Crack lengths at the stud feet after the cyclic loading phase Preparation stages for examination purposes . . . . . . . . . . . . . . . . 3.115 Representation of diﬀerent failure surfaces in the principal strain space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.116 Stressstrain diagrams for uniaxial compressive and tensile loading obtained from the damage model by Mazars . . . . . . . . 3.117 Anisotropic damage model by [604]: Illustration of the failure surface in the principal stress space, see eq. (3.29) . . . . . . . . . . . . 3.118 Deﬁnition of a local coordinate system and decomposition of the traction vector t = into the normal part tn and the tangential part tm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.119 Anisotropic elastoplastic damage model by [534]: Inﬂuence of the scalar coupling parameter β on the stressstrain diagram . . 3.120 Yield conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.121 Stressstrain relation of concrete . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.122 Discrete representation of cracks: Traction separation law of the format t = t( u ) across the crack surface . . . . . . . . . . . . . . 3.123 Strong Discontinuity Approach: Additive decomposition of the displacement ﬁeld u (equation (3.84)) . . . . . . . . . . . . . . . . . . . 3.124 Strong Discontinuity Approach: Strain ﬁeld resulting from ¯ (x) + u ˆ (x) . . . . . . . . . . . . . . . . . . . the displacement ﬁeld u(x) = u 3.125 Modelbased concept for life time assessment of metallic structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.126 Numerical and experimental data for (a) material softening and (b) ratcheting eﬀect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.127 Low Cycle Fatigue in metals: Numerical and experimental results for cyclically loaded round notched bar . . . . . . . . . . . . . . . 3.128 Low Cycle Fatigue in metals: Damage accumulation and predicted damage in a cyclically loaded round notched bar . . . . 3.129 SN approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.130 Degradation of compressive strength and sequence eﬀects . . . . . 3.131 Evaluation of the approach for sequence eﬀects . . . . . . . . . . . . . . 3.132 Rheological element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.133 Fatigue strain evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.134 Split of fatigue strains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
230 231 234 234 235 236 237 239 240 242
243 246 247 249 253 254 254 257 259 260 261 263 263 264 265 267 268
XXXIV
List of Figures
3.135 Evaluation of the split variable β fat . . . . . . . . . . . . . . . . . . . . . . . . 3.136 Kinked crack and its equivalent elliptical crack . . . . . . . . . . . . . . . 3.137 Growth of the circular crack and its equivalent elliptical crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.138 Order of the considered sequential loading . . . . . . . . . . . . . . . . . . . 3.139 Evolution of the geometry and the orientations of the equivalent elliptical crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.140 Evolution of the stiﬀness components in the principle directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.141 Specimen geometry and diﬀerent mesh patterns . . . . . . . . . . . . . . 3.142 Loadcycle curves for diﬀerent mesh patterns . . . . . . . . . . . . . . . . 3.143 Chemomechanical damage of porous materials within the Theory of Mixtures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.144 Conductivity of the pore ﬂuid D0 and macroscopic conductivity of nonreactive porous media φD0 . . . . . . . . . . . . . . 3.145 Chemical equilibrium function by G´ erard [307, 308] and Delagrave et al. [232] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.146 Microstructure, constituents and volume fractions of concrete as a partially saturated porous media . . . . . . . . . . . . . . . . . . . . . . . 3.147 Chemical material parameters k and u / r − 1 and of their dependence on the liquid saturation sl . . . . . . . . . . . . . . . . . . . . . . 3.148 Theoretical model for the prediction of the mean value of the ultimate shear resistance according [684] . . . . . . . . . . . . . . . . . . . . 3.149 Result of the statistical analysis of the results of 101 statically loaded pushout tests according to EN 1990 [16] . . . . . . . . . . . . . 3.150 Comparison of the result of the statistical analysis with the rules in current German and European standards . . . . . . . . . . . . 3.151 Preparation stages for examination purposes . . . . . . . . . . . . . . . . 3.152 Failure modes A and B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.153 Weld collar  Closeup view of the crack shown in Figure 3.152 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.154 Correlation between reduced static strength and damage at the stud feet based on the fatigue fracture area . . . . . . . . . . . . . . 3.155 Correlation between reduced static strength and damage at the stud feet based on crack lengths . . . . . . . . . . . . . . . . . . . . . . . . 3.156 Comparison of fatigue test results with the prediction in Eurocode 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.157 Model for the prediction of the fatigue life of a headed shear stud in a pushout test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.158 (a) Reduced static strength over lifetime, (b) Comparison of the reduced static strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.159 Loadslip curve of headed shear studs  load deﬂection behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.160 Eﬀect of highcycle loading on the loadslip behaviour . . . . . . . . 3.161 Elastic stiﬀness and accumulated plastic slip . . . . . . . . . . . . . . . . .
268 277 279 280 281 282 283 284 295 299 302 303 312 317 322 324 324 325 326 327 328 329 331 331 332 333 334
List of Figures
XXXV
3.162 Relationship between crack velocity, crack propagation and reduction of static strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.163 Fatigue strength and lifetime of cyclic loaded shear studs . . . . . 3.164 Comparison between the test results with the lifetime prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.165 Damage accumulation considering the load sequence eﬀects . . . . 3.166 Damage accumulation in the case of multiple block loading tests with decreasing peak loads . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.167 Comparison between the test results with the results of the lifetime prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.168 Ductility after high cycle loading . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.169 Comparison between test results and ﬁnite element calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.170 Comparison between test results and ﬁnite element calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.171 Test series S9  Eﬀect of control mode  Eﬀect of low temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.172 Failure surface of the improved material model CONCRETE . . 3.173 Comparison between the results of numerical simulations and test results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.174 Test beam VT1  Eﬀect of high cycle loading on load bearing capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.175 Cyclic behaviour of test beam VT1 . . . . . . . . . . . . . . . . . . . . . . . . . 3.176 Test beam VT2  Eﬀect of high cycle loading  Typical crack formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.177 Geometry of a tunnel lining subjected to cyclic hygral and thermal loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.178 Evolution of the crack width w of a tunnel lining subjected to cyclic hygral and thermal loading . . . . . . . . . . . . . . . . . . . . . . . . 3.179 Scalar damage measure d at the crown of a tunnel lining subjected to cyclic hygral and thermal loading . . . . . . . . . . . . . . . 3.180 Liquid saturation Sl at the crown of a tunnel lining subjected to cyclic hygral and thermal loading . . . . . . . . . . . . . . . . . . . . . . . . 3.181 Simulation of a cementitious beam exposed to calcium leaching and mechanical loading . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.182 Temporal evolution of the vertical displacement us of the cementitious beam and prediction of the collapse . . . . . . . . . . . . . 3.183 Chemomechanical analysis of a concrete panel: Conditions . . . . 3.184 Chemomechanical analysis of a concrete panel: Results I . . . . . 3.185 Chemomechanical analysis of a concrete panel: Results II . . . . . 3.186 Numerical simulation of a concrete beam aﬀected by alkalisilica reaction: Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.187 Numerical simulation of a concrete beam aﬀected by alkalisilica reaction: Results I . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
335 336 338 339 340 340 341 342 343 345 347 348 348 350 351 352 352 353 354 355 355 356 358 359 360 361
XXXVI
List of Figures
3.188 Numerical simulation of a concrete beam aﬀected by alkalisilica reaction: Results II . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.189 Numerical simulation of a concrete beam aﬀected by alkalisilica reaction: Results III . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.190 Low Cycle Fatigue Model: (a) Spherical pressure vessel, (b) Vertical displacementtime plot of the El Centro earthquake . . . 3.191 Low Cycle Fatigue Model: (a) Damage accumulation (El Centro earthquake), (b) Temporal evolution of the maximal void volume fraction f . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13
4.14 4.15 4.16 4.17 4.18 4.19 4.20 4.21
Overview of the methodological implementation of lifetime oriented design concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical modeling and general multiphysics problem . . . . . . . . Modeling and numerical analysis of multiphysics problems . . . . Illustration of isotropic Lagrange shape functions . . . . . . . . . . . Illustration of anisotropic Lagrange shape functions . . . . . . . . Computation of generalized element tensors of multiphysics pﬁnite elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sinusoidial loading of a truss member and rel. error of internal energy plotted over the number of dof . . . . . . . . . . . . . . . Modiﬁed Legendrepolynomials . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of high order shape function concepts . . . . . . . . . . . . Comparison of the structure of element vectors and matrices for the Legendre and Lagrangeconcept . . . . . . . . . . . . . . . . . 3Dpelement: deﬁnition and numbering of element vertices (Ni ), edges (Ei ) and faces (Fi ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Dpshape functions: nodal, edge, face and internal modes for diﬀerent polynomial degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . Structure types, corresponding classical ﬁnite element models and 3Dp ﬁnite element models with spatially anisotropic approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hygrothermomechanical loading of a structural segment, Fieldwise anisotropic discretization using the pFEM . . . . . . . . . Discretization of the standard structures (truss, slab, shell) into an inﬁnite numbers of elements . . . . . . . . . . . . . . . . . . . . . . . . Relative reduction of system nodes/dof for diﬀerent structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Strategy for solving nonlinear vector equation ri (u) = r . . . . . . Control of load factor and NewtonRaphson iteration . . . . . . Algorithmic setup of the load controlled NewtonRaphson scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Illustration of arclength methods and predictor step calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Algorithmic setup of the arclength controlled NewtonRaphson scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
362 363 363
364
366 375 376 381 382 387 388 390 391 392 393 395
396 398 399 402 404 404 406 407 410
List of Figures
4.22 4.23 4.24 4.25 4.26 4.27 4.28 4.29 4.30 4.31 4.32 4.33 4.34 4.35 4.36 4.37 4.38 4.39 4.40 4.41
4.42
4.43 4.44 4.45
XXXVII
Design of Newmark type time integration schemes . . . . . . . . . . Illustration of Newmark and generalized midpoint approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Algorithmic setup of Newmarkα schemes including error controlled adaptive time stepping . . . . . . . . . . . . . . . . . . . . . . . . . . Galerkin time integration schemes . . . . . . . . . . . . . . . . . . . . . . . . Algorithmic setup of discontinuous and continuous Galerkin time integration schemes . . . . . . . . . . . . . . . . . . . . . . . . Modular concept for multiphysics ﬁnite element programs . . . . . Example geometry and warpingbased error criterion . . . . . . . . . Twoelement example with two hanging nodes . . . . . . . . . . . . . . . Beam 1: Geometry and boundary conditions . . . . . . . . . . . . . . . . . Beam 1: Loaddisplacement curve for tolerr = 10−5 and crit1 (various nGP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Beam 1: Diﬀerent states of mesh reﬁnement (Q1SPs/o, 16El.), contours: accumulated plastic strain . . . . . . . . . . . . . . . . . . . . . . . . Beam 1: Loaddisplacement curve and number of elements for tolerr = 10−7 and crit1 (various nGP0) . . . . . . . . . . . . . . . . . . . Beam 1: Loaddisplacement curve and number of elements for diﬀerent tolerances and crit2 (Q1SPs/o, nGP0 = 16) . . . . . . . Beam 2: Loaddisplacement curve and number of elements for diﬀerent tolerances and crit2 (Q1SPs/o, nGP = 16) . . . . . . . . Beam 2: Diﬀerent states of mesh reﬁnement (Q1SPs/o, 16 El.), contours: accumulated plastic strain . . . . . . . . . . . . . . . . . Plate 1: Geometry and boundary conditions . . . . . . . . . . . . . . . . . Plate 1: Loaddisplacement curve and number of elements for diﬀerent tolerances and crit2 (Q1SPs, nGP = 8) . . . . . . . . . . . . . . Plate 1: Loaddisplacement curve for diﬀerent tolerances and crit2 (Q1SPs, nGP = 8) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plate 1: Diﬀerent states of mesh reﬁnement (Q1SPs/o, 16 El.), contours: accumulated plastic strain . . . . . . . . . . . . . . . . . Plate 1: Loaddisplacement curve and number of elements for diﬀerent load steps and crit2 (Q1SPs/o, nGP = 8, tolerr = 0.01) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plate 1: Loaddisplacement curve and number of elements for diﬀerent load steps and crit2 (Q1SPs, nGP = 8, tolerr = 0.0001) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Illustration of h and pmethod error estimates and indicators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Algorithmic setup for the error controlled adaptive time integration by Newmarkα schemes . . . . . . . . . . . . . . . . . . . . . . . Algorithmic setup for the error controlled adaptive time integration by Newmarkα or pGalerkin methods and hmethod error estimates/indicators . . . . . . . . . . . . . . . . . . . . . . . .
413 414 417 418 423 425 432 434 435 435 436 437 438 438 439 440 440 441 441
442
442 443 447
447
XXXVIII
4.46
4.47 4.48 4.49 4.50 4.51 4.52 4.53 4.54 4.55 4.56 4.57 4.58 4.59 4.60 4.61 4.62 4.63 4.64 4.65 4.66 4.67 4.68 4.69 4.70
4.71 4.72 4.73 4.74 4.75 4.76 4.77 4.78
List of Figures
Algorithmic setup for the error controlled adaptive time integration by pGalerkin methods and pmethod error estimates/indicators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Function to be approximated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Approximation of equation (4.147) . . . . . . . . . . . . . . . . . . . . . . . . . Normal and tangential vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Four crack tip functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Crack with one kink . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Crack after mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiple kinked crack . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiple kinked crack after the ﬁrst mapping . . . . . . . . . . . . . . . . ˆ.............................. Point x and mirrored point x Strain ε from equation (4.173) for the integral (4.175) . . . . . . . . Number of integration points used in the numerical integration of (4.174) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Strain ε from equation (4.173) for the integral (4.177) . . . . . . . . Number of integration points used in the numerical integration of (4.176) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Strain ε from equation (4.173) for the integral (4.179) . . . . . . . . Number of integration points used in the numerical integration of (4.178) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Strain ε from equation (4.173) for the integral (4.181) . . . . . . . . Number of integration points used in the numerical integration of (4.180) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Strain ε from equation (4.173) for the integral (4.183) . . . . . . . . Number of integration points used in the numerical integration of (4.182) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Strain ε from equation (4.173) for the integral (4.185) . . . . . . . . Number of integration points used in the numerical integration of (4.184) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tension test conﬁguration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Displacements ux for the deformed system using bilinear shape functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Displacements ux for the deformed system, left: using biquadratic shape functions, right: using quadratic hierarchical shape functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Diﬀerences of displacements inside the 1st blending element . . . Diﬀerences of displacements inside the 2nd blending element . . Diﬀerences of displacements inside the 3rd blending element . . . Diﬀerences of displacements inside the 4th blending element . . . Diﬀerences of displacements inside the 5th blending element . . . Numerical integration in the context of XFEM: Subdivision of the continuum element into six subtetrahedrons . . . . . . . . . . . Separation of a subtetrahedron by a plane crack segment . . . . . C0 crack plane evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
448 450 451 452 453 454 456 456 457 458 462 462 463 463 465 465 466 466 467 467 468 468 469 470
470 471 471 472 472 473 475 475 476
List of Figures
4.79 4.80
XXXIX
Deﬁnition of the crack plane by point P and normal vector n . . Constant strain triangular element cut by means of a planar internal boundary ∂s Ω; see [745] . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.81 Enhanced discontinuous displacement ﬁeld ru (Hs − ϕ): (a) bilinear approximation (2 nodes in Ω + ); (b) biquadratic approximation (1 node in Ω + ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.82 Numerical study of a notched concrete beam: dimensions (in [cm]) and material parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.83 Numerical study of a notched concrete beam using the proposed multiple crack concept and the rotating crack approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.84 Sketch for the computation of the SIF for a kinking crack with r → 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.85 Schematic ﬁgure for the calculation of the SIF with constant radius for kinking cracks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.86 Sketch of KII (left) and KII  (right) depending on the angle θ for a three point bending test . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.87 Energy function Πtot for a three point bending test . . . . . . . . . . 4.88 Crack simulation of a double notched slab: System, material data and ﬁnite element mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.89 Crack simulation of a double notched slab: Visualization of the crack topology by the φ = 0level set . . . . . . . . . . . . . . . . . . . . 4.90 Crack simulation of a double notched slab: Comparison of crack topology and of loaddisplacement curves . . . . . . . . . . . . . . 4.91 Bumerical investigation of crack propagation of an anchor pullout test: System and ﬁnite element mesh (N E = 996) . . . . 4.92 Numerical investigation of crack propagation of an anchor pullout test: Crack topology and displacement u3 in pullout direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.93 Numerical investigation of crack propagation of an anchor pullout test: Stress σ 33 at the beginning and the end of the crack process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.94 Numerical investigation of crack propagation of an anchor pullout test: Loaddisplacement curve . . . . . . . . . . . . . . . . . . . . . . 4.95 Concept for the eﬃcient simulation of dynamic, partially damaged structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.96 Decomposition of the structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.97 Geometry and loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.98 Exploded view of the bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.99 Damage evolution in the largest two hangers . . . . . . . . . . . . . . . . 4.100 Displacement in X2 direction in point B . . . . . . . . . . . . . . . . . . . . 4.101 Mean relative displacementbased error in point B . . . . . . . . . . . 4.102 Comparison of a pure implicit and an explicit calculation of accumulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
477 481
482 486
488 491 491 492 493 494 495 495 496
497
497 498 501 507 513 514 515 516 516 518
XL
List of Figures
4.103 General deﬁnition of the failure domain depending on scattering resistance (R) and stress (S) values . . . . . . . . . . . . . . . 4.104 Standardization of an exemplary 2D joint distribution function for a subsequent FORM/SORM analysis . . . . . . . . . . . . 4.105 Comparison of Latin Hypercube Sampling and MonteCarlo Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.106 Parallel execution of stochastically independent DCMCS of fatigue analyses on a distributed memory architecture [824] . . . 4.107 Parallel software framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.108 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.109 Damage equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.110 Singular values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.111 1’st eigenfrequency and mode shape . . . . . . . . . . . . . . . . . . . . . . . . 4.112 2’nd eigenfrequency and mode shape . . . . . . . . . . . . . . . . . . . . . . . 4.113 3’rd eigenfrequency and mode shape . . . . . . . . . . . . . . . . . . . . . . . . 4.114 4’th eigenfrequency and mode shape . . . . . . . . . . . . . . . . . . . . . . . . 4.115 Cut modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.116 Optimization topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.117 The new 3series convertible . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.118 3series convertible with battery . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.119 Battery as vibration absorber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.120 FE model of the shaker test arrangement . . . . . . . . . . . . . . . . . . . 4.121 Measured acceleration data for the ydirection . . . . . . . . . . . . . . . 4.122 Power spectral density function of the resulting von Mises stress for the elements of Figure 4.119, load direction y . . . . . . . 4.123 Dirlik distribution function of the stress amplitudes . . . . . . . . 4.124 Typical stress picture for load in ydirection (Time History Analysis) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.125 Expected life time in arbitrary time units for the Time History calculation (acceleration load in ydirection) . . . . . . . . . . 4.126 Hygromechanically loaded concrete shell structure: System geometry and material data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.127 Hygromechanically loaded concrete shell structure: Hygral boundary conditions of the inner and outer surface of the shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.128 Hygromechanically loaded concrete shell structure: Finite element mesh of the numerical analysis . . . . . . . . . . . . . . . . . . . . . 4.129 Hygromechanically loaded concrete shell structure: Deformation and stresses due to dead load . . . . . . . . . . . . . . . . . . 4.130 Hygromechanically loaded concrete shell structure: Distribution of the saturation Sl . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.131 Hygromechanically loaded concrete shell structure: Damage evolution at the support area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.132 Hygromechanically loaded concrete shell structure: Damage zone and accelerated transport process in the area of cracks . . .
529 532 536 545 561 562 563 564 564 565 565 566 566 570 573 574 574 575 576 577 579 581 582 584
584 585 586 587 588 588
List of Figures
4.133 Hygromechanically loaded concrete shell structure: Distribution of saturation Sl and damage variable d across the shell thickness (I) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.134 Hygromechanically loaded concrete shell structure: Distribution of saturation Sl and damage variable d across the shell thickness (II) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.135 Calcium leaching of a cementitious bar and a cementitious beam: Geometry, FE mesh and chemical loading history . . . . . . 4.136 Calcium leaching of a cementitious bar: Numerical results obtained from the cG(1) method . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.137 Calcium leaching of a cementitious bar: Numerical results and time integration error obtained from adaptive Newmark integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.138 Calcium leaching of a cementitious bar: Time histories c(t, X1 )/c0 obtained from dG(p)integration (t [108 s], X1 [mm]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.139 Calcium leaching of a cementitious bar: Time histories c(t, X1 )/c0 obtained from cG(p)integration (t [108 s], X1 [mm]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.140 Calcium leaching of a cementitious bar: Spatial local and global error estimates for Newmark time integrations . . . . . . . . 4.141 Calcium leaching of a cementitious bar: Logarithm of error estimates eΔt/5 for dGmethods with diﬀerent time steps Δt . . . 4.142 Calcium leaching of a cementitious bar: Logarithm of error estimates ep/p+1 for dGmethods with diﬀerent time steps Δt . . 4.143 Calcium leaching of a cementitious bar: Logarithm of error estimates ep/p+1 and eΔt/5 for cGmethods with diﬀerent time steps Δt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.144 Calcium leaching of a cementitious bar: Average relative errors of the Newmark method and Galerkin methods . . . . . 4.145 Calcium leaching of a cementitious beam: Numerical results obtained from cG(1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.146 Calcium leaching of a cementitious beam: Investigation of the oscillations in the results of cG(1) and cG(2)solutions . . . . 4.147 Calcium leaching of a cementitious beam: Investigation of the robustness of the cG(1)solution for small Tc . . . . . . . . . . . . 4.148 Pictures of damaged road bridge in M¨ unster (Germany) and correspondent FE models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.149 Reﬁned FE models of a connecting plate and the correspondent welding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.150 Eﬀective stress values of a connecting plate under a constant rod deﬂection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.151 Representative surface of partial damage values for varying wind and initial displacements at the critical tie rod . . . . . . . . . .
XLI
589
590 591 593
595
596
597 598 599 600
601 603 604 605 606 607 608 610 611
XLII
List of Figures
4.152 Timedependent evolution of the failure probability of critical material points in the welding region and the bulk material . . . 4.153 Optimization model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.154 Optimization results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.155 The road bridge at H¨ unxe (Germany) shortly before its deconstruction in 2006 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.156 Location of prestressing tendons and crack pattern observed on the bridges main girders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.157 Location of drilling cores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.158 Comparison of stressstrain curves between bridge concrete and laboratory concretes with diﬀerent strengths [193] . . . . . . . . 4.159 LMmicrograph of insitu concrete . . . . . . . . . . . . . . . . . . . . . . . . . 4.160 Total longitudinal (left) and fatigue strain (right) at Smax . . . . . 4.161 Correlation between fatigue strain and the residual stiﬀness for Smax /Smin = 0.675/0.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.162 Three dimensional Finite Element model of the road bridge at H¨ unxe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.163 Applied corrosion model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.164 Modiﬁed SN curves for steel and fatigue damage evolution function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.165 Higher order statistical moments . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.166 Validation of input data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.167 Evolution of compressive strength and histogram of concrete strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.168 Random ﬁeld dependency on correlation length and eigenvalues used for reconstruction of correlation matrix . . . . . . 4.169 Load deﬂection diagram and time deﬂection diagram 3D . . . . . . 4.170 Load deﬂection curves and lifetime distribution and estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.171 State space model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.172 Impulse excitation in laboratory . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.173 Comparison between measured signals and signals from identiﬁed model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.174 Cantilever bending beam used for experiments in laboratory . . . 4.175 Drawing from the cantilever bending beam with the location of saw cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.176 Markov parameters for damage detection . . . . . . . . . . . . . . . . . . . 4.177 Bridge near H¨ unxe / Germany (span: 62.5m) . . . . . . . . . . . . . . . . 4.178 System modiﬁcation: hanger cut through . . . . . . . . . . . . . . . . . . . . 4.179 Torsional mode from reference system and after cut hanger . . . . 4.180 Recalculation of a centrifuge model test of Helm et al. [365] . . . 4.181 Parametric studies of shallow strip foundations under cyclic loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.182 FE calculations with stochastically ﬂuctuating void ratio . . . . . .
612 613 614 616 617 618 620 621 622 623 624 626 627 628 628 629 632 633 633 636 639 639 641 641 642 643 643 644 646 647 648
List of Figures
4.183 FE calculation of vibratory compaction and bridge settlements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.184 Calculation of pore water pressure accumulation due to earthquake loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.185 FE calculation of a geogridreinforced embankment . . . . . . . . . . . 4.186 FE calculation of a monopile foundation of an oﬀshore wind power plant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 5.2 5.3 5.4 5.5 5.6
Reinforced concrete column under fatigue loading . . . . . . . . . . . . Loadcarryingcapacity and response surface method . . . . . . . . . Timedependent hazard function and timedependent reliability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multilevel system approach followed during the lifetime analysis of the arched steel bridge [826] . . . . . . . . . . . . . . . . . . . . . Multiscale modeling and analysis of fatiguerelated structural problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of resulting timedependent failure probabilities of the researched connection plate . . . . . . . . . . . . . . . . . . . . . . . . . .
XLIII
649 650 651 651 654 654 655 656 657 658
List of Tables
2.1 2.2 2.3 2.4
2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14
3.1 3.2 3.3
Conversion of the wind data of the observation station at the airport of Hannover into data for the building location . . . . . . . . 20 Determination of a reduced characteristic suction force on the fa¸cade element after Figure 2.8 . . . . . . . . . . . . . . . . . . . . . . . . . 24 Statistical parameters of the traﬃc records of Auxerre (1986) . . 49 Relation between gross weight of the heavy vehicles and the axle weights of the lorries of types 1 to 4 in % (mean values and standard deviation) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 Distance of axles in [m] of the diﬀerent types of vehicles (mean values and standard deviation) . . . . . . . . . . . . . . . . . . . . . . 50 Statistical parameters of the corrected static traﬃc records of Auxerre (1986) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Diﬀerent crosssections and traﬃc types for the random generations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 Traﬃc data of diﬀerent locations and characteristic values of gross and axle weight [720] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Diﬀerent design situations and corresponding return periods and fractiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 Factors Ψ for the determination of the representative values for serviceability limit states acc. to [9] . . . . . . . . . . . . . . . . . . . . . 63 Traﬃc categories acc. to Eurocode 12 . . . . . . . . . . . . . . . . . . . . . . 66 Statistical parameters of the traﬃc records at highway A61 (2004) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Relation between gross weight of the heavy vehicles and the axle weights of the lorries of types 1 to 5 (mean values) . . . . . . . 76 Readings: winter 05/06 and winter 06/07; ﬁeld station Meißen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Classiﬁcation of pore sizes in concrete according to [724] . . . . . . 151 Inﬂuences on the degree of chemical attack . . . . . . . . . . . . . . . . . . 152 Changes of concrete properties due to cyclic loading . . . . . . . . . . 185
XLVI
3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17 3.18 3.19 3.20 3.21 3.22 3.23 3.24 3.25 3.26 3.27 3.28
4.1 4.2 4.3
List of Tables
Crack characteristics at certain number of cycles Smax /Smin = 0.675/0.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Correlation between frost suction and internal damage due to freezethaw testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary of the single level tests . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary of the tests with multiple blocks of loading . . . . . . . . . Mean values of material properties of concrete according to EN 2061 [12] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mean values of material properties of steel members . . . . . . . . . . Average test results per stud . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Loading parameters and results of the tests with two blocks of loading (series S5) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Loading parameters and results of the tests with four blocks of loading (series S6) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Average test results per stud in series S9 . . . . . . . . . . . . . . . . . . . . Measured mean values of the peak load and the load range at discrete number of load cycles in tests S9 4 . . . . . . . . . . . . . . . Loading parameters and block lengths in tests S9 5 . . . . . . . . . . . Average test results per stud in series S11 and S13 . . . . . . . . . . . Mean values of material properties of concrete according to EN 2061 [12] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mean values of material properties of steel members . . . . . . . . . . Main test results of beams VT1 and VT2 . . . . . . . . . . . . . . . . . . . Parameter of the elastoplastic micropore damage model for 20MnMoNi55 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Low Cycle Fatigue in metals: Number of load cycles until failure obtained . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Characteristics of the applied sequential loading . . . . . . . . . . . . . Summary of the functions, material constants and reference quantities of the highcycle model . . . . . . . . . . . . . . . . . . . . . . . . . . Summary of the statically loaded pushout tests with decisive criterion “failure of the concrete” (tests 1  27) . . . . . . . . . . . . . . Summary of the statically loaded pushout tests with decisive criterion “failure of the concrete” (tests 28  58) . . . . . . . . . . . . . Summary of the statically loaded pushout tests with decisive criterion “shear failure of the stud” . . . . . . . . . . . . . . . . . . . . . . . . . Result of the statistical analysis according EN 1990, Annex D [16] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mean values of the crack length ah (see Figure 3.155) in test series S11 and S13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
191 197 213 214 218 219 219 222 222 223 223 224 225 231 232 233 260 261 281 314 319 320 321 323 337
Multidimensional Lagrange shape functions . . . . . . . . . . . . . . . 382 Total number of geometric entities (vertices, edges, faces) of the discretizations with an inﬁnite number of elements . . . . . . . . 399 Convergence criteria of iterative solution methods . . . . . . . . . . . . 405
List of Tables
4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14
4.15
4.16 4.17
4.18
4.19 4.20 4.21 4.22
XLVII
Comparison of iteration methods . . . . . . . . . . . . . . . . . . . . . . . . . . . Constraints and load factor increments of selected arclength methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Error indicators for Newmark type time integration schemes for nonlinear second order initial value problems . . . . . . . . . . . . Error indicators for Newmark type time integration schemes for nonlinear ﬁrst order initial value problems . . . . . . . . . . . . . . . Equivalent square sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modal Assurance Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . GaussNewton (cp /cd =800/1smm) . . . . . . . . . . . . . . . . . . . . . . . GaussNewton iteration (cp /cd =1400/1mm) . . . . . . . . . . . . . . . Results for an early design proposal . . . . . . . . . . . . . . . . . . . . . . . . Standard parameter set [307, 454, 457] . . . . . . . . . . . . . . . . . . . . . . Calcium leaching of a cementitious bar: Average relative errors of the Newmark method, discontinuous Galerkin methods and continuous Galerkin methods . . . . . . . . . . . . . . . . Type of random variables (RV) included in the reliability problem used to describe the scatter of wind load parameters as well as material properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparison of resulting runtime values analyzing the connecting plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dynamic elastic moduli Edyn (mean) and their standard deviations (SD) of the concrete after a service life of 50 years . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relevant mechanical concrete properties Estat , u and fc (mean values) as well as their standard deviations (SD) after a service life of 50 years . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Number of elements of structural members . . . . . . . . . . . . . . . . . . Determination of compressive strength at time of construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concrete strength grades according to German standards . . . . . Summary of the results of the FE calculations of strip foundations under cyclic loading . . . . . . . . . . . . . . . . . . . . . . . . . . .
406 409 445 446 514 569 571 571 582 592
602
609 615
619
619 624 629 630 647
1 LifetimeOriented Design Concepts
Authored by Friedhelm Stangenberg
1.1 LifetimeRelated Structural Damage Evolution Authored by Friedhelm Stangenberg Structures deteriorate during their lifetimes, e.g. their original quality decreases. In terms of structural safety, this reduces the original safety margin, a process, which also can be described as an increase of structural damage. If, in such deterioration, the safety parameter decreases below the admissible safety limit, or the structural damage parameter increases beyond the admissible damage limit, then the structural service life will be terminated. If the failure safety value or the structural damage parameter both reach unity, the structure (theoretically) will fail. The initial structural properties must have suﬃcient reserves, in order to compensate future reductions of safety against failure and of safety against reaching the limits of serviceability. The ﬁnal structural properties, at the end of the service life or at the end of the relevant inspection interval, respectively, must include a minimum resistance safety, a minimum serviceability level, and other minimum qualities. Lifetimerelated deteriorations can happen in various forms and can consist of various components. For example for structural concrete, the lifetimerelated aspects with inﬂuence on the long time structural behavior are listed in Figure 1.1. Deteriorations therein can be induced mechanically, e.g. by load cycles leading to fatigue eﬀects. They can also be induced nonmechanically, e.g. by corrosion or other chemical processes. In combination, deterioration eﬀects can be superimposed by addition, if there is no interaction between them. In case of interactions, the superposition can be more than additive because of ampliﬁcation eﬀects due to inﬂuences
2
1 LifetimeOriented Design Concepts
Appropriate Quality Assurance for structural design, detailing and execution QA is a very important overhead necessity, also for lifetimeoriented design concepts, in order to eliminate big mistakes and big errors a priori, as well as to make sure that certain tolerable deviations of structural qualities are not exceeded. This is a fundamental requirement for servicelife control.
Durability Integrated from outside into SFB 398 models and concepts Resistance against abrasion
Chemical durability against substances penetrating from outside
Own basic SFB 398 research
particularly: corrosion protection of reinforcing steel in concrete: • carbonation • penetration of chlorides • further deteriorations
Resistance against frost (thaw cycles)
Resistance against fatigue (cyclic loading)
Simulations for combined mechanical and chemical processes and propagations
Integrated design concepts
Fig. 1.1. Lifetimerelated aspects of structural concrete
between each other. E.g. the longtime increase of concrete crack widths leads to increased penetration of chemically aggressive media, through the cracks
damage phenomena d
mechanically induced degradation
d
Ti
d1
T
Te
Ti d
Te
interactions included d2
d2
Te
Ti : intermediate time Te : end of observation time Fig. 1.2. Evolution of degradation
d1
T Ti
T
Te
interaction
nonmechanically induced degradation, e.g. corrosion
Ti
without interactions d2
d1
d
superimposed
T
1.2
TimeDependent Reliability of Ageing Structures
3
into the structure, causing increased chemical deteriorations, e.g. increased corrosion in the reinforcing steel. On the other hand, this leads to increased steel weakening and, thus, viceversa to a higher ampliﬁcation of concrete crack widths, and so on. Therefore, combinations of interactive eﬀects can lead to additional deteriorations, which would be underestimated by additive superposition only. This is illustrated, in principle, in Figure 1.2.
1.2 TimeDependent Reliability of Ageing Structures and Methodological Requirements Authored by Friedhelm Stangenberg In the beginning of the structural lifetime, the safety against failure and against losses of other important structural qualities must have suﬃcient reserves. In the early lifetime, maybe e.g. concrete posthardening may improve the safety situation for a while, but later on, deteriorations lead to safety reductions. At the end of the planned service period, a remaining minimum safety is still required. This time variant safety problem or reliability problem, respectively, is presented in Figure 1.3, in principle, where the time histories of resistances and actions, together with their statistical distributions, are plotted in relation to each other. Degradations of the resistances and maybe certain increases of the actions eﬀect timedependent safety losses. For analytical predictions of these developments, methods for timedependent stochastic calculations are needed. resistance R
R, S
actions S
R(t1 ) R(td )
Safety margin (fractilebased)
S(t1 )
S(td )
time t t1
td = planned service period
Fig. 1.3. Timedependent reliability of structures
mean lifetime (50% probability)
4
1
LifetimeOriented Design Concepts
R, S R(t1 )
upgrading by repairs
resistance R actions S
¯ 1) R(t safety margin (fractilebased)
S(t1 )
t1
R(td )
S(td )
time t td = planned service period
Fig. 1.4. Timedependent reliability of structures with upgrading by repairs
In Figure 1.4, two alternatives of servicelife control are compared. The planned service period td can be obtained, by starting on a high level of structural reserves R(t1 ), high enough to reach td , with enough remaining safety, and without repairs during the full service life. The other alternative is: to ¯ 1 ), but to upgrade by restart at a lower level of initially invested reserves R(t pairs, i.e. by additional investments, before td , before the minimum acceptable safety is undergone. The success at time td may be the same for both alternatives. The second alternative means initial savings, but additional investments later, maybe combined with temporary restrictions or losses of use during the repairs. It depends on the special conditions, what alternative should have preference.
1.3 Idea of WorkingLife Related Building Classes Authored by Friedhelm Stangenberg Current design standards do not provide a satisfactory basis or procedure to ensure expected structural lifetimes. These may vary from only a few years— for temporary structures—to more than a century for tunnels, dams of water reservoirs, or nuclear repositories. There is an urgent demand for handling this wide spectrum of lifetimes, in structural design and maintenance. An appropriate diﬀerentiation of the design service lives of diﬀerent building classes is necessary. A proposal for such a diﬀerentiation of expected working lives is given in Figure 1.5.
1.4 Economic and Further Aspects of ServiceLife Control
5
expected working life [years] ≥ 150
dams of water reservoirs
≥ 80
bridges
≥ 60
residential and buisness buildings
30–40
industrial buildings
≈ 10
temporary buildings
Fig. 1.5. Workinglife related building classes
1.4 Economic and Further Aspects of ServiceLife Control Authored by Friedhelm Stangenberg Servicelife control is successful, if all initial and later investment costs are minimized, and if all aspects of sustainability are respected, from the beginning of the construction, over all the lifetime with perhaps intervals of repair, including ﬁnal removal or perhaps recycling, revitalization etc. Lifetimeoriented strategies aim at successful investment economy combined with sustainability. Building investment costs comprise • • • •
costs of initial construction, maybe costs of periodic inspection, of maintenance and strengthening, costs consequent to temporary losses or restrictions of use (in cases of interruptions for maintenance), costs of ﬁnancing for the initial construction and, in case, for later maintenance and repair.
6
1 P
LifetimeOriented Design Concepts
costs constant
designed for no maintenance
service life (without maintenance measures)
P
costs
savings at the beginnig
inspections and maintenance (eventually temporary loss of use)
Fig. 1.6. Service Life control and economic aspects
Life Cycle Assessment of Structures via Innovative Monitoring
Materials and Structures in Revitalization of Buildings Fig. 1.7. Related Collaborative Research Centers
Aspects of sustainability comprise • • •
saving of natural resources, e.g. by economizing raw materials, energy etc., prevention of future negative consequences and remains as well as harmony with nature and human life qualities, now and in future.
For the partial aspects of economic servicelife control see the Figure 1.6. Servicelife control with high probability of success, in future will also bring up new ideas in the ﬁeld of warranty. Duration time of warranty for structural qualities is diﬀerent in diverse countries (according to warranty laws in European countries: 5 or 10 years or other). New law aspects will perhaps follow in new design concepts making a successful service life more reliable. Duration time of warranty according to law should be in correlation with the degree of realizability of structural
1.5 Fundamentals of LifetimeOriented Design
7
lifetimes. The progress in lifetimeoriented design concepts can contribute to an international harmonization of warranty law.
1.5 Fundamentals of LifetimeOriented Design Authored by Friedhelm Stangenberg Current structural design concepts are oriented towards serviceability as well as towards safety against failure. They are based on structural virgin states, largely excluding predamage as well as later damage accumulation. Design standards use limit state formats in terms of load scenarios, material behavior, and required (partial) safeties. With respect to servicelife control, such formats should include quantiﬁed damage predictions and assumptions for future tendencies of relevant action scenarios. Such anticipations are necessary for realistic simulations of future lifetime inﬂuences. Thus, expected or monitored structural safety and serviceability evolutions can be described properly. Such concepts are created presently and named lifecycle design. Methods for describing damage evolutions, resistance alterations, increases of actions etc. are needed as well as simulations techniques and methods for the estimation of the reliability of structures with respect to these phenomena. This book provides contributions to these topics, which form the basis for lifetimeoriented design.
2 DamageOriented Actions and Environmental Impact on Materials and Structures
Authored by R¨ udiger H¨ oﬀer Mechanical loading and ambient actions on civil engineering structures and components cause lifetimerelated deteriorations. Not the rare extreme loading events are in the ﬁrst place responsible for the evolution of structural degradation but the ensemble of load eﬀects during the lifetime of the structure. It is of major importance to have models at hand which adequately reﬂect the experienced time histories of impacts, and which can include justiﬁed predictions of future trends. Leading types of loading and loadeﬀects with relation to mechanical fatigue as well as damages due to hygrothermal and chemical impacts are considered in this chapter. Selected contributions from wind and temperature eﬀects with certain meteorological characteristics as well as from traﬃc loads on roads and railway lines are modeled as typical examples of contributions to mechanically induced degradations of structures. A speciﬁc aspect is the permanent settlement of soil due to highcyclic, longterm loading, for which novel representations are developed. The attack of freezethaw circles in diﬀerent environments and of chemical impacts leading to solving, swelling and leaching processes in concrete including principle interactions are discussed as examples for the main types of nonmechanically induced degradations.
2.1 Wind Actions Authored by R¨ udiger H¨ oﬀer Windinduced cyclic motions of structures can generate deterioration of constructions and materials. It is therefore required to check the exposure of structures and structural components regarding the probability and accumulation of such damages. Cyclic responses due to wind buﬀeting as one of the aerodynamic loading mechanisms are evaluated based on the concept of gust response factors. In general gustresponse factors relate equivalent static loads
10
2 DamageOriented Actions and Environmental Impact
12
4.0
W [kN]
a[m]
a[m] 0.002
8
t[s]
4 0
20
40
60
2.0
0.001
80
20 0
4
20
40
60
40
60
80
80
8 2.0
12
f=0.18 Hz
16
(a)
(b)
f=3.56 Hz
(c)
Fig. 2.1. Typical wind load process (a), and related low frequency (b) and high frequency (c) response of a structure [572]
to maximal dynamic eﬀects. In the presented approach frequently repeated gust eﬀects of lower levels than the extreme eﬀect with a return period of 50 years are modeled using the statistical distributions of ensembles of registered wind speeds. The inﬂuence of the wind direction on a structural failure probability is included via advanced directional factors. Resonant vortex excitation of slender cylindrical structures is a classical aerodynamic interaction mechanism which can cause highcyclic responses with large amplitudes. A time domain model is applied and validated using data from wind tunnel investigations and from an experiment in full scale. A novel representation in the micro and the macro time domain is developed. The succession of damage events is modeled through the adaption of renewal processes and pulse processes. 2.1.1 Wind Buﬀeting with Relation to Fatigue Authored by R¨ udiger H¨ oﬀer Buﬀeting loading on structures consist of mean and ﬂuctuating components. The mean wind load is the static load component, which is related to the mean wind speed in a deterministic manner. The second, ﬂuctuating load component is primarily due to wind gustiness. Both parts are required for the evaluation of aerodynamic forces. The time histories of forces are derived from a superposition of the eﬀect of wind buﬀeting and of bodyinduced turbulence, from vortex forces and from aeroelastic interaction forces (s. e.g. [200]). The forces vary randomly in space and time. Time domain or frequency domain methods are available to calculate the stochastic response. However, they are cumbersome in view of the input data required and the computer time needed. For practical design it is suﬃcient to apply equivalent static gust wind loads.
2.1 Wind Actions
11
They are based on the socalled gust response factor G that incorporates the most adverse gust eﬀect on a structural response, which dominates in the design, the socalled leading response. Aeroelastic oscillations, such as galloping or ﬂutter, are usually not object of a fatigue analysis as the associated, often continuously growing oscillation amplitudes can lead rapidly to a structural failure from an overload breakage. Structures which are prone to such type of excitations are dynamically detuned, damping devices or other dissipative mechanism are arranged. An important exception is the structural oscillation due to vortex resonance which  depending on the damping ration  can generate resonant amplitudes of diﬀerent levels and also the lockin phenomenon due to synchronized vortex seperations (s. e.g. [740]). Such fatigue behaviour must always be analysed. Often scruton coils against regular vortex separations are installed. In contrast, gustinduced oscillations of structures are unavoidable if the structure exhibits eigenfrequencies in the limits of up to ca. 2 Hz, Fig. 2.1 (b). The reason is that higher energy levels below 2 Hz are introduced into the ﬂexible structure and are ampliﬁed due to the resonant behavior of the structure [571]. Above 2 Hz windinduced oscillations are often marginal because the spectral energy of wind turbulence is minor. Here, the excitation process is not ampliﬁed but uniformly transferred into structural responses. However, in such case the structure follows the wind gusts quasistatically, Fig. 2.1 (a), which alone by itself acts as a fatigue loading, Fig. 2.1 (c). 2.1.1.1 Gust Response Factor Authored by R¨ udiger H¨ oﬀer and Norbert H¨ olscher The gust response factor G is the magniﬁcation of the static, mean reaction Ym of a structure to the ﬂuctuations of the wind load. Stresses, internal forces, and displacements are the responses of interest for which a speciﬁc gust response factor is evaluated. The factor is applied to generate an equivalent static force FE which refers to the maximal dynamical eﬀect. The maximal eﬀect YP is composed from the mean response Ym , which is deﬁned as the mean value of responses in a time window of 10 minutes, and the standard deviation of the reaction σY with the peak factor g as a weight. Yp = Ym ± g · σY
(2.1)
The total standard deviation of response σY results from quasistatic variance 2 2 σYQ and resonant contributions σYR which can originate from one or more resonant contributions (Figure 2.2). The assumed maximum reaction is given by 2 + σ2 = Y · Yp = Ym ± g · σYQ m YR
σYQ 1+g· · Ym
σ2 1 + YR 2 σYQ
= Ym · G (2.2)
12
2 DamageOriented Actions and Environmental Impact SM ⋅ f σ 2M
um
0.8
1.2
1.6
2.0
2.4
M
resonance 2 MR
σ
quasistatic
0.4
σ 2MQ
0.005
0.01
0.02
0.05
0.1
0.2
0.3
0.5
1.0
frequency in Hz
Fig. 2.2. Curve of the total variance of the base bending moment of a cantilever due to buﬀeting excitation plotted over frequencies [572]
which also yields the gust response factor. The equivalent quasistatic load is then deﬁned as FE = G · Fm = G · (caero · qm · Aref )
(2.3)
The gust response factor G raises the original static wind load Fm which is calculated from the aerodynamic coeﬃcient caero , the stagnation pressure qm due to the mean velocity vm , and the reference area Aref , which is in general the projection area of the component or structure in direction of the attack of the wind. The distribution of qm over the building height depends on the proﬁle of the mean wind. For stiﬀ building structures the lowfrequency components of the excitation are not ampliﬁed but quasistatically transferred into responses. The quantity of the quasistatic component is controlled through two sets of parameters of wind turbulence, the three components Ii , i = u, v, w, of turbulence intensity, and the integral length scales. The turbulence intensity Ii is a measure for the ﬂuctuations of wind speed due to turbulence at a ﬁxed point in space. Realizations of the turbulent ﬂuctuations exist in longitudinal (Iu ), lateral (Iv ), and vertical (Iw ) directions. In a boundary layer ﬂow the turbulence intensity is diﬀerent for each of the three components, and it decays with increasing height above ground. For each component the intensity is estimated from the r.m.s. value of the associated statistically stationary process of ﬂuctuations and the mean wind speed in longitudinal direction, Iu = σu /¯ u, Iv = σv /¯ u, Iw = σw /¯ u. The intensity in longitudinal directions is the largest of the three measures. Turbulence intensities of a wind ﬂow over rough terrain are higher than those of a wind ﬂow over a smoother terrain. The integral length scales Lij are statistical measures of turbulence which describe the mean spatial extent of the quantity of air which is homogenously moved
2.1 Wind Actions
13
by a wind gust. The length scales depend on the direction of observation x, y and z and on the observed wind speed components u, v and w. For example the integral length scale Lux describes the correlation of the longitudinal wind speed components in xdirections. In an atmospheric boundary layer ﬂow Lux increases with height above ground. It results smaller in a wind ﬂow over a rough terrain compared to a wind ﬂow over a smoother terrain. Such behavior is also characteristic for the other components of length scales. Equivalent static forces are usually determined assuming a linear load bearing behavior of structures. The quasistatic reaction is then deﬁned by σYQ ∼ σF = ¯ · Q0 Ym F
(2.4)
σF and F¯ are local wind loads applied at a characteristic point of the structure. Q0 is a proportionality factor which quantiﬁes the inhomogeneity of the gust body along the surface of the structure. The r.m.s. value of a quasistatic structural response can directly be derived from the turbulence intensity and the proportionality factor Q0 (s. [572]). Simplifying it can be assumed after [32] that Q0 describes the eﬀect of the longitudinal component u(t) = um + u (t) of the vector of wind speeds by Q20 =
1 1 + 0.9 · ((b + h)/Lux )0.63
(2.5)
b is the width and h is the height of the windexposed area. Dominant resonant gust eﬀects are generated if a ﬂexible structure is prone to oscillatory excitation. A required condition for that is the vicinity of both, eigenfrequencies of bending or torsion modes of the structure, and the highenergy range of the excitation, and low or moderate damping. It is often suﬃcient to consider the mechanical impedance of the structure at the fundamental mode only. In a modal analysis of windinduced oscillations the transfer function Gν (f ) is formulated in terms of the modal ampliﬁcation function Vν (f ). 2 2
kν kν 1 2 2 (2.6) · Vν (f ) = · Gν (f ) = mν mν [ων2 − ω 2 ]2 + 4 ξ 2 ω 2 ων2 kν mν Gν (f )2
active stiﬀness in the νth eigenform oscillating mass in the νth eigenform square of the modulus of the mechanical transfer function in the νth eigenform ω = 2 π · f circular frequency νth circular eigenfrequency ων
14
2 DamageOriented Actions and Environmental Impact
It can be assumed
σYR σYQ
2
≤
ν σYR ν σYQ
2
∞ = 0 ∞
SFν (f ) · Gν (f )2 df SFν (f )
· Gν
(0)2
∼ =
df
0
SFν (fν ) ∞
π 2 · fν 2·δ
(2.7)
SFν (f ) df
0
2
ν (σYR )
variance of the resonant component of the structural reaction in the νth eigenform
ν 2 σYQ variance of the quasistatic component of the structural reaction in the νth eigenform SFν (f ) modal power spectral density function of the modal wind load SFν (fν ) power spectral density function of the modal wind load at fν δ logarithmic damping decrement The resonant wind eﬀects are superimposed from one or more modal eﬀects which result from the mechanical transfer of the respective power spectral density function of the modal wind loads. The latter function results from SFν (f ) = ηiν · ηkν · SFi Fk (f ) (2.8) i
k
ηiν and ηkν are the modal ordinates of the eigenform ν at the nodal points i and k of the oscillating structure. The correlation of the wind loading in physical coordinates is introduced through the cross spectral density function SFi Fk (f ). The complete construction of the cross spectral density function follows sophisticated concepts (s. e.g. [379] and [385]). A simpliﬁed approach for vertical structures is realized and implemented in [32]. 2.1.1.2 Number of Gust Eﬀects Authored by R¨ udiger H¨ oﬀer and Volkmar G¨ ornandt The eﬀect Y of actions due to gust loading reaches or exceeds the characteristic value of the action Yk once in 50 years in the statistical mean. The number Ng of exceedances of a lower level Y (Ng ) < Yk is higher. The curve drawn with a fat line in Figure 2.3 is implemented into the European standards, s. [32]. The value Y (Ng ) of the gust eﬀect under consideration is related to the peak eﬀect Yk due to a gust in a storm of a return period of 50 years, and represents the general behaviour of Y (Ng )/Yk through the equation Y (Ng ) 2 = 1 − 0.174 · log(Ng ) + 0.007 · (log(Ng )) Yk
(2.9)
Setting Equation 2.9 equal to (c1 − c2 log(Ng ))2 = c21 − 2 c1 c2 log(Ng ) + c22 (log(Ng ))2
(2.10)
(a)
Y(Ng) / Yk
2.1 Wind Actions
15
1.00 Y(Ng) / Yk after European standards BerlinTempelhof Berus Braunschweig Bremgarten
0.75
FrankfurtFlughafen HamburgFuhlsbüttel HannoverLangenhagen Laupheim Memmingen
0.50
NürnbergFlughafen SaarbrückenEnsheim Stötten Wasserkuppe
0.25
0.00 101
1
(b)
10 2
10 3
10 4
10 5
10 6
10 7
10 8
10 9
Ng
0.30 Y(Ng) / Yk after European standards BerlinTempelhof Bremgarten HannoverLangenhagen NürnbergFlughafen
0.25
0.20
0.15
0.10
0.05
0.00 0.0
2.0
4.0
6.0
8.0 v (m/s)
10.0
12.0
14.0
Fig. 2.3. Comparison of the occurence of repeated wind eﬀects at diﬀerent locations in Germany and a codiﬁed representation. (a) Number Ng of exceedances of an eﬀect Y (Ng ) ≤ Yk . (b) Probability density functions of ensembles of mean wind speeds
and adding a correction term Y (Ng ) 2 = (c1 − c2 log(Ng ))2 − c23 (log(Ng )) Yk
(2.11)
 where c1 , c2 and c3 are identiﬁcation parameters  yields the form Y (Ng ) = Yk
2 0.174 2 (log(Ng )) − 0.000569 (log(Ng )) 1− 2
(2.12)
The statistical property of the quotient Y (Ng )/Yk is dominated from the probabilistic distribution of the square of the mean wind speeds vi (10minutesmean). The return period of the mean velocity vi = vm is smaller or equal
16
2 DamageOriented Actions and Environmental Impact
than the return period of 50 years which applies for vm , the characteristic mean wind speed. The variable Ng can be interpreted as a measure for an exceedance probability of vi . Typically, a Weibulldistribution is applied to describe the probabilistic behavior of the complete ensemble of mean wind speeds. The cumulative distribution is v k W (v) = exp − (2.13) A with the probability density function k v k−1 v k w(v) = exp − A A A
for v ≥ 0
(2.14)
in which k is called the Weibull shape parameter, and A is the Weibull shape parameter. Troen and Petersen comment in [795] that extreme values are insuﬃciently represented in such distribution type. In alternative, a Gumbeldistribution or a logistic distribution is proposed. A logistic distribution with two parameters [263] which is comparatively referred to yields as the cumulative distribution L(v) =
1 v−a 1 + exp − b
The inverse function results from this as 1 −1 v(L) = a − b · ln L 2 1 v 2 (L) 1 − 1 = · a − b · ln 2 2 vm vm L
(2.15)
(2.16)
(2.17)
The two parameters a,b are extracted from meteorological data bases, e.g. [196]. Equation 2.17 is normalized with respect to v(L = 0.98) = vm and represents a ﬁrst approximation of Y (Ng )/Yk . Equations 2.12 and 2.17 can directly be compared. Figure 2.3(a) shows the relation between Y (Ng ) as a fraction of Yk and the number Ng of exceedances at various locations in Germany. Figure 2.3(b) compares the density functions after Equation 2.14 of some of the locations to the assumed density function which corresponds to Y (Ng )/Yk after [32]. Equations 2.12 and 2.17 are applicable for the analysis of fatigue. A possible resonant contribution is included if Yk contains a resonant contribution as well. Figure 2.4 represents the absolute frequency Ni of normalized gust responses Yi which are partitioned into i classes of diﬀerent response level and equal class widths of 0.1 times Yk . The double amplitude of the extreme stress
2.1 Wind Actions
0.95
2.9
0.85
12
0.75
59
0.65
2
3.1x10
0.55
Y(Ni)/Yk
17
3
1.9x10
0.45
4
1.4x10
0.35
5
1.4x10
0.25
6
2.1x10
0.15
7x10
7
11
3.1x10
0.05 10
0
10
1
10
2
10
3
10
4
10
5
10
6
N
10
7
10
8
10
9
10
10
10
11
10
12
10
13
Fig. 2.4. Distribution of absolute frequencies of normalized gust responses into subsequent classes of diﬀerent levels of eﬀect
Δσ = 2 · (
Nk Mk + ) W A
(2.18)
is the reference value, Mk an Nk are characteristic internal forces of a construction component, W is the elastic section modulus, A is the loaded area. Stress levels between 0.9 · Δσ and 1.0 · Δσ can occure 2.9 times in 50 years in the statistical mean. A damage accumulation after PalmgrenMiner D = i (Ni /Nci ) is performed in order to assess resistance of the considered component with respect to fatigue. Figure 2.5 shows an example taken from a fatigue analysis of the
S N c u rv e (W ö h le r c u rv e ) o f s tre s s c o n c e n tra tio n c a te g o ry 3 6 *
Fig. 2.5. Comparison of the distribution of cyclic stress amplitudes with the SN curve (W¨ ohler curve) of stress concentration category 36* after [30]
18
2 DamageOriented Actions and Environmental Impact
gust responses of steel archs of a road bridge. The considered cerb is suﬃcient to resist the repeated gust impacts. The application of the Equations 2.12 or 2.17 permits a detailed and safe method for the fatigue analysis of gustinduced eﬀects at building structures. 2.1.2 Inﬂuence of Wind Direction on Cycles of Gust Responses Authored by R¨ udiger H¨ oﬀer and HansJ¨ urgen Niemann Meteorological observations document that the intensity of a storm is strongly related to its wind direction. Figure 2.6(a) shows the wind rosette of the airport Hannover, Germany, as an example. The probability of the ﬁrst passage of the same threshold value can strongly vary for diﬀerent sectors of wind direction. That means that the risk of a high wind induced stressing of a structural component is diﬀerent between the wind directions. The failure risk
(a)
0◦
(b)
35 m/s
25 m/s
25 m/s
15 m/s
270◦
5 m/s
15 m/s
90◦
5 m/s
0
90◦
270◦
180◦
(c)
0◦ 35 m/s
180◦
◦
(d)
35 m/s
0◦
25 m/s
15 m/s
270◦
5 m/s
180◦
90◦
270◦
0.25
0.50
90◦ 0.75
1.00
180◦
Fig. 2.6. Rosettes of wind quantities at Hannover (12 sectors, 50 years return period) (a) extremes of 10minutes means of wind velocities at the airport of Hannover at reference height of 10 m above ground (b) extremes of 10minutes means of wind velocities at a building location at building height of 35 m above ground (c) extremes of gust wind speeds at a building location at building height of 35 m above ground (d) comparison of the load factors of the sectors; the largest load factor is valid for the design of the fa¸cade element after Figure 2.8
2.1 Wind Actions
19
of the structure or structural components is determined by the superposition of all probability fractions originating from the sectors of wind direction. Usually, codes follow the conservative approach to assume the same probability of an extreme wind speed for all wind directions. In general, more realistic and very often also more economic results can be achieved if the eﬀect of wind direction is considered. This can be done by employing wind speeds for the structural loading which are adjusted in each sector with a directional factor. Such procedure is in principle permitted by the Eurocode [32]. It is left to the national application documents to regulate the procedures. The wind load is a nonpermanent load; within statical proofs of the load bearing capacity it is employed using a characteristic value, which is deﬁned as a 98% fractile, and an associated safety factor of 1.5. A load level is required which is exceeded not more than 0.02 times a year in a statistical sense. Such value is statistically evaluated from the collective of yearly extremes of the wind speeds. The intensity of the wind load is deduced from the level of the wind speed, or more exact, from its dynamic pressure. The related statistical parameters are used to determine the characteristic value of the load. The wind load depends on the wind direction as the wind speed is diﬀerently distributed regarding their compass, and as the aerodynamic coeﬃcients varies with respect to the angle of ﬂow attack. Taking this into account the most unfavourable load can originate from combining a lower characteristic value of the wind speed, which might be associated to a directional sector, and the related aerodynamic coeﬃcient for this sector. In order to evaluate completely the eﬀect of the inﬂuence of the wind direction it is required to take the structural response into account, e.g. after [227]. In such procedure a response quantity, which is a representative value of the wind action, is evaluated with the restriction to limit its exceedance probability of its yearly extremes to a value lower than 0.02 instead of focussing on loads. Using this requirement the characteristic wind velocities related to the diﬀerent sectors can be deduced. 2.1.2.1 Wind Data in the Sectors of the Wind Rosette The maximum wind load eﬀect on a structural component is resulting from the most unfavourable superposition of the function of the aerodynamic coeﬃcient and the dynamic pressure. Both variables are independent and functions of the direction of mean wind. The usual zoning in statistical meteorology into twelve sectors of 30◦ each is a suﬃcient resolution in order to include distribution eﬀects. The prediction of the risk requires an analysis of the extreme wind velocities for each sector at the building location. If available a complete set of data is taken from a local station for meteorological observations near the considered building location. The wind statistics of a considered building location in the city of Hannover in Germany is shown in Figure 2.6(a) as an example. The wind rosette is evaluated from data collected at the observation station at the airport of Hannover. The terrain in the environment of the station is plain with a relatively homogeneous surface represented by a roughness
20
2 DamageOriented Actions and Environmental Impact
Table 2.1. Conversion of the wind data of the observation station at the airport of Hannover into data for the building location Sectors of wind directions airport: 1 z0 = 0.05 m: vm (z = 10 m) in m/s arena: 2 z0 in m 3 kr · ln(
z
)
z0 4 vm (z = 35 m) 5 Iu (z = 35 m) v 6 gust factor vm 7 v(z = 35 m)
0◦
30◦
60◦
12.1
11.7
17.4
90◦ 120◦ 150◦ 180◦ 210◦ 240◦ 270◦ 300◦ 330◦
13.0
15.2
15.9
17.1
20.5
23.0
20.6
16.7
12.5
0.44
0.27
0.31
0.24
0.24
0.08
0.10
0.11
0.36
0.36
0.36
0.35
0.96
1.03
1.02
1.05
1.05
1.20
1.16
1.15
1.00
1.00
1.00
1.00
11.7 12.1 17.7 13.7 16.0 19.0 19.9 23.7 22.9 20.5 16.6 12.5 0.229 0.206 0.212 0.201 0.201 0.164 0.171 0.174 0.218 0.218 0.218 0.217 1.540 1.494 1.506 1.485 1.485 1.409 1.423 1.429 1.520 1.520 1.520 1.518 18.0
18.1
26.7
20.3
23.7
26.8
28.3
33.8
34.8
31.2
25.3
19.0
of ca. z0 = 0.05 m in all of the sectors. The measurements have been conducted in a standard height of 10 m above ground level, cf. J. Christoffer and M. UlbrichtEissing [196]. N yearly extremes of the mean wind velocity vm are ranked in each sector F , and respective probability distributions are identiﬁed. In the presented example distributions of Gumbeltype were adapted. The occurrence probability of an extreme value in a year, which is lower than a reference value vm,ref , is calculated from −a(vm,ref −U )
P (vm ≤ vm,ref ) = F (vm,ref ) = e−e
(2.19)
In Equation 2.19 U is the modal parameter, and the parameter a describes the diﬀusion. The wind velocities with return periods of 50 years for all sectors are listed in Table 2.1, line 1. In opposite to the conditions at the observation station, the building location is surrounded by a terrain with strongly nonhomogeneous surface roughnesses. The eﬀect of the varying roughnesses superpose the undisturbed conditions evaluated for the location of the observation station. These additional eﬀects inﬂuence the wind velocity in reference height, its proﬁle and the proﬁle of gustiness over height, which vary between the directions according to the respective roughness conditions of a sector. The surface roughnesses for each sector are required. The local roughness lengths z0 of the surface roughness is analysed from aerial photographs over a radius of 50 to 100 times the height of the considered building, e.g. ca. 5 km in case of the considered stadium, Figure 2.7. Mixed proﬁles are evaluated for those sectors with signiﬁcantly changing surface roughnesses; for approximation an equivalent roughness length is adapted. The results are shown in line 2 of Table 2.1; the conditions within each sector are described by conversion factors related to the undisturbed wind rosette. The factor in line 3 of Table 2.1 relates the mean wind speeds with a return period of 50 years at the building
2.1 Wind Actions
21
cp=1.4
0° b/5
90°
b
Fig. 2.7. Roughness lengths of the terrain in the farther vicinity of the building location [771]
Fig. 2.8. Sketch of a building contour (top view) with b < 2 h and fa¸cade element exposed to a pressure coeﬃcient cp = −1.4 [32] at the eastern fa¸cade in the case of winds from 0◦
location at a building height of 35 m of the stadium and the reference wind speed of the same return period at the location of the observation station in reference height of 10 m. The logarithmic law for the proﬁle of the mean wind velocities is applied (Equation 2.20). The terrain factor kr is evaluated using an empirical relation (Equation 2.21). vm (z, z0 ) z = kr · ln( ) vm (zref , z0ref ) z0
kr = (
z0 0,07 1 ) · z0ref ln(zref /z0ref )
(2.20)
(2.21)
The wind velocities at the building location with a return period of 50 years are evaluated for each sector and are listed in line 4 of Table 2.1. As shown before, mean and gust wind speeds and the respective dynamic pressures are applied to determine equivalent loads which represent the resulting wind loading for design procedures. The dynamic gust pressure is calculated from the mean dynamic pressure qm and the turbulence intensity Iu . q = (1 + 2g · Iu · Q0 ) · qm
(2.22)
The gust velocity in the last row of Table 2.1 is calculated from Equation 2.23, where g is the peak factor and Q0 is the quasistatic gust reaction. Q20 is also called background response factor after [32].
22
2 DamageOriented Actions and Environmental Impact
v=
1 + 2 g Q0 Iu · vm
(2.23)
For simplicity Q0 can consistently be determined from 2 g Q0 = 6 assigning to Q0 its maximum value 1. It has to be pointed out that the surface roughness is also aﬀecting the turbulence intensity, as shown in line 5 of Table 2.1. The statistical evaluation for all sectors leads to a mean wind of 50 years return period of 23.8 m/s at the building location. Figure 2.6(b) represents the rosette of mean wind speeds at the building location. In comparison of both wind rosettes, representing the building location and the location of the observation station, it can be concluded that the main character of the local wind climate is preserved but relevant changes due to the terrain roughness are introduced. 2.1.2.2 Structural Safety Considering the Occurrence Probability of the Wind Loading The wind load eﬀect on a structure can be expressed in terms of a response quantity Y . For a linear, stiﬀ structure without dynamic ampliﬁcation, Y is calculated from: 1 2 Y (Φ) = ρvΦ · ηp (r) · cp (r, Φ) · dA (2.24) 2 A in which: ηp inﬂuence factor for the pressure p acting at the point on the surface of the structure; r  local vector; cp pressure coeﬃcient at a point of the surface of the structure for a given wind direction Φ; ρ  mass density of air; A  pressure exposed inﬂuence area. A certain response force Y forms the basis for the determination of a characteristic wind velocity vik , which is valid over the sector with the central wind direction Φi . The starting point is vi,lim : 1 2 Yi,lim (Φi ) = CY (Φi ) · ρ · vi,lim 2
(2.25)
In Equation 2.25 the response Yi,lim is determined as an equivalent wind eﬀect by use of the gust velocity v. The wind eﬀect admittance depending on the wind direction Φ, CY = CY (Φ), is identical to the integral in Equation 2.24. It covers the distribution and the value of the aerodynamic coeﬃcient within the inﬂuence area of the load as well as the mechanical admittance, which is the transfer from the dynamic pressure into the response quantity. This operation is conducted for a selected wind direction Φi . In a second step the complete risk is evaluated as the exceedance probability of the response quantity Y , which adds up from the contributions from each sector. The safety requirements are met if the total risk has a value smaller than 0.02. In case of a risk larger 0.02 an increased value of the vi,lim enters into the iteration until a value smaller 0.02 is achieved. In an analogeous manner a
2.1 Wind Actions
23
decreased value of vi,lim is introduced aiming on an economical optimization if the ﬁrst iteration yields a value much smaller than 0.02. The total risk of exceeding the bearable response quantity Yi,lim , or as complementary formulation, the probability of nonexceedance of Yi,lim , is proved within the following steps. The main idea of the procedure is to make 2 use of combinations CY (Φ) · 12 ρ · vΦ,lim instead of a global CY · 12 ρ · v 2 . A probability of nonexceedance of 0.98 of the applied force must be guaranteed for both in the sectors and in total. 1 1 2 2 CY (Φi ) · ρ · vi,lim = CY (Φ) · ρ · vΦ,lim 2 2 The velocity limit vΦ,lim for a sector Φ results as CY (Φi ) 1 · vi,lim = · vi,lim vΦ,lim = CY (Φ) a(Φ)
(2.26)
(2.27)
The eﬀect of the direction of the wind on the wind eﬀect is expressed through a directional wind eﬀect factor: a(Φ) =
CY (Φ) CY (Φi )
(2.28)
The probability P (v ≤ vΦ,lim ) = FΦ (vΦ,lim ) of the nonexceedance of vΦ,lim within the sector Φ also applies for the response Y ≤ Yi,lim . F (vΦ,lim ) can be calculated from the probability distribution of the mean wind velocity in the sector as given by Equation 2.19. The probability of the nonexceedance of the limit Yi,lim after Equation 2.25 under the condition of a certain vi,lim in sector Φi is satisﬁed from a product (Equation 2.29) of all nonexceedance probabilities under the condition that the yearly extremes in the diﬀerent sectors are statistically independent. P (Y ≤ Yi,lim ) = P ((v ≤ v1,lim ) (v ≤ v2,lim ) · · · (v ≤ v12,lim )) 12 = 1 FΦ (vΦ,lim ) ≥ 0.98 (2.29) The considered value of the gust speed is adequate if the exceedance probability P (Y > Yi,lim ) is less or equal 0.02 which corresponds to the probability of nonexceedance of (1 − 0.02) = 0.98, Equation 2.29. Obviously, the condition P (Y = Yi,lim ) ≥ 0.98 must be observed in any sector. 2.1.2.3 Advanced Directional Factors The responses of a structure must be taken into consideration for the determination of the relevant wind speeds and wind loads for each sector. This
24
2 DamageOriented Actions and Environmental Impact
Table 2.2. Determination of a reduced characteristic suction force on the fa¸cade element after Figure 2.8 through the consideration of the eﬀect of wind direction on loading. line 1: extreme gust speed at a building location at Hannover at building height of 35 m; line 2: cp,10 values at the considered fa¸cade element for wind ﬂow from the respective directions; line 3: directional wind eﬀect factor after Equation 2.8; line 4: iterative determination of applicable wind speeds in sectors and associated nonexceedance probabilities in sectors; line 5: applicable fraction of codiﬁed standard load after the proposed method Sectors of wind directions 0◦ 1 2 3 4
30◦
18.0 18.1 1.4 1.4 1 1 18.0 18.1 0.98 0.98 18.2 18.3 0.9985 0.9985
5 0.194
0.196
60◦
90◦
120◦
26.7 – 0 ∞ 1.0 ∞ 1.0
20.3 – 0 ∞ 1.0 ∞ 1.0
23.7 – 0 ∞ 1.0 ∞ 1.0
–
–
–
150◦
180◦
210◦
240◦
270◦
300◦
330◦
26.8 28.3 33.8 34.8 31.2 25.3 19.0 0.8 0.8 0.8 0.6 0.6 0.6 1.4 0.57 0.57 0.57 0.36 0.36 0.36 1 35.5 37.5 44.8 58.0 52.0 42.2 19.0 0.98 0.98 0.98 0.98 0.98 0.98 0.98 36.0 38.1 45.5 59.0 52.8 42.9 19.2 0.9985 0.9985 0.9985 0.9985 0.9985 0.9985 0.9985 0.434 0.486 0.694 0.874 0.701 0.462 0.216
can be achieved using the values of the wind eﬀect admittance CY (Φ) for the respective sectors. The procedure of calculating the characteristic wind speed in the sectors is exempliﬁed in Table 2.2 for a building located at Hannover, Germany. The ﬁxing forces of fa¸cade claddings due to suction is considered. Figure 2.6 shows a topview sketch of a building cubus of 35 m height with fa¸cades oriented in northern, eastern, southern and western directions. The question is if reduced values of the suction forces at the cladding elements at the edge of the eastern fa¸cade can be adopted as the wind rosettes clearly indicate diﬀerent wind extremes when comparing the sectors, cf. line 1 in Table 2.2. Wind from eastern directions generate pressure forces at the element, whereas suction forces at the same element are generated through winds from all other sectors. Suction coeﬃcients from [26], Table 3, are used to describe the aerodynamic admittance in simpliﬁed terms. An element size of more than 10 m2 is assumed. The pressure minimum — or maximum suction — occurs for northern directions and is described through the pressure coeﬃcient cp = −1.4 for h/b ≥ 5, h = 35 m. Southern wind directions generate a coeﬃcient of cp = −0.8, cp = −0.6 is inserted for western wind directions (cf. line 2 in Table 2.2). The directional wind eﬀect factor a(φ) in line 3 after Equation 2.28 is calculated refering the sectorial pressure coeﬃcents to the minimum pressure coeﬃcient cp = cp,min = −1.4. The results of two iterations are listed in line 4. The ﬁrst two rows represent vΦ,lim = vi,lim and the corresponding probability of nonexceedance FΦ,lim (vΦ,lim ) which remains 0.98 according to the probability of nonexceedance of the values given in line 1, or it is 1 in sectors 0◦ , 60◦ and 90◦ as only pressure instead of suction can occur here. The application of Equation 2.29 leads to P = 0.8171 < 0.98. In a second iteration the extreme wind speeds are increased in such a way that the total probability
2.1 Wind Actions
25
of nonexceedance after Equation 2.29 results to be larger or equal to 0.98. The third and fourth row in line 4 of Table 2.2 represent a valid solution for which P = 0.9866 and results larger than the required value of P = 0.98. The codiﬁed standard design procedure requires a reference wind speed of vref = 25 m/s irrespective the wind direction. The calculation of a gust speed after the wind proﬁle for midlands ([26], Table B.3) leads to a characteristic gust speed of v = 41.3 m/s at building height of 35 m. The standard suction force for the considered element — without any consideration of the inﬂuence of wind directions — must be calculated as Y = 12 ρ · v 2 · cp · A. The applicable characteristic suction force after Equation 2.25 — with consideration of the inﬂuence of wind directions — can be calculated as a fraction 2 2 (cp (φ) · vφ,lim )/(cp,min · vref ) of the standardized characteristic value. The quotient is listed in line 5 of Table 2.2, and it is represented in Figure 2.6, (d). The largest factor in line 5 must be applied. The respective characteristic velocity is ca. 59 m/s but the associated characteristic suction force after Equation 2.26 is lower than the standard suction force after the code. The reason is in the application of the much higher pressure coeﬃcent — or lower suction coeﬃcient — of cp = −0.5 for wind in the sector 240◦ instead of cp = −1.4. The procedure can also be adopted for a fatigue analysis after Equation 2.9. 2.1.3 Vortex Excitation Including LockIn Authored by J¨ org Sahlmen and M´ ozes G´ alﬀy Vortex excitations represent an aerodynamic load type which can cause vibrations leading to fatigue, especially for slender bluﬀ cylindrical structures — bridge hangers, towers or chimneys. The nature of air ﬂow around the structure depends strongly on the wind velocity and on the dimensions of the structure. Accordingly, diﬀerent wind velocity ranges can be deﬁned, depending on the value of a nondimensional parameter called the Reynoldsnumber Re =
u ¯D . ν
(2.30)
Here, u ¯ represents the mean wind velocity, D is the signiﬁcant dimension of the body in the acrosswind direction — for cylindrical structures, the diameter — and ν = 1.5 · 10−5 m2/s is the kinematic viscosity of air. In the Reynoldsnumber range between 30 and ca. 3 · 105 , vortices are formed and alternately shed in the wake of the cylinder creating the von ´rma ´n vortex trail (Figure 2.9) and giving rise to the lift force — an alterKa nating force which acts on the structure in the acrosswind direction. The nature of the vortex shedding and of the lift force is considerably inﬂuenced by the wind turbulence σu Iu = , (2.31) u ¯
26
2 DamageOriented Actions and Environmental Impact
´ rma ´ n vortex trail formed by vortex shedding Fig. 2.9. Von Ka
where σu denotes the standard deviation of the stochastically ﬂuctuating wind velocity u. In a smooth wind ﬂow, i. e. if the wind turbulence is low (Iu ≤ 0.03), the acrosswind force is a harmonic function of the time t: Fl (t) =
ρ¯ u2 DCl sin 2πfs t. 2
(2.32)
Here, Fl denotes the lift force per unit span, ρ = 1.25 kg/m3 is the density of air, Cl is the dimensionless lift coeﬃcient and fs = S
u ¯ D
(2.33)
is the frequency of the vortex shedding, also called the Strouhalfrequency. The nondimensional coeﬃcient S in (2.33) is the Strouhalnumber which depends on the shape of the structure; its value for cylinders is S ≈ 0.2. In a turbulent ﬂow, the excitation frequencies are distributed in an interval around the mean frequency, the width of the interval depending on the turbulence. When the Strouhalfrequency approaches one of the natural frequencies fn of the structure1 and the structure begins to oscillate at higher amplitudes because the resonance, an aeroelastic phenomenon, the socalled lockin eﬀect occurs. This results in the synchronization of the vortex shedding process to the motion of the excited structure (Figure 2.10), acting as a negative aerodynamic damping, and can lead to very large oscillation amplitudes. Consequently, the lockin eﬀect can play an essential role in the evolution of the fatigue processes in the damagesensitive parts of the structure. The width of the lockin range is zero for a ﬁxed system and increases with increasing oscillation amplitude. As the amplitude depends on mass and damping, these systemparameters have a large inﬂuence on the lockin eﬀect. This inﬂuence can be numerically catched by introducing the dimensionless Scrutonnumber Sc =
2μδ , ρD2
(2.34)
where μ denotes the mass of the structure per unit length, and δ is the structural logarithmic damping decrement. The width of the lockin range is 1
Generally only the ﬁrst natural frequency is of practical importance.
2.1 Wind Actions
27
Fig. 2.10. Dependence of the vortex shedding frequency fv on the wind velocity u ¯. fn is the natural frequency of the structure
reduced with increasing Scrutonnumber, and for very large values of Sc, no lockin eﬀect occurs at all. In the case of a uniform smooth ﬂow, the lift force per unit span acting on a circular cylinder ﬁxed in both the alongwind and acrosswind directions is given by (2.32). However, the force is not fully correlated along the cylinder span. If the cylinder is allowed to oscillate, the magnitude of the lift force and also the correlation increases. The equation of motion of the cylinder is given by m¨ y + cy˙ + ky = Fl (u, D, y, y, ˙ y¨, t),
(2.35)
where y denotes the acrosswind displacement, m, c and k are the mass, the damping coeﬃcient and the stiﬀness of the cylinder per unit span. As the lift force per unit span Fl depends not only on the wind velocity, on the cylinder diameter and on time, but also on the displacement, on the velocity and on the acceleration of the structure2 , it is not a trivial task to establish its explicite expression. Furthermore, the wind velocity u(t) is a stochastic variable which generally describes a turbulent wind process, and consequently a suitable wind load model must also correctly describe the oscillations in turbulent ﬂow. Much eﬀort has been done in order to ﬁnd an expression for the acrosswind force which ﬁts the experimentally observed facts. However, all of the windload models developed up to the present can only describe the experimentally observed oscillations correctly if some limiting conditions are fulﬁlled. 2.1.3.1 Relevant Wind Load Models The Ruscheweyhmodel [695], which is implemented in the German Codes DIN 4131 (Steel radio towers and masts) and DIN 4133 (Steel stacks), describes the acrosswind oscillations in the time domain. The lift force per unit 2
The lift force also depends on the roughness of the cylinder surface, which is here not explicitely shown.
28
2 DamageOriented Actions and Environmental Impact
span is given by (2.32). The lift coeﬃcient is Cl = 0.7 for Re ≤ 3 × 105 , for higher Reynoldsnumbers, Cl decreases. It is assumed that the lift force acts over the correlation length Lc , which is the lengthscale of the synchronized vortex shedding along the cylinder span. The increase of correlation with increasing oscillation amplitude Ay is described by the function ⎧ ⎪ for Ay ≤ 0.1D ⎨6D Lc = 4.8D + 12Ay for 0.1D < Ay < 0.6D (2.36) ⎪ ⎩ 12D for Ay ≥ 0.6D The width of the lockin range is set to ±15 % around the critical velocity uc =
Dfn , S
(2.37)
which leads to a Strouhalfrequency equal to the natural frequency: fs = fn . This model predicts the oscillation amplitudes of slender cylindrical structures in a smooth wind ﬂow for constant mean wind velocities within and outside of the lockin range with a remarkable accuracy. Large estimation errors occur, however, in the case of high turbulence, or if the mean wind velocity considerably varies in time — especially in the case of entering or exiting the lockin range. The Vickerymodel [811] uses a frequencydomainapproach to describe the acrosswind vibrations. Assuming a Gaussian distribution for the spectral density of the lift force, the standard deviation (rmsvalue or eﬀective value) of the acrosswind deﬂection is obtained as √ πLc h fs3 −( 1−fBn/fs )2 CLσ ρ D3 σy = e . (2.38) 2 2 8π S me 2Bξ fn3 Here, CLσ is the lift coeﬃcient expressed as rmsvalue, me and ξ are the eﬀective mass and damping ratio of the structure, h is the height of the cylinder and B is a dimensionless parameter which describes the relative width of the Gaussian spectral peak of the lift force. The parameters CLσ ≈ 0.1, Lc ≈ 0.6 D and B are obtained from ﬁts to experimental data; obviously, B depends on the wind turbulence. The model is suitable for predicting the oscillation amplitudes, both in smooth and turbulent ﬂow, but it is limited to the case of stationary ﬂow, i. e. to constant mean wind velocities, and it doesn’t take the lockin eﬀect into consideration. The model of Vickery and Basu [810] describes the acrosswind oscillations in smooth or turbulent ﬂow, with mean wind velocities outside or within the lockin range. The lift force is written as the sum of two forces: a narrowband stochastic term with a normal distribution of the spectral density and a motion dependent term — negative aerodynamic damping — which describes the lockin eﬀect. For the lockin range, the rmsvalue of the displacement is obtained as
2.1 Wind Actions
σy = 2.5
Cl ρD3 Lc 16π 2 S 2
π me (μe ξ + μξa )
h 0
ψ 2 (z) dz
,
29
(2.39)
where μ and μe are mass and eﬀective mass of the structure per unit span, ψ(z) is the value of the normalized mode shape at height z, and ξa is the aerodynamic damping ratio. The aerodynamic damping is negative in the lockin range, and it depends on the ratio u ¯/uc, on the turbulence and on the Reynoldsnumber. Additionally, a dependence on the oscillation amplitude is deﬁned in such a way that it limits the amplitude to a predeﬁned value. The most exhaustive model of vortexinduced acrosswind vibrations has been developed by ESDU [262], mainly based on the work of Vickery and Basu [810]. The response equations give the standard deviation of the oscillation amplitude and incorporate the inﬂuences of turbulence and of the lockin eﬀect. The system response is obtained from the superposition of a broadand of a narrowband term. A very large variety of parameters, such as the surface roughness or the integral length of the turbulent wind, is included in the calculation. Also, the dependence of the lockin range width on the oscillation amplitude is taken into consideration. Because of their complexity, the response equations will not be presented here. Like all the models presented above, also this model is only suitable to describe the acrosswind vibrations in a stationary or quasistationary ﬂow, i. e. if the mean wind velocity doesn’t change too rapidly and if there is no transition into or from the lockin range. Based on the normal distribution of the lift force spectral density SF , suggested by Vickery and Clark [811], Lou has developed a convolution model [507] which describes the lift force in the timedomain, for a stationary turbulent ﬂow, outside of the lockin range: t ρ ¯¯ ) Fl (t) = DCl βu2 (τ ) e−ξ ω(t−τ cos ω ¯ (t − τ ) dτ, (2.40) 2 0 ω ¯ = 2πS u ¯/D denoting the Strouhal circular frequency corresponding to the mean wind velocity u ¯. From the assumption of the normal distribution for SF , the parameters β and ξ¯ can be determined as √ √ 2π ln 2 Iu ω ¯ (2 + 2 ln 2 Iu2 ) , ξ¯ = ln 4 Iu , β=u ¯ (2.41) 2 Su (¯ ω )(1 + 2 ln 2 Iu ) where Su is the spectral density of the wind velocity. 2.1.3.2 Wind Load Model for the Fatigue Analysis of Bridge Hangers In the project C5 of the Collaborative Research Center (SFB) 398, the vortexinduced acrosswind vibrations of the vertical tie rods of an arched steel bridge in M¨ unsterHiltrup have been analysed for the purpose of a fatigue analysis of
30
2 DamageOriented Actions and Environmental Impact
Fig. 2.11. Wind velocity, measured and simulated deﬂection vs. time for the bridge hanger 1 (left) and 2 (right). The horizontal lines in the upper panels show the mean width of the lockin range
their extremely damagesensitive welded connections. Therefor, the vibrations of two hangers have been ﬁlmed by digital cameras, and the time histories of the deﬂections have been extracted from the videos by means of a Java program. Simultaneously, the ﬂuctuating wind velocity has been recorded with an ultrasonic 3Danemometer. The mean wind velocity varied with time in such a way, that one of the hangers entered and exited the lockin range several times during the measurement, while the other one stayed outside of the lockin range, see Figure 2.11. Because of the low oscillation amplitude, the lockin range of the second hanger was very narrow; it lies within the width of the horizontal line in the upper right panel. In order to check the validity of the previously presented wind load models for bridge hangers, the amplitudes measured on hanger 1 in the lockin range, in the timeinterval between 8.5–12.5 min have been compared to the predictions of the Ruscheweyh [695] and ESDUmodels [262]. The experiment shows a peak amplitude of ca. 9 mm and an rmsamplitude of ca. 6 mm, while the Ruscheweyhmodel predicts peak amplitude of about 5 mm and the ESDUmodel an rmsamplitude of ca. 30 mm. As both models show a substantial discrepancy compared to the measured values, a new wind load model for the acrosswind vibrations of bridge tierods in nonstationary, turbulent ﬂow, including the lockin eﬀect, has been developed [296], based on the model by Lou [507]. For this purpose, a
2.1 Wind Actions
31
powerfunction dependence of the parameter β in (2.40) on the ﬂuctuating wind velocity u has been supposed: β = Kun−2 ,
(2.42)
with the ﬁtparameters K and n. Furthermore, in order to describe the nonstationary wind process, the mean values in (2.40) have been replaced by the corresponding timedependent quantities; only the wind turbulence Iu is supposed to be constant. The lift force per unit span obtained this way is: t ρ Fl (t) = DCl K un (τ ) eα(t,τ ) cos ϕ(t, τ ) dτ, (2.43) 2 0 with
√ α(t, τ ) = ln 4 Iu
τ
ω(θ) dθ,
t
ϕ(t, τ ) =
τ
ω(θ) dθ + ϕ0 (t).
(2.44)
t
ω(θ) = 2πSu(θ)/D is the Strouhal circular frequency corresponding to the ﬂuctuating wind velocity u at the time moment θ. It is supposed that the lift force acts over the correlation length Lc which can be determined from equation (2.36). The phase angle ϕ0 in (2.44) describes the lockin eﬀect. For wind velocities in the lockin range, it is set in phase with the rod motion: ϕ0 (t) = π + arctan
y(t) ˙ , ωn y(t)
(2.45)
outside of the lockin range, it is set to 0. ωn = 2πfn denotes the angular natural frequency of the rod. The increase of the force amplitude caused by the phasesynchronization is compensated by the reduction of the multiplicative parameter K in equation (2.43) for the lock in range. It has been assumed that the lockin range is symmetric with respect to the critical wind velocity (2.37) with a half width Δu depending on the oscillation amplitude Ay according to a simple parabolic function (Figure 2.12). The parabola is deﬁned by three points, P1 , P2 and P3 , obtained from ﬁts to the experimental data. The ﬁt of the model parameters to the experimental data has been performed by simulating the vortexinduced vibrations in the time domain, on a ﬁniteelement model of the hanger, which has been excited by the force calculated using equation (2.43) applied to the experimental wind data u(τ ). The time dependent deﬂections have been calculated using the NewmarkWilson timestep method, applying Cl = 0.5 and Lc = 6D. The time histories obtained for the ﬁtted values of the model parameters, K = 175 m−1 and n = 3, are shown in the lower panels of Figure 2.11. For the lockin range, the multiplicative parameter K has been reduced by a factor 4.
32
2 DamageOriented Actions and Environmental Impact
Fig. 2.12. Width of the lockin range for bridge tie rods
The time history of the measured and simulated oscillation amplitudes shows a remarkable similarity for both hangers (Figure 2.11). Furthermore, the averaged rmsamplitudes of the simulated deﬂections are very close to the values determined from the experiment: for hanger 1, 3.81 mm is obtained for both the measured and simulated data, while for hanger 2, measurement and simulation yield 0.133 mm and 0.130 mm respectively. The wind load model has also been validated by wind tunnel measurements, carried out on a rigid cylinder, elastically suspended in such a way that it could oscillate only in the acrosswind direction. Wind velocity and displacement have been simultaneously recorded for 17 ﬁxed values of the mean wind velocity. The displacements of both ends have been averaged in order to eliminate the rotational vibration of the cylinder around the axis parallel to the wind direction. The ﬁt of the model parameters has been performed analogously to the full scale case, applying the same values for the parameters Cl and Lc , obtaining K = 23 m−1 and n = 3. The values for the full scale and the wind tunnel experiments diﬀer because K obviously depends on the wind turbulence (see eq. (2.41) and (2.42)). Again, for the lockin range, the parameter K has been reduced by the factor 4. The measured and simulated time histories of the amplitudes are shown in Figure 2.13 for a representative measurement within and another outside of the lockin range. In both cases, the measured and the simulated data show timedependent amplitudes with qualitatively and quantitatively similar characteristics. The ratio of the simulated to measured rmsamplitudes of the displacement varies between 0.47 and 1.95 for the diﬀerent ﬁxed mean wind velocities, which can be considered as a good agreement between model and experiment, in comparison to other models: The amplitudes are overestimated by a factor of ca. 7 by the Ruscheweyh and by a factor of ca. 11 by the ESDUmodel.
2.1 Wind Actions
33
Fig. 2.13. Measured and simulated amplitude of the displacement within and outside of the lockin range
2.1.4 Micro and Macro Time Domain Authored by M´ ozes G´ alﬀy and Andr´es Wellmann Jelic In modeling stochastic, especially timevariant fatigue processes, commonly the time scale is split into a micro and a macro time domain. In the micro time domain, loading events and resulting fatigue events are simulated. Theoretically, the loading and fatigue process can be considered as continuous in the micro time domain, but for practical calculations discrete realizations of these processes are used, which are separated in time by a constant increment called time step. The macro time domain is used for estimating the lifetime of the structure, taking into consideration the succession of fatigue events in time. The splitting procedure is applicable to any stochastic loading which causes fatigue — e. g. wind, traﬃc, seawaves, etc. The reasons for splitting the time scale are: •
Within the micro time domain, the system properties, and in most cases also the excitation process, can be considered timeindependent. Consequently, the simulation of a fatigue process in this time domain — a fatigue event — can be performed using timeindependent stiﬀness, damping and
34
2 DamageOriented Actions and Environmental Impact
Fig. 2.14. Sample realizations of a renewal process (left) and of a pulseprocess (right)
•
mass matrices and an excitation force derived from a stationary random function (generally white noise). The numerical simulation of the fatigue process over the macro time domain would result in unacceptably large computation times, especially for complex structures, where the solution of the equation of motion implies a laborious ﬁniteelement calculation at every timestep.
The advantage of the time scale splitting is that the fatigue results obtained for a load event in the micro time domain (e. g. using the rainﬂow cycle counting method) can be used in the macro time domain several times, without the need of recalculating the time histories of the loads and of the system responses. Generally, the length of the micro time domain is chosen btw. 1 ms and 1 s, depending on the properties of the structure and the loading. For some applications, however, considerably larger durations are needed, e. g. for the lifetime analysis of bridge hangers, performed in the project C5 of the Collaborative Research Center (SFB) 398. Because of the large mass and small damping of the tie rods (logarithmic damping decrement δ ≈ 6 × 10−4 ), the system answer to changes in the nature of the excitation force (e. g. on entering or exiting the lockin range, see Section 2.1.3) is very slow and consequently it was necessary to choose a duration of ca. 1.5 hours for the micro time domain. Another uncommon feature of this application is that because of the lockin eﬀect, the stochastic excitation force cannot be considered stationary, even in the micro time domain [295]. The macro time domain spans the whole lifetime of the structure, implying an order of magnitude of several years. 2.1.4.1 Renewal Processes and Pulse Processes In the macro time domain, the succession of the fatigue events is numerically represented by discrete processes which occur at certain moments of time
2.2 Thermal Actions
35
ti , called renewal points. Each process causes a jump in the fatigue function, between the renewal points the function remains constant. The processes with constant height are called renewal processes, and those with variable height are called pulse processes. Renewal processes can be characterized by one single stochastic variable representing the length of the renewal period (the period between two successive renewal points). For pulse processes, a second stochastic variable is needed for the full description: the pulse height. Figure 2.14 presents the time dependence of the state function (e. g. fatigue) for a renewal process, represented by the number N of the occured processes, and of a pulse process, characterized by the pulse height X.
2.2 Thermal Actions Authored by J¨ org Sahlmen and Anne Spr¨ unken Climatic conditions (e.g. air temperature, solar radiation, wind velocity) cause a nonlinear temperature proﬁle within a structure or a structural component and stress due to thermal actions is induced. For the design and lifetime analysis of many engineering structures (e.g. bridges, cooling towers, tall buildings, etc.) thermal eﬀects, in combination with moisture and chemical actions, remain an important issue. 2.2.1 General Comments Authored by J¨ org Sahlmen and Anne Spr¨ unken Temperature changes generate expansions or contractions, hence considerable stress may occur. The amount of stress is depending on the magnitude of loading. In the elastic range of deformation the material returns to its original dimension or shape when the load is removed. When subjected to sustained or longterm loading, many building materials experience additional deformation, which does not fully disappear when the loading is removed. Due to this special load cracks may occur and deterioration starts or proceeds. As a consequence the deterioration over time leads to a reduction of stiﬀness of the structure. The implementation of aﬀected nonlinearities due to thermal loads in the design process and lifetime analysis is still part of ongoing research. The numerical modelling of the temperature eﬀects on structures based on experimental results are in the focus of this chapter. 2.2.2 Thermal Impacts on Structures Authored by J¨ org Sahlmen and Anne Spr¨ unken Permanent change of meteorological conditions (e.g. cloudiness, rain, sunny periods, etc.) leads to nonstationary und locale sitedominated loads on a structure. For the optimization of lifetime analysis a numerical algorithm is
36
2 DamageOriented Actions and Environmental Impact
needed to describe the physical thermal load scenario on an observed structure or structural component. A realistic temperature ﬁeld, based on experimental data, has to be modelled to simulate the thermal transmission and moisture ﬂux within a material with the ﬁnal aim to determine the time dependent stress acting. Parameters like heat transfer and heat storage as well as the content of moisture have to be considered [517, 74, 704, 463]. Further more material and site conditions of the observed structure (location, climate, orientation, surrounding properties, etc.) have to be implemented in a numerical optimization model of thermal actions [518]. The process of heat transmission in materials is elementary controlled by three phenomena [807]: • • •
heat conduction natural convection thermal radiation
In the following the physical fundamentals of heat transmission are brieﬂy described. Material properties and structure dimensions are aﬀecting directly the heat conduction and the storage capacity. The rate at which heat is conducted through a material is proportional to the area normal to the heat ﬂow and the temperature gradient along the heat ﬂow path. For a one dimensional, steadystate heat ﬂow the rate is expressed by Fourier’s diﬀerential equation: Q = −λ dT /dh = −λ grad T
(2.46)
with: T = T (x = h) − T (x = 0) and assuming stationary heat transfer the formula rearranges to: Q = −λ A(δT /h)
(2.47)
where: λ = thermal conductivity [W/mK] Q = rate of heat ﬂow [W] δT = temperature diﬀerence [K] A = contact area [m2 ] h = thickness layer [m] Thermal conductivity λ is an intrinsic property of a homogeneous material which describes the material ability to conduct heat. This property is independent of material size, shape or orientation. For nonhomogeneous materials, those having glass mesh or polymer ﬁlm reinforcement, the term relative thermal conductivity is used because the thermal conductivity of these materials depends on the relative thickness of the layers and their orientation with respect to the heat ﬂow direction. The thermal resistance R is another material property which describes the measure of how a material of a speciﬁc thickness resists to the ﬂow of heat. This parameter is deﬁned as follows:
2.2 Thermal Actions
R = A(δT /Q)
37
(2.48)
Hence, the relationship between λ and R is shown by the substitution of 2.47 and 2.48 and rearranging to the form: λ = h/R
(2.49)
Equation 2.49 reﬂects that for homogeneous materials, thermal resistance is directly proportional to the thickness. For nonhomogeneous materials, the resistance generally increases with thickness but the relationship is maybe nonlinear. Following this relation Eurocode 1 [19] is using a concept for the determination of the total resistance value as follows: Rtot = Rin + (hi /λi ) + Rout (2.50) where: Rin = thermal resistance at inner surface [m2 K/W] Rout = thermal resistance at outer surface [m2 K/W] λi = thermal conductivity of layer i [W/m K] hi = thickness of layer i [W/m K] The process of convection is dominated by the climatic conditions like wind, temperature, humidity, etc. Convection describes the transfer of heat energy by circulation and diﬀusion of the heated material. The ﬂuid motion of the surrounding air is caused only by buoyancy forces set up by the temperature diﬀerences between the outer surface of the structure and the air temperature. The basic equation for the convective heat transfer is given as follows: Qconv = αconv (Tair − Tsurf ace )
(2.51)
where: = convection heat transfer coeﬃcient [W/m2 K] αconv = air temperature [K] Tair Tsurf ace = surface temperature of the structure [K] Thermal radiation, essentially induced by the visible and nonvisible light of the sun, consists of electromagnetic waves with diﬀerent wavelengths (see Figure 2.15). The energy which a wave is able to transport is related to its wavelength. Shorter wavelengths carry more energy than longer wavelengths. The transported energy is released when these waves are absorbed by an object or structure. Due to solar radiation thermal actions on structures could be subdivided into two general types of solar impact depending on the wavelength: •
Short wave radiation with the highest heat energy content is described as global radiation. It includes the direct and the diﬀuse part of the thermal action on a structure as well as the reﬂected solar radiation from the immediate vicinity (see Figure 2.16) of the observed object.
38
2 DamageOriented Actions and Environmental Impact
Fig. 2.15. Wavelength of the visible light diﬀuse
direct wind atmospheric antiradiation
reﬂection airtemperature
reﬂection of radiation of atmospheric immediate antiradiation vicinity
reﬂection of global solar radiation
Fig. 2.16. Climatic load on a structure
•
Long wave radiation contains the atmospheric antiradiation with its reﬂection to the surrounding area and to the atmosphere.
Additionally to the described external actions, the reﬂection of radiation at the structure is inﬂuencing the thermal stress. Figure 2.16 shows all types of radiation having a part on the thermal impact of a structure [517, 74, 286]. Heat transfer due to solar radiation is expressed by Boltzmann’s equation as follows: Qrad = αrad (Temitter − Tabsorber )
(2.52)
2.2 Thermal Actions
39
where: = heat transfer coeﬃcient due to radiation [W/m2 K] αrad Temitter = absolute temperature of the emitter [K] Tabsorber = absolute temperature of the absorber [K] All parts of thermal radiation are directly aﬀected by external interference effects. The local climatic conditions at the site (e.g. airtemperature, surface temperature, humidity, cloudiness, etc.) as well as the properties of the observed structural component control the intensity of the total thermal action. Surface colour and characteristic (colour, roughness, layer thickness of the wall, etc.) for example control absorption, reﬂection and transmission process. In addition to that the complete mechanism of heat transmission is considerably in dependency on the moisture content in the material of the structure and from other parameters like evaporation or condensation as well as special weather conditions like rain, snow and frost (see Section 2.4). Against this background longterm experiments are helpful to understand the complicated nature of the mechanisms involved. To give more precise recommendations for the reduction or elimination of cracking and failure of building materials better numerical models are needed where the interaction of all discussed parameters are implemented and nonstationary eﬀects are taken into account.
2.2.3 Test Stand Authored by J¨ org Sahlmen and Anne Spr¨ unken For the analysis of thermal actions on structural elements under free atmospheric conditions a test stand with diﬀerent test objects is performed. On the roof of the IABuilding of the RuhrUniversity Bochum three diﬀerent test plates, made of concrete, are installed (see Figure 2.17). Each test plate spans an area of 0.7 × 0.7 m2 (thickness: 0.1 m). Plate 1 is made of pure concrete whereas plate 2 and 3 contain two layers of reinforcement. The plates are mounted in the centre of the ﬂat building roof to provide an undisturbed and direct solar radiation for the test bodies. Plate 1 and 2 are situated horizontally and parallel to the building roof in a height of 0.3 m above the ground. Whereas test object 3 is positioned in a height of 0.1 m above the building roof in vertical direction. The front side of this test plate is oriented to the south to get the maximal solar radiation impact at noon time. All test plates are equipped with thermo sensors on the front and the back side of the bodies to observe the outside surface temperature. Further more, simultaneous to this temperature measurement the basic atmospheric conditions are monitored. The wind speed and direction is measured next to the plates by an ultrasonic anemometer. The global radiation is recorded with a CM3pyranometer which is connected to the top side of plate 3 and the atmospheric temperature is measured by a thermo sensor (type k, class 2) at the feet of the ultrasonic anemometer.
40
2 DamageOriented Actions and Environmental Impact
CM3
@
Usonicanemometer data logger plate 1
thermo sensor Ts,pl3 plate 3 plate 2
thermo sensor Tair
Fig. 2.17. Test stand for the analysis of thermal actions on concrete specimen
A data logger in the centre of the test stand is used to collect all measured data in terms of time histories. For the measurements a sampling rate of one Hz is used for all sensors and the total time period of measurements is scheduled for one year. 2.2.4 Modelling of Short Term Thermal Impacts and Experimental Results Authored by J¨ org Sahlmen and Anne Spr¨ unken Seasonal and daily ﬂuctuations in solar radiation, cloudiness and spacious air exchange due to global weather conditions cause a permanent change in the air temperature. Hence, in a ﬁrst step of analysis the basic load of the thermal impact is subdivided in short term (daily) and long term (annually) actions. For the assessment of the short term action of the temperature on structures the ﬁeld experiment provides a fundamental data base and is helpful to understand the physical causal relations between atmospheric conditions and surface temperature at the test plates. The measurements at the RuhrUniversity Bochum have shown that the extreme values for the daily air temperatures can be found close before sunrise (minimum) and two to four hours after high noon
2.2 Thermal Actions
41
Fig. 2.18. Measured temperature proﬁle during a summer day
(maximum). Thereby the amplitudefrequency characteristic in general is sinusoidal over the day and the daily extremes are characterized by the location and the season. Figure 2.18 shows the measured daily characteristic of the surface temperature for the three test plates. The surface temperatures, measured every second, are plotted against a 24h period. The documented temperature distributions represent the typical behaviour of the airtemperature versus surface temperature on a structure during a summer day. Alternatively to the measurements the daily proﬁle of the air temperature can be approximately described with the following idealized approach [286]: t1 ≤ t ≤ t 2 : ϑair (t) = 0.5 · (ϑair,max + ϑair,min ) + 0.5 · (ϑair,max − ϑair,min ) · sin(π · (
2t − (t1 + t2 ) ) 2(t2 − t1 )
(2.53)
t2 ≤ t ≤ t3 : ϑair (t) = 0.5 · (ϑair,max + ϑair,min ) + 0.5 · (ϑair,max − ϑair,min ) · sin(−π · (
2t − (t2 + t3 ) ) 2(t3 − t2 )
(2.54)
42
2 DamageOriented Actions and Environmental Impact
Fig. 2.19. Rainﬂow analysis of the macroscopic temperature behaviour
where ϑair,max and ϑair,min describe the maximum and minimum air temperature to the time t1 and t2 , t3 is to calculate with t3 = t1 + 24. This model is useful for the assessment of the macroscopic temperature behaviour of the air temperature. Sudden temperature changes in a short time period (microscopic temperature behaviour) occur due to special weather events (e.g. strong rain, thunderstorm, etc.). The time duration for a sudden temperature jump induced by such weather phenomena is in the frame of 0.5 to 1 hour. In contrast to that long term drops in temperature need one or two days to increase visible the air temperature. Figure 2.19 gives an example for the macroscopic temperature behaviour in the Northern part of Germany based on a rainﬂow analysis. For the investigation a time period of 30 years, based on DWD dataset, is used. The temperature is plotted in dependency on mean value and amplitude. Output of the analysis is the number of events for each temperature class. Additionally a time history of the temperature for a 2years period is plotted. The ﬁgure shows a scattering for the mean values combined with relatively low amplitudes or rather low temperature diﬀerences. The daily rate is marginal whereas the annually behaviour is characterized by some single events with higher amplitudes. For the thermal loads on structures it can be presumed that the occurrence of lowcycle fatigue is low due to the fact that amplitude and number of events are not very signiﬁcant. Based on this result the numerical model is concentrated on the shortterm thermal actions. Especially, the nonlinear temperature impacts due to sudden changes in solar radiation are in the focus of the modelling. In Figure 2.20 the inﬂuence of the solar radiation on the air and surface temperatures for a summer period (2days term) is shown. Additional to the measured temperatures the corresponding calculated values are plotted for veriﬁcation.
2.2 Thermal Actions
43
Fig. 2.20. Temperature behaviour due to a sudden change in solar radiation
The strong correlation between the air and surface temperatures and the intensity of the solar radiation can be seen in Figure the plot. The temperature proﬁles follow with a time delay of about 3 to 4 hours the time history of the global solar radiation. For the calculated temperature trends over the observed time period a good estimation is found for the outer surface temperature. The numerical plot matches nearly exactly the measured distribution. In comparison to that the inner surface temperature is slightly overestimated by the numerical analysis, but the general trend is captured. 2.2.5 Application: Thermal Actions on a Cooling Tower Shell Authored by J¨ org Sahlmen and Anne Spr¨ unken As an application of thermal loads on structures under atmospheric conditions a cooling tower with a shell thickness of 0.15 m is analysed for an extreme winter situation, where the temperature diﬀerence between inner and outer shell could rise to 40 − 50 ◦ C. Main issue for the calculations in this step is to ﬁnd out how long it needs to get stationary temperature conditions in the shell for constant external actions. For the numerical analysis a constant temperature diﬀerence of ΔT = 45 K and a thermal inner transmission coeﬃcient αi = 20 W/(m2 K) [235] are assumed. At the outer surface the thermal transmission coeﬃcient is a function of wind speed. To minimise the calculation eﬀort a mean wind speed of vmean = 4 m/s is selected where both coeﬃcients have the same absolute value in terms of αi = αa = 20 W/(m2 K) [235]. Figure 2.21 presents as a result from the calculations the temperature distribution across the cooling tower shell (subdivided into 16 layers). The plotted shell temperature distributions for the layers is imaged with the inner temperature of the cooling tower and the outer atmospheric temperature as well as
44
2 DamageOriented Actions and Environmental Impact
Fig. 2.21. Temperature distributions determined at 16 layers within a cooling tower shell under constant external load actions
the intensity of the global radiation for a 4days period with a constant wind speed of 4 m/s. It can be seen that the given external impacts need about 40 to 60 hours to reach a stationary temperature distribution within the shell. For shorter time periods a nonstationary thermal reaction can be observed. In a further step the variation of the wind speed and the induced thermal distribution within the cooling tower shell is in the focus of the numerical studies. The surface temperature diﬀerence ΔT is calculated under the action of ten diﬀerent wind velocities, again for a 4day time period. Figure 2.22 shows the results of the calculations in comparison to the value given by the VGBguideline BTR [235]. Longtime observations of the surface temperatures of cooling towers have shown that the inﬂuence of the solar radiation in winter periods reduces the thermal actions due to the fact of a lower temperature diﬀerence between inner and outer shell in comparison to summer periods. The results in Figure 2.22 show for the variation of the wind speed a reverse trend for the temperature diﬀerences. It seems that an increase of the mean wind speed leads to an increase of the temperature diﬀerence ΔT . The reason for the rising is the cooling eﬀect at the outer surface of the cooling tower. This eﬀect is obviously visible for wind speeds between 0 and 10 m/s. For higher velocity steps the trend shows a lower signiﬁcance. In comparison with the constant temperature
2.2 Thermal Actions
45
Fig. 2.22. Eﬀect of the mean wind speed on the development of the temperature diﬀerence at the inner and outer surface of a cooling tower shell under external thermal load actions (4days load period [235])
diﬀerence given by the VGBguideline BTR [235] for a stationary status an underestimation of the temperature behaviour for higher wind velocities can be clearly seen. For lower wind speeds the guideline approach tends to be conservative. The results of the numerical analysis show for a strong intensity of solar radiation a reduction of the nonstationary behaviour of the temperature within a structure. The nonstationarities increase with lower global radiation action. This relation may lead to an increase of the thermal load action like it is shown in the comparison in Figure 2.22. Many load models used in the codes don’t include nonstationarities. In the current Eurocode [19] nonstationary thermal actions are included in chapter 6 for bridge decks, only. For buildings just the common rules are established. For advanced lifetime analysis concepts it is recommended to integrate nonlinear and timedependent temperature effects to precise the calculation of the stresses in the structure and hence, to optimize the detection of deterioration eﬀects and lifetime estimates. Uncertainties of representative values of temperature, of initial conditions and of material parameters as well as uncertainties of the numerical model have to be included for the deﬁnition of safety factors and combination coeﬃcients.
46
2 DamageOriented Actions and Environmental Impact
Recommendations for the implementation of such factors into Euro and national codes for bridges have been done by Lichte [495] and Mangerig [518].
2.3 Transport and Mobility Authored by Gerhard Hanswille and HansJ¨ urgen Niemann Highways, federal roads and high speed railway lines will in the future remain to be the most important parts of traﬃc infrastructure in Europe. All forecasts show that especially the amount of freight traﬃc on roads will continue to increase extremely over the next years. For the transport of persons over long distances especially the number of high speed railway lines will increase also signiﬁcantly. These developments have to be considered for realistic lifetime oriented design concepts especially with regard to the fatigue damage of structures and structural members. The basis of such design concepts is the realistic modelling of actions. The clauses 2.3.1 and 2.3.2 exemplify the modelling of actions with regard to the static resistance and to the realistic lifetime oriented fatigue resistance of structures. Clause 2.3.1 gives the basis of the development of load models for traﬃc loads on road bridges and clause 2.3.2 shows in principal the procedure of the development of the load models for noise barrier walls to take into account the aerodynamic loads caused by high speed trains. 2.3.1 Traﬃc Loads on Road Bridges Authored by Gerhard Hanswille For a realistic lifetime oriented design especially with regard to the fatigue damage of structures and structural members realistic models for traﬃc loads are needed. These models have to cover several special aspects, because long time prognoses for the whole design life of a structure are necessary, e.g. 100 years for bridges. Traﬃc loads on road bridges are a good example, where several aspects must be considered for the development of lifetime oriented design concepts. 2.3.1.1 General In this case it should be pointed out, that especially for actions on bridges the models must cover current national and European traﬃc data and future developments due to the cross border trade. The main inputs required for the development of realistic traﬃc load models for bridges are: •
the currently available traﬃc data in Europe with information about the axle and vehicle weights, the diﬀerent types of lorries and information of the European traﬃc composition,
2.3 Transport and Mobility
• • • • •
47
information about the inﬂuence of the dynamic behaviour of the vehicles and the bridge structures including information about the pavement quality, information about the diﬀerent types of bridge structures and the corresponding inﬂuence surfaces, principles for the model calibration for ultimate limit and fatigue limit states and the damage accumulation under consideration of diﬀerent materials, methods for the exploitation of the currently available traﬃc data, development of large capacity and heavy load transports not covered by the normal traﬃc models, the inﬂuence of future political decisions with regard to new traﬃc concepts.
2.3.1.2 Basic European Traﬃc Data With regard to the cross border trade, load models must be based on traﬃc data which are representative for the European traﬃc. For example the development of the models in Eurocode 12 [9] is based on data collected from 1977 to 1990 in several European countries [487, 720, 530, 37, 157, 361, 158]. The main data basis with information about the axle weights of heavy vehicles, about the spacing between axles and between vehicles and about the length of the vehicles came from France, Germany, Italy, United Kingdom and Spain. Most of the data relate to the slow lane of motorways and main roads and the duration of records varied from a few hours to more than 800 hours. Another important point is the medium ﬂow of heavy vehicles per day on the slow lane. In order to analyse the composition of the traﬃc for the development of the load model in [9] four types of vehicles were deﬁned for the European load model for bridges. Type 1 is a doubleaxle vehicle, Type 2 covers rigid vehicles with more than two axles, Type 3 articulated vehicles and Type 4 draw bar vehicles. Figure 2.23 shows the typical frequency distribution of these four types resulting from traﬃc records of the Auxerre traﬃc in France. The data base of diﬀerent countries shows that the traﬃc composition is not identical in various European countries. The most frequent types of heavy vehicles are 1 and 3. Especially in Germany the traﬃc records in 1984 show that lorries with trailers (Type 4) dominated the traﬃc composition at that time. The traﬃc records of the Auxerre traﬃc (Motorway A6 between Paris and Lyon) gave a full set of the required information for the development of an European load model. In addition the Auxerre traﬃc includes a high percentage of heavy vehicles and gives a representative data base for the development of a realistic European load model. Figure 2.23 shows the distribution of the above explained types of heavy vehicles based on the Auxerre traﬃc records. Figure 2.24 shows the gross vehicle weight and the axle load distributions for the representative traﬃc in Auxerre and Brohltal (Germany) where n30 is the number of lorries with G ≥ 30 kN and n10 the number of axles with PA ≥ 10 kN. Especially for the development of models for the fatigue resistance
48
2 DamageOriented Actions and Environmental Impact N
N
800 700 600 500 400 300 200 100
80 70 60 50 40 30 20 10
Type 1 G(kN) 120
240
360
480
600
Type 2 G(kN)
720
120
N
240
360
480
600
720
N
1400
160 140 120 100 80 60
1200 1000 800 600 400 200
Type 3
G(kN) 120
240
360
480
600
Type 4
40 20
G(kN)
720
120
240
360
480
600
720
Fig. 2.23. Frequency distribution of the total weight G of the representative lorries per 24 hours based on traﬃc data of Auxerre in France (1986)
of structures further traﬃc records regarding the number of heavy vehicles per day are needed. These data were taken for the load model in [9] from several traﬃc records in Europe. From all the traﬃc records only the record locations
G[kN] 750 600 450
PA[kN]
total weight of heavy vehicles
Périphérique Brohltal Doxey
300
150
Brohltal Forth
100
Doxey
Auxerre
Forth
50
150 104
axle loads
200
Auxerre
103
102
101
1,0
n n30
104
103
102
101
n 1,0 n10
Fig. 2.24. Gross vehicle and axle weight distribution of recorded traﬃc data from England, France and Germany
2.3 Transport and Mobility
49
Table 2.3. Statistical parameters of the traﬃc records of Auxerre (1986) mean value P of the total vehicle weight kN
standard deviation V kN
relative frequency %
Lane 1
Lane 2
Lane 1
Lane 2
Lane 1
Lane 2
Type 1
Go Gl
74 183
64 195
35 28
33 34
13,3 9,4
17,2 10,4
Type 2
Go Gl
123 251
107 257
46 38
45 43
0,3 1,0
1,3 2,2
Type 3
Go Gl
265 440
220 463
60 54
78 79
17,1 48,1
28,0 30,4
Type 4
Go Gl
254 429
196 443
45 68
69 78
3,6 7,2
4,1 6,4
N
Type 4
1500
Go G1 10,8%
10,5%
65,2 %
58,4%
1,3 %
3,5 %
22,7 %
27,6 %
lane 1
lane 2
Type 3 1000 Type 3 Type 2
500
G1
Go 120
240
360
G(kN) 480
600
Type 1
720
Fig. 2.25. Histogram of vehicle Type 3 and approximation by two separate distribution functions based on traﬃc data of Auxerre in France (1986 ) and frequency of the diﬀerent vehicle types in the lanes 1 and 2
with a high rate of heavy vehicle in the total traﬃc are of interest, for example the traﬃc records of Brohltal and Auxerre in Figure 2.24. The histograms acc. to Figure 2.23 can be subdivided into two separated density functions, where the mean values correspond to loaded and unloaded vehicles. The statistical parameters of these distribution functions are given in Table 3.6. For the vehicle of Type 3 the distributions are shown examplarily in Figure 2.25. Furthermore for the development of the load model the frequency of the diﬀerent vehicle types in the lanes 1 and 2 is needed. The records based on the Auxerre traﬃc are given in Figure 2.25. The number of axles per vehicle varies widely depending on the diﬀerent vehicle manufactures. Nevertheless the frequency distributions of the axle
50
2 DamageOriented Actions and Environmental Impact
Table 2.4. Relation between gross weight of the heavy vehicles and the axle weights of the lorries of types 1 to 4 in % (mean values and standard deviation) Type of vehicle Type 1
Go Gl Go Gl Go Gl Go Gl
Type 2 Type 3 Type 4
Axle 1 m V 50,0 8,0 35,0 7,0 40,5 8,4 29,4 5,7 30,6 5,8 17,1 2,4 31,7 5,7 18,5 4,1
Axle 2 m V 50,0 8,0 65,0 7,0 36,2 8,8 42,8 4,2 27,5 4,4 26,9 4,4 31,3 5,8 29,1 4,2
Axle 3 m V
23,7 27,8 16,2 19,9 13,4 18,9
7,3 5,3 3,6 3,0 4,1 3,6
Axle 4 m V
13,6 19,0 13,7 18,3
Axle 5 m V
3,1 12,1 2,8 16,7 3,5 9,9 3,4 15,2
3,1 3,8 3,3 4,3
Table 2.5. Distance of axles in [m] of the diﬀerent types of vehicles (mean values and standard deviation) Type of vehicle Type 1 Type 2 Type 3 Type 4
Axle 12 m V
Axle 23 m V
Axle 34 m V
Axle 45 m V
3,71
1,1
3,78
0,71
1,25
0,03
3,30
0,26
4,71
0,78 1,22 0,13 1,23
0,14
4,27
0,40
4,12
0,31 4,00 0,42 1,25
0,03
pacings show three cases with peak values nearly constant and very small standard deviations (vehicles of types 2, 3 and 4 with a space of 1.3 m corresponding to double and triple axles and with a space of 3.2 m corresponding to tractor axles of the articulated lorries). For the other spacings widely scattered distributions were recorded resulting from the diﬀerent construction types of vehicles. As mentioned before, the traﬃc data given in Figures 2.23 and 2.24 are based on the traﬃc records of the Auxerre traﬃc in France. These data gave no suﬃcient information about the distribution of gross vehicle weight G on the single axles. Additional information from the traﬃc records of the Brohltal Traﬃc in Germany (Highway A61) was used to deﬁne single axles weights and the spacing of the axles. These data (mean values of axle weight and axle spacing and corresponding standard deviations) are given in Tables 3.7 and 3.8. A further important parameter is the description of diﬀerent traﬃc situations. For the development of load models the normal free ﬂowing traﬃc as
2.3 Transport and Mobility
f(a)
51
f(a)
0,005 D 90
0,004 0,003
O(1 D )
0,002 0,001
a[m]
a[m] 200
400
600
20
100
Fig. 2.26. Comparison of measured and theoretical values for the density function of intervehicle distances
well as condensed traﬃc and traﬃc jam have to be distinguished. The main parameters of the probability density functions for the distance are the lorry traﬃc density per lane (lorries per hour), the ratio between lorries and motorcars, the mean speed and the probability of occurrence of lorry distances less than 100 m to cover the development of convoys. A typical example for the distribution of distances measured at motorway A7 near Hamburg is given in Figure 2.26 and compared with an analytical function for high traﬃc densities given in [720]. The density function is approximated by a linear increase up to 20 m due to the minimum distance, a constant part up to a distance of 100 m because of convoys and an exponentially decreasing part for distances greater than 100 m for covering free ﬂowing traﬃc. Another possibility is the approximation of the intervehicle distance by a lognormal distribution [305] which is based on new traﬃc data [314]. In Figure 2.26 the value α of the constant part between 20 and 100 m, giving the probability of occurrence for lorry distances less than 100 m, and the value λ were obtained from traﬃc records of 24 representative traﬃcs in Germany. Additional information regarding the probability of occurrence of convoys are given in [267]. These accurate models apply mainly to the development of fatigue load models. Regarding load models for ultimate and serviceability limit states simpliﬁed models for the vehicle distances can be used on the safe side. In case of ﬂowing traﬃc the distance between lorries is given by a minimum distance required, which results from a minimum reaction time of a driver to avoid a collision with the front vehicle in case of braking. On the safe side a minimum braking reaction time Ts of the driver of one second is assumed. Then the minimum distance a is given by a = v · (Ts ) where v is the mean speed of the vehicles. With this assumption also convoys are covered. The distance is limited to a minimum value of 5 m in case of jam situations.
52
2 DamageOriented Actions and Environmental Impact
2.3.1.3 Basic Assumptions of the Load Models for Ultimate and Serviceability Limit States in Eurocode As mentioned before, the load model in Eurocode 1 is mainly based on the traﬃc records of the A6 motorway near Auxerre with 2 × 2 lanes because these measurements were performed over long time periods in both lanes of the Highway and because these data represent approximately the current and future European traﬃc with a high rate of heavy vehicles related to the total traﬃc amount and also with a high percentage of loaded heavy vehicles (see also Figure 2.24). The European traﬃc records had been made on various locations and at various time periods. For the deﬁnition of the characteristic values of the load model therefore the target values of the traﬃc eﬀects have to be determined. For Eurocode 12 it was decided, that these values correspond to a probability p = 5% of exceeding in a reference period RT = 50 years which leads to a mean return period of 1000 years. For the determination of target values of the traﬃc eﬀects additional aspects have to be considered. The measurements of the moving traﬃc (e.g. by piezoelectric sensors) include some dynamic eﬀect depending on the roughness proﬁle of the pavement and the dynamic behaviour of the vehicles which has to be taken into account for modelling the traﬃc. The dynamic eﬀects of the vehicles can be modelled acc. to Figure 2.27 taking into account the mass distribution of the vehicle, the number and spacing of axles, the axle characteristic (laminated spring, hydraulic or pneumatic axle suspension), the damping characteristics and the type of tires [720, 530, 238, 99, 330, 331]. The normal surface roughness can be modelled by a normally distributed stationary ergodic random process. The roughness is a spatial function h(x) and the relation between the spatial frequency Ω and the wave length L is given by Ω = 2π/L [1/m]. In the literature many surfaces have been classiﬁed by power spectral densities Φh (Ω) acc. to Figure 2.27. Increasing exponent w results in a larger number of wave length and increasing Φh (Ω) results in larger amplitudes of h(x). For modelling the surface roughness of road bridges w = 2 can be assumed. The quality of the pavement of German roads can be classiﬁed for motorways as ”very good”, for federal road as ”good” and for local roads as ”average”. While for the global eﬀects of bridge structures an average roughness proﬁle can be assumed, for shorter spans up to 15 m local irregularities (e.g. located default of the carriageway surface, special characteristics at expansion joints and diﬀerences of vertical deformation between end cross girders and the abutment) have to be taken into account. These irregularities were modelled in Eurocode 12 by a 30 mm thick plank as shown in Figure 2.27. As mentioned above, the axle and gross weights of the vehicles of the Auxerre traﬃc were measured by piezoelectric sensors. The calculations with ﬁxed base and the vehicle model acc. to Figure 2.27 showed for good pavement quality, that the characteristic values determined from the measured gross and axle weights include a dynamic ampliﬁcation of approximately 15% of
2.3 Transport and Mobility
ª:º ) h ( : ) ) h ( : o )« » ¬ :o ¼
101
100
101
w
102
102
spatial frequency :=2S/L [m1]
z
30 200 300
103 102 101 100 101
power spectral density )h(: ) [cm3]
102
Model for irregularities
6 )=1 (: o 4 )h )= nt (: o me )h ave ent ep )=1 em rag (: o pav )h od ent go em pav
unevenness of the carriageway
ave
spring and damper of the tyre
:o=1 m1 w=2
ood
h(x)
spring and damper of the vehicle body mass of the axle
yg
M
v er
mA,TA
S
103
PSD spectras acc. to ISOTC 108
Modelling of the vehicles
x
53
L
200
+h
+x[m] h
Fig. 2.27. Model for the vehicles and local irregularities and power spectral density of the pavement
the axles weights and 10% of the vehicle gross weight. The ﬁltering of the dynamic eﬀects leads in comparison to the measured values to a reduced standard deviation. The corrected data of the static vehicle weights are given in Table 3.9. The dynamic behaviour of the bridge structure is mainly inﬂuenced by the span length and the dynamic characteristics of the structure [169] (eigenvalues acc. to Figure 2.28 and the damping characteristics). With the vehicle model and the modelling of the roughness of pavement surface acc. Table 2.6. Statistical parameters of the corrected static traﬃc records of Auxerre (1986)
Type 1 Type 2 Type 3 Type 4
Go Gl Go Gl Go Gl Go Gl
mean value P of the total standard deviation V vehicle weight [kN] [kN] lane 1 lane 2 lane1 lane 2 74 64 31 29 183 195 23 28 123 107 40 39 251 257 31 35 265 220 51 68 440 463 42 65 254 196 37 60 429 443 55 64
2 DamageOriented Actions and Environmental Impact
10
f
Comparison of calculated and measured dynamic amplification
Eigenvalues (1. mode)
f [Hz]
95,4
1 L0,933
70
V r 0,81 Hz
dynamic amplification in [%]
54
8
6
4
2
10
20
30
40 50 60
70 80
span length in [m]
90
60
36,95
calculated values
41,0m 32,35
50 measured values
40 30 20 10
10
20
30
40
50
60
70 80
vehicle speed [km/h]
Fig. 2.28. Measurements of the eigenvalues of the ﬁrst mode of steel and concrete Bridges [169], and comparison of theoretically determined dynamic ampliﬁcations with measurements
to Figure 2.27 results can be obtained by dynamic calculations of the bridge and be compared with measurements at bridges. Figure 2.28 shows an example of the calculated and measured dynamic ampliﬁcation of the DeibelBridge [720]. With the assumptions and models explained above, a realistic determination of the dynamic and static action eﬀects due to traﬃc loads is possible. In a ﬁrst step random generations of load ﬁles and roughness proﬁles of the pavement surface can be produced. Each load ﬁle consists of lorries with distances based on constant speed per lane. The main input parameters are the number and types of lorries, the probability of occurrence of each lorry type, the histogram of the static lorry weights of each type, the distribution of lorries to several lanes. For the load ﬁles simply supported and continuous bridges with one, two and four lanes and diﬀerent span lengths between 1 and 200 m with a representative dynamic behaviour (mass, ﬂexural rigidity, mean frequency acc. to Figure 2.28 and damping) have to be investigated in order to get results which are representative for the dynamic ampliﬁcation of action eﬀects of common bridges. Three diﬀerent types of bridges with crosssections with one, two and four lanes were investigated for the load model in Eurocode 12. For the diﬀerent lanes the traﬃc types acc. to 3.10 were assumed, where traﬃc type 1 is a heavy lorry traﬃc for which motorcars were eliminated from the measured Auxerre traﬃc. The traﬃc type 2 is the measured traﬃc of lane
2.3 Transport and Mobility
55
Table 2.7. Diﬀerent crosssections and traﬃc types for the random generations
number of lanes
type of cross section
traffic types of the different lanes
3,0 m
1
Type 1 3,0 m 3,0 m
Lane 1: Type 1 Lane 2: Type 2
2
3,0 m 3,0 m
3,0 m
Lane 1: Type 1 Lane 2: Type 3 Lane 3:Type 3 Lane 4: Type 2
3,0 m
4
1 in Auxerre, including motorcars and traﬃc type 3 is the measured traﬃc of lane two in Auxerre. Detailed information about the generation of these load ﬁles are given in [720, 530]. With random load ﬁles the static and the dynamic action eﬀects of the diﬀerent bridge types can be determined. The comparison of the static and dynamic action eﬀects gives information about the dynamic ampliﬁcation and the dynamic factor Φ, inﬂuenced by the dynamic behaviour of the lorries, the bridge structure and by the quality of the pavement. The results of the simulations can be plotted in diagrams which give the cumulative frequency of the action eﬀects. A typical example is given in Figure 2.29 for a bridge with
cumulative frequency [%]
99,9 97
traffic jam
50 convoy v= 80 km/h
ME
convoy v= 60 km/h convoy v= 40 km/h
action effect 700
1000
1300
ME [kNm]
Fig. 2.29. Cumulative frequency of the action eﬀects for diﬀerent vehicle speeds [530]
56
2 DamageOriented Actions and Environmental Impact M 2,2 2,0 1,8
pavement irregularities (30 mm thick plank)
1,6
flowing traffic and average pavement quality
1,4 1,2
flowing traffic and good pavement quality
1,0 0,8
L [m] 10
20
30
40
50
60
70
80
Fig. 2.30. Inﬂuence of the quality of the pavement on the dynamic ampliﬁcation factor ϕ[530]
one lane, good pavement quality and a span of 20 m. It can be seen that for this example the increase of the vehicle speed leads also to an increase of the dynamic action eﬀects. Furthermore the dynamic ampliﬁcation is extremely inﬂuenced by the roughness of the pavement and also by the span of the bridge. The inﬂuences of the pavement quality and traﬃc in more than one lane are shown in Figures 2.30 and 2.31. The results of the simulations show for condensed traﬃc no signiﬁcant inﬂuence of the span length and the number of loaded lanes on the dynamic ampliﬁcation. In case of ﬂowing traﬃc the dynamic ampliﬁcation of action eﬀects depends signiﬁcantly on the quality of the pavement, the number of loaded lanes, the span length and the type of the inﬂuence line of the action eﬀect considered.
M
M
1,8
1,8
1,6
1,6
bending moment
1,4
1,4
vertical shear
1,2
bending moment
1,2 L [m]
L [m] 5 10
15
20
25
30 35
10
20 30 40 50 60
70 80
Fig. 2.31. Inﬂuence of the span length and the number of loaded lanes on the dynamic ampliﬁcation factor ϕ
2.3 Transport and Mobility
57
'M 1,3 1,2 1,1
30
1,0 L[m] 5
10 15
20
25
30
200 300
200
Model for irregularities
Fig. 2.32. Additional dynamic factor Δϕ taking into account irregularities of the pavement [9]
Figure 2.31 shows the envelope of the calculated dynamic factors ϕ for ﬂowing traﬃc as a function of the span length. For the development of the load model in Eurocode 12 it was decided, that the dynamic ampliﬁcation of the action eﬀects should be included in the load model because otherwise diﬀerent parameters like the traﬃc situation (ﬂowing traﬃc or traﬃc jam, the quality of the pavement, the number of loaded lanes and the type of the inﬂuence line) had to be considered separately. The calculations show additionally, that the dynamic ampliﬁcation due to ﬂowing traﬃc is only relevant for shorter span length up to 50 m because for greater span length the condensed traﬃc with low vehicle spacings or the traﬃc jam lead to extreme action eﬀects. As explained above the dynamic eﬀects due to local irregularities were modelled by a 30 mm thick plank, which leads especially for shorter spans to a signiﬁcant additional dynamic ampliﬁcation factor. Figure 2.32 gives the additional dynamic factor Δϕ due to irregularities which has to be considered especially for fatigue veriﬁcations for short spans, e.g. for end cross girders and members near expansion joints (see Figure 2.32). With the random load ﬁles the static and the dynamic action eﬀects and the characteristic values of the action eﬀects can be determined. As mentioned above, the characteristic values in Eurocode 12 correspond to a probability p = 5% of exceeding in a reference period R = 50 years which leads to a return period of TR = 1000 years. The procedure for the determination is shown in Figure 2.33. The simulation of diﬀerent bridge types gives a cumulative frequency of the considered action eﬀects. The characteristic values can be determined by extrapolation. Finally these characteristic values can be compared with a simpliﬁed characteristic load model. The load model for global eﬀects in Eurocode 12 [9] consists of uniformly distributed loads and simultaneously acting concentrated loads, so that global eﬀects in large spans and the local eﬀects in short spans can be covered by
2 DamageOriented Actions and Environmental Impact
99,90 99,00
50,00
extrapolation for the determination of the characteristic values
dynamic amplification factor: I
static values of simulations
E k ,dyn E k ,stat.
ME
dynamic values of simulations
Ek,dyn
cumulative frequency [%]
99,9999
Ek,stat.
58
action effect
ME
influence line for ME
Fig. 2.33. Determination of the characteristic values of the action eﬀects from the random generations of loads
the same model taking into account the dynamic ampliﬁcation, where average pavement quality is expected. The carriageway with the width w is measured between kerbs or between the inner limits of vehicle restraint systems. For the notional lanes a width of wl = 3,0 m is assumed, and the greatest possible number nl of such lanes on the carriageway has to be considered. The locations of the notional lanes are not be necessarily related to their numbering. The lane giving the most unfavourable eﬀect is numbered as Lane Number 1, the lane giving the second most unfavourable eﬀect is numbered as Lane Number 2 and so on. For each individual veriﬁcation the load models on each notional lane and on the remaining area outside the notional lanes have to be applied on such a length and longitudinally located so that the most adverse eﬀect is obtained. The Load Model 1 in Eurocode 12 is shown in Figure 2.34. It consists of a double axle as concentrated loads (Tandem System TS) and uniformly distributed loads (UDLSystem). For the veriﬁcation of global eﬀects it can be assumed that each tandem system travels centrally along the axes of notional lanes. For local eﬀects the tandem system has to be located at the most unfavourable location and in case of two neighbouring tandem systems they have to be taken closer, with a distance between wheel axles not smaller than 0,5 m. With the adjustment factors αQi and αqi the expected traﬃc on diﬀerent routes can be taken into account. The last step in the development of the load model is the comparison of the characteristic action eﬀects caused by the normative load model with the characteristic values of the dynamic values of the real traﬃc simulations. Figure 2.35 shows this comparison for a three span bridge girder with one, two and four lanes. For the veriﬁcation of local eﬀects a Load Model 2 is given in Eurocode 12. This model consists of a single axle load equal to 400 kN, where the
2.3 Transport and Mobility
Application of the Tandem System for local verifications
Application of the Tandem System for global verifications DQi Qik
1,20m
Dqi qik
2,00m 2,00m > 0,50m
DQi Qik
0,50
w1
Lane number 1: Q1k = 300 kN aQ1q1k = 9 KN/m²
2,00 0,50 0,50
w2
Lane number 2: Q2k= 200 aQ2 q2k = 2,5 KN/m²
2,00 0,50 0,50
w3
59
contact area of the wheel loads
Lane number 3: Q3k= 100 aQ2 q2k = 2,5 KN/m²
2,00 0,50
0,4 m Lane number 4 and further lanes as well as remaining areas: aQ3 q3k = 2,5 KN/m²
wi
0,4 m
Fig. 2.34. Load Model 1 according to Eurocode 12
ME/L 500
Load Model 1 acc. to Eurocode 1 400
simulation 300
200
ME
100
L
L 20
40
span length
60
80
L
L
Fig. 2.35. Comparison of the Load Model 1 in Eurocode 2 with the characteristic values obtained from real traﬃc simulations
dynamic ampliﬁcation for average pavement quality is included. In the vicinity of expansion joints an additional dynamic ampliﬁcation has to be applied for
60
2 DamageOriented Actions and Environmental Impact
Table 2.8. Traﬃc data of diﬀerent locations and characteristic values of gross and axle weight [720] number nl of lorries per day
weight of one axle kN
tandem axles kN
tridem axles kN
gross weight of vehicle kN
1984
4793
211
357
434
853
Chamonix
1987
1204
192
355
480
724
Auxerre
1986
2630
245
397
527
811
France
Angers
1987
1272
192
340
456
670
France
Lyon
1987
1232
267
450
475
930
country
location
year
Germany
Brohltal
Belgium France
Table 2.9. Diﬀerent design situations and corresponding return periods and fractiles
Design situation
Return period TR
infrequent frequent quasi  permanent
1 year 1 week 1 day
Fractile of the distribution of action effects in % 99,997 99,891 99,240
taking into account the local irregularities at expansion joints. The contact surface of each wheel can be taken into account as a rectangle of sides 0,35 m and 0,6 m. The evaluation of the traﬃc data of diﬀerent locations lead to static characteristic axle values Qk given in Table 3.11, where the characteristic values relate to a return period TR of 1000 years (probability p of 5% in 50 years). It can be seen that the characteristic values are depending on the location. Taking into account the dynamic ampliﬁcation for short spans (see Figure 2.31), this leads to the axle weight given in Eurocode 12. For serviceability limit states like limitation of deﬂections, crack width control and limitation of stresses to avoid inelastic behaviour, diﬀerent design situations have to be distinguished. The Eurocodes distinguish between infrequent, frequent and quasi permanent design situations characterised by diﬀerent return periods. The return periods and the corresponding fractile of the distribution of the dynamic action eﬀects are given in Table 3.12. A change of the return period is equivalent with a change of the fractile of the distribution (see Figure 2.36). The representative values Frep of the action eﬀects can then be written as Frep = ψ Fk , where Fk is the characteristic value. As explained above, the characteristic values were determined with adverse assumptions regarding the quality of the pavement Φ(Ωh ) = 16 acc. to
2.3 Transport and Mobility
representative valuesErep=\ Ek
characteristic values:
static values of simulations
I
Ek,dyn
Ek,stat.
50,00
Erep,dyn.
dynamic values of simulations Erep,stat.
cumulative frequency [%]
99,90 99,00
dynamic amplification factor
characteristic values Ek
99,9999
61
E k ,dyn E k ,stat.
representative values: E rep,dyn I E rep,stat. action effect E
ME Fig. 2.36. Determination of the representative values and the corresponding dynamic factors
2 lanes
condensed traffic and traffic jam (v= 10km/h)
1,0 0,8
4 lanes
0,6
ME
0,4
flowing traffic (v= 80km/h)
0,2
L
L
L
L[m] 10
20
30
40
50
60
70
80
Fig. 2.37. Factors ψT R for frequent design situations acc. to [37] for average pavement quality with Φ(Ωh ) = 16
Figure 2.27, the composition of the traﬃc (100% lorries in the ﬁrst lane) and a probability of traﬃc jam of 100%. The combination values taking into account these assumptions lead to values ΨT R , which only cover the inﬂuence of the return period TR . Figure 2.37 shows an example for the frequent design situation [37] for average pavement quality. It can be seen that the values ΨT R are dependent on the span length, the traﬃc situation and the number of lanes. The condensed traﬃc and traﬃc jam give the greatest values ΨT R . The values ΨT R can be reduced by additional factors to be more close to reality. As mentioned before the quality of the pavement has a signiﬁcant inﬂuence on the dynamic action eﬀects. On the basis of a good pavement quality with Φ(Ωh ) = 4 acc. to Figure 2.27 which can be assumed e.g. for
62
2 DamageOriented Actions and Environmental Impact
condensed traffic and traffic jam (v= 10km/h)
0,8
average pavement quality )(:h) =16
0,6
good pavement quality )(:h) =4
ME
flowing traffic (v= 80km/h)
0,4
0,2
L
L
L
L[m] 10
20
30
40
50
60
70
80
Fig. 2.38. Inﬂuence of the pavement quality on the factor ΨT R for frequent design situations
highways and federal roads in Germany, a reduction factor for the dynamic action eﬀects of ΨΩ ≈ 0.89 results from the simulations. The inﬂuence of the pavement quality is shown in Figure 2.38 as a function of the span length. A second reduction factor covers the inﬂuence of the probability of traﬃc jams. Based on the evaluations in [267] with a rate of ν = 3 × 10−3 traﬃc jams per kilometre and day a percentage of traﬃc jam of 6% to 9% of the annual traﬃc results in comparison with the Auxerre traﬃc. This leads to the reduction factor Ψν ≈ 0.95. A further reduction factor Ψv covers the eﬀect of the realistic traﬃc composition (percentage of the lorry on each lane). For a mean rate of 32% of lorries related to the total traﬃc the factor Ψv for bridges with one lane is approximately 0.9 for a return period of 1 week and 0.96 for a return period of 1 year. For bridge structures with two lanes values of 0.74 and 0.76 result from the calculations in [530, 37]. With the additional reduction factors values Ψ = ΨT R ΨΩh Ψν result which are in the range of 0.8 for the infrequent and 0.7 for the frequent design situations of bridges with small spans up to 40 m where the single axle loads dominate the actions eﬀects. For spans exceeding 40 m the ﬂowing traﬃc with mainly uniform distributed loads gives values Ψ ≈ 0.8 for the infrequent and 0.45 for frequent design situations. These values correspond to the values in Eurocode 12 (Table 3.13). 2.3.1.4 Principles for the Development of Fatigue Load Models Fatigue is the progressive, localized and permanent structural change occurring in materials subjected to ﬂuctuating stresses initiating and propagating cracks through a structural part after a suﬃcient number of load cycles. Fatigue is induced in bridges mainly by heavy vehicles. The development of appropriate load models and veriﬁcation concepts is a main topic in modern
2.3 Transport and Mobility
63
Table 2.10. Factors Ψ for the determination of the representative values for serviceability limit states acc. to [9] Load Model 1 uniform distributed tandem system loads infrequent design situations frequent design situations quasi permanent design situations
Load Model 2 single axle
0,8
0,8
0,8
0,75
0,4
0,75
0
0
0
bridge design. As mentioned above the load model in Eurocode 12 is based on the Auxerre traﬃc which covers heavy European continental traﬃc. Therefore also for the fatigue load models in Eurocode 12 the Auxerre traﬃc was used for the prenormative investigations. For typical bridges, random ﬁles of the traﬃc loads, the traﬃc ﬂow and the intervehicle distances were produced for the determination of the dynamic action eﬀects, which take into account the pavement quality and the dynamic behaviour of the vehicles and the bridge structure (Figure 2.39). From this calculation the time history of stresses or internal forces is obtained and with the rainﬂowcounting or reservoir method [201] the spectrum of the action eﬀects can be determined. The next step is the determination of the damage accumulation based on given fatigue strength curves based on a damage accumulation hypothesis. For steel structures and for reinforcement and prestressing steel the fatigue strength curves acc. to Figure 2.40 can be used, where the fatigue strength curve for steel structures [30] is deﬁned by the fatigue strength category (fatigue strength at two millions cycles) and the constant amplitude fatigue limit ΔσD at 5 million cycles. For stress ranges above ΔσD the slope m of the curve in a double logarithmic scale is equal 3 and for stress ranges less than ΔσD a slope m = 5 can be assumed. The fatigue strength curves were derived from international evaluations of fatigue tests with large scale test specimen. For stress ranges of the design spectrum which are below the cutoﬀ limit ΔσL at 108 load cycles it may be assumed that these stress ranges do not contribute to the calculated cumulative damage. Typical examples for fatigue strength categories in steel and composite bridges are shown in Figure 2.41. The fatigue strength curves in Figures 2.40 and 2.41 relate to nominal stresses. For steel reinforcement bars the fatigue strength curve is given in [28, 437] and described by a two linear function in the double logarithmic scale without any constant amplitude fatigue limit (Figure 2.40). While for steel structures normally a linear relation can be assumed between the fatigue loading and the stresses, for concrete structures the non linear behaviour due to cracking of concrete has to be taken into account for the determination of the time history of the stresses. In this case in addition to the fatigue loading also the dead load and eﬀects due to climate temperature actions have to be considered [437].
64
2 DamageOriented Actions and Environmental Impact
traffic loading and traffic composition
stress history of the dynamic action effects and cycle counting (e.g. reservoir or rain flow V method) 'V1
'V2
'V3
influence line for the stress Vi 'V4
time t
stress Vi
'V(log)
damage accumulation D
¦ Di
'Vi
stress range spectrum
'V
n ¦ i d 1,0 N Ri
'V1 'V2
fatigue strength curve
'V3 'V4
N (log) ni
N n1
NRi
n2
n3
n4
Fig. 2.39. Determination of stress spectra and damage accumulation due to fatigue loading
Fatigue strength curves for structural steel
'VR log)
1
'VR(log) 'V Ri
1 'Vc
Fatigue strength curves for reinforcement and prestressing steel
m
ªN ºm 'V D « D » ¬ Ni ¼
1
i
'V Rs,i
m1
1
m1=3
m
'VRsk
ª N º m 'V Rsk « » ¬ Ni ¼
m2
m2=5
'VD 'VL
NR(log) N*
NR(log) Nc
ND
i
NL
'VC  Detail Category NC= 2 · 106 'VD  Constant amplitude fatigue limit ND= 5 · 106 'VL  Cut off limit NL= 108
type of reinforcement
N*
'VRsk at N* cycles [N/mm2]
m1
m2
5
9
straight bars
106
162,5
welded bars and wire fabric
107
58,5
3
5
splicing devices
107
35
3
5
Fig. 2.40. Fatigue strength curves for structural steel and reinforcement
The main issues in the development of fatigue load models is the damage accumulation hypothesis. In civil engineering normally a linear damage
2.3 Transport and Mobility
4
1
'V
Category 125 'Vc= 125 N/mm2
4
'V Category 80 'Vc= 80 N/mm2
65
automatic butt weld carried out from both sides
'V
'V
L Category 71 'Vc= 71 N/mm2 for Ld 50mm
2 'V
'V >10mm
2 3
3
L
'V
L Category 56 'Vc= 56 N/mm2 for L>100mm
1
'V
Fig. 2.41. Typical examples for fatigue strength categories
accumulation according to Miner [543] is used (Figure 2.40). Based on this assumption a realistic fatigue load model must fulﬁl the condition, that the cumulative damage produced by the real traﬃc must be equal to the cumulative damage caused by the load model. The main parameters which have to be considered are the design fatigue life, the type and number of lorries crossing the bridge, the traﬃc composition and the number of lanes with heavy traﬃc and in addition the quality of the pavement and the dynamic behaviour of the vehicles and the bridge. For fatigue problems of bridges only the traﬃc situation of ﬂowing traﬃc has to be considered because the number of traﬃc jams is negligible during the design life. Furthermore the inﬂuence of motorcars can be neglected, because the stress ranges caused by motorcars do not reach the cut oﬀ limit of the fatigue strength curves. For the development of fatigue load models further considerations are necessary. For Eurocode 12 e.g. it was decided that the load model should include the dynamic ampliﬁcation of the real traﬃc. Regarding the modelling several strategies are possible. One possibility is to consider only one type of vehicle in veriﬁcations and to take into account all other eﬀects resulting from the real traﬃc by damage equivalent factors. This is the basis of the Load Model 3 in Eurocode 12. An other possibility is the deﬁnition of a set of lorries which together produce eﬀects equivalent to those of typical traﬃc on European roads. An example for such a model is the Load Model 4 in Eurocode 12 (Figure 2.42). The fatigue models 3 and 4 are intended to be used for fatigue life veriﬁcations by reference to a fatigue strength curve. For the fatigue life veriﬁcation it has to be distinguished between diﬀerent traﬃc categories. The category is deﬁned by the number of slow lanes, the number Nobs of heavy vehicles with a maximum gross weight more than 100 kN which was observed
66
2 DamageOriented Actions and Environmental Impact
Table 2.11. Traﬃc categories acc. to Eurocode 12
Traffic category
Nobs per year and per slow lane
1
Roads and motorways with 2 or more lanes per direction with high flow rates of lorries
2,0 x 106
2
Roads and motorways with medium flow rates of lorries
0,5 x 106
3
Main roads with low flow rates of lorries
0,125 x 106
4
Local roads with low flow rates of lorries
0,05 x 106
or estimated per year and per slow lane. Typical traﬃc categories acc. to Eurocode 12 are given in Table 3.14. Fatigue Load Model 4 in Eurocode 12 consists of a set of standard lorries (Figure 2.42) which together produce eﬀects equivalent to those of typical traﬃc on European roads. This model is intended to determine stress range spectra resulting from the passage of lorries on bridge. The equivalent lorries are deﬁned by the number of axles and the axle spacing, the equivalent load of each axle, the contact surface of the wheels, the transverse distance of the wheels and the percentage of each standard lorry in the traﬃc ﬂow. For the veriﬁcation of global action eﬀects the model can be placed centrally on the notional lanes acc. to Figure 2.34. For local members (e.g. concrete slabs or orthotropic decks) the model has to be centred on notional lanes assumed to be located anywhere on the carriageway. Where the transverse location of the fatigue load model is signiﬁcant for the action eﬀects e.g. in orthotropic decks, a statistical distribution of this transverse location acc. to Figure 2.43 has to be taken into account. As mentioned above, the fatigue load models in Eurocode 12 include a dynamic load ampliﬁcation ϕf at . An additional dynamic load ampliﬁcation factor Δϕf at acc. to Figure 2.43 has to be taken into account near expansion joints to allow for the eﬀects of local irregularities in this regions. For the other regions of the bridge the dynamic load ampliﬁcation factor must take into account the high number of relative small load cycles. This can be achieved by introducing a damage equivalent dynamic ampliﬁcation factor acc. to Figure 2.44 which results from the comparison of the cumulative damage calculated with and without dynamic ampliﬁcation of the Auxerre traﬃc. The procedure for the determination of ϕf at is shown in Figure 2.44. Because most of the stress ranges are below the fatigue strength limit ΔσD , the dynamic factor can be determined with a constant value of m = 5 for the slope of the fatigue strength curve. In Eurocode 12 good pavement quality acc. to Figure 2.27 was assumed. The inﬂuence of the pavement quality on the dynamic ampliﬁcation factor ϕf at can be seen from Figure 2.45. For good pavement qualities the dynamic
vehicle type Lorry
long distance
medium distance
local traffic
wheel type
2.3 Transport and Mobility
20
40
80
A B
traffic type and lorry percentage axle spacing [m]
axle loads [kN]
4,50
70 130
4,20 1,30
70 120 120
5
10
5
A B B
3,20 5,20 1,30 1,30
70 150 90 90 90
50
30
5
A B C C C
3,40 6,00 1,80
70 140 90 90
15
15
5
A B B B
4,80 3,60 4,40 1,30
70 130 90 80 80
10
5
5
A B C C C
67
wheel types and dimensions of the wheel contact surface in mm 320
320
2,0 m
220
x
Type A
320 220
270
220
Type B
Type C
Fig. 2.42. Set of lorries of Fatigue Load Model 4 in Eurocode 2 and contact surfaces of the wheels
Dynamic load amplification factor near expansion joints
Distribution of transverse location of centre line of vehicle 50% 18%
D
7%
'Mfat 1,3 1,2 1,1 1,0
5 x 0,1 m
D [m] 2,0
4,0
6,0
Fig. 2.43. Distribution of transverse location of centre line of vehicles and dynamic load ampliﬁcation factor near expansion joints
68
2 DamageOriented Actions and Environmental Impact 'V(log)
damage:
1
'V Ri
ªN ºm 'V D « D » ¬ Ni ¼
100
i
category 160
1 NC
ª 'Vi º « 'V » ¬ C¼
Di
ni N Ri
1 ND
ª 'Vi º « 'V » ¬ D¼
category 36
10
NL
NC ND 105
ni N Ri
Di
'Vi
106
107
108
m1
Di
for 'Vi t 'V D m2
for 'V D t 'Vi t 'V L
0
for 'Vi V L
Linear damage accumulation: D
N(log)
¦ Di
D Auxerre d 1,0
109
Damage equivalent dynamic amplification factor: i
m ¦ n i,dyn 'Vi,dyn i
m m ¦ n i,dyn 'Vi,dyn ¦ n i,stat Mfat 'Vstat Mfat
mi
i
¦
i n i,stat 'Vim,stat
mi
D Auxerre,dyn D Auxerre,stat
Fig. 2.44. Linear damage accumulation and damage equivalent dynamic ampliﬁcation factor ϕf at
Mfat 1,8 1,6
average pavement quality )h(:o)=16
good pavement quality )h(:o)=4
flowing traffic with v= 80 km/h
ME
1,4 1,2 L [m]
1,0 10
20
30
40
50
60
70
L
L
L
80
Fig. 2.45. Inﬂuence of the pavement quality on the damage equivalent dynamic ampliﬁcation factor [530]
factor ϕf at is in the range of 1.2, which is included in the load model in Figure 2.44. For average pavement qualities a mean increase in the range of 20% was obtained which leads to an increase of the damage D by a factor of 2.5 and a decrease of the fatigue life to 0.4 when for the slope of the fatigue strength curve m = 5 is assumed. This demonstrates that the authorities have the responsibility for a careful maintenance of the roads. As mentioned above, Fatigue Load Model 3 (Figure 2.46) consists of a single vehicle with four axles, each of them having two identical wheels with a squared surface contact area of each wheel with the side lenght of 0.4 m. The weight of the axles is equal to 120 kN and includes the dynamic ampliﬁcation factor ϕf at . The damage of the real traﬃc is taken into account by a damage
2.3 Transport and Mobility Fatigue verification
Axle loads of Fatigue Load Model 3 120 kN 120 kN
69
fatigue strength curve
'VC
120 kN 120 kN
O 'VLM 1,20m
'VLM 0,4 m
0,4 m
3,00 m
6,00 m
lane width
2,00 m
1,20m
'Vi(ni) NC J F,fat O 'V LM d
ND 'VC J M ,fat
Fig. 2.46. Fatigue Load model 3 in Eurocode 12 and fatigue veriﬁcation for steel structures
equivalent stress range λ · ΔσLM [31, 327] with λ = λ1 · λ2 · λ3 · λ4 ≤ λmax The factor λ1 takes into account the damage eﬀect of traﬃc depending on the length of the critical inﬂuence length, λ2 is a factor for the traﬃc volume, λ3 allows for diﬀerent design life and λ4 takes into account the traﬃc on other lanes. The fatigue veriﬁcation can then be performed according to Figure 2.46, where ΔσLM is the stress range caused by the load model, ΔσC is the reference strength at 2 million load cycles and γF,f at and γM,f at are the partial safety factors for the equivalent constant amplitude stress λ · ΔσLM and the fatigue strength ΔσC . The damage equivalent factors must be determined from the real traﬃc, where for Eurocode 12 the Auxerre traﬃc was used. In a ﬁrst step it is assumed that for the determination of λ1 the factor λ2 is equal to 1.0 for No = 0.5 × 106 lorries per year and slow lane and that a design life Tso = 100 years corresponds to a factor λ3 = 1.0. Furthermore only one slow lane is investigated which gives λ4 = 1.0. Then the factor λ1 must fulﬁl the condition, that the damage of the load model DLM is equal to the damage of the Auxerre traﬃc DAuxerre = Σ Di . For the calculation of λ1 at ﬁrst random load ﬁles based on the Auxerre traﬃc and the corresponding stress range spectra acc. to Figure 2.44 have to be determined. The corresponding accumulative damage caused by the number nLs of simulated lorries results in DAuxerre = Σ Di . As the damage caused by the load model has to be equal to the cumulative damage DAuxerre , a correction factor λe for the stress range ΔσLM of the load model has to be introduced (see Figure 2.48). For the same number of lorries nLs the equivalent damage of the load model DLM and the factor λe is given by m nLs λe · ΔσLM ΔσD m ND · DAuxerre λe = (2.55) DLM = ND ΔσD ΔσLM nLS
70
2 DamageOriented Actions and Environmental Impact flowing traffic with v= 80 km/h
Oe
Oe static action effects
1,4 1,2
1,4
dynamic action effects
1,0
1,2 1,0
0,8
0,8
0,6
0,6
ME
0,4
L
L
0,2
L [m] 10
20
30
40
50
ME
0,4
L
60
L
L
L
0,2
70 80
L [m] 10
20
30
40
50
60 70 80
Fig. 2.47. Example for the damage equivalent factor λe [530]
'V(log)
m1=3 D
'VC O1 'VLM
¦ Di
D Auxerre
m2=5
'Ve=Oe'VLM 'VLM
'Vi(ni) N(log) NC
N D nS
NL
Fig. 2.48. Determination of the damage equivalent factor λ1
A typical example for the damage equivalent factor for a three span bridge is given in Figure 2.47. It can be seen that the dynamic ampliﬁcation leads to a signiﬁcant increase of the factor λe . Furthermore the factor depends on the type of the inﬂuence line and the assumption for the quality of the pavement. The values in Figure 2.47 were determined for a good pavement quality. For the fatigue veriﬁcation acc. to Figure 2.46 it has to be taken into account that the veriﬁcation is based on the fatigue strength ΔσC at NC = 2×106 load cycles and that in addition the relevant number of lorries during the design life Tso is given by NT O = No ·Tdo . This leads to a further transformation for the damage equivalent stress range Δσe = λe · ΔσLM (see Figure 2.48).
2.3 Transport and Mobility O1 2,8 2,6
O1 midspan regions 2,55
2,4
2,2
2,2
2,0
2,0 1,85
1,4
L [m] 10
20
30
L1
40
50 60
70
80
L2 2,2
2,0
1,8 1,6
1,70
1,4
L
1,2
internal supports
2,8 2,6
2,4
1,8 1,6
71
L= ½ (L1+L2)
1,2 10
20
30
40
50 60
70
L [m] 80
Fig. 2.49. Factors λ1 for steel bridges given in Eurocode 32
Because NT o is greater than ND in the ﬁrst step a correction factor α for the damage equivalent stress related to ND is determined using the slope of the fatigue strength curve m2 = 5. NT o 5 5 (2.56) NT o [λe · ΔσLM ] = ND [α · λe · ΔσLM ] ⇒ α = 5 ND In the second step the transformation of the equivalent stress range related to NC follows using the slope m1 = 3 (see Figure 2.48) ND 3 3 (2.57) ND [α · λe · ΔσLM ] = NC [α · β · λe · ΔσLM ] = 3 NC The damage equivalent factor λ1 is then given by: ND 5 NT o · 3 λ1 = λe · α · β = λe · ND NC
(2.58)
The equivalent damage factor λ1 depends on the damage equivalent factor λe , the type of the fatigue strength curve (slopes m1 and m2 and the fatigue strength ΔσD and ΔσD respectively) and the relevant numbers NT o of lorries during the design life assumed for λ2 = 1.0. Therefore the factor diﬀers for structures and structural members with diﬀerent materials (e.g. structural steel, reinforcement, shear connectors). Figure 2.49 shows the λ1 values for steel bridges which are an envelope of the most adverse values determined for diﬀerent types of inﬂuence lines. For concrete and composite bridges corresponding values are given in Eurocode 22 [29] and Eurocode 42 [31], respectively. As explained above, the factor λ1 was determined for the reference value No = 0.5 × 106 , where No corresponds to the traﬃc category 2 in Table 3.14. Furthermore for the design life a reference value Tso = 100 years was assumed. In case of another traﬃc category or design life the damage equivalent factor
72
2 DamageOriented Actions and Environmental Impact
has to be modiﬁed with the factors λ2 for the traﬃc category and λ3 for the design life. Regarding the traﬃc category it also has to be considered, that on special routes the mean gross weight of the lorries can be higher or less than the average gross weight of the Auxerre traﬃc. For the factor λ2 results Qml 5 Nobs (2.59) λ2 = Qo N0 where Nobs is the relevant number of lorries per year for the relevant traﬃc category given in Table 3.14, Qo = 480 kN is the reference value for the gross weight of the heavy vehicles and Qm1 is the damage equivalent gross weight of the lorries in the slow lane speciﬁed by the competent authority by the number ni of lorries and the corresponding gross weight Qi in the slow lane. Qml =
Σni · Qi Σni
1/5 (2.60)
With the reference value NT O = No ·Tdo the factor λ3 is given by equation 2.61 Td λ3 = 5 (2.61) Tdo The factor λ1 in Figure 2.49 is determined for lorries only in the slow lane of the bridge. In case of more than one heavy lane on the bridge the eﬀect is taken into account by the factor λ4
N2 λ4 = 1 + N1
η2 · Qm2 η1 · Qm1
m
N3 + N1
η3 · Qm3 η1 · Qm1
m
Nk + ... + N1
ηk · Qmk η1 · Qm1
m m1 (2.62)
where Qmi
is the average gross weight of lorries in lane j
Nj
is the number of lorries per year in lane j
k
is the number of lanes with heavy traﬃc
m
is the slope of the fatigue strength curve (e.g. m = 5 for structural steel, m = 9 for reinforcement (straight bars) and m = 8 for headed stud shear connectors)
ηj
is the value of the transverse inﬂuence line for the internal force that produces the stress range in the middle of lane j acc. to Figure 2.50 and to be inserted in equation 8 with positive sign.
2.3 Transport and Mobility
73
Lane 2
Lane 1
K1
K2
transverse influence line
1,0
Fig. 2.50. Assumptions for the factor λ4
Omax
Omax 2,8 2,6
internal supports
2,8
midspan regions 2,55
2,4
2,4
2,2
2,2
2,05
2,0
2,0
1,8
1,80
L= ½ (L1+L2)
1,8 1,6
1,6 1,4
1,4
L
1,2
L [m] 10
2,70
2,6
20
30
40
50 60
70
L1
L2
1,2
80
L [m] 10
20
30
40
50 60
70
80
Fig. 2.51. Damage equivalent factor λmax
For materials (e.g. structural steel) with a fatigue limit for constant amplitude stress ranges the damage equivalent factor λ is limited to a value λmax . Where all stress ranges caused by the real traﬃc do not exceed the fatigue limit (Δσmax ≤ ΔσD ) the fatigue life is unlimited. In this case results from the condition Δσmax = λmax · ΔσLM λmax =
Δσmax ΔσLM
(2.63)
where Δσmax can be determined from the traﬃc simulations of the Auxerre traﬃc. Figure 2.51 shows the values λmax given in [31] for steel bridges. 2.3.1.5 Actual Traﬃc Trends and Required Future Investigations For the transport of persons and goods bridges are an important part of the infrastructures in Europe. As explained above the load models for bridges in Eurocode 12 cover the European traﬃc of the year 2000. Contrary to all forecasts the amount of heavy traﬃc on motorways has increased in the last
74
2 DamageOriented Actions and Environmental Impact billion to km 600 500
forecast
400 300 200 100
2010
2000
1998
1996
1994
1992
1988
1990
1986
1984
1982
1980
year
Fig. 2.52. Development of the freight traﬃc on roads, railways and ships
years and it is expected that this increase will continue in future. Figure 2.52 shows the development of the freight traﬃc on roads, railways and ships in Germany [563]. In the last 20 years the total freight traﬃc on roads has increased signiﬁcantly after the German reuniﬁcation in comparison with traﬃc on rails and ships. It can be expected that a further increase of traﬃc will take place due to the increasing transit trade and the aﬃliated cross border traﬃc. This can also be seen from Figure 2.53 which gives the recorded and expected number of heavy vehicles per day [563]. The comparison with Table 3.14 shows that with regard to fatigue at present the number of heavy vehicles exceeds the values assumed for category 1 in Eurocode 12. In order to optimize the transport capacities and minimize the transport costs there is a strong tendency to produce vehicles with higher gross weights. This results from Figure 2.53 giving the relative frequency of articulated vehicle with two driving axles and triple axle semitrailer, which is the most frequent type on German roads at present. Table 3.15 shows the results of actual traﬃc records (2004) at the Highway A61 near Brohltal [314]. The table demonstrates that the articulated vehicles (Type 5) dominate the traﬃc composition with a percentage of nearly 60%. The comparison with traﬃc data of the Auxerre traﬃc recorded in 1986 (see Table 1) shows, that presently the mean values of the gross vehicle weights in Germany are nearly conform with the values of the Auxerre traﬃc. For loaded articulated vehicles Table 3.15 gives a mean value of the gross weight of 405 kN which corresponds to a mean value of 463 kN of the Auxerre traﬃc, but with the addition that the standard deviation of the actual records is higher. The increasing of the standard deviation is mainly based on the fact that there is an increasing number of overloaded lorries.
2.3 Transport and Mobility number of heavy vehicles per day 7
12000
6
10000
5
8000
4 3
forecast
14000
6000 4000
2015
1998
1994
1996
1990
year 1992
f[%]
2
2000 1988
75
1
G[kN] 100
200
300
400
500
Fig. 2.53. Development of the number of heavy vehicles per day on highways and relative frequency of the gross weight for articulated vehicles with two driving axles and triple axle semitrailer
Table 2.12. Statistical parameters of the traﬃc records at highway A61 (2004)
type of vehicle Type 1 Type 2 Type 3 Type 4 Type 5
Go Gl Go Gl Go Gl Go Gl Go Gl
mean value P of the standard deviation relative frequency % V total vehicle weight kN kN 59,6 14,6 5 91,7 44,0 6 190,3 23,2 1 208,4 73,9 4 276,8 59,5 12 414,5 32,5 5 156,7 18,8 3 211,4 52,8 5 259,6 92 37 405,3 24,8 22
The distribution of the gross weight to the single axles as recorded in 2004 is shown in Table 3.17. These new data are comparable with the values measured in Auxerre in 1986. Furthermore new data of the density function of the intervehicle distances [305] show in comparison with old traﬃc records that there is a trend to lower intervehicle distances mainly based on the fact that at present the number of convoys increases conditioned by modern breaking systems. This is especially important for the fatigue resistance of bridges with longer spans. In summary it can be stated, that the Auxerre traﬃc which was the basis of the load models in Eurocode 12 covers presently the actual traﬃc in Germany.
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2 DamageOriented Actions and Environmental Impact
Table 2.13. Relation between gross weight of the heavy vehicles and the axle weights of the lorries of types 1 to 5 (mean values) Axle 1
Axle 2
Type 1
Type of vehicle
44,9 %
55,1 %
Type 2
25,8 %
Type 3 Type 4 Type 5
Axle 3
Axle 4
37,2 %
18,9 %
18,1 %
20,9 %
25,8 %
16,1 %
19,5 %
30,6 %
30,9 %
19,1 %
19,4 %
20,8 %
28,1 %
17,0 %
17,0 %
Axle 5
17,7 %
17,1 %
Nevertheless Figure 2.53 and Table 3.15 indicate that in the near future an adjustment of the load models in the codes is necessary. The data given in Tables 3.15 and 3.17 cover the normal traﬃc on highways, which do not include abnormal and heavy load transports. For such transports normally a special permit by the authorities is necessary. Figure 2.54 shows the development of the number of applications to authorisation for heavy load transports in Bavaria and NorthRhine Westphalia. The diagram demonstrates the signiﬁcant increase in the number of such transports. At present in Germany it is considered to permit heavy load transports with gross weights up to 720 kN (see Figure 2.54) for deﬁned routes over limited
n
(number of permits of heavy transports per year)
heavy vehicle with G= 520 kN
80000 70000 90 kN
60000
North Rhine Westphalia
50000
130 kN
3 x 100 kN
heavy vehicle with G= 550 kN
40000 90 kN
30000
North Bavaria
20000
2 x 130 kN
2 x 100 kN
crane with G= 720 KN
10000
2004
2002
1998
2000
1994
1996
1992
1988
1990
1984
1986
1982
1980
year
6 x 120 kN
Fig. 2.54. Development of the number of permits of heavy transports in Bavaria and NorthRhine Westphalia and examples for vehicles for heavy transports
2.3 Transport and Mobility
77
G [kN] 1270 kN
1300 1200 1100 1000 900
vehicle weight
800 700 600 500 400 axle weight
300 200 100
n
102
103
104
105
106
107
number of vehivles and axles per year
Fig. 2.55. Traﬃc records from the Netherlands recorded in 2006
periods of one or two years. In this case a further increase of these transports can be expected and it cannot be excluded that a signiﬁcant percentage of these transports is overloaded. A possible increase in the number of such vehicles in combination with a possible overloading has especially to be considered for the development of future fatigue load models. A comparable development takes place in other European countries. Figure 2.55 shows the vehicle weight and axle load distributions recorded in 2006 near the harbour of Rotterdam in the Netherlands. It can be seen that the extreme values of the gross weight and also the extreme values of the axle loads are signiﬁcant higher than the values of the Auxerre traﬃc (see Figure 2.24). The shape of the distribution shows that the heavy load transports lead in comparison with the Auxerre traﬃc to a new shape of the distribution which could be taken into account by splitting the distribution into a distribution for normal traﬃc and a distribution for heavy load transports. Additionally the transport industry is extremely interested in new transport concepts at present. In some European countries and also in some German federal states ﬁeld trials take place with modular vehicle concepts, the so called GigaLiners with gross weight up to 600 kN and a total length of 25.25 m [314]. Typical vehicles and the corresponding allowable axle loads are shown in Figures 2.56 and 2.57. These types of vehicles have signiﬁcant higher transport capacities and can reduce the transport cost. At present it cannot be foreseen how the future traﬃc composition will change. Some people argue that the new modular concept will reduce the total number of lorries on roads due to the higher transport capacity. On the other hand it has to be
78
2 DamageOriented Actions and Environmental Impact Current trucks in Germany (gross weight 400kN)
Vehicles acc. to the modular concept (gross weight up to 600kN)
25,25 m
16,5018,25 m
Fig. 2.56. Heavy vehicles on the basis of the modular concept (GigaLiners)
1,475
5,10 m
1,35
4,65
1,35
78 kN
78 kN 5,965 m
78 kN
54 kN
54 kN
92 kN
92 kN
74 kN
Giga – Liner with gross weight of 600 kN
1,36 1,36
2,64
1,475 3,215 1,36
5,965 m
1,36 1,36
90 kN 90 kN
65 kN 65 kN 65 kN
74 kN
74 kN
57 kN
Giga – Liner with gross weight of 580 kN
6,27 m
1,36
2,88
25,25 m
Fig. 2.57. Axle spacing and allowable axle weights of ”GigaLiners”
considered that this new type of vehicle can not be loaded on trains, so that it can be expected that no signiﬁcant reduction of the total road traﬃc will occur. First investigations [201] show that especially for bridges with longer spans the current European load model has to be modiﬁed, when the percentage of the new vehicles reaches 20% to 40% related to the total heavy traﬃc. Furthermore at present no information is available regarding the driving of such vehicles in convoys, especially on routes with acclivities, and the possible overloading and wrong loading which can lead to higher axle weights.
2.3 Transport and Mobility
79
The new traﬃc concepts and development regarding heavy transports need new technologies to get more detailed information about the actual traﬃc situation and also a more close cooperation between the car industry and the authorities and experts for the development of realistic traﬃc models. The Weight in Motion (WIM) is a technology [407, 588] for the determination of the weight of vehicles without requiring it to stop for weighting. The system uses automated vehicle identiﬁcation to classify the type of the vehicle and measures the dynamic tyre force of the moving vehicle when the vehicle drives over a sensor. From the dynamic tyre load then the corresponding tyre load of a static vehicle is estimated. The most common WIM device is a piezoelectric sensor embedded in the pavement which produces a charge that is equivalent to the deformation induced by the tyre loads on the pavements surface. Normally two inductive loops and two piezoelectric sensors in each monitoring lane are used. The system can be used in combination with an automatic vehicle classiﬁcation system (AVC). Vehicles which do not meet the gross weight and axle weight requirements are notiﬁed with dynamic message signs. While in the USA this systems are used in some states all over the country, in Europe only in some countries these systems are used on special routes. First ﬁeld trials with combined WIM and AVC methods take place presently in the Netherlands. The records demonstrate that besides the problem that the total weight of the vehicles exceed the permissible total weight there are also cases where the permissible total weight is not exceeded, but due to wrong loading of the vehicles the weight of single axles is signiﬁcantly higher than the permissible axle weight. This can lead to excessive fatigue damage especially in orthotropic decks of steel bridges and also in concrete decks. These new traﬃc records demonstrate that in the future a better cooperation between bridge designers and truck producers is necessary. Strategies to avoid such overloading of single axles could be the implementation of immobiliser systems in trucks if single axles or the total gross weight of the truck are exceeded. 2.3.2 Aerodynamic Loads along HighSpeed Railway Lines Authored by HansJ¨ urgen Niemann Shelter walls often accompany highspeed railway lines for noise protection or to provide wind shelter for the trains. The walls consist of vertical cantilevered beams connected by horizontal panels. The pressure pulses from head and tail of the train induce a pressure load on the walls, which is in general smaller than the wind load. However, the load is dynamic which may cause resonant ampliﬁcation. The load is furthermore frequent which may require design for fatigue. These issues are the topic of the following chapter.
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2 DamageOriented Actions and Environmental Impact
Fig. 2.58. Pressure time history at the trackside face of a 8 m high wall; at a ﬁxed position; V = 234.3 km/h, [573]
2.3.2.1 Phenomena As a train passes, a sudden rise and drop of the static pressure occurs. Structures at the trackside, such as noise barrier or wind shelter walls, in turn experience a time variant aerodynamic load [777]. It is caused by the pressure diﬀerence over the wall sides facing the track and the rear face. The load intensity of this aerodynamic loading is proportional to the square of the train speed. Figure 2.58 shows a pressure time history measured at a ﬁxed position at the trackside surface of a wall, 1.65 m above rail level. The wall distance to the track axis is ag = 3.80 m. Typically, the head pulse starts with a positive pressure which is followed by a negative pressure approximately identical in magnitude. The subsequent tail pulse is reversed and its amplitudes are smaller unless the train is short. For short vehicles, head and tail pulse may merge and the negative pressure may dominate. Additional pulses occur at intercar gaps with amplitudes much smaller than head and tail pulses. The measured time history clearly depends on the train speed. If instead of the time history the load pattern along the wall is considered, it becomes independent of the train speed. Figure 2.59 gives an example. The pattern of the pulse sequence travels along the wall at the train speed. It provides a dynamic load on the wall structure within a narrow bandwidth of frequencies determined by the train speed V . Furthermore, a spectral decomposition shows that the distance Δx of the positive and negative pulses is related to the prevailing frequency. Figure 2.59 gives two values of Δx measured at a track distance of ag = 3.80 m at two diﬀerent train speeds. The eﬀect of the train speed is within the scatter of the experimental results.
2.3 Transport and Mobility
81
Fig. 2.59. Pressure distribution along the trackside face of a wall at two diﬀerent train speeds [573]
(a)
(b)
Fig. 2.60. Full scale tests performed along the high speed line CologneRhine/Main: view of the trough; (a) measuring the train speed, (b) with measurement setup at the eastern wall
A spectral decomposition shows that the prevailing frequency fp is in the order of fp ≈
V 2.7Δx
(2.64)
Depending on the natural frequencies fn of the wall or any other trackside structure resonance may occur at a critical train speed Vres ≈ 2.7Δxfn , which in turn may cause considerable fatigue at rather few train passages. The
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2 DamageOriented Actions and Environmental Impact
maximal pressure amplitude measured at a train speed of 304 km/h = 84.6 m/s is ca. 0.550 kN/m2. Typical wind loads are larger by a factor of 2 to 4. It has been argued that the load eﬀect will become important only at very high speeds beyond 300 km/h (see [617]). In fact, the aerodynamic load does not dominate the design as long as the train speed is suﬃciently below the critical. If however the critical speed is lower than the maximal track speed, resonant ampliﬁcation will provide the dominant design situation. Fatigue damage occurred at protection walls along a high speed railway line in 2003. Previous investigations e.g. [36] had dealt with the static eﬀect of the pulse and developed simpliﬁed design loads which cover the static action eﬀect. However, they did not consider to model the loading process in view of the dynamic load eﬀects. Therefore, additional investigations became necessary with a focus on the dynamic nature of the load. One issue concerned fullscale measurements of the aerodynamic load patterns along the wall and over the wall height, and the relation of natural wall frequency to the critical train speed. The following ﬁndings rely on the results of a campaign performed in 2003, see [573]. The measurements were performed along a concrete wall in order to avoid disturbances coming from the strong deformations of some of the walls. 2.3.2.2 Dynamic Load Parameters The streamlined shape of nose and tail, as well as the frontal area do not only determine the drag of the train but also the pulse amplitudes. As well, the nose length aﬀects the distance between the pressure peaks. The ERRIreport [36] identiﬁes three typical train nose shapes and gives load reduction factors as follows: freight trains express trains with Vmax = 220 km/h high speed trains (TGV, ICE, ETR)
k1 = 1, 00; k1 = 0, 85; k1 = 0, 60.
The dynamic stagnation pressure of the train speed clearly governs the aerodynamic pressures. Figure 2.61 is based on the pressures at the trackside wall surface. The diagram relates the measured pressure peaks of the head pulse, positive and negative, to the dynamic head of the train speed: q=
1 ρV2 2
(2.65)
The relation is linear with a high degree of correlation, and it follows that pressure coeﬃcients may be introduced as cp =
p q
(2.66)
2.3 Transport and Mobility
83
(a)
(b)
Fig. 2.61. 3 Eﬀect of train speed stagnation pressure on the head pulse acting at the trackside face of a wall; (a) positive pressure; (b) negative pressure
Figure 2.62 shows the pattern of the head pulse in terms of pressure coeﬃcients. The peak coeﬃcients of ±0.15 are typical for the well shaped, slender nose of the ICE 3 train. The mean values are somewhat smaller. The detailed coeﬃcients cp obtained for 152 train passages are: peak pressure maximum mean pressure maximum lowest pressure maximum
cp = 0, 1499 cp = 0, 1380 cp = 0, 1049
peak pressure minimum mean pressure minimum highest pressure minimum
cp = −0, 1520 cp = −0, 1419 cp = −0, 1041
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2 DamageOriented Actions and Environmental Impact
Fig. 2.62. Pressure coeﬃcients of the head pulse from 34 passages (at the trackside wall face) at 1.65 m above track level
Fig. 2.63. Distance between the pulse peaks and the zero crossing (ΔL1 = pressure maximum, ΔL2 = pressure minimum)
The dynamic eﬀect is related to the distance between the pulse peaks. As is seen in Figure 2.63 a mean distance of Δx = 6.9 m is typical for the ICE 3 passing at a track distance of 3.80 m. At a train speed of 300 km/h, the related frequency is fp = 4.5 Hz. Natural frequencies of light protection walls are in the same order of magnitude. Obviously, the critical train speed may happen and its dynamic eﬀect may become important.
2.3 Transport and Mobility
85
Fig. 2.64. Head pulse in a free ﬂow at various distances from the track axis [98]
Fig. 2.65. Head pulse in the presence of a wall
The results refer to a distance between the wall and the track axis of ag = 3.80 m. This parameter plays an important role both for the amplitude of and the distance between peaks. Figure 2.64 shows the result obtained theoretically regarding the pressure pulse in a free ﬂow. As the track distance ag increases, the peak amplitudes max p and min p decrease whereas the separation Δx between the pulse peaks increases. Theory predicts that in free ﬂow without walls, the separation Δx depends linearly on the track distance ag , see e.g. [98] √ (2.67) Δx = 2 ag
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2 DamageOriented Actions and Environmental Impact
Experimental results can best be ﬁtted by a slight modiﬁcation: Δx = 1.424 a1.029 g
(2.68)
Figure 2.65 shows the head pulse in the presence of a wall for two diﬀerent distances. The measurements at a track distance of 3.80 m and 8.30 m were performed simultaneously i.e. at identical train speeds at diﬀerent walls, both 8 m high. The distance of the peaks at the wall decreases similar to the free ﬂow case. However, the results indicate that the eﬀect of the track distance becomes nonproportional in the presence of a wall. An analogous approximation matches the test results 0.653 ag Δx(ag ) = 6.9 (2.69) ag,ref in which ag,ref = 3.8 m is used as reference. The pressure amplitudes decrease with the inverse of the square of the track distance. Various empirical expressions take account of this theoretical result. The following formula developed in [36] is widely accepted: 2.5 cp,max = k1 + 0.025 (2.70) (ag + 0.25)2 Introducing the pressure at ag = 3.80 m as a reference, the peak pressure amplitude at any distance becomes 14.1 cp,max (ag ) = ca · cp,max (3.8) = + 0.14 cp,max (3.8) (2.71) (ag + 0.25)2 For ag = 8.3 m, the formula gives a wall distance factor of ca = 0.333. The experimental result is in this case a decrease by a mean factor of 0.3. The formula presented is a conservative estimate. The pressure varies over the wall height. Figure 2.66 is an example of a pressure pattern measured at a wall, 8 m high. The pressure intensity decreases at the upper end. This end eﬀect coincides with a shift of the pulse peaks between wall foot and top, meaning that they do not occur simultaneously at each level. Figure 2.67 shows the time lag between head pulse maximum and minimum as it varies over the height of a 3.5 m wall. The measurements include various train speeds, the time lag has been transformed to V = 300 km/h. The maxima occur simultaneously at each level, whereas the minimum is not simultaneous but lags increasingly at higher levels. This will in general diminish the dynamic load eﬀect. A conservative approximation is to assume identical and simultaneous pulse patterns at each level. Finally, the pressure magnitudes depend on the wall height. The experiments show that the pressures measured at low levels are higher in magnitude at high walls compared to lower walls. The pulse between the walls apparently levels out more rapidly when the walls are low. A convenient wall height factor is:
2.3 Transport and Mobility
87
Fig. 2.66. Load pattern over the height of the wall
cWH =
1 − 0.03 HW ref , 1 − 0.03 HW
2 m < HW ≤ 5 m
(2.72)
where HW is the height of the wall above the track level in m is, and HW ref the reference wall height, for which the pressure coeﬃcients have been determined. The results refer here to HW ref = 3.50 m. 2.3.2.3 Load Pattern for Static and Dynamic Design Calculations The following expression summarizes the observed eﬀects and may be applied to static and in particular to dynamic design calculations: q1k (x, z, ag ) = cWH (HW ) ca (ag ) cz (z) cp (x) ρ
V2 2
(2.73)
where: q the pressure at a distance x from the train nose, at a level z above track height; cWH factor accounting for the wall height; cp pattern of the pressure coeﬃcient at low levels acc. to Figure 2.69;
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2 DamageOriented Actions and Environmental Impact
Fig. 2.67. Variation of the time lag between maxima and minima of the head pulse over the wall height transformed to V = 300 m/s
Fig. 2.68. Load factor for the load distribution over the height of the wall
cz ca ρ V ag x z
load factor accounting for the pressure variation over the wall height acc. to Figure 2.68; load factor accounting for the wall distance from the track axle; mass density of air; train speed in m/s; track axle distance; distance from zerocrossing of the head pulse; height above rail level.
2.3 Transport and Mobility
89
(a)
(b)
Fig. 2.69. Pattern of pressure coeﬃcients cp for the ICE3 train: (a) pressure diﬀerence between trackside and rearside faces of the wall; (b) pressure at the trackside face
The speed of an adverse wind has to be added to the train speed where required. The load factor cz in ﬁg 2.68 neglects the phase shift occurring towards the top and is valid for any wall height. Figure 2.69 shows the reference load pattern. The stochastic component superimposed on the pressures by the boundary layer turbulence has been smoothed out by averaging. The head pulse at the trackside face (b) is symmetric. Considering the net pressure, the rearside pressure has to be included. The measurements in ref. [229] include the required data. They show that the pressure maximum on the rear side precedes the trackside maximum. Therefore, regarding the net pressure the pulse maximum increases whereas the minimum decreases. The eﬀect on the remaining load pattern is not noticeable.
90
2 DamageOriented Actions and Environmental Impact
(a)
(b)
Fig. 2.70. Noise protection wall (a): height 3.50 m above track level; post distance 5.00 m; lightweight panels (b) Mode shape of the 1st mode; natural frequency f1 = 4.67 Hz
The formula includes the wall distance eﬀect on the pressure amplitude as a constant factor. It does not include the increasing distance between pressure maximum and minimum. In general, calculations of the dynamic load eﬀect may be restricted to the head pulse. It governs the dynamic ampliﬁcation of the response. A simple and suﬃcient approximation applicable to the symmetric load pattern is 2x x cp (x) = cp,max exp 1 − (2.74) Δx Δx The expression includes the eﬀect of the track distance as well with regard to the pressure amplitude as to the distance of positive and negative peaks. 2.3.2.4 Dynamic Response A typical wall structure consists of concrete panels or lightweight metal panels ﬁlled with mineral wool. The panels are supported by steel posts at a distance of 2.00 or 5.00 m. Figure 2.70 (a) shows an example. It is rather laborious to model the dynamic behaviour of the structure. The transient response involves large parts of the wall between recesses. The attempt was misleading to identify the dynamic response at a single pole in a 1D model. Similarly, the natural frequencies and the relevant mode shapes cannot be identiﬁed realistically in a simpliﬁed model: as an example, the panels have to be included as 2D plates since their torsional stiﬀness contributes considerably to the system stiﬀness. Figure 2.70 (b) shows the 1st mode shape which is excited dominantly by the pulse load.
2.3 Transport and Mobility
91
displacement in m
×10−1
time Fig. 2.71. Time history of post top displacement calculated for a post in the middle of the wall; displacement in m, positive direction outward
The natural frequencies are not well separated. For the wall shown above, the ﬁrst 4 modes range from 4.67 Hz to 4.90 Hz, the 12th mode shape has a natural frequency of 6.04 Hz which is still rather close to the ﬁrst one. The post top displacement from time history calculations, s. Figure 2.71 indicates that the wall moves outward at the pulse maximum. As it swings back, the negative pulse ampliﬁes the movement: the 1st inward amplitude is ca. twice the 1st outward. This is a consequence of resonance. The eﬀect of natural frequencies on the resonant ampliﬁcation of the displacement may be studied in a simpliﬁed manner using modal decomposition. The response time history is calculated for a static behaviour and for various natural frequencies. A critical damping ratio of D = 0.05 was adopted independent of the natural frequency. The dynamic ampliﬁcation of the response r is characterized by two resonant ampliﬁcation factors: max ϕdyn =
max r rstat
min ϕdyn =
min r rstat
(2.75)
The Figures 2.72 and 2.73 show how the resonance factors depend on the natural frequency and the train speed, i.e. the pulse time lag. Both factors display identically that the maximal ampliﬁcation is independent of the natural frequency with a value of max ϕdyn = 2.0 and min ϕdyn = 2.6. The range of natural frequencies where peak resonance occurs is however not identical in the two cases. At a train speed of 300 km/h, a natural frequency of 3.8 Hz provides the highest ampliﬁcation of the outward displacement whereas the inward displacement is ampliﬁed most strongly at a natural frequency of 4.6 Hz. The wall considered suﬀers strong resonant vibrations.
92
2 DamageOriented Actions and Environmental Impact
Fig. 2.72. Resonant ampliﬁcation of the displacement maximum vs. the natural frequency at train speeds between 200 and 300 km/h
Fig. 2.73. Resonant ampliﬁcation of the displacement minimum vs. the natural frequency at train speeds between 200 and 300 km/h
2.4 LoadIndependent Environmental Impact Authored by Ivanka Bevanda and Max J. Setzer During their serviceable life, concrete structures are exposed to various environmental inﬂuences which aﬀect their durability to diﬀering degrees. Ensuring durability is understood to mean that the loadindependent inﬂuences which occur in the course of its serviceable life do not reduce the useful properties and the loadbearing capacity of the concrete structure. This means that a structure is suﬃciently stable to be able to absorb the expected loads
2.4 LoadIndependent Environmental Impact
93
(e.g.traﬃc, wind) on the one hand and at the same time that the loadbearing capacity is not reduced by environmental inﬂuences. An overview of the practical observations for the frost attack and a ﬁrst introduction into external chemical attack are given in the following sections. 2.4.1 Interactions of External Factors Inﬂuencing Durability Authored by Ivanka Bevanda and Max J. Setzer The DIN EN 2061 [1] introduces mechanismrelated exposure classes which describe and account for environmental inﬂuences which are not directly taken into account as loads for constructional measurement (Figure 2.74). From a technological point of view, durability is determined by minimum concrete composition requirements (water/cement ratio, cement content). The design concept was derived from current knowledge of deterioration mechanisms and correlations between exposure and resistance. This simple approach does, however, have the major disadvantage that the application of new materials and concrete types for which there are as yet no empirical values is limited. Furthermore, it is not possible to evaluate existing structures whose composition is not known. Chronological changes in resistance to a diﬀerent behavior compared with the original exposure are also not recorded. A durability prognosis of a concrete structure requires that the expected environmental conditions to which the structure will be exposed can be reasonably reliably predicted. The causes and correlations which lead to damage must be clearly recognized and understand. Knowledge of damage mechanisms and the complex interactions of external inﬂuences, transport and degradation process is necessary for forecasting durability and serviceable life (Figure 2.74).
Performance Concept
intensity
Environmental Impact (classification of EN 2061) reinforcement corrosion
concrete corrosion
carbonation
chloride
frost attack with/
chemical
process
penetration
without deicing agent
attack effect
Incubation Time
temperature and moisture 㩳 transport and/or reaction parameters
Serviceable Time
Climatic Conditions
Degradation Process Damage
damage
limit
criterion
Fig. 2.74. Schematic diagram  Interaction of climate, environmental attack and damage process  basis for the perfomance concept
94
2 DamageOriented Actions and Environmental Impact
(a)
(b)
(c)
Fig. 2.75. Reinforcement corrosion (above): (a) due to the inﬂuence of chloride (b) due to carbonation [400]; Concrete corrosion (below): (c) combined attack  AAR intensiﬁed by alternating frost and thawing [529]
The damage process depends on the transport process. The eﬃciency of the transport mechanisms is in turn dependent on moisture and/or temperature. Moisture is necessary as a transport and reaction medium and the external temperature works as a reaction accelerator. For example, the maximum carbonation speed occurs at humidities between 60% and 80% and the extent of sulfate corrosion rises with sinking temperatures. In case of frost attack the damage mechanisms only become active after the concrete texture is critically saturated through frost suction (transport mechanism). At the same time, the ”real” environmental attack is a complex strain, the sum total of several, sometimes simultaneous partial attacks which mutually inﬂuence one another. For example, weathering with deep craters can lead to increased chloride penetration of the concrete by deicing salt or a deeper carbonation of the concrete. This causes faster depassivation of the reinforcement, which causes more rapid corrosion of the outer reinforcement (Figure 2.75 (b)). A further example is the additional strain caused by temperature cycles, especially the alternation of frost and thawing of the alkaliaggregate reaction (AAR). These aid the development of the AAR by either leading to cracks in the concrete so that it can be better penetrated by moisture and an AAR can be initiated, or they lead to the expansion of existing AARrelated cracks (Figure 2.75 (c)).
2.4 LoadIndependent Environmental Impact
Concrete Corrosion
95
Physical Action
thermal (e.g. freezethaw, freezedeicing salt)
Chemical Action
dissolution (e.g. leaching, acid) expansion (e.g. sulfates, alkaliaggregate reaction)
Combined Action
e.g. alkaliaggregate reaction + freezethaw
Fig. 2.76. Attacks on concrete (in imitation of [872])
Figure 2.76 shows examples of physical and chemical environmental inﬂuences which cause concrete corrosion . The frost attack, the calcium leaching, the sulfate attack and the alkaliaggregate reaction were processed as part of SFB 398. It should be noted that in SFB 398 no practical examination of the listed chemical attacks was performed and a summary of the practical examinations in the literature can be found in Chapter 3. The laboratory tests are accordingly also listed in Chapter 3. In addition, more detailed summaries of the relevant aspects of durability in concrete structures can be found in e.g. [770],[702]. 2.4.2 Frost Attack (with and without Deicing Agents) Authored by Ivanka Bevanda and Max J. Setzer Frost and deicing salt attack are under the most detrimental environmental phenomena to be taken into account for durability design of concrete. Frost attack with and without the presence of deicing salt is a dynamic effect that involves both a transport mechanism and a damage mechanism. Setzer coined the term frost suction for the transport mechanism, and explained this phenomenon by surface physics described by the microicelens model (see Subsection 3.1.2.2.3). During the freezethaw cycle, external water is sucked inward by the action of the microicelens pump; the pore structure becomes saturated. Only once the critical degree of saturation is exceeded does ice expansion cause damage. Since there is not enough space in the concrete microstructure for lateral yield, critical internal stresses build up during the process of ice formation, and then abate again as microcracks form. The result of this is internal and/or external damage to the concrete structure. External damage known as scaling (Figure 2.77) can be recognized as (1) sandy decay and (2) local scaling of the hardened cement paste, and in the case of aggregaterelated damage as (3) popouts and (4) Dcracking.
96
2 DamageOriented Actions and Environmental Impact
Fig. 2.77. Surface of frost damaged concrete in situ [20]
External damage is the most frequently observed frost damage. It starts oﬀ as an aesthetic fault, but then the surface destruction can lead to limitation and loss of the function of the component, although structural stability is still assured (e.g. in the case of airport taxiways). Internal damage is characterized by microstructural damage arising from microcracks (Figure 2.78), which inﬂuence the mechanical and physical properties of the concrete structure, and its structural integrity as a result. While both types of damage go handinhand with the critical degree of saturation and ice expansion, they still must be treated as separate phenomena, since they appear not to be strictly related. In the case of external damage, dissolved substances (salts) add their own damaging eﬀect to the equation. This is an especially important factor in the case of deicing salt attack, and is discussed at length in literature. New ﬁndings, including those from SFB 398/ Project A11, show that even the inﬂuence of commonly ignored salt concentrations increases weathering in what is regarded as ”pure” frost attack. References in literature and our own investigations [21] show how diverse the possible variations of alternating frost and deicing salt stressing of concrete components can be. One actual overview is given in the progress report DAfStb3 [737]. The progression of damage following pure frost attack was also investigated under real climatic conditions (in situ) and under laboratory conditions in SFB 398/ Project A11. The essential results and their signiﬁcance are summarized below. 2.4.2.1 The ”Frost Environment”: External Factors and Frost Attack Details on the composition and properties of the tested concretes are given in [119],[120]. In order to emulate the conditions as realistically as possible, the ﬁeld samples were sealed and insulated on the sides, since moisture and 3
German Committee for Reinforced Concrete.
2.4 LoadIndependent Environmental Impact
97
0,125 mm
Fig. 2.78. Microcracking of cement paste(left); ESEM image of frost damaged concrete (right) [20]
Fig. 2.79. Field exposure (left); Modiﬁed multiring electrode (right)
heat transport through components is typically onedimensional in real applications. A side overlapping edge for catching rainwater was attached onto the test surfaces Figure 2.79. This way, a persistent water layer was simulated. In real situations, this type of frost attack typically occurs on horizontal components directly exposed to weathering, which are classiﬁed as exposure class XF3 according to DIN EN 2061 (frost attack without deicing salt, high water saturation) [1]. Under the climatic conditions, there were alternating periods of wetness and dryness, i.e. periods with dynamic moisture entry and redistribution inside the specimen. Climatically induced humidity and temperature stressing of the component is the most important factor to consider when investigating frost damage. As such, it was decided to obtain information on the changes in moisture content and concrete temperature using a modiﬁed multiring electrode4 (MRE) Figure 2.79. The modiﬁed MRE is a humidity/temperature sensor. Detailed information on its construction and function are given in [660],[762]. 4
Humidity sensor with integrated thermometers, pursuant to the Aachen patent.
98
2 DamageOriented Actions and Environmental Impact 9,0 o
T >0 C
8,5 8,0
8,0
moisture pentetration
drying
7,0
7,5
6,5 6,0 5,5
o
T >0 C
8,5
ln (R)
ln (R)
7,5
9,0
o
T <0 C
b = 4271 2 R = 0.97
5,0 3,40E03
8.1110.11 10.11 11.11 15.11 16.11 10.1116.11
3,50E03
b = 2048 2 R = 0.95
6,0 5,5
3,60E03
1/T [K1]
3,70E03
3,80E03
b = 4167 2 R = 0.91
7,0 6,5
0.7cm 0.7cm 0.7cm 0.7cm 0.7cm 3.4cm
o
T <0 C
0.7 cm 10.11 0.7 cm 15.11 0.7 cm 18.11
5,0 3,40E03
3,50E03
3,60E03
3,70E03
3,80E03
1
1/T [K ]
Fig. 2.80. Eﬀects at speciﬁc depths of water penetration, logarithm of resistance as a function of reciprocal ground temperature (left); Dependence of Arrhenius factor b on moisture content (right)
The resistance of concrete is dependent on both temperature and humidity. Therfore, humidity changes and distribution can be derived from the resistances measured if the temperature eﬀect is taken into account. The temperature dependence follows an Arrhenius equation5 . The Arrhenius factor b required for temperature compensation can be determined by taking the logarithm of the exponential correlation between the reciprocal ground temperature and the resistance with linear regression. What we ﬁnd most commonly in literature is that this temperature compensation is done by using a constant Arrhenius factor b. Our own tests conﬁrmed the situation found in [165],[701] namely that the activation energy depends on both temperature and less pronounced on moisture content (Figure 2.80, right). In order to determine the resistances precisely, the two inﬂuences should be decoupled, and the moisture and temperaturedependence of the Arrhenius factor clearly deﬁned. The dependency on moisture content can be given only in approximation. Therefore, moisture measurement is limited to a qualitative or semiquantitative level. Even if the temperature dependency of resistance could be evaluated only in a fair approximation of moisture content its results allowed a clear deﬁnition of the point when ice formation sets in since here the resistance increases at the same moisture content disproportionately, since the ice basically acts as an insulator. A new, automatic data analysis system was developed for analyzing the phase change from water to ice. That way, the strong dependency of resistance on temperature was used in the data analysis to analyze the number of phase changes, or the number of frost periods. The data analysis system was veriﬁed by experimental laboratory events. 5
Ri = Ro ∗ e
b T1 − T1 o
i
; Ri,o  electrical resistance at temperature Ti,o.
2.4 LoadIndependent Environmental Impact
20
2
air temperature
10 5 0
1
5 10
Percipitation [l/m²]
percipitation
15
Temperature [°C]
99
15 20 8.11
10.11
12.11
14.11
16.11
18.11
20.11
22.11
24.11
26.11
0 28.11
Days
Fig. 2.81. Air temperature and rainfall; ﬁeld station Meißen, local weather station, 11/08/0511/28/05
An online monitoring system allowed continual recording. Under the existing exposure conditions the humidity readings and concrete quality were not strictly correlated. Additionally, the strong dependence of the resistance on temperature allowed only semiquantitative conclusions on the moisture content. However, by analyzing the relative change in humidity distribution in the exposed concrete specimens in correlation with rainfall events, our tests also conﬁrmed the ﬁndings of [701] who studied the moisture penetration into concrete under natural weathering conditions above freezing point. Schiegg deﬁnes two types of incidents, depending on eﬀects at speciﬁc times and effects at speciﬁc depths: small incidents (transport zone <20 mm, time of eﬀect over a number of days) and large incidents (transport zone >40 mm, longterm eﬀect over several months, moisture penetration occurring in multiple phases). The temperature and humiditydependence of resistance can be seen clearly in Figure 2.80. This partially shows the moisture penetration to depth level 3.4 cm into a specimen directly after ﬁeld exposure (Field Station Meißen, East Germany). Following Arrhenius equation the logarithm of resistance is presented as a function of the reciprocal ground temperature. On November 10, we see that the resistance at a depth of 0.7 cm drops, since the ﬁrst moisture penetration occurred at that time. This also correlates with the recorded rainfall event on that day (Figure 2.81). After that, there was a dryout until November 15. Then, on November 15, a freezethaw cycle was recorded, but still no ice formation process had taken place yet. While the resistance rises as temperature drops, it does so in linear fashion, and not in jumps as it characteristically does right at the watertoice phase change (see Figure 2.82). On November 15, there was further moisture penetration, which again resulted in a drop in resistance. In the same period, there was no change in moisture content recorded
2 DamageOriented Actions and Environmental Impact 9,0
0.7 cm 1.7 cm 6.6 cm
8,5
o
T >0 C
5
o
T <0 C
4 3
8,0
ln (R)
7,5 7,0 6,5 6,0
frost suction
freeze thaw cycles
5,5 5,0 3,45E03
Temperature [°C]
100
2 1 0 1 2 3
air temperature 1.7 cm 3.4 cm 6.6 cm
4 3,55E03
3,65E03 1
1/T [K ]
3,75E03
3,85E03
5 4:48 AM
9:36 AM
2:24 PM
7:12 PM
Time []
Fig. 2.82. Freezethaw cycle illustrated by example (left); Temperature curve during thaw phase on November 26 (right)
at 3.4 cm depth, and the change in resistance is attributed to temperature alone. The temperaturedependence of resistance can be compensated for using the Arrhenius equation. Greater moisture penetration into the specimen interiors occurred in both winters before and/or at the beginning of the ”frost period”. The concrete surface zone is essentially saturated before the actual freezing phase (see Figure 2.80). Moisture absorption inside the specimens after a freezethaw cycle at the beginning of the ”cold period” can be attributed to frost suction according to the microicelens model. Figure 2.82 shows an example illustrating the change in moisture content after two successive freezethaw cycles (Nov. 24/25 and Nov. 25/26). Resistance at all depth levels increases with a jump when the temperature drops below the ”0o C transition”. Field and laboratory results show that ice formation sets in at about 0.5o C. The resistance curve also shows that the water continually freezes as temperature drops. After the thaw process on November 26, the resistance at depth levels 0.7 and 1.7 cm drops back down to the original value. At depth level 6.6 cm, on the other hand, the resistance drops as it would for an increase in moisture content. A detailed description of frost suction and the microicelens model is discussed in (see Subsection 3.1.2.2.3). Here, it is of relevance that the frost pump is activated during the thaw phase, with a penetrating melting front. External water can be sucked inward together with this penetrating melting front. The temperature curve shown in Figure 2.82 (right) shows the penetrating melting front at each point in time. The change in resistance in the winter phase reveals the following: (1) the moisture penetration into the specimen can be attributed to individual events at the beginning of the frost phase and (2) has a longterm action of several months. This is illustrated by the example given for depth level 6.6 cm (specimen core) in Figure 2.83. The most moisture penetration took place up
2.4 LoadIndependent Environmental Impact
9,0 8,5 8,0
ln (R)
7,5
o
6.6 cm 6.6 cm 6.6 cm 6.6 cm 6.6 cm
11.11. 26.11. 02.12. 20.02. 22.03.
T >0 C
101
o
T <0 C
7,0 6,5 6,0 5,5 5,0 3,40E03 3,50E03 3,60E03 3,70E03 3,80E03
1/T [K1]
Fig. 2.83. Exemplary illustration of the change in resistance at depth level 6.6 cm in the winter of 05/06; ﬁeld station Meißen
until November 30. In the phase after that, up until February 20, no change in moisture content at this depth was recorded. From the end of February 06, it can be seen that the specimens started drying out. A diagram of the resistance on March 22 is shown as an example. The moisture content at this time practically matches the initial moisture content. In the analysis of the data, a process is counted as a freeze or thaw phase according to a combination of criteria  predeﬁned minimum temperature and signal drop  which in turn seems to depend on moisture content or ice formation. The intensity of the frost attack distinguishes itself the most by the minimum temperature and number of freezethaw cycles. Also, the damage is increased by high cooling rates. Accordingly, the developed data analysis system analyzes each freeze event individually and delivers the following data for each depth level: minimum temperature, maximum and averaged cooling and thawing rates, and time and duration of the individual phase changes. The relevant data for the winter of 05/06 and 06/07 are summarized in Table 2.14. Figure 2.84 shows the frequency of freezethaw cycles depending on minimum temperature (left) and maximum cooling and thawing rates (right) for measuring point 0.7 cm over the exposure period. Measuring point 0.7 cm is especially of interest in connection with the observed surface damage to the exposed concretes, which we shall discuss later. There is a diﬀerence between the individual winters regarding the number of cycles and the minimum temperatures (see Table 2.14). Nevertheless, the greatest number of freezethaw events in both winter periods happened in the temperature range between 2 and 10o C. Between the individual specimens, there is no signiﬁcant diﬀerence in the number of freezethaw cycles, the deviation being only 1 ftc. As expected, the number of ftc drops in proportion to the depth level. In the winter of 05/06, there were 40 ftc recorded at a depth of 0.7 cm, 39 at 1.7 and
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2 DamageOriented Actions and Environmental Impact
Table 2.14. Readings: winter 05/06 and winter 06/07; ﬁeld station Meißen MP [cm] 0.7 1.7 3.4 6.6 11.8 MP [cm] 0.7 1.2 1.7 3.4 6.6
number of ftc [] 40 39 39 36 31
min. T
number of ftc [] 10 10 10 9 9
min. T
[oC] 19.3 18.7 18.7 18.6 18.5
[oC] 8.7 8.7 8.7 8.5 8.6
max. av. max. av. cooling rate cooling rate melting rate melting rate [K/h] [K/h] [K/h] [K/h] 4.6 1.7 5.7 2.4 3.5 1.6 4.9 2.2 2.5 1.4 4.7 2.2 2.6 1.4 4.9 2.2 2.3 1.2 3.7 2.2 max. av. max. av. cooling rate cooling rate melting rate melting rate [K/h] [K/h] [K/h] [K/h] 2.7 1.0 4.0 1.8 2.7 1.0 3.8 1.9 2.4 1.0 3.8 1.9 2.5 1.0 3.8 1.9 2.6 1.0 3.8 1.9
3.4 cm, 36 at 6.6 cm and 31 at 11.6 cm. In the winter of 06/07, there were comparably only very few ftc. At measuring points 0.7, 1.2 and 1.7 cm, there were 10 ftc recorded, and at 3.4 and 6.6 cm, there were 9 ftc recorded (Table 2.14). The tests show that the formation of ice begins on the surface and that an ice front forms, which then penetrates into the interior. The depthdependent diﬀerences can be attributed to two types of frost attack: (1) the freeze process only takes place in the concrete surface zone, and (2) after a frost phase, the melting process only occurs in the concrete surface zone while the concrete core remains frozen. The winter averages of cooling and thawing rates are practically the same (see Table 2.14). The maximum cooling and thawing rate recorded in the winter of 05/06 were 4.6 K/h and 5.5 K/h, and in the winter of 06/07 were 2.7 K/h and 4.0 K/h. In both cases, the extreme values were measured at depth level 0.7 cm. As concerns frostrelated environmental factors, the dominating factor in the present case study was the temperature stress (number of ftc and minimum temperature). The ﬁndings on moisture and temperature stress, and freezing and thawing processes in situ provide information for numerical simulation of durabilityrelated processes, in that they can be worked into the models as boundary conditions. With the considerations of [701], [803] in mind, the ﬁndings also provide a basis for elaborating possible frost scenarios with regard to effects at speciﬁc depths and duration of the individual frost phase: (1) fastacting incident  short term eﬀect immediately at the concrete surface (eﬀect at depth of <7 mm, not recorded metrologically), (2) small incident  eﬀects of several hours due to the day/night cycle, (3) large incident  corresponding to the ice phases, the duration goes beyond the day/night cycle, eﬀects at depths down to interior of concrete, possibly intermittent thawing in the concrete surface zone.
o
Min. temperature [ C]
< 20
Cooling/ melting rate class [K/h]
2.4 LoadIndependent Environmental Impact
winter 05/06 winter 06/07
10 / 20 5 / 10 2 / 5 0 / 2 0
5
10
15
20
Number of ftc []
103
cooling rate w05/06 cooling rate w06/07 melting rate w05/06 melting rate w06/07
5 4 3 2 1 0
10
20
30
40
Number of ftc []
Fig. 2.84. Frequency of freezethaw cycles depending on minimum temperature (left) and maximum cooling and thawing rates (right); depth level 0.7 cm; ﬁeld station Meißen
Fig. 2.85. External damage of concrete specimens after one winter at ﬁeld station Meißen; scaling 260 g/m2 , visual degree of damage 21%
2.4.2.2 Damage Due to Frost Attack In keeping with the objective, both the external damage and the internal microstructural damage to the exposed concrete samples were determined. The internal damage was determined by ultrasound pulse transit time measurement pursuant to [729]. After two winters, there was no internal damage, or drop in the relative dynamic Young’s modulus of elasticity, detected in either specimen. The external damage occurred in the form of scaling of thin cement paste layers (Figure 2.85).The damaged parts of the surface were quantitatively evaluated by analytical processing of photographs and deﬁned as visual degree of damage [%] in relation to the exposed test surface. Also, the
104
2 DamageOriented Actions and Environmental Impact
Visuel damaging degree [%]
25
20
y = 0.0745x + 1.5736 2 R = 0.97
15
10
5
0 0
100
200
300
400
500
Surface scaling [g/m²]
Fig. 2.86. Correlation between surface scaling and degree of visual damage on ﬁeld exposed specimens
weathered material on the test surface of the specimens  exposure batch W06/07 was collected after the ﬁrst winters. Even assuming the recorded weathering of the concretes conditioned in situ does not correspond to the absolute values, a good functional correlation between the visual degree of damage and the sampled weathering can be seen (Figure 2.86). Figure 2.87 shows a comparison of the surface damage and progression of damage depending on concrete quality and exposure site. The frost data presented above refer to ﬁeld station II (Meißen, East Germany). The frost damage to the exposed concretes of ﬁeld station I (Holzkirchen, South Germany) can only be estimated from the weather data (measured at the exposure site), and are therefore not discussed. Nevertheless, all exposed specimens can be precisely classiﬁed according to the degree of damage and concrete quality (Figure 2.87, left). The bigger the w/cratio, and thus the more porous the hardened cement paste and cement pasteaggregate6 interface zone, the greater the surface damage. Even a dependency on the type of cement was observed; concretes containing blast furnace slag tended to suﬀer greater surface damage. Furthermore, the diﬀerent climatic stresses are reﬂected clearly in the degree of damage observed. Concretes of exposure batch 05/06 demonstrated greater damage than the concretes of exposure batch 06/07. When we compare the humidity and temperature stresses to the concretes in Meißen (ﬁeld station II) over the two winters, the greater damage after the ﬁrst winter can be attributed mainly to the stronger temperature stress during freezethaw stressing. We even see diﬀerences within exposure batch 05/06 depending on the exposure site  generally, the specimens in Meißen (ﬁeld station II) were more damaged than those in Holzkirchen (ﬁeld station I). Nevertheless, the gradation of the damage is identical and correlates, as already mentioned, to 6
The concrete was produced with frostresistant aggregate.
2.4 LoadIndependent Environmental Impact
Field station I; 1st winter (05/06) Field station II; 1st winter (05/06) Field station II; 1st winter (06/07)
25 20 15 10 5
30
Visuel damaging degree [%]
0
25
Field station I; 1st winter (05/06) Field station I; 2th winter (06/07) Field station II; 1st winter (05/06) Field station II; 2th winter (06/07)
20 15 10 5
C III/B 295kg/m³ w/c= 0.61
C III/B 330kg/m³ w/c= 0.55
C I 280kg/m³ w/c= 0.64
C I 295kg/m³ w/c= 0.61
C III/B 295kg/m³ w/c= 0.61
C III/B 330kg/m³ w/c= 0.55
C III/B 360kg/m³ w/c= 0.50
C I 260kg/m³ w/c= 0.70
C I 280kg/m³ w/c= 0.64
C I 295kg/m³ w/c= 0.61
C I 330kg/m³ w/c= 0.55
0
C I 330kg/m³ w/c= 0.55
Visuel damaging degree [%]
30
105
Fig. 2.87. Left: surface damage after 1 winter’s exposure; right: progression of damage with increasing number of ftc (after 2 winters); ﬁeld station I  Holzkirchen (South Germany), ﬁeld station Meißen (East Germany)
the concrete quality, independent of the climatic stressing (winter 05/06 or 06/07) or exposure site. The damage gradation observed under real conditions is also reﬂected in the damage to laboratory concretes. Working from the observed damage conditions in the ﬁeld tests and the ﬁndings in [69],[769] frost tests were performed in the laboratory (CIF test, see Subsection 3.2.1.3) with test solutions that simulated the rainwater and similar factors. Details on how these tests were performed are given in [120]. In [69],[769], it was established that diﬀerent drinking water qualities led to diﬀerences in the amount of weathering in the same quality of concrete. The laboratory results from Project A11 show that even typically ignored quantities of salt ions caused a considerable increase in weathering. This observed situation cannot be adequately explained by the laws of classical chemistry (pessimal deicing salt agent concentrations are in the range of 13 M.%) or macroscopic physical laws such as supercooling or nucleation. The gel structure of the cement paste suggests surface physical causes or surface chemical causes and calls for further research. The increase in weathering in the investigated concentration range is proportional to the ion concentration of the test solution. The trend regarding the degree of damage observed in situ was simulated in the laboratory in the CIF test with a 0.2 M.% calcium nitrate solution (Figure 2.88). Overall, we can state that in the present case study, the frost damage determined by CIF testing in the laboratory is portable to the frost damage occurring under real conditions, with regard to weathering conditions. A
106
2 DamageOriented Actions and Environmental Impact
5000
1000 app.0.4M.% CN
4000
800
app.0.8M.% CN field test
3000
600
2000
400
1000
200
C III/B 295kg/m³ w/c= 0.61
C III/B 330kg/m³ w/c= 0.55
C III/B 360kg/m³ w/c= 0.50
C I 280kg/m³ w/c= 0.70
C I 295kg/m³ w/c= 0.61
0 C I 330kg/m³ w/c= 0.55
0
Scaling [g/m²] field: 10 natural ftc
Scaling [g/m²] laboratory: 28 standardised ftc
app.0.2M.% CN
Fig. 2.88. Comparison of the surface scaling obtained in laboratory and in ﬁeld; CNcalcium nitrates [120]
semiquantitative correlation was derived between the laboratory and real results. After a winter phase, the damage to the in situ concrete is proportionally comparable to the damage to the laboratory concretes after 28 standardized ftc (CIF test). In light of the measurement of frost resistance according to the ”performance concept”, this is signiﬁcant. Namely, it demands test methods that correctly reﬂect the corresponding attack and predict the durability of concrete even under the obligatory time lapse during simulation. For damages in situ, the ascertained weathering in connection with the visible degree of damage is decisive. These ﬁndings could be worked into evaluation systems for construction concrete, seeing as there are currently no speciﬁcations for evaluating the condition of frostdamaged buildings in the present national or international bodies of rules [737]. 2.4.3 External Chemical Attack Authored by Ivanka Bevanda and Max J. Setzer Substances which attack concrete can be present in surfaces, in ground and waste water and in rainwater. They can equally occur in damp ground. Gases (e.g. exhaust gases) can also attack concrete. The chemical damage process can be (1) swelling or (2) dissolvent in nature. The swelling attack is characterised by the rapid failure of the concrete after a comparatively long time with slight changes of the macroscopic properties. In chemically dissolvent concrete, attacks components of the cement stone in the pore ﬂuid
2.4 LoadIndependent Environmental Impact
107
are dissolved. This leads directly to an increase in porosity and permeability and also to a loss of stability. As a rule, no direct deformation of the aﬀected structural element is observed in case of a dissolvent process. 2.4.3.1 Sulfate Attack An external sulfate attack is caused by water and soil layers containing sulfate or SO2 in the air. The sulfate attack can only occur if damp is present. The formation of reaction products (see Subsection 3.1.2.3.3) which cause swelling in the concrete in suﬃcient quantities is decisive for a swelling attack following sulfate penetration. The resulting compressive stress due to expansion causes swelling, crack formation and ultimately leads to a loss of stability and damage to the cement matrix. Due to its great technological signiﬁcance on account of the prevalence of concrete structures and the sulfate compounds which occur almost everywhere (e.g in ground water, seepage water and soil layers), a large number of investigations into sulphate attack have been performed in the past. Current knowledge has been integrated into rules and standards [6],[14]. No damage has been reported in Germany for concrete with a high sulfate resistance where the measures deﬁned in the standards have been adhered to [147]. For a number of years there have been international reports of a new form of sulphate damage to concrete structures; this is described as the thaumasite form of sulphate attack (Subsection 3.1.2.3.3). Unlike the generally known forms of sulfate attack which lead to the formation of cracks and thus to a decrease in stability of the concrete through swelling reactions (ettringite swelling and, at high sulphate concentrations, also gypsum swelling), a damaging formation of thaumasite leads to weakening; the strengthforming CSHphases of the cement matrix are degraded. In general, concrete foundations of bridges and structures which were exposed to a strong sulfate attack in the ground are aﬀected (e.g tunnel shells), Figure 2.89. Current knowledge of the most important damagerelevant factors and the overview of thaumasite damage in Germany and abroad are summarised in the progress report DAfStb7 [147] entitled ”Sulfate attack on concrete”. In the [147] special interest is taken in the damage potential of pyritecontaining soils in Germany. The oxidation of pyritecontaining minerals (Subsection 3.1.2.3.3) in the adjacent stone or soil has been determined in several cases as the cause of the thaumasite form of sulphate attack [291]. 2.4.3.2 Calcium Leaching If the surface of a concrete structural element is in contact with soft water over extended periods the calcium hydroxide is broken down hydrolytically and calcium in the pore liquid is released (see Subsection 3.1.2.3.2). As a result, the porosity and the permeability of the structure are increased and can 7
German Committee for Reinforced Concrete.
108
2 DamageOriented Actions and Environmental Impact
Fig. 2.89. Concrete damage caused by thaumasite (taken from [151], origin: left BRE; right  FA FingerInstitute, Weimar)
Fig. 2.90. Corrosion on mortar coatings in two drinking water reservoirs. The coating shown on the right has been almost completely destroyed after about 10 years [138]
ultimately lead to a loss of stability. The progression of the dissolving and thus of the damage front takes place very slowly with calcium leaching, especially under environmental conditions which are not constantly damp (several centimetres per decade). For normal structures, calcium leaching can generally be classiﬁed as uncritical. This environmental attack is, however, signiﬁcant for structures which are in direct contact with soft water for an extended period of time, such as the inside of cooling tower shells and cementitious layers of drinking water reservoirs (Figure 2.90). In addition, calcium leaching is a decisive damaging mechanism for concrete constructions of nuclear disposal sites, as the assessment periods for these are several hundred years [804]. Further structures for which calcium leaching can be a stability problem are dams, tunnels and water pipes.
2.5 Geotechnical Aspects
109
2.5 Geotechnical Aspects Authored by Theodoros and Andrzej Niemunis
Triantafyllidis,
Torsten
Wichtmann
This section deals with the eﬀect of a highcyclic (longterm) loading on possible ”deterioration” eﬀects in a soil. It is worth to be noticed that a highcyclic loading does not cause ”damage”, ”fatigue” or ”deterioration” in a soil in the common sense, as it is observed for steel or concrete materials. Eﬀects like abrasion of the soil particles or even fragmentation of the grains are not considered here because the design of a foundation usually exclude such states. Furthermore, within the framework of a continuum approach the permanency of the soil particles is assumed. However, a highcyclic loading may change the soil fabric and may lead to an accumulation of permanent deformations. Thus, the serviceability of a foundation is the main concern if it is subjected to a highcyclic loading. In a constitutive relation for soils under highcyclic loading (see Section 3.3.3) the development of these permanent deformations may be modelled similar to a ”fatigue” in steel or concrete materials. Section 2.5.1 discusses possible sources of a highcyclic loading of soils. It deals with the diﬀerent appearance of the ”accumulation” phenomenon in dependence of the boundary conditions (e.g. drained or undrained cyclic loading) and outlines the possible consequences for structures. Section 2.5.2 presents a novel deﬁnition of an amplitude capturing a multidimensional cyclic excitation. The deﬁnition is applicable not only to soils but also to any other material (e.g. steel or concrete) under multiaxial loading conditions. 2.5.1 Settlement Due to Cyclic Loading Authored by Theodoros and Andrzej Niemunis
Triantafyllidis,
Torsten
Wichtmann
Structures are interacting with the soil. The stiﬀness of the soil depends on the loading of the foundation and in turn the behaviour of the structure is inﬂuenced by the stiﬀness of the subsoil. The design of foundations depends in a great extent on the conditions of the underlain soil and in this way the soil is forming a part of the building. Uniform settlements of foundations do not produce any structural damage. The admissible settlement may be restrained by serviceability requirements only. Diﬀerential settlements are much more important. They may be caused by local variations of the geotechnical conditions such as a variation of the thickness or the depth of the settlementsensitive layers, inclusions of soft materials or nonhomogeneities of the void ratio or of the fabric of the soil. Diﬀerential settlements may also occur due to diﬀerent foundation schemes (pile and shallow foundations side by side) and diﬀerent loadings arising from the superstructure design (despite design eﬀorts to avoid this).
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2 DamageOriented Actions and Environmental Impact
traffic loading, e.g. high speed or magnetic leviation trains
crane rails
wind power plants offshore
watergates
tanks, silos
onshore
surface compaction, vibrocompaction
Fig. 2.91. Sources of cyclic loading of soils
Soil compaction, soil replacement or the choice of a more appropriate foundation design are possible measures prior to the construction to minimize diﬀerential settlements. Such procedures have been developed in the past and are not subject of the present study. While diﬀerential settlements that occur during the construction process due to unpredictable soil inhomogeneities can be counteracted to some extent (by ground improvement or a change of the method of construction), such measures are diﬃcult and expensive during the lifetime of a structure. With reference to the subsoil, life time oriented design concepts focus on permanent deformations in the subsoil which occur due to repeated loading during the operating time of a structure. Examples for such cyclic loading caused by traﬃc (highspeed trains, magnetic leviation trains), industrial sources (crane rails, machine foundations), wind and waves (onshore and oﬀshore wind power plants) or repeated ﬁlling and emptying processes (watergates, tanks and silos) are given in Figure 2.91. Furthermore, construction processes (e.g. vibration of sheet piles) and mechanical compaction (e.g. vibratory compaction) introduce cyclic loads into the soil. They cause a densiﬁcation at the required position which is usually desired for the future construction but may cause some detrimental eﬀects for the existing neighbours. A stress path due to a wheel passing on the ground surface is given in Figure 2.92a. In statically indeterminate structures the diﬀerential settlements may cause changes of internal forces which may slow down or accelerate the process of deterioration in the structure. Vice versa, a change of the reaction forces leads to a diﬀerent rate of settlement accumulation. In statically indeterminate structures under monotonic loading the loading of more compliant foundations decreases due to a redistribution of internal forces. The loading of the less compliant foundations increases and this may cause plastic deformations in the subsoil, i.e. the settlements of these foundations increase. Thus, the diﬀerential
2.5 Geotechnical Aspects
a)
111
traffic loading
τ,σ σv0
σv
σh0 = K0 σv0
γ
σh
τ
σv
σv
σh
σh
γ
t τ
b) GW
GW σv0 σh0 = K0 σv0
GW σv τ γ
σh
τ
σv
τ σh
γ
t
wave propagation
soil rock
ground shaking
ground shaking
Fig. 2.92. Cyclic stresses in a soil element a) due to a passing wheel load and b) due to an earthquake loading
ampl
av
s t
s
Fig. 2.93. Accumulation of settlement due to cyclic loading
settlement is reduced. For a cyclic loading this smoothing does not always work due to the decrease of the accumulation rate with the average pressure (Section 3.2.2). A life time oriented design concept for structures should include a joint analysis of the structure and the inhomogeneous subsoil. The settlements (Figure 2.93) due to cyclic loading occur since in an element of soil closed stress loops, resulting from external loading, lead to not perfectly closed strain loops. An irreversible deformation remains in the soil, caused by particle rearrangement due to changes of the intensity and the distribution of the contact forces between the particles. This permanent deformation is accumulated with the number of cycles. Even small amplitudes can signiﬁcantly contribute if the number of cycles is high. Such a loading with small amplitudes and large numbers of cycles (Nc > 103 ) is called poly or highcyclic loading. As conﬁrmed by the element tests presented in Section 3.2.2 and also by parametric studies outlined in Section 4.6.6 the amount of residual settlement depends on the loading of the foundation (average load, load amplitude) and on the current state of the soil (void ratio, cyclic preloading).
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2 DamageOriented Actions and Environmental Impact
Unfortunately, as demonstrated in Section 4.6.6 diﬀerential settlements due to cyclic loading are much more sensitive (by a factor 3) to inhomogeneities in the subsoil than those due to monotonic loading. In the context of foundations subjected to cyclic loading, one may distinguish between the shortterm and the longterm behaviour. Studies of the shortterm behaviour deal with the deformation of the structure and the subsoil within a few cycles (e.g. examinations of the dynamic characteristics of a system). In the majority of such studies a linear response is assumed considering no changes of the soil parameters during the event. In the case where a nonlinear behaviour of the soil has to be considered an implicit calculation can be performed as outlined in Section 4.2.11. In longterm studies the accumulation of settlements or changes of the soilstructure interaction are the main concern. This book is dedicated to the longterm behaviour. If the load cycles are applied at a low amplitude and low frequency f = ω/(2π), the inertial forces are negligible and it is spoken of a quasistatic cyclic loading. If the frequency is large, inertial forces are relevant and the loading is dynamic. A harmonic excitation with the displacement u = uampl cos(ωt) can be considered as quasistatic, if uampl ω 2 is small compared to the acceleration of gravity g. Often the amplitudedependence is ignored and the borderline to dynamic loading is said to lay above f ≈ 5 Hz. As reported by the literature and conﬁrmed also by tests of the authors (with f < 2 Hz and εampl ≤ 10−3 , [835]) the loading frequency f does not inﬂuence the rate of strain accumulation as long as the strain amplitude εampl is constant. In order to estimate settlements due to cyclic loading and in order to incorporate them into a life time oriented design concept for engineering structures one needs special calculation strategies and a constitutive description for the soil. Such a strategy and a highcycle model have been developed and are presented in Sections 3.3.3 and 4.2.11. In Section 3.2.2 it is demonstrated for uniaxial cycles with a constant polarization that having packages of cycles with diﬀerent amplitudes their sequence does not play a signiﬁcant role for the ﬁnal value of the permanent deformation. It is further assumed that a transient or periodic signal can be decomposed into a series of cyclic signals with diﬀerent frequencies (Section 2.5.2). Afterwards these signals are grouped into packages in which the amplitude is constant (Figure 2.94). The analysis of the permanent soil deformation can then be performed as given in Sections 3.3.3 and 4.2.11. If the cyclic stresses in the soil are not too close to the failure criterion and if the amplitudes are below εampl ≈ 10−5 the accumulation rate can be expected to become very small or even vanish after a suﬃciently large number of cycles (adaptation, ”shakedown”). Having reached such asymptotic state the soil behaviour is almost linear elastic during the subsequent cycles. In such cases accumulation eﬀects need not to be considered in the design of structures. In Section 3.2.2 it is demonstrated that polarization changes lead to a temporary increased accumulation rate. Having reached an asymptotic
2.5 Geotechnical Aspects
σ
h(σ) t
113
σ t
Fig. 2.94. Decomposition of a signal with varying amplitudes into packages of cycles with constant amplitude
state a restart of the accumulation and adaptation process may occur after a sudden change of the polarization. However, no sound experimental studies exist on the accumulation at such small amplitudes. Thus, the eﬀect cannot be validated or quantiﬁed yet. Another asymptotic state may be observed in saturated cohesive soils supporting a foundation which are subjected to a cyclic loading. The accumulation of pore water pressure (see remarks below) is very small, if the excitation frequency f is below the ratio cv /b2 with cv being the coeﬃcient of consolidation and b the width of the foundation (almost drained conditions) and if the strain amplitude is below εampl ≤ 10−2 [329]. If the cyclic stress path repeatedly reaches the failure criterion an incremental soil collapse may occur. An application of cyclic loading with smaller amplitudes after a strong event (e.g. a storm in the case of oﬀshore foundations) can lead at least hypothetically to a ”healing eﬀect”, i.e. to a reduction of deformations imposed by the strong event. A cyclic loading may not only cause permanent deformations. Depending on the boundary conditions it may also result in a change of the average stress. In watersaturated soils under partly drained or undrained conditions the pore water pressure uav may accumulate with the number of cycles due to the contracting soil behaviour. Thus, the eﬀective mean pressure pav , the shear strength and the stiﬀness decrease or even vanish (socalled ”liquefaction” or ”cyclic mobility” in case of temporary loss of shear strength). Such eﬀects are observed e.g. during earthquakes (Figure 2.92b). While a ”manmade” highcyclic loading on structures is associated with small amplitudes and a high number of cycles the number of cycles is small in the case of a seismic loading but the amplitudes are large. The drainage conditions play a signiﬁcant role. Usually undrained conditions are considered for an earthquake loading because of the great intensity and the short duration of action. In contrast, a highcyclic loading is calculated assuming drained conditions because of the long duration and the small intensity of action. In the undrained case a pore water pressure accumulation takes place and as a consequence eﬀects like liquefaction, phase or layer separation and spontaneous densiﬁcation (during reconsolidation) may be observed.
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2 DamageOriented Actions and Environmental Impact
These eﬀects can be utilized for an intelligent foundation design in order to establish a passive screening, i.e. to reduce the seismic loading acting on the structure and thus to prevent it from damage during an earthquake. A popular example for a passive screening are the foundations of the Higashi temple in Kyoto. In the case of an earthquake layers of ﬁne grained material are brought to liquefaction in order to avoid the passage of shear waves to overlain layers or structures (so called ”Hanchikueﬀect”). The liquefaction phenomenon is also utilized for soil improvement techniques (deep vibratory compaction). However, if the described phenomena under an undrained cyclic loading are not well understood by the design engineer a nonappropriate design of the foundation may be chosen. A more detailed discussion of the eﬀect of a cyclic loading under various boundary conditions is given in Section 3.1.3. Another source of cyclic loading of soil, which is not discussed in detail in the present book, is caused by climatic changes and seasonal eﬀects. Such loading is connected with changes of the portions of the three phases (solid particles, pore water, air) of a soil and may lead to changes of its fabric and its mechanical properties. The cyclic change of the water table e.g. leads to an accumulation of water content (degree of saturation) in the transition zone and an alteration of the eﬀective stress and the suction. This cyclic change of the eﬀective stress acting on the solid phase may cause permanent deformations. In the case of cohesive soils permanent deformations are generally associated with wetting and drying processes leading to swelling and shrinkage. Clusters of tension cracks may occur inﬂuencing the hydraulic and mechanical properties of the soil. Such kind of cyclic loading referring to hydromechanical coupling and partial saturation of soils is of great importance for water reservoirs, dam embankments, dykes, etc. Sources and eﬀects of cyclic loading are maningfold. In a life time oriented design all relevant inﬂuences and boundary conditions a soil may be exposed to (depending in turn on the design solution) have to be kept in mind. 2.5.2 Multidimensional Amplitude for Soils under Cyclic Loading Authored by Andrzej and Theodoros Triantafyllidis
Niemunis,
Torsten
Wichtmann
A cycle is understood as a path (a trajectory parametrized by time) which is recurrently passed through by a state variable (like strain or stress). For a scalar or tensorial variable we may deﬁne its average value av to be the centre of the smallest (hyper)sphere that encompasses all states upon the cycle. For a scalar variable one obtains av = 12 (max + min ) and the amplitude is ampl = max −av . For tensorial variables, apart from the size of the (hyper)sphere, we want to convey some information on the polarization and the ovality of the path, which renders the amplitude to become a tensor. Further we consider strain cycles ε(t) only, with ε = ln U where U is the right stretch tensor. We distinguish between inphase (=IP) strain cycles
2.5 Geotechnical Aspects
a) uniaxial IP  cycles ampl = 1, ampl = 0 1 3
1.0
0.5
0.5
c) OOP  cycles: ampl = 1, ampl = 1, 1 3
b) multiaxial IP  cycles ampl = 1, ampl = 1 1 3
ε3 1.0
ε1
1.0
ε3
ε3
1.0
1.0
0.5
0.5
0.5
0.5
1.0
115
ε1
1.0
0.5
0.5
0.5
0.5
1.0
1.0
= /4
1.0
ε1
Fig. 2.95. Distinction between uniaxial IP, multiaxial IP and OOPcycles ampl εij = εav ij + εij f (t),
−1 ≤ f (t) ≤ 1
(2.76)
for which the variability of all components in time can be described by a common function f (t) and outofphase (=OOP) cycles which cannot be expressed in this way, e.g. ampl ε = εav + diag(εampl 11 sin(ωt + ϕ11 ), ε22 sin(ωt + ϕ22 ), 0)
ϕ11 = ϕ22 (2.77)
and which require individual time tracking fij (t) of various εij components. The collection εampl of the amplitudes of the individual components in (2.76) ij should not be mixed up with the tensorial deﬁnition of the amplitude A ε which will be proposed further. The IPcycles that have only one nonzero eigenvalue of εampl are termed uniaxial, ε = εav + diag(εampl , 0, 0) f (t) , 1
(2.78)
otherwise they are multiaxial ε = εav + diag(εampl , εampl , εampl ) f (t) . 1 2 3
(2.79)
All OOP cycles are multiaxial too. All deﬁnitions are illustrated in Figure 2.95. A harmonic OOPcycle (Figure 2.95c) the components of which diﬀer by the phaseshifts ϕij but not by the angular frequency ωij = ω = const is termed harmonic oscillation and forms a 6d ellipse in the strain space. This concept is useful in the Fourier analysis of the deformation treated as a 6d signal. Arguments for expressing the amplitude and the accumulation in terms of strain rather than stress in the highcycle model for soils (Section 3.3.3) have been discussed in [578]. Given from laboratory tests a cycle in form of a stress path σ(t) or a mixed path we must evaluate all unknown (if any) components
116
2 DamageOriented Actions and Environmental Impact
of the strain path ε(t) solving σ(t) = E : ε(t) for the unknown εij (t) using the secant elastic stiﬀness E of the cycle. Similarly, given the cumulative rates from experiments (measured (= m ) or prescribed), namely pseudorelaxation σ˙ m (Nc ) and pseudocreep Dm (Nc ), the constitutive strain accumulation rate D acc is obtained by solving the material equation σ˙ m = E m : (D m − Dacc ). Note that all rates are meant as increments or residuals after a single cycle in the highcyclic context. A description of polarization must involve all 6 components of the strain path ε(t) because the strain states need not be coaxial upon a cycle. In order to evaluate the tensorial strain amplitude A ε from a discrete path ε(t1 ), ε(t2 ), . . . obtained from laboratory tests or from FEcalculations one should avoid using the ﬁrst cycle (= irregular cycle discussed in Section 4.2.11). From a representative (recorded or calculated) cycle ε(t) we extract the resilient strain path εe (t). It is done by subtracting the residual (cumulative) portion (pseudocreep) from it. This operation is called detrending. The proposed detrending procedure consists of four steps: • • • •
˙ Calculate the hodograph D(t) ≈ ε(t), Fig. 2.96 Find the shortest period T from the requirement
D(t) − D(t + T ) dt → min Find the average D av and interpret it as the rate of accumulation wrt time, Dav = D acc /T Subtract the cumulative portion from the original path: εe (t) = ε(t)−D av t with t ∈ [0, T ].
a)
D2 2
D
acc
D1
1
b)
D2
2
D 1
acc
D1
˙ Fig. 2.96. A hodograph is a trajectory of D(t) ≈ ε(t) parametrized with time t, analogously to the strain path ε(t). The rate of accumulation can be easily identiﬁed as a drift rate (denoted with arrow) of the average strain upon a cycle. Note that the strain rate is an exactly periodic function D(t) = D(t + N T ) whereas the strain ε(t) is not. The distinction between a) the cycles encompassing some area (outofphase cycles (= OOP) and b) the opencurve cycles is of importance
2.5 Geotechnical Aspects
Projection of
ε(t)
117
from 3D to 2D
R (3)
Projection of
r (3)
r (2) R
ε(t)
from 2D to 1D
r (2) (1)
R
(2)
r
(1)
Fig. 2.97. Determination of spans R(3) with r (3) , R(2) with r (2) and R(1) with r (1) for a 3d loop
In the following sections the index e is omitted for brevity. A tensorial deﬁnition of the strain amplitude A ε has been proposed [575] to consider the observations (Section 3.2.2) that apart from the size of the strain cycle also its ovality, deplanation and the changes of its polarization can strongly inﬂuence the rate of accumulation Dacc which is the most important element of the highcycle model (Section 3.3.3). The proposed deﬁnition has proven to work well with various convex strain cycles similar to harmonic oscillations and to consider the experimental observation that the change of circulation of circular cycles does not aﬀect the rate of accumulation. Suppose, we are given a detrended strain cycle in form of a sequence of discrete strains ε(tk ), k = 1, . . . , M recorded (smartly to save the computer memory) by an FE program at a Gauss integration point. From ε(tk ) with k = 1, . . . , M we determine the pair of the two most distant states, say ε(ta ) and ε(tb ). The socalled span of the cycle is quantiﬁed by its size 2R(6) = −−−−−−−−→ → −
ε(ta ) − ε(tb ) and its orientation r (6) = ε(ta ) − ε(tb ), wherein = /
(i) denotes normalization. The upper index corresponds to the dimensionality of the strain cycle. For example the original strain path (before socalled ﬂattening, see below) can be at most sixdimensional, ε(6) (t) = ε(t). In order to ﬁnd the second longest span the original strain path is projected onto the hyperplane perpendicular to r(6) . The resulting ﬂattened strain trajectory ε(5) = ε(6) − r(6) : ε(6) ⊗ r(6) has at most ﬁve dimensions. The span of the ﬂattened trajectory can be determined analogously and described by R(5) and r (5) . The ﬂattened loop is subjected to the subsequent projection, this time along r (5) , etc. Of course R(6) ≥ R(5) ≥ · · · ≥ R(1) holds. The tensorial amplitude A ε is proposed to be the following sum of dyadic products Aε =
6 i=1
R(i) r (i) ⊗ r(i)
(2.80)
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2 DamageOriented Actions and Environmental Impact
collecting all spans. Summing up, the method consists in the stepwise evaluation of spans and the degeneration (ﬂattening) of the strain path. The sense of the direction of r(i) is of no importance, which is obvious from (2.80). Projections from a 3dimensional path to the 1dimensional path are shown in Figure 2.97. For the 4th order tensor A ε the list of radii R(6) ≥ R(5) ≥ · · · ≥ R(1) can be seen as the spectrum of eigenvalues and the mutually orthonormal orientations r(6) , r (5) , . . . , r (1) are the eigentensors. The normalized amplitude − → Aε
A ε = A ε / A is called the polarization and the norm 2 2 2
Aε = R(6) + R(5) + · · · + R(1) εampl = A
(2.81)
(2.82)
is the scalar amplitude. For the special case of a harmonic oscillation εij (t) = εampl sin(ωt + ϕij ) with equal or diﬀerent phase shifts ϕij the size of the amij ampl plitude is εampl = εampl and for the 1d case it is identical with the ij εij conventional deﬁnition. Using the quadratic dependence Dacc ∼ (εampl )2 discussed in Section 3.2.2, the accumulation model (Section 3.3.3) predicts a twice larger accumulation rate from a circular cycle than from a uniaxial one along the diameter. This is in good agreement with the experiments presented in Section 3.2.2. If several sources of cyclic loading are acting simultaneously, complex strain loops may result from diﬀerent polarizations and frequencies of the cycles. A procedure for the determination of the strain amplitude and the number of cycles is discussed in the following. Some problems related to the deﬁnition of the amplitude are: •
•
•
After a full cycle the strain path does not exactly pass through the same strain state (due to accumulation). Moreover the strain loop may intersect itself which does not indicate that the loop is over. It is evident that a mathematical tool is required in order to detect the period, i.e. when a strain loop is ﬁnished. Suppose a strain loop has been prescribed by two spans with slightly diﬀerent frequencies so that a slow rotation of polarization occurs, Fig. 2.98a,b. If two spans were equally polarized the beat would occur. The hitherto hypothesis either ignores the small spans or overestimates their eﬀect describing such loading as distinct packages with alternating polarization. It is not clear if smaller but faster cycles in plane with the dominant cycle or out of this plane may be ignored, Fig. 2.98c,d.
For practical applications in soil mechanics the detrended strain path εij (t) can be assumed to be a superposition of individual harmonic oscillations. The harmonic oscillations can be distinguished judging by the frequency fK (or angular velocity ωK = 2πfK ). From each of six components εij (t) of the
2.5 Geotechnical Aspects
119
b)
a)
0.5 0 0.5 1
1 1
e)
0.5
1
0.5
0.5
Acceleration aNS [m/s2]
0.5
0
1
0.5
0.5
1 1
1 0.5 0 0.5 1
d)
c) 1
0.5
1
0.5
0.5 0.5
1
1
0.2 0 0.2
0.5
2 N 1 0
W
E
1 S 2 2 1 0 1 2 Acceleration aWE [m/s2]
0
1 0.5
0.5
0 1 0.5 1
1
Fig. 2.98. Strain paths (Lissajous curves) obtained from the superposition of sine functions a),b) with slightly diﬀerent frequencies and amplitudes or c),d) with very diﬀerent frequencies and diﬀerent amplitudes. e) Acceleration measured on ground surface during the Niigata Earthquake (1964)
strain path we pick up a portion which corresponds to a common dominant frequency fK . The componentwise sum of these six signals constitutes a harmonic oscillation. In general it is a 6dimensional ellipse in the strain space. In this Section the oscillations are numbered with the capital letter K. We will try to approximate the signal εij (t) as a sum of M oscillations: εij (t) ≈
M
K εampl sin(ω K t + ϕK ij ) ij
(2.83)
K=1
The essential purpose of the present spectral analysis is ﬁltering out the portions of the individual strain components εij corresponding to the same angular velocity ω K and gathering them into common oscillations. This needs to be done only for several dominant frequencies f K with K = 1, 2, . . . for which ampl K the strain amplitudes εij are large. Since the square of the amplitude dictates the accumulation rate (Section 3.2.2) the impact of smaller amplitudes becomes negligible (assuming the superposition of their eﬀects). Each component function εij (t) is treated as a series of discrete values εij(k) given at k = 0, 1, . . . , N − 1 (N is an even number) points equally distributed along the time axis over the time window from t = 0 to t = (N − 1)Δ. Denoting the sampling interval as Δ we ﬁnd the Nyquist frequency fc = 1/(2Δ) and the (complex valued) discrete Fourier transform (DFT) Yij(n) of the discrete
120
2 DamageOriented Actions and Environmental Impact
strain path εij(k) with frequencies fn = ωn /(2π) such that −fc < fn < fc . The frequencies are indexed with n = −N/2, . . . , −1, 0, 1, . . . N/2 and Yij(n) ≡
N −1
εij(k) e2πIkn/N
(2.84)
k=0
where I 2 = −1. This DFT approximates the Fourier transform Yij (f ) at discrete frequencies fn according to Yij (fn ) ≈ Yij(n) Δ but we will use the DFT only. We assume that our sampling interval Δ is suﬃciently small to capture all strain oscillations of importance and that no higher frequencies can leak into the (−fc , fc ) range (no aliasing). Plotting the tensorial norm (sum over ij only) we obtain the periodogram
Y(n)
⎧ ⎪ ⎨Y ij(n) Y ij(n) + Y ij(−n) Y ij(−n) = Y ij(0) Y ij(0) ⎪ ⎩ Y ij(N/2) Y ij(N/2)
for n = 1, 2 . . . , N/2 − 1 for n = 0 for n = N/2 (2.85)
and among the frequencies fn = n/(N Δ) we may ﬁnd the one for which the (real valued) function Y(n) has its maximum. This frequency is denoted as fK and the corresponding angular velocity ωK = 2πfK enters (2.83). Technically, since the original signal εij (t) may be a pure sine function with a frequency lying in the middle between two adjacent fn s one needs data windowing (apodization, e.g. Hann or Barlett window) in order to reduce the leakage of frequency. Having found the dominant frequency fK we ﬁlter out a band around this frequency from each component of the strain. For this purpose K we simply multiply each DFT Yij (fn ) by the bandpass ﬁlter HBP (an even function equal to unity in the vicinity of ±fK and to zero elsewhere) in the frequency domain. We obtain six fK bandpass ﬁltered transforms K K Yij(n) = HBP Yij(n)
(2.86)
ampl K which constitute the DFT of the Kth oscillation. The amplitudes εij of the strain components are obtained from the discrete inverse Fourier transform (DIFT)
εij(k) =
N −1 1 K −2πIkn/N Y e N n=0 ij(n)
(2.87)
Among all kindexed values we ﬁnd the diﬀerence between the maximum and the minimum of εij(k) (for each component ij separately). These diﬀerences K correspond to the double amplitudes 2εampl of the oscillation K and enter ij (2.83). The expression for DIFT contains an nsum from 0 to N (instead of from −N/2 to N/2) thanks to the N periodicity of the DFT, i.e. Yij(−n) =
2.5 Geotechnical Aspects
121
Yij(N −n) . The phase shift ϕK ij is calculated from the correlation of individual K components ij . For example, we may assume ϕK 11 = 0 and the phase shift ϕ22 K ∗K is calculated in the frequency domain using the product Y11 Y22 wherein the asterisk denotes the complex conjugation. The phase shift follows from the time lag τ22 obtained as the time shift for which the DIFT of the above K product has its maximum. Finally we have ϕK 22 = ω τ22 . The Kth oscillation is completely determined by repeating analogous calculations of correlation for all strain components εij . The remaining oscillations are selected analogously using the reduced signal K Yij(n) = HN O Yij(n) ,
(2.88)
K where HN O denotes the notch ﬁlter (an even function equal to zero in the vicinity of ±fK and to unity elsewhere). Currently, the fatigue load contributions from the individual oscillations i.e. the size of the amplitude εamplK =
εamplK and the number of cycles Nc enter the fatigue loading independently. 2 A single load package of duration T is calculated as T K fK εamplK i.e. without considering the mutual polarizations of diﬀerent oscillations within the package. An experimental proof of the proposed analysis will be given in future.
3 Deterioration of Materials and Structures: Phenomena, Experiments and Modelling
Authored by Otto T. Bruhns and G¨ unther Meschke Reliable computational prognoses of the structural integrity and serviceability throughout the lifetime of structures require the realistic consideration of the damage behaviour of the construction materials for various loading scenrios including static and cyclic loading, environmental loading processes such as moisture and heat transport, corrosion processes, freezethaw actions and possible interactions between these long and shortterm processes. Both, loadinduced damage mechanisms such as evolving microcracks and physically and chemically induced deterioration originate from mechanical, physical and chemical processes starting at lower scales of the microstructure of the materials. Investigating and understanding these processes acting at various scales is a prerequisite for the development of adequate and suitable material models suitable for lifetime oriented simulations. Accordingly, this chapter is organized within three subchapters covering • •
•
the most relevant phenomena obeserved in diﬀerent materials such as metals, cementitious materials and soils (Subchapter 3.1), results and insights gained from laboratory investigations on the fatigue behaviour of concrete, metals and soils, nondestructive testing of microcrack evolution in metals and structural testing of composite structures performed within the Collaborative Research Center SFB 398 (Subchapter 3.2), numerical models developed within the SFB 398 for cementitious materials, metallic materials and soils suﬀering from diﬀerent forms of (accumulating) damage and deterioration mechanisms when subjected to external and environmental loadings. This includes damage accumulation resulting from quasistatic and cyclic external loading as well as deterioration phenomena in concrete resulting from time variant drying and wetting processes, dissolution and chemically expansive processes such as the AlkaliSilica reaction (Subchapter 3.3). This Subsection also contains selected applications to structural durability analyses.
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3.1 Phenomena of Material Degradation on Various Scales Authored by Otto T. Bruhns and G¨ unther Meschke Initiation and evolution of damage and deterioration in solids may have various origins depending on the speciﬁc microstructure of the material and the type of loading. This Subchapter provides an overview of diﬀerent damage and deterioration phenomena of metals, cementitious materials and soils. It constitutes, in association with laboratory investigations reported in Subchapter 3.2, the basis for the deveolpment of respective material models in Subchapter 3.3. A hierarchical mode of classiﬁcation of material damage phenomena is adopted. It is characterized, on the highest level, by a diﬀerentiation into material damage caused by external loading and degradation caused by environmental (nonmechanical) actions such as thermal, hygral and chemical loadings. Load induced damage mechanisms are further diﬀerentiated into damage resulting from quasistatic (Section 3.1.1.1) and from cyclic loading (Section 3.1.1.2), respectively. In both cases two basic modes of damage are described separately: Ductile damage in metals, characterized by the nucleation and growth of micropores (Section 3.1.1.2.1, and brittle damage in cementitious and metallic materials resulting from the propagation of microcrcacks (Sections 3.1.1.1, 3.1.1.2.2). The accumulation of deformations in soils during cyclic loading is addressed in Section 3.1.3. 3.1.1 Load Induced Degradation Authored by Rolf Breitenb¨ ucher and G¨ unther Meschke 3.1.1.1 Quasi Static Loading in Cementitious Materials Authored by Rolf Breitenb¨ ucher, and G¨ unther Meschke
Otto
T.
Bruhns,
Hursit
Ibuk
Concrete as the most important cementitious material is of heterogeneous character consisting of coarse aggregates and a ﬁne mortar matrix. Hence, the deterioration of this composite material depends mainly on the behaviour of the mortar matrix, the aggregates and the interaction between both components in addition to the type of loading. In this Subsection, the typical failure processes in concrete acting at the mesoscale are ﬁrst described according to uniaxial compression experiments followed by uniaxial tension tests as well as triaxial loadings. For a more elaborate description of the failure mechanisms in concrete on diﬀerent scales, see [542, 406, 142, 540, 180, 406]. 3.1.1.1.1 Fracture Mechanism of Concrete Subjected to Uniaxial Compression Loading The fracture mechanism of concrete subjected to uniaxial loading is mainly controlled by the growth of microcracks, which often already exist before
3.1 Phenomena of Material Degradation on Various Scales
s/fc 100 7090
125
unstable crack propagation à sustained loading strength
stable crack propagation à increasing plastic deformation 30 marginal change of the microcracks à minor plastic deformation e Fig. 3.1. Schematic stressstrain diagram of cementitous materials subjected to uniaxial compression [867]
any load is applied [393, 567]. In particular along the interface between coarse aggregates and the cement paste microcracks can be induced by shrinkage and thermal strains restrained by the heterogeneity of the concrete meso structure. When an external load is applied these microcracks remain stable up to a level of about 30 % of the ultimate load. As a consequence, the stressstrain curve is approximately as linear within this range. When the loading exceeds this range up to approximately 70 to 90 % of the ultimate load these cracks tend to increase in length, width and number (Figure 3.1). This is the stage of the so called slow crack propagation. This crack propagation results in a decreasing macroscopic stiﬀness and a nonlinear stressstrain curve. However, the observed nonlinearity is marginal at this level of loading. Above a stress level of about 70 to 90 % of the ultimate strength, cracks start to open through the mortar, bridging the bond cracks so that a continuous crack pattern is formed [393, 567]. This is the fast crack propagation stage leading to material failure with time if the load is sustained [567]. Ultimately, material failure is characterised by the formation of continuous macroscopic cracks. Further loading in displacement controlled tests is connected with a fast reduction of the residual stresses (material softening). 3.1.1.1.2 Fracture Mechanism of Concrete Subjected to Uniaxial Tension Loadings The fracture mechanism and the tensile strength of concrete subjected to tension in most cases is investigated by means of indirect tests such as bending or splitting tests. Only in some few special cases direct tension tests are performed due to the problematic load transmisson [265, 362, 538, 429]. A typical stressstrain diagram obtained from a displacement controlled direct
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tension test is illustrated in Figure 3.2. To obtain insight into damage processes occuring within the microstructure, nondestructive testing methods such as measurements of acoustic emmission during the loading process and Electronic Speckle Pattern Interferometry (ESPI) are frequently used (see, e.g. [672, 360]). As mentioned above, microcracks can already exist in the interface zone between aggregate particles and cement paste prior to its ﬁrst exposure to external mechanical loading. Initially the stressstrain curvature is nearly linear up to a stress level of about 80 % of the tensile strength fct (range 1 and 2 in Figure 3.2). At this stage, preexisting microcracks remain stable. Further increasing tensile stresses are linked to an enlargement of the preexisting bond cracks resulting in a reduction in the macroscopic stiﬀness, which becomes obvious in a deviation of the stressstrain curve from linearity. When the tensile strength (range 3) is exceeded coalescence of microcracks is starting to form localized macrocracks. After this stage the stresses decrease gradually with increasing strain combined with increasing crack propagation. The descending branch of the stressstrain curve (softening) in range 4 is associated with progressive localisation and accumulation of micro and mesocracks leading ﬁnally to the fracture of the structural member by a discrete crack. A lower content of coarse aggregates as well as smaller maximum aggregate sizes result in a more abrupt decrease of the stressstrain curve compared to larger contents of aggregate and larger aggregate sizes. The behaviour of concrete during un and reloading tensile cycles is illustrated in a stressdisplacement diagram from a uniaxial cyclic tensile test (Figure 3.3). It shows, that the unloading curves  approximated by the dashed lines  do not pass the origin and that their slopes are decreasing from the initial stiﬀness with increasing cycles. Hence, the material behaviour of concrete under tensile loading is characterized by permanent deformations as well as by stiﬀness degradation. Secondly, the envelope of the diagram in Figure 3.3, representing cyclic loadings, correlates with the curvature in Figure 3.2 representing monotonically increasing tensile loadings. In contrast to the mechanical behaviour of concrete subjected to alternating compressive and tensile loadings, these two conclusions also are valid for purely compressive loading states [381]. 3.1.1.1.3 Concrete under Multiaxial Loadings The strength of concrete highly depends on the multiaxiality of the applied stresses. Figure 3.4 shows a typical biaxial strength envelope. Under biaxial compressive stresses σ1 = σ2 the strength is approximately by about 15 to 22 % higher compared to the uniaxial strength. On the other hand the biaxial tensile strength is marginally smaller than the uniaxial tensile strength [464, 567, 564, 786]. The shape of the biaxial failure envelope is not sensitive with respect to the level of uniaxial compressive strength [464, 567]. The behaviour of concrete in triaxial compression as documented in [700, 541, 848, 698, 516] is illustrated
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127
Fig. 3.2. Schematic stressstrain diagram of cementitous respectively geological materials due to tension [538]
Fig. 3.3. Stressdisplacement diagram of a concrete specimen subjected to cyclic tensile loading [381]
in Figure 3.5. It contains stressdisplacement diagrams from triaxial tests at three diﬀerent levels of conﬁning pressure. At moderate conﬁning pressure, the failure mode is characterised by the formation of inclined shear planes.
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uniaxial tension
+s2/fcm
uniaxial compression
+s1/fcm
1.0
0.5
s1/fcm
s2 0.5
s1
s1 s2
max s1/fcm = 1.27 ... 1.42
1.0 uniaxial compression
s2/fcm
s1 = s2 = 1.15 ... 1.22 fcm
Fig. 3.4. Biaxial failure envelope for concrete [464, 567]
Fig. 3.5. Stressdisplacement diagrams obtained from triaxial compression tests for three levels of conﬁning pressure σ2
As the ratio σ1 /σ2 increases, the triaxial strength of concrete also increases and a transition from a brittle to ductile type of failure is observed. For the limit case σ1 /σ2 = 1, characterising a purely hydrostatic stress state, no peak strength can be observed. The failure doesn’t occur due to cracking but by the crushing of the single particles. The strength of concrete under triaxial stresses is deﬁned by means of a failure envelope or limit surface (Figure 3.6). The limit surface contains all possible failure points under any triaxial state of stress. Figure 3.6 also
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129
Fig. 3.6. Failure surface of concrete in principal stress space and crack patterns corresponding to diﬀerent triaxial loading conditions [541]
contains the failure mechanisms of concrete when subjected to diﬀerent states of triaxial stresses. Depending on the loading condition, the failure behaviour of concrete changes from a brittle to a ductile failure. 3.1.1.2 Cyclic Loading Authored by Otto T. Bruhns and G¨ unther Meschke 3.1.1.2.1 Ductile Mode of Degradation in Metals The lifetime of steel structures such as tanks, pressure vessels, steel frame or steel girder structures as well as steel bridges subjected to cyclic loading may be limited by material fatigue. While High Cycle Fatigue (HCF) is associated with the evolution of damage resulting from a large number of cycles at low and moderate levels, Low Cycle Fatigue (LCF) is a frequently observed mode of failure in structural components made of ductile metals subjected to repeated loading at high stresses and high stress amplitudes. The damage mechanisms for LCF can be classiﬁed broadly into two major groups: transgranular fracture and ductile fracture. The former is caused by the initiation and growth of microcracks through the metallic grains where potential crack initiation sites are the interfaces between intrusions and extrusions of PSB (persistent slip bands) which are formed during local plastic shearing. However, some metals develop diﬀerent textures e.g. cell structures instead of planar bundle structures like the PSB. The kind of texture depends on the stacking fault energy of the metal considered [537]. Normally, the crack initiates at the surface and advances into the interior of the member.
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F a ilu r e a fte r 3 0 c y c le s N o tc h r a d iu s 2 m m
Fig. 3.7. Ductile fracture surfaces of a round notched bar with Al2024T351 material after 30 cycles with notch radius 2mm. (Reprinted with permission of GKSS research institute, Geesthacht, Germany.)
(a )
(b )
(c )
Fig. 3.8. (a) Void nucleation due to fracture of inclusions, (b) partition of inclusionmatrixarea, (c) void coalescence
In zones of very large plastic deformations, low cycle fatigue damage may be initiated by void nucleation, continuous growth and ﬁnally coalescence of micropores leading eventually to macroscopic cracks and structural failure (Figs.3.8, 3.7). This damage phenomenon highly depends on the stress triaxiality, which is the ratio between the mean and equivalent stress [524]. Since nucleation and growth of micropores requires more energy than transgranular fracture, often only very few cycles are suﬃcient for material failure. This phenomenon is also know as ultra or extreme low cycle fatigue [465, 279] since it is, in contrast to classical low cycle fatigue (<5·104 load cycles) characterized by a much smaller number (<100) of load cycles to failure.
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Fig. 3.9. Schematic SN curves for concrete (W¨ ohler curves)
3.1.1.2.2 QuasiBrittle Damage 3.1.1.2.2.1 Cementitious Materials Authored by Rolf Breitenb¨ ucher and Hursit Ibuk When exposed to cyclic loadings concrete may fail after a number of load cycles at a maximum stress level, which is lower than the static compressive strength. However, already during cyclic loading, before the ﬁnal fatigue failure occurs, damaging processes in the concrete microstructure results in a degradation. First investigations concerning fatigue problems were carried out in the 1830’s by W¨ ohler [852]. He performed fatigue tests on railroad axles made of steel and created the concept of the ”fatigue endurance limit” leading to the widely accepted stresslife, SN or W¨ohler curves (Figure 3.9). At the beginning of the 20th century also the fatigue of cementitious materials was investigated [603]. Van Ornum characterised the mechanisms causing the ﬁnally fatigue fracture as gradually progressive. Up to now in most of the investigations dealing with fatigue of concrete the maximal bearable number of cycles Nf is determined for various stressstrength ratios (Figure 3.10 and 3.11). Representative for such investigations tests carried out by Holmen [383], Weigler [822], Kim [432] and Oh [595] can be mentioned, in which the fatigue state of normalweight concrete in terms of curves by W¨ ohler are described. However, before the ﬁnal failure occurs the stressstrain relation determined between the lower and the upper stresslevels varies for the diﬀerent number of cycles (Figure 3.12). With increasing number of cycles the stressstrain curve changes from a concave form towards the strain axis to a straight line and eventually to a convex form. The characteristic form of this relation is an indication for the current state of the degradation process. Transferred to practical application, fatigue failure will take place only if the upper stress level Smax exceeds a certain limit failure, the socalled endurance
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Fig. 3.10. Fatigue fracture of concrete specimens due to cyclic compression load
Stress level
1.0 smax/fc 0.9 0.8
Assimacopoulos 1959, Antrim 1959, Bennet 1967, Do 1991, Gaede 1962, Galloway 1979, Graf 1936, Gray 1961, Holmen 1979, Kessler 1958, Kim 1996, Oh 1991, Ople 1966, Weigler 1981, Williams 1943 (783 tests)
0.7 0.6 2 4 6 Number of cycles to failure
8 log Nf
Fig. 3.11. Number of cycles to failure Nf for diﬀerent load levels and their variation [627]
limit (or fatigue limit). If Smax is below this endurance limit the stressstrain curve will remain straight, and failure of concrete due to the cyclic loading will not occur within the lifetime. Up to now, the existence of such a value for the endurance limit of concrete has not been explicitly determined as it has been investigated e.g. for steel already. The range of the stress levels between the endurance limit and the short term strength is called ﬁnitelife fatigue strength [567].
3.1 Phenomena of Material Degradation on Various Scales
0.7
133
smax/fc
0.1 0
0.5
0 ,0 0 40 =6 F
0.3
N
N= N=1 50, 000 N= 200 ,00 0 N= 500 ,00 0
0.5
1.0 1.5 2.0 2.5 Total longitudinal strain
3.0 e [‰]
Fig. 3.12. Stressstrain relation of concrete measured after diﬀerent number of cycles
Uniaxial Cyclic Compression Loads: Strains due to Cyclic Loading Experiments to investigate the timedependent development of concrete due to cyclic compression loading were carried out by various scientists [70, 411, 383, 148]. It reveals that the development of the longitudinal strain versus the time or the number of cycles follows a typical Sshape, as it is illustrated in Figure 3.13 exemplarily. In general, strain develops with the cycle ratio (N/Nf ) in three domains: a rapid increase up to about 10 per cent of total life Nf , a second uniform increase from 10 to about 80 per cent of Nf with a signiﬁcant lower slope and ﬁnally a rapid increase to failure. The total maximum strain max can be divided into two parts. The ﬁrst one is raised by a quasistatic loading at the ﬁrst cycle (initial strain) and the second part describes the real fatigue strain caused by the subsequent cycles. The increase in total longitudinal strains during cyclic loading is coupled with an initiation and/or growth of microcracks. Previous to its ﬁrst exposure to external mechanical loading, concrete can already exhibit a cracked microstructure especially in the interface zone between aggregates and cement paste (see also 3.1.1.1.1). These cracks result from a superposition of three decisive eﬀects: heterogeneity of concrete composition, the conditions during concrete placement and the permanent interactions of concrete structures with their environment. In concretes of normal strength microcracking starts primarily in the interface zone between cement paste and the coarse grains. Such cracks can have a width of about 1 μm and a crack length of about 50 to 420 μm [466]. Further cyclic loading lead to an extended opening and prolongation of these microcracks as well as to the initiation of new cracks and so ﬁnally to the formation of macrocracks and failure. In this context it has to be considered furthermore, that the total strains observed during cyclic loadings are the sum of two types. On one hand side the deformations are raising already by the cyclic loading itself, while on
3 Deterioration of Materials and Structures
Total strain e [‰]
134
C 30/37 Smax = 0.75 Smin = 0.05 frequency 5 Hz
3.0
emax
2.0 emin
1.0 1
0.0
0
2
3
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Related cycle ratio N/Nf
Fig. 3.13. Development of total longitudinal strain with the cycle ratio (N/Nf ) [383]
the other hand also deformations are developing due to creep eﬀects by the quasicontinuous applied averaged load. This phenomenon was illustrated in W¨ ohler tests performed by Holmen, where besides the numbers of cycles to failure Nf on 6 specimens also the longitudinal strains during the tests were determined. The variation of Nf ranges from 100,000 to 3.3 Mio cycles. It is evident, that the tests with Nf = 3.3 Mio cycles belong to a longer period than those which failed already after 100,000 cycles. Within this longer test period (about 8 days) in most cases also a higher total strain could be observed than in the tests lasting only a few hours [383, 628]. This phenomenon only can be explained by creep eﬀects. Uniaxial Cyclic Compression Loads: Eﬀects of Degradation on Stiﬀness A couple of test results are described in the literature concerning the eﬀect of degradation processes on the concrete’s stiﬀness due to fatigue loading. The degradation of the stiﬀness can be deduced from direct and indirect measurements. Holmen described the change of the stiﬀness by calculating the secant modulus whithin the measured maximal and minimal strains of one load cycle (Figure 3.14). Ec,s = (σmax − σmin )/(max − min )
(3.1)
Globally, these test results show a Sshape form versus the related cycle ratio (N/Nf ). In the investigations of Breitenb¨ ucher & Ibuk [148] where the degradation of the stiﬀness was investigated by nondestructive ultrasonic (US) measurements after a deﬁned number of load cycles, the similar trend could be proved (3.2.1.2).
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135
Residual strength f c/f c,28d
Fig. 3.14. Change of secant modulus of elasticity [383]
Related cycle ratio N/Nf Fig. 3.15. Development of the value of the residual strength [70]
Uniaxial Cyclic Compression Loads: Eﬀects of Degradation on Strength Only a small number of investigations concerning the changing in stressstrain relations and strength under cyclic compression load are available, as for such a large number of extensive testseries are necessary. The current results are still inconsistent to each other. Awad & Hilsdorf [70] observed at very high stress levels (Smax = 0.90) a moderate increase in strength within the about ﬁrst quarter of the lifetime. This is explained by compacting eﬀects in the microstructure due to cyclic loading. Afterwards a signiﬁcant loss in strength was established (Figure 3.15). On the other hand investigations by Shah & Chandra [733] revealed already up to 10,000 cycles a reduction in strength. However, these results do not allow extrapolations to higher cyclic numbers. Contrary to these results in current investigations [374, 148] no impact of cyclic compression loadings on the concrete strength has been determined.
3 Deterioration of Materials and Structures
Stress level
136
90 % confidence interval
Number of cycles to failure Fig. 3.16. W¨ ohler curves for tensile loads [207]
Uniaxial Cyclic Compression Loads: Multistage Cyclic Loadings While in most of the investigations discussed up to now only constant stress levels within one cyclic test were considered, constructions and structural elements in practice normally are exposed during their lifetime to quite varying cyclic stress levels and regimes, which can be pooled by means of damage accumulation hypothesis in multistage loading histories. Nevertheless, there are a couple of investigations dealing with the degradation due to such multistage loading histories. Weigler & Klausen [823] observed the fatigue behaviour due to such loadings by acoustic emission analysis and measurements of the volume changes within the cyclic loadings. In those multistage tests with load levels within the ﬁnitelife fatigue strength the validity of PalmgrenMiner’s hypothesis in this range was conﬁrmed. The PalmgrenMiner hypothesis assumes a linear increase of the damaging process with increasing number of cycles. Each several loading at a certain stress level within the total loadhistory until failure causes a part of the total damage. Nevertheless, the fatigue due to multistage loading on the basis of the damage accumulation hypothesis of PalmgrenMiner [543] diﬀers signiﬁcantly from the real fatigue loading scenarios. Cyclic Tensile and Flexural Loads To estimate the fatigue life of concrete specimens by tensile and ﬂexural loads a large number of W¨ ohler curves are avilable in the literature [207, 696, 865, 595] (Figure 3.16 and 3.17). Similar to cyclic compression tests also under cyclic tensile loading (Figure 3.18) as well as deﬂections in cyclic ﬂexural loadings (Figure 3.19) deformations develop in form of a Sshape, also characterized by the typical three faces [208, 662] (Figure 3.20 and 3.21). After applying a deﬁned number of load cycles destructive tests were carried out in order to study the changes in stressdeformation relations in comparison
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137
Stress level
min max
Number of cycles to failure Fig. 3.17. W¨ ohler curves for ﬂexural loads [865]
Strain
min max
Related cycle ratio Fig. 3.18. Development of strains in tensile loading [207]
to the unloaded state. By cyclic tensile as well as by cyclic ﬂexural loadings so induced microcracks lead to a progressive degradation in the concrete strength (Figure 3.20 and 3.21) and fracture energy with increasing number of cycles [430]. 3.1.1.2.2.2 Metallic Materials Authored by Otto T. Bruhns, Gerhard Hanswille and Henning Sch¨ utte Also in metallic materials loaded cyclically at nominal stresses below the static yield strength undergo progressive, localized, and permanent structural changes, i.e. also in these materials fatigue takes place. The structural change thus can be described on the macroscopic level as brittle, due to the fact that under these conditions there is only microplasticity. In general, this fatigue
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Measuring point 2
Strain
Measuring point 1
Measuring point 3
Number of cycles
Stress
Fig. 3.19. Development of strains in bending [662]
Deformation Fig. 3.20. Degradation process of relevant concrete properties due to tensile loadings [429]
process in metals consists of a microcrack initiation and afterwards microcrack propagation phase. The fatigue degradation may culminate into macrocracks and cause fracture after an adequate number of ﬂuctuations or loads cycles. The part of fatigue modeled here is the propagation of the microcracks. Altough most materials on a scale of a few dozen grains are still anisotropic, and microplasticty certainly plays a role in the propagation of such microcracks in most cases it is tried to build up a phenomenological continuum damage model based on the small scale yielding approach of linear elastic fracture mechanics. So the propagation of microcracking can be described with stress intensity factors of the cracks near tip ﬁeld embedded in an isotropic material with the properties of the macroscopic scale.
139
Flexural strength
3.1 Phenomena of Material Degradation on Various Scales
Related cycle ratio Fig. 3.21. Degradation process of relevant concrete properties due to ﬂexural loadings [866]
Fig. 3.22. Stiﬀness reduction by high cycle fatigue
Brittle Damage by Microcracks As in this context only damaging processes caused by microcracks, which are triggered by elastic stresses, are regarded no macroscopic plasticity has to be considered. Imagine such a member with growing microcracks undergoing a process, in which it is deformed by a total deformation, a certain part of this will be elastically recoverable, and another part can be induced by damage. When these loads are released, the member will have, in contrast to plasticity, not any remaining permanent deformation. Nevertheless, the state of the member could have changed; its elastic stiﬀness could have been reduced by the growth of microcracks. For a process which involves no further damaging, the total deformation is an elastic one, but starting from a state with
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Fig. 3.23. Model for brittle damage by microcrack growth
changed elastic properties. The underlying micromechanics for a continuum point and the corresponding macrostresses and strains are sketched in Figure 3.22. Starting with an unstressed member, containing a crack of length 2a and the resulting average stiﬀness E (the stiﬀness of the matrix remains unchanged by crack growth), up to a certain load the crack will not grow in length but only open its width. Beyond this threshold the crack length will increase and the average stiﬀness decreases. When the member is unloaded again the crack will close and no further growth occurs. For the same stress a greater strain will result, due to the reduced stiﬀness. In the stress free state there is  as already mentioned  no permanent deformation. Only the stiﬀness remains on a lower level than in the initial stage. The free energy W stored at the end of the process and the energy dissipated by crack growth Wcrack are also sketched in Figure 3.22. This means the process of stiﬀness degradation can be modelled by ﬁnding a correct representation for the energy dissipated by crack growth. This will be the basis for the continuum damage model for highcycle fatigue of metals presented in Section 3.3.1.2.2.2. 3.1.2 Nonmechanical Loading Authored by Otto T. Bruhns and G¨ unther Meschke 3.1.2.1 Thermal Loading Authored by Rolf Breitenb¨ ucher and Hursit Ibuk 3.1.2.1.1 Degradation of Concrete Due to Thermal Incompatibility of Its Components If the thermal behaviour and the thermal properties of the various concrete constituents are quite diﬀerent from each another, in cases of signiﬁcant
3.1 Phenomena of Material Degradation on Various Scales
141
temperature changes, incompatibilities in the deformations of the diﬀerent materials cause internal stresses between the aggregates and the cement paste, wich further can result in internal cracking. For this purpose the coeﬃcients of thermal expansion (αT value) of concrete constituents can become important. However, under normal conditions, in practice, diﬀerences in the thermal expansion coeﬃcient are not necessarily deleterious when the temperature does not exceed the temperature range of about 4 to 60 ◦ C. However, if the two relevant αT values (aggregates, cement paste) diﬀer seriously (much more than 5.5 · 10−6 K −1 ) from each another the durability of concrete concerning freezing and thawing may be aﬀected [567]. 3.1.2.1.2 Stresses Due to Thermal Loading Much more important for microcracking and degradation processes in concrete structures are restraint stresses, caused by restraining of thermal deformations (T = αT · Δt). Such restraint can be external as well as internal. In most cases temperature proﬁles over a crosssection are not constant or linear, but more or less stochastic and nonlinear (Figure 3.24). Thus, the resulting stresses can be divided into longitudinal, warping and internal stresses. For longitudinal stresses with constant magnitude an external restraint and constant temperature changes over the crosssection of concrete are responsible. Such an external restraint is caused in practice e.g. by bond to a stiﬀ foundation or an already hardened concrete member. A linear distributed temperature gradient results in warping stresses, since the bending deformations usually are restrained already by the deadload or also by external restraint. Internal stresses are formed by the restraint of nonlinear thermal deformations. In this case the restraint is internal, as the crosssection cannot deform unevenly (Bernoullihypothesis). In context with restraint stresses due to restrained thermal deformations especially in thicker concrete members often already the loadcase ”heat of hydration” becomes relevant.
Fig. 3.24. Stresses in a concrete slab at onesided, nonlinear cooling from the top [145]
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3.1.2.1.3 Temperature and Stress Development in Concrete at the Early Age Due to Heat of Hydration When concrete has been placed, initially the temperature remains unchanged, because the hydration process is still in its rest period (stage I) (Figure 3.25). A few hours after starting of hydration also a moderate temperature raise can be observed, however (also in case of restraint) without developing signiﬁcant compressive stresses. At this stage II the concrete has not yet set and is therefore still plastically deformable. Along with further hydration the stiﬀness of the concrete increases and may lead – if the deformations are restrained – to compressive stresses (stage III). The concrete temperature at the beginning of this third stage is called the ﬁrst zerostress temperature (1. Tz ). However, also in this stage the relaxation of the young concrete is still high, so that in spite of a signiﬁcant temperature rise only small compressive stresses are raised. In the consequence of this high relaxation at the end of stage III the maximum of the compressive stresses is obtained in general some time before the temperature maximum. After exceeding the temperature maximum the remaining compressive stresses decrease rapidly (stage IV). Only a few degrees below the temperature maximum the second zerostress temperature (2. Tz ) is obtained. Already starting from this point tensile stresses are caused during
Temperature [°C]
2. TZ Tconcrete 1. TZ
Tcrack Time [t]
Tair = Tfresh concrete Stage
I
II
III
IV
V
Longitudinal stress
s
Time [t]
+s
Fig. 3.25. Temperature and stress development during the ﬁrst hydration phase in restrained concrete elements [763, 145, 466]
3.1 Phenomena of Material Degradation on Various Scales
143
further cooling (stage V). When the not yet signiﬁcantly developed tensile strength is exceeded in this cooling period, at an age of only a few days, ﬁrst cracks will be formed at the socalled crackingtemperature (Tcrack ). [145]. Especially in mass concrete structures the internal restraint and thus the resulting internal stresses can become a dominant cause for thermal cracking. If the heat of hydration is not controlled and large temperature diﬀerences between the inner core and the surface are raised, internal stresses with tension at the surface develop in the concrete member. Thus, a surface mapcracking in the surfacezone can occur, whereby the crackwidth usually is very small. It is evident, that the described cracking also at such thermal loadings doesn’t develop suddenly. Furthermore it has to be considered, that also in such cases a complex microcracking is preceding the macrocracking formation. Thus also by this way degradation processes can take place, even if the tensile strength is not exceeded, i.e. when the ambient temperature is achieved before macrocracks could be formed. In this case the concrete structure remains on a high tensile stress level and micro cracks (with resulting degradation) develop. 3.1.2.2 ThermoHygral Loading Authored by Max J. Setzer and Rolf Breitenb¨ ucher 3.1.2.2.1 Hygral Behaviour of Hardened Cement Paste Authored by Max J. Setzer and Christian Duckheim Due to its nano and microporous structure hardened cement paste interacts strongly with its environmental humidity. This gain or loss of water has a deep impact on durability and material properties below 0◦ C (e.g. frost) as well as above 0◦ C (e.g. creep and shrinkage) [633]. Even if further research is required, freezethawresistance of concrete structures and the corresponding mechanisms have been investigated extensively in the last years and can be explained well today. In contrast, despite numerous diﬀerent analyses creep and shrinkagemechanisms are only fragmentarily understood up to now. Amongst others, this fact can be attributed to the manifold parameters which inﬂuence experimental results (such as sample composition and shape or the measuring setup and procedure) but most of all to the complex colloidal structure formed by nanosized CSHparticles, where only complicated ascertainable surface interactions play a decisive role. Drying shrinkage and swelling as a basic hygric property of hardened cement paste (w/c = 0, 35; 0, 40; 0, 50 and 0, 60) has been investigated over the complete humidity range by means of a newly developed laser supported measuring principle. This new technique allows the speedy, precise measurement of the pure material characteristic of several ﬁligree samples with an accuracy of about 20 nm. Further mainly novel methods have been applied for examining sorption behaviour as well as inner volume and density change. Measurement data have been analysed with
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0
10
20
30
relative humidity (%) 40 50 60
70
80
90
100
0 1 2 3 4 5 6 7 8 9
w/c = 0,35 (1st Des.)
w/c = 0,35 (1st Ads.)
w/c = 0,40 (1st Des.)
w/c = 0,40 (1st Ads.)
w/c = 0,50 (1st Des.)
w/c = 0,50 (1st Ads.)
w/c = 0,60 (1st Des.)
w/c = 0,60 (1st Ads.)
10
Fig. 3.26. Hygric strains vs. relative humidity
0
2
4
6
water content (%) 8 10 12
14
16
18
20
0 1 2 3 4 5 6
w/c = 0,40 (1st Des.)
7
w/c = 0,40 (1st Ads.)
8
w/c = 0,40 (2nd Des.)
9
w/c = 0,40 (2nd Ads.)
10
Fig. 3.27. Hygric strains vs. relative humidity & vs. water content
respect of the prevailing mechanisms on the nano and microscale as surface energy, disjoining pressure and capillary tension. In Figure 3.26 hygric strains of four samples with diﬀerent w/cratios during ﬁrst de and adsorption are illustrated. Figure 3.27 shows the relation between the measured deformations (w/c = 0, 40) and water content of
3.1 Phenomena of Material Degradation on Various Scales
145
surface free energy change (J/g) 30
25
20
15
10
5
0 3
4
5
6
w/c = 0,35 (1st Des.)
w/c = 0,35 (1st Ads.)
w/c = 0,40 (1st Des.)
w/c = 0,40 (1st Ads.)
w/c = 0,50 (1st Des.)
w/c = 0,50 (1st Ads.)
w/c = 0,60 (1st Des.)
w/c = 0,60 (1st Ads.)
7
8
9
Fig. 3.28. Hygric strains vs. surface free energy change. For further details (calculation of surface energy and deformations due to capillary tension) see [239]
the structure including a second desorptionadsorptioncycle. The total deformation, which grows with increasing w/cratio, lies in between 7 mm/m and 9 mm/m. Examining the results, in the range from 0 % r. h. to 100 % r. h. diﬀerent sections (desorption: 100% → 35% → 25% → 0%; adsorption: 0% → 60% → 100%) with each varying dominating mechanisms can be found. A close connection between water content of the structure and studied properties is demonstrated with only a marginal hysteresis between drying and wetting as well as the inﬂuence of capillary condensation. It could be proved that in the lower humidity range shrinkage and swelling are indeed proportional to changes in the surface free energy indeed (Figures 3.28 to 3.31). However, an energy reduction during adsorption does not lead to an expansion as assumed up to now (Munich Model), but rather to a contraction of cshparticles (Figure 3.31), while the pore volume increases simultaneously and vice versa during desorption. Solid density which is nearly independent from w/c varies between about 2.3 g/cm3 (dry) and 2.5 g/cm3 (wet). For this reason the inﬂuence of surface energy has to be attributed to the dispersive component of disjoining pressure which prevails in the lower humidity range, whereas in the range of condensation repulsiv components (electrostatic and structural component) and capillary tension dominate the processes in hardened cement paste. Consequently here a distinct linear relationship exists between hygric strains and water content (Figure 3.27). Irreversible strains have to be attributed merely to ﬁrst drying.
146
3 Deterioration of Materials and Structures relative humidity (%)
0
10
20
30
40
50
60
70
80
90
100 0
1
2
w/c w/c w/c w/c w/c w/c w/c w/c
= = = = = = = =
0,35 0,35 0,40 0,40 0,50 0,50 0,60 0,60
(calculated)
3
(calculated) (calculated)
4
(calculated) 5
Fig. 3.29. Hygric strains vs. surface free energy change & comparison between measured hygric strains and hygric strains calculated by capillary tension. For further details (calculation of surface energy and deformations due to capillary tension) see [239]
30
25
w/c = 0,35 (1st Des.)
w/c = 0,35 (1st Ads.)
w/c = 0,40 (1st Des.)
w/c = 0,35 (1st Ads.)
w/c = 0,50 (1st Des.)
w/c = 0,50 (1st Ads.)
w/c = 0,60 (1st Des.)
w/c = 0,60 (1st Ads.)
20
15
10
5
0 0
10
20
30
40 50 60 relative humidity (%)
Fig. 3.30. Sorption isotherms vs. relative humidity
70
80
90
100
3.1 Phenomena of Material Degradation on Various Scales
147
2,6
2,5
2,4
w/c w/c w/c w/c w/c w/c w/c w/c
2,3
2,2 0
10
20
30
40 50 60 relative humidity (%)
= = = = = = = =
70
0,35 0,35 0,40 0,40 0,50 0,50 0,60 0,60
(1st (1st (1st (1st (1st (1st (1st (1st
80
Des.) Ads.) Des.) Ads.) Des.) Ads.) Des.) Ads.) 90
100
Fig. 3.31. Solid density vs. relative humidity
The presented ﬁndings and additional results are merged in a schematic diagram (Figure 3.32) which describes the change of various hygric properties qualitatively and illustrates the eﬀects of the two diﬀerent mechanisms (disjoining pressure and capillary tension) on the system of hardened cement paste during ﬁrst desorption and adsorption. Elaborate explanations and further details can be found in [239]. 3.1.2.2.2 Inﬂuence of Cracks on the Moisture Transport Authored by G¨ unther Meschke Cracks, irrespective of their origin, have a considerable inﬂuence on the moisture permeability of cementitious materials. As a consequence, the transport of aggressive substances is promoted and the degradation process is further accelerated. The signiﬁcant inﬂuence of fracture on the transport properties of porous materials was ﬁrst recognized in the context of the coupled mechanical and hydraulic behavior of fractured rock masses. Experiments by Zoback & Byerlee [874] indicate an increase of the permeability of granite caused by microcracking. Particularly for materials with very low moisture permeabilities, such as granite and shale, ﬂow through the connected pore space was found to be insigniﬁcant compared to ﬂow through fracture zones. The role of cracks on the transport properties of cementbased materials has been investigated in e.g. [92, 155, 309, 41, 310], see Breysse and G´erard [153] for a stateoftheart survey. It has been shown, that the problems of moisture transfer change the scale, in fact that the permeability is increased by several orders of magnitude, when cracking is considered.
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Fig. 3.32. Schematic diagram of hygric mechanisms and properties of hardened cement paste
3.1.2.2.3 Freeze Thaw Authored by Max J. Setzer and Ivanka Bevanda Deterioration under freeze thaw attack has been discussed in literature under various aspects. The most recent development deals with the aspects of fracture mechanics. But for understanding the damage mechanism of frost and frost deicing salt attack, it is important to understand the following: (1) freeze thaw cycles (with and without deicing salt) acted as a micro pump and (2) a distinction between the forgoing transport process and the following damage process and ﬁnal degradation is essential. The unusual freezing behaviour of the pore solution in cement paste i.e. the special pore system is responsible that water and solution will be sucked up during freeze thaw cycles [723],[725]. This phenomena: (1) frost suction and (2) following deterioration after reaching the critical degree of saturation explained by Setzers modul of the microicelens [725],[727],[728],[730]. This model, based on the fact that within the concrete matrix water, vapour and ice can coexist in a wide temperature range, as concrete deviates from macroscopic behaviour due to its nanostructure. This leads to shrinkage of the gel during
3.1 Phenomena of Material Degradation on Various Scales approx. 150 nm macro gel
I: Cooling approx. 150 mm External heat
x
149
Vapor
Compression zone Triple phase gelwater
Freezing zone only water  vapor 0
0
T
bulkice
matrix
'p
II: Heating
x External heat
External water
external water
Melting zone
Expansion zone 0
T
0
expansion / contraction liquid water flow
'p pressure due to triple phase condition internal vapor transport
Fig. 3.33. Comparison of macroscopic and microscopic situation of the microicelens model during the heating and cooling phase [731]. Because of a pressure diﬀerence between the unfrozen gel water and ice in larger pores, water transport occurs, when freezing starts. Water from the micro pores is transported to existing ice in capillary pores. Simultaneously shrinkage of smaller pores can be registered. The water, i.e. ice, content of the macro pores increases. During thawing the gel tries to expand be again sucking water, which is available not from the still frozen ice in larger pores but from external sources. Independently with the 9% expansion of volume of ice, frost damage occurs in completely ﬁlled pores
freezing and to transport process within the pore system during melting (see Figure 3.33). Powers [644] and Fagerlund [270] particularly describe models of retarded ice formation which leads to hydraulic pressure. Powers and Helmunt were the ﬁrst who discussed the problems arising from transport phenomena, osmotic and hydraulic pressure [644],[645],[366]. Fagerlund [269] reﬁned this and stated the distinction between the critical degree and when damage started. Some models discussed the submicroscopic stress which is generated by surface interaction and by curved surfaces. Everentt and Haynes [266] described this phenomena as an ice crystal which is successively penetrating a micro pore. Scherer [699] uses similar semimacroscopic explanation for the damage process. Litvan [503] explained the transport from unfrozen pore water to ice due to diﬀusion. The impact of surface forces on frost damage was taken into
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3 Deterioration of Materials and Structures
account by the thermodymamic model developed by Setzer [723]. One group of damage models was explained by macroscopically generated stress. Podvalnyi [640] discussed the inﬂuence of thermal expansion of the matrix and the ¨mel and Springenschmidt [131] the stress due aggregate, Snyder [759], Blu ¨ sli and Harnik [689] the to temperature and salt concentration gradients, Ro temperature and stress gradient due to sudden melting of ice by deicing agents. Besides this, physical models a chemical model for the damage mechanism under frost deicing salt attack is described in [509],[508]. Ludwig has shown a preferred formation of ettringite under low temperatures. It should be noted that it was found in the SFB 398/ TP A11 that small amounts of dissolved ions increase the surface scaling dramatically (see Subsection 2.4.2, [120]). This can neither be explained by macroscopic or semimacroscopic physics  concentrations are much too small  nor by pure chemical eﬀects  the phenomena reach a maximum between 0.2 mol/l and 0.5 mol/l. Similar to chromatographic eﬀects during transport both must probably be attributed to superimposed eﬀects of surface physics. 3.1.2.3 Chemical Loading Authored by Rolf Breitenb¨ ucher, Hursit Ibuk and G¨ unther Meschke 3.1.2.3.1 Microstructure of Cementitious Materials Concrete is a nanoporous multicomponent system composed of aggregates and cement matrix. The cement matrix is consisting of a heterogeneous system of nonhydrated cement, hydration products, pores and pore solutions. It should be noted, that during the hydration process of cement both large calcium hydroxide crystals and CSHgel are generated simultaneously resulting in a pore structure characterised by a pore size distribution ranging from capillary pores to gel pores (Figure 3.34). Diﬀerent mechanical, physical and chemical processes which may considerably aﬀect the durablity of the material are caused and controlled by the pore size distribution, the ﬂuid saturation, the mechanically and chemically induced changes in the porosity and the chemical composition of the pore ﬂuid and the matrix. Durability of concrete is highly aﬀected by the transport of moisture and ionic (corrosive) species eventually leading to damage processes caused by chemically expansive reactions as well as by dissolution of load bearing constituents. The pore size distribution of hardened cement paste covers a large spectrum of pores extending over 7 orders of magnitude, see Figure 3.34. The smallest pores are smaller than one nanometre and the sizes of voids due to noncompleted compaction might exceed some millimetres. Furthermore, at least two diﬀerent kinds of pores have to be distinguished: The gel pores resulting from the cement hydration within these CSHproducts and the larger capillary pores in the cementpaste between the original cement particles. The diﬀerent pores also may
3.1 Phenomena of Material Degradation on Various Scales
151
Fig. 3.34. Volume fractions of constituents of hardened cement paste as a function of the water cement ratio [448]
be distinguished according to their behaviour and/or to the necessary time to obtain a speciﬁc capillary pore system [724]. This yields a classiﬁcation into coarse pores, capillary pores, meso and microgel pores, see Table 3.1. Table 3.1. Classiﬁcation of pore sizes in concrete according to [724] Type of pore Coarse Macrocapillary
Hydr. Characteristics radius ≥ 1 mm empty < 1 mm sucking, immediately reﬁllable
Meso< 30μm sucking within minutes, reﬁllcapillary able within weeks Micro < 1μm capillary Mesogel
Microgel
no stationary state
< 30 nm Transition from macroscopic behaviour to surface physics; ﬁlled by condensation at 50 % to 98 % rel. humidity < 1 nm surface physics, ﬁlled by sorption below < 50 % relative humidity
Type of pore water
free macroscopic water, freezable, highly mobile, small capillary rise. free macroscopic water, freezable, considerable capillary rise within a few days macroscopic water, freezable, strong capillary attraction, but increasing internal friction prestructured, condensed water, evaporation below 50 % relative humidity, not freezable beyond −23◦ C structured surface water, strongly disturbed, not freezable
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Table 3.2. Inﬂuences on the degree of chemical attack Acid attack increases with  increase in acid concentration and decrease in pHvalue  constant and fast renewal of acidic solution at the concrete/liquid interface  higher temperatures  higher pressure
Environmentally induced deterioration of cementitious materials and consequently the lifetime of concrete structures are to a large extent controlled by transport processes within the pore system. In particular, the accumulation of environmentally induced deterioration processes such as dissolution processes (e.g. calcium leaching), chemical expansive reactions (e.g. sulfate attack) or the transport of chlorides, which interacts with damage caused by time variant external loading, may limit the durability of concrete and reinforced concrete structures. Main concrete constituents subjected to aggressive substances also may be chemically damaged by calcium leaching which is controlled by the dissolution and decalciﬁcation of diﬀerent cement phases and the diﬀusion of dissolved species through the pore system. The physical and chemical processes strongly interact with mechanical deformations and degradations of concrete structures, such as microcracks and hence the increase of the pore spaces due to the additional mechanical load. Considerable progress was achieved in materialoriented research on environmentally induced degradation mechanisms, which led to a better understanding of the microstructural mechanisms and to analysis tools to simulate the relevant processes. In Subsections 3.1.2.3.2, 3.1.2.3.3, 3.1.2.2.2 the main experimental ﬁndings associated with longterm degradation processes caused by environmental loading and their interactions with external loading are summarized. 3.1.2.3.2 Dissolution When concrete structures remain in continuous contact with acidic ﬂuids, exchangeable salt solutions or softened water with a low content of alkaline earth ions (e.g. Ca++) chemical dissolution processes lead to a continuous deterioration of the material. The degree of such dissolutions depends on chemical conditions of the ﬂuid as well as on environmental conditions (Table 3.2). It should be noted, that the chemical dissolution strongly can interact with mechanically induced micro and macrocracks caused, e.g. by external loading (see Figure 3.35). This may considerably aﬀect the long term serviceablilty and the integrity of concrete structures. Cooling towers, containments for nuclear or other waste disposal, cementbound coatings of drinking water reservoirs, grouted anchors and tunnel linings are examples for structures and structural
3.1 Phenomena of Material Degradation on Various Scales
(a)
(b)
(c)
(d)
153
Fig. 3.35. Schematic illustration of the dissolution and loading induced longterm deterioration of concrete: (a) Reduction of Ca2+ concentration along the outside surface of the structure, (b) Outward diﬀusion of Ca2+ ions within the pore ﬂuid, (c) Dissolution of the components of the skeleton (portlandite, ettringite, CSH), (d) Increase of pore space results in decreasing stiﬀness and strength and the increase of the permeability
components, respectively, potentially exposed to aggressive environments connected with dissolution processes. The dissolution process can be interpreted as microdiﬀusion, where the size of the pores are increased by the dissolution of the surrounding cement substances. The dissolved reaction products of the dissolution are, driven by resulting the concentration gradient, diﬀusing outwards (Figure 3.35). This dissolution process strongly changes the microstructure and the chemical composition of the cementitious skeleton. As a consequence, the macroscopic mechanical and transport properties of concrete also change. As soon as the calcium hydroxide in the pores is more or less dissolved what in practice can belong to some decades  the reduced Ca2+ concentration in the pore ﬂuid starts to extract the calcium bound within the CSHphase. In the pioneering experiments by Berner [114, 115] on the dissociation of cement paste the strong inﬂuence of chemistry of the attacking water as well as of the cement paste on the longterm behaviour of cementitious materials have been investigated. The experimental data give evidence for the dependence of the decalciﬁcation of cement constituents on the calcium ion concentration of the pore ﬂuid. States of chemical equilibrium between solid and ionic solute species obtained from these experiments are illustrated in Figure 3.36. In particular, two dissolution fronts, representing the dissolution of portlandite and the CSH phases, can be distinguished in this ﬁgure. Based on Berner’s experimental data, G´ erard [307, 308] and Delagrave et al. [232] propose an empirical function s(c) relating the calcium
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3 Deterioration of Materials and Structures
e x p e r im e n ta l d a ta
1 5
É R A R D
1 0
5
d is s o lu tio n o f p o r tla n d ite
d is s o lu tio n o f C S H p h a s e s
C a lc iu m
c o n c e n tr a tio n s [m o l/m
3
]
m o d e l b y G
d is s o lu tio n 0 0
5
1 5
1 0 C [m o l/m 3
2 0
2 5
]
Fig. 3.36. Equilibrium states between the calcium concentration s in the cementitious skeleton and the ratio c/s of the calcium concentration c in the pore solution and in the skeleton: Experimental data by [114, 115] and analytical description by [307, 308]
concentration of the pore ﬂuid c to the calcium concentration of the cementitious skeleton s for instantaneous dissolution processes, i.e. assuming chemical equilibrium between the solid and ﬂuid phase (see Figure 3.36). In [232] it was observed, that the calcium content of a cement sample is primarily reduced in two steps, which can be identiﬁed as the dissolution fronts of portlandite and CSHphases. Since the calcium leaching kinetics in water is very slow, the majority of experiments on calcium leaching of cement samples are performed by means of accelerated test methods using strongly concentrated ammonium nitrate solutions instead of deionized water (Carde et al. [170, 173, 174], Carde & Franc ¸ ois [171], Heukamp et al. [367] and Ulm et al. [799]). The equivalence of the leaching process in samples exposed to deionized water and to ammonium nitrate solution is shown by Carde et al. [170] by means of chemical analyses of standard and accelerated leached cement samples. Only the dissolution of ettringite is not captured by this accelerated test method. According to [170], however, this mineral has only a marginal inﬂuence on the mechanical properties. To the best knowledge of the authors, the only real life time experiment ¨g˚ documented in the open literature has been performed by Tra ardh & Lagerblad [794]. They investigated a concrete sample subjected to deionized water from a water reservoir for 90 years. In accordance with [170, 171, 173, 174],
3.1 Phenomena of Material Degradation on Various Scales
155
In c r e a s e in p r o s ity D f
0 .2 5
0 .2
P u re c e m e n t p a s te e x p e r im e n ta l r e s u lts P a s te w ith s ilic a fu m e e x p e r im e n ta l r e s u lts
0 .1 5 0 .1 0 .0 5 0 0
0 .1
0 .2
0 .3
0 .4
0 .5
0 .6
0 .7
0 .8
L o s s o f c o m p r e s s iv e s tr e n g th D fc u /fc u
Fig. 3.37. Decrease of compressive strength as a function of the increase in porosity resulting from calcium leaching [172]
they observed an increased porosity resulting from the dissolution of cement phases within a degradation zone of approximately 9 mm thickness separated from the sound material by the dissolution front of portlandite crystals. The inﬂuence of calcium leaching on the mechanical properties and the porosity of cement paste is investigated on micro cylinder cement paste samples subjected after accelerated leaching to uniaxial compression and water porosity tests [170, 171, 173, 174], respectively. The total leaching of portlandite and the progressive decalciﬁcation of CSHphases leads to a linear dependence of the porosity and the strength on the ratio between the degraded and the sound cross sections. Compared to the virgin material the ductility of the chemically degraded material is larger because the micro structure of the material is modiﬁed by leaching. Furthermore, stressstraindiagrams given by [171, 174] illustrate, that the stiﬀness of the material is signiﬁcantly reduced due to calcium leaching. From triaxial tests of cylindrical cement samples subjected to accelerated leaching [367, 799] a strong dependence of the mechanical properties on the pore pressure due to the increased pore space and the reduction of the materials frictional performance of the leached cement paste is found. Calcium dissolution increases the porosity of the cementitious material, and, consequently, leads to a decrease in stiﬀness, strength and fracture energy of the material. Depending on the immersion time of calcium leaching, also a decrease of the internal length lc related to the fracture process (see Section 3.1.1.1) was observed, whereas the brittleness of the failure is increased [174, 171, 479, 478]. Experimental investigations by [171, 172] show that the strength reduction is almost linearly related to the increase of porosity (Figure 3.37).
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From extensive experimental investigations using accelerated test methods [170, 173, 174, 171, 367, 799], it was observed, that the amount of calcium dissolved with time depends almost linearly upon the squareroot of time. This implies that leaching of the cement paste is governed by a diﬀusiondissolution process with almost instantaneous dissolution kinetics. In addition to the reduction of the stiﬀness and the strength, the conductivity of cementitious materials is considerably aﬀected by calcium leaching (Figure 3.35), which in turn has an inﬂuence on the dissolution process. According to experiments performed by [38, 232], the conductivity depends nonlinearly on the porosity and on the calcium concentration of the pore solution. The conductivity increases with progressive chemical degradation. On the other hand, loadinginduced micro and macrocracks also increase the conductivity and, consequently, accelerate the chemical degradation of concrete [478]. An interesting aspect concerning a comparison between experimentally and numerically obtained propagation of the portlandite dissolution front is addressed in [232]. According to this work, the numerical analysis is only able to ﬁt the experiments, if only 50% of the conductivity of the pore ﬂuid with respect to calcium ions in pure water is taken. This indicates, that the calcium ion conductivity of the pore ﬂuid increases as the calcium concentration c decreases with propagating chemical dissolution. The inﬂuence of chemical degradation on the structural behaviour of concrete specimens have been investigated by Le Bell´ ego et al. [477, 479, 478]. In these experiments mortar beams have been exposed to ammonium nitrate solution on the front and back face to introduce unidirectional leaching fronts moving through the thickness of the beam. At diﬀerent stages of the chemical attack the beam was subjected to a vertical load up to ultimate failure. Depending on the immersion time of calcium leaching the decrease in stiﬀness, strength, fracture energy and, consequently, internal length has been recorded. In Le Bell´ ego et al. [478] an alternative threepoint bending test has been proposed to evaluate the chemomechanically coupled degradation. In these test, simultaneously to the mechanical loading a part of the bottom surface of the beam was exposed to an aggressive ammonium nitrate solution. After increasing the displacement controlled loading up to a certain level, the displacement was held constant while the dissolution processes were progressing. In this phase of the test the decreasing reaction force due to calcium leaching has been recorded. Subsequently, the mechanical loading has been increased up to ultimate failure. These test results have been compared with respective results from a mortar beam subjected in the ﬁrst phase of the experiment only to chemical attack. The resulting loaddisplacementdiagrams clearly demonstrate, that the reduction in stiﬀness and strength is signiﬁcantly larger in the coupled chemomechanical experiments. This indicates, that micro and macrocracking increases the conductivity and, consequently, accelerates the chemical degradation.
3.1 Phenomena of Material Degradation on Various Scales
157
3.1.2.3.3 Expansion 3.1.2.3.3.1 Sulphate Attack on Concrete and Mortar Ettringite Formation by Sulphate Attack A frequent cause of expansive attacks on concrete results from the penetration of sulphate ions into the microstructure and their reaction with certain concrete constituents to the formation of expansive minerals such as gypsum or secondary ettringite. Thus concrete members, which are exposed to external sulfatesources, e.g. groundwater or soil containing dissolved calcium, sodiumand/or magnesiumsulfates often are subjected to such an attack. However, also other speciﬁc sources like in sewers, cooling towers of coalpowerplants etc. can become relevant. All of the mentioned sulfates primarily will react with portlandite (Ca(OH)2 ) to gypsum (CaSO4 · 2 H2 O) or with tricalciumaluminates C3 A, hydrated aluminates or monosulfate (C3 A · CaSO4 · 12 H2 O) to ettringite (C3 A · 3 CaSO4 · 32 H2 O). Both the formation of gypsum and ettringite is combined with a large volumeincrease (expansive reaction), that may result at the beginning in microcracking, i.e. ﬁrst degradation process, and ﬁnally in extensive cracking and spalling of concrete. Comparing the diﬀerent cations with respect to their inﬂuence on the severeness of attack magnesium sulphates are signiﬁcantly more aggressive towards concrete than e. g. calcium or sodium sulfates. Magnesium sulphate does not only react with the aluminates, but also with the calciumsilicatehydrates (CSH) in the matrix by forming brucite and hydrous silica. This furthermore results in an additional loss in strength and a softening in the aﬀected areas in concrete [787]. The risk of concrete damaging processes by such ettringite formations mainly is increased in grounds since there normally also the optimal temperature conditions for such reactions in a range of about 5 to 10 ◦ C are present (Figure 3.38). Furthermore the degree of expansion is inﬂuenced by the concrete’s w/cratio. Thaumasite Formation by Sulfate Attack (TSA) Additionally or alternatively to the expansive ettringite formation a deleterious thaumasite formation can take place in case of sulphate attack under special conditions. In various regions, especially in UK, however, in the meantime also in some parts of Germany, damages in some concrete structures in contact with speciﬁc sulﬁdic soils have been observed [199, 792]. In most cases such soils contain minerals as pyrite (F eS2 ). Pyrite itself does not attack the concrete matrix. However, when exposed to oxygene and sufﬁcient moisture provided, which can be assumed within natural soils, pyrite oxidizes to sulphuric acid and iron sulphate. Both of these reaction products may exhibit a combined attack on adjacent concrete microstructures. If the concrete contains also carbonates, e.g. limestone ﬁller, thaumasite can be formed. Thaumasite is a calciumsilicatesulfatecarbonate hydrate (CaSiO3 · CaCO3 · CaSO4 · 15 H2 O) that is formed in the concrete by
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Expansion [mm/m]
5 4 3 w/c = 0.60; 8 °C w/c = 0.60; 20 °C w/c = 0.40; 8 °C w/c = 0.40; 20 °C
2 1 0 0
50
100
150
200
Storage [d] Fig. 3.38. Expansion behaviour of ﬂat mortar prisms (1x4x16 cm) with Portland cement (11 wt.% aluminate) during storage in sodium sulfate solution (29,800 mg sulfate/l), considering diﬀerent water/cement ratios (0.60; 0.40) as well as diﬀerent storage conditions (8 ◦ C; 20 ◦ C) [502]
reaction of the CSH with sulphates in the presence of carbonate ions. This reaction becomes very critical due to the fact, that the loadbearing CSHstructure is destroyed in the cement paste. Similar to the ettringite formation the optimal temperatures for a thaumasite formation are in a range between about 5 and 10 ◦ C. 3.1.2.3.3.2 AlkaliAggregate Reaction in Concrete In most applications aggregates used in concrete mixtures are more or less chemically inert. However, some aggregates can react with the alkalines (potassium, sodium) in the cement paste of a concrete, combined with a signiﬁcant expansion, which normally result in a deleterious cracking and ﬁnally  within a period of some years or even decades  to a complete destruction of the concrete structure respectively. Such alkaliaggregate reactions can follow two forms: on one hand the most known alkalisilica reaction (ASR) and on the other hand in special cases also the alkalicarbonate reaction (ACR). In both cases the service lifetime of the concrete structure is reduced signiﬁcantly. AlkaliSilica Reaction in Concrete (ASR) The alkalisilica reaction in concrete is a chemical reaction between amorphous forms of silica, present in some aggregates (e.g. chert, quartzite, opal, strained quartz crystals), and the alkalines within the pore solution. Two main mechanisms constitute the alkalisilica reaction. Firstly, silica is dissolved from the aggregates and forms a gel. Secondly, the combined swelling of the gel by imbibition of water, may cause an expansion and consequently a deterioration
3.1 Phenomena of Material Degradation on Various Scales
159
Fig. 3.39. Alkalisilica reaction damage
of the aﬀected concrete (Figure 3.39). Thus, the ability of moisture is a necessary condition for this deleterious reaction. Nevertheless, normal moisture as available in normal outdoor exposure is suﬃcient. Typical indicators of ASR are at the beginning random map cracking and in advanced states attendant spalling of concrete. Petrographic examination can conclusively identify ASR [146, 768]. The basic structural unit of all forms of silica is a silicon ion (Si4+ ) surrounded by four oxygen ions (O2− ) with the arrangement of a tetrahedron [403]. In crystalline, the rather low reactive form of silica, these tetrahedra are linked to produce a dense threedimensional network. In contrast, amorphous silica consists of tetrahedra that are joint in a random, more spacious network with a large speciﬁc surface. The latter results in a substantially enhanced reactivity towards alkaline solutions like the pore liquid in concrete [646], which contains relatively high concentrations of hydroxyl ions (OH − ). The charge of these hydroxyl ions is mainly balanced by alkali ions (N a+ , K + ), which are usually provided by sodium oxide (N aO2 ) and potassium oxide (K2 O) in the cement due to dissolution during the process of hydration. Considering the alkalisilica reaction as a multistage process, it starts with the dissolution of silica on the surface of the aggregates as a topochemical reaction, whereby silanol groups are formed. In a second step, these silanol groups react with further hydroxyl ions to negative charged ions, which attract positively charged sodium, potassium and calcium ions (N a+ , K + , Ca2+ ) present in the pore liquid. As more siloxane bonds are attacked by the dissolution process, a gellike layer forms at the surface of the aggregates. Some silica may even pass into solution as monomeric species (H4 SiO4 , H3 SiO4− , H2 SiO42− ) depending on the pH value of the pore liquid [234]. The reaction velocity mainly depends on the reactivity of silica, the alkalinity of the pore liquid, the temperature and the available amount of water.
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The gel formed by this process is hydrophilic, see e.g. Poole [642]. This means, that in a suﬃciently humid environment the gel imbibes water, which results in the swelling of the gel. The swelling of the gel exerts a pressure on the concrete skeleton and the material deforms. The swelling pressure is variable within a wide range depending on the moisture content and the type and proportions of the reacting materials. It can belong up to 20 MPa [146, 768]. Thus, it is often suﬃciently large to induce expansions in localized regions which, in turn, lead to the opening and propagation of cracks and to the disruption of the aﬀected concrete. This results in a drastic reduction of the mechanical properties and consequently to structural degradation [373]. By comparing the time scales of the swelling of synthetic gels (e.g. Struble & Diamond [779]) and of concrete specimens (e.g. Larive [469], it can be concluded, that the imbibition of water by a gel is much faster than the progress of deterioration in ASR aﬀected concrete. This leads to the conclusion, that the imbibition of water by the gel can be regarded as an instantaneous process in comparison to the formation of the gel by the dissolution of silica. Since in a multistage process the slowest process controls the overall kinetics, it is reasonable to assume, that the alkalisilica reaction (formation of gel + swelling of gel) is governed by the noninstantaneous kinetic law of the gel formation. AlkaliCarbonate Reaction in Concrete (ACR) Alkalicarbonate reactions (ACR) are observed in concretes with certain dolomitic rocks. Dedolomitization, the breaking down of dolomite, is normally associated with expansion. This reaction and subsequent crystallization of brucite (magnesium hydroxide M g(OH)2 ) may cause considerable expansion. The deterioration caused by ACR is similar to that caused by ASR; however, ACR is relatively rare because aggregates susceptible to this phenomenon are less common and are usually unsuitable for use in concrete for other reasons. Aggregates susceptible to ACR tend to have a characteristic texture that can be identiﬁed by petrographers. 3.1.3 Accumulation in Soils Due to Cyclic Loading: A Deterioration Phenomenon? Authored by Theodoros and Andrzej Niemunis
Triantafyllidis,
Torsten
Wichtmann
In an element of soil, depending on the boundary conditions, cyclic loading can lead to residual strains and/or changes in stress. Closed stress loops result in not perfectly closed strain loops or vice versa (Figure 3.40 a,b). Therefore, strictly speaking, the term ”cyclic” in the sense of ”mathematically periodic” is appropriate only to the strain rates (see the hodograph in Fig. 2.96). For the strain the term ”almost cyclic” may be more suitable. In the laboratory an accumulation of strain is observed in stresscontrolled drained cyclic triaxial tests. In straincontrolled cycles the average stress changes, which manifests itself as a relaxation. A special case is the displacementcontrolled undrained
3.1 Phenomena of Material Degradation on Various Scales
a)
b)
c)
ε2
ε2
prescribed
ε0
σ2
σ2
ε2
ε0
ε1
ε1
161
σ0
ε1 σ2 σ0
prescribed
σ1
σ1
σ1
Fig. 3.40. Accumulation of stress or strain, illustrated for the twodimensional case
cyclic triaxial test on fully watersaturated specimens (constant volume). A simultaneous accumulation of stress and strain is also possible (Figure 3.40c). It occurs in case of mixed boundary value problems. Contrarily to the deterioration or fatigue in metals or concrete the strength and stiﬀness of sand usually increase during a highcyclic loading under drained conditions. This is due to the compaction of the granular material which takes place if the average stress ratio η av = q av /pav (ratio of deviatoric and isotropic stress components) is not too large (i.e. not surpassing the critical state line known from monotonic tests) and the strain amplitudes are relatively small (i.e. the cyclic stress path does not touch the failure line). Usually these conditions are fulﬁlled for soils under highcyclic loading since the foundations are designed to keep the stress path (including the amplitudes) away from the failure condition. The eﬀects of compaction due to cyclic loading result in strengthening and stiﬀening of the soil. They are even used in soil improvement techniques and therefore it is somewhat misleading to use the term ”deterioration” or ”fatigue” for soil. However, excessive settlements of foundations under cyclic loadings may result from unsuitably designed foundations (e.g. inappropriate dimensions, missing soil improvement). Especially, in statically indeterminate structures large diﬀerential settlements may accelerate the deterioration processes in the structure (Section 2.5.1). In Section 3.2.2 the direction and the intensity of strain accumulation under drained conditions are discussed based on laboratory tests on granular material. The ”cyclic ﬂow rule” has been shown to be approximately equal to ﬂow rules for monotonic loading. Beside the inﬂuence of void ratio and average stress it is important that the intensity of accumulation increases with the number
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of dimensions penetrated by the strain path (see also Section 2.5.2). A twodimensional circular strain path e.g. produces twice larger accumulation rates than a onedimensional cyclic shearing with an amplitude which is equal to the radius of the circles. Similarly, changes of the polarization lead to temporary accelerations of accumulation. Finally, it can be shown that some more subtile fabric eﬀects resulting from the cyclic history play a signiﬁcant role for the further accumulation. All these eﬀects complicate the formulation of a constitutive model (Section 3.3.3). Fortunately, given several packages of cycles with diﬀerent amplitudes their sequence does not signiﬁcantly aﬀect the ﬁnal permanent strain. Therefore complicated strain paths consisting of several amplitudes and frequencies will be decomposed into packages of cycles of constant amplitude (Section 2.5.1) and calculated one by one in any sequence. In studies on the life time of structures calculations with large number of cycles Nc are necessary. The highcycle tests on a mediumcoarse sand reveal (Section 3.2.2) that the proportionality εacc ∼ ln Nc (which has been reported by several authors for small Nc values) does not hold above Nc = 104 . A part of the accumulation turns out to be proportional to the number of cycles, i.e. εacc ∼ Nc and this additional overlogarithmic part becomes dominant for large Nc values (Nc > 104 ). This eﬀect may be explained (at least partly) by abrasion and fragmentation of particle corners, i.e. by the nonpermanency of grains [426]. The linear dependence εacc ∼ Nc is wellknown from abrasion experiments on ballast material. Thus, the assumption of a permanent material usually made by theories for monotonic loading (plasticity, hypoplasticity) does not apply to cyclic loading, at least for very high numbers of cycles. However, in recent experiments on wellgraded granular material the accumulation has been observed to run signiﬁcantly faster than according to εacc ∼ ln(N ) already from the beginning of the tests [841]. This cannot be explained only by abrasion. The term ”deterioration” can be applied to stress relaxation observed in saturated soils. Under nearly undrained conditions (e.g. during an earthquake) the strength and the stiﬀness of the material decrease because the eﬀective stress does. The pore pressure buildup is equivalent to a reduction of the eﬀective stress (Figure 3.40b). In the extreme case σ av = 0 the stiﬀness and strength may completely vanish. The soil is then said to be ”liqueﬁed” and the soil skeleton is temporarily ”deteriorated” until the pore water is squeezed out and volume changes occur. The contact loss of the grains gives rise to phenomena like cyclic mobility (Section 3.2.2), phase separation between soil layers and spontaneous densiﬁcation during reconsolidation. Usually a few fast and strong cycles are necessary to reach a liquefaction and a collapse of a foundation. The number of cycles leading to liquefaction (Nc = 10  100) is much smaller than the Nc values usually considered in studies on the life time of structures. Therefore, this ”deterioration” phenomenon is irrelevant for lifetime oriented design concepts. The liquefaction phenomenon may also be utilized for passive isolation of structures or for soil improvement techniques (Section 2.5.1). Under drained conditions a ”deterioration” may take place if large amplitudes are applied, i.e. if the cyclic stress path signiﬁcantly surpasses the
3.2 Experiments
163
critical state lines, or if small cycles are applied with an average stress ratio above the critical one. In both cases dilatancy occurs leading to a reduction of the strength and the stiﬀness of the soil. However, such stress paths should be avoided by an appropriate design of foundations considering also the cyclic part of the loading. While the extensive experimental study presented in Section 3.2.2 has signiﬁcantly improved the understanding of the accumulation phenomenon during a drained cyclic loading, the postcyclic behaviour is not well understood yet. The undrained displacementcontrolled cyclic triaxial tests presented in Section 3.2.2 (Figure 3.92) reveal that a latent accumulation in the soil skeleton takes place even if the cycles are applied at σ av = 0 (i.e. after liquefaction). This latent accumulation becomes visible as a permanent deformation during reconsolidation. Similar observations have been reported also e.g. by Shamoto et al. [734] and Sento et al. [721]. Sento et al. have performed straincontrolled cyclic tests and could correlate the volume change during reconsolidation with the length of the shear strain path γ ˙ dt. Despite these recent ﬁndings, the postcyclic behaviour of soils needs further experimental studies.
3.2 Experiments Authored by Otto T. Bruhns and G¨ unther Meschke This Section contains results from laboratory investigations of damage evolution in materials and structures performed in the conetxt of the Joint Collaborative Center SFB 398 at Ruhr University Bochum. It comprises non destructive investigations of crack propagation in metallic materials (Subsection 3.2.1.1), laboratory tests of concrete and soil subjected to cyclic loading (Subsections 3.2.1.2 and (Subsections 3.2.2) and structural experiments of steelconcrete composite structures (Subsection 3.2.3). 3.2.1 Laboratory Testing of Structural Materials Authored by Otto T. Bruhns and G¨ unther Meschke 3.2.1.1 Micromacrocrack Detection in Metals Authored by Henning Sch¨ utte and Otto T. Bruhns 3.2.1.1.1 Electric Resistance Measurements 3.2.1.1.1.1 Introduction Crack growth due to fatigue and stress corrosion is normally a slow degradation process up to a point, beyond which failure may be sudden and catastrophic. For this reason early detection of crack growth during this initial
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period is essential for the prevention of failure, especially for metallic structures subjected to cyclic loading. In fracture mechanics, experiments involving stable crack growth and measurement of the instantaneous crack length are essential. That is a reason why many experimental methods have been developed to monitor such crack growth. One of the most powerful nondestructive methods for monitoring crack initiation and measurement of crack growth is the electrical resistance method, in which a constant direct current is passed through the specimen and electrical resistance between two measurement points located on opposite sides of the crack is monitored. As the crack propagates, the uncracked crosssectional area of the test specimen decreases, and electrical resistance between two points on either side of the crack rises. The advantage of the electrical resistance method lies in the simplicity of its system compared with other nondestructive techniques such as acoustic emission, vortex current, ultrasonic crack detection, etc. Also, the electrical resistance method has many advantages over the optical measurement of the crack propagation. It provides a total measurement, and, because it does not require visual accessibility, tests may be conducted in any sealed environment. The output is continuous which permits automated data collection and processing together with all day usage of testing machine capacity. The technique is capable to detect small increments of crack growth, which cannot be resolved optically. In addition to the majority of fatigue testing methods, which concentrate on intermediate and high stress intensities it is particularly useful in the study of fatigue thresholds and very slow crack propagation rates, which occur by high cycle fatigue tests. For a test specimen under cyclic fatigue loading the electrical resistance increases (Ra ), and comparing it with a reference electrical resistance (R0 ) measured on the test specimen before the beginning of the test, the crack length (a) or the crack lengthtospecimen width ratio (a/w) may be determined. This determination is possible by making use of suitable calibration curves. In practice, the accuracy of an electrical resistance measurement of the crack length may be limited by a number of factors: • • • •
The electrical stability and resolution of the electrical resistance measurement system. Electrical contact between crack surfaces where the fracture morphology is rough or where signiﬁcant crack closures are present. Changes in electrical resistance with plastic deformation. Changes in electrical resistance with temperature change.
This accuracy limitations are present in particular by the determination of calibration curves relating changes in electrical resistance across crack length (Ra ) to the crack length (a). In most cases, experimental calibration curves have been obtained by measuring the electrical resistance across:
3.2 Experiments
• • •
165
Machined slots of increasing length in a single test specimen. A growing fatigue crack, where the length of the crack at each point of measurement is marked on the fracture surface by a single overload cycle or by a change in the mean stress. A growing fatigue crack in thin specimens where the length of the crack is measured by surface observation.
3.2.1.1.1.2 Measurement of the Electrical Resistance The measurement of the electrical resistance was performed using an industry device Buster Resistomat 2304 with a maximal resolution of 10nΩ in the testing range up to 200μΩ and a constant direct current of 10A through the specimen. The specimen form and the distance between two points in which the electrical potential was measured were adjusted to reach this high resolution. The measurement of the electrical resistance was performed on two types of specimen: plain specimen and circular specimen, shown in the Figures 3.48 and 3.49 respectively. The measurement of the electrical resistance Rx in the observed area of the specimen was achieved using a Kelvinprocedure. In this procedure, a regulated constant direct electrical current of 10A is driven through the specimen and an internal reference resistance (Rk ) is measured. A diﬀerence in the electrical potential at any two points of the specimen can be measured, and the electrical resistance can be calculated. The potential drop over the testing range of the specimen can be related to the potential drop over the highprecision internal reference resistance and can be calculated using the known value of this reference resistance. The advantage of this procedure is the independence of the measured values of the parasitic resistance in the measuring system, such as contact resistance and transition resistance in the measurement connection. The measured values depend only on the quality of the internal reference resistance. To eliminate possible electrostatic eﬀects or diﬀerences of the two ampliﬁers, the direction of the driven current and the polarity of the potential measurement are chanced within one period of the resistance measurement. For this reason the measuring value is the middle value from 4 diﬀerent measurements. For all tests, the measuring time for one value was set to 1s. One value of the electrical resistance represents an average value of the electrical resistance over 9 loading cycles for the plain specimen and over 6 cycles for the circular specimen. This way, eﬀects of elastic elongation of the specimen during the measuring time for one value of the electrical resistance are eliminated. The measurement of the diﬀerence in electrical potential was obtained using bypassed thermocouples, which are welded onto the specimen. Welding the spherical thermocouples onto the specimen surface also prevents the friction noise in the acoustic emission measurement. The temperature of the specimen is measured using the same bypassed thermocouple, which is welded onto the specimen between two thermocouples
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3 Deterioration of Materials and Structures
for the measuring of the diﬀerence in the electrical potential. The measured temperature is used for the elimination of the temperature based eﬀects in the electrical resistance, which are a result of the change of the speciﬁc resistance (resistivity) and the crosssection. 3.2.1.1.1.3 Calculation of the Electrical Resistance The measured values of the electrical resistance are resolved in the postprocessing using the inﬂuence of the temperature measured at the same time. A variation of the electric resistance can be calculated as a function of the possible temperature change of the specimen, using the conditional temperature change of the speciﬁc resistance ρ(Θ) = ρ20 (1 + αp ΔΘ20 ),
(3.2)
the temperature dependence of the length of the specimen l(Θ) = l20 (1 + α ΔΘ20 ),
(3.3)
the temperature dependence of the cross sectional area of the specimen A(Θ) = A20 (1 + α ΔΘ20 )2
(3.4)
and relation R(Θ) = R20
1 + αp ΔΘ20 . 1 + α ΔΘ20
(3.5)
Here, R20 is the electrical resistance at 20◦ C, αp is the temperature coeﬃcient of speciﬁc electrical resistance, α is the linear thermal expansions coeﬃcient and ΔΘ20 is the temperature diﬀerence between the current temperature and the reference temperature of 20◦ C. 3.2.1.1.1.4 Experiments Experiments were performed by a hydrodynamic tensiontorsion testing system (Schenck/Instron Fast Track 8800). The displacement was measured by a real time analogbuilt mean of 3 displacement transducers (HBMW5TK), and the load was measured by a 160kN load cell. The fatigue load was deﬁned as cyclic sinusoidal load in tension range. Tests were performed either with constant amplitude for the entire duration of the test, or as a blocktest with the amplitude which is constant inside the block and diﬀers between the blocks. Stressratio range was R=0.05 and R=0.25. Test frequency was 9Hz for plain specimen and 6Hz for circular specimen. The electrical resistance is measured using an industry device Buster Resistomat 2304 with a constant direct current of 10A through the specimen, maximal resolution of 10nΩ and the testing range until 200μΩ.
3.2 Experiments
1.2
PS16 PS17 PS20 PS22 PS31
1.15 1.1
elect. Resistance R/Ro
elect. Resistance R/Ro
1.2
1.05 1 0.95
167
CS02 CS05
1.15 1.1 1.05 1 0.95
0
150000
300000 Cycles
450000
600000
0
50000
100000 150000 Cycles
200000
Fig. 3.41. Evolution of the electrical resistance vs. number of cycles during fatigue  plain and circular specimen
1.2
PS24
elect. Resistance R/Ro
elect. Resistance R/Ro
1.2 1.15 1.1 1.05 1 0.95
PS29
1.15 1.1 1.05 1 0.95
0
200000
400000 Cycles
600000
800000
0
300000
600000 900000 Cycles
1.2e+06
Fig. 3.42. Evolution of the electrical resistance during fatigue  plain specimen blocktest
3.2.1.1.1.5 Experimental Results The electrical resistance is normalised using average electrical resistance for the service life between 30% and 70%. The evolution of the normalised electrical resistance over the service life is shown in Figures 3.41 and 3.42. As can be seen in Figure 3.41, after a stabilisation during the ﬁrst couple of load cycles, the electrical resistance is constant over a long time of the service life. To the end of the service life, an exponential growth of the electrical resistance is to be observed. Depending on the level of the cyclic load amplitude, an exponential evolution of the electrical resistance happens between 60% 80% of the service life. Figure 3.42, in which results of the blocktest are represented shows that the normalised electrical resistance is not constant between the blocks. Although one value of the electrical resistance represents an average value of electrical resistance over 9 respectively 6 loading cycles, the inﬂuence of the diﬀerent load amplitude level between the blocks on the electrical resistance can be seen clearly. The dependency between the evolution of the electric resistance and the crack propagation is calculated using ﬁnite element analysis software ANSYS.
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3 Deterioration of Materials and Structures
Fig. 3.43. Electrical potential  plain specimen
Fig. 3.44. Electrical potential  circular specimen
The distribution of the electrical potential across the plain and circular specimen for diﬀerent crack length is given in Figures 3.43 and 3.44. The results of both the ﬁnite element analysis and experiments for both types of specimen are represented in Figure 3.45 as a dimensionless plot of electrical resistance ratio R/R0 against a/w where R0 is the electrical resistance at the initial crack length a0 , R is the electrical resistance at the crack length a and w is half width of the specimen at the crack height. A good agreement between experimental result and the numerical solution of evolution of the electrical resistance with the crack propagation can be observed. The obtained evolution of the electrical resistance shows at the same time a high level of similarity to the measured crack propagation behaviour under cyclic fatigue load. Based on this results, it can be concluded that the method of the resistance measurement detects the appearance of the damage in the early phase, and it conﬁrms the development of a damage evolution on the basis of microscopic crack incubation and initiation.
3.2 Experiments
1.5
ANSYS PS Exper PS
elect. Resistance R/Ro
elect. Resistance R/Ro
1.6
1.4 1.3 1.2 1.1 1 0.1
0.2
0.3
0.4
0.5
0.6
0.7
Crack length a/w
0.8
0.9
1
169
1.8 ANSYS CS Exper CS 1.6 1.4 1.2 1 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Crack length a/w
Fig. 3.45. Evolution of electrical resistance vs. crack length during fatigue  plain and circular specimen
3.2.1.1.2 Acoustic Emission The mechanisms by which metals absorb and release strain energy under stress, the modelling of which is the basis of fracture mechanics analysis, can be diﬀerent and complicated. Acoustic emission is the elastic energy that is spontaneously released by materials when they undergo deformation. The stress waves which result from this sudden release of elastic energy due to microfracture events are of most interest to the structural engineer. These events are typically 10μm to 100μm in linear dimension. Sources of acoustic emission include many diﬀerent mechanisms of deformation and fracture. Sources that have been identiﬁed in metals include crack growth, moving dislocations, slip, twinning, grain boundary sliding and the fracture and decohesion of inclusion. Other mechanisms fall within the deﬁnition and are detectable with acoustic emission equipment. These include leaks and cavitation, friction (as in rotating bearings), liquefaction and solidiﬁcation, solidsolid phase transformation. Sometimes these sources are called secondary sources to distinguish them from the classic acoustic emission due to mechanical deformation of stressed materials. Acoustic emission examination is a nondestructive testing method with demonstrated capabilities for monitoring structural integrity, detecting leaks and incipient failures in mechanical equipment. Acoustic emission diﬀers from most other nondestructive methods in two signiﬁcant respects: • •
The detected energy is released from within the test object rather than being supplied by the nondestructive method, as in ultrasonics or radiography. The acoustic emission method is capable of detecting the dynamic processes associated with the degradation of structural integrity.
Acoustic emission expected in fatigue studies is primarily of the burst type. Burst type emission signals originate from sources such as intermittent
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3 Deterioration of Materials and Structures
Rise Time Decay Time Volts Amplitude
Energy
Threshold
Threshold Crossing
Time
Counts
Duration
Time
Fig. 3.46. Deﬁnition of simple waveform parameters for a burstsignal
dislocation motion and crack growth in metals. A Burst signal, given in the Figure 3.46, has the following characteristic parameters: • • • • • • • • •
•
threshold: A preset voltage level that has to be exceeded before an acoustic emission signal is detected and processed. This threshold is independent for every sensor, and must be chosen depending on the background noise. burst: A signal whose oscillations have a rapid increase in amplitude from an initial reference level, followed by a decrease to a value close to the initial value. hit: Total signal from the ﬁrst to the last threshold crossing. amplitude: Maximum signal amplitude within duration of the burst. duration: The interval between the ﬁrst and the last time the threshold was exceeded by the burst. counts: The number of times the signal amplitude exceeds the preset threshold. rise time: The time interval between the ﬁrst threshold crossing and maximum amplitude of the burst. decay time: The time interval between the maximum amplitude of the burst and the last threshold crossing. event: A microstructural displacement that produces elastic waves in material under load or stress, which are detected by several AEtransducer. Using time analysis the origin of acoustic emission signal can then be detected. event counts: Counts which belong to an event.
3.2 Experiments
171
3.2.1.1.2.1 Location of Acoustic Emission Sources The ability to locate the sources of acoustic emission is one of the most important functions of the multichannel instrumentation system used in ﬁeld application. One of the methods for detecting the emission source is the measurement of the time diﬀerences in reception of the stress waves at a number of sensors in an array. Depending on the sensor location linear (1D), two and three dimensional problems can be deﬁned. 3.2.1.1.2.2 Linear Location of Acoustic Emission Sources Consider the situation where two sensors are mounted on a linear structure. Assume that an acoustic emission event occurs somewhere on the structure, and that the resulting stress waves propagate in both directions at the same velocity. Using the measurement of the time diﬀerences between hits it is possible to locate position of acoustic emission source. If the time diﬀerence between the hits of both sensors is zero, it would indicate a site precisely midway between the sensors. In general, for the case of constant velocity, the source location is given by: d=
1 (D − V Δt) 2
(3.6)
where D is the distance between sensors, V is the constant wave velocity, Δt is the time deference and d is the distance from the ﬁrst hit sensor. If the source is outside the sensor array, the time diﬀerence measurement corresponds to the time of ﬂight between outer sensor pair and remains constant. 3.2.1.1.2.3 Location of Sources in Two Dimensions The case of location of sources in two dimensions requires a minimum of three sensors. The input data now include a sequence of three hits and two time diﬀerence measurements (between the ﬁrst and second hit sensors and the ﬁrst and third hit sensors), as can be seen in the Figure (3.47). Then: Δt1 V = r1 − R
Δt2 V = r2 − R
(3.7)
which yields R=
D12 − Δt21 V 2 1 2 2 Δt1 V + D1 cos(Θ − Θ1 )
(3.8)
D22 − Δt22 V 2 1 2 2 Δt2 V + D2 cos(Θ3 − Θ)
(3.9)
and R=
Equations (3.8) and (3.9) can be solved simultaneously to provide the location of a source in two dimensions.
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3 Deterioration of Materials and Structures
Sensor 3 X3 , Y 3
r2 D2 Z2 R
Source XS , Y S Z1
Θ Sensor 1 X1 , Y 1
Sensor 2 X2 , Y 2
r1
Θ1
D1 Θ3
Reference
Fig. 3.47. Location of the source in two dimensions
3.2.1.1.2.4 Kaiser Eﬀect Kaiser eﬀect is the phenomenon that a material under load emits acoustic waves only after a primary load level is exceeded. During reloading these materials behave elastically and little or no acoustic emission will be recorded before the previous maximum stress level is achieved. This is true only for materials in which no change in microstructure, such as dislocation movement or crack initiation, can be observed. The case when acoustic emission is recorded before the previous maximum load is reached is known as felicity eﬀect and describes the breakdown of the Kaiser eﬀect. If we deﬁne the ratio between the load level at which the acoustic emission appears and previous maximum load level as felicity ratio, it can be used as the associated quantitative measure of the felicity eﬀect. In the case of the Kaiser eﬀect the value of the felicity ratio is 1. 3.2.1.1.2.5 Experimental Procedures Experiments were performed by a hydrodynamic tensiontorsion testing system (Schenck/Instron Fast Track 8800. The displacement was measured by a real time analogbuilt mean of 3 displacement transducers (HBMW5TK), and the load was measured by a 160kN load cell. The fatigue load was deﬁned as cyclic sinusoidal load in tension range. Tests were performed either with constants amplitude for the entire duration of the test, or as a blocktest with the amplitude which is constant inside the block and diﬀers between the blocks. Stressratio range was R=0.05 and R=0.25. Test frequency was 9Hz for plain specimen and 6Hz for circular specimen. Acoustic emission was detected on two types of specimens. The ﬁrst type is a plain specimen with thickness of 5mm shown in Figure 3.48. The second type is a circular specimen with inconstant thickness and outer diameter
3.2 Experiments
173
R5
330 280 230 225,5
50
26,2
10
22
R3
85
40 30
R1 0
t=5
Fig. 3.48. Geometry of the plain specimen (dimension in mm)
4,847
4,668
R757,25
0,8
R2 0,9
R40
19 9,701
166 150 146 111,883 108,169
Fig. 3.49. Geometry of the circular specimen (dimension in mm)
166mm shown in Figure 3.49. All specimens were made from heattreatable steel 42CrMo4 (No. 1.7225). Two types of AEtransducers with appropriate preampliﬁers were used for the detection of acoustic emission. The ﬁrst set was 4 piezoelectric transducers R15 with resonant frequency 150kHz and 4 single InLine 40dB preampliﬁers with a 100300kHz bandpass ﬁlter. The second set was 4 wideband piezoelectric transducers WD with operating frequency range 1001000kHz and 4 voltage preampliﬁers with 20/40/60dB selectable gain and 1001200kHz bandpass ﬁlter. All acoustic emission signals were ampliﬁed with 40dB and recorded using Physical Acoustics software on a two TwoChanelAEBoards. The sample rate for all measurements was 10MHz. All AEtransducers were clamped to the specimen with a spring clamp. The coupling between the specimen and the transducers was made with a silicon gel. The acoustic emission threshold was set to 35dB, as a compromise between eﬀectively avoiding background noise and cutting oﬀ low level signals during damage evolution. The threshold setting depended on the experimental conditions.
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117 x
3
4
1
2
60
67
40
Fig. 3.50. The position of AEtransducers on the plain specimen
4
1
2
90
3
98
Fig. 3.51. The position of AEtransducers on the circular specimen
The position of AEtransducers on the plain and circular specimen is given in Figures 3.50 and 3.51 respectively. The position of AEtransducers on the plain specimen was given in such a way, that transducers 1 and 4 are socalled guard transducers with the function to eliminate signals originating outside the specimen test section from the recorded data. Thus, extraneous signals such as those emanating from loadchain noise or from servovalves and hydraulic pump were avoided without loss of data. The AEtransducer 2 and 3 were used as measuring sensors. In the case of the circular specimen all 4 transducers are measuring sensors. 3.2.1.1.2.6 Experimental Results Acoustic emission recorded during the test is represented using acoustic emission events counts per cycle over the whole stress range and cumulative acoustic emission event counts during fatigue damage. For all experiments, the load was in the range, which leads to high cycle fatigue with brittle damage. This type of load and the brittle damage behaviour lead to the signiﬁcant
3.2 Experiments
175
1400 PS16 PS17 PS20 PS22 PS24 PS29 PS31
1200
Event Counts
1000
800
600
400
200
0 0
200000
400000
600000 800000 Cycles
1e+06
1.2e+06
Fig. 3.52. Acoustic emission event count rate during fatigue  plain specimen
8000 CS02 CS05 7000
Event Counts
6000 5000 4000 3000 2000 1000 0 0
50000
100000 Cycles
150000
200000
Fig. 3.53. Acoustic emission event count rate during fatigue  circular specimen
increase in the acoustic emission output, as the crack advances towards ﬁnal failure. The rate of acoustic emission in the form of acoustic emission events counts per cycle over the whole stress range is given in the Figures 3.52 and 3.53. When the load was in the elastic range, the low acoustic emission output was evident during initial cycles, due mostly to microscopic dislocation dynamics. This stage was followed by a dead period with almost no acoustic emission. In this stage of fatigue damage accumulation results in the long crackinitiation lifetime. Low energy dislocation motion, which generates acoustic emission
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PS16 PS17 PS20 PS22 PS24 PS29 PS31
50000
Total Event Counts
40000
30000
20000
10000
0 0
200000
400000
600000 800000 Cycles
1e+06
1.2e+06
Fig. 3.54. Acoustic emission total event counts during fatigue  plain specimen
1.1e+07 CS02 PS05
1e+07 9e+06
Total Event Counts
8e+06 7e+06 6e+06 5e+06 4e+06 3e+06 2e+06 1e+06 0 0
50000
100000 Cycles
150000
200000
Fig. 3.55. Acoustic emission total event counts during fatigue  circular specimen
waves, is frequently under background noise and is relatively hard to detect. Only discrete acoustic emission events counts represent the existence of acoustic emission and consequential evolution of fatigue damage. The crack propagation, which is connected with the high rate of the acoustic emission, occurs in the third stage. The release of the elastic energy due to the crack propagation has a signiﬁcant level and the detection of the acoustic emission is not inﬂuenced by the background noise as in the second stage of the fatigue. Evolution of cumulative acoustic emission event counts during fatigue damage is given in the Figures 3.54 and 3.55 and represent a cumulative fatigue
3.2 Experiments
Total Event Counts
Total Event Counts
2500
PS16
2500 2000 1500 1000 500
PS20
2000 1500 1000 500
0
0 0
60 Location [mm]
2500
128
168
0
60 Location [mm]
400
PS29 Total Event Counts
Total Event Counts
177
2000 1500 1000 500 0
128
168
PS31
300 200 100 0
0
60 Location [mm]
128
168
0
60 Location [mm]
128
168
Fig. 3.56. The location of the origin of acoustic emission for the plain specimen
Fig. 3.57. The location of the origin of acoustic emission for the circular specimen
damage process. This process can be divided into the same three stages as indicated for the rate of acoustic emission: (i) microscopic dislocation dynamics; (ii) microscopic crack incubation and initiation; (iii) macroscopic crack propagation. The location of the origin of acoustic emission was computed using time diﬀerence measurement methods described earlier. The obtained distance represents the distance between origin of acoustic emission and AEtransducers. The location of the origin of acoustic emission for the plain specimen is given
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PS17 Total Event Counts
Total Event Counts
200 150 100 50
6000 4000 2000
0
0 0
30
60 Amplitude [dB]
5000
90
120
0
30
60 Amplitude [dB]
90
250
PS24 Total Event Counts
Total Event Counts
PS22
8000
3750 2500 1250 0
120
PS31
200 150 100 50 0
0
30
60 Amplitude [dB]
90
120
0
30
60 Amplitude [dB]
90
120
Fig. 3.58. Acoustic emission event counts against amplitude for the plain specimen
500000
CS02 Total Event Counts
Total Event Counts
1.2e+06 900000 600000 300000 0
CS05
375000 250000 125000 0
0
30
60 Amplitude [dB]
90
120
0
30
60 Amplitude [dB]
90
120
Fig. 3.59. Acoustic emission event counts against amplitude for the circular specimen
in the Figure 3.56 and for the circular specimen in Figure 3.57. As can be seen in the Figures, the computed location of the origin is in good agreement with the real position of the fatigue damage. Most events are located at the position where the crack is present, conﬁrming that the acoustic emission is from the crack. As a consequence of the specimen size, there are inherent problems with the location due to the size of the AEtransducers compared to their spacing. Namely, spacing between AEtransducers was 68mm for the plain specimen and between 9098mm for the circular specimen and the diameter of transducers is 18mm, which excludes a point representation of AEtransducers. For this reason, signals which are located within 10mm of the real crack position are regarded as well located. Acoustic emission event counts with respect to the amplitude are given in the Figures 3.58 and 3.59. As can be seen most of acoustic emission
3.2 Experiments
PS17 Total Event Counts
Total Event Counts
300 225 150 75
PS22
10000 7500 5000 2500
0
0 0
200
400 600 Frequency [kHz]
5000
800
1000
0
200
400 600 Frequency [kHz]
PS24
800
1000
PS31 Total Event Counts
Total Event Counts
179
3750 2500 1250 0
300 200 100 0
0
200
400 600 Frequency [kHz]
800
1000
0
200
400 600 Frequency [kHz]
800
1000
Fig. 3.60. Acoustic emission event counts against frequency for the plain specimen
event counts occur with the amplitude between 40 and 70dB. This behaviour could be used as an additional condition in the elimination of the extraneous noise. Using wideband AEtransducers gives a possibility to investigate the frequency at which the acoustic emission occurs. In the Figures 3.60 and 3.61 acoustic emission event counts versus the frequency are given. Since almost all acoustic emission counts have a frequency between 50300kHz, resonant piezoelectric transducers R15 with resonant frequency 150kHz can be used.
Total Event Counts
Total Event Counts
750000
CS02
1e+06 750000 500000 250000 0
CS05
600000 450000 300000 150000 0
0
200
400 600 Frequency [kHz]
800
1000
0
200
400 600 Frequency [kHz]
800
1000
Fig. 3.61. Acoustic emission event counts against frequency for the circular specimen
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3 Deterioration of Materials and Structures
3.2.1.2 Degradation of Concrete Subjected to Cyclic Compressive Loading Authored by Rolf Breitenb¨ ucher and Hursit Ibuk A great number of concrete structures is exposed to cyclic mechanical loading scenarios. Therefore, the reliability of such structures depends among other inﬂuences also on the degree of structural degradation due to fatigue loading. In order to estimate the state of a structure it is necessary to know the development of the degradation of the material properties during its lifetime. However, up to now a statistically based description of the degradation processes and their eﬀects on the compressive strength, stiﬀness and fracture energy referring to pure cyclic compression loads of plain concrete are still missing. For this purpose, within the large joint research project (Collaborative Research Center 398) extensive experimental investigations were carried out at the RuhrUniversity in Bochum. In order to get information about the degradation processes with suﬃcient reliability a large number of specimens were tested by measuring ultrasonic transmission time, longitudinal strains and stressstrain curves during cycling loadings [148, 149, 402]. 3.2.1.2.1 Test Series and Experimental Strategy Most of the extensive cyclic tests were performed on normalweight concrete of grade C 30/37. Furthermore also high strength concrete of grade C 70/85 as well as airentrained concrete (grade C 30/37) were investigated. Within the C 30/37 series the types of the coarse aggregates (quartzite, basalt and sandstone) were varied. Additionally concretes with diﬀerent coarse grading curves were tested, whereby in these cases the matrix of mortar was kept constant. For all the tests cylindrical specimens with a diameter d of 100 mm and a height h of 350 mm were used. These specimens were taken as cores, drilled from concrete blocks at an age of about 25 days. In comparison to specimens made in separate formworks, the drilled specimens have no accumulations of ﬁnegrained mortar along the surface and represent a part of a real concrete structure in a better way. Thereby additional impairment of the peripheral zone can be prevented. The cyclic tests normally started at a concrete age of about 40 days. Previous the specimens remained on air; within these about two weeks between coredrilling and teststart nearly constant hygral conditions in the concrete specimens (equilibrium moisture content) could be reached. Most of the cyclic tests were performed as singlestage tests at constant stress levels. In some few series also twostage tests with an alternating upper stress level were carried out. For all tests a hydraulic cylinder pulsators were used; the frequency of the cyclic loading was constantly f = 7 Hz.
3.2 Experiments Singlestage tests
Twostage tests Variation of load sequence
S Smax= const.
N1 N2 N3 N... emax at Smax
fc, Estat., Edyn., etc.
181
S S1,max
Smin= const. N
S2,max Smin Ntotal N
S N
S2,max
S1,max
Sequence 2 Smin Ntotal N
N Aim: Revealing the degradation process (by USmeasurement + serelation)
Sequence 1
Aim: Revealing the influence of different sequences at Ntotal on the changes in fc, Estat., Edyn (by USmeasurement + serelation)
Fig. 3.62. Execution of the singlestage and twostage test
In the singlestage test series the specimens were loaded within a deﬁned stressrange Smax /Smin (Figure 3.62, left). In all of these tests the lower stress level Smin was adjusted constantly at 0.10 (i.e 10 per cent of fc ), while the upper stress level Smax was varied from series to series between 0.75, 0.675 and 0.60, however, within one series the Smax was kept also constant. Furthermore interruptions within these singlestage tests were investigated, mainly with the aim to check, to which extent degradations can recover in such rest periods between various load scenarios. Concerning the twostage tests the lower stress level was also kept constant at Smin = 0.10, whereas the upper stress levels were varied within one series after a deﬁned number of load cycles (Figure 3.62, right). In these series various sequences for the upper stress level were considered (Smax = 0.60 → Smax = 0.675 and v.v.; Smax = 0.75 → Smax = 0.675 and v.v.). So in both cases the tests were started with a higher Smax and then reduced to lower one as well as they started at a lower Smax and afterwards the upper stress level was increased. During these cyclic tests the longitudinal strains were measured continuously by two strain gauges (50 mm in length) which were applied in axial direction on opposite sides of each specimen. Additionally microdefects and their development were investigated by nondestructive ultrasonic (US) measurements perpendicular to the main direction of stress after applying a deﬁned number of load cycles. Furthermore at the same stages also the static Young’s modulus was determined, however, in this
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case in the main loaddirection, by carrying out an additional load step at a Smax /Smin ratio of 0.30 / 0.10 during a short interruption of the cyclic test. In addition to these nondestructive tests, the development of the mechanical properties  especially the changes in the stressstrain curve and strength  were investigated by destructive tests on specimens taken out of the setup after certain deﬁned numbers of cycles. By comparison of the so obtained various stressstrain curves the development of the strength, stiﬀness and fracture energy of the respective concrete could be described. 3.2.1.2.2 Degradation Determined by Decrease of Stiﬀness The proceeding in the degradation process is mainly described by the changes in the stiﬀness. This can be demonstrated also within the performed testseries by means of the Young’s modulus Estat (Figure 3.63) as well as the dynamic Emodulus (3.64). For all considered stress levels a typical sharp decrease could be observed after only a few number of cycles. Exemplarily at the stress regime Smax /Smin = 0.675/0.10 the Young’s modulus Estat as well as the dynamic elastic modulus Edyn decreased within the ﬁrst 180,000 load cycles by about 12.5 per cent (averaged value). A following steady decline with a signiﬁcant lower slope was – also typically – observed between 180,000 and 400,000 cycles. After applying about 400,000 cycles some specimens showed an accelerated decrease in these characteristic values. Additionally an increasing scatter of the measured Estat and Edyn by increasing number of cycles was determined. So, e.g., the coeﬃcient of variation in the Edyn grows up from 40.1 per cent after 10,000 cycles to 79.8 per cent after 600,000 cycles. Especially after about 400,000 cycles a signiﬁcant increment in the standard deviation was observed.
Fig. 3.63. Decrease and scatter of Estat at Smax /Smin = 0.675/0.10 (singlestatetests)
3.2 Experiments
183
Fig. 3.64. Decrease and scatter of Edyn at Smax /Smin = 0.675/0.10 (singlestatetests)
Stress level
1.0 smax/fc 0.9 0.8
Assimacopoulos 1959, Antrim 1959, Bennet 1967, Do 1991, Gaede 1962, Galloway 1979, Graf 1936, Gray 1961, Holmen 1979, Kessler 1958, Kim 1996, Oh 1991, Ople 1966, Weigler 1981, Williams 1943 (783 tests)
0.7 0.6 2 4 6 Number of cycles to failure
8 log Nf
Fig. 3.65. Variation of maximal bearable number of load cycles to failure Nf [627]
As already mentioned the specimens were not loaded until failure. Therefore the results in Figure 3.63 and 3.64 could not be referred to their speciﬁc failure cycles (Nf ). Moreover it is shown by many of W¨ ohler tests that the maximal bearable number of load cycles up to failure (Nf ) vary widely ([627]). Therefore, it is not suitable to predict the speciﬁc state of failure (Nf ) of the obtained specimens with an adequate accuracy (Figure 3.70). 3.2.1.2.3 Degradation Determined by Changes in StressStrain Relation The evolution of deformations due to the cyclic loadings was observed simultaneously by continuous measuring the longitudinal strain. The characteristic development of longitudinal strain at the upper stress level Smax is representatively illustrated for the stress regime Smax /Smin = 0.60/0.10 in
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3 Deterioration of Materials and Structures
Fig. 3.66. Measured longitudinal strain at Smax (Smax /Smin = 0.60/0.10)
Figure 3.66. Under these conditions until the ﬁrst 2.0 million cycles the longitudinal strain in general increased faster than in the phase after 2.0 million cycles. Here also up to 25.55 million cycles no failure due to cyclic loading could be observed. For the same test series (Smax /Smin = 0.60/0.10) the characteristic averaged results of the stressstrain relations after deﬁned load cycles, determined in the destructive tests, are illustrated in Figure 3.67. Herein also the residual irreversible strains  caused by the cyclic loading itself, determined after unloading  are also considered at the beginning of the stressstrain curves. These residual deformations increased signiﬁcantly with increasing numbers of load cycles. The ascending branches of the stressstrain curves themselves also changed with increasing cycle numbers from concave shape (towards the strain axis)
Fig. 3.67. Stressstrain curves at diﬀerent number of cycles (Smax /Smin = 0.60/0.10)
3.2 Experiments
185
Table 3.3. Changes of concrete properties due to cyclic loading determined by changes in stressstrain relation at Smax /Smin = 0.60/0.10 Number of cycles N [Mio.] 0 1.8 4.15 25.5
Young’s modulus Estat kN/mm2 28.4 26.8 25.6 24.4
Compressive strength fc N/mm2 40.2 41.6 42.0 39.6
Compressive strain
u 0/00 2.3 2.0 1.9 1.8
Fracture energy gc kJ/m3 63.8 47.5 43.8 36.8
to a straight line and further to a convex shape in Figure 3.67. A similar development also was already observed in orientating investigations by e.g. Holmen [383]. For such a development mainly microcracking is responsible. This obviously correlates with the decreasing Young’s modulus (Estat ) – at least at lower stresslevels – with increasing number of cycles and parallel a reduction in the fracture energy (gc ) and the ultimate compressive strain (u ) (Table 3.3). According to the increase in the longitudinal strain the concrete properties Estat , gc and u decreased also faster up to the ﬁrst 2.0 million cycles than afterwards, whereas the compressive strength (fc ) almost remained constant. 3.2.1.2.4 Adequate Description of Degradation by Fatigue Strain Investigations by Holmen [383] demonstrated the diﬃculties to formulate the state of concrete degradation and damage resp. depending on the ratio of already applied number of cycles N related to the maximal bearable number of load cycles Nf . So for example, at a deﬁned ratio of Smax /Smin(0.675/0.050) some specimens failed already after only 105 cycles, whereas other ones under the same conditions exceeded 3 x 106 cycles [383]. This quite diﬀerent behaviour indicates that each examined specimen has its own speciﬁc Nf value. Hence an alternative approach was used to characterise the state of damage more precisely by a diﬀerentiation of the measured total longitudinal strain max into one part caused by the pure static loading 0 and in a second part derived by the cyclic loadings (Figure 3.68). In the following the second part is deﬁned as fatigue strain f at,max (Figure 3.69). In Figure 3.71 these fatigue strains are illustrated for the same tests as in Figure 3.68. The increase in strain due to the cyclic loading, which takes place only within the fatigue strains, can also be explained in a stressstrain diagram of the complete cyclic test (Figure 3.70). (Within a simpliﬁcation the stressstrain relation are linearised). Already the ﬁrst single load (N1 ) leads directly to the initial strain 0,max at σmax . The values of the total strain max at σmax increase by increasing number of cycles (Ni ), i.e. that the cyclic loading cause further strains – namely the fatigue strain f at,max – exceeding the initial strain signiﬁcantly. During the unloading at the end of the cyclic test (Ni )
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3 Deterioration of Materials and Structures
Fig. 3.68. Total strain at Smax /Smin = 0.675/0.10
Fig. 3.69. Calculation of fatigue strain at Smax
from σmax to σmin and further to the unstressed state, the strain decreases on a signiﬁcant lower rate as expected according to the deformation at the ﬁrst loading 0,max . In Figure 3.72 the averaged values of the residual Young’s moduli are plotted against the corresponding fatigue strains f at,max at Smax for all 3 investigated Smax levels. In general for all series a linear relationship between these two parameters could be proved. It became obvious that the approximation lines shift together. The trendlines for upper stress levels Smax = 0.60 and Smax = 0.675 almost were identical, while at Smax = 0.75 the residual Young’s modulus decreases somewhat more than expected ahead the achieved fatigue strain. So the process of fatigue degradation seems to be mainly coupled with the evolution of fatigue strain. Thus the parameters stress level and number of cycles seem to be negligible to describe the degradation process, especially at lower stress levels. In a global evaluation the residual Young’s
3.2 Experiments
emax efat,max cyclic loading
e0,max smax
s
N1
187
Ni
smin e0,max
eplastic
e
efat,max Fig. 3.70. Formation of fatigue strain (schematically)
Fig. 3.71. Fatigue strain at Smax /Smin = 0.675/0.10
modulus of all investigated stress levels of normal strength concrete can be resumed to just one trendline (Figure 3.72). 3.2.1.2.5 Behaviour of High Strength Concrete and AirEntrained Concrete In comparison to the investigated normal strength concrete, the results of the high strength concrete followed in general almost the same trendline (Figure 3.73). In opposite to this airentrained concrete showed a quite diﬀerent relation between fatigue strain f at,max and the residual Young’s modulus in comparison with (nonairentrained) normal strength concrete (Figure 3.74). After an accelerated drop at the beginning up to about f at,max = −0.20/00 ,
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3 Deterioration of Materials and Structures
Fig. 3.72. Correlation between the fatigue strain and the residual stiﬀness for diﬀerent load levels
the airvoids reduce the crack propagation so that the decline of the residual Young’s modulus continued only with a signiﬁcant lower slope. This speciﬁc behaviour can be explained by the entrained micro airvoids into the concrete microstructure. Due to the socalled ”buttonholeeﬀect”, raised by the microvoids, microcracks, which penetrate up to such voids, require a much
Fig. 3.73. Correlation between the fatigue strain and the residual stiﬀness of normal and high strength concrete
3.2 Experiments
189
Fig. 3.74. Correlation between the fatigue strain and the residual stiﬀness of normal and airentrained concrete
higher fracture energy for their further propagation than in a concrete matrix without microvoids. 3.2.1.2.6 Inﬂuence of Various Coarse Aggregates and Diﬀerent Grading Curves The inﬂuence of diﬀerent aggregates and diﬀerent grading curves on the degradation process and the residual Young’s modulus with increasing fatigue strain is illustrated in Figure 3.75 and 3.76. For the specimens containing sandstone Fatigue strain ?fat,max [‰] 1.5
1.0
0.5
0
Residual Young`s modulus [%]
100 90 80 70 60 50 basalt
quartzite
sandstone
Fig. 3.75. Correlation between the fatigue strain and the residual stiﬀness subjected to diﬀerent aggregates in concrete (C 30/37)
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1.0
Fatigue strain ?fat,max [‰] 0.8 0.6 0.4 0.2
0
Residual Young`s modulus [%]
100 90 80 70 60 50
high
middle
low
Fig. 3.76. Correlation between the fatigue strain and the residual stiﬀness subjected to diﬀerent grading curves in concrete (C 30/37)
as coarse aggregate the changes in Young’s modulus (E˙ = dE/df at,max ) is about three times lower than that of specimens containing basalt or quartzite (Figure 3.75). This mainly can be explained by the ratio of the Young’s moduli of the aggregates and the hardened cement paste. The Young’s modulus of sandstone can be assumed between 21,000 N/mm2 until 58,000 N/mm2 and is therefore closer to the Young’s modulus of the hardened cement paste (about 20,000 N/mm2 ), whereas the corresponding values of basalt or quartzite exceed 60,000 N/mm2 signiﬁcantly. Thus, in case of sandstone the concrete’s stiﬀness is a much more uniform within the crosssection. Therefore lower and less stress peaks within the crosssection, especially in the contactzone coarse aggregate – cementmatrix, lead to less degradation in the concrete microstructure compared to the concrete with basalt or quartzite. The inﬂuence of diﬀerent grading curves on the development of the Young’s modulus with increasing fatigue strain is shown in Figure 3.76. A high fraction of coarse grains (high) leads almost to a decisive higher velocity in the degradation process compared to the others with lower fraction of coarse grain (middle, low). Analogue to the inﬂuence of the type of aggregate, the stresses are spread more uniformly in the crosssection of concrete with middle or low fraction of coarse grains than in the specimens with high ones. 3.2.1.2.7 Cracking in the Microstructure Due to Cyclic Loading With the intention to determine the crack formation due to cyclic loading from some specimens, cyclic tested at the stress regime of Smax /Smin = 0.675/0.10, after certain cycles, subspecimens were prepared for LMMicroscopy (light microscopy) analyses. In order to get representative results the microcrack characteristics were determined within a test area of 48 by 48 mm2 (polished surface) in the crosssection of the subspecimens.
3.2 Experiments
191
Fig. 3.77. Light microscopy micrographs
Table 3.4. Crack characteristics at certain number of cycles Smax /Smin = 0.675/0.10 Number of cycles Crack width ∗ Crack amount Crack area ∗ averaged values
0 [μm] 4.0 [–] 2 [μm2 ] 3,400
1 8.0 1 2,900
180,000 7.0 5 28,400
600,000 11.0 14 179,200
The left photo in Figure 3.77 illustrates exemplarily the situation at an unloaded specimen (N = 0) without any cracks, while in the right photo after 600,000 cycles already a characteristic formation of microcracks with width of only a few μm could be proved. In order to quantify the crack formation and to enable a comparison between various concretes or load scenarios, the widths, lengths and number of cracks within the test area were determined as crack characteristics. From these values the crack area, i.e. the integrative product of crack width and length, was calculated (Table 3.4). It became obvious that the observed increase of the crack area is mainly governed by the increase of the crack amount. Further, it could be proved that the ﬁrst load cycle (N = 1)  which corresponds to a static load  normally leads only to marginal changes in the microstructure in comparison with the further cyclic loads at the same stress level. Furthermore, this shows that the microcracking and therefore the degradation is mainly caused by the cyclic loading. 3.2.1.2.8 Inﬂuence of Single Rest Periods As already mentioned before, cyclic loading leads to some degradations and in consequence to an increase in strain and to a reduction in stiﬀness. The inﬂuence of single rest periods within periods of cyclic loading on the
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3 Deterioration of Materials and Structures
Stress level Smax
N1
Rest period duration tp
N2
Smin
Investigated durations of the rest periods:
t respectively N 1. tp = 20 minutes 2. tp = 2 hours 3. tp = 72 hours
Fig. 3.78. Load history with various rest periods [150]
development of strains and stiﬀness has not been clariﬁed up to now. Thus, the inﬂuence of single rest periods (RP) within the cyclic compression loadings at the stress regime of Smax /Smin = 0.675/0.10 on strain and stiﬀnessbehaviour has been investigated also within this research project [150] (Figure 3.78). In these investigations also the durations of rest periods were modiﬁed. During the whole test periode the total longitudinal strains as well as the changes in stiﬀness by ultrasonic methods were determined in the same manner as described before. Initially, an increase in the total longitudinal strains up to 600,000 cycles at Smax = 0.675 (Figure 3.79) was observed (averaged: about +40.5 per cent). Because of the large scattering in the strain values already in the forefront of the rest periods, the development of the longitudinal strains up to N1 =
Fig. 3.79. Behaviour of the longitudinal strain at Smax /Smin = 0.675/0.10
3.2 Experiments
193
Fig. 3.80. Related longitudinal strain at Smax /Smin = 0.675/0.10
600,000 are related on the adequate value of the longitudinal strain at N1 to obtain a comparable basis (Figure 3.80). Immediately after unloading at the end of the ﬁrst cyclic loading N1 resiliences were observed which correspond more or less to the elastic part of the strains at the beginning of the tests (by 50 to 60 per cent). These instantaneous deformations are followed by a gradual decrease in strain within the rest period. The longer the rest period, the higher the increase in strain. A similar behaviour has been observed also already by other scientists [693]. After starting the second cyclic loading N2 the strains increase more or less immediately up to the state at the end of the ﬁrst loading period N1 . The further development obviously is not inﬂuenced by the rest period. Thus from these results it can be concluded, that under such conditions as investigated here the load histories and rest periods do not inﬂuence the strain behaviour of concrete under cyclic load signiﬁcantly. Minor recoveries due to the gradual decrease in the strainbehaviour of the concrete within the rest periods were quickly used up after restarting the load regime. A similar behaviour was observed for the stiﬀness of the concrete [150]. 3.2.1.2.9 Sequence Eﬀect Determined by TwoStage Tests In order to simulate more practical load scenarios twostage tests additionally were carried out. Within these twostage tests the lower stress level was also kept constant on 0.10. The upper stress level was varied once after several numbers of cycles (Figure 3.81). Sequence 1 means that the cyclic loading started with the higher upper stress level, in sequence 2 the tests were started with the lower one. For both sequences a linear relationship of the fatigue strains at Ntotal against the residual Young’s modulus in case of the load parameters S1,max /S2,max = 0.75/0.675 could be observed (Figure 3.81). Additionally these trendlines are nearly identical. Thus it can be assumed in this case
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3 Deterioration of Materials and Structures
Fig. 3.81. Correlation between the fatigue strain and the residual stiﬀness subjected to diﬀerent sequences of cyclic loading for S1,max /S2,max = 0.75/0.675
that the load sequence did not aﬀect the behaviour of changes in the residual Young’s modulus against the fatigue strain. 3.2.1.3 Degradation of Concrete Subjected to Freeze Thaw Authored by Max J. Setzer and Ivanka Bevanda Damage caused to constructions due to frost and deicing salt attack led to intensive research on development of reproducible practical test procedures. The durability of concrete against a frost and deicing salt attack is linked to both damage eﬀects the surface scaling and the internal damage inside the material. An overview of the existing test procedures for determining frost and deicing salt resistance of concrete can be found in Auberg [69]. RILEM TC 117 FDC recommends the CDF test1 [732] for determining deicing salt resistance and RILEM TC 119 recommends the CIF test2 [729] for frost resistance. The CDF test is also one procedure in prEN 123909 [17]. The CIF test is subject to the standard CEN committee TC 51 and the BAW3 speciﬁcation [24]. Both test methods are based on the microicelens model (refer to Subsection 3.1.2.2.3). The test procedure is described in detail in [732],[729]. With the CDF/CIF test, it is possible to simultaneously measure external and internal damage as well as moisture uptake (see Figure 3.82). The following data have been investigated using the CIF test in accordance with the RILEM Recommendations. In [69] was shown that the internal damage of concrete and moisture uptake correlate directly with each other. In his experiments with normal concrete, 1 2 3
C apillary suction of Deicing solution and F reezethaw test. C apillary suction, I nternal damage and F reezethaw test. German Federal Waterways Engineering and Research Institute.
3.2 Experiments
Isothermal Isothermal Isothermes suction +20 °C suction +20 °C
195
Freezethaw cycles Freezethaw cycles

Tau Zyklen
CIF/CDF test chest CDF/CIF test chest
0
Time Time[h] [h] 4 7 12
Temperature
Fros t
Saugen 20°C
20 °C 0 °C 20 °C
Tempering (+20°C°C...20°C) ...20°C) Tempering bath bath (+20
Ultrasonic transit time Ultrasonic timedynamic modulustransit of elasticity
Ultrasonic cleaning Ultrasonic cleaning bath bath
Mass Mass gain gain
Surfacescaling scaling Surface
dynamic modulus of elasticity
Mass Mass gain gain 
Internal damage Internal damage
Fig. 3.82. Steps of exposure and measuring during CDF/CIF test [731]
Auberg determined the dependence between frost suction and internal damage as well as length changes. Studies performed by Heine [363] and Palecki [610] show a direct materialspeciﬁc connection between pore structure, moisture uptake and internal damage. Palecki discovered that there is a correlation between the rate of saturation, pore radius ratio and frost resistance, and divided concrete into damage types depending on moisture uptake behaviour. For example, the Figure 3.83 demonstrates the correlation between internal damage (calculated from ultrasonic pulse transit time) and moisture uptake by high performance concrete (hpc). The moisture uptake (frost Phases I
II
1
2
2,5%
III critical saturation in a depth of 35 mm  start of damage 
100%
2,0%
rapid decrease of RDM
1,5%
80%
0,30+7%SF+20%FA SERVICE LIFE
60%
1,0%
exponential moisture uptake
Incubation time
40%
0,5% damage has reached a depth of 35 mm
20%
0,0%
Rel.moisture uptake in M.%
Rel. dyn. Modulus of elasticity in %
120%
first microcracking at surface layer  damage induction 
freezing at the surface layer
0%
0,5% 7
0
7
14
21
28
35
42
Test duration in days [1d=2ftc]
Fig. 3.83. Example relationship between RDM and relative moisture uptake  concrete type 2 [610]
196
3 Deterioration of Materials and Structures
suction) rises continuously with an increasing number of freezethaw cycles (ftc). The nonsaturated pores which cannot be sucked in capillary will be ﬁlled up to approx. 95 % of the original value of the relative dynamic modulus of elasticity (RDM=100%); no irreversible microcracks are observed, despite a very much increased moisture uptake [69]. Only after a critical saturation of these pores does the damage of the concrete structure start as a result of ice expansion. The critical saturation degree marks a nick point in the moisture curve i.e. in the RDM curve. The ”incubation period” up until the concrete does not show any damage is very striking. It eﬀectively prolongs the service life. The last phase is distinguished by a rapid drop in the RDM and increased moisture uptake within a few ftc. Not only are the pores ﬁlled with moisture, but also the cracks will be ﬁlled due to frost attack. For the service life of a concrete structure under frost attack, this means that the saturation speed is the decisive parameter and not the amount of moisture [69],[610]. The aim of the tests in SFB 398/Project A11 was the nondestructive measurement of moisture and damage progress, working from the frostexposed concrete surface. By applying multispherical electrodes (MSE), the moisture distribution was determined at several depths of specimen during the ftc; and the moisture volume calculated from the amount of moisture uptake (Figure 3.84, Table 3.5). The multispherical electrode is a moisture/ temperature sensor according to [660] improved by Xu [861] and Bevanda (see Subsection 2.4.2) during the SFB 398 project. Internal damage progress was detected by measuring the change of RDM (calculated from ultrasonic pulse transit time) also at several depths of the specimen. Figure 3.84 shows the internal damage as a function of moisture at corresponding ftc. Comparing the damage proﬁle to the moisture proﬁle, it follows that at an MSE signal of 80 %, the RDM is 95 % at all measured depths. If one considers the volume of moisture uptake at that time (Table 3.5), the volume does not exceed the concrete capillary porosity (rh >30 nm) of 3.5 % as measured by mercury intrusion porosimetry. Comparing the results found at a later time (Figure 3.84, 90 % of MSE signal), the volume of moisture uptake exceeds the capillary porosity. At the same time, the RDM decreases to <95 %, which corresponds to crack formation as a result of frost damage to the structure [69]. This correlation between moisture uptake and the start of microcrack formation can be observed at all measuring depths of the specimen. If, on the other hand, one considers the saturation process until critical saturation is reached (MSE signal <80%), the suction rate increases. This can be explained by the greater permeability of the matrix as a result of crack formation in the lower part of the specimen. The data attained are another step towards deﬁning input parameters for modelling frost attack and the microicelens model [448]. Furthermore, the data are interesting for monitoring real concrete structures under frost attack. Additionally, important for durability is the surface scaling. Both internal damage and surface scaling are linked to the damage by ice expansion after critical degree of saturation has been reached and to the forgoing transport process following the microicelens model by Setzer. However, surface
3.2 Experiments
64
100
56 48
Moisture [%]
90
RDM [%]
2,0 1 cm 2 cm 3 cm 5 cm frost suction
80 70
1 cm 2 cm 3.5cm 5 cm
60 50
40
1,8 1,6 1,4 1,2
32
1,0
24
0,8 0,6
16
0,4
8
40
0,2
0
40
50
60
70
80
90
Moisture [%]
100
110
Moisture uptake [kg/m²]
110
197
0,0 0
8
16
24
32
40
48
56
64
Number of ftc [1d=2 ftc]
Fig. 3.84. Internal damage as a function of moisture to the corresponding freezethaw cycles at several depths of the specimen (left); internal damage is demonstrated by RDM [%] and moisture of MSE output signal [%]; Moisture uptake versus number of ftc (right); yaxis changes of moisture distribution (MSE), y‘axis total amount of moisture (weight of specimen)
Table 3.5. Measurement results: MSE signal at several depths at diﬀerent points in time of CIF test related to moisture uptake (frost suction) and RDM MSEsignal
height
time
moisture uptake
[m] [ftc] [kg/m²] [vol.%]1) 0.012 6 0.16 1 0.020 11 0.23 1 0.028 20 0.45 2 0.052 36 1.05 2 80 0.012 16 0.35 3 0.020 28 0.63 3 0.028 36 0.89 3 0.052 51 1.61 3 90 0.012 30 0.70 5 0.020 40 1.04 5 0.028 44 1.28 5 0.052 54 1.65 3 1) relating to specimen volume: 0.11 x 0.15 x adequate height [m³]. [%] nick point
RDM
suction rate
[%] 97 99 98 98 96 95 95 95 89 90 93 95
[kg/m² x d]
0.038 0.047 0.055 0.075
scaling takes place in a very thin layer between 0.1 mm and 3 mm at most. Here dissolved ions, chromatographic eﬀects and surface physical eﬀects like disjoining pressure and ions adsorbed at internal surfaces of the gel matrix must not be neglected. In SFB 398/Project A11 it was found that an increase of ion concentration from zero (demineralised water) to 0.02 mol/l (approx.
198
3 Deterioration of Materials and Structures
0.1M.% sodium chloride) increases the scaling by more than a factor of 4 (see Subsection 2.4.2). 3.2.2 HighCycle Laboratory Tests on Soils Authored by Torsten and Theodoros Triantafyllidis
Wichtmann,
Andrzej
Niemunis
The accumulation phenomenon has been studied systematically on quartz sand with subangular grain shape. Most of the tests were performed on a grain size distribution curve (No. 3 in Figure 3.85a) with a mean grain size d50 = 0.55 mm and a uniformity index Cu = 1.8. Numerous cyclic triaxial tests and cyclic multidimensional simple shear tests have been performed. A scheme of the cyclic triaxial device (four similar devices were available in the present study) is given in Figure 3.85b. Details are explained e.g. in [837, 835]. Cylindrical specimens with 10 cm diameter and 20 cm height were used. They were prepared by dry pluviation and afterwards they were watersaturated in order to measure volume changes via the pore water. The tests were performed under drained conditions. In the stresscontrolled tests the axial eﬀective stress σ1 and the lateral eﬀective stress σ3 could be varied simultaneously using pneumatic loading devices. In Figure 3.85c typical stress paths with inphase and outofphase cycles are presented in the pqplane. p = −(σ1 + 2σ3 )/3 and q = −(σ1 − σ3 ) av denote Roscoe’s invariants. The average stress is described by pav and √ q or av av av the ratio η = q /p . Alternatively, the isomorphic variables P = 3p and Q = 2/3q are used in the following. In most tests the cycles were applied with a frequency f ≤ 1 Hz, i.e. inertial forces were negligible small. The axial strain ε1 and the volumetric strain εv = ε1 + 2ε3 were calculated from the measured changes of specimen height and volume. The deviatoric strain εq = 23 (ε1 −ε3 ), the total strain ε = (ε1 )2 + 2(ε3 )2 and the isomorphic strain variables εP = 1/3εv and εQ = 3/2εq are derived quantities. In a cyclic test the residual strain in the ﬁrst cycle may diﬀer signiﬁcantly from the strain in the subsequent cycles (Figure 3.85e). Thus, it is distinguished between the ﬁrst ”irregular” cycle and the subsequent ”regular” ones. The high cycle accumulation model described in Section 3.3.3 describes only the regular cycles (the ﬁrst cycle is calculated implicitly, see Section 4.2.11). Thus, in the following only test results for the regular cycles are presented. The strain is composed of a residual (or cumulative) portion εacc and an elastic, resilient portion εampl (strain amplitude, Figure 3.85e). In the context of polycyclic loading, ”rate” means a derivative with respect to the number of cycles Nc , i.e. ˙ = ∂ /∂Nc. The device used for the cyclic multidimensional simple shear tests is presented in Figure 3.85d. An arbitrary displacement of the base plate in both horizontal directions is possible while the top cap is guided vertically. The horizontal movement is generated by an electrical motor and an eccentric.
3.2 Experiments
b)
a) fine
100 Finer by weight [%]
Sand medium coarse
1
3 4
2
back pressure u
8
7 80
5
axial load Fav+ Fampl (pneumatic loading system)
cell pressure + σampl σav 3  3
Gravel fine
199
displacement transducer
6
60
soil specimen (d = 10 cm, h = 20 cm)
40 20 0
0.1
0.2 0.6 1 2 Grain diameter [mm]
pressure transducers (u, σ3)
6 diff. pressure transducer 2σ1 σ1av
drainage
load cell
q = (σ1σ3)
σav
m
ax .s tre
c)
ng th CS L
ampl
inphase (IP) cycles
d)
displ. transducer ball bearing (vert. guidance) top cap aluminium rings guidance rods soil specimen base plate drainage ball bearing (hor. guidance) displ. transducer eccentric rod electric motor
F
σ3av 2σ ampl 3
ηav = 1 qav / pav
qav
outofphase (OOP) cycles pav
e)
p = (σ1+2σ3)/3
ε
2εampl 2ε1ampl
εacc
ε1acc ε3acc 2ε3ampl
average accumulation curve (described by the highcycle model)
εirreg t,Nc
first "irregular" cycle "regular" cycles
Fig. 3.85. a) Tested grain size distribution curves, b) Scheme of the cyclic triaxial device, c) Stress paths of cyclic triaxial tests in the pqplane, d) Scheme of the cyclic multidimensional simple shear (CMDSS) device, e) Course of strain in a cyclic triaxial test
The eccentric runs in a cutout of the base plate. Diﬀerent paths of horizontal deformations can be tested by using diﬀerent eccentrics and cutouts. Lateral deformations of the specimens (diameter 10 cm, height 20 cm) are prevented by a stack of 200 aluminium rings which are guided by vertical rods in order to guarantee a linear deformation of the specimen boundaries. However, in
av
η
75
0.
=0
.5 5
0.2 v ηa =
0.5
ηav = 0 η av = 0 .25
0
η av
0.4
0
0.4
200
0.8
1.2
η = η 1.3 = M 1 .3 75 c (ϕ 13 c)
= 1. 25
0.7
0
0.5
100
5
η = 0.2
0
η = 0.125
100
200
Accumulated volumetric strain εacc v [%]
5
η= η=
=
0.5
1.
300
η = 0 .25 η= 0.3 75 di η= la 0 ta nc η = .50 η y 0. = εqacc 6 0 .8 η = 25 8 εvacc = 0.7 5 M e (ϕ c) 100 200 300 400
) (ϕ p Me
1.5
5 .62 0 5 av = 0.7 η av = η 5 0.81 v ηa = ηav = 0.88
1.0
300
=
contractancy
η
=
η p)
av
1.0
0.5
00
η
5 87 0. η av =
1.5
acc. up to cycle: Nc = 100,000 Nc = 10,000 Nc = 1,000 Nc = 100 Nc = 20
M c (ϕ
1.12 5
=1 .0
η av =
η av = 1.25
η av
Accumulated deviatoric strain εacc q [%]
2.0
5 1.37 v ηa =
2.5
ηav = 1.313
3.0
=
all tests: b) 4 5 Nc,max = 10  10 , pav = 200 kPa, ζ = qampl/pav = 0.2  0.3 ID0 = 0.57  0.69, f = 0.1  1 Hz
a)
dil ata nc y
3 Deterioration of Materials and Structures
Average deviatoric stress qav [kPa]
200
0
Average mean pressure pav [kPa]
acc Fig. 3.86. a) εacc strain paths in tests with diﬀerent average stress ratios η av q εv at pav = constant, b) Direction of strain accumulation presented as a vector in the pqplane
simple shear tests the distribution of stress and strain within the specimen is not homogeneous [162, 840]. Thus, the CMDSS test results are rather of a qualitative nature. In all CMDSS tests specimens were tested in the airdry condition. First, the cyclic triaxial test results are discussed concerning the direction of accumulation Dqacc /Dvacc . Tests were performed with diﬀerent average stresses, some of them with triaxial compression and others with triaxial exacc tension. Figure 3.86a presents the εacc q εv strain paths in tests with an average av mean pressure p = 200 kPa but with diﬀerent average stress ratios η av . For an isotropic average stress (η av = 0) the accumulation is purely volumetric (Dqacc ≈ 0). With increasing stress ratio η av  the direction of accumulation becomes more deviatoric. If the average stress lies on the critical state line (CSL, known from monotonic tests) a purely deviatoric accumulation takes place. With increasing number of cycles a slight increase of the volumetric portion of the direction of accumulation was observed. For average stresses between the critical state lines a cyclic loading causes a compaction of the sand and a dilative behaviour is observed in the overcritical regime. It could be demonstrated that several other parameters (average mean pressure pav , stress amplitude q ampl , void ratio e, polarisation, shape of the cycles, static preloading, grain size distribution curve) do not inﬂuence the direction
3.2 Experiments
Q [kPa]
b) 2.0
qampl
[kPa] = 80 70 60 51
1.5
42 31 22 12
1.0 5
all tests: Nc,max = 10 , pav = 200 kPa, ηav = 0.75, ID0 = 0.58  0.61, f = 1 Hz
0.5
0
0
0.2
0.4
0.6
0.8
Acc. volumetric strain
1.0
εacc v
[%]
1.2
1.4
Acc. deviatoric strain εacc q [%]
Acc. deviatoric strain εacc q [%]
a)
201
80
1.2
82 80
1.0
346 P [kPa] 100 100
0.8 0.6 0.4
4
all tests: Nc,max = 10 , pav = 200 kPa, ηav = 0.5, ID0 = 0.56  0.64, f = 0.05 Hz
0.2 0
0
0.2
0.4
0.6
0.8
1.0
Acc. volumetric strain εacc v [%]
acc Fig. 3.87. a) εacc strain paths in tests with diﬀerent stress amplitudes q ampl , q εv acc acc b) εq εv strain paths in tests with diﬀerent polarizations
of accumulation [838, 835]. A constant direction of accumulation (= direction acc ampl of the εacc in q εv strain path) is shown for diﬀerent stress amplitudes q ampl ampl Figure 3.87a and for diﬀerent stress polarisation Q /P in Figure 3.87b. The direction of strain accumulation (”cyclic ﬂow rule”) has been found to be almost exclusively inﬂuenced by the stress ratio η av . The cyclic ﬂow rule can be clearly illustrated in the pqplane (Figure acc 3.86b). For this purpose an εacc q εv arrow is plotted from the average stress av σ of a test. It could be demonstrated [838, 835] that the ratio Dqacc /Dvacc can be approximated by the ﬂow rule for the monotonic loading of clay (modiﬁed Cam Clay model) or by the hypoplastic ﬂow rule (Section 3.3.3). The intensity of accumulation is a function of several parameters. Figure 3.88a presents typical accumulation curves εacc (Nc ) in tests with IP cycles and diﬀerent stress amplitudes q ampl . The intensity Dacc of accumulation increases with increasing q ampl . If the residual strain εacc is plotted versus the square of the strain amplitude (¯ εampl )2 linear curves are obtained independently of the number of cycles (Figure 3.88b). Since in the stresscontrolled tests the strain amplitude εampl varies slightly with Nc , a mean value of the strain amplitude was used on the abscissa in Figure 3.88b (here and in the following a bar over ¯ = N1 0Nc dNc ). a quantity denotes that a mean value over Nc is used, i.e. c The division of εacc by a void ratio function f¯e (Table 3.23) on the ordinate considers slightly diﬀerent initial void ratios e0 and diﬀerent compaction rates e. ˙ The proportionality between Dacc and the square of the strain amplitude (εampl )2 has been described by the function fampl (Table 3.23) and holds up to εampl = 10−3 [835]. Another test series was performed in order to study the inﬂuence of the polarization of the cycles in the stress or strain space. Onedimensional stress cycles with six diﬀerent polarizations (tan αP Q = Qampl /P ampl ) in the
3 Deterioration of Materials and Structures a) 2.0
all tests: pav = 200 kPa, ηav = 0.75, ID0 = 0.58  0.61, f = 1 Hz
80 70 60 51 42 31 22 12
1.6
εacc [%]
b) 10
qampl [kPa] =
1.2 0.8
Nc = 100,000 Nc = 50,000 Nc = 10,000 Nc = 1,000 Nc = 100 Nc = 20
8
εacc / fe [%]
202
6 4 2
0.4
0
0 100
101
102
103
104
105
0
0.5
d)
αPQ =
4
30˚ 10˚ 0˚
2
0
0
90˚ 75˚ 54.7˚
1
all tests: pav = 200 kPa, ηav = 0.5, ID0 = 0.56  0.64, f = 0.05 Hz
2
3
7
Residual strain εacc [%]
Strain amplitude
6
γ13
4 εampl
5
[104]
13
5
ampl = 6.5 103, γ13 ID0 = 0.56
x3 x1
3 γ13 ampl = 5.8 103, γ13 ID0 = 0.61
1 0
0
200
400
0.68  0.72
4
0.81  0.81 all tests: pav = 16 kPa, γ ampl = 5.8 103
2
x3
13
1,000
2,000
3,000
4,000
5,000
x2 x1
x2
0.3 0.2
Nc = 100,000 Nc = 50,000 Nc = 10,000 Nc = 1,000 Nc = 100 Nc = 20
all tests: pav = 200 kPa, ηav = 0.75, ζ = 0.3, f = 1 Hz
0.1
γ ampl
γ23
13
600
0.61  0.63
0.4
13
2
6
f)
ampl γ ampl γ23 =
4
ID0 = 0.53  0.54
Number of cycles Nc []
γ23
γ ampl
2.0
change of the polarization
0 0
6
εacc / fampl [%]
εacc / fe [%]
346 P [kPa] 100 100
8
Residual strain εacc [%]
80
6
e)
Nc = 104
80
82
1.5 2
Q [kPa]
8
1.0
( ε ampl) [107]
Number of cycles Nc []
c)
all tests: pav = 200 kPa, ηav = 0.75, ID0 = 0.58  0.61, f = 1 Hz
800
Number of cycles Nc []
1,000
0 0.56
0.60
0.64
0.68
0.72
0.76
0.80
Void ratio e []
Fig. 3.88. Results of drained cyclic triaxial tests: a) Accumulation curves εacc (Nc ), b) Dependence of Dacc on strain amplitude εampl and c) on the polarization tan αP Q = Qampl /P ampl of the cycles, d) Inﬂuence of polarization changes, e) Dependence of Dacc on the shape of the strain loop and f) on void ratio e
isomorphic PQplane were tested. For each polarization tests with diﬀerent amplitudes were performed. In Figure 3.88c the residual strain after 10,000 cycles is plotted versus a mean value of the strain amplitude. For a given ε¯ampl the residual strain does not signiﬁcantly depend on the polarization of the cycles (as long as the polarization does not change, see below). The eﬀect of changes of the polarization was studied in the multidimensional simple shear device. 1,000 cycles with a certain polarization were followed by 4,000 cycles with a perpendicular polarization. Figure 3.88d reveals that a sudden change of the polarization causes a temporary increase of
3.2 Experiments
203
the accumulation rate. In the highcycle model (Section 3.3.3) this eﬀect is described by a function fπ . The inﬂuence of the shape of the strain loop has also been studied in CMDSS tests (Figure 3.88e). A circular cycle compared with a onedimensional cycle with identical maximum span (i.e. identical shear strain amplitude in the γ13 direction) causes an approximately twice larger accumulation rate. Thus, the shape of the strain loop signiﬁcantly inﬂuences the accumulation rate. In the accumulation model (Section 3.3.3) the shape of the strain loop has been captured by a tensorial deﬁnition of the amplitude (Section 2.5.2). Figure 3.88f presents results of cyclic triaxial tests with identical stresses but diﬀerent initial void ratios. The residual strain εacc after diﬀerent values of Nc has been normalized by the amplitude function fampl (Table 3.23) in order to consider slightly diﬀerent strain amplitudes and was plotted versus a mean value of the void ratio e¯. The increase of Dacc with increasing void ratio may be described by a hyperbolic function fe as given in Table 3.23. Figure 3.89a presents the dependence of Dacc on the average mean pressure av p . Interestingly, the intensity of accumulation increases with decreasing pav . The data of tests with diﬀerent stress ratios η av are plotted in Figure 3.89b. A normalized stress ratio Y¯ av ≈ η av /M (ϕc ) has been used on the xaxis with η = M (ϕc ) being the inclination of the CSL. The accumulation rate increases with increasing stress ratio. The stressdependence of Dacc may be captured by the functions fp and fY which are given in Table 3.23. Figure 3.89c contains the accumulation curves from the diﬀerent test series normalized by the functions f¯ampl , f¯e , fp , fY and fπ (Table 3.23). The curves for diﬀerent stress amplitudes, initial densities, average mean pressures and average stress ratios fall together into a band which can be approximated by the historiotropic function fN (Table 3.23). It consists of a logarithmic and a linear portion. The logarithmic portion is predominant up to Nc = 104 while the linear portion is necessary to describe the curves εacc (Nc ) for larger numbers of cycles. The large inﬂuence of a cyclic preloading is illustrated in Figure 3.89d. It presents the evolution of void ratio e with the number of cycles Nc in three cyclic triaxial tests with identical stresses but slightly diﬀerent initial values of e. Considering a state with identical void ratio and identical stress (as marked by the horizontal line in Figure 3.89d) the rate of compaction e˙ of a freshly pluviated sample (No. 1) is signiﬁcantly larger than the rate of a sample (No. 3) which was preloaded by 40,000 load cycles. Thus, the accumulation rate is signiﬁcantly reduced by a cyclic preloading. For this reason a prediction of accumulation with a highcycle model the knowledge of void ratio and stress alone is not suﬃcient. Information about the cyclic preloading of the soil is indispensable. Unfortunately, the cyclic preloading cannot be directly measured in situ. It has to be determined by correlations. Despite considerable eﬀorts a clear correlation of cyclic preloading with dynamic soil properties (e.g. P and Swave velocities) could not be established [845, 846]. A correlation
3 Deterioration of Materials and Structures
εacc / (fampl fe) [%]
a) 2.0
Nc = 1,000 Nc = 100 Nc = 20
1.5
b) 3.0
Nc = 100,000 Nc = 50,000 Nc = 10,000
all tests: ηav = 0.75, ζ = 0.3, ID0 = 0.61  0.69, f = 1 Hz
1.0
0.5
0 0
100
200
Nc = 100,000 Nc = 50,000 Nc = 10,000 Nc = 1,000 Nc = 100 Nc = 20
2.5
εacc / (fampl fe) [%]
204
2.0 1.5 1.0 0.5 0 0
300
0.2
tests on fampl tests on fp tests on fY tests on fe
Void ratio e []
εacc / (fampl fe fp fY fe fπ) [%]
d)
fN 0.4
0.8
1.0
0.65
1.2
[]
all tests: pav = 100 kPa, qav = 77 kPa, qampl = 55 kPa
0.64 despite identical void ratio and identical stress: 0.63
e2
e1
e1 > e 2 > e 3
e3
0.62
0.2 0 0 10
101
102
103
104
0.61
105
2 104
0
150
2 q
20 40
150
20
150
20 40
150
q 20
80
40 60
t
5 q60 150
20 40
t
6 q 80 60
80
150 t
105
t
4
80 60
8 104
2.0
t
3 q
6 104
all tests: pav = 200 kPa, ηav = 0.75, ID0 = 0.58  0.63, f = 0.25 Hz
2.4
60 40 80
εacc [%]
qampl = 60 80
1 q
4 104
Number of cycles Nc []
Number of cycles Nc []
e)
0.6
Average stress ratio Y
c)
0.6
0.4
av
Average mean pressure pav [kPa]
0.8
all tests: pav = 200 kPa, ζ = 0.3, ID0 = 0.57  0.67, f = 1 Hz
1.6 1.2 qampl [kPa] = 80 60 40 20
0.8 0.4
40 20 t
0
0
25,000
50,000
75,000
100,000
Number of cycles Nc []
Fig. 3.89. Results of drained cyclic triaxial tests: a) Dependence of Dacc on average mean pressure pav , b) on average stress ratio Y¯ av ≈ η av /Mc (ϕc ), c) on the number of cycles Nc and d) on cyclic preloading. e) Tests with packages of cycles with diﬀerent amplitudes
with the liquefaction resistance, however, could be formulated [844] but its practical application has still to be proven. A correlation of cyclic preloading with acoustic emissions seems to be rather insuﬃcient [581]. As an alternative, the cyclic preloading could be determined by cyclic test loadings in situ (some ideas are explained in [835]). In many practical problems the amplitude of the cycles is not constant but varies with the cycles. Such random cyclic loadings could be replaced by
Gravel Sand fine med. coarse fine
80
6
60
1.2
40
1.0
20 0 0.06
2
0.2
2 3 4 5
1
0.2 0.6 2 Grain diameter [mm]
6
3 5
0.1
0.8
205
Gravel Sand fine med. coarse fine 7 8 3
100 80 60 40 20
8
0 0.06
0.2 0.6 2 Grain diameter [mm]
6
0.6 7
0.4
4
after Nc = 105 0
Finer by weight [%]
1
100
εacc / fampl [%]
εacc / fampl [%]
0.3
Finer by weight [%]
3.2 Experiments
0.1
0.2
0.5
1
2
5
Mean grain diameter d50 [mm]
3
0.2
6
10
0
1
after Nc = 105 2
3
4
5
Uniformity coefficient Cu = d60/d10 []
Fig. 3.90. Inﬂuence of the grain size distribution curve on Dacc
packages of cycles each with a constant amplitude if the sequence of application would not aﬀect the residual strain that means if Miner’s rule [543] were applicable to soil. In order to examine the inﬂuence of the order of packages cyclic triaxial tests were performed [839, 835]. In each test four packages each with 25,000 cycles were applied. The amplitudes q ampl = 20, 40, 60 and 80 kPa were applied in diﬀerent sequences. Figure 3.89e presents the accumulation curves. Irrespectively of the sequence the residual strains at the end of the tests are quite similar. Thus, for a constant polarization of the cycles and as a ﬁrst approximation Miner’s rule can be assumed to be valid for sand. In cyclic triaxial tests with diﬀerent frequencies 0.05 ≤ f ≤ 2 Hz no inﬂuence of the loading frequency could be detected [835]. Thus, in this range the loading frequency does not need to be considered in a highcycle model. In [835] also the inﬂuence of a static preloading was studied and found small. The grain size distribution curve has also a signiﬁcant inﬂuence on the accumulation rate. In order to develop a simpliﬁed procedure for the determination of the material constants of the highcycle model presented in Section 3.3.3, approx. 200 cyclic triaxial tests have been performed on eight diﬀerent grain size distribution curves (Figure 3.85a) of a natural quartz sand. The results have been documented in detail in [841]. Figure 3.90 compares the strain remaining in the eight sands for similar test conditions (similar values of εampl , ID0 , σ av ). The accumulation rate increases with decreasing mean grain size d50 and grows signiﬁcantly with increasing coeﬃcient of uniformity Cu . In the accumulation model (Section 3.3.3) the inﬂuence of the grain size distribution curve has to be considered by diﬀerent sets of material constants which enter the f functions. Correlations of these constants with index properties (d50 , Cu , emin) are discussed in [841]. In contrast to drained cyclic tests the pore water pressure u accumulates in tests without drainage. Results of a typical test with an isotropic initial
206
3 Deterioration of Materials and Structures
a)
b)
500
Vertical strain ε1 [%]
Stresses σ3, u [kPa]
400 300 u 200
σ3 100 0
0
10 Nc,preload = 10
σ3 + u = const.
5
ID = 0.66 qampl = 45 kPa
5
0
1,000 2,000 3,000 4,000 5,000 6,000
Time [s]
Time [s]
d)
60 Nc = 142 43 44 45 46
40
60 47
40
20
q [kPa]
Deviatoric stress q [kPa]
c)
47 46 45 44 Nc = 43
0
10
1,000 2,000 3,000 4,000 5,000 6,000
ampl = 50 kPa qpreload
0
20 0 20
20
40
40 Nc = 46 45 44 43 42 141
60 10 8
6
4
2
0
2
60 4
Vertical strain ε1 [%]
6
8
10
0
20
40
60
80
100
p [kPa]
Fig. 3.91. Results of an undrained cyclic triaxial test (after a drained cyclic preloadampl , see [844]): a) excess pore pressure ing with Nc,preload cycles at amplitude qpreload accumulation u(t), b) vertical strain ε1 (t), c) stressstrain hysteresis, d) eﬀective stress path in the pqplane
stress are presented in Figure 3.91. The axial cyclic loading was applied stresscontrolled. The excess pore water pressure increased with each cycle (Figure 3.91a). When after some cycles the condition u ≈ −σ1 = −σ3 was approached (i.e. the eﬀective stress was σ ≈ 0, socalled ”initial liquefaction”), the strain amplitude εampl started to grow rapidly (Figure 3.91b). During the subse1 quent cycles the stressstrainhystereses (Figure 3.91c) showed no shearing resistance over a wide range of ε1 and the stress path in the pqplane followed a butterﬂylike curve (Figure 3.91d). After several such ”cyclic mobility loops” the specimens failed during triaxial extension. A ”full liquefaction” is often quantiﬁed with a double amplitude 2εampl above approx. 10 %. 1 The stiﬀness E of a highcycle model (Section 3.3.3) may be developed by comparing e.g. the accumulation of strain in drained cyclic triaxial tests and the relaxation of stress in undrained cyclic triaxial tests with similar initial conditions. From the rate of pore water pressure u˙ in undrained cyclic triaxial tests and the rate of volumetric strain accumulation Dvacc in drained cyclic triaxial tests one can derive the bulk modulus K = u/D ˙ vacc . A study of the stiﬀness E is documented in [841].
3.2 Experiments
207
undrained cycles package No. 2, Nc = 100 at σav = 0
380 340 0.64
60 80
0
40
80
p [kPa]
120
0.62
~ ~ ~ ~
~ ~
Δe1 = 0.010 Δe2 = 0.021
0.60 Δe3 = 0.024
0.58 0
500
1000
~ ~
40
~ ~
0 20
undrained cycles package No. 3, Nc = 900 at σav = 0
Recons. 3
420
20
Void ratio e []
q [kPa]
40
~ ~
u [kPa]
60
Recons. 2
Recons. 1
~ ~
460 80
~ ~
undrained cycles package No. 1, Nc = 0 at σav = 0
1500 3500
4000
4500 23000 23500
Time t [s]
Fig. 3.92. Change of void ratio Δe during reconsolidation after diﬀerent numbers of cycles in the liqeﬁed state (σ = 0)
An unsolved problem in connection with the postcyclic behaviour is depicted in Figure 3.92 [579]. It presents displacementcontrolled tests with three test phases. In the ﬁrst phase the specimen was reconsolidated directly after a zero eﬀective stress (σ = 0) was reached. In the second and third phase the specimen was monotonically reconsolidated after diﬀerent numbers of cycles at σ = 0. Evidently the compaction during reconsolidation increases with the number of cycles at σ = 0. Similar observations were made by Shamoto et al. [734]. A latent accumulation in the grain skeleton seem to take place during the undrained cycles and it becomes visible during reconsolidation. Future investigations on this phenomenon are necessary. 3.2.3 Structural Testing of Composite Structures of Steel and Concrete Authored by Gerhard Hanswille and Markus Porsch 3.2.3.1 General In recent decades as a result of the beneﬁts of combining the advantages of its components, steelconcrete composite beams have seen widespread use in buildings and bridges. The composite action of the components steel and concrete is realized by the shear connectors welded on the steel ﬂange. Because of its economic and fast application headed shear studs are the most common used type of shear connectors in steelconcrete composite constructions. Typical examples of applications of headed studs in composite bridges are given in Figure 3.93.
208
3 Deterioration of Materials and Structures
Fig. 3.93. Application of headed shear studs in composite bridges
Due to the high initial stiﬀness headed shear studs eﬀect nearly rigid composite action under service loads. At ultimate loads they allow for redistribution of longitudinal shear forces in the interface of steel and concrete due to their high ductility, which leads to robust composite structures. Especially in bridges and in industrial buildings with traﬃc actions or vibrations of machines these studs are subjected to highcycle fatigue loading. In order to ensure high lifetimes and high reliability indices over lifetime it must be made sure, that the stiﬀness properties and the shear resistance of the studs remain suﬃciently high. The behaviour of studs in solid concrete slabs under a static shear force is illustrated in Figure 3.94 [682, 686, 684]. At the beginning the shear force P acts mainly on the weld collar (component PW ). Under further increase of the load the concrete compressions at the weld collar lead to local destruction of the concrete in this region and thus to redistribution of the shear force into the stud shank. The stud shank is increasingly subjected to bending (component PB ) causing plastic deformations in its lower part. Due to the restraint of the stud head by the surrounding concrete further loading leads to tensile forces (Z) in the stud shank and a respective compressive force (D) in the concrete between the stud head and the ﬂange of the steel beam. The horizontal component of the tensile force of the stud causes component PZ . The compressive force in the concrete activates additional frictional forces in the interface between concrete and steel ﬂange, resulting in component PR . Because of the complex load deﬂection behaviour of headed studs embedded in solid slabs so far no design formula exists, which describes the ultimate shear resistance and the amount of each component by means of a mechanical model. The lifetime of cyclic loaded headed shear studs is eﬀected by the load range, by its load bearing capacity and by the peak load. In current German
3.2 Experiments
209
P PR P
D PZ
D Z PR
PB PR
P G
PZ
PB PW
PW G
Fig. 3.94. Loaddeﬂection behaviour of headed shear studs embedded in solid concrete slabs under static loading
design codes for bridges (DINFachbericht 104) [18], which is based on the design rules of the Eurocode 4 [22, 23] the peak load level under service loads is limited to 60 % of the design value of the shear resistance of a stud. Due to this limitation mainly the components PW and PB are activated at studs of cyclic loaded composite structures. As the component PB leads to bending in the stud shank and thus to tension stresses the magnitude of this component has a signiﬁcant inﬂuence on the development of cracks through the steel at the stud feet. In case of static loading even large bending moments can be resisted suﬃciently by plastiﬁcation of the shank cross section. Repeated loading causes a constant increase of the damage of the concrete in front of the stud, which leads to a steadily increase of the bending moments in the stud shank and to a gradual change of the stress state at the restraint during lifetime. Because of its costs and diﬃculties arising in the interpretation of the results of the fullscale beam tests, the evaluation of the behaviour of the shear studs generally takes place with standard pushout test specimen. Since 1960s various researchers conducted a great number of cyclic pushout tests under force control to determine the fatigue life of shear connectors. Some of them are given in [756, 513, 336, 681, 40, 591, 793, 687, 683]. As shown in Figure 3.95 by means of these investigations a fatigue strength curve for headed shear studs embedded in normal weight concrete was derived [685] based on the nominal stress concept. The scope of application is limited to studs in concrete ﬂanges under compressive stressing and to peak loads less than 60 % of the ultimate stud shear resistance. In regions subjected to tensile stressing additionally the interaction between the shear stresses Δτ and the tensile stresses Δσ in the steel ﬂange has to be considered [341, 594]. In the double logarithmic scale this model represents a linear relationship between the stress range Δτ in the stud shank and the number of cycles to failure N with the slope 1/m. It was taken as the basis for the design rules of
210
3 Deterioration of Materials and Structures
'WR (log) 1000
'W R
§N · ¨ c¸ ¨ N¸ © ¹
1m
'Wc N
§ 'W ¨ c ¨ 'W R ©
m
· ¸ N c ¸ ¹
test results: m = 8.658 Eurocode 4: m = 8
'Wcm = 110 N/mm²
100
5%fractile
'P
d
'P 'W
'Wck = 90 N/mm²
4 'P Sd2
N (log)
10 104
105
106 Nc = 2x106
107
Fig. 3.95. Fatigue strength curve for cyclic loaded headed shear studs according [685]
cyclic loaded headed shear studs in Eurocode 4 in association with the linear damage accumulation rule of PalmgrenMiner [611, 543] in order to allow for load sequence eﬀects. The magnitude of the value m (test results: m = 8.658) of the fatigue strength curve lies signiﬁcantly above the values of m for details in steel structures (m = 5 or m = 3). This results in considerable deviations from the real lifetime, if the shear forces are already calculated slightly inexactly. Thus, particularly in hogging moment regions due to the inﬂuence of the axial stiﬀness of the concrete slab on the magnitude of the longitudinal shear forces the eﬀect of the tension stiﬀening of the concrete between cracks, is of main interest [341]. In all before mentioned pushout tests the fatigue resistance was based on the total fracture of the studs but not on crack initiation being the main reason for the small slope of the fatigue strength curve. On this background one of the remarkable points in earlier investigations was that the repeated loading causes a reduction of the static strength of the shear studs not only at the end of their lifetime but within [593]. This indicates that the design concepts in the current codes [22, 23] does not describe suﬃciently the real behaviour, because the determination of the ultimate load bearing capacity and the fatigue resistance takes place with separate and independent veriﬁcations at the ultimate limit state and the fatigue limit state.
3.2 Experiments
211
slip G P
G
P number of load cycles N
R, E ultimate load bearing capacity
R, E P
R resistance G
E
fatigue resistance 'W
effect of actions
N
a)
lifetime
b)
td
fatigue life tm
Fig. 3.96. (a) Safety concept to determine the lifetime of composite structures subjected to high cycle loading according to present codes, (b) Actual inﬂuence of high cycle loading on lifetime
In Figure 3.96 the design concept of the present codes is compared with the actual inﬂuence of high cycle loading on the lifetime of composite structures. So far the deterioration in strength of stud shear connectors and the change of the deformation behaviour due to cyclic loading remain unconsidered. In general this may result in a reduction of the reliability index of the composite structure so that it may fall below the target values in codes. The eﬀect of the reduction of the static strength of the headed shear studs on the global load bearing capacity of a composite member depends on the magnitude of the loading and the capability of the redistribution of the shear forces. Further more it has to be considered that in ﬂanges under tension the damage process at the stud welds can lead to a damage of the ﬂange, on which the studs are welded. Except the test conducted by Oehlers [592] there were no further tests where the reduced static strength after highcycle preloading was investigated. In the light of the information gained from previous researches a comprehensive program of more than 90 standard EC4pushout test specimens and two fullscale beam tests were developed [349, 352, 347] which consider the crack propagation through the stud foot and the local damage of concrete surrounding the studs as relevant consequences of high cycle loading. In detail the research on the pushout tests were mainly focused on
212
• • • • • • • • •
3 Deterioration of Materials and Structures
the static strength Pu,0 of headed shear studs without any predamage the number of cycles to failure Nf taking into account the relevant (loading) parameters given as static strength Pu,0 , peak load Pmax and load range ΔP the reduced static strength Pu,N of headed shear studs after high cycle loading crack initiation and crack propagation through the stud feet as the main reason for the decrease of the static strength over lifetime load sequence eﬀects regarding nonlinearities caused by cracking of the steel and crushing of the concrete static load deﬂection behaviour of undamaged and damaged shear studs as well as the cyclic load deﬂection behaviour the ductility behaviour of studs after predamage eﬀect of the mode control on the results of cyclic loaded pushout tests (force control versus displacement control) eﬀect of low temperature on the load deﬂection behaviour of damaged headed shear studs
Considering the interaction between the local damage and the behaviour of the global structure, these research results were subsequently taken as the basis to simulate the cyclic behaviour of composite beams by means of damage accumulation method. In order to verify the theoretical models derived from the pushout tests additionally two fullscale beam tests were performed, in which the eﬀect of the deterioration of the properties of the interface between steel and concrete on the loaddeﬂection behaviour, on the redistribution of the inner forces and on the reduced static strength after high cyclic loading were investigated. Regarding typical loading conditions in composite structures it was distinguished between a beam subjected to a sagging moment and a beam subjected to a hogging moment. 3.2.3.2 Basic Tests for the Fatigue Resistance of Shear Connectors 3.2.3.2.1 Test Program The experimental program of the pushout tests reported in the following chapters consists of a total of 9 series (S1  S6, S9, S11 and S13) [352, 353]. The ﬁrst four series S1S4 deal with constant amplitude tests where the eﬀect of unidirectional cyclic loading on the static strength and the fatigue life of the pushout specimen were investigated with the varying loading parameters peak load Pmax and the loading range ΔP . In each series initially three static tests were performed to determine the mean value of the ultimate static load P u,0 of the pushout specimen. The mean value of the ultimate static load represents the reference parameter for the relative values of loading parameter required for cyclic tests. Using the relative loading parameters three load controlled cyclic tests were performed to determine the mean fatigue life N f
3.2 Experiments
213
Table 3.6. Summary of the single level tests
series
'P Pu ,0
Pmax Pu ,0
S1 S2 S3 S4 S5E
0.20 0.25 0.25 0.20 0.25
0.44 0.71 0.44 0.71 0.30
Pu ,0
Nf
3 3 3 3 3
3 3 3 3 1
Pu,0
P
number of tests N ~ 0.3 N f 3 3 3 3 1
Pu,N
Pmax,4
Pmax,2 Pmax,1
3 3 3 3 1
Pu,0
P
Pu,N1 + N2 = Pmax,2
N ~ 0.7 N f
1 + N2 + N3 + N4
= Pmax,4
Pmax,1
'P
'P N1
N2
'P
N1 N2 N3
N4
load cycles load cycles
'P = constant
a)
Gu,N
1 + N2
G
'P = constant
b)
G
Gu,N
1 + N2 + N3 + N4
Fig. 3.97. Tests with multiple blocks of loading: (a) two blocks, (b) four blocks (increasing peak load)
of the pushout specimen. Subsequently six cyclic tests were conducted for approximately 30 and 70 percent of the mean fatigue life N f . After reaching the corresponding number of cycles each of these six test specimens was statically loaded to failure under displacement control to obtain the reduced static strength after high cycle preloading. The chosen loading parameters and number of performed tests for the ﬁrst four series are summarized in Table 3.6. Based on the results of the constant amplitude tests in the series S5 and S6 tests with two and four blocks loading sequence were performed in order to check the validity of the linear damage accumulation according to the PalmgrenMiner rule in the case of headed shear studs embedded in solid slabs. In these tests the peak load was increased or decreased subsequently while the loading range was held constant (see Figure 3.97). Like the constant amplitude tests the series S5 and S6 initiate with shorttime static tests to determine the ultimate static load. During the tests with multiple blocks of loading the necessity of performing further constant amplitude tests with not investigated low peak load (Pmax / P u,0 = 0.3) was arisen.
214
3 Deterioration of Materials and Structures
Table 3.7. Summary of the tests with multiple blocks of loading series
number of tests
S5  2 S5  3 S5  4 S5  6 S6  3 S6  4
3 1 4 4 3 3
'P Pu ,0
0.25
0.20
Pmax Pu ,0 block 1 0.71 0.44 0.44 0.30 0.44 0.74
block 2 0.44 0.71 0.30 0.44 0.54 0.64
block 3 0.64 0.54
block 4 0.74 0.44
Thus, 3 complementary constant amplitude cyclic tests were additionally performed in these series, referred as series S5E. The chosen loading parameters and number of performed tests for tests with multiple blocks of loading are summarized in Table 3.7. The loading parameters of series S5E are also listed in Table 3.6. Regarding the simulation of the cyclic behaviour of composite beams by means of the results gained from force controlled pushout tests, in series S9 nine tests were performed in which the inﬂuence of the control mode was investigated (Figure 3.98). After experimental determination of the mean static strength P u,0 (test specimen S9 1b and S9 1c) three cyclic displacement controlled pushout tests with constant values of the maximum slip smax and the slip range Δs (initial peak load approximately 0.65 P u,0 ) subjected to 5 × 106 load cycles were conducted (S9 4). The resulting loading history of the varying peak load Pmax (N) and the load range ΔP (N) were classiﬁed in eight blocks with constant loading parameters and subsequently taken as input values for four additional cyclic force controlled pushout tests. The number of load cycles Ni in a successive block was chosen twice the number of load cycles in the former block. With the exception of test specimen S9 5d after each cyclic loading phase the reduced static strength was experimentally determined at a temperature of T = 20◦ C. In order to investigate the inﬂuence of cracks through the stud feet on both, the reduced static strength and the ductility test specimen S9 5d was tested after cooling down to T = −40◦ C. During the cooling process and subsequently in the test rig, the temperature of the concrete slabs were measured permanently by means of 2 temperature sensors PT 100, each embedded in the middle of one slab. By using thermal insulation it could be realized, that the core temperature of the concrete did not fall below a value of T = −38◦ C at the end of the static test procedure. Due to the notch eﬀect of the stud welds in all test specimens in series S1 to S9 subjected to fatigue loading damage occurred at the stud feet in form of cracks through the steel leading to a reduction of the static shear resistance of the shear studs. On the background of decoupling of the ultimate limit state and the fatigue limit state in test series S11 and S13 7 additional
3.2 Experiments
215
P [kN] 1000
7 tests (S9_4, S9_5)
load applied displacement controlled
temperature sensor PT 100 's=0.08mm
smax
800
smin
P
P
600
s 400 Pmax
'P(N)
200
Pmin N [x 106]
load applied force controlled
0
2
1 specimen parameter Pu,0 S9_1 bc 

S9_4 c–e
's [mm] block
S9_5 a–d

4
3
5
static strength T = 20°C
S9_5d during the cooling process in the climate chamber and in the test rig
cyclic loading  N = 5x 106 0.08 (initial load ~ 0.65 Pu,0 ) (displacement control)
Pmax Pu,0
1 0.54
2 0.51
3 0.48
4 0.44
5 0.41
6 0.36
7 0.27
8 0.21
'P Pu,0
0.23
0.23
0.22
0.21
0.20
0.17
0.15
0.12
load cycles 'N1 = 2 'N1 2 'N2 2 'N3 2 'N4 2 'N5 2 'N6 ~2 'N7 19608 reduced static strength 
S9_4 c–e S9_5 a–c
Pu,N
T = 20°C
S9_5 d
Pu,N
T = 40°C
Fig. 3.98. Tests to compare the eﬀect of the mode control  force control vs. displacement control  and the eﬀect of low temperature
tests in accordance with Figure 3.99 were carried out, in which the eﬀect of extreme low peak loads on the crack initiation and the crack propagation at the stud feet was investigated. Due to the low cyclic loading parameters the cyclic loading phases were ﬁnished in each case after a certain number of load cycles, before determining the residual static strength. In most cases these numbers were only a small fraction of the mean number of cycles to failure according to the current design concept [685], taking into account a slope m = 8. 3.2.3.2.2 Test Specimens The specimen used in the pushout tests consists of a 650 mm high HEB260 proﬁle and two 650 mm high, 600 mm wide and 150 mm thick concrete slabs. The slabs are connected to the steel beam by means of four headed shear studs of 22 mm diameter welded on each side of the beam. The weld collars complied with the requirements of EN ISO 13918 [10]. The mean height of the weld collars was 8.4 mm and the mean diameter 28.9 mm. The height of the welded stud was 125 mm. For casting the steel beams were cut into two halves and the concrete slabs were casted horizontally. The steel ﬂanges were greased
216
3 Deterioration of Materials and Structures
Pu,0
P
Pu,N
test
Pmax / Pu,0 'P / Pu,0
Pmax
'P
[kN]
[kN]
S111a
Pmax 'P N
G
N / Nf,cal
(m = 8.0)
[]
[]
[]
Nf,cal
N
[]
[]
static strength Pu,0 (valid for S11 and S13)
S114a
240
200
0.16
0.13
7.00 x 106
1.22 x 108
1 / 17.4
S114b
360
320
0.24
0.21
4.27 x 106
2.86 x 106
1 / 0.67
S114c
120
80
0.08
0.05
9.41 x 106
1.87 x 1011 1 / 19872
S132a
250
200
0.16
0.13
3.80 x 106
1.22 x 108
1 / 32.1
S132b
350
200
0.23
0.13
4.10 x 106
1.22 x 108
1 / 29.8
S132c
450
200
0.30
0.13
4.08 x 106
1.22 x 108
1 / 29.9
Gu,0 Gu,N
Fig. 3.99. Tests to investigate the eﬀect on the duration of the crack initiation phase and crack growth velocity due to very low cyclic loads (Nf,cal according [685])
section BB
section CC
B
mortar
B
L120x120x12
mortar
130
150
A
B
L120x120x12
A
200
50
100
50
820
560
260
A
150
730
650
C
12 10
C
stud 22/125
108
C
HEB 260
30 20 200
C
HEB 260 (S235 J2G3)
250
150
50
HEB 260
80
A
10 12
30
108
B
head plate
130
section AA
200
600
Fig. 3.100. Details of the pushout test specimen
prior to casting to remove friction between the concrete and steel. The two halves were subsequently welded together. This test specimen complies with the standard pushout specimen according to Eurocode 4 with an exception of lateral restraints of the concrete slabs at the specimen bottom. The lateral restraints avoid especially in the lower row of shear studs the introduction of additional tensile forces resulting from the moment of eccentricity and enable a better simulation of the real behaviour in composite beams. Details of the pushout specimens are given in Figure 3.100. 3.2.3.2.3 Test Setup and Loading Procedure Generally cyclic and monotonic loading was applied by a 2500 kN servo hydraulic actuator. Only in particular cases, for instance in order to determine eﬀects of relaxation on the load bearing capacity, additionally a 2000 kN and a 10000 kN servo hydraulic actuator were used to apply static loading. The load
3.2 Experiments
2000 kN
2500 kN
217
10000 kN
Fig. 3.101. Servo hydraulic actuators
was introduced into the steel beam by means of a head plate welded on the cross section of the beam and an additional centring bar. All actuators were controlled by an Instron 8800 controller which allowed load and displacement control. For the cycling testing sinusoidal control waveforms were utilized. With the exception of test specimen S9 4c  S9 4e displacement control was used for the monotonic tests and load control was used for the cyclic phases. The actuators used in the experiments are shown in Figure 3.101. The monotonic tests were conducted at a displacement rate of 0.004 mm/sec. The time taken to reach the ultimate load was typically of the order of 50 minutes. After reaching the ultimate load the displacement rate was increased up to 0.008 mm/sec. Cyclic tests were conducted with a load frequency of 3  5.5 Hz. In order to collect data about the stiﬀness and plastic deformation, cyclic tests were held after speciﬁc number of cycles and the specimens were released and reloaded monotonically. During the tests the time, load from the actuator load cell, ram displacement from the builtin transducer in the actuator, vertical slip between the concrete slabs and steel beam (WV1a, WV1b, WV2a and WV2b) and horizontal displacement of the slabs (WHa and WHb) were measured (see Figure 3.102). The ram displacement included movement due to the compliance of the test rig, and therefore it was not used in any subsequent data analysis. In the case of specimen S9 4c  S9 4e also the cyclic test phase was performed under displacement control. For this purpose the measuring equipment was supplemented by two additional LVDT’s (WA20/1a and WA20/1b) in parallel connection placed directly above the before mentioned transducer (one at each ﬂange) in the interface between the steel beam and the concrete slabs in order to control the actuator. 3.2.3.2.4 Material Properties At various intervals during the curing process at 28 days as well as at the beginning, in the middle and at the end of each series cylinder compression
218
3 Deterioration of Materials and Structures
Section AA
Section CC
Section BB B
A
side b
side a
B side b
side a
WV20/1a WA20/1b
WV2a
WA20/1a
WV1b WV2b
WV2b
WV1a
80 45 45 80
WHa
WV2a WHb
WV2b WHa
WHb
B
WV1b
side b
A
WV20/1b
B
Fig. 3.102. Position of transducers
Table 3.8. Mean values of material properties of concrete according to EN 2061 [12]
property fc [N/mm2] Ecm [N/mm2]
series S1
S2
S3
44.2 –
42.8 –
53.9 –
51.7
45.0
56.2
36400
33800
39000
S4
S5
S6
43.4
42.9
45.8
33900
33050
33700
S9 47.2 – 54.8 32250
S11, S13 38.5 30635
tests were carried out to determine the compressive cylinder strength and elastic stiﬀness of the concrete. Standard cylinders of 150 mm in diameter and 300 mm in length were used. Table 3.8 presents the interval of the mean concrete strength and the mean modulus of elasticity (taken as secant modulus Ecm according to EC 2[33]) associated with the test begin for each series according to EN 2061 [12]. In test series S1  S6 on one side and in series S9, S11 and S13 otherwise structural steel beams, headed shear studs and reinforcing bars were from the same batch. Structural steel beams of HEB 260 section with the material quality S235 J2G3 were used in each test. Stud shear connectors, which were welded automatically onto the steel beam ﬂange, had a material quality of S235 J2G3+C450. As reinforcing steel standard deformed bars with diameters of 10 mm and 12 mm were used in the concrete slabs. In order to obtain detailed data about material properties tensile tests of all steel members were conducted according to the requirements of DIN EN 10002 [15]. The results of the corresponding values of each yield strength, tensile strength and modulus of elasticity are summarized in Table 3.9.
3.2 Experiments
219
Table 3.9. Mean values of material properties of steel members member yield strength, tensile strength and modulus of elasticity
steel beams
headed studs
reinforcement Ø 10
reinforcement Ø 12
property
S1 – S6
S9, S11, S13
fy fu Ea fy fu Es Rp0.2 Rm Es Rp0.2 Rm Es
337 448 210000 440 528 215000 549 606 197800 501 561 204500
413 516 204000 448 538 207800 543 586 192300 545 588 196500
[N/mm2]
3.2.3.2.5 Results of the PushOut Tests 3.2.3.2.5.1 General In the following the main results of the pushout tests of series S1  S6, S9, S11 and S13 are reported. They mainly concern load bearing capacities and number of load cycles. Further information especially results regarding simulation of cyclic loaded beams are presented in the associated Chapter 3.3.4. 3.2.3.2.5.2 Results of the Constant Amplitude Tests (S1  S4, S5E) Table 3.10 shows the results of the static strength, the fatigue life N f and the reduced static strength after high cycle preloading for series S1  S5E. The limit state of fatigue is given, when the reduced strength has reached the value of the peak load. Because of diﬀerent static strength within each test Table 3.10. Average test results per stud series
Pu ,0
Nf
[]
[kN]
[]
S1 S2 S3
205 184 201
Pu ,N1
Pu ,N1 Pu ,0
N1 Nf
Pu ,N 2
Pu ,N 2 Pu ,0
N2 Nf
[kN]
[]
[]
[kN]
[]
[]
6.2x10
6
154
0.75
0.32
129
0.63
0.90
1.2x10
6
174
0.95
0.32
154
0.84
0.70
5.1x10
6
133
0.66
0.24
123
0.61
0.69
6
181
1.00
0.29
156
0.86
0.72
111
0.59
0.19
114
0.60
0.73
S4
181
3.5x10
S5E
189
6.4x106
220
3 Deterioration of Materials and Structures
series the absolute values of Pmax and ΔP diﬀer slightly. All data given in this Table represent generally the mean values of three comparable tests. The strength data are based on short time behaviour. From the statically loaded pushout test specimens without any predamage it could be found out, that if the position of the hydraulic actuator is held constant during a test phase the test specimen typically relaxes and withdraws the applied loading. Near to failure the loss of loading amounts up to 10% of the applied load. On this background it is necessary to reduce the values given in Table 3.10 by 10% [345] when relaxation must be taken into account. In contrast to series S2 and S4 the low peak loads in series S1, S3 and S5E led to very high fatigue lives N f . This shows the signiﬁcant inﬂuence of the relative peak load Pmax / P u,0 on the lifetime. On the background of a redistribution of inner forces due to a deterioration of the mechanical properties of the connectors under service loads not only the reduction of the static strength of the headed shear studs is of main interest but also the development of the deformation under repeated loads. In the case of high cyclic loading the loaddeformation behaviour is characterized by an increasing plastic slip δ and a decreasing elastic stiﬀness Kel . In Figure 3.103 the inelastic displacement δ i related to the plastic slip in the ﬁrst cycle δ 1 in the steelconcrete interface is plotted against the number of cycles over the fatigue life N i / N f for series S1  S4. The beginning and the end of the lifetime are associated with a steep increase in the plastic slip with the number of cycles while in the remaining part of the lifetime a nearly linear increase of the plastic slip occurs with the number of cycles. Due to the
series S1
Gi G1
20
series S2
Gi G1
25
25 G1 0.06 mm
G1 0.52 mm
20
15
15
10
10 5
5 Ni Nf 0
0 0.0
0.2
0.4
1.0
0.8
1.0
Ni Nf
0.0
0.2
0.4
1.0
1.0
0.8
Pmax
P 25 20
Gi G1
series S3
G1 0.06 mm
25 20
15
15
10
10
5 0 0.0
Gi G1
series S4 c w,
G1 0.55 mm
w,1
Kel,1 G1
d0
Kel,i
G
Gi
5 Ni Nf
0.2
0.4
1.0
0.8
1.0
0 0.0
Ni Nf
0.2
0.4
1.0
0.8
1.0
Fig. 3.103. Development of plastic slip over the fatigue life in series S1  S4
3.2 Experiments
Pu / Pu,0
221
Pu,0
P
Pu VX
1.0
0.03
VX
0 .03
Pmax
2 0.8
0 .09
Pmin
'P G
N
1 3
0.6
VX
0 .10
5E Stages of the strength reduction: I no reduction II non linear  declining III linear (stable) IV non linear  progressive
0.4
0.2
I
0
VX
4
0
static tests
3
4
5E
ǻP Pu,0
series
0.20 0.25
0.25
0.20
0.25
Pmax Pu,0
0.44 0.71
0.44
0.71
0.30
N f (x106)
6.2
5.1
3.5
6.4
2
1.2
III
II
0.2
1
0.4
0.6
tests with cyclic preloading
IV
0.8
N / Nf 1.0
fatigue tests
Fig. 3.104. Decrease of static strength vs. lifetime due to high cycle loading
increasing compliance of headed shear studs with higher load levels the mean value of the initial plastic slip δ 1 in the ﬁrst cycle is in the test series with high peak loads approximately 8 times greater than in series with low peak loads. The inﬂuence of the cyclic loading becomes evident, when the static strengths are plotted versus number of load cycles. This is shown in Figure 3.104, where the results are related to the mean static strength and to the mean fatigue life of each series, respectively. Especially in series S1, S3 and S5E with low peak loads the rapid decrease of the static strength within the ﬁrst 20 percent of the fatigue life is noteworthy. The reduction of the static strength over the lifetime is considered in four stages. In stage I there is no signiﬁcant damage in form of cracks through the stud feet and therefore no noticeable reduction of the shear resistance. In stage II and IV in each case a more or less nonlinear reduction of the static strength could be observed. In the case of lower peak loads the decrease is disproportionately high. The range within stage II and stage IV shows a linear reduction of the static strength with the number of cycles. In the cases of low peak loads the decrease of static strength is disproportionately low. In Figure 3.104 the coeﬃcient of variation Vx of the static strengths gained from three similar tests is marked out exemplarily for the series S4. The scatter of the results increases with the degradation of the strength of the shear studs. The
222
3 Deterioration of Materials and Structures
Table 3.11. Loading parameters and results of the tests with two blocks of loading (series S5) block 1
test
block 2
N1 + N2
Pmax,1
'P
N1
Pmax,2
'P
N2
[]
[kN]
[kN]
[ u106 ]
[kN]
[kN]
[ u106 ]
[ u106 ]
S52a
133
0.204
83
0.792
0.996
S52c
133
0.198
83
1.440
1.638
S53a
83
1.099
133

1.099
S54a
83
0.473
56
1.365
1.838
S54b
83
0.517
56
0.772
1.289
S54c
83
0.544
56
S54d
83
0.542
S56a
56
S56b S56c S56d
47
47
0.735
1.279
56
3.396
3.938
0.537
83
5.821
6.358
56
1.223
83
0.761
1.984
56
1.295
83
1.744
3.039
56
1.277
83
3.206
4.483
comparison of the series with the same related load range makes clear that the static strength and the fatigue life are aﬀected not only from the load range but also from the peak load of the cyclic loading. Furthermore it is also obvious that for the same amount of change the eﬀect of the load range on the static strength is much greater than of the peak load. 3.2.3.2.6 Results of the Tests with Multiple Blocks of Loading (S5 and S6) The results of the two and four blocks loading sequence (see Figure 3.97) are given in Table 3.11 and 3.12 respectively. The mean value of the reference Table 3.12. Loading parameters and results of the tests with four blocks of loading (series S6) test
'P
block 1 Pmax,1
block 2
N1
Pmax,2 6
block 3
N2
Pmax,3 6
4
block 4
N3
Pmax,4 6
¦ Ni
N4
i 1
6
[]
[kN]
[kN]
[ u10 ]
[kN]
[ u10 ]
[kN]
[ u10 ]
[kN]
[ u10 ]
[ u106 ]
S63a
38
83
0.756
101
0.768
120
0.770
139
0.868
3.162
S63b
38
83
0.765
101
0.804
120
0.785
139
0.324
2.678
S63c
38
83
0.754
101
0.759
120
0.750
139
0.449
2.712
S64a
38
139
0.550
120
0.763
101
0.754
83
0.583
2.650
S64b
38
139
0.550
120
0.758
101
0.750
83
0.756
2.815
S64c
38
139
0.540
120
0.753
101
0.753
83
1.208
3.254
3.2 Experiments
223
ultimate static strength P u,0 of the shear studs is 186 kN for the tests with two blocks of loading in series S5 and 196 kN for the tests with four blocks of loading in series S6. It is to be noticed that in tests with multiple blocks of loading the failure of the shear studs can occur on one hand during the cyclic loading by the decrease of static strength to the peak load and on the other hand during switching to the next block with a higher peak load by exceeding the reduced static strength. A typical example for the last case occurred in the test S5 3a during switching to the second block with a peak load of 133 kN by exceeding the reduced static strength of 124 kN per stud. 3.2.3.2.7 Results of the Tests Regarding the Mode Control and the Eﬀect of Low Temperature (Series S9) The results of the tests of series S9 are given in Table 3.13, the corresponding loading histories are shown in Table 3.14 and Table 3.15. The mean value Table 3.13. Average test results per stud in series S9
N
Pu,0
Pu,N
T
[]
[kN]
[kN]
°C
S9_1b

197

S9_1c

208

S9_4c
5019542

112
S9_4d
4966493

121
S9_4e
5031757

123
S9_5a
4964001

119
S9_5b
4999716

117
S9_5c
5457200

111
S9_5d
5000000

150
test
20
40
Table 3.14. Measured mean values of the peak load and the load range at discrete number of load cycles in tests S9 4 test S9_4ac
test S9_4ac
Pmax,1
'P 1
N1
Pmax,2
'P 2
[kN] 873
N2
Pmax,3
'P3
[kN]
[]
[kN]
375
9804
822
N3
Pmax,4
'P 4
[kN]
[]
[kN]
366
39216
777
[kN]
[]
[kN]
[kN]
[]
356
98040
719
340
215688
N4
Pmax,5
'P 5
N5
Pmax,6
'P 6
N6
Pmax,7
'P7
N7
Pmax,8
'P 8
N8
[kN]
[kN]
[]
[kN]
[kN]
[]
[kN]
[kN]
[]
[kN]
[kN]
[]
660
318
450984
584
279
921576
443
236
1862760
339
190
3745108
224
3 Deterioration of Materials and Structures
Table 3.15. Loading parameters and block lengths in tests S9 5 cyclic phase 1
cyclic phase 2
cyclic phase 3
cyclic phase 4
test 
Pmax,1
'P 1
N1
Pmax,2
'P 2
N2
Pmax,3
'P3
N3
Pmax,4
'P 4
N4
[kN]
[kN]
[]
[kN]
[kN]
[]
[kN]
[kN]
[]
[kN]
[kN]
[]
S9_5a
19607
41408
76683
S9_5b
20209
37928
79566
S9_5c
873
375
S9_5d
19609
822
366
19616 cyclic phase 5
39209
777
356
39221 cyclic phase 6
78437
156421 719
340
78419 cyclic phase 7
156024 157160 156922
cyclic phase 8
test 
Pmax,5
'P 5
N5
Pmax,6
'P 6
N6
Pmax,7
'P7
N7
Pmax,8
'P 8
N8
[kN]
[kN]
[]
[kN]
[kN]
[]
[kN]
[kN]
[]
[kN]
[kN]
[]
S9_5a
313724
627452
1255227
2473479
S9_5b
313433
627744
1259583
2505229
S9_5c S9_5d
660
318
313429 313949
584
279
627452 627167
443
236
1257995 1256006
339
190
2963909 2508700
of the reference ultimate strength P u,0 of the shear studs is 202 kN. After 5 × 106 load cycles the load bearing capacity of a stud decreases to a mean value P u,N of 118 kN due to the displacement controlled high cycle loading and to an only slightly diﬀerent mean value P u,N of 116 kN in the case of the force controlled cycle loading phase, respectively. As it can be seen from the test results of specimen S9 5d the lower temperature causes a remarkable increase of the reduced static strength compared to tests at a temperature of T = 20◦ C. Metallurgical investigation revealed that due to the loading history nearly the same size of crack lengths at the stud feet as in all other cases of series S9 were induced. In order to ﬁnd the main reason for the higher reduced static strength further concrete cylinder compression tests of the same batch were carried out to determine the material properties of the concrete at a temperature T = −40◦ C. In these tests the compressive cylinder strength and the modulus of elasticity (taken as secant modulus) of specimens stored according to the requirements of EN 206 [12] were determined as fc = 67 N/mm2 and Ecm = 39250 N/mm2 . The cyclic loading parameters in the phases 1 to 8 of the force controlled tests were determined on the basis of the loading history of the displacement controlled tests. As shown in Table 3.14 and Table 3.15 for this purpose the measured values Pmax,i and ΔPi (i = 1 to 8) at 8 discrete numbers of load cycles of the displacement controlled tests were taken as the mean values for the corresponding blocks of loading for the force controlled tests.
3.2 Experiments
225
Table 3.16. Average test results per stud in series S11 and S13
N
Pu,0
Pu,N
[]
[kN]
[kN]
S11_1a

190

S11_4a
7000600

197
S11_4b
4272816

157
S11_4c
9416000

205
S13_2a
3801147

202
S13_2b
4100300

206
S13_2c
4086509

189
test
3.2.3.2.8 Results of the Tests Regarding Crack Initiation and Crack Propagation in the Case of Low Peak Loads (Series S11 and S13) The results of the tests of series S11 and S13 are given in Table 3.16. The test specimens of both test series were casted at the same time using the same concrete mix, so that the reference ultimate strength Pu,0 of test S11 1a is also valid for test series S13. Although in most cases the static strength after high cyclic preloading were in the same order or even higher than the reference static strength without predamage Pu,0 the stud welds showed partly considerable fatigue fracture areas due to crack propagation during the cycling loading phase. 3.2.3.3 Fatigue Tests of FullScale Composite Beams 3.2.3.3.1 General Regarding the transition of results gained from pushout tests to the global behaviour of composite structures tests of composite beam are an essential part of experimental research. Within the framework of the project ”Modelling of damage mechanism to describe the fatigue life of composite steelconcrete structures”[348, 350] the test program was extended by 2 fullscale composite beams subjected to unidirectional high cyclic loading. In order to investigate the interaction between the local damage at the studs and the global behaviour of beams at the serviceability limit state and at the ultimate limit state these tests were mainly focused on • •
the increase of vertical deﬂections, the development of the elastic stiﬀness of the composite beams,
226
• • • •
3 Deterioration of Materials and Structures
the development of the plastic slip in the interfaces between steel and concrete, the crack propagation at the stud feet and the weld collars, the redistribution of the sectional forces of the concrete slabs and the steel cross sections, the reduced static strength of each beam after high cyclic preloading.
3.2.3.3.2 Test Program The test program consists of 2 simply supported beams, each subjected to a point load at midspan causing a sagging moment in the case of beam VT1 and a hogging moment in the case of beam VT2. As shown in Figure 3.105 the beams were loaded by 1.38 × 106 and 2.10 × 106 load cycles before testing their reduced static strengths. This procedure is in accordance with the principles of the constant amplitude tests of series S1S4 of the test program of the pushout tests, in which also the eﬀect of the high cyclic loading on the static strength was investigated. The cyclic loading phases should lead to noticeable reductions of the ultimate load bearing capacities but in no case to a complete
P
loading parameters
Pu,0 Pu,N 0.9 Pu,N
Gu2
Gu1
Pmax 'P
test
Pmax
'P
[]
[kN]
[kN]
[]
VT1
450
265
1.37x106
VT2
250
100
2.10x106
1. unloading 1. reloading
P N
N
VT1 and VT2 during testing
6m G Gu Gu,0
0 Pmax cyc G res Gcyc G LC Pmax , lc N res G ini
54 studs HEA300
load distribution plates (280x280) placed on mortar
b/h = 1500/150
VT1 VT1
(rb) HEA300
roller bearing (rb) b/h = 1500/150
0 G LC res
54 studs
load distribution plates (280x280)
(rb) 300
3000
3000
[mm]
VT2
(rb) 300
VT2
Fig. 3.105. Test programme and loading parameters of the composite beam tests VT1 and VT2
3.2 Experiments
227
fatigue shear failure of all studs in order to obtain clearly deﬁned damage states between states with nearly no damage and states, in which the concrete slabs makes no contribution to ultimate load bearing capacity. During the cyclic loading phase the test beams were released and reloaded monotonically at speciﬁc number of load cycles, in order to collect data about accumulated plastic deformations and irreversible strains. 3.2.3.4 Test Specimen The specimens consist of a 6600 mm long HEA 300 proﬁle and a likewise 6600 mm long, 1500 mm wide and a 150 mm thick concrete slab. In both cases the slabs were connected to the steel beam by means of 54 studs (in 2 rows with 27 studs per row) of 22 mm diameter and a height of 125 mm after welding. The weld collars complied with the requirements of EN ISO 13918 [10]. The mean height as well as the mean diameter of the weld collars were in the same magnitude as in the pushout tests reported in Chapter 3.2.3.2. Prior to casting the steel ﬂanges were greased in order to remove friction in the interface between steel and concrete. Moreover thin sliding layers composed of steel plates and PTFEfoils were placed at midspan (VT1) and at each support (VT2) in order to avoid additional restraints caused by the introduction of the vertical forces directly into the concrete slabs at these sections. The longitudinal reinforcement was chosen to 18 bars with diameters of 12 mm (VT1) and 30 bars with diameters of 16 mm (VT2). As transversal reinforcement in both cases 52 closed stirrups with diameters of 12 mm were used. Like bridges the beams were casted in horizontal position with the concrete slabs at the upper side. Details of the test beams can be taken from Figure 3.106 and Figure 3.107. 3.2.3.5 Test Setup Cyclic and monotonic loading was applied by a 2000 kN servo hydraulic actuator, controlled by an Instron 8800 controller. The load from the actuator was introduced by means of 4 load distribution plates, placed directly on the steel ﬂange (VT1) and on a mortar layer (VT2), respectively, at midspan of each test beam. The peak load and the load range during the cyclic loading phase was applied under force control with a frequency of 0.4 Hz. For all other loadings  ﬁrst loading and ﬁrst release, releases and reloadings during the cyclic loading phase and loading up to failure  displacement control was used. The time taken to release and to reload the test beams was typically of the order of 8 (VT1) and 4 (VT2) minutes. Regarding the evaluation of the tests extensive monitoring systems consisting of 62 measuring points at each beam according to Figure 3.108 were set up. By means of these systems data of the vertical deformations, of the slip
3 Deterioration of Materials and Structures
125
146
2
290
10
18.8 12.5 656
12.5
10
18.8 5 6 x 18
18 3
material properties: reinforcement: S500 structural steel: S460 sliding layer: PTFE (greased), S235
concrete: C35/45
8.6 cm
[cm] 15
3 18 [mm]
2 11 2
75 150 75
1 18 Ø 10 – spacing 18 cm – length 693.2 cm
47 x 12.5
concrete slab – reinforcement (sliding layer not shown)
6600
300
2650
700
2650
HEA 300
300
concrete slab – 15/150/660 [cm]
5
11 52 Ø 12 – spacing 12.5 and 18.8 cm – length 334 cm (alternately) 8.6
262x140x10 (stiffener)
100
sliding plate (700x300) on PTFEfoil (each 1mm thick – PTFE double sided greased – cutout Ø 40 mm)
200 54 headed studs Ø22  h/d = 125mm / 22mm (1a 27a, 1b27b) 200
250
22 x 250
11 4 1 side b
side a
steel beam – headed studs – sliding layer
17
24
250
27
100
290
10
11
228
studs: S235 J2G3 + C450
Fig. 3.106. Details of test beam VT1
in the interfaces between steel and concrete and data of the strain states of the steel beams and the concrete slabs were collected continuously during all test phases.
125
146
2
290
10
18.8 12.5
675
concrete slab – 15/150/660 [cm]
5
11 52 Ø 12 – spacing 12.5 and 18.8 cm – length 334 cm (alternately) 8.6
656
12.5
10
18.8 5 5x9
18 18
5x9
material properties: reinforcement: S500 structural steel: S460 sliding layer: PTFE (greased), S235
concrete: C35/45
8.6 cm
[cm] 2 11 2
93
15
39 [mm] 262x140x10 (stiffener)
75 150 75
1 30 Ø 16 – spacing 9 cm and 18 cm – length 693.2 cm
47 x 12.5
concrete slab – reinforcement (sliding layer not shown)
500 675
2375
6600
2375
HEA 300
200 250 54 headed studs Ø22  h/d = 125mm / 22mm (1a 27a, 1b27b) 200 100
side a
side b
1
250
22 x 250
24 17 4
11
steel beam – headed studs – sliding layer
sliding plate (675x300) on PTFEfoil (each 1mm thick – PTFE double sided greased – cutout Ø 40 mm)
27
100
290
229
10
11
3.2 Experiments
studs: S235 J2G3 + C450
Fig. 3.107. Details of test beam VT2
Without additional measurements or detailed monitoring it is not possible to determine the failure of studs. As known from the pushout tests the damage process in the interface between steel and concrete proceeds continuously and
230
3 Deterioration of Materials and Structures
300
1500
[mm]
145
50 (QS2, QS6: 20)
side A
50
290
50 (QS2, QS6: 20)
440
150
300
VT1
transducers
strain gauges (oriented in longitudinal direction / QS1 – QS7 ) 1500
side A
50
transducers (side A )
1125
675 QS0
QS1
1125 QS2
375 QS3
375
QS4
QS5
strain gauges (oriented in longitudinal direction / QS1 – QS7)
1125 QS6
QS8
QS7
transducers
VT2
side A 440
290
50 (QS2, QS6: 20)
675
50
145
side A
50
1125
150
50 (QS2, QS6: 20)
1500
300
[mm]
300
horizontal transducers (side A)
400
400
400
400
400
400
400
vertical transducers (side A)
1125
675 QS0
QS1
1125 QS2
375 QS3
375
QS4
1125
QS5
1125 QS6
675 QS7
QS8
Fig. 3.108. Test setup of test beams VT1 and VT2
so no signiﬁcant change in properties of a beam can be observed after single stud failure. In order to avoid a complete shear failure of studs the studs of one row of each beam were coupled in an electric circuit. According to the circuit shown in Figure 3.109 shear failure during a cyclic loading phase can be detected, when the corresponding LED starts to ﬂicker or extinguishes.
3.2 Experiments
231
VT1
LED with series resistance
3B
LED with series resistance
2B
LED with series resistance
1B 0V
LED with series resistance
LED with series resistance
LED with series resistance
+12V
25B
26B
27B
HEA 300
Fig. 3.109. Electric circuit to detect complete shear failure of headed studs
3.2.3.6 Material Properties At the beginning, in the middle and at the end of the test procedure of the test beams cylinder compression tests at standard cylinders according to EN 206 [12] (height 300 mm, diameter 150 mm, cured 28 days in water) were carried out. The results are shown in Table 3.17. Due to the high age of the test beams during the test procedure no increase of the concrete strength and the modulus of elasticity (taken as secant modulus according to EC 2 [33]) could be observed. In both tests structural steel beams of HEA 300 section with the material quality S460 were used. The stud shear connectors welded automatically onto the steel beam ﬂanges had a material quality of S235 J2G3+C450. As reinforcing steel standard deformed bars with diameters of 10 mm, 12 mm and Table 3.17. Mean values of material properties of concrete according to EN 2061 [12]
test beam
VT1, VT2
fc [N/mm2] 2
Ecm [N/mm ]
45.0 32040
232
3 Deterioration of Materials and Structures
Table 3.18. Mean values of material properties of steel members
steel beams
headed studs
longitudinal reinforcement
Ø 10 VT1  Ø 16 VT2
transverse reinforcement
Ø 12 VT1 and VT2
yield strength, tensile strength and modulus of elasticity
member
property
VT1
VT2
fy
460
461
fu
527
531
Ea
203500
203500
fy
448
448
fu
538
538
207800
207800
Rp0.2
620
536
Rm
707
607
Es
216000
200000
Rp0.2
614
613
Rm
653
653
Es
203500
206500
Es
[N/mm2]
16 mm were used in the concrete slabs. In order to obtain detailed data about material properties tensile tests of all steel members were conducted according to the requirements of DIN EN 10002 [15]. The results of the corresponding values of each yield strength, tensile strength and modulus of elasticity are summarized in Table 3.18. 3.2.3.7 Main Results of the Beam Tests Table 3.19 gives an overview about the cyclic loading parameters, the number of load cycles, the reduced static strengths (as short time load bearing capacities) and of main deﬂections measured at midspan. Denotations are explained in Figure 3.105. During the static test phases the loss of the load bearing capacities near the ultimate loads was in the order of 5 % while holding the position of the actuators constant for visual inspection and checking the eﬀects of relaxation. In the case of test beam VT1 cyclic loading caused an increase of the irreversible vertical deﬂections at midspan from 1.0 mm to 4.0 mm. Over the same period of time the vertical deﬂections at the peak load level rose from 18.9 mm to 25.4 mm. Despite these very high increments no apparent damage in the interface of steel and concrete could be observed. This changed when the load was increased up to the ultimate load. At a level of 650 kN the slab lifted from the steel ﬂange by 0.75 mm on both sides of the load introduction area. This clearly indicated a high damage level of the studs which was also noticed near to fatigue failure in the case of the cyclic loaded pushout tests.
3.2 Experiments
233
Table 3.19. Main test results of beams VT1 and VT2 loading parameter test
Pmax
'P
[kN] [kN]
number of load cycles
reduced static strength
N
Pu,N
G iniPmax
[]
[kN]
[mm] [mm]
[mm]
[mm]
18.9
25.4
4.0
VT1
450
265
1372194
756
VT2
250
100
2100000
625
deflections at midspan LC 0 G Pcyc G res max ,lc N
1.0
' = 17.9 18.9
5.1
' = 13.8
cyc G res
' = 21.4 22.5
7.3
' = 15.2
Gu
Gu1
Gu2
[mm] [mm] [mm] 80.0
38.6
10.1
90.0
27.9
> 27
Up to this time the maximum crack width at the bottom of the slab was 0.15 mm and the average distance between the cracks measured 12 cm. Crack formation at the bottom side of the concrete ﬂange was ﬁnished to almost 80 % after initial loading to the peak load level. After development of a plastic hinge at midspan the beam failed at a maximum deﬂection of 80 mm at a load level of 756 kN caused by crushing of the concrete. After applying the initial peak load to test beam VT2 the slab was cracked nearly over the whole length between cross section 1 (QS1) and cross section 7 (QS7). The maximum crack width was 0.2 mm. The distance between two cracks measured 10 cm. Due to these cracks the subsequent reloading lead to very high irreversible vertical deﬂection at midspan of 5.1 mm, slightly increasing during the cyclic loading phase to 7.3 mm. In this period of time the vertical deﬂections at the peak load level grew from 18.9 mm to 22.5 mm. Unlike test beam VT1 the interface of steel and concrete showed no visible damage up to the end of the static test after cyclic preloading. The eﬀect of repeated loading on the vertical deﬂections during the cyclic loading phases and the loadslip behaviour of both test beams in the subsequently performed static tests are shown in Figure 3.110 and Figure 3.111. By comparing the size of grey coloured areas surrounded by two related deﬂection curves in Figure 3.110 it becomes clear that the increase of the vertical deﬂections under the peak load level is signiﬁcantly higher than the increase of the irreversible deﬂections. Consequently repeated loading not only causes an increase of plastic deformations but additionally a reduction in each elastic beam stiﬀness. In the case of test beam VT1 the reduction is in the order of approximately 20 %, in the case of test beam VT2 of approximately 10 %. This indicates that a remarkable redistribution of the inner forces had occurred. In order to allow for plastic deformations of the steel section near to midspan during the static test after cyclic preloading 4 transverse stiﬀeners were provided in a distance of 25 cm from the centre. The top ﬂange was additionally welded to the lowest load introduction plate. At a load level of 580 kN one of the connection on side A between the top ﬂange of the steel
234
3 Deterioration of Materials and Structures
2.4
3.0
1.8
1.2
0.6
0.6
0.0
distance from midspan [m] 2.4 3.0 1.8
1.2
0 3.0
5
15
unloading level (10 kN) 13.8 peak load level
20 3.6
15.2 (+10%)
10
21.4 (+20%)
17.9
2.2
6.5
VT2
25
VT1
P 30
increase of deflection due to cycling loading  VT1
w 6 m
w [mm]
increase of deflection due to cycling loading  VT2
Fig. 3.110. Change of initial deﬂections due to cyclic loading
P [kN] 800
buckling of the top flange
756 kN
700 580 kN
600
VT 2
VT 1
625 kN
state after testing
500 VT 2
400
crushing of concrete
lifting of the slab
300 200 650 kN
100
w [mm]
0 0
20
40
60
80
100
120
756 kN
VT 1
140
Fig. 3.111. Loaddeﬂection behaviour of test beams VT1 and VT2 in the static tests after cyclic loading
section and the load introduction plate were torn oﬀ unintentionally when the top ﬂange began to buckle. This situation is shown in Figure 3.112 a). After this failure the composite beam was unloaded. As it can be seen in Figure 3.112 b) the top ﬂange was subsequently straightened and the steel beam was stiﬀened by 4 additional massive round bars adjusted between the ﬂanges. Although it must be mentioned that the top ﬂange was not completely even after repairing the ultimate load bearing capacity could be signiﬁcantly increased in the following static test phase. After reloading the beam failed at
3.2 Experiments
a)
235
b)
side A
(1) (2)
(1) side A
buckling of the top flange ( P ~ 580 kN)
straightened top flange and strengthened load introduction area by four massive round bars (state after first unloading)
c)
d)
side B
(1)
side A
twosided buckling of the top flange (state after finishing the static test)
side B
buckling of the web in the load introduction area (state after finishing the static test)
Fig. 3.112. Steel section near midspan at diﬀerent point of times during experimental determination of the reduced static strength after high cycle preloading
a maximum deﬂection of 90 mm at a load level of 625 kN. At this time the failure was primarily caused by local buckling of the top ﬂange on side A between stiﬀener (1) and the adjacent round bar (2) followed by buckling of the top ﬂange on the opposite side B and by buckling of the web beneath the load introduction plates (Figure 3.112 c) and d)). It cannot be excluded, that the experimental observed ultimate load was slightly aﬀected by the ﬁrst buckling at a load level of 580 kN. Because of the interaction between local stud behaviour and global beam behaviour the change of the deﬂections of the test beams during the cyclic loading phases decisively depends on the deterioration of the properties of the interface of steel and concrete. Analogous to the eﬀect of cyclic loading on the behaviour of headed studs in pushout test specimens the repeated longitudinal shear forces lead to irreversible deformations at each stud and to a reduction of their elastic stiﬀness due to local crushing of the concrete and due to crack initiation at each stud foot. Thus the experimental observed loadbearing capacities given in Figure 3.111 are signiﬁcantly aﬀected by the stud damage and lie below corresponding ultimate load bearing capacities without any damage caused by cyclic preloading. The measured values of the irreversible part of the slip as well as the slip at the peak load level along each interface between the steel ﬂange and the concrete slab at the beginning and at the end of the cyclic loading phases
236
3 Deterioration of Materials and Structures
3.0 1.0 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1.0
2.4
1.8
1.2
0.6
0.0
0.6
1.2
distance from midspan [m] 2.4 3.0 1.8 VT 1
G
unloading level (10 kN) (f)
(c)
G (c)
(f)
peak load level (450 kN) (f): peak load level after first loading / unloading level after first loading
G [mm] 0.6 0.4 0.2 0
(c): peak load level at the end of the cyclic loading phase / unloading level after cyclic loading
VT 2
G
unloading level (10 kN) (f)
(c)
0.2 0.4 0.6
(c)
(f)
G
peak load level (250 kN)
Fig. 3.113. Slip along the interfaces of steel and concrete after ﬁrst loading and after cyclic loading
can be taken from Figure 3.113. Comparable to the observations regarding the vertical deﬂections the increase of the slip under the peak load levels due to cyclic loading is signiﬁcantly higher than the increase of the plastic slip at each unloading level. In Figure 3.114 the mean values of the crack lengths of two adjacent studs caused by the cyclic loading phases are given.
3.3 Modelling Authored by Otto T. Bruhns and G¨ unther Meschke This section contains numerical models for the description of long and shortterm damage in metallic and cementitious materials as well as in soil, developed within the Collaborative Reseacrch Center SFB 398 at Ruhr University Bochum. Following the classiﬁcation of damage phenomena in Section 3.1 the structure of the section is diﬀerentiated into quasistatic and cyclic loading, in loadinduced and environmentally induced damage and into ductile and brittle damage of metallic and cementitious materials as well as of soils. In Section 3.4 selected models are applied to lifetime oriented ﬁnite element simulations of structures subjected to short and longterm degradation.
3.3 Modelling
237
a [mm] 35 30 stud B4 (VT2)
25
a
25
av
15
a = ah
stud B25 (VT1)
10 5
(1)
(27)
(14)
0 2.4 3.0 1.8 (..) stud pair number
1.2
0.6
0.0
0.6
1.2
2.4 3.0 1.8 distance from midspan [m]
stu
VT1
ds
1 2, 31
5 42
VT1
Fig. 3.114. Crack lengths at the stud feet after the cyclic loading phase  Preparation stages for examination purposes
3.3.1 Load Induced Damage Authored by Otto T. Bruhns and G¨ unther Meschke 3.3.1.1 Damage in Cementitious Materials Subjected to Quasi Static Loading 3.3.1.1.1 ContinuumBased Models Authored by Tobias Pﬁster and G¨ unther Meschke This Subchapter provides a concise summary of continuumbased models for brittle damage of concrete subjected primarily to tensile stresses. After a short review of scalar damage models, anisotropic damage models are described. Although plasticity theory is a versatile concept for describing ductile material behavior, it is also frequently used for the modeling of the more or less ductile behavior of concrete subjected to uni and triaxial compressive states of stresses. Hence, a concise overview over multisurface plasticity and combined plasticdamage models for concrete is provided in Subsections 3.3.1.1.1.2
238
3 Deterioration of Materials and Structures
and 3.3.1.1.1.3. Without use of regularization techniques, results from the ﬁnite element analyses exhibit a mesh dependency. For a review of existing regularization methods, the reader is referred to [141]. 3.3.1.1.1.1 Damage MechanicsBased Models On the basis of onedimensional damage models ﬁrst proposed by [422] and [652], threedimensional damage models were developed by [488, 489, 439] and [183]. Because of the conceptual simplicity and algorithmic robustness of these models, they are widely applied to numerical analyses of concrete despite the fact that cracks induce a signiﬁcant material anisotropy. Starting with the strain energy density Ψ0 of the uncracked material, the free energy of the cracked material can be formulated as Ψ (ε, d) = (1 − d) Ψ0 (ε) = (1 − d)
1 ε : C0 : ε 2
(3.10)
with the scalar damage parameter d and the elastic constitutive tensor C 0 of the virgin material. From (3.10), using standard thermodynamic arguments, the stress tensor σ is obtained as σ = ∂ε Ψ (ε, d) = (1 − d) C 0 : ε. ! =: C
(3.11)
Frequently, the space of admissible states is controlled by the strainlike internal variable κ ≥ 0 deﬁned in the strain space E ε := {(ε, κ) ∈ S × R +  f (ε, κ) ≤ 0} ,
(3.12)
where S is the space of symmetric second order tensors and R is the space of positive rational numbers. In Equation (3.12) f represents a failure surface. The evolution of the admissible strain space E ε is controlled by the strainlike internal variable κ. From the KuhnTucker conditions, f (ε, κ) ≤ 0
κ˙ ≥ 0
κf ˙ = 0,
(3.13)
the current threshold of the equivalent strain κ is deﬁned by the failure criterion f (ε, κ) = η(ε) − κ ⇒ κ = max {κ0 , max η(ε)} .
(3.14)
In general, the numerical implementation of material models within the ﬁnite element codes is strain driven. Hence, the algorithmic formulation of scalar damage models is relatively simple. The constitutive model is completed with the deﬁnition of a damage law d(κ) relating the equivalent strain κ to the damage parameter d: d(κ) = 0 ∀ κ ≤ κ0 , d(κ) > 0 ∀ κ > κ0 .
(3.15)
3.3 Modelling
239
If the fracture energy concept is used to avoid meshdependent results, d(κ) has to be related to the fracture energy Gf of concrete and to the size of the ﬁnite element [596]. For the approximation of brittle material characteristics of concrete under tensile loadings an equivalent strain corresponding to the Rankine criterion 1 max σ ˜i (ε) H(˜ σi (ε)), E
η(ε) =
(3.16)
with ˜ = C0 : ε = σ σ ˜i = 2μεi + γ
3 i=1 3
σ ˜i ni ⊗ ni ,
(3.17)
εk ,
(3.18)
i=k
making use of the Heaviside function H, can be applied (H(x) = 1 ∀x > 0, H(x) = 0 ∀x ≤ 0). The strain space illustration of Equation (3.16) is given in Figure 3.115b. A scalar damage model for the numerical analysis of concrete structures was proposed by [522]. Mazars introduced two damage parameters dt and dc , corresponding to tension and compression, respectively, to account for the diﬀerent material behavior under compressive and tensile loadings. One of the key assumptions of this model is the additive decomposition of the damage variable d = dt αt + dc αc
(3.19)
1
ε2 τ
ν
2
√
ε:ε
ε2 τ2
Eτ 3κ
ε1
ε1
a) η(ε) =
ε2
b) η(ε) =
1 E
max σ ˜i H(˜ σi )
ε1
c) η(ε) =
3 i=1
ε2i H(εi)
Fig. 3.115. Representation of diﬀerent failure surfaces f (η, κ) = η(ε) − κ = 0 in the principal strain space
240
3 Deterioration of Materials and Structures
2
σ11 [N/mm ]
0.0
−10.0
−20.0
−30.0 −0.005
−0.003
−0.001
ε11
0.001
Fig. 3.116. Stressstrain diagrams for uniaxial compressive and tensile loading obtained from the damage model by Mazars (Material parameters: E = 35000 N/mm2 , ν = 0.2, κ0 = 10−4 , At = 0.81, B − t = 1.045 · 104 , Ac = 1.34, B − c = 2.537 · 103 )
into a part (•)t corresponding to tensile loading and one (•)c associated with compressive states. In Equation (3.19) αt and αc represent weighting functions. The equivalent strain is deﬁned in the format " # 3 # H(εi ) ε2i . (3.20) η(ε) = $ i=1
Figure 3.115c illustrates Equation (3.20) in the ε1 − ε2 space. The weighting functions αt , αc are assumed to depend upon the state of the strain. The model is completed by the deﬁnition of the damage laws for dt (κ) and dt (κ) [522]. The stressstrain diagrams obtained from the analysis of concrete subjected to uniaxial tensile and compressive loading are illustrated in Figure 3.116. Several models have been proposed to extend the isotropic damage theory to capture anisotropic failure mechanisms. These models can be subdivided into formulations based on damage vectors (see [441]), formulations based on secondorder damage tensors (see [523, 116, 187]) and formulations based on fourthorder damage tensors (see [604, 744, 178, 318, 60, 179]), respectively. In what follows, attention is restricted to models considering the fourthorder compliance tensor or the stiﬀness tensor as the fundamental internal variable. In an attempt to represent the anisotropic character of brittle failure of concrete within a continuum damage model formulated in the stressspace, [604] considered the complementary energy Ψ (σ, D , χ) =
1 σ : D : σ − Ψin (χ) 2
(3.21)
3.3 Modelling
241
D := with ζ(χ)2 = 2 ∂χ Ψin . In Equation (3.21) D is the compliance tensor (D C −1 ) and ζ(χ) a stresslike internal variable. From standard arguments of thermomechanics follows ε = ∂σ Ψ = D : σ.
(3.22)
From the rate form of Equation (3.22), an additive split of the strain rate ε˙ into an elastic part ε˙ e and an inelastic part ε˙ i results in ε˙ = D : σ˙ + D˙ : σ =: ε˙ e + ε˙ i .
(3.23)
One key point of this concrete model is the assumption of an additive split of the compliance tensor D D = D0 + D c
(3.24)
into the compliance tensor D 0 of the virgin material and the damage tensor D c associated with additional ﬂexibility corresponding to active microcracks. Consequently, the total strains ε can be rewritten into the format
ε = D 0 + D c : σ =: ε0 + εc . (3.25) Crack closure is taken into account by the restriction that the eigenvalues of εc must be positive. Tensile and compressive portions of the stresses can be rewritten in the format σ+ = P+ : σ
and σ − = P − : σ.
(3.26)
using the projection tensors P+ = −
P =
3 k=1 3
H(σk ) nk ⊗ nk ⊗ nk ⊗ nk , (3.27) H(−σk ) nk ⊗ nk ⊗ nk ⊗ nk .
k=1
The space of admissible states is the stress space E σ = {(σ, χ) ∈ S × R  f (σ, χ) ≤ 0} ,
(3.28)
where the failure surface is deﬁned as [604] f (σ, χ) =
1 1 1 + σ : σ + + c σ − : σ − − ζ(χ)2 . 2 2 2
(3.29)
In Equation (3.29), the parameter c represents a coeﬃcient accounting for the crosseﬀect between compression and tension. The failure function (3.29)
242
3 Deterioration of Materials and Structures
σ2 q(α) σ1
elastic region √ q(α)/ c
Fig. 3.117. Anisotropic damage model by [604]: Illustration of the failure surface in the principal stress space, see eq. (3.29)
is illustrated in Figure 3.117. The dependence of the stress like internal variable ζ(χ) can be deﬁned on the basis of uniaxial tensile tests [604]. From Equation (3.29) the evolution of the compliance tensor is derived exploiting the postulate of maximum dissipation. Adopting the idea to directly include the stiﬀness tensor (or as in [604] the compliance tensor) as arguments within the function of free energy, an anisotropic damage model was proposed by [318]. The model is based on the free energy Ψ (ε, C , χ) =
1 ε : C : ε + Ψin (χ). 2
(3.30)
The admissible stress space E σ is deﬁned by a set of damage functions fk of the form fk = σkeq (σ) − fk + ζk (χ)
(3.31)
with ∂σ σkeq : σ = σkeq (σ) k ∈ {1, 2, · · · M }.
(3.32)
The evolution equations for the compliance tensor D and the internal variable χ are obtained in an associated format as D˙ =
M k=1
γk
∂σ fk ⊗ ∂σ fk , ∂σ fk : σ
χ˙ =
M k=1
γk ∂ζk fk .
(3.33)
3.3 Modelling
tm
243
t
m
tn n
crack Fig. 3.118. Concrete specimen with a crack: Deﬁnition of a local coordinate system by means of the crack normal vector n and the tangential vector m. Decomposition of the traction vector t = σ · n into the normal part tn = (n ⊗ n) : σ and the tangential part tm = (n ⊗ m) : σ
In Equation (3.31), the failure surfaces are deﬁned as f1 (σ, χ) =
(n ⊗ n) : σ
−fn + kn ζ(χ),
f2 (σ, χ) =(n ⊗ m1 )sym : σ −fs + ks ζ(χ),
(3.34)
f3 (σ, χ) =(n ⊗ m2 )sym : σ −fs + ks ζ(χ), with n representing the normal vector of the crack surface and mα two vectors orthogonal to n (m1 × m2 = n and mα · n = 0) (see Fig 3.118). fn and fs denote the tensile and the shear strength, kn a parameter which deﬁnes the residual tensile strength and ks the residual shear strength, respectively. Since the vectors n and mα are assumed as constant during the deformation, the model falls within the class of ﬁxed crack approaches (see [381]). Since t = σ·n, the failure surfaces (3.34) control all components of the traction vector t. For a detailed comparison between diﬀerent interface models, reference is made to [434]. The integration of the rate of the stress tensor %
σ˙ = C : ε˙ −
&
M
γk ∂σ fk
(3.35)
k=1
within a time interval [tn , tn+1 ] by means of an implicit backward Euler scheme yields an algorithmic formulation σ n+1 = σ trial n+1 −
M
Δγk
C n : ∂σ fk ,
(3.36)
k=1
which is formally identical to standard plasticity. Hence, the standard return mapping algorithm can be used for the integration of the diﬀerential equations. In Equation (3.36) the deﬁnition of the trial stress tensor
244
3 Deterioration of Materials and Structures
σ trial n+1 = C n : εn+1
(3.37)
at the pseudo time tn+1 is used. The update of the compliance tensor results in ' M ∂σ fk ⊗ ∂σ fk '' D n+1 = D n + Δγk . (3.38) ∂σ fk : σ 'n+1 k=1
3.3.1.1.1.2 Elastoplastic Models In plasticity theory, cracking of hydrates is described by means of inelastic strains at the macroscale, denoted as εp . In addition, internal variables χ are employed to monitor evolution of inelastic mechanisms occurring at the microlevel. They represent microstructural changes caused by cracking of concrete. The energetically conjugated thermodynamic quantities are the hardening/softening forces ζ. They are related to the internal variables via the state equation ζ = ζ(χ). The hardening forces represent the actual strength of the material, deﬁning the space of admissible stress states, C E . In case of multisurface plasticity, C E is deﬁned by M yield surfaces: σ ∈ C E ⇔ fk = fk (σ, ζ(χ)) ≤ 0
(3.39)
for k ∈ [1, 2, . . . , M ], where fk denotes the kth yield function. The internal variables and the tensor of plastic strains are obtained by means of evolution equations, reading ˙ = χ
k∈Jact
γ˙ k
∂hk , ∂ζ
ε˙ p =
k∈Jact
γ˙ k
∂gk , ∂σ
(3.40)
where γk denotes the plastic multiplier of the kth yield function. gk and hk are potentials which generally depend on σ and ζ. In the Equations (3.40), Jact denotes the set of active yield surfaces. It is deﬁned as Jact := {k ∈ [1, 2, . . . , M ]fk (σ, ζ) = 0} .
(3.41)
A wide range of plasticity models for concrete (see, e.g., [339, 648, 276, 531, 166, 381] exists and the reader is referred to the respective references. A model representative for the class of single surface models applicable to the simulation of failure of concrete under triaxial stress states is the Extended Leon Model (ELM) [264]. Detailed information on the model and its algorithmic treatment is contained in [639], [516]. 3.3.1.1.1.3 Coupled ElastoplasticDamage Models Failure of concrete in tension as well as in compression for low levels of conﬁnement is associated with stiﬀness degradation as well as with inelastic deformations. To take account of both phenomena models characterized by a coupling between damage and plasticity theory have been developed.
3.3 Modelling
245
The simplest mode of coupling between damage and plasticity is a scalar damage elastoplastic model [420] based on the eﬀective stress concept. Considering plastic strains in Equation (3.10) yields the stressstrain relation σ = (1 − d(κ)) C e : (ε − εp ) .
(3.42)
According to the eﬀective stress concept, the plastic yield function fp is forˆ mulated in terms of eﬀective stresses σ: σ ˆ − ζ(χ) with σ ˆ= fp = σeq (σ) . (3.43) 1−d The evolution of the damage parameter is controlled by a damage surface formulated in the strain space fd = η(ε) − κ.
(3.44)
An anisotropic elastoplasticdamage model for plain concrete using the principle of maximum dissipation has been proposed by Meschke, Lackner & Mang [534]. In contrast to [420], the stiﬀness degradation is taken into account by means of additional cracking strains. Conceptually, this model is an extension of the anisotropic damage model proposed by [318]. In contrast to [318], a rotating crack formulation is considered in [534]. The model is based on a function of free energy of the form C, ε, εp , χ) = Ψ (C
1 (ε − εp ) : C : (ε − εp ) − Ψin (χ). 2
(3.45)
From the postulate of maximum dissipation, the evolution equations are obtained as ε˙ p + D˙ : σ =
M
γk ∂σ fk (σ, χ)
(3.46)
k=1
and χ˙ =
M
γk ∂qk fk (σ, χ),
(3.47)
k=1
respectively. With the introduction of the rate of damage strains, ε˙ d := D˙ : σ, and a scalar coupling parameter β ∈ [0, 1], damageinduced and plastic strains are additively decomposed into ε˙ p = (1 − β) d
ε˙ = β
M k=1
M
γk ∂σ fk (σ, χ),
k=1
γk ∂σ fk (σ, χ),
(3.48)
246
3 Deterioration of Materials and Structures
σ
σ
ε
β = 0.5
β = 1.0
ε
Fig. 3.119. Anisotropic elastoplastic damage model by [534]: Inﬂuence of the scalar coupling parameter β on the stressstrain diagram obtained from cyclic uniaxial tensile loading
with β ∈ [0, 1]. Since the governing equations of the coupled model are formally identical with plasticity models, the standard return mapping can be used for the algorithmic formulation. Figure 3.119 illustrates the inﬂuence of the coupling parameter β on the un and reloading paths. The limit case (β = 1.0) corresponds to a damage model, ignoring plastic deformations. For β = 0.5, permanent deformation as well as stiﬀness degradation are taken into account. 3.3.1.1.1.4 Multisurface ElastoplasticDamage Model for Concrete To account for brittle material response under tension and for the more ductile mode of damage under compression a 3D elastoplastic multisurface model for concrete is described in this Subsection. The model is designed such that the parameters can be derived from standard uniaxial tests or estimated on basis of the compressive strength [641, 444]. Applications and enhancements can be found in [133, 622, 628]. The model is characterized by a DruckerPrager potential active in case of predominantely compressive states of stresses combined with Rankine damage surfaces governing the softening response in tension. The evolution laws for both loading states are derived from the free Helmholtz energies. In accordance with the proposal made in [318] the free energy ψc is deﬁned for compression as ρ0 ψc (ε, εp , D d , χc ) =
−1 1 [ε − εp,c ] : D 0 + D d : [ε − εp,c ] + ρ0 ψc,in (χc ) 2 (3.49)
and for tension as ρ0 ψt (ε, D d,t , χt ) =
−1 1 ε : D 0 + D d,t : ε + ρ0 ψt,in (χt ). 2
(3.50)
3.3 Modelling
DruckerPrager
247
−σII σIII Rankine DruckerPrager
−σI
−σIII Rankine
σI
Fig. 3.120. Multisurface model for concrete: Yield and failure surfaces in the stress space
The compliance relation reads −1 σ = D 0 + D d + D d,t : [ε − εp,c ] .
(3.51)
The evolution laws for inelastic strains, for the damage compliance tensor and for the internal variables will be introduced in the following, separately for concrete under compression and under tension. The DruckerPrager potential (see Fig. 3.120) and its derivatives are given as ( ) 1 φc (σ, ζ) = 1 μ I1 + J2 − ζc (χc ) , (3.52) √ −μ 3 ∂φc = ∂σ
1 s + sT √ , μ 1 + √1 − μ 4 J2 3
∂φc = −1 . ∂ζc
(3.53)
Using an associated ﬂow rule and the split of inelastic strains into damaging and plastic parts according to [534], the evolution laws for the plastic and the damaging strains result in ε˙ p,c = β γ˙ c
∂ φc , ∂σ
ε˙ d,c = (1 − β) γ˙ c
d,c ∂ φc = D˙ : σ. ∂σ
(3.54)
The latter relation is the basis for the formulation of the rate of the damage d,c compliance tensor D˙ : d,c μ 1 β 1 ¯ ˙ 1⊗1+ √ I + I γ˙ c , − √ (3.55) D = 1 √ I1 6 J2 4 J2 3−μ
248
3 Deterioration of Materials and Structures
where isotropic damage evolution is assumed. Exploiting the symmetry of this tensor, which results from the postulation of the free Helmholtz energy according to Eq. (3.49), this fourth order tensor can be expressed by means of two independent variables: d,c d,c d,c D˙ I + ¯I . = D˙ s1 1 ⊗ 1 + D˙ s2
(3.56)
The evolution laws of the scalar variables are found by comparison of the coeﬃcients of Eq. (3.55) and Eq. (3.56) as μ 1 β 1 β d,c d,c √ D˙ s1 γ˙ c , D˙ s2 γ˙ c . = 1 − √ = 1 (3.57) √ − μ I1 √ − μ 4 J2 6 J2 3 3 The rate of the internal variable χc is redeﬁned in the format √1 − μ 2 2β 3 . χ˙ ∗c = γ˙ c with ψ(σ) = μ 1 √ ψ(σ) + I1 3 J
(3.58)
2
This deﬁnition enables the interpretation of the rate of the internal variable χ˙ ∗c as the rate of the uniaxial compliance χ˙ ∗c = D˙ d,c,1d ,
(3.59)
which is the basis for the derivation of the evolution law for the internal variable from the uniaxial stressstrain relation. Using the parameter b, the uniaxial inelastic strains may be decomposed into plastic and damaging parts: εp,c = b εin ,
εd,c = (1 − b) εin .
(3.60)
Together with Eq.3.59 the uniaxial stressstrain relation can be expressed as 1 1 ∗ ε= + χ σ. Ec 1−b c
(3.61)
In this equation, σ can be replaced by any stressstrain relation σ(ε) for concrete and the resulting expression can then be rearranged to obtain a relation ε(χ∗c ). Inserted this relation into the given expression σ(ε), the relation ζc (χ∗c ) = σ(ε(χ∗c ))
(3.62)
can be found. The softening/hardening modulus then reads Hc =
dζc dσ dσ dε = = = dχ∗c dχ∗c dε dχ∗c
(1 − b) 1 −
ζc . dσ 1 χ∗c + Ec 1 − b dε
(3.63)
3.3 Modelling
2
1
3
2
stress σ
stress −σ
1
249
εp,c
εd,c
εel
εd,t
εel
strain −ε
strain ε
Fig. 3.121. Stressstrain relation of concrete in compression and in tension
The relation for the scalar variable β, which divides the inelastic strain rate into a rate for the plastic and the damaging strains, respectively, reads β(χ∗c ) = 1 +
−1 χ∗ dζc 1 1+ c . 1−b ζc dχ∗c
(3.64)
A particular stressstrain relation will be introduced in the following. It is subdivided into a linear elastic part for stresses below 13 of the compressive strength, a hardening part, which is a slight modiﬁcation of the relation suggested in [182] and a softening part. This is illustrated in the left diagram in Fig. 3.121. The equations σ(ε) for these three domains read as follows:
σc1 = Ec ε ,
σc2
with Eci =
1 2 Ec
fc εc
σc3
2
−
2 Eci fεc + εεc = ε c 1 − Eci fcε−2 εc fc
fc 3 + Ec , εc 2
2 + γc f c ε c γc = + γc ε + 2 γc 2 εc ε2
with γc =
( 2 gcl,e −
(3.65)
(3.66)
−1
π 2 fc εc )2 . fc 1 ε (1 − b) + b c 2 Ec
(3.67)
(3.68)
250
3 Deterioration of Materials and Structures
In the softening domain, γc adjusts the area under the stressstrain relation. It is a function of the localised crushing energy gcl,e , which has to be adjusted to the element size in order to avoid illposedness and mesh dependency of the simulation results due to localisation. The equation for the Rankine damage potential for the tensile domain reads φt,i (σ, ζ t ) = ξi (σ, ζ t ) − fct ≤ 0 ,
(3.69)
as illustrated in Fig. 3.120. ζ t is called the backstress tensor, that quantiﬁes the internal (softening) state of the material. ξ is the diﬀerence between the stress and the backstress tensor: ξ = σ − ζt ,
(3.70)
ξi is its projection into the ith crack direction: ξi = ξ : M i .
(3.71)
The crack direction is, diﬀerent from the original description of the model in [641, 444], assumed to be determined from the strain tensor ε [628]. Thus, the eigenvalue basis M i is derived as the dyadic product of the eigenvectors xi of ε: M i = xi ⊗ xi .
(3.72)
The derivatives of the Rankine potential with respect to the stress and the backstress tensor result in ∂φt,i = Mi , ∂σ
∂φt,i = −M i . ∂ζ t
(3.73)
In tension, the inelastic strains are assumed only to be associated with damage processes. The evolution law for the damaging strains εda and the internal variable χt read: ε˙ d,t =
3
γ˙ t,i
i=1
˙t = χ
3 i=1
γ˙ t,i
d,t ∂φt,i = D˙ : σ, ∂σ
∂φt,i . ∂ζ t
(3.74)
(3.75)
The evolution law of the backstress tensor ζ t is derived from its total diﬀerential dζ t ∂φt,i dζt ˙t = ζ˙ t = :χ : γ˙ t,i . dχt dχt ∂ζ t i=1 3
(3.76)
3.3 Modelling
251
As the evolution of the softening behaviour is formulated by means of a kinematic softening law independently for each crack, the fourth order tensor dζ t /dχt can be replaced by three scalar softening moduli Ht,i , which will be derived from a uniaxial stressstrain relation. The evolution law for the backstress tensor than simpliﬁes to ζ˙ t =
3
∂φt,i γ˙ t,i = −Ht,i M i γt,i . ∂ζ t i=1 3
Ht,i
i=1
(3.77)
This relation can be transformed into the crack directions, so that the scalar values of ζ t result in ζ˙t,i = −Ht,i γ˙ t,i = Ht,i χ˙ t,i .
(3.78)
Hence, the evolution of ζ t can be evaluated separately for each crack direction with scalar variables, which then are assembled to the tensor ζ t using Eq. (3.77). The evolution law for the damage compliance tensor reads d,t D˙ =
3 1 γt,i M i ⊗ M i σ i=1 i
with σi = σ : M i .
(3.79)
This relation has also been used in [318]. It satisﬁes Eq. (3.74) as well the major symmetries of D d,t . The softening behaviour is again derived from uniaxial stressstrain relations. To this end, the uniaxial strain rate is expressed as 1 ε 1 ε dε ε˙ = − = − σ˙ ⇔ . (3.80) σ Ht dσ σ Ht In this formulation, an arbitrary stressstrain relation σ(ε) for concrete under tension is applied and the expression is rearranged for Ht . A speciﬁc stressstrain relation will be introduced in the following. It is decomposed into a linear elastic part before and an exponential softening part after the peak (see the diagram on the right hand side of Fig. 3.121): σt1 = Ec ε ,
1
σt2 = fct e γt
(εct −ε)
.
(3.81)
In analogy to the formulation of the softening branch in compression, the variable γt adjusts the area under the curve according to the fracture energy gﬂ,e and the element size in order to avoid mesh dependent solutions: γt =
gﬂ,e 1 fct − . fvt 2 Ec
(3.82)
252
3 Deterioration of Materials and Structures
3.3.1.1.2 Embedded Crack Models Since the early 1990s, attempts were made to represent the discontinuous character of cracks by incorporating the failure kinematics directly into the ﬁnite element formulation. From a mathematical point of view, the crack width can be interpreted as a jump of the displacement ﬁeld. This idea has been pioneered by Dvorkin, Cuitino & Gioia [248], who enriched a twodimensional ﬁnite element by a kinematics representing a discontinuous displacement ﬁeld. These approaches account for the multiscale character of the problem (the failure zone is several dimensions smaller than the structure). They allow to use relatively large elements compared to the width of the localization zone. Hence, these methods are suitable for large scale structural applications. Enhanced ﬁnite element models considering discrete crack propagation can generally be categorized into elementbased formulations, generally denoted as Embedded Crack Models (see [746, 598, 413, 59, 244], among others) and nodalbased formulations, e.g. the Extended Finite Element Method (XFEM) (see [545, 544, 828, 243, 532]). For a comparative assessment of both approaches we refer to [412, 244]. In the Strong Discontinuity Approach (SDA), suggested by Simo, Oliver & Armero [746] and further elaborated in [597, 58, 671, 413, 559], among many others, the failure kinematics of solids, i.e. crack opening in brittle materials or sliding along shear failure zones in metallic materials, is approximated by means of discontinuous displacement ﬁelds locally embedded within the ﬁnite elements undergoing localization. With the exception of the “rotating” crack formulation of the Strong Discontinuity Approach [559], the topology of crack segments is held ﬁxed once they are signaled to open. Furthermore in the framework of the Extended Finite Element Method crack path continuity is required. Hence, the correct prediction of the direction of new crack segments is crucial for the reliability as well as for the robustness of the numerical analysis. If the predicted propagation direction is incorrect, locking occurs, which generally leads to unreasonable results and eventually may cause failure of the analysis. A discrete representation of cohesive cracks based on a relation between the crack opening w and the normal component σn of the traction vector t was formulated in the context of the ﬁctitious crack concept proposed by [369]. Starting with the deﬁnition of the fracture energy w max
Gf =
σn dw
(3.83)
w0 =0
in terms of σn of the traction vector t of a crack surface and the crack width w, a linear softening law of the form σn (w) was postulated. The underlying idea is illustrated in Figure 3.122. Since the softening law is formulated in terms of the traction vector and the crack displacement, the resulting load displacement diagram obtained from a computation is independent of the reﬁnement of the respective ﬁnite element mesh.
3.3 Modelling
253
t
u
Gf
crack
Fig. 3.122. Discrete representation of cracks: Traction separation law of the format t = t( u ) across the crack surface
The socalled Strong Discontinuity Approach (SDA) was proposed by [746]. In this concept, the localization zone is represented as a surface of discontinuous displacements within the respective ﬁnite elements. This method accounts for the multiscale character of the problem. In the SDA, displacement jumps are embedded locally in the respective ﬁnite elements without aﬀecting neighboring elements. In accordance with the underlying enhanced assumed strain (EAS) concept [747], only the (enhanced) strains resulting from the discontinuous displacement ﬁeld appear explicitely in the formulation. The idea to enhance standard ﬁnite element models by additional modes to capture displacement jumps has been already suggested in the early work [417]. Since then, several variants of this concept have been proposed (see [105, 248, 435]). A more detailed insight into the incorporation of strong displacement discontinuities into classical (local) elastoplastic and elastodamage continuum models was provided by [746]. This formulation corresponds to a PetrovGalerkin method in the sense that the weak form involves test functions diﬀerent from the variations of the enhanced strains. Details of the corresponding algorithmic formulations are given in [598, 57, 112, 757, 413] for constant strain elements and in [671, 599, 559] for the more general case. Based on the assumption of a jump in the displacement ﬁeld across the crack surface, an additive decomposition of the displacement ﬁeld ¯ (x) + u ˆ (x), u(x) = u
∀ x ∈ Ω,
(3.84)
with
ˆ (x) = u MS (x). u
(3.85)
254
3 Deterioration of Materials and Structures
u¯
u
uˆ u
x
x
x
Fig. 3.123. Strong Discontinuity Approach: Additive decompositionofthe dis¯ and a part u ˆ containing a jump u (Equaplacement ﬁeld into a smooth part u tion (3.84))
u
¯ u
∞ x
ˆ u
x
∞ x
Fig. 3.124. Strong Discontinuity Approach: Strain ﬁeld resulting from the displace¯ (x) + u ˆ (x) ment ﬁeld u(x) = u
¯ (x) and a jump term u ˆ (x) is motivated. The function into a regular part u MS (x) can be decomposed into a Heaviside function HS (x) and a smooth function ϕ(x) MS (x) = HS (x) − ϕ(x),
∀ x ∈ Ω.
(3.86)
Figure 3.123 illustrates the additive decomposition of the displacement ﬁeld according to Equation (3.84). Restricting the model to the geometrically linear theory and assuming ∇ u = 0 (see [139]), the strain ﬁeld is obtained as ¯ + ∇sym u ˆ = ∇sym u ¯ − ˜ε, ε = ∇sym u = ∇sym u
(3.87)
with ˜ε =
sym sym u ⊗ ∇ϕ − u ⊗n δs ,
(3.88)
see Figure 3.124. In Equation (3.88) n represents the normal vector of the crack surface and δs is the Diracdelta distribution. The displacement discontinuities u follow from a traction separation law t∂s Ω = ts ( u ) to be deﬁned along the crack surface ∂s Ω. From traction continuity follows the compatibility condition
3.3 Modelling
ts ( u ) = t+ = σ + · n.
255
(3.89)
The ﬁnite element formulation of the SDA according to [745] is based on the weak form of the equilibrium equations divσ + b = 0 ⇒ BT σ dV = r (3.90) Ω
as well as on the weak form of the traction equilibrium GT σ dV = 0. σ + · n = ts ( u ) ⇒
(3.91)
Ω
According to Equation (3.91), the softening behavior is controlled by means of a traction separation law ts ( u ). From Equations (3.90) and (3.91) the analogy with the enhanced assumed strain concept becomes obvious. An alternative implementation was proposed by [139] and [559], where the amplitude of the displacement jump is condensed out on the material level leaving the element routines of the 3D formulation unaﬀected. Consequently, the standard return mapping algorithms can be applied for the integration of the governing equations [560]. 3.3.1.2 Cyclic Loading 3.3.1.2.1 MechanismOriented Simulation of Low Cycle Fatigue of Metallic Structures Authored by JanHendrik Hommel and G¨ unther Meschke Classical phenomenological approaches for an assessment of damage accumulation of metallic structures subjected to fatigue loading are based on ¨ hlertype relations between the number of cycles to failure and quantiWo ties such as stress, strain amplitudes or amplitudes of dissipated energy (see [257] for an overview). In contrast, modelbased approaches are based on constitutive models accounting for the accumulation of damage on a macroscopic level by means of appropriate internal variables representing the growth of damage at the microlevel (e.g. [186]). In [389, 388, 390] a mechanismoriented concept for lifetime prediction of metallic structures under (ultra) low cycle fatigue loading is proposed. It is characterized by combining two diﬀerent strategies for the modelling of damage accumulation in highly plasticized zones of metallic materials related to two diﬀerent levels of observation: One approach is based on unitcell analyses representing, in a simpliﬁed manner, the microstructure characterized by a regular distribution of microvoids embedded within the matrix material. The matrix material is modelled by means of a suﬃciently sophisticated elastoplastic material model suitable for the representation of large plastic deformations under cyclic loading conditions as described in Section 3.3.1.2.1.1.
256
3 Deterioration of Materials and Structures
This model has to allow for the simulation of all relevant phenomena associated with cyclic plasticity such as the Bauschingereﬀect, ratcheting or mean stress relaxation, cyclic hardening or softening. In addition to a nonlinear isotropic hardening rule for the modelling of cyclic plasticity a BariHassantype kinematic hardening law [87], which represents a superposition of several kinematic hardening laws according to ArmstrongFrederick [62], is implemented in a ﬁnite deformation continuum mechanics framework. At the scale of the microstructure the growth of the initially spherical micropore represents the accumulated damage. On a macroscopic level, a Gursontype micropore damage model is employed to represent evolving damage. This micropore damage model is extended to cyclic loading and combined with isotropic and kinematic hardening similar to [63]. Within the presented lifetime concept for metallic structures failure analyses are based upon the micropore damage model at the macrolevel, which is calibrated to the speciﬁc characteristics of the metallic alloy by means of unit cell analyses performed on the microlevel [386]. The computational concept can be divided into three parts as shown in Figure 3.125. •
•
•
At ﬁrst, numerical analyses of the material under investigation subjected to monotonic, constant stress triaxiality loading are performed both by means of a unitcell model as well as by means of the macroscopic model to calibrate the parameters of the Gursontype micropore model. This procedure is demonstrated in [386] for low alloy steel typically used for structures such as pressure vessels and is not part of this paper. After the calibration procedure, the micropore damage model is validated by means of typical fatigue tests, e.g. on notched specimens. To this end, results from cyclic tests on two diﬀerent round notched bars performed by [638] are reanalyzed in Section 3.3.1.2.1.2. To study the damage evolution due to low cycle fatigue, the results are compared in terms of the degradation of the peak reaction force. Finally, the macroscopic micropore damage model is used for ﬁnite element analyses of a spherical pressure vessel supported by cylindrical columns subjected to earthquake loading (Section 3.4), which represents a typical loading scenario that may lead to structural failure induced by low cycle fatigue at highly stressed locations of the structures.
3.3.1.2.1.1 Macroscopic ElastoPlastic Damage Model for Cyclic Loading The ﬁnite strain elastoplastic damage model is based on the multiplicative decomposition of the deformation gradient into elastic and plastic parts F=Fe Fp . The elastic material response is described by means of the compressible NeoHookemodel [750]. The Gursontype yield condition, introduced by [332] and modiﬁed by [482], is formulated in the intermediate conﬁguration ˆ2 ˆm Σ Σ 3 ln(f ∗ qA ) eq ∗ F = 2 + 2 f qA cosh qB − 1 − (f ∗ qA )2 ≤ 0 , (3.92) 2 ln(f ) qˆ ¯ qˆ ¯
3.3 Modelling
257
(I ) m ic r o s tr u c tu r e : d e te r m in a tio n o f m o d e l p a r a m e te r s (i) e la s tic m o d e l p a r a m e te r s fro m
(ii) p la s tic m o d e l p a r a m e te r s e x p e rim e n ta l a n a ly s is
d a ta ta k e n lite ra tu re
s
s
e e
e la s to p la s tic c a lc u la tio n c a lib ra tio n
s e
(iii) d a m a g e m o d e l p a r a m e te r s e la s to p la s tic u n it c e ll a n a ly s is h o m o g e n is e d d a m a g e a n a ly s is v o id v o lu m e fra c tio n f(t)
c a lib ra tio n
v o id v o lu m e fra c tio n f(t)
(I I ) m a c r o s tr u c tu r e : v a lid a tio n o f m o d e l p a r a m e te r s s p e c im e n te s t
e x p e rim e n ta l a n a ly s is lo a d c y c le s to fa ilu re s e c u rv e
v a lid a tio n
n u m e ric a l a n a ly s is lo a d c y c le s to fa ilu re s e c u rv e
(I I I ) e n g in e e r in g s tr u c tu r e : life tim e a s s e s s m e n t g e o m e try &
lo a d in g
lo a d
s tr u c tu r a l a n a ly s is n u m e ric a l a n a ly s is
tim e
lo a d c y c le s to fa ilu re
Fig. 3.125. Modelbased concept for life time assessment of metallic structures
ˆ eq = (G( ˆ Σ ˆ −κ ˆm = (BG( ˆ Σ ˆ −κ ˆ ˆ¯ ))m denotes the equivalent where Σ ¯ ))eq and Σ and the mean part of the relative stress, respectively. In (3.92) qA and qB represent the material parameters and the scalar damage variable f ∗ describes ˆ G ˆ −1 BC ˆ is a symmetric Mandeltype stress the void volume fraction. Σ= ˆS eT e ˆ tensor where BC ˆ =GF F is the elastic right CauchyGreentensor and p ˆ = Fp G (C ) denotes the metric of the intermediate conﬁguration. Note that the deviatoric part of (3.92) reduces for vanishing f ∗ to a von Misesyield condition. The additional term ln(f ∗ qA )/ln(f ) introduced by
258
3 Deterioration of Materials and Structures
[386] within the mean part of the yield condition is a scaling factor, that leads to total material softening for f ∗ qA =1. Nonlinear isotropic hardening is considered by the following relation ˆ + β (σY∞ − σY0 )(1 − exp(−δiso α))] ˆ , qˆ ¯ = J p J e [σY0 + β Hiso α
(3.93)
ˆ ]/[(1 − f ∗ qA ) qˆ ˆ−κ ˆ ¯) : N ¯] describes the evolution of the scalar where α= ˆ˙ γ[( ˙ Σ isotropic hardening parameter α ˆ and the determinants J e and J p transform from Cauchy to Kirchhoffstresses. Cyclic loading is accounted for by an advanced kinematic hardening model [87] using a superposition of at most four 4 ˆ ˆ i with the following kinematic hardening tensors κ ¯ =J p J e (1 − f ∗ qA ) i=1 κ assumption for the material time derivative of the back stress tensor ˜ˆ D ˆ p − b ζ δ γ˙ κ ˆp ˆ + (1 − δ )(ˆ κ ˆ˙ = (1 − β) c D , (3.94) κ : N) i
i
i
kin
i
kin
i
κ ¯ ˆ˜ is the ˆ p =γ˙ F, ˆ > for i=4, D =γ˙ N ˆ) (Σ −κ ˆ κ4  ˆ˜ describes the symmetric ˆ p =G ˆ −1 D ˆ pG ˆ −1 , N symmetric plastic strain rate, D ˆ ) is the norm of the back gradient of the yield surface, ˆ κ= 3/2 tr(ˆ κκ ˆ ˆ κ ˆ and κ ˆ =Gˆ κG, ¯ , δkin are model parameters and β controls stress tensor κ the decomposition of isotropic and kinematic hardening. The evolution of the void volume fraction f is described by
where ζ=1 for i=1,2,3 or < 1 −
ˆ pG ˆ −1 ) + f˙nucl , f˙ = f˙growth + f˙nucl = qC (1 − f ) tr(D
(3.95)
∗
which is related to f in (3.92) according to (3.98). Note that an additional material coeﬃcient qC is introduced in (3.95), which is necessary to calibrate the Gurson model according to the results from unit cell analyses [387]. f˙nucl in (3.95) represents a nucleation law according to [197] with % p 2 & − 1 f n n ˙p , exp − (3.96) f˙nucl = √ 2 sn sn 2 π where fn , sn and n are model parameters and ˙p is given by p ˆ pG ˆ −1 )2 : ˆI . ˙ = 3/2 (D
(3.97)
To describe the physical process of void nucleation adequately, the evolution of ˙p is only deﬁned for loading. In case of unloading no nucleation of micropores is considered. For the consideration of the coalescence of the micropores the phenomenological law according to [797] is used, ⎧ ⎪ ⎨ f for f ≤ fc f∗ = ⎪ ⎩ fc + K(f − fc ) for f > fc (3.98) with
K=
fu∗ − fc 1 and fu∗ = , ff − fc qA
3.3 Modelling
259
Fig. 3.126. Numerical and experimental data for (a) material softening and (b) ratcheting eﬀect
wherein f is transformed into f ∗ and an acceleration of the evolution of f is driven by a scalar factor K deﬁned by fc and ff according to (3.98). In addition to the void volume fraction f ∗ , an additional variable S is used to characterize the void shape. The evolution of S is described by an equation proposed by [258]. It is implemented into the Gursonmodel by a modiﬁcation of the material parameter qB [386]. The implementation of the model is based upon the returnmap algorithm and a consistent linearization procedure [743]. Because of the anisotropy induced by the kinematic hardening, the iterative solution involves 8 unknowns ˆ ˜ the plastic multiplier γ˙ and (the components of the symmetric gradient N, the void volume fraction f ). The proposed macroscopic elastoplastic damage model has the ability to replicate all typical phenomena of cyclic plasticity such as the Bauschingereﬀect, ratcheting or mean stress relaxation, cyclic hardening or softening [386]. In the following, a comparison of numerical and experimental data shows the eﬃciency in case of simulating the eﬀects of material softening and ratcheting. To this end, a cyclically loaded hollow cylindrical specimen of CS 1026 [86] is reanalysed numerically. Using the isotropic hardening law according to (3.93) the stress amplitude of the ﬁrst 25 load cycles can be simulated in good agreement to the experimental results, as Figure 3.126(a) shows. The use of the BariHassantype of kinematic hardening rule allows for the simulation of the ratcheting eﬀect, which is demonstrated by the evolution of the radial strain in case of biaxial loading in Figure 3.126(b). 3.3.1.2.1.2 Model Validation The following analysis are performed for 20MnMoNi55, a low alloy steel typically used for structures such as pressure vessels. For this special type of material a calibration leads to the model parameters presented in Table 3.20 [386]. After the calibration procedure, the micropore damage model is validated according to Figure 3.125 by means of fatigue tests. Therefore, results from
260
3 Deterioration of Materials and Structures
Table 3.20. Parameter of the elastoplastic micropore damage model for 20MnMoNi55 E=204 [GPa]
β=0.5
ν=0.3 [] b1 =25000 []
σY0 =220 [MPa] δiso =25 [] σY∞ =410 [MPa] Hiso =0 [MPa]
b2 =500 []
b3 =5 []
b4 =5000 []
δkin =0.18 []
c1 =500000 [MPa] c2 =60000 [MPa] c3 =3000 [MPa] c4 =100000 [MPa] κ ˘ =0 [MPa] f0 =0.01 []
S0 =0.0 []
qA =1.85 []
qB =0.48 []
qC =1.4 []
fn =0.08 []
n =3.0 []
sn =1.0 []
fkrit =0.09 []
fBruch =0.14 []
Fig. 3.127. Low Cycle Fatigue in metals: Numerical and experimental results for cyclically loaded round notched bar with (a) 2mm notch radius and (b) 10mm notch radius
cyclically loaded round notched bars with two diﬀerent notch radii (2mm and 10mm) performed by [638] are reanalyzed. To study the damage evolution due to low cycle fatigue, the results are compared in terms of the degradation of the peak reaction force in Figure 3.127. The chosen set of material speciﬁc model parameters result in a good agreement of experimental and numerical results. In particular the strong change of the slope of the reaction force curve, when coalescence becomes the dominant damage mechanism, is simulated by the micropore damage model in a good manner for both cases. This ﬁnal change state can be correlated to the life time of the structure, which is reasonably well predicted (see Table 3.21). It should be noted that he numerical simulations also allow a localization of the position of damage accumulation in accordance with the experimental observations. For the smaller notch radius the micropore damage initiates and starts to accumulate from the notch root (Fig.3.128(a,b)). In contrast, for the specimen with the larger notch radius a nearly homogeneous damage accumulation initiating from the interior of the specimen is observed(Fig.3.128(c,d)).
3.3 Modelling
261
Table 3.21. Low Cycle Fatigue in metals: Number of load cycles until failure obtained from numerical simulations and experiments 2 mm notch radius
10 mm notch radius
Experiment
Num. Model
Experiment
Num. Model
Failure initiation
17 cycles
24 cycles
31 cycles
35 cycles
Lifetime
23 cycles
28 cycles
43 cycles
40 cycles
(a )
(b )
(c )
(d )
Fig. 3.128. Low Cycle Fatigue in metals: Damage accumulation and numerically predicted damage in a cyclically loaded round notched bar: (a,b) 2 mm notch radius, (c,d) 10 mm notch radius
3.3.1.2.2 QuasiBrittle Damage in Materials 3.3.1.2.2.1 Cementitious Materials Authored by Tobias Pﬁster and Friedhelm Stangenberg Concept In highcycle fatigue processes, a large number of load cycles under moderate stress level leads to increase of strains and degradation of material properties. Diﬀerent from approaches for lowcycle fatigue, the simulation of every single load cycle is too timeconsuming for practical application. Therefore, degradation, damage and strainevolutions are modelled indirectly, depending on the applied increment of loadcycles ΔN . For a standardised evaluation and formulation, the load cycles are related to the ultimate number of loadcycles Nf according to the SN approach. This leads to the standardised time scale n ∈ [0, 1] with increments Δn = ΔN Nf . This standardised time scale is
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3 Deterioration of Materials and Structures
related to the time scale via Nf and the frequency f by: Δn =
ΔN Δt · f = Nf Nf
⇔
Δt =
Nf · Δn . f
(3.99)
This concept has already been applied in [627, 628, 629, 630]. SN Approach Thus, one basic quantity for the fatigue model is the fatigue lifetime Nf of concrete. Generally, any SN curve can be applied for its evaluation. In the following the approach presented in [392] will be reused. It has been compared to other approaches and to a large number of experiments in [627] and proved to be well suitable. It takes the loading frequency into account and distinguishes between highcycle and lowcylce fatigue: ⎧ tfat ⎪ ⎪ 1.0 − 0.0294 log + smax ⎪ ⎪ tref ⎪ ⎨ 0.062 (1 − 0.556 rfat ) . log Nf = max tfat ⎪ ⎪ 1.2 − 0.2 r + s − 0.053 (1 − 0.445 r ) log ⎪ fat max fat ⎪ ⎪ tref ⎩ 0.133 (1 − 0.778 rfat) (3.100) In this expression, tfat is the duration of one load cycle, which is the inverse of the frequency: tfat = 1/f , tref is a reference time of the same unit as tfat : tref = 1[tfat ], smax and smin are the related upper and lower stress limits, respectively: smax = σmax /fc , smin = σmin /fc and rfat is the relationship of lower to upper fatigue stress: rfat = σmin /σmax . The approach, together with a large number of experiments from the literature, is illustrated in Figure 3.129. The evaluation of the 0.05 and of the 0.95quantile of Nf , which is also shown in this diagram, will be introduced later in this section. Degradation of the Compressive Strength The degradation of the compressive strength is formulated empirically with a direct approach in the time scale of the related number of load cycles n, as introduced above. To quantify the degradation, the variable dfc is introduced and the resulting compressive strength reads as follows: fc (n) = fc,28 · (1 − dfc (n))
(3.101)
According to the experimental results presented in [70, 374], the degradation process starts very slowly. Nevertheless, fatigue failure is associated with a drop of the compressive strength onto the level of the upper fatigue stress. Thus, the value of dfc results in dfc ,fail = 1 − smax 
(3.102)
3.3 Modelling
263
related stress smax
1.0 0.9 0.8 0.7 0.6 0.5 100
102
104
106
108
1010
lifetime Nf Fig. 3.129. SN approach (0.05, 0.50, and 0.95quantiles) with experimental results
for the state of fatigue failure, n = 1. Based on an exponential approach, suggested in [370] for the description of sequence eﬀects, dfc is substructured into dfc = nafc · dfc ,fail
with afc = 26.5 − 25.0
σmax  . fc
(3.103)
This implies a faster degradation (in the time scale n = N/Nf ) of the compressive strength for higher stresses. The evaluation of the degradation is shown exemplarily for three diﬀerent load levels in the left diagram in Figure 3.130.
1.0
relative strength fc /fc,28
relative strength fc /fc,28
1.2
1.0
0.8
0.6
0.4 0.0
0.2
0.4
0.6
0.8
related cycles n = N/Nf
1.0
0.6
0.7
0.8
related cycles n = N/Nf
Fig. 3.130. Degradation of compressive strength and sequence eﬀects
0.9 0.9
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3 Deterioration of Materials and Structures
2.0
} }
N2,res /Nf,2
1.5
1.0
0.5
0.0 0.0
0.2
0.4
0.6
0.8
1.0
N1 /Nf ,1
Fig. 3.131. Evaluation of the approach for sequence eﬀects an comparison with single simulation results from [383]
Sequence Eﬀects This direct formulation of the degradation of the compressive strength includes indirectly the formulation of sequence eﬀects. As introduced, the degradation process depends on the applied upper fatigueload level. When this load level is changed, the degradation curve changes, too. As illustrated in the right diagram of Figure 3.130, this requires a modiﬁcation of the related number of load cycles n = N/Nf . Keeping n constant would imply a sudden drop or increase of the compressive strength, which is physically nonsensical. Thus, n has to be changed. The updated value of n can be evaluated from the approach for the description of the degradation process: !
dfc ,n = dfc ,n+1
⇒
n=
dfc ,n dfc ,fail,n+1
1/afc ,n (3.104)
Fig. 3.131 shows an evaluation of the presented approach together with the results of two stage tests from [383]. The mean values of the single results are plotted by solid lines. It can be seen, that the approach delivers qualitatively reasonable results, for quantitative evaluations, the data basis is too small. Strain Evolution As introduced in Section 3.1.1.2.2.1, the strain evolution in concrete under fatigue loading can be interpreted as the sum of creep and cyclic strain evolution. This interpretation is picked up for the modelling approach. In the following, the approaches for the creep strain evolution as well as for the evolution of cyclic strains is introduced.
3.3 Modelling
spring
265
frictional element
σ
σ
dashpot Fig. 3.132. Rheological element for the description of nonlinear creep processes
Creep Strain Evolution The creep strain evolution is modelled with rheological elements. The model is based on an approach presented in [735], which has been enhanced in [133, 134]. To account for the nonlinear relation of stress and creep strain rate, especially for concrete under higher stresses than approximately 0.4 fc , nonlinear rheological elements are used. They consist of a nonlinear spring with a friction element (to describe plastic deformations) and a nonlinear dashpot. Fig. 3.132 shows such elements. The nonlinear behaviour of the spring is described with the stressstrain relation for concrete under compression given in [182], with the compressive strength fc replaced by fc,T = 0.8 fc, taking longterm eﬀects into account. Thus, the stressstrain relation for the spring reads: cr 2 cr Ec fεc,T + εεc fc,T . σs = εcr c 1 − Ec fεc,T −2 εc
(3.105)
In incremental formulation, this equation can be approximated as s σn+1 ≈ σns + Etan Δεcr n+1 ,
(3.106)
where Etan is the tangent d σ s /d εcr . Analogue to the smeared crack model described earlier in this chapter, strains which result from the nonlinearity of the spring are regarded dissipative. For the modelling of creep they are regarded as plastic strains, indicated by the friction element in Figure 3.132. For linear descriptions of dashpots, the viscosity is coupled linearly via the retardation time τ with the stiﬀness: η = τ Ec . This relation is now enhanced and is reformulated dependent on time and on the applied stress: η = τ Ec
t − t0 τ
12
σd 1− fc,T
ncr .
(3.107)
Herein, σ d is the stress in the dashpot and ncr a material parameter. According to [133, 134], for concrete it takes values between 1.5 and 2.0.
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3 Deterioration of Materials and Structures
The classical relation between stress and strain rate, σ d = η ε˙cr ,
(3.108)
is now replaced by an incremental formulation: d d σn+1 ≈ σnd + Δt σ˙ n+1 .
(3.109)
The stress rate can be found as time derivative of eq. (3.108) to σ˙ d =
d (η ε˙cr ) = η˙ ε˙cr + η ε¨ , dt
(3.110)
the time derivative of eq. (3.107) results in the rate of viscosity: η˙ =
Ec 2
t + t0 τ
12 ncr σd 1− . fc,T
(3.111)
The equation for the resulting stress σ cr = σ s + σ d
(3.112)
yields a diﬀerential equation, which can be solved numerically. In [133] the Newmark method according to [569, 198, 409] is suggested. Cyclic Strain Evolution The rate of cyclic fatigue strains, in the time scale of related load cycles n:
ε˙fat,∗ =
∂εfat , ∂n
(3.113)
is formulated empirically on the basis of the experiments documented in [383]. Therefore, the typical Sshaped evolution curve of fatigue strains is devided into three parts, where a constant strain rate is assumed within each domain. The strain rates are evaluated from the experiments by linear regression, like illustrated in the left diagram of Figure 3.133 together with the experimental results from [383]. The borders between these domains are assumed at n = 0.1 and n = 0.9. In order to approximate the fatigue strain rates as functions of the applied load level, a scalar measure for the fatigue loading, taking upper and lower fatigue stress into account, is introduced as the product of mean stress and stress diﬀerence: s=
smax + smin · [smax − smin ] . 2
(3.114)
The evaluation of the experimental results of [383, 70] are shown in the right diagram of Figure 3.133. The experiments, that exhibit signiﬁcant creep
3.3 Modelling
2.0
¯ smax /smin = 0.80/0.20 ¯ smax /smin = 0.75/0.05 ¯ smax /smin = 0.95/0.05
¯ ¯
4.0
Holmen Awad & Hilsdorf approach
3.0
1.5
2.0 1.0 dom. 3 0.5
1.0
domain 2 domain 1
0.0 0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.1
related cycles n = N/Nf
0.2
0.3
0.4
strain rate −ε˙fat,∗ [10−3 ]
fatigue strain −εfat [10−3 ]
2.5
267
0.0 0.5
stress measure s
Fig. 3.133. Fatigue strain evolution (stress measure vs strain rate by [383, 70])
strains due the test duration are plotted in white. These experiments have not been used for the evolution of cyclic strain evolution. Those ones that exhibit pure cyclic strain are plotted in black. By least square ﬁtting, second order polynomials are evaluated to approximate the (pure cyclic) fatigue strain rate as function of the stress measure s: 2 ε˙fat,∗ 1,3 = −113.189 s + 67.5492 s − 4.50913 ,
(3.115)
ε˙fat,∗ = −6.54818 s2 + 4.55811 s − 0.268655 . 2
(3.116)
The right diagram in Figure 3.133 shows the polynomial as well as experimental results, exemplarily for domain 2. The results of [383] have been used for the evaluation of the polynomial. For additional proof, the results of [70] are plotted in the diagram. These values have been evaluated graphically. They are therefore regarded as too imprecise and where not taken into account for the evaluation. Fatigue Damage For the evaluation of fatigue damage, again the tests of [383] deliver the experimental basis. In these experiments, not only the evolution of the maximum, but also of the minimum4 fatigue strains is reported. That can be utilised for the subdivision of the total fatigue strains into damaging and plastic parts. At ﬁrst, the measured total strains are reduced by the initial ones. Assuming linear unloading and reloading, according to the damage theory, the fatigue strains corresponding to σ = 0 can be extrapolated, like illustrated in Figure 3.134. This yields reversible and irreversible parts of the total fatigue strains, which are interpreted as damaging and plastic, respectively. 4
That means the strains, that correspond to the lower fatigue stress level.
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3 Deterioration of Materials and Structures
σ
σ
stress
ε
0
ε − ε0 εfat
ε εir
εrev
strain Fig. 3.134. Split of total fatigue strains into reversible and irrversible parts
To distinguish the damaging part from the total fatigue strains, the variable β fat is introduced as β fat =
εda,fat . εfat
(3.117)
The evaluation of β fat , corresponding to n, is shown in Figure 3.135. It is evident, that β fat = 0.35 is a reasonable assumption. The curves of β fat for the series 0.675/0.05 (with smax /smin  = 0.675/0.05) and 0.80/0.05 exhibit signiﬁcant smaller values. The reason for these values for the series 0.80/0.20 is to be found in the relative high lower stress limit. The linear extrapolation leads,
split parameter β fat
0.8 smax /smin = 0.675/0.05 smax /smin = 0.80/0.20
0.6
0.4
0.2
0.0 0.0
0.2
0.4
0.6
0.8
related cycles n = N/Nf Fig. 3.135. Evaluation of the split parameter β fat
1.0
3.3 Modelling
269
due to the convex curvature of the unloading path, to an overestimation of the irreversible strains. The series 0.675/0.05 exhibits signiﬁcant creep strains, which are regarded as plastic. Thus, the reduction of the damaging part of total measured timedependent strains proves the assumption, that the total fatigue strains can be interpreted as the sum of time and cycledependent parts, see Section 3.1.1.2.2.1. MultiAxial Stress States To enhance the uniaxial approach for fatigue and creep strain evolution, a potential and a ﬂow rule, analogous to the classical timeindependent description of the behaviour of concrete according to the smeared crack approach, which has been introduced earlier in this section, is assumed: ε˙ fat,∗ =
∂φc ˙ fat,∗ . λ ∂σ c
(3.118)
In this equation, the derivative ∂φc /∂σ describes the direction of the strain evolution, λ˙ fat,∗ the norm. The latter one is derived from the uniaxial strain c rate and the assumption of energetic equality of uniaxial and multiaxial stress states: σeq ε˙fat,∗ 1d λ˙ fat,∗ = , c ∂φc σ: ∂σ
(3.119)
where the scalar equivalent stress σV is evaluated from the DruckerPrager potential according to eq. (3.52): φc (σ, α) =
( ) 1 μ I + J2 −αc (qc ) . 1 √1 − μ 3 ! σeq
(3.120)
Fatigue damage is quantiﬁed by a fourth order compliance tensor, analogously to the smeared crack model described earlier in this section. The evolution laws of the two independent variables read μ β 1 da,fat,∗ ˙ Ds1 λ˙ fat,∗ = 1 − √ , c √ − μ I1 6 J 2 3
da,fat,∗ D˙ s2 =
1 β √ λ˙ fat,∗ , c √1 − μ 4 J2 3
(3.121)
which is similar to eq. (3.57). Analogously to eq. (3.56), the relation for the rate of the fatigue damage compliance tensor reads
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3 Deterioration of Materials and Structures
da,fat,∗ da,fat,∗ da,fat,∗ D˙ I +¯ I , = D˙ s1 1 ⊗ 1 + D˙ s2
(3.122)
to evaluate increments of this tensor, it has to multiplied with the increment Δn of the number of related load cycles: da,fat,∗ Dda,fat = D˙ ΔD Δn .
(3.123)
Thus, the ﬁnal compliance relation, which takes damage and plastic strains due to timeindependent and timedependent modelling of the material behaviour of concrete into account, reads −1 σ = D 0 + D da,c + D da,t + D da,fat : ε − εpl,c − εpl,fat − εcr .
(3.124)
Scatter of Basic Model Properties In order to account for the scatter, which is inherent to fatigue processes, the scatter of the basic quantity Nf is investigated. According to the observations in e.g. [738, 821], the scatter of the fatigue lifetime is related to the scatter of the compressive strength fc of concrete. An approach presented in [738] takes this into account; it describes the standard deviation of log Nf as a function of the standard deviation of fc and of the applied fatigue load: s(fc ) m(fc ) s(log Nf ) = √ −σmax . 1 − rfat s(fc ) 5.714
(3.125)
In order to avoid physically nonsensical solution, the values for the upper fatigue load and for the relation of lower to upper fatigue load are limited to −σmax ≥ 0.5 fc ,
rfat ≥ 0.75 .
(3.126)
This is a reasonable suggestion that is hardly to proof experimentally: At load levels below 0.5 fc, very high fatigue lifetimes Nf occur, which makes the realisation of a large number of experiments, the basis for the evaluation of the scatter, diﬃcult. 3.3.1.2.2.2 Metallic Materials Authored by Henning Sch¨ utte It is known from experiments that most materials, and in particular brittle and quasibrittle materials, under general loading conditions develop anisotropic damage [441]. For a given stress state, materials damaged by microcracks in general accumulate additional damage through the growth of these microcracks. Considering this and the mentioned points, the concern of this paper is to provide a consistent, continuum damage model based on the micromechanical framework and the local anisotropy induced by kinking
3.3 Modelling
271
and growing elliptical and/or circular microcracks. The reason for considering elliptical and circular cracks is that these geometries are good approximations for the shape of a ﬂaw in many engineering materials. For clarity purposes and to explain the main issues of the proposed model in a more clear mathematical way, the complexity of the proposed damage model is reduced here by leaving out the thermal eﬀects and other nonmechanical phenomena. Strains and rotations are assumed to be small, hence the framework of linear elastic fracture mechanics can be applied. Furthermore, viscous eﬀects and permanent deformations are neglected and the material behavior is assumed to be linear elastic in its pristine state. The small strain assumption, and the lack of permanent deformations in this model makes it suitable to show the evolution of damage in structures with brittle and quasibrittle fracture behavior experiencing highcycle fatigue. Eﬀective Continuum Elastic Properties of Damaged Media The micromechanical models are commonly referred to a class of analytical models which give the relation between the macroscopic state of a specimen and its microstructure [164]. One of the goals of the micromechanical models is to provide relatively simple constitutive laws. Within this approach, the eﬀective elastic properties are derived by using the pertinent results of microconstituent analysis, such as that of a planar crack embedded in an inﬁnite medium. Using the concept of micromechanics, continuum damage models based on the framework of fracture mechanics and elasticity can give the local details of the damage response within a representative volume element. These class of models are based on the hypothesis of statistical homogeneity and weak interaction of defects, which are justiﬁable for reasonably modest concentration of heterogeneities [565]. In this respect, the ﬁrst step in the formulation of the proposed continuum damage model requires the formulation of the change of continuum elastic properties due to the presence, kinking and growth of elliptical and/or circular microcracks. Applying the approach of micromechanics, the components of the eﬀective compliance tensor of an inﬁnite, homogeneous, isotropic (in its pristine state) and elastic continuum damaged by a single internal circular crack is given by [440]. Here, applying the same method, the components of the tensor for the change of compliance due to the presence of a single internal elliptical crack are derived, from which the results corresponding to a single circular crack can be reproduced (see also [163]). Within the approach of micromechanics, the eﬀective elastic properties of a solid damaged by a planar internal elliptical crack are derived from the contribution to the complementary strain energy corresponding to the quasistatic, selfsimilar growth of the crack. For this, the stress intensity factors suﬃce to give the energy released during the quasistatic, selfsimilar growth of the crack. However, for the formulation of the complementary strain energy corresponding to the kinking of a crack, the analytical expressions for the so called Tstresses are required as well. The complete set for the Tstresses for internal elliptical and circular cracks
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3 Deterioration of Materials and Structures
embedded in a homogenous isotropic inﬁnite solid have been addressed by [549] and [716], respectively. Kinking of an Internal Elliptical Crack To study the degradation of the elastic material properties due to the kinking and growth of elliptical and/or circular microcracks, consider a single elliptical crack in an inﬁnite, homogeneous, isotropic and elastic continuum subjected to mechanical loads applied at inﬁnity. For a stress state outside the damage surface, the considered crack will kink and propagate to a new geometry, and the local kinking angle and the local crack extension length can be calculated from the considered fracture criterion coupled with a fatigue crack evolution law. In an analogous manner to the previous section, this problem can be decomposed into two subproblems: that of the continuum without a crack subjected to the remote traction ﬁeld, and that of the same continuum, where only the crack faces are subjected to the traction ﬁeld. In the framework of linear elasticity, the compliance tensor of a material containing a kinked crack can be decomposed into three parts [715] * = S Matrix + ΔS SCrack + ΔS S Kink , S
(3.127)
SCrack refer respectively to the compliance tensor of the where S Matrix and ΔS matrix material in its pristine state and the change of compliance due to the SKink is the change of compliance due to presence of the microcrack, and ΔS the kinking and growth of the microcrack. The analytical expression for the tensor of the change of compliance due to the presence of a single active elliptical or circular microcrack was given by [163]. In a similar way, the tensor of the change of compliance due to the kinking of a crack can be calculated from the contribution to the complementary strain energy corresponding to the kinking of the crack, which is the energy released during the kinking growth of the crack. The rate of the change of the compliance tensor for a volume element V of elastic material, attributable to the extension rate s, ˙ through which a point on the perimeter of a single crack kinks to a new position and integrating this along the crack perimeter, the rate of the change of compliance due to the growth of an internal crack is resulting + 2 Kink 1 ∂ G(s) S˙ ijmn = s˙ dl , (3.128) V c ∂σij ∂σmn where ψ ∗∗ is the complementary energy associated with the kinking of the crack and G(s) is the energy released during the kinking of crack with a local extension of s, and is given by 1 − ν2 G(s) = E
2 KI2 (s) + KII (s) +
1 2 K (s) = Mαβ Kα (s) Kβ (s) , 1 − ν III (3.129)
3.3 Modelling
273
with Kα (s) being the stress intensity factors at the propagated crack front, given by (see [45, 481]) √ Kα (s) = Kα + Kα(1/2) s + O(s) , (3.130) with Kα = Fαβ (φ) Kβ ,
Kα(1/2) = Gαβ (φ) Tβ + a Hαβ (φ) Kβ ,
(3.131)
where Fαβ , Gαβ , and Hαβ are universal functions of the kinking angle φ, Kβ and Tβ are the stress intensity factors and the Tstresses of the crack prior to kinking, and a is the curvature parameter of the crack extension which for a straight extension vanishes (a = 0). Unlike the case of a single elliptical crack, the stress intensity factors alone would not suﬃce to give the complementary strain energy corresponding to the kinking of the crack, and the analytical expressions for the Tstresses of the elliptical crack prior to kinking are required as well. [549] and [716] derived the asymptotic solutions for the stress components, based on the potential method and a transformation technique, from which the Tstresses for mixedmode internal elliptical and circular cracks in homogeneous, isotropic linear elastic solids are resulting. Considering the expansion of the stress intensity factors in terms of the extension length and the crack tip parameters prior to kinking, Eq. (3.128) can be rewritten as + Kink ∂Kα (s) ∂Kβ (s) 1 S˙ ijmn = Mαβ s˙ dl , (3.132) V c ∂σij ∂σmn with √ ∂Kα (s) ∂ (Fαλ (φ) Kλ + Gαλ (φ) Tλ s) = ∂σij ∂σij ∂Kλ ∂Tλ √ = Fαλ (φ) + Gαλ (φ) s. ∂σij ∂σij
(3.133)
The local propagation rate s, ˙ measured in the direction normal to the crack front at a considered point, can be calculated using a fatigue crack evolution law coupled with the selected fracture criterion. For example considering the following modiﬁed Paris’ law [490, 713], which is combined with the fracture criterion of maximum driving force [476], and setting a threshold value for crack growth, it results [713] s˙ =
√ η ds =C G∗ − Gth , dN
(3.134)
where C and η are constants for the considered fatigue crack evolution law, N represents the number of load cycles, G∗ is the maximum driving force acting at the (inﬁnitesimal) kinked crack tip
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3 Deterioration of Materials and Structures
G = max lim G(s) = φ
s→0
' 1 − ν2 1 2 [(KI )2 + (KII (KIII ) + )2 ]'φ=φ E 1−ν (3.135)
and Gth is the threshold value, below which there is no damage growth. Here, the threshold value Gth is considered to take into account the load history and is varying by crack growth. Indeed, its value is decreasing by damage growth for brittle and quasibrittle materials. The expression for the fourthrank tensor of the rate of the change of compliance due to the kinking of an elliptical crack with a local growth rate s˙ is derived by substituting relations (3.133) and (3.134) into (3.132), and performing the requisite integration along the crack front in the following form Kink 1 2π S˙ ijmn = Mαβ α2 sin2 ϕ + β 2 cos2 ϕ 0 V , ∂Kλ ∂Kμ ∂Kλ ∂Tμ Fαλ Fβμ + Fαλ Gβμ + ∂σij ∂σmn ∂σij ∂σmn ∂Tλ ∂Kμ √ ∂Tλ ∂Tμ Gαλ Fβμ s + Gαλ Gβμ s s˙ dϕ , ∂σij ∂σmn ∂σij ∂σmn (3.136) where the Tα and Kα are given by [549]. A Fatigue Fracture Based Anisotropic Continuum Damage Model Generally, it is impossible to formulate a damage model covering the mixedmode propagation of microcracks in a fully traceable way. Considering microcracks in the form of elliptical and circular cracks, one may calculate the kinking of the initial cracks analytically only for the ﬁrst growth increment. After the kinking of the initial cracks, however, the mathematical formulation of the next kinking steps is no longer possible. To overcome this diﬃculty, some researchers have introduced models based on simplifying assumption such as ﬁxing the plane of crack growth, so that microcracks may only grow in a selfsimilar and/or coplanar manner [612, 860, 101]. This assumption may be acceptable for the case of monotonic loading and proportional loading, but for loads with changing direction and amplitude, as is the case for nonproportional loads or even sequential loads, this assumption leads to underestimating the damage, since it does not allow for crack kinking. The assumption of selfsimilar growth of mixedmode cracks, in general, results in a smaller damage accumulation than what the real mixedmode kinking results in. To overcome the diﬃculties in the formulation of a damage model, which accounts for the kinking and growth of microcracks in a mathematical traceable manner, the objective of this section is to propose a micromechanical based continuum damage model, which is based on the reduction of stiﬀness due to kinking elliptical microcracks. To be able to formulate the model in a
3.3 Modelling
275
fully mathematical traceable way, the concept of an equivalent elliptical crack is introduced in the sense that a kinked crack is replaced with an equivalent elliptical crack, resulting in an equivalent dissipation of energy. Basically, eight degrees of freedom can be considered for each equivalent elliptical microcrack replacing the kinking one. These are the major and minor axes of the ellipse (2 unknowns), orientation of the microcrack given by three Eulerian rotation angles (3 unknowns), and the position of the crack in the space (3 unknowns). Considering the concept of unit cell and assuming that the microcrack is located in the center of the cell, the position of the microcrack can be ﬁxed and may be left out of the formulation. This is because in the case of noninteracting cracks, the position of the crack does not have an impact on the elastic properties of the material. To calculate the other ﬁve unknowns characteristics of the equivalent crack, the following postulates are proposed • •
equivalent type of damage induced anisotropy, and .Crack = ΔS SCrack + ΔS SKink , equivalent change of compliance tensors ΔS ijmn
ijmn
ijmn
where quantities with a hat indicate the ones corresponding to the equivalent replacement crack. With this, the geometry and the orientation of the equivalent elliptical crack replacing the kinked one then result by considering two optimization problems. The ﬁrst postulate takes into account the fact that the local damage associated with a single planar elliptical crack results in an orthotropic material symmetry [163], thus it can be argued that changing the type of material symmetry from isotropy to orthotropy may imply the existence of local damage due to an elliptical crack. [714], have shown that the damage associated with a growing mixedmode elliptical or circular crack also changes the virgin isotropic material into approximately orthotropic one. Considering the mentioned points, it is deduced that the orientation of the equivalent crack replacing the kinked one is such that the resulting orthotropy axes are aligned with the ones due to the damage associated with the kinked crack. It is reminded that for a single elliptical crack in an initially isotropic material, the axes of orthotropy are aligned with the axes of the ellipse, i.e. two axes of orthotropy are aligned with the major and minor axes of the ellipse and the third one is the normal to the plane of the ellipse. The second postulate results in the geometry of the equivalent elliptical crack in the sense that the components of the change of compliance due to the kinked crack and the equivalent crack are approximately identical. Such a formulation of the dissipative damage process due to kinking elliptical microcracks, taking into account the damage induced orthotropy of an elliptical crack in a local sense, results in a consistent damage model capturing the load history through the local orthotropic degradation of the mechanical properties.
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Optimization Subroutines As mentioned in the previous section, based on the assumption of equivalent type of damage induced anisotropy, here orthotropy, the orientations of the equivalent crack are resulting. For this purpose the following optimization procedure is considered. Considering the fact that the damage associated with a growing elliptical or circular crack changes the virgin isotropic material properties into approximately orthotropic one [714], it is possible to ﬁnd the orthotropy axes using an optimization method. The idea of the optimization procedure is to ﬁnd the coordinate transformation that yields the best representation of a tensor with a known type of symmetry. Thus, the optimization algorithm consists of ﬁnding the appropriate coordinate transformation for the measured compliance tensor in a known coordinate system that yields the best orthotropic representation. Considering the transformation law for fourthrank tensors, one possible way is considering the following optimization procedure which looks for the best Eulerian rotation angles minimizing the nonorthotropic components of the corresponding compliance tensor F1
=
non.ORT i,j,m,n
2 S ijmn (R(γx , γy , γz ))
S1213 )2 + (S S1223 )2 + (S S1323 )2 + = (S Sii12 )2 + (S S ii23 )2 + (S Sii13 )2 , + 3i=1 (S 0 / ' RORT (. γx , γ .y , . γz ) = arg min F1 (R) ' det R = 1, RT = R−1 , ⇒ where
(3.137)
0 / ORT S ORT , ijmn = ORT S ijmn R
S ijmn = Rir Rjs Rmt Rnu S rstu ,
where R is an orthogonal transformation including all rotations about the three cartesian axes (R(γx , γy , γz )), Figure 3.136, ORT (•) is the operator that nulls out the nonorthotropic components of S ijmn deviating from zero, and S and S are the compliance tensors in the considered global coordinate system and the local orthotropic coordinate system, respectively. The geometry of the equivalent crack is resulting from the postulate of equivalent tensors for the change of compliance. This is done by a direct comparison of the change of compliance tensors associated with the damages due to the kinked crack and due the equivalent elliptical crack, given with respect to a known coordinate system (local coordinates of the equivalent crack may be a good choice). In this regard, the following optimization problem should be solved
3.3 Modelling
y
da
y
γy
γ ^y
2 α
2 α ^
γx β
3 z
x γz
1
277
γ ^x
^ β 3 z
x γ ^z
φ
1
Fig. 3.136. Kinked crack and its equivalent elliptical crack
F2 =
2 3 .ijmn α S ., β. − S ORT , ijmn
i,j,m,n
/ 0 / ' 0 α ., β. = arg min F2 'α . ≥ αo , β. ≥ βo ,
(3.138)
. is the tensor of the change of compliance due to the equivalent elwhere S . = β/. . α, the components of which are liptical crack with an aspect ratio of λ given in the local crack coordinates in Eq. (3.136), and αo and βo are the semimajor and semiminor axes of the elliptical crack prior to kinking. The proposed continuum damage model based on the reduction of stiﬀness due to the kinking of equivalent elliptical microcracks results in the eﬀective elastic properties of a damaged material volume element in a consistent way. Based on the incremental analysis of the eﬀective elasticity tensor for the given current values of the local stress and strain tensors, and taking the load history into account by introducing the concept of an equivalent elliptical crack, the propagation of microcrack is calculated by considering a crack evolution law. In this study, the propagation of microcracks is governed by the modiﬁed Paris’ law given by Eq. (3.134) coupled with the fracture criterion of maximum driving force [476]. In the incremental formulation, to have a more stable algorithm and for a faster convergence, the tangent stiﬀness of the damage material, the so called algorithmic tangent [749], should be introduced. The proposed damage model can also capture the unilateral eﬀect observed in tensioncompression tests, observed for a certain classes of materials including ceramics and concrete [156, 391], provided that for a passive crack the corresponding components of the compliance components are recovered, and they return to their degraded state upon the activation of the crack.
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3 Deterioration of Materials and Structures
Numerical Examples The proposed continuum damage model, based on the reduction of stiﬀness due to kinking elliptical microcracks can be easily implemented in a ﬁnite element code. This has been performed here in the commercial ﬁnite element analysis software Ansys as a user material subroutine. In the following sections, a variety of numerical examples is presented. The ﬁrst two examples are illustrative examples, which explain the proposed model in a fully analytical manner. The ﬁrst example addresses the determination of the equivalent crack after a single kink step. As the second example, a unit cell damaged by a single mixedmode circular crack is subjected to four stages of cyclic loadings with changing directions. With this, the degradation of the material properties and the evolution of the considered damage parameters are presented. This example is especially important, since it provides the reader with the details of the irreversible damage process due to growing microcracks subjected to cyclic loading with changing directions. The objective of the other examples is to show the mesh sensitivity of the model. All experiments are conducted on AISI 4130 steel with the mechanical properties and the chemical composition given in [394], except the ﬁrst illustrative example. Example1 The objective of this section is to explain the proposed continuum damage model by giving an illustrative example in a fully analytical manner. The considered crack problem is a circular crack with γz = 45◦ and γx = γy = 0◦ as the initial orientations. The loading level is σ ∞ /E = 1/1000, the Poisson’s ratio ν = 1/3, and the initial crack size is considered to be αo /L = 1/150 with L being the characteristic length of the volume element. To avoid more complexity, it is assumed that the stress state is outside the damage surface, thus the considered circular microcrack will propagate, and at this step due to simplicity purposes, the W¨ohler’s limit stress (i.e. also the threshold for crack growth) is assumed to be negligible. The ﬁrst step in the model is calculating the propagation of the crack under the given local stress state. Considering Eq. (3.134), the rate of crack growth is √ η s˙ = C G∗ , (3.139) where C and η are the Paris’ parameters. For clarity purposes, the propagation of the crack is exaggerated by selecting these parameters as C = 10−2 and η = 2. This and the considered loading level and material parameters result in crack extension length of approximately 12% of initial crack radius αo . The propagation parameters, i.e. the extension length and the kinking angle are resulting as functions of the geometrical angle ϕ along the crack front. With these the new crack geometry is resulting (Figs.3.137left). The next step is to calculate the inﬂuence of crack growth on the compliance tensor through relations (3.136). These result in the eﬀective compliance tensor modiﬁed by
3.3 Modelling
σ∞
σ∞
y 45
y
◦
z
1
2.65◦
^ 2
2
2
x 3
279
x 3 z
1
Fig. 3.137. Growth of the circular crack and its equivalent elliptical crack
the damage due to the propagated circular crack. In sixdimensional tensorial notation, normalizing the components of the compliance tensor with the tensile compliance component of the pristine material 1/E, and the factor (αo /L)3 , where αo and L are respectively the characteristic initial size of the crack and the length of the unit cell, results in: ⎞ 0.0527 0.0185 −0.0034 0.1884 0 0 ⎟ ⎜ ⎟ ⎜ ⎜ 0.0185 5.5727 0.0092 0.2988 0 0 ⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎜ 0 ⎟ 1 αo 3 ⎜−0.0034 0.0092 0.0193 0.0074 0 ⎟ SC = ΔS ⎟. ⎜ ⎟ ⎜ E L ⎜ 0.1884 0.2988 0.0074 3.7958 0 0 ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ 0 0 0 0 3.5995 0.1220⎟ ⎟ ⎜ ⎠ ⎝ 0 0 0 0.1220 0.0396 ⎛
(3.140) To ﬁnd the equivalent elliptical crack which replaces the propagated crack, it suﬃces to perform two further steps. The ﬁrst step is ﬁnding the axes of orthotropy due to the damage induced by crack growth which can be deduced as the local coordinates of the equivalent crack. The second step is to ﬁnd the geometry of the equivalent crack, the local axes of which are aligned with the axes of orthotropy calculated at the previous step. These two steps are performed by solving the optimization problems, given by Eqs. (3.137) and (3.138), respectively The application of this procedure to the considered
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example results in the following parameters for the equivalent elliptical crack replacing the current propagated crack γ .x 0◦ ,
γ .y 0◦ ,
γ .z −2.65◦ ,
β. 1.00 , βo
α . = 1.14755 , αo
where γ .’s are given with respect to the local coordinates of the crack prior to propagation. The resulting local Eulerian angles result in the best approximate orthotropic representation of tensor (3.139), which is indeed given in the local coordinate system of the equivalent elliptical crack replacing the propagated one ⎞ ⎛ 0.0441 0.0071 −0.003 0 0 0 ⎟ ⎜ ⎟ ⎜ ⎜ 0.0071 5.6042 0.0097 0 0 0 ⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎜ α 3 −0.003 0.0097 0.0193 0 0 0 ⎟ ORT 1 ⎟ ⎜ o . ΔS = ⎟. ⎜ C ⎟ ⎜ E L ⎜ 0 0 0 3.7729 0 0 ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ 0 0 0 0 3.6032 0 ⎟ ⎟ ⎜ ⎠ ⎝ 0 0 0 0 0.0359 (3.141) Example2 The objective of this example is to provide a better insight to the irreversible process of brittle damage in a local sense. For this a representative volume element of AISI 4130 steel is considered which embeds a single circular crack. The circular crack in its initial conﬁguration has an inclination angle of 45◦ with respect to the 2axis (Figure 3.138). The initial size of the crack is considered to
σ1
σ4
2 3
2 1
3
2 1
3
2 1
σ3
3
σ2 stage 1
stage 2
stage 3
Fig. 3.138. Order of the considered sequential loading
stage 4
1
3.3 Modelling
281
Table 3.22. Characteristics of the applied sequential loading stage 1
stage 2
stage 3
stage 4
stress σ1 cycles (MPa) N1
stress σ2 cycles (MPa) N2
stress σ3 cycles (MPa) N3
stress σ4 cycles (MPa) N4
1500
1400
1350
1450
100,000
Α Αo
250
100
100,000
Γz Γy
crack orientation
150
100,000
40
Β Βo
200 crack size
100,000
Γx
30
20
10
50 0 0 0
100000 200000 300000 400000 number of loadcycle N
0
100000 200000 300000 number of loadcycle N
400000
Fig. 3.139. Evolution of the geometry and the orientations of the equivalent elliptical crack
be L/500, L being the characteristic size of the unit cell. The constants in the modiﬁed Paris’ equation (3.134) are chosen as C = 10−5 and η = 2. To study the local degradation of the considered material under fatigue conditions with the help of the proposed model, the unit cell is subjected to four stages of cyclic loading as presented in Table 3.22 and Figure 3.138. In the load stage 1, a stress of 1500 (MPa) is applied in the direction of 2axis for 100, 000 cycles, and in the subsequent stages the direction of the applied stress is parallel to the 3axis, 1axis, and 2axis, respectively. The applied stresses are 1400 (MPa), 1350 (MPa), 1450 (MPa) and the corresponding number of cycles are the same as for the stage 1. The proposed optimization subroutines are solved at each load increment, in order to calculate the geometry and the rotation of the equivalent elliptical crack replacing the kinked one. Figure 3.139 shows the evolution of the geometrical parameters and the orientation of the equivalent elliptical crack with respect to the number of cycles. It is observed that up to approximately 300, 000 loading cycles, the
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3 Deterioration of Materials and Structures
evolution of the corresponding damage is relatively smooth, and at this moment application of the load stage 4 accelerates the accumulation of damage. To explain this, lets review each load stage separately. During load stage 1, it is observed that by crack growth the equivalent elliptical crack rotates to become perpendicular to the direction of σ1 . This is more evident in Figure 3.139right, where at N = 100, 000 the equivalent crack has an inclination of approximately γz = 10◦ with direction1. Changing the load to the load stage 2 results in no observable damage accumulation in the material, because at this stage, the equivalent crack is parallel to σ2 . Hence, very small changes in the presented parameters are induced, which due to the scaling of the curves is not observable. The load stage 3, similar to the load stage 2 has small contribution to the process of damage growth, since at this stage, plane of the crack is inclined nearly 10◦ with respect to the direction of σ3 (identical to the direction1). This small angle induces a relatively small normal stress on the plane of the equivalent crack, leading to a relatively small driving force along the crack front, which consequently results in a slow propagation of the crack. This is observed between cycles 200, 000 and 300, 000 in Figure 3.139, as γz grows again bigger. At the end of this stage, γz has reached a value of approximately 15◦ . Application of the load stage 4, however, accelerates the propagation of the crack, since at this stage the orientation of the equivalent crack is such that the local normal stress acting on the plane of the crack induces a strong driving force on the crack front. As can be deduced from Figure 3.139, the dimensions of the crack evolve very fast to damage the whole material volume. Figure 3.140 shows the degradation of the stiﬀness components in the principle loading directions. It is observed that at the end of the load stage 4, the
1 0.95 0.9 E Eo 0.85
principle direction 1
0.8
principle direction 2
0.75
principle direction 3
0.7 0
100000 200000 300000 number of loadcycle N
400000
Fig. 3.140. Evolution of the stiﬀness components in the principle directions
3.3 Modelling
283
u, F
r2 h2 R h1
r1
mesh A
mesh B
mesh C
Fig. 3.141. Specimen geometry and diﬀerent mesh patterns
stiﬀness in direction3 does not show a considerable change, while in directions 1 and 2, the degradation of stiﬀness is obvious. Mesh Sensitivity The objective of this example is to show the degree of the mesh dependency for the proposed damage model. For this, the specimen of the form given in Figure 3.141 is considered, where r1 = 10 mm , r2 = 20 mm , h1 = 100 mm , h2 = 20 mm , R = 505 mm . Displacement controlled analyses have been performed for diﬀerent discretization levels (Figure 3.141), where a constant displacement with the magnitude of 2% of the specimen’s initial length h1 + h2 is applied for 2, 000, 000 cycles. All models are meshed with the help of hexahedral elements with quadratic displacement behavior. Considering mesh pattern A, 48 hexahedral elements are generated, and mesh patterns B and C result in 384 and 1728 elements, respectively. The corresponding constants of the damage model are chosen as η = 2 , C = 7.39 × 10−9. The resulting forcecycle curves for these experiments are given in Figure 3.142, where FN and Fo are the resultant forces at the end cross section of the specimen (r2 ), which correspond respectively to the current loadcycle and the initial loadcycle. The good agreement between the results corresponding to diﬀerent discretization levels demonstrate the mesh independency of the model. This is due to the fact that in the considered fatigue microcrack evolution law, the rate of the driving force is not appearing, as the process is parametrized with the number of cycles as a timelike parameter.
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3 Deterioration of Materials and Structures
1 0.95 FN Fo
0.9 mesh A mesh B
0.85
mesh C
0
0.5 1 1.5 number of loadcycle N 106
2
Fig. 3.142. Loadcycle curves for diﬀerent mesh patterns
Summary and Conclusions A micromechanical based continuum damage model based on the reduction of stiﬀness due to kinking elliptical microcracks has been proposed to show the anisotropic irreversible process of damage accumulation due to microcrack kinking and growth in brittle and quasibrittle materials. The model is formulated consistently in a fully analytical way and the degradation of the elastic properties is associated with the irreversible process of crack kinking and growth. In order to make the formulation of the model mathematically traceable, the concept of an equivalent elliptical crack is proposed. The geometry and the orientation of the equivalent cracks are resulting from the postulates of equivalent dissipation and equivalent type of damage induced anisotropy. The proposed formulation yields a consistent damage model suitable for predicting the failure of structures and mechanical components subjected to fatigue conditions, independent of the type of loading. Accounting for the kinking and growth of microcracks and the type of damage induced anisotropy in the formulation of damage models is especially important in the case of nonproportional loads or even sequential loads, since the assumption of selfsimilar growth of mixedmode cracks may underestimate the accumulated damage. The small strain assumption, and the lack of permanent deformations in this model makes it suitable to show the evolution of damage in structures with brittle and quasibrittle fracture behavior experiencing highcycle fatigue. The proposed damage model can also capture the unilateral eﬀect observed in tensioncompression tests, observed for a certain classes of materials including ceramics and concrete, provided that for a passive crack the corresponding components of the compliance components are recovered, and they return to their degraded state upon the activation of the crack.
3.3 Modelling
285
3.3.2 Nonmechanical Loading and Interactions Authored by Otto T. Bruhns and G¨ unther Meschke 3.3.2.1 ThermoHygroMechanical Modelling of Cementitious Materials  Shrinkage and Creep Authored by Max J. Setzer and G¨ unther Meschke 3.3.2.1.1 Introductory Remarks While in engineering practice drying shrinkage is accounted for by means of shrinkage strains εs depending on an empirically determined shrinkage coeﬃcient [182], recent progress in computational durability mechanics (see, e.g. [800, 142, 82]), together with appropriate numerical methods open the perspective of a more fundamental approach to obtain not only a better insight into the degradation mechanisms resulting from the interaction between mechanical, thermal and hygral loading but also to provide more reliable estimates for the (residual) lifetime of concrete structures. In particular, thermodynamics of deformable porous media according to the BiotCoussytheory [122, 211] provides a suitable framework for the homogenization of microscopic or submicroscopic quantities to describe coupled hygromechanical mechanisms and damage on a macroscopic level. Within this framework, a nonlinear poroelastic model was employed in [212] for the modeling of drying shrinkage of concrete structures. In recent extensions of the theory of porous media, damage mechanics [89, 167, 177, 144, 533, 320, 533] and the viscous behavior of brittle materials [176] were taken into account. For saturated porous media the concept of plastic eﬀective stress has been introduced at a macroscopic level as the thermodynamic force associated with the plastic strain of the skeleton [211]. This concept has been extended to partially saturated materials by [167] and, using micromechanical considerations, by [320, 321, 533]. In attempts to describe drying creep processes in concrete structures, the coupling between spatially and temporarily varying moisture distributions and the extent of long term creep have been considered by [110] and [320]. Cracks, irrespective of their origin, have a considerable inﬂuence on the moisture permeability of cementitious materials. As a consequence, the transport of aggressive substances may be promoted and the degradation process is further accelerated (see Subsection 3.1.2.3). In this Subchapter, a 3D coupled thermohygromechanical elastoplastic damage model for durabilityoriented ﬁnite element analyses of concrete structures, formulated within the framework of the BiotCoussytheory [122, 211] that has been developed in [321, 533, 320] is presented. In accordance with the hygral processes acting on the various levels of the nanoporous skeleton (nano, micro and capillary pores) (see Subchapter 3.1.2.2.1 the eﬀect of shrinkage is
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3 Deterioration of Materials and Structures
taken into consideration by means of a macroscopic capillary pressure which represents, on a macroscopic level, hygrally induced stresses of various sources [533]. In addition to deformations and cracking resulting from drying shrinkage, the eﬀect of cracks on the moisture transport, the moisturedependence of the strength and stiﬀness of concrete and deformations resulting from longterm creep are considered in this model. Since longterm creep is associated with dislocationlike processes in the nanopores of the cement gel which are prestressed by hygrally induced stresses (disjoining pressure) [93], creep depends on the moisture content. This coupling between moisture transport and creep deformations is also considered in the model. The speciﬁc coupled thermohygromechanical material behaviour of concrete described in the Subsection if formulated within the context of thermodynamics of deformable porous media and based on the BiotCoussytheory [122, 211]. In the model, concrete is assumed to consist of the matrix material (subscript s) – a mixture of cement paste and the aggregates – and the pores, which are partially saturated by liquid water (subscript l) and an ideal mixture of water vapour and dry air (subscript g). Provided that there is thermodynamic equilibrium between the mixture of water vapour and dry air and the external atmosphere, it is often assumed that the gaseous phase is at constant atmospheric pressure, taken as zero [97]. Therefore, for the sake of simplicity, the capillary pressure is expressed as pc = −pl in what follows. 3.3.2.1.2 State Equations Coupled phenomena on the microlevel of cementitious materials are described in a macroscopic framework using state variables. For the present 3D model, the function of free energy Ψ Ψ = W(ε − εp − εf , ml − ρl φpl , ψ, T ) + U(αR , αDP ),
(3.142)
depends on three external variables (ε, ml , T ) and six internal variables (εp , εf , φpl , ψ, αR , αDP ) [211, 213]. In Equation (3.142) ε denotes the linearised strain tensor, εp is the tensor of plastic strains, εf are the ﬂow strains corresponding to longterm creep eﬀects, ml denotes the liquid mass content variation, ρl is the mass density of the liquid phase, φpl stands for the nonrecoverable portion of the porosity occupied by the liquid phase, ψ = 1 − d is the integrity with d denoting the isotropic damage parameter 0 ≤ d ≤ 1, T denotes the absolute temperature and αR and αDP characterise the inelastic pre and postfailure behaviour of concrete in tension (subscript R) and compression (subscript DP). From the entropy inequality, the state equations are obtained as ∂W ∂W σ= ; p l = ml ; ∂(ε − εp − εf ) ∂( − φpl ) (3.143) ρl ∂U ∂U ∂W ; qR = − ; qDP = − , S=− ∂T ∂αR ∂αDP
3.3 Modelling
287
where σ is the total stress tensor, pl is the liquid pressure and S is the entropy. qR and qDP are the thermodynamic forces conjugate to αR and αDP , which determine the damagedependent size of the damage (fR ) and loading (fDP ) surface in the stress space. The diﬀerentiatial form of (3.143)) is obtained as dml p ed p f dσ = C u : (dε − dε − dε ) − ψM B (3.144) − dφl ρl + Λu dψ − Au dT, dpl = ψM
dml − dφpl ρl
− ψM B : (dε − dεp − dεf )
(3.145)
+ Ξdψ + LdT, Cu dT + Au : (dε − dεp − dεf ) + sl (dml − ρl dφpl ) T0 dml p −L − dφl + Πdψ. ρl
dS =
(3.146)
The symbols introduced in (3.144) – (3.146) represent the mixed partial derivatives of the free energy and can be interpreted as follows: Cu denotes the undrained fourthorder stiﬀness tensor, • C ed u = ψC • the term ψM B represents the hygromechanical couplings with M as the isotropic Biot modulus and B = b1 as the secondorder tensor of tangential Biot coeﬃcients b, • Λu is the undrained secondorder tensor describing the coupling mechanisms between damage evolution and the total stress increment, • Au = C ed u : 1αt,u denotes the undrained secondorder thermomechanical coupling tensor with αt,u as the undrained thermic dilatation coeﬃcient, • Ξ is a coupling coeﬃcient connected with the change of the liquid pressure due to damage evolution, • L = 3ψM αt,u characterises the thermohygral coupling mechanisms, • Cu denotes the undrained volume heat capacity and T0 the reference temperature, • sl is the internal entropy of the liquid phase and • T0 Π represents the latent heat due to damage evolution. Inserting (3.145) into (3.144) yields an alternative drained formulation for the diﬀerential stress tensor as dσ = C ed : (dε − dεp − dεf ) − Bdpl + Λdψ − AdT, with the drained stiﬀness tensor C, C ed = ψ C u − M b2 (1 ⊗ 1) = ψC
(3.147)
(3.148)
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3 Deterioration of Materials and Structures
the drained thermomechanical coupling tensor A = Au − 3αt,u M B = C ed : 1αt ,
(3.149)
and the drained tensor Λ = Λu + ΞB,
(3.150)
respectively [321]. 3.3.2.1.3 Identiﬁcation of Coupling Coeﬃcients According to [321] the poroelastic hygromechanical coeﬃcients b and M can be determined by relating diﬀerential stress and diﬀerential strain quantities deﬁned on the mesolevel to respective homogenised quantities on the macrolevel. The soobtained tangential Biot coeﬃcient is determined as K b = Sl 1 − ψ , (3.151) Ks which includes the expression b = Sl suggested by [211] for the special case of poroelastic materials with incompressible matrix behaviour. An expression for the Biot modulus M = ψM is obtained as −1 Sl pl ∂Sl φSl Sl (b − φSl ) M = φ 1− + + (3.152) Ks ∂pl Kl Ks see [705, 493] for a similar formulation. For cementitious materials, expression (3.152) can be replaced by −1 ∂Sl M≈ φ . (3.153) ∂pl In the special case of a fully saturated material (Sl = 1), (3.152) yields the classical relation [211, 493] −1 φ (b − φ) M Sl =1 = + . (3.154) Kl Ks The coeﬃcients related to damage phenomena Λ and Ξ are identiﬁed by exploiting the symmetry relations that are connected to the existence of a macroscopic potential. Using the Maxwell symmetries, the drained tensor Λ can be expressed as [533] K p f Λ = C : (ε − ε − ε ) + Sl dpl 1 − C : 1αt T, (3.155) Ks pl and the coupling coeﬃcient Ξ is obtained as M Sl K Ξ= Sl dpl ≈ 0. Ks2 pl
(3.156)
3.3 Modelling
289
3.3.2.1.4 Eﬀective Stresses The concept of eﬀective stress [281, 791] is a generally accepted approach in soil mechanics for the determination of stresses in the skeleton of fully saturated soils. In addition to the original proposal of [791], several alternative suggestions for the deﬁnition of eﬀective stresses exist, taking the compressibility of the matrix material or the porosity into account (see e.g. [123, 587, 128]). Based on the relevance of the concept of eﬀective stress for the analysis of fully saturated soils, this concept has also been adapted for the description of partially saturated soils. Early formulations introduced the capillary pressure in the (elastic) eﬀective stress deﬁnition [127]. However, diﬃculties to obtain satisfactory agreements with experimental results have motivated the use of two independent stress ﬁelds for the constitutive modelling of unsaturated soils (see e.g. [129, 44]). As far as the numerical modelling of partially saturated cementbased materials is concerned, the assumption of (elastic) eﬀective stresses seems not to be well suited for the description of shrinkageinduced cracks using stressbased crackmodels. However, the concept of plastic eﬀective stress ﬁrst introduced at a macroscopic level by [210] for saturated porous media (see [211] for details), allows to overcome these diﬃculties in the framework of poroplasticity – porodamage models. The proposed form of the plastic eﬀective stress is the same as the classical Biottype, however, a plastic eﬀective stress coeﬃcient is used. A similar form has been derived from micromechanical considerations by [510]. This concept has been recently extended to partially saturated materials [167, 533], and is also adopted in the present formulation. From the coupled relations between total stresses, strains, liquid saturation and temperature C : (ε − εp − εf ) σ = ψC
K + 1−ψ Ks
(3.157) Sl (pc )dpc 1 − AT,
pc
the following deﬁnition of the elastic eﬀective stress tensor C : (ε − εp − εf ) − AT, σ e = ψC
(3.158)
with
K Sl (pc )dpc 1. σ = σ e + 1 − ψ Ks pc
(3.159)
is obtained. The plastic eﬀective stress tensor σ p = σ , deﬁned as σ = σ − bp pc 1,
(3.160)
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3 Deterioration of Materials and Structures
characterises the thermodynamic force associated with the plastic strain rate [211]. In contrast to the elastic eﬀective stress tensor, σ represents the macroscopic counterpart to matrixrelated microstresses with the coeﬃcient bp as the plastic counterpart of the Biot coeﬃcient b. By relating stress quantities on the mesoscale to respective macroscopic quantities, a possible identiﬁcation of bp as a function of the integrity ψ, the porosity φ and the liquid saturation Sl can be accomplished as bp = ψφSl (pc ) ,
(3.161)
see [321] for details. 3.3.2.1.5 Multisurface DamagePlasticity Model for Partially Saturated Concrete According to the concept of multisurface damageplasticity theory, mechanisms characterised by the degradation of stiﬀness and inelastic deformations are controlled by four threshold functions deﬁning a region of admissible stress states in the space of plastic eﬀective stresses σ E = {(σ , qk ) fk (σ , qk (αk )) ≤ 0, k = 1, ..., 4} .
(3.162)
In (3.162), the index k = 1, 2, 3 stands for an active cracking mechanism associated with the damage function fR,k (σ , qR ) and k = 4 represents an active hardening/softening mechanism in compression associated with the loading function fDP (σ , qDP ). Cracking of concrete is accounted for by means of the Rankine criterion, employing three failure surfaces perpendicular to the axes of principal stresses fR,A (σ , qR ) = A − qR (αR ) ≤ 0,
A = 1, 2, 3.
(3.163)
In (3.163), the subscript A refers to one of the three principal directions and qR (αR ) = −∂U/∂αR denotes the softening parameter. The ductile behaviour of concrete subjected to compressive loading is described by a hardening/softening DruckerPrager plasticity model qDP (αDP ) fDP (σ , qDP ) = J2 − κDP I1 − ≤ 0, (3.164) βDP with qDP (αDP ) = −∂U/∂αDP as the hardening/softening parameter. The determination of the model parameters κDP and βDP is based on the ratio of the biaxial and the uniaxial compressive strength of concrete fcb /fcu as [534] fcb /fcu − 1 1 , (3.165) κDP = √ 3 2fcb /fcu − 1 √ 2fcb /fcu − 1 βDP = 3 , (3.166) fcb /fcu
3.3 Modelling
291
whereby fcb /fcu is approximately equal to 1.16. The fracture energy concept is employed to ensure meshobjective results in the postpeak regime. Details of the material model are found in [534]. For an eﬃcient implementation of the multisurface model based on an algorithmic formulation in the principal stress space reference is made to [531]. The evolution equations of the tensor of plastic strains ε˙ p , of the reciprocal value of the integrity (ψ −1 )˙, of the plastic porosity occupied by the liquid phase φ˙ pl and of the internal variables α˙ R and α˙ DP are obtained from the postulate of stationarity of the dissipation functional [318] as ε˙ = (1 − β) p
4 k=1
γ˙ k
∂fk , ∂σ
(3.167)
∂fk ∂fk : Cu : ∂σ ∂σ γ˙ k , (ψ )˙ = β ∂fk k=1 : σ ∂σ 4 ∂fk γ˙ k : 1bp , φ˙ pl = ∂σ −1
4
(3.168)
(3.169)
k=1
α˙ R =
3 A=1
γ˙ R,A
∂fR,A , ∂qR
α˙ DP = γ˙ DP
∂fDP , ∂qDP
(3.170)
together with the loading/unloading conditions fk (σ , qk ) ≤ 0; γ˙ k ≥ 0; γ˙ k fk (σ , qk ) = 0.
(3.171)
The parameter 0 ≤ β ≤ 1 contained in (3.167) and (3.168) allows a simple partitioning of eﬀects associated with inelastic deformations due to the crackinduced misalignment of the asperities of the crack surfaces, resulting in an increase of inelastic strains εp , and deterioration of the microstructure, resulting in a decrease of the integrity ψ. An elastoplastic model ((ψ −1 )˙ = 0, ε˙ p = 0) and a damage model ((ψ −1 )˙ = 0, ε˙ p = 0) are recovered as special cases by setting β = 0 and β = 1, respectively. 3.3.2.1.6 LongTerm Creep Consideration of longterm or ﬂow creep eﬀects is accomplished in the framework of the microprestresssolidiﬁcation theory [93]. The evolution law of the ﬂow strains is based on a linear relation between the rate ε˙ f and the stress tensor σ as 1 ε˙ f = G ed : σ, (3.172) ηf (Sf ) −1 with the fourthorder tensor G ed = E C ed and Young’s modulus E. The viscosity ηf is a decreasing function of the microprestress Sf and can be written as [93]
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3 Deterioration of Materials and Structures
1 = cpSfp−1 , ηf (Sf )
(3.173)
where c and p > 1 are positive constants. According to [93], the microprestress relaxation is connected to changes of the disjoining pressure. Consequently, variations of the internal pore humidity h due to drying, which entail a changing disjoining pressure, lead to a change of the microprestress Sf . This mechanism partially explains the Pickett eﬀect [631], also called drying creep. 3.3.2.1.7 Moisture and Heat Transport Starting with a simpliﬁed nonlinear diﬀusion approach, in which the diﬀerent moisture transport mechanisms in liquid and in vapour form are represented by means of a single macroscopic moisturedependent diﬀusivity [94], the relation between the moisture ﬂux q l and the spatial gradient of the capillary pressure ∇pc is given by ql =
k · ∇pc . μl
(3.174)
In (3.174), k denotes the intrinsic liquid permeability tensor and μl is the viscosity of water. According to the hypothesis of dissipation decoupling [212], possible couplings between heat and moisture transport are disregarded in the present formulation. In order to account for the dependence of the moisture transport properties on the nonlinear material behaviour of concrete, k is additively decomposed into two portions as k = kr (Sl ) [kt (T ) kφ (φ) k 0 + kd (αR )] ,
(3.175)
one related to the moisture ﬂow through the partially saturated pore space and one related to the ﬂow within a crack, respectively [758]. This approach is consistent with the smeared crack concept. In (3.175), k0 denotes the initial isothermal permeability tensor, kr is the relative permeability, kt accounts for the dependence of the isothermal moisture transport properties on the temperature and kφ describes the relationship between the permeability and the porosity. Furthermore, kd is the permeability tensor relating plane Poiseuille ﬂow through discrete fracture zones to the degree of damage in the continuum model, see [533, 319] for details. Using again the hypothesis of dissipation decoupling, the relation between the heat ﬂux q t and the gradient of the temperature ∇T can be described by a linear heat conduction law reading q t = −Dt ∇T, whereby Dt (T, Sl , φ) denotes the eﬀective thermal conductivity.
(3.176)
3.3 Modelling
293
3.3.2.1.7.1 Freeze Thaw Authored by Max J. Setzer and Jens Kruschwitz The main reason for frost damage in porous materials is the expansion by 9 Vol.% in the transition from water to ice, if a critical degree of saturation in the pores is exceeded. This artiﬁcial saturation, e.g. observed by Auberg & Setzer [69], is as well a multi scaling as a coupled phenomenon. The scaling problem is characterised by the existence of two scales, which should be separated when modelling frost processes in hardened cement paste. Most relevant for the distinction between these scales are of course the macroscopic temperature changes and their typical time constants compared to the time necessary to obtain equilibrium within a certain scale. On the macroscopic scale transient conditions have to be modeled, i.e. mass transport due to viscous ﬂuid ﬂow is slow. On this scale the model deals with bigger volumes than on the microscale. In the big macroscopic volumes thermodynamic processes need a large time span to obtain equilibrium. This can be observed in practise as well as in standard experiments. The second part of the theory in this contribution is restricted to the nanoscopic CSH gel system consisting of solid CSH, pore water and air ﬁlled gelpores with adsorbed water ﬁlms. The liquid water ﬁlm is an essential part of the Setzers model [726], which was determined by [812] experimentally. By going down in length scales it adopts primarily surface thermodynamics and the theory of disjoining pressure. At least thermal or thermodynamic equilibrium is established under normal conditions. This can be assumed for cubes of length up to 120 μm [731]. At constant temperature, the nonfreezing interlayers and ﬁlms are in equilibrium with ice and vapour. The temperature of the bulk ice governs the pressure and by this the equilibrium. Experiments have shown that the ice freezes in situ, referring to [778]. That means on the submicroscopic scale the motion of the pore water to the ice is highly dynamic. However, the response time for movement from gel to ice and the ﬂow distance is rather small. Nevertheless, the pressure gradient is extremely high. By a combination of the Theory of porous Media (TPM), mainly inﬂuenced by de Boer [135], Ehlers [252], Bluhm [130], etc., and a micromechanical theory of surface forces developed by Setzer [723] the artiﬁcial saturation phenomenon can be described [448]. Basis of this model is the work of Kruschwitz & Setzer [450] and Kruschwitz & Bluhm [449] respectively. Last describe the frost heave of a critical ﬁlled cementitious matrix. In the mentioned combination the macroscopic, thermodynamic aspects of the model base on the Theory of Porous Media. This theory is a combination of the mixture theory and the concept of volume fractions. The interactions of the nanostructure of the hardened cement paste are modelled by a smeared micromechanical model. This part of the model is characterised by the properties of the two phase system solid and pore liquid. The transport on the micro structure and the unfrozen, adsorbed water ﬁlm between matrix and ice are included.
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3 Deterioration of Materials and Structures
3.3.2.2 ChemoMechanical Modelling of Cementitious Materials It has been shown in Subsections 3.1.2.3, 3.1.2.3.2, 3.1.2.3.3, 3.1.2.2.2 that the main microstructural mechanisms of environamentally induced corrosion and deterioration processes are by now fairly well understood. There seems to exist, however, a gap between research focused on the material level and durability oriented computational analysis of concrete structures. Although considerable progress has been achieved in the modeling of the mechanical behavior of concrete subjected to various loading conditions (see Subsection 3.1.1.1), environmental inﬂuences aﬀecting the durability of concrete structures are stilc l accounted for by more or less heuristic evaluations of the degradation process and its inﬂuence on the residual structural safety. Recent progress in computational durability mechanics (see e.g. [75, 211, 800, 798, 214]), together with appropriate numerical discretization methods in space and time [460, 453] (see also Chapter 4) open the perspective of a more fundamental approach to obtain not only estimates for the lifetime, but also to provide insight into the degradation mechanisms as a result of the interaction between mechanical and environmental loading. Using a continuum mechanicsbased mode of description, concrete subjected to mechanical and nonmechanical loading is generally described as a multiphase material whose behaviour is inﬂuenced by the interaction of the solid skeleton containing the cementititious matrix and the aggregates and the liquid and gaseous pore ﬂuids. To this end, the scale of observation may either take the microscale or macroscale as a point of departure. In the framework of a microscale approach the individual constituents are described by means of classical continuum mechanics for onephase materials, formulating appropriately the interactions between the constituents and the contact conditions, respectively. To this end, the exact knowledge of the morphology of the material, in particular of the geometry of the pore space, is required. This is, however, not available in general. This diﬃculty motivates the description of porous materials on the basis of a macroscopic approach. The Theory of Mixtures (see e.g. [254] for more details) has been established as a suitable homogenisation procedure, which allows to treat multiphase materials as a continuum while each constituent may be described by its own kinematics and balance equations. The interactions between the constituents are included by production terms within the balance equations. Since the Theory of Mixtures contains no microscopic information of the mixture it need to be complemented by the concept of volume. This leads to the well established concept of the Theory of Porous Media (TPM). It deﬁnes the volume fraction of each constituent dv α and the volume of the mixture dv, which provides a representation of the local microscopic composition of multiphase materials: φα = dv α /dv. The sum of the volume fractions of all constituents has to be equal to one α φα = 1. The TPM provides a general continuum mechanically and thermo dynamically established concept for the macroscopic description of multiphase materials like concrete.
3.3 Modelling
φ0 1 − φ0
mech. damage ← virgin material chem. dissolution ← virgin material
chem. dissolution ← virgin material
virgin material → mech. damage
295
dm φm
s˙ φc
φm φ0 φc 1−φ
Representative Elementary Volume (REV)
Theory of Mixture  material point
Fig. 3.143. Chemomechanical damage of porous materials within the Theory of Mixtures. Three types of deterioration are illustrated: virgin material, mechanically damaged material, chemically damaged material and chemomechanically damaged material
3.3.2.2.1 Models for Ion Transport and Dissolution Processes Authored by Detlef Kuhl and G¨ unther Meschke 3.3.2.2.1.1 Introductory Remarks Based on insights and data obtained from experimental investigations on calcium dissolution and coupled chemomechanical damage processes (see Subesection 3.1.2.3.2) constitutive models formulated on a macroscopic level of observation have been developed for the analysis of the time dependent dissolution process of concrete and concrete structures. One class of models is based on a phenomenological chemical equilibrium model relating the calcium concentration of the skeleton and the pore solution s(c) in conjunction with the concept of isotropic damage mechanics [422], as proposed by G´ erard [307] and subsequent publications (G´ erard [308], G´ erard et al. [311], PijaudierCabot et al. [635, 634, 636] and Le Bell´ ego et al. [477, 479, 478]). Ulm et al. [801] and Ulm et al. [799] have proposed a chemoplasticity model formulated within the BiotCoussyTheory of porous media [211]. This model is also based on a chemical equilibrium model, using empirical relations for the conductivity and aging. In both models, the irreversible character of skeleton dissolution is not accounted for. Hence, chemical unloading or cyclic chemical loading processes cannot be described. From the experiments the keyrole of the porosity for the changing material and transport properties of chemomechanically loaded cementitious materials becomes obvious. Based on this observation and in order to consider
296
3 Deterioration of Materials and Structures
the interaction phenomena of chemical and mechanical material degradation described in Subsection 3.1.2.3.2 a fully coupled chemomechanical damage model has been developed in [454, 455] within the framework of the Theory of Porous Media. The material is described as ideal mixture of the fully saturated pore space and the matrix. In this model, the pore ﬂuid acts as a transport medium for calcium ions. The pore pressure, however, is not accounted for in the present version of the model. The changing mechanical and transport properties are related to the total porosity deﬁned as the sum of the initial porosity, the chemically induced porosity and the apparent mechanical porosity. Together with the assumptions of chemical and mechanical potentials the need for further assumptions or empirical models is circumvented. Microcracks are interpreted according to Kachanov [422] as equivalent pores aﬀecting, on a macroscopic level, the conductivity and stiﬀness but not the mass balance. The evolution of the mechanically and chemically induced porosities are both controlled by internal parameters. This enables the modeling of cyclic loading conditions and allows a consistent thermodynamic formulation of the coupled ﬁeld problems [454]. The link between the mechanical and the chemical ﬁeld equations is accomplished by the deﬁnition of the total porosity φ as the sum of the initial porosity φ0 , the porosity due to matrix dissolution φc and the apparent mechanical porosity φm : φ = φ0 + φc + φm .
(3.177)
The chemically induced porosity φc can be calculated by multiplying the amount of dissolved calcium of the skeleton s0 − s by the averaged molar volume of the skeleton constituents M/ρ φc =
M [s0 − s] , ρ
(3.178)
where s0 denotes the initial skeleton concentration. The apparent mechanically induced porosity φm considers the inﬂuence of mechanically induced micro pores and micro cracks on the macroscopic material properties of the porous material. It is obtained by multiplying the scalar damage parameter dm by the current volume fraction of the skeleton 1 − φ0 − φc φm = [1 − φ0 − φc ] dm .
(3.179)
This deﬁnition of the mechanical porosity φm takes into account that microcracking is restricted to the solid matrix material. 3.3.2.2.1.2 Initial Boundary Value Problem The coupled system of calcium diﬀusiondissolution, mechanical deformation and damage is characterized by the concentration ﬁeld c of calcium ions
3.3 Modelling
297
in the pore solution and the displacement ﬁeld u as external variables and a set of internal variables concerning the irreversible material behavior. The macroscopic balance of linear momentum is given by: div σ = 0 .
(3.180)
The matrix dissolutiondiﬀusion problem is governed by the macroscopic balance of the calcium ion mass in the representative elementary volume div qc + [[φ0 + φc ] c ]· + s˙ = 0 ,
(3.181) ·
whereby qc is the mass ﬂux of the solute. The term [[φ0 + φc ] c ] accounts for the change of the calcium mass due to the temporal change of the porosity and the concentration, which is up to one dimension smaller compared to the calcium mass production resulting from the dissolution of the skeleton s˙ [452]. The system of diﬀerential equations (3.180)(3.181) is completed by boundary conditions on the boundary Γ given by σ · n = t , qc · n = qc , u = u , c = c
(3.182)
and initial conditions in the domain Ω given by u(t = 0) = u0 , c(t = 0) = c0 ,
(3.183)
where qc is the calcium ion mass ﬂux across the boundary and c is the prescribed concentration. 3.3.2.2.1.3 Constitutive Laws The elastodamage constitutive law is characterized by the free energy function Ψm : Ψm =
1−φ ε : Cs : ε . 2
(3.184)
Herein, ε denotes the linearized strain tensor and C s is the fourth order elasticity tensor of the the skeleton. The derivative of the free energy function Ψm with respect to the strain tensor ε yields the stress tensor σ: σ=
∂Ψm = [1 − φ] C s : ε . ∂ε
(3.185)
The diﬀusiondissolution problem is deﬁned by the dissipation potential Ψc of the calcium ions in the representative elementary volume Ψc =
φ Dl γ·γ, 2
(3.186)
298
3 Deterioration of Materials and Structures
where γ = −∇c is the negative gradient of the concentration ﬁeld. The derivative of the dissipation potential Ψc with respect to the negative concentration gradient γ results in the calcium ion mass ﬂux vector qc qc =
∂Ψc = φ Dl γ ∂γ
(3.187)
of the pore ﬂuid, which is discussed in the next subsection. In eqs. (3.186) and (3.187) D 0 denotes the second order conductivity tensor of the pore ﬂuid. Consequently, the macroscopic conductivity of the porous material is given by D = φD 0 . φ = φ0 deﬁnes the chemical and mechanical sound macroscopic material (φc = φm = 0), characterized by the subscript s. Hence, the macroscopic conductivity of the virgin material is given by D s = φ0 D 0 . In contrast to existing reactiondiﬀusion models describing calcium leaching, the dependence of D 0 on the square root of the calcium concentration within the pore ﬂuid is considered. This dependency follows from Kohlrausch’s law, describing the molar conductivity of strong electrolytes (see Atkins [66] and Section 3.3.2.2.1.4), using NernstEinstein’s relation. In the isotropic case, the conductivity tensor D 0 is given in terms of the second order identity tensor 1, the calcium ion conductivity for the inﬁnitely diluted solution D00 ≥ 0 and the constant D0c ≥ 0. √ D0 = D0 1 = D00 − D0c c 1 (3.188) It can be observed, that the conductivity decreases with an increasing calcium concentration. This follows from the interaction of moving cations Ca2+ and anions OH− by electrostatic forces and viscous forces. For D0c = 0 Kohlrausch’s law (3.188) degenerates to Fick’s law [280] of independent diﬀusing particles. 3.3.2.2.1.4 Migration of Calcium Ions in Water and Electrolyte Solutions The molar conductivity Λ of a strongly electrolyte solution is given as function of the calcium ion concentration in the pore ﬂuid c by the empirical Kohlrausch law, see Kohlrausch [436]. D 0 = D0 1 =
RT Λ1 z2 F 2
Λ = Λ0 − Λc
√ c
(3.189)
Herein, R = 8.31451J/Kmol is the universal gas constant, T is the total temperature chosen as T = 298K, z = 2 is the number of elementary charges of a cation Ca2+ , F = 9.64853 ·104C/mol is the Faraday constant, Λ0 = 11.9 · 10−3Sm2/mol is the molar conductivity at inﬁnite dilution and Λc is the Kohlrausch constant of the molar conductivity. Based upon the ¨ckelOnsager theory (Debye & model of ionic clouds, the DebyeHu ¨ckel [230] and Onsager [602]) veriﬁes Kohlrausch’s law and allows to Hu determine the Kohlrausch constant Λc ,
3.3 Modelling ﬂuid conductivity D0
macroscopic conductivity φD0
8
8
6
6 κc c0
323K φD0
D0
2
D0c = 0, T = 298 D0c = 0, T = 298 T = 273K, T = 323K
0 0
5
10 c 15
299
0.0
κc /c0 0.0 0.2 0.4 0.6 0.8 1.0
φ 0.725 0.497 0.457 0.430 0.414 0.200
0.2 273K
20
2 0 0
25
0.8 5
1.0 10 c 15
20
25
Fig. 3.144. Conductivity of the pore ﬂuid D0 [10−10 m2 /s] as function of the calcium concentration c [mol/m3 ] and the total temperature T [K]. Macroscopic conductivity of nonreactive porous media φD0 [10−10 m2 /s] as function of the calcium concentration c [mol/m3 ] and the porosity φ [−] with φ = φ(κc , dm = 0)
2 z2 e F 2 A= 3πη R T 2 q z3 e F 2 B= 24 π η R T RT
Λ c = A + B Λ0
(3.190)
where the constants A and B account for electrophoretic and relaxation eﬀects associated with the ionion interactions. These constants are given in terms of the universal gas constant, the total temperature, the elementary charge e = 1.602177 ·10−19C, the constant q = 0.586, the electric permittivity = 6.954·10−10C 2/Jm and the viscosity η = 0.891·10−3kg/ms of water (see e.g. Atkins [66]). From comparing equations (3.188) and (3.189) the macroscopic diﬀusion constants D00 and D0c can be determined. D00 =
2 RT −12 m Λ = 791.8·10 0 s z2F 2
(3.191)
D0c
m2 RT = 2 2 Λc = 96.85·10−12 s z F
m3 mol
(3.192)
In the present model the macroscopic diﬀusion coeﬃcient φD0 can be determined for any state of chemomechanical degradation characterized by the history variables κc and κm and for any concentration c. Figure 3.144 contains plots of the conductivity D0 in the pore ﬂuid and the macroscopic conductivity φD0 vs. the calcium concentration within the
300
3 Deterioration of Materials and Structures
range c = 0 corresponding to pure water and c = c0 corresponding to the concentrated pore solution of the virgin material. The diagrams on the left hand side of Figure 3.144 underline the relevance of using higher order ion transport models, considering electrophoretic and relaxation eﬀects, as a basis for realistic calcium leaching models. Standard and higher order transport models are characterized by D0c = 0 and D0c = 0, respectively. A pronounced change of the conductivity D0 proportional to the square root of the concentration c can be observed within the considered concentration range. The ratio of the conductivities related to the fully degraded material D0 (0) and the virgin material D0 (c0 ) is approximately 9 : 4. The average value of the conductivity D0 is approximately D0 ≈ 4 ·10−10 m/s2 . This is in accordance with the suggestion by Delagrave et al. [232], that in numerical analyses D0 /2 should be used as the macroscopic conductivity in order to ﬁt experimental results. Standard transport models are not capable to capture the signiﬁcant increase of the conductivity with a decreasing calcium ion concentration corresponding to propagating chemical damage in reactive porous media. As expected, the results of the standard model and the present model are identical in the case of inﬁnitely diluted solutions (c = 0). The sensitivity of the ion transport with regards to temperature changes is studied by including plots of the conductivity for T = 273 K, corresponding to the freezing point of water (no calcium ion transport occurs below this temperature), and for T = 323 K, representing approximately a desert climate, in Figure 3.144. According to equations (3.188), (3.191) and (3.192), the conductivity D0 depends linearly on the total temperature T . Within the considered temperature interval D0 is only changed by approximately 16%. Compared to the inﬂuence of the concentration the inﬂuence of the temperature plays a minor role in the transport process of ions within the pore water of cementitious materials. On the right hand side of Figure 3.144, the macroscopic conductivity is plotted for various values of the threshold calcium concentration κc and the corresponding values of the porosity, respectively, assuming a nonreactive porous material. 3.3.2.2.1.5 Evolution Laws According to Simo & Ju [744] the evolution of the damage parameter dm (κm ) is described by the damage criterion Φm = η(ε) − κm ≤ 0 ,
(3.193)
where η and κm are the equivalent strain function and the internal variable deﬁning the current damage threshold. From the KuhnTucker loading/unloading conditions and the consistency condition Φm ≤ 0 , κ˙ m ≥ 0 , Φm κ˙ m = 0 , Φ˙ m κ˙ m = 0 ,
(3.194)
3.3 Modelling
301
follows, that κm is unchanged for Φm < 0 and calculated as κm = η otherwise. The description of the elastodamage material model is completed by the deﬁnition of the equivalent strain η and the damage function dm . Here the equivalent strain measure proposed by de Vree et al. [814] is used ks − 1 [ks − 1]2 1 12ks η= I1 + I2 + J2 , (3.195) 2 2ks [1 − νs ] 2ks [1 − 2νs ] [1 + νs ]2 in which I1 = tr[ε], I2 = [tr2 [ε] − ε : ε]/2 and J2 = [εdev : εdev ]/2 are the ﬁrst and the second invariant of the strain tensor ε and the second invariant of the strain deviator εdev , respectively. The parameter ks denotes the ratio of tensile to compressive strength and νs the Poisson’s ratio of the skeleton. The exponential damage function is given by dm = 1−
κ0m 1−αm +αm exp[βm [κ0m −κm ]] , κm
(3.196)
where κ0m is the initial damage threshold and αm , βm are material parameters. The state of the chemically induced degradation of the porous material is characterized by the chemical porosity φc (s). Starting from a chemical equilibrium state between the calcium solved in the pore ﬂuid and the calcium bound in the skeleton, the dissolution process requires a decreasing concentration c in the pore ﬂuid. Otherwise, if c is increased, the structure of the skeleton is unchanged. In order to describe chemically induced degradation similarly to the elastodamage problem, an internal variable κc is introduced, which corresponds to the current chemical equilibrium state. Based on this internal variable κc , the chemical reaction criterion Φc is formulated as Φc = κc − c ≤ 0 .
(3.197)
According to the KuhnTucker conditions and the consistency condition Φc ≤ 0 ,
κ˙ c ≤ 0 ,
Φc κ˙ c = 0 ,
Φ˙ c κ˙ c = 0 ,
(3.198)
the process of matrix dissolution is associated with a decreasing chemical equilibrium calcium concentration (κ˙ c ≤ 0). The dissolution threshold κc is unchanged for Φc < 0 and equal to the current calcium concentration of the pore ﬂuid (κc = c) otherwise. The conditions (3.197) and (3.198) for the occurence of chemical reactions are identical to those given by Mainguy & Coussy [512]. This identity is shown in Kuhl et al. [454]. As already mentioned, the current state of the calcium concentration in the skeleton s is controlled by the spontaneous calcium dissolution. It can be described as a function of the chemical equilibrium threshold κc given by G´ erard [307, 308] and Delagrave et al. [232])
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3 Deterioration of Materials and Structures
Skeleton Concentration s [kmol/m3 ]
16 s0
14 12 10
sh
8 6 4 ccsh
2
dissolution cp
c0
0 0
5
10
15
20
25
Poreﬂuid Concentration c [mol/m3 ]
Fig. 3.145. Chemical equilibrium function by G´ erard [307, 308] and Delagrave et al. [232]
1 1 2 κc + κc − s = s0 − [1 − αc ] sh 1 − 10 400
s0 − sh αc sh m n − κc κc 1+ 1+ cp ccsh (3.199)
for 0 < κc < c0 and s = s0 for κc ≥ c0 . αc , n and m are model parameters. c0 and s0 are the initial equilibrium concentrations of the sound material, cp and ccsh are material constants related to the averaged ﬂuid calcium concentration of the progressive dissolution of the portlandite and the CSH phases, sh is the solid calcium concentration related to the portlanditefree cement matrix. A plot of function (3.199) and an illustration of the material parameters are given in Figure 3.145. 3.3.2.2.2 Models for Expansive Processes Authored by Falko Bangert and G¨ unther Meschke 3.3.2.2.2.1 Introductory Remarks Several numerical models have been developed in order to characterize the observed behavior of concrete aﬀected by the AlkaliSilica Reaction (ASR) on a material level or even a structural level. Depending on the level of observation these models follow either a mesoscopic or a macroscopic approach. A mesoscopic approach involves the analysis of a single representative aggregate particle and its vicinity, whereby the kinetics of the chemical and diﬀusional processes involved are described on the scale of the aggregates, see e.g. Baˇzant & Steﬀens [96]. On the other hand, in a macroscopic approach concrete is described at the scale of laboratory specimens, see e.g. Larive &
Volume fractions
Microstructure
3.3 Modelling t=0
t>0
303
t→∞
ϕr g
ϕ ϕl ϕu
φs = φu
⎫ ⎪ φr ⎪ ⎬ ⎪ ⎪ φ ⎭
φs = φ r + φu
φs = φr
u
φg
φg
φg
φl
φl
φl
Fig. 3.146. Microstructure, constituents and volume fractions of concrete as a partially saturated porous media: light gray → unreacted part of the skeleton, dark gray → reacted part of the skeleton, white → pore gas, black → pore liquid
Coussy [470]. In these models, the main characteristics of ASR are incorporated phenomenologically on the macroscopic level. Hence, they can directly be used for numerical analysis of concrete structures [798]. Concluding from Subsection 3.1.2.3.3, there are two main mechanisms that have to be taken into account for a computational model which allows for realistic predictions of concrete deterioration caused by ASR. Firstly, the gel formation by the noninstantaneous dissolution of silica and secondly the swelling of the gel by the instantaneous imbibition of water. Both processes strongly depend on the moisture content within the concrete since water acts as a transport medium of ions and as a necessary compound for the formation of the swollen gel. Only very limited information on the properties of the individual constituents on the microscale, in particular of the gel, is available. In a chemohygromechanical damage model for the simulation of damage induced by the AlkaliSilica Reaction of concrete developed by [83, 81] the Theory of Porous Media (see e.g. Ehlers [254], Lewis & Schreﬂer [493]) together with a geometrically linear kinematics is used as the macroscopic continuum mechanics framework for the numerical simulation of concrete structures aﬀected by the AlkaliSilica Reaction. Concrete is modeled as a partially saturated porous material consisting of a mixture of three main superimposed and interacting constituents ϕα , namely the nonporous skeleton (index α = s), the pore liquid (index α = l) and the pore gas (index α = g). When the alkalisilica reaction has not yet started (t=0), the skeleton represents a mixture of the unreacted aggregates and the hydration products. During the alkalisilica reaction, mass of the aggregates passes noninstantaneously into mass of the gel. The model formulation is based on the idea, that the gel formation is initiated at the surface of the aggregate particles and progresses
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3 Deterioration of Materials and Structures
from the surface inward the particles. It is assumed, that the gel, which is responsible for the pressure buildup and the macroscopic expansion, is trapped at the reaction sites inside the reacting aggregate particles. The possibility, that the expansive gel may permeate in pores and cracks in the cement paste located near the surface of the aggregate particles or even may diﬀuse away through the connected pores space according to a throughsolution mechanism is not explicitly considered in this model [83, 81]. Thus, for an instant t > 0 the skeleton ϕs is regarded as a mixture of the unreacted portion of the aggregates, the gel and the hydratation products. ϕs = ϕu + ϕr
(3.200)
At the same time, the pore space is solely saturated by the pore liquid ϕl and the pore gas ϕg . The unreacted phase ϕu represents the unreacted, unswollen skeleton material before it was aﬀected by the alkalisilica reaction. On the other hand the reacted phase ϕr represents the reacted, swollen skeleton material after completion of ASR. The reactive aggregates of the reacted phase ϕr are completely converted into a gel. During the alkalisilica reaction, mass of the unreacted phase ϕr passes noninstantaneously into mass of the reacted phase ϕr . The mass exchange, which phenomenologically represents the gel formation by the dissolution of silica, is illustrated in Figure 3.146. At time t = 0 the skeleton is not aﬀected by ASR as is indicated by the light gray color corresponding to unreacted material. At an instant t > 0 the dark gray part of the skeleton has already been aﬀected by ASR. Finally, for t → ∞ the entire skeleton is aﬀected by ASR. Following the standard concepts of the Theory of Porous Media, it is assumed, that the constituents ϕα are homogenized over a representative volume element, which is occupied by the mixture ϕ = ϕs + ϕl + ϕg . Therefore, material points of each constituent ϕα exist at each geometrical point x. Hence, the local composition of the mixture ϕ is described by the volume fraction φα , which is deﬁned as the ratio of the volume element dv α occupied by the individual constituent ϕα and the volume element dv occupied by the mixture ϕ (see Figure 3.146): φα =
dv α . dv
(3.201)
Since the solid skeleton is regarded as a binary mixture, the respective volume fraction φs is given as the sum of the volume fraction of the unreacted volume fraction φu and the reacted volume fraction φr : φs = φu + φr .
(3.202)
It follows from deﬁnition (3.201), that the saturation condition must hold: φs + φl + φg = 1 .
(3.203)
3.3 Modelling
305
The material density $α and the partial density ρα of the constituent ϕα are introduced as $α =
dmα , dv α
ρα =
dv α dmα dmα = = φα $α . dv dv dv α
(3.204)
Herein, dmα denotes the local mass of the volume element dv α . The partial density of the skeleton is assumed to be composed by an unreacted and a reacted part: ρs = φs $s = φu $u + φr $r .
(3.205)
For the material densities of the unreacted and the reacted material the relationship $u > $ r
(3.206)
is assumed. By means of equation (3.206) it is considered, that during the noninstantaneous gel formation represented by the mass exchange between ϕu and ϕr the gel swells instantaneously. In other words, the ratio $u /$r represents phenomenologically the volume increase of the gel by the imbibition of water. The amount of water imbibed by the gel and consequently the ratio $u /$r strongly depend on the moisture content of the concrete. Since according to equation (3.206) the material densities of the unreacted and the reacted material are diﬀerent, a variation of the volume fractions φu and φr due to the aforementioned mass exchange results in a variation of the material density of the skeleton $s , see equations (3.202) and (3.205). Thus, the ASRinduced swelling of the skeleton is associated with the variation of the material density $s . 3.3.2.2.2.2 Balance Equations Investigations on the role of water in the alkalisilica reaction have shown, that reactive concrete specimens do not absorb signiﬁcantly more water than nonreactive ones, when they are stored under the same hygral conditions [469, 471]. Thus, no speciﬁc model needs to be developed to predict water movement in ASR aﬀected concrete and it is reasonable to neglect any mass exchange between the skeleton and the pore ﬂuids. In doing so, the mass balance equation of the skeleton ϕs as binary mixture reads [254] (φs $s )s + φs $s div(xs ) =
∂[φs $s ] + div(φs $s xs ) = 0, ∂t
(3.207)
where (•)α = ∂(•)/∂t + grad(•) · xα denotes the material time derivative of the quantity (•) following the individual motion of the respective constituent ϕα .
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3 Deterioration of Materials and Structures
Assuming incompressible constituents ϕu and ϕr of the skeleton ϕs (→ $ =const., $r =const.), the associated partial mass balance equations of the unreacted and reacted phase result in the following volume balance equations: u
∂φu ∂φu→r + div(φu xs ) = , ∂t ∂t
∂φr ∂φr←u + div(φr xs ) = . ∂t ∂t
(3.208)
The terms $u ∂φu→r /∂t and $r ∂φr←u /∂t represent the mass exchange between the phases ϕu and ϕr due to the dissolution process. Since the summation of the partial balances (3.208)1 and (3.208)2 must result in the mixture balance equation (3.207), the following constraint must hold: $u
∂φu→r ∂φr←u + $r = 0. ∂t ∂t
(3.209)
Proceeding with the assumption, that the kinetics of the dissolution of silica and consequently the mass exchange between the constituents ϕu and ϕr follow a ﬁrst order kinetic law, one may write (e.g. Atkins [66]) $u ∂φu→r $u ∂φr←u =− r = r k φu , ∂t $ ∂t $
∂φu→r = −k φu , ∂t
(3.210)
whereby the parameter k is the reaction velocity. Inserting the equations (3.210) into the volume balance equations (3.208) and neglecting the skeleton velocity xs ≈ 0 ,
(3.211)
yields: ∂φu = −k φu , ∂t
$u ∂φr = r k φu . ∂t $
(3.212)
For constant environmental conditions the volume balance equations (3.212) can be integrated analytically with the initial value φu0 = φs0 leading to φu = φu0 [1 − ξ] ,
φr =
$u u φ ξ, $r 0
(3.213)
where ane overall reaction extent ξ has been used. Finally, inserting (3.213) into (3.205) yields the material density of the skeleton $s as a function of the reaction extent ξ: $s =
$u $ r . $r + ξ [$u − $r ]
(3.214)
Thus, expression (3.214) reﬂects the swelling state of the skeleton ranging from an unswollen state (ξ = 0 ⇒ $s = $u ), if the alkalisilica reaction has
3.3 Modelling
307
not yet started, to a fully swollen state (ξ = 1 ⇒ $s = $r ) after the ASR process has come to an end. In analogy to equation (3.207), the mass balance equations of the pore ﬂuids ϕβ (index β = l → liquid phase, index β = g → gas phase) are given by: ∂[φβ $β ] + div(φβ $β xβ ) = 0 . ∂t
(3.215)
Neglecting the material compressibility of the pore liquid in comparison to the material compressibility of the pore gas (→ $l = const.), and using the assumption (3.211) one obtains from equation (3.215): ∂φl + div(φl wl ) = 0 , ∂t
∂[φg $g ] + div(φg $g wg ) = 0 . ∂t
(3.216)
The partial momentum balances for the quasistatic case with the body forces neglected are given by: ˆα = 0 . div(σ α ) + p
(3.217)
ˆ α the momentum production, Herein, σ α is the partial stress tensor and p which can be interpreted as the local interaction force per unit volume between ϕα and the other constituents. Thereby the following constraint ˆs + p ˆl + p ˆg = 0 p
(3.218)
must hold due to the overall conservation of momentum div(σ) = 0 ,
(3.219)
with the overall stress tensor σ = σ s + σ l + σ g . 3.3.2.2.2.3 Constitutive Laws The pore space of concrete φl + φg is partially saturated with liquid and partially with gas, see Figure 3.146. The degree of liquid and gas saturation sβ , respectively, is given by: sβ =
φβ . φl + φg
(3.220)
The pore liquid and the pore gas are separated by a curved interface (meniscus) because of surface tensions. The radius of curvature of this interface depends on the pressure jump across the interface expressed by the socalled capillary pressure pc : pc = pg − pl .
(3.221)
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3 Deterioration of Materials and Structures
In what follows, however, the capillary pressure pc will be interpreted as a macroscopic pressure representing all hygrally induced stresses acting on various scales of the nanoporous cementitious material, see e.g. [97]. There exists a relationship between the water content of the porous medium expressed by the liquid saturation sl and the capillary pressure pc . In [83] the following expression for the capillary pressure pc as a function of the liquid saturation sl is used pc = pr
( − 1 ) n1 sl m − 1 ,
(3.222)
according to van Genuchten [306]. In equation (3.222), pr , n, m denote material parameters, which have to be determined experimentally. The relation (3.222) has been originally proposed for soils. However, the pc (sl )relations determined experimentally by BaroghelBouny et al. [88] for diﬀerent types of cementitious materials by means of water vapor sorption isotherms are well ﬁtted by expression (3.222). From thermodynamical considerations follows that the stress state of the skeleton and the ﬂuid constituents is separated into two parts, where the ﬁrst part is governed by the skeleton deformation and the pore ﬂuid ﬂow, respectively, while the second part is governed by the pore pressures (see e.g. [254]):
σ s = σ s − φs p 1 ,
σ β = σ β − φβ pβ 1 .
(3.223)
Therein, the pore pressure p is given by Dalton’s law p = sl p l + s g p g ,
(3.224)
where pl denotes the unspeciﬁed liquid pressure, whereas the gas pressure pg is related to the material density $g by the following constitutive law for an ideal gas: $g =
Mg g p . RT
(3.225)
In equation (3.225), Mg denotes the molar mass of the pore gas, R the universal gas constant and T the absolute temperature. The overall stress tensor σ of the porous material is given by the sum of the partial stress tensors σ α according to (3.223):
σ = σs + σl + σg − p 1 .
(3.226)
In the Theory of Porous Media the ﬂuid frictional stresses σ β are usually neglected (σ β ≈ 0), yielding the well known concept of eﬀective stress (see Bishop [127]): σ = σ s − p 1 .
(3.227)
3.3 Modelling
309
For the modeling of brittle failure of the skeleton (reduction of stiﬀness and strength) the continuum damage theory proposed by Kachanov [422] is employed. According to the eﬀective area concept by Kachanov [422], the scalar damage parameter d can be interpreted as the ratio of the damaged cross section and the initial cross section. Thus, the undamaged material is characterized by d = 0, while d = 1 corresponds to the complete loss of integrity. Since the stresses in the skeleton are transferred by the intact, undamaged cross section, the eﬀective stress reads
σ s = [1 − d] φs0 C s : [εs − εas 1] ,
(3.228)
with the eﬀective elasticity tensor of the skeleton C s = E s [II + ν s /[1 − 2ν s]1 ⊗ 1]/[1 + ν s ], deﬁned in terms of the Young’s modulus E s and the Poisson’s ratio ν s . In equation (3.228), the volumetric expansion resulting from ASR is considered by the volumetric strain εas . As mentioned above, the ASR swelling of the skeleton results from the variation of the material density $s of the skeleton, compare equation (3.214). Therfore, the volumetric expansion εas is deﬁned as εas =
$s0 −1, $s
(3.229)
where $s0 denotes the initial material density of the skeleton. There is an ongoing debate whether the gel formed by the dissolution of silica initially saturates the pores in the cement paste located near the surface of the aggregates before a expansive pressure builds up (see Section 3.1.2.3.3). The deﬁnition 3.229 of the ASRexpansion implies, that the gel is trapped at the reaction sites inside the aggregates, thus representing a part of the skeleton. Consequently, the local ASRprogress directly results in the deformation of the skeleton. However, an initiation period due to a ﬁlling process can be considered by the model by introducing an initiation threshold for the ASRexpansion as e.g. suggested by Steﬀens et al. [772]. In the model proposed by [83], an isotropic damage model characterized by a single damage parameter d and a strain based description of the damage evolution in the sense of Simo & Ju [744] is used (see Section 3.3.1.2.2). Since the local continuum description of material degeneration suﬀers from the loss of wellposedness beyond a certain level of accumulated damage resulting in unphysical numerical results, a gradient enhanced damage formulation as proposed by Peerlings et al. [614] is used as a means of regularization. The evolution of damage is governed by the deformation of the skeleton. According to Simo & Ju [744] an internal variable κ is introduced, which represents the most severe deformation the skeleton material has experienced in the previous loading history and which acts as a threshold below which there is no further damage evolution. The damage parameter d is an explicit function of the internal variable κ. The evolution of κ is governed by the damage criterion
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3 Deterioration of Materials and Structures
Φ = η¯ − κ ≤ 0 ,
(3.230)
where η¯ denotes the nonlocal equivalent strain. From the KuhnTucker loading / unloading conditions and the consistency condition ∂κ ≥ 0, ∂t
Φ ≤ 0,
Φ
∂κ = 0, ∂t
∂Φ ∂κ =0 ∂t ∂t
(3.231)
follows, that κ is unchanged for Φ < 0 and calculated by κ = η¯ otherwise. The nonlocal equivalent strain η¯ in equation (3.230) is calculated on the basis of the following partial diﬀerential equation: η = η¯ − div(g grad(¯ η )) .
(3.232)
In this equation, η denotes the (local) equivalent strain representing a scalar measure of the local deformation state. Due to the gradient parameter g with the dimension of length squared an internal length scale is present in the formulation, which avoids the loss of wellposedness mentioned above. Finally, the equivalent strain η and the damage parameter d must be speciﬁed. Here, the equivalent strain measure corresponding to the Rankine criterion of maximal principal stress is used [83]: η=
1 max < σ ˜is > , s E
i = 1, 2, 3 ,
(3.233)
with max < σ ˜is > denoting the positive part of the largest eigenvalue of the ˜s . undamaged eﬀective stress tensor σ The deﬁnition of the eﬀective stress tensor σ s according to equation (3.228) is based on the assumption, that deterioration due to ASR only takes place if the ASR expansion εas is hindered. If the concrete can expand freely (σ s = 0 → εs = εas 1), the stiﬀness and strength are not aﬀected by the alkalisilica reaction. This assumption, which is used for most model formulations in the literature [632, 798, 772], implies, that the degradation of concrete caused by ASR is mainly induced by structural eﬀects. These structural eﬀects may result from hindered deformations due to geometrical constraints or from gradients in the ASR expansion following from a nonuniform moisture distribu tion. It should be mentioned, that even under stressfree conditions (σ s = 0) microcracks can develop in the vicinity of the aggregate particles e.g. due to geometrical incompatibilities. However, on the macroscopic level the structural eﬀects have a much more severe inﬂuence on the deterioration of concrete structures than these microcracks on the level of the aggregate particles. Although the ﬂuid frictional stresses σ β are neglected, the ﬂuid viscosity ˆ β in the partial momentum is included via the momentum production terms p balance equations (3.217). These are chosen as 2 μβ ˆ β = pβ grad(φβ ) − φβ p wβ , kβ
(3.234)
3.3 Modelling
311
with the dynamic viscosity μβ and the permeability k β [254]. In turn, the permeability k β depends on the intrinsic permeability k0 and on the nondimensional scaling factor krβ , which takes the dependence of the permeability k β on the saturation into account: k β = krβ k0 .
(3.235)
The intrinsic permeability k0 represents the permeability of the fully saturated porous material, which is independent of the saturating ﬂuid phase. The inﬂuence of the saturation is considered according to van Genuchten [306] krl
( √ l m1 )m 2 l = s 1− 1− s ,
krg =
( 1 )2m 1 − sl 1 − sl m , (3.236)
where m is the same material parameter as used in the capillary pressure relation (3.222). ˆ β (3.234) into the related Finally, inserting the momentum productions p momentum balance equations (3.217) yields Darcy’s law: φβ wβ = −
kβ grad(pβ ) . μβ
(3.237)
By inserting the result into the partial mass balance equations of the pore ﬂuids (3.216), the seepage velocities wβ can be eliminated as primary variables. 3.3.2.2.2.4 Model Calibration In this paragraph, the calibration of the chemical material parameters $u , $ and k, which control the deterioration caused by the alkalisilica reaction is described. First, a stressfree expansion test (σ = 0) of a reactive concrete specimen carried out at a certain temperature and humidity is considered. Inserting equation (3.228) into equation (3.227) yields after rearrangement: r
εs = εas 1 +
1 Cs ]−1 : p 1 . [C [1 − d] φs0
(3.238)
For isohydrothermal laboratory conditions the second part on the right hand side of equation (3.238) is almost constant since the pore pressure p does not change signiﬁcantly in the course of the alkalisilica reaction. Hence, in laboratory tests on ASR aﬀected concrete only the part εas of the strain tensor εs is measured, which is governed by the chemical reactions. Inserting the material density of the skeleton $s according to equation (3.214) and the initial value $s0 = $u into the ASR expansion εas deﬁned in equation (3.229) yields for constant environmental conditions: u $ a εs = r − 1 ξ . (3.239) $
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3 Deterioration of Materials and Structures
Diﬀerentiation of equation (3.239) with respect to time results in: u ∂εas ∂ξ $ ∂εas = = k [1 − ξ] r − 1 . ∂t ∂ξ ∂t $
(3.240)
From equations (3.239) and (3.240) the following values are obtained for the onset (t = 0 ⇒ ξ = 0) and for the completion of the alkalisilica reaction (t → ∞ ⇒ ξ = 1): u $ ∂εas ξ = 0 ⇒ εas = 0 , = k − 1 , ∂t $r u (3.241) $ ∂εas ξ = 1 ⇒ εas = r − 1 , = 0. ∂t $
0.4
Expansion εas [%]
1/k 0.3
0.2 u
r
/ − 1 0.1 Test results 0 0
100
200 Time t [d]
300
400
0.6
120 1/k
0.45
90
0.3
60 u
r
/ − 1 0.15
30 Test results Model results
0 0.7
Inverse velocity 1/k [d]
Asymptotic expasion u /r − 1 [%]
From equations (3.241) together, with the left diagram in Figure 3.147, which shows a typical strain evolution in a stressfree expansion test, the meaning of the chemical material parameters becomes clear: The parameter $u /$r − 1 represents the asymptotic strain in a stressfree expansion test. Furthermore, the parameter k controls the slope of the respective expansiontimerelation at the onset of ASR. Hence, the chemical material parameters $u /$r − 1 and k are welldeﬁned and can be easily determined by means of macroscopic strain measurements on reactive concrete specimens. Both chemical material parameters ($u /$r − 1 and k) depend on the concrete mix design, the type of aggregates, the temperature and the moisture content. In particular, the moisture dependence plays a dominant role in the ASR deterioration. The role of moisture within the alkalisilica reaction has been studied in detail in an extensive test campaign at the Laboratoire Central des Ponts et Chauss´ees by Larive [469]. In these tests, cylindrical concrete
0 0.8
0.9
1
Liquid saturation sl []
Fig. 3.147. Illustration of the chemical material parameters k and u /r − 1 and of their dependence on the liquid saturation sl according to experimental results by Larive [469] and to model results by Steﬀens et al. [772]
3.3 Modelling
313
specimens of a certain mix design were stored under diﬀerent hygral conditions (immersed in water, exposed to diﬀerent relative humidities, wrapped in aluminum foil), whereby the temperature was kept constant at 38◦ C [471]. For each specimen, the macroscopic expansion and the weight change have been measured. The left diagram in Figure 3.147 shows a typical result of the expansion measurements. It turns out from the test results, that the asymptotic expansion $u /$r − 1 and the reaction velocity k increase with an increasing moisture content. In a recent study based on Larive’s results by Steﬀens et al. [772], the weight change of the specimens has been converted into an averaged liquid saturation sl within the specimens. An almost linear relation between the liquid saturation sl and the asymptotic expansion $u /$r − 1 as well as the inverse of the reaction velocity 1/k has been observed, which can be expressed by (Figure 3.147, right): $u − 1 = 1.27 sl − 0.754 [%], $r
1 = −300 sl + 326 [d]. k
(3.242)
Hence, the chemical material parameters of the proposed chemohygromechanical model controlling the deterioration caused by the alkalisilica reaction are known. 3.3.3 A HighCycle Model for Soils Authored by Andrzej and Theodoros Triantafyllidis
Niemunis,
Torsten
Wichtmann
Based on the tests presented in Section 3.2.2 a highcycle accumulation model [578] has been developed. The basic assumption of the model is that the strain path and the stress path that result from a cyclic loading can be decomposed into an oscillating part and a trend (accumulation) which can be treated separately. The oscillating part is described by the strain amplitude (Section 2.5.2). The model follows the trend using an empirical expression for the rate of strain accumulation D acc which depends (among others) on the oscillating portion of strain and enters the main constitutive equation σ˙ = E : (D − D acc − D pl )
(3.243)
with the Jaumann stress rate σ˙ of the eﬀective stress σ, the strain rate D, the plastic strain rate D pl and the (barotropic) elastic stiﬀness E . In the highcyclic context ”rate” means the derivative with respect to the number of cycles Nc , i.e. ˙ = ∂ /∂Nc , or the increment per cycle. In cyclic element tests D pl = 0 holds and we obtain pseudocreep D = D acc at σ˙ = 0 or pseudoE : D acc at D = 0. In FEcalculations, apart from the highrelaxation σ˙ = −E cyclic loading, the elements are subjected to monotonic loading which requires a conventional plastic strain rate Dpl with the yield condition of Matsuoka and Nakai [520] (trσ tr(σ −1 ) < const) in order to restrict the admissible stress
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3 Deterioration of Materials and Structures
ratios. For instance, such monotonic loading may be caused by accumulation eﬀects in neighbouring elements. The direction m (a unit tensor) of D acc Dacc = Dacc m
(3.244)
turns out to be almost independent of the amplitude or the polarization of the cycles and independent of the density of the soil. The ﬂow rule m is a function of the stress ratio only. The experiments (Section 3.2.2) show that the ﬂow rule for monotonic loading proposed in the modiﬁed Cam clay (MCC) model → 1 q2 3 m= − p− 2 1 + 2 σ∗ 3 M p M
(3.245)
can be directly adopted to predict a suﬃciently exact direction of accumulation, at least for the triaxial (axisymmetric) compression and extension. The asterisk ∗ denotes the deviatoric part of , the superposed arrow → denotes Euclidean normalization, and M (σ) describes in (3.245) the presumed (could not be fully tested in the conventional apparatus) dependence of m √ →∗ triaxial → → on the Lode angle θ = arccos(− 6 − σ ·− σ∗ : − σ ∗ ). The intensity of strain accumulation Dacc in Eq. (3.244) is calculated as a product of six empirical functions: Dacc = fampl f˙N fe fp fY fπ
(3.246)
The multiplicative combination of these functions has been also found empirically. Each function (see Table 3.23) considers separately the inﬂuence of a diﬀerent parameter. The function fampl describes the proportionality between Dacc and the square of the strain amplitude (εampl )2 . The stressdependence Table 3.23. Summary of the functions, material constants and reference quantities of the highcycle model Inﬂuencing parameter Strain amplitude
Function
fampl = min
εampl ampl εref
2
Void ratio
fe =
(Ce −e)2 1+eref 1+e (Ce −eref )2
Reference quantities εampl = 10−4 ref
; 100
A B f˙N = f˙N + f˙N ) ( A A f˙N = CN1 CN2 exp − CN 1g f ampl B = CN1(CN3 f˙N ) av Average mean pressure fp = exp −Cp ppref − 1
Average stress ratio fY = exp CY Y¯ av
Cyclic preloading
Material constants
CN1 CN2 CN3 Cp
pref = 100 kPa
CY Ce
eref = emax
3.3 Modelling
315
(increase of Dacc with decreasing pav and with increasing stress ratio σ∗ /trσ) is captured by the functions fp and fY while fe expresses the increase of the rate with increasing void ratio. The increase of the accumulation rate due to changes in polarization is → − described by the factor fπ which compares the current polarization A of the amplitude with the socalled ’back polarization’ π , see [578]. If a package of cycles is directly followed by another package with the same polarization, i.e. →(1) − − →(2) A :: A = 1, then no correction of the accumulation rate is needed and fπ = 1. However, if the polarization has changed then the above product may become signiﬁcantly smaller or even zero and then the rate of accumulation is increased up to fπ = 1 + Cπ1 . The 4th rank back polarization tensor π represents the polarization in the recent history of cyclic deformation. The factor fπ entering Eq. (3.246) fπ = 1 + Cπ1 (1 − cos α)
(3.247)
→ − is proposed to be a function of the angle α = arccos( A :: π ) between the → − → − current polarization A and π . During cycles with A = const the tensor π → − is evolving (rotating) towards the current polarization, π → A . The angle α is proposed to evolve according to α˙ = −Cπ2 α (εampl )2
(3.248)
The constant Cπ2 > 0 inﬂuences the rate of this rotation. In order to perform π = R :: π by the angle Δα = αΔN the rotation π + Δπ ˙ we need to introduce the rotation operator (8th rank tensor) → → → → → → → → R = (cos Δα − 1)(− μ ⊗− μ +− ν ⊗− ν ) + sin Δα(− ν ⊗− μ −− μ ⊗− ν ) +JJ (3.249) → − → − where μ = A + π and ν = A − π denote mutually orthogonal tensors constructed on the hyperplane perpendicular to the rotation axis. J denotes the 8th rank identity tensor. The function f˙N describes the subtle problem of the dependence of Dacc on cyclic preloading (historiotropy). The cycles in the past are represented by their number Nc weighted by their amplitude [578]. For the sand samples in the laboratory the cycles can be counted from the beginning i.e. from their preparation, e.g. by pluviation. In situ the loading history is unknown and we must conclude the preloading from the behaviour of the soil under a cyclic test loading or from some empirical correlations. The cyclic loading in the past can be termed ”fatigue” which in case of soil means that preloaded soils respond with less accumulation (say of deformation) to the current cyclic loading than the freshly pluviated ones (at the same stress and density, see Figure 3.89d). The preloading can be quantiﬁed with g A deﬁned via the following evolution equation gA A (3.250) g˙ = fampl CN 1 CN 2 exp − CN 1 fampl
316
3 Deterioration of Materials and Structures
The physical meaning (on the granular level) of this variable is not clear. Presumably it is related to the coordination number, to the statistical distribution of contact normals or to the spatial ﬂuctuation of stress. The recent tests [581] could not deﬁnitively answer this question. The tensorial deﬁnition of the amplitude A ε for multidimensional strain loops is explained in Section 2.5.2. The determination of the material constants of the model has been discussed in detail in [842]. A simpliﬁed procedure proposed in [841] uses correlations of the constants with index properties (e.g. mean grain size d50 , uniformity coeﬃcient Cu , minimum void ratio emin ). These correlations are based on approx. 200 cyclic triaxial tests. 3.3.4 Models for the Fatigue Resistance of Composite Structures Authored by Gerhard Hanswille and Markus Porsch 3.3.4.1 General Most design codes consider the static and fatigue resistance of composite steelconcrete structures with separate veriﬁcations for the ultimate limit state and the limit state of fatigue. For headed shear studs both veriﬁcations are based on world wide performed experimental investigations with pushout specimens. The determination of the fatigue life of headed shear studs in recent European codes is based on the SN curve developed from the statistical analysis of a great number of cyclic pushout tests. The SN curve considers the shear stress range only. In addition the maximum shear force is limited at serviceability limit states in order to avoid signiﬁcant inelastic behaviour. However, recent researches showed that beside the shear stress range also the peak load and the static strength of the shear studs inﬂuence the fatigue life. On this background a comprehensive test program of more than 90 standard EC4pushout test specimens and two fullscale beam tests was developed, in order to investigate the interaction between high cyclic loading and static strength as well as the eﬀect of cyclic loading on the loaddeﬂection behaviour of studs. This test program and the results are described in Chapter 3.2.3. More detailed information are given in [352, 349, 347]. The results clearly show that cyclic loading of headed shear studs leads to a substantial decrease of static strength of stud connectors during their lifetime, so that the assumption for independent limit states under service loads and under ultimate loads is not given[351, 354, 355, 353, 346, 643, 805]. In the following the results of the test program are evaluated and new models are presented considering the interaction between cyclic loading and a reduced static resistance in ultimate limit state. These models are not only focused on the local behaviour of headed shear studs, but also on the eﬀect of the local damage on the global behaviour of composite beams.
3.3 Modelling
317
concrete failure
steel failure
concrete failure:
Pt ,c k c,m D d 2
steel failure:
Pt ,s
d fcm fu Ecm D kc,m , ks,m
E cm f cm
k s,m f u (S d 2 4)
diameter of the shank (16 d d d 22mm) mean value of the cylinder compressive strength mean value of the tensile strength of the stud shank mean value of the modulus of elasticity for concrete (secant modulus) = 0.2 [(h/d) + 1] for 3 d h/d d 4; = 1.0 for h/d > 4 coefficients to fit the theoretical model kc,m = 0.374, ks,m = 1.0 (mean prediction)
pushout test mean prediction: Pt = min (Pt,c , Pt,s)
Fig. 3.148. Theoretical model for the prediction of the mean value of the ultimate shear resistance according [684]
3.3.4.2 Modelling of the Local Behaviour of Shear Connectors in the Case of Cyclic Loading 3.3.4.2.1 Static Strength of Headed Shear Studs without Any Predamage Figure 3.148 shows the semiempirical model for the prediction of the mean value of the ultimate shear resistance of headed studs [684], which was taken as the basis for the design rules in current national and international codes. It was derived for headed studs with a diameter of 16 mm to 22 mm embedded in solid slabs of normal weight concrete on the basis of the results of 76 statically loaded pushout test specimen, which were already in accordance to the version of the Eurocode 4 of the year 1994 [8]. The resistance is given by the minimum value of two equations, which describe the the failure mode ”shear failure of the stud” and ”concrete failure”, respectively. The model is based on the assumption, that in the case of low concrete strength the shear resistance is determined only by the failure of concrete in the lower part of the shank. In the case of high concrete strength it is assumed, that the shear resistance is determined by the shear resistance of the stud shank. It is known that this model is a simpliﬁed empirical approximation, but so far due to the complex interaction between the stud and the local concrete surrounding the stud it was not possible to ﬁnd a veriﬁed mechanical model. The comparison of the static tests results (Pe ) for series S1  S6 with the corresponding predictions according to Figure 3.149 shows that this model can also be applied on the new tests. Because of the relative high concrete strengths for all these cases the prediction according to the case ”shear failure of the stud” was decisive. Due to an erroneous speciﬁcation of the secant modulus of elasticity Ecm in the earlier national and international concrete codes [5, 13] the evaluation of the statically loaded pushout tests was unintentionally based on both, the secant modulus Ecm and the tangent modulus Ec0m , although it was only
318
3 Deterioration of Materials and Structures
intended to use the secant modulus of elasticity. On this background the statistical analysis of [684] was repeated when specifying the national parameters of the German Annex of Eurocode 4 [34]. In this analysis additionally the new results of the static tests of series S1  S6 and the results of larger headed studs with a diameter of 25 mm [343] were considered taking into account the revised secant modulus of elasticity Ecm according to the edited version of DIN 1045 [25]. In total 101 pushout tests could be included, which are summarized for the diﬀerent failure modes in Table 3.24, Table 3.25 and Table 3.26. In these tables n means the number of studs per test specimen and h/d the ratio of the height of each stud (after welding) to its shank diameter. In 58 cases the criterion ”failure of the concrete” and in 43 cases the criterion ”shear failure of the stud” was relevant. Further information regarding specimen geometry and determination of the material properties are given in [345]. The result of the reanalysis according to EN 1990 [16] are shown in Table 3.27 and Figure 3.149. In accordance with the background report [684] the following coeﬃcients of variation Vx were chosen. • • • •
Vx Vx Vx Vx
= = = =
3 % for the stud diameter d, 20 % for the modulus of elasticity (secant modulus) Ecm , 15 % for the cylinder compressive strength fcm , 5 % for the tensile strength of the headed stud fu .
In the case of relation of the equations of the theoretical model (Pt,c and Pt,s ) to the characteristic values (Xk ) of the cylinder compressive strength fck and the tensile strength of the headed studs fuk instead of each mean value (Xm ) the required partial safety factors γR shown in Table 3.27 can be reduced by the correction factors Δkc and Δks according equation 3.251. In the case of ”failure of the concrete” Δkc lies between 0.84 and 0.94 for a compressive strength range 20 ≤ fck ≤ 60 N/mm2 , thus a value of Δkc = 0.94 can be applied on the safe side. In the case of ”shear failure of the stud” Δk can be assumed constant equal to 0.92 for tensile strengths fuk between 400 and 620 N/mm2 . Δkc =
Pt,c (Xk ) Pt,c (Xm )
Δks =
Pt,s (Xk ) Pt,s (Xm )
(3.251)
Because of ∗ γR = Δkc · γR = 0.94 · 1.318 = 1.239
(3.252)
( γR according Table 3.27, column 3 ) and ∗ γR = Δks · γR = 0.92 · 1.198 = 1.102
(3.253)
( γR according Table 3.27, column 4 ) the design value of the shear resistance of a headed stud in concrete slabs with normal weight concrete as a short time static strength is given to:
3.3 Modelling
319
Table 3.24. Summary of the statically loaded pushout tests with decisive criterion ”failure of the concrete” (tests 1  27) reference []
[601]
[590]
PRd =
≤
test
no.
Pe
n
fcm
Ecm
fu
d
h/d
Pt,c
[]
[]
[kN]
[]
[N/mm²]
[N/mm²]
[N/mm²]
[mm]
[]
[kN]
SA1
1
88.5
8
28.2
25200
493
16
4.75
80.7
SA2
2
94.4
8
28.2
25200
493
16
4.75
80.7
SA3
3
90.3
8
28.2
25200
493
16
4.75
80.7
SB1
4
82.6
8
28.3
22300
493
16
4.75
76.1
SB2
5
76.7
8
28.3
22300
493
16
4.75
76.1
SB3
6
85.3
8
28.3
22300
493
16
4.75
76.1
A1
7
132.9
8
35.7
26300
499
19
4.00
130.8
A2
8
147.4
8
35.7
26300
499
19
4.00
130.8
A3
9
138.8
8
35.7
26300
499
19
4.00
130.8
LA1
10
111.1
8
25.6
24700
499
19
4.00
107.4
LA2
11
120.2
8
25.6
24700
499
19
4.00
107.4
LA3
12
112.0
8
25.6
24700
499
19
4.00
107.4
B1
13
124.3
8
33.6
22400
499
19
4.00
117.1
B2
14
115.2
8
33.6
22400
499
19
4.00
117.1
B3
15
115.2
8
33.6
22400
499
19
4.00
117.1
LB1
16
83.0
8
18.8
15400
499
19
4.00
72.6
LB2
17
82.1
8
18.8
15400
499
19
4.00
72.6
LB3
18
78.5
8
18.8
15400
499
19
4.00
72.6
2B1
19
118.4
8
33.6
22400
499
19
4.00
117.1
2B2
20
115.7
8
33.6
22400
499
19
4.00
117.1
2B3
21
113.4
8
33.6
22400
499
19
4.00
117.1
RSs1
22
135.0
2
27.0
24549
620
19
5.26
109.9
RSs2
23
133.0
2
27.0
24549
620
19
5.26
109.9
RSs3
24
122.0
2
21.8
22546
620
19
5.26
94.7
RSs4
25
131.0
2
21.8
22546
620
19
5.26
94.7
RSs5
26
133.0
2
25.5
23990
620
19
5.26
105.6
RSs6
27
142.0
2
25.5
23990
620
19
5.26
105.6
0.721 0.374 d2 α Ecm fck = 0.218 d2 α Ecm fck 1.239
(3.254)
d2 d2 0.811 1.000 π fuk = 0.736 π fuk 1.239 4 4
(3.255)
Due to short time relaxation eﬀects in static tests under displacement control with structural composite members of steel and concrete a partly signiﬁcant
320
3 Deterioration of Materials and Structures
Table 3.25. Summary of the statically loaded pushout tests with decisive criterion ”failure of the concrete” (tests 28  58) reference
test
no.
Pe
n
fcm
Ecm
fu
d
h/d
Pt,c
[]
[]
[]
[kN]
[]
[N/mm²]
[N/mm²]
[N/mm²]
[mm]
[]
[kN]
[513]
S3 S4 S5 S6 S8 S11 S16 S19 S22 S26 S29
28 29 30 31 32 33 34 35 36 37 38
96.2 100.1 106.7 126.2 121.4 112.7 115.0 115.0 106.9 99.1 104.1
4 4 4 4 4 4 4 4 4 4 4
29.0 28.3 27.7 29.1 30.7 29.6 31.3 32.0 34.7 24.9 27.1
25273 25022 24805 25309 25873 25486 26081 26322 27233 23763 24586
600 600 600 600 600 600 600 600 600 600 600
19 19 19 19 19 19 19 19 19 19 19
5.33 5.33 5.33 5.33 5.33 5.33 5.33 5.33 5.33 5.33 5.33
115.6 113.6 111.9 115.9 120.3 117.3 122.0 123.9 131.2 103.9 110.2
[528]
P1 P2 P3 P4 P5 P6
39 40 41 42 43 44
97.5 96.5 97.0 127.0 127.0 127.0
4 4 4 4 4 4
16.6 16.6 16.6 40.8 40.8 40.8
20302 20302 20302 29196 29196 29196
600 600 600 600 600 600
19 19 19 19 19 19
5.33 5.33 5.33 5.33 5.33 5.33
78.4 78.4 78.4 147.4 147.4 147.4
[862]
D1/1 D1/2 D2/1 D2/2 D2/3 D3/1 D3/2 D3/3
45 46 47 48 49 50 51 52
99.0 94.0 123.0 128.8 126.5 148.5 148.0 146.8
4 4 4 4 4 4 4 4
30.2 30.2 30.2 30.2 30.2 30.2 30.2 30.2
25698 25698 25698 25698 25698 25698 25698 25698
580 580 500 500 500 548 548 548
16 16 19 19 19 22 22 22
6.25 6.25 5.26 5.26 5.26 4.54 4.54 4.54
84.3 84.3 118.9 118.9 118.9 159.5 159.5 159.5
[371]
2A
53
141.0
4
40.3
29040
485
19
3.68
136.7
[343]
I/1 I/2 I/3 I/4 I/5
54 55 56 57 58
179.5 183.0 180.4 183.1 178.6
8 8 8 8 8
23.7 23.7 23.7 23.7 23.7
29445 29445 29445 29445 29445
468 468 468 468 468
25 25 25 25 25
5.00 5.00 5.00 5.00 5.00
195.3 195.3 195.3 195.3 195.3
loss of load bearing capacity can be observed, when the actuator is held in constant position. In pushout tests near ultimate load this loss amounts approximately 10% [345], even if the tests are carried out with a very low displacement rate as in the present cases. In order to allow for these eﬀects as a result of the test procedure the short time static strengths according equation (3.254) and (3.255) have to be reduced by an additional reduction factor in the order of 0.9. Thus on the basis a uniform partial safety factor γv = 1.25 for both failure modes the design value of the shear resistance of a single
3.3 Modelling
321
Table 3.26. Summary of the statically loaded pushout tests with decisive criterion ”shear failure of the stud” reference
test
no.
Pe
n
fcm
Ecm
fu
d
h/d
Pt,s
[]
[]
[]
[kN]
[]
[N/mm²]
[N/mm²]
[N/mm²]
[mm]
[]
[kN]
T1/1 T1/2 T1/3 T1/4 T1/5 T3/1 T3/2 T4/1 T4/2 T4/3 T2/1 T2/2 T2/3 T2/4 T2/5 T5/1 T5/2 T6/1 T6/2 T6/3 3A 4A 5A II/1 II/2 II/3 II/4 II/5 S11a S11b S11c S21a S21b S21c S31a S31b S31c S41a S41b S41c S51a S51b S61a
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43
144.5 147.8 135.5 148.9 137.8 140.1 145.1 137.3 133.7 137.7 170.1 168.1 165.9 170.6 168.8 176.3 177.5 166.1 159.9 177.9 166.0 160.0 172.0 233.0 238.0 234.9 243.5 232.8 191.3 211.3 213.0 201.3 173.3 175.3 216.0 200.6 201.0 186.8 176.5 179.1 184.6 186.8 196.0
8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 4 4 4 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8
36.7 36.7 36.7 38.3 38.3 44.7 44.7 44.7 44.7 44.7 36.3 36.3 36.3 36.3 36.3 59.0 59.0 57.3 57.3 57.3 39.1 47.1 57.5 41.3 41.3 41.3 41.3 41.3 44.2 49.0 49.7 44.7 42.8 42.8 56.2 53.9 53.9 43.4 43.4 43.4 42.9 42.9 45.8
27890 27890 27890 28405 28405 30397 30397 30397 30397 30397 27759 27759 27759 27759 27759 34546 34546 34069 34069 34069 28661 31119 34126 34687 34687 34687 34687 34687 36400 36400 36400 33800 33800 33800 39000 39000 39000 33900 33900 33900 33050 33050 33700
460 460 460 460 460 460 460 460 460 460 471 471 471 471 471 471 471 471 471 471 485 485 485 468 468 468 468 468 528 528 528 528 528 528 528 528 528 528 528 528 528 528 528
19 19 19 19 19 19 19 19 19 19 22 22 22 22 22 22 22 22 22 22 19 19 19 25 25 25 25 25 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22
5.26 5.26 5.26 5.26 5.26 5.26 5.26 5.26 5.26 5.26 4.50 4.50 4.50 4.50 4.50 4.50 4.50 4.50 4.50 4.50 5.26 5.26 5.26 5.00 5.00 5.00 5.00 5.00 5.68 5.68 5.68 5.68 5.68 5.68 5.68 5.68 5.68 5.68 5.68 5.68 5.68 5.68 5.68
130.4 130.4 130.4 130.4 130.4 130.4 130.4 130.4 130.4 130.4 179.0 179.0 179.0 179.0 179.0 179.0 179.0 179.0 179.0 179.0 137.5 137.5 137.5 229.8 229.8 229.8 229.8 229.8 200.7 200.7 200.7 200.7 200.7 200.7 200.7 200.7 200.7 200.7 200.7 200.7 200.7 200.7 200.7
[682]
[371]
[343]
[352]
stud connector considering time dependent eﬀects due to high local concrete pressure in front of the studs is ﬁnally given by the minimum of equation (3.256) and (3.257).
322
3 Deterioration of Materials and Structures
Pe [kN]
Pe [kN] Pt,c
250
Pt ,c 0.374 d
2
200.7 Pt,s
250
E cm f cm
Pt ,s
200
PRk
VR = 0.19
150
PRd
200
Sd2 fu 4
PRk
VR = 0.12
PRd
150
100
100
50
50
S1S6 PRk = 0.721 Pt,c
0 0
50
100
150
PRk = 0.811 Pt,s
Pt,c [kN]
PRd = 0.547 Pt,c
200
250
Pe experimental shear resistance
Pt,s [kN]
PRd = 0.678 Pt,s
0 0
50
100
150
200
250
PRk characteristic value of the shear resistance according
Pt,c mechanical model (concrete failure) (mean value)
EN 1990 (5%fractile)
Pt,s mechanical model (steel failure) (mean value)
PRd design value of the shear resistance according EN 1990
Fig. 3.149. Result of the statistical analysis of the results of 101 statically loaded pushout tests according to EN 1990 [16]
PRd = 0.245 d2 α ≤ 0.83 π fuk
Ecm fck
d2 1 4 γv
1 γv
(γv = 1.25)
(3.256)
(γv = 1.25)
(3.257)
This result is nearly coincident to the original evaluation [684] and it conﬁrms the use of the secant modulus of elasticity Ecm [33, 25] as one of the main material properties of the concrete in equation (3.256). In Figure 3.150 the result of the statistical reanalysis according EN 1990 is compared to the design rules of the German and the European rules. The design rules of DIN 188005 [27] are nearly identical to the result of the statistical reanalysis, whereas in the Eurocode 4 [22, 23] a signiﬁcant higher shear resistance can be taken into account. In order to compensate this lower safety level in the German Annex of Eurocode 4 [34] a partial safety factor γv,c = 1.5 for the mode ”failure of the concrete” was introduced. 3.3.4.2.2 Failure Modes of Headed Shear Studs Subjected to HighCycle Loading The test results given in Chapter 3.2.3 clearly indicate, that the mechanical properties of headed shear studs under static loading can not be applied without restrictions on the properties of headed shear studs subjected to
3.3 Modelling
323
Table 3.27. Result of the statistical analysis according EN 1990, Annex D [16] Table 3.24, Table 3.25 Pt,c
Table 3.26
58
43
¦ (P P ) ¦P
1.0
1.0
Pei b Pti


ln (G i )


0.035
0.012
0.124
0.087
0.124
0.088
0.139
0.078
VG2 Vrt2
0.187
0.117
test according theoretical model ("failure" mode) n
number of tests ei
b
b
Gi
Gi
'i
'i
1 n
'
'
1 n 1
s '2
s'
V G2
VG Vrt2
Vrt
6
V r2
Vr
¦'
¦ ('
i
i
') 2
exp (s 2' ) 1
ª 1 « « Pt ( X 1 ¬«
n i
ti
2 ti
º wPt Vi » » m ) wX i ¼»
Pt,s
2
QG
QG
ln (VG2
1)
0.124
0.087
Qrt
Qrt
ln ( Vrt2 1)
0.138
0.078
Q
Q
ln(Vr2 1)
0.185
0.117
Qrt2 Q2 k n G 0. 5 Q 2 ) Q Q
0.721 Pt,c
0.811 Pt,s
Q2 Qrt2 k d ,n G 0.5 Q 2 ) Q Q
0.547 Pt,c
0.678 Pt,s
1.318
1.198
PRk
PRk
b Pt (X m ) exp (1.645
PRd
PRd
b Pt (X m ) exp ( 3.04
JR
JR
PRk / PRd
kn
Vx unknown – 5%fractile – (n)
1.694
1.713
k d ,n
Vx unknown – (n)
3.28
3.366
highcyclic preloading. High cyclic loading leads to a reduction of the stiﬀness of the interface between steel and concrete due to the irreversible slip and moreover it results in an early reduction of the static strength. In order to ﬁnd the reasons for the signiﬁcant eﬀect of highcyclic loading, the concrete slabs were separated from the steel beams and the fractured surfaces at the
324
3 Deterioration of Materials and Structures
concrete failure: PRd ,c k c,d D d 2 steel failure: d fuk fck Ecm D kc,d , ks,d Jv,c , Jv,s
PRd,s
design value: PRd = min (PRd,c , PRd,s)
E cm f ck J v,c
kc,d / ks,d []
k s,d f uk (S d 2 4) J v,s
diameter of the shank (16 d d d 25mm) characteristic value of the ultimate tensile strength of the stud shank characteristic value of the compressive cylinder strength (according EN 206) mean value of the modulus of elasticity for concrete (secant modulus) (according EN 206) = 0.2 [(h/d) + 1] for 3 d h/d d 4; = 1.0 for h/d > 4 coefficients to fit the theoretical model partial safety factors for the design shear resistance
statistical analysis 0.245 / 0.83 (EN 1990)
Jv,c / Jv,s []
fu [N/mm²]
1.25 / 1.25
460  620
DIN 188005
0.25 / 0.80
1.25 / 1.25
< 450
EN 199411
0.29 / 0.80
1.25 / 1.25
< 500
EN 199411 incl. National Annex
0.29 / 0.80
1.50 / 1.25
< 500
Fig. 3.150. Comparison of the result of the statistical analysis with the rules in current German and European standards
metallurgical investigations
microstructure
forced fracture area and fatigue fracture area
Fig. 3.151. Preparation stages for examination purposes
foot of each headed stud of each test specimen were examined. Figure 3.151 shows in detail the stages of preparation of the test specimens after the test phases for examination purposes. In two speciﬁc cases additional metallurgical investigations of the microstructure were carried out. The exposed fracture surfaces at each stud foot consisted of a typical smooth fatigue fracture zone and a partly coarse forced fracture zone as shown in Figure 3.152. In nearly all cases these zones could be clearly distinguished
3.3 Modelling Mode A
Mode B
stud shank
fatigue fracture (with arrest lines)
325
weld collar
fatigue fracture P1
crack tip P2 forced fracture
mode B
forced fracture
Mode A: crack initiation at point P1 followed by a horizontal crack propagation through the shank
P1: transition between the stud shank and the weld collar P2: transition between the weld collar and the flange
Mode B: crack initiation at point P1 or at P2 followed by a crack propagation headed through the flange
Fig. 3.152. Failure modes A and B
from each other because of the diﬀerent surface structures, so that it was possible to determine clearly the size and the geometry of the exposed fatigue fracture areas. The fatigue fracture area was in all cases caused by cracks at the stud foot, initiated at the points P1 or P2 and then propagating horizontal through the shank or headed through the ﬂange. The corresponding forced fracture area was caused by a combination of a bendingshear failure of the residual cross section. This kind of failure occurred at the end of a fatigue test at which due to crack propagation the static strength was reduced to the applied peak load or during the static loading phase after high cyclic preloading, which was carried out in order to determine the residual strength. The failure modes were closely correlated with the peak load Pmax . For high peak loads such in series S2 and S4 only mode A occurred. For lower peak loads such in series S1, S3, S5E in most cases mode B occurred. Nevertheless in some cases two cracks of mode A and mode B were detected at the same time at a stud foot, which means, that two cracks grew directly above each other and both could initiate forced fracture. The investigations of the microstructure revealed that both points, P1 and P2, show exceptionally high geometrical and metallurgical notch eﬀect due to welding technique. This is in no case in agreement with the requirement of common arcwelded joints in structural steelwork regarding the quality level according to [35]. Both sharp transitions are typical results of the drawn arc stud welding process. The process begins with presetting the current time and the welding time and placing the stud on the ﬂange. Upon triggering a pilot arc occurs after lifting the stud to a preset height. Subsequently the main arc is ignited which melts the end of the stud and the ﬂange on the opposite side. By means of a spring force ﬁnally the stud is forged into the molten ﬂange.
326
3 Deterioration of Materials and Structures
crack initiation point (P2)
proper stud weld
200:1 crack propagation inter and transcristallin
voids with rough surfaces and transitions
200:1
corresponding crack tip
Fig. 3.153. Weld collar (exterior appearance and inner state)  Closeup view of the crack shown in Figure 3.152 at the starting point (P2) and at the corresponding crack tip
This forces excessive material out into the ceramic ferrule shaping the weld collar. Due to the diﬀerent aggregate states this does not lead to a fusion between the inside of the weld collar and the outside of the stud base and results in sharp edged transitions in P1 and P2. These two points coincide with the points of the highest stress levels and the crack growth consequently starts at these notches. Moreover Figure 3.153 (left) illustrates, that the drawn arc welding process leads to an apparent faultless weld collar on the outside, but on the inside it may contain voids due to the degassing process during welding. So contrary to the outside appearance the weld collar is generally not homogeneous and of lower strength compared to the stud and the base material. Figure 3.153 (right) shows the crack initiation point P2 and the corresponding crack tip of the crack in Figure 3.152 enlarged 200 times. It illustrates, that the transition between the weld collar and the ﬂange is not smooth but undercut, being an ideal condition for early crack initiation in the case of high cycle loading. In the present case the crack propagated both transcrystalline and intercrystalline. Beginning near the line of fusion at the transition between the collar and the ﬂange the crack grew through the ﬁne grained structure of the heat aﬀected zone, working its way through the coarse grained structure
3.3 Modelling
weld collar
327
crack orientation
shank
fatigue fracture area (AD)
crack front
mode B
forced fracture area (AG)
mode A
point C
circular
Pu / Pu,0
1.0
Pu AD  1 ( Eq . B) Pu,0 AD AG
0.8
62 tests Æ 496 studs
Pu AD (Eq. A)  1 0 . 6 Pu,0 AD AG
0.6
0.4
0.2
failure
cyclic loading
series
static
constant
S2, S4
fatigue static
constant constant
S2, S4 S1, S3, S5E
fatigue
constant
S1, S3, S5E
fatigue fatigue static
variable variable variable
S5 S6 S9
only mode A within a specimen
mode A and mode B within a specimen
AD/(AD+AG)
0.0
0.0
0.2
0.4
0.6
0.8
1.0
Fig. 3.154. Correlation between reduced static strength and damage at the stud feet for failure modes A and B based on the fatigue fracture area for av < ah
of the heat aﬀected zone and ending at the nonaﬀected base material of the ﬂange. 3.3.4.2.3 Correlation between the Reduced Static Strength and the Geometrical Property of the Fatigue Fracture Area In order to detail the crack development, the test specimen were released and reloaded periodically during the cyclic loading phases. As shown in Figure 3.154 in the case of mode A it was possible to produce arrest line by means of this test procedure, which could be used for information about the number of load cycles causing crack initiation and about the crack propagation. Probably due to diﬀerent microstructure no usable stop marks could be observed in the case of mode B although the testing procedure was always the same. However, in all cases geometrical properties of each fatigue fracture area (such as outline, size (area AD ) , extension in the direction of the loading (crack length ah ), extension into the base material (crack depths av )) can be used for evaluation purposes. The relationship between the reduced static strength and the relative size of the fatigue fracture zone can be assumed to be linear as a good approximation independently of the modes. This is illustrated in Figure 3.154, which shows the result of an evaluation of 496 studs of 62 pushout tests.
328
3 Deterioration of Materials and Structures
In Figure 3.154 AD is the area of the fatigue cracking zone and AG the area of the forced shear fracture, both taken as the horizontal projections. In the case of mode A the whole fracture area (AD + AG ) corresponds to the stud area, which is for this reason clearly deﬁned. In the case of mode B due to the crack propagation into the ﬂange the whole fracture area can be much larger than the stud area. In order to interpret the test results in a deﬁnite way and additionally allow for situations, in which only the fatigue fracture area AD (e.g. from nondestructive measurements) is known, it is necessary to make reasonable assumptions concerning the deﬁnition of the shape of the forced fracture area. Based on the observations of the failure modes the size of the forced fracture area was determined by assuming, that this area is bounded by the crack front and by a circular border passing through the outer diameter of the weld collar on the opposite side (given as point C in Figure 3.154). The coeﬃcient of correlation of the linear relationship is 0.96 for series S2 and S4, in which due to the high peak loads of 0.70 Pu,0 exclusively mode A occurred. Except for very high degrees of damage of more than 90 % it can be deduced that the crack propagation in the shear stud independently of the modes has approximately 60 percent attribution in the reduction of the static strength. Regarding the reduced static strength the loading history during the cyclic loading phase (force controlled, displacement controlled, one block of loading and multiple blocks of loading) has only a minor inﬂuence. In the case of mode B (test series S1, S3 and S5E) the reduction of the static strength is very small for damage grades AD / (AD + AG ) between 35 % and 80 %. For estimations on the safe side the dotted relationship according equation (B) in Figure3.154 can be applied. For practical applications in which (e.g. from nondestructive inspection like ultrasonic) only the crack initiation point and the crack length at a stud foot instead of the whole outline of the fatigue fracture area is known, the relationships Eq. C and Eq. D according to Figure 3.155 can be used. If crack initiation starts at the outer edge of the weld collar (mode B) the horizontal crack length ah should be referred to the diameter dW of the stud weld. If crack initiation starts at the transition between the stud shank and the inner Pmax, Pu crack initiation at Point P2
dW
av
P2
Pmax, Pu crack initiation at Point P1
a ah
ah
d
av § 0 P1
Pmax, Pu
Eq . C
Pu Pu,0  1 0 .6 a h d W
ah
ah
Pmax, Pu
Eq . D
Pu Pu,0  1 0.6 a h d
Fig. 3.155. Correlation between reduced static strength and damage at the stud feet for failure modes A and B based on crack lengths and crack initiation points for av < ah
3.3 Modelling
329
edge of the stud weld (mode B) the crack length a (a ∼ ah ) should be referred to the stud shank diameter d. According to Figure 3.154 on the safe side the coeﬃcient 0.6 can be substituted by 1.0. 3.3.4.2.4 Lifetime  Number of Cycles to Failure Based on Force Controlled Fatigue Tests In Figure 3.156 the results of the fatigue tests of series S1 to S4 and S5E are compared with the corresponding test results, from which the fatigue strength curve in Eurocode 4 was derived [685]. In this concept the prediction of the number of cycles to failure depends on the nominal shear stress in the shank of the studs, provided that the peak load Pmax is smaller than 0.6 Pu,0 [685]. It can be seen, that the lifetimes of the fatigue tests of series S1, S3 and S5E, which lie in the scope of application of the fatigue strength curve, are predicted very well. One of the reason is obviously the additional lateral supporting of the concrete slabs shown in Figure 3.100 of Chapter 3.2.3, which was not used in the tests on which the fatigue curve is based. However, the results of the fatigue tests clearly show the inﬂuence of the peak load Pmax on the life time. In the case of an identical relative load range ΔP / P u,0 it can be observed that if the relative peak load Pmax / P u,0 is increased the number of cycles to failure decreases from 6.2 × 106 to 3.5 × 106 load cycles (series S1 and S4) and from 6.4×106 over 5.1×106 to 1.2×106 load cycles (series S5E, S3 and S2), respectively. In order to develop a theoretical model for the prediction of the fatigue life, in which not only the eﬀect of the load range ΔP can be taken into account, but also the eﬀects of the static
'WR (log) 1000
'W R
§N · ¨ c¸ ¨ N ¸ © ¹
test results: m = 8.658 Eurocode 4: m = 8
1m
'Wc
S2, S4
(Pmax = 0.71 Pu,0 )
S1, S3, S5E (Pmax 0.44 Pu,0 ) 'Wcm = 110 N/mm²
100
5%fractile 'Wck = 90 N/mm²
'P
d
'P 'W
10 104
105
4 'P Sd2
N (log) 106 Nc = 2x106
107
Fig. 3.156. Comparison of fatigue test results with the prediction in Eurocode 4
330
3 Deterioration of Materials and Structures
strength Pu,0 and the peak load Pmax , national and international fatigue tests of pushout test specimens subjected to unidirectional cyclic loading were reanalysed in the view of these parameters. To achieve comparable results great importance was attached to the geometry of the specimen, the number of welded studs and the lateral supporting condition of the concrete slabs. In this analysis only those tests were included, in which the requirements of the Eurocode 4 regarding geometry and test conditions were met. Thus the specimen had to consist of one steel beam and two lateral concrete slabs with four headed studs on each ﬂange. The slabs had to be casted in horizontal position and the studs had to be welded with an adequate welding procedure ensuring the formation of weld collar in accordance with EN13918 [10] and EN14555 [11]. These requirements were fulﬁlled by 26 tests. In the case of 13 specimen the concrete slabs were additionally laterally supported. Among the group of test specimen without lateral supporting count the tests of Oehlers [591] and Hanswille [342] of 1989 and 1999. Among the other group count the fatigue tests listed in Table 3.6 (Chapter 3.2.3) and a fatigue test of Velkovic et al. [809] of 2003. In the case of [342] short time static tests were not carried out, so the reference value of the static resistance was calculated with the model given in Figure 3.148. In the 26 tests the concrete cylinder compressive strength fc according to EN 206 [12] varied between 31.0 N/mm2 and 54.3 N/mm2 . The range of the diameter d of the stud shanks was 13 mm to 25 mm and the tensile strength fu of the studs lied between 450 N/mm2 and 528 N/mm2 . For evaluation purposes the test were sorted in two groups each with identical supporting condition and evaluated by means of a common theoretical model according to Figure 3.157 giving the value of the fatigue life of a headed shear stud embedded in solid concrete slabs subjected to unidirectional cyclic loading. The free parameters K1 and K2 are to be chosen in dependence of the lateral supporting condition. In the case of additional lateral support, the parameters can be chosen to K1 = 0.1267 and K2 = 0.1344. In the case of no lateral support the parameters are to be chosen to and to K1 = 0.1483 and K2 = 0.1680. 3.3.4.2.5 Reduced Static Strength over Lifetime As it can be seen from the tests the static strength reduces with increasing number of cycles. The failure envelope, i.e. static strength over the number of cycles, is characterized by a sigmoidal shape as shown in Figure 3.158 (a). The results of the ﬁve more tests given in [809] with exact the same specimen geometry and supporting condition and a diﬀerent relative peak load show the same characteristics and are also illustrated in Figure 3.158 (a). The sigmoidal relationship between the relative values of the static strength and the load cycles can be described with the equation given in Figure 3.158 (b). This equation is the result of a parametric study of totally 60 tests. It is to mention that the relative load range chosen in the tests was between 0.2 and 0.25. If further tests with diﬀerent values of the load range are available
3.3 Modelling
331
Nf : number of load cycles to failure in a forcecontrolled pushout fatigue test
1 max Pu,0
10
theoretical model
Nf
P 0.5 'P K1 K 2 max Pu,0
with lateral restraint
Nf,t
108
P
without lateral restraint
107 26 tests 106
105 K1 = 0,1267
K1 = 0,1483
K2 = 0,1344
K2 = 0,1680
104 104
with without lateral restraint
105
106
107
108
Nf,e
experimental results
Fig. 3.157. Theoretical model for the prediction of the fatigue life of a headed shear stud in a pushout test  relationship between experimental and theoretical fatigue life a)
b) Pu Pu,0
1.2
(Pu Pu,0 ) theor
1.0 60 tests 0.8
1.0
0.6 0.8 0.4 0.6
0.2 series
0.4
0.2
0
0
1 2 3
0.20 0.25 0.25
0.44 0.71 0.44
4
0.20
0.71
5E
0.25
0.30
Vel
0.20
0.60
0.2
(Pu Pu,0 ) exp
ǻP Pu,0 Pmax Pu,0
0.4
0.6
0
Pu
N Nf 0.8
Pu,0
0
0.2
0.74
Pmax 'P Pu,0
0.4
0.6
0.8
1.0
d 1 § · ° 1 ¨ ¸ 0.54  0.04 ln ¨ 1¸ °® P ¨1 N N f ¸ °t max © ¹ °¯ Pu,0
1.0
Fig. 3.158. Analytical description of the reduced static strength over lifetime (a) Comparison of the theoretical and experimental values of the reduced static strength (b)
332
3 Deterioration of Materials and Structures
the parametric study should be repeated in order to extend the scope of application. 3.3.4.2.6 LoadSlip Behaviour Regarding the numerical simulation of composite beams the load deﬂection behaviour of headed shear studs under static and cyclic loading is of main interest. These results should not be neglected but be comprised as fundamental research results. According to diﬀerent stages of the test procedure it was possible for the tests reported in Chapter 3.2.3 to deduce the loaddeﬂection behaviour of headed studs embedded in normal weight concrete during initially static loading, during cyclic loading (including phases of releasing and reloading) and during static loading after high cycle preloading. As already known the initial static loadslip behaviour of headed shear studs embedded in normal weight concrete is characterized by a high initial stiﬀness and high ductility. Based on a statistical analysis of 15 comparable static pushout tests of the series S1S6 and S9 the mean behaviour can be described by the exponential function, given in Figure 3.159, which can be applied up to mean value of the slip at ultimate load δu of 7.5 mm. The associated coeﬃcient of variation Vx of the slip depends on the load level and P/Pu,0 []
P/Pu,0 []
scatter band
1.0
1.0 0.8
P Pu,0  (1 e1.22 į
PRd = 0.68 Pu,0 (su)
0.6
)
mean behaviour
0.6
PRd = 0.55 Pu,0 (cu) 0.6 PRd = 0.41 Pu,0 (sf) 0.6 PRd = 0.33 Pu,0 (cf)
0.4 0.2
0.2
Gu = 7.5 mm
G [mm]
Gcf = 0.15 mm
0
1
2
3
G
0.4
Gsf = 0.24 mm
0
P
0.8
0.59
Vx []
0 4
5
6
7
8
10
9
11
12
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7
a)
abbreviations: s: "steel failure"  c: "concrete failure"  u: ultimate limit state  f: fatigue limit state
1.0
P/Pu,0 []
P/Pu,0 []
0.8
1. loading
ln (1 P Pu ,0 )1/ 0.59 1.22 ª1 º G K1 K1 « (P Pu ,0 )1.5 » «¬1.1 »¼
0.6
G
0.4
G pl
0.2
G 2.loading
1. unloading
2. loading
G K2 K 2
ª º 1 (P Pu ,0 ) 0.5 » «1 ¬« 7.5 ¼»
4
6
G [mm]
0 0
1
2
3
5
7
8
9
Gpl G G2. loading
G [mm]
b)
Fig. 3.159. Standardised loadslip curve of headed shear studs in normal weight concrete  load deﬂection behaviour after ﬁrst unloading and successive reloading
3.3 Modelling
P / Pu,0
P/ Pu,0
(1 e 1.22 G
0.59
333
) (mean value) 7.5 mm
1.0 Pmax / Pu,0 = 0.71
0.8
P
0.6
G
0.4 0.2
Pmax / Pu,0 = 0.44
G [mm] 0
1
2
3
4
5
6
7 Gu 8
9
10
11
12
Fig. 3.160. Eﬀect of highcycle loading on the loadslip behaviour
varies between 0.65 at low levels and 0.25 at higher levels. In addition to the initial static behaviour the large number of tests under the same conditions were used for the evaluation of both, the magnitude of the plastic slip after ﬁrst unloading and the magnitude of the slip after successive reloading to the preceding load. As shown in Figure 3.159 these values can be calculated by multiplication the initial static slip δ taken as a reference value with two simple functions η1 and η2 . Provided that the initial static load is less than the ultimate load ﬁrst unloading and successive reloading only leads to an increase of the elastic and the accumulated plastic slip. However, high initial loading to the ultimate load followed by a hysteresis may additionally result in a lower load bearing capacity at the end of the reloading. The functions η1 and η2 illustrate that the secant stiﬀness during unloading and reloading are naturally slightly diﬀerent and that their magnitudes decrease with increasing load level because of disproportionate increase of plastic slip. One additional important question in the concept of the numerical simulation of cyclic loaded beams is the inﬂuence of the cyclic loading on the loadslip behaviour of the studs after cyclic preloading. In Figure 3.160 the grey shaded area again shows the range of the loadslip curves of all statically loaded pushout tests without any predamage. The mean value of this range is given by the marked continuous blue line within the shaded area. All other curves show the static behaviour after diﬀerent numbers of load cycles. The loadslip behaviour of a stud without any predamage can be interpreted as an envelope for all other cases, as all other curves lie within or signiﬁcantly below the shaded area.
334
3 Deterioration of Materials and Structures
elastic stiffness
P
Pu,N
0.8Pu,N
P = Kel,N Pu,N (G – Gpl,N ) for P 0.8 Pu,N Kel,N =1.4 [1/mm]
G pl, N
Kel,N Gpl,N
G
§ N/N f D1 ¨ ¨ 1 N / Nf ©
1/ D 2
· ¸ ¸ ¹
5.04 ( Pmax / Pu , 0 )
D1
0.049 e
D2
§ P 0.5 'P · § · ¸ 11.522 ¨ Pmax 0.5 'P ¸ 3.054 24.865 ¨¨ max Pu , 0 Pu , 0 ¨ ¸¸ ¨¨ ¸¸ © ¹ © ¹
2
plastic slip
Fig. 3.161. Elastic stiﬀness and accumulated plastic slip after N number of load cycles  each based on the results of test series S1  S4 and S5E
Highcyclic loading results in a linearization of the static loadslip behaviour up to approximately 80% of each corresponding reduced static strength. The elastic stiﬀness after high cyclic preloading can be assumed as being constant if the stiﬀness is determined on the basis of the reduced static strength Pu,N . The mathematical functions for both, the elastic stiﬀness and the accumulated plastic slip δpl,N after N numbers of load cycles are given in Figure 3.161. 3.3.4.2.7 Crack Initiation and Crack Development The eﬀect of force controlled high cyclic preloading on the static strength as shown in Figure 3.104 in Chapter 3.2.3 is mainly caused by an early crack initiation followed by a long phase of crack propagation. Due to the reduction of the static strength under cyclic loading, the mechanical properties of headed shear studs under static loading cannot be adopted on the behaviour of studs under fatigue loading. In order to assess existing design concepts of current national and international codes and in order to develop new concepts based on based on crack propagation the knowledge of the exact time of crack initiation is of main interest. Another important question is the question about the cutoﬀ limit. In comparison with typical welding details in steel structures, the sharp notches in the welding area lead to the conclusion that there is only a very low load limit or no load limit, where a crack initiation can be excluded. In the case of horizontal cracks of type A it is possible to produce systematically arrest lines (visible to the naked eye) on the fatigue fracture areas by releasing and reloading the test specimens during the cycle loading phases. As shown in Figure 3.162 for a representative stud of test specimen S24b (fatigue test in series S2) the arrest lines provide important details of crack initiation and crack growth velocity. By means of the correlation between the reduced static strength and the damage at the stud feet given in Figure 3.154 the reduced static strength can be determined if the fatigue fracture area AD is known. In the present case the crack velocity function shows a nearly
3.3 Modelling
Pu / Pu,0
reduction of static strength 2
1.0
AD AG S d 4
0.8 0.6
crack propagation
AD(N/Nf) [mm²] 200 test S24b stud D2b
150 Pu AD ~ 1 0.60 Pu,0 AD AG
335
100
0.4
50
0.2
crack type A
AD/(AD+AG)
0 0
0.1
0.2
0.3
0.4
0.5
0.6
N / Nf
0 0
0.7
0.2
0.4
0.6
0.8
1.0
dAD/dNi [103 mm²/load cycle] 0.4 fatigue fracture area (AD)
0.3
crack velocity
test S24b stud D2b
0.2 force fracture area (AG)
P Pmax
arrest lines caused by unloadings and reloadings during the cyclic loading phase load cycle
0.1 0 0
N / Nf 0.2
0.4
0.6
0.8
1.0
Fig. 3.162. Relationship between crack velocity, crack propagation and reduction of static strength for test series S2
sigmoidal characteristic. Similar to the development of plastic slip the crack velocity increases disproportionately at the end of the fatigue life reaching a critical crack length depending on the peak load level. The arrest lines clearly show that the start of the crack growth nearly from the beginning of the cyclic loading is possible. This observation led to the question whether in real composite structures signiﬁcant cracking at the stud feet due to cyclic loading can be avoided if they are designed on the basis of current national and international codes. Based on the results explained above [351, 354, 355], in current German codes [27, 18, 34] the safety level for headed shear studs subjected to cyclic loading was increased compared to the safety level in other international codes based on Eurocode 4. The partial safety factor γMf,v was changed from 1.0 to 1.25 in the design model for the fatigue resistance of headed shear studs by means of the characteristic value of the fatigue resistance curve (5%fractile) shown in Figure 3.156. The eﬀect of this approach is summarized in Figure 3.163. Due to the slope m = 8 an increase of the partial safety factor from 1.0 to 1.25 results in a decrease of the design value of the fatigue life Nf m of cyclic loaded headed shear studs by factor 6 (γMf,v = 1.258 ). On the other hand the characteristic value (5%fractile) of the fatigue resistance curve used in the codes leads to a theoretical life time which is 5times lower than the lifetime according to the mean value of the fatigue strength derived in [685]. Hence in current German codes in a design only 1/30 of the mean value of the
336
3 Deterioration of Materials and Structures
curve 1: fatigue strength curve – mean value (test results m = 8.658 ~ 8) curve 2: fatigue strength curve – characteristic value (5%fractile) (JMf,v = 1.0) curve 3: fatigue strength curve – design value (JMf,v = 1.25)
'WR (log) 103 §N · ¨ c¸ ¨ N ¸ © ¹
'W R
1m
'W c N
§ 'W ¨ c ¨ 'W © R
m
'P
'P
· ¸ N c ¸ ¹
4 'P S d2
'W 'Wc = 110 N/mm²
1/ N f ,13 ~ (1/ 30 )
d
1/ N f ,12 ~ (1/ 5 )
1/ N f ,23 ~ (1/ 6)
curve 1 curve 2 curve 3
65.76 (tests S13_2)
101
'Wc = 72 N/mm² N f ,12 N f ( 'Wc
110 N / mm²) / N f ('W c
N f , 23 N f ( 'W c
90 N / mm ²) / N f ( 'Wc
N f ,13 N f ( 'Wc
110 N / mm²) / N f ('W c
104
105
90 N / mm²) 72 N / mm²) 72 N / mm²)
(110 / 90) 8 (90 / 72) 8 (110 / 72) 8
}
design codes m = 8
102
'Wc = 90 N/mm²
4.98 ~ 5 5.96 ~ 6
N (log)
29.7 ~ 30
106 Nc = 2x106
4.1x106
107 (tests S13_2)
Fig. 3.163. Fatigue strength and lifetime of cyclic loaded shear studs according diﬀerent design concepts depending on the safety levels  curve 1 [685]  curve 2 [22, 23] (European codes)  curve 3 (German codes) [27, 18, 34]
lifetime given in [685] can be adopted, whereas in international codes based on Eurocode 4 1/5 can be adopted. In order to investigate, if these safety margins are suﬃciently high to avoid signiﬁcant cracking at the stud feet during lifetime test series S11 and S13 were performed as reported in Chapter 3.2.3. In the case of test series S13 the cyclic loading phases were aborted just after subjecting 1/30 (N = 4.1 × 106 load cycles) of the mean value of the number of cycles to failure according to [685] (Δτ = 65.76 N/mm2  m = 8  Nf = 1.22 × 108 load cycles) before determining the reduced static strength. In the cases of test specimens S114a and S114c the cyclic loading phases were completed after applying 1/17.4 and 1/19872 of the corresponding values according [685] (m = 8). In all cases the peak loads Pmax were lower than 0.6 PRd , so that the requirement of Eurocode 4 regarding the peak load level under service loads were fulﬁlled. In Table 3.28 the experimental observed crack lengths in test series S11 and S13 are listed. In the cases of test specimens S132b and S132c, in which the cyclic loading phase was aborted just after subjecting 1/30 of the mean value of the number of cycles to failure according to [685] the crack length was of remarkable size. According to Figure 3.155 the observed cracks result in a reduction of the static strength of approximately 1015%. This can be accepted
3.3 Modelling
337
Table 3.28. Mean values of the crack length ah (see Figure 3.155) in test series S11 and S13 test
S114a
S114b
S114c
S132a
S132b
S132c
ah [mm]
~ 0.8
20.7
~ 0.7
~ 0.8
5.3
5.4
at the end of the numerical design life. However, it must be stated, that with the current design concepts cracks at the stud feet cannot be avoided. 3.3.4.2.8 Improved Damage Accumulation Model PalmgrenMiner cumulative linear damage rule [611, 543] provides a simple criterion for predicting the extent of fatigue damage induced by a particular block of constant amplitude cyclic loads in a loading sequence with diﬀerent stress amplitudes. This linear damage rule assumes that the number of cycles imposed on a component, expressed as a percentage of the total number of cycles of the same amplitude to cause failure, gives the part of damage and the order of the loading blocks does not inﬂuence the fatigue life. If Ni is the number of cycles corresponding to the ith block of constant loading amplitude in a sequence of m blocks with Nf,i as the number of cycles to failure, the failure occurs, if condition 3.258 is fulﬁlled. m Ni =1 N f,i i=1
(3.258)
Evaluation of the tests with multiple blocks of loading on the basis of the linear damage accumulation hypothesis of Palmgren and Miner, on which the present design codes are based, is shown in Figure 3.164. The fatigue life Nf,i corresponding to each block of cyclic loading is gained from the results of the constant amplitude tests of series S1 to S4 and S5E. The missing values of number of cycles to fatigue for the blocks 2 and 3 in the test with four blocks of loading are determined by means of a linear interpolation from the results of series S1 and S4. Thus, for the peak loads of 101 kN and 120 kN per stud the fatigue life Nf is determined as 5.3 × 106 and 4.4 × 106 number of cycles, respectively. It is obvious that except for one test in Figure 3.164 all results of the lifetime prediction according to Palmgren and Miner lie on the unsafe side. Main reason for this is neglecting of the eﬀects due to crack propagation in the shank of the stud and the increasing local crushing of concrete surrounding the stud weld. An improvement of the prediction is succeeded by the introduction of an additional damage term Δnf i in equation (3.259), which considers the eﬀects resulting from crack propagation in the stud and steady increasing of crushing of concrete due to cyclic loading;
338
3 Deterioration of Materials and Structures
Nfe [x 106] 7 Ni
¦N
6
test results
1.0
fi
Ni
¦N
5
!
Ș N ft
fi
1 ¦ Ni Ș
4 3 2
test
K
test
K
S52a S52c
0.21
S56c
0.52
0.33
S56d
0.75
S53a
0.21
S63a
0.686
S54a
0.31
S63b
0.549
S54b
0.22
S63c
0.564
S54c
0.22
S64a
0.566
S54d
0.64
S64b
0.592
S56a
1.01
S64c
0.670
S56b
0.36
1 Nft [x 106]
damage accumulation according to Palmgren and Miner
0
0
1
2
4 5 3 lifetime prediction
6
7
Fig. 3.164. Comparison between the test results with the results of the lifetime prediction according to PalmgrenMiner m m−1 Ni + Δnf,i = 1 Nf,i i=1 i=1
(3.259)
Figure 3.165 explains this method by means of a cyclic test with two blocks of loading where the peak load of the ﬁrst block is increased in the second block while the load range was held constant. The two curves 1 and 2 give the relationship between the relative static strength and relative number of load cycles and corresponding to the cyclic loading parameter of each block. After applying N1 number of cycles the static strength reduces to the value Pu,N 1 on curve 1 (Point B). The relative damage until this point can be expressed with the term N1 / Nf 1 based on the PalmgrenMiner rule. The point C on curve 2 corresponds to the same damage state, i.e. the same reduced static strength for the loading parameters of the second block and thus points up the starting value for the subsequent course of the reduction of the static strength along the path of curve 2. The horizontal oﬀset Δnf between the damage equivalent points B and C can be interpreted as the loss of the lifetime and is introduced to the damage sum in the new model. The remaining lifetime is then governed by the value of N2 / Nf 2 until the failure of the specimen due to the decrease of the static strength to the value of peak load Pmax,2 . In Figure 3.165 the fatigue fracture zones corresponding to reduced static strength are depicted for diﬀerent states from points A to E. As a consequence
3.3 Modelling
339
fatigue fracture area AD
P / Pu,0
A
1.0
1
AD,A = 0
0.8
Ni
¦N
D
f,i
¦ ǻn f,i d 1
C
B AD,B
AD,C ~ AD,B
2 Pmax,2 / Pu,0
D
0.6 Pmax,1 / Pu,0
'P / Pu,0
0.4
E
'P / Pu,0
'nf,1
N1 /Nf,1
0.2
'AD,ED
AD,D
Pmax,1 / Pu,0 AD,E
N2 /Nf,2 Ni / Nf,i
0
0
0.2
0.4
0.6
0.8
1.0
Fig. 3.165. Damage accumulation considering the load sequence eﬀects
of the correlation between the fatigue fracture area AD and the relative value of the reduced static strength Pu,N / Pu,0 the fracture areas in states D and E diﬀer from each other due to diﬀerent peak loads. Due to the raising of the peak load in the second block of loading, fatigue fracture area at the end of the lifetime can not be shaped corresponding to the fatigue fracture area AD,E of the ﬁrst block of loading. The fatigue failure occurs by a rather smaller fatigue fracture area AD,D . In this case the damage term Δnf,1 considers the shortening of the fatigue life in consequence of the reduction of the fatigue fracture area to an extent of ΔAD,E−D . The results of the tests S52, S53 and series S6 showed that the loading sequence (i.e. increasing of decreasing the peak load) has a subsidiary eﬀect on the fatigue life of a cyclic test with multiple blocks of loading. For an improved damage accumulation hypothesis before the analysis the load collectives with decreasing loadings must be resorted to collectives with increasing loadings. This procedure is shown exemplarily for the tests S64 in Figure 3.166. In such case the additional damage terms Δnf,i can be interpreted as the eﬀect of concrete damage on the extent of the crack velocity. In test with increasing peak load the failure occurs by a fatigue fracture area AD,A and in the case of decreasing peak load by a fatigue fracture area AD,B which is greater then the AD,A by an extent of ΔAD,B−A . Considering identical lifetimes due to bending stresses in the stud shank resulting from local concrete damage and consequential high notch stresses at the crack tip, in the case of decreasing peak load the crack velocity is greater than in the case of increasing peak load.
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3 Deterioration of Materials and Structures
Pu / Pu,0 N1 / Nf,1 'nf,1
'nf,2
N2 / Nf,2
'nf,3
N3 / Nf,3
N4 / Nf,4
1.0
4
AD,A
A
0.8
0.74 ~ 0.71
1
0.64 0.6
tserVe i M
:.1 6 3 0 8 5 7 .1 0 8 5 7
Pmax,4
0.4
P
4
Pmax,4
Pmax,4
2 . 1 .
0
0
20 + .1 7 + 5 0 4 .1 2 7 + 5 0 4
4 .1 0 2
6
.2 0
8 3
.1 2 0
0 .3
0 5 .4
.+ 4 0 6 7 4 9 6 4 7 . 0 + 9 6
1 7 + 6 0 + 5 3 8 6 7 1 + 0 5 3 8 6 7
.1 5 2 = 5 6
8 2 0 5
3 = 8 2
8 5 1
4 .1 0
5 )(c l a
8 3 2 1 5
0 5 .1
8 7 4 5 +
5(e ) p x
. 9 0 8
. 1 0
.+ 8 4 2 1 5
1 S 6 7 .1 0
2 "S 3 "S 4 S
0 4 7 .1
0 .6
.7 0
/ .2 (0
x p (e
)
x p (e
)
(l ca
)
). 4
/ .2 (0
). 4 5
/ .2 (0
.6 ) 4
(.2 /0
1 .7 )~
4 1 9 6 3
6
9 6 0 3 4 15
1 5
f,1 N =
N f,2 N = N = 4 ,f3
0 5 3
0
(.2 /0
4 7 )
fN
0.54 0.44
4
B
Pmax,1
AD,B
1 N1 N2 N3
0.2
2 0 .1 9 . 8 . 7 . 6 . 5 . 4 . 3 .
0 0
Pu,0
1 N4 N3 N2 N1
N4 'P konst.
0
rtueShw l ce
u P 1 0 0 0 0 0 0 0
Pu,0
P
0
0.2
'AD,BA
G
Ni / Nf,i
G
0.4
0.6
0.8
1.0
Fig. 3.166. Damage accumulation in the case of multiple block loading tests with decreasing peak loads
Nfe [x 106]
7new damage accumulation rule
7
6
6
5
5
4 3 2 1
4 3 2 Nft [x
0 0
D
experimental results
experimental results
Nfe [x 106]
4 1 2 5 6 7 3 predicted lifetimes – single values
106]
(2)
1
¦
Ni ¦ ' n f, i d 1 N f, i
(3)
(3) (3) (4)
(1) ( ) number of tests
0 0
(1)
Nft [x 106]
4 1 2 5 6 7 3 predicted lifetimes – mean values
Fig. 3.167. Comparison between the test results with the results of the lifetime prediction according to the improved damage accumulation hypothesis
In Figure 3.167 the comparison of the results of multiple block loading tests with the theoretical values according the improved damage accumulation method is illustrated. With the exception of two tests nearly all single values
3.3 Modelling
341
Gu, 0.9 Pu [mm] 35
Pu 0.9 Pu
30
Pmax / Pu,0 = 0.71
P
type B
Gu, 0.9Pu
25
type A
d
20
3
15
1 2
10
Pmax / Pu,0 0.44
4 type A
5 0
0
0.2
0.4
0.6
0.8
1.0
type B 1.2
N / Nf
Fig. 3.168. Ductility after high cycle loading
of the test results are well predicted. In these two tests presumably due to worse compaction of the concrete a larger slip development at the stud feet was noticed. This shows the importance of the detailing of the shear connection as given in Eurocode 4 [22, 23]. Looking at similar tests it can be observed that the prediction matches the mean values leading to a signiﬁcant improvement compared with the PalmgrenMiner rule. 3.3.4.2.9 Ductility and Crack Formation High initial stiﬀness and high ductility are main advantages of headed studs embedded in normal weight concrete. From the static tests carried out after cyclic preloading it could be found, that the load deﬂection behaviour is signiﬁcantly aﬀected by the crack formation, which is itself closely correlated to the peak load level. Very high peak loads cause horizontal cracks through the stud foot like crack type A shown in Figure 3.154. As shown in Figure 3.168 this formation results in a gradual decrease of ductility during lifetime and the values may fall below the target values of the codes. In the case of lower peak loads the cracks propagate into the ﬂange like crack type B and ductility increases. This behaviour is of great importance regarding the capability of redistribution of shear forces in the interface between steel and concrete of composite beams subjected to fatigue loading. 3.3.4.2.10 Finite Element Calculations of the (Reduced) Static Strength of Headed Shear Studs in PushOut Specimens The experimental results of the pushout tests can be taken as the basis for further theoretical investigations regarding the eﬀect of cracks at the stud feet on the static strength of headed shear studs given in Figure 3.154. As shown
342
3 Deterioration of Materials and Structures
without predamage
with predamage (here: crack of type B at each stud foot)
P = 0.2
FEmodel of a pushout test specimen (quarter)
P = 0.1
local deformation of concrete and steel at the stud feet at ultimate limit state P [kN] fc = 30 N/mm² Ecm = 27960 N/mm²
Pu = 1440 kN
1400 1200
test
'Pu,FEM ~ 0.26 Pu
FEM
fu = 528 N/mm E = 210000 N/mm²
1600
fu = 235 N/mm
1000 fu = 528 N/mm
800 600 400 200 0
crack pattern (here: type B) fy = 337 N/mm  fu = 448 N/mm
AD  0.5 'Pu , exp  0.30  'Pu , FEM AD AG 1
2
3
4
5
6
7
G [mm]
crack pattern (here: type B)
Distribution of main material properties at the stud feet
8
Fig. 3.169. Comparison between test results and ﬁnite element calculations of statically loaded pushout test specimens
in Figure 3.169 for this purpose a comprehensive threedimensional FEModel  using the ﬁnite element programme ANSYS  of a statically loaded pushout test specimen with lateral support of the concrete slabs according to Figure 3.100 of Chapter 3.2.3 has been built up in order to simulate he loaddeﬂection behaviour of headed shear studs embedded in solid slabs without any predamage. Concerning the numerical simulation the material properties of the steel members and the concrete members are of main interest. So far no detailed information about the precise material properties of the steel in the heated aﬀected zones, in the weld collar and in the melted zone at each stud are available. For this reason the material properties of the steel beam and the studs (determined by means of tensile tests) were taken as the basis of the material properties of the steel aﬀected by the welding process. Microscopic examinations of the steel structure at the stud feet given in Figures 3.151  3.153 were performed in order to consider suﬃciently the weld formations regarding the assignment of the main material properties in the FEmodel. Metal plasticity behaviour of the steel was simulated by using the vonMises criterion. The concrete behaviour was modelled elastic  perfectly plastic taking into account a yielding surface according to DruckerPrager (DP) with an associated ﬂow rule. The two parameters of the DPyield surface were adjusted to the uniaxial (1.0 fc ) and to the biaxial compressive strength (taken as 1.2 fc ) of the concrete, obtained from concrete cylinders cured at air as the corresponding test specimens. As shown in Figure 3.153 the weld collar is commonly non
3.3 Modelling
343
homogeneous and of low strength, therefore the tensile strength of the collars was only chosen to fu = 235 N/mm2 . Figure 3.169 shows the results of a numerical simulation of a pushout test, considering concrete strength properties fc = 30 N/mm2 and Ecm = 27960 N/mm2 (cured at air), which are typical values like e.g. in test series S2 or S4. For the coeﬃcient of friction in the interface between steel and concrete a very low value of μ = 0.2 was chosen, because the steel ﬂanges were greased before casting. The FEmodel leads to good agreement between the calculated load deﬂection curve, the ultimate static strength and the deformation of the studs with the corresponding experimental results of test series S1 to S9. After calibrating the FEmodel of a statically loaded pushout specimen without damage subsequently cracks of diﬀerent geometries were implemented in the FEmodel at each stud foot and the calculations were repeated in order to check the inﬂuence of the steel damage on the reduced static strength. One typical crack pattern and the corresponding results are shown as an example in Figure 3.169. The coeﬃcient of friction in the interface between steel and concrete was slightly reduced to μ = 0.1 in comparison to the nondamaged state in order to take the sliding in the interface due to cyclic preloading into account. The eﬀect of three crack patterns with nearly half the stud foot size on the reduction of the static strength is shown in Figure 3.170. For the investigated sizes of the cracks, the results of the numerical simulations are in the same order as observed in the tests (see Figure 3.154). Regarding the shear resistance of predamaged studs the local crushing of the concrete due to cyclic loading is insigniﬁcant compared to the predamage in form of cracks in the steel at the stud feet.
Pu / Pu,0
crack pattern (FEM)
1.0 type B
0.8
FEM  type B
Pu / Pu,0
FEM  type A
0.6
experiment type A
0.4
Pu AD  1 0 .6 Pu,0 all series A D A G
0.2 0.0
0.0
0.2
AD /(AD+AG ) 0.4
0.6
0.8
1.0
Fig. 3.170. Comparison between test results and ﬁnite element calculations of statically loaded pushout test specimens damaged by cracks in the steel at the stud feet
344
3 Deterioration of Materials and Structures
3.3.4.2.11 Eﬀect of the Control Mode  Eﬀect of Low Temperatures In force controlled pushout tests monotonic high cyclic loading leads to a successive increase of the accumulated plastic slip at the stud feet and to a gradual decrease of the elastic stiﬀness. In force controlled composite beams these eﬀects cause successive redistribution of the longitudinal shear forces in the interface between steel and concrete and reduction of the normal forces in the steel and the concrete members. The stiﬀness of the interface between steel and concrete gradually decreases and the composite action deteriorates. In general the peak load Pmax and the load ranges ΔP of each stud decreases and the value of the maximum slip at the peak load level smax and the slip range Δs within a load cycle increases. Only in the cases with a very low load level the increase of the slip values smax and Δs at each stud foot over lifetime can be neglected. The deterioration of the mechanical properties of each stud in the interface between steel and concrete can then be predicted directly by means of displacement controlled pushout tests [527, 312, 278, 483]. In this context it is of main interest, which control mode should be considered in order to simulate cyclic behaviour of composite beams by means of results gained by pushout tests. In order to answer this question in series S9 nine tests according Figure 3.98 of Chapter 3.2.3 were performed in which the inﬂuence of the control mode on the results of cyclic loaded pushout test specimen was investigated. As shown in Figure 3.171 after 5 × 106 load cycles the loadslip behaviour and the mean values of the reduced static strength P u,N of the compared cyclic loaded pushout test specimens were nearly coincident. Moreover the mean values of the crack lengths acr through the stud feet were almost identical. Independent from the control mode in both cases the static shear resistance was reduced from P u,0 = 1620 kN to approximate P u,N = 930 kN. This shows that displacement controlled behaviour of headed studs can also be simulated by the results of force controlled pushout tests with multiple blocks of loading, taking into account an appropriate damage accumulation hypothesis in order to consider sequence eﬀects. Regarding the numerical simulation of composite beams it appears, that the eﬀect of the cyclic loading can be simulated by the results of force controlled pushout tests as gained from test series S1  S6. Notchedbar impact tests carried out with test specimens taken from welded joints of headed shear studs show that the notchedbar impact values significantly fall below commonly accepted values of steel structures. As given in Figure 3.171 for a temperature of T = −20◦C in the tests the notched bar impact values decreased to 9 J near the melted line adjacent to the material of the steel ﬂange. In order to investigate the interaction between the predamage and the toughness the reduced static strength of test specimen S95d (beforehand also being subjected to 5×106 load cycles under force control with multiple blocks of loadings) was tested after cooling down to T = −40◦ C. In comparison with the before mentioned cyclic loaded test specimens, in which the reduced static strength was tested at a temperature of T = 20◦ C, the
3.3 Modelling
P [kN]
T = 40°C (S9_5d)
1200
'Pu, N  'f c ('T) 'E c ('T)
'Pu,N
1000
T = 20°C
800 600
Cyclic loading applied: displacement controlled force controlled
9J
type B reduced static strength after 5x106 load cycles 5
10
24J 19 J
acr
200
KV 150 / T = 20°C
notchedbar impactbending test
Pu,0 1620kN
400
0
345
15
6J
G [mm]
Fig. 3.171. Test series S9  Eﬀect of control mode  Eﬀect of low temperature on the loadslip behaviour of predamaged studs with cracks of type B
reduced static strength increases in accordance with the increase of the relevant material properties (fc , Ecm ) of the concrete at lower temperatures. The ductility decreases as a matter of course but compared to the requirements of the codes the ductility remains suﬃciently high. In the present case the cracks propagated slightly into the steel ﬂanges. For cracks of type A it must be presumed, that in the case of low temperatures the ductility lies below the corresponding values shown in Figure 3.168. This is again an import reason, why cracks of type A must be strongly avoided by means of limiting the peak load in the limit state of serviceability. 3.3.4.3 Modelling of the Global Behaviour of Composite Beams Subjected to Cyclic Loading 3.3.4.3.1 Material Model for the Concrete Behaviour Considering the interaction between the local damage of headed shear studs and the behaviour of the global structure, the research results based on the pushout tests were taken as the basis for numerical simulations of the static behaviour (with and without any predamage eﬀects) and the cyclic behaviour of steelcomposite beams subjected to fatigue loads. For this purpose at ﬁrst the commercial ﬁnite element programme ANSYS, R7.1 was improved by implementing a modiﬁed version of the threedimensional structural solid element SOLID65 [643]. In original ANSYS versions this element is not implemented satisfactorily, because the corresponding material model CONCRETE shows some severe errors. Due to numerical instabilities and inaccurate calculation results this material model is not applicable for typical problems of structural concrete members. The element behaviour has been updated in this manner that the primarily requested material failure modes of brittle materials subjected to static loading are enabled [48]. The element is deﬁned by eight nodes having three
346
3 Deterioration of Materials and Structures
degrees of freedom at each node (translations in the nodal x, y, and z directions) based on linear shape functions. The solid is capable of cracking (in three orthogonal directions) in tension and crushing in compression. It can be adjusted to the behaviour of concrete by giving typical concrete material data like the shear transfer coeﬃcient, ultimate uniaxial tensile strength, ultimate uniaxial compressive strength and ultimate compressive strength for states of multiaxial compression. Typical shear transfer coeﬃcients range from 0.0 to 1.0, with 0.0 representing a smooth crack (complete loss of shear transfer) and 1.0 representing a rough crack (no loss of shear transfer). These speciﬁcations are possible for both a closed and an open crack. The new implemented failure surface in the compression domain (compression  compression  compression) is oriented at the 5parameter failure criterion of Willam and Warnke [849], but having an open failure surface [190, 189, 191]. Moreover nonlinear material properties such as creep and plastic deformation can be treated by incorporating additional creep and plasticity options. In the case of the numerical simulation of structural composite members of steel and concrete, like composite beams and composite columns subjected to static loading up to failure, it is reasonable to incorporate the Multilinear Isotropic Hardening (MISO) option with yield surface according to von Mises in order to consider suﬃciently the typical nonlinear stressstrain relationship of the concrete under compression. Figure 3.172 shows the implemented failure surface of the material model CONRETE corresponding to the structural element SOLID65 and a combination of this failure surface with a yield surface according to von Mises. As shown in Figure 3.173 it is possible to predict the loaddeﬂection behaviour and the load bearing capacity of typical composite members of steel and concrete [342] under static loading up to failure by means of the improved material model. The properties of the studs were modelled by means of discrete nonlinear spring elements, taking into account the analytical expressions developed from the statically loaded pushout tests of series S1  S9. Similar good predictions were achieved in the case of numerical simulations of the behaviour of statically loaded composite columns of high strength steel and concrete within the framework of a research project at the University of Wuppertal, Germany [344, 501]. 3.3.4.3.2 Eﬀect of HighCycle Loading on Load Bearing Capacity of Composite Beams For real structures the eﬀect of the local damage of the headed shear studs on the global behaviour and the resistance of beams subjected to fatigue loading is of main interest. The deterioration of the mechanical properties of the studs due to cyclic loading leads to a successive decrease of the composite action, resulting in a decrease of load bearing capacity and in a successive loss of elastic stiﬀness. The extent mainly depends the contribution of the steal beam to the load bearing capacity and to the stiﬀness of the composite beam.
347
improved failure surface of the material model CONCRETE incorporated by a yield surface according to von Mises
improved failure surface of the material model CONCRETE
3.3 Modelling
Fig. 3.172. Failure surface of the improved material model CONCRETE  Failure surface incorporated by a yield surface according to von Mises
For this reason the pushout test program was extended by two simply supported fullscale cyclic loaded composite beam tests (VT1 and VT2). In these tests the eﬀect of the deterioration of the properties of the interface between steel and concrete on the loaddeﬂection behaviour, on the redistribution of the inner forces and on the reduced static strength for two typical cyclic loading conditions (VT1: sagging moment  VT2: hogging moment) were investigated. The main results are given in Chapter 3.2.3. In both cases the test beams were statically loaded up to failure after the cyclic loading phase and subsequently checked for cracks in the steel at the stud feet. Figure 3.174 summarizes the main results of test beam VT1 regarding the interaction between the local damage of the studs and the loaddeﬂection behaviour after high cyclic loading. The residual static strength of the beam was determined after 1.37 × 106 loadings. Based on the theoretical model given by Equation A in Figure 3.154 due to crack growth at the stud feet during the cyclic loading phase the static strengths of the studs were partly decreased by up to 65% of its original value. The value of the reduced static strength of all studs averages approximately 52% of the undamaged state. Numerical
348
3 Deterioration of Materials and Structures
Maximum concrete stresses at ultimate load (beam test 1)
min Vx = 43.0 MPa
HEA300S460
failure surface (CONCRETE)
max Vx = 1.4MPa
F [kN]
700
beam test 1
600 test
500
VII
Von Mises
fc
HEA300S460
FEM
400 300
ft
200
VI
fc
ft
beam test 4
uz
F 100
uz [mm]
6m
CONCRETE
0
0
10
20
30
40
50
60
70
80
90
100
Fig. 3.173. Comparison between the results of numerical simulations and test results of typical composite members of steel and concrete
shear resistance of the studs after 1.370.000 load cycles
1000
1.0
F [kN]
0.8
'F = 60 kN (8%)
0.65
0.6
800
Fu,N = 756 kN
0.4 0.2
0
600
Pu,N /Pu,0 54 studs
450 kN
1.370.000 cycles
FEM
F
400 200
reduced static strength first loading
3.0 [m] 1.0 0.5 0
0.0 0.52
3.0 0.47
0.48 Pu,N /Pu,0
0 1.0 0.89
0.53
uz
6.0 m
185 kN
20
uz [mm]
experiment (VT1)
60
40
80
100
M/Mu ' = 11%
0.8 (VT1)
0.6 0.4 b/h = 1500/150
0.2 HEA300 fy = 452N/mm²  fc = 36N/mm²
K
0.52
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Fig. 3.174. Test beam VT1  Eﬀect of high cycle loading on load bearing capacity
3.3 Modelling
349
investigations considering the experimental observed values of the accumulated plastic slip and the damage of each stud indicate that the deterioration of the strength of the interface between steel and concrete causes a loss of the load bearing capacity of the composite beam of nearly 8% compared to the undamaged beam. This result is in good agreement with the result obtained by applying partialinteraction theory taking into account the average damage of the studs along the interface instead of discrete local damage of each stud. In the case of test beam VT1 the reduction of the global load bearing capacity due to cyclic loading is considerably low compared to the reduced static strength of the studs. However, the eﬀect of the local damage of the studs increases if the contribution of the steel beam on the load bearing capacity of the composite beam is low. Typical examples are composite girders with very small or even no steel top ﬂanges, in which the steel ﬂange is almost completely substituted by the concrete ﬂange. Other typical structural details are cyclic loaded load introduction areas with headed shear studs, where no redistribution of the shear forces is possible. In these cases the reduction of the static strength of the studs directly aﬀects the shear carrying capacity and with this the load bearing capacity of the load introduction area. On this background the design rules in current national and international design codes must be reconsidered. 3.3.4.3.3 Cyclic Behaviour of Composite Beams  Development of Slip In general the redistribution of the inner forces in the concrete slab and in the steel beam due to the decrease of the elastic stiﬀness and the plastic slip in the interface between concrete and steel is of main interest, where local buckling of the steel beam governs the design. In order to simulate the behaviour of cyclic loaded composite beams the ﬁrst generation of incremental algorithms was developed, which are based on the damage accumulation method according to Figure 3.165. These algorithms are incremental processes in which the characteristics of each stud such as ”plastic slip”, ”elastic stiﬀness” and ”reduced static strength” are updated step by step regarding the accumulated damage D, the current peak load Pmax and current load range ΔP after each step. In each increment it is assumed that the loading parameters of each stud remain constant and thus the stud behaviour during the increment can directly be taken from appropriate forcecontrolled pushout test results as reported in Chapter 3.3.4.2. In Figure 3.175 for test beam VT1 the development of the plastic slip during the cyclic loading phase is given. Due to the high loading parameters the test beam shows a considerable increase of plastic slip during the cyclic loading phase. In the present case for the recalculation of the plastic slip the total number of load cycles Nk was split in 20 increments and after each FEanalysis, representing an increase of Nk / 20 numbers of load cycles, the relevant mechanical properties of each headed shear stud (plastic slip, elastic stiﬀness and reduced static strength) were updated. Although further research is needed the so far achieved numerical results are in agreement with the experimental observations in the beam test.
350
3 Deterioration of Materials and Structures
450 kN
Pu,N
1.370.000 cycles
P
185 kN
Kel,N
3.0 [m]
Gpl,N
3.0
0.0
G
G [mm] 1.0
D
0.8 0.6
1.370.000 load cycles
Pu,N
1. loading
Gpl
0.4 0.2 0
simulation
0.2
20 increments with 68.500 load cycles
0.4
P
0.6
G
0.8 1.0
Nf
3.0
2.0
1.0
0
1.0
2.0
3.0
[m]
Pmax, 'P
Fig. 3.175. Cyclic behaviour of test beam VT1  Veriﬁcation of the concept
3.3.4.3.4 Eﬀect of Cyclic Loading on Beams with Tension Flanges The occurrence of cracks at the stud feet and the early crack initiation has to be assessed a in diﬀerent way, if the ﬂange, on which the studs are welded, is not in compression but in tension. For this purpose beam test VT2 in hogging bending was subjected to 2.1 million load cycles before the residual static strength of the beam was determined. In ﬂanges under compression the cracks typically grow horizontal leading to a deterioration of the properties of the interface between steel and concrete. However, in tension ﬂanges the direction of the cracks at the stud feet is additionally inﬂuenced by the tensile stresses in the steel ﬂange. In test beam VT2 it could be observed, that the cracks partly propagate nearly vertical through the ﬂange. In this case not only the properties of the interface are aﬀected, but also the load bearing capacity of the cross sections. This is pointed out in Figure 3.176, in which the inﬂuence of predamage on the static loaddeﬂection behaviour after high cyclic loading and a typical crack formation at a highly stressed stud (stud B9) is shown. The test shows that also in hogging bending the signiﬁcant local damage causes only a small global reduction of the ultimate load, provided that the bending stiﬀness of the steel member is suﬃciently high compared to the bending stiﬀness of the composite section. With regard to fatigue cracks growing in vertical direction through the top ﬂange of the steel girder further research is needed.
3.4 Numerical Examples
351
tension flange
250 kN
2.100.000 cycles 150 kN
54 studs
stud B9
3.0 [m]
3.0
0.0
VT 2 – tension flange
F [kN] 800 'P = 47 kN
700
Pu,N = 625 kN
FEM, undamaged
600
a stud B9
500 400
6.0 m
200 100 0
VT 1 – compression flange
P
300
experiment (VT2)
0
20
40
60
80
uz
a
VT 2
100
120
140
uz [mm]
stud A20
Fig. 3.176. Test beam VT2  Eﬀect of high cycle loading on the load bearing capacity  typical crack formation in tension ﬂanges and in compression ﬂanges
3.4 Numerical Examples Authored by G¨ unther Meschke 3.4.1 Durability Analysis of a Concrete Tunnel Shell Authored by Stefan Grasberger and G¨ unther Meschke Large tunnels often exhibit crown cracks in the inner linings caused by restrained stresses due to cooling and drying of the surface. Using the hygromechanical model described in Subchapter 3.3.2.1 a durability analysis of the longterm degradation of an inner tunnel lining is presented in this section. The eﬀect of creep is disregarded in this analysis. Figure 3.177 contains the dimensions of a cross section of the analysed tunnel. Exploiting symmetryconditions, only one half of the tunnel lining is discretised by means of 1832 ﬁnite volume elements. Details on the material and model parameters are contained in [322, 692]. The initial temperature is speciﬁed as T0 = 17.0◦ C, the initial pore humidity as h0 = 0.93 and the initial porosity as φ0 = 0.25. After application of the selfweight of the construction (g = 25 kN/m3 ), the inner tunnel lining is exposed to cyclic changes of the hygral and thermal environmental conditions based on meteorological data according to [680]. Elastic support conditions are assumed a long the interface of the tunnel shell to the surrounding soil.
3 Deterioration of Materials and Structures
9 5
8 3 5
9 3 0
352
5 4 0
Fig. 3.177. Numerical simulation of a tunnel lining subjected to cyclic hygral and thermal loading: Geometry of the investigated tunnel (dimensions in [cm])
1.2
Crack width Z [mm]
1 0.8 0.6 0.4 0.2 0
Model prediction
1
2
3
4
5
Time t [a]
Fig. 3.178. Numerical simulation of a tunnel lining subjected to cyclic hygral and thermal loading: Comparison of the calculated evolution of the crack width w at Point A at the crown and measurements by [764]
Figure 3.179 illustrates the computed evolution of the crown cracks represented by the scalar damage parameter d after 1, 2 and 5 years, respectively.
3.4 Numerical Examples
353
b A
a) t = 1 a
b) t = 2 a
c) t = 5 a 0.000
d []
1.000
Fig. 3.179. Numerical simulation of a tunnel lining subjected to cyclic hygral and thermal loading: Computed distribution of the scalar damage measure d at the crown
During the ﬁrst year some cracks start to open at the top of the inner lining due to the arising temperature and moisture gradients. As the drying proceeds, these cracks are further opening, whereby the predicted crack spacing is about 20 ∼ 25 cm. According to the stress distribution owing to the selfweight of the construction, the lower part of the tunnel lining remains undamaged. Figure 3.178 contains a comparison of the predicted evolution of the crack width at the top of the inner lining (Point A in Figure 3.179) with reported insitu measurements by [764]. The numerical results demonstrate that the ﬁnite element model is not only capable of reproducing shrinkageinduced cracks but also to predict the time of opening of the cracks, the location of the cracks and the width of the cracks in accordance with insitu observations.
354
3 Deterioration of Materials and Structures
a) t = 1 a
b) t = 2 a
c) t = 5 a 0.900
Sl []
0.400
Fig. 3.180. Numerical simulation of a tunnel lining subjected to cyclic hygral and thermal loading: Computed distribution of the liquid saturation Sl at the crown after 1, 2, and 5 years
Figure 3.180 illustrates the predicted evolution of the drying process by means of the liquid saturation Sl . According to the strong interdependencies between moisture transport and the nonlinear material behaviour of concrete, the drying process at the top of the inner tunnel lining is clearly dominated by the inﬂuence of the growing crown cracks. 3.4.2 Durability Analysis of a Cementitious Beam Exposed to Calcium Leaching and External Loading Authored by G¨ unther Meschke The chemomechanical model described in Subchapter 3.3.2.2.1 is applied for durability analyses of a cement paste beam (Figures 3.181 and 3.182)
Ca 2+ Concentration s h [kmol/m 3 ]
3.4 Numerical Examples
16 14 12 10 8 6 4 2 0
355
moving front of portlandite dissolution
P
moving front of CSH dissolution
z cattack
t = 1600d, 3200d,... t = 9120d 0
10
20
30
40
50
Coordinate z [mm]
Displacement u s [mm]
Fig. 3.181. Simulation of a cementitious beam exposed to calcium leaching and mechanical loading. Distribution of the calcium concentration sh of the solid skeleton along the symmetry axis z
0.075 0.07 0.065 0.06 0.055 0.05 0.045 0.04
P us A
Collapse
cattack
0
2000
4000
6000
8000
10000
Time t [d]
Fig. 3.182. Temporal evolution of the vertical displacement us in point A (P = const.). Collapse is predicted after approx. 25 years
exposed simultaneously to chemical dissolution and external loading. In the central part of the bottom face of the beam, the calcium concentration ci of the pore solution is decreased from an initial value to zero in order to simulate the contact with deionized water. Simultaneously, a constant load P (75% of ultimate load) is applied. The diagram in Figure 3.181 shows the distribution of the calcium concentration sh in the solid skeleton along the symmetry axis z at diﬀerent time intervals. A decrease of sh below the initial value sh0 = 15 kmol/m3 is associated with both an increase of the porosity and a decrease of the stiﬀness and strength. The two fronts propagating from the bottom to the top are associated with the dissolution of portlandite and CSH. In diagram in Figure 3.182 the degrading structural behavior of the beam is represented by the evolution of the vertical displacement us in point A
356
3 Deterioration of Materials and Structures
square panel
mechanical load bearing behaviour
loading history
4 t 2 [N/mm]
point A
loading
40 mm
d = 1 mm
t 2,max
3 2 1
x2
Ω x1
40 mm
0 0 0.01 0.02 0.03 0.04 displacement u s, 2 in A [mm]
c /c i0 und t 2 /t 2,max
t2
c
c /c i0 t 2 /t 2,max
1
0 0
100 200 300 400 · · · ratio t/Δt
Fig. 3.183. Chemomechanical analysis of a concrete panel: a) Geometry, ﬁnite element mesh and boundary conditions, b) loaddisplacement curve when subjected to external loading only, c) chemomechanical loading history
along time. Since the leaching process results in a drastic reduction of the local and global stiﬀness, the displacement us increases in spite of a constant mechanical load P . The collapse of the structure is indicated at t = 9120 d ≈ 25 years, by a sudden increase of the displacement us . It is worth noting that in a purely mechanical analysis no collapse would be predicted. 3.4.3 Durability Analysis of a Sealed Panel with a Leakage Authored by Falko Bangert and G¨ unther Meschke As a second numerical application of the coupled chemomechanical model for concrete described in Subchapter 3.3.2.2.1 a panel made of cement paste, which is simultaneously subjected to external loading and exposed to calcium dissolution is investigated. Details of the material and model parameters and of the discretiztion are contained in [81]. Figure 3.183a illustrates the geometry, the ﬁnite element discretization, the boundary conditions of the concrete slab, which is discretized by 40 × 40 biquadratic ﬁnite elements Elements for the interpolation of the displacment ﬁeld us and of the calcium concentration ci . For the nonlocal equivalent strains η¯ a bilinear interpolation is used (see Subchapter 3.3.2.2.2). Plane stress conditions are assumed, using the algorithm proposed in [140]. The temporal evolution of the the external and chemical loading is shown in Figure 3.183c. In the upper left corner the calcium concentration c in the pore ﬂuid is decreased from the initial value c = c0 corresponding to a state of chemical equilibrium to zero in order to simulate contact with deionized water. The rest of the surface of the panel is assumed to be sealed. This chemical loading scenario shall represent a very localized leakage of a sealing of an underground structure permanently exposed to deionized water. Simultaneousely, the panel is subjected to a time invariant external distributed loading
3.4 Numerical Examples
357
t∗ =2.5 N/mm. This level corresponds to 75 % of the ultimate load carrying capacity (tu = 3.4 N/mm) if this external loading would be monotonously increased up to structural failure (see the loaddisplacement curve in Figure 3.183b). Figure 3.184 illustrates the distribution of the calcium concentration c in the pore ﬂuid, of the concentration s of the calcium bound in the skeleton, the rate of calcium dissolution s˙ and of the scalar damage parameter dm at four stages of the loading history. The results at t/Δt = 100 correspond to the end of the chemical loading at the top left corner of the panel (see Figure 3.183c), inducing a strong gradient of the calcium ion concentration grad(c) in the vicinity of this corner. This concentration gradient activates the calcium dissolution of the hydration products of the cement paste and the diﬀusion of the dissolved calcium ions towards the top left corner. The temporal evolution of the dissolution process is well represented by the contour plots showing the stepwise distribution of calciumconcentration sh of the hydratation products emanating from the top left corner, where the leakage is assumed. The third row in Figure 3.184 shows a dissolution front associated with the dissolution of the calciumhydroxide moving from the left top corner into the structure. The speed of propagation of the Ca(OH)2 dissolution front is shown in Figure 3.185 In contrast to the dissolution of the portlandite, the solution front associated with the dissolution of the CSHphases starts at a much later stage and only moves marginally. It is more or less restricted to the vicinity of the top left corner and propagates slowly into the direction of the opening vertical crack (see Figure 3.184 at t/Δt = 400 and t/Δt = 428). The sequence of contour plots on the right hand side of Figure 3.184 illustrates that the external loading in conjunction with the propagating chemical dissolution initiates a crack at the top left corner which propagates vertically along the ﬁxed support. Although the external load is kept constant, the chemical attack continues to contribute to the weakening of the structure, resulting in an increase of the width of the crack and its propagation into the structure. The location of the crack tip propagates more or less simultaneousely with the dissoultion front. The presence of the crack accerlerates the dissolution process considerably. This is illustrated by the contour plots in Figure 3.184 at t/Δt = 300, 400), where the Ca(OH)2 dissolution progresses faster along the vertical crack compared to the horizontal direction. This chemomechanical interaction is also demonstrated by the results shown in Figure 3.185a which contains the temporal evolution of the propagating Ca(OH)2 dissolution front along the vertical direction as obtained from a fully chemomechanically coupled analysis compared to results from an uncoupled chemical analysis, in which the eﬀect of the damage is disregarded. While for a purely chemical load√ ing the Ca(OH)2 dissolution front moves √ with a velocity of ac = 3.0 mm/ a, the speed increases to acm = 4.5 mm/ a if chemomechanical interactions are considered. This is an increase of 50%.
358
3 Deterioration of Materials and Structures
t CaConcentration ci CaConcentration sh rate log(−∂sh/∂t) Δt [1] 21
[mol/m3 ]
015
[kmol/m3 ]
02[kmol/[m3 d]] ≤−61
Damage dm [1]
0
100
200
300
400
428
Fig. 3.184. Chemomechanical analysis of a concrete panel: Distribution of the calcium concentration c (pore ﬂuid), the calcium concentration s (skeleton), the dissolution rate s˙ and the scalar damage parameter dm at diﬀerent states of exposure
40
displacement us,2 in point A [10−3 mm]
position of the Ca(OH)2 front x2 [mm]
3.4 Numerical Examples
chemomechanical chemical
35 30 25 20 15
acm
10
ac
5 failure 0 0
50
100
150
200 250 √ √ square root of time t [ d]
300
359
4 chemomechanical mechanical
6 8 10 12 14
failure
16 0
200
400
600
800
1000
ratio t/Δt [1]
Fig. 3.185. Chemomechanical analysis of a concrete panel: a) Temporal propagation of the Ca(OH)2 dissolution front along the clamped support, b) Deformationtime relation
As a consequence of this (longterm) chemomechanical material degradation and the continuous propagation of the crack the structural stiﬀness decreases. This connected with a progressive growth of the vertical displacments u of the right edge of the panel (Figure 3.185b). 94 years after the start of the exposure to deionized water the resistance of the structure is ﬁnally exhausted. Without consideration of chemomechanical couplings no structural failure would have been predicted. 3.4.4 Numerical Simulation of a Concrete Beam Aﬀected by AlkaliSilica Reaction Authored by Falko Bangert and G¨ unther Meschke The chemohygromechanical model for chemically expansive processes presented in Subchapter 3.3.2.2.2 is applied to the durability analysis of an ASRaﬀected concrete beam to predict the temporal deterioration of this structure. Details of the analysis including the chosen model parameters are contained in [83]. The left hand side of Figure 3.186 contains the geometry, the support conditions, the hygral Dirichlet boundaries Γp1l and Γp2l and the spatial discretization with ﬁnite elements. To avoid stress oscillations, the interpolation for the displacements us is chosen as one order higher than the interpolation of the nonlocal equivalent strain η¯, see Peerlings et al. [614]. The displacements us and the liquid pressure pl are interpolated with quadratic serendipity shape functions [782], while linear interpolations are used for the nonlocal equivalent strain η¯.
360
3 Deterioration of Materials and Structures Loading history
u
R
Γp2l
pl
pl on Γp1l
5
p
l
on Γp2l
10
16 cm
d = 0.1 cm
Liquid pressure pl [N/mm2]
0
Ω Γ 64 cm
Γp1l
pl
15 20 25 30 35 0
50
100 150 200 250 300 Time t [d]
Fig. 3.186. Numerical simulation of a concrete beam aﬀected by alkalisilica reaction: Geometry, mechanical and hygral boundary conditions, ﬁnite element mesh and hygral loading history
The diagram on the right hand side of Figure 3.186 illustrates the hygral loading history. The initial conditions are given by us0 = 0 for the displacements and pl0 = −15 N/mm2 for the liquid pressure corresponding to an initial liquid saturation of sl0 = 0.8. In an initial phase (Phase I), drying is simu lated by decreasing the liquid pressure from the initial value pl = pl0 to 2 pl = −29 N/mm corresponding to a liquid saturation of sl = 0.6. After 20 days of drying, the liquid pressure pl at the bottom of the beam is raised to 2 pl = −9 N/mm corresponding to a liquid saturation of sl = 0.9 (Phase II). In the third loading phase starting at t = 130 d, all hygral boundary conditions are reset to the initial value pl = pl0 and held constant afterwards. Figure 3.187 illustrates, from the left to the right, the distribution of the liquid saturation sl , the ASR expansion rate ∂εas /∂ t, the ASR expansion εas and the damage parameter d at selected stages of the hygral loading history. The ﬁrst row of Figure 3.187 (t = 20 d) corresponds to the end of the initial drying process. Due to drying of the surface, the more humid inner part is characterized by larger values of the ASR expansion εas and the respective expansion rate ∂εas /∂t. Since the ASRinduced expansion of the core is at least partially hindered, the inner part of the beam is subjected to compression while the outer part is subjected to tension. These tensile stresses cause cracking along the surface. The opening of such surface cracks is frequently observed in ASRaﬀected concrete structures, see e.g. Poole [642] and Hobbs [373]. The contour plots in Figure 3.187 for t = 75 d and t = 130 d correspond to the wetting process (Phase II) characterized by a moisture transport oriented from the bottom towards the top. Accordingly, the ASRinduced expansion accelerates at the bottom face of the beam. At this stage, the ASR expansion
3.4 Numerical Examples
t [d]
Saturation sl
361
Expansion rate ε˙as Damage parameter d
20
0.6 75
ASR damage front
sl []
0.9
0.0 ε˙as [10−5 /d] 4.5 0.0
d []
1.0
130
Fig. 3.187. Numerical simulation of a concrete beam aﬀected by alkalisilica reaction: Distribution of the liquid saturation sl , the expansion rate ∂εas /∂t, the expansion εas and the damage parameter d at diﬀerent stages of the loading history
εas within the inner part of the beam is still larger compared to the expansion within the outer parts of the beam. The simultaneous moisture diﬀusion into the beam and the activation of reaction kinetics within the more humid zone near the surface leads to the formation of an ASR front which starts to propagate from the bottom to the top In front of the highly aﬀected lower part of the beam the concrete is subjected to tensile stresses, which exhaust the tensile strength and lead to substantial cracking in the vicinity of the ASR front. At the top of the beam, cracks associated with the spatial gradient of the expansion εas due to drying of the upper part of the beam has penetrated about 5 cm into the beam. It should be noted, that in a purely hygromechanical analysis (i.e. εas = 0) corresponding to a beam made of concrete containing no reactive silica within the aggregates, no damage would be observed (i.e. d = 0) for the hygral loading scenario considered in this analysis. The left diagram in Figure 3.188 shows the evolution of the expansion εas in three points located along the axis of symmetry. The results clearly show, that the onset of moisture transport from the bottom into the beam (ﬁrst vertical line at t = 20 d) considerably accelerates the ASRexpansion at the bottom side. At the beginning of Phase III (second vertical line at t = 130 d), the ASR reaction is accelerated at the top and decelerated at the bottom of the beam, respectively. However, at this relatively late stage the inﬂuence of varying hygral conditions on the kinetics of the ASR expansion is comparatively small since the sensitivity of the reaction kinetics on the moisture content reduces with an increasing reaction extent. Although the moisture content is uniformly distributed within the beam at the end of the numerical simulation (see Figure 3.187, t = 300 d), the expansion εas within
362
3 Deterioration of Materials and Structures
Evolution of ASR expansion
Load displacement diagrams 180
point A point B point C
0.3
C B A
0.2
loading at t = 0 d loading at t = 300 d
160 Reaction force R [N]
Expansion εas [%]
0.4
0.1
140
R0u
120 u R300
100 80 E0
60 40
E300
20 0
0 0
50
100
150
200
Time t [d]
250
300
0
0.2
0.4
0.6
0.8
1
Load factor λ [1]
Fig. 3.188. Numerical simulation of a concrete beam aﬀected by alkalisilica reaction: Evolution of the ASR expansion εas in three points along the axis of symmetry and load displacement diagrams R(λ) of the virgin concrete beam (ultimate load analysis at t = 0 d) and the ASRaﬀected concrete beam (ultimate load analysis at t = 300 d)
the three points A, B and C diﬀers considerable. This observation reﬂects the fact, that the ASRinduced expansion and, consequently, the deterioration of concrete structures caused by the alkalisilica reaction strongly depends not only on the moisture content but also on the hygral loading history. The right diagram in Figure 3.188 contains the load displacement relations R(λ) from ultimate load analyses performed for the ASRaﬀected beam and, for comparison, for an identical beam without being aﬀected by the alkalisilica reaction. During 300 days of ASRexpansion, a considerable structural degradation is observed: the structural stiﬀness is reduced by [1 − E300 /E0 ] = 55% due to the alkalisilica reaction and the ultimate load is decreased by u [1−R300 /R0u ] = 27%. Furthermore, at time t = 300 d a more brittle structural response is observed in the very early postpeak regime compared to t = 0 d. 3.4.5 Lifetime Assessment of a Spherical Metallic Container Authored by JanHendrik Hommel and G¨ unther Meschke This Subsection contains an application of the concept for lifetime predictions of metallic structues subjected to cyclic loading [390] and described in Subsection 3.3.1.2.1. Figure 3.190(a) shows the spherical pressure vessel of the ICIWilhelmshaven supported by cylindrical columns. This structure is assumed to be subjected to earthquake loading, which causes settlements of the columns. In this prototype application of the life time assessment concept proposed in Section 3.3.1.2.1 the base displacements according to the El Centro
3.4 Numerical Examples
363
Damage parameter d
t [d]
0
[1]
1
0
300
λ=0
λ = 1/3
λ = 2/3
λ=1
Fig. 3.189. Numerical simulation of a concrete beam aﬀected by alkalisilica reaction: Distribution of the damage parameter d at diﬀerent stages of the mechanical loading test of the virgin concrete beam (ultimate load analysis at t = 0 d) and the ASRaﬀected concrete beam (ultimate load analysis at t = 300 d)
v e rtic a l d is p la c e m e n t [c m ]
2 0 1 0 0 1 0 2 0
(a )
0
1 0
2 0 tim e t [s ]
3 0
4 0
(b )
Fig. 3.190. Application of Low Cycle Fatigue Model: (a) Spherical pressure vessel of ICIWilhelmshaven, (b) vertical displacementtime plot of the El Centro earthquake
earthquake in 1940 (magnitude 6.9) is assumed as the relevant quasistatic loading scenario. The vertical displacement is illustrated for the ﬁrst 32 seconds in Figure 3.190(b) [113]. Further details of the analysis are given in [386]. In the analysis, the structure is discretized by 7parametric shell elements using a quadratic MindlinReissnertype kinematics. A special updatedrotational formulation allows the simulation of the intersection of the spherical
3 Deterioration of Materials and Structures
v o id v o lu m e fra c tio n f []
0 .0 1 6 0 .0 1 5 0 .0 1 4 0 .0 1 3 0 .0 1 2 0 .0 1 1 0 .0 1 0 0 .0 0 9 (a )
m a x . v o id v o lu m e fra c tio n f []
364
0 .0 1 6 0 .0 1 4 0 .0 1 2 0 .0 1 0 0 .0 0 8 0
1 0
2 0 tim e t [s ]
3 0
4 0
(b )
Fig. 3.191. Application of Low Cycle Fatigue Model: (a) Damage accumulation resulting from the El Centro earthquake, (b) Temporal evolution of the maximal void volume fraction f
vessel and the cylindrical columms. To avoid locking phenomena an Enhanced Assumed Strain as well as an Assumed Natural Strain concept is implemented in the ﬁnite element procedure. Assuming the structue consistes of the steel alloy 20MnMoNi55 [390], the GTNmodel with the calibrated model is used for the following numerical studies. Further details of the analysis are given in [386]. Figure 3.191(a) shows the damage accumulation in the vicinity of the vesselcolumn intersections. The maximum value of damage is observed at the highest point of the vesselcolumn intersection. An evolution of this maximum damage is plotted for the complete loading history in Figure 3.191(b). It is noted, that the maximum value of damage is not of critical level. It can be concluded, that this engineering structure would have survived the El Centro earthquake without failure.
4 Methodological Implementation
Authored by Dietrich Hartmann and Detlef Kuhl Based on the modeling of external as well as internal actions with respect to lifetimeoriented design (see Chapter 2), and by either applying or adapting various deterioration models of materials and structures (see Chapter 3), a methodological approach and implementation of appropriate concepts, numerical methods and solution paradigms are required to facilitate practical design to reality tasks. For this purpose, having addressed fundamental aspects of problem classiﬁcation, numerical concepts, uncertainty and design methodology, details of speciﬁed solution methods for structural design are introduced. In this context, single and multiﬁeld numerical methods, reliability analysis methods and optimization methods are dealt with, where particularly timeinvariant behavior is taken into account. Finally, sophisticated practical applications are to demonstrate the capabilities of the established lifetimeoriented design concepts.
4.1 Fundamentals Authored by Dietrich Hartmann and Detlef Kuhl The previous chapters are dealing with the scientiﬁc bases required to establish lifetimeoriented design models for a solution by means of modern computer facilities. For that, the external impacts on materials and structures due to deterioration phenomena on micromacro levels, to some extent even on a nanomicro level, have been scrutinized. As a result, an appropriate theoretical fortiﬁcation of lifetimeoriented design concepts on the structural system level has been achieved, based on solid veriﬁcation and, where required, on validation through experiments.
366
4 Methodological Implementation optimization and design reliability analysis numerical methods modeling of deterioration materials and structures
external attack
Chapter 3 • spatial discretization • stationary problems • temporal discretization • adaptivity • discontinuities • model reduction • reliability analysis • 1. and 2. order reliability • ν (t)approximation • covariance analysis • Monte Carlo • response surface • parallelization • optimization problems • derivative based numerical solution methods • evolution strategies • parallelization Fig. 4.1. Overview of the methodological implementation of lifetime oriented design concepts
4.1.1 Classiﬁcation of Deterioration Problems Authored by Dietrich Hartmann and Detlef Kuhl Analyzing and evaluating the evolution of the mechanical behavior of projected structures over short and long time spaces are the contemporary concern of research worldwide. To structure the current research endeavors, and to integrate the own work carried out in this ﬁeld, a classiﬁcation of the potential deterioration problems is taken with respect to the structural system level, in particular. Given that structures in civil engineering have to be designed, analyzed and constructed such that they are satisfying all requirements, imposed to them during their service performance, deteriorations and the following damages have to be scheduled within design concepts in a systemic fashion. According to the deﬁnitions introduced in the previous chapters, deteriorations are understood as irreversible microscopic alterations in a structure leading to macroscopic degradations on the structure level. In this connection, the characteristic mechanisms inducing the deteriorations and subsequent damages allow the classiﬁcation as follows: • • • •
brittle deteriorations without creation of signiﬁcant plastic strains ductile deteriorations due to pores during plastic eﬀects creep damage caused by intergranular decohesion along with high temperature loading fatigue damage excited by means of transgranular fracture phenomena due to high as well as low loadcycles
4.1 Fundamentals
• •
367
chemomechanical, hygromechanical and thermomechanical damage chemically caused aging
Mathematical models describing these phenomena on the continuum mechanical level can be classiﬁed based on the character of the underlying diﬀerential equations: • • • • •
single ﬁeld problems multi ﬁeld problems stationary second order partial diﬀerential equations second order partial diﬀerential equations including derivatives with respect to spatial coordinates and time nonlinear diﬀerential equations
Derivatives with respect to spatial coordinates are solved numerically by using the ﬁnite element method. As a result of this discretization process the deterioration mechanisms are described on the structural level. The associated semidiscrete structural equations are strongly nonlinear vector equations depending on nodal values of time independent as well as time dependent state variables. Applying time integration schemes and consistent linearization, the approximated nodal solution of deterioration processes can be generated by the recurring solution of linear systems of equations. To allow for a goaloriented and straightforward design procedure on the structural system level it is customary to represent the intricate internal deterioration and damage processes in terms of a set of appropriate limit states, either ultimate limit states or serviceability limit states. Both can be formulated as constraints to a structural design problem that must be satisﬁed necessarily. Also, dependent on the view a structural designer wants to take on the realistic representation of his design problem, the limit states may be deﬁned in a deterministic or stochastic fashion. Obviously, the stochastic mapping of quantities as well as processes, describing the individual limit states, leads to more accurate, reliable and robust design results, however, the complexity and the computational eﬀort increase drastically. In most of the cases, the approaches based upon stochastic problem representations require a multilevel solution philosophy with respect to space and time scales. Common ultimate limit states applied for civil engineering structures are: • • • • • •
loss of equilibrium change of position in the total structure or its parts (overtuning, sliding, lifting, etc.) fracture excessive displacement or twists passage to a kinematic mechanism instability
While ultimate limit states are serious threats to life or physical conditions of human beings the serviceability limit states constitute a danger to the
368
4 Methodological Implementation
utilization of structures or buildings, respectively, and are described by means of limits for • • •
displacements which have to be below practical bounds natural frequencies whose exceedance avoid an usage of the structure stresses or strains which would jeopardize the durability of structures
In the particular cases, the nature of the design problem determines how many and which limit states are mandatory and have to be included in the lifetimeoriented design approach. As a matter of course, the computation of the quantities as well as processes incorporated in the individual limit states, according to the recent advances in computer technologies, are based on or aided by powerful numerical methods. 4.1.2 Numerical Methods Authored by Detlef Kuhl According to the character of mathematical models describing deterioration processes of concrete material and concrete structures several numerical methods are combined for their approximative solution and the prognosis of long term behavior. Spatial derivatives of single and multi ﬁeld problems are approximated by using the spatial ﬁnite element method. The time variant formulation of durability models requires additionally the numerical integration by time integration methods. Adaptive methods in space and time supplement the basic discretization methods such that the quality of the numerical results can be controlled and the numerical eﬀort is reduced. Finally, the numerical solution of nonlinear diﬀerential equations includes the NewtonRaphson iteration based on the consistent linearization of partial diﬀerential equations, semidiscrete ordinary diﬀerential equations and fully discrete algebraic vector equations. The broad variety of mathematical models describing deterioration processes is numerically discretized by the ﬁnite element method. In order to simplify the development and coding process of numerical methods they are expressed within the framework of a generalized multi ﬁeld problem containing all investigated models as particular examples. This generalized multiﬁeld problem is prepared for the ﬁnite element procedure by the weak formulation and the consistent linearization. Within the next development step a general ﬁnite element method for multiﬁeld processes is generated using a multidimensional spatial ﬁnite element concept. Approximations of state variables and test functions are evolved from onedimensional Lagrange and Legendre polynomials of arbitrary polynomial degree, respectively. Both types of ansatz functions allow for the structured progress of two and threedimensional shape functions of arbitrary polynomial degree. Legendre and Lagrange polynomial based hierarchical shape functions provide additionally the generation of anisotropic ansatz functions. Furthermore, eﬀectively formulated structural ﬁnite elements as special kinds of volume elements as for example beams, plates and shells are used for advanced structural calculations. The numerical simulation of fracture
4.1 Fundamentals
369
processes, as one of the main aspects of deterioration mechanics, can only be described with the standard ﬁnite element methods in combination with interface elements and remeshing in every load or time step. In order to avoid this ineﬃcient computational technique, ﬁnite element methods including discontinuous displacement approximations are used. In particular, the embedded discontinuity approach and the extended ﬁnite element method are used for the computation of crack propagations. After the ﬁnite element discretization, semidiscrete ordinary diﬀerential equations are obtaind. The integration of these vector valued diﬀerential equations is realized by time integration schemes originally developed for structural dynamics. Furthermore continuous Galerkin methods, discontinuous Galerkin methods and generalized Newmark methods are used. Advantages of Newmark type integration schemes are the moderate numerical effort and the controllable numerical dissipation. Galerkin integration schemes are developed for arbitrary polynomial degree in time. Consequently, the order of accuracy overcomes the upper bound two of Newmark type integration schemes. Controlled by the polynomial degree, Galerkin integration schemes possessing an arbitrary order of accuracy, can be designed. Continuous and discontinuous Galerkin integration schemes are energy conserving and energy decaying methods, respectively. As a result of this, discontinuous Galerkin schemes are well suited for the calculation of transport processes including strong variations of source terms and boundary conditions. Both, Newmark and Galerkin type integrations schemes are supplemented by error based adaptive time stepping schemes. Because almost all present deterioration processes of material and structures in this book are highly nonlinear, iterative schemes complete the collection of numerical methods solving these kind of problems. Exclusively, the pure NewtonRaphson scheme based on the consistent linearization, is adopted. The applied numerical methods are prepared for the iterative procedure in every development step. In particular, the underlying diﬀerential equations, weak forms, semidiscrete diﬀerential equations and algebraic equations are linearized. 4.1.3 Uncertainty Authored by Dietrich Hartmann Uncertainties, in general, have been of central interest in the solution of civil engineering problems over the past decades. Since the quantiﬁcation of uncertainties in engineering is extremely relied on computing methods, it is not surprising that structural engineering, i. e. structural analysis as well as design, being the pathﬁnders for traditional deterministic computing in engineering, also substantially promoted the quantiﬁcation of uncertainties using various computational approaches. In this context uncertainty should be understood according to the deﬁnition of Bothe [143] as the gradual assessment of the truth content of a proposition. Consequently, three categories of uncertainty can be
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4 Methodological Implementation
identiﬁed in engineering problems that need to have a computational representation if not only qualitative evaluation are to be carried out. These are • • •
stochastic uncertainty informal uncertainty lexical uncertainty
Depending on the nature of a problem considered the individual categories of uncertainty may occur separately from each other or in hybrid fashion. Stochastic uncertainty which are exclusively dealt with in the frame of this book is applicable when the facts of cases (either in terms of systems or processes) show a clear random behavior. This means that the principles, laws as well as rules of randomness, in harmony with the theory of probability, are completely satisﬁed. Characteristic assumptions that should theoretically apply (but are not always met in reality!) contain requirements such as (i) a sufﬁciently large universe (sample sizes), (ii) to specify the relevant stochastic parameters or functions correctly, (iii) the applicability of the describing parameters from reference cases to a new problem given (reproduction conditions) and (iv) the appropriateness of probability density (pdf) or distributed functions. Also, the often assumed identical independently distributed paradigm must apply accordingly. If the aforementioned premises do not or can not be satisﬁed because of lacking or unavailable information, e. g. if only undervalued sample sizes exist, then, strictly speaking, informal uncertainty models should or must be created. In such cases the fuzzy set theory, based upon pure mathematics, serves as a relatively new and powerful methodology to cope with such information deﬁcits. The fuzzy set theory can also be applied if the third category of uncertainty occurs where uncertainty is quantiﬁed in terms of linguistic variables which are mapped to an adequate numerical scale. Hence, uncertainties stemming from information deﬁcits or linked to linguistic variables are characterized by virtue of nonstatistical quantities which, to a large extend, exhibit subjective instead of of objective views on uncertainty. It should be mentioned that there are further mathematical quantiﬁcation concepts for describing uncertainty. Although, in the following, the focus is placed solely on stochastic uncertainties some potential main concepts are to be named brieﬂy; they include the method of subjective probability [231], [857], convex modelling [256] interval algebra [42] and chaos theory [425]. 4.1.4 Design Authored by Dietrich Hartmann The design of systems, in particular, the process of designing and erecting structures with respect to all the demands during their lifetime is a grand challenge problem. Complex structures, such as buildings, bridges, industrial plants and many others, are an excellent testimonial for the engineering design performance. However, if failures and casualties are reported one gets aware of the dangers and risks associated with stability, deteriorations damages, failures or inserviceability of structures. Then, the question arises if a
4.1 Fundamentals
371
structure could not have been designed better than the existing one to avoid the observed damages occurred or occurring. It is, therefore, obvious that the design process in civil engineering is a fairly complex process. Many assumptions must be made to develop structural models that can then be subjected to analysis by the available methods which are today, according to the revolution in computer technology and numerical computations, computerbased methods. Many possibilities and factors with respect to material properties, load actions and construction techniques must be considered during the design formulation phase. The larger the scheduled lifetime of a structure must be, the more is the increase in complexity and the more uncertainties have to be scrutinized and integrated into the design. Of course, also economic considerations play an important role in designing costeﬀective structures. Despite the importance of costs, the lifetimeoriented design is assumed here to be represented by means of cost equivalent physical of structural quantities. This premise allows for a crisp structural mechanicsoriented formulation of the design process and avoids the consideration of further uncertain cost models. The design process begins by analyzing various options. Subsystems and their components are identiﬁed, preliminarily designed, evaluated, redesigned in more detail and then realized (fabricated). As a result, design represents an iterative process based upon the designer’s intuition and ingenuity as well as engineering competence and experience, in association with intensive structural analysis using computer facilities. In this context, the diﬀerence between structural analysis and structural design should be recognized. The analysis problem clearly addresses determining the static and dynamic behavior of a structural system designed for a given task. Hereby, the structural response due to speciﬁed actions (loadings) in terms of displacements, stresses, cracks, kinematic quantities (velocities, accelerations, frequencies, etc.) plays the dominant role. To compute these quantities the topology, geometry and sizes of the structural data of the various parts of a structure are totally known. By contrast, the design process calculates the sizes, shapes and, where applicable, even the topology of a structure to meet given performance requirements. Therefore, designs are estimated and then analyzed to check if they perform according to given speciﬁcations. If these speciﬁcations are satisﬁed, an acceptable or a feasible design is created; although it is attempted to change the design to improve its performance. If the design fails a redesign is mandatory to come up with a feasible solution. Both cases intensively rely on (computational) analysis capabilities, but, also represent more than pure analysis. According to the nature inherent in design problems, design processes must be understood as a ”synthesis (composition) approach”. Evidently, synthesis problems are more complex and cause greater eﬀorts than analysis problems alone. Scarcity and eﬃciency in today’s competitive world force engineers to streamline and accelerate their design processes. Here, the computeraided design provides means and ways to master complexity and time eﬀorts. In
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4 Methodological Implementation
particular, those design problems either in preliminary or detailed design, where the conceptual decisions have been come already and an abstract formulation of design in terms of design variables, constraints and assessment (objective) criterion is available, can be transferred to an equivalent structural optimization problem (SOP). Based on such an optimization problem or model, numerical optimization strategies can be applied to solve design problems rapidly, within acceptable response times. This rigorous computeroriented design approach also helps a designer to gain a indepth insight and understanding of the design problem, which is of particular use in lifetimeoriented design. To accomplish this, however, the structural optimum design process forces to identify explicitly a set of reasonable design variables, relevant constraints and signiﬁcant and powerful optimization criterions.
4.2 Numerical Methods Authored by Detlef Kuhl The description of numerical methods, solving mathematical models for the prognosis of deterioration processes of materials and structures, contains the following aspects: • • • • • • • •
Deterioration models described in Chapter 3 are generalized in an abstract multi ﬁeld problem. The solution strategy of highly nonlinear and time dependent coupled ﬁeld problems is outlined. General weak forms of coupled problems are generated and consistently linearized with respect to the ﬁeld variables. Spatial ﬁnite element discretization techniques, using Lagrange or Legendre polynomial based multidimensional ansatz functions, are presented and discussed. Solution methods for stationary problems are brieﬂy reviewed. Newmark and Galerkin type time integration schemes are presented. Adaptive methods in space and time as well as several underlying error measures and solution strategies are compared with attention to the algorithmic formulation. Standard ﬁnite elements methods are enriched for the description of discontinuities.
4.2.1 Generalization of Single and MultiField Models Authored by Detlef Kuhl Mathematical models of single ﬁeld and coupled ﬁeld problems discussed in Chapter 3 are constituted by general balance equations, constitutive equations including history variables, driving forces and boundary conditions. In general, coupled processes in materials can be modeled by an abstract set of
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373
interacting balance equations in local or, alternatively, global formulations. The local balances represent coupled partial diﬀerential equations which can be decomposed in the following parts: 1. 2. 3. 4.
time derivatives of the primary variables, divergenz of the balance quantity ﬂuxes, constitutive laws connecting ﬂux and driving forces and, ﬁnally, the driving forces expressed in terms of spatial gradients of primary variables. 5. Furthermore, interactions of the balance equations can be manifested in all parts of the diﬀerential equations. The modeling of generalized multiphysics problems is based on the generalized formulation of local and global balance equations by [253]. In order to prepare the general multiphysics model for its numerical solution, the strong form is weakly formulated and linearized. Before going into the technical aspects of this section, it is worth to embed the modeling of multiphysics proplems between the real physical problem and its virtual counterpart, compare Figure 4.2. This collection of the two step procedure from the original physical problem to its numerical simulation by means of modeling and numerical solution is denoted as numerical modeling. It is essential for reliable simulation of physical processes to develop adequate mathematical models capturing the main character of real phenomena and to apply numerical methods which solve the mathematical model preferably accurate and eﬃcient. As a result of this statement, the modeling of multiphysics problems yielding partial diﬀerential equations and numerical methods, namely ﬁnite element spatial discretization methods and time integration methods, are discussed separately in the following sections. 4.2.1.1 Integral Format of Balance Equations The integral format of the balance equation related to the tensor ﬁeld f ∈ [1, NF ] ∂ ˆ f dV Θ f dV = Φf · ndA + Σ f dV + Θ (4.1) ∂t Ω
ΓΦf
Ω
Ω
is described in terms of the volume speciﬁc density of a sf th order tensor valued balance quantity Θ f , the [sf + 1]th order tensor valued ﬂux density Φf , the sf th order tensor valued production term Σ f and the balance quantity ˆ f . For the explanation of the exchange between the involved tensor ﬁelds Θ domain Ω and the boundaries ΓΦf see the example of a two ﬁeld problem given in Figure 4.2. Since the model for ion transport, dissolution processes and mechanically caused material failure in Section 3.3.2.2 and [454] is one of the simplest
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multiﬁeld problems investigated in the present book, it is intensively used as illustrative example for the abstract theory of numerical methods in computational durability mechanics. The chemomechanical damage model constitutes a two ﬁeld multiphysics problem NF = 2 with tensor valued primary variables u1 = u and u2 = c of order s1 = 1 and s2 = 0, respectively. The associated balance equations are deﬁned in terms of the stress tensor Φ1 = σ, ˙ 1 = 0, the calvanishing volume forces Σ 1 = 0 and linear momentum ﬂux Θ cium ion mass ﬂux tensor Φ2 = q, a vanishing production term Σ2 = 0, the storage/production term Θ2 = [[φ0 +φ2 ]c]˙ + s˙ and vanishing exchange terms ˆ 2 = 0. ˆ 1 = 0 and Θ Θ 4.2.1.2 Strong Form of Individual Balance Equations The local format of balance equations according to tensor ﬁelds f ˆf ˙ f = divΦf + Σ f + Θ Θ
˙ f = ∂Θf Θ ∂t
(4.2)
is given in terms of sf th order tensor valued ﬁeld variables uf and associated internal variables κf . The time derivative of balance quantity Θ is in general deﬁned in terms of all ﬁeld variables uNF , time derivatives of ﬁeld variables u˙ NF and internal variables κNF . ˙ f (u˙ 1 , . . . , uNF , u1 , . . . uNF , κ1 , . . . , κNF ) = Θ ˙ f (u˙ NF , uNF , κNF ) ˙f =Θ Θ (4.3) The constitutive law deﬁning the ﬂux density Φf is formulated as a function of the driving force γ f and the potential Ψf .
γ f = γ f (∇uf ), Φf (γ NF , uNF , κNF ) =
∂Ψf (γ NF , uNF , κNF ) ∂γ f
(4.4)
Dirichlet boundary conditions, Neumann boundary conditions uf = uf ∀ X ∈ Γuf
Φf · n = φf ∀ X ∈ ΓΦf
(4.5)
u˙ f (t0 ) = u˙ 0f
(4.6)
and initial conditions uf (t0 ) = u0f
complete the generalized initial boundary value problem for multiphysics analyses with Γ = Γuf ∪ΓΦf and Γuf ∩ΓΦf = ∅ (compare Figure 4.2).
4.2 Numerical Methods physical problem
375
abstract model of coupled processes φ1 n ΓΦ1 ,ΓΦ2
modeling
mathematical model
•
Γu 2
Γu 1 ˆ 1 = −Θ ˆ2 Θ
˙1 Θ Ω
•
numerical solution
˙2 Θ
φ2 Γ
simulation of physical problem
Fig. 4.2. Numerical modeling via modeling and numerical solution. General multiphysics problem with ﬁeld variables u1 and u2
Applied to the representative example of coupled mechanically and chemically induced damage [454], the associated balance equations DIV[σ] = 0
DIV[q] + [[φ0 +φ2 ]c]˙+ s˙ = 0
(4.7)
and constitutive laws Ψ1 =
1−φ ε : C0 : ε 2
Ψ2 =
φ γ · D0 · γ 2
(4.8)
are based on the deﬁnition of the driving forces γ 1 = ε = ∇sym u
γ 2 = γ = −∇c
(4.9)
in terms of the elasticity tensor of the skeleton C 0 , the linear strain measure ε, the conductivity tensor of the pore ﬂuid D 0 , the concentration gradient γ and the total porosity φ = φ0 + φ1 + φ2 . φ includes the initial porosity φ0 , the mechanically induced porosity φ1 = [1 − φ0 − φ2 ]d and the chemical porosity φ2 . Consequently, the stress tensor and the calcium ion ﬂux tensor are obtained as follows: σ = [1 − φ] C 0 : ε
q = φ D0 · γ
(4.10)
Internal variables κ1 and κ2 , evolution equations and phenomenological models for mechanically
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4 Methodological Implementation
φ1 = φ1 (d, φ2 )
d = d(κ1 (η(ε)))
(4.11)
and chemically caused damage s = s(κ2 (c)), s˙ =
φ2 = φ2 (s)
∂s ∂κc c˙ ∂κc ∂c
(4.12)
complete the continuum mechanical description of chemomechanical damage. 4.2.2 Strategy of Numerical Solution Authored by Detlef Kuhl The strategy of the numerical solution of generalized multiﬁeld problems is designed by the character of the underlying diﬀerential equations. In general the coupled ﬁeld problems are described by highly nonlinear partial diﬀerential equations. This diﬀerential equations include second order spatial derivatives and up to second order temporal derivatives. The resulting concept for the spatial and temporal discretization of nonlinear multiﬁeld problems is illustrated in Figure 4.3. 1. For the preparation of the numerical solution the highly nonlinear and time dependend diﬀerential equations are weakly formulated and linearized. 2. The spatial ﬁnite element discretization is realized by multidimensional multiﬁeld elements of arbitrary polynomial degree.
a. Ψf = divΦf +Σf
1. b. δW = 0 2. M, D K
ri = r
c. semdiscrete balance d. discrete balance
3.
δW(◦) = 0
d. ri (◦) = rn+1−αf f. Newton correction e. linearized balance
ΔδW
c.
a. strong form b. weak form
1. weak formulation 2. spatial discretization 3. temporal discretization 4. linearization
ΔδW(◦)
e. K Δu dG dG 4. = rdG
5.
5. iterative solution
f. uk+1 n+1 = ukn+1+Δu
Fig. 4.3. Modeling and numerical analysis of multiphysics problems
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377
3. The time derivatives within the underlying diﬀerential equations are numerically solved either by Newmark or Galerkin type time integration schemes. 4. Finally, the nonlinear format of resulting algebraic vector equations is solved by the NewtonRaphson procedure. Spatial and temporal discretization methods are the main tasks of the numerical solution strategy. They are enriched by adaptive methods in order to control the numerical eﬀort and the solution quality of structural reliability simulations. 4.2.3 Weak Formulation Authored by Detlef Kuhl Within the present section the weak formulation of generalized multiphysics problems is presented. In order to prepare the weak form also for iterative solution methods, the linearized weak form is additionally generated. 4.2.3.1 Weak Form of Coupled Balance Equations The weak form of individual balance equations and Neumann boundary conditions is generated by the contraction with the test function δuf , integration over the domain Ω and the boundary ΓΦf
˙ f dV − δuf ◦ Θ
δWf (•) =
δuf ◦ divΦf Ω
Ω
δuf ◦ Σ dV +
− Ω
dV
δuf ◦ Φf · n−φf dA = 0
(4.13)
ΓΦf
and application of the divergence theorem.
˙ f dV + δuf ◦ Θ
δWf (•)=
Ω
Ω
δuf ◦ Σ dV −
− Ω
◦
δ∇uf ◦ Φf dV δuf
◦ φf dA =0
(4.14)
ΓΦf
According to the multiphysics problems considered in the present book ˆ f = 0 is assumed for the sake of simplicity. In equation (4.14) ◦ and ◦◦ Θ represent the sf th and [sf + 1] tensor contraction, respectively, and argument
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4 Methodological Implementation
(•) represents all state variables u˙ NF , uNF and spatial gradients ∇uNF . On the basis of equation (4.14) the weak form of the coupled multiphysics system δW is generated in equation (4.15) by the weighted summation of the individual weak forms δWf , whereby Af is introduced to adapt physical units and dimensions of coupled ﬁelds.
δW (•) =
NF
Af δWf (•) = 0
(4.15)
f =1
4.2.3.2 Linearized Weak Form of Coupled Balance Equations In order to prepare the weak form for the numerical solution with the NewtonRaphson scheme, the weak form is expanded in a Taylor series about the trial solution of all state variables δW (•k+1 ) = δW (•k ) + ΔδW (•k ) = 0
(4.16)
and spatial gradients characterized by •k .
¨ NF , u˙ NF , uNF ) = ΔδW (•) = ΔδW (∇uNF , u
NF
Af ΔδWf (•) = 0
f =1
(4.17)
The increment of weak forms is generated by summation of individual portions in terms of the increments of ﬁeld variables Δug and gradients of ﬁeld variables ∇Δug with g ∈ [1, NF ].
ΔδWf (•) =
NF ∂δWf g=1
∂∇ug
◦
◦ Δ∇ug +
∂δWf ∂δWf ◦ Δu˙ g + ◦ Δug ∂ u˙ g ∂ug
(4.18)
It is worth to mention that the derivative with respect to gradient ∇ug is performed explicitly in oder to obtain an advantageous format for
4.2 Numerical Methods
379
the ﬁnite element discretization discussed in Section 4.2.4.2. The individual terms in equation (4.18) are expressed as follows: ∂δWf ◦ ◦ Δ∇ug = ∂∇ug ∂δWf ◦ Δu˙ g = ∂ u˙ g ∂δWf ◦ Δug = ∂ug
δuf ◦ Ω
Ω
˙f ◦ ∂Θ ◦Δ∇ug dV + ∂∇ug
˙f ∂Θ δuf ◦ ∂ u˙ g δuf ◦
˙f ∂Θ ∂ug
◦
∂Φf ◦ ◦Δ∇ug dV ∂∇ug
◦
∂Φf ∂ug
δ∇uf ◦ Ω
◦ Δu˙ g dV ◦ Δug dV +
Ω
δ∇uf ◦
◦ Δug dV
Ω
(4.19)
4.2.4 Spatial Discretization Methods Authored by Detlef Kuhl and Christian Becker Within the framework of the semdiscretization technique applied to solve durability single and multiphysics problems, the spatial discretization is realized by the ﬁnite element method (see e.g. [90, 106, 223, 224, 870, 871]). The scientiﬁc and industrial oriented literature documents the broad range of applications of this method for the spatial discretization of diﬀerential equations. Highly nonlinear problems, stationary and transient problems as well as single and multiﬁeld problems can by solved adopting the ﬁnite element method. 4.2.4.1 Introduction Standard ﬁnite elements for one, two and threedimensional problems are using low order approximations of the primary variables. The great advantage of these ﬁnite element methods is the eﬀective formulation and implementation in ﬁnite element codes, see as a particular example the popular constant strain triangle [796]. The drawback of such kind of ﬁnite elements is the low computation accuracy because of only linear ansatz functions and locking phenomena. Basically ﬁnite element solutions can be improved by reducing the element size (hmethod) and increasing the polynomial degree of ansatz functions (pmethod), respectively. Furthermore, special element techniques can be used to overcome the well known locking problem. In summary, two main philosophies are used in the context of structural mechanics, to obtain high quality numerical results applying the ﬁnite element method. •
Low order ﬁnite element methods combined with methods to prevent locking [637, 790]:
380
•
4 Methodological Implementation
◦ selective reduced integration [397, 649] and hourglass control [283] ◦ assumed natural strain concept [398, 247] ◦ Bbar methods [850, 789] ◦ enhanced assumed strain concept [747, 742] Higher order ﬁnite element methods using diﬀerent kind of higher order ansatz polynomials: ◦ multidimensional Lagrange polynomials [870, 452] ◦ Legendre based hierarchical one, two and threedimensional shape functions [72, 717, 246]
A literature review about high quality, low order ﬁnite element methods makes clear that the related element techniques are separately developed for selected applications in structural mechanics. But computational durability mechanics is characterized by manifold various underlying diﬀerential equations. Therfore, the more general higher order ﬁnite element method is presented in the following. 4.2.4.2 Generalized Finite Element Discretization of Multiﬁeld Problems The numerical analysis of nonlinear multiphysics problems can be subdivided in the spatial ﬁnite element discretization, the temporal discretization and the iterative solution of the resulting nonlinear algebraic equation. In the present section a detailed description of the spatial pﬁnite element discretization of generalized multiphysics problems is given. 4.2.4.2.1 Approximations The ﬁnite element formulation of the original and the linearized weak forms of multiphysics problems is based on the approximation of test functions, state variables and their gradients by shape functions and nodal values of state variables. Therefore, NDdimensional anisotropic shape functions of arbitrary polynomial degrees pd for the ND spatial directions d are designed based on onedimensional Lagrange shape functions. Furthermore, the approximation of state variables is given and the transformation between natural and physical element coordinates is performed by the Jacobi transformation. Onedimensional Lagrange shape functions of polynomial degrees pd can be generated for every spatial direction d ∈ [1, ND] by the product N i (ξd ) =
p: d +1 k=1 k=i
ξdk − ξd ξdk − ξdi
ξdi =
2[i − 1] −1 pd
(4.20)
in terms of the natural coordinate ξd and the natural nodal positions ξdi and ξdk with i, k ∈ [1, pd + 1]. Consequently, the derivatives of shape functions are also calculated for arbitrary polynomial degrees pd .
4.2 Numerical Methods
i N;d (ξd ) =
p: p d +1 d +1 −1 ξdk − ξd ∂N i (ξd ) = ∂ξd ξdl − ξdi k =1 ξdk − ξdi l= 1
381
(4.21)
k = i k = l
l= i
NDdimensional isotropic or anisotropic shape functions are generated by the product of onedimensional shape functions N i (ξ) =
ND :
ND d−1 : i = k1 + [kd −1] [pl +1]
N kd (ξd )
d=1
d=2
(4.22)
l=1
and their derivatives with respect to natural coordinates ξm with m ∈ [1, ND] kd analog by the product of the derivatives N;m . ND :
i kd N;m (ξ) = N;m (ξm )
N kd (ξd )
(4.23)
d=1 d=m
In equations (4.22) and (4.23) the counter kd ∈ [1, pd + 1] for onedimensional shape functions is used. Representative examples for twodimensional isotropic and anisotropic Lagrange shape functions as well as partial derivatives of Lagrange shape functions are given in Figures 4.4 and 4.5. Furthermore, the general design of one, two and threedimensional shape functions is illustrated in Table 4.1. Shape functions (4.22) and and spatial derivatives of shape functions (4.23) are applied for the approximation of state variables
N24 (ξ2 )
N12 (ξ1 )
ξ2
N 11 (ξ1 , ξ2 )
ξ1 shape function N 11
N24 (ξ2 )
11 N;1 (ξ1 , ξ2 )
ξ2
ξ2
ξ1
ξ1 2 (ξ1 ) N1;1
ξ1 11 derivative N;1
Fig. 4.4. Illustration of isotropic Lagrange shape functions and derivatives of Lagrange shape functions by means of a cubic planar ﬁnite element
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4 Methodological Implementation
N22 (ξ2 )
N 8 (ξ1 , ξ2 )
ξ2
N12 (ξ1 )
2 N2;2 (ξ2 )
8 (ξ1 , ξ2 ) N;2
ξ2
ξ2
ξ1
ξ1 N12 (ξ1 )
ξ1
ξ1
shape function N 8
8 derivative N;2
Fig. 4.5. Illustration of anisotropic Lagrange shape functions and derivatives of Lagrange shape functions by means of a cubiclinear planar ﬁnite element Table 4.1. Multidimensional Lagrange shape functions and specialization to one, two and threedimensional ﬁnite elements element type
position vector
ﬁeld variable
X
u
integral • dV Ωe
ND = 1
( ) X = X1
( ) u = u1
1 −1
dVξ
truss element %
ND = 2
X1 X2
X=
ND = 3
X1 ⎢ ⎥ X = ⎣ X2 ⎦ X3
⎡
volume element
NN
%
&
plane element
uf ≈
• dξ1 A !
u= ⎤
N i uei f
i=1
u1 u2
⎡
&
−1−1
⎤
u1 ⎢ ⎥ u = ⎣ u2 ⎦ u3
u˙ f ≈
NN
1 1
• dξ1 dξ2 H ! dVξ
1 1 1 −1−1−1
• dξ1 dξ2 dξ3 ! dVξ
N i u˙ ei f
(4.24)
∂N i ∂X
(4.25)
i=1
and the gradient of state variables ∇uf ≈
NN i=1
i uei f ⊗∇N
∇N i (ξ) =
4.2 Numerical Methods
383
˙ ei in terms of nodal state variables uei f . Furthermore, equations (4.24) f and u and (4.25) are used for the approximation of δuf , Δuf , δ∇uf and Δ∇uf δuf (ξ) ≈
NN
N i (ξ) δuei f
δ∇uf (ξ) ≈
i=1
Δuf (ξ) ≈
NN
NN
i δuei f ⊗∇N (ξ)
i=1
N i (ξ) Δuei f
Δ∇uf (ξ) ≈
i=1
NN
i Δuei f ⊗∇N (ξ)
i=1
(4.26) ˙ f , Φf and their as well as for the implicit approximation of resulting terms Θ derivatives. Finally, the Jacobi tensor J(ξ) ≈
NN
X ei ⊗ ∇ξ N i (ξ)
∇ξ N i (ξ) =
i=1
∂N i (ξ) ∂ξ
(4.27)
allows for the transformation of the gradient of shape functions ∇N i (ξ) =
∂N i (ξ) ∂ξ · = J −T (ξ) · ∇ξ N i (ξ) ∂ξ ∂X
and the diﬀerential volume element. ⎧ ⎨ dξ1 dξ2 dξ3 for ND = 3 dV = J  dVξ , dVξ = dξ1 dξ2 H for ND = 2 ⎩ for ND = 1 dξ1 A
(4.28)
(4.29)
4.2.4.2.2 NonLinear Semidiscrete Balance Inserting the approximations discussed above into the weak form of multiphysics problems (4.14) and (4.15) for individual ﬁnite elements e ∈ [1, NE] yields the discretized weak form on the element level
δW e ≈
NF NN
ei ei δuei f ◦ r if − r f = 0
(4.30)
f =1 i=1
˙ eNF , ueNF ) and the in terms of the generalized internal force tensor r ei if (u ei generalized external force tensor r f according to the nodal values of the test function δuei f .
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4 Methodological Implementation
Ω e
r ei f
Φf · ∇N i dV
˙ f dV + Af Ni Θ
r ei if = Af
e Ω
N i φf
i
= Af
N Σ dV + Af
(4.31) dA
e ΓΦ
Ωe
f
After assembling the element quantities over element nodes, elements and tensor ﬁelds, application of the fundamental lemma of variational calculus and consideration of initial values the semidiscrete initial value problem ˙ u) = r ri (¨ u, u,
¨0 u(t0 ) = u0 , u˙ 0 , u
(4.32)
is obtained. In the nonlinear second order vector diﬀerential equation ¨ represent the generalized vectors of (4.32) the vectors ri , r, u, u˙ and u internal forces, external forces, primary variables, ﬁrst and second temporal rates of primary variables, respectively, whereby every structural vector contains the nodal values of all contributing ﬁelds f . As particular example the assembling of the generalized vector of internal forces ri based on the nodal element internal force tensors r ei i is given. ˙ u) = ri (¨ u, u,
NE,N AN
˙ eNF , ueNF ) rei i (u
(4.33)
e,i
It is worth to mention that the second time derivative only exists, if the ˙ f is identiﬁed with a second order temporal change of the balance quantity Θ time derivative as for example for the modeling of structural dynamics or wave propagation problems (see e.g [452]). The application of the present generalized multiphysics ﬁnite element concept for the discretization of the chemomechanical damage model yields the ei element tensors of internal and external forces (r ei i1 and r 1 ) and internal and ei ei external calcium ion mass ﬂuxes (r i2 and r2 ). ei ∇N i · σ dV ri1 = Ω ei ri2
=− ∇N · q dV + i
Ω
rei 1 =
Ω i
N t dA Γσ
r2ei
N i q dA
= Γq
N i [[φ0 +φ2 ]c+s]˙ dV (4.34)
4.2 Numerical Methods
385
4.2.4.2.3 Linearized Semidiscrete Balance Applying the approximation procedure to the linearized weak form (4.174.19) on the element level yields its discrete counterpart
ΔδW e ≈
NN NF NF NN
eij eij ej ej ˙ δuei ◦ d ◦ Δ u + k ◦ Δu f g g fg fg
(4.35)
f =1 g=1 i=1 j=1
˙ eNF , ueNF ) and in terms of generalized tangent stiﬀness tensors keij f g (u ˙ eNF , ueNF ) according to the test generalized tangent damping tensors deij f g (u ej functions δuei f and increments Δug . keij f g = Af
Ni Ω
+ Af
N
i
˙f ∂Θ ∂ug
N
i
˙f ∂Θ ∂ u˙ g
Ω
deij fg
= Af
˙f ∂Θ · ∇N j dV + Af ∂∇ug
∇N i ·
∂Φf · ∇N j dV ∂∇ug
∇N i ·
∂Φf ∂ug
Ω
j
N dV + Af
N j dV
Ω j
N dV
Ω
(4.36) On the structural level the linearized discrete balance equation is obtained accordingly: ˙ u)Δ¨ ˙ u)Δu˙ + K(¨ ˙ u)Δu = r − ri (¨ ˙ u) (4.37) M(¨ u, u, u + D(¨ u, u, u, u, u, u, ˙ u), D(¨ ˙ u) and K(¨ ˙ u) are the generalized tanHerein M(¨ u, u, u, u, u, u, ˆteaux derivative of the generalized gents which can be deﬁned by the Ga ¨. internal force vector with respect to the state variables u, u˙ and u
˙ u) = K(¨ u, u,
˙ u) u, u, ∂ri (¨ ∂u
D=
∂ri ∂ri , M= ¨ ∂ u˙ ∂u
(4.38)
As particular example the generalized tangent tensors associated with the chemomechanical model of coupled mechanical damage and calcium leaching are given. The generalized tangent stiﬀness tensors are given in terms of the second order mechanical tangent stiﬀness tensor,
386
4 Methodological Implementation
keij 11
∇N i ·
=
∂σ · ∇N j dV ∂ε
(4.39)
Ω
the scalar valued chemical tangent c ∂q eij = − ∇N i · N j dV + ∇N i · D0 φ ·∇N j dV k22 ∂c Ω Ω (4.40) ∂ 2s ∂κc i ∂s ∂φ2 j j 2 cN ˙ φ N dV + N i c ˙ N dV + s ∂c ∂s ∂κ22 ∂c Ω
Ω
and the ﬁrst order chemomechanical tangent coupling tensors. ∂q eij i ∂σ j keij N · ∇N j dV = ∇N · dV, k = − ∇N i · 12 21 ∂c ∂ε Ω
(4.41)
Ω
Furthemore, the damping tensor of the chemomechanical model of calcium leaching is computed. i ∂s j φ deij = N N dV + N i φ0 + φ2 N j dV (4.42) s 22 ∂c Ω
Ω
eij , Because of the time independent balance of momentum (4.7) the tensors d11 eij eij d12 and d21 vanish. In equations (4.40) and (4.42) the following abbreviations are used: c ∂q ∂φ2 ∂D 0 ∂φ2 D0 · γ + φ · γ, φs = 1 + c (4.43) = [1−d] ∂c ∂c ∂c ∂s
4.2.4.2.4 Generation of Element and Structural Quantities The integrals in equations (4.31) and (4.36) are computed by the GaußND Legendre quadrature with NG = d=1 NGd integration points ξdld , ld ∈ [1, NGd ] contained in vectors ξ l and the weights αl based on the onedimensional integration rule, see e.g. [870]. f (ξ)J (ξ)dVξ ≈ Ωξ
NG 1 NG 2 NG 3
αl f (ξ l )J (ξ l ), αl =
l1 =1 l2 =1 l3 =1
ND :
αld
(4.44)
ld =1
Figure 4.6 illustrates the ﬁnal calculation of element quantities of generalized multiphysics pﬁnite elements. It is obvious that speciﬁc multiphysics problems as described in Chapter 3 can be implemented on the model or Gauß point level marked within the algorithmic setup. Internal variables κf are just managed on the ﬁnite element level. Manipulations of internal variables take exclusively part on the material point level.
4.2 Numerical Methods select element nodal values for i ∈ [1, NN ]
387
˙ ei X ei , uei f , u f
ND loops over Gauss points ld ∈ [1, NGd ] coordinates and weight of Gauß point
ξ = [ξ1l1 ξ2l2 ξ3l3 ]T , α = αl1 αl2 αl3
loop over element nodes i ∈ [1, NN ] N i (ξ), ∇ξ N i (ξ)
shape functions and natural derivatives
J (ξ)
Jacobi transformation tensor next node i
J (ξ), J −1 (ξ)
Jacobi determinant and inverse Jacobi tensor loop over element nodes i ∈ [1, NN ]
X(ξ), uf (ξ), u˙ f (ξ)
state variables physical gradient of shape functions physical gradient of tensor ﬁelds
∇N (ξ) = J −T(ξ)·∇ξ N i (ξ) i
∇uf
next node i model level
˙ f ,Φf ,∂ Θ ˙ f/∂∇ug ,∂ Θ ˙ f/∂ug ,∂ Θ ˙ f/∂ u˙ g ,∂Φf/∂∇ug , . . . Θ
loop over element nodes i ∈ [1, NN ] generalized external load vector
r ei f
generalized internal force vector
r ei if
summation and assembly
ei αJ r ei f → r, αJ r if → ri
loop over element nodes j ∈ [1, NN ] generalized tangent damping tensor
deij fg
generalized tangent stiﬀness tensor
k eij fg
summation and assembly
eij αJ deij f g → D, αJ k f g → K
next node j next node i next Gauß point ld Fig. 4.6. Computation of generalized element tensors of external and internal forces and generalized tangent damping and stiﬀness tensors of multiphysics pﬁnite elements
4.2.4.3 pFinite Element Method The pﬁnite element method (pFEM) is the exact counterpart to the hﬁnite element method (hFEM). Whereas in the hFEM a mesh of loworder elements is reﬁned by increasing the number of elements that are naturally
388
4 Methodological Implementation
1
p(X ) = sin(−8 X )
X
1
L = 1, 0
log rel. error [−]
0 1
1 2 3 4
pmesh hmesh
5 6 0
5
10
15
20
25
30
DOF [−]
Fig. 4.7. Sinusoidal loading of a truss member (left), Rel. error of internal energy plotted over the number of degrees of freedom (right)
smaller than the original ones, the pFEM approach is similiar to the wellknown Ritzmethod. The basic idea of the approach is to use a relative coarse ﬁnite element mesh. If the quality of the numerical solution has to be improved the mesh remains unchanged but the polynomial degree of the shape functions, and therefore the polynomial degree of the approximation, is increased subsequently. In the limit case of p → ∞ the exact solution is gained. A considerable advantage of the pﬁnite element method in the case of a smooth solution is the exponential convergency compared to the almost linear convergence rate for the hmethod. This is illustrated by an example that was analyzed in [245] and deals with a truss member that is loaded by a sinusoidal force that is acting along the member’s length (see Figure 4.7). The internal energy and the relativ error are deﬁned as: 1 FE REF Wint − Wint  Wint = σ : ε dΩ E= . (4.45) 2 REF W  int Ω A standard happroach and papproach is used for the numerical analysis. The reference solution is the solution of the pdiscretization with p = 11. In the ﬁgure the convergency rate with respect to the relative error in energy norm is ploted logarithmical over the degrees of freedom. The exponential convergence of the pmethod becomes obvious in contrast to the almost linear convergency of the hmethod. Concluding, if a certain degree of accuracy is required, the pFEM is reaching this accuracy with less degrees of freedom than the hFEM. Basically, there are two concepts of higher order shape function concepts within the pFEM: the nonhierarchical Lagrange concept and the hierarchical concept based on the Legendre polynomials.
4.2 Numerical Methods
389
4.2.4.3.1 Onedimensional HigherOrder Shape Function Concepts Before dealing with details on threedimensional shape function concepts the basic concept of higher order approximations is illustrated in the onedimensional case. The generation of shape functions of the LagrangeType has already been introduced in Section 4.2.4.2.1. 4.2.4.3.1.1 Shape Functions of the LegendreType Legendre polynomials are the solutions Pn (ξ) of the homonymous diﬀerential equation [1 − ξ 2 ] Pn (ξ)− 2 ξ Pn (ξ)+ n [n+ 1] Pn(ξ) = 0 ,
ξ ∈ R,
n ∈ {0, 1, 2, 3...}. (4.46)
The solutions Pn (ξ) can be generated by the recursive formula by Bonnet Pn (ξ) =
1 [[2n − 1] ξ Pn−1 (ξ) − [n − 1] Pn−2 (ξ)] , n ≥ 2 n (4.47)
chosing P0 (ξ) = 1 and P1 (ξ) = ξ as starting polynomials. Before the polynomials (4.47) are applied within a shape function concept, they are modiﬁed such that their value vanish at the outer vertices of an onedimensional element: 1 Φi (ξ) = [Pi (ξ) − Pi−2 (ξ)] , 2 (2 i − 1)
i ≥ 2.
(4.48)
Figure 4.8 shows a selection of some modiﬁed higherorder Legendre polynomials. Within an onedimensional hierarchical shape function concept the ﬁrst two shape functions are the standard linear Lagrange shape functions N1 (ξ) =
1 [1 − ξ] 2
N2 (ξ) =
1 [1 + ξ], 2
(4.49)
representing the linear approximation of the respective ﬁeld variable, compare Section 4.2.4.2.1. The higherorder shape functions are taken from the modiﬁed Legendre polynomials (4.48): Nj (ξ) = Φj−1 (ξ) ,
3 ≤ j ≤ p + 1.
(4.50)
390
4 Methodological Implementation
Φj [ξ i ] [−]
0.2 0 0.2 0.4 0.6 1
0
1
i
ξ [−] Fig. 4.8. Modiﬁed Legendrepolynomials for polynomial degrees p = 2, 3, 4, ..., 7
4.2.4.3.1.2 Comparison of Both Shape Function Concepts In general the numerical solutions of both concepts are identical if the same approximation degree is used. This is because hierarchical and nonhierarchical approximation describe the same polynomial Lp of degree p Lp (ξ) = a0 + a1 ξ 1 + a2 ξ 2 + ... + ap ξ p .
(4.51)
Solely the meaning of the discrete degrees of freedom diﬀer from each other. This topic will be dealt with in a subsequent paragraph. Figure 4.9 shows the sets of shape functions for the Lagrange and for the Legendre concept respectively for the approximations p = 1, 2, 3. As it can be seen easily, the shape functions of the Lagrange type fullﬁll the interpolation property Ni (ξk1 ) = δik whereas this is only partially the case for the shape functions of the Legendre type (just for the linear ones). Within the Lagrange concept  contrary to the Legendre concept  the interpolation property can be used for an isoparametric approximation of the geometry, because all nodes are uniquely deﬁned in space. A drawback of this concept is that all the shape functions have to be of the same degree of approximation and therefore have to be newly generated when the approximation order is increased. Regarding the hierarchical concept it can be seen that an increase of the approximation order from p to p + 1 results in generating only one new shape function of degree p+1 whereas the existing shape functions remain unchanged. This is the main characteristic of the hierarchical concept. Figure 4.10 illustrates the structure of the element stiﬀness matrices K and of the internal load vector r of both concepts for diﬀerent polynomial degrees within an arbitrary linear problem. It can be seen that the bandwith of the stiﬀness matrix is smaller in the Legendre concept. Furthermore, the hierarchical structure of the matrices and vectors within the Legendre concept is obvious. In linear problems the
4.2 Numerical Methods 1 Nj [ξ i ] [−]
p=1
Nj [ξ i ] [−]
1 0.5 0 0.5 1
0
0.5 0 0.5
1
1
ξ i [−]
Nj [ξ i ] [−]
Nj [ξ i ] [−]
0.5 0 0.5 0
0.5 0 0.5
1
1
ξ i [−]
0
1
ξ i [−]
1
1 Nj [ξ i ] [−]
Nj [ξ i ] [−]
1
1
1
p=3
0 ξ i [−]
1
p=2
391
0.5 0 0.5 1
0 i
ξ [−] Legendre concept
1
0.5 0 0.5 1
0
1
i
ξ [−] Lagrange concept
Fig. 4.9. Set of hierarchically organized shape functions of the Legendre type for polynomial degrees p = 1, 2, 3 (left), Shape functions of the Lagrange type holding for the interpolation property for polynomial degrees p = 1, 2, 3 (right)
hierarchical concept can be used in adaptive preﬁnements where the quality of the approximation is controlled by an errorestimator or indicator. If a preﬁnement is needed most parts of the existing matrices and vectors can be used and only few entries have to be newly calculated. Furthermore it can be seen that within an onedimensional hierarchical concept the higher order degrees of freedom are decoupled from the other ones, because there are only entries on the main diagonal of the element stiﬀness matrix. In general, in a hierarchical concept the resulting equation system has a better conditioning [245]. The meaning of the nodal degrees of freedom within both concepts is as follows: In the Lagrange concept the nodal values represent the value of the
392
4 Methodological Implementation
p=1 p1 p1
0
0
p1
p1 p1
0
0
p1
0
0 p2
0
p2
0
0
0 p3
p3
p=2
p=3
Legendre discretization
Lagrange discretization
Fig. 4.10. Comparison of the structure of element vectors and matrices for the Legendre and Lagrangeconcept for polyomial degrees p = 1, 2, 3
ﬁeld variable at the corresponding node of the element. Because of the lack of the interpolation property the nodal values of the nodes within the Legendre concept  except the values at the vertices of a brick element  can not be interpreted that way. The higherorder degrees of freedom represent  multiplied with the corresponding shape function  a pfraction of the resulting distribution of the solution. 4.2.4.3.2 3DpFinite Element Method Based on Hierarchical Legendre Polynomials After illustrating the main properties of the onedimensional hierarchical shape function concept in comparison to the Lagrange concept, the approximation techniques of the threedimensional hierarchical shape function concept are described subsequently. The basis of the threedimensional implementation is the brick element illustrated in Figure 4.11. Therein the deﬁnition and numbering of element vertices, edges and faces is given. 4.2.4.3.2.1 Generation of 3DpShape Functions The threedimensional shape functions result from a spatial mutliplication of the onedimensional ones (4.49,4.50), leading to: i ∈ {1, 2, 3, ..., pξ1 + 1} Nl3D (ξ) = Ni1D (ξ 1 )Nj1D (ξ 2 )Nk1D (ξ 3 )
j ∈ {1, 2, 3, ..., pξ2 + 1} k ∈ {1, 2, 3, ..., pξ3 + 1} l ∈ {1, 2, 3, ..., NN 3D }.
(4.52)
4.2 Numerical Methods E3
N8 E6
F6 E4
N5
N6
E12 ξ 3 E9
F1 N4 E5 F5
N1
N7 E7
ξ2 F4 ξ1 E10E 2
E11
N3
F2 E 8
F3 E1
393
N2
Fig. 4.11. 3Dpelement: deﬁnition and numbering of element vertices (Ni ), edges (Ei ) and faces (Fi )
The pξi represent the maximal approximation degree in the direction of the natural element coordinates ξ i . The total number of element nodes NN 3D can be calculated as NN 3D = [pξ1 + 1][pξ2 + 1][pξ3 + 1].
(4.53)
According to the number and type of involved onedimensional shape functions, the threedimensional shape functions can be classiﬁed into four diﬀerent shape function mode groups. Firstly, nodal modes are identical to the standard trilinear Lagrange shape functions
Ni (ξ) =
1 1 + ξi1 ξ 1 1 + ξi2 ξ 2 1 + ξi3 ξ 3 , 8
(4.54)
with ξij = ±1 corresponding to the natural coordinate ξ j of vertice node i, and their value correspond to the value of the ﬁeld variable at the element vertices. Edge modes are a product of two linear and one higherorder polynomial, describing a polynomial approximation at the corresponding edges 2 2 3 3 NpE,1−4 (ξ) = 14 Np1 +1 (ξ 1 ) 1 + ξE ξ 1 + ξE ξ 1 −1 1 1 3 3 NpE,5−8 (ξ) = 14 Np2 +1 (ξ 2 ) 1 + ξE ξ 1 + ξE ξ 2 −1 1 1 2 2 NpE,9−12 (ξ) = 14 Np3 +1 (ξ 3 ) 1 + ξE ξ 1 + ξE ξ , 3 −1
(4.55)
394
4 Methodological Implementation
with pi ∈ {2, 3, ..., pξi }. Furthermore face modes result from a multiplication of one linear function and two higherorder polynomials. Therewith a bipolynomial approximation is realized on the respective faces N F,1−2 (ξ) = 12 Np2 +1 (ξ 2 )Np3 +1 (ξ 3 ) 1 + ξF1 ξ 1 [pξ2 −1][p3 −2]+[p2 −1] (ξ) = 12 Np1 +1 (ξ 1 )Np3 +1 (ξ 3 ) 1 + ξF2 ξ 2 N F,3−4 [pξ1 −1][p3 −2]+[p1 −1] (ξ) = 12 Np1 +1 (ξ 1 )Np2 +1 (ξ 2 ) 1 + ξF3 ξ 3 . N F,5−6 [pξ1 −1][p2 −2]+[p1 −1]
(4.56)
Finally, functional values of internal modes I NNN (ξ) = Np1 +1 (ξ 1 ) Np2 +1 (ξ 2 ) Np3 +1 (ξ 3 ) I
(4.57)
with NNI = pξ1 − 1 pξ2 − 1 [p3 − 2] + pξ1 − 1 [p2 − 2] + [p1 − 1]
(4.58)
vanish at the boundaries of the ﬁnite brick element because they are a product of higherorder polynomials only. In some formulations the latter property is used to eliminate these nodes at the structural level by a static condensation technique to achieve a better condition of the matrices and a better performance at the structural level [245]. The number of nodes within each mode class depends on the order of polynomial degree in the natural coordinates and is calculated by: nodal modes
8
edge modes
4 · [pξ1 − 1] + 4 · [pξ2 − 1] + 4 · [pξ3 − 1]
face modes
2 · [pξ2 − 1] · [pξ3 − 1] + 2 · [pξ1 − 1] · [pξ3 − 1]+
(4.59)
2 · [pξ1 − 1] · [pξ2 − 1] internal modes [pξ1 − 1] · [pξ2 − 1] · [pξ3 − 1]. Figure 4.12 shows several higherorder modes for diﬀerent polynomial degrees. 4.2.4.3.2.2 Spatially Anisotropic Approximation Orders In the subsequent paragraphs approximation orders are written as (•)i,j,k . Therein (•) represents an arbitrary ﬁeld variable, whereas the ith index shows the degree of approximation in the direction of the natural coordinate ξ i .
395
nodal modes
4.2 Numerical Methods
N6
N7
N8
p2
p3
p4
p5
p2,2
p3,2
p3,3
p4,3
p2,2,2
p3,2,2
p3,3,3
p4,3,2
internal modes
face modes
edge modes
N5
Fig. 4.12. 3Dpshape functions: nodal, edge, face and internal modes for diﬀerent polynomial degrees
The 3Dpformulation should allow for the numerical simulation of a wide range of structures, in particular shell structures. This is realized by applying the anisotropic shape functions for the approximation of the displacement ﬁeld. Figure 4.13 illustrates some basic types of structures and how they are modelled by means of the classical ﬁnite element method and by a 3Dpformulation.
396
4 Methodological Implementation
l >> (b, h)
ξ1 up,1,1
truss ξ2 ξ1 h << (l, b) slab
up,p,1
ξ2 ξ1 h << (l, b) plate
up,p,2 / up,p,3
ξ2 ξ1
shell
up,p,2 / up,p,3
compact bodies
up1,p2,p3
structural level
classical FEM
3Dpdiscretization
Fig. 4.13. Structure types and corresponding ﬁnite element models of the classical ﬁnite element approach and using 3Dpelements with spatially anisotropic approximations
On the one hand the classical approach simulates the real structure by special types of ﬁnite elements that represent the structure in an ideal way. Therefore generated ﬁnite elements are used (truss, beam or slab and
4.2 Numerical Methods
397
shell elements). Additionally, there are assumptions concerning kinetics or kinematics that aim towards a further idealization of the mechanical behaviour of the structural element. Wellknown representatives of these element types are Timoshenko or Bernoulliformulations for beams and MindlinReissner or KirchhoffLoveformulations for plates. On the other hand every type of structure can be simulated with threedimensional elements. But a major drawback regarding standard higherorder elements is the considerable increase of element nodes. At that point the beneﬁt of anisotropic ansatz functions becomes clear. In order to simulate the structural element in a more eﬃcient way only the characteristic deformation behaviour is approximated by higherorder approximations. Speaking in the terms of [126], discretizing only the relevant ﬁeld variables by higherorder approximations leads to shelllike, slablike or trusslike solid formulations. For example, a shelllike solid formulation is obtained when the displacement ﬁeld u is approximated either u ≈ up,p,2
or
u ≈ up,p,3 .
(4.60)
Thus in thickness direction only quadratic or cubic approximations are used whereas an even higherorder bipolynomial approximation is used inplane of the shell structure. As an other example, in a trusslike solid formulation the displacement ﬁeld could be approximated as: u ≈ up,1,1 .
(4.61)
Normally in the classical approach truss elements have a constant distribution of all variables across the cross section. Of course the lowest approximation within the solidformulation is a linear one. Additional eﬃciency can be gained by linking the relevant degrees of freedom. In the subsequent paragraphs the beneﬁts on the eﬃciency of this approximation technique will be illustrated. 4.2.4.3.2.3 FieldWise Choice of the Approximation Order After showing how spatially anisotropic shape functions can be applied to simulate diﬀerent types of structures eﬃciently, this subsection is about how environmental loads in the form of additional independent ﬁeld variables, like temperature or pressure ﬁelds, can be eﬃciently discretized and incorporated into the ﬁnite element. The participating ﬁeld variables can have diﬀerent requirements concerning the approximation. As an example, a shell structure loaded thermohygromechanically is investigated. Figure 4.14 points out the subsequent considerations by means of an illustration of an environmentally loaded segment of a structure (left) and an illustration of a possible discretization according solely to the individual ﬁeld requirements. As described in the previous section, the approximation of the displacement ﬁeld for a shell structure should be chosen to up,p,2 or up,p,3 respectively with p > 3. Consequently the inplane approximation is higher than the approximation
398
4 Methodological Implementation moisture cracks
mechanical load
Θ, u, pl u
temperature
Θ, pl
Fig. 4.14. Hygrothermomechanical loading of a structural segment (left), Fieldwise anisotropic discretization using the pﬁnite element method (right)
in thickness direction. Considering the environmental ﬁelds it can be vice versa. Mostly shell structures are subjected to a heat or moisture transport in thickness direction. Therefore the approximation in this primary transport direction has to be higher than the inplane approximation. An anisotropic approximation like Θ1,1,q or pc 1,1,q is recommended. Therein q >> 1 and mostly even q > p to capture the fronts of the environmental variables adequately which play an important role in the transient transport process. At this point it can be seen  and it will be shown subsequently  that by using ﬁeldwise anisotropic approximations a considerable reduction of computing time can be achieved compared to equally high spatially isotropic approximations for all ﬁelds. Up to this point only the individual requirements of all incorporated ﬁelds were taken into account. But concerning the numerical simulation of multiphysics of structures by using an approximative method like the ﬁnite element method, it is required that the FEformulation converges to an unique solution. This requires to fullﬁll the wellknown infsub or rather Babuˇ skaBrezziconditions [71, 154]. In accordance with [124], proving the fullﬁlling of these conditions mathematically is a major task and regarding the numerical level it can only be tested if the numerical model does not fullﬁll the conditions [188]. Consequently the ﬁeldwise anisotropic approximation order has to be wellbalanced. A suitable approach for wellbalanced discretizations are TaylorHoodlike approximations [255, 847]. Within this concept the approximations of environmental variables like temperature or capillary pressure are chosen one degree lower than the approximation of the displacement ﬁeld: pθ = ppc = pu − 1.
(4.62)
399
n
→
∞
4.2 Numerical Methods
n→∞
n→∞
n→
∞
n→∞
Fig. 4.15. Discretization of the standard structures (truss, slab, shell) into an inﬁnite numbers of elements Table 4.2. Total number of geometric entities (vertices, edges, faces) of the discretizations with an inﬁnite number of elements
NNV Eξ1 Eξ2 Eξ3 Fξ1 ξ2 Fξ2 ξ3 Fξ1 ξ3
trussdiscretization 4(n + 1) ≈ 4n 4n 2(n + 1) ≈ 2n 2(n + 1) ≈ 2n 2n n+1≈n 2n
slab/plate/shelldiscretization 2(n + 1)2 ≈ 2n2 2n(n + 1) ≈ 2n2 2n(n + 1) ≈ 2n2 (n + 1)2 ≈ n2 2n2 n(n + 1) ≈ n2 n(n + 1) ≈ n2
At this point some considerations are made concerning the eﬃcieny of the ﬁeldwise spatially anisotropic approximation technique. As a qualitativ indicator of the eﬃciency, the relation between the number of nodes or rather degrees of freedom of a structure is investigated for the proposed discretization method and the standard fully isotropic approximation technique. Therefore, archetypes of structures (truss, slab, plate/shell structures) are considered which are meshed with an inﬁnite numbers of pelements in the dominating spatial dimensions. Figure 4.15 illustrates these archetypes. For slablike, platelike or slablikestructures only one element is used in thickness direction. For plate and shell problems this would be compensated by using a higherorder kinematic in thickness direction. An inﬁnite number is chosen to make the relative values independent of the number of elements. Table 4.2 gives the total numbers of vertice nodes NNV , edges in ξ i direction Eξi and faces in ξ i ξ j plane Fξi ξj for the archetype discretizations. These geometric entities are used to calculate the total number of nodes in the (•) respective mesh. In the following, NNp1,p2,p3 represents the total number of nodes when dealing with the ﬁeld (•) with the underlying approximation order p1, p2, p3.When dealing with the eﬀect of a ﬁeldwise choice of the
400
4 Methodological Implementation
approximation order only one environmental ﬁeld variable will be added (temperature θ), leading to a maximum of 4 degrees of freedom (dof). TrussLike Solid Discretizations Here, the polynomial degree is chosen to be p > 2. The total number of nodes for the spatially anisotropic approximation is u NNp,1,1 = 4n + 4n[p − 1]
(4.63)
whereas the full ansatz space leads to u NNp,p,p = 4n + 8n[p − 1] + 5n[p − 1]2 + n[p − 1]3
(4.64)
nodes. Because the quality of the numerical solution is not changed when using a spatially anisotropic approximation the eﬃciency of this method is indicated by the ratio of the total number of system nodes/dof. The structural eﬃciency and the combined ﬁeldwise and structural eﬃciency for a truss structure is Eﬀu struc =
u NNp,1,1 u NNp,p,p
Eﬀu,θ TH =
u θ 3NNp,1,1 + NNp−1,1,1 , u 4NNp,p,p
(4.65)
whereas TaylorHoodlike approximations were used for the coupled discretization. SlabLike Solid Discretizations Also in this case the approximation order is chosen as p > 1. The total number of nodes for the anisotropic approximation is u NNp,p,1 = 2n2 + 4n2 [p − 1] + 2n2 [p − 1]2
(4.66)
and for the isotropic case: u NNp,p,p = 2n2 + 5n2 [p − 1] + 4n2 [p − 1]2 + n2 [p − 1]3 .
(4.67)
Once more the eﬃciency can be indicated by: Eﬀu struc =
u NNp,p,1 u NNp,p,p
Eﬀu,θ TH =
u θ 3NNp,p,1 + NNp−1,p−1,1 . u 4NNp,p,p
(4.68)
ShellLike Solid Discretizations Concerning shellstructures, possible approximations in thickness direction are ps = 2 or ps = 3. The approximation degree in plane is p > 1 or rather p > ps . As mentioned in the previous section the approximation degree of the environmental variable pe should be chosen relatively high in thickness direction. So, pe >> ps and pe >> p. For simplicity, pe = pθ = 10 subsequently. The total number of nodes for the shelllike solid formulation is:
4.2 Numerical Methods u 2 2 2 s NNp,p,p s = 2n + 4n [p − 1] + n [p − 1]
401
(4.69)
+ 2n [p − 1] + 2n [p − 1][p − 1] + n [p − 1] [p − 1], 2
2
2
s
2
2
s
whereas for the full ansatz space it is: u = 2n2 + 5n2 [p − 1] + 4n2 [p − 1]2 + n2 [p − 1]3 . NNp,p,p
(4.70)
The anisotropic approximation for the thermal ﬁeld needs only Θ u NN1,1,p θ = NN1,1,pθ
(4.71)
nodes. The pure structural eﬃciency can be calculated to Eﬀu struc =
u NNp,p,p s . u NNp,p,p
(4.72)
The TaylorHooddiscretizations are investigated for an approximation degree of ps in thickness direction and the necessary TaylorHood approximation in thickness direction of pθ + 1: Eﬀu,θ =
u θ 3NNp,p,p s + NN 1,1,pθ
4NNpuθ ,pθ ,pθ
Eﬀu,θ TH =
u θ 3NNp,p,p θ +1 + NN1,1,pθ
4NNpuθ +1,pθ +1,pθ +1
.
(4.73) Figure 4.16 shows the aforementioned relations of system nodes/dof for diﬀerent types of structural discretizations. Regarding aspects of eﬃciency, it is stressed here, that the number of system nodes or dof is just an indicator for the real reduction of computing time. In fact, the reduction of nodes is at least proportional to the squared number of nodes because of the quadratic composition of the stiﬀness matrix. Obviously, the largest reduction of computational eﬀorts can be gained with the trusslike solid or slablike solid formulations, because trusslike solid discretization only lead to nodes at the vertices and edges of the element and slablike solid formulations do not contain any internal modes. Furthermore, there is an additional reduction of computing time regarding multiphysics discretizations, which however is getting smaller with increasing p. A considerable reduction of computing time can be gained in analyses of shellstructures when using shelllike solid formulations with a quadratic ps = 2 or cubic ps = 3 approximation of the displacements in thickness direction. This is mainly caused by the considerable reduction of internal modes. Regarding the multiphysics approximation degrees it can be seen in Figure 4.16 d) u,θ that the function of the eﬃciencies EﬀTH1 and Eﬀu,θ TH2 (4.73) in the range θ of p < p is increasing with increasing p, but we also see in this case a considerable reduction of system nodes/dof. For the uncommon case that p > pe = pθ the curves would look similiar to the curves a) to c). Considering these results it can be stated that using a ﬁeldwise, spatially anisotropic
402
4 Methodological Implementation
u,θ Eﬀu struc , EﬀTH
u,θ Eﬀu struc , EﬀTH
0.5 Eﬀu struc Eﬀu,θ TH
0.4 0.3 0.2 0.1
a)
Eﬀu struc Eﬀu,θ TH
0.4 0.3 0.2 0.1
2
3
4 p
5
6
b)
2
3
4 p
5
6
4 p
5
6
0.8 ps = 2 ps = 3 TH
0.2 Eﬀu,θ TH
Eﬀu struc
ps = 2 ps = 3 0.6
0.1
0.4 c)
4
5
6 p
7
8
d)
2
3
Fig. 4.16. Relative reduction of system nodes/dof for a) trusslike solid, b) slablike solid, c) mechanical shelllike solid and d) multiphysics shelllike solid formulation
approximation technique is an adequate way of simulating coupled multiphysics of structures. 4.2.4.3.2.4 Geometry Approximation Inherently meshes for the pﬁnite element method appear to be relatively coarse. Therefore an adequate method for the description of the geometry is needed. An exact representation of the element geometry can be realized by using the blending function method [317, 782]. If this method is coupled with a professional CADenvironment a powerful tool is developed in conjunction with the pﬁnite element method [246]. Despite all the eﬃciency there is a lot of extra work to be done concerning the generation of the connectivities between structural and geometrical entities. Therefore in the presented formulation a subparametric concept is used. The geometry is described by a twentynoded continuum element which allows for a quadratic Serendipity approximation of the element geometry. The data of the geometry approximation is independent of the approximation of the ﬁeld variables. Consequently nodes used for the geometry are not identical to the nodes of the pﬁnite element formulation. This approximation in general holds for accurate results.
4.2 Numerical Methods
403
4.2.5 Solution of Stationary Problems Authored by Detlef Kuhl In the following section the solution of nonlinear time independent durability mechanics is discussed. In particular, iteration methods and control strategies are combined in order to solve the nonlinear vector equation obtained as result of the spatial discretization process. Since this procedures are intensively discussed in textbooks, see e.g. [106, 224, 225, 855], the present section is restricted to a brief summary. 4.2.5.1 Numerical Solution Technique Nonlinear static systems can be expressed in terms of the generalized internal force vector ri , the generalized external force vector r and generalized vector valued variables or displacements u. ri (u) = r
(4.74)
As a basis for the numerical solution of equation (4.74), the consistent linearization of the generalized internal force vector, deﬁning the generalized tangent stiﬀness matrix K, is given. ∂ri (u) = K(u) ∂u
(4.75)
In general, the numerical solution of nonlinear vector equation (4.74) is realized by the combination of two algorithms: • •
The ﬁrst one controls the application of generalized loads r, whereas the second one solves the resulting nonlinear equation ri (u) = r of the particular load step.
In order to control the application of loads, the load factor λ is induced.
ri (un+1 ) = λn+1 r
[0, λ] =
NT −1 A
λn+1 − λn
(4.76)
n=0
Within every load step the nonlinear vector equation (4.76) is solved iteratively. Figures 4.17 and 4.18 illustrate the resulting combination of computational methods to control the application or generalized external loads and the iterative solution of single load steps. 4.2.5.2 Iteration Methods For the explanation of iteration methods it is assumed that the load factor λn+1 is prescribed and un+1 is demanded. This represents the classical
404
4 Methodological Implementation
nonlinear vector equations ri (u) = r parameterized nonlinear vector equations  load factor λ ri (u) = λ r control of load parameter λ NTB −1 [0, λ] = λn+1 − λn n=0
control of
iterative solution of nonlinear vector equations, if λn is changed to λn+1
• • •
• • •
load displacement arclength
NewtonRaphson modiﬁed NewtonRaphson quasi Newton
Fig. 4.17. Strategies for solving nonlinear vector equations ri (u) = r λ λn+1
λ λn+1
iteration
λn
λ2 control parameter load displacement λn arclength
λ1
λ0 u0 u1
u2
un
un+1
u
un = u0n+1
u1n+1
k+1
ukn+1un+1 un+1u
Fig. 4.18. Control of load factor and NewtonRaphson iteration
load controlled analysis. The consideration of variable load factors within the framework of arclength methods is discussed in Section 4.2.5.3. In general, iterative solution methods are based on the Taylor expansion of the vector of generalized internal forces about the trial solution ukn+1 . k ri (uk+1 n+1 ) = ri (un+1 ) +
∂ri (ukn+1 ) k+1 un+1 − ukn+1 + · · · = λn+1 r k ∂un+1
(4.77)
Herein k counts the number of iterations. If the Taylor expansion is truncated after the linear term, the Newton correction k Δu = uk+1 n+1 − un+1
(4.78)
4.2 Numerical Methods
405
Table 4.3. Convergence criteria of iterative solution methods method
residuum
displacement
norm absolute
η¯rk+1 = λn+1 r − ri (uk+1 ¯r η¯uk+1 = n+1 ) ≤ η
norm relative
ηrk+1 =
Δu
≤ η¯u
λn+1 r − ri (uk+1
Δu n+1 ) ≤ ηr ηuk+1 = ≤ ηu
ri (ukn+1 )
uk+1 n+1 − un
components j k+1 η¯r = λn+1 r j − rij (uk+1 ¯r η¯uj k+1 = n+1 ) ≤ η absolute
Δuj 
≤ η¯u
components j k+1 λn+1 r j − rij (uk+1 Δuj  n+1 ) ≤ ηr ηuj k+1 = ≤ ηu ηr = j k j k+1 relative ri (un+1 ) un+1 − ujn 
can be calculated as result of a linear system of equations. ∂ri (ukn+1 ) ∂ukn+1
Δu = K(ukn+1 ) Δu = λn+1 r − ri (ukn+1 )
(4.79)
Finally, the sequence of solutions constitutes an iterative solution of the nonlinear vector equation (4.76). The quality of the approximative solution is checked by several alternative convergence criteria collected in Table 4.3. In summary the iterative solution of a single load step consists of the successive solution of linear systems of equation (4.79) and the convergence check. Several iteration methods are distinguishable by means of the used coeﬃcient matrix K. If K is identiﬁed, as already derived in equation (4.79), by the current generalized tangent stiﬀness matrix, the pure NewtonRaphson method is obtained. If, in contrast to this, K is substituted by the initial tangent stiﬀness matrix of the current load step and an approximated tangent stiﬀness matrix, the modiﬁed NewtonRaphson method and a quasi Newton method are obtained, respectively. Alternative iteration methods ﬁtting equation (4.79) are summarized in Table 4.4. It is obvious that the NewtonRaphson scheme is advantageous with respect to the number of iterations and the modiﬁed NewtonRaphson method is more eﬀective within every iteration step. Since robust and reliable numerical methods are essential to solve highly nonlinear multiphysics problems of the present book, the original NewtonRaphson scheme is preferred. In Figure 4.19 the algorithmic setup of the load controlled NewtonRaphson scheme distinguishing between predictor and corrector iterations is given. According to the solution of the last load step ri (u0n+1 ) = λn r the implementation of the algorithm can be simplyﬁed by using identical predictor and corrector step. Therefore, in Figure 4.19 the frames ri (u0n+1 ) = λn r to u1n+1 = u0n+1 + Δu have to be deleted and the initial iteration counter k = 1 should be substituted by k = 0.
406
4 Methodological Implementation
Table 4.4. Comparison of iteration methods based on the Taylor expansion of nonlinear vector equations ri (un+1 ) = λn+1 r method
iteration with convergence
pure NewtonRaphson
current tangent K(ukn+1 )
quadratic
modiﬁed NewtonRaphson
initial tangent K(un )
non quadratic
quasi Newton
approx. inverse tangent ˆk K
non quadratic
eﬀort of iteration
number of iterations
high generation K(ukn+1 ), ri (ukn+1 ) solution KΔu =λn+1 r−ri low generation ri (ukn+1 ) back substitution KΔu = λn+1 r−ri moderate generation ˆ k , ri (ukn+1 ) K back substitution ˆ k [λn+1 r−ri ] Δu = K
low
high
moderate
loop over load steps n = 0, N T − 1 external load vector and trial solution predictor internal force vector predictor tangent stiﬀness matrix solution of displacement increment update displacement vector
λn+1 r and u0n+1 = un ri (u0n+1 ) = λn r K(u0n+1 ) K(u0n+1 ) Δu = λn+1 r − ri (u0n+1 ) u1n+1 = u0n+1 + Δu
loop over iteration steps k = 1, · · · internal force vector
ri (ukn+1 )
tangent stiﬀness matrix
K(ukn+1 )
solution of displacement increment update displacements check for convergence, e.g.
K(ukn+1 ) Δu = λn+1 r − ri (ukn+1 ) k uk+1 n+1 = un+1 + Δu
ηuk+1 ≤ ηu
k + 1 −→ k n + 1 −→ n Fig. 4.19. Algorithmic setup of the load controlled NewtonRaphson scheme distinguishing between predictor and corrector iterations
4.2 Numerical Methods
407
λ λn +1
λ
pre dic tor
λn+1
constraint f (un+1 , λn+1 ) = 0 equilibrium path ri (u) − λr = 0
s0 s
1 Δλ K(un )
λn
λn Δu Δuλ un+1
un
u
un
u
u1n+1 un + Δuλ
Fig. 4.20. Illustration of arclength methods and predictor step calculation
4.2.5.3 ArcLength Controlled Analysis Arclength controlled iteration methods include the standard displacement and load controlled analyses as special cases. Consequently, only a brief summary of textbooks [832, 817, 224, 673, 855] and papers [91, 219, 222, 220, 221, 303, 304, 438, 655, 656, 677, 678, 679, 719, 829] is given. As a basis of arclength methods the equilibrium path ri (un+1 ) = λn+1 r is enriched by a constraint f in terms of the displacements un+1 and the load factor λn+1 , compare Figure 4.20. ri (un+1 ) − λn+1 r = 0
f (un+1 , λn+1 ) = 0
(4.80)
This means that the load factor is also a variable and the solution should fulﬁll both nonlinear equations. In Figure 4.20 the solution of the extended system (4.80) is given by the intersection of the equilibrium path and the constraint. In consequence of the consistent linearization of equation (4.80) the linear system of equations ⎡ ⎣
K(ukn+1 )
−r
T f,u (ukn+1 , λkn+1 )
f,λ (ukn+1 , λkn+1 )
⎤⎡ ⎦⎣
⎤ Δu
⎡
⎦=⎣
Δλ
λkn+1 r − ri (ukn+1 )
⎤ ⎦
(4.81)
−f (ukn+1 , λkn+1 )
for the solution of the Newton corrections k Δu = uk+1 n+1 − un+1
k Δλ = λk+1 n+1 − λn+1
(4.82)
408
4 Methodological Implementation
is obtained. Since the linear system of equations (4.81) is nonsymmetric and looses, furthermore, the band structure of the generalized tangent matrix, it is solved by applying the partitioning technique [91]. Therefore, the partial incremental solutions Δur and Δuλ are calculated in advance. K−1 (ukn+1 ) Δur = λkn+1 r − ri (ukn+1 ), K−1 (ukn+1 ) Δuλ = r
(4.83)
Afterwards the increments Δu = Δur + Δuλ Δλ
(4.84)
and Δλ = −
f (ukn+1 , λkn+1 ) + f,u (ukn+1 , λkn+1 ) · Δur f,u (ukn+1 , λkn+1 ) · Δuλ + f,λ (ukn+1 , λkn+1 )
(4.85)
are computed. Since this procedure is restricted to the corrector iteration, a specialized predictor step, adopting an user deﬁned step length s, is implemented. As shown in Figure 4.20, the load factor is increased by one and the resulting displacement increment Δuλ and step length s0 are calculated. Δuλ = K−1 (un ) r, s0 = Δuλ · Δuλ + 1 (4.86) Afterwards the increments of the displacement vector and the load factor are scaled, such that the user deﬁned step length s is obtained. Δλ =
s 1 s0
Δu =
s Δuλ s0
(4.87)
Selected constraints within the framework of the present generalized arclength method are summarized in Table 4.5. It is worth to mention that the standard control algorithms used in the present book, namely the displacement and load controlled analyses, are also included in Table 4.5 and the load controlled NewtonRaphson scheme has already been discussed in Section 4.2.5.2. As a particular example of the generalized path following method, the a