Finite-element modelling of structural concrete: short- term static and dynamic loading conditions

Structural Engineering

The book presents a finite-element model of structural concrete under short-term

loading, covering the whole range of short-term loading conditions, from static

(monotonic and cyclic) to dynamic (seismic and impact) cases. Experimental data

on the behaviour of concrete at both the material and structural levels reveal the

unavoidable development of triaxial stress conditions prior to failure which dictate

the collapse and ductility of structural concrete members. Moreover, and in contrast

with generally accepted tenets, it can be shown that the post-peak behaviour of

concrete as a material is realistically described by a complete and immediate loss of

load-carrying capacity. Hence rational analysis and design of concrete components

in accordance with the currently prevailing limit-state philosophy requires the use

of triaxial material data consistent with the notion of a fully brittle material; and this

approach is implemented in the book by outlining a finite-element method for the

prediction of the strength, deformation and cracking patterns of arbitrary structural

concrete forms.

Numerous examples are given that show both the unifying generality of this proposed

approach and the reliability of the ensuing numerical procedure for which the sole

input is the specified uniaxial cylinder compressive strength of concrete and the yield

stress of the steel. This not only offers a better understanding of the phenomenology

of structural concrete behaviour but also illustrates, by means of suitable examples,

the type of revision required for improving design methods in terms of both safety

and economy.

Michael Kotsovos is a senior research fellow, and former professor and head of the

Structures Department, at the National Technical University of Athens.

A SPON PRESS BOOK

Finite￾Element

Modelling

of

Structural

Concrete

Short-Term

Static and

Dynamic Loading

Conditions

Michael D. Kotsovos

Kotsovos Finite-Element Modelling of Structural Concrete

www.sponpress.com

ISBN: 978-1-4987-1230-9

9 781498 712309

90000

K25110

6000 Broken Sound Parkway, NW

Suite 300, Boca Raton, FL 33487

711 Third Avenue

New York, NY 10017

2 Park Square, Milton Park

Abingdon, Oxon OX14 4RN, UK

an informa business

www.crcpress.com

Cover image courtesy of Panayotis Carydis

K25110 mech rev.indd 1 4/23/15 11:36 AM

Finite-Element

Modelling of

Structural Concrete

Short-Term Static and

Dynamic Loading Conditions

A SPON PRESS BOOK

Finite-Element

Modelling of

Structural Concrete

Short-Term Static and

Dynamic Loading Conditions

National Technical University of Athens

Michael D. Kotsovos

CRC Press

Taylor & Francis Group

6000 Broken Sound Parkway NW, Suite 300

Boca Raton, FL 33487-2742

© 2015 by Taylor & Francis Group, LLC

CRC Press is an imprint of Taylor & Francis Group, an Informa business

No claim to original U.S. Government works

Version Date: 20150421

International Standard Book Number-13: 978-1-4987-1231-6 (eBook - PDF)

This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been

made to publish reliable data and information, but the author and publisher cannot assume responsibility for the valid￾ity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright

holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this

form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may

rectify in any future reprint.

Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or uti￾lized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopy￾ing, microfilming, and recording, or in any information storage or retrieval system, without written permission from the

publishers.

For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://

www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923,

978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For

organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged.

Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for

identification and explanation without intent to infringe.

Visit the Taylor & Francis Web site at

http://www.taylorandfrancis.com

and the CRC Press Web site at

http://www.crcpress.com

In memory of Jan Bobrowski

vii

Contents

Preface xiii

Author xv

1 Need for a reappraisal 1

1.1 Physical modelling of structural concrete 1

1.1.1 Typical models 1

1.1.2 Underlying concepts 3

1.1.2.1 Load transfer 3

1.1.2.2 Load-carrying capacity 3

1.1.3 Shortcomings 4

1.1.3.1 Shear capacity 4

1.1.3.2 Flexural capacity 9

1.2 Constitutive modelling 10

1.2.1 Underlying concepts 10

1.2.2 Inconsistencies of concepts underlying constitutive modelling 14

1.3 Concluding remarks 15

References 15

2 Main behavioural characteristics of concrete 19

2.1 Cylinder test 19

2.1.1 Underlying considerations 19

2.1.2 Test results and required clarifications 20

2.2 Post-peak behaviour 22

2.2.1 Uniaxial-compression tests 22

2.2.1.1 Behaviour of a test specimen under compressive load 22

2.2.1.2 Experimental evidence for the brittle nature

of concrete 24

2.2.2 Triaxial compression tests and the effect of tensile stresses 32

2.2.3 Concrete: A brittle fracturing material 34

2.3 Fracture processes in concrete 36

2.3.1 Non-linear behaviour of concrete materials 36

2.3.2 Causes of fracture 37

2.3.3 Fracture mechanism of concrete 37

2.3.3.1 Fracture processes under deviatoric stress 40

viii Contents

2.3.3.2 Fracture processes under hydrostatic stress 45

2.3.3.3 Fracture processes under generalised stress 48

2.4 Failure mechanism in concrete structures 49

2.4.1 A fundamental explanation of failure initiation

based on triaxial material behaviour 49

2.4.2 Triaxiality and failure initiation by macro-cracking:

Some experimental and analytical evidence 53

2.5 Summary of characteristic features of concrete

relevant to modelling material behaviour 65

References 66

3 Modelling of concrete behaviour 69

3.1 Constitutive relations for concrete 69

3.1.1 Experimental data on observed behaviour 71

3.1.1.1 Deformational behaviour during loading 71

3.1.1.2 Deformational behaviour during unloading 76

3.1.2 Mathematical description of deformational behaviour 76

3.1.2.1 Three-moduli approach 76

3.1.2.2 Internal-stress approach 79

3.1.2.3 Combined approach 86

3.1.3 Accuracy of the mathematical model

for the constitutive relations 88

3.2 Strength envelopes for concrete 93

3.2.1 Experimental data on, and mathematical

description of, failure surfaces 93

3.2.2 Accuracy of the mathematical model for the failure surfaces 102

3.3 Deformational and yield characteristics of reinforcing steel 105

3.4 A summary of characteristic features of concrete

relevant to modelling of material behaviour 107

References 110

4 Structure modelling for static problems 113

4.1 Finite-element method 113

4.1.1 Direct formulation of FE characteristics 113

4.1.2 Generalisation to the whole structure 116

4.1.3 Finite elements selected 117

4.2 Non-linear analysis 125

4.2.1 Direct iteration method 125

4.2.2 Newton–Raphson method 126

4.2.3 Modified 127

4.2.4 Generalised Newton–Raphson method 128

4.2.5 Concluding remarks 129

4.3 Non-linear finite element model for structural concrete 130

4.3.1 Background and scope 130

4.3.2 Incremental formulation up to macro-cracking 131

Contents ix

4.3.2.1 Incremental Newton–Raphson method 131

4.3.2.2 Incremental strain–displacement relationships 134

4.3.2.3 Incremental stress–strain relationships

for uncracked concrete 135

4.3.2.4 Incremental force–displacement relationships 136

4.3.2.5 Residual forces 137

4.3.2.6 Convergence and divergence criteria 139

4.3.3 Modelling of macro-cracking 141

4.3.3.1 Nature of structural cracking 141

4.3.3.2 Smeared-crack approach versus the

discrete-crack approach 146

4.3.3.3 Incremental stress–strain relationships

for cracked concrete 148

4.3.3.4 Shear-retention factor: Its role in the conditioning

of stiffness matrices and actual physical evidence 151

4.3.3.5 Macro-crack closure 153

4.3.4 Description of the reinforcing steel 154

4.3.4.1 Incremental relations for the three-node

uniaxial element 154

4.3.4.2 Concrete–steel interaction 155

4.3.5 Overall non-linear strategy 160

4.4 Material and procedural factors influencing FE predictions 161

4.4.1 Material parameters 162

4.4.2 Procedural parameters 163

4.4.3 Tentative recommendations 166

4.5 A brief outline of the smeared-model package 167

References 174

5 Finite-element solutions of static problems 177

5.1 Effect of crack closure on predictions of structural￾concrete behaviour under monotonic loading 178

5.1.1 Background 178

5.1.2 RC beams in shear 178

5.1.3 RC slabs in punching 179

5.1.4 RC structural walls under combined edge

compressive and shear stresses 183

5.2 Performance of structural-concrete members exhibiting

points of contra-flexure under sequential loading 186

5.2.1 Introduction 186

5.2.2 Beam designed in compliance with the Greek code 189

5.2.3 Beam designed in partial compliance with the CFP method 190

5.2.4 Beam designed in full compliance with the CFP method 191

5.2.5 Beams designed in accordance with the European code 192

5.2.6 Safeguarding against shear types of failure 194

5.2.7 Causes of premature failure of beams

designed to the European code 196

x Contents

5.3 RC beam–column joints under cyclic loading 196

5.3.1 Background 196

5.3.2 Structural form investigated 197

5.3.3 FE discretisation 199

5.3.4 Results of analysis and discussion 199

5.3.5 Concluding remarks 205

5.4 Structural walls under cyclic loading 206

5.4.1 Wall details 206

5.4.2 FE modelling 207

5.4.3 Numerical results 209

5.4.4 Discussion of numerical results and

comparison with experimental data 209

5.4.5 Conclusions 215

5.5 Numerical experiments on flat slabs 216

5.5.1 Slabs investigated 216

5.5.2 Mesh discretisation adopted 218

5.5.3 Results of analysis and discussion 219

5.5.4 Concluding remarks 224

References 225

6 Extension of finite element modelling to dynamic problems 227

6.1 Background 227

6.2 Equation of motion 227

6.3 Numerical solution of the equation of motion 227

6.3.1 Explicit method 228

6.3.2 Implicit method 229

6.4 Numerical procedure adopted for structural concrete 231

6.4.1 Newmark family of approximations 231

6.4.2 Stability conditions 233

6.5 Implementation of the dynamic scheme 233

6.6 Verification studies for the dynamic scheme 235

6.7 General remarks 238

References 243

7 Reinforced concrete structural members under earthquake loading 245

7.1 Introduction 245

7.2 Application of the earthquake load 245

7.3 RC columns 247

7.3.1 Design details 247

7.3.2 FE discretisation 248

7.3.3 Static loading 249

7.3.4 Dynamic loading 250

7.3.5 Discussion of the numerical results 251

7.4 RC frames 253

7.4.1 Design details 253

Contents xi

7.4.2 FE discretisation 255

7.4.3 Results of the analysis 255

7.4.3.1 Static loading 256

7.4.3.2 Dynamic loading 257

7.4.3.3 Discussion of results 260

7.5 Three-storey RC wall 262

7.5.1 FE discretisation 265

7.5.2 Results of analysis 266

7.5.2.1 Static loading 266

7.5.2.2 Dynamic loading 266

7.6 Two-level RC frame under seismic action 271

7.6.1 Design details 271

7.6.2 FE modelling 273

7.6.3 Results 274

7.6.4 Discussion of the results 275

7.6.5 Concluding remarks 279

7.7 Effect of the confinement of reinforcement in boundary-column elements

on the behaviour of structural-concrete walls under seismic excitation 279

7.7.1 Background 279

7.7.2 Design details 285

7.7.3 Loading regimes 286

7.7.4 Mesh discretisation adopted 287

7.7.5 Results 288

7.7.6 Discussion of results 289

7.7.6.1 Walls W1 289

7.7.6.2 Walls W2 290

7.7.7 Conclusions 294

7.8 Concluding remarks 294

References 295

8 Structural concrete under impact loading 297

8.1 Introduction 297

8.2 Structural concrete under compressive impact loading 297

8.2.1 Background 297

8.2.2 Experimental information 298

8.2.3 Structural form investigated 300

8.2.4 Numerical modelling of the dynamic problem 301

8.2.5 Numerical predictions 302

8.2.6 Validation of numerical predictions 309

8.2.7 Causes of the loading-rate effect on the behaviour of the specimen 317

8.2.8 Conclusions 320

8.3 Structural concrete under tensile impact loading 321

8.3.1 Background 321

8.3.2 Review of experimental data 321

8.3.3 Structural form investigated and FE modelling 322

8.3.4 Presentation and discussion of the numerical predictions 323

xii Contents

8.3.5 Validation of the numerical predictions 328

8.3.6 Parametric investigation 329

8.3.7 Conclusions 331

8.4 RC beams under impact loading 332

8.4.1 Background 332

8.4.2 Review of experimental data 332

8.4.3 Structural form investigated 335

8.4.4 FE modelling of the problem 338

8.4.5 Static loading 338

8.4.6 Impact loading 338

8.4.6.1 Predicted behaviour 339

8.4.6.2 Causes of beam behaviour 341

8.4.7 Conclusions 342

8.5 Concluding remarks 345

References 345

Appendix A: Octahedral formulation of stresses and strains 349

Appendix B: Coordinate transformations 353

xiii

Preface

Over the years, in collaboration with Milija N. Pavlovic, the author has steadily conducted a

programme of research aimed at rationalising the analysis and design of structural concrete.

This programme of work was preceded by a decade of experimental research at the material

level which furnished the key data input for structural considerations, namely the funda￾mental behaviour of concrete materials under multiaxial stress conditions. Thus, the latter

provided the starting point for a comprehensive investigation into the response of concrete

to loading up to collapse, a study that encompassed the following three basic structural

aspects: numerical modelling by the finite-element method; laboratory testing of structural

members and a consistent design methodology.

This book is concerned with the first of these three fundamental approaches to structural

behaviour; it extends the range of application of the part of the work carried out in the mid￾1990s (Kotsovos M. D. and Pavlovic M. N., Structural Concrete: Finite-Element Analysis

for Limit-State Design, Thomas Telford, 1995) so as to cover structural concrete behaviour

within the whole spectrum of short-term loading ranging from static (monotonic and cyclic)

to dynamic (seismic and impact). The characteristic feature of the work described is that it

contrasts widely accepted tenets in that concrete is considered to be a brittle material and it

is demonstrated that the ductility of concrete structures or members of structures is dictated

by triaxial stress conditions, the latter invariably developing in concrete when its strength is

approached, rather than strain-softening material properties. Some elements of the remain￾ing two structural aspects, namely laboratory testing of structural members and a consistent

design methodology, will also be mentioned whenever the necessary practical supporting

evidence to the theoretical findings may be deemed appropriate; a full outline of the pro￾posed unified design methodology (also based on the concepts underlying the numerical

modelling of structural concrete) has formed the subject of a recent publication (Kotsovos

M. D., Compressive Force-Path Method: Unified Ultimate Limit-State Design for Concrete

Structures, Springer, 2014).

This book, therefore, not only provides the theoretical background of, and justification

for, the latter design approach for ultimate strength but also affords a powerful tool for

both analysis and design of the more complex structural elements for which hand calcula￾tions and/or simplified design rules are not sufficient and which therefore demand a more

formal, rigorous and, indeed, sophisticated computational model such as the one based on

non-linear finite-element analysis described here.

The book is divided into eight chapters. The need for a reappraisal of the concepts that

underlie the methods adopted in practical structural analysis and design is demonstrated

in Chapter 1 through the use of typical study cases highlighting the significance of valid

experimental information on the behaviour of concrete under triaxial stress conditions for

interpreting structural behaviour. Such information is presented in Chapter 2, where the

techniques developed for obtaining valid test data are also discussed, and used for modelling