Computational Structural Concrete: Theory and Applications [2 ed.] 3433033102, 9783433033104

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Ulrich Häussler-Combe

Computational Structural Concrete Theory and Applications Second Edition

Computational Structural Concrete

Computational Structural Concrete Theory and Applications

Ulrich Häussler-Combe

Second enlarged and improved Edition

Author Univ.-Prof. Dr.-Ing. habil. Ulrich Häussler-Combe

Technische Universität Dresden Faculty of Civil Engineering Institute of Concrete Structures 01069 Dresden Germany

All books published by Ernst & Sohn are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate. Library of Congress Card No.:

Cover and copyright:

Cut Concrete Structure (archive Ernst & Sohn GmbH); Interaction Layers (With the use of a figure in “Microplane model m4 for concrete. I. Formulation with work conjugate deviatoric stress, II: Algorithm and calibration” by Zdenek P. Bažant et al., Journal of Engineering Mechanics 126 (2000), pp. 944–980, ASCE.). Photo editing:

Petra Franke, Ernst & Sohn GmbH

applied for British Library Cataloguing-in-Publication Data:

A catalogue record for this book is available from the British Library. Bibliographic information published by the Deutsche Nationalbibliothek

The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.d-nb.de. © 2023 Ernst & Sohn GmbH, Rotherstraße 21, 10245 Berlin, Germany All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form – by photoprinting, microfilm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law. Print ISBN 978-3-433-03310-4 ePDF ISBN 978-3-433-61025-1 ePub ISBN 978-3-433-61024-4 oBook ISBN 978-3-433-61026-8 Typesetting

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Leipzig Printing and Binding Cover design Petra Franke/Ernst & Sohn GmbH

using a design by Sophie Bleifuß, Berlin, Germany Printed on acid-free paper.

V

Preface This book grew out of lectures that the author gave at the Technische Universität Dresden. These lectures were entitled “Computational Methods for Reinforced Concrete Structures” and “Design of Reinforced Concrete Structures.” Reinforced concrete is a composite of concrete and reinforcement connected by bond. Bond is a key item for the behaviour of the composite, which utilises the compressive strength of concrete and the tensile strength of reinforcement while allowing for controlled crack formation. This makes reinforced concrete unique compared to other construction materials such as steel, wood, glass, masonry, plastic materials, fibre reinforced plastics, geomaterials, etc. The theory and use of reinforced concrete in structures falls in the area of structural concrete. Numerical methods like the finite element method, on the other hand, basically allow for a realistic computation of the behaviour of all types of structures. But the implementations are generally presented as black boxes in the view of the users. Input is fed in and the output has to be trusted. The assumptions and methods in-between are not transparent. This book aims to provide transparency with special attention being paid to the unique properties of reinforced concrete structures. Corresponding methods are described with their potentials and limitations while integrating them into the larger framework of computational mechanics connected to reinforced concrete. This is aimed at advanced students of civil and mechanical engineering, academic teachers, designing and supervising engineers involved in complex problems of structural concrete, and researchers and software developers interested in the broader picture. Most of the methods described are complemented with examples computed with a Python software package developed by the author and coworkers. Program package and example data should be available at https://www.concrete-fem.com. The package exclusively uses the methods described in this book. It is open for discussion with the disclosure of the source code and should give stimulation for alternatives and further developments. This book represents a fundamental revision of the book Computational Methods for Reinforced Concrete Structures, which was published in 2014. In particular, the chapter on multi-axial concrete material laws was expanded, and the topics of crack formation and the regularisation of material laws with strain softening were dealt with in a separate chapter. Thanks are given to the publisher

VI

Preface

Ernst & Sohn, Berlin, and in particular to Mrs Claudia Ozimek for the engagement in supporting this work. My education in civil engineering and my professional and academic career were guided by my academic teacher Prof. Dr.-Ing. Dr.-Ing. E.h. Dr. techn. h.c. Josef Eibl 1), former Head of the Department of Concrete Structures at the Institute of Concrete Structures and Building Materials at the Technische Hochschule Karlsruhe (nowadays KIT – Karlsruhe Institute of Technology). Further thanks are given to former coworkers Patrik Pröchtel, Jens Hartig, Mirko Kitzig, Tino Kühn, Joachim Finzel, Tilo Senckpiel-Peters, Daniel Karl, Ahmad Chihadeh, Ammar Siddig Ali Babiker, Evmorfia Panteki, and Alaleh Sehni for their specific contributions. I deeply appreciate the inspiring and collaborative environment of the Institute of Concrete Structures at the Technische Unversität Dresden, which is directed by Prof. Dr.-Ing. Dr.-Ing. E.h. Manfred Curbach. It was my pleasure to teach and research at this institution. And I have to express my deep gratitude to my wife Caroline for her love and patience. Dresden, Spring 2022

1) He passed away in 2018.

Ulrich Häussler-Combe

VII

Contents Preface

V

List of Examples Notation

XIII

XV

1

Introduction 1 Why Read This Book? 1 Topics of the Book 3 How to Read This Book 5

2

Finite Elements Overview 7 Modelling Basics 7 Discretisation Outline 9 Elements 13 Material Behaviour 19 Weak Equilibrium 20 Spatial Discretisation 22 Numerical Integration 24 Equation Solution Methods 26 Nonlinear Algebraic Equations 26 Time Incrementation 28 Discretisation Errors 31

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.8.1 2.8.2 2.9 3

3.1 3.2 3.3 3.4 3.5 3.6 3.7

Uniaxial Reinforced Concrete Behaviour 37 Uniaxial Stress–Strain Behaviour of Concrete 37 Long–Term Behaviour – Creep and Imposed Strains 45 Reinforcing Steel Stress–Strain Behaviour 52 Bond between Concrete and Reinforcement 53 Smeared Crack Model 56 Reinforced Tension Bar 59 Tension Stiffening of Reinforced Bars 64

VIII

Contents

4

4.1 4.1.1 4.1.2 4.1.3 4.2 4.3 4.3.1 4.3.2 4.4 4.4.1 4.4.2 4.4.3 4.4.4 4.5 4.6 4.7 4.8 4.9 4.10 5

5.1 5.2 5.3 5.4 5.5 6

6.1 6.1.1 6.1.2 6.2 6.2.1 6.2.2 6.2.3 6.3 6.3.1 6.3.2 6.3.3

Structural Beams and Frames 67 Cross-Sectional Behaviour 67 Kinematics 67 Linear Elastic Behaviour 70 Cracked Reinforced Concrete Behaviour 71 Equilibrium of Beams 81 Finite Elements for Plane Beams 85 Timoshenko Beam 86 Bernoulli Beam 88 System Building and Solution 91 Integration 91 Transformation and Assembling 93 Kinematic Boundary Conditions and Solution 95 Shear Stiffness 98 Creep of Concrete 101 Temperature and Shrinkage 105 Tension Stiffening 109 Prestressing 112 Large Displacements – Second-Order Analysis 118 Dynamics 126 Strut-and-Tie Models 133 Elastic Plate Solutions 133 Strut-and-Tie Modelling 136 Solution Methods for Trusses 138 Rigid Plastic Truss Models 145 Application Aspects 147

151 Basics 151 Continua and Scales 151 Characteristics of Concrete Behaviour 153 Continuum Mechanics 154 Displacements and Strains 154 Stresses and Material Laws 156 Coordinate Transformations and Principal States 157 Isotropy, Linearity, and Orthotropy 159 Isotropy and Linear Elasticity 159 Orthotropy 161 Plane Stress and Strain 162 Multi-Axial Concrete Behaviour

Contents

6.4 6.4.1 6.4.2 6.4.3 6.4.4 6.5 6.5.1 6.5.2 6.6 6.7 6.8 6.9

Nonlinear Material Behaviour 164 Tangential Stiffness 164 Principal Stress Space and Isotropic Strength 165 Strength of Concrete 168 Nonlinear Material Classification 172 Elasto-Plasticity 173 A Framework for Multi-Axial Elasto-Plasticity 173 Pressure-Dependent Yield Functions 178 Damage 183 Damaged Elasto-Plasticity 190 The Microplane Model 192 General Requirements for Material Laws 199

7

201 Basic Concepts of Crack Modelling 201 Mesh Dependency 205 Regularisation 209 Multi-Axial Smeared Crack Model 216 Gradient Methods 223 Gradient Damage 223 Phase Field 228 Assessment of Gradient Methods 235 Overview of Discrete Crack Modelling 236 The Strong Discontinuity Approach 237 Kinematics 237 Equilibrium and Material Behaviour 240 Coupling 242

7.1 7.2 7.3 7.4 7.5 7.5.1 7.5.2 7.5.3 7.6 7.7 7.7.1 7.7.2 7.7.3 8

8.1 8.1.1 8.1.2 8.1.3 8.2 8.3 8.4 8.5

Crack Modelling and Regularisation

Plates 249 Lower Bound Limit State Analysis 249 General Approach 249 Reinforced Concrete Resistance 250 Reinforcement Design 255 Cracked Concrete Modelling 261 Reinforcement and Bond 266 Integrated Reinforcement 273 Embedded Reinforcement with a Flexible Bond 275

IX

X

Contents

9

9.1 9.2 9.2.1 9.2.2 9.3 9.3.1 9.3.2 9.3.3 9.4 9.5 9.5.1 9.5.2 9.6 9.7 9.7.1 9.7.2 9.8 9.8.1 9.8.2 9.8.3 9.9

285 Classification 285 Cross-Sectional Behaviour 286 Kinematics 286 Internal Forces 288 Equilibrium of Slabs 290 Strong Equilibrium 290 Weak Equilibrium 292 Decoupling 294 Reinforced Concrete Cross-Sections 296 Slab Elements 298 Area Coordinates 298 Triangular Kirchhoff Slab Element 299 System Building and Solution Methods 301 Lower Bound Limit State Analysis 305 Design for Bending 305 Design for Shear 311 Nonlinear Kirchhoff Slabs 314 Basic Approach 314 Simple Moment–Curvature Behaviour 315 Extended Moment–Curvature Behaviour 318 Upper Bound Limit State Analysis 322 Slabs

10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.7.1 10.7.2

Shells 329 Geometry and Displacements 329 Deformations 332 Shell Stresses and Material Laws 335 System Building 337 Slabs and Beams as a Special Case 339 Locking 341 Reinforced Concrete Shells 344 Layer Model 344 Slabs As a Special Case 347

11

Randomness and Reliability

10

11.1 11.2 11.2.1 11.2.2 11.2.3 11.3 11.3.1 11.3.2

353 Uncertainty and Randomness 353 Failure Probability 356 Linear Limit Condition 356 Nonlinear Limit Condition 362 Multiple Limit Conditions 367 Design and Safety Factors 369 Safety Factor Basics 369 Partial Safety Factor Application 373

12

Concluding Remarks

377

Contents

A.1 A.2 A.3 A.4

Appendix A Solution Methods 381 Nonlinear Algebraic Equations 381 Transient Analysis 384 Stiffness for Linear Concrete Compression 386 The Arc Length Method 388 Appendix B Material Stability

391

Appendix C Crack Width Estimation

395

Appendix D Transformations of Coordinate Systems Appendix E Regression Analysis References Index

417

407

405

401

XI

XIII

List of Examples

3.1 3.2 3.3 3.4

Tension bar with localisation Tension bar with creep and imposed strains Simple uniaxial smeared crack model Reinforced concrete tension bar

4.1 4.2 4.3 4.4 4.5

Moment–curvature relations for given normal forces Simple reinforced concrete (RC) beam Creep deformations of RC beam Effect of temperature actions on an RC beam Effect of tension stiffening on an RC beam with external and temperature loading Prestressed RC beam Stability limit of cantilever column Ultimate limit for RC cantilever column Beam under impact load

4.6 4.7 4.8 4.9

Page 41 50 58 61 80 96 103 107 110 116 122 123 128

5.1 Continuous interpolation of stress fields with the quad element 5.2 Deep beam with strut-and-tie model 5.3 Corbel with an elasto-plastic strut-and-tie model

135 141 143

6.1 6.2 6.3 6.4

Mises elasto-plasticity for uniaxial behaviour Uniaxial stress–strain relations with Hsieh–Ting–Chen damage Stability of Hsieh–Ting–Chen uniaxial damage Microplane uniaxial stress–strain relations with de Vree damage

175 186 188 196

7.1 7.2 7.3 7.4 7.5 7.6

Plain concrete plate with notch Plain concrete plate with notch and crack band regularisation 2D smeared crack model with elasticity Gradient damage formulation for the uniaxial tension bar Phase field formulation for the uniaxial tension bar Plain concrete plate with notch and SDA crack modelling

207 210 218 226 232 244

8.1 8.2 8.3 8.4

Reinforcement design for a deep beam with a limit state analysis Simulation of cracked reinforced deep beam Simulation of a single fibre connecting a dissected continuum Reinforced concrete plate with flexible bond

258 269 278 280

XIV

List of Examples

9.1 9.2 9.3 9.4 9.5 9.6 9.7

Linear elastic slab with opening and free edges Reinforcement design for a slab with opening and free edges with a limit state analysis Computation of shear forces and shear design Elasto-plastic slab with opening and free edges Simple RC slab under concentrated loading Simple RC slab with yield line method and distributed loading Simple RC slab with yield line method and concentrated loading

Page 303 309 313 317 320 325 326

10.1 Convergence study for linear simple slab 10.2 Simple RC slab with interaction of normal forces and bending

343 347

11.1 Analytical failure probability of cantilever column 11.2 Approximate failure probability of cantilever column with Monte Carlo integration 11.3 Simple partial safety factor derivation

360 365 373

XV

Notation The same symbols may have different meanings in some cases. But the different meanings are used in different contexts, and misunderstandings should not arise. firstly used General

∙T ∙−1 𝛿∙ 𝛿∙ ∙̃ ∙̇ ∙𝑒

transpose of vector or matrix ∙ inverse of quadratic matrix ∙ virtual variation of ∙, test function solution increment of ∙ within iterations ∙ transformed in (local) coordinate system time derivative of ∙ ∙ related to single finite element

Eq. (2.5) Eq. (2.13) Eq. (2.5) Eq. (2.75) Eq. (6.14) Eq. (2.4) Eq. (2.18)

Normal lowercase italics

𝑎𝑠 𝑏 𝑏𝑤 𝑑 𝑒 𝑓 𝑓𝑐 𝑓𝑐𝑡 𝑓𝑡 𝑓𝑦 𝑓𝐸 , 𝑓𝑅 𝑔𝑓 ℎ 𝑚𝑥 , 𝑚𝑦 , 𝑚𝑥𝑦 𝑛 𝑛𝐸 𝑛𝑖 𝑛𝑁 𝑛𝑥 , 𝑛𝑦 , 𝑛𝑥𝑦 𝑝 𝑝𝑓 𝑝𝑥 , 𝑝𝑧

reinforcement cross-section per unit width cross-section width crack band width cross-section effective height element index strength condition uniaxial compressive strength of concrete (unsigned) uniaxial tensile strength of concrete uniaxial failure stress of reinforcement uniaxial yield stress of reinforcement probability density functions of random variables 𝐸, 𝑅 specific crack energy per unit volume cross-section geometric height moments per unit width total number of degrees of freedom in a discretised system total number of elements order of Gauss integration total number of nodes normal forces per unit width pressure failure probability loading distributed along beam

Eq. (9.61) Eq. (4.9) Eq. (3.6) Eq. (9.67) Eq. (2.18) Eq. (6.48) Eq. (3.2) Eq. (3.4) Eq. (3.41) Eq. (2.48) Eqs. (11.2), (11.3) Eq. (3.7) Eq. (4.10) Eq. (9.7) Eq. (2.70) Section 4.3 Eq. (2.69) Section 4.3 Eq. (9.7) Eq. (6.8) Eq. (11.19) Eq. (4.49)

XVI

Notation

𝑟, 𝑠, 𝑡 𝑠 𝑠𝑏𝑓 𝑠𝑏 max 𝑡 𝑡𝑥 , 𝑡𝑦 , 𝑡𝑥𝑦 𝑢𝑖 𝑣𝑥 , 𝑣𝑦 𝑤 𝑤 𝑤𝑐𝑟 𝑥, 𝑦, 𝑧 𝑥, 𝑥 𝑧, 𝑧

local spatial coordinates slip slip at residual bond strength slip at bond strength clock time or loading time couple force resultants per unit width 𝑖-th displacement component shear forces per unit width deflection fictitious crack width critical crack width global spatial coordinates compression zone height internal lever arm

firstly used Eq. (2.15) Section 3.4 Section 3.4 Section 3.4 Eq. (2.4) Eq. (9.58) Eq. (6.1) Eq. (9.7) Eq. (2.56) Eq. (3.5) Eq. (3.9) Eq. (2.14) Eqs. (4.29), (9.66) Eqs. (4.115), (9.58)

Bold lowercase roman

b f p n s t t𝑏 t𝑐𝐿 t𝑐 u 𝝊 𝝊𝑒 w𝑐𝐿 w𝑐

body forces internal nodal forces external nodal forces normal vector slip surface tractions bond force crack traction in local system crack traction in global system displacement field nodal displacement vector nodal displacement vector related to a single element fictitious crack width in local system fictitious crack width in global system

Eq. (2.5) Eq. (2.9) Eq. (2.9) Eq. (6.5) Eq. (8.53) Eq. (2.5) Eq. (8.54) Eq. (7.3) Eq. (7.133) Eq. (2.1) Eq. (2.1) Eq. (2.18) Eq. (7.2) Eq. (7.133)

Normal uppercase italics

𝐴 𝐴𝑠 𝐴𝑡 𝐴𝑢 𝐶 𝐶𝑇 𝐷 𝐸 𝐸0 𝐸𝑐 𝐸𝑠 𝐸𝑇 𝐹 𝐹 𝐹𝐸 𝐺

cross-sectional area of a bar or beam cross-sectional area reinforcement part of surface with prescribed tractions part of surface with prescribed displacements material stiffness coefficient tangential material stiffness coefficient scalar damage variable Young’s modulus initial Young’s modulus initial Young’s modulus of concrete initial Young’s modulus of steel tangential hardening material stiffness coefficient yield function damage function distribution function of random variable 𝐸 shear modulus

Eq. (2.54) Section 3.6 Eq. (2.5) Eq. (2.53) Eq. (3.35) Eq. (3.37) Eq. (6.105) Eq. (2.43) Eq. (3.16) Eq. (3.1) Eq. (3.41) Eq. (3.41) Eq. (6.64) Eq. (6.108) Eq. (11.1) Eq. (4.8)

Notation

𝐺 𝐺𝑓 𝐼1 𝐽 𝐽2 , 𝐽3 𝐾 𝐿𝑐 𝐿𝑒 𝑀 𝑁 𝑃 𝑇 𝑉 𝑉

flow potential specific crack energy per surface first invariant of stress determinant of Jacobian matrix second, third invariant of stress deviator slab bending stiffness characteristic length of an element length of bar or beam element bending moment normal force probability natural period shear force volume

firstly used Eq. (6.63) Eq. (3.8) Eq. (6.19) Eq. (2.37) Eq. (6.19) Eq. (9.12) Eq. (7.18) Eq. (2.23) Eq. (4.9) Eq. (4.9) Eq. (11.1) Eq. (4.209) Eq. (4.9) Eq. (2.5)

Bold uppercase roman

B C C𝑇 C𝑐𝐿𝑇 D D𝑇 D𝑐𝑇 D𝑐𝐿𝑇 E G1 , G2 , G3 G1 , G2 , G3 I J K K𝑇 M N Q T V𝑛 V𝛼 , V𝛽

matrix of spatial derivatives of trial functions material stiffness matrix tangential material stiffness matrix tangential local crack stiffness matrix material compliance matrix tangential material compliance matrix tangential crack band compliance matrix tangential local crack compliance matrix isotropic linear elastic material stiffness matrix unit vectors of covariant system unit vectors of contravariant system unit matrix Jacobian matrix stiffness matrix tangential stiffness matrix mass matrix matrix of trial functions stress / strain rotation matrix element rotation matrix shell director unit vectors of local shell system

Eq. (2.2) Eq. (2.47) Eq. (2.50) Eq. (7.9) Eq. (2.51) Eq. (2.51) Eq. (7.38) Eq. (7.9) Eq. (6.23) Eq. (10.16) Eq. (10.17) Eq. (6.100) Eq. (2.20) Eq. (2.11) Eq. (2.67) Eq. (2.61) Eq. (2.1) Eq. (6.14) Eq. (4.105) Eq. (10.2) Eqs. (10.2), (10.3)

Normal lowercase Greek

𝛼 𝛼𝐸 , 𝛼𝑅 𝛼𝑠 𝛽 𝛽𝑡 𝜖 𝜖 𝜖1 , 𝜖2 , 𝜖3 𝜖𝑐𝑡

for several purposes in a local context sensitivity parameters shear retention factor for several purposes in a local context tension stiffening coefficient uniaxial strain strain of a beam reference axis principal strains concrete strain at uniaxial tensile strength

Eq. (11.14) Eq. (7.7) Eq. (3.65) Eq. (2.43) Eq. (4.4) Section 6.2.3 Figure 3.3

XVII

XVIII

Notation

𝜖𝑐𝑢 𝜖𝑐1 𝜖𝑐𝑢1 𝜖𝐼 𝜖𝑉 𝜙 𝜙 𝜑 𝜑𝑐 𝜑𝑠 𝛾 𝛾𝐸 , 𝛾𝑅 𝜅 𝜅𝑝 𝜅𝑑 𝜇𝑅 , 𝜇𝐸 𝜈 𝜈𝑅 , 𝜈𝐸 𝜃 𝜗 𝜌 𝜌𝑠 𝜚𝑠 𝜎 𝜎1 , 𝜎2 , 𝜎3 𝜎𝑅 , 𝜎𝐸 𝜏 𝜏 𝜏𝑏𝑓 𝜏𝑏 max 𝜔 𝜔 𝜉

concrete failure strain at uniaxial tension concrete strain at uniaxial compressive strength (signed) concrete failure strain at uniaxial compression (signed) imposed uniaxial strain volumetric strain cross-section rotation angle of external friction for several purposes in a local context orientation of concrete principal compression orientation of reinforcement shear angle partial safety factors curvature internal state variable for plasticity internal state variable for damage means of random variables 𝑅 and 𝐸 Poisson’s ratio coefficients of variation Lode angle angle of internal friction deviatoric length reinforcement ratio specific mass uniaxial stress principal stresses standard deviations of random variables 𝑅, 𝐸 bond stress for several purposes in a local context residual bond strength bond strength circular natural frequency related crack width hydrostatic length

firstly used Figure 3.3 Eq. (3.1) Eq. (3.1) Eq. (3.35) Eq. (6.101) Eq. (4.1) Eq. (6.90) Eq. (8.5) Eq. (8.6) Eq. (4.1) Eqs. (11.58), (11.59) Eq. (4.4) Eq. (6.64) Eq. (6.107) Eqs ((11.3), (11.6)) Eq. (2.44) Eq. (11.60) Eq. (6.45) Eq. (6.88) Eq. (6.44) Eq. (8.8) Eq. (2.52) Eq. (2.43) Section 6.2.3 Eqs. (11.3), (11.6) Eq. (3.47) Figure 3.13 Figure 3.13 Eq. (4.209) Eq. (7.5) Eq. (6.43)

Bold lowercase Greek

𝝐 𝝐 𝝐𝑝 𝜿 𝝈 𝝈 𝝈′

small strain generalised strain small plastic strain vector of internal state variables Cauchy stress generalised stress deviatoric part of Cauchy stress

Eq. (2.2) Eq. (2.33) Eq. (6.61) Eq. (6.38) Eq. (2.3) Eq. (2.34) Eq. (6.9)

Uppercase Greek

Φ 𝚺

standardised normal distribution function stress extension

Eq. (11.20) Eq. (2.82)

1

1 Introduction

Why Read This Book? Concrete is by far the most used building material in the world. Concrete can be given arbitrary forms, its basic constituents are available everywhere, its processing is basically simple, and it is inexpensive. Furthermore, concrete can be customised to fulfil special requirements – e.g. high strength, resistance in rough environments, impermeability, ductility – through adjustment of binder, aggregates, fibres, and additives. Its major characteristic from a mechanical point of view is given by a relatively high compressive strength but a low tensile strength. Thus, it is reinforced with bars, wire mats, fabrics of steel, carbon, glass, and more, which leads to an immense variety of composite building materials. With this we see architectural landmark buildings like the television tower in Stuttgart, Germany, the first of this type designed and engineered by Fritz Leonhardt and built in 1956, Figure 1.1a, the Palazzetto dello Sport in Rome, Italy, a coliseum for the Olympic games 1960 built in 1956 and engineered by Pier Lucri Nervi, Figure 1.1b, the Ganter bridge within the access road to the Simplon pass in the Swiss Alps built in 1980 and designed and engineered by Christian Menn, Figure 1.2a, and

(a)

(b)

Figure 1.1 (a) Stuttgart television tower, from Kleinmanns and Weber (2009), photography: Landesmedienzentrum Baden-Württemberg: Albrecht Brugger. (b) Palazetto dello Sport, from Ehmann and Pfeffer (1999).

Computational Structural Concrete – Theory and Applications, Second Edition. Ulrich Häussler-Combe. © 2023 Ernst & Sohn GmbH. Published 2023 by Ernst & Sohn GmbH.

2

1 Introduction

(a)

(b)

Figure 1.2 (a) Ganter bridge, from Billington (2014), photography: Nicolas Janberg. (b) National Veterans Memorial and Museum, from Helbig et al. (2020), photography: Knippers Helbig Stuttgart – New York – Berlin.

(a)

(b)

Figure 1.3 (a) Office building: Züblin-Haus, from Bachmann et al. (2021). (b) High-speed railway viaduct over the valley Unstruttal, Germany, photomontage, from Schenkel et al. (2009).

the National Veterans Memorial and Museum, Columbus, Ohio, USA, built in 2018 and engineered by Knippers Helbig, Figure 1.2b, to mention only a few. A countless number of concrete buildings contribute to everyday life; for example, office buildings, Figure 1.3a (Züblin headquarters, Stuttgart, Germany; precast concrete with steel–glass atrium), railway bridges, Figure 1.3b (Unstruttal viaduct, Thuringia, Germany), power plants, Figure 1.4a (RWE, Niederaußem, Germany), station concourses, Figure 1.4b (Stuttgart 21, Germany; final state visualisation, still under construction). This demonstrates some visible contributions of the application of concrete. Indispensable infrastructures providing freshwater, drainage, and wastewater processing, waste disposal processing in general, generation and provision of electricity, support of transport via vehicles, trains, ships, and airplanes are generally hidden from immediate visibility. To sum it up, today’s civilisation would be unthinkable without concrete as a building material. ◀

It can be stated that reinforced concrete is the building material of the twentieth century. But will it also be the building material of the twenty-first century?

Presumably yes, due to its advantages listed above. But sustainability has to become a predominant topic also for reinforced concrete besides bearing capacity, usabili-

1 Introduction

(a)

(b)

Figure 1.4 (a) Power plant, RWE, Niederaußem, Germany, from Krätzig et al. (2007), photography: RWE. (b) Underground station concourse, Stuttgart 21, from Bechmann et al. (2019), visualisation: Ingenhoven Architekten, Düsseldorf.

ty, and durability. Production of cement – the predominant binder for concrete – causes a high output of CO2 due to its energy consumption on the one hand and chemical conversion processes on the other hand. The same also applies to reinforcing steel whereupon its contribution to reinforced concrete is relatively small measured by weight ratio. Construction waste makes up the largest proportion of the total amount of waste. What is the conclusion? ◀

We have to use less concrete and fewer reinforcement materials and at the same time achieve a higher quality of building components.

Structural design plays a key role to reach this goal. We should gain a better understanding of load carrying mechanisms of building components in order to fully utilise load bearing potentials and to optimise structural forms and materials. There is still a lot of room for improvement in this regard. Computational methods are an extremely important tool for this. Numerical simulation in combination with experimental investigations allows for a comprehensive understanding of the deformation behaviour, force flow, and failure mechanisms of building components. This permits weak points to be identified and eliminated in a targeted manner. New concepts may be initiated, and a simulation-based rapid prototyping may be performed for initial assessments of new innovative structural forms and materials. On the basis of the knowledge gained from this, the design and elaboration of components in building practice can be carried out more efficiently and with higher quality using computational methods.

Topics of the Book Such methods are generally demanding with respect to methodology, implementation, and application. This is especially true for nonlinear problems as are typical for structural concrete. Computational methods for nonlinear structural analysis offer a wide range of capabilities. But they are made available to users as black boxes.

3

4

1 Introduction

This hides the fact that numerical methods usually have application limits. If these are not observed, the results become questionable. Often, this is not obvious to users providing input for black boxes and accepting output without hesitation. This motivates the goals and contents of this textbook about computational methods – in particular, the finite element method (FEM) – for reinforced concrete (RC): • Survey of the key aspects of the FEM. • Understanding of basic mechanisms of RC regarding interaction of concrete and reinforcement through bond. • Specifics of FEM regarding structural elements like RC-beams, plates, slabs, and shells. • Essential characteristics of the multi-axial mechanical behaviour of concrete. • Pitfalls related to FEM treating structural concrete and in particular the failure behaviour. Knowing these issues, the black boxes should become more transparent, and their results should be better comprehensible. The finite element method is the preferred method also for the computation of reinforced concrete structures due to its versatility and adaptability. Chapter 2 gives an overview of modelling in general and summarises items of FEM as far as is required for its application to reinforced concrete structures. Chapter 3 describes basic mechanisms of structural concrete, which relies on the interaction of concrete and reinforcement by continuous transfer of forces through bond. This is restricted to uniaxial behaviour in a first approach to point out essential properties and describes the mechanisms of the reinforced uniaxial tension bar as prototype of structural concrete. In Chapter 4, this is extended to reinforced concrete beams and frames, which are characterised by bending that may be superimposed with normal forces whereby still basing on uniaxial behaviour of materials. This also includes first aspects of creep, temperature, and shrinkage. Furthermore, prestressing of beams is treated, which is an important technology to extend the application range of reinforced concrete. The chapter closes with the analysis of large displacements and dynamics, exemplarily in each case with their application to beams. A first extension of bending of beams to high beams and plates is given in Chapter 5 with strut-and-tie models, which utilise the uniaxial behaviour of concrete and reinforcement for a design of plane structures with in-plane loading. Furthermore, limit theorems of plasticity – which are an important basis for design in structural concrete – are exemplarily developed within this context. Chapter 6 treats multi-axial concrete behaviour as extension of the uniaxial approach applied in the foregoing chapters. Multi-axial material concrete models are the basis for the structural models for plates, slabs, and shells treated in the following chapters. Basic topics of continuum mechanics are described in as far as they are necessary to understand multiaxial nonlinear stress–strain and failure behaviour of concrete. Material models like elasto-plasticity, damage, and microplane are applied with respect to concrete modelling. A major item regarding material modelling occurs with strain softening – increasing strains with decreasing stresses – which requires a regularisation to reach reliable numerical solution. A further major item concerns the cracking of concrete,

1 Introduction

which separates parts of a continuum into a discontinuum. This couples discretisation issues with material modelling and is described in Chapter 7. Chapter 8 treats design and simulation of reinforced concrete plates with high beams as a special but common case. In this respect, the design is considered separately, as it may be based on linear solutions for plate stresses utilizing a limit theorem of plasticity. On the other hand, simulation considers nonlinear stress–strain relations additionally leading to solutions for the deformation behaviour. Reinforced concrete slabs, which are treated in Chapter 9, extend uniaxial 1D-bending of beams into biaxial 2D-bending. As before with plates, aspects of design and simulation may be separated in an analogous manner. The most general approach for structural analysis is given with shells, which combine in-plane actions of plates and transverse actions of slabs whereby extending flat geometries to folded or curved geometries. Shells require complex mechanical models, which is exemplarily treated in Chapter 10 together with the application to reinforced concrete. Chapter 11 treats first aspects of randomness, which is a major topic regarding structural concrete behaviour. Deterministic models – however sophisticated they may be – always give a more or less restricted view of the real world. First notions of an extended view are given in this chapter. Finally, a number of topics are treated in the appendices insofar they are reasonable for better understanding of the main text but might disturb the line of concise arguing therein.

How to Read This Book The treatment of the above combines methods of mechanics, structural analysis, and applied mathematics. This recourse should be self-explanatory and conclusive to a large degree, so that a study of accompanying literature is generally not required. In doing so, essential lines of development are worked out on the one hand, but on the other hand, the available concepts and methods cannot be described with all details. Furthermore, not every problem addressed is provided with a comprehensive solution. The book is intended to encourage the reader to deepen and explore such topics independently. Nevertheless, the book involves a large volume. Proposals for shorter tracks are given in the following thereby also enlightening the structure of the book content and the relations between sections. Major groups are characterised as • • • •

FEM and reinforced concrete bases, see Figure 1.5a. Uniaxial structures, see Figure 1.5b. Multi-axial concrete and its implications for numerical methods, see Figure 1.6a. Multi-axial structures such as plates, slabs, and shells, see Figure 1.6b.

This includes a short track (left column) and branches (right column) for each of these. Chapter 11.1 Randomness and Reliability falls out of this scheme. Nevertheless, basic knowledge of stochastics related to reinforced concrete is considered necessary. Many topics are illustrated with examples. Most of them are computational and are processed with the Python 3.6 program package ConFem. A few are performed

5

6

1 Introduction 4.1 Cross-sectional Behavior

2.1 Modeling Basics 2.2 Discretisation Outline

2.3 Elements

2.5 Weak Equilibrium 3.1 Uniaxial Stress-Strain of Behavior of Concrete 3.3 Reinforcing Steel Stress-Strain Behavior

4.2 Equilibrium of Beams 4.3 Finite Elements for Beams

2.4 Material Behavior 2.6 2.7 2.8 3.2

Numerical Integration Eq. Solution Methods Discretisation Errors Creep and Imposed Strains

3.4 Bond between Concrete and Reinforcement

3.5 Smeared Crack Model

3.6 The Reinforced Tension Bar

3.7 Tension Stiffening of Reinforced Bar

4.4 System Building & Solution

4.5 Creeo of Concrete 4.6 Temperature and Shrinkage 4.7 Tension Stiffening 4.8 Prestressing 4.9 Large Displacements 4.10 Dynamics

5.1 Elastic Plate Solutions 5.2 Strut & Tie Modeling

(a)

5.3 Solution Methods Trusses

5.4 Rigid Plastic Models 5.5 Application Aspects

(b)

Figure 1.5 (a) FEM and reinforced concrete bases. (b) Uniaxial structures. 6.1 Basics Concrete Behaviour 6.3 Isotropy, Linearity, Orthotropy 6.4 Nonlinear Material Behaviour 6.6 Damage 6.7 Damaged Elasto-plasticity 7.1 Basic Concepts Crack Modeling 7.2 Mesh Dependency 7.3 Regularization

(a)

Plates

6.2 Continuum Mechanics 6.5 Elasto-plasticity

6.8 Microplane Model 6.9 General Requirements

7.4 Multi-axial Smeared Crack 7.5 Gradient Methods Requirements 7.6 Strong Discontinuity Approach

8.1 Lower Bound Analysis 8.3 Reinforcement and Bond 9.1 9.2 9.3 9.5 9.6

8.2 Cracked Concrete 8.4 Integrated Reinforcement 8.5 Embedded Reinforcement

Slabs Classification Cross-sectional Behaviour 9.4 Slab Elements Equilibrium of Slabs System Building & Solution 9.7 Nonlinear Kirchhoff Lower Bound Analysis

10.1 Geometry and Displacements 10.4 System Building

Slabs 9.8 Upper Bound Analysis

Shells

10.2 Deformations 10.3 Stresses and Material 10.5 Special Cases 10.6 Locking

10.7 Reinforced Concrete Shells

(b)

Figure 1.6 (a) Multi-axial concrete and its implications. (b) Multi-axial structures.

with stand-alone Python scripts or are short, illustrating theoretical derivations. Environments to perform Python are freely available on the internet for all common platforms. ◀

All Python sources for ConFem, a basic documentation, example input data, and reference result data are available at https://www.concrete-fem.com under open-source conditions.

Thus, all book examples should be reproducible by the reader. But the ConFem project is not finished and may be subject to continuous development. The user should see it as an inspiring challenge to master this tool. The interplay of theory, implementation, and application – possibly with overcoming resistance – ultimately leads to a deeper understanding of numerical methods,structural concrete, and their dependencies.

7

2 Finite Elements Overview Numerical methods like the finite element method are outstanding as engineering tools but actually have to be embedded in a larger frame of modelling. Discretisation is a key therein, whereby an infinite number of unknowns is reduced to a finite number. On the one hand, this is based on cornerstones of structural analysis like equilibrium, kinematic compatibility, and material behaviour, and on the other hand, methods of numerical mathematics have to be used to reach solutions. An overview is given in the following in as far it is necessary for the application to structural concrete.

2.1 Modelling Basics There are no exact answers. Just bad ones, good ones and better ones. Engineering is the art of approximation. Approximation is performed with models. We consider a reality of interest, e.g. a concrete beam. In a first view, it has properties such as dimensions, colour, and surface texture. From the view of structural analysis, the latter are irrelevant. A more detailed inspection reveals a lot more properties: weight, displacements, stiffness, strength, temperatures, conductivities, capacities, and so on. Only a part of these is essential from the structural point of view. We combine those essential properties to form a conceptual model. Whether a property is essential is obvious for some, but the valuation of others might be doubtful. We have to choose. By choosing properties our model becomes an approximation compared to reality. Approximations are more or less accurate. On the one hand, we should reduce the number of properties of a model. Any reduction of properties will make a model less accurate. Nevertheless, it might remain a good model. On the other hand, an over-reduction of properties will make a model inaccurate and therefore useless. Furthermore, maybe properties that have no counterparts in the reality of interest are introduced. Conceptual modelling is the art of choosing properties. As all other arts, it cannot be performed guided by strict rules. The chosen properties have to be related to each other in a quantitative manner. This leads to a mathematical model. In many cases, we have systems of differenComputational Structural Concrete – Theory and Applications, Second Edition. Ulrich Häussler-Combe. © 2023 Ernst & Sohn GmbH. Published 2023 by Ernst & Sohn GmbH.

8

2 Finite Elements Overview

(a)

(b)

Figure 2.1 Modelling. (a) Type of models following Schwer (2007). (b) Relations between model and reality.

tial equations relating variable properties or variables. After prescribing appropriate boundary and initial conditions an exact, unique solution should exist for variables depending on spatial coordinates and time. Thus, a particular variable forms a field. Such fields of variables are infinite, as space coordinates and instants of time are infinite although being bounded. As analytical solutions are not available in many cases, a discretisation is performed to obtain approximate numerical solutions. Discretisation reduces underlying infinite space and time into a finite number of supporting points in space and time and maps differential equations into algebraic equations connecting a finite number of variables. This leads to a numerical model. A numerical model needs a completion by means of programming to form a computational model. Finally, programs yield solutions through processing by computers. The whole cycle is shown in Figure 2.1. The sequence of partial models forms the model as a whole. A final solution provided after computer processing is approximate compared to the exact solution of the underlying mathematical model. This is caused by discretisation and round-off errors. Let us assume that we can minimise this mathematical approximation error in some sense and consider the final solution as a model solution. Nevertheless, the relation between the model solution and the underlying reality of interest remains an issue. Both – model and reality of interest – share the same properties by definition or conceptual modelling, respectively. Let us also assume that the real data of properties can be objectively determined, e.g. by measurements. Thus, real data of variables should be approximated by their computed model counterparts at least. The difference between model solution data and real data yields a modelling error. In order to distinguish between bad (inaccurate), good (accurate), and better model solutions, we have to choose a reference to measure the modelling error. This choice has to be made within a larger context, allows for a margin of discretion, and again is not guided by strict rules. Furthermore, the reference may shift while getting improved model solutions due to methodical progress or a better data survey. A bad model solution may be caused by a bad model – bad choice of properties, poor relations of properties, insufficient discretisation, programming errors – or by incorrect model parameters. Parameters are those properties that are assumed to be known in advance for a particular problem and are not subject to a computation.

2.2 Discretisation Outline

Under the assumption of a good model, the model parameters can be corrected by calibration. This is based upon appropriate problems from the reality of interest with known real data. On the one hand, calibration minimises the modelling error by adjustment of parameters. On the other hand, validation examines similar problems with modified parameters and known real data and assesses the modelling error. A proper calibration generally does not guarantee a successful validation. Regarding reinforced concrete structures, calibrations usually involve the adaption of material parameters like strength and stiffness as part of material models. These parameters are chosen such that the behaviour of material specimen observed in experiments is reproduced. A validation is usually performed with structural elements such as bars, beams, plates, and slabs. Computational results of structural models are compared with the corresponding experimental data. This may lead to basic peculiarities. Reproducible experiments performed with structural elements are of a small, simplified format compared with complex unique buildings. Furthermore, repeated experimental tests with the same nominal parameters exhibit scattering results. Standardised benchmark tests carving out different aspects of reinforced concrete behaviour are required. Actually, agreements about such benchmark tests exist only to a limited extent. Regarding a particular structural problem a corresponding model has to be validated on a case-by-case strategy using adequate experimental investigations. Again, there are no strict rules like for the preceding arts. Complex proceedings have been sketched hitherto outlining a model of modelling; see also Babuska and Oden (2004) for a more comprehensive discussion. Some benefit is finally expected. Thus, a model that passes validations is usable for predictions. Structures built along such predictions, hopefully, prove their high quality in the reality of interest. This textbook covers the range of conceptual models, mathematical models, and numerical models with special attention being paid to reinforced concrete structures. The computational model with Python-sources is available under opensource conditions at https://www.concrete-fem.com. A major aspect of the following is the modelling of ultimate limit states: states with maximum bearable loading or acceptable deformations and displacements in relation to failure. Another aspect is given with serviceability: deformations, and in some cases oscillations, of structures have to be limited to allow their proper usage and fulfilment of intended services. Durability is a third important aspect: deterioration of materials through, e.g. corrosion, has to be controlled. This is connected to cracking and crack width in the case of reinforced concrete structures. These topics are also treated in the following.

2.2 Discretisation Outline The finite element method (FEM) is the predominant method to derive numerical models from mathematical models. Its basic theory is described in the following sections of this chapter insofar as it is needed for its application to different types of structures with reinforced concrete in the following chapters.

9

10

2 Finite Elements Overview

Figure 2.2 Model of a plate.

(a)

(b)

Figure 2.3 (a) Elements and nodes (deformed). (b) Nodal quantities.

The underlying mathematical model is defined in one-, two-, or three-dimensional fields of space related to a body and one-dimensional space of time. A body undergoes deformations during time due to loading. We consider a simple example with a plate defined in 2D space, see Figure 2.2. Loading is generally defined depending on time, whereby time may be replaced by a loading factor or an equivalent loading time in the case of quasi-static problems. Field variables depending on spatial coordinates and time are, e.g. given by the displacements. • Such fields are discretised by dividing space into elements that are connected by nodes, see Figure 2.3a. Elements adjoin but do not overlap and fill out the space of the body under consideration. • Discretisation basically means interpolation, i.e. displacements within an element are interpolated using the values at nodes belonging to the particular element. In the following, this is written as u=N⋅𝝊

(2.1)

with the displacements u depending on spatial coordinates and time, a matrix N of trial functions depending on spatial coordinates, and a vector 𝝊 depending on time and collecting all displacements at nodes.

2.2 Discretisation Outline

The number of components of 𝝊 is 𝑛. It is two times the number of nodes in the case of the plate, as the displacement u has components 𝑢𝑥 , 𝑢𝑦 . Generally some values of 𝝊 may be chosen such that the displacement or Dirichlet boundary conditions of the problem under consideration are fulfilled by the displacements interpolated by Eq. (2.1). This is assumed for the following. Strains are derived from displacements by differentiation with respect to spatial coordinates. In the following, this is written as 𝝐 = B⋅𝝊

(2.2)

with the strains 𝝐 depending on spatial coordinates and time, a matrix B of spatial derivatives of trial functions depending on spatial coordinates, and the vector 𝝊 as was used in Eq. (2.1). The first examples for Eqs. (2.1) and (2.2) are given in Section 2.3. ◀

Field variables u, 𝝐 are discretised with Eqs. (2.1) and (2.2), i.e. infinite fields in space are reduced into a finite number n of variables defined in spatial points or nodes and collected in 𝝊.

Thereby kinematic compatibility should be assured regarding interpolated displacements, i.e. generally formulated, a coherence of displacements and deformations should be given. Strains 𝝐 lead to stresses 𝝈. A material law connects both. Material laws for solids are a science in themselves. This textbook mainly covers the options for reinforced concrete structures. To begin with such laws are abbreviated as 𝝈 = 𝑓(𝝐)

(2.3)

Besides total values of stress and strain, their changes in time 𝑡 have to be considered. Time may be a clock time in dynamics or a loading time in quasi-statics. Changes are measured with time derivatives 𝝐̇ =

𝜕𝝐 , 𝜕𝑡

𝝈̇ =

𝜕𝝈 𝜕𝑡

(2.4)

Nonlinear material behaviour is mainly formulated as a relation between 𝝐̇ and 𝝈. ̇ The first concepts about material laws are given in Section 2.4, and more details are given in Chapter 6. An equilibrium condition is the third basic element of structural analysis in addition to kinematic compatibility and material laws. It is advantageously formulated as a principle of virtual work leading to ∫ 𝛿𝝐 T ⋅ 𝝈 d𝑉 = ∫ 𝛿uT ⋅ b d𝑉 + ∫ 𝛿uT ⋅ t d𝐴 𝑉

𝑉

(2.5)

𝐴𝑡

for quasi-static cases with the volume 𝑉 of the solid body of interest, its body forces b, its boundary 𝐴, and its surface tractions t, which are prescribed at a part 𝐴𝑡 of the

11

12

2 Finite Elements Overview

whole boundary 𝐴. Furthermore, virtual displacements 𝛿u and the corresponding virtual strains 𝛿𝝐 are introduced. They are arranged as vectors and 𝛿uT , 𝛿𝝐 T indicate their transposition into row vectors to have a proper scalar product with 𝝈, b, t, which are also arranged as vectors. Fields of b and t are generally prescribed for a problem under consideration, while the field of stresses 𝝈 remains to be determined. Surface tractions t constitute the force or Neumann boundary conditions. ◀

Stresses 𝝈 and loadings b, t are in static equilibrium if Eq. (2.5) is fulfilled for arbitrary virtual displacements 𝛿u and the corresponding virtual strains 𝛿𝝐 .

Thereby, 𝛿u can be assumed as zero at the part 𝐴𝑢 of the whole boundary 𝐴 with prescribed displacement boundary conditions. Applying the displacement and strain interpolation Eqs. (2.1) and (2.2) to virtual displacements and strains leads to 𝛿u = N ⋅ 𝛿𝝊 ,

𝛿𝝐 = B ⋅ 𝛿𝝊

(2.6)

and using this with Eq. (2.5) leads to ⎡ ⎤ ⎡ ⎤ 𝛿𝝊T ⋅ ⎢∫ BT ⋅ 𝝈 d𝑉 ⎥ = 𝛿𝝊T ⋅ ⎢∫ NT ⋅ b d𝑉 + ∫ NT ⋅ t d𝐴⎥ ⎢ ⎥ ⎢ ⎥ 𝐴𝑡 ⎣𝑉 ⎦ ⎣𝑉 ⎦

(2.7)

with transpositions 𝛿𝝊T , BT , NT of the vector 𝛿𝝊 and the matrices B, N. As 𝛿𝝊 is arbitrary, a discretised condition of quasi-static equilibrium is derived in the form f =p

(2.8)

with the vector f of internal nodal forces and the vector p of external nodal forces f = ∫ BT ⋅ 𝝈 d𝑉 𝑉

(2.9)

p = ∫ NT ⋅ b d𝑉 + ∫ NT ⋅ t d𝐴 𝑉

𝐴𝑡

The vectors f, p have 𝑛 components corresponding to the length of the vector 𝝊. ◀

By means of 𝝈 = f (𝝐) and 𝝐 = B ⋅ 𝝊, Eq. (2.8) constitutes a system of n nonlinear algebraic equations, whereby the nodal displacements 𝝊 have to be determined such that – under the constraint of displacement boundary conditions – internal nodal forces f are equal to prescribed external nodal forces p.

Nonlinear stress–strain relations, i.e. physical nonlinearities, are always an issue for reinforced concrete structures. But it is a good practice in nonlinear simulation to start with a linearisation to have a reference for the refinements of a conceptual model. Physical linearity is described with 𝝈=C⋅𝝐

(2.10)

2.3 Elements

with a constant material stiffness matrix C. Thus, using Eq. (2.2) internal forces f (Eq. (2.9)) can be formulated as f =K⋅𝝊,

K = ∫ BT ⋅ C ⋅ B d𝑉

(2.11)

𝑉

with a constant stiffness matrix K leading to K⋅𝝊=p

(2.12)

This allows for a direct determination of nodal displacements, which is symbolically written as 𝝊 = K−1 ⋅ p

(2.13)

Actually, the solution is not determined with a matrix inversion but with more efficient techniques, e.g. a Gauss triangularisation or LU-decomposition. Stresses 𝝈 and strains 𝝐 follow with a solution 𝝊 given. A counterpart of physical linearity is geometric linearity. ◀

Small displacements and geometric linearity are assumed throughout the following if not otherwise stated.

This closes a fast track for the finite element method. The rough outline will be filled out in the following. Comprehensive descriptions covering all aspects are given in, e.g. Zienkiewicz and Taylor (1989, 1991); Belytschko et al. (2000); Bathe (2001). The special aspects of reinforced concrete structures are also treated in CEB-FIP (2008); Hofstetter and Mang (1995); Rombach (2006).

2.3 Elements Interpolation performed with finite elements is described with more details in the following. We consider the mechanical behaviour of material points within a body. A material point is identified by its spatial coordinates. It is convenient to use different coordinate systems simultaneously. First of all, the global Cartesian coordinate system, see Appendix D, which is shared by all material points of a body, is used. Thus, a material point is identified by global Cartesian coordinates ( x= 𝑥

𝑦

𝑧

)T

(2.14)

in 3D space. We assume that the space occupied by the body is divided into finite elements. Thus, a material point may alternatively be identified by the label 𝑒 of the element it belongs to and its local coordinates ( r= 𝑟

𝑠

𝑡

)T

(2.15)

13

14

2 Finite Elements Overview

related to a local coordinate system belonging to the element 𝑒. A material point undergoes displacements. In the case of translations, they are measured in the global Cartesian system by ( u= 𝑢

𝑣

𝑤

)T

(2.16)

Displacements in a general sense may also be measured by means of rotations ( 𝝋 = 𝜑𝑥

𝜑𝑦

𝜑𝑧

)T

(2.17)

if we consider a material point embedded in some neighbourhood of surrounding points. The indices indicate the respective reference axes of rotation. Isoparametric interpolation is used in the following. The general interpolation form (Eq. (2.1)) is specified as u = N(r) ⋅ 𝝊𝑒

(2.18)

whereby the global coordinates of the corresponding material point are given by x = N(r) ⋅ x𝑒

(2.19)

The vector 𝝊𝑒 collects all nodal displacements of all nodes belonging to the element 𝑒 and the vector x𝑒 all global nodal coordinates of that element. Isoparametric interpolation is characterised by the same interpolation for geometry and displacements with the same trial functions N(r). This is essential, as it allows for the definition of general element types. Global and local coordinates are related by the Jacobian matrix J=

𝜕x 𝜕r

(2.20)

which may be up to a 3 × 3 matrix for 3D cases. Strains are derived with displacements related to global coordinates through isoparametric interpolation. Their definition depends on the type of the structural problem. A general formulation 𝝐 = B(r) ⋅ 𝝊𝑒

(2.21)

is used. Strains 𝝐 finally lead to stresses 𝝈. Some element types that will be used later are described below. • A two-node bar element along a line (1D) The line is measured by a coordinate 𝑥. Each coordinate has a cross-section with a cross-sectional area. The kinematic assumption of a bar is that every material point in the cross-section has the same displacement in the line direction. A bar element 𝑒 has nodes 𝐼, 𝐽 with coordinates 𝑥𝐼 , 𝑥𝐽 . The nodes have the displacements 𝑢𝐼 , 𝑢𝐽 along the line. The origin of the local coordinate 𝑟 is placed in

2.3 Elements

the centre between the two nodes. Regarding Eqs. (2.18) and (2.19) we obtain ( ) ( ) x= 𝑥 , u= 𝑢 ] [1 1 (1 + 𝑟) N = (1 − 𝑟) 2 2 (2.22) ⎛𝑥𝐼 ⎞ ⎛ 𝑢𝐼 ⎞ x𝑒 = ⎜ ⎟ , 𝝊𝑒 = ⎜ ⎟ 𝑥 𝑢 ⎝ 𝐽⎠ ⎝ 𝐽⎠ This leads to a scalar Jacobian 𝐽=

𝐿𝑒 𝜕𝑥 = 2 𝜕𝑟

(2.23)

with the element length 𝐿𝑒 . Strains are uniaxial and defined by 𝜖=

𝜕𝑢 𝜕𝑢 𝜕𝑟 = 𝜕𝑥 𝜕𝑟 𝜕𝑥

(2.24)

2 [ 1 − 2 𝐿𝑒

(2.25)

leading to B=

1

]

2

with a bar length 𝐿𝑒 = 𝑥𝐽 − 𝑥𝐼 , and finally, regarding Eq. (2.3), to uniaxial strains and stresses ( ) ( ) 𝝐= 𝜖 , 𝝈= 𝜎 (2.26) which are constant along the element. • A two-node bar element in a plane (2D) The plane is measured by coordinates 𝑥, 𝑦. The centre axis of a bar is a line in this plane. Each point of the centre axis again has a cross-sectional area, and again the kinematic assumption of this bar is that every material point in the cross-section has the same displacement in the direction of the centre axis. A bar element 𝑒 has nodes 𝐼, 𝐽 with coordinates 𝑥𝐼 , 𝑦𝐼 , 𝑥𝐽 , 𝑦𝐽 . The nodes have the displacements 𝑢𝐼 , 𝑣𝐽 , 𝑢𝐼 , 𝑣𝐽 in a plane. The origin of the local coordinate 𝑟 is placed in the centre between the two nodes. Regarding Eqs. (2.18) and (2.19) we obtain ⎛𝑥 ⎞ ⎛𝑢⎞ x=⎜ ⎟ , u=⎜ ⎟ 𝑦 𝑣 ⎝ ⎠ ⎝ ⎠ 1 0 ⎡ (1 − 𝑟) N = ⎢2 1 (1 − 𝑟) 0 ⎣ 2 ⎛𝑥𝐼 ⎞ ⎜ ⎟ 𝑦𝐼 x𝑒 = ⎜ ⎟ , ⎜𝑥𝐽 ⎟ ⎜ ⎟ 𝑦 ⎝ 𝐽⎠

⎛ 𝑢𝐼 ⎞ ⎜ ⎟ 𝑣𝐼 𝝊𝑒 = ⎜ ⎟ ⎜𝑢𝐽 ⎟ ⎜ ⎟ 𝑣 ⎝ 𝐽⎠

1 2

(1 + 𝑟) 0

⎤ ⎥ (1 + 𝑟) ⎦ 2 0

1

(2.27)

15

16

2 Finite Elements Overview

Uniaxial strain is measured in the direction of the bar centre axis, i.e. in a rotated coordinate system 𝑥′ , 𝑦 ′ with 𝑥′ being aligned to the centre axis. The rotation angle 𝛼 (counterclockwise positive) and the transformation matrix T for global coordinates and displacements are given by, see Appendix D ⎡ cos 𝛼 T=⎢ − sin 𝛼 ⎣

sin 𝛼 ⎤ 𝑦𝐽 − 𝑦𝐼 𝑥𝐽 − 𝑥𝐼 , sin 𝛼 = (2.28) ⎥ , cos 𝛼 = 𝐿 𝐿𝑒 cos 𝛼 𝑒 ⎦ √ with a bar length 𝐿𝑒 = (𝑦𝐽 − 𝑦𝐼 )2 + (𝑥𝐽 − 𝑥𝐼 )2 . The scalar Jacobian is similar to before 𝐽=

𝐿𝑒 𝜕𝑥′ = 2 𝜕𝑟

(2.29)

Strains are again uniaxial and defined by 𝜖=

𝜕𝑢′ 𝜕𝑟 𝜕𝑢′ = 𝜕𝑥′ 𝜕𝑟 𝜕𝑥′

(2.30)

leading to B=

2 [ 1 − 2 𝐿𝑒

] ⎡cos 𝛼 ⋅⎢ 2 0 ⎣ 1

sin 𝛼

0

0

cos 𝛼

0 ⎤ ⎥ sin 𝛼 ⎦

(2.31)

with global displacements rotated into the local 𝑢′ regarding Eqs. (2.222 ) and (2.28). Uniaxial strains and stresses have the form given by Eq. (2.26). • A two-node spring element along a line (1D) The line is measured by a coordinate 𝑥. A spring element 𝑒 has nodes 𝐼, 𝐽 with coordinates 𝑥𝐼 , 𝑥𝐽 . The nodes may have the same initial coordinates but must not. A kinematic assumption for springs is that only the displacement difference of two nodes is relevant, irrespective of their original distance. Springs are an abstract concept and do not occupy a space. They lack material points, local coordinates, and a Jacobian. Thus, regarding Eq. (2.21) ( ) 𝝐 = Δ𝑢 ,

[ B = −1

] 1 ,

⎛ 𝑢𝐼 ⎞ (2.32) 𝝊𝑒 = ⎜ ⎟ 𝑢𝐽 ⎝ ⎠ whereby this particular strain 𝝐 = (Δ𝑢) corresponds to a difference in displacements of nodes and leads to a force 𝝈 = (𝐹). The relation between Δ𝑢 and 𝐹 or spring characteristics may be linear or nonlinear. • A two-node spring element in a plane (2D) The plane is measured with coordinates 𝑥, 𝑦. A spring element 𝑒 has nodes 𝐼, 𝐽 with coordinates 𝑥𝐼 , 𝑦𝐼 , 𝑥𝐽 , 𝑦𝐽 , which may again coincide initially but must not. In analogy to Eq. (2.32) ⎛Δ𝑢⎞ 𝝐=⎜ ⎟ , Δ𝑣 ⎝ ⎠

⎡−1 B=⎢ 0 ⎣

−1

0

0

1

0⎤ ⎥, 1 ⎦

⎛ 𝑢𝐼 ⎞ ⎜ ⎟ 𝑣𝐼 𝝊𝑒 = ⎜ ⎟ ⎜𝑢𝐽 ⎟ ⎜ ⎟ 𝑣 ⎝ 𝐽⎠

(2.33)

2.3 Elements

Generalised strain 𝝐 leads to a generalised stress ⎛𝐹𝑥 ⎞ (2.34) 𝝈=⎜ ⎟ 𝐹𝑦 ⎝ ⎠ The relation between 𝝐 and 𝝈 may again be linear or nonlinear. It may be appropriate to transform 𝝐 to a rotated coordinate system before evaluating 𝝈 using a transformation matrix as given by T in Eq. (2.28). This requires a back transformation of 𝝈 to the original coordinate system with the transposed TT , see Appendix D. • A four-node continuum element in a plane or quad element(2D) The plane is measured with coordinates 𝑥, 𝑦. A continuum element has nodes 𝐼, 𝐽, 𝐾, 𝐿 with coordinates 𝑥𝑖 , 𝑦𝑖 , 𝑖 = 𝐼, … , 𝐿. They span a quad and are ordered counterclockwise. The following local coordinates are assigned: 𝐼 ∶ 𝑟𝐼 = −1, 𝑠𝐼 = −1; 𝐽 ∶ 𝑟𝐽 = 1, 𝑠𝐽 = −1; 𝐾 ∶ 𝑟𝐾 = 1, 𝑠𝐾 = 1; 𝐿 ∶ 𝑟𝐿 = −1, 𝑠𝐿 = 1. The kinematic assumption of a continuum is that displacements are continuous, i.e. no gaps or overlapping occur. Regarding Eqs. (2.18) and (2.19) ⎛𝑥 ⎞ ⎛𝑢⎞ x=⎜ ⎟ , u=⎜ ⎟ 𝑦 𝑣 ⎝ ⎠ ⎝ ⎠ ⎤ 0 1 ⎡(1 + 𝑟𝑖 𝑟)(1 + 𝑠𝑖 𝑠) N𝑖 (𝑟, 𝑠) = ⎢ ⎥ 4 0 (1 + 𝑟𝑖 𝑟)(1 + 𝑠𝑖 𝑠) ⎦ ⎣ ⎛𝑥𝑖 ⎞ ⎛𝑢𝑖 ⎞ x𝑒,𝑖 = ⎜ ⎟ , 𝝊𝑒,𝑖 = ⎜ ⎟ 𝑦 𝑣 ⎝ 𝑖⎠ ⎝ 𝑖⎠ with 𝑖 = 𝐼, … , 𝐿 and ∑ ∑ x(𝑟, 𝑠) = N𝑖 (𝑟, 𝑠) ⋅ x𝑒,𝑖 , u(𝑟, 𝑠) = N𝑖 (𝑟, 𝑠) ⋅ 𝝊𝑒,𝑖 𝑖

𝑖

(2.35)

(2.36)

This leads to a Jacobian matrix and a scalar Jacobian determinant 𝜕𝑥

𝜕𝑦

⎤ ⎡ 𝜕𝑟 𝜕𝑟 ⎥ , 𝐽 = det J (2.37) J(𝑟, 𝑠) = ⎢ ⎢ 𝜕𝑥 𝜕𝑦 ⎥ ⎣ 𝜕𝑠 𝜕𝑠 ⎦ Its components are determined by Eq. (2.361 ) depending on the nodal coordinates. The Jacobian matrix relates the partial derivatives of a function ∙ with respect to local and global coordinates ⎛ 𝜕∙ ⎞ ⎛ 𝜕∙ ⎞ ⎛ 𝜕∙ ⎞ ⎛ 𝜕∙ ⎞ 𝜕𝑥 𝜕𝑥 𝜕𝑟 −1 ⎜ ⎟ = J ⋅ ⎜ ⎟ → ⎜ ⎟ = J ⋅ ⎜ 𝜕𝑟 ⎟ ⎜ 𝜕∙ ⎟ ⎜ 𝜕∙ ⎟ ⎜ 𝜕∙ ⎟ ⎜ 𝜕∙ ⎟ 𝜕𝑦 𝜕𝑠 ⎝ ⎠ ⎝ 𝜕𝑦 ⎠ ⎝ 𝜕𝑠 ⎠ ⎝ ⎠ with the inverse J−1 of J. Small strains are defined by ⎛ 𝜖𝑥 ⎞ ⎛ 𝜕𝑢 ⎜ ⎟ ⎜ 𝜕𝑥 𝜕𝑣 𝝐 = ⎜ 𝜖𝑦 ⎟ = ⎜ 𝜕𝑦 ⎜ ⎟ ⎜ ⎜ ⎟ ⎜ 𝜕𝑢 + 𝛾𝑥𝑦 ⎝ ⎠ ⎝ 𝜕𝑦

⎞ ⎛ ⎟ ⎜ ⎟=⎜ ⎟ ⎜ 𝜕𝑣 ⎟ ⎜ 𝜕𝑢 𝜕𝑟 + 𝜕𝑥 ⎠ ⎝ 𝜕𝑟 𝜕𝑦

𝜕𝑢 𝜕𝑟 𝜕𝑟 𝜕𝑥 𝜕𝑣 𝜕𝑟 𝜕𝑟 𝜕𝑦 𝜕𝑢 𝜕𝑠 𝜕𝑠 𝜕𝑦

+

𝜕𝑢 𝜕𝑠

+

𝜕𝑣 𝜕𝑠

+

𝜕𝑣 𝜕𝑟

𝜕𝑠 𝜕𝑥 𝜕𝑠 𝜕𝑦 𝜕𝑟 𝜕𝑥

(2.38)

⎞ ⎟ ⎟ ⎟ 𝜕𝑣 𝜕𝑠 ⎟ + 𝜕𝑠 𝜕𝑥 ⎠

(2.39)

17

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2 Finite Elements Overview

leading to 𝝐(𝑟, 𝑠) =

∑ 𝑖

B𝑖 (𝑟, 𝑠) ⋅ 𝝊𝑒,𝑖

(2.40)

with 𝑖 = 𝐼 … 𝐿 and 𝜕𝑟

𝜕𝑠

⎡𝑟𝑖 (1 + 𝑠𝑖 𝑠) + 𝑠𝑖 (1 + 𝑟𝑖 𝑟) 𝜕𝑥 𝜕𝑥 ⎢ 1⎢ 0 B𝑖 (𝑟, 𝑠) = ⎢ 4⎢ ⎢ 𝑟 (1 + 𝑠 𝑠) 𝜕𝑟 + 𝑠 (1 + 𝑟 𝑟) 𝜕𝑠 𝑖 𝑖 𝑖 𝑖 𝜕𝑦 𝜕𝑦 ⎣

⎤ ⎥ 𝜕𝑠 ⎥ + 𝑠𝑖 (1 + 𝑟𝑖 𝑟) ⎥ 𝜕𝑦 ⎥ 𝜕𝑠 + 𝑠𝑖 (1 + 𝑟𝑖 𝑟) ⎥ 𝜕𝑥 ⎦ (2.41) 0

𝑟𝑖 (1 + 𝑠𝑖 𝑠) 𝑟𝑖 (1 + 𝑠𝑖 𝑠)

𝜕𝑟 𝜕𝑦 𝜕𝑟 𝜕𝑥

The partial derivatives 𝜕𝑟∕𝜕𝑥 … are given by the components of the inverse Jacobian J−1 . Matrices N𝑖 , B𝑖 related to single nodes are assembled in larger matrices to yield N, B. Finally, stresses ⎛ 𝜎𝑥 ⎞ 𝝈 = ⎜ 𝜎𝑦 ⎟ (2.42) ⎜ ⎟ ⎝𝜎𝑥𝑦 ⎠ correspond to strains in a plane. Lateral strains 𝜖𝑧 or stresses 𝜎𝑧 come into play with the distinction of plane stress, that is, 𝜎𝑧 = 0, which may lead to a lateral strain 𝜖𝑧 ≠ 0, or plane strain, that is, 𝜖𝑧 = 0, which may lead to a lateral stress 𝜎𝑧 ≠ 0. The particular values in the 𝑧-direction have to be determined indirectly with a material law, see Section 2.4. All mentioned stresses and the corresponding strains are conjugate with respect to energy, i.e. the product 𝝈 ⋅ 𝝐̇ corresponds to a rate of specific internal energy. The concept of stress may be generalised. ◀

Depending on the type of structural element, 𝝈 may stand for components of Cauchy stresses, see Section 6.2.2, or for components of forces or for components of internal forces in a beam cross-section, see Section 4.1.1. Strain 𝝐 is generalised in a corresponding way to obtain the internal energy, involving displacements in the case of forces or curvature in the case of moments.

A basic property of the aforementioned elements is that they approximate coordinates and displacements by interpolation: nodal values and interpolated values are identical at nodes. For instance, for the four-node continuum element we get u = 𝝊𝑒,𝑖 for 𝑟 = 𝑟𝑖 , 𝑠 = 𝑠𝑖 𝑖 = 𝐼, … , 𝐿. This property is shared by all types of finite elements. Another issue concerns continuity. Regarding, e.g. the four-node quad the interpolation is continuous between adjacent elements along their common boundary. One-sided first derivatives of interpolation exist for each element along the boundary but are different for each element. Thus, the four-node continuum element has 𝐶 0 -continuity, and the integrals for internal and external nodal forces (Eq. (2.9)) are evaluable. Other element types may require higher orders of continuity for nodal forces to be integrable.

2.4 Material Behaviour

Finally, the issue of element locking has to be mentioned. The quad element, e.g. does not allow us to model the behaviour of incompressible solids. Constraining Eqs. (2.41) with the condition of incompressibility 𝜖𝑥 + 𝜖𝑦 + 𝜖𝑧 = 0 makes the element much too stiff if internal nodal forces are integrated exactly (Belytschko et al. 2000, 8.4). First hints regarding locking are given in Section 2.9. The locking problem is exemplarily treated for shells in Section 10.6. Only a few element types have been touched up to now. Further elements often used are 3D-continuum elements, 2D- and 3D-beam elements, slab elements 1) and shell elements. Furthermore, elements imposing constraints like contact conditions have become common in practice. For details, see, e.g. Bathe (2001). Regarding the properties of reinforced concrete more details about 2D-beam elements including Bernoulli beams and Timoshenko beams are given in Section 4.3, about slabs in Section 9.5, and about shells in Chapter 10.

2.4 Material Behaviour From a mechanical point of view, material behaviour is primarily focused on strains and stresses. The formal definitions of strains and stresses assume a homogeneous area of matter (Malvern 1969). Regarding the virgin state of solids their behaviour can initially be assumed as linear elastic in nearly all relevant cases. Furthermore, the behaviour can be initially assumed as isotropic in many cases, i.e. the reaction of a material is the same in all directions. The concepts of isotropy and anisotropy are discussed in Section 6.3 with more details. The following types of elasticity are listed exemplarily: • Uniaxial elasticity 𝜎=𝐸 𝜖

(2.43)

with uniaxial stress 𝜎, Young’s modulus 𝐸, and uniaxial strain 𝜖. • Isotropic plane strain 𝜈

0 ⎤ ⎛ 𝜖𝑥 ⎞ ⎡ 1 ⎛ 𝜎𝑥 ⎞ 1−𝜈 𝐸(1 − 𝜈) ⎢ 𝜈 ⎥ ⎜𝜎 ⎟= 1 0 ⎥ ⋅ ⎜ 𝜖𝑦 ⎟ (2.44) ⎢ 1−𝜈 𝑦 ⎜ ⎟ (1 + 𝜈)(1 − 2𝜈) ⎢ ⎟ 1−2𝜈 ⎥ ⎜ 𝛾 0 0 ⎝𝜎𝑥𝑦 ⎠ ⎣ 2(1−𝜈) ⎦ ⎝ 𝑥𝑦 ⎠ with stress components 𝜎𝑥 , 𝜎𝑦 , 𝜎𝑥𝑦 , Young’s modulus 𝐸, Poisson’s ratio 𝜈, and strain components 𝜖𝑥 , 𝜖𝑦 , 𝛾𝑥𝑦 . This is a subset of the triaxial isotropic linear elastic law as described in Section 6.3. • Isotropic plane stress ⎡1 𝜈 0 ⎤ ⎛ 𝜖𝑥 ⎞ ⎛ 𝜎𝑥 ⎞ ⎢ ⎥ 𝐸 ⎜𝜎 ⎟= 0 ⎥ ⋅ ⎜ 𝜖𝑦 ⎟ ⎢𝜈 1 𝑦 ⎟ ⎜ ⎟ 1 − 𝜈2 ⎢ 1−𝜈 ⎥ ⎜ 0 0 𝛾𝑥𝑦 ⎠ ⎝ ⎝𝜎𝑥𝑦 ⎠ 2 ⎣ ⎦ ensuring 𝜎𝑧 = 0 for every combination 𝜖𝑥 , 𝜖𝑦 , 𝛾𝑥𝑦 1) In the following, slabs are expressly differentiated from plates, see Figure 9.1.

(2.45)

19

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2 Finite Elements Overview

• Plane bending 𝑀 = 𝐸𝐽 𝜅

(2.46)

with the moment 𝑀, curvature 𝜅, Young’s modulus 𝐸, and cross-sectional moment of inertia 𝐽. This is covered by the concept of generalised stresses with 𝝈 = (𝑀) and generalised strains 𝝐 = (𝜅). Equations (2.43)–(2.45) are a special case of 𝝈=C⋅𝝐

(2.47)

with the constant material stiffness matrix C describing linear material behaviour. At the latest upon approaching material strength, the behaviour becomes physically nonlinear. A simple case is given by the uniaxial elasto-plastic law

𝜎=

⎧ 𝐸 (𝜖 − 𝜖𝑝 ) for − 𝜖𝑝 ≤ 𝜖 ≤ 𝜖𝑝 ⎨sign 𝜖 𝑓𝑦 otherwise ⎩

(2.48)

and 𝜖̇ 𝑝 = 𝜖̇

for |𝜎| = 𝑓𝑦

(2.49)

with a yield stress 𝑓𝑦 (unsigned) and an internal state variable 𝜖𝑝 . The internal state variable indicates the actual permanent strain upon unloading, i.e. 𝜎 = 0 for 𝜖 = 𝜖𝑝 . An internal state variable captures the preceding load history. The approach covers elastic loading, yielding, elastic unloading and re-loading, and ongoing yielding in the opposite uniaxial range. This cycle may be repeated, see Figure 3.11. Equation (2.49) is a simple evolution law for internal state variables. More details about elasto-plasticity are given in Section 6.5. In the case of nonlinear material laws, at least an incremental form 𝝈̇ = C𝑇 ⋅ 𝝐̇

(2.50)

should exist with the tangential material stiffness matrix C𝑇 , which is no longer constant but might depend on stress, strain, and internal state variables. On occasion, the compliance is needed, as a counterpart of stiffness, i.e. 𝝐 = D ⋅ 𝝈 or 𝝐̇ = D𝑇 ⋅ 𝝈̇

(2.51)

whereby compliance matrices are inverses of stiffness matrices: D = C−1 , D𝑇 = C−1 𝑇 .

2.5 Weak Equilibrium The preceding sections gave an introduction to (1) kinematic compatibility within the context of spatial discretisation and of (2) material laws. The third cornerstone of

2.5 Weak Equilibrium

structural mechanics is equilibrium, which is formulated in a weak form as a principle of virtual work. Boundary conditions have to be regarded in advance. Given a point on a boundary of a body, either a displacement (Dirichlet) boundary condition or a force (Neumann) boundary condition (zero force is also a condition) has to be prescribed for this point. Let us assume that displacements are prescribed with u on a surface part 𝐴𝑢 , tractions are prescribed with t on a surface part 𝐴𝑡 , while 𝐴𝑢 together with 𝐴𝑡 contain the whole surface 𝐴 but do not overlap. Thus, equilibrium is given by ∫ 𝛿𝝐 T ⋅ 𝝈 d𝑉 + ∫ 𝛿uT ⋅ ü 𝜚 d𝑉 = ∫ 𝛿uT ⋅ b d𝑉 + ∫ 𝛿uT ⋅ t d𝐴 𝑉

𝑉

𝑉

(2.52)

𝐴𝑡

under the conditions u = u on 𝐴𝑢 ,

𝛿u = 0 on 𝐴𝑢

(2.53)

and 𝛿u arbitrary otherwise. The meaning of the symbols is summarised as follows: (∙)T u ü 𝛿u 𝛿𝝐 𝝈 𝜚 b t 𝑉 𝐴 𝐴𝑢 𝐴𝑡

transpose of column vector (∙) leading to row vector field of displacement vector field of acceleration vector field of test functions or virtual displacement vector field of virtual strain vector corresponding to 𝛿u field of stress vector specific mass prescribed field of loads distributed in the body prescribed field of tractions distributed over the body surface body volume body surface part of surface with prescribed displacements part of surface with prescribed tractions

Equation (2.52) describes structural dynamics and includes quasi-statics as a special case. Concentrated loads are not explicitly noted. For extensions with generalised variational principles, see Wunderlich and Pilkey (2003). All parameters listed have to be considered as generalised. The following specifications are listed exemplarily: • In the case of a uniaxial bar, Eq. (2.52) becomes ∫ 𝛿𝜖 𝜎 𝐴 d𝑥 + ∫ 𝛿𝑢 𝑢̈ 𝜚𝐴 d𝑥 = ∫ 𝛿𝑢 𝑏 d𝑥 + [𝛿𝑢 𝑡]𝐿0 𝐿

𝐿

(2.54)

𝐿

with 0 ≤ 𝑥 ≤ 𝐿 under the conditions 𝑢0 = 𝑢0 , 𝛿𝑢0 = 0 and/or 𝑢𝐿 = 𝑢𝐿 , 𝛿𝑢𝐿 = 0

(2.55)

21

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2 Finite Elements Overview

with a cross-sectional area 𝐴 and a load per length 𝑏 in the bar direction, whereby the formulation of the last term indicates the boundary term of a partial integration. Surface tractions degenerate to end forces 𝑡, which are prescribed at either 𝑥 = 0 or 𝑥 = 𝐿 (or none, but not both at the same time). • In the case of a plane Bernoulli beam, Eq. (2.52) becomes ∫ 𝛿𝑤 𝑤̈ 𝑚 d𝑥 + ∫ 𝛿𝜅 𝑀 d𝑥 = ∫ 𝛿𝑤 𝑏 d𝑥 − [𝛿𝜑 𝑀]𝐿0 + [𝛿𝑤 𝑉]𝐿0 𝐿

𝐿

(2.56)

𝐿

with 0 ≤ 𝑥 ≤ 𝐿, the deflection 𝑤, the beam’s slope 𝜑, moment 𝑀, shear force 𝑉, a distributed mass 𝑚 per length, and a distributed lateral load 𝑏 per length. Two boundary conditions can be given at each end 𝑥 = 0 and 𝑥 = 𝐿 with corresponding pairs (𝜑, 𝑀) and (𝑤, 𝑉). Only one quantity out of a pair – 𝜑 or 𝑀 and 𝑤 or 𝑉 – can be prescribed at a boundary. Furthermore, at least two displacement boundary conditions should be given with at least one deflection 𝑤 0 and/or 𝑤 𝐿 to avoid rigid body displacements. The principle of virtual work or weak integral forms of equilibrium conditions treat a body as a whole. Strong differential forms consider forces applied to infinitesimally small sections or differentials of a body and lead to differential equations. Both are equivalent from a mechanical point of view. This is exemplarily demonstrated for beams in Section 4.2 and for slabs in Section 9.3.

2.6 Spatial Discretisation Weak forms are the starting point for a discretisation of a whole domain with finite elements. This has the following steps with respect to Eq. (2.52): 1. Mesh generation The respective body has to be to filled with elements. No gaps between elements and no overlapping of elements are allowed in the body interior. Elements may form facets or polygons on the exterior. Proportions and geometric distortions of a single element may have a considerable influence on the mathematical approximation error. 2. Spatial interpolation of displacements with Eq. (2.18) An infinite number of degrees of freedom u is reduced to a finite number of nodal degrees of freedom 𝝊𝑒 with trial functions according to Eq. (2.18). This leads to discretised strains 𝝐 with Eq. (2.21). 3. Spatial interpolation of virtual displacements Interpolation of virtual displacements 𝛿u is performed with test functions. The method of Bubnov–Galerkin is generally used with the same functions as trial functions and test functions implying virtual nodal degrees of freedom 𝛿𝝊𝑒 𝛿u = N ⋅ 𝛿𝝊𝑒 ,

𝛿𝝐 = B ⋅ 𝛿𝝊𝑒

whereby virtual 𝛿𝝐 strains are determined in the same way as strains.

(2.57)

2.6 Spatial Discretisation

4. Evaluation of stresses 𝝈 from stains 𝝐 according to a prescribed material law This has to be performed by the integration of the incremental form Eq. (2.50). The details depend on the material and structural type and are a major issue in all that follows. 5. The evaluation of integrals is performed element by element ∫ 𝛿𝝐 T ⋅ 𝝈 d𝑉 = 𝛿𝝊T𝑒 ⋅ f𝑒 ,

f𝑒 = ∫ BT ⋅ 𝝈 d𝑉

𝑉𝑒

𝑉𝑒

∫ 𝛿uT ⋅ ü 𝜚 d𝑉 = 𝛿𝝊T𝑒 ⋅ M𝑒 ⋅ 𝝊̈ 𝑒 , 𝑉𝑒

M𝑒 = ∫ NT ⋅ N 𝜚 d𝑉 𝑉𝑒

∫ 𝛿uT ⋅ b d𝑥 = 𝛿𝝊T𝑒 ⋅ b𝑒 ,

(2.58)

b𝑒 = ∫ NT ⋅ b d𝑉

𝑉𝑒

𝑉𝑒

t𝑒 = ∫ NT ⋅ t d𝐴

∫ 𝛿uT ⋅ t d𝐴 = 𝛿𝝊T𝑒 ⋅ t𝑒 , 𝐴𝑒,𝑡

𝐴𝑒,𝑡

with an element index 𝑒. This includes the element internal nodal forces f𝑒 , its mass matrix M𝑒 , and its external nodal forces or loadings b𝑒 , t𝑒 , which are collected with p𝑒 = b𝑒 + t𝑒

(2.59)

For integration methods, see Section 2.7. Internal nodal forces in the end are functions of nodal displacements f𝑒 = f𝑒 (𝝊𝑒 ). 6. Assembling of element contributions Regarding global internal nodal forces f, the vector has entries for every degree of freedom of every global node. An element node is simultaneously a global node from a system point of view. Furthermore, every meshing should have a table that connects an element to the global nodes belonging to it. This table relates the position of the entries in f𝑒 from Eq. (2.58) to a position in f from Eq. (2.9). As a node generally gets contributions from more than one element, the value of an entry in f𝑒 has to be added to the corresponding entry in f. This is symbolically written as ∑ ∑ ∫ 𝛿𝝐 T ⋅ 𝝈 d𝑉 = 𝛿𝝊T ⋅ f = 𝛿𝝊T𝑒 ⋅ f𝑒 , f = f𝑒 (2.60) 𝑒

𝑒

𝑉

The same procedure is applied to 𝛿𝝊𝑒 → 𝛿𝝊, 𝝊𝑒 → 𝝊, M𝑒 → M, p𝑒 → p. Global internal nodal forces in the end are a function of global nodal displacements f = f(𝝊). 7. With respect to arbitrary values of 𝛿𝝊, a spatially discretised system M ⋅ 𝝊̈ + f(𝝊) = p

(2.61)

finally results whereby including the system mass matrix M, its internal nodal forces f, and its loading p. This is a set of ordinary differential equations of sec-

23

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2 Finite Elements Overview

ond order in time 𝑡 for nodal displacements 𝝊. It might be nonlinear due to the nonlinear dependence of internal nodal forces f on 𝝊. This procedure allows for physical nonlinearities. In the special case of physical linearity, the linear material stiffness 𝝈 = C ⋅ 𝝐 yields internal nodal forces f𝑒 = ∫ BT ⋅ C ⋅ B d𝑉 ⋅ 𝝊𝑒 = K𝑒 ⋅ 𝝊𝑒

(2.62)

𝑉𝑒

see Eqs. (2.581 ) and (2.21), with a constant element stiffness matrix K𝑒 . Assembling leads to a system stiffness matrix K f(𝝊) = K ⋅ 𝝊

(2.63)

and with respect to Eq. (2.61) to M ⋅ 𝝊̈ + K ⋅ 𝝊 = p

(2.64)

which is a system of linear ordinary differential equations of second order in time 𝑡. To treat nonlinearities the system tangential stiffness is involved. The tangential stiffness matrix is needed for the solution of the nonlinear system and furthermore reveals characteristic properties, e.g. with respect to stability behaviour. The tangential stiffness of an element is derived with df𝑒 =

𝜕f𝑒 ⋅ d𝝊𝑒 = K𝑇𝑒 ⋅ d𝝊𝑒 𝜕𝝊𝑒

or ḟ 𝑒 = K𝑇𝑒 ⋅ 𝝊̇ 𝑒

(2.65)

𝜕𝝈 𝜕𝝐 d𝑉 = ∫ BT ⋅ C𝑇 ⋅ B d𝑉 ⋅ 𝜕𝝐 𝜕𝝊𝑒

(2.66)

with K𝑇𝑒 = ∫ BT ⋅ 𝑉𝑒

𝑉𝑒

see Eqs. (2.581 ), (2.50), and (2.21). A tangential stiffness matrix K𝑇 ∑ df = K𝑇 ⋅ d𝝊 or ḟ = K𝑇 ⋅ 𝝊̇ , K𝑇 = K𝑇𝑒 𝑒

(2.67)

is assembled from the element contributions. Finally, the system – Eq. (2.61) or Eq. (2.64) – should be constrained with appropriate conditions with respect to 𝝊 to prevent rigid body displacements.

2.7 Numerical Integration The integral formulation of equilibrium conditions requires the evaluation of integrals as given by Eq. (2.58). The evaluation is performed element by element. The integration of a quad element (Section 2.3) is exemplarily discussed in the following. This may be reduced for a 1D-domain as is required for bar or beam elements (Eqs. (4.98)–(4.101)) or extended to a 3D-domain as is required for volume elements or continuum-based shell elements (Section 10.4).

2.7 Numerical Integration

Table 2.1 Sampling points and weights for Gauss integration (15 digits shown). ni

𝝃i

𝜼i

0 1 2

0.0 ±0.577 350 269 189 626 ±0.774 596 669 241 483 0.0 ±0.861 136 311 594 053 ±0.339 981 043 584 856 ⋮

2.0 1.0 0.555 555 555 555 556 0.888 888 888 888 889 0.347 854 845 137 454 0.652 145 154 862 546 ⋮

3 ⋮

A general function 𝑓(𝑥, 𝑦) indicates the integrand. The isoparametric quad element has a local coordinate system 𝑟, 𝑠 with −1 ≤ 𝑟, 𝑠 ≤ 1. Thus, integration is performed by +1 +1

∫ 𝑓(𝑥, 𝑦) d𝑉 = ∫ ∫ 𝑓(𝑟, 𝑠) 𝐽(𝑟, 𝑠) 𝑏 d𝑟 d𝑠 𝑉𝐼

(2.68)

−1 −1

with the determinant 𝐽 of the Jacobian (Eq. (2.37)) and a thickness 𝑏. ◀

The Jacobian J maps the local coordinates into global coordinates and is a key quantity for isoparametric finite elements.

As closed analytical forms generally are not available for 𝑓(𝑟, 𝑠) a numerical integration has to be performed +1 +1

∫ ∫ 𝑓(𝑟, 𝑠) 𝐽(𝑟, 𝑠) 𝑏 d𝑟 d𝑠 = 𝑏 −1 −1

𝑛𝑖 𝑛𝑖 ∑ ∑ 𝑖=0 𝑗=0

𝜂𝑖 𝜂𝑗 𝑓(𝜉𝑖 , 𝜉𝑗 ) 𝐽(𝜉𝑖 , 𝜉𝑗 )

(2.69)

with integration order 𝑛𝑖 , sampling points 𝜉, and weighting factors 𝜂. An appropriate scheme is given by the Gauss integration. Its sampling points and weighting factors are listed in Table 2.1 up to an integration order 𝑛𝑖 = 3. Weighting factors obey a rule ∑𝑛𝑖 𝜂 = 2. Accuracy of integration is a key issue. 𝑖=0 𝑖 ◀

Integration accuracy increases with increasing integration order. On the other hand, numerical integration leads to a major contribution to computational costs.

Gauss integration is generally the most efficient compared to other numerical integration schemes. An integration order 𝑛𝑖 gives exact results for polynomials of order 2𝑛𝑖 + 1 disregarding round-off errors, e.g. a uniaxial integration of order 1 with two sampling points exactly integrates a polynomial of the order 3. Alternative numerical integration schemes are given by schemes of Simpson, Newton–Cotes, and Lobatto.

25

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2 Finite Elements Overview

2.8 Equation Solution Methods The majority of the example problems are nonlinear due to nonlinear material behaviour. Nonlinear material behaviour is not only characteristic of reinforced concrete but also occurs with all other solid materials at the latest with approaching strength. A solution approach like Eq. (2.13) for quasi-static problems can no longer be applied. Incrementally iterative methods are generally used in structural analysis. A loading is applied in a number of steps reducing a ‘stronger nonlinearity’ into a sequence of ‘weaker nonlinearities’. Each step yields a nonlinear algebraic problem. Thus, everything is based upon nonlinear algebraic equation solving, see the following Section 2.8.1. This is embedded in time incrementation, see Section 2.8.2.

2.8.1 Nonlinear Algebraic Equations Discretisation of quasi-static nonlinear problems leads to the nonlinear algebraic form r(𝝊) = p − f(𝝊) = 0 ,

r, p, f, 𝝊, 0 ∈ ℝ𝑛

(2.70)

with a residual r, internal nodal forces f depending on displacements 𝝊, external nodal loads p independent of 𝝊, and the total number of degrees of freedom 𝑛 for the discretised system, see Eq. (2.8) or Eq. (2.61), with inertial terms disregarded. We assume that Eq. (2.70) has a unique solution for 𝝊, i.e. the underlying structural system has to be supported such that rigid body displacements are prevented, and the loading of a system should not exceed its load bearing capacity. To start with, the Newton–Raphson method to solve nonlinear algebraic equations is introduced with the scalar form 𝑛 = 1 𝑟(𝜐) = 𝑝 − 𝑓(𝜐) = 0

(2.71)

A first guess of the unknown should be on hand with 𝜐 (0) with a residual 𝑟(𝜐 (0) ) ≠ 0. Equation (2.71) is expanded with a linear Taylor row 𝑟(𝜐 (𝜈) + 𝛿𝜐) ≈ 𝑟(𝜐 (𝜈) ) +

d𝑟(𝜐) ||| | 𝛿𝜐 d𝜐 |||𝜐=𝜐 (𝜈)

(2.72)

to determine a correcting change 𝛿𝜐. The condition 𝑟(𝜐 (𝜈) + 𝛿𝜐) = 0 leads to −1

d𝑟(𝜐) ||| | 𝑟(𝜐 (𝜈) ) d𝜐 |||𝜐=𝜐 (𝜈) [ (𝜈) ]−1 d𝑓(𝜐) ||| (𝜈) | 𝑟(𝜐 (𝜈) ), 𝐾𝑇 = = 𝐾𝑇 d𝜐 |||𝜐=𝜐 (𝜈)

𝛿𝜐 = −

(2.73)

with [∙]−1 = 1∕∙, and an improved solution should be given by 𝜐 (𝜈+1) = 𝜐 (𝜈) + 𝛿𝜐

(2.74)

2.8 Equation Solution Methods

Figure 2.4 Newton–Raphson method.

Equations (2.73), (2.74) define an iteration sequence with an index (𝜈) starting with 𝜈 = 0. This is illustrated in Figure 2.4. The iteration may stop if 𝛿𝑣 → 0 and the iterated residual is small compared to the initial residual 𝑟(0) . This is extended to the case 𝑛 > 1 with (𝜈)

r(𝝊(𝜈) + 𝛿𝝊) ≈ r(𝝊(𝜈) ) + K𝑇 ⋅ 𝛿𝝊 =0

(2.75)

with a tangential stiffness matrix (Eq. (2.66)) 𝜕r ||| 𝜕f ||| (𝜈) (𝜈) || | K𝑇 = − = , K𝑇 ∈ ℝ𝑛×𝑛 | 𝜕𝝊 |𝝊=𝝊(𝜈) 𝜕𝝊 |||𝝊=𝝊(𝜈)

(2.76)

leading to [ (𝜈) ]−1 ⋅ r(𝝊(𝜈) ) 𝛿𝝊 = K𝑇

(2.77)

𝝊(𝜈+1) = 𝝊(𝜈) + 𝛿𝝊

Iteration may stop with convergence if ‖r(𝝊(𝜈+1) )‖ ≪ ‖r(𝝊(0) )‖ and ‖𝛿𝝊‖ → 0 with a suitable norm ‖ ⋅ ‖ transforming a vector into a scalar. A rule of thumb is ‖r(𝝊(𝜈+1) )‖ < 10−3 ‖r(𝝊(0) )‖; see also Zienkiewicz and Taylor (1991, 7.5), Bathe (1996, 8.4.4), Belytschko et al. (2000, 6.3.9), and de Borst et al. (2012, 4.5) for a discussion of equilibrium convergence criteria. (𝜈) The reciprocal of the scalar stiffness 𝐾𝑇 becomes the inverse of the tangential (𝜈)

stiffness matrix K𝑇 . But the inversion is quite expensive from a computational point of view for large values of 𝑛. A LU-decomposition is performed instead using a Gaussian elimination, which requires considerably fewer operations compared to an inversion. A matrix K𝑇 is decomposed into K𝑇 = L ⋅ U ,

L, U ∈ ℝ𝑛×𝑛

(2.78)

with a lower triangular matrix L with components 𝐿𝑖𝑗 = 0, 𝑗 > 𝑖 and 𝐿𝑖𝑖 = 1 and an upper triangular matrix U with components 𝑈𝑖𝑗 = 0, 𝑗 < 𝑖. The task of an iteration step is reformulated as L⋅𝝎=r,

U ⋅ 𝛿𝝊 = 𝝎 ,

𝝎 ∈ ℝ𝑛

(2.79)

27

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2 Finite Elements Overview

or a sequence of forward and backward substitutions, which for given L, U, r, is computationally inexpensive due to the triangular structure of L, U. Convergence of the Newton–Raphson method is not guaranteed. The residual r(𝝊) should be a ‘smooth’ function of 𝝊 and the nonlinearity should not be to ‘strong’. This is explicated to some extent in Appendix A.1. Furthermore, the computational costs of the Newton–Raphson method is still relatively high, as the LU-decomposition is performed for each iteration step (𝜈). Alternatives are also given in Appendix A.1.

2.8.2 Time Incrementation Loading is given as a history p = p(𝑡). An appropriate choice is 0 ≤ 𝑡 ≤ 1 for the scaling of the load history time, which is different from clock time in the case of a quasi-static analysis. The following steps are performed in the incrementally iterative scheme: 1. Discrete time values 𝑡𝑖 are regarded with a time step Δ𝑡 = 𝑡𝑖+1 − 𝑡𝑖 and an initial time 𝑡0 = 0. A loading p𝑖 = p(𝑡𝑖 ) is prescribed for all time steps. The incremental material law Eq. (2.50) 𝝈(𝑡) ̇ = C𝑇 ⋅ 𝝐(𝑡) ̇

(2.80)

is integrated by a numerical integration of stresses using an implicit Euler scheme 𝝈𝑖+1 = 𝝈𝑖 + C𝑇,𝑖+1 ⋅ (𝝐 𝑖+1 − 𝝐 𝑖 )

(2.81)

with 𝝈𝑖 = 𝝈(𝑡𝑖 ), 𝝐 𝑖 = 𝝐(𝑡𝑖 ), whereby the algorithmic tangential material stiffness matrix or algorithmic modulus C𝑇,𝑖+1 depends on 𝝈𝑖+1 and/or 𝝐 𝑖+1 . Thus, Eq. (2.81) is generally nonlinear. The algorithmic modulus represents the adaptation of the tangential material stiffness matrix (Eq. 2.50) to the implicit discretisation in time (Belytschko et al. 2000, 5.9.4). 2. The initial state described by 𝝊0 , 𝝐 0 , 𝝈0 , f0 is assumed to be known, and the initial equilibrium is given by r0 = p0 − f0 = 0. 3. Unknowns are nodal displacements 𝝊𝑖+1 = 𝝊(𝑡𝑖+1 ) leading to strains 𝝐 𝑖+1 , stresses 𝝈𝑖+1 , and internal nodal forces f𝑖+1 (Eq. (2.58)) except the initial state. 4. The solution starts with 𝑡1 and 𝝊1 has to be determined. This is performed with an (0) (𝜈) iteration 𝝊1 , … , 𝝊1 with, e.g. the Newton–Raphson method (Eq. (2.77)) using (0)

(𝜈)

(𝜈)

(𝜈)

an initial 𝝊1 = 𝝊0 . The iteration involves 𝝐 1 , 𝝈1 , C𝑇,1 according to Eq. (2.81). 5. A converged 𝝊1 – see the equilibrium convergence criteria Section 2.8.1 – and the corresponding strains 𝝐 1 and stresses 𝝈1 serve as a base for 𝑡2 and so on, until a target time is reached. The procedure is illustrated in Figure 2.5 and combined with integrations according to Eqs. (2.58), (2.66), (2.67) and assembling according to Eq. (2.60). A scaling of time, i.e. multiplying time with a constant factor in each occurrence, will not have any influence upon the results.

2.8 Equation Solution Methods

Figure 2.5 Flow of displacementbased nonlinear calculation.

This starts to become different with a transient analysis. Material behaviour like creep (Section 3.2) has to be regarded as transient. Such behaviour is modelled by including viscosity (Malvern 1969, 6.4). Hence, the incremental material law (Eq. (2.80)) is extended as 𝝈̇ = C𝑇 ⋅ 𝝐̇ + 𝚺

(2.82)

with a stress extension term 𝚺 depending on stress 𝝈(𝑡) and strain 𝝐(𝑡). The time integration procedure described above has to be extended regarding clock time dependence of 𝚺. This is described in Appendix A.2. Clock time 𝑡 is also a key factor for a dynamic analysis with respect to inertia. Based on Eq. (2.61) we obtain in analogy to Eq. (2.70) r = p(𝑡) − M ⋅ 𝝊̈ − f = 0

(2.83)

Equation (2.83) is discretised in the spatial domain but not yet in the time domain, i.e. it is a system of ordinary differential equations of second order in time. In addition to displacement boundary conditions, this problem needs initial conditions for the displacements 𝝊0 = 𝝊(0) and velocities 𝝊̇ 0 = 𝝊(0). ̇ A popular approach for the time discretisation of accelerations and velocities is given with the Newmark method 𝝊̇ 𝑖+1 = 𝝊̇ 𝑖 + Δ𝑡 [𝛾 𝝊̈ 𝑖+1 + (1 − 𝛾)̈𝝊𝑖 ] 1 𝝊𝑖+1 = 𝝊𝑖 + Δ𝑡 𝝊̇ 𝑖 + Δ𝑡2 [𝛽 𝝊̈ 𝑖+1 + ( − 𝛽) 𝝊̈ 𝑖 ] 2

(2.84)

̇ 𝑖+1 ), 𝝊̈ 𝑖+1 = 𝝊̈ (𝑡𝑖+1 ), a time step length Δ𝑡 = 𝑡𝑖+1 − 𝑡𝑖 with 𝝊𝑖+1 = 𝝊(𝑡𝑖+1 ), 𝝊̇ 𝑖+1 = 𝝊(𝑡 and time integration parameters 𝛾, 𝛽. These are extensions of the Euler schemes, e.g. Eq. (2.81), for time integration. From Eqs. (2.84), the acceleration 𝝊̈ 𝑖+1 =

1 [𝝊𝑖+1 − 𝝊̃ 𝑖+1 ] 𝛽Δ𝑡2

(2.85)

29

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2 Finite Elements Overview

is obtained with an auxiliary quantity Δ𝑡2 (1 − 2𝛽) 𝝊̈ 𝑖 2

𝝊̃ 𝑖+1 = 𝝊𝑖 + Δ𝑡 𝝊̇ 𝑖 +

(2.86)

and the velocity 𝝊̇ 𝑖+1 =

𝛾 𝛾 𝛾 ) 𝝊̈ 𝑖 [𝝊𝑖+1 − 𝝊𝑖 ] + (1 − ) 𝝊̇ 𝑖 + Δ𝑡 (1 − 𝛽Δ𝑡 𝛽 2𝛽

(2.87)

Finally, dynamic equilibrium Eq. (2.83) is applied for the time step 𝑖 + 1 with acceleration according to Eq. (2.85) r𝑖+1 = p𝑖+1 −

1 M ⋅ [𝝊𝑖+1 − 𝝊̃ 𝑖+1 ] − f𝑖+1 = 0 𝛽Δ𝑡2

(2.88)

With given parameters 𝛾, 𝛽, Δ𝑡, a previous state 𝝊𝑖 , 𝝊̇ 𝑖 , 𝝊̈ 𝑖 , mass matrix M, and load p𝑖+1 , the Eq. (2.88) has to be solved for 𝝊𝑖+1 , whereby the dependence of f𝑖+1 on 𝝊𝑖+1 might be nonlinear. We again apply the Newton–Raphson method (Eq. (2.77)). An extended tangential stiffness (Eq. (2.76)) is given by (𝜈)

A𝑇 =

1 𝜕f ||| 1 (𝜈) | M+ = M + K𝑇 2 𝜕𝝊 |||𝝊=𝝊(𝜈) 𝛽Δ𝑡 𝛽Δ𝑡2 𝑖+1

(2.89)

leading to an iteration scheme (𝜈+1)

𝝊𝑖+1

]−1 [ (𝜈) (𝜈) = 𝝊𝑖+1 + A𝑇 ⋅ (p𝑖+1 −

) ( 1 (𝜈) (𝜈) M ⋅ 𝝊𝑖+1 − 𝝊̃ 𝑖+1 − f𝑖+1 ) 2 𝛽Δ𝑡

(2.90)

This includes the linear case with (𝜈)

(𝜈)

f𝑖+1 = K ⋅ 𝝊𝑖+1 ,

(𝜈)

A𝑇 = A =

1 M+K 𝛽Δ𝑡2

(2.91)

and Eq. (2.90) simplifies to 𝝊𝑖+1 = A−1 ⋅ (p𝑖+1 +

1 M ⋅ 𝝊̃ 𝑖+1 ) 𝛽Δ𝑡2

(2.92)

with no iteration necessary (Bathe 1996, 9.2.4). Numerical integration parameters 𝛾, 𝛽 rule the numerical consistency and stability of the method. • Stability means that an amount of error introduced due to a finite time step length Δ𝑡 does not build up in a sequence of steps. • Consistency means that the iteration scheme converges to the differential equation for Δ𝑡 → 0. Stability and consistency are necessary to ensure that the error of the time integra1 tion remains within some bounds for a finite time step length Δ𝑡. A choice 𝛽 = , 𝛾 = 1

4

is reasonable for the Newmark method (Bathe 1996, 9.4). For more details about 2 time integration methods, see Zienkiewicz and Taylor (1991, 10), Bathe (1996, 9), Belytschko et al. (2000, 6), and de Borst et al. (2012, 5).

2.9 Discretisation Errors

2.9 Discretisation Errors Numerical methods like the finite element method (FEM) yield approximate solutions for differential equations. Estimation of mathematical approximation errors is essential and needs a theoretical base. In the following, this is given for FEM with some generalizing theoretical background. The description is related to linear problems and cannot be strictly applied to nonlinear problems. But the conclusions are also reasonable for nonlinear and dynamic problems. The major contribution to the mathematical approximation error (Section 2.1) is the discretisation error arising from the difference between mathematical and numerical model (Figure 2.1). This difference should become smaller with a mesh refinement, i.e. the solution of the numerical model should converge to the solution of the related mathematical model. Under the assumption of linearity, the convergence behaviour of FEM can be analysed theoretically. Quasi-static problems are considered in the following. The following mathematical symbols are used in this section: ∀ ∈ ⊂ ∃ ∩ ∪

for all element of subset of there exists intersection union Given a linear material law 𝝈=C⋅𝝐

(2.93)

the condition of weak, integral equilibrium (Eq. (2.52)) can be written as ∫ 𝛿𝝐 T ⋅ C ⋅ 𝝐 d𝑉 = ∫ 𝛿uT ⋅ b d𝑉 + ∫ 𝛿uT ⋅ t d𝐴 𝑉

𝑉

(2.94)

𝐴𝑡

with a given body geometry 𝑉 and given values for C, b, and t. The boundary 𝐴 of 𝑉 is composed of 𝐴𝑢 and 𝐴𝑡 , whereby 𝐴 = 𝐴𝑡 ∪ 𝐴𝑢 and 𝐴𝑡 ∩ 𝐴𝑢 = 0. Displacement boundary conditions or Dirichlet conditions are prescribed on 𝐴𝑢 and force boundary conditions or Neumann conditions on 𝐴𝑡 with t = n ⋅ 𝝈 with the boundary normal n. Displacement boundary conditions have to prevent rigid body displacements. Generalised strains 𝝐, 𝛿𝝐 are derived from the generalised displacements u, 𝛿u by a differential operator depending on the type of the structural problem under consideration. The trial functions according to Eq. (2.18) and test functions according to Eq. (2.57) are assumed to belong to a Sobolev function space 𝐻 (→ square integrable functions (Bathe 1996, 4.3.4)) defined over the body 𝑉 and to fulfil the displacement boundary conditions. Equation (2.94) can be written in a general form as 𝑎(u, v) = (f, v) ∀v ∈ 𝐻

(2.95)

31

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2 Finite Elements Overview

with a symmetric, bilinear operator 𝑎(⋅, ⋅), a further linear operator (f, ⋅), and v formally replacing 𝛿u. This has the following properties: • Symmetry 𝑎(u, v) = 𝑎(v, u)

(2.96)

• Bilinearity 𝑎(𝛾1 u1 + 𝛾2 u2 , v) = 𝛾1 𝑎(u1 , v) + 𝛾2 𝑎(u2 , v) 𝑎(u, 𝛾1 v1 + 𝛾2 v2 ) = 𝛾1 𝑎(u, v1 ) + 𝛾2 𝑎(u, v2 )

(2.97)

• Linearity (f, 𝛾1 v1 + 𝛾2 v2 ) = 𝛾1 (f, v1 ) + 𝛾2 (f, v2 )

(2.98)

A norm maps a function v into a non-negative number. Sobolev norms ||v||𝑖 of order 𝑖 are used in this context (Bathe 1996, 4.3.4, (4.76)). Sobolev norms are built from integration of squares of functions and squares of their derivatives up to order 𝑖. It is assumed that 𝑖 = 1 is appropriate for the following. It can then be shown that 𝑎 has the properties • Continuity ∃ 𝑀 > 0∶

||𝑎(v , v )|| ≤ 𝑀 ‖v ‖ ‖v ‖ 1 1 2 1 | 1 2 |

∀v1 , v2 ∈ 𝐻

(2.99)

• Ellipticity ∃ 𝛼 > 0∶

2

𝑎(v, v) ≥ 𝛼 ‖v‖1

∀v ∈ 𝐻

(2.100)

• whereby 𝑀, 𝛼 depend on problem type and material values but not on v1 , v2 , v. Due to Eq. (2.100) 𝑎(v, v) ≥ 0, i.e. 𝑎 is a norm and may be physically interpreted as energy. It is twice the internal strain energy. It can be shown that the problem Eq. (2.95) – i.e. determine a function u ∈ 𝐻 such that Eq. (2.95) is fulfilled for all v ∈ 𝐻 – has a unique solution u, see, e.g. Bathe (1996, 4.3). This is the exact solution of the mathematical model (Figure 2.1). Discretisation uses trial and test functions uℎ , vℎ ∈ 𝐻ℎ of a subset 𝐻ℎ ⊂ 𝐻 based upon the concept of meshes and interpolation with elements and nodes (Section 2.3). To simplify the derivations, a uniform mesh of elements is assumed with a mesh size parameter ℎ, e.g. a diameter or length of a generic element. For non-uniform meshes, see Bathe (1996, 4.3.5). The approximate solution uℎ ∈ 𝐻ℎ of Eq. (2.95) is determined by 𝑎(uℎ , vℎ ) = (f, vℎ ) ∀vℎ ∈ 𝐻ℎ

(2.101)

The difference between approximate and exact solution yields the discretisation error e ℎ = u − uℎ

(2.102)

The approximation uℎ is known for 𝐻ℎ given as it can be determined according to the procedure described in Section 2.6. The error eℎ has to be estimated. The approximate solution has the following properties:

2.9 Discretisation Errors

• The orthogonality of error, see Bathe (1996, (4.86)) 𝑎(eℎ , vℎ ) = 0 ∀vℎ ∈ 𝐻ℎ

(2.103)

• The energy of approximation is smaller than exact energy (Bathe 1996, (4.89)) 𝑎(uℎ , uℎ ) ≤ 𝑎(u, u)

(2.104)

• The energy of error is minimised (Bathe 1996, (4.91)) 𝑎(eℎ , eℎ ) ≤ 𝑎(u − vℎ , u − vℎ ) ∀vℎ ∈ 𝐻ℎ

(2.105)

Combination of Eqs. (2.100), (2.105), and (2.99) leads to 2

2

𝛼 ‖eℎ ‖1 = 𝛼 ‖u − uℎ ‖1 ≤ 𝑎(eℎ , eℎ ) 2

= inf 𝑎(u − vℎ , u − vℎ ) ≤ 𝑀 inf ‖u − vℎ ‖1 vℎ ∈𝐻ℎ

vℎ ∈𝐻ℎ

(2.106)

where inf is infimum, the largest lower bound 2). This is rewritten as ‖u − uℎ ‖1 ≤ 𝑐 𝑑(u, 𝐻ℎ )

(2.107)

with 𝑑(u, 𝐻ℎ ) = inf ‖u − vℎ ‖1 , vℎ ∈𝐻ℎ

𝑐=

√ 𝑀∕𝛼

(2.108)

The quantity 𝑑 is a ‘distance’ of functions in 𝐻ℎ to the exact solution u, 𝑐 depends on the structural problem type and the values of its parameters but not on 𝐻ℎ . ◀

Convergence means uh → u or ‖u − uh ‖1 → 0 with mesh size h → 0.

Convergence can be reached with an appropriate selection of function spaces 𝐻ℎ , thereby reducing the distance 𝑑(u, 𝐻ℎ ). A more precise statement is possible using interpolation theory. This introduces the interpolant 3) u𝑖 ∈ 𝐻ℎ of the exact solution u. Complete polynomials 4) of degree 𝑘 are used for discretisation and interpolation. Interpolation theory estimates the interpolation error with ‖u − u𝑖 ‖1 ≤ 𝑐̂ ℎ𝑘 ‖u‖𝑘+1

(2.109)

with the mesh size ℎ and a constant 𝑐, ̂ which is independent of ℎ (Bathe 1996, (4.99)). The quantity ‖u‖𝑘+1 is the 𝑘 + 1-order Sobolev norm of the exact solution. On the 2) ‖u − vℎ ‖1 , vℎ ∈ 𝐻ℎ is a subset of real numbers. inf vℎ ∈𝐻ℎ ‖u − vℎ ‖1 is the largest number less or equal to the numbers in this subset. 3) u, and u𝑖 coincide at nodes but generally not apart from nodes with u𝑖 ≠ uℎ . 4) A polynomial in 𝑥, 𝑦 is complete of order 1 if it includes 𝑥, 𝑦, complete of order 2 if of order 1 and including 𝑥 2 , 𝑥𝑦, 𝑦 2 , complete of order 3 if complete of order 2 and including 𝑥 3 , 𝑥 2 𝑦, 𝑥𝑦 2 , 𝑦 3 , and so on.

33

34

2 Finite Elements Overview

other hand, a relation inf vℎ ∈𝐻ℎ ‖u − vℎ ‖1 ≤ ‖u − u𝑖 ‖1 must hold as u𝑖 ∈ 𝐻ℎ . Using this and Eqs. (2.107) and (2.109) yields ‖u − uℎ ‖1 ≤ 𝑐𝑐̂ ℎ𝑘 ‖u‖𝑘+1

(2.110)

The value 𝑐𝑐̂ can be merged to 𝑐, which depends on the structural problem type and the values of its parameters but not on ℎ. A further merging of 𝑐 and ‖u‖𝑘+1 leads to the well-known relation ‖u − uℎ ‖1 ≤ 𝑐 ℎ𝑘

(2.111)

The following conditions for convergence can be derived (Bathe 1996, 4.3.2): • A prerequisite is theoretical integrability of all quantities. This leads to requirements for the integrands of the energy 𝑎 and the arguments of the Sobolov norms, which are uℎ , vℎ , u or derivatives thereof. This corresponds to the requirement of compatibility or continuity – with a different meaning compared to Eq. (2.99) – of finite element interpolation functions – generally displacement interpolations – along inter-element boundaries. • According to Eq. (2.111), a sequence of approximate solutions uℎ with ℎ → 0 will converge 5) with respect to ‖u − uℎ ‖1 if 𝑘 ≥ 1. The case 𝑘 = 1 is treated by the patch test, i.e. the ability to model fields with constant first derivatives of finite element interpolation functions in arbitrary element configurations (Belytschko et al. 2000, 8.3.2). • The convergence rate will be higher for larger values of 𝑘, i.e. if the finite element interpolation has a higher order of completeness. Limitations of these arguments have to be mentioned. Under certain conditions the coefficient 𝑐 may become so large that acceptable solutions, i.e. a sufficiently small value ||u − uℎ ||, cannot be reached with realisable values ℎ small enough. Such a case is given, e.g. by locking of approximate solutions with incompressible or nearly incompressible materials. The locking problem motivates the inclusion of extended weak forms of equilibrium conditions. Equations (2.94) and (2.95) are weak forms of displacement-based methods, as a solution is given by a displacement field. Strains and stresses are derived from this solution. Extended weak forms allow us to involve fields for stresses and strains as independent solution variables. The most prominent are the principles of Hu–Washizu and Hellinger–Reissner (Bathe 1996, 4.4.2). An abstract extended problem definition analogous to Eq. (2.101) is given by Bathe (2001, (16)) 𝑎(uℎ , vℎ ) + 𝑏(𝝐 ℎ , vℎ ) = (f, vℎ )

∀vℎ ∈ 𝐻ℎ

𝑏(wℎ , uℎ ) − 𝑐(𝝐 ℎ , eℎ ) = 0

∀wℎ ∈ 𝑊ℎ

(2.112)

in which 𝑎, 𝑐 are symmetric bilinear operators, 𝑏 is a general bilinear operator, (f, ⋅) is a linear operator, 𝐻ℎ , 𝑊ℎ are appropriate function spaces, and uℎ ∈ 𝐻ℎ , 𝝐 ℎ ∈ 𝑊ℎ 5) Convergence with respect to the first-order Sobolev norm ‖u − uℎ ‖1 may not be sufficient if generalised strains are derived from higher derivatives of displacements, e.g. for beams, slabs, and shells. The theory has to be extended for this case.

2.9 Discretisation Errors

are the approximate solutions. In most cases, 𝝐 ℎ denotes an independent field of strains or stresses. Such an approach requires an extension of the foregoing derivations using the widely referenced inf-sup condition (Bathe 2001). The extended framework including independent interpolations for displacements, strains, and stresses may avoid locking problems to a large degree. Cases of locking risks will be discussed individually, if necessary, in the following.

35

37

3 Uniaxial Reinforced Concrete Behaviour The study of uniaxial tension gives an understanding of basic characteristics of reinforced concrete behaviour. This depends on the constituents concrete, rebars, and their interaction via bond. Bond is essential and is activated by concrete tensile cracking. Details are described in the following and demonstrated with examples.

3.1 Uniaxial Stress–Strain Behaviour of Concrete Structural elements such as bars, beams, and columns are characterised by uniaxial states of stress and strain. Thus, it is sufficient to describe the material behaviour by uniaxial stress–strain relations for such elements. This simplifies stress–strain relations to a large extent. On the other hand, different time scales have to be regarded with respect to the behaviour of concrete. This is related to the type and duration of the load application. We assume a monotonic loading, which is applied uniformly from zero to a final value and then held constant. Another aspect to be considered is the loading speed, which is the load magnitude related to clock time. Material specimens with states of stress and strain basically constant or continuous in space, i.e. homogeneous states of stress and strain, are considered. The material behaviour may be classified as follows with respect to time scales. • Short-term behaviour is observed as the immediate response of a material specimen exposed to a loading, whereby the loading speed is slow. A slow loading speed does not have any influence on the stress–strain relation. It is only influenced by the magnitude of loading. In the case of concrete, the corresponding time horizon typically ranges from minutes up to days. • Long-term behaviour extends the time horizon beyond the application time of loading. The material behaviour is observed as the delayed response of a material specimen after load application. Corresponding phenomena are creep and relaxation. The time horizon typically ranges from weeks up to years. • Highly dynamical behaviour is related to high loading speeds that influence stress– strain relations in addition to the magnitude of loading. A corresponding phenomenon is the strain rate effect, where the concrete strength is increased due to high Computational Structural Concrete – Theory and Applications, Second Edition. Ulrich Häussler-Combe. © 2023 Ernst & Sohn GmbH. Published 2023 by Ernst & Sohn GmbH.

38

3 Uniaxial Reinforced Concrete Behaviour

strain rates. The time horizon is restricted to milliseconds, whereby very high loading is reached within very short times with a high loading speed. The strain rate effect is still an open object of research. A first approach is given in CEB-FIP2 (2012). Observation of uniaxial short-term behaviour under monotonic loading is the first approach to characterise the behaviour of materials. A typical stress–strain relation as is observed for concrete with a cylindrical or cubic specimen under uniaxial compression is shown in Figure 3.1a. The relation has the following parts: • The initial linear part with the initial Young’s modulus 𝐸𝑐 . • A nonlinear hardening part with decreasing tangential material stiffness but still increasing stress. • A nonlinear softening part with negative tangential material stiffness and decreasing stress. Characteristic values are given by the maximum stress, which corresponds to the compressive strength 𝑓𝑐 (unsigned), the corresponding strain 𝜖𝑐1 (signed), and the ultimate failure strain 𝜖𝑐𝑢1 (signed). A large variety of analytical forms is available for such a course. The approach of Saenz (Chen and Saleeb 1994, 8.8.1) is chosen as an example. Stress 𝜎 depending on strain 𝜖 is given by 𝐸𝑐 𝜖

𝜎= 1+(

𝐸𝑐 𝐸𝑐1

− 2)

𝜖 𝜖𝑐1

+(

𝜖 𝜖𝑐1

2

,

𝜖𝑐𝑢1 ≤ 𝜖 ≤ 0

(3.1)

)

with the secant modulus 𝐸𝑐1 = −

𝑓𝑐 𝜖𝑐1

(3.2)

at compressive strength 𝑓𝑐 . Equation (3.1) yields 𝜎 = −𝑓𝑐 for 𝜖 = 𝜖𝑐1 . Alternatives are described by Modelcode 2010 (CEB-FIP2 2012, 5.1.8.1) and Eurocode 2 (EN 1992-1-1 2004, 3.1.5). The tangential material stiffness matrix (Eq. (2.50)) is given

Figure 3.1 Uniaxial compressive stress–strain curve of concrete.

3.1 Uniaxial Stress–Strain Behaviour of Concrete

by d𝜎 𝐶𝑇 = = d𝜖

𝐸𝑐 (1 − (1 + (

𝐸𝑐 𝐸𝑐1

− 2)

𝜖2 2 𝜖𝑐1

𝜖 𝜖𝑐1

) +

𝜖2 2 𝜖𝑐1

2

(3.3)

)

with 𝐶𝑇 = 𝐸𝑐 for 𝜖 = 0. Furthermore, 𝐶𝑇 = 0 for 𝜖 = 𝜖𝑐1 and 𝐶𝑇 < 0 for 𝜖 < 𝜖𝑐1 , |𝜖| > |𝜖𝑐1 |. Uniaxial tension in a simplified first approach is described by

𝜎=

⎧

𝐸𝑐 𝜖 , 𝜖 ≤ 𝑓𝑐𝑡 ∕𝐸𝑐 ⎨0, else ⎩

(3.4)

with the uniaxial tensile strength 𝑓𝑐𝑡 . The tangential stiffness corresponds to the initial stiffness 𝐶𝑇 = 𝐸𝑐 for 𝜖 ≤ 𝑓𝑐𝑡 ∕𝐸𝑐 . ◀

These types of uniaxial stress–strain relations are the basis for the modelling and design of structural elements like bars and beams.

Spatial dimensions of reinforced concrete structures have a scale of [m] for spans or cross-sections. A lower bound of this is marked by the dimension of cylindrical or cubic concrete specimen with dimensions of 0.1−0.3[m]. This corresponds to the macroscale. A material description as is given by Eqs. (3.1)–(3.4) is derived from the macroscale specimen and is applicable to macroscale structures only. Other spatial scales are the mesoscale with spatial dimensions of millimetres [m−3 ] and the microscale with spatial dimensions of micrometres [m−6 ]. A strict demarcation between spatial scales is not possible but depends on the particular problem under consideration. Further aspects regarding material models, scales, and the fundamental concept of homogeneity are described in Section 6.1.1. Concrete obviously cannot be considered as homogeneous in the mesoscale. A spatial resolution of millimetres reveals its composition of aggregates and mortar leading to discontinuities in material properties. But the comprehension of the mesoscale is necessary to understand the failure of concrete. A model for the composition of aggregates and mortar within a concrete specimen is shown in Figure 3.2. Furthermore, this figure schematically shows the mechanism of compressive force transfer that concentrates upon aggregates that are relatively stiff compared to mortar. This is obviously connected with a redirection of forces leading to lateral tensile forces in local regions that have to be resisted by the mortar. ◀

Thus, concrete will fail due to internal lateral tension under homogeneous uniaxial compressive loading.

This is a diffuse failure – in contrast to local failure under tensile loading, see the following – as failure spreads throughout a whole specimen and is not localised.

39

40

3 Uniaxial Reinforced Concrete Behaviour

Figure 3.2 Simplified model for force transfer in the composition of aggregates and mortar.

(a)

(b)

Figure 3.3 (a) Cracking in mesoscale with the process zone. (b) Uniaxial tensile stress– strain behaviour.

A formation of cracks occurs with crack directions aligned to the load direction. But this requires an adequate experimental set-up with minimised lateral constraints on the loaded faces of a plain concrete specimen. We also consider uniaxial tensile failure in the mesoscopic scale. The process is shown in Figure 3.3a. The following successive stages can be distinguished: • Development of microcracks with random distribution but progressing crack alignment normal to the stress direction. • Coalescence of microcracks to larger cracks that form several mesoscopic branches and are still bridged by aggregates acting as crack bridges. • Breaking of crack bridges and fusion of branched cracks into a macrocrack. On the one hand, notions of stresses and strains obviously become questionable in this view. On the other hand, macroscale and homogenisation (Section 6.1.1) allow us to derive stress–strain relations. With respect to tension, a typical course is shown in Figure 3.3b. This is again characterised by an initial linear part, a hardening part with increasing strains and stresses, and a tension softening part with increasing strains and decreasing stresses. Uniaxial tension occurs with tension bars. The phenomenon of softening leads to a localisation of strains within tension bars. Due to scatter, the tensile strength measured over cross-sections varies along a bar. Tensile failure will start in the cross-

3.1 Uniaxial Stress–Strain Behaviour of Concrete

Figure 3.4 Scheme of homogenised localisation within a crack band in a tension bar.

section with the smallest tensile strength. With ongoing tension softening, the tensile force applied to the bar has to be reduced to avoid a sudden failure. • The failing cross-section reaches a point 𝐵 on the stress–strain curve (Figure 3.3b), while other cross-sections, which are still in the hardening range, unload to point 𝐴. • Strains increase in the failing cross-section, while strains decrease in the other cross-sections with ongoing unloading. • Softening spreads over a process zone with some thickness 𝑏𝑤 as cracking involves microcracking, mesoscopic branches, and crack bridges (Figure 3.3a). Synonyms for process zone are crack band and cohesive zone. • Relatively high homogenized strains with mean 𝜖𝑐 develop in the process zone compared to decreasing strains 𝜖𝑢 in the rest of the bar; see Figure 3.4. • The process ends with a macrocrack in the failing cross-section, while the rest of the bar has zero stresses and strains. This qualitative description is illustrated with the following example. Example 3.1: Tension Bar with Localisation

A scheme of system and discretisation is shown in Figure 3.5. The following properties are chosen: • Uniaxial two-node bar elements are used for discretisation (Section 2.3). The length of the whole bar is chosen with 𝐿 = 0.5 m and its cross-section with 𝐴𝑐 = 0.1 ⋅ 0.1 m2 . The element length has to be small compared to the length of the crack band to resolve the high strain gradient within it. Thus, the element length is assumed with 𝐿𝑒 = 0.001 m, leading to 500 elements. • The material properties are chosen according to a concrete grade C 40 as is described in CEB-FIP2 (2012, 5.1.5, 5.1.7) with an initial Young’s modulus 𝐸𝑐 = 36 000 MN∕m2 and a tensile strength 𝑓𝑐𝑡 = 3.5 MN∕m2 . The damage formulation (Section 6.6) is used to model material behaviour. This formulation allows us to describe tension softening.

41

42

3 Uniaxial Reinforced Concrete Behaviour

Figure 3.5 Example 3.1. Scheme of system and discretisation.

(a)

(b)

Figure 3.6 Example 3.1. (a) Reaction force displacement curve. (b) Strains along the bar.

On the other hand, modelling of tension softening with finite elements needs a regularisation (Section 7.3). The gradient damage approach (Section 7.5.1) is used for the example. This requires a characteristic length 𝑅 as material parameter that corresponds to the length of the crack band and is chosen with 𝑅 = 0.03 m. • Boundary conditions are prescribed with zero displacements on the left-hand side and with a prescribed displacement 𝑢𝑁 = 0.1333 ⋅ 10−3 m on the right corresponding to a mean strain 𝜖 = 0.2667 ⋅ 10−3 . A clamping is assumed at both ends, which prevents damage at the ends only. • An incrementally iterative approach (Section 2.8.2) is used for nonlinear problem solving. The size of loading increments is determined with arc length control; the prescribed displacement increment size is chosen such that a norm of the vector of local displacements has a prescribed fixed value (Section A.4). The computation leads to the following results: • The load displacement curve is shown in Figure 3.6b. This starts with a linear behaviour followed by nonlinear hardening while reaching the tensile strength. But in contrast to Figure 3.3b, which exposes material behaviour, this curve shows a snap-back after a short range of softening. A snap-back is characterised by decreasing reaction forces and simultaneously decreasing displacements. • In the case of a relatively short length of the softening crack band, its elongation cannot compensate for elastic shortening of the elastic parts with the reduction

3.1 Uniaxial Stress–Strain Behaviour of Concrete

of stress. The prescribed displacement of the right end has to be reduced in order to maintain equilibrium. The resulting bar contraction proceeds when stress is reduced to zero. The snap-back characteristics depend on the ratio of the length of the crack band compared to the total length of the bar. With smaller ratios, the snap-back behaviour becomes more pronounced. The whole process may be unstable under quasi-static conditions and is difficult to realise experimentally, as displacement control of the softening area is required. • Strain distributions along the bar for two loaded states are shown in Figure 3.6b. Curve 𝐴 shows the strain distribution before reaching the tensile strength in the hardening range, that is, point 𝐴 in Figure 3.6a. Strain moderately increases in the mid-range of the bar due to prescribed zero non-local damage on both ends. Curve 𝐵 in Figure 3.6b shows the strain distribution in the softening range, that is, point 𝐵 in Figure 3.6a. Strains strongly increase as a localisation occurs within a short length. In a real specimen, the localisation will not centre exactly in the mid-point but in the weakest cross-section. This cross-section will arise due to the random variation of material strength. Computation of whole structural elements has to rely on the macroscale and the homogenisation of concrete behaviour. A corresponding course 1) of strains 𝜖𝑐 (𝑥) across the process zone with 𝑥1 ≤ 𝑥 ≤ 𝑥2 and a crack band width 𝑏𝑤 = 𝑥2 − 𝑥1 is shown in Figure 3.4. The size of the crack band width 𝑏𝑤 is estimated from experimental observation with two to three times the largest aggregate size (Figure 3.3a). This leads to the fictitious crack with a fictitious crack width (Figure 3.4) 𝑥2

𝑤 = ∫ 𝜖𝑐 (𝑥) d𝑥

(3.5)

𝑥1

as the difference between the displacements of the left and right-hand cross-sections bounding the process zone. A further simplification leads to 𝑤 = 𝑏𝑤 𝜖𝑐

(3.6)

with the mean value 𝜖𝑐 of 𝜖𝑐 (𝑥). The crack strain 𝜖𝑐 corresponds to the strain of the stress–strain relation shown in Figure 3.3b. Thereby the softening range is bounded by the strain 𝜖𝑐𝑡 corresponding to the tensile strength 𝑓𝑐𝑡 and the strain 𝜖𝑐𝑢 where no stresses are transferred anymore: 𝜎𝑐 = 0 for 𝜖𝑐 ≥ 𝜖𝑐𝑢 . ◀

A stress transfer across a crack in progress related to a fictitious crack width is denoted as cohesive crack; see also Section 7.1.

As cracking is an irreversible process leading to the creation of new surfaces some amount of energy is dissipated within the process zone. The volume specific crack 1) Actually this is a calculational course under the assumption of a homogenised material.

43

44

3 Uniaxial Reinforced Concrete Behaviour

energy is given by 𝜖𝑐𝑢

𝑔𝑓 = ∫ 𝜎(𝜖𝑐 ) d𝜖𝑐

(3.7)

𝜖𝑐𝑡

with 𝜎(𝜖𝑐 ) according to Figure 3.3b. Its integration along the thickness of the process zone leads to the surface specific crack energy or simply crack energy (Figure 3.4) 𝑥2

𝐺𝑓 = ∫ 𝑔𝑓 (𝑥) d𝑥 ≈ 𝑏𝑤 𝑔𝑓

(3.8)

𝑥1

With respect to Eq. (3.6), an alternative form is given with 𝑤𝑐𝑟

𝐺𝑓 = ∫ 𝜎(𝑤) d𝑤

(3.9)

0

with 𝜎 depending on the fictitious crack width 𝑤, and the critical crack width 𝑤𝑐𝑟 corresponding to 𝜖𝑐𝑢 . The crack energy 𝐺𝑓 indicates energy dissipation due to the crack surface creation. Due to current state of knowledge it is assumed as a material parameter so that Eq. (3.8) leads to a constraint for a 𝜎−𝜖𝑐 or a 𝜎−𝑤 relation. ◀

Crack energy or energy dissipation due to cracking of concrete contributes to the ductility of concrete structures, i.e. the ability to deform while internal forces retain their level and insofar is a significant property of structures.

The behaviour of stresses in the softening branch has been investigated to a large extent (Petersson 1981; Gopalaratnam and Shah 1985; Reinhardt, Cornelissen and Hordijk 1986). A typical parabolic approximation for stresses depending on the fictitious crack width is shown in Figure 3.7 and is described with ) ( 𝜎 = 𝑓𝑐𝑡 𝛿2 − 2𝛿 + 1 ,

0≤𝛿=

𝑤 ≤1 𝑤𝑐𝑟

(3.10)

1

leading to a fracture energy 𝐺𝑓 = 𝑓𝑐𝑡 𝑤𝑐𝑟 . This yields a value 𝑤𝑐𝑟 for prescribed val3 ues 𝐺𝑓 , 𝑓𝑐𝑡 generally in accordance with experimental data, which for 𝑤𝑐𝑟 are anyway hard to estimate regarding the background given in Figure 3.3. The extension

Figure 3.7 Approximation of stresses with the fictitious crack width.

3.2 Long–Term Behaviour – Creep and Imposed Strains

of the uniaxial case to modelling of multi-axial cracking is described in Section 7.1 and a simple biaxial implementation in Section 8.2. While uniaxial material modelling of concrete is sufficient for structural elements like bars and beams, a multi-axial approach is necessary for plates, slabs, and shells. Multi-axial material modelling of concrete is described in Chapter 6.

3.2 Long–Term Behaviour – Creep and Imposed Strains Creep occurs as a delayed response of a material specimen after load application. A concrete specimen exposed to a loading within minutes will have increasing strains within months, while its loading is held constant. The complementary phenomenon to creep is relaxation. Deformations imposed to a concrete specimen within minutes will lead to immediate stresses, but these stresses will decrease within months, while the imposed deformation is held constant. Mechanisms of creep and relaxation have to be treated in the microscale of materials and are connected to a slow redistribution in the arrangement of microstructures and, in the case of cement, water. All solids undergo creep and relaxation, but its extent is different for different materials. It is relatively large for cementitious materials and, thus, for concrete. A first approach to describe the development of uniaxial strain 𝜖 with time 𝑡 for a constant uniaxial stress 𝜎0 is given by 𝜖(𝑡) = 𝐽(𝑡) 𝜎0

(3.11)

with a creep function 𝐽(𝑡). The creep function is specific for every material. The creep strain is proportional to the applied stress with this approach. Such a linear characteristic with respect to stress or linear creep is valid for moderate stress levels relative to strength. A qualitative curve of a uniaxial creep strain derived from experimental data is shown in Figure 3.8. Equation (3.11) is generalised as 𝑡

̇ d𝜏 , 𝜖(𝑡) = ∫ 𝐽(𝑡, 𝜏) 𝜎(𝜏)

0≤𝜏≤𝑡

(3.12)

0

for stresses 𝜎 variable in time 𝑡 with a stress time derivative 𝜎(𝜏) ̇ and an extended creep function 𝐽(𝑡, 𝜏). It describes the effect of a stress increment occurring in a previous time 𝜏 on strain in the current time 𝑡. The following general approach is appropriate within this context 𝐽(𝑡, 𝜏) =

∑𝑁𝜇

𝐽 (𝑡, 𝜏) 𝜇=0 𝜇

(3.13)

with 𝐽𝜇 (𝑡, 𝜏) =

) 1 ( 1 − e−[𝑦𝜇 (𝑡)−𝑦𝜇 (𝜏)] , 𝐸𝜇

𝑦𝜇 (𝜏) = (

𝜏 ) 𝜏𝜇

𝑞𝜇

(3.14)

45

46

3 Uniaxial Reinforced Concrete Behaviour

Figure 3.8 Uniaxial strain depending on time for a material with creep.

and material parameters 𝜏𝜇 , 𝑞𝜇 , 𝐸𝜇 . For alternative approaches see, e.g. Bažant and Baweja (1995), Jirásek and Bažant (2001, Sec. 28). The parameter 𝜏𝜇 has a dimension of time and 𝐸𝜇 a dimension of stress. Equations (3.12)–(3.14) are written as 𝜖(𝑡) =

𝑡

∑𝑁𝜇

𝜖 (𝑡) 𝜇=0 𝜇

,

𝜖𝜇 (𝑡) = ∫ 𝐽𝜇 (𝑡, 𝜏) 𝜎(𝜏) ̇ d𝜏

(3.15)

0

A time parameter 𝜏𝜇 approaching zero (𝜏𝜇 → 0) yields a constant compliance 𝐽𝜇 = 1∕𝐸𝜇 as a special case of creep. A choice 𝑁𝜇 = 0, 𝜏0 = 0, 𝐸0 = 𝑐𝑜𝑛𝑠𝑡. reproduces linear elasticity 𝜖(𝑡) =

1 𝜎(𝑡) 𝐸0

(3.16)

This has at least to be extended with 𝑁𝜇 = 1, 𝜏𝜇 > 0 to gain a qualitative curve as is shown in Figure 3.8. We consider a sudden jump of stress at a time 𝜏0 from zero to a value 𝜎0 . The Dirac–Delta function 𝛿(𝜏 − 𝜏0 ) is used to describe the time derivative of stress for this case. It is defined as 𝛿(𝜏 − 𝜏0 ) =

⎧ ∞ 𝜏 = 𝜏0 , ⎨0 𝜏 ≠ 𝜏0 ⎩

∞

∫

−∞

𝛿(𝜏 − 𝜏0 ) d𝜏 = 1

(3.17)

Its integral is given by the Heaviside function

𝐻(𝜏 − 𝜏0 ) =

⎧

0 𝜏 < 𝜏0 ⎨ 1 𝜏 ≥ 𝜏0 ⎩

(3.18)

The Dirac–Delta function has the property 𝜏2

∫

𝜏1

𝐹(𝜏) 𝛿(𝜏 − 𝜏0 ) d𝜏 = 𝐹(𝜏0 ) ,

𝜏1 < 𝜏0 < 𝜏2

(3.19)

3.2 Long–Term Behaviour – Creep and Imposed Strains

We write 𝜎(𝜏) = 𝜎0 𝐻(𝜏 − 𝜏0 ), 𝜎(𝜏) ̇ = 𝜎0 𝛿(𝜏 − 𝜏0 ) and obtain from Eqs. (3.15) and (3.19) 𝑡

𝜖𝜇 (𝑡) = 𝜎0 ∫ 𝐽𝜇 (𝑡, 𝜏) 𝛿(𝜏 − 𝜏0 ) d𝜏 = 𝜎0 𝐽𝜇 (𝑡, 𝜏0 )

(3.20)

0

Finally, regarding Eq. (3.14) with 𝜏0 = 0 𝜖𝜇 (𝑡) = 𝜎0 𝐽𝜇 (𝑡, 0) = 𝜎0

) 1 ( 1 − e−𝑦𝜇 (𝑡) , 𝐸𝜇

( )𝑞𝜇 𝑦𝜇 (𝑡) = 𝑡∕𝜏𝜇

(3.21)

Thus, the creep strain 𝜖𝜇 starts with zero for 𝑡 = 0 and has an asymptotic value 𝜎0 ∕𝐸𝜇 . The approach to the asymptotic value during time is ruled by the parameters 𝜏𝜇 , 𝑞𝜇 . A variety of functions 𝐽𝜇 allows us to adapt to experimental creep data with any desired accuracy with a calibration of 𝑁𝜇 times a set 𝐸𝜇 , 𝜏𝜇 , 𝑞𝜇 . The assumption 𝑞𝜇 = 1 allows for more simplifications. Regarding Eq. (3.14) all terms in Eq. (3.15) are trivially integrated, leading to −

𝑡

𝑡

𝜏

𝜎(𝑡) e 𝜏𝜇 ∫ 𝜎(𝜏) 𝜖𝜇 (𝑡) = − ̇ e 𝜏𝜇 d𝜏 𝐸𝜇 𝐸𝜇

(3.22)

0

The time derivative of this strain is given by −

𝑡

𝑡

𝜏

1 e 𝜏𝜇 ∫ 𝜎(𝜏) ̇ e 𝜏𝜇 d𝜏 𝜖̇ 𝜇 (𝑡) = 𝜏𝜇 𝐸𝜇

(3.23)

0

Thus, the strain 𝜖𝜇 (𝑡) fulfils the differential equation 𝜂𝜇 𝜖̇ 𝜇 (𝑡) + 𝐸𝜇 𝜖𝜇 (𝑡) = 𝜎(𝑡) ,

𝜂𝜇 = 𝐸𝜇 𝜏𝜇

(3.24)

with the viscosity 𝜂. Equation (3.24) describes a Kelvin–Voigt element with a spring and a viscous damper in parallel; see Figure 3.9a. A simple superposition according to Eq. (3.151 ) yields a Kelvin–Voigt chain, i.e. a spring and a Kelvin–Voigt element in a row (Figure 3.9b). An alternative basic combination with spring and damper in a row is given by the Maxwell element (Figure 3.9a). The Maxwell element can be treated in analogy to the Kelvin–Voigt element by formally interchanging stress and strain and using a relaxation function 𝑅(𝑡, 𝜏) instead of a creep function 𝐽(𝑡, 𝜏). Creep and relaxation functions can be inverted into each other mathematically. A simple combination of a spring and a Maxwell element in parallel leads to a Maxwell series (Figure 3.9b). ◀

Kelvin–Voigt elements and Maxwell elements are the basic blocks of visco-elasticity. Together with springs they may be combined in series and/or chains to form rheological models to describe creep and relaxation.

47

48

3 Uniaxial Reinforced Concrete Behaviour

(a)

(b)

Figure 3.9 (a) Kelvin–Voigt and Maxwell elements. (b) Chain and series.

The Kelvin–Voigt chain is used in the following. The stress is the same in each member of the chain, but the member strains 𝜖0 and 𝜖1 add up to the total strain 𝜖(𝑡) = 𝜖0 (𝑡) + 𝜖1 (𝑡) ,

𝜖(𝑡) ̇ = 𝜖̇ 0 (𝑡) + 𝜖̇ 1 (𝑡)

(3.25)

The relations for the partial strains are obtained with 𝐸0 𝜖0 (𝑡) = 𝜎(𝑡) ,

𝜂1 𝜖̇ 1 (𝑡) + 𝐸1 𝜖1 (𝑡) = 𝜎(𝑡)

(3.26)

The combination of Eqs. (3.25) and (3.26) results in 𝜖1 (𝑡) = 𝜖(𝑡) −

𝜎(𝑡) , 𝐸0

𝜖̇ 1 (𝑡) = 𝜖(𝑡) ̇ −

𝜎(𝑡) ̇ 𝐸0

(3.27)

This is inserted in Eq. (3.262 ) to yield, after rearrangement, 𝜎(𝑡) ̇ +

𝐸0 + 𝐸1 𝐸0 𝐸1 𝜎(𝑡) = 𝐸0 𝜖(𝑡) ̇ + 𝜖(𝑡) 𝜂1 𝜂1

(3.28)

We introduce a final stiffness 1∕𝐸 = 1∕𝐸0 + 1∕𝐸1 , consider 𝐸0 as an initial stiffness, and introduce a dimensionless creep coefficient 𝜑 defined by 𝜑 = 𝐸0 ∕𝐸 − 1. This leads to 𝐸1 = 𝐸0 ∕𝜑, and Eq. (3.28) may be reformulated as 𝜎(𝑡) ̇ +

1+𝜑 𝐸0 𝜎(𝑡) = 𝐸0 𝜖(𝑡) 𝜖(𝑡) , ̇ + 𝜁 𝜁

𝜁=

𝜑 𝜂1 𝐸0

(3.29)

The parameter 𝜁 has the dimension of time and is called creep time in the following. The differential equation Eq. (3.29) is a mathematical model for the conceptual model (Figure 2.1a) of the Kelvin–Voigt chain (Figure 3.9b). ◀

The Kelvin–Voigt chain is a simple model for uniaxial linear creep and relaxation characterised by an initial Young’s modulus E0 , a creep coefficient 𝜑, and a creep time 𝜁 . It can be used to model creep for bars and beams.

The mathematical model allows for variable stresses and strains. It is rewritten in a more general form as ̇ + 𝑉 𝜖(𝑡) − 𝑊 𝜎(𝑡) 𝜎(𝑡) ̇ = 𝐸0 𝜖(𝑡)

(3.30)

3.2 Long–Term Behaviour – Creep and Imposed Strains

for use in numerical solution methods, see Section 2.8.2 and Appendix A.2, whereby 𝑉=

𝐸0 , 𝜁

𝑊=

1+𝜑 𝜁

(3.31)

according to Eqs. (2.82) and (A.23). Its application is demonstrated with Examples 3.2 and 4.3. To get a better insight of the meaning of 𝜑 and 𝜁 the Eq. (3.29) may be analytically solved for a constant stress 𝜎(𝑡) = 𝜎0 , 𝜎̇ = 0 with an initial strain 𝜖(0) = 𝜎0 ∕𝐸0 . The solution is 𝜖(𝑡) =

𝑡 𝜎0 − [1 + 𝜑 (1 − e 𝜁 )] 𝐸0

(3.32)

The asymptotic strain is 𝜖asym = (1 + 𝜑)𝜎0 ∕𝐸0 with a creep portion 𝜑 𝜎0 ∕𝐸0 , i.e. 𝜑times the initial strain. The value of 𝜑 has to be determined from experimental data or see EN 1992-1-1 (2004, 3.1.4), CEB-FIP2 (2012, 5.9.1.4). Experimental data also provide a time 𝑡⋆ , where a fraction 𝛼 of the asymptotic creep part occurs with 0 ≤ 𝛼 < 1. This leads to 1 − e−𝑡

⋆ ∕𝜁

=𝛼

→

𝜁=−

𝑡⋆ ln(1 − 𝛼)

(3.33)

If, e.g. half of asymptotic creep occurs at time 𝑡⋆ with 𝛼 = 0.5, then 𝜁 ≈ 1.44 𝑡⋆ . Thus, at least three points of an experimental uniaxial stress–strain relation are reproduced by the Kelvin–Voigt chain: the immediate strain after load application, the final asymptotic strain and an intermediate value at a time 𝑡⋆ . A better approximation of experimental data requires extended Kelvin–Voigt chains or the combination of Maxwell elements. Strains resulting from mechanical loading have been discussed up to now. But strains are also imposed by a change of temperature or, in the case of concrete, shrinkage. Shrinkage results from enduring drying of concrete. Concrete shrinkage strains 𝜖𝑐𝑠 mainly depend on time, humidity conditions, and the ratio of surface to volume (EN 1992-1-1 (2004, 3.1.4), CEB-FIP2 (2012, 5.9.1.4)). Uniaxial temperature strains are given by 𝜖𝑇 = 𝛼𝑇 Δ𝑇

(3.34)

with the thermal expansion coefficient 𝛼𝑇 and a temperature change Δ𝑇 (signed). The total measurable strain results from stresses and imposed strains. For the uniaxial case, it is given by 𝜖=

𝜎 + 𝜖𝐼 , 𝐶

𝜖𝐼 = 𝜖𝑇 + 𝜖𝑐𝑠

(3.35)

with the uniaxial stress 𝜎 and a scalar material stiffness 𝐶 (Eq. (2.47)). This leads to the basic form of a material law regarding imposed strains 𝜎 = 𝐶 (𝜖 − 𝜖𝐼 ) ,

𝜖𝐼 = 𝜖𝑇 + 𝜖𝑐𝑠

(3.36)

49

50

3 Uniaxial Reinforced Concrete Behaviour

with an incremental formulation (Eq. (2.50)) 𝜎̇ = 𝐶𝑇 (𝜖̇ − 𝜖̇ 𝐼 ) ,

𝜖̇ 𝐼 = 𝜖̇ 𝑇 + 𝜖̇ 𝑐𝑠

(3.37)

A bar that is fully constrained, i.e. measurable strain is zero (𝜖 = 0 in Eq. (3.36)), gets stresses 𝜎𝐼 = −𝐶 𝜖𝐼 , that are tensile stresses 𝜎𝐼 with an imposed contraction 𝜖𝐼 < 0 and compressive stresses 𝜎𝐼 with an imposed elongation 𝜖𝐼 > 0. ◀

Constraint stresses resulting from imposed strains are proportional to the stiffness of the material or the stiffness of the structure.

Aspects of temperature loading for beams are discussed in Section 4.6 and with Example 4.4. Stresses from constraints may be reduced by creep or relaxation. To model this behaviour Eq. (3.30) describing creep and relaxation is extended in analogy to Eq. (3.37) to include imposed strains 𝜎(𝑡) ̇ = 𝐶𝑇 [𝜖(𝑡) ̇ − 𝜖̇ 𝐼 (𝑡)] + 𝑉 [𝜖(𝑡) − 𝜖𝐼 (𝑡)] − 𝑊 𝜎(𝑡)

(3.38)

Solution methods for this type of material model leading to a transient problem are described in Appendix A.2. This is also demonstrated with the following example. Example 3.2: Tension Bar with Creep and imposed Strains

The following properties are chosen: • A homogeneous uniaxial state of stress and strain is assumed to expose pure material behaviour. Thus, the absolute spatial dimensions and number of finite elements are irrelevant. Nevertheless, some data have to be chosen with a bar length of 𝐿 = 1.0 m, a cross-sectional area 𝐴𝑐 = 1.0 m2 and a discretisation with five uniaxial two-node bar elements (Section 2.3). • Material properties are the focus of the problem. Concrete is chosen with a Young’s modulus 𝐸𝑐 = 30 000 MN∕m2 . Creep properties are assumed with a creep coefficient 𝜑 = 2.0 and a time 𝑡⋆ = 100 [d] for 𝛼 = 0.5, i.e. half of total creep occurs after 100 days for a constant stress load. With Eq. (3.33), creep time is given by 𝜁=−

100 = 144 d ln 0.5

(3.39)

Equation (3.31) leads to 𝑉 = 207.94

MN , m2 d

𝑊 = 0.020 794

1 d

(3.40)

• With respect to boundary conditions, the displacement of one bar end is prescribed with zero, while a stress or a prescribed displacement is applied at the other bar end. • An incrementally iterative approach according to a transient analysis (Appendix A.2) is chosen for problem solving. The time step is assumed with Δ𝑡 = 10 days while a period of 500 days is considered.

3.2 Long–Term Behaviour – Creep and Imposed Strains

(a)

(b)

Figure 3.10 Example 3.2. Time dependencies. (a) Strain. (b) Stress.

The following cases are considered for the computation: 1. A loading 𝜎0 = 3.0 MN∕m2 constant in time. The computed strain depending on time is shown in Figure 3.10a. An exact solution for this case is given by Eq. (3.32). Differences between the exact solution and the numerically computed solution are small and are not visible in the figure. 2. An immediate right-end displacement corresponding to an immediate strain 𝜖0 = 𝜎0 ∕𝐸0 = 0.1‰ is applied and held constant in time. The computed stress depending on time is shown in Figure 3.10b, case 2. In contrast to the case before, relaxation occurs with stress decreasing to an asymptotic value. 3. A slow imposed contraction of 0.15o/oo is linearly increased over a period of 𝑡 = 100 days and then held constant. The displacements of both ends are prescribed with zero. As total strain 𝜖 is prescribed with zero, a tensile constraint stress is induced. The computed stress depending on time is shown in Figure 3.10b, case 3. It becomes obvious that a constraint stress due to a slowly increasing imposed strain is reduced already during its initiation. While analytical exact solutions are available for cases 1 and 2, the numerical approach is necessary for arbitrarily prescribed loads or displacements. Another creep example for reinforced concrete beams only accessible numerically is given with Example 4.3. More complex creep models, see, e.g. Malvern (1969, 6.4) and Jirásek and Bažant (2001, 28, 29), can also only be solved with numerical models.

51

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3 Uniaxial Reinforced Concrete Behaviour

3.3 Reinforcing Steel Stress–Strain Behaviour Reinforcing steel has to be regarded as a second basic component besides plain concrete. Steel can be considered as homogeneous already in the mesoscale – with spatial dimensions of millimetres – in contrast to concrete. Furthermore, steel has the same behaviour under compression and tension. Typical uniaxial stress–strain relations as are derived from experimental data are shown in Figure 3.11a. The relation has the following major parts: • • • •

The initial linear elastic part. The transition part with the initiation of yielding. The yielding part with high strains and slightly increasing stresses. A relatively short softening part closed by failure.

The characteristic of these parts varies with different types of steel. Relevant design properties of reinforcing steel are given in EN 1992-1-1 (2004, 3.2), CEB-FIP2 (2012, 5.2). A bilinear approximation of uniaxial stress–strain relations is generally used for the design and computation of reinforced concrete structures. It is characterised by the initial Young’s modulus 𝐸𝑠 , an initial yield stress 𝑓𝑦𝑘 (unsigned), a failure stress 𝑓𝑡 (unsigned), and a corresponding failure strain 𝜖𝑢 (unsigned). As 𝑓𝑡 > 𝑓𝑦𝑘 , hardening occurs, i.e. the material gains strength. The yield strain and tangential material stiffness in the hardening range are given by 𝜖𝑦 =

𝑓𝑦𝑘 𝐸𝑠

,

𝐸𝑇 =

𝑓𝑡 − 𝑓𝑦𝑘

(3.41)

𝜖𝑢 − 𝜖𝑦

Nonlinear steel behaviour is characterised by elasto-plasticity. Such nonlinear material behaviour becomes obvious with unloading from the yielding part; see Section 6.4.4. ◀

Elasto-plasticity is characterised by approximately the same material stiffness for initial elastic loading and unloading. Plastic strains remain as permanent strains during unloading from yielding to zero stresses.

(a)

(b)

Figure 3.11 Reinforcing steel. (a) Uniaxial stress–strain behaviour. (b) Cyclic isotropic hardening.

3.4 Bond between Concrete and Reinforcement

This phenomenon results from sliding in the crystal microstructure. It is schematically illustrated in Figure 3.11b with a cyclic behaviour. Unloading from a tensile regime may proceed to re-loading into the compressive regime while crossing a zero stress. With the maximum stress 𝑓𝑦 reached for tensile hardening the material behaves linear elastically during re-loading until stress reaches −𝑓𝑦 . Plastic yielding continues with further hardening in the compressive range. The cycle of loading– unloading–re-loading–unloading with isotropic hardening is shown in Figure 3.11b. The uniaxial elasto-plastic stress–strain relation for each branch is described by

𝜎=

⎧𝐸 (𝜖 − 𝜖 ) if 𝜖 − 𝑠 𝑝 𝑝 ⎨sign 𝜖 𝑓 else 𝑦 ⎩

𝑓𝑦 𝐸𝑠

≤ 𝜖 ≤ 𝜖𝑝 +

𝑓𝑦 𝐸𝑠

(3.42)

with the sign function, the current yield stress 𝑓𝑦 and the actual plastic strain 𝜖𝑝 (signed) as an internal state parameter; see also the remarks following Eq. (2.48). The evolution law for the internal state variable and the rule for hardening are given by 𝜖̇ 𝑝 = 𝜖̇

⎫

̇ ⎬ 𝑓̇ 𝑦 = 𝐸𝑇 |𝜖| ⎭

if 𝜖 𝜖̇ > 0 and |𝜎| = 𝑓𝑦

(3.43)

with a tangential material stiffness or hardening modulus 𝐸𝑇 (Eq. (3.412 )). Finally, 𝜎 = 0 if |𝜖| > 𝜖𝑢 and for all strains following

(3.44)

The hardening under consideration is isotropic, as hardening in the tensile range also leads to a compressive strength increase and vice versa. Equations (3.42) and (3.43) are an extension of Eqs. (2.48) and (2.49), as the latter do not cover hardening. This yields a hardening modulus 𝐸𝑇 = 0 , which might lead to a singular tangential material stiffness C𝑇 (Eq. (2.50)) and, finally, to a singular tangential stiffness matrix K𝑇𝑒 (Eq. (2.66)). This prevents a solution; see Eq. (2.77). It is appropriate to assume some amount of hardening from a numerical point of view, and the stress–strain relations (Eqs. (3.42) and (3.43)) are used for reinforcing steel in the following. Elasto-plasticity allows for cycles of stress–strain behaviour (Figure 3.11b). The area within such a cycle amounts to the specific internal dissipated energy. On the other hand, energy dissipation in a structure contributes to its ductility, i.e. its ability to deform while its internal forces retain their level.

3.4 Bond between Concrete and Reinforcement Due to the limited tensile strength of concrete, a reinforcement has to take over tensile forces. An experimental set-up to expose transmission of forces between concrete and a rebar is shown in Figure 3.12a: a single rebar is pulled out of a concrete block.

53

54

3 Uniaxial Reinforced Concrete Behaviour

(a)

(b)

Figure 3.12 (a) Basic bond set-up. (b) Main bond mechanism.

The system is characterised by the relative displacement of the rebar compared to the concrete block and furthermore by the force system of rebar tension and concrete block support. Transmission of forces relies on three mechanisms. • Adhesion as a rigid connection of boundary layers of concrete and steel. • Friction as slip between the boundary layers of concrete and steel combined with lateral pressure. • Rebars have ribbed surfaces with ribs or dents acting like toothwork. The last mentioned toothwork mechanism contributes the most to the rebar force for a given relative rebar displacement. Such an interaction due to ribbed surfaces leads to a triaxial state of stresses within the concrete body immediately surrounding the rebar. This is schematically illustrated in Figure 3.12b. A system of skew concrete struts braces against rebar ribs in the projection of a plane cross-section. These concrete struts form a cone in the spatial view. A circumferential tensile ring is necessary to redirect the cone compression into a kind of a cylinder compression aligned to the rebar force. The tensile cylinder around the rebar is activated through tensile stresses within the concrete body. A bond failure may occur with concrete splitting along a rebar in the case that such tensile stresses exceed the limited tensile strength of concrete. This can be prevented by placing a lateral secondary reinforcement or through reducing tensile stresses by increasing the radial concrete cross-section or by providing sufficient concrete cover, respectively. Bond is a complex mechanical problem that requires the mesoscale view for a thorough understanding and analysis where both rebar and concrete have to be considered as three-dimensional solids with nonlinear material behaviour. A simplified model is necessary to make the bond treatable from a macroscopic view. Such a model is shown in Figure 3.13a. A cut in the interface between a simplified cylindrical rebar and the concrete body exposes a bond force flow 𝑇, which is a force related to length and follows a relation 𝑇=

d𝐹𝑠 d𝐹𝑐 = d𝑥 d𝑥

(3.45)

3.4 Bond between Concrete and Reinforcement

(a)

(b)

Figure 3.13 (a) Schematic bond equilibrium. (b) Typical bond law.

with the rebar force 𝐹𝑠 and the resulting force 𝐹𝑐 in the concrete body. The relative displacement between rebar and concrete is measured by a slip 𝑠. The notion of slip assumes the deformation of concrete in a cross-section (Figure 3.13a) as approximately homogeneous beyond the immediate surroundings of the rebar and defines slip 𝑠 as the difference between the longitudinal displacements of the outer concrete area and the centre axis of the rebar. Thus, slip has a dimension of length. The force variable 𝑇 and kinematic variable 𝑠 are connected by a bond law for a flexible bond 𝑇 = 𝑓𝑇 (𝑠)

(3.46)

An alternative formulation assumes a constant circumference 𝑈 of a rebar and derives a bond stress 𝜏 = 𝑇∕𝑈 with the dimension of stress leading to 𝜏 = 𝑓𝜏 (𝑠)

(3.47)

Such a formulation is generally used, as it is independent from specific geometric properties and may be considered as a special case of a material law. A characteristic smoothed course of a bond law is shown in Figure 3.13b. It has the following parts (CEB-FIP2 2012, 6.1.1): • An initial part with increasing nonlinear mechanisms due to nonlinear behaviour of concrete and reinforcing steel. • A point or range of maximum bond stress corresponding to bond strength 𝜏𝑏max , which is mainly related to the tensile strength of concrete; see the foregoing toothwork mechanism. • A softening part with increasing slip and decreasing bond stress due to softening in the concrete tensile range (Figure 3.3b). • A final horizontal part with approximately constant bond stress 𝜏𝑏𝑓 and increasing slip due to the friction of sheared concrete corbels. The particular curve of Figure 3.13b is composed of a quadratic, cubic, and linear polynomial with continuous derivatives at the connection points, which improves convergence when applying, e.g. the Newton–Raphson method (Eq. (2.76)) for nonlinear problem solving. This course is ruled by the values of 𝜏bmax , 𝜏𝑏𝑓 , and the

55

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3 Uniaxial Reinforced Concrete Behaviour

corresponding slip values 𝑠bmax , 𝑠𝑏𝑓 . These characteristic values of a bond law have to be determined from experimental data. Some rules are given in CEB-FIP2 (2012, 6.1.1.1). Analytical or numerical simulation models that realistically represent the complex bond situation and yield reliable bond laws from a theory are not available regarding the current state of knowledge. ◀

Phenomenological, integrating bond laws are generally used in numerical macroscale models in the case that a flexible bond has to be considered.

Bond stress changes direction in the case of a reversed slip, i.e. the curve of Figure 3.13b can be mirrored for values 𝑠, 𝜏 < 0. Monotonic loading with increasing slip was described up to now. Bond behaviour in the case of unloading and re-loading is discussed in CEB-FIP2 (2012, 6.1.1.4) and Hartig et al. (2008). A simpler, more convenient approach based on the concept of damage is as follows: begin with unloading from a point (𝑠′ , 𝜏′ ) on a curve as given in Figure 3.13b and the respective 𝜏 − 𝑠-relation goes through the origin on a straight line with reloading until a point (−𝑠′ , −𝜏′ ) is reached. The mirrored curve is subsequently followed. The application of flexible bond is illustrated with the following Example 3.4 and in Section 8.5. Basic ideas of damage are described in Section 6.6.

3.5 Smeared Crack Model Cracking, as is described in Section 3.1, in its final stage of a macrocrack leads to a discontinuity of displacements regarding the crack surfaces. In the case of a uniaxial concrete tension bar, the displacement field 𝑢(𝑥) has a jump at a macrocrack. On the other hand, common finite elements (Section 2.3) do not allow for displacement discontinuities. The smeared crack model combines macrocracking and continuous displacement fields. In the following, this is treated for the uniaxial two-node bar element (Section 2.3) and includes not only the final macrocrack but also the foregoing tension softening process. We consider the cracked element as a black box whose inner state cannot be inspected in detail. The mean strain of such an element is assumed as 𝜖=

1 [(𝐿𝑒 − 𝑏𝑤 ) 𝜖𝑢 + 𝑏𝑤 𝜖𝑐 ] 𝐿𝑒

= (1 − 𝜉) 𝜖𝑢 + 𝜉 𝜖𝑐

,

𝜉=

𝑏𝑤 , 𝐿𝑒

0 1, i.e. 𝐿𝑒 < 𝑏𝑤 in calculations. The softening tangential stiffness is given by −𝑓𝑐𝑡 ∕(𝜉 𝜖𝑐𝑢 − 𝜖𝑐𝑡 ). A choice 𝜉 𝜖𝑐𝑢 = 𝜖𝑐𝑡 should obviously be avoided. A choice 𝜉 𝜖𝑐𝑢 < 𝜖𝑐𝑡 reverses the negative stiffness

Figure 3.15 Linearised cohesive crack model for the 1D smeared crack (see Figure 3.3b).

3.6 Reinforced Tension Bar

into a positive stiffness and results in snap-back behaviour (Example 3.1) within one element. The outcome of an equilibrium iteration might become questionable. This imposes an element length requirement 𝜉 𝜖𝑐𝑢 > 𝜖𝑐𝑡

→

𝐿𝑒 < 𝑏𝑤

𝜖𝑐𝑢 𝜖𝑐𝑡

(3.62)

but 𝜖𝑐𝑢 ∕𝜖𝑐𝑡 is generally a large number. With the choice 𝜉 = 1 → 𝐿𝑒 = 𝑏𝑤 , the crack band occupies the whole element, and the crack width Eq. (3.6) is obtained as 𝑤 = 𝐿𝑒 𝜖

for 𝜉 = 1

(3.63)

Neglecting the crack energy 𝐺𝑓 = 0 – this might be appropriate for reinforced structures – leads to 𝜖𝑐𝑢 = 𝜖𝑐𝑡 . Thus, in the case of cracking, 𝜎 = 0, 𝜖𝑢 = 0 is obtained with a smeared strain 𝜖 = 𝜉 𝜖𝑐 according to Eqs. (3.58), (3.57), and (3.48). This yields a crack width 𝑤 = 𝑏𝑤 𝜖𝑐 = 𝑏𝑤 𝜖𝑐 ∕𝜉 = 𝐿𝑒 𝜖. A stress–strain relation like Eq. (3.61) blends or smears strains of uncracked parts and cracking strains into a unified continuous strain. Thus, common finite element interpolations (Section 2.3) may still be used while regarding cracking of elements. But a side effect arises. ◀

The smeared crack model introduces the length of the respective finite bar element into the stress–strain relations.

This property makes the smeared crack model in combination with the cohesive crack suitable for a regularisation (Section 7.3). It may be extended to two and three dimensions. This is demonstrated for the 2D case in Section 8.2.

3.6 Reinforced Tension Bar Basic components of reinforced concrete have been described up to now. Their interaction is demonstrated in the following with the reinforced tension bar. The basic set-up is shown in Figure 3.16. A single reinforcement bar is embedded in a concrete bar. Both are connected by bond springs acting longitudinally. The left end of the rebar is fixed, while a displacement is prescribed for its right-hand end. The basic scheme for a discretisation is also indicated together with this conceptual model.

Figure 3.16 Scheme of a reinforced tension bar model.

59

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3 Uniaxial Reinforced Concrete Behaviour

The model is one-dimensional with a coordinate 𝑥 and displacements 𝑢. The concrete part and the reinforcement part are discretised in the same way but each with its own nodes and elements. Concrete nodes and rebar nodes share the same 𝑥-position initially. They separate from each other in the course of a calculation according to the prescribed bond law or bond element characteristics, respectively. The following properties are given for each part: • A linear elastic behaviour is assumed for the tensile behaviour of concrete with a limited tensile strength 𝑓𝑐𝑡 ⎧𝐸 𝜖 , 𝜖 ≤ 𝑓𝑐𝑡 𝑐 𝐸𝑐 𝜎= ⎨0, else ⎩

•

• • •

•

(3.64)

with a concrete Young’s modulus 𝐸𝑐 . As the focus is laid upon the interaction of concrete and rebar, a tensile strain softening of concrete is disregarded with a crack energy 𝐺𝑓 = 0. The interpolation of displacements is performed with the two-node uniaxial bar element (Section 2.3) for both the concrete part and the rebar part. This leads to a constant strain 𝜖 within an element. The smeared crack model (Section 3.5, Eq. (3.63)) leads to a crack width 𝑤 = 𝐿𝑒 𝜖 with an element length 𝐿𝑒 . The rebar is modelled with the uniaxial elasto-plastic material law with hardening, see Eqs. (3.42) and (3.43). The bond law is shown in Figure 3.13b and described in Section 3.4. It is characterised by the bond strength 𝜏bmax , the residual strength 𝜏𝑏𝑓 , and the corresponding slip values 𝑠bmax , 𝑠𝑏𝑓 . The interpolation of slips is performed with the two-node spring element along a line (Section 2.3). The variable Δ𝑢 corresponds to the slip 𝑠 and the variable 𝐹 to the bond force flow 𝑇 times element length 𝐿𝑒 . The bond force flow is derived from the bond law by multiplying bond stress with the rebar circumference.

With given element types – leading to forms for N, B (Eqs. (2.18) and (2.21)) – and given material behaviour – leading to stresses 𝝈 and tangential material stiffness C𝑇 – the numerical model is built according to the procedure described in Section 2.6. Numerical integration is performed with the one-dimensional form of Eq. (2.69) with the integration order 𝑛𝑖 = 0 (Table 2.1). The lowest order is sufficient due to constant strains and stresses within an element. The most simple element types are used for the tension bar model to interpolate displacements and to perform a discretisation. A model complexity arises from the particular material laws and the interaction of parts leading a nonlinear behaviour. An incrementally iterative scheme has to be used for the solution (Figure 2.5). The application of the tension bar model is demonstrated with the following example.

3.6 Reinforced Tension Bar

Example 3.4: Reinforced Concrete Tension Bar

Geometry and discretisation are chosen as follows: • Bar length 𝐿 = 1.0 m, cross-sectional area of concrete 𝐴𝑐 = 0.1 ⋅ 0.1 m2 , reinforcement 1 ⊘ 16 with cross-sectional area 𝐴𝑠 = 2.01 cm2 and circumference 𝑈𝑠 = 5.02 cm. • The concrete and the reinforcement part are each discretised with 100 bar elements with 101 nodes and an element length of 𝐿𝑒 = 0.01 m. This leads to 101 bond elements connecting concrete nodes and reinforcement nodes, 301 elements totally and 202 nodes totally with the same number of degrees of freedom. The chosen material properties are shown in Table 3.1. The corresponding equations for the material behaviour are Eqs. (3.42), (3.43), and (3.64). Some scatter is allowed for the concrete tensile strength with a Gaussian random value for each concrete bar element with a small variance. The bond law is derived from the characteristic values as a sequence of parabola, cubic polynomial, and horizontal line with the same slope in the connecting points (Figure 3.13b). The bond strength is related to concrete tensile strength by the empirical relation 𝜏max ≈ 1.8 𝑓𝑐𝑡 . The loading of the tension bar is applied with prescribed displacement boundary conditions: zero displacement for the left-hand reinforcement node, a prescribed displacement 𝑢𝑁 = 2.4 mm for the reinforcement node on the right-hand side incrementally applied in 100 steps during a loading time 0 ≤ 𝑡 ≤ 1. This leads to a final mean strain 𝜖mean = 2.4‰. The BFGS method (Section A.1) is used to determine a solution within a loading step as the Newton–Raphson method fails due to discontinuities in the tangent stiffness. The computed relation between reaction force and displacement is shown in Figure 3.17. The following states, which are characteristic for reinforced concrete behaviour, can be seen: Table 3.1 Material parameters Example 3.4. Concrete

Young’s modulus 𝐸𝑐 Mean tensile strength 𝑓𝑐𝑡

[MN/m2 ] [MN/m2 ]

35 000 3.5

Reinforcing steel

Young’s modulus 𝐸𝑠 Yield strength 𝑓𝑠𝑦

[MN/m2 ] 200 000 [MN/m2 ] 500

Bond

Strength 𝜏bmax Slip at strength 𝑠bmax Residual strength 𝜏𝑏𝑓 Slip at residual 𝑠𝑏𝑓

[MN/m2 ] [mm] [MN/m2 ] [mm]

6.0 0.1 3.0 1.0

61

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3 Uniaxial Reinforced Concrete Behaviour

Figure 3.17 Example 3.4. Reaction force displacement curve.

(a)

(b)

Figure 3.18 Example 3.4. (a) Concrete stresses. (b) Rebar stresses.

• Uncracked state I with concrete stresses below tensile strength. • Crack formation state IIa. Cracks develop in a sequence whereby each sudden reaction force decrease corresponds to a crack. The reaction force decrease results from a stiffness reduction of the bar due to cracking. A characteristic sawtooth pattern develops. • Stabilised cracking state IIb before rebar yielding. No new cracks occur while the bar’s stiffness is significantly reduced compared to state I. • Limit state III with rebar yielding. The slight increase in reaction forces results from rebar hardening. A numerical solution cannot be determined without the assumption of hardening as a singularity of the system’s stiffness matrix would otherwise occur. The stress distributions along the bar are shown in Figures 3.18 and 3.19a for halfloading 𝑡 = 0.5, 𝑢𝑁 = 1.2 mm and for full loading 𝑡 = 1.0, 𝑢𝑁 = 2.4 mm. • Concrete stresses are shown in Figure 3.18a with zero stresses in a cracked element. Four cracks occur according to four peaks in the load displacement curve (Figure 3.16). Concrete tensile stresses for full loading are only slightly larger compared to half loading and remain below concrete tensile strength. • Reinforcement stresses are shown in Figure 3.18b with peak stresses at a crack and the characteristic garland pattern. Yielding of the reinforcement occurs in

3.6 Reinforced Tension Bar

(a)

(b)

Figure 3.19 Example 3.4. (a) Bond stresses. (b) Displacements in the final state (values must be divided by scale. The dimension is [m]).

each cracked element for full loading with stresses slightly above the yield limit and high strains compared to all uncracked elements. • Bond stresses transfer forces between concrete and reinforcement between cracks; see Figure 3.19a. They have maximum absolute values near cracks and change sign across a crack. Maximum values are also given for the left and right-hand stress free concrete surfaces. • The bond strength is reached already after half loading. A more rectangular shape is given for full loading with a slight softening at cracks due to large slip and the prescribed bond law (Figure 3.13b). Displacements 𝑢 along the bar axis 𝑥 are shown in Figure 3.19b for all parts at full loading. • Displacements of concrete elements and reinforcement elements are different due to the flexible bond. The difference of the values results in slip leading to bond stresses. • Five nearly horizontal plateaus of concrete displacements arise with four cracks in between. The jumps in concrete displacements correspond to the crack widths, which are also given by the difference of the nodal displacements of a cracked element in accordance with Eq. (3.63). A typical value is 𝑤 ≈ 0.5 mm (consider scale!) for full loading. • In any case, the computed displacement field in cracked concrete elements is continuous. Cracks are in a black box (Figure 3.14), and the crack width is derived indirectly. • Rebar displacements are given by an approximately straight line with small kinks at cracks. These kinks with increased displacement slope correspond to high strains for rebar elements corresponding to cracked concrete elements. These high rebar strains are connected with the yielding of rebar elements. The model for the reinforced concrete tension bar already shows the typical characteristics of reinforced concrete behaviour. It gives the basis for the understanding of other structures like reinforced concrete beams, plates, slabs, and shells.

63

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3 Uniaxial Reinforced Concrete Behaviour

3.7 Tension Stiffening of Reinforced Bars The tension bar has a lower limiting case in the load displacement behaviour: a rebar without concrete contributing to load transfer. This is indicated in Figure 3.17. The difference between the computed behaviour and load transfer by the rebar alone indicates the tension stiffening effect. ◀

Tension stiffening results from the contribution of concrete between cracks to load bearing. This leads to a larger stiffness of a cracked reinforced concrete element compared to the corresponding rebar stiffness.

Tension stiffening will not increase the ultimate load of a reinforced tension bar, i.e. its ultimate load will be the same as for the corresponding rebar. An elaborated model is derived in Section 3.6 to describe the contribution of concrete between cracks. This contribution leads to concrete tensile stresses below tensile strength and to reduced reinforcement stresses between cracks (Figure 3.18). A simplified model to quantify tension stiffening can be derived based on the concept of mean stresses between cracks. The resulting displacement of a tension bar can be determined from a mean rebar strain 𝜖𝑠𝑚 multiplied by the bar length 𝐿. It is assumed that the reinforcement has not yet reached its yield strength and that rebar strains have the same shape as rebar stresses. The bar force is ruled by the peak stresses 𝜎𝑠𝑐 of the rebar at a crack – concrete does not contribute here by definition – multiplied by the rebar cross-sectional area 𝐴𝑠 . Thus, the load displacement behaviour is determined by a relation deriving 𝜎𝑠𝑐 from 𝜖𝑠𝑚 . But some minimum length of a structural element is required to make such a relation applicable, which is basically the crack distance or a small multiple of it, respectively. A key to quantify a relation 𝜎𝑠𝑐 and 𝜖𝑠𝑚 is to consider the characteristics of stresses in crack situations. This is illustrated in Figure 3.20a, which gives a schematic cutout of Figure 3.18. The mean value of reinforcement strain between cracks is estimated with 𝜖𝑠𝑚 =

(a)

𝜎𝑠𝑚 1 = (𝜎 − 𝛽𝑡 Δ𝜎𝑠 ) 𝐸𝑠 𝐸𝑠 𝑠𝑐

(3.65)

(b)

Figure 3.20 (a) Cracks and stresses. (b) Tension stiffening model.

3.7 Tension Stiffening of Reinforced Bars

with reinforcement Young’s modulus 𝐸𝑠 , reinforcement stress decline Δ𝜎𝑠 from crack to the minimum value between cracks, and a tension stiffening coefficient 0 < 𝛽𝑡 < 1 for the shape of reinforcement stress distribution. ◀

The value of 𝛽t characterises the quality of the bond. A value 𝛽t = 0 indicates no stress transfer from rebar to concrete, i.e. no bond effective; a value 𝛽t = 1 indicates immediate stress transfer from rebar to concrete, i.e. rigid bond with no slip between concrete and rebar.

Values are assumed to be in the range 0.4 ≤ 𝛽𝑡 ≤ 0.6. The validity of such a range is determined by computations as shown in Example 3.4, even more sophisticated models, and especially pure observation or phenomenology, respectively. Code provisions are also given for the choice of 𝛽𝑡 ; see for 𝑘𝑡 in EN 1992-1-1 (2004, 7.3.4) and for 𝛽 in CEB-FIP2 (2012, 7.6.4.4). This value will be assumed as given in the following. A further parameter arises with the stress decline parameter Δ𝜎𝑠 . Two states of cracking have to be distinguished to derive an estimation for this parameter: stabilised cracking and crack formation. Cracks occur one after the other during loading and not simultaneously (Figure 3.16). But crack formation generally occurs within a relatively small band of deformations and an even smaller band of corresponding forces or stresses. Passing through these bands ends with stabilised cracking. Basically, no further cracks arise if cracking is stabilised. Decrease of rebar stresses is transferred into increase of tensile concrete stresses. But concrete stresses are limited by concrete tensile strength. Hence, the following relation is assumed for reinforcement stress decline in the case of stabilised cracking 𝐴𝑠 Δ𝜎𝑠 = 𝐴𝑐,eff 𝑓𝑐𝑡

(3.66)

with the effective concrete cross-sectional area 𝐴𝑐,eff . This value may be smaller than a concrete cross-sectional area 𝐴𝑐 as not all parts of a larger concrete cross-section take part in the exchange of stresses with the reinforcement. Code provisions are given for the choice of 𝐴𝑐,eff (EN 1992-1-1 2004, 7.3.2). Equation (3.66) leads to Δ𝜎𝑠 =

𝑓𝑐𝑡 𝜚eff

(3.67)

with the effective reinforcement ratio 𝜚eff = 𝐴𝑠 ∕𝐴𝑐,eff . Combining Eqs. (3.65) and (3.67) the relation for the reinforcement stress in cracks depending on the mean reinforcement strain for stabilised cracking is given by 𝜎𝑠𝑐 = 𝐸𝑠 𝜖𝑠𝑚 + 𝛽𝑡

𝑓𝑐𝑡 𝜚eff

(3.68)

This corresponds to a shift of the pure rebar course to the left; see Figure 3.20b. On the one hand, a given stress has a lower strain, and, on the other hand, a given strain has a higher stress. This is limited by the yielding level of the reinforcement. The corresponding relation for crack formation has two limiting points within the rebar stress–mean strain relation. The first point is related to initial cracking, and

65

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3 Uniaxial Reinforced Concrete Behaviour

the second point is the initial point of stabilised cracking. The first point is assumed with a rebar stress 𝜎𝑠𝑐,𝑖 = 𝑓𝑐𝑡 ∕𝜚eff – force from concrete tensile strength is assigned to the rebar – and a corresponding strain 𝜖𝑠𝑐,𝑖 = 𝑓𝑐𝑡 ∕𝐸𝑐 with Young’s modulus 𝐸𝑐 of concrete. Using this as starting point the following assumption is made regarding crack formation ( ) 𝑓𝑐𝑡 𝑓𝑐𝑡 )+ , 𝜎𝑠𝑐 = 𝑘 𝜖𝑠𝑚 − 𝜖𝑠𝑐,𝑖 + 𝜎𝑠𝑐,𝑖 = 𝑘 (𝜖𝑠𝑚 − 𝐸𝑐 𝜚eff

𝜖𝑠𝑚 ≥

𝑓𝑐𝑡 𝐸𝑐

(3.69)

with a slope 𝑘 that remains to be determined. Equation (3.69) should connect to Eq. (3.68) with a stress 𝛼 𝜎𝑠𝑐,𝑖 leading to 𝑘 = 𝐸𝑐

𝛼−1 , 𝛼 − 𝛽𝑡 − 𝑛 𝜚eff

𝑛=

𝐸𝑠 𝐸𝑐

(3.70)

The coefficient 𝛼 marks the final stress point of crack formation and the initial stress point of stabilised cracking, i.e. the increase factor of the reinforcement stress during crack formation. ◀

In analogy to the stiffening coefficient 𝛽t , a heuristic approach – which is not untypical for practical reinforced concrete design – leads to 𝛼 ≈ 1.3. ′ The corresponding strain 𝜖𝑠𝑚 can be determined with Eq. (3.68) and 𝜎𝑠𝑐 = 𝛼 𝜎𝑠𝑐,𝑖 , leading to ′ = 𝜖𝑠𝑚

1 𝑓𝑐𝑡 (𝛼 − 𝛽𝑡 ) 𝐸𝑠 𝜚eff

(3.71)

A simplified version might straighten the initial kink (Figure 3.20b) with 𝜎𝑠𝑐 =

𝛼𝑓𝑐𝑡

′ 𝜚eff 𝜖𝑠𝑚

𝜖𝑠𝑚

(3.72)

in the initial range up to the beginning of stabilised cracking. Further reference for tension stiffening and its comprehension into analysis are given in Belarbi and Hsu (1994); Bischoff (2001); Fields and Bischoff (2004) and CEB-FIP2 (2012, 7.6.5.2, 7.6.7.2). Tension stiffening is included as an option for reinforced concrete beams (Section 4.7).

67

4 Structural Beams and Frames Beam theory is an extremely powerful tool. On the one hand it covers a wide range of structural engineering problems. On the other hand it is relatively simple from a mathematical point of view and only needs uniaxial material laws. Cracking of concrete remains a prominent topic in the case of reinforced concrete beam bending. The following describes how it is considered on the cross-sectional level, integrated into the FEM and applied to typical structural engineering problems. Furthermore, implementation procedures common to all types of finite elements are exemplarily introduced.

4.1 Cross-Sectional Behaviour 4.1.1 Kinematics Kinematic assumptions characterise the different types of models for structures whereby ensuring kinematic compatibility. Different types of such assumptions have already been discussed for bar, spring, and continuum elements in Section 2.3. Kinematic assumptions for beams need more elaboration but form the base for the structural beam theory. Plane beams will be considered in the following which are straight in their undeformed configuration. Small displacements are assumed if not otherwise stated. A beam first of all is characterised by a longitudinal direction with a reference axis and a longitudinal coordinate 𝑥. Every reference coordinate 𝑥 has a cross-section with a transverse height coordinate 𝑧, see Figure 4.1. Every height coordinate 𝑧 has a width which may be variable. Height and width form a cross-section which is perpendicular to the reference axis in the undeformed configuration. It is not necessary to assume that the reference axis coincides with the centre of gravity of cross-sections. The following kinematic assumption constrains the description of displacements: ◀

The Bernoulli–Navier hypothesis states that undeformed plane cross-sections of a beam remain plane during a displacement.

Computational Structural Concrete – Theory and Applications, Second Edition. Ulrich Häussler-Combe. © 2023 Ernst & Sohn GmbH. Published 2023 by Ernst & Sohn GmbH.

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4 Structural Beams and Frames

Figure 4.1 Kinematics of a plane beam.

The displacements of every material point of a beam with the coordinates 𝑥, 𝑧 are given by the longitudinal displacement 𝑢(𝑥, 𝑧) in the 𝑥-direction and the lateral displacement 𝑤(𝑥, 𝑧) in the 𝑧-direction. The Bernoulli–Navier hypothesis is formulated as 𝑤(𝑥, 𝑧) = 𝑤(𝑥, 0) = 𝑤(𝑥) 𝑢(𝑥, 𝑧) = 𝑢(𝑥) − 𝑧 𝜙(𝑥) = 𝑢(𝑥) − 𝑧 [

(4.1)

𝜕𝑤(𝑥) − 𝛾(𝑥)] 𝜕𝑥

with a cross-sectional rotation angle 𝜙(𝑥), a shear angle 𝛾(𝑥), and longitudinal 𝑢(𝑥) and lateral displacements 𝑤(𝑥) of the reference axis. • Eq. (4.11 ) states that every material point in a cross-section has the same lateral displacement but this may change with the longitudinal coordinate. • Eq. (4.13 ) states that a cross-section rotates by an angle 𝜙 during deformation. • Eq. (4.14 ) decouples the rotation of the cross-section 𝜙 and the slope of the reference axis 𝜕𝑤∕𝜕𝑥 by the angle 𝛾. The relation is 𝜙=

𝜕𝑤 −𝛾 𝜕𝑥

(4.2)

The significance of 𝛾 becomes evident with its relation to the shear strain, see Eq. (4.33 ). • The case 𝛾 ≪ 𝜙 with the assumption 𝛾 = 0 leads to the Bernoulli beam where cross-sections remain rectangular to the reference axis after deformation. The inclusion of shear deformation leads to the Timoshenko beam. Cross-sections after a displacement remain plane but are not rectangular to the reference axis for the Timoshenko beam. The Timoshenko beam theory is treated first in the following and the Bernoulli beam will be derived as a special case. Beam kinematics may be considered as constrained kinematics of plate kinematics (Section 2.3). Thus, strains are defined according to

4.1 Cross-Sectional Behaviour

Eq. (2.39) with 𝑦, 𝑣 replaced by 𝑧, 𝑤. This yields 𝜖𝑥 (𝑥, 𝑧) =

𝜕𝑢 𝜕𝑥

=

𝜕𝑢 𝜕 2 𝑤 𝜕𝛾 −𝑧 [ 2 − ] 𝜕𝑥 𝜕𝑥 𝜕𝑥 (4.3)

𝜖𝑧 (𝑥, 𝑧) = 0 𝛾𝑥𝑧 (𝑥, 𝑧) =

𝜕𝑢 𝜕𝑤 + 𝜕𝑧 𝜕𝑥

=−

𝜕𝑤 𝜕𝑤 +𝛾+ =𝛾 𝜕𝑥 𝜕𝑥

regarding Eq. (4.1). A notation 𝜕 ∙ ∕𝜕𝑥 = ∙′ , 𝜕 2 ∙ ∕𝜕𝑥2 = ∙′′ is used for abbreviation. Furthermore, the overbars on 𝑢, 𝑤 will be omitted in the following. To simplify the notation, 𝑢, 𝑤 will be written instead and refer to the reference axis. It is appropriate to introduce the variables 𝜖(𝑥) = 𝑢′ ,

𝜅(𝑥) = 𝜙′ = 𝑤 ′′ − 𝛾 ′

(4.4)

with the curvature 𝜅. The variable 𝜖 has the meaning of a strain of the reference axis in the context of beams while 𝜖𝑥 indicates a longitudinal strain in a cross-section varying with 𝑧. With Eq. (4.3) this leads to longitudinal strains 𝜖𝑥 (𝑥, 𝑧) = 𝜖(𝑥) − 𝑧 𝜅(𝑥)

(4.5)

linearly varying along the beam height with extreme values on the top and bottom of the cross-section. ◀

The variables 𝜖, 𝜅, 𝛾 are chosen as generalised strains for beams whereby 𝜖 indicates the longitudinal strain of the reference axis, 𝜅 the curvature related to deformed crosssections and 𝛾 the shearing angle of deformed cross-sections relative to the reference axis.

The curvature 𝜅 = 𝜙′ is different compared to the second derivative 𝑤 ′′ of the lateral displacement 𝑤 of the reference axis. Both are related according to Eq. (4.4). To describe material behaviour, deformation variables have to be connected to force variables which are moment 𝑀, normal force 𝑁 and shear force 𝑉 in the case of plane beams. The following dependencies are assumed 𝑀 = 𝑀(𝜖, 𝜅) ,

𝑁 = 𝑁(𝜖, 𝜅) ,

𝑉 = 𝑉(𝛾)

(4.6)

Particular forms are derived in the following. Basics of beam theory look conclusive but there are some inconsistencies. • A shear strain 𝛾𝑥𝑧 , which is constant along the cross-section leads to non-vanishing shear stress at the lower and upper side of a beam. But this contradicts to the local equilibrium conditions. • On the other hand, a parabolic or other nonlinear course of shear stresses – according to equilibrium conditions with linear normal stresses – leads to a curved course of shear strains with vanishing values on top and bottom sides. These contradictions can be resolved with the plate theory. Plane beam theory is its limiting case and a very useful approximation.

69

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4 Structural Beams and Frames

4.1.2 Linear Elastic Behaviour As longitudinal strains are given by Eq. (4.5) and shear strains by Eq. (4.33 ) depending on generalised strains the corresponding stresses may be determined with a material law. We start with linear elastic behaviour (Section 2.4) and refer to isotropic plane stress Eq. (2.45). The coordinate direction 𝑦 has to be replaced by 𝑧 according to beam conventions. In addition to the kinematic assumptions, Poisson’s effect has to be neglected for beams with Poisson’s ratio assumed as 𝜈 = 0. Combining Eqs. (2.45), (4.3), and (4.5) results to stresses 𝜎𝑥 = 𝐸 𝜖𝑥

= 𝐸 [𝜖(𝑥) − 𝑧 𝜅(𝑥)]

𝜎𝑧 = 𝐸 𝜖𝑧

=0

𝜎𝑥𝑧 = 𝐺 𝛾𝑥𝑧

(4.7)

=𝐺𝛾

with Young’s modulus 𝐸 and the beam’s shear modulus 1 𝐸 (4.8) 2 Internal forces of a beam – not to be confused with internal nodal forces (Eq. (2.58)) – are derived by the integration of stresses in a cross-section leading to the normal force 𝑁, the bending moment 𝑀, and the shear force 𝑉 𝐺=

𝑧2

𝑧2

𝑁 = ∫ 𝜎𝑥 𝑏 d𝑧

𝑧2

= 𝐸 ∫ 𝑏 d𝑧 𝜖 − 𝐸 ∫ 𝑧 𝑏 d𝑧 𝜅

𝑧1

𝑧1 𝑧2

𝑧1 𝑧2

𝑀 = − ∫ 𝜎𝑥 𝑧 𝑏 d𝑧

𝑧2

= −𝐸 ∫ 𝑧 𝑏 d𝑧 𝜖 + 𝐸 ∫ 𝑧2 𝑏 d𝑧 𝜅

𝑧1

𝑧1

𝑧2

𝑧2

𝑉 = ∫ 𝜎𝑥𝑧 𝑏 d𝑧

(4.9)

𝑧1 𝑧2

= 𝐺 ∫ 𝛾𝑥𝑧 𝑏 d𝑧 = 𝛼 𝐺 ∫ 𝑏 d𝑧 𝛾

𝑧1

𝑧1

𝑧1

with the coordinate 𝑧1 of the cross-section bottom line, the coordinate 𝑧2 of the top line, the cross-section height ℎ = 𝑧2 − 𝑧1

(4.10)

and the cross-section width 𝑏. A shear correction factor 𝛼 is introduced for the shear force 𝑉 to compensate for the difference between mean shearing strain/stress over the cross-section – see concluding remarks of Section 4.1.1 – and the shearing strain/stress 𝛾∕𝐺𝛾 in the reference point 𝑧 = 0. Its value is 𝛼 = 5∕6 in the case of a rectangular cross-section shape with a reference axis through the centre of gravity (Gruttmann and Wagner 2001). The evaluation of integrals in Eq. (4.9) leads to geometric section properties with cross-sectional area 𝐴, first moment of area 𝑆 and second moment of area 𝐽 𝑧2

𝐴 = ∫ 𝑏 d𝑧 , 𝑧1

𝑧2

𝑆 = ∫ 𝑧 𝑏 d𝑧 , 𝑧1

𝑧2

𝐽 = ∫ 𝑧2 𝑏 d𝑧 𝑧1

(4.11)

4.1 Cross-Sectional Behaviour

In the case that the reference axis 𝑥 coincides with the centre of gravity the first moment of area vanishes 𝑧2

𝑆 = ∫ 𝑧 𝑏 d𝑧 = 0

(4.12)

𝑧1

which formally simplifies the linear elastic case but is not mandatory. Finally, the linear elastic case with 𝑆 = 0 yields the well-known relations 𝑁 = 𝐸𝐴 𝜖 ,

𝑀 = 𝐸𝐽 𝜅 ,

𝑉 = 𝛼 𝐺𝐴 𝛾

(4.13)

The sign of the moment is different compared to classical structural beam theory. The difference results from a different orientation of the 𝑧-axis (Figure 4.1). According to Eq. (2.47) these relations are collected in 𝝈=C⋅𝝐

(4.14)

with ⎛𝑁 ⎞ 𝝈 = ⎜𝑀 ⎟ , ⎜ ⎟ ⎝𝑉⎠

⎡𝐸𝐴 ⎢ C=⎢ 0 ⎢ 0 ⎣

0 𝐸𝐽 0

0 ⎤ ⎥ 0 ⎥, ⎥ 𝛼 𝐺𝐴 ⎦

⎛𝜖 ⎞ 𝝐 = ⎜𝜅 ⎟ ⎜ ⎟ ⎝𝛾 ⎠

(4.15)

with the generalised stresses 𝝈, generalised strains 𝝐, and a material stiffness matrix C. The tangential material stiffness matrix C𝑇 (Eq. (2.50)) is identical to C for linear materials. The subset ⎛𝑁 ⎞ 𝝈=⎜ ⎟ , 𝑀 ⎝ ⎠

⎡𝐸𝐴 C=⎢ 0 ⎣

0⎤ ⎥, 𝐸𝐽 ⎦

⎛𝜖⎞ 𝝐=⎜ ⎟ 𝜅 ⎝ ⎠

(4.16)

is applied for the Bernoulli beam.

4.1.3 Cracked Reinforced Concrete Behaviour 4.1.3.1 Compression Zone and Internal Forces

Linear elasticity assumes unlimited strength of materials both for compression and tension. This assumption is not valid for reinforced concrete (RC), in particular not for the limited tensile strength of concrete. Thus, cracked concrete cross-sections have to be considered in which longitudinal beam strains 𝜖𝑥 exceed the tensile limit strain 𝜖𝑐𝑡 of concrete. Cross section properties as have been defined by Eq. (4.11) and also the notion of a centre of gravity loose their immediate applicability. Nevertheless, the position of cross-section points relative to a reference axis has to be defined.

71

72 ◀

4 Structural Beams and Frames

The reference axis is placed in the mid-point of a cross-section irrespective of the shape of the cross-section. 1) The bottom side has the coordinate z = z1 = −h∕2 and the top side has the coordinate z = z2 = h∕2 with a cross-section height h.

Using Eq. (4.5) the strains 𝜖𝑥 = 𝜖1 on the bottom and 𝜖𝑥 = 𝜖2 on the top side are given by 𝜖1 = 𝜖 − 𝑧1 𝜅 = 𝜖 +

ℎ 𝜅, 2

𝜖2 = 𝜖 − 𝑧2 𝜅 = 𝜖 −

ℎ 𝜅 2

(4.17)

This leads to a relation for the curvature 𝜅=

𝜖1 − 𝜖2 , ℎ

ℎ = 𝑧2 − 𝑧1

(4.18)

Correct signs for strains have to be considered, e.g. 𝜖1 > 0, 𝜖2 < 0 in bending with compression on the top side. The coordinate 𝑧 of a line with a given strain 𝜖𝑥 = 𝜖𝑥′ is also determined by Eq. (4.5) 𝑧′ =

𝜖 − 𝜖𝑥′ 𝜅

(4.19)

with the strain 𝜖 of the reference axis. The the line with zero strains or zero line is determined as a special case with 𝜖𝑥′ = 0 and 𝑧0 =

𝜖 𝜅

(4.20)

The 𝑧-coordinates of the concrete tensile limit strain 𝜖𝑐𝑡 or compressive limit strain 𝜖𝑐𝑢1 (Figure 3.1) are determined in a similar way. A value 𝜖𝑐𝑡 = 0 with an exclusion of concrete tensile stresses is assumed as a first approach. This assumption is not mandatory but follows general conventions. ◀

The contribution of concrete to cross-sectional behaviour is basically determined through the compression zone. Its extent is determined by the position of the zero line.

Edge strains according to Eq. (4.17) and the position of the zero line according to Eq. (4.20) allow a classification: 𝑧0 < −ℎ∕2 and 𝜖1 < 0 𝑧0 < −ℎ∕2 and 𝜖1 ≥ 0 −ℎ∕2 ≤ 𝑧0 ≤ ℎ∕2 and 𝜖2 < 0 −ℎ∕2 ≤ 𝑧0 ≤ ℎ∕2 and 𝜖1 < 0 𝑧0 > ℎ∕2 and 𝜖2 < 0 𝑧0 > ℎ∕2 and 𝜖2 ≥ 0

1) This is not mandatory. Other choices are possible.

cross-section totally under compression totally under tension upper bending compression zone lower bending compression zone totally under compression totally under tension

4.1 Cross-Sectional Behaviour

(a)

(b)

Figure 4.2 Reinforced concrete cross-section. (a) Geometry. (b) Internal forces.

A precise localisation is given by the lower and upper compression zone coordinates 𝑧𝑐1 , 𝑧𝑐2 : cross-section totally under compression totally under tension upper bending compression zone lower bending compression zone

𝑧𝑐1 = 𝑧1 , 𝑧𝑐2 = 𝑧2 no concrete contribution 𝑧𝑐1 = 𝑧0 , 𝑧𝑐2 = 𝑧2 𝑧𝑐1 = 𝑧1 , 𝑧𝑐2 = 𝑧0

with cross-section bottom and top coordinates 𝑧1 , 𝑧2 . This approach may be extended to consider a restricted compression limit strain 𝜖𝑐𝑢1 or a tensile limit strain 𝜖𝑐𝑡 larger than zero through Eq. (4.19) with 𝜖𝑥′ = 𝜖𝑐𝑡 and/or 𝜖𝑥′ = 𝜖𝑐𝑢1 and using corresponding 𝑧′ -values as edges for the contribution of the concrete part. Nominal strain values 𝜖𝑥 > 𝜖𝑐𝑡 have to be considered regarding integrated crosssectional behaviour, e.g. for the inclusion of a reinforcement as shown in Figure 4.2. A lower rebar has a coordinate 𝑧𝑠1 = −ℎ∕2 + 𝑑1 and an upper rebar has a coordinate 𝑧𝑠2 = ℎ∕2 − 𝑑2 where 𝑑1 , 𝑑2 are edge distances. Equation (4.5) leads to rebar strains 𝜖𝑠1 = 𝜖 − (

ℎ − 𝑑1 ) 𝜅 , 2

𝜖𝑠2 = 𝜖 + (

ℎ − 𝑑2 ) 𝜅 2

(4.21)

This approach is not restricted to an upper and a lower reinforcement but may be extended to an arbitrary number of rebar layers. On the other hand the reinforcement may also be omitted on one side. The strain of rebars with tension and the concrete strain within the compression zone share the same straight line according to the linear strain distribution following Eq. (4.5). This implies a rigid bond with disregarded slip between concrete and reinforcement in contrast to flexible bond (Sections 3.4, 3.7, 8.5, Examples 3.4, 8.3, 8.4). ◀

Thus, the extent of the compression zone, the concrete strains within it and the reinforcement strains are completely determined by the generalised beam strains 𝜖 and 𝜅.

With longitudinal strains given the concrete stresses 𝜎𝑐 are determined by, e.g. Eqs. (3.1), and (3.4) and rebar stresses by Eq. (3.42). The corresponding internal

73

74

4 Structural Beams and Frames

forces are determined similar to Eq. (4.9) with a normal force 𝑧𝑐2

𝑁 = 𝐴𝑠1 𝜎𝑠1 + 𝐴𝑠2 𝜎𝑠2 + ∫ 𝜎𝑐 𝑏 d𝑧

(4.22)

𝑧𝑐1

and a moment 𝑧𝑐2

𝑀 = −𝐴𝑠1 𝜎𝑠1 𝑧𝑠1 − 𝐴𝑠2 𝜎𝑠2 𝑧𝑠2 − ∫ 𝜎𝑐 𝑧 𝑏 d𝑧

(4.23)

𝑧𝑐1

The sign conventions follow Figure 4.2. Stresses in all cases have to be signed as positive for tension and negative for compression. The integration limits 𝑧𝑐1 , 𝑧𝑐2 may be variable due to the variable values of 𝜖, 𝜅. An analytical evaluation of integrals in Eqs. (4.22) and (4.23) is generally not possible. 2) A numerical integration in one spatial dimension has to be used instead. As the course of concrete stresses is smooth, a simple integration scheme like the trapezoidal rule (Kreyszig 2006) with an interval number of the order of 10–50 is sufficient for the compression zone and – where applied – additionally for the tensile zone. A variable cross-section width 𝑏(𝑧) is easily regarded within a numerical integration scheme. A relation between shear force 𝑉 and shear angle 𝛾 cannot be derived within the isolated scope of a cracked cross-section shown in Figure 4.2. An approach based on the truss model for shear is described in Section 4.4.4. 4.1.3.2 Linear Concrete Compressive Behaviour

Equations (4.22) and (4.23) leading to internal forces are still open for various forms of uniaxial stress-strain relations. A linear relation may be assumed for concrete under compression 𝜎𝑐 = 𝐸𝑐 𝜖𝑥 ,

𝜖𝑥 ≤ 0

(4.24)

with the initial Young’s modulus 𝐸𝑐 (Figure 3.1) while excluding a tensile strength. Exclusion of tensile strength is a first distinctive feature compared to the linear elastic approach (Eq. (4.7)). The assumption of linear compressive stress is appropriate for moderate stress levels up to roughly 60% of the strength. Corresponding states of loading are investigated in the context of serviceability of structures, e.g. for the calculation of deformations. A cross-section with height ℎ and width 𝑏 is considered. The width may basically be variable along the cross-section. A compression zone is determined with given generalised beam strains 𝜖, 𝜅 according to Section 4.1.3.1. It has an extension 𝑧𝑐1 ≤ 𝑧 ≤ 𝑧𝑐2 with a compression zone height 𝑥 = 𝑧𝑐2 − 𝑧𝑐1 and edge strains 𝝐 𝑐𝑒 = B𝜖 ⋅ 𝝐 2) An exception is described in the following Section 4.1.3.2.

(4.25)

4.1 Cross-Sectional Behaviour

Figure 4.3 Reinforced concrete cross-section with linear compressive concrete.

with ⎛𝜖𝑐1 ⎞ 𝝐 𝑐𝑒 = ⎜ ⎟ , 𝜖 ⎝ 𝑐2 ⎠

⎡1 B𝜖 = ⎢ 1 ⎣

−𝑧𝑐1 ⎤ ⎥, −𝑧𝑐2 ⎦

⎛𝜖 ⎞ 𝝐=⎜ ⎟ 𝜅 ⎝ ⎠

(4.26)

according to Eq. (4.5). This leads to edge stresses ⎛𝜎𝑐1 ⎞ 𝝈𝑐𝑒 = ⎜ ⎟ = 𝐸𝑐 𝝐 𝑐𝑒 = 𝐸𝑐 B𝜖 ⋅ 𝝐 , 𝜎 ⎝ 𝑐2 ⎠

(4.27)

In the range 𝑧𝑐1 ≤ 𝑧 ≤ 𝑧𝑐2 concrete stresses are linearly interpolated with 𝜎𝑐1 𝑧𝑐2 − 𝜎𝑐2 𝑧𝑐1 𝜎𝑐2 − 𝜎𝑐1 + 𝑧 𝑧𝑐2 − 𝑧𝑐1 𝑧𝑐2 − 𝑧𝑐1

𝜎𝑐 (𝑧) =

(4.28)

Concrete contributions to internal forces are given by (Eq. (4.9)), see Figure 4.3 𝑧𝑐2

𝑁𝑐 = ∫ 𝜎𝑐 (𝑧) 𝑏 d𝑧

=

𝑧𝑐1

𝜎𝑐1 𝑧𝑐2 − 𝜎𝑐2 𝑧𝑐1 𝜎𝑐1 − 𝜎𝑐2 𝐴𝑐 − 𝑆𝑐 𝑥 𝑥 (4.29)

𝑧𝑐2

𝑀𝑐 = − ∫ 𝜎𝑐 (𝑧)𝑧 𝑏 d𝑧 𝑧𝑐1

𝜎 𝑧 − 𝜎𝑐2 𝑧𝑐1 𝜎 − 𝜎𝑐2 = − 𝑐1 𝑐2 𝑆𝑐 + 𝑐1 𝐽𝑐 𝑥 𝑥

with cross-sectional values 𝑧𝑐2

𝐴𝑐 = ∫ 𝑏 d𝑧 , 𝑧𝑐1

𝑧𝑐2

𝑆𝑐 = ∫ 𝑧 𝑏 d𝑧 , 𝑧𝑐1

𝑧𝑐2

𝐽𝑐 = ∫ 𝑧2 𝑏 d𝑧

(4.30)

𝑧𝑐1

A matrix notation of Eq. (4.29) is given by 𝝈𝑐 = A𝜎 ⋅ 𝝈𝑐𝑒

(4.31)

75

76

4 Structural Beams and Frames

with ⎛ 𝑁𝑐 ⎞ 𝝈𝑐 = ⎜ ⎟ , 𝑀 ⎝ 𝑐⎠

A𝜎 =

1 ⎡ 𝑧𝑐2 𝐴𝑐 − 𝑆𝑐 𝑥 ⎢−𝑧𝑐2 𝑆𝑐 + 𝐽𝑐 ⎣

−𝑧𝑐1 𝐴𝑐 + 𝑆𝑐 ⎤ ⎥, 𝑧𝑐1 𝑆𝑐 − 𝐽𝑐 ⎦

(4.32)

whereby 𝝈𝑐 and 𝝈𝑐𝑒 have to be clearly distinguished: 𝝈𝑐 is the vector of internal forces and 𝝈𝑐𝑒 the vector of edge stresses. The matrix A𝜎 is decomposed through A𝜎 = A ⋅ B𝜎 ,

⎡ 𝐴𝑐 A=⎢ −𝑆 ⎣ 𝑐

−𝑆𝑐 ⎤ ⎥, 𝐽𝑐 ⎦

B𝜎 =

1 ⎡𝑧𝑐2 ⎢ 𝑥 1 ⎣

−𝑧𝑐1 ⎤ ⎥ −1 ⎦

(4.33)

The combination of Eqs. (4.27) and (4.31) leads to 𝝈𝑐 = C𝑐 ⋅ 𝝐

(4.34)

with C𝑐 = 𝐸𝑐 A𝜎 ⋅ B𝜖 = 𝐸𝑐 A ⋅ B𝜎 ⋅ B𝜖 = 𝐸𝑐 A

(4.35)

as B𝜎 ⋅ B𝜖 = I with the unit matrix I. The matrix C𝑐 is the material stiffness regarding the compression zone only. It is similar to Eq. (4.16) but Eq. (4.34) couples normal force 𝑁𝑐 as well as moment 𝑀𝑐 each to both 𝜖 and 𝜅. ◀

Besides the coupling effect the material stiffness of a cracked concrete cross-section is nonlinear as the compression zone extension depends on the generalised beam strains 𝜖, 𝜅.

In the case of rectangular cross-sections with the constant width 𝑏 the cross𝑏 2 𝑏 3 2 sectional values are evaluated to 𝐴𝑐 = 𝑏(𝑧𝑐2 − 𝑧𝑐1 ), 𝑆𝑐 = (𝑧𝑐2 − 𝑧𝑐1 ), 𝐽𝑐 = (𝑧𝑐2 − 2

3

3 ), and in the case of 𝑧𝑐1 = −ℎ∕2, 𝑧𝑐2 = ℎ∕2 to 𝐴𝑐 = 𝑏ℎ, 𝑆𝑐 = 0, 𝐽𝑐 = 𝑏ℎ3 ∕12. 𝑧𝑐1 The incremental form of Eq. (4.34) has to consider the change of the extension of the compression zone due to the change of 𝜖, 𝜅. Its derivation is given in Appendix A.3 and yields

𝝈̇ 𝑐 = [ C𝑐 + (A𝑧 − 𝜅𝐸𝑐 A𝜎 ) ⋅ B𝑧 ] ⋅ 𝝐̇ = C𝑐𝑇 ⋅ 𝝐̇

(4.36)

with a tangential cross-sectional stiffness C𝑐𝑇 and the matrices A𝜎 , A𝑧 , B𝑧 according to Eqs. (4.32), (A.33), and (A.37). The contribution of the reinforcement is derived from Eqs. (4.22) and (4.23) by 𝝈𝑠 = A𝑠 ⋅ 𝝈𝑠𝑒

(4.37)

4.1 Cross-Sectional Behaviour

with ⎛ 𝑁𝑠 ⎞ 𝝈𝑠 = ⎜ ⎟ , 𝑀 ⎝ 𝑠⎠

⎡ 𝐴𝑠1 A𝑠 = ⎢ −𝐴 𝑧 ⎣ 𝑠1 𝑠1

⎤ ⎥, −𝐴𝑠2 𝑧𝑠2 ⎦ 𝐴𝑠2

⎛𝜎𝑠1 ⎞ 𝝈𝑠𝑒 = ⎜ ⎟ 𝜎 ⎝ 𝑠2 ⎠

(4.38)

Reinforcement strains are given with Eq. (4.21) 𝝐 𝑠 = B𝑠 ⋅ 𝝐

(4.39)

with ⎛𝜖𝑠1 ⎞ 𝝐𝑠 = ⎜ ⎟ , 𝜖 ⎝ 𝑠2 ⎠

⎡1 B𝑠 = ⎢ 1 ⎣

−𝑧𝑠1 ⎤ ⎥, −𝑧𝑠2 ⎦

⎛𝜖 ⎞ 𝝐=⎜ ⎟ 𝜅 ⎝ ⎠

(4.40)

A linear elastic reinforcement behaviour 𝜎𝑠 = 𝐸𝑠 𝜖𝑠 is assumed according to the assumption of the moderate loading level. Finally, this leads to 𝝈𝑠 = 𝐸𝑠 A𝑠 ⋅ B𝑠 ⋅ 𝝐 = C𝑠 ⋅ 𝝐

(4.41)

which has to be superposed to the concrete part 𝝈𝑐 𝝈 = (C𝑐 + C𝑠 ) ⋅ 𝝐

(4.42)

The tangential cross-sectional stiffness equals the cross-sectional stiffness C𝑠𝑇 = C𝑠 in the case of linear elastic reinforcement behaviour. 4.1.3.3 Nonlinear Behaviour of Concrete and Reinforcement

Nonlinear stress-strain relations have to be regarded besides a variable concrete compression zone to model limit states of structures. Such relations are already introduced with Eq. (3.1) for the compression of concrete and with Eq. (3.42) for the reinforcement. This can be used in Eqs. (4.22) and (4.23) to determine internal forces for beams. They should not be confused with internal nodal forces (Eq. (2.58)). Uniaxial strains serve as input. They are derived from the generalised beam strains by Eqs. (4.5) and (4.21) whereby the extent of the compression zone is determined as is described in Section 4.1.3.1. The evaluation of integrals in Eqs. (4.22) and (4.23) has to be performed numerically and leads to nonlinear relations 𝑀(𝜖, 𝜅) and 𝑁(𝜖, 𝜅). In contrast to the linear elastic relations Eq. (4.16) moments 𝑀 also depend on the strain 𝜖 of the reference axis besides curvature 𝜅 and normal forces 𝑁 also depend on 𝜅 besides 𝜖. This effect already arises with the case described in the foregoing Section 4.1.3.2. Nonlinear material behaviour leads to nonlinear system behaviour which may be solved as is described in Section 2.8.2. Thus, a tangential material stiffness is required for a tangential system stiffness matrix. General forms for the derivatives of internal forces with respect to generalised strains are described in the following. Normal force derivatives are derived from

77

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4 Structural Beams and Frames

Eq. (4.22) leading to 𝑧𝑐2

𝜕𝜎𝑠1 𝜕𝜖𝑥 𝜕𝜎𝑠2 𝜕𝜖𝑥 𝜕𝜎𝑐 𝜕𝜖𝑥 𝜕𝑁 = 𝐴𝑠1 + 𝐴𝑠2 +∫ 𝑏 d𝑧 𝜕𝜖 𝜕𝜖𝑥 𝜕𝜖 𝜕𝜖𝑥 𝜕𝜖 𝜕𝜖𝑥 𝜕𝜖 𝑧𝑐1

𝑧𝑐2

= 𝐴𝑠1

𝜕𝜎𝑠1 𝜕𝜎𝑠2 𝜕𝜎𝑐 + 𝐴𝑠2 +∫ 𝑏 d𝑧 𝜕𝜖𝑥 𝜕𝜖𝑥 𝜕𝜖𝑥 𝑧𝑐1

(4.43)

𝑧𝑐2

𝜕𝜎 𝜕𝜖 𝜕𝜎 𝜕𝜖 𝜕𝜎𝑐 𝜕𝜖𝑥 𝜕𝑁 = 𝐴𝑠1 𝑠1 𝑥 + 𝐴𝑠2 𝑠2 𝑥 + ∫ 𝑏 d𝑧 𝜕𝜅 𝜕𝜖𝑥 𝜕𝜅 𝜕𝜖𝑥 𝜕𝜅 𝜕𝜖𝑥 𝜕𝜅 𝑧𝑐1

𝑧𝑐2

= −𝐴𝑠1

𝜕𝜎𝑠1 𝜕𝜎𝑠2 𝜕𝜎𝑐 𝑧 − 𝐴𝑠2 𝑧 −∫ 𝑧 𝑏 d𝑧 𝜕𝜖𝑥 𝑠1 𝜕𝜖𝑥 𝑠2 𝜕𝜖𝑥 𝑧𝑐1

Moment derivatives are derived from Eq. (4.23) leading to 𝑧𝑐2

𝜕𝜎𝑠1 𝜕𝜖𝑥 𝜕𝜎𝑠2 𝜕𝜖𝑥 𝜕𝜎𝑐 𝜕𝜖𝑥 𝜕𝑀 = −𝐴𝑠1 𝑧 − 𝐴𝑠2 𝑧 −∫ 𝑧 𝑏 d𝑧 𝜕𝜖 𝜕𝜖𝑥 𝜕𝜖 𝑠1 𝜕𝜖𝑥 𝜕𝜖 𝑠2 𝜕𝜖𝑥 𝜕𝜖 𝑧𝑐1

𝑧𝑐2

= −𝐴𝑠1

𝜕𝜎𝑠1 𝜕𝜎𝑠2 𝜕𝜎𝑐 𝑧 − 𝐴𝑠2 𝑧 −∫ 𝑧 𝑏 d𝑧 𝜕𝜖𝑥 𝑠1 𝜕𝜖𝑥 𝑠2 𝜕𝜖𝑥 𝑧𝑐1

=

𝜕𝑁 𝜕𝜅

(4.44) 𝑧𝑐2

𝜕𝜎𝑠1 𝜕𝜖𝑥 𝜕𝜎𝑠2 𝜕𝜖𝑥 𝜕𝜎𝑐 𝜕𝜖𝑥 𝜕𝑀 = −𝐴𝑠1 𝑧𝑠1 − 𝐴𝑠2 𝑧𝑠2 − ∫ 𝑧 𝑏 d𝑧 𝜕𝜅 𝜕𝜖𝑥 𝜕𝜅 𝜕𝜖𝑥 𝜕𝜅 𝜕𝜖𝑥 𝜕𝜅 𝑧𝑐1

𝑧𝑐2

= 𝐴𝑠1

𝜕𝜎𝑠1 2 𝜕𝜎𝑠2 2 𝜕𝜎𝑐 2 𝑧𝑠1 + 𝐴𝑠2 𝑧𝑠2 + ∫ 𝑧 𝑏 d𝑧 𝜕𝜖𝑥 𝜕𝜖𝑥 𝜕𝜖𝑥 𝑧𝑐1

This already includes uniaxial strain dependence on generalised beam strains from Eq. (4.5) with 𝜕𝜖𝑥 =1, 𝜕𝜖

𝜕𝜖𝑥 = −𝑧 𝜕𝜅

(4.45)

The variation of integration borders 𝑧𝑐1 , 𝑧𝑐2 , which has been considered before (Eq. (A.32)), is disregarded in this approach to simplify the description and to avoid an inflation of relations. ◀

The key for the derivatives of internal forces is given with the derivatives

𝜕𝜎s1 𝜕𝜎s2 𝜕𝜖x

,

𝜕𝜖x

,

𝜕𝜎c 𝜕𝜖x

of stresses with respect to strains. They are derived according to the description of uniaxial concrete behaviour in Section 3.1 and reinforcing steel behaviour in Section 3.3.

4.1 Cross-Sectional Behaviour

The evaluation of integrals has to be performed numerically in one spatial dimension, see also remarks following Eq. (4.23). The resulting terms are collected in a tangential material stiffness matrix ⎡ 𝜕𝑁 ⎢ 𝜕𝜖 C𝑇 = ⎢ ⎢ 𝜕𝑀 ⎣ 𝜕𝜖

𝜕𝑁 ⎤ 𝜕𝜅 ⎥ ⎥ 𝜕𝑀 ⎥ 𝜕𝜅 ⎦

The linear elastic, diagonal system is derived as a special case with 𝐸𝑠 ,

𝜕𝜎𝑐 𝜕𝜖𝑥

(4.46) 𝜕𝜎𝑠1 𝜕𝜖𝑥

=

𝜕𝜎𝑠2 𝜕𝜖𝑥

=

= 𝐸𝑐 and ∫ 𝑏 d𝑧 = 𝐴, ∫ 𝑧 𝑏 d𝑧 = 0, ∫ 𝑧2 𝑏 d𝑧 = 𝐽. To include shear forces,

Eq. (4.46) has to be extended to ⎡ 𝜕𝑁 𝜕𝑁 0 ⎤ ⎢ 𝜕𝜖 𝜕𝜅 ⎥ ⎢ 𝜕𝑀 𝜕𝑀 ⎥ ⎢ 0 ⎥ (4.47) C𝑇 = ⎢ 𝜕𝜖 ⎥ 𝜕𝜅 ⎢ ⎥ 𝜕𝑉 ⎥ ⎢ 0 0 𝜕𝛾 ⎦ ⎣ whereby a coupling of shear with longitudinal actions has been neglected. The re𝜕𝑉 maining coefficient may be determined according to Section 4.4.4. 𝜕𝛾

Regarding an isolated cross-section the general nonlinear approach allows for several calculation types: • Specification of strain 𝜖, curvature 𝜅 and calculation of 𝑀, 𝑁. This is a standard procedure as has been described in connection with Eqs. (4.22) and (4.23). • Specification of curvature 𝜅 and normal force 𝑁 and calculation of moment 𝑀. A result for 𝜖 occurs as side effect. This is a procedure to derive moment-curvature relations parameterised by a normal force. With 𝑁, 𝜅 given the nonlinear equation 𝑓(𝜖, 𝜅) − 𝑁 = 0 has to be solved for 𝜖. This is efficiently done with a Newton–Raphson iteration, i.e. ) ( (𝜈) 1 (4.48) 𝑓(𝜖 , 𝜅) − 𝑁 𝜖 (𝜈+1) = 𝜖 (𝜈) − 𝜕𝑓 | || | 𝜕𝜖 |𝜖=𝜖(𝜈) see Eq. (2.77). The starting value is 𝜖 (0) = 0. The derivative 𝜕𝑓∕𝜕𝜖 is given by Eq. (4.431 ). With 𝜅 given and 𝜖 calculated, 𝑀 = 𝑓(𝜖, 𝜅) can be determined. • Specification of moment 𝑀, normal force 𝑁 and calculation of 𝜅 and 𝜖. This is the inverse to the standard procedure to derive deformations from given internal forces. From Eqs. (4.22) and (4.23) a nonlinear algebraic system arises for unknowns 𝜅, 𝜀 connected with the evaluation of integrals over 𝜎𝑠1 , 𝜎𝑠2 , 𝜎𝑐 . This may also be solved with the Newton–Raphson method. The second calculation type leads to moment-curvature relations which is demonstrated with the following example.

79

80

4 Structural Beams and Frames

Example 4.1: Moment-Curvature Relations for given Normal Forces

An RC cross-section of rectangular shape is given with the following properties: • Cross section height ℎ = 0.4 m and width 𝑏 = 0.2 m. • Concrete grade C 30/37 according to EN 1992-1-1 (2004, Table 3.1) with a strength 𝑓𝑐 = 38 MN∕m2 and characteristic strains 𝜖𝑐1 = −2.2‰, 𝜖𝑐𝑢1 = −3.5‰ (Figure 3.1). The stress-strain curve is chosen as proposed by EN 1992-1-1 (2004, 3.1.5) whereby basically following Figure 3.1. The initial Young’s modulus is 𝐸𝑐 = 33 000 MN∕m2 . A tensile strength is not considered. • Reinforcement behaviour is assumed according to Section 3.3 and EN 1992-1-1 (2004, 3.2.7) with 𝑓𝑦𝑘 = 500 MN∕m2 , 𝑓𝑡 = 525 MN∕m2 , 𝜖𝑦0 = 2.5‰ and 𝜖𝑢 = 25‰ (Figure 3.11a). • Upper and lower reinforcement each with a geometry 4 ⊘ 20, 𝐴𝑠2 = 𝐴𝑠1 = 12.57 cm2 , 𝑑2 = 𝑑1 = 5 cm (Figure 4.2a). These values are not modified by safety factors. The moment—-curvature relations are computed for 𝑁 = 0, −1, −2 MN. The curvature 𝜅 is increased starting from zero and 𝑁, 𝜅 serve as input values for the nonlinear computation. The computed moments are of primary interest and lead to moment-curvature curves as shown in Figure 4.4. • The stiffness is ruled by the following factors: (1) height of the compression zone, (2) tangential material stiffness of concrete, (3) tangential material stiffness of steel. The cases with normal compression initially have a higher compression zone leading to a higher stiffness. • The final points indicate the state when the ultimate concrete compressive strain of 𝜖𝑐1𝑢 = −0.0035 is reached. • Ultimate moments increase with moderate normal compression due to a larger concrete force resultant acting with an lever arm (Figure 4.2b). But the ductility – range of curvature with sustained moment – decreases. • Kinks for 𝑁 = 0, 𝑁 = −1 MN correspond to beginning reinforcement yielding on the tensile side.

Figure 4.4 Example 4.1. Moment-curvature curves.

4.2 Equilibrium of Beams

• Reinforcement yielding does not occur for 𝑁 = −2 MN but concrete stresses extend into the compressive softening range (Figure 3.1) which causes an initial softening of the cross-section behaviour. The case of the uncracked, linear elastic cross-section is shown as reference, furthermore the case with linear elastic reinforcement alone without concrete contribution. The initial bending stiffness of concrete alone is 𝐸𝐽𝑐 = 32.5 MN∕m2 , the initial bending stiffness of reinforcement alone is 𝐸𝐽𝑠 = 11.3 MN∕m2 and the total initial stiffness is 𝐸𝐽 = 43.8 MN∕m2 . A generalisation of RC is given with fibre models. Every line along the beam axis cut by the cross-section may be regarded as a fibre. Each fibre is strained according to beam kinematics (Eq. 4.5) which leads to a longitudinal stress. With the integration of stresses resulting to internal forces any type of a uniaxial material law may be used for fibres. The quality of such fibre models is influenced by numerical integration methods. Simple methods like the trapezoidal rule or the Simpson rule may be used with a sufficient number of intervals. Gauss integration is not optimal, as the important upper and lower edges are not captured. Lobatto integration might be an alternative.

4.2 Equilibrium of Beams Kinematic compatibility and material laws for beams are discussed in the preceding sections. Equilibrium remains to be added. We regard a loading 𝑝𝑥 (𝑥, 𝑡), 𝑝 𝑧 (𝑥, 𝑡) acting at an infinitesimal section of a plane beam varying with place 𝑥 and time 𝑡, see Figure 4.5. Inertial effects are considered. Therefore, the bar (inertial) mass per length 𝑚 and an inertial mass moment 𝛩 have to be regarded. The strong differential formulation

Figure 4.5 Equilibrium of an infinitesimal beam element.

81

82

4 Structural Beams and Frames

of dynamic equilibrium is given by differential equations (𝜕 ∙ ∕𝜕𝑥 = ∙′ ) 𝑝 𝑥 + 𝑁 ′ = 𝑚 𝑢̈

𝑝𝑧 + 𝑉 ′ = 𝑚 𝑤̈ 𝑉+

𝑀′

(4.49)

= 𝛩 𝜙̈

according to Newton’s law – force = mass × acceleration – with the longitudinal acceleration 𝑢, ̈ the lateral acceleration 𝑤̈ and the acceleration 𝜙̈ of the cross-sectional rotation angle. To begin with, the linear elastic case is mentioned to connect these equations to well-known formulations. From Eq. (4.13) 𝑁 = 𝐸𝐴 𝜖 = 𝐸𝐴 𝑢′ ,

𝑀 = 𝐸𝐽 𝜅 = 𝐸𝐽 𝜙′

(4.50)

The combination of Eqs. (4.491 ) and (4.501 ) leads to (𝜕 2 ∙ ∕𝜕𝑥2 = ∙′′ , …) 𝑚 𝑢̈ − 𝐸𝐴 𝑢′′ = 𝑝 𝑥

(4.51)

This represents the one-dimensional wave equation. We set 𝑚 = 𝜚 𝐴 with the specific mass 𝜚 and 𝑝𝑥 = 0 to gain the familiar form. Furthermore, the term 𝛩 𝜙̈ is generally neglected as 𝛩 ≪ 1. Thus, from Eqs. (4.492,3 ) 𝑚 ⋅ 𝑤̈ + 𝑀 ′′ = 𝑝 𝑧 . In the case of a slender beam the shear rotation 𝛾 is small compared to the total rotation 𝑤 ′ and 𝜙 ≈ 𝑤 ′ . The same holds true for the derivatives and Eq. (4.502 ) yields 𝑀 = 𝐸𝐽 𝑤 ′′ . Finally, we get 𝑚 𝑤̈ + 𝐸𝐽 𝑤 ′′′′ = 𝑝𝑧

(4.52)

representing dynamic beam bending including the quasi-static case with 𝑚 = 0. To revisit the general approach, consider the relationship between cross-sectional angles Eq. (4.2) 𝜙 = 𝑤′ − 𝛾

(4.53)

to include shear deformations again for the following. Equilibrium has to be reformulated as weak integral formulation as a base for finite element discretisations. This uses test functions or virtual displacements 𝛿𝑢, 𝛿𝑤, 𝛿𝛾, 𝛿𝜙 = 𝛿𝑤 ′ − 𝛿𝛾 (Section 2.5) which are assumed as independent from each other. These functions must be kinematically compatible, i.e. be continuous and first derivatives must exist. Regarding a bar of finite length 𝐿 with a longitudinal coordinate 0 ≤ 𝑥 ≤ 𝐿 and admitting all such functions an equivalent to the strong form (Eq. (4.49)) is given

4.2 Equilibrium of Beams

by 𝐿

𝐿

∫ 𝛿𝑢 𝑚𝑢̈ d𝑥 − ∫ 0

𝐿

𝛿𝑢 𝑁 ′

0

𝐿

d𝑥 + ∫ 𝛿𝑤 𝑚𝑤̈ d𝑥 − ∫ 𝛿𝑤 𝑉 ′ d𝑥 0

𝐿

0

𝐿

+ ∫ 𝛿𝜙 𝛩𝜙̈ d𝑥 − ∫ 𝛿𝜙 𝑀 ′ d𝑥 0 𝐿

0 𝐿

𝐿

= ∫ 𝛿𝑢 𝑝𝑥 d𝑥 + ∫ 𝛿𝑤 𝑝𝑧 d𝑥 + ∫ (𝛿𝑤 ′ − 𝛿𝛾) 𝑉 d𝑥 0

0

(4.54)

0

whereby each differential equation is multiplied by its own test function, the product integrated over the beam length and finally all parts are added. ◀

The solutions of Eq. (4.54) also solve Eq. (4.49) and vice versa.

Those terms with derivatives of internal forces are partially integrated in a further step 𝐿

∫

𝐿

𝛿𝑢 𝑁 ′ d𝑥

= [𝛿𝑢(𝐿) 𝑁(𝐿) − 𝛿𝑢(0) 𝑁(0)]

− ∫ 𝛿𝜖 𝑁 d𝑥

0

0

𝐿

𝐿

∫ 𝛿𝜙 𝑀 ′ d𝑥 = [𝛿𝜙(𝐿) 𝑀(𝐿) − 𝛿𝜙(0) 𝑀(0)]

− ∫ 𝛿𝜅 𝑀 d𝑥

0

0

𝐿

∫

(4.55)

𝐿

𝛿𝑤 𝑉 ′ d𝑥

= [𝛿𝑤(𝐿) 𝑉(𝐿) − 𝛿𝑤(0) 𝑉(0)]

− ∫ 𝛿𝑤 ′ 𝑉 d𝑥

0

0

with virtual generalised strains 𝛿𝜖 = 𝛿𝑢′ , 𝛿𝜅 = 𝛿𝜙′ . Combining this with Eq. (4.54) results to 𝐿

𝐿

𝐿

∫ 𝛿𝑢 𝑚𝑢̈ d𝑥 + ∫ 𝛿𝜙 𝛩𝜙̈ d𝑥 + ∫ 𝛿𝑤 𝑚𝑤̈ d𝑥+ 0

0 𝐿

0 𝐿

𝐿

+ ∫ 𝛿𝜖 𝑁 d𝑥 + ∫ 𝛿𝜅 𝑀 d𝑥 + ∫ 𝛿𝛾 𝑉 d𝑥 0 𝐿

0

0

𝐿 𝐿

𝐿

𝐿

= ∫ 𝛿𝑢 𝑝𝑥 d𝑥 + ∫ 𝛿𝑤 𝑝𝑧 d𝑥 + [𝛿𝑢 𝑁]0 + [𝛿𝜙 𝑀]0 + [𝛿𝑤 𝑉]0 0

0

(4.56)

83

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4 Structural Beams and Frames

whereby the boundary terms in Eq. (4.55) have been abbreviated. This is interpreted as virtual work principle and has the following parts: • • • •

Inertial forces with the first three terms of the left-hand side. Internal forces with the last three terms of the left-hand side. Distributed loading with the first two terms of the right-hand side. Boundary terms with beam end forces with the last three terms of the right-hand side.

A generalizing matrix notation of Eq. (4.56) is given by 𝐿

∫

𝐿

𝛿𝝐 T

⋅ 𝝈 d𝑥 + ∫

0

𝐿

𝛿uT

⋅ m ⋅ ü d𝑥 = ∫ 𝛿uT ⋅ p d𝑥 + 𝛿UT ⋅ c

0

(4.57)

0

see also Eq. (2.52). In the case of beams the vector and matrix quantities have the components ( )T ( )T 𝝐= 𝜖 𝛾 𝜅 , 𝝈= 𝑁 𝑉 𝑀 ⎡𝑚 0 0 ⎤ ⎛𝑝 𝑥 ⎞ ⎛𝑢⎞ ⎢ ⎥ u = ⎜𝑤 ⎟ , m = ⎢ 0 𝑚 0 ⎥ , p = ⎜ 𝑝𝑧 ⎟ ⎜ ⎟ ⎜ ⎟ ⎢ ⎥ 0 0 𝛩 ⎝0⎠ ⎝𝜙 ⎠ ⎣ ⎦ ( )T U = 𝑢0 𝑤0 𝜙0 𝑢𝐿 𝑤𝐿 𝜙𝐿 )T ( c = 𝑁0 𝑉0 𝑀0 𝑁𝐿 𝑉𝐿 𝑀𝐿

(4.58)

with ∙𝐿 = ∙(𝐿), ∙0 = ∙(0) and generalised strains 𝜖, 𝛾, 𝜅, see Figure 4.1 and Eq. (4.4). The boundary force c would need a sign convention )T ( t = −𝑁0 −𝑉0 −𝑀0 𝑁𝐿 𝑉𝐿 𝑀𝐿 to be consistent with Eq. (4.55) and Figure 4.5, but regarding a finite element discretisation a global sign convention is more appropriate: internal forces at the l.h.s. are assumed as positive with the same orientation as at the r.h.s. and the above is reversed into Eq. (4.584 ). Regarding the beam ends, i.e. its boundary conditions, one quantity has to be prescribed from each pair (𝑢, 𝑁), (𝑤, 𝑉), (𝜙, 𝑀) for every end. 3) Shear deformations are still included covering the Timoshenko beam. In the case of a slender beam the shear rotation 𝛾 is small compared to the total rotation 𝑤 ′ and 𝜙 = 𝑤′ ,

𝜅 = 𝜙′ = 𝑤 ′′

(4.59)

may be assumed. With 𝛾 = 0, the contribution 𝛿𝛾 𝑉 vanishes in Eq. (4.56). Thus, we obtain ⎛𝜖 ⎞ 𝝐=⎜ ⎟ , 𝜅 ⎝ ⎠

⎛𝑁 ⎞ 𝝈=⎜ ⎟ 𝑀 ⎝ ⎠

3) A zero value is also a prescription.

(4.60)

4.3 Finite Elements for Plane Beams

and get the formulation for the Bernoulli beam. With 𝑀 given the shear force 𝑉 is determined by postprocessing using Eq. (4.493 ) which still is valid for 𝛾 = 0 whereby an assumption 𝛩𝜙̈ = 0 is generally appropriate. Apart from 𝝈, 𝝐 all other quantities in Eq. (4.58) remain unchanged. Mass inertia is still included and the formulations apply to dynamics. A simplification is taken as the inertial mass moment 𝛩 generally is relatively small and may be neglected with 𝛩 = 0. This applies to the Timoshenko beam as well as for the Bernoulli beam and we obtain ⎛𝑢 ⎞ ⎛𝑝 ⎞ ⎡𝑚 0 ⎤ 𝑥 u=⎜ ⎟ , m=⎢ (4.61) ⎥ , p=⎜ ⎟ 𝑝𝑧 𝑤 0 𝑚 ⎣ ⎦ ⎝ ⎠ ⎝ ⎠ as modification of Eq. (4.58). Finally, quasi-statics is included for both beam types with a mass inertia 𝑚 = 0 leading to a mass matrix m = 0. The boundary terms U, c of Eq. (4.57) remain unchanged for all cases mentioned. The conditions Eq. (2.53) concerning the prescribed boundary displacements also have to be applied for the case of beams. It is generally easy to choose displacement trial functions (Section 2.6) such that they fulfil prescribed displacements on boundaries. Thus, test functions or virtual displacements may be set to zero along boundaries with kinematic boundary conditions. As a further consequence such end force components in c (Eq. (4.584 )) belonging to prescribed displacement components in U (Eq. (4.583 )) result as reaction forces whereby corresponding to internal nodal forces (Eq. (2.581 )).

4.3 Finite Elements for Plane Beams A general form of displacement interpolation with finite elements is given by Eq. (2.18). This is specified for the interpolation of the beam displacement variables 𝑢, 𝑤, 𝜙 in the following. A beam with a coordinate range 0 ≤ 𝑥 ≤ 𝐿 is subdivided into a number of 𝑛𝐸 elements. Each element has two nodes in a first approach leading to 𝑛𝑁 = 𝑛𝐸 + 1 nodes. An element 𝑒 has the global nodal coordinates 𝑥𝐼 , 𝑥𝐽 , a length 𝐿𝑒 = 𝑥𝐽 − 𝑥𝐼 and a local coordinate 𝑟. The relation between local and global coordinates is given by ] ⎛𝑥𝐼 ⎞ (1 + 𝑟) ⋅ ⎜ ⎟ , −1 ≤ 𝑟 ≤ 1 (4.62) 𝑥𝐽 ⎝ ⎠ with a local coordinate 𝑟 and 𝑥(−1) = 𝑥𝐼 , 𝑥(1) = 𝑥𝐽 . This yields a Jacobian 𝐽 (Eq. (2.20)) 𝑥=

[1 2

(1 − 𝑟)

1 2

𝐿𝑒 𝜕𝑥 = (4.63) 2 𝜕𝑟 which is needed for the numerical integration (Eq. (2.68)). The inverse relation 𝑟 = (2𝑥 − 𝑥𝐽 − 𝑥𝐼 )∕𝐿𝑒 leads to 𝐽=

𝜕𝑟 2 = 𝐽 −1 = 𝐿𝑒 𝜕𝑥

(4.64)

85

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4 Structural Beams and Frames

Polynomial forms ∑𝑞 𝑎𝑖 (𝑡) ⋅ 𝑟 𝑖 𝑦(𝑟, 𝑡) =

(4.65)

𝑖=0

of order 𝑞 are chosen to interpolate displacement variables within an element. 4) The coefficients 𝑎𝑖 are functions of time 𝑡 while 𝑦 represents the displacement variables 𝑢, 𝑤, and 𝜙 (Figure 4.1). A particular formulation yields a trial function. A trial function is constrained by the interpolation property to reproduce nodal values as is demonstrated in the following sections. According to the method of Bubnov– Galerkin test functions are chosen in the same way as the trial functions ∑𝑞 𝛿𝑎𝑖 (𝑡) ⋅ 𝑟 𝑖 (4.66) 𝛿𝑦(𝑟, 𝑡) = 𝑖=0

Deformation variables 𝜖, 𝛾, 𝜅 and their variations 𝛿𝜖, 𝛿𝛾, 𝛿𝜅 are derived from displacement variables according to Section 4.1.1 as spatial derivatives of Eqs. (4.65) and (4.66). Derivatives should be finite across a domain of elements to ensure the integrability of Eq. (4.57). Specifications of Eq. (4.65) are derived in the following based on these general remarks.

4.3.1 Timoshenko Beam Nodal degrees of freedom of the two-node Timoshenko beam element are collected with )T ( 𝝊𝑒 = 𝑢𝐼 𝑤𝐼 𝜙𝐼 𝑢𝐽 𝑤𝐽 𝜙𝐽 (4.67) The dependence on time 𝑡 is not explicitly written for convenience. A first reasonable interpolation approach for the beam displacement variables is given by 𝑢=

𝑤=

𝜙=

[1 2

[1 2

[1 2

(1 − 𝑟)

(1 − 𝑟)

(1 − 𝑟)

] ⎛ 𝑢𝐼 ⎞ (1 + 𝑟) ⋅ ⎜ ⎟ 2 𝑢 ⎝ 𝐽⎠ ] ⎛ 𝑤𝐼 ⎞ 1 (1 + 𝑟) ⋅ ⎜ ⎟ 2 𝑤 ⎝ 𝐽⎠ ] ⎛ 𝜙𝐼 ⎞ 1 (1 + 𝑟) ⋅ ⎜ ⎟ 2 𝜙 ⎝ 𝐽⎠ 1

(4.68)

straightly following the isoparametric concept. This is equivalent to 𝑦(𝑟) = 𝑎0 + 𝑎1 𝑟 ,

𝑎0 =

𝑦𝐼 𝑦 + 𝐽 , 2 2

𝑎1 = −

𝑦𝐼 𝑦 + 𝐽 2 2

(4.69)

regarding Eq. (4.65) and reproduces nodal values: 𝑦(−1) = 𝑦𝐼 , 𝑦(1) = 𝑦𝐽 . Thus, the continuity is given along the whole beam as nodal values are shared by neighboured elements. Equation (4.68) again is abbreviated as u = N(𝑟) ⋅ 𝝊𝑒 4) We see an extension of the isoparametric concept Eqs. (2.18) and (2.19).

(4.70)

4.3 Finite Elements for Plane Beams

with 𝝊𝑒 according to Eq. (4.67) and ⎡1 − 𝑟 1 ⎢ N(𝑟) = ⎢ 0 2 ⎢ 0 ⎣

⎛𝑢 ⎞ u = ⎜𝑤 ⎟ , ⎜ ⎟ ⎝𝜙⎠

0

0

1+𝑟

0

1−𝑟

0

0

1+𝑟

0

1−𝑟

0

0

0 ⎤ ⎥ 0 ⎥ (4.71) ⎥ 1+𝑟 ⎦

Generalised strains are determined using Eqs. (4.4), (4.2), and (4.63)

𝜖=

𝜕𝑢 𝜕𝑟 𝜕𝑟 𝜕𝑥

=

1 [ −1 𝐿𝑒

] ⎛ 𝑢𝐼 ⎞ 1 ⋅⎜ ⎟ 𝑢 ⎝ 𝐽⎠ ] ⎛𝜙𝐼 ⎞ 1 ⋅⎜ ⎟ 𝜙 ⎝ 𝐽⎠

𝜅=

𝜕𝜙 𝜕𝑟 𝜕𝑟 𝜕𝑥

=

1 [ −1 𝐿𝑒

𝛾=

)T ] ( 𝜕𝑤 𝜕𝑟 1 [ 𝐿 𝐿 −1 − 𝑒 (1 − 𝑟) 1 − 𝑒 (1 + 𝑟) ⋅ 𝑤𝐼 𝜙𝐼 𝑤𝐽 𝜙𝐽 −𝜙 = 2 2 𝐿𝑒 𝜕𝑟 𝜕𝑥 (4.72)

This is abbreviated with 𝝐 = B(𝑟) ⋅ 𝝊𝑒

(4.73)

using ⎛𝜖 ⎞ 𝝐 = ⎜𝜅⎟ , ⎜ ⎟ ⎝𝛾 ⎠

⎡−1 1 ⎢ B(𝑟) = 0 𝐿𝑒 ⎢ ⎢ 0 ⎣

0

0

1

0

−1

0

0

− (1 − 𝑟)

0

1

0 −1

𝐿𝑒 2

⎤ ⎥ 1 ⎥ ⎥ 𝐿𝑒 − (1 + 𝑟) 2 ⎦ 0

(4.74)

This approach exposes a 𝐶 0 -continuity as discontinuities of the 1st derivative arise at nodes or between neighboured elements, respectively. Regardless of this, the integration Eq. (4.57) can be performed. One more item has to be mentioned. In the case of pure bending or slender beams with a low bending stiffness compared to shear stiffness the shear angle is evaluated with 𝛾 ≈ 0. Thus, Eq. (4.723 ) imposes a constraint on the nodal variables which is not justified by physics but might lead to a severe artificial stiffening or locking effects. This is analysed in the context of thin shells elements in Section 10.6. The artificial locking can be reduced with the introduction of additional degrees of freedom to relax the artificial stiffness. The previous vector of nodal degrees of freedom according to Eq. (4.67) is extended with translational components 𝑢𝐾 , 𝑤𝐾 as ( 𝝊 𝑒 = 𝑢𝐼

𝑤𝐼

𝜙𝐼

𝑢𝐾

𝑤𝐾

𝑢𝐽

𝑤𝐽

𝜙𝐽

)T

(4.75)

87

88

4 Structural Beams and Frames

Trial and test function of the Timoshenko beam element are enhanced with 𝑟(𝑟−1)

⎡ 2 ⎢ N(𝑟) = ⎢ 0 ⎢ ⎣ 0

0

0

𝑟(𝑟−1) 2

1 − 𝑟2

0 (1−𝑟)

0

2

0

𝑟(1+𝑟) 2

0

1 − 𝑟2

0

0

0

0

0 𝑟(1+𝑟) 2

0

0 ⎤ ⎥ 0 ⎥ (1+𝑟) ⎥ 2 ⎦ (4.76)

leading to ⎤ ⎡2𝑟 − 1 0 0 −4𝑟 0 2𝑟 + 1 0 0 ⎥ 1 ⎢ B(𝑟) = 0 −1 0 −4𝑟 0 0 1 ⎥ ⎢ 0 𝐿𝑒 ⎢ ⎥ 𝐿 𝐿 0 0 2𝑟 + 1 − 𝑒 (1 + 𝑟) 0 2𝑟 − 1 − 𝑒 (1 − 𝑟) 0 2 2 ⎦ ⎣ (4.77) Vectors u (Eq. (4.71)) and 𝝐 (Eq. (4.74)) remain unchanged. The additional degrees of freedom correspond to bubble functions (Bathe 2001). Another and popular remedy for locking is given with reduced integration using an integration order 𝑛𝑖 = 0 (Table 2.1) while evaluation beam integrals Eqs. (4.98)–(4.101). For a further discussion of locking problems with Timoshenko beam elements, see Reddy (1997).

4.3.2 Bernoulli Beam Nodal degrees of freedom of the two-node element for the Bernoulli beam are again given by ( 𝝊 𝑒 = 𝑢𝐼

𝑤𝐼

𝜙𝐼

𝑢𝐽

𝑤𝐽

𝜙𝐽

)T

(4.78)

The dependence on time 𝑡 is not explicitly notified. A physical constraint relating lateral displacement 𝑤 and cross-section rotation 𝜙 is given by Eq. (4.59) 𝑤′ = 𝜙

(4.79)

An approach like Eq. (4.682,3 ) would lead to an artificial constraint for nodal degrees of freedom −

1 1 1 1 𝑤𝐼 + 𝑤𝐽 = (1 − 𝑟)𝜙𝐼 + (1 + 𝑟)𝜙𝐽 𝐿𝑒 𝐿𝑒 2 2

(4.80)

which makes the approach useless. The constraint has to be dissolved by additional degrees of freedom. Regarding Eq. (4.65) we choose 𝑤 = 𝑎0 + 𝑎1 𝑟 + 𝑎2 𝑟 2 + 𝑎3 𝑟 3

(4.81)

The interpolation property requires that nodal degrees of freedom are reproduced whereby considering Eq. (4.79): 𝑤(−1) = 𝑤𝐼 , 𝑤(1) = 𝑤𝐽 , 𝑤 ′ (−1) = 𝜙𝐼 , 𝑤 ′ (1) = 𝜙𝐽 . This yields 𝑎0 = (4𝑤𝐼 + 4𝑤𝐽 + 𝐿𝑒 𝜙𝐼 − 𝐿𝑒 𝜙𝐽 )∕8, 𝑎1 = (−6𝑤𝐼 + 6𝑤𝐽 − 𝐿𝑒 𝜙𝐼 − 𝐿𝑒 𝜙𝐽 )∕8,

4.3 Finite Elements for Plane Beams

𝑎2 = (−𝐿𝑒 𝜙𝐼 + 𝐿𝑒 𝜙𝐽 )∕8, 𝑎3 = (2𝑤𝐼 − 2𝑤𝐽 + 𝐿𝑒 𝜙𝐼 + 𝐿𝑒 𝜙𝐽 )∕8 or in matrix-vector notation 𝑟3

3𝑟

4

4

𝑤=[ − ( ⋅ 𝑤𝐼 𝑤′ =

+

1

𝐿𝑒 𝑟3

2

8

𝜙𝐼

𝑤𝐽

− 𝜙𝐽

𝐿𝑒 𝑟2 8 )T

𝜕𝑤 𝜕𝑟 2 3𝑟2 3 [ − = 𝐿𝑒 4 4 𝜕𝑟 𝜕𝑥 )T ( ⋅ 𝑤𝐼 𝜙𝐼 𝑤𝐽 𝜙𝐽

−

𝐿𝑒 𝑟 8

+

𝐿𝑒

−

2𝐿𝑒 𝑟

3𝐿𝑒 𝑟2 8

𝑟3

3𝑟

4

4

𝐿𝑒

−

− +

8

8

−

+

1

𝐿𝑒 𝑟3

2

8

3𝑟2

8

4

+

+

𝐿𝑒 𝑟2

−

8

3

3𝐿𝑒 𝑟2

4

8

+

𝐿𝑒 𝑟 8

2𝐿𝑒 𝑟 8

−

−

𝐿𝑒

𝐿𝑒 8

8

]

]

(4.82) The longitudinal displacement 𝑢 is not involved in constraints and simply interpolated with 𝑢=

[1 2

] ⎛ 𝑢𝐼 ⎞ (1 + 𝑟) ⋅ ⎜ ⎟ 2 𝑢 ⎝ 𝐽⎠ 1

(1 − 𝑟)

(4.83)

yielding 𝑢(−1) = 𝑢𝐼 , 𝑢(1) = 𝑢𝐽 . Eqs. (4.82) and (4.83) are abbreviated as u = N(𝑟) ⋅ 𝝊𝑒

(4.84)

with 𝝊𝑒 according to Eq. (4.78) and 1

⎛𝑢 ⎞ u=⎜ ⎟ , 𝑤 ⎝ ⎠

⎡ (1 − 𝑟) N(𝑟) = ⎢ 2 ⎣ 0

0 𝑟3

−

4

3𝑟 4

0 +

1

𝐿𝑒 𝑟3

2

8

−

𝐿𝑒 𝑟2 8

−

𝐿𝑒 𝑟 8

+

𝐿𝑒 8

⋯⎤ ⎥ ⋯⎦ (4.85)

The deformation variables are derived from Eqs. (4.4), (4.59), (4.64), (4.82), and (4.85) 𝜖=

𝜕𝑢 𝜕𝑟 𝜕𝑟 𝜕𝑥

𝜅=

𝜕𝑤 ′ 𝜕𝑟 𝜕𝑟 𝜕𝑥

] ⎛ 𝑢𝐼 ⎞ 1 [ −1 1 ⋅ ⎜ ⎟ 𝐿𝑒 𝑢 ⎝ 𝑗⎠ [ 4 6𝑟 6𝐿𝑒 𝑟 2𝐿𝑒 − = 2 8 8 𝐿𝑒 4

=

−

6𝑟

6𝐿𝑒 𝑟

4

8

+

2𝐿𝑒 8

] ( )T ⋅ 𝑤𝐼 𝜙𝐼 𝑤𝐽 𝜙𝐽 (4.86)

This is abbreviated as 𝝐 = B(𝑟) ⋅ 𝝊𝑒

(4.87)

using ⎛𝜖 ⎞ 𝝐=⎜ ⎟ , 𝜅 ⎝ ⎠

B(𝑟) =

1 ⎡−1 ⎢ 𝐿𝑒 0 ⎣

0 6𝑟 𝐿𝑒

0 3𝑟 − 1

1 0

0 −

6𝑟 𝐿𝑒

⎤ ⎥ 3𝑟 + 1 ⎦ 0

(4.88)

89

90

4 Structural Beams and Frames

These formulations of N and B establish a continuity for the longitudinal displacements 𝑢 between neighboured elements and a discontinuity for the derived strains 𝜖, i.e. 𝐶 0 -continuity. Regarding the lateral displacements 𝑤 a continuity between neighboured elements is given up to curvature or 2nd derivative, i.e. 𝐶 2 -continuity. Thus, different polynomial orders are given for 𝜖 and 𝜅. This is not a problem in the case of a linear elastic material behaviour (Eq. (4.16)) with 𝑁 decoupled from 𝜅 and 𝑀 decoupled from 𝜖. But it may lead to artificial constraints in case of coupled material behaviour as is characteristic for the cracked cross-section of reinforced concrete (Eq. (4.46)). This is discussed with a simple test. A single element is regarded with 𝑢𝐼 = 𝑤𝐼 = 𝜙𝐼 = 0 and a material stiffness matrix ⎡𝐶11 C=⎢ 𝐶 ⎣ 21

𝐶12 ⎤ ⎥ 𝐶22 ⎦

(4.89)

leading to 6 𝐶 𝑟 𝑤𝐽 + 𝐶12 (3𝑟 + 1) 𝜙𝐽 𝐿𝑒 12 6 𝑀 = 𝐶21 𝑢𝐽 − 𝐶 𝑟 𝑤𝐽 + 𝐶22 (3𝑟 + 1) 𝜙𝐽 𝐿𝑒 22 6 𝑉 = −𝑀 ′ = 2 𝐶22 (2𝑤𝐽 − 𝜙𝐽 𝐿𝑒 ) 𝐿𝑒 𝑁 = 𝐶11 𝑢𝐽 −

(4.90)

The conditions 𝑁 = 0, 𝑀 = 0, 𝑉 = 𝑉 are prescribed on the right-hand side 𝑟 = 1 and results in displacements 𝑢𝐽 = 0, 𝜙𝐽 = 1∕(2𝐶22 ) 𝑉𝐿𝑒 , 𝑤𝐽 = 1∕(3𝐶22 ) 𝑉𝐿𝑒2 and internal forces 𝐶12 (1 − 𝑟) 𝑉 𝐿𝑒 2𝐶22 1 𝑀 = (1 − 𝑟) 𝑉 𝐿𝑒 2 𝑉=𝑉 𝑁=

(4.91)

While 𝑀, 𝑉 are determined correctly an error in 𝑁 arises due to the coupling material component 𝐶12 . ◀

Discretised integral equilibrium conditions are nevertheless fulfilled as no normal force components arise from this particular N(r ) for integrated internal nodal forces (Eqs. (2.58) and (4.56)).

The artificial normal forces can be reduced with the introduction of an additional degree of freedom for the longitudinal displacement. The previous vector of nodal degrees of freedom according to Eq. (4.78) is extended with a component 𝑢𝐾 as ( 𝝊 𝑒 = 𝑢𝐼

𝑤𝐼

𝜙𝐼

𝑢𝐾

𝑢𝐽

𝑤𝐽

𝜙𝐽

)T

(4.92)

4.4 System Building and Solution

Trial and test function of the Bernoulli beam element are enhanced with 1

⎡ 𝑟(𝑟 − 1) N(𝑟) = ⎢ 2 0 ⎣

0 𝑟3 4

−

3𝑟 4

1 − 𝑟2

0 +

1

𝐿𝑒 𝑟3

2

8

−

𝐿𝑒 𝑟2 8

−

𝐿𝑒 𝑟

+

8

𝐿𝑒

0

8

⋯⎤ ⎥ ⋯⎦ (4.93)

This approach replaces the first row of the previous N with quadratic functions whereby the trial function corresponding to the additional degree of freedom corresponds to a bubble function (Bathe 2001). This degree of freedom is not ‘visible’ from the ‘outside’ and not used for the interpolation of geometry and lateral displacements. The approach leads to B(𝑟) =

1 ⎡2𝑟 − 1 ⎢ 𝐿𝑒 0 ⎣

0 6𝑟 𝐿𝑒

0 3𝑟 − 1

−4𝑟

2𝑟 + 1

0

0

0 −

6𝑟 𝐿𝑒

⎤ ⎥ 3𝑟 + 1 ⎦ 0

(4.94)

and effects a linear longitudinal strain within an element corresponding to the linear curvature. Vectors u (Eq. (4.85)) and 𝝐 (Eq. (4.88)) remain unchanged. The simple test is also performed with the enhanced Bernoulli beam element. Internal forces are determined with 𝑁 = −4𝐶11 𝑟 𝑢𝐾 + 𝐶11 (2𝑟 + 1) 𝑢𝐽 − 6𝐶12 𝑟∕𝐿𝑒 𝑤𝐽 + 𝐶12 (3𝑟 + 1) 𝜙𝐽 𝑀 = −4𝐶21 𝑟 𝑢𝐾 + 𝐶21 (2𝑟 + 1) 𝑢𝐽 − 6𝐶22 𝑟∕𝐿𝑒 𝑤𝐽 + 𝐶22 (3𝑟 + 1) 𝜙𝐽 𝑉 = 2(4𝐶21 𝑢𝐾 𝐿𝑒 − 2𝐶21 𝑢𝐽 𝐿𝑒 + 6𝐶22 𝑤𝐽 −

(4.95)

3𝐶22 𝜙𝐽 𝐿𝑒 )∕𝐿𝑒2

An additional condition 𝑁 = 0 is prescribed on the left-hand side besides 𝑁 = 𝑀 = 0, 𝑉 = 𝑉 on the right-hand side. This results to displacements 𝑢𝐾 = 3𝐶12 𝑉𝐿𝑒 ∕8, 𝑢𝐽 = 𝐶12 𝑉𝐿𝑒 ∕2, 𝑤𝐽 = −𝐶11 𝑉𝐿𝑒2 ∕3, 𝜙𝐽 = −𝐶11 𝑉𝐿𝑒 ∕2 with 𝑐 = 1∕(𝐶12 𝐶21 − 𝐶11 𝐶22 ) and to internal forces 𝑁 =0,

𝑀=

1 (1 − 𝑟) 𝑉 𝐿𝑒 , 2

𝑉=𝑉

(4.96)

All internal forces are correctly determined for the simple test. The enhanced Bernoulli beam element will be predominantly used for the following examples but generally it is not part of commercial finite element packages.

4.4 System Building and Solution 4.4.1 Integration Basics of structural analysis for beams are derived with beam kinematics in Section 4.1.1, material behaviour in Sections 4.1.2 and 4.1.3 and equilibrium in Section 4.2. They are combined with the evaluation of the integral Eq. (4.57). This is done element by element, see Section 2.6, Eq. (2.58). The integration is performed numerically according to the one-dimensional version of Eq. (2.68). In the case of

91

92

4 Structural Beams and Frames

structural beam elements the Jacobian is given by Eq. (4.63). Internal nodal forces (Eq. (2.581 )) are determined with 1

f𝑒 = ∫

BT (𝑥)

𝐿 ⋅ 𝝈(𝑥) d𝑥 = 𝑒 ∫ BT (𝑟) ⋅ 𝝈(𝑟) d𝑟 2

𝐿𝑒

(4.97)

−1

where B is given according to the element type chosen and 𝝈 according to Eq. (4.581 ) or Eq. (4.60). The internal nodal force dimension has to match the row dimension of B. According to Eq. (2.69) the numerical integration is performed with f𝑒 =

𝑛 𝐿𝑒 ∑𝑖 𝜂 BT (𝜉𝑖 ) ⋅ 𝝈(𝜉𝑖 ) 2 𝑖=0 𝑖

(4.98)

with integration order 𝑛𝑖 , local sampling points 𝜉𝑖 and sampling weights 𝜂𝑖 (Table 2.1). This is again applied to determine the tangential element stiffness matrix (Eq. (2.66)) K𝑇𝑒 =

𝑛 𝐿𝑒 ∑𝑖 𝜂 BT (𝜉𝑖 ) ⋅ C𝑇 (𝜉𝑖 ) ⋅ B(𝜉𝑖 ) 2 𝑖=0 𝑖

(4.99)

with the tangential material stiffness according to Eqs. (4.46) or (4.16) in the case of Bernoulli beams, or Eqs. (4.47), (4.15) in the case of Timoshenko beams. Furthermore, the element mass matrix (Eq. (2.582 )) is given by M𝑒 =

𝑛 𝐿𝑒 ∑𝑖 𝜂 NT (𝜉𝑖 ) ⋅ m ⋅ N(𝜉𝑖 ) 2 𝑖=0 𝑖

(4.100)

with m according to Eq. (4.58), and the element distributed loading by t𝑒 =

𝑛 𝐿𝑒 ∑𝑖 𝜂 NT (𝜉𝑖 ) ⋅ p(𝜉𝑖 ) 2 𝑖=0 𝑖

(4.101)

with p according to Eq. (4.58). A concentrated loading can be applied to element nodes in analogy to Eq. (4.58) ( c𝑒 = 𝑁 𝐼

𝑉𝐼

𝑀𝐼

𝑁𝐽

𝑉𝐽

𝑀𝐽

)T

(4.102)

Distributed and concentrated loadings must be prescribed – marked by overbars –, be it formally with the value zero. Finally, the element loading vector or external nodal forces (Eq. (2.59)) are given by p𝑒 = t𝑒 + c𝑒

(4.103)

The numerical integration has to be precise to minimise the discretisation error. The integration error of the Gauss integration (Table 2.1) is determined by the integration order 𝑛𝑖 (Eq. (2.69)). Disregarding round-off errors an integration order 𝑛𝑖 gives exact results for polynomials of order 2𝑛𝑖 + 1.

4.4 System Building and Solution

• The Bernoulli beam element with strains according to Eq. (4.88) in the case of a constant material stiffness has a stiffness matrix K𝑇𝑒 with highest polynomial degree 2 within an element. This requires 𝑛𝑖 = 1 and two sampling points with the Gauss integration. • The same argument holds for the Timoshenko beam element (Eq. (4.74)) due to the shear deformation parts. Reduced integration 𝑛𝑖 = 0 neglects the linear contribution of shear deformations but might solve the locking problem in the case of slender beams. Finally, integration of an ordinary Bernoulli or Timoshenko two-node beam element ends up with a vector of internal nodal forces ( f𝑒 = 𝑁𝐼

𝑉𝐼

𝑀𝐼

𝑁𝐽

𝑉𝐽

𝑀𝐽

)T

(4.104)

The number of components corresponds to the components of nodal displacements (Eqs. (4.78) and (4.67)) with 𝑁𝑖 ↔ 𝑢𝑖 , 𝑉𝑖 ↔ 𝑤𝑖 , 𝑀𝑖 ↔ 𝜙𝑖 , 𝑖 = 𝐼, 𝐽. According to these dimensions the element stiffness matrix K𝑇𝑒 and the element mass matrix M𝑒 have a dimension 6 × 6 for the ordinary two-node beam elements. Additional degrees of freedom arise with the enhanced elements (Eqs. (4.92) and (4.75)). They may be condensed on the element level and then will not arise as additional unknowns on the system level. Alternatively they can be carried over to the system level and are connected with the corresponding internal and external nodal force components and the stiffness and mass matrices components.

4.4.2 Transformation and Assembling The longitudinal axis of a beam element and the global 𝑥-axis have the same direction up to now. But regarding frame systems a 2D beam may have an orientation in the 2D space. Thus, we have to consider a transformation of vectors in 2D Cartesian coordinate systems. The direction of a straight element is assumed with the first element node as starting and the last element node as ending point. A rotation angle 𝛼 is given from the global 𝑥-direction to the local 𝑥 ˜-direction of an element, see Figure 4.6. Regarding Eq. (D.8) the transformation of nodal displacements (Eq. (4.78)), internal nodal forces (Eq. (4.104)) and external nodal forces (Eq. (4.103)) from the global

Figure 4.6 Beam orientation in 2D space.

93

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4 Structural Beams and Frames

into the local system is performed with ˜ 𝝊𝑒 = T ⋅ 𝝊𝑒 ,

˜ f𝑒 = T ⋅ f𝑒 ,

˜ p𝑒 = T ⋅ p𝑒

(4.105)

and from the local into the global system 𝝊𝑒 , 𝝊𝑒 = TT ⋅ ˜

f𝑒 = TT ⋅ ˜ f𝑒 ,

⎡ cos 𝛼 ⎢ ⎢− sin 𝛼 ⎢ ⎢ 0 T=⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ 0 ⎣

sin 𝛼

0

0

0

cos 𝛼

0

0

0

0

1

0

0

0

0

cos 𝛼

sin 𝛼

0

0

− sin 𝛼

cos 𝛼

0

0

0

0

p𝑒 = TT ⋅ ˜ p𝑒

(4.106)

with 0⎤ ⎥ 0⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ 1 ⎦

(4.107)

for two-node elements whereby ∙T denotes the transpose of ∙. Nodal rotations 𝜙𝐼 , 𝜙𝐽 and moments 𝑀𝐼 , 𝑀𝐽 remain unchanged with this transforf𝑒 , df𝑒 . The mation. Eqs. (4.105) and (4.106) are also valid for increments d˜ 𝝊𝑒 , d𝝊𝑒 , d˜ evaluation of internal and external nodal forces is initially performed in the local system of an element whereby each element may have its own orientation. Thus, Eqs. (4.1062,3 ) have to be applied before assembling the element contributions to the global system. Nodal displacements are computed in the global system. They have to transformed to local displacements using Eq. (4.1051 ) such that the beam generalised strains and stresses can be computed. The tangential element stiffness matrix is also initially defined in the local system by ˜ 𝑇𝑒 ⋅ d˜ d˜ f𝑒 = K 𝝊𝑒

(4.108)

Using the transformation rules of Eq. (4.105) and considering T−1 = TT yields ˜ 𝑇𝑒 ⋅ T ⋅ d𝝊𝑒 T ⋅ df𝑒 = K

→

˜ 𝑇𝑒 ⋅ T ⋅ d𝝊𝑒 df𝑒 = TT ⋅ K

(4.109)

and finally results in a transformation rule for the tangential element stiffness matrix ˜ 𝑇𝑒 ⋅ T K𝑇𝑒 = TT ⋅ K

(4.110)

Similar arguments lead to the transformation rule for the element mass matrix ˜𝑒 ⋅T M𝑒 = TT ⋅ M

(4.111)

4.4 System Building and Solution

In the case of the enhanced Bernoulli beam element (Section 4.3.2) the transformation matrix T is given by ⎡ cos 𝛼 sin 𝛼 0 0 0 0 0⎤ ⎢ ⎥ 0 0 0⎥ ⎢− sin 𝛼 cos 𝛼 0 0 ⎢ ⎥ 0 1 0 0 0 0⎥ ⎢ 0 ⎢ ⎥ T=⎢ 0 (4.112) 0 0 1 0 0 0⎥ ⎢ ⎥ 0 0 0 0 cos 𝛼 sin 𝛼 0 ⎢ ⎥ ⎢ ⎥ 0 0 0 − sin 𝛼 cos 𝛼 0⎥ ⎢ 0 ⎢ ⎥ 0 0 0 0 0 0 1 ⎣ ⎦ The central 1 belongs to the additional degree of freedom of the interior node. This value should not be rotated as it corresponds to a local longitudinal displacement without relating to the global system. Apart from the change in T all transformations remain unchanged. Element contributions f𝑒 , p𝑒 , K𝑇𝑒 , M𝑒 must be assembled to system vectors and matrices. The procedure is basically the same for all element types, see Section 2.6, item 6 and Eq. (2.67). Regarding a particular node which is shared by several elements, the element contributions to internal nodal forces and nodal loads have to sum up to zero with respect to equilibrium.

4.4.3 Kinematic Boundary Conditions and Solution Displacements at boundaries or kinematic boundary conditions still have to be considered. There should be enough boundary conditions to prevent rigid body displacements. They are applied at the global system level by prescribing nodal values for the respective global displacements 𝑢, 𝑤, 𝜙. The transformation Eq. (4.105) may be applied in case that local values are prescribed. To simplify the description the quasi-static linear case K⋅u=p

(4.113)

is considered (Eq. (2.63)). The displacement 𝑢 with a global index 𝑘 with a prescribed value 𝑢 is treated exemplarily. Let 𝑛 be the total number of degrees of freedom. To apply the particular boundary condition we may set 𝑝𝑖 ∶= 𝑝𝑖 − 𝐾𝑖𝑘 𝑢

𝑖 = 1, … , 𝑘 − 1, 𝑘 + 1, … , 𝑛

𝑝𝑘 ∶= 𝑢 𝐾𝑘𝑘 ∶= 1

(4.114)

𝐾𝑘𝑗 ∶= 0 𝑗 = 1, … , 𝑘 − 1, 𝑘 + 1, … , 𝑛 𝐾𝑖𝑘 ∶= 0 𝑖 = 1, … , 𝑘 − 1, 𝑘 + 1, … , 𝑛 for the components of the right-hand side and the matrix in Eq. (4.113). With 𝑝𝑘 = 𝑢 and 𝐾𝑘𝑘 = 1 this must lead to 𝑢𝑘 = 𝑢 after solving the system of equations. The additional terms 𝐾𝑖𝑘 𝑢 on the right-hand side apply constraint forces in the case 𝑢 ≠ 0.

95

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4 Structural Beams and Frames

The degree of freedom 𝑘 has to be excluded from balancing equilibrium by summing up nodal force contributions from elements and loadings (Eq. (2.70)). The corresponding internal nodal forces result in a support reaction. The procedure as has been described for the linear case may also be applied for every iteration step within an incrementally iterative scheme (Section 2.8.2). Thus, fulfilment of kinematic boundary conditions is reached in the same way for nonlinear and dynamic cases. Nonlinear systems are given for beams with cracked reinforced concrete (RC) cross-sections due to the nonlinear relations between moment, normal force, curvature, and longitudinal strain. The incrementally iterative scheme is appropriate to solve such systems. A loading history is followed whereby a clock time (→ dynamic or transient case) or a loading time (→ quasi-static case) is used to control the load. Solution increments correspond to time steps. The Newton–Raphson method (Eq. (2.77)) is generally appropriate for equilibrium iteration within each increment. Alternative methods like the BFGS method (Eq. (A.15)) may be required in some cases (Section A.1). A first application example is demonstrated in the following. Example 4.2: Simple Reinforced Concrete (RC) Beam

Geometry, discretisation, and boundary conditions are chosen as follows: • Single span beam with 𝐿 = 5.0 m, square cross-section with width 𝑏 = 0.2 m, and height ℎ = 0.4 m, see Figure 4.7a. • Discretisation with 𝑛𝐸 = 10 enhanced two-node Bernoulli beam elements (Section 4.3.2) with 11 full nodes and 10 extension nodes. This includes 11 ⋅ 3 + 10 = 43 degrees of freedom. The integration of elements is performed with the Gauss integration (Table 2.1) and integration order 𝑛𝑖 = 2 with three integration points per element as material coefficients are not constant within an element. • Hinge bearing of left-hand and right-hand node, i.e. lateral displacements are zero but rotations are not restricted. The longitudinal displacement of the left node is zero, the right node displacement is not restricted in the longitudinal direction.

(a)

(b)

Figure 4.7 Example 4.2. (a) System. (b) Load factor lf depending on mid-span deflection w.

4.4 System Building and Solution

Material properties and loading are assumed as follows: • Concrete grade C30/37 according to EN 1992-1-1 (2004, Table 3.1) with an initial Young’s modulus 𝐸𝑐 = 33 000 MN∕m2 . The compressive strength of the concrete is chosen with 𝑓𝑐 = 38 MN∕m2 with 𝜖𝑐1 = −0.0023, 𝜖𝑐𝑢1 = −0.0035 (Figure 3.1). A tensile strength is assumed with 𝑓𝑐𝑡 = 3 MN∕m2 which allows to distinguish uncracked state I from cracked state II. The uniaxial stress–strain curve is chosen according to EN 1992-1-1 (2004, 3.1.5). • Reinforcing steel according to EN 1992-1-1 (2004, 3.2.7) with 𝑓𝑦𝑘 = 550 MN∕m2 , 𝑓𝑡 = 600 MN∕m2 , 𝜖𝑦0 = 0.002 75, 𝜖𝑢 = 0.05 (Figure 3.11a). Reinforcement with 4 ⊘ 20, 𝐴𝑠1 = 12.57 cm2 , 𝑑1 = 5 cm. No compression reinforcement. • In the same way as demonstrated in Example 4.1 an ultimate moment 𝑀𝑢 ≈ 0.20 MNm is determined with 𝑁 = 0. This leads to a load of 𝑞 = 8𝑀𝑢 ∕𝐿2 ≈ 65 kN∕m which is chosen as reference for the computation and scaled with a loading factor 𝑙𝑓. An incrementally iterative scheme with Newton–Raphson iteration within each loading increment (Section 2.8.2, Figure 2.5) is used as solution method whereby loading steps are controlled with the arc length method with a prescribed arc length size of 𝛾 = 1 ⋅ 10−3 (Section A.4, Eq. (A.51)). The computed load factor-displacement curve is shown in Figure 4.7b. It has four states: (I) uncracked beam, (IIa) crack formation, (IIb) stabilised cracking with elastic reinforcement, (III) yielding of reinforcement. This corresponds to the uniaxial tension bar (Example 3.4, Figure 3.17). More detail results are available with numerical methods. A selection is described in the following. • Deformations of cracked concrete beams can no longer be determined with linear structural analysis and require numerical methods. Figure 4.8a shows the computed nodal displacements for the final loading state. They are connected by straight lines in the figure and insofar slightly deviate from trial functions (Eq. (4.93)). The uncracked linear elastic case with a bending stiff-

(a)

(b)

Figure 4.8 Example 4.2. Final loading state. (a) Deflection of reference axis (scaled). (b) Strain of reference axis.

97

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4 Structural Beams and Frames

(a)

(b)

Figure 4.9 Example 4.2. Final loading state. (a) Reinforcement strain. (b) Upper edge concrete strain.

ness 𝐸𝐽 = 33 000 ⋅ 0.2 ⋅ 0.43 ∕12 = 35.2 MNm is shown as comparison. The difference has a factor of roughly 2.7. • A special property of RC cross-sections is given by a reference axis length change even if there is no resulting normal force. The reference axis becomes slightly longer for the example during loading (Figure 4.8a). This is connected with longitudinal strains of the reference axis shown in Figure 4.8b. Longitudinal strains are caused by a coupling of normal forces to curvature, i.e. 𝑁 = 𝑁(𝜅, 𝜖) (Section 4.1.3) which leads to 𝜖 ≠ 0 for 𝜅 ≠ 0 even in the case 𝑁 = 0. • Finally, the computed strains of the upper compressed concrete edge and the strains of the reinforcement, as they are determined with Eq. (4.5), are shown in Figure 4.9 for the final state. Regarding concrete they have to be compared to the strength strain 𝜖𝑐1 = −0.0023 and regarding reinforcement to yield strain 𝜖𝑦 = 𝑓𝑦 ∕𝐸𝑠 = 0.002 75. Failure obviously occurs in the centre region. The example has a coarse discretisation, but the computed load-displacement behaviour will not change to a larger extent with finer discretisations. However, finer discretisations will yield a more pronounced resolution of the failing centre area.

4.4.4 Shear Stiffness Up to now a shear stiffness has been derived for the linear elastic material behaviour only (Eq. (4.15)). This bases upon the linear elastic relation between shear stresses and shear strains derived from plane elasticity (Eq. (4.73 )). The shear behaviour of cracked RC sections has to be derived from the truss model for shear instead (EN 1992-1-1 2004, 6.2.3). A square cross-section is assumed with geometric height ℎ and width 𝑏 to simplify the description. The model has concrete struts and reinforcement ties. The geometry of struts is shown in Figure 4.10a. The strut geometry is characterised by an orientation inclined with an angle 𝜃 against the reference axis within a sector 𝑧 of the cross-section height. The length 𝑧 corresponds to the internal lever arm of bending as the distance of compression chord and tensile chord. The undeformed length of the strut is given

4.4 System Building and Solution

(a)

(b)

Figure 4.10 Shear stiffness. (a) Struts. (b) Ties.

by 𝑙𝑢 =

𝑧 sin 𝜃

(4.115)

A rotation 𝜙 of the cross-section is not regarded. Thus, the inclination of the reference axis corresponds to the shear angle 𝛾 (Figure 4.10a). With a reference to a longitudinal coordinate 𝑥1 a cross-section moves laterally over a length 𝑧∕ tan 𝜃 by 𝛾 𝑧∕ tan 𝜃 and the deformed length of the strut is given by √ 𝑙𝑑 =

2

(

2

𝑧 𝑧 ) + (𝑧 + 𝛾) tan 𝜃 tan 𝜃

(4.116)

with the sign convention for 𝛾 (Figure 4.10) considered. With 𝛾 ≪ 1 this may be written as √ (4.117) 𝑙𝑑 = 𝑙𝑢 1 + 2𝛾 sin 𝜃 cos 𝜃 whereby 𝛾 2 is neglected. The strut strain is given by 𝜖1 =

𝑙𝑑 − 𝑙𝑢 √ = 1 + 2𝛾 sin 𝜃 cos 𝜃 − 1 𝑙𝑢

(4.118)

The root term is expanded with a Taylor series. As 𝛾 ≪ 1, this leads to 𝜖1 = sin 𝜃 cos 𝜃 𝛾

(4.119)

A corresponding strut force has to be determined next. A linear material behaviour with Young’s modulus 𝐸𝑐 is assumed. Furthermore, a strut has a width 𝑏 and a height ℎ1 in its own cross-section (Figure 4.10a) and the strut force is given by 𝐹1 = 𝑏ℎ1 𝐸𝑐 𝜖1 = 𝑏ℎ1 𝐸𝑐 sin 𝜃 cos 𝜃 𝛾 A cross-section of the beam involves 𝑛 = of a cross-section leads to 𝐹strut = 𝑛 𝐹1 =

(4.120) 𝑧

ℎ1 ∕ cos 𝜃

struts. Summing up all strut forces

𝑧 𝐹1 = 𝑏𝑧 𝐸𝑐 sin 𝜃 cos2 𝜃 𝛾 ℎ1 ∕ cos 𝜃

(4.121)

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4 Structural Beams and Frames

To have equilibrium a strut force and a corresponding shear force are related by 𝑉 = 𝐹1 sin 𝜃. Thus, shear forces from struts are given by 2

𝑉strut = 𝑏𝑧 𝐸𝑐 sin 𝜃 cos2 𝜃 𝛾

(4.122)

and finally a shear stiffness from struts 𝜕𝑉strut 2 = 𝐸𝑐 𝑏𝑧 sin 𝜃 cos2 𝜃 𝜕𝛾

(4.123)

is obtained. The contribution of shear reinforcement or ties to the shear stiffness can be derived in the same way. A tie with an inclination 𝛼 is considered, see Figure 4.10b. It has an undeformed length 𝑙𝑢 =

𝑧 sin 𝛼

(4.124)

a deformed length √ ( 𝑧 )2 ( √ 𝑧 )2 𝑙𝑑 = + 𝑧− 𝛾 = 𝑙𝑢 1 − 2𝛾 sin 𝛼 cos 𝛼 tan 𝛼 tan 𝛼

(4.125)

whereby the sign convention for 𝛾 again has to be considered. This yields a strain 𝜖2 =

𝑙𝑑 − 𝑙𝑢 √ = 1 − 2𝛾 sin 𝛼 cos 𝛼 − 1 ≈ − sin 𝛼 cos 𝛼 𝛾 𝑙𝑢

(4.126)

A linear elastic behaviour with a Young’s modulus 𝐸𝑠 is assumed. With a crosssection 𝐴𝑠 of the tie or rebar the force is given by 𝐹2 = 𝐴𝑠 𝐸𝑠 𝜖2 = −𝐴𝑠 𝐸𝑠 sin 𝛼 cos 𝛼 𝛾

(4.127)

The spacing 𝑠𝑐 of rebars in a beam cross-section and their longitudinal spacing 𝑠 are related by 𝑠𝑐 ∕𝑠 = tan 𝛼. Thus, a cross-section involves 𝑛 = 𝑧∕𝑠𝑐 = 𝑧∕(𝑠 tan 𝛼) rebars. Summing up all ties in a cross-section leads to 𝐹𝑡𝑖𝑒 =

𝐴 𝑧 2 𝐹2 = −𝑧 𝑠 𝐸𝑠 sin 𝛼 cos 𝛼 𝛾 𝑠 𝑠𝑐 ∕ tan 𝛼

(4.128)

A tie force and a corresponding shear force and are related by 𝑉 = −𝐹2 sin 𝛼 for equilibrium. Thus, shear forces from ties are given by 3

𝑉𝑡𝑖𝑒 = 𝑧 𝑎𝑠 𝐸𝑠 sin 𝛼 cos 𝛼 𝛾

(4.129)

with 𝑎𝑠 = 𝐴𝑠 ∕𝑠 and finally a shear stiffness from ties is 𝜕𝑉𝑡𝑖𝑒 3 = 𝑧 𝑎𝑠 𝐸𝑠 sin 𝛼 cos 𝛼 𝜕𝛾

(4.130)

Finally, the total shear stiffness is given by ) ( 𝜕𝑉strut 𝜕𝑉𝑡𝑖𝑒 𝜕𝑉 2 3 = + = 𝑧 𝑏 𝐸𝑐 sin 𝜃 cos2 𝜃 + 𝑎𝑠 𝐸𝑠 sin 𝛼 cos 𝛼 𝜕𝛾 𝜕𝛾 𝜕𝛾

(4.131)

4.5 Creep of Concrete

As a special case we consider stirrups as ties with 𝛼 = π∕2 and a strut inclination 𝜃 = π∕4. This leads to a shear stiffness 𝜕𝑉 1 1 = 𝑧𝑏 𝐸𝑐 = 𝑧𝑏 𝐺𝑐 4 2 𝜕𝛾

(4.132)

with the shear modulus 𝐺𝑐 according to Eq. (4.8). We compare this with the linear elastic case Eq. (4.133 ) with a geometry coefficient 𝛼. Here a relation 𝑧𝑏 = 𝛼𝑐 𝐴, 𝛼𝑐 = 𝑧∕ℎ is used with a modified geometry coefficient 𝛼𝑐 . Thus, Eqs. (4.133 ) and (4.132) differ by a factor of 2. In the current set-up, concrete tensile struts are disregarded due to the limited tensile strength of concrete and the shear reinforcement was assumed as vertical stirrups, which do not directly contribute to 𝑉. This yields the bisection of shear strength. There is some margin to choose the concrete strut angle (EN 1992-1-1 (2004, 6.2.3), CEB-FIP2 (2012, 7.3.3.3)). The limits are roughly in a range 20° ≤ 𝜃 ≤ 45°. As a first estimation, the same strut angle should be used as for the design of the stirrups.

4.5 Creep of Concrete Creep leads to increasing deformations of concrete structures regarding the long term behaviour. This concerns the serviceability of structures with moderate levels of loading. Thus, the assumption of linear concrete compressive behaviour whereby disregarding concrete tensile stresses (Section 4.1.3.2) is appropriate. Furthermore, visco-elastic material laws according to Section 3.2 can be used to model the creep behaviour of RC beams. We start with collecting previous items. Eqs. (3.30) and (3.31) are applied to concrete 𝜎̇ 𝑐 = 𝐸𝑐 𝜖̇ 𝑥 +

1 𝐸𝑐 𝜖𝑥 − 𝜓𝑐 𝜎𝑐 , 𝜁𝑐

𝜓𝑐 =

1 + 𝜑𝑐 𝜁𝑐

(4.133)

with longitudinal strains 𝜖𝑥 and 𝐸𝑐 according to Eq. (4.24). Reinforcing steel is assumed as linear for moderate loading levels 𝜎𝑠 = 𝐸𝑠 𝜖𝑥 ,

𝜎̇ 𝑠 = 𝐸𝑠 𝜖̇ 𝑥

(4.134)

Kinematics is ruled by Eq. (4.5) 𝜖𝑥 = 𝜖 − 𝑧 𝜅 ,

𝜖̇ 𝑥 = 𝜖̇ − 𝑧 𝜅̇

(4.135)

Strain of reinforcement is considered with 𝜖𝑠1 = 𝜖 − 𝑧𝑠1 𝜅 ,

𝜖̇ 𝑠1 = 𝜖̇ − 𝑧𝑠1 𝜅̇ ,

𝜖𝑠2 = 𝜖 − 𝑧𝑠2 𝜅 ,

𝜖̇ 𝑠2 = 𝜖̇ − 𝑧𝑠2 𝜅̇ (4.136)

101

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4 Structural Beams and Frames

according to Eq. (4.21). Finally, regarding Eqs. (4.22) and (4.23) internal forces are 𝑧𝑐2

𝑁𝑐 = ∫ 𝜎𝑐 𝑏 d𝑧 𝑧𝑐1

𝑁𝑠 = 𝐴𝑠1 𝜎𝑠1 + 𝐴𝑠2 𝜎𝑠2

(4.137)

𝑧𝑐2

𝑀𝑐 = − ∫ 𝜎𝑐 𝑧 𝑏 d𝑧 𝑧𝑐1

𝑀𝑠 = −𝐴𝑠1 𝜎𝑠1 𝑧𝑠1 − 𝐴𝑠2 𝜎𝑠2 𝑧𝑠2 and 𝑁 = 𝑁𝑐 + 𝑁𝑠 , 𝑀 = 𝑀𝑐 + 𝑀𝑠 . The coordinates 𝑧𝑐1 , 𝑧𝑐2 indicate the range of the )T ( concrete compression zone. Internal forces of concrete 𝝈𝑐 = 𝑁𝑐 𝑀𝑐 have to be )T ( connected to the concrete edge stresses of the compression zone 𝝈𝑐𝑒 = 𝜎𝑐1 𝜎𝑐2 (Section 4.1.3.2). A combination of Eqs. (A.38), (A.39), and (4.133) yields rates of edge stresses including visco-elasticity or creep 𝝈̇ 𝑐𝑒 = 𝐸𝑐 (B𝜖 − 𝜅 B𝑧 ) ⋅ 𝝐̇ +

𝐸𝑐 B𝜖 ⋅ 𝝐 − 𝜓𝑐 𝝈𝑐𝑒 𝜁𝑐

(4.138)

This is connected to rates of internal forces by Eq. (A.32) 𝝈̇ 𝑐 = A𝜎 ⋅ 𝝈̇ 𝑐𝑒 + A𝑧 ⋅ ż 𝑐

(4.139)

)T ( with z𝑐 = 𝑧𝑐1 𝑧𝑐2 . Combining edge stresses (Eq. (4.138)), internal forces (Eq. (4.139)) and regarding Eq. (A.37), which relates rates of z𝑐 and the generalised beam strains 𝝐, results to 𝐸𝑐 B𝜖 ⋅ 𝝐 − 𝜓𝑐 𝝈𝑐𝑒 ] + A𝑧 ⋅ B𝑧 ⋅ 𝝐̇ 𝜁𝑐 𝐸𝑐 A𝜎 ⋅ B𝜖 ⋅ 𝝐 − 𝜓𝑐 A𝜎 ⋅ 𝝈𝑐𝑒 = [𝐸𝑐 A𝜎 ⋅ B𝜖 + (A𝑧 − 𝜅𝐸𝑐 A𝜎 ) ⋅ B𝑧 ] ⋅ 𝝐̇ + 𝜁𝑐 (4.140)

𝝈̇ 𝑐 = A𝜎 ⋅ [𝐸𝑐 (B𝜖 − 𝜅B𝑧 ) ⋅ 𝝐̇ +

The term in the parenthesis in the second line corresponds to the tangential material stiffness C𝑐𝑇 for elastic behaviour in the compression zone (Eq. (4.36)). Furthermore, with A = A𝜎 ⋅ B𝜖 , C𝑐 = 𝐸𝑐 A and 𝝈𝑐𝑒 = A−1 𝜎 ⋅ 𝝈𝑐 (Eqs. (4.31), (4.32), and (4.35)) the Eq. (4.140) is written as 𝝈̇ 𝑐 = C𝑐𝑇 ⋅ 𝝐̇ +

1 C𝑐 ⋅ 𝝐 − 𝜓𝑐 𝝈𝑐 𝜁𝑐

(4.141)

whereby C𝑐𝑇 , C𝑐 , and 𝝈𝑐 are functions of the generalised beam strains 𝝐. The material properties are given by the initial Young’s modulus 𝐸𝑐 of the compression zone and the creep parameters 𝜁𝑐 and 𝜓𝑐 (Section 3.2).

4.5 Creep of Concrete

( The rate of the internal forces of the reinforcement 𝝈𝑠 = 𝑁𝑠 by a combination of Eqs. (4.134), (4.136), and (4.137) 𝝈𝑠 = C𝑠 ⋅ 𝝐 ,

𝝈̇ 𝑠 = C𝑠 ⋅ 𝝐̇

𝑀𝑠

)T

is determined

(4.142)

with ⎡ 𝐴𝑠1 + 𝐴𝑠2 C𝑠 = 𝐸𝑠 ⎢ −𝐴 𝑧 − 𝐴𝑠2 𝑧𝑠2 ⎣ 𝑠1 𝑠1

−𝐴𝑠1 𝑧𝑠1 − 𝐴𝑠2 𝑧𝑠2 ⎤ 2 2 ⎥ 𝐴𝑠1 𝑧𝑠1 + 𝐴𝑠2 𝑧𝑠2 ⎦

(4.143)

Adding concrete and reinforcement contributions 𝝈𝑐 = 𝝈 − 𝝈𝑠 , 𝝈̇ = 𝝈̇ 𝑐 + 𝝈̇ 𝑠 yields 𝝈̇ = [ C𝑐𝑇 + C𝑠 ] ⋅ 𝝐̇ + [

1 C𝑐 + 𝜓𝑐 C𝑠 ] ⋅ 𝝐 − 𝜓𝑐 𝝈 𝜁𝑐

(4.144)

This forms a system of ordinary differential equations of first order for 𝝈 depending on time 𝑡 driven by 𝝐(𝑡), 𝝐(𝑡). ̇ Strains generally come from a superordinated calculation (Figure 2.5). Eq. (4.144) is a specification of Eqs. (2.82) and (A.23) 𝝈̇ = C𝑇 ⋅ 𝝐̇ + 𝚺 ,

𝚺=V⋅𝝐 −W⋅𝝈

(4.145)

with ⎛𝑁 ⎞ ⎛𝜖 ⎞ 𝝈=⎜ ⎟ , 𝝐 =⎜ ⎟ 𝑀 𝜅 ⎝ ⎠ ⎝ ⎠ C𝑇 = C𝑐𝑇 + C𝑠 1 V= C𝑐 + 𝜓𝑐 C𝑠 , W = 𝜓𝑐 I 𝜁𝑐

(4.146)

with the unit matrix I. Using Eqs. (A.26)–(A.29) this is embedded into the incrementally iterative solution method for transient analysis, see Section 2.8.2 and Appendix A.2. The application is demonstrated with the following example. Example 4.3: Creep Deformations of RC Beam

Geometry, boundary conditions, and discretisation are adopted from Example 4.2 (Figure 4.7a). The following material properties are assumed: • Concrete grade C 30/37 according to EN 1992-1-1 (2004, Table 3.1) without tensile strength with a Young’s modulus 𝐸𝑐 = 33 000 MN∕m2 in the compressive range. • Creep properties with 𝜑 = 2.0 and 𝑡⋆ = 100 [d] for 𝛼 = 0.5, i.e. half of total creep occurs after 100 days for a constant stress load. With Eq. (3.39) 𝜁 = 144 d and with Eq. (4.133) 𝜓 = 0.020 793 1∕d. • Lower reinforcement with 4 ⊘ 20, 𝐴𝑠1 = 12.57 cm2 , 𝑑1 = 5 cm and a Young’s modulus 𝐸𝑠 = 200 000 MN∕m2 for the reinforcing steel.

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4 Structural Beams and Frames

(a)

(b)

Figure 4.11 Example 4.3. (a) Mid-span deflection during time. (b) Concrete and reinforcement strains along beam for certain time steps.

The loading and the resulting loading level are determined as follows: • According to Example 4.2 the ultimate limit moment of the cross-section is given by 𝑀𝑢 ≈ 0.20 MNm corresponding to a uniform load of about 60 kN∕m. Roughly a third of this is assumed to occur as a permanent load under service conditions, therefore 𝑝 = 20 MN∕m. This leads to a maximum bending moment of 𝑀 = 0.0625 MNm. • The absolute value of the corresponding concrete strain is estimated as a third of the maximum absolute concrete strain of Example 4.2, Figure 4.8b, leading to 𝜖𝑐 ≈ −0.9‰. This has to be compared with the strain at strength 𝜖𝑐1 = −2.3‰ (Figure 3.1) for C 30/37. A linear concrete behaviour can be assumed as approximation in the compressive range and the loading level can be regarded as moderate. Thus, the prerequisite of creep modelling is fulfilled. A time step Δ𝑡 = 10 days is chosen for time discretisation. An incrementally iterative approach with Newton– Raphson iteration within each time increment is used for solving as is described in Section 2.8.2 and Appendix A.2. A period of 500 days is considered. This leads to the following results: • Figure 4.11a shows the computed mid-span deflections in the course of time. The short-term uncracked linear elastic deflection value (𝐸𝐽 = 33 900 ⋅ 0.2 ⋅ 0.43 ∕12 = 35.2 MNm) is given for comparison to RC short-term and long-term deflection values. • The contraction of concrete becomes larger with a constant concrete stress due to creep while reinforcement extension basically remains the same, see Figure 4.11b. This leads to an increasing curvature (Eq. (4.18)) and an increasing deflection. But the increase factor is less than 1 + 𝜑 due to bending and reinforcement. Small variations of stresses and reinforcement strain result from a small change of the internal lever arm. • As a variation, the influence of an upper compression reinforcement 2 ⊘ 20, 𝐴𝑠2 = 6.28 cm2 , 𝑑2 = 5 cm (Figure 4.2a) is taken into account. From Figure 4.11a can be seen that a compression reinforcement further reduces long-term creep deflections, as concrete deformations are constrained to some extent.

4.6 Temperature and Shrinkage

For a comprehensive treatment of creep problems see Jirásek and Bažant (2001, Chapter 29). The creep model may be combined with modelling of imposed strains (Section 4.6) and prestressing (Section 4.8).

4.6 Temperature and Shrinkage Imposed deformations due to temperature or shrinkage may lead to constraint stresses. This has to be considered for ultimate limit states and serviceability. Uniaxial considerations for imposed deformations (Section 3.2) can be directly transferred to strains and stresses of beams. According to Eq. (3.35) rates of measurable longitudinal strains are given by 𝜖̇ 𝑥 =

1 𝜎̇ + 𝜖̇ 𝐼𝑥 , 𝐶𝑇 𝑥

𝜖̇ 𝐼𝑥 = 𝜖̇ 𝑇𝑥 + 𝜖̇ 𝑐𝑠,𝑥

(4.147)

with the longitudinal stress 𝜎𝑥 , the tangential modulus 𝐶𝑇 , temperature strains 𝜖𝑇𝑥 (Eq. (3.34)) and shrinkage strains 𝜖𝑐𝑠,𝑥 . A linear variation of imposed longitudinal strains is assumed over the height of a beam cross-section according to Eq. (4.5) 𝜖𝐼𝑥 (𝑧) = 𝜖𝐼 − 𝑧 𝜅𝐼

(4.148)

without explicitly indicating dependence on the longitudinal coordinate 𝑥. Furthermore, the beams reference axis (Figure 4.1) is placed in the centre of a cross-section height without loss of generality (Section 4.1.3.1). Thus, the imposed strain of the reference axis is given by 𝜖𝐼 =

𝜖𝐼1 + 𝜖𝐼2 2

(4.149)

with index 1 for the lower and index 2 for the upper edge. The imposed curvature is obtained by 𝜅𝐼 =

𝜖𝐼1 − 𝜖𝐼2 ℎ

(4.150)

with the cross-section height ℎ in accordance to Eq. (4.18). Equation (4.148) yields 𝜖𝐼𝑥 (−ℎ∕2) = 𝜖𝐼1 , 𝜖𝐼𝑥 (ℎ∕2) = 𝜖𝐼2 with these definitions. Regarding cracked reinforced cross-sections (Figure 4.2) a variation is described with an upper compression side and a lower tension side leading to an imposed concrete strain 𝜖𝐼2 = 𝜖𝐼 − (ℎ∕2) 𝜅𝐼 and an imposed reinforcement strain 𝜖𝐼𝑠1 = 𝜖𝐼 − (ℎ∕2 − 𝑑1 ) 𝜅𝐼 . This is resolved with 𝜖𝐼 =

1 ℎ 𝜖 + 𝜖𝐼2 ) , ( 2 ℎ − 𝑑1 𝐼𝑠1

𝜅𝐼 =

𝜖𝐼𝑠1 − 𝜖𝐼2 ℎ − 𝑑1

(4.151)

A reversal of tension and compression sides is resolved in analogy. ◀

Be it a strain 𝜖I2 , 𝜖I1 or 𝜖Is they should be prescribed – caused, e.g. by temperature Eq. (3.34) and/or shrinkage – to prescribe imposed generalised strains 𝜖I , 𝜅I to derive a loading.

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4 Structural Beams and Frames

To start with imposed internal forces a linear elastic behaviour Eq. (3.36) is assumed with Young’s modulus 𝐸 leading to 𝜎𝑥 = 𝐸 (𝜖𝑥 − 𝜖𝐼𝑥 ) = 𝐸 [(𝜖 − 𝜖𝐼 ) − 𝑧 (𝜅 − 𝜅𝐼 )]

(4.152)

whereby considering Eqs. (4.5) and (4.148). Internal forces are derived by integrating stresses over a cross-section height (Eq. (4.9)). We obtain 𝝈 = C ⋅ (𝝐 − 𝝐 𝐼 )

(4.153)

as extension of Eq. (4.14) with generalised imposed strains ⎛ 𝜖𝐼 ⎞ 𝝐 𝐼 = ⎜𝜅𝐼 ⎟ ⎜ ⎟ ⎝0⎠

(4.154)

and generalised forces 𝝈, material stiffness C and measurable generalised strains 𝝐 according to Eq. (4.15). Furthermore, imposed or internal constraint forces 𝝈𝐼 = C ⋅ 𝝐 𝐼

(4.155)

are derived. As internal forces lead to internal nodal forces (Eq. (4.97)) internal constraint forces lead to constraint nodal forces f𝑒𝐼 = ∫ BT ⋅ 𝝈𝐼 d𝑥

(4.156)

𝐿𝑒

Imposed strains 𝝐 𝐼 and therefore also internal constraint forces are prescribed as, e.g. function of time. Thus, nodal constraint forces are also prescribed. ◀

Nodal constraint forces from imposed strains may be shifted from the left-hand side of the discretised equilibrium condition Eq. (2.61) to the right-hand loading side whereby becoming part of the external nodal forces.

This approach can also be used for a nonlinear material behaviour. Rates of internal constraint forces are given by 𝝈̇ 𝐼 = C𝑇 ⋅ 𝝐̇ 𝐼

(4.157)

corresponding to Eq. (2.50) and are regarded within the incrementally iterative scheme (Section 2.8.2). The time integration of 𝝈̇ 𝐼 is performed in the same way as for 𝝈̇ using Eq. (2.81), but is considered as part of the loading side. These constraint forces are larger for stiff materials or smaller for soft materials through the influence of C𝑇 . This also implies some nonlinearity in the constraint loadings as C𝑇 might be nonlinear. The influence of imposed strains in form of temperature strains is demonstrated with the following example.

4.6 Temperature and Shrinkage

Example 4.4: Effect of Temperature Actions on an RC Beam

We refer to Example 4.2 with the same system with the exception of boundary conditions. The original system is statically determinate and thus will not have imposed forces in the case of temperature actions. It will be changed into a statically indeterminate system through changed boundary conditions. Furthermore, an upper reinforcement is additionally arranged. The following data are changed compared to Example 4.2: • Left and right node are totally constrained, i.e. lateral and longitudinal displacements and rotations are prescribed with zero. • Upper and lower reinforcement with 𝐴𝑠1 = 𝐴𝑠2 = 12.57 cm2 , 𝑑1 = 𝑑2 = 5 cm (Figure 4.2). • The thermal expansion coefficient is chosen with 𝛼𝑇 = 1 ⋅ 10−5 K−1 both for concrete and reinforcement. The same type of nonlinear material behaviour is assumed for concrete and reinforcement as in Example 4.2 but with a concrete tensile strength neglected. A corresponding linear elastic case is regarded as reference case with a Young’s modulus 𝐸 = 33 000 MN∕m2 and a bending stiffness 𝐸𝐽 = 35.2 MNm2 . Two load cases are investigated: 1. Dead and service load with 𝑞 = 20 kN∕m as in Example 4.3. 2. Dead and service load together with temperatures at the lower edge 𝑇1 = −10 K and at the upper edge 𝑇2 = 10 K. Such a temperature gradient causes an upward movement of a statically determinate system. A positive constraint moment has to be applied to reach compatibility with the zero rotation of boundaries. In the case of nonlinearities all loadings must be applied simultaneously as a superposition is not allowed. The incrementally iterative scheme (Section 2.8.2, Figure 2.5) is chosen as solution method with the BFGS method (Appendix A.1) for the equilibrium iteration in each load increment. This leads to the following computation results: • Figure 4.12a shows the bending moments for the linear elastic reference case whereby the reinforcement is neglected. Load case 1 has end moments 𝑀𝑒 = −0.0402 MNm and a mid-span moment 𝑀𝑓 = 0.0209, which agrees with the analytic solution −𝑞𝐿2 ∕12 and 𝑞𝐿2 ∕24, respectively. Small differences may arise as moments are determined in integration points of elements in finite element calculations. Load case 2 can be superposed due to linearity with a constant 𝑀 tem = −𝐸𝐽 𝜅tem with 𝜅tem = 𝛼𝑇 (𝑇1 − 𝑇2 )∕ℎ = −0.5 ⋅ 10−3 leading to 𝑀 tem = 0.0176. • Figure 4.12b shows the bending moments for the case of reinforced concrete (RC). First of all distribution of moments in a statically indeterminate system depend on stiffness relations. Those depend on loading in the case of RC. The bending stiffness is generally lower for RC compared to the linear elastic case but the stiffness relations do not differ very much. Thus, we get 𝑀𝑒 = −0.0391, 𝑀𝑓 = 0.0220 in load case 1 for RC, i.e. the mid-span area is somehow stiffer and attracts bending moments.

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4 Structural Beams and Frames

(a)

(b)

Figure 4.12 Example 4.4. Bending moments. (a) Linear elastic. (b) RC (different scale!).

The computed mid-span deflection is 𝑤max = 1.23 cm in load case 1 for RC and is higher compared to the linear elastic case with 𝑤max = 𝑞𝐿4 ∕384𝐸𝐽 = 0.92 cm due to the overall reduced stiffness. Superposing is not allowed for load case 2. Computed total moments are 𝑀𝑒 = −0.0281, 𝑀𝑓 = 0.0330 leading to imposed moments 𝑀𝑒tem = 0.0391 − 0.0281 = 0.0110, 𝑀𝑓tem = −0.0220 + 0.0330 = 0.0114. The additional temperature moment is lower compared to the linear elastic case. A computation with temperature loading alone without other loading would lead to a constant 𝑀 tem = 0.0143, i.e. a superposition actually is not correct. • Figure 4.13 shows the RC deformation state for load case 2. Figure 4.13a shows the curvature along the beam. The course is not scaled to the course of the bending moment anymore as it would be in the linear elastic case. Figure 4.13b shows the strain of the reference and centre axis, respectively. Such strains arise in contrast to the linear elastic case as RC beams tend to elongate without longitudinal displacement restrictions, compare Example 4.2, Figure 4.8. An overall elongation is not allowed in this example due to boundary conditions, i.e. a normal compression force is induced on the one hand, and on the other hand strain values occur depending on 𝑀∕𝑁-ratio. But the integral of longitudinal strains must sum up to zero.

(a)

(b)

Figure 4.13 Example 4.4. RC. (a) Curvature. (b) Strain of the reference axis.

4.7 Tension Stiffening

As is shown in Example 3.2 constraint forces are reduced by creep in view of longterm behaviour. Regarding beams this may be modelled by a combination of methods of Examples 4.3 and 4.4. This requires the prescription of temperature and/or shrinkage histories.

4.7 Tension Stiffening A model for the tension zone of a bending beam (Figure 4.2) is given by the reinforced tension bar which is described in Section 3.6. It shows a cracking pattern whereby cracks arise with specific distances. Such a pattern will also arise in the tension zone of RC beams. A simple example is shown in Figure 4.14 with a single span beam with a constant moment in its centre area and stabilised cracking. Crack spacing in connection with bond, i.e. a transmission of forces between reinforcement and concrete, leads to the effect of tension stiffening. Due to bond concrete bears tensile stresses between cracks up to its tensile strength (Figure 3.18a). These mechanisms are already discussed with Example 3.4. The same mechanisms are active in the tensile zone of beams. A quantitative model works with the reduction of reinforcement strains between cracks whereby reinforcement strains in cracked regions are replaced by mean reinforcement strains and the peak values of reinforcement stresses in cracks are still utilised. A corresponding approach is developed in Section 3.7 leading to the following modified stress-strain relations for the reinforcement according to Eqs. (3.68) and (3.72) – assuming the simplified straightened version – whereby 𝜎𝑠𝑐 is replaced by 𝜎𝑠 and 𝜖𝑠𝑚 is replaced by 𝜖𝑠 𝛼𝑓 ⎧ 𝑐𝑡′ 𝜖𝑠 𝜚 ⎪ eff 𝜖𝑠

for 𝜖𝑠 ≤ 𝜖𝑠′

𝜎𝑠 = 𝐸 𝜖 + 𝛽 𝑓𝑐𝑡 ⎨ 𝑠 𝑠 𝑡 𝜚 ⎪ ( eff ′ ) 𝑓 + 𝐸 𝜖 𝑇 𝑠 − 𝜖𝑦 ⎩ 𝑦

(4.158)

𝜖𝑠′ < 𝜖𝑠 ≤ 𝜖𝑦′ 𝜖𝑦 < 𝜖𝑠

with the reinforcement yield stress 𝑓𝑦 and 𝜖𝑠′ =

1 𝑓𝑐𝑡 (𝛼 − 𝛽𝑡 ) , 𝐸𝑠 𝜚eff

𝜖𝑦′ =

𝑓𝑐𝑡 1 (𝑓𝑦 + 𝛽𝑡 ) 𝐸𝑠 𝜚eff

(4.159)

The relation is shown in Figure 3.20b, straightened course, and leads to a nominal stiffening of the rebar stress-strain relation. For the discussion of the parameters

Figure 4.14 Crack pattern of RC beam with constant moment.

109

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4 Structural Beams and Frames

𝛽𝑡 , 𝜚eff , 𝛼 compare Section 3.7. A basic parameter is given by the tension stiffening coefficient 𝛽𝑡 , which controls the quality of bond. The value 𝛽𝑡 = 0, e.g. indicates no stress transfer from rebar to concrete. This leads to 𝜖𝑠′ = 𝛼𝑓𝑐𝑡 ∕𝜚eff , 𝜖𝑦 = 𝑓𝑦 ∕𝐸𝑠 and restores the relation 𝜎𝑠 = 𝐸𝑠 𝜖𝑠 . A reasonable choice for this parameter is in the range 𝛽𝑡 = 0.4 … 0.6. It has to be regarded that the tension zone of a beam cross-section exposed to bending is not homogeneous as tension on one side gradually changes into compression on the other side. This is considered through the effective concrete cross-sectional area 𝐴𝑐,eff and the effective reinforcement ratio 𝜚eff = 𝐴𝑠 ∕𝐴𝑐,eff with the reinforcement cross-sectional area 𝐴𝑠 . The value of 𝐴𝑐,eff is smaller than the area of the tensile zone as not all of its parts contribute to the exchange of stresses with the reinforcement but only some neighbourhood around the main tensile reinforcement. Code provisions are given for the choice of 𝐴𝑐,eff , see EN 1992-1-1 (2004, 7.3.2). Finally, the parameter 𝛼 considers the range of crack formation starting with the first crack and ending with stabilised cracking. The parameter indicates the increase factor of the reinforcement stress during crack formation and can be assumed with 𝛼 ≈ 1.3. Equation (4.158) may replace 𝜎𝑠 = 𝐸𝑠 𝜖𝑠 in Eqs. (4.22) and (4.23) in the case of tension and all which is derived from these equations. The application is demonstrated with the following example. Example 4.5: Effect of Tension Stiffening on an RC Beam with External and Temperature Loading

We refer to Example 4.2 with the same data. Additionally the following values are assumed to model tension stiffening 𝛽𝑡 = 0.6 ,

𝛼 = 1.3

(4.160)

The effective cross-sectional area is determined with an effective height of ℎ𝑐,𝑒𝑓 = 0.1 m according to EN 1992-1-1 (2004, 7.3.2) leading to 𝐴𝑐,eff = 𝑏 ℎ𝑐,𝑒𝑓 = 0.02 m2 and further to 𝜚eff =

0.1257 ⋅ 10−2 = 0.063 , 0.02

𝜖𝑠′ = 0.215 ⋅ 10−3

(4.161)

with 𝜖𝑠′ according to Eq. (4.159). Three loading values are examined: 𝑞 = 60 kN∕m, which is near to the ultimate limit load, furthermore 𝑞 = 40 MN∕m and finally 𝑞 = 20 MN∕m which roughly corresponds to dead load and service load. The mid-span deflections with and without tension stiffening are determined with 𝑞 = 20 𝑞 = 40 𝑞 = 60 with tension stiffening 0.78 1.87 2.97 without tension stiffening 0.96 2.01 3.15 with a unit of [cm]. The absolute deflection differences with and without tension stiffening differs in a small range and are basically independent of the load level. This corresponds to the constant horizontal offset of the stress-strain relation for the reinforcement.

4.8 Prestressing

(a)

(b)

Figure 4.15 Example 4.5. (a) Moment. (b) Normal force.

As beam stiffness is increased due to tension stiffening an influence on constraint forces (Eq. (4.155)) may be supposed. To examine this effect, we refer to Example 4.4 with the RC case. All system and loading parameters are kept but with tension stiffening included. The tension stiffening parameters Eqs. (4.160) and (4.161) are used. Furthermore, a concrete tensile strength of 𝑓𝑐𝑡 = 3.0 MN∕m2 is assumed for tension stiffening parameters only as a value is required for Eqs. (4.158) and (4.159). Concrete tensile strength does not contribute to bending in the context of tension stiffening but indirectly controls stress transfer between concrete and rebars (Figure 3.12b). Load case 2 – dead/service load and temperature – is considered. Figure 4.15a shows the computed bending moments for the cases with and without tension stiffening. The difference is quite small as the stiffness relations basically do not change even if the absolute values of stiffness change. The induced compression normal force is shown as a further result in Figure 4.15b. A compression is caused from the potential elongation of the reference axis of cracked RC sections (Figure 4.8a) which is prevented by boundary displacement conditions. The potential elongation is smaller in case with tension stiffening as the same moment has a smaller mean rebar strain compared to the case without tension stiffening. This induces a smaller compression in case with tension stiffening. Some oscillating scatter is seen in the normal forces especially in those areas with low moments. This scatter is caused by the coupling of both normal forces and moments to the strain of the reference axis 𝜖 and the curvature 𝜅 (Eq. (4.46)). This scatter effect is erroneous and can be reduced with the enhanced Bernoulli beam element (Section 4.3.2) to a large extent but cannot be totally avoided. It is compensated in the balance of nodal forces as opposite peak values are computed in neighboured integration points which compensate in the average of an element. The effect can be further reduced with a finer discretisation.

111

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4 Structural Beams and Frames

4.8 Prestressing Prestressing applies lateral redirection forces and normal forces on a beam, see Figure 4.16a. While the redirection forces act against dead and variable loads a moderate normal force may increase the bearing capacity for moments (Example 4.1.3.3, Figure 4.4). But these positive effects require prestressing tendons. A concrete beam and its ordinary reinforcement on the one hand and the tendons with high strength steel on the other hand are regarded as separated structural elements in the following. We consider the Bernoulli beam. Primarily, the generalised stress 𝝈 (Eq. 4.60)) is formulated as a function of the generalised strains 𝝈=C⋅𝝐

(4.162)

with, e.g. C according to Eq. (4.16) or Eq. (4.42). A linear C is not necessarily required. This concept is extended with respect to prestressing: an additional part is assigned to the generalised stresses resulting from prestressing tendons 𝝈 = C ⋅ 𝝐 + 𝝈𝑝

(4.163)

This additional part 𝝈𝑝 depends on the tendon profile ⎛𝑁𝑝 ⎞ 𝝈𝑝 = ⎜ 𝑝 ⎟ = −𝐹 𝑝 𝑀 ⎝ ⎠

⎛ cos 𝛼𝑝 ⎞ ⎜ ⎟ −𝑧 cos 𝛼𝑝 ⎝ 𝑝 ⎠

(4.164)

with the prestressing force 𝐹 𝑝 , the height coordinate or lever arm 𝑧𝑝 of the tendon and the inclination 𝛼𝑝 = d𝑧𝑝 ∕ d𝑥 of the tendon against the beam reference axis, see Figure 4.16b. This may be extended with respect to shear forces in combination with the Timoshenko beam. Using the extended generalised stresses Eq. (4.163) for the internal nodal forces Eq. (4.97) leads to a split 𝑝

f𝑒 = ∫ BT ⋅ 𝝈 d𝑥 = ∫ BT ⋅ C ⋅ 𝝐 d𝑥 + ∫ BT ⋅ 𝝈𝑝 d𝑥 = f𝑒𝜖 + f𝑒 𝐿𝑒

𝐿𝑒

(4.165)

𝐿𝑒

𝑝 −f𝑒

may be regarded as a further contribution to the load vector (Eq. The part (2.59)) or external nodal forces. This approach integrates prestressing in the given

(a)

(b)

Figure 4.16 (a) Redirection forces from prestressing. (b) Internal forces with prestressing.

4.8 Prestressing

framework whereby all procedures but for a part of load evaluation remain unchanged. An alternative and common view of prestressing of beams is based on Eqs. (4.492,3 ). We consider the quasi-static case, split internal forces into a part ∙𝜖 from beam deformation, a part ∙𝑝 from prestressing and eliminate shear forces ′′

′′

−𝑀 𝜖 − 𝑀 𝑝 = 𝑝 𝑧

(4.166)

and furthermore Eq. (4.164) is used leading to ( )′′ ′′ −𝑀 𝜖 = 𝑝𝑧 + 𝑧𝑝 𝐹 𝑝 cos 𝛼𝑝

(4.167)

A common approximation is 𝐹 𝑝 ≈ 𝑐𝑜𝑛𝑠𝑡., cos 𝛼𝑝 ≈ 1 resulting in ′′

−𝑀 𝜖 = 𝑝𝑧 + 𝑧𝑝′′ 𝐹 𝑝

(4.168)

wherein 𝑧𝑝′′ 𝐹 𝑝 is a lateral redirection force in the 𝑧-direction from the curvature 𝑧𝑝′′ of the tendon geometry. This term may be seen as an additional lateral loading counteracting the other loadings (Figure 4.16). Some characteristic properties of prestressing have to be regarded for the evaluation of 𝝈𝑝 or f 𝑝 , respectively. • Tendon profile parameters 𝑧𝑝 , 𝛼𝑝 may vary with the beam coordinate 𝑥 according to prestressing design. • The prestressing force may vary due to the loss of prestress from friction of the tendon in a conduit. • Furthermore, a beam deformation may lead to a change in the tendon profile. Two types have to be considered in this context: 1. Unbonded prestressing: total length of the tendon changes. This leads to a global change of the prestressing force. 2. Bonded prestressing: length of the tendon changes locally to keep the geometric compatibility with the concrete. This leads to locally varying changes in the prestressing force. Two subsequent stages have to be considered for prestressing: 𝑝

• Set-up stage of prestressing with the prescribed prestressing force 𝐹0 Prestressing is gradually applied at the beam ends through anchors. The value 𝑝 of 𝐹0 may vary along the longitudinal beam coordinate 𝑥 due to friction losses. Such losses have to be determined from prescribed friction coefficients and the curvature of the tendon geometry. • Tendons may be grouted at the end of the set-up stage leading to post-bonding. This is generally denoted as bonded prestressing although this is not entirely precise. Grouting is omitted for unbonded prestressing. • Service stage of prestressing with locked anchors 𝑝 The prestressing force 𝐹0 changes into 𝐹𝑝 depending on loading and prestressing type.

113

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4 Structural Beams and Frames

The tendon geometry plays a key role. It is described for each finite beam element in analogy to the Bernoulli beam trial function (Eq. (4.82)) by 𝑟3

3𝑟

4

4

𝑧𝑝 = [ −

+

1

𝐿𝑒 𝑟3

2

8

−

𝐿𝑒 𝑟2 8

( ⋅ 𝑧𝑝𝐼

−

𝐿𝑒 𝑟 8

𝛼𝑝𝐼

+

𝐿𝑒 8

𝑧𝑝𝐽

𝑟3

3𝑟

4

4 )T

− + 𝛼𝑝𝐽

+

1

𝐿𝑒 𝑟3

2

8

+

𝐿𝑒 𝑟2 8

−

𝐿𝑒 𝑟 8

−

𝐿𝑒 8

]

(4.169)

with the element length 𝐿𝑒 and the tendon inclination 𝛼𝑃 =

𝜕𝑧𝑝

=

𝜕𝑥

𝜕𝑧𝑝 𝜕𝑟 = 𝑧𝑝′ 𝜕𝑟 𝜕𝑥

(4.170)

Lateral tendon position 𝑧𝑝 and inclination 𝛼𝑝 at the left-hand and right-hand element nodes are given by 𝑧𝑝𝐼 , 𝛼𝑝𝐼 and 𝑧𝑝𝐽 , 𝛼𝑝𝐽 . The local element coordinate is in the range −1 ≤ 𝑟 ≤ 1. This approach reproduces 𝑧𝑝 (−1) = 𝑧𝑝𝐼 , 𝑧𝑝′ (−1) = 𝛼𝑝𝐼 and 𝑧𝑝 (1) = 𝑧𝑝𝐽 , 𝑧𝑝′ (1) = 𝛼𝑝𝐽 . The geometric length of a tendon within an element 𝑒 is given by 1

𝐿𝑒 √ ′ 2 ∫ (𝑥𝑝 ) + (𝑧𝑝′ )2 d𝑟 𝐿𝑒𝑃 = 2

(4.171)

−1

whereby the derivative of the tendon position 𝑥𝑝 in the longitudinal direction has also to be regarded. We obtain 𝑥𝑝′ = 1

(4.172)

for the nominal undeformed tendon geometry and 𝑥𝑝′ = 1 + 𝜖

(4.173)

considering beam deformations with the longitudinal strain 𝜖 of the reference axis. Furthermore, on the one hand Eq. (4.169) is applied to the nominal undeformed tendon geometry according to design with ) ( ) ( (4.174) 𝑧𝑝𝐼 𝛼𝑝𝐼 𝑧𝑝𝐽 𝛼𝑝𝐽 = 𝑧𝑝0𝐼 𝛼𝑝0𝐼 𝑧𝑝0𝐽 𝛼𝑝0𝐽 with prescribed values 𝑧𝑝0𝐼 , 𝛼𝑝0𝐼 , 𝑧𝑝0𝐽 , 𝛼𝑝0𝐽 . On the other hand, Eq. (4.169) gives the tendon geometry considering beam deformations with ( ) ( ) 𝑧𝑝𝐼 𝛼𝑝𝐼 𝑧𝑝𝐽 𝛼𝑝𝐽 = 𝑧𝑝0𝐼 + 𝑤𝐼 𝛼𝑝0𝐼 + 𝜙𝐼 𝑧𝑝0𝐽 + 𝑤𝐽 𝛼𝑝0𝐽 + 𝜙𝐽 (4.175) with the beam nodal displacements 𝑤𝐼 , 𝜙𝐼 , 𝑤𝐽 , 𝜙𝐽 . This yields 𝑧𝑝′ from Eq. (4.169) according to Eq. (4.822 ). The Eq. (4.171) has to be integrated numerically for each element, e.g. with a Gauss integration (Section 2.7). The total length 𝐿𝑃 of a tendon is given by adding all element contributions. Regarding prestressing without bond the tendon length can be determined separately with a value 𝐿0𝑃 for the set-up stage and with a value 𝐿𝑃 for the service stage whereby Eq. (4.175) is used with the actually calculated nodal displacements for each stage. This has a side effect.

4.8 Prestressing ◀

Strains of unbonded tendons do not follow the Bernoulli–Navier hypothesis (Section 4.1.1). 𝑝

The prestressing force is prescribed with 𝐹0 for the set-up stage and is determined with 𝐹𝑝 =

𝐿𝑃 𝑝 𝐹 𝐿0𝑃 0

(4.176)

in the service stage of prestressing. Regarding prestressing with post-bonding a tendon gets a local elongation after locking of prestressing anchors and grouting due to bond between tendons and concrete. This local elongation is ruled by the beam deformation kinematics (Eq. (4.5)) and the additional strain of the tendon is given by Δ𝜖𝑝 (𝑥) = Δ𝜖(𝑥) − 𝑧𝑝 Δ𝜅(𝑥)

(4.177)

with the difference Δ𝜖, Δ𝜅 of generalised strains between set-up and service stage whereby varying along the beam axis. ◀

Strains of post-bonded tendons follow the Bernoulli–Navier hypothesis during the service stage although not during the set-up stage.

This leads to a prestressing force 𝑝

𝐹 𝑝 (𝑥) = 𝐹0 + 𝐸𝑝 𝐴𝑝 Δ𝜖𝑝 (𝑥)

(4.178)

in the service stage with Young’s modulus 𝐸𝑝 of the prestressing steel – elastic behaviour is assumed to simplify – and the cross-sectional area 𝐴𝑝 of tendons. Finally, internal prestressing forces contributing to loadings are determined using Eq. (4.164). A loading from prestressing distinguishes from self weight, service load, temperature as it depends on displacements and yields a nonlinear load con𝑝 tribution. But the computations show that 𝐹 𝑝 and 𝐹0 generally do not differ by large amounts. Thus, the particular procedures concerning prestressing can be summarised with • • • • •

define the tendon geometry and prestressing force, compute internal forces from prestressing, compute nodal forces from prestressing and apply as loads, compute system reaction, iterate if necessary to consider a change in prestressing forces

which seamlessly fits into the incrementally iterative approach (Section 2.8.2). The application is demonstrated with the following example.

115

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4 Structural Beams and Frames

Example 4.6: Prestressed RC beam

We refer to Example 4.2 with basically the same system, but the span is doubled to 𝐿 = 10 m. Thus, load bearing capacity is strongly reduced. Prestressing is used to maintain this capacity. The relevant system parameters are as follows: • A concrete cross-section 𝑏 = 0.2, ℎ = 0.4, a compressive strength 𝑓𝑐𝑑 = 38 MN∕m2 and a lower and upper reinforcement 𝐴𝑠1 = 𝐴𝑠2 = 12.57 cm2 , 𝑑1 = 𝑑2 = 5 cm yield an ultimate bending moment 𝑀𝑢 ≈ 0.20 MNm with 𝑁 = 0 (Example 4.1, Figure 4.4). This corresponds to an uniform loading 𝑞𝑢 = 8𝑀𝑢 ∕𝐿2 = 15.2 kN∕m which should be increased by prestressing. • A nominal uniform concrete prestressing stress of 𝜎𝑐0 = −10 MN∕m2 is chosen 𝑝 in a first approach leading to 𝐹0 = 0.8 MN. The nominal tendon geometry of the whole beam is given by a parabola starting and ending in the centre line with a downward catenary ℎ𝑝 . This is described by 𝑧𝑝 = 4ℎ𝑝 (

𝑥2 𝑥 − ) 𝐿2 𝐿

(4.179)

A value ℎ𝑝 = 0.15 m is chosen in this example. • Prestressing tendon and steel properties are chosen with cross-sectional area 𝐴𝑝 = 6 cm2 , elastic limit 𝑓𝑝0,1 = 1600 MN∕m2 , strength 𝑓𝑝 = 1800 MN∕m2 , Young’s 𝑝 modulus 𝐸𝑝 = 200 000 MN∕m2 , nominal initial steel stress 𝜎0 = 1333 MN∕m2 𝑝 with a strain 𝜖0 = 6.67 o/oo. • A dead load is assumed with 𝑞 = 5 kN∕m. Loading is applied in two steps: (1) set-up of prestressing and dead load, (2) locking of prestressing and additional application of a service load 𝑞𝑝 = 25 kN∕m. Frictional losses are neglected to simplify this example. Both cases – prestressing with and without bond – are alternatively regarded for the service stage of prestressing. The solution method is incrementally iterative with Newton–Raphson iteration within increments. This leads to the following results for prestressing without bond: • The computed increase in prestressing force after load step 2 according to 𝑝 Eq. (4.176) is minimal with 𝐹 𝑝 ∕𝐹0 = 1.002. This results from the low ratio ℎ𝑝 ∕𝐿 = 1∕67. • For the computed mid-span displacements, see Figure 4.17a. The deflection starts with an uplift during application of prestressing. The final mid-span deflection in load step 2 is quite large with 0.107 m (≈ 1/90 of the span), but the load-carrying capacity is not yet exhausted with an upper mid-span concrete compressive strain of −2.0 o/oo (limit strain is −3.5 o/oo). Serviceability is presumably not given without further provisions due to the high slenderness (1/25). • For the bending moment 𝑀𝑐 in the RC cross-section only see Figure 4.18a. The total moment from the dead load and the service load is 𝑀𝑞 = 0.03 ⋅ 102 ∕8 = 0.375 MNm. The computed RC mid-span contribution is 𝑀𝑐 = 0.255 and the contribution from prestressing 𝑀𝑝 = 0.120. The increased RC moment compared to the initial estimation results from the compressive normal force.

4.8 Prestressing

(a)

(b)

Figure 4.17 Example 4.6. (a) System. (b) Mid-span load–deflection curve.

(a)

(b)

Figure 4.18 Example 4.6. Final stage. (a) RC bending moment Mc . (b) Prestressing force F p .

Furthermore, the results for prestressing with post-bonding: • The tendon gets a local additional strain due to the locally varying deformation of the beam (Eq. (4.177)). This leads to an additional prestressing force 𝐹 𝑝 , see Eq. (4.178) and Figure 4.18b, and to a higher contribution of prestressing to the load bearing capacity whereby reducing concrete demand. • More significant results are given with the final mid-span deflection reduced to 0.086 m (Figure 4.17b) compared to the unbonded case, the RC moment contribution reduced to 𝑀𝑐 = 0.220 (Figure 4.18a) and the prestressing moment contribution increased to 𝑀𝑝 = 0.155. Prestressing roughly leads to a doubling of ultimate limit loads in this example. Aspects of serviceability have to be treated separately. The comparison between prestressing with and without bond in this example is somehow academic, as in practice prestressing with bond is exposed to more non-mechanical effects which might lead to some restrictions to fully utilise the load carrying capacities of the prestressing steel. Details are ruled in codes.

117

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4 Structural Beams and Frames

All methods and procedures may be also applied to material laws of a rate type (Eq. (2.50)) instead of Eq. (4.162). Multiple tendons may be arranged. A particular tendon may be limited to a part of a beam. Prestressing may be combined with creep of concrete (Section 4.5), temperature and shrinkage (Section 4.6) and tension stiffening (Section 4.7). All these approaches may be used in any combination. Example 4.6 demonstrates the application for a single-span beam which is statically determinate. ◀

All methods and procedures in same way may be applied to statically indeterminate systems as solution procedures always simultaneously regard equilibrium, material behaviour, and kinematic compatibility.

High strength steel is used for prestressing with roughly three times the strength of ordinary reinforcing steel. Relaxation (Section 3.2) occurs for such types of steel. The approach of Eq. (3.30) can basically be used for the phenomena of creep and relaxation and for steel as well as for concrete. This can be applied to the prestressing force 𝐹 𝑝 (Eq. (4.164)) leading to a transient analysis as in a similar way has already been discussed in Section 4.5.

4.9 Large Displacements – Second-Order Analysis Up to now, we considered the equilibrium of structures in their undeformed configuration. The displacements of a structure were neglected in the balance of external and internal nodal forces and a geometrically linear analysis was performed. This is justified for RC beams which are exposed to predominant bending or tension. Deformations generally will not have an appreciable influence on internal forces in such cases. This might change for structural members exposed to compression. Depending on their slenderness internal forces may considerably increase due to deformations and they have to be considered regarding equilibrium. This leads to geometrical nonlinearities. We consider a section of a plane beam in some deformed configuration, see Figure 4.19. A quasi-static analysis is performed whereby equilibrium should be given

Figure 4.19 Equilibrium of beam section in the deformed configuration.

4.9 Large Displacements – Second-Order Analysis

in the deformed configuration. Thus, the integration of the equilibrium condition (Eq. (4.57)) of a section of length 𝐿 𝐿

∫

𝐿

𝛿𝝐 T

⋅ 𝝈 d𝑠 = ∫ 𝛿uT ⋅ p d𝑠 + 𝛿UT ⋅ t

0

(4.180)

0

has to be performed in the deformed configuration with the coordinate 𝑠 along the beam axis. It is appropriate to relate the generalised stresses 𝝈 and the generalised strains 𝝐 (Eqs. (4.58) and (4.60)) to a local co-rotational coordinate system. A single element – straight in its undeformed configuration – is considered in the following. It is assumed that the curvature of a single element is still small in the deformed configuration and that its deformed geometry can be approximated with a straight line connecting its end nodes. This assumption is appropriate for RC structures failing with relatively small strains. Furthermore, the deformed geometry can be approximated with any desired accuracy by refining the discretisation. The corresponding element orientation is given by the angle 𝛼𝑒 . Global nodal degrees of freedom 𝝊𝑒 are transformed to the local system with Eq. (4.105) ˜ 𝝊𝑒 = T(𝛼𝑒 ) ⋅ 𝝊𝑒

(4.181)

Local displacements and strains are given according to Eqs. (4.84) and (4.87) u ˜(𝑟) = N(𝑟) ⋅ ˜ 𝝊𝑒 ,

𝝐(𝑟) = B(𝑟) ⋅ ˜ 𝝊𝑒

(4.182)

leading to virtual strains 𝛿𝝐 T = 𝛿𝝊T𝑒 ⋅ TT (𝛼𝑒 ) ⋅ BT (𝑟)

(4.183)

Internal nodal forces of a beam element are derived by Eqs. (4.97) and (4.106) 1

f𝑒 =

TT (𝛼𝑒 ) ⋅

𝐿𝑒 ∫ BT (𝑟) ⋅ 𝝈(𝑟) d𝑟 2

(4.184)

−1

with the actual element length 𝐿𝑒 whereby the orientation 𝛼𝑒 depends – besides the initial coordinates of the element nodes – on the values of the nodal degrees of freedom or displacements, respectively. ◀

With the internal nodal forces depending on the deformation the problem becomes geometrically nonlinear. This is combined with physical nonlinearity in the case of RC beams due to the nonlinear behaviour between the generalised strains and stresses.

The evaluation of the tangential stiffness is mandatory for such problems. The tangential element stiffness matrix is determined by (Eq. (2.65)) K𝑇𝑒 =

𝜕f𝑒 𝜕𝝊𝑒

(4.185)

119

120

4 Structural Beams and Frames

Applying the product rule of differentiation to Eq. (4.184) yields K𝑇𝑒 = K𝑇𝑀𝑒 + K𝑇𝐺𝑒

(4.186)

with the tangential stiffness contribution from material behaviour (tangential physical stiffness) 1

K𝑇𝑀𝑒

𝐿𝑒 𝝊𝑒 𝜕𝝈 𝜕𝝐 𝜕˜ ∫ BT ⋅ = TT ⋅ ⋅ d𝑟 ⋅ 2 𝜕𝝐 𝜕˜ 𝝊𝑒 𝜕𝝊𝑒 −1 1

=

TT

𝐿𝑒 ∫ BT ⋅ C𝑇 ⋅ B d𝑟 ⋅ T ⋅ 2

=

TT

˜ 𝑇𝑒 ⋅ T ⋅K

−1

(4.187)

according to Eqs. (2.66) and (4.110) and furthermore with the tangential stiffness contribution from geometry (tangential geometrical stiffness) T

K𝑇𝐺𝑒 = (

𝜕𝛼𝑒 𝜕TT ˜ ⋅f )⋅( ) 𝜕𝛼𝑒 𝑒 𝜕𝝊𝑒

(4.188)

with the local internal nodal forces 1

𝐿𝑒 ˜ ∫ BT ⋅ 𝝈 d𝑟 f𝑒 = 2

(4.189)

−1

Regarding the two-node Bernoulli beam element (Section 4.3.2, Eq. (4.107)) the parts of Eq. (4.188) are given by

𝜕TT 𝜕𝛼𝑒

˜ f𝑒

⎡− sin 𝛼𝑒 − cos 𝛼𝑒 ⎢ − sin 𝛼𝑒 ⎢ cos 𝛼𝑒 ⎢ 0 ⎢ 0 =⎢ 0 0 ⎢ ⎢ 0 ⎢ 0 ⎢ 0 0 ⎣ ( = 𝑁𝐼 𝑉𝐼 𝑀𝐼 𝑁𝐽

0

0

0

0

0

0

0

0

0

0

− sin 𝛼𝑒

− cos 𝛼𝑒

0

cos 𝛼𝑒

− sin 𝛼𝑒

0

0

0

𝑉𝐽

𝑀𝐽

0⎤ ⎥ 0⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ 0 ⎦

)T

(4.190)

and 𝜕𝛼𝑒 1 ( = sin 𝛼𝑒 𝐿𝑒 𝜕𝝊𝑒

− cos 𝛼𝑒

0

− sin 𝛼𝑒

cos 𝛼𝑒

)T 0

(4.191)

4.9 Large Displacements – Second-Order Analysis

This leads to a – generally unsymmetric – tangential geometrical element stiffness

K𝑇𝐺𝑒

⎡ −𝐴𝐼 sin 𝛼𝑒 ⎢ ⎢ 𝐵𝐼 sin 𝛼𝑒 ⎢ 0 ⎢ =⎢ ⎢−𝐴𝐽 sin 𝛼𝑒 ⎢ ⎢ 𝐵𝐽 sin 𝛼𝑒 ⎢ 0 ⎣

𝐴𝐼 cos 𝛼𝑒

0

𝐴𝐼 sin 𝛼𝑒

−𝐴𝐼 cos 𝛼𝑒

−𝐵𝐼 cos 𝛼𝑒

0

−𝐵𝐼 sin 𝛼𝑒

𝐵𝐼 cos 𝛼𝑒

0

0

0

0

𝐴𝐽 cos 𝛼𝑒

0

𝐴𝐽 sin 𝛼𝑒

−𝐴𝐽 cos 𝛼𝑒

−𝐵𝐽 cos 𝛼𝑒

0

−𝐵𝐽 sin 𝛼𝑒

𝐵𝐽 cos 𝛼𝑒

0

0

0

0

0⎤ ⎥ 0⎥ ⎥ 0⎥ ⎥ (4.192) 0⎥ ⎥ 0⎥ ⎥ 0 ⎦

with coefficients depending on internal forces 𝐴𝑖 = sin 𝛼𝑒 𝑁𝑖 + cos 𝛼𝑒 𝑉𝑖 ,

𝐵𝑖 = cos 𝛼𝑒 𝑁𝑖 − sin 𝛼𝑒 𝑉𝑖 ,

𝑖 = 𝐼, 𝐽

(4.193)

Internal nodal forces from Eq. (4.184) have to be in equilibrium with the external nodal forces. External nodal forces for the two-node Bernoulli beam element are determined in analogy to Eq. (4.101) leading to 1

𝐿𝑒 ∫ TT (𝛼𝑒 ) ⋅ NT (𝑟) ⋅ Q(𝛼𝑒 ) ⋅ p(𝑟) d𝑟 t𝑒 = 2

(4.194)

−1

for a distributed loading with the vector rotation matrix Q (Eq. (D.6)) and ( c𝑒 = 𝑁 𝐼

𝑉𝐼

𝑀𝐼

𝑁𝐽

𝑉𝐽

𝑀𝐽

)T

(4.195)

for the boundary terms (Eq. (4.102)). The components of c𝑒 and p (Eq. (4.58)) are related to the global coordinate system (Figure 4.19). The enhanced Bernoulli beam element (Section 4.3.2) and the Timoshenko beam elements (Section 4.3.1) can be treated in the same way with the adaption of 𝝊𝑒 , 𝝐, 𝝈 and T. The system equations are assembled from the element contributions (Section 2.6, item 6) leading to a condition for quasi-static equilibrium (Eq. (2.70)) f(𝝊) = p

(4.196)

The solution is determined with an incrementally iterative scheme (Section 2.8.2) whereby the tangential stiffness should include physical and geometrical stiffness as are given for single elements with Eqs. (4.187) and (4.188). The described approach corresponds to a co-rotational updated Lagrangian discretisation (Belytschko et al. 2000, 4.4, 4.6). It may be applied to cases with large displacements and small strains. A second-order analysis is included as a special case whereby displacements are linearised with respect to an initially undeformed configuration. A first validation is given by the following example.

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4 Structural Beams and Frames

Example 4.7: Stability Limit of Cantilever Column

We consider a simple cantilever column with linear elastic behaviour and a discretisation with one two-node Bernoulli beam element, see Figure 4.20. It has a vertical concentrated load at its top acting along the centre line of gravity. A buckling instability from the initial configuration will occur within a theory regarding equilibrium in deformed configurations. The buckling load is determined for this model. It has three degrees of freedom with 𝑢, 𝑤, 𝜙 at the top node. The initial configuration is given by 𝛼𝑒 = π∕2. The physical stiffness is determined by Eq. (4.187) with B from Eq. (4.88), C𝑇 = C from Eq. (4.16) and T from Eq. (4.107) with 𝛼𝑒 replacing 𝛼. This yields 12𝐸𝐽

K𝑇𝑀

⎡ 3 ⎢ 𝐿 =⎢ 0 ⎢ 6𝐸𝐽 ⎣ 𝐿2

0 𝐸𝐴 𝐿

0

6𝐸𝐽

⎤ ⎥ 0 ⎥ 4𝐸𝐽 ⎥ 𝐿 ⎦ 𝐿2

(4.197)

for the actual degrees of freedom. The geometric stiffness Eq. (4.192) is determined by 1

K𝑇𝐺 =

𝑁 𝐽 K0𝑇𝐺

,

K0𝑇𝐺

⎡ ⎢𝐿 = ⎢0 ⎢ 0 ⎣

0 0 0

0⎤ ⎥ 0⎥ ⎥ 0 ⎦

(4.198)

The matrix K0𝑇𝐺 is independent from loading and considers the geometry of the initial undeformed configuration. The incremental system behaviour is described by Eq. (2.67) (K𝑇𝑀 + K𝑇𝐺 ) ⋅ d𝝊 = df ◀

(4.199)

An instability is given by non-zero increments of displacements d 𝝊 in connection with zero increments of nodal forces df.

This leads to the generalised eigenvalue problem K𝑇𝑀 ⋅ d𝝊 = −𝑁𝐽𝑏 K0𝑇𝐺 ⋅ d𝝊

(4.200)

Figure 4.20 Cantilever column.

4.9 Large Displacements – Second-Order Analysis

with the buckling load 𝑁𝐽𝑏 . The solution for this model is 3 𝐸𝐽 (4.201) 𝐿2 whereby the relation between the eigenvector components or lateral displacement and rotation is determined with 9𝑤˜𝐽 + 6𝐿 𝜙 = 0. The exact solution is given by π2 𝐸𝐽∕4𝐿2 ≈ 2.47 𝐸𝐽∕𝐿2 according to the well-known Euler cases for buckling. The error in 𝑁𝐽𝑏 results from the discretisation with one element which is a rough approximation of the exact cosine solution. A refinement of the discretisation should improve the solution for 𝑁𝐽𝑏 . 𝑁𝐽𝑏 = −

The method described for the stability analysis of a cantilever column is based on the generalised eigenvalue problem Eq. (4.200). It may be applied to all types of columns and frames with compression members in the undeformed configuration. Furthermore, it may be generalised for the stability analysis of all types of structures (Belytschko et al. 2000, 6.5). For more aspects of stability, bifurcation and uniqueness of solutions, see Zienkiewicz and Taylor (1991, 8.2), Bathe (1996, 6.8.2), de Borst et al. (2012, 4.4) We regarded linear elastic material behaviour up to now. The nonlinear behaviour of RC cross-sections (Section 4.1.3) has to be considered in a next step. This is described for Bernoulli beams in the following. A deformed configuration is given within an incremental iterative scheme (Section 2.8.2) with an orientation angle 𝛼𝑒 and an actual length 𝐿𝑒 for each element. Generalised strains 𝝐 and generalised stresses 𝝈 or internal forces (Eq. (4.60)) are determined in the local co-rotational system. With the generalised strains given the internal forces are determined from Eqs. (4.22) and (4.23) and the tangential material stiffness matrix C𝑇 from Eq. (4.46). This yields the basic quantities required for the internal nodal forces in the global system (Eq. (4.184)), the tangential physical stiffness (Eq. (4.187)) and the tangential geometric stiffness (Eq. (4.188)). The external nodal forces in the global system are finally determined from Eqs. (4.194) and (4.195) for a new loading target. Prescribed displacements are treated as described in Section 4.4.3. The nonlinear problem – including the continuing updating of 𝛼𝑒 , 𝐿𝑒 – is solved by, e.g. the Newton–Raphson method (Eqs. (2.76) and (2.77)) whereby the tangential system stiffness is given by the tangential physical stiffness and the tangential geometric stiffness. The application is demonstrated with the following example. Example 4.8: Ultimate Limit for RC Cantilever Column

The stability of the cantilever column has already been treated in Example 4.7 as a generalised eigenvalue problem whereby resembling the classical Euler case. This assumes a centred load without eccentricity and a linear elastic material behaviour and leads to an upper bound for the load carrying capacity. A load eccentricity and the nonlinear behaviour of RC cross-sections are considered while regarding large displacements and small strains. Geometry, discretisation, and boundary conditions are as follows:

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4 Structural Beams and Frames

(a)

(b)

Figure 4.21 Example 4.8. (a) System. (b) Vertical load-horizontal displacement curve.

• Cantilever column with a height 𝐿 = 5.0 m, square cross-section with width ℎ = 0.4 m and a depth 𝑏 = 0.2 m, see Figure 4.7a. • Discretisation with 𝑛𝐸 = 10 enhanced Bernoulli beam elements (Section 4.3.2). • The bottom node is clamped with zero displacements and zero rotations. The column is assumed to be stabilised in the out-of-plane direction. The material properties and the reinforcement are chosen following Example 4.1 with some additional reinforcement. • Concrete grade C30/37 according to EN 1992-1-1 (2004, Table 3.1) with an initial Young’s modulus 𝐸𝑐 = 33 000 MN∕m2 . The compressive strength of the concrete is assumed with 𝑓𝑐 = 38 MN∕m2 with 𝜖𝑐1 = −0.0023, 𝜖𝑐𝑢1 = −0.0035 (Figure 3.1). A tensile strength is disregarded. The uniaxial stress–strain relation is chosen according to EN 1992-1-1 (2004, 3.1.5). • Reinforcement behaviour is assumed according to Section 3.3 and EN 1992-1-1 (2004, 3.2.7) with 𝑓𝑦𝑘 = 500 MN∕m2 , 𝑓𝑡 = 525 MN∕m2 , 𝜖𝑦0 = 2.5‰ and 𝜖𝑢 = 25‰ (Figure 3.11a). • Left and right reinforcement each with 4 ⊘ 20, 𝐴𝑠2 = 𝐴𝑠1 = 12.57 cm2 , 𝑑2 = 𝑑1 = 5 cm. The elastic in-plane stability point or buckling load is determined as π2 𝐸𝐽 π2 33 000 ⋅ 0.001 067 = = 3.47 MN (4.202) 2 4 𝐿 4 5.02 The vertical downward load target is chosen with 𝑃 = 2 MN with an eccentricity of 𝑒 = 0.032 m (Figure 4.21a). A comparable moment-curvature relation for this compression force has been determined in Example 4.1 and is shown in Figure 4.4. An incrementally iterative scheme with arc length control to determine load increments (Appendix A.4) and Newton–Raphson iteration within each loading increment (Section 2.8.2) is used as the solution method. The computation leads to the following results: 𝑃𝑏 =

• The computed relation between horizontal top displacement and vertical load is shown in Figure 4.21b. It is nonlinear due to both geometrical and physical nonlinearities. The prescribed maximum is reached with a horizontal displacement

4.9 Large Displacements – Second-Order Analysis

(a)

(b)

Figure 4.22 Example 4.8. (a) Moments along column for different loading factors. (b) Strains along column in final stage.

of 𝑢𝑢 = 0.071 m. The vertical load has to be reduced for larger horizontal displacements to maintain equilibrium in the deformed configuration. The arc length method is mandatory to model this structural softening behaviour effect. A load value of 𝑃 = 2 MN cannot be reached with initial eccentricities 𝑒 > 0.032 m as the structural softening behaviour will start with lower loads. • The computed moments along the column are shown in Figure 4.22a for different loading factors. The finally computed top moment 𝑀𝑧=4.94 = 0.067 MNm corresponds to the prescribed eccentricity moment of 𝑒 𝑃. In the same way the computed bottom moment 𝑀𝑧=0.06 = 0.206 MNm corresponds to the prescribed plus deformation eccentricity moment (𝑒 + 𝑢𝑢 ) 𝑃. This exceeds the moment from firstorder theory by the factor of 3.2. Using a linear elastic material behaviour with 𝐸 = 33 000 MN∕m2 would lead to a stiffer behaviour with a moment 𝑀𝑧=0.06 = 0.172 MNm. • The maximum bottom bending moment connected to the ultimate structural load is considerably lower compared to the ultimate cross-sectional moment of 𝑀𝑢 ≈ 0.26 MNm (Example 4.1, Figure 4.4) indicating a structural instability. • Final strain distributions along the column for the left reinforcement and the right concrete edge are shown in Figure 4.22b. The materials limit strains – e. g. 𝜖𝑐𝑢1 = −0.0035 for concrete – are not reached although the system has reached its limit load. The cantilever column is the most important case for large displacement computations for RC beams. But the described approach may also be applied to other types of columns and to all kinds of plane multi-story frames. Considering certain effects like contributions of tensile strength or tension stiffening will increase the stiffness of the model. Tension stiffening may be regarded with a modification of the stress-strain relation for the reinforcement (Section 4.7). A value for the tensile strength is considered with the lower and upper compression zone coordinates 𝑧𝑐1 , 𝑧𝑐2 (Section 4.1.3.1) with an extended definition of the zero line 𝑧0 (Eqs. (4.19) and (4.20)).

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On the other hand, an effect like creep will increase the deformations and also the internal forces in the deformed configuration. But the load level should be lower as only a permanent or quasi-permanent loading will lead to creep. The approach for creep as is described in Section 4.5 can be combined with large displacements to investigate creep effects.

4.10 Dynamics Dynamic actions on structures may arise from walking pedestrians, vehicular traffic, rotating machines, wind, seismicity, impact and explosions. Decisive is a structure’s largest natural period compared to a characteristic time of an action. Such characteristic times may come from the step frequency of pedestrians, velocity of vehicles, rotational frequency of machines, frequency of gusts or ground motions, duration of impact and explosions. If the largest natural period of a structure is not considerably smaller than the characteristic action time a structure’s inertia comes into play. The manifold aspects of dynamics of civil engineering structures are treated by, e.g. Biggs (1964); Bachmann (1995); Krätzig and Niemann (1996). Basics of inertia for beams are given with Eqs. (4.49) and (4.57) which introduce the inertial mass 𝑚 per unit length of a beam and the inertial mass moment 𝛩. In case that the centre line of gravity coincides with the beam’s reference axis (Section 4.1.1) they are given by 𝑚 = 𝜚𝐴,

𝛩 = 𝜚𝐽

(4.203)

with the material specific mass 𝜚, the cross-sectional area 𝐴 and the second moment of area 𝐽. If the centre line of gravity does not coincide with the reference axis the definition of 𝛩 has to be modified. On the other hand, contributions connected with the inertial mass moment 𝛩 are often neglected as they are relatively small. Some care should be give to the definition of mass. ◀

Specific mass 𝜚 has to be distinguished from specific weight: a given specific weight has to be divided by earth acceleration.

Appropriate methods to solve dynamic problems are already laid down. The basic approach for a discretised system is given by Eq. (2.83) M ⋅ 𝝊̈ (𝑡) + f(𝑡) = p(𝑡)

(4.204)

with the clock time 𝑡 and element contributions from • • • •

element mass matrices M𝑒 according to Eq. (4.100), element nodal displacements 𝝊𝑒 according to Eqs. (4.78) or (4.92), element internal nodal forces f𝑒 according to Eq. (4.98), element nodal loads p𝑒 = t𝑒 + c𝑒 according to Eqs. (4.101) and (4.102)

in the case of beams. Nodal loads p(𝑡) are defined as a function of clock time 𝑡. The Eq. (4.204) forms a system of ordinary differential equations of second order in

4.10 Dynamics

time 𝑡. Thus, initial conditions have to be prescribed for the nodal displacements and velocities at a time 𝑡 = 0. Internal nodal forces are given by f(𝑡) = K ⋅ 𝝊(𝑡)

(4.205)

in the case of linear material behaviour and small displacements (Eq. (2.63)). This leads to M ⋅ 𝝊̈ (𝑡) + K ⋅ 𝝊(𝑡) = p(𝑡)

(4.206)

instead of Eq. (4.204). Two fundamental solution approaches are given • Modal decomposition • Direct integration in time Modal decomposition presupposes constant symmetric matrices M, K and at first neglects loading p(𝑡). An oscillation may occur with such a system due to initially prescribed displacements or velocities. The homogeneous system M ⋅ 𝝊̈ (𝑡) + K ⋅ 𝝊(𝑡) = 0

(4.207)

of ordinary differential equations of second order in time 𝑡 is solved by 𝝊 = 𝝃 sin 𝜔𝑡

(4.208)

with a constant vector 𝝃 and a constant number 𝜔. They will come out as eigenvector and circular natural frequency. Circular natural frequency and natural period are related by 𝑇=

2π 𝜔

(4.209)

A generalised eigenvalue problem is derived from Eqs. (4.207) and (4.208) with K ⋅ 𝝃 = 𝜔2 M ⋅ 𝝃

(4.210)

This has 𝑛 solutions 𝝃 𝑖 , 𝜔𝑖 , 𝑖 = 1, … , 𝑛 for a system with 𝑛 nodal degrees of freedom whereby the length of a vector 𝝃 𝑖 remains undetermined. For solution methods for the generalised eigenvalue problem, see Bathe (1996, 10., 11.). The eigenvectors 𝝃 𝑖 constitute a matrix 𝜩 used for transformations into the modal space. Firstly, the transformation ˜ 𝝊 = 𝜩 ⋅ 𝝊 is applied. Secondly, the transformations ˜ = 𝜩 T ⋅ M ⋅ 𝜩 each lead to a diagonal matrix. Thirdly, multiplying ˜ = 𝜩 T ⋅ K ⋅ 𝜩 and M K Eq. (4.206) from left with 𝜩 T decouples this set of equations into 𝑛 single degree of freedom systems of second differential order in time. This decoupling facilitates the solution. The short description outlines a modal decomposition. Only first aspects of the modal decomposition, which is a powerful tool for all types of structures, can be given. For a comprehensive presentation see, e.g. He and Fu (2001). A few basic items of linear structural dynamics remain to be added. The smallest circular natural frequency 𝜔1 is determined through the Rayleigh quotient √ T ⎛√ √ 𝝊 ⋅ K ⋅ 𝝊 ⎞ 2π (4.211) 𝜔1 = min ⎜ ⎟= 𝑇 T 1 𝝊 ⋅ M ⋅ 𝝊 ⎠ ⎝

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4 Structural Beams and Frames

out of a set of appropriate displacement vectors 𝝊. This also gives the largest natural period 𝑇1 which is relevant for the estimation of dynamic situations. In the case of a single degree of freedom system – a mass 𝑚 connected with a base through a spring with stiffness 𝑘 – this leads to the well-known relation √ √ 𝑘 𝑚 (4.212) , 𝑇 = 2π 𝜔= 𝑚 𝑘 In the case of multi-degree of freedom systems with 𝑛 > 1 the quantities 𝜔1 and 𝑇1 may be approximately determined with a 𝝊 resulting from a quasi-static analysis with dead loading. Finally, a relation is derived for the largest natural period of a single span hinged beam. The base is given with the homogeneous differential equation of beam bending (Eq. (4.52)) 𝑚 𝑤̈ + 𝐸𝐽 𝑤 ′′′′ = 0

(4.213)

with a bending stiffness 𝐸𝐽. This is solved by 𝑤(𝑥, 𝑡) = sin

π𝑥 sin 𝜔𝑡 𝐿

(4.214)

with a span 𝐿 and leads to 𝑤̈ =

π𝑥 𝜕2 𝑤 = −𝜔2 sin sin 𝜔𝑡 , 𝐿 𝜕𝑡2

𝑤 ′′′′ =

𝜕4 𝑤 π4 π𝑥 = 4 sin sin 𝜔𝑡 (4.215) 4 𝐿 𝜕𝑥 𝐿

Thus, boundary conditions 𝑤(0, 𝑡) = 𝑤(𝐿, 𝑡) = 0 and 𝐸𝐽 𝑤 ′′ (0, 𝑡) = 𝐸𝐽 𝑤 ′′ (𝐿, 𝑡) = 0 are fulfilled. Furthermore, combining Eqs. (4.213) and (4.215) results to √ π2 𝐸𝐽 (4.216) 𝜔= 2 𝑚 𝐿 and with Eq. (4.209) the longest natural period of a single span hinged beam √ 𝑚 2𝐿2 (4.217) 𝑇= π 𝐸𝐽 is obtained. This is a useful relation to derive reference values for the largest natural period of beams. The reciprocal 𝜈 = 1∕𝑇 gives the largest natural frequency, i.e. the number of cycles of a structure free oscillations per time unit. The application of the modal decomposition is restricted to linear or linearised problems. Such a restriction does not apply if the basic Eq. (4.204) is directly integrated in time. A well-known method for numerical integration in time is given with the Newmark method (Section 2.8.2). This may immediately be applied to Eq. (4.204) and is demonstrated with the following example. Example 4.9: Beam under Impact Load

We refer to Example 4.2 with the same geometry and boundary conditions. A sudden concentrated point load or impact is applied in mid-span, see Figure 4.23a. A linear elastic behaviour is assumed in a first approach to demonstrate basic characteristics of dynamic behaviour under impact. Following data are chosen:

4.10 Dynamics

• Young’s modulus is assumed with 𝐸 = 33 000 MN∕m2 and the specific weight with 25 kN∕m3 . With an earth acceleration 𝑔 ≈ 10 m∕s2 this leads to a specific mass 𝜚 = 0.025∕10 = 2.5 ⋅ 10−3 MNs2 ∕m4 and with the cross-sectional area 𝐴 = 0.2 ⋅ 0.4 m2 to a beam mass per length 𝑚 = 0.2 ⋅ 10−3 MNs2 ∕m2 . √ • The longest natural period is determined with 𝑇 = quency 𝜈 = 26 Hz.

2𝐿2

𝑚

π

𝐸𝐽

= 0.038 s and a fre-

The point load is characterised by an amplitude 𝑃0 and a time variation function 𝑃(𝑡) = 𝑃0 𝑓(𝑡)

(4.218)

The time function is chosen as a step function with limited duration 𝑓(𝑡) =

⎧

1 for 𝑡 ≤ 𝑡𝑑 ⎨0 𝑡 > 𝑡𝑑 ⎩

(4.219)

Thus, loading is characterised by the parameters 𝑃0 , 𝑡𝑑 . Values 𝑃0 = −0.07 MN and 𝑡𝑑 = 0.1 s are used for the linear elastic reference case. The spatial discretisation is performed with 𝑛𝐸 = 20 enhanced two-node Bernoulli beam elements (Section 4.3.2). The Newmark method (Section 2.8.2) is used for time integration with a time step Δ𝑡 = 0.001s. The computed time span is chosen with 0.06 s which has to be related to the longest natural period. Figure 4.23b shows the computed mid-span deflection during time with the following characteristics: • A cosine-shaped oscillation occurs. • The maximum deflection doubles with an absolute value of 0.0106 m compared to the deflection value 0.0053 m caused by a quasi-static loading 𝑃0 . In the same way maximum internal forces are doubled compared to the quasi-static case. Figure 4.24a shows the bending moments along the beam in certain time steps up to the time 0.02 s when the first mid-span maximum is reached. This shows the following characteristics:

(a)

(b)

Figure 4.23 Example 4.9. (a) System and loading. (b) Linear elastic mid-span deflection with time.

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4 Structural Beams and Frames

(a)

(b)

Figure 4.24 Example 4.9. Linear elastic. (a) Moments along beam until the first maximum deflection for sequence of time steps. (b) Related maximum deflection depending on related load duration time.

• Internal forces do not immediately follow the load due to inertial effects. • There is no longer a triangular course due to a wave propagation effect of moments. In the beginning bending moments are initiated in the impact point while the support areas are basically unaffected. Impact loads have a short duration. Thus, a parameter study is performed with varying load duration 𝑡𝑑 and constant load amplitude 𝑃0 = −0.07 MN. Figure 4.24b shows the computed maximum mid-span deflections related to the quasi-static deflection depending on 𝑡𝑑 related to the longest natural period. If we consider a very short load duration time, e.g. 𝑡𝑑 = 0.001 s, the beam gets only roughly 20% of the quasi-static moment. Or in other words, it may sustain five times the original load to have the same internal forces. This may be generalised. ◀

Very short loadings are compensated by inertia and only partially yield internal forces.

Finally, we consider the original reference case 𝑃0 = −0.07 MN and 𝑡𝑑 = 0.1 s but with RC cross-sections instead of linear elastic behaviour. Material properties and reinforcement are chosen as in Example 4.2 with an additional reinforcement 𝐴𝑠2 = 12.57 cm2 , 𝑑2 = 5 cm on the upper side and concrete tensile strength neglected. The computation leads to the following results: • Figure 4.25 shows the mid-span displacement during time. Due to the reduced stiffness the period of the oscillation increases, compare Figure 4.23b. Maximum mid-span displacement grows to 0.019 m, roughly a doubling occurs compared to the linear elastic case. • Figure 4.26a shows the bending moments along the beam up to the time 0.027 s when the first mid-span maximum is reached. The same moments occur with some time shift compared to the linear elastic case (Figure 4.24a). • Some normal forces arise for RC without normal force loading, see Figure 4.26b. It is caused by the beam movement in the longitudinal direction due to cracking restrained by beam inertia. The longitudinal movement effect is already discussed with Example 4.2.

4.10 Dynamics

Figure 4.25 Example 4.9. RC mid-span deflection with time.

(a)

(b)

Figure 4.26 Example 4.9. RC. (a) Moments along beam until first maximum displacement for sequence of time steps. (b) Normal forces along beam for sequence of time steps.

• The ultimate limit state is not reached for the given load with maximum absolute values of concrete strain 𝜖𝑐 ≈ −1‰ and 𝜖𝑠 ≈ 2‰. The characteristics of the beam response are to a large extent determined by the time variation function 𝑓(𝑡) (Eq. (4.218)). The direct integration in time allows for arbitrary characteristics. A harmonic type would potentially lead to a resonance if the excitation frequency is near the natural periods. This effect may occur both for linear and nonlinear behaviour. Another aspect of dynamics is damping. Damping leads to a dissipation of energy. Dissipation may be treated on the material level and on the system level. A first model of dissipation on the material level is given by viscosity (Section 3.2, Figure 3.9). An further example with the cyclic behaviour of steel (Figure 3.11b) with an incremental material description (Eq. (2.50)) leading to different tangential stiffness in the case of loading and unloading and thus to energy dissipation within cycles of stress-strain histories. Dissipation on the system level can be treated in analogy to a Kelvin–Voigt element (Figure 3.9a) which is extended with a term for mass inertia. This extension is

131

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4 Structural Beams and Frames

introduced in the system’s dynamic equilibrium conditions Eq. (4.206) M ⋅ 𝝊̈ (𝑡) + C ⋅ 𝝊(𝑡) ̇ + K ⋅ 𝝊(𝑡) = p(𝑡)

(4.220)

with a viscosity matrix C. For aspects determining C in the case of structural systems and solution methods see Bathe (1996, 9.3).

133

5 Strut-and-Tie Models Up to now, bars have been treated with a much longer span dimension compared to cross-section dimensions. This allows for relatively simple uniaxial material laws. Strut-and-tie models keep the uniaxial material description but allow us to model reinforced concrete plates with the same dimensions for span and height. This is embedded in the FEM framework in the following. Furthermore, the important limit theorems of plasticity are introduced with the concept of rigid plasticity.

5.1 Elastic Plate Solutions Beams are characterised insofar as height and width of their cross-section are small compared to their span. This allows applying the Bernoulli–Navier hypothesis: undeformed plane cross-sections remain plane during a deformation. This kinematic assumption is no longer valid for plates in a 2D space where height has the same dimension as span. An example is given with the single-span deep beam with an opening; see Figure 5.1a, as a special case of a plate. It is supported by a column on the lower lefthand side and fixed in a larger shear wall on the right-hand side. The loading is given by a distributed load and a concentrated load at the upper side. Even if linear elastic material behaviour is assumed, an analytic solution is not available for such a system. A numerical method has to be used instead for the

Figure 5.1 The deep beam system. Computational Structural Concrete – Theory and Applications, Second Edition. Ulrich Häussler-Combe. © 2023 Ernst & Sohn GmbH. Published 2023 by Ernst & Sohn GmbH.

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5 Strut-and-Tie Models

Figure 5.2 Deep beam principal stresses.

solution, and the finite element method is appropriate. The discretisation may be performed with the plane four-node quad element (Section 2.3). The displacement field has two degrees of freedom 𝑢, 𝑣 (Eq. (2.35)). Strains and stresses have the components (Eqs. (2.39) and (2.42)) ⎛ 𝜖𝑥 ⎞ 𝝐 = ⎜ 𝜖𝑦 ⎟ , ⎜ ⎟ ⎝𝛾𝑥𝑦 ⎠

⎛ 𝜎𝑥 ⎞ 𝝈 = ⎜ 𝜎𝑦 ⎟ ⎜ ⎟ ⎝𝜎𝑥𝑦 ⎠

(5.1)

The steps of the discretisation follow the general outline described in Section 2.6. The evaluation of integrals is performed with a Gauss integration (Section 2.7, Eq. (2.69)) and an integration order 2 × 2 with 𝑛𝑖 = 𝑛𝑗 = 1. With respect to the material, a linear elastic behaviour should be assumed in a first approach to assess such a problem. Plane strain or plane stress conditions (Eqs. (2.44) and (2.45)) can be assumed for a plate depending on the depth. With a depth of 𝑏 = 0.6 m, for the given example, a plane stress condition is more appropriate, but the differences do not have a major effect. After prescribing boundary conditions for 𝑢, 𝑣 and a loading, the solution can be computed in one step according to Eq. (2.13), and an equilibrium iteration is not necessary. Results for the problem described in Figure 5.1 are shown in Figure 5.2. They are given as principal stresses in each integration point of each element. A principal stress is characterised by two principal stress values 𝜎1 , 𝜎2 and an orientation angle 𝜑, which indicates the inclination of the principal stress component 𝜎1 against the global 𝑥-axis. The orientation of the stress component 𝜎2 is perpendicular. Due to their inclination, principal stresses 𝜎1 , 𝜎2 are locally in equilibrium to stresses 𝜎𝑥 , 𝜎𝑦 , 𝜎𝑥𝑦 . More details regarding principal values are given in Sections 6.2.3 and 8.1.2. The varying orientation of principal stresses leads to a flow of forces. As an example, Figure 5.2 indicates such a flow force. A compression arch arises from the point of the concentrated load to the lower left and right bearing. The lower left bearing, which may displace horizontally, is tied to the right end through a tension band. The

5.1 Elastic Plate Solutions

upper right-hand side of the deep beam has a tension band as it is fixed in the shear wall. Furthermore, the compression arch is disturbed by the opening, which leads to a secondary system of compression and tension bands. The rough outline of force flow from immediate inspection may be stated more precisely through a construction of principal stress trajectories. Let 𝑦(𝑥) be the function describing the course of a stress trajectory in the 𝑥, 𝑦-plane and 𝑥1 , 𝑦1 be an arbitrary starting point. Such a starting point should lie on a bounding edge. The fields of stress components 𝜎𝑥 (𝑥, 𝑦), 𝜎𝑦 (𝑥, 𝑦), 𝜎𝑥𝑦 (𝑥, 𝑦) have to be known. As discrete values are given with results in integration points, an interpolation has to be used to construct fields leading to results for every desired coordinate. Such a method is described in Example 5.1. With this prerequisite, the equation d𝑦 = 𝜑 = 𝑓(𝜎𝑥 (𝑥, 𝑦), 𝜎𝑦 (𝑥, 𝑦), 𝜎𝑥𝑦 (𝑥, 𝑦)) d𝑥

(5.2)

provides an ordinary differential equation of first order as an initial value problem, where the orientation angle 𝜑 of principal stresses is determined through Eq. (8.2). This problem can at least be solved numerically for 𝑦(𝑥). The inspection of a number of starting points on the boundary finally leads to a pattern of trajectories. This procedure is not yet consistent with finite quad element interpolation yielding a state of strain 𝜖𝑥 , 𝜖𝑦 , 𝛾𝑥𝑦 for every integration point of every finite element. These pointwise strain values may be extrapolated to every desired element position using Eq. (2.40). But the result is generally discontinuous along element boundaries, as only the continuity of displacements is required. As a consequence, stresses derived from strains will also be discontinuous. This impedes the determination of stress trajectories. Continuous stress fields can be derived with so-called mixed formulations. An example is given in the following. Example 5.1: Continuous Interpolation of Stress Fields with the Quad Element

Stress fields in a whole domain are generally discontinuous along element boundaries. A method to derive a continuous field from a discontinuous field is derived following Zienkiewicz and Taylor (1989, 12.6). We consider a given stress field 𝝈 determined from a computation with quad elements (Section 2.3). Another stress field 𝝈 is constructed according to the interpolation of a displacement field (Eq. (2.18)) leading to 𝝈 = N(r) ⋅ 𝝈𝑒

(5.3)

where the vector 𝝈𝑒 collects the stresses of all nodes belonging to an element 𝑒. ◀

Stress fields interpolated in the same way as displacement fields are continuous throughout a whole domain using an interpolation with quad elements.

135

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5 Strut-and-Tie Models

The question remains how to determine the nodal stress values 𝝈𝑒 . An approach is to minimise the difference between 𝝈 and 𝝈 in an integral sense. This leads to ∫ 𝛿𝝈

T

) ( 𝝈 − 𝝈 d𝑉 = 0

(5.4)

𝑉

with the test functions 𝛿𝝈 = N ⋅ 𝛿𝝈𝑒 (Eq. (2.57)) where 𝑉 indicates the whole domain, including a discretisation with finite elements. The integration is performed element by element and using Eq. (5.3) results in ∫ NT ⋅ N d𝑉 ⋅ 𝝈𝑒 = ∫ NT ⋅ 𝝈 d𝑉 𝑉𝑒

(5.5)

𝑉𝑒

as 𝛿𝝈𝑒 is arbitrary. The left-hand integral corresponds to a unit mass matrix (Eq. (2.582 )). The left and right-hand integrals are evaluated with a numerical integration procedure (Section 2.7). Furthermore, element contributions are assembled into a whole system, as is done with finite element computations. This yields a system of linear algebraic equations for the unknown nodal values 𝝈𝑒 . The solution costs for this system may be minimised with a lumped unit mass matrix (Zienkiewicz and Taylor 1989, (12.69)). Finally, the continuous stress field may be computed using Eq. (5.3). Actually, the continuous field is more accurate than the original stress field with respect to an exact solution. The elaborate procedure from above may be simplified for practical purposes. ◀

A manual sketch of curves along principal stress orientations underlaid by a diagram of principal stresses is generally sufficient to approximate trajectories in order to understand the force flow in a plate.

Some trajectories have to be chosen out of a more or less dense band such that they give a characteristic pattern of force flow or – more generally – a particular conceptual model (Section 2.1, Figure 2.1). This task often does not have a unique solution but is essential for strut-and-tie models.

5.2 Strut-and-Tie Modelling The survey of trajectories and a potential force flow are a basis for the design of reinforced concrete plates with strut-and-tie models. • Trajectories under compression are assigned to concrete struts. • Trajectories under tension are assigned to reinforcement ties. Trajectories or force flow itself may be derived by a reasonable estimation. However, trajectories are curved, but struts and ties should be straight by definition. Thus,

5.2 Strut-and-Tie Modelling

(a)

(b)

Figure 5.3 (a) Example truss system. (b) Compression field as a strut-and-tie model.

the geometries of the chosen characteristic trajectories are approximated by straight line segments. A rough approximation with only a few line segments is generally sufficient. This approach leads to a truss with • Compression members corresponding to struts • Tensile members corresponding to ties • Nodes connecting struts and ties, which altogether form a strut-and-tie model. A possible model for the example of Figure 5.1 is shown in Figure 5.3a (CEB-FIP 2008, 8.8). The determination of a member layout often has a margin of discretion. An alternative for the same system is given in CEB-FIP (2011, 6). The approach using trajectories as orientation (trajectory method) has the load path method as an alternative. A load path connects the points where loads are applied to a structure to its support points by straight line segments with minimised length. From a structural point of view, these line segments correspond to compression members or struts in most cases. A load path necessarily involves redirections with redirection forces required to ensure equilibrium. Such redirection forces are compressive, corresponding to struts, or tensile, corresponding to ties. Both the load path method and the trajectory method should lead to similar results for a given structural task. Both methods may be combined with criteria of total energy minimisation to optimise the member layout of a strut-and-tie modelling. The result of strut-and-tie modelling is used for reinforcement design. This leads to a new aspect in the model of modelling, as was discussed in Section 2.1. ◀

The layout of a strut-and-tie model to some extent influences the behaviour of the corresponding plate or the reality of interest, respectively – in contrast to the intention of usual modelling.

Plates like nearly all structures of civil engineering, initially behave linearly elastically, as is sketched in Section 5.1. This changes during crack formation of reinforced concrete plates: the force flow changes according to the design of rebars and their interaction with compression zones. The strut-and-tie model is ‘good’ if some extent

137

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5 Strut-and-Tie Models

of ‘agreement’ is given between compression fields of the reality of interest and the struts of the model. Further properties have to be assigned to a strut-and-tie model besides its member layout. Struts and ties are uniaxial structural elements that must be given a crosssection. While this is naturally provided for ties through rebar cross-sectional areas, the assignment of cross-sections to struts is artificial to some extent. Struts may be regarded as models of compression fields, as is exemplarily shown in Figure 5.3b with a characteristic oval shape and a fan-out and fan-in of stresses. A single strut obviously lends itself to a secondary strut-and-tie model. The strut outer oval may be replaced by a rectangle of equal area leading to a width. A cross-sectional area is determined in combination with the plate depth. This outlines an appropriate approach for the cross-section design of struts. Material properties have to be assigned in addition to geometric properties. They can be derived according to Section 3.1 for the uniaxial behaviour of concrete and according to Section 3.3 with respect to rebars. Tension stiffening effects with rebars may be treated as is described in Section 3.7. Connection of struts and ties in nodes is outlined in the following Section 5.5. Code provisions for strut-and-tie models are given in EN 1992-1-1 (2004, 5.6.4,6.5), CEB-FIP2 (2012, 7.3.6). Reductions of uniaxial concrete strength due to lateral tensile stresses – frequently occurring in biaxial stress fields but unseizable with a truss – have to be regarded in particular. The foregoing reasoning should lead to a strut-and-tie model for a given plate problem together with boundary conditions and loading. A more detailed description is given in CEB-FIP (2008, 8). Comprehensive collections of examples are provided with CEB-FIP (2011); Reineck (2002); Reineck and Novak (2010). Methods to determine forces in struts and ties and the load carrying capacity of the model are described in the following.

5.3 Solution Methods for Trusses A strut-and-tie model has to be analysed as a truss system. A plane truss with 𝑛𝐸 bars and 𝑛𝑁 nodes is considered. Every node has two degrees of freedom 𝑢, 𝑣. Some nodes must be supported and degrees of freedom constrained to keep the truss in position. This leads to a total number of 𝑛 degrees of freedom. The structural behaviour of trusses is again ruled by kinematic compatibility, equilibrium, and material behaviour. We start by considering kinematic compatibility or the kinematic assumption for a truss. A truss member 𝑒 is given by the end nodes 𝐼, 𝐽. It has an orientation angle 𝛼𝑒 from 𝐼 to 𝐽 and node displacements 𝑢̃ 𝐽𝑒 , 𝑢̃ 𝐼𝑒 in the longitudinal member direction; see Figure 5.4. The strain 𝜍 of member 𝑒 is defined as 𝜍𝑒 = 𝑢 ˜𝐽𝑒 − 𝑢 ˜𝐼𝑒

(5.6)

This differs from the conventional definition of strain but is more convenient for the following. Displacements in the longitudinal bar direction have to be transformed

5.3 Solution Methods for Trusses

(a)

(b)

Figure 5.4 Truss. (a) Member strain. (b) Member force and nodal forces.

into the global system leading to global displacements of nodes 𝐼, 𝐽 ⎛𝑢𝐼 ⎞ ⎛cos 𝛼𝑒 ⎞ ˜𝐼𝑒 , 𝝊𝐼 = ⎜ ⎟ = ⎜ ⎟𝑢 𝑣𝐼 sin 𝛼𝑒 ⎠ ⎝ ⎠ ⎝ resulting in 1) ( 𝑢 ˜𝐼𝑒 = cos 𝛼𝑒

⎛𝑢𝐽 ⎞ ⎛cos 𝛼𝑒 ⎞ ˜𝐽𝑒 𝝊𝐽 = ⎜ ⎟ = ⎜ ⎟𝑢 𝑣𝐽 sin 𝛼𝑒 ⎠ ⎝ ⎠ ⎝

) sin 𝛼𝑒 ⋅ 𝝊𝐼 ,

(5.7)

) sin 𝛼𝑒 ⋅ 𝝊𝐽

( 𝑢 ˜𝐽𝑒 = cos 𝛼𝑒

(5.8)

Combining Eqs. (5.6) and (5.8) yields the strain of member 𝑒 depending on global displacements ) ) ( ( (5.9) 𝜍 𝑒 = − cos 𝛼𝑒 − sin 𝛼𝑒 ⋅ 𝝊𝐼 + cos 𝛼𝑒 sin 𝛼𝑒 ⋅ 𝝊𝐽 Collection of all bars in a vector gives a form

⎛⋮⎞ ⎡ ⎜𝜍 𝑒 ⎟ = ⎢ ⎢⋯ ⎜ ⎟ ⎢ ⋮ ⎝ ⎠ ⎣

⋮

⋮

⋮

⋮

− cos 𝛼𝑒

− sin 𝛼𝑒

cos 𝛼𝑒

sin 𝛼𝑒

⋮

⋮

⋮

⋮

⋯

⎛⋮⎞ ⎜𝑢 ⎟ 𝐼 ⎜ ⎟ ⎤ ⎜ 𝑣𝐼 ⎟ ⎥ ⋯⎥ ⋅ ⎜ ⋮ ⎟ ⎥ ⎜ ⎟ 𝑢 ⎦ ⎜ 𝐽⎟ ⎜𝑣 ⎟ 𝐽 ⎜ ⎟ ⋮ ⎝ ⎠

(5.10)

or in matrix notation e = B⋅𝝊,

B ∈ ℝ𝑛𝐸 ×𝑛 , 𝝊 ∈ ℝ𝑛 , e ∈ ℝ𝑛𝐸

(5.11)

where 𝑑 in ℝ𝑑 indicates the dimension of matrices and vectors. The vector e collects all 𝑛𝐸 member strains and 𝝊 all 𝑛 nodal displacements. There is a similarity compared to Eq. (2.2) for strains within finite elements. 1) Multiply the first rows by cos and the second ones by sin and add them.

139

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5 Strut-and-Tie Models

Figure 5.5 Truss system types.

Equilibrium has to be considered next. A member 𝑒 has end nodes 𝐼, 𝐽 and a member force 𝑠𝑒 (tension positive, compression negative). Member 𝑒 contributes forces to both the nodes 𝐼, 𝐽 (Figure 5.4b) leading to global nodal forces ⎛− cos 𝛼𝑒 ⎞ ⎛𝑓 𝑒 ⎞ f𝐼𝑒 = ⎜ 𝐼𝑥 = 𝑠𝑒 ⎜ ⎟, 𝑒 ⎟ 𝑓𝐼𝑦 − sin 𝛼𝑒 ⎝ ⎠ ⎝ ⎠

⎛cos 𝛼𝑒 ⎞ ⎛𝑓 𝑒 ⎞ f𝐽𝑒 = ⎜ 𝐽𝑥 = 𝑠𝑒 ⎜ ⎟ 𝑒 ⎟ 𝑓 sin 𝛼𝑒 ⎝ ⎠ ⎝ 𝐽𝑦 ⎠

(5.12)

The local equilibrium condition for nodes 𝐼, 𝐽 with external nodal loads p𝐼 , p𝐽 requires ∑ 𝑒 ∑ 𝑒 f𝐼 = p𝐼 , f𝐽 = p𝐽 (5.13) 𝑒

𝑒

The global equilibrium condition is given through assembling all node contributions ⎡ ⋮ ⎢ ⎢⋯ − cos 𝛼𝑒 ⎢ ⎢⋯ − sin 𝛼𝑒 ⎢ ⋮ ⎢ ⎢ ⎢⋯ cos 𝛼𝑒 ⎢ sin 𝛼𝑒 ⎢⋯ ⎢ ⋮ ⎣ or in a matrix notation f =p,

⎤ ⎛ ⋮ ⎞ ⎥ ⎜𝑝 ⎟ ⋯⎥ 𝐼𝑥 ⎜ ⎟ ⎥ ⋯⎥ ⎛ ⋮ ⎞ ⎜ 𝑝𝐼𝑦 ⎟ ⎥ ⎜ 𝑒⎟ ⎜ ⎟ ⎥⋅ 𝑠 = ⋮ ⎥ ⎜ ⎟ ⎜ ⎟ ⋯⎥ ⎝ ⋮ ⎠ ⎜𝑝𝐽𝑥 ⎟ ⎥ ⎜𝑝 ⎟ ⋯⎥ 𝐽𝑦 ⎜ ⎟ ⎥ ⋮ ⎝ ⎠ ⎦

f, p ∈ ℝ𝑛

(5.14)

(5.15)

with f = BT ⋅ s ,

BT ∈ ℝ𝑛×𝑛𝐸 , s ∈ ℝ𝑛𝐸

(5.16)

where the vector s collects all 𝑛𝐸 member forces, f the nodal forces, and p all 𝑛 nodal loads. Here, we again see the transposed matrix B of Eq. (5.11). There is a similarity compared to Eq. (2.581 ) for internal nodal forces of finite elements. A case 𝑛𝐸 < 𝑛 indicates an statically under-determinate system or kinematic chain; see Figure 5.5, 𝑛𝐸 = 𝑛 a statically determinate and 𝑛𝐸 > 𝑛 a statically indeterminate system. Member forces for a statically determinate system are determined directly from Eq. (5.16) by s = B−1 ⋅ p in the case B is not singular, and the inverse B−1 exists.

5.3 Solution Methods for Trusses

A singular B indicates an dysfunctional system. Statically underdeterminate systems are possible for strut-and-tie models of plates, as stability is reached through the concrete body surrounding the struts and ties. Member forces may be determined regarding equilibrium of one node after another in a sequence of neighbouring nodes for such cases. The introduction of blind members should be considered to make an underdeterminate system determinate. Member forces calculation of indeterminate systems requires an implication of stiffness and kinematic compatibility. This is resolved in the following. Material behaviour has to be considered in the last step. To begin with, uniaxial linear elastic material behaviour is assumed (Eq. (2.43)). A quantity 𝐴𝑒 is used for the cross-sectional area of a member 𝑒, 𝐿𝑒 as its length, and 𝐸𝑒 as its Young’s modulus. Using 𝑠𝑒 = 𝐴𝑒 𝜎𝑒 with the stress 𝜎𝑒 of member 𝑒 and 𝜖𝑒 = 𝜍 𝑒 ∕𝐿𝑒 , where 𝜖𝑒 is the physical strain of member 𝑒, member forces are given by 𝑠𝑒 = 𝐶𝑒 𝜍𝑒 ,

𝐶𝑒 =

𝐸𝑒 𝐴𝑒 , 𝐿𝑒

𝑒 = 1, … , 𝑛𝐸

(5.17)

In matrix notation this can be written as s=C⋅e,

C ∈ ℝ𝑛𝐸 ×𝑛𝐸

(5.18)

where the material stiffness matrix C is diagonal with coefficients 𝐶𝑒 . The combination of Eqs. (5.16), (5.18), and (5.11) leads to K⋅𝝊=p,

K = BT ⋅ C ⋅ B ,

K ∈ ℝ𝑛×𝑛

(5.19)

with a constant symmetric stiffness matrix K. There is a similarity of Eq. (5.19) compared to Eq. (2.62) for the linear stiffness matrix of finite elements. This completes the linear elastic framework for trusses. Obviously, strong similarities to finite element approaches are given. Actually, a plane truss member corresponds to a twonode 2D bar element (Section 2.3) with a one-point integration. An application is demonstrated with the following example. Example 5.2: A Deep Beam with the Strut-and-Tie Model

We refer to the problem given in Figure 5.1 and the linear elastic principal stresses shown in Figure 5.2. This problem is also discussed in CEB-FIP (2008, 8.8). The chosen strut-and-tie model is shown in Figure 5.3a. Obviously, there is some effort to circumvent the central rectangular hole. The resulting total load has been distributed to upper nodes according to the loading scheme of Figure 5.1. The system has 𝑛𝐸 = 36 bars and 21 nodes; 7 nodal degrees of freedom are restrained by boundary conditions leading to 𝑛 = 2 ⋅ 21 − 7 = 35. Thus, we ‘nearly’ have a statically determinate system. Internal forces are influenced by the stiffness of the members to a small extent. A preliminary computation reveals a distinction between struts (→ compressive) and ties (→ tensile). The following reference values for Young’s modulus and crosssectional area are chosen for the ties: 𝐸𝑠 = 200 000 MN∕m2 , 𝐴𝑠 = 10 ⋅ 10−3 m2 ;

141

142

5 Strut-and-Tie Models

(a)

(b)

Figure 5.6 Example 5.2. (a) Member stresses [MN∕m2 ]. (b) Proposed reinforcement scheme (CEB-FIP 2008, 8.8).

and for the struts: 𝐸𝑐 = 30 000 MN∕m2 , 𝐴𝑐 = 0.12 m2 . The last value corresponds to a deep beam depth 𝑏 = 0.6 m and an assumed strut width of 0.2 m. Computed stresses of struts and ties are shown in Figure 5.6a. Negative values correspond to struts and positive values correspond to ties. According to each individual stress value, the reference value of the cross-sectional area may be increased or reduced for the corresponding member to gain some target stress. Target stress values may be derived from strength values in codes, see, e.g. EN 1992-1-1 (2004, 6.5). A reinforcement scheme as proposed in CEB-FIP (2008, Figure 8.49) is shown in Figure 5.6b. Single vertical ties of the model are split as stirrups with a small spacing. The width of cracks occurring with ties may be estimated according to Appendix C. An iteration has to be considered in the case of statically indeterminate systems as a revaluation of cross-sectional areas of rebars influences the stiffness of ties and insofar the results of a statically indeterminate calculation. On the other hand, assumptions about the stiffness of struts are basically uncertain. Thus, statically determinate systems are preferred for strut-and-tie models. A nonlinear material behaviour of trusses is considered in the following. Actually, struts and ties have limited bearing capacities. In a first approach, this may be described with an uniaxial elasto-plastic material law (Section 3.3) applied to both the reinforcement and the compressive concrete. The linear elastic material law (Eq. (5.17)) changes into the incremental form 𝑠̇ 𝑒 = 𝐶𝑇𝑒 𝜍̇ 𝑒 ,

𝑒 = 1, … , 𝑛𝐸

(5.20)

or in matrix notation ṡ = C𝑇 ⋅ ė

(5.21)

with a diagonal tangential material stiffness matrix C𝑇 . The coefficients 𝐶𝑇𝑒 are determined according to Section 3.3, whereby the material model of elasto-plasticity is also applied to uniaxial concrete behaviour with adapted material parameters. The combination of Eqs. (5.11), (5.16), and (5.21) leads to rates of nodal forces ḟ = K𝑇 ⋅ 𝝊̇

(5.22)

5.3 Solution Methods for Trusses

with a tangential system stiffness in analogy to Eq. (5.19) K𝑇 = BT ⋅ C𝑇 ⋅ B ,

K𝑇 ∈ ℝ𝑛×𝑛

(5.23)

One has to distinguish an upper index T for transposition of a matrix and a lower index 𝑇 for a tangential stiffness matrix. Equilibrium has a condition r(𝝊) = p − f(𝝊) = 0

(5.24)

in analogy to Eq. (2.70). The solution may be determined along the outline for the incrementally iterative scheme as is described in Section 2.8.2. The application of strut-and-tie models is not restricted to deep beams. Every plane structure that does not behave according to the Bernoulli–Navier hypothesis of plane deformed cross-sections lends itself to strut-and-tie models. This is demonstrated with the following corbel example in connection with an elastoplastic analysis. Example 5.3: Corbel Geometry with an Elasto-Plastic Strut-and-Tie Model

Corbel geometry, loading, and strut-and-tie model are shown in Figure 5.7. The model has 𝑛𝐸 = 14 members and 𝑛 = 12 degrees of freedom, i.e. the system is twofold statically indeterminate. A provisional first calculation has to be performed to distinguish between tensile and compressive members. Basing upon this calculation member properties are chosen as follows: • Tensile members 1–7 are chosen as elasto-plastic ties with a Young’s modulus 𝐸 = 200 000 MN∕m2 , an initial yield strength 𝑓𝑦𝑘 = 500 MN∕m2 , and a reference cross-sectional area 𝐴𝑠,ref = 10 cm2 = 10 ⋅ 10−4 m2 . The cross-sectional area of each particular member is modified by a factor that is also shown in Figure 5.7b. • Compressive members 8–14 are chosen as struts with a Young’s modulus 𝐸 = 30 000 MN∕m2 . Concrete is assumed as elasto-plastic in this context, with an initial yield limit 𝑓𝑦𝑘 = 40 MN∕m2 . The assumed value is relatively large. Code provisions restrict allowable concrete stresses to some extent; see, e.g. EN 1992-1-1 (2004, 6.5). The cross-sectional area of all struts is assumed with a width ℎ = 0.05 m and a corbel depth of 𝑏 = 0.5 m leading to 𝐴𝑐 = 0.025 m2 .

(a)

(b)

Figure 5.7 Corbel example 5.3. (a) System. (b) Strut-and-tie model.

143

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5 Strut-and-Tie Models

(a)

(b)

Figure 5.8 Example 5.3. (a) Load displacement curve. (b) Member stresses [MN∕m2 ].

• The entity of all struts in the model Figure 5.7a corresponds to a single oval strut (Figure 5.3b). The redirection forces within this larger oval strut are assigned to the intermediate horizontal and vertical ties or stirrups. • A slight hardening (Section 3.3) is assumed for the yielding of the reinforcing steel and the concrete to avoid singularities of the tangential stiffness. An approach for system building and solution is given by Eqs. (5.22)–(5.24). The finite element method with 2D-bar elements in a plane (Section 2.3), a material law according to Section 3.3, and system building according to Section 4.4 may be applied alternatively, which leads to the same results. Loading is applied with prescription of the vertical displacement of the load bearing node. The solution is determined incrementally iterative with the Newton–Raphson approach (Section 2.8.2) and arc length control (Appendix A.4). The computed load displacement curve is shown in Figure 5.8a. The deformation behaviour can be described as follows: • A linear elastic behaviour is given initially up to reaching the first yielding of a tie. • A strongly decreased stiffness occurs with the first tie yielding, but some more loading can be applied. • The ultimate limit state is reached with the yielding of other ties in the statically indeterminate system, such that the system becomes ‘nearly kinematic’. The system does not become ‘really kinematic’ with the hardening assumption. Without hardening, the tangential system stiffness would become singular, and a unique solution can no longer be determined. The computed member stresses in the final state are shown in Figure 5.8b. The yield limit is reached in most ties and is slightly exceeded due to hardening. Struts are not critical in this particular case but are not far away from reaching the nominal compressive yield limit. An inclusion of limited strength with elasto-plasticity allows for a direct determination of ultimate loads of strut-and-tie models. The computed deformation behaviour has to be judged with some caution due to the limited ability of struts to model the behaviour of compression fields. The actual deformations of a plate are generally much smaller.

5.4 Rigid Plastic Truss Models

5.4 Rigid Plastic Truss Models Strut-and-tie models offer a simple way to determine the relation between loading, on the one hand, and reinforcement stresses and estimated demands of concrete on the other. The ability of strut-and-tie models to determine the deformation behaviour of real structures is limited, as the material laws (Eqs. (5.17) and (5.21)) are principally limited in their capability to capture in particular the deformation behaviour of compressions fields. There is no easy way to overcome this drawback, and we accept it as a characteristic property of strut-and-tie models. However, this opens up simplifying the description of the material behaviour. The problem of elasto-plasticity for trusses is reformulated. The conditions for kinematic compatibility according to Eq. (5.11) (𝑛𝐸 equations) and equilibrium according to Eq. (5.16) (𝑛 equations) are kept. But the material behaviour is described with • A limit state or strength condition |𝑠𝑒 | ≤ 𝑠𝑒𝑢 ,

𝑠𝑒𝑢 = 𝑓𝑒𝑦 𝐴𝑒 , 𝑒 = 1, … , 𝑛𝐸

(5.25)

with signed member forces 𝑠𝑒 , unsigned bearing capacities 𝑠𝑒𝑢 , yield strength 𝑓𝑒𝑦 , and cross-sectional areas 𝐴𝑒 of members. • An assumption about strains or flow rule |𝜍𝑒 |

⎧

= 0 for |𝑠𝑒 | < 𝑠𝑒𝑢

⎨> 0 ⎩

|𝑠𝑒 | = 𝑠𝑒𝑢

,

𝑒 = 1, … , 𝑛𝐸 .

(5.26)

with the member strain according to Eq. (5.6). • A dissipation condition or Kuhn–Tucker conditions 𝑠𝑒 𝜍𝑒 ≥ 0 ,

s⋅e≥0,

s, e ∈ ℝ𝑛𝐸

(5.27)

Elastic deformations are no longer considered in this rigid plastic approach. This is convenient with respect to the uncertain estimation of elastic stiffness properties of struts. The balance of equations versus unknowns for a truss with 𝑛𝐸 members and 𝑛 nodal degrees of freedom shows the following: on the one hand, there are 𝑛𝐸 member forces s, 𝑛 nodal displacements u, and 𝑛𝐸 bar strains e as unknowns; on the other hand, there are 2 ⋅ 𝑛𝐸 + 𝑛 equations Eqs. (5.11), (5.16), and (5.26) plus 2 ⋅ 𝑛𝐸 constraint equations Eqs. (5.25) and (5.27). The solution to this problem is not straightforward. It is provided by methods of optimisation and linear programming (Luenberger 1984). We restrict ourselves to a loading type p = 𝜆 p0 ,

𝜆>0

(5.28)

with a unit load p0 , which is fixed, and a loading factor 𝜆. The basic solution idea concerns a maximisation problem 𝜆 → max

(5.29)

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5 Strut-and-Tie Models

while fulfilling equilibrium and the strength condition, i.e. constraints BT ⋅ s = 𝜆 p0

and |𝑠𝑒 | ≤ 𝑠𝑒𝑢 , 𝑒 = 1, … , 𝑛𝐸

(5.30)

The quantities B, s𝑢 , p0 are known, and s, 𝜆 are unknown in this formulation. Equation (5.29) can be interpreted as the maximisation of loading. Furthermore, linear programming theory states a so-called dual problem associated with the maximisation problem. This is a minimisation problem s𝑢 ⋅ e → min

(5.31)

where e collects all |𝜍𝑒 | and s𝑢 all 𝑠𝑒𝑢 . The notation ∙ should be interpreted as operator on ∙ in this context. The minimisation has constraints B ⋅ 𝝊 = e and p0 ⋅ 𝝊 = 1

(5.32)

i.e. kinematic compatibility (Eq. (5.11)) and bounding of displacements. The quantities B, s𝑢 , p0 are known and e, 𝝊 are unknown in this formulation. Equation (5.31) can be interpreted as minimisation of energy. Linear programming theory finally states that solutions s⋆ , 𝜆 ⋆ exist for the maximisation, and e⋆ , 𝝊⋆ exist for the minimisation. The target functions (Eqs. (5.29) and (5.31)) have the same optimal value ⋆

𝜆 ⋆ = s𝑢 ⋅ e

⋆

(5.33)

with 𝜆 ⋆ > 0. It can be shown that the solutions fulfil the conditions 𝑒𝑒⋆ = 0 if |𝑠𝑒⋆ | < 𝑠𝑒𝑢

and s⋆ ⋅ e⋆ > 0

(5.34)

This is summarised as follows: ◀

The solutions s⋆ , 𝜆 ⋆ , e⋆ , 𝝊⋆ of the associated optimisation problems for load maximisation and dissipated energy minimisation are also the solutions of the rigid plastic problem.

Furthermore, the rigid plastic problem only is also fulfilled by s⋆ , 𝜆 ⋆ , 𝛽e⋆ , 𝛽𝝊⋆ with an arbitrary scalar 𝛽 > 0. But this may violate the additional constraint p0 ⋅ 𝝊 = 1, and the minimisation problem is no longer solved. With respect to the rigid plastic problem, the absolute values of the deformations are indeterminate. Only the relations between the deformation components are determined to some extent. The rigid plastic problem and the corresponding elasto-plastic problem – with the same parameters but additional elasticity before yielding without hardening – share basically the same solution. The vectors s = s⋆ , p = 𝜆 ⋆ p0 fulfil Eqs. (5.15) and (5.16) and furthermore Eqs. (5.22) and (5.24) are fulfilled for arbitrary values of Young’s modulus for members. Thus, the elastic stiffness does not influence the limit state for forces and loads. But elasticity influences the deformations of the elasto-plastic problem due to elastic deformations prior to plastic deformations.

5.5 Application Aspects

The solution method for the optimisation problems still needs to be discussed. Primarily we are interested in loads 𝜆 p0 and member bar forces s. Thus, we have to solve the maximisation problem defined in Eqs. (5.29) and (5.30). The solution is simple for statically determinate systems with 𝑛 = 𝑛𝐸 and a square and non-singular B. The maximum value of 𝜆 can be easily found from s = 𝜆 B−1 ⋅ p0 (Eq. (5.30)) and the condition |𝑠𝑒 | ≤ 𝑠𝑒𝑢 , 𝑒 = 1, … , 𝑛𝐸 . The loading factor 𝜆 has to be scaled such that the latter condition in not violated. The member force with the largest amount is decisive. For small statically indeterminate systems, a solution may be found by inspection or trial and error with assumptions about member forces and applying them to Eq. (5.30) to reach square submatrices of B. Larger problems require systematic methods of linear programming, e.g. the simplex method or general methods of optimisation. The simplex method is a standard method to solve linear constrained optimisation problems in all disciplines of science, technology, and economics; for a detailed description see Luenberger (1984). The analogy of the optimisation problems with the rigid plastic structural problem leads to the limit theorems of plasticity: 1. Any equilibrium state that fulfils the limit state condition, see Eqs. (5.29) and (5.30), gives a lower bound for the loading factor 𝜆. 2. The work s𝑢 ⋅ e of load bearing capacities on kinematically admissible deformations with a constraint p0 ⋅ 𝝊 = 1; see Eqs. (5.31) and (5.32), yields an upper bound value for the loading factor 𝜆. An unconstrained formulation of this theorem is given by s𝑢 ⋅ e = 𝜆 p0 ⋅ 𝝊 or internal work = external work leading to the upper bound of loading factor 𝜆 →

s𝑢 ⋅ e p0 ⋅ 𝝊

(5.35)

whereby magnitudes of e, 𝝊 are arbitrary but proportional. These theorems are generally stated as postulates and are widely used for the analysis of all types of structures whereby variables are adapted. Actually, their exact formulation and proof corresponds to linear programming theory.

5.5 Application Aspects The focus of strut-and-tie models is on the ultimate limit loads of reinforced concrete structures. This primarily depends on the strength of the ties and struts. Furthermore, the strength of members depends on cross-sectional areas and the material strength of materials. While both can be reliably – within usual scatter – determined for the reinforcement, some uncertainties remain for concrete struts.

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5 Strut-and-Tie Models

(a)

(b)

(c)

Figure 5.9 Nodes. (a) Compression. (b) Compression–tension. (c) Reinforcement redirection.

The estimation of the compressive strength of concrete struts is insofar difficult as pure uniaxial compression states are generally not given in plates. In many cases, a principal compression is combined with a more or less pronounced lateral principal tension. An example can be seen in the introductory example and Figure 5.2, where the diagonal compression fields with their major principal compression have a minor principal tension. Such a lateral tension reduces the compressive strength. Thus, the strength for struts should be reduced compared to the uniaxial compressive strength. Recommendations are given in EN 1992-1-1 (2004, 6.5.2), CEB-FIP2 (2012, 7.3.6.2). A conservative rule to determine the width of struts can be derived from the inspection of nodes connecting the struts and ties. Some hints are given in the following. Nodes are an essential part of strut-and-tie models. This term gets a specific meaning in this context, differently from a node in a finite element discretisation. Nodes connect struts and ties and permit an exchange of forces. Characteristic types are given by: • A compression node without ties; see Figure 5.9a. • A compression tension node with one tie; see Figure 5.9b. • A compression tension node with two ties; see Figure 5.9c. A characteristic width is given for each of the three types, e.g. the length of a bearing or the length of a diagonal connecting the radius of the curvature of a rebar. This length determines the initial width of struts. The width may increase due to a fanout of a compression field (Figure 5.3b), but this may be neglected for a conservative estimation of a strut width. The above-mentioned characteristic width also defines a space for the node itself. Multi-axial stress states are given within this space to reach a balance of forces from ties, struts, and supporting forces, respectively. Failure of concrete may occur due to this stress state, and a check has to be performed. The uniaxial compressive strength 𝑓𝑐 serves as basic value. A reference value for the strength of nodes is derived from 𝜎𝑅 = 𝑘𝑐 𝜈𝑓𝑐

(5.36)

with a general reduction factor 𝜈 ruled by the material ductility, which is different for different concrete grades, and a factor 𝑘𝑐 with respect to the type of node. Recom-

5.5 Application Aspects

mendations for the choice of 𝜈, 𝑘𝑐 are given in EN 1992-1-1 (2004, 6.5.4), CEB-FIP2 (2012, 7.3.6.4). Nominal uniaxial stresses 𝜎𝑐,𝑖 derived from given strut forces 𝐹𝑐,𝑖 (Figure 5.9) should fulfil |𝜎𝑐,𝑖 | < 𝜎𝑅 , whereby a safety margin has to be regarded in practical design. Deformations of strut-and-tie models have been a minor aspect up to now because the determination of the ultimate limit loads was the major concern. It was demonstrated that the ultimate limit load is reached in a process where certain members reach their yielding strength in a sequence. Yielding of subsequent members in this sequence requires sufficient plastic deformations of prior members. Thus, sufficient plastic deformations without failure in the sense of collapse are required to reach the ultimate limit load. The whole structure will become unstable or not reach the potential ultimate limit load in the case of premature failure of a single member due to a lack of ductility. ◀

The application of the strut-and-tie model to a real structure requires a sufficient ductility for the whole structure.

Ductility in this context means that all rebars and compression fields and the areas where they exchange forces (→ nodes) may have large enough strains with a nearly constant level of stresses. While this is generally fulfilled for the reinforcement, it is not necessarily given for concrete. Thus, concrete should predominantly not be utilised to its limits, and a minimum orthogonal reinforcement net is required irrespective of tensile stresses determined with models.

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151

6 Multi-Axial Concrete Behaviour Although uniaxial material laws are extremely powerful for many problems of structural analysis, they reach their limits with respect to plates beyond strut-and-tie models and further with slabs and shells (Figure 9.1a). An immediate transition from uniaxial to triaxial models is appropriate, and biaxial behaviour is derived as a special case from 3D. With common construction materials – concrete, steel, wood, glass – the behaviour of concrete reveals to be most complex. Thus, the following chapter summarises the essential aspects of multi-axial concrete behaviour within the context of continuum mechanics and fracture mechanics and connects them to numerical methods for structural analysis.

6.1 Basics 6.1.1 Continua and Scales Continuum mechanics provides a framework to describe the behaviour of solids. A basic assumption within this framework is homogeneity, i.e. material parameters are constant or continuous in space. Two scales are considered to describe concrete with continuum mechanics: mesoscale and macroscale. ◀

The mesoscale distinguishes the cement matrix, aggregates, and the interfacial transition zone (ITZ). Each of these material phases is regarded as a homogeneous solid with its own material law and its own material parameters.

The ITZ generally forms the weakest link; see Liao et al. (2004) for its characterisation. Spatial dimensions of the mesoscale have to be chosen on the scale of [mm] to resolve the phases spatially. A continuous displacement is assumed along contact surfaces of different phases in the case of deformations. This leads to discontinuities of strains along contact surfaces due to different stiffness of phases. The internal geometric characteristics of a concrete specimen composed of phases are random as a matter of principle; e.g. the size, shape, position, and orientation of aggregates are random.

Computational Structural Concrete – Theory and Applications, Second Edition. Ulrich Häussler-Combe. © 2023 Ernst & Sohn GmbH. Published 2023 by Ernst & Sohn GmbH.

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6 Multi-Axial Concrete Behaviour

(a)

(b)

Figure 6.1 (a) Concrete at mesoscale. (b) Microcracking.

Thus, two samples chosen out of a collection of specimens of the same geometry and with the same material parameters of phases will show different reactions under the same imposed action. The variation of reactions depends on the size of the specimen relative to the size of the largest aggregates and the type of the action. The variation of reactions tends to become smaller for larger specimen sizes. If they are considered to be negligible for the relevant types of action, the specimen size constitutes a representative volume element(RVE). With concrete the spatial dimensions of an RVE are on a scale of a few times of the largest dimension of the aggregates or on a scale of [cm], respectively. This corresponds to the macroscale. ◀

Concrete, cement matrix, aggregates, and the interfacial transition zone are homogenised within the macroscale with a single type of material law and a corresponding set of material parameters.

The material parameters of the homogenised continuum may be determined analytically from the material parameters of the phases by mixture theories (Mori and Tanaka 1973; Wriggers and Moftah 2006). But such approaches are limited with respect to the mesoscale randomness of concrete. As an alternative, numerical multi-scale methods may provide macroscale parameters derived from numerical mesoscale calculations considering particular actions and phase properties. Finally, parameters of macroscale material laws may be directly chosen according to the experimental behaviour of specimens with at least RVE size using parameter identification and calibration methods. Initiation and propagation of cracks may occur in a continuum with limited tensile strength. Thus, a continuum may become a discontinuum along curves or surfaces indicating a crack geometry. Cracks are macrocracks within the framework of continuum mechanics with a defined boundary. Microcracks (Figure 6.1b), on the other hand, with a spatial extent of 𝜇m are modelled inherently through material laws. The phase between micro and macrocracking (Figure 3.3a) is treated with the cohesive crack model; see Section 7.1. This model assigns surface tractions along fictitious crack boundaries to model mesoscopic branching and crack bridging within the framework of continuum mechanics under the assumption of homogeneous materials.

6.1 Basics

Each scale requires an adapted grade of spatial discretisation. Mesoscale discretisation of an RVE needs a number of finite elements for each piece of aggregate and a corresponding finite element resolution for the cement matrix. The number of degrees of freedom for a mesoscale discretisation of larger structures currently exceeds computation capabilities. Thus, numerical calculations of larger structures are performed on a macroscale using homogenised material laws.

6.1.2 Characteristics of Concrete Behaviour The macroscale is used in the following to describe the stress–strain behaviour of concrete. Due to current state of knowledge the only reliable way to access the characteristics of concrete behaviour is given through observation in general and experiments performed under controlled conditions in particular. Sizes of specimens at least have to correspond to an RVE to yield reproducible results. The following major characteristics of concrete behaviour – within the short-term time scale (Section 3.1) – are derived under this premise: • The tensile strength is low compared to the compressive strength. Uniaxial concrete strength was already discussed in Section 3.1; see Figures 3.1 and 3.3b. Uniaxial compressive strength 𝑓𝑐 is roughly 10–15 times higher than the uniaxial tensile strength 𝑓𝑐𝑡 for normal graded concrete (EN 1992-1-1 2004, Table 3.1). Such a relation is still valid for multi-axial tensile states compared to multi-axial compressive states; see Section 6.4.3. • The compressive strength increases with multi-axial stress states. Specimens exposed to large (hydrostatic) pressures sustain a compaction but will not fail under ideal conditions. A lower subordinated lateral confining compression supports a material compressive bearing capacity in a principal orientation. Furthermore, strains corresponding to states of multi-axial strength increase compared to the uniaxial case. Modelling of multi-axial strength is described in Section 6.4.3. A number of experimental data are available, see, e.g. Kupfer et al. (1969); Gerstle et al. (1980); Hampel et al. (2009). • A reduced stiffness and permanent strains occur with unloading. The microstructure of a solid material is changed with the application of loads. Two major effects can be distinguished: sliding within crystalline structures and the formation of microcracks (Figure 6.1b). These effects lead to a different response with the removal of loads. Sliding leads to permanent strains, which can be modelled with plasticity (Section 6.5). Microcracking leads to a reduced material stiffness compared to the virgin unloaded state and can be modelled with damage(Section 6.6). • A softening behaviour develops in the post-peak regime. The post-peak regime comprises material behaviour after the material strength has been utilised. Due to ongoing microcracking stresses decrease while strains increase, whereby the solid material coherence is still preserved. Uniaxial softening behaviour is discussed in Section 3.1.

153

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Nevertheless, in the end, every solid material will lose its coherence and will ultimately fail. Ductile materials fail with relatively large strains and brittle materials with relatively small strains. Concrete is classified in between as quasi-brittle. Experimental investigations are given with, e.g. van Mier (1986); van Mier et al. (1997). • A volume dilatancy occurs under compression in the post-peak regime. High compressive stress levels during post-peak regimes lead to a disaggregation of inhomogeneous materials like concrete (Figure 6.1a). Stiff aggregates dislocate in the relatively soft cement matrix. This is observed as dilatancy on the macroscale. Experimental data are given in Kupfer et al. (1969). • Anisotropy is induced through loading. The effect of load-induced anisotropy becomes particularly evident with the formation of cracks (Figure 3.3a). Less or no stress can be transmitted across a developing crack, but at least compressive stresses can still be transmitted in the direction of cracks. Hence, macroscopic relations between stresses and strains depend on the orientation within a material whereby the evolution of material orientations depend on the loading history. • The activation of crack energy leads to size effects. Crack bands (Figures 3.3a and 3.4) basically have the same width in small concrete structures and large concrete structures with the same concrete. Thus, for the same crack pattern crack bands occupy relatively more volume in small structures, and crack energy or energy dissipation is relatively larger in small structures. As a consequence, small concrete structures behave in a more ductile manner regarding their load displacement relations compared to large concrete structures. This issue is strongly related to fracture mechanics and cohesive cracks (Section 7.1). These observations indicate the complexity of the mechanical behaviour of concrete compared to other common building materials. Further characteristic effects of concrete behaviour arise with the long-term time scale on the one hand, and the highly dynamical time scale on the other.

6.2 Continuum Mechanics 6.2.1 Displacements and Strains Items of continuum mechanics are introduced in the following, as they are necessary for the formulation of material laws for solids like concrete. A comprehensive description of continuum mechanics is given in, e.g. Malvern (1969); Ziegler (1977). We consider a solid body in space. Space is measured in a 3D Cartesian coordinate system in the following, if not stated otherwise. A space point x has a vector ( )T of coordinates 𝑥1 𝑥2 𝑥3 . The indication of coordinate directions is changed compared to in Section 2.3 and is performed with integer numbers to facilitate the notation. A body occupies an area of space in a configuration; see Figure 6.2a. This configuration changes with time 𝑡 due to a loading history. A material point is iden-

6.2 Continuum Mechanics

(a)

(b)

Figure 6.2 (a) Body in reference and deformed configuration. (b) Infinitesimal stress tetrahedron.

tified by the space point X that it occupies in a reference configuration at a time 𝑡0 and X = x for 𝑡 = 𝑡0 . Displacements are defined with u = x − X and have a vector of components u = ( )T 𝑢1 𝑢3 𝑢3 . They become zero in the reference configuration per definition. Displacements are considered to be small in the following, if not stated otherwise. The notion of small is that they have a magnitude of millimetres, while the body has dimensions in the magnitude of metres. Strain is derived from displacements with small strain components 𝜖𝑖𝑗 =

𝜕𝑢𝑗 1 𝜕𝑢𝑖 + ( ) = 𝜖𝑗𝑖 , 2 𝜕𝑥𝑗 𝜕𝑥𝑖

𝑖, 𝑗 = 1, … , 3

(6.1)

Strain components involve two directions: a displacement direction 𝑢𝑖 , 𝑢𝑗 and a reference direction 𝑥𝑖 , 𝑥𝑗 . Hence, small strain components form a symmetric tensor of the second order. The components are arranged in a matrix

𝝐𝑀

⎡𝜖11 ⎢ = ⎢𝜖12 ⎢ 𝜖 ⎣ 13

𝜖12 𝜖22 𝜖23

𝜖13 ⎤ ⎥ 𝜖23 ⎥ ⎥ 𝜖33 ⎦

(6.2)

or in a vector in Voigt notation for strains ( 𝝐 = 𝜖11 ( = 𝜖11

𝜖22

𝜖33

𝛾23

𝜖22

𝜖33

2𝜖23

𝛾13 2𝜖13

𝛾12

)T )T

(6.3)

2𝜖12

utilizing the symmetry of shear strain components. The Voigt notation also introduces the engineering notation for shear strain components 𝛾𝑖𝑗 = 2𝜖𝑖𝑗 , 𝑖 ≠ 𝑗, which simplifies the writing of stress–strain relations. The components of a second-order tensor obey particular transformation laws based upon transformation laws for vectors (→ first-order tensors) in the case of a transformation of the underlying coordinate system; see Appendix D.

155

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6 Multi-Axial Concrete Behaviour

6.2.2 Stresses and Material Laws Strain leads to stress. It is at first defined through Cauchy stress ϖ∖𝑏𝑜𝑙𝑑𝑠𝑦𝑚𝑏𝑜𝑙{∖𝑠𝑖𝑔𝑚𝑎}ϖ with reference to an infinitesimal tetrahedron at a position x with a tetrahedron base )T ( area d𝐴, a base normal n = 𝑛1 𝑛2 𝑛3 exposing a force vector df and a traction vector t = df∕ d𝐴. The force df is equilibrated by three force vectors df𝑖 on each coordinate plane 𝑖; see Figure 6.2b. Each of these forces corresponds to a traction vector t𝑖 , and each traction has three Cauchy stress components 𝜎𝑖𝑗 . Thus, nine stress components are given with 𝑖, 𝑗 = 1, … , 3. The first index denotes the coordinate plane normal and the second index the global direction. Thus, stress components form a tensor of second order. The components are arranged in a matrix as

𝝈𝑀

⎡𝜎11 ⎢ = ⎢𝜎21 ⎢ 𝜎 ⎣ 31

𝜎12 𝜎22 𝜎32

𝜎13 ⎤ ⎥ 𝜎23 ⎥ ⎥ 𝜎33 ⎦

(6.4)

In order to have equilibrium a relation t = 𝝈𝑀 ⋅ n

(6.5)

must hold with the traction vector t on the tetrahedron base and the matrix vector product of the matrix 𝝈𝑀 and the tetrahedron base normal n. Considering the equilibrium of an infinitesimal cube it can be shown that 𝜎𝑖𝑗 = 𝜎𝑗𝑖

(6.6)

and stress components also form a symmetric tensor of second order. In a similar way to strain components, Cauchy stress components can be arranged in a vector with Voigt notation for stresses ( 𝝈 = 𝜎11

𝜎22

𝜎33

𝜎23

𝜎13

𝜎12

)T

(6.7)

utilizing symmetry of Cauchy stress. Pressure is derived from stress components as 1 𝑝 = − (𝜎11 + 𝜎22 + 𝜎33 ) 3

(6.8)

and the deviatoric stress 𝝈′ as 2

1

⎛ 𝜎11 − 𝜎22 − ⎛𝜎′ ⎞ ⎛𝜎11 + 𝑝⎞ 3 3 11 ⎜2 1 ⎜ ′ ⎟ ⎜ ⎟ ⎜ 𝜎22 − 𝜎11 − ⎜𝜎22 ⎟ ⎜𝜎22 + 𝑝⎟ 3 3 ⎜ ′ ⎟ ⎜ ⎜ ⎟ 𝜎 + 𝑝 𝜎 1 2 33 ⎜ =⎜ 𝝈′ = ⎜ 33 ⎟ = 3 𝜎33 − 3 𝜎11 − ′ ⎟ ⎜ 𝜎 𝜎 ⎜ 23 ⎟ ⎜ 23 ⎟ 𝜎23 ⎜ ′ ⎟ ⎜ 𝜎13 ⎟ ⎜𝜎13 ⎜ 𝜎13 ⎜ ′ ⎟ ⎜ ⎟ ⎜ 𝜎12 𝜎12 ⎝ ⎠ ⎝ ⎠ 𝜎12 ⎝

1

𝜎33 ⎞ ⎟ 1 𝜎33 ⎟ 3 ⎟ 1 𝜎22 ⎟ 3 ⎟ ⎟ ⎟ ⎟ ⎠ 3

(6.9)

6.2 Continuum Mechanics

Similar relations may be derived for strain. Incremental changes of strain and stress with progressing time 𝑡 are defined with the strain and stress rate 𝝐̇ =

Δ𝝐 𝜕𝝐 , = lim 𝜕𝑡 Δ𝑡→0 Δ𝑡

𝝈̇ =

Δ𝝈 𝜕𝝈 = lim Δ𝑡→0 Δ𝑡 𝜕𝑡

(6.10)

The Cauchy stress and small strain are conjugate with respect to energy as 𝑢̇ = 𝝈 ⋅ 𝝐̇

(6.11)

denotes the rate of internal energy per volume for material points within a continuous body or rate of specific internal energy. This includes both recoverable and dissipated contributions (Section 6.9). Other measures for stresses and conjugated strains, which are relevant for large deformations, are discussed in Belytschko et al. (2000); Malvern (1969). The short-term stress–strain relation of a specific material may generally be described through an incremental material law 𝝈̇ = C𝑇 ⋅ 𝝐̇

(6.12)

whereby a material specific tangential material stiffness matrix C𝑇 depends on the loading history. As stress and strain are each a tensor of second order, the material stiffness has to be a tensor of fourth order. The Voigt notation of stresses and strains allows us to arrange the components of C𝑇 in a matrix. If the components of C𝑇 are constant with C𝑇 = C, Eq. (6.12) may be integrated in time 𝑡 to give 𝝈=C⋅𝝐

(6.13)

i.e. a linear material law.

6.2.3 Coordinate Transformations and Principal States Transformations of coordinate systems clarify characteristics of material states and material behaviour. A further Cartesian coordinate system is regarded in addition to the basic global system of the same origin but different orientation or different directions of axes, respectively. The relation between these systems is ruled by three rotation angles, see Figure 6.3a, and a transformation matrix Q depending on the rotation angles. The matrix Q is orthogonal: Q−1 = QT . Values of the components of stress and strain differ in the two systems. A strain ˜ 𝝐 = Q⋅𝝐

(6.14)

is given in the transformed system corresponding to the strain 𝝐 in the initial system. A stress 𝝈 in the initial system is related by ˜ 𝝈 = QT ⋅ 𝝈

(6.15)

to the stress 𝝈 ˜ in the transformed system. In the case of 2D states with one rotation ˜ ⋅˜ angle, the exact form of Q is given in Appendix D. With 𝝈 ˜=C 𝝐 , the material stiffness transforms according to ˜ = Q ⋅ C ⋅ QT , C

˜⋅Q C = QT ⋅ C

(6.16)

157

158

6 Multi-Axial Concrete Behaviour

(a)

(b)

Figure 6.3 (a) Rotation of coordinate system. (b) Principal stress space.

This completes the transformation rules. Principal stress states are based on a matrix eigenvalue problem 𝝈𝑀 ⋅ n = 𝜎 n

(6.17)

derived from Eq. (6.5). Due to the symmetry of 𝝈𝑀 , this problem is solved through real eigenpairs n𝑖 , 𝜎𝑖 with 𝑖 = 1, 2, 3. The three orthogonal unit vectors n𝑖 form principal directions and a distinguished Cartesian coordinate system or principal system, respectively. A transformation matrix Q relating the initial system and the principal system is constructed from the direction cosines of the principal directions and the initial coordinate axes. The transformed stress state has principal stresses 𝜎˜11 = 𝜎1 , 𝜎˜22 = 𝜎2 , 𝜎˜33 = 𝜎3 and vanishing shear stress components 𝜎˜12 = 𝜎˜13 = 𝜎˜23 = 0. Principal stresses and principal directions form the principal stress state, which is still described by six values: the three principal stress values and three values for the angles specifying the principal directions. In the case of 2D states with only one rotation angle 𝜑, the angle determining principal directions is given by Eq. (8.2) with 𝜎𝑥 = 𝜎11 , 𝜎𝑦 = 𝜎22 , 𝜎𝑥𝑦 = 𝜎12 , and the principal stress values are given by Eq. (8.1). The same approach as for stresses is valid for strains with 𝝐𝑀 ⋅ n = 𝜖 n

(6.18)

leading to a principal strain state with principal strains 𝜖1 , 𝜖2 , 𝜖3 and their own corresponding principal directions n1 , n2 , n3 . Tensorial shear strain components 𝜖13 , 𝜖23 , 𝜖12 must be used for 𝝐 𝑀 in Eq. (6.18). The principal directions of strain do not necessarily have to coincide with the principal directions of stress. This depends on the type of the material law that connects strain and stress and is discussed in Section 6.3. Principal states are distinguished as all strains ˜ 𝝐 transformed according to Eq. (6.14) have the same principal values and principal directions for arbitrary rotations Q. The same holds for all stresses 𝝈 ˜ transformed according to Eq. (6.15). ◀

Principal states are independent of the choice of coordinate systems. Hence, principal states indicate a physical point of view of stress and strain.

6.3 Isotropy, Linearity, and Orthotropy

A material point’s state of stress and strain is preferably characterised by the corresponding principal values and directions. Invariants are used as an alternative formulation for principal values. In the same way as principal values they do not change with a change of coordinate system. In the case of stress, its first invariant is 𝐼1 , and the second and third invariants of its deviator are predominantly used. They are defined as 𝐼1 = 𝜎11 + 𝜎22 + 𝜎33 = 𝜎1 + 𝜎2 + 𝜎3 ] 1[ (𝜎11 − 𝜎𝑚 )2 + (𝜎22 − 𝜎𝑚 )2 + (𝜎33 − 𝜎𝑚 )2 + 𝜎23 𝜎32 + 𝜎13 𝜎31 + 𝜎12 𝜎21 𝐽2 = 2 ] 1[ = (𝜎1 − 𝜎2 )2 + (𝜎2 − 𝜎3 )2 + (𝜎3 − 𝜎1 )2 6 𝐽3 = (𝜎1 − 𝜎𝑚 )(𝜎2 − 𝜎𝑚 )(𝜎3 − 𝜎𝑚 ) (6.19) with 𝜎𝑚 = (𝜎1 + 𝜎2 + 𝜎3 )∕3. Due to their inherent structure, an important property of invariants is given by their exchange isotropy. An exchange is given with any 𝜎𝑖 ↔ 𝜎𝑗 , 𝑖 ≠ 𝑗 and their arbitrary successive combinations. ◀

An invariant does not change its value in the case of exchange of arguments 𝜎1 , 𝜎2 , 𝜎3 .

The first invariant corresponds to hydrostatic pressure (Eq. (6.8)) with 𝑝 = −𝐼1 ∕3. A demonstrative interpretation of 𝐽2 , 𝐽3 is shown in Section 6.4.2 with Eqs. (6.44) and (6.46). A detailed description of stress and strain invariants is given in Malvern (1969, 3.3).

6.3 Isotropy, Linearity, and Orthotropy 6.3.1 Isotropy and Linear Elasticity The isotropy of materials is connected with various aspects of directions. The following types of directions have to be distinguished: • Action directions: principal directions of given strains • Reaction directions: principal directions of response stresses • Material directions: a coordinate system spanned by four material points in the reference configuration. An isotropic material behaves in the same way in every action direction. This can be stated more precisely as follows: • The reaction directions coincide with the action directions. • Principal stress values are independent of action directions, i.e. principal stress values do not change with a change of action directions relative to material directions but unchanged principal strain values.

159

160

6 Multi-Axial Concrete Behaviour

This descriptive definition of isotropy also has to be formulated mathematically. A given strain or action 𝝐 with an associated response 𝝈 = C ⋅ 𝝐 is considered. The action directions are rotated into other arbitrary directions ˜ 𝝐 with ˜ 𝝐 = Q ⋅ 𝝐 according to Eq. (6.14). Isotropy requires that the rotated associated response 𝝈 ˜ has the same material law, i.e. 𝝈 ˜ = C⋅˜ 𝝐 , while rotating 𝝈 according to Eq. (6.15). With 𝝈 = QT ⋅ 𝝈 ˜ = QT ⋅ C ⋅ ˜ 𝝐 = QT ⋅ C ⋅ Q ⋅ 𝝐 a requirement C = QT ⋅ C ⋅ Q

(6.20)

follows for arbitrary rotations Q. This imposes restrictions for C. A general form ⎡𝐶11 ⎢ ⎢𝐶21 ⎢ ⎢𝐶31 C=⎢ ⎢𝐶41 ⎢ ⎢𝐶51 ⎢ 𝐶 ⎣ 61

𝐶12

𝐶13

𝐶14

𝐶15

𝐶22

𝐶23

𝐶24

𝐶25

𝐶32

𝐶33

𝐶34

𝐶35

𝐶42

𝐶43

𝐶44

𝐶45

𝐶52

𝐶53

𝐶54

𝐶55

𝐶62

𝐶63

𝐶64

𝐶65

𝐶16 ⎤ ⎥ 𝐶26 ⎥ ⎥ 𝐶36 ⎥ ⎥ 𝐶46 ⎥ ⎥ 𝐶56 ⎥ ⎥ 𝐶66 ⎦

(6.21)

is assumed. It can be shown that for arbitrary rotations Q, the isotropy requirement (Eq. (6.20)) can only be fulfilled with a form ⎡𝐶1 ⎢ ⎢𝐶2 ⎢ ⎢𝐶2 C=⎢ ⎢0 ⎢ ⎢0 ⎢ 0 ⎣

𝐶2

𝐶2

0

0

𝐶1

𝐶2

0

0

𝐶2

𝐶1

0

0

1

(𝐶1 − 𝐶2 )

0

0

0

0

0

0

0

0

2

⎤ ⎥ ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 1 (𝐶1 − 𝐶2 ) ⎦ 2 0 0

0 1 2

(𝐶1 − 𝐶2 ) 0

(6.22)

This has further implications. ◀

Stresses and strains have the same principal directions if they are related by an isotropic material stiffness matrix ϖ∖mathbf {C }ϖ like Eq. (6.22).

Assuming constant values for 𝐶1 , 𝐶2 , Eq. (6.22) corresponds to the triaxial isotropic linear elastic material law ⎡ 𝐸(1−𝜈) ⎢ (1+𝜈)(1−2𝜈) ⎢ 𝐸𝜈 ⎢ (1+𝜈)(1−2𝜈) ⎢ ⎢ 𝐸𝜈 E = ⎢ (1+𝜈)(1−2𝜈) ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ 0 ⎣

𝐸𝜈

𝐸𝜈

(1+𝜈)(1−2𝜈)

(1+𝜈)(1−2𝜈)

𝐸(1−𝜈)

𝐸𝜈

(1+𝜈)(1−2𝜈)

(1+𝜈)(1−2𝜈)

0

0

0

0

0

0

𝐸𝜈

𝐸(1−𝜈)

(1+𝜈)(1−2𝜈)

(1+𝜈)(1−2𝜈)

0

0

0

0

0

0

0

0

𝐸 2(1+𝜈)

0 𝐸 2(1+𝜈)

0

0 ⎤ ⎥ ⎥ 0 ⎥ ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 𝐸 ⎥ 2(1+𝜈) ⎦

(6.23)

6.3 Isotropy, Linearity, and Orthotropy

with Young’s modulus 𝐸 and Poisson’s ratio 𝜈, whereby the lower three diagonal components fulfil the requirement prescribed by the lower three diagonal entries of Eq. (6.22).

6.3.2 Orthotropy Particular choices of Q in Eq. (6.20) applied to Eq. (6.21) lead to different types of anisotropy. An orthotropic material is given if Eq. (6.20) is fulfilled for three rotations Q each with an angle 180° around each coordinate axis (→ spatial mirroring). This can be fulfilled with a form ⎡𝐶11 ⎢ ⎢𝐶21 ⎢ ⎢𝐶31 C=⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ 0 ⎣

𝐶12

𝐶13

0

0

𝐶22

𝐶23

0

0

𝐶32

𝐶33

0

0

0

0

𝐶44

0

0

0

0

𝐶55

0

0

0

0

0 ⎤ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 𝐶66 ⎦

(6.24)

with 12 non-zero independent components Malvern (1969, (6.2.24)). This form requires the coincidence of coordinate axes and the material symmetry directions. The material symmetry directions are given through distinguished material directions forming a Cartesian coordinate system in the case of an orthotropic material. The orthotropic form Eq. (6.24) allows us to model a different material behaviour for each of the material symmetry directions. The same material may be rigidly rotated in space, leading to a deviation of the material symmetry directions from the spatial coordinate directions. The application of Eq. (6.20) to Eq. (6.24) then leads to a fully occupied matrix C. Formal differences arise compared to Eq. (6.24), but 12 independent material parameters remain. The relations between principal directions weaken in the case of orthotropy. ◀

Stresses and strains have the same principal directions if they coincide with the material symmetry directions, otherwise they have not.

Thermodynamic postulates about a non-negative product 𝝐 ⋅ 𝝈 = 𝝐 ⋅ C ⋅ 𝝐 or a positive definite matrix C (Section 6.9) require the symmetry of the material stiffness matrix. In the case of orthotropy with material symmetry directions aligned to coordinate axes, this yields ⎡𝐶11 ⎢ ⎢𝐶12 ⎢ ⎢𝐶13 C=⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ 0 ⎣

𝐶12

𝐶13

0

0

𝐶22

𝐶23

0

0

𝐶23

𝐶33

0

0

0

0

𝐶44

0

0

0

0

𝐶55

0

0

0

0

0 ⎤ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 𝐶66 ⎦

(6.25)

161

162

6 Multi-Axial Concrete Behaviour

with nine non-zero independent components. A compliance form (Eq. (2.51)) with physical evidence – material coefficients are directly taken from experimental results – is given by 1

−

⎡ 𝐸 ⎢ 1 ⎢− 𝜈21 ⎢ 𝐸2 ⎢ 𝜈 ⎢− 31 ⎢ 𝐸 D=⎢ 3 ⎢ 0 ⎢ ⎢ ⎢ 0 ⎢ ⎢ 0 ⎣

𝜈12 𝐸1 1

𝐸2

−

−

𝜈13

−

𝜈23

𝐸1 𝐸2

0

0

0

0

0

0

𝜈32

1

𝐸3

𝐸3

0

0

0

0

0

0

0

0

1 𝐺4

0 1 𝐺5

0

0⎤ ⎥ 0⎥ ⎥ ⎥ 0⎥ ⎥ ⎥ 0⎥ ⎥ ⎥ 0⎥ ⎥ 1 ⎥ 𝐺6 ⎦

(6.26)

whereby 𝐸𝑖 indicates uniaxial stiffness, 𝐺𝑖 the pure shear stiffness, and 𝜈𝑖𝑗 is a measure for the deformation in the 𝑖-direction caused by a stress in the 𝑗-direction. Relations 𝜈12 ∕𝐸1 = 𝜈21 ∕𝐸2 , …, and so on must hold to preserve symmetry.

6.3.3 Plane Stress and Strain With respect to the isotropic case, simplified forms can be derived from Eq. (6.23) for plane strain or plane stress, whereby 𝛾23 = 𝛾13 = 0 in both cases. Plane strain is determined with 𝜖33 = 0 and a direct subset of Eq. (6.23) may be used for ⎛𝜎11 ⎞ ⎜𝜎 ⎟ = 𝐸 22 ⎜ ⎟ 1+𝜈 ⎝𝜎12 ⎠

𝜈

1−𝜈

⎡ ⎢ 1−2𝜈 ⎢ 𝜈 ⎢ 1−2𝜈 ⎢ 0 ⎣

1−2𝜈 1−𝜈 1−2𝜈

0

The lateral stress is given by 𝜎33 =

0⎤ 𝜖 ⎥ ⎛ 11 ⎞ ⎥ ⎜ ⎟ 0 ⎥ ⋅ ⎜ 𝜖22 ⎟ 1⎥ ⎝𝛾12 ⎠ 2⎦ 𝐸𝜈 (1+𝜈)(1−2𝜈)

(6.27)

(𝜖11 + 𝜖22 ). Plane stress is determined

with 𝜎33 = 0. Using this condition with Eqs. (6.13) and (6.23) leads to ⎛𝜎11 ⎞ ⎜𝜎 ⎟ = 𝐸 22 ⎜ ⎟ 1 − 𝜈2 𝜎 ⎝ 12 ⎠

⎡1 ⎢ ⎢𝜈 ⎢ 0 ⎣

𝜈 1 0

0 ⎤ ⎛ 𝜖11 ⎞ ⎥ 0 ⎥ ⋅ ⎜ 𝜖22 ⎟ ⎟ 1−𝜈 ⎥ ⎜ 𝛾 2 ⎦ ⎝ 12 ⎠

(6.28)

𝜈

The lateral strain is given by 𝜖33 = − (𝜎11 + 𝜎22 ). 𝐸 Plane stress is also treated for the symmetric orthotropic case where material symmetry directions coincide with the coordinate directions. With 𝜎33 = 0 and 𝛾23 = 𝛾13 = 0, the compliance form Eq. (6.26) leads to 1

⎡ ⎛ 𝜖11 ⎞ ⎢ 𝐸1 ⎜ 𝜖 ⎟ = ⎢− 𝜈21 22 ⎜ ⎟ ⎢ 𝐸2 ⎢ 𝛾 0 ⎝ 12 ⎠ ⎣

−

𝜈12 𝐸1 1

𝐸2

0

0⎤ ⎥ ⎛𝜎11 ⎞ ⋅⎜ ⎟ 0⎥ ⎥ ⎜𝜎22 ⎟ 1 ⎥ ⎝𝜎12 ⎠ 𝐺6 ⎦

(6.29)

6.3 Isotropy, Linearity, and Orthotropy

The inversion yields the stiffness form ⎛𝜎11 ⎞ 1 ⎜𝜎 ⎟ = 22 ⎜ ⎟ 1 − 𝜈12 𝜈21 ⎝𝜎12 ⎠

⎡ 𝐸1 ⎢ ⎢𝜈21 𝐸1 ⎢ 0 ⎣

𝜈12 𝐸2 𝐸2 0

⎤ ⎛ 𝜖11 ⎞ ⎥ ⎜ ⎟ 0 ⎥ ⋅ 𝜖22 ⎥ ⎜ ⎟ (1 − 𝜈12 𝜈21 )𝐺 ⎝𝛾12 ⎠ ⎦ 0

(6.30)

with 𝐺 = 𝐺6 (Chen and Saleeb 1994, (6.110)). A notation 𝜈1 = 𝜈12 (→ deformation in 1-direction caused by a lateral stress in 2-direction) and 𝜈2 = 𝜈21 (→ deformation in 2-direction caused by a lateral stress in 1-direction) is used in the following. Requiring symmetry 𝜈1 𝐸2 = 𝜈2 𝐸1 and using a modified Poisson’s ratio √ √ 𝜈 𝐸1 𝐸2 𝜈 𝐸1 𝐸2 𝜈2 𝐸1 𝜈1 𝐸2 𝜈= √ = √ → 𝜈2 = , 𝜈1 = (6.31) 𝐸1 𝐸2 𝐸 𝐸 𝐸 𝐸 1 2

1 2

finally leads to √ 𝜈 𝐸1 𝐸2

𝐸

⎡ 1 ⎛𝜎11 ⎞ ⎢ 1−𝜈2 ⎜𝜎 ⎟ = ⎢ 𝜈√𝐸1 𝐸2 22 ⎜ ⎟ ⎢ 1−𝜈2 ⎢ 𝜎 ⎝ 12 ⎠ ⎣ 0

1−𝜈

0⎤ 𝜖 ⎥ ⎛ 11 ⎞ ⎥ ⎜ ⎟ 0 ⎥ ⋅ ⎜ 𝜖22 ⎟ ⎥ 𝛾 𝐺 ⎦ ⎝ 12 ⎠

2

𝐸2 1−𝜈

2

0

(6.32)

where 𝐸1 , 𝐸2 ≥ 0 is assumed. The material coefficients 𝐸1 , 𝐸2 , 𝜈 may be relatively easily approximated from experiments with two sets of uniaxial stress–strain data determined from orthogonal directions. A problem might remain with the experimental determination of the shear modulus 𝐺. The invariance of shear flexibility is postulated to circumvent experimental inconvenience. With respect to Eq. (6.20), it is assumed that the shear coefficient 𝐺 in Eq. (6.32) does not change for all plane transformations. A plane transformation has a rotation angle 𝜑, and the rotation matrix is defined by ⎡ cos2 𝜑 ⎢ Q = ⎢ sin2 𝜑 ⎢ 2 cos 𝜑 sin 𝜑 ⎣

2

sin 𝜑 cos2

𝜑

−2 cos 𝜑 sin 𝜑

− cos 𝜑 sin 𝜑 ⎤ ⎥ cos 𝜑 sin 𝜑 ⎥ 2 ⎥ cos2 𝜑 − sin 𝜑 ⎦

(6.33)

leading to a rotation of stresses and strains around the plane normal with an arbitrary angle 𝜑 (Appendix D). The requirement of shear invariance can be fulfilled with a form ⎡ 𝐸1 2 ⎢ √1−𝜈 ⎢ C = ⎢ 𝜈 𝐸12𝐸2 ⎢ 1−𝜈 ⎢ 0 ⎣

√ 𝜈 𝐸1 𝐸2 1−𝜈

2

𝐸2 1−𝜈

0

2

⎤ ⎥ ⎥ ⎥ 0 ⎥ √ 𝐸1 +𝐸2 −2𝜈 𝐸1 𝐸2 ⎥ 2 4(1−𝜈 ) ⎦ 0

(6.34)

The other coefficients corresponding to 𝐶11 , 𝐶12 , 𝐶22 are not necessarily invariant with respect to coordinate transformations. This matrix includes Eq. (6.28) for the isotropic plane stress state as special case with 𝐸 = 𝐸1 = 𝐸2 , 𝜈 = 𝜈.

163

164

6 Multi-Axial Concrete Behaviour

Finally, the uniaxial stress–strain relations are derived for the two material symmetry directions. The 1-direction has 𝜎22 = 0, leading to 𝜎11 = 𝐸1 𝜖11 ,

𝜈 𝜖22 = − √ 𝜎11 𝐸1 𝐸2

(6.35)

With given values for 𝜎11 , 𝜖11 , 𝜖22 from a test 𝐴, two equations are given for three unknowns 𝐸1 , 𝐸2 , 𝜈. The 2-direction has 𝜎11 = 0 leading to 𝜎22 = 𝐸2 𝜖22 ,

𝜈 𝜖11 = − √ 𝜎22 𝐸1 𝐸2

(6.36)

With given values for 𝜎22 , 𝜖22 , 𝜖11 from a test 𝐵 – strains from test 𝐴 and test 𝐵 are not the same – two further equations are given for the unknown material parameters. Thus, the set of four equations (6.35) and (6.36) is overdetermined for the unknowns 𝐸1 , 𝐸2 , 𝜈. A best fit may be found with a least squares approach (Appendix E, Eqs. (E.2)–(E.6)).

6.4 Nonlinear Material Behaviour 6.4.1 Tangential Stiffness With respect to the uniaxial case, nonlinear material behaviour of concrete is characterised by a decreasing tangential material stiffness (Figure 3.1). This property is transferred to the multi-axial case. A general formulation of nonlinear material behaviour is given by Eq. (6.12) 𝝈̇ = C𝑇 ⋅ 𝝐̇

(6.37)

With respect to the initial behaviour of previously unloaded concrete, it can be assumed that it initially behaves as a linear elastic isotropic material. The initial tangential material stiffness matrix C𝑇 is given according to Eq. (6.23). Values for the initial Young’s modulus 𝐸𝑐 and the initial Poisson’s 𝜈 ratio depending on concrete grade are given by EN 1992-1-1 (2004, 3.1.3), CEB-FIP2 (2012, 5.1.7). But a tangential stiffness matrix C𝑇 is subject to change after the initial state and may depend on stress 𝝈, strain 𝝐 and internal state variables 𝜿 C𝑇 = C𝑇 (𝝈, 𝝐, 𝜿)

(6.38)

Internal state variables 𝜿 comprise a loading history. They are necessary, as an actual state 𝝈, 𝝐 may lead to different responses 𝝈̇ for different loading histories with a given 𝝐. ̇ Internal state variables require evolution laws 𝜿̇ = F(𝝈, 𝝐, 𝜿)

(6.39)

describing their rates depending on stress, strain, and values of internal state variables.

6.4 Nonlinear Material Behaviour

Aspects of isotropy and anisotropy as were described in Section 6.3 are also an issue for nonlinear behaviour. Isotropic nonlinear behaviour is characterised in the same way as was formulated in Section 6.3.1. The previous reasoning regarding 𝝈, 𝝐 in the same way applies to rates 𝝈, ̇ 𝝐, ̇ leading the same conclusion about the principal directions of the stress and strain rates and to the same restrictions for the coefficients of an isotropic tangential material stiffness C𝑇 ◀

A nonlinear isotropic material behaves in the same way in every action direction. Principal directions of stress increments coincide with principal directions of strain increments for a given material state. The principal stress increments have the same values for all principal strain directions with given strain increment values.

In analogy to Eq. (6.20), an isotropic tangential material stiffness matrix obeys a relation C𝑇 = QT ⋅ C𝑇 ⋅ Q

(6.40)

for arbitrary rotations Q. As a consequence, the tangential material stiffness matrix C𝑇 has to follow a form like Eq. (6.22), which allows only for two independent coefficients. Materials initially isotropic may become anisotropic in higher loading levels. In the case of concrete a load-induced anisotropy especially arises with cracking whereby the direction normal to a crack has a reduced capacity to transmit tensile stresses while stiffness and strength may remain unaffected in the direction of a crack (Figure 3.3a). Orthotropic forms may be used to model a load-induced anisotropy due to cracking. The tangential material stiffness matrix C𝑇 then has to obey to a form like Eq. (6.25) or Eq. (6.34) in the case of plane stress and shear isotropy. The respective matrix coefficients may depend on stress, strain, and loading history according to Eq. (6.38). The orthotropic tangential material flexibility D𝑇 for 3D states gets a form according to Eq. (6.26) with varying coefficients. Corresponding forms are derived in Section 7.4 within the framework of 2D smeared crack models (Section 3.5).

6.4.2 Principal Stress Space and Isotropic Strength Stress limit states mark the other end compared to initial states. They describe the strength of materials. For initially isotropic materials like concrete, such stress limit states are described by an isotropic strength condition 𝑔(𝐼1 , 𝐽2 , 𝐽3 ) = 0

(6.41)

using stress invariants 𝐼1 , 𝐽2 , 𝐽3 derived from principal stress values 𝜎1 , 𝜎2 , 𝜎3 by Eq. (6.19). Stress states with 𝑔(𝐼1 , 𝐽2 , 𝐽3 ) ≤ 0 are admissible; states 𝑔(𝐼1 , 𝐽2 , 𝐽3 ) > 0 cannot be sustained. This has some implications. • The orientation of principal stress directions relative to material directions (Section 6.3) has no influence on the strength condition.

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(a)

(b)

Figure 6.4 (a) Hydrostatic length and deviatoric plane. (b) Deviatoric length and Lode angle in the deviatoric plane.

• Furthermore, an exchange of principal stress values (Section 6.2.3) does not have any influence on the strength condition either. • Admissible forms for equivalent functions 𝑓(𝜎1 , 𝜎2 , 𝜎3 ) derived from 𝑔(𝐼1 , 𝐽2 , 𝐽3 ) by using Eq. (6.19) are constrained by the inherent structure of the invariants. We assume the admissible forms 𝑓(𝜎1 , 𝜎2 , 𝜎3 ) for the following. The stress–strain behaviour described by Eq. (6.37) is sometimes separated from the strength limit states described by 𝑓(𝜎1 , 𝜎2 , 𝜎3 ), and both are treated independently. To have a consistent material description the integration of stresses 𝝈̇ from Eq. (6.37) during the loading history driven by a time 𝑡 should not lead to stress states violating the strength condition Eq. (6.41). The elasto-plastic material models described in Section 6.5, the damage models described in Section 6.6, or the microplane models described in Section 6.8 combine stress–strain relations and strength in such a way that the consistency of the material description is ensured. An isotropic strength condition generally becomes active with one predominant principal stress component. Let us assume that 𝜎1 activates the tensile strength in the 1-direction. The respective material point is generally not considered to fail in every aspect. A utilisation of strength in the remaining directions is still allowed in the following loading history. Thus, e.g. a load-induced anisotropy – tensile strength is reached in one direction, while compressive strength is utilised in orthogonal directions – may be combined with an isotropic strength condition. ◀

Isotropic strength conditions basically do not exclude anisotropic stress–strain relations.

Such isotropic strength conditions 𝑓(𝜎1 , 𝜎2 , 𝜎3 ) are used for concrete and will be considered in the following. Principal stress values span a triaxial Cartesian coordinate system (→ principal stress space), and the corresponding stress state is given by a vector. This has the following significant elements, see also Figure 6.4:

6.4 Nonlinear Material Behaviour

( )T √ • The hydrostatic axis as a space diagonal with a direction n𝜉 = 1 1 1 ∕ 3. A direction is a vector of length 1 by definition. ( )T • The projection of a stress vector 𝝈 = 𝜎1 𝜎2 𝜎3 on the hydrostatic axis: ( )T √ √ 𝝃 = 𝜉 1 1 1 ∕ 3 with hydrostatic length 𝜉 = (𝜎1 + 𝜎2 + 𝜎3 )∕ 3. • The deviatoric plane with origin at 𝝃 and a normal n𝜉 . It is spanned by all vectors starting in 𝝃 with zero hydrostatic length 𝜉. ( )T • The projection of a stress vector 𝝈 = 𝜎1 𝜎2 𝜎3 on its deviatoric plane ⎛ 2𝜎1 − 𝜎2 − 𝜎3 ⎞ 1⎜ −𝜎 + 2𝜎2 − 𝜎3 ⎟ 3⎜ 1 ⎟ ⎝−𝜎1 − 𝜎2 + 2𝜎3 ⎠ with 𝜌1 + 𝜌2 + 𝜌3 = 0. 𝝆=𝝈−𝝃 =

(6.42)

( )T • The projection of the particular vector 𝝈 = 1 0 0 on the deviatoric plane ac) ( 1 1 T cording to Eq. (6.42): 𝝆1 = 2∕3 1 − − . It has a direction 𝝆1 = 2 2 ) ( √ 1 1 T 2∕3 1 − − called the Rendulic direction in the following. 2

2

An isotropic strength condition like Eq. (6.41) forms a strength surface in the principal stress space and defines triaxial strength. A stress vector and, in particular, a point on this surface can be described by Haigh–Westergaard coordinates with the following components: • The hydrostatic length 𝜉 is already introduced as length of the stress vector on the hydrostatic axis √ 𝐼1 1 (6.43) → 𝐼1 = 3 𝜉 𝜉 = √ (𝜎1 + 𝜎2 + 𝜎3 ) = √ 3 3 • The deviatoric length 𝜌 results from the length of the vector 𝝆 from Eq. (6.42). This leads to the second invariant of the stress deviator (Eq. (6.19)) √

𝜌2 (6.44) 2 • The Lode angle 𝜃 spans between the Rendulic direction and deviatoric direction 𝜌 = |𝝆| =

cos 𝜃 =

2𝐽2

→

𝐽2 =

1 𝝆 ⋅ 𝝆1 𝜌

A common alternative of this formulation is given by √ 3 3 𝐽3 3 cos 3𝜃 = 4 cos 𝜃 − 3 cos 𝜃 = √ 2 𝐽23

(6.45)

(6.46)

with the second and third invariants 𝐽2 , 𝐽3 of the stress deviator (Eq. (6.19)). Equation (6.46) yields one solution in the range 0° ≤ 𝜃 ≤ 60°. But this is not a restriction, as any interchanging of 𝜎1 , 𝜎2 , 𝜎3 is equivalent; see the remarks following Eq. (6.41).

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A convention 𝜎1 ≥ 𝜎2 ≥ 𝜎3

(6.47)

(signed!) is generally used without loss of generality with respect to isotropic strength. Thus, a state of stress is uniquely determined by 𝜉, 𝜌 and, finally, 𝜃 in the range 0 ≤ 𝜃 ≤ 60°.

6.4.3 Strength of Concrete The strength of solid materials is determined experimentally with specimens of at least the size of an representative volume element (RVE) (Section 6.1.1). Cylindrical specimens are often used for concrete. A typical experimental set-up is shown in Figure 6.5 with the triaxial cell. It allows applying longitudinal and radial pressure independently from another. The radial pressure is connected with a circumferential pressure of the same value to establish equilibrium. Both form the confining pressure. A first principal stress directly corresponds to the longitudinal pressure; the confining pressure leads to the identical second and third principal stress. A test is started with identical longitudinal and confining pressures. After that, the longitudinal pressure is changed until it reaches an extremal value corresponding to strength. Such a set-up has the following locations in the principal stress space: • The compressive meridian with 𝜎1 = 𝜎2 > 𝜎3 (signed): a cylindrical specimen with compression 𝜎3 < 0 in the longitudinal direction and circumferential confining pressure 𝜎1 = 𝜎2 < 0, |𝜎1 | < |𝜎3 |. Equation (6.19) yields 𝐽2 = (𝜎1 − 𝜎3 )2 ∕3 and 𝐽3 = −2(𝜎1 − 𝜎3 )3 ∕27 and Eq. (6.46) cos 3𝜃 = −1 or 𝜃 = 60°. • The tensile meridian with 𝜎1 > 𝜎2 = 𝜎3 (signed): a cylindrical specimen with circumferential confining pressure 𝜎2 = 𝜎3 < 0 and a longitudinal compression 𝜎1 < 0, |𝜎1 | < |𝜎3 |. Equation (6.19) yields 𝐽2 = (𝜎1 − 𝜎3 )2 ∕3 and 𝐽3 = 2(𝜎1 − 𝜎3 )3 ∕27 and Eq. (6.46) cos 3𝜃 = 1 or 𝜃 = 0°. The compressive and tensile meridians form particular curves within the strength surface as is shown in Figure 6.6. They are determined as the intersection of the strength surface with the deviatoric planes with Lode angles 𝜃 = 60° and 𝜃 = 0°, re-

Figure 6.5 Triaxial cell.

6.4 Nonlinear Material Behaviour

(a)

(b)

Figure 6.6 Strength surfaces. (a) General view direction. (b) Pressure axis view direction.

spectively. Strength surfaces of concrete themselves form a smoothed, curved tetrahedron (Figure 6.6). Its tip is located in the positive octant (𝜎1 > 0, 𝜎2 > 0, 𝜎3 > 0) near the origin. The origin indicates the triaxial tensile strength. The strength surface opens in the negative octant (𝜎1 < 0, 𝜎2 < 0, 𝜎3 < 0) and can be specified as follows: • Many experimental data exist for the compressive and the tensile meridians due to the relatively simple triaxial cell set-up (Figure 6.5). Both meridians are slightly curved. The tensile meridian falls below the compressive meridian, if both are sketched in a plane. • The range between compressive and tensile meridians with a Lode angle 0° ≤ 𝜃 ≤ 60° is sufficient to describe the whole strength surface. This section replicates for the remaining range full range of 𝜃 due to the isotropic strength condition; see the remarks following Eq. (6.41). This also replicates the meridians. • The range 0° ≤ 𝜃 ≤ 60° has three different principal stresses, which cannot be realised with the conventional triaxial cell according to Figure 6.5. True triaxial cells are required with a much higher experimental effort, and experimental data are rare in this range (Hampel et al. 2009). • Strength ‘increases’ under hydrostatic pressure. Or more precisely, the admissible deviatoric length increases with increasing pressure for a certain range of pressures. This also depends on the Lode angle. • The deviatoric concrete strength under very high pressures is not really known yet. From a theoretical point of view, there is no strength limit for a pure pressure. Practically, pure pressure is not reachable in experimental set-ups. Small deviatoric parts cannot be avoided. • The tensile strength in multi-axial tension does not significantly differ from uniaxial tensile strength. Thus, it should be possible to reach the uniaxial tensile strength in three directions simultaneously. But this has not yet been proved experimentally up to now. A stress–strain relation has to be defined for all states within the strength surface. This may be assumed as isotropic linear elastic initially and become increasingly

169

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nonlinear when approaching the strength surface. Basic approaches to describe nonlinear material behaviour are given with elasto-plasticity described in Section 6.5, damage described in Section 6.6, and microplane described in Section 6.8. A selection of widely referenced formulations for the strength surface of concrete is given in the following. • The strength surface of Ottosen (Ottosen 1977) √ 𝐽2 𝐽2 𝐼 𝑓 =𝑎 2 +𝜆 +𝑏 1 −1=0 𝑓 𝑓 𝑓𝑐 𝑐 𝑐

(6.48)

with the uniaxial compression strength 𝑓𝑐 (unsigned), constants 𝑎, 𝑏, and 1 𝜆 = 𝑘1 cos [ arccos(𝑘2 cos 3𝜃)] , 3 π 1 𝜆 = 𝑘1 cos [ − arccos(−𝑘2 cos 3𝜃)] , 3 3

for cos 3𝜃 ≥ 0 (6.49) for cos 3𝜃 ≤ 0

with further constants 𝑘1 , 𝑘2 . The four material constants 𝑎, 𝑏, 𝑘1 , 𝑘2 are determined from the tensile strength 𝑓𝑐𝑡 , the biaxial strength, and points on the compressive meridian. • The strength surface of Hsieh–Ting–Chen (Hsieh et al. 1982) √ 𝐽2 𝐽2 𝜎 𝐼 𝑓 =𝑎 2 +𝑏 +𝑐 1 +𝑑 1 −1=0 (6.50) 𝑓𝑐 𝑓𝑐 𝑓𝑐 𝑓𝑐 with constants 𝑎, 𝑏, 𝑐, 𝑑 and the largest principal stress 𝜎1 . This is rewritten as 𝑓 = 𝑎(

) 𝜌 𝜌 2 ( 𝜉 ) + 𝑏 cos 𝜃 + 𝑐 +𝑑 −1 =0 𝑓𝑐 𝑓𝑐 𝑓𝑐

(6.51)

with the hydrostatic length 𝜉 (Eq. (6.43)), the deviatoric length 𝜌 (Eq. (6.44)), and Lode angle 𝜃 (Eq. (6.46)). Hence, the largest principal stress 𝜎1 is replaced by invariants. • The strength surface of Willam/Warnke (Willam and Warnke (1975), Chen and Saleeb (1994, Section 5.5)).

𝜌=

√ 2𝜌𝑐 (𝜌𝑐2 − 𝜌𝑡2 ) cos 𝜃 + 𝜌𝑐 (2𝜌𝑡 − 𝜌𝑐 ) 4(𝜌𝑐2 − 𝜌𝑡2 ) cos2 𝜃 + 5𝜌𝑡2 − 4𝜌𝑡 𝜌𝑐 4(𝜌𝑐2 − 𝜌𝑡2 ) cos2 𝜃 + (𝜌𝑐 − 2𝜌𝑡 )2 (6.52)

with 𝜉 = 𝑎0 + 𝑎1 𝜌𝑡 + 𝑎2 𝜌𝑡2 ,

𝜉 = 𝑏0 + 𝑏1 𝜌𝑐 + 𝑏2 𝜌𝑐2

(6.53)

and 𝜉 = 𝜉∕𝑓𝑐 , 𝜌 = 𝜌∕𝑓𝑐 . The parameters 𝜌𝑡 describe the normalised tensile meridian, i.e. 𝜃 = 0° and 𝜌𝑐 the normalised compressive meridian, i.e. 𝜃 = 60°. The parameters 𝑎0 , 𝑎1 , 𝑎2 , 𝑏0 , 𝑏1 , 𝑏2 are material constants. As the compressive and

6.4 Nonlinear Material Behaviour

(a)

(b)

Figure 6.7 (a) Biaxial strength. (b) Stress paths.

tensile meridians should coincide at the same point on the 𝜉-axis, one obtains 𝑎0 = 𝑏0 . With the hydrostatic length 𝜉 given, the values of 𝜌𝑡 , 𝜌𝑐 are determined from Eq. (6.53). This may be used to determine the deviatoric length 𝜌by Eq. (6.52) depending on 𝜃. At first glance, these approaches provide the same shapes of strength surfaces (Figure 6.6). Differences are given with details, e.g. the simplicity of formulation, the number of material constants, the exact course of the compressive and tensile meridians, and the occurrence of sharp edges. Sharp edges are curves on the strength surface with a non-unique normal. The formulation of Hsieh–Ting–Chen has a sharp compressive meridian, while the formulation of Willam/Warnke has no sharp edges. The triaxial strength includes the biaxial strength as a special case with one zero principal stress or a plane stress state, respectively. Biaxial strength is determined through the intersection of the triaxial strength surface with any of the planes 𝜎1 = 0 or 𝜎2 = 0 or 𝜎1 = 0. We assume 𝜎3 = 0 irrespective of the convention Eq. (6.47). This leads to a closed curve in the 𝜎1 − 𝜎2 stress plane instead of a surface in the stress space (Figure 6.6b). A curve with biaxial strength related to the uniaxial compressive strength is shown in Figure 6.7a, which is valid for normal graded concrete (Kupfer et al. 1969). It has the following characteristics: • A lateral compression leads to an increased compressive strength in the higher loaded direction. The largest strength is roughly given for 𝜎2 ≈ 0.5 𝜎3 with |𝜎3 | > |𝜎2 | or vice versa. • The tensile strength of a direction is only influenced to a minor extent by lateral compression or tension. The biaxial strength includes the uniaxial strength as a further special case with two zero principal stresses. The uniaxial strength is determined through the intersection of the biaxial limit curve with a stress coordinate axis intersecting the uniaxial compressive and tensile strength. It is generally assumed that multi-axial strength is independent of the stress path. Different loading histories with different paths aiming at the same point of the

171

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strength surface or strength curve should have the same ultimate point or strength, respectively. But the stress–strain behaviour may be different. Regarding, e.g. the stress paths D1 and D2 of Figure 6.7b, the path D1 will lead to a load-induced anisotropy due to approaching the tensile strength and cracking, while a major amount of compressive strength and stiffness remains in the orthogonal direction. On the other hand, the path D2 aiming at the same point has a more or less isotropic stress– strain behaviour. Cracking issues are treated in Section 7.

6.4.4 Nonlinear Material Classification Driven by strains or a strain path defined by strain components (Eq. (6.3)) the integration of an incremental material law (Eq. (6.37)) leads to a path of stresses in stress space defined by the stress components (Eq. (6.7)). Let us assume a proportional strain path 𝝐(𝑡) = 𝑔(𝑡) 𝝐 0

(6.54)

with a constant strain direction 𝝐 0 and a scalar function 𝑔 depending on a loading time 𝑡 and 𝑔(0) = 0. Let us further assume 𝑔(𝑡) = 𝑔0 𝑡 with a constant 𝑔0 . As the materials treated here initially behave isotropically elastic (Eq. (6.23)), the initial stress path is given with 𝝈(𝑡) = 𝑔0 𝝈0 𝑡 ,

𝝈0 = E ⋅ 𝝐 0

(6.55)

which is a proportional stress path. For 𝑡 exceeding some value depending on 𝑔0 , the resulting stress path derived by integration of Eq. (6.37) will deviate from a proportional path in the case of a nonlinear material behaviour. This is related to the limited strength of the material. A particular point in the nonlinear range is marked by 𝑡𝑢 with strains 𝝐 𝑢 and integrated stress 𝝈𝑢 . At this time, a strain path reversal or unloading path 𝝐(𝑡) = 𝑔0 (2𝑡𝑢 − 𝑡) 𝝐 0 ,

𝑡 ≥ 𝑡𝑢

(6.56)

is applied. A nonlinear material classification may be taken with unloading. ◀

Elasto-plastic materials regain their initial elastic stiffness E with unloading and leave residual plastic strains when zero stresses are reached during unloading. Damaged materials show a degraded stiffness and have zero strains when zero stresses are reached.

This is illustrated with the uniaxial case; see Figure 6.8. With respect to elastoplasticity, the uniaxial unloading stress is given with (𝜖(𝑡) ≤ 𝜖𝑢 ) 𝜎(𝑡) = 𝜎𝑢 + 𝐸 [𝜖(𝑡) − 𝜖𝑢 ] ,

𝑡 ≥ 𝑡𝑢

(6.57)

with Young’s modulus 𝐸. The state 𝜎(𝑡) = 0 corresponds to a residual plastic strain 𝜖𝑝 = 𝜖𝑢 −

𝜎𝑢 𝐸

(6.58)

6.5 Elasto-Plasticity

Figure 6.8 Nonlinear material classification.

With respect to damage, the unloading stress is given with 𝜎(𝑡) = 𝜎𝑢 +

𝜎𝑢 [𝜖(𝑡) − 𝜖𝑢 ] , 𝜖𝑢

𝑡 ≥ 𝑡𝑢

(6.59)

with a degraded modulus 𝐸𝑢 = 𝜎𝑢 ∕𝜖𝑢 , and the state 𝜎(𝑡) = 0 corresponds to 𝜖(𝑡) = 0. Elasto-plasticity and damage mark limiting cases of nonlinear solid material behaviour. Real materials are characterised by a combination 𝜎(𝑡) = 𝜎𝑢 + 𝐸𝑑 [𝜖(𝑡) − 𝜖𝑢 ] ,

𝑡 ≥ 𝑡𝑢 ,

𝐸𝑢 < 𝐸𝑑 < 𝐸

(6.60)

with a partially degraded modulus 𝐸𝑑 revealed upon unloading. This is also the case for concrete. Both approaches are described in the following with exemplary specifications.

6.5 Elasto-Plasticity 6.5.1 A Framework for Multi-Axial Elasto-Plasticity The basic properties of elasto-plasticity have already been described in Sections 3.3, 6.4.4. They become evident with the unloading of a material: the same stiffness occurs for unloading as the initial stiffness for loading and permanent strains remain for a stress-free state. The general triaxial stress–strain relation for such behaviour is given by ) ( (6.61) 𝝈 = E ⋅ 𝝐 − 𝝐𝑝 with the isotropic linear elastic material stiffness matrix E according to Eq.(6.23), total strains 𝝐, and permanent plastic strains 𝝐 𝑝 . This leads to zero stresses 𝝈 = 0 in the case of total strains equal to plastic strains 𝝐 = 𝝐 𝑝 . The rate form ( ) 𝝈̇ = E ⋅ 𝝐̇ − 𝝐̇ 𝑝 (6.62) with variable plastic strains has to be used for general purposes. They are derived with a flow rule 𝝐̇ 𝑝 = 𝜆̇

𝜕𝐺 𝜕𝝈

(6.63)

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6 Multi-Axial Concrete Behaviour

using a flow potential 𝐺(𝝈, 𝜅𝑝 ), the rate 𝜆̇ of a plastic multiplier 𝜆, and an internal state variable 𝜅𝑝 comprising the loading history. Equation (6.63) may lead to shear components 𝜖23 , 𝜖13 , 𝜖12 instead of 𝛾23 , 𝛾13 , 𝛾12 (Eq. (6.3)) depending on the formulation of 𝐺. An increment of plastic strains or plastic flow occurs in the case of yielding. Yielding is ruled by a yield function 𝐹(𝝈, 𝜅𝑝 ) = 0

(6.64)

Yielding may only occur for states of loading. Loading is distinguished from unloading by the Kuhn–Tucker conditions 𝐹 ≤ 0,

𝜆̇ ≥ 0 ,

𝐹 𝜆̇ = 0

(6.65)

• In the case 𝐹 < 0 is 𝜆̇ = 0: elastic loading, unloading, or re-loading occurs, and plastic strains will not change. • In the case 𝐹 = 0 is 𝜆̇ ≥ 0: plastic loading occurs, and plastic strains change. This implies a consistency condition 𝜕𝐹 𝜕𝐹 𝐹̇ = 𝜅̇ = 0 ⋅ 𝝈̇ + 𝜕𝝈 𝜕𝜅𝑝 𝑝

(6.66)

Finally, the formalism has to be completed with an evolution law for the internal state variable. This is assumed with 𝜅̇ 𝑝 = 𝜆̇ 𝐻(𝝈, 𝜅𝑝 )

(6.67)

The functions 𝐹(𝝈, 𝜅𝑝 ), 𝐺(𝝈, 𝜅𝑝 ), 𝐻(𝝈, 𝜅𝑝 ) are functions specific for a particular material. They have to be known and must be defined in advance. To treat initially elastic behaviour the state variable, 𝜅𝑝 generally has a threshold value 𝜅𝑝0 > 0 for the initially unloaded material. The actual state of the material is described by a given stress 𝝈 and a given state variable 𝜅𝑝 . ◀

Elasto-plasticity is generally formulated as stress-based. as is demonstrated with the foregoing relations. This allows for a straightforward definition of yield functions F.

The different states for plastic loading and all other are evaluated as follows. • In the case of plastic loading, it is 𝐹 = 0, and Eq. (6.66) is combined with Eqs. (6.62) and (6.67) to obtain )] 𝜕𝐹 𝜕𝐹 [ ( ⋅ E ⋅ 𝝐̇ − 𝝐̇ 𝑝 + 𝜆̇ 𝐻 = 0 𝜕𝝈 𝜕𝜅𝑝

(6.68)

̇ leading to Using Eq. (6.63) this can be solved for 𝜆, 1 𝜕𝐹 ⋅ E ⋅ 𝝐̇ , 𝜆̇ = 𝐴 𝜕𝝈

𝐴=−

𝜕𝐹 𝜕𝐹 𝜕𝐺 𝐻+ ⋅E⋅ 𝜕𝜅𝑝 𝜕𝝈 𝜕𝝈

(6.69)

6.5 Elasto-Plasticity

Combining Eqs. (6.69), (6.63), and (6.62) leads to the incremental material law 𝝈̇ = C𝑇 ⋅ 𝝐̇

(6.70)

with a tangential material stiffness matrix C𝑇 = E −

1 𝜕𝐹 𝜕𝐺 ⊗ ⋅E E⋅ 𝐴 𝜕𝝈 𝜕𝝈

(6.71) 𝜕𝐺

𝜕𝐹

with E according to Eq. (6.23). The form ⊗ represents an outer or dyadic 𝜕𝝈 𝜕𝝈 product of two vectors. An outer or dyadic product a ⊗ b results in a matrix c with components 𝑐𝑖𝑗 = 𝑎𝑖 𝑏𝑗 . This is generally not symmetric. • In all other cases with 𝐹 < 0, the stress increment is given by 𝝈̇ = E ⋅ 𝝐̇

(6.72)

according to Eq. (6.62) as 𝝐̇ 𝑝 = 0. An initial isotropy (Section 6.3.1) is generally not preserved, as will be demonstrated with the following Example 6.1. In any case, a correct evaluation of the tangential material stiffness is essential for the solution of nonlinear equations (Section 2.8.1). With respect to incremental approaches or discretisation in time, algorithms such as radial return connected to an algorithmic tangential material stiffness (Eq. (2.81)) with a consistent linearisation are appropriate (Belytschko et al. (2000, 5.9), de Borst et al. (2012, 7.3,7.4)). An associated plasticity with the identity 𝐺 = 𝐹 of the flow potential and yield condition simplifies the formalism to a large extent. A relatively simple form of elasto-plasticity is demonstrated with the following example. Example 6.1: Mises Elasto-Plasticity for Uniaxial Behaviour

Mises elasto-plasticity is used to demonstrate the general procedures of elastoplasticity. It has an associated flow rule with a yield function limiting the deviatoric length (Eq. (6.44)). The Mises yield function and flow rule are given by √ √ 3 (6.73) 𝐹=𝐺= 𝜌𝜎 − 𝜅𝑝 = 3𝐽2 − 𝜅𝑝 2 with the second invariant 𝐽2 of the stress deviator (Eq. (6.192 )). This leads to partial derivatives √ 𝜕𝐹 3 ′ 𝜕𝐹 𝜕𝐺 1 𝝈 , = −1 (6.74) = = 2 𝐽2 𝜕𝝈 𝜕𝝈 𝜕𝜅𝑝 with the stress deviator 𝝈′ (Eq. (6.9)). The tensor components 𝜎23 , 𝜎32 … have to be distinguished while computing 𝜕𝐹∕𝜕𝝈 irrespective of symmetry. With respect to the tangential stiffness according to Eq. (6.71), it can be shown that √ 3 ′ 𝐸 𝜕𝐹 𝝈 , 𝐺= (6.75) ⋅E = 𝐺 𝐽2 𝜕𝝈 2(1 + 𝜈)

175

176

6 Multi-Axial Concrete Behaviour

with the shear modulus 𝐺 derived from Young’s modulus 𝐸 and Poisson’s ratio 𝜈 (Eq. (6.23)). With respect to Eq. (6.69), 𝐴 = 𝐻 + 3𝐺

(6.76)

finally leads to C𝑇 = E −

𝐺 1+

𝐻

1 ′ 𝝈 ⊗ 𝝈′ 𝐽2

(6.77)

3𝐺

Full tensor notations are required in this context to derive Eq. (6.76). The product 𝝈′ ⊗ 𝝈′ is an outer or dyadic product of second-order tensors leading to a fourthorder tensor. The case of uniaxial stress with 𝜎22 = 𝜎33 = 𝜎23 = 𝜎13 = 𝜎12 = 0 yields 𝐽2 =

1 2 𝜎 , 3 11

𝐹 = 𝜎11 − 𝜅𝑝

(6.78)

After some rearrangement, the tangential material stiffness under uniaxial stress conditions is evaluated as 1 (3𝛼+1−3𝜈𝛼+𝜈)

⎡ 3 (1−2𝜈)(1+𝛼) ⎢ ⎢ 1 (3𝜈𝛼+𝜈+1) ⎢ 3 (1−2𝜈)(1+𝛼) ⎢ 𝐸 ⎢ 1 (3𝜈𝛼+𝜈+1) C𝑇 = ⎢ 1 + 𝜈 ⎢ 3 (1−2𝜈)(1+𝛼) 0 ⎢ ⎢ 0 ⎢ ⎢ 0 ⎣

1 (3𝜈𝛼+𝜈+1)

1 (3𝜈𝛼+𝜈+1)

3 (1−2𝜈)(1+𝛼)

3 (1−2𝜈)(1+𝛼)

1 (6𝛼+5−6𝜈𝛼−4𝜈)

1 (6𝜈𝛼+8𝜈−1)

(1−2𝜈)(1+𝛼)

6

6 (1−2𝜈)(1+𝛼)

1 (6𝜈𝛼+8𝜈−1)

1 (6𝛼+5−6𝜈𝛼−4𝜈)

6 (1−2𝜈)(1+𝛼)

6

(1−2𝜈)(1+𝛼)

0

0

0

0

0

0

1

0

0

0

0

0

0

0

0

2

0 1 2

0

0⎤ ⎥ 0⎥ ⎥ ⎥ 0⎥ ⎥ ⎥ 0⎥ ⎥ 0⎥ 1⎥ 2⎦ (6.79)

2 𝐻

with 𝛼 = (1 + 𝜈) . This does not correspond to the form Eq. (6.22) required for 3 𝐸 isotropy, and an anisotropy will occur. The case of uniaxial stress has 𝜎̇ 22 = 𝜎̇ 33 = 𝜎̇ 23 = 𝜎̇ 13 = 𝜎̇ 12 = 0. Thus, regarding Eqs. (6.70) and (6.79) with the Voigt notation (Eqs. (6.3) and (6.7)) leads to 𝛾23 = 𝛾13 = 𝛾12 = 0 and 𝜖̇ 22 = 𝜖̇ 33 = −

3𝜈𝛼 + 1 + 𝜈 𝜖̇ 3𝛼 + 2 + 2𝜈 11

(6.80)

This is used to determine 𝜎̇ 11 = 𝐶𝑇,11 𝜖̇ 11 + 𝐶𝑇,12 𝜖̇ 22 + 𝐶𝑇,13 𝜖̇ 33 =

𝛼𝐸 2

𝛼 + (1 + 𝜈) 3

𝜖̇ 11 =

𝐻 1+

𝐻

𝜖̇ 11 (6.81)

𝐸

in the case of plastic loading. The elastic part of longitudinal strain is determined with 𝜖̇ 𝑒𝑙,11 = 𝜎̇ 11 ∕𝐸 yielding a plastic strain 𝜖̇ 𝑝,11 = 𝜖̇ 11 − 𝜖̇ 𝑒𝑙,11 =

1 𝜎̇ 𝐻 11

→

𝜎̇ 11 = 𝐻 𝜖̇ 𝑝,11

(6.82)

6.5 Elasto-Plasticity

The material function 𝐻 is assumed as constant for Mises plasticity and corresponds to the hardening modulus. Considering Eq. (6.782 ) with 𝐹 = 0 results in 𝜎11 = 𝜅𝑝 in the case of plastic loading. But the longitudinal stress 𝜎11 also corresponds to a current uniaxial yield stress 𝑓𝑦 . Thus, 𝜅𝑝 = 𝑓𝑦 , and 𝑓𝑦 replaces 𝜅𝑝 for Mises plasticity. Pure shear with 𝜎11 = 𝜎22 = 𝜎33 = 𝜎23 = 𝜎13 = 0 and 𝜎12 ≠ 0 may be treated in an analogous way, leading to 𝜎̇ 12 =

𝐻 3+

𝐻

𝛾̇ 12 ,

𝛾̇ 𝑝,12 = 𝛾̇ 12 − 𝛾̇ 𝑒𝑙,12 =

𝐺

3 𝜎̇ 𝐻 12

(6.83)

This employs the engineering notation for shear strains (Eq. (6.3)), as the inverse of Eq. (6.77) is used instead of Eq. (6.63) to derive the relation for the plastic shear strain increment. Mises elasto-plasticity is characterised by four material constants: the initial Young’s modulus 𝐸, Poisson’s ratio 𝜈, hardening modulus 𝐻, and the initial uniaxial yield stress 𝑓𝑦𝑘 . The value of 𝑓𝑦𝑘 may be directly taken from a given uniaxial bilinear stress–strain relation (Figure 3.11a). The value of 𝐻 is indirectly determined from such a relation by transforming Eq. (6.81) into Δ𝜎11

𝐻=

Δ𝜖11

1−

(6.84)

1 Δ𝜎11 𝐸 Δ𝜖11

The current yield stress 𝑓𝑦 starts with the initial value 𝑓𝑦𝑘 and changes in the case of plastic loading, as is ruled by the hardening modulus. This applies in the same way to tension and compression indicating an isotropic hardening. The values for 𝐻, 𝑓𝑦𝑘 derived from a uniaxial bilinear stress–strain relation may also be used for the general triaxial case, whereby 𝑓𝑦 corresponds to 𝜅𝑝 and 𝑓𝑦𝑘 to the initial 𝜅𝑝0 . Ideal Mises elasto-plasticity is given by 𝐻 = 0 and 𝑓𝑦 = 𝑓𝑦𝑘 = 𝑐𝑜𝑛𝑠𝑡. as a special case. Example 6.1 is restricted to the uniaxial case. For the triaxial extension of Mises elasto-plasticity, see Bathe (1996, 6.6.3), for its algorithmic tangential material stiffness matrix (Eq. (2.81)) with a consistent linearisation, which might be required for the radial return algorithm; see de Borst et al. (2012, 7.4)). With respect to stress-based elasto-plasticity, the material functions 𝐹 (Eq. (6.64)), 𝐺 (Eq. (6.63)), and 𝐻 (Eq. (6.67)) are generally assumed to depend on stress invariants 𝐹 = 𝐹(𝐼1 , 𝐽2 , 𝐽3 , 𝜅𝑝 ) ,

𝐺 = 𝐺(𝐼1 , 𝐽2 , 𝐽3 , 𝜅𝑝 ) ,

𝐻 = 𝐻(𝐼1 , 𝐽2 , 𝐽3 , 𝜅𝑝 )

(6.85)

with 𝐼1 , 𝐽2 , 𝐽3 according to Eq. (6.19). Hence, the orientation of principal stress directions relative to material directions has no influence on these material functions; see also the remarks following Eq. (6.41). This allows the representation of yield functions as surfaces in principal stress space. Yield functions 𝐹 are related to multi-axial strength (Eq. (6.41)), as the latter form a boundary for yield criteria with 𝐹 ≤ 𝑔.

177

178 ◀

6 Multi-Axial Concrete Behaviour

The multi-axial strength is constant for a certain material and does not depend on state variables in contrast to yield functions.

In contrast to strength, the isotropy with respect to the stress–strain behaviour is generally lost as a tangential material stiffness C𝑇 no longer follows the form required by Eq. (6.22); see Example 6.1. Deviating principal directions of stresses and strains already become evident with Eq. (6.61) with the bias of strains 𝝐 by plastic strains 𝝐 𝑝 .

6.5.2 Pressure-Dependent Yield Functions Mises elasto-plasticity is characterised through its limitation of the deviatoric length Eq. (6.44). This is independent of hydrostatic pressure and treats compression and tension in the same way. Thus, Mises elasto-plasticity is not adequate to describe multi-axial concrete behaviour. As an alternative, the yield function of Drucker–Prager introduces pressure through the first stress invariant 𝐼1 √ 𝐹 = 𝜅𝑝 𝑎 𝐼1 + 3𝐽2 − 𝜅𝑝 (6.86) with stress invariants 𝐼1 , 𝐽2 according to Eq. (6.19) and the internal state variable 𝜅𝑝 . Using Haigh–Westergaard coordinates (Eqs. (6.43)–(6.46)) the Eq. (6.86) is reformulated as √ √ 𝐹 = 𝜅𝑝 𝑎 3 𝜉 + 3∕2 𝜌 − 𝜅𝑝 (6.87) With respect√ to 𝐹 = 0, a circular cone in the principal stress space is obtained with a radius 𝜌 = 2∕3 𝜅𝑝 in the deviatoric plane 𝜉 = 0 and an apex (→ 𝜌 = 0) located at √ 𝜉0 = 1∕( 3 𝑎); see Figure 6.9. The cone opens in the compressive octant with 𝑎 > 0. Material parameters are given with the value of 𝑎 and the initial value 𝜅𝑝0 of the internal state variable.

Figure 6.9 Surfaces of Mohr–Coulomb and Drucker–Prager yield functions in principal stress space.

6.5 Elasto-Plasticity

Figure 6.10 Mohr circle for the Mohr–Coulomb yield type.

An angle of internal friction 𝜗 is defined with the ratio of deviatoric length to total hydrostatic length measured from the cone apex. It is given by tan 𝜗 =

𝜌 𝜉0 − 𝜉

(6.88)

This can be written as √ tan 𝜗 = 2 𝑎 𝜅𝑝

(6.89)

√ √ using 𝐹 = 0 from Eq. (6.86) or 𝜉 = 1∕(𝑎 3) − 𝜌∕(𝜅𝑝 𝑎 2), respectively. Starting from √ a value tan 𝜗0 = 2 𝑎 𝜅𝑝0 the angle of internal friction changes with the internal state parameter 𝜅𝑝 . The Mohr–Coulomb yield function based on Coulomb friction assumes the shear stress 𝜏 in a plane as depending on the plane normal stress 𝜎 𝜏 = 𝑐 − 𝜎 tan 𝜙

→

𝐹 = 𝑐 − (𝜏 + 𝜎 tan 𝜙)

(6.90)

with a cohesion 𝑐 and an angle of external friction 𝜙. Regarding triaxial stress states the maximum shear stress is given by 𝜏𝑚 = (𝜎1 − 𝜎3 )∕2 under the condition of Eq. (6.47) and its attached normal stress by 𝜎𝑚 = (𝜎1 + 𝜎3 )∕2 (Malvern 1969, 3.4). They are related to a pair 𝜏, 𝜎 with the largest ratio |𝜏∕𝜎|, see Figure 6.10, by 𝜎 = 𝜎𝑚 + 𝜏𝑚 sin 𝜙, 𝜏 = 𝜏𝑚 cos 𝜙. This leads Eq. (6.90) to 𝐹 = cos 𝜙 (𝜏 + 𝜎 tan 𝜙 − 𝑐) = 𝜏𝑚 + sin 𝜙 𝜎𝑚 − 𝑐 cos 𝜙 =

sin 𝜙 1 (𝜎1 − 𝜎3 ) + (𝜎1 + 𝜎3 ) − 𝑐 cos 𝜙 2 2

(6.91)

with 𝐹 = 0 initially multiplied by cos 𝜙. The condition 𝐹 = 0 spans a 3D plane between the compressive meridian 𝜎2 = 𝜎1 > 𝜎3 (signed!) and the tensile meridian𝜎2 = 𝜎3 < 𝜎1 ; see Section 6.4.3 for meridians. An alternative formulation deriving hydrostatic length 𝜉 from the deviatoric length 𝜌 is given for the compressive meridian with √ √ 3 12𝑐 cos 𝜙 − 6(3 − sin 𝜙) 𝜌comp (6.92) 𝜉= 12 sin 𝜙

179

180

6 Multi-Axial Concrete Behaviour

and for the tensile meridian with √ √ 3 12𝑐 cos 𝜙 − 6(3 + sin 𝜙) 𝜌tens 𝜉= 12 sin 𝜙

(6.93)

This yields a relation 𝜌comp 𝜌tens

=

3 + sin 𝜙 3 − sin 𝜙

(6.94)

indicating the different slopes of compressive and tensile meridians against the hydrostatic axis. A value of, e.g. 𝜙 = 30 ° leads to 𝜌comp = 7∕5 𝜌tens . This ratio increases for increasing values of 𝜙. A cyclic interchanging of principal stresses leads to a total of six planes forming √ the hexagonal cone with an apex at 𝜎1 = 𝜎2 = 𝜎3 = 𝑐 cot 𝜙 and 𝜉0 = 3𝑐 cot 𝜙 (Figure 6.9). Angles of internal friction 𝜗 are derived according to Eq. (6.88). With respect to the Mohr–Coulomb compressive meridian, the value is given with tan 𝜗 =

𝜌comp 𝜉0 − 𝜉

√ =2 2

sin 𝜙 3 − sin 𝜙

(6.95)

As an example, an angle of external friction 𝜙 = 30 ° leads to an angle of internal friction 𝜗 = 29.5 ° on the compressive meridian. The angle of external friction may be subject to change due to the loading history. It is determined with 𝜙 = 𝜅𝑚𝑐 𝜙0

(6.96)

with the internal state variable 𝜅𝑚𝑐 and an initial angle of internal friction 𝜙0 . An initial value 𝜅𝑚𝑐,0 = 1 is appropriate for the internal state variable. With a fixed apex position 𝜎1 = 𝜎2 = 𝜎3 = 𝑓𝑡 = 𝑐 cot 𝜙, the cohesion 𝑐 (Eq. (6.90)) is determined with 𝑐 = 𝑓𝑡 tan 𝜙

(6.97)

The parameters 𝜙0 , 𝑓𝑡 serve as material constants for the Mohr–Coulomb yield function, where 𝑓𝑡 corresponds to a tensile strength. The yield function has the Rankine limit function as a special case with tan 𝜙 = 1. This restricts sustainable tensile stresses but allows for unbounded compressive stresses. It is the stress-based counterpart of the Rankine damage function (Eq. (6.108)). The actual size of the yield functions of Drucker–Prager and Mohr–Coulomb in the principal stress space is ruled by the value of the respective internal state variables 𝜅𝑝 , 𝜅𝑚𝑐 . An evolution law has to be defined for each according to Eqs. (6.67) and (6.85). The limits are defined by initial values 𝜅𝑝,0 , 𝜅𝑚𝑐,0 , which mark the elastic range and final values 𝜅𝑝,max , 𝜅𝑚𝑐,max when the surface of a yield function reaches a strength surface. This may be followed by a decrease of 𝜅𝑝 , 𝜅𝑚𝑐 , leading to a softening behaviour. The parameters 𝜅𝑝0 , 𝜅𝑝,max or 𝜅𝑚𝑐0 , 𝜅𝑚𝑐,max have to be defined as further material parameters. The topic of flow rules remains to be treated. In the case of Drucker–Prager, a variation of the yield function (6.86) √ (6.98) 𝐺 = 𝜅𝑝𝑔 𝑎 𝐼1 + 3𝐽2

6.5 Elasto-Plasticity

is often used as flow rule with an own internal state parameter 𝜅𝑝𝑔 leading to a nonassociated plasticity with 𝐺 ≠ 𝐹. In order to derive plastic strain increments according to Eq. (6.63) 𝝐̇ 𝑝 = 𝜆̇

𝜕𝐺 𝜕𝝈

(6.99)

the derivative with respect to stress is required. With respect to Eqs. (6.19) and (6.74), the derivatives of the flow function are given by 1 𝜕𝐺 = 𝜅𝑝𝑔 𝑎 I + 2 𝜕𝝈

√ 3 ′ 𝝈 𝐽2

(6.100)

with the unit matrix I. Volume change is of special interest in the following. It is defined with the volumetric strain 𝜖𝑉 as 𝜖𝑉 = 𝜖11 + 𝜖22 + 𝜖33

(6.101)

Using Eqs. (6.99) and (6.100) the rate of plastic volume change is determined as ̇ 𝑝𝑔 𝑎 𝜖̇ 𝑝𝑉 = 𝜖̇ 𝑝11 + 𝜖̇ 𝑝22 + 𝜖̇ 𝑝33 = 3𝜆𝜅

(6.102)

′ ′ ′ as 𝜎11 + 𝜎22 + 𝜎33 = 0 for deviatoric stresses by definition (Eq. (6.9)). As 𝑎 > 0 and ̇𝜆 ≥ 0 due to the Kuhn–Tucker conditions (Eq. (6.65)), a value 𝜅𝑝𝑔 > 0 indicates a dilatation of volume or dilatancy, while 𝜅𝑝𝑔 < 0 indicates a compaction of volume. In analogy to Eq. (6.89), an angle of dilatancy is defined as

tan 𝜗 ′ =

√

2 𝑎 𝜅𝑝𝑔

(6.103)

For concrete, it has to be considered that its most relevant states lie in the compressive octant of the principal stress space. ◀

Regarding experimental data a compaction is seen for concrete for moderate compressive stress levels. This turns into dilatancy for larger ratios of deviatoric length 𝜌 to hydrostatic length 𝜉 while approaching the strength surface (Chen and Saleeb 1994, 6.2.2).

The evolution law for the internal state parameter 𝜅𝑝𝑔 for plastic flow has to be formulated in analogy to Eq. (6.67). A similar approach for non-associated plasticity may be followed concerning Mohr–Coulomb type of plasticity. More details are given in Chen and Saleeb (1994, 6.4.4). The deviatoric projections with the intersections of yield or strength surfaces of Drucker–Prager and Mohr–Coulomb with the deviatoric plane are shown in Figure 6.11. The values of 𝜗 for Drucker–Prager and 𝜙 for Mohr–Coulomb are chosen such that the compressive meridian of Mohr–Coulomb coincides with the surface of Drucker–Prager. Both strength surfaces form limiting cases for the observed behaviour of concrete.

181

182

6 Multi-Axial Concrete Behaviour

Figure 6.11 Intersections of Mohr–Coulomb, Drucker–Prager, and Willam–Warnke surfaces with the deviatoric plane.

A sketch of the deviatoric projection of the Willam–Warnke strength surface Eq. (6.52) is also given in Figure 6.11. Drucker–Prager has identical compressive and tensile meridians. Mohr–Coulomb has sharp edges along the meridians with undefined yield surface gradients. Other projections show straight compressive and tensile meridians for both, in contrast to experimental data. The more realistic Willam–Warnke surface may be used a base for a yield function, whereby the material parameters of the strength surface have to be formulated as functions of internal state parameters. The tangential material stiffness according to Eq.(6.70) for both Drucker–Prager and Mohr–Coulomb in the case of plastic loading is given by Eq. (6.71). This uses the elasticity matrix E and the derivatives of yield function 𝜕𝐹∕𝜕𝝈 and flow function 𝜕𝐺∕𝜕𝝈, respectively. The scalar parameter 𝐴 (Eq. (6.692 )) includes the evolution law 𝐻 (Eq. (6.67)) for the internal state variables 𝜅𝑝 and 𝜅𝑚𝑐 used in the flow functions. Flow functions like Mises, Drucker–Prager, or Mohr–Coulomb cover plasticity due to increasing deviatoric parts of the stress state. No plasticity will arise due to increasing hydrostatic compression with these types of predominantly deviatoric plasticity. But experimental data indicate a plastic compaction of a concrete grain structure under increasing hydrostatic compression. This effect may be treated with a predominantly volumetric plasticity. Hence, a flow function is given by a sphere section in the principal stress space. Volumetric and deviatoric plasticity may be combined with an extension of Eq. (6.62) 𝝈̇ = E ⋅ (𝝐̇ − 𝝐̇ 𝑑 − 𝝐̇ 𝑣 )

(6.104)

Each plastic part has its own yield function 𝐹 (Eq. (6.64)), flow function 𝐺 (Eq. (6.63)), and evolution laws 𝐻 for internal state variables (Eq. (6.67)). This leads to the following cases:

6.6 Damage

• • • • •

Loading in the elastic range without yielding. Loading in the elasto-plastic range with predominantly deviatoric yielding. Loading in the elasto-plastic range with predominantly volumetric yielding. Loading in the elasto-plastic range with both deviatoric and volumetric yielding. Unloading in the elastic range without yielding.

The occurrence of cases is ruled by the particular Kuhn–Tucker conditions (Eq. (6.65)). In the case of both deviatoric and volumetric yielding, the tangential material stiffness matrix C𝑇 is determined with an extended set of equations based on Eqs. (6.68)–(6.71). Implementations of the foregoing concepts are given in, e.g. Han and Chen (1985); Grassl et al. (2002); Folino and Etse (2012).

6.6 Damage Elasto-plasticity is characterised by the evolution of permanent strains with a case by case constant material stiffness. In contrast, damage assumes a degrading material stiffness without permanent strains upon unloading (Section 6.4.4). The approach for isotropic damage is 𝝈 = (1 − 𝐷) E ⋅ 𝝐

(6.105)

with the isotropic linear elastic material stiffness E according to Eq. (6.23). This is applicable for triaxial behaviour and includes biaxial and uniaxial behaviour as special cases. Equation (6.105) introduces a state variable 𝐷. By definition, this scalar damage variable has a range 0≤𝐷≤1

(6.106)

where 𝐷 = 0 denotes a fully undamaged and 𝐷 = 1 a fully damaged material leading to 𝝈 = 0 for every 𝝐. The value of 𝐷 is not allowed to decrease. It may retain its value or increase during a loading process, i.e. 𝐷̇ ≥ 0. The damage variable 𝐷 needs a law describing its development from 0 to 1. It is generally coupled to an internal state variable 𝜅𝑑 , which comprises the loading history but does not have a condition 0 ≤ 𝜅𝑑 ≤ 1. Stress-based damage uses the stress history to drive 𝜅𝑑 , while strainbased damage uses the strain history. In the following, relatively simple forms of evolution laws are given for 𝜅𝑑 . The conventions 𝜎1 ≥ 𝜎2 ≥ 𝜎3 or 𝜖1 ≥ 𝜖2 ≥ 𝜖3 with signed values have to be followed. The exemplary approach for strain-based damage may start with a relation between damage variable 𝐷 and the state variable 𝜅𝑑 𝐷(𝜅𝑑 ) =

⎧0 ⎨ 1−e ⎩

𝑔𝑑 𝜅 −𝑒 −( 𝑑 0 ) 𝑒𝑑

𝜅𝑑 ≤ 𝑒0

(6.107)

𝜅𝑑 > 𝑒0

see Figure 6.12, with constant material parameters 𝑒0 , 𝑒𝑑 , 𝑔𝑑 . This form guarantees the condition 0 ≤ 𝐷 ≤ 1 for arbitrary values 𝜅𝑑 ≥ 0. The internal state variable 𝜅𝑑

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6 Multi-Axial Concrete Behaviour

Figure 6.12 Damage variable D depending on equivalent strain 𝜅d .

is considered as an equivalent strain for strain-based damage. The equivalent strain is related to the strain 𝝐 with a damage function 𝐹. Damage functions 𝐹 should be isotropic: 𝐹 = 𝐹(𝐼1 , 𝐽2 , 𝐽3 , 𝜅𝑑 ) according to the isotropic strength of concrete (Section 6.4.3). Its formulation has a range of alternatives. A short selection is given in the following: • The Rankine damage function 𝐹=

⎧

𝛼𝜖1 − 𝜅𝑑 ⎨0 ⎩

𝜖1 > 0 else

(6.108)

with the largest principal strain 𝜖1 and a material constant 𝛼. The largest principal strain 𝜖1 may be replaced by strain invariants; compare Eqs. (6.50) and (6.51). This models tensile failure in the direction of the largest principal strain. • The Hsieh–Ting–Chen damage function √ 𝐹 = 𝑐1 𝐽2,𝜖 + 𝜅𝑑 (𝑐2 𝐽2,𝜖 + 𝑐3 𝜖1 + 𝑐4 𝐼1,𝜖 ) − 𝜅𝑑2 (6.109) with the largest principal strain 𝜖1 , the first strain invariant 𝐼1,𝜖 of 𝝐, and the second invariant 𝐽2,𝜖 of the deviator of 𝝐, whereby Eq. (6.19) is applied to tensorial strain components. The coefficients 𝑐1 … 𝑐4 are further constant material parameters to be calibrated from prescribed components for strain or stress, respectively. This is demonstrated with the following Example 6.2. The damage function includes uniaxial tension with 𝜖2 = 𝜖3 = −𝜈 𝜖1 as a special case leading to 1 (1 + 𝜈)2 𝜖12 , 𝐼1,𝜖 = (1 − 2𝜈) 𝜖1 3 (1 + 𝜈)2 2 1+𝜈 𝐹 = 𝑐1 𝜖1 + 𝜅𝑑 (𝑐2 √ + 𝑐3 + 𝑐4 (1 − 2𝜈)) 𝜖1 − 𝜅𝑑2 3 3

𝐽2,𝜖 =

𝜅𝑑 = 𝛼𝑡 𝜖1

(6.110)

for 𝐹 = 0

with 𝛼𝑡 depending on the material parameters. The case of uniaxial compression 𝜖1 = 𝜖2 = −𝜈 𝜖3 yields

6.6 Damage

1 (1 + 𝜈)2 𝜖32 , 𝐼1,𝜖 = (1 − 2𝜈) 𝜖3 3 (1 + 𝜈)2 𝜖32 1+𝜈 + 𝜅𝑑 (−𝑐2 √ − 𝑐3 𝜈 + 𝑐4 (1 − 2𝜈)) 𝜖3 − 𝜅𝑑2 𝐹 = 𝑐1 3 3

𝐽2,𝜖 =

𝜅𝑑 = 𝛼𝑐 𝜖3

(6.111)

for 𝐹 = 0

with 𝛼𝑐 again depending on the material parameters. Equation (6.109) has a formal similarity to Eq. (6.50) of the Hsieh–Ting–Chen strength surface. Actually, the damage function Eq. (6.109) combined with Kuhn–Tucker conditions Eq. (6.112) lead to multi-axial stress conditions as is formulated with Eq. (6.50), including multi-axial strain hardening and strength and strain softening (Häussler-Combe and Hartig 2008). The expression as strength lends itself to a calibration of the material parameters. • More damage function alternatives are described in, e.g. CEB-FIP (2008, 6.2.3). The equivalent strain 𝜅𝑑 may be connected to damage 𝐷 by Eq. (6.107) for both the above-mentioned damage functions. Other formulations for the relation between 𝐷 and 𝜅𝑑 are possible but should have the characteristics shown in Figure 6.12. Similar to elasto-plasticity loading states have to be distinguished from unloading states for damage. This is again reached with the Kuhn–Tucker conditions 𝐹 ≤ 0,

𝐷̇ ≥ 0 ,

𝐹 𝐷̇ = 0

(6.112)

similar to Eq. (6.65). • In the case 𝐹 < 0 is 𝐷̇ = 0, i.e. unloading occurs, and damage will not change. • In the case 𝐹 = 0 is 𝐷̇ ≥ 0, i.e. loading occurs, and damage increases. This implies a consistency condition similar to Eq. (6.66) 𝜕𝐹 𝜕𝐹 𝐹̇ = 𝜅̇ = 0 ⋅ 𝝐̇ + 𝜕𝝐 𝜕𝜅𝑑 𝑑

→

𝜅̇ 𝑑 = −

1 𝜕𝐹 ⋅ 𝝐̇ 𝜕𝝐

𝜕𝐹

(6.113)

𝜕𝜅𝑑

leading to the evolution law for the equivalent strain. Stress-based damage is only mentioned briefly. The internal state variable 𝜅𝑑 becomes an equivalent stress in the case of stress-based damage. A damage function may again be used to connect the equivalent stress to the stress state 𝝈. Yield functions as have been previously derived for elasto-plasticity – Drucker–Prager, Mohr– Coulomb or more complex types to cover characteristics of concrete behaviour – may be used for this purpose, whereby the internal state parameter of elasto-plasticity 𝜅𝑝 is replaced by an equivalent stress measure. The relation between the damage variable 𝐷 and the equivalent stress has to have another characteristic, as is shown in Figure 6.12 for strain-based damage, as measures of stress in general and equivalent stress in particular have upper limits due to limited strength. The relations are no longer straightforward and have to be formulated implicitly. Finally, stress-based damage has to be completed by the Kuhn–Tucker conditions in the same way as Eq. (6.112) to distinguish loading from unloading. We will refer to strain-based damage in the following and demonstrate its application with the following example.

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Example 6.2: Uniaxial Stress–Strain Relations with Hsieh–Ting–Chen Damage

We assume strain-based damage with damage variables according to Eq. (6.107) and the damage function according to Hsieh–Ting–Chen Eq. (6.109). The latter has a unique mapping to the strength condition Eq. (6.50) connecting the coefficients 𝑐1 … 𝑐4 to 𝑎, 𝑏, 𝑐, 𝑑 (Häussler-Combe and Hartig 2008). The coefficients 𝑎, 𝑏, 𝑐, 𝑑 are be determined by prescribing four points on the strength surfaces or curves, respectively (Figures 6.6 and 6.7a). Appropriate points are • • • •

Uniaxial compressive strength 𝑓𝑐 . Uniaxial tensile strength 𝛼 related to 𝑓𝑐𝑡 . Biaxial strength 𝛽 with 𝜎2 = 𝜎3 (Figure 6.7a) related to 𝑓𝑐 . Compressive meridian strength 𝛾 with 𝜎1 = 𝜎2 = 0.2 𝜎3 (Figure 6.6b) – all negative by value – related to 𝑓𝑐 .

These choices are not mandatory but reasonable. Each of these points has a unique set of Haigh–Westergaard coordinates (Section 6.4.2) resulting in a linear system of equations regarding Eq. (6.51). The solution for 𝑎, 𝑏, 𝑐, 𝑑 via 𝑎, 𝑏, 𝑐, 𝑑 leads to solutions 𝑐1 … 𝑐4 for Eq. (6.109). Furthermore, this procedure yields 𝛼𝑡 = 1∕𝛼 (Eq. (6.1103 )) and 𝛼𝑐 = −1 (Eq. (6.1113 )). This completes the multi-axial part of the calibration. The material parameters 𝑒0 , 𝑒𝑑 , 𝑔𝑑 for scalar damage 𝐷 depending on the equivalent damage strain 𝜅𝑑 must still be determined. Equations (6.105) and (6.107) result in the uniaxial compressive stress–strain relation 𝜎3 = (1 − 𝐷) 𝐸0 𝜖1 = e

−(

𝑔 −𝜖3 −𝑒0 𝑑 ) 𝑒𝑑

𝐸 𝜖3 ,

𝜖3 < 𝑒0

(6.114)

in the case of loading 𝜖̇ 3 < 0, 𝐷̇ > 0, considering 𝜖3 = −𝜅𝑑 from Eq. (6.1113 ) and 𝛼𝑐 = −1. A general choice 𝑔𝑑 = 2 proves to be appropriate. Thus, two unknowns 𝑒0 , 𝑒𝑑 remain to reproduce 𝜎3 = −𝑓𝑐 at a prescribed strain 𝜖3 = 𝜖𝑐1 (Figure 3.1). This forms a small nonlinear problem, which may be solved with the Newton–Raphson method (Eq. (2.77)). Another option is choosing 𝑒0 = 0, i.e. damage starts from the beginning, which allows for an immediate solution for 𝑒𝑑 but excludes prescribing 𝜖𝑐1 , which comes as a result. This option is followed and completes the calibration. All values involved are listed in Table 6.1. Table 6.1 Material parameters of Example 6.2. 𝐸0 [MN∕m2 ] 𝜈 [−] 𝑓𝑐 [MN∕m2 ] 𝑓𝑐𝑡 [MN∕m2 ] 𝛼 [−] 𝛽 [−] 𝛾 [−]

30 000 0.2 30 3 0.1 1.2 2.0

𝑒0 [−] 𝑒𝑑 [−] 𝑔𝑑 [−] 𝜖𝑐1 [−] 𝛼𝑐 [−] 𝛼𝑡 [−]

0 2.332 ⋅ 10−3 2 −1.65 ⋅ 10−3 −1 10

𝑐1 [−] 𝑐2 [−] 𝑐3 [−] 𝑐4 [−]

1.764 0.706 7.575 3.085

6.6 Damage

(a)

(b)

Figure 6.13 Example 6.2. (a) Uniaxial stress–strain curves. (b) Loading, unloading, and re-loading.

With respect to continuously increasing tension with 𝜖̇ 1 > 0, 𝐷̇ > 0, Eq. (6.110) is again derived from Eqs. (6.105) and (6.107) considering 𝜅𝑑 = 𝛼𝑡 𝜖1 leading to 𝜎1 = (1 − 𝐷) 𝐸0 𝜖1 = e

𝑔𝑑 𝛼 𝜖 −𝑒 −( 𝑡 1 0 ) 𝑒𝑑

𝐸0 𝜖1 ,

(6.115)

in the case of loading. The stress–strain relation from Eqs. (6.114) and (6.115) is shown in Figure 6.13a. • The stress–strain curve reproduces the empirical curves in Figure 3.1 for compression and Figure 3.3b for tension. • The computed initial value of Young’s modulus corresponds to the prescribed value 𝐸0 . • The uniaxial compressive strength is determined with 𝑓𝑐 = 30 MN∕m2 at a strain 𝜖𝑐1 = −0.0015 = −1.5‰ and the uniaxial tensile strength with 𝑓𝑐𝑡 = 3 MN∕m2 . Materials with damage differ from elasto-plastic materials in the case of unloading. This is demonstrated with a loading history: • Compression loading in a range −1.4 ⋅ 10−3 ≤ 𝜖3 ≤ 0 with 𝜖̇ 3 < 0, 𝐷̇ > 0, 𝐹 = 0. • Unloading in a range −1.4 ⋅ 10−3 ≤ 𝜖3 ≤ 0 with 𝜖̇ 3 > 0, 𝐷̇ = 0, 𝐹 = −𝜖3 − 𝜅𝑑′ < 0, i.e. 𝜖3 > −𝜅𝑑′ with 𝜅𝑑′ = 1.4 ⋅ 10−3 . • Change of index in stress and strain from 3 to 1 due to change from compressive into tensile regime. The physical direction is actually not changed. • Elastic re-loading ranging from 0 ≤ 𝜖1 ≤ 𝜅𝑑′ ∕𝛼𝑡 with 𝜖̇ 1 > 0, 𝐷̇ = 0, 𝐹 = 𝜖1 𝛼𝑡 − 𝜅𝑑′ < 0. • Resumed loading ranging from 𝜅𝑑′ ∕𝛼𝑡 ≤ 𝜖1 ≤ 0.5 ⋅ 10−3 with 𝜖̇ 1 > 0, 𝐷̇ > 0, 𝐹 = 0. The resulting stress–strain curve is shown in Figure 6.13b. In contrast to elastoplasticity, there are no permanent strains after unloading to zero stress. But the unloading stiffness – actually a secant stiffness – is reduced in contrast to the unloading stiffness of elasto-plasticity. Although a uniaxial application is demonstrated, the forms Eqs. (6.105), (6.107), and (6.109) with the material parameters from Table 6.1 immediately allow for a multi-axial usage.

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The incremental form of the material law for damage (Eq. (6.105)) remains to be added. The derivative with respect to time 𝑡 is given by 𝝈̇ = (1 − 𝐷) E ⋅ 𝝐̇ − E ⋅ 𝝐 𝐷̇ = (1 − 𝐷) E ⋅ 𝝐̇ − 𝝈0 𝐷̇

(6.116)

To derive the rate of 𝐷̇ we consider 𝐷 as the function of equivalent strain 𝜅𝑑 (Eq. (6.107)). A general form is determined by d𝐷

d𝐷 d𝜅𝑑 𝜕𝐹 𝐷̇ = 𝜅̇ 𝑑 = − 𝜕𝐹 ⋅ 𝝐̇ 𝜕𝝐 d𝜅𝑑

(6.117)

𝜕𝜅𝑑

using Eq. (6.113) for 𝜅̇ 𝑑 . Thus, Eq. (6.116) can be written as 𝝈̇ = C𝑇 ⋅ 𝝐̇

(6.118)

with

C𝑇 =

⎧ ⎪(1 − 𝐷) E + ⎨ ⎪(1 − 𝐷) E ⎩

The form 𝝈0 ⊗

𝜕𝐹 𝜕𝝐

d𝐷 d𝜅𝑑 𝜕𝐹 𝜕𝜅𝑑

𝝈0 ⊗

𝜕𝐹 𝜕𝝐

for loading

(6.119)

unloading

is again an outer or dyadic product of two vectors in Voigt nota-

tion. The quantities

𝜕𝐹 𝜕𝝐

,

𝜕𝐹

,

d𝐷

𝜕𝜅𝑑 d𝜅𝑑

have to be computed from the forms for 𝐹 and 𝐷.

Similarly to elasto-plasticity (Section 6.5.1) the isotropy is generally lost with deviating principal directions of stress and strain, as a tangential material stiffness matrix C𝑇 no longer follows the form required by Eq. (6.22). At first glance, this seems to contradict Eq. (6.105), which provides a scaling of the isotropic elasticity matrix Eq. (6.23). But the scaling factor changes during the loading history, which might disturb isotropy. Considering Eq. (6.116) the isotropy is only preserved in the case of proportional strains and strain increments, i.e. 𝝐̇ = 𝑓 𝝐 with a scalar function 𝑓 of the loading history. A further topic concerns material stability. Materials with limited strength might reach states that allow for a bifurcation of strain increments for a given stress increment. The theoretical background is treated in Appendix B. The previous Example 6.2 is complemented with the following. Example 6.3: Stability of Hsieh–Ting–Chen Uniaxial Damage

We refer to Example 6.2 where each incremental loading step is examined with regard to its stability behaviour. This is ruled by the tangential material stiffness (Eq. (6.119)) applied to the Hsieh–Ting–Chen damage. The respective functions 𝐷 and 𝐹 are given by Eqs. (6.107) and (6.109). A biaxial plane stress is considered to reduce the problem to 2D. The onset of bifurcation or weak discontinuity is ruled by Eqs. (B.17) with the determinant of the

6.6 Damage

acoustic matrix Q from Eq. (B.14). One should keep in mind that numerics of material stability might involve pitfalls, and the numerical results should be treated with caution. Uniaxial tension is considered first. With respect to Example 6.2, a tensile bifurcation is indicated with 𝜎 = 3 MN∕m2 corresponding to the tensile strength. This is connected with symmetric orientations for the line of discontinuity with 𝜙 = ±17.2° and leads to a normalised strain discontinuity tensor ⎡ 0.3102 ∓0.4040⎤ Δ𝝐 = ⎢ (6.120) ⎥ ∓0.4040 −0.2790 ⎣ ⎦ derived from Eqs. (B.6) and (B.18). This indicates a mixture of shear with slightly dominating tension with principal strains 𝜖1 = 0.5156, 𝜖2 = −0.4844 and an orientation 𝜑1 = 27.0° of the major principal strain. A compressive bifurcation is indicated with 𝜎 = −28.5 MN∕m2 , which is slightly below the prescribed uniaxial compressive strength 𝑓𝑐 = 30 MN∕m2 (Figure 6.13a). This is connected with discontinuity line orientations 𝜙 = ±43.3° and normalised strain discontinuity tensors ⎡−0.3549 ±0.1508⎤ Δ𝝐 = ⎢ (6.121) ⎥ ±0.1508 0.5985 ⎣ ⎦ indicating a mixture of shear strain and compression strain with principal strains 𝜖1 = −0.3782, 𝜖2 = 0.6218 and an orientation 𝜑1 = 8.8° of the compressive principal strain. The foregoing descriptions cover basic ideas of damage. A comprehensive treatment is given in Lemaitre and Desmorat (2005). Several refinements have been developed as concerns the characteristics of concrete behaviour. • Tensile states are distinguished from compressive states using positive and negative projections of strain. Such projections are determined using the spectral decomposition of the strain tensor. Tensile and compressive damage is assigned to each projection, and each damage type acts independently. This considers the effect that concrete retains its compressive stiffness and strength after tensile loading. • Isotropic damage assigns the same stiffness degradation in every material orientation. Such an approach cannot capture a load-induced anisotropy (Section 6.1.2). Anisotropic damage introduces a stiffness degradation, which depends on the orientation within a material. A special but relatively convenient form of anisotropic damage is given by orthotropic damage. This may be realised with the degradation of Young’s moduli of the orthotropic compliance (Eq. (6.26)). Corresponding implementations are given in, e.g. Govindjee et al. (1995); Carol et al. (2001); Desmorat et al. (2007); Kitzig and Häussler-Combe (2011); Desmorat (2016).

189

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6 Multi-Axial Concrete Behaviour

6.7 Damaged Elasto-Plasticity An uniaxial form of damaged elasto-plasticity has already been given with Eq. (6.60). This is extended to a multi-axial form of isotropically damaged plasticity by ( ) 𝝈 = (1 − 𝐷) E ⋅ 𝝐 − 𝝐 𝑝

(6.122)

with stresses 𝝈 (Eq. (6.7)), strains 𝝐 (Eq. (6.3)), elasticity matrix E (Eq. (6.23)), damage variable 𝐷 with the range 0 ≤ 𝐷 ≤ 1, and multi-axial plastic strains 𝝐 𝑝 . We described stress-based elasto-plasticity (Section 6.5) and strain-based isotropic damage (Section 6.6) above. A strain-based damage is generally preferred due to its conceptual simplicity compared to stress-based damage (Section 6.6). An approach combining strain-based damage with strain-based plasticity is given by HäusslerCombe and Hartig (2008). On the other hand, stress-based elasto-plasticity is generally preferred. A general form combining strain-based damage and stress-based elasto-plasticity is described in the following. To start with, we summarise isotropic damage with the inclusion of plastic strains 𝝐 𝑝 . The scalar damage variable 𝐷 depends on an equivalent damage strain 𝜅𝑑 𝐷 = 𝐻(𝜅𝑑 )

(6.123)

with a material function 𝐻 and the constraints 𝜅𝑑 ≥ 0 and 0 ≤ 𝐷 ≤ 1. The equivalent strain is ruled by multi-axial strains with a damage function 𝐹𝑑 (c𝑑 , 𝜅𝑑 , 𝝐, 𝝐 𝑝 ) = 0

(6.124)

with a set of material parameters c𝑑 that characterise a material grade, e.g. strength, of a material type, e.g. concrete. Damage evolution is ruled by the Kuhn–Tucker conditions 𝐹𝑑 ≤ 0 ,

𝐹𝑑 𝜅̇ 𝑑 = 0 ,

𝜅̇ 𝑑 ≥ 0

(6.125)

with the time derivative 𝜅. ̇ Due to this condition damage will only increase with continuing 𝐹𝑑 = 0, which implies 𝐹̇ 𝑑 = 0 and a consistency condition 𝜕𝐹𝑑 𝜕𝐹𝑑 𝜕𝐹𝑑 𝜅̇ 𝑑 + ⋅ 𝝐̇ = 0 ⋅ 𝝐̇ + 𝜕𝜅𝑑 𝜕𝝐 𝜕𝝐 𝑝 𝑝

(6.126)

This yields the evolution law for the equivalent damage strain 𝜅𝑑 𝜕𝐹𝑑 1 𝜕𝐹𝑑 𝜅̇ 𝑑 = − 𝜕𝐹 [ ⋅ 𝝐̇ ] ⋅ 𝝐̇ + 𝑑 𝜕𝝐 𝜕𝝐 𝑝 𝑝

(6.127)

𝜕𝜅𝑑

depending on the change of strains 𝝐, ̇ 𝝐̇ 𝑝 with gradients 𝜕𝐹𝑑 ∕𝜕𝝐, 𝜕𝐹𝑑 ∕𝜕𝝐 𝑝 . The implementation within a time integration scheme (Section 2.8.2) results in scalar damage 𝐷, where the plastic strain 𝝐 𝑝 is assumed to be known. Actually, a trial value is used.

6.7 Damaged Elasto-Plasticity

An improved value for 𝝐 𝑝 is derived from the elasto-plastic stress-based approach. This introduces effective stresses. Damage ‘homogenises’ microcracks (Figure 6.1b). Effective stresses are defined with 𝝈=

1 𝝈 1−𝐷

(6.128)

and nominally exclude microcracks from stress transfer, whereas stresses act on the faultless continuum only. Hence, we summarise elasto-plasticity with the inclusion of effective stresses. Multi-axial plastic strains 𝝐 𝑝 correspond to a scalar equivalent plastic strain 𝜅𝑝 . This internal state variable is again derived in a sequence of steps in analogy to damage. The equivalent plastic strain is ruled by effective stresses with a yield condition 𝐹𝑝 (c𝑝 , 𝜅𝑝 , 𝝈) = 0

(6.129)

with a further set of material parameters c𝑝 that characterise a material grade of a material type. A yield condition becomes a strength limit condition when it reaches its largest extension. Plastic evolution is again ruled by the Kuhn–Tucker conditions 𝐹𝑝 ≤ 0 ,

𝐹𝑝 𝜅̇ 𝑝 = 0 ,

𝜅̇𝑝 ≥ 0

(6.130)

Due to this condition plastic deformations will only occur with continuing 𝐹𝑝 = 0, which implies 𝐹̇ 𝑝 = 0 and a consistency condition 𝜕𝐹𝑝 𝜕𝜅𝑝

𝜅̇ 𝑝 +

𝜕𝐹𝑝 𝜕𝝈

( ) 𝝈̇ = E ⋅ 𝝐̇ − 𝝐̇ 𝑝

⋅ 𝝈̇ = 0 ,

(6.131)

Plastic strains are derived from a flow rule 1) 𝝐̇ 𝑝 =

𝜕𝐺(𝝈) 𝜅̇ 𝑝 𝜕𝝈

(6.132)

Equations (6.131) and (6.132) are combined for the plastic strain evolution 𝐴=−

𝜕𝐹𝑝 𝜕𝜅𝑝

+

𝜕𝐹𝑝

⋅E⋅

𝜕𝐺 𝜕𝝈

𝜕𝝈 𝜕𝐹 𝑝 1 𝜅̇ 𝑝 = ⋅ E ⋅ 𝝐̇ 𝐴 𝜕𝝈 𝜕𝐹𝑝 1 𝜕𝐺 𝝐̇ 𝑝 = ⊗ ⋅ E ⋅ 𝝐̇ 𝐴 𝜕𝝈 𝜕𝝈

(6.133)

with the dyadic product ⊗. This leads to a corrector 𝝐̇ 𝑝 for plastic strains and, furthermore, to damage 𝐷 (Eq. (6.123)) and effective stress 𝝈 (Eq. (6.128)). This is a first sketch. A greater degree of complexity is behind it. Nested iterations on a material point level are involved: in inner iteration to determine damage parameters embedded in outer iteration for plastic strains. Convergence issues arise, 1) This deviates from Eq. (6.63) to simplify the approach.

191

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and divergence of iterations regarding systems of nonlinear equations on the material point level are not exceptional. The definition of material functions 𝐻, 𝐹𝑑 , 𝐹𝑝 , 𝐺 for a material type and the calibration of their respective material grade parameters are required as a prerequisite. We omit details of the derivation of the tangential material stiffness for brevity. It basically follows from the time derivation of Eq. (6.122) ) ( )] [ ( 𝝈̇ = E ⋅ −𝐷̇ 𝝐 − 𝝐 𝑝 + (1 − 𝐷) 𝝐̇ − 𝝐̇ 𝑝

(6.134)

with 𝐷̇ specified according to Eq. (6.117) and 𝝐̇ 𝑝 according to Eqs. (6.63) and (6.69). A widely used proposal for strain-based damage combined with stress-based elasto-plasticity is fully explicated by Lubliner et al. (1989); Lee and Fenves (1998). For further approaches, see, e.g. Grassl and Jirásek (2006); Voyiadjis et al. (2008); Grassl et al. (2013).

6.8 The Microplane Model A deeper inspection of the mesoscales or microscales of materials (Section 6.1.1) reveals that stiff parts embedded in a matrix interact in microscopic boundary layers; see Figure 6.14a. The microplane concept represents these layers by a variety of microscopic planes. Stress and strain vectors are defined with respect to these planes, and material laws are formulated to relate them. A superposition of microplanes finally yields homogenised relations between strains and stresses. From another viewpoint, microplanes may be imagined as the tangent planes of a sphere surrounding every continuum point; see Figure 6.14b. A basic question of the microplane concept is how stress and strain tensors are related to stress and strain vectors. A local approach for stresses is given by Eq. (6.5) t = 𝝈𝑀 ⋅ n

(a)

(6.135)

(b)

Figure 6.14 Microplane. (a) Interaction layers (Bažant et al. 2000), and (b) Unit sphere (Kuhl, Ramm and Willam 2000).

6.8 The Microplane Model

called the static constraint with the microplane normal n. The same approach can be used for strains leading to a strain vector e = 𝝐𝑀 ⋅ n

(6.136)

called the kinematic constraint with the strain tensor 𝝐 𝑀 according to Eq. (6.2). As local behaviour is generally driven by prescribed strains derived from a global FEM calculation, the kinematic constraint is chosen as a basis. The corresponding V-D split (Leukart and Ramm 2006) is exemplarily described. Tensorial strain components (Eq. (6.2)) are involved in the following. The Einstein summation convention 2) is used in this section to allow for a compact notation. Lower indices have a range 1 … 3. The macroscopic strain tensor 𝝐 is given with components 𝜖𝑖𝑗 (Eq. (6.1)). It is split into volumetric and deviatoric parts vol dev + 𝜖𝑖𝑗 , 𝜖𝑖𝑗 = 𝜖𝑖𝑗

vol 𝜖𝑖𝑗 =

1 𝜖 𝛿 , 3 𝑘𝑘 𝑖𝑗

dev 𝜖𝑖𝑗 = 𝜖𝑖𝑗 −

1 𝜖 𝛿 3 𝑘𝑘 𝑖𝑗

(6.137)

with the Kronecker delta 𝛿𝑖𝑗 ⎧ 1 𝑖=𝑗 𝛿𝑖𝑗 = ⎨0 𝑖 ≠ 𝑗 ⎩

(6.138)

Deviatoric strains are rewritten as dev = Idev 𝜖𝑖𝑗 𝑖𝑗𝑘𝑙 𝜖𝑘𝑙 ,

dev

I𝑖𝑗𝑘𝑙 = (

) 1 1( 𝛿𝑖𝑘 𝛿𝑗𝑙 + 𝛿𝑖𝑙 𝛿𝑗𝑘 − 𝛿𝑖𝑗 𝛿𝑘𝑙 ) 2 3

(6.139)

with the fourth-order deviatoric unit tensor Idev with components Idev 𝑖𝑗𝑘𝑙 . A plane is considered with a normal vector n with components 𝑛𝑖 . In Eq. (6.136), the strain vector on this plane can be split into components ) ( 1 vol dev 𝑛𝑗 = 𝜖𝑘𝑘 𝑛𝑖 + D𝑖𝑘𝑙 𝜖𝑘𝑙 = 𝑒 𝑉 𝑛𝑖 + 𝑒𝑖𝐷 𝑒𝑖 = 𝜖𝑖𝑗 + 𝜖𝑖𝑗 3

(6.140)

whereas D𝑟𝑖𝑗 =

] 1 1[ 𝑛 𝛿 + 𝑛𝑗 𝛿𝑟𝑖 − 𝑛𝑟 𝛿𝑖𝑗 2 𝑖 𝑟𝑗 3

(6.141)

with the scalar volumetric strain 𝑒 𝑉 and the deviatoric strain e𝐷 vector with components 𝑒𝑖𝐷 related to the plane with normal n; see Figure 6.15a. Thus, the projection of the macroscopic strain tensor into microplane strains is given by 𝑒𝑖𝐷 = D𝑖𝑟𝑠 𝜖𝑟𝑠 ,

𝑒𝑉 =

1 𝜖 = 𝑉𝑟𝑠 𝜖𝑟𝑠 , 3 𝑘𝑘

𝑉𝑟𝑠 =

1 𝛿 3 𝑟𝑠

(6.142)

which yields the volumetric-deviatoric (V-D) split. 2) This convention stipulates that whenever a letter index appears twice in an expression, the sum is to be taken over this index, e.g. 𝑎𝑘𝑘 = 𝑎11 + 𝑎22 + 𝑎33 or 𝑎𝑖𝑟 𝑏𝑟𝑗 = 𝑎𝑖1 𝑏1𝑗 + 𝑎𝑖2 𝑏2𝑗 + 𝑎𝑖3 𝑏3𝑗 .

193

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6 Multi-Axial Concrete Behaviour

(a)

(b)

Figure 6.15 (a) V-D-split. (b) Microplanes by triangularisation of the unit sphere.

Microplane strains lead to microplane stresses 𝑡𝑉 = 𝑓 𝑉 (𝑒𝑉 , e𝐷 ) ,

𝑡𝑖𝐷 = 𝑓𝑖𝐷 (𝑒𝑉 , e𝐷 )

(6.143)

with microplane material laws 𝑓 𝑉 , 𝑓𝑖𝐷 . The macroscopic stress tensor 𝝈 is related to microplane stresses with the virtual work principle (Eq. (2.5)) within a unit sphere of radius 1 with a surface area 𝛤 ( ) ( ) 4π 𝜎 𝛿𝜖 = ∫ 𝑡𝑉 𝛿𝜖 𝑉 + 𝑡𝑟𝐷 𝛿𝜖𝑟𝐷 d𝛤 = ∫ 𝑡𝑉 𝑉𝑖𝑗 + 𝑡𝑟𝐷 D𝑟𝑖𝑗 d𝛤 𝛿𝜖𝑖𝑗 3 𝑖𝑗 𝑖𝑗 𝛤

(6.144)

𝛤

This has to be fulfilled by arbitrary 𝛿𝜖𝑖𝑗 . Furthermore, regarding the symmetry of the projection of the macroscopic strain tensor with respect to n and −n results in 𝜎𝑖𝑗 =

( ) 3 ∫ 𝑡𝑉 𝑉𝑖𝑗 + 𝑡𝑟𝐷 D𝑟𝑖𝑗 d𝛤 2π

(6.145)

𝛤ℎ

with the unit half-sphere surface area 𝛤ℎ . This is integrated numerically with the unit half-sphere approximated by, e.g. triangles forming microplane samples; see Figure 6.15b. ◀

Each microplane sample has its own particular microplane strain and stress. This naturally models anisotropic behaviour in general and load-induced anisotropy in particular.

For numerical integration schemes, see Bažant and Oh (1985). Integration order is an issue that might influence the results significantly. Alternatives for the V-D-split are available with the normal-tangential (N-T) split (Bažant and Oh 1985) and the volumetric-deviatoric-tangential (V-D-T) split (Bažant et al. 2000). Specifications of microplane material laws are exemplarily described in the following. Microplane elasticity is assumed with 𝑡𝑉 = 𝐸 𝑉 𝑒𝑉 ,

𝑡𝑖𝐷 = 𝐸 𝐷 𝑒𝑖𝐷

(6.146)

6.8 The Microplane Model

with constant elastic modules 𝐸 𝑉 , 𝐸 𝐷 . Inserting Eqs. (6.146) and (6.142) into Eq. (6.145) yields 𝜎𝑖𝑗 = [𝐸𝑉

3 3 ∫ 𝑉 𝑉 d𝛤 + 𝐸𝐷 ∫ D D d𝛤] 𝜖𝑟𝑠 2π 𝛤 𝑖𝑗 𝑟𝑠 2π 𝛤 𝑘𝑖𝑗 𝑘𝑟𝑠 ℎ

(6.147)

ℎ

The evaluation of the integrals results in 3 ∫ 𝑉 𝑉 d𝛤 = Ivol 𝑖𝑗𝑟𝑠 , 2π 𝛤 𝑖𝑗 𝑟𝑠 ℎ

3 ∫ D D d𝛤 = Idev 𝑖𝑗𝑟𝑠 2π 𝛤 𝑘𝑖𝑗 𝑘𝑟𝑠

(6.148)

ℎ

with the fourth-order deviatoric unit tensor Idev (Eq. (6.139)) and the volumetric fourth-order unit tensor Ivol with components Ivol 𝑖𝑗𝑟𝑠 = 𝛿𝑖𝑗 𝛿𝑟𝑠 ∕3. This leads to 𝜎𝑖𝑗 = 𝐾𝑖𝑗𝑟𝑠 𝜖𝑟𝑠 ,

𝐷 dev 𝐾𝑖𝑗𝑟𝑠 = 𝐸 𝑉 Ivol 𝑖𝑗𝑟𝑠 + 𝐸 I𝑖𝑗𝑟𝑠

(6.149)

and finally to 𝐸 𝑉 = 3𝐾 =

𝐸 , 1 − 2𝜈

𝐸 𝐷 = 2𝐺 =

𝐸 1+𝜈

(6.150)

with the bulk modulus 𝐾, shear modulus 𝐺, Young’s modulus 𝐸, and Poisson’s ratio 𝜈 of an isotropic linear elastic material (Eq. (6.23)). Thus, microplane elastic modules are uniquely determined from macroscopic elastic modules with the V-Dsplit. This approach reproduces isotropic linear elastic behaviour by the definitions of Eq. (6.150). This is extended with respect to microplane damage with the approach 𝑡𝑉 = (1 − 𝐷) 𝐸 𝑉 𝑒 𝑉 ,

𝑡𝑖𝐷 = (1 − 𝐷) 𝐸 𝐷 𝑒𝑖𝐷

(6.151)

with a common damage variable 0 ≤ 𝐷 ≤ 1 for volumetric and deviatoric parts 𝐷(𝜅) =

⎧0 ⎨ 1−e ⎩

𝑔 𝜅−𝑒0 𝑑 −( ) 𝑒𝑑

𝜅 ≤ 𝑒0 (6.152) 𝜅 > 𝑒0

with the equivalent microplane damage strain 𝜅 and material parameters 𝑒𝑑 , 𝑒0 , 𝑔𝑑 . This follows Eq. (6.107) for isotropic macroscopic damage but is individual for each microplane. The equivalent strain is related to microplane strains by a damage function (de Vree et al. 1995) √ 3 2 (6.153) 𝐹(𝜅, 𝑒 𝑉 , e𝐷 ) = 𝑘0 3𝑒 𝑉 + (𝑘1 3𝑒 𝑉 ) + 𝑘2 e𝐷 ⋅ e𝐷 − 𝜅 2 whereby 𝑘0 = 𝑘1 =

𝑘−1 , 2𝑘(1 − 2𝜈)

𝑘2 =

3 𝑘(1 + 𝜈)2

(6.154)

195

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6 Multi-Axial Concrete Behaviour

with a material parameter 𝑘 ruling the ratio of uniaxial compressive and tensile strength. Using Eqs. (6.142) such a microplane form can be translated into an isotropic – assuming the same behaviour of all microplanes – macroscopic form. This exposes the invariants 𝐼1 , 𝐽2 (Eq. (6.19)), i.e. an influence of hydrostatic and deviatoric length (Eqs. (6.43) and (6.44)). The Lode angle (Eq. (6.46)) is not considered. Thus, a different strength along compressive and tensile meridians (Figures 6.6 and 6.11) cannot be described. This also becomes evident when regarding individual microplanes and is linked to the V-D-split. The approach is completed with the Kuhn– Tucker-conditions 𝐹 ≤ 0,

𝜅̇ ≥ 0 ,

𝐹 𝜅̇ = 0

(6.155)

ruling loading and unloading in analogy to Eq. (6.112). A macroscopic tangential material stiffness matrix C𝑇 is needed for the tangential element stiffness matrix (Eq. (2.66)) assembled to a system stiffness matrix (Eq. (2.67)) for iterative nonlinear equation solving (Eq. (2.76)). This is based on the time derivation of Eq. (6.145) 𝜎̇ 𝑖𝑗 =

( ) 3 ∫ 𝑡̇ 𝑉 𝑉𝑖𝑗 + 𝑡̇ 𝑘𝐷 D𝑘𝑖𝑗 d𝛤 2π

(6.156)

𝛤ℎ

and is elaborated using Eqs. (6.142), (6.152), and (6.153) as 𝑡̇ 𝑉 = (1 − 𝐷) 𝐸 𝑉 𝑒̇ 𝑉 − 𝐷̇ 𝐸 𝑉 𝜖 𝑉 𝑡̇ 𝐷 = (1 − 𝐷) 𝐸 𝐷 𝑒̇ 𝐷 − 𝐷̇ 𝐸 𝐷 𝑒 𝐷 𝑖

𝑖

𝐷̇ =

d𝐷 𝜅̇ d𝜅

𝜅̇ =

𝜕𝐹 𝑉 𝜕𝐹 𝑒̇ + 𝐷 𝑒̇ 𝑖𝐷 𝑉 𝜕𝑒 𝜕𝑒𝑖

𝑖

= (1 − 𝐷) 𝐸 𝑉 𝑉𝑟𝑠 𝜖̇ 𝑟𝑠 − 𝑡0𝑉 𝐷̇

𝐷 ̇ 𝐷 = (1 − 𝐷) 𝐸 𝐷 D𝑖𝑟𝑠 𝜖̇ 𝑟𝑠 − 𝑡0𝑖

(6.157) =(

𝜕𝐹 𝜕𝐹 𝑉𝑟𝑠 + 𝐷 D𝑖𝑟𝑠 ) 𝜖̇ 𝑟𝑠 𝑉 𝜕𝑒 𝜕𝑒𝑖

with 𝐷̇ > 0 in the case of loading and 𝐷̇ = 0 in the case of unloading. After calculation and numerical integration over sample microplanes, a form 𝜎̇ 𝑖𝑗 = 𝐶𝑇,𝑖𝑗𝑟𝑠 𝜖̇ 𝑟𝑠

(6.158)

is obtained, which is again rearranged into vectors and a matrix 𝝈̇ = C𝑇 ⋅ 𝝐̇

(6.159)

using the Voigt notation (Eqs. (6.3) and (6.7)). The uniaxial stress–strain behaviour of this microplane model is demonstrated with the following example. Example 6.4: Microplane Uniaxial Stress–Strain Relations with De Vree Damage

We assume strain-based damage with the damage variable according to Eq. (6.152) and the damage function according to de Vree Eq. (6.153). The elastic properties are derived from Eq. (6.150).

6.8 The Microplane Model

This leaves Young’s modulus 𝐸 and Poisson’s ratio 𝜈, furthermore the damage parameters 𝑒0 , 𝑒𝑑 , 𝑔𝑑 , and finally the strength ratio 𝑘 as parameters to be prescribed. With respect to damage, the direct application of Eq. (6.152) for uniaxial tension 𝜎(𝜖) = 𝐸

⎧𝜖,

𝜖 ≤ 𝑒0

⎨ e ⎩

𝜖 , 𝜖 > 𝑒0

𝑔 𝜖−𝑒0 𝑑 −( ) 𝑒𝑑

(6.160)

is used for a calibration. As for isotropic damage (Example 6.2), it is generally found appropriate to assume 𝑔𝑑 = 2. The remaining parameters 𝑒0 , 𝑒𝑑 may be determined from the condition that a prescribed uniaxial tensile strength 𝑓𝑐𝑡 is reached at a prescribed corresponding strain. The latter is determined with d𝜎∕ d𝜖 = 0 and used with Eq. (6.160) to obtain 𝑒𝑑 from 𝑓𝑐𝑡 . This may be further simplified assuming 𝑒0 = 0, i.e. with initial onset of damage leading to a direct relation between 𝑒𝑑 and 𝑓𝑐𝑡 , whereby obtaining the strength strain 𝜖𝑐𝑡 (Figure 3.3b) as result. Finally, the strength ratio 𝑘 must be determined by an iterative inverse approach to obtain a prescribed uniaxial compressive strength. The whole set of parameters is given in Table 6.2. In contrast to uniaxial isotropic damage (Example 6.2), the uniaxial microplane stress–strain relations cannot be specified directly, as they relate to multiple planes. Thus, a set-up with one plane stress quad element (Section 2.3) is used. The uniaxial stress–strain curves computed are shown in Figure 6.16a. This basically reproduces typical uniaxial concrete behaviour (Figures 3.1 and 3.3b) and corresponds to uniaxial isotropic damage behaviour (Figure 6.13a). The uniaxial tensile strength is nuTable 6.2 Example 6.4. Material parameters 𝐸0 [MN∕m2 ] 𝜈 [−] 𝑓𝑐 [MN∕m2 ] 𝑓𝑐𝑡 [MN∕m2 ] 𝑘 [−]

(a)

30 000 0.2 30 3 13

𝑒0 [−] 𝑒𝑑 [−] 𝑔𝑑 [−] 𝜖𝑐𝑡 [−]

0 2.332 ⋅ 10−4 2 −1.65 ⋅ 10−4

(b)

Figure 6.16 Example 6.4. (a) Uniaxial stress–strain curves. (b) Damage on microplanes.

197

198

6 Multi-Axial Concrete Behaviour

merically determined with 2.99 MN∕m2 , which fits to the prescribed value although using the simplified calibration described above. The computed uniaxial compressive strength 30.3 MN∕m2 deviates slightly from the prescribed value. This can be corrected by more inverse iterations with respect to the strength ratio 𝑘. The computed damage for microplanes – each represented by a 3D arrow – upon reaching the uniaxial tensile strength is shown in Figure 6.16b. Only one quarter of the half-sphere is included due to symmetry reasons, which leaves nine microplanes out of 21. The first three values of each arrow indicate the normal direction of the respective microplane, the last value the computed damage. The latter also scales the arrow length. In contrast to isotropic damage, a load-induced anisotropy becomes evident with different – apart from symmetry matches – damage values for microplanes. Finally, the issue of material stability should be noted. The theoretical background is described in Appendix B, a first Example 6.3 is given regarding isotropic damage. The introductory remarks from there also apply here. A tensile bifurcation is indicated with 𝜎 = 2.97 MN∕m2 , which is slightly below the tensile strength. This is connected with a unique orientation for the line of discontinuity with 𝜙 = 0° and leads to a normalised strain discontinuity tensor ⎡1 Δ𝝐 = ⎢ 0 ⎣

0⎤ ⎥ 0 ⎦

(6.161)

derived from Eqs. (B.6) and (B.18) and corresponds to an extensional strain jump in the direction of loading. A discussion of the different tensile stability behaviours of isotropic damage (Eq. (6.120) exceeds the current scope and is not pursued further. A compressive bifurcation is indicated with 𝜎 = −29.9 MN∕m2 slightly below the uniaxial compressive strength. This is connected with discontinuity line orientations 𝜙 = ±40.1° and normalised strain discontinuity tensors ⎡−0.3881 Δ𝝐 = ⎢ ±0.1659 ⎣

±0.1659⎤ ⎥ 0.5552 ⎦

(6.162)

indicating a mixture of contracting and shear strain bifurcation with principal strains 𝜖1 = −0.4164, 𝜖2 = 0.5836, and orientations 𝜑1 = ±9.7° of the compressive principal strain. This is similar to the compressive bifurcation behaviour of isotropic damage (Eq. (6.121)). The formats of elasticity, plasticity, damage, and combinations thereof may be used to formulate relations between microplane strains and microplane stresses. Further damage approaches for the microplane model are given in Kuhl, Ramm and de Borst (2000); Leukart and Ramm (2006), an elasto-plastic approach in Kuhl, Ramm and Willam (2000). For a later advanced state of the popular M-models, see Caner and Bažant (2013). The microplane model shows similarities to the smeared crack model; see Section 7.4. Both work with vector or scalar measures of stress and strain. While the

6.9 General Requirements for Material Laws

smeared crack model considers at most very few planes to apply scalar relations, the microplane model basically uses all orientations. Thus, it does not need the tensorial relations that the smeared crack model does in order to have an embedding frame. The microplane model has the major advantage that it models initial or load-induced anisotropy in a natural and relatively simple way.

6.9 General Requirements for Material Laws General requirements for material laws fall under the two categories objectivity and thermodynamic restrictions. With respect to solids, the basics of thermodynamics are given with the following postulates and neglecting temperature aspects. 1. The change of internal energy 𝑈̇ of a solid body should be equal to the power 𝑃̇ applied by external forces plus the change in kinetic energy 𝐾̇ 𝑈̇ = 𝑃̇ + 𝐾̇

(6.163)

This is stated for a whole body and may be transformed into a local formulation using the Gauss divergence theorem (Kreyszig (2006, 10.7), Malvern (1969, 5.4)) 𝑢̇ = 𝝈T ⋅ 𝝐̇

(6.164)

with the specific internal energy rate 𝑢, ̇ the Cauchy stress 𝝈, and the strain rate 𝝐. ̇ The first postulate provides a definition for the specific internal energy. 2. Diverse formulations exist for the second postulate. For solid materials, the specific internal energy is split into a recoverable Helmholtz energy 𝜓(𝝐) depending on the strain 𝝐 only and into a dissipated energy 𝑑 with 𝑢̇ = 𝜓̇ + 𝑑̇

(6.165)

The Clausius–Duhem inequality postulates 𝑑̇ = 𝑢̇ − 𝜓̇ ≥ 0

(6.166)

i.e. energy dissipated in materials should not be negative. In contrast to the first postulate, the second postulate might impose restrictions for the formulation of stress–strain relations. A stress–strain relation not fulfilling the Clausius–Duhem inequality is not regarded as suitable to describe local material behaviour. The fulfilment can generally be proven for simple nonlinear stress–strain relations. This is shown for isotropic damage (Section 6.6). The key is the formulation of the specific recoverable energy 𝜓. In the case of isotropic damage with a stress–strain relation according to Eq. (6.105) it is given by 𝜓=

1 (1 − 𝐷) 𝝐 T ⋅ E ⋅ 𝝐 2

(6.167)

199

200

6 Multi-Axial Concrete Behaviour

with the scalar damage parameter 𝐷 and the linear elastic material stiffness E according to Eq. (6.23). The rate is given by 𝐷̇ 𝜓̇ = − 𝝐 T ⋅ E ⋅ 𝝐 + (1 − 𝐷) 𝝐 T ⋅ E ⋅ 𝝐̇ 2 𝐷̇ T = − 𝝐 ⋅ E ⋅ 𝝐 + 𝝈T ⋅ 𝝐̇ 2

(6.168)

The combination with Eqs. (6.164) and (6.165) results in a dissipation rate 𝐷̇ 𝑑̇ = 𝝐 T ⋅ E ⋅ 𝝐 2

(6.169)

Thus, 𝑑̇ ≥ 0 as 𝐷̇ ≥ 0 by definition (Eq. (6.112)) and 𝝐 T ⋅ E ⋅ 𝝐 ≥ 0, like the matrix E, is positive definite. A further restriction may be formulated with the Drucker stability postulate (Malvern (1969, 6.6 Part 2), Bower (2010, 3.1)). It states that T

(6.170)

T

(6.171)

𝝐̇ ⋅ 𝝈̇ ≥ 0 or 𝝐̇ ⋅ C𝑇 ⋅ 𝝐̇ ≥ 0

using the tangential material stiffness matrix C𝑇 (Eq. (2.50)) and implying its positive definiteness. This is obviously not fulfilled for softening materials of damage type. In Example 6.2 with uniaxial compressive concrete behaviour, we see 𝜎̇ > 0 and 𝜖̇ < 0 (Figure 6.13a), and Drucker’s postulate is violated. This corresponds to the fact that softening materials need a regularisation (Section 7.3) to reach discretisation objectivity when used in numerical methods. A regularisation is not required for materials fulfilling the Drucker stability postulate. Objectivity or material frame indifference of stress–strain relations must be given with respect to rotating and translating coordinate systems (Malvern 1969, 6.7). We consider a solid body with boundary conditions preventing rigid body displacements and with some loading applied in a quasi-static state. This leads to strains 𝝐 and stresses 𝝈. The stress–strain relations are denoted with 𝝈 = f(𝝐). An arbitrary rotation is applied to this set-up. Depending on the reference configuration – Lagrangian or Eulerian (Malvern 1969, 4.5) – such rotations may be considered as equivalent with particular types of coordinate transformations and require a transformation of the position of the body, of the applied forces, and of strains and stresses. Transformation rules for plane states are given in Appendix D. The transformation of strains is denoted with 𝝐 → ˜ 𝝐 and the transformation of stresses with 𝝈 → 𝝈 ˜. Objectivity requires that f(˜ 𝝐 ) yields 𝝈 ˜ with unchanged function f. This issue is relevant for Eulerian reference configurations with incremental material laws according to Eq. (6.12) and large displacements leading to the formulation of co-rotational or objective stress rates (Belytschko et al. 2000, 3.7). This issue is of minor relevance within the context of reinforced concrete structures.

201

7 Crack Modelling and Regularisation As concerns multi-axial macroscopic behaviour, up to now we restricted ourselves to material samples exposed to continuous – preferably homogenous – deformations. But due to its limited tensile strength concrete is characterised by cracking as a regular behaviour in the tensile range. Concrete cracking is not a sudden event but occurs in a process that gradually leads from continuous to discontinuous displacement fields connected with a softening behaviour. This implies two fundamental aspects about the finite element method: mesh dependency and discontinuity modelling due to cracking. Both need special treatment for us to reach valid simulation results.

7.1 Basic Concepts of Crack Modelling The topic of crack propagation is part of fracture mechanics. Linear elastic fracture mechanics (LEFM) forms the core (Knott 1973; Anderson 2017; Dharan and Kang 2016). LEFM analyses given cracks in homogeneous elastic bodies, where cracks are surfaces or planes within 3D bodies or curves or lines within 2D bodies defining internal boundaries allowing for discontinuities of displacements. LEFM distinguishes three basic fracture modes that are amenable to analytical treatment within the framework of elasticity, see Figure 7.1a: • Mode-I: opening arising from a tensile stress normal to he crack plane. • Mode-II: sliding from a shear stress parallel to the crack plane but normal to the front of the crack plane. • Mode-III: tearing from a shear stress parallel to the crack plane and parallel to the front of the crack plane. Failure types are another category in addition to fracture modes. We distinguish brittle failure, quasi-brittle failure, and ductile failure; see Figure 7.1b. Uniaxial stress–strain relations are considered to characterise them. Behaviour before failure is assumed to be elastic with respect to the following scheme.

Computational Structural Concrete – Theory and Applications, Second Edition. Ulrich Häussler-Combe. © 2023 Ernst & Sohn GmbH. Published 2023 by Ernst & Sohn GmbH.

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7 Crack Modelling and Regularisation

σ

ductile

fct quasi brittle

brittle Mode-I opening

(a)

Mode-II sliding

Mode-III tearing

εct

εcu

ε

(b)

Figure 7.1 (a) Fracture modes. (b) Material failure types.

• Brittle failure is connected with a sudden drop of stress after reaching strength. The internal elastic energy is transformed into energy to form new surfaces. This type of failure is typical for glass. • Quasi-brittle failure is connected with decreasing stress after reaching strength. The internal energy is transformed into the process zone creation (Figure 3.3a). This type of failure is typical for concrete and many geomaterials. • Ductile failure is connected with yielding and hardening, i.e. with a slightly increasing stress after the strain passes the point of yielding. Yielding and hardening go on for a relatively long range of strain before a localisation starts ending with rupture. The internal energy is predominantly transformed into crystalline sliding. This is typical for metals. The application of LEFM is basically restricted to cases with brittle failure. Quasibrittle failure will be considered in the following, and LEFM is no longer directly applicable. The formation of a process zone or crack band ending up in macrocracking is discussed in Section 3.1. Continuum mechanics is not appropriate for a detailed microscopic or even mesoscopic description of the complex mechanisms during the formation of the process zone. Furthermore, the macroscale viewpoint (Section 6.1.1) requires homogenisation. The transmission of forces via crack bridges and mesoscopic branches (Figure 3.3a) is represented by the cohesive zone model (Hillerborg et al. 1976; Elices et al. 2002). A synonym is the cohesive crack model, which is used in the following. This model implicates the fictitious crack (Eq. (3.5)). ◀

The cohesive crack model assigns surface tractions along fictitious crack boundaries. A cohesive crack law relates such a – crack – traction and a fictitious crack width. Thus, it allows us to model crack propagation for quasi-brittle materials based on continuum mechanics and within the macroscale.

The fictitious crack width is conjugate to crack tractions with respect to energy. Both can be considered as generalised strain or generalised stress, respectively. The cohesive crack model is illustrated in Figure 7.2 assuming mode-I. The fictitious crack width 𝑤 has a physical meaning insofar as the traction corresponds to the

7.1 Basic Concepts of Crack Modelling

F σ = f (w)

σ = f (ε)

wcr stress-free fictitious crack crack

continuum Figure 7.2 Cohesive crack model with fictitious crack.

F

tensile strength 𝑓𝑐𝑡 for 𝑤 = 0, and the traction becomes zero for 𝑤 = 𝑤𝑐𝑟 whereby the critical crack width 𝑤𝑐𝑟 corresponds to the width of the beginning of the macrocrack. Such a value can be estimated from experimental data. Furthermore, the cohesive crack is strongly connected to the crack energy 𝐺𝑓 (Eqs. (3.8) and (3.9)). A cohesive characteristic length (Elices et al. 2002) 𝑙𝑐ℎ =

𝐸 𝐺𝑓 2 𝑓𝑐𝑡

(7.1)

with the initial Young’s modulus 𝐸 gives a measure for the size of the fully developed process zone (Figure 3.3); see also Eq. (7.56). The fictitious crack implies two crack surfaces as opposite boundaries. These crack surfaces change their relative position during cracking, and the relative position change is used as a fictitious crack width. One of these surfaces is chosen as the reference. Each position on the reference surface has a tangential cracking plane supporting a local Cartesian coordinate system with a normal n. The distance between the crack surfaces is measured by a normal component 𝑤1 or crack width and two sliding components 𝑤2 , 𝑤3 in the local coordinate system. They form a fictitious crack width vector ⎛𝑤1 ⎞ w𝑐𝐿 = ⎜𝑤2 ⎟ ⎜ ⎟ ⎝𝑤3 ⎠

(7.2)

The cohesive crack model assumes a crack traction vector t𝑐𝐿 . Due to reasons of equilibrium, its components are connected to the local Cauchy stress (Eqs. (6.5) and (6.15)) by

t𝑐𝐿

⎛𝑡1 ⎞ ⎛𝜎˜11 ⎞ = ⎜𝑡2 ⎟ = ⎜𝜎˜12 ⎟ ⎜ ⎟ ⎜ ⎟ ⎝𝑡3 ⎠ ⎝𝜎˜13 ⎠

(7.3)

in the tangential cracking plane. A material law relates stresses and strains in a continuum. In the same way, a material law relates the crack traction and the fictitious

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7 Crack Modelling and Regularisation

crack width in a crack. A general approach is formulated as 𝑡1 = 𝑓𝑛 (𝑤1 ) ,

𝑡2 = 𝑓𝑠 (𝑤2 , 𝑤1 ) ,

𝑡3 = 𝑓𝑠 (𝑤3 , 𝑤1 )

(7.4)

with different relations 𝑓𝑛 for the normal component and 𝑓𝑠 for the shear or frictional component. ◀

The relations between crack tractions and the fictitious crack width in all that follows will also be referred to as traction–separation relations, as is common practice.

The dependence of 𝑓𝑠 on 𝑤1 is included as friction basically decreases with increasing crack width. The relation 𝑓𝑛 for the normal component corresponds to mode-I or uniaxial cracking in the softening range (Figure 3.7). An appropriate form for 𝑓𝑛 is given with Figure 3.7 and Eq. (3.10) and is adapted as ) ( 𝑡1 = 𝑓𝑐𝑡 𝜔2 − 2𝜔 + 1 ,

0≤𝜔=

𝑤1 ≤1 𝑤𝑐𝑟

(7.5)

with a tangential stiffness 2𝑓𝑐𝑡 𝜕𝑡1 = (𝜔 − 1) 𝑤𝑐𝑟 𝜕𝑤1

(7.6)

In contrast to the normal component less knowledge is available for the sliding component 𝑓𝑠 (Walraven 1981; Divakar et al. 1987; Martın-Pérez and Pantazopoulou 2001). A simple shear retention approach is given with 2

𝑡𝑖 = 𝛼𝑠 (1 − 𝜔) 𝐸 𝑤𝑖 ,

𝑖 = 2, 3

(7.7)

with the initial Young’s modulus 𝐸 of the embedding material and a shear retention factor 𝛼𝑠 . A value 𝛼𝑠 > 0 generally proves to be necessary regarding convergence and stability of numerical computations. A range 0 < 𝛼𝑠 ≤ 0.01 looks appropriate, but this should be subject to parameter variations. The tangential stiffness is derived with 𝜕𝑡𝑖 2 = 𝛼𝑠 (1 − 𝜔) 𝐸 , 𝜕𝑤𝑖

𝑖 = 2, 3

(7.8)

The contribution 𝜕𝑡𝑠 ∕𝜕𝑤1 is generally neglected in order to preserve symmetry. Irrespective of these proposals, basically all material frameworks like plasticity or damage may be used to formulate relations connecting the generalised strain w𝑐𝐿 with the generalised stress t𝑐𝐿 . The corresponding functions 𝑓𝑛 , 𝑓𝑠 and their derivatives generally refer to the local tangential cracking plane. Tangential behaviour in a general form is written as ṫ 𝑐𝐿 = C𝑐𝐿𝑇 ⋅ ẇ 𝑐𝐿 ,

ẇ 𝑐𝐿 = D𝑐𝐿𝑇 ⋅ ṫ 𝑐𝐿

(7.9)

with a local tangential crack stiffness C𝑐𝐿𝑇 and a local tangential crack compliance D𝑐𝐿𝑇 . This sets a base for the following. For a comprehensive treatment of fracture mechanics, cohesive cracks, size effects (Section 6.1.2), and other related topics, see Karihaloo (1995); Bažant and Planas (1998).

7.2 Mesh Dependency

Originally, the cohesive crack model is not connected to numerical methods. It serves to treat complex process zones (Figure 3.3a) within the framework of continuum mechanics. ◀

With standard FEM we see an incompatibility to the cohesive crack model, as displacement interpolations with standard finite elements are continuous by definition and exclude discontinuities.

But finite elements are ‘flexible’. With a focus on structural behaviour – not on particular local points – discontinuities may be modelled by relatively large strains connected with decreasing stresses in cracked regions, maintaining the continuity of interpolated displacements. But such an approach may lead to the effect of mesh dependency (Section 7.2) requiring a so-called regularisation. Common regularisation methods are given with the crack band approach (Section 7.3) and non-local methods (Section 7.3). A specification of the crack band approach is given with the multi-axial smeared crack model (Section 7.4), which has already been discussed for 1D (Section 3.5) and explicitly includes traction–separation relations depending on the crack width. ◀

It should be noted that the terms in the literature are not always uniform. The term smeared crack may also mean stress–strain relations with strain softening, whether or not they are regularised. This is a much more general class.

A fine resolution of localisation zones with continuous displacement fields including regularisation may be reached with gradient methods (Section 7.5). Furthermore, finite elements may be extended for modelling of discontinuous displacement fields with explicit crack surfaces. They allow for a direct application of crack traction, fundamentally avoiding mesh dependency issues (Section 7.6). Such finite element extensions are exemplarily described with the strong discontinuity Approach (SDA) (Section 7.7).

7.2 Mesh Dependency A basic property of the stress–strain behaviour of concrete in its limit state is given by strain softening; stress decreases while strain increases (Figure 3.3b). The effects of strain softening on structural behaviour are demonstrated for a tension bar with Example 3.1. A zone of localisation develops in a relatively small area of a structure, leading to a snap-back in the load–displacement behaviour. This type of behaviour is again discussed for a simplified configuration. We consider a bar model according to Figure 7.3a. The bar consists of three sections. A linear elastic relation 𝜎 = 𝐸 𝜖 is assumed for the first and third sections. The centre section of length 𝐿𝑒 has a linear elastic behaviour initially up to a maximum stress value or strength followed by a linear stress decrease; see Figure 7.3b. This material

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σ L1=1/2 (1-α)L

Le=α L

fct quasi-brittle strain softening

L2=1/2 (1-α)L

L εct (a)

εcu

ε

(b)

Figure 7.3 (a) Model for a softening bar. (b) Material model for a softening bar.

softening behaviour is described by ⎧𝐸 𝜖 ⎪ 𝜖𝑐𝑢 − 𝜖 𝜎 = 𝑓𝑐𝑡 ⎨ 𝜖𝑐𝑢 − 𝜖𝑐𝑡 ⎪0 ⎩

𝜖 ≤ 𝜖𝑐𝑡 𝜖𝑐𝑡 < 𝜖 ≤ 𝜖𝑐𝑢

(7.10)

𝜖𝑐𝑢 < 𝜖

with 𝑓𝑐𝑡 = 𝐸 𝜖𝑐𝑡 . Softening is characteristic for quasi-brittle failure (Figure 7.1b). In the case of concrete 1 (7.11) 𝐺𝑓 = 𝐿𝑒 𝑓𝑐𝑡 (𝜖𝑐𝑢 − 𝜖𝑐𝑡 ) , 𝐿𝑒 = 𝛼 𝐿 2 is dissipated as crack energy 𝐺𝑓 (Eq. (3.8)) with Eq. (7.10), as 𝜖 ≥ 𝜖𝑐𝑢 corresponds to a fully opened crack, and the initial uncracked state cannot be restored. The bar is loaded with a stress 𝜎 at the right-hand end and fixed at the left-hand end. All bar sections have the stress 𝜎 due to equilibrium. The displacement of the right-hand end is given by 𝑢 = 𝜖1 𝐿1 + 𝜖𝐿𝑒 + 𝜖2 𝐿2

(7.12)

with the strains 𝜖1 , 𝜖2 for the lateral sections and the strain 𝜖 for the centre section. At first this leads to a right-hand end displacement 𝑢=

𝜎 𝐿, 𝐸

𝜖 ≤ 𝜖𝑐𝑡

(7.13)

Furthermore, we consider the softening of the centre section and use Eq. (7.10) for 𝜖𝑐𝑡 ≤ 𝜖 ≤ 𝜖𝑐𝑢 . Sideward strains are 𝜖1 = 𝜖2 = 𝜎∕𝐸 in this strain range. With 𝐿𝑒 = 𝛼 𝐿 and 𝐿1 + 𝐿2 = (1 − 𝛼) 𝐿 and with Eqs. (7.10) and (7.12), the stress 𝜎 can be determined depending on the displacement. This yields 𝑢

𝛼 𝜖𝑐𝑢 − 𝜎 𝐿 = 𝛼 𝜖𝑐𝑢 − 𝜖𝑐𝑡 𝑓𝑐𝑡

(7.14)

We assume 𝜖𝑐𝑡 = 0.01, 𝜖𝑐𝑢 = 0.03 for an example. Results for the related loading 𝜎∕𝑓𝑐𝑡 depending on related displacement 𝑢∕𝐿 are shown in Figure 7.4 for different values of the softening length ratio 𝛼.

7.2 Mesh Dependency

σ/fct α=0.8 α=0.5 α=0.2

u/L Figure 7.4 Load-displacement relations for a softening bar.

The bar behaviour after reaching the strength 𝑓𝑐𝑡 depends on the softening length. A decreasing loading occurs with an increasing strain in the softening centre section, while strains decrease elastically in the remaining parts. A relatively small softening length leads to a snap-back behaviour. Decreasing elastic strains overcompensate increasing softening strains, leading to a decreasing displacement. We transfer this set-up to a bar discretised with a number 𝑛𝐸 of two-node bar elements along a line (Section 2.3). All elements are chosen with the same length and a material law according to Eq. (7.10). But one element is assumed with a slightly reduced strength 𝑓𝑐𝑡 or slightly reduced cross-section compared to all other 𝑛𝐸 − 1 elements. Thus, the softening length ratio is given by 𝛼 = 1∕𝑛𝐸 and the snap-back behaviour will be more pronounced for a finer discretisation. The same effect will occur with a homogeneous system with the same strength and cross-section for all elements but a non-homogeneous stress state, e.g. due to a distributed loading along the bar. The same phenomenon can be seen for 2D states along curves that connect 2D elements and for 3D states along surfaces that connect 3D elements. The spatial thickness of the softening band will spread over one element – maybe two or three in cases when the softening band does not align to single element boundaries – and will reduce with finer discretisations. The implications are exemplarily demonstrated with the following example. Example 7.1: Plain Concrete Plate with Notch

We consider a simple, small plate with plain concrete, which is typical for an experimental set-up; see Figure 7.5a. The plate has a notch at the lower side mid-span. The isotropic damage model (Section 6.6) with the Hsieh–Ting-Chen damage function (Eq. (6.109)) is used for the concrete part. Its uniaxial application is demonstrated in Example 6.2. The uniaxial tensile failure behaviour (Figures 3.3b and 6.13a) is crucial for this example. The same material parameters are used as with Example 6.2. Plane stress conditions are assumed. Flat steel plates are used for lower support and upper application of prescribed displacements. They need some extension in order to avoid stress concentrations that might lead to a premature failure. The discretisation is performed with regular meshes with square 2D quad elements (Section 2.3).

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7 Crack Modelling and Regularisation

(a)

(b)

Figure 7.5 Example 7.1. (a) System. (b) Load displacement curves. Table 7.1 Discretisation refinement of Example 7.1. No.

Edge length [m] Number of Elements

Number of Nodes

Degrees of Freedom

1 2 3

0.02500 0.01250 0.00625

350 1290 4946

700 2580 9892

a) b)

308 1208 4784

Includes discretisation of concrete and steel plates. Degrees of freedom are twice the number of nodes. Nodes constrained due to boundary conditions are formally kept as unknowns (Section 4.4.3).

Three discretisation refinements, see Table 7.1, are examined. Element shapes are generally nearly square by a ratio of 1.25. The larger vertical edge lengths are listed in Table 7.1. The notch has a height of 0.025 m and is modelled by disregarding elements in the vertical centre line: 1 for discretisation No. 1, 2 for No. 2, and 4 for No.3. The loading is applied in small steps as the prescribed displacement in the upper edge of the upper centred steel plate. A viscous stabilisation is applied (Appendix A.1) with an artificial viscosity 𝜂 = 1 MN∕m2 ∕s. The loading time is 1 s with a time step of Δ𝑡 = 0.001 s. This has a negligible influence on the results before reaching the peak load but allows obtaining solutions for the softening range at all. Figure 7.5b shows the computed load–displacement behaviour with the total reaction force along the upper edge of the upper steel plate versus its prescribed vertical displacement. • Minor differences occur in the initial stiffness due to discretisation refinement, whereby convergence is indicated with decreasing differences in the refinement sequence 1–2–3. • The smaller elements favor an earlier localisation due to strain softening starting from the lower notch. This reduces the computed peak loading, whereby the peak values are significantly different. • A softening range can be computed due to viscous stabilisation, ending up in a horizontal plateau.

7.3 Regularisation

A significant mesh dependency occurs as different element sizes share the same tensile stress–strain relations in the softening range but with different extensions within the spatial discretisations. With the underlying partial differential equations (PDEs) – combining equilibrium, kinematic compatibility, and stress–strain relations – in the case of softening, a loss of ellipticity occurs from a mathematical point of view. The PDEs become hyperbolic with softening material models violating the Drucker stability postulate (Eq. (6.170)). This yields a fundamental change in the solution characteristics (de Borst et al. 2012, 6.4). Elliptic PDEs generally involve smooth solutions, while hyperbolic PDEs – depending on boundary conditions – allow for non-unique and discontinuous solutions. With standard finite elements such discontinuities are smoothed within element bands of basically one-element width. Thus, the geometric width of a strain softening band decreases with finer discretisations. ◀

A discretisation may have a considerable influence on the numerical results for structures with softening materials. Results are mesh dependent. As concerns peak load and post-peak regimes, convergent solutions cannot be reached with a refinement of discretisations.

In the same way, the dissipated crack energy approaches zero with refining discretisations. This contradicts experimental data of real systems with softening materials, which show a localisation (Figure 3.4) with considerable energy dissipation while passing the softening process up to an ultimate failure.

7.3 Regularisation Strain softening is connected with dissipation of energy in zones of localisation and, in particular, within crack bands of concrete structures. On the other hand, a mesh dependency of discretised systems with softening materials becomes evident – using the standard material and element formulations considered up to now – with dissipated energy going to zero with the refinement of the discretisation. Thus, energy dissipation can be used as an indicator for mesh dependency. Concrete dissipated energy in process zones (Figure 3.3a) corresponds to the crack energy in crack bands (Eq. (3.8)). Its value is assumed as a constant material parameter with the current state of knowledge. This suggests a concept to avoid mesh dependency. ◀

The crack energy should be reproduced for systems with softening materials independent of discretisation. The corresponding methods are regarded as regularisation. 1)

1) To be more general, regularisation classifies methods to keep the ellipticity of PDEs in the case of strain softening material behaviour.

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We consider again the simple three-section bar shown in Figure 7.3 and the stress– strain relations Eq. (7.10). The centre section is modified with the application range of the softening stress–strain relation ⎧𝐸 𝜖 ⎪ 𝛼(𝜖 − 𝜖 ) − (𝜖 − 𝜖 ) 𝑐𝑢 𝑐𝑡 𝑐𝑡 𝜎 = 𝑓𝑐𝑡 ⎨ 𝛼(𝜖𝑐𝑢 − 𝜖𝑐𝑡 ) ⎪ 0 ⎩

𝜖 ≤ 𝜖𝑐𝑡 𝜖𝑐𝑡 < 𝜖 ≤ 𝛼(𝜖𝑐𝑢 − 𝜖𝑐𝑡 ) + 𝜖𝑐𝑡

𝛼=

𝛼(𝜖𝑐𝑢 − 𝜖𝑐𝑡 ) + 𝜖𝑐𝑡 < 𝜖

𝑏𝑤 𝐿𝑒 (7.15)

with the concrete tensile strength 𝑓𝑐𝑡 , the crack band width 𝑏𝑤 (Figure 3.4), and the section or element length 𝐿𝑒 , respectively. The range of strain in the softening range is scaled with a parameter 𝛼 according to the ratio of crack the band width to element length. After this modification of the stress–strain relations,the corresponding crack energy is given by 𝐺𝑓 =

1 𝑏 𝑓 (𝜖 − 𝜖𝑐𝑡 ) 2 𝑤 𝑐𝑡 𝑐𝑢

(7.16)

This yields the correct crack energy independent of the element length 𝐿𝑐 in contrast to Eq. (7.11). Thus, scaling of softening strains compensates the influence of element size while integrating stress–strain relations for the dissipated energy in a softening zone localizing in an element. Small elements gain a large integration range of strains, large elements a low integration range. ◀

A regularisation can be reached with a scaling of strains in the softening range of the stress–strain relations. This scaling depends on the ratio of the crack band width to a characteristic element length and leads to the crack band approach (Bažant and Oh (1983); Jirásek and Bauer (2012)), also called softening modulus regularisation. But this yields stress–strain relations depending on a discretisation while blending finite elements and material laws.

This approach may also be extended to multi-axial cases with the modification of stress–strain relations by a characteristic element length. But such rules are not straightforward for 3D continuum-based stress–strain relations for damage and elasto-plasticity, and their combinations. This is exemplarily demonstrated in the following for the damage approach (Section 6.6). Example 7.2: Plain Concrete Plate with Notch and Crack Band Regularisation

We refer to Example 7.1 with the same geometry, material, and boundary conditions. The same cases of discretisation refinement are examined. A viscous stabilisation is not used (𝜂 = 0). A crack band regularisation is applied instead. This includes the crack energy (Eq. (3.8)) as a further material parameter. It is assumed with 𝐺𝑓 = 50 ⋅ 10−6 [MN∕m] as a typical value. On the other hand, the volume-specific crack energy 𝑔𝑓 (Eq. (3.7)) is determined with the original stress–

7.3 Regularisation

strain relations; see Example 6.2, Figure 6.13. Its integration according to Eq. (3.7) results in 𝑔𝑓 = 0.495 ⋅ 10−3 [MN∕m2 ], leading to a crack band width (Eq. (3.8)) 𝑏𝑤 =

𝐺𝑓 𝑔𝑓

≈ 0.1 m

(7.17)

which indicates whether the set of chosen material parameters is reasonable. This value has to be related to the characteristic length of the given quad elements. It is generally appropriate that the geometric shape of quad elements should not be too far away from squares in the global system. Thus, the characteristic element length is generally defined with √ (7.18) 𝐿𝑐 = 𝐴 with the geometric area 𝐴 of an element. This may be different for each element in a discretisation. Referring to Table 7.1, the values are derived with 𝐿𝑐 = 0.022 36, 0.011 78, 0.006 06 m for the example due to the edge length ratio of 1.25. As 𝐿𝑐 is quite different compared to the crack band width 𝑏𝑤 , a scaling of the stress–strain relations has to be performed. This is reached with a scaling of the equivalent damage strain 𝜅𝑑 , which is proportional to the uniaxial longitudinal strain (Eq. (6.1103 )). With small elements or larger values 𝑏𝑤 ∕𝐿𝑐 , an approach 𝜅𝑑,scaled = (1 − 𝛽) 𝜅𝑑,lim ln (

𝜅𝑑 − 𝛽 𝜅𝑑,lim ) (1 − 𝛽) 𝜅𝑑,lim

) + 𝜅𝑑,lim ,

𝜅𝑑 ≥ 𝜅𝑑,lim

(7.19)

is appropriate (Kitzig and Häussler-Combe 2011). This includes the equivalent damage strain 𝜅𝑑 derived from the state of strain according to Eq. (6.109), its value 𝜅𝑑,lim belonging to strength – it corresponds to |𝜖𝑐1 | in Table 6.1 but is also valid for multiaxial states – and, finally, a scaling factor 𝛽, to be discussed in the following. A scaling according to Eq. (7.19) is recommended for ratios 𝑏𝑤 ∕𝐿𝑐 >≈ 3 only 2) and will not reproduce the originate uniaxial stress–strain behaviour for any value 𝛽. The approach Eq. (7.19) provides a downscaling with reduced values 𝜅𝑑,scaled compared to 𝜅𝑑 . The scaled equivalent damage strain is used in Eq.(6.107) instead of 𝜅𝑑 to derive the scalar damage 𝐷. This is the only item that changes for processing the stress– strain relations. The influence of the scaling factor on the uniaxial tensile stress–strain behaviour for the current material properties (Table 6.1) is shown in Figure 7.6a. This also indicates the scaled volume specific crack energies 𝑔𝑓,scaled (Eq. (3.7)) and leads to a scaled crack band width 𝑏𝑤,scaled (Eq. (3.8)) 𝑏𝑤,scaled =

𝐺𝑓

(7.20)

𝑔𝑓,scaled

( ) 2) An alternative 𝜅𝑑,scaled = 𝛾1 𝜅𝑑 − 𝜅𝑑,lim +

1−𝛾1 𝛾2

( ) 1 − e−𝛾2 (𝜅𝑑 −𝜅𝑑,lim ) + 𝜅𝑑,lim with a pair of scaling factors

𝛾1 , 𝛾2 is proposed for larger elements (Häussler-Combe and Hartig 2008). This reproduces the original uniaxial stress–strain behaviour but will not work for small elements. The question arises as to how to choose such approaches. Actually, it is a matter of trial and error with the current state of knowledge.

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(a)

(b)

Figure 7.6 Example 7.2. (a) Scaled tensile uniaxial stress relations. (b) Scaling factor 𝛽 depending on characteristic element length Lc .

Figure 7.7 Example 7.2. Load–displacement curves.

The scaling factor 𝛽 has to be adjusted such that 𝐿𝑐 ≈ 𝑏𝑤,scaled to yield a prescribed crack energy 𝐺𝑓 within a single element according to the crack band approach for regularisation. As 𝐿𝑐 is given from a discretisation and 𝐺𝑓 from material properties, Eq. (7.20) has to be solved for the scaling factor 𝛽 𝑔𝑓,scaled (𝛽) =

𝐺𝑓 𝐿𝑐

(7.21)

With the processing sequence Eqs. (7.19), (6.115), and (3.7) an inverse approach is required. Figure 7.6b shows the solution 𝛽 for the current material depending on the characteristic element length 𝐿𝑐 . The figure also indicates the solution values for the discretisation refinements (Table 7.1). As in Example 7.1, the loading is applied in small steps as a prescribed displacement of the upper edge of the upper centred steel plate. The target displacement is increased to 0.1 mm. Figure 7.7 shows the computed load–displacement behaviour with the total reaction force along the upper edge of the upper steel plate versus its prescribed vertical displacement. The corresponding data from Example 7.1, Figure 7.5b, without regularisation but a viscous stabilisation are also shown for comparison.

7.3 Regularisation

Figure 7.8 Example 7.2. Principal stresses on a deformed structure (scaling factor 500).

• Regularized peak loads considerably exceed corresponding values of Example 7.1. • The coarse discretisation No. 1 (Table 7.1) becomes unstable shortly after reaching the peak load. The prescribed target displacement is reached with the finer discretisations without convergence issues. • There are still significant differences in the peak loadings for the regularised cases, but a convergence tendency is evident. Figure 7.8 shows the computed principal stresses on the deformed discretisation near peak loading for the coarse discretisation No. 1. • Large deformations with low stresses arise in the vertical element row next to the centre with the upper end indicating the actual process zone. • The principal tensile stresses in the process zone area reach the prescribed tensile strength (Table 6.1) but do not exceed it. • Principal compressive stresses near the loading and supporting plates are low compared to the compressive strength. The example demonstrates the basic suitability of a crack band regularisation, although a final level of mesh independence is not yet reached. We refer to the companion Example 7.6, which describes an alternative for the crack band approach. Furthermore, it becomes obvious that a scaling of multi-axial stress–strain relations is not trivial. This issue is also addressed by Jirásek and Bauer (2012). Parameter studies regarding varying spatial discretisations are generally appropriate in the case of structures with softening behaviour. The crack band approach can be applied more smoothly to microplane models due to their scalar stress–strain relations (Eq. (6.151)) comparable to uniaxial stress–strain relations, which are easily amenable to scaling. A special case of the crack band approach is given with the smeared crack model, which is described in Section 3.5 for the uniaxial case. This is generalised for multi-axial states; see Section 7.4.

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214

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ε, ε

ε

ε x

Figure 7.9 Non-local uniaxial strain.

The reproduction of crack energy has been defined as major criterion for regularisation. The crack energy contributes to the ductile behaviour of whole structures or structural elements. This concerns the relations between loads and displacements. Insofar a crack band regularisation should first of all ensure the correct modelling of the overall behaviour of structures with softening materials. But such methods will not reproduce local behaviour. The spatial thickness of the softening band will basically spread over one element and reduces with finer discretisations (Section 7.2), whereby strains within softening elements are scaled according to element size in order to retain the crack energy. The real behaviour within crack bands is different (Figures 3.3a, 3.4, Example 3.1). Non-local methods might resolve the local crack band behaviour additionally to a regularisation. We consider again the simple three-section bar shown in Figure 7.3 with the stress–strain relations Eq. (7.10). The relation for the centre section is modified regarding its variable strain argument 𝜖 𝜎 = 𝑓𝑐𝑡

𝜖𝑐𝑢 − 𝜖 , 𝜖𝑐𝑢 − 𝜖𝑐𝑡

𝜖𝑐𝑡 < 𝜖 ≤ 𝜖𝑐𝑢

(7.22)

with the non-local integral strain 𝜖 𝜖(𝑥) =

1 ∫ 𝑔(𝑠) 𝜖(𝑥 + 𝑠) d𝑠 𝑆

(7.23)

𝐿

with a fixed coordinate 𝑥, a variable coordinate 𝑠, a bar length 𝐿, and a weighting function 𝑔(𝑠) = e

−

𝑠2 2𝑅2

,

𝑆 = ∫ 𝑔(𝑠) d𝑠

(7.24)

𝐿

The weighting function 𝑔(𝑠) corresponds to a bell-shaped curve with maximum 1 for 𝑠 = 0. The parameter 𝑅 determines the lateral decline of 𝑔. Small values cause a steep decline with a small range, while large values cause a flat decline with a large range. The non-local approach Eq. (7.23) decreases the local extremal value of 𝜖 and broadens the base, depending on the parameter 𝑅, see Figure 7.9. Spatially constant strains would remain constant in contrast.

7.3 Regularisation

The length 𝑅 is assumed as constant and independent of a discretisation. Thus, Eq. (7.23) spans over a number of neighboured elements depending on the ratio of 𝑅 to a single element length. This enforces a smoothing of strain peak values and leads to a localisation extension that does not depend on the discretisation. Furthermore, the value of 𝑅 and the crack band width 𝑏𝑤 are approximately proportional and have the same magnitude. This corresponds to the physical significance of the crack band width, which primarily depends on the concrete aggregate size. ◀

Non-local methods introduce a characteristic length R as a material parameter. This characteristic material length is a measure for the heterogeneity of the material, as it is observed in the mesoscale (Figure 6.1a) and compensates for a ‘loss of information’ due to homogenisation.

The non-local integral approach will not be pursued with details and applications. For a comprehensive treatment, see Pijaudier-Cabot and Bažant (1987); Jirásek (1998); Bažant and Planas (1998); Bažant and Jirázek (2002). An alternative for the non-local integral formulation Eq. (7.23) is given with the non-local differential form (Peerlings et al. 1996). A second-order Taylor expansion of 𝜖(𝑥 + 𝑠) yields 𝜖(𝑥 + 𝑠) ≈ 𝜖(𝑥) +

𝜕𝜖 1 𝜕2 𝜖 2 𝑠 𝑠+ 2 𝜕𝑥2 𝜕𝑥

(7.25)

Insertion into Eq. (7.23) results in a differential non-local formulation ∞

∞

1 1 𝜕2 𝜖 𝑅2 𝜕 2 𝜖 ∫ 𝑔(𝑠) d𝑠 𝜖 + ∫ 𝑔(𝑠) 𝑠2 d𝑠 𝜖≈ =𝜖+ 2 2 𝜕𝑥2 𝑆 2𝑆 𝜕𝑥 −∞

(7.26)

−∞

under the simplifying assumption of an infinite bar length 𝐿 and the antisymmetry of the linear 𝑠-term. This explicit form is not yet appropriate for common finite element formulations due to the second derivative 𝜕 2 𝜖∕𝜕𝑥2 . Thus, the second derivative of Eq. (7.26) 𝜕2 𝜖 𝑅2 𝜕 4 𝜖 𝜕2 𝜖 = + 2 𝜕𝑥4 𝜕𝑥2 𝜕𝑥2

(7.27)

is multiplied by 𝑅2 ∕2 and subtracted from Eq. (7.26). This results in 𝜖−

𝑅2 𝜕 2 𝜖 𝑅4 𝜕 4 𝜖 =𝜖− 2 𝜕𝑥2 4 𝜕𝑥4

(7.28)

Neglecting the last term with the assumption 𝑅 < 1, 𝑅4 ≪ 𝑅2 leads to the implicit differential form to describe non-local strains 𝜖(𝑥) −

𝑅2 𝜕 2 𝜖(𝑥) = 𝜖(𝑥) 2 𝜕𝑥2

(7.29)

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7 Crack Modelling and Regularisation

This differential equation is driven by a given strain function 𝜖(𝑥) determined in a superordinated calculation. Its solution 𝜖(𝑥) is used as the strain to determine the stress according to Eq. (7.22). Integral or differential non-local forms like Eq. (7.23) or Eq. (7.29) may be generalised for multi-axial cases and applied to internal state variables connected to strain states. This is demonstrated with gradient damage in Section 7.5.1.

7.4 Multi-Axial Smeared Crack Model The smeared crack model has already been discussed in Section 3.5 for the uniaxial case. A first question concerns the onset of cracking or material failure, respectively. A range of corresponding criteria is described in Willam (2002, 4.) from a general point of view. The loss of material stability plays a significant role in this issue. This particular topic is treated in Appendix B and exhibits a certain complexity with respect to its implementation. An application is demonstrated with the Examples 6.3 and 6.4. Instead, this is simplified in the following with the application of the Rankine criterion, as it is generally used for cracking criteria in the analysis of concrete structures. Cracking starts in the uniaxial case when the stress reaches the uniaxial tensile strength 𝑓𝑐𝑡 . This is extended to the multi-axial case with the Rankine criterion. We consider a material point with a given multi-axial stress state. ◀

Cracking occurs when the largest principal stress (Section 6.2.3) reaches the uniaxial tensile strength fct .

Using the uniaxial strength as a criterion for a multi-axial case has been justified by experimental data indicating that for concrete the values of multi-axial tensile strength do not differ significantly from the uniaxial value 𝑓𝑐𝑡 (Figures 6.6 and 6.7). A crack direction must also be addressed in multi-axial cases. ◀

The crack direction is assumed as normal to the direction of the principal stress inducing the crack.

Thus, principal direction and the normal n of the local tangential cracking plane (Section 7.1) coincide. The local cracking plane provides a local coordinate system for the components of the fictitious crack width vector w𝑐𝐿 (Eq. (7.2)). In analogy to Eq. (3.6) a local crack strain for the multi-axial case is defined as ˜ 𝝐𝑐 =

1 L ⋅ w𝑐𝐿 , 𝑏𝑤

w𝑐𝐿 = 𝑏𝑤 LT ⋅ ˜ 𝝐𝑐

(7.30)

7.4 Multi-Axial Smeared Crack Model

Figure 7.10 Crack band (2D).

with the crack band width 𝑏𝑤 (Figure 3.4) and an incidence matrix L ⎡1 ⎢ ⎢0 ⎢ ⎢0 L=⎢ ⎢0 ⎢ ⎢0 ⎢ 0 ⎣

0 0 0 0 0 1

0⎤ ⎥ 0⎥ ⎥ 0⎥ ⎥ 0⎥ ⎥ 1⎥ ⎥ 0 ⎦

(7.31)

assigning differences of displacements to nominal strains in Voigt notation (Eq. (6.3)). The fictitious crack width w is connected to crack tractions t according to Eq. (7.4) t𝑐𝐿 = f𝑐 (w𝑐𝐿 )

(7.32)

The crack tractions are related to the local Cauchy stress by Eq. (7.3) to maintain local equilibrium. This is reformulated as ˜, t𝑐𝐿 = LT ⋅ 𝝈

𝝈 ˜ = L ⋅ t𝑐𝐿

(7.33)

Equations (7.30)–(7.33) set the local stage for the multi-axial crack band; see Figure 7.10. As the orientation of the cracking plane in the global coordinate system is known, the local crack strains and local stresses can be transformed into the global system according to Eqs. (6.14) and (6.15). In analogy to Eq. (3.48), for the multi-axial case, we set 𝝐 = (1 − 𝜉) 𝝐 𝑢 + 𝜉 QT ⋅ ˜ 𝝐𝑐 ,

0≤𝜉=

𝑏𝑤 ≤1 𝐿𝑐

(7.34)

with the total strain 𝝐, the strain 𝝐 𝑢 of the uncracked material, and a characteristic length 𝐿𝑐 (Eq. (7.18)) of the respective element. 3) Finally, we assume a relation 𝝈 = f𝑢 (𝝐 𝑢 )

(7.35)

3) Depending on 𝜉, this may put some artificial constraints on 𝝐 𝑢 , as some ˜ 𝝐 𝑐 -components are prescribed with zero (Eq. (7.31)) Thus, Eq. (7.34) has to be relaxed case by case, which is exemplarily shown in Example 7.3

217

218

7 Crack Modelling and Regularisation

for stresses related to the strains for the uncracked material. The total strain 𝝐 is generally prescribed from a superordinated computation. Thus, with the crack plane orientation and the related crack band width 𝜉 given, the quantities 𝝈, 𝝐 𝑢 , ˜ 𝝐 𝑐 , … may be determined from the – generally nonlinear – system of Eqs. (7.30)–(7.35) regarding the transformation rules Eqs. (6.14) and (6.15). These equations also serve as a basis to derive the material tangential stiffness for the multi-axial smeared crack. The time derivatives 1 L ⋅ ẇ 𝑐𝐿 ˜ 𝝐̇ 𝑐 = 𝑏𝑤 ˜̇ ṫ 𝑐𝐿 = LT ⋅ 𝝈

(7.36)

𝝐̇ = (1 − 𝜉) 𝝐̇ 𝑢 + 𝜉 QT ⋅ ˜ 𝝐̇ 𝑐 have to be considered. The combination of Eqs. (7.363 ), (7.361 ), (7.92 ), (7.362 ), and (6.15) yields 𝝐̇ = (1 − 𝜉) 𝝐̇ 𝑢 + 𝜉 D𝑐𝑇 ⋅ 𝝈̇

(7.37)

with the crack compliance D𝑐𝑇 =

1 T Q ⋅ L ⋅ D𝑐𝐿𝑇 ⋅ LT ⋅ Q 𝑏𝑤

(7.38)

The material behaviour of the uncracked material still has to be considered. The combination of the uncracked material compliance Eq. (2.51) with Eq. (7.37) leads to 𝝐̇ = [(1 − 𝜉) D𝑇 + 𝜉 D𝑐𝑇 ] ⋅ 𝝈̇

(7.39)

which has to be inverted for 𝝈̇ = C𝑇 ⋅ 𝝐̇ = [(1 − 𝜉) D𝑇 + 𝜉 D𝑐𝑇 ]

−1

⋅ 𝝐̇

(7.40)

The computational inversion effort is manageable, as the matrix has a 6× 6 size at most. The general approach is illustrated with the following example. Example 7.3: 2D Smeared Crack Model with Elasticity

A biaxial plane stress elasticity (Eq. (6.28)) is assumed for the uncracked material behaviour. A discretisation is generally connected with 2D quad elements (Section 2.3). The incidence matrix Eq. (7.31) reduces to ⎡1 ⎢ L = ⎢0 ⎢ 0 ⎣

0⎤ ⎥ 0⎥ ⎥ 1 ⎦

(7.41)

The Rankine criterion is assumed for crack initiation with a normal of the crack plane in the global 𝑥 or 1-direction. Thus, the transformation matrix Q becomes

7.4 Multi-Axial Smeared Crack Model

Figure 7.11 Linearised cohesive crack model with crack energy Gf .

a unit matrix. This is not mandatory in any way but simplifies the following relations to a large degree. The 2D traction–separation relations according to Eqs. (7.4) and (7.32) are assumed with ⎛⎧𝑓 (1 − 𝑤1 − 𝑤𝑐𝑡 ) , 𝑤 < 𝑤 < 𝑤 ⎞ 𝑐𝑡 1 𝑐𝑟 𝑤𝑐𝑟 − 𝑤𝑐𝑡 ⎛𝑡1 ⎞ ⎜ 𝑐𝑡 ⎟ ⎨ ⎟ ⎜ ⎟ = ⎜ 0, 𝑤 ≤ 𝑤 𝑐𝑟 1 𝑡 ⎟ ⎝ 2 ⎠ ⎜⎩ 0 ⎠ ⎝

(7.42)

see Figure 7.11, with 4) 𝑤𝑐𝑡 = 𝑏𝑤 𝜖𝑐𝑡 ,

𝑤𝑐𝑟 = 𝑏𝑤 𝜖𝑐𝑢 ,

𝜖𝑐𝑢 = 2

𝐺𝑓 𝑏𝑤 𝑓𝑐𝑡

+ 𝜖𝑐𝑡

(7.43)

see Figure 3.3 for 𝜖𝑐𝑡 , 𝜖𝑐𝑢 . This follows the bilinear uniaxial approach Eq. (3.58) in the crack plane normal direction, reproduces the uniaxial crack energy 𝐺𝑓 (Eq. (3.8)), and neglects a friction along the crack tangential plane for simplicity. Crack tractions are related to stresses and, furthermore, the fictitious crack width to crack strains by 𝑡1 = 𝜎11 ,

𝑡2 = 𝜎12

𝑤1 = 𝑏𝑤 𝜖̃𝑐,11 = 𝑏𝑤 𝜖𝑐,11 ,

𝑤2 = 𝑏𝑤 𝜖̃𝑐,12 = 𝑏𝑤 𝜖𝑐,12

(7.44)

Strains 𝝐 𝑢 are determined by the inversion of the linear elastic law for plane stress Eq. (6.28) ⎡1 ⎛𝜖𝑢,11 ⎞ ⎜𝜖 ⎟ = 1 ⎢ ⎢−𝜈 𝑢,22 ⎜ ⎟ 𝐸 ⎢ 1 ⎝𝜖𝑢,12 ⎠ ⎣

−𝜈 1 0

⎤ ⎛𝜎11 ⎞ ⎥ 0 ⎥ ⋅ ⎜𝜎22 ⎟ ⎥ ⎜ ⎟ 2 + 2𝜈 ⎝𝜎12 ⎠ ⎦ 0

(7.45)

4) A crack width 𝑤𝑐𝑡 = 𝑏𝑤 𝜖𝑐𝑡 is formally assumed with the crack initiation to reach a consistent exchange of crack width and crack strain. This also conforms to a crack width definition as the distance of bounding edges of the crack bandwidth (Figure 3.14).

219

220

7 Crack Modelling and Regularisation

Finally, a relaxed form of Eq. (7.34) – using 𝜖𝑐,22 = 𝜖𝑢,22 – ⎛𝜖11 ⎞ ⎜𝜖 ⎟ = (1 − 𝜉) 22 ⎜ ⎟ 𝜖 ⎝ 12 ⎠

⎛𝜖𝑢,11 ⎞ ⎜𝜖 ⎟ + 𝜉 𝑢,22 ⎜ ⎟ 𝜖 ⎝ 𝑢,12 ⎠

⎛ 𝜖𝑐,11 ⎞ ⎜𝜖 ⎟ 𝑢,22 ⎜ ⎟ 𝜖 ⎝ 𝑐,12 ⎠

(7.46)

leads to the 11 equations Eqs. (7.42)–(7.46) for a number of 11 unknowns for crack quantities 𝑡1 , 𝑤1 , 𝑤2 , strains 𝜖𝑢,11 , 𝜖𝑢,22 , 𝜖𝑢,12 , 𝜖𝑐,11 , 𝜖𝑐,12 , and stresses 𝜎11 , 𝜎22 , 𝜎12 as 𝜖11 , 𝜖22 , 𝜖12 are given from a superordinated calculation, and 𝜉 = 𝑏𝑤 ∕𝐿𝑐 , 𝐺𝑓 , 𝐸, 𝜈 are prescribed as discretisation or material parameters, respectively. The solutions for stresses are 𝜉𝜖𝑐𝑢 − 𝜖11 − (1 − 𝜉)𝜈𝜖22 𝑑 (𝜉𝜖𝑐𝑢 − 𝜖𝑐𝑡 ) 𝜖22 + 𝜈𝜖𝑐𝑡 (𝜉𝜖𝑐𝑢 − 𝜖11 ) =𝐸 𝑑 =0

𝜎11 = 𝑓𝑐𝑡 𝜎22 𝜎12

(7.47)

𝑑 = 𝜉𝜖𝑐𝑢 − 𝜖𝑐𝑡 + 𝜈2 𝜖𝑐𝑡 (1 − 𝜉) in the range 𝑎 ≤ 𝜖11 ≤ 𝑏 ,

⎧ 𝑎 ⎨𝑏 ⎩

= (1 − 𝜈2 + 𝜉𝜈2 )𝜖𝑐𝑡 − (1 − 𝜉)𝜈 𝜖22 = 𝜉𝜖𝑐𝑢 − (1 − 𝜉)𝜈 𝜖22

(7.48)

and for the crack width (𝜖𝑐𝑢 − 𝜖𝑐𝑡 )𝜖11 + 𝜈(1 − 𝜉)(𝜖𝑐𝑢 − 𝜖𝑐𝑡 )𝜖22 − (1 − 𝜈2 − 𝜉 + 𝜈2 𝜉)𝜖𝑐𝑢 𝜖𝑐𝑡 𝑑 𝜖12 (7.49) 𝑤2 = 𝑏𝑤 𝜉 𝑤1 = 𝑏𝑤

These solutions reproduce uncracked elastic stress–strain behaviour for 𝜎1 , 𝜎2 with 𝜉 = 0. A related crack band width 𝜉 > 1 is nominally allowed but should be checked with respect to side effects. In the case of decoupling with 𝜈 = 0, the stress component 𝜎11 agrees with the uniaxial case (Example 3.3, Eq. (3.61)) and leads to a crack width 𝑤1 = 𝑏𝑤

(𝜖𝑐𝑢 − 𝜖𝑐𝑡 ) 𝜖11 − (1 − 𝜉) 𝜖𝑐𝑢 𝜖𝑐𝑡 𝜉𝜖𝑐𝑢 − 𝜖𝑐𝑡

(7.50)

which yields 𝑤1 = 𝑏𝑤 𝜖𝑐𝑡 for 𝜖11 = 𝜖𝑐𝑡 , as is required by definition. Furthermore, a ‘full’ crack without crack traction, where 𝑤1 ≥ 𝑤𝑐𝑟 (Figure 7.2), should be considered. This is connected with 𝑡1 = 𝜎11 = 0 (Eq. (7.441 )) and leads to 𝜎22 = 𝐸 𝜖22 𝑤1 =

𝑏𝑤 (𝜖11 + 𝜈(1 − 𝜉)𝜖22 ) = 𝐿𝑐 (𝜖11 + 𝜈(1 − 𝜉) 𝜖22 ) . 𝜉

with the characteristic element length 𝐿𝑐 .

(7.51)

7.4 Multi-Axial Smeared Crack Model

The tangential stiffness is at first derived according to Eq. (7.40). The uncracked material compliance is given with the matrix from Eq. (7.45). The tangential matrix core D𝑐𝐿𝑇 follows Eq. (7.9). The resolution of Eq. (7.42) with respect to 𝑤1 yields 𝑤1 = (𝑤𝑐𝑟 − 𝑤𝑐𝑡 )(1 − 𝑡1 ∕𝑓𝑐𝑡 ) + 𝑤𝑐𝑡 . Its time derivative is 𝑤̇ 1 = −(𝑤𝑐𝑟 − 𝑤𝑐𝑡 )∕𝑓𝑐𝑡 𝑡̇ 1 , leading to ⎡− D𝑐𝐿𝑇 = ⎢ ⎢ ⎣

𝑤𝑐𝑟 − 𝑤𝑐𝑡 𝑓𝑐𝑡 0

0⎤ ⎥ ⎥ 0 ⎦

(7.52)

which further results in 𝑓𝑐𝑡 C𝑇 = 𝑑

⎡−1 ⎢ ⎢−𝜈 ⎢ ⎢ 0 ⎣

0⎤ ⎥ 0⎥ ⎥ ⎥ 0 ⎦

−𝜈 𝜉𝜖𝑐𝑢 − 𝜖𝑐𝑡 𝜖𝑐𝑡 (1 − 𝜉) 0

(7.53)

according to Eq. (7.40) and with 𝑤𝑐𝑟 = 𝑏𝑤 𝜖𝑐𝑢 , 𝑤𝑐𝑡 = 𝑏𝑤 𝜖𝑐𝑡 . Comparing to the time derivatives of Eq. (7.471,2 ) this deviates in the second column by factors (1 − 𝜉), as Eq. (7.34) is now strictly applied. The form derived from Eq. (7.47) relaxing the above-mentioned constraint is recommended, although it misses symmetry. The strain energy within an element connected with strain softening is obtained with 𝑏

𝑏

𝑊 = 𝐿 ∫ 𝜎11 d𝜖11 = 𝐿𝑐 𝑓𝑐𝑡 ∫ 𝑎

𝑎

𝐿𝑐 𝑓𝑐𝑡 𝑏 − 𝜖11 (𝑏 − 𝑎) d𝜖11 = 2 𝑏−𝑎

𝐿𝑐 𝑓𝑐𝑡 = (𝜉𝜖𝑐𝑢 − 𝜖𝑐𝑡 (1 − 𝜈2 (1 − 𝜉))) 2 𝐿𝑐 𝑓𝑐𝑡 2𝐺𝑓 + 𝜖𝑐𝑡 𝜈2 (1 − 𝜉)) = ( 2 𝐿𝑐 𝑓𝑐𝑡

(7.54)

considering Eq. (7.433 ). The quantity 𝜖𝑐𝑡 𝜈2 is a small number in the range of 10−5 for concrete, which generally should be significantly smaller than 𝐺𝑓 ∕(𝐿𝑐 𝑓𝑐𝑡 ). Thus, 𝑊 ≈ 𝐺𝑓 , and the crack energy is basically reproduced within a single element. A key property is given with the crack band width 𝑏𝑤 or its related value 𝜉 = 𝑏𝑤 ∕𝐿𝑐 , respectively. According to Eq. (3.8) it has to be determined from the crack energy 𝐺𝑓 and the volume specific crack energy 𝑔𝑓 (Eq. (3.7)), whereby 𝑔𝑓 follows from the softening tensile stress–strain relation (Figures 3.3b and 3.15). We set 𝜖𝑐𝑢 = 𝛼𝜖 𝜖𝑐𝑡 ,

𝛼𝜖 > 1

(7.55)

which results in a volume-specific crack energy 𝑔𝑓 = (𝛼𝜖 − 1) 𝑓𝑐𝑡 𝜖𝑐𝑡 ∕2 with the bilinear approach (Figure 3.15). This yields 𝑏𝑤 =

2𝐺𝑓 (𝛼𝜖 − 1) 𝑓𝑐𝑡 𝜖𝑐𝑡

=

1 𝐺𝑓 𝐸 1 = 𝑙 𝛼𝜖 − 1 𝑓 2 𝛼𝜖 − 1 𝑐ℎ 𝑐𝑡

(7.56)

221

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7 Crack Modelling and Regularisation

with the cohesive characteristic length 𝑙𝑐ℎ (Eq. (7.1)). Furthermore, the critical crack width (Figure 7.11) is derived by 𝑤𝑐𝑟 = 𝑏𝑤 𝜖𝑐𝑢 =

𝛼𝜖 2𝐺𝑓 𝛼𝜖 − 1 𝑓𝑐𝑡

(7.57)

With some margin of discretion, we generally use 𝛼𝜖 = 5, leading to a magnitude 𝑤𝑐𝑟 ≈ 0.1 mm for typical values of 𝐺𝑓 , 𝑓𝑐𝑡 . Such a magnitude is supported by experimental observations. A bilinear softening law (Figure 3.15) was used for the example in order to avoid nonlinear equations. More adequate nonlinear approaches (Figure 3.7 and Eq. (7.5)) follow the same scheme but lead to nonlinear systems to be solved for the unknown quantities. Crack tractions with shear components along a crack plane can be treated as an extension of the equations, but the required description for the shear traction depending on crack sliding is rare. A first proposal is given with Eq. (7.7). A further elaboration of the smeared crack model with elasticity is described in Section 8.2. A reproduction of crack energy within a single element irrespective of the element size is defined as a criterion for the crack band approach for regularisation (Section 7.3). The smeared crack model fulfils this criterion, as is exemplarily demonstrated with Example 7.3, Eq. (7.54). Both are compatible with the standard FEM, as they allow for continuous displacement interpolations. Insofar the crack band approach and the smeared crack model are similar. But in contrast to the crack band approach, the smeared crack model introduces a crack orientation, leading to a load-induced anisotropy. Furthermore, the smeared crack model is not restricted to a single crack. Multiple cracking can be treated with an extension of Eq. (7.34) ( ) 𝝐̇ = (1 − 𝜉) 𝝐 𝑢 + 𝜉 QT1 ⋅ ˜ 𝝐 𝑐,1 + QT2 ⋅ ˜ 𝝐 𝑐,2 + ⋯ (7.58) where QT𝑖 denotes the orientation of a crack 𝑖. This leads to extended forms of the above. Furthermore, basically any type of material law – elasticity, elasto-plasticity, damage – may be used for the material description of the uncracked material. For a combination of smeared cracking and triaxial elasto-plasticity, see Cervenka and Papanikolaou (2008). The incremental stress–strain relation Eq. (7.40) derived under the premise of Eq. (7.34) represents a superposition – smearing – of cracks and uncracked material covering some area of material represented by a single material point, i.e. an integration point with respect to finite elements. This does not require an explicit modelling of displacement discontinuities. ◀

Crack propagation is determined by a temporal sequence of spatial integration points fulfilling the Rankine criterion with respect to the tensile strength and crack orientation.

Due to smearing a crack geometry cannot be precisely determined unlike discrete crack modelling (Section 7.6) but only some area of cracking is indicated and its propagation during a load history.

7.5 Gradient Methods

A final question concerns the evolution of crack orientations for a given cracked point. With respect to the Rankine criterion, principal stress orientations may change during a load history. This leads to two alternative concepts of crack orientations: • A fixed crack: the crack orientation is fixed with the position, which occurs at the onset of cracking. This corresponds to clearly separated opposite crack surfaces or macrocracks (Figure 3.3a). • A rotating crack: The crack orientation follows the direction of principal stresses, and it may change during a load history. This corresponds to crack bands and microcracking, where the orientation of a bunch of microcracks may change with changing principal stress directions. Cracking is put down to mode-I cracking (Figure 7.1a) when using rotating cracks. Although the fixed crack concept seems to be more realistic from a phenomenological point of view, complications may be involved. In many applications, it yields to a stiff behaviour of the cracked material. Furthermore, it introduces a relative sliding of crack surfaces with sliding components 𝑤2 , 𝑤3 leading to in-plane crack tractions 𝑡2 , 𝑡3 (Eq. (7.4)). Concepts of crack orientation may be combined, starting with a rotating crack and switching over to a fixed crack in the case when some threshold crack width value is exceeded. But this introduces one more estimate coefficient. Finally, the effect of crack closure has to be considered. A cracked material more or less regains stiffness after a reduction of the crack width to zero or a relatively small value. This is exemplarily treated for the strong discontinuity approach (Section 7.7.2, Figure 7.16). Such an approach is also applicable for the smeared crack due to the similarity of Eq. (7.32) with Eq. (7.133) and is described with a further elaboration of the smeared crack model with elasticity in Section 8.2.

7.5 Gradient Methods 7.5.1 Gradient Damage Non-local methods allow for a regularisation including the resolution of localisation zones – the latter in contrast to the crack band and smeared crack models. The uniaxial non-local differential form Eq. (7.29) is extended to multi-axiality and applied to damage in the following. The approach is known as gradient damage. The damage material model described in Section 6.6 is used as the frame of application. The generalised gradient damage form is given by 𝜅(x) − 𝑐 Δ𝜅(x) = 𝜅(x) ,

𝑐=

𝑅2 2

(7.59)

with the spatial coordinate x, the non-local equivalent damage strain 𝜅(x), the Laplace differential operator Δ, the local equivalent damage strain 𝜅(x), and a characteristic material length 𝑅. We replace 𝜅𝑑 from Section 6.6 with 𝜅 to simplify the notation.

223

224

7 Crack Modelling and Regularisation

The gradient damage approach has to be integrated in the finite element method. Thus, Eq. (7.59) must be transformed into a weak integral formulation. Such a procedure has already been demonstrated for beams in Section 4.2, where a standard way was outlined. In the case of Eq. (7.59), this starts with [ ] ∫ 𝛿𝜅 𝜅 − 𝜅 + 𝑐 Δ𝜅 d𝑉 = ∫ 𝛿𝜅 𝜅 d𝑉 − ∫ 𝛿𝜅 𝜅 d𝑉 + ∫ 𝑐 𝛿𝜅 Δ𝜅 d𝑉 = 0 𝑉

𝑉

𝑉

𝑉

(7.60) with the volume 𝑉 of the body under consideration and a test function 𝛿𝜅. The product rule of differentiation leads to ( ) (7.61) 𝛿𝜅 Δ𝜅 = div 𝛿𝜅 ∇𝜅 − ∇𝛿𝜅 ⋅ ∇𝜅 with the scalar product ⋅, the divergence operator div, and the nabla operator ∇. Application of the Gauss divergence theorem (Kreyszig 2006) yields ( ) ∫ div 𝛿𝜅 ∇𝜅 d𝑉 = ∫ 𝛿𝜅 n ⋅ ∇𝜅 d𝐴 𝑉

(7.62)

𝐴

with the surface 𝐴 of the body and the outer surface normals n. Thus, Eq. (7.60) can be written in the form ∫ 𝛿𝜅 𝜅 d𝑉 + ∫ 𝑐 ∇𝛿𝜅 ⋅ ∇𝜅 d𝑉 = ∫ 𝛿𝜅 𝜅 d𝑉 + ∫ 𝑐 𝛿𝜅 n ⋅ ∇𝜅 d𝐴 𝑉

𝑉

𝑉

(7.63)

𝐴

This form is suited for discretisation with respect to the non-local equivalent damage strain 𝜅, while the local value 𝜅 is given as a driving part. The surface integral part remains to be discussed. Additional boundary conditions for the non-local equivalent damage strain are required, either 𝜅 or the normal derivative n ⋅ ∇𝜅 have to be specified in every point of the surface 𝐴. It is generally assumed that the gradient ∇𝜅 at boundaries yields major gradients in the direction of boundary tangents only, i.e. zones of localisation are approximately normal to boundaries. Thus, n ⋅ ∇𝜅 = 0 can be set along the whole boundary 𝐴 (Peerlings et al. 1996), and Eq. (7.63) is simplified as ∫ 𝛿𝜅 𝜅 d𝑉 + ∫ 𝑐 ∇𝛿𝜅 ⋅ ∇𝜅 d𝑉 = ∫ 𝛿𝜅 𝜅 d𝑉 𝑉

𝑉

(7.64)

𝑉

This is used in the following and combined with isotropic damage (Section 6.6). To begin with, the local 𝜅 in Eq. (6.107) is replaced by the non-local 𝜅 𝐷(𝜅) = 1 − e

−(

𝑔 𝜅−𝑒0 𝑑 ) 𝑒𝑑

,

𝜅 ≥ 𝑒0

(7.65)

and we obtain the increment of damage d𝐷 depending on the increment d𝜅 of the non-local equivalent damage d𝐷 1 d𝜅 = d𝜅 , d𝐷 = ℎ d𝜅

1 = ℎ

𝑔𝑑 (

𝜅−𝑒0 𝑒𝑑

𝑔𝑑

)

𝜅 − 𝑒0

e

−(

𝑔 𝜅−𝑒0 𝑑 ) 𝑒𝑑

(7.66)

7.5 Gradient Methods

Using Eq. (6.116) the stress increment is given by d𝝈 = (1 − 𝐷) E ⋅ d𝝐 − E ⋅ 𝝐 d𝐷 1 = (1 − 𝐷) E ⋅ d𝝐 − 𝝈0 d𝜅 , ℎ

𝝈0 = E ⋅ 𝝐

(7.67)

completing the material and gradient damage parts. Static equilibrium has a condition ∫ 𝛿𝝐 T ⋅ 𝝈 d𝑉 = ∫ 𝛿uT ⋅ b d𝑉 + ∫ 𝛿uT ⋅ t d𝐴 𝑉

𝑉

(7.68)

𝐴𝑡

according to Eq. (2.52). The weak forms (Eqs. (7.64) and (7.68)) are discretised by u = N𝑢 ⋅ 𝝊 ,

𝜅 = N𝜅 ⋅ 𝜿

(7.69)

with the matrices N𝑢 , N𝜅 of the trial functions and the vectors 𝝊, 𝜿 of the nodal values of displacement, and non-local equivalent damage strain as global unknowns. Strain measures and their increments are interpolated by 𝝐 = B𝑢 ⋅ 𝝊 , ∇𝜅 = B𝜅 ⋅ 𝜿 ,

d𝝐 = B𝑢 ⋅ d𝝊

(7.70)

d ∇𝜅 = B𝜅 ⋅ d𝜿

The test functions 𝛿u, 𝛿𝜅 are discretised in the same way. Using Eqs. (7.69) and (7.70) together with the weak forms (Eqs. (7.64) and (7.68)) leads to an extension of the condition of discretised equilibrium Eq. (2.70) r = p − f(a) = 0

(7.71)

with ⎛p𝑢 ⎞ p=⎜ ⎟ , 0 ⎝ ⎠

⎛ f𝑢 ⎞ f =⎜ ⎟ , f ⎝ 𝜅⎠

⎛𝝊 ⎞ a=⎜ ⎟ 𝜿 ⎝ ⎠

(7.72)

with the external nodal forces p𝑢 = ∫ NT𝑢 ⋅ b d𝑉 + ∫ NT𝑢 ⋅ t d𝐴 𝑉

(7.73)

𝐴𝑡

and the extended internal nodal forces f𝑢 = ∫ BT𝑢 ⋅ 𝝈 d𝑉 ,

f𝜅 = ∫

𝑉

𝑉

] [ T N𝜅 (𝜅 − 𝜅) + 𝑐 BT𝜅 ⋅ ∇𝜅 d𝑉

(7.74)

with f𝑢 depending on 𝝊 → 𝝐 → 𝝈 and 𝜿 → 𝐷 → 𝝈 and f𝜅 depending on 𝜿 and on the local 𝝊 → 𝝐 → 𝜅. Thus, f𝑢 and f𝜅 are coupled, and Eq. (7.71) forms a system of non-linear algebraic equations. The solution requires a tangential stiffness matrix 𝜕f𝑢

⎡ 𝜕f 𝜕𝝊 = ⎢ 𝜕f K𝑇 = 𝜅 𝜕a ⎣ 𝜕𝝊

𝜕f𝑢

⎤

⎡K𝑢𝑢 =⎢ K ⎣ 𝜅𝑢 𝜕𝜿 ⎦

𝜕𝜿 𝜕f𝜅 ⎥

K𝑢𝜅 ⎤ ⎥ K𝜅𝜅 ⎦

(7.75)

225

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7 Crack Modelling and Regularisation

Equations (7.741 ) and (7.67) yield K𝑢𝑢 = ∫ (1 − 𝐷) BT𝑢 ⋅ E ⋅ B𝑢 d𝑉 ,

K𝑢𝜅 = − ∫

𝑉

𝑉

1 T B ⋅ 𝝈 ⋅ N𝜅 d𝑉 ℎ 𝑢 0

(7.76)

and in the case of loading, Eqs. (6.113) and (7.742 ) lead to K𝜅𝑢 = − ∫ 𝑉

with d𝜅 =

1 𝐻

1 T T N ⋅ n ⋅ B𝑢 d𝑉 , 𝐻 𝜅

nT ⋅ d𝝐 and 𝐻 = −

𝜕𝐹 𝜕𝜅

, n=

K𝜅𝜅 = ∫

𝑉

𝜕𝐹 𝜕𝝐

( T ) N𝜅 ⋅ N𝜅 + 𝑐 BT𝜅 ⋅ B𝜅 d𝑉 (7.77)

and the damage function 𝐹 (Eq. (6.109)).

Furthermore, in the case of unloading, K𝑢𝜅 = K𝜅𝑢 = 0 and d𝜿 = 0, according to the Kuhn–Tucker conditions Eq. (6.112). The tangential system stiffness matrix K𝑇 is unsymmetric, but this generally occurs for damage formulations not derived from potentials with the principle of maximum dissipation. The iteration rule is [ (𝜈) ]−1 ⋅ r(a(𝜈) ) 𝛿a = K𝑇 a(𝜈+1) = a(𝜈) + 𝛿a

(7.78)

according the Newton–Raphson method (Eq. (2.77)). This is embedded in an incremental load application (Section 2.8.2). A unified formalism allowing for a seamless integration into standard FEM algorithms is exemplarily described in the following example. Example 7.4: Gradient Damage Formulation for the Uniaxial Tension Bar

The gradient damage formulation is illustrated with the simple uniaxial tensile case in connection with the uniaxial two-node bar element (Section 2.3). The same interpolation is chosen for displacements and non-local damage strain N𝑢 = N𝜅 = N with N from Eq. (2.22). A single element is considered. Assembling follows as is outlined in Section 2.6, item 6. With Eqs. (7.67) and (7.77) we set [ ] E= 𝐸 ( ) 𝝈0 = 𝐸 𝜖 𝐹 = 𝑐1 n=

(1 + 𝜈)2 2 1+𝜈 𝜖 + 𝜅 (𝑐2 √ + 𝑐3 + 𝑐4 (1 − 2𝜈)) 𝜖 − 𝜅2 3 3

(7.79)

(1 + 𝜈)2 1+𝜈 𝜕𝐹 𝜖 + 𝜅 (𝑐2 √ + 𝑐3 + 𝑐4 (1 − 2𝜈)) = 2𝑐1 3 𝜕𝜖 3

The uniaxial two-node element has the nodal degrees of freedom ( a𝑒 = 𝑢 𝐼

𝜅𝐼

𝑢𝐽

𝜅𝐽

)T

(7.80)

7.5 Gradient Methods

Forms for the internal nodal forces and tangential element stiffness matrix are prepared with ⎡𝐵𝐼

0

𝐵𝐽

0

𝐵𝐼

0

B=⎢

⎣

0⎤ ⎥, 𝐵𝐽 ⎦

𝐵𝐼 =

2 1 2 1 (− ) , 𝐵𝐽 = ( ), 𝐿𝑒 2 𝐿𝑒 2

0⎤ ⎥ 𝑐 ⎦ (7.81)

⎡(1 − 𝐷) 𝐸

C=⎢

⎣

0

with 𝑐 from Eq. (7.59), leading to BT ⋅ C ⋅ B whereby accounting for the stiffness matrix part K𝑢𝑢 (Eq. (7.76)) and the second part of K𝜅𝜅 (Eq. (7.77)). The matrix C is variable due to variable damage 𝐷. Non-local damage strains and nodal displacements are still uncoupled with this approach. Coupling is performed using ⎡𝐵𝐼 0 𝐵𝐽 0 ⎤ N=⎢ ⎥, 0 𝑁𝐼 0 𝑁𝐽 ⎦ ⎣

1 1 𝑁𝐼 = (1 − 𝑟), 𝑁𝐽 = (1 + 𝑟) , 2 2

⎡ 0 C = ⎢ 1 𝜕𝐹 − ⎣ 𝐻 𝜕𝜖 ′

−

𝐸𝜖

⎤ ⎥ 1 ⎦ (7.82) ℎ

with 𝐵𝐼 , 𝐵𝐽 as in Eq. (7.81), leading to N T ⋅ C′ ⋅ N, whereby accounting for K𝑢𝜅 , K𝜅𝑢 (Eqs. (7.76) and (7.77)) and the first part of K𝜅𝜅 (Eq. (7.77)). The numerical integration is performed with one-point integration leading to an tangential element stiffness matrix (Eq. (7.75)) K𝑇 =

] 𝐿𝑒 [ T T ′ B ⋅C⋅B+N ⋅C ⋅N 2

(7.83)

To derive the extended internal nodal forces a generalised strain is defined including ∇𝜅 = d𝜅∕ d𝑥 ⎛𝜖⎞

E = ⎜ d𝜅 ⎟ = B ⋅ a𝑒

(7.84)

⎝ d𝑥 ⎠

leading to a generalised stress S=C⋅E

(7.85)

and further to BT ⋅ S contributing to the extended internal nodal forces (Eq. (7.74)). Its coupling part NT (𝜅 − 𝜅) is still missing. Another general stress-like parameter is introduced as ′

⎛ 0 ⎞ ⎟ 𝜅−𝜅 ⎠ ⎝

S =⎜

(7.86)

and with one-point integration finally leading to element internal nodal forces (Eq. (7.72)) f=

) 𝐿𝑒 ( T T ′ B ⋅S+N ⋅S 2

(7.87)

227

228

7 Crack Modelling and Regularisation

A scaling of the components of the components K𝜅𝑢 , K𝜅𝜅 of K𝑇 (Eq. (7.75)) or of the second rows in C, C′ may be necessary to avoid large differences of values in K𝑇 and a bad numerical condition. Young’s modulus 𝐸 is appropriate as a scaling factor. This 1D implementation is applied with Example 3.1. The approach demonstrated in Example 7.4 can be transferred to other element types in an analogous manner. More aspects concerning gradient damage are given in, e.g. Geers et al. (1998); Kuhl, Ramm and de Borst (2000); Peerlings et al. (2001); Pamin (2005).

7.5.2 Phase Field An alternative to gradient damage is given with the phase field. ◀

Phase field solutions are derived from the minimisation of potentials, which makes them attractive from a mathematical point of view regarding, e.g. convergence behaviour, and might yield advantageous properties from a numerical point of view.

The phase field approach is described regarding a 1D-bar and following Miehe et al. (2010). The spatial coordinate is given with 𝑥 along the bar length 0 ≤ 𝑥 ≤ 𝐿. The concept of damage is again involved with a degradation function 𝜔(𝑑) =

⎧ (1 − 𝑑)2 ⎨0 ⎩

0≤𝑑≤1 𝑑≥1

(7.88)

with the phase field variable 𝑑 considered as damage. This degradation function is applied to the linear elastic material law 𝜎 = 𝜔(𝑑) 𝐸 𝜖 with Young’s modulus 𝐸. A sharp crack in a position 𝑥 = 𝑥𝑐 is described with 𝑑(𝑥) =

⎧

1 for 𝑥 = 𝑥𝑐 ⎨0 otherwise ⎩

(7.89)

see Figure 7.12a. This discontinuity is regularised with 𝑑(𝑥) = 𝑒

−

|𝑥−𝑥𝑐 | 𝑙

(7.90)

whereby 𝑙 is a measure of spatial range (Figure 7.12a) and may be considered as smoothly smearing a sharp peak. This function minimises the functional 𝐷=

( ) 𝑐 ∫ 𝑑2 + 𝑙2 𝑑′2 d𝑥 2𝑙

(7.91)

𝐿

with 𝑑′ = d𝑑∕ d𝑥 with an arbitrary constant 𝑐. It is chosen with 𝑐 = 𝐺𝑓 𝐴 with the bar cross-sectional area 𝐴. For the following, 𝐺𝑓 is given a physical significance

7.5 Gradient Methods

(a)

(b)

Figure 7.12 (a) Phase field regularisation (Miehe et al. 2010). (b) Reference bar for phase field.

as the fracture toughness 𝐺𝑓 with a unit of [force × length∕length2 ]. The quantity 𝐷 is assumed to be a dissipation potential. This is complemented with the usual potentials, as is the internal energy assuming damaged elasticity 𝛹=

𝐸𝐴 ∫ 𝜔(𝑑) 𝜖 2 d𝑥 2

(7.92)

𝐿

with the bar longitudinal displacement 𝑢, its strain 𝜖, and its cross-sectional area 𝐴, and furthermore with the potential of external loading 𝑃 = ∫ 𝑝 𝑢 d𝑥 − 𝑡𝑎 𝑢𝑎 + 𝑡𝑏 𝑢𝑏

(7.93)

𝐿

see Figure 7.12b. The total energy functional is given by 𝛱(𝑢, 𝜖, 𝑑, 𝑑′ ) = 𝛹(𝜖, 𝑑) + 𝐷(𝑑, 𝑑′ ) − 𝑃(𝑢)

(7.94)

to be minimised in order to derive solutions for the displacement field and the damage field. ◀

‘Crack propagation results from the competition between the bulk (𝛹 ) and surface (D) energy terms. On the one hand, deformations of an elastic body under load increase the elastic energy. When this value approaches a critical value in a region, it is energetically favorable for the system to decrease the elastic energy by increasing the crack phasefield towards unity. On the other hand, increasing the value of d leads to an increase in the surface energy.’ (Wu et al. 2020)

This concept originates from the pioneering work of Griffith (1920). Variational calculus yields the Euler–Lagrange equations of 𝛱 ′

𝐸𝐴 (𝜔(𝑑) 𝜖) + 𝑝 = 0 ,

𝑑 − 𝑙2 𝑑′′ = −

𝑙𝐸 ′ 𝜔 (𝑑) 𝜖 2 2𝐺𝑓

(7.95)

229

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7 Crack Modelling and Regularisation

with 𝜔′ (𝑑) = d𝜔∕ d𝑑. Using the bar force 𝐹 = 𝐴 𝜎 = 𝐸𝐴 𝜔(𝑑) 𝜖 and Eq. (7.88) for the degradation function 𝜔(𝑑) we obtain 𝐹′ + 𝑝 = 0 𝑑 − 𝑙2 𝑑′′ = F ,

F=

𝑙𝐸 (1 − 𝑑) 𝜖 2 𝐺𝑓

(7.96)

The first equation states equilibrium, the second the evolution law for the damage variable 𝑑 with a driving force. Equation (7.962 ) classifies the approach as a nonlocal gradient method with some formal agreement to gradient damage Eqs. (7.29) and (7.59). It has to be pointed out that F = 0 for 𝑑 = 1, i.e. the constraint 0 ≤ 𝑑 ≤ 1 should automatically be fulfilled, and 𝑑 − 𝑙2 𝑑′′ = 0 is solved by 𝑑(𝑥) with Eq. (7.90). The functional 𝛱 Eq. (7.94) is minimised using disturbances 𝛼 applied with displacement test functions 𝛿𝑢, 𝛿𝜖 and 𝛽 applied with damage field test functions 𝛿𝑑, 𝛿𝑑′ , which leads to a variational fracture model (Bourdin et al. 2008) 𝜕𝛱 = 0 → − ∫ 𝛿𝑢 𝑝 d𝑥 + 𝛿𝑢𝑎 𝑡𝑎 − 𝛿𝑢𝑏 𝑡𝑏 + 𝐸𝐴 ∫ 𝛿𝜖 𝜔(𝑑) 𝜖 d𝑥 = 0 𝜕𝛼 𝐿

𝐿

𝐸𝐴 𝜕𝛱 ∫ 𝛿𝑑 𝜔′ (𝑑) 𝜖 2 d𝑥 + =0→ 2 𝜕𝛽

𝐺𝑓 𝐴 𝑙

𝐿

∫ 𝛿𝑑 𝑑 d𝑥 + 𝐺𝑓 𝐴 𝑙 ∫ 𝛿𝑑′ 𝑑′ d𝑥 = 0 𝐿

𝐿

(7.97) This is used for a finite element discretisation with trial functions for the displacement (Eq. (2.57)) 𝑢 = N𝑢 ⋅ 𝝊 𝑒 ,

𝛿𝑢 = N𝑢 ⋅ 𝛿𝝊𝑒 → 𝜖 = B𝑢 ⋅ 𝝊𝑒 ,

𝛿𝜖 = B𝑢 ⋅ 𝛿𝝊𝑒

(7.98)

and the damage field 𝑑 = N𝑑 ⋅ d 𝑒 ,

𝛿𝑑 = N𝑑 ⋅ 𝛿d𝑒 → 𝑑′ = B𝑑 ⋅ d𝑒 ,

𝛿𝑑′ = B𝑑 ⋅ 𝛿d𝑒

(7.99)

Internal nodal forces are derived with f𝑢 = 𝐸𝐴 ∫ BT𝑢 𝜎 d𝑥 ,

𝜎 = 𝜔(𝑑) 𝜖 = (1 − 𝑑)2 𝜖

𝐿

f𝑑 =

𝐺𝑓 𝐴 𝐸𝐴 ∫ NT𝑑 𝜔′ (𝑑) 𝜖 2 d𝑥 + ∫ NT𝑑 𝑑 d𝑥 + 𝐺𝑓 𝐴 𝑙 ∫ BT𝑑 𝑑′ d𝑥 2 𝑙 𝐿

𝐿

= −𝐸𝐴 ∫ NT𝑑 (1 − 𝑑) 𝜖 2 d𝑥 + 𝐿

𝐺𝑓 𝐴 𝑙

(7.100)

𝐿

∫ NT𝑑 𝑑 d𝑥 + 𝐺𝑓 𝐴 𝑙 ∫ BT𝑑 𝑑′ d𝑥 𝐿

𝐿

whereby the degradation function 𝜔(𝑑) (Eq. (7.88)) is involved with 𝜔(𝑑) = (1 − 𝑑)2 ,

𝜔′ (𝑑) = −2 (1 − 𝑑) ,

𝜔′′ (𝑑) = 2

(7.101)

7.5 Gradient Methods

and the discretised equilibrium condition is given with ⎛p⎞ ⎛f𝑢 ⎞ r = ⎜ ⎟−⎜ ⎟ = 0 f 0 ⎝ ⎠ ⎝ 𝑑⎠

(7.102)

with prescribed external nodal forces p (Eq. (2.59)) applied to 1D. This constitutes a non-linear problem due to the overall 𝜔(𝑑)-dependence of Eq. (7.100). The incremental change required for an equilibrium iteration is given with ⎛df𝑢 ⎞ ⎡K𝑢𝑢 ⎜ ⎟=⎢ df K ⎝ 𝑑 ⎠ ⎣ 𝑑𝑢

K𝑢𝑑 ⎤ ⎛ d𝝊𝑒 ⎞ ⎥⋅⎜ ⎟ K𝑑𝑑 dd ⎦ ⎝ 𝑒⎠

(7.103)

for a tangential stiffness with K𝑢𝑢 = 𝐸𝐴 ∫ 𝜔(𝑑) BT𝑢 ⋅ B𝑢 d𝑉 = 𝐸𝐴 ∫ (1 − 𝑑)2 BT𝑢 ⋅ B𝑢 d𝑉 𝐿

𝐿

K𝑢𝑑 = 𝐸𝐴 ∫ 𝜔′ (𝑑) 𝜖 BT𝑢 ⋅ N𝑑 d𝑉 = −2𝐸𝐴 ∫ (1 − 𝑑) 𝜖 BT𝑢 ⋅ N𝑑 d𝑉 𝐿

𝐿

K𝑑𝑢 = 𝐸𝐴 ∫

𝜔′ (𝑑) 𝜖 NT𝑑

⋅ B𝑢 d𝑉 = −2𝐸𝐴 ∫ (1 − 𝑑) 𝜖 NT𝑑 ⋅ B𝑢 d𝑉

𝐿

K𝑑𝑑

𝐿

𝐸𝐴 ∫ 𝜖 2 𝜔′′ (𝑑) NT𝑑 ⋅ N𝑑 d𝑥 = 2 𝐿

+

𝐺𝑓 𝐴 𝑙

∫ NT𝑑 ⋅ N𝑑 d𝑥 + +𝐺𝑓 𝐴 𝑙 ∫ BT𝑑 ⋅ B𝑑 d𝑥 𝐿

𝐿

= 𝐸𝐴 ∫ 𝜖 2 NT𝑑 ⋅ N𝑑 d𝑥 + 𝐿

𝐺𝑓 𝐴 𝑙

∫ NT𝑑 ⋅ N𝑑 d𝑥 + +𝐺𝑓 𝐴 𝑙 ∫ BT𝑑 ⋅ B𝑑 d𝑥 𝐿

𝐿

(7.104) with K𝑑𝑢 = KT𝑢𝑑 , leading to a symmetric stiffness matrix. Apart from symmetry, there is some formal agreement of this tangential stiffness with the tangential stiffness of gradient damage (Eqs. (7.75)–(7.77)). A condition to discriminate loading from unloading remains to be defined. This has generally been accomplished with the Kuhn–Tucker conditions; see, e.g. Eqs. (6.65), (6.112), and (6.155) and is adapted with a strain function 𝐹 = |𝜖| − max |𝜖|

(7.105)

for the current uniaxial set-up with the largest amount of strain max |𝜖| ever reached in the loading history. The Kuhn–Tucker conditions are defined with 𝐹 ≤ 0,

𝑑̇ ≥ 0 ,

𝐹 𝑑̇ = 0

(7.106)

231

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7 Crack Modelling and Regularisation

with the damage field variable 𝑑, where 𝑑̇ > 0 indicates loading and |𝜖| < max |𝜖| unloading. One has to set 𝜖 = sign(𝜖) max |𝜖| in Eqs. (7.1002 ) and (7.1042–4 ) in the case of unloading to decouple the damage field from strains and maintain it frozen where required. Appropriate boundary conditions have to be supplemented. Considering the bar ends these are prescribed with displacements and appropriate loadings and generally 𝑑′ = 0 with respect to the damage field, as is also derived from the functional minimisation. The solution again follows the incrementally iterative scheme (Section 2.8.2) extended with the damage field. The parameter 𝑙 (Figure 7.12a) rules the width of the localisation zone. A convergence to brittle fracture (Figure 7.1b) with a sharp discontinuity can be proven for 𝑙 → 0 (𝛤-convergence) thereby reproducing a prescribed fracture toughness 𝐺𝑓 . The application is demonstrated with the following example where the limits of numerical calculation are also shown. Example 7.5: Phase Field Formulation for the Uniaxial Tension Bar

The phase field method is illustrated again with the simple uniaxial tensile case in connection with the uniaxial two-node bar element (Section 2.3). The same interpolation is chosen for displacements and the damage variable N𝑢 = N𝑑 = N with N from Eq. (2.22). A single element is considered. Assembling follows, as is outlined in Section 2.6, item 6. The uniaxial two-node element has the nodal degrees of freedom )T ( a𝑒 = 𝑢𝐼 𝑑𝐼 𝑢𝐽 𝑑𝐽 (7.107) Forms for the internal nodal forces and tangential element stiffness matrix are prepared with ⎡𝐵𝐼

0

𝐵𝐽

0

𝐵𝐼

0

B=⎢

⎣

0⎤ ⎥, 𝐵𝐽 ⎦

𝐵𝐼 =

2 1 2 1 (− ) , 𝐵𝐽 = ( ) 𝐿𝑒 2 𝐿𝑒 2

(7.108)

and ⎡(1 − 𝑑)2 𝐸

C=𝐴⎢

⎣

0

0 ⎤ ⎥ 𝐺𝑓 𝑙 ⎦

(7.109)

leading to BT ⋅ C ⋅ B, thereby accounting for the stiffness matrix part K𝑢𝑢 and the second part of K𝑑𝑑 (Eq. (7.104)). The matrix C is variable due to a variable damage 𝑑. Damage and nodal displacements are still uncoupled. Coupling is performed using ⎡𝐵𝐼

0

𝐵𝐽

0

𝑁𝐼

0

N=⎢

⎣

0⎤ ⎥, 𝑁𝐽 ⎦

𝑁𝐼 =

1 1 (1 − 𝑟), 𝑁𝐽 = (1 + 𝑟) 2 2

(7.110)

and ⎡

˜ = 𝐴⎢ C ⎣

0 −2𝐸 (1 − 𝑑) 𝜖

−2𝐸 (1 − 𝑑) 𝜖⎤ ⎥ 𝐺 𝐸 𝜖2 + 𝑓 𝑙 ⎦

(7.111)

7.5 Gradient Methods

˜ ⋅ N, thereby accounting for K𝑢𝑑 (Eqs. (7.103) and (7.104)) and the leading to N T ⋅ C first part of K𝑑𝑑 . The numerical integration is performed with one-point integration leading to an tangential element stiffness matrix according to Eq. (7.103) K𝑇 =

] 𝐿𝑒 [ T T ˜ B ⋅C⋅B+N ⋅C ⋅N 2

(7.112)

with the element length 𝐿𝑒 . To derive the extended internal nodal forces a generalised strain is defined with ⎛𝜖⎞

E = ⎜ ⎟ = B ⋅ a𝑒 ′

(7.113)

𝑑 ⎝ ⎠

leading to a generalised stress S=C⋅E

(7.114)

and to BT ⋅ S contributing to the extended internal nodal forces (Eq. (7.1002 )). Another general stress-like parameter is introduced as ⎛

˜=𝐴⎜ S

0

−𝐸 (1 − 𝑑) 𝜖 2 + ⎝

𝐺𝑓 𝑙

⎞ ⎟ 𝑑 ⎠

(7.115)

and with one-point integration finally leads to element internal nodal forces (Eq. (7.100)) f=

) 𝐿𝑒 ( T T ˜ B ⋅S+N ⋅S 2

(7.116)

These approaches are applied to the same problem as in Example 3.1 with the uniaxial concrete tension bar. System and discretisation are unchanged; see Figure 3.5 with 𝜅 replaced by 𝑑. Whereas the gradient damage approach of Example 3.1 requires the specification of the initial Young’s modulus 𝐸, tensile strength 𝑓𝑐𝑡 , and the characteristic length 𝑅, the phase field approach requires 𝐸, the fracture toughness 𝐺𝑓 , and the characteristic length 𝑙 (Figure 7.12a). The initial Young’s modulus is retained with 𝐸 = 36 000 MN∕m2 . A fracture toughness is chosen with 𝐺𝑓 = 100 ⋅ 10−6 MNm∕m2 , which is a typical value for the crack energy of concrete. With respect to the characteristic length, it has to be anticipated that 𝑙 determines the resulting tensile strength for a given value 𝐺𝑓 . Decreasing 𝑙 leads to a higher strength, increasing 𝑙 lead to a lower strength. By an inverse iterative approach it is chosen with 𝑙 = 0.03 m to reach a maximum tensile strength of ≈ 3.5 MN∕m2 as for Example 3.1. Figure 7.13 shows a comparison of the computed load–displacement curves for gradient damage and phase field. A similar behaviour can be seen in the hardening range, but a continuation of the phase field cannot be reached after the turning point of the prescribed displacement to initiate a snap-back, not even with extremely small prescribed values for the arc length control. Equation (7.112) with kernels ˜ includes entries that are different in the order of 𝐸 𝑙∕𝐺𝑓 , which yields ≈ 107 C, C

233

234

7 Crack Modelling and Regularisation

Figure 7.13 Example 7.5. Load displacement curves.

(a)

(b)

Figure 7.14 Example 7.5. (a) Strains 𝜖 along the bar. (b) Phase fields d along the bar.

for the material values chosen. The resulting matrix condition numbers are in the order 1010 , which might indicate convergence problems for the Newton–Raphson method in connection with negative eigenvalues of the tangential stiffness as they arise with the onset of softening. On the other hand, a preconditioning of the phase field contributions to the system equations – paid for by the loss of symmetry – does not have any influence on the convergence behaviour either. A potential snap-back is estimated as being extremely sharp. A dissipation-based arc length control might be a remedy (Verhoosel et al. 2009). This issue is not further pursued within this limited scope. Regardless of this, a localisation is already emerging for the last calculated state. This can be seen with the sequence of strain and damage fields along the bar evaluated for equidistant loading steps up to the last converged step; see Figure 7.14. The extension to 2D and 3D follows the same reasoning as described for 1D (Miehe et al. 2010). Phase fields are currently still an emerging field of research, and the standard approach described above is subject to a range of further developments, e.g. among others: • Anisotropic phase fields with discrimination of tensile and compressive states, whereby compression is excluded from damage (Miehe et al. 2010).

7.5 Gradient Methods

• Consideration of quasi-brittle crack behaviour (Figure 7.1b) with a cohesive crack (Figure 7.2) (Verhoosel and de Borst 2013). • Dynamic crack propagation (Hofacker and Miehe 2012). • Mixed mode I-II fracture (Figure 7.1a) (Wang et al. 2020). For a detailed phase field overview see Wu et al. (2020) and the references therein.

7.5.3 Assessment of Gradient Methods Gradient methods – non-local differential forms (Section 7.3) and phase fields – introduce further degrees of freedom in addition to the standard displacement degrees of freedom. This allows for a regularisation with mesh independence without further provisions and, furthermore, for a resolution of localisation zones. Fracture criteria like the Rankine criterion (Section 7.4) are not required to initiate localisation evolving into cracking or to follow the propagation of localisation zones. But the objective requires a discretisation of localisation zones across their thickness direction with a sufficient number of elements. Common crack band widths with the magnitude of 10−2 m require very fine discretisations and large discretised systems. This might become prohibitive for very sharp localisation areas as may, in particular, basically be modelled with phase fields. The non-local differential approach was demonstrated with gradient damage (Section 7.5.1). Gradient damage has lot in common with phase fields (Section 7.5.2). Both introduce internal length scales – gradient damage with 𝑅 (Eq. (7.59)) and the phase field with 𝑙 (Eq. (7.91)) – whereby the question of whether these have to be considered as material parameters is still under discussion. If so, it should be assumed that they have different values for the same material and the same problem. There are more significant differences. Gradient damage tends to enlarge the width of a localisation zone during a loading history, whereas the phase field keeps this basically constant. This can be explained by the respective driving forces, which are the right-hand sides of Eq. (7.59) for gradient damage and Eq. (7.962 ) for a phase field. The latter becomes zero with full damage 𝑑 = 1, while the former still contributes to the equivalent strain even for damage 𝐷 approaching 1, as the equivalent damage strain 𝜅 is not bounded by definition. Another issue concerns the mapping of fracture energy. While it has an explicit equivalent with the fracture toughness 𝐺𝑓 of a phase field, its assignment is not directly possible for gradient damage. It should be ruled by the internal length scale 𝑅. On the other hand, strength can be directly prescribed for gradient damage, which is not possible for the phase field. An inverse analysis has to be used in each case in order to achieve a target value for crack energy or strength by adapting the respective characteristic lengths. Finally, it should be mentioned that the application of gradient approaches is not restricted to damage. Basically, all variable quantities may be subject to non-local integration like Eq. (7.23) or non-local differentiation like Eq. (7.29). Non-local methods are also applied, e.g. with gradient plasticity (de Borst and Pamin 1996; de Borst et al. 1999; Vrech and Etse 2012).

235

236

7 Crack Modelling and Regularisation

7.6 Overview of Discrete Crack Modelling Discrete crack models explicitly describe displacement discontinuities and the crack width in contrast to smeared crack models, which replace the crack width by a crack strain (Eq. (7.36)). With respect to finite elements, discrete crack modelling may be performed on the element level or on the system level. Discontinuities on the element level are given with embedded discontinuity formulations or the strong discontinuity approach (SDA) (Dvorkin et al. 1990; Simo et al. 1993; Jirásek 2000; Oliver et al. 2003; Alfaiate et al. 2003; Mosler and Meschke 2004). The same criteria for crack initiation may be used as for smeared cracks with the Rankine criterion (Section 7.4) applied to the integration points of finite elements. Some constraints are formulated by Oliver et al. (2012) for the seamless transition from continuum to discontinuity. In contrast to smeared cracks, the traction– separation relations (Eq. (7.9)) are directly implied. Similarly to smeared cracks, a crack propagation is indicated by a temporal sequence of spatial integration points fulfilling the Rankine criterion. But this does not directly guarantee a continuous crack contour and displacement compatibility across elements. This is only achieved through additional constraints; see, e.g. Alfaiate et al. (2003). An example of the SDA is given in Section 7.7. Discontinuities on the system level involve an explicit description of a crack contour – a curve in 2D or a surface in 3D – connected with a displacement jump of the global displacement field. Such contours may be captured with zero thickness interface elements linking continuum elements (Khisamitov and Meschke 2018). But to some extent the crack contour is constrained to the discretisation, which might require re-meshing. Re-meshing, which is generally elaborate, can be avoided with the extended finite element method (XFEM) (Moës et al. 1999; Stolarska et al. 2001; Karihaloo and Xiao 2007; Fries and Belytschko 2010). XFEM allows for crack contours intersecting elements, thereby introducing additional degrees of freedom for jump quantities on the system level. Meshless methods like the element-free Galerkin (EFG) method (Belytschko et al. 1996; Lee et al. 2003) avoid an element intersection problem by avoiding elements at all. Displacement interpolation functions are aligned with nodes only, whereby a local support is reached with node distancedependent weighting functions instead of element boundaries. This allows for an unconstrained consideration of given crack contours by a local adaptation of weighting functions without additional degrees of freedom. The mentioned methods belong to computational fracture mechanics. For a placement within a corresponding taxonomy, see Ingraffea and Wawrzynek (2004). Modelling of crack propagation with crack contours on the system level using methods like XFEM or EFG requires an explicit crack path tracking algorithm. Such explicit crack path tracking may also provide constraints for crack modelling with SDA to reach a continuous crack path. Explicit crack tracking involves the following: • Under which conditions will a crack contour start and extend. • Which direction its extension will have. • How long its extension will be.

7.7 The Strong Discontinuity Approach

The first two may be treated with the Rankine criterion on a first impression. But in contrast to element-level approaches, it has to be applied to the tip of a crack contour (point in 2D, spatial curve in 3D). This requires that well-founded finite stress values are computed for the crack tip (Figure 7.2) based on continuum material laws (Sections 6.4–6.8). Linear elasticity in the first instance is excluded, as it yields stress singularities with infinite values at the crack tip. Crack propagation with crack tip singularities has to be treated with the LEFM (Knott 1973; Anderson 2017; Sih 1991; Dharan and Kang 2016) or comparable theories transforming singularities into equivalent bounded quantities like the stress intensity factor or configurational/material forces (Özenç et al. 2014). Elasto-plastic and damage material models have finite stress values due to their intrinsic strength conditions, and the Rankine criterion is basically applicable for crack initiation or extension, whereby regarding the first two items above. Rules for determining an extension length are trial and error – search for a length extension with a crack tip no longer fulfilling the criterion for crack extension – or may rely again on corresponding criteria of LEFM as were cited above. The computation of crack contours may also be decoupled to some extent from local considerations whereby potential crack contours are determined as isolines (2D) or isosurfaces (3D) to prescribe directions – derived from cracking criteria for a given state of stress – similar as with heat conduction problems (Oliver et al. 2004; Feist and Hofstetter 2006). For more details regarding explicit description of crack contours and crack path tracking see Rabczuk (2013). Modelling of crack propagation with crack contours on the system level will not be pursued in the following.

7.7 The Strong Discontinuity Approach The strong discontinuity approach (SDA) treats displacement discontinuities within a continuum on the element level. The following examplarily describes SDA for 2D continuum quad elements (Section 2.3). Basic principles for kinematics, equilibrium and material behaviour are applied again. This is complemented with a coupling of system and element levels. Crack tractions are explicitly regarded along a fictitious crack width (Figure 7.2). Thus, the crack energy is taken into account in a natural way according to prescribed traction–separation relations (Eq. (7.4)). An additional regularisation is not required.

7.7.1 Kinematics Regarding discontinuity kinematics the total displacement is decomposed into u(x) = u(x) + H ‖u(x)‖

(7.117)

237

238

7 Crack Modelling and Regularisation

Figure 7.15 Kinematics of the discontinuous part Eq. (7.117).

with a continuous part u and a discontinuous part composed of a rigid body displacement term ⎡1 ‖u(x)‖ = ⎢ 0 ⎣

0 1

⎛𝑢0,𝑥 ⎞ −𝑦 + 𝑦0 ⎤ ⎜ ⎥ ⋅ 𝑢0,𝑦 ⎟ = N𝑤 (x) ⋅ w𝑒 ⎟ ⎜ 𝑥 − 𝑥0 ⎦ ⎝ 𝛽 ⎠

(7.118)

with a reference point 𝑥0 , 𝑦0 , translational degrees of freedom 𝑢0,𝑥 , 𝑢0,𝑦 , and a rotational degree of freedom 𝛽, see Figure 7.15, multiplied by a term to split a discontinuity H(x) = 𝐻(x) − 𝜑(x)

𝐻(x) =

⎧

1, x ∈ 𝑉 +

(7.119)

⎨0, 𝑒𝑙𝑠𝑒 ⎩ with the Heaviside function 𝐻 and with 𝜑(x) summing up local values N𝐼 (x) of those trial functions that belong to nodes in the positive domain 𝑉 + ∑ 𝜑(x) = 𝑁(x) (7.120) + 𝑉

Thus, H(x) = 0 at nodes and ΔH(x) = 1 along the discontinuity 𝐴𝑑 . Equation (7.117) leads to a wedged crack with centre x0 without contributions to nodal displacements. ◀

The crack orientation is ruled by the element splitting into domains V + , V − by a line Ad . This line has to be chosen according to a cracking criterion, e.g. the Rankine criterion (Section 7.4). The centre point x0 is an arbitrary point on Ad .

Strains are derived from Eq. (7.117) with 𝝐 = ∇sym u = ∇sym u + (∇sym 𝐻 − ∇sym 𝜑) ‖u‖ + H ∇sym ‖u‖

(7.121)

with the symmetric Nabla operator ∇sym and omitting written x-dependence of all quantities for brevity. Equation (7.118) yields ∇sym ‖u‖ = 0, and the total strain is subdivided into 𝝐 = 𝝐 +ˆ 𝝐

(7.122)

7.7 The Strong Discontinuity Approach

with a strain from continuous displacement field contributions 𝝐 = ∇sym u − ∇sym 𝜑 ‖u‖

(7.123)

and a strain from a discontinuous displacement field contributions sym

ˆ 𝝐 = ∇sym 𝐻 ‖u‖ = 𝛿 [∇n ⊗ ‖u‖]

(7.124)

with the Dirac delta function 𝛿(Eq. (3.17)) along the discontinuity 𝐴𝑑 , the normal n along the discontinuity 𝐴𝑑 , and the dyadic product ⊗ (Section 6.5.1). The form 1 sym = (∙ + ∙T ) transforms an unsymmetric second-order tensor ∙ into a symmet[∙] 2 ric one. ◀

The quantity ˆ 𝝐 marks a strain singularity. For the following, it is substituted by a fictitious crack width (Figure 7.2) related to H ‖u(x)‖ (Eq. (7.117)) along Ad .

Displacements and strain 𝝐 remain to be interpolated. As u is assumed as a continuous field, its interpolation on the element level is adopted from the quad element (Eqs. (2.18) and (2.36)) u = N ⋅ 𝝊𝑒

(7.125)

Strain interpolation is derived from Eq. (7.123), also including the strain field interpolation for quad elements (Eqs. (2.21) and (2.40)), which yields 𝝐 = B ⋅ 𝝊𝑒 − B𝑤 ⋅ w𝑒

(7.126)

with ∇sym 𝜑 =

∇sym 𝜑

∑ 𝑉+

⎛𝐵𝑥 ⎞ ⎜ ⎟ 𝐵 ⎝ 𝑦⎠

sym

⎡⎛𝐵𝑥 ⎞ ⎤ ‖u‖ = ⋅ w𝑒 ⊗ N𝑤 ⎥ ⎢ 𝑉 + ⎜𝐵 ⎟ ⎣⎝ 𝑦 ⎠ ⎦ ⎤ ⎡𝐵 0 𝐵𝑥 (−𝑦 + 𝑦0 ) ∑ ⎢ 𝑥 ⎥ = 𝐵𝑦 (𝑥 − 𝑥0 ) ⎥ ⋅ w𝑒 ⎢ 0 𝐵𝑦 𝑉+ ⎥ ⎢ 𝐵𝑥 (𝐵𝑦 (−𝑦 + 𝑦0 ) + 𝐵𝑥 (𝑥 − 𝑥0 )) 𝐵 ⎦ ⎣ 𝑦 = B𝑤 ⋅ w𝑒 ∑

(7.127)

with 𝐵𝑥 , 𝐵𝑦 from the first two rows of Eq. (2.41) and N𝑤 from Eq. (7.118). The term B𝑤 ⋅ w𝑒 in Eq. (7.126) relieves stresses due a fictitious crack width.

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7 Crack Modelling and Regularisation

The approach may be extended to multiple discontinuities. A secondary discontinuity is introduced with the extension of Eq. (7.117) u(x) = u + H1 ‖u‖1 + H2 ‖u‖2 = u + H1 N𝑤,1 ⋅ w𝑒1 + H2 N𝑤,2 ⋅ w𝑒2 [ = u + H1 N𝑤1

] ⎛w𝑒1 ⎞ ⎟ w ⎝ 𝑒2 ⎠

H2 N𝑤,2 ⋅ ⎜

𝐼𝐼 = u + N𝐼𝐼 𝑤 ⋅ w𝑒

(7.128)

𝐼𝐼 The extensions to N𝐼𝐼 𝑤 (Eq. (7.118)), (Eq. (7.119)), and B𝑤 (Eq. (7.126)) are straightforward, with the number of discontinuity degrees of freedom doubled on the element level, and each discontinuity with its particular 𝑉 + (Figure 7.15). The application of SDA to other than quad elements, in particular triangular elements and 3D elements, is also straightforward. The latter introduces six discontinuity degrees of freedom for each discontinuity according to spatial rigid body displacements and has a plane convex polygon separating an element. A constant positive domain 𝑉 + defines a constant discontinuity contour, i.e. a fixed crack (Section 7.7). A rotating crack with changing a discontinuity contour is described with modifications of 𝑉 + during a loading history.

7.7.2 Equilibrium and Material Behaviour Discontinuity equilibrium has to be formulated next. A weak form (Alfaiate et al. 2003) T

∫ H 𝛿 ‖∇sym u‖ ⋅ 𝝈 d𝑉 = 𝑉𝑒 ∖𝐴𝑑

T

∫ H 𝛿 ‖u‖ ⋅ b d𝑉 𝑉𝑒 ∖𝐴𝑑 T

T

+ ∫ H 𝛿 ‖u‖ ⋅ t d𝐴 + ∫ 𝛿 ‖u‖ ⋅ t𝑐 d𝐴 𝐴𝑒,𝑡

𝐴𝑑

(7.129) applied on the element level is used in analogy to Eq. (2.52) – inertia terms disregarded – with 𝛿 indicating test functions, the symmetric Nabla operator ∇sym , the path 𝐴𝑑 , and the traction t𝑐 along the discontinuity (Figure 7.15), whereby t𝑐 is related to the global system as is w𝑒 . The splitting term H is omitted in the 𝐴𝑑 -integral, as t𝑐 is a counteracting pair on 𝐴𝑑 . The discontinuity interpolation Eq. (7.118) yields ‖∇sym u‖ = 0 ,

𝛿 ‖u‖ = N𝑤 ⋅ 𝛿w𝑒

(7.130)

and leads to a discretised discontinuity equilibrium 𝛿w𝑒T ⋅ (p𝑤 + f𝑤 ) = 0

(7.131)

7.7 The Strong Discontinuity Approach

fct

t

Gf

loading

w un/re-loading

wcr

crack closure Figure 7.16 Normal components of traction–separation relations with unloading and re-loading.

with f𝑤 = ∫ NT𝑤 ⋅ t𝑐 d𝐴 𝐴𝑑

(7.132)

p𝑤 = ∫ H NT𝑤 ⋅ b d𝑉 + ∫ H NT𝑤 ⋅ t d𝐴 𝑉

𝐴𝑡

In contrast to Eqs. (2.58), f𝑤 , p𝑤 are not nodal forces assigned to nodal degrees of freedom but generalised discontinuity forces assigned to the discontinuity degrees of freedom. Furthermore, a generalised material description has to be taken into account. A generalised incremental form is introduced within the framework of the cohesive crack model (Section 7.1, Eq. (7.9)) aligned to the crack orientation. Specifications for normal and shear components are given with Eqs. (7.5)–(7.8). The state of loading has been treated up to now. Unloading is possible from all points of the loading path and follows a branch indicated in Figure 7.16 with the point of initial unloading (𝑤𝑛,𝑢 , 𝑡𝑛,𝑢 ) connected to the origin. The pair (𝑤1,𝑢 , 𝑡1,𝑢 ) is an internal state variable comprising the preceding load history. The ratio (𝑡1,𝑢 ∕𝑤1,𝑢 ) indicates the unloading stiffness and (𝑡1,𝑢 𝑤1 )∕𝑤1,𝑢 the crack traction with the current 𝑤1 on the unloading path. Crack closure is modelled with a penalty approach with a high constant stiffness in the range 𝑤1 ≤ 0. Re-loading retraces crack closure elastically and unloading up to the current (𝑤1,𝑢 , 𝑡1,𝑢 ) and continues along the loading path. The local approach has to be transformed into the global system and vice versa, according to transformation rules (Eq. (D.8)). This corresponds to global values ṫ 𝑐 = C𝑐𝑇 ⋅ ẇ 𝑐 ,

w𝑐 = ‖u‖

(7.133)

with a global tangential crack stiffness C𝑐𝑇 = QT ⋅ C𝑐𝐿𝑇 ⋅ Q (Eq. (7.9)). With respect to non-linear equations solving the internal discontinuity forces, Eq. (7.132) is lin-

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7 Crack Modelling and Regularisation

earised with the time derivative ḟ 𝑤 = ∫ NT𝑤 ⋅ C𝑐𝑇 ⋅ N𝑤 d𝐴 ⋅ ẇ 𝐴𝑑

= K𝑇,𝑛𝑛 ⋅ ẇ 𝑒

(7.134)

using Eq. (7.133). This results in a tangential discontinuity stiffness matrix K𝑇,𝑛𝑛 to be used for the iterative solution of the coupling of weak equilibrium conditions (Eq. (7.145)).

7.7.3 Coupling Finally, a coupling of weak equilibrium conditions for continuous and discontinuous approaches must be established. An equilibrium condition Eq. (2.52), as is applied for a whole domain, may also be applied to subdomains or single elements T

∫ 𝛿𝝐 ⋅ 𝝈 d𝑉 = 𝑉𝑒 ∖𝐴𝑑

∫ 𝛿uT ⋅ b d𝑉 + ∫ 𝛿uT ⋅ t d𝐴

(7.135)

𝐴𝑡,𝑒

𝑉𝑒 ∖𝐴𝑑

again disregarding inertial parts. The corresponding test functions are given according to Eqs. (7.117), (7.125), (7.118), and (7.126) 𝛿u = N ⋅ 𝛿𝝊𝑒 + H N𝑤 ⋅ 𝛿w𝑒 ,

𝛿𝝐 = B ⋅ 𝛿𝝊𝑒 − B𝑤 ⋅ 𝛿w𝑒

(7.136)

Thus, Eq. (7.135) is rewritten as 𝛿𝝊T𝑒 ⋅ ∫ BT ⋅ 𝝈 d𝑉 − 𝛿w𝑒T ⋅ ∫ BT𝑤 ⋅ 𝝈 d𝑉 𝑉𝑒 ∖𝐴𝑑

𝑉𝑒 ∖𝐴𝑑

= 𝛿𝝊T𝑒 ⋅ ∫ NT ⋅ b d𝑉 + 𝛿w𝑒T ⋅ ∫ HNT𝑤 ⋅ b d𝑉 𝑉𝑒 ∖𝐴𝑑

(7.137)

𝑉𝑒 ∖𝐴𝑑

+ 𝛿𝝊T𝑒 ⋅ ∫ NT ⋅ t d𝐴 + 𝛿w𝑒T ⋅ ∫ HNT𝑤 ⋅ t d𝐴 𝐴𝑡

𝐴𝑡

or 𝛿𝝊T𝑒 ⋅ (p𝑒 + f𝑒 ) + 𝛿w𝑒T ⋅ (p𝑤 + f𝑐 ) = 0

(7.138)

with p𝑒 , f𝑒 from Eqs. (2.59) and (2.58), p𝑤 from Eq. (7.132) and coupling discontinuity forces f𝑐 =

∫ BT𝑤 ⋅ 𝝈 d𝑉

(7.139)

𝑉𝑒 ∖𝐴𝑑

Subtracting Eq. (7.131) from Eq. (7.138) yields two sets of combined discretised equilibrium conditions for a particular element p𝑒 − f𝑒 = 0 ,

f𝑐 − f𝑤 = 0

(7.140)

7.7 The Strong Discontinuity Approach

as 𝛿𝝊𝑒 , 𝛿w𝑒 are independent and arbitrary. The terms of Eq. (7.140) have to be linearised for nonlinear equation solving (Section 2.8.1). With a general incremental stress–strain relation 𝝈̇ = C𝑇 ⋅ 𝝐̇

(7.141)

a linearisation of internal nodal forces with respect to time is obtained with ḟ 𝑒 =

𝑉𝑒 ∖𝐴𝑑

=

∫ BT ⋅ C𝑇 ⋅ 𝝐̇ d𝑉

∫ BT ⋅ 𝝈̇ d𝑉 =

𝑉𝑒 ∖𝐴𝑑

∫ BT ⋅ C𝑇 ⋅ B d𝑉 ⋅ 𝝊̇ 𝑒 − ∫ BT ⋅ C𝑇 ⋅ B𝑤 d𝑉 ⋅ ẇ 𝑒 𝑉𝑒 ∖𝐴𝑑

𝑉𝑒 ∖𝐴𝑑

= K𝑇,𝑢𝑢 ⋅ 𝝊̇ 𝑒 − K𝑇,𝑢𝑤 ⋅ ẇ 𝑒

(7.142)

with the tangential stiffness matrices K𝑇,𝑢𝑢 , K𝑇,𝑢𝑤 . Coupling discontinuity forces (Eq. (7.139)) are linearised with ḟ 𝑐 =

∫ BT𝑤 ⋅ C𝑇 ⋅ B d𝑉 ⋅ 𝝊̇ 𝑒 − ∫ BT𝑤 ⋅ C𝑇 ⋅ B𝑤 d𝑉 ⋅ ẇ 𝑒 𝑉𝑒 ∖𝐴𝑑

(7.143)

𝑉𝑒 ∖𝐴𝑑

= K𝑇,𝑤𝑢 ⋅ 𝝊̇ 𝑒 − K𝑇,𝑤𝑤 ⋅ ẇ 𝑒 with further tangential stiffness matrices K𝑇,𝑤𝑢 , K𝑇,𝑤𝑤 . The linearisation of the internal discontinuity forces ḟ 𝑤 is given with Eq. (7.134). The time derivative of p𝑒 is ̇ t.̇ Thus, incremental equilibrium ruled by prescribed body forces and tractions b, conditions ṗ 𝑒 − ḟ 𝑒 = 0 ,

ḟ 𝑐 − ḟ 𝑤 = 0

(7.144)

corresponding to Eq. (7.140) are written as ⎡ K𝑇,𝑢𝑢 ⎢ −K𝑇,𝑤𝑢 ⎣

⎤ ⎛ 𝝊̇ 𝑒 ⎞ ⎛ṗ 𝑒 ⎞ ⎥⋅⎜ ⎟= ⎜ ⎟ K𝑇,𝑤𝑤 + K𝑇,𝑛𝑛 ẇ 0 ⎦ ⎝ 𝑒⎠ ⎝ ⎠ −K𝑇,𝑢𝑤

(7.145)

This is decoupled with the second row to be solved first on the element level [ ] K𝑇,𝑤𝑤 + K𝑇,𝑛𝑛 ⋅ ẇ 𝑒 = K𝑇,𝑤𝑢 ⋅ 𝝊̇ 𝑒 (7.146) and the solution ẇ 𝑒 to be used for the first row K𝑇,𝑢 ⋅ 𝝊̇ 𝑒 = ṗ 𝑒

(7.147)

with [

K𝑇,𝑢 = K𝑇,𝑢𝑢 − K𝑇,𝑢𝑤 ⋅ K𝑇,𝑤𝑤 + K𝑇,𝑛𝑛

]−1

⋅ K𝑇,𝑤𝑢

(7.148)

where the latter is assembled on the system level and solved for the nodal displacement increments of the whole system.

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7 Crack Modelling and Regularisation

In the case of small displacements, a nonlinearity is introduced with nonlinear material behaviour by variable tangential material stiffness matrices C𝑇 and C𝑐𝑇 . A discretisation leads to systems of nonlinear algebraic equations on the element level according to Eq. (7.146) and on the system level – after assembling – according to Eq. (7.147). They have to be resolved in a nested approach within an incrementally iterative solution scheme (Section 2.8.2). For details regarding crack initiation, adjusting of element crack geometries, and integration schemes with a similar approach, see Häussler-Combe et al. (2020). Example 7.6: Plain Concrete Plate with Notch and SDA Crack Modelling

We refer to Example 7.2 with the same geometry and boundary conditions. A different material model is assumed with isotropic linear elasticity with limited tensile strength. The Rankine criterion is used for crack initiation occurring when a reference value exceeds the uniaxial tensile strength. This reference value is determined as the mean of the largest principal stresses of the four integration points of the quad element. The loading is applied incrementally. The Rankine criterion is checked after each load step and, if applicable, applied to the following load step. Crack modelling is performed with SDA as described above. A reduced one-point Gauss integration (Table 2.1, 𝑛𝑖 = 0) is applied together with SDA. This avoids a moderate locking, which would occur with full four-point integration (Table 2.1, 𝑛𝑖 = 1). The elements’ mean point (𝑟, 𝑠 = 0, Eq. (2.35)) is chosen as the crack reference point 𝑥0 , 𝑦0 (Eq. (7.118)). The same cases of discretisation refinement are examined as with Example 7.1, Table 7.1. Explicit crack modelling with SDA yields a 2D crack contour with crack width values assigned. Thus, traction–separation relations are directly applied using Eqs. (7.5) and (7.7), and a regularisation like the crack band approach for Example 7.2 is not necessary. Crack energy (Eq. (3.8)) and the uniaxial tensile strength are required as material parameters for this approach. The critical crack width 𝑤𝑐𝑟 in Eq. (7.5) is obtained by the combination of Eqs. (3.8) and (7.5) 𝐺𝑓 =

1 𝑓 𝑤 3 𝑐𝑡 𝑐𝑟

→

𝑤𝑐𝑟 = 3

𝐺𝑓 𝑓𝑐𝑡

(7.149)

The values of all material parameters are listed in Table 7.2 and correspond to the material properties of Example 7.2 or – neglecting compressive strength – Table 6.1, respectively. Table 7.2 Material parameters of Example 7.6. Young’s modulus Poisson’s ratio Tensile strength Crack energy

𝐸 𝜈 𝑓𝑐𝑡 𝐺𝑓

Critical crack width 𝑤𝑐𝑟 Shear retention factor 𝛼𝑠

[MN∕m2 ] — [MN∕m2 ] [MNm∕m2 ]

30 000 0.2 3.0 50 ⋅ 10−6

[mm] —

0.0167 0.01

7.7 The Strong Discontinuity Approach

Figure 7.17 Example 7.6. Load displacement curves.

Figure 7.18 Example 7.6. Principal stresses on deformed structure (scaling factor 500).

The loading is applied in small steps as the prescribed displacement of the upper edge of the upper centred steel plate. Figure 7.17 shows the computed load– displacement curve with the total reaction force along the upper edge of the upper steel plate versus its prescribed vertical displacement for all three discretisations. We see the following: • The initial stiffness and peak loads agree well. • The formation of cracks is not a continuous process but occurs as discrete events of element cracks. This leads to spikes in the load–displacement curve, which is pronounced for the rough discretisation and reduces with the finer discretisations. • The load–displacement behaviour becomes unstable for all three discretisations shortly after reaching the peak loads. Loadings fall down without an increase of displacements. Figure 7.18 shows the computed principal stresses on the deformed structure for the coarse discretisation near peak loading. Cracked elements transformed into SDA elements are indicated with a bullet, crack orientation with a centred line.

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Figure 7.19 Example 7.6. Principal stresses on deformed structure (scaling factor 500).

Figure 7.20 Example 7.6. Load displacement curves.

• Stresses of SDA elements are only evaluated in the centre due to one-point integration. Their horizontal stresses correspond to crack tractions with vertical crack orientations. • SDA crack tractions and stresses decrease with increasing deformations from the upper SDA element down to the lower one. • Principal tensile stresses do not exceed the tensile strength throughout the plate. The last statement is illustrated with a further representation. Figure 7.19 shows principal stresses in a cutout around the notch for the finest discretisation. It has to be observed that the element length is only one-quarter compared to Figure 7.17 and a finer resolution of cracking is given with smaller elements. The scaling of stresses is changed accordingly by a factor of 0.25. Finally, the load–displacement behaviour is compared to that of Example 7.2 in Figure 7.20. Both examples differ as the latter assumes isotropic damage with crack regularisation and the current linear elasticity with limited tensile strength. Regardless of this, an improved convergence or mesh independence is obtained for the current example with SDA. However, this should not simply be generalised.

7.7 The Strong Discontinuity Approach

SDA basically allows us to include arbitrary material laws for the continuum stress– strain and the traction–separation relations, as no specific assumption was made regarding the material tangential stiffness Eqs. (7.133) and (7.141). For multi-axial stress conditions, this basically allows for a strain softening without reaching the uniaxial tensile strength. Thus, the Rankine stress criterion would fail. A Rankine strain criterion might be an alternative. For alternative approaches, see, e.g. Huespe et al. (2006). A comprehensive description of the SDA with its variety of developments is given with the references in Section 7.6.

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8 Plates With multi-axial stress–strain relations at hand the view may be extended beyond trusses and beams. In the following, this is first of all described with biaxial material models for plates and, in particular, for deep beams. A span is treated ranging from reinforcement design based on linear elastic plate solutions to advanced models including cracked concrete, reinforcement, and their interaction through bond.

8.1 Lower Bound Limit State Analysis 8.1.1 General Approach Plates are plane surface structures in contrast to bars and beams, which are line structures. Plates are characterised by in-plane loading, see Figure 9.1. Length and height are significant as geometrical properties. The state of stress and strain is assumed as constant throughout the thickness. A corresponding kinematic assumption is given in Section 2.3. Plates were already treated in Section 5.1 with strut-and-tie models, which regard uniaxial stress–strain relations. This is a simple but crude model to describe the material behaviour of concrete. Actually, biaxial states are characteristic for plates (Figure 5.1). Thus, with respect to biaxial plates, design and simulation methods for reinforced concrete plates are described in the following. The limited tensile strength of concrete still plays a predominant role. We start with a design method considering local equilibrium on the one hand and the limited strength of concrete and reinforcement on the other to perform a limit state analysis based on the first limit theorem of plasticity (Section 5.4). This theorem is basically valid for all types of structures. Local equilibrium, which is the first condition of a limit state analysis, can be determined with a linear elastic analysis with a unit loading applied, i.e. the load or the largest load in a combination has a value of 1. Stresses resulting from a linear analysis due to a unit loading can be linearly scaled with a loading factor and maintaining equilibrium. As a further condition, the loading factor is scaled such that the stresses utilise the strength of the materials in at least one point of the plate and do not exceed strength anywhere. This includes the strength of concrete as well as the strength of reinforcement. The two condiComputational Structural Concrete – Theory and Applications, Second Edition. Ulrich Häussler-Combe. © 2023 Ernst & Sohn GmbH. Published 2023 by Ernst & Sohn GmbH.

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8 Plates

tions – equilibrium and strength – of a limit state analysis fulfil the assumptions of the first limit theorem. ◀

Thus, the corresponding loading factor determines a lower bound for the admissible loading with respect to the unit loading.

The analysis conforms to a safe proof of the admissible load for a given system. The key point is to combine the strength of concrete and the reinforcement in an appropriate way. This is described in Section 8.1.2. A variant is given by the design procedure for a given load, i.e. the compressive strength of concrete or the amount of reinforcement are adjusted such that the strength conditions are not violated. The shape and the dimensions of the concrete body are assumed as prescribed in this context. The procedure is based upon a linear elastic analysis. Analytical solutions for plates are available for simple cases (Girkmann 1974). Finite element solutions are appropriate for more complex situations; an example is demonstrated in Section 5.1. A suitable element type is given by the quad element (Section 2.3). Plane stress conditions (Eq. (2.45)) are assumed, but in the same way, the approach may be applied to plane strain conditions (Eq. (2.44)). For now, it is sufficient to describe the material properties by the concrete initial Young’s modulus 𝐸𝑐 , its Poisson’s ratio 𝜈, and, most importantly, by the uniaxial strength of concrete and reinforcement.

8.1.2 Reinforced Concrete Resistance The key item of the limit state set-up outlined in the previous section is to find an appropriate combination of concrete and reinforcement strength as reference values for the calculated stress state. This is based on the modified compression field theory (Vecchio and Collins 1986). The determination of principal stresses (Section 6.2.3) is needed as a prerequisite. A given plane stress state 𝜎𝑥 , 𝜎𝑦 , 𝜎𝑥𝑦 has principal stresses 𝜎1 , 𝜎2 𝜎1 =

𝜎𝑥 + 𝜎𝑦 2

√ +

𝜎𝑥 + 𝜎𝑦 − 𝜎2 = 2

( √

𝜎𝑥 − 𝜎𝑦 2

2 2 , ) + 𝜎𝑥𝑦

𝜎𝑥 − 𝜎𝑦 2 2 ) + 𝜎𝑥𝑦 ( 2

(8.1)

with an angle of orientation 𝜑 𝜎𝑥 − 𝜎𝑦 2

cos 2𝜑 = √ (

𝜎𝑥 − 𝜎𝑦 2

(8.2) 2 2 ) + 𝜎𝑥𝑦

which is positive from the 𝑥-axis in the counterclockwise direction. Equation (8.2) has one solution for 𝜑 in the range 0 … π∕2. The value of 𝜑 multiplied by the sign

8.1 Lower Bound Limit State Analysis

Figure 8.1 Mohr circle.

of 𝜎𝑥𝑦 , indicates the direction of 𝜎1 . The directions of 𝜎1 and 𝜎2 are perpendicular. The relations have a well-known representation with the Mohr circle; see Figure 8.1. An alternative representation of stresses derived from the rules for plane coordinate transformations (Eq. (D.11)) is given by 2

𝜎1 = 𝜎𝑥 cos2 𝜑 + 𝜎𝑦 sin 𝜑 + 𝜎𝑥𝑦 2 cos 𝜑 sin 𝜑 2

𝜎2 = 𝜎𝑥 sin 𝜑 + 𝜎𝑦 cos2 𝜑 − 𝜎𝑥𝑦 2 cos 𝜑 sin 𝜑

(8.3)

2

0 = (𝜎𝑦 − 𝜎𝑥 ) cos 𝜑 sin 𝜑 + 𝜎𝑥𝑦 (cos2 𝜑 − sin 𝜑) under the condition of vanishing shear stresses, where 𝜑 indicates the direction of 𝜎1 and 𝜑 + π∕2 the direction of 𝜎2 . This form allows us to control the principal stress values depending on the orientation angle 𝜑. Equations (8.3) may be solved for 𝜎𝑥 , 𝜎𝑦 , 𝜎𝑥𝑦 depending on 𝜎1 , 𝜎2 , 𝜑, which yields 1) 2

𝜎𝑥 = 𝜎1 cos2 𝜑 + 𝜎2 sin 𝜑 2

𝜎𝑦 = 𝜎1 sin 𝜑 + 𝜎2 cos2 𝜑

(8.4)

𝜎𝑥𝑦 = sin 𝜑 cos 𝜑 (𝜎1 − 𝜎2 ) Furthermore, the relation 𝜎1 ≥ 𝜎2 resulting from Eqs. (8.1) is considered for the following cases: • Pure compression with the principal stresses 𝜎1 ≤ 0, 𝜎2 ≤ 0. • Mixed tension–compression with the principal stresses 𝜎1 > 0, 𝜎2 ≤ 0. • Pure tension with the principal stresses 𝜎1 > 0, 𝜎2 > 0. We assume that pure compression does not require resistance from a reinforcement. A reinforcement is required for mixed tension–compression and pure tension. The contributions of reinforcement and concrete are treated separately in the following. Concrete has its own principal stress state with values 𝜎𝑐1 , 𝜎𝑐2 , with an orientation 𝜑 of 𝜎𝑐1 , and 𝜎𝑐1 > 𝜎𝑐2 (signed!). It is assumed that it can sustain compression in an orientation 𝜑 + π∕2, but no stresses are transferred in the orientation 𝜑. This implies 1) Equations in the set Eqs. (8.1) and (8.2) should not be mixed with or complemented to equations in the set Eqs. (8.4).

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cracking in the orientation 𝜑 + π∕2 and principal stresses 𝜎𝑐1 = 0, 𝜎𝑐2 < 0. A notation 𝜎𝑐2 = 𝜎𝑐 and 𝜑𝑐 = 𝜑 + π∕2 is used from now on. ◀

The orientation 𝜑c of concrete principal compression and the corresponding compressive concrete stress, 𝜎c serve as first variables for the limit state analysis.

Using Eqs. (8.4) and regarding sin 𝜑 = − cos 𝜑𝑐 , cos 𝜑 = sin 𝜑𝑐 the concrete resistance contribution in the global directions is given by 𝜎𝑐,𝑥 = 𝜎𝑐 cos2 𝜑𝑐 2

𝜎𝑐,𝑦 = 𝜎𝑐 sin 𝜑𝑐

(8.5)

𝜎𝑐,𝑥𝑦 = 𝜎𝑐 sin 𝜑𝑐 cos 𝜑𝑐 Further contributions come from the reinforcement. The content of reinforcement is measured by the ratio 𝜌𝑠 of the reinforcement cross-sectional area related to the total cross-sectional area, whereby a reinforcement mesh is regarded as ‘smeared’ or as a sheet, respectively. In general, more than one reinforcement orientation is used. Every orientation 𝑖 is directed with an angle 𝜑𝑠𝑖 . Corresponding rebars form a reinforcement group with a reinforcement ratio 𝜌𝑠𝑖 . The reinforcement stress is uniaxial by definition. The principal stress state of a reinforcement group 𝑖 is determined through a single non-zero value, that is 𝜎𝑠𝑖,1 ≠ 0, while the other principal stress component vanishes, i.e. 𝜎𝑠𝑖,2 = 0. A notation 𝜎𝑠𝑖 = 𝜎𝑠𝑖,1 is used in the following. ◀

Reinforcement group orientation 𝜑si and group stress 𝜎si serve as further variables for the limit state analysis.

Although an arbitrary number of reinforcement orientations may be allowed, their number is restricted to two, 𝑖 = 1, 2 to simplify the notation. Reinforcement resistance contributions in the global directions are given by 𝜎𝑠1,𝑥 = 𝜎𝑠1 cos2 𝜑𝑠1 2

𝜎𝑠1,𝑦 = 𝜎𝑠1 sin 𝜑𝑠1

(8.6)

𝜎𝑠1,𝑥𝑦 = 𝜎𝑠1 sin 𝜑𝑠1 cos 𝜑𝑠1 and 𝜎𝑠2,𝑥 = 𝜎𝑠2 cos2 𝜑𝑠2 2

𝜎𝑠2,𝑦 = 𝜎𝑠2 sin 𝜑𝑠2

(8.7)

𝜎𝑠2,𝑥𝑦 = 𝜎𝑠2 sin 𝜑𝑠2 cos 𝜑𝑠2 in analogy to the concrete contribution Eq. (8.5). Summed contributions from concrete and reinforcement groups are assumed to be in equilibrium with the given stress state 𝜎𝑥 , 𝜎𝑦 , 𝜎𝑥𝑦 determined from a superordinated analysis. Thus, with re-

8.1 Lower Bound Limit State Analysis

(a)

(b)

Figure 8.2 (a) sin 𝜑c cos 𝜑c and solution range of Eq. (8.10). (b) Mohr circles collecting contributions.

spect to also the reinforcement ratios 𝜌𝑠1 , 𝜌𝑠2 , the equilibrium conditions 𝜌𝑠1 𝜎𝑠1,𝑥 + 𝜌𝑠2 𝜎𝑠2,𝑥 + 𝜎𝑐,𝑥 = 𝜎𝑥 𝜌𝑠1 𝜎𝑠1,𝑦 + 𝜌𝑠2 𝜎𝑠2,𝑦 + 𝜎𝑐,𝑦 = 𝜎𝑦

(8.8)

𝜌𝑠1 𝜎𝑠1,𝑥𝑦 + 𝜌𝑠2 𝜎𝑠2,𝑥𝑦 + 𝜎𝑐,𝑥𝑦 = 𝜎𝑥𝑦 must hold in every position of the plate. ◀

The reinforcement ratios 𝜌si of every reinforcement group complete the set of variables for the limit state analysis.

Inserting Eqs. (8.5)–(8.7) into Eq. (8.8) leads to three equations for eight variables 𝜑𝑐 , 𝜎𝑐 , 𝜑𝑠1 , 𝜎𝑠1 , 𝜑𝑠2 , 𝜎𝑠2 , and 𝜌𝑠1 , 𝜌𝑠2 . The values of the reinforcement orientations 𝜑𝑠1 , 𝜑𝑠2 are prescribed without loss of generality, as they are ruled by practical considerations or construction site constraints. As a consequence, six variables finally remain with three equilibrium equations with a prescribed local ‘loading’ 𝜎𝑥 , 𝜎𝑦 , 𝜎𝑥𝑦 . Thus, solutions cannot yet be derived with the current formulation. The conditions of kinematic compatibility with measures of strain and material laws relating stresses with strains are still missing. Approaches to consider these are discussed in Hsu (1993, Chapters 6,7), Vecchio and Collins (1986), introducing strains 𝜖𝑥 , 𝜖𝑦 , 𝛾𝑥𝑦 and principal strains 𝜖1 , 𝜖2 as further variables together with uniaxial material laws for reinforcement and concrete. This yields an extended, complete set of equations. But the corresponding values 𝜖𝑥 , 𝜖𝑦 , 𝛾𝑥𝑦 will more or less deviate from those directly determined in the superordinated finite element computation. Thus, kinematic compatibility is limited with this approach, and it will not be pursued in the following. A usual simplification is utilised instead. ◀

A limit state analysis – supported by the first limit theorem of plasticity – does not regard kinematic compatibility and replaces stress–strain relations with material strength conditions leading to a lower bound for an admissible loading.

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Furthermore, a special but very common case is considered. It is assumed that reinforcement directions are aligned to global coordinate axes, leading to 𝜑𝑠1 = 0, 𝜑𝑠2 = π∕2. Equations (8.6) and (8.7) yield 𝜎𝑠1,𝑥 = 𝜎𝑠1 𝜎𝑠2,𝑦 = 𝜎𝑠2

(8.9)

𝜎𝑠1,𝑦 = 𝜎𝑠1,𝑥𝑦 = 𝜎𝑠2,𝑥 = 𝜎𝑠2,𝑥𝑦 = 0 Combining Eqs. (8.53 ), (8.83 ), and (8.9) leads to sin 𝜑𝑐 cos 𝜑𝑐 =

𝜎𝑥𝑦 𝜎𝑐

(8.10)

with 𝜎𝑐 < 0 by definition, and 𝜎𝑥𝑦 has to be signed correctly. With a prescribed value of 𝜎𝑥𝑦 and an assumed value of 𝜎𝑐 , Eq. (8.10) basically has two solutions for 𝜑𝑐 in the range −π∕2 ≤ 𝜑𝑐 ≤ π∕2 under the condition |𝜎𝑥𝑦 ∕𝜎𝑐 | ≤ 0.5; see Figure 8.2a. Using a notation 𝜎𝑠1,𝑥 = 𝜎𝑠𝑥 , 𝜎𝑠2,𝑦 = 𝜎𝑠𝑦 and 𝜌𝑠1 = 𝜌𝑥 , 𝜌𝑠2 = 𝜌𝑦 , Eqs. (8.81,2 ) and (8.51,2 ) finally result in 𝜌𝑥 𝜎𝑠𝑥 = 𝜎𝑥 − 𝜎𝑐 cos2 𝜑𝑐

(8.11)

2

𝜌𝑦 𝜎𝑠𝑦 = 𝜎𝑦 − 𝜎𝑐 sin 𝜑𝑐

with concrete stress 𝜎𝑐 , concrete stress orientation 𝜑𝑐 , reinforcement stresses 𝜎𝑠𝑥 , 𝜎𝑠𝑦 , and reinforcement ratios 𝜌𝑥 , 𝜌𝑦 in the global 𝑥 and 𝑦-orientation. Their relations can again be illustrated with Mohr circles; see Figure 8.1b. Equations (8.10) and (8.11) involving prescribed values 𝜎𝑥 , 𝜎𝑦 , 𝜎𝑥𝑦 and variables 𝜌𝑥 , 𝜌𝑦 , 𝜎𝑠𝑥 , 𝜎𝑠𝑦 , and 𝜎𝑐 , 𝜑𝑐 are used for design considerations with further appropriate assumptions. Similar approaches are described in CEB-FIP (2008, Section 2.2), Hsu (1993, Chapter 4). A further special case is added for completion. We consider again the general formulation Eqs. (8.5)–(8.8) applied to the cross-section of a web of a beam with a height 𝑧 between chords as a limiting case of a plate. The conditions 𝜎𝑦 = 0 (no vertical ‘loading’), 𝜌𝑠2 = 0 and variable parameters 𝜌𝑠1 , 𝜎𝑠1 , 𝜎𝑐 , 𝜑𝑐 and 𝜑𝑠1 are assumed. Equations (8.8), (8.5), and (8.6) result in 𝜌𝑠 cos2 𝜑𝑠 𝜎𝑠 + 𝜎𝑐 cos2 𝜑𝑐 = 𝜎𝑥 2

2

𝜌𝑠 sin 𝜑𝑠 𝜎𝑠 + 𝜎𝑐 sin 𝜑𝑐 = 0

(8.12)

𝜌𝑠 sin 𝜑𝑠 cos 𝜑𝑠 𝜎𝑠 + 𝜎𝑐 sin 𝜑𝑐 cos 𝜑𝑐 = 𝜎𝑥𝑦 with the index omitted for the reinforcement group. Furthermore, we use a shear force 𝑉 = −𝑧 𝑏 𝜎𝑥𝑦 with a cross-section width 𝑏, a difference of normal force Δ𝑁 = 𝑧 𝑏 𝜎𝑥 along the web height, and an admissible reinforcement stress 𝜎𝑠 = 𝑓𝑦𝑘 with a yield stress 𝑓𝑦𝑘 . Equations (8.12) lead to 𝜎𝑐 = − 𝜌𝑠 =

𝑉 2

𝑧 𝑏 (sin 𝜑𝑐 cos 𝜑𝑐 − cot 𝜑𝑠 sin 𝜑𝑐 ) 𝑉 2

𝑓𝑦𝑘 𝑧 𝑏 (sin 𝜑𝑠 cot 𝜑𝑐 − sin 𝜑𝑠 cos 𝜑𝑠 )

Δ𝑁 = −𝑉 (cot 𝜑𝑐 + cot 𝜑𝑠 )

(8.13)

8.1 Lower Bound Limit State Analysis

where the concrete stress orientation is generally chosen with a range 0 < 𝜑𝑐 < π∕2 and the reinforcement orientation with a range π∕2 ≤ 𝜑𝑠 ≤ π for 𝑉 > 0. In the case of stirrups with 𝜑𝑠 = π∕2, this simplifies to 𝜌𝑠 =

𝑉 , 𝑓𝑦𝑘 𝑧𝑏 cot 𝜑𝑐

𝜎𝑐 = −

𝑉 (cot 𝜑𝑐 + tan 𝜑𝑐 ) , 𝑧𝑏

Δ𝑁 = −𝑉 cot 𝜑𝑐 (8.14)

and in the optimal case of diagonal concrete compression 𝜑𝑐 = π∕4 and shear reinforcement 𝜑𝑠 = 3π∕4 to 𝜎𝑐 = −

𝑉 , 𝑧𝑏

𝜌𝑠 =

𝑉 , 𝑓𝑦𝑘 𝑧𝑏

Δ𝑁 = 0

(8.15)

These relations are identical to the well-known design rules for the shear reinforcement of beams (EN 1992-1-1 2004, 6.2.3). A shifting distance 𝑎 for the bending reinforcement in the upper and lower chords is defined with 𝑎 𝑉 = 𝑧 Δ𝑁∕2. This yields another well-known equation 𝑎=

𝑧 𝑧 (cot 𝜑𝑐 + cot 𝜑𝑠 ) = (cot 𝜑𝑐 − cot 𝛼) 2 2

(8.16)

with 𝛼 = 2π − 𝜑𝑠 (EN 1992-1-1 2004, 9.2.1.3). The previous case assumes a constant stress state along the cross-section of a web and concentrated forces in the upper and lower chords. Let us consider a set-up with four of such webs in a spatial configuration forming a quadrangular cross-section connected with four chords, leading to a hollow box girder. A loading is given by the prescribed resulting internal forces – two bending moments, a torsional moment, a normal force, and two shear forces – derived from external actions. The loading has to be in equilibrium with chord forces and web stresses. But the equilibrium conditions will not suffice to determine these from the resulting internal forces. Compatible web strains and relations connecting strains with stresses again have to be introduced (Rahal and Collins 1995, 2003). With the finite element method at hand, shell elements allowing for membrane forces in space combined with local bending (Chapter 10) using stress–strain relations for reinforced concrete (Section 10.7) are another option.

8.1.3 Reinforcement Design An arbitrary point is regarded within a plate with a local load given as stress state 𝜎𝑥 , 𝜎𝑦 , 𝜎𝑥𝑦 determined by a superordinated analysis. The corresponding principal stresses are given by 𝜎1 , 𝜎2 . A design approach is derived as follows. Concrete is assumed as isotropic (Section 6.4.2). Its biaxial strength is shown in Figure 6.7a. A lower strength bound for all compressive states is given by the uniaxial compressive strength 𝑓𝑐 , i.e. |𝜎𝑐2 | < 𝑓𝑐 ; see Figure 8.3. Tensile stress states with 𝜎𝑐1 > 0 are excluded for concrete within the current scope. Regarding the pure compression state given the principal stresses should fulfil the conditions 𝜎1 ≤ 𝑓𝑐 and 𝜎2 ≤ 𝑓𝑐 . In the case that they do not, the loading has to be scaled down (Section 8.1.1) or plate dimensions have to be changed to fulfil the

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Figure 8.3 Strength square for biaxial concrete strength.

strength conditions. A compressive reinforcement might be another option. This alternative can be treated as an extension of the tension–compression case. Mixed tension–compression and pure tension require a reinforcement. A special, but common, case is given with the orientations of the reinforcement aligned to the 𝑥 and 𝑦-axis of the global coordinate system, which is ruled by Eqs. (8.10) and (8.11). The amount of reinforcement has to be determined for both orientations, while the concrete strength should not be exceeded. Regarding the concrete compressive orientation 𝜑𝑐 the Eq. (8.10) leads to a first constraint 𝜑𝑐 ≤ 0 for 𝜎𝑥𝑦 ≥ 0 or 𝜑𝑐 > 0 for 𝜎𝑥𝑦 < 0

(8.17)

and regarding the compressive strength to a further constraint || || 𝜎𝑥𝑦 || ≤ 𝑓 |𝜎𝑐 | = ||| 𝑐 || sin 𝜑𝑐 cos 𝜑𝑐 |||

(8.18)

The best case or largest admissible shear stress |𝜎𝑥𝑦 | = 0.5 𝑓𝑐 is connected with |𝜑𝑐 | = π∕4 (Figure 8.2a). The given shear stress component 𝜎𝑥𝑦 must not exceed this value. If it does, the loading again has to be scaled down or plate dimensions have to be changed to fulfil the constraint. We assume that some margin of discretion is possible for 𝜑𝑐 . ◀

According to general design practice for reinforced concrete, the concrete compressive orientation 𝜑c is prescribed as a design parameter.

This approach deserves a remark. The reality has a unique solution for 𝜑𝑐 . We might miss this solution with the current approach, as the deformation behaviour is not taken into account. An estimation of the 𝜑𝑐 value should not be too far away from a comprehensive solution or sound empirical values. Under this premise, we expect to reach a reliable design. To begin with, 𝜑𝑐 may be more or less freely chosen within the range not violating the constraint Eq. (8.17). This determines 𝜎𝑐 by Eq. (8.18), and Equations (8.11) may be used to determine the values 𝜌𝑥 𝜎𝑠𝑥 and 𝜌𝑦 𝜎𝑠𝑦 for given values of 𝜎𝑥 , 𝜎𝑦 . The uniaxial yield stress 𝑓𝑦𝑘 (Figure 3.11a) is used as a strength for the reinforcement,

8.1 Lower Bound Limit State Analysis

leading to 𝜎𝑠𝑥 = 𝜎𝑠𝑦 = 𝑓𝑦𝑘 and 𝜌𝑥 =

) 1 ( 𝜎𝑥 − 𝜎𝑐 cos2 𝜑𝑐 , 𝑓𝑦𝑘

𝜌𝑦 =

) 1 ( 2 𝜎𝑦 − 𝜎𝑐 sin 𝜑𝑐 𝑓𝑦𝑘

(8.19)

This is the required result for a reinforcement design in the case of computed values 𝜌𝑥 > 0 and 𝜌𝑦 > 0. The total amount of reinforcement is given by ) 1 ( 2 𝜎𝑥 − 𝜎𝑐 cos2 𝜑𝑐 + 𝜎𝑦 − 𝜎𝑐 sin 𝜑𝑐 𝜌tot = 𝜌𝑥 + 𝜌𝑦 = 𝑓𝑦𝑘 ) 1 ( 𝜎𝑥 + 𝜎𝑦 − 𝜎𝑐 (8.20) = 𝑓𝑦𝑘 The contribution −𝜎𝑐 is positive, as 𝜎𝑐 is negative by definition. Thus, for a given value 𝜎𝑥𝑦 , the concrete stress |𝜎𝑐 | = |𝜎𝑥𝑦 ∕sin 𝜑𝑐 cos 𝜑𝑐 | has to be minimised to minimise the total amount of reinforcement. This is reached with 𝜑𝑐 = ±π∕4 (Figure 8.2a) depending on the sign of 𝜎𝑥𝑦 . ◀

A concrete compressive orientation 𝜑c = ±π∕4 leads to optimal results for the utilisation of concrete strength and the value of the reinforcement required.

The minimum reinforcement amount is determined with ) 1 ( 𝜎𝑥 + 𝜎𝑦 + 2|𝜎𝑥𝑦 | 𝜌tot,min = 𝑓𝑦𝑘

(8.21)

These relations are valid only for orthogonal reinforcement meshes with coordinate directions aligned to reinforcement directions. The cases with values 𝜌𝑥 < 0 or 𝜌𝑦 < 0 computed from Eq. (8.19) are still open. Such values may be interpreted as required compression reinforcement with a reinforcement stress 𝜎𝑠𝑥 = −𝑓𝑦𝑘 and/or 𝜎𝑠𝑦 = −𝑓𝑦𝑘 (Eqs. (8.11)). But this should generally be avoided for plates. The case 𝜌𝑥 < 0, 𝜌𝑦 > 0 is treated exemplarily. The idea is to prescribe the value of 𝜌𝑥 𝜎𝑠𝑥 instead of 𝜑𝑐 . Prescribing 𝜌𝑥 𝜎𝑠𝑥 = 0 leaves (𝜌𝑦 𝜎𝑠𝑦 ), 𝜎𝑐 , and 𝜑𝑐 as unknowns to be determined from the three Eqs. (8.10) and (8.11). This set of equations is nonlinear and cannot be solved directly. A numerical method like the Newton–Raphson method (Eq. (2.77)) is used instead. Collecting Eqs. (8.11) and (8.10) by ⎛ ⎞ 𝜎𝑥 − 𝜎𝑐 cos2 𝜑𝑐 ⎜ ⎟ 2 f(u) = ⎜𝜎𝑦 − 𝜎𝑐 sin 𝜑𝑐 − 𝜌𝑦 𝜎𝑠𝑦 ⎟ = 0 𝜎𝑥𝑦 ⎜ − sin 𝜑𝑐 cos 𝜑𝑐 ⎟ ⎝ 𝜎𝑐 ⎠ leads to the iteration rule ⎛ 𝜎𝑐 ⎞ u = ⎜𝜌𝑦 𝜎𝑠𝑦 ⎟ , ⎜ ⎟ ⎝ 𝜑𝑐 ⎠

𝜕𝑓

u(𝜈+1)

⎡ 1 ⎢ 𝜕𝑢1 ⎢ 𝜕𝑓2 = u(𝜈) − ⎢ ⎢ 𝜕𝑢1 ⎢ 𝜕𝑓3 ⎣ 𝜕𝑢1

𝜕𝑓1

𝜕𝑓1

𝜕𝑢2

𝜕𝑢3 ⎥

⎤

𝜕𝑓2

𝜕𝑓2 ⎥

𝜕𝑢2

𝜕𝑢3

𝜕𝑓3 𝜕𝑢2

(8.22)

−1

⋅ f(u(𝜈) ) ⎥ ⎥ 𝜕𝑓3 ⎥ 𝜕𝑢3 ⎦u=u(𝜈)

(8.23)

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This generally converges with an appropriate start value u(0) and provides the required solution for 𝜎𝑐 , 𝜌𝑦 𝜎𝑠𝑦 , 𝜑𝑐 . The analogous method can be used in the case with prescribed 𝜌𝑦 𝜎𝑠𝑦 = 0 and unknown values for 𝜌𝑥 𝜎𝑠𝑥 , 𝜎𝑐 , 𝜑𝑐 . The design procedure has to be performed at representative positions of the plate under consideration. Every position has its particular result for the reinforcement required. This is demonstrated with the following example of a deep beam. Example 8.1: Reinforcement Design for a Deep Beam with a Limit State Analysis

We refer to the example of Figure 5.1 with the same system and loading. Young’s modulus and Poisson’s ratio are chosen with 𝐸 = 31 900 MN∕m2 , 𝜈 = 0.2, for a linear elastic calculation. The reinforcement yield strength is assumed with 𝑓𝑦𝑘 = 435 MN∕m2 . The depth of the deep beam is 𝑏 = 0.6 m. All units are in [MN] and [m]. Plane stress conditions are assumed. A state of stress with components 𝜎𝑥 , 𝜎𝑦 , 𝜎𝑥𝑦 is calculated for each integration point of each element, and the design procedure is performed for all these points. This gives a representative assessment of the whole deep beam. Four characteristic points with different types of stress state are considered; see Figure 8.4a. • Biaxial principal compression. Element 62 is considered with the lower right-hand integration point. Computed stresses are 𝜎𝑥 = −0.86, 𝜎𝑦 = −3.66, 𝜎𝑥𝑦 = −0.87. This leads to principal compressive stresses with values −3.91, −0.61. A reinforcement is not necessary from a structural point of view. • Biaxial principal tension. Element 123 is considered with the upper right-hand integration point with 𝜎𝑥 = 3.90, 𝜎𝑦 = 2.27, 𝜎𝑥𝑦 = 2.76 with principal tensile stresses 5.96, 0.21. The direction of concrete compressive stress is chosen with 𝜑𝑐 = −π∕4. Using Eq. (8.10) leads to 𝜎𝑐 = −2𝜎𝑥𝑦 = −5.52 MN∕m2 , and Eq. (8.19) yields 1 𝑓𝑦𝑘 1 𝜌𝑦 = 𝑓𝑦𝑘

𝜌𝑥 =

( ) 𝜎𝑥 − 𝜎𝑐 cos2 𝜑𝑐 (

2

𝜎𝑦 − 𝜎𝑐 sin 𝜑𝑐

)

= 0.0153 (8.24) = 0.0116

This results in reinforcement cross-sections 𝑎𝑠𝑥 = 𝑡 𝜌𝑥 = 0.009 18 m2 ∕m (→ 91.8 cm2 ∕m) and 𝑎𝑠𝑦 = 𝑡 𝜌𝑦 = 0.006 96 cm2 ∕m (→ 69.6 cm2 ∕m). These related values are required locally only, not along a whole 𝑥∕𝑦-length of 1 m. A concrete compression part is obviously necessary even in the case of biaxial principal tension to ensure equilibrium. To reach the state with reinforcement yielding in the global 𝑥 and 𝑦-directions and a concrete diagonal under compression some amount of redistribution of internal forces might be necessary. This should be connected with cracking, with the crack directions roughly aligned to the concrete diagonal. • Mixed principal stresses: element 92 is considered with the lower left-hand integration point with 𝜎𝑥 = −0.52, 𝜎𝑦 = 0.15, 𝜎𝑥𝑦 = 2.40 and principal stresses −2.61, 2.24. The concrete compressive direction is again chosen with 𝜑𝑐 = −π∕4 leading to 𝜎𝑐 = −4.80 MN∕m2 . This results in 𝜌𝑥 = 0.0043, 𝜌𝑦 = 0.0059 in the same way as Eq. (8.24).

8.1 Lower Bound Limit State Analysis

• Mixed principal stresses with initially negative reinforcement ratio: element 49 is considered with the upper left-hand integration point with 𝜎𝑥 = 4.01, 𝜎𝑦 = −0.11, 𝜎𝑥𝑦 = 0.02 and principal stress values 4.01, −0.12. A concrete compressive direction 𝜑𝑐 = −π∕4 leads to 𝜌𝑦 𝜎𝑠𝑦 < 0. Thus, 𝜌𝑦 𝜎𝑠𝑦 = 0 is prescribed, and an iteration is performed according to Eq. (8.23). This results in 𝜎𝑐 = −0.10 MN∕m2 , 𝜑𝑐 = −78°and 𝜌𝑥 = 0.0092 with 𝜎𝑠𝑥 = 𝑓𝑦𝑘 . Local results for the required reinforcement have to be transformed into a general reinforcement layout. A minimum reinforcement ratio is necessary to compensate for effects that are not explicitly regarded, like temperature and shrinkage, and to control the width of cracks. The minimum ratio is chosen with 𝜌𝑥,min = 𝜌𝑦,min = 𝜌min = 0.75 ⋅ 10−3 = 0.075% (EN 1992-1-1 2004, 9.7). This minimum ratio serves as a basis for supplementary reinforcement. Figure 8.4b shows the computed principal stresses for the upper right portion of the deep beam and the corresponding values for the required reinforcement – horizontal %-ratios for the 𝑥-reinforcement and vertical %-ratios for the 𝑦-reinforcement – in the case they exceed 𝜌min. The final reinforcement layout might use graded

(a)

(b)

Figure 8.4 Example 8.1. (a) Characteristic stress points with required reinforcement. (b) Reinforcement of the upper right-hand part.

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maximum values to cover larger areas with a constant reinforcement. Single rebars of larger diameter may also be used to treat smaller areas with higher reinforcement demand. Their contribution may also be calculated as a reinforcement ratio within small areas. The basic procedure – perform a linear elastic calculation for internal forces, followed by a reinforcement design and concrete proof with methods of limit analysis – shows analogies to the common practice for the design and proof of reinforced concrete beams considering bending, shear, and torsion. Furthermore, analogies are given to strut-and-tie models (Section 5.1), as a local system of reinforcement ties and a concrete strut are regarded in a plate position under consideration. Thus, similar remarks as are appropriate for strut-and-tie models have to be added. • Compressive strength of concrete The concrete contribution within this scope is basically uniaxial for principal stress states with tensile components. But lateral tension may actually lead to a decrease in the compressive concrete strength. This is taken into consideration by reduction factors applied on 𝑓𝑐 in the same way as for strut-and-tie models (Section 5.5); consider also CEB-FIP (2008, Section 2.2.7). • Ductility requirements In order to reach stress limit states a redistribution of internal forces may be necessary. This may require larger deformations or a sufficient ductility of the whole system, respectively. • Serviceability The current analysis does not consider deformations. But each local tie may be regarded as a small uniaxial tension bar. Thus, a crack width may be estimated according to Appendix C. The reinforcement stresses 𝜎𝑠𝑥 , 𝜎𝑠𝑦 required for such an estimation are determined using Eqs. (8.10) and (8.11) with given reinforcement ratios 𝜌𝑥 , 𝜌𝑦 and given stresses 𝜎𝑥 , 𝜎𝑦 , 𝜎𝑥𝑦 . These local ‘loadings’ for serviceability are generally derived with different global loadings compared to ultimate limit states. The concept of added contributions of concrete and reinforcement – each with its own principal orientation – to reach equilibrium with a local stress state given from a superordinated calculation may be transferred from biaxial states to triaxial states (Vecchio and Selby 1991; Foster et al. 2003; Nielsen and Hoang 2010). This corresponds to spatial strut-and-tie models with a spatial system of reinforcement ties and a spatial reinforcement strut, leading to design rules as a spatial extension of Eqs. (8.19). A limit state analysis is based upon stresses derived with linear elastic stress–strain relations. Such linear elastic relations may lead to stress concentrations with small areas of high stresses characterised by distinct peak values. If these high values are tensile, they may be crucial for the respective local reinforcement design. On the other hand, nonlinear stress–strain relations result in smoothed stress states, i.e. stress peaks are smoothed, while stresses moderately increase in the surrounding.

8.2 Cracked Concrete Modelling

The stress–strain behaviour of the (concrete) continuum and the reinforcement are decoupled with the current approach, and a realistic description of the overall deformation behaviour cannot be reached. A coupling with the concrete stress– strain behaviour, its cracking, and its interaction with reinforcement has to be taken into account to reach a consistent description of the structural behaviour. This is exemplarily described in the following sections.

8.2 Cracked Concrete Modelling For crack modelling, basic approaches have already been described with the following: • The cohesive crack (Section 7.1), which embeds concrete cracking into the framework of continuum mechanics. • The crack band approach (Section 7.3), which treats the crack as a large strain arising with strain softening. • The smeared crack model (Section 7.4), which transforms a crack width and orientation into an equivalent strain and blends this with the strain of the continuum. • Discrete crack models (Sections 7.6, 7.7), which explicitly describe crack width with discontinuous displacement fields. This offers a range of options to model the stress–strain behaviour of cracked concrete. For plates, we will exemplarily pursue the smeared crack model and, in particular, focus on the limited tensile strength of concrete. The major items within this set-up are summarised as follows. • Crack initiation is determined with the Rankine criterion (Section 7.4). A crack starts when the largest principal stress 𝜎1 reaches the tensile strength 𝑓𝑐𝑡 . The crack direction is given with the direction perpendicular to the direction 𝜑 of 𝜎1 (Eq. (8.2)). • A mesh-based scan method scans a whole domain in discrete points with respect to the Rankine criterion. For finite elements, such a testing mesh is naturally given by the element integration points. • Principal stress orientations may change with a load history. This leads to two alternative concepts for crack orientations: the fixed crack and the rotating crack (Section 7.4). The crack orientation is constant for the fixed crack with the value occurring at crack initiation, while the crack orientation follows the direction of principal stresses for the rotating crack, i.e. it may change during a loading history where crack shear tractions (Eq. (7.4)) vanish by definition. • If a discretising mesh is sufficiently dense, crack propagation is described by an increasing number of cracked points, e.g. points with a stress state fulfilling the Rankine criterion. Thus, a propagating crack geometry is not precisely captured, but the cracking of concrete is initially a diffuse matter anyway. The approach of the rotating, cohesive, smeared crack is exemplarily elaborated in the following. Furthermore, a linear elastic, biaxial material behaviour with a plane

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Figure 8.5 Cohesive crack model with loading, unloading, and re-loading stages.

stress state according to Eq. (2.45) and a limited tensile strength will be assumed. This has already been treated with Example 7.3 but will be carried forward. Each component represents a simplified model within its scope. This allows us to derive relatively simple stress–strain relations. More realistic models – for, e.g. nonlinear stress–strain behaviour of concrete or nonlinear relations for the crack–traction depending on the crack width – basically follow the same procedure. A 2D state of plane stress is considered, with given strain components 𝜖𝑥 , 𝜖𝑦 , 𝛾𝑥𝑦 determined from a superordinated calculation. The notation of indices is adapted for the current context. Principal strains are derived in analogy to principal stresses with Eqs. (8.1) and (8.2), where 𝜖𝑥𝑦 = 𝛾𝑥𝑦 ∕2 has to be used as the shear component. This leads to principal strains 𝜖1 , 𝜖2 with 𝜖1 ≥ 𝜖2 and the orientation angle 𝜑 of 𝜖1 . The following derivations are performed with respect to the principal strain system. Several states have to be considered; see Figure 8.5: 1. Initial tensile loading up to tensile strength and crack initiation followed by softening. 2. Further tensile elongation (→ loading) in the strain softening range. 3. Unloading until a stress free state is reached. 4. Crack closure with loading in the compressive range. 5. Re-loading into the tensile range. 6. Further loading until a macrocrack with a traction free state (→ 7). We start with 2. loading in the strain softening range. A solution has already been derived with Example 7.3. A key property is given with the crack band width 𝑏𝑤 (Eq. (7.56)) or the related 𝜉 = 𝑏𝑤 ∕𝐿𝑐 , respectively. As the crack normal direction is aligned to the principal 1-direction by definition, Eqs. (7.47)–(7.49) for stresses and crack width depending on strains lead to 𝜉𝜖𝑐𝑢 − 𝜖1 − (1 − 𝜉)𝜈 𝜖2 𝑑 (𝜉𝜖𝑐𝑢 − 𝜖𝑐𝑡 )𝜖2 + 𝜈𝜖𝑐𝑡 (𝜉𝜖𝑐𝑢 − 𝜖1 ) 𝜎2 = 𝐸 𝑑 (𝜖𝑐𝑢 − 𝜖𝑐𝑡 ) 𝜖1 + 𝜈(1 − 𝜉)(𝜖𝑐𝑢 − 𝜖𝑐𝑡 ) 𝜖2 − (1 − 𝜈2 − 𝜉 + 𝜉𝜈2 )𝜖𝑐𝑢 𝜖𝑐𝑡 𝑤1 = 𝑏𝑤 𝑑 𝑑 = 𝜉𝜖𝑐𝑢 − 𝜖𝑐𝑡 + 𝜈2 𝜖𝑐𝑡 (1 − 𝜉) 𝜎1 = 𝑓𝑐𝑡

(8.25)

8.2 Cracked Concrete Modelling

and a tangential stiffness C𝑇 =

1 ⎡ −𝑓𝑐𝑡 ⎢ 𝑑 −𝜈𝑓𝑐𝑡 ⎣

−(1 − 𝜉)𝜈𝑓𝑐𝑡 ⎤ ⎥ (𝜉𝜖𝑐𝑢 − 𝜖𝑐𝑡 ) 𝐸 ⎦

(8.26)

3. unloading is assumed to have a damage characteristic (Section 6.6). It starts from a state 𝑤1 = 𝑤 1 , 𝑡1 = 𝑡1 in the softening range (Figure 8.5) and follows the ‘damage’ path to a zero-state with zero strains, no deformations, and initiated crack closure. The traction–separation relation is given by 𝑡1 = 𝑡 1

𝑤1

(8.27)

𝑤1

and the largest crack width 𝑤 1 becomes an internal state variable with 𝑡1 assigned. Uncracked material behaviour is ruled with (compare Eq. (7.45)) ⎛𝜖𝑢,1 ⎞ 1 ⎡ 1 ⎜ ⎟= 𝐸 ⎢ 𝜖 −𝜈 ⎣ ⎝ 𝑢,2 ⎠

−𝜈⎤ ⎛𝜎1 ⎞ ⎥⋅⎜ ⎟ 𝜎 1 ⎦ ⎝ 2⎠

(8.28)

Furthermore, strains are connected by (compare Eq. (7.46)) ⎛ 𝜖1 ⎞ ⎜ ⎟ = (1 − 𝜉) 𝜖 ⎝ 2⎠

⎛𝜖𝑢,1 ⎞ ⎜ ⎟+𝜉 𝜖 ⎝ 𝑢,2 ⎠

⎛ 𝜖𝑐,1 ⎞ ⎜ ⎟ 𝜖 ⎝ 𝑢,2 ⎠

(8.29)

and crack quantities given with (compare Eq. (7.44)) 𝑡1 = 𝜎1 ,

𝑤1 = 𝑏𝑤 𝜖𝑐,1

(8.30)

leading to seven equations for seven unknowns 𝜎1 , 𝜎2 , 𝑡1 , 𝑤1 , 𝜖𝑢,1 , 𝜖𝑢,2 , 𝜖𝑐,1 . This is solved by 𝛽𝑡1 (𝜖 + 𝜈(1 − 𝜉) 𝜖2 ) 𝑑 1 𝛽𝑡1 𝐸 𝜎2 = (𝜈 𝜖1 + (1 − 𝜉 + 𝜉 ) 𝜖2 ) 𝑑 𝛽𝑡1

𝜎1 =

𝑏𝑤 𝑤1 = (𝜖1 + 𝜈(1 − 𝜉) 𝜖2 ) 𝑑 𝛽𝑡1 +𝜉 𝑑 = (1 − 𝜈2 )(1 − 𝜉) 𝐸

(8.31)

with 𝛽 = 𝑏𝑤 ∕𝑤 1 and a tangential stiffness C𝑇 =

𝛽𝑡1 ⎡1 ⎢ 𝑑 𝜈 ⎣

𝜈(1 − 𝜉) ⎤ 𝐸 ⎥ 1−𝜉+𝜉 𝛽𝑡 1 ⎦

(8.32)

4. Crack closure followed by compressive loading is treated as isotropic linear elastic with stress–strain relations according to Eq. (2.45). Unloading in the compressive

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range follows the same relations until the zero-state with 𝜖1 = 0. Regarding 5. tensile re-loading the ‘damage’ path is again followed with stress–strain relations and crack width according to Eqs. (8.31) until the crack width 𝑤1 reaches the value of the state variable 𝑤 1 . A state of 6. loading follows with ongoing softening ruled by Eqs. (8.25) with increasing values of 𝑤 1 and decreasing values of 𝑡1 . Finally, the state of 7. macrocracking is reached with 𝑤1 = 𝑤𝑐𝑟 and 𝜎1 = 0 𝜎2 = 𝐸 𝜖2

(8.33)

𝑤1 = 𝐿𝑐 ((𝜖1 + 𝜈(1 − 𝜉) 𝜖2 ) according to Eq. (7.51) with the characteristic element length 𝐿𝑐 (Eq. (7.18)). A further potential unloading will occur with zero stresses 𝜎1 until the zero state is again reached with a crack closure. Compression states are again ruled by the linear elastic isotropic law. A further re-loading follows the previous path in the opposite direction. Considering a point with a crack in the principal 1-direction, a further crack may arise in the principal 2-direction in the case of biaxial principal tension, leading to multiple cracking. This is treated by Eq. (7.58) for general cases but is reformulated in the current context. Multiple cracks are dual cracks in 2D. Dual cracks are orthogonal in the case of rotating cracks with the Rankine cracking criterion. The application of Eq. (7.58) yields ⎛ 𝜖1 ⎞ ⎛𝜖𝑢,1 ⎞ ⎡⎛˜ 𝜖𝑐1,1 ⎞ ⎛ 0 ⎞⎤ ⎟+⎜ ⎜ ⎟ = (1 − 𝜉) ⎜ ⎟ + 𝜉 ⎢⎜ ⎟⎥ 𝜖 𝜖 0 ˜ 𝜖 ⎣⎝ ⎠ ⎝ 𝑐2,1 ⎠⎦ ⎝ 2⎠ ⎝ 𝑢,2 ⎠ as Q1 (Eq. (7.58)) is a unit matrix and

(8.34)

⎡0 1⎤ Q2 = ⎢ (8.35) ⎥ 1 0 ⎣ ⎦ according to Eq. (D.11) with 𝜑 = π∕2. The states listed above (page ) are also applied to the second crack. A further assumption is made. ◀

In the case of dual cracking, a decoupling of principal directions with neglecting Poisson’s ratio 𝜈 = 0 simplifies the relations for stresses, crack widths, and tangential stiffness considerably.

This should be justified under the assumption that dual cracking is dominated by the cohesive crack behaviour in both directions. For loading and dual cracking, the following relations are assembled in analogy to Eqs. (7.42)–(7.46) 𝑡𝑖,1 𝑡𝑖,1

⎧𝑓 (1 − 𝑤𝑖,1 − 𝑤𝑐𝑡 ) , 𝑤 < 𝑤 < 𝑤 𝑐𝑡 𝑐𝑡 𝑖,1 𝑐𝑟 𝑤𝑐𝑟 − 𝑤𝑡 = ⎨ 0, 𝑤𝑐𝑟 ≤ 𝑤𝑖,1 , ⎩ = 𝜎𝑖

𝑤𝑖,1 = 𝑏𝑤 ˜ 𝜖𝑐𝑖,1

𝑖 = 1, 2

(8.36)

8.2 Cracked Concrete Modelling

involving 10 unknowns 𝜖𝑢,𝑖 , ˜ 𝜖𝑐𝑖,1 , 𝜎𝑖 , 𝑡𝑖,1 , 𝑤𝑖,1 , 𝑖 = 1, 2 ruled by 10 Eqs. (8.28), (8.34), and (8.36) with 𝜖1 , 𝜖2 given from a superordinated calculation. This is solved by (𝜈 = 0) 𝜉𝜖𝑐𝑢 − 𝜖𝑖 𝜉𝜖𝑐𝑢 − 𝜖𝑐𝑡 , (𝜖𝑐𝑢 − 𝜖𝑐𝑡 ) 𝜖𝑖 − (1 − 𝜉) 𝜖𝑐𝑡 𝜖𝑐𝑢 = 𝑏𝑤 𝜖𝑐𝑢 𝜉 − 𝜖𝑐𝑡

𝜎𝑖 = 𝑓𝑐𝑡 𝑤𝑖,1

𝑖 = 1, 2

(8.37)

with 𝑖 indicating the respective crack. This replicates Eqs. (8.251,3 ) with respect to 𝜈 = 0. In the case of unloading, Eqs. (8.361,2 ) are replaced by 𝑡𝑖,1 = 𝑡𝑖,1

𝑤𝑖,1 𝑤 𝑖,1

,

𝑖 = 1, 2

(8.38)

leading to (𝜈 = 0) 𝜎𝑖 =

𝑡𝑖,1 𝛽𝑖 𝜖𝑖 (1 − 𝜉)

𝑤𝑖,1 = 𝑏𝑤

𝑡𝑖,1 𝛽𝑖 𝐸

+𝜉

𝜖𝑖

(1 − 𝜉)

𝑡𝑖,1 𝛽𝑖 𝐸

,

𝑖 = 1, 2

(8.39)

+𝜉

with 𝛽𝑖 = 𝑏𝑤 ∕𝑤 𝑖,1 . This replicates Eq. (8.311,3 ) with respect to 𝜈 = 0. Crack closure and compression is again treated as an uncracked continuum using Eq. (8.28). Reloading is treated as unloading and further loading (Figure 8.5). A full crack follows Eq. (8.33) with 𝜈 = 0 𝑤𝑖 = 𝐿𝑐 𝜖𝑖 ,

𝑖 = 1, 2

(8.40)

It has to be considered that states may be mixed in the case of dual cracking, i.e. loading occurs in one direction, while the other direction is exposed to unloading, crack closure, or re-loading. Such mixtures will occur in plates due to a considerable redistribution of stresses during the formation of cracks. Decoupling of directions with Poisson’s ratio 𝜈 = 0 is also helpful in implementing these mixtures occurring with dual cracking. All these relations are derived in a local coordinate system aligned to the principal axes of strain. Principal stresses arise in the same system by definition due to the Rankine crack criterion and the rotating crack approach. Stresses and tangential material stiffness matrix have to be transformed to the global system to derive internal nodal forces (Eqs. (2.581 )) and the tangential element stiffness (Eq. (2.66)). The transformation is performed using Eqs. (6.15) and (6.16) with the transformation matrix Q for 2D states according to Eq. (D.12), with the orientation angle 𝜑 of the principal strain state. This is connected with a load-induced anisotropy (Section 6.1.2). All relations include the related crack band width 𝜉 = 𝑏𝑤 ∕𝐿𝑐 with the characteristic length 𝐿𝑐 of a particular quad element according to Eq. (7.18). This ensures the prescribed crack energy and accomplishes the regularisation as with the crack band approach (Section 7.3).

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8.3 Reinforcement and Bond To start with, a single reinforcement bar may be modelled with 2D bar elements (Section 2.3). The material behaviour of rebars is uniaxial and is described by the uniaxial elasto-plastic law (Section 3.3). Thus, modelling of rebars may be considered as a special case of a elasto-plastic truss (Section 5.3). But this truss is embedded in concrete and interacts with it through bond. Basic mechanisms of bond are discussed in Section 3.4; see also Example 3.4. Two approaches have to be distinguished with respect to modelling with finite elements. • Rigid bond It is assumed that slip between concrete and reinforcement can be disregarded. As a consequence, finite elements for reinforcement on the one hand and concrete on the other share the same nodes; see Figure 8.6a. This enforces the same displacements of concrete and reinforcement in nodes. There might be minor displacement differences along a rebar axis between nodes, but this is not significant, if the mesh is not too coarse. • Flexible bond with node connection Slip between concrete and reinforcement in the longitudinal direction is considered while both have the same displacement in the lateral direction. As a consequence, finite elements for reinforcement on the one hand and concrete on the other have their own nodes; see Figure 8.6b. Concrete nodes and reinforcement nodes initially share the same position and have to be connected by a special type of spring elements (Section 2.3). Such a spring should be constrained or very stiff in the lateral rebar direction and have a bond law characteristics (Figure 3.12b) in the longitudinal direction. • Flexible bond with embedded bars While the former two approaches may lead to constraints with respect to the nodal positions of concrete and rebar elements, the approach with embedded bars allows for an arbitrary placement of bars independent of a surrounding continuum. This is reached with special bond elements and is described in Section 8.5. Modelling of reinforcement meshes – often used for plates in addition to single rebars of larger diameter – with single truss elements might be inconvenient. In many

(a)

Figure 8.6 Bond (a) Rigid. (b) Flexible.

(b)

8.3 Reinforcement and Bond

cases, a reinforcement mesh consists of two orthogonal reinforcement groups. The rebars of a group have a small diameter and a narrow spacing. To model each bar of a reinforcement group with a number of finite element bars is elaborate. A smeared model is used instead, i.e. reinforcement bars of a group are modelled as a reinforcement sheet. The cross-section of a reinforcement sheet is given by the cross-section 𝐴𝑠 of a single bar and the bar spacing 𝑠. This leads to a sheet thickness 𝑡𝑠 =

𝐴𝑠 𝑠

(8.41)

and a reinforcement ratio 𝜌𝑠 =

𝑡𝑠 𝑏

(8.42)

with a thickness 𝑏 of a plate. ◀

A reinforcement sheet is regarded as a plate. Quad elements may be used for a discretisation. This has to be combined with uniaxial behaviour as a special case of anisotropy.

It is assumed that a reinforcement sheet has an orientation 𝜑𝑠 given by the direction of its bars. This is measured positive counterclockwise starting from the global 𝑥-direction and a rotated Cartesian coordinate system with the 𝑥 ˜-axis in the bar di)T ( rection. A given global state of strain 𝝐 = 𝜖𝑥 𝜖𝑦 𝛾𝑥𝑦 is transformed into the rotated system using Eq. (6.14) ˜ 𝝐 = Q⋅𝝐

(8.43)

with (Eq. (D.13)) ⎡ cos2 𝜑𝑠 ⎢ 2 Q=⎢ sin 𝜑𝑠 ⎢ −2 cos 𝜑𝑠 sin 𝜑𝑠 ⎣

2

sin 𝜑𝑠 cos2 𝜑𝑠 2 cos 𝜑𝑠 sin 𝜑𝑠

cos 𝜑𝑠 sin 𝜑𝑠 ⎤ ⎥ − cos 𝜑𝑠 sin 𝜑𝑠 ⎥ ⎥ 2 cos2 𝜑𝑠 − sin 𝜑𝑠 ⎦

(8.44)

This leads to a strain ˜ 𝜖𝑥 , which can be used to determine a stress 𝜎˜𝑥 according to the uniaxial material law (Eqs. (3.42) and (3.43)) appropriate for the reinforcement. )T ( The rotated stress state is given by 𝝈 ˜ = 𝜎˜𝑥 0 0 . This is transformed back to the global system with Eq. (6.15) 𝝈 = QT ⋅ 𝝈 ˜

(8.45)

The tangential material stiffness matrix in the rotated system is obtained by ⎡𝐶𝑇 ˜𝑇 = ⎢ C ⎢0 ⎢ 0 ⎣

0 0 0

0⎤ ⎥ 0⎥ ⎥ 0 ⎦

(8.46)

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Figure 8.7 Overlay of elements.

with 𝐶𝑇 = 𝐸𝑠 in the case of loading below the yield limit and unloading and 𝐶𝑇 = 𝐸𝑇 at the yield limit (Figure 3.11). This is transformed to the global system (Eq. (6.16)) ⎡ cos4 𝜑𝑠 2 ˜ 𝑇 ⋅ Q = 𝐶𝑇 ⎢ C𝑇 = QT ⋅ C ⎢cos2 𝜑𝑠 sin 𝜑𝑠 ⎢ cos3 𝜑𝑠 sin 𝜑𝑠 ⎣

2

cos2 𝜑𝑠 sin 𝜑𝑠 4

sin 𝜑𝑠 3

sin 𝜑𝑠 cos 𝜑𝑠

cos3 𝜑𝑠 sin 𝜑𝑠 ⎤ ⎥ 3 sin 𝜑𝑠 cos 𝜑𝑠 ⎥ ⎥ 2 cos2 𝜑𝑠 sin 𝜑𝑠 ⎦ (8.47)

leading to a symmetric but fully occupied matrix C𝑇 . With respect to Eqs. (8.43)– (8.47), the required components are provided for the modelling of a reinforcement sheet using quad elements or other element types for 2D states. The procedure for a reinforcement sheet concerns a particular reinforcement group out of multiple groups of a reinforcement mesh. The set of reinforcement sheets and the model for concrete with stress–strain relations as are described, e.g. in the previous Section 8.2 have to be combined to complete the model for a reinforced concrete plate. This is performed with an overlay of elements; see Figure 8.7. Each part is modelled separately with its own finite elements. But these parts must be coupled, which is a matter of bond. The different groups of a mesh are connected by welding spots, whereby each group impedes the sliding of rebars of a group with crosswise orientation. Thus, the bond is assumed as rigid for reinforcement sheets. Furthermore, elements for reinforcement sheets may share the same geometry and the same nodes. Their contributions are simply added in the assembling process (Section 2.6). Finally, bar elements for single reinforcement bars may be superposed. Cracking of the concrete part may be considered with the smeared crack model also in the case of overlays with reinforcement sheets and single rebars. The increased smeared strains due to concrete cracking are coupled to the corresponding reinforcement strains. Concrete and reinforcement parts interact, and a consistent deformation modelling is accomplished in cracked and uncracked regions. The application of the basic set of constituents – limited tensile strength of concrete, load-induced anisotropy, the smeared crack model, reinforcement sheets with rigid bond – is demonstrated with the following example.

8.3 Reinforcement and Bond

Example 8.2: Simulation of a Cracked, Reinforced Deep Beam

We refer to the deep beam example given in Figure 5.1 and treated in Example 8.1 with a limit state analysis. In contrast to Example 8.1, the nonlinear stress–strain behaviour of concrete and reinforcement will be considered. System and loading are shown in Figure 5.1a. The following material properties are assumed: • Concrete with Young’s modulus 𝐸𝑐 = 33 600 MN∕m2 , Poisson’s ratio 𝜈 = 0.2, compressive strength 𝑓𝑐 = 38 MN∕m2 , and tensile strength 𝑓𝑐𝑡 = 2.9 MN∕m2 according to concrete grade C30 (CEB-FIP2 2012, 5.1). The concrete crack energy is assumed with 𝐺𝑓 = 150 Nm∕m2 (CEB-FIP2 2012, 5.1.5.2) • The reinforcement material model (Section 3.3) is chosen with the parameters Young’s modulus 𝐸𝑠 = 200 000 MN∕m2 , initial yield limit 𝑓𝑦𝑘 = 500 MN∕m2 , ultimate strength 𝑓𝑡 = 550 MN∕m2 , and strain at ultimate strength 𝜖𝑢 = 0.05. Plane stress conditions are assumed for the concrete part. Three cases of material models are assumed for the concrete as alternatives: 1. The biaxial linear elastic case with limited tensile strength and smeared cracking (Section 8.2, abbreviated as ‘limited elastic’) with a crack band width 𝑏𝑤 = 0.13 m using Eq. (7.56). 2. Isotropic damage (Section 6.6) with the Hsieh–Ting–Chen damage function Eq. (6.109). The regularisation is performed with the crack band approach (Section 7.3), scaling the stress–strain relations in the tensile softening range by the characteristic element length (Eq. (7.18)). 3. Microplane (Section 6.8) with the concrete damage formulation (Eq. (6.151)) allowing for a load-induced anisotropy where the regularisation is again performed with the crack band model applied to each microplane (Figure 6.15b). Damage 2. and microplane 3. are stabilised by an artificial viscosity (Appendix A.1) to facilitate stable equilibrium iterations with the incrementally iterative solution scheme (Section 2.8.2). With 1. limited elastic, the artificial viscosity concept is applied analogously for the crack tractions (Eq. (7.42)). The viscosity parameter 𝜂 in each case is selected such that the influence on the load–displacement behaviour is not significant. This requires a small parameter study about 𝜂, which is not explicitly described. The stabilisation is supported by applying a dynamic analysis (Appendix A.1 and Section 2.8.2), where a quasi-static behaviour is preserved by a slow load application. The reinforcement is given by two orthogonal reinforcement sheets with indices 𝑖 = 1, 2, with orientations 𝜑𝑠1 = 0, 𝜑𝑠2 = π∕2 and thickness 𝑏𝑠1 = 𝑏𝑠2 = 0.006 m leading to reinforcement ratios 𝜌𝑠1 = 𝜌𝑠2 = 1%. Additionally, the upper left-hand and the lower right-hand edges of the opening each are additionally strengthened with single rebars with a cross-section sum of 25cm2 modelled as 2D bar elements (Section 2.3), sharing nodes with concrete elements. The loading is applied by displacement control of the node exposed to the concentrated load; see Figure 5.1a. Distributed loads are neglected. The actual loading is given by the reaction force of the displaced node. In contrast to load control, the

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(a)

(b)

Figure 8.8 Example 8.2. (a) Discretisation. (b) Load–displacement curve.

control of displacements allows us to model the limit states of a structure, i.e. limited loading with increasing displacements. The target displacement of the loaded node is prescribed with 0.04 m. The discretisation is shown in Figure 8.8a. With respect to the concrete and reinforcement parts, it consists of 528 quad elements (Section 2.3) of uniform size and quadratic shape. Thereof, 176 elements are used for the concrete layer and 176 overlayed elements each for the two reinforcement directions, whereby sharing nodes. This yields 207 nodes with 414 nodal degrees of freedom in total. The integration order (Eq. (2.69)) is generally chosen with 𝑛𝑖 = 1, leading to a 2 × 2 integration within elements. The characteristic element length is determined with 𝐿𝑐 = 0.5 m (Eq. (7.18)). Special care has to be taken for the lower left-hand support. A nodal support is not appropriate, as this leads to concentrated high tensile stresses and premature failure of the concrete part. A more realistic approach is given with a support by an extra linear elastic triangular plate element – 𝐸 = 33 600 M∕m2 , 𝜈 = 0.2 – with a vertical nodal support of its lower node. This support may move in the horizontal direction. The same consideration applies to the point of concentrated loading. Two triangular plate elements with the same properties as for the support element are used for the load distribution (Figure 8.8a), whereby the prescribed displacement is applied at the top node. Computational results are described starting with case 1. limited elastic. The computed load–displacement curve for the loaded node is shown in Figure 8.8b. It becomes unstable shortly before reaching the target displacement due to the spread of dual cracking. Nevertheless, four typical states can again be seen: • Uncracked state I. • Crack formation state IIa with elastic reinforcement behaviour. • Stabilised cracking state IIb with elastic reinforcement behaviour, with gradual transition to yielding. • Stabilised cracking state III with yielding reinforcement. This is basically the same behaviour as for the reinforced concrete tension bar (Example 3.4) and for the simple reinforced concrete beam (Example 4.2). But the transitions are gradual in this example, as a larger number of elements are involved in a state change, each with their own specific behaviour.

8.3 Reinforcement and Bond

(a)

(b)

Figure 8.9 Example 8.2. Principal stresses on the deformed structure (scale 30). (a) Reinforcement – tensile only. (b) Concrete – compressive only.

The computed reinforcement stresses in the last computed step are shown on the deformed structure in Figure 8.9a. Horizontal lines belong to the stresses of the horizontal reinforcement direction, vertical lines to the stresses of the vertical reinforcement direction. Horizontal stresses reach the yield strength in the lower midspan area and the upper right-hand side support area, vertical reinforcement stresses in the areas near the upper left-hand and lower right-hand vertices of the opening. Small reinforcement areas – not shown – are in compression due to the bond with concrete. The computed concrete stresses of the last computed step are shown in Figure 8.9b. Low tensile stresses – not shown – arise according to the prescribed tensile strength. The minimum concrete stress (→ maximum compression) is computed with −65 MN∕m2 in the right-hand end lower edge. This computed value exceeds the compressive strength of normal graded concrete, as linear elastic behaviour is assumed for the compressive concrete material behaviour of case 1. The limited compressive strength of concrete is considered by the alternative isotropic and microplane damage models. A comparison of the load–displacement behaviour of the three concrete models is shown in Figure 8.10 with all other parameters unchanged.

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Figure 8.10 Example 8.2. Comparison of load–displacement behaviour.

Figure 8.11 Example 8.2. Principal stresses of concrete for the microplane model on the deformed structure (scale 30) near maximum loading.

This exhibits large differences. The isotropic and microplane damage models both limit concrete compressive stresses, but the microplane model allows for a loadinduced anisotropy – the concrete’s compressive strength is maintained in a direction, while the tensile strength at the same position is exhausted in an orthogonal direction – while the isotropic damage model does not allow for this. The microplane model is presumably more realistic for the example. While the limited elastic model overestimates the bearing capacity to a large extent, the isotropic damage model leads to a considerable underestimation. A crack formation behaviour is not as pronounced for isotropic damage and microplanes as it is for limited elasticity. This is caused by the different specifications of the crack band approach (Section 7.3 versus Section 8.2). For completion, Figure 8.11 shows the computed concrete principal stresses near maximum loading for the microplane damage model. The concrete tensile strength is slightly exceeded in a few points due to the viscous stabilisation. Reinforcement stresses are basically similar to those computed for the limited elastic model (Figure 8.9b). In its highly stressed areas, the reinforcement is near the yielding range. The system failure after peak loading is initiated by a concrete failure above the lower left-hand supporting triangular element.

8.4 Integrated Reinforcement

The model of Example 8.2 combines equilibrium, nonlinear material behaviour with limited material strength, and – in contrast to the companion Example 8.1 – kinematic compatibility. Furthermore, the reinforcement data have to be given as input, leading to the ultimate load as result, while loading is assumed, and reinforcement determined in the former example. The results of Example 8.2 suggest that a higher loading can be sustained with approximately the same reinforcement compared to the loading assumed for the design method demonstrated with Example 8.1. This is even the case with the least favorable isotropic damage model for concrete. On the other hand, the current calculation is very elaborate and shows a pronounced model uncertainty. The topic of model uncertainty due to different concrete models is again treated with Example 10.2.

8.4 Integrated Reinforcement The combination of concrete and reinforcement for the case of a rigid bond is demonstrated with the overlay approach (Figure 8.7). This can be applied for reinforcement sheets as well as for bar elements representing single rebars. Another type of rigid bond is given with the integration of reinforcement. This is again exemplarily described for quad elements (Section 2.3). The overlay approach introduces position constraints, insofar as nodes of the concrete part and the reinforcement part coincide. An integrated reinforcement may be arbitrarily embedded within an element where its position is given as a line with arbitrary orientation; see Figure 8.12a. It must not share nodes. A rigid bond enforces the same displacement field for whole domain, including the reinforcement. Thus, the standard displacement interpolation according to Eqs. (2.18) and (2.35) is appropriate, leading to continuous strains within an element according to Eq. (2.40). Strains are connected with stresses, but these stresses will no longer be continuous with the reinforcement contribution. Regardless of this, the stresses have to be transformed into nodal forces by integration (Eq. (2.9)).

(a)

(b)

Figure 8.12 Reinforcement embeddings. (a) 2D. (b) 3D.

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Standard numerical integration, e.g. with Gauss integration (Eq. (2.69)) relies on smooth stress fields to yield reliable integration results. This is generally the case for homogeneous materials but not for materials with integrated reinforcement with a different material model. ◀

Stress discontinuities arise within elements with integrated reinforcement.

Such stress discontinuities are given with the stress along the rebar line. The position of the rebar line using global coordinates 𝑥, 𝑦 is assumed with 𝑦 = 𝑎𝑥 +𝑏

(8.48)

with constant coefficients 𝑎, 𝑏. With respect to a particular element, this can be transformed into a rebar line position described with local coordinates 𝑟, 𝑠 𝑠=𝑎 ˜ 𝑟 + 𝑏˜ ,

−1 ≤ 𝑟 ≤ 1

(8.49)

using Eq. (2.36). Strains along the rebar line are determined by Eq. (2.40) with respect to local coordinates. As rebar stresses are uniaxial, the global strains 𝜖𝑥 , 𝜖𝑦 , 𝛾𝑥𝑦 have to be transformed into the direction 𝜑𝑠 of the rebar according to Eq. (8.43). The 𝜖𝑦 , 𝛾˜𝑥𝑦 are not principal strains (˜ 𝛾𝑥𝑦 ≠ 0), but only the comporesulting strains ˜ 𝜖𝑥 , ˜ ˜𝑥 ≠ 0 and 𝜎˜𝑦 = 𝜎˜𝑥𝑦 = 0. The stress component ˜ 𝜖𝑥 is relevant for a rebar stress state 𝜎 nent 𝜎˜𝑥 is derived from the material law for the reinforcement (Section 3.3). Finally, the stress discontinuity along the rebar line is determined, transforming the stress state 𝝈 ˜ back to the global system using Eq. (8.45), leading to a rebar stress 𝝈𝑠 . As the position of the rebar line is arbitrary, its contribution to internal forces (Eq. (2.9)) is generally not considered with a Gauss integration according to Eq. (2.69), which considers the concrete part only. ◀

The contribution of rebar stresses to the internal nodal forces have to be superposed to the contribution of concrete stresses with their own integration procedure.

An additional Gauss integration along a line in analogy to Eq. (2.69) +1

∫ 𝝈𝑠 (𝑟) 𝐽(𝑟) 𝐴𝑠 d𝑟 = 𝐴𝑠 −1

𝑛𝑖 ∑ 𝑖=0

𝜂𝑖 𝝈𝑠 (𝜉𝑖 ) 𝐽(𝜉𝑖 )

(8.50)

is used for this purpose, with the rebar cross-sectional area 𝐴𝑠 , sampling points 𝜉𝑖 , sampling weights 𝜂𝑖 according to Table 2.1, and a Jacobian 𝐽(𝑟) (Eqs. (2.20) and (2.23)) relating the local coordinate 𝑟 to the global coordinate 𝑥. An integration order 𝑛𝑖 = 1 with two integration points should be adequate for integrated rebars in quad elements with integration order 1 and 2 × 2 integration points. Contributions of rebar lines to tangential stiffness matrices (Eq. (2.66)) can be considered in an analogous way. In contrast to reinforcement overlays, which basically allow also for a flexible bond (Figure 8.6b), an integrated reinforcement is restricted

8.5 Embedded Reinforcement with a Flexible Bond

to a rigid bond only. On the other hand, an overlay of rebars with a rigid bond may be considered as a special case of an integrated reinforcement where the rebar line coincides with element boundaries. The integration approach may be generalised for 3D states; see Figure 8.12b. A special implementation for slabs and shells is given with the layer model (Section 10.7.1).

8.5 Embedded Reinforcement with a Flexible Bond The previous approaches imply restrictions, be it that they require adaptations of rebar discretisations to continuum discretisations or that they enforce a rigid bond. A more general approach is described in the following with an arbitrary placement of rebars discretised by truss or beam elements, allowing for a flexible bond by special bond elements; see Figure 8.13. Displacements of truss or beam elements are interpolated by u𝑡 (x) = N𝑡 (x) ⋅ 𝝊𝑡𝑒

(8.51)

according to Eq. (2.18) but with their specific trial functions N𝑡 . In the same way, displacements of continuum elements are interpolated using Eqs. (2.18) and (2.35) u(x) = N(x) ⋅ 𝝊𝑒

(8.52)

A bond is discretised with respect to so-called anchor points x𝑎,𝑖 placed along the reinforcement elements (Figure 8.13). This yields a difference of displacements or slip for a particular anchor point ( ) ( ) [ s = u𝑡 x𝑎,𝑖 − u x𝑎,𝑖 = N𝑡 (x𝑎,𝑖 ) ⎛𝝊𝑡𝑒 ⎞ = B𝑏 (x𝑎,𝑖 ) ⋅ ⎜ ⎟ 𝝊 ⎝ 𝑒⎠

] ⎛𝝊𝑡𝑒 ⎞ −N(x𝑎,𝑖 ) ⋅ ⎜ ⎟ 𝝊 ⎝ 𝑒⎠ (8.53)

derived from the nodal displacements of the respective reinforcement element and the nodal displacements of the underlying continuum elements. The forms N𝑡 , N are generally formulated as functions of local coordinates r (Eqs. (2.18) and (2.15)). Thus, a mapping r ←→ x𝑎 is required based on the isoparametric geometry interpolation (Eq. (2.19)). Slip induces the bond forces t𝑏 = t𝑏 (s)

(8.54)

as a counteracting couple for the reinforcement element on the one hand and the underlying continuum element on the other. Such couples may also be seen as springs; see Figure 8.13b. Their behaviour is generally nonlinear and may be considered as a further material relation in analogy to Eq. (2.47). The bond law Eq. (8.54) is assumed to be known. An incremental form ṫ 𝑏 = C𝑇𝑏 (s) ⋅ ṡ

(8.55)

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(a)

(b)

Figure 8.13 Embedded reinforcement. (a) Spatial discretisation. (b) Bond discretisation.

is used in the following in analogy to Eq. (2.50). This is advantageously formulated in a local system aligned to the reinforcement direction. A slip derived from Eq. (8.53) with 𝝊𝑡𝑒 , 𝝊𝑒 related to the global system has to be transformed into the local aligned system according to the transformation rule Eq. (D.8) with the inclination angle 𝜑 (Figure 8.13a). This leads to a modified B′𝑏 = Q ⋅ B𝑏 to be applied with Eq. (8.53). The local slip is immediately suitable for a bond law. ◀

The bond–slip relation is assumed such that a lateral slip or intersection is basically avoided by a penalty term. Furthermore, the longitudinal slip reproduces a prescribed nonlinear bond behaviour that is generally connected with an upper limit for the longitudinal bond force or bond strength.

With respect to the discretisation, a bond force couple – one part applied to a reinforcement element, the other part applied to a continuum element – is introduced as point load at x𝑎,𝑖 , leading to nodal forces for the reinforcement element f𝑏𝑡 on the one hand and for the underlying neighbour continuum element f𝑏𝑐 on the other ⎛f𝑏𝑡 ⎞ T f𝑏 = ⎜ ⎟ = B′𝑏 ⋅ t𝑏 f𝑏𝑐 ⎝ ⎠

(8.56)

already related to the global system. This is in analogy to Eq. (2.581 ) where the integration over a domain is not required as bond is implemented through single forces. Equation (8.56) is linearised with ⎛𝝊̇ 𝑡𝐼 ⎞ ⎛𝝊̇ 𝑡𝐼 ⎞ T ḟ 𝑏 = B′𝑏 ⋅ C𝑇𝑏 ⋅ B′𝑏 ⋅ ⎜ ⎟ = K𝑇,𝑏 ⋅ ⎜ ⎟ 𝝊̇ 𝝊̇ ⎝ 𝐼⎠ ⎝ 𝐼⎠

(8.57)

for a particular bond element with the tangential bond stiffness C𝑇𝑏 . As embedded rebars do not have displacement boundary conditions explicitly prescribed, at least two anchor points x𝑎,𝑖 are required with two bond elements per reinforcement element to avoid rigid body displacements. They are conveniently placed in reinforcement element integration points. Nodal forces f𝑏 and the tangential stiffness K𝑇,𝑏 can be assembled on the system level for each bond element: f𝑏 is added to

8.5 Embedded Reinforcement with a Flexible Bond

the assembled f (Eq. (2.60)), and K𝑇,𝑏 is added to the assembled K𝑇 (Eq. (2.67)) according to the mappings of the underlying reinforcement and continuum elements into the system. A specification for the 2D truss element as reinforcement element is given in the following. A standard 2D truss element is regarded with the isoparametric two-node element with four nodal degrees of freedom (Section 2.3). The trial function ⎡𝑁 (𝜉) 0 ˜ 𝑡 = ⎢ 𝑡1 N 0 𝑁𝑡1 (𝑟) ⎣ 1 𝑁𝑡1 (𝑟) = (1 − 𝑟) 2 1 𝑁𝑡2 (𝑟) = (1 + 𝑟) 2

𝑁𝑡2 (𝑟) 0

⎤ ⎥ 𝑁𝑡2 (𝑟) ⎦ 0

(8.58)

is formulated in a local coordinate system, with the first displacement direction 𝑢 ˜𝑥 aligned to the truss element axis (Figure 8.13) with the local coordinate −1 ≤ 𝑟 ≤ 1. This implies constant longitudinal strains and stresses. An enhanced approach is chosen for quadratic longitudinal displacements 𝑢 ˜𝑥 with ⎡𝑁 (𝑟) 0 𝑁𝑡4 (𝑟) ˜ 𝑡 = ⎢ 𝑡3 N 0 0 𝑁𝑡1 (𝑟) ⎣ 1 1 𝑁𝑡3 (𝑟) = (1 − 𝑟) − (1 − 𝑟 2 ) 2 2 1 1 𝑁𝑡4 (𝑟) = (1 + 𝑟) − (1 − 𝑟 2 ) 2 2 𝑁𝑡5 (𝑟) = 1 − 𝑟 2

0 𝑁𝑡2 (𝑟)

𝑁𝑡5 (𝑟)⎤ ⎥ 0 ⎦ (8.59)

introducing a fifth degree of freedom for longitudinal displacements leading to an aligned B-matrix ⎡2𝑟 − 1 ˜𝑡 = 1 ⎢ B 𝐿𝑒 0 ⎣

0

2𝑟 + 1

0

0

0

0

−4𝑟 ⎤ ⎥ 0 ⎦

(8.60)

for a linear interpolation of longitudinal strains within the element of length 𝐿𝑒 and improving the modelling of the bond behaviour. A transformation of global nodal displacements into the local system according to Eq. (D.8) is again required, whereby the additional fifth degree of freedom remains unaffected. The additional degree of freedom for the enhanced truss element can be resolved on the element level or carried to and solved on the system level. For embedded truss elements, this enhanced approach yields the same behaviour as fully quadratic elements but with fewer degrees of freedom. The application is demonstrated with the following example.

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Example 8.3: Simulation of a Single Fibre Connecting a Dissected Continuum

We consider two continuum blocks separated by a small gap and connected by a single fibre; see Figure 8.14a. The blocks are cubes of dimension 20 × 10 × 20 mm3 each. This is a typical set-up for an experimental investigation of the bond behaviour of a fibre in a fine-grained cement matrix. The problem is treated in 2D to demonstrate the special bond treatment. The material behaviour of the blocks is assumed as linear elastic, with material properties given in Table 8.1. The movement of the lower block is constrained at its lower edge, while some vertical displacement is prescribed for the displacement of the upper edge of the upper block. The fibre connecting the block is given an arbitrary orientation with arbitrary lengths of embedment. Its material behaviour is also assumed as linear elastic. The geometry and material properties are listed in Table 8.1. The assumed bond law is shown in Figure 8.14b. It relates a bond stress 𝜏 to a slip 𝑠, which acts over the fibre’s circumference. The bond stress is advantageously transformed into a stress flow 𝑇 with a unit [force∕length] by multiplying it by fibre circumference 𝑈𝑓 √ 𝑇 = 𝑈𝑓 𝜏 = 2 π𝐴𝑓 𝜏

(8.61)

(a)

(b)

Figure 8.14 Example 8.3. (a) System. (b) Bond law between fibre and concrete continuum. Table 8.1 Parameters of Example 8.3. Continuum Young’s modulus Poisson’s ratio

𝐸 𝜈

[MN∕m2 ] 30 000 — 0.2

Fibre Young’s modulus Diameter Cross-sectional area Length

𝐸 ⊘ 𝐴𝑓 𝐿𝑓

[MN∕m2 ] [mm] [mm2 ] [mm]

800 000 0.02 3.14 ⋅ 10−4 10

8.5 Embedded Reinforcement with a Flexible Bond

(a)

(b)

Figure 8.15 Example 8.3. (a) Discretisation. (b) Load–displacement curve.

(a)

(b)

Figure 8.16 Example 8.3. Near maximum load (displacements scaled by 5). (a) Continuum principal stresses. (b) Fiber stresses (bond stress: curve unsigned, values signed).

derived from, e.g. the fibre cross-sectional area 𝐴𝑓 . This is applied for nodal forces in Eq. (8.56) and compensates for arbitrary thickness values of an underlying 2D continuum, whereby the fibre acts as a thin sheet through the thickness of the continuum. Quad elements (Section 2.3) and the enhanced embedded truss element (Eqs. (8.58) and (8.59)) are used for the discretisation with 153 nodes and 296 nodal degrees of freedom; see Figure 8.15a. The computed load–displacement behaviour – the reaction force at the upper edge depending on the upper edge displacement – is dominated by the bond law. It is shown in Figure 8.15b and is characterised by maximum loading followed by a softening behaviour. This is a mirror of the assumed bond law, with further influence of the fibre embedment length and orientation. The computed load–displacement behaviour may be compared to experimental results and serve for an inverse analysis of bond laws. But experiments hardly allow for detailed results like the development of bond and fibre stresses and stresses in the underlying continuum. This is accessible with numerical simulations. The principal stresses of the continuum are shown in Figure 8.16a in the deformed configuration for a state near the maximum reaction force. A horizontal displacement of the upper block is enforced, as the fibre truss element

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crossing the gap is embedded in the lower as well as in the upper block, whereby lateral deformations or bending effects, respectively, are not allowed between nodes. Thus, lateral bond springs enforce the horizontal movement due to fibre inclination. This effect can be avoided with a fibre node placed in the initial gap. The fibre is isolated in Figure 8.16b with longitudinal and bond stresses. Maximum bond stresses reach the prescribed bond strength (Figure 8.14b). The truss element crossing the gap has a stress 𝜎 ≈ 1700 N∕mm2 corresponding to a fibre force 𝐹𝑓 = 0.53 N. Its vertical component shows a considerable difference compared the maximum system reaction force (Figure 8.15b). As before with the upper block horizontal displacement, the connecting truss element is laterally rigid and enforces spurious lateral forces acting on the connecting truss element, leading to additional vertical force contributions. This is another locking effect and requires further provisions. Truss elements are a simple way to model reinforcement explicitly. But their combination with bond elements may lead to locking effects if the underlying continuum is exposed to shear relative to truss orientations, as is demonstrated with Example 8.3. This can be avoided with embedded beam elements for reinforcement discretisation, which allow for a lateral flexibility by introducing additional rotation degrees of freedom. Furthermore, reinforcement beam elements allow us to model effects like dowel action if the reinforcement crosses cracks. The extension of the above to embedded beam elements is straightforward. With the Bernoulli beam (Section 4.3.2), local forms of shape functions like Eq. (4.85) may be used in place of Eq. (8.51) – after transformation into the global system – thereby introducing additional rotational degrees of freedom for the beam element. The rotational degrees of freedom are independent from the embedding continuum. The bond elements coupling beam elements and continuum elements are applied in the same way as for truss elements, leading to force couples implementing a prescribed longitudinal bond law, thereby preventing lateral penetration by a penalty approach. Bending moments and normal forces of the beam elements are ruled by Eps. (4.87), (4.14), and (4.46). Nodal forces of beam elements (Eq. (2.58)) are seamlessly integrated into the combined system, whereby nodal forces corresponding to bending moments form a self-equilibrating system. The application is demonstrated with the following example. Example 8.4: Reinforced Concrete Plate with Flexible Bond

We refer to Examples 7.1 and 7.2 for an experimental plate (Figure 7.5a). Here, however, the lower notch is omitted, and a lower reinforcement is inserted instead; see Figure 8.17. A single steel rebar diameter ⊘ 14 mm is assumed, with material properties as in Table 3.1. Otherwise, geometry, concrete model, concrete material parameters and boundary conditions are unchanged. Loading is applied with a prescribed downward displacement of the upper edge of the upper steel plate. Plane stress conditions are assumed. The discretisation of the concrete continuum is exemplarily performed with the coarse discretisation No.1 of Examples 7.1, 7.2; see Table 7.1. The crack band ap-

8.5 Embedded Reinforcement with a Flexible Bond

(a)

(b)

Figure 8.17 Example 8.4. (a) System. (b) Load-displacement curve.

proach is used for regularisation in the same way as for Example 7.2 complemented with a viscous stabilisation as for Example 7.1, with 𝜂 = 1 MN∕(m2 s), a loading time 1 s, and a time step of Δ𝑡 = 0.001 s. The discretisation of the rebar with enhanced Bernoulli beam elements (Eqs. (4.93) and (4.94)) is independently chosen with an element length of 𝐿𝑒 = 0.02 m, leading to 30 elements with 31 full and 30 enhancing nodes and 123 additional degrees of freedom. Rebar nodes and concrete nodes are spatially independent and do not share initial positions. The bond law is the same as that already used with Example 3.4, as is shown in Figure 3.13 with data in Table 3.1, whereby slip at bond strength is increased to 0.2 mm. Figure 8.17b shows the computed load–displacement curve with the total reaction force along the upper edge of the upper steel plate versus its prescribed vertical displacement. • The three typical stages of reinforced concrete behaviour arise with uncracked state I, cracked state II, and failure state III. • A decrease in reaction force in the transition from state I to state II results from a sudden stiffness decrease, which can also be seen with the uniaxial tension bar (Example 3.4). • The failure state III has a quasi-brittle behaviour due to premature concrete failure prior to reinforcement yielding. This is described in more detail below. Two points of the load–displacement curve – indicated in Figure 8.17b – are selected to show principal stress fields, rebar stresses, and bond stresses. For point 𝐴 near maximum loading, these are shown in Figures 8.18 and 8.19. The principal stress field in Figure 8.18 illustrates a compression arch tied by the reinforcement bar. • The lower concrete elements in the centre line exceed the range of concrete strain softening with zero stresses and indicate cracking. • Concrete elements with tensile stresses in the upper centre line correspond to the concrete process zone or strain softening, respectively. This is terminated with a horizontal zone of concrete compression. • Concrete principal tensile stresses do not exceed the prescribed tensile strength of 3.0 MN∕m2 , maximum principal compressive stresses are in the range of −27 MN∕m2 .

281

282

8 Plates

Figure 8.18 Example 8.4. Point A (Figure 8.17b). Principal stress field.

Figure 8.19 Example 8.4. Point A (Figure 8.17b). Rebar stresses [MN∕m2 ].

Typical rebar behaviour is shown with rebar stresses in Figure 8.19. Rebar normal stresses at its upper and lower edge are derived from the Bernoulli beam normal forces and bending moments. Thick line sections connect the respective integration values of an element, thin lines connect neighboured elements. • The rebar collects increasing tension due to bond. Pure tension is disturbed by some bending in the centre area. • The computed bond stress basically reaches the bond strength near the ‘cracked’ centre line. • The rebar yield limit is not reached with maximum loading, indicating a system failure due to concrete failure. The final failure mechanism in the softening range following the peak loading is shown in Figures 8.20 and 8.21.

8.5 Embedded Reinforcement with a Flexible Bond

Figure 8.20 Example 8.4. Point B (Figure 8.17b). Principal stress field.

Figure 8.21 Example 8.4. Point B (Figure 8.17b). Rebar stresses [MN∕m2 ].

• Lateral cracks develop additionally, as is shown by the principal stress field indicating a shear failure. • This is connected with additional intermediate concrete struts supported by the rebar, whereby mobilizing the rebar bending resistance in the laterally cracked regions. The corresponds to the well-known dowel effect of rebars. • The dowel effect is connected to the rebar stress double peak at a lateral crack corresponding to negative downward bending, immediately followed by upward bending (left-hand side values Figure 8.21). • Bond is highly utilised at the plate ends with values near the bond strength in order to transfer stresses from concrete into the rebar. • Everywhere principal concrete stresses remain within the bound prescribed by uniaxial compressive and tensile strength.

283

284

8 Plates

Figure 8.22 Example 8.4. Deformed mesh (scaled by 50).

Finally, the deformed mesh for this stage is shown in Figure 8.22. This points out the final shear failure mechanism of the reinforced plate. Also note the independent discretisations of rebar and concrete with beam elements embedded in the continuum. The rebar evades the full load in the longitudinal direction due to the flexible bond. In the transverse direction, it follows the deformation of the continuum. This approach for an embedded reinforcement with flexible bond covers a wide range of reinforcement types, from long single rebars arranged in a regular pattern, each discretised with a large number of elements, up to a large number of short fibres with random orientation, each discretised with a small number of elements (Häussler-Combe et al. 2020). Nonlinear material laws may be used for embedded reinforcement elements, as well as for the embedding continuum, in addition to a nonlinear bond law. Embedded reinforcement with flexible bond works in 2D and 3D and basically with all types of finite elements, including the SDA approach (Section 7.7).

285

9 Slabs Slabs are plane surface structures like plates. But slabs are exposed to loading normal to their plane in contrast to plates. At a first glance, slabs may be considered as a generalisation of bending beams, where basic assumptions regarding cross-section behaviour and integration of stresses into internal forces are transferred to multiple plane directions. This concerns a range from finite element modelling of slabs, reinforcement design based on linear elastic slab solutions and the lower bound limit state analysis, elasto-plastic solutions for nonlinear behaviour, and, finally, the upper bound limit state analysis.

9.1 Classification The structural types treated up to now are bars, beams, and plates. First of all, these types are characterised by their geometric properties. Two geometric dimensions of bars and beams – height and width – are small compared to the length. One geometric dimension of plates – width or thickness, respectively – is small compared to height and length or span, respectively. Slabs, as a further structural type have height as a small geometric dimension compared to ‘width’ and span. The placement in a geometric frame is shown in Figure 9.1a. The name width is not really appropriate to denote the second long

(a)

(b)

Figure 9.1 (a) Structural types. (b) Coordinate system for slabs.

Computational Structural Concrete – Theory and Applications, Second Edition. Ulrich Häussler-Combe. © 2023 Ernst & Sohn GmbH. Published 2023 by Ernst & Sohn GmbH.

286

9 Slabs

dimension. Actually, it becomes another span in a ground view. Due to their geometric properties, plates and slabs may be classified as plane surface structures, while bars and beams are line structures. The difference between plates and slabs is given with load application: the loading direction for a plate is in-plane, while the loading direction of a slab is normal to the slab plane. A position on the surface area of a slab is determined through coordinates 𝑥, 𝑦; see Figure 9.1b. A position has cross-sections with a 𝑧-direction and a corresponding coordinate. These coordinates form a Cartesian system. In contrast to beams, an indefinite number of cross-sections exists in a position 𝑥, 𝑦, which are characterised by an in-plane direction angle 𝜑 (Figure 9.1). Two cross-sections are regarded as representative: the first with 𝜑 = 0 and a normal in the 𝑥-direction and the second with 𝜑 = π∕2 and normal in the 𝑦-direction.

9.2 Cross-Sectional Behaviour 9.2.1 Kinematics Each structural type is characterised by a kinematic assumption constraining the description of its displacements. The kinematic assumption for slabs is similar to that for beams (Section 4.1.1). ◀

The Bernoulli–Navier hypothesis states that undeformed plane cross-sections of a slab remain plane during a deformation.

A reference axis is chosen for beams as a reference to describe the displacement of plane cross-sections. A plane surface area or reference plane is used for the same purpose with slabs. A coordinate system was defined before (Figure 9.1b). The reference plane is placed in the midst of a slab. Bottom and top coordinates are given by 𝑧1 = −ℎ∕2, 𝑧2 = ℎ∕2 with the slab height ℎ. A synonym for the height of slabs is thickness. The displacements of every material point of a slab with the coordinates 𝑥, 𝑦, 𝑧 are given with the horizontal displacements 𝑢(𝑥, 𝑦, 𝑧) in the 𝑥-direction and 𝑣(𝑥, 𝑦, 𝑧) in the 𝑦-direction, and furthermore with the lateral displacement or deflections 𝑤(𝑥, 𝑦, 𝑧) in the 𝑧-direction. The Bernoulli–Navier hypothesis for slabs is included with the approach 𝑤(𝑥, 𝑦, 𝑧) = 𝑤(𝑥, 𝑦, 0)

= 𝑤(𝑥, 𝑦)

𝑢(𝑥, 𝑦, 𝑧) = 𝑢(𝑥, 𝑦) − 𝑧 𝜙𝑦 = 𝑢(𝑥, 𝑦) − 𝑧 [

𝜕𝑤(𝑥, 𝑦) − 𝛾𝑥 (𝑥, 𝑦)] 𝜕𝑥

𝑣(𝑥, 𝑦, 𝑧) = 𝑣(𝑥, 𝑦) − 𝑧 𝜙𝑥 = 𝑣(𝑥, 𝑦) − 𝑧 [

𝜕𝑤(𝑥, 𝑦) − 𝛾𝑦 (𝑥, 𝑦)] 𝜕𝑦

(9.1)

9.2 Cross-Sectional Behaviour

with cross-section rotation angles 𝜙𝑥 (𝑥, 𝑦), 𝜙𝑦 (𝑥, 𝑦), shear angles 𝛾𝑥 (𝑥, 𝑦), 𝛾𝑦 (𝑥, 𝑦) and displacements 𝑢(𝑥, 𝑦), 𝑣(𝑥, 𝑦), 𝑤(𝑥, 𝑦) of the reference plane. This implies the following properties: • Equation (9.11 ): every material point in a cross-section has the same deflection, but it may change with the reference plane coordinates 𝑥, 𝑦. • Equation (9.12 ): a cross-section with 𝑥-normal rotates by an angle 𝜙𝑦 during a deformation. • Equation (9.13 ): a cross-section with 𝑦-normal rotates by an angle 𝜙𝑥 during a deformation. • Equations (9.12,3 ) decouple the rotations of the cross-sections 𝜙𝑦 , 𝜙𝑥 and the slopes of the reference plane 𝜕𝑤∕𝜕𝑥, 𝜕𝑤∕𝜕𝑦 by 𝛾𝑥 , 𝛾𝑦 𝜙𝑦 =

𝜕𝑤 − 𝛾𝑥 , 𝜕𝑥

𝜙𝑥 =

𝜕𝑤 − 𝛾𝑦 𝜕𝑦

(9.2)

The relation of 𝛾𝑥 , 𝛾𝑥 to shear strains is evident with Eqs. (9.34,5 ). • The case 𝛾𝑥 ≪ 𝜙𝑦 , 𝛾𝑦 ≪ 𝜙𝑥 with the assumption 𝛾𝑥 , 𝛾𝑦 = 0 leads to the Kirchhoff slab, where cross-sections remain rectangular to the reference plane after deformation. The inclusion of shear deformation leads to the Reissner–Mindlin slab. Cross-sections remain plane after deformation but are not rectangular to the reference plane after a deformation for the Reissner–Mindlin slab. Slab kinematics may be considered as an extension of beam kinematics (Eq. (4.3)) in two directions. Thus, slab strains are defined as 𝜖𝑥 =

𝜕𝑢 𝜕𝑥

=

𝜕𝑢 𝜕 2 𝑤 𝜕𝛾 −𝑧 [ 2 − 𝑥] 𝜕𝑥 𝜕𝑥 𝜕𝑥

𝜖𝑦 =

𝜕𝑣 𝜕𝑦

=

𝜕𝑣 𝜕 2 𝑤 𝜕𝛾𝑦 −𝑧 [ 2 − ] 𝜕𝑦 𝜕𝑦 𝜕𝑦

𝜕𝑢 𝜕𝑣 + 𝜕𝑦 𝜕𝑥

=

𝜕𝛾𝑥 𝜕𝛾𝑦 𝜕𝑢 𝜕𝑣 𝜕2 𝑤 + − 𝑧 [2 − − ] 𝜕𝑦 𝜕𝑥 𝜕𝑥𝜕𝑦 𝜕𝑦 𝜕𝑥

𝛾𝑥𝑦 =

𝜕𝑢 𝜕𝑤 + 𝜕𝑧 𝜕𝑥 𝜕𝑣 𝜕𝑤 = + 𝜕𝑧 𝜕𝑦

𝛾𝑥𝑧 = 𝛾𝑦𝑧

𝜕𝑤 + 𝛾𝑥 + 𝜕𝑥 𝜕𝑤 =− + 𝛾𝑦 + 𝜕𝑦

=−

(9.3)

𝜕𝑤 = 𝛾𝑥 𝜕𝑥 𝜕𝑤 = 𝛾𝑦 𝜕𝑦

It should be noted that 𝛾𝑥𝑦 is in plane, while 𝛾𝑥𝑧 , 𝛾𝑦𝑧 are transverse to the plane. The strains of the reference plane are given by 𝜖𝑥 =

𝜕𝑢 , 𝜕𝑥

𝜖𝑦 =

𝜕𝑣 , 𝜕𝑦

𝛾 𝑥𝑦 =

𝜕𝑢 𝜕𝑣 + 𝜕𝑦 𝜕𝑥

(9.4)

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9 Slabs

With the inclusion of shear deformations, curvature is defined as 𝜕 2 𝑤 𝜕𝛾𝑥 − 𝜕𝑥 𝜕𝑥2 𝜕 2 𝑤 𝜕𝛾𝑦 − 𝜅𝑦 = 𝜕𝑦 𝜕𝑦 2 2 𝜕𝛾𝑥 𝜕𝛾𝑦 𝜕 𝑤 − − 𝜅𝑥𝑦 = 2 𝜕𝑥𝜕𝑦 𝜕𝑦 𝜕𝑥 𝜅𝑥 =

(9.5)

The quantities 𝜖 𝑥 , 𝜖 𝑥 , 𝜖 𝑥𝑦 and 𝜅𝑥 , 𝜅𝑦 , 𝜅𝑥𝑦 each form the components of a symmetric second-order tensor defined by a displacement direction and the direction of a reference cross-section. ◀

The variables 𝜖 x , 𝜖 x , 𝜖xy , 𝜅x , 𝜅y , 𝜅xy , 𝛾x , 𝛾y , which depend on x, y only, are chosen as generalised strains for slabs, whereby 𝜖 x , 𝜖x , 𝜖xy indicate the strain of the reference plane, 𝜅x , 𝜅y , 𝜅xy the curvatures of deformed cross-sections, and 𝛾x , 𝛾y the shearing angles of deformed cross-sections relative to the reference plane.

Combining Eqs. (9.3) and (9.5) leads to strain formulations 𝜖𝑥 = 𝜖 𝑥 − 𝑧 𝜅𝑥 ,

𝜖𝑦 = 𝜖𝑦 − 𝑧 𝜅𝑦 ,

𝛾𝑥𝑦 = 𝛾 𝑥𝑦 − 𝑧 𝜅𝑥𝑦

(9.6)

with 𝜖𝑥 , 𝜖𝑦 , 𝛾𝑥𝑦 linearly varying along the slab height with extremal values at the top and bottom of the cross-section and constant 𝛾𝑥𝑧 = 𝛾𝑥 , 𝛾𝑦𝑧 = 𝛾𝑦 along the slab height. Nevertheless, all these strains vary with the reference plane coordinates 𝑥, 𝑦.

9.2.2 Internal Forces Generalised slab strains have to be connected to generalised slab stresses or internal forces to describe the material behaviour.

Figure 9.2 Stresses at the slab element.

9.2 Cross-Sectional Behaviour

At a slab position 𝑥, 𝑦 such internal forces are defined as ℎ∕2

𝑛𝑥 = ∫ 𝜎𝑥 d𝑧 ,

ℎ∕2

ℎ∕2

𝑛𝑦 = ∫ 𝜎𝑦 d𝑧 ,

−ℎ∕2

𝑛𝑥𝑦 = ∫ 𝜎𝑥𝑦 d𝑧

−ℎ∕2

−ℎ∕2

ℎ∕2

ℎ∕2

ℎ∕2

𝑚𝑥 = − ∫ 𝜎𝑥 𝑧 d𝑧 ,

𝑚𝑦 = − ∫ 𝜎𝑦 𝑧 d𝑧 ,

𝑚𝑥𝑦 = − ∫ 𝜎𝑥𝑦 𝑧 d𝑧

−ℎ∕2

−ℎ∕2

−ℎ∕2

ℎ∕2

𝑣𝑥 = ∫ 𝜎𝑥𝑧 d𝑧 ,

ℎ∕2

𝑣𝑦 = ∫ 𝜎𝑦𝑧 d𝑧

−ℎ∕2

−ℎ∕2

(9.7) see Figures 9.2 and 9.3. In analogy to generalised strains, the quantities 𝑛𝑥 , 𝑛𝑦 , 𝑛𝑥𝑦 and 𝑚𝑥 , 𝑚𝑦 , 𝑚𝑥𝑦 each form a symmetric second-order tensor defined by a force direction and the direction of a reference plane. Equation (9.7) is a generalisation of the corresponding relations for beams (Eq. (4.9)), and generalised pairs (𝑛𝑥 , 𝜖 𝑥 ), …, (𝑚𝑥 , 𝜅𝑥 ), … and (𝑣𝑥 , 𝛾𝑥 ), … are conjugate with respect to internal energy. Cauchy stresses 𝜎𝑥 , 𝜎𝑦 , 𝜎𝑥𝑦 , 𝜎𝑥𝑧 , 𝜎𝑦𝑧 depend on 𝜖𝑥 , 𝜖𝑦 , 𝛾𝑥𝑦 , 𝛾𝑥𝑧 , 𝛾𝑦𝑧 through a material law. Biaxial stress–strain relations are at least required in contrast to bars and beams. Such laws may be derived as special cases of multi-axial stress–strain relations (Chapter 6). The corresponding relations between internal forces and generalised slab strains are derived through integration along the slab thickness. This is exemplarily demonstrated for linear elastic behaviour. We consider linear isotropic elasticity under plane stress conditions (Eq. (2.45)). The combination with Eq. (9.6) yields normal stresses 𝜎𝑥 =

𝐸 (𝜖𝑥 + 𝜈 𝜖𝑦 ) 1 − 𝜈2

𝜎𝑦

𝐸 1 − 𝜈2 𝐸 = 1 − 𝜈2

=

[

] (𝜖𝑥 + 𝜈𝜖 𝑦 ) − 𝑧 (𝜅𝑥 + 𝜈 𝜅𝑦 )

[

] (𝜖𝑦 + 𝜈𝜖 𝑥 ) − 𝑧 (𝜅𝑦 + 𝜈 𝜅𝑥 )

(9.8)

and as a shear stress in planes parallel to the reference plane 𝜎𝑥𝑦 = 𝐺 𝛾𝑥𝑦 = 𝐺 (𝛾 𝑥𝑦 − 𝑧 𝜅𝑥𝑦 )

(9.9)

with 𝐺 = 𝐸∕2(1 + 𝜈). The out-of-plane shear stress components are derived from Eq. (6.23) 𝜎𝑥𝑧 = 𝐺 𝛾𝑥𝑧 = 𝐺 𝛾𝑥 ,

𝜎𝑦𝑧 = 𝐺 𝛾𝑦𝑧 = 𝐺 𝛾𝑦

(9.10)

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9 Slabs

Stresses from Eqs. (9.8)–(9.10) are used to determine internal forces from Eq. (9.7), leading to ℎ∕2

𝐸 𝑛𝑥 = (𝜖𝑥 + 𝜈 𝜖𝑦 ) ∫ d𝑧 1 − 𝜈2

= 𝐾𝑛 (𝜖𝑥 + 𝜈 𝜖𝑦 )

𝑛𝑦

= 𝐾𝑛 (𝜖𝑦 + 𝜈 𝜖𝑥 )

−ℎ∕2

ℎ∕2

𝑛𝑥𝑦 =

𝐸 ∫ d𝑧 𝛾 2(1 + 𝜈) 𝑥𝑦

= (1 − 𝜈) 𝐾𝑛

𝛾 𝑥𝑦 2

−ℎ∕2

ℎ∕2

𝐸 (𝜅𝑥 + 𝜈 𝜅𝑦 ) ∫ 𝑧2 d𝑧 𝑚𝑥 = 1 − 𝜈2

= 𝐾 (𝜅𝑥 + 𝜈 𝜅𝑦 )

𝑚𝑦

= 𝐾 (𝜅𝑦 + 𝜈 𝜅𝑥 )

−ℎ∕2

ℎ∕2

𝑚𝑥𝑦 =

𝜅𝑥𝑦 𝐸 𝜅𝑥𝑦 ∫ 𝑧2 d𝑧 = (1 − 𝜈) 𝐾 2 2(1 + 𝜈)

= (1 − 𝜈) 𝐾

−ℎ∕2

(9.11)

𝜕2 𝑤 𝜕𝑥𝜕𝑦

ℎ∕2

𝑣𝑥 = 𝛼𝐺 𝛾𝑥 ∫ d𝑧

= 𝛼 𝐺ℎ 𝛾𝑥

−ℎ∕2

𝑣𝑦

= 𝛼 𝐺ℎ 𝛾𝑦 ℎ∕2

with respect to ∫−ℎ∕2 𝑧 d𝑧 = 0 with the slab stiffness 𝐾𝑛 =

𝐸ℎ , 1 − 𝜈2

𝐾=

𝐸 ℎ3 12(1 − 𝜈2 )

(9.12)

with Young’s modulus 𝐸, Poisson’s ratio 𝜈, and the shear correction factor 𝛼 (Section 4.1.2). This is again an extension of the corresponding beam relations (Eqs. (4.9)–(4.13)).

9.3 Equilibrium of Slabs 9.3.1 Strong Equilibrium Equilibrium has to be regarded as the third item besides kinematic compatibility and material laws. The symbols 𝑢, 𝑣 will be used for the displacements of the reference plane instead of 𝑢, 𝑣 in the following to simplify the notation. We consider a position 𝑥, 𝑦 of the reference plane with a distributed loading 𝑝𝑥 , 𝑝𝑦 , 𝑝𝑧 . Other descriptions use an opposite sign convention for the 𝑧-coordinate and the corresponding displacement 𝑤. This reverses the sign in the moment–curvature relations. The strong differential formulation of static equilibrium for an infinitesimal

9.3 Equilibrium of Slabs

Figure 9.3 Slab equilibrium.

section of a slab is considered first. The internal forces are shown in Figure 9.3. Equilibrium of normal forces in the 𝑥 and 𝑦-directions is given by 𝜕𝑛𝑦

𝜕𝑛𝑥 𝜕𝑛𝑥𝑦 + + 𝑝𝑥 = 0 , 𝜕𝑥 𝜕𝑦

𝜕𝑦

+

𝜕𝑛𝑥𝑦 𝜕𝑥

+ 𝑝𝑦 = 0

(9.13)

equilibrium of shear forces in the 𝑧-direction by 𝜕𝑣𝑥 𝜕𝑣𝑦 + + 𝑝𝑧 = 0 𝜕𝑥 𝜕𝑦

(9.14)

and equilibrium of moments with 𝜕𝑚𝑥 𝜕𝑚𝑥𝑦 + + 𝑣𝑥 = 0 , 𝜕𝑥 𝜕𝑦

𝜕𝑚𝑦 𝜕𝑦

+

𝜕𝑚𝑥𝑦 𝜕𝑥

+ 𝑣𝑦 = 0

(9.15)

These equilibrium conditions are independent of kinematic assumptions and material laws. The case of plates is included as a special case with 𝑝𝑧 = 0. This leaves Eqs. (9.13) only. We consider the linear elastic Kirchhoff slab to connect to a well-known theory. The combination of Eqs. (9.15) and (9.14) yields 𝜕 2 𝑚𝑥𝑦 𝜕 2 𝑚𝑦 𝜕 2 𝑚𝑥 + 2 = 𝑝𝑧 + 𝜕𝑥𝜕𝑦 𝜕𝑥2 𝜕𝑦 2

(9.16)

Shear deformations are neglected. Thus, Eq. (9.5) leads to 𝜅𝑥 =

𝜕2 𝑤 , 𝜕𝑥2

𝜅𝑦 =

𝜕2 𝑤 , 𝜕𝑦 2

𝜅𝑥𝑦 = 2

𝜕2 𝑤 𝜕𝑥𝜕𝑦

(9.17)

Using this for Eqs. (9.114-6 ) results in 𝜕 2 𝜅𝑦 𝜕 2 𝑚𝑥 𝜕 2 𝜅𝑥 = 𝐾 + 𝜈 ) ( 𝜕𝑥2 𝜕𝑥2 𝜕𝑥2 𝜕 2 𝑚𝑦 𝜕𝑦 2 𝜕 2 𝑚𝑥𝑦 𝜕𝑥𝜕𝑦

=𝐾 (

𝜕 2 𝜅𝑦 𝜕𝑦 2

= (1 − 𝜈)

+𝜈

𝜕 2 𝜅𝑥 ) 𝜕𝑦 2

2 𝐾 𝜕 𝜅𝑥𝑦 2 𝜕𝑥𝜕𝑦

=𝐾 (

𝜕4 𝑤 𝜕4 𝑤 +𝜈 2 2) 4 𝜕𝑥 𝜕𝑥 𝜕𝑦

=𝐾 (

𝜕4 𝑤 𝜕4 𝑤 +𝜈 2 2) 4 𝜕𝑦 𝜕𝑦 𝜕𝑥

= (1 − 𝜈) 𝐾

𝜕4 𝑤 𝜕𝑥2 𝜕𝑦 2

(9.18)

291

292

9 Slabs

The combination of Eqs. (9.16) and (9.18) finally gives the well-known relation 𝑝𝑧 𝜕4 𝑤 𝜕4 𝑤 𝜕4 𝑤 +2 2 2 + = 4 4 𝐾 𝜕𝑥 𝜕𝑥 𝜕𝑦 𝜕𝑦

(9.19)

with the stiffness 𝐾 from Eq. (9.12). Boundary conditions are regarded with appro𝜕𝑤

priate combinations of 𝑤, tions like 𝑚𝑥 = 𝐾 (

𝜕2 𝑤 𝜕𝑥 2

,

𝜕2 𝑤 𝜕3 𝑤

𝜕𝑥 𝜕𝑥 2 𝜕2 𝑤

+𝜈

𝜕𝑦 2

,

𝜕𝑥 3

. Their physical meaning comes from rela-

), 𝑣𝑥 = − (

𝜕𝑚𝑥 𝜕𝑥

+

𝜕𝑚𝑥𝑦 𝜕𝑦

) and similar relations for the

other internal forces (Eqs. (9.11)). The determination of analytical solutions for the partial differential equation of fourth order Eq. (9.19) is challenging from a mathematical point of view. A comprehensive discussion is given in Girkmann (1974).

9.3.2 Weak Equilibrium Equilibrium again has to be reformulated as weak integral equilibrium. To start with, the kinematic relations Eqs. (9.2) and (9.5) are reconsidered as 𝜕𝑤 − 𝛾𝑥 → 𝜕𝑥 𝜕𝑤 𝜙𝑥 = − 𝛾𝑦 → 𝜕𝑦 𝜕 2 𝑤 𝜕𝛾𝑥 𝜅𝑥 = − = 𝜕𝑥 𝜕𝑥2 𝜕 2 𝑤 𝜕𝛾𝑦 − = 𝜅𝑦 = 𝜕𝑦 𝜕𝑦 2 𝜕𝜙𝑦 𝜕𝜙𝑥 + 𝜅𝑥𝑦 = 𝜕𝑦 𝜕𝑥 𝜙𝑦 =

𝜕𝑤 − 𝜙𝑦 𝜕𝑥 𝜕𝑤 𝛾𝑦 = − 𝜙𝑥 𝜕𝑦 𝜕𝜙𝑦 𝛾𝑥 =

(9.20)

𝜕𝑥 𝜕𝜙𝑥 𝜕𝑦

An equivalent to the strong differential equilibrium formulation (Eqs. (9.13)–(9.15)) is given by the following integral formulation for a slab of area 𝐴 with a potential coupling of normal forces and moments ∫ 𝛿𝑢 ( 𝐴

+ ∫ 𝛿𝑣 ( 𝐴

𝜕𝑛𝑥 𝜕𝑛𝑥𝑦 + + 𝑝𝑥 ) d𝐴 𝜕𝑥 𝜕𝑦 𝜕𝑛𝑦 𝜕𝑦

+ ∫ 𝛿𝜙𝑦 ( 𝐴

+

𝜕𝑛𝑥𝑦 𝜕𝑥

+ 𝑝𝑦 ) d𝐴 + ∫ 𝛿𝑤 ( 𝐴

𝜕𝑣𝑥 𝜕𝑣𝑦 + + 𝑝𝑧 ) d𝐴 𝜕𝑥 𝜕𝑦

𝜕𝑚𝑦 𝜕𝑚𝑥𝑦 𝜕𝑚𝑥 𝜕𝑚𝑥𝑦 + + 𝑣𝑥 ) d𝐴 + ∫ 𝛿𝜙𝑥 ( + + 𝑣𝑦 ) d𝐴 = 0 𝜕𝑥 𝜕𝑦 𝜕𝑦 𝜕𝑥 𝐴

(9.21) with test functions 𝛿𝑢, 𝛿𝑣, 𝛿𝑤, 𝛿𝜙𝑦 , 𝛿𝜙𝑥 . This is a generalisation of the approach for beams (Section 4.2). Terms with derivatives of internal forces are partially integrat-

9.3 Equilibrium of Slabs

ed, leading to ∫ 𝛿𝑢 ( 𝐴

𝜕𝑛𝑥 𝜕𝑛𝑥𝑦 𝜕𝛿𝑢 𝜕𝛿𝑢 + 𝑛 + 𝑛 ) d𝐴 ) d𝐴 = − ∫ ( 𝜕𝑥 𝜕𝑦 𝜕𝑥 𝑥 𝜕𝑦 𝑥𝑦 𝐴

+ ∮ 𝛿𝑢(𝑛𝑥 𝑒𝑥 + 𝑛𝑥𝑦 𝑒𝑦 ) d𝑆 𝑆

∫ 𝛿𝑣 (

𝜕𝑛𝑦

𝐴

𝜕𝑦

+

𝜕𝑛𝑥𝑦 𝜕𝑥

) d𝐴 = − ∫ ( 𝐴

𝜕𝛿𝑣 𝜕𝛿𝑣 𝑛 + 𝑛 ) d𝐴 𝜕𝑦 𝑦 𝜕𝑥 𝑥𝑦

+ ∮ 𝛿𝑣(𝑛𝑥𝑦 𝑒𝑥 + 𝑛𝑦 𝑒𝑦 ) d𝑆 𝑆

𝜕𝑣𝑦 𝜕𝑣 𝜕𝛿𝑤 𝜕𝛿𝑤 ∫ 𝛿𝑤 ( 𝑥 + 𝑣𝑥 + 𝑣 ) d𝐴 ) d𝐴 = − ∫ ( 𝜕𝑥 𝜕𝑦 𝜕𝑥 𝜕𝑦 𝑦 𝐴

𝐴

+ ∮ 𝛿𝑤(𝑣𝑥 𝑒𝑥 + 𝑣𝑦 𝑒𝑦 ) d𝑆 𝑆

∫ 𝛿𝜙𝑦 ( 𝐴

𝜕𝛿𝜙𝑦 𝜕𝛿𝜙𝑦 𝜕𝑚𝑥 𝜕𝑚𝑥𝑦 + + 𝑣𝑥 ) d𝐴 = − ∫ ( 𝑚 + 𝑚𝑥𝑦 − 𝛿𝜙𝑦 𝑣𝑥 ) d𝐴 𝜕𝑥 𝜕𝑦 𝜕𝑥 𝑥 𝜕𝑦 𝐴

+ ∮ 𝛿𝜙𝑦 (𝑚𝑥 𝑒𝑥 + 𝑚𝑥𝑦 𝑒𝑦 ) d𝑆 𝑆

∫ 𝛿𝜙𝑥 ( 𝐴

𝜕𝑚𝑦 𝜕𝑦

+

𝜕𝑚𝑥𝑦 𝜕𝑥

+ 𝑣𝑦 ) d𝐴 = − ∫ ( 𝐴

𝜕𝛿𝜙𝑥 𝜕𝛿𝜙𝑥 𝑚𝑦 + 𝑚𝑥𝑦 − 𝛿𝜙𝑥 𝑣𝑦 ) d𝐴 𝜕𝑦 𝜕𝑥

+ ∮ 𝛿𝜙𝑥 (𝑚𝑥𝑦 𝑒𝑥 + 𝑚𝑦 𝑒𝑦 ) d𝑆

(9.22)

𝑆

with line integrals ∮ along the slab boundary 𝑆 and a unit normal vector e with components 𝑒𝑥 , 𝑒𝑦 along the boundary. Combining Eqs. (9.21) and (9.22) yields ( ) ∫ 𝛿𝜖𝑥 𝑛𝑥 +𝛿𝜖𝑦 𝑛𝑦 +𝛿𝛾𝑥𝑦 𝑛𝑥𝑦 +𝛿𝜅𝑥 𝑚𝑥 +𝛿𝜅𝑦 𝑚𝑦 +𝛿𝜅𝑥𝑦 𝑚𝑥𝑦 +𝛿𝛾𝑥 𝑣𝑥 +𝛿𝛾𝑦 𝑣𝑦 d𝐴 𝐴

( ) = ∫ 𝛿𝑢 𝑝𝑥 + 𝛿𝑣 𝑝𝑦 + 𝛿𝑤 𝑝𝑧 d𝐴 𝐴

[ + ∮ 𝛿𝑢(𝑛𝑥 𝑒𝑥 + 𝑛𝑥𝑦 𝑒𝑦 ) + 𝛿𝑣(𝑛𝑥𝑦 𝑒𝑥 + 𝑛𝑦 𝑒𝑦 ) + 𝛿𝑤(𝑣𝑥 𝑒𝑥 + 𝑣𝑦 𝑒𝑦 ) 𝑆

] +𝛿𝜙𝑦 (𝑚𝑥 𝑒𝑥 + 𝑚𝑥𝑦 𝑒𝑦 ) + 𝛿𝜙𝑥 (𝑚𝑥𝑦 𝑒𝑥 + 𝑚𝑦 𝑒𝑦 ) d𝑆

(9.23)

293

294

9 Slabs

with virtual strains – with overbars omitted – 𝛿𝜖𝑥 =

𝜕𝛿𝑢 , 𝜕𝑥

𝛿𝜖𝑦 =

𝜕𝛿𝑣 , 𝜕𝑦

𝛿𝛾𝑥𝑦 =

𝜕𝛿𝑢 𝜕𝛿𝑣 + 𝜕𝑦 𝜕𝑥

(9.24)

virtual curvatures 𝛿𝜅𝑥 =

𝜕𝛿𝜙𝑦 𝜕𝑥

𝛿𝜅𝑦 =

,

𝜕𝛿𝜙𝑥 , 𝜕𝑦

𝜕𝛿𝜙𝑦

𝛿𝜅𝑥𝑦 =

𝜕𝑦

+

𝜕𝛿𝜙𝑥 𝜕𝑥

(9.25)

and virtual shear deformations 𝛿𝛾𝑥 =

𝜕𝛿𝑤 − 𝛿𝜙𝑦 , 𝜕𝑥

𝛿𝛾𝑦 =

𝜕𝛿𝑤 − 𝛿𝜙𝑥 𝜕𝑦

(9.26)

A generalizing matrix notation of Eq. (9.23) is obtained by ∫ 𝛿𝝐 T ⋅ 𝝈 d𝐴 = ∫ 𝛿uT ⋅ p d𝐴 + ∮ 𝛿UT ⋅ t d𝑆 𝐴

𝐴

(9.27)

𝑆

with ( 𝝐 = 𝜖𝑥 ( 𝝈 = 𝑛𝑥 ( u= 𝑢 ( p = 𝑝𝑥 ( U = 𝑢𝑠 ( t = 𝑛𝑠𝑥

𝜖𝑦

𝛾𝑥𝑦

𝑛𝑦 𝑣

𝜅𝑥

𝑛𝑥𝑦 𝑤

𝑝𝑦 𝑣𝑠 𝑛𝑠𝑦

𝑚𝑥 𝜙𝑥

𝜙𝑦 𝑝𝑧

𝑤𝑠

𝜅𝑦

0

𝑚𝑦

)T

𝛾𝑥

𝑚𝑥𝑦

𝛾𝑦 𝑣𝑥

)T 𝑣𝑦

)T

(9.28)

)T 0

𝜙𝑠𝑦

𝑣𝑠

𝜅𝑥𝑦

𝑚𝑠𝑥

𝜙𝑠𝑥

)T

𝑚𝑠𝑦

)T

with the displacement boundary values 𝑢𝑠 , 𝑣𝑠 , 𝑤𝑠 , 𝜙𝑠𝑦 , 𝜙𝑠𝑥 and forces 𝑛𝑠𝑥 = 𝑛𝑥 𝑒𝑥 + 𝑛𝑥𝑦 𝑒𝑦 ,

𝑛𝑠𝑦 = 𝑛𝑥𝑦 𝑒𝑥 + 𝑛𝑦 𝑒𝑦 ,

𝑚𝑠𝑥 = 𝑚𝑥 𝑒𝑥 + 𝑚𝑥𝑦 𝑒𝑦 ,

𝑚𝑠𝑦 = 𝑚𝑥𝑦 𝑒𝑥 + 𝑚𝑦 𝑒𝑦

𝑣𝑠 = 𝑣𝑥 𝑒 𝑥 + 𝑣𝑦 𝑒 𝑦

(9.29)

at boundaries. A coupling of normal forces and moments – as basically arises with reinforced cracked concrete (Section 4.1.3.2) – occurs with a dependence of normal forces on strains and additionally curvatures and also of moments on curvature and strains.

9.3.3 Decoupling A decoupling is generally applied for plane surface structures. In the case that normal forces do not depend on curvature, and moments do not depend on strains –

9.3 Equilibrium of Slabs

e.g. for a linear elastic material behaviour – weak equilibrium is formulated independently for Eq. (9.13) and for Eqs. (9.14) and (9.15). Taking only normal forces into consideration reduces Eq. (9.28) to )T ( 𝝐 = 𝜖𝑥 𝜖𝑦 𝛾𝑥𝑦 , ( )T u= 𝑢 𝑣 , )T ( U = 𝑢 𝑠 𝑣𝑠 ,

( 𝝈 = 𝑛𝑥 ( p = 𝑝𝑥 ( t = 𝑛𝑠𝑥

𝑛𝑦 𝑝𝑦

)T

𝑛𝑠𝑦

𝑛𝑥𝑦

)T (9.30)

)T

to be used for Eq. (9.27). This describes plates with biaxial plane strain or plane stress. Taking only bending into consideration reduces Eq. (9.28) to )T ( 𝝐 = 𝜅𝑥 𝜅𝑦 𝜅𝑥𝑦 𝛾𝑥 𝛾𝑦 , ) ( u = 𝑤 𝜙𝑦 𝜙𝑥 , )T ( U = 𝑤𝑠 𝜙𝑠𝑦 𝜙𝑠𝑥 ,

)T ( 𝝈 = 𝑚𝑥 𝑚𝑦 𝑚𝑥𝑦 𝑣𝑥 𝑣𝑦 ) ( p = 𝑝𝑧 0 0 )T ( t = 𝑣𝑠 𝑚𝑠𝑥 𝑚𝑠𝑦

(9.31)

to be used for Eq. (9.27). These relations correspond to the Reissner–Mindlin slab. It is characterised by decoupling the deflection 𝑤 from the slopes 𝜙𝑦 , 𝜙𝑥 by the shear deformation 𝛾𝑥 , 𝛾𝑦 (Eq. (9.20)). Thus, 𝑤, 𝜙𝑦 , 𝜙𝑥 may be interpolated as independent displacement variables within the finite element approach. As a further consequence, 𝐶 0 -continuity is sufficient for the interpolation of 𝑤, 𝜙𝑦 , 𝜙𝑥 . A common approach is given with the Kirchhoff slab, which disregards shear de𝜕𝑤 𝜕𝑤 formations. It has assumptions = 𝜙𝑦 → 𝛾𝑥 = 0 and = 𝜙𝑥 → 𝛾𝑦 = 0 and is 𝜕𝑥

𝜕𝑦

a special case of the Reissner–Mindlin slab. Thus, Eq. (9.31) is further reduced to )T ( 𝝐 = 𝜅𝑥 𝜅𝑦 𝜅𝑥𝑦 , ) ( u = 𝑤 𝜙𝑦 𝜙𝑥 )T ( U = 𝑤𝑠 𝜙𝑠𝑦 𝜙𝑠𝑥

)T ( 𝝈 = 𝑚𝑥 𝑚𝑦 𝑚𝑥𝑦 ) ( p = 𝑝𝑧 0 0 )T ( t = 𝑣𝑠 𝑚𝑠𝑥 𝑚𝑠𝑦

(9.32)

to be used for Eq. (9.27). The relations between curvatures and deflections are obtained by 𝜅𝑥 =

𝜕2 𝑤 , 𝜕𝑥2

𝜅𝑦 =

𝜕2 𝑤 , 𝜕𝑦 2

𝜅𝑥𝑦 = 2

𝜕2 𝑤 𝜕𝑥𝜕𝑦

(9.33)

according to Eq. (9.5). 𝐶 1 -continuity is required for finite element interpolation with test and trial functions in analogy to Bernoulli beams (Section 4.3.2). Finally, a linear elastic material behaviour (Eqs. (9.114-6 )) is assumed. Thus, the relation between generalised stresses and generalised strains for the Kirchhoff slab is given with

𝝈=C⋅𝝐 ,

⎡1 ⎢ C = 𝐾 ⎢𝜈 ⎢ 0 ⎣

𝜈 1 0

0 ⎤ ⎥ 0 ⎥, 1−𝜈 ⎥ 2 ⎦

𝐾=

𝐸 ℎ3 12(1 − 𝜈2 )

(9.34)

295

296

9 Slabs

with Young’s modulus 𝐸, Poisson’s ratio 𝜈, and the slab height ℎ. The set of Eqs. (9.27), (9.32), and (9.34) in the format of weak equilibrium is equivalent to the partial differential equation for the Kirchhoff slab Eq. (9.19). But the former allows for arbitrary geometries and boundary conditions in combination with numerical methods, while the latter is restricted to relatively simple cases due the mathematical challenge it involves.

9.4 Reinforced Concrete Cross-Sections First a general approach is developed, which is later restricted again with regard to the slab theory. Nonlinear behaviour and limited strength of concrete have to be considered once more. The uniaxial case treated in Section 4.1.3 is to be generalised for the biaxial case. A layer model is applied in a slab position 𝑥, 𝑦 with respect to cross-sections. The slab thickness is subdivided into stacked layers, see Figure 9.4. The strain of each layer is determined by the kinematic relations Eqs. (9.3). The stress of each layer is derived from strains with the following assumptions: • The in-plane reaction – stresses from 𝜖𝑥 , 𝜖𝑦 , 𝛾𝑥𝑦 – is decoupled from vertical shear stresses resulting from 𝛾𝑥𝑧 , 𝛾𝑦𝑧 , which corresponds to the approach for beams (Sections 4.1.3.1, 4.4.4). • Each layer is regarded as a plate with a biaxial behaviour. Thus, a biaxial material law can be used. • Each layer has its own material law that allows us to distinguish between concrete layers and reinforcement layers. This implies a rigid bond due to the Bernoulli– Navier hypothesis for slab cross-sections (Eqs. (9.1)). • With respect to concrete, a first approach to model cracks in a biaxial setting is described in Section 8.2. This includes limited tensile strength, the Rankine crack criterion, smeared cracks, crack orientation, crack width, crack tractions, and softening. • Differently to plates cracking states will generally vary for different layers along the slab thickness in a slab position. This concerns the event of cracking itself but, furthermore, actual values for crack orientation and crack width. • With the reinforcement, smeared layers or reinforcement sheets are appropriate as described in Section 8.3.

Figure 9.4 Layer model.

9.4 Reinforced Concrete Cross-Sections

This approach yields stresses 𝜎𝑥 , 𝜎𝑦 , 𝜎𝑥𝑦 for every layer in a slab position 𝑥, 𝑦. Internal forces 𝑛𝑥 , 𝑛𝑦 , 𝑛𝑥𝑦 and 𝑚𝑥 , 𝑚𝑦 , 𝑚𝑥𝑦 are determined through integration according to Eqs. (9.7). One-dimensional integration has to be performed numerically with schemes like Gauss, Simpson, Newton–Cotes, or Labatto. But a layer is not capable of resisting transverse shear with stress components 𝜎𝑥𝑧 , 𝜎𝑦𝑧 . Transverse shear for reinforced concrete is described for beams in Section 4.4.4. With respect to the transverse shear behaviour of slabs, a substantially better approach than the strut-and-tie model for beam shear is not yet available. This leads to a form adopted from Eq. (9.11) 𝑣𝑥 = 𝛼 𝐺ℎ 𝛾𝑥 ,

𝑣𝑦 = 𝛼 𝐺ℎ 𝛾𝑦 ,

𝐺=

𝐸𝑐 2(1 + 𝜈)

(9.35)

with a shear modulus 𝐺, a reduction factor 𝛼, and the initial values of Young’s modulus 𝐸𝑐 and Poisson’s ratio 𝜈 of the respective concrete. The reduction factor may be chosen with 𝛼 = 0.5 according to Eq. (4.132). A tangential material stiffness matrix might be derived in analogy with the tangential material stiffness for reinforced concrete beams (Eqs. (4.43)–(4.47)). The general form for an incremental internal forces–generalised strain relation is given by ⎛ 𝑛̇ 𝑥 ⎞ ⎡𝐶𝑇11 ⎜ ⎟ ⎢ 𝐶𝑇21 ⎜ 𝑛̇ 𝑦 ⎟ ⎢ ⎢ ⎜ 𝑛̇ 𝑥𝑦 ⎟ ⎢𝐶𝑇31 ⎜ ⎟ ⎢ 𝐶𝑇41 ⎜ 𝑚̇ 𝑥 ⎟ = ⎢ ⎢ ⎜ 𝑚̇ 𝑦 ⎟ ⎢𝐶𝑇51 ⎜ ⎟ ⎢ 𝐶𝑇61 ⎜𝑚̇ 𝑥𝑦 ⎟ ⎢ ⎢ ⎜ 𝑣̇ 𝑥 ⎟ ⎢ 0 ⎜ ⎟ ⎢ 𝑣̇ 0 ⎝ 𝑦 ⎠ ⎣

𝐶𝑇12

𝐶𝑇13

𝐶𝑇14

𝐶𝑇15

𝐶𝑇16

0

𝐶𝑇22

𝐶𝑇23

𝐶𝑇24

𝐶𝑇25

𝐶𝑇26

0

𝐶𝑇32

𝐶𝑇33

𝐶𝑇34

𝐶𝑇35

𝐶𝑇36

0

𝐶𝑇42

𝐶𝑇43

𝐶𝑇44

𝐶𝑇45

𝐶𝑇46

0

𝐶𝑇52

𝐶𝑇53

𝐶𝑇54

𝐶𝑇55

𝐶𝑇56

0

𝐶𝑇62

𝐶𝑇63

𝐶𝑇64

𝐶𝑇65

𝐶𝑇66

0

0

0

0

0

0

𝐶𝑇77

0

0

0

0

0

0

0 ⎤ ⎛ 𝜖̇ 𝑥 ⎞ ⎥ ⎜ ⎟ 0 ⎥ ⎜ 𝜖̇ 𝑦 ⎟ ⎥ 0 ⎥ ⎜𝛾̇ 𝑥𝑦 ⎟ ⎥ ⎜ ⎟ 0 ⎥ ⎜ 𝜅̇ 𝑥 ⎟ ⎥⋅ 0 ⎥ ⎜ 𝜅̇ 𝑦 ⎟ ⎥ ⎜ ⎟ 0 ⎥ ⎜𝜅̇ 𝑥𝑦 ⎟ ⎥ 0 ⎥ ⎜ 𝛾̇ 𝑥 ⎟ ⎥ ⎜ ⎟ 𝛾̇ 𝐶𝑇88 ⎦ ⎝ 𝑦⎠ (9.36)

within the current framework. This includes a coupling between internal normal forces and internal moments, which basically occurs for cracked reinforced concrete cross-sections (Section 4.1.3.2). The coefficients 𝐶𝑇𝑖𝑗 of the tangential material stiffness matrix have to be derived each with numerical integration. The above approach is obviously elaborate due to the full coupling of normal force and moment quantities. As will be shown later, this effect can be represented much more simply and conclusively within the framework of the continuum-based shell theory (Chapter 10) with a specific layer model. This circumvents the elaborate explicit formulation of relations between normal forces and moments depending on reference plane strains and curvatures. In the following, we limit ourselves to the slab theory in its known and proven form, neglecting the coupling effect. 1) This is done with the Kirchhoff slab described 1) With respect to beams, the coupling of bending moments and normal forces is demonstrated in Example 4.2. It is shown that the activation of normal forces in the case of transverse loading depends on

297

298

9 Slabs

by Eqs. (9.27) and (9.32) because of its practical importance. The internal forces– generalised strain relations Eq. (9.36) reduce to ⎛ 𝑚̇ 𝑥 ⎞ ⎡𝐶𝑇11 ⎜ 𝑚̇ ⎟ = ⎢ ⎢𝐶𝑇21 𝑦 ⎜ ⎟ ⎢ ⎝𝑚̇ 𝑥𝑦 ⎠ ⎣𝐶𝑇31

𝐶𝑇12 𝐶𝑇22 𝐶𝑇32

𝐶𝑇13 ⎤ ⎛ 𝜅̇ 𝑥 ⎞ ⎥ 𝐶𝑇23 ⎥ ⋅ ⎜ 𝜅̇ 𝑦 ⎟ ⎥ ⎜ ⎟ 𝐶𝑇33 ⎝𝜅̇ 𝑥𝑦 ⎠ ⎦

(9.37)

On the other hand, this is a generalisation of Eq. (9.34).

9.5 Slab Elements 9.5.1 Area Coordinates Equations (9.32) and (9.33) define the variables of the Kirchhoff slab. The definition of generalised strains (Eq. (9.33)) requires a 𝐶 1 -continuity of the interpolation functions for 𝑤 to ensure compatibility and integrability of Eq. (9.27). This corresponds to the continuity of slopes 𝜙𝑦 , 𝜙𝑥 and some more constraints along inter-element boundaries. Such a requirement is not trivial to fulfil within the two-dimensional setting (Zienkiewicz and Taylor 1991, 1.1,1.2). Thus, the continuity requirement has to be relaxed, which leads to non-conforming element formulations. Three independent interpolation variables are given by 𝑤, 𝜙𝑦 , 𝜙𝑥 , (Eq. (9.322 )). Triangular elements are a first choice to approach continuity requirements. Usage of area coordinates 𝐿𝑖 , 𝑖 = 1, … , 3 instead of Cartesian coordinates is appropriate to mark a position within a triangle. They are defined with 𝑥 = 𝐿1 𝑥1 + 𝐿2 𝑥2 + 𝐿3 𝑥3 ,

𝑦 = 𝐿1 𝑦1 + 𝐿2 𝑦2 + 𝐿3 𝑦3 ,

𝐿1 + 𝐿2 + 𝐿3 = 1 (9.38)

with the coordinates 𝑥𝐼 , 𝑦𝐼 of a node 𝐼 = 1, … , 3. The values of 𝐿𝑖 denote relative areas; see Figure 9.5. The definition leads to 𝐿1 = 𝑎1 + 𝑏1 𝑥 + 𝑐1 𝑦 ,

𝐿2 = 𝑎2 + 𝑏2 𝑥 + 𝑐2 𝑦 ,

𝐿3 = 𝑎3 + 𝑏3 𝑥 + 𝑐3 𝑦

(9.39)

with 𝑦3 𝑥2 − 𝑦2 𝑥3 , 2Δ 𝑦1 𝑥3 − 𝑦3 𝑥1 , 𝑎2 = 2Δ 𝑦2 𝑥1 − 𝑦1 𝑥2 𝑎3 = , 2Δ

𝑎1 =

𝑦2 − 𝑦3 , 2Δ 𝑦3 − 𝑦1 𝑏2 = , 2Δ 𝑦1 − 𝑦2 𝑏3 = , 2Δ 𝑏1 =

𝑥3 − 𝑥2 2Δ 𝑥1 − 𝑥3 𝑐2 = 2Δ 𝑥2 − 𝑥1 𝑐3 = 2Δ 𝑐1 =

(9.40)

the displacement boundary conditions. In most cases, restrained horizontal displacements – characterised by moderate compression zone heights – lead to a slight normal compression of cracked reinforced cross-sections. This is favorable with respect to the bending bearing capacity.

9.5 Slab Elements

Figure 9.5 Triangular element and area coordinates.

and || ||1 || 1 Δ = det |||1 2 || ||1 |

𝑥1 𝑥2 𝑥2

| 𝑦1 ||| || 𝑦3 ||| = area of triangle 123 || 𝑦3 |||

(9.41)

1 (𝑦 𝑥 + 𝑦3 𝑥2 + 𝑦2 𝑥1 − 𝑦2 𝑥3 − 𝑦3 𝑥1 − 𝑦1 𝑥2 ) 2 1 3 Area coordinates of nodes are given by =

𝑥 = 𝑥1 , 𝑦 = 𝑦1

→

𝐿1 = 1, 𝐿2 = 0, 𝐿3 = 0

𝑥 = 𝑥2 , 𝑦 = 𝑦2

→

𝐿1 = 0, 𝐿2 = 1, 𝐿3 = 0

𝑥 = 𝑥3 , 𝑦 = 𝑦3

→

𝐿1 = 0, 𝐿2 = 0, 𝐿3 = 1

(9.42)

and derivatives of area coordinates with respect to Cartesian coordinates 𝜕𝐿𝑖 = 𝑏𝑖 , 𝜕𝑥

𝜕𝐿𝑖 = 𝑐𝑖 𝜕𝑦

(9.43)

with 𝑏𝑖 , 𝑐𝑖 according to Eq. (9.40).

9.5.2 Triangular Kirchhoff Slab Element We consider a triangular element with three corner nodes and nine nodal degrees of freedom 𝑤𝑖 , 𝜙𝑖𝑥 , 𝜙𝑖𝑦 , 𝑖 = 1, 2, 3. An interpolation approach for the deflections 𝑤 depending on area coordinates is assumed with 𝑤 = 𝛼1 𝐿1 + 𝛼2 𝐿2 + 𝛼3 𝐿3 + 𝛼4 𝐿1 𝐿2 + 𝛼5 𝐿2 𝐿3 + 𝛼6 𝐿3 𝐿1 + 𝛼7 𝐿12 𝐿2 + 𝛼8 𝐿22 𝐿3 + 𝛼9 𝐿32 𝐿1

(9.44)

leading to 𝜙𝑦 =

𝜕𝑤 𝜕𝑤 𝜕𝐿1 𝜕𝑤 𝜕𝐿2 𝜕𝑤 𝜕𝐿3 = + + 𝜕𝑥 𝜕𝐿1 𝜕𝑥 𝜕𝐿2 𝜕𝑥 𝜕𝐿3 𝜕𝑥 ( ) = 𝛼1 + 𝛼4 𝐿2 + 𝛼6 𝐿3 + 2𝛼7 𝐿1 𝐿2 + 𝛼9 𝐿32 𝑏1 ( ) + 𝛼2 + 𝛼4 𝐿1 + 𝛼5 𝐿3 + 𝛼7 𝐿12 + 2𝛼8 𝐿2 𝐿3 𝑏2 ( ) + 𝛼3 + 𝛼5 𝐿2 + 𝛼6 𝐿1 + 𝛼8 𝐿22 + 2𝛼9 𝐿1 𝐿3 𝑏3

(9.45)

299

300

9 Slabs

and 𝜙𝑥 =

𝜕𝑤 𝜕𝑤 𝜕𝐿1 𝜕𝑤 𝜕𝐿2 𝜕𝑤 𝜕𝐿3 = + + 𝜕𝑦 𝜕𝐿1 𝜕𝑦 𝜕𝐿2 𝜕𝑦 𝜕𝐿3 𝜕𝑦 ( ) = 𝛼1 + 𝛼4 𝐿2 + 𝛼6 𝐿3 + 2𝛼7 𝐿1 𝐿2 + 𝛼9 𝐿32 𝑐1 ( ) + 𝛼2 + 𝛼4 𝐿1 + 𝛼5 𝐿3 + 𝛼7 𝐿12 + 2𝛼8 𝐿2 𝐿3 𝑐2 ( ) + 𝛼3 + 𝛼5 𝐿2 + 𝛼6 𝐿1 + 𝛼8 𝐿22 + 2𝛼9 𝐿1 𝐿3 𝑐3

(9.46)

For node 1 with 𝐿1 = 1, 𝐿2 = 𝐿3 = 0, this yields (Figure 9.5) 𝑤1 = 𝛼1 𝜙1𝑦 = 𝛼1 𝑏1 + (𝛼2 + 𝛼4 + 𝛼7 ) 𝑏2 + (𝛼3 + 𝛼6 ) 𝑏3

(9.47)

𝜙1𝑥 = 𝛼1 𝑐1 + (𝛼2 + 𝛼4 + 𝛼7 ) 𝑐2 + (𝛼3 + 𝛼6 ) 𝑐3 and for node 2 with 𝐿2 = 1, 𝐿2 = 𝐿3 = 0, 𝑤2 = 𝛼2 𝜙2𝑦 = (𝛼1 + 𝛼4 ) 𝑏1 + 𝛼2 𝑏2 + (𝛼3 + 𝛼5 + 𝛼8 ) 𝑏3

(9.48)

𝜙2𝑥 = (𝛼1 + 𝛼4 ) 𝑐1 + 𝛼2 𝑐2 + (𝛼3 + 𝛼5 + 𝛼8 ) 𝑐3 and, finally, for node 3 with 𝐿3 = 1, 𝐿1 = 𝐿2 = 0, 𝑤3 = 𝛼3 𝜙3𝑦 = (𝛼1 + 𝛼6 + 𝛼9 ) 𝑏1 + (𝛼2 + 𝛼5 ) 𝑏2 + 𝛼3 𝑏3

(9.49)

𝜙3𝑥 = (𝛼1 + 𝛼6 + 𝛼9 ) 𝑐1 + (𝛼2 + 𝛼5 ) 𝑐2 + 𝛼3 𝑐3 These nine equations may be solved for 𝛼1 … 𝛼9 depending on the nodal degrees of freedom. Employing them for Eq. (9.44) results in 𝑤 = −𝐿1 (−1 + 𝐿2 − 2𝐿1 𝐿2 − 𝐿3 + 2𝐿32 )𝑤1 + 𝐿1 (𝑐3 𝐿1 𝐿2 + 𝐿32 𝑐2 − 𝐿3 𝑐2 )𝜙1𝑦 − 𝐿1 (𝑏2 𝐿32 + 𝑏3 𝐿1 𝐿2 − 𝑏2 𝐿3 )𝜙1𝑥 + 𝐿2 (−2𝐿12 − 𝐿3 + 𝐿1 + 1 + 2𝐿2 𝐿3 )𝑤2 + 𝐿2 (−𝑐3 𝐿1 + 𝑐3 𝐿12 + 𝑐1 𝐿2 𝐿3 )𝜙2𝑦 + −𝐿2 (−𝑏3 𝐿1 + 𝑏3 𝐿12 + 𝑏1 𝐿2 𝐿3 )𝜙2𝑥 + 𝐿3 (𝐿2 + 2𝐿3 𝐿1 − 2𝐿22 + 1 − 𝐿1 )𝑤3 + 𝐿3 (𝑐1 𝐿22 + 𝑐2 𝐿3 𝐿1 − 𝑐1 𝐿2 )𝜙3𝑦 − 𝐿3 (𝑏1 𝐿22 + 𝑏2 𝐿3 𝐿1 − 𝑏1 𝐿2 )𝜙3𝑥 (9.50) Generalised strains or curvatures (Eq. (9.33)) are derived with 𝜕2 𝑤 𝜕𝑥2 𝜕2 𝑤 𝜕𝑦 2 𝜕2 𝑤 𝜕𝑥𝜕𝑦 𝜕2 𝑤 𝜕𝑦𝜕𝑥

𝜕 2 𝑤 𝜕𝐿1 𝜕 2 𝑤 𝜕𝐿2 𝜕𝑤 2 𝜕𝐿3 + + 𝜕𝑥𝜕𝐿1 𝜕𝑥 𝜕𝑥𝜕𝐿2 𝜕𝑥 𝜕𝑥𝜕𝐿3 𝜕𝑥 𝜕 2 𝑤 𝜕𝐿1 𝜕 2 𝑤 𝜕𝐿2 𝜕𝑤 2 𝜕𝐿3 = + + 𝜕𝑦𝜕𝐿1 𝜕𝑦 𝜕𝑦𝜕𝐿2 𝜕𝑦 𝜕𝑦𝜕𝐿3 𝜕𝑦 2 2 𝜕 𝑤 𝜕𝐿1 𝜕 𝑤 𝜕𝐿2 𝜕𝑤 2 𝜕𝐿3 = + + 𝜕𝑥𝜕𝐿1 𝜕𝑦 𝜕𝑥𝜕𝐿2 𝜕𝑦 𝜕𝑥𝜕𝐿3 𝜕𝑦 𝜕2 𝑤 = 𝜕𝑥𝜕𝑦 =

= 𝜅𝑥 = 𝜅𝑦 =

𝜅𝑥𝑦 2

(9.51)

9.6 System Building and Solution Methods

or 𝜅𝑥 = 𝐵11 𝑤1 + 𝐵12 𝜙1𝑦 + 𝐵13 𝜙1𝑥 + 𝐵14 𝑤2 + 𝐵15 𝜙2𝑦 + 𝐵16 𝜙2𝑥 + 𝐵17 𝑤3 + 𝐵18 𝜙3𝑦 + 𝐵19 𝜙3𝑥 𝜅𝑦 = 𝐵21 𝑤1 + 𝐵22 𝜙1𝑦 + 𝐵23 𝜙1𝑥 + 𝐵24 𝑤2 + 𝐵25 𝜙2𝑦 𝜅𝑥𝑦 2

+ 𝐵26 𝜙2𝑥 + 𝐵27 𝑤3 + 𝐵28 𝜙3𝑦 + 𝐵29 𝜙3𝑥

(9.52)

= 𝐵31 𝑤1 + 𝐵32 𝜙1𝑦 + 𝐵33 𝜙1𝑥 + 𝐵34 𝑤2 + 𝐵35 𝜙2𝑦 + 𝐵36 𝜙2𝑥 + 𝐵37 𝑤3 + 𝐵38 𝜙3𝑦 + 𝐵39 𝜙3𝑥

with 𝐵11 = −8𝑏3 𝑏1 𝐿3 + 4𝑏12 𝐿2 + 2(4𝑏2 𝑏1 − 2𝑏32 )𝐿1 + 2(−𝑏2 𝑏1 + 𝑏3 𝑏1 ) 𝐵12 = 4𝑏1 𝑐2 𝑏3 𝐿3 + 2𝑐3 𝑏12 𝐿2 + 2(2𝑐3 𝑏1 𝑏2 + 𝑐2 𝑏32 )𝐿1 − 2𝑏1 𝑐2 𝑏3

(9.53)

… This demonstrates the basic procedure. Obviously, the issue becomes elaborate. The selection of higher-order polynomial terms becomes a delicate question with several options (Zienkiewicz and Taylor 1991, 1.5). The particular approach Eq. (9.44) belongs to the class of non-conforming elements, as a 𝐶 1 -continuity is not strictly given. This must not be a criterion for exclusion. A severe drawback is that the patch test (Section 2.9) is not fulfilled for arbitrary element shapes. Improved forms with efficient performance are proposed by Specht (1988), using fourth-order terms instead of cubic terms in Eq. (9.44).

9.6 System Building and Solution Methods General aspects of system building have already been treated in Section 2.6. Particular items regarding beams are described in Section 4.4. They are shortly rephrased and specified with respect to slabs. Generalised strains are approximated with finite element approximations (Eq. (2.21)) 𝝐 = B ⋅ 𝝊𝑒

(9.54)

with 𝝊𝑒 according to Eq. (9.322 ) applied to all nodes of an element and 𝝐 according to Eq. (9.321 ) for the Kirchhoff slab. The matrix B follows Eq. (9.52) as an example. Numerical integration of triangular elements has to be treated differently compared to quadrilaterals. Area coordinates (Section 9.5.1) are used to mark a position within an triangular element. Sampling points and weights (Section 2.7) are used for the numerical integration of internal nodal forces, external loads, and stiffness matrices. Area coordinates are addressed in Table 9.1 up to the integration order 2; compare also Table 2.1. The integration of the weak equilibrium condition (Eq. (9.27)) has

301

302

9 Slabs

Table 9.1 Sampling points and weights for triangular numerical integration. ni

L1

L2

L3

𝜼i

0 1

1/3 1/2 0 1/2 1/3 6/10 2/10 2/10

1/3 1/2 1/2 0 1/3 2/10 6/10 2/10

1/3 0 1/2 1/2 1/3 2/10 2/10 6/10

1 1/3 1/3 1/3 −27∕48 25/48 25/48 25/48

2

to be performed element by element according to Eq. (2.58). This is followed by assembling (Eq. (2.60)) and evaluation of the discretised system equilibrium condition (Eq. (2.70)). Boundary conditions have to provide a stable support and prevent rigid body displacements. Boundary forces are given by Eq. (9.29) with a unit normal vector e with components 𝑒𝑥 , 𝑒𝑦 along the boundary. To simplify a boundary edge parallel to the global 𝑦-axis with 𝑥 = 𝑐𝑜𝑛𝑠𝑡., 𝑒𝑥 = 1, 𝑒𝑦 = 0 (Figure 9.1b) is considered exemplarily, i.e. 𝑤 = 𝑤𝑠 ,

𝜙𝑥 = 𝜙𝑠𝑥 ,

𝜙𝑦 = 𝜙𝑠𝑦

𝑣𝑠 = 𝑣𝑥 ,

𝑚𝑠𝑥 = 𝑚𝑥 ,

𝑚𝑠𝑦 = 𝑚𝑥𝑦

(9.55)

along the boundary edge. The following cases are considered: • Free edge: values 𝑣𝑠 , 𝑚𝑠𝑥 , 𝑚𝑠𝑦 are prescribed – in most cases with 0 – and go directly into the boundary force vector t; see Eqs. (9.313 ) and (9.323 ). Their mutual dependency given by Eq. (9.15) must be considered, if necessary, in cases where they are not zero. This corresponds to the notion of compensatory shear forces within the theory of Kirchhoff slabs. • Simply supported edge: with respect to Eqs. (9.313 ) and (9.323 ), kinematic boundary conditions 𝑤𝑠 = 0, 𝜙𝑠𝑥 = 0 characterise the simply supported edge. On the other hand, 𝜙𝑠𝑦 is not prescribed, and the corresponding force 𝑚𝑠𝑥 = 𝑚𝑥 has to be prescribed, generally with a value 0. But as 𝜙𝑠𝑥 is prescribed, the corresponding boundary force 𝑚𝑠𝑦 = 𝑚𝑥𝑦 has to result from the computation and is transmitted to the slab support. It is connected to a compensatory shear force, which is combined with the internal shear force 𝑣𝑥 . • Clamped edge: with respect to Eqs. (9.313 ) and (9.323 ), kinematic boundary conditions 𝑤𝑠 = 0, 𝜙𝑠𝑦 = 0, 𝜙𝑠𝑥 = 0 are given. All corresponding boundary forces are determined from the computation where 𝑚𝑠𝑦 = 𝑚𝑥𝑦 = 0 as 𝜕𝜙𝑦 ∕𝜕𝑦 = 0. A boundary edge parallel to the global 𝑥-axis with 𝑦 = 𝑐𝑜𝑛𝑠𝑡. is treated in the same way with indices exchanged. A straight skew or curved boundary edge with, e.g. simple support leads to a prescribed coupling of 𝜙𝑦 , 𝜙𝑥 and 𝑚𝑥 , 𝑚𝑦 , 𝑚𝑥𝑦 along the

9.6 System Building and Solution Methods

boundary. These may be considered as additional constraint conditions. Kinematic boundary conditions are applied to nodes and directly implemented upon assembling the system by the modification of the system stiffness matrix and load vector; see Section 4.4.3 for the basic approach. Boundary forces or reactions, respectively, are computed as internal nodal forces for those boundary degrees of freedom that have kinematic boundary conditions prescribed. These particular internal nodal forces are not equilibrated by external nodal loads. The computation of slabs arises as an everyday task. Linear elastic behaviour is generally assumed in practical problems. Nevertheless, geometric properties and boundary conditions prevent an analytical treatment, and only numerical methods may provide useful solutions according to the general forms Eqs. (2.11)–(2.13). This is demonstrated with the following example. Example 9.1: Linear Elastic Slab with Opening and Free Edges

The slab geometry with opening and boundary conditions is shown in Figure 9.6a. A single support is given in the lower left-hand corner. The left-hand and lower edges are not supported. The upper and right-hand edges are simply supported (hinged). The material properties are assumed with a Young’s modulus 𝐸 = 31 900 MN∕m2 and a Poisson’s ratio 𝜈 = 0.2. The slab thickness is ℎ = 0.3 m. A uniform loading is given by 𝑝 = 10 kN∕m2 , which does not act in the opening area. Kirchhoff slab theory is assumed, and only moments will be directly considered as internal forces (Eq. (9.321 )). A triangular slab element is chosen according to Section 9.5.2 with the implementation proposed by Specht (1988). The discretisation is given by 132 elements, which are shown in Figure 9.6b. The computation leads to the following results: • Principal moments – derived from 𝑚𝑥 , 𝑚𝑦 , 𝑚𝑥𝑦 in analogy to principal stresses (Eqs. (8.1) and (8.2)) – provide instructive information about the load bearing behaviour. The computed values are shown in Figure 9.6b, where the arrow direction indicates the stress direction caused by the respective moment. Positive values have compressive stresses on the upper side and tensile stresses on the lower side, negative values the opposite. Principal moments are generally aligned to free edges and connected with a uniaxial behaviour. • A skew orientation of principal moments with opposite sign and approximately the same absolute value arises in the upper right-hand corner and are reversed near the lower left-hand single support. This indicates the effect of twisting. • The computed deflections are shown in Figure 9.7a. They conform to prescribed boundary conditions. The maximum deflection is in the range of 3 mm with 𝑤max ∕𝐿 ≈ 1∕2300. In the case of linear elastic behaviour, this is still a stiff system, even with an opening and two free edges. • The computed reaction forces in the boundary nodes are listed in Figure 9.7b. The sum of all reaction forces is 330 kN and equals the total loading. The by far largest reaction force is given with the lower left-hand point support. The upper righthand reaction force corresponds to uplift. This conforms to the theory of Kirchoff slabs with twisting stiffness.

303

304

9 Slabs

(a)

(b)

Figure 9.6 Example 9.1. (a) System. (b) Discretisation and principal moments.

(a)

(b)

Figure 9.7 Example 9.1. (a) Deflections [m]. (b) Boundary support reactions [MN].

A redistribution of moments generally occurs with nonlinear material behaviour. An approach for nonlinear behaviour for slabs is described in Section 9.8.

9.7 Lower Bound Limit State Analysis

9.7 Lower Bound Limit State Analysis Principles of a lower bound limit state analysis are already described in Section 8.1.1 for plates. Internal forces are determined with a linear elastic computation with a unit load applied. Internal forces can be scaled with a loading factor. The loading factor is adapted such that the cross-sectional resistance resulting from thickness, concrete strength, and the strength of reinforcement – basically by its amount – is not exceeded in any position of the slab reference plane (→ admissible load proof), or the design of the thickness, concrete strength, and amount of reinforcement is adapted to a given loading factor (→ design procedure). The key item is again to find an appropriate combination of concrete and reinforcement strength as reference values for the computed state of internal forces.

9.7.1 Design for Bending The bending design of slabs generally follows that of beams in engineering practice. An approach combining an admissible load proof for a given concrete together with a design procedure to determine the required reinforcement is described in the following, which deviates on a first view of beam design. An accordance is shown at the end of the section. Moments 𝑚𝑥 , 𝑚𝑦 , 𝑚𝑥𝑦 defined by Eq. (9.7) and Figure 9.3 behave as components of a second-order tensor in the same way as the components of a plane stress tensor (Eq. (6.4)) do, as they are directly derived from stress components. The transformation of tensorial moments into a coordinate system rotated by an angle 𝜑 (positive counterclockwise (Eq. (D.11))) is given by m ˜ =Q⋅m

(9.56)

in the same way as for stresses with ⎡ cos2 𝜑 ˜𝑥 ⎞ ⎛𝑚 ⎢ 2 ⎜ ⎟ m ˜= 𝑚 ˜𝑦 , Q = ⎢ sin 𝜑 ⎜ ⎟ ⎢ ˜𝑥𝑦 ⎠ − cos 𝜑 sin 𝜑 ⎝𝑚 ⎣

2

sin 𝜑 cos2 𝜑 cos 𝜑 sin 𝜑

2 cos 𝜑 sin 𝜑 ⎤ ⎛ 𝑚𝑥 ⎞ ⎥ −2 cos 𝜑 sin 𝜑 ⎥ , m = ⎜ 𝑚𝑦 ⎟ ⎜ ⎟ 2 ⎥ cos2 𝜑 − sin 𝜑 ⎝𝑚𝑥𝑦 ⎠ ⎦ (9.57)

Thus, principal moments with principal directions may be derived in analogy to Eqs. (8.1) and (8.2) under the condition of vanishing components 𝑚 ˜𝑥𝑦 . According to the definition of Eq. (9.7) a moment 𝑚𝑥 leads to stresses 𝜎𝑥 , 𝑚𝑦 to 𝜎𝑦 , and 𝑚𝑥𝑦 to 𝜎𝑥𝑦 . Assigning an internal lever arm 𝑧 yields the corresponding force couple, 𝑡𝑥 = ±

𝑚𝑥 𝑧

,

𝑡𝑦 = ±

𝑚𝑦 𝑧

,

𝑡𝑥𝑦 = ±

𝑚𝑥𝑦 𝑧

(9.58)

with a positive sign for the lower layer force component and a negative sign for the upper layer force (Figure 9.4), according to the sign conventions of Figure 9.3.

305

306 ◀

9 Slabs

All considerations with respect to the reinforcement design of plates (Section 8.1) are applied to the respective layer forces and the required upper or lower reinforcement, respectively.

Upper and lower reinforcements both have to be explicitly treated, whereby each of them is connected with respective layer forces for a given position of the reference plane. The procedure is the same for both. Layer forces have principal values 𝑡1 , 𝑡2 like the moments from which they are derived. A reinforcement is not required in the biaxial compressive case 𝑡1 < 0, 𝑡2 < 0. The following treats all the other cases. The values 𝑡𝑥 , 𝑡𝑦 , 𝑡𝑥𝑦 are connected to the principal values by 2

𝑡𝑥 = 𝑡1 cos2 𝜑 + 𝑡2 sin 𝜑 2

𝑡𝑦 = 𝑡1 sin 𝜑 + 𝑡2 cos2 𝜑

(9.59)

𝑡𝑥𝑦 = (𝑡1 − 𝑡2 ) sin 𝜑 cos 𝜑 in analogy to Eq. (8.4) with the principal direction angle 𝜑 measured from the 𝑥axis to the 1-axis (counterclockwise positive). Two reinforcement directions 𝜑𝑠1 , 𝜑𝑠2 and a concrete direction 𝜑𝑐 are considered. These directions describe the principal 1-directions for each part. The corresponding principal stress values are transformed to the global directions applying Eq. (9.59) to each part, whereby the stresses in the principal 2-directions vanish due to the uniaxial behaviour of each part. This leads to (Eqs. (8.5) and (8.6)) 𝑡𝑐,𝑥 = 𝑡𝑐,1 cos2 𝜑𝑐 , 𝑡𝑠1,𝑥 = 𝑡𝑠1,1

cos2

𝜑𝑠1 ,

𝑡𝑠2,𝑥 = 𝑡𝑠2,1 cos2 𝜑𝑠2 ,

2

𝑡𝑐,𝑦 = 𝑡𝑐,1 sin 𝜑𝑐 ,

𝑡𝑐,𝑥𝑦 = 𝑡𝑐,1 sin 𝜑𝑐 cos 𝜑𝑐

2

𝑡𝑠1,𝑥𝑦 = 𝑡𝑠1,1 sin 𝜑𝑠1 cos 𝜑𝑠1

2

𝑡𝑠2,𝑥𝑦 = 𝑡𝑠2,1 sin 𝜑𝑠2 cos 𝜑𝑠2

𝑡𝑠1,𝑦 = 𝑡𝑠1,1 sin 𝜑𝑠1 , 𝑡𝑠2,𝑦 = 𝑡𝑠2,1 sin 𝜑𝑠2 ,

(9.60) The notation 𝑡𝑐 = 𝑡𝑐,1 , 𝑡𝑠1 = 𝑡𝑠1,1 , 𝑡𝑠2 = 𝑡𝑠2,1 is used in the following. Reinforcement forces are connected to reinforcement stresses by reinforcement cross-sections 𝑎𝑠1 , 𝑎𝑠2 per unit width with 𝑡𝑠1 = 𝑎𝑠1 𝜎𝑠1 ,

𝑡𝑠2 = 𝑎𝑠2 𝜎𝑠2

(9.61)

The different parts contribute to total forces in analogy to Eq. (8.8) 𝑡𝑐,𝑥 + 𝑡𝑠1,𝑥 + 𝑡𝑠2,𝑥 = 𝑡𝑥 𝑡𝑐,𝑦 + 𝑡𝑠1,𝑦 + 𝑡𝑠2,𝑦 = 𝑡𝑦

(9.62)

𝑡𝑐,𝑥𝑦 + 𝑡𝑠1,𝑥𝑦 + 𝑡𝑠2,𝑥𝑦 = 𝑡𝑥𝑦 The usage of Eqs. (9.60) and (9.61) with Eq. (9.62) first of all yields three equations for the variables 𝑡𝑐 , 𝜑𝑐 , 𝜎𝑠1 , 𝑎𝑠1 , 𝜑𝑠1 , 𝜎𝑠2 , 𝑎𝑠2 , 𝜑𝑠2 . Three further Eqs. (9.58) connect 𝑡𝑥 , 𝑡𝑦 , 𝑡𝑥𝑦 to moments 𝑚𝑥 , 𝑚𝑦 , 𝑚𝑥𝑦 by the internal lever arm 𝑧, where 𝑚𝑥 , 𝑚𝑦 , 𝑚𝑥𝑦 are given from a linear elastic FE-calculation performed in advance. It is reasonable

9.7 Lower Bound Limit State Analysis

Figure 9.8 Slab reinforcement with positive bending.

to prescribe the four values 𝜎𝑠1 , 𝜑𝑠1 , 𝜎𝑠2 , 𝜑𝑠2 . Thus, the parameters 𝑡𝑐 , 𝜑𝑐 , 𝑎𝑠1 , 𝑎𝑠2 and 𝑧 remain open for proof or design purposes. A special, but common, case is again considered. It is assumed that the reinforcement directions are aligned to global coordinate axes, i.e. 𝜑𝑠1 = 0, 𝜑𝑠2 = π∕2, and, therefore, sin 𝜑𝑠1 = 0, cos 𝜑𝑠1 = 1, sin 𝜑𝑠2 = 1, cos 𝜑𝑠2 = 0. This leads to 𝑡𝑠1,𝑥 = 𝑡𝑠𝑥 = 𝑎𝑠𝑥 𝜎𝑠𝑥 ,

𝑡𝑠1,𝑦 = 0 ,

𝑡𝑠1,𝑥𝑦 = 0

𝑡𝑠2,𝑦 = 𝑡𝑠𝑦 = 𝑎𝑠𝑦 𝜎𝑠𝑦 ,

𝑡𝑠2,𝑥 = 0 ,

𝑡𝑠2,𝑥𝑦 = 0

(9.63)

with 𝜎𝑠𝑥 = 𝜎𝑠1 , 𝑎𝑠𝑥 = 𝑎𝑠1 , 𝜎𝑠𝑦 = 𝜎𝑠𝑦 , 𝑎𝑠𝑦 = 𝑎𝑠𝑦 . Insertion into Eq. (9.62) together with Eq. (9.601 ) results in 𝑡𝑐 cos2 𝜑𝑐 + 𝑎𝑠𝑥 𝜎𝑠𝑥 = 𝑡𝑥 2

𝑡𝑐 sin 𝜑𝑐 + 𝑎𝑠𝑦 𝜎𝑠𝑦 = 𝑡𝑦

(9.64)

𝑡𝑐 sin 𝜑𝑐 cos 𝜑𝑐 = 𝑡𝑥𝑦 The concrete force 𝑡𝑐 – negative by definition – is reformulated as 𝑡𝑐 =

𝑡𝑥𝑦 sin 𝜑𝑐 cos 𝜑𝑐

(9.65)

and is ruled by 𝜑𝑐 (Figure 8.2a). The bearing capacity of the compression zone can be estimated with some assumptions about the values and the distribution of concrete stresses whereby a compressive reinforcement is not taken into account. A constant concrete stress distribution is assumed with a value 𝜎𝑐 = 𝜒 𝑓𝑐 (unsigned), a zero line of bending 𝑥 measured from the compressed side, and the compression stress zone height 𝑘 𝑥; see Figure 9.8. 2) Values 𝜒 = 0.95, 𝑘 = 0.8 are appropriate (EN 1992-1-1 2004, 3.1.7). This leads to a concrete force 𝑡𝑐 = −𝜒𝑓𝑐 𝑘 𝑥 (negative by definition) on the compressive side and a compression zone height 𝑥=−

𝑡𝑐 𝜒𝑓𝑐 𝑘

(9.66)

2) Another familiar approach assumes nonlinear concrete stresses (Figure 3.1) according to a linear compressive strain course. But this involves nonlinear equations for the determination of the internal lever arm. It is not followed here for simplicity.

307

308

9 Slabs

Using Eqs. (9.66), (9.65), and (9.58) an internal lever arm is derived with 𝑧=𝑑−

𝑡𝑐 |𝑚𝑐 | 𝑥 1 =𝑑+ , =𝑑− 2 2𝜒𝑓𝑐 𝑘 2𝜒𝑓𝑐 𝑘 𝑧

𝑚𝑐 =

𝑚𝑥𝑦 sin 𝜑 cos 𝜑

(9.67)

with the effective height 𝑑. This is solved with 𝑧=

) √ 𝑑 ( 1+ 1−𝛼 , 2

𝛼=

2|𝑚𝑐 | 𝑘𝜒𝑓𝑐 𝑑2

(9.68)

where 𝛼 is a measure for the utilisation of the concrete compressive zone with a condition 𝛼 < 1. The angle 𝜑𝑐 (Figure 8.2a) has to be chosen such that 𝑡𝑐 , 𝑚𝑐 < 0. It is assumed that the reinforcement stresses exploit the load carrying capacity with 𝜎𝑠𝑥 = 𝜎𝑠2 = 𝑓𝑦𝑘 with the reinforcement yield limit 𝑓𝑦𝑘 . Thus, using Eqs. (9.64) and (9.58) the required reinforcement is determined with 𝑎𝑠𝑥 =

𝑡𝑥 − 𝑡𝑐 cos2 𝜑𝑐 ±𝑚𝑥 − 𝑚𝑐 cos2 𝜑𝑐 = 𝑓𝑦𝑘 𝑧 𝑓𝑦𝑘 2

𝑎𝑠𝑦 =

𝑡𝑦 − 𝑡𝑐 sin 𝜑𝑐 𝑓𝑦𝑘

2

=

±𝑚𝑦 − 𝑚𝑐 sin 𝜑𝑐

(9.69)

𝑧 𝑓𝑦𝑘

where the positive sign is used for 𝑚𝑖 > 0, 𝑖 = 𝑥, 𝑦 for a lower reinforcement (coordinate 𝑧 < 0) and the negative sign for 𝑚𝑖 < 0 for an upper reinforcement (coordinate 𝑧 > 0). The 𝑚𝑐 -contribution increases the required reinforcement as 𝑚𝑐 < 0 by definition. The internal lever arm might be slightly different for 𝑥 and 𝑦-directions, as the respective reinforcement cannot share the same 𝑧-position. This may be regarded with adjusted 𝑑-values (Eq. (9.67)) and is disregarded here. Setting 𝑧 = 1, 𝑚 = 𝜎 reproduces the approach for the reinforcement design of plates (Section 8.1.3). Summarizing, three Eqs (9.67) and (9.69) are available for four remaining variables 𝑎𝑠𝑥 , 𝑎𝑠𝑦 , 𝑧, 𝜑𝑐 . For 𝜑𝑐 , a first choice ⎧− π 𝜑𝑐 = π 4 ⎨ ⎩4

for 𝑚𝑥𝑦 ≥ 0

(9.70)

𝑚𝑥𝑦 < 0

is appropriate, as is shown in Section 8.1.3. This leads to 𝑚𝑐 = −2|𝑚𝑥𝑦 |, and the required reinforcement is given with (Eqs. (9.67) and (9.69)) 𝑎𝑠𝑥 =

±𝑚𝑥 + |𝑚𝑥𝑦 | 𝑧 𝑓𝑦𝑘

,

𝑎𝑠𝑦 =

±𝑚𝑦 + |𝑚𝑥𝑦 | 𝑧 𝑓𝑦𝑘

(9.71)

A computed value 𝑎𝑠𝑖 < 0, 𝑖 = 𝑥 or 𝑦 indicates that the angle 𝜑𝑐 has to be modified to reach 𝑎𝑠𝑖 = 0. This is treated by the nonlinear system of three Eqs. (9.67) and (9.69) with 𝑎𝑠𝑖 = 0 and (9.68), for three unknowns 𝜑𝑐 , 𝑧, and 𝑎𝑠𝑗 with 𝑖 ≠ 𝑗. This system has to be solved iteratively in a similar way as was demonstrated for plates (Section 8.1.3). Equations (9.71) lead to the slab reinforcement that is always required on both sides due to |𝑚𝑥𝑦 |. Actually, this contribution is neglected for small values of |𝑚𝑥𝑦 |

9.7 Lower Bound Limit State Analysis

ruled by the respective code provisions. On the other hand, certain slab areas may be exposed to |𝑚𝑥 | ≪ |𝑚𝑥𝑦 |, |𝑚𝑦 | ≪ |𝑚𝑥𝑦 |. This yields a twisting reinforcement twist = 𝑎𝑠𝑥

|𝑚𝑥𝑦 | 𝑧 𝑓𝑦𝑘

,

twist 𝑎𝑠𝑦 =

|𝑚𝑥𝑦 |

(9.72)

𝑧 𝑓𝑦𝑘

to be arranged on both slab sides. The concrete compression height has to be restricted to assure a sufficient rotation capacity. The zero line 𝑥 of bending should not exceed ≈ 𝑑∕2 (EN 1992-1-1 2004, 5.6.3). With respect to Eq. (9.66), a condition 𝑥 ≤ 𝑑∕2 leads to 𝑓𝑐 𝜒𝑓𝑐 𝑘 𝑎𝑠𝑖 = 0.38 , < 𝑑 2𝑓𝑦𝑘 𝑓𝑦𝑘

𝑖 = 𝑥, 𝑦

(9.73)

with 𝑘 = 0.8, 𝜒 = 0.95. This corresponds to an upper bound for the reinforcement ratio. A sufficient rotation capacity enables deformations that are necessary for a redistribution of internal forces to adjust to the conditions caused by a prescribed value of 𝜑𝑐 . The design approach is demonstrated with the following example. Example 9.2: Reinforcement Design for a Slab with Opening and Free Edges with a Limit State Analysis

We refer to Example 9.1 with the same system and loading. Moments have been calculated for each element integration point with a linear elastic calculation. The reinforcement strength is assumed with 𝑓𝑦𝑘 = 435 MN∕m2 and the uniaxial compressive strength of the concrete (unsigned) with 𝑓𝑐 = 17.0 MN∕m2 , 𝑓𝑐 = 0.8 ⋅ 0.95 ⋅ 17 = 12.9 MN∕m2 . The slab has an effective height of 𝑑 = 0.26 m. Safety factors are not explicitly regarded for the loading. The computed required sums 𝑎𝑠𝑥 + 𝑎𝑠𝑦 of reinforcement are shown in Figure 9.9 for the bottom and top sides as a raw contour 3) plot. Each element has four inte-

(a)

(b)

Figure 9.9 Example 9.2. Sum asx + asy of reinforcement [cm2 ∕m] (a) Bottom side. (b) Top side. 3) No image postprocessing is performed to smooth rough isolines resulting from the relatively coarse discretisation.

309

310

9 Slabs

Figure 9.10 Example 9.2. Required reinforcement of selected points [cm2 ∕m].

gration points. The top side reinforcement corresponds to twisting moments and skew principal moments, compare Figure 9.6b. The bottom reinforcement covers the whole area and also reflects principal moment distributions. Lower reinforcement (first number in a pair) and upper reinforcement (second number in a pair) are shown for selected points in Figure 9.10 together with underlying principal moments. • Position 𝑥 = 0.17, 𝑦 = 2.5 near mid-span left, free edge The computed moments are 𝑚𝑥 = 0.002 MNm∕m, 𝑚𝑦 = 0.042, 𝑚𝑥𝑦 = 0.015, leading to principal moments 𝑚1 = 0.047, 𝑚2 = −0.003 with a direction 𝜑 = 72°. Lower reinforcement: assuming 𝜑 = −45° yields 𝑎𝑠𝑥 = 1.50 cm2 ∕m, 𝑎𝑠𝑦 = 5.12 with 𝑧∕𝑑 = 0.98 and 𝑥∕𝑑 = 0.03. Upper reinforcement: 𝜑𝑐 = 45° leads to 𝑎𝑠𝑦 < 0. Thus, an iteration has to be performed with a result 𝑎𝑠𝑥 = 0.28 cm2 ∕m, 𝑎𝑠𝑦 = 0 with 𝑧∕𝑑 = 0.97, 𝑥∕𝑑 = 0.06 and 𝜑𝑐 = 71° computed. • Position 𝑥 = 6.5, 𝑦 = 4.17 near the upper right corner of a simple line support Computed moments are 𝑚𝑥 = 0.004 MNm∕m, 𝑚𝑦 = 0.004, 𝑚𝑥𝑦 = −0.023, leading to 𝑚1 = 0.027, 𝑚2 = −0.019 with 𝜑1 = −45°. The curvature is negative in the corner’s diagonal direction and positive laterally to it. This indicates a load transfer ‘over edge’ supporting a load transfer in the corner diagonal and corresponds to a twisting effect. Lower reinforcement: 𝜑 = 45° yields 𝑎𝑠𝑥 = 2.45 cm2 ∕m, 𝑎𝑠𝑦 = 2.41 with 𝑧∕𝑑 = 0.0.98, 𝑥∕𝑑 = 0.05. Upper reinforcement: 𝜑 = −45° yields 𝑎𝑠𝑥 = 1.73 cm2 ∕m, 𝑎𝑠𝑦 = 1.77 with 𝑧∕𝑑 = 0.98, 𝑥∕𝑑 = 0.05. • Position 𝑥 = 3.5, 𝑦 = 0.17 near mid-span lower free edge Computed moments are 𝑚𝑥 = 0.043 MNm∕m, 𝑚𝑦 = 0.006, 𝑚𝑥𝑦 = 0.006, leading to 𝑚1 = 0.044, 𝑚2 = 0.005 with 𝜑1 = 9°. Lower reinforcement: 𝜑 = 45° yields 𝑎𝑠𝑥 = 4.59 cm2 ∕m, 𝑎𝑠𝑦 = 1.14 with 𝑧∕𝑑 = 0.99, 𝑥∕𝑑 = 0.002. An upper reinforcement is not necessary, as both principal stresses are positive and lead to a upper biaxial compressive stress state. • Position 𝑥 = 0.5, 𝑦 = 0.17 near lower left single support Computed moments are 𝑚𝑥 = 0.013 MNm∕m, 𝑚𝑦 = 0.007, 𝑚𝑥𝑦 = 0.028, leading to 𝑚1 = 0.038, 𝑚2 = −0.018 with 𝜑1 = 42°. The curvature is positive in the corner diagonal direction and negative laterally to it. This corresponds to the major load transfer along the free edges compared to the diagonal. Lower reinforce-

9.7 Lower Bound Limit State Analysis

ment: 𝜑𝑐 = −45° yields 𝑎𝑠𝑥 = 3.75 cm2 ∕m, 𝑎𝑠𝑦 = 3.17 with 𝑧∕𝑑 = 0.97, 𝑥∕𝑑 = 0.07. Upper reinforcement: 𝜑𝑐 = 45° yields 𝑎𝑠𝑥 = 1.32 cm2 ∕m, 𝑎𝑠𝑦 = 1.90 with 𝑧∕𝑑 = 0.97, 𝑥∕𝑑 = 0.07. As compressive heights are computed with 𝑥∕𝑑 < 0.5 for all points, the required load bearing capacity of concrete (Eq. (9.73)) is provided. The same comments as for plates with respect to concrete strength, ductility requirements, and crack width estimation (Section 8.1.3) are also valid for slabs. The question of how Eqs. (9.71) and (9.72) relate to current design practice deserves a remark. In most cases – regarding a particular slab position – either |𝑚𝑥 |, |𝑚𝑦 | have significant values or |𝑚𝑥𝑦 |, but not both concurrently. Maximum values of |𝑚𝑥 |, |𝑚𝑦 | in most cases correspond to principal values |𝑚1 |, |𝑚2 | with |𝑚𝑥𝑦 ≈ 0|. Thus, Eq. (9.71) corresponds to a beam reinforcement design. Equation (9.72) leads to additional twisting reinforcement, in particular for edges of simply supported slabs.

9.7.2 Design for Shear Shear has to be considered as further action in addition to bending. In the case of Kirchhoff slabs, shear forces have to be calculated as derivatives of bending moments (Eq. (9.15)) 𝑣𝑥 = −

𝜕𝑚𝑥 𝜕𝑚𝑥𝑦 − , 𝜕𝑥 𝜕𝑦

𝑣𝑦 = −

𝜕𝑚𝑦 𝜕𝑦

−

𝜕𝑚𝑥𝑦 𝜕𝑥

(9.74)

Bending moments themselves are determined from curvature using Eq. (9.34) in the case of linear elastic material behaviour. The curvature is determined from finite element interpolation (Eq. (9.54)). Thus, another derivative has to be determined from B ⋅ 𝝊𝑒 . This is elaborate for relations like Eqs. (9.52) and (9.53). Furthermore, the accuracy decreases with the computation of higher displacement derivatives. A separate interpolation or approximation of moments within elements should be less expensive and yield sufficiently reliable shear force values. This is based on a linear approach 𝑚 = 𝑎𝑥 +𝑏𝑦 +𝑐

(9.75)

𝜕𝑚 =𝑎, 𝜕𝑥

(9.76)

and 𝜕𝑚 =𝑏 𝜕𝑦

where 𝑚 stands for 𝑚𝑥 , 𝑚𝑦 , 𝑚𝑥𝑦 . The coefficients 𝑎, 𝑏, 𝑐 need at least three computed values, e.g. from three integration points within triangular elements and the integration order 𝑛𝑖 = 1 (Table 9.1). In the case of more integration points, a linear regression analysis may be used to determine the three coefficients (Appendix E). With respect to Eqs. (9.74) and (9.76), these coefficients immediately lead to shear force values 𝑣𝑥 , 𝑣𝑦 , which are constant within the interpolation area.

311

312

9 Slabs

Figure 9.11 The mechanism of shear.

Similarly to the transformation rules for moments (Eq. (9.57)) a transformation rule for shear forces is required. A cross-section with normal vector e = ( )T cos 𝜑 sin 𝜑 is used, where 𝜑 denotes the angle with the global 𝑥-axis. The shear force in this cross-section is calculated from 𝑣𝑥 , 𝑣𝑦 (Figure 9.3) by 𝑣𝜑 = 𝑣𝑥 cos 𝜑 + 𝑣𝑦 sin 𝜑

(9.77)

corresponding to Eq. (9.291 ). Shear forces become relevant near supported edges or supported points. Shear arises from change of longitudinal cross-sectional forces resulting from change of moments. This change of forces in adjacent positions leads to forces in horizontal cross-sections and adjoined shear forces in vertical cross-sections; see Figure 9.11. In the case of reinforced concrete, these forces are resisted by concrete struts and reinforcement ties. The mechanisms are basically the same for beams and slabs. Principal longitudinal forces 𝑡1 , 𝑡2 computed from 𝑡𝑥 , 𝑡𝑦 , 𝑡𝑥𝑦 are used in the case of slabs. The force 𝑡1 acts at a cross-section inclined with an angle 𝜑 against the 𝑥-axis, the force 𝑡2 at a cross-section perpendicular to it. Each of these has an opposite longitudinal force – both forming a couple for moments 𝑚1 , 𝑚2 – and a shear force 𝑣𝜑 computed with Eq. (9.77). This completes the principal cross-section of a slab from an internal force point of view. ◀

Shear design of the principal cross-section of a slab is treated like a beam cross-section of unit width.

Thus, the strut-and-tie approach for beam shear should also be applicable for principal cross-sections of slabs. The value of 𝑣𝜑 is used for an appropriate procedure; see Eqs. (8.13)–(8.15), where 𝑉 is replaced by 𝑣𝜑 . The proof and design of shear have to be performed for both principal directions. The respective shear reinforcement has to be superposed. Following EN 1992-1-1 (2004, 6.2.1) shear reinforcement is not necessary for slabs in the case when 𝑣𝜑 does not exceed a threshold value 𝑣𝑅𝑑,𝑐 . Due to the current state of knowledge this threshold depends on the concrete strength, the slab thickness, and the amount of longitudinal reinforcement. The amount of longitudinal reinforcement needs some specification in the context of principal cross-sections. Reinforcement relations are determined by 𝑡𝑠𝑥 = 𝑎𝑠𝑥 𝑓𝑦𝑘 , 𝑡𝑠𝑥 = 𝑎𝑠𝑥 𝑓𝑦𝑘 (Eq. (9.63)). The transformation of reinforcement forces to the principal system is performed ac-

9.7 Lower Bound Limit State Analysis

cording to Eq. (9.57) and leads to 2

𝑡𝑠𝜑 = 𝑡𝑠𝑥 cos2 𝜑 + 𝑡𝑠𝑦 sin 𝜑

(9.78)

This transformation differs from Eq. (9.77), as 𝑡𝑠𝑥 , 𝑡𝑠𝑦 are parallel to the reference plane and subject to a transformation of both direction and reference length, while 𝑣𝑥 , 𝑣𝑦 are orthogonal to the plane and subject to transformation of the reference length only. A principal reinforcement amount is defined in analogy to Eq. (9.78) as 2

𝑎𝑠𝜑 = 𝑎𝑠𝑥 cos2 𝜑 + 𝑎𝑠𝑦 sin 𝜑

(9.79)

The value can be used to determine the threshold value for 𝑣𝑅𝑑,𝑐 according to EN 1992-1-1 (2004, 6.2.2). Larger values 𝑎𝑠𝜑 lead to larger threshold values 𝑣𝑅𝑑,𝑐 . An abundant value 𝑎𝑠𝜑 might be useful to avoid a slab shear reinforcement, which is generally considered as inconvenient on construction sites. Example 9.3: Computation of Shear Forces and Shear Design

We refer to Examples 9.1 and 9.2 with the same system and loading. With four integration points for a triangular element (Table 9.1), four sets of 𝑚𝑥 , 𝑚𝑦 , 𝑚𝑥𝑦 are computed for each element. A linear approximation of moments is performed within each element according to Eq. (9.75), with details given in Appendix E. Derivatives of moments are determined according to Eqs. (E.15), which are constant within an element. Thus, shear forces 𝑣𝑥 , 𝑣𝑦 computed with Eq. (9.74) are also constant within an element but may differ from element to element. The results are shown in Figure 9.12a. Shear forces within elements adjacent to the right-hand and upper boundary edge correspond to support reactions with – according to Kirchhoff slab theory – modifications due to the change of twisting moments 𝑚𝑥𝑦 . The contribution of the latter leads to a uplift force in the upper right-hand edge (Figure 9.7b), where 𝑣𝑥 = 𝑣𝑦 = 0 locally. Nodal force support reactions from Figure 9.7b have to be related to length to make them comparable to internal shear forces. The positive sign of shear forces, see Figure 9.3 for sign conventions, corresponds to the negative (downward) external loading. Larger negative shear forces around the left-hand lower single support

(a)

(b)

Figure 9.12 Example 9.3. (a) Computed shear forces vx , vy . (b) Sum |v1 | + |v2 |[MN∕m] of principal shear forces.

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correspond to the positive point support reaction. Shear forces distribution somehow looks confusing around the central opening. A finer discretisation presumably leads to more evidence regarding this area. A shear design can be performed according to the outline given before. Principal moment calculations determine an orientation 𝜑 of the larger principal moment 𝑚1 . Principal shear forces 𝑣1 , 𝑣2 are determined using Eq. (9.77) with 𝜑, 𝜑 + π∕2. Results for the sum of their absolute values are shown in Figure 9.12b as a raw contour plot. This, again, proves a too coarse discretisation – shown in the figure, see also Figure 9.6b – indicated by a bias of element shear forces, in particular along the upper edge. In general, the approximation quality deteriorates with derivatives of field variables. Stirrups should be used for a slab shear reinforcement. Thus, Eq. (8.14) is appropriate 𝑎𝑠𝑣,𝑖 =

|𝑣𝑖 | 𝑓𝑦𝑘 𝑧 cot 𝜙𝑐

,

𝑖 = 1, 2

(9.80)

which yields a stirrup cross-sectional area per unit base area with a concrete stress orientation 𝜙𝑐 (Figure 9.11) – 𝜙𝑐 = π∕4, cot 𝜙𝑐 = 1 is appropriate –, a reinforcement design strength 𝑓𝑦𝑘 , and the internal lever arm 𝑧 for bending (Eq. (9.67)). It must be checked whether this is actually required with the threshold value 𝑣𝑅𝑑,𝑐 mentioned above. From EN 1992-1-1 (2004, 6.2.2) it is determined – based on empirical values and depending on units – with 𝑣𝑅𝑑,𝑐 = 0.10 ⋅ 𝜅 ⋅ (100𝜌𝑠 ⋅ 𝑓𝑐𝑘 )1∕3 ⋅ 𝑑

(9.81) √

with 𝑓𝑐𝑘 = 30 MN∕m2 , 𝑑 = 0.26 m and 𝜅 = 1 + 200∕𝑑[mm] ≤ 2 = 1.88 for the example and the prefactor from EN 1992-1-1 (2004, 6.2.2) and DIN EN 1992-1-1 (2011, 6.2.2). The parameter 𝜌𝑠 denotes the reinforcement ratio 𝜌𝑠 = 𝑎𝑠𝜑 ∕𝑑 with 𝑎𝑠𝜑 from Eq. (9.79). A typical value is 𝑎𝑠𝜑 = 5 cm2 ∕m (Figure 9.9) and 𝜌𝑠 = 5 ⋅ 10−4 ∕0.26 = 0.002. This leads to 𝑣𝑅𝑑,𝑐 = 0.089 MN∕m2 . The actual example values are not very precise but clearly fall below this value (Figure 9.12b). Insofar, a slab shear reinforcement is not required. In some cases it is, and |𝑣1 | > 𝑣𝑅𝑑,𝑐 and |𝑣2 | > 𝑣𝑅𝑑,𝑐 corresponding values 𝑎𝑠𝑣,1 , 𝑎𝑠𝑣,2 have to be superposed. A special occurrence of shear arises with punching, i.e. high shear forces near single points like supports. A design value for punching results from a computed support reaction; see, e.g. Figure 9.7b from Example 9.1. Design for punching follows common methods of reinforced concrete (EN 1992-1-1 2004, 6.4).

9.8 Nonlinear Kirchhoff Slabs 9.8.1 Basic Approach The analysis of Kirchhoff slabs is based on Eqs. (9.27) and (9.32). A linear elastic material behaviour according to Eq. (9.34) was used in the preceding examples.

9.8 Nonlinear Kirchhoff Slabs

Nonlinearity of moments may be treated within the framework of multi-axial elastoplasticity (Section 6.5.1) with 𝝈, 𝝐 according to Eq. (9.32), C from Eq. (9.34) applied for E in Eq. (6.61), and matched formulations 4) for yield function 𝐹 and flow potential 𝐺. A simplified approach to describe nonlinear moment–curvature behaviour is used in the following. Nonlinear behaviour in a general form is treated by the tangential material stiffness matrix (Eq. (9.37)). An incremental approach formally following Eq. (9.34) with 𝜈 = 0 is chosen as 𝝈̇ = C𝑇 ⋅ 𝝐̇ ,

⎡𝐾𝑇𝑥 ⎢ C𝑇 = ⎢ 0 ⎢ 0 ⎣

0 𝐾𝑇𝑦 0

0 ⎤ ⎥ 0 ⎥ ⎥ 𝐾𝑇𝑥𝑦 ⎦

(9.82)

with the bending stiffness 𝐾𝑇𝑥 , 𝐾𝑇𝑦 according to uniaxial beam behaviour. This corresponds to an orthotropic characteristic. An approach for the twisting stiffness 𝐶𝑇33 = 𝐾𝑇𝑥𝑦 is based on the theory of orthotropic slabs (Girkmann 1974, 119.) leading to √ (9.83) 𝐾𝑇𝑥𝑦 = 𝛼twist 𝐾𝑇𝑥 𝐾𝑇𝑦 with 𝛼twist = 0.5 recommended. This implies the isotropic case with 𝜈 = 0 and 𝐾𝑇𝑥 = 𝐾𝑇𝑦 (Eqs. (9.114-6 )). A value 𝛼twist = 0 yields a slab without twisting stiffness. The slab cross-sections with normals in the 𝑥 and 𝑦-directions (Figure 9.1b) are treated separately to derive 𝐾𝑇𝑥 , 𝐾𝑇𝑦 , and each is determined as for a beam crosssection of unit width. This implies that an orthogonal reinforcement is arranged in the coordinate directions, or that coordinate directions are aligned to an orthogonal reinforcement the other way around. Elasto-plastic moment–curvature relations are assumed for the following, see Figures 3.11 and 4.4 with 𝑁 = 0, applied to each direction. The material behaviour is described by • • • •

Initial bending stiffness 𝐾0 Initial yielding moment 𝑚𝑦𝑘 Hardening bending stiffness 𝐾𝑇 Current plastic curvature 𝜅𝑝 as a state parameter.

These parameters behave in analogy to uniaxial elasto-plastic behaviour (Eqs. (3.42) and (3.43)). With respect to reinforced concrete, they may be determined with a numerical analysis of moment–curvature relations, as is demonstrated with Example 4.1. Analytical approaches for moment–curvature relations based on simplifying assumptions are derived in the following.

9.8.2 Simple Moment–Curvature Behaviour Relations for a simplified case with reinforcement on the tension side only have already been described in Section 9.7.1. The internal moment for bending is obtained 4) The plastic state parameter 𝜅𝑝 should be distinguished from plastic curvature if necessary.

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with (Figure 9.8) 𝑚 = 𝑡𝑠 𝑧 ,

𝑡𝑐 = 𝑡𝑠

(9.84)

with a reinforcement tension force 𝑡𝑠 , a concrete compression force 𝑡𝑐 (unsigned), and the internal lever arm 𝑧. A force 𝑡𝑠 = 𝑎𝑠 𝑓𝑦𝑘 is obtained in the case of reinforcement yielding with the reinforcement cross-sectional area per unit width 𝑎𝑠 and the yield strength of the reinforcement 𝑓𝑦𝑘 . Thus, using Eqs. (9.66) and (9.67) the compression zone height is derived with 𝑥=

𝑎𝑠 𝑓𝑦𝑘

(9.85)

𝜒𝑓𝑐 𝑘

and the internal lever arm with 𝑧=𝑑−

1 𝑎𝑠 𝑓𝑦𝑘 𝑘𝑥 =𝑑− 2 2 𝜒𝑓𝑐

(9.86)

with an effective height 𝑑, compression zone height 𝑥, concrete strength 𝑓𝑐 , and coefficients 𝜒, 𝑘 (Section 9.7.1). This implicitly assumes that the strength of concrete and reinforcement are utilised simultaneously and results in a yield moment 𝑚𝑦𝑘 = 𝑎𝑠 𝑓𝑦𝑘 𝑧

(9.87)

A common case is yielding of reinforcement without fully utilizing the concrete strength. But with a small compression zone height 𝑥 – this is also a common case – the difference is generally small for the current assumptions. Furthermore, considering Eq. (9.71) twisting moments are neglected with this approach. This is justified insofar as twisting moments are generally small when ‘normal stress’ moments are large by absolute value. With respect to Eq. (4.18), the yield curvature is obtained with 𝜅𝑦 =

𝜖𝑦 𝑑−𝑥

,

𝜖𝑦 =

𝑓𝑦𝑘 𝐸𝑠

(9.88)

with Young’s modulus 𝐸𝑠 of the reinforcement and the initial bending stiffness with 𝐾0 =

𝑚𝑦𝑘 𝜅𝑦

(9.89)

A hardening can be considered with a hardening of the reinforcement stress up to reaching its strength 𝑓𝑡 at a strain 𝜖𝑡 . This leads to 𝑚𝑡 , 𝜅𝑡 (Eqs. (9.87) and (9.88)) and to a hardening bending stiffness 𝐾𝑇 =

𝑚𝑡 − 𝑚𝑦𝑘 𝜅𝑡 − 𝜅𝑦

(9.90)

These equations can be applied to each cross-section direction with its own 𝑎𝑠 and 𝑑 to derive the coefficients of Eqs. (9.82) and (9.83). The application is demonstrated with the following example.

9.8 Nonlinear Kirchhoff Slabs

Example 9.4: Elasto-Plastic Slab with Opening and Free Edges

We refer to Example 9.1 with the same geometry, boundary conditions, and loading. Furthermore, reinforced concrete is assumed with the following parameters: • Concrete 𝑓𝑐 = 17 MN∕m2 , 𝜒 = 0.95, 𝑘 = 0.8. • Reinforcement 𝐸𝑠 = 200 000 MN∕m2 , 𝑓𝑦𝑘 = 435 MN∕m2 , 𝑓𝑡 = 480 MN∕m2 . • Lower rebar cross-sectional areas and positions 𝑎𝑠𝑥 = 𝑎𝑠𝑦 = 5.13 cm2 ∕m, 𝑑𝑥 = 𝑑𝑦 = 0.26 m throughout the whole ground view. • The same upper rebar layout for appropriate areas (Figure 9.9b). This leads to a compression zone height and internal lever arm (Eq. (9.86)) 𝑥𝑥 = 𝑥𝑦 = 0.0173 m

𝑧𝑥 = 𝑧𝑦 = 0.2531 m

(9.91)

and, furthermore, to a yield moment (Eq. (9.87)) for both directions 𝑚𝑦𝑘,𝑥 = 𝑚𝑦𝑘,𝑦 = 5.13 ⋅ 10−4 ⋅ 435 ⋅ 0.2531 = 0.056 MNm∕m

(9.92)

and to a yield curvature (Eq. (9.88)) 435

𝜅𝑦𝑘,𝑥 = 𝜅𝑦𝑘,𝑦 =

200 000

0.26 − 0.0173

= 0.896 ⋅ 10−2 m−1

(9.93)

The initial bending stiffness is given with 𝐾0,𝑥 = 𝐾0,𝑦 =

0.056 = 6.30 MNm2 ∕m 0.896 ⋅ 10−2

(9.94)

according to Eq. (9.89), and in a similar way, the hardening bending stiffness with 𝐾𝑇 = 0.03 MNm2 ∕m (Eq. (9.90)). The twisting coefficient (Eqs. (9.82) and (9.83)) is chosen with 𝛼twist = 0.5 as a sufficient upper reinforcement and is arranged in areas with high twisting moments, which are indicated in Figures 9.6b and 9.9b. The same discretisation is used as in Example 9.1. An incrementally iterative scheme with Newton–Raphson iteration within each loading increment (Section 2.8.2) is used as solution method. The computed principal moments for the final loading are shown in Figure 9.13a. Slight redistribution of moments can be

(a)

(b)

Figure 9.13 Example 9.4. (a) Discretisation and principal moments. (b) Deflections [m].

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9 Slabs

observed; compare Figure 9.13a with Figure 9.6b for the linear elastic case. But moments 𝑚𝑥 , 𝑚𝑦 do not exceed the prescribed yield limit of 𝑚𝑦,𝑖 = 0.056 MNm∕m. The computed deflections are shown in Figure 9.13b. The maximum displacements increase by a factor of approximately 11 compared to the linear elastic case of Example 9.1. This corresponds to the bending stiffness relations, which are 74.8 MNm2 ∕m (Eq. (9.12)) for the linear elastic case and 6.3 MNm2 ∕m for the example. It has to be noted that the displacement behaviour is sensitive with respect to the twisting stiffness (Eq. (9.83)). This topic is treated in Example 9.5. Regardless of this, the computed displacements have to be considered as an upper bound for realistic values. The current approach neglects that • A relevant portion of the slab with lower moments will not crack due a activated tensile strength of concrete. • Tension stiffening (Sections 3.7 and 4.7) is not considered in the case of cracking. The simple bilinear moment–curvature relation should be extended with an initial uncracked stage, leading to a trilinear form starting with two elastic parts followed by an elasto-plastic part. Furthermore, the reinforcement stress–strain relation for its second cracked elastic part should be shifted according to Eq. (4.158). These extensions require some effort, but the implementation is straightforward and is not pursued further. The proof and design for bending are coupled in Example 9.4, as the chosen reinforcement may influence the computed moment to some degree. This is in contrast to Example 9.2, where a reinforcement is not a parameter for the computation of moments in the linear elastic analysis. On the other hand, the proof and design for shear can be performed according to Section 9.7.2, whereby the elastic moments in Eq. (9.74) have to be replaced by elasto-plastic moments. A shear reinforcement is not a parameter for the moment computation of Example 9.4. The approach for moments depending on curvature according to Eq. (9.82) is not isotropic as is defined in Section 6.3.1. A theoretical transformation of, e.g. the data of Example 9.4 in a rotated coordinate system followed by applying Eq. (9.82) in the diagonal form and concluded by back transformation of results in analogy to Eqs. (9.56) and (D.12), will lead to different moments compared to Example 9.4. Such differences should not be too large.

9.8.3 Extended Moment–Curvature Behaviour An extension is made by regarding a compressive reinforcement and a bilinear course of concrete compression stresses, see Figure 9.14, as proposed by CEB-FIP2 (2012, 7.2.3.1.5), where a concrete tensile strength is neglected. This introduces a strain 𝜖𝑐3 as a material parameter. In contrast to the simple approach (Section 9.8.2), a full utilisation of the concrete strength in the compression zone is not implicitly assumed but the actual concrete utilisation comes as a result. Furthermore, a compressive reinforcement may be considered.

9.8 Nonlinear Kirchhoff Slabs

Figure 9.14 Concrete bending with tensile and compressive reinforcement.

The reinforcement strain 𝜖𝑠1 on the tensile side and the compression zone height 𝑥 are used as key values. As longitudinal strains are assumed as linear along the cross-section height, the maximum concrete strain (unsigned) and the compressive reinforcement strain (unsigned) are given by 𝜖𝑐 =

𝑥 𝜖 , 𝑑 − 𝑥 𝑠1

𝜖𝑠2 =

𝑥 − 𝑑𝑠2 𝜖 𝑑 − 𝑥 𝑠1

(9.95)

with the effective height 𝑑 and the compressive reinforcement edge distance 𝑑𝑠2 . The tensile reinforcement is assumed to start yielding with a force 𝑡𝑠1 = 𝑎𝑠1 𝑓𝑦𝑘

(9.96)

with the yield strength 𝑓𝑦𝑘 . A constant concrete stress is given as extending over a length 𝑥1 𝑥1 = 𝑥 − 𝑥2 ,

𝑥2 =

𝜖𝑐3 (𝑑 − 𝑥) 𝜖𝑠1

(9.97)

with a prescribed strain 𝜖𝑐3 , leading to resulting forces 𝑡𝑐1 = 𝑥1 𝑓𝑐 ,

𝑡𝑐2 =

1 𝑥 𝑓 , 2 2 𝑐

𝑡𝑠2 = 𝑎𝑠2 𝐸𝑠 𝜖𝑠2

(9.98)

with the compressive strength of concrete 𝑓𝑐 and the reinforcement Young’s modulus 𝐸𝑠 . The upper reinforcement is assumed to behave as elastic with respect to 𝑡𝑠2 . This should be subject to a control and modifications when appropriate. The condition for pure bending 𝑡𝑐1 + 𝑡𝑐2 + 𝑡𝑠2 = 𝑡𝑠1 results in a quadratic equation in 𝑥 𝑎𝑠1 𝑓𝑦𝑘 − 𝑎𝑠2 𝐸𝑠 𝜖𝑠1

𝑥 − 𝑑𝑠2 1 𝜖𝑐3 (𝑑 − 𝑥)] = 0 − 𝑓𝑐 [𝑥 − 2 𝜖𝑠1 𝑑−𝑥

(9.99)

A state of yielding is determined by prescribed values 𝑓𝑐 , 𝑓𝑦𝑘 , 𝜖𝑐3 and 𝜖𝑠1 = 𝜖𝑦 with the reinforcement yield strain 𝜖𝑦 (Eq. (9.88)). With 𝑎𝑠1 , 𝑎𝑠2 , 𝑑, 𝑑𝑠1 given, a solution for the compression zone height 𝑥 is obtained, generally the smaller one of two. Thus,

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the resulting yield moment – with reference to the tension reinforcement layer – is determined with 𝑚𝑦𝑘 = 𝑡𝑠2 (𝑑 − 𝑑𝑠2 ) + 𝑡𝑐1 (𝑑 −

𝑥1 𝑥 ) + 𝑡𝑐2 (𝑑 − 𝑥1 − 2 ) 2 3

(9.100)

to be applied to each of the 𝑥 and 𝑦-directions with 𝑡𝑐1 , 𝑡𝑐2 , 𝑡𝑠2 given from Eq. (9.98). The yield curvature 𝜅𝑦 is determined as with Eq. (9.88). We obtain the initial bending stiffness 𝐾0 by Eq. (9.89) and the hardening stiffness by Eq. (9.90), whereby 𝑓𝑦𝑘 , 𝜖𝑦 have to be replaced by reinforcement ultimate values 𝑓𝑡 , 𝜖𝑡 to determine 𝑚𝑡 , 𝜅𝑡 . A special case has to be treated if the concrete load is relatively low, with 𝜖𝑐 ≤ 𝜖𝑐3 , 𝑥1 = 0, 𝑡𝑐1 = 0. Equations (9.95) and (9.96) are still valid, but the remaining relations reduce to 𝑡𝑐 =

1 𝑥 𝐸𝑐 𝜖𝑐 , 2

𝑡𝑠2 = 𝑎𝑠2 𝐸𝑠 𝜖𝑠2

𝑥 − 𝑑𝑠2 𝑥 1 𝜖 − 𝑥 𝐸𝑐 𝜖 =0 𝑑 − 𝑥 𝑠1 2 𝑑 − 𝑥 𝑠1 ) ( 𝑥 (𝑑 − 𝑑𝑠2 ) + 𝑡𝑐 𝑑 − 3

𝑎𝑠1 𝑓𝑦𝑡 − 𝑎𝑠2 𝐸𝑠 𝑚𝑦𝑘 = 𝑡𝑠2

(9.101)

to be solved for 𝑥, further for 𝑡𝑠1 , 𝑡𝑠2 , 𝑡𝑐 , and finally for 𝜅𝑦 , 𝑚𝑦𝑘 . The application is demonstrated with the following example, whereby also performing a small parameter study about the influence of the twisting stiffness (Eq. (9.83)). Example 9.5: Simple RC Slab under Concentrated Loading

We consider a simulation of a quadratic slab exposed to displacement controlled concentrated loading in its centre, as is typical for an experimental investigation. The influence of the twisting parameter 𝛼twist (Eq. (9.83)) on the load bearing behaviour is to be examined. • The span and geometric height are given with 𝐿 = 8 m and ℎ = 0.25 m. • The concrete’s material properties are 𝑓𝑐 = 38 MN∕m2 , 𝐸𝑐 = 33 600 MN∕m2 , 𝜖𝑐3 = 𝑓𝑐 = 1.13 ⋅ 10−3 for concrete grade C 30 (CEB-FIP2 2012, 5.1). 𝐸𝑐

• Ultimate concrete failure is assumed when the upper edge concrete strain reaches a value of 𝜖𝑐 = 𝜖𝑐𝑢 = 3.5 ⋅ 10−3 (CEB-FIP2 2012, 7.2.3.1). This has to be considered in addition to the ultimate moment 𝑚𝑡 due to a tension side reinforcement failure. • The lower reinforcement is chosen with 𝑎𝑠1 = 10 ⋅ 10−4

m2 m

with a effective height

𝑑 = 0.225 m and a concrete cover 𝑑𝑠1 = 0.025 m (Figure 9.14). The same reinforcement is chosen for 𝑥 and 𝑦-directions. • The upper reinforcement is chosen with 𝑎𝑠2 = 5 ⋅ 10−4 in both directions.

m2 m

, 𝑑 = 0.225, 𝑑𝑠2 = 0.025

Two sets of reinforcement material properties have to be regarded for the yield limit and the ultimate limit. With the aforementioned parameters, they lead to limit moments, limit curvatures, and furthermore to initial bending stiffness and hardening bending stiffness (Eqs. (9.89) and (9.90)). The example values are listed in

9.8 Nonlinear Kirchhoff Slabs

Table 9.2 Example 9.5. Reinforcement material parameters and derived quantities.

Strength

𝑓𝑖

Tens. reinf. strain 𝜖𝑖 Comp. concrete strain 𝜖𝑐 Comp. zone height 𝑥∕𝑑 Limit moment

𝑚𝑖

Limit curvature

𝜅𝑖

Bending stiffness

𝐾0

Bending stiffness

𝐾𝑇

[ MN ] m2

— — — [

MNm

]

[ 1 m] m

[ MNm2 ] m

Positive moment i=t i=y

Negative moment i=y i=t

500

550

500

0.0025 0.62 ⋅ 10−3 0.2000

0.025 0.0025 0.025 2.51 ⋅ 10−3 0.42 ⋅ 10−3 2.04 ⋅ 10−3 0.0913 0.1445 0.0756

0.1047

0.1202

0.0533

0.0637

0.0139

0.1223

0.0130

0.1202

7.54

550

4.10

[ MNm2 ]

0.14

m

0.10

Table 9.2, separated for positive and negative bending. The ultimate limit 𝑖 = 𝑡 results in 𝑥 < 𝑑𝑠2 for both, i.e. the ‘compressive’ reinforcement is under tension, which leads to negative values 𝑡𝑠2 in the above relations with very small compression zone heights. A quarter symmetry is utilised for a discretisation with triangular slab elements (Section 9.5.2) with the improved form proposed by Specht (1988). The discretisation with 64 elements, 41 nodes, and 103 unconstrained nodal degrees of freedom is shown in Figure 9.15a. The loading is given with a prescribed centre deflection with a target value 𝑤 = 0.2 m, which corresponds to a span–deflection ratio of 40. The computation is stopped if an ultimate limit moment is reached at any position. Figure 9.15b shows the computed load deflection curves for 𝛼twist = 0.5, 0.25, 0. Ending bullets mark the last computation steps. The value 𝛼twist = 0 indicates the totally lost twisting stiffness and the value 𝛼twist = 0.5 the full twisting stiffness. This has a significant influence on the results. Computed principal moments are shown in Figure 9.16. Twisting moments related to the 𝑥, 𝑦-system are indicated by diagonal principal moment directions and values with opposite sign. They arise in the

(a)

(b)

Figure 9.15 Example 9.5. (a) Discretisation. (b) Load deflection–behaviour.

321

322

9 Slabs

(a)

(b)

(c)

Figure 9.16 Example 9.5. Principal moments; for magnitudes of values, see Table 9.2. (a) 𝛼twist = 0.5. (b) 𝛼twist = 0.25. (c) 𝛼twist = 0.

lower left-hand corner of the lower left-hand slab quarter investigated. Large differences can be seen for twisting moment values and their spatial extension with respect to full twisting stiffness 𝛼twist = 0.5, reduced stiffness 𝛼twist = 0.25, and no stiffness 𝛼twist = 0 without twisting moments. A value 𝛼twist = 0.25 seems to be appropriate, following the remarks related to Eq. (9.83). These conclusions are essentially correct, but there is a pitfall with respect to the result values. This becomes obvious when considering a yield line approach as with Example 9.7 to determine an upper loading limit. It is evaluated with 𝑃 = 0.96 [MN] and falls below the loading for 𝛼twist = 0.5; see Figure 9.15b. Actually, the discretisation (Figure 9.15a) is too coarse and yields a system that is too stiff. A system that is too stiff is connected with moments that are too large for a prescribed displacement. A refined discretisation would scale the curves of Figure 9.15b below the upper limit. For the sake of brevity, no convergence study is presented here. The above conclusions remain unchanged from a qualitative point of view. Again, it proves useful to include alternative models in order to control the results of nonlinear calculations. Again we see a considerable model uncertainty with Example 9.5, as is also demonstrated with the Examples 8.2 and 10.2. But now it does not concern different multiaxial stress–strain models for concrete, but the twisting stiffness of slabs based on integrated uniaxial laws for concrete and reinforcing steel.

9.9 Upper Bound Limit State Analysis Nonlinear calculations for cracked reinforced concrete slabs may become complex even for simple geometries. A rigid-plastic analysis is also permitted by current codes (EN 1992-1-1 2004, 5.6). This is based upon the limit theorems of plasticity (Section 5.4). Theory and application are provided with • Rigid-plastic strut-and-tie models (Section 5.4). • Lower bound limit state analysis for plates (Section 8.1, Example 8.1). • Lower bound limit state analysis for slabs (Section 9.7, Examples 9.2, 9.3).

9.9 Upper Bound Limit State Analysis

Another application is given with the yield line method. This method divides a slab into rigid parts connected by lines, where moments have their ultimate limit value when exposed to an external loading. The choice of such lines is basically arbitrary. The external loading is given by 𝑝 = 𝜆 𝑝0 with a distributed reference loading 𝑝0 and a scalar loading factor 𝜆. The corresponding system of external loading and internal forces generally cannot be assumed to be in equilibrium. It is difficult to find an equilibrating system for slabs, as they are highly statically indeterminate regarding internal forces. Thus, the lower bound theorem 1 (Section 5.4) cannot be applied. On the other hand, the lines of yielding act as plastic hinges, and the kinematic compatibility is preserved. Thus, the upper bound theorem 2 may be applied. The procedure for slabs with yield lines is outlined as follows: • With thickness, concrete cover, reinforcement, and concrete grade given, limit moments 𝑚𝑢 can be derived. • A yield line geometry is assumed parameterised by shape parameters, which serve as minimisation variables. The path along the yield lines is measured with a variable 𝑠. With shape parameters given, the deflection is ruled by the deflection 𝑤 ⋆ of one distinct point in the slab reference plane. • Assuming small displacements the rotations 𝜃 along the yield lines depend on 𝑤 ⋆ in a linear way. Furthermore, the integration of ∫ 𝑚𝑢 𝜃 d𝑠 leads to an internal work 𝑊𝐼 . • The deflections 𝑤 of all rigid parts of the slab also depend linearly on 𝑤 ⋆ . Furthermore, integration of ∫ 𝑝0 𝑤 d𝑎 with 𝑎 indicating the ground area leads to an external work 𝑊𝐸 . • In analogy to Eq. (5.35), an upper limit of the admissible loading factor 𝜆 is given by upper limit of 𝜆 =

𝑊𝐼 ∫ 𝑚𝑢 𝜃 d𝑠 = 𝑊𝐸 ∫ 𝑝0 𝑤 d𝑎

(9.102)

The upper limit has to be minimised with respect to the shape parameters to gain a reliable estimation of the loading factor 𝜆. These items are explained with a simple set-up in the following. The determination of limit moments is already exemplarily demonstrated in the Examples 9.4 and 9.5 with Table 9.2. Furthermore, a rectangular slab is considered exemplarily (Marti 2013, 24.5.3); see Figure 9.17. It is simply supported along its edges, i.e. hinged without vertical displacements. A constant distributed reference loading 𝑞0 is used. Figure 9.17 also shows a sketch of the assumed yield lines, which are ruled to a large extent by symmetry in this simple case. A single shape parameter is given with the distance 𝑐. The deflection 𝑤 ⋆ is determined with the displacement of the centre line. It is connected with rotations 𝜙1 = 2

2𝑤 ⋆ , 𝑏

𝜙2−5 =

2𝑤 ⋆ 1 , 𝑏 cos 𝜑

𝜑 = arctan

𝑏 2𝑐

(9.103)

323

324

9 Slabs

Figure 9.17 Simple slab with yield lines.

along the yield lines. The lengths of the yield lines are given by 𝑠1 = 𝑎 − 2𝑐 ,

𝑠2−5 = 𝑐

1 cos 𝜑

(9.104)

and, finally, the directions of the yield lines by 𝜑1 = 0 ,

𝜑2 = 𝜑4 = 𝜑 ,

𝜑3 = 𝜑5 =

π +𝜑 2

(9.105)

The ultimate limit moments 𝑚𝑢,𝑥 , 𝑚𝑢,𝑦 of the 𝑥 and 𝑦-directions have to be transformed into a moment with stress components normal to the cross-section inclined with an angle 𝜑 against the 𝑥-axis. Thus, they have to be transformed with a rotation 𝜑 − π∕2. The transformation rules are given by Eq. (9.57) with 𝑚𝑢,𝑥𝑦 = 0. This results in (Marti 2013, 24.5.2.1, Figure 24.27) ( ( π) π) 2 𝑚𝑢,𝑥 + sin 𝜑𝑖 − 𝑚𝑢,𝑦 𝑚𝑢,𝑖 = cos2 𝜑𝑖 − 2 2 2 = sin 𝜑𝑖 𝑚𝑢,𝑥 + cos2 𝜑𝑖 𝑚𝑢,𝑦 𝑖 = 1, … , 5 for yield lines (9.106) and further to 𝑚𝑢,1 = 𝑚𝑢,𝑦 2

𝑚𝑢,2−5 = sin 𝜑 𝑚𝑢,𝑥 + cos2 𝜑 𝑚𝑢,𝑦

(9.107)

Thus, the internal work is given by 𝑊𝐼 =

5 ∑ 𝑖=1

𝑚𝑢,𝑖 𝜙𝑖 𝑠𝑖 = (

4𝑎 2𝑏 𝑚 ) 𝑤⋆ 𝑚 + 𝑐 𝑢,𝑥 𝑏 𝑢,𝑦

(9.108)

The determination of the external work 𝑊𝐸 requires the ground areas of the rigid parts 𝑎1,3 =

1 (𝑎 − 𝑐) 𝑏 , 2

𝑎2,4 =

1 𝑏𝑐 2

(9.109)

9.9 Upper Bound Limit State Analysis

The loading assigned to each part has a resultant in its centre of gravity (Figure 9.17). The deflections of these centres are given by 𝑤1,3 =

1 3𝑎 − 4𝑐 ⋆ 𝑤 , 6 (𝑎 − 𝑐)

𝑤2,4 =

1 ⋆ 𝑤 3

(9.110)

and the external work is determined by 𝑊𝐸 = 𝑝0

4 ∑ 𝑖=1

𝑎𝑖 𝑤𝑖 = 𝑝0 𝑏

(𝑎 2

−

𝑐) ⋆ 𝑤 3

(9.111)

Equation (9.102) finally provides the upper limit of the loading 4𝑎 2𝑏 𝑚 𝑚 + 𝑐 𝑢,𝑥 𝑏 𝑢,𝑦 upper limit of 𝑝 = upper limit of 𝜆 𝑝0 = (𝑎 𝑐 ) 𝑏 − 2 3

(9.112)

The minimum of the upper limit should yield a reasonable approximation for the admissible loading. The minimum value of Eq. (9.112) is derived by differentiation with respect to the shape parameter and equating the result to zero. This leads to √ 2 2 𝑏 𝑏 𝑚𝑢,𝑥 𝛽 + 3𝑚𝑢,𝑦 𝑚𝑢,𝑥 − 𝑚𝑢,𝑥 𝛽 , 𝛽= (9.113) 𝑐= 2 𝑚𝑢,𝑦 𝑎 This value is inserted into Eq. (9.112) to gain the minimum of the upper value. The application is demonstrated with the following example. Example 9.6: Simple RC Slab with a Yield Line Method and Distributed Loading

We consider a quadratic slab with spans and thickness 𝑎 = 𝑏 = 8.0 m, ℎ = 0.25 m. The slab is simply supported along its edges, i.e. hinged with zero vertical displacements. The material properties are given with a concrete grade C30/37 according to EC2 (EN 1992-1-1 2004, 3.1) with a characteristic concrete strength of 𝑓𝑐𝑘 = 30 MN∕m2 . Furthermore, reinforcing steel properties from EC2 (EN 1992-1-1 2004, 3.2) with a yield strength 𝑓𝑦𝑘 = 500 MN∕m2 . This yields design values 𝑓𝑐 = 17 MN∕m2 ,

𝑓𝑦 = 435 MN∕m2

(9.114)

for concrete and reinforcement strength. Reinforcement and effective height are assumed as 𝑎𝑠𝑥 = 𝑎𝑠𝑦 = 6.0 cm2 ∕m ,

𝑑 = 0.22 m

(9.115)

Resisting design limit moments are determined with Eqs. (9.85)–(9.87) and 𝑓 𝑐 = 𝜒𝑓𝑐 𝑘, 𝜒 = 0.95, 𝑘 = 0.8, 𝑓 𝑐 = 12.9 MN∕m2 , and 𝑥 = 𝑎𝑠

𝑓𝑦 𝑓𝑐

= 0.02 m ,

𝑧=𝑑−

𝑥 = 0.21 m 2

(9.116)

325

326

9 Slabs

is obtained as compression zone height, and 𝑚𝑢 = 𝑎𝑠 𝑓𝑦 𝑧 = 0.055 MNm∕m

(9.117)

as a limit moment. The shape parameter 𝑐 is determined by Eq. (9.113) and 𝑚𝑢𝑥 = 𝑚𝑢𝑦 = 𝑚𝑢 and 𝛽 = 1, leading to 𝑐 = 𝑏∕2 = 4.0 m. Finally, the admissible loading is given by inserting 𝑐 into Eq. (9.112) ⎛ 4𝑎 + 2𝑏 ⎞ 𝑐 𝑏 𝑝 = ⎜ ( 𝑎 𝑐 ) ⎟ 𝑚𝑢 = 0.0205 MN∕m2 ⎜𝑏 ⎟ − ⎝ 2 3 ⎠

→

20.5 kN∕m2

(9.118)

This value has to be compared to a design loading. Self-weight is given by 𝑔 = 0.25 ⋅ 25 = 6.25 kN∕m2 , and a variable service load of 𝑞 = 5.0 kN∕m2 is assumed. These are characteristic values, see Section 11.3, and are multiplied by safety factors for a design load of, e.g. 𝑝 = 1.35 ⋅ 6.25 + 1.50 ⋅ 5.0 = 16 kN∕m2 . This provides some margin. Serviceability or deflections, respectively, are not considered. This is out of scope for the yield line method. A calculation of the yield line method has to be performed with some care. An assumed yield line scheme may deviate from the respective real failure mechanism. Problems are generally not as simple as with Figure 9.17. Thus, even large variations of shape parameters may not allow modelling the real mechanism, leading to an overestimation of the admissible loading even in the case of correct determination of the minimum of an optimisation. In the following, another example is described as a reference for alternative simulation models, i.e. Examples 9.5 and 10.2. Example 9.7: Simple RC Slab with Yield Line Method and Concentrated Loading

We refer to Example 9.5 with the same system, materials, and boundary conditions. The yield line method is used to determine an upper limit for a concentrated midspan loading. The ultimate limit moment is adopted from Table 9.2 (pos. moment, 𝑖 = 𝑡) with 𝑚𝑢,𝑥 = 𝑚𝑢,𝑦 = 𝑚𝑢 = 0.12 MNm∕m

(9.119)

The quadratic slab is considered with the geometry 𝑎 = 𝑏 = 8.0 m, 𝑎 = 𝑏, 𝑐 = 𝑎∕2. The internal work is derived from Eq. (9.108) with 𝑊𝐼 = (

4𝑎 2𝑏 𝑚 ) 𝑤 ⋆ = 8 𝑚𝑢 𝑤 ⋆ = 0.96 𝑤 ⋆ [MNm] 𝑚 + 𝑐 𝑢,𝑥 𝑏 𝑢,𝑦

(9.120)

with a unit [𝑚] for 𝑤 ⋆ . The external work of a concentrated mid-span unit loading is given by 𝑊𝐸 = 1 𝑤 ⋆ MNm

(9.121)

9.9 Upper Bound Limit State Analysis

Equation (9.102) yields an upper limit for the loading factor upper limit of 𝜆 =

𝑊𝐼 = 0.96 𝑊𝐸

(9.122)

i.e. an upper limit for the ultimate loading 𝑃 = 0.96 [MN]. This is noteworthy because it does not depend on the absolute span values. A larger span increases the acting moment on the one hand, but also the total internal work due to increasing length to yield lines, on the other. A systematic approach for slabs is described in Damkilde and Krenk (1997), which fulfils both the upper bound theorem and the lower bound theorem. Thus, rigidplastic solutions are derived that fulfil equilibrium, kinematic compatibility, and material limit states – compare also Section 5.4 for rigid-plastic solutions for strutand-tie models. In the case that the method leads to a correct estimation of the admissible loading, a sufficient ductility has to be provided by reinforcement design to allow for the rotations that are necessary to reach limit moments. This issue has already been addressed for strut-and-tie models in Section 5.5 and for plates in Section 8.1.3. The yield line method is applicable with respect to limit loads of structures only. Aspects of serviceability are not covered. A comprehensive treatment of yield line methods is given in Nielsen and Hoang (2010).

327

329

10 Shells Whereas plates are characterised by membrane actions and slabs by bending actions, shells combine both. As a further extension, shells may be curved in space. A proven finite shell element allowing for these features within a continuum-based approach is described in the following. This approach permits the explicit modelling of reinforcement layers combined with a variety of nonlinear multi-axial material laws for concrete. Its application is demonstrated with reinforced concrete slabs exposing self-equilibrating membrane forces in addition to bending moments due to concrete cracking. This also highlights current potential and limitations of numerical simulation for reinforced concrete structures.

10.1 Geometry and Displacements Thin shells are treated as a further structural element type. Shell kinematics is quite complex (Green and Zerna 1954). Thus, deviating from what has been the standard way for structural elements up to now – kinematics, generalised material behaviour, equilibrium formulated in generalised forces, appropriate element types – a short track coupled to a standard finite shell element is described in the following. We use the continuum-based four-node shell element (Dvorkin and Bathe (1984), Bathe (1996, 5.4.2)). Shells include plates and slabs as special cases. In particular, they can model slabs exposed to the combined action of lateral and in-plane actions. This effect has already been discussed for cracked reinforced concrete beams in a simpler set-up, see Examples 4.2 and 4.4, and will be extended to cracked reinforced surface structures. A reference plane of a plate or a slab becomes a simply or doubly curved reference surface in the case of shells. The geometry of a surface in space is described by coordinates 𝑥1 = 𝑥1 (𝑟, 𝑠) ,

𝑥2 = 𝑥2 (𝑟, 𝑠) ,

𝑥3 = 𝑥3 (𝑟, 𝑠)

(10.1)

in a global Cartesian system with base vectors e1 , e2 , e3 . The indication of the global coordinate directions is changed compared to previous sections to facilitate the notation. Isoparametric coordinates 𝑟, 𝑠 serve as independent variables. A pair 𝑟, 𝑠 identifies a point of the reference surface or a shell position. Every shell position has Computational Structural Concrete – Theory and Applications, Second Edition. Ulrich Häussler-Combe. © 2023 Ernst & Sohn GmbH. Published 2023 by Ernst & Sohn GmbH.

330

10 Shells

(a)

(b)

Figure 10.1 Shell element (Dvorkin and Bathe 1984). (a) Geometry. (b) Local coordinate system.

a thickness ℎ. The reference surface and thickness occupy a shell body. Furthermore, every shell position has a shell director. This is a unit vector V𝑛 describing the direction of a cross-section. ◀

The validity of the Bernoulli–Navier hypothesis – stating that undeformed plane crosssections remain plane during a deformation – in the case of thin shells is assumed for cross-sections defined by shell directors.

A shell director may be chosen independently from the geometry definition given by Eq. (10.1). But generally it coincides with the normal of the reference surface in the case of smooth geometries. A local Cartesian coordinate system is defined for a shell position using, e.g. the unit vector e2 of the global coordinate system and the director multiplied with the vector cross product ×, leading to a unit vector V𝛼 ⎛ 𝑉𝑛3 ⎞ e2 × V𝑛 = ⎜ 0 ⎟ ⎜ ⎟ ⎝−𝑉𝑛1 ⎠ see Figure 10.1, and another unit vector V𝛽 e2 × V𝑛 V𝛼 = , |e2 × V𝑛 |

(10.2)

𝑉𝛼3 𝑉𝑛2 ⎛ ⎞ (10.3) V𝛽 = V𝑛 × V𝛼 = ⎜𝑉𝛼1 𝑉𝑛𝑧 − 𝑉𝛼3 𝑉𝑛1 ⎟ ⎜ ⎟ −𝑉𝛼1 𝑉𝑛2 ⎝ ⎠ Vectors V𝛼 , V𝛽 , V𝑛 in this sequence form an orthogonal, normalised, right-handed coordinate system, which adapts to the reference surface. The shell geometry is approximated by an isoparametric finite element interpolation. Nodes are placed on the reference surface spanning a mesh of quadrilateral elements, where each element has four nodes. The reference surface of a quadrilateral element must not be plane in space. The geometry of the undeformed shell is interpolated by ∑4 𝑡 ∑4 𝑥𝑖 (𝑟, 𝑠, 𝑡) = 𝑁 (𝑟, 𝑠) 𝑥𝑖𝐾 + ℎ 𝑁 (𝑟, 𝑠) 𝑉𝑛𝑖𝐾 , 𝑖 = 1, … , 3 𝐾=1 𝐾 𝐾=1 𝐾 𝐾 2 (10.4)

10.1 Geometry and Displacements

see Figure 10.1a, with 𝑥𝑖 𝑟, 𝑠 𝑡 𝑁𝐾 (𝑟, 𝑠)

𝑖-th coordinate of shell body local isoparametric coordinates within the reference surface local isoparametric coordinate transverse to the reference surface 1 = (1 + 𝑟𝐾 𝑟)(1 + 𝑠𝐾 𝑠) according to Eq. (2.19)

𝑟𝐾 , 𝑠𝐾 𝑥𝑖𝐾 ℎ𝐾 𝑉𝑛𝑖𝐾

local isoparametric coordinates of node 𝐾 𝑖-th coordinate of node 𝐾 shell thickness at node 𝐾 𝑖-th component of director at node 𝐾

4

The quantities 𝑥𝑖 , 𝑥𝑖𝐾 , ℎ𝐾 have dimensions of [length], while −1 ≤ 𝑟, 𝑠, 𝑡 ≤ 1 and 𝑟𝐾 , 𝑠𝐾 = ±1 are dimensionless. Equation (10.4) leads to a Jacobian similar to Eq. (2.37) ⎡𝐽11 ⎢ J = ⎢𝐽21 ⎢ 𝐽 ⎣ 31

𝜕𝑥

1 𝐽13 ⎤ ⎡ 𝜕𝑟 ⎥ ⎢ 𝜕𝑥2 𝐽23 ⎥ = ⎢ 𝜕𝑟 ⎥ ⎢ ⎢ 𝜕𝑥3 𝐽33 ⎦ ⎣ 𝜕𝑟

𝐽12 𝐽22 𝐽32

𝜕𝑥1

𝜕𝑥1

𝜕𝑠 𝜕𝑥2

𝜕𝑡 ⎥ 𝜕𝑥2 ⎥

𝜕𝑠 𝜕𝑥3

⎤ (10.5)

𝜕𝑡 ⎥ 𝜕𝑥3 ⎥

𝜕𝑠

𝜕𝑡

⎦

connecting global coordinates with local isoparametric coordinates. Its components are 𝜕𝑥𝑖 ∑4 𝑡 ∑4 𝑏 𝑥 + 𝑏 ℎ 𝑉 = 𝐾=1 𝑟𝐾 𝑖𝐾 𝐾=1 𝑟𝐾 𝐾 𝑛𝑖𝐾 2 𝜕𝑟 𝜕𝑥𝑖 ∑4 𝑡 ∑4 𝑏 𝑥 + 𝑏 ℎ 𝑉 = 𝐾=1 𝑠𝐾 𝑖𝐾 𝐾=1 𝑠𝐾 𝐾 𝑛𝑖𝐾 2 𝜕𝑠 𝜕𝑥𝑖 1 ∑4 𝑁 ℎ 𝑉 = 𝑘=1 𝐾 𝐾 𝑛𝑖𝐾 2 𝜕𝑡

𝑖 = 1, … , 3

(10.6)

with 𝑏𝑟𝐾 =

𝜕𝑁𝐾 1 = 𝑟𝐾 (1 + 𝑠𝐾 𝑠) , 4 𝜕𝑟

𝑏𝑠𝐾 =

𝜕𝑁𝐾 1 = 𝑠𝐾 (1 + 𝑟𝐾 𝑟) 4 𝜕𝑠

(10.7)

Shell displacements have to be interpolated in the next step. For this purpose, a small rotation 𝛼 around the vector V𝛼 (Eq. (10.2)) and a small rotation 𝛽 around the vector V𝛽 are introduced (Eq. (10.3)). This leads to a vector v v = −𝛼 V𝛽 + 𝛽 V𝛼

(10.8)

lying in the plane spanned by V𝛼 , V𝛽 ; see Figure 10.1b. In particular, the vector v is given at nodes with v𝐾 = −𝛼 V𝛽𝐾 + 𝛽 V𝛼𝐾 and V𝛽𝐾 , V𝛼𝐾 determined from Eqs. (10.2) and (10.3), with the particular director V𝑛𝐾 . A vector v𝐾 with components 𝑣𝑖𝐾 is used to displace a director V𝑛𝐾 , leading to a superposed interpolation of translational displacements 𝑢𝑖 (𝑟, 𝑠, 𝑡) =

∑4 𝐾=1

𝑁𝐾 (𝑟, 𝑠) 𝑢𝑖𝐾 +

𝑡 ∑4 ℎ 𝑁 (𝑟, 𝑠) 𝑣𝑖𝐾 , 𝐾=1 𝐾 𝐾 2

𝑖 = 1, … , 3 (10.9)

331

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in the same way as coordinates, whereby 𝑢𝑖 𝑖-th component of displacement 𝑢𝑖𝐾 𝑖-th component of displacement of node 𝐾 𝑣𝑖𝐾 = −𝛼𝐾 𝑉𝛽𝑖𝐾 + 𝛽𝐾 𝑉𝛼𝑖𝐾 ; 𝑖-th component of director change at node 𝐾 This realises the Bernoulli–Navier hypothesis for cross-sections defined by shell directors as the kinematic assumption for this type of shell.

10.2 Deformations Shell deformations are derived from shell displacements by their derivatives with respect to spatial coordinates. We start by taking into account the local isoparametric coordinates ⎛ 𝜕𝑢𝑖 ⎞ ⎜ 𝜕𝑟 ⎟ ∑ 4 ⎡𝑏𝑟𝐾 ⎢ 𝜕𝑢 𝑖 ⎟= ⎜ ⎢𝑏𝑠𝐾 ⎜ 𝜕𝑠 ⎟ 𝐾=1 ⎢ 0 ⎜ 𝜕𝑢𝑖 ⎟ ⎣ 𝜕𝑡 ⎠ ⎝

𝑡 𝑔𝛼𝑖𝐾 𝑏𝑟𝐾 𝑡 𝑔𝛼𝑖𝐾 𝑏𝑠𝐾 𝑔𝛼𝑖𝐾 𝑁𝐾

𝑡 𝑔𝛽𝑖𝐾 𝑏𝑟𝐾 ⎤ ⎛𝑢𝑖𝐾 ⎞ ⎥ 𝑡 𝑔𝛽𝑖𝐾 𝑏𝑠𝐾 ⎥ ⋅ ⎜ 𝛼𝐾 ⎟ , ⎥ ⎜ ⎟ 𝑔𝛽𝑖𝐾 𝑁𝐾 𝛽 ⎦ ⎝ 𝐾⎠

𝑖 = 1, … , 3

(10.10)

with rotation axes scaled by shell thickness ℎ𝐾 at nodes ⎛𝑔𝛼1𝐾 ⎞ ⎜𝑔 ⎟ = − 1 ℎ𝐾 𝛼2𝐾 2 ⎜ ⎟ 𝑔 ⎝ 𝛼3𝐾 ⎠

1 g𝛼𝐾 = − ℎ𝐾 V𝛽𝐾 , 2

g𝛽𝐾

⎛𝑉𝛽1𝐾 ⎞ ⎜𝑉 ⎟ 𝛽2𝐾 ⎜ ⎟ 𝑉 ⎝ 𝛽3𝐾 ⎠

⎛𝑔𝛽1𝐾 ⎞ ⎛𝑉𝛼1𝐾 ⎞ ⎜𝑔 ⎟ = 1 ℎ𝐾 ⎜𝑉 ⎟ 𝛽2𝐾 𝛼2𝐾 ⎜ ⎟ 2 ⎜ ⎟ 𝑔 𝑉 ⎝ 𝛽3𝐾 ⎠ ⎝ 𝛼3𝐾 ⎠

1 = ℎ𝐾 V𝛼𝐾 , 2

(10.11)

Equation (10.10) is transformed considering derivatives to with respect to global coordinates with the inverse of the Jacobian matrix 𝜕𝑢 ⎛ 𝑖⎞ ⎛ 𝜕𝑢𝑖 ⎞ ⎜ 𝜕𝑥1 ⎟ ⎜ 𝜕𝑟 ⎟ ⎜ 𝜕𝑢𝑖 ⎟ = J−1 ⋅ ⎜ 𝜕𝑢𝑖 ⎟ ⎜ 𝜕𝑥2 ⎟ ⎜ 𝜕𝑠 ⎟ ⎜ 𝜕𝑢𝑖 ⎟ ⎜ 𝜕𝑢𝑖 ⎟ ⎝ 𝜕𝑥3 ⎠ ⎝ 𝜕𝑡 ⎠

𝑖 = 1, … , 3

(10.12)

with

J−1

−1 ⎡𝐽11 ⎢ −1 = ⎢𝐽21 ⎢ −1 𝐽 ⎣ 31

−1 𝐽12 −1 𝐽22 −1 𝐽32

𝜕𝑟 ⎡ −1 ⎤ 𝐽13 ⎢ 𝜕𝑥1 −1 ⎥ = ⎢ 𝜕𝑠 𝐽23 ⎥ ⎢ ⎢ 𝜕𝑥1 −1 ⎥ 𝜕𝑡 𝐽33 ⎦ ⎢ ⎣ 𝜕𝑥1

𝜕𝑟 𝜕𝑥2 𝜕𝑠 𝜕𝑥2 𝜕𝑡 𝜕𝑥2

𝜕𝑟 ⎤ 𝜕𝑥3 ⎥ 𝜕𝑠 ⎥ ⎥ 𝜕𝑥3 ⎥ 𝜕𝑡 ⎥ 𝜕𝑥3 ⎦

(10.13)

10.2 Deformations

Equations (10.10), (10.12), and (10.13) are used to obtain the interpolation of the small strain tensor components 𝜖𝑖𝑗 =

𝜕𝑢𝑗 1 𝜕𝑢𝑖 + ( ), 2 𝜕𝑥𝑗 𝜕𝑥𝑖

𝑖, 𝑗 = 1, … , 3

(10.14)

This is identical to the strains of a three-dimensional body. The second-order strain tensor as a whole is given by E=

∑3 ∑3 𝑖=1

𝑗=1

𝜖𝑖𝑗 e𝑖 e𝑗

(10.15)

where e𝑖 e𝑗 is the tensor product of the global system of unit vectors e𝑖 and e𝑗 . Deformations according to Eq. (10.14) are measured in the global Cartesian system and are inconvenient for thin curved shell bodies. A covariant or so-called natural coordinate system is more suitable (Bathe 1996, p. 425). Its base vectors are formed by the Jacobian J as ⎛G11 ⎞ ⎛𝐽11 ⎞ G1 = ⎜G12 ⎟ = ⎜𝐽21 ⎟ , ⎜ ⎟ ⎜ ⎟ ⎝G13 ⎠ ⎝𝐽31 ⎠

⎛G21 ⎞ ⎛𝐽12 ⎞ G2 = ⎜G22 ⎟ = ⎜𝐽22 ⎟ , ⎜ ⎟ ⎜ ⎟ ⎝G23 ⎠ ⎝𝐽32 ⎠

⎛G31 ⎞ ⎛𝐽13 ⎞ G3 = ⎜G32 ⎟ = ⎜𝐽23 ⎟ ⎜ ⎟ ⎜ ⎟ ⎝G33 ⎠ ⎝𝐽33 ⎠

(10.16)

with the 𝑗-th component G𝑖𝑗 of base vector 𝑖 measured in the global Cartesian system. ◀

G1 is the tangential vector along the space curve given with varying r, while s, t are held constant, G2 the tangential vector along the curve with varying s and r , t constant and G3 with varying t and r , s constant.

The covariant system generally is skew and not normalised, i.e. G𝑖 ⋅ G𝑗 ≠ 0 for 𝑖 ≠ 𝑗 and G𝑖 ⋅ G𝑗 ≠ 1 for 𝑖 = 𝑗 with the vector dot product ⋅. Thus, a contravariant coordinate system is introduced with the base vectors 11 −1 ⎞ ⎛G ⎞ ⎛𝐽11 12 1 −1 ⎟ , G = ⎜G ⎟ = ⎜𝐽12 ⎜ 13 ⎟ ⎜ −1 ⎟ ⎝G ⎠ ⎝𝐽13 ⎠

21 −1 ⎞ ⎛G ⎞ ⎛𝐽21 22 2 −1 ⎟ , G = ⎜G ⎟ = ⎜𝐽22 ⎜ 23 ⎟ ⎜ −1 ⎟ ⎝G ⎠ ⎝𝐽23 ⎠

31 −1 ⎞ ⎛G ⎞ ⎛𝐽31 32 3 −1 ⎟ (10.17) G = ⎜G ⎟ = ⎜𝐽32 ⎜ 33 ⎟ ⎜ −1 ⎟ ⎝G ⎠ ⎝𝐽33 ⎠

utilizing the inverse J−1 of the Jacobian. Due to the definitions of G𝑖 , G𝑗 the properties G𝑖 ⋅ G𝑗 = 0 hold for 𝑖 ≠ 𝑗 and G𝑖 ⋅ G𝑗 = 1 for 𝑖 = 𝑗. The components G𝑖𝑗 of the covariant base form a second-order tensor. But it is not symmetric, i.e. G𝑖𝑗 ≠ G𝑗𝑖 . The same holds for the contravariant base: G𝑖𝑗 ≠ G𝑗𝑖 . Contravariant and covariant systems may also be formally derived for Cartesian coordinate systems, but then they coincide due to normalisation and orthogonality. Following Bathe (1996, 2.4, 6.5.2), Dvorkin and Bathe (1984) the strain as a unity is also written as E=

∑3 ∑3 𝑖=1

𝑗=1

˜ 𝜖𝑖𝑗 G𝑖 G𝑗

(10.18)

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with covariant strain components ˜ 𝜖𝑖𝑗 or natural strains. Writing indices of ˜ 𝜖𝑖𝑗 as subscripts and, thus, making these quantities ‘contravariant’ is pure convention but quite convenient in the context of tensor calculus. Another view on strains is 𝜖𝑖𝑗 =

3 3 ∑ ∑ 𝑖=1 𝑗=1

e𝑖 ⋅ E ⋅ e𝑗 ,

˜ 𝜖𝑖𝑗 =

3 3 ∑ ∑ 𝑖=1 𝑗=1

G𝑖 ⋅ E ⋅ G𝑗

(10.19)

Natural strain components have a dimension of [length2 ], as G𝑖 , G𝑗 each have a dimension of [length−1 ]. The identity of Eqs. (10.15) and (10.18) together with Eq. (10.14) yields ˜ 𝜖𝑖𝑗 =

3 3 ∑ ∑ 𝑟=1 𝑠=1

G𝑖𝑟 G𝑗𝑠 𝜖𝑟𝑠 =

3 𝜕𝑢𝑘 𝜕𝑢𝑘 1 ∑ + G𝑗𝑘 (G ), 2 𝑘=1 𝑖𝑘 𝜕𝜉𝑗 𝜕𝜉𝑖

𝑖, 𝑗 = 1, … , 3

(10.20)

with 𝜉1 = 𝑟, 𝜉2 = 𝑠, 𝜉3 = 𝑡. This may be shown using (1) G𝑖 ⋅ G𝑗 = 0 for 𝑖 ≠ 𝑗, (2) e𝑗 ⋅ G𝑖 = G𝑖 ⋅ e𝑗 = G𝑖𝑗 and (3) G𝑖𝑗 = 𝐽𝑗𝑖 = 𝜕𝑥𝑗 ∕𝜕𝜉𝑖 . Combining Eqs. (10.10) and (10.20) the interpolation of contravariant strain components is obtained by ˜ 𝝐=

4 ∑ 𝐾=1

B𝐾 ⋅ u𝐾

(10.21)

with ( ˜ 𝝐= ˜ 𝜖11

˜ 𝜖22

˜ 𝜖33

⎡G11 𝑏𝑟𝐾 G12 𝑏𝑟𝐾 ⎢ B𝐾 = ⎢G21 𝑏𝑠𝐾 G22 𝑏𝑠𝐾 ⎢ ⋮ ⋮ ⎣ ( u𝐾 = 𝑢1𝐾 𝑢2𝐾 𝑢3𝐾

2˜ 𝜖23

2˜ 𝜖13

2˜ 𝜖12

)T

G13 𝑏𝑟𝐾

𝑡 𝑏𝑟𝐾 H11𝐾

G23 𝑏𝑠𝐾

𝑡 𝑏𝑠𝐾 H21𝐾

⋮

⋮

𝛼𝐾

𝛽𝐾

)T

𝑡 𝑏𝑟𝐾 H12𝐾 ⎤ ⎥ 𝑡 𝑏𝑠𝐾 H22𝐾 ⎥ ⎥ ⋮ ⎦

(10.22)

and H11𝐾 = G11 𝑔𝛼1𝐾 +G12 𝑔𝛼2𝐾 +G13 𝑔𝛼3𝐾 ,

H12𝐾 = G11 𝑔𝛽1𝐾 +G12 𝑔𝛽2𝐾 +G13 𝑔𝛽3𝐾

H21𝐾 = G21 𝑔𝛼1𝐾 +G22 𝑔𝛼2𝐾 +G23 𝑔𝛼3𝐾 ,

H22𝐾 = G21 𝑔𝛽1𝐾 +G22 𝑔𝛽2𝐾 +G23 𝑔𝛽3𝐾

⋮

⋮

⋮

⋮ (10.23)

with G𝑖𝑗 according to Eqs. (10.5) and (10.16), 𝑏𝑟𝐾 , 𝑏𝑠𝐾 according to Eq. (10.7), both depending on 𝑟, 𝑠, and 𝑔𝑖𝑥𝐾 , 𝑔𝑖𝑦𝐾 according to Eq. (10.11), where 𝐾 indicates the element nodes. Equation (10.22) still includes a non-zero strain component ˜ 𝜖33 normal to the shell reference surface. This results from the continuum-based approach. Its absolute value should be considerably smaller compared to the other components in practical applications. With Eq. (10.22), the discretised strain state of every position in the shell body is ruled by five degrees of freedom per node. A so-called five-parameter shell model that corresponds to the Reissner–Mindlin shell kinematics is given.

10.3 Shell Stresses and Material Laws

10.3 Shell Stresses and Material Laws Strains have to be related to stresses to derive a structural resistance. The concept of Cauchy stresses (Section 6.2.2) is used for continuum-based shell formulations in contrast to beams and slabs. Stress components are introduced with respect to the global Cartesian system. The stress tensor as a unity is given by S=

3 3 ∑ ∑ 𝑖=1 𝑗=1

𝜎𝑖𝑗 e𝑖 e𝑗

(10.24)

in analogy to Eq. (10.15). Within the context of shells, it is appropriate to use the covariant system (Eq. (10.16)) as a base for stress components S=

3 3 ∑ ∑ 𝑖=1 𝑗=1

𝜎˜𝑖𝑗 G𝑖 G𝑗

(10.25)

with so-called contravariant stress components 𝜎˜𝑖𝑗 . The identity of Eqs. (10.24) and (10.25) yields 3 3 ∑ ∑

𝜎˜𝑖𝑗 =

𝑟=1 𝑠=1

𝑖𝑟 𝑗𝑠

G G 𝜎𝑟𝑠

(10.26)

This may be shown using (1) G𝑖 ⋅ G𝑗 = 0 for 𝑖 ≠ 𝑗, (2) e𝑗 ⋅ G𝑖 = G𝑖 ⋅ e𝑗 = G𝑖𝑗 . The motivation for introducing contravariant stress components is given by formulating the rate of the internal specific strain energy. It is defined as 𝑢̇ =

3 3 ∑ ∑ 𝑖=1 𝑗=1

𝜎𝑖𝑗 𝜖̇ 𝑖𝑗

(10.27)

in the global Cartesian system (Section 6.9). This particular formulation of the strain energy establishes the formulation of the principle of virtual displacements (Eq. (2.52)), which is a basis for the finite element method. Using the transformation rules (Eqs. (10.19) and (10.26)) and regarding G𝑖 ⋅ G𝑗 = 0 for 𝑖 ≠ 𝑗 and G𝑖 ⋅ G𝑗 = 1 for 𝑖 = 𝑗 it can be shown that 3 3 ∑ ∑ 𝑖=1 𝑗=1

𝜎𝑖𝑗 𝜖̇ 𝑖𝑗 =

3 3 ∑ ∑ 𝑖=1 𝑗=1

𝜎 ˜𝑖𝑗 ˜ 𝜖̇ 𝑖𝑗

(10.28)

i.e. contravariant stress components are complementary to covariant strain components from an energetic point of view. ◀

Using covariant or natural strain components to describe shell deformations it is mandatory to use contravariant stress components for weak equilibrium conditions like the principle of virtual displacements.

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To enable an adequate description of the material behaviour of shells it is appropriate to use the local system V𝛼 , V𝛽 , V𝑛 , as is introduced with the shell director V𝑛 and Eqs. (10.2) and (10.3). The local system is an orthogonal, normalised, and right-handed or Cartesian coordinate system, respectively. On the one hand, it leans against the shell reference surface, and it may change with every point of the reference surface, and in this way forms a local co-rotational coordinate system. On the other hand, it is appropriate for the description of material behaviour due to its normalisation and orthogonality. To facilitate the notation V1 = V𝛼 , V2 = V𝛽 , V3 = V𝑛 is used in the following for the local Cartesian coordinate system. ◀

Local stress components 𝜎ij and local strain components 𝜖 ij related to the local corotational Cartesian system are appropriate to formulate the stress–strain relations for thin shells.

In analogy to Eqs. (10.15) and (10.24), these stress and strain components are given by S=

3 3 ∑ ∑ 𝑖=1 𝑗=1

𝜎𝑖𝑗 V𝑖 V𝑗 ,

E=

3 3 ∑ ∑ 𝑖=1 𝑗=1

𝜖𝑖𝑗 V𝑖 V𝑗

(10.29)

As they are referenced in a Cartesian system, no distinction between contravariant and covariant components is necessary. To determine local material behaviour the natural strains ˜ 𝝐 , as they are derived from nodal displacements using Eq. (10.21), have to be transformed into local strains 𝝐. The identity of Eqs. (10.291 ) and (10.18) leads to 𝜖 𝑖𝑗 =

3 ∑ 3 ∑ 𝑟=1 𝑠=1

T𝑖𝑟 T𝑠𝑗 ˜ 𝜖𝑟𝑠

(10.30)

with T𝑖𝑗 = V𝑖 ⋅ G𝑗 ,

T𝑖𝑗 ≠ T𝑗𝑖

(10.31)

This may be written as a matrix operation 𝝐 = T ⋅˜ 𝝐

(10.32)

with 𝝐, ˜ 𝝐 ordered according to Eq. (6.3) and the components of T derived from Eq. (10.31). A linear elastic material behaviour is considered to begin with. The shell body differs from the three-dimensional continuum ruled by the linear elastic law (Eq. (6.23)) insofar as its normal stress in a plane normal to the reference surface should be negligible compared to all other stress components. Thus, we use 𝝈=C⋅𝝐

(10.33)

10.4 System Building

with 𝝈 ordered as in Eq. (6.7) and 𝐸𝜈

𝐸(1−𝜈)

⎡ ⎢ (1+𝜈)(1−2𝜈) ⎢ 𝐸𝜈 ⎢ (1+𝜈)(1−2𝜈) ⎢ ⎢ 0 C=⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ 0 ⎣

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

(1+𝜈)(1−2𝜈) 𝐸(1−𝜈) (1+𝜈)(1−2𝜈)

𝐸 2(1+𝜈)

0 0 𝐸 2(1+𝜈)

0

0 ⎤ ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 𝐸 ⎥ 2(1+𝜈) ⎦

(10.34)

according to Eq. (6.23), where the influence of the transverse components 𝜖 33 , 𝜎33 has been neglected due to the assumption of thin shells. Equation (10.33) may be generalised in incremental form as 𝝈̇ = C𝑇 ⋅ 𝝐̇

(10.35)

(Eq. (6.12)) with the tangential material stiffness matrix C𝑇 describing nonlinear behaviour as required. Local stress components 𝜎𝑖𝑗 have to be transformed into contravariant components 𝜎˜𝑖𝑗 . It can be shown that 𝝈 ˜ = TT ⋅ 𝝈

(10.36)

with 𝝈 ˜ ordered as in Eq. (6.7) and the transposed of the transformation matrix T from Eq. (10.32). This is based on the identity of Eqs. (10.291 ) and (10.25). Finally, the combination of Eqs. (10.36), (10.33), and (10.32) leads to ˜ ⋅˜ 𝝐=C 𝝐 𝝈 ˜ = TT ⋅ C ⋅ T ⋅ ˜

(10.37)

with a transformation law ˜ = TT ⋅ C ⋅ T C

(10.38)

for the material stiffness matrix or the tangential one, respectively.

10.4 System Building The current theory treats the shell body as a continuum with constraints considering the shell deformations. Thus, the general form (Eq. (2.52)) is used to describe weak equilibrium T

∫ 𝛿˜ 𝝐 ⋅𝝈 ˜ d𝑉 + ∫ 𝛿uT ⋅ ü 𝜚 d𝑉 = ∫ 𝛿uT ⋅ p d𝑉 + ∫ 𝛿uT ⋅ t d𝐴 𝑉

𝑉 T

𝑉

(10.39)

𝐴𝑡

where the product 𝛿˜ 𝝐 ⋅𝝈 ˜ replaces 𝛿𝝐 T ⋅ 𝝈 with ˜ 𝝐 according to Eq. (10.21) and 𝝈 ˜ according to Eq. (10.36). For the evaluation of integrals, see Eqs. (2.58)–(2.61). Integration is performed by numerical methods; the basic approach is described in

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Section 2.7. It is extended to the case of continuum-based shells. The integration of internal nodal forces is performed with +1 +1 +1

f𝐼 = ∫

BT (𝑟, 𝑠, 𝑡)

𝑉𝐼

⋅𝝈 ˜ (𝑟, 𝑠, 𝑡) d𝑉 = ∫ ∫ ∫ BT (𝑟, 𝑠, 𝑡) ⋅ 𝝈 ˜ (𝑟, 𝑠, 𝑡) 𝐽(𝑟, 𝑠, 𝑡) d𝑡 d𝑟 d𝑠 −1 −1 −1

(10.40) with B assembled with the B𝐾 ’s from Eq. (10.22) and 𝝈 ˜ from Eq. (10.36). The local isoparametric coordinates are given by 𝑟, 𝑠, 𝑡 and the determinant 𝐽 = det J of the Jacobian from Eq. (10.5). Internal nodal forces are determined numerically by f𝐼 =

𝑛𝑢 ∑ 𝑛𝑣 𝑛𝑢 ∑ ∑ 𝑖=0 𝑗=0 𝑘=0

𝜂𝑖 𝜂𝑗 𝜂𝑘 BT (𝑟𝑖 , 𝑠𝑗 , 𝑡𝑘 ) ⋅ 𝝈 ˜ (𝑟𝑖 , 𝑠𝑗 , 𝑡𝑘 ) 𝐽(𝑟𝑖 , 𝑠𝑗 , 𝑡𝑘 )

(10.41)

with integration orders 𝑛𝑢 , 𝑛𝑣 , sampling points 𝑟𝑖 , 𝑠𝑗 , 𝑡𝑘 , and weighting factors 𝜂𝑖 , 𝜂𝑗 , 𝜂𝑘 . Let us assume that a Gauss integration is used. It may be appropriate to use different integration orders 𝑛𝑢 across the reference surface with local coordinates 𝑟, 𝑠 and 𝑛𝑣 along the transverse direction with the local coordinate 𝑡. ◀

The transverse direction of thin shells needs a different treatment compared to the in-surface directions due to shell bending with transverse linear strains.

In the case of linear elastic material behaviour, integration orders 𝑛𝑢 = 𝑛𝑣 = 1 with two sampling points in each direction (Table 2.1) are appropriate, as stresses appear linearly. In the case of nonlinear material behaviour, stresses may vary nonlinearly with kinks and jumps. This occurs, in particular, for the cross-sections of cracked reinforced concrete and requires a higher integration order for the transverse direction, e.g. 𝑛𝑣 = 4 with five sampling points, while the in-surface directions may remain with 𝑛𝑢 = 1 and two sampling points in each direction. It may also be appropriate to choose a different integration scheme for the transverse directions, e.g. a Lobatto scheme, which yields a higher accuracy in cases with extreme integrand values at boundaries. As every node has five kinematic degrees of freedom, Eq. (10.40) leads to five components for internal forces f𝐼 at every node 𝐼, i.e. three force components with respect to the global coordinate system and two bending moment components with respect to the local directions V𝛼 , V𝛽 (Eqs. (10.2) and (10.3)). Prescribed distributed loads p (Eq. (10.39)) are given as forces per volume and prescribed surface tractions t as forces per area, each with directions related to the global coordinate system. The corresponding nodal forces again have five components for each node. Element stiffness and mass matrices have 20 × 20 entries with the four-node element. Assembling of element contributions is performed in the standard way (Section 9.3). Due to the continuum-based approach structural response is described by strains and stresses varying with the position in the reference surface and the transverse direction distance. Regarding shells and slabs, a more familiar approach is given with

10.5 Slabs and Beams as a Special Case

internal forces such as normal forces, moments, and shear forces. It is appropriate to refer them to the local co-rotational system. In analogy to slabs (Eq. (9.7)), resulting local internal forces are derived from local stresses 𝝈 (Eq. (10.33)) by 1

1

ℎ 𝑛1 = ∫ 𝜎 11 d𝑡 , 2

ℎ 𝑛2 = ∫ 𝜎22 d𝑡 , 2

−1

−1 1

ℎ2 ∫ 𝜎 11 𝑡 d𝑡 , 𝑚1 = − 4

−1

−1

ℎ2 ∫ 𝜎22 𝑡 d𝑡 , 𝑚2 = − 4

−1

ℎ ∫ 𝜎 13 d𝑡 , 2

𝑛12

1

−1

1

𝑣1 =

1

ℎ = ∫ 𝜎12 d𝑡 2 1

𝑚12

ℎ2 ∫ 𝜎12 𝑡 d𝑡 =− 4 −1

1

𝑣2 =

ℎ ∫ 𝜎23 d𝑡 2 −1

(10.42) with the local shell thickness ℎ and the isoparametric local coordinate −1 ≤ 𝑡 ≤ 1. ◀

Internal forces – not to be confused with internal nodal forces (Eq. (10.40)) – are a byproduct determined by postprocessing.

In practice, the integration is again performed numerically, and nonlinear material relations are automatically regarded. For the integration method and order see the remarks above concerning integration of system integrals.

10.5 Slabs and Beams as a Special Case A rectangular slab element of constant thickness ℎ is considered as a special case of the general shell element; see Figure 10.2. The director indicating cross-sectional ( )T directions is given by V𝑁 = V3 = 0 0 1 and completed to a local coordinate sys( )T ( )T tem by V𝛼 = V1 = 1 0 0 , V𝛽 = V2 = 0 1 0 . The local system coincides with the global coordinate system. Thus, after some calculations, the matrix B of

Figure 10.2 Slab element as a special case of a shell element.

339

340

10 Shells

interpolation functions (Eq. (10.22)) is written as ⎡𝐽11 𝑏𝑟𝐾 ⎢ ⎢ 0 ⎢ ⎢ ⎢ 0 B𝐾 = ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ 𝐽 𝑏 ⎣ 11 𝑠𝐾

0

0

ℎ 𝐽 𝑏 ⎤ 2 11 𝑟𝐾 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ ℎ 𝐽11 𝑁𝐾 ⎥ ⎥ 2 ⎥ ℎ 𝑡 𝐽11 𝑏𝑠𝐾 ⎦ 2

𝑡

0

ℎ 𝐽 𝑏 2 22 𝑠𝐾 0 0 0 (10.43) ℎ 0 𝐽33 𝑏𝑠𝐾 − 𝐽22 𝑁𝐾 2 0 0 𝐽33 𝑏𝑟𝐾 ℎ 𝐽22 𝑏𝑟𝐾 0 −𝑡 𝐽22 𝑏𝑟𝐾 2 with the components 𝐽𝑖𝑗 of the Jacobian matrix according to Eq. (10.5), 𝑏𝑠𝐾 , 𝑏𝑟𝐾 according to Eq. (10.7), and 𝑁𝐾 according to Eq. (10.4). This element formulation is suitable to treat coupled normal forces and bending moments for slabs. A plane beam element is derived as a special case, assuming that a displacement )T ( in the 𝑥1 − 𝑥3 -plane is applied with u1 = u4 = 𝑢1 0 𝑤1 0 𝛽1 , u2 = u3 = ( )T 𝑢2 0 𝑤2 0 𝛽2 leading to 𝐽22 𝑏𝑠𝐾

−𝑡

0

ℎ ℎ 1 1 ⎞ ⎛˜ − 𝐽11 𝑢1 − 𝑡 𝐽11 𝛽1 + 𝐽11 𝑢2 + 𝑡 𝐽11 𝛽2 𝜖11 ⎞ ⎛ 2 2 4 4 ⎟ ⎜ ⎟ ⎜ 𝜖22 ⎟ ⎜ 0 ⎟ ⎜˜ ⎟ ⎜˜ 𝜖33 ⎟ ⎜ 0 ⎟ ⎜ ⎟=⎜ 𝜖23 ⎟ ⎜ 0 ⎟ ⎜2˜ ⎜2˜ 𝜖13 ⎟ ⎜− 1 𝐽33 𝑤1 + ℎ 𝐽11 (1 − 𝑟)𝛽1 + 1 𝐽33 𝑤2 + ℎ 𝐽11 (1 + 𝑟)𝛽2 ⎟ 2 4 4 ⎟ ⎜ ⎟ ⎜ 2 2˜ 𝜖12 0 ⎠ ⎝ ⎠ ⎝

(10.44)

(Eqs. ((10.4)–(10.7) and 10.21)). Furthermore, 𝐽11 = 𝐿1 ∕2, 𝐽33 = ℎ∕2 (Eq. (10.5)), with the element length 𝐿1 (Figure 10.2). It is convenient to transform natural strain components ˜ 𝜖𝑖𝑗 back into the global Cartesian coordinate system. Transformation rules are again derived by the identity of Eqs. (10.15) and (10.18), and the transformation is given by ( )2 4 4 𝜖11 = 2 ˜ 𝜖11 , 𝜖13 = G11 G33 ˜ 𝜖13 = (10.45) 𝜖11 = G11 ˜ ˜ 𝜖13 𝐿 𝐿1 1ℎ This finally yields ⎡ ⎛ 𝜖11 ⎞ 1 ⎢−1 0 ⎜ ⎟= 𝐿 ⎢ 2𝜖 1 0 −1 ⎝ 13 ⎠ ⎣ ( ⋅ 𝑢1 𝑤 1 𝛽 1

ℎ 2 𝐿1 (1 − 𝑟) −𝑡

2

𝑢2

𝑤2

1

0

0

1

𝛽2

)T

ℎ ⎤ 2 ⎥ 𝐿1 (1 + 𝑟)⎥ 2 ⎦ 𝑡

(10.46)

and corresponds to the interpolation of strains of the 2D Timoshenko beam element (Eq. (4.74)). This becomes obvious with (1) setting 2𝜖13 = 𝛾, (2) considering the reversed orientation of rotations (Eq. (10.8)), and (3) adapting Eq. (4.5) ruling beam ℎ kinematics to the current case with 𝜖11 = 𝜖 + 𝑡 𝜅, −1 ≤ 𝑡 ≤ 1. 2

10.6 Locking

10.6 Locking Problems of artificial stiffening or locking were already mentioned for the Timoshenko beam element (Section 4.3.1). They will be demonstrated within the context of the shell element using the aforementioned simplified cases to make the locking problem comprehensible within a limited frame. A state of uniform bending in the longitudinal 𝑥1 -direction is applied to Eq. (10.46) with 𝛽2 = −𝛽1 = 𝛽∕2 and 𝑢1 = 𝑤1 = 𝑢2 = 𝑤2 = 0 and Eq. (10.46) yields 𝜖11 = 𝑡

ℎ 𝛽 , 2 𝐿1

𝛾13 =

1 𝛽𝑟, 2

−1 ≤ 𝑟, 𝑡 ≤ 1

(10.47)

with a longitudinal local coordinate 𝑟 and a transverse local coordinate 𝑡. The term ℎ 𝑧 = 𝑡 with the thickness ℎ describes the distance from the reference plane and 𝛽 𝐿1

2

corresponds to a curvature. With 𝜈 = 0, a linear elastic behaviour is assumed to

simplify, leading to 𝜎11 = 𝐸 𝜖11 = 𝐸 𝑡

𝛽 ℎ 𝛽 = 𝐸𝑧 , 2 𝐿1 𝐿1

1 𝐺 𝛽𝑟 2

𝜎13 = 𝐺 𝛾13 =

(10.48)

with a normal stress 𝜎11 in the longitudinal 𝑥1 -direction, a shear stress 𝜎13 in the 𝐸 vertical 𝑥3 -direction, Young’s modulus 𝐸, and 𝐺 = (Eqs. (6.3), (6.7), (6.13), and 2

(6.23)). This corresponds to stresses in beams with 𝜅 =

𝛽 𝐿1

(Eqs. (4.7) and (4.8)) and

leads to a resulting moment and shear force per unit width 𝑚=𝐸

ℎ3 𝛽 , 12 𝐿1

𝑣=

1 𝐺 ℎ 𝛽𝑟 2

(10.49)

The applied deformation results in a constant bending moment and a linearly varying shear force along the element. This obviously violates the equilibrium conditions locally, as a zero shear force is required throughout the element in the case of constant bending moment. A spurious transverse shear force arises with this type of element. The local error is measured by 𝐿1 𝑣 =3 2𝑟, 𝑚 ℎ

−1 ≤ 𝑟 ≤ 1

(10.50)

It becomes larger with more slender elements, i.e. with decreasing thickness or increasing element length. Local or strong equilibrium is not enforced within the finite element method but weak or integral equilibrium is (Section 2.5). As a consequence, nodal forces resulting from the integration of internal forces have to be in equilibrium. For the case

341

342

10 Shells

under consideration, nodal forces according to Eq. (10.40) are given by ⎡ −1 ⎢ ⎢ 0 1 1 ⎢ ℎ 1 ⎢−𝑡 2 r= ∫ ∫ ⎢ 𝐿1 ⎢ 1 −1 −1 ⎢ ⎢ 0 ⎢ ℎ 𝑡 ⎣ 2

⎤ ⎥ −1 ⎥ ℎ 𝛽 ⎥ 𝐿1 ⎞ ℎ (1−𝑟)⎥ ⎛𝐸 𝑡 𝐿1 2 ⋅ d𝑟 = 𝐸 ⎥ ⎜ 1 2 𝐿1 ⎟ d𝑡 2 0 ⎥ 𝐸𝛽𝑟 2 ⎠ ⎥ ⎝4 1 ⎥ ⎥ 𝐿1 (1+𝑟) 2 ⎦ 0

0 ⎛ ⎞ ⎜ ⎟ 0 ⎜ ℎ3 𝛽 ℎ ⎟ ⎜− 12 𝐿 − 24 𝐿1 𝛽 ⎟ 1 ⎜ ⎟ 0 ⎜ ⎟ ⎜ ⎟ 0 ⎜ ℎ3 𝛽 ℎ ⎟ + 𝐿 𝛽 ⎝ 12 𝐿1 24 1 ⎠ (10.51)

The non-zero nodal forces are conjugate to the applied rotation angle −𝛽∕2 on the left-hand side, 𝑟 = −1, and 𝛽∕2 on the right-hand side 𝑟 = 1. Insofar, moments are justified from a mechanical point of view whereby forming an equilibrated system. As was mentioned, 𝛽∕𝐿1 is a curvature. Furthermore, 𝐸ℎ3 ∕12 is the bending stiffness per unit width. Thus, the first term in each non-zero entry corresponds to a reasonable mechanical behaviour. The second part ±𝐸ℎ𝐿1 𝛽∕24 leads to an additional spurious moment resulting from the spurious shear force. This moment corresponds to an additional spurious stiffness of this element. The influence of these effects increases with decreasing thickness ℎ and a constant element length 𝐿1 , i.e. with increasing element slenderness. The spurious effects reduce with finer discretisations, i.e. decreasing 𝐿1 for a constant ℎ. A convergence is basically given, but it is reached very slowly with a large parameter 𝑐 (Eq. (2.111)). ◀

The thin continuum-based shell element (Section 10.1) yields models that are too stiff in practical applications due to spurious transverse shear forces. A transverse shear locking occurs.

A number of locking phenomena are known, e.g. • Transverse shear locking: spurious transverse shear forces in the case of transverse bending. • In-plane shear locking: spurious in-plane shear forces in the case of in-plane bending. • Membrane locking: spurious membrane forces in the case of transverse bending. As well as other ones (Bischoff 1999, 6.4). Transverse shear locking is a major cause for deficiencies of slab and shell elements. But it has the property that spurious shear stresses disappear in distinguished points of an element, e.g. in the point 𝑟 = 0 for the case in Section 10.5. Furthermore, shear force values that are reasonable from a mechanical point of view are given in these distinguished points. This motivates established approaches to avoid locking.

10.6 Locking

• Reduced integration of system integrals (Eqs. (2.58), (2.66), and (2.69)). This corresponds to an integration at 𝑟 = 0, 𝑠 = 0 for the element, where the reduced integration does not affect the integration order along the local 𝑡-axis. Albeit, a numerical instability of results as so-called hour glassing may occur with reduced integration. Whether hour glassing actually occurs depends on the discretised geometry and the boundary and loading conditions applied. • Interpolation of transverse shear strains with own fields applying a mixed interpolation. These fields are connected to the fields given by Eq. (10.21) through the values of ˜ 𝜖13 , ˜ 𝜖23 in those distinguished points with vanishing spurious transverse shear forces. Such points are given by the local element coordinates 𝐴 ∶ 𝑟 = 0, 𝑠 = 1, 𝐵 ∶ 𝑟 = −1, 𝑠 = 0, 𝐶 ∶ 𝑟 = 0, 𝑠 = −1 and 𝐷 ∶ 𝑟 = 1, 𝑠 = 0. The particular strains 𝐶 𝐴 𝐵 𝐷 ,˜ 𝜖13 , ˜ 𝜖13 ,˜ 𝜖13 and with respect to ˜ 𝜖23 in determined by Eq. (10.21) are given by ˜ 𝜖13 a corresponding way. Anchored by these values the fields for transverse shear strains are assumed with 1 𝐴 𝜖13 + (1 + 𝑠) ˜ 2 1 𝐷 𝜖23 ˜ 𝜖23 (𝑟, 𝑠) = (1 + 𝑟) ˜ + 2

˜ 𝜖13 (𝑟, 𝑠) =

1 𝐶 𝜖13 (1 − 𝑠) ˜ 2 1 𝐵 𝜖23 (1 − 𝑟) ˜ 2

(10.52)

The approach is called assumed natural strain (ANS) method (Dvorkin and Bathe 1984) and leads to a modification of rows 4 and 5 of the matrix B𝐾 (Eq. (10.22)). The modification is straightforward when evaluating Eq. (10.21) in points A, B, C, D and combining it with Eq. (10.52). Assuming strains as partially independent from displacements and applying mixed interpolations is not covered by the principle of virtual displacements (Eqs. (2.52) and (2.53)). Thus, an extended weak form like the principle of Hu-Washizu (Bathe 1996, 4.4.2) is required (Section 2.9). This involves fields for stresses and strains as independent solution variables. The application of the assumed strains requires an additive split of the matrix B from Eq. (10.22). The current approach of Eq. (10.52), which leads to an extension of Eq. (10.22) as indicated above, allows for such a split. This also allows for an elimination of the independent parts of the stress field in advance on the element level under reasonable assumptions. Thus, the mixed interpolation may finally be applied in the framework of the principle of virtual displacements with respect to the system level. The effects of shear locking and the correction with ANS are demonstrated with the following example. Example 10.1: Convergence Study for Linear Simple Slab

We consider a quadratic linear elastic slab with a span 𝐿 = 𝐿1 = 𝐿2 = 8.0 m, a thickness ℎ = 0.25 m, and material parameters 𝐸 = 33 000 MN∕m2 , 𝜈 = 0.2. The slab is simply supported along its edges, i.e. hinged with zero vertical displacements. A constant vertical loading is assumed with 𝑞 = 16 kN∕m2 downward.

343

344

10 Shells

Figure 10.3 Example 10.1. Quarter slab discretisation.

Assuming Kirchhoff theory (Section 9.3.3) an exact solution for this problem is described in Girkmann (1974, 78.b) with neglecting shear deformations. The maximum deflection in the centre point is given by an infinite double sum 𝑛π

𝑒 = 𝑤max

𝑚π

16𝑞 𝐿4 ∑ ∑ sin 2 sin 2 , , 𝐾 π6 𝑚 𝑛 𝑚𝑛(𝑚2 + 𝑛2 )2

𝐾=

1 𝐸 ℎ3 1 − 𝜈2 12

(10.53)

𝑒 = 5.95 ⋅ 10−3 m. This results in a converged value 𝑤max A convergence study is performed with meshes of 1 element up to 16 elements whereby quarter symmetry is used. Figure 10.3 shows the system. Boundary conditions of nodes along symmetry axes are given by prescribing appropriate zero rotations. The maximum deflection 𝑤max arises at the right-hand lower corner node. Computed values are

Discretisation 1 × 1 2×2 3×3 5.39 ⋅ 10−3 5.87 ⋅ 10−3 5.95 ⋅ 10−3 𝑤max [m]

4×4 5.98 ⋅ 10−3

where the ANS method (Eq. (10.52)) has been applied to avoid transverse shear locking. If it is not applied and Eq. (10.22) is used as is without modifications regarding 𝜖13 , the computed maximum deflection is 𝑤max = 1.44 ⋅ 10−3 m the entries for ˜ 𝜖23 , ˜ for the 4 × 4-discretisation with an error of roughly 80%. Inclusion of shear deformations leads to slightly larger converged displacement compared to Kirchhoff theory.

10.7 Reinforced Concrete Shells 10.7.1

Layer Model

Aspects of the local behaviour of cracked reinforced concrete have been described from several points of view up to now with • • • • •

Cross-sectional behaviour of tension bars in Section 3.6. Cross-sectional behaviour of beams in Section 4.1.3. Biaxial behaviour of plates in Sections 8.2 and 8.3. elasto-plastic behaviour of Kirchhoff slabs in Section 9.8. The cross-sectional behaviour of slabs based on the layer model is outlined in Section 9.4.

10.7 Reinforced Concrete Shells

The layer model is also suitable for continuum-based thin shells. In contrast to slabs, aspects of reference or coordinate systems have to be considered. A local corotational system (Section 10.3) is appropriate, implying local stress components 𝜎𝑖𝑗 and local strain components 𝜖 𝑖𝑗 . The approach Eq. (10.33) 𝝈(𝑟, 𝑠, 𝑡) = C ⋅ 𝝐(𝑟, 𝑠, 𝑡)

(10.54)

with local isoparametric coordinates 𝑟, 𝑠, 𝑡 or its incremental form ̇ 𝑠, 𝑡) ̇ 𝑠, 𝑡) = C ⋅ 𝝐(𝑟, 𝝈(𝑟, 𝑇

(10.55)

allow for an arbitrary material behaviour with variable material stiffness C or tangential material stiffness matrix C𝑇 within the context of continuum mechanics (Section 6.2.2). Local isoparametric coordinates translate into global coordinates (Eq. (10.4)). For a shell position 𝑟, 𝑠 on the reference surface, it is suitable to use the concept of layers. ◀

A layer is a plane through the point r , s, t with the shell director Vn (r , s) as normal. Every thickness coordinate t has its own layer, where it shares the normal with all layers of the same shell position r , s.

A layer is comparable to a slab sector, as indicated in Figures 9.2 and 9.4, where stress and strain component indices are replaced with 𝑥 → 1, 𝑦 → 2, 𝑧 → 3. The material behaviour with respect to a layer is based on stress components 𝜎11 , 𝜎22 , 𝜎12 and strain components 𝜖 11 , 𝜖22 , 𝛾 12 . Furthermore, for reinforced cracked concrete, it is appropriate to decouple transverse shear characterised by 𝜎13 , 𝜎23 , 𝛾 13 , 𝛾23 from layer behaviour. These assumptions motivate a generalisation of Eq. (10.34) ⎡𝐶11 ⎢ ⎢𝐶21 ⎢ ⎢ 0 C=⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ 𝐶 ⎣ 61

𝐶12

0

0

0

𝐶22

0

0

0

0

0

0

0

0

0

𝑐44

0

0

0

0

𝑐55

𝐶62

0

0

0

𝐶16 ⎤ ⎥ 𝐶26 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 𝐶66 ⎦

(10.56)

with component ordering according to Eqs. (6.3) and (6.7). Uppercase coefficients mark layer behaviour, while lowercase coefficients mark transverse shear behaviour. The form of Eq. (10.56) – after moving the entries for transverse shear – corresponds to the form of Eq. (9.36) proposed for the layer model of reinforced concrete slabs. But the former is much more comprehensible because the detour taken with internal forces is avoided. The transverse shear stiffness has already been discussed in Section 4.4.4 in the context of structural beams. This was rephrased for slabs in Section 9.4. The approach for slabs (Eq. (9.35)) is also used for the layers of shells that lead to 𝑐44 = 𝑐55 = 𝛼 𝐺 ,

𝐺=

𝐸𝑐 2(1 + 𝜈)

(10.57)

345

346

10 Shells

with a reduction factor 𝛼 and the initial values of Young’s modulus 𝐸𝑐 of concrete, its Poisson’s ratio 𝜈, and the shear modulus 𝐺. The reduction factor may be chosen with 𝛼 = 0.5 according to Section 4.4.4. The in-plane stress–strain relations in a general form are given by ⎛𝜎11 ⎞ ⎡𝐶11 ⎜𝜎 ⎟ = ⎢ ⎢𝐶21 22 ⎜ ⎟ ⎢ 𝜎 ⎝ 12 ⎠ ⎣𝐶61

𝐶12 𝐶22 𝐶62

𝐶16 ⎤ ⎛ 𝜖11 ⎞ ⎥ 𝐶26 ⎥ ⋅ ⎜ 𝜖22 ⎟ ⎥ ⎜ ⎟ 𝐶66 ⎝𝛾 12 ⎠ ⎦

(10.58)

or the related incremental form. This corresponds to a biaxial plane stress state and is exemplarily discussed for plates in Section 8.2. The ‘thickness’ of a shell layer corresponds to the thickness of the counterpart plate. Such a thickness is a matter of 𝑡-integration of stresses into nodal forces (Eq. (10.40)) or resulting internal forces (Eq. (10.42)). The thickness is implicitly included in a numerical integration process. Thus, the following procedure is appropriate for reinforced concrete shell layers. a) Modelling of cracks due to limited tensile strength of concrete is exemplarily treated in Section 8.2. This yields a stress–strain relation in a principal local coordinate system (Eqs. (8.25)–(8.32)). These material relations are based on the principal strains 𝜖1 , 𝜖2 derived from 𝜖11 , 𝜖 22 , 𝛾12 . The principal strain directions also rule the crack directions. b) The application of such relations requires the coordinate system transformations ‘natural system’ → ‘local system’ → ‘principal system’ and backwards. This looks elaborate but is justified because of geometrical and physical complexities. The back transformation to the local system leads to a form corresponding to Eq. (10.58). This is transferred into Eq. (10.56) and finally used in Eq. (10.35) for a tangential material stiffness matrix C𝑇 . c) Such calculations are performed for each layer, indicated by the local transverse coordinate 𝑡. A principal strain direction may change with 𝑡 for a given shell point 𝑟, 𝑠, i.e. faces of cracked concrete layers may become curved surfaces in space. d) Reinforcement and bond are treated in Section 8.3 with respect to plates. Rigid bonds are assumed for this. For thin reinforcement meshes, the same procedure as described for plates is applied for reinforced shell layers. A difference is given, as shell reinforcement layers are not implemented as separate elements but subject to integration according to the approach described in Section 8.4 extended to sheets. A reinforcement sheet of thickness ℎ𝑅 (Eq. (8.41)) is regarded with a transverse coordinate −1 ≤ 𝑡𝑅 ≤ 1. According to Eq. (10.40), the contribution of a sheet to internal nodal forces is determined with +1 +1

˜ (𝑟, 𝑠, 𝑡𝑅 ) 𝐽(𝑟, 𝑠, 𝑡𝑅 ) d𝑟 d𝑠 f𝑅,𝐼 = ∫ ∫ BT (𝑟, 𝑠, 𝑡𝑅 ) ⋅ 𝝈 −1 −1

(10.59)

10.7 Reinforced Concrete Shells

with contravariant reinforcement sheet stresses 𝝈 ˜ . Its contribution to internal forces is determined according to Eq. (10.42) with 𝑛𝑅1 = ℎ𝑅 𝜎𝑅11 , 𝑚𝑅1

ℎ = − 𝑡𝑅 𝑛𝑅1 , 2

𝑛𝑅2 = ℎ𝑅 𝜎𝑅22 , 𝑚𝑅2

ℎ = − 𝑡𝑅 𝑛𝑅2 , 2

𝑛𝑅12 = ℎ𝑅 𝜎𝑅12 𝑚𝑅12 = −

ℎ 𝑡 𝑛 2 𝑅 𝑅12

(10.60)

with local reinforcement stresses 𝜎𝑅11 , 𝜎𝑅22 , 𝜎𝑅12 . Thin reinforcement sheets do not contribute to transverse shear forces. e) Item d. may be applied to multiply stacked reinforcement layers with arbitrary orientations while adding up their contributions. Reinforcement strains and stresses are basically also subject to the transformations indicated in item b. Larger reinforced concrete shells were built in large numbers for a wide span of applications during the 1920s to 1960s (Joedicke 1962). Due to expensive formwork and other upcoming restrictions their application have mainly been reduced to cooling towers of large power plants nowadays. A future perspective may arise with new composite cement-based materials like, e.g. carbon concrete (Scheerer et al. 2017; Bielak et al. 2018; Müller et al. 2019). Shells elements may also be used for folded plates, where multiple shell directors (Figure 10.1) arise for nodes along folded plate edges. A further special, but significant, case is given with slabs exposed to normal forces and bending.

10.7.2 Slabs As a Special Case The relevance of the shell approach for reinforced concrete arises with the combination of bending with normal or membrane forces, respectively. This has to be considered for folded plates or plates with an orientation in 3D space. A special case of these set-ups is given with widely used T-beams. Plates act simultaneously as slabs in these cases, and combined actions will arise anyway. With respect to the structural behaviour of simple slabs, bending and membrane forces interact in the case of cracked reinforced concrete sections. This is shown for beams in Example 4.2 and is demonstrated for slabs with the following example. Example 10.2: Simple RC Slab with Interaction of Normal Forces and Bending

We consider the system of Example 10.1 with the same dimensions. Material properties and reinforcement are chosen as follows: • A concrete grade C30 according to CEB-FIP2 (2012, 5.1) with Young’s modulus 𝐸𝑐 = 33 600 MN∕m2 and a compressive strength 𝑓𝑐 = 38 MN∕m2 . A tensile strength is mandatory for the material models employed and is assumed with 𝑓𝑐𝑡 = 2.9 MN∕m2 . • Reinforcing steel properties with Young’s modulus 𝐸𝑠 = 200 000 MN∕m2 , yield strength 𝑓𝑦 = 500 MN∕m2 , and tensile strength 𝑓𝑡 = 550 MN∕m2 at a strain 𝜖𝑢𝑘 = 50 ⋅ 10−3 . • The amount of reinforcement has to be determined as input for a nonlinear computation. A reasonable loading is given with 𝑞 = 20 kN∕m2 . Standard design

347

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10 Shells upper

upper

lower lower methods result in a reinforcement 𝑎𝑠𝑥 = 𝑎𝑠𝑦 = 10 cm2 ∕m, 𝑎𝑠𝑦 = 𝑎𝑠𝑦 = 5 cm2 ∕m. An upper reinforcement is required for twisting moments in the outer corner regions (Section 9.7.1, Eq. (9.72)). For simplicity, this is arranged over the entire base area. • Reinforcement cover 𝑐 = 0.02 m and effective depth of cross-section with 𝑑 = 0.225 m in both directions.

This is a set-up for a small study about advanced modelling and simulation of reinforced concrete structures. Similarly to experimental investigations, the slab is exposed to a prescribed centre deflection in the simulation whereby recording the corresponding centre reaction force. Three alternative concrete models are investigated. 1. Biaxial linear elasticity with limited tensile strength and smeared cracking (Section 8.2, abbreviated to ‘elastic limited’) applied to each layer within the framework of the layer model (Section 10.7.1). This allows for dual orthogonal cracking and varying cracking directions along a cross-sectional height. 2. Isotropic damage (Section 6.6) with the Hsieh–Ting–Chen damage function Eq. (6.109), where local stress components 𝜎33 are neglected as with Eq. (10.34). The regularisation is performed with the crack band approach (Section 7.3) with a scaling of the stress–strain relations in the tensile softening range depending on the characteristic element base area length (Eq. (7.18)). 3. Microplane (Section 6.8) with the concrete damage formulation (Eq. (6.151)) allowing for a load-induced anisotropy, where local stress components 𝜎33 are again neglected. The regularisation is again performed with the crack band approach applied to each microplane (Figure 6.15b). The respective material parameters are chosen such that the prescribed uniaxial behaviour is reproduced, if applicable 1). In addition, a number of other model parameters must be specified. • Reinforcement layers with Sheet thickness 𝑎𝑅 [m] Sheet height coordinate 𝑡𝑅 [m] −0.10 Lower 𝑥∕𝑦-direction 1.0 ⋅ 10−3 0.10 Upper 𝑥∕𝑦-direction 0.5 ⋅ 10−3

each with a uniaxial behaviour. Sheet height coordinates 𝑡𝑅 are the same in the 𝑥 and 𝑦-directions to preserve symmetry. • A quarter slab is discretised with 4 × 4 elements without further convergence study. • A Gaussian scheme (Section 2.7) is used for numerical integration. Integration orders (Eq. (10.41)) are chosen with 𝑛𝑢 = 1 in the reference surface directions and with 𝑛𝑣 = 4 in the transverse direction, leading to 2 × 2 × 5 = 20 continuum integration points per element. • The additional transverse integration of reinforcement contributions is performed separately while considering their discrete positions 𝑡𝑅 (Section 10.7.1). 1) In the case of the ‘elastic limited’ material model for tension only.

10.7 Reinforced Concrete Shells

(a)

(b)

Figure 10.4 Example 10.2. (a) Load displacement curve (simulation result ×4). (b) Principal strains in upper and lower quarter concrete layers in the final state (thickness direction with larger scale).

A target centre deflection 𝑤max = 0.2 m is prescribed, which is applied incrementally. A dynamic calculation (Appendix A.1 and Section 2.8.2) with slow load application is performed with the incrementally iterative scheme. This facilitates Newton– Raphson equilibrium iterations while preserving quasi-static behaviour. The simulation results of the case ‘elastic limited’ are described first as a reference. The computed load–displacement curve is shown in Figure 10.4a. • The characteristic states of reinforced concrete occur again: (I) uncracked, (IIa) crack formation, (IIb) stabilised cracking with elastic reinforcement, and (III) yielding of reinforcement. The transitions are gradual compared to former cases (Examples 3.4, 4.2). • Crack formation starts with a strong load decrease due the relative low reinforcement ratio. Furthermore, later, intermediate jags indicate cracking events of single integration points, which stand out due to the coarse discretisation of the whole system.

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• Although the limit load does not seem to be reached on reaching the target deflection, the lower central reinforcement is in the yielding range, whereby concrete compressive stresses are in the range of 30 MN∕m2 . Smeared strains in cracked integration points (Section 7.4) in the final state are shown in Figure 10.4b. Integration points are indicated by dots. • The top and bottom layers are chosen out of the five concrete layers. The length and orientation of lines indicate the size and orientation of the principal values of the smeared strains. Cracks are not shown explicitly but arise orthogonally to the strain directions. Double crosses indicate dual orthogonal cracking. • The top and bottom show a different behaviour due to bending. The bottom layer has two orthogonal cracks in the central region from bending moments and single cracks in diagonal direction in the outer corner regions from twisting moments. The corresponding principal strains are across the diagonal direction for the latter. • The top layer has no cracks in the central region because it is under compression. But it has cracks across the corner diagonal direction from twisting moments. The corresponding principal strains occur in the diagonal direction. Bending moments 𝑚1 , 𝑚2 , 𝑚12 get contributions from the concrete according to Eq. (10.42) and from the reinforcement according to Eq. (10.60). The resulting principal moments in the final state are shown in Figure 10.5a. Line directions indicate the directions of the corresponding stresses. A positive moment has tension on the lower surface and compression on the upper surface. • Skew principal moment directions with opposite signs in the corner region correspond to twisting moments. The negative moment in the diagonal direction has diagonal tension on the upper side, which is compensated by upper 𝑥 and 𝑦-reinforcement. The positive moment across the diagonal direction has the corresponding tension on the lower side. It is compensated by lower 𝑥 and 𝑦-reinforcement. • Central areas also have skew principal moments but with the same sign and nearly the same size. Thus, there will only be minor twisting moments. Lower surface tension is compensated by lower 𝑥 and 𝑦-reinforcement.

(a)

(b)

Figure 10.5 Example 10.2. Quarter slab in final state. (a) Principal moments. (b) Principal membrane forces.

10.7 Reinforced Concrete Shells

Normal or membrane forces 𝑛1 , 𝑛2 , 𝑛12 arise, although a slab is given with lateral loading only. They are determined from concrete stresses according to Eq. (10.42) superposed with reinforcement stresses according to Eq. (10.60). • Membrane forces are caused by the elongation effect of cracked reinforced crosssections; see Example 4.2 for beams. In contrast to beams, an elongation may have a different direction for every point of the slab reference surface because the crack directions may be different. • This leads to eigenstresses, i.e. self-equilibrating internal forces without reaction forces on supports, where kinematic compatibility is maintained over all local elongations or contractions. • The corresponding principal membrane forces are shown in Figure 10.5b. A centred – upper right-hand corner for quarter symmetry – rudimentary compression field from the concrete compression zone is tied by an outer circumferential tension field. For smaller loads, this is more pronounced. • If the horizontal displacements of the slab are prevented on the supporting edges, horizontal reaction forces will arise. The alternative material models ‘isotropic damage’ and ‘microplane’ show the same qualitative behaviour with respect to moments and eigenstress normal forces but with different values. A comparison of the load–deflection behaviour depicts considerable differences; see Figure 10.6. While characteristic states (I), (IIa,b), and (III) are still given for all cases, the points of (I → IIa)-transition and, in particular, final loads are different. Yielding of lower centre reinforcement occurs for all of them. But the material models ‘isotropic damage’ and ‘microplane damage’ consider the nonlinear compressive behaviour of concrete in contrast to the ‘elastic limited’ material model. This causes considerably lower limit loads. Again, there is a pronounced model uncertainty. The same problem has already been treated with the elasto-plastic Kirchhoff approach for slabs (Example 9.5, Figure 9.15b). This yields a final loading of ≈ 0.8MN for a twisting parameter value estimated as realistic and ranges between ‘elastic limited’ and ‘microplane’. Furthermore, the yield line approach again provides an upper limit 𝑃 ≈ 1.0 [MN] for the loading; see Example 9.6. It does not seem quite clear

Figure 10.6 Example 10.2. Load– deflection behaviour for material models (simulation result ×4).

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10 Shells

whether ‘elastic limited’ exceeds this value. However, the ultimate limit moment for ‘elastic limited’ is not correctly determined with Eq. (9.119). This leads to the question of which of these models represents the right solution. At the current state of the discussion, this question is not to be answered. Further variations of the respective model parameters are to be treated. Sensitivity studies should determine material parameters and parameters for reinforcement layout, as well as discretisation parameters with a strong influence on model behaviour. Such significant parameters may give lead to reach a better model ‘convergence’. A crucial reference is given by the corresponding experimental results. But one should not underestimate the complexity involved in experimental research to get reliable data from that side. Model uncertainty appears to be depressing. But its source is mainly identified with the uncertainty of models for multi-axial stress–strain behaviour of concrete. Example 10.2 can only exemplarily demonstrate a limited spectrum of what material models are available for concrete; see Section 6. Finite element technology for shells, multi-axial nonlinear concrete behaviour, concrete cracking, and explicit anisotropic reinforcement modelling are combined. Each of these topics is still a field of ongoing research. Thus, the example becomes quite complex regardless of its simple set-up with respect to geometry, loading, and boundary conditions. On the other hand, it might provide a comprehensive amount of detailed result data that cannot be presented for reasons of limited scope. This qualifies complex shell modelling, e.g. as with Example 10.2, primarily for research leading to a better understanding of structural concrete mechanisms. From this, in turn, simplified models can be derived that are suitable for practical application.

353

11 Randomness and Reliability So far, everything has been treated from a deterministic point of view. But the reality of interest (Figure 2.1) is more or less random. One might consider the deterministic view as representative, but this should be put more precisely in order to, e.g. estimate risks or undesired behaviour of structures in a rational manner. The objective might fill textbooks, see, e.g. Ayyub and McCuen (2003); Schneider (2006); Bucher (2009); Nowak and Collins (2012); CEB-FIP (2018), but only first aspects can be given in the following.

11.1 Uncertainty and Randomness We rely on models relating properties by equations (Section 2.1). A property is either an input parameter or a response variable. A response variable is derived with the solution for the mathematical model, whereby parameters determine the model coefficients. It is assumed that the mathematical model can be solved with a desired degree of accuracy. Thus, the solution solely depends on the parameters. We use parameters like material properties, geometric dimensions, loadings, and constraints and up to now have assumed that determined parameter values yield deterministic predictions for variables. But this assumption does not take uncertainty into account. ◀

Predictions will be uncertain due to deviations of real parameter values from assumed parameter values.

An input parameter may be identified as response variable in another context. Material stiffness and strength, e.g. may be solutions in a microscopic material model. On the other hand, we do not have models in the sense of Section 2.1 for, e.g. construction site processes leading to deviations of real geometric dimensions from dimensions in design documents. Thus, uncertainty is classified as epistemic or aleatoric (Roy and Oberkampf 2011). • Aleatoric uncertainty results from fundamental unpredictability of single phenomena or events, respectively. • Epistemic uncertainty results from insufficient knowledge. Computational Structural Concrete – Theory and Applications, Second Edition. Ulrich Häussler-Combe. © 2023 Ernst & Sohn GmbH. Published 2023 by Ernst & Sohn GmbH.

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11 Randomness and Reliability

Both types are more or less mixed in real phenomena and will not be distinguished in the following. Furthermore, we restrict uncertainty to randomness, which is strongly connected to the probability 𝑃 of events. Randomness is quantitatively treated with stochastics, which comprises probabilistic and statistics (Kreyszig 2006, G). The difference between input parameters and response variables will generally not be pursued in the following: both are subsumed as random variables. The subject of model uncertainty, see Examples 8.2, 9.5, and 10.2, remains to be classified. The term is generally usual but is misleading in the current context. A model (Section 2.1) corresponds to a set of equations. The models used for the mentioned examples differ – regardless of parameter values – by different forms of equations related to the material behaviour. This is not suitable for a treatment with methods of stochastics; it is pure assumption. The best predictive model is qualified by references to reality or experimental investigations, respectively. This might be seen as epistemic but does not provide access by stochastics. 1) To start with randomness of input parameters and result variables, we exemplarily regard a population of cantilever columns exposed to a concentrated top load due to supporting weights of other parts. This set-up is also used with Examples 11.1 and 11.2. The loading in a first approach has two parameters: the magnitude of the load and the magnitude 𝑒 of the eccentricity with respect to the centre of gravity of the cross-sectional area. Both are basically random. We consider, e.g. the population 𝐸 of eccentricities as a univariate random variable. Its probability is described by a distribution function 𝐹𝐸 (𝑒) = 𝑃(𝐸 ≤ 𝑒)

(11.1)

where 0 ≤ 𝑃 ≤ 1 denotes the portion of events from 𝐸 with eccentricities smaller than 𝑒 in the case of a complete population. 2) The distribution function is the integral of the probability density function 𝑓𝐸 (𝑒) 𝑒

𝐹𝐸 (𝑒) = ∫ 𝑓𝐸 (𝑒) d𝑒 , −∞

𝑓𝐸 (𝑒) =

d𝐹𝐸 (𝑒) d𝑒

(11.2)

The probability density function is determined from measurements on samples out of the population. Such samples should yield a sound assessment of the distribution type and its parameters like mean 𝜇𝐸 and standard deviation 𝜎𝐸 . A prominent type is given with the normal distribution; see Figure 11.1a. Samples also mark the range of reasonable realisations. With randomness for structures, multi-variate random variables or random vectors have to be considered. A simple one is given by a bivariate random variable with random components; let’s say populations 𝑅 and 𝐸 with realisations 𝑟, 𝑒. The concepts of distribution functions and probability density functions are generalised 1) This statement is disputable (Ditlevsen 1982). 2) Actually, we never have a complete population but a more or less large number of samples out of it. This creates a stochastics of statistics with parameter estimation, confidence intervals, testing of hypotheses, and more.

11.1 Uncertainty and Randomness

(a)

(b)

Figure 11.1 (a) Normal distribution. (b) Correlation.

Figure 11.2 Uniaxial random fields with five samples each.

with joint functions having vector arguments, see e.g. Eq. (11.9). The components of a random vector may exhibit a correlation. This is measured by a scalar −1 ≤ 𝜌 ≤ 1 for bivariate random variables and indicates a degree of dependence. A value 𝜌 = −1 indicates full reverse correlation, 𝜌 = 0 independent or uncorrelated variables, and 𝜌 = 1 full concordant correlation; see Figure 11.1b. An extension of random variables is given by random fields, whereby randomness of a property extends over space, i.e. along a line, a plane, or over the volume of a structure (Spanos and Zeldin 1998). An example is shown for the tensile strength of samples of concrete bars in Figure 11.2. Strength is a one-dimensional random field within a single bar, which stands for the population of cross-sectional strength. Furthermore, bars also form a population leading to – in some sense – a two-dimensional random field. It is generally assumed that the distribution is the same in both dimensions. The correlation of the strength of neighbour points along a particular bar is described by the correlation length. Samples of two bar populations with different correlations length are shown in Figure 11.2. A random field may be seen as a multidimensional random vector in the case of spatially discretised systems, whereby the correlation between spatially neighboured property values is described by the correlation length. The correlation length may basically vary with respect to position and direction.

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356 ◀

11 Randomness and Reliability

Regardless of this, randomness of model input parameters will lead to randomness of response variables or model solutions. The model may be regarded as a filter transforming distributions of parameters into distributions of responses.

With respect to structural analysis, the filter may become complex when using the finite element method. A stochastic analysis of filtering is performed with stochastic FEM (Stefanou (2009), Bucher (2009, 5.3)). This provides comprehensive stochastic data but is elaborate and is not pursued in the following. Fundamental aspects of structural behaviour concern ultimate limit states, serviceability, and durability (CEB-FIP2 2012). All are subject to randomness. Ultimate limit states potentially involve the greatest impact. A survey of real life failure cases reveals the human factor as the major source, e.g. by ignoring or misunderstanding design documents on a construction site, and much more. This is beyond the scope of randomness as is understood here. Ultimate limit states quantified by a failure probability as are described with simple cases in the following, generally ignore the human factor 3) and should not be confused with what has to be expected in the real world. It is an operative failure probability to compare particular cases against reference values.

11.2 Failure Probability 11.2.1 Linear Limit Condition A population of systems is considered with random variables for the action 𝐸 and for the resistance 𝑅, reducing a system to a critical spot. This is obvious for, e.g. a singlespan beam with the cross-section exposed to the largest bending moment but might be no longer be straightforward for statically indeterminate systems, plates, and slabs. A linear structural analysis is often used to determine critical spots, although this does not account for internal redistribution due to nonlinear characteristics and redundancies of the bearing capacities. Regardless of this, a normal distribution is assumed for a scalar resistance 𝑅 with a probability density function 𝑓𝑅 (𝑟) =

1 √

𝜎𝑅 2π

e

1 𝑟−𝜇𝑅 ) 2 𝜎𝑅

− (

2

(11.3)

with realisations 𝑟, a mean 𝜇𝑅 , and a standard deviation 𝜎𝑅 . It should be kept in mind that a structure may undergo deterioration during its service life. ◀

The distribution of the resistance may change during time, i.e. a mean 𝜇R will generally decrease, and the standard deviation 𝜎R will generally increase.

3) The human factor is generally considered by, e.g. training, quality assurance manuals, four-eye principles, and, if it gets worse, at court.

11.2 Failure Probability

(a)

(b)

Figure 11.3 (a) Actions and their extreme values. (b) Joint probability density of resistance and extreme actions.

The distribution Eq. (11.3) is standardised with a transformation 𝑟=

𝑟 − 𝜇𝑅 , 𝜎𝑅

(11.4)

leading to a transformed mean 𝜇𝑅 = 0, a transformed standard deviation 𝜎𝑅 = 1, and a probability density function 2

𝑟 1 − 𝑓𝑅 (𝑟) = √ e 2 , 2π

(11.5)

Furthermore, a reference period 𝑇𝐸 that might be the anticipated service life of a structure is considered. Each sample of the population is exposed to the random action during the reference period. This yields an extreme maximum scalar action 𝐸 during the reference period, see Figure 11.3a, which is a random variable as is its base action. To simplify we also assume a normal distribution for the extreme action 𝐸 𝑓𝐸 (𝑒) =

1 √

𝜎𝐸 2π

e

1 𝑒−𝜇𝐸 ) 2 𝜎𝐸

− (

2

(11.6)

with realisations 𝑒, a mean value 𝜇𝐸 , and a standard deviation 𝜎𝐸 . ◀

The mean and the standard deviation of the extreme action depends on the value of the reference period TE . Larger values lead to larger means and larger standard deviations.

On the other hand, a remaining anticipated service life reduces mean and standard deviations of extreme actions and may allow accepting a resistance deterioration during a previous service life. For the following, we assume time independence in order to simplify. The action 𝐸 is also standardised with a transformation 𝑒=

𝑒 − 𝜇𝐸 𝜎𝐸

(11.7)

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11 Randomness and Reliability

leading to 𝜇𝐸 = 0, 𝜎𝐸 = 1 and a probability density function 2

𝑒 1 − 𝑓𝐸 (𝑒) = √ e 2 2π

(11.8)

For structural systems, it may be assumed that 𝑅 and 𝐸 are stochastically independent. This yields a joint probability density function as a product of the single probability densities 𝑓𝑅𝐸 (𝑟, 𝑟) = 𝑓𝑅 (𝑟) 𝑓𝐸 (𝑒) =

1 𝑟−𝜇𝑅 𝑒−𝜇𝐸 + ) 𝜎𝑅 𝜎𝐸

1 −2( e 2π

(11.9)

see Figure 11.3b, or 1

𝑓𝑅𝐸 (𝑟, 𝑒) = 𝑓𝑅 (𝑟) 𝑓𝐸 (𝑒) =

1 − 2 (𝑟 e 2π

2

2

+𝑒 )

(11.10)

with a rotational symmetry with respect to the origin 𝑟 = 0, 𝑒 = 0. A limit state condition is described with the limit state function 𝑔(𝑟, 𝑒), which is assumed as linear 𝑔(𝑟, 𝑒) = 𝑟 − 𝑎 𝑒 − 𝑏 = 0

(11.11)

with constants 𝑎, 𝑏, separating the failure domain 𝑔 ≤ 0 in the 𝑅, 𝐸-plane from the safe domain. The limit state condition is also standardised with Eqs. (11.4) and (11.7), leading to 𝛽 + 𝛼𝑅 𝑟 + 𝛼𝐸 𝑒 = 0

(11.12)

in the 𝑟, 𝑒-plane with the reliability index 𝜇𝑅 − 𝑎𝜇𝐸 − 𝑏 𝛽= √ 𝜎𝑅2 + 𝑎2 𝜎𝐸2

(11.13)

and the sensitivity parameters 𝜎𝑅 , 𝛼𝑅 = √ 𝜎𝑅2 + 𝑎2 𝜎𝐸2

−𝑎𝜎𝐸 𝛼𝐸 = √ 𝜎𝑅2 + 𝑎2 𝜎𝐸2

(11.14)

with 𝛼𝐸2 + 𝛼𝑅2 = 1

(11.15)

The sensitivity parameters 𝛼𝑅 , 𝛼𝐸 indicate the influence of 𝑅 and 𝐸 on the system randomness. In the case where 𝜎𝑅 ≫ 𝜎𝐸 is 𝛼𝑅 → 1, 𝛼𝐸 → 0, and the randomness depends on 𝑅 only. In the case where 𝜎𝑅 ≪ 𝜎𝐸 is 𝛼𝑅 → 0, 𝛼𝐸 → −1, and the randomness depends on 𝐸 only. The sensitivity parameters correspond to an angle 𝜑

11.2 Failure Probability

with cos 𝜑 = 𝛼𝑅 , sin 𝜑 = 𝛼𝐸 , where 𝜑 indicates the inclination of the linear standardised limit state condition (Eq. (11.12)) against the 𝑒-axis. A further simplification is reached with a transformation to variables ˜ 𝑟 , 𝑒˜ 𝑟=˜ 𝑟 𝛼𝑅 − 𝑒˜𝛼𝐸 ,

𝑒=˜ 𝑟 𝛼𝐸 + 𝑒˜𝛼𝑅

(11.16)

corresponding a rotation of the 𝑟, 𝑒-coordinate system with the angle 𝜑 (Eq. (D.7)). The limit state condition (Eq. (11.12)) can be written as ˜ 𝑟+𝛽 =0

(11.17)

in the ˜ 𝑟, 𝑒˜-coordinate system, whereby the failure domain is given by ˜ 𝑟 ≤ −𝛽

(11.18)

The failure probability for the population of systems regarding the reference period is derived from the integration of the joint probability density function 𝑓𝑅˜𝐸˜ in the range defined by Eq. (11.18). The functions 𝑓𝑅𝐸 (Eq. (11.10)) and 𝑓𝑅˜𝐸˜ are basically the same due to their rotational symmetry. Hence, the failure probability is given by ∞ −𝛽

−𝛽

−𝛽

𝑟 d˜ 𝑒 = ∫ 𝑓𝑅˜ d˜ 𝑟 = ∫ 𝑓𝑅 d𝑟 𝑝𝑓 = ∫ ∫ 𝑓𝑅˜𝐸˜ d˜ −∞ −∞

−∞

(11.19)

−∞

with 𝑓𝑅 according to Eq. (11.5), leading to 𝑝𝑓 = Φ(−𝛽)

(11.20)

with the distribution function Φ of the standardised normal distribution. The failure probability depends only on the reliability index 𝛽 for normal distributions of extreme actions 𝐸 and resistances 𝑅 and linear limit states (Eq. (11.11)). Larger values of 𝛽 lead to a smaller failure probability. The reliability index increases with increasing distance between mean values of resistances and extreme actions and decreases with increasing standard deviations of resistances and extreme actions. A distinguished realisation of the random variables is key for the following. ◀

The design point (rd , ed ) is the realisation of random variables with the highest probability or the largest value of the joint probability density function, fulfilling the limit state condition g(rd , ed ) = 0.

With the current assumptions the design point is located in the ˜ 𝑟, 𝑒˜-coordinate system with the condition ˜ 𝑟𝑑 = −𝛽 ,

𝑒˜𝑑 = 0

(11.21)

It is transformed into the plane of standardised 𝑟, 𝑒-variables using Eq. (11.16) 𝑟 𝑑 = −𝛽 𝛼𝑅 ,

𝑒𝑑 = −𝛽 𝛼𝐸

(11.22)

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11 Randomness and Reliability

Finally, using Eqs. (11.4) and (11.7) the design point for the origin variables is given by 𝑟𝑑 = 𝜇𝑅 + 𝜎𝑅 𝑟 𝑑

= 𝜇𝑅 − 𝛽𝛼𝑅 𝜎𝑅

𝑒𝑑 = 𝜇𝐸 + 𝜎𝐸 𝑒 𝑑

= 𝜇𝐸 − 𝛽𝛼𝐸 𝜎𝐸

(11.23)

This realisation of the random variables fulfils the limit state condition (Eq. (11.11)) with 𝑔(𝑟𝑑 , 𝑒𝑑 ) = 0 as is required. The notion of a design point is also a major element for more general cases, e,g. with non-normal distributions, stochastic dependence, nonlinear limit conditions, and semi-probabilistic design concepts using partial safety factors. But its determination is generally not as straightforward as with Eq. (11.23). These topics are touched upon in the following section. Regarding the current assumptions the determination of failure probability is demonstrated with the following simple example. Example 11.1: Analytical Failure Probability of Cantilever Column

We consider a population of reinforced concrete cantilever columns with a height of 𝐿 = 5 m and a constant cross-section. The dimensions of the cross-section and the material properties are chosen as in Example 4.1. The corresponding moment—curvature relation is shown in Figure 4.4. The maximum moment 𝑀 that can be sustained by this particular cross-section is assumed as a random variable. A concentrated permanent downward loading of 𝑃 = 2 MN is given on top of the cantilever beam. It has some eccentricity 𝑒𝑐𝑐 with respect to the centred reference axis, whereby 𝑒𝑐𝑐 is oriented along the larger cross-sectional dimension of ℎ = 0.4 m. This eccentricity is assumed as a random variable for the action leading to a realised random moment 𝑃 𝑒𝑐𝑐. Second-order effects are not regarded in a first approach. The limit state condition is given by (Eq. (11.11)) with 𝑎 = 𝑃, 𝑏 = 0) 𝑔 = 𝑀 − 𝑃 𝑒𝑐𝑐 = 0

(11.24)

A normal distribution is assumed for resistance (→ 𝑀) and action (→ 𝑒𝑐𝑐), with parameters given in Table 11.1. The fractile value 𝑆𝑘,𝑥% of a random variable 𝑆 has the following property: 𝑥 percent of all samples of 𝑆 have a value below 𝑆𝑘,𝑥% . On Table 11.1 Parameters of Example 11.1. r→M

Mean μ Standard deviation 𝜎 10%-fractile 90%-fractile

e → ecc

[MNm]∕[m] 0.27 [MNm]∕[m] 0.015 [MNm] 0.251 [m]

0.126

Sensitivity parameter 𝛼 (Eq. (11.14)) — 0.351 Design point (Eq. (11.23)) [MNm]∕[m] 0.261

–0.936 0.131

0.1 0.02

11.2 Failure Probability

the other hand, 100 − 𝑥% of all samples of 𝑆 have a value above 𝑆𝑘,𝑥% . Fractile values are derived from the inverse of the probability function of 𝑆 with 𝑆𝑘,𝑥% = 𝐹𝑆−1 (𝑥). The reliability index is determined from Eq. (11.13) with 𝛽 = 1.6386 resulting in a design point also given in Table 11.1 and a failure probability of (Eq. (11.20)) 𝑝𝑓 = 0.0506

(11.25)

According to this model, a portion of 5 % of the population of cantilever columns under consideration will fail within the reference period or the anticipated service life, respectively. This is obviously quite high for building structures, even considering a service life of, e.g. 50 years. Following provisions will reduce the failure probability by • Increasing resistance mean 𝜇𝑀 by increasing the nominal material strength or cross-section dimensions. • Increasing construction quality by decreasing standard deviations of resisting moments and/or load eccentricity. • Reduction of loading. The example itself is academic, whereas the proposed provisions in a general sense are realistic. The easiest way practically always followed is given with increased nominal material strength within a semi-probabilistic design; see Section 11.3. We restricted ourselves to two random variables up to now. This is extended to multiple dimensions with random variables 𝑋1 … 𝑋𝑛 without further specifications about their meaning. To abbreviate we directly refer to Eq. (11.10) for standardised variables with the extension of the joint probability density function 𝑓𝑋

1

…𝑋 𝑛 (𝑥 1 … 𝑥 𝑛 ) =

1

1 𝑛

e

2

2

− (𝑥1 +…+𝑥 𝑛 ) 2

(11.26)

(2π) 2

The limit state condition Eq. (11.11) is extended with 𝑔(𝑥1 … 𝑥𝑛 ) = 𝑎0 + 𝑎1 𝑥1 + … + 𝑎𝑛 𝑥𝑛 = 0

(11.27)

and its standardised form Eq. (11.12) 𝛽 + 𝛼1 𝑥1 + … + 𝛼𝑛 𝑥 𝑛 = 0

(11.28)

with extended reliability index and sensitivity parameters (Eqs. (11.13) and (11.14)) ∑𝑛 𝑎0 + 𝑖=1 𝑎𝑖 𝜇𝑖 𝑎𝑖 𝜎𝑖 , 𝛼𝑖 = √ (11.29) 𝛽= √ ∑𝑛 ∑ 𝑛 2 2 2 2 𝑎 𝜎 𝑎 𝜎 𝑖=1 𝑖 𝑖 𝑖=1 𝑖 𝑖 with means 𝜇𝑖 and standard deviations 𝜎𝑖 . The sensitivity parameters fulfil conditions 𝛼12 + … + 𝛼𝑛2 = 1 ,

−1 < 𝛼𝑖 < 1

(11.30)

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11 Randomness and Reliability

A hyper-rotation in analogy to Eq. (11.16) and integration of the probability density function in the failure domain again leads to a failure probability 𝑝𝑓 = Φ(−𝛽). The design values are derived with 𝑥𝑖𝑑 = 𝜇𝑖 − 𝛽𝛼𝑖 𝜎𝑖

(11.31)

in extension of Eq. (11.23). We will refer to Eq. (11.31) with regards to partial safety factors in Section 11.3.

11.2.2 Nonlinear Limit Condition Stochastics of structural systems is often characterised by non-normal distribution, e.g. log-normal distributions on the resistance side and extreme value distributions on the action side. Furthermore, random variables may be correlated (Figure 11.1). This prevents the application of the previous approach at a first glance. But transformation rules are available in the case that stochastic properties of variables are provided with their marginal distributions 4) and their correlation data. This is performed as follows. (Bucher 2009, 2.2.6) 1. Marginal distributions are transformed into marginal standard normal distributions (Eq. (11.61)). 2. Corresponding correlation coefficients are transformed using the Nataf model (Liu and Kiureghian 1986). 3. A joint probability density – correlated standard normal – is derived according to the Nataf model. 4. Correlated variables are transformed into uncorrelated ones on the basis of known correlation coefficients, utilizing the principal space defined by the covariance matrix. 5. This should yield a form given according to Eq. (11.26). This looks straightforward but might be laborious in a concrete situation, in particular regarding item 2. Correlation data are rarely available in the case of structural concrete. Thus, it is often assumed from the start that the included variables are uncorrelated. Regardless of this, the limit state function must be included . ◀

A sequence of transformations applied to random variables also has to be followed by the corresponding limit state condition.

The sequence of steps is referred to as N-standardisation. A limit state condition generally has to be assumed as nonlinear, at least after being processed by an N-standardisation. An approximate evaluation of failure probabilities requires a linearisation for such cases. Again, we exemplarily consider the bivariate case in the following, sketching the first-order reliability method (FORM). A linear or nonlinear limit state function (Eq. (11.11) for a linear) yields a scalar value 𝑧 = 𝑔(𝑟, 𝑒) with realisations 𝑟, 𝑒, which 4) A marginal distribution of a random variable is sampled and parameterised as univariate irrespective of whether it is correlated to other random variables; see Figure 11.1b.

11.2 Failure Probability

may be seen as a safety margin in the case 𝑧 > 0. The relation between 𝑧 and 𝑟, 𝑒 is approximated with a Taylor expansion. The design point that marks the failure realisation with the highest probability is the most appropriate expansion point 𝑧𝑑 = 𝑔(𝑟𝑑 , 𝑒𝑑 ). Thus, it should be known for general cases. Analytical relations like Eq. (11.23) are generally no longer available. With a joint probability density of random variables 𝑅, 𝐸 given, its determination may be formulated as an optimisation problem with the constraint of the limit state function. In the standardised space, the design point is also the point with the shortest distance to the origin. Under this condition, the optimisation problem can be solved more easily. We assume that the design point is known or a least a good estimation of it. Thus, the Taylor expansion is given with 𝑧 = 𝑧𝑑 +

𝜕𝑔 ||| 𝜕𝑔 ||| || | (𝑟 − 𝑟𝑑 ) + (𝑒 − 𝑒𝑑 ) + ⋯ | 𝜕𝑟 |𝑟𝑑 ,𝑒𝑑 𝜕𝑒 |||𝑟𝑑 ,𝑒𝑑

(11.32)

The safety margin corresponds to a (𝑅, 𝐸)-dependent random variable 𝑍. Based on the linearised Taylor expansion its mean 𝜇𝑍 is determined by the means 𝜇𝑅 , 𝜇𝐸 𝜇𝑍 = 𝑧𝑑 +

𝜕𝑔 ||| 𝜕𝑔 ||| || | (𝜇𝑅 − 𝑟𝑑 ) + (𝜇 − 𝑒𝑑 ) 𝜕𝑟 ||𝑟𝑑 ,𝑒𝑑 𝜕𝑒 |||𝑟𝑑 ,𝑒𝑑 𝐸

and its standard deviation 𝜎𝑍 by the standard deviations 𝜎𝑅 , 𝜎𝐸 √ 2 2 √ √ 𝜕𝑔 || 𝜕𝑔 || 𝜎𝑍 = √( ||| ) 𝜎𝑅2 + ( ||| ) 𝜎𝐸2 𝜕𝑟 ||𝑟𝑑 ,𝑒𝑑 𝜕𝑒 ||𝑟𝑑 ,𝑒𝑑

(11.33)

(11.34)

A normal distribution is assumed for the safety margin assured by N-standardisation. The failure probability is determined as an integral over the failure domain 𝑧≤0 0

𝑝𝑓 = ∫ √ −∞

1 2𝜎𝑍

e

1 𝑧−𝜇𝑍 ) 2 𝜎𝑍

− (

2

d𝑧 = Φ (−

𝜇𝑍 ) 𝜎𝑍

(11.35)

with the probability function Φ of the standardised normal distribution. Comparing Eqs. (11.11) and (11.32) we see an analogy 𝜇𝑍 ←→ 𝛽 𝜎𝑍

(11.36)

with the reliability index 𝛽 from Eq. (11.13) used for Eq. (11.20). Depending on the curvature of the limit state function at the design point, the linearisation will overestimate – convex/bowed in – or underestimate – concave/bowed out – the failure probability. Second-order terms of the Taylor expansion Eq. (11.32) are considered to improve the estimation, leading to the second-order reliability method (SORM). The methods based on expansions of the limit state function require its differentiability. This might be a limitation, e.g. when a discrimination of cases has to be

363

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11 Randomness and Reliability

Figure 11.4 Nonlinear limit state functions and linearisation.

regarded or when a limit state condition cannot be explicitly defined, which is generally the case with nonlinear finite element models. ◀

A given state (r , e) as such should at least be identifiable as a failure or a non-failure state.

Thus, a general formulation for the limit state condition ⎧

1 if (𝑟, 𝑒) → failure ⎨0 else ⎩ allows a generalisation. The failure probability is determined as 𝐼𝑔 (𝑟, 𝑒) =

(11.37)

∞ ∞

𝑝𝑓 = ∫ ∫ 𝑓𝑅𝐸 (𝑟, 𝑒) 𝐼𝑔 (𝑟, 𝑒) d𝑟 d𝑒

(11.38)

−∞ −∞

with the joint probability density function 𝑓𝑅𝐸 of the basic random variables. This is approximately evaluated with 𝑝𝑓 ≈

𝑚 1 ∑ 𝐼 (𝑟 , 𝑒 ) 𝑚 𝑖=1 𝑔 𝑖 𝑖

(11.39)

using 𝑚 sample pairs (𝑟𝑖 , 𝑒𝑖 ) according to a joint distribution probability 𝑓𝑅𝐸 . This type of numerical evaluation of the failure probability belongs to Monte Carlo simulations. A variant is given by importance sampling ∞ ∞

𝑝𝑓 = ∫ ∫ −∞ −∞

𝑓𝑅𝐸 (𝑟, 𝑒) ℎ𝑅𝐸 (𝑟, 𝑒) 𝐼𝑔 (𝑟, 𝑒) d𝑟 d𝑒 ℎ𝑅𝐸 (𝑟, 𝑒)

(11.40)

extending Eq. (11.38) with another arbitrary joint distribution probability function ℎ𝑅𝐸 and 𝑝𝑓 ≈

𝑚 1 ∑ 𝑓𝑅𝐸 (𝑟𝑖 , 𝑒𝑖 ) 𝐼 (𝑟 , 𝑒 ) 𝑚 𝑖=1 ℎ𝑅𝐸 (𝑟𝑖 , 𝑒𝑖 ) 𝑔 𝑖 𝑖

(11.41)

11.2 Failure Probability

where samples are determined according to the distribution probability ℎ𝑅𝐸 . It should be chosen such that most samples are taken in the area with the largest probability of failure, i.e. around the design point. The application is demonstrated with the following example. Example 11.2: Approximate Failure Probability of a Cantilever Column with Monte Carlo Integration

We refer to Example 11.1 and treat the same problem. The limit state function from Eq. (11.24) is not changed at first. The samples are generated on the basis of a random number generator 𝐺[0, 1], which provides real, equally distributed random numbers in the interval [0, 1]. Random numbers of a normal distribution with mean 𝜇 and standard deviation 𝜎 are determined with 𝜇 + 𝜎 Φ−1 (𝐺[0, 1]) → sample

(11.42)

with the inverse distribution function Φ−1 of the standardised normal distribution. This is applied with two independent generators 𝐺 to each of 𝑀 and 𝑒𝑐𝑐 with the parameters of Table 11.1. The normalised histograms of a set with 𝑚 = 50 samples are shown in Figure 11.5a with corresponding normal probability densities overlaid. Figure 11.5b shows the samples in the (𝑟, 𝑒)-plane together with the linear limit state function 𝑔. The evaluation of Eq. (11.39) delivers 𝑝𝑓 = 0.08, i.e. 4 samples out of 𝑚 = 50 are detected in the failure domain. This indicates a weakness of the method. The event of failure has a small probability, and only a relatively few numbers of samples determine the approximate value of the failure probability. This leads to relatively large errors. Importance sampling may be used to shift the area of sampling initially centred around the mean (𝜇𝑒𝑐𝑐 , 𝜇𝑀 ). A shift to the design point leads to a relatively large number of samples from the failure domain. This is reached using the joint probability density function 𝑓𝑀,𝑒𝑐𝑐 , where the design values 𝑒𝑐𝑐𝑑 and 𝑀𝑑 replace the mean values 𝜇𝑒𝑐𝑐 and 𝜇𝑀 to derive the joint probability density ℎ𝑀,𝑒𝑐𝑐 (Eq. (11.41)). An example with 𝑚 = 50 samples is shown in Figure 11.5b, where the evaluation of Eq. (11.41) yields

(a)

(b)

Figure 11.5 Example 11.2. (a) Normalised histograms. (b) Sampling around mean.

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11 Randomness and Reliability

(a)

(b)

Figure 11.6 Example 11.2. (a) Importance sampling. (b) Importance sampling with nonlinear limit state condition.

𝑝𝑓 = 0.049. The number of samples in the failure region is nearly the same as in the safe region. But the first ones get a much lower weighting with Eq. (11.41). Importance sampling generally leads to a significant improvement of the failure probability estimation for a given number 𝑚. Second-order effects may have to be considered for a cantilever column (Example 4.8). We assume a simplified quadratic limit state condition 𝑔(𝑀, 𝑒𝑐𝑐) = 𝑀 − |𝑃| 𝑒𝑐𝑐1 (

𝑒𝑐𝑐 𝑒𝑐𝑐2 𝑒𝑐𝑐2 − ) − 𝑀1 2 𝑒𝑐𝑐1 𝑒𝑐𝑐 𝑒𝑐𝑐12 1

(11.43)

which resembles the linear limit state condition for 𝑒𝑐𝑐 = 0 with the same tangent but otherwise approximately considers second-order effects with a representative limit state 𝑒𝑐𝑐 = 𝑒𝑐𝑐1 , 𝑀 = 𝑀1 for the behaviour of the cantilever column model. In contrast to Example 4.8, where the structural limit load is reached before a local bearing capacity of cross-sections is exhausted, the quantity 𝑀1 corresponds to a cross-sectional failure. The values are assumed with 𝑒𝑐𝑐1 = 𝜇𝑒𝑐𝑐 , 𝑀1 = 0.23, i.e. a 15 % increase of moment at mean eccentricity (Table 11.1) – compared to 𝑃 𝜇𝑒𝑐𝑐 – due to second-order effects. The nonlinear limit state condition is shown in Figure 11.6. The corresponding design point is determined with the standardised point nearest to the origin whereby using an iteration. The corresponding solution is 𝑒𝑐𝑐𝑑 = 0.116, 𝑀𝑑 = 0.273, which is used for importance sampling. The results for a set with 𝑚 = 50 samples are shown in Figure 11.6b. The failure probability is determined with 𝑝𝑓 = 0.288 according to Eq. (11.41). Thus, a second-order analysis results in a considerably higher failure risk for the current set-up. Sets with 𝑚 = 50 samples are considered. Actually, this is generally too small to reproduce the results for the failure probability by applying the Monte Carlo simulation. A convergence study is necessary to determine a magnitude of 𝑚, which yields the same failure probability for each set of samples within a given tolerance. This exceeds the scope of the current example.

11.2 Failure Probability

Monte Carlo simulations have the following properties for the evaluation of failure probabilities: • Model behaviour is ruled through a limit state condition that relates the model’s properties and variables and yields a result 1/0 or failure/non-failure, respectively. • Multi-variate random variables composed of a larger number of model parameters are allowed. • Any type of joint probability density function is possible with random variables correlated. • Random fields (Figure 11.2) may be considered; they are generally synthesised based on correlation length estimates (Vorechovsky 2008). On the other hand, a large number of included model parameters require a very large number of samples, and the demand for computational resources may become very high due to the large number of evaluations of the limit state function. Each evaluation becomes computationally expensive for complex nonlinear models, which might be prohibitive for a Monte Carlo simulation. Response surface methods or neural networks might offer a potential workaround. Another issue concerns the availability of distribution data or samples of random variables in the case of building structures. Often, the number of available samples is not large enough to gain reliable results. Concepts of confidence have to be incorporated, and statements about a failure probability are supplied with a probability. Randomness, which is quantified with statistical methods by definition, expands into a wider scope of uncertainty. Concepts of fuzziness may be used to quantify this (Möller and Beer 2004).

11.2.3 Multiple Limit Conditions Up to now, we restricted ourselves to a single critical spot of a structure, e.g. a midspan cross-section of a single-span beam with a single limit state function related to the bending capacity of a cross-section. There might be a number of critical spots, e.g. cross-sections near mid-spans and in supporting points of continuous beams. A cross-section in a supporting point may comprise two critical spots, 5) one related to bending and another one related to shear. To simplify we assume that a critical spot 𝑖 is characterised by a single 𝑔𝑖 (𝑟𝑖 , 𝑒𝑖 )-limit state function, be it linear (Eq. (11.11)) or nonlinear. 6) A two-span beam is exemplarily considered with the same spans, constant crosssection, and constant loading; see Figure 11.7a. A joint random vector is given by ) ( (11.44) V = 𝑅1 𝑅2 𝑅3 𝐸 with resistance index 1 for near-mid-span bending 𝑅1 , 2 for support point bending 𝑅2 , 3 for support point shear 𝑅3 , a common loading random variable 𝐸, and limit 5) A geometric position is allowed to have more than one ‘spot’. 6) Treatment of multiple limit conditions 𝑔1 (𝑟, 𝑒), 𝑔2 (𝑟, 𝑒) … for the same random variables 𝑅, 𝐸 require generalisations that are straightforward and are not explicitly described.

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11 Randomness and Reliability

(a)

(b)

Figure 11.7 (a) Two-span beam. (b) Abstract failure model for the two-span beam.

state functions 𝑔𝑖 (𝑟𝑖 , 𝑒), 𝑖 = 1…3 describing failure mechanisms. The format (→ unit) of the action variable 𝑒 is adapted to the respective format of the limit state function. Known marginal distributions for each random variable are assumed and, if available, correlation data. Uncorrelated variables are assumed otherwise. Applying the Nataf model (Section 11.2.2) with the N-standardisation leads to a joint probability density function according to Eq. (11.26). The variables span a hyperspace with limit state conditions as hypersurfaces separating the failure domain from the safe domain. The hyperspace includes three design points according to three limit state conditions. Partial failure probabilities 𝑝𝑓𝑖 related to each failure mechanism are determined applying FORM (Eq. 11.35) under the premise of bivariate limit state functions 𝑔𝑖 (𝑟𝑖 , 𝑒𝑖 ). But this does not directly yield a joint probability failure 𝑝𝑓 for the whole system. A Monte Carlo approach (Eq. 11.39) is still possible, but the computational costs may become prohibitive for systems with a larger number of variables. For further considerations, only partial failure probabilities are considered. ◀

As the failure probability is regarded as operational within the current context (Section 11.1) an estimation of the joint failure probability derived from the partial failure probabilities is generally accepted in structural analysis.

This requires abstract models for the failure of structures, like series, parallel systems, and combinations thereof (Schneider 2006, 4.7). The two-span beam has three failure mechanisms ordered as a combination of a parallel system of bending failures in a series with shear failure, as is shown in Figure 11.7b. Failure is assumed to occur in the case of bending failure or shear failure. Bending failure will occur if both critical spots for bending fail together under the assumption of sufficient ductility. The design of such orderings combining series and parallel systems becomes exponentially elaborate with an increasing number of failure mechanisms and generally yields multiple results. This is only justified for selected structures with a high risk potential in the case of failure. Thus, a series model – system failure occurs if already one series member fails – is chosen to allow for general statements. This does not account for, e.g. stress redistribution regarding statically indeterminate structures with ductile materials. But it is on the safe side. The task now is to specify the limits of the probability of failure. Further alternative assumptions with respect to the statistical correlations (Figure 11.1) between failure mechanisms have to be made for this. A series of 𝑛 failure mechanisms is

11.3 Design and Safety Factors

considered. The first assumption is that all failure mechanisms are uncorrelated, which yields the upper limit of the joint failure probability 𝑝𝑓 ≤ 1 −

𝑛 𝑛 𝑛 ∏ ∑ ∑ (1 − 𝑝𝑓𝑖 ) ≈ 𝑝𝑓𝑖 = Φ(−𝛽𝑖 ) 𝑖=1

𝑖=1

(11.45)

𝑖=1

with the partial failure probability 𝑝𝑓𝑖 ≪ 1 and the reliability index Eqs. (11.20) and (11.13) of each failure mechanism. Actually, this assumption is not realistic. For the simple two-span beam from above, the limit state conditions include the same action random variable 𝐸. Thus, they cannot be uncorrelated by definition. The opposite assumption states that all failure mechanisms are fully correlated, i.e. the case of failure occurs simultaneously for all failure mechanisms. This yields a lower limit for the joint failure probability ruled by the failure mechanism with the largest failure probability ( ) 𝑛 𝑛 𝑝𝑓 ≥ max 𝑝𝑓𝑖 = Φ − min𝑖=1 𝛽𝑖 𝑖=1

(11.46)

The respective assumption is also only realistic to a limited extent. For the twospan beam, the random resistances are not fully correlated, e.g. for shear resistance and bending resistance, which have different failure mechanisms from a mechanical point of view. Furthermore, bending resistances in different cross-sections will not be fully correlated due to random stochastic fields (Figure 11.2) of material and geometric properties. To narrow the wide range between these upper and lower limits is not trivial. We only mention an approach by Ditlevsen (1979) 𝑝𝑓 ≤

𝑛 ∑ 𝑖=1

Φ(−𝛽𝑖 ) −

𝑛 ∑ 𝑖=2

max Φ2 (−𝛽𝑖 , 𝛽𝑗 , 𝜚𝑖𝑗 ) 𝑗1, 𝑟𝑑

𝛾𝐸 =

𝑒𝑑 >1 𝑒𝑘

(11.56)

leading to a semi-probabilistic design rule 𝑟𝑘 ≥ 𝛾 𝑒𝑘

→

𝑟𝑘 ≥ 𝛾𝐸 𝑒𝑘 𝛾𝑅

(11.57)

Assuming normal distributions the partial safety factor for the resistance is given by (Eq. (11.54) and Eq. (11.23)) 𝛾𝑅 =

𝜇𝑅 − 𝛿 𝜎𝑅 1 − 𝛿 𝜈𝑅 = 𝜇𝑅 − 𝛽𝛼𝑅 𝜎𝑅 1 − 𝛽 𝜈𝑅 𝛼𝑅

(11.58)

and for the action by 𝛾𝐸 =

𝜇𝐸 − 𝛽𝛼𝐸 𝜎𝐸 1 − 𝛽 𝜈𝐸 𝛼𝐸 = 𝜇𝐸 + 𝛿 𝜎𝐸 1 + 𝛿 𝜈𝐸

(11.59)

with the coefficients of variation 𝜈𝑅 =

𝜎𝑅 , 𝜇𝑅

𝜈𝐸 =

𝜎𝐸 𝜇𝐸

(11.60)

Some care has to be taken with 𝛾𝑅 according to Eq. (11.58) with large values 𝜈𝑅 and/or small values 𝛼𝑅 , which might lead to values 𝛾𝑅 < 1. This is not reasonable. There is no similar risk regarding 𝛾𝐸 , as 𝛼𝐸 < 0 by definition (Eq. (11.14)). The approach conforms to FORM (Section 11.2.2), as a linear limit state function (Eq. (11.48)) is assumed. The safety factors depend on the reliability index 𝛽 or the failure probability 𝑝𝑓 , the coefficients of variation 𝜈𝑅 , 𝜈𝐸 , the sensitivity parameters 𝛼𝑅 , 𝛼𝐸 and the parameter 𝛿 for fractile values. Within limits the choice for fractiles or 𝛿 is a matter of convention only. The choice for the failure probability or reliability index 𝛽 (Eq. (11.52)) reflects a desired overall safety standard. ◀

General data assessment shows that the sensitivity parameters 𝛼R , 𝛼E exhibit a low variability. They are assumed as constants for the semi-probabilistic design. This decouples action from resistance, which are treated independently.

To a large extent, this simplifies design, as different categories of resistance – due to, e.g. different materials (concrete, steel, wood) – share an exposition to a common framework of actions. Typically assumed values are 𝛼𝑅 = 0.8, 𝛼𝐸 = −0.7, (EN 1990 2010, C7) which violates the constraint Eq. (11.15) on the safe side. The partial safety factors 𝛾𝑅 , 𝛾𝐸 only depend on the respective coefficients of variation 𝜈𝑅 , 𝜈𝐸 with

11.3 Design and Safety Factors

these common assumptions. These should be available based on systematic data acquisition for the specific types of the resistance 𝑅 and the action 𝐸. A simplified application is demonstrated with the following example Example 11.3: Simple Partial Safety Factor Derivation

We refer to Example 11.1 of a population of reinforced cantilever columns with parameters in Table 11.1. The action side must have the same measure as the resistance side. Thus, the eccentricity 𝑒𝑐𝑐 of Example 11.1 is multiplied by the assumed downward loading 𝑃 = 2 MN to derive the corresponding parameters, as given in Table 11.2. The limit state function (Eqs. (11.11) and (11.50)) has coefficients 𝑎 = 1, 𝑏 = 0. Table 11.2 Parameters of Example 11.3.

Mean μ Standard deviation 𝜎 Coefficient of variation 𝜈 (Eq. (11.60)) Sensitivity parameter 𝛼 (EN 1990 2010, C7) Partial safety factor 𝛾 (Eqs. (11.58), (11.59))

[MNm]∕[m] [MNm]∕[m] — — —

r → Mr

e → Me

0.27 0.015 0.056 0.8 ≈ 1.1

0.20 0.04 0.200 −0.7 ≈ 1.2

The required reliability index is generally assumed with 𝛽 = 3.8 leading to a failure probability of 𝑝𝑓 ≈ 7 ⋅ 10−5 during a lifetime of 50 years (EN 1990 2010, B3.2). This results in the partial safety factors also listed in Table 11.2 to be applied to characteristic values according to Eq. (11.57). Note that this is exemplary and cannot be applied to actual situations. The Example 11.3 has its focus on the bending of cross-sections regardless of cantilever columns. With the given conventions with respect to the reliability index, the influence parameters, and fractiles it depends only on the coefficients of variation.

11.3.2 Partial Safety Factor Application The current assumptions have to be extended. This starts with distribution types. A transformation is used to consider other than normal distributions. An arbitrary random variable 𝑋 with distribution function 𝐹𝑋 according to Eq. (11.1) and realisations 𝑥 is transformed into a standardised normal random variable 𝑌 with realisations 𝑦 by Φ(𝑦) = 𝐹𝑋 (𝑥)

→

𝑦 = Φ−1 (𝐹𝑋 (𝑥))

(11.61)

with the distribution function Φ of the standardised normal distribution. The design value (Section 11.2.1) of a standardised normal 𝑦 of 𝑌 with mean 𝜇𝑌 = 0 and standard

373

374

11 Randomness and Reliability

deviation 𝜎𝑌 = 1 is given according to Eq. (11.23) 𝑦𝑑 = −𝛽 𝛼𝑌

(11.62)

This leads to a condition for the design value 𝑥𝑑 of the source 𝑋 −𝛽 𝛼𝑌 = Φ−1 (𝐹𝑋 (𝑥𝑑 ))

(11.63)

and is applied for a normal distribution with 𝐹𝑋 = Φ((𝑥 − 𝜇𝑋 )∕𝜎𝑋 )) for control purposes −𝛽 𝛼𝑌 = Φ−1 (Φ (

𝑥𝑑 − 𝜇𝑋 )) 𝜎𝑋

→

𝑥𝑑 = 𝜇𝑋 − 𝛽𝛼𝑌 𝜎𝑋

(11.64)

which conforms to Eq. (11.23), where 𝛼𝑌 indicates the sensitivity parameter after N-standardisation (Section 11.2.2). Log-normal distributions are often used for resistances. Applying Eq. (11.63) with 𝐹𝑅 = Φ((ln 𝑟 − 𝜇𝑅 )∕𝜎 𝑅 ) yields as standardised normalised resistance design point ( ) 𝜎𝑅 2 1 2 𝑟𝑑 = e𝜇𝑅 −𝛽𝛼𝑅 𝜎𝑅 , 𝜇𝑅 = ln 𝜇𝑅 − 𝜎 𝑅 , 𝜎𝑅 = ln 1 + 𝜈𝑅2 , 𝜈𝑅 = 2 𝜇𝑅

(11.65)

with a sensitivity parameter 𝛼𝑅 after N-standardisation. Assuming 𝜈𝑅 < 0.2, applying a row expansion to ln(1 + 𝜈𝑟2 ), and neglecting contributions of 𝜈𝑅2 ∕2 or higher, Eq. (11.65) is simplified with 𝑟𝑑 = 𝜇𝑅 e−𝛽𝛼𝑅 𝜈𝑅

(11.66)

Actions are often represented by extreme value distributions, e.g. a Type 1 Gumbel distribution with 𝐹𝐸 = exp(− exp(−𝑠𝐸 (𝑒 − 𝑢𝐸 ))) with exp(∙) = e∙ leading to a standardised normalised action design point 𝑒 𝑑 = 𝑢𝐸 −

1 ln(− ln(Φ(−𝛽𝛼𝐸 ))) , 𝑠𝐸

𝑢𝐸 ≈ 𝜇 𝐸 −

0.5772 π , 𝑠𝐸 = √ (11.67) 𝑠𝐸 𝜎𝐸 6

with a sensitivity parameter 𝛼𝐸 after N-standardisation. The corresponding characteristic values – normalised and standardised – are given with −𝛿 for the resistance and with +𝛿 for the action (Eq. (11.54)). They replace the term −𝛽 𝛼𝑌 on the left-hand side of Eq. (11.63) or −𝛽𝛼𝑅 in Eq. (11.66) to determine 𝑟𝑘 and −𝛽𝛼𝐸 in Eq. (11.67) for 𝑒𝑘 . Alternatively, characteristic values may be directly estimated from statistical sampling. Partial safety factors are again determined with Eq. (11.56). A further extension concerns a specification of random variables that were recapitulated with 𝑅, 𝐸 before. Actions as well as resistances may arise from several sources. Their specific sensitivity parameters Eq. (11.29) used by Eq. (11.28) rule their respective contribution to the reliability index or the failure probability. The sensitivity parameters also rule the respective partial safety factor, as is described above. They are generally assumed as fixed values depending on the general type

11.3 Design and Safety Factors

of action or resistance. Thus, the respective partial safety factor determination follows the rules above for each random contribution to resistance or action. This also decouples all random design variables regarding the determination of a failure probability. The procedures described above have to be transferred to a concrete semi-probabilistic design task with a specific structural system, material grades, and loading. This is sketched as follows with • Appropriate characteristic loadings from code provisions. • Structural analysis with critical spot identification leading to characteristic action 𝑒𝑘 in the critical spot. • Characteristic resistance 𝑟𝑘 in the critical spot based on values of material grades – according to code provisions – and its geometric properties. • Design rule Eq. (11.57) whereby applying partial safety factors. General design practice insofar deviates as a) safety factors for actions are applied to the characteristic loading, and b) safety factors for resistances are applied to characteristic material properties. This will basically not change the coefficients of variation accounting for the respective partial safety factor in the case of a) linear structural analysis and b) linear relations between material properties and critical spot resistances, e.g. resisting bending moments. Nonlinear structural analysis and nonlinear resistances each act as filtering and more or less cause deviations of coefficients of variations between a) loading and critical spot actions and b) material properties and critical spot resistance. This is generally neglected for a) in order to avoid a re-coupling of actions with structural system characteristics, or considered, if at all, by additional overall coefficients. Nonlinear filtering is also generally neglected in common design practice 8) with b). Regardless of this, design standards for reinforced concrete provide rules to take this into account, providing a general framework for nonlinear computations ((CEB-FIP 2008, 3.6), CEB-FIP2 (2012, 7.11.3), EN 1992-2 (2005, 5.7)). Such rules are based on the mean values of the material strength of concrete and reinforcement. These mean values are used for a nonlinear structural analysis leading to a ‘mean’ resistance in the critical spot on the one hand, and, on the other hand, the corresponding integrated resistance is divided by a single partial safety factor 𝛾𝑅 covering the included material components. In addition, taking the nonlinear material behaviour into account in a structural analysis also accounts for the nonlinear filtering with a). An alternative approach first performs a nonlinear structural analysis with the mean values of material properties, leading to a critical spot resistance 𝑟𝑚 , and then performs a second analysis with characteristic values of material properties leading to a critical spot resistance 𝑟𝑘 (ECOV) (Cervenka 2008). Assuming a log-normal

8) This has as side effect that the design rule Eq. (11.57) becomes nonlinear in 𝑟𝑘 and has to be solved iteratively. Standard design tables provide solutions in order to minimise the difference between 𝑟𝑘 ∕𝛾𝑅 and 𝛾𝐸 𝑒𝑘

375

376

11 Randomness and Reliability

distribution for both, the Eq. (11.66) is utilised with 𝑟𝑘 = e−𝛿 𝜈𝑅 𝑟𝑚

(11.68)

with the fractile coefficient 𝛿 (Eq. (11.54)). Using 𝛿 = 𝛿5% = 1.65 finally yields 𝜈𝑅 =

𝑟𝑚 1 ln ( ) 1.65 𝑟𝑘

(11.69)

for the resistance coefficient of variation. It is used with Eq. (11.66) to determine a resistance design value 𝑟𝑑 for a design rule 𝑟𝑑 ≥ 𝛾𝐸 𝑒𝑘 as a variation of Eq. (11.57). This is restricted to one loading case and might require some iteration to find a value 𝑟𝑑 that does not exceed 𝛾𝐸 𝑒𝑘 by too much.

377

12 Concluding Remarks To start with general conclusions let us first point out the key results of the chapters. • Chapter 1 describes concrete as the building material of the twentieth century and states that concrete is also likely to be the building material of the twenty-first century. But sustainability has to become a key factor in addition to limit states, serviceability, and durability. • Chapter 2 introduces models as mirrors of reality, which should include essential aspects and disregard everything else. A particular problem may have a number of reasonable models. It is not a matter of true or false but of appropriateness. Discretised equilibrium conditions are described in the main part of the chapter: internal nodal forces have to be equal to external nodal forces. Both are vectors according to the discretised degrees of freedom. External nodal forces are a projection of a loading into nodes. They are generally known a priori. Internal nodal forces are a projection of the internal state of stress into nodes. They depend on the deformations of a structure. Such a dependence is generally nonlinear in the case of structural concrete. Thus, a discretised equilibrium condition is solved by nonlinear equation solvers – typically Newton–Raphson – embedded in an incremental load application. • Nonlinear uniaxial behaviour of plain concrete and reinforcing materials is described in Chapter 3 as a basis to understand their interaction in a compound material. Interaction is a matter of stress transfer between concrete and reinforcement utilizing bond. Bond is a mandatory requirement to make structural concrete work. Furthermore, bond controls the cracking of concrete and implicitly allow us to control crack formation as an ordinary property of structural concrete. FEM gives a comprehensive understanding of this process. • Chapter 4 treats 2D beams and frames with an exemplary description of the whole range of applications typical for structural analysis in civil engineering. Uniaxial material behaviour is basically sufficient to model the structural behaviour of beams and frames due to the Bernoulli–Navier hypothesis. Stress–strain relations are generalised to relations between moments, normal forces, curvatures, and strains of a reference axis.

Computational Structural Concrete – Theory and Applications, Second Edition. Ulrich Häussler-Combe. © 2023 Ernst & Sohn GmbH. Published 2023 by Ernst & Sohn GmbH.

378

12 Concluding Remarks

•

•

•

•

These generalised relations are nonlinear and fully coupled in the case of reinforced concrete beam and frame structures. They also form the core to derive finite element models that are applied to topics like creep of concrete, effects of temperature and shrinkage, prestressing, large displacements, and dynamics. Characteristics of structural concrete are demonstrated for each case. Uniaxial stress–strain relations are still utilised to model reinforced concrete deep beams or plates, respectively. This is described in Chapter 5 with strut-and-tie models. On the one hand, such models seamlessly embed into the context of linear and nonlinear FEM, and, on the other hand, they are an important tool for practical design of reinforced concrete plate structures. Limit theorems of plasticity are introduced in the context of strut-and-tie models. In particular, the lower limit theorem constitutes a fundamental basis for the practical design of all types of reinforced concrete structures. Chapter 6 treats triaxial stress–strain relations and strength conditions for concrete in the framework of continuum mechanics. Uniaxial and biaxial relations can be derived as special cases. Material models like elasto-plasticity and damage should ensure that the particular stress–strain relations and strength limit conditions are consistent, i.e. stresses derived from arbitrary strains will always conform to limit strength conditions. Plasticity and damage may be combined to account for permanent strains and degraded stiffness simultaneously. Load-induced anisotropy is a topic, as concrete may sustain compressive stresses in some direction, while tensile failure occurs in perpendicular directions. This is implicitly provided by the microplane model. Limits of concrete, in particular in the tensile range, are characterised by quasibrittle behaviour connecting increasing strains with decreasing stresses. This chapter in itself is independent from numerical methods. But stress–strain relations constitute the core of such methods for structural analysis. Strain softening and cracking of concrete involve a mutual dependency of stress– strain relations derived from continuum mechanics and numerical methods or discretisation, respectively. The topic includes regularisation requirements and is treated in Chapter 7. Smeared crack models map crack discontinuities into continuous forms, where mesh dependency due to strain localisation is avoided by crack band approaches to preserve crack energy. A localisation band geometry may be resolved by gradient methods and phase fields with regularisation included. Displacement discontinuities are explicitly modelled by, e.g. FEM extended with the strong discontinuity approach, whereby modelling quasi-brittle behaviour by traction–separation relations, which naturally reproduce a crack energy. Mesh independence is implicitly ensured without explicit regularisation. Biaxial stress–strain relations account for the treatment of deep beams/plates in Chapter 8. A design of reinforced concrete plates can be performed with stresses derived from linear elastic FEM whereby regarding the limited strength of concrete and reinforcement based on the lower limit theorem of plasticity. But this disregards kinematic compatibility and deformations.

12 Concluding Remarks

Nonlinear biaxial stress–strain relations for concrete have to be included for a consistent deformation treatment. A number of alternatives exist to study concrete interaction with reinforcement. The common approach is to assume a rigid bond, but a flexible bond is also possible with a self-contained placement of rebars within a concrete plate. Enhanced modelling allows us to combine nonlinear characteristics of concrete and reinforcement, quasi-brittle failure behaviour of concrete, and bond in a consistent way. But numerical computations may become elaborate within this set-up. Furthermore, a model uncertainty may also arise due to alternate models for nonlinear concrete behaviour. • Chapter 9 treats slabs as a biaxial extension of beams. Stresses and strains are generalised by bending moments and curvatures. As with plates, a design of reinforced concrete slabs can be performed with linear elastic FEM results, again based upon the lower limit theorem of plasticity. This again disregards deformation behaviour. Nonlinear moment—curvature relations have to be involved to consider slab deformations. This is simplified to some extent by deriving bending of slabs in analogy to the bending of beams. But assumptions have to made with respect to the twisting behaviour of slabs. This again leads to a model uncertainty for numerical computations. The chapter also describes an application of the upper limit theorem of plasticity to slabs with respect to ultimate limit states, which may be used as a reference for nonlinear numerical computations. • Shell structures, as they are treated in Chapter 10, are generally associated with curved geometries. But first and foremost, they combine bending with membrane or normal forces, respectively. Thus, they form a generalisation including both plates and slabs. Slabs treated with a shell theory allow us to model the interaction of bending with normal forces as is characteristic for cracked reinforced concrete cross-sections. Nonlinear triaxial material models as they are described in Chapter 6 are employed with continuum-based finite shell elements. This may be superposed with reinforcement layers with uniaxial nonlinear stress–strain behaviour in arbitrary stacking and orientation and marks some kind of high end of the modelling of structural concrete. However, the computational effort becomes elaborate, and studies with a range of concrete material models also show that model uncertainties are to be expected. • Chapter 11 points out that the parameters of the models described above cannot be determined with fixed values in reality and are subject to randomness. This is highlighted with basic concepts of statistics and probabilistic, which are used to estimate operative failure probabilities to quantify the reliability of structures against references. Operative failure probabilities serve to derive partial safety factors on a rational basis. Partial safety factors allow us to decouple actions from resistances and different categories of resistance regarding structural reliability. Thus, they are the predominant approach to reach the objectives of safety levels in design practice.

379

380

12 Concluding Remarks

We face the topic of model uncertainty regarding nonlinear reinforced concrete plates, slabs, and shells with the Examples 8.2, 9.5, and 10.2. This is generally connected to a range of material models for concrete behaviour. Model uncertainty is a phenomenon often seen with blind benchmarks 1) for reinforced concrete plate and slab structures with complex simulation models; see, e.g. Ghavamian and Carol (2003); Collins et al. (2015). But one should keep in mind that experimental data that serve as a reference for competing models are also subject to uncertainty; different measurement results are obtained for nominally identical test specimens. This may pose a problem for large structural elements that are only examined as individual pieces. Regardless of this, model uncertainties should be qualified more precisely. As far as can be assessed, they are related to the modelling of crack formation, localisation, strain softening, and load-induced anisotropy. From a mathematical point of view, these aspects may lead to ill-posed problems, and applied numerical methods will not yield reliable solutions, if there is any solution at all. Thus, methods of regularisation are required. Numerical mechanics provides a span of regularisation methods. But their application requires a good comprehension in order to assess the quality of results of numerical computations. This raises the question as to which extent nonlinear numerical methods can be used in design practice. Models including uniaxial concrete behaviour are generally reliable. But in our opinion, some caution should be exercised when including nonlinear multi-axial material models. Research and development have to considered as another field in contrast to design practice. Despite of model uncertainty, the application of sophisticated models for structures including multi-axial concrete models yields invaluable data, leading to answers as well as to new questions. A researcher should not expect reliable solutions to be found in the first trial and should always consider alternatives. The reference to experimental investigations – despite their own uncertainty – generally proves to be very fruitful in obtaining better answers and improved models. Better models ultimately lead to the goal stated at the beginning (Chapter 1) that ◀

we have to use less concrete and less reinforcement material and at the same time achieve a higher quality of building components.

This textbook described topics of numerical methods applied to structural concrete in order to contribute to this goal. A complete description of the field exceeds the scope of a single book with limited volume. But the reader should have gained the necessary comprehension to continue with further studies.

1) A blind benchmark poses a problem for numerical simulation supported by experimental data. The data (Section 2.1) are published but not the experimental outcome initially. Competing simulations provide results to be rated against the ‘real’ behaviour afterward.

381

Appendix A Solution Methods A.1 Nonlinear Algebraic Equations The Newton–Raphson (NR) method, described in Section 2.8.1, can be considered as a standard method for the solution of nonlinear algebraic systems and is used for most of the example problems. But it will not always converge. This is illustrated with a few examples. First of all, the tangential stiffness 𝐾𝑇 is not allowed to be zero or singular regarding K𝑇 , as can be seen from Eqs. (2.73) and (2.77). But also very small values or ‘near singularity’ will prevent NR convergence due to roundoff errors with arithmetic computer operations. With respect to systems of equations, a single degree of freedom with near zero stiffness might impair the convergence of the whole system. There are more conditions for NR convergence from a mathematical point of view, which can only be indicated exemplarily. • An iteration starting point should be near enough to the solution point. • A second derivative should not change its characteristic, as it does for the scalar example shown in Figure A.1a; the curvature of 𝑓(𝜈) changes its sign leading to a cycle iteration. • Another type of cycle iteration is demonstrated with a simple two-degree of freedom system with two bar elements is shown in Figure A.1b. This has the tangen-

(a)

(b)

Figure A.1 NR cycle iteration examples. (a) Curvature change. (b) Two dof system with softening Computational Structural Concrete – Theory and Applications, Second Edition. Ulrich Häussler-Combe. © 2023 Ernst & Sohn GmbH. Published 2023 by Ernst & Sohn GmbH.

382

A Solution Methods

tial bar stiffness 𝐾𝑇1 , 𝐾𝑇2 , leading to a tangential stiffness matrix ⎡𝐾𝑇1 + 𝐾𝑇2 K𝑇 = ⎢ −𝐾𝑇2 ⎣

−𝐾𝑇2 ⎤ ⎥ 𝐾𝑇2 ⎦

(A.1)

as is derived for the 1D two-node bar element with Eqs. (2.26) and (2.66). Both are assigned an initial tangential stiffness 𝐾𝑇ℎ > 0. A limited strength is assumed for bar 2, reached with Δ𝜐2 > Δ𝜐𝑡 , followed by a softening with a tangential stiffness 𝐾𝑇𝑠 < 0, |𝐾𝑇𝑠 | < |𝐾𝑇ℎ |. This occurs, e.g. for smeared cohesive crack models, as is shown with Example 3.3 and results in the tangential stiffness matrices

K𝑇 =

⎧⎡ ⎪⎢ 2𝐾𝑇ℎ > 0 ⎪ −𝐾 < 0 ⎪⎣ 𝑇ℎ

−𝐾𝑇,ℎ < 0⎤ ⎥, 𝐾𝑇ℎ > 0 ⎦

⎨ ⎪⎡𝐾𝑇ℎ + 𝐾𝑇𝑠 > 0 ⎪⎢ ⎪ −𝐾𝑇𝑠 > 0 ⎩⎣

Δ𝜐2 ≤ Δ𝜐𝑡 (A.2)

−𝐾𝑇𝑠 > 0⎤ ⎥ , Δ𝜐2 > Δ𝜐𝑡 𝐾𝑇𝑠 < 0 ⎦

With appropriate values for a loading 𝑃 and strength deformation Δ𝜐𝑡 , the NR method according to Eq. (2.77) leads to a cycle iteration without convergence. We assume 𝐾𝑇ℎ = 10, 𝐾𝑇𝑠 = −2, 𝑃 = 1, Δ𝜐𝑡 = 0.05 as example values, resulting in ( )T ( )T (𝜈) a cycling sequence 𝝊(𝜈) = 0.1 0.2 , r(𝜈) = −1.2 1.2 , Δ𝜐2 = 0.1 followed ( )T ( )T (𝜈+1) = −0.5. by 𝝊(𝜈+1) = 0.1 −0.4 , r(𝜈+1) = −6.0 6.0 , Δ𝜐2 The sign change in the first derivative prevents convergence from a mathematical point of view. This may be overcome by adding ‘positive’ contributions to the tangential stiffness matrix and can be performed by adding stabilizing parts to the tangential material stiffness or to the tangential system stiffness. With respect to the first, we refer to Eq. (2.81), extended by an artificial viscosity 𝝈𝑖+1 = 𝝈𝑖 + C𝑇,𝑖+1 ⋅ (𝝐 𝑖+1 − 𝝐 𝑖 ) + 𝜂 𝝐̇ 𝑖+1

(A.3)

with a viscosity parameter 𝜂. The strain velocity is discretised in time using the Newmark method Eq. (2.87) 𝝐̇ 𝑖+1 =

𝛾 𝛾 𝛾 ) 𝝐̈ 𝑖 (𝝐 𝑖+1 − 𝝐 𝑖 ) + (1 − ) 𝝐̇ 𝑖 + Δ𝑡 (1 − 𝛽Δ𝑡 𝛽 2𝛽

(A.4)

With 𝛾 = 1∕2, 𝛽 = 1∕4, the time integration parameters are generally chosen to have unconditional stability independent of Δ𝑡 (Bathe 1996, 9.4). This leads to 𝝐̇ 𝑖+1 =

2 (𝝐 − 𝝐 𝑖 ) − 𝝐̇ 𝑖 Δ𝑡 𝑖+1

(A.5)

and is inserted into Eq. (A.3) to obtain 𝝈𝑖+1 = 𝝈𝑖 + C𝑇,𝑖+1 ⋅ (𝝐 𝑖+1 − 𝝐 𝑖 ) +

2𝜂 (𝝐 − 𝝐 𝑖 ) − 𝜂 𝝐̇ 𝑖 Δ𝑡 𝑖+1

(A.6)

A.1 Nonlinear Algebraic Equations

(a)

(b)

Figure A.2 (a) Modified Newton–Raphson method (b) Secant method.

The extended tangential material stiffness matrix is derived with 𝝈̇ 𝑖+1 = (C𝑇,𝑖+1 +

2𝜂 I) ⋅ 𝝐̇ 𝑖+1 Δ𝑡

(A.7)

with the unit matrix I. Thus, the extended tangential material stiffness matrix is always ‘positive’ with an appropriate choice of 𝜂∕Δ𝑡. This transfers to the tangential element stiffness matrix (Eq. (2.66)). But the artificial viscosity 𝜂 might introduce an unphysical behaviour. There is no patent remedy to resolve the conflict, and parameter studies are required to choose appropriate values. Stabilizing extensions on the system level can also be reached with a mass matrix scaled by an adaptable parameter. The implicit Newmark method (Eq. (2.89)) is again appropriate but implies a dynamic calculation. A loading has to be applied very slowly to approach quasi-static conditions. This may be supported by Rayleigh damping on the system level (Bathe (1996), Section 4.10). Convergence considerations are obsolete for explicit dynamics (Belytschko et al. 2000, 6.2). Any calculated material response is equilibrated by inertia, whether it models reality or something else. A secant method as is described in the following might also solve the NR convergence problem in the case of a quasi-static approach. There are alternatives for the NR method. The computational costs of NR are still relatively high, as the LU-decomposition (Eq. (2.78)) is performed at each iteration step. A cost reduction can be reached with the modified Newton–Raphson method. The LU-decomposition is performed only once for an iteration sequence. The scalar version derived from Eq. (2.73) is given by [ ]−1 (0) 𝛿𝜐 = 𝐾𝑇 𝑟(𝜐 (𝜈) )

(A.8)

and is illustrated in Figure A.2a. On the other hand, the modified NR method requires more iterations to converge, if this is reached at all. A further alternative is given with secant methods. We consider again the scalar forms Eqs. (2.71)–(2.74), which are summed up as [ (𝜈) ]−1 𝛿𝜐 = 𝐾𝑆 𝑟(𝜐 (𝜈) ) 𝜐 (𝜈+1) = 𝜐 (𝜈) + 𝛿𝜐

(A.9)

383

384

A Solution Methods

with 𝐾𝑆 replacing 𝐾𝑇 . The choice of 𝐾𝑆 is basically arbitrary, as long as it leads to a convergence of the sequence 𝜐 (𝜈+1) to the solution of 𝑟(𝜐) = 𝑝 − 𝑓(𝜐) = 0. A secant method defines a sequence of secant stiffness 𝐾𝑆 with (𝜈+1)

𝐾𝑆

𝛿𝜐 = 𝜐 (𝜈+1) − 𝜐 (𝜈) , 𝛿𝑟 = 𝑟(𝜐 (𝜈) ) − 𝑟(𝜐 (𝜈+1) )

𝛿𝜐 = 𝛿𝑟 ,

(𝜈+1)

see Figure A.2b. The secant stiffness 𝐾𝑆

(A.10)

is used for the following iteration with

[ (𝜈+1) ]−1 𝑟(𝜐 (𝜈+1) ) 𝜐 (𝜈+2) = 𝜐 (𝜈+1) + 𝐾𝑆

(A.11)

The generalisation of the condition Eq. (A.10) for 𝑛 > 1 is given by (𝜈+1)

K𝑆

𝛿𝝊 = 𝝊(𝜈+1) − 𝝊(𝜈) ,

𝛿𝝊 = 𝛿r ,

𝛿r = r(𝝊(𝜈) ) − r(𝝊(𝜈+1) )

(A.12)

But in contrast to the scalar case 𝑛 = 1, this no longer has a unique solution for (𝜈+1) K𝑆 , as is demonstrated for 𝑛 = 2: assuming a symmetric but otherwise unknown secant stiffness matrix and known vectors 𝛿𝝊, 𝛿r, there are two equations for three unknown components of the secant stiffness. This gives room for further alternatives. A popular approach is given with the BFGS method, which defines the secant stiffness matrix by (𝜈+1)

K𝑆

(𝜈)

= K𝑆 +

𝛿r ⋅ 𝛿r

(𝜈)

T

T

𝛿r ⋅ 𝛿𝝊

−

T

(𝜈)

K𝑆 ⋅ 𝛿𝝊 ⋅ 𝛿𝝊 ⋅ K𝑆 T

(𝜈)

𝛿𝝊 ⋅ K𝑆 ⋅ 𝛿𝝊

(A.13)

fulfilling Eq. (A.121 ). Furthermore, the inverse is given with the BFGS normal form T T T [ (𝜈+1) ]−1 [ (𝜈) ]−1 𝛿𝝊 ⋅ 𝛿r 𝛿r ⋅ 𝛿𝝊 𝛿𝝊 ⋅ 𝛿𝝊 K𝑆 = (I − T ⋅ (I − T ) ⋅ K𝑆 )+ T 𝛿r ⋅ 𝛿𝝊 𝛿r ⋅ 𝛿𝝊 𝛿r ⋅ 𝛿𝝊

with the unit matrix I. This is used with the iteration rule [ (𝜈+1) ]−1 ⋅ r(𝝊(𝜈+1) ) 𝝊(𝜈+2) = 𝝊(𝜈+1) + K𝑆

(A.14)

(A.15)

as the generalisation of Eq. (A.11). The computation of the sequence of the BFGS normal forms may be efficiently implemented on the basis of the LU-decomposition (0) of a given K𝑆 – the initial tangential stiffness is appropriate – with a recursion on T

vector products like 𝛿𝝊 ⋅ 𝛿r (Luenberger (1984, S. 269), Matthies and Strang (1979, (5))). The computational costs are generally considerably lower compared to NR. For more details, see also Zienkiewicz and Taylor (1991, 7), Belytschko et al. (2000, 6.3), Bathe (2001, 8.4), and de Borst et al. (2012, 4.6). The BFGS method is used for Examples 3.4 and 8.2, which are characterised by a strong influence of cracking on their structural behaviour.

A.2 Transient Analysis The transient analysis has to be considered as a third type besides quasi-static analysis and dynamic analysis. Static or dynamic analysis will not work for phenomena

A.2 Transient Analysis

like creep (Section 3.2). Creep is modelled by including viscosity into stress–strain relations. This requires an extension of the incremental material law (Eq. (2.80)) 𝝈̇ = C𝑇 ⋅ 𝝐̇ + 𝚺

(A.16)

with a stress extension term 𝚺 depending on stress 𝝈(𝑡) and strain 𝝐(𝑡) with clock time 𝑡. Similarly to Eq. (2.80) leading to Eq. (2.81), this is integrated in time with 𝝈𝑖+1 = 𝝈𝑖 + C𝑇,𝑖+1 ⋅ (𝝐 𝑖+1 − 𝝐 𝑖 ) + Δ𝑡 𝚺𝑖+1

(A.17)

Internal nodal forces are determined according to Eq. (2.581 ) f𝑖+1 = ∫ BT ⋅ 𝝈𝑖+1 d𝑉 = f𝑖 + K𝑇,𝑖+1 ⋅ Δ𝝊 + Δ𝑡 f 𝑖+1

(A.18)

𝑉

with element index 𝑒 omitted and Δ𝝊 = 𝝊𝑖+1 − 𝝊𝑖 and f𝑖 = ∫ BT ⋅ 𝝈𝑖 d𝑉 𝑉

K𝑇,𝑖+1 = ∫ BT ⋅ C𝑇,𝑖+1 ⋅ B d𝑉

(A.19)

𝑉

f 𝑖+1 = ∫ BT ⋅ 𝚺𝑖+1 d𝑉 𝑉

The contributions K𝑇,𝑖+1 , f 𝑖+1 generally involve nonlinearities due to the dependence of C𝑇,𝑖+1 , 𝚺𝑖+1 on strains and stresses. Equilibrium at a time 𝑡𝑖+1 has the condition r𝑖+1 = p𝑖+1 − f𝑖+1 = p𝑖+1 − f𝑖 − K𝑇,𝑖+1 ⋅ Δ𝝊 − Δ𝑡 f 𝑖+1 =0

(A.20)

according to Eqs. (2.70) and (A.18). The Newton–Raphson method is applied to solve this system of algebraic equations within an incrementally iterative scheme (Eq. (2.77)). An extended tangential stiffness (Eq. (2.76)) is given by (𝜈) A𝑇,𝑖+1

=

(𝜈) K𝑇,𝑖+1

| 𝜕f ||| | + Δ𝑡 𝜕𝝊 ||| (𝜈) |𝝊=𝝊𝑖+1

(A.21)

with the iteration counter (𝜈) leading to an iteration scheme (𝜈+1)

𝝊𝑖+1

[ (𝜈) ]−1 (𝜈+1) (𝜈) = 𝝊𝑖+1 + A𝑇,𝑖+1 ⋅ r𝑖+1

(A.22)

The exact formulation of the extended tangential stiffness depends on the specific form of 𝚺.

385

386

A Solution Methods

A common case is given with the visco-elasticity of materials (Section 3.2), leading to 𝚺 =V⋅𝝐 −W⋅𝝈

(A.23)

with constant material terms V and W (Eq. (3.30)). This yields 𝚺𝑖+1 = V ⋅ 𝝐 𝑖+1 − W ⋅ 𝝈𝑖+1 , and the stress Eq. (A.17) becomes 𝝈𝑖+1 = 𝝈𝑖 + C𝑇,𝑖+1 ⋅ (𝝐 𝑖+1 − 𝝐 𝑖 ) + Δ𝑡 V ⋅ 𝝐 𝑖+1 − Δ𝑡 W ⋅ 𝝈𝑖+1

(A.24)

leading to −1

𝝈𝑖+1 = [I + Δ𝑡 W]

⋅

([ ] ) C𝑇,𝑖+1 + Δ𝑡 V ⋅ (𝝐 𝑖+1 − 𝝐 𝑖 ) + 𝝈𝑖 + Δ𝑡 V ⋅ 𝝐 𝑖

(A.25)

with the unit matrix I. Internal nodal forces from Eq. (A.181 ) are reformulated as f𝑖+1 = ∫ BT ⋅ 𝝈𝑖+1 d𝑉 = f 𝑖 + K𝑇,𝑖+1 ⋅ Δ𝝊

(A.26)

𝑉

with Δ𝝊 as before and −1

f 𝑖 = [I + Δ𝑡 W]

∫ BT ⋅ (𝝈𝑖 + Δ𝑡 V ⋅ 𝝐 𝑖 ) d𝑉 𝑉

−1

K𝑇,𝑖+1 = [I + Δ𝑡 W]

∫

BT

[ ] ⋅ C𝑇,𝑖+1 + Δ𝑡 V ⋅ B d𝑉

(A.27)

𝑉

The residual (Eq. (A.20)) is given by r𝑖+1 = p𝑖+1 − f 𝑖 − K𝑇,𝑖+1 ⋅ Δ𝝊

(A.28)

leading to an iteration scheme (𝜈+1)

𝝊𝑖+1

(𝜈)

(𝜈)

= 𝝊𝑖+1 + [K𝑇,𝑖+1 ]

−1

(𝜈+1)

⋅ r𝑖+1

(A.29)

All quantities at time 𝑡𝑖 can assumed to be known during time incrementation. A potential source of nonlinearity is still given with C𝑇,𝑖+1 . The approach following Eqs. (A.27)–(A.29) is used as solution method for Examples 3.2 and 4.3.

A.3 Stiffness for Linear Concrete Compression A correct stiffness evaluation is essential to reach convergence for equilibrium iterations. As a special case, we consider concrete with linear stress–strain behaviour in the compressive range but without tensile strength (Section 4.1.3.2). For the crosssectional behaviour of beams, the contribution of the concrete part to the generalised forces depending on generalised strains is determined with Eq. (4.34) 𝝈𝑐 = C𝑐 ⋅ 𝝐 = 𝐸𝑐 A ⋅ 𝝐

(A.30)

A.3 Stiffness for Linear Concrete Compression

with 𝝈𝑐 from Eq. (4.32), concrete Young’s modulus 𝐸𝑐 (Eq. (4.24)), A from Eq. (4.33), and 𝝐 according to Eq. (4.26). The incremental form is derived in the following, considering that the extension of the compression zone depends on the generalised strains. As 𝝈𝑐 is a function of edge stresses 𝜎𝑐1 , 𝜎𝑐2 (Eq. (4.29)) its rate is written as 𝜕𝑁𝑐 ⎛ 𝑁̇ 𝑐 ⎞ ⎡ 𝜕𝜎 ⎢ 𝑐1 ⎜ ̇ ⎟ = ⎢ 𝜕𝑀𝑐 𝑀𝑐 ⎝ ⎠ ⎣ 𝜕𝜎𝑐1

𝜕𝑁𝑐

⎤ ⎛𝜎̇ ⎞ ⎡ 𝜕𝑁𝑐 𝑐1 𝑐1 ⋅ ⎜ ⎟ + ⎢ 𝜕𝑧 ⎢ 𝜕𝑀𝑐 𝜎̇ 𝑐2 𝜕𝜎𝑐2 ⎦ ⎝ ⎠ ⎣ 𝜕𝑧𝑐1 𝜕𝜎𝑐2 ⎥ 𝜕𝑀𝑐 ⎥

𝜕𝑁𝑐

⎤ ⎛𝑧̇ ⎞ 𝑐1 ⋅⎜ ⎟ 𝑧̇ 𝑐2 𝜕𝑧𝑐2 ⎦ ⎝ ⎠ 𝜕𝑧𝑐2 ⎥ 𝜕𝑀𝑐 ⎥

(A.31)

or 𝝈̇ 𝑐 = A𝜎 ⋅ 𝝈̇ 𝑐𝑒 + A𝑧 ⋅ ż 𝑐

(A.32)

with A𝜎 according to Eq. (4.32). The second term A𝑧 ⋅ ż 𝑐 considers the change of integration borders. To simplify A𝑧 a linear variation of width 𝑏 is assumed with 𝑏1 = 𝑏(𝑧𝑐1 ), 𝑏2 = 𝑏(𝑧𝑐2 ) and yields ⎡−𝜎𝑐1 𝑏2 −2𝜎𝑐1 𝑏1 −2𝜎𝑐2 𝑏2 −𝜎𝑐2 𝑏1 1⎢ (𝑏2 𝑧𝑐1 +𝑧𝑐2 𝑏2 +𝑧𝑐1 𝑏1 )𝜎𝑐2 A𝑧 = ⎢ 6⎢ +(𝑏2 𝑧𝑐1 −𝑧𝑐2 𝑏1 +3𝑧𝑐1 𝑏1 )𝜎𝑐1 ⎣

𝜎𝑐1 𝑏2 +2𝜎𝑐1 𝑏1 +2𝜎𝑐2 𝑏2 +𝜎𝑐2 𝑏1 ⎤ ⎥ (𝑏2 𝑧𝑐1 −3𝑧𝑐2 𝑏2 −𝑧𝑐2 𝑏1 )𝜎𝑐2 ⎥ ⎥ −(𝑧𝑐1 𝑏1 +𝑧𝑐2 𝑏2 +𝑧𝑐2 𝑏1 )𝜎𝑐1 ⎦ (A.33)

The variables 𝑧𝑐1 , 𝑧𝑐2 stand for lower or upper edges of the concrete compression zone (Figure 4.3). The following cases have to be considered: 1. Dominating bending with lower compression zone 𝑧𝑐1 = −ℎ∕2, 𝑧𝑐2 = 𝑧0 < ℎ∕2 and 𝑧̇ 𝑐1 = 0, 𝜎𝑐2 = 0, 𝜎̇ 𝑐2 = 0 and ℎ

⎛𝑧𝑐1 ⎞ ⎛− ⎞ 2 ⎜ ⎟=⎜ 𝜖 ⎟ , 𝑧𝑐2 ⎝ ⎠ ⎝ 𝜅 ⎠

⎛𝑧̇ 𝑐1 ⎞ ⎡ 0 ⎜ ⎟ = ⎢1 𝑧̇ ⎝ 𝑐2 ⎠ ⎣ 𝜅

0 ⎤ ⎛ 𝜖̇ ⎞ 𝜖 ⎥⋅⎜ ⎟ − 2 𝜅̇ 𝜅 ⎦ ⎝ ⎠

(A.34)

2. Dominating normal forces with fully compressed cross-section 𝑧𝑐1 = −ℎ∕2, 𝑧̇ 𝑐1 = 0, 𝑧𝑐2 = ℎ∕2, 𝑧̇ 𝑐2 = 0, 𝑥 = ℎ and ⎛𝑧𝑐1 ⎞ ⎛− ℎ ⎞ 2 ⎜ ⎟=⎜ ℎ ⎟ , 𝑧𝑐2 ⎝ ⎠ ⎝ 2 ⎠

⎛𝑧̇ 𝑐1 ⎞ ⎡0 ⎜ ⎟=⎢ 𝑧̇ 0 ⎝ 𝑐2 ⎠ ⎣

0⎤ ⎛ 𝜖̇ ⎞ ⎥⋅⎜ ⎟=0 0 𝜅̇ ⎦ ⎝ ⎠

(A.35)

3. Dominating bending with upper compression zone 𝑧𝑐1 = 𝑧0 > −ℎ∕2, 𝑧𝑐2 = ℎ∕2 and 𝑧̇ 𝑐2 = 0, 𝜎𝑐1 = 0, 𝜎̇ 𝑐1 = 0 and ⎛𝑧𝑐1 ⎞ ⎛ 𝜖 ⎞ 𝜅 ⎜ ⎟ = ⎜ℎ⎟ , 𝑧𝑐2 ⎝ ⎠ ⎝2⎠

⎛𝑧̇ 𝑐1 ⎞ ⎡ 1 ⎜ ⎟ = ⎢𝜅 𝑧̇ 0 ⎝ 𝑐2 ⎠ ⎣

𝜖

⎤ ⎛ 𝜖̇ ⎞ ⋅⎜ ⎟ 𝜅̇ 0 ⎦ ⎝ ⎠

−

𝜅2 ⎥

(A.36)

Independently of this, we set ż 𝑐 = B𝑧 ⋅ 𝝐̇

(A.37)

387

388

A Solution Methods

with B𝑧 according to one of these cases. The rate of the edge stresses 𝝈𝑐𝑒 in Eq. (A.32) is given by 𝝈̇ 𝑐𝑒 = 𝐸𝑐 𝝐̇ 𝑐𝑒

(A.38)

Equations (4.25), (4.26), and (A.37) result in the rate of the edge strains 𝝐̇ 𝑐𝑒 = B𝜖 ⋅ 𝝐̇ + Ḃ 𝜖 ⋅ 𝝐 = B𝜖 ⋅ 𝝐̇ − 𝜅 ż 𝑐 = (B𝜖 − 𝜅B𝑧 ) ⋅ 𝝐̇

(A.39)

Finally, Eq. (A.32) is combined with Eqs. (A.38), (A.39), and (A.37), taking Eq. (4.35) into consideration 𝝈̇ 𝑐 = 𝐸𝑐 A𝜎 ⋅ (B𝜖 − 𝜅B𝑧 ) ⋅ 𝝐̇ + A𝑧 ⋅ B𝑧 ⋅ 𝝐̇ ( ) ] [ = C𝑐 + A𝑧 − 𝜅𝐸𝑐′ A𝜎 ⋅ B𝑧 ⋅ 𝝐̇ = C𝑐𝑇 ⋅ 𝝐̇

(A.40)

leading to the tangential cross-sectional stiffness C𝑐𝑇 for concrete with linear compressive behaviour but without tensile strength.

A.4 The Arc Length Method All the described solution methods are embedded in an incrementally iterative scheme (Section 2.8.2), where the load terms p or 𝑝 are prescribed with small increments to reach a target value. Such a type of fixed incrementing will not work with, e.g. snap-back behaviour of structures; see Example 3.1 with Figure 3.6a. A loading has to be reduced after reaching the peak in order to follow the system load–displacement characteristics. The arc length method can be used to reach variable loading increments combined with loading reductions. The quasi-static case Eq. (2.70) is considered and modified as r(𝜆, 𝝊) = 𝜆 p0 − f(𝝊) = 0

(A.41)

with a target load vector p0 and a scalar loading factor 𝜆. The loading history is discretised with 𝜆𝑖+1 = 𝜆𝑖 + Δ𝜆 ,

𝑖 = 0, 1, 2, …

(A.42)

with 𝜆0 = 0 and a variable Δ𝜆. In the same way, nodal displacements 𝝊 are discretised with respect to loading time 𝝊𝑖+1 = 𝝊𝑖 + Δ𝝊

(A.43)

and 𝝊0 = 0 is assumed initially. The vector increment Δ𝝊 related to an increment of loading indicates an arc in the vector space ℝ𝑛 . Starting from a known state 𝑖 the application of Eq. (A.41) on the following unknown state 𝑖 + 1 yields r(𝜆𝑖+1 , 𝝊𝑖+1 ) = 𝜆𝑖+1 p0 − f(𝝊𝑖+1 ) = 0

(A.44)

A.4 The Arc Length Method (0)

(1)

This generally nonlinear equation is solved with iteration sequences 𝝊𝑖+1 , 𝝊𝑖+1 , … and (0)

(1)

𝜆𝑖+1 , 𝜆𝑖+1 , …, where

(𝜈+1)

(𝜈)

𝛿𝝊 = 𝝊𝑖+1 − 𝝊𝑖+1 (𝜈+1)

Δ𝝊(𝜈+1) = 𝝊𝑖+1 − 𝝊𝑖 = Δ𝝊(𝜈) + 𝛿𝝊

(A.45)

and (𝜈+1)

(𝜈)

𝛿𝜆 = 𝜆𝑖+1 − 𝜆𝑖+1

(A.46)

(𝜈+1)

Δ𝜆 (𝜈+1) = 𝜆𝑖+1 − 𝜆𝑖 The iteration rule is given in analogy to Eq. (2.77) with ( (𝜈+1) (𝜈) ) −1 𝛿𝝊 = [K] ⋅ r 𝜆𝑖+1 , 𝝊𝑖+1 ( (𝜈) ) −1 −1 (𝜈+1) = 𝜆𝑖+1 [K] ⋅ p0 − [K] ⋅ f 𝝊𝑖+1 ( (𝜈) ) ( (𝜈) ( (𝜈) (𝜈) )) −1 −1 = 𝜆𝑖+1 + 𝛿𝜆 [K] ⋅ p0 − [K] ⋅ 𝜆𝑖+1 p0 − r 𝜆𝑖+1 , 𝝊𝑖+1 ( (𝜈) (𝜈) ) −1 −1 = 𝛿𝜆 [K] ⋅ p0 + [K] ⋅ r 𝜆𝑖+1 , 𝝊𝑖+1

(A.47)

or 𝛿𝝊 = 𝛿𝜆 𝝊0 + 𝛿𝝊(𝜈)

(A.48)

with −1

𝝊0 = [K] 𝛿𝝊(𝜈)

−1

= [K]

⋅ p0 ( (𝜈) (𝜈) ) ⋅ r 𝜆𝑖+1 , 𝝊𝑖+1

(A.49)

with a stiffness K according to the Newton–Raphson method, the modified Newton– Raphson method, or a secant method. The inversion is not explicitly performed, (0) but a LU-decomposition is used instead. The iteration starts with 𝝊𝑖+1 = 𝝊𝑖 , Δ𝝊(0) = (0)

0, 𝜆𝑖+1 = 𝜆𝑖 . A further condition is needed to determine the variable 𝜆𝑖+1 . It is derived from T

Δ𝝊(𝜈+1) ⋅ Δ𝝊(𝜈+1) = 𝛾 2

(A.50)

with Δ𝝊(𝜈+1) according to Eq. (A.45) and a scalar 𝛾 controlling the arc length size. The evaluation yields a quadratic equation for the corrector 𝛿𝜆 𝑎 𝛿𝜆 2 + 𝑏 𝛿𝜆 + 𝑐 + 𝑑 = 𝛾 2

(A.51)

with 𝑎 = 𝝊0 T ⋅ 𝝊0 T

T

𝑏 = Δ𝝊(𝜈) ⋅ 𝝊0 + 𝛿𝝊(𝜈) ⋅ 𝝊0 T

T

𝑐 = 2Δ𝝊(𝜈) ⋅ 𝛿𝝊(𝜈) + 𝛿𝝊(𝜈) ⋅ 𝛿𝝊(𝜈) T

𝑑 = Δ𝝊(𝜈) ⋅ Δ𝝊(𝜈)

(A.52)

389

390

A Solution Methods

These coefficients are determined with the results of the 𝜈-iteration, where 𝑏 = 𝑐 = (0) (0) 𝑑 = 0 and 𝑎 > 0 for 𝜈 = 0. Furthermore, r(𝜆𝑖+1 , 𝝊,𝑖+1 ) = r(𝜆𝑖 , 𝝊𝑖 ) ≈ 0 is reasonable. Equation (A.51) can be further simplified with 𝑑 = 𝛾 2 for 𝜈 > 0, as the prescribed arc length size should not change during the iteration. Two real solutions are given with √ −𝑏 ± 𝑏2 − 𝑎𝑐′ (A.53) 𝛿𝜆 = 𝑎 in the case 𝑏2 − 𝑎𝑐′ > 0 with 𝑐′ = 𝑐 + 𝑑 − 𝛾 2 . The scalar product ) T T ( 𝑔 = Δ𝝊(𝜈) ⋅ Δ𝝊(𝜈+1) = Δ𝝊(𝜈) ⋅ Δ𝝊(𝜈) + 𝛿𝜆 𝝊0 + 𝛿𝝊(𝜈)

(A.54)

may be used to determine the solution to choose. The larger value 𝑔 brings the actual arc Δ𝝊(𝜈+1) closer to the previous arc Δ𝝊(𝜈) and is generally the choice. It is finally used in Eq. (A.48) to determine the corrector 𝛿𝝊. In the case of 𝜈 = 0 with Δ𝝊(0) = 0, the value Δ𝝊 of the √ previous loading increment may be included. For 𝑖 = 0, 𝜈 = 0 the choice 𝛿𝜆 = +𝛾∕ 𝑎 is appropriate. The case 𝑏2 − 𝑎𝑐′ < 0 remains to be treated. The prescribed arc length 𝛾 cannot be reached with any length of 𝛿𝜆 for given directions 𝝊0 and 𝛿𝝊(𝜈) . In most cases, it may be reached with an increased 𝛾. An alternative approach is to minimise the left-hand side of Eq. (A.51). This yields the smallest value 𝛾 that can be reached. For more details, see Belytschko et al. (2000, 6.5.3), Bathe (2001, 8.4.3), and de Borst et al. (2012, 4.2). The arc length method is used for the Examples 3.1, 4.2, 4.8, and 5.3.

391

Appendix B Material Stability Stability – or more precisely its loss – has several aspects in structural analysis. • A system with small variations with respect to its parameters and initial conditions exhibits large differences in the respective reactions during a loading history (→ Lyapunov instability). • A loaded system in quasi-static equilibrium exhibits more than one solution for further displacements (→ bifurcation). • A system in quasi-static equilibrium cannot maintain a position at rest (→ structural instability or snap through). This enumeration is exemplary. There may be alternate definitions and more stability specifications. Bifurcation is of particular interest for the stability of materials. As the Voigt notation (Sections 6.2.1, 6.2.2) does not allow for the analytic description of what follows, the meanings of products that connect vectors, stress and strain tensors, and material tensors has to be extended. Employing the Einstein summation convention (Section 6.8) we apply a ⋅ v → 𝑎𝑖𝑗 𝑣𝑗 ,

a ⋅ b → 𝑎𝑖𝑗 𝑏𝑗𝑟 ,

C ∶ a → 𝐶𝑖𝑗𝑟𝑠 𝑎𝑟𝑠 ,

u ⊗ v → 𝑢 𝑖 𝑣𝑗

(B.1)

with first-order tensors u, v with components 𝑢𝑖 , 𝑣𝑖 , second-order tensors a, b with components 𝑎𝑖𝑗 , 𝑏𝑖𝑗 , and fourth-order tensors C with components 𝐶𝑖𝑗𝑟𝑠 . To this end, variations are to be used accordingly. With this at hand, a material point within a continuum is considered. The state of stress is connected to the state of strain by an incremental relation (Eq. (2.50)) 𝝈̇ = C𝑇 ∶ 𝝐̇

(B.2)

A bifurcation may occur if a given stress increment 𝝈̇ is no longer uniquely connected to a strain increment 𝝐, ̇ or the other way around, if multiple solutions for strain increments are possible. This depends on the structure of the tangential material stiffness matrix C𝑇 . This point is made more precise by opening up the possibility of a discontinuity or a jump of strains. A specification of such a jump requires an orientation n separating the continuum into two parts named 𝛺− and 𝛺+ ; see Figure B.1. This extends the material point into a discontinuity curve (2D) or surface (3D). For the normal direction n along 𝑆, incremental equilibrium is also described according Computational Structural Concrete – Theory and Applications, Second Edition. Ulrich Häussler-Combe. © 2023 Ernst & Sohn GmbH. Published 2023 by Ernst & Sohn GmbH.

392

B Material Stability

Figure B.1 Discontinuity curve in 2D.

to Eq. (6.5) + − ṫ + + ṫ − = 𝝈̇ ⋅ n − 𝝈̇ ⋅ n = 0

(B.3)

with the index 𝑚 omitted. Furthermore, the same tangential material stiffness matrix is assumed for both parts such that +

+

𝝈̇ = C𝑇 ∶ 𝝐̇ ,

−

𝝈̇ = C𝑇 ∶ 𝝐̇

−

leading to [ ] + − C𝑇 ∶ 𝝐̇ − C𝑇 ∶ 𝝐̇ ⋅ n = 0

(B.4)

(B.5)

for the incremental equilibrium condition. The crucial point now is that a strain jump described by +

−

𝝐̇ − 𝝐̇ =

1 𝛾̇ (m ⊗ n + n ⊗ m) 2

(B.6)

is allowed with the unit vector m indicating the orientation of the jump and 𝛾 indicating the jump size. Such an approach preserves the continuity of the underlying displacement field but produces a discontinuity of strains across 𝛺+ and 𝛺− . ◀

A discontinuity of strains – weak discontinuity – is assumed to trigger a discontinuity of displacements – strong discontinuity – or cracks, respectively

The orientation of the vector m relative to the orientation n of the discontinuity surface or line indicates • A shear deformation jump of 𝛺+ relative to 𝛺− in the case when m is perpendicular to n. • An extensional deformation jump of 𝛺+ relative to 𝛺− in the case when m and n have the same direction. • A mixed mode deformation jump for all orientations of m in between. The combination of Eqs. (B.5) and (B.6) yields 1 𝛾̇ [C𝑇 ∶ (m ⊗ n + n ⊗ m)] ⋅ n = 0 2

(B.7)

and utilizing (minor-)symmetry properties of C𝑇 leads to Q ⋅ 𝛾m ̇ =0

(B.8)

B Material Stability

with the so-called second-order acoustic tensor Q Q = n ⋅ C𝑇 ⋅ n

(B.9)

with components 𝑄𝑖𝑗 = 𝑛𝑟 𝐶𝑇,𝑟𝑖𝑗𝑠 𝑛𝑠

(B.10)

According to Eq. (B.8) a strain discontinuity or bifurcation 𝛾m ̇ ≠ 0 occurs with det Q = 0

(B.11)

A state of stress and strain given, i.e. C𝑇 given; det Q is a function of the direction n only. This unit vector has three directional variables in 3D and one directional variable in 2D. The 2D case is exemplarily described further with ⎛cos 𝜙⎞ n=⎜ ⎟ sin 𝜙 ⎝ ⎠

(B.12)

with the orientation angle 𝜙. Utilizing minor symmetry of the fourth-order tensor C𝑇 , its tensorial components 𝐶𝑇,𝑖𝑗𝑟𝑠 , 𝑖, 𝑗, 𝑟, 𝑠 = 1, 2 are mapped into the commonly used form ⎡𝐶𝑇,1111 → 𝐶𝑇,11 ⎢ C𝑇 = ⎢𝐶𝑇,2211 → 𝐶𝑇,21 ⎢ 𝐶 → 𝐶𝑇,31 ⎣ 𝑇,1211

𝐶𝑇,1122 → 𝐶𝑇,12 𝐶𝑇,2222 → 𝐶𝑇,22 𝐶𝑇,1222 → 𝐶𝑇,32

𝐶𝑇,1112 → 2𝐶𝑇,13 ⎤ ⎥ 𝐶𝑇,2212 → 2𝐶𝑇,23 ⎥ ⎥ 𝐶𝑇,1212 → 2𝐶𝑇,33 ⎦

(B.13)

according to the Voigt notation (Eqs. (6.3), (6.7), and (6.21)) reduced to 2D with components including shear 𝐶𝑇,𝑖3 , 𝐶𝑇,3𝑖 , 𝑖 = 1, 2, 3 and taking the difference between tensorial and engineering notations of shear strains (Eq. (6.3)) into account. From Eqs. (B.9)–(B.13), we obtain det Q = 𝑎4 𝑥4 + 𝑎3 𝑥3 + 𝑎2 𝑥2 + 𝑎1 𝑥 + 𝑎0

(B.14)

with (Ortiz et al. (1987) with reversing 𝑎4 … 𝑎0 ) 𝑥 = tan 𝜙 𝑎4 = −2𝐶𝑇,32 𝐶𝑇,23 + 2𝐶𝑇,33 𝐶𝑇,22 𝑎3 = −2𝐶𝑇,12 𝐶𝑇,23 + 2𝐶𝑇,13 𝐶𝑇,22 + 𝐶𝑇,31 𝐶𝑇,22 − 𝐶𝑇,32 𝐶𝑇,21 𝑎2 = −2𝐶𝑇,33 𝐶𝑇,21 − 2𝐶𝑇,12 𝐶𝑇,33 − 𝐶𝑇,12 𝐶𝑇,21 + 2𝐶𝑇,31 𝐶𝑇,23 + 2𝐶𝑇,13 𝐶𝑇,32 + 𝐶𝑇,11 𝐶𝑇,22 𝑎1 = 2𝐶𝑇,11 𝐶𝑇,23 + 𝐶𝑇,11 𝐶𝑇,32 − 𝐶𝑇,12 𝐶𝑇,31 − 2𝐶𝑇,13 𝐶𝑇,21 𝑎0 = −2𝐶𝑇,13 𝐶𝑇,31 + 2𝐶𝑇,11 𝐶𝑇,33 (B.15)

393

394

B Material Stability

To start with a simple example, plane stress elasticity is considered with C𝑇 = C according to Eq. (6.28). The application of Eq. (B.14) yields det Q =

) ( 4 𝐸2 (𝑥 + (1 − 𝜈)𝑥2 + 1 2 (𝜈 + 1)(1 − 𝜈 )

(B.16)

with Young’s modulus 𝐸 and Poisson’s ratio 𝜈. We obtain det Q > 0 for all 𝑥 or orientations 𝜙, respectively. After that, as was to be expected, there is always stability regarding linear elasticity. This should initially occur for all concrete material models. But in the case of softening, a point of instability might be reached during loading with the evolution of the tangential material stiffness matrix C𝑇 . The instability is indicated when det Q touches zero the first time and has the conditions det Q = 0 ,

(det Q)′ =

ddet Q =0 d𝑥

(B.17)

The zeros of the third-order polynomial (det Q)′ may be analytically determined using Cardano’s formula. They mark potential directions of instability, which may continuously be traced for switching det Q from positive to negative. This is exemplarily demonstrated for damage with Examples 6.3 and 6.4. The first zero value 𝜙0 is inserted into Eq. (B.9) to determine the acoustic tensor Q. Although it is singular, the jump orientation m (Eq. (B.8)) is determined with Q

⎛− 12 ⎞ ⎛ 𝑚1 ⎞ 1 Q m=⎜ ⎟= √ ⎜ 11 ⎟ 2 𝑚2 1 Q ⎝ ⎠ ⎠ 1 + ( 12 ) ⎝

(B.18)

Q11

which in turn may be used to derive the normalised strain jump according to Eq. (B.6). The approach may basically be applied to an arbitrary tangential material stiffness matrix C𝑇 . For the extension to 3D with triaxial stress–strain relations, see Ortiz et al. (1987). Such a determination of points of material instability in the case of a finite element discretisation has to be performed for all integration points of all continuum elements during a whole loading history. This turns out to be elaborate. Furthermore, small differences of large numbers have to be determined for Eqs. (B.14) and (B.15), which is known to be error-prone in numerics. After all, variations of det Q are high with small variations in the loading history, which impedes tracing the zero of det Q. It would be desirable to have analytical solutions available. For some general classification of stress–strain relations and the corresponding analytical stability solutions, see Oliver and Huespe (2004). With regard to the indication of crack initiation, it should be noted that the Rankine criterion (Section 7.4) is generally given preference over the stability criterion.

395

Appendix C Crack Width Estimation The formation of cracks is a characteristic property of structural reinforced concrete. It influences the local stiffness of structures, which is reduced in areas with high tension. This leads to a redistribution of stresses within statically indeterminate structures. Softer areas are relieved, while stiffer areas gain stresses compared to a linear elastic calculation. Furthermore, stresses from constraints like settlements or temperatures are reduced. Insofar, crack formation leads to favorable effects. On the other hand, the overall deformation of cracked structures increases, the corrosion risk of steel 1) reinforcement grows in cracked areas, and the visual impression suffers with larger cracks. Thus, the crack width has to be controlled, and methods to predict crack width are required. The direct computation of crack width is demonstrated with Example 3.4 with a fine discretisation using an element length in the order of 1 cm. This actually leads to a discrete modelling of cracks. But such fine discretisations are not appropriate for the modelling of structures like plates, slabs, and shells. Crack modelling is generally performed with the smeared crack model for such structures (Sections 7.4 and 8.2) with a characteristic length of elements considerably larger than 1 cm. Thus, an alternative model for crack width estimation is required. This is described in the following. The model is based on the uniaxial tension bar, where analytical crack width relations are derived with some simplifying assumptions. This leads to crack width estimations that primarily depend on the reinforcement stress and bond characteristics. On the other hand, crack situations in, e.g. plates may be considered as uniaxial in the context of the Rankine criterion (Section 7.4). ◀

Crack width estimations for uniaxial tension bars may also be applied to cracked plates locally with rebar stresses and reinforcement parameters given.

Furthermore, such estimations may also be transferred to slabs and shells in the context of layer models (Sections 9.4 and 10.7.1). The simplified model for crack 1) Alternative reinforcement materials like carbon, which are not exposed to corrosion risks, are emerging. Regardless of this, all approaches are basically the same with material and bond parameters adapted. Computational Structural Concrete – Theory and Applications, Second Edition. Ulrich Häussler-Combe. © 2023 Ernst & Sohn GmbH. Published 2023 by Ernst & Sohn GmbH.

396

C Crack Width Estimation

width estimations is also suitable for reinforced concrete beams when applied to the tension zone or the immediate surrounding of rebars in cracked cross-sections. Details of the uniaxial tension bar are described in Section 3.6. The current setup refers to Figure 3.20a with a cracked reinforced cross-section and is shown in Figure C.1a. Assuming a centred symmetry the width 𝑤 of the crack is determined as 𝑙𝑡

𝑤 = 2 ∫ (𝜖𝑠 (𝑥) − 𝜖𝑐 (𝑥)) d𝑥

(C.1)

0

with the force transfer length 𝑙𝑡 , the rebar strain 𝜖𝑠 (𝑥), and the concrete strain 𝜖𝑐 (𝑥), which are variable in the longitudinal direction. The transfer length corresponds to the length from the crack to the position with constant strains or local extrema of strains. Mean strains within the transfer length are given by 𝑙𝑡

𝜖𝑠𝑚

𝑙𝑡

1 = ∫ 𝜖𝑠 (𝑥) d𝑥 , 𝑙𝑡

𝜖𝑐𝑚

0

1 = ∫ 𝜖𝑐 (𝑥) d𝑥 𝑙𝑡

(C.2)

0

and the crack width can be rewritten as 𝑤 = 2𝑙𝑡 (𝜖𝑠𝑚 − 𝜖𝑐𝑚 )

(C.3)

Strains 𝜖𝑠 (𝑥), 𝜖𝑐 (𝑥) are related to rebar stresses 𝜎𝑠 (𝑥) and concrete stresses 𝜎𝑐 (𝑥). The following stress–strain relations are assumed

𝜎𝑠 = 𝐸𝑠 𝜖𝑠 ,

𝜎𝑐 =

⎧ 𝐸𝑐 𝜖𝑐 ⎨0 ⎩

𝜖𝑐 ≤ 𝑓𝑐𝑡 ∕𝐸𝑐 else

(C.4)

With respect to the crack position 𝑥 = 0, the concrete stress is zero, and the rebar stress has its maximum value 𝜎𝑠 (0) = 𝜎𝑠𝑐 . The rebar stress decreases with 𝑥 and reaches its minimum value with 𝑥 = 𝑙𝑡 . A rebar stress difference Δ𝜎𝑠 = 𝜎𝑠𝑐 − 𝜎𝑠 (𝑙𝑡 )

(C.5)

is used to determine the mean rebar stress along 𝑙𝑡 with 𝜎𝑠𝑚 = 𝜎𝑠𝑐 − 𝛽𝑡 Δ𝜎𝑠

(C.6)

compare Figure 3.20a. The parameter 𝛽𝑡 was already used in Section 3.7 as a tension stiffening coefficient. It encompasses the extent of load transfer between rebar and concrete or bond quality, respectively. A value 𝛽𝑡 = 0 denotes no stress transfer and no effective bond, a value 𝛽𝑡 = 1 immediate stress transfer and rigid bond. The values are generally assumed in the range 0.4 ≤ 𝛽𝑡 ≤ 0.6 (EN 1992-1-1 (2004, 7.3.4), CEBFIP2 (2012, 7.6.4.4)).

C Crack Width Estimation

(a)

(b)

Figure C.1 (a) Strains at cracked cross-section. (b) Equilibrium with bond stresses.

For reasons of equilibrium (Figure C.1b), the concrete stress along the longitudinal direction is given by 𝜎𝑐 (𝑥) = 𝜌eff [𝜎𝑠𝑐 − 𝜎𝑠 (𝑥)]

(C.7)

where 𝜌eff = 𝐴𝑠 ∕𝐴𝑐,eff is the effective reinforcement ratio with the cross-sectional area 𝐴𝑠 of the rebar and the effective cross-sectional area 𝐴𝑐,eff of the concrete. The cross-sectional area 𝐴𝑐,eff is described in Section 3.7. The concrete stress has its maximum at 𝑥 = 𝑙𝑡 with a value 𝜎𝑐 ,max = 𝜌eff Δ𝜎𝑠

(C.8)

Using Eqs. (C.6) and (C.7) the mean concrete stress is obtained with 𝑙𝑡

𝜎𝑐𝑚

1 = ∫ 𝜎𝑐 (𝑥) d𝑥 = 𝜌eff 𝛽𝑡 Δ𝜎𝑠 𝑙𝑡

(C.9)

0

The load transfer between concrete and reinforcement is connected with bond stresses 𝜏(𝑥). For reasons of equilibrium, see Figure C.1b, it is related to the rebar stress difference Δ𝜎𝑠 by 𝑙𝑡

𝐴𝑠 Δ𝜎𝑠 = 𝐶𝑠 ∫ 𝜏(𝑥) d𝑥

(C.10)

0

with the reinforcement circumference 𝐶𝑠 . Using the mean bond stress 𝜏𝑚 yields Δ𝜎𝑠 =

𝐶𝑠 4𝑙𝑡 𝑙 𝜏 = 𝜏 𝐴𝑠 𝑡 𝑚 𝑑𝑠 𝑚

(C.11)

with the reinforcement diameter 𝑑𝑠 and 𝐶𝑠 ∕𝐴𝑠 = 4∕𝑑𝑠 . The mean bond stress 𝜏𝑚 is assumed to be proportional to the concrete tensile strength 𝑓𝑐𝑡 (EN 1992-1-1 2004, 8.4.2) 𝜏𝑚 = 𝜂 𝑓𝑐𝑡

(C.12)

397

398

C Crack Width Estimation

with 𝜂 in the magnitude of 2 (CEB-FIP2 2012, 6.1.1). Equation (C.11) leads to a relation for the stress transfer length 𝑙𝑡 =

𝑑𝑠 Δ𝜎𝑠 4𝜏𝑚

(C.13)

This yields the first part of Eq. (C.3) for the crack width determination. The mean reinforcement strain (Eq. (C.6)) 𝜖𝑠𝑚 =

𝜎𝑠𝑚 1 = (𝜎 − 𝛽𝑡 Δ𝜎𝑠 ) 𝐸𝑠 𝐸𝑠 𝑠𝑐

(C.14)

and the mean concrete strain (Eq. (C.9)) 𝜖𝑐𝑚 =

𝜎𝑐𝑚 1 = 𝜌 𝛽 Δ𝜎 𝐸𝑐 𝐸𝑐 eff 𝑡 𝑠

(C.15)

complete the second part. This finally leads to 𝑤=

𝑑𝑠 Δ𝜎𝑠 [𝜎𝑠𝑐 − 𝛽𝑡 (1 + 𝛼𝑒 𝜌eff ) Δ𝜎𝑠 ] 2𝜏𝑚 𝐸𝑠

(C.16)

with a-priori known or assumed parameters 𝑑𝑠 , 𝜌eff , 𝜏𝑚 and the stiffness ratio 𝛼𝑒 = 𝐸𝑠 ∕𝐸𝑐 . The rebar stress 𝜎𝑠𝑐 in the cracked cross-section is determined according to the external loading. The rebar stress difference Δ𝜎𝑠 remains to be determined. Thus, two limiting cases are considered: the state with single cracks and the state with stabilised cracking. A tension bar is uncracked at load initiation, where the concrete properties are subject to scatter. A first single crack will arise in the crosssection with the smallest overall tensile strength due to increasing loading starting from zero. The softening behaviour (Figure 3.7) is neglected for crack width estimation, as the critical crack width 𝑤𝑐𝑟 generally does not exceed 0.1 mm, while – within the strain softening context – the relevant crack width is in the range considerably above 0.1 mm. Perfect brittle tensile behaviour of concrete behaviour (Figure 7.1b) is assumed for the following. A single crack is characterised by 𝜖𝑠 (𝑙𝑡 ) = 𝜖𝑐 (𝑙𝑡 )

(C.17)

where strains of concrete and reinforcement match beyond the stress transfer length (Figure C.2a.) Replacing strains with stresses using Eqs. (C.5) and (C.7) leads to 1 1 (𝜎 − Δ𝜎𝑠 ) = 𝜌 Δ𝜎𝑠 𝐸𝑠 𝑠𝑐 𝐸𝑐 eff

(C.18)

and Δ𝜎𝑠 =

𝜎𝑠𝑐 1 + 𝛼𝑒 𝜌eff

(C.19)

and the crack width of the first single crack can finally be determined with Eq. (C.16). Further cracks develop with slightly increasing loading in the cross-sections with the currently smallest tensile strength. The measure of crack spacing 𝑠𝐶 has to be

C Crack Width Estimation

(a)

(b)

Figure C.2 Crack states. (a) Single cracks. (b) Stabilised cracks.

introduced with a rising number of cracks. Crack spacing, on the one hand, has the condition 𝑠𝑐 ≥ 𝑙𝑡

(C.20)

as a crack is not possible within 𝑙𝑡 because of the decreasing concrete stresses towards the crack. On the other hand, crack spacing reduces with an increasing number of cracks (Figure C.2a). New cracks will occur as long as 𝑠𝑐 ≥ 2𝑙𝑡 , with slightly increasing loading, whereby reducing crack spacing. Cracking will finally reach a stabilised state with 𝑠𝑐 ≤ 2𝑙𝑡 : the concrete stress will no longer exceed the tensile strength under this condition. The process is demonstrated in Example 3.4 with all its stages. The state of stabilised cracking is characterised by 𝑙𝑡 ≤ 𝑠𝑐 < 2𝑙𝑡

(C.21)

Loading may increase further, but the maximum concrete stress (Eq. (C.8)) does not exceed 𝑓𝑐𝑡 . This leads to a condition Δ𝜎𝑠 ≤ 𝑓𝑐𝑡 ∕𝜌eff for stabilised cracking. We assume Δ𝜎𝑠 =

𝑓𝑐𝑡 𝜌eff

(C.22)

for usage with the crack width relation (C.16). The crack width estimation for stabilised cracking is finally obtained with 𝑤=

𝑑𝑠 𝑓𝑐𝑡 1 𝑓𝑐𝑡 [𝜎 − 𝛽𝑡 (1 + 𝛼𝑒 𝜌eff ) ] 2𝜏𝑚 𝜌eff 𝐸𝑠 𝑠𝑐 𝜌eff

(C.23)

Stabilised cracking is characterised by the following properties: • Increased loading is completely taken by the reinforcement until its load bearing capacity is reached. • Between two cracks, a position exists where the slip between concrete and reinforcement is equal to zero. This position margins the force transfer lengths of the cracks to the left and to the right; see Figure 3.19b for an example. • At this position, the reinforcement stress has a minimum, while the concrete stress has a maximum but does not exceed the concrete tensile strength (Figure 3.18a).

399

400

C Crack Width Estimation

• The reinforcement strain increases compared to the concrete strain that holds its level. This distinguishes the state of stabilised cracking from the state of single cracks (Figure C.2). The crack width of stabilised cracking has to be considered as relevant as the values of the reinforcement stress in the stage of crack formation – from first cracking to reaching the final crack pattern – is generally relatively short compared to later values with ongoing loading under stabilised cracking conditions. Thus, variations of Eq. (C.23) are used in code provisions for a crack width estimation (EN 1992-1-1 (2004, 7.3.4), CEB-FIP2 (2012, 7.6.4.4)). For a discussion of the influence of imposed strains from temperature and shrinkage on the crack width estimation, see HäusslerCombe and Hartig (2012). The application of these rules is generally stipulated for reinforced concrete beams. The predominant variable is given with the rebar stress 𝜎𝑠𝑐 in the cracked cross-section (Figure 4.2), where the parameters 𝑑𝑠 , 𝜏𝑚 , 𝑓𝑐𝑡 , 𝜌eff , 𝐸𝑠 , 𝛽𝑡 , 𝛼𝑒 are assumed to be known. The estimation may also be directly applied for cracked positions in plates or layers of slabs or shells when the reinforcement direction is orthogonal to the crack direction. This should be the case for the positions with the largest stresses. For the case that principal tension or crack normals deviate by more than 15° from reinforcement directions, rules for crack width estimations are given in EN 1992-1-1 (2004, 7.3.4(4)) or CEB-FIP2 (2012, 7.6.4.4.3).

401

Appendix D Transformations of Coordinate Systems Cartesian coordinate systems are used throughout to describe space, if not otherwise stated. Coordinates 𝑥𝑖 are measured with respect to three (3D space) or two (2D plane) orthogonal base vectors e𝑖 of unit length. Orthogonality of unit vectors leads to e𝑖 ⋅ e𝑗 = e𝑗 ⋅ e𝑖 =

⎧

1 𝑖=𝑗 ⎨0 𝑖 ≠ 𝑗 ⎩

(D.1)

with the vector product ⋅. The sequence of base vectors has a right-hand orientation. We restrict ourselves to the 2D plane in the following. A vector x describing a position is written as r⃗ = 𝑥1 e1 + 𝑥2 e2

(D.2)

or, alternatively, in another Cartesian system (Figure D.1) e1 + 𝑥˜2 ˜ e2 r⃗ = 𝑥 ˜1 ˜

(D.3)

rotated by an angle 𝜑 positive in the counterclockwise direction.

Figure D.1 Plane coordinate transformation.

Computational Structural Concrete – Theory and Applications, Second Edition. Ulrich Häussler-Combe. © 2023 Ernst & Sohn GmbH. Published 2023 by Ernst & Sohn GmbH.

402

D Transformations of Coordinate Systems

The base vectors ˜ e1 , ˜ e2 are still orthogonal according to Eq. (D.1). The rotated coordinates are given by 𝑥˜1 = 𝑥1 e1 ⋅ ˜ e1 + 𝑥2 e2 ⋅ ˜ e1 = 𝑥1 cos 𝜑 + 𝑥2 cos(π∕2 − 𝜑) = 𝑥1 cos 𝜑 + 𝑥2 sin 𝜑

(D.4)

𝑥˜2 = 𝑥1 e1 ⋅ ˜ e2 + 𝑥2 e2 ⋅ ˜ e2 = 𝑥1 cos(π∕2 + 𝜑) + 𝑥2 cos 𝜑 = −𝑥1 sin 𝜑 + 𝑥2 cos 𝜑 after applying Eq. (D.1). This is written as ˜ x=Q⋅x

(D.5)

with ⎛𝑥˜1 ⎞ ˜ x=⎜ ⎟ , 𝑥˜ ⎝ 1⎠

⎡ cos 𝛼 Q=⎢ − sin 𝛼 ⎣

sin 𝛼 ⎤ ⎥, cos 𝛼 ⎦

⎛𝑥1 ⎞ x=⎜ ⎟ 𝑥 ⎝ 1⎠

(D.6)

collecting the coordinate components in vector notation. The back transformation is obtained with x x = QT ⋅ ˜

(D.7)

as Q−1 = QT . Any other vector – e.g. displacement, velocity, force – of a form corresponding to Eq. (D.2) has the same transformation rules for its components ˜ a =Q⋅a,

a, a = QT ⋅ ˜

(D.8)

)T )T ( ( with ˜ a= 𝑎 ˜𝑦 and a = 𝑎𝑥 𝑎𝑦 . ˜𝑥 𝑎 Second-order tensors have to be considered in addition to vectors or first-order tensors. A second tensor is given, e.g. with the Cauchy stress 𝝈 (Section 6.2.2). For plane states, the stress tensor is written as S = 𝜎11 e1 e1 + 𝜎12 e1 e2 + 𝜎21 e2 e1 + 𝜎22 e2 e2

(D.9)

with an outer or dyadic base e1 e1 , e1 e2 , e2 e1 , e2 e2 ;compare Eq. (10.24). The dyadic base emerges from the force direction and the direction of the normal of the reference plane (Eq. (6.5)). This tensor is alternatively written in a Cartesian system rotated by an angle 𝜑 e1 ˜ e1 + 𝜎˜12 ˜ e1 ˜ e2 + 𝜎˜21 ˜ e2 ˜ e1 + 𝜎˜22 ˜ e2 ˜ e2 S = 𝜎˜11 ˜

(D.10)

The identity of these formulations is used to determine the values of the tensor components in the rotated system. Applying vector products with ˜ e1 , ˜ e2 from left

D Transformations of Coordinate Systems

and right, exploiting their orthogonality and the symmetry of tensor components, 𝜎12 = 𝜎21 leads to ⎛𝜎˜11 ⎞ ⎡ cos2 𝜑 2 ⎜𝜎˜ ⎟ = ⎢ ⎢ sin 𝜑 22 ⎜ ⎟ ⎢ ⎝𝜎˜12 ⎠ ⎣− cos 𝜑 sin 𝜑

2

sin 𝜑 cos2 𝜑 cos 𝜑 sin 𝜑

2 cos 𝜑 sin 𝜑 ⎤ ⎛𝜎11 ⎞ ⎥ −2 cos 𝜑 sin 𝜑 ⎥ ⋅ ⎜𝜎22 ⎟ ⎟ 2 ⎥ ⎜ cos2 𝜑 − sin 𝜑 ⎝𝜎12 ⎠ ⎦

(D.11)

in analogy to Eq. (D.4). The inversion is derived in the analogous way and yields ⎛𝜎11 ⎞ ⎡ cos2 𝜑 2 ⎜𝜎 ⎟ = ⎢ ⎢ sin 𝜑 22 ⎜ ⎟ ⎢ ⎝𝜎12 ⎠ ⎣cos 𝜑 sin 𝜑

2

sin 𝜑 cos2 𝜑 − cos 𝜑 sin 𝜑

−2 cos 𝜑 sin 𝜑 ⎤ ⎛𝜎˜11 ⎞ ⎥ 2 cos 𝜑 sin 𝜑 ⎥ ⋅ ⎜𝜎˜22 ⎟ ⎟ 2 ⎥ ⎜ cos2 𝜑 − sin 𝜑 ⎝𝜎˜12 ⎠ ⎦

(D.12)

Strain tensor components 𝜖11 , 𝜖12 = 𝜖21 , 𝜖22 as a subset of Eq. (6.2) are transformed in the same way. This results to 𝜖11 ⎞ ⎡ cos2 𝜑 ⎛˜ ⎢ 2 ⎜˜ 𝜖22 ⎟ = ⎢ sin 𝜑 ⎜ ⎟ ⎢ ⎝𝛾˜12 ⎠ ⎣−2 cos 𝜑 sin 𝜑

2

sin 𝜑 cos2 𝜑 2 cos 𝜑 sin 𝜑

cos 𝜑 sin 𝜑 ⎤ ⎛ 𝜖11 ⎞ ⎥ − cos 𝜑 sin 𝜑 ⎥ ⋅ ⎜ 𝜖22 ⎟ ⎟ 2 ⎥ ⎜ cos2 𝜑 − sin 𝜑 ⎝𝛾12 ⎠ ⎦

(D.13)

with respect to the definition 𝛾𝑖𝑗 = 2𝜖𝑖𝑗 , 𝑖 ≠ 𝑗 for strain components in engineering notation. This transformation matrix is the transpose for the inversion of stresses (Eq. (D.12)). The inversion of strains is again derived in the analogous way and yields ⎧ ⎫ ⎡ 2 ⎪ 𝜖11 ⎪ ⎢ cos 𝜑 2 = 𝜖 sin 𝜑 ⎨ 22 ⎬ ⎢ ⎪𝛾12 ⎪ ⎢2 cos 𝜑 sin 𝜑 ⎩ ⎭ ⎣

2

sin 𝜑 cos2 𝜑 −2 cos 𝜑 sin 𝜑

⎧ ⎫ 𝜖11 ⎪ − cos 𝜑 sin 𝜑 ⎤ ⎪˜ ⎥ cos 𝜑 sin 𝜑 ⎥ ⋅ ˜ 𝜖 ⎨ 22 ⎬ 2 ⎥ cos2 𝜑 − sin 𝜑 ⎪𝛾˜12 ⎪ ⎦ ⎩ ⎭

(D.14)

This back transformation matrix is the transpose of that for transformation of stresses (Eq. (D.11)). Cartesian transformations of vectors and second-order tensors in 3D space are derived in a similar way based on extended formulations of Eqs. (D.2), (D.3), and (D.9), (D.10), involving three rotation angles (Figure 6.3) and again exploiting orthogonality of base vectors.

403

405

Appendix E Regression Analysis Regression analysis leads to an approximation of multiple data by combining functions with a few degrees of freedom. We consider 𝑛 discrete data 𝑓𝑖 determined from a survey over a plane in positions 𝑥𝑖 , 𝑦𝑖 . A linear approximation leads to errors 𝑟𝑖 = 𝑎 𝑥𝑖 + 𝑏 𝑦𝑖 + 𝑐 − 𝑓𝑖 ,

𝑖 = 1, … , 𝑛

(E.1)

We search for the optimal coefficients 𝑎, 𝑏, 𝑐 to minimise the error. The problem may be also written as r=X⋅a−f

(E.2)

with ⎡ 𝑥1 ⎢ ⎢ 𝑥2 X=⎢ ⎢⋮ ⎢ 𝑥 ⎣ 𝑛

1⎤ ⎥ 1⎥ ⎥, ⋮⎥ ⎥ 1 ⎦

𝑦1 𝑦2 ⋮ 𝑦𝑛

⎛𝑎 ⎞ a = ⎜𝑏 ⎟ , ⎜ ⎟ ⎝𝑐⎠

⎛ 𝑓1 ⎞ ⎜ ⎟ 𝑓2 f =⎜ ⎟ ⎜⋮⎟ ⎜ ⎟ 𝑓 ⎝ 𝑛⎠

(E.3)

The error should be zero r = 0, i.e. X⋅a =f

(E.4)

This is an overdetermined set of linear equations for 𝑛 > 3 and will not have a solution for a. Thus, the error is minimised regarding the error length 𝑒 = rT ⋅ r with T

𝑒 = (X ⋅ a − f) ⋅ (X ⋅ a − f) = aT ⋅ XT ⋅ X ⋅ a − 2aT ⋅ XT ⋅ f + f T ⋅ f

(E.5)

The error length is minimised under the condition 𝜕𝑒∕𝜕a = 0, leading to XT ⋅ X ⋅ a = XT ⋅ f

(E.6)

This corresponds to the method of least squares. The basic approach allows for many specifications. The dimension of the underlying space is arbitrary. In the following, area coordinates 𝐿1 , 𝐿2 , 𝐿3 (Section 9.5.1, Eq. (9.39)) are used for a linear approximation of a given field 𝑚 𝑟(𝐿1 , 𝐿2 , 𝐿3 ) = 𝑎 𝐿1 + 𝑏 𝐿2 + 𝑐 𝐿3 − 𝑚(𝐿1 , 𝐿2 , 𝐿3 )

(E.7)

Computational Structural Concrete – Theory and Applications, Second Edition. Ulrich Häussler-Combe. © 2023 Ernst & Sohn GmbH. Published 2023 by Ernst & Sohn GmbH.

406

E Regression Analysis

defined over a triangle (Figure 9.5). Four survey points are used due to four sampling points L𝑖 utilised for a numerical integration. The sampling points have the coordinates (Table 9.1, 𝑛𝑖 = 2) ⎛1∕3⎞ L1 = ⎜1∕3⎟ , ⎜ ⎟ ⎝1∕3⎠

⎛0.6⎞ L2 = ⎜0.2⎟ , ⎜ ⎟ ⎝0.2⎠

⎛0.2⎞ L3 = ⎜0.6⎟ , ⎜ ⎟ ⎝0.2⎠

⎛0.2⎞ L4 = ⎜0.2⎟ ⎜ ⎟ ⎝0.6⎠

(E.8)

The error vector is written as r=X⋅a−m

(E.9)

with ⎡1∕3 ⎢ ⎢ 0.6 X=⎢ ⎢ 0.2 ⎢ 0.2 ⎣

1∕3 0.2 0.6 0.2

1∕3⎤ ⎥ 0.2 ⎥ ⎥, 0.2 ⎥ ⎥ 0.6 ⎦

⎛𝑎 ⎞ a = ⎜𝑏 ⎟ , ⎜ ⎟ ⎝𝑐⎠

⎛𝑚1 ⎞ ⎜ ⎟ 𝑚2 m=⎜ ⎟ ⎜𝑚3 ⎟ ⎜ ⎟ 𝑚 ⎝ 4⎠

(E.10)

and Eq. (E.6) yields A ⋅ a = XT ⋅ m

(E.11)

with ⎡124 1 ⎢ T A = X ⋅X = 88 225 ⎢ ⎢ 88 ⎣

88 124 88

88 ⎤ ⎥ 88 ⎥ , ⎥ 124 ⎦

53

⎡ 1⎢ 2 −1 A = ⎢−11 6⎢ −11 ⎣

−11 53 2

−11

−11⎤ ⎥ −11⎥ (E.12) 53 ⎥ 2 ⎦

and, finally, ⎡1 ⎛𝑎⎞ 1⎢ ⎜ ⎟ a = 𝑏 = ⎢1 ⎜ ⎟ 4⎢ 1 ⎝𝑐⎠ ⎣

23 3

− −

−

7

7

3 23

3 7

3

3

−

7 3

7

− ⎤ 3 7⎥ − ⎥⋅m 3 23 ⎥ 3 ⎦

(E.13)

where m is given from the field 𝑚 evaluated in the sampling points. The three rows of the matrix each add up to 1 with the prefactor 1∕4. The linear approximation 𝑚(𝐿1 , 𝐿2 , 𝐿3 ) ≈ 𝑎 𝐿1 + 𝑏 𝐿2 + 𝑐 𝐿3

(E.14)

is obtained according to Eq. (E.7). Finally, the derivatives of the field 𝑚 with respect to global coordinates are determined by 𝜕𝑚 𝜕𝑚 𝜕𝐿1 𝜕𝑚 𝜕𝐿2 𝜕𝑚 𝜕𝐿3 = + + 𝜕𝑥 𝜕𝐿1 𝜕𝑥 𝜕𝐿2 𝜕𝑥 𝜕𝐿3 𝜕𝑥 = 𝑎 𝑏1 + 𝑏 𝑏2 + 𝑐 𝑏3 𝜕𝑚 = 𝑎 𝑐1 + 𝑏 𝑐2 + 𝑐 𝑐3 𝜕𝑦 with 𝑏𝑖 , 𝑐𝑖 according to Eq. (9.40).

(E.15)

407

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415

417

Index

a acoustic tensor 393 algorithmic modulus 28 analysis dynamic 21, 29, 85, 269 inverse 279, 370 quasi-static 28, 85 rigid-plastic 322 second-order 121, 366 transient 29, 50, 103, 384 angle cross-section rotation 68, 82, 287 dilatancy 181 external friction 179 internal friction 179 shear 68, 74, 87, 99, 287 anisotropy 161, 176, 267 load-induced 154, 166, 172, 189, 194, 198, 222, 265, 268, 272, 348 arc length method 42, 97, 124, 144, 233, 388 area coordinates 298, 299, 301, 405

b benchmark 9 blind 380 Bernoulli beam 68, 71, 85, 112, 123, 280 element 88, 93, 120, 281 enhanced element 91, 95, 96, 111, 124, 129 Bernoulli–Navier hypothesis 67, 115, 286, 296, 330 BFGS method 61, 107, 384

bond 266 flexible 55, 63, 266, 280, 284 law 55, 60, 266, 275, 280, 284 rigid 65, 73, 266, 268, 273, 296, 346, 396 boundary conditions Dirichlet 11, 31 displacement 11, 95 forces 12, 31 Neumann 12, 31 Bubnov–Galerkin 22, 86

c calibration 9, 47, 186, 197 characteristic length 42 cohesive 203, 222 element 211, 217, 265 non-local 215, 233 characteristic value 371 Clausius–Duhem inequality 199 cohesive zone 41, 202 compliance 20, 46, 57, 162, 189, 221 compression field theory 250 consistency condition 174, 185, 190, 191 constraint force 106, 111 microplane kinematic 193 multi-plane static 193 stress 50, 51 continuity 34 𝐶 0 18, 87, 90 𝐶 1 295, 298, 301 𝐶 2 90

Computational Structural Concrete – Theory and Applications, Second Edition. Ulrich Häussler-Combe. © 2023 Ernst & Sohn GmbH. Published 2023 by Ernst & Sohn GmbH.

418

Index

convergence equilibrium iteration 27 FEM 31, 33 𝛤 232 Newton–Raphson 27, 381 coordinate system Cartesian 13, 401 contravariant 333 corotational 119, 336, 339, 345 covariant 333 local 14 transformation 157, 402 correlation 355, 362, 367, 369 crack bridge 40, 202 closure 223, 241, 262, 263, 265 cohesive 43, 56, 59, 152, 202, 241, 264, 382 compliance 204, 218 dual 264, 270, 348 fictitious 43, 152, 202, 203 fixed 223, 240, 261 formation 62, 65, 97, 110, 270, 349, 395, 400 initiation 218, 236, 261, 394 macro 40, 56, 152, 202, 223, 262, 264 micro 40, 152, 153, 191, 223 path tracking 236 propagation 201, 222, 229, 236, 261 rotating 223, 240, 261, 264, 265 smeared 56, 59, 198, 205, 213, 216, 222, 261, 268 spacing 109, 398 stabilised 62, 65, 97, 109, 270, 349, 398, 399 strain 56, 216, 219 tangential plane 203, 216 crack band 41 approach 205, 210, 213, 222 width 43, 56, 210, 215, 218, 221, 235, 269 crack energy surface 44, 58, 154, 203, 206, 209, 219, 233, 269 volume 44, 221

crack width critical 44, 203, 222 fictitious 43, 57, 202, 216, 237 creep 45 coefficient 48, 50 function 45, 47 linear 45, 47 time 48, 50

d damage 153, 172 anisotropic 189 gradient 42, 223, 226, 235 isotropic 183, 189, 207, 224, 269, 348 microplane 195, 271, 351 orthotropic 189 strain-based 183, 185, 190, 192, 196 stress-based 183, 185, 190 variable 183, 190, 196 damage function 184 Hsieh–Ting–Chen 184, 188, 207, 269, 348 microplane 195 Rankine 184 damping 131, 383 design deterministic 370 probabilistic 370 semi-probabilistic 361, 371, 372 design point 359, 363, 365 deviatoric length 167, 169, 175, 178, 181, 196 plane 167, 168, 178, 181 projection 181 dilatancy 154, 181 Dirac delta function 46, 239 discontinuity strong 392 weak 392 discretisation 8 spatial 22, 32 time 29 updated Lagrangian co-rotational 121 dissipation potential 229 distribution function 354, 359, 365, 373 Drucker stability postulate 200, 209

Index

ductility 44, 53, 80, 148, 149, 260, 311, 327, 368 dyadic product 175, 188, 239

e eigenstress 351 eigenvalue problem generalised 122, 127 matrix 158, 234 Einstein summation convention 193, 391 elasto-plasticity 52, 172 damaged 190 Drucker–Prager 178 Mises 175, 177 Mohr–Coulomb 179 multi-axial 173 uniaxial 53, 142, 172 element bar 1D 14 bar 2D 15 Bernoulli beam 88 continuum quad 2D 17 continuum-based shell 329 Kirchhoff slab 299 locking 19, 34, 87, 93, 280, 341, 344 overlay 268 spring 1D 16 spring 2D 16 Timoshenko beam 86 ellipticity loss 209 energy dissipation 44, 53, 131, 154, 200, 209 internal 18, 157, 199, 229, 289 equilibrium dynamic 30 incrementally iterative 26, 42, 50, 96, 104, 107, 116, 232, 349 strong differential 22, 82, 290 weak integral 22, 82, 292 error approximation 8, 22, 31 discretisation 31, 32 modelling 8 evolution law 20, 53, 164, 174, 182, 185, 190, 230

f failure brittle 202 ductile 202 quasi-brittle 202, 206 failure probability 359, 363, 368, 370 fibre model 81, 278, 284 flow potential 174, 175 rule 145, 173, 175, 180, 191 fracture mode 201 toughness 229, 235 variational model 230 fracture mechanics computational 236 linear elastic 201 function test 21, 22, 31, 82, 85, 136, 224, 230, 292 trial 10, 14, 22, 31, 86, 114, 277

h Haigh–Westergaard coordinates 167, 178, 186 hardening 38, 52 isotropic 53, 177 modulus 53, 177 Heaviside function 46 homogeneity 151 hour glassing 343 hydrostatic axis 167, 180 length 167, 179, 181 pressure 153, 156, 159, 169, 178

i impact 128 importance sampling 364, 365 instability 122, 391 material 391 numerical 343 structural 125, 391 integration Gauss 25, 92, 134, 175 numerical 25 triangular elements 301

419

420

Index

internal forces beam 70 shell 339 slab 288 internal state variable 20, 53, 164, 174, 178, 180, 183, 241, 263 interpolation 10, 18 isoparametric 14, 330 mixed 343 invariant 159, 165, 177 isotropy 159, 160, 165, 178

j Jacobian matrix 14, 17, 340 scalar 15, 17, 25, 85, 274, 331, 338

k Kelvin–Voigt chain 47 element 47, 131 kinematic assumption bar 14, 15 beam 67 continuum 17 plate 249 shell 332 slab 286 spring 16 truss 138 Kuhn–Tucker condition damage 185, 196 phase field 231 plasticity 145, 174, 191

l layer model 296, 306, 344, 348, 395 limit state condition 358, 360, 361, 364, 367, 368 function 358 lower bound 147, 249, 305 ultimate 9 upper bound 147, 323 linearity geometrical 13, 118 physical 12, 24 localisation 40, 205, 215, 223, 232, 235

Lode angle 167, 168, 170 LU decomposition 27, 389

m mass element matrix 92 matrix 23, 30, 94, 383 moment 85, 126 per length 22, 81, 126 specific 21, 82, 126 material algorithmic tangential stiffness matrix 28, 177 damaged 172 elasto-plastic 172 isotropic linear elastic 160 law 11 stiffness matrix 13, 20, 90, 141, 160, 173 tangential stiffness matrix 20, 38, 79, 142, 157, 164, 175, 182, 297, 382, 391 Maxwell element 47 series 47 meridian compressive 168, 170, 179, 186 tensile 168, 170, 179 mesh dependency 205, 209 generation 22 modal decomposition 127 model computational 8 conceptual 7 mathematical 7 numerical 8 Monte Carlo simulation 364, 367

n natural circular frequency 127 period 126, 127 Newmark method 29, 128, 382, 383 consistency 30 stability 30 Newton–Raphson method 26, 381 modified 383

Index

nodal forces external 12, 23 internal 12, 23 nonlinearity geometrical 118 physical 12 non-local method 214 differential 215, 223, 235 integral 214 norm 27, 32, 42 Sobolev 32 N-standardisation 362, 368

o operator bilinear 32 linear 32 orthotropy 161

p parameter 8 patch test 34 phase field 228, 232, 235 plasticity 153 associated 175 deviatoric 182 limit theorems 147, 249, 322 non-associated 181 rigid 145 volumetric 182 prediction 9 prestressing bonded 113 post-bonded 113 tendon 112 unbonded 113 principal strain 158, 262 stress 134, 158 stress space 166, 168, 177, 178 system 158 probability 354 density function 354, 356, 358 joint density function 358, 361 process zone 41, 202 property 7 punching 314

r random variable 357, 360, 363, 373 bivariate 354, 370 fractile 360 multi-variate 354 univariate 354 randomness 354, 356, 367 Rankine criterion 216, 218, 222, 236, 261, 394 limit function 180 Rayleigh quotient 127 reference axis 67 configuration 155, 159 period 357, 359 plane 286 surface 329 regularisation 200, 205, 209, 223, 237, 265 softening modulus 210 reinforcement group 252 integrated 273, 346 mesh 252, 266 sheet 267, 296, 346 relaxation 45, 118 function 47 Rendulic direction 167 representative volume element RVE 152, 168 residual 26 resonance 131 rheological model 47

s safety global factor 370, 372 margin 363, 370 partial factor 372, 374 sample 354, 365 scale macro 39, 43, 56, 152, 202 meso 39, 151, 152, 215 micro 39, 45, 192 secant method 383, 389 serviceability 9, 74, 101, 260

421

422

Index

shear correction factor 70, 290 modulus 70, 176, 195, 297, 346 retention factor 204 stiffness 100 shell 329 director 330 displacement 331 five parameter model 334 geometry 330 shrinkage 49, 105, 259 simplex method 147 size effect 154 slab 285 Kirchhoff 287, 291, 295 Reissner–Mindlin 287, 295 snap-back 42, 205, 388 Sobolev function space 31 norm 32, 34 split normal-tangential 194 volumetric-deviatoric 193 volumetric-deviatoric-tangential 194 stiffness matrix 13 tangential 24, 27 stochastic FEM 356 strain 155 assumed natural 343 covariant, natural 334 equivalent 184, 190, 195 generalised 17, 18, 20, 31, 69, 71, 202, 227, 288, 295 imposed 49, 105, 400 local 336 measurable 49, 105 non-local 215 plane 18, 162 rate 157 softening 38, 205, 209, 221 volumetric 181, 193 strength biaxial 171 condition 145, 165, 253 surface 167, 170, 180, 185

triaxial 167, 171 uniaxial 171, 216 stress Cauchy stress 156, 199, 203, 289, 335, 402 contravariant 335 deviatoric 156 effective 191 extension 29 generalised 17, 18, 20, 71, 202, 288, 295 local 336, 339 plane 18, 162 rate 157 trajectory 135 strut 98, 136, 260 strut-and-tie model 137, 249, 260, 297

t temperature 49, 105, 110, 259 tension softening 40, 41, 56 stiffening 64, 66, 109, 110, 396 stiffening coefficient 65 tie 98, 136, 260 time clock 11 loading 11 Timoshenko beam 68, 84, 85, 112 element 86, 93, 340 enhanced element 88 traction–separation relation 204, 219, 236, 244, 263 transfer length 396, 399 triaxial cell 168

u uncertainty aleatoric 353 epistemic 353 model 273, 322, 351, 352, 354

v validation 9 variable 8 virtual work principle

11, 21, 84, 194

Index

visco-elasticity 47 viscosity 47 artificial 208, 269, 382 Voigt notation strain 155 stress 156

y yield function 174, 177, 180 Drucker–Prager 178 Mises 175 Mohr–Coulomb 179 yield line method 323, 325, 326

423