0_Contents and Preliminary Pages

7/28/2019 0_Contents and Preliminary Pages

http://slidepdf.com/reader/full/0contents-and-preliminary-pages 1/15

Finite element analysis ingeotechnical engineering

Theory

David M. Potts and Lidija Zdravkovic

Imperial College of Science, Technology and Medicine

Thomas Telford



Published by Thomas Telford Publishing, Thom as Telford Ltd, 1 Heron Quay, London

E14 4JD.URL: http://www.t-telford.co.uk

Distributors for Thomas Telford books are

USA: AS CE P ress, 1801 Alexander Bell Drive, Reston, VA 20191-4400, U SA

Japan: Maruzen Co. Ltd, Book Department, 3-10 Nihonbashi 2-chome, Chuo-ku,

Tokyo 103

Australia: DA Books and Journals, 648 Whitehorse Road, Mitcham 3132, Victoria

First published 1999

Also available from Thomas Telford Books

Finite element analysis in geotechnical engineering: application. ISBN 0 7277 2783 4

A c atalogue record for this book is available from the British Library

ISBN: 0 7277 2753 2

© Da vid M . Potts and Lidija Zdravkovic, and Thoma s Telford Limited, 1999

All rights, including translation, reserved. Except for fair copying, no part of this

publication may be reproduced, stored in a retrieval system or transmitted in any form

or by any means, electronic, mechanical, photocopying or otherwise, without the prior

written permission of the Books Publisher, Thomas Telford Publishing, Thomas

Telford Ltd, 1 Heron Q uay, Londo n E14 4JD.

This book is published on the understanding that the author is/authors are solely

responsible for the statements made and opinions expressed in it and that itspublication does not necessarily imply that such statements and/or opinions are or

reflect the views or opinions of the publishers.



Contents

Preface xi

1.

2.

Geotechmcal analysis1.1 Synopsis

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

1.10

1.11

Finite

2.12.2

2.3

2.4

2.5

Introduction

Design objectives

Design requirements

Theoretical considerations

1.5.1 Requirem ents for a general solution

1.5.2 Equilibrium

1.5.3 Compatibility

1.5.4 Equilibrium and compatibility equations1.5.5 Constitutive behaviour

Geometric idealisation

1.6.1 Plane strain

1.6.2 Axi-symmetry

Methods of analysis

Closed form solutions

Simple methods

1.9.1 Limit equilibrium

1.9.2 Stress field solution1.9.3 Limit analysis

1.9.4 Comments

Numerical analysis

1.10.1 Beam-spring approach

1.10.2 Full numerical analysis

Summary

element theory for linear materials

SynopsisIntroduction

Overview

Element discretisation

Displacement approximation

1

12

2

3

4

4

4

5

67

8

8

9

10

11

12

12

1415

18

19

19

20

21

23

2323

23

24

27



ii / Finite element analysis in geotechnical engineering: Theory

2.5.1 Isoparametric finite elements 29

2.6 Elemen t equations 31

2.6.1 Num erical integration 34

2.7 Global equations 362.7.1 The direct stiffness assembly method 36

2.8 Boundary conditions 39

2.9 Solution of global equations 39

2.9.1 Storage of global stiffness matrix 40

2.9.2 Triangular decomposition of the global stiffness matrix41

2.9.3 Solution of the finite element equations 43

2.9.4 Modification due to displacement boundary conditions 45

2.10 Calculation of stresses and strains 47

2.11 Example 472.12 Axi-symm etric finite element analysis 49

2.13 Summ ary 50

Appendix II. 1 Triangular finite elements 51

11.1.1 Derivation of area coordinates 51

II. 1.2 Isoparametric formulation 53

3. Geotechnical con siderations 55

3.1 Synopsis 55

3.2 Introduction 553.3 Total stress analysis 56

3.4 Pore pressure calculation 58

3.5 Finite elements to model structural components 61

3.5.1 Introduction 61

3.5.2 Strain definitions 62

3.5.3 Constitutive equation 63

3.5.4 Finite element formulation 64

3.5.5 Mem brane elements 67

3.6 Finite elements to model interfaces 683.6.1 Introduction 68

3.6.2 Basic theory 69

3.6.3 Finite element formulation 70

3.6.4 Comments 72

3.7 Boundary conditions 72

72

73

74

7678

80

82

83

3.7.13.7.2

3.7.3

3.7.43.7.5

3.7.6

3.7.7

3.7.8

IntroductionLocal axes

Prescribed displacements

Tied degrees of freedomSprings

Boundary stresses

Point loads

Body forces



4.

5.

3.8

Real

4.1

4.2

4.3

4.4

4.5

4.6

4.7

3.7.9

3.7.10

3.7.11

Summary

Contents

Construction

Excavation

Pore pressures

soil behaviour

Synopsis

Introduction

Behaviour

4.3.1

4.3.2

4.3.3

4.3.4

4.3.5

4.3.6

Behaviour

4.4.1

4.4.2

4.4.3

4.4.44.4.5

of clay soils

Behaviour under one dimensional compression

Behaviour when sheared

Effect of stress path direction

Effect of the magnitude of the intermediateprincipal stress

Anisotropy

Behaviour at large strains

of sands

Behaviour under one dimensional compression

Behaviour when sheared

Effect of the magnitude of the intermediate

principal stress

AnisotropyBehaviour at large strains

Behaviour of soils containing both clay and sand

4.5.1

4.5.2

4.5.3

Comparison of sedimentary soils

Residual soils

Residual strength

Concluding remarks

Summary

Elastic constitutive models5.1

5.2

5.3

5.4

5.5

5.6

5.7

Synopsis

Introduction

Invariants

Elastic behaviour

Linear isotropic elasticity

Linear anisotropic elasticity

Nonlinear elasticity

5.7.1

5.7.25.7.3

5.7.4

5.7.5

5.7.6

Introduction

Bi-linear modelK - G model

Hyperbolic model

Small strain stiffness model

Puzrin and Burland model

/iii

84

86

87

89

90

90

90

91

91

92

94

95

97

97

99

99

100

103

104105

105

105

110

111

112

112

114114

114

114

118

118

120

122

122

123123

124

125

127



iv / Finite element analysis in geotechnical engineering: Theory

5.8 Sum mary 131

6. Elasto-plastic behaviour 132

6.1 Synopsis 132

6.2 Introduc tion 132

6.3 Uniaxial behaviour of a linear elastic perfectly plastic

material 133

6.4 Uniaxial behaviour of a linear elastic strain hardening

plastic material 134

6.5 Un iaxia l behaviou r of a linear elastic strain softening


6.6 Relevance to geotechnical engineering 135

6.7 Extension to general stress and strain space 135

6.8 Basic concepts 136

6.8.1 Coin cidenc e of axes 13 6

6.8.2 A yield function 136

6.8.3 A plastic poten tial function 137

6.8.4 The hardening/softening rules 138

6.9 Two dimension al behavio ur of a linear elastic perfectly


6.10 Two dimensional behaviour of a linear elastic hardening

plastic material 1406.11 Two dimension al behaviour of a linear elastic softening

plastic ma terial 141

6.12 Comparison with real soil behaviou r 142

6.13 Form ulation of the elasto-plastic constitutive matrix 143

6.14 Summ ary 146

7. Simple elasto-plastic constitutive models 147

7.1 Synopsis 147

7.2 Introduction 1477.3 Tresca mod el 148

7.4 Von Mises mod el 150

7.5 Mohr-Co ulomb model 151

7.6 Druck er-Prager mod el 155

7.7 Com men ts on simple elastic perfectly plastic mod els 157

7.8 An elastic strain hardening/softening Mo hr-Cou lomb model 158

7.9 Dev elopm ent of the critical state mod els 160

7.9.1 Basic formulation in triaxial stress space 161

7.9.2 Exten sion to general stress space 1667.9.3 Undrained strength 168

7.10 Modifications to the basic formulation of critical state mo dels 169

7.10.1 Yie ld surface on the superc ritical side 169

7.10.2 Yie ld surface for Kn consolidated soils 171



Contents / v

7.10.3 Elastic component of the model 172

7.10.4 Plastic behaviour inside the main yield surface 173

7.11 Alternative shapes for the yield and plastic potential

surfaces for critical state models 1757.11.1 Introduction 175

7.11.2 Developm ent of a new expression in triaxial

stress space 176

7.11.3 Generalisation of the expression 181

7.12 The effect of the plastic potential in plane strain deformation 181

7.13 Summ ary 185

Appendix VII. 1 Derivatives of stress invariants 186

Appendix VII.2 Analytical solutions for triaxial test on

modified Cam clay 187VII.2.1 Drained triaxial test 188

VII.2.2 Undrained triaxial test 192

Appendix VII.3 Derivatives for modified Cam clay model 195

Appendix VII.4 Undrained strength for critical state models 197

8. Advanced constitutive models 200

8.1 Synopsis 200

8.2 Introduction 200

8.3 Modelling of soil as a limited tension material 2018.3.1 Introduction 201

8.3.2 Model formulation 202

8.3.2.1 Yield surface 202

8.3.2.2 Plastic potential 203

8.3.2.3 Finite element implementation 204

8.4 Formulation of the elasto-plastic constitutive matrix when

two yield surfaces are simultaneously active 205

8.5 Lad e's double hardening model 208

8.5.1 Introduction 2088.5.2 Overview of model 208

8.5.3 Elastic behaviour 209

8.5.4 Failure criterion 209

8.5.5 Conical yield function 210

8.5.6 Conical plastic potential function 210

8.5.7 Conical hardening law 210

8.5.8 Cap yield function 211

8.5.9 Cap plastic potential function 211

8.5.10 Cap hardening law 2118.5.11 Comments 211

8.6 Bounding surface formulation of soil plasticity 212


8.6.2 Bounding surface plasticity 213



MIT soil models8.7.1

8.7.2

8.7.38.7.4

8.7.5

8.7.6

Bubble

8.8.1Q Q OO . O . Z

Introduction

Transformed variables

Hysteretic elasticityBehaviour on the bounding surface

Behaviour within the bounding surface

Comments

models

Introduction

Behaviour of a kinematic yield surface

Al-Tabbaa and Wood model

8.9.1

8.9.28.9.3

8.9.4

Bounding surface and bubble

Movement of bubbleElasto-plastic behaviour

Comments

Summary

vi / Finite element analysis in geotechnical engineering: Theory

8.7 MIT soil models 215

215

215

216

218

223

226

227

227

227

8.9 Al-Tabbaa and Wood model 229

229

230

231

232

8.10 Summary 232

Appendix VIII. 1 Derivatives for Lad e's double hardening model 233

9. Finite element theory for nonlinear materials 237

9.1 Synopsis 237

9.2 Introduction 237

9.3 Nonlinear finite element analysis 2389.4 Tangent stiffness method 238


9.4.2 Finite element implementation 239

9.4.3 Uniform compression of a Mohr-Coulomb soil 240

9.4.4 Uniform compression of modified Cam clay soil 245

9.5 Visco-plastic method 246


9.5.2 Finite element application 247

9.5.3 Choice of time step 2509.5.4 Potential errors in the algorithm 251

9.5.5 Uniform compression of a Mohr-Coulomb soil 251

9.5.6 Uniform compression of modified Cam clay soil 252

9.6 Modified Newton-Raphson method 256


9.6.2 Stress point algorithms 257

9.6.2.1 Introduction 257

9.6.2.2 Substepping algorithm 258

9.6.2.3 Return algorithm 2589.6.2.4 Fundamental comparison 259

9.6.3 Convergence criteria 260

9.6.4 Uniform compression of Mohr-Coulomb and

modified Cam clay soils 260



10.

Contents

9.7 Com pariso n of the solution strategies

9.7.1

9.7.2

9.7.3

9.7.4

9.7.5

9.7.6

9.8 Summary

Append ix IX. 1

IX . 1

IX.

IX.

IX.l

IX.l

LI

L2

1.3

.4

1.5

IX. 1.6

Append ix IX.2

IX.2.1

1X22

IX.2.3

IX.2.4Appendix IX.3

IX.3.1

IX.3.2

IX.3.3

IX.3.4

IX.3.5

Introduction

Idealised triaxial test

Footing problemExcavation problem

Pile problem

Comments

Substepping stress point algorithm

Introduction

Overview

Modified Euler integration scheme with

error controlRunge-Kutta integration scheme

Correcting for yield surface drift in

elasto-plastic finite element analysis

Nonlinear elastic behaviour

Return stress point algorithm

Introduction

Overview

Return algorithm proposed by Ortiz & Simo (1986 )

Return algorithm proposed by Borja & Lee (1990)Comparison of substepping and return algorithms

Introduction

Fundamental comparison

IX.3.2.1 Undrained triaxial test

1X322 Drained triaxial test

Pile problem

Consistent tangent operators

Conclusions

Seepage and consolidation

10.1 Synopsis

10.2 Introduction

10.3 Finite element formulation for coupled problem s

10.4 Finite element implementation

10.5 Steady state seepage

10.6 Hydraulic

10.6.1

10.6.210.6.3

10.6.4

10.6.5

10.6.6

boundary conditions

Introduction

Prescribed pore fluid pressuresTied degrees of freedom

Infiltration

Sources and sinks

Precipitation

/ VII

261

261

263

267270

273

275

276

277

277

278

280283

283

286

286

286

286

287

290296

296

296

296

299

301

303

304

305

305

305

306

311

312

313

313

313314

315

316

316



viii / Finite element analysis in geotechnical engineering: Theory

10.7 Permeability models 318


10.7.2 Linear isotropic permeability 318

10.7.3 Linear anisotropic permeability 31910.7.4 Nonlinear permeability related to void ratio 319

10.7.5 Nonlinear permeability related to mean effective

stress using a logarithmic relationship 320

10.7.6 Nonlinear permeability related to mean effective

stress using a power law relationship 320

10.8 Unconfined seepage flow 320

10.9 Validation example 321

10.10 Summary 323

11. 3D finite element analysis 325

11.1 Synopsis 325

325

326

332

332

332

334

336

337

341

342

342

12. Fourier series aided finite element method (FSAFEM) 344

12.1 Synopsis 344

12.2 Introduction 34412.3 The continuous Fourier series aided finite element method 345

12.3.1 Formulation for linear behaviour 345

12.3.2 Symmetrical loading conditions 352

12.3.3 Existing formulations for nonlinear behaviour 354

12.3.4 New formulation for nonlinear behaviour 355

12.3.5 Formulation for interface elements 359

12.3.6 Bulk pore fluid compressibility 361

12.3.7 Formulation for coupled consolidation 364

12.4 Implementation of the CFSAFEM 37012.4.1 Introduction 370

12.4.2 Evaluating Fourier series harmonic coefficients 371

12.4.2.1 The stepwise linear method 372

12.4.2.2 The fitted method 373

11.211.3

11.4

11.5

IntroductionConventional 3D finite element analysis

Iterative

11.4.1

11.4.2

11.4.3

11.4.411.4.5

11.4.6

11.4.7

solutions

Introduction

General iterative solution

The gradient method

The conjugate gradient methodComparison of the conjugate gradient and

banded solution techniques

Normalisation of the stiffness matrix

Comments

Summary



Contents / ix

12.4.3 The modified New ton-Raphson solution strategy 374


12.4.3.2 Right hand side correction 375

12.4.4 Da tastorag e 37612.4.5 Boundary conditions 377

12.4.6 Stiffness matrices 377

12.4.7 Simplification due to symm etrical

boundary conditions 378


12.4.7.2 Examples of problem s associated with

parallel and orthogonal analysis 380

12.5 The discrete Fourier series aided finite element method 385

12.5.1 Introduction 38512.5.2 Description of the discrete FSAFEM method 386

12.6 Comparison between the discrete and the continuous FSAFEM 391

12.7 Comparison of CFSAFEM and the 3D analysis 396

12.8 Summ ary 398

Appendix XII. 1 Harmonic coefficients of force from harmonic

point loads 399

Appendix XII.2 Obtaining the harmonics of force from harmonic

boundary stresses 400

Appendix XII.3 Obtaining the harmonics of force fromelement stresses 401

Appendix XII.4 Resolving harmonic coefficients of nodal force 403

Appendix XII.5 Fourier series solutions for integrating the product

of three Fourier series 404

Appendix XII.6 Obtaining coefficients for a stepwise linear

distribution 405

Appendix XII.7 Obtaining harmonic coefficients for the

fitted method 407

References 411

List of symbols 425

Index 435



Preface

While the finite element method has been used in many fields of engineering

practice for over thirty years, it is only relatively recently that it has begun to be

widely used for analysing geotechnical problems. This is probably because there

are many complex issues which are specific to geotechnical engineering and which

have only been resolved relatively recently. Perhaps this explains why there are

few books which cover the application of the finite elem ent method to geotechnical

engineering.

For over twenty years we, at Imperial College, have been working at the

leading edge of the application of the finite element method to the analysis of

practical geotechnical problems. Consequently, we have gained enormous

experience of this type of work and have shown that, when properly used, this

method can produce realistic results which are of value to practical engineeringproblems. Because we have w ritten all our own computer code, we also have an

in-depth understanding of the relevant theory.

Based on this experience we believe that, to perform useful geotechnical finite

element analysis, an engineer requires specialist know ledge in a range of subjects.

Firstly, a sound understanding of soil mechanics and finite element theory is

required. Secondly, an in-depth understanding and appreciation of the limitations

of the various constitutive models that are currently available is needed. Lastly,

users must be fully conversant with the manner in which the software they are

using works. Unfortunately, it is not easy for a geotechnical engineer to gain allthese skills, as it is vary rare for all of them to be part of a single undergraduate or

postgraduate degree course. It is perhaps, therefore, not surprising that many

engineers, who carry out such analyses and/or use the results from such analyses,

are not aware of the potential restrictions and pitfalls involved.

This problem was highlighted when we recently gave a four day course on

numerical analysis in geotechnical engineering. Although the course was a great

success, attracting many participants from both industry and academia, it did

highlight the difficulty that engineers have in obtaining the necessary skills

required to perform good numerical analysis. In fact, it was the delegates on thiscourse who urged us, and provided the inspiration, to write this book.

The overall objective of the book is to provide the reader with an insight into

the use of the finite element method in geotechnical engineering. More specific

aims are:



xii / Finite element analysis in geotechnical engineering: Theory

To present the theory, assumptions and approximations involved in finite

element analysis;

- To describe some of the more popular constitutive models currently available

and explore their strengths and weaknesses;- To provide sufficient information so that readers can assess and compare the

capabilities of available commercial software;

- To prov ide sufficient information so that readers can make judgem ents as to the

credibility of numerical results that they may obtain, or review, in the future;

- To show, by means of practical exam ples, the restrictions, pitfalls, advantages

and disadvantages of numerical analysis.

The book is primarily aimed at users of commercial finite element software both

in industry and in academia. However, it will also be of use to students in theirfinal years of an undergraduate course, or those on a postgraduate course in

geotechnical engineering. A prime objective has been to present the material in the

simplest possible way and in manner understandable to most engineers.

Consequently, we have refrained from using tensor notation and presented all

theory in terms of conventional matrix algebra.

When we first considered writing this book, it became clear that we could not

cover all aspects of numerical analysis relevant to geotechnical engineering. We

reached this conclusion for two reasons. Firstly, the subject area is so vast that to

adequately cover it would take many volumes and, secondly, we did not haveexperience with all the different aspects. Consequently, we decided only to include

material which we felt we had adequate experience of and that was useful to a

practising engineer. As a result we have concentrated on static behaviour and have

not considered dynamic effects. Even so, we soon found that the material we

wished to include would not sensibly fit into a single volume. The material has

therefore been divided into theory and application, each presented in a separate

volume.

Volume 1 concentrates on the theory behind the finite element method and on

the various constitutive models currently available. This is essential reading for anyuser of a finite element package as it clearly outlines the assumptions and

limitations involved. Volume 2 concentrates on the application of the method to

real geotechnical problems, highlighting how the method can be applied, its

advantages and disadvantages, and some of the pitfalls. This is also essential

reading for a user of a software package and for any engineer who is

commissioning and/or reviewing the results of finite element analyses.

This volume of the book (i.e. Volume 1) consists of twelve chapters. Chapter

1 considers the general requirements of any form of geotechnical analysis and

provides a framework for assessing the relevant merits of the different methods ofanalysis currently used in geotechnical design. This enables the reader to gain an

insight into the potential advantage of numerical analysis over the more



Preface / xiii

'conventional' approaches currently in use. The basic finite element theory for

linear material behaviour is described in Chapter 2. Emphasis is placed on

highlighting the assumptions and limitations. Chapter 3 then presents the

modifications and additions that are required to enable geotechnical analysis to beperformed.

The main limitation of the basic finite element theory is that it is based on the

assumption of linear material behav iour. Soils do not behave in such a manner and

Chapter 4 highlights the important facets of soil behaviour that ideally should be

accounted for by a constitutive model. Unfortunately, a constitutive model which

can account for all these facets of behaviour, and at the same time be defined by

a realistic number of input parameters which can readily be determined from

simple laboratory tests, does not exist. Nonlinear elastic constitutive models are

presented in Chapter 5 and although these are an improvement over the linearelastic models that were used in the early days of finite element analyses, they

suffer severe limitations. The majority of constitutive models currently in use are

based on the framework of elasto-plasticity and this is described in Chapter 6.

Simple elasto-plastic m odels are then presented in Chapter 7 and more complex

models in Chapter 8.

To use these nonlinear constitutive models in finite element analysis requires

an extension of the theory presented in Chapter 2. This is described in Chapter 9

where some of the most popular nonlinear solution strategies are considered. It is

shown that some of these can result in large errors unless extreme care is exercisedby the user. The procedures required to obtain accurate solutions are discussed.

Chapter 10 presents the finite element theory for analysing coupled problems

involving both deformation and pore fluid flow. This enables time dependent

consolidation problems to be analysed.

Three dimensional problems are considered in Chapter 11. Such problems

require large amounts of computer resources and methods for reducing these are

discussed. In particular the use of iterative equation solvers is considered. While

these have been used successfully in other branches of engineering, it is shown

that, with present computer hardware, they are unlikely to be economical for themajority of geotechnical problem s.

The theory behind Fourier Series Aided Finite Element Analysis is described

in Chapter 12. Such analysis can be applied to three dimensional problem s which

possess an axi-symmetric geometry but a non axi-symmetric distribution of

material properties and/or loading. It is shown that analyses based on this approach

can give accurate results with up to an order of magnitude saving in computer

resources compared to equivalent analyses performed with a conventional three

dimensional finite element formulation.

Volume 2 of this book builds on the material given in this volume. H owever,the emphasis is less on theory and more on the application of the finite element

method in engineering prac tice. Topics such as obtaining geotechnical parameters



xiv / Finite element analysis in geotechnical engineering: Theory

from standard laboratory and field tests and the analysis of tunne ls, earth re taining

structures, cut slopes, embankments and foundations are covered. A chapter on

benchmarking is also included. Emphasis is placed on explaining how the finite

element method should be applied and what are the restrictions and pitfalls. Inparticular, the choice of suitable constitutive models for the various geotechnical

boundary value problems is discussed at some length. To illustrate the material

presented, examples from the authors experiences with practical geotechnical

problems are used. Although we have edited this volume, and written much of the

content, several of the chapters involve contributions from our colleagues at

Imperial College.

All the numerical exam ples presented in both this volume and Volume 2 of this

book have been obtained using the Author s' own computer code. This software is

not available commercially and therefore the results presented are unbiased. Ascommercial software has not been used, the reader must consider what implications

the results may have on the use of such software.

London David M. Potts

November 1998 Lidija Zdravkov ic

Documents

0_Contents and Preliminary Pages