Transcript
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
7/28/2019 0_Contents and Preliminary Pages
http://slidepdf.com/reader/full/0contents-and-preliminary-pages 2/15
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.
7/28/2019 0_Contents and Preliminary Pages
http://slidepdf.com/reader/full/0contents-and-preliminary-pages 3/15
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
7/28/2019 0_Contents and Preliminary Pages
http://slidepdf.com/reader/full/0contents-and-preliminary-pages 4/15
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
7/28/2019 0_Contents and Preliminary Pages
http://slidepdf.com/reader/full/0contents-and-preliminary-pages 5/15
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
7/28/2019 0_Contents and Preliminary Pages
http://slidepdf.com/reader/full/0contents-and-preliminary-pages 6/15
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
plastic material 134
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
plastic material 139
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
7/28/2019 0_Contents and Preliminary Pages
http://slidepdf.com/reader/full/0contents-and-preliminary-pages 7/15
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
8.6.1 Introduction 212
8.6.2 Bounding surface plasticity 213
7/28/2019 0_Contents and Preliminary Pages
http://slidepdf.com/reader/full/0contents-and-preliminary-pages 8/15
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
9.4.1 Introduction 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
9.5.1 Introduction 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
9.6.1 Introduction 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
7/28/2019 0_Contents and Preliminary Pages
http://slidepdf.com/reader/full/0contents-and-preliminary-pages 9/15
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
7/28/2019 0_Contents and Preliminary Pages
http://slidepdf.com/reader/full/0contents-and-preliminary-pages 10/15
viii / Finite element analysis in geotechnical engineering: Theory
10.7 Permeability models 318
10.7.1 Introduction 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
7/28/2019 0_Contents and Preliminary Pages
http://slidepdf.com/reader/full/0contents-and-preliminary-pages 11/15
Contents / ix
12.4.3 The modified New ton-Raphson solution strategy 374
12.4.3.1 Introduction 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
12.4.7.1 Introduction 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
7/28/2019 0_Contents and Preliminary Pages
http://slidepdf.com/reader/full/0contents-and-preliminary-pages 12/15
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:
7/28/2019 0_Contents and Preliminary Pages
http://slidepdf.com/reader/full/0contents-and-preliminary-pages 13/15
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
7/28/2019 0_Contents and Preliminary Pages
http://slidepdf.com/reader/full/0contents-and-preliminary-pages 14/15
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
7/28/2019 0_Contents and Preliminary Pages
http://slidepdf.com/reader/full/0contents-and-preliminary-pages 15/15
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
top related