Finite Element Analysis in GeotechnicalEngineering Application by David M. Potts and Lidija Zdravkov

Finite element analysis ingeotechnical engineering

Finite element analysis ingeotechnical engineering

Application

I

David M. jPotts and lidija ZdravkovicImperial College ojScience, Technology and Medicine

With contributions from:

Trevor I. AddenbrookeKelvin G. HigginsNebojsa Kovacevic

-.~I ThomasTelford

Published by Thomas Telford Publishing, Thomas Telford Ltd, 1 Heron Quay, LondonE144JD.URL: http://www.thomastelford.com

----_.._--------------------------------_._-----------._---------

Contents

Distributors for Thomas Telford books areUSA: ASCE Press, 1801 Alexander Bell Drive, Reston, VA 20191-4400, USAJapan: Maruzen Co. Ltd, Book Department, 3-10 Nihonbashi 2-chome, Chuo-ku,Tokyo 103Australia: DA Books and Journals, 648 WhitehoR"se Road, Mitcham 3132, Victoria

First published 2001

Also available from Thomas Telford BooksFinite element analysis in geotechnical engineering: theory. ISBN 0 7277 2753 2

A catalogue record for this book is available from the British Library

ISBN: 0 7277 2783 4

David M. Potts and LidijaZdravkovic, and Thomas Telford Limited, 2001

All rights, including translation, reserved. Except as permitted by the Copyright, Designsand Patents Act 1988, no part of this publication may be reproduced, stored in a retrievalsystem or transmitted in any form or by any means, electronic, mechanical, photocopying orotherwise, without the prior written permission of the Publishing Director, Thomas TelfordPublishing, Thomas Telford Ltd, 1 Heron Quay, London EI4 410.

This book is published on the understanding that the author are solely responsible for thestatements made and opinions expressed in it and that its publication does not necessarilyimply that such statements and/or opinions are or reflect the views or opinions of thepublishers. While every effort has been made to ensure that the statements made and theopinions expressed in this publication provide a safe and accurate guide, no liability orresponsibility can be accepted in this respect by the authors or publishers.

1.

2.

Preface

Authorship

Acknowledgements

Obtaining geotechnical parameters1.1 Synopsis1.2 Introduction1.3 Laboratory tests

1.3 .1 Introduction1.3.2 Oedorneter test1.3 .3 Triaxial test1.3 A True triaxial test1.3.5 Direct shear test1.3.6 Simple shear test1.3.7 Ring shear test1.3.8 Hollow cylinder test1.3.9 Directional shear cell1.3.10 Geophysical techniques1.3.11 Penneameters

I A In-situ tests104.1 Introduction104.2 Standard penetration test (SPT)1.4.3 Cone penetration test (CPT)10404 Pressuremeter testing104.5 The plate loading test104.6 Pumping tests

1.5 Summary

Tunnels2.1 Synopsis2.2 Introduction2.3 Tunnel construction

xi

xvi

XVI

1I12236

111214151620202223232327303235

3538383839

ii I Finite element analysis in geotechnical engineering: Application Contents I iii

2.3.1 Introduction 39 3.4.5 Choice of constitutive models 842.3.2 Open faced shield tunnelling 40 3.4.5.1 Structural components 842.3.3 Tunnel Boring Machines (TBM), including slurry 3.4.5.2 Soil 85

shields and Earth Pressure Balance (EPB) tunnelling 40 3.4.6 Initial ground conditions 882.3.4 The sprayed concrete lining (SCL) method 41 3.4.6.1 General 882.3.5 Ground response to tunnel construction 41 3.4.6.2 'Greenfield' conditions 88

2.4 Simulation of the construction process 43 3.4.6.3 Modified initial soil stresses 892.4.1 Introduction 43 3.4.7 Construction method and programme 912.4.2 Setting up the initial conditions 44 3.4.7. I General 912.4.3 Important boundary conditions 45 3.4.7.2 Construction method 912.4.4 Modelling tunnel excavation 45 3.4.7.3 Time related movements 922.4.5 Modelling the tunnel lining 48 3.4.7.4 Ground water control 93

2.5 Modelling time dependent behaviour 52 3.5 Gravity walls 932.5.1 Introduction 52 3.5. I Introduction 932.5.2 Setting up the initial conditions 52 3.5.2 Earth pressure due to compaction 942.5.3 Hydraulic boundary conditions 54 3.5.3 Finite element analysis 952.5.4 Permeability models 55 3.6 Reinforced earth walls 962.5.5 A parametric study of the effect of permeable and 3.6.1 Introduction 96impermeable tunnel linings 57 3.6.2 Finite element analysis 99

2.6 Choice of soil model 59 3.7 Embedded walls 1032.6.1 Introduction 59 3.7.1 Introduction 1032.6.2 Results from a parametric study 59 3.7.2 Installation effects 1042.6.3 Devices for improving the surface settlement 3.7.2.1 General 104prediction 60 3.7.2.2 Field measurements 1042.7 Interaction analysis 63 3.7.2.3 Analysis 1052.7.1 The influence of building stiffness on tunnel-induced 3.7.2.4 Comments 106ground movements 63 3.7.3 Modelling of walls 1072.7.2 The Treasury building - a case study 66 3.7.3.1 Element type 1072.7.3 Twin tunnel interaction 70 3.7.3.2 Wall stiffness 1092.8 Summary 72 3.7.3.3 Interface behaviour III

3.7.3.4 Wall permeability III3. Earth retaining structures 74 3.7.4 Support systems 1123.1 Synopsis 74 3.7.4.1 Introduction 1123.2 Introduction 74 3.7.4.2 Support stiffness 1123.3 Types of retaining structure 75 3.7.4.3 Connection details 1133.3.1 Introduction 75 3.7.4.4 Active support systems 1143.3.2 Gravity walls 75 3.7.4.5 Berms 1153.3.3 Reinforced/anchored earth wall 76 3.7.4.6 Ground anchors 1153.3.4 Embedded walls 76 3.7.4.7 Relieving slabs 1163.4 General considerations 77 3.7.5 Long term behaviour and post construction effects 1183.4. I Introduction 77 3.7.6 Adjacent structures 1193.4.2 Symmetry 77 3.8 Summary 1223.4.3 Geometry of the finite element model 79 Appendix m.1 1233.4.4 Support systems 82

iv / Finite element analysis in geotechnical engineering: Application Contents / v

4. Cut slopes 125 5.3.3.2 'Power law' models 1694.1 Synopsis 125 5.3.3.3 Hyperbolic model 1704.2 Introduction 125 5.3.3.4 K-G model 1714.3 'Non-softening' analyses 126 5.3.3.5 Elasto-plastic models 171

4.3.1 Introduction 126 5.3.4 Layered analysis, stiffuess of the simulated layer4.3.2 Cut slopes in stiff 'non-softening' clay 127 and compaction stresses 173

4.3.2.1 Introduction 127 5.3.5 Example: Analysis of Roadford dam 1754.3.2.2 Soil parameters 127 5.3.5.1 Introduction 1754.3.2.3 Finite element analyses 127 5.3.5.2 Material parameters 1754.3.2.4 Results of analyses 128 5.3.5.3 Finite element analysis 177

4.3.3 Cut slopes in soft clay 131 5.3 .5.4 Comparison with observations 1794.3.3.1 Introduction 131 5.3.6 Example: Analysis of old puddle clay core dams 1804.3.3.2 Soil parameters 132 5.3.6.1 Introduction 1804.3.3.3 Finite element analyses 136 5.3.6.2 Dale Dyke dam 1814.3.3.4 Results of analyses 138 5.3.6.3 Ramsden dam 183

4.4 Progressive failure 141 5.4 Finite element analysis of earth embankments 1854.5 'Softening' analyses 145 5.4.1 Introduction 185

4.5.1 Introduction 145 5.4.2 Modelling of earthfill 1864.5.2 Choice of constitutive model 146 5.4.3 Example: Road embankments on London Clay 1864.5.3 Implications for convergence 147 5.4.3.1 Introduction 1864.5.4 Cut slopes in London Clay 147 5.4.3.2 Material properties 187

4.5.4.1 Introduction 147 5.4.3.3 Finite element analyis 1884.5.4.2 Soil parameters 148 5.4.4 Example: Failure of Carsington embankment 1894.5.4.3 Finite element analyses 150 5.4.4.1 Introduction 1894.5.4.4 Results of a typical analysis 150 5.4.4.2 Material parameters and soil model used 1904.5.4.5 Effect of coefficient of earth 5.4.4.3 Finite element analysis 191

pressure at rest 153 5.4.4.4 Original Carsington section 1914.5.4.6 Effect of surface boundary suction 155 5.4.4.5 Effect of the core geometry on4.5.4.7 Effect of slope geometry 155 progressive failure 1924.5.4.8 Effect of surface cracking 156 5.4.4.6 Effect of berm in improving the stability 1934.5.4.9 Effect of subsequent changes to slope 5.5 Finite element analysis of embankments on soft clay 194

geometry 158 5.5.1 Introduction 1944.5.4.10 Further discussion 160 5.5.2 Typical soil conditions 195

4.6 Construction of cut slope under water 162 5.5.3 Choice of constitutive model 1964.7 Summary 163 5.5.4 Modelling soil reinforcement 1985.5.5 Example: Effect of a surface crust 198

5. Embankments 166 5.5.5.1 Introduction 1985.1 Synopsis 166 5.5.5.2 Soil conditions 1985.2 Introduction 166 5.5.5.3 Finite element analysis 1995.3 Finite element analysis of rockfill dams 167 5.5.5.4 Results 200

5.3.1 Introduction 167 5.5.6 Example: Effect of reinforcement 2005.3.2 Typical stress paths 167 5.5.6.1 Introduction 2005.3.3 Choice of constitutive models 168 5.5.6.2 Soil conditions 201

5.3.3.1 Linear elastic analysis 169 5.5.6.3 Results 201

vi / Finite element analysis in geotechnical engineering: Application Contents / vii

6.

5.5.7 Example: Staged construction5.5.7.1 Introduction5.5.7.2 Soil conditions5.5.7.3 Finite element analysis5.5.7.4 Results

5.5.8 Example: Effect of annsotropic soil behaviour5.5.8.1 Introduction5.5.8.2 Geometry5.5.8.3 Soil conditions5.5.8.4 Finite elemlent analysis5.5.8.5 Results

5.6 Summary

Shallow foundations6.1 Synopsis6.2 Introduction6.3 Foundation types

6.3.1 Surface foundations6.3.2 Shallow foundations

6.4 Choice of soil model6.5 Finite element analysis of surface foundations

6.5.1 Introduction6.5.2 Flexible foundations6.5.3 Rigid foundations6.5.4 Examples of vertical loading

6.5.4.1 Introduction6.5.4.2 Strip footings on undrained clay6.5.4.3 Effect offooting shape on the bearing

capacity of undrained clay6.5.4.4 Strip footings on weightless drained soil6.5.4.5 Strip footings on a drained soil6.5.4.6 Circular footings on a weightless drained

soil6.5.4.7 Circular footings on a drained soil

6.5.5 Undrained bearing capacity of non-homogeneousclay6.5.5.1 Introduction6.5.5.2 Constitutive model6.5.5.3 Geometry and boundary conditions6.5.5.4 Failure mechanisms

6.5.6 Undrained bearing capacity ofpre-Ioaded stripfoundations on clay6.5.6.1 Introduction6.5.6.2 Constitutive model

202202203204205206206206207207208211

214214214215215215215216216218218219219219

223225227

230232

233233234236236

238238239

7.

6.5.6.3 Geometry and boundary conditions6.5.6.4 Results of the analyses6.5.6.5 Concluding remarks

6.5.7 Effect of anisotropic strength on bearing capacity6.5.7.1 Introduction6.5.7.2 Soil behaviour6.5.7.3 Behaviour of strip footings6.5.7.4 Behaviour of circular footings

6.6 Finite element analysis of shallow foundations6.6.1 Introduction6.6.2 Effect of foundation depth on undrained bearing

capacity6.6.3 Example: The leaning Tower of Pisa

6.6.3.1 Introduction6.6.3.2 Details of the Tower and ground profile6.6.3.3 History of construction6.6.3.4 History of tilting6.6.3.5 The motion of the Tower foundations6.6.3.6 Stability of tall towers6.6.3.7 Soil properties6.6.3.8 Finite element analysis6.6.3.9 Simulation of the history of inclination6.6.3.10 Temporary counterweight6.6.3.11 Observed behaviour during application

of the counterweight6.6.3.12 Permanent stabilisation of the Tower6.6.3.13 Soil extraction6.6.3.14 The response of the Tower to

soil extraction6.6.3.15 Comments

6.7 Summary

Deep foundations7.1 Synopsis7.2 Introduction7.3 Single piles

7.3.1 Introduction7.3.2 Vertical loading7.3.3 Lateral loading

7.4 Pile group behaviour7.4.1 Introduction7.4.2 Analysis of a pile group7.4.3 Superposition

7.4.3.1 Simple superposition

240240243243243244246247248248

248252252253254255256256259263265267

269271271

275276278

280280280282282282287289289291291292

viii I Finite element analysis in geotechnical engineering: Application

7.4.3.2 Pile displacements with depth7.4.4 Load distribution within a pile group

7.4.4.1 Obtaining an initial trial division of theapplied loads

7.4.4.2 Evaluating pile head displacements7.4.4.3 Checking the rigid pile cap criterion

7.4.5 Pile group design7.4.5.1 Matrix formulation of the pile group

response7.4.5.2 Superposition ofloads7.4.5.3 Evaluating the solution displacements

and rotations7.4.6 Magnus

7.4.6.1 Introduction7.4.6.2 Soil properties and initial conditions7.4.6.3 Finite element analyses7.4.6.4 Design of Magnus foundations7.4.6.5 Environmental loading

7.5 Bucket foundations7.5.1 Introduction7.5.2 Geometry7.5.3 Finite element analysis7.5.4 Modelling of the interface between top cap and soil7.5.5 Isotropic study

7.5.5.1 Soil conditions7.5.5.2 Parametric studies7.5.5.3 Results

7.5.6 Anisotropic study7.5.6.1 Introduction7.5.6.2 Results

7.5.7 Suction anchors7.5.7.1 Introduction7.5.7.2 Geometry7.5.7.3 Results

7.6 Summary

8. Benchmarking8.1 Synopsis8.2 Definitions8.3 Introduction8.4 Causes of errors in computer calculations8.5 Consequences of errors8.6 Developers and users

8.6.1 Developers

Contents I ix

293 8.6.2 Users 337294 8.7 Techniques used to check computer calculations 339

8.8 Benchmarking 339296 8.8.1 General 339297 8.8.2 Standard benchmarks 340297 8.8.3 Non-standard benchmarks 341298 8.9 The INTERCLAY II project 341

8.10 Examples of benchmark problems - Part I 342298 8.10.1 General 342299 8.10.2 Example 1: Analyses of an ideal triaxial test 343

8.10.3 Example 2: Analysis of a thick cylinder 344302 8.10.4 Example 3: Analyses of an advancing tunnel304 heading 346304 8.10.5 Example 4: Analysis ofa shallow waste disposal 348304 8.10.6 Example 5: Simplified analysis of a shallow waste 351308 8.11 Examples of benchmark problems - Part 11309 (German Society for Geotechnics benchmarking exercise) 353314 8.1 I.l Background 353317 8.11.2 Example 6: Construction of a tunnel 353317 8.11.3 Example 7: Deep excavation 355318 8.11.4 General comments 356318 8.12 Summary 357320 Appendix VIII. 1 Specification for Example I: Analyses of an321 idealised triaxial test 358321 VIII. I.l Geometry 358322 VIII. 1.2 Material properties and initial stress conditions 358322 VIIl.I.3 Loading conditions 358326 Appendix VIII.2 Specification for Example 2: Analysis of a thick326 cylinder 358326 VII1.2. I Geometry 358327 V1I1.2.2 Material properties 358327 VIII.2.3 Loading conditions 359327 Appendix VIII.3 Specification for Example 3: Analysis of an329 advancing tunnel heading 359329 VIII.3.1 Geometry 359

VIII.3.2 Material properties 359332 VIII.3.3 Loading conditions 359332 Appendix VIllA Specification for Example 4: Analysis of a shallow332 waste disposal 360333 VIII.4.1 Geometry 360334 VIIIA.2 Material properties 360335 VIII.4.3 Loading conditions 361336 Appendix VIII.5 Specification for Example 5: Simplified analysis336 of a shallow waste disposal 361

x / Finite element analysis in geotechnical engineering: Application

VIII.5.1 Geometry 361VIII.5.2 Material properties 361VIII.5.3 Loading conditions 361VIII.5.4 Additional boundarY conditions 362

Appendix VIII.6 Specification for Example 6: Construction of a tunnel 362VIII.6.1 Geometry 362VIII.6.2 Material properties 362

Appendix VIII.7 Specification for Example 7: Deep excavation 362VIII.7.1 Geometry 362VIII.7.2 Material properties 362VIII.7.3 Construction stages 363

9. Restrictions and pitfalls 3649.1 Synopsis 3649.2 Introduction 3649.3 Discretisation errors 3659.4 Numerical stability of zero thickness interface elements 368

9.4.1 Introduction 3689.4.2 Basic theory 3689.4.3 Ill-conditioning 3709.4.4 Steep stress gradients 373

9.5 Modelling of structural members in plane strain analysis 3769.5.1 Walls 3769.5.2 Piles 3779.5.3 Ground anchors 3789.5.4 Structural members in coupled analyses 3809.5.5 Structural connections 3&99.5.6 Segmental tunnel linings 381

9.6 Use of the Mohr-Coulomb model for undrained analysis 3829.7 Influence of the shape of the yield and plastic potential

surfaces in the deviatoric plane 3849.8 Using critical state models in undrained analysis 3869.9 Construction problems 3879.10 Removal of prescribed degrees of freedom 3889.11 Modelling underdrainage 3899.12 Summary 394

References 396

List of symbols 410

Index 415

Preface

While the finite element method has been used in many fields of engineeringpractice for over thirty years, it is only relatively recently that it has begun to bewidely used for analysing geotechnical problems. This is probably because thereare many complex issues which are specific to geotechnical engineering and whichhave only been resolved relatively recently. Perhaps this explains why there arefew books which cover the application ofthe fmite element method to geotechnicalengineering.

For over twenty years we, at Imperial College, have been working at theleading edge of the application of the fmite element method to the analysis ofpractical geotechnical problems. Consequently, we have gained enormousexperience of this type of work and have shown that, when properly used, thismethod can produce realistic results which are of value to practical engineeringproblems. Because we have written all our own computer code, we also have anin-depth understanding of the relevant theory.

Based on this experience we believe that, to perform useful geotechnical finitee~ement analysis, an engineer requires specialist knowledge in a range ofsubjects.FIrst~y, a sound under~tanding of soil mechanics and finite element theory isreqUIred. Secondly, an m-depth understanding and appreciation ofthe limitationsof the various constitutive models that are currently available is needed. Lastly,

us~rs must be fully conversant with the manner in which the software they areusmg w~rks. ~n~ortunately, it is not easy for a geotechnical engineer to gain allthese SkIlls, as It IS vary rare for all of them to be part ofa single undergraduate orpos:graduate degree course. It is perhaps, therefore, not surprising that manyengmeers, who carry out s~ch analyses and/or use the results from such analyses,are not aware of the potentIal restrictions and pitfalls involved.

This problem was highlighted four years ago when we gave a four day courseon numerical analysis in geotechnical engineering. Although the course was a greatsuccess, attracting many participants from both industry and academia it didhighlight the difficulty that engineers have in obtaining the necessar; skillsrequired 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 intot~e use of the finite element method in geotechnical engineering. More specificaIms are:

xii I Finite element analysis in geotechnical engineering: Application

To present the theory, assumptions and approximations involved in finiteelement analysis;To describe some of the more popular constitutive models currently availableand explore their strengths and weaknesses;To provide sufficient information so that readers can assess and compare thecapabilities of available commercial software;To provide sufficient information so that readers can make judgements as to thecredibility of numerical results that they may obtain, or review, in the future;To show, by means of practical examples, the restrictions, pitfalls, advantagesand disadvantages of numerical analysis.

The book is primarily aimed at users of commercial finite element software bothin 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 ingeotechnical engineering. A prime objective has been to present the material in thesimplest possible way and in manner understandable to most engineers.Consequently, we have refrained from using tensor notation and have presented alltheory in terms of conventional matrix algebra.

When we first considered writing this book, it became clear that we could notcover all aspects of numerical analysis relevant to geotechnical engineering. Wereached this conclusion for two reasons. Firstly, the subject area is so vast that toadequately cover it would take many volumes and, secondly, we did not haveexperience with all the different aspects. Consequently, we decided only to includematerial which we felt we had adequate experience of and that was useful to apractising engineer. As a result we have concentrated on static behaviour and havenot considered dynamic effects. Even so, we soon found that the material wewished to include would not sensibly fit into a single volume. The material hastherefore been divided into theory and application, each presented in a separatevolume.

Volume I concentrates on the theory behind the finite element method and onthe various constitutive models currently available. This is essential reading for anyuser of a finite element package as it clearly outlines the assumptions andlimitations involved. Volume 2 concentrates on the application of the method toreal geotechnical problems, highlighting how the method can be applied, itsadvantages and disadvantages, and some of the pitfalls. This is also essentialreading for a user of a software package and for any engineer who iscommissioning and/or reviewing the results of finite element analyses.

Volume I of this book consists of twelve chapters. Chapter I considers thegeneral requirements of any form of geotechnical analysis and provides aframework for assessing the relevant merits of the different methods of analysiscurrently used in geotechnical design. This enables the reader to gain an insightinto the potential advantage of numerical analysis over the more 'conventional'approaches currently in use. The basic finite element theory for linear materialbehaviour is described in Chapter 2. Emphasis is placed on highlighting the

Preface I xiii

assumptions and limitations. Chapter 3 then presents the modifications andadditions that are required to enable geotechnical analysis to be performed.

The main limitation of the basic finite element theory is that it is based on theassumption oflinear material behaviour. Soils do not behave in such a manner andChapter 4 highlights the important facets of soil behaviour that ideally should beaccounted for by a constitutive model. Unfortunately, a constitutive model whichcan account for all these facets of behaviour, and at the same time be defined bya realistic number of input parameters which can readily be determined fromsimple laboratory tests, does not exist. Nonlinear elastic constitutive models arepresented in Chapter 5 and although these are an improvement over the linearelastic models that were used in the early days of finite element analyses, theysuffer severe limitations. The majority ofconstitutive models currently in use arebased on the framework of elasto-plasticity and this is described in Chapter 6.Simple elasto-plastic models are then presented in Chapter 7 and more complexmodels in Chapter 8.

To use these nonlinear constitutive models in finite element analysis requiresan extension of the theory presented in Chapter 2. This is described in Chapter 9where some of the most popular nonlinear solution strategies are considered. It isshown that some ofthese 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 problemsinvolving both deformation and pore fluid flow. This enables time dependentconsolidation problems to be analysed.

Three dimensional problems are considered in Chapter 11. Such problemsrequire large amounts of computer resources and methods for reducing these arediscussed. In particular the use of iterative equation solvers is considered. Whilethese have been used successfully in other branches of engineering, it is shownthat, with present computer hardware, they are unlikely to be economical for themajority of geotechnical problems.

The theory behind Fourier Series Aided Finite Element Analysis is describedin Chapter 12. Such analysis can be applied to three dimensional problems whichpossess an axi-symmetric geometry but a non axi-symmetric distribution ofmaterial properties and/or loading. It is shown that analyses based on this approachcan give accurate results with up to an order of magnitude saving in computerresources, compared to equivalent analyses performed with a conventional threedimensional finite element formUlation.

This volume ofthe book (Le Volume 2) builds on the material given in VolumeI. However, the emphasis is less on theory and more on the application ofthe finiteelement method in engineering practice. It consists of nine chapters.

Chapter I considers the problems involved in obtaining geotechnicalparameters. These are necessary to define the constitutive models and initialconditions for an analysis. The relative merits of laboratory and field testing arediscussed and the parameters that can be obtained from the various tests examined.

The analyses of tunnel construction is considered in Chapter 2. Emphasis is

xiv / Finite element analysis in geotechnical engineering: Application Preface / xv

All the numerical examples presented in both this volume and Volume I ofthisbook have been obtained using the Authors' own computer code. This software isnot available commercially and therefore the results presented are unbiased. Ascommercial software has not been used, the reader must consider what implicationsthe results may have on the use of such software.

placed on simulating the construction process and how this can be achieved in atwo dimensional analysis. Modelling of the tunnel lining, the choice of anappropriate constitutive model for the soil and the selection of appropriatehydraulic boundary conditions are considered.

Chapter 3 considers the analysis of earth retaining structures. In particular theanalysis of gravity, embedded and reinforced/anchored walls are examined.Emphasis is placed on modelling the structural elements, choosing appropriateconstitutive models and simulating construction.

Cut slopes are considered in Chapter 4.. The concepts behind progressive failureare introduced. Its role in slope stability is then examined and in particular itsinteraction with the long term dissipation of excess pore water pressures.

The analysis ofembankments is discussed in Chapter 5. Embankments built ofearthfill and rockfill and those built on weak and strong foundations areconsidered. The choice of appropriate constitutive models is discussed at somelength as are the appropriate hydraulic boundary conditions and the role ofprogressive failure. For embankments on soft ground, single and multi-stagedconstruction and the benefits of reinforcement are examined.

Chapter 6 considers shallow foundations. To begin with, simple surfacefoundations are considered and comparisons with the classical bearing capacitysolutions made. The ability ofnumerical analysis to advance the current state oftheart is then demonstrated by considering SOme of the weaknesses in current bearingcapacity theory. For example, the effect ofselfweight on drained bearing capacity,the effect of foundation shape and its depth below the soil surface are considered.The effects ofanisotropic soil strength and of pre-Ioading on bearing capacity arealso examined. The analysis of tall towers and the difference between bearingcapacity failure and leaning instability is discussed. Analysis ofthe leaning Towerof Pisa is then used to demonstrate the power of numerical analysis.

Deep foundations are considered in Chapter 7. The analyses ofsingle piles andpile groups subjected to combined vertical, lateral and moment loading areconsidered. The behaviour ofsuction caissons and the possible detrimental effectsof neglecting anisotropic soil strength are discussed.

Benchmarking and validation of numerical analyses are discussed in Chapter8. The various options, their deficiencies and results from some recentbenchmarking exercises are described.

Chapter 9 describes many of the restrictions and pitfalls that the authors haveexperienced. In particular, restrictions implicit in modelling problems as planestrain, problems associated with initial conditions and pitfalls associated with theuse of some of the more common constitutive models are discussed.

Emphasis throughout this volume of the book is placed on explaining how thefinite element method should be applied and what are the restrictions and pitfalls.In particular, the choice ofsuitable constitutive models for the various geotechnicalboundary value problems is discussed at some length. To illustrate the materialpresented, examples from the authors experiences with practical geotechnicalproblems are used.

LondonMarch 2001

David M. PottsLidija Zdravkovic

AuthorshipThis volume has been edited and much of its content written by David Potts andLidija Zdravkovic. Several of the chapters involve contributions from colleaguesat Imperial College and the Geotechnical Consulting Group (GCG). In particular:Dr Trevor Addenbrooke (Imperial College) wrote a large part of Chapter 2

(Tunnels);

Mr Kelvin Higgins (GCG) wrote Chapter 8 (Benchmarking) and contributed toChapter 3 (Earth retaining structures);

Dr Nebojsa Kovacevic (GCG) wrote large parts of Chapter 4 (Cut slopes) andChapter 5 (Embankments).

AcknowledgementsWe would like to acknowledge our colleagues Professors John Burland and PeterVaughan and all the past and present research students at Imperial College, withoutwhose interest and involvement this book would not have been possible. Inparticular we would like to acknowledge Dr Dennis Ganendra whose PhD work onpile groups (supervised by David Potts) forms part of Chapter 7 (Deepfoundations).

David PottsLidija Zdravkovic

1. Obtaining geotechnical parameters

1.1 SynopsisOne of the essential ingredients for a successful finite element analysis of ageotechnical problem is an appropriate soil constitutive model. As explained inVolume I of this book, there is not a single constitutive model currently availablewhich can reproduce all aspects ofreal soil behaviour. It is therefore important torecognise in the analysis what aspects ofthe problem are of major interest and tochoose a model accordingly. However, to employ a particular soil model in ananalysis, appropriate laboratory and/or field tests are required from which to derivethe necessary model parameters. This chapter describes the standard laboratory andfield experiments used in geotechnical practice and the parameters that can beobtained from each of them.

1.2 IntroductionSince none of the currently available soil constitutive models can reproduce all ofthe aspects of rea! soil behaviour, it is necessary to decide which soil featuresgovern the behaviour of a particular geotechnical problem (e.g. stiffness,deformation, strength, dilation, anisotropy, etc.) and choose a constitutive modelthat can best capture these features. Another factor that governs the choice of soilmodels for finite element analysis is the availability of appropriate soil data fromwhich to derive the necessary model parameters. This often limits the use ofsophisticated soil models in practice, because their parameters cannot be readilyderived from standard laboratory or field tests.

The aim of this chapter is to give a brief description of standard and speciallaboratory and field tests and to review the soil parameters that can be derived fromeach of them. Due to restrictions on space it is not possible to give a detailedaccount of how the parameters are deduced frornthe raw laboratory and/or fielddata. For this information the reader is referred to the specialist texts on thissubject. It is recognised that a good site investigation should combine the strengthsof both laboratory and field testing.

2 / Finite element analysis in geotechnical engineering: Application Obtaining geotechnical parameters / 3

Figure 1.1: Schematic presentationof an oedometer apparatus

(I.l)

Confining ring

Sample

cr'a

Top cap

Porous stones

c = !1ec !1(1oger;)

To examine the swelling potential of a soil, the oedometric sample can beunloaded axially, which causes an increase in the void ratio (i.e. path 'ce' in Figure1.2). In the same e-loga,,' diagram this line is called a swelling line and again isusually assumed to be straight, with a gradient c., which is called the swellingindex.

1.3.2 Oedometer testA standard oedometer apparatus (seeFigure 1.1) consists of a circularmetal ring containing a soil samplewhich is usually 70mm in diameterand 20mm high. Porous discs areplaced at the top and bottom end ofthe sample, thus allowing freedrainage of the sample in the verticaldirection. Radial drainage is notpossible because the metal ring isimpermeable. The sample is loadedonly in the vertical direction, via thetop platen, by applying increments ofload until a desired stress level isreached. After each load increment isapplied sufficient time is usuallyallowed for full dissipation of any excess pore water pressures.

The complete stress state ofa sample in the oedometer apparatus is not known.The only known stress is the axial total stress, aa . Radial stress and pore waterpressure are not normally measured in the standard oedometer. Regarding strains,the axial strain ea is measured as the displacement of the top cap divided by thesample height, while the lateral deformation does not exist because of theconfinement imposed by the metal ring (i.e. e,=O). Therefore the total volumetricstrain of the sample equals the axial strain (i.e. ev=e). Due to these constraintssamples are said to undergo one dimensional compression.

Results from oedometer tests on clay soils are normally presented in the formofa void ratio-axial effective stress (i.e. e-logaa') diagram, see Figure 1.2. As notedabove, sufficient time is allowed after the application ofeach increment of load toensure full dissipation of any excess pore water pressures. Consequently, at thisstage a"'=a,,. With increasing axial effective stress the volume ofa sample (Le. itsvoid ratio) reduces and the loading curve from the initial sample conditions followsthe path' abcd', see Figure 1.2. This path is usually assumed to become straightonce the pre-consolidation pressure is exceeded (Le. aa'>(aa')b) and is called thevirgin compression line. The slope of this line is known as the compression indexCc, which is calculated as:

1.3 Laboratory tests1.3.1 IntroductionA laboratory investigation is a key feature ofalmost all geotechnical projects. Theactivities in the laboratory can be divided into several groups:

Soil profiling, which involves soil fabric studies, index tests (e.g. water content,grading, Atterberg limits), chemical ffind organic content, mineralogy, etc.;Element testing to characterise the mechanical behaviour of the soils (e.g.stress-strain properties, yield, strength, creep, permeability, etc.);Derivation of parameters for empirical design methods (e.g. CBR for roadpavements);Physical modelling (e.g. centrifuge tests, calibration chamber tests, shakingtable tests)For the purpose ofderiving parameters for constitutive models, element testing

is the most appropriate investigation. However, limited use is also made of theresults from soil profiling. The aim of these element tests is to measure, asaccurately as possible and in a controlled manner, the response of a soil elementto imposed changes in stresses, strains and/or pore pressures. This ability ofaccurate control and measurement of soil response is the main advantage oflaboratory over field testing. On the other hand, the main disadvantage oflaboratory element testing is the difficulty in obtaining undisturbed samples (i.e.samples with preserved initial fabric and state). However, this can be overcome,to some extent, by the use ofadvanced sampling techniques (e.g. LavaI samplers,Hight (1993). Difficulties, however, still remain ifsandy and weakly bonded soilsare to be sampled.

The essential requirement for element testing is to simulate field conditions asclosely as possible. This involves simulation of:

Initial stresses (e.g. state of consolidation);Imposed stress changes;Sequence and rate of changes;Field drainage conditions.

If a natural material is tested, experiments are normally performed on a set of soilsamples from different depths across a particular site, so that a full range of soilresponse parameters can be collected. In a similar manner, if reconstituted soil istested, a group ofsamples is normally prepared with an identical set-up procedureand tested at different initial stress levels so that again a full range of materialresponse parameters can be obtained.

The following paragraphs of this section will describe the majority of soillaboratory equipment currently available and the experiments that can beperformed in it.

4 I Finite element analysis in geotechnical engineering: Application Obtaining geotechnical parameters I 5

The slope of the swelling line, K, in the same diagram is calculated in a similarmanner. These two parameters, Aand K, are essential for critical state type models,such as modified Cam clay, bounding surface plasticity and bubble models (seeChapters 7 and 8 of Volume 1).

The difficulty is estimating the value of Ko which is not measured in thestandard oedometer test. When the soil is on the virgin compression line it istermed normally consolidated and it is often assumed that Ko=KoNC=(1-sinlp') (i.e.laky's formula), where lp' is the angle of shearing resistance. If lp' is known, thenKo and hence p' can be estimated. As K/c is approximately constant along thevirgin compression line, then ifthis line is straight in e-loga,,' space it will also bestraight in v-lnp' space. In fact, it can be easily shown that Cc=2.3A. However, ona swelling line Ko is not constant but increases as the sample is unloaded. Whilethere are some empirical equations describing how K"oC varies, there is nouniversally accepted expression. Consequently it is difficult to calculate p' on aswelling line. In addition, ifKooC varies and the swelling line is straight in e-logaa'space, then it will not be straight in v-lnp' space and vice versa. Clearly thedetermination of K from an oedometer test is therefore difficult and considerablejudgement is required.

log cr.'

Swelling line

loose

dense

The third parameter, which is needed to define the position of the virgincompression and swelling curves in a v-lnp' diagram, is the value of the specificvolume at unit mean effective stress or, for the MIT models, at p'=lOOkPa. Caremust be exercised here as usually this parameter is required for the virgincompression line obtained under conditions of isotropic compression (i.e. nodeviatoric stress). As noted above in the oedometer test the horizontal and verticalstresses are unlikely to be equal (Le. Ko '* 1), consequently the virgin consolidationline, although parallel to the equivalent line obtained under isotropic conditions,will be shifted towards the origin of the v-lnp'diagram. For most constitutivemodels it is possible to transform the values of the specific volume on theoedometer virgin consolidation line to the appropriate values on the isotropicvirgin consolidation line.

To complicate matters even further some of the more advanced constitutivemodels plot isotropic consolidation data iri Inv-Inp' space and assume that thevirgin consolidation and swelling lines are straight in this space, for example theAITabbaa and Wood model described in Section 8.9 of Volume 1 of this book.

The results from the oedometer test can also be used to estimate the coefficientof permeability in the vertical direction, kv' If, after each increment of load isapplied, the change in sample height is monitored with time this data can becombined with one dimensional consolidation theory to estimate kv (Head (1994.

For sands the behaviour in thee

oedometer test is more complex, as Nonnal compression linesthe initial density of the sampleaffects its behaviour. If two samplesof the same sand are placed in anoedometer, one in a loose and one ina dense state, their behaviour underincreased vertical loading might be asindicated in Figure 1.3. As thesamples have different initial densitiesthey will have different initial voidsratios. With increasing verticaleffective stress the samples willinitially follow different normal Figure 1.3: Typical result from an

oedometer test on sand soilcompression lines. At some pointthese normal compression lines will merge with a unique virgin compression line,see Figure 1.3. This will happen at a lower vertical effective stress the looser thesample. The magnitude of the vertical effective stress at which the virain

bcompression line is reached is much higher than for clay soils and is oftenconsiderably larger than the stress levels experienced in many practical situations.Ifunloaded at any time the sample will follow a swelling/reloading line. Behaviouris therefore characterised either as being on a normal compression line, on thevirgin compression line or on a swelling/reloading line. These lines are oftenassumed to be straight in either e-logaa', v-Inp' or Inv-lnp' space.

lnp'

(1.2)

le

Swellingline

v

d

log (1:

b Virgin consolidationline

C, /

A= ~VL1(lnp')

The pre-consolidation ea

pressure is usually associatedwith the maximum previousvertical effective stress that thesample has ever beensubjected to. Hence theoriginal overconsolidationratio of the sample, OCR, canbe calculated by dividing (au')bby the vertical effective stress Figure 1.2: Typical results from anexisting in the field at the .I t th 1 t k oedometer test on clay sOiloca IOn e samp e was a enfrom. It should be noted that for structured soils this approach may lead to anoverestimate ofOCR, as the process ofageing has the effect ofincreasing the stressassociated with point 'b'.

To be able to use the oedometer data in general stress space, its is necessary toexpress the gradients of the compression and swelling lines in terms of invariants.By knowing the ratio of the radial to axial effective stress in one dimensionalcompression (i.e. Ko = ar'lao'), it is possible to estimate the mean effective stressp' (=(a,,'+2Ko a,,')/3) and replot compression and swelling behaviour in terms ofspecific volume, v (= l+e), and mean effective stress, p'(Le. v-Inp' diagram). Thegradient of the virgin compression line is then usually denoted as A and iscalculated as:

6 I Finite element analysis in geotechnical engineering: Application Obtaining geotechnical parameters I 7

(1.7)

F.er,

Figure 1.5: Stresses actingon a triaxial sample

(1.6)O"~ = O"u - PI0"; = 0", - PI

If 0"(/ and 0", are increased together such that the radial strain ,=0, the samplewill deform in one dimensional compression, similar to oedometer conditions.The advantage over the oedometer test is that, by measuring pore waterpressure and radial stress, effective stresses can be calculated and thecoefficient of earth pressure at rest, KoNc, estimated as:

In a similar manner, if (7(/ and 0", are reduced together such that =0 thecoefficient of earth pressure in overconsolidation, Ko(x:, can be estim~ted fordifferent OCRs. The results ofsuch one dimensional compression and swellingtests can be plotted in a e-logO"a' diagram and parameters Cc and C., estimatedas in Equation (1. 1). The total volumetric strain, v, in both cases equals theaxial strain, a' It should be noted that these values can only be related to in-situ conditions if the OCR is known. An alternative procedure for estimatingthe value of Ko from undisturbed sample is discussed subsequently.If the loading frame is locked such that it is not in contact with the top platenof the sample and the sample is loaded only with an all-around cell pressure(i.e. 0"(/=0",), it is deforming under isotropic compression. If the all-around cellpressure is reduced, the sample is said to undergo isotropic swelling. The

Pore pressure can be measured in the endplatens adjacent to the bottom and/or top end ofthe sample, or by a probe placed atapproximately mid-height of the sample.

Originally axial strains were deduced frommeasurements ofaxial displacements made withdial gauges positioned outside the triaxial celland therefore remote from the sample. However, strain measurement has advancedin the last twenty years so that it is now possible to attach instruments on thesample itself. This avoids errors associated with the compliance of the testingsystem. Both axial and radial strains can be measured in this way.

Triaxial apparatus can be used to perform several different tests. Some of themore common tests are:

It is important to note that the force in theloading frame is not equivalent to the axialstress, but to the stress equal to (O"u-0",), which isknown as the deviator stress, q. The pore waterpressures can be measured in the triaxial sampleand therefore effective axial and radial stressescan be evaluated:

(1.5)

(104)

(1.3)

Back pressure

To loading frame

Figure 1.4: Schematicpresentation of a triaxial

apparatus

E = E(l- ,u)c (1-,u-,u2)

1.3.3 Triaxial testThe triaxial apparatus is the most widely usedpiece of laboratory equipment for soil testing. Itis described in detail by Bishop and Henkel(1962) in their standard text book on triaxialtesting of soil. A more advanced version of thetriaxial apparatus is the stress path cell describedby Bishop and Wesley (1975).

A conventional triaxial apparatus (Figure 1.4)incorporates a cylindrical soil sample which hasa diameter ofeither 38mm or 100mm. The largerdiameter apparatus is usually employed fortesting natural clays, because ofthe fissuring thatis often present in these materials. With a smallerdiameter sample this feature might be missed.The sample is enclosed in a thin rubbermembrane which is sealed at the top and bottomplatens by rubber a-ring seals. The membranegives flexibility to radial deformation of the Cell pressuresample. It also separates pore pressuresgenerated inside the sample from total radialstresses applied to the outside of the sample.

The sealed sample is placed on a pedestal ina water-filled cell. An all-around cell pressure,0"" applies radial total stress, 0"" to the vertical sides of the sample and a uniformvertical stress to the top rigid platen, see Figure 1.5. An additional axial force, F,,,is applied to the top platen via a loading frame. If the cross-sectional area of thesample is A, then the total axial stress on the sample is:

If a value for the Poisson's ratio J.J is assumed, it is possible to estimate theYoung's modulus E from Ec

If the soil is assumed to be isotropic elastic, this can be equated to the onedimensional modulus to give:

The data from the oedometer test can also be used to calculate the constrainedmodulus as:

8 1 Finite element analysis in geotechnical engineering: Application Obtaining geotechnical parameters 1 9

(1.10)

(1.12)

(1.11)

Figure 1. 7: Soil stiffness curve

where: AI is the incremental invariant deviatoric stress:

M = *~(O": - O"n2+ (O"~ - 0"))2 + (0": - 0"))2 = JJ (O"~ - 0";) = q /.J3!:l.Ed is the incremental invariant deviatoric strain:

Md =*~(I - 2)2 +(2 - )2 +(1 - )2 = 1J (a - er)

The triaxial test provides a complete E'history of the degradation of soil (E.)stiffness with increase in strain level, E_as shown in Figure 1.7. If theresolution of the local straininstrumentation is sufficiently high sothat it can detect the initial elasticbehaviour of a material, the stiffnesscurve will have an initial horizontalplateau associated with the maximumstiffness value. This small strainbehaviour is an essential part of constitutive modelling if soil deformations are ofinterest in the analysis, because soil stiffness is much higher in the early stages ofdeformation than it is at large strains. Ignoring this can result in the prediction ofpatterns of movement considerably different to those observed in the field, seeChapters 2 and 3.

If the small strain Young's modulus, E'maXl can be obtained, the other elasticparameter that can be calculated from a drained triaxial compression test isPoisson's ratio Par' for straining in the radial direction due to changes in axialstress, which is calculated as:

where: !:l.p' is the incrmental mean effective stress andt' = (!l0": +2!l0")) / 3 =(!l0"~ +2!l0";) / 3

The ability to measure the radial strain to a very high resolution is essential for thisparameter to be estimated.

However, instead ofusing parameters E' and P, soil elasticity can be expressedin terms of shear modulus G and bulk modulus K, see Section 5.5 of Volume 1.Both can be calculated from triaxial compression or extension tests, although Konly from a drained test:

and

p' (s')

(1.9)

( 1.8)

q(t)

&.

Figure 1.6: Schematic presentationof stress paths and stress-strain

curves from a triaxial test

, .-l( 0": - O"~)({l =sm0": + O"~

Another parameter of interest in geotechnical practice is the stiffness ofthe soil.From triaxial tests both drained, E', and undrained, E", Young's moduli can beestimated in both triaxial compression. and extension. From stress-stain curvesthese parameters can be calculated as either secant or tangent values, see Figure1.6:

results of such tests can be plotted in a v-lnp' diagram and used to obtain valuesfor the parameters Aand K, see Equation (1.2).After initial isotropic or anisotropic compression to a certain stress level,samples can be subjected to either drained or undrained shearing. In this casethe total radial stress is held constant, while the axial stress is either increasedor decreased. Ifaa>ar the sample is undergoing triaxial compression and in thiscase a,,=a l and ar=ae=a2=a). The parameter b (=(a2-a))/(a l -a))), whichaccounts for the effect ofthe intermediate principal stress, is therefore zero. Onthe other hand, if the sample is axially unloaded, such that aa


b.e" is the incremental volumetric strain and

where A, accounts for the change in pore water pressure due to the reduction indeviatoric stress which occurs during sampling in the field. A value of A,,=1/3corresponds to an isotropic elastic material, whereas a stiffclay (e.g. London clay)typically has a value of approximately 1/2; (I: is the original vertical effectivestress in the sample in-situ and can be estimated knowing the depth of the sample,the bulk unit weights of the overlying materials and the pore water pressure.

Accurate estimates of K" from this approach require that the sample remainsundrained at all times. On setting up the sample in the triaxial cell care must beexercised so that it does not come into contact with water. This can be difficult asit is usual practice to saturate the porous stones at the top and bottom ofthe sampleand consequently special testing techniques are required, see Burland and

6&. =6&1 +26&3 =6&" +2/).&rAgain, the complete degradation of shear and bulk stiffness from small to largestrains can be obtained from a triaxial test.

It is worth noting that it is not possible to obtain many anisotropic stiffnessparameters from a triaxial test. For example, if the parameters for the lineartransversely isotropic model were required, see Section 5.6 ofVolume I, then onlythe values of the axial Young's modulus, E"=b.(I,,'/b.e,, and Poisson's ratioP"r=b.e/b.Cc, can be obtained from the standard tests described above. If ~ynamicprobes such as bender elements, see Section 1.3.10, are mounted on the Sides andends ofthe sample it is possible to obtain estimates ofa further two parameters, butit is still not possible to obtain a value for the fifth parameter.

The angle of shearing resistance and stiffness parameters are essential fornearly all of the currently existing constitutive models.

Although not common, the triaxial cell can be used to obtain an estimate of in-situ values of K" in clay soils. Usually, when soil samples are taken during a siteinvestigation, they are immediately sealed to ensure no transfer of~ater ~rom orto the sample. This seal is only broken when a smaller test sample IS reqUIred fortesting. If it is assumed that during the whole sampling and storage process, fr~mextraction from the ground until placement in the triaxial cell, the sample remamsundrained then the sample will retain its mean effective stress, p/. The sample onplacemen; in the triaxial cell will have a zero mean total stress applied to it andconsequently the initial pore water suction will be equal to p/. If a cell pressu~e(i.e. (I) is then applied to the sample. until a positive pore water pressure ISrecorded, (note that most pore water pressure transducers used for triaxial testingcannot measure suction), the initialp/ can be calculated as the difference betweenthe applied cell pressure and measured pore water pressure. This can then be usedto determine the in-situ value of K" using Equation (I.l3) (see Burland andMaswoswe (1982)):

Figure 1.8: Stressescontrolled in a true triaxial

apparatus

1.3.4 True triaxial testIn contrast to a conventional triaxialapparatus, the true triaxial equipment tests acubical soil sample. The difference betweenthe two experiments is that in the true triaxialequipment all three principal stresses, (It, (12and (13 , can be applied and controlledindependently, see Figure 1.8, while in aconventional triaxial cell two principalstresses are always equal (Le. (I,=(le).

The advantage of this experiment is thatthe values ofthe angle ofshearing resistance,qJ', can be obtained for the full range of bbetween 0 and I, while conventional triaxialtests with a cylindrical sample can only haveb=O (triaxial compression) or b=l(triaxialextension). This is useful ifthe value of rp' isneeded for conditions in the ground otherthan triaxial (for example plane strain, forwhich O.2

12 I Finite element analysis in geotechnical engineering: Application Obtaining geotechnical parameters / 13

cr

(1.14)

"""--Original soilsurface

Disp~ntu

Y>y=u/h ; ,=vlh ; ex=0

Figure 1. 11: Simple shearconditions

't..,. ---------------------------- .H-IIII\IIII

conditions. The behaviour ofa typicalelement is indicated. The soil issubjected to a prescribed shear strainYxy and is constrained to have zerodirect strain in the x direction (ex=O). LAs a result of these prescribedboundary conditions, the soil issubjected to a shear stress i xy , a directstrain ey and a change in stress in thex direction (b.11x'''0). Becausehorizontal planes remain horizontaland are inextensible (ex=O), atultimate plastic failure these planes

~are a set of velocity characteristics(Davis (1968. Such planes areinclined at (45- v/2) to the directionof the major principal plastic strainincrement. In an isotropic elasto-plastic material the directions of theprincipal stresses and the principalplastic strain increments coincide.Consequently at failure, when alldeformation is plastic, any horizontalplane is a velocity characteristic and Figure 1. 12: Mohr's circle of stressis inclined at (45-v/2) to the at ultimate conditionsdirection ofthe major principal stress111 and hence is inclined at (45+v/2) to the plane on which crI acts. The Mohrcircle ofstress for this state is shown in Figure 1.12. Point P is the pole ofthe circleand point H represents the stresses (ixy)r and 11y' which act on the horizontal plane.From the geometry of the circle it can be shown that:

(Z'xy) _ sinq)' cos va; r - 1- sin lp' sinv

which reduces to (ixjl1y')r=sinq/ when v=O and (rxjl1y')r=tancp' when v=cp'. It shouldbe noted that (ixjo'y')r=sincp' is the interpretationadvocated by Hill (1950), whereas(rxjl1y')r=tancp' is the more common interpretation. In reality, due to the non-uniformities inherent in the direct shear box test, it is unlikely thatthe condition ofidealised simple shear exists throughout much ofthe sample.

The usual presentation of direct shear results is in terms of a shear stress-horizontal displacement diagram (Le. i-d), see Figure 1.13. The shear stressnormally increases up to a peak value during the early stages of horizontaldisplacement, and then gradually decreases until reaching a residual value after alarge horizontal displacement of the shear box. From this test both peak andresidual angles ofshearing resistance can be evaluated, although to obtain the latter

1.3.5 Direct shear testThe direct shear test is performed inthe shear box apparatus,schematically illustrated in Figure 1.9.The sample is typically 60x60mm inplane and about 25mm high. It isplaced between rigid and rough topand bottom platens, in a square boxwhich is split horizontally. Verticalload is applied via the top cap and isnormally kept constant during the Shear planetest. A horizontal force is applied tothe top half of the box with the Figure 1.9: Schematic presentationbottom half fixed, thus forcing the

of a direct shear box apparatustwo halves of the box to movehorizontally relative to each other.Rough top and bottom platens transfershear stress into the soil. The mainpurpose of the test is to examine soil l'strength. It is unsuitable for ....--'---measuring soil stiffness because ofthe non-uniformities imposed in thes'ample by the loading arrangement.

The only known stresses on thesample are the average vertical stressand the average shear stress, seeFigure 1.10. The application of a Figure 1. 10: Stresses acting on ashear stress causes the major principal sample in a direct shear boxstress, 111, to rotate continually awayfrom the vertical, but there is no means of measuring this rotation. Because thesample is confined in a box, it is subjected to significant stress and strain non-uniformities during shearing.

Interpretation ofthe direct shear box is based on assuming that the deformationof the sample conforms to that of ideal simple shear, see Potts et al. (1987). Suchan idealised condition is indicated in Figure 1.11 which shows the top boundaryof a layer of soil displaced parallel to the bottom boundary, under plane strain

are installed in the sample it is also possible to obtain an estimate for the fifthparameter.

While clearly an advantage over the conventional triaxial equipment, it is noteasy to obtain and set up the samples. Consequently true triaxial apparatus arerarely used for commercial testing. Only a limited number of apparatus exist andthese are mainly located in universities.


Displacement

b) Cambridge simple shear device

~cr, Top platen

Wife Pore water Bottom platenpressure

a) NOI simple shear device

1.3.7 Ring shear testThe ring shear apparatus is another device suitable only for measuring soilstrength. The principle of operation is similar to the direct shear box apparatus,except that the sample has the shape ofa squat hollow cylinder and the top halfofthe box is rotated with respect to the bottom. The sample, about IOOmm in outerdiameter,. a~out 20mm thick and of 15mm wall thickness (see Figure l.l5), isplaced wlthm rough top and bottom platens in a horizontally split metal ring.~on~tant vertica~ load is applied via the top platen, while the bottom part of thermg IS rotated wIth respect to the top part, thus applying a horizontal shear stressand creating a horizontal slip surface in the sample. No other stresses or strains canbe measured in the standard ring shear apparatus and consequently interpretationof the results is difficult.

The results from a ring shear test are normally plotted in terms of shear stressversus displacement (i.e. r-cl) diagram, as for the direct shear box, see Figure 1.13.

Figure 1.14: Schematic presentation of a) NGIand b} Cambridge simple shear devices

The Cambridge University device is schematically presented in Figure 1.14b.The sample here is a rectangular prism, lOO by lOOmm in plan and about 20mmthick. It is placed within rough rigid top and bottom platens which allow verticaldeformation and rotation of the side platens. From the measurements of theseexternal deformations it is possible to estimate the average strains within thesample. As the side platens are designed to allow vertical movement (i.e. they aresmooth) they do not provide complementary shear stresses on the verticalboundaries of the sample. Consequently the stress and strain state is not uniformwithin the sample. Usually normal and shear stress transducers are placed in theplatens and stresses are measured over the middle third ofthe sample. As it is moredifficult to obtain, test and instrument these prismatic samples only a few piecesof equipment of this type exist.

Both types of simple shear apparatus are mainly used for measuring soilstrength (i.e. undrained strength or angle of shearing resistance). Due to the moreuniform behaviour of the sample in the Cambridge apparatus, it can also be usedto obtain pre-failure stress strain data.

Figure 1. 13: Typical shear stress-displacement diagram from direct

shear box test

several reversals of loading may benecessary. Residual strength can bevery important in some boundary valueproblems involving high plasticityclays, see Chapter 4. Because clayparticles are elongated and, in theirnatural condition, oriented in somestructure, after large deformations theycan become aligned with the directionof shearing. Such a situation couldoccur, for example, during a landslide.Low plasticity clays and granular materials have a less pronounced drop fromcritical state to residual strength.

The direct shear box apparatus is also used for interface shear testing. In thissituation soil is sheared against some other material (e.g. concrete, steel, etc.) inorder to examine the angle offriction, 6, atthe interface of the two materials. Thisparameter is used, for example, in pile or retaining wall analysis, where it isnecessary to know the maximum value of the angle offriction between the pile orwall material (concrete or steel) and soil (Jardine and Chow (1996), Day and Potts(1998)). For such investigations the interface between the soil and the materialagainst which it is to be sheared is positioned at the split between the upper andlower halves of the shear box.

1.3.6 Simple shear testThe simple shear apparatus works on a similar principle to the direct shear box inthat the top platen is moved horizontally with respect to the bottom platen.However, the apparatus is designed so that the sample is allowed to deform moreuniformly. There are currently two designs of simple shear apparatus: onedeveloped at Cambridge University and the other at the Norwegian OeotechnicalInstitute (NGI) in Oslo. The NGI device is the more common design and can befound in many laboratories, both academic and commercial, around the world.

The NOI device is schematically drawn in Figure l.l4a. It has a cylindricalsample, 80mm in diameter and about IOmm thick, contained within a wire-reinforced rubber membrane. This membrane maintains a constant samplecircumference, but allows uniform vertical deformation and rotation ofthe verticalsides of the sample. The whole sample is placed between top and b()ttom platens.Axial and shear forces are applied via the top platen. Usually the axial force is keptconstant, while the shear force increases during the test, similar to the direct shearbox test. The vertical and shear strains ofa sample can be measured by observingthe vertical and horizontal displacement of the top platen. Although the porepressure can be measured, the full stress and strain states of the sample are notknown in the NOI device and again the results are interpreted assuming idealsimple shear conditions, as discussed above in Section 1.3.5.


Element principalstresses:

Element componentstresses:

samples in the other laboratory Hollow cylinder coordinates:devices. These loads are the innercell pressure, Pi , the outer cellpressure, Po , the vertical load, W,and the torque, Mr This rcombination of loads allowscontroloffourcomponentstresses,namely the normal stresses az , a,and all and the torsional shearstress TZ11 (note the remaining shearstresses T" and T,II are zero). Thisthen permits control over themagnitudes and directions a oftheprincipal stresses. Because all three Element componentmajor stresses can be controlled, it strains:is also possible to independentlycontrol the parameter b. So, with awell designed hollow cylinderapparatus it is possible to performa set of shear tests with a constantvalue of b while changing a, andvice versa, with a constant value of Figure 1. 16: Stresses and strains in aa while changing b, thus enabling ho/Iow cylinder apparatusan independent evaluation of theeffects of each of these two parameters on soil strength, stiffness, yielding, etc.

All component strains can be measured locally in a hollow cylinder device: thethree normal strains ez , e, and ell and the torsional shear strain '/zll' so that the strainstate in the sample is completely known (note '/"='/'11=0).

Because of the flexibility of a hollow cylinder apparatus, a variety of stresspaths can be applied to soil samples. However, for design purposes its use isusually most valuable for investigating initial soil anisotropy, where samples areinitially Ko consolidated to a certain stress level (thus simulating initial green-fieldconditions in the ground after the sedimentation/erosion process) and then shearedwith different orientations ofthe major principal stress (i.e. a set ofsamples whereeach sample is sheared with a different a value in the range from 0 to 90). Theresults from these tests are normally plotted in terms of stress-strain (J-Ed) andstress path (J-p') diagrams, the latter being a two dimensional projections of thethree dimensional J-p'-a stress space, see Figure 1.17. This figure shows resultsfrom a typical series of hollow cylinder tests performed on a granular soil (seeZdravkovic and Jardine (2000)). In particular, the results show how the peakdeviator stress reduces with an increase ofa. Typical results derived from such aset ofexperiments gives the variation ofboth drained, (fJp', and undrained, SI/, soilstrength with a, shown in Figure 1.18. These results clearly indicate a degree ofsoil anisotropy.

Figure 1.15: Schematicpresentation of a ring shear

apparatus

1.3.8 Hollow cylinder testFrom the previous discussion it can be c.oncluded tha~ in !he ~edomete~, triaxialand true triaxial apparatus the principal stress and stram directIOns applied to !hesample are parallel with the sample boundaries. It is only possibl~ fo~ the.relatlVemagnitudes of the principal stresses and strains to change. Their dlre~tlons arefixed. In contrast the direct, simple and ring shear apparatus allow rotatIOn of theprincipal stress and strain directions, but because not all of~he stress comp~n:ntscan be measured it is not possible to measure the magmtude of the pnnclpalstresses, or more importantly control their directions. The only exception migh~ bea highly instrumented Cambridge type simple shear apparatus, where suffiCientload cells are used to enable sufficient stress components to be measured tocalculate the principal effective stresses, however, it is still not possible to controlor alter the direction of the principal stresses, as this is implicitly set by the mannerin which the apparatus works. .

The hollow cylinder apparatus does not suffer from the above shortcommgs,as it allows full control ofboth the magnitude and the rotation ofprincipal stresses.It is therefore extremely suitable for investigating soil anisotropy. The sample isa hollow cylinder (see Figure 1.16) which can have various dimensions. Forexample, the large and small hollow cylinder apparatus in the Imp~rial Coll~gelaboratory test samples with the following dimensions: the large ~ne .IS 25cm h~gh,25cm outer diameter and 2.5cm wall thickness; the smaller deVice IS 19cm high,10cm outer diameter and l.5cm wall thickness.

Four independent loads can be applied to a hollow cylindrical sample (seeFigure 1.16), as opposed to only the one, two or three that can be applied to

The shear stress can experience a peakvalue (i.e. associated with q>P'), after whichthe soil strength drops and reaches aresidual value (i.e. (fJ,.,'). Again strengthparameters are determined assumingbehaviour is similar to ideal simple shear.The difference between this test and shearbox test is that much larger displacementscan be achieved in a ring shear device (e.g.several metres) and therefore a betterestimate of the residual strength can beobtained. Another advantage of the ringshear apparatus is that it can be used toexamine the effects of very high rates ofshearing on residual strength, which isuseful when investigating soil behaviourunder earthquake loading.

The ring shear apparatus can also be used for interface testing.

72 I Finite element analysis in geotechnical engineering: Application Tunnels I 73

~.3L-_.l-_..L-_..L-_--l-_--Jo 2 4 6 8 10

Pillar depth (number of diameters)

\ lengthening Piggy. back data:Kimmance et al. (1996)

\. vertical diameter _ horizontal diameter

H~:~;~~:;:;:=-...~.2

alternative will alter the long term pore water pressure regime, for the samehydraulic boundary conditions. This will alter the ground response duringconsolidation and sweIling.

5. The intermediate and long term behaviour is governed by many factors. Inparticular, whether the tunnel acts as a drain or is impermeable, and whether theinitial pore water pressure profile is close to hydrostatic or not. It is importantto be aware of the dependencies, and so to view any prediction of intermediateand long term behaviour with a critical eye.

6. It is important to select constitutive models capable of reproducing fieldbehaviour. For example, in a situation where pre-yield behaviour dominates theground response, it is essential to model the nonlinear elasticity at smaIl strains.

7. Devices for improving settlement predictions can be developed. Thesequestions must be asked: What is the influence of this adjustment on the soilbehaviour? What are the knock on effects? For example, if one is adopting adevice to match a surface settlement profile, how does this alter any predictionofsub-surface movement, the pore pressure response, or the lining stresses anddeformations.

8 The finite element method can be used to quickly assess the impact of differentinfluences on tunnelling-induced ground movements. Parametric studies canprove extremely useful in the development of design charts and interactiondiagrams.

9. One of the great benefits ofnumerical analysis to the tunnel engineer is that ananalysis can incorporate adjacent influences. For example existing surfacestructures, or existing tunnels. It is also possible to reproduce the effects ofcompensation grouting to protect surface structures during tunnelling projects.This chapter has demonstrated the power of the finite element method in thisrespect.

Lower tunnel distortion due to passage ofupper:o vertical diameter horizontal diameter

Upper tunnel distortion due to passage of lower:Q vertical diameter horizontal diameter

shortening

0.3

excavation inducing a lengtheningof the vertical diameter andshortening of the horizontaldiameter. It is clear that the closerthe two tunnels, the greater theinduced distortion. For all theanalyses, the lengthening of thevertical diameter is greater thanthe associated shortening of thehorizontal diameter, and theinduced distortions are never assevere as those for side-by-sidetunnels at the same pillardimension (the curves for side- F.' 2 43 D + t' f 1st t IIgure . : etorma IOn 0 unneby-side tunnels are reproduced . nd fi . 2 42) S 'bl In response to 2 tunnel for piggy backrom Figure . . ensl e t

. h . I geome ryextrapolatIOns of t e numencapredictions for the distortion to the upper tunnel on passage ofthe lower tunnel arein good agreement with the field data from Kimmance et al. (1996). Notsurprisingly this construction sequence causes more distortion than the alternativeof the second tunnel passing above the existing tunnel. Indeed, it is clear that thereis very little influence when the second tunnel passes above the existing tunnel, andfor piIlar depths greater than 3 diameters the influence is effectively zero.

The resu Its presented here demonstrate that the relative position (above, below,or to the side) and the physical spacing ofthe tunnels has a significant effect on thetunnel lining response. The results presented quantify the expected distortioninduced in an existing tunnel lining when an adjacent tunnel is excavated.Addenbrooke and Potts (200 I) draw further conclusions with respect to the surfacesettlement profiles expected above twin tunnel projects, and the influence of therest period between construction of the two tunnels.

2.8 SummaryI. Tunnel construction is a three dimensional engineering process. Ifrestricted to

two dimensional analysis, then one must consider either plane strain or axiaIlysymmetric representations, depending on what the analysis aims to achieve.

2. Methods of simulating tunnel construction in plane strain require at least oneassumption: the volume loss to be expected; the percentage of load removalprior to lining construction; or the actual displacement ofthe tunnel boundary.

3. If severe distortions of the tunnel lining are expected (i.e. in response to thepassage of an adjacent tunnel in close proximity) then a model can be usedwhich aIlows segmental linings to open or rotate at their joints, or aIlowssprayed concrete linings to crack.

4. The recognition of permeability's dependence on stress level has led tononlinear models for permeability. Using such a model in place of a linear

Earth retaining structures / 75

Figure 3. 1: Types of retainingstructure (cont... )

Counterfort

Reverse cantilever

Gravity

Cantilever

3.3 Types of retaining structure3.3.1 IntroductionThe complexity and uncertainty involved in design and analysis increase with thedegree of soil-structure interaction and thus depend on the type of retainingstructure to be employed. It is therefore appropriate to categorise the types ofretaining structure on the basis of the soil structure interaction problems that arisein design. In Figure 3.1 the main walltypes are shown in order of increasingcomplexity of soil-structureinteraction.

statically indeterminate nature ofmultiple propped (or anchored) walls, these haveoften been dealt with using empirical approaches such as those suggested by Peck(1969). Simplified methods ofanalysis are also available for reinforced/anchoredearth, for example see BS 8006 (1995).

Because all ofthese traditional design methods are based on simplified analysisor empirical rules, they cannot, and do not, provide the engineer with all thedesired design information. In particular, they often only provide very limitedindications of soil movements and no information on the interaction with adjacentstructures.

The introduction of inexpensive, but sophisticated, computer hardware andnumerical software has resulted in considerable advances in the analysis and designof retaining structures over the past ten years. Much progress has been made inattempting to model the behaviour of retaining structures in service and toinvestigate the mechanisms of soil-structure interaction. This chapter will reviewsome of these advances and discuss some of the important issues that must beaddressed when performing numerical analysis of earth retaining structures. Itbegins by describing the different types ofretaining structure and the general issueswhich must be considered before starting an analysis. It then goes on to considerthe three main categories of retaining structures (i.e. gravity, embedded andreinforced/anchored earth walls) in more detail.

3.3.2 Gravity wallsGravity, counterfort and cantileverrwalls are stiffstructures for which thesoil-structure interaction is relativelysimple. For overall stability, the earthforces on the back of a wall have tobe balanced by normal and shearstresses at its base. The magnitudes ofthese resisting forces are, to a largeextent, controlled by the weight ofthe

3.1 SynopsisThis chapter discusses the analysis of earth retaining structures of various forms.It draws heavily on issues that have to be considered when analysing realstructures. After a brief description of the main types of retaining structures incurrent use, general considerations, such as choice of constitutive model,construction method, ground water control and support systems, are discussed.Attention is then focussed on the specific analysis of gravity, embedded andreinforced/anchored retaining walls.

3. Earth retaining structures

3.2 IntroductionThe purpose of an earth retaining structure is, generally, to withstand the lateralforces exerted by a vertical or near vertical surface in natural ground or fill. Thestructural system usually includes a wall, which may be supported by otherstructural members such as props, floor slabs, ground anchors or reinforcing strips.Alternatively, or additionally, the wall may be supported by ground at its base orinto which it penetrates. In most situations the soil provides both the activating andresisting forces, with the wall and its structural support providing a transfermechanism.

The design engineer must assess the forces imposed on the wall and otherstructural members, and the likely movements of both the structure and retainedmaterial. Usually these have to be determined under working and ultimate loadconditions, see Chapter 1 of Volume 1 of this book. In addition, estimates of themagnitude and extent ofpotential groundmovements arising from construction ofthe structure, both in the short term and in the long term as drainage within theground occurs, are required. This may be because of the effect construction mayhave on existing, or planned, services or structures (buildings, tunnels, foundations,etc.) in close proximity. Potential damage could occur which has to be assessed andmethods of construction considered which minimise these effects.

Design of retaining walls has traditionally been carried out using simplifiedmethods ofanalysis (e.g. limit equilibrium, stress fields) or empirical approaches.Simplified methods have been developed for free standing gravity walls, embeddedcantilever walls, or embedded walls with a single prop or anchor. Some of theseare described in BS 8002 (1994) and Padfield and Mair (1984). Because of the

76 / Finite element analysis in geotechnical engineering: Application Earth retaining structures / 77

//A" IIIIIIII

3.4.2 SymmetryIn reality all geotechnical problems involving retaining structures are threedimensional and, ideally, three dimensional analyses, fully representing thestructure's geometry, loading conditions and variations in ground conditions acrossthe site, should be undertaken. With current computer hardware this is not apractical proposition for all, but a number of very limited and extremely simplecases. To analyse any structure it is therefore necessary to make a number ofsimplifying assumptions. Most commonly two dimensional plane strain or axi-symmetric analyses are undertaken, see Chapter I of Volume I of this book.

For two dimensional and axi-symmetricanalysis the assumption is frequently madethat there is an axis of symmetry about thecentre line of an excavation and that only a -...,.".,~,,-....'half section' needs to be modelled. In thecase of a three dimensional analysis twoplanes of symmetry are often assumed and a'quarter section' is considered. For example,Figure 3.2 shows a plane strain excavationsupported by two parallel walls propped nearto the ground surface. If there is symmetryabout the vertical line passing through thecentre of the excavation, it is only necessary Figure 3. 2: ~xamp/e ofaxi-to analyse halfthe problem, either that halfof symmetflc geometrythe problem to the right of the plane of symmetry or that half to the left. Thisclearly reduces the size of the problem and the number of finite elements neededto represent it. However, for such an analysis to be truly representative there mustbe complete symmetry about the centre line of the excavation. This symmetryincludes geometry, construction sequence, soil properties and ground conditions.[n practice it is rarely the case that all of these have symmetry and thereforeanalyses using a 'half section' usually imply further approximations.

3.4 General considerations3.4.1 IntroductionBefore starting any numerical analysis it is important to address a number of issuesto ensure that the most appropriate methods ofmodelling the soil and structure areused. It is also important that the correct boundary conditions (e.g. displacements,pore water pressures, loads, etc.) are applied in any analysis. This section brieflyoutlines some of the more important issues. A number of these are discussed ingreater detail in subsequent sections.

increases with the number of levels of props and/or anchors and hence thestructural redundancy.

Multi-propped

Soil nails

Multi-anchored

Propped cantilever

Figure 3.1: (... cont) Types ofretaining structure

mg

structure, hence the term gravity wall.Such walls can be formed of massconcrete, reinforced concrete, or pre-cast units such as crib walls orgabions. Movement arising fromdeformation of the ground beneaththe wall is often negligibly small, but~exceptions occur if the wall and Reinforced earthbackfill are constructed on a deeplayer of compressible soil.

3.3.3 Reinforced/anchoredearth walls

Embedded cantileverReinforced earth walls and wallsinvolving soil nails essentially act as~.large gravity walls. However, theirinternal stability relies on a complexinteraction between the soil and thereinforcing elements (or nails). These Propped at excavationwalls generally have non-structural levelmembrane facing units which areintended to prevent erosion, or arepurely aesthetic. The membrane hasto be designed to resist any bendingmoments and forces that occur, butthis is not the primary method ofearthretention. The resistance to ground Anchored cantilevermovement is provided by soil nails,anchors and ties ofvarious forms (notground anchors) or strips of metal orgeo-fabric.

3.3.4 Embedded wallsWhen soil movements are important and/or construction space is limited,embedded cantilever walls may be used, with or without props or anchors. Sheetpile walls, diaphragm walls, contiguous bored pile walls and secant pile walls areall examples of this type of wall. To maintain stability the walls rely on theresistance of the ground below excavation level and on the resistance forcesprovided by any props or anchors. The flexibility of embedded walls vary withina wide range and this has considerable effect on the distribution ofearth pressures.The more flexible walls often have smaller bending moments in the structuralelements, but may lead to larger deformations, particularly for embeddedcantilevers with no props or anchors. The complexity of soil-structure interaction

78 I Finite element analysis in geotechnical engineering: Application Earth retaining structures I 79

E "'" 28.106 kPaEl= 2.310' kNm1/m

~ =O.IS

wall properties'

lOOm

,l //~.

SQil properties'Z

E' = SOOO + Sooo z (kPa)Kt = looOK..., (kPa)K.=2YlaI =20 kN/ml

~'=2S"v= 2S"c= 0

~'=O.2

ProD

20m

Figure 3.6: Typical finite elementmesh for excavation problem

3.4.3 Geometry of the finite element modelAnother decision that arises whenperforming a numerical analysis is the rtchoice of the depth and lateral extentof the finite element mesh. Forexample, when considering the simpleexcavation problem shown in Figure3.5, a decision must be made as to thedepth of the mesh below the groundsurface and the lateral extent of theright hand boundary of the mesh. As,in this case, the excavation issymmetric, the position of the lefthand boundary of the mesh is fixed. Figure 3.5: Propped retaining wall

When considering the bottomboundary ofthe mesh, the soil stratigraphy often provides an obvious location. Forexample, the occurrence of a very stiff and strong layer (e.g. rock) at depthprovides an ideal location for the bottom boundary ofthe mesh. Also, because soilstrength and stiffness usually increase with depth, analyses are not so sensitive tothe location of the bottom boundary as long as it is not unreasonably close to thebottom of the wall. The location of the far field vertical (i.e. the right hand)boundary is more problematic and will, in part, depend on the constitutive modelemployed to represent soil behaviour.

To illustrate these points analyseshave been performed for theexcavation problem shown in Figure3.5. A typical mesh is shown inFigure 3.6. Also shown are thedisplacement boundary conditionsimposed on the bottom and verticalboundaries of the mesh. Both the soiland the wall are modelled using eight ~noded isoparametric elements. Widermeshes were obtained by addingelements to the right side of the meshshown in Figure 3.6. Shallowermeshes were obtained by deletingelements from the bottom of themesh. In the vicinity ofthe excavationall meshes were similar.

Obviously, no analysis can model geometry in detail, or for that matter everyconstruction activity, but this example illustrates the care that must be taken whenmaking simplifying assumptions of any form.

"R"wall,sccantpile

Roof slab

"L"waJl,secnntpile

As an example consider thesituation shown in Figure 3.3.This shows the cross section of a

fi -=:S=i;;~31~.5m~;f_ii.'f=::ZJ:i' ];4mOOiJODroad tunnel orming part of a road lO.6~improvement scheme. The tunnelis formed of two secant pileretaining walls, a roof slab whichis connected to the outer walls bya full moment connection, a baseslab which is not connected to thewalls and a central wall that alsoprovides support to the roof slab. Figure 3.3: Cross-section of a roadThe central piles support the tunnelcentral wall, but neither areconnected to the base slab.

The road alignment at the location of the tunnel is on a bend and consequentlythe tunnel roofand base slab are inclined to the horizontal. However, the fall acrossthe structure does not appear great, less than lm across the excavation which is

b~tween 25 to 26m wide. Consequently, it is tempting to simplify the analysis byignoring the fall across the structure and assuming symmetry about its centre. Onlya 'half section' would then need to be considered in any analysis.

To investigate whether such anapproximation is reasonable, a finiteelement analysis was undertakenconsidering the full cross section of Lh ~t e tunnel as shown in Figure 3.3. L_

Figure 3.4 shows the predicted I1--,displaced shape of the two retaining ~ _:walls at an exaggerated scale (solid ~_'

I Iprofile; the undeformed profile of the I I1--:walls, as represented in the finite ~::element mesh, is shown as a dotted I It::line). It is apparent that there is some I Isway of the tunnel structure and wall L-:

I Idisplacements are not the same on "L_,either side of the excavation. The top :_..!of the right hand wall of the tunnel ispushed back into the soil behind it.. .

A I f 'h If t' ,FIgure 3.4: DIsplaced shape of wallsn ana YSIS 0 a a sec Ionwould not have predicted this of road tunnelasymmetrical behaviour. It would have predicted that both walls movesymmetrically inwards towards the centre ofthe excavation. Because the displacedshape of one, or both, walls was not correct, the predicted bending moments andforces in the walls would also have been incorrectly predicted.

80 / Finite element analysis in geotechnical engineering: Application Earth retaining structures / 81

x. coordinate from the wall to the far side of the mesh (m)a) Linear elastic-plastic model

340

140lOO

260 300

260

220

180 220

180'40

140loo

100

60

60

x - coordinate from the wall to the far side of the mesh (m)a) Linear elastic-plastic model

x .. coordinate from the wall to the far side of the mesh (m)b) Small strain stiffuess - plastic model

IFar boundaty of mesh:-

Finite Element Analysis in GeotechnicalEngineering Application by David M. Potts and Lidija Zdravkov

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