CAVITY EXPANSION METHODS IN GEOMECHANICS …978-94-015-9596-4/1.pdf · Cavity Expansion Methods in Geomechanics by Hai-Sui Yu School ofCivil Engineering, University ofNottingham,

CAVITY EXPANSION METHODS IN GEOMECHANICS

Cavity Expansion Methods in Geomechanics

by

Hai-Sui Yu School ofCivil Engineering,

University ofNottingham, U. K

Springer-Science+Business Media, B.Y.

Library of Congress Cataloging-in-Publication Data

ISBN 978-90-481-4023-7 ISBN 978-94-015-9596-4 (eBook) DOI 10.1007/978-94-015-9596-4

Cover illustration: Circular opening with an internal support pressure

Printed on acid-free paper

All Rights Reserved © 2000 Hai-Sui Yu

Originally published by Springer Science+Business Media Dordrecht in 2000.

Softcover reprint ofthe hardcover Ist edition 2000 No part of the material protected by this copyright notice may be reproduced

or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and

retrieval system, without written permission from the copyright owner.

This book is dedicated to my late parents

for many sacrifices they made in supporting my education

FOREWORD

by J ames K. Mitchell

University Distinguished Professor Emeritus, Virginia Tech

Analysis of the expansion of cylindrical and spherical cavities in soil and rock pro­vides a surprisingly versatile and accurate geomechanics approach for study of im­portant problems in geotechnical engineering. Among them are the axial and lateral capacity of deep foundations, interpretation of pressuremeter and cone penetration tests for determination of soil state and properties in-situ, and analysis of stability and deformations associated with excavation and tunnelling. A realisation in the latter part of the Twentieth Century that improved understanding and quantification of many problems in soil and rock mechanics could be obtained by application of cavity expansion theory marks a milestone in the development of our field.

In this very thorough and comprehensive treatment of the subject, Professor Yu has provided a valuable resource for students, researchers, and engineering practi­tioners alike. A full range of assumptions and solution procedures has been ex­amined. In the first part of the book analytical solutions for stresses and deforma­tions are developed using elastic theory, theory of elastic-perfectly plastic behaviour, critical state theory, and for the cases of strain hardening and strain soft­ening materials. Numerical analysis using the finite element method is also de­scribed. The assumptions are c1early stated, and the details of the derivations are presented for all of the cases analysed. Limitations in the results are identified. Like all theories in geomechanics, idealisations of geometry, soil properties and behav­iour are necessary; nonetheless, the results using cavity expansion theory refIect field behaviour as weIl or better than many other soil and rock mechanics theories in common use in geotechnical practice.

For those readers interested primarily in the results of the derivations and their applications, Professor Yu has inc1uded a summary at the end of each chapter in which what has been done is listed, the key equations are identified, and where the solutions are used in the geotechnical applications presented in the second part of the book is stated. A very comprehensive list of references makes it possible to trace all the information to its source.

The second part of the book, Geotechnical Applications, contains numerous il­lustrations ofhow the theoretical solutions can be applied to real problems in in-situ testing, foundation engineering, and underground construction, as weIl as many comparisons between predicted and measured behaviour. The ability to treat the re­sults of cone penetration and pressuremeter tests in sand and c1ay on a realistic theo-

VB

viii Foreword

retical basis enhances their value for site characterisation and determination of rel­evant soil mechanical properties.

I am unaware of any other treatment of cavity expansion analysis that is so com­plete and definitive as in this book by Professor Yu.

J.K.M. April 28, 2000

PREFACE

Cavity expansion theory is concerned with the stress and displacement fields around cavities embedded in either linear or nonlinear media. The procedures that make use of cavity expansion theory in solving practical problems are termed cavi­ty expansion methods. No mathematical theories can completely describe the com­plex world around us, and therefore each theory is only aimed at a certain type of problem, describes its essential features, and ignores what is of minor importance. As a result, the theory will be invalid or inaccurate when a neglected influence be­comes important. Over the last few decades, cavity expansion theory has found many applications in the analysis and design of a variety of geotechnical problems. This book arose from my belief that there is a need for the geotechnical community to have a unified presentation of cavity expansion theory and its applications in geo­mechanics. Accordingly, the book attempts to summarise and present the major de­velopments in the field of cavity expansion theory and its geotechnical applica­tions. Much of this research monograph is based on work carried out over the last two decades.

The book is intended primarily as a reference book for civil, mining and petro­leum engineers who are interested in cavity expansion methods and their applica­tions. As cavity expansion problems have long been used as a classic example in the teaching of elasticity and plasticity theories, the solutions presented in the book will also be of interest to students and researchers in the fields of applied mechanics and mechanical engineering.

As indicated by its table of contents, the book is divided into two parts. The first part, Chapters 2 to 7, presents fundamental solutions for the expansion and contrac­tion of cavities in soil and rock. Whilst Chapters 2 to 6 cover some ofthe key ana­lytical solutions for cavity expansion in geomaterials modelled by elastic, elasto­plastic and viscoelastic/viscoplastic theories, Chapter 7 also provides abrief summary of the basic numerical formulations for finite element analysis of cavity expansion problems. To facilitate the use and application of cavity expansion theory, a finite element program, CAVEXP, will be made available for readers of this book.

The second part of the book, containing Chapters 8 to 11, summarises the major applications of cavity expansion methods in soil and rock mechanics. While new areas of application may continue to emerge, the well-established areas of applica­tion are in the fields of pile foundations and earth anchors, tunnels and underground excavations, in-situ soil testing, and wellbore instability. All these applications

IX

x Preface

have been covered in some detail. Due to space limitations, it is not possible to in­clude every single application that has been published in the literature. Instead the aim of the application section is to present typical examples to show how cavity expansion solutions can be used to provide simple and useful frameworks for the analysis and design of complex geotechnical problems.

The preparation of the book reflects many years of study and research. In this process, I have benefited much from discussions and collaborations with many col­leagues. Those concemed will know that I appreciate their help and assistance with considerable gratitude.

In particular, I wish to thank Professor Jim Mitchell for his constant support over the years and also for his many constructive comments on the original manuscript. His contribution in the form of a Foreword to the book is deeply appreciated.

I am grateful to the late Professor Peter Wroth, Professors Ted Brown and Guy Houlsby for introducing me to the challenging field of soil and rock mechanics. Professor Ted Brown also read some chapters of the original manuscript and I thank hirn for his detailed comments.

I am indebted to Professor Scott Sloan and Professor Ian Collins for being my mentors in the early stages of my academic career. Their advice and encouragement in those early years has been most valuable.

I also want to thank Professor Kerry Rowe, Professor W.F. Chen, Professor J ohn Carter, Professor Bruce Kutter, and Professor Mark Randolph for having been a constant source of inspiration and encouragement.

A few sections ofthe book were prepared in the summer of 1999 when I was on sabbatical leave at MIT. I would like to record my thanks to Professor Andrew Whittle for his hospitality during my visit.

I am very grateful to Miss K y lie Ebert for her excellent proofreading of the manu­script. Dr. Mark Allman and Dr. Bailin Wu also read some chapters of the book and I thank them for their comments. Special thanks are also extended to Ms. Petra van Steenbergen and Ms. Manja Fredriksz ofKluwer Academic Publishers for their as­sistance during the final stage of this project.

Finally I like to say 'thankyou' to my wife, Xiu-li, and children, Christina and Thomas, for their love and support without which this book could not have been written.

Hai-Sui Yu Newcastle, Australia

April 2000

TABLE OF CONTENTS

Foreword

Preface

1 INTRODUCTION

1.1 SCOPE AND AIMS

1.2 CAVITY EXPANSION THEORY

1.3 APPLICATION TO GEOMECHANICS

1.3.1 In-situ soil testing

1.3.2 Pile foundations and earth anchors

1.3.3 Underground excavations and tunnelling

1.3.4 Wellbore instability

1.4 SIGN CONVENTIONS

1.5 SUMMARY

REFERENCES

Part I: Fundamental Solutions

2 ELASTIC SOLUTIONS

2.1 INTRODUCTION

2.2 ELASTIC SOLUTIONS IN ISOTROPIC MEDIA

2.2.1 Expansion of a hollow sphere

2.2.2 Expansion of a thick-walled cylinder

2.2.3 Cylindrical cavity subject to biaxial in-situ stresses

2.3 ELASTIC SOLUTIONS IN ANISOTROPIC MEDIA

2.3.1 Expansion of a hollow sphere

2.3.2 Expansion of a thick-walled cylinder

2.4 ELASTIC SOLUTIONS IN A SEMI-INFINITE HALF-SPACE

2.4.1 Cylindrical cavity in a half-space

2.4.2 Spherical cavity in a half-space

2.5 SUMMARY

Xl

vii

ix

1

2

2

2

3

4

4

5

5

6

9

9

9

9

12

14

18

18

21

23

24

27

30

Xll Table of Contents

REFERENCES 30

3 ELASTIC-PERFECTLY PLASTIC SOLUTIONS 32

3.1 INTRODUCTION 32

3.2 SOLUTIONS FOR TRESCA CRITERION 32

3.2.1 Expansion of a spherical cavity in a finite medium 33

3.2.2 Expansion of a cylindrical cavity in a finite medium 38

3.2.3 Contraction of cavities in an infinite medium 43

3.3 SOLUTIONS FOR MOHR-COULOMB CRITERION 50

3.3.1 Expansion of a spherical cavity in a finite medium 50

3.3.2 Expansion of a cylindrical cavity in a finite medium 57

3.3.3 Expansion of cavities in an infinite medium 65

3.3.4 Contraction of cavities in an infinite medium 72

3.3.5 Expansion of cavities from zero initial radius 84

3.4 SUMMARY 91

REFERENCES 93

4 CRITICAL STATE SOLUTIONS 95

4.1 INTRODUCTION 95

4.2 CAVITY EXPANSION FROM A FINITE INITIAL RADIUS 95

4.2.1 Undrained expansion of cavities in clays 95

4.2.2 Undrained contraction of cavities in clays 112

4.2.3 Drained expansion of a cylindrical cavity in NC clays 116

4.2.4 Drained expansion of cavities in heavily OC clays 121

4.3 CAVITY EXPANSION FROM ZERO INITIAL RADIUS 125

4.3.1 Drained expansion of cavities in sands 125

4.3.2 Undrained expansion of a cylindrical cavity in a rate-type clay l31

4.4 SUMMARY l36

REFERENCES l37

5 FURTHER ELASTOPLASTIC SOLUTIONS 139

5.1 INTRODUCTION l39

5.2 CAVITY EXPANSION IN HARDENING/SOFTENING SOlLS l39

5.2.1 Undrained expansion of a cylindrical cavity in strain hardening/softening clays 139

Table of Contents xm

5.2.2 Undrained cavity expansion from zero radius in clays 142

5.3 CAVITY CONTRACTION IN BRITTLEIPLASTIC ROCK 144

5.3.1 Cavity unloading in brittle-plastic rock using the Mohr-Coulomb criterion 145

5.3.2 Cavity unloading in brittle-plastic rock using the Hoek-Brown criterion 150

5.4 SOLUTIONS FOR PIECE-WISE MOHR-COULOMB CRITERION 155

5.5 INVERSE CAVITY EXPANSION PROBLEMS 163

5.5.1 Cavity expansion in undrained clay 163

5.5.2 Cavity expansion in cohesionless sand 165

5.6 SUMMARY 166

REFERENCES 168

6 TIME-DEPENDENT SOLUTIONS 170

6.1 INTRODUCTION 170

6.2 VISCO-ELASTIC SOLUTIONS 170

6.2.1 Visco-elastic models and method of stress analysis 170

6.2.2 Solutions for two simple cavity problems 174

6.3 ELASTIC-VISCOPLASTIC SOLUTIONS 176

6.3.1 Elastic-viscoplastic stress-strain relations 177

6.3.2 Stresses and displacement in the initial plastic zone 177

6.3.3 Stresses and displacement in the time-dependent plastic zone 179

6.4 CONSOLIDATION SOLUTIONS 181

6.4.1 Consolidation of soil around an expanding cavity 181

6.4.2 Consolidation of soil around a contracting cavity 185

6.5 SUMMARY 188

REFERENCES 189

7 FINITE ELEMENT SOLUTIONS 190

7.1 INTRODUCTION 190

7.2 UNCOUPLED DRAINED AND UNDRAINED ANALYSIS 190

7.2.1 Finite element formulation 190

7.2.2 Plasticity models for soils 193

7.2.3 Finite element program 201

XIV Table of Contents

7.3 COUPLED CONSOLIDATION ANALYSIS 202

7.3.1 Finite element formulation 202

7.3.2 The modified Cam c1ay model 204

7.3.3 Finite element pro gram 205

7.4 SUMMARY 205

REFERENCES 206

Part 11: Geotechnical Applications

8 IN-SITU SOlL TESTING 209

8.1 INTRODUCTION 209

8.1.1 The principle of pressuremeter testing 209

8.1.2 Types of pressuremeter 209

8.1.3 Cone penetrometer testing 210

8.2 SELF-BORING PRESSUREMETER TESTS IN CLAY 211

8.2.1 Shear modulus 212

8.2.2 In-situ total horizontal stress 213

8.2.3 Undrained shear strength 213

8.2.4 Consolidation coefficient 223

8.2.5 Effects of finite pressuremeter length and initial stress state 224

8.3 SELF-BORING PRESSUREMETER TESTS IN SAND 227

8.3.1 Shear modulus 227

8.3.2 In-situ total horizontal stress 228

8.3.3 Drained shear strength 228

8.3.4 State parameter 233

8.3.5 Effect of finite pressuremeter length 243

8.4 CONE PRESSUREMETER TESTS IN CLAY AND SAND 245

8.4.1 Cone pressuremeter testing in c1ay 245

8.4.2 Cone pressuremeter testing in sand 247

8.5 CONE PENETRATION TESTS IN SOlL 255

8.5.1 Cone penetration in cohesive soils 256

8.5.2 Cone penetration in cohesionless soils 260

8.5.3 GeneraIremarks 266

8.6 SUMMARY 267

Table of Contents xv

REFERENCES 267

9 PILE FOUNDATIONS AND EARTH ANCHORS 275

9.1 INTRODUCTION 275

9.2 AXIAL CAPACITY OF DRIVEN PILES IN CLAY 276

9.2.1 Shaft capacity of piles: effect of the installation on soil stress 276

9.2.2 End bearing capacity of dri yen piles 282

9.2.3 Increase in capacity of piles with time: effect of consolidation 283

9.3 AXIAL CAPACITY OF DRIVEN PILES IN SAND 285

9.3.1 End bearing capacity of piles in sand 285

9.3.2 End bearing capacity of piles in crushable sands 289

9.4 LATERAL CAPACITY OF PILES 289

9.4.1 Limiting lateral pressures in c\ay 289

9.4.2 Limiting lateral pressures in sand 290

9.4.3 Limiting lateral pressures in rock 291

9.5 BEARING CAPACITY OF SAND WITH A SURCHARGE 293

9.6 UPLIFT CAPACITY OF PLATE ANCHORS IN SOlLS 295

9.6.1 Plate anchors in c\ay 296

9.6.2 Plate anchors in sand 300

9.7 SUMMARY 303

REFERENCES 305

10 UNDERGROUND EXCAVATIONS AND TUNNELLING 309

10.1 INTRODUCTION 309

10.2 EXCAVATION DESIGN IN MASSIVE ROCK 310

10.2.1 Elastic stress analysis 311

10.2.2 Elastic-plastic (fracture) stress analysis 316

10.3 ROCK SUPPORT IN UNDERGROUND EXCAVATIONS 319

10.3.1 The principle of rock support and reinforcement 319

10.3.2 Ground response curves 321

10.4 TUNNELS IN COHESIVE SOlLS 324

10.4.1 Settlements due to tunnelling - total stress analysis 327

xvi Table of Contents

10.4.2 Settlements due to tunnelling - effective stress analysis

10.4.3 Stability of tunnels

10.5 TUNNELS IN COHESIVE-FRICTIONAL SOlLS

10.5.1 Settlements due to tunnelling

10.5.2 Stability of tunnels

10.6 SUMMARY

REFERENCES

11 WELLBORE INSTABILITY

11.1 INTRODUCTION

11.2 ELASTIC ANALYSIS OF WELLBORE INSTABILITY

11.2.1 Stress analysis using constant stiffness elasticity

332

346

348

348

352

354

356

360

360

361

362

11.2.2 Analysis using pressure-dependent elasticity 364

11.2.3 Effect of stress-induced anisotropy on wellbore instability 366

11.3 POROELASTIC ANALYSIS OF WELLBORE INSTABILITY 369

11.3.1 Semi-analytical solutions

11.3.2 Application to wellbore instability prediction

11.4 PLASTIC ANALYSIS OF WELLBORE INSTABILITY

11.4.1 Stability criteria

1l.4.2 Stability analysis using critical state models

11.5 SUMMARY

REFERENCES

INDEX

369

372

376

376

377

379

380

383