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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
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