Engineering. Lectu re Notes in. W S. Venturini. Boundary Element Method in Geomechanics. Springer-Verlag Berlin Heidelberg New York Tokyo 1983

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1 Lectu re Notes in Engineering Edited by C.A. Brebbia and S.A. Orszag 4 W S. Venturini Boundary Element Method in Geomechanics Springer-Verlag Berlin Heidelberg New York Tokyo 1983

2 Series Editors C.A Brebbia. S.A Orszag Consulting Editors J. Argyris. K.-J. Bathe' AS. Cakmak J. Connor' R. McCrory C.S. Desai' K.-P. Holz. F.A Lecki. F. Pinder' AR.S. Pont J. H. Seinfeld. P. Silvester' W. Wunderlich' S. Yip Author W. S. Venturini Departamento de Estruturas Escola de Engenharia de Sao Carlos Universidade de Sao Paulo Av. Dr. Carlos Botelho, Sao Carlos - SP Brazil Formerly Southampton University, England With 114 Figures ISBN-13: e-isbn-13: : / Library of Congress Cataloging in Publication Data Venturini, W.S., Boundary element method in geomechanics. (Lecture notes in engineering; 4) Bibliography: p. Soil mechanics. 2. Boundary value problems. I. Title. II. Series TA710)J '51' This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprillting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to 'VerWertungsgeselischaft Wort!, Munich. Springer-Verlag Berlin Heidelberg 1983 Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2061/

3 CXNl'ENTS NOl'ATION VI CHAPTER 1 lntrodocticn CHAPTER 2 MATERIAL BEHAVIOUR AND NUMERICAL T.EX::HNIQUES Introduction Linear Elastic Material Problems NonLinear Elastic Material Problems Inelastic Material Problems Tme-Dependent Problems 23 CHAPTER 3 BOUNDARY l:n'i'&;ral muaticns IntrOduction Governing Equations and FundaIrental Solutions Integral Equations Body Force Problem Prestress Force Problem TeIrperature Shrinkage and SWelling 46 CHAPTER 4 BOUNDARY l:n'i'&;ral muaticns FOR COMPLEl'E PLANE STRAIN PROBLEMS Introduction Governing Equations and FundaIrental Solutions Integral Equations for Interior Points Boundary Integral Equation 58 CHAPTER 5 BOUNDARY ELEMENT ME'IHOD Introduction Discretization of the Integral Equations 61

4 IV 5.3 Subregions 5.4 Traction Discontinuities 5.5 '!hin Subregions 5.6 Solution Technique 5.7 Practical Application of Boundary Element on Linear Problems CHAPTER 6 N<rTENSICN BOUNDARY ELEMENl'S Introduction 6.2 Rock Material Behaviour 6.3 Method of Solution 6.4 Application of No-Ten.sion in Rock ~ics CHAPTER 7 DIsc:<:NTlNUITY ProBLEMS Introduction 7.2 Plane of weakness 7.3 Analysis of Discontinuity Problems 7.4 NuIrerical Applications CHAPTER 8 BOUNDARY ELEMENT 'l'.ochnlque FOR PIASTICITY P:ROBLEMS Introduction Elastoplastic Problems in One Dimension 8.3 '!heory of Plasticity for Continuum Problems 8.4 NuIrerical Approach for the Plastic Solution 8.5 Practical Applications in Geomechanics CHAPTER 9 ELAS'IO/VISCC>PLASTIC BOUNDARY ELEMENT APPRJ1ICH 9.1 Introduction 9.2 TiIre-Dependent Behaviour in One Dimension 9.3 Elasto/Viscoplastic Constitutive Relations for Continuum Problems

5 v CHAPl'ER OUU.ine of the Solution Teclmique 9.5 Tille Interval Selection and Convergence 9.6 Elasto/Viscoplastic Applications APPLICATICNS OF THE NONLINEAR l3ou.nd1iry ELEMENT FOmJIATICliI 10.1 Introduction 10.2 Strip Footmg Problem 10.3 Slope Stability Analysis 10.4 Tunnellmg Stress Analysis CHAPl'ER 11 APPENDICES CCNCLUSICNS

6 NOTATION A bi B c matrix of the final system of equations body forces body force vector cohesion of the material c.. coefficients of the free term D. 'k S. 'k ~1mk ~1mk components of tensors corresponding to derivatives of the fundamental solution. Eijmk F ijmk E F G modulus of elasticity independent vector of the final system of equations shear modulus Jacobian (for boundary elementsl H' slope of the uniaxial stress strain curve H G D matrices for the boundary equation E H' D' matrices for stress determination E' I IJI k M N unit matrix Jacobian (for internal cellsl hardening parameter elastic boundary solution; vectorial form elastic stress solution; vectorial form

7 VII p. p p fraction components traction vector nodal traction vector traction fundamental solution matrix of fundamental tractions q Q r s s t t m u. u u u*. u* x. x. y r (s,q) e.. e e.. o e. e...vp e.. Tli field point inside the domain field point on the boundary distance between the field and load points load point inside the domain load point on the boundary time overlay thickness displacement components displacement vector nodal displacement vector displacement fundamental solution matrix of fundamental displacements coordinates of the global system coordinates of the local system unit weight viscosity parameter boundary of the body Kronecker delta Dirac delllai.function strain components elastic strain components initial strain components strain rate components viscoplastic strain rate components direction cosines

8 VIII v Poisson's ratio I; homogeneous coordinate over a boundary element 1;. (].. 0 (].. e (].. homogeneous trian.gular coordinates stress components initial stress components elastic stress components 4> angle of internal friction t ~ } interpolation functions