NUMERICAL SIMULATION OF EXCAVATIONS WITHIN JOINTED ROCK OF INFINITE EXTENT by ALEXANDROS I . SOFIANOS (M.Sc.,D.I.C.) May 1984- - A thesis submitted for the degree of Doctor of Philosophy of the University of London Rock Mechanics Section, R.S.M.,Imperial College London SW7 2BP -2- A bstract The subject of the thesis is the development of a program to study i the behaviour of stratified and jointed rock masses around excavations. The rock mass is divided into two regions,one which is. supposed to * exhibit linear elastic behaviour,and the other which will include discontinuities that behave inelastically.The former has been simulated by a boundary integral plane strain orthotropic module,and the latter by quadratic joint,plane strain and membrane elements.The # two modules are coupled in one program.Sequences of loading include static point,pressure,body,and residual loads,construetion and excavation, and quasistatic earthquake load.The program is interactive with graphics. Problems of infinite or finite extent may be solved. Errors due to the coupling of the two numerical methods have been analysed. Through a survey of constitutive laws, idealizations of behaviour and test results for intact rock and discontinuities,appropriate models have been selected and parameter ♦ ranges identif i ed. The representation of the rock mass as an equivalent orthotropic elastic continuum has been investigated and programmed. Simplified theoretical solutions developed for the problem of a wedge on the roof of an opening have been compared with the computed results.A problem of open stoping is analysed. * ACKNOWLEDGEMENTS The author wishes to acknowledge the contribution of all members of the Rock Mechanics group at Imperial College to this work, and its full financial support by the State Scholarship Foundation of * G reece. Furthermore thanks are due to: * Dr. J. 0-Watson, for his supervision of this study,and for introducing me to the boundary element method. Dr. J.W.Bray,for discussions. Messrs S.Budd and T.Sippel,for suggestions on programming. » Mr. J.A.Samaniego,for his friendship and exchange of ideas. Last but not least I whish to express my thanks to my wife for her patience during the hard period of the study, and to my mother for dealing with all my interests during my absence from home. * * TABLE OF CONTENTS Page ABSTRACT 2 ACKNOWLEDGEMENTS 3 TABLE OF CONTENTS 4 LIST OF FIGURES 7 LIST OF TABLES 11 NOTATION AND CONVENTIONS 14 CHAPTER 1 - INTRODUCTION 20 CHAPTER 2 - NUMERICAL MODELLING OF JOINTED ROCK 23 2.0 Distribution of stresses and displacements 23 2.1 The continuum 24 2.1.1 Mechanical properties 24 2.1.2 Simulation 26 2.2 Discontinuities - A literature survey 34 2.2.1 Mechanical properties 34 2.2.2 Sim ulation 39 2.3 The joint element 47 2.3.1 The element 47 2.3.2 The constitutive law 50 2.3.3 Iterative solution 63 2.3.4 Examples 72 2.4 Change of the geometry 77 2.4.1 Excavation 77 2.4.2 Construction 79 2.5 Types of activities 80 - 5 - Page CHAPTER 3 - THE ELASTIC REGION 81 3.0 General 81 3.1 Equivalent elastic properties of a jointed rock mass 82 3.1.1 Three orthogonal sets of joints - 82 3.1.2 Two oblique sets of joints 84 3.2 Implementation of the direct boundary integral method 86 3.2.1 The integral equation for the complementary function 87 3.2.2 Kernels U and T 89 3.2.3 Isoparametric element 91 * 3.2.4 Nodal collocation 93 3.2.5 Numerical integration 95 3.2.6 Rotation of axes 96 3.2.7 Particular integral 97 3.2.8 Infinite domain 98 3.3 Example 103 CHAPTER 4 - COUPLING REGIONS WITH CONTINUOUS AND DISCONTINUOUS DISPLACEMENT FIELDS 107 4.0 General 107 4.1 Symmetric coupling 108 * 4.2 V alidation 110 4.3 Inherent erro rs 125 4.3.1 Causes of erro rs 125 ** 4.3.2 Examples 130 -o-/ Page CHAPTER 5 - STABILITY OF AN OVERHANGING ROCK WEDGE IN AN EXCAVATION 142 5.0 General 142 5.1 Idealised behaviour 142 5.1.1 Symmetric wedge 144 5.1.2 Asymmetric wedge 161 5.2 Numerical solution 169 5.2.1 Symmetric wedge 170 5.2.2 Asymmetric wedge 182 CHAPTER 6 - APPLICATION OF THE PROGRAM TO ORE STOPING 188 CHAPTER 7 - SUMMARY AND CONCLUSIONS 200 APPENDIX 1 - Description of input for program AJROCK 206 APPENDIX 2 - Overall structure of the program 231 APPENDIX 3 - Additional information relevant to Chapter 3 233 A3.1 O rthotropic kernels 233 A3.2 Integration of kernel - shape function products over an element containing the first argument 243 A3.3 Particular integral 248 APPENDIX 4 - Estimate of error due to the assumption of continuous tractions at nodes 252 APPENDIX 5 - Graphs for estimating the stability of a wedge in a tunnel roof. 254 REFERENCES 279 - 7 - LIST OF FIGURES Page CHAPTER 2_ % Fig. 2.1 Eight node serendipity element 27 Fig. 2.2 Axes for transverse isotropy and global cartesian system 29 * Fig. 2.3 Sign convention for internal forces 29 Fig. 2.4 Isoparametric three node membrane element 31 Fig. 2.5 Peak shear strength 36 Fig. 2.6 Peak shear strength 37 Fig. 2.7 First joint element 43 Fig. 2.8 Three dimensional joint elements 43 Fig. 2.9 Isoparametric quadratic joint element 48 * Fig. 2.10 Failure criteria and parameters 51 Fig. 2.11 Load history effect on current peak shear strength 55 Fig. 2.12 Normal stress vs normal strain law 57 Fig. 2.13 Shear strain vs shear stress 58 * Fig. 2.14 Three dimensional sketch for e s ,a , T . 60 Fig. 2.15 Dilation vs shear strain law for the two models 62 Fig. 2.16 Iterative process (for compression) - Joint 1, ♦ no dilation 65 Fig. 2.17 Iterative process - Joint 2 without dilation 66 Fig. 2.18 Iterative process - Joint 1 with dilation 67 * Fig. 2.19 Iterative process - Joint 2 with dilation 68 Fig. 2.20 Iterations for simple examples 71 Fig. 2.21 Strain softening joints (examples) 73 CHAPTER 3_ Fig. 3.1 Three orthogonal sets of joints 83 Fig. 3.2 Two oblique sets of joints 83 Fig. 3.3 Conventions for kernel arguments 90 Fig. 3.4 Isoparametric boundary element 92 Fig- 3.5 Coordinate systems H.V and 1,2 92 Fig. 3.6 Integration over remote boundary 99 Fig. 3.7 Initial meshes for the examples of Section 3.3 104 Fig. 3.8 Boundary element region subjected to gravitational fie ld 105 Fig. 3.9 Plane strain and joint elements subjected to gravitational field 106 CHAPTER 4_ Fig. 4.1 Square block in tension 111 Fig. 4.2 A circular hole under pressure 111 Fig. 4.3 Hole within infinite rock mass modelled by boundary elem ents only 113 Fig. 4.4 Hole within infinite rock mass modelled by boundary and finite elements 113 Fig. 4.5 Tension of a long plate 116 Fig. 4.6 Lined opening 119 Fig. 4.7 Excavation of a circular tunnel 122 Fig. 4.8 Excavation of a circular tunnel 122 Fig. 4.9 Various methods to determine the limiting values of tractions at the two sides of a corner 126 Fig- 4.10 Two boundary element regions 131 Fig. 4.11 Circular disc 131 Fig. 4.12 Square block modelled by 32 boundary elements 134 Fig. 4.13 Square block modelled by boundary and plane strain elements 134 Fig. 4.14 Large problem with boundary and finite elements 138 CHAPTER _5 Fig. 5.1 Wedge id e a liz a tio n 143 Fig. 5.2 Symmetric wedge - Friction angle greater than a 145 Fig. 5.3 Symmetric wedge - Friction angle less than a 149 Fig. 5.4 Examples for very low stiffness ratio joints 151 Fig- 5.5 Behaviour of a symmetric rigid wedge 152 Fig- 5.6 Effect of intact rock flexibility 154 Fig. 5.7 Models for elastic wedge 155 Fig. 5.8 Stress redistribution 159 Fig. 5.9 Asymmetric rigid wedge 162 Fig. 5.10 Oblique wedge 167 F ig. 5.11 Wedge with ro tatio n 167 Fig. 5.12 Symmetric flexible wedge within rigid rock 171 Fig. 5.13 Symmetric elastic wedges within elastic rock ; a=20^ 175 Fig. 5.14 Example of stress redistribution in a symmetric wedge 181 Fig. 5.15 Asymmetric wedge 183 - 1 0 - Page CHAPTER 6 Fig. 6.1 Stope and drive geometry 189 % Fig. 6.2 Stope.drive,and surrounding rock discretization 189 Fig. 6.3 I n itia l mesh 193 Fig- 6.4 Gravitational loading 194 Fig. 6.5 Excavation of the drive 195 ♦ Fig. 6.6 First level ore excavation 196 Fig. 6.7 Second level ore excavation 197 Fig. 6.8 Third level ore excavation 198 APPENDICES Fig. A l.l Boundary element convention 213 * Fig. A1.2 Plane strain element convention 219 F ig . Al-3 Joint element convention 219 Fig. A2.1 Flow chart of program AJROCK 232 Fig. A3.1 Lines on which the orthotropic kernel U is undefined 240 * Fig. A3.2 Analytical integration of a logarithm - polynomial product over a straight line element 244 Fig. A3.3 Spiral method used for the determination of the diagonal terms of matrix T 244 * - 11 - LIST OF TABLES Page CHAPTER 2 Table 2.1 Shear strain regions 59 Table 2.2 Two plane strain and two joint elements 74 Table 2.3 One plane strain and three joint elements 76 CHAPTER .3 Table 3.1 Integration of kernel shape function products over an element containing the first argument 95 Table 3.2 Displacements at the nodes of a brick (example) 103 CHAPTER 4_ Table 4.1 Square block in tension 112 Table 4.2 Circular hole modelled by boundary elements only 114 Table 4.3 Circular hole modelled by boundary and finite elements - Displacements at nodes 114 Table 4.4 Circular hole modelled by boundary and finite elements - Stresses within plane strain elements 114 Table 4.5 Tension of a long plate modelled by symmetric mesh 115 Table 4.6 Tension of a long plate modelled by asymmetric mesh 117 Table 4.7 Lined circular tunnel with full adhesion on interface 123 Table 4.8 Lined circular tunnel with free slip on interface 123 Table 4.9 Excavation of a circular tunnel 124 Table 4.10 Prescribed values in finite and boundary elements 129 - 12 - Page Table 4.11 Equivalent nodal forces and displacements