BOUNDARY ELEMENT METHODS FOR MINE DESIGN by BARRY HUGH GARNET BRADY M. A thesis submitted to the University of London (Imperial College of Science and Technology) for the Degree of Doctor of Philosophy in the Faculty of Engineering July 1979 2 ABSTRACT The subject addressed in the thesis is the design of mine structures in hard rock generated by underground mining methods. Issues to be resolved in the design of supported mine structures are identified, and currently available techniques for analysis and prediction of the performance of these structures are reviewed briefly. Fundamental and operational limitations of the various techniques are assessed.
The inherent advantages of Boundary Element Methods for mine design applications are discussed. Several different formulations of the Boundary Element Method are presented. Indirect formulations for analysis of stress and displacement distributions around long openings inclined in a triaxial stress field are described. It is shown that concentrated singularities, which form the basis of an indirect formulation for analysis of problems involving long, narrow, parallel-sided openings, can be constructed readily by coupling line load singularities.
An indirect formulation for three-dimensional analysis of tabular orebody extraction is developed by taking account of the procedures established in the two-dimensional, complete plane strain analysis of long slits. A direct formulation of the Boundary Element Method is developed for the complete plane strain analysis of structures in non-homogeneous media. The main advantage of the direct formulation over the indirect formulations is shown to be the capacity to handle a wide range of excavation cross-sectional geometries. A simple technique is established for estimation of pillar and mine stiffness properties, using the Boundary Element direct formulation.
The technique is applied, in 3 conjunction with data obtained from the literature, to assessment of the stability of pillars in a series of hypothetical stoping layouts. It is demonstrated that pillar stability is sensitive to the pattern of natural fractures in the rock mass. It is concluded that the absence of field data on the post-peak performance of hard rock masses prevents proper evaluation of the proposed technique for pillar stability analysis. 4 ACKNOWLEDGEMENTS The author records his gratitude to the people who advised him during the execution of the work reported in the thesis, and assisted him during thesis preparation.
He would like to thank his supervisor, Dr E.Brown, for general advice and guidance throughout the work programme, for information and discussion on the strength and deformation characteristics of rock masses, and for critical assessment of the draft of the thesis. He is grateful to Dr J. Bray for the interest taken in the work, for unpublished information on a number of topics, and for discussion on a wide range of issues in mechanics. The author was fortunate to have a number of prolonged debates with Dr G.Hocking and Dr J.Watson on problems associated with the Boundary Element Method.
Colleen Brady provided consolation during several desperate stages of the enterprise. The author recognizes the achievements of Miss Jennifer Wills and her assistants, who typed the thesis. The work was conducted during the author's tenure of a Lectureship in Rock Mechanics in the Department of Mineral Resources Engineering. He is grateful to Professor R.
Pryor and the Imperial College of Science and Technology for providing a rewarding teaching and research environment. Some of the work reported in the thesis was conducted in the course of a project at Imperial College supported by member companies of the Australian Mineral Industries 5 Research Association Limited. The author thanks the Management of Mount Isa Mines Limited for permission to use information on operations at the Mount Isa Mine, Australia. The author is pleased to record the useful advice and generous assistance given by Mr Barrie Holt and members of his section in production of the thesis.
6 CONTENTS Page ABSTRACT 2 ACKNOWLEDGEMENTS 4 LIST OF FIGURES 11 LIST OF TABLES 16 NOTATION 17 PREFACE 20 CHAPTER 1.1 Underground mining methods 22 1.2 Techniques for design of supported mine structures 24 1.3 Energy changes accompanying underground mining 32 1.4 Stability of mine pillars and mine structures 38 1.5 Information for design of stable pillars 49 CHAPTER 2. THE BOUNDARY ELEMENT METHOD FOR ELASTOSTATICS 2.1 Principles and limitations of the method 53 2.2 Indirect Boundary Element formulations 58 2.3 Direct Boundary Element formulations 64 2.4 Displacement Discontinuity Method 70 2.5 Required developments in Boundary Element solution procedures 77 7 Page CHAPTER 3. COMPLETE PLANE STRAIN AND COMPLETE PLANE STRESS 3.1 Problem specification and definitions 79 3.3 Complete plane strain 81 3.4 Complete plane stress 87 CHAPTER 4. INDIRECT FORMULATION OF THE BOUNDARY ELEMENT METHOD FOR COMPLETE PLANE STRAIN 4.1 Description of method of analysis 91 4.2 Antiplane line and strip loads 95 4.3 Boundary Element solution procedure 98 4.4 Validation of Boundary Element program 101 CHAPTER 5.
INDIRECT FORMULATION OF THE BOUNDARY ELEMENT METHOD FOR NARROW EXCAVATIONS AND COMPLETE PLANE STRAIN 5.1 Objectives and scope of work 105 5.2 Development of singularities for modelling contiguous parallel surfaces 108 5.3 Optimum distribution of singularities for modelling single slits 113 5.4 Boundary Element solution procedure 121 8 Page 5.5 Validation of Boundary Element program 126 5.6 Assessment of slit modelling procedure 133 CHAPTER 6. THREE-DIMENSIONAL ELASTIC ANALYSIS OF TABULAR OREBODY EXTRACTION 6.1 Problem description for three- dimensional analysis 135 6.2 Development of compressive and shear singularities 138 6.3 Imposed distributions of singularity intensity on excavation segments 144 6.4 Three-dimensional Boundary Element solution procedure 151 6.5 Validation of Boundary Element program 154 6.6 Assessment of slot modelling procedure 164 CHAPTER 7. DIRECT FORMULATION OF THE BOUNDARY ELEMENT METHOD FOR COMPLETE PLANE STRAIN 7.1 Objectives in development of direct formulation 168 7.2 Establishment of boundary constraint equations 169 7.3 Solution of boundary constraint equations 176 7.5 Displacements and stresses at internal points 180 7.6 Symmetry code 182 9 Page 7.7 Validation of Boundary Element program 185 7.8 Use of higher order singularities in the Boundary Element algorithm 188 7.9 Non-homogeneous media 195 7.10 Appraisal of Boundary Element direct formulation 205 CHAPTER 8. MINE DESIGN APPLICATIONS OF THE BOUNDARY ELEMENT METHOD 8.2 Design problems requiring complete plane strain analysis 208 8.3 Study of pillar stability 213 8.4 The Mount Isa lead orebodies 234 CHAPTER 9.
SUMMARY AND CONCLUSIONS 252 REFERENCES. Stresses and displacements induced by a point load in an infinite, isotropic, elastic continuum (Kelvin Equations) 266 APPENDIX II. Stresses and displacements induced by infinite line loads in an infinite, isotropic, elastic continuum 267 APPENDIX III.Stresses and displacements due to infinite strip loads 269 10 Page APPENDIX IV. Stresses and displacements due to infinite line quadrupoles and dipoles 272 APPENDIX V.
Stresses and displacements due to a point hexapole and a point shear quadrupole 274 APPENDIX VI. User information and input specifications for Boundary Element programs 276 11 LIST OF FIGURES Ficture No.1 (a) Pre-mining conditions in a body of rock; (b) tractions and displacements induced within the surface Sr 1.2 Correlation between frequency of rock bursts, ground conditions and rate of energy release during mining (from Cook, 1978) 1.4 Schematic representation of pillar loading by the country rock, and cases of stable and unstable pillar loading (from Starfield and Fairhurst, 1968) 1.5 Replacement of underground pillars (a) by equivalent forces (b) (from Salamon, 1970) .6 Stress-strain curves for specimens of Tennessee Marble with various length/diameter ratios (from Starfield and Wawersik, 1968) 2.2 (a) Surface S subject to imposed tractions or displacements; (b),(c) Distributions of normal and shear singularities on S; (d),(e) Normal and shear singularity intensities on element of surface S 2.3 (a),(b) Load cases for establishment of Boundary Integral Equation; (c) Method of handling singularity in range of integration 2.3 Boundary conditions on coupled half spaces for generation of normal displacement discontinuity Dz (after Crouch, 1976b) 12 Figure No.5 Boundary conditions on coupled half spaces for generation of shear displacement discontinuity Dx (after Crouch, 1976b) 3.1 Plane (p ) and out-of-plane (p x, pz' pzx xY' pYz) stress components for a long opening excavated in a medium subject to a triaxial state of stress 4.1 (a) Long excavation in a medium subject to initial stress; (b),(c) Resolution into component problems; (d) Geometric parameters for discretized problem 4.2 Uniformly distributed transverse, longitudinal and normal strip loads ,and geometric parameters determining the effect of strip loads on element. j at the point i (xi , zi ) 4.3 Problem geometry for determining stresses and displacements due to an infinite, Y-directed line load.4 Stress distribution around a circular hole in a triaxial stress field, from Boundary Element analysis and analytical solution 4.5 Excavation-induced displacements around a circular hole in a triaxial stress field, from Boundary Element analysis and analytical solution 5.1 Discretization of long narrow opening into segments 5.2 Resolution of real problem into uniformly stressed medium and subsidiary problem 5.3 Construction of compressive quadrupole singularity 5.4 Construction of shear quadrupole singularity from counteracting couples 5.5 Construction of antiplane dipole from opposing line loads 5.6 Stress distribution in the plane of a slit, and parabolic and elliptical distributions of singularity intensity 5.7 Stress and displacement distribution in the plane of a slit in a uniaxial compressive field (a),(b) and quasi-elliptical distribution of singularity intensity (c) 13 Figure No.8 Geometric parameters determining influence coefficients for uniformly loaded element (a) and edge element (b) 5.9 Stress and displacement distributions around a slit in a uniaxial compressive field, modelled with three segments 5.10 Stress distribution along ray AB for a slit, in a unit shear field 5.11 Displacement distribution over excavated area, and stress distribution in pillar area, for row of slits in a uniaxial field 6.1 Isolated pillar generated during room-and-pillar mining 6.2 Single narrow opening in a medium subject to triaxial loading (a), discretization into segments (b), and a typical excavation segment(c) 6.3 Construction of a compressive dipole (a), and a compressive hexapole from three dipole singularities (b) 6.4 Construction of a shear dipole (a), and a shear quadrupole from counteracting shear dipoles(b) 6.5 Axes of symmetry for a square excavation, along which elliptical variation of singularity intensity is inferred 6.6 Distributions of singularity intensity over internal, edge and corner excavation segments • 6.7 Stresses in the plane of, and perpendicular to the plane of a penny shaped crack in a uniaxial field (a),(b), displacement distribution over the crack(c), and singularity distribution which models crack formation (d) 6.8 Distribution of shear stress around square openings with various span/height ratios in a unit shear field 6.9 Stress and displacement distributions around square openings with various span/height ratios in a uniaxial compressive field 6.10 Stress and displacement distributions around a square room with a central square pillar in a uniaxial compressive field 14 Figure No.1 slice of the surface of an opening in a medium subject to triaxial stress, and problem specification for complete plane strain analysis 7.2 Load cases for establishing boundary integral equation 7.3 Geometric parameters for determination of directional derivatives of displacement at excavation boundary 7.4 Load cases for determining displacements at internal points in the medium 7.5 Problem specification for an opening which is symmetric about the Z-axis 7.6 Stress distribution around a circular hole in a triaxial stress field 7.7 Displacement distribution around a circular hole in a triaxial stress field 7.8 Displacement and stress distributions around a narrow excavation in a uniaxial stress field 7.9 Displacement and stress distributions around a narrow excavation in a longitudinal shear stress field 7.10 Problem specification for a non-homogeneous medium 7.11 Stress distribution in and around a solid cylindrical inclusion in a triaxial stress field 7.12 Stress distribution around a circular hole in a circular inclusion in a medium subject to plane strain 8.1 Problem geometry for assessing the significance of the antiplane component of complete plane strain 8.2 Representation of interaction between country rock and pillar and country rock and abutment in a supported mine structure 8.3 Application of uniformly distributed load at a pillar position to determine mine local stiffness 8.4 Pillar and mine performance characteristics based on convergences at the centre line of the pillar 15 Figure No.
