Modelling hard rock pillars using a Synthetic Rock Mass approach

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Modelling Hard Rock Pillars Using a

Synthetic Rock Mass Approach

by

Yabing Zhang

M.Eng., Northeastern University, 2008

B.Eng., Northeastern University, 2005

Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of

Doctor of Philosophy

in the

Department of Earth Sciences Faculty of Science



Yabing Zhang 2014

SIMON FRASER UNIVERSITY

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Approval

Name: Yabing Zhang

Degree: Doctor of Philosophy

Title: Modelling Hard Rock Pillars Using a Synthetic Rock

Mass Approach

Examining Committee: Chair: Dan Marshall

Professor Doug Stead Senior Supervisor Professor Matt Pierce Supervisor Principal Engineer Itasca Consulting Ltd Don Roberts Supervisor Principal Engineer Golder Associates Ltd Davide Elmo Supervisor Assistant Professor

University of British Columbia

Glyn Williams-Jones Supervisor Associate Professor Andy Calvert Internal Examiner Professor Vahid Hajiabdolmajid External Examiner Engineer

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Abstract

Rock pillar strength and the characterisation of pillar failure mechanisms are of major importance in mine design. The recently developed Synthetic Rock Mass (SRM) approach provides a state-of-the-art numerical technique to more accurately characterize the mechanical properties of rock pillars. The SRM approach used in this thesis is based on a combination of two well accepted numerical methods, a Particle Flow Code (PFC3D) incorporating a Discrete Fracture Network (DFN). This research presents the results of a systematic study of the use of SRM modelling for hard rock pillars. The effect of assumed joint set characteristics (orientation and persistence) is first investigated through comparison of the numerical results from a series of conceptual pillar models. The joint set properties are shown to have important controls on the pillar peak strength, deformation modulus, lateral stiffness and the pillar strain-softening gradient in the post-peak stage. The effect of pillar confinement is then examined using two conceptual pillar models with varied slenderness (Width/Height ratio). The pillar confinement effect is investigated by comparing the axial and lateral stresses at the pillar core and pillar boundaries, and this effect attributed to the lateral restraint due to the loading platens. The confinement effect is further examined using a series of triaxial compression test simulations in which the pillar peak strength, residual strength and post-peak strain-softening gradient are quantified. Simulations of the development of 3D cracks in two jointed pillar models, including wing cracks, large scale crack coalescence and step path failure are presented. A 3D visualisation of internal pillar failure mechanisms is illustrated by examining crack development and the changes in the localised stresses within the pillar model. Research presented will contribute significantly to the development of a more robust SRM approach for rock pillar design.

Keywords: rock pillar; Synthetic Rock Mass; Particle Flow Code; Discrete Fracture Network; numerical modelling

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Dedication

I dedicate this thesis to my family, for

unconditional love.

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Acknowledgements

I take this opportunity to acknowledge several individuals who were involved in my PhD research at Simon Fraser University from 2009 to 2013.

I would like to express my sincere gratitude to the senior supervisor Doug Stead, for his guidance and insightful comments at all stages of this thesis.

I acknowledge all technical contributions from the committee members: Matt Pierce and Davide Elmo for SRM simulations, Don Roberts for pillar design and Glyn William-Jones for geological field survey.

I thank examinations and suggestions from the two thesis reviewers: Andy Calvert and Vahid Hajiabdolmajid.

I also appreciate helpful discussions with Jim Hazzard, Xavier Garcia and Steve Rogers regarding the SRM simulations.

I thank all EASC folks for being with me during the entire PhD program. I enjoy the positive and academic atmosphere in the Earth Sciences department.

I acknowledge the financial support from NSERC Discovery grant, FRBC Endowment and Itasca Education Partnership Program.

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List of Tables

Table 2.1. Database summary for compiled cases (Salamon and Munro, 1967). ... 10

Table 2.2. Empirical parameters suggested by Salamon and Munro (1967). ... 10

Table 2.3. Empirical formulae for hard rock pillars (Martin and Maybee, 2000). ... 12

Table 3.1. Mechanical parameters for SRM pillars. ... 44

Table 3.2. Detailed pillar IDs and joint set configurations*. ... 47

Table 3.3. Normalized peak strength (%) for jointed pillars, from Fig 3.9. ... 57

Table 3.4. Selected empirical formulae for hard rock pillars. ... 58

Table 3.5. Two stage post-peak strain-softening gradients for the joint-free and jointed R1, R2 and R3 pillars, and the axial stress drops over the same stages. .... 66

Table 3.6. Normalized deformation modulus (%) for jointed pillars, from Fig 3.16. ... 69

Table 3.7. Normalized lateral stiffness ratio (%) for jointed pillars, from Fig 3.18. ... 71

Table 3.8. Normalized tensile crack number (%) for jointed pillars, from Fig 3.25, with simulations ended at an axial strain of 0.262%. ... 83

Table 3.9. DFN configuration for joint set persistence study on R2-30*. ... 86

Table 3.10. Normalized peak strength (%) for jointed pillars, from Fig 3.33. ... 95

Table 3.11. Two stage post-peak strain-softening gradients for the joint-free and jointed LP, R2 and HP pillars, and the axial stress drops over the same stages. .... 96

Table 3.12. Normalized deformation modulus (%) for jointed pillars, from Fig 3.34. ... 97

Table 3.13. Normalized lateral stiffness ratio (%) for jointed pillars, from Fig 3.35. ... 98

Table 4.1. Statistical parameters for two sub-vertical joint sets. ... 107

Table 4.3. Localised pillar strengths monitored from the wide and slender pillars. ... 110

Table 4.4. Results from uniaxial and biaxial simulations in Figs 4.18-19. ... 132

Table 4.5. Results from uniaxial and true triaxial simulations in Figs 4.26-27. ... 143

Table 4.6. Results from uniaxial and conventional triaxial simulations in Fig 4.34. ... 153

Table 4.7. Cohesion and friction angle calculated from the fitting curves in Fig 4.38. . 158

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List of Figures

Fig 2.1. Structure-controlled failures of jointed pillars; a, rock block sliding; b, through-going shear failure; c, shear failure along transgressive joints; d, buckling failure (Brady and Brown, 2007). ... 6 Fig 2.2. Pillar rating system; Stage 1, intact pillar; Stage 2, minor spalling and short axial fractures; Stage 3, substantial spalling, axial fracture length shorter than the half pillar height; Stage 4, continuous open fractures cutting towards pillar core, formation of the "hour-glass" shape; Stage 5, large continuous open fractures, well developed “hour-glass” shape; Stage 6, failed pillar by either extreme “hour-glass” shape or necking (Roberts et al., 2007). ... 7 Fig 2.3. Close-range photogrammetry to capture the "hour-glass" shape of a pillar; a, average pillar surface; b, artificial plane; c, cross section calculation; d, surface profile (Styles et al., 2010). ... 8 Fig 2.4. Tributary area analysis for estimating stresses on a pillar (Brady and Brown, 2007). ... 9 Fig 2.5. Modelling wing cracks in a through-going jointed model. The properties of Section AB and CD are calibrated to fit the property of the intact rock; the property of Section BC is defined by the joint. ... 14 Fig 2.6. Pillar failure simulations using Phase2; crosses and circles represent shear and tensile damage respectively (Martin and Maybee, 2000). ... 17 Fig 2.7. Progressive failures of a rock pillar using the local degradation method (Fang and Harrison, 2002). ... 18 Fig 2.8. Mechanical parameters associated with the pillar w/h ratios; a, pillar strength vs w/h ratio; b, post-peak modulus vs w/h ratio (Jaiswal and Shrivastva, 2009). .. 19 Fig 2.9. UDEC Voronoi model and the simulated fractures in the post-peak stage (Alzo'ubi, 2009). ... 21 Fig 2.10. Progressive pillar failures in the hybrid ELFEN models (Pine et al., 2006). ... 22 Fig 2.11. Crack formation in the pre- and post-peak stages; a, dynamic simulation; b, static simulation. Green cracks are formed early and red cracks are formed late (Hazzard et al., 2000). ... 29 Fig 2.12. Distribution of the force chains in the post-peak stage of a uniaxial compressive test on a PFC2D model with six initial holes; blue lines denote particle-particle compression; black and red lines denote compression and tension in the bonds (Potyondy and Cundall, 2004). ... 30 Fig 2.13. Notch failure near a circular tunnel, a, shear failure allowed (3267 cracks); b, shear failure excluded (3800 cracks), by Potyondy and Cundall, 2004. ... 30 Fig 2.14. Damage modes of a PFC2D model in biaxial tests; a, σ3 = 0.1MPa; b, σ3 = 10.0MPa; c, σ3 = 70.0MPa. Red and blue dots denote tensile and shear cracks respectively (Potyondy and Cundall, 2004). ... 31

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Fig 2.15. Comparisons of fracture modes between PFC2D models and concrete samples; a, uniform aggregates; b, long aggregates; c, crack branching; d, aggregates breaking the edge; e, crack path; f, AE events and simulated micro cracks (Katsaga, 2010). ... 33 Fig 2.16. "Smooth joint" logic used in SRM pillar models; Nc is the original particle contact unit-normal vector; Nj and Tj are unit-normal and shear vectors of the joint (Itasca, 2010). ... 34 Fig 2.17. The SRM model for scale effect investigations; a, inserted veins; b, initial clusters formed by veins (Pierce et al., 2009). ... 35 Fig 2.18. SRM models for estimating the Representative Element Volume (REV) of a jointed rock mass (Esmaieli et al., 2010). ... 36 Fig 2.19. Jointed SRM rock masses; a, the host model; b, the 80mx40mx40m model in Y direction; c, eight subdivided models (40mx20mx20m) based on b model; d, eight subdivided models (20mx10mx10m) based on U2 model (Mas Ivars et al., 2011). ... 37 Fig 3.1. Joint-free pillar model; a, 3D pillar scale and measurement spheres; b, 2D pillar scale and measurement spheres. ... 43 Fig 3.2. Three selected SRM pillars and the measurement sphere; a, R2-30; b, R2-60; c, R2-90. The mean radius of each joint set is 0.5m; the P32 values are 3.49,

3.23 and 3.57. Units of the pillar scale and P32 are meter and m2/m3

respectively. ... 46 Fig 3.3. Axial stress vs axial strain recorded using the internal-strain and regular platen loading methods. V is the axial velocity assigned to the loading platens in the regular loading methods; ε is the axial strain. ... 50 Fig 3.4. Particle displacement distribution at an axial strain of 0.20%; a, V = 0.40m/s; b, V = 0.10m/s; c, internal-strain loading. V is the axial velocity applied to the loading platens. ... 52 Fig 3.5. Axial stress vs axial strain for the joint-free pillar; the loading platens are fixed and free to move laterally. ... 53 Fig 3.6. Failure modes of the joint-free pillar when the platens are: a, fixed in the X and Y directions; b, free to move in the X and Y directions. ... 53 Fig 3.7. Numerical results of the joint-free pillar; a, axial stress vs axial strain curves recorded by measurement spheres; b. axial strain curves recorded by measurement spheres; the platen-based strain is plotted against itself as a reference. ... 54 Fig 3.8. Tensile and shear crack number over the uniaxial loading process for the

joint-free pillar; the loading platens are fixed laterally. ... 55 Fig 3.9. Pillar peak strength influenced by joint set orientation and size; the average joint set radius is 0.375m, 0.500m and 0.625m in R1, R2 and R3 groups respectively. ... 56 Fig 3.9. Numerical SRM pillar strength vs empirical pillar strength for pillars having a W/H ratio of 0.57 (4.0m/7.0m). ... 59

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Fig 3.10. Axial stress vs axial strain and the change in tensile and shear crack number with axial strain for the jointed pillar R2-90. ... 60 Fig 3.11. Axial stress and tensile crack number vs axial strain for the six R1 pillars (R = 0.375m). ... 63 Fig 3.12. Axial stress and tensile crack number vs axial strain for the six R2 pillars (R = 0.500m). ... 64 Fig 3.13. Axial stress and tensile crack number vs axial strain for the six R3 pillars (R = 0.625m). ... 65 Fig 3.14. Axial stress vs axial strain and the simulated stage AB; a, R1-45, R2-45, R3-45; b, R1-90, R2-90, R3-90. ... 67 Fig 3.15. Pillar deformation modulus influenced by joint set orientation and size; the average joint set radius is 0.375m, 0.500m and 0.625m for R1, R2 and R3 pillars respectively. ... 68 Fig 3.16. Lateral strain in the X and Y directions vs axial strain for pillar R1-15. ... 70 Fig 3.17. Lateral stiffness ratio influenced by joint set orientation and size; the average joint set radius is 0.375m, 0.500m and 0.625m for R1, R2 and R3 pillars. ... 71 Fig 3.18. Acoustic Emission method for characterizing stress thresholds; A = first crack; B = systematic crack initiation; C = crack damage (Diederichs et al., 2004). ... 73 Fig 3.19. Strain method for crack initiation and damage stress identification (Martin and Chandler, 1994). ... 75 Fig 3.20. Elastic lateral (L), total lateral (L) and total volumetric (V) strain vs axial strain for the six R1 pillars (R = 0.375m). The axial strain corresponding to σci is

identified when the elastic L strain becomes lower than the total L strain. The axial strain corresponding to σcd is identified when the total V strain curve

reverses. The lateral strains are plotted as positive values for clarity. ... 78 Fig 3.21. Elastic lateral (L), total lateral (L) and total volumetric (V) strain vs axial strain for the six R2 pillars (R = 0.500m). The axial strain corresponding to σci is

reverses. The lateral strains are plotted as positive values for clarity. ... 79 Fig 3.22. Elastic lateral (L), total lateral (L) and total volumetric (V) strain vs axial strain for the six R3 pillars (R = 0.625m). The axial strain corresponding to σci is

reverses. The lateral strains are plotted as positive values for clarity. ... 80 Fig 3.23. Ratio of the crack initiation stress σci and crack damage stress σcd to the pillar

peak strength (σp) for R1, R2 and R3 pillars; a, σci/σp; b, σcd/σp ... 81

Fig 3.24. Tensile crack number influenced by joint set orientation and size at the end of the simulations (axial strain 0.262%); the average joint set radius is 0.375m, 0.500m and 0.625m for the R1, R2 and R3 pillars. ... 82 Fig 3.25. Ultimate cracking identified at the end of the simulations, a, R1-30; b, R1-60; c, R1-90. Tensile and shear cracks are denoted by red and black dots. ... 84

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Fig 3.26. Selected pillar models incorporating the R2-30 based DFNs; a, removal of the longest 50 joints in each set (DFN2); b, removal of the longest 100 joints in each set (DFN3); c, removal of the shortest 50 joints in each set (DFN4); d, removal of the shortest 100 joints in each set (DFN5). The P32 is 3.49m2/m3 for

the four DFN models. ... 87 Fig 3.27. Axial stress vs axial strain for R2-30 and the pillars incorporating DFN1-DFN5 (Table 3.8); the peak strength and deformation modulus are marked for each pillar model. ... 88 Fig 3.28. Axial stress vs axial strain for LP pillars with low persistence joint sets. ... 90 Fig 3.29. Axial stress vs axial strain for HP pillars with high persistence joint sets. ... 91 Fig 3.30. Elastic lateral (L), total lateral (L) and total volumetric (V) strain vs axial strain for the six LP pillars (low persistence). The axial strain corresponding to σci is

captured when the elastic L strain becomes lower than the total L strain. The axial strain corresponding to σcd is captured when the total V strain curve

reverses. The lateral strains are plotted as positive values for clarity. ... 92 Fig 3.31. Elastic lateral (L), total lateral (L) and total volumetric (V) strain vs axial strain for the six HP pillars (high persistence). The axial strain corresponding to σci is

identified when the elastic L strain becomes lower than the total L strain. The axial strain corresponding to σcd is captured when the total V strain curve

reverses. The lateral strains are plotted as positive values for clarity. ... 93 Fig 3.32. Pillar peak strength influenced by joint set orientation and persistence; LP and HP denote low and high joint set persistence. ... 95 Fig 3.33. Pillar deformation modulus influenced by joint set orientation and persistence; LP and HP denote low and high joint set persistence. ... 97 Fig 3.34. Lateral stiffness ratio influenced by joint set orientation and persistence; LP and HP denote low and high joint set persistence. ... 98 Fig 3.35. Ratio of the crack initiation stress σci and crack damage stress σcd to the pillar

peak strength (σp) for LP, R2 and HP pillars; a, σci/σp; b, σcd/σp. ... 99

Fig 4.1. Idealized platen-specimen interactions; a, the platens stiffer than the specimen; b, the platens softer than the specimen (Tang et al., 2000). ... 104 Fig 4.2. SRM pillars for w/h ratio effect characterizations; a, wide pillar, w/h = 15.0m/10.0m (1.5); b, nine measurement spheres in the wide pillar; c, slender pillar, w/h = 5.0m/10.0m (0.5); d, single measurement sphere in the slender pillar; unit: meter. ... 108 Fig 4.3. Platen configuration for the slender pillar; a, laterally fixed loading platens; b, laterally free loading platens. ... 109 Fig 4.4. Localised axial stress in the wide pillar, w/h = 15.0m/10.0m (1.5); a, axial stress vs axial strain at spheres 1-3; b, axial stress vs axial strain at spheres 4-6; c, axial stress vs axial strain at spheres 7-9. The lateral freedom of the platens is fixed over the loading process. ... 112

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Fig 4.5. Localised X stress in the wide pillar, w/h = 15.0m/10.0m (1.5); a, X stress vs axial strain at spheres 1-3; b, X stress vs axial strain at spheres 4-6; c, X stress vs axial strain at spheres 7-9. The lateral freedom of the platens is fixed over the loading process. ... 113 Fig 4.6. Localised Y stress in the wide pillar, w/h = 15.0m/10.0m (1.5); a, Y stress vs axial strain at spheres 1-3; b, Y stress vs axial strain at spheres 4-6; c, Y stress vs axial strain at spheres 7-9. The lateral freedom of the platens is fixed over the loading process. ... 114 Fig 4.7. Localised axial stress in the wide pillar, w/h = 15.0m/10.0m (1.5); a, axial stress vs axial strain at spheres 1-3; b, axial stress vs axial strain at spheres 4-6; c, axial stress vs axial strain at spheres 7-9. The platens are free to deform laterally during the loading process. ... 116 Fig 4.8. Localised X stress in the wide pillar, w/h = 15.0m/10.0m (1.5); a, X stress vs axial strain at spheres 1-3; b, X stress vs axial strain at spheres 4-6; c, X stress vs axial strain at spheres 7-9. The platens are free to deform laterally during the loading process. ... 117 Fig 4.9. Stresses monitored in the slender pillar, w/h = 5.0m/10.0m (0.5); a, axial stress vs axial strain; b, lateral stress vs axial strain, fixed platens; c, lateral stress vs axial strain, free platens. ... 119 Fig 4.10. Simulated failure of the wide pillar loaded with laterally fixed loading platens; a, global particle displacements; b, localised zone for damage characterization; c, localised cracks and displacements; d, "hour-glass" shape. ... 121 Fig 4.11. Simulated failure of the wide pillar loaded with laterally free loading platens; a, global particle displacements; b, localised zone for damage characterization; c, localised cracks and displacements; d, through-going failure zones. ... 122 Fig 4.12. Simulated failure of the slender pillar loaded with laterally fixed loading platens; a, global particle displacements; b, localised zone for damage characterization; c, localised cracks and displacements; d, through-going shear failure. ... 123 Fig 4.13. Simulated failure of the slender pillar loaded with laterally free loading platens; a, global particle displacements; b, localised zone for damage characterization; c, localised cracks and displacements; d, through-going shear failure. ... 124 Fig 4.14. Pillar model for triaxial compression tests; a, the jointed pillar; b, platens; the platens are free to displace in the directions shown. ... 127 Fig 4.15. Particle velocity distributions for initial stress installation; a, velocity distribution in X direction; b, velocity distribution in Y direction; c, velocity distribution in Z direction. ... 128 Fig 4.16. Pillar model for Fishtank-based triaxial compression tests; a, the wall-confined pillar; b, three measurement spheres laterally across the pillar. ... 129 Fig 4.17. Lateral and axial stress vs axial strain monitored from the walls and measurement spheres in the triaxial test; a, lateral stress; b, axial stress. The confining stresses are 10.0MPa in the X and Y directions. ... 130 Fig 4.18. Axial stress vs axial strain in the uniaxial and biaxial compression tests; the pillar in the biaxial tests is confined only in the X direction. ... 133

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Fig 4.19. Axial stress vs axial strain in the uniaxial and biaxial compression tests; the pillar in the biaxial tests is confined only in the Y direction. ... 134 Fig 4.20. Simulated cracking in the uniaxial test and in the biaxial tests confined in the X direction; a, uniaxial test; b, σx = 1.0MPa; c, σx = 5.0MPa; d, σx = 10.0MPa; e,

σx = 20.0MPa; f, σx = 30.0MPa. Red and black dots denote tensile and shear

cracks respectively. ... 136 Fig 4.21. Simulated cracking and particle displacements in a localised layer parallel to the Y direction in the uniaxial and biaxial tests; a, uniaxial test; b, σx = 1.0MPa;

c, σx = 5.0MPa; d, σx = 10.0MPa; e, σx = 20.0MPa; f, σx = 30.0MPa; g, applied

confining stress. Red and black dots denote tensile and shear cracks respectively. ... 137 Fig 4.22. Simulated cracking and particle displacements in a localised layer parallel to the X direction in the uniaxial and biaxial tests; a, uniaxial test; b, σx = 1.0MPa;

c, σx = 5.0MPa; d, σx = 10.0MPa; e, σx = 20.0MPa; f, σx = 30.0MPa; g, applied

confining stress. Red and black dots denote tensile and shear cracks respectively. ... 138 Fig 4.23. Simulated cracking in the uniaxial test and in the biaxial tests confined in the Y direction; a, uniaxial test; b, σy = 1.0MPa; c, σy = 5.0MPa; d, σy = 10.0MPa; e,

σy = 20.0MPa; f, σy = 30.0MPa. Red and black dots denote tensile and shear

cracks respectively. ... 139 Fig 4.24. Simulated cracking and particle displacements in a localised layer parallel to the Y direction in the uniaxial and biaxial tests; a, uniaxial test; b, σy = 1.0MPa;

c, σy = 5.0MPa; d, σy = 10.0MPa; e, σy = 20.0MPa; f, σy = 30.0MPa; g, applied

confining stress. Red and black dots denote tensile and shear cracks respectively. ... 140 Fig 4.25. Simulated cracking and particle displacements in a localised layer parallel to the X direction in the uniaxial and biaxial tests; a, uniaxial test; b, σy = 1.0MPa;

c, σy = 5.0MPa; d, σy = 10.0MPa; e, σy = 20.0MPa; f, σy = 30.0MPa; g, applied

confining stress. Red and black dots denote tensile and shear cracks respectively. ... 141 Fig 4.26. Axial stress vs axial strain in the uniaxial and true triaxial tests, constant X stress of 5.0MPa. ... 143 Fig 4.27. Axial stress vs axial strain in the uniaxial and true triaxial tests, constant X stress of 20.0MPa. ... 144 Fig 4.28. Simulated cracking in the uniaxial test and the true triaxial tests; a, uniaxial test; b, σy = 1.0MPa; c, σy = 5.0MPa; d, σy = 10.0MPa; e, σy = 20.0MPa; f, σy =

30.0MPa; constant confining X stress of 5.0MPa. Red and black dots denote tensile and shear cracks respectively. ... 146 Fig 4.29. Simulated cracking and particle displacements in a localised layer parallel to the Y direction in the uniaxial and true triaxial tests; a, uniaxial test; b, σy =

1.0MPa; c, σy = 5.0MPa; d, σy = 10.0MPa; e, σy = 20.0MPa; f, σy = 30.0MPa; g,

applied confining stresses; constant confining X stress of 5.0MPa. Red and black dots denote tensile and shear cracks respectively. ... 147

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Fig 4.30. Simulated cracking and particle displacements in a localised layer parallel to the X direction in the uniaxial and true triaxial tests; a, uniaxial test; b, σy =

applied confining stresses; constant confining X stress of 5.0MPa. Red and black dots denote tensile and shear cracks respectively. ... 148 Fig 4.31. Simulated cracking in the uniaxial test and the true triaxial tests; a, uniaxial test; b, σy = 1.0MPa; c, σy = 5.0MPa; d, σy = 10.0MPa; e, σy = 20.0MPa; f, σy =

30.0MPa; constant confining X stress of 20.0MPa. Red and black dots denote tensile and shear cracks respectively. ... 149 Fig 4.32. Simulated cracking and particle displacements in a localised layer parallel to the Y direction in the uniaxial and true triaxial tests; a, uniaxial test; b, σy =

applied confining stresses; constant confining X stress of 20.0MPa. Red and black dots denote tensile and shear cracks respectively. ... 150 Fig 4.33. Simulated cracking and particle displacements in a localised layer parallel to the X direction in the uniaxial and true triaxial tests; a, uniaxial test; b, σy =

applied confining stresses; constant confining X stress of 20.0MPa. Red and black dots denote tensile and shear cracks respectively. ... 151 Fig 4.34. Axial stress vs axial strain in the uniaxial and conventional triaxial tests, with σx

= σy. ... 152

Fig 4.35. Simulated cracking in the uniaxial test and the conventional triaxial tests; a, uniaxial test; b, σx = σy = 1.0MPa; c, σx = σy = 5.0MPa; d, σx = σy = 10.0MPa; e,

σx = σy = 20.0MPa; f, σx = σy = 30.0MPa. Red and black dots denote tensile and

shear cracks respectively. ... 154 Fig 4.36. Simulated cracking and particle displacements in a localised layer parallel to the Y direction in the uniaxial and conventional triaxial tests; a, uniaxial test; b, σx = σy = 1.0MPa; c, σx = σy = 5.0MPa; d, σx = σy = 10.0MPa; e, σx = σy =

20.0MPa; f, σx = σy = 30.0MPa; g, applied confining stresses. Red and black

dots denote tensile and shear cracks respectively. ... 155 Fig 4.37. Simulated cracking and particle displacements in a localised layer parallel to the X direction in the uniaxial and conventional triaxial tests; a, uniaxial test; b, σx = σy = 1.0MPa; c, σx = σy = 5.0MPa; d, σx = σy = 10.0MPa; e, σx = σy =

20.0MPa; f, σx = σy = 30.0MPa; g, applied confining stresses. Red and black

dots denote tensile and shear cracks respectively. ... 156 Fig 4.38. Maximum principal stresses vs minimum principal stresses derived from the σx

= σy conventional triaxial tests. ... 157

Fig 5.1. Three fundamental crack modes defined in classical fracture mechanics: tension, shear and distortion (Anderson, 2005). ... 163 Fig 5.2. Nine failure modes observed in laboratory experiments; S, M and W denote shear, mixed shear/tensile and wing crack coalescence (Wong and Chau, 1998). ... 164 Fig 5.3. Pillar model incorporating DFN1; a, the jointed pillar model with w/h = 4.0m/7.0m; b, location of the measurement sphere. ... 168

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Fig 5.4. Stress and strain monitored in the DFN1 pillar model; a, axial stress vs axial strain; b, lateral strain vs axial strain. ... 169 Fig 5.5. DFN1 pillar and crack distribution at stress level E; a, the original pillar; b, 3D crack distribution; c, 3D crack distribution observed from the Y direction; d, 3D crack distribution observed from the X direction. Red and black dots denote tensile and shear cracks respectively. ... 171 Fig 5.6. Wing crack initiation and propagation; a, localised zone; b, three key joints for the wing cracks; c-g, wing crack propagation at stress levels A-E (Fig 5.4a). The units of D30 and D32 are number/m3 and m2/m3 respectively. ... 173

Fig 5.7. 3D views of the wing crack shape at axial stress level E (Fig 5.4a). ... 174 Fig 5.8. Geometrical relationship used to calculate the crack radius R between two particles; R1 and R2 are the radius of the two selected particles; G is the half length of the particle gap. ... 174 Fig 5.9. Localised "Damage Intensity" (D30, D32) for the wing crack model during the

loading process; stress levels A-E are denoted in Fig 5.4a. ... 175 Fig 5.10. Wing cracks in a rock pillar in Missouri, USA; a, location; b, closer view. ... 176 Fig 5.11. Crack coalescence; a, location of the monitored zone; b, key joints for the crack coalescence; c-g, progressive crack coalescence at stress levels A-E (Fig 5.4a). The units of D30 and D32 are number/m3 and m2/m3 respectively. ... 177

Fig 5.12. Localised "Damage Intensity" (D30, D32) for the large scale crack coalescence

model during the loading process; stress levels A-E denoted in Fig 5.4a. ... 178 Fig 5.13. Crack coalescence in 3D views at axial stress level E. ... 179 Fig 5.14. Crack coalescence in a hard rock pillar in Missouri, USA; a, location; b, closer view. ... 179 Fig 5.15. Pillar model incorporating DFN2; a, the jointed pillar; b, layout of the 25 measurement spheres with ID numbers. ... 180 Fig 5.16. Axial stress vs axial strain monitored from the DFN2 pillar model. ... 181 Fig 5.17. Lateral strain vs axial strain monitored from the DFN2 pillar model. ... 182 Fig 5.18. Localised mechanical behaviour at spheres 11 and 13; a, axial stress vs axial strain; b, "Damage Intensity" values at spheres 11 and 13. ... 184 Fig 5.19. Localised axial stress vs axial strain at spheres 12, 14 and 15. ... 185 Fig 5.20. Localised pillar peak strength monitored at spheres 11-15; a, peak strengths; b, sphere layout. ... 185 Fig 5.21. Lateral stress changes at spheres 11 and 13; a, X stress vs axial strain; b, Y stress vs axial strain. ... 187 Fig 5.22. Lateral stress changes at spheres 12, 14 and 15; a, X stress vs axial strain; b, Y stress vs axial strain. ... 188 Fig 5.23. Crack distribution at spheres 11 and 12 at a global axial strain of 0.178%; a, global distribution; b and c, localised distribution from different viewpoints. ... 189

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Fig 5.24. Crack distribution and particle displacements at spheres 13-15 at a global axial strain of 0.05%; a, cracks at the ends of a large joint; b, relative location of the spheres and cracks; c, particle displacements; d, particle displacements at sphere 13; e, particle displacements at spheres 14-15. ... 190 Fig 5.25. Crack distribution and particle displacements at spheres 14 and 15; a, particle displacements with no crack formation (global axial strain 0.05%); b, particle displacements with cracks (global axial strain 0.09%); c and d, 3D splitting fracture. ... 191 Fig 5.26. Splitting fracture in the DFN2 pillar at a global axial strain of 0.09%; 2D plots below are derived by clipping the 3D pillar in thin layers. ... 192 Fig 5.27. Localised mechanical behaviour at spheres 3, 8, 13, 18, 23; a, axial stress vs axial strain; b, "Damage Intensity" values at spheres 13 and 23. ... 195 Fig 5.28. Lateral stress vs axial strain at spheres 3, 8, 13, 18, 23; a, X stress; b, Y stress. Positive and negative stresses denote compression and tension. ... 196 Fig 5.29. Localised axial stress vs global axial strain for the DFN2 pillar; the global axial stress peak occurs at an axial strain of 0.118%. ... 199 Fig 5.30. Localised X stress vs global axial strain for the DFN2 pillar. Negative and positive stresses denote tension and compression respectively. ... 200 Fig 5.31. Localised Y stress vs global axial strain for the DFN2 pillar. Negative and positive stresses denote tension and compression respectively. ... 201

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Chapter 1.

Introduction

1.1. Thesis background

Pillar strength is a major concern in underground pillar design. To date, there is however arguably no accurate and efficient methodology with which to characterize the pillar strength. Due to the impracticality of direct loading tests on a rock pillar, existing analytical methods are largely based on empirical estimation and back-analysis. Empirical methods provide an important reference for the pillar strength, however, these methods inevitably contain uncertainties, for instance most empirical methods are only applicable for specific geological situations and rock types. Using these methods without due care may lead to serious safety problems. Numerical methods provide an alternative for pillar strength characterization and deformation simulation. Reliable pillar strength can be achieved when the in situ geological structures in the pillars are appropriately considered. Nevertheless most numerical methods are limited due to their inability to incorporate pre-existing structures and simulate new fractures. More realistic numerical models depend both on the explicit input of the pre-existing structures in the pillar, and the consideration of 3D effects to eliminate limitations due to plane stress or plane strain assumptions.

In order to overcome the limitations of the current empirical and numerical methods for rock pillars, this thesis introduces a Synthetic Rock Mass (SRM) approach to characterize the strength and failure mechanism of jointed pillars. The SRM is a state-of-the-art numerical method incorporating a Particle Flow Code in 3 dimensions (PFC3D, Itasca, 2010) and a Discrete Fracture Network (DFN). The PFC3D-based particle assembly serves as the fundamental element to represent intact rock in a pillar model, and the DFN inserted into the particle assembly simulates pre-existing joint sets. A

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series of applications using the SRM technique for hard rock pillar simulations are presented in this research.

1.2. Thesis objectives

This thesis used the state-of-the-art SRM method to provide a quantitative numerical technique for analysis of hard rock pillars. The principal objectives are:

• To investigate the effect of pre-existing joint sets on the mechanical behaviour of hard rock pillars.

• To characterize the confinement effect on the mechanical behaviour of hard rock pillars.

• To introduce a new numerical method for simulating 3D failure mechanisms in hard rock pillars.

1.3. Thesis structure

This thesis consists of six chapters and two appendixes. Chapter 1 presents the thesis objectives and outline of each chapter. Chapter 6 provides a summary of the entire thesis, together with recommendations for future work. The contents of the other chapters and appendixes are introduced below:

Chapter 2 reviews the widely used analytical and numerical methods for rock pillars. This chapter introduces the failure modes of hard rock pillars in practice, and the empirical methods for pillar designs. Applications of the numerical models which have already been used or potentially can be used for hard rock pillars, including Finite Element Method (FEM), Finite Difference Method (FDM), Discrete Element Method (DEM) and Hybrid Method (HM) are reviewed. Chapter 2 finally highlights the basic concepts and applications of the Particle Flow Code and the Synthetic Rock Mass technique used in this thesis.

Chapter 3 focuses on the effect of joint set orientation, size and persistence on rock pillars. The effects of these factors are demonstrated by comparing the simulated peak strength, deformation modulus, lateral stiffness ratio and post-peak response of a

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joint-free intact pillar and a series of jointed pillars. Internal-strain loading is optimized to load the pillars for more realistic numerical results. Crack initiation stress and crack damage stress thresholds, are characterized for the jointed pillars.

Chapter 4 investigates the confinement effect on rock pillars using the SRM method. The confinement effect is initially characterized using two SRM pillars with varied slenderness (width/height ratio). The confinement effect on the wide pillar is reflected in the high axial and lateral stresses recorded at the pillar core, and is attributed to end restraints due to the loading platens. The confinement effect on the slender pillar, as expected, is reduced considerably. A series of triaxial compression tests are then conducted and the confinement effect on pillar strength and post-peak strain-softening gradient systematically examined. The pillar failure modes in these triaxial compression tests are discussed.

Chapter 5 addresses the simulation and characterization of 3D crack propagation in two jointed pillars. The statistical parameters of the DFNs in the two pillar models are identical, but the actual generated fracture distributions are different. Wing crack and large scale crack coalescence are characterized in one of the pillars with DFN1 and damage quantified using a proposed "Damage Intensity" parameter. More localised failure mechanisms are characterized using 25 measurement spheres in the second pillar, using the stress-strain relationship, "Damage Intensity", crack distribution and particle displacement. The axial and lateral stress distributions in the 25 measurement spheres are presented and discussed.

Appendix A presents a full set of the jointed pillar models discussed in Chapter 3. Appendix B shows a further study on the damping effect on SRM pillars. Large scale pillar deformation is generally difficult to simulate in a uniaxial compressive test, unless the default damping force is removed from the SRM models. The damping-reduction method provides an unconventional perspective on the SRM simulation, and the results may be useful to simulate large scale displacements in jointed pillars.

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Chapter 2.

Literature review

2.1. Introduction

In this chapter a systematic literature review on pillar failure modes, followed by empirical and numerical methods for pillar designs, is presented. The literature review also highlights basic concepts and applications of the Particle Flow Code, PFC2D/3D and the Synthetic Rock Mass (SRM) approach. The specific topics covered are:

• Failure modes of hard rock pillars, including structure-controlled and stress-controlled pillar failures.

• Tributary area analysis and empirical formulae for pillar strength estimation. • Numerical methods for pillar design, including state-of-the-art continuum,

discontinuum and hybrid numerical methods used for pillar strength and deformability analysis.

• Basic concepts and applications of PFC2D/3D and SRM. The basic concepts presented include the contact model, law of motion, damping and "smooth joint" logic. Applications include simulations of laboratory scale rock samples and rock masses using PFC2D/3D and SRM.

2.2. Failure modes of hard rock pillars

The failure mode of a hard rock pillar is controlled by several factors, including rock type, geological structure, pillar slenderness, host rock and in-situ stress conditions (Brady and Brown, 2007). The failure modes of underground tunnels, originally proposed by Hoek et al. (1995), are suitable to describe hard rock pillar failures, where the failure modes are considered as structure-controlled and stress-controlled types. In a low in-situ stress environment, the existing geological structures generally control the pillar failure, behaving as unstable rock block sliding along discontinuities. As the in-situ stress

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increases with mining depth, the effect of pre-existing discontinuities becomes less significant and the failure generally behaves as stress-controlled tensile fractures parallel to the maximum principal stress.

Brady and Brown (2007) presented four pillar failure modes commonly observed in the field. Fig 2.1a shows a rock block sliding along pre-existing discontinuities. This type of failure occurs when relatively persistent joints exist in the pillar. Fig 2.1b shows a through-going shear failure across the entire pillar. The slenderness of the pillar favours the formation of the global shear band. Fig 2.1c and d demonstrate two additional structure-controlled failure modes when a persistent joint set exists in the pillar. Shear sliding occurs along each through-going plane when the joint set is inclined; and buckling failure occurs when the joint set is vertically orientated.

Lunder (1994) proposed a four-stage progressive failure process for hard rock pillars. The failure is essentially stress-controlled and usually occurs in wide pillars in a high in-situ stress condition. The initial damage occurs at the pillar corners. Surface spalling and isolated axial fractures then follow as a sign of a higher damage level. At this stage the pillar has partially failed but the pillar core remains intact. As the axial stress increases, extensive axial fractures develop and the pillar core is progressively degraded. As the surface spalling becomes deeper, a typical “hour-glass” shape is formed and eventually results in a through-going shear band and the complete pillar failure.

Roberts et al. (2007) developed a 6-stage pillar rating system to describe progressive stress-induced pillar failures and used it as the primary reference to constrain numerical models. The pillar rating system is based on many years of observations on the Doe Run mine hard rock pillars so it represents closely the observed pillar failure characteristics. As shown in Fig 2.2, the entire pillar failure process is divided into 6 stages, from an intact pillar (Stage 1) to a failed pillar (Stage 6). The surface spalling and fractures parallel to the axial loading are typical signs of the stress-induced pillar failures. The spalling depth increases with a higher external loading and transforms the pillar's original prismatic shape into an "hour-glass" shape. Axial fractures

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meanwhile propagate longer and toward the pillar core. The pillar eventually fails due to either the extreme "hour-glass" shape or necking.

a. b.

c. d.

Fig 2.1. Structure-controlled failures of jointed pillars; a, rock block sliding; b, through-going shear failure; c, shear failure along transgressive joints; d, buckling failure (Brady and Brown, 2007).

Styles et al. (2010) developed a close-range digital photogrammetry approach for hard rock pillar damage characterization. The typical "hour-glass" shape is captured by inserting a parallel plane on the pillar surfaces in a 3D photogrammetric model. The intersection lines between the plane and the pillar surfaces are captured, as shown in Fig 2.3. The photogrammetry technique offers an efficient way to characterize pillar damage without direct access to the pillar surface.

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Stage 1 Stage 2

Stage 3 Stage 4

Stage 5 Stage 6

Fig 2.2. Pillar rating system; Stage 1, intact pillar; Stage 2, minor spalling and short axial fractures; Stage 3, substantial spalling, axial fracture length shorter than the half pillar height; Stage 4, continuous open fractures cutting towards pillar core, formation of the "hour-glass" shape; Stage 5, large continuous open fractures, well developed “hour-glass” shape; Stage 6, failed pillar by either

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Fig 2.3. Close-range photogrammetry to capture the "hour-glass" shape of a pillar; a, average pillar surface; b, artificial plane; c, cross section calculation; d, surface profile (Styles et al., 2010).

Esterhuizen et al. (2011) conducted detailed field characterization for underground stone pillars in the United States. Three factors were suggested from the field survey that may influence the pillar stability: a, large angular discontinuities extending from the roof to floor; b, extrusion from weak bands; c, high stress-induced spalling.

2.3. Tributary area analysis

Tributary area analysis is a practical and simple method to estimate the in situ axial stress on a rock pillar. Details of this method are illustrated in Fig 2.4 (Brady and Brown, 2007). Assuming the pillar length and width are a and b; the room span is c; the tributary area representing the pillar is (a+c) (b+c); the vertical equilibrium requires:

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1/ 1 (2.2) Where σp is the average axial stress on the pillar; Pzz is the axial component of

pre-mining stress and r is the area extraction ratio, yielding:

/ (2.3)

Fig 2.4. Tributary area analysis for estimating stresses on a pillar (Brady and Brown, 2007).

2.4. Empirical formulae

Many formulae have been developed to empirically estimate the pillar strength. The parameters usually include the pillar's width/height (w/h) ratio, and some additional coefficients. These empirical formulae are based on the "back analysis" of direct field observations and have been successfully applied in many mining systems in the past decades (Sakurai, 1993). Existing formulae are summarized below:

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Salamon and Munro (1967) systematically investigated 125 coal pillars and proposed the pillar strength be calculated using Eq 2.4:

(2.4)

Where: σ is the pillar strength (psi), K is the strength of the unit volume of coal (psi), and W and H are pillar width and height (m) respectively. Table 2.1 and Table 2.2 list the field observation data and suggested coefficients for Eq 2.4. Note these parameters are only suitable for coal pillars in South Africa room-and-pillar mines, and may not be appropriate for hard rock pillars due to different in situ geological conditions. Table 2.1. Database summary for compiled cases (Salamon and Munro, 1967).

Group Stable Collapsed

Number cases 98 27

Depth (feet) 65-720 70-630

Pillar height (feet) 4-16 5-18

Pillar width (feet) 9-70 11-52

Extraction ratio 37-89 45-91

Width/Height ratio 1.2-8.8 0.9-3.6

Table 2.2. Empirical parameters suggested by Salamon and Munro (1967).

K α β

1322 psi 0.6609 0.459

Hedley and Grant (1972) conducted a case study in Canadian Elliot Lake room-and-pillar uranium mines. They analyzed 28 rib-pillars (3 failed, 2 partially failed and 23 stable), where the long-axis of these pillars is parallel to the dip direction of the in situ quartzites. They used Eq 2.5 to calculate the pillar strength:

.

. (2.5)

Where K is set as 179.0MPa. Lunder (1994) argued that Eq 2.5 can be applied to underground hard rock pillars, but the database contains only three failed rock pillars so the application of only three failed rock pillars to develop a strength relationship may not

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be robust. A larger database including more failed and stable rock pillars may be required.

Hustrulid (1976) proposed to use Eq 2.6 to estimate the strength of prism pillars. This formula considers the scale effect of the laboratory sample strength using two additional parameters h and hcrit. Here h is the cylinder height of the laboratory sample,

and hcrit is the minimum height of a cubic pillar material, where the volume of the cube is

equal to the Representative Element Volume (REV). The REV is the minimum volume of a rock sample/ mass where the peak strength and deformation modulus remain constant if the volume becomes larger (Hudson and Harrison, 1997).

σ 0.875 0.25 q (2.6)

σ is the pillar strength (MPa), W and H are the pillar width and height (m), qu is

the uniaxial compressive strength of the laboratory sample (MPa).

Galvin et al. (1999) and Galvin (2006) developed a strength formula (Eq 2.7) for coal pillars in Australia. An updated empirical formula (Eq 2.8) was proposed later by the same authors based on a larger database. It was concluded that the discontinuities within a coal pillar control the pillar strength when the pillar's w/h ratio is less than 2.0, and the pillar strength is higher than the calculated strength using Eq 2.8 when the w/h ratio is higher than 4.0.

8.60 _..

(2.7)

6.88 _..

(2.8)

Martin and Maybee (2000) summarized additional formulae corresponding to rock pillar strength with various rock types and pillar shapes (Table 2.3). They emphasized that for all pillar strength formulae, except that suggested by Lunder and Pakalnis (1997), the effect of σ is ignored. While Lunder and Pakalnis (1997) attempted to include the effect of σ using a parameter K; the formula predicts similar strength

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values as the other formulae in Table 2.3. The effect of σ hence is proposed to be ignored by empirical formulae in matching the observed pillar failures.

Table 2.3. Empirical formulae for hard rock pillars (Martin and Maybee, 2000).

Pillar strength

formulae (MPa) strength (MPa) Rock sample Rock type pillar number Investigated References

133 _.. 230 Quartzites 28 Hedley and Grant, ₁₉₇₂

65 _.. 94 Metasediments 57 Kimmelmann et al, ₁₉₈₄

35.4 0.778

0.222 100 Limestone 14 Krauland and Soder, 1987 0.42σ — Canadian Shield 23 Potvin et al., 1989 74 0.778

0.222 240 Limestone/Skarn 9 Sjoberg, 1992

0.42σ 0.68

0.52k — Hard rocks 178 Pakalnis, 1997 Lunder and Unit of W and H is meter.

Roberts et al. (2007) argued that the traditional empirical strength formulae fail to match the field observations of Doe Run pillars and proposed adoption of the Confinement Method (Lunder, 1994) to calculate the pillar strength:

1 2 (2.9)

Where: σ = pillar ultimate strength (MPa); K = pillar strength size factor; UCS = pillar uniaxial compressive strength (MPa); κ = pillar friction term; C1, C2 = empirical rock mass constants. κ is calculated by:

tan

(2.10)

0.75 . /

_(2.11)

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Existing empirical formulae provide a valuable reference for underground pillar designs, however, Kaiser et al. (2010) emphasized that the data resource of failed pillars is limited and in most cases is from shallow depth slender pillars (depth < 600m; w/h < 1.5). The formulae may be inaccurate for pillars at depth > 1000m and with w/h > 1.5-2.0. The implied increase in pillar strength with a higher w/h ratio becomes unrealistic for an extremely wide pillar, as the emprical formulae suggest that a very wide pillar has a finite strength which essentially can be treated as infinite (Kaiser et al., 2010). In addition, these formulae are dependent on site specific factors (rock types) in different mines, and potential uncertainties may arise if adopted without care in other mining systems (Mortazavi et al., 2009). Furthermore, although these formulae provide strength of a rock pillar under complex in-situ conditions, they do not involve the deformation characteristic in the pillar failure process.

2.5. Numerical methods for pillar designs

The numerical methods reviewed in this section include continuum, discontinuum and hybrid pillar simulations. The continuum method is generally based on a triangular or quadrilateral mesh assembly. Due to the difficulty of explicitly incorporating a complex DFN in a mesh assembly, most continuum methods use equivalent material with reduced strength and deformation modulus (compared to laboratory samples) to implicitly involve discontinuities. The heterogeneity of a pillar is simulated by assigning varied material properties among elements, usually following a Gaussian or Weibull distribution. Most continuum methods are capable of simulating pillar failures with in-situ stress changes, not specific failures along pre-existing discontinuities. The discontinuum method, in contrast, supports explicit insertion of the discontinuities, so that rock block shear failure can be simulated. This makes the discontinuum method a better option for rock pillar failure simulations as discontinuities generally impose significant effects on pillar strength and deformability, especially under a low in-situ stress environment. The hybrid methods are developed to overcome some disadvantages of the discontinuum methods. For instance, conventional discontinuum methods (such as UDEC, Itasca, 2012) cannot support non-persistent discontinuities in the model, which leads to a difficulty of simulating a rock block falling when part of the block is still connected to the

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host rock. This limitation also makes the simulation of crack initiation and nucleation between non-persistent discontinuities more complex (a suggested solution is to assign different mechanical properties to different sections in a through-going discontinuity, as shown in Fig 2.5). The hybrid methods allow improved simulation of pillar failures, where tensile and shear failures are explicitly simulated between inserted non-persistent discontinuities. More importantly, the rock blocks, once formed by discontinuities and step paths, can slide and rotate during the pillar failure process. A more realistic pillar failure simulation can then be expected.

Fig 2.5. Modelling wing cracks in a through-going jointed model. The properties of Section AB and CD are calibrated to fit the property of the intact rock; the property of Section BC is defined by the joint.

2.5.1.

Continuum methods

The continuum methods consist of Boundary Element Method (BEM), Finite Element Method (FEM) and Finite Difference Method (FDM). The principles of these numerical methods are reviewed by Jing and Hudson (2002), Jing (2003). Applications of these methods are reviewed below.

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Boundary Element Methods (BEM)

The main advantage of the BEM for rock mass modeling is the reduction of computational expense, with much simpler mesh generation and input data preparation, compared with full domain discretization methods such as FEM and FDM (Jing, 2003). The BEM only divides boundaries of a model into elements and mathematically leaves the interior as an infinite continuum. The basis of the BEM is the definition and solution of a problem entirely in terms of surface values of traction and displacements. The BEM is particularly useful when linear elastic behaviour can be assumed for a rock mass, or when continuous weak planes separate an elastic domain (Elmo, 2006).

Three different modes of BEMs are applicable for rock mass modeling, including Direct Method, Indirect Method and Displacement Discontinuity Method (DDM). Yan (2008) summarized the characteristics of these BEMs:

• The Direct Method explicitly describes stresses and/or displacements and solves them directly from the specific boundary conditions;

• In the Indirect Method, stress conditions on the boundaries are solved first and separate relations are applied to calculate boundary displacements;

• The Displacement Discontinuity Method (DDM) represents the result of pulling apart a "slit" and uses these "slits" as discontinuities for jointed rock mass models.

The DDM code NFOLD has been consistently used for stability analysis of Doe Run pillars (Lane et al., 2001). In the NFOLD model, the ore body is located within an infinite elastic material (the host rocks) and the elements characterizing the mining horizon may be either elastic, nonlinear with post-peak properties, or representative of backfill material. Roberts et al. (2007) described the application of NFOLD for modeling Doe Run pillars. The geometry of the NFOLD model is created by placing several 2.4m grids originally generated in AutoCAD. The central elements are assigned a higher strength to consider the lateral confinement effect. The pillar strength is calculated in Eq 2.12. This formula is suggested to be consistent with observed responses of Missouri lead belt pillars between 10–15m high, but for pillars higher than 15m, it provides conservative results. Pillar strength/stress ratios (FOS) predicted from NFOLD models are linked to the pillar rating system (Section 2.2) for an integrated pillar design

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/ _/ /

_(2.12)

σp is ultimate pillar strength (MPa); K is constant; UCS is laboratory sample

uniaxial compressive strength (MPa); W and H are pillar width and height (m).

Finite Element / Finite Difference Methods (FEM/FDM)

The difference between the FEM and FDM lies in the methodology of calculating nodal displacements in each element. The FEM assembles local element equations and solves these equations in a global matrix system, whilst FDM approximates the related partial derivatives directly with differences at regular or irregular grids (Jing, 2003). In practice, these two methods provide similar results. The modelling applications are therefore reviewed together in this section.

Hoek and Brown (1980) presented the applications of using FEM for rock pillar stress characterizations. They concluded that as the pillar becomes taller and narrower, the stress distribution across the mid-height of the pillar becomes more uniform. In the case of a very slender pillar, the stress distribution across the centre of the pillar is very close to a uniaxial compressive condition: and 0, where σp is the average

external stress; σ1 and σ3 are the maximum and minimum principal stresses in the pillar.

Tang and Kaiser (1998) presented the applications of a FEM code RFPA2D to simulate crack initiation and crack propagation in rock samples. Kaiser and Tang (1998) made a further effort to use RFPA2D to simulate damage accumulation and seismic energy release during the failure process in rib pillars. They confirmed that a "soft loading system" promotes unstable pillar failures, and weak pillar foundations drastically reduce the pillar's load bearing capacity. Tang et al. (2000) simulated the wing crack initiation and coalescence between two/three artificial flaws. Although these simulations are at the laboratory sample scale, the modelling principles can be applied to a realistic pillar. More applications of RFPA2D for pillars are presented by Wang et al. (2011), where the failure mechanism of serial and parallel rock pillars is characterized.

Martin and Maybee (2000) proposed the use of brittle Hoek-Brown parameters for hard rock pillar failure simulations, as the traditional Hoek-Brown criterion

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over-predicts the rock mass strength under low confinement. A Phase2 model was used to simulate progressive spalling damages, Fig 2.6. They emphasized that the brittle Hoek-Brown parameters are suitable for slender pillars but may be inappropriate for wide pillars.

Fig 2.6. Pillar failure simulations using Phase2; crosses and circles represent shear and tensile damage respectively (Martin and Maybee, 2000).

Fang and Harrison (2002) developed a local degradation model to simulate rock pillar failures. The model is based on a "degradation index" factor, where the rock fracture at the microscopic level is classified into brittle and ductile components. The brittle component results in degradation of local material elasticity and strength, whilst the ductile component only leads to plastic deformation. This model is verified using a FDM code FLAC2D (Itasca, 2013a), as shown in Fig 2.7.

Feng et al. (2006) introduced a continuum Elasto-Plastic Cellular Automaton (EPCA) numerical method for hard rocks. The EPCA, with consideration of initial rock material heterogeneity, is capable of realistically simulating fracture localization and parallelization under the uniaxial compression process.

Sainsbury et al. (2008) proposed a Strain Softening Ubiquitous Joint Rock Mass (SUJRM) constitutive model to calculate the jointed rock mass strength. The SUJRM allows for installation of ubiquitous joints at the element level, where joint dips and dip

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directions are based on the macroscopic discontinuity orientations. Using the SUJRM technique, realistic mechanical responses such as non-linear strain hardening/softening can be simulated.

Fig 2.7. Progressive failures of a rock pillar using the local degradation method (Fang and Harrison, 2002).

Jaiswal and Shrivastva (2009) described a 3D FEM method for modelling softening behaviour of coal pillars. They treated the pillar as a Hoek-Brown strain-softening material and concluded that the pillar yielding starts at 2/3 of the peak strength. For Indian coal pillars having a w/h ratio less than 5.0, the pillar strength is almost linearly dependent on the w/h ratio (Fig 2.8a), and non-linearly dependent on UCS of the coal specimen. The post failure modulus of the coal pillar is non-linearly dependent on the w/h ratio, and not dependent on the UCS (Fig 2.8b).

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a.

b.

Fig 2.8. Mechanical parameters associated with the pillar w/h ratios; a, pillar strength vs w/h ratio; b, post-peak modulus vs w/h ratio (Jaiswal and Shrivastva, 2009). Tesarik et al. (2003, 2009) adopted a 3D FEM code UTAH3 to model the excavation and backfill for Doe Run pillars. The yield criterion in UTAH3 is Drucker-Prager, in which strength is dependent on three principal stresses. An associated flow rule is applied to calculate strains in the yielded elements.

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Mortazavi et al. (2009) investigated the mechanical behaviour of hard rock pillars using FLAC2D (Itasca, 2013a). The application of the elastic-perfect-plastic model, suitable for relatively "soft" rock pillars, results in an overestimation of the pillar confinement effect and pillar strength. For a large w/h ratio, the pillar demonstrates a hardening behaviour and when the w/h ratio is higher than 5.0, the pillar does not fail in practice. Therefore, the strain-softening constitutive model in FLAC2D can be used for simulating hard rock pillar deformation.

2.5.2.

Discontinuum methods

In this section, the applications of the Universal Distinct Element Code (UDEC) Voronoi for hard rock samples are reviewed. The modelling principles can be used for rock pillars. The concepts and applications of the Particle Flow Code (PFC3D), as the fundamental numerical tool in this research, are reviewed in a separate Section 2.6.

UDEC Voronoi (Itasca, 2012) is a powerful numerical method for rock failure modeling. Unlike a traditional UDEC model where rock blocks are solid and the block failure behaves as deformed zones, UDEC Voronoi generates randomly sized polygonal blocks and represents fractures as flaw failures between these polygons. The polygonal blocks can also be defined as deformable, and the normal or shear failures of the blocks are simulated.

Christianson et al. (2004) proposed a "standard procedure" for calibrating parameters for UDEC Voronoi models and simulated the mechanical responses of Lithophysal Tuff in a series of compression tests. Kazerani and Zhao (2010) discussed the advantages and disadvantages of using UDEC Voronoi for rock brittle failure simulations, and systematically investigated the influences of the Voronoi parameters on macro rock properties (cohesion, friction angle and UCS).

Alzo’ubi (2009) adopted the UDEC Voronoi to simulate the mechanical behaviour of plaster of Paris at the laboratory scale. The numerical results suggest that UDEC Voronoi is capable of explicitly simulating fracture propagations in the pre-peak and post-peak stages (Fig 2.9).

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Fig 2.9. UDEC Voronoi model and the simulated fractures in the post-peak stage (Alzo'ubi, 2009).

2.5.3.

Hybrid methods

In this section, simulations using hybrid FEM/DEM code ELFEN (Rockfield, 2007) for hard rocks are reviewed. The ELFEN model supports explicit installation of non-persistent discontinuities and can explicitly simulate fracture coalescence between these discontinuities. The results of the ELFEN models are suggested to be more accurate for simulations of integrated pillar failures involving rock bridge failures and subsequent shear sliding along weak zones.

Pine et al. (2006) used ELFEN to simulate the failure process of mine pillars. An example of the progressive pillar failures is presented in Fi

Modelling hard rock pillars using a Synthetic Rock Mass approach