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Hard rock pillar strength estimation an applied empirical approach Lunder, Per John 1994

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HARD ROCK PILLAR STRENGTH ESTIMATIONAN APPLIED EMPIRICAL APPROACHByPER JOHN LUNDERB.A.Sc., The University of British Columbia, 1983A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinTHE FACULTY OF GRADUATE STUDIESDEPARTMENT OF MINING AND MINERAL PROCESS ENGINEERINGWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRiTISH COLUMBIAAugust 18, 1994© Per John Lunder, 1994In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Ubrary shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature)______________Department of Mi vzo c&SThe University of British ColumbiaVancouver, CanadaDate PRJ’1 tQ ) t)’-1.DE$ (2)88)ABSTRACTPillars are present in all hard rock mining operations and in order to effectively design these pillars, anestimate of the pillar strength is required. Two new pillar strength estimation methods for hard rock minepillars are presented in this thesis. 31 pillar case histories of the database that was used to develop these newformulae were acquired during a cooperative study, entitled “Ground Stability Guidelines for the Extraction ofBarrier Pillars in Hard Rock Mines”, between Westmin Resources Ltd. and The Canadian Centre for Mineraland Energy Technology (CANMET). 147 additional case histories were acquired from six documented hardrock pillar case studies in the literature, resulting in a combined database of 178 case histories.The combined database is comprised mainly of massive sulphide pillars with rock mass ratings ofbetween 60% and 85%. Major structural features were not deemed to be an influence in pillar instability.Pillar stressess were calculated using either tributary area theory or numerical modelling methods. The factorsdetermined to influence pillar strength for the combined database therefore are:• the average pillar confinement (which is a function of pillar geometry)• the unconfined compressive strength of the intact pillar material• the stresses that a pillar is subjected toThe degree to which a pillar has failed is quantifiable using a pillar stability classification index whichranges from “1” (stable) to “5” (failed). The estimation of pillar stresses is preferably determined using three-dimensional numerical modelling, but in some situations two-dimensional numerical modelling or tributaryarea theory may provide adequate results. It was concluded that the full size unconfined compressive strengthof a pillar can be approximated by a strength size factor of 44 percent of the small scale unconfinedcompressive strength of intact pillar material.Two pillar strength formulae have been developed from the combined pillar database: “The Log-PowerShape Effect Formula” and “The Confinement Formula”. Both of the methods utilize the average pillarconfinement. “The Log-Power Shape Effect Formula” is a refined shape effect formula which has a formsimilar to that proposed by researchers in the past. “The Confinement Formula” has a form that resembles theMohr-Coulomb shear strength formula.The combined database was analyzed and the predicted strengths from “The Confinement Formula”was compared to the results for existing pillar strength methods (Hedley & Grant (1972), Bieniawski (1975),Salamon & Munro (1967), Obert & Duvall (1967), Hoek & Brown (1980)). “The Confinement Formula” isshown statistically to be the most reliable method of estimating the strength of the pillars that make up thecombined database.11TABLE OF CONTENTSABSTRACTTABLE OF CONTENTSLIST OF TABLESLIST FIGURES xLIST OF PHOTOS xviACKNOWLEDGMENTS xvii1. INTRODUCTION 11.1 CONTENTS OF THESIS 11.2 PILLAR STRENGTH ASSESSMENT - AN OVERVIEW 21.3ROLEOFPILLARS1NMINLNG 32. PILLAR DESIGN METHODOLOGY 52.1 PILLAR STRESS DETERMINATION 72.1.1 Tributary Area Theory 82.1.1.1 Pariseau (1982) Inclined Stress Formulae 122.1.1.2 Szwilski (1982) Chain Pillar Formula 132.1.1.3 Hedley & Grant’s (1972) Formula for Inclined Pillars 132.1.1.4 Subsidence Formula 142.1.2 Numerical Methods 152.1.3 Discussion - Pillar Stress Determination 162.2 PILLAR STRENGTH DETERMiNATION 172.2.1 Empirical Design Methods 172.2.1.1 Linear Shape Effect Formula 182.2.1.1.1 Obert&Duvall (1967) 182.2.1.1.2 Bieniawski (1975) 192.2.1.1.3 Hudyma (1988) 202.2.1.2 Power Shape Effect Formula 202.2.1.3 Effective Pillar Width 212.2.1.4Hoek & Brown (1980) Failure Criteria 222.2.1.5 The Size Effect Formula 252.2.1.5.1 Salamon & Munro (1967) 262.2.1.5.2 Hedley & Grant (1972) 272.2.1.5.3 Sheorey et al. (1987) 302.2.1.6 Discussion - Empirical Strength Formulae 302.2.2 Theoretical Design Methods 302.2.2.1 Wilson’s (1972) Confined Core Method 30Lu2.2.2.2 Coates (1965) .322.2.2.3 Panek (1979) 322.2.2.4 Grobbelaar (1970) 332.2.2.5 Discussion - Theoretical Strength Formulae 342.2.3 Heuristic Methods 342.2.3.1 Mines Inspector Formula 342.2.3.2 Holland (1964) Formula 352.2.3.3 Morrison et al. (1961) 352.2.3.4 Barrier Pillar Formula 352.3 NUMERICAL MODELLING TECHNIQUES 362.3.1 Pillar Design Using Numerical Models 362.3.2 Types of Modelling Methods 372.3.2.1 Continuum Methods 372.3.2.2 Integral Methods 372.3.2.3 Discontinuum Methods 382.3.2.4 Hybrid Methods 382.3.3 Choice ofMethod 402.3.4 Pillar Failure Assessment using Numerical Models 422.3.5 Discussion - Numerical Methods 422.4 PILLAR STRENGTH ESTIMATION METHODOLOGY ATWESTMIN RESOURCES LTD.’S H-WMINE 452.5 CHAPTER SUMMARY 453. PILLAR FAILURE MECHANISM 463.1 PILLAR STABILiTY ASSESSMENT 473.2 FACTORS RELATED TO PILLAR STABILITY 473.2.1 Intact Rock Strength 493.2.2 Pillar Stress 493.2.3 Pillar Shape 493.2.4 Pillar Volume 503.2.5 Pillar Modulus 503.2.6 Constitutive Relationship 503.2.7 Pillar Confmement 513.2.8 Structural Features 523.3 PILLAR STABILITY CLASSIFICATION METHOD AT WESTMIN RESOURCES LTD. H-W MINE533.4 CHAPTER SUMMARY 544. PILLAR STRENGTH ESTIMATION AT WESTMIN RESOURCES LTD.’S, H-W MINE, - A CASESTUDY 55iv4.1 GEOLOGY OF THE H-W MINE 554.2 MINING PRACTICE AT THE H-W MINE 584.3 WESTMIN IN-SITU DATABASE 614.3.1 Intact Strength Analysis 614.3.2 Fabric Analysis 614.3.3 Rock Mass Classification 634.3.4 Geometry 634.3.5 In-Situ Stress Determination 644.4 PILLAR STRENGTH ESTIMATION ATWESTMIN RESOURCE LTD.’S H-W MINE 664.4.1 Numerical Modelling 664.4.1.1 Map3D Numerical Modelling Program 674.4.1.2 Westmin Map3D Modelling Sessions 684.4.1.2.1 Sensitivity Analysis 694.4.1.3 Map3D - Core Barrier Pillar Stress vs. Average Barrier Pillar Stress 704.4.1.4 Determination Of Average “Small” Pillar Stresses at Westmin Resources 704.4.1.5 Model Calibration to Actual Mining Conditions 734.4.1.6 Numerical Modelling Summary 734.4.2 Pillar Stability Classification 734.4.3 Pillar Geometry 744.4.4 Westmin Pillar Database 744.4.5 Development of a Pillar Strength Relationship 754.4.5.1 Pillar Stability Plots 804.4.5.1.1 Excluded Pillars 804.4.5.2 Division of Pillar Classes 824.4.5.3 Factor of Safety 844.4.5.4 Stability Line with Individual Data Sets 844.4.5.5 Pillar Strength Relationship 864.5 CHAPTER SUMMARY 875. NEW PILLAR STRENGTH FORMULAE 885.1 PILLAR DATABASE 885.1.1 Westmin Database 895.1.2 Hudyma (1988) 895.1.3 Von Kimmelman et al. (1984) 945.1.4 Hedley & Grant (1972) 955.1.5 Sjoberg (1992) 955.1.6 Krauland & Soder (1987) 965.1.7 Brady (1977) 100v5.1.8 DatabaseSummary.ioo5.2 HARD ROCK PILLAR SThENGTH FORMULAE DERIVATION .1025.2.1 Empirical Strength Formula Derivation Methodology 1025.2.2 Requirements for a Strength Formula 1025.2.3 Pillar Strength Variables 1035.2.3.1 Pillar Geometry 1045.2.3.2 Pillar Confinement 1045.2.3.2.1 Detailed Modelling 1055.2.3.2.2 Additional Modelling 1055.2.3.2.3 Discussion - Pillar Confinement 1075.2.3.3 Intact Rock Strength 1105.2.3.4 Influence Of Pillar Volume On Strength 1105.2.3.5 Pillar Stress Determination 1115.2.3.6 Pillar Stability Classification 1125.2.4 Refmed Empirical Strength Formulae 1135.2.4.1 Linear Shape Effect 1135.2.4.2 Refined Power Shape Effect 1175.2.4.3 Log-Power Shape Effect 1175.2.4.4 Discussion - Refined Empirical Strength Formulae 1195.2.5 A New Strength Hypothesis - The Confinement Formula 1205.2.5.1 Mohr-Coulomb-Navier’s Theory 1215.2.5.2 Frictional Effect of Mine Pillars 1215.2.5.3 The Confinement Formula 1245.3 SUCCESS OF THE EMPIRICAL AN]) PROPOSED FORMULAE 1275.3.1 Success Matrix Methodology 1275.3.2 Success of Confmement and Refined Empirical Formulae 1285.3.3 Success of Confmement Formula against Past Formulae 1295.3.3.1 Hock & Brown (1980) Pillar Curves 1295.3.3.2 Other Methods with Original Strength Size Coefficient 1295.3.3.3 Revised Methods with New Strength Size Coefficient 1295.4 CHAPTER SUMMARY 1306. SUMMARY and CONCLUSIONS 1366.1 SUMMARY 1366.1.1 Current Pillar Strength Determination Methods 1366.1.2 Assessment of Pillar Failure 1366.1.3 Pillar Strength Estimation Methods 1376.1.3.1 Refined Empirical Strength Formulae 137vi6.1.3.2 ANew Strength Hypothesis .1376.1.4 Pillar Strength Calibration 1386.2 FUTURE WORK 1386.3 CONCLUSIONS 138BIBLIOGRAPHY 139APPENDIX A - TWO - DIMENSIONAL PARAMETRIC MODELLING RESULTS 147APPENDIX B - METRIC TO IMPERIAL CONVERSION OF UNITS USED IN THESIS 166viiLIST OF TABLESTable I: “Linear Shape Effect” emprirical constants, “A” and “B”, from various authors 19Table 2: Approximate relationship between rock mass quality and material constants (after Hoek & Brown,1988) 23Table 3: “Size Effect Formulae” empirical constants, “a” and “b”, from various authors 26Table 4: Salamon & Munro (1967) database summary for compiled case histories 27Table 5: “Size Effect” empirical constants determined by Salamon & Munro (1967) 27Table 6: Pillar failure classification method (after Krauland & Soder, 1987) 48Table 7: Westmin Resources Ltd.’s pillar stability classification method 54Table 8: Westmin intact rock properties for the H-W Main Zone 62Table 9: Summary of joint features or the Westmin H-W Main Zone 64Table 10: In-situ triaxial stress measurement results at the H-W Mine 66Table 11: In-situ biaxial stress measurement results at the H-W Mine 66Table 12: Pillar stability classification method for Westmin Resources Ltd.’s H-W Mine 74Table 13: Original pillar classification database for Westmin Resources Ltd.’s H-W Mine 76Table 14: Factor of safety determined for pillar stability classification division lines 84Table 15: Westmin Resources Ltd., summary pillar database 90Table 16: Hudyma (1988) original geometric and pillar stability classification data (after Hudyma, 1988) 91Table 17: Hudyma (1988) original stress data (after Hudyma, 1988) 92Table 18: Hudyma (1988), summary pillar database 93Table 19: Summary of rock properties from the Selbi-Philcwe Mines (after Von Kimmelman et al. 1984) 95Table 20: Original pillar classification data for square pillars from the Selbi-Phikwe Mines (after VonKimmelman et al., 1984) 97Table 21: Von Kimmelman et al. (1984), summary pillar database 98Table 22: Hedley & Grant (1972) original pillar database. (after Hedley & Grant, 1972) 99Table 23: Hedley & Grant (1972), summary pillar database 100Table 24: Sjoberg (1992), summary pillar database 101Table 25: Krauland & Soder (1987), summary pillar database 101Table 26: Brady (1977), summary pillar database 101Table 27: Results of two-dimensional parametric modelling sessions to investigate average pillarconfinement 106Table 28: Results of two-dimensional parametric modelling sessions investigating the effects of differingmodelled extraction ratios on the average pillar confinement 108Table 29: Common pillar stability assessment designation for each individual database in the combineddatabase 112VIIITable 30: Linear shape effect constants and strength size factor determined for each of the individual databasesin the combined database 114Table 31: Pillar strength prediction success matrix methodology 127Table 32: Summary of success matrix prediction methodology 127ixLIST OF FIGURESFigure 1: Sketch of streamlines in a smoothly flowing stream obstructed by three bridge piers (after Hoek &Brown, 1980) 8Figure 2: Layout of barrier pillars and panel pillars in a laterally extensive orebody (after Brady & Brown,1985) 9Figure 3: Pillar layout for extraction of an inclined orebody, showing biaxially confined transverse andlongitudinal pillars, ‘A’ and ‘B’, respectively (after Brady & Brown, 1985) 9Figure 4: Typical pillar layouts showing loads carried by variouspillars assuming total rock load to beuniformly distributed over all pillars (after Hoek & Brown, 1980) 11Figure 5: Redistribution of stress in the axial direction of a pillar accompanying stope development (afterBrady & Brown, 1985) 12Figure 6: Basis of the tributary area method for estimating average axial pillar stress in an extensive minestructure, exploiting long rooms and rib pillars (after Brady & Brown, 1985) 13Figure 7: The “Chain Pillar Formula” layout (after Szwilski, 1982) 14Figure 8: The Hudyma (1988) pillar stability graph showing the stable, transition, and failed zones (afterHudyma, 1988) 20Figure 9: Idealized illustration of the transition from intact rock to a heavily jointed rock mass with increasingsample size (after Hoek & Brown, 1980) 22Figure 10: Hoek & Brown (1980) pillar strength curves for igneous crystalline rock (after Hock & Brown,1980) 25Figure 11: Histogram showing frequencies of intact pillar performance, and pillar failure, for South Africancoal mines (after Salamon & Munro, 1967) 28Figure 12: Hedley & Grant’s (1972) method for the determination of pillar stress (after Hedley & Grant, 1972).29Figure 13: Hedley & Grant’s (1972) estimation of pillar stresses and strengths (after Hedley & Grant, 1972). 29Figure 14: Conceptual models relating rock structure and rock response to excavation (after Brown, 1987). ...36Figure 15: Development of a finite element model of a continuum problem, and specification of elementgeometry and loading for a constant strain, triangular finite element (after Brady & Brown, 1985).38Figure 16: Simplified finite element and boundary element problem formulation for the same excavationgeometry (after Brown, 1987) 39Figure 17: Superposition scheme demonstrating that generation of an excavation is mechanically equivalent tointroducing a set of traction’s on a surface in a continuum (after Brady & Brown, 1985) 39Figure 18: Surface, element and load distribution description for development of a quadratic, isoparametric,indirect boundary element formulation (after Brady & Brown, 1985) 39Figure 19: Principal stress distribution in a rib pillar defined by a ratio of pillar width to height of 1.0. Thecontour values are given by the ratio of major and minor principal stresses to the average pillarxstress. Plane strain analysis for uniformly distributed vertical applied stress (after Hoek & Brown,1980) 40Figure 20: An idealized sketch showing the principle of numerical modelling of underground excavations(after Hudyma, 1988) 41Figure 21: Nonnal and shear modes of interaction between distinct elements (after Brady & Brown, 1985). ...41Figure 22: Modelled pillar failure as a result of pillar reduction (after Brady, 1977) 43Figure 23: The peak strength, deformation characteristics, and effect of location used for investigating a pillarcase history with a displacement discontinuity program (after Maconochie et al., 1981) 43Figure 24: The normal stress and the failed regions estimated with the displacement discontinuity program,NFOLD, for a sill pillar case history (after Maconochie et al., 1981) 44Figure 25: The distribution of normal stress in a mining block was estimated for two different miningsequences to determine the best extraction sequence using displacement discontinuity methods (afterBywater et al., 1983) 44Figure 26: Principal modes of deformation behaviour of mine pillars (after Brady & Brown, 1985) 47Figure 27: Fracturing of rock specimens at various stages of loading (after Krauland & Soder, 1987) 48Figure 28: Relationship between pillar shape and pillar strength for constants suggested by various authors(after Hock & Brown, 1980) 49Figure 29: Relationship between pillar volume and pillar strength for constants suggested by various authors(after Hock & Brown, 1980) 50Figure 30: Influence of specimen size upon the strength of intact rock (after Hock & Brown, 1980) 51Figure 31: Effect of specimen size on compressive strength (after Kostak, 1971) 52Figure 32: Size effect in modulus of deformation, E (after Kostak & Bielenstein, 1970) 52Figure 33: Stress distribution in a coal pillar at various stages of loading (after Wagner, 1974) 53Figure 34: Stress - strain curves for laboratory specimens loaded under increasing confining stresses show anincrease in peak load and an increase in the post-peak load bearing capacity (after Starfield &Fairhurst, 1968) 53Figure 35: Schematic illustration of Westmin Resources Ltd.’s pillar stability classification method - Class “1”- “5” 54Figure 36: Location plan of Westmin Resources Ltd.’s Myra Falls operations 56Figure 37: A simplified cross-section of the H-W Mine showing the location of the shaft, lateral developmentand the major orebodies 57Figure 38: Schematic of mining method and the respective location within the orebody used in the H-W Mine.58Figure 39: Schematic layout for sublevel open stoping with ring-drilled blast holes (after Hamrin, 1982) 59Figure 40 Schematic layout for room-and-pillar mining (after Hamrin, 1982) 60Figure 41: Elements of a supported method of mining (after Hamrin, 1982) 60Figure 42: Isometric view showing the major joint sets identified within the H-W Main Zone 63Figure 43: Isometric view showing the location of stopes with the H-W Main Zone 65xiFigure 44: Isometric view showing the location of the barrier pillars within the H-W Main Zone 65Figure 45: Polar stereonet plots of the results of the triaxial and the biaxial in-situ stress measurement programswithin the H-W Main Zone 67Figure 46: Simplified barrier pillar schematic showing the development of the “small” pillars 68Figure 47: Typical Map3D stress modelling output showing the location at which the core stress wasdetermined 69Figure 48: Barrier pillar stress comparison, average Map3D barrier pillar stresses vs. core Map3D barrier pillarstresses 71Figure 49: Plan showing the stope and pillar configurations that were used to perform the parametric modellingsessions to justify the use of the tributary area method for determining the stresses on the “small”pillars within the barrier pillars 72Figure 50: Definition of pillar geometry terms used at Westmin Resources Ltd.’s H-W Mine 75Figure 51: Definition of pillar types for classification purposes 77Figure 52: Stability plot of massive sulphide, drawpoint and rib pillars from the H-W Mine 81Figure 53: Stability plot of massive suiphide, nose pillars from the H-W Mine 81Figure 54: Stability plot with the stability classification division lines, for massive sulphide, drawpoint and ribpillars from the H-W Mine. The excluded pillars are not plotted on this figure 83Figure 55: Stability plot with the stability classification division lines, for massive sulphide, drawpoint and ribpillars from the H-W Mine. All drawpoint and rib pillars are included on this figure 85Figure 56: Stability plot with the stability classification division lines, for massive sulphide, nose pillars 85Figure 57: Hudyma pillar stability graph with all the pillars that made up more than one case history joined toindicate loading paths that pillars were subjected to (after Hudyma, 1988) 94Figure 58: Plan of classified mine pillars at the Selbi-Phikwe Mines (after Von Kimmelman et al., 1984) 96Figure 59: Relationship determined between average pillar confinement and pillar width / height ratio for amodelled extraction ratio of 99.5% 108Figure 60: Relationship between the average pillar confinement and pillar width / height ratio for differingmodelled extraction ratios 109Figure 61: Plot of the variation in the Cp coefficient for different modelled extraction ratios 109Figure 62: Histogram showing the variation in the influence of the pillar volume on pillar strength for all datain the combined database 111Figure 63: Stability graph for the Westmin database showing the range of slopes for the stability lines and thevalid range for pillar width / height ratios 115Figure 64: Stability graph for the Hudyma (1988) database showing the range of slopes for the stability linesand the valid range for pillar width / height ratios 115Figure 65: Stability graph for the Von Kimmelman et al. (1984) database showing the range of slopes for thestability lines and the valid range for pillar width / height ratios 116xiiFigure 66: Stability graph for the Hedley & Grant (1972) database showing the range of slopes for the stabilitylines and the valid range for pillar width / height ratios 116Figure 67: Stability graph showing the stability lines over the valid width I height ratio ranges for each of theindividual databases plotted alongside the strength formulae subsequently developed 118Figure 68: Stability graph for the “Refined Power Shape Effect” formula with a power coefficient, “&‘, of 0.45,plotted along with all of the case histories in the combined database 118Figure 69: Plot of the preferred value of the refined power coefficient, “Ct”, for differing values of pillar width Iheight ratio plotted along with the relationship represented by Equation 44 119Figure 70: Stability graph for the “Log Power Shape Effect” formula with a power coefficient represented byEquation 45, plotted along with all of the case histories in the combined database 120Figure 71: Construction of a Mohr’s failure envelope with reference to average pillar confinement, Cpa, 122Figure 72: Construction of Mohr’s failure envelopes for differing values of average pillar confinement, Cp. 123Figure 73: Mohr-Coulomb pillar strength envelope for any value of average pillar confinement, Cp 123Figure 74: Stability graph for “The Confinement Formula” as described by Equation 48, plotted along with allof the case histories in the combined database 125Figure 75: Confinement graph for “The Confinement Formula” as described by Equation 48 with pillar width /height ratio replaced by average pillar confmement on the x-axis and plotted along with all of thecase histories in the combined database 125Figure 76: Stability graph comparing the newly developed pillar strength formulae 126Figure 77: Confinement graph comparing the newly developed pillar strength formulae 126Figure 78: Pillar strength prediction success statistics for the newly developed formulae for all data in thecombined database 128Figure 79: Stability graph with the Hoek & Brown (1980) pillar curves for good to very-good rock massesplotted over all of the data in the combined database 131Figure 80: Pillar strength prediction success statistics for “The Confinement Formula” and the Hoek & Brown(1980) pillar curves for differing rock mass quality 131Figure 81: Stress - strength plot for all data in the combined database using the Salamon & Munro (1967)strength formula and “K” — 0.7.UCS 132Figure 82: Stress - strength plot for all data in the combined database using the Hedley & Grant (1972) strengthformula and “K” — 0.7.UCS 132Figure 83: Stability graph for Bieniawski’s (1975) strength formula using “K” — 0.7.UCS plotted over all ofthe data in the combined database 133Figure 84: Pillar strength prediction success statistics for “The Confinement Formula” vs. Bieniawski (1975),Hedley & Grant (1972), and Salamon & Munro (1967) using “K” 0.7.UCS 133Figure 85: Stability graph for the original “Power Shape Effect Formula” using “K” — 0.44.UCS plotted overall of the data in the combined database 134xliiFigure 86: Stability graph for Obert & Duvall’s (1967) and Bieniawski’s (1975) strength formulae using “K” —0.44.UCS plotted over all of the data in the combined database 134Figure 87: Pillar strength prediction success statistics for “The Confinement Formula” vs. the original; “PowerShape Effect Formula”, Bieniawski (1975), and Obert & Duvall (1967) using “K” — 0.44.UCS. .135Figure 88: Plot of pillar factor of safety for width / height ratios of 0.25 and 0.5. Hoek & Brown (1980) rockmass constants used: m 10, s — 0.1. Modelling performed using Examine2D 148Figure 89: Plot of pillar factor of safety for width / height ratios of 0.75 and 1.0. Hock & Brown (1980) rockmass constants used: m — 10, s 0.1. Modelling performed using Examine2D 149Figure 90: Plot of pillar factor of safety for width / height ratios of 1.25 and 1.5. Hock & Brown (1980) rockmass constants used: m — 10, s — 0.1. Modelling performed using Examine2D 150Figure 91: Plot of pillar factor of safety for width / height ratios of 1.75 and 2.0. Hock & Brown (1980) rockmass constants used: m — 10, s — 0.1. Modelling performed using Exainine2D 151Figure 92: Plot of pillar factor of safety for width / height ratios of 2.5 and 3.0. Hock & Brown (1980) rockmass constants used: m — 10, s — 0.1. Modelling performed using Examine2D 152Figure 93: Plot of pillar factor of safety for width / height ratios of 3.5 and 4.0. Hock & Brown (1980) rockmass constants used: m — 10, s —0.1. Modelling performed using Examine2D 153Figure 94: Principal stress plots for pillar width I height ratio of 0.25 using two-dimensional boundary elementmodelling. Modelling performed using Exaxnine2D 154Figure 95: Principal stress plots for pillar width / height ratio of 0.5 using two-dimensional boundary elementmodelling. Modelling performed using Examine2D 155Figure 96: Principal stress plots for pillar width / height ratio of 0.75 using two-dimensional boundary elementmodelling. Modelling performed using Exaniine2D 156Figure 97: Principal stress plots for pillar width I height ratio of 1.0 using two-dimensional boundary elementmodelling. Modelling performed using Examine2D 157Figure 98: Principal stress plots for pillar width / height ratio of 1.25 using two-dimensional boundary elementmodelling. Modelling performed using Examine2D 158Figure 99: Principal stress plots for pillar width / height ratio of 1.5 using two-dimensional boundary elementmodelling. Modelling performed using Examine2D 159Figure 100: Principal stress plots for pillar width I height ratio of 1.75 using two-dimensional boundaryelement modelling. Modelling performed using Exaxnine2D 160Figure 101: Principal stress plots for pillar width / height ratio of 2.0 using two-dimensional boundary elementmodelling. Modelling performed using Examine2D 161Figure 102: Principal stress plots for pillar width! height ratio of 2.5 using two-dimensional boundary elementmodelling. Modelling performed using Examine2D 162Figure 103: Principal stress plots for pillar width! height ratio of 3.0 using two-dimensional boundary elementmodelling. Modelling performed using Examine2D 163xivFigure 104: Principal stress plots for pillar width / height ratio of 3.5 using two-dimensional boundaiy elementmodelling. Modelling performed using Examine2D 164Figure 105: Principal stress plots for pillar width / height ratio of 4.0 using two-dimensional boundary elementmodelling. Modelling performed using Examine2D 165xvLIST OF PHOTOSPhoto 1: A class “5” discrete pillar located within the 366 barrier pillar 78Photo 2: A class “4” discrete pillar located within the 383 barrier pillar 78Photo 3: A class “3” discrete pillar located within the 383 barrier pillar 79Photo 4: A class “2” discrete pillar located within the 383 barrier pillar 79Photo 5: A class “1.5” discrete pillar located within the 383 barrier pillar 80xviACKNOWLEDGMENTSThe author must thank certain individuals and groups for their contribution in bringing this work tocompletion.• Westmin Resources Ltd., Myra Falls Operations, and CANMET for providing both the fundingand the mining environment that made the collection of the data necessary for this projectpossible.• The Cy and Emerald Keyes Scholarship fund for financial support.• The proof readers, you know who you are.• My advisor, Dr. Rimas Pakalnis, for his questions, comments, criticism, and hands on approach tomining related problems, without whom, this project would not have turned out as it did.• My parents, Jean and Jakob Lunder for their support and encouragement,• and finally, I must thank my wife Kaarina and my children, Angus and Emma, for their supportand patience through the disruption in their lives that this work has imposed upon them.xvii1. INTRODUCTIONPillars are present in mining operations to provide support for mine openings. In order to design andutilize pillars effectively, it is required that an estimate of the pillar strength be made. Pillars are often locatedwithin “ore grade” material and as a result, mine operators strive to maximize the extraction of this valuableresource while maintaining overall mine stability. Economic conditions therefore dictate that the optimumpillar is the smallest one that will meet the load bearing requirements. Failure to properly design andimplement pillar strategies may result in either pillar failure or over-designed pillars. Both of these cases mayleave valuable “ore grade” material unrecoverable.In 1991, The Canadian Centre for Mineral and Energy Technology, Energy, Mines and ResourcesCanada (CANMET) and Westmin Resources Ltd., Myra Falls Operations, initiated a research project toinvestigate ground stability entitled: “Ground Stability Guidelines for the Extraction ofBarrier Pillar’s in HardRock Mines”. The scope of this project was to investigate strategies for the extraction of barrier pillars at theculmination of mining operations at Westmin Resources Ltd. ‘s H-W Mine.This thesis presents an analysis of data collected for the estimation of the strength of hard rock minepillars at Westmin Resources Ltd.’s H-W Mine. A pillar strength relationship for mine pillars at WestminResources Ltd. is developed. This data is analyzed in conjunction with published hard rock mine pillar datafrom six additional sources and two new pillar strength estimation formulae, “The Log-Power Shape EffectFormula” and “The Confinement Formula”, are proposed.1.1 CONTENTS OF THESISThis thesis deals with the estimation of the strength of hard rock mine pillars. The data used to derivethe new strength formulae was obtained by the author from Westmin Resources Ltd.’s H-W Mine and isassimilated with the available published hard rock pillar case histories. This thesis is divided into six chaptersas follows:1. INTRODUCTION: This chapter provides a thesis overview and a discussion of the role of pillarsin mining operations.2. PILLAR DESIGN METHODOLOGY: This chapter reviews the current state-of-the-art of pillardesign in both hard rock and soft rock mining situations. These methods are reviewed in and thelimitations or benefits of each method are discussed. Numerical modelling techniques for stressanalysis are reviewed.3. PILLAR FAILURE MECHANISM: This chapter investigates the mechanism of pillar failure andidentifies the variables that influence the stability of mine pillars. A description of the pillar stabilityclassification method developed for use at Westmin Resources Ltd.’s H-W Mine is presented.4. PILLAR STRENGTH DETERMINATION AT WESTMIN RESOURCES LTD.’S, H-W MINE.- A CASE STUDY.: This chapter examines the methodology that was utilized to develop a pillarstrength relationship for Westmin Resources Ltd.’s H-W Mine. An overview incorporating adescription of geology and mining methods is included. An empirical strength formula, presented inthe form of a pillar stability graph and a “Linear Shape Effect” formula, was derived from 31 of 65pillar case histories collected by the author at the H-W Mine. The variables used to derive the pillarstrength relationship are: pillar stability classification, average pillar stress, and pillar geometry. Thedata is normalized to the intact unconfmed compressive strength of pillar material so that it iscomparable to the additional data presented in Chapter 5.5. NEW PILLAR STRENGTH FORMULAE: This Chapter augments the Westmin H-W Minepillar database with hard rock pillar data from six additional sources. This comprehensive database of178 case histories is used to develop refined empirical strength formulae in the form of a “ShapeEffect Formula” as proposed by other researchers and discussed in Chapter 2. It was concluded thatpreviously developed strength determination methods did not fully describe the pillar behaviour of thecase histories in the combined database. Two-dimensional numerical modelling has been used toderive a relationship between pillar width / height ratio and the average pillar confinement, Cp. Theaverage pillar confinement is subsequently used in the development of new empirical pillar strengthformulae. Two new pillar strength formulae are developed: “The Log-Power Shape Effect Formula”,a purely empirical “Shape Effect” formula, and “The Confinement Formula”, a new empiricalhypothesis that relates the strength of mine pillars to average pillar confinement.6. SUMMARY and CONCLUSIONS: Recommendations regarding the use of “The ConfinementFormula” and the use of numerical modelling to assess induced pillar loads are presented in thischapter.7. APPENDIX A: The results of two-dimensional parametric modelling sessions that were used todevelop the relationship between pillar width / height ratio and average pillar confinement arepresented in this appendix.1.2 PILLAR STRENGTHASSESSMENT - AN OVERVIEWThe design of mine pillars is an essential part of all mining operations and the methods that may beused for design are quite varied. Hedley & Grant (1972) and Hudyma (1988) have proposed empirical pillarstrength determination methods for hard rock mine pillars. A large amount of research into pillar strength hasbeen based upon coal pillars in the United States and South Africa. An alternative strength criteria for jointedrock masses was presented by Hock & Brown (1980) and was developed into a series of pillar strength curves.2The above methods can be used for the estimation of pillar strength, however, not with a high degreeof confidence in hard rock mining operations. There exists a need to develop a more reliable method ofdesigning hard rock mine pillars in underground mining operations. The benefits of a more reliable method ofpillar design may result in the following:• increased ore recovery• improved safety through better pillar design• improved knowledge of pillar loading and failure mechanism such that modifications to miningplans can be quantifiedIt is critical that pillars can be designed with confidence for a varying range of rock types, pillarshapes, pillar sizes, and varying in-situ stress regimes.1.3 RoLE OF PILLARS INMININGPillars are found in all underground mines and play a wide and varied role depending on the situationin which they are used. Pillar types can be:• protection pillars surrounding mine shafts• temporary pillars that allow quick exploitation of mineable reserves• barrier pillars that must remain stable for the duration of a mine’s life.Pillars may be designed such that failure will occur, while other pillars may require that they remainstable for the duration of their life. In general, the role of a mine pillar is to support the adjacent rock mass fora given period of time while mining takes place. In order for a pillar to perform its designed role, the strengthof a mine pillar and the load acting upon it must be assessed. If these two factors are not adequatelydetermined, the pillar may not perform as desired.Salamon (1983) lists three major categories of pillar’s that can be classified: “support pillars”,“protective pillars”, and “control pillars” as described below:1. “Support pillars” include all pillars that are used in situations where undermined hangingwallrock support is provided by a series of pillars. They are usually laid out in a systematic manner.Examples in hard rock mining operations include room-and-pillar stope pillars, post-pillars andrib pillars.2. “Protection pillars” are employed to safeguard installations for which failure is intolerable.Examples of installations to be protected are surface buildings, mine shafts, and boundary pillarsbetween two adjacent mining operations. These pillars can also be referred to as “shaft pillars”,“roadway pillars”, or “boundary pillars”. A significantly high factor of safety is used in these3situations to compensate for potential errors associated with the assumptions made in pillarstrength estimation.3. “Control pillars” are employed in situations where rockburst activity is anticipated orexperienced. These pillars are designed so that failure will not occur. They are designed toreduce the magnitude of stress changes in a mine environment and to alleviate the risk of rockbursting.In consideration of these pillar types, this thesis will be concerned with “support pillars” only. Whendesigning “support pillars” in hard rock mines, the material that comprises the pillar is commonly ore-grade.For this reason it is desirable for the mine operator to extract as much of this resource as possible. However,pillars must be of sufficient size so that they can support the induced loads throughout their design life. Thesetwo contradictory factors require that the amount of ore left in pillars and the amount of extraction be balancedto optimize the exploitation of the orebody for profitability, while still operating in a safe and efficient manner.The impact of poorly designed pillars can result in the mine being deemed uneconomic because of an overlyconservative design. Conversely, overly optimistic strength estimates can result in local or regional failure inthe mine horizon, making a portion or all of the mineable resource unrecoverable.42. PILLAR DESIGNMETHODOLOGYA literature review of pillar design methods has been undertaken to assess the current state-of-the-artin pillar design. The following sections define the procedures employed in pillar design and pillar strengthestimation. The majority of the research work in published literature has been performed in horizontallybedded coal deposits, and as a result, these techniques are primarily applicable to similar deposits.The function of pillars in mining is to maintain the stability of adjacent strata for the design life of thepillars. Equation 1 is the primary, although simplistic, form of the pillar strength equation. The premise thatwhen pillar stress exceeds pillar strength, a pillar fails, forms the basis of all strength formulae. The factor ofsafety can subsequently be used to compensate for errors in estimation of the input parameters used within thestrength formulae. This requires that strength and stress estimates be determined with the associated variabilityin each.F S — Strength/ 1— /Stresswhere:F.S. Factor of safety against pillar failure.Strength Pillar strength.Stress = Applied pillar stress.The assessment of pillar stress in non-tabular or irregular dipping deposits is a complex task. Theintact strength of a sample of rock can be determined reasonably accurately testing laboratory samples,however, the correct method of applying this intact strength to make an assessment of the strength of a full sizesample (a pillar) is complex. The relationship between intact rock strength and in-situ pillar strength has beenthe primary goal of many researchers investigating pillar strength.A common approach for pillar design is to use experiences under similar conditions. This trial anderror method may have success on occasion, but is generally not based upon fundamental engineeringprinciples. A number of empirical and deterministic methods of estimating pillar stress and strength arepresented in this chapter.Pillar design follows the premise that in most cases it is desirable to design pillars that will maintaintheir load bearing capacity throughout their design life. In order to achieve this, the pillar strength must besufficient to support the stresses that the pillar will be subjected to. The two primary factors that must beconsidered in designing mine pillars are pillar strength and pillar stress.5A rock mass is generally not a homogeneous isotropic medium and as such the determination of pillarstrength is highly dependent on the factors that affect the strength of a rock mass including, but not necessarilylimited to:• the intact strength of pillar material• the pillar geometry (width, height, width / height ratio)• the structural features within the pillar• the material properties of the pillar, such as deformational characteristics• the effects of blasting on the pillarDesign methods are largely based on equating stress to strength so that a stable equilibrium exists.This requires that an estimate of stress be made with levels of accuracy commensurate to strength estimates.The actual pillar stress is dependent on, but not necessarily limited to:• the in-situ stress conditions• the mining induced stress changes• the effects of geological features, such as faults and jointing• the shape and orientation of pillars• the spatial relationship between pillars and mine openings• the effects of ground waterPotvin (1985) presents pillar design as being divided into four broad groups: heuristic, empirical,theoretical, and computer methods. These broad categorizations represent the methods that have been used inthe past to design pillars and to assess the pillar strength.Potvin (1985) states that heuristic methods are the most widely and least sophisticated methods used todesign mine pillars. Pillar design is generally based on the principle that “what worked before could workagain”. This technique does not consider the strength or loading conditions of pillars and as such is notrecommended.Empirical, by definition, is the method of relying on experiment or experience. The primarydifference from heuristic methods is that case histories are studied and then applied to the future design of minepillars. Numerous researchers have undertaken empirical studies and developed empirical strength formulaefor the design of mine pillars. A drawback of this work is that the majority of the studies were performed incoal mines. This work has been extrapolated to hard rock underground mines but not in a comprehensivemanner.Theoretical methods of pillar design attempt to utilize mathematical concepts and input parameters todesign pillars based upon a rigorous formulation. Rock mass conditions can be highly variable and thedetermination of the critical variables that would be used to design mine pillars are difficult, if not impossible,6to obtain. The complexity of theoretical approaches make their use both difficult and time consuming. Workby Wilson (1972), Coates (1965), Grobbelaar (1970), and Panek (1979) will be discussed.Numerical Methods have become a popular method of making an assessment of the stresses on pillars.With relatively inexpensive computing power available to rock mechanics engineers, mine layouts can beanalyzed and predicted pillar stresses determined with a minimum of effort.2.1 PILLAR STRESS DETERMINATIONThe determination of the actual stresses acting on a pillar is difficult. As mentioned previously, thepillar stress depends on a number of factors which include:• the in-situ stress conditions• the mining induced stress changes• the effects of geological features, such as faults and jointing• the shape and orientation of pillars• the spatial relationship between pillars and mine openings• the effects of ground waterFigure 1 is a simplified illustration of the theory of stress redistribution as being analogous to a streamflowing around bridge piers. Two methods of calculating pillar stress are described in the literature. They aretributary area theory (and its variations) and numerical modelling methods. Tributary area theory utilizes asimplified approach to stress determination while numerical modelling relies on computational techniques todetermine stress redistribution around mine openings. Numerical modelling techniques can only be performedon computers due to the associated complexities. Numerical models have become increasingly complex andintricate as desktop computer power has increased and computers have become more affordable.The importance of valid determination of the in-situ stresses must not be overlooked. It is common toassume that the vertical in-situ stress component will be equal to the weight of the overlying strata. Inhorizontally bedded deposits this may be adequate for determining normal stresses acting on pillars. Inirregular, non-tabular deposits, however, the induced pillar stress is a factor of the three principal stresses, notjust the vertical component. In-situ stresses in a particular locality may not vary greatly on average, but theycan vary significantly on a local scale as a result of geological structure and the proximity to surface. In-situstresses are generally of similar magnitudes and orientation over a large area, as in the Canadian shield forexample (Herget, 1987). All of the pillar stress determination techniques available are dependent upon thevalues of in-situ stresses used.7rFigure 1: Sketch of streamlines in a smoothly flowing stream obstructed by three bridge piers (after Hoek &Brown, 1980).2.1.1 Tributary Area TheoryBabcock et al. (1981) state that Bunting (1911) was the first author to introduce the tributaiy areamethod for the determination of average pillar stress. Agapito (1972) however stated that the thbutary areatheory stemmed from an investigation into stress analysis utilizing photoelastic techniques. Photoelastic studiesshowed that the stress concentration in pillars increased with the number of openings in the plate and with theincrease in the opening size to pillar size ratio. It was also shown that the stress concentration in central pillarsreaches an upper limit and becomes constant as the number of openings increases to five or more (Obert &Duvall, 1967).Tributary area theory assumes that a pillar will support its “share” of the applied load. Tributary areatheory is applicable to situations where similarly sized pillars are developed in a large regular array and is notparticularly applicable to:• irregular and dipping deposits• inconsistent or irregular mining patterns• complex triaxial stress fieldsTributary area theory has been used with the greatest success in horizontally bedded deposits whichare uniform and cover a large area, such as horizontally bedded coal deposits or room-and-pillar mines.Equation 2 is the equation for tributary area theory in a room-and-pillar mine utilizing equally sizedrectangular pillars. Figure 2 and Figure 3 illustrate some sample mining pillar layouts.8DDEEELDEEDEEDDstope. or minedroom panel pillar barrier pillarFigure 2: Layout of barrier pillars and panel pillars in a laterally extensive orebody (after Brady & Brown,1985).Figure 3: Pillar layout for extraction of an inclined orebody, showing biaxially confined transverse andlongitudinal pillars, ‘A’ and ‘B’, respectively (after Brady & Brown, 1985).Y Cross section Y — Yorehodyboundary Y9W0a=yH. (1÷—). (1÷—) (2)Wp Lpwhere:Average pillar stress (MPa)Unit weight of rock (MN / m3)H — Depth of overburden cover (m)W0 — Width of opening (m)L3 Length of opening (m)W, — Pillar width (m)— Pillar length (m)Tributary area theory has also been referred to as the extraction ratio formula. The stress on the pillarcan be approximated based on the ratio of the amount of extraction around an array of pillars. Tributary areatheory can be written, in this case, in the form of Equation 3.(3)where:— Average pillar stress (MPa)aa In-situ stress normal to mining horizon (MPa)R — Extraction RatioTributary area theory is intended for use in tabular deposits where an estimation of the stress normal tothe orebody can be made with a high degree of confidence. It is used extensively in coal deposits, whereoverburden stresses can be readily estimated. Figure 4, Figure 5, and Figure 6 illustrate pillar configurationsand the respective formulae that can be used to calculate the pillar stress using tributary area theory.Variations of tributary area theory have been proposed to account for inclined mining geometries andtriaxial stress conditions. These are presented in the following sections.10Plan area of pillar on surface+___SQUARE PILLARS-yz(l + 1e/Wp>2Figure 4: Typical pillar layouts showing loads carried by various pillars assuming total rock load to beuniformly distributed over all pillars (after Hoek & Brown, 1980).+ w)woRIB PILLARS— 0p yz(1+ Wo/w)________E LL9Pillar area Rock columnarea0IRREGULAR PILLARS -Rock column areaPillar areaRECTANGULAR PILLARS - oyz(l +Wo/w)(l +Lo/L)11post-mining pillarStress distributionpost-miningabutment Stressdistribution(1+Ko)+(1—Ko)cos(2ct)yII.2(l-R)= Average normal pillar stress (MPa)— Average shear pillar stress (MPa)— Unit weight of overburden (MN / m3)Depth below surface (m)— Ratio of in-situ horizontal to vertical stressExtraction ratioDip of seam (degrees)Figure 5: Redistribution of stress in the axial direction of a pillar accompanying stope development (afterBrady & Brown, 1985).2.1.1.1 Pariseau (1982) Inclined Stress FormulaePariseau (1982) proposed a method of detennining the stresses acting on pillars in dipping seams thatwould account for both the vertical and horizontal stress components. Equations 4 and 5 are the formulae forthe normal and shear stresses respectively, acting on inclined pillars.(1— Ko)sifl(2cL)‘yH.2tp =(1 -R)(4)(5)where:a’,tpHK0RaThis is a variation on the tributary area theory approach that makes an estimate of normal and shearstresses on a pillar, taking into account both horizontal and vertical components of in-situ stress. Onelimitation of these formulae is that Pariseau (1982) assumes the magnitudes of the horizontal stresses are equalin the horizontal plane. This is not necessarily the case as it has been shown that the magnitude of horizontalstresses can be variable (Herget, 1987).121< 4c P(a)Ii___I I(b),— I (C)(w0 + w)Figure 6: Basis of the tributary area method for estimating average axial pillar stress in an extensive minestructure, exploiting long rooms and rib pillars (after Brady & Brown, 1985).2.1.1.2 Szwilski (1982) Chain Pillar FormulaSzwilski (1982) presented a modification to the tributary area theory for use in longwall coal minescalled the “Chain Pillar Formula”. The stress equation includes a term that considers the added load due tothe cantilevering effect of the immediate roof over the panel being mined. Figure 7 illustrates the terms used bySzwilski (1982) in the “Chain Pillar Formula”. Equation 6 is the “Chain Pillar Formula”.a =yH.(Lp+S)(Wf +2Wp+3S)(6)2WpLpwhere:Average pillar stress (MPa)— Unit weight of overburden (MN / m3)H Depth of overburden (m)W, = Pillar width (m)— Pillar length (m)S Spacing of chain pillars (m)Wf Width of face (m)2.1.1.3 Hedley & Grant’s (1972) Formulafor Inclined PillarsThis variation of tributary area theory, to account for inclined geometry in a triaxial stress field, waspresented by Hedley & Grant (1972), in conjunction with their pillar strength formula (Section 2.2.1.5.2). Thismethod considers the dip of the seam and the values of the vertical and average horizontal stresses acting in thevicinity of the area under investigation. Equation 7 is Hedley & Grant’s (1972) stress formula for inclinedpillars.13Average normal pillar stress (MPa)— Unit weight of overburden (MN / m3)— Depth below surface (m)Extraction ratioHorizontal component of in-situ stress (MPa)Dip of orebody from the horizontal (degrees)I/{_Gob ‘•-‘S\_•\ c c---I Area of StrataLp+S Load on Chain Pillars_.,, —-‘ \_•( — ‘IGobI-rCoal Face(Coal PanelCoalRibsideFigure 7: The “Chain Pillar Formula” layout (after Szwilski, 1982).=yHcos2(a)÷ohsinc )(l-R)where:- Wf-- Tail EntryChain Pillarsop7HRa.(7)2.1.1.4 Subsidence FormulaWhittaker & Singh (1979) developed Equation 8 and Equation 9 to determine barrier pillar stresses inlongwall mining situations. This formula assumes that the “goaf’ area behind the longwall is loaded by atriangular roof mass sheared at an angle, “0”, from the edge of the barrier pillar as measured from the vertical.The load outside the triangular roof region is assumed to be transferred to the barrier pillars.9.817.{[(w+W).H— w2c0t0] .W} (8)1000w 4for W/H < 2(tan)14o=9.81’yw.(wH+H2tanØ) (9)for WIH> 2(tan4))where:ap — Pillar stress (MPa)= Density of overburden material (MN /m3)4) = Angle of shear of roof strata at edge of longwall extraction measured from thevertical (degrees)w Width of barrier pillar (m)W — Width of longwall extraction (m)H = Depth of overburden material (m)This technique is based on the research of King & Whittaker (1971) to determine pillar stress.Whittaker & Singh (1979) use the estimated pillar stress and combine it with pillar strength estimation afterSalamon & Munro (1967) to determine a critical mining depth for a given width of barrier pillar.2.12 Numerical MethodsNumerical modelling is the technique of applying numerical methods to solve problems that involvethe response of a rock mass to loading. The loading is generally a result of mining excavations, causing aredistribution of stress within the rock mass. Brown (1987) states that “it is only rarely that analyticalsolutions can be found to rock mechanics problems ofpractical concern”. This can be attributed to the factthat boundaiy problems associated with complex mining geometries cannot be described by simplemathematical functions. The governing equations are generally non-linear, the problem domain heterogeneousand anisotropic, and the constitutive relations for the rock mass non-existent.The use of numerical models can be two-fold:• Numerical methods can be used to determine pillar stress distributions in the place of the tributaryarea theory methods. This approach is used in situations where the conditions around a miningscenario are sufficiently complex that tributary area theory cannot be relied upon to provideadequate results.• Numerical methods can also be used as a design tool. Failure criteria and strength parameters canbe utilized in various modelling programs and the predicted rock mass response can be analyzed.Failure analysis can take place as either a post processing routine or by an interactive method.Numerical modelling has distinct advantages over tributary area theory for the determination of pillarstresses. Numerical modelling generally uses elastic theory to determine stress redistribution within a domainof material of different elastic properties. It is possible to estimate stresses in complex mining geometries15where the tributary area theory formulae would not provide acceptable results. The use of numerical modelsdoes however require that generalizations be made about the area of interest.Listed below are the types of numerical models and programs that are available for use in the minedesign process. Details of each model type are discussed in Section 2.3. For more detailed informationregarding numerical modelling techniques, the reader is directed to Brown (1987) and Zienkiewicz (1977)regarding boundary element and finite element numerical modelling techniques respectively.The types of numerical models available include:• finite element models• finite difference models• distinct element models• fictitious force boundary element models• displacement discontinuity boundary element models• hybrid models that combine two or more of the above methods2.1.3 Discussion - Pillar Stress DeterminationAll of the methods for determining pillar stresses require that the premining (in-situ) state of stressaround excavations be known. In horizontally bedded deposits, only the vertical stress acting upon a seam needbe considered. This value can be approximated by the density of the overburden material multiplied by thedepth below surface. In areas that are mountainous or that have been subjected to glaciation, the overburdenstress value is less reliable. In inclined orebodies the values of the in-situ stresses become more difficult todetermine without resorting to in-situ stress measurement programs. Panseau’s (1982) and Hedley & Grant’s(1972) inclined stress formulae use terms that account for the average in-situ horizontal stress. However, inrock it must be recognized that the horizontal components of in-situ stress are generally not equal, and whenusing Pariseau’s (1982) or Hedley & Grant’s (1972) inclined formulas, the user must consider the impact thatdifferent horizontal stresses will have on the pillar stresses.It must be recognized that the only technique for determining the in-situ stress magnitudes accuratelyis to perform in-situ stress measurements. These tests, however, are expensive and can have variable successrates (Pakalnis et al., 1985). This means that a number of tests must be performed in order to obtain reliableresults. These results subsequently make up the basic input data for all methods of computing the inducedpillar stresses.162.2 PILLAR STRENGTHDETERMINATIONPillar strength determination methods can be divided into three groups:• empirical methods• theoretical methods• heuristic methodsEmpirical methods rely on experience, combined with geotechnical terms related to pillar stability, todevelop a strength formula. Theoretical methods are derived mathematically to describe the expectedperformance of mine pillars subject to loading for a given set of input variables. Heuristic methods cangenerally be considered as “rule of thumb” techniques for designing pillars that may disregard many of thevalid input parameters related to pillar strength. In this section we will investigate each of these methods asproposed by various authors for the estimation of pillar strength.2.2.1 Empirical Design MethodsA number of empirical methods for pillar strength determination have been developed by variousresearchers as follows:• the “Linear Shape Effect Formula”• the “Power Shape Effect Formula”• the “Size Effect Formula”• the Hock & Brown (1980) Empirical Rock mass Failure CriteriaThese techniques relate pillar width, pillar height, intact rock strength, and factor of safety to estimatepillar strength. The width of a pillar is measured normal to the major principal stress induced in the pillar andthe height is measured parallel to the major principal stress induced in the pillar. With the exception of Hock& Brown (1980), these formulae all take the general form of Equation 10.WaP3=K.[A+B(-)] (10)where:P — Pillar strength (Mpa, psi)K — Term related to the strength of pillar material (Mpa, psi)w — Pillar width (m, ft)h — Pillar height (m, ft)A, B, a, b = Empirically derived constants17This generalized equation has been divided into two well known empirical methods, “Size EffectFormulae” and “Shape Effect Formulae”. The “Shape Effect Formula” uses empirical constants “a” and “b”that are equal, meaning that the pillar strength is independent of pillar volume. The “Size Effect Formula”uses empirical constants “a” and “b” that are unequal, meaning that for pillars of the same shape, the pillarstrength will decrease as pillar volume increases. This is after the work of various researchers who have shownthat with increasing sample size, there will be a corresponding decrease in strength. This is thought to be as aresult of an increase in the number of structural defects in larger sample specimens. Kostak & Bielenstein(1971) showed that a decrease in strength resulted from an increase in sample size based upon laboratory sizedspecimens. Bieniawsld (1975) and Denkhaus (1962) suggest that there is a critical sample size above whichthe influence of added structural defects, with the exception of major structural features, will not have anydetrimental effect on pillar strength.2.2.1.1 Linear Shape Effect FormulaThe “Linear Shape Effect Formula” assumes that pillars of equal width / height ratios will have equalstrength, independent of the pillar volume, and that the relationship between pillar strength and pillar width /height ratio will be of a linear fonn. The “Linear Shape Effect Formula” is defined by Equation 11.P=K.[A÷B()] (11)where:P = Pillar strength (MPa)K = Strength constant related to pillar material (MPa)w = Pillar width (m)h = Pillar height (m)A, B = Empirically derived constants which when added, equal 1.0The constants “A” and “B” determined by various researchers for this formula are listed in Table 1.The work of Obert & Duvall (1967), Bieniawski (1975) and Hudyma (1988) is discussed in the followingsections.2.2.1.1.1 Obert & Duvall (1967)Obert & Duvall (1967) reported data from a series of compressive strength tests performed by Obert etal. (1946) on specimen coal pillars of varying shapes (width / height ratios). Obert & Duvall (1967)determined that Equation 12 could be used to estimate coal pillar strength. Obert & Duvall (1967) suggest thatthe strength term “K” that should be used in this formula is the strength of a specimen of pillar material with awidth / height ratio of one. This formula does not include a term to account for the size effect on strength18Table 1: “Linear Shape Effect” emprirical constants, “A” and “B”, from various authors.Source A B w/h rangeBunting (1911) 0.700 0.300 0.5-1.0Obert & Duvall (1967) 0.778 0.222 0.5-2.0Bieniawski (1968) 0.556 0.444 1.0-3.1van Heerden (1974) 0.704 0.296 1.1-3.4Bieniawski (1975) 0.64 0.36 1.0-3.1Sorenson & Pariseau (1978) 0.693 0.307 0.5-2.0Obert & DuvalI (1967) also made no recommendation about the size of the specimen to be used for thedetermination of the “K” value, but they do however suggest that a factor of safety between two and four couldbe used to account for the size effect on strength.= K. [0.778 + 0.222(-.)] (12)where:P Pillar strength (MPa)K — Unconfmed compressive strength of a cubical pillar specimen (MPa)w - Pillar width (m)h = Pillar height (m)2.2.1 .1 .2 Bieniawski (1975)Bieniawski (1975) concluded, based on tests of large scale coal specimens, that the strength of coalpillars could be described by Equation 13. This formula was a result of performing in-situ tests on large scalecoal specimens over a period of eight years. A total of 66 in-situ tests were performed on samples that varied inside length from 0.6 to 2.0 metres and had width I height ratios varying from 0.5 to 3.4. Bieniawski (1968)originally proposed that empirical constants of 0.556 and 0.444, “A” and “B” respectively in Equation 11,could be used to describe pillar strength.P =K.[0.64+0.34()] (13)where:P — Pillar strength (MPa)K — Unconfined compressive strength of a cubical pillar specimen 30 cm square (MPa)w = Pillar width (m)h = Pillar height (m)19OPEN STOPE RIB PILLAR DATA0.60 -0.50 -0.40 -U,C.)0.30S0.200.10 -0.00- I0.0 0.4 0.8 1.2 1.6 2.0 2.4PILLAR WIDTH/PILLAR HEIGHT0 STABLE + SLOUGHING 0 FAILUREFigure 8: The Hudyma (1988) pillar stability graph showing the stable, transition, and failed zones (afterHudyma, 1988).2.2.1 .1 .3 Hudyma (1988)Hudyma (1988) presented a method entitled the “Pillar Stability Graph Method” for determining thestrength of open stope rib pillars based upon data derived from Canadian hard rock underground miningoperations. Data was collected on 47 case histories of pillars that had been classified as being either stable,sloughing, or failed. The geometric data along with predicted pillar loads were related to derive the “PillarStability Graph”. Three distinct regions were defined based on pillar observations as being either failed,sloughing, or stable. Hudyma’s (1988) “Pillar Stability Graph” is presented in Figure 8 and is discussed ingreater detail in Chapter 5. The valid range of pillar width / height ratios for this method is 0.5 to 1.4.2.2.1.2 Power Shape Effect FormulaThe “Power Shape Effect Formula” assumes that the strength of a pillar is governed by the squareroot of the width / height ratio of the pillar. The formula is defined by the Equation 14. This relationship hasbeen proposed by Zern (1926), Holland (1956), and Hazen & Artler (1976).00000// 0/0 / / 0+ / /+0 00aaao a+0+10 0/20P5=K.Ji (14)where:Pillar strength (MPa)K — Unconfined compressive strength of a 30 cm cubical pillar specimen (MPa)w = Pillar width (m)h — Pillar height (m)2.2.1.3 Effective Pillar WidthIn most of the cases listed in the previous sections, strengths have been assessed from pillars that aresquare in plan cross-section. Several authors have suggested that rectangular pillars will have a higher strengththan their square counterparts because of increased confinement from the long dimension, in plan, of the pillar.The proposed modifications to the “Shape Effect Formula” to account for this increase in confinement over asquare pillar are listed below. In all cases the pillar width in the strength formulae is replaced by an effectivepillar width term.Sheorey & Singh (1974) proposed that the effective pillar width to be used in place of the width termin the shape effect formula be the average of the length of the two pillar sides. This work was based on thetesting of small scale samples of various rectangular dimensions. Wagner (1980) and Stacey & Page (1986)both proposed that the width term could be replaced by an effective pillar width term defined by Equation 15.We=4. (15)Rwhere:We — Effective pillar width (m)Cross-sectional area of pillar (m2)R = Pillar circumference (m)These methods may have merit for extrapolating the strength of a square pillar to a rectangular pillar,however it is evident that there must be an upper limit on the increase in strength that can be attributed to anincrease in pillar side length. Wagner’s (1980) and Stacey & Page’s (1986) approach to effective pillar widthhas an upper limit of two times the minimum pillar width, while Sheorey & Singh’s (1984) approach has anupper limit of one half the length of the long side of the pillar for very long pillars, making it the leastconservative for strength estimation. Salamon (1983) states that Wagner’s (1980) formula is “putforth onlytentatively at this stage” and that “Limited experience appears to suggest that it does give a reasonableestimate of the effective pillar width”.212.2.1.4 Hoek & Brown (1980) Failure CriteriaHoek & Brown (1980) have proposed an empirical strength criteria for rock masses as given inEquation 16. Hock & Brown (1980) propose that the influence of structural defects and pillar volume can bequantified through the use of rock mass classification parameters. These classifications result in empiricalparameters “m” and “s” for a given rock type, and are used in Equation 16 to detennine the correspondingrock mass strength. This value is then compared to the calculated pillar stress and a factor of safety can then bedetermined. Table 2 lists the values of the empirical rock mass constants ‘em” and “s” for various rock typesand rock mass classification ratings (Hock & Brown, 1988). Figure 9 illustrates the conceptual transition of arock mass from an intact condition to a heavily jointed condition.‘i=o3+4[(m.a.ai)+(s.a)] (16)where:a1 — Major principal stress (MPa)a3 — Minor principal stress (MPa)o Unconfined compressive strength of intact rock mass material (MPa)m ,s — Empirically derived constants based on rock mass quality of pillar materialHock & Brown (1980) developed pillar strength curves based upon this strength criteria and thedistribution of stresses inside modelled pillars. Figure 10 is a plot of the Hock & Brown (1980) pillar strengthcurves for igneous crystalline rock. The stress results were determined using two-dimensional boundaryelement modelling methods. Modelled pillar failure was assumed to have occurred when the average factorintact rocksingle discontinuitytwo discontinuitiesseveral discontinuitiesrock massFigure 9: Idealized illustration of the transition from intact rock to a heavily jointed rock mass withincreasing sample size (after Hock & Brown, 1980).undergroundexcavation- -—‘ _—_.22Table 2: Approximate relationship between rock mass quality and material constants (after Hock &Brown, 1988).Disturbed rock mass m and a values undisturbed rock mass m and a values..J)<EMPIRICAL FAILURE CRITERION < 0“ ‘> 0 ><.8 o.Lu0<E LLJ.O— 0oj = major principaleffectivestress0 —.U<.(2 00:9°’uI—’_ >‘.0ç)_u, .J= minor principal effective stress t.j . 0 < 8 0 <O 0.I-• . UJOo = uniaxial compressive strength — 0 .<Mof intact rock, and 0 —6m and $ are empirical constants. 0 6 Cgg <Zs 2zou. 0’-ot>I x.8 Iu>ujuJ8 <.‘<uJ °.8 <u,0,INTACT ROCK SAMPLESLaboratory size specimens free m 7.00 10.00 15.00 17.00 25.00from discontinuities $ 1.00 1.00 1.00 1.00 1.00CSIR rating: RMR = 100 m 7.00 10.00 15.00 17.00 25.00NGI rating: Q = 500 s 1.00 1.00 1.00 1.00 1.00VERY GOOD QUALITY ROCK MASSTightly interlocking undisturbed rock m 2.40 3.43 5.14 5.82 8.56with unweatheredjoints at 1 to 3m. a 0.082 0.082 0.082 0.082 0.082CSIR rating: RMR = 85 m 4.10 5.85 8.78 9.95 14.63NGI rating: Q = 100 a 0.189 0.189 0.189 0.189 0.189GOOD QUALITY ROCK MASSFresh to slightly weathered rock, slightly m 0.575 0.821 1.231 1.395 2.052disturbed with joints at 1 to 3m. $ 0.00293 0.00293 0.00293 0.00293 0.00293CSIR rating: RMR = 65 m 2.006 2.865 4.298 4.871 7.163NGI rating: Q = 10 s 0.0205 0.0205 0.0205 0.0205 0.0205FAIR QUALITY ROCK MASSSeveral sets of moderately weathered m 0.128 0.183 0.275 0.311 0.458joints spaced at 0.3 to lm. a 0.00009 0.00009 0.00009 0.00009 0.00009CSIR rating: RMR = 44 m 0.947 1.353 2.030 2.301 3.383NGI rating: Q = 1 a 0.001 98 0.00198 0.00198 0.001 98 0.00198POOR QUALITY ROCK MASSNumerous weathered joints at 30.500mm, m 0.029 0.041 0.061 0.069 0.102some gouge. Clean compacted waste rock s 0.000003 0.000003 0.000003 0.000003 0.000003CSIR rating: RMR = 23 m 0.447 0.639 0.959 1.087 1.598NGI rating: Q = 0.1 a 0.00019 0.00019 0.00019 0.00019 0.00019VERY POOR QUALITY ROCK MASSNumerous heavily weathered joints spaced m 0.007 0.010 0.015 0.017 0.025<50mm with gouge. Waste rock with fines. S 0.0000001 0.0000001 0.0000001 0.0000001 0.0000001CSIR rating: RMR = 3 m 0.219 0.313 0.469 0.532 0.782NGI rating: Q = 0.01 a 0.00002 0.00002 0.00002 0.00002 0.00002of safety across the center of the pillar fell below 1.0 based upon the modelled stresses. Hock & Brown (1980)propose that for a given pillar width / height ratio and a rock mass rating, the average pillar strengthnormalized to the unconfined compressive strength of intact pillar material can be determined.The pillar strength curves presented by Hock & Brown (1980) are, as stated, for igneous crystallinerock. In order to use this technique for differing rock types, a series of curves utilizing the correct values of23“rn” and “s” should be developed. In an attempt by the author to reproduce the Hoek & Brown (1980) pillarcurves for differing rock types, it was determined that the construction of the pillar strength curves appears tobe flawed. This is discussed below.Equation 16, where for a given rock type “rn” and “s” are constant, can be divided into 3 generalizedterms as follows:(i) =03(ii) 4m.o.o3(iii) =At pillar width / height ratios of less than 0.6, it can be shown that the strength will be controlled byterm (iii). As pillar width / height ratio increases, the ratio between term (ii) and term (iii) gradually increasesuntil, at width / height ratios above 1.0, the pillar strength is controlled predominantly by term (ii). Term (i)has negligible (< 5%) effect on strength. This can be generalized as follows:• A pillar with a width I height ratio of less than one will have strength that is approximately afunction of the addition of o, and 03.• A pillar with a width / height ratio greater than one will have strength that is approximately afunction the square root of the multiplication of o and 03.Considering the strength of a pillar with a width / height ratio greater than 1.0, it can be seen that,from Equation 16 and term (ii) above, strength is controlled by a function of the square root of o and thesquare root of 03, the minor principal stress (which, based upon pillar geometry, is a function of applied pillarstress, of). The pillar strength as determined from Figure 10, however, is a function of o and the pillar width/ height ratio only. It is unclear as to why the pillar curves are constructed using o, and not the square root ofthe o, and why the minor principal stress ,03 , is excluded.Since the pillar curves are a function of o, only term (iii) can be accurately represented and this termignores the influence of confinement. Hock & Brown (1980) show that the confining stress, 03, in a pillarincreases with the applied stress, a. Using Figure 10, the strength for a pillar of a given width / height ratiowill be constant, dependent only on o, regardless of the applied pillar stresses (or depth below surface). The isin direct contrast to Equation 16 which clearly shows that as stress (03) increases, so to will the calculatedstrength (os).A second point to note with this method is that the unconfined compressive strength is included in twoplaces if the CSIR-RMR rock mass classification system (Bieniawski, 1973) is used:• the CSIR-RMR (Bieniawski, 1973) strength parameter• the unconfined compressive strength normalizing term240a0)0.001C4)0C4)0)4)0.E0I.;0xI,C.001C4,C,aSC01C4,>Figure 10: Hock & Brown (1980) pillar strength curves for igneous crystalline rock (after Hock & Brown,1980).A lower unconfined compressive strength will result in a lower RMR, which will place the pillar on aweaker pillar line. The pillar strength will then be reduced again when normalizing the pillar strength to theunconfined compressive strength on the vertical scale on the pillar strength graph.There is no report of this technique being used with success to design pillars in literature. Page &Brennen (1982) report that this method approximated observed pillar strength for a good quality rock mass forpillar with width / height ratios of 0.7.2.2.1.5 The Size Effect FormulaEmpirical size effect strength formulae have been developed by a number of researchers in the form ofEquation 17. The effect of this type of formula is that as pillar size increases, the strength of pillars of equalshape will decrease.(17)where:= Pillar strength (MPa, psi)K — Strength term related to pillar material (MPa, psi)w Pillar width (m, ft)3.02.5Intactm 8.5. s 0.1Good quality rock mass— 1.7, S — 0.004Fair quality rock massrn 0.34, s — 0.0001Poor quality 000k massrn 0.09, 5 — 0.000010 3Pillar width/height Wp/h25h — Pillar height (m, ft)a, b — Empirically derived constantsA number of researchers have proposed strength formulae in this form and the values of the empiricalconstants suggested by each author are presented in Table 3.The methods presented in Table 3 represent the accumulated knowledge with respect to pillar designbased upon the “Size Effect Formula”. The work of Salamon & Munro (1967), Hedley & Grant (1972) andSheorey et al. (1987) is discussed in the following sections. It must be noted that due to the dimensionallyunbalanced nature of the “Size Effect Formula” , the pillar width and height must be in feet and the strengthmust be in pounds I square inch. In order to convert this formula to SI units, the “K” term must be reduced tocompensate for the dimensionally unbalanced conversion, as a result of the differing exponents, of feet tometres in the width / height term.Table 3: “Size Effect Formulae” empirical constants, “a” and “b”, from various authors.Source a bSteart (1954) 0.5 1.0Holland-Gaddy (1962) 0.5 1.0Greenwald et al. (1939) 0.5 0.833Hedley & Grant (1972) 0.5 0.75Salamon & Munro (1967) 0.46 0.66Bieniawski (1968) 0.16 0.55Sheorey et al. (1987) 0.5 0.86(for slender pillars)2.2.1 .5.1 Salamon & Munro (1967)Salamon & Munro (1967) conducted an investigation into the strength of square pillars in SouthAfrican coal mines. Questionnaires were sent to mine operators requesting observations of current workingsand areas of collapsed workings. An observation represented a mine or portion of a mine, where miningdimensions were essentially constant and pillars were square in cross-section. The database consisted of 125case histories, of which 98 cases were classified as “stable” and 27 cases were classified as “collapsed”. Thisdata is presented in Table 4. Pillar loads were calculated using tributary area theory. Statistical analysis onthis data yielded the empirical strength constants presented in Table 5 for the “Size Effect Formula”1.The benefit of this work is that the data used to determine the strength relationship was obtained fromactual mine pillar case histories as reported by mine operators. It should be noted that Salamon & MunroThe pillar width and pillar height are presented here in imperial units because of the dimensionally unbalanced nature of the Size EffectFormula” as discussed in Section 2.3.1.5.26(1967) used one coal strength constant “K” for all of the pillar case histories in the database. This value wasdetermined statistically from all of the case histories in the database without reference to the actual intact coalstrength at each mining operation.Salamon & Munro (1967) calculated that the average factor of safety for stable pillars was 1.6 basedupon the histogram presented in Figure 11. Budavari (1983) reported that South African coal mines almostexclusively use the Salamon & Munro (1967) formula with a factor of safety of 1.6 to design pillar layouts.Table 4: Salamon & Munro (1967) database summary for compiled case histories.Group Stable CollapsedNumber Cases 98 27Depth (feet) 65-720 70-630Pillar Height (feet) 4-16 5-18Pillar Width (feet) 9-70 11-52Extraction Ratio 37-89 45-91Pillar Width / Height ratio 1.2-8.8 0.9-3.6TableS: “Size Effect” empirical constants determined by Salamon & Munro (1967).K(strength) a b1322 psi 0.6609 0.45902.2.1.5.2 Hedley & Grant (1972)Hedley & Grant (1972) proposed a pillar design method based upon data obtained from uranium minesin the Elliot Lake district of Ontario, Canada. The database consists of 28 pillar case histories (23 stablepillars, two partially failed pillars , and three failed pillars). It was proposed that the pillar strengthrelationship could be approximated by a “Size Effect Formula” described by Equation 18.(18)where:2P Pillar strength (psi)K Strength of 30 cm cubic sample0.7.UCS (50 mm diameter. sample)179 MPa (26,000 psi) for Elliot Lake rocks.w Pillar width (ft)h Pillar height (ft)2The pillar width and pillar height are presented here in imperial units because of the dimensionally unbalanced nature of the “Size EffectFormuia’ as discussed in Section 2.3.1.5.27a5)IUU00UaVVU.Figure 11: Histogram showing frequencies of intact pillar performance, and pillar failure, for South Africancoal mines (after Salamon & Munro, 1967).Pillar stress was determined using the tributary area theory with modifications to account for theaddition of horizontal in-situ stresses as described in Section 2.1.1.3. This is because the Elliot Lake uraniummines occur in dipping orebodies. In-situ measurements of stress were made at two mines in order to verify theuse of tributary area theory. The authors determined that the average value of measured stress and tributaryarea theory compared favourably, however there was a wide scatter in the individual measurements. Hedley &Grant (1972) state that “.. the measurements themselves cannot be taken as accurate values and can vary by ±50 per cent”.This work represents one of the few instances where hard rock pillar data has been used to develop apillar strength formula. Figure 12 and Figure 13 are Hedley & Grant’s (1972) plots of pillar stress versus depthand extraction ratio method and pillar stress versus calculated pillar strength using Equation 18. Hedley &Grant (1972) have used a “Size Effect Formula”, which they stated was based upon the work of Salamon &Munro (1967), to derive their strength relationship. The development of a “Size Effect Formula” for pillarstrength, however, requires a database that contains pillars of a wide range of sizes. The pillars that make upthe Hedley & Grant (1972) database are all of similar size and therefore cannot fully justify the use of a “SizeEffect Formula”.It should also be noted that this method is used in underground hard rock mine design, yet thedatabase contains only three failed pillar case histories. The use of only three failed pillars to develop astrength relationship leaves the potential for a wide margin of error.12 -(0 -86420 0.5Keycollapsedcasesstable casesrhalf the stable cases are mostdensely concentrated in range Rlower limit of R F1 = 131mean of R Fm 1.57upper limit of R F,, = 1.881.0 I 2.0 2.5FSafety factor3.0 3.5285000 /. ExtractionDipd.g070203040E310Figure 12: Hedley & Grant’s (1972) method for the determination of pillar stress (after Hedley & Grant,1972).SF 1.0 SF 1.3ZSF 15• Failed Pillarso Partially Foiled PillarSo Stable PillarsSF=Safety FactorSF2.0UNSTALE : TRANSITIONAL STA&E SF 2.500. °SF3.00Q 0 0o SF3.500roóoo ‘,zbooEstimated Pillar Strength psi.Figure 13: Hedley & Grant’s (1972) estimation of pillar stresses and strengths (after Hedley & Grant,1972).50 6065 70 75 8000851.10900‘0isàoo 2000 250004doo lox 200 idoo a sOoo toxoDepth Below Surface ft Pillar Stress psi1aa.0aE750001000050000 5000 20000 25000292.2.1 .5.3 Sheorey et al. (1987)Sheorey et al. (1987) investigated stable and failed pillars for coal mines in India and proposed anempirical strength formula as presented in Equation 19. The database is comprised of 23 failed and 20 stablepillar observations. They also proposed a second strength formula for slender pillars (pillar width / height ratioless than 4) as presented in Equation 20.Pillar stresses for the majority of the pillar case histories used to derive these formulae were calculatedusing tributary area theory. Numerical modelling was used to calculate pillar stresses for the remaining pillarcase histories.0.27a H w= h 0.36(19)Pr = 0.27a.h°•86(20)where:= Pillar strength (MPa)— Unconfined compressive strength of a 25 mm cube of pillar materialh = Pillar height (m)w — Pillar width (m)H — Depth below surface (m)2.2.1.6 Discussion - Empirical Strength FormulaeAn advantage of empirical strength formulae is that they are generally based on observations thatincorporate full size mine pillars. They do not, however, make any attempt to explain the mechanism of pillarloading and pillar failure. In the next section, theoretical strength determination methods for mine pillars thathave been suggested by various researchers are investigated.222 Theoretical Design MethodsA number of researchers have presented theoretical formulae to estimate the strength of mine pillarsthat will be covered in the following section. Potvin (1985) states that Wilson’s (1972) confined core method isthe only method to have been used to design pillars in Canadian mining operations.2.2.2.1 Wilson’s (1972) Confined Core MethodWilson’s (1972) confined core method attempts to evaluate mathematically the primary factors thatinfluence pillar strength and is designed to be used to determine the strength of wide pillars (width I heightratio greater than 4.5). Wilson (1972) states that when moving from the pillar boundary to the pillar core,transition from yielded zone will progress into an undisturbed elastic zone towards the pillar core. Hisproposed relationship is given in Equations 21 and 22.301•ln() (21)h 4tan()(tfl(1)) 0tan(13)= 1 +sin(Ø)(22)1—sin(4))where:3Y = Depth of yield zone into pillar from ribside (ft)h Seam height (ft)0v — Maximum pillar stress which occurs at the boundary between theyield zone and pillar core (psi)00 Unconfined compressive strength of pillar material (psi)tan(13) = Triaxial stress coefficient4) — Angle of internal friction of the pillar material (degrees)Wilson (1972) develops equations for pillar load based on the above hypothesis for wide pillars andslender pillars. Slender pillars are defined as those meeting the criteria defined in Equation 23. When thiscriteria is met, there will be no confined pillar core and no elastic triaxial zone. Equations 24 and 25 arestrength formulae for wide and slender square pillars respectively. Wilson (1972) also developed similarformulae for rectangular and long pillars which are not included here.wcO.003.hH (23)L = 4pH . (w — 0.003 • whH + 0.000003.h2H) (24)L=444.p—-- (25)where:L = Pillar load (tons)p Average density of rock (tons / ft3)H — Depth of cover (ft)w — Width of pillar (ft)h — Height of pillar or seam (ft)It should be noted that the slender pillar equation, Equation 25, is a variation of the “Shape EffectFormula”. By moving “w2” from the right to the left side of the equation, we generate a stress term in place ofthe load, “L”. We are left with a “width I height” term and a density term on the right side of the equationwhich is in the form of a “Shape Effect Formula”.Imperial units are presented here as originally presented by Wilson (1972).312.2.2.2 Coates (1965)Coates (1965) proposed a method for predicting pillar loads based upon relating the staticallyindeterminate deflection of pillar walls to the increase in stress due to mining. This method was applied to ribpillar mining and involves ten major variables and predicts lower pillar stresses than tributary area theory.This method appears impractical since it seems that for pillar wall deflection to be measurable, the pillarmaterial would have to be intact and behave elastically. Jointed rock masses would more than likely degradeprogressively such that deflection would be immeasurable. Agapito (1972) states that Coates & Ignatieff (1966)appeared unable to verify this approach in practice.2.2.2.3 Panek (1979)Panek (1979) developed a general strength equation for the load bearing capacity of rock pillars ofdifferent cross-sectional shapes. The goal of his work was to develop a means of using laboratory compressiontests to estimate the strength of full scale mine pillars. Panek (1979) derives an expression for pillar strength ofbrittle material that is expressed as a mathematical product of the following factors:• a size effect• a shape effect• a function of the mechanical properties (deformability, joint spacing, and friction) of the pillar• a function of the mechanical properties (deformability, joint spacing, and friction) of the floor androof materialsPanek (1979) deals solely with the maximum resistance offered by a pillar to a specific load and makesno reference to the method with which the pillar load is to be determined. Panek (1979) presented a formula ofthe general form as described by Equation 26.= )C6(L (26)E5 w h w Es Es 5 ;Vj ‘Or Vf Vs d2 d3where:h,w,b Pillar dimensions height, width and length (m)d1,23 Representative parameters of the frequency distribution of the defects (joints,cleats) in the mined seamModulus of elasticity of the seam, roof, floor material (GPa)Poison’s ratio of seam, roof, floor materialJLsir.s/r Coefficient of friction between seam and roof, between seam and floor1(0 = Maximum resistance of pillar to compressive force (MPa)S — Pillar compressive stress at maximum resistance (MPa)— Constants32Panek (1979) proposed that this form of equation could be used to introduce new variables into thepillar strength formula if required. In order to utilize this equation in practice, Panek (1979) suggests using thetheory of similitude and that Equation 27 equation would be used to determine the pillar strength.d1 c1 w c2r Cr, . [(—) (—) .. . . Ipredicted[)jpredzced— w h 7[S]known — CIICI 7C]known(2 )w hEquation 27 would be used to relate the results obtained from testing a small sample, known, in a steelplatened testing machine to the actual large scale pillar, predicted, strength. In a response to this method,Sheorey (1980) suggested that two additional terms, time dependence of strength and the influence of mineenvironment (moisture and weathering), could be included in this equation. There is no reference available inliterature that suggests that this technique has been used in practice.2.2.2.4 Grobbelaar (1970)Grobbelaar (1970) proposed a theoretical method of determining the strength of coal pillars. Heproposed the conversion of the strength of small cubes of pillar material to larger cubical specimens using agraph that related the normalized strength of the pillar to the normalized cube side length along with thenumber and the standard deviation of the number of flaws in the larger cubical specimen. He also postulatesthat the strength of non-cubical samples is dependent on the properties of the pillar material and the frictionbetween the pillar and the surrounding rock. He states that for most mine pillars the friction component at thepillar end is sufficient to implement the derived theoretical expression. The resulting expression for pillarstrength for a square pillar is given by Equation 28.— (Cw+WDD_,V)_WD)28)avGuW2(C+D2(D_y)_WD1)where:= Average stress on the pillar at failure (MPa)— Strength of a cubical pillar specimen of height equal to the actual height of thepillar (m)W — Width / height ratio of the pillarC k1*k2 where k1 is the ratio of lateral stress (developed within the pillar) to thevertical stress and equal to k2D = logC33Grobbelaar (1970) states that for a pillar that has an elastic core and peripheral zone of fractured rock,a sample calculation shows that the average stress at which failure will propagate into the pillar core is afunction of ten parameters, five of which are material constants. Grobbelaar (1970) states that this approachcorresponds to the strength measured on model pillars comprised of sand, coal, sandstone, and rocksalt but hasyet to be verified by case studies. It should be noted that Grobbelaar (1970) is attempting to determine stressesin pillars that have a wide cross-section and can be considered to be under triaxial stress conditions within thepillar core.2.2.2.5 Discussion - Theoretical Sfrength FormulaeTheoretical strength formula have been derived by a number of researchers. They have, however, notbeen verified in practice. As mentioned previously, Potvin (1985) states that in a survey of Canadian miningoperations, only Wilson’s (1972) confined core method had been used. .The benefit of this theoretical work,however, is that it may provide us with insight into the mechanism of pillar loading and failure.2.23 HeuristicMethodsA number of heuristic pillar design methods have been used in the past. These “rule of thumb”methods include the “Mines Inspector Formula” (Ashley, 1930), the “Holland Formula” (Holland, 1964),Morrison et al. (1961), and the “Barrier Pillar Formula” (King & Whittaker, 1971). These approachesoversimplify the pillar design process and will be presented, but not be discussed in detail.2.2.3.1 Mines Inspector FormulaAshley (1930) presented the Equation 29 based on experiments on Pennsylvania coal beds andassumes that an arch equal to half of the panel width be stable. No reference is made to pillar material oroverburden density although it is safe to assume that this method was designed to be used for coal pillars. Asthe formula shows, this is designed to dimension pillars but makes no assessment of pillar strength.w=20+4.h+0.1.H (29)where:4w — Pillar width (ft)h Pillar height (ft)H Depth of overburden cover (ft)4These units axe presented in imperial as originally presented by Ashley (1930).342.2.3.2 Holland (1964) FormulaPotvin (1985) reported on the work of Holland (1964). The Holland Formula is a dimensioningformula for pillar design based on convergence studies by Belinski & Borecki (1964). This is anothersimplistic method for pillar design in coal mines.w=15.h or= log(2W2)Klog(e)where:w — Width of barrier pillar (m)h — Thickness of pillar (m)W2 Estimated convergence on high stress side of pillar (mm)K — Constant- 0.09 if caving after mining is permitted- 0.08 if strip packs are built- 0.07 if hydraulic stowage is used2.2.3.3 Morrison et al. (1961)Potvin (1985) reported the work of Morrison et al. (1961). The authors presented an overly simplisticformula for barrier pillar dimensioning based upon research into mine safety in Canada as follows:w=.Hfor H< 1200 m (31)where:w = Width of pillar (m)H = Depth of overburden cover (m)2.2.3.4 Barrier Pillar FormulaKing & Whittaker (1971) developed this rule of thumb approach for the design of barrier pillars andagain is a simplistic pillar design method.w=-’-.H+4.5 (32)10where:w — Width of pillar (m)H = Depth of overburden cover (m)352.3 NUMERICALMODELLING TECHNIQUESNumerical modelling techniques are able to determine stress redistribution around mine openingswithin a triaxial stress field employing either two-dimensional or three-dimensional methods. This alternativeto tributary area theory for complex geometrical or geological environments makes pillar stress determinationmore accurate.Numerical modelling is a technique that uses mathematical formulae to solve stress related problems.It is used in many fields of engineering and has gained considerable acceptance. Models are becomingincreasingly complex, yet there are limitations on what can be analyzed. At the heart of any numerical modelis the numerical code and the algorithm that define the model. These are fixed in the solution of miningrelated problems. These codes are coupled with input data to generate results, which will be poor if the inputdata is not valid. Input data consists of in-situ stress conditions (either measured or estimated), mininggeometry, rock mass characteristics, and elastic constants. Most models require making simplifiedassumptions, such as homogeneous, isotropic, and elastic rock mass conditions, which is not the case inmining.Even with the generalizations that are necessary to perform numerical modelling, the results obtainedwill still be of higher quality than those which might be estimated using tributary area theory. Figure 14illustrates the conceptual models we can attempt to solve using numerical modelling techniques. The scope ofthis thesis is not to perform an in-depth analysis of numerical modelling codes and techniques. It is howeveruseful to discuss the types of models and their applicability to mining conditions.L_______JLbFigure 14: Conceptual models relating rock structure and rock response to excavation (after Brown, 1987).2.3.1 Pillar Design Using Numerical ModelsThere are a two ways that the results from numerical modelling programs can be used to design pillarsin underground mines. The results can be used to predict pillar stresses, or a failure criteria can beincorporated to predict the extent of failed areas. One of the inherent problems is the difficulty of knowingContinuous planesof weakness(C) di36with confidence what is happening within the rock mass. Observations of conditions at the boundary are easyto obtain, but just how far that condition extends into the rock mass is difficult to assess. These points shouldbe kept in mind while reading the following section on the use of numerical modelling results.2.3.2 Types ofModelling MethodsBrown (1987) divides current modelling methods applicable to mining into four separate groups:• differential continuum methods• integral methods• differential discontinuum methods• hybrid or linked methods (that combine integral and differential methods)2.3.2.1 Continuum MethodsThe continuum methods are the “Finite Element” and the “Finite Difference” methods. Continuummethods require that mathematical and physical approximations be made throughout the region of interest (theproblem domain). Finite difference methods utilize approximate numerical solutions for the problemsgoverning equations at an array of points within the problem domain. This gives an approximate solution to anexact problem. The finite element method discretizes the entire problem domain area into a series of discreteelements which provide a physical approximation to the continuity of displacements and stresses within thecontinuum. The governing equations for the problem are solved exactly at nodes at which adjacent elementsconnect. This results in an exact solution for a differential approximation to the problem. A major limitationof the application of finite element methods to exterior problems is the fact that an arbitrary outer boundary tothe problem must be defined. The time requirements for continuum methods for data preparation and solutiontime can be extremely large if the problem domain is extended sufficiently such that far field stress conditionsare satisfied. Figure 15 illustrates a finite element problem construction. For an in-depth discussion of finiteelement methods, the reader is referred to Zienkiewicz (1977).2.3.2.2 Integral MethodsIntegral methods replace the physical problem to be modelled with an equivalent solution using“fictitious forces” and “fictitious displacements” on a boundary. The two integral methods used in rockmechanics, which are referred to collectively as boundary element methods, are the “Boun&#y Element”method and the “Displacement Discontinuity” method. Figure 16 illustrates the conceptual difference betweenfinite element and boundary element problem construction. Figure 16 (a) represents a finite element problemthat has a finite boundary and problem domain defined by elements. Figure 16 (b) illustrates a boundaryelement problem where the problem boundary is infinite and boundary elements are utilized on the excavationboundaries only. Figure 17 and Figure 18 give examples of the boundary element problem formulation.37(a) t (b) (c)4-4-4-y0‘Jr ‘JrFigure 15: Development of a finite element model of a continuum problem, and specification of elementgeometry and loading for a constant strain, triangular finite element (after Brady & Brown,1985).Figure 19 shows example output from a modelled pillar using two-dimensional boundary element modelling.The contour values are for the major and minor principal stresses developed within the pillar. Figure 20illustrates a simplified three-dimensional boundary element problem defmition.2.3.2.3 Discontinuum MethodsDiscontinuum methods are used where it is not possible to replace a rock mass with an equivalentcontinuum material. Problems involve a finite number of elements of which the ratio of element size toproblem domain is such that an equivalent material cannot be used. The most powerful and versatile methodavailable is the distinct element method originally developed by Cundall (1971) as a means of modellingprogressive failure of a rock slope. This method utilizes a dynamic relaxation technique to solve Newton’s lawsof motion to determine the forces and the displacements of elements that result during the large scaleprogressive deformation of discontinuities. Figure 21 illustrates the distinct element method.2.3.2.4 HybridMethodsHybrid methods combine different numerical methods to solve mining problems. Boundary elementmethods are linked to finite element or distinct element methods. The problem domain around mine openingsis discretized and defined using either a distinct element or fmite element method and the exterior of theproblem domain is defined using the boundary element method. Hybrid methods make it possible to resolvestresses in close proximity to mine openings in detail while maintaining boundary conditions using boundaryelement methods.--,-4— S..Uf fQk/’%’ Uk/etqr38(a) (b)Figure 16: Simplified finite element and boundary element problem formulation for the same excavationgeometry (after Brown, 1987).Figure 17: Superposition scheme demonstrating that generation of an excavation is mechanically equivalentto introducing a set of traction’s on a surface in a continuum (after Brady & Brown, 1985).,) ((S)1)xFigure 18: Surface, element and load distribution description for development of a quadratic, isoparametric,indirect boundary element formulation (after Brady & Brown, 1985).Free surfaceF,n,lebotindaryInfàniteboundary4—4-4-(a) (b) (c)S.s L.L ‘“ -.. - ...,. h(S) — —t(S),(a)(c) q3=+i 3j(x,y)-Y2)(x1, Yi)39Figure 19: Principal stress distribution in a rib pillar defined by a ratio of pillar width to height of 1.0. Thecontour values are given by the ratio of major and minor principal stresses to the average pillarstress. Plane strain analysis for uniformly distributed vertical applied stress (after Hoek &Brown, 1980).2.3.3 Choice ofMethodThe choice of which numerical method to use to solve stress related problems in mining depends uponthe desired results. It is the author’s opinion that finite element methods may provide a result that has a highdegree of accuracy, however the required input parameters are difficult to assess commensurate with thisaccuracy. lii addition, the amount of time required to define and solve finite element problems can beextremely large in relation to the solution time required for boundary element methods. It is the author’sopinion that the boundary element method is the most desirable numerical method in mining situations wheregeneral stress solutions are desired. Boundary element techniques are currently used extensively andsuccessfully to solve mining related stress problems.401RL_Figure 20: An idealized sketch showing the principle of numerical modelling of underground excavations(after Hudyma, 1988).(a) (b) (c)‘ftnewn2Figure 21: Normal and shear modes of interaction between distinct elements (after Brady & Brown, 1985).412.3.4 Pillar Failure Assessment using Numerical ModelsIt is possible to utilize failure criteria in numerical modelling methods. Two forms of failureassessment are possible, post processing and interactive. Post processing methods involve applying a failurecriteria after the solution of the model to assess whether modelled failure has occurred or not. The two mostcommon failure criteria applied are the Mohr-Coulomb and the Hock & Brown (1980) methods. A drawbackof using post-processing methods that the stresses are not shed from the failed portions of the model whileprocessing is taking place. As a result, the modelling results may not accurately reflect actual failed conditions.Figure 22 is an example of a modelling session where a failure criteria was applied to a pillar mining problemto predict failure (Brady, 1977). Figure 22 shows that as pillar size was reduced, the predicted failure zoneextended across the entire pillar width. The failure criteria used was based on the work of Murrell (1965).The interactive method has been used in displacement discontinuity models (NFOLD, ref. Maconochieet al., 1981) whereby non-linear elements are used. Model elements are assigned a peak and post-peak strengthas illustrated in Figure 23. The numerical model is solved through an iterative process. Elements that havemodelled stresses that exceed the peak strength of the element are assigned lower stress values during eachmodel iteration, to a minimum value of the post-peak strength value. The iterative process continues until themodel reaches equilibrium. Figure 24 and Figure 25 show the stress results along with failed areas for amodelling sessions performed, using displacement discontinuity methods, by Maconochie et al. (1981) andBywater et al. (1983) respectively.2.3.5 Discussion - NumericalMethodsNumerical methods provide us with a number of codes that can be used to solve stress related miningproblems. These codes have an advantage over the simplified assumptions made with tributary area theory,resulting in the ability to analyze complex mining geometries in three dimensions. It must be recognizedhowever, that numerical models are not a panacea for all stress related mining problems. This is due to thedifficulty in identifying and assigning the critical input parameters that are required when dealing withheterogeneous anisotropic media, the rock mass.Numerical models can be used successfully for comparative and parametric studies provided themodelling technique employed can consistently assess various geometries. Errors in the computed stresses canbe accounted for in the design process, provided the errors are consistent. There is little point, however, inaccurately determining induced pillar stresses with expensive and time consuming modelling techniques whenthere is no accurate method of determining the absolute pillar stress and strength.42Figure 22: Modelled pillar failure as a result of pillar reduction (after Brady, 1977).BRITTLENESS PEAKMODULUS (MPa) STRENGTH (MPa)I bi dv’MINED IaibIbIbIcV/EI.ASTICAREA IbIdIdid/bidididlalbi bibI bi dV,’Figure 23: The peak strength, deformation characteristics, and effect of location used for investigating apillar case history with a displacement discontinuity program (after Maconochie et al., 1981).FFFFFFFPPFFFFPPFFPFFFFFFFFPFFFFPPFF‘F’ denotes modelled failure.0FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFPFFFFPFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFPFFFFFFFFFFFFPFFFFFFFFFFPFFFFFFFFFFFFFFFFFFFFFFFFFFFCF F F F FFFFFFFFFFFFPPFFFFFFFFFFFFFFFFFFFFFFFFFFFPFFFFFFFFFFFFFFFFFFFFFFFFFFFFPFFFFFFFFFFFFFFFFFFFFFFFFFFFF:FFFFFF:FFFFFFFFFFFFFFFFFPFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF‘F’ denotes modelled failure.a EXPOSED CORNERb EXPOSED SIDEo RE-ENTRANT CORNERd ONE BEHIND FREE SIDE12.58.37.06.37290126156&CU 0.10 0.20 0.30 0.40 0.50 0.60 CONVERGENCE(m)43Figure 24: The normal stress and the failed regions estimated with the displacement discontinuity program,NFOLD, for a sill pillar case history (after Maconochie et al., 1981).NORMAL .70 MN/rn2STRESS60 - 69 MN/rn250 - 59 MN/rn240. 49 MN/rn2Figure 25: The distribution of normal stress in a mining block was estimated for two different miningsequences to determine the best extraction sequence using displacement discontinuity methods(after Bywater Ct al., 1983).ELASTICBRITrLE RANGE29FAILED310032I3334353615/I.1GB02550 100METRES02550 100METRES17 D17/I.18 B19015/I-1GB17 D17/L1GB19C3138442.4 PILLAR STRENGTHESTIMATIONAT WESTMIN RESOURCES LTD. ‘S H-WMINEThe approaches to pillar design presented in this chapter may be valid in the environment in whichthey were derived, however, they do not present a pillar design method that can be used for all pillar situations,particularly underground hard rock mines. In order to make an assessment of pillar strength at WestminResources Ltd.’s H-W Mine, the following approach was taken:1. Determine the material properties of the rock mass at Westmin Resources Ltd. ‘s H-W Mine suchas intact rock strength, rock mass characteristics, and structural features.2. Perform in-situ stress measurements in the vicinity of the H-W Mine as a primary input parameterfor three-dimensional boundary element modelling.3. Classify all pillars in the study domain according to their observed stability condition.4. Perform numerical modelling simulating mining and pillar extraction sequencing in an attempt todetermine pillar stresses and pillar stress history. Calibrate the numerical model with observationof pillar failures.5. Analyze and correlate the pillars classified with regard to the geometric properties of the pillar,the stress conditions from modelling, the structural and rock mass conditions in the pillar vicinity,and visual assessment of pillar stability.6. Augment the Westmin pillar database with available published hard rock pillar data and develop acomprehensive pillar strength relationship.2.5 CHAPTER SUMMARYThis chapter has reviewed the state-of-the-art regarding pillar stress and strength determination andhas presented the methodology for pillar strength estimation at Westmin Resources Ltd.’s H-W Mine.Empirical strength formulae that relate pillar strength to pillar width I height ratios have been successfully usedin the literature for pillar strength estimation. Theoretical methods have been developed by various researchersbut these techniques have not been widely used in practice. The majority of pillar strength analysis has beenconducted for coal pillars in horizontally bedded deposits and this work must be augmented by hard rock pillardata in order to advance the state-of-the-art in pillar design for hard rock mining operations.453. PILLAR FAILURE MECHANISMThe mechanism of pillar failure must be understood in order to adequately assess the strength of minepillars. Two basic forms of pillar instability are possible:• a progressive or controlled failure mode• a bursting or unstable failure modeProgressive failure occurs over an extended period of time and is the result of the gradual release ofenergy by a pillar. Progressive pillar failure is characterized by the increase in the number and size of fracturesalong with blocks falling out of the pillar and the gradual hour-glassing of pillar walls.Bursting pillar failure is characterized by a rapid release of energy and pillar mass. Severe damage tonearby infrastructure and life can result due to flying rock from a burst type failure. The degree to which a rockmass system is susceptible to bursting failures is described by Budavari (1983) as follows: In a system wherethe host rock mass is less stiff than the pillar rock mass, the host rock mass can store up loaded energy similarto a spring. When the host rock mass (the spring) releases the stored energy, the resulting failure results in aviolent release of energy as the “stiff’ pillar fails rapidly.Much of the research into pillar failure and pillar strength has been undertaken on coal pillars. Thewidth / height ratios of these coal pillars is generally higher than the width / height ratios of hard rock pillars.Numerical modelling shows that when a pillar has a width / height ratio greater than one, the effect of theminor principal stress on pillar strength is significant (this is discussed in Section 5.2.3.2). This has a dramaticimpact on the type of failure that one would observe.Figure 26 (a-e) are representations of various progressive pillar failure modes that are possible and aredescribed as follows:• Figure 26 (a) illustrates spalling of pillar walls into the mine opening, this type of failure is of aprogressive nature.• Figure 26 (b) illustrates failure along a discrete failure plane that develops within the core of thepillar.• Figure 26 (c) illustrates internal splitting as the result of pillar movement along soft partings(clay) at the top and bottom of the pillar.• In a rock mass that has a pronounced structural presence, failure can occur as illustrated Figure 26(d, e). Where structure is oriented along the vertical axis of the pillar, failure can assume abuckling mode. In pillars that are transected by inclined structures, the failure is similar to that ofa sliding deck of cards.46(a) (b) (C)original pillar surface— soft partings_L _internal splittingFigure 26: Principal modes of deformation behaviour of mine pillars (after Brady & Brown, 1985).3.1 PILL4R STABILITYASSESSMENTKrauland & Soder (1987) presented a method for characterizing the progressive failure of mine pillarsat Boliden Minerals Black Angel Mine, Greenland. They extended the small scale work of Hallbauer et al.(1973) and John (1971) which showed that zones of extensive fracturing, on a microscopic scale within asample specimen, occurred at load levels in excess of 95% of the maximum stress as shown in Figure 27.Using this general methodology, Krauland & Soder (1987) developed the six stage pillar failure classificationmethod presented in Table 6. Using this method, relative pillar stability can be classified and the availablepillar load bearing capacity estimated.3.2 FACTORS RELATED TO PILLAR STABILITYOne must investigate the factors that contribute to pillar stability and ultimately pillar strength whendiscussing pillar failure mechanism. The major variables are discussed in the following section. This does notpreclude that in some cases there may be additional variables that have an effect on the stability of mine pillars.(d) (e)47C0.6 am. 095 am. 0.98o1nc. arn tam.Region III 4 Region I Region VFigure 27: Fracturing of rock specimens at various stages of loading (after Krauland & Soder, 1987).Table 6: Pillar failure classification method (after Krauland & Soder, 1987).Pillar Stability Pillar ConditionsClassification0 No fractures1 Slight spalling of pillar corners and pillar walls, with short fracturelengths_in relation to pillar height,_sub parallel_to_pillar walls.2 One or a few fractures near surface, distinct spalling.3 Fractures also appear in central parts of the pillar.4 One or few fractures occur through central parts of the pillar.Fracture may be parallel to pillar walls or diagonal, indicatingemergence of an hour-glass-shaped pillar.5 Disintegration of pillar. Major blocks fall out and / or the pillar iscut off by well defined fractures. Alternatively, a well developedhour-glass shape may emerge, with central parts completelycrushed.°doI483.2.1 Intact Rock StrengthOne of the governing factors on the strength of mine pillars is the unconfined compressive strength ofthe intact pillar material. This value can be obtained through the testing of small scale laboratory samples, thepoint load index or the hammer test. The drawback of using a laboratory strength for pillar strength estimationis that the effects of structural defects and variability of strength associated within the rock mass are ignored. Itmust be noted that in many mining situations, the unconfined compressive strength is the only valuerepresentative of rock mass strength available to mine engineers. The unconfined strength does not howeverrepresent the true unconfined compressive strength of the rock mass. “Size Effect” formulae or strengthcorrection factors are used to attempt to relate laboratory values of intact rock strength to in-situ rock massstrength estimates.3.2.2 Pillar StressPillar failure occurs when pillar stresses exceed the pillar strength. This is visually observed byfracturing and spalling of pillar material. In order to make an assessment of pillar strength, it is necessary tohave an estimate of pillar stress to compare with the calculated pillar strength. Therefore, pillar stress is acritical component in the assessment of pillai strength.3.2.3 Pillar ShapePillar shape plays an important role in pillar strength estimation as illustrated by the researcherswhose work was presented in Chapter 2. It is acknowledged that the more slender a pillar, the lower the pillarstrength. Figure 28 is a plot of curves that Hock & Brown (1980) presented which illustrate the effect of pillarshape on pillar strength based upon the work of various researchers. It can be seen that as pillar width / heightratio increases, the pillar strength increases at a decreasing rate, for all methods presented.2.01.81.6Salaien and Munro141 2 Greenwald et al‘Average, c 0.6, d — -0.1aHolland and Gaddyo.8CBieniawski0.60.40.20 1 2 3Pillar width/height ratio — Wp/hFigure 28: Relationship between pillar shape and pillar strength for constants suggested by various authors(after Hock & Brown, 1980).Kvdfor V 5000 ft34 549.614... 1.2n 1.00.8C.30.60.4.3.0.200.1Figure 29: Relationship between pillar volume and pillar strength for constants suggested by various authors(after Hoek & Brown, 1980).3.2.4 Pillar VolumeA pillar with a larger volume will contain more structural defects and consequently have a lowerstrength for a given width / height ratio. It has been shown, however, that there is an upper boundary to thisrelationship. Figure 29 illustrates the effect of volume on pillar strength as determined by various researchers.Figure 30 presents a curve relating sample strength to sample size for specimens of various sizes. The curvesshow that as sample size reaches a critical limit, the loss of strength becomes negligible with an increase insample volume. This is observed to occur when the sample diameter is greater than 1.0 to 1.5 metres. Figure31 is a figure presented by Kostak (1971) that shows the effect of sample size on the strength testing of smallscale intact samples.3.2.5 Pillar ModulusThe elastic modulus of a mine pillar is a combination of the intact elastic modulus of the pillar rockand the degree of fracturing that is present in the pillar. Work by Wagner (1974), Hallbauer et al. (1973), andJohn (1971) suggests that the elastic modulus of a pillar or laboratory specimen does not begin to decrease untilapproximately 90-95% of the ultimate load of a sample is reached. Kostak & Bielenstein (1971) present arelationship between pillar modulus and pillar size in Figure 32.3.2.6 Constitutive RelationshipThe constitutive relationship of a mine pillar dictates how it will perform when subjected to loading.Figure 33 shows the results of large scale loading performed on a coal pillar by Wagner (1974). Results of thistest showed that a pillar develops higher stresses in the core and lower stresses at the perimeter. At pillar— (Wp/h)C vdfor W/h — 1and Munro10Pillar volume V - ft3100 1000 10000 10000050failure, the centre of the pillar is still subjected to stress, however the entire pillar can be considered to be in astate of failure. Figure 34 shows a graph which represent the constitutive relationship of a pillar under varyingdegrees of confinement presented by Starfield & Fairhurst (1968). This relationship shows that as pillarconfinement increases, the pillar strength also increases.3.2.7 Pillar ConfinementPillar confinement plays a role in defining strength and predicting failure type. Pillars that areexposed on four sides (square pillars) may be weaker than pillars that are longer in one dimension (rib pillars)because of a lower average confinement within the square pillar. Researchers have proposed (Section 2.2.1.3) ameans to compensate for this variation in geometry and have verified it with laboratory sized samples.Empirical strength formulae routinely use pillar width I height ratio as an input variable, however this is onlyan indirect measure of pillar confinement. A confinement factor is introduced in Chapter 5 that maycompensate for the effects of different shapes of pillars.CSCUCC4;CCaUVCeC UN4-00.0.00,CC54;SC>>54;ammmCCFigure 30: Influence of specimen size upon the strength of intact rock (after Hoek & Brown, 1980).1.31.2Symbol Rook Tested byo Marble Mogia Limestone KoifmanGranite Burchartz et alBasalt KoifmanBasalt-andesite lava MelekidzeGabbro llnickaya• Marble llnickaya• Monte BieniawskiGranite Hoskins & HorinoV Quartz dionite Pratt et al(mc/°C5Q) (50/d)0’8I.00.0.50 100Specimen diameter d - m25051Sptcim.n Volume cu inFigure 31: Effect of specimen size on compressive strength (after Kostak, 1971)4,0.Li04-0E0004•00Figure 32: Size effect in modulus of deformation, E (after Kostak & Bielenstein, 1970)3.2.8 Structural FeaturesDenkhaus (1962) and Bieniawski (1968) postulate that rock mass strength is dependent on the amountof stnictural defects and that there must be a certain dimension that above which, the effect of sample size is nolonger significant. This approach suggests that the intact pillar strength should be determined on a largeenough sample size so that it will be representative of mine size pillars. Based upon in-situ tests, Bieniawski(1968), suggests that the representative sample size for coal is a cube with a side length of 1.5 metres..5 .5 .5 ,— 05 .,•R ..0b 0 4 1.4,(I— F•_ 4000C— — — —4,0. --0 — — —3000C__; _—.b2000C — — —mwI I0 700 7000 70000 70000 100000Volume Ioq10V. in3522Figure 33: Stress distribution in a coal pillar at various stages of loading (after Wagner, 1974).Figure 34: Stress - strain curves for laboratory specimens loaded under increasing confining stresses show anincrease in peak load and an increase in the post-peak load bearing capacity (after Starfield &Fairhurst, 1968).3.3 PILLAR STABILITYCLASSIFICATIONMETHODAT WESTMIN RESOURCES LTD. H-WMINEA pillar stability classification method has been developed for use at Westmin Resources Ltd.’s H-WMine. The pillar stability classification is an integral part of the pillar strength relationship that is developed inChapter 4. Pillar stability classification is a tool that can be used to assess a pillars’ current stability conditionand is related to results from numerical modelling and pillar geometry to derive a pillar strength relationship.PIar compression (m)53Table 7 is the pillar stability classification method that was developed for use at the H-W Mine. Thismethod was based upon the methodology presented by Krauland & Soder (1987) and upon the observation ofpillar conditions by the author within the H-W Mine. Figure 35 schematically illustrates each of the pillarclasses as presented in Table 7.3.4 CHAPTER SUMMARYThis chapter reviewed pillar failure mechanisms with particular reference to progressive pillar failure.The significant variables related to pillar strength have been identified and summarized. The pillar stabilityclassification method used at Westmin Resources Ltd.’s H-W Mine has been presented.OpeningClass 1jOpeningClass 2OpeningClass 3)J\,\i/ OpeningClass 4OpeningClass 5Figure 35: Schematic illustration of Westmin Resources Ltd.’s pillar stability classification method - Class— “5”.Table 7: Westmin Resources Ltd.’s pillar stability classification method.Pillar Stability Observed Pillar ConditionsClassification1 No Sign of Stress Induced Fracturing2 Corner Breaking Up Only3 Fracturing in Pillar WallsFractures < ½ Pillar Height in LengthFracture Aperture <5 mm4 Fractures> ½ Pillar Height in LengthFracture Aperture> 5_mm, <_10 mm5 Disintegration of PillarBlocks Falling OutFracture aperture> 10 mmFractures Through Pillar Core544. PILLAR STRENGTH ESTIMATIONAT WESTMINRESOURCES LTD. ‘S.H-WMINE, - A CASE STUDY.An objective of the Westmin - CANMET project, and consequently this thesis, was to develop amethod for determining the strength of massive sulphide pillars at Westmin Resources Ltd. ‘s, H-W Mine. Thiswas accomplished through the analysis of collected geological data, pillar stability classification, and numericalmodelling results. This data was analyzed in conjunction with the known mining history, resulting in thedevelopment of a strength relationship for use at Westmin Resources Ltd.’s H-W Mine.Westmin Resources Ltd. operates a base metal mine at the south end of Buttle Lake on VancouverIsland, approximately 90 km. west of Campbell River, B.C. Figure 36 is a location plan of Westmin ResourcesLtd.’s Myra Falls Operations. Mining has been ongoing at the Myra Falls property since 1967 when WesternMines Ltd. started the Lynx open pit mine. In 1979 the H-W orebody was discovered and, through the period1981-1985, underground development was carried out and the construction of a 2700 mtpd mill was completed.The current rated mill production at the Myra Falls Operations is 3650 mtpd with the bulk of the productiontonnage being supplied by the H-W Mine.4.1 GEOLOGYOF THE H-WMINE5The ore deposits of Westmin Resources Ltd.’s Myra Falls Operations occur as individual orebodiesgrouped into several major zones. This case study is based on the Main Zone of the H-W orebody. The MyraFalls ore deposits are polymetallic massive sulphide deposits associated with felsic volcanic rocks. The oreformed as sedimentary lenses on the sea floor, precipitated from metal bearing hot springs contemporaneouswith felsic volcanic rocks. Hot spring activity produced widespread hydrothermal alteration of walirocks,particularly below the ore lenses. Hydrothermal alteration is represented by sericitization, silicification, andpryritization. A generalized cross-section of the H-W Mine with the larger ore lenses is shown in Figure 37.The ore minerals are: pyrite, sphalerite, chalcopyrite, galena and barite, all of which vary widely intheir proportions. Occurrences of rhyolite, sulphides and altered rocks are distributed vertically and laterallywithin a stratigraphic zone approximately 400 to 500 metres thick. The mine sequence lies within the Myraformation of the Palaeozoic Sicker Group. The mine sequence is comprised of massive volcanic and coarse tofine volcaniclastic rocks, which include basalt, andesite, dacite and rhyolite, as well as subordinate sedimentaryrocks which include chert, carbonaceous argillite, suiphides and barite. The mine sequence is internally beddedand is predominantly mafic and volcaniclastic. Lithologic units are laterally discontinuous with a distinctnorthwest trend which parallel the trend of the individual orezones.The geological description presented in this section is reproduced predominantly from Walker, 1983.55Figure 36: Location plan of Westmin Resources Ltd.’s Myra Falls operations.The H-W orebody and associated lenses occur at the base of, and within, the H-W rhyolite unit whichlies at the bottom of the mine sequence. The mine sequence has been folded and metamorphosed in the lowergreenschist facies. Deformational fabrics are variably developed with widespread occurrence of schistose andlineated rocks. Schistosity is most intense in sericitic rhyolites and altered rocks. Schistosity strikes northwestand dips steeply northeast. Lineations, as well as fold hinges, trend northwest with flat to very shallow plunge.Post metamorphic faults offset the ore zones and zones of broken ground are common along major faults andore contacts. Most notable is the east-west striking flat fault against which the Main Zone of the H-W orebodyterminates to the north. A series of east-west trending faults are present in the southern portion of the orebody.56The H-W orebody has a thickness of 60 metres at the core and tapers at the margins. The majority ofthe orebody is between 10 metres and 40 metres thick. The orebody exhibits strong lateral zoning ranging froma very massive, pyrite core with high copper-zinc ratios, to zinc and barite rich margins with low copper-zincratios.20 Level(2962.8)21 Level(2918.0)Figure 37: A simplified cross-section of the H-W Mine showing the location of the shaft, lateral developmentand the major orebodies.HW MineHeodframe13 Level(3289.4)NorthPricePortal18 Level23 Level(2826.3)24 Leve.l(27793)25 Level(2719.6)26 Level(2683.3)27 Level(2652.1)Ore BinNorth Lens0 100 metres0 400 feet-574.2 MINING PRACTICE AT THE 11WMINEVariations of several different mining methods have been used at the H-W Mine. Listed below are themethods that have accounted for the bulk of the production from the H-W Mine:• blasthole open stoping• cut-and-fill post-pillar• cut-and-fill longitudinal• room-and-pillarThe more steeply dipping, southern flank of the Main Zone has been primarily mined withlongitudinal cut-and-fill stopes, augmented by longitudinal blasthole stoping. The core of the orebody has beenmined primarily with cut-and-fill post pillar stopes and the north flank has been mined using transverseblasthole open stopes. The smaller, gently dipping tabular lenses to the north of the Main Zone have beenmined using room-and-pillar methods. Figure 38 is a simplified cross-section through the orebody lookingFigure 38: Schematic of mining method and the respective location within the orebody used in the H-WMine.58west showing the location within the orebody of each mining method used. Hydraulically placed cementedbackfill is used in all stopes. Figure 39, Figure 40, and Figure 41 schematically illustrate open stoping androom-and-pillar mining methods.In 1989, the mining practices at the H-W Mine were abruptly changed from 80% cut-and-fill post-pillar and 20% blasthole methods to 90% blasthole and 10% cut-and-fill post-pillar methods. This resulted inlower mining costs through the reduction of manpower and equipment requirements. Blasthole stopes aredrilled off using either 57 mm or 89 mm blastholes up to 20 metres in length. Stoping blocks are variable insize and shape, with the largest blasthole stope created to-date being 25,600 m3. Smaller blocks ofapproximately 7,500 m3 are more frequently mined. The mining methods employed have necessitated thedevelopment of a number of barrier pillars in the orebody. These barrier pillars contain numerous openingswhich include: drifts, drawpoints, ore passes and raises.Figure 39: Schematic layout for sublevel open stoping with ring-drilled blast holes (after Hamrin, 1982).59Figure 40: Schematic layout for room-and-pillar mining (after Hamrin, 1982).Figure 41: Elements of a supported method of mining (after Hamrin, 1982).604.3 WESTMIN IN-siTu DATABASEA data collection program was undertaken in order to determine the distribution of the in-situ rockmass properties at Westmin Resources Ltd.’s H-W Mine. This section presents the results of the data collectionprogram.4.3.1 Intact Strength AnalysisIn order to assess the strength of a rock mass, knowledge of the intact rock strength is required.Samples were collected from five locations throughout the H-W Mine, representing massive suiphide ore,hangingwall volcanics, footwall volcanics, altered footwall volcanics, and felsic dyke in order to assess theintact strength properties. Samples were obtained at depths between 4.40 metres and 540 metres below surface.Testing was carried out to ISRM standards at the Mining Research Laboratories (Gorski & Conlon, MRLReport 92-027(Int.)), Ottawa to determine the following intact rock properties:• unconfmed compressive strength• tensile strength• elastic modulus• poisson’s ratioThe results of this testing program are presented in Table 8.4.3.2 Fabric AnalysisGeotechnical mapping was undertaken at all accessible locations within the barrier pillars throughoutthe H-W Mine - Main Zone and limited mapping was undertaken in the footwall volcanics. The geotechnicaldata collected included structural features and rock mass classification. The parameters collected for eachstructure mapped were:• orientation• infilling• roughness• length• continuity• openness• hardness• water conditions• planarity61Table 8: Westmin intact rock properties for the H-W Main Zone.Unconfined Average Standard CountCompressive DeviationStrength (MPa) (MPa)Massive Suiphide Ore 172 4 3Footwall Rhyolite 70 17 4Hangingwall Rhyolite 91 31 4Feldspar Porphyry Dyke 147 12 3Altered Footwall Andesite 29 13 6TensileStrengthMassive Suiphide OreFootwall RhvoliteAverage(MPa)124StandardDeviationMPa)12n/a13166Count728Hangingwall RhyoliteFeldspar Porphyry DykeAltered Footwall Andesite131613136ElasticModulusMassive Suiphide OreFootwall RhyoliteAverage(GPa)16663457124StandardDeviation(GPa)23Count34536Haningwall RhyoliteFeldspar ‘orphyry DykeAltered Footwall Andesite10510Poisson’s Average Standard CountRatio DeviationMassive Sulphide Ore 0.15 0.06 3Footwall Rhyolite 0.12 n/aHangingwall Rhyolite 0.26 n/a 2Feldspar Porphyry Dyke 0.25 0.04 3Altered Footwall Andesite 0.32 0.3 3Three joint sets were identified, one major, one intermediate, and one minor. The major joint setcorresponds to the orientation of the foliation of the volcanics and the major faults in the mine area. Theorientations of the joints sets are presented graphically on Figure 42 and are tabulated Table 9.62Joint Set A251/57 NJoint Set058/68EIev.NorthEastFigure 42: Isometric view showing the major joint sets identified within the H-W Main Zone.4.3.3 RockMass ClassWcationRock mass classification according to the CSIR-RMR method, (Bieniawski, 1973), was performedthroughout the barrier study area at approximately ten metre intervals along drifts within the barrier pillars.Rock mass quality within the barrier pillars varied from a fair-good (R14R 60%) to good-very good quality rockmass (RMR 80%). It must be noted that, due to the extent of mining, the RMR values recorded representinduced rock mass ratings within the pillars and not the pre-mining rock mass quality. Areas of the minesubject to lower stress conditions, as determined by modelling, were observed to have higher rock massclassification ratings.4.3.4 GeometryThe H-W Mine orebodies consist of the Main Zone, the North Zone and a number of smaller zones.The case study domain was confined to the Main Zone orebody, which has approximate dimensions of 850metres in strike length and 220 metres in width. The Main Zone orebody varies in dip from 20° to 50° to thenorth and plunges at approximately 10° to the west. Mineable thickness varies from approximately 10 metresin the south to 40 metres in the north-west.The study required a detailed understanding of the mine geometry in three dimensions. This wasaccomplished by transforming two-dimensional mine sections and plans into a three-dimensional model withinAutoCAD. This three-dimensional computer model formed the framework for the numerical modelling phaseof the project. Figure 43 and Figure 44 are isometric views of the three-dimensional geometric model showingstoping blocks and barrier pillars respectively.Joint Set B159/75 W63Table 9: Summary of joint features or the Westmin H-W Main Zone.Length (m) 4.7I3.04.3.5 In-Situ Stress DeterminationTriaxial overcoring stress measurements to determine in-situ stress conditions were performed at twolocations in the H-W Mine by the Mining Research Laboratories, Elliot Lake Ont. (Arjang & Stevens, MRLReport 91- 1 44(TR)). The in-situ test was performed at the shaft station on 23 Level, at a depth of 600 metres,to determine the current state of in-situ stress in the vicinity of the H-W Mine. The results of the in-situ triaxialtest are presented in Table 10. The results of the biaxial in-situ stress measurement program, which wasperformed previously at the H-W Mine, are presented in Table 11.Set A - 172 obs. Average Std. Dev.Strike (deg.) 251 18Dip (deg.) 57 N 17Length (m) 7.1 3.3End Visible 0Planarity Planar -WavyRoughness 9Openness tightSet B - 172 obs. Average Std. Dev.Strike (deg.) 159 19Dip (deg.) 75 W 16L.ength (m) 3.3 2.1End Visible 1Planarity PlanarRoughness 12Openness tightSet C - 39 obs. Average Std. Dev.Strike (deg.) 058 20Dip (deg.) 68 S 17End VisiblePlanarity PlanarRoughness 7Openness tight64Figure 44: Isometric view showing the location of the barrier pillars within the H-W Main Zone.S388S368S356Figure 43: Isometric view showing the location of stopes with the H-W Main Zone.65Table 10: In-situ triaxial stress measurement results at the H-W Mine.Principal Multiplier Magnitude OrientationStress (MPa) Direction / Plungea 1.902 36 N1l4°E/04°(‘2 1.lGv 20 N025°E/ 18°03 (‘V 19 VerticalTable 11: In-situ biaxial stress measurement results at the H-W Mine.Principal Multiplier Magnitude OrientationStress (MPa) Direction I PlungeCi 2.25a,, 34.2 N021°E/27°02 1.8102 27.5 N290°E/01°03 j 02 15.2 N108°E/63°Figure 45 (a) is a stereonet plot of the triaxial in-situ stress measurements and Figure 45 (b) is a plot ofthe historical measurements (biaxial doorstopper). Figure 45 shows that the orientations of the principalstresses determined from the two programs compare favorably. The major difference between these twoprograms is the magnitude of the horizontal components of the in-situ stress. The ratio of horizontal to verticalstress for the triaxial program were lower than those of the biaxial program. This was most pronounced for(‘N-S , which was 2.25 o for the biaxial program and 1.1 o for the triaxial program. The results of the triaxialprogram were used for numerical modelling purposes, however, the impact of the potentially higher stresses aspredicted by the biaxial program was investigated with a parametric modelling session and is discussed inSection 4.4.1.2.1.4.4 PILLAR STRENGTHESTIMATIONAT WESTMIN RESOURCE LTD. ‘S H-WMINEThe H-W Mine, Main Zone includes ten barrier pillars, which vary in width between 15 and 25metres. These barrier pillars have numerous secondary openings within them which have resulted in thedevelopment of many “small” pillars (drawpoints & ribs). A typical simplified cut-away barrier pillarschematic with development openings and “small” pillars is shown in Figure 46. It is the “small” pillarswhich have been used to develop the Westmin pillar strength relationship.4.4.1 NumericalModellingNumerical modelling for the prediction of mining induced stresses in the barrier pillars was anintegral part of this thesis. Numerical modelling of the H-W Mine was performed using Map3D, a threedimensional, fictitious force boundary element program, that models elastic stress conditions. Excavations aredefined as a series of boundary elements and stresses are calculated away from the excavations on user definedstress grids.66Figure 45: Polar stereonet plots of the results of the triaxial and the biaxial in-situ stress measurementprograms within the H-W Main Zone.4.4.1.1 Map3D Numerical Modelling ProgramMap3D is a commercially available three-dimensional boundary element modelling programdeveloped by Terry Wiles of Mine Modelling Ltd., Copper Cliff, Ont. This program was chosen over othernumerical modelling programs because of its ease of use and IBM PC compatibility. Building a model inMap3D involves defining a series of blocks to be excavated (stopes or portions of stopes) using conventionalnumerical modelling techniques. Element discretization takes place within the model based upon a number ofdiscretization parameters supplied by the user to the program. Figure 47 shows a portion of the WestminMap3D model with a vertical stress grid running through the centre of the 366 south barrier pillar. Figure 47also shows the results for the induced on the stress grid and the location at which the core pillar stresses aremeasured.Using Map3D, stresses are calculated only on grids of field points as specified by the user. Thisapproach to three-dimensional modelling results in reduced computation times over other modelling programs.A typical model of the Westmin orebody with a single grid of field points requires two to four hours ofcomputational time on an IBM Intel 8046 computer with four megabytes of random access memory and a 650megabyte hard disk. The model construction was accomplished by exporting information from AutoCADdirectly to a Map3D input data file. The geometric input required for the model was extracted from the three-dimensional AutoCAD model as shown in Figure 43 and Figure 44.In—Situ Stress Plot — Triaxial Overcore 23 Level StationSIn—Situ Stress — Biaxial Measurements67HANGING WALL4.4.1.2 Westmin Map3D Modelling SessionsNumerical modelling of the H-W Mine using Map3D was completed as follows:modelling of the mining sequence of the H-W Mine - Main Zone from the beginning of mining tothe present time, utilizing the in-situ stress conditions determined from the in-situ stressmeasurement program as discussed in Section 4.3.5• modelling of the planned barrier pillar extraction sequence• sensitivity analyses based upon modelling input parametersThe input parameters that were investigated in the sensitivity analyses are:• the in-situ stress parameters• the effects of modelling the orebody as a stiff inclusion within a soft hostThe goals of the numerical modelling sessions for the Westmin-CANMET report were different thanthose of this thesis. For the purpose of this thesis, it was only required that an assessment of the stresses on theclassified pillars at the time of pillar stability classification be made.It was not feasible to model every opening in the H-W Mine for the purpose of stress determinationand all of the modelling sessions have considered the barrier pillars intact. However, these pillars havedevelopment openings within them in the form of drifts, drawpoints, and raises. The pillars that have beenclassified for use in pillar strength determination are “small” pillars developed within these barrier pillars. Inorder to assess the stresses on these “small” pillars, the average core stress (a1) of the barrier pillar in whichthe “small” pillar lies has been used and then scaled to determine the stress on the “small” pillar. The locationEXTRACTIC - -LEVEL“SMALL” PILLARFigure 46: Simplified barrier pillar schematic showing the development of the “small” pillars.68Figure 47: Typical Map3D stress modelling output showing the location at which the core stress wasdetermined.at which the core stress (Of) is measured and the method used to determine the “small” pillar stresses isdiscussed further in Section 4.4.1.4.4.4.1.2.1 Sensitivity AnalysisThe results of the modelling sessions that investigated the model sensitivity to the two in-situ stressregimes and the differing rock mass stiffness’ are presented below.4.4.1.2.1.1 In-Situ Sfress VariationTwo in-situ stress measurement programs have been perfonned at the H-W Mine. A triaxial CSIRprogram was performed as part of this project and a biaxial program had been performed earlier as discussed inSection 4.3.5. While the stress orientations, as illustrated in Figure 45, determined in these two programsprovided similar results, the magnitude of the horizontal principal stresses in the north-south direction variedby approximately a factor of two. Modelling was performed using both of these two in-situ stress conditions asmodel input parameters to ascertain what the impact of the two different in-situ stress regimes would be on thepredicted barrier pillar stresses. Results showed a 10 - 20% increase in induced stresses in the barrier pillars asa result of increasing the aNS component of in-situ stress from 1.1 ato 2.25 0. It was observed from69modelling that pillars oriented in the north-south direction experienced a greater percentage stress increasethan pillars oriented in the east-west direction.4.4.1 .2.1 .2 StiffOrebodyThe H-W orebody was modelled as a stiff body surrounded by a soft host. This modelling wasperformed because the massive sulphide ore is significantly stiffer than the surrounding volcanic host rock asreported in Section 4.3.1. Modelling results showed that the predicted core stresses (a1) in the barrier pillarswere 10-30% higher for a two material model than for a single material model. The stress increases were mostpronounced in the barrier pillars that had a high percentage of mining extraction within their vicinity.4.4.1.3 Map3D - Core Barrier Pillar Stress vs. Average Barrier Pillar StressThe average pillar stress (ar) of the barrier pillar (as opposed to the core pillar stress) is the mainstress input parameter used to make an assessment of the stress on the “small” pillars developed within thebarrier pillars. The core stress value is obtained from the modelling results at the location within the model asshown in Figure 47 and modified according to the extraction ratio in the vicinity of the “small” pillar. Thiscore stress is used because it is difficult to obtain the three-dimensional average stress in the barrier pillars. Itwould be necessary to take a three-dimensional stress average, which could only be obtained with extensivemodelling sessions utilizing stress grids of many different orientations.In order to utilize the core stress in comparison with other data which uses average pillar stress, it wasnecessary to develop a relationship between “core” barrier pillar stress and “average” barrier pillar stress.Average barrier pillar stresses (a1) parallel to the direction of loading were detennined at a number of locationswithin the model and compared to the “core” stresses (as) at the same location. These results are presented inFigure 48. Although there is significant scatter in the results, the average barrier pillar stress (a1) can beapproximated by 85% of the core stress (a1) at the same location. This average stress value of 85% of the corestress (a1) is used for the construction of the pillar stability plots in order to be consistent with the method ofrepresenting pillar stresses as averages, as used by the author’s whose databases will be presented in Chapter 5.4.4.1.4 Determination OfAverage “Small” Pillar Stresses at Westmin ResourcesThe method used to calculate the average “small” pillar stresses for use with the pillar strengthdetermination methods, presented in Chapter 5, is as follows:• determine the core barrier pillar stress (a1), calculated using Map3D, at the location of the“small” pillar within the barrier pillar.• modify this value according to the previous section, whereby the average barrier stress equals 85%of the core barrier stress, to determine the average barrier pillar stress at that location.70• finally, scale the average barrier pillar stress based upon the extraction ratio within the barrierpillar at the “small” pillars’ location within the barrier pillar, resulting in the average “small”pillar stress.Estimating the overall barrier pillar stress based upon Map3l) modelling realistically represents themine geometry, ignoring “secondary” development. Using the extraction ratio to determine the local “small”pillar stresses provides an effective means of determining the average “small” pillar stresses. This method ofestimating the “small” pillar stress was verified utilizing a parametric numerical modelling study of asimplified barrier pillar geometry as illustrated in Figure 49.The following parametric modelling sessions were performed:A. a simplified barrier pillar separating two stopes without developmentB. a single drift being placed at the centre of the simplified barrier pillarC. a single drift and drawpoints on one side of the simplified barrier pillar807060---4:20Average Barrier Pillar Stress —0.85 * Core Barrier Pillar Stress1065 Observations00 10 20 30 40 50 60 70Average Map3D Barrier Stress (MPa)Observed vaiJFigure 48: Barrier pillar stress comparison, average Map3D barrier pillar stresses vs. core Map3D barrierpillar stresses.71The results obtained from the simplified barrier pillar with development, cases B & C, and the intactbarrier pillar, case A with tributary area approach applied, agreed to within 10%. It is the author’s opinion thatthis amount of error is acceptable, making this approach valid for the detennination of the average “small”pillar stresses. An example calculation of this method of determining average “small” pillar stress from the“core” barrier pillar stress is given as follows:O bamer core01 banier averageExtraction Ratio01 small average01 small average3OMPa— 0.85.a barrier core— 0.40l barrier average(I/(l_EXlracttOn Ratio))42.5 MPa(b)Barer PUI________________DriftFigure 49: Plan showing the stope and pillar configurations that were used to perform the parametricmodelling sessions to justify the use of the tributary area method for determining the stresses onthe “small” pillars within the barrier pillars.25.5 MPa(a)Drift & Drawpoints(c)‘-cQzcQQQcQwQjBarrier PillarF;fl724.4.1.5 Model Calibration to Actual Mining ConditionsIt is important to calibrate numerical modelling to actual mining conditions in order to haveconfidence in the modelling results. Calibration can be accomplished using two different methods:• instrumentation can be used to measure stress change as mining progresses and can then becompared to the modelled stress changes for the same mining sequence• modelled stress increases can be verified visually through observation of such occurrences as anincreased frequency of ground falls or deterioration of mine openings in areas of high modelledstressThe Westmin model was calibrated through the verification that the high stress regions in the modelcorresponded to the areas of the mine having the greatest pillar deterioration as described by Table 7. Therewere also a number of local failures in the mine that were correlated to areas of high modelled stresses. It wasconcluded that the modelling program approximated actual stress conditions, as a result of mining in the H-WMine - Main Zone, to a high degree of confidence and commensurate with the stress estimates subsequentlyemployed.4.4.1.6 NumericalModelling SummaryThe sensitivity analyses performed showed that the numerical modelling results obtained are highlydependent on the input parameters utilized. Modifying the input parameters for the Westmin model, within therange of the collected data, resulted in predicted pillar stresses could be varied, in theory, by as much as 50%.It is therefore, the author’s opinion that modelled stresses should be recognized as variable and not absolutevalues. It must also be recognized that the difficulty in obtaining an accurate measurement of the in-situ stressconditions can have a large impact on the variability of modelled stresses. The results from stress analysis aredependent on both the modelling technique and the input parameters used. In spite of these limitations, themodelling results obtained compared favourably with the conditions observed in the H-W Mine, leading one toconclude that the correct input parameters were selected.4.4.2 Pillar Stability ClassificationThe pillar stability classification method developed for use at the H-W Mine was discussed in Section3.3 and is reproduced in Table 12. Pillar stability classification methods can be used to quantify a pillar’sstability condition. This classification can then be used to determine the amount of available load bearingcapacity that a pillar has, provided a pillar strength relationship exists. Pillar stability classification can also beused to monitor a pillars’ gradual deterioration as mining proceeds and if required, measures can be taken toensure that the pillar behaves as designed.734.4.3 Pillar GeometiyThe review of pillar strength determination methods (Chapter 2) showed that the critical geometricpillar variables are the pillar width and the pillar height. Pillar height is measured in the direction of the majorinduced principal stress (01) within the pillar, and pillar width is measured in the direction perpendicular topillar height. The pillar geometry definition terms are defined in Figure 50.Although it has been suggested (Wagner (1980), Sheorey & Singh (1974), Stacey & Page (1986)) thatfor non-square pillars, the pillar width can be replaced by an “effective” pillar width, the width used for thedetermination of the Westmin pillar strength relationship is the minimum width of the pillar. This follows thesuggestion of Salamon (1983) that there are no accepted rules about using an effective pillar width in place ofthe minimum pillar width.Table 12: Pillar stability classification method for Westmin Resources Ltd.’s H-W Mine.Pillar stability Observed Pillar Conditionsclassification1 No Sign of Stress Induced Fracturing2 Corner Breaking Up Only3 Fracturing in Pillar WallsFractures < ½ Pillar Height in LengthFracture Aperture <5 mm4 Fractures> 1/2 Pillar Height in LengthFracture Aperture> 5 mm, < 10 mmS Disintegration of PillarBlocks Falling OutFracture aperture> 10 mmFractures Through_Pillar Core4.4.4 Westmin PillarDatabase65 discrete pillars were classified at all accessible locations within the barrier pillar study domainaccording to Table 12. Of the 65 discrete pillars in the database, eight are rib pillars (long side length greaterthan twice the short side length), 32 are drawpoint pillars (long side I short side ratio less than two) and 25 arenose pillars (one or more confined sides). Figure 51 illustrates the plan views of the different pillar typesclassified. Nose pillars were classified with the intent of using this data to augment the drawpoint and rib pillardatabase. The determination of which value to use for the width of the nose pillars proved difficult and after anexamination of these pillars in conjunction with the other data, it was concluded that these pillars should beexcluded from the database. Table 13 is the original discrete pillar database for the Westmin H-W Mine whichincludes:74• pillar orientation• pillar type• rocktype• unconfined compressive strength• extraction ratio in the barrier at the small pillar location• pillar stability classification• pillar width• pillar height• pillar width / height ratio• core stress calculated with Map3D• stress ratio for the scaled core stress divided by unconfined compressive strength• stress ratio for the scaled average stress divided by unconfined compressive strengthAll of the discrete pillars in the database were photographed in order to develop a visual database toaid in the assessment of pillar ratings. Photo 1 - Photo 5 show mine pillars for pillar stability classifications of— “15”4.4.5 Development ofa Pillar Strength RelationshipA pillar strength relationship has been developed for Westmin mine pillars using the pillar width Iheight ratio, the predicted average “small” pillar stress from numerical stress analysis using Map3D, and thepillar stability classification. It was not possible to include the nose pillars in the final database because of thedifficulty in assessing an “effective” pillar width of these pillars.DIRECTIONOF LOADINGminedI-wi’___Imined H mined W ----—-—_________I DIRECTION IOF LOADING H[_minedj2D Pillar Geometry 3D Pillar GeometryFigure 50: Definition of pillar geometry terms used at Westmin Resources Ltd.’s H-W Mine.DIRECTIONOF LOADING Breadth /75w‘t‘w00000a3333333‘C‘C‘C‘C‘C‘C‘C‘C‘C‘C‘C‘C‘C‘C‘C‘C‘C‘CLt1LzzzLLTL7zzzLzz71L1L7L1(11L11L1çtiLTLitititritiicriEEEEEEoEEEEEzWI)—0‘C00‘-40%VI—-J‘C00-40%Vi.‘t.J—-Cl)Cl)Cl)(6)Cl)Cl)Cl)Cl)Cl)Cl)Cl)Cl)Cl)(6’)Cl)Cl)(6)Cl)Cl)aaaaaCl)(6)-c,.,Cl)Cl)Cl)Cl)C’)Cl)Cl)Cl)Cl)C’)COOOOOOOOOOOOOOOOOOOOOOOloooOOOOoOOOoooooOozzI.)t’.).)I.).4.)tJt)t’JL.)Is)ts)Is)Is)Is)Is)Is)Is)Is)Is)Is)Is)-JIs)Is)Is)Is)Is)Is)Is)1.3Is)Is)Is)Is)Is)1.41.4Is)Is)ViViViW.....))...ViC)VI)U)U)U)U)U)..U)..ViViViUIViViViVIViViViVi0000000000000000000000000000000000000000000000E‘0000000ViVI0000000000000UIViViViViViVi0000000000000000000U)Is)—Is)—1.3—————O—U)l3——U)—0—QIs)OtLi‘ob000.)M61UI.Vi1.)01.).#—0-.400‘C0‘CIs)1.4—0%.0U)00Vi.0‘CU)00000000000a00000000000000000000pppp00pppppp%.ö00Is)0%0%.‘C‘C‘C‘C‘C..‘C.I.....-4Vi‘C—-)‘.3‘C-)-3000000000U)..‘.4-1-)Is).U)U)U)‘00000000000000000000000pp0000p0ppppppppppppppp>.—‘C0%—‘CU)U)ViUlVi‘.3U)—)-)‘C’C-).Is.Vi-3.——Vi——U)00.--—...oC’)0%U)C,CD I-.CD -o 0 -‘ z 0 C,) CD -o 0 -‘rj 0 0 I() -o Co-.oD 0.0co3: (n_’—D00OD-oCDo :.C< CDQ. 0 D 0’DO a-CD (0c3. -o 0wcwzzzzzz111-aac’ccc,c’CCDcc.CCDg0000000000CD00000t)(.)I-.)t)()I-.)C’.)C’.)C’)—)C’)C’)C’)C’)C’).-.-vvvuw-(A.(A(A(A(A(A(A0Q0Ui00000000000000(A(A(A—(A)(A(A(A(AVE-‘).0%0WC’)C’)—.J.WW-)(A(A-.IAo.W..(A(A(A.-4P.P.P.P.(ACD•0A0(A000(A(A0000(A(A(A(A0————000——00t--p——CCWWW(A(A(A(A(AW(A(A(A(A(A(A(A(A(A(A(A(A(A(A(A(A(A00(A(A(A(A(A(A(A(AtI.CD0000000000000ppo00Ot.b%oIAb%b%0%0%C-.)-0%(A—)0%000%0%-P.-P.00-P.-P..P..P.LA00-P.C..)(AC-P.--P.-00000000pppppppppp>IAIAIAIAIAC’)-P.(A40%(A(A(A(A(APhoto 1: A class “5” discrete pillar located within the 366 barrier pillar.Photo 2: A class “4” discrete pillar located within the 383 barrier pillar.78Photo 3: A class “3” discrete pillar located within the 383 barrier pillar.FPhoto 4: A class “2” discreto pillar located within the 383 barrier pillar.4,K794.4.5.1 Pillar Stability PlotsThe average pillar stress, calculated as described earlier, normalized to unconfined compressivestrength of the intact pillar material has been plotted against pillar width / height ratio with a different symbolfor each pillar stability classification. Figure 52 and Figure 53 are stability plots of massive sulphide drawpointand rib pillars and massive sulphide nose pillars, respectively. It can be seen from these plots that pillars ofhigher pillar stability classification plot farther to the left and higher on the graphs, in locations of higherstresses and/or lower width I height ratios.4.4.5.1 .1 Excluded PillarsSeven “small” drawpoint or rib pillars were excluded from the database for the construction of thefinal stability graph, Figure 54. These pillars were excluded because it was concluded that the pillar stresses inthese instances could not be assessed with confidence using the numerical modelling methods employed. Thesepillars were all located in the same barrier pillar and were excluded for one of the two following reasons:;..•r•1’ ‘st)4.-,.Photo 5: A class “1.5” discrete pillar located within the 383 barrier pillar.800.70.600 v-- -v0 X0 0C.., x xv V a.----------“-------00ea)x<0.2---.1•38 Observations0.0• I I I I I I I0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2Pillar Width / Height RatioPillar Stability Classification5 0 4 v 3 x 2Figure 52: Stability plot of massive sulphide, drawpoint and rib pillars from the H-W Mine.0.70.60 0C)zx0 0 VVz 0<0.20.118 Observations0.0• I I I I I I0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2Pillar Width / Height RatioPillar Stability Classification[0 o v x 2 a.Figure 53: Stability plot of massive suiphide, nose pillars from the H-W Mine.81• The H-W hangingwall fault is in close proximity to the orebody in the vicinity of the barrier pillar.It was concluded that the assessment of the pillar stress and the effects of stress redisthbutioncould not be adequately compensated for using the methods previously developed for pillar stressdetermination.• In portions of the barrier pillar there are up to three levels of “stacked” development openingsand it was concluded that the methodology used to estimate the “small” pillar stress wasinadequate to account for these geometric conditions.4.4.5.2 Division ofPillar ClassesIn order to develop the Westmin pillar strength relationship, the pillar database from the massivesuiphide drawpoint and rib pillars has been analyzed. Only the massive suiphide pillars were used in theanalysis because there are insufficient cases from pillars consisting of dyke (two cases) and hangingwallvolcanics (seven cases) to justify their inclusion. The strength relationship developed also does not consider the“excluded” pillars as mentioned previously. A total of 31 pillar case histories are used to locate the pillarstability class division lines.Pillar stability classifications were analyzed and the pillar stability classification division lines weredetermined according to the following methodology. Systematic analysis of the database involved comparingall pillars above a particular class division to all of the pillars below the same class division. For example: tolocate the class “3” - “4” division line, a “bestfit” line dividing the class “4” and “5” pillars and the class“1”, “2” and “3” pillars was drawn, based upon the author’s judgment, on the pillar stability plot. Thisprocedure was repeated for each of the classification divisions resulting in four “preliminary” classificationdivision lines.After the “preliminary” classification division lines were determined, the division lines were reviewedas a group to ensure that the slopes’ of each of the division lines were consistent with each other. It wasconcluded that the ratio of the slope between the different classification division lines should be constant fordiffering width I height ratios, i.e. the strength of a class “3” - “2” pillar is always a fixed ratio (as anexample) higher than the strength of a class “2” - “1” pillar. Using this approach, “refined” stability lineswere located for the selected massive suiphide drawpoint and rib pillars. These “refined” stabilityclassification division lines are presented along with the selected pillar database on Figure 54.820.7-—F.S.—1.0F—11• —— 0 —- — —- F.S.—1.2C., — —__-_- __‘v--——_— _— — — F.S.—1.4— — —— z —— — —— — —0.4 P--- —C.., — — x— ——0.3x0.20.1Ps_UCS*(0.25+0.18*(w/h))31 Observations0.0- I I I I I0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2Pillar Width / Height RatioPillar Stability Classification[ 5 4 V 3 2 jJFigure 54: Stability plot with the stability classification division lines, for massive sulphide, drawpoint andrib pillars from the H-W Mine. The excluded pillars are not plotted on this figure.There are a three salient points that must be considered when reviewing Figure 54 as follows:• There are ten class “5” pillars that plot well above the class “4” - “5” division line. It is theauthor’s conclusion that these pillars show exaggerated modelled stresses because an elasticnumerical model has been used. Modelled pillars have an infinite strength and as a result, theycontinue to take stresses in the model beyond the point at which they would have failed in themine. In reality these pillars would have failed earlier in the mining sequence than the miningstage that was modelled, and consequently would have lower predicted stresses. This would placethese pillars at a lower point on the stability graph, in closer proximity to the class “4” - “5”division line.• In consideration of the previous point, 31 pillars have been used to locate the stability lines,however only 21 can be used with confidence if the previous 10 are removed. It can also be seenfrom Figure 54, that there is a wide margin for locating the stability lines. Statistical methodscould have been used to locate the stability lines, however it is the author’s conclusion that there isinsufficient data (21 cases for 5 regions) to use conventional statistical methods to locate thestability classification division lines and have confidence in their placement.83• The placement of the stability lines and the slopes chosen may seem arbitrary, however, in thenext chapter an in depth analysis is performed with a comprehensive database which will supportthe placement of the lines in this manner.4.4.5.3 Factor ofSafetyEach of the stability classification division lines has been assigned a factor of safety based upon theultimate pillar strength (failure). The calculation of factor of safety is based upon the assumption that the “4” -“5” stability classification division line represents failure and therefore has a factor of safety of 1.0. Using thisas a baseline, the location of each of the other stability classification division lines were determined to havecalculated factors of safety as presented in Table 14.Having assigned a factor of safety to each pillar stability classification, the pillar stability classificationmethod can be used as a means of assessing the current factor of safety for any given pillar. This enablesWestmin Resources Ltd. to quickly determine the condition of pillars and their proximity to failure through theuse of a pillar stability classification method.Table 14: Factor of safety determined for pillar stability classification division lines.Pillar Factor of SafetyStability Classification5 F.S.<l.O4 l.0<F.S.<1.l3 l.1<F.S.<1.22 l.2<F.S.<1.41 F.S.>l.44.4.5.4 Stability Line with Individual Data SetsThe final stability classification division lines are plotted against subsets of the pillar database which were notincluded in the analysis. The classification division lines are those which were derived from the selectedmassive sulphide drawpoint and rib pillar data. Figure 55 is a plot of the pillar database that includes theexcluded pillars in the database. It can be seen that the seven excluded pillars all plot in regions of lower pillarstability classification than their respective classification predicts. Figure 56 is a plot of the massive sulphidenose pillar data that was excluded from the selected pillar stability graph. These pillars were excluded basedupon the difficult in assessing a pillar width, however, it is interesting to note that even though the data in thissubset does not fit the stability lines, it still follows the same general form, with pillars of higher stabilityclassification falling higher and farther to the left on the graph, and pillars of lower stability classificationfalling to the right and lower on the graph.840.7F.S.—1.0— ps—Il• -- —__.I__ - — — — F.S.—l.20.5 - _—,Z__— — F.S.—1.4- — — —— — —— — —0.4 _.. _-C’, — — — I ——— — — V V03 •--•I0.1Ps_UCS*(0.25+O.18*(w/h))38 Observations0.0 I I I I0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2Pillar Width / Height RatioPillar Stability Classification5 0 4 V 3 1 2Figure 55: Stability plot with the stability classification division lines, for massive suiphide, drawpoint andrib pillars from the H-W Mine. All drawpoint and rib pillars are included on this figure.0.7 -F.S.—1.0—- F.S.—l.10.6--_- - _- F.S.-I.2C’, — —z — — — _. F.S.—1.4— — — —— — — —— — — ——.w— —_— — _-— — I• •—•—0.1---Ps=UCS*(0.25+0.18*(w/h)) 18 Observations0•0 I I I I I0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2Pillar Width / Height RatioPillar Stability Classification5 0 4 V 3 I 2Figure 56: Stability plot with the stability classification division lines, for massive sulphide, nose pillars.854.4.5.5 Pillar Strength RelationshipIt was shown in the previous section that pillars that are of the same rock type, similar pillar type, andsimilar size can be combined to form a pillar stability graph as in Figure 54. The stability graph developed forthe Westmin pillar database has been constructed using straight lines and as discussed in Section 2.2.1.1, apillar stability graph that is constructed using straight lines is a “Linear Shape Effect” relationship. One canrecall that the “Linear Shape Effect” is defined by the following equation:Ps=(K.UCS).[A+B()] (33)where:= Pillar strength (MPa)K = Strength size constantUCS = Unconfined compressive strength of pillar material (MPa)w = Pillar width (m)h = Pillar height (m)A, B = Empirically derived constants which when added equal 1.0The class “4” - “5” division can, based upon its linearity, be defined by a relationship in the sameform as Equation 33. The empirical coefficients, “A” “B” & “K”, calculated for this division line arepresented in Equation 34.P = (0.45. UCS) . [0.6 + 0.4 (-i)] (34)where:P = Pillar strength (MPa)UCS = Unconfined compressive strength = 172 MPaw = Pillar width (m)h = Pillar height (m)It should be noted that the pillar stress on Figure 54 has been normalized to the unconfinedcompressive strength of intact pillar material. This may be slightly misleading, as all of the pillars included onFigure 54, the “Selected Pillar Stability Graph “, are massive sulphide with an unconfined compressivestrength of 172 MPa. Hoek & Brown (1980) and Hudyma (1988) both suggest that the unconfined compressivestrength of intact pillar material can be used as a normalizing term for computing pillar strength. Thisapproach is used so that pillars of different rock types can be compared with each other. The data is presentednormalized at this point in preparation for the integration of this data with data from literature that will bediscussed in Chapter 5.864.5 CHAPTER SUMMARYPillar stability classification and numerical modelling techniques have been applied successfully todevelop a strength relationship for massive sulphide drawpoint and rib pillars at Westmin Resources Ltd.’s HW Mine. The relationship developed is a “Linear Shape Effect Formula” as discussed in Section 2.2.1.1. Thefive stage failure classification developed for the assessment of pillar stability has been used successfully andcan now be used to assess factor of safety of pillars that are in a state of pre-failure. The results presented inthis chapter can now be used at Westmin Resources Ltd.’s H-W Mine to design mine pillars based upon localexperience.875. NEW PILLAR STRENGTH FORMULAEIn Chapter 4 a strength relationship for pillars at Westmin Resources Ltd.’s H-W Mine was developed.A limitation of this relationship is that it is based upon a small number of pillars of similar’ dimension and thesame rock type. In this chapter we will examine the “Westmin Database” along with other available publishedstrength information for hard rock mine pillars. Refined empirical “Shape Effect” strength formulae will bederived and a new hypothesis for pillar strength estimation will be presented. Average pillar confinement isexamined and an alternative term to width / height ratio is proposed based upon two-dimensional elasticboundary element modelling. Empirical strength constants for the “Linear Shape Effect” and the “PowerShape Effect” formulae are derived for the combined pillar database. A refined “Power Shape Effect” formula,“The Log-Power Shape Effect Formula” is developed. In addition a new pillar strength formula, “TheConfinement Formula”, is developed that has a form similar to the Mohr-Coulomb shear strength formula.The basic components of the pillar strength formulae are examined. The proposed strength formulae are becompared against existing formulae and are shown to have the highest success in the prediction of pillarstability for the case histories in the combined database.This chapter is divided into four major sections as follows:• the database’s that are used to develop the new strength formulae• an investigation of the variables and methodology used to develop the new strength formulae• the development the new strength formulae• verification that the new strength formulae outperform the methods that have preceded them5.1 PILLAR DATABASEIn addition to the data that was presented in Chapter 4, there are six sources of pillar strength dataavailable in literature. The published comprehensive data which is available is the work of Hudyma (1988),Von Kimnielman et al. (1984), and Hedley & Grant (1972). The published supporting data which is availableis the work of Krauland & Soder (1987), Sjoberg (1992) and Brady (1977). These databases are discussed inthe following section. Each of the databases are presented along with the “common” pillar stabilityclassification that has been developed for the combined database. The assignment of the “common” pillarstability classification is discussed in Section 5.2.3.6.885.1.1 Westmin DatabaseThe Westmin database was discussed in detail in Chapter 4 and will be presented here in summaryform for consistency. The information in the Westmin database was collected throughout the Main Zoneorebody of the H-W Mine of Westmin Resources Ltd., Myra Falls Operations. Data from 31 selected pillar casehistories, from a total database of 65 pillar case histories, has been used to derive a pillar strength relationship.The Westmin orebody is a massive sulphide, copper-zinc orebody located at an average depth of 600metres below surface. Pillars have been classified according to the pillar stability classification methodpresented in Section 3.3. The classified pillars are rib pillars and drawpoint pillars that have been developedwithin larger barrier pillars, as illustrated in Figure 46. The orebody rock has a unconfined compressivestrength of 172 MPa and a Young’s modulus of 166 GPa.Stresses on individual “small” pillars were determined using three-dimensional boundary elementmethods to determine stresses on the larger barrier pillars, and then employing tributary area theory, asdiscussed Section 4.4.1.4, to estimate the average “small” pillar stresses.Mining methods employed at Westmin have been blasthole open stoping, cut-and-fill, and room-andpillar. Blasthole open stoping methods are used predominantly at the present time. Hydraulically placedbackfill is used in all stopes, however, all of the pillars that have been used for strength determination arelocated on development horizons and as such are not overly influenced by backfill placement. The summarydatabase is presented in Table 15.5.1.2 Hudyma (1988)Hudyma (1988) presented data on 47 pillar case histories which were collected as part of “TheIntegrated Mine Design Project” commissioned by NSERC, Noranda Research and Falconbridge Ltd. in 1986.Twelve failed, nine sloughing and 26 stable pillar case histories were presented from thirteen different mines todevelop the “Pillar Stability Graph” method of designing open stope rib pillars. Thirty of the case historieswere generated from thirteen pillars. All of the pillars within the database are open stope rib pillars.Assessment of pillar stability was generally obtained from observation and description by on-site staff.Sloughing pillars were defined as showing one or more of the following signs:• cracking and spalling in development and raises within the rib pillar• audible noise in the pillar• deformed drill holes• excess muck being pulled from stopes (dilution)• cracking of pillars• major displacements within the pillar89Pillar failure was defined as showing large displacements and severe signs of instability that may havenecessitated immediate stope backfilling to prevent total pillar collapse. The unconfined compressive strengthof the pillar material ranged between 70 MPa and 316 MPa. Rock mass ratings (RMR) varied from60 -78% for all of the case histories. Premining stresses varied between 12 and 46 MPa. Pillar stresses werecalculated using two-dimensional boundary element methods. Both fictitious force and displacementdiscontinuity methods were used depending upon the mining geometry.The original Hudyma (1988) database is presented in Table 16 and Table 17. Figure 57 is the “PillarStability Graph” developed by Hudyma (1988) showing the pillar loading paths of pillars that made up morethan one case history in the database. Table 18 is the summary database for Hudyma (1988).Table 15: Westmin Resources Ltd., summary pillar database.Pillar Pillar Pillar Cpav UCS Average Common PillarWidth Height Width/Height Pillar Stress Stability(m) (m) Ratio (MPa) (MPa) Classification5.9 4.0 1.48 0.17 172 49.6 Stable7.2 4.0 1.80 0.23 172 56.7 Stable7.8 4.0 1.95 0.25 172 99.9 Unstable4.8 4.0 1.20 0.11 172 77.9 Unstable4.6 4.0 1.15 0.10 172 70.8 Unstable3.5 4.0 0.88 0.04 172 70.8 Unstable8.7 5.0 1.74 0.22 172 85.0 Unstable7.6 4.0 1.90 0.24 172 91.8 Unstable5.1 4.0 1.28 0.13 172 63.8 Unstable12.1 4.0 3.03 0.36 172 91.8 Unstable5.9 4.0 1.48 0.17 172 63.8 Unstable12.0 4.0 3.00 0.36 172 42.5 Unstable7.8 4.0 1.95 0.25 172 77.9 Unstable4.6 4.5 1.02 0.07 172 93.5 Failed4.6 4.5 1.02 0.07 172 93.5 Failed5.6 4.0 1.40 0.15 172 93.5 Failed6.3 4.0 1.58 0.19 172 93.5 Failed5.4 5.0 1.08 0.08 172 93.5 Failed4.7 5.0 0.94 0.05 172 93.5 Failed3.9 4.5 0.87 0.04 172 93.5 Failed5.8 4.5 1.29 0.13 172 93.5 Failed4.0 4.0 1.00 0.06 172 93.5 Failed5.9 4.0 1.48 0.17 172 105.4 Failed5.5 4.0 1.38 0.15 172 93.5 Failed5.7 4.0 1.43 0.16 172 105.4 Failed5.2 4.0 1.30 0.13 172 98.6 Failed5.3 4.0 1.33 0.14 172 91.8 Failed4.8 5.0 0.96 0.05 172 88.4 Failed7.1 4.0 1.78 0.22 172 98.6 Failed4.3 4.0 1.08 0.08 172 93.5 Failed3.1 5.0 0.62 0.01 172 93.5 Failed90000000000= .oc0,v)r/)C)Cl)Cl)(/)Cl)00000’0’0’00’0’0’0’0%0%0’0%00000000%0%O%Oo00000WIfl—Cc’lNNOW‘rvV‘flfl00000000000000000——‘flV)‘fl‘fI(0Cl)%0’0%0%0NN%0NNrNNNNNNNNr-t-N%0’0’O%0%0’0NNN%0%0’0’00%0%O’DNNNNNNNN00vcnv.%0N‘t‘000‘0NN‘00000vN‘0‘000N.nCVN‘0‘0‘0fl‘0‘0‘0NN()0O00000000O00000000)OO000000000000000000Ec’iC-——‘0NN00000(4C’C’lC’0’0%00000C’l-0000(4C’’0%0’0’0flfl——16)16)5’)———to00...,0000000000000000000._.._.,_._0000.2.2.2.200000,-.——r’.26100Q0000000000000000000000000000000000000000000c-!0%0%0%0%0000‘00%0%0%0%0%0%0’0%0%0%0%0%0%0’0%0%0%0%0%0%0%0%0%0%NNNNO’0%0%0%0%0%0%0%0s0%0%0%16)0%0000N—“)00000C’N0%NC’—000—000C’16)fl0000000000000%0e’C400It)(qC’)C.)———-—C.)C’)—C.)C’)—C’)C’)—C’)C’)——‘)C’)C’)C’)——It)It)C’)0’.2C’)C.)‘0‘000000——‘.0‘0‘0‘0‘0‘0‘0‘.0‘0‘0‘0‘0NNNNNN0’0’0’0’————C’)C.)C.)cim0z(‘)‘)C’)C’’)C.)C’)ciC’.)C.)4ciC’N000000C’E>1I%0II)—————UIUIUIUI————UI000000000UUIViUIlUIUIUIUIUIUIViViUIUI444-40%UI0pIi.UI—ViZZt9VIUIViVIt-)—t40UIVieZ-Z‘o——-———-00-UIZZ..U-4J%C-UJ0%4Z4UIW44%0‘04t-JI’.)0%Vi0%UI094000‘0.W>‘-0—J00%UI%0000——000%00%.‘0-U00‘000—0‘0‘000‘0Vi—Z00ppcp-ZZZ00pp—.p-—eoo000w0——0000-41-44*00-4OseOs—4ViUI—IU0t.3—0004—000000-444-4000‘0-4UIOs0’sZUIUI-.-UWWZZZZoUI0%—UW-3WW.ie’-).0%0%t)4i-.)000%UIU.UI0%X0%0%0%00%O—>.W%O000UII)000000——000—4-O%00Li0%4U0UI-4AVVAAAAAAASAAAAbAAAAAVAAA1AAISSSSSSSb0000j±05>V)VCfjfl00000CC-0C0CCC-C0C00_C—-——-CC——-rtOtOtoto-to(0(0(0(0tO0000 ITable 18: Hudyma (1988), summary pillar database.Pillar Pillar Pillar Cpav UCS Average Common PillarWidth Height Width/Height Pillar Stress Stability(m) (m) Ratio (MPa) (MPa) Classification24.0 52.0 0.46 0.00 265 38.0 Stable21.0 39.0 0.54 0.00 176 26.0 Stable27.0 40.0 0.68 0.01 176 28.0 Stable30.0 44.0 0.68 0.01 265 40.0 Stable30.0 40.0 0.75 0.02 176 33.0 Stable30.0 40.0 0.75 0.03 176 29.0 Stable45.0 53.0 0.85 0.03 200 51.0 Stable21.0 24.0 0.88 0.04 176 29.0 Stable21.0 21.0 1.00 0.06 100 31.0 Stable21.0 21.0 1.00 0.06 100 26.0 Stable32.0 28.0 1.14 0.10 90 30.0 Stable15.0 12.0 1.25 0.12 176 37.0 Stable15.0 12.0 1.25 0.12 176 33.0 Stable24.0 18.0 1.33 0.14 72 36.0 Stable33.0 23.0 1.43 0.16 316 75.0 Stable12.0 8.0 1.50 0.17 215 28.0 Stable33.0 20.0 1.65 0.20 121 55.0 Stable17.0 10.0 1.70 0.21 310 46.0 Stable15.0 7.0 2.14 0.28 215 29.0 Stable24.0 11.0 2.18 0.28 148 66.0 Stable33.0 15.0 2.20 0.38 316 76.0 Stable20.0 8.0 2.50 0.32 310 46.0 Stable17.0 6.0 2.83 0.34 72 31.0 Stable35.0 12.0 2.92 0.35 148 63.0 Stable21.0 5.0 4.20 0.41 72 39.0 Stable18.0 4.0 4.50 0.42 72 48.0 Stable24.0 52.0 0.46 0.00 265 72.0 Unstable15.0 27.0 0.56 0.00 176 28.0 Unstable27.0 46.0 0.59 0.00 265 59.0 Unstable24.0 38.0 0.63 0.01 160 70.0 Unstable30.0 44.0 0.68 0.01 265 82.0 Unstable15.0 18.0 0.83 0.03 100 31.0 Unstable25.0 28.0 0.89 0.04 90 32.0 Unstable25.0 27.0 0.93 0.05 70 29.0 Unstable15.0 15.0 1.00 0.06 176 43.0 Unstable15.0 49.0 0.31 0.00 200 64.0 Failed9.0 20.0 0.45 0.00 100 38.0 Failed11.0 23.0 0.48 0.00 316 99.0 Failed15.0 30.0 0.50 0.00 100 38.0 Failed14.0 28.0 0.50 0.00 90 49.0 Failed11.0 20.0 0.55 0.00 121 69.0 Failed15.0 27.0 0.56 0.00 176 31.0 Failed11.0 18.0 0.61 0.00 316 102.0 Failed27.0 40.0 0.68 0.01 176 38.0 Failed19.0 28.0 0.68 0.01 90 41.0 Failed30.0 40.0 0.75 0.02 176 57.0 Failed15.0 18.0 0.83 0.03 100 40.0 Failed930.60OPEN STOPE RIB PU..LAR DATA0.50 -0.40(I)C)0.30 -0-j0.20 -0.10 -0.00 -0.00 STABLE 0 FAILUREFigure 57: Hudyma pillar stability graph with all the pillars that made up more than one case history joinedto indicate loading paths that pillars were subjected to (after Hudyma, 1988).5.1.3 Von Kimmelman et al. (1984)Von Kimmelman et al. (1984) presented data on pillar stability and numerical modelling techniquesfrom the Selbi-Phikwe Mines of BCL Ltd. in South Africa. The orebody is a strata bound massive sulphidedeposit located between 80 and 400 metres in depth. Figure 58 is a plan view of the layout of the pillar area forthe case histories. The intact rock mass strength properties are given in Table 19. Data was collectedregarding pillar stability from forty seven square pillars and ten long pillars.A three stage pillar stability classification system was used to categorize pillar stability as follows:• “A” - minor spalling, no joint opening• “B” - prominent spalling• “C” - severe spalling, pronounced opening of joints, deformation of drill holesOnly the square pillars were included in the development of the new strength formulae because of thedifficulty of assessing an “effective width” of long pillars as previously discussed.Twenty two pillars were classified as “C”, six pillars were classified as “B/C”, fourteen pillars wereclassified as “B”, and fifteen pillars were classified as “A”. Pillar stresses were calculated usinga//a/0 0a aI I I I I I I I I I I0.4 0.8 1.2 1.6 2.0 2.4PILLAR WIDTH/PILLAR HEIGHT+ SLOUGHING94MINSIMINFOLD, a two-dimensional displacement discontinuity boundary element modelling program. VonKimmelman et al.’s (1984) original database is presented in Table 20 and the summary database is presented inTable 21.Table 19: Summary of rock properties from the Selbi-Phikwe Mines (after Von Kiminelman et al. 1984)Rock Type UCS Young’s Modulus Poissons Ratio(MPa) (GPa)Hangingwall Rock 89.0 88.8 0.25Massive Sulphide 94.1 81.1 0.24Footwall Rock 189.1 90.7 0.265.1.4 Hedley & Grant (1972)Hedley & Grant (1972) published a study on hard rock pillar stability assessment from data collectedat uranium mines in the Elliot lake district of Ontario. Twenty eight pillar case histories were used to developan empirical strength formula based upon the work of Salamon & Munro (1967). Hedley & Grant’s (1972)data was comprised of 23 stable, two partially failed and three crushed pillars. Extensive work by Kostak &Bielenstein (1971) on small scale strength testing, resulted in unconfined compressive strength estimates ofbetween 210 and 275 MPa for 50 mm samples. Pillar stresses were determined using Hedley & Grant’s (1972)modification of tributary area theory as presented in Section 2.1.1.3.In order to utilize this data with the rest of the combined database, a unconfined compressive strengthof 210 MPa was used as the strength of Elliot Lake rocks. This was based upon the use of a common strengthsize factor, “K”, which is discussed in detail in Section 5.2.4.1. This is the only database for which inputparameters, the unconfined compressive strength, have been modified from the originally published values.The original and summary data are presented in Table 22 and Table 23 respectively.5.1.5 Sjoberg (1992)Sjoberg (1992) presented a summary of rock mechanics data from the Zinkgruvan Mine in southcentral Sweden. Included in this was data on nine pillars, five of which were classified as failed and four ofwhich were classified as being in a pre-failure stage but subjected to severe spalling. The Zinkgruvan rockmass is homogeneous and massive, exhibiting a low joint frequency. The intact rock stength is 215-265 MPa(Sjoberg & TilIman, 1990) and the Young’s modulus is 75-85 GPa. Pillar stresses were calculated usingMINSJM-2D, two-dimensional displacement discontinuity stress modelling program. The summary pillardatabase is presented in Table 24.95Figure 58: Plan of classified mine pillars at the Selbi-Phikwe Mines (after Von Kimmelman et al., 1984).5.1.6 Krauland & Soder (1987)Krauland & Soder (1987) presented information on a number of pillars from the Black Angel Mine, aroom-and-pillar-operation, of Boliden Minerals AB at Marmolik, Greenland. Data was presented on 14 class“3” pillars (refer to Section 3.1) from “Area E” of the Black Angel Mine. Pillar stresses were calculated usingNFOLD (as described previously), a two-dimensional displacement discontinuity program. The intact strengthof the Black Angel rock is 100 MPa (Golder Associates, 1983). These pillars were approximately square indimension. The summary pillar database is presented in Table 25.F o.o £o-40 XX o-eo__ __IIPO UPe__960 0 El — 00 e) 0 I IJ’0-.%D0VI0%0000’C00-J—’0W0%00bo0(I)—000000000000000000000000000000000000000000000000000000CO00000boboooboooo,ooobojoboocoo00oPPPPt--000000008888888SSS0ppppppppppppppppppppppppppppppppppppppppppppp..JVi4.Vi——0‘.00%0%0%0%0%00%-‘C—VI0%VIbt..)000.—VIc.WVI——.1I-.)..t)0%W..VI‘00%I-.)——‘0‘0‘.0‘.0‘.0‘.0‘.0‘.0‘0‘0‘0‘0‘0‘0‘0‘0‘0‘.0‘0‘0‘0‘0‘0‘0‘0‘0‘.0‘0‘0‘0‘0‘0‘0‘0‘0‘.0‘.0‘0‘0‘0‘0‘0‘.0‘0‘0‘0SOSS•000000bbb000000000000000000000000000000000000000, :- 0) CD I 000Table 22: Hedley & Grant (1972) original pillar database. (after Hedley & Grant, 1972)Depth Dip Extraction Width Height Stress Strength(feet) (degrees) (%) (feet) (feet) (psi) (psi)Stable Pillars500 17 85 10 10 5000 14.600700 17 85 10 10 6400 14600800 26 65 20 18 3800 13300850 20 85 10 10 7600 14600950 II 85 10 10 7500 146001000 22 65 20 18 4000 133001050 15 85 10 10 8500 146001200 18 85 10 10 9400 146001300 20 65 20 20 4600 123001600 20 60 20 18 4800 133001600 20 65 18 18 5400 126001600 22 75 20 14 7600 160001700 22 65 40 20 5800 174001700 22 60 22 20 5000 129001700 12 75 20 14 7600 160001800 5 75 20 14 8000 160001900 23 65 19 18 6400 130002200 25 65 20 20 7200 123002400 1 I 65 20 8 7600 244002500 9 65 20 8 7900 244002700 13 65 20 8 8600 244002900 12 70 15 9 10500 194002900 12 75 20 9 12600 22400Depth(feet)Partially Failed Pillars14001Dip Extraction Width Height Stress Strength(degrees) (%) (feet) (feet) (psi) (psi)24001• 20 85 10 10 11400 14.60018 80 10 9 13400 15800Depth Dip Extraction Width Height Stress Strength(feet) (degrees) (%) (feet) (feet) (psi) (psi)Crushed Pillars2800 12 80 10 9 15200 158002900 12 80 10 9 15700 158003400 5 80 15 10 18500 1790099Table 23: Hedley & Grant (1972), summary pillar database.Pillar Pillar Pillar Cpa,, UCS Average Common PillarWidth Height Width/Height Pillar Stress Stability(m) (m) Ratio (MPa) (MPa) Classification3.0 3.0 1.00 0.06 210 34.5 Stable3.0 3.0 1.00 0.06 210 44.1 Stable6.1 5.5 1.11 0.09 210 26.2 Stable3.0 3.0 1.00 0.06 210 52.4 Stable3.0 3.0 1.00 0.06 210 51.7 Stable6.1 5.5 1.11 0.09 210 27.6 Stable3.0 3.0 1.00 0.06 210 58.6 Stable3.0 3.0 1.00 0.06 210 64.8 Stable6.1 6.1 1.00 0.06 210 31.7 Stable6.1 5.5 1.11 0.09 210 33.1 Stable5.5 5.5 1.00 0.06 210 37.2 Stable6.1 4.3 1.43 0.16 210 52.4 Stable12.2 6.1 2.00 0.26 210 40.0 Stable6.7 6.1 1.10 0.09 210 34.5 Stable6.1 4.3 1.43 0.16 210 52.4 Stable6.1 4.3 1.43 0.16 210 55.2 Stable5.8 5.5 1.06 0.08 210 44.1 Stable6.1 6.1 1.00 0.06 210 49.7 Stable6.1 2.4 2.50 0.32 210 52.4 Stable6.1 2.4 2.50 0.32 210 54.5 Stable6.1 2.4 2.50 0.32 210 59.3 Stable4.6 2.7 1.67 0.21 210 72.4 Stable6.1 2.7 2.22 0.29 210 86.9 Stable3.0 3.0 1.00 0.06 210 78.6 Unstable3.0 2.7 1.11 0.09 210 92.4 Unstable3.0 2.7 1.11 0.09 210 104.8 Failed3.0 2.7 1.11 0.09 210 108.3 Failed4.6 3.0 1.50 0.17 210 127.6 Failed5.1.7 Brady (1977)Brady (1977) presented data from a sill pillar at Mt. Isa Mines located in Australia. Two stableobservations and one failed observation of the pillar were made as mining progressed towards the pillar. Finiteelement modelling was used to determine stresses on the sill pillar. The summary pillar database is presentedin Table 26.5.1.8 Database SummaryAll of the data included in the combined database represents mines that have pillars developed withingood to very good quality rock masses (RMR 60-85%). Major structural failure was not recognized as acontributing factor to pillar instability. Five of the seven databases are from within massive suiphide orebodies.Pillar stresses were calculated using either tributary area theory or numerical modelling techniques for alldatabases. A pillar strength relationship can therefore be developed for good quality, hard rock (predominantlymassive sulphide) orebodies where structurally influenced failure is not a factor.100Table 24: Sjoberg (1992), summary pillar database.Pillar Pillar Pillar Cpav UCS Average Common PillarWidth Height Width/Height Pillar Stress Stability(m) (m) Ratio (MPa) (MPa) Classification3.8 6.0 0.63 0.01 244) 68.0 Unstable6.0 6.0 1.00 0.06 240 84.0 Unstable6.2 6.0 1.03 0.07 240 74.0 Unstable7.3 6.0 1.22 0.11 240 67.0 Unstable4.7 6.0 0.78 0.02 240 95.0 Failed7.5 6.0 1.25 0.12 240 83.0 Failed7.5 6.0 1.25 0.12 240 100.0 Failed8.5 6.0 1.42 0.16 240 82.0 Failed10.5 6.0 1.75 0.22 240 92.0 FailedTable 25: Krauland & Soder (1987), summary pillar database.Pillar Pillar Pillar Cpav UCS Average Common PillarWidth Height Width/Height Pillar Stress Stability(m) (m) Ratio (MPa) (MPa) Classificationn/a n/a 0.88 0.04 100 39.3 Unstablen/a n/a 0.88 0.04 100 47.5 Unstablen/a n/a 0.83 0.03 100 31.0 Unstablen/a n/a 0.74 0.02 100 29.0 Unstablen/a n/a 0.74 0.02 100 31.0 Unstablen/a n/a 0.66 0.01 100 47.5 Unstablen/a n/a 0.62 0.01 100 31.0 Unstablen/a n/a 0.59 0.00 tOO 29.0 Unstablen/a n/a 0.55 0.00 100 29.0 Unstablen/a n/a 0.53 0.00 100 31.0 Unstablen/a n/a 0.51 0.00 100 37.0 Unstablen/a n/a 0.47 0.00 100 41.4 Unstablen/a n/a 0.45 0.00 100 25.0 UnstableTable 26: Brady (1977), summary pillar database.Pillar Pillar Pillar CPav UCS Average Common PillarWidth Height Width/Height Pillar Stress Stability(m) (m) Ratio (MPa) (MPa) Classificationn/a n/a 2.33 0.30 170 39.3 Stablen/a n/a 2.33 0.30 170 55.6 Stablen/a n/a 1.00 0.06 170 88.7 Failed1015.2 HARD ROCKPILLAR STRENGTH FORMULAEDERIVATIONIn this section the databases presented in the previous section will be investigated along with thevariables that are necessary to make estimations of pillar strength. A refined empirical strength formula, “TheLog-Power Shape Effect Formula”, and a new pillar strength formula, “The Confinement Formula”, arepresented.5.2.1 Empirical Strength Formula Derivation MethodologyIn order to derive the empirical strength formulae presented in this section, each database wasinvestigated individually to determine individual empirical constants for the “Linear Shape Effect” formula.The formulae developed will be of the form presented in Equation 35. The methodology used to develop theempirical strength formulae is as follows:• Specify the requirements that the empirical formula should meet.• Show that the variation in volume between the largest and smallest pillars in the database is smallenough that it will not have to be dealt with in the shape term in Equation 36.• For each of the detailed databases, derive the empirical constants “A” and “B” of the “LinearShape Effect” formula (Equation 11). Based on this information, a common strength size factor,“K”, for full size mine pillars is determined.• Using the empirical constants derived and the valid width I height ratio ranges for each of thedatabases, show that pillar strength is driven by a non-linear function of the pillar width / heightratio.• Refine the power function using the average pillar confinement to derive a descending power or“Log-Power Shape Effect” formula.• Introduce a new pillar term, “kappa”, which is dependent on pillar confinement and isfundamental to the development of a new pillar strength hypothesis, “The Confinement Formula”.This methodological approach to the derivation of empirical constants for the “Shape Effect” formulawill be investigated in detail in the next section.5.2.2 Requirementsfor a Strength FormulaIn defining a strength formula that is to be used by practicing mine engineers and designers, thefollowing criteria should be met:• Input parameters should be readily available and not require an extraneous amount of effort todetermine.• The method should be readily calibrated to a particular mme site.• The strength formula should be dimensionally balanced.102The strength formulae developed here will be in the general form of Equation 35.P5=SizexShape (35)where:P = Pillar strength (MPa)Size — Strength term that incorporates the size effect and strength of intact pillarmaterial (MPa)Shape Geometric term that incorporates the shape effect of the pillarAs discussed in Chapter 2, pillar strength formulae have been derived empirically in the past using the“Linear Shape Effect”, “Power Shape Effect” and “Size Effect” formulae. In review, the general form of theseformulae is presented in Equation 36.P=K.(A÷B) (36)where:P — Pillar strength (MPa)K — Strength pillar material (MPa)w = Pillar width (m)h — Pillar height (m)A, B, a, b — Empirically derived constantsIn most mining operations there is neither the manpower nor the financial resources available toexamine the strength characteristics of mine pillars in detail. The data available generally consists of:• geometric characteristics of pillars• predicted pillar stresses• estimates of pillar stability• the intact strength of the rock mass materialEach of these variables as they relate to pillar strength is examined in the following section.5.2.3 Pillar Strength VariablesThe input variables that have been deemed to control pillar strength are presented in the followingsections. It is concluded that the critical variables required to make an assessment of pillar strength, asdiscussed in previous sections, include the following:103• pillar geometry / pillar confinement• intact rock strength• pillar volume• applied pillar stress• pillar stability classificationThese variables are summarized in the following sections. A relationship between pillar geometry andpillar confinement is developed based upon two-dimensional boundary element modelling.5.2.3.1 Pillar GeometryPillar geometry is collected at mining operations on a regular basis in order to quantify the amount ofmaterial removed through the course of mining operations. This data is generally collected by mine surveyorsand engineers and recorded on mine plans and sections. A typical three-dimensional pillar is shown in Figure50. The pillar has a width, length, and height. These dimensions, for the sake of pillar strength estimation, aremade in the axes of the principal stresses acting on the pillars. Some authors (Wagner (1980), Sheorey &Singh (1974), Stacey & Page(1986)) have suggested that the width can be replaced by an effective width termthat encompasses both the length and the width. The author could find no reason to use an “effective” pillarwidth, based on the data presented in Chapter 4, and has concluded that the width used should be the minimumpillar width through the pillar centre as illustrated in Figure 50.5.2.3.2 Pillar ConfinementExisting empirical strength formula relate the geometric characteristics, width and height, of a pillarto the pillar strength. Hoek & Brown (1980) proposed that pillar width / height ratios could be replaced withan equivalent term equal to the average (73/01 across the mid-height centerline of the pillar. This term will becalled the average pillar confinement, Cp. In this section a relationship between average pillar confinementand pillar width / height ratio is derived based upon the results of two-dimensional numerical modelling. Thisterm will be used in the “Log-Power Shape Effect Formula” and the new strength formula, “The ConfinementFormula”.In order to investigate how average pillar confinement is related to pillar geometry, a detailedparametric numerical modelling session was undertaken. Two-dimensional boundary element modelling wasperformed, using Examine2D, to model pillars of varying width / height ratios. Detailed modelling wasperformed with an extraction ratio of 99.5%. In addition, less comprehensive modelling sessions wereperformed with decreasing extraction ratios, down to 66.7%. The results of these modelling sessions arediscussed in the following section and are presented in Appendix A.1045.2.3.2.1 DetailedModellingA single pillar was modelled with width I height ratios varying from 0.25 to 10. Homogeneousconditions in a constant stress field were simulated. Examination of the zones of failure using both MohrCoulomb and Hoek & Brown (1980) failure criteria revealed that predicted failure always occurred initially atthe mid-height of the pillar as illustrated in Figure 88 to Figure 93. In order to assess modelled pillar failure,“m” & “s” were fixed at 10 and 0.1 respectively. These are representative values for a good quality rock mass.The model unconfined compressive strength was then varied in each model until a zone with a factor of safetyof less than one was generated at any location across the pillar. In all cases the failure zone first materialized atthe mid height of the pillar, which corresponds to the point in the pillar that has the lowest average ratio of 03to 01. Average a and 03 values were tabulated and are presented in Table 27 along with the average pillarconfinement, Cpa,,, calculated for each width / height ratio in the modelling session.Figure 59 shows the relationship between pillar width / height ratio and average calculated pillarconfinement for an extraction ratio of 99.5%. It can be seen from Figure 59 is that the average confinement isextremely low for pillar width I height ratios of 0 to 0.6, increases rapidly for width / height ratios from 0.6 to2.5, and becomes a constant slope above width / height ratios of 4.0. Equation 37 was determined through trialand error to relate average pillar confinement determined from modelling and pillar width / height ratio forwidth / height ratios less than 4.5 and for a modelled extraction ratio of 99.5%.Cpav = 0.34. [log(- + 0.75)] (w/h)where:Cpa,, Average pillar confinementw — Pillar width (m)h — Pillar height (m)5.2.3.2.2 Additional ModellingAdditional modelling was performed to determine the sensitivity of the average pillar confinement,Cpa,,, to various input parameters. The effects of altering the in-situ stress ratio and the modelled extractionratio were investigated.5.2.3.2.2.1 Variable In-situ Stress RatioThe in-situ stress conditions were modified for a limited number of modelled width / height ratios todetermine what the influence of the in-situ stress ratio (01/03) would have on the modelled pillar confinement.These points are plotted on Figure 59 and the values are reported in Table 27. Figure 59 shows105Table 27: Results of two-dimensional parametric modelling sessions to investigate average pillarconfinementthat the in-situ stress ratio appears to have little effect on the modelled pillar confinement. This session wasperformed using a modelled extraction ratio of 99.5%.52.3.2.22 Variable Extraction RatioThe detailed modelling session was originally performed using an extraction of 99.5%. This value isnot representative of most mining situations so additional modelling sessions were performed to determine ifthe average pillar confinement is dependent on the extraction ratio. Table 28 lists the results from modellingdifferent extraction ratios for a limited number of pillar width / height ratios. The results show that a lowerextraction ratio results in a higher confinement value for a given width I height ratio. The results for two of thesessions are presented in Figure 60 with the original 99.5% line along with the proposed confinement lines formodelled extraction ratios of 98.3% and 88.3%. Based upon the relationship determined between pillar width Iheight ratio and Cpa. for 99.5% extraction, it was determined that for various extraction ratios can beapproximated by Equation 38. This relationship is valid for width / height ratios between 0 and 3.0. The valueof the “coefj” in Equation 38 as determined for each of the modelling sessions is presented in Figure 61.Pillar In-Situ Stress In-Situ Stress In-Situ StressW/H 03=01 0301/2 03012Ratio 03 Oj Cpav 03 (1 Cp 03 Ui Cpav025 03 9000 000 SlIt SliW SlIl St. IItWa0.50 0.3 1078.9 0.00 0.6 1072.3 0.00 0.3 1074.9 0.000.60 0.5 1108.7 0.00 0.7 1111.2 . i”’ 4 (‘“00.65 3.7 1117.9 0.000.70 8.7 1130.9 0.010.75 16.9 1136.6 0.011.0 58.0 1173.0 0.05 58.7 1158.6 0.05 58.3 1168.3 0.051.25 110.9 1195.4 0.09 JJJ1.5 159.8 1223.0 0.13 156.1 1227.6 0.13 155.3 1214.6 0.131.75 202.2 1232.2 0.16 ,2.0 239.8 1254.9 0.19 236.5 1257.9 0.19 247.7 1254.6 0.202.5 302.4 1291.6 0.233.0 354.5 1343.8 0.26 351.5 1342.8 0.26 365.3 1342.1 0.273.5 394.5 1398.5 0.284.0 435.6 1406.7 0.315.0 478.6 1513.0 0.326.0 521.6 1570.0 0.337.0 565.6 1647.0 0.348.0 61.1 1674.0 0.379.0 652.0 1702.0 0.3810.0 651.3 1611.1 0.40106Cpav = coeff . [log(- + 0.75)](W/h) (38)where:Cp Average pillar confinementcoeff = Coefficient of pillar confinementw Pillar width (m)h = Pillar height (m)5.2.3.2.3 Discussion - Pillar ConfinementThe mining scenarios used in hard rock mining generally will have overall extraction ratios between65% and 90%. Reviewing Figure 61, we can see that the confinement “coeff’ does not begin to decreasedramatically until the extraction ratio is above 95%. Based upon this, an “average” value for the confinement“coeff’ in the strength formula will be used, as opposed to using a relationship for the confinement “coeff’based upon extraction ratio. The confinement “coeff’ varies between 0.445 and 0.465 for extraction ratiosbetween 90% and 65%. This range represents a maximum potential error of 5% in the value of the confinement“coeff’ for the expected extraction ratios in most pillar mining situations. Therefore an average confinement“coeff’ of 0.46, corresponding to an extraction ratio of 72%, as selected from Figure 61 will be used. It is theauthor’s opinion that the magnitude of error associated with the confinement “coeff’ is acceptable given thevariability of the other factors that contribute to pillar strength. The formula for average pillar confmement forextraction ratios between 65% and 95% can then be approximated by Equation 39.Cpav = 0.46. [log(. + 0.75)]) (39)where:Cp — Average pillar confinementw - Pillar width (m)h — Pillar height (m)It is important to understand why it is suggested that the average pillar confinement, Cp, be used inplace of the pillar width / height ratio in strength formulae. Pillar width / height ratio has been used as primaryinput variable in the past by a number of researchers. This is primarily applicable to pillars that are of regularshape and as such, the pillar width / height ratio can be determined readily. However, in situations where pillarshapes are irregular in plan, the assessment of pillar width I height ratio becomes more difficult.1070.45Cpav —0.23 + 0.017*(w/h)0.401-I - - - - -- - - ---0.35 ---0.15------ -0.10--(1.4/(w/h))Cpav — O.34*(Iog((w/h)+0.75)))0.05-----a--I I I I I I I I I I I0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0Pillar Width / Height RatioModelled In-situ Stress ConditionsSig3 — Sigi Sig3 — Sigl/2 Sig3 — Sig1*2)Figure 59: Relationship determined between average pillar confinement and pillar width I height ratio for amodelled extraction ratio of 99.5%.Table 28: Results of two-dimensional parametric modelling sessions investigating the effects of differingmodelled extraction ratios on the average pillar confinement.Modelled Pillar W/H Ratio Pillar W/H RatioExtraction 1:1 2:1Ratio 03 01 CPav 03 01 CPay99.5% 58.0 1173.0 0.05 239.8 1254.9 0.1999.0% 32.0 648.7 0.05 142.8 686.7 0.2198.4% 22.1 433.0 0.05 99.2 454.9 0.2295.2% 9.6 183.4 0.05 60.3 267.5 0.2390.9% 5.9 111.2 0.05 29.3 125.7 0.2383.3% 3.5 73.9 0.05 18.0 80.2 0.2266.7% 2.0 47.6 0.4 12.7 51.6 0.25Modelled Pillar W/H Ratio Pillar W/H RatioExtraction 3:1 4:1Ratio 03 0j 03 Oi CPav99.5% 354.5 1343.8 0.26 435.6 1406.7 0.3199.0% 215.7 725.8 0.30 271.0 760.3 0.3698.4% 135.2 414.1 0.33 181.9 480.4 0.3895.2% 66.6 198.2 0.34 81.2 201.7 0.4090.9% 44.8 128.6 0.35 51.5 129.9 0.4083.3% 27.8 82.5 0.34 34.6 84.3 0.4166.7% 20.2 53.8 0.38 24.1 53.4 0.45108. 0.420Q>a 0.38L)0.400.350.300.250.20c-)0.150.10-0.05-j-co-eff —0.45 —co-eff —0.40- ----— co-eff -0.34o —0.007/7/7,,—(1.4/(w/h)) / / o-_Cpav — <coeff.*(1og((w/h)+O.75))) / / 7V 7____V1.5 2.0Pillar Width! Height Ratio VModelled Extraction Ratio88.3 % 0 98.3 99.5Figure 60: Relationship between the average pillar confinement and pillar width I height ratio for differingmodelled extraction ratios.0.0 0.5 1.0 2.5 3.00.50 -0.46 -For an average mining extraction ratio of 72%,the corresponding Cpav coeff is 0.46.NI ‘72%0.34 LI I I I0.65 0.70 0.75 0.80 0.85 0.90 0.95Extraction RatioFigure 61: Plot of the variation in the Cp coefficient for different modelled extraction ratios.1.00109Utilizing a new term, Cp or average pillar confinement, allows the assessment of the strength ofpillars that have irregular shapes. The potential exists that pillar confinement can be determined in the futurefrom numerical modelling and not from the pillar width / height ratio, as has been done here. The pillar width/ height ratio is used in the strength formulae subsequently developed, however, because the pillars that makeup the database have not been modelled individually to determine the average pillar confinement of each. Allgeometric observations were of pillar width and height only.5.2.3.3 Intact Rock StrengthThe intact rock strength is commonly determined from diamond drill core samples. The mostcommon strength term derived is the unconfined compressive strength. Estimates of the unconfinedcompressive strength can be made in a number of manners such as:• laboratory testing• point load index• hammer testKostak & Bielenstein (1971) determined that there was considerable scatter in sample strengths for agiven size of sample. Kostak & Bielenstein’s (1971) research for 50 mm samples suggests that for a given rocktype, unconfined compressive strength can vary by ± 10%. This variability must be considered when using theunconfined compressive strength as an input parameter for estimating pillar strength.5.2.3.4 Influence OfPillar Volume On StrengthIt has been recognized by researchers that as sample size of a given rock type increases, the strengthdecreases. This is the result of having an increased number of structural defects present within larger samplesand is discussed in Section 3.2.4. An example of this phenomenon is shown in Figure 29, Figure 30 and Figure31. We can see that small intact samples have a higher strength than a rock mass of the same materialcontaining a large number of structural features. Work by Kostak & Bielenstein (1971) showed the influence ofstrength for samples ranging from diameters of 45 mm - 240 mm as shown in Figure 31. Agapito & Hardy(1982) suggest a correction formula for the adjustment of strength between two different sample sizes asdescribed by Equation 40.CL(40)S2 Viwhere:S1. S2 = Strength of samples I and 2 respectivelyV1, V2 — Corresponding volume of samples I and 2CL — Exponent that ranges from 0.06 for hard rock to 0.18 for soft rock110Based upon 162 observations0.7-0.9 0.9-1.1 1.1-1.3Normalised Co-efficient RangeVolume Coefficient Normalised to the Mean Volume Coefficient for the Total Database.Figure 62: Histogram showing the variation in the influence of the pillar volume on pillar strength for alldata in the combined database.The combined database was investigated with respect to sample (pillar) volume utilizing Agapito &Hardy’s (1982) formula and it was determined that 62% of the case histories fell within ±10% of the strengthvolume factor for a five metre cubical pillar specimen, a representative pillar size for the pillars in thecombined pillar database. In addition 94% of the case histories fell within ± 30% of the mean strength volumefactor. It is concluded that this variation in the effect of volume on pillar strength is sufficiently small that itcan be disregarded for the case histories in the combined database. Figure 62 is a histogram showing thedistribution of the volume coefficient for the pillar in the combined database.5.2.3.5 Pillar Stress DeterminationFor the case histories included in the combined pillar database, average pillar stress has beendetermined using a number of different methods. The hierarchy of complexity of pillar stress determinationgenerally follows, from simplest to most complex, the following:1. tributary area theory2. two-dimensional boundary element modelling3. two-dimensional finite element modelling4. three-dimensional boundary element modelling5. three-dimensional finite element modellingPillar UCSI / Pillar UCS2 — (V2 / VI)0.060%1.3- 1.5IllIn the case of the combined database, methods 1, 2, 3, and 4 have been used to calculate average pillarstresses. It is recognized that errors can be realized in stress determination, however the process of calibratinga numerical model should reduce much of the uncertainty that accompanies numerical modelling in general. Itis the author’s opinion that, in spite of the different stress determination methods used, the values obtained foreach database are comparable. In general there is no need to use a more complex method than a problemrequires. The effort required to obtain results using a more complex method, that may be of marginallyimproved accuracy, may require a significant amount of additional work.5.2.3.6 Pillar Stability Classf1cationIn all of the cases in the combined database, pillar stability assessments, which range from a simpleassessment of “Stable / Failed” to a more rigorous approach based upon a five or six stage stabilityclassification method, have been made. Reviewing the combined database, the author concludes that aminimum of a three stage failure assessment should be used, in the form of:• stable pillars• unstable pillars• failed pillarsThe above assessment was made during the process of reviewing each of the summary database’s inthe derivation of the empirical strength constants discussed in the following sections. The correlation of thethree stage pillar stability classification with the classification’s originally used for each of the database’s ispresented in Table 29.Unstable pillars are classified as showing any visible signs of pillar degradation. The unstable rangemay, as with the Westmin database and the Krauland & Soder (1987) database, be subdivided into moredetailed levels of instability, however the basic premise of “failed”, “unstable” and “stable” provides adequateresults for the combined database.Table 29: Common pillar stability assessment designation for each individual database in the combineddatabase.Combined Westmin Hudyma Von Kimmelman et Hedley & Sjoberg (1992) Krauland &Database Resources (1988) al. (1984) Grant Soder(l987)(1972)Failed Class “5” Failed Class “C” Crushed Failed NAUnstable Class “2”-”4” Sloughing Class “B”, “B/C” Partially Pre-failure, Class “3”Failed severe spallingStable Class “1” Stable Class “A” Stable NA NA1125.2.4 Refined Empirical Strength FormulaeThe databases from Westmin Resources Ltd.’s H-W Mine, Hudyma (1988), Von Kimmelman et al.(1984), and Hedley & Grant (1972) have been used to develop a refined empirical strength formula for hardrock mine pillars based upon back analysis. The databases from Sjoberg (1992), Krauland & Soder (1987), andBrady (1977) have been used to corroborate the formulae developed. Empirical strength formula have takenthree forms to date as discussed in Chapter 2. The general formulae are the “Linear Shape Effect”, “PowerShape Effect” and the “Size Effect” formulae. The only published strength formulation for hard rock minepillars was a size effect formula presented by Hedley & Grant (1972) based upon the Salamon & Munro (1967)strength formula for coal pillars in South Africa.52.4.1 Linear Shape EffectThe Shape Effect Formula can be described by Equation 41 and Equation 42 based upon the work byresearchers discussed in Chapter 2.Ps = UCS.(A’+B’(-)) (41)horPs=(K.UCS).(A+B()) (42)where:— Pillar strength (MPa)w Pillar width (m)h Pillar height (m)UCS — Unconfined compressive strength of intact sample of pillar material (MPa)A’, B’ Empirical constantsK Strength size factor equal to A’ plus B’A, B Empirical constants where A + B = 1.0The empirical constants, “A”, “B”, and “K”, for the linear shape effect formula were determined foreach of the individual databases and the results are presented in Table 30. These variables were determined for“bestfit” steep and flat lines that could be used to sub-divide the pillar stability classification groups for each ofthe databases individually. The criteria for the placement of these “bestfit” lines was that the predictabilitysuccess for the full range between the flat and steep lines remained constant and was the highest predictabilitypossible. This is discussed in detail in Section 5.3. The coefficients generated for the Westmin database areslightly different than those reported in Chapter 4 because the techniques used to place the stability lines was113Table 30: Linear shape effect constants and strength size factor determined for each of the individualdatabases in the combined database.Case A B A B K Value Valid W/ll(steep) (steep) (flat) (flat) (range) (range)Westmin 0.66 0.34 0.77 0.23 0.45-0.49 1.3-1.9Hudyma (1988) 0.0 1.0 0.25 0.75 0.45 0.4-1.0Von Kimmelmanetal. 0.35 0.65 0.65 0.35 0.48-0.51 1.3-1.9(1984)Hedley & Grant (1972) 0.25 0.75 0.75 0.25 0.3-0.5 0.9-1.2slightly different and was based upon three generalized pillar stability classifications instead of five detailedstability classifications.Figure 63 - Figure 66 are stability plots for each of the individual databases showing the respectivepillar case histories, the valid range of slopes for the linear shape effect, and the valid ranges, as shown by thesolid segments of the lines, of width / height ratios based upon the database distribution. A “K” factor of 0.44has been used for these plots. In order to use the “Linear Shape Effect” formula it was necessary to determinea common strength size effect coefficient, “K”, that could be used for each of the databases. The value of “K”ranges from 0.3 to 0.51 and the values for each of the databases are presented in Table 30. It can be noted thatthe values of “K” derived for each of the individual databases have small variability with the exception Hedley& Grant (1972). This is because the Hedley & Grant (1972) data was supplied with a range of potentialunconfined compressive strength values that could be used (210 - 275 MPa). In comparison to the otherdatabases, it is determined that when using the average “K” value of 0.44, the corresponding unconfmedcompressive strength of the Hedley & Grant (1972) database is 210 MPa.A.n average value of 0.44 was chosen for the strength size coefficient, “K”, to be used for all of thedatabase’s. This value was determined by taking the average of the “K” values for the database’s as presentedin Table 30. This value will be used as the common strength size factor for the subsequent pillar strengthformulations. This value compares favourably with methods of determining a strength size factor employed byother researchers. The work of Kostak & Bielenstein (1971) suggests that “K” for a full size pillar would be0.5, based upon Figure 31. Equation 40, presented by Agapito & Hardy (1982), results in “K” 0.4 for acubical pillar that is 5 metres in dimension on each side, a representative size for the pillars in the combineddatabase, using “a.” 0.06, as suggested for hard rock.The valid range of width / height ratios for each of the databases are also presented in Table 30. Thevalid range is defmed as: the range of width / height ratios for which there is sufficient data to subdivide thedifferent pillar stability classifications with a high degree of confidence. Each database is only valid over alimited range of width I height ratios. Figure 67 is plot of the “bestfit” steep and flat “Linear Shape Effect”1140.70.6C’, 0.5z0.40.30.1 -0.0— F.S. — 1.0—— —Ps — 0.44*UCS*(A + B*(w/h))31 observations— I I I I I0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2Pillar Width / Height RatioPillar Stability Classification[ Failed • Unstable A Stable — Valid W/H RangeJStability graph for the Westmin database showing the range of slopes for the stability lines and thevalid range for pillar width / height ratios.Figure 63:0.70.6Ct,0.5z0.30.20.10.00.0.//// //F.S. — 1.0 / - F.S. — 1.4/ / /// /7// V/,--A-----V/ /_-_/ A A/___,;—-AAPs0.44*UCS*(A+B*(w/h))47 observationsAI I I I I0.4 0.8 1.2 1.6 2.0Pillar Width / Height RatioPillar Stability Classification2.4 2.8 3.2[ Failed • Unstable A Stable — Valid W/H Range)Figure 64: Stability graph for the Hudyma (1988) database showing the range of slopes for the stability linesand the valid range for pillar width / height ratios.1150.60.5z0.40.3S..> 0.20.10.7F.S. = 1.0—- >-_-/_— _z •——A.—2--,, — — A7— — —- - —7> - - -—————A APs — 0.44*IJCS*(A + B*(wlh))47.0.00.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2Pillar Width / Height RatioPillar Stability ClassificationFailed • Unstable A Stable — Valid W/H RangedFigure 65: Stability graph for the Von Kimmelman et al. (1984) database showing the range of slopes for thestability lines and the valid range for pillar width / height ratios.0.70.6 -C’, 0.5.. 0.30.20.10.0/ F.S. — 1.0 -/——// .-.-- - F.S. —1.4/ —------—— — — —7 — —- —--14--- 1 JIll 11111/ -._ A Ps 0.44*UCS*(A + B*(wfh))28 observationsI I I I I II I I I I0.0 0.4 0.8 1.2 1.6 2.0Pillar Width! Height RatioPillar Stability Classification2.4 2.8 3.2( Failed • Unstable A Stable — Valid W/H RangedFigure 66: Stability graph for the Hedley & Grant (1972) database showing the range of slopes for thestability lines and the valid range for pillar width / height ratios.116lines, for the valid width /height ratios, for each of the databases. It can be observed from Figure 67 that theslopes of the “Linear Shape Effect” lines are steeper at lower width I height ratios and flatter at higher width Iheight ratios. These curves will be shown to follow a non-linear relationship with pillar width / height ratio inthe next section.5.2.4.2 Refined Power Shape EffectAll comprehensive and supporting data has been used to develop a power coefficient for the “PowerShape Effect” formula. It can be seen from Figure 67 that the pillar strength is a non-linear function of pillarwidth / height ratio and it was determined that the strength function resembles a power function as described bythe following formula:Ps = (0.44. UCS)• (W)CL(43)hwhere— Pillar strength (MPa)w — Pillar width (m)h = Pillar height (m)UCS Unconfined compressive strength of intact sample of pillar material (MPa)a. — Empirical power coefficientAnalysis of the data from the combined database for varying values of “a.” resulted in a value of “a” of0.45 for optimum strength prediction results. This “Refined Power Shape Effect” curve is presented in Figure68 along with all of the combined database case histories. In the derivation of the power coefficient, “a.” of0.45, it was observed that higher values of “a.” gave better strength estimation results at lower width / heightratios and that an “a.” value of 0.4 gave better results at width I height ratios of 2.5 and greater. This lead tothe development of the “Log-Power Shape Effect Formula”, as presented in the next section.52.4.3 Log-Power Shape EffectA method of making “a.” a function of width / height was investigated. A means of reducing “a” withincreasing width I height ratios was developed through a trial and error approach utilizing the value of Cp. Inthe course of the investigation of the optimum power coefficient in the previous section, “bestfit” powercoefficients for small ranges of width I height ratio were tabulated. These desired “a.” values, for a given widthI height ratio, are presented in Figure 69. It was determined that the optimum empirical success rate inpredicting pillar strength was driven by an “a.” term defined by Equation 44.1170.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2Pillar Width / Height RatioConfinement Formula —-— Log - Power Shape EffectStrength FormulaeRefined Power Shape Effect — Valid Linear Shape Effect RangesFigure 67: Stability graph showing the stability lines over the valid width I height ratio ranges for each of theindividual databases plotted alongside the strength formulae subsequently developed.0.7—F.S. — 1.0A0.70.6 -C-,0.5zP.. 0.35)<0.20.10.0000I0.60.5zC’,0.. 0.30)0.20.1A•0ACA Ao AAA AAA AAA SAA A 0.45A Ps — 0.44*UCS*(wlh)178 observationsA0.00.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2Pillar Width I Height RatioPillar Stability ClassificationC Failed • Unstable A Stable DFigure 68: Stability graph for the “Refined Power Shape Effect” formula with a power coefficient, “a”, of0.45, plotted along with all of the case histories in the combined database.1181.0[0.35Figure 69: Plot of the preferred value of the refined power coefficient, “a”, for differing values of pillar widthI height ratio plotted along with the relationship represented by Equation 44.where:0.1a =l.31—Cpava = Log-power coefficientCpa,, Average pillar confinement(44)Although derived through a trial and error approach, this formula for the power coefficient fits thedesired “a” data very well. Figure. 70 is a plot of the “Log-Power Shape Effect Formula” with all of the casehistories in the database.5.2.4.4 Discussion - Refined Empirical Strength FormulaeRefined empirical strength formulae have been derived for hard rock mine pillars. Individual “LinearShape Effect” constants have been derived for each of the comprehensive databases and it was determined thatthe unconfined compressive strength of a full scale mine pillar can be approximated by a strength size factor,“K”, of 0.44 of the small scale unconfined compressive strength (50 mm sample). This was developed fromthe investigation of the pillar case histories that make up the combined database. When assimilating the linearstrength constants, it became evident that pillar strength could not be represented by a linear relationship over0.90.80.70.69. 0.50.tAlpha— 1.31 -Cpav0.00.00 0.05 0.10 0.15Cpav coeff - 0.460.20Cpav: Observed Values0.25 0.301190.7F.S. — 1.00.6 -0.5C’,0.4-- - -5)Ce1-.5)>0.1 •- - -0.00.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2Pillar Width / Height RatioPillar Stability ClassificationFailed • Unstable a’ StableFigure 70: Stability graph for the “Log Power Shape Effect” formula with a power coefficient represented byEquation 45, plotted along with all of the case histories in the combined database.a wide range of pillar width / height ratios. A refined empirical power coefficient, “Ct”, of 0.45 for the “PowerShape Effect” formula has been determined, however it was clear that the “Power Shape Effect” did notrepresent the optimum empirical strength formula. A refined empirical strength formula, “The Log-PowerShape Effect Formula” has been developed where the power coefficient is shown to be a function of the averagepillar confinement. The prediction success of these refined empirical formulae, as related to the combineddatabase, is discussed in Section 5.3.5.2.5 A New Strength Hypothesis - The Confinement FormulaA fundamentally derived - empirically verified strength estimation formula, “The ConfinementFormula”, for hard rock mine pillars is presented in this section. Wilson (1972) and Sheorey et al. (1987) haveboth suggested that the strength of mine pillars is controlled by confinement, which is a function of pillar width/ height ratio. While pillar strength is commonly presented using pillar width / height ratios, it is actuallypillar confinement that controls the strength of mine pillars.A new method of estimating the strength of hard rock mine pillars that uses pillar confinement as afundamental input variable in place of pillar width / height ratio is proposed. This approach is verified usingthe case histories that have been tabulated and presented in Section 5.1. It will be shown that the proposedstrength formula predicts pillar stability for the case histories in the database as well as the best empiricalD •-- --•.•••••F.S. = 1.4a’A• AAAAa’AA AAA•Ps — 0.44*UCS*(w/h)0.I(I.31-CpavCpav coeff — 0.46l78observations120strength formula, yet it follows a form of the fundamental theory of the shear strength of rock. It is proposedthat the pillar strength is comprised of an unconfined strength and a confined strength. At very low width /height ratios the strength is controlled by the unconfined term, and at high width I height ratios the strength iscontrolled jointly by the confined strength and the unconfined strength.5.2.5.1 Mohr-Coulomb-Navier’s TheoryLundborg (1968) presented a paper on the strength of rock and investigated the various failure criteriathat have been applied to rock. One of the topics covered was Mohr-Coulomb-Navier’s theory (1773), wherethe strength of a plane through a sample subjected to a triaxial stress condition can be determined using Mohr’scircle diagrams. A typical Mohr’s circle diagram is shown in Figure 71. Coulomb (1773) presented thefollowing equation to describe the shear strength of a material which represents the rupture envelope of amaterial.tn=t0+J.Lp (45)where:— Shear strength (MPa)— Cohesive shear strength (MPa)— Coefficient of slopePo — Normal stress on the plane being considered (MPa)The second term resembles a frictional effect, yet when dealing with rock there is no discrete surfacebeing considered. Lundberg (1968) states that this description of strength shows satisfactory agreement forrock at low pressure and is often quoted when describing the variation of rock strength at a normal pressure.5.2.5.2 Frictional Effect ofMine PillarsIn Section 5.2.3.2, it was shown that the width / height ratio of a pillar can be approximated by anaverage pillar confinement term, C , that is the average a3 divided by the average a. Mohr’s circle diagramscan be constructed for any average pillar confinement and a friction term, kt2ppa (K), can be determined.Figure 71 is a Mohr’s circle construction showing how average pillar confinement, Cpa,,, is related to the anglebeta. The friction term, kappa (ic), can be defined from basic trigonometry by Equation 46.K = tan[cos1(1 — CPaV)] (46)1+ Cpav121Cl)C/)ci)(1)01)(I)(sg1 —sig3)cos(beta)2 (sigl—sig3) — (1—Cpov)(siq1—sg3) (sigl+sig3) — (1+Cpav)__________÷ sig32Cpcv = sig3/ig1SigmaFigure 71: Construction of a Mohr’s failure envelope with reference to average pillar confinement, Cp.where:IC — kappa, mine pillar friction termCpa, — Average pillar confinementAs pillar confinement increases, the slope of the strength envelope constructed using Mohr’s circlesdecreases. However, as the pillar width I height ratio increases, the strength of a mine pillar will also increase.It was subsequently determined that the friction term kappa, (K), could be used in “The Confinement Formula”to estimate pillar strength. Figure 72 is a construction of a series of Mohr’s strength envelopes for variousvalues of Cp showing how, for increasing values of Cp, the slope of the strength envelope changes. Figure73 shows the relationship between kappa and Cp for the full range of Cpa,, values. An interesting point ofnote is that Wilson (1972) also derived a term that is similar to kappa. His term, called the triaxial stresscoefficient (Equation 22), was dependent on the friction angle of pillar material only and was used in histheoretical strength formula (Section 2.2.2.1).122Sigma Sigma 0iCpav = 0.1 Cpav = 0.2° SigmaCpav = 0.3Figure 72: Construction of Mohr’s failure envelopes for differing values of average pillar confinement, Cp.2.52.0--1.5cos(beia) — (1-Cpav) / (1+Cpav)0.50.0 I I I I I0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40CpavFigure 73: Mohr-Coulomb pillar strength envelope for any value of average pillar confinement, Cpa.,.1235.2.5.3 The Confinement FormulaA new strength formula, “The Confinement Formula”, is proposed for hard rock mine pillars in theform of Equation 47.P3=(K.UCS).(Ct+C21 ) (47)where:P Pillar strength (MPa)K Pillar strength size factor — 0.44 (as previously determined)UCS Unconfined compressive strength pillar material, 50 mm sample (MPa)C1, C2 — Empirical rock mass constantsIC = Pillar confinement / friction term (kappa)The basic form of this equation is based upon a modifications to Equation 45 after investigating howthe combined database performed for various equation configurations. The empirical constants “C1” and “C2”determined for the pillars in the combined database are 0.68 and 0.52 respectively. These values weredetermined by varying the coefficients in order to maximize the predictiQn success rate for the combineddatabase. The strength equation for the combined database can now be represented as Equation 48.= (0.44. UCS).(0.68 + 0.52K) (48)where:Pillar strength (MPa)UCS — Unconfined compressive strength pillar material, 50 mm sample (MPa)IC Pillar confinement / friction term (kappa)This formula is represented graphically in Figure 74 along with the all the case histories in thedatabase. This formula is also presented with the width / height ratio replaced by Cp on the horizontal axis ofthe graph in Figure 75. The new formula is plotted against the “Power Shape Effect” and “Log-Power ShapeEffect” formulae for a factor of safety of 1.0 in Figure 76. We can see that for width / height ratios greater than0.5, the proposed strength formula is essentially the same as both of the refined empirical formulae. Figure 77is a confinement plot of a comparison of the various strength formula.It is concluded that this form of equation could be used to predict pillar strength of rock types otherthan “hard” rock. The empirical coefficients “C1” and “C2”, as derived, are for hard rock mine pillars only.The general shape of the confinement curve suggests that different coefficients could be derived for differentrock types, such as coal, to aid in pillar strength estimation.1240.7 -0.6 -Clj -0.5z0.3 -I.< 0.20.1 -0.0- i I I I I I I0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2Pillar Width / Height RatioPillar Stability ClassificationFailed • Unstable A StablejFigure 74: Stability graph for “The Confinement Formula” as described by Equation 48, plotted along withall of the case histories in the combined database.0.70.6•FSi.00.1Cpvcofl-0.6Ps — 0.44*UCS*(0.68 + 0.52k) 178 observations0.0 I I I I I I0.00 0.04 0.08 0.12 0.16 0.20 0.24 0.28 0.32CpavPillar Stability ClassificationFailed • Unstable A StabZFigure 75: Confinement graph for “The Confinement Formula” as described by Equation 48 with pillarwidth / height ratio replaced by average pillar confinement on the x-axis and plotted along withall of the case histories in the combined database.125F.S. — 1.0- ; : -•.;A • AA•F.S. — 1.4 - -•Cpav coeff — 0.46178 observationsAA AAAA AAA AAPs — 0.44*UCS*(0.68 + 0.52k)0.70.6o 0.5‘1,0.4C 0.3UI..a)0.20.10.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8Pillar Width / Height RatioStrength FonnulaeConfinement Formula —°--- Log - Power Shape Effect —--- Refined Power Shape Effect]Figure 76: Stability graph comparing the newly developed pillar strength formulae.0.73.2C’, 0.5‘a0.4Q.. 0.34)<0.20.10.04 0.08 0.12 0.16 0.20 0.24 0.280.00.00CpavStrength Formulae[—a— Confinement Formula —°-- Log - Power Shape Effect —a—- Refined Power ShapeFigure 77: Confinement graph comparing the newly developed pillar strength formulae.0.321265.3 SUCCESS OF THE EMPIRICAL AND PROPOSED FORMULAEIn assessing the validity of “The Confinement Formula”, a statistical analysis has been performed tomeasure the success of each of the potential strength formulas. The individual “Linear Shape EffectFormulas”, Figure 63 - Figure 66, for each of the databases is used as the baseline for comparison with each ofthe strength formulae. This baseline data represents the optimum results we could expect to achieve whenpredicting pillar strength for the case histories in the combined database.5.3.1 Success Matrix MethodologyA success matrix method was developed whereby the number of each pillar stability classification, andthe stability classification that they fell under on the graph, were tabulated. The success matrix methodology toassess the prediction success of each of the respective strength methods is presented in Table 31 and Table 32.The optimum pillar stability classification method will be one where all of the pillar stabilityclassifications fall in their predicted (or calculated) categories, represented by the central diagonal of Table 31.The next best result we could hope for is that a pillar falls in a higher stability classification than where itshould be: i.e. classified stable pillar falling in predicted unstable or failed range. The impact would be aconservative design and at worst would result in loss of ore but not the potential catastrophic results of pillarfailure. The worst scenario would be one in which a classified pillar falls in a lower stability classification thanpredicted: i.e. a classified failed pillar being predicted stable or unstable. In this case, failure would not beexpected and could have severe consequences both from a safety and stability standpoint. In determining whereeach of the pillars fell on the stability graph, a margin of 5% was used to define the boundaries between each ofthe stability classification regions. Pillars that fell in the 5% boundary region were deemed to be in either ofthe adjacent regions.Table 31: Pillar strength prediction success matrix methodology.Predicted -> Predicted Predicted PredictedActual Failed Unstable StableClassified Failed correct one class two classesunconserv alive unconservativeClassified Unstable one class correct one classconservative unconservativeClassified Stable two classes one classconservative conservativeTable 32: Summary of success matrix prediction methodology.Statistic Classification +2 Classes +1 Classes Perfect -l Classes -2 ClassesSum of Each group £ two classes Z one class £ Correct E one class Z two classesfrom Table 31 conservative conservative unconservative unconservative127The prediction statistics are interpreted from the figures as follows. The sum of the columns for eachmethod totals 100%. A perfect prediction formula would have a column height of 100% in the perfect columnand 0% in the other 4 columns. The assessment of the “best” formula, from a prediction standpoint whencomparing different formula, will be the formula that has the highest prediction percentage in the perfectcolumn. In addition to it being desirable to have a large percentage of pillars in the perfect column, it is moredesirable to have the additional pillars in the “+“ classes on the left side of the perfect column rather than the“-“ classes on the right side of the perfect column. These “+“ columns represent pillars that have beenpredicted conservatively, i.e. pillar predicted failed but classified as either unstable or stable.5.3.2 Success ofConfinement and Refined Empirical FormulaeThe results of the predictability analysis of the combined database are presented in Figure 78 and showthat “The Confinement Formula” marginally outperforms both of the two refined empirical formulae andperforms equally as well as the “best” “Linear Shape Effect Formula” for each of the individual databases,when derived in isolation. This supports the concept that shear strength theory is applicable to the strengthdetermination of mine pillars.1UJ7050%30%------I I I+1 Class Perfect -1 Class -2 ClassesPredicted Pillar ClassificationRefined Power Formula Log-Power Formula Confinement Formula [] Best Individual Suc)Figure 78: Pillar strength prediction success statistics for the newly developed formulae for all data in thecombined database.Based upon 178 observations90%80%70%60%10%20%+2 Classes1285.3.3 Success ofConfinement Formula against Past FormulaeIn order to assess the effectiveness of the new strength formulae, the prediction success of “TheConfinement Formula” is compared to the more prevalent strength formulae used in the past, and to the Hock& Brown (1980) pillar curves. The pillar database is presented along with curves representing the work ofHock & Brown (1980), Salamon & Munro (1967), Hedley & Grant (1972), Obert & Duvall (1967), andBieniawski (1975) in Figure 79 to Figure 86. These stability plots were analyzed according to the successmethodology described previously. The success plots for these curves are plotted in comparison to the successplot for “The Confinement Formula” in Figure 80 to Figure 87. We can see that in each case “TheConfinement Formula” predicts the stability of the pillars in the database with greater success than thepreceding formula.53.3.1 Hoek & Brown (1980) Pillar CurvesFigure 79 is the combined database plotted with the Hock & Brown (1980) pillar curves plotted forrock mass qualities of good to very-good. It is clear from this figure that the Hock & Brown (1980) pillarcurves are overly optimistic for pillars that have width / height ratios greater than 0.5 and a rock mass that islocated between good and very-good quality. It is also clear that the use of Hock & Brown curves are extremelysusceptible to the assessment of rock mass quality, evident by the wide variation of the strength for good andvery-good rock masses. Figure 80 shows the success statistics for the Hock & Brown (1980) pillar curvesplotted against “The Confinement Formula”. This figure shows how the variability in the prediction success ofthe Hock & Brown (1980) curves is large when compared to “The Confinement Formula”.53.3.2 OtherMethods with Original Strength Size CoefficientFigure 81, Figure 82 and Figure 83 are plots of the combined database for the strength methodsproposed by Salamon & Munro (1967), Hedley & Grant (1972) and Bieniawski (1975) respectively utilizing astrength size coefficient, “K”, proposed by various researchers of 0.7. Figure 84 is the prediction success plotfor these methods plotted against “The Confinement Formula”. This figure show that of these methods,Salamon & Munro’s (1967) method provides the highest prediction success. It can be noted that Bieniawski’s(1975) method provides extremely optimistic estimates of pillar strength.53.3.3 RevisedMethods with New Strength Size CoefficientThe prediction success has been determined for a number of methods using the newly developedstrength size factor, “K”, of 0.44. Figure 85 and Figure 86 are plots of the original “Power Shape EffectFormula” and the original “Linear Shape Effect Formula”, utilizing the empirical constants determined byBieniawski (1975) and Obert & Duvall (1967), respectively. Figure 87 is a plot of the success of these methodsplotted against “The Confinement Formula”. It can be seen that these methods all perform reasonably well.This can be attributed to the fact that the strength size coefficient, “K”, previously determined has been used.129The “Power Shape Effect Formula” performs the best of these methods which is understandable as theoptimum power coefficient previously determined, of 0.45, is comparable to a square root which has a powercoefficient of 0.5.5.4 CHAPTER SUMMARYRefined empirical coefficients have been derived for the “Power Shape Effect Formula” for minepillars, using the combined database of information collected at Westmin Resources Ltd. and other publishedhard rock pillar data. Two new empirial strength formulae, “The Log-Power Shape Effect” and “TheConfinement Formula”, have been developed. “The Confinement Formula” formula follows the shear strengthapproach of Coulomb (1773) and is determined to be as successful as even the most refined empirical formulaefor predicting the strength of the pillars that make up the combined database. Pillar width / height ratio isreplaced with the average pillar confinement in “The Confinement Formula” in order to estimate pillarstrength.It is concluded that pillar instability begins to exhibit visible characteristics at a factor of safety of 1.4,assuming that pillar failure occurs at a factor of safety of 1.0. Pillar stability classification has been used tosuccessfully assess the strength of pillars and it was determined that, using “The Confinement Formula”, noobserved stable pillars fell in the failed region on the graph. However observed failed pillars did fall in thestable region of the graph. “The Confinement Formula” predicts the stability of the pillars in the database aswell as the refined empirical formulae. In addition it was determined that these new formulae performsubstantially better than the methods that have preceded them.130F.S. — 1.4 F.S. — 1.0 F.S. — 1.40.7- FS. - i.o/ — - - FS. -1.0/ — A FS. - 1.40.6. — - --:•-- AA/ - — • A- A AV•/‘ A —AA Ae,•__//. A • A AV.Good / • / /A -RMR // / .‘ - - AA A A A0.2---/- A AV.Good-Good1 +—L A A A A A0.1+Good RMR178 observations0.0 I I I I I I0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8Pillar Width / Height RatioPillar Stability ClassificationFailed • Unstable A Stable—_VeryGood RMR — - Very Good - Good RMR — - Good RMRFigure 79: Stability graph with the Hoek & Brown (1980) pillar curves for good to very-good rock massesplotted over all of the data in the combined database.÷1 Class Perfect -1 ClassPredicted Pillar ClassificationGood RMR Good - Very Good RMR Very Good RMR Confinement Formula JFigure 80: Pillar strength prediction success statistics for “The Confinement Formula” and the Hoek &Brown (1980) pillar curves for differing rock mass quality.1313.2IBased upon 178 observations+2 Gasses -2 ClassesF.S. — 1.4•-— AAA A•0.5 0.75A Ps0.7*UCS*(w ,h )160140120100800.4020.•-- -__F.S. — 1.4•• --— A• --• A AAA A-AA - -Ps0.7*UCS*(w /h )A00 20 40 60 80 100 120 140 160Calculated Pillar Strength (MPa)Pillar Stability CalculationFailed • Unstable A StableFigure 81: Stress - strength plot for all data in the combined database using the Salamon & Munro (1967)strength formula and “K” O.7.UCS.140120COl0080604020•• •- - - -• A0E0 20 40 60 80 100 120 140 160Calculated Pillar Strength (MPa)Pillar Stability Calculation• Failed • Unstable A StableFigure 82: Stress - strength plot for all data in the combined database using the Hedley & Grant (1972)strength formula and “K” — O.7.UCS.1320.70.6 -C,,0.5z0.40.3. 0.20.1 -0.0 —0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2Pillar Width / Height RatioPillar Stability ClassificationFailed • Unstable A StableFigure 83: Stability graph for Bieniawski’s (1975) strength formula using “K”.’O.7.UCS plotted over all ofthe data in the combined database.Based upon 178 observationsSalamon & Munro (1967) and Hedley &Grant (1972) based upon 162 observations.F.S. - 1.0 - F.S. - 1.4— — AV.._.__•• A • A• . AF3_.• ; • • A A A— A— A AA A••A • A A A• AAAAA AAAA A• A AAA A A AA APs 0.7 *UCS * (0.64 + 0.36 (w/h)).178 observationsI+2 Classes +1 Class PerfectPredicted Pillar Classification-1 Class -2 Classes( Bieniawski (1975) E Hedley & Grant (1972) Salamon & Munro (1967) Confinement Formula JFigure 84: Pillar strength prediction success statistics for “The Confinement Formula” vs. Bieniawski(1975), Hedley & Grant (1972), and Salamon & Munro (1967) using “K”-O.7.UCS.1330.70.6 -0.5 -0.4.. 0.30.20.10.0F.S. = 1.0•AAA A - AAA AAA A0.5Ps0.44*UCS *(w/h)F.S. — 1.4178 observations0.0 0.4 0.8 1.2 1.6 2.0Pillar Width / Height RatioPillar Stability Classification2.4 2.8 3.2[ aiiec • Unstable A Stabl)Stability graph for the original “Power Shape Effect Formula” using “K”-O.44.UCS plotted overall of the data in the combined database.Figure 85:0.7F.S.-l.0AAA ABieniawski (1975) Oben & Duvall (1961)Ps 0.44 *UCS * (0.64 + 0.36 (w/h)) Ps 0.44 *UCS * (0.778 + 0.222 (w/h)) 178 observations0.0 I I I I I I0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2Pillar Width! Height RatioPillar Stability Classificationf’ Failed • Unstable StableL - Bieniawski (1975) Obert & DuvalI (1967)Figure 86: Stability graph for Obert & Duvall’s (1967) and Bieniawski’s (1975) strength formulae using“K”=O.44.UCS plotted over all of the data in the combined database.134Based upon 178 observations90%-80%70%-4o%30%-20%-10%-I —-2 Classes+1 Class Perfect -l ClassPredicted Pillar ClassificationI+2 ClassesOrg. Power Shape Effect Bieniawski (1975) Obert & Duvall (1967) Confinement FormulaFigure 87: Pillar strength prediction success statistics for “The Confinement Formula” vs. the original;“Power Shape Effect Formula”, Bieniawski (1975), and Obert & Duvall (1967) usingO.44.UCS.1356. SUMMARYAND CONCLUSIONS6.1 SUMMARYThis chapter summarizes the practice of using rock mechanics data obtained through observation,measurement and modelling to implement methods for pillar strength estimation. The previous chapters enablemining practitioners to develop an understanding of the critical variables that affect and ultimately control thebehaviour of pillars in hard rock mining operations. The methods that have been introduced in the precedingchapters have given us a pillar design guidelines as follows:• pillar stability assessment using pillar stability classification• numerical modelling strategies• assessment of full-size unconfined compressive strength size factor, “K”• a refined purely empirical strength formula, “The Log-Power Shape Effect Formula”• a new pillar strength hypothesis, “The Confinement Formula”These design methods are intended to be used for hard rock pillars with rock mass ratings of between60 and 85 and to be implemented in operations where there exists a knowledge of pillar stability and loadingconditions. The pillar strength formulae should be used with caution in mine operations where a local databasefor calibration is not available.It has been shown that the results from numerical modelling for stress analysis can be highly variable.Because of the inherent variability in the pillar design variables, caution must be exercised in implementingdesign guidelines of this nature in new mining operations. It is therefore recommended that a factor of safetyof 1.4, representing the transition from a stable to an unstable pillar, could be used initially for pillar design.After the pillar design guidelines have been calibrated to a particular operation, this methodology can be usedwith a higher degree of confidence and with reduced factors of safety, for designing mine pillars. It is alwaysdesirable to design as close to a factor of safety of 1.0 as possible while maintaining the maximum level ofconfidence in design.6.1.1 Current Pillar Sfrength Determination MethodsNumerous researchers have proposed strength formulae. These formulae have been shown to beinapplicable for use in hard rock mining operations. Much of this work was derived in soft rock coaloperations with pillars that have significantly higher width / height ratios than comparable hard rock minepillars. Wherever possible, the data from hard rock mine operations was included in the development of thenew pillar strength formulae.6.1.2 Assessment ofPillar FailureThe pillar stability classification method developed can be used to assess a pillar’s current stabilitycondition. Pillars that are classified with a high stability classification number will have less load bearing136with numerical modelling results to determine if a pillar will perform as desired through the life of a miningoperation. The anticipated design guidelines suggest that a factor of safety of 1.2, representing a moderatelyunstable pillar, could be used for design in situations where limited support is desired. Pillars that have a factorof safety of 1.0 or less will invariably fail and require significant support as well as having limited load bearingcapacity.6.1.3 Pillar Strength Estimation MethodsIn addition to the collected data, published case histories data from around the world have beensubsequently used to develop improved empirical strength formulae and a new empirical strength hypothesis.These relationships are shown in Figure 68, Figure 70, & Figure 74 and are determined to successfully predictthe stability for the combined database with a higher success than any preceding pillar strength estimationmethods. Two new strength formulae have been derived in this thesis, “The Log-Power Shape EffectFormula” and “The Confinement Formula”. Both of these formulae plot in essentially the same location onthe pillar stability plot, however “The Confinement Formula” predicts pillar strength from a more theoreticalapproach.6.1.3.1 RefinedEmpirical Strength FormulaeRefined empirical strength constants for the “Power Shape Effect FQrmula” have been derived forhard rock mine pillars to advance the state-of-the-art in mine pillar design. It was .determined that a variablepower exponent, based upon average pillar confinement, provided the optimum prediction success for thedesign of mine pillars when compared to previous pillar strength estimation methods. This formula is calledthe “Log-Power Shape Effect Formula”.6.1.3.2 A New Strength HypothesisA new strength hypothesis pertaining to the strength of mine pillars has been proposed and is verifiedby the combined hard rock pillar database. “The Confinement Formula” proposes that pillar strength isdependent on average pillar confinement as a driving term. This confinement term can be derived from thewidth I height ratio of a mine pillar or potentially from numerical modelling. The use of pillar confinement isan area where further investigation should be applied. Average pillar confmement was derived from two-dimensional modelling using variable extraction ratios. For extraction ratios of between 65% and 90%, thevariability in the value of the Cp coefficient represents a potential error of ±5% and the author concludes thatfor most mining situations this potential error is acceptable.“The Confinement Formula” can be used in two situations; to design pillars in a new mine that has alimited database, and within an existing mine where there is experience related to how pillars will performunder load. In the existing mine situation, “The Confinement Formula” can be used to design new pillarsbased on the behaviour of existing pillars and to make assessments of the stability of existing pillars. In a new137mine situation, some form of calibration must be performed before a strength formula can be used withconfidence.6.1.4 Pillar Strength CalibrationIn order to use any pillar strength estimation method, it must be calibrated to existing mine pillars.Calibration using pillars that are at a known stability rating and have had loads determined through numericalmodelling can be used to calibrate the “The Confinement Formula” and give an operator an indication ofwhether or not mine pillars are performing as predicted. Calibration can be accomplished in two ways.Numerical modelling input parameters, i.e. in-situ stress values, can be altered so that the predicted loads onthe classified mine pillars place those pillars in the correct stability region on the pillar stability graph. Thesecond approach is to modify the unconfined compressive strength, within acceptable ranges, of the intact pillarmaterial so that classified pillars fall within their predicted stability range6.2 FUTURE WORKIn order to advance the results that have been presented in this thesis, the author recommends that anyfuture work in pillar strength estimation and failure assessment should be directed to the following topics:• Verification that the methodology of pillar stability classification is valid in other miningoperations. This should be accomplished through the monitoring of pillars from the originalcreation, throughout their life.• Follow-up work to verify that the use of average pillar confinement reasonably represents theresults of two dimensional modelling presented here. Three-dimensional modelling of variouspillar shapes would be required.• Collection of additional case histories to be added to the pillar stability database and the values ofthe empirical strength constants revised if required.6.3 CONCLUSIONSThe above pillar strength determination methods will enable hard rock mine operators to improvepillar design methodology which will increase the design success and level of safety in areas where pillars areused. It should be noted that this approach is applicable only to the progressive failure of pillars that are notsubject to other failure modes such as discrete failure planes transecting pillars. The methods applied in thisthesis have shown that the estimation of pillar strength is not a “Black Art”. Various stress determination andpillar stability classification methods have been applied to a large database, which has subsequently been usedto develop the results presented in this thesis. 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Mi J. 169 No. 5, 78-84.Starfield, A.M., Wawersik, W.R. 1968. Pillars as structural components in room and pillar mines. In Basic andapplied rock mechanics, Proc. 10th U.S. symp. rock mech., Gray (ed.), 793-809. New York: Soc. Mm.Engrs, AJME.Steart, F.A. 1954. Strength and stability of pillars in coal mines. J. Chem. Metall. Mm. Soc. S. Afr. 54, 309-35.Stephansson, 0. 1985. Pillar design for large open stoping. Proc. mt. symp. on large scale undergroundmining, Almgren (ed.), 185-97. Lulea: Centek.Szwilski, A.B. 1980. Influence of coal rib pillar width on the stability of strata around the face end andgateroad in longwall mining. In The state of the art in rock mechanics, Proc. 21st U.S. symp. rock mech.,Summers (ed.), 285-98. Rolla: Univ. Missouri.Szwilski, A.B. 1982. Sizing of chain pillars around longwall panels. Proc. 1st mt. conf. on stability inunderground mining, Brawner (ed.), 535-58. New York: Soc. Mi Engrs, AIME.Townsend, P. 1982. Pillar stability at Elliot Lake, a comparison of empirical and analytical methods ofprediction. In Rock breaking and mechanical excavation, Proc. 14th Can. symp. rock mech.,Baumgartner (ed.), 133-7. Montreal: Can. Inst. Mm. Metall.Tsur-Lavie, Y., Denekamp, S. 1977. Application of stress measurements to the solution of undergroundstability problems at the Timna Copper Mine. Proc. mt. symp. on field measurements in rock mech.,Zurich, Kovari (ed.), 209-17. Rotterdam: A.A. Balkema.Tsur-Lavie, Y., Denekamp, S.A. 1982. Size and shape effect in pillar design. Proc. symp. on strata mech.,Newcastle upon Tyne, Farmer (ed.), 245-8. Amsterdam: Elsevier Sci. Pub. Co.van Heerden, W.L. 1970. Stress measurements in coal pillars. Proc. 2nd congr. mt. Soc. Rock Mech. 2, 4.16.Belgrad: “Privredni Pregled”.145van Heerden, W.L. 1974. In-situ determination of complete stress-strain characteristics of 1.4 m. squarespecimens with width to height ratios up to 3.4. CSIR Research Rept, ME 1265.Von Kimmelman, M.R., Hyde, B., Madgwick, RJ. 1984. The use of computer applications at BCL Limited inplanning pillar extraction and the design of mining layouts. In Design and performance ofundergroundexcavations, mt. Soc. Rock Mech. symp., Brown & Hudson (eds.), 53-63. London: Brit. Geotech. Soc.Wagner, H. 1974. Determination of the complete load-deformation characteristics of coal pillars. In Advancesin rock mechanics, Proc. 3rd congr., mt. Soc. Rock Mech. IIB, 1076-81. Washington, DC: Nat. Acad.Sci.Wagner, H. 1975. The application of rock mechanics principles to strata control in South African mines. Proc.10th Can. symp. rock mech., Kingston, 247-80. Dept. Mi Engrng, Queens Univ.Wagner, H. 1980. Pillar design in coal mines. J. S. Afr. Inst. Mm. Metal!. 80, 37-45.Wagner, H. 1983. Principles of support of tabular excavations. In Rock mechanics in mining practice, Budavari(ed.), 15 1-71. Johannesburg: SAIMM.Walker R.R. Westmin Resources’ Massive Sulfide Deposits, Vancouver Island. in Mineral deposits ofVancouver island Fleming 3., Walker R. and Wilton P. eds. (GAC-MAC-CGU, 1983), 5-19.Wang, F.D., Panek, L.A., Sun, M.C. 1971. Stability analysis of underground openings using a Coulomb failurecriterion. Trans. Soc. Mm. Engrs, AIME 250, 317-21.Westmin Resources Ltd. 1993, Ground stability guidelines for the extraction of barrier pillars in hard rockmines. Project Number: 1-9045. Canadian Centre for Mineral and Energy Technology.Whittaker, B.N., Singh, R.N. 1979. Design and stability of pillars in longwall mining. Mm. Engr 138, 59-73.Wiles, T.D., 1988. Accuracy and applicability of elastic stress analysis methods in mining. In Rock engineeringfor underground excavations. Proc. 15th Can. symp. rock mech., Toronto, 207-19.Toronto: Dept. Civ.Engng, Univ. Toronto.Wiles, T.D. 1990. Predication of ground instability using numerical modelling. Can. Inst. Mm. Metal!., 92ndAGM, Ottawa, Paper 46.Wiles, T.D. 1991. Map3D - users manual.Wilson, A.H. 1977. The effect of yield zones on the control of ground. Proc. 6th mt. strata control conf., Banif,Paper 3. Ottawa: Can. Center Mm. Energy Tech.Wilson, A.H., 1972. Research into the determination of pillar size, part 1: A hypothesis concerning pillarstability. Mm. Engr 131, 409-416.Wilson, A.H., 1982. Pillar stability in longwall mining. In State-of-the-art ofground control in longwallmining and mining subsidence, Chugh & Karmis (eds.), 85-95. New York: Soc. Mi Engrs, AIME.Zern, E.N. 1926. Coal miners pocketbook, 11th edition, New York: McGraw Hill.Zienkiewicz, O.C. 1977. The finite element method. London: McGraw-Hill.146APPENDIXA - TWO - DIMENSIONAL PARAMETRICMODELLING RESULTSThis appendix contains results of the two-dimensional parametric modelling session that wasperformed in conjunction with the determination of the “Average Pillar Confinement” relationship in chapter5. Figure 88 - Figure 93 are factor of safety plots of various pillar width / height ratios using the Hock andBrown failure criteria and “m” and “s” values of 10 and 0.1 respectively. The model unconfined compressivestrength was modified in each case such that a zone of failure existed in each of the models. In all cases thepredicted zone of failure initially materialized at the mid-height of the pillar. Figure 94- Figure 105 are plotsof major and minor principal stresses in the modelled pillars. Contour intervals were chosen such that theaverage pillar stresses could be calculated with the greatest ease at the mid-height of the pillar, the point ofinitial predicted failure. The stress values presented do not represent actual mining conditions. The valuesused were for comparative purposes only in the derivation of the average pillar confinement.147Will = 0.25TENSION 9.7Horizontal scale exaggerated by 6%Modelled extraction ratio 99.5%1. • 3G..W/H = 0.5TENSION Ø.8 9.9 i.ø L•1 j. j.3I I • •Figure 88: Plot of pillar factor of safety for width! height ratios of 0.25 and 0.5. Hoek & Brown (1980) rockmass constants used: m — 10, s = 0.1. Modelling performed using Examine2D.L.i i.2.e 0. 9 1.00..148W/H = 0.75Horizontal scale exaggerated by 6%Modelled extraction ratio = 99.5%W/H = 1.09.89 9.99 1.99 1.19 1.29 1.39• •Figure 89: Plot of pillar factor of safety for width I height ratios of 0.75 and 1.0. Hoek & Brown (1980) rockmass constants used: m 10, s = 0.1. Modelling performed using Examine2D.TENSION 9.79 9.89 9.90 1.99 1.19 1.20 1.39I ..TENSION 9.79I —149W/H = 1.25TENSION 9.7wHorizontal scale exaggerated by 6%Modelled extraction ratio 99.5%.BG 1. i.i 1. i.31 Vii ‘ri’1 • •Figure 90: Plot of pillar factor of safety for width I height ratios of 1.25 and 1.5. Hoek & Brown (1980) rockmass constants used: m — 10, s — 0.1. Modelling performed using Examine2D.W/ll = 1.5TENSION ø.6I — a.7 .8O.9g 1.GG j.jgF IVIR Vi VI ••150W/H = 175IHorizontal scale exaggerated by 6%Modelled extraction ratio — 99.5%Will = 2.0Ià.70 .9ø i.O 1. 1.2IWIll.I!:::ITENSION .6W .Ba .9G i.aa I.I 1 •Figure 91: Plot of pillar factor of safety for width / height ratios of 1.75 and 2.0. Hoek & Brown (1980) rockmass constants used: m — 10, s 0.1. Modelling performed using Examine2D.TENSION4151W/H=2.5Horizontal scale exaggerated by 6%Modelled extraction ratio = 99.5%WIN = 3.0TENSION I. 1.L 1.2wI ITENSION 9.5 g.7’.BO i. i.iFigure 92: Plot of pillar factor of safety for width / height ratios of 2.5 and 3.0. Hoek & Brown (1980) rockmass constants used: m 10, s = 0.1. Modelling performed using Examine2D.152W/H=3.5Horizontal scale exaggerated by 6%Modelled extraction ratio — 99.5%Will = 4.0TENSION 9.5 .6U .99 1. 1.1wF I —TENSIONI—_g.7g .$G ø.9g 1. 1.iFigure 93: Plot of pillar factor of safety for width I height ratios of 3.5 and 4.0. Hoek & Brown (1980) rockmass constants used: m — 10, s — 0.1. Modelling performed using Examine2D.153Sigma 1Sigma 3Horizontal scale exaggerated by 6%Modelled extraction ratio — 99.5%TENSION aee.ew 859.UU 9G.US 95U.SO Loee.se iese.oe nea.ceI I aFigure 94: Principal stress plots for pillar width I height ratio of 0.25 using two-dimensional boundaryelement modelling. Modelling performed using Examine2D.TENSION 0.99 1.50 3.00 4.50 6.00 - 7.50 9.90I I154TENSION 0.90 1.50 3.99 4.50 6.09 7.59I 1Horizontal scale exaggerated by 6%Modelled extraction ratio = 99.5%Figure 95: Principal stress plots for pillar width / height ratio of 0.5 using two-dimensional boundary elementmodelling. Modelling performed using Examine2D.Sigma 1Sigma 34TENSION 999.00 958.99 1909.09 1050.08 1199.09 1150.09 1299.99I I •19 • 99I..155Sigma 3Horizontal scale exaggerated by 6%Modelled extraction ratio 99.5%Sigma 1TENSION i75. 11B. Ii2. i175.ITENSION e.uuI — ieu 2Ø. 39. 4e.90 I • •Figure 96: Principal stress plots for pillar width / height ratio of 0.75 using two-dimensional boundaryelement modelling. Modelling performed using Examine2D.156Sigma 3Horizontal scale exaggerated by 6%Modelled extraction ratio = 99.5%Sigma 1aTENSION 19.U 1e5.Ø9 1i5.O 125.9 i3W.I ITENSION Z5.WU 75. 1Z5.Figure 97: Principal stress plots for pillar width I height ratio of 1.0 using two-dimensional boundary elementmodelling. Modelling performed using Examine2D.157Horizontal scale exaggerated by 6%Modelled extraction ratio — 99.5%ITTENSION 1109.00 1150.90 1290.09 1259.90 1300.00 1350.00 1400.I I . .Sigma 1Sigma 3Figure 98: Principal stress plots for pillar width / height ratio of 1.25 using two-dimensional boundaryelement modelling. Modelling performed using Examine2D.158Horizontal scale exaggerated by 6%Modelled extraction ratio 99.5%TEION 1159. i25.9U i39.ø i35U.9 I4UU.I I ..::: iI hihI ••Sigma 1Sigma 3Figure 99: Principal stress plots for pillar width / height ratio of 1.5 using two-dimensional boundary elementmodelling. Modelling performed using Examine2D.ITENSION 49.9w S9.ø iae. i6.U 2ø.G9I I ..: ....I-. ‘ij159Sigma 3Horizontal scale exaggerated by 6%Modelled extraction ratio — 99.5%Sigma 1TENSION 11.9 1.15.9 i2W. 125.ø 139.9 i35.I I • •I 1TENSION ø.9G 6U.G 24.UU 3.8ø 36.øI I —Figure 100: Principal stress plots for pillar width I height ratio of 1.75 using two-dimensional boundaryelement modelling. Modelling performed using Examine2D.160Sigma 1— IHorizontal scale exaggerated by 6%Modelled extraction ratio 99.5%Sigma 3TENSION jg5g.9 1i2. i.L9O. ia6.Bg 133.O i49.e i47ø.I I • •Figure 101: Principal stress plots for pillar width / height ratio of 2.0 using two-dimensional boundary elementmodelling. Modelling performed using Examine2D.ITENSION 9. 79. 2i.SS 2Be. 35.B 42W.UI I161Sigma 1Horizontal scale exaggerated by 6%Modelled extraction ratio = 99.5%Sigma 3TENSION 195.U ii5. i25.9 i35.99 145.U 155.9 i65.I I—TENSIONI I —LI=1Figure 102: Principal stress plots for pillar width / height ratio of 2.5 using two-dimensional boundary elementmodelling. Modelling performed using Examine2D.9U.eg i8.e 54.9162Sigma 3Horizontal scale exaggerated by 6%Modelled extraction ratio 99.5%Figure 103: Principal stress plots for pillar width I height ratio of 3.0 using two-dimensional boundary elementmodelling. Modelling performed using Examine2D.Sigma 1. .TENSION 15Q. ii75. L3GW. i425.ø 155.G 1675. L899.TENSION ø.gI I — 36G.u 4Z.US 54.a163Horizontal scale exaggerated by 6%Modelled extraction ratio — 99.5%Sigma 1Sigma 3ITENSION 98. iB9.O 27.U 36.BI IFigure 104: Principal stress plots for pillar width / height ratio of 3.5 using twodimensional boundary elementmodelling. Modelling performed using Examine2D.TENSION 1.25.e 145.ø i65ø. 1B5. 225W.I I •I164Sigma 1Horizontal scale exaggerated by 6%Modelled extraction ratio 99.5%Sigma 3tENSION 15ø. 125.I I-1.i45G.9 i65G9 1B59. 2Ø5. 225.•TENSION U.I I —Figure 105: Principal stress plots for pillar width / height ratio of 4.0 using two-dimensional boundary elementmodelling. Modelling performed using Examine2D.1BU. 4OU.U165APPENDIXB - METRIC TO IMPERICAL CONVERSIONOF UNITS USED INTHESISMETRIC IMPERIAL1 metre (m) = 3.28 feet (ft) = 39.3 inches (in)1 centimetre (cm) = 0.0328 feet (ft) = 0.394 inches (in)1 millimetre (mm) = 0.00328 feet (ft) = 0.0394 inches (in)1 megapascal (MPa) = 145.5 pounds force per square inch (psi)1 gigapascal (GPa) = 145,500 pounds force per square inch (psi)1 meganewton (MN) = 224.7 x i03 pounds force (lbf)166

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