Rock Mechanics.pdf - Mining and Blasting

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CYLINDRICAL POLAR CO-ORDINATES coincides with the x, y plane. The elastic constitutive equations for this material are given by ⎡ ⎢ ⎣ εxx εyy εzz xy yz zx ⎤ ⎥ = ⎥ ⎦ 1 ⎡ 1 ⎢−1 ⎢ ⎢−2 ⎢ E1 ⎢ 0 ⎣ 0 −1 1 −2 0 0 −2 −2 E1/E2 0 0 0 0 0 2(1 + 1) 0 0 0 0 0 E1/G2 0 0 0 0 0 ⎤ ⎡ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎦ ⎣ 0 0 0 0 0 E1/G2 xx yy zz xy yz zx ⎤ ⎥ ⎦ (2.42) It appears from equation 2.42 that five independent elastic constants are required to characterise the elasticity of a transversely isotropic medium: E1 and 1 define properties in the plane of isotropy, and E2, 2, G2 properties in a plane containing the normal to, and any line in, the plane of isotropy. Inversion of the compliance matrix in equation 2.42, and putting E1/E2 = n, G2/E2 = m, produces the elasticity matrix given by E2 [D] = (1 + 1) 1 − 1 − 2n 2 ⎡ n ⎢ ⎢ 2 ⎢ ⎣ 1 − n 2 2 n 1 + n 2 2 n22(1 + 1) 0 0 0 n 1 − n 2 2 n22(1 + 1) 0 0 0 1 − 2 1 0 0 0 0.5 ∗ n∗ symmetric ∗ 1 − 1 − 2n 2 2 0 0 m(1 + 1)∗ 0 ∗ 1 − 1 − 2n2 2 m(1 + 1)∗ ∗ 1 − 1 − 2n2 ⎤ ⎥ ⎦ 2 Although it might be expected that the modulus ratios, n and m, and Poisson’s ratios, 1 and 2, may be virtually independent, such is not the case. The requirement for positive definiteness of the elasticity matrix, needed to assure a stable continuum, restricts the range of possible elastic ratios. Gerrard (1977) has summarised the published experimental data on elastic constants for transversely isotropic rock materials and rock materials displaying other forms of elastic anisotropy, including orthotropy for which nine independent constants are required. 2.11 Cylindrical polar co-ordinates A Cartesian co-ordinate system does not always constitute the most convenient system for specifying the state of stress and strain in a body, and problem geometry may suggest a more appropriate system. A cylindrical polar co-ordinate system is used frequently in the analysis of axisymmetric problems. Cartesian (x, y, z) and cylindrical polar (r, , z) co-ordinate systems are shown in Figure 2.11, together with an elementary free body in the polar system. To operate in the polar system, it is necessary to establish equations defining the co-ordinate transformation between Cartesian and polar co-ordinates, and a complete set of differential equations of equilibrium, strain displacement relations and strain compatibility equations. 37
Figure 2.11 Cylindrical polar coordinate axes, and associated free-body diagram. STRESS AND INFINITESIMAL STRAIN and The co-ordinate transformation is defined by the equations. r = (x 2 + y 2 ) 1/2 y = arctan x x = r cos y = r sin If R, , Z are the polar components of body force, the differential equations of equilibrium, obtained by considering the condition for static equilibrium of the element shown in Figure 2.11, are ∂rr ∂r 1 ∂r ∂rz + + r ∂ ∂z + rr − + R = 0 r ∂r 1 ∂ ∂z 2r + + + + = 0 ∂r r ∂ ∂z r ∂rz ∂r 1 ∂z ∂zz zz + + + + Z = 0 r ∂ ∂z r For axisymmetric problems, the tangential shear stress components, r and z, and the tangential component of body force, , vanish. The equilibrium equations reduce to ∂rr ∂r ∂rz + ∂z + rr − + R = 0 r ∂rz ∂r + ∂zz ∂z + rz r + Z = 0 For the particular case where r, ,zare principal stress directions, i.e. the shear stress component rz vanishes, the equations become ∂rr ∂r + rr − + R = 0 r ∂zz + Z = 0 ∂z Displacement components in the polar system are described by ur, u, uz. The elements of the strain matrix are defined by 38 εrr = ∂ur ∂r ε = 1 ∂u ur + r ∂ r εzz = ∂uz ∂z
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Rock Mechanics
Page 3 and 4: Rock Mechanics for underground mini
Page 5 and 6: Contents Preface to the third editi
Page 7 and 8: CONTENTS 9 Excavation design in blo
Page 9 and 10: CONTENTS ix Appendix A Basic constr
Page 11 and 12: PREFACE TO THE THIRD EDITION Mining
Page 13 and 14: PREFACE TO THE SECOND EDITION In th
Page 15 and 16: PREFACE TO THE FIRST EDITION design
Page 17 and 18: ACKNOWLEDGEMENTS Safety in Mines Re
Page 19 and 20: Figure 1.1 (a) Pre-mining condition
Page 21 and 22: ROCK MECHANICS AND MINING ENGINEERI
Page 25 and 26: Figure 1.4 Principal features of a
Page 27 and 28: 10 Figure 1.5 Definition of activit
Page 31 and 32: Figure 1.7 Components and logic of
Page 35 and 36: Figure 2.1 (a) A finite body subjec
Page 37 and 38: Figure 2.2 Free-body diagram for es
Page 39 and 40: STRESS AND INFINITESIMAL STRAIN As
Page 41 and 42: STRESS AND INFINITESIMAL STRAIN In
Page 43 and 44: Figure 2.3 Free-body diagram for de
Page 45 and 46: Figure 2.5 Problem geometry for det
Page 47 and 48: Figure 2.7 Rigid-body rotation of a
Page 49 and 50: STRESS AND INFINITESIMAL STRAIN the
Page 51 and 52: STRESS AND INFINITESIMAL STRAIN str
Page 53: ⎡ ⎢ ⎣ xx yy zz xy yz zx STRES
Page 57 and 58: STRESS AND INFINITESIMAL STRAIN fre
Page 59 and 60: Figure 2.13 Construction of a Mohr
Page 61 and 62: STRESS AND INFINITESIMAL STRAIN fun
Page 63 and 64: 3 Rock Figure 3.1 Sidewall failure
Page 65 and 66: Figure 3.2 Jointing in a folded str
Page 67 and 68: Figure 3.5 Diagrammatic longitudina
Page 69 and 70: Figure 3.7 Discontinuity spacing hi
Page 71 and 72: Figure 3.9 Illustration of persiste
Page 73 and 74: Figure 3.11 Typical roughness profi
Page 75 and 76: ROCK MASS STRUCTURE AND CHARACTERIS
Page 81 and 82: Figure 3.17 Sample number vs. preci
Page 83 and 84: Figure 3.19 Diagrammatic illustrati
Page 87 and 88: Figure 3.20 Computerised depiction
Page 89 and 90: Figure 3.23 Stereographic projectio
Page 91 and 92: Figure 3.26 Polar stereographic net
Page 93 and 94: Figure 3.28 Contours of pole concen
Page 99 and 100: Figure 3.30 Geological Strength Ind
Page 103 and 104: Figure 4.1 Idealised illustration o
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ROCK STRENGTH AND DEFORMABILITY wit
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Figure 4.4 Influence of end restrai
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ROCK STRENGTH AND DEFORMABILITY whe
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Figure 4.8 Principle of closed-loop
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Figure 4.12 Two classes of stress-
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Figure 4.14 Point load test apparat
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Figure 4.15 Biaxial compression tes
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Figure 4.18 Results of triaxial com
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ROCK STRENGTH AND DEFORMABILITY was
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Figure 4.23 Coulomb strength envelo
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Figure 4.25 Extension of a preexist
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Figure 4.29 The three basic modes o
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Figure 4.30 Normalised peak strengt
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ROCK STRENGTH AND DEFORMABILITY Tab
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Figure 4.32 The normality condition
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Figure 4.33 Variation of peak princ
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Figure 4.35 Direct shear test confi
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Figure 4.37 Shear stress-shear disp
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Figure 4.40 Peak and residual effec
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Figure 4.43 Effect of shearing dire
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Figure 4.45 Relations between norma
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Figure 4.47 Coulomb friction, linea
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ROCK STRENGTH AND DEFORMABILITY whe
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Figure 4.49 Composite peak strength
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Figure 4.50 Hoek-Brown peak strengt
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Figure 4.52 Determination of the Yo
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ROCK STRENGTH AND DEFORMABILITY 4 A
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5 Pre-mining Figure 5.1 Method of s
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Figure 5.2 The effect of irregular
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PRE-MINING STATE OF STRESS surround
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PRE-MINING STATE OF STRESS induced
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Figure 5.5 (a) Definition of hole l
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Figure 5.6 (a) Core drilling a slot
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Figure 5.7 Principles of stress mea
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PRE-MINING STATE OF STRESS strength
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PRE-MINING STATE OF STRESS A second
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PRE-MINING STATE OF STRESS by the e
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PRE-MINING STATE OF STRESS extend i
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PRE-MINING STATE OF STRESS (d) Dete
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METHODS OF STRESS ANALYSIS quantita
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METHODS OF STRESS ANALYSIS It is in
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Figure 6.2 A thick-walled cylinder
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METHODS OF STRESS ANALYSIS For the
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Figure 6.3 Problem geometry, coordi
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Figure 6.4 Problem geometry, coordi
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METHODS OF STRESS ANALYSIS When the
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Figure 6.5 Superposition scheme dem
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METHODS OF STRESS ANALYSIS The disc
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Figure 6.7 Development of a finite
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METHODS OF STRESS ANALYSIS Solution
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Figure 6.8 A simple finite element
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Figure 6.9 A schematic representati
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METHODS OF STRESS ANALYSIS block ce
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METHODS OF STRESS ANALYSIS where ˚
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METHODS OF STRESS ANALYSIS The prin
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EXCAVATION DESIGN IN MASSIVE ELASTI
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Figure 7.2 A logical framework for
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Figure 7.3 (a) Axisymmetric stress
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Figure 7.6 A plane of weakness, ori
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Figure 7.8 A flat-lying plane of we
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Figure 7.10 Shear stress/normal str
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Figure 7.12 Ovaloidal opening in a
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Figure 7.15 States of stress at sel
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Figure 7.16 Prediction of the exten
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Figure 7.18 Contour plots of princi
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Figure 7.19 Problem geometry for de
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8 Excavation Figure 8.1 An excavati
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EXCAVATION DESIGN IN STRATIFIED ROC
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Figure 8.4 Experimental apparatus f
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Figure 8.7 Free body diagrams and n
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Figure 8.8 Assumed distributions of
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Figure 8.9 Flow chart for the deter
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Figure 8.10 Normalised arch thickne
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EXCAVATION DESIGN IN STRATIFIED ROC
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Figure 8.11 Normalised deflection a
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9 Excavation Figure 9.1 Generation
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Figure 9.3 (a) A finite, non-tapere
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(a) (b) Figure 9.4 (a) Vertical cro
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(a) (b) (c) EP EP Reference circle
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Figure 9.10 JP 100 is the only JP w
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Figure 9.12 Traces of the views of
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EXCAVATION DESIGN IN BLOCKY ROCK In
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Figure 9.14 Free-body diagrams of a
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EXCAVATION DESIGN IN BLOCKY ROCK di
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Figure 9.16 Symmetrical wedge in th
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Figure 9.17 (a) Geometry for determ
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Figure 9.18 Problem geometry demons
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Figure 9.20 Cut-and-fill stope mine
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Figure 9.22 Chart to determine fact
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EXCAVATION DESIGN IN BLOCKY ROCK Th
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Figure 10.1 (a) Pre-mining state of
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Figure 10.3 (a) Dynamic loading of
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Figure 10.5 (a) Pre-mining and (b)
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Figure 10.6 Problem definition and
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ENERGY, MINE STABILITY, MINE SEISMI
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Figure 10.9 Force and stress compon
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Figure 10.12 Distribution of radial
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Figure 10.15 Problem geometry for d
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Figure 10.17 (a) Schematic represen
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Figure 10.20 Elastic/post-peak stif
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Figure 10.24 Relation between frequ
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Figure 10.28 Six possible ways that
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Figure 10.29 First motions for P an
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11Rock support and reinforcement 11
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Figure 11.1 (a) Hypothetical exampl
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Figure 11.4 Non-linear support reac
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Figure 11.5 Idealised elastic-britt
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Figure 11.6 Calculated required sup
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ROCK SUPPORT AND REINFORCEMENT The
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Figure 11.9 Ground reaction curves
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Figure 11.12 Use of grouted reinfor
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ROCK SUPPORT AND REINFORCEMENT If,
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Figure 11.16 Local reinforcement ac
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Figure 11.18 Typical working sketch
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Figure 11.19 Permanent support and
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Figure 11.22 Basis of natural coord
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Figure 11.24 Distributions of (a) s
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Figure 11.26 Resin grouted rockbolt
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Figure 11.28 Alternative methods of
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ROCK SUPPORT AND REINFORCEMENT Tabl
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Figure 11.31 Toussaint-Heintzmann y
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MINING METHODS AND METHOD SELECTION
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Figure 12.2 Elements of a supported
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Figure 12.6 Schematic layout for bi
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Figure 12.8 Layout for shrink stopi
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Figure 12.9 Schematic layout for VC
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Figure 12.11 Key elements of longwa
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Figure 12.13 Mining layout for tran
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13 Figure 13.1 Schematic illustrati
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Figure 13.3 Layout of barrier pilla
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Figure 13.5 Principal modes of defo
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Figure 13.8 Geometry for tributary
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PILLAR SUPPORTED MINING METHODS str
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Figure 13.10 Distribution of vertic
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Figure 13.12 Pillar behaviour domai
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PILLAR SUPPORTED MINING METHODS Lun
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Figure 13.15 Options in the design
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Figure 13.17 Relation between yield
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Figure 13.19 Model of yield of coun
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Figure 13.20 North-south vertical c
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Figure 13.23 Stope-and-pillar layou
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Figure 13.25 Calibrated stability c
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PILLAR SUPPORTED MINING METHODS wor
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Figure 13.28 Pillar performance, de
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Figure 13.29 (a) Stope and pillar l
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Figure 13.31 (a) Plane strain analy
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PILLAR SUPPORTED MINING METHODS Pan
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14 Artificially supported mining me
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ARTIFICIALLY SUPPORTED MINING METHO
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Figure 14.2 Simplified view of stru
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Figure 14.5 Confined block model fo
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Figure 14.7 Crown and sidewall stre
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Figure 14.10 Sublevel open stoping
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Figure 14.12 Some applications of c
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15 Longwall and caving mining metho
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Figure 15.2 Shear stress drop in th
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LONGWALL AND CAVING MINING METHODS
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Figure 15.6 Hydraulic prop reaction
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Figure 15.7 Development and extract
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Figure 15.8 Vertical stress redistr
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Figure 15.11 Distribution of observ
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Figure 15.13 Plan view of microseis
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Figure 15.16 Ground-support interac
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Figure 15.18 Roadway support and re
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Figure 15.25 Comparison of isolated
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Figure 15.26 Geometry of a sublevel
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Figure 15.28 Theoretical determinat
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Figure 15.31 Deterioration of a cro
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Figure 15.32 Distinct element simul
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Figure 15.34 Extended Mathews stabi
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Figure 15.36 Comparison of postand
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Figure 15.39 Idealised plan illustr
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Figure 15.41 Idealised vertical sec
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Figure 15.42 Vertical slice through
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16 Figure 16.1 Trough subsidence ov
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MINING-INDUCED SURFACE SUBSIDENCE c
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Figure 16.4 North-south section, At
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Figure 16.6 (a) Rectangular block g
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MINING-INDUCED SURFACE SUBSIDENCE f
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Figure 16.8 Relation between stope
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MINING-INDUCED SURFACE SUBSIDENCE M
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MINING-INDUCED SURFACE SUBSIDENCE
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Figure 16.14 Chart developed to est
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Figure 16.16 Progressive hangingwal
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Figure 16.19 Idealised model used i
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Figure 16.21 Longitudinal section,
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MINING-INDUCED SURFACE SUBSIDENCE t
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MINING-INDUCED SURFACE SUBSIDENCE w
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MINING-INDUCED SURFACE SUBSIDENCE F
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Figure 16.25 Subsidence troughs pre
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Figure 16.28 Predicted and measured
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17 Blasting mechanics 17.1 Blasting
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Figure 17.1 An empirical matching o
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Figure 17.2 A finite difference mod
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Figure 17.4 Reflection of a cylindr
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BLASTING MECHANICS means that no ci
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Figure 17.8 Layout of blast holes i
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Figure 17.9 Influence of field stat
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Figure 17.11 Generation of surface
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BLASTING MECHANICS The components o
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BLASTING MECHANICS amplitudes of th
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BLASTING MECHANICS 17.9 Evaluation
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Figure 17.15 (a) Schematic cross se
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BLASTING MECHANICS in Figure 17.17,
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MONITORING ROCK MASS PERFORMANCE (a
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MONITORING ROCK MASS PERFORMANCE su
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MONITORING ROCK MASS PERFORMANCE Ta
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Figure 18.2 The Distometer ISETH, a
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Figure 18.5 Self-inductance multipl
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MONITORING ROCK MASS PERFORMANCE is
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Figure 18.9 Biaxial vibrating wire
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MONITORING ROCK MASS PERFORMANCE me
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Figure 18.12 Cross section at 6650N
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Figure 18.13 Examples of convergenc
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Figure 18.15 Longitudinal section l
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Figure 18.16 (Cont.) MONITORING ROC
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Appendix A Basic constructions usin
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Figure A.3 Determining the angle be
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APPENDIX A USE OF HEMISPHERICAL PRO
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APPENDIX B STRESSES AND DISPLACEMEN
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Figure A.6 Axisymmetric tunnel prob
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Figure A.9 Bolt load-extension curv
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APPENDIX D LIMITING EQUILIBRIUM ANA
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ANSWERS TO PROBLEMS 2 (a) 0.087 - 0
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ANSWERS TO PROBLEMS 3 wp = 38.6 m,
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REFERENCES Symp. & 17th Tunn. Assn
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REFERENCES Brady, B. H. G. and Bray
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REFERENCES Collier, P. A. (1993) De
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REFERENCES Drescher, A. and Vardoul
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REFERENCES Gustafsson, P. (1998) Wa
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REFERENCES Hood, M. and Brown, E. T
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REFERENCES Kaiser, P. K. and Tannan
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REFERENCES Lorig, L. J. and Brady,
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REFERENCES Ortlepp, W. D. (1994) Gr
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REFERENCES Rojas, E., Molina, R. an
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REFERENCES Spottiswoode, S. M. and
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REFERENCES Villaescusa, E., Windsor
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Index Page numbers appearing in bol
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INDEX Coulomb (cont.) parameters 96
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INDEX Excavation (cont.) support ra
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INDEX Jaeger’s plane of weakness
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INDEX Panel caving 470-2, 473, 474,
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INDEX Seismic (cont.) moment 306, 3
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INDEX Strength (cont.) residual 86,
Page 645:
INDEX United States (USA) 395, 396,
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