Rock Mechanics.pdf - Mining and Blasting

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THE FINITE ELEMENT METHOD The components of [N], i.e. the terms Ni, are prescribed functions of position, and [u e ] is a column vector listing the nodal displacements uxi, u yi, uxj . . . etc. The interpolation functions which constitute the elements of [N] must be chosen to return the nodal displacements at each of the nodes. This requires that [Ni]xi,yi = [I] [Ni]xi,yj = [0], etc. where [I] and [0] are the identity and null matrices respectively. Also, since both components of displacement at a point are to be interpolated in the same way, it is clear that [Ni] = Ni[I] where Ni is a scalar function of position within the element. A simple development of a linear interpolation function is demonstrated by representing the displacements in terms of linear functions of position, i.e. ux = 1 + 2x + 3y u y = 4 + 5x + 6y (6.37) The six interpolation constants are determined by ensuring that the displacements ux, u y assume the nodal values when nodal co-ordinates are inserted in equation 6.37. Thus 1, 2, 3 are determined by solving the simultaneous equations uxi = 1 + 2xi + 3yi uxj = 1 + 2x j + 3yj uxk = 1 + 2xk + 3yk Solution for 1, 2, 3 and some rearrangement, produces ux = 1 2 [(ai + bi x + ci y)uxi + (a j + b j x + c j y)uxj + (ak + bkx + ck y)uxk] where ai = x j yk − xk y j bi = y j − yk ci = xk − x j with cyclic permutation of i, j, k to obtain a j, etc., and 185 2 = 2 × area of the triangular element ⎡ ⎤ 1 xi yi = 2 ⎣1 x j y j⎦ 1 xk yk (6.38)
METHODS OF STRESS ANALYSIS Solution for 4, 5, 6 yields an interpolation function for u y identical to equation 6.38, with u yi replacing uxi, etc. The variation of displacements throughout an element is therefore described by ux [u] = = [N][u e ] = [NiI, N jI, NkI][u e ] (6.39) u y where Ni = (ai + bi x + ci y)/2, with similar expressions N j, Nk and I isa2× 2 identity matrix. By defining the displacement field in an element in terms of the nodal displacements, the interpolation procedure ensures continuity of the displacements both across an element interface with an adjacent element, and within the element itself. Once the displacement field in an element is defined, the state of strain can be established from the strain–displacement relations. For plane strain problems, a strain vector may be defined by ⎡ ⎤ εxx ⎢ ⎥ [ε] = ⎢ εyy ⎥ ⎣ ⎦ xy = ⎡ ∂ux ⎢ ∂x ⎢ ∂u y ⎢ ∂y ⎢ ⎣ ∂ux ∂y + ∂u ⎤ ⎡ ⎤ ∂ ⎥ ⎢ 0 ⎥ ⎢∂x ⎥ ⎥ ⎢ ⎥ ⎢ ∂ ⎥ ux ⎥ ⎥ = ⎢ 0 ⎥ ⎥ ⎢ ∂y ⎥ ⎥ ⎢ ⎥ u y y ⎦ ⎣ ∂ ∂ ⎦ ∂x ∂y ∂x or [ε] = [L][u] (6.40) Since displacements are specified by equation 6.39, equation 6.40 becomes where ⎡ ∂ Ni ⎢ ∂x ⎢ [B] = ⎢ 0 ⎢ ⎣∂ Ni ∂y [ε] = [L][N][u e ] = [B][u e ] 0 ∂ Ni ∂y ∂ Ni ∂x ∂ N j ∂x 0 ∂ N j ∂y For the case of linear displacement variation, the terms ∂ Ni/∂x, etc., of the B matrix are constant, and thus the strain components are invariant over the element. 6.6.2 Stresses within an element The state of total stress within an element is the sum of the induced stresses and the initial stresses. Ignoring any thermal strains, total stresses, for conditions of plane 186 0 ∂ N j ∂y ∂ N j ∂x ∂ Nk ∂x 0 ∂ Nk ∂y 0 ∂ Nk ∂y ∂ Nk ∂x ⎤ ⎥ ⎦
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Rock Mechanics
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Rock Mechanics for underground mini
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Contents Preface to the third editi
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CONTENTS 9 Excavation design in blo
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CONTENTS ix Appendix A Basic constr
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PREFACE TO THE THIRD EDITION Mining
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PREFACE TO THE SECOND EDITION In th
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PREFACE TO THE FIRST EDITION design
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ACKNOWLEDGEMENTS Safety in Mines Re
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Figure 1.1 (a) Pre-mining condition
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ROCK MECHANICS AND MINING ENGINEERI
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Figure 1.4 Principal features of a
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10 Figure 1.5 Definition of activit
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Figure 1.7 Components and logic of
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Figure 2.1 (a) A finite body subjec
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Figure 2.2 Free-body diagram for es
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STRESS AND INFINITESIMAL STRAIN As
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STRESS AND INFINITESIMAL STRAIN In
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Figure 2.3 Free-body diagram for de
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Figure 2.5 Problem geometry for det
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Figure 2.7 Rigid-body rotation of a
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STRESS AND INFINITESIMAL STRAIN the
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STRESS AND INFINITESIMAL STRAIN str
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⎡ ⎢ ⎣ xx yy zz xy yz zx STRES
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Figure 2.11 Cylindrical polar coord
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STRESS AND INFINITESIMAL STRAIN fre
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Figure 2.13 Construction of a Mohr
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STRESS AND INFINITESIMAL STRAIN fun
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3 Rock Figure 3.1 Sidewall failure
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Figure 3.2 Jointing in a folded str
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Figure 3.5 Diagrammatic longitudina
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Figure 3.7 Discontinuity spacing hi
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Figure 3.9 Illustration of persiste
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Figure 3.11 Typical roughness profi
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ROCK MASS STRUCTURE AND CHARACTERIS
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Figure 3.17 Sample number vs. preci
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Figure 3.19 Diagrammatic illustrati
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Figure 3.20 Computerised depiction
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Figure 3.23 Stereographic projectio
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Figure 3.26 Polar stereographic net
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Figure 3.28 Contours of pole concen
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Figure 3.30 Geological Strength Ind
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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
Page 151 and 152: Figure 4.49 Composite peak strength
Page 153 and 154: Figure 4.50 Hoek-Brown peak strengt
Page 155 and 156: Figure 4.52 Determination of the Yo
Page 157 and 158: ROCK STRENGTH AND DEFORMABILITY 4 A
Page 159 and 160: 5 Pre-mining Figure 5.1 Method of s
Page 161 and 162: Figure 5.2 The effect of irregular
Page 163 and 164: PRE-MINING STATE OF STRESS surround
Page 165 and 166: PRE-MINING STATE OF STRESS induced
Page 167 and 168: Figure 5.5 (a) Definition of hole l
Page 169 and 170: Figure 5.6 (a) Core drilling a slot
Page 171 and 172: Figure 5.7 Principles of stress mea
Page 173 and 174: PRE-MINING STATE OF STRESS strength
Page 175 and 176: PRE-MINING STATE OF STRESS A second
Page 177 and 178: PRE-MINING STATE OF STRESS by the e
Page 179 and 180: PRE-MINING STATE OF STRESS extend i
Page 181 and 182: PRE-MINING STATE OF STRESS (d) Dete
Page 183 and 184: METHODS OF STRESS ANALYSIS quantita
Page 185 and 186: METHODS OF STRESS ANALYSIS It is in
Page 187 and 188: Figure 6.2 A thick-walled cylinder
Page 189 and 190: METHODS OF STRESS ANALYSIS For the
Page 191 and 192: Figure 6.3 Problem geometry, coordi
Page 193 and 194: Figure 6.4 Problem geometry, coordi
Page 195 and 196: METHODS OF STRESS ANALYSIS When the
Page 197 and 198: Figure 6.5 Superposition scheme dem
Page 199 and 200: METHODS OF STRESS ANALYSIS The disc
Page 201: Figure 6.7 Development of a finite
Page 205 and 206: Figure 6.8 A simple finite element
Page 207 and 208: Figure 6.9 A schematic representati
Page 209 and 210: METHODS OF STRESS ANALYSIS block ce
Page 211 and 212: METHODS OF STRESS ANALYSIS where ˚
Page 213 and 214: METHODS OF STRESS ANALYSIS The prin
Page 215 and 216: EXCAVATION DESIGN IN MASSIVE ELASTI
Page 217 and 218: Figure 7.2 A logical framework for
Page 219 and 220: Figure 7.3 (a) Axisymmetric stress
Page 221 and 222: Figure 7.6 A plane of weakness, ori
Page 223 and 224: Figure 7.8 A flat-lying plane of we
Page 225 and 226: Figure 7.10 Shear stress/normal str
Page 227 and 228: Figure 7.12 Ovaloidal opening in a
Page 229 and 230: Figure 7.15 States of stress at sel
Page 231 and 232: Figure 7.16 Prediction of the exten
Page 233 and 234: Figure 7.18 Contour plots of princi
Page 235 and 236: Figure 7.19 Problem geometry for de
Page 241 and 242: 8 Excavation Figure 8.1 An excavati
Page 243 and 244: EXCAVATION DESIGN IN STRATIFIED ROC
Page 245 and 246: Figure 8.4 Experimental apparatus f
Page 247 and 248: Figure 8.7 Free body diagrams and n
Page 249 and 250: Figure 8.8 Assumed distributions of
Page 251 and 252: 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,
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INDEX United States (USA) 395, 396,
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