Rock Mechanics.pdf - Mining and Blasting

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Figure 10.7 Geometry describing longitudinal wave transmission and reflection in a two-component bar. ENERGY TRANSMISSION IN ROCK co-ordinate direction, i.e. a backward progressive wave. Each of the functions f1 and f2 is individually a solution to the wave equation, and since the constitutive behaviour of the system is linear, any linear combination of f1 and f2 also satisfies the governing equation. During the propagation of the elastic wave, represented by equation 10.10, along a bar, each particle executes transient motion about its equilibrium position. The transient velocity, V, of a particle is associated with a transient state of stress, xx, which is superimposed on any static stresses existing in the bar. For uniaxial longitudinal stress and using Hooke’s Law, dynamic stresses and strains are related by or, from equation 10.10 Transient particle velocity is defined by or, from equation 10.10 xx = Eεxx =−E∂ux/∂x xx =−E[ f ′ 1 (x − CBt) + f ′ 2 (x + CBt)] (10.12) ˙ux = V = ∂ux/∂t V = (−CB) f ′ 1 (x − CBt) + CB f ′ 2 (x + CBt) (10.13) Considering the forward progressive wave, the relevant components of equations 10.12 and 10.13, together with equation 10.11, yield or V = CBxx/E = xx/CB xx = CBV (10.14) Thus the dynamic longitudinal stress induced at a point by passage of a wave is directly proportional to the transient particle velocity at the point. In equation 10.14, the quantity CB is called the characteristic impedance of the medium. For the backward wave, it is readily shown that xx =−CBV (10.15) A case of some practical interest involves a forward wave propagating in a composite bar, as indicated in Figure 10.7. The bar consists of two components, with 279
ENERGY, MINE STABILITY, MINE SEISMICITY AND ROCKBURSTS different material properties, denoted by subscripts 1 and 2. The interface between the component bars is welded. The forward, incident wave in medium 1 impinges on the interface, and is partly transmitted, generating a forward wave in medium 2, and partly reflected, generating a backward wave in medium 1. The quantities to be determined are the relative intensities of longitudinal stress in the forward and reflected waves. Suppose the longitudinal stress and particle velocity in the forward incident wave are defined by magnitudes 0, V0, and the corresponding magnitudes in the transmitted and reflected waves are given by t, Vt and r, Vr. The ratio of the characteristic impedances of the two media is defined by n = 2CB2/1CB1 = 2C2/1C1 (10.16) The conditions to be satisfied at the interface between the bar components are continuity of longitudinal stress and displacement (and therefore particle velocity). These continuity conditions are expressed by the equations 0 + r = t V0 + Vr = Vt (10.17) (10.18) Introducing equations 10.14 and 10.15 to relate stresses and velocities for incident, reflected and transmitted waves, equation 10.18 becomes and introducing equation 10.17 On rearrangement, this yields It is then readily shown that 0/1C1 − r/1C1 = t/2C2 0/1C1 − r/1C1 = (0 + r)/2C2 r = [(n − 1)/(n + 1)]0 t = [2n/(n + 1)]0 Similarly, the relation between particle velocities may be shown to be Vr =−[(n − 1)/(n + 1)]V0 Vt = [2/(n + 1)]V0 (10.19) (10.20) (10.21) (10.22) It is useful to explore, briefly, some of the consequences of these expressions governing wave transmission and reflection. Suppose 0 is compressive. For the case n > 1, the reflected wave is characterised by a compressive stress. For n < 1, the reflected wave induces a tensile stress. Thus an important general point to note is that internal reflections of a compressive wave in a medium may give rise to tensile stresses. Preceding chapters have described the low tensile strength of rock masses, and their inability to sustain tensile stress. 280
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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
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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
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
Page 253 and 254: Figure 8.10 Normalised arch thickne
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Page 257 and 258: Figure 8.11 Normalised deflection a
Page 259 and 260: 9 Excavation Figure 9.1 Generation
Page 261 and 262: Figure 9.3 (a) A finite, non-tapere
Page 263 and 264: (a) (b) Figure 9.4 (a) Vertical cro
Page 265 and 266: (a) (b) (c) EP EP Reference circle
Page 267 and 268: Figure 9.10 JP 100 is the only JP w
Page 269 and 270: Figure 9.12 Traces of the views of
Page 271 and 272: EXCAVATION DESIGN IN BLOCKY ROCK In
Page 273 and 274: Figure 9.14 Free-body diagrams of a
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Page 277 and 278: Figure 9.16 Symmetrical wedge in th
Page 279 and 280: Figure 9.17 (a) Geometry for determ
Page 281 and 282: Figure 9.18 Problem geometry demons
Page 283 and 284: Figure 9.20 Cut-and-fill stope mine
Page 285 and 286: Figure 9.22 Chart to determine fact
Page 287 and 288: EXCAVATION DESIGN IN BLOCKY ROCK Th
Page 289 and 290: Figure 10.1 (a) Pre-mining state of
Page 291 and 292: Figure 10.3 (a) Dynamic loading of
Page 293 and 294: Figure 10.5 (a) Pre-mining and (b)
Page 295: Figure 10.6 Problem definition and
Page 299 and 300: Figure 10.9 Force and stress compon
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Page 307 and 308: Figure 10.12 Distribution of radial
Page 309 and 310: Figure 10.15 Problem geometry for d
Page 311 and 312: Figure 10.17 (a) Schematic represen
Page 315 and 316: Figure 10.20 Elastic/post-peak stif
Page 319 and 320: Figure 10.24 Relation between frequ
Page 325 and 326: Figure 10.28 Six possible ways that
Page 327 and 328: Figure 10.29 First motions for P an
Page 329 and 330: 11Rock support and reinforcement 11
Page 331 and 332: Figure 11.1 (a) Hypothetical exampl
Page 333 and 334: Figure 11.4 Non-linear support reac
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Page 337 and 338: Figure 11.6 Calculated required sup
Page 339 and 340: ROCK SUPPORT AND REINFORCEMENT The
Page 341 and 342: Figure 11.9 Ground reaction curves
Page 343 and 344: Figure 11.12 Use of grouted reinfor
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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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