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Introduction to
Rock Mechanics
Second Edition

Richard E. Goodman
University of California at Berkeley

Jahn Wiley & Sans

New York I Chichester I Bri.bane I Toronto I Singapore
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Page 144

272 Applications of Rock Mechanics in Engineering for Underground Openings

E

w E

lb)

Figure 7.28 (b) Projection in the tunnel
section of the maximum remoyable block
corresponding to JP 010.

limit plane must pass through both the comer and its opposite. Ifthe comer has
distance x from the center of the reference circle, its opposite has distance R2/X
from the center in the opposite direction (where R is the radius ofthe reference
circle). In the case of Figure 7.28a, the lP lies inside each of the enveloping
great circles, so the lP is above both of them. The angles of inclination of the
orthographic projections of the comers in the tunnel section are transferred to
the tunnel section in Figure 7.28b, and the removable block is found as the
region simultaneously aboye each. In this case, the maximum removable block
is not a key block if gravity is the main contributor to the resultant force
direction in the block.

Figure 7.29a shows a similar construction but for lP 001. The comer 1,2 of
this lP is off the paper so its opposite -1'2 has been drawn. (The opposite is the
upper hemisphere line that is perpendicular to both ofthe normal s n, and n2 and
therefore perpendicular to the great circle connecting normals n, and n2.) Great
circles through each comer in tum intersect the tunnel section at the points
shown and the limiting great circles are established as those through lz3 and 113 ,
intersecting the tunnel section respectively 24° aboye east and 83° aboye west.
The lP is outside the first of these and inside the second so JP 001 is below the
first and aboye the second. Figure 7.29b transfers this information to the tunnel
section.

7.S Underground Openings in Blocky Rock-"Block Theory" 273

Having performed an analysis of maximum key block s for two lPs, we now
proceed to find all the rest. Since JPs 000 and 111 contain the tunnel axis, they
have no maximum key block areas in the tunnel section. The maximum key
block regions of all the other lPs are shown on Figure 7.30, where each tunnel
section is drawn in the curved polygon corresponding to the lP in question. We

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/ I \ ... ~----

/ I ,
I I \

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JP 001

~----7-- Tunnel
section

Limit plane 2 ,

~
• I
~ I

'. ;<.......
.... -1 1" ;1' " "'- .' .7 •• -----

•••• 13 ." "" Lim it plane 1
_ 11'"' , •• '

: A
123 .-
~.~----~~-----~--

/ .'

Reference
circle

/{/ ..... 1" ....
.... J .."",. ~ ..... . ~-- .

~···· •••••••••••••••••• II ••• ••••

(a)

JP 001 is below limit plane 1
and aboye limit plane 2

Figure 7.29 (a) Construction for the projections of the edges of the maximum
remoyable block in the tunne! section, corresponding to JP 001.

Page 145

274 Applications of Rock Mechanics in Engineermgfor Underground Openings

W------~~--~~--~--------E

(b)

Figure 7.29 (h) Projection in the tunnel
section of the maximum removable block
corresponding to lP 001.

see at a glance that under gravity loading JPs 101, 100, and 001 may generate
block s that require support. A1though these drawings are two dimensional, the
three-dimensional maximum keyblocks can be drawn corresponding to each of
thesejoint pyramids and subjected to a limit equilibrium analysis. The direction
of sliding under gravity is determined by a mode analysis, based upon the
direction of the resultant force and the description of the joint pyramid, as
discussed by Goodman and Shi (1985). Then when friction angles are input on
each face, the support force vector can be computed for each maximum key
block using methods of analysis discussed in the next chapter. It may be that
none of the blocks require support; on the other hand, in certain tunnel direc-
tions, the same set of joint planes and friction angles may create the need for a
large support force. This is largely due to the changing size of the maximum key
block s as the direction of the tunnel is varied.

CASE HISTORY--"TUNNEL SUPPORT SPECTRUM"

The effect of tunnel direction on rock mass support requirements will be illus-
trated for the rock mas s of Figure 7.20 by studying the SUpport force for

7.S Underground Openings in Blocky Rock-"Block Theory" 275

Figure 7.30 Tunnel sections showing maximum removable block s for each lP super-
imposed on the stereographic projection of the lPs.

equilibrium of the most critical block as a function of a complete sweep of
tunnel directions. AH the examples considered thus far correspond to a hori-
zontal tunnel axis in direction N 21 0 E. Consid~ring only horizontal tunnels, for
the time being, we can observe the relative size of the most critical key block,
for five tunnel directions, in Figure 7.31. The maximum key block s become
d.ramaticaHy larger as the direction 3150 (N 450 W) is approached. Limit equilib-
num analysis of this system, for a horseshoe tunnel 6 m wide and 5.4 m high,
yields required support forces for the maximum k~y block varying from a low
of less than 1 me trie ton per meter length of tunnel, to a maximum of 33 metric

Page 288

560 Inde~

Photoelastic stress gage, 116
Piles and piers, 346

settlement of, 376, 377

Pillars, 231
Plane sliding, see Block sliding
Plane stress and plane strain, 452
Plastic analysis of tunnels, 242-249, 454-464
Plate bearing test, 188-190
Poisson's ratio, 67, 181

measurement, 184
for permanent and total strain, 220
typical values, 186
variation with deviatoric compression,

70, 185
Pore "crush-up", 69
Pore pressure:

and fault slip, 171-173
and rock failure, 85, 86
and stress measurement, 140

Porosity:
effect of age on rocks, 29
measurement, 31
types of pore spaces, 13, 28, 29
values for selected rocks, 29, 83
and water content, 32

Poulos, H. G., 352, 370
Pratt, H. R., 91, 92
Principal strains, 412
Principal stresses, 133, 134, 393,402,403
Principal stress ratio:

effect on strength, 74-76, 87
initial values, 106-113

Principal symmetry directions, 182
Punching failure of footing, 351, 352

, Q system (classification), 47-49
Quartzite, 142, 144

compressive and tensile strength, 61
Hoek-Brown constants, 99
Lundborg's strength parameters, 90
modulus ratio and Poisson's ratio, 186
Mohr-Coulomb parameters, 83

Raleigh, C. B., 172
Rangeley oil field, 172
Raphael, J., 363
Reichmuth, D. R., 62
Relaxation, 203

Removable blocks, 258
Rengers, N., 166, 167, 169
Reservoir induced seismicity, 86, 171-172
Residual strength, 81, 161
Residual stresses, 116
Rhyolite, 20

density, 33
Riley, P. B., 116
Ring shear test, 66, 88-90
Roberts, A., 116
Rocha, M., 36, 116, 121, 134
Rock boIt design:

to control plastic zone of tunnels, 246
to prevent slip of joints in tunnels, 241
for rock slope stability, 279, 283, 317-319
to secure roof beams in layered rock, 236,

237, 281, 290, 334, 387
Rock bursts, 102
Rock classification:

comparison of Q and RMR, 47
fabrics, 20, 21
fissuring, 39-42
geological, 19, 419-425
NGI (Q) system, 47
RMR (Geomechanics) system, 43-46
for tunneling, 43-47

Rock loosening, 53, 247
Rock mas s rating, see Geomechanics

classification
Rock noise, 69

design
Rock slopes:

buckling of, 337
types, 293, 294

Rock socket, 345, 371, 377
Rock test preparation equipment, 58, 59
Roof beams, buckling of, 233
Roof design, underground, 233-237
Rosettes, strain gages, 411, 412
RQD,44 '
Rummel, F., 63
Rutter, E. H., 206

Safety, see Factor of safety
Salem, A., 190
Sallstrom, S., 419
Salt, 15

creep properties, 252
density, 252
ductility, 74

Sandstone, 143, 144, 146
compressive and tensile strength, 61

(

foundations, 349, 350
Hoek-Brown constants, 99
joints in, 146
modulus ratio and Poisson's ratio, 186
Mohr-Coulomb parameters, 83
permeability, 35
porosity, 29, 83
in triaxial compression, 75

Schist, 8, 20, 21, 161
bearing capacity, 350
compressive and tensile strength, 61
density, 33
modulus ratio and Poisson's ratio, 186
permeability, 35

Schistosity, 13
and strength, 61, 83

Schneider, B., 147, 190, 199-201
Schneider, H. J., 157, 168, 169
Scratch hardness, 416-418
"Seamy rock" in building code, 349
Seismicity, reservoir induced, 87, 172
Serpentinite, 20, 21
Shale, 144-146

bearing capacity, 350
compressive and tensile strength, 37, 61
ductility, 74
Hoek-Brown constants, 99
modulus ratio and Poisson's ratio, 140, 186
Mohr-Coulomb parameters, 83
permeability, 35
pore water pressure, induced, 85
porosity, 29, 83
residual strength, 81
swelling pressure, 380

Shear failure, 56
Shear modulus, 182

of jointed rock, 196
Shear zones, 141
Shear stiffness, 163, 196
Shear strength intercept, 81

typical values, 83
variation with direction, 93

Shi, G. H., 257
Shi's theorem, 259, 260
Shield, R. T., 362
Shtenko, V. W., 382
Siltstone, 145, 146

compressive and tensile strength, 61
Mohr-Coulomb parameters, 83
porosity, 83

Size effect on strength, 90-93, 229
tests, 90, 91, 231, 233

Skinner, E. H., 42
Slake durability classification, 38
Slaking,37
Slate,20

anisotropy in strength, 93
bearing capacity, 350

Inde~ 561

Lundborg's strength parameters, 90
Mohr-Coulomb strength parameters, 83

Snow, D. T., 35
Sonic velocity, 39-42
Sowers, G., 39, 378
Specific gravity, 30-33
Spillways, 9
Splitting tension test, see Brazilian test
Squeezing ground, 104, 286
Stability analysis:

of block on aplane, 312-319
of general polyhedron, 325-331
of 2d active/passive system, 333-334
of 2d block, 287
of wedge, 319-320

Stagg, K. G., 189
Standards, 17

for compression tests, 60
Stephenson, D. E., 117
Stereographic projection, 475-493

equations for drawing great circles, 261
joint orientation analysis, 150-154
of joint system, 263
kinematic analysis of slopes, 301-310
limit equilibrium analysis for a given JP,

325-331
roughness angles of joints, 167
stability analysis of slopes, 312-320

Stereonet, 479, 480
Stiffness of joints, 196

of testing machines, 76
Strain in two dimensions, 409-413
Strength, 34, 36, 37, 76
Stress concentrations, 225, 228-231
Stresses, manipulation and operations,

389-408
Stress history related to sedimentation and

erosion, 176
Stresses in the crust map, 112
Stress measurement, 116, 117

complete state of stress, 132-135
doorstopper test, 127-129
flat jack method, 121-123
hydraulic fracturing, 117-121
inclusion methods, 116, 127
overcoring, 123-127
undercoring, 130, 131

Page 289

562 Inde~

Stress-strain relations:
deviatoric values in terms of, 218
elastic, 181
for jointed rock, 195-197, 220
viscoelastic, 206

Strike and dip, defined, 400
Subsidence over old mines, 379
Swelling rock, 380-382

mineral s responsible for, 343, 419
Syenite, 33

Ta ehien dam site, 147
Taylor, H. W., 46
Tensile failure, 56, 57
Tensile strength, 82, 119
Terratek, Inc., 12,64,92
Terzaghi, K., 242
Terzaghi, R., 153
Three-dimensional analysis of stresses in situ,

134, 135
Tiedemann, H. R., 42
Timoshenko, S., 189, 468
Tincelin, M. E., 121
Toews, N. A., 126, 135
Topographic effects on stresses, 102,

104, 105, 107
Toppling failure, 298-300

kinematic analysis, 304-306
Tourenq, e., 65
Townsend, F. e., 81
Transversely isotropic rock, 183
Triaxial compression:

equipment, procedures, 61-65
for joints, 160-164

Tributary area, 231, 232
Trona, 20
Tuff,20

compressive and tensile strength, 61
modulus ratio and Poisson's ratio, 186
porosity, 29

Tunnel support spectrum, 274-280
Twin gliding, 74
Two block sliding analysis of slopes, 333, 334

Unconfined compressive strength, see
eompressive strength

Undercoring, 130, 131
Underground openings:

circular, elastic rock, 220, 225
circular, plastic rock, 242-249, 454-464

elliptical, oval, rectangular, 230
spherical, 230
time dependency, creep, 250-256

U.S. Bureau of Mines gage, 124

Van Heerden, W. L., 91
Velocity of sound in rocks, 39-42

longitudinal ve!ocity in minerals, 40
Virgin stresses, see Initial stress
Viscoelasticity:

spring and dashpot models, 204-206
and stress measurement, 117

Viscosity, 202
for water, 34

Vogler, U. W., 61
Voight, B., 116
Volcanic rocks, 20. See alsa individual

rack type

Volumetric strain, 68

Wagner, P. L., 546
Water, effect on rock strength, 84-87
Water loss coefficient, 52
Wave equation, 442
Wave velocity, 39-42
Wawersik, W. R., 69, 71
Weathering:

effect on foundations, 341, 343
effect on porosity, 24, 28, 30

Wedge slides, 295-297
kinematic analysis, 304
stability analysis, 319, 320

Wickham, G. E., 42
Willett, D. e., 224
Wilson, L. e., 385
Wittke, W., 318
Woodward, R. J., 370, 377

X-ray measurement of stresses, 117

Young's modulus, 181
fractured or jointed rocks, 195-202
measurement, 183-195
ratio to compressive strength, 186
variation with strain, 185

Yudhbir,99

Zaruba, Q., 132
Zienkiewicz, O. e., 189,352
Zoback, M., 112