Engineering Rock Mechanics

368 Design and analysis of underground excavations

Sliding

A = area of sliding surface

= Coulomb strength parameters

‘1 9

CA + (Wcos $+ Tcos e) tan +

W sin + - T sin 8

F=

The effect of stresses in the rock has been ignored, and this analysis is

for sliding on one discontinuity surface only.

Figure 20.9 Analysis of a tetrahedral block sliding on one face.

as illustrated in Fig. 20.9. This calculation is for sliding on a discontinuity
surface, and hence is analogous to the plane sliding instability for slopes
discussed in Chapters 17 and 18. The formula for the rock bolt force, F, in
Fig. 20.9 is a modification to that presented in Section 17.1.2 and, from
Section 16.3.2, this is minimized for these idealized circumstances when
e = 90 - 4.

Symmetric and asymmetric triangular roof prisms. To incorporate the effect
of the stresses in the rock surrounding the excavation, Fig. 20.10 shows an
extended analysis considering the forces in a symmetric triangular roof
prism. The purpose of this analysis is to establish the influence of a hori-
zontal force Ho on the block, and its influence on the support force necessary
to hold the block in place. The presence of any such compressive stress in
the rock (and hence compressive force on the block) will affect the support
force required: depending on the wedge apical angle and other mechanical
parameters, the block will either tend to be squeezed out or be constrained.
The support force, R, is given by W-P, where W is the block weight and P
is the resultant of the frictional forces acting on the block, expressed as
positive downwards. The expressions for P are given in Fig. 20.10.

Assuming S = N tan + (Le. no cohesion) and resolving forces vertically

P= 2N sin (+-a)

cos +

IfN> 0, then P>Oonly if a< +; if a > 4, thenR> W

Introducing discontinuity stiffnesses and rock stress:

(k, cos2 a + kn sin2 a)

P = 2H0 sin (+-a)

(ks cos a cos + + kn sin a sin +)

Figure 20.10 Analysis of a symmetric triangular roof prism.