Page 274 - Rock Mechanics For Underground Mining

EXCAVATION DESIGN IN BLOCKY ROCK

Figure 9.14 Free-body diagrams of
a crown prism (a) subject to surface
forces (N and S), its own weight (W)
and a support force (R), and (b) in a
state of limiting equilibrium.

boundary of the opening. In the following discussion, attention is confined to anal-
ysis of crown stability problems. Methods for analysis of sidewall problems may be
established by some minor variations in the proposed procedures.
A rock block in the crown of an excavation is subject to its own weight, W,
surface forces associated with the prevailing state of stress, and possibly fissure water
pressure and some support load. Assume, for the moment, that fissure water pressure is
absent, that the block surface forces can be determined by some independent analytical
procedure, and that the block weight can be determined from the joint orientations
and the excavation geometry.
Figure 9.14a represents the cross section of a long, uniform, triangular prism gener-
ated in the crown of an excavation by symmetrically inclined joints. The semi-apical
angle of the prism is . Considering unit length of the problem geometry in the an-
tiplane direction, the block is acted on by its weight W, a support force R, and normal
and shear forces N and S on its superficial contacts with the adjacent country rock.
The magnitude of the resultant of W and R is P. To assess the stability of the prism
under the imposed forces, replace the force P by a force P , as shown in Figure 9.14b,
and find the magnitude of P required to establish a state of limiting equilibrium of
the block.
From Figure 9.14b, the equation of static equilibrium for the vertical direction is
satisfied if
P = 2(S cos − N sin ) (9.17)

If the resistance to slip on the surfaces AB, AC is purely frictional, in the limiting
equilibrium condition,

S = N tan (9.18)

and equation 9.17 becomes
P = 2N(tan cos − sin )
(9.19)
= 2N sec sin( − )

Thus for N > 0, the condition P > 0 can be satisfied only if < . Thus if >
, P < 0, and even in the absence of its own weight, the prism would be displaced
from the crown under the influence of the joint surface forces. For the case < ,
the prism is potentially stable, but stability can only be assured by a more extensive
analysis.
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