Page 277 - Rock Mechanics For Underground Mining

SYMMETRIC TRIANGULAR ROOF PRISM

which, on rearrangement, yields

H 0 2 2
N = (K s cos + K n sin ) cos
D
(9.26)
H 0 2 2
S = (K s cos + K n sin ) sin
D
where

D = K s cos cos + K n sin sin

Introducing these expressions for N and S into equation 9.24, and simplifying,
H 0 2 2
H = (K s cos + K n sin ) cos( − ) (9.27)
D
The expression for N in equation 9.26, when substituted in equation 9.19 for the
limiting vertical force, yields
2H 0 2 2
P = (K s cos + K n sin ) sin( − ) (9.28)
D
For the case where K n K s , which is the usual condition in practice, equation 9.28
becomes

2H 0 sin sin( − )
P = (9.29)
sin
This analysis indicates that when the elastic state of stress has been determined, the
net vertical external load required to produce a state of limiting equilibrium can be
estimated, from equation 9.29, using the known prism geometry and joint frictional
properties. The factor of safety against roof collapse can then be calculated from
equation 9.20. In particular, if the result of the determination of P is that P > W,
the weight of the prism, the analysis suggests that the wedge is stable in the absence
of any installed support. If P < W, stability of the prism can be assured only by the
application of a positive support load.
It is instructive to examine the relation between the limiting vertical load, P , and
the horizontal force components on the prism surfaces. Introducing equation 9.20 in
equation 9.24, rearranging, and then substituting the resulting expression for N in
equation 9.19, gives

P = 2H tan( − ) (9.30)

Equations 9.29 and 9.30 emphasise the important role of the horizontal force com-
ponents acting on the prism surfaces. It is clear that any process which acts to reduce
these surface forces, applied by the adjacent rock, will reduce the limiting vertical
load. This is equivalent, of course, to increasing the tendency for collapse of the prism
from the roof. Excavation development procedures may affect these internal forces.
For example, uncontrolled blasting practice near the excavation periphery will inject
high-pressure gases directly into the joints, promote vertical displacement greater than

259