Page 313 - Rock Mechanics For Underground Mining

INSTABILITY DUE TO PILLAR CRUSHING

For equilibrium at some stage of loading of the specimen through the spring, the net
force at the rock–spring interface (i.e. P r − P s ) must be zero. Suppose the equilibrium
is probed by applying a small external force at the point O 2 in Figure 10.17a, causing
an incremental convergence S. From equations 10.75 and 10.76, the incremental
changes in the forces in the rock and the spring are given by

P r = f (S) S = S
where is the slope of the specimen force–displacement characteristic, defined in
Figure 10.17c, and

P s =−k S

Thus the net probing force causing an incremental displacement S is given by
P = P r − P s = (k + ) S (10.77)

Equation 10.77 relates applied external force to the associated convergence in the
system, so that (k + ) can be interpreted as the effective stiffness of the spring–
specimen system. The criterion for stability, defined by inequality (a) in expression
¨
10.74, is that the virtual work term W, given by
1 1
¨ 2
W = P S = (k + ) S (10.78)
2 2
be greater than zero. Thus equation 10.78 indicates that stable equilibrium of the
spring–specimen system is assured if
k + > 0 (10.79)
Figure 10.18 A mine pillar treated
as a deformable element in a soft load- A similar procedure may be followed in assessing the global stability of a mine
ing system, represented by the country structure, as has been described by Brady (1981). Figure 10.18 represents a simple
rock. stoping block, in which two stopes have been mined to generate a single central pillar.
The mine domain exists within an infinite or semi-infinite body of rock, whose remote
surface is described by S ∞ . Suppose a set of probing loads [ r] is applied at various
points in the pillar and mine near field, inducing a set of displacements [ u] at these
points. The global stability criterion expressed by the inequality (a) in equation 10.74
then becomes
T
[ u] [ r] > 0 (10.80)
In general, incremental displacements [ u] at discrete points in a rock mass may
be related to applied external forces [ r] by an expression of the form

[K g ][ u] = [ r] (10.81)
where [K g ] is the global (or tangent) stiffness matrix for the system. The stability
criterion given by equation 10.80 is then expressed by

T
[ u] [K g ][ u] > 0 (10.82)
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