Page 309 - Rock Mechanics For Underground Mining

GENERAL DETERMINATION OF RELEASED AND EXCESS ENERGY

Figure 10.14 Problem geometry
for determination of released en-
ergy from excavation-induced trac-
tions and displacements.

The nature of the problem to be solved is illustrated in Figure 10.14. Figure 10.14a
illustrates a body of rock, subject to field stresses p xx , p yy , p xy , in which it is proposed
to excavate an opening whose surface is S. Prior to excavation, any point on the surface
S is subject to tractions t x , t y . For any corresponding point on the complementary

surface S which lies immediately within S and includes the material to be excavated,

as shown in Figure 10.14b, the pre-mining tractions are t , t , and these are related to
x y
t x , t y by the equations

t x + t = 0, t y + t = 0 (10.69)
x
y
Excavation of the material within S is mechanically equivalent to applying a set

of tractions t xi , t yi on S, and simultaneously inducing a set of displacements u xi , u yi .
The magnitudes of the applied tractions are such as to make the surface S traction-free
after excavation; i.e.
t xi + t x = 0, t yi + t y = 0 (10.70)

When the excavation is created gradually, the surface S does work on the material

within S , as the tractions on S are gradually reduced to zero. The work, W i , done
by S against the tractions applied by S is calculated from the average force and the

displacement through which it acts; i.e.
1 %

W i = (t u xi + t u yi )dS (10.71)
x
y
2 s
From equations 10.69 and 10.70 it is seen that

t = t xi , t = t yi
x y
so that equation 10.71 becomes
1 %
W i = (t x u xi + t y u yi )dS
2 s

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