Page 318 - Rock Mechanics For Underground Mining

ENERGY, MINE STABILITY, MINE SEISMICITY AND ROCKBURSTS

lower surfaces of the excavation are given by
2
2 1/2
u y =±[(1 − )/G]p[(L /4) − x ] (10.84)
where the negative sign corresponds to the upper excavation surface, and L is the
stope span.
Using the methods described previously, it is readily shown that the released energy
and the excess energy are given by
2 2
W r = W e = [(1 − )/8G] L p (10.85)
Bray (1979) showed that the increase in static strain energy, W s , is also given by
equation 10.85.
The expressions for W r , W e and W s (equation 10.85) apply up to the stage where
the excavation remains open, i.e. until convergence between the footwall and hang-
ingwall sides of the stope produces contact. This occurs when the convergence at
midspan is equal to the mined stope height H. From equation 10.84, the critical span
L 0 at which contact occurs is given by

H = [2(1 − )/G]pL 0 /2

or
L 0 = GH/(1 − )p (10.86)

At this stage, the released energy W r and stored strain energy W s are given by
W r = W s = ( /8)pL 0 H

As was noted earlier, the total released energy and strain energy increase are of
limited practical significance, since a complete stope is not generated instantaneously.
Mining interest is, instead, in the energy changes for incremental extension of the
stope. Thus, while a stope remains open

dW r /dL = [(1 − )/4G] Lp 2 (10.87)

For stoping spans greater than the critical span L 0 , Bray (1979) showed that W r
approaches asymptotically to the expression

W r = LHp (10.88)
while W s approaches the maximum value

W s max = HL 0 P/4 (10.89)
Therefore the incremental rates of energy storage and release are given by

dW r /dL = Hp
(10.90)
d W s /dL = 0
The nature of equations 10.90 indicates the key mechanical principle involved in
longwall mining. If it were possible to mine a narrow orebody as a partially closed,

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