Page 600 - Rock Mechanics For Underground Mining

APPENDIX D LIMITING EQUILIBRIUM ANALYSIS

Mining depth for new failure. Substitution for W, A, T, V and U from equations
D.1, 2, 3, 6 and 7 into equation D.9 and rearrangement gives a quadratic equation for
H 2 , the new mining depth at which failure will occur

2

H 2 sin( + 0 ) sin( p2 + 0 ) sin( p2 − )
2
c sin 0
2

H 2 sin( + 0 ) cos w Z w sin( + 0 ) sin
−2 −
c sin 0 2c sin 0

2
H 1 sin( + 0 ) sin( p1 + 0 ) sin( p2 − ) sin( p2 − )
− 2
c sin 0 sin( p1 − )
2

Z 1 cos cos p1 sin( p2 − ) sin( p2 − )
+
c sin( p1 − )
2

Z 2

− cos cos p2 sin( p2 − )
c
2

c H c

+ K sin(
− ) sin( p2 − )
c
2 Z 2 cos w Z w

+ cos − sin
c 2c
2

w Z w

+ cos( p2 − ) sin( p2 − ) = 0 (D.12)
c
Equation D.12 gives a solution for the dimensionless group H 2 /c in the form

H 2 2 1/2
= a + (a + b) (D.13)
c
where
w Z w sin

sin 0 cos −
2c
a = (D.14)
sin( p2 + 0 ) sin( p2 − )

and

2
H 1 sin( p1 + 0 ) sin( p2 − )
b =
c sin( p1 − ) sin( p2 + 0 )
2

2
Z 1 cos cos p1 sin( p2 − ) sin 0
−
c sin( + 0 ) sin( p2 − 0 )

2 2
Z 2 cos cos p2 sin 0
+
c sin( + 0 ) sin( p2 − 0 )
2

2
c H c K sin(
− ) sin( p2 − ) sin 0
−
c sin( + 0 ) sin( p2 + 0 ) sin( p2 − )

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