Page 601 - Rock Mechanics For Underground Mining

DERIVATION OF EQUATIONS

w Z w 2

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

2

2
w Z w cos( p2 − ) sin( p2 − ) sin 0
− (D.15)
c sin( + 0 ) sin( p2 + 0 ) sin( p2 − )

Critical tension crack depth. The left-hand side of equation D.12 can be differen-
tiated with respect to Z 2 holding p2 constant; the result is equated to zero to obtain
the critical value of Z 2 as
cos
Z 2
= (D.16)
c cos p2 sin( p2 − )

Critical failure plane inclination. By holding Z 2 constant, differentiating equation
D.12 with respect to p2 , putting ∂ H 2 /∂ p2 = 0 and rearranging, an expression for
the critical failure plane angle may be obtained as
1 −1 X

p2 = + cos (D.17)
2 1/2
2
2 (X + Y )
where
2

sin( + 0 ) sin( p1 + 0 ) cos
H 1
X =
2
c sin 0 sin( p1 − )

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

cos p1 cos
Z 1 c H c
− − K cos( p1 + − w )
c sin( p1 − ) c
2

w Z w
+ cos (D.18)
c
and
2 2

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

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

c H c w Z w
− K sin( p1 + − w ) + sin (D.19)
c c
Angle of break. By simple trigonometric manipulation it is found that
Z 2 sin( p2 − )
tan b = tan p2 + (D.20)
H 2 sin( + 0 )
cos p2 − Z 2 cos
sin 0
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