Page 599 - Rock Mechanics For Underground Mining

DERIVATION OF EQUATIONS

The weight of caved material below the level of point B has been ignored in this
calculation.

Inclination of thrust to failure surface. It is assumed that the thrust T is transmitted
through the wedge BCDNML to the failure surface without loss or deviation. Hence,
the inclination of T to the normal to the failure surface LM is

(D.5)

= p2 + w − p1
As shown in Figure 16.19, the angle may be either positive or negative. If
is
negative, the thrust T has a shear component that acts up the failure plane, and tends
to stabilise rather than activate slip of the wedge.

Water-pressure forces. The water-pressure force due to water in the tension crack
is

1
V = w Z 2 (D.6)
2 w
The water-pressure force U that acts normal to the failure surface is

1
U = w Z w A
2
1 H 2 (sin cot 0 + cos ) − Z 2 cos
= w Z w (D.7)
2 sin( p2 − )

Conditions of limiting equilibrium. It is assumed that the shear strength of the
rock mass in the direction of failure is given by the linear Coulomb criterion

= c + tan (D.8)

n
The effective normal and shear stresses acting on the failure surface are

W cos p2 + T cos
− U − V sin p2

= (D.9)
n
A
and
W sin p2 + T sin
− V cos p2
= (D.10)
A

The conditions for limiting equilibrium are found by substituting for and into
n
equation D.8, which, on rearrangement, gives
W cos ( p2 − ) + T sin(
− ) + V cos( p2 − ) + U sin

−c A cos = 0 (D.11)

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