Rock Mechanics.pdf - Mining and Blasting

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
= p2 + w − p1
(D.5)
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
V = 1
2 wZ 2 w
The water-pressure force U that acts normal to the failure surface is
U = 1
2 wZw A
= 1
2 wZw
H2(sin cot 0 + cos ) − Z2 cos
sin(p2 − )
(D.6)
(D.7)
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 ′ + ′ n tan ′
The effective normal and shear stresses acting on the failure surface are
and
′ n = W cos p2 + T cos − U − V sin p2
A
= W sin p2 + T sin − V cos p2
A
The conditions for limiting equilibrium are found by substituting for ′ n
equation D.8, which, on rearrangement, gives
581
W cos (p2 − ′ ) + T sin( − ′ ) + V cos(p2 − ′ ) + U sin ′
(D.8)
(D.9)
(D.10)
and into
−c ′ A cos ′ = 0 (D.11)