Page 180 - Rock Mechanics For Underground Mining

PRE-MINING STATE OF STRESS

extend infinite distances horizontally from the toe of the 100 m high cliff. The effect

of the ground above the elevation D CD can be treated as a wide surcharge load on a
half-space.
(a) The stress components due to a line load of magnitude P on a half space are
given by

2P sin
rr =
r

= r
= 0

where r and
are defined in figure (b).
Relative to x, z reference axes, show that the stress components due to the
strip load defined in figure (c) are given by
p
xx = [2(
2 −
1 ) + (sin 2
2 − sin 2
1 )]
2
p
zz = [2(
2 −
1 ) − (sin 2
2 − sin 2
1 )]
2
p
zx = [cos 2
1 − cos 2
2 ]
2
−3
(b) If the unit weight of the rock is 27 kN m , calculate the stress components
induced at a point P, 80 m vertically below the toe of the cliff, by the surcharge
load.
(c) If the state of stress at a point Q remote from the toe of the cliff and on the same
elevation as P, is given by xx = 2.16 MPa, zz = 3.24 MPa, zx = 0, estimate
the magnitudes and orientations of the plane principal stresses at P.

2 An element of rock 800 m below ground surface is transgressed by a set of parallel,
smooth continuous joints, dipping as shown in the figure below. The fissures are
water filled below an elevation 100 m below the ground surface. The vertical stress
component p zz is a principal stress, and equal to the depth stress. From the calculated
depth stress, p zz , calculate the range of possible magnitudes of the horizontal stress
−3
component, p xx . The unit weight of the rock mass is 26 kN m , and the unit weight
162