Engineering Rock Mechanics

306 SurFace excavation instability mechanisms

approximate, because once an inter-layer slip failure criterion has been
applied, the elastic Boussinesq solution itself is not valid.
In applying these ideas in practice, it is prudent to study the influence
of rock anisotropy. Dr Bray developed a solution for an ’equivalent isotrop
ic medium’ for a line load inclined at an arbitrary angle to the surface. The
solution is developed by considering the effect of a single set of dis-
contintuies which have been subsumed into an equivalent transversely
isotropic rock-but the solution does explicitly include the normal and
shear stiffnesses and mean spacing of the discontinuities. The solution is
given below and the geometry illustrated in Fig. 17.21:

h

m (cos2 p-gsin2 p)’ +h2 sin2 pcos2 p

X cos p + Yg sin

0, = 0, z,, = 0, or =-I

where

and where k, and k, are the normal and shear discontinuity stiffnesses,

respectively, and X is the mean discontinuity spacing.

The resulting contours of radial stress for an equivalent isotropic medium

with the plane of anisotropy at various angles to the surface of the half-

space are shown in Fig. 17.22 (note that the forms of these contours

will vary with the exact values of all the elastic constants, including the

normal and shear discontinuity stiffnesses). Experimental data produced

by Gaziev and Erlikhman (1971) are shown in Fig. 17.23 for comparative

purposes.

The significance of Figs 17.22 and 17.23 is clear: contours of radial stress

can be deeper than those predicted with a CHILE solution; and they can

be severely distorted, so that they are not only extended downwards

X \ IP

\

Figure 17.21 Geometry of Bray’s equivalent continuum solution (from Goodman,

1989).