144 Rock masses
diagram, the minimum moduli are in a direction perpendicular to the sets,
and the maximum moduli are at a direction of 45" to the sets. Like the
discontinuity frequency, the directions of maximum and minimum
moduli are not perpendicular.
A most interesting case occurs when k = 1, i.e. the normal and shear
stiffnesses are equal, and the modulus is isotropic. The significance of even
this very simple case of rock mass deformability for in situ testing and
numerical modelling is apparent.
8.2 Strength
In the same way as we have considered the deformability of a rock mass,
expressions can be developed indicating how strength is affected by the
presence of discontinuities, starting with a single discontinuity and then
extending to any number of discontinuities. The initial approach is via the
'single plane of weakness' theory, attributable to Jaeger, whereby the
strength of a sample of intact rock containing a single discontinuity
can be established. Basically, the stress applied to the sample is resolved
into the normal and shear stresses on the plane of weakness and the
Mohr-Coulomb failure criterion (discussed in Chapter 6) applied to
consider the possibility of slip.
The strength of the sample depends on the orientation of the discon-
tinuity. If the discontinuity is, for example, parallel or perpendicular to the
applied loading, it will have no effect on the sample strength. At some
angles, however, the discontinuity will significantly reduce the strength of
the sample. This is illustrated in Fig. 8.4 which shows that the lowest
strength occurs when the discontinuity normal is inclined at an angle of
45" + (@"/2) to the major applied principal stress. The formula for the
reduction in strength is found by establishing the normal and shear stress
on the plane passing through the specimen and substituting these into the
Mohr-Coulomb failure criterion.
Given the geometry of the applied loading conditions in Fig. 8.4,
IzI = Yz (q - 03) sin 2&,
3 0L
Figure 8.4 Effect of a discontinuity on the strength of a rock sample.