Engineering Rock Mechanics

Deformability 143

The case illustrated via the mathematics above and shown in Fig. 8.2 only

involves loading parallel to the discontinuity normals. Clearly, even in

these idealized circumstances, we need to extend the ideas to loading at

any angle and the possibility of any number of non-parallel sets. An

argument similar to that given above can be invoked in the derivation of

shear loading parallel to the discontinuities, as succinctly described by

Goodman (1989), to give

The mathematics associated with further extensions to account for
discontinuity geometry rapidly becomes complex. A complete solution has
been provided by Wei (1988), which can incorporate the four stiffnesses of
a discontinuity (normal, shear and the two cross terms), any number of sets
and can approximate the effect of impersistent discontinuities.
In the stress transformations presented in Chapter 3, the resolution of
the stress components involves only powers of two in the trigonometrical
terms, because theforce is being resolved and the urea is also being resolved.
However, for the calculation of the deformability modulus, powers of four
are necessary because of the additional resolution of the discontinuity
frequency (explained in Chapter 7) and the displacements. An example
equation from Wei's theory, the roots of which provide the directions of
the extreme values of the modulus, is

(Atan4a-Btan3a-Ctan2a-Dtana-F)cos4a= 0

where A, B, C, D, E and F are constants formed by various combinations

of the discontinuity stiffnesses and a is the angle between the applied stress

and one of the global Cartesian axes. The reader is referred to Wei's work

for a complete explanation.

The utility of this type of analysis is illustrated by the polar diagrams in

Fig. 8.3 representing the moduli variations for two discontinuity sets in two

dimensions. (It is emphasized that this figure is one example of a general

theory.) When k is high, as in the left-hand diagram, the lowest moduli are

in a direction at 45" to the discontinuity sets, and the highest moduli are

perpendicular to the sets. Conversely, when k is low, as in the right-hand

k=5 k=2 k=l k = 0.25

k is the ratio of the shear stiffnesses to normal stiffnesses

Figure 8.3 Variation in rock mass modulus for two orthogonal discontinuity sets

with equal frequencies and equal stiffnesses (from Wei, 1988).