Engineering Rock Mechanics

Geometrical properties of discontinuities 1 23

and minimum discontinuity frequency are perpendicular for the case of
one set: this is the only circumstance in which the two are perpendicular.
If now two discontinuity sets are considered, as illustrated in Fig. 7.9, the
contribution from each set is resolved onto the sampling line as

It can be seen from the associated polar plot that the rock mass is now far
less geometrically anisotropic and that the directions of maximum and
minimum discontinuity frequency are no longer orthogonal.
This procedure can be readily extended to any number of discontinuity
sets, n, with the result

where 4 and Oi are the fundamental set frequency and the angle between
the set normal and the sampling direction, respectively, for the ith set. In
Fig. 7.10, the progression from one set to an infinite number of sets is
illustrated for mutually symmetric set normals and where each set
frequency is the reciprocal of the number of sets present (in order to keep
the areal density of discontinuities constant).
The progression from anisotropy to isotropy is elegantly demonstrated
in the top row of polar plots as the discontinuity frequency locus changes
from two circles (for one set), through various lobular shapes, to a single
circle (for an infinite number of sets). The diagram shows how a rock mass
with, say, four or five similar and equally spaced sets may be regarded as
effectively isotropic. The discontinuity frequency variation has been shown
for symmetrically orientated sets possessing equal frequencies: there will
obviously be more anisotropy in the less-ideal and real cases.
The lower row of stereographic plots in Fig. 7.10 shows the same
variation for the same sets, but for the sampling direction being variable
in three dimensions. The diagrams are contoured lower-hemisphere
projections: a single set results in a single 'peak', and an infinite number

180" 'ma

Figure 7.9 Variahon in discontinuity frequency for a sampling line passing through

two sets of discontinuities-two-dimensional case.

.,sin0

270"