Engineering Rock Mechanics

Geometrical properties of discontinuities 1 3 1

along a scanline is independent of the position of other discontinuities), we
can give the probability that k discontinuities will intersect a scanline
interval of length x as

~(k, x) = e-IL”(Ax)%!

For example, if the discontinuity frequency, A, along the scanline is
8.43 m-’, then the probability that exactly two discontinuities will be
intersected in a 0.3 m scanline interval is given by this equation as

P(2,0.3) = e-(8.43’0.3)(8.43.0.3)2/2! = 0.255.

Hence, in about a quarter of all such possible intervals, two discontinuities
will be intersected. Clearly, by repeated use of this formula we can
determine the probability of such events as ’two or less’, ’less than two’ or
‘more than two’ discontinuities being intersected:

k k= 1 k

P(Ik,x)=CP(l,x) P(<k,x)=CP(l,x) P(>k,x)=l-CP(1,x).

I=O 14 I=O

The usefulness of such calculations in designing, say, the length of rock
bolts using criteria such as ’rockbolts should intersect more than three
discontinuities’ is evident.

Cumulative probability distributions and the central limit theorem. The same
type of information is of interest in other areas where we may wish to
know, for example, for the design of tunnel boring machine bearings, what
is the probability that the compressive strength of the rock will exceed a
certain value. This can be established directly from a data set using the
cumulative probability distribution, as illustrated in Fig. 7.17. In the left-
hand part of this figure, the results of a strength testing programme are
tabulated, and these results are shown in the form of a cumulative
probability distribution in the central part of the figure. Extracting values
from the distribution allows statistical questions to be answered: in the
example shown here, what is the probability that the strength will be less

IO0 rock samples

are tested for uC

P(uc < n) P(uc s 50) = I.0

Number

with

P(u, < 25) = 0.56

P(uC > 25) = 1 - P(u, s 25)

= 1 - 0.56 = 0.44

o(MPa) CT~ c (r Proportion - - - - - - -.

IO 6 0.06

P(25 < oc s 30)

= P(uc s 30) - P(u, S 25)

= 0.73 - 0.56

2 0 40 0.40 0.2

30 73 0.73

30 91 0.91

50 100 I .u0

0 IO 20 30 40 XI

u(MPa)

= 027

Figure 7.17 Estimating the probability that the compressive strength will be greater

or less than certain values.