Techniques for incorporating variations in rock and site 335Modern computational facilities are such that large numbers of sim-
ulations can be conducted in a short period of time on a desktop computer.
To perform Monte Carlo simulation, generation of random variables for the
pen probability density distribution is required. These are elegantly
generated by consideration of a cumulative distribution plot. Every point
on the vertical axis of a cumulative distribution plot has equal probability
of occurring: thus, to generate the random variable, the equation for the
cumulative distribution function is inverted so that the random variable is
expressed as a function of the cumulative probability. Then, inserting a
random number, taken from a uniform distribution, with a value between
0 and 1 for P will give a random variable for the distribution desired.
For example, with the negative exponential distribution, the cumulative
probability, P, is given by P = 1 - e-h, which upon inversion produces x
= -(l/A)log,(l - P). Uniformly distributed random variables may now be
substituted for P to provide x variables from a negative exponential
distribution. This technique is valid for all probability density functions,
although the inversion is not always as easy as the one demonstrated here.
lnterpretation of probabilistic analyses. Having conducted a probabilistic
analysis in the manner just described, the resulting histogram of factor of
safety values has to be interpreted for engineering purposes. The
interpretation must take into account both the mean factor of safety and
the spread of values about the mean. In Fig. 18.19, two tables are shown
which can be used to assist in this interpretation.
The first table categorizes slopes in terms of the mean factor of safety
and the probability of the actual factor of safety being less than a specific
value, in this case 1.0 and 1.5. These last two conditions are used to take
into account the spread of the histogram about the mean. The second table
in Fig. 18.19 considers the engineering consequences of the various
combinations in which the three probabilistic criteria might be satisfied.
There is a degree of subjectivity in the levels at which the various prob-
abilistic criteria are set and the associated interpretation. In practice, an
enpeer would have to consider the site-specific circumstances.
To use these tables, the engineer initially assesses the consequences of
failure of the slope and hence establishes the slope category (the first two
columns of the upper table in Fig. 18.19). This sets values for the minimal
mean factor of safety and the maximal probabilities of not exceeding a
factor of safety of 1.0 or 1.5 (the three right-hand columns). Having
established these criteria, and the degree to which they are satisfied for a
specific slope (through the use of Monte Carlo analysis and compared with
the left-hand column of the lower table in Fig. 18.19), the engineer can
utilize the interpretation provided (the right-hand column).
There are many potential variations on this probabilistic theme and
many design techniques that can be based on alternative approaches to
assessing instability. However, the basic methodology has been explained
in this section and thus, by extrapolation, the reader can conceive how
similar probabilistic approaches can be developed and adopted. We
concentrate next on an alternative technique for assessing variations in rock
and site factors using fuzzy mathematics.