Engineering Rock Mechanics

Techniques for incorporating variations in rock and site 333

question. Instead, it is said that the parameter can take on a range of values

defined by a probability density function, with the result that statements

can be made about the probability that the parameter will take on values

within a certain range. Thus, given any mechanical model, the effect of the

various parameters in the model can be considered as random variables

taken from probability density functions.

In those cases where only one or two parameters are considered as

random variables, it is possible to use probabilistic statements to examine

the system, and the method of solution may be by so-called direct methods.

In those cases where a large number of parameters are considered to be

random variables from different types of distribution, the mathematics

associated with the direct probabilistic analysis becomes intractable, and

so a numerical technique, e.g. the Monte Carlo method, must be used.

Monte Carlo simulation involves repeatedly substituting generated ran-

dom variables into a deterministic model and collation of the results into

a histogram.

Direct approach. The direct approach is demonstrated in Fig. 18.16 for
the simple case of a block sliding on a plane, where the angle of friction
is considered as a random variable. Considering the left-hand histogram
to represent the results of 133 shear box tests to determine the angle of
friction, the histogram can either be used directly, or a normal distribution
(for example) can be fitted to the results. In the former case, the probability
density histogram is defined by the class intervals; in the latter case, it
is defined by the mathematical expression for the function in question, e.g.
the normal distribution with particular values of mean and standard
deviation.
The important distinction between the deterministic and probabilistic
methods is illustrated by the fact that the class intervals in the probability
density histogram are used and not the actual separate 133 test results.
Inserting the mean value of each of the class intervals into the deterministic
model in turn allows a cumulative distribution function to be generated, as
illustrated in the right-hand graph of Fig. 18.16. Probabilistic statements
can then be made about the factor of safety, e.g. what is the probability
that the factor of safety will be above 1.25 for a case when the angle of
friction is a random variable from the same population as that determined
by the sample tests. The probabilistic analysis can be initiated by assuming
a continuous probability density function, with or without reference to
test data.

Monte Carlo simulation. Monte Carlo simulation is a procedure which

permits the variation in many parameters to be considered simultaneously.

The calculation is performed many times for repeatedly generated sets of

input data. Each calculation produces one value of the factor of safety, from

which a histogram or cumulative distribution of factors of safety is generat-

ed. Figure 18.17 demonstrates the principle of the simulation, and Fig. 18.18

shows how it may be applied to the analysis of a curvilinear slip in a poor-

quality rock mass, using the method of slices and following the procedure

outlined by Priest and Brown (1983).