Engineering Rock Mechanics

Extensions to rock mass classification techniques 203

maximum and minimum values, and the manner in which it varies
between the two. Hence, fuzzy mathematics permits the ‘uncertainty’
surrounding the assessment of parameters to be included. Also, the appli-
cation of this technique in rock mass classification is straightforward and
direct, because fuzzy numbers may be assigned easily to the parameters in
a rock mass classification scheme.
Some examples of fuzzy numbers and fuzzy arithmetic are shown in
Fig. 12.3. It is important to realize that the distributions of uncertainty illus-
trated in Fig. 12.3 ure not probability density distributions. The parameters A
and B illustrated in Fig. 12.3 are uncertain numbers: we know that A varies
between^3 and 10, with a most likely value of 7-but we are uncertain as
to which precise value it will take; similarly, B will vary between 12 and 20
with a most likely value of 14. The fuzzy sum and fuzzy multiplicand of
two fuzzy numbers are shown in Fig. 12.3.
In Fig. 12.4, as an example we have applied fuzziness to. the six para-
meters of the Q-system, presenting illustrations of the nature of fuzzy
assessment of parameters. The assessment of RQD and Jr are straight-
forwardly analogous to A and B in Fig. 12.3. However, in the case of
parameter Jw, its most likely value coincides with its maximum value-with
the result that the skewed number shown is generated. Similarly, for Jn its
most likely value also coincides with its minimum value. The two
parameters Ja and SRF have been assessed as having only one value: these
are crisp, i.e. conventional, numbers.
Applying fuzzy arithmetic to the basic formula of the Q-system, given
in Section 12.2, results in the fuzzy classification value shown in Fig. 12.4.
Taking all of the most likely values of the individual parameters, and
combining them, gives a value of 5.8, which is the most likely value of the
classification value. Similarly, the maximum and minimum values of the

Addition of two fuzzy numbers

Multiplication of two fuzzy numbers

Figure 12.3 Fuzzy numbers and their additive and multiplicative forms.