Engineering Rock Mechanics

Symmetry of interaction mutrices 229

In Fig. 14.6, we illustrate another 2 x 2 interaction matrix in which the
parameters point load strength and compressive strength appear on the leading
diagonal. In calculating the regression line linking these two parameters,
we could consider either one as the independent variable, as is illustrated
in the off-diagonal boxes. The data points are the same in each case: the
axes have simply been exchanged, and a different regression line is
obtained in each of the two cases. Regression analysis using these^13 data
points produces

and

0, = -3.13 + 23.511,

1, = 0.563 + 0.03670,.

These two equations are not the same line. They are plotted in the off-
diagonal boxes, with the solid line being the appropriate one in each case.
The two off-diagonal boxes are not the same: the matrix is asymmetric-
caused by the two estimation directions. The point is not the accuracy with
which these two parameters are related, but the asymmetry of the matrix
resulting from the path-dependency.
In Fig. 14.7, there is a 4 x 4 matrix illustrating the inter-relations between
the orientation, spacing, persistence and roughness of a discontinuity.
These are four of the primary geometrical properties of discontinuities
illustrated in Fig. 7.3. On examining the schematic in Fig. 7.3, it might
appear at first sight that the four parameters are independent: however,
as Fig. 14.7 illustrates, all four of these parameters are interdependent. This
interaction matrix demonstrates the comprehensive identification of all of
the binary interactions-some of which might otherwise be overlooked.

14.4 Symmetry of interaction matrices

We have seen that the stress tensor is symmetrical about the leading

diagonal of the matrix because of the equilibrium conditions, as illustrated

in Fig. 3.6. Both the stress and strain tensors, and other similar second-order

Compressive

strength is

the

independent

variable.

Point

strength

(MN/m2)

Compressive

Point load

strength is

. the
independent
variable

Asymmetric because of estimation direction

Figure 14.6 Point load strength-compressive strength relations, illustrating

asymmetry of the interaction matrix induced by the estimation direction.