Engineering Rock Mechanics

(^232) Rock mechanics interactions and rock engineering systems (RES)
Asymmetric because there is a rotation direction
Y
Y'
t4 X' = xcos0 - ysin0
4
Interaction with other axis
Figure 14.9 Transformation matrix for rotation of axes.
the old axes: the matrix is still asymmetric. On rotating through 180", as
shown in the third case, the axes remain in the same orientation, with only
the positive directions changing. Thus, the primary operation is a multi-
plication by -1, and there is no interaction between the axes. This matrix
is symmetric, because it does not matter whether we rotate clockwise or
anticlockwise through 180": i.e. there is no path-dependency in the
transformation for this specific angle of rotation.
This example has been included because of the importance of under-
standing the nature of the off-diagonal terms in the rock mechanics inter-
action matrices. Consideration of whether path dependency is inherent in
any matrix will assist in determining the symmetry or otherwise of the terms.
14.5 A rock mechanics-rock engineering
interaction matrix
Referring again to Fig. 1.12 and the three-tier approach to all rock engi-
neering problems, the analysis of coupled mechanisms is necessary in
extrapolating the analysis of individual subjects to the solution of
complete rock engineering problems. In Fig. 14.11, the primary parameters
on the leading diagonal are rock mass structure, in situ stress, water flow
and construction. This is a 4 x 4 matrix, with four primary variables and 12
interaction terms. The indentification and location of these interactions
within the matrix is of help when interpreting the rock mechanics and rock
engineering components. If we were to add two more leading diagonal
terms-for example, thermal effects and time-dependency-the matrix
would increase to a 6 x 6 size with six primary parameters and 30 inter-
action terms. It is unlikely that these interactions could be coherently
identified and studied without the use of the interaction matrix or some
similar approach.
There are several points to be noted about this matrix. First, the boxes
have been numbered according to conventional matrix notation, with the