Engineering Rock Mechanics

52 In situ rock stress

A4.8 The stress transformation equations are given by

which can be written as Olmn = RuXy,RT.
This means that, if we know the stresses relative to the xyz axes (i.e.
uxyz) and the orientation of the lmn axes relative to the xyz axes (i.e. R),
we can compute the stresses relative to the lmn axes (Le. ulm,).
However, in this problem we have been given the principal stresses,
which is a stress state relative to some lmn system (i.e. ulmn), where
the lmn axes correspond to the principal directions. As we know the
principal directions relative to the xyz axes, we are able to compute R.
Thus, we need to evaluate uxy,, and we do this using the inverse of the
stress transformation equations: uxyz = RTalmn R.
Notice that we have not had to use the inverse of R, i.e. R-*. The
rotation matrix is orthogonal, and this property means that R-' = RT.
With the given data for the principal directions:

Bl = 35" pm = 43" Bn = 27"

the matrix R is computed as

COS~ICOSBI sinwcosB1 sin"] = [ 0.071 0.816 0.5741

COS am COS pm sin am COS Bm sin Bm -0.584 -0.440 0.682

COS an COS Bn sin an COS #?,, sin Bn 0.808 -0.377 0.454

and the matrix ulmn is given by

Oxy, =

-0.44 2.65 11.23

A useful check of the calculations involved in stress transformation is
obtained by computing the stress invariants. The first and third invari-
ants are easily obtained using standard matrix functions. For example,
the first invariant is computed as the sum of the leading diagonal ele-
ments (see A3.8); this sum is also known as the trace of the stress