Engineering Rock Mechanics

Questions and answers: in situ rock stress 53

matrix. The third invariant is computed as the determinant of the stress

matrix.

11 = tr(qrnn) = 33.0 MPa, and 11 = &(axyz) = 33.0 MPa.

13 = IUlmnl = 12000 MPa3 and 13 = luxyzI = 1199.7 MPa3.

The orientations of the axes used above have been rounded to the

nearest whole degree, and so the axes are not exactly orthogonal. The

effect of this is to introduce errors in the computation, which can be

seen in the values of the third invariant. Here we can use orientations

that are correct to two decimal places of degrees, such that the ortho-

gonality condition is improved and hence a more accurate answer is

obtained, but in general these data will not be available from the field

information.

The orientations correct to two decimal places are

= 85.50' urn = 216.65" a,, = 334.81"

= 35.06" prn = 43.15" pn = 26.72"

the matrix R is

1

0.064 0.816 0.574

-0.585 -0.435 0.684

0.808 0.380 0.450

from which we find

[ 8.71 0.88 -0.541

a,,, = 0.88 13.04 2.69 MPa.

-0.54 2.69 11.24

Calculating the invariants again gives

1, = tr(alrnn) = 33.00000 MPa, and ZI = &(axyz) = 33.00000 MPa.

13 = lulrnnl = 1200.000 MPa3 and 13 = laxyzI = 1200.000 ma3.

showing that the invariants now agree exactly to 7 significant figures.

44.9 A fault is present in the same rock (continuing from 44.8)
with an orientation of 295O/5Oo. Determine the stress components
in a local co-ordinate system aligned with the fault. Assume for this
question that the presence of the fault does not affect the stress
field.

A4.9 Here we use the methodology given in A4.8 to find the 3-D stress
tensor in an lmn co-ordinate system where the rz axis coincides with the
normal to the fault and the Z axis coincides with the strike of the fault.
We need to determine ulmn, where Zmn are given by the orientation of the
fault.