Page 84 - Statistics and Data Analysis in Geology

Statistics and Data Analysis in Geology- Chapter 3

point in a positive coordinate direction, or both components point in a negative
coordinate direction. Otherwise, the shear stress is negative. In order for the cube
to be in rotational equilibrium, shear stresses on adjacent faces must balance; so,
for example, uxy = urx. This means that the stress matrix is symmetric about the
diagonal:

Turcotte and Schubert (1982) provide a more detailed discussion of stress in the
subsurface and the measurement of stress components.
By finding the eigenvalues and eigenvectors of the 3 x 3 stress matrix, we can
rotate the imaginary cube into a coordinate system in which all the shear stresses
will be zero. The eigenvalues represent the magnitudes of the three orthogonal
stresses. Their associated eigenvectors point in the directions of the stresses. The
largest eigenvalue, hl, represents the maximum normal stress and the smallest,
h3, represents the minimum normal stress. The maximum shear stress is given by
(Al - h3)/2 and occurs along a plane oriented perpendicular to a line that bisects
the angle between the directions of maximum and minimum normal stress (that
is, between the first and third eigenvectors). In a homogenous, isotropic material,
failure (te., faulting) will tend to occur along this plane. The orientation of this plane
can be determined from the elements of the first eigenvector. In the conventional
notation used by geologists, the strike of the first eigenvector is tan-l (Y~z/YI~)
and its dip is

(Here, Vij refers to the jth element of the ith eigenvector.) The strike and dip of the
second and third eigenvectors can be found in the same manner.
Three-dimensional stress measurements have been made in a pillar in a deep
mine, yielding the following stress matrix:

1
61.2 4.1 -8.2
4.1 51.5 -3.0
-8.2 -3.0 32.3

The data are given in megapascals (MPa) and were recorded by strain gauges placed
so the measurements have the same orientation as our imaginary cube (X increasing
to the east, Y to the north, and Z increasing upward). Find the principal stresses
and their associated directions. What is the maximum shear stress and what is the
strike and dip of the plane on which this stress occurs?

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