Page 37 - Rock Mechanics For Underground Mining

STRESS TRANSFORMATION

In the doubly-subscripted notation for stress components, the first subscript indicates
the direction of the outward normal to the surface, the second the sense of action of
the stress component. Thus xz denotes a stress component acting on a surface whose
outward normal is the x axis, and which is directed parallel to the z axis. Similarly,
for the other cases where the normals to elements of surfaces are oriented parallel
to the y and z axes respectively, stress components on these surfaces are defined in
terms of the respective traction components on the surfaces, i.e.

yx = t x , yy = t y , yz = t z (2.2)
zx = t x , zy = t y , zz = t z (2.3)

The senses of action of the stress components defined by these expressions are shown
in Figure 2.1c, acting on the visible faces of the cubic free body.
It is convenient to write the nine stress components, defined by equations 2.1, 2.2,
2.3, in the form of a stress matrix [ ], defined by

⎡ ⎤
xx xy xz
[ ] = ⎣ yx yy yz ⎦ (2.4)
zx zy zz

The form of the stress matrix defined in equation 2.4 suggests that the state of stress
at a point is defined by nine independent stress components. However, by consider-
ation of moment equilibrium of the free body illustrated in Figure 2.1c, it is readily
demonstrated that

xy = yx , yz = zy , zx = xz

Thus only six independent stress components are required to define completely the
state of stress at a point. The stress matrix may then be written

⎡ ⎤
xx xy zx
[ ] = ⎣ xy yy yz ⎦ (2.5)
zx yz zz

2.3 Stress transformation

The choice of orientation of the reference axes in specifying a state of stress is
arbitrary, and situations will arise in which a differently oriented set of reference axes
may prove more convenient for the problem at hand. Figure 2.2 illustrates a set of
old (x, y, z) axes and new (l, m, n) axes. The orientation of a particular axis, e.g. the
l axis, relative to the original x, y, z axes may be defined by a row vector (l x , l y , l z )
of direction cosines. In this vector, l x represents the projection on the x axis of a unit
vector oriented parallel to the l axis, with similar definitions for l y and l z . Similarly,
the orientations of the m and n axes relative to the original axes are defined by row
vectors of direction cosines, (m x , m y , m z ) and (n x , n y , n z ) respectively. Also, the state
of stress at a point may be expressed, relative to the l, m, n axes, by the stress matrix
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