Page 45 - Rock Mechanics For Underground Mining

PLANE PROBLEMS AND BIAXIAL STRESS

Figure 2.4 A long excavation, of
uniform cross section, for which a
contracted form of the stress transfor-
mation equations is appropriate.

Suppose a set of reference axes, x, y, z, is established for such a problem, with the long
axis of the excavation parallel to the z axis, as shown in Figure 2.4. As shown above,
the state of stress at any point in the medium is described by six stress components.
For plane problems in the x, y plane, the six stress components are functions of (x, y)
only. In some cases, it may be more convenient to express the state of stress relative
to a different set of reference axes, such as the l, m, z axes shown in Figure 2.4. If the
angle lOx is , the direction cosines of the new reference axes relative to the old set
are given by

l x = cos , l y = sin , l z = 0
m x =− sin , m y = cos , m z = 0

Introducing these values into the general transformation equations, i.e. equations
2.14 and 2.15, yields

2
2
ll = xx cos + yy sin + 2 xy sin cos
2
2
mm = xx sin + yy cos − 2 xy sin cos
2
2
lm = xy (cos − sin ) − ( xx − yy ) sin cos (2.22)
mz = yz cos − zx sin
zl = yz sin + zx cos
and the zz component is clearly invariant under the transformation of axes. The
set of equations 2.22 is observed to contain two distinct types of transformation:
those defining ll , mm , lm , which conform to second-order tensor transformation
behaviour, and mz and zl , which are obtained by an apparent vector transformation.
The latter behaviour in the transformation is due to the constancy of the orientation
of the element of surface whose normal is the z axis. The rotation of the axes merely
involves a transformation of the traction components on this surface.
For problems which can be analysed in terms of plane geometry, equations 2.22
indicate that the state of stress at any point can be defined in terms of the plane
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