Page 41 - Rock Mechanics For Underground Mining

PRINCIPAL STRESSES AND STRESS INVARIANTS

Expressions for the other four components of the stress matrix are readily obtained
from these equations by cyclic permutation of the subscripts.

2.4 Principal stresses and stress invariants

The discussion above has shown that the state of stress at a point in a medium may
be specified in terms of six components, whose magnitudes are related to arbitrarily
selected orientations of the reference axes. In some rock masses, the existence of a
particular fabric element, such as a foliation or a schistosity, might define a suitable
direction for a reference axis. Such a feature might also determine a mode of defor-
mation of the rock mass under load. However, in an isotropic rock mass, any choice of
a set of reference axes is obviously arbitrary, and a non-arbitrary way is required for
defining the state of stress at any point in the medium. This is achieved by determining
principal stresses and related quantities which are invariant under any rotations of
reference axes.
In section 2.2 it was shown that the resultant stress on any plane in a body could
be expressed in terms of a normal component of stress, and two mutually orthogonal
shear stress components. A principal plane is defined as one on which the shear stress
components vanish, i.e. it is possible to select a particular orientation for a plane such
that it is subject only to normal stress. The magnitude of the principal stress is that
of the normal stress, while the normal to the principal plane defines the direction of
the principal stress axis. Since there are, in any specification of a state of stress, three
reference directions to be considered, there are three principal stress axes. There are
thus three principal stresses and their orientations to be determined to define the state
of stress at a point.
Suppose that in Figure 2.2, the cutting plane abc is oriented such that the resultant
stress on the plane acts normal to it, and has a magnitude p . If the vector ( x , y , z )
defines the outward normal to the plane, the traction components on abc are defined
by
⎡ ⎤ ⎡ ⎤
t x x
⎣ t y ⎦ ⎣ y ⎦ (2.16)
= p
t z z
The traction components on the plane abc are also related, through equation 2.7, to
the state of stress and the orientation of the plane. Subtracting equation 2.16 from
equation 2.7 yields the equation
⎡ ⎤ ⎡ ⎤
xx − p xy zx x
yy − p yz
⎣ xy ⎦ ⎣ y ⎦ = [0] (2.17)
zx yz zz − p z
The matrix equation 2.17 represents a set of three simultaneous, homogeneous,
linear equations in x , y , z . The requirement for a non-trivial solution is that the
determinant of the coefficient matrix in equation 2.17 must vanish. Expansion of the
determinant yields a cubic equation in p , given by
2
3
− I 1 + I 2 p − I 3 = 0 (2.18)
p
p
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