Page 44 - Rock Mechanics For Underground Mining

STRESS AND INFINITESIMAL STRAIN

Figure 2.3 Free-body diagram for
development of the differential equa-
tions of equilibrium.

Figure2.3showsasmallelementofabody,inwhichoperatebodyforcecomponents
with magnitudes X, Y, Z per unit volume, directed in the positive x, y, z co-ordinate
directions. The stress distribution in the body is described in terms of a set of stress
gradients, defined by ∂ xx /∂x,∂ xy /∂y, etc. Considering the condition for force
equilibrium of the element in the x direction yields the equation

∂ xx ∂ xy ∂ zx
· dx · dy dz + · dy · dx dz + · dz · dx dy + X dx dy dz = 0
∂x ∂y ∂z
Applying the same static equilibrium requirement to the y and z directions, and
eliminating the term dx dy dz, yields the differential equations of equilibrium:

∂ xx ∂ xy ∂ zx
+ + + X = 0
∂x ∂y ∂z
∂ xy ∂ yy ∂ yz
+ + + Y = 0 (2.21)
∂x ∂y ∂z
∂ zx ∂ yz ∂ zz
+ + + Z = 0
∂x ∂y ∂z
These expressions indicate that the variations of stress components in a body under
load are not mutually independent. They are always involved, in one form or another,
in determining the state of stress in a body. A purely practical application of these
equations is in checking the admissibility of any closed-form solution for the stress
distribution in a body subject to particular applied loads. It is a straightforward matter
to determine if the derivatives of expressions describing a particular stress distribution
satisfy the equalities of equation 2.21.

2.6 Plane problems and biaxial stress

Many underground excavation design analyses involving openings where the length
to cross section dimension ratio is high, are facilitated considerably by the relative
simplicity of the excavation geometry. For example, an excavation such as a tunnel of
uniform cross section along its length might be analysed by assuming that the stress
distribution is identical in all planes perpendicular to the long axis of the excavation.
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