Page 187 - Rock Mechanics For Underground Mining

PRINCIPLES OF CLASSICAL STRESS ANALYSIS

or

2 2
∂ ∂
+ ( xx + yy ) = 0 (6.5)
∂x 2 ∂y 2
Equation 6.5 demonstrates that the two-dimensional stress distribution for isotropic
elasticity is independent of the elastic properties of the medium, and that the stress
distribution is the same for plane strain as for plane stress. The latter point validates
the use of photoelastic plane-stress models in estimating the stress distribution in
bodies subject to loading in plane strain. Also, as noted in section 6.1, equation 6.5
demonstrates that the sum of the plane normal stresses, xx + yy , satisfies the Laplace
equation.
The problem is to solve equations 6.1 and 6.5, subject to the imposed boundary
conditions. The method suggested by Airy introduces a new function U(x, y), in
terms of which the stress components are defined by
2
∂ U
xx =
∂y 2
2
∂ U
yy = (6.6)
∂x 2
2
∂ U
xy =−
∂x ∂y
These expressions for the stress components satisfy the equilibrium equations 6.1,
identically. Introducing them in equation 6.5 gives

4
∇ U = 0
where

∂ 2 ∂ 2
2
∇ = 2 + 2 (6.7)
∂x ∂y
Equation 6.7 is called the biharmonic equation.
Several methods may be used to obtain solutions to particular problems in terms
of an Airy stress function. Timoshenko and Goodier (1970) transform equations
6.5 and 6.6 to cylindrical polar co-ordinates, and illustrate a solution procedure by
reference to a thick-walled cylinder subject to internal and external pressure, as shown
in Figure 6.2. For this axisymmetric problem, the biharmonic equation assumes the
form
2
4
3
d U 2d U 1 d U 1 dU
+ − + = 0
3
2
dr 4 rdr 3 r dr 2 r dr
for which a general solution for U is given by
2
2
U = A nr + Br nr + Cr + D
In this expression, the constants A, B, C, D are determined by considering both the
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