Page 189 - Rock Mechanics For Underground Mining

PRINCIPLES OF CLASSICAL STRESS ANALYSIS

functions and of a complex variable z, in the form

U = R[z (z) + (z)]
1
= [z (z) + z (z) + (z) + (z)] (6.9)
2
Expressions for the stress components may then be established from U (equation
6.9) by successive differentiation. The displacements are obtained by setting up ex-
plicit expressions for the normal strain components ε xx and ε yy in terms of the stress
components, and integrating. It is then found that stresses and displacements are given
by

xx + yy = 4R[ (z)]
− xx + yy + 2i xy = 2[¯ z (z) + (z)] (6.10)

2G(u x + iu y ) =−[
(z) − z (z) − (z)]

where

(z) = (z)
and

= 3 − 4 for plane strain
In applying these results, it is often useful to invoke the transformation between the
rectangular Cartesian and cylindrical polar co-ordinates, given in complex variable
form by

rr +

= xx + yy
(6.11)
− rr +

+ 2i r
= [− xx + yy + 2i xy ]e i2

The solution to particular problems in two dimensions involves selection of suit-
able forms of the analytic functions (z) and (z). Many useful solutions involve
−1
polynomials in z or z . For example, one may take
d
(z) = 2cz, (z) = (6.12)
z
where c and d are real.
Using the relations 6.9 and 6.10, equation 6.11 yields
rr +

= 2c
2d
− rr +

+ 2i r
=−
r 2
so that
d
rr = c + (6.13)
r 2
d

= c − 2
r
r
= 0
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