Page 190 - Rock Mechanics For Underground Mining

METHODS OF STRESS ANALYSIS

For the axisymmetric problem defined by Figure 6.2, introducing the boundary
conditions rr = p i when r = a, and rr = p o when r = b, into equations 6.13 yields
2
p o b − p i a 2
c =
2
2
(b − a )
(6.14)
2 2
a b (p i − p o )
d = 2 2
(b − a )
Equations 6.13 and 6.14 together are identical in form to equations 6.8. Thus
the choice of the analytic functions in the form given by equation 6.12 is sufficient
to represent conditions in a thick-walled cylinder subject to internal and external
pressure. Expressions for the displacements induced in the cylinder by application of
the internal and external pressures are obtained directly from (z), (z) and the third
of equations 6.10.
It is clear that expressions for the stress and displacement distributions around
openings of various shapes may be obtained by an heuristic selection of the forms
of the analytic source functions. For example, for a circular hole with a traction-free
surface, in a medium subject to a uniaxial stress p xx at infinity, the source functions
are
1 A
1 B C
(z) = p xx z + , (z) =− p xx z + + (6.15)
4 z 2 z z 3
The real constants A, B, C are then selected to satisfy that known boundary condi-
tions. These conditions are that, for all
, rr = r
= 0at r = a (the hole boundary),
and rr → p xx for
= 0 and r →∞. The resulting equations yield

2
2
A = 2a , B = a , C =−a 4
and the stress components are given by
1 a 2
1 4a 2 3a 4
rr = p xx 1 − 2 + p xx 1 − 2 + 4 cos 2
2 r 2 r r
1 a 2
1 3a 4

= p xx 1 + − p xx 1 + cos 2
(6.16)
2 r 2 2 r 4
1 2a 2 3a 4
r
=− p xx 1 + − sin 2
2 r 2 r 4
In spite of the apparent elegance of this procedure, it appears that seeking source
analytic functions to suit particular problem geometries may be a tedious process.
However, the power of the complex variable method is enhanced considerably by
working in terms of a set of curvilinear co-ordinates, or through a technique called
conformal mapping. There is considerable similarity between the two approaches,
which are described in detail by Muskhelishvili (1963) and Timoshenko and Goodier
(1970).
A curvilinear co-ordinate system is most conveniently invoked to match the shape
of a relatively simple excavation cross section. For example, for an excavation of
elliptical cross section, an orthogonal elliptical ( , ) co-ordinate system in the z
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