Page 196 - Rock Mechanics For Underground Mining

METHODS OF STRESS ANALYSIS

When the axes of the ellipse are oriented parallel to the field stress directions, equation
6.22 reduces to
p 2 2
= {(1 + K)[(1 + q ) + (1 − q ) cos 2 ]
2q
2 2
− (1 − K)[(1 + q) cos 2 + (1 − q )]} (6.23)
In assessing the state of stress in the sidewall and crown of an elliptical excavation,
i.e. for points A and B ( = /2, = 0) in Figure 6.4c, it is useful to introduce the
effect of the local boundary curvature on boundary stress. For an ellipse of major and
minor axes 2a and 2b, the radius of curvature at points A and B, A and B , is found
from simple analytical geometry (Lamb, 1956) to be

b 2 a 2
A = , B =
a b
Since q = W/H = a/b, it follows that

$ $
W 1 H
q = , =
2 A q 2 B
Sidewall and crown stresses in the ellipse boundary, for the problem defined in Figure
6.4c, may then be expressed as

$ !
2W
A = p(1 − K + 2q) = p 1 − K +
A
(6.24)
$
!
2K 2H

B = p K − 1 + = p K − 1 + K
q B
It will be shown later that the formulae for stress distribution about ideal excavation
shapes, such as a circle and an ellipse, can be used to establish useful working ideas
of the state of stress around regular excavation shapes.

6.4 Computational methods of stress analysis

The preceding discussion indicated that even for a simple two-dimensional excavation
geometry, such as an elliptical opening, quite complicated expressions are obtained for
the stress and displacement distributions. Many design problems in rock mechanics
practice involve more complex geometry. Although insight into the stress distributions
around complex excavation shapes may be obtained from the closed form solutions for
approximating simple shapes, it is sometimes necessary to seek a detailed understand-
ing of stress distribution for more complicated configurations. Other conditions which
arise which may require more powerful analytical tools include non-homogeneity of
the rock mass in the problem domain and non-linear constitutive behaviour of the
medium. These conditions generally present difficulties which are not amenable to
solution by conventional analysis.
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