Page 195 - Rock Mechanics For Underground Mining

CLOSED-FORM SOLUTIONS FOR SIMPLE EXCAVATION SHAPES

on the point of interest. The various geometric parameters are defined as follows:

(W + H)
e 0 =
(W − H)
2
2
4(x + z )
1
1
b = 2 2
(W − H )
2
2
8(x − z )
1
1
d = − 1
(W − H )
2
2
e 0 2 1/2
u = b + (b − d)
|e 0 |
e 0 2 1/2
e = u + (u − 1)
|e 0 |

e+1 z 1
= arctan
e − 1 x 1
" #

2
e+1 z 1

= arctan
e − 1 x 1
C = 1 − ee 0
2
J = 1 + e − 2e cos 2
The stress components are given by
p(e 0 − e) 2 C
ll = (1 + K)(e − 1)
J 2 2e 0
J

+ (1 − K) (e − e 0 ) + Ce cos 2( + ) − C cos 2
2
p 2
mm = {(1 + K)(e − 1) + 2(1 − K)e 0 [e cos 2( + ) − cos 2 ]}− ll
J
p(e 0 − e) Ce
lm = (1 + K) sin 2 + (1 − K) e(e 0 + e) sin 2
J 2 e 0

J 2
+ e sin 2( − ) − (e 0 + e) + e e 0 sin 2( + ) (6.21)
2
In applying these formulae, it should be noted that the angle
, defining the orientation
of the local reference axes l, m relative to the ellipse local axes x 1 , z 1 , is not selected
arbitrarily. It is defined uniquely in terms of the ellipse shape and the point’s position
co-ordinates.
The boundary stresses around an elliptical opening with axes inclined to the field
stress directions are obtained by selecting values of x 1 , z 1 which fall on the boundary
contour. For this case, e = e 0 , ll = lm = 0, and the l axis is directed normal to the
boundary. For the problem geometry defined in Figure 6.4b, the boundary stress is
given by
p 2 2
= {(1 + K)[(1 + q ) + (1 − q ) cos 2( − )]
2q
2 2
− (1 − K)[(1 + q) cos 2 + (1 − q ) cos 2 ]} (6.22)
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