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EXCAVATION SHAPE AND BOUNDARY STRESSES
Figure 7.11 Definition of nomen-
clature for an elliptical excavation
with axes parallel to the field stresses.
7.4 Excavation shape and boundary stresses
The previous discussion has indicated that useful information on boundary stresses
around a mine opening can be established from the elastic solution for the particular
problem geometry even in the presence of discontinuities. It is now shown that simple,
closed form solutions have greater engineering value than might be apparent from a
first inspection.
Figure 7.11 illustrates a long opening of elliptical cross section, with axes parallel
to the pre-mining stresses. For the particular cases of = 0, = 0, and = 0, =
/2, equation 6.21 reduces to
2W
A = p(1 − K + 2q) = p 1 − K + (7.6)
A
2K 2H
B = p K − 1 + = p K − 1 + K (7.7)
q B
where A and B are boundary circumferential stresses in the sidewall (A) and crown
(B) of the excavation, and A and B are the radii of curvature at points A and
B. Equation 7.6 indicates that if A is small, A is large. Equation 7.7 defines a
similar relation between B and B . A generalisation drawn from these results is
that high boundary curvature (i.e. 1/ ) leads to high boundary stresses, and that
boundary curvature can be used in a semi-quantitative way to predict boundary
stresses.
Figure 7.12 shows an ovaloidal opening oriented with its major axis perpendicular
to the pre-mining principal stress. The width/height ratio for the opening is three, and
the radius of curvature for the side wall is H/2. For a ratio of 0.5 of the horizontal and
vertical field principal stresses, the sidewall boundary stress is given, by substitution
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