380 Design and analysis of underground excavafions
WIH=2 k=0.5
WIH=2 k=0.1
Figure 20.22 Illustration of the variation in the 5% stress perturbation contours for
different ratios of vertical to horizontal stress for an elliptical opening (from chapter
by J. W. Bray in Brown, 1987).giving contour values of 0.45 and 0.55 in the left-hand diagram, and 0.05
and 0.15 in the right-hand diagram.20.2.2 Approximations for other excavation shapes
The closed form solutions presented above, namely for circular and ellip-
tical openings, can be used to give valuable engineering approximations
for stress distributions in two important classes of problem: shapes other
than truly circular or elliptical; and complicated boundary profiles.Other excavation shapes. In Fig. 20.23, the upper diagram illustrates an
ovaloid opening, in which the roof and floor are planar, and the ends are
semi-cylindrical (but note that a vertical cross-section is being considered
through a long excavation). Then, W/H =^3 and radii of curvature pA = H/2
and p~ = -. As a method of approximately determining the circumferential
stresses at A and B (and hence an indication of the maximum and minimum
induced boundary stresses), the equations shown in Fig. 19.16 can be
applied which give the stresses induced on the boundary of an elliptical
excavation in terms of the radius of curvature of the boundary. For the
stress at point A in terms of the radius of curvature at that point, the
magnitude of the circumferential stress is 3.96~.
By similar means, at point B the value is -0.17~ if we take a value for the
radius of curvature appropriate for the ellipse inscribed to the ovaloid. As
a means of determining a more exact answer to the boundary stresses for
this geometry, the boundary element method was applied, with the result
that the stresses at A and B were found to be 3.60~ and -0.15p, respectively.
Thus, the approximation is seen to be good for a preliminary estimation.