Engineering Rock Mechanics

354 Underground excavation instability mechanisms

maximum stress concentration is minimized. Fig. 19.17 shows how the
stress concentrations at A and B vary with 9/ and demonstrates that the two
concentrations are equal when 9 = k. Thus, an elliptical excavation has an
optimal shape when the eccentricity of the ellipse is harmonized with the
ratio of field stresses-an elegant result.
Elliptical openings in anisotropic rock. In order to allow for the real nature
of rock, the solution for the stresses around an elliptical opening can be
extended to take into account the case of transversely isotropic rock. The
extension is realistic for many rock types, because transverse isotropy is a
good representation of sedimentary or metamorphic rocks. There are now

1 J.p'l 1 1

where the following geometrical parameters are defined

0 = arctan[( -) e+l 3,

e- 1

Figure 19.14 Stresses induced around an elliptical excavation in plane strain for a
CHILE material (after Bray, 1977; from Brady and Brown, 1985).