Engineering Rock Mechanics

390 Design and analysis OF underground excavations

dimensional geometries. This is elegantly demonstrated by the stress
distribution around a spherical opening in a uniaxial stress field, in which
the magnitude of the induced stress on the boundary is given by the
equation shown in Fig. 20.31. As an analogue to the maximum stress
around a circular opening in a uniaxial stress field, the stress at

8=Oiso,=- -

2 3 [9 7-5v - 5viP

with numerical values of 2.00~ when v = 0.20 and 2.02~ when v = 0.25.

There are two key points to note. First, the stress concentration depends

on one of the elastic constants, i.e. Poisson’s ratio (note that in the two-

dimensional case the maximum stress concentration was 3.00 for any

isotropic elastic material, and independent of all elastic properties).

Second, the stress concentration in the three-dimensional case is signif-

icantly different from that of the two-dimensional case. This means that one

cannot validly approximate the three-dimensional geometry by a two-

dimensional geometry-unless part of the three-dimensional geometry is

well represented in two dimensions, which has tacitly been assumed in all

of the two-dimensional solutions presented heretofore.

Even so, in cases where the geometry more accurately reflects

engineering structures, and is therefore more complex, two-dimensional

approximations can be successfully used in locations where these are likely

to be valid. Two such cases are shown in Fig. 20.32.

The first of these, in the upper diagram, is a T-shaped intersection

between two circular tunnels. At a distance of 3r from the centreline of the

branch tunnel, the magnitude of the discrepancy between the maximum

boundary stress computed using a three-dimensional analysis and a two-

dimensional plane strain analysis is less than 10%. Moving further away

from the intersection, to a distance of 5r from the centreline of the branch

tunnel, the magnitude of the discrepancy has reduced to less than 5%. So,

the two-dimensional approximation will be sufficient for engineering

purposes at sufficiently large distances from the line of intersection.

,x

Y

u, =

3 (10 COS%-1-5U)

• 2 (7-5) P

p,=p =o

P,=P

ur

Figure 20.31 Boundary stress around a spherical opening in an isotropic material

subjected to a uniaxial stress field.