Engineering Rock Mechanics

Design against stress controlled instabiliv 391

Effect of intersection

& 5r 4

I I I

Effect of the end of a circular tunnel

7

v = 0.25, 0 < k < 2

& At y = 0.7%

Y

At y = 4r

Io3D-ops 1 < 0.05

Figure 20.32 Comparison between two-dimensional and three-dimensional stress
analyses for two engineering geometries.

The second example, shown in the lower diagram, represents the con-
ditions at the end of a borehole or circular tunnel. At a distance of 0.75~ from
the end of the tunnel, the discrepancy between the plane strain two-
dimensional solution and the full three-dimensional solution is already less
than 20%. At a distance of 4r from the end of the tunnel, this discrepancy
is reduced to less than 5%. So, in the latter case, not only does the two-
dimensional approximation provide an excellent estimate of the stresses
over most of the tunnel length, it also indicates directly how rapidly the
three-dimensional geometry effectively changes to a two-dimensional
geometry during tunnel construction. This can be of use in determining
design aspects such as the installation time of tunnel support elements and
instrumentation.
In cases such as that shown in Fig. 20.32, where a two-dimensional
approximation is adequate, there is benefit in restricting the analyses to two
dimensions. However, there are circumstances where the intersections of
such underground excavations-and the general engineering layout-
cannot be adequately represented in two dimensions. For example, com-
plex engineering structures such as hydroelectric schemes and most
methods of mining cannot be adequately represented in two dimensions.
We are fortunate today to have full three-dimensional analysis capabilities,
both for discontinuous and continuous materials, readily available on
desktop computers.
There are off-the-shelf codes now available for three-dimensional
discrete element, finite element and boundary element methods of analy-