Questions and answers: underground excavation instability mechanisms 363a m
I 6 1500 ~(^4) =
cn
E" 1000 1
D
F
included in each solution. The final stress state is then given by
point A lefttunnel right tunnel field stresses
which evaluates to
Kirsch approximation crosses
[z
= [10.01 0 ] + [ 11.28 -0.951 - [11.0 0 ]
0 11.99 -0.95 10.72 0 11.0
and from this the friction angle required to maintain stability is com-
puted as
-0.95
= arctan (:) = arctan (-) = -4.6"
11.71
where the negative sign is an indicator of the sense of shearing. As the
friction angle for the fault is 20", it is therefore stable.
We can perform the calculation sequence given above for a number
of different points (say, by using a computer spreadsheet) in order to
examine the stress distribution along the fault. The results of this are
shown in the figure below, together with stress distributions computed
using a boundary element program. Notice that there is very good
agreement between the two methods.
25.00 1
Distance from LH tunnel centreline, m
-5.00
The figure below shows the variation of required friction angle, com-
puted using both the Kirsch approximation and a boundary element
program. The errors involved are relatively small - no more than half
of one degree - and show how the Kirsch approximation can be used
for initial assessments of stability.