364 Underground excavation instability mechanisms
I/)
a, m
-0
(^6) m
m
8.0 1
numerical analysis
- Kirsch approximation
Distance from LH tunnel centreline, m20 100 120 140 0a
-6 0-8.0 ’
Q19.9 An ovaloid excavation at a depth of 750 m has in vertical
section its maior axis horizontal, and the ratio of its width to height
is 4. The radius of curvature of its ends is equal to half its height.
Assume that the in situ stress state can be calculated on the basis of
complete lateral restraint in a CHILE medium (ERM 1, Section 4.6.2)
with y = 28.0 kN/m3 and v = 0.3.
An elastic boundary element analysis for k = 0 shows that the
stress in the centre of the roof is -20.5 MPa, and in the side wall
is 104 MPa. An analysis with k = 1 gives corresponding stresses of
4.59 MPa and 84.2 MPa. What stresses would the boundary element
analysis give for the in situ stress state?
Using the equations for stresses in terms of radius of curvature:
(a) compare the stress in the centre of the roof of the excavation
with that for an elliptical excavation with the same width/height
ratio; and
(b) compare the stress in the side wall with that for a boundary of
equal radius of curvature.A19.9 As the numerical analysis used an elastic model, we can de-
termine the stresses induced for any value of k, the ratio of horizontal
to vertical stress magnitudes, pro rata. For a case of complete lateral
restraint, the ratio of horizontal to vertical stress magnitude is given by
OH V
k=-=-
av 1-u
which means that in this case we have k = 0.3/(1 - 0.3) = 0.429. As we
already have results for k = 0 and k = 1, the stress magnitudes for the inThis simple formula is a traditional one in rock mechanics but note that, in a specific
field case, the ratio between the horizontal and vertical stress components cannot be
estimated using this formula, because of the influence of other factors, such as erosion
and tectonic stresses as discussed in Chapter 4.