Engineering Rock Mechanics

358 Underground excavation instability mechanisms

419.7 At a depth of 450 m, a 3-m diameter circular tunnel is driven
in rock having a unit weight of 26 kN/m3 and uniaxial compressive
and tensile strengths of 60.0 MPa and 3.0 MPa, respectively. Will the
strength of the rock on the tunnel boundary be reached if
(a) k = 0.3, and
(b) k = 2.5?
A second tunnel, of 6 m diameter, is subsequently driven parallel
to and at the same centre line level as the first, such that the centre
line spacing of the two tunnels is 10 m. Comment on the stability of
the tunnels for the field stresses given by (a) and (b) above.

A19.7 This is a problem to be
solved using the Kirsch solution
for circular openings in a state
of plane strain. We are asked
to examine the stability of the
rock on the boundary of the tun-
nel. As the tunnel has neither a
support pressure nor an internal
pressure applied to it, the rock
on the boundary is subjected to
a uniaxial state of stress, with
the local a3 = a, = 0 and local
a1 =OB.
The Kirsch solution for the cir-
cumferential stress is

and for a location on the tunnel boundary, where a = r, this reduces to

00 = a, [ (1 + k) + 2( 1 - k) cos 281.

We assume that the vertical stress is caused by the weight of the

overburden, in which case we have

a, =y.z=0.026-450= 11.70MPa.

The extreme values of induced stress occur at positions aligned with

the principal in situ stresses, and so in order to compute the stress

induced in the crown and invert of the tunnel we use 8 = 90", and for

the sidewalls we use 8 = 0". For the case of k = 0.3, the stresses are then

found to be -1.17 MPa (i.e. tensile) and 31.59 ma, respectively. For the

case of k = 2.5, the induced stresses are 76.05 MPa and 5.85 MPa. Thus,

we see that the compressive strength of the rock, 60 MPa, is reached at

the crown and invert of the tunnel for the case of k = 2.5.

After the second tunnel has been driven, we can find an approximate

solution to the problem through a multiple application of the Kirsch

solution. We start by determining whether the tunnels are inside each

other's zone of influence. Given the approximate nature of the solution,

a rigorous computation of this is not justified. Instead, we take the