Engineering Rock Mechanics

360 Underground excavation instability mechanisms

The discretization of the problem is shown below. We have taken
advantage of vertical symmetry, and used four boundary elements on
the half-boundary of each tunnel.

curved, 3-node isopammeiric boundary elements

fi ;m:;:::nGi between upper and lower

half-spaces of problem

We initially examine the stresses induced around each of the tunnels

individually, and the stresses induced when they both exist. The results

are given below.

3.0 m diameter tunnel 6.0 m diameter tunnel 6 Kirschsolution

k = 0.3 k = 2.5 k = 0.3 k = 2.5 k = 0.3 k = 2.5

node circum- node circum- node circum- node circum- CireUm-

ferential ferential ferential ferential ferential

stress stress stress stress stress

(ma) (ma) (ma) (ma) (Mpa)

1 31.50 1 5.90 13 31.50 13 5.90 0" 31.59 5.85

9 31.50 9 5.90^21 31.50^21 5.90 180" 31.59 5.85

5 -1.14 5 76.00 17 -1.14 17 76.00 90" -1.17 76.05

These results show how the induced stresses are independent of the

size of the tunnel, and how the induced stresses are symmetric for each

tunnel. They are also essentially identical with the results obtained using

the Kirsch solution.

We now analyse the problem when both tunnels are in place. The

results are shown in the table below and in the two graphs following.

Boundary element Kirsch-based approximation

3.0 m and 6.0 m diameter tunnel

k = 0.3 k = 2.5 k = 0.3 k = 2.5

3.0 m and 6.0 m diameter tunnel

node circum- node circum- node circum- node circum-

ferential ferential ferential ferential

stress (MPa) stress (MPa) stress (MPa) stress (MF'a)

1 33.10 1 12.20 1 33.25 1 15.53

5 -0.26 5 62.50 5 0.12 5 60.05

9 33.70 9 19.70 9 33.25 9 15.53

13 32.10 13 10.90 13 31.59 13 5.85

17 -1.12 17 73.90 17 -1.17 17 76.05

21 31.90 21 6.94 21 31.59 21 5.85

The results here are revealing and instructive. Two points need to be

emphasized.

(a) In a general sense, the approximation gives reasonable results (it does,

for example, correctly predict zones of tension and compression).

(b) The approximation predicts the stresses around both tunnels equally

well.