Analysis of twin tunnels in a gravitating half-space

Abstract

An analytical model is developed to solve, for the first time, the stress problem of two equal circular horizontal tunnels in an isotropic elastic gravitating half-space. The method used employs the Airy Stress function, which is decomposed into three parts. The first part corresponds to the infinity conditions, the second part to line loads acting at the tunnel centres and the third to the residual stresses at the tunnel surfaces. The final stress function is defined in terms of a set of unknowns which may be determined by solving a set of quadruply infinite linear simultaneous equations. The solution converges within a region around the tunnels. Full numerical results are presented, along with a Finite Element verification. The Boundary Element method is used to obtain further results for twin tunnels in isotropic and cross-anisotropic materials. The formulation used has linear and circular arc segments, piecewise constant boundary values and exploits the symmetry of the problem. Displacement results, and stress solutions for pressure tunnels, elliptic tunnels and tunnels under a different type of in situ loading condition, are given. The results show a large dependence on the in situ stress condition, but the interference between the two tunnels is less than expected. The solutions are very sensitive to the degree of anisotropy and the shear modulus.