Analytical solution for determining the plastic zones around two unequal circular tunnels

Abstract

An analytical solution is proposed for the plastic zones developed around two circular tunnels with unequal size excavated in intact rock which is governed by the Mohr-Coulomb yield criterion in failure. The tunnels are deep buried and subjected to remote uniform in-situ stresses. The elastic–plastic interfaces on the physical plane are mapped onto an annulus on the image plane by using the conformal transformation. The solution is obtained by the stress continuity condition along the interfaces. Correspondingly, the stress field in elastic zone is given by the complex variable method and the stress field in plastic zone is statically determined. Then a set of nonlinear equations involving the mapping function coefficients is constructed, which reduces the problem of finding the elastic–plastic interfaces to be the problem of solving the mapping function coefficients. The differential-evolution algorithm with the coefficients as design variables is used to solve the nonlinear equations. Numerical solution is made to verify the presented solution. Parametric investigation is performed for varying relative size of tunnels and in-situ stresses.