Complex variable solution of elastic tunneling problems

Abstract

The ground loss problem occurs when a cylindrical tunnel is constructed in a soil with the radius of the tunnel being somewhat smaller than the radius of the cavity. The method used in this paper is Muskhelishvilli’s complex variable method considering conformal mapping of the elastic region onto a circular ring. The problem of an elastic half plane with a circular cavity was investigated, regarding the case that along the boundary of the cavity, the surface tractions were prescribed. The computer program (ground loss) was used. The program worked interactively, on the basis of values of Poisson's ratio and the ratio of the radius of the cavity to its depth (r/h). It was investigated whether certain problems of stresses and deformations caused by deformation of a tunnel in an elastic half plane could be solved by the complex variable method. For this purpose, two elementary boundary value problems were considered in detail. These include the problem of a half plane with a circular cavity loaded by a uniform radial stress, and the problem in which a uniform radial displacement is imposed on the cavity boundary (this is usually called the ground loss problem). It was concluded that the displacement of the bottom of the tunnel was always smaller than the value, uo (the displacement of the cavity). For large values of r/h, the displacement may even be negative, that is, downward. The displacement of the bottom was always equal to the average displacement of the tunnel plus a constant value Mo which is the imposed radial displacement. Key words: Complex variable, tunnel, boundary value problem, elastic.