The solution is obtained through the use of complex variables and a conformal mapping onto a circular ring. The boundary conditions are that the surface of the half plane is completely free of stress and that the tunnel undergoes a prescribed displacement corresponding to an ovalization of the tunnel. When the solution is compared to a previously published approximation, it appears that the approximation suffers from an inaccuracy in the boundary conditions for compressible soils. From the solution it is evident that the settlement trough at the soil surface is much narrower than in the case of a uniform contraction of the tunnel cavity. For a shallow tunnel, the stresses accommodating an ovalization of the tunnel cavity do not correspond to an ovalization of the tunnel lining, suggesting that in practice pure ovalization of shallow tunnels is unlikely to occur.


In this paper an exact analytical solution for the ovalization of a tunnel in a linearly elastic half-plane is presented. Although soil behavior can only be roughly approximated by linearly elastic models, this solution can be used as a tool to gain insight into the fundamental nature of the problem and as a benchmark for numerical procedures based on more sophisticated models of soil behavior. The paper is divided into five sections; the first two sections deal with the mathematical details of the solution, the third section discusses issues encountered during the validation, and the last two sections present the results and conclusions.

General Solution

The problem of a circular tunnel in an elastic half-plane with a stress free upper surface and a prescribed displacement at the tunnel boundary was solved by Verruijt (1997) through use of the complex variable method (Muskhelishvili, 1964).

This content is only available via PDF.
You can access this article if you purchase or spend a download.