The implementation of closed-form solutions for stress and displacement fields around tunnels with arbitrary geometry, often based on the complex variable theory and the method of conformal mapping, can be quite challenging from a mathematical point of view. In this paper a solution strategy for the implementation of a chosen closed-form solution from literature is presented, including the possibility to account for rock mass anisotropy and arbitrary tunnel geometries. The evaluation of the involved elastic potential functions is described, respectively derivatives thereof, in terms of solving non-linear constrained optimization problems. To validate our approach, the analytical results for stresses and displacements around a tunnel with semicircular geometry are compared to numerical results from finite element computations. The outcome of the study should be regarded as a basis for the development of refined analytical solutions within anisotropic rock masses considering more realistic boundary conditions and effects such as material non-linearity.
For the derivation of closed-form planar elasticity solutions for stress and displacement fields around arbitrarily shaped tunnels in isotropic or anisotropic grounds the complex variable theory, as initiated by Kolosov (1909), in combination with the conformal mapping method (Muskhelishvili 1953) can be used. Thereby, the exterior of the original problem configuration is mapped to the outside or inside of a fictitious unit circle. Despite the complexity of the involved mapping procedure, the associated difficulties are outweighed by the simpler definitions of the elastic potential functions influencing the amount of generated stresses and displacements.
One elastic closed-form solution based on the complex variable theory and the method of conformal mapping is the solution by Tran Manh et al. (2015). It accounts for arbitrary tunnel shapes and elastic rock mass anisotropy by the assumption over transverse isotropy. Mathematical optimization problems need to be solved in the course of determining the elastic potential functions, which is not a straightforward process. Consequently, in this paper a solution strategy to overcome such difficulties in connection with complex variable solutions is provided. The assumption is made that the mapping coefficients of the conformal mapping function are already known, e.g. from an iterative scheme as presented by Winkler et al. (2023). Finally, the solutions for stresses and displacements of a specific case are compared with the results from finite element calculations. Section 2 refers to the problem definition. In section 3 the solution procedure for the determination of the derivatives of the elastic potentials and the retrieval of the final solutions are described. In section 4 the analytical solution is compared against results from finite element computations and section 5 presents the conclusions.
