An analytical approximate solution for stress and displacement around a shallow circular tunnel excavated under concave slope topography

For shallow buried tunnels under slope, overloading on the ground surface and unloading of excavation both significantly influence the stability of the tunnel surrounding rock and slope. This study proposes an analytical approximate solution for the stress and displacement around shallow buried circular tunnel under slope boundary. In derivation process, the original problem is divided into two models: Model A is a slope plane without any holes subjected to gravity and surcharge load along the ground surface, and Model B is a tunnel contained in a slope plane loaded by released stresses along the tunnel boundary. Furthermore, Model B is decomposed into another two models: Model B1 represents a slope without holes subjected to virtual surface force along the ground surface, and Model B2 is an infinite domain containing a tunnel loaded by another virtual surface force along its boundary. Two sets of virtual surface forces are then achieved by an iterative calculation. The solution, which strictly meets all stress boundary and compatibility conditions of the original problem, are ultimately solved by superposing the solution of Models A, B1 and B2, respectively obtained by the method proposed by the authors (Models A, B1) and complex variable theory (Model B2). The convergence of iteration process and good agreement between the derived and numerical results under the same assumptions verify the correctness of the proposed solution. Furthermore, a parametric investigation is performed to investigate the influence of surcharge load position, slope topography and tunnel position on the stress and displacement around the tunnel and near the ground. The proposed solution is of great value for the conceptual understanding of mechanical behavior of shallow tunneling in slope terrain and for verifying the numerical models. Moreover, the solution is useful for the preliminary design of tunnels at shallow depth under slope topography.