A Complex Variable Solution for the Stress and Displacement Around the Shallow Rectangular Tunnel Under Slope Condition Considering the Gravity Effect

This paper presents an analytical method for solving the problem of shallow rectangular tunnel excavation, explicitly accounting for both the surface slope condition and the soil gravity effect. Firstly, a conformal mapping function for an asymmetric arbitrary cavity is employed to transform the semi‐infinite domain containing the rectangular cavity into an annular region in the image plane, simplifying the solution process for the irregular region. Then, utilizing the complex variable method, the fundamental governing equations for the stress and displacement of soil are established, with the stress and displacement boundary condition equations expanded via Fourier series. Subsequently, all the coefficients in the potential function are determined by the boundary condition equations, yielding a complex variable solution for the stress and displacement of soil around the shallow rectangular tunnel under slope condition. Finally, the accuracy of the proposed method is validated through numerical analysis, and the influence of tunnel and soil parameters on the stress and displacement around the tunnel is investigated. The proposed solution can be utilized as a tool to evaluate soil disturbances induced by tunnel excavation during the preliminary design stages.