Analytical solutions of non-circular deep tunnels incorporating thermal stresses: from analytical solutions to its application in shape optimization

Abstract

The influence of the temperature field on mechanical behaviours is significant when performing stability analysis of underground structures. This article concentrates on investigating the stress and displacement of arbitrary non-circular tunnels constructed in the elastic ground, considering thermal effects. To achieve this, the complex variable function method and the conformal transformation technique are employed to derive the analytical solutions. Meanwhile, the temperature boundary and tunnel inner boundary in the physical plane are mapped into concentric annuli in the complex plane, and two potential functions are used to determine the analytical expressions of thermal stress. Additionally, the potential function forms of the boundary conditions in the complex plane are derived. Analytical solutions are then obtained to calculate the thermal stress of arbitrarily shaped, deeply buried tunnels, based on the temperature field expression in the complex plane. A good agreement is observed between analytical solutions and numerical predictions, verifying the correctness of the developed analytical theory and obtained analytical solutions.

After that, the developed analytical solutions are employed to optimize the design of tunnel shapes, taking into account the effect of the temperature field on the mechanical responses. The study found that the optimal tunnel shapes for minimizing maximum hoop stress at the hole boundary are elliptical, with or without temperature effects. Additionally, a closed-form formula for the optimal shape ratio has been proposed, considering all mechanical and physical parameters except Poisson's ratio. This approach serves as a valuable tool with significant potential for application to various engineering cases, such as optimizing the design of high-temperature tunnels and nuclear waste disposal systems.