Analytical solutions for rock stress around square tunnels using complex variable theory

• An analytical solution for rock stress around square tunnels is presented.
• An application is provided to illustrate how the analytical solution works.
• The rules of stress distribution of square tunnels are obtained.
• A universal framework is provided to analyze rock stability for any shape of holes.

The complex variable theory is employed to find the analytical solution for rock stress around square tunnels in a homogeneous, isotropic and elastic rock mass. The solution is more accurate than previous available solutions, because the first three terms of transformation function are taken in the early deduction. We find that in situ stress and coefficients of lateral pressure play a crucial role in stress distribution. High compressive stress concentrations are found to exist at the four square corners. The surrounding rock is compressed over the complete square periphery when the pressure coefficients with values between 0.8 and 1.2. The boundary stress is gradually converted from tensile stress to compressive stress for the two sidewalls with the increasing pressure coefficient, whereas the opposite situation occurs for the roof and floor. In order to avoid the stress concentration, a square tunnel should choose a rounded corner and take certain support reinforcement measures. Besides, the results provide a theoretical basis on support design for deep square tunnels, and a universal framework to analyze surrounding rock stability for other tunnels of noncircular shape.