Based on the plane elastic complex function method, theoretical analysis on the stress field around a circular tunnel with ring lining subjected to in situ stresses and inner hydrostatic pressure is carried out. The results of two extreme boundary conditions cases, pure slip and pure bond, are compared with each other. The displacement release coefficient is considered. The calculation results show that the magnitude of the deformation completed before lining installation has great effects on the stress distribution in the lining. If the lining is installed at a proper moment, the degree of stress concentration in the lining can be minimized. When the Young's modulus of the lining is much lower than that of the rock, stress solutions for two extreme boundary condition cases are roughly the same. The smaller the lateral pressure coefficient λ, the larger the differences between the corresponding stress solutions for two extreme boundary condition cases.
Underground tunnels are widely used in hydropower, traffic, mining and military engineering to ensure the tunnel safety, lining is applied in the tunnel. The closed-form solutions for unsupported tunnel at great depth have been proposed for single circular tunnel (Kirsch 1898, Hefny 1999) and tunnels with complex geometry, such as elliptic tunnel, rectangular tunnel, semi-circular tunnel, inverted U-shaped tunnel and notched circular openings (Lu & Zhang 2007). The complex function method developed by Muskhelishvili (1963) is especially suitable for solving underground tunnel problems. The complex function method can be applied to find the stress solutions not only for single tunnel in arbitrary shape, but also for multiple tunnels with arbitrary shape (Lu & Zhang 1997, Zhang et al. 2001).
As for elastic solutions for deep supported tunnels, the plane strain problem associated with a single circular tunnel with ring lining in an infinite domain has been studied in depth (Bulychev 1982, Lu et al. 2011). Theoretically, the solutions based on plane strain can only be applicable to the situation that lining is applied immediately after excavation and no deformation occurs in the surrounding rock mass before that. In fact, after the tunnel is excavated by a certain length and before the lining is applied, some deformation has occurred in the surrounding rock mass. After lining installation, as the working face is advanced, the surrounding rock mass experiences further deformation and leads to forces on the lining. Although the support delay process was considered by Bulychev (1982) and Wang & Li (2009), some limitations still exist (Lu et al. 2011).
