ABSTRACT
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). After the tunnel is excavated, some instantaneous
elastic deformation occurs in the surrounding rock mass, however, the deformation is not finished. The amount of deformation completed is related to the distance between the working face and the supported section. This is due to the spatial constraint effect exerted by the unexcavated rock mass in front of the working face. Within a certain distance from the working face, the displacement in the surrounding rock mass varied with the distance (Li & Liu 1982, Zhu & Lin 1987). Lu et al. (2011) considered the displacement release coefficient according to the distance between the working face and the supported section. If the deformation in the surrounding rock mass before the lining installation is known, the analytic stress solutions for a circular pressure tunnel at great depth can be obtained based on the assumptions of plane strain. The pure bond condition where there is continuity
of the normal stresses and displacements as well as the tangential shear stresses and displacements along the lining/rock interface was implemented by Bulychev (1982), Lu et al. (2011).
