Analytical solution for longitudinal bending stiffness of shield tunnels

The longitudinal bending stiffness of tunnels is one of the most important parameters for the structural design of the tunnel longitudinal deformations during seismic loading. The aim of the paper is to derive an analytical solution for the longitudinal bending stiffness of a segmental liner, typically used on tunnels built with a shield. For the derivation, it is assumed that the tunnel liner includes bolts, segments and rubber gaskets. All the elements are considered as linear elastic. The bolts are assumed to work only in tension and the rubber gaskets are assumed to work only in compression. The deformation of any tunnel cross section complies with the assumption that normals to the neutral axis remain normal and thus the neutral axis does not change along the tunnel during deflection. Based on these assumptions, a longitudinal equivalent model for shield tunnels is developed and the governing equations are obtained. A closed-form analytical solution for the longitudinal bending stiffness is developed and verified by providing comparisons between its results and those from the Finite Element program ABAQUS. A parametric analysis, using the new analytical solution, is included to investigate the influence of key parameters such as the thickness of the rubber gaskets, the ratio of the liner thickness to the internal tunnel diameter, and the transverse bending stiffness of the tunnel cross section on the longitudinal bending stiffness of the tunnel.