Analytical analysis of the longitudinal response of shield tunnel lining considering ring-to-ring interaction

This study presents an analytical solution for the longitudinal response of shield tunnel lining with an explicit consideration of ring-to-ring interaction represented by two-directional joint spring, based on the state space method. The analytical expressions of joint spring stiffnesses as the key parameters used in the longitudinal beam spring model are derived along with their conversion relationship with those of the equivalent stiffnesses in the longitudinal equivalent continuous model. The present solution can easily degenerate into that of the longitudinal equivalent continuous model and its capacity is demonstrated by several numerical examples through comparison with other existing analytical solutions and model tests. By virtue of the state space method, arbitrary loads and ring-to-ring interaction can be conveniently formulated into the analytical solutions. In addition, the present method is applicable to not only homogeneous deposit condition and simple multiple discontinuities in strata, but also more general scenarios such as the longitudinal variation of in-between ring joint stiffness and spatial soil variability in the longitudinal domain.