Analytical solution for longitudinal dynamic response of shield tunnel linings using the state-space method

Using the state-space method and D'Alembert's principle, a novel analytical solution was derived for the longitudinal dynamic response of shield tunnel linings. Timoshenko beams are used to model the tunnel lining rings, the joints connecting the rings are idealized by a series of springs, and the lining-surrounding soil interaction is described by distributed reaction springs with viscous dampers. By applying the state-space method, the governing equations along with the boundary conditions can be concisely expressed in matrix form. A variable separation method, in conjunction with a frequency-scanning strategy, was employed to obtain the natural frequencies and corresponding mode shapes of the tunnel. The orthogonal relationships of the mode shapes were also demonstrated by introducing the symplectic inner product. The tunnel responses under arbitrary dynamic loads were obtained using the mode-superposition method. Finally, numerical examples are presented to verify the efficiency and robustness of the proposed solutions and to predict the dynamic responses of tunnel linings to moving subway trains. The results demonstrate that the proposed method can solve the longitudinal dynamic response of shield tunnel linings by considering the joint effect with high efficiency and accuracy.