Analytical solution for dynamic response of curved tunnels under travelling wave effect

The tunnel section with curvature variation that restricts the seismic safety of a tunnel is one of the critical sections for structural design. However, the current seismic design of tunnel structures only focuses on the shear deformation of the tunnel cross-section and the longitudinal curvature is not considered. The most important issue is that there are no available simplified methods in the current literatures for longitudinal seismic analysis of curved tunnels. It is therefore necessary to solve the forward problem with the purpose of obtaining an analytical solution for the dynamic response of curved tunnels under travelling loads. Firstly, a curved tunnel is assumed as a finite homogeneous beam with variable curvature resting on a viscoelastic foundation, and the governing differential equation and boundary conditions of the dynamic problem are established based on the Hamilton principle and the viscoelastic foundational beam theory. Then, the modal superposition method is employed to solve the dynamic problem, and thus the analytical solutions of dynamic responses for curved tunnels subjected to arbitrary dynamic loads are derived. Finally, the degraded solution for travelling loads is obtained with the proposed solution. The solutions of tunnel responses investigated are deflection, velocity, acceleration, bending moment, and shear force. The validation of the analytical solution is verified by providing comparisons between its results and those from the finite element method. The parametric analyses are performed to investigate the influences of the radius of curvature, the velocity and frequency of travelling loads and soil-structure relative stiffness ratio on the dynamic responses of the curved tunnel.