Abstract: How to estimate the seismic response of shield tunnels reasonably has been a significant issue in both industry and academia. Compared with the transverse seismic response, the longitudinal seismic response is more complex. The shield tunnel is modeled as a Timoshenko beam on Winkler foundations, considering both residual axial force and additional axial force due to longitudinal seismic. A theoretical model was presented to consider the longitudinal and transverse stratum displacements. The longitudinal seismic response of shield tunnels was solved using the finite difference method. The theoretical model and calculation method were validated through case studies of stagger-assembled shield tunnels with a 6.2 m diameter. Considering axial force increased the overall stiffness of shield tunnel, resulting in decreased internal force and deformation response. The proposed method was degraded to the traditional one when the axial force is neglected. The influences of residual axial force, seismic wavelength, seismic incidence angle and foundation reaction coefficient on the longitudinal seismic response were further explored. With the increase of residual axial force, the overall stiffness of shield tunnel increased, and the peak seismic response of shield tunnel decreased. When the incident angle is less than 45°, the influences of residual axial force and foundation reaction coefficient on the peak response were more significant. With the increase of the foundation reaction coefficient, the tunnel deflection and the discontinuous deformation between joints increased. Wavelengths between 20 m and 100 m may lead to greater opening and dislocation between joints. These studies can provide theoretical support for the longitudinal seismic design of shield tunnels.

Key words: shield tunnel, longitudinal seismic response, Timoshenko beam, finite difference solution