Analytical solution for longitudinal seismic responses of long tunnels crossing soil-rock stratum

The contact between soil and rock is the controlling point in the seismic design of long tunnels, which has not been considered in any design methods yet. The mechanical model for a long tunnel crossing soil-rock stratum is proposed with assumptions based on its dynamic response characteristics, in which two semi-infinite elastic foundation beams in different soils are assumed to model the behaviors of the tunnel. An analytical solution for longitudinal seismic responses of tunnels crossing soil-rock stratum is derived by solving the displacement differential governing equations. The simplified method for longitudinal seismic responses of tunnels crossing soil-rock stratum is proposed. The traveling wave effect can be considered by introducing the displacement phase angle. A benchmark model, a finite element model for a long tunnel crossing soil-rock stratum, is used to calibrate the proposed method. Comparison under the same parameters shows a high degree of consistency between them, which verifies the efficiency of the innovative method. Finally, sensitivity analyses of parameters are carried out using the proposed analytic formulas, and the influences of parameters on seismic response of tunnels crossing soil-rock stratum are revealed. The results suggest that with the increase of the structural stiffness, the internal force response of the tunnel becomes larger, the difference of the values of the peak internal forces becomes greater, and the zone of influence of the