Analytical solutions for seismic responses of shaft-tunnel junction under travelling SH-wave

When the shaft-tunnel junction is subjected to vertically propagating SH-waves, the dominant factor is the discrepant responses between the shaft and the tunnel. However, in the case of inclined SH-waves, the travelling wave effect also has a significant influence, especially on the tunnel. To address this issue in an analytical fashion, the displacement field of the ground under travelling SH-wave is described by the stiffness matrix method and then incorporated into the solutions for the shaft and the tunnel. Some of the major assumptions include: 1) the SH-wave is a plane wave with a given incident angle; 2) the shaft is regarded as a rigid body; and 3) the tunnel is simplified into a massless Euler-Bernoulli beam. Two scenarios are considered according to the direction of the tunnel axis. When the tunnel axis is parallel to the shaking direction of the ground, the travelling wave effect is rather irrelevant, and the solutions for shaft-tunnel junction under uniform longitudinal excitations are still valid with some minor modifications. Therefore, this scenario is only briefly discussed. In the second scenario, when the tunnel axis is parallel to the horizontal propagation of the SH-wave, the travelling wave effect is no longer negligible, and a new set of solutions are offered. Validation of the solutions is made by comparison with finite element method. As demonstrated by the solutions, the displacements of the shaft and the tunnel are both affected by the travelling wave effect. The seismic distortions originate from the nonuniformities of the structure and the seismic excitation at the same time. Great internal forces at the intersection point are inevitable regardless of the direction of wave propagation, but the influence of the shaft on the tunnel diminishes exponentially with increasing distance to the shaft. Beyond the scope of the shaft influence, the responses of the tunnel are dominated by the travelling wave effect. In the case of vertically propagating SH-wave, the internal forces of the tunnel would approach to zero towards the free end of the tunnel. However, under travelling SHwave, even at a great distance to the shaft, the tunnel would still have significant internal forces due to the travelling wave effect.