Analytical solutions for seismic responses of shaft-tunnel junction under longitudinal excitations

Analytical solutions are deduced for seismic responses of shaft-tunnel junction under longitudinal excitations. The influence of the tunnel is incorporated by introducing terms of shaft-tunnel and soil-tunnel interactions into equations originally developed for rigid caissons. The shaft is simplified into a rigid body. The tunnel is represented by a continuous beam perpendicularly fixed onto the shaft. Firstly, solutions for displacements of the shaft are given. Then, solutions for the three major internal forces of the tunnel are proposed. Validity of the proposed solutions is examined by finite element method in respect of the shaft and the tunnel separately. The comparisons confirm that the proposed solutions could successfully predict the displacements of the shaft and the three major internal forces of the tunnel at the vicinity of the shaft-tunnel junction. However, they become less accurate with increasing distance to the shaft. Interactions among the shaft, the tunnel, and the soil are discussed based on the proposed solutions. A special emphasis is placed on the mutual influences between the shaft and the tunnel. Displacements of the shaft, especially the rotational displacements, are likely to decrease under longitudinal excitations when connected to tunnels. Shaft-soil relative displacement and soil-tunnel relative stiffness are the two key factors affecting the seismic responses of the tunnel. The former determines the amplitudes of the internal forces, while the latter governs how the internal forces distribute along the tunnel axis. A conceptual aseismic measure is studied by setting a pin joint at the shaft-tunnel junction. Theoretically, it could eliminate the influence imposed on the tunnel by the rotational displacement of the shaft.