Analytical solutions for seismic responses of the tunnel in a shaft-tunnel junction under transverse excitations

The analytical solutions are proposed for seismic responses of the tunnel in a shaft-tunnel junction under transverse excitations. The vertical shaft in the junction is regarded as a rigid body. The tunnel is simplified into a massless beam. It is connected to the shaft perpendicularly in a fixed end. The surrounding soil is represented by distributed springs and dashpots. Displacements of the tunnel can be obtained by adopting displacements of the shaft as boundary conditions for the governing equations. Thereby, analytical solutions for the three major internal forces of the tunnel are derived. Validity of the proposed solutions is examined by finite element method. The comparison shows that the analytical solutions could provide accurate results of internal forces of the tunnel at the portion near the junction. However, they become less accurate when the distance to the shaft increases. Relations between the internal forces and physical parameters of the soil-shaft-tunnel system are discussed. Distributions of the internal forces along the tunnel axis are highly dependent on the stiffness of the soil and the tunnel, while amplitudes of the internal forces are also determined by the amount of shaft-soil relative displacements. When the connection between shaft and tunnel is replaced by a pin joint as a conceptual aseismic measure, the maximum shearing force and the maximum bending moment of the tunnel would be both significantly reduced.