Dynamic analytical model for shaft-tunnel junction under P-SV waves

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Shaft-tunnel junction is a typical case of critical underground joint structures. The abrupt structural change between the shaft and the tunnel makes it particularly susceptible to seismic damages. The focal point of this paper is the dynamic responses of shaft-tunnel junction under P-SV waves, for which a dynamic analytical model is developed. The shaft is assumed to be conditionally rigid, and the tunnel is modeled as a Euler-Bernoulli beam. The ground motion is calculated using the stiffness matrix method for layered media. The soil-structure interactions are mediated by a series of supporting springs and dashpots. Within the plane of wave propagation, the shaft has three generalized coordinates, and the tunnel is described by two displacement functions. Their analytical solutions are therefore derived in the frequency domain. Validity of the solutions is examined by comparison with the numerical results of two validation models. According to the solutions, responses of the shaft are controlled by the impedance of the surrounding ground, its own inertia properties, and the shaft-tunnel interactions. Among them, the impedance of the surrounding ground is by far the predominant factor. Responses of the tunnel are the superpositions of the shaft displacements and the free field displacements. The influence of the shaft attenuates exponentially with increasing distance from it. If the distance is sufficiently large, responses of the tunnel are dominated by the propagation of the P-SV waves.