Closed-form solution for dynamic responses of shaft-tunnel junction under uniform P waves

Abstract

This paper discusses the special situation when a shaft-tunnel junction is subject to vertically propagating P waves. For a complicated system like the shaft-tunnel junction, it is necessary to investigate the most basic cases firstly, and the responses induced by uniform P waves are one of them. To this end, a closed-form solution is derived from the classical differential equation describing the forced vibration of a single degree-of-freedom system. The key issue is the characterization of the coupled dynamic interactions among the shaft, the tunnel, and the ground. The paradigm regards the three of them as a massive rigid body, a continuous Euler-Bernoulli beam, and a set of supporting elements, respectively. The final solution is to calculate the displacements of the shaft and the tunnel in the form of two compact equations, and the internal forces of the tunnel are therefore easily obtainable. By means of a finite element simulation, the validity as well as the accuracy of the closed-form solution is verified. The solution provides an explicit comprehension of the coupled dynamic interactions and thus enhances the understanding of the dynamic mechanics of the shaft-tunnel junction.

Keywords

• Shaft-tunnel junction
• Dynamic responses
• Closed-form solution
• P waves