Flexural wave propagation and localization in periodic jointed tunnels subjected to moving loads

Wave propagation and localization in the ordered and randomly disordered periodic jointed tunnel under moving loads are investigated. The periodic tunnel is approximated as a pipe-beam model with periodic joints on elastic foundations. By using the differential equations governing the flexural vibration of the pipe-beam on elastic foundations as well as the force equilibrium of the joints, the dynamic stiffness matrix is developed and the transfer matrix of the periodic tunnel is formulated in a coordinate system moving with the load. A critical velocity and the dynamic response of a uniform tunnel are investigated. For the ordered periodic tunnel, the dynamics of wave propagation induced by the moving loads are evaluated by analyzing the propagation constants. For the disordered periodic tunnel, a random disorder is introduced therein and the localization factors are calculated to examine the wave localization according to the Wolf’s algorithm. The effects of various controlling parameters on the wave propagation and localization are assessed. The obtained results show that the flexural wave is always localized near the moving load when the velocity is less than some critical velocity of a uniform tunnel on elastic foundations. Accordingly, the critical velocity is identified as the minimum velocity at which wave can propagate along the uniform tunnel. Furthermore, the travel velocity of the moving load drastically affects the amplitude of the transverse response. In the periodic tunnel, the velocity stop bands and pass bands exist alternately. The band width and location depend on structural parameters. As the level of disorder increases, the localization degree is strengthened. For the same disorder level, the phenomenon of wave localization is more pronounced at the low velocity range. The validity of the proposed methodology is confirmed by evaluating the transfer property and the deformation of the periodic tunnel through the finite element simulations.