Nonlinear interaction between a moving vehicle and a plate elastically mounted on a tunnel

Abstract The paper presents a theoretical study of the interaction between a nonlinear model of a moving vehicle (velocity v ) and a plate elastically mounted in a tunnel. An efficient approach to the solution of the problem of vehicle–slab-track–tunnel–soil interaction is developed on the basis of a coupling of the Finite Element and Integral Transform methods (FEM and ITM). According to this approach the tunnel which may have an arbitrary shape and a portion of the surrounding soil is modelled by Finite Elements while the soil (half-space) is described by the ITM. The corresponding solution is found using the solutions for the uniform half-space and for the continuum with a cylindrical cavity. To exploit the invariance of the structure in longitudinal direction x, for this direction additionally to the time–frequency transform ( t ω ) a space–wavenumber transform ( x k x ) is used. The case of a half-space is analyzed using a Fourier transform also in the second horizontal direction ( y k y ) and an analytical solution for the z-direction on the basis of exponential functions. In the case of the infinite continuum with the cavity a Fourier series (for the circumferential direction) and a series of cylindrical functions (for the radial direction) are used for the solution regarding the cross-sectional coordinates ( y , z ) . The tunnel structure, that may have an arbitrary shape, and a portion of the surrounding soil will be modelled by Finite Elements in a ( x k x , t ω ) transformed domain. In order to observe the boundary conditions at the surface of the half-space as well as at the surface of the cavity, the superposition of the two solutions has to be performed after an inverse Fourier transform (IFT) in the ( k x , y , z , ω ) -domain. The solution for the complete system floating slab-track–tunnel–half-space is obtained in the ( k x , y , z , ω ) -domain. Once the system transfer function H ( ω ′ ) (with ω ′ = ω + vk x ) for this complete coupled system is found, the displacements of the plate can be calculated in time domain by the IFT of the product p ( ω ′ ) H ( ω ′ ) , i.e. the convolution of the loading p ( t ) with the impulse response function h ( t ) , which completely represents the behavior of the coupled system. In the context of the coupling of systems in a relative movement the problems of differential algebraic equations (DAE) have to be observed.