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

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.