Keywords: boundary elements, high speed train, wave propagation, transient vibrations, experimental validation.

High speed trains are becoming a usual mean of transportation for intermediate distances in many countries. New lines have been constructed in recent years. Factors such as: vibrations in the train, wave propagation in the soil next to the track and dynamic effects in nearby construction, are much more important in high speed trains than in conventional ones. These effects require analysis in order to maintain security and comfort in the trains and to avoid problems in nearby construction as a result of vibrations induced by waves transmitted through the soil. Particularly serious would be situations in which the train speed may be higher than that of the surface waves in the underneath soil. This possibility was completely unthinkable in conventional trains but it is now something that should be taken into account when high speed trains operate at locations with particularly soft soils or underground discontinuities that may result in a relatively low surface wave speed.

The study of these problems requires comprehensive models that take into account many factors related to the characteristics of the train, the track, and the particular soil properties. Nearby structures should also be modeled in case the effects on these structures are being analyzed. Different analytical, semi-analytical or numerical methods have been developed in recent years. Numerical methods are intended to reproduce more precisely the particular conditions of the problem taking into account the important dynamic interaction effects. The finite element method (FEM) and the boundary element method (BEM) have been used to study the problem of high speed trains passing taking into account the propagation of waves in the soil and dynamic soil-structure interaction [1,2]. The main differences between these models are based on the way in which excitation and interaction effects are considered.

It is known that boundary elements are very well suited for dynamic soil-structure interaction problems. They are able to represent unbounded regions in a natural way as a result of the fact that the fundamental solution and, therefore the solution of the problem at hand, satisfies the radiation conditions.

A three dimensional time domain formulation of the BEM with a full-space fundamental solution is used in this paper. Nine node quadrilateral and six node triangular quadratic elements are used. Piecewise linear and piecewise constant time interpolation are used for displacements and tractions, respectively. Special attention is paid to stabilizing algorithms and element subdivision to improve efficiency, stability and accuracy of the procedure. A decaying law for wave amplitude is introduced in order to represent internal damping in the soil.

The soil is assumed to be a uniform viscoelastic half-space and is discretized into quadratic elements in a zone of the soil surface around the points of interest. The passing of a high speed Thalys train is assumed.

Two symmetry axes of the problem were considered in order to reduce the number of degrees of freedom of the problem. A track length of 86.4 m was discretized. The discretized surface has a width of 25m.

During 1997 the Structural Mechanics Group of the Catholic University of Leuven obtained a series of experimental results of soil induced vibrations due to the passing of a Thalys train at different speeds. In the present paper, results for the type of train and soil conditions of those experiments are computed using the time domain BEM. The soil properties are as proposed by Degrande and Schillemans [3] taking into account the measured values of P and S-wave speeds and the internal damping. Those values are: , (top layer) and an internal damping higher than 3% (we consider ). Krylov's loading distribution has been used to represent the constant load transmitted by each train wheel. The coefficient controlling stability of the time domain BEM is set equal to 0.3 in relation to the smallest elements of the mesh.

It can be seen that there is a rather good agreement between the numerical results obtained by the model proposed and the experimental results obtained by Degrande and Schillemans [3]. Both sets of results have rather close values of amplitude and the main frequency and show a similar decaying law with the distance to the track. More results corresponding to different train velocities and other locations will be presented in a forthcoming paper.

1

J.Domínguez,"Boundary elements in dynamics", Computational Mechanics Publications and Elsevier Applied Science, Southampton, Boston, London, New York, 1993.

2

G.Lombaert,"Development and experimental validation of a numerical model for the free field vibrations induced by road traffic", Ph. D. Thesis, Katholieke Universiteit Leuven, Belgium, 2001.

3

G.Degrande, L.Schillemans,"Free field vibrations during the passage of a Thalys HST at variable speed", Journal of Sound and Vibration, 247, 131-144, 2001. doi:10.1006/jsvi.2001.3718

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