Keywords: ground vibration, wave propagation, moving loads, numerical methods, finite element method, special finite elements.

The problem of determining the response of a solid medium under the action of dynamic loads has received considerable attention in the past. Ground vibrations due to seismic, rail or road sources can lead to discomfort and, in some cases, cause damage to structures. It is thus important to study these phenomena theoretically in order to envisage ways of reducing their impact on the built environment. Consequently, work in this area is indispensable with the need to determine the vibratory motion on the ground surface and at depth caused by these sources. This is particularly true with regard to rail and road traffic, which is becoming faster and heavier. Combining with this fact and the observation that Rayleigh wave speeds are slower in soft soils, we see that the study of moving loads is important for environmental and geotechnical engineers. For example, some problems of high vibrations (displacements higher than 15 mm) induced by a train traffic have been observed in several countries.

Previously, for simple problems, analytical methods of calculating wave fields were investigated for a soil treated as an elastic isotropic and homogeneous half- space. These methods can explain the kinematics and generation of waves, and also the primary and secondary interferences in wavefields due to the superposition of Rayleigh, and waves in the nearfield, in the case of a stationary periodic excitation [1]. In the nearfield, the interferences due to the superposition of the three wave types are controlled by frequency and Poisson's ratio of soil.

For more complex problems, these methods are no more sufficient and available. In fact, actual conditions imply particularly two special data : first generally a soil medium is constituted with a lot of layers (not a half-space), and secondly we often deal with moving loads (for example railway or rail traffic). To take into account these peculiarities and through the development of computers, now it is possible to call up numerical methods allowing a better simulation of the wave propagation in soil. Among these methods, we can select a semi-analytical method using a Fourier transform and for conclusion a numerical calculation based on a FFT algorithm. In particular, the results deduced with this method can explain the formation of Mach cones observed for displacements [2]. Other methods rest on finite element modelling or infinite element modelling, but these ones require a meshing needing at least ten nodal points per wavelength with consequently foregone difficulties for high frequency problems. To solve it, a special element method is investigated [3] and now the recent developments are very promising.

The goal of this review is to present briefly any methods with a limitation at a general presentation, by leaning on significant results. Of course, some methods (for example boundary element methods) will not be approached, because it is not possible to present here an exhaustive viewpoint.

1

Prange B., "Primary and secondary interferences in wavefields", Proceedings of DMSR 77, Karlsruhe, 5-16 September, A.A. Balkema, vol. 1, pp. 281-308, 1977.

2

Lefeuve-Mesgouez G., Le Houédec D., Peplow A.T., "Ground vibration due to a high-speed moving harmonic load", "Structural Dynamics - EURODYN'99", A.A. Balkema, Ed. L. Fryba & J. Naprstek, vol. 2, pp. 963-968, 1999.

3

Laghrouche O., Bettess P., Sugimoto R., "Analysis of the conditioning number of the plane wave approximation for the Helmholtz equation", Proceedings of the European Congress on Computational Methods in Applied Sciences and Engineering, ECCOMAS 2000, 1-14, 2000.

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