Keywords: Rayleigh waves, critical speed, resonance, high-speed trains, coupled train-track interaction, railways, geo-dynamics, undrained shear strength, non-linearity.

It is known that as trains traverse soft soils a critical point can be reached whereby the train speed will exceed the Rayleigh ground wave velocity leading to critical velocity issues. If track structures are present, such as stiffened embankments, increases in the train's critical speed can occur. This higher ground wave velocity is termed the critical track velocity. Approaching a critical speed will lead to an increase in the transient track deflection when compared to the quasi-static case. As resonant conditions develop, very high transient track displacements will occur, including the potential for track uplift. In the past, these sites were few in number as the train speeds were relatively low, however as train speeds continue to increase critical speed issues will become more of an issue. This paper first presents simple graphs and equations of the permissible train speed versus Young's modulus and undrained shear strength for typical homogenous clay subgrades. However, for nonhomogenous soils (layered soils), it is necessary to understand the development of the dynamic interaction and hence determine the most cost effective mitigation strategy. For layered soils, this can become difficult to determine without more sophisticated analysis.

The development of critical velocities is studied in this paper using a threedimensional time domain finite element program in which the train and track are coupled and the soil is simulated using non-linear soil constitutive models where the stiffness is a function of the isotropic and deviatoric stress, and the damping ratio is a function of the deviatoric strain. This paper benchmarks the analyses using measured data from the Ledsgard site and shows that this approach is able to model the development of track dynamic conditions with train speed through plots of the transient sleeper deflection, and displacement contour plots of the overall ground surface behaviour. The non-linear variation in the different layers' stiffness is highlighted through plots of the time histories of Young's modulus with train speed and depth.

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