Proof of critical speed of high-speed rail underlain by stratified media

Abstract

As is well known, the safe travel velocity of high-speed rail must remain below a well-defined limit commonly referred to as the critical speed . This upper bound depends, in turn, on the speed at which waves propagate in the ground. But as models for trains in motion have increased in complexity, over the years the concept of critical speed has grown in obscurity. It was not until late in the twentieth century that it was realized that the critical speed was somehow connected to the so-called dispersion spectrum for the complete system, but until now, the justification for its application in practice has remained empirical and lacking in rigorous mathematical explanations. This task is taken up in this article, where for the first time a most general proof is provided for the problem at hand that is applicable to any layered soil configuration when one such system is subjected to one or more loads in motion.

Moving load in an ice channel with a crack

“…This suggests that ω (1) (k) ∼ c (1) (0)k for small wavenumbers k. Then the corresponding group speed, dω (1) (k)/dk, at k = 0 is equal to the phase speed c (1) (0), which is the condition for this speed to be a critical speed. A critical speed of a moving load is defined as a speed at which the corresponding group and phase speeds are equal, see Kausel et al [28]. The phase speed is the speed of the load and the speed of the waves generated by the load.…”

“…once the functions F R n (ξ) and F I n (ξ) have been determined as solutions of the system (27) for any ξ ≥ 0. We denote the integrals in ( 28) by W n (x), and retain N lowest modes in the series (28) in our numerical results. The integrals W n (x) are evaluated numerically by trapezoidal rule for a finite interval which is large enough.…”

“…Both the interval and the step of integration in W n (x), and the number of retained modes N are carefully selected to guarantee that (29) well approximates the deflection (28) for 0 ≤ y ≤ 1 and an x-interval of interest. The second derivatives of the plate deflection in the strain tensor (15) are obtained by differentiation of the integrals in (28). The convergence test for the second derivatives has been also performed.…”

Analysis of embankment underlain by elastic half-space: 2.5D model with paralongitudinal approximations to the half-space

“…Then again, when applying the proposed approximations in the context of moving loads that move with velocity below the critical speed at which the response grows without any bound [18], it will be found that there are no waves that propagate either down into the half-space or laterally along the surface, and this is the reason for why the proposed method is applicable in that case. Indeed, if such waves existed, they would radiate energy away from the moving load and damp out the…”

Protection of pipeline below pavement subjected to traffic induced dynamic response

“…Apart from full scale experimental and three-dimensional numerical analyses, another alternative way of solving complex soil-structure interaction related problem in a simplified way is the analytical approach which is based on beam spring concept. For instance, Kausel et al 22 investigated the critical speed of high-speed rail considering the theory of beam on elastic foundation. The limitation of the study is that it did not incorporate the system damping and any shear interaction between elastic springs.…”