Abstract
A periodically supported beam on a visco-elastic half-space is considered to model the vibration of railway tracks. The viscosity of the half-space is assumed to be of the Kelvin-Voigt type. Making use of the concept of equivalent dynamic stiffness, the reaction of the half-space to the sleepers is replaced by a system of identical spring located under each sleeper. The frequency-dependent equivalent stiffness of the springs is a function of the phase shift of vibrations of neighbouring supports. The equivalent stiffness is derived analytically employing the contour integration technique, resulting in a comprehensive expression for different phase velocities of the waves in the beam with respect to the wave speeds of the half-space. Apart from the Rayleigh type surface wave (quasi-elastic wave), an extra visco-elastic surface wave may exist in a visco-elastic half-space depending on the parameters of the half-space and the frequency range. The existence of this second surface wave has not been addressed within the field of train-induced ground vibration. The importance of this wave for the equivalent stiffness is analysed. An effective method to determine the frequency range for the visco-elastic surface wave to exist is proposed.
Highlights
With increasing computational power, in recent years numerical methods such as FEM, BEM and hybrid methods are commonly used in modelling train-induced ground vibrations (Hall, 2003; Sheng et al, 2006; Degrande et al, 2006; Yang and Hung, 2009; Galvín et al, 2010; Triepaischajonsak and Thompson, 2015)
Apart from the Rayleigh type surface wave, an extra visco-elastic surface wave may exist in a visco-elastic half-space depending on the parameters of the half-space and the frequency range
A uniform expression is obtained for the entire velocity range of the moving load regardless of the ratio be tween the load speed and the wave speeds of the half-space
Summary
In recent years numerical methods such as FEM, BEM and hybrid methods are commonly used in modelling train-induced ground vibrations (Hall, 2003; Sheng et al, 2006; Degrande et al, 2006; Yang and Hung, 2009; Galvín et al, 2010; Triepaischajonsak and Thompson, 2015). Analytical methods retain their significance since they are apt to reveal the underlying mechanisms of the generation of ground vibrations caused by moving trains. Various analytical/semi-analytical models for ground vibration induced by moving trains on open tracks (Sheng et al, 1999; Karlstro€m and Bostro€m, 2006) and in tunnels (Forrest and Hunt, 2006; Metrikine and Vrouwenvelder, 2000; Yuan et al, 2015; Di et al, 2018; Zhou et al, 2020) can be found in the literature. A comprehensive review can be found in (Lombaert et al, 2015) which covers various prediction methods and mitigation mea sures for train-induced ground vibration
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