On the Equivalence between Static and Dynamic Railway Track Response and on the Euler-Bernoulli and Timoshenko Beams Analogy

Wlodzimierz Czyczula,Piotr Koziol,Dorota Blaszkiewicz

doi:10.1155/2017/2701715

Wlodzimierz Czyczula, Piotr Koziol + Show 1 more

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https://doi.org/10.1155/2017/2701715

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Abstract

The paper tries to clarify the problem of solution and interpretation of railway track dynamics equations for linear models. Set of theorems is introduced in the paper describing two types of equivalence: between static and dynamic track response under moving load and between the dynamic response of track described by both the Euler-Bernoulli and Timoshenko beams. The equivalence is clarified in terms of mathematical method of solution. It is shown that inertia element of rail equation for the Euler-Bernoulli beam and constant distributed load can be considered as a substitute axial force multiplied by second derivative of displacement. Damping properties can be treated as additional substitute load in the static case taking into account this substitute axial force. When one considers the Timoshenko beam, the substitute axial force depends additionally on shear properties of rail section, rail bending stiffness, and subgrade stiffness. It is also proved that Timoshenko beam, described by a single equation, from the point of view of solution, is an analogy of the Euler-Bernoulli beam for both constant and variable load. Certain numerical examples are presented and practical interpretation of proved theorems is shown.

Highlights

The problem of track dynamic response under moving load is the subject of many theoretical and experimental investigations
For the linear model of the track, if load is varying in time, the steady-state solution for the Timoshenko beam (see (38a)) is equivalent to the steady-state solution for the Euler-Bernoulli beam (see (30)), with accuracy determined by approximation of rail displacements by using Fourier series in finite interval [0, λ], in terms of the following substitution in matrix (35) and (36): PjE fl PjT; j = 1, . . . , 6, a0 fl aT0; (47)
Analytical solutions of railway track response to moving forces are studied in the case of linear models

Summary

Introduction

The problem of track dynamic response under moving load is the subject of many theoretical and experimental investigations. The beam on elastic foundation can be considered as a typical track model It is worth mentioning the initial study of beams on the Winkler foundation subjected to a concentrated force moving with constant speed that was initiated by Timoshenko [1]. In all described generalizations of classical approach, the track response model was composed of rail (as the EulerBernoulli or Timoshenko beam) and viscoelastic or elastic foundation. Hunt [24] presents numerical approach to solve the inverse Fourier transform in the case of certain simplified equation for a beam on viscoelastic foundation loaded by a single oscillating force. One can observe a lack of simple interpretation of the dynamic railway track response under moving distributed load in terms of differences between static and dynamic solutions and between the Euler-Bernoulli and Timoshenko beams. One can observe a lack of such papers trying to formulate basics of railway track analysis in a way similar to subjects recognized more systematized research fields

Track Response to a Set of Constant Forces Moving in Longitudinal Direction

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