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Shock and Vibration
Volume 2017, Article ID 2701715, 13 pages
Research Article

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

Faculty of Civil Engineering, Cracow University of Technology, Kraków, Poland

Correspondence should be addressed to Piotr Koziol;

Received 31 May 2017; Revised 14 August 2017; Accepted 19 September 2017; Published 30 October 2017

Academic Editor: Giuseppe Piccardo

Copyright © 2017 Wlodzimierz Czyczula et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.