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

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...

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Main Authors:	Wlodzimierz Czyczula, Piotr Koziol, Dorota Blaszkiewicz
Format:	Article
Language:	English
Published:	Hindawi Limited 2017-01-01
Series:	Shock and Vibration
Online Access:	http://dx.doi.org/10.1155/2017/2701715

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spelling	doaj-9c888c1f2d42460c8867377b1abdb7ba2020-11-24T22:43:56ZengHindawi LimitedShock and Vibration1070-96221875-92032017-01-01201710.1155/2017/27017152701715On the Equivalence between Static and Dynamic Railway Track Response and on the Euler-Bernoulli and Timoshenko Beams AnalogyWlodzimierz Czyczula0Piotr Koziol1Dorota Blaszkiewicz2Faculty of Civil Engineering, Cracow University of Technology, Kraków, PolandFaculty of Civil Engineering, Cracow University of Technology, Kraków, PolandFaculty of Civil Engineering, Cracow University of Technology, Kraków, PolandThe 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.http://dx.doi.org/10.1155/2017/2701715
collection	DOAJ
language	English
format	Article
sources	DOAJ
author	Wlodzimierz Czyczula Piotr Koziol Dorota Blaszkiewicz
spellingShingle	Wlodzimierz Czyczula Piotr Koziol Dorota Blaszkiewicz On the Equivalence between Static and Dynamic Railway Track Response and on the Euler-Bernoulli and Timoshenko Beams Analogy Shock and Vibration
author_facet	Wlodzimierz Czyczula Piotr Koziol Dorota Blaszkiewicz
author_sort	Wlodzimierz Czyczula
title	On the Equivalence between Static and Dynamic Railway Track Response and on the Euler-Bernoulli and Timoshenko Beams Analogy
title_short	On the Equivalence between Static and Dynamic Railway Track Response and on the Euler-Bernoulli and Timoshenko Beams Analogy
title_full	On the Equivalence between Static and Dynamic Railway Track Response and on the Euler-Bernoulli and Timoshenko Beams Analogy
title_fullStr	On the Equivalence between Static and Dynamic Railway Track Response and on the Euler-Bernoulli and Timoshenko Beams Analogy
title_full_unstemmed	On the Equivalence between Static and Dynamic Railway Track Response and on the Euler-Bernoulli and Timoshenko Beams Analogy
title_sort	on the equivalence between static and dynamic railway track response and on the euler-bernoulli and timoshenko beams analogy
publisher	Hindawi Limited
series	Shock and Vibration
issn	1070-9622 1875-9203
publishDate	2017-01-01
description	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.
url	http://dx.doi.org/10.1155/2017/2701715
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On the Equivalence between Static and Dynamic Railway Track Response and on the Euler-Bernoulli and Timoshenko Beams Analogy

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