Shock and Vibration

Volume 2017, Article ID 2701715, 13 pages

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

## 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; lp.ude.kp@loizokp

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.

#### 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.

#### 1. Introduction

The problem of track dynamic response under moving load is the subject of many theoretical and experimental investigations. Under some assumptions, 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]. The first solution to a simple stationary case of the Euler-Bernoulli beam on elastic foundation was properly obtained by Ludwig [2]. Mathews formulated and partly solved the case of moving and oscillating force [3]. The case of varying moving force was studied, for example, by Fryba [4] and by Bogacz and Krzyzynski [5].

Many papers are devoted to study various effects of generalized models:(1)Analysis of Timoshenko beam under moving constant and varying loads (presented, e.g., in [6–9])(2)Analysis of a beam on elastic half-space [10, 11](3)Response of beam on nonlinear foundation (e.g., [8, 12–14])(4)Dynamic response of beam on random foundation; see [15–17](5)Dynamic response of track as multilayered structure (see [18, 19], analytical approach; [20–22], numerical approach);(6)Analysis of set of distributed moving forces, described by Heaviside functions (e.g., [7]), rectangular function [9, 19], cosine square formula [8, 12, 13], or Gauss function [19](7)Effect of axial force on dynamic response [19, 23](8)Analysis of set of forces varying harmonically and associated with track imperfections including the phase of sine function for particular axles (numerically [20–22]) and analytical approach [13, 19]

In all described generalizations of classical approach, the track response model was composed of rail (as the Euler-Bernoulli or Timoshenko beam) and viscoelastic or elastic foundation. The sleepers and the ballast were modelled as additional layers.

Analytical closed form solutions were obtained for infinite Euler-Bernoulli beam on elastic foundation under single concentrated load moving along the beam with constant speed using so-called matching conditions for homogeneous solution in the point related to a load position [2, 3]. Using the same approach, the damping properties of foundation or axial force in beam (rail) were also included in the closed form solutions (comp., e.g., [4, 5, 23]). For oscillating force or set of oscillating forces, the solution was obtained by using the Fourier transform and the inverse Fourier transform [5, 6, 8]. Difficulties arising from the integration of solution in the frequency domain lead to the various method of approximation of the transformed function. An interesting approach based on wavelets application is developed by Koziol and coauthors (comp. [8, 10]). Applications of Fourier series to obtain the solution of railway track response are practically limited to the case of bridges vibrations so far [4, 22]. An example of use of the Fourier series in bounded interval to study the railway track response is presented in [19]. It is shown that solutions obtained in this manner can approximate the track response with very high accuracy depending on a number of Fourier series coefficients and the assumed length of interval.

Analysis of Timoshenko beam is mainly carried out for a set of equations describing coupled beam displacements and rotations [6, 8]. The characteristics of the dynamic response of Timoshenko beam described by a single equation are rarely analysed. 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. In [7], the problem of critical speeds for both constant and varying distributed loads is studied analytically without detailed interpretation of final solution. Single equation of dynamics of Timoshenko beam without foundation was analysed by Majkut [25].

The above-mentioned papers present many interesting results. However, 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 also between the Euler-Bernoulli and Timoshenko beams. This paper tries to clarify the formulated problem described by linear equation of rail motion. The equivalence between static and dynamic railway track response for foundation without damping is proved. Using the Fourier series in finite interval for both distributed load and solution it is shown that damping properties of rail foundation can be interpreted as additional, substitute load. The equivalence of dynamic track solution, in terms of mathematical physics equations, between the Euler-Bernoulli and Timoshenko beams is also proved. Numerical examples are presented leading to practical interpretation of the formulated theorems in wide range of train speed and foundation damping properties. The introduced set of theorems can be treated as an important contribution to a proper arrangement and classification of knowledge that can be recognized as an essential part of fundamentals of the theory of railway track dynamics. 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.

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

##### 2.1. Rail Modelled by the Euler-Bernoulli Beam Equation

Basic equation of motion of track modelled by the Euler-Bernoulli beam on viscoelastic foundation has the following form (comp., e.g., [4]):where is rail Young modulus [], is moment of inertia of beam (rail) in vertical plane [], is axial force in rail (positive sign means compressive force) [], is unit mass of beam [], is damping coefficient of foundation [], is foundation stiffness [], and is distributed load [].

In the moving coordinate system (), if load is constant in time, (1) may be transformed to an ordinary differential equation:

Theorem 1. *For linear model of the track, described by (1), if load does not change in time, the steady-state solution of (2) is an equivalent static case for damping by the substitution:where is substitute axial force [N]. Effect of foundation damping, with an accuracy determined by approximation of rail displacements by Fourier series in finite interval , can be considered as a static case with substitution (3) and additional (substitute) load described as follows:where is substitute load [N/m]; is function depending on real load q and track model parameters P (i.e., EI; N; m; etc.).*

*Proof. *The proof of the first part of the theorem (see (3)) is relatively simple. The equation of static deflection of beam on elastic foundation (parameter ) with compressive axial force and load has the following form (see, e.g., [26]):As can be seen, for assumed train speed and unit mass of beam , in the moving coordinate system, the sum of realistic axial force and in (2) for expresses the substitute axial force for the beam on elastic foundation in (5), that is, .

For proving second part of the theorem one assumes that realistic load and the solution can be described by the Fourier series in the assumed interval [], that is,where are the coefficients of Fourier series.

Then (2) leads to the following formulas (for cosine and sine coefficients of the Fourier series):where (comp. (3)). One can derive the form of parameters and by solving the system of (7):whereand the constant value* y*_{0} is described by formula (8).

Let other coefficients and , together with the same value (formula (8)), describe the cosine and sine coefficients of solution of (2) in the static case with substituted axial force and the damping coefficient . In this case, the solution of (7) has a simple form:If the solution of dynamic case, for , and static solution are denoted by and , respectively, then the following formula can be written in terms of Fourier series in the assumed interval [] (comp. (6)):and the difference between dynamic and static solutions with isThis difference represents function which can be considered as linked to an additional load in the static case with . Then one can obtain the cosine and sine parts of substitute load in the static case for any index : with constant value equal to zero due to the fact that the difference between dynamic and static solution includes only cosine and sine Fourier series coefficients (the constant is the same for both dynamic and static solutions (comp. (2))). Using formulas (9), (11), and (14), one can obtain the following:The above formulas (15) describe the cosine and sine coefficients of substitute load:It means that substitute load which comes from damping properties of the rail foundation is determined as a function of realistic load (coefficients , and speed ) and model parameters and which describe beam and foundation properties (see notation (10)). The second part of the theorem is then proved.

For practical interpretation of Theorem 1, certain remarks should be formulated.

() The equivalence between static and dynamic response of railway track described by simple model, that is, the Euler-Bernoulli beam on viscoelastic foundation under moving load invariant in time, concerns only mathematical solutions.

() In the case when damping properties of rail foundation can be neglected, due to the load being distributed on very small spans and the distance between wheels being relatively large (practically more than about 5-6 m), dynamic factor for maximum rail displacements can be described by simple formula (based on [19, 23]):where formula (5), is critical value of axial force.

The sign of the substitute axial force depends on relation between realistic compression/tension force in rail and the term which has positive value. Knowing that real axial force in rail depends mainly on the rail temperature changes in relation to the laying rail temperature ,where is coefficient of thermal expansion of rail steel [1/°C] and the part of substitute axial force (depending on speed) can be calculated in relation to the equivalent rail temperature increase . For typical rail, being in use in Europe, that is, 60E1, and unit mass of rail 60 kg/m (with the unit mass of half sleeper, usually equal to around kg/m), one can obtain the following equivalent rail temperature increase: or ; see [26]. It means that(i)°C or °C for km/h (27.77 m/s);(ii)°C or °C for km/h (55.55 m/s);(iii)°C or °C for km/h (83.33 m/s).

For typical track structure, the critical axial force (formula (17)) is around 25–35 MN. The equivalent rail temperature increase for these values is on the level of 1500°C. Using expression (17), it follows that, for realistic rail temperature increase (maximum 50–60°C) and speed up to 300 km/h, the dynamic factor for track without imperfections and damping is lower than 1%, in the case of typical track structure.

() The dynamic factor associated with damping properties of rail foundation in relation to the static response with substitute axial force cannot be described by simple formula. Also, the substitute load linked to the foundation damping (formulas (15)) cannot be simply explained. Currently, these problems can be numerically solved. The following track and load parameters are assumed in further study [19]:(i)Rail: 60E1, Young modulus, N/m^{2}; cross-sectional area, m^{2}; thermal expansion coefficient, 1/°C; unit mass, kg/m (or kg/m if rail and a half of concrete sleeper PS-94 type are taken into account); rail temperature increase, °C(ii)Rail foundation: substitute foundation stiffness due to fasteners and sleeper foundation elastic properties N/m^{2}, damping properties – fasteners Ns/m^{2} (reference value obtained during laboratory tests and also and ), sleepers foundation Ns/m^{2} (reference value and )(iii)Load: 4 wheels of EMU-250 Pendolino train with configuration: 2.7 m, 4.5 m, and 2.7 m; see [27], wheel load 80 kN; load distribution represented by the Gauss function with the parameter m, number of Fourier series coefficients equal to 3000

Figure 1 presents the dynamic factor due to rail foundation damping for two options: higher values describe dynamic factor for viscous properties of fasteners and sleeper foundation properties ; lower values describe dynamic factor for the reference values and . As can be observed, these relations have various curvatures and show sensitivity related to the effect of damping on dynamic factor. In the considered range of operating train speeds and viscous coefficients, the maximum value of dynamic factor can reach 4%.