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

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.

A Validated Train-Track-Bridge Model with Nonlinear Support Conditions at Bridge Approaches

“…The equations of motion for vehicle components, crossties and ballast masses listed in Equations ( 16)- (18), respectively, are all used to assemble the final matrix. To be concise, one car of the train, consisting of one car body, two bogies, and four wheelsets, were considered here to illustrate the basic form of the final matrix.…”

“…Using this approach, the rail is modeled as Euler-Bernoulli beam or Timoshenko beam laying on a Winkler foundation. Czyczula et al showed that both Euler-Bernoulli beam and Timoshenko beam gave almost the same results [18]. To study the time-dependent behavior of track under dynamic loading, often the finite element method (FEM) has been adopted and used by researchers as the numerical modeling method of choice.…”

Influence of Non-Uniform Rail Loads on the Rotation of Railway Sleepers

“…Most of the works related to railway dynamics deal with the development of the concept of a beam on an elastic foundation. A broad review of these works was presented, e.g., [5,6]. Generalisations were applied in relation to introducing an axial force in rails [7,8]; a mobile load; and, first, one oscillating force, e.g., [9,10], and then multiple forces, e.g., [11].…”

“…Apart from linear models, cases of the unevenness of the rail foundation were also analysed, e.g., [1,10]. In the featured works, the rail was modelled as the Euler-Bernoulli beam; the analysis of the Timoshenko beam was presented, e.g., in [6,20]. Various methods were employed to analytically solve the problem of a variable mobile force.…”

Archives of Civil Engineering

“…The beam can be described either by the Euler-Bernoulli equation or the Timoshenko beam model. The beam foundation can be also viscoelastic with linear or nonlinear stiffness [3,4,5].…”