Influence of beam models on dynamic responses of ballasted railway track subjected to moving loads

Abstract

This research presents a development of the dynamical model for ballasted railway tracks by combining two models: a periodically supported beam model for the rail and model of beam posed on a viscoelastic foundation for the sleeper. In this development, two beam theories can be used to describe the dynamic behavior of beam: Euler–Bernoulli and Timoshenko. In steady-state condition, a relation between the reaction force applied on the support and displacement of rail is established in the frequency domain. In the other way, these two variables are linked with the help of Green’s function for the sleeper. By performing the two previous analyses, the dynamic responses of rails are obtained analytically in the frequency domain. The numerical examples show the influence of beam models for the rails and sleeper on the dynamic responses. This work provides analytical choices to calculate the dynamic track responses where the rails and sleeper can be described by the two beam theories.

Appendix A: Green’s function for the model of sleeper

Equation (17) describes a dynamic responses of railway sleeper for both theories where the coefficients for each beam model are expressed in Table 3. This equation is a 4$^{\textrm{th}}$ linear differential equation, and the solution is given by using Green’s function:

$$\begin{aligned} G(x_s,\omega ;a) = \hat{M}_1 \cosh (\hat{\beta }_1 x_s) + \hat{M}_2 \sinh (\hat{\beta }_1 x_s) + \hat{N}_1 \cos (\hat{\beta }_2 x_s) + \hat{N}_2 \sin (\hat{\beta }_2 x_s) + \hat{D}(x_s,\omega ;a) \end{aligned}$$

(A.1)

where

• Function $\hat{D}(x_s,\omega ;a)$ which denotes the discontinuity of the shear force due to the application of the unit load at $x_s = a$ is determined as:

  $$\begin{aligned} \hat{D}(x_s,\omega ;a) = \dfrac{H(x_s-a)}{\hat{\beta }_1^2+\hat{\beta }_2^2} \left( \hat{P} \dfrac{\sinh \hat{\beta }_1(x_s - a)}{\hat{\beta }_1}- \hat{Q} \dfrac{\sin \hat{\beta }_2(x_s - a)}{\hat{\beta }_2}\right) \end{aligned}$$

  (A.2)

  where $H(x_s-a)$ is the Heaviside step function and:

  $$\begin{aligned} \left\{ \begin{aligned}&\hat{\beta }_{1, 2}^2 = \alpha _s^2 \\&\hat{P} = \hat{k}_1 + \hat{\beta }_1^2 \hat{k}_2 \\&\hat{Q} = \hat{k}_1 - \hat{\beta }_2^2 \hat{k}_2 \end{aligned} \right. \end{aligned}$$

• Four coefficients $\hat{M}_i$, $\hat{N}_i$ with $i = [1, 2]$ are evaluated to satisfy the four conditions for free-free beam at each end $x_s = \pm L$:

  $$\begin{aligned} \left\{ \begin{aligned}&\dfrac{\partial ^2 G(-L,\omega ; a)}{\partial x_s^2} - \left( \dfrac{k_b+\textrm{i} \omega \zeta _b - \omega ^2 \rho _s A_s}{\kappa _s A_s G_s} \right) G(-L, \omega ;a)&= 0 \\&\dfrac{\partial ^3 G(-L,\omega ; a)}{\partial x_s^3} - \left[ \dfrac{E_s(k_b+\textrm{i} \omega \zeta _b)-\omega ^2\rho _s A_s (E_s + \kappa _s G_s)}{\kappa _s A_s G_s} \right] \dfrac{\partial G(-L, \omega ;a)}{\partial x_s}&= 0 \\&\dfrac{\partial ^2 G(L,\omega ; a)}{\partial x_s^2} - \left( \dfrac{k_b+\textrm{i} \omega \zeta _b - \omega ^2 \rho _s A_s}{\kappa _s A_s G_s} \right) G(L, \omega ;a)&= 0 \\&\dfrac{\partial ^3 G(L,\omega ; a)}{\partial x_s^3} - \left[ \dfrac{E_s(k_b+\textrm{i} \omega \zeta _b)-\omega ^2\rho _s A_s (E_s + \kappa _s G_s)}{\kappa _s A_s G_s} \right] \dfrac{\partial G(L, \omega ;a)}{\partial x_s}&= 0 \end{aligned} \right. \end{aligned}$$

  (A.3)

  The four unknowns can be determined analytically or numerically by solving this system of 4 equations.