In this study, a novel model for a periodically supported beam subjected to moving loads was developed using a periodicity condition on reaction forces. This condition, together with Fourier transforms and Dirac combs properties, forms a relation between the beam displacement and support reaction forces. This relation explains the force distribution to the supports, and holds for any type of support and foundation behaviors. Based on this relation, a system equivalence for a periodically supported beam is presented in this paper. An application to nonballasted viscoelastic supports is presented as an example and the results clearly match the existing model. Next, an approximation of realtime responses was developed for the moving loads as periodical series. The comparison shows that this approximation can be used for a limited number of loads if the distances between loads are sufficiently large. The system equivalence for a periodically supported beam is efficient for supports with linear behavior, and could be developed for other behaviors.

Existing analytical models for railway tracks consider only one rail supported by a continuous foundation or periodic concentrated supports (called the periodically supported beam). This paper presents an analytical model for a railway track which includes two rails connected by sleepers. By considering the sleepers as Euler–Bernoulli beams resting on a Kelvin–Voigt foundation, we can obtain a dynamic equation for a sleeper subjected to the reaction forces of the rails. Then, by using the relation between the rail forces and displacements from the periodically supported beam model, we can calculate the sleeper responses with the help of Green's function. The numerical applications show that the sleeper is in flexion where the displacement at the middle of the sleeper is greater than those at the rail seats. Moreover, the deformed shape of the sleeper is nonsymmetric when the loads on the two rails are different. The model result agrees well with measurements performed using instrumented sleeper in situ

The influence of a nonlinear foundation on the dynamics of a periodically supported beam has been investigated by a novel model. By using Fourier transforms and Dirac comb properties, a relation between the displacement of the beam and the reaction forces of its supports in steady-state has been established from the Euler-Bernoulli beam's equation. This relation holds for any foundation behaviours. Therefore, the dynamic equation of a support has been built by combining this relation and the constitutive law of the foundation and the supports. This equation describes a forced nonlinear oscillator provided that the moving loads are a periodical series. Then, an iteration procedure has been developed to compute the periodic solution. This procedure has been demonstrated converging to the analytic solution for linear foundations. The applications to bilinear and cubic nonlinear foundations have been performed as examples. Moreover, the influences of non-linearity on the dynamic responses have been investigated by parametric studies.