Transition radiation in a piecewise-linear and infinite one-dimensional structure–a Laplace transform method

Transition zones in railway tracks are areas with considerable variation of track properties (i.e., foundation stiffness) encountered near structures such as bridges and tunnels. Due to strong amplification of the response, transition zones require frequent maintenance. To better understand the underlying degradation mechanisms, a one-dimensional model is formulated, consisting of an infinite Euler-Bernoulli beam resting on a locally inhomogeneous and nonlinear Winkler foundation, subjected to a moving load. The nonlinearity is characterized by a piecewise linear stiffness, and the system thus behaves piecewise linearly. Therefore, the solution can be obtained by sequentially applying the Laplace Transform combined with the Finite Difference Method for the spatial discretization and derived non-reflective boundary conditions. Results show that the plastic deformation is a consequence of constructive interference of the excited waves and the response to the load's deadweight, particularly for the soft-to-stiff transition. The plastic deformation area decreases quasi-monotonically with increasing transition length, and for super-critical velocities, small transition lengths and/or large stiffness dissimilarities, parts of the foundation experience plastic deformation even in the second loading-unloading cycle. Furthermore, the nonlinearity causes the maximum energy associated with the waves radiated forward and the maximum energy input not to occur for the smallest transition length, contrary to findings in corresponding linear systems. Moreover, the energy input drastically increases for the second passage of the moving load, making it a possible indicator of the damage in the supporting structure. The novelty of the current work lies in the computationally efficient solution method for an infinite system which locally exhibits nonlinear behaviour and in the study into the influence of the foundation’s nonlinear behaviour on the generated waves (i.e., transition radiation), and on the resulting plastic deformation. The model presented here can be used for the preliminary design of transition zones in railway tracks.