A lattice model for transition zones in ballasted railway tracks

The procedure presented in this work aims at providing a framework for studying settlement of ballast in zones with stiffness variation of the railway track support. The proposed procedure results from expanding an existing infinite periodic model of a railway track to account for variations in the stiffness of the foundation. Ballast is simulated via a linear lattice, whose dynamic response differs from that of a continuum. The expanded model is composed of three regions: a left region, which is semi-infinite and periodic; a mid-region, of finite length and where the properties of the foundation can change; and a right region, which is also semi-infinite and periodic and whose properties can differ from those of the left region. The equations of motion of the mid-region are written directly in the time domain, with the rail being described by finite elements. On the other hand, the left and right semi-infinite regions are treated semi-analytically in the frequency domain, and afterwards their responses are converted to the time domain, resulting in convolution integrals prescribed at the boundaries of the mid-region that simulate non-reflective boundaries. The final model only contains the degrees of freedom corresponding to the mid-region (which can be as short as the region where the stiffness variation is present), and that leads to faster calculations than if the boundaries were placed further away to dissipate undesired reflections. The method is cast in the time domain, and all elements are assumed to behave linearly. In the future, the model will be expanded to incorporate non-linear behaviour of the ballast. The presented method is validated by means of simple examples, and afterwards applied to a real scenario in which a culvert crosses a railway track. As presented, the method can be used to study the linear dynamics of transitions zones, study mitigation measures, and infer about indicators like force transmitted and energy dissipated, which might be useful to assess the development of settlement of the ballast.