An enhanced three-dimensional reduced-order track model for predicting differential settlement in railway transition zones on stratified soils

Railway transition zones between two distinct track forms are prone to differential settlement, which can undermine track performance and increase maintenance demands over time. Accurately predicting the long-term evolution of settlement in these zones remains challenging due to the complex interplay between dynamic vehicle–track interaction, changing support conditions, and the gradual development of track geometry irregularities. This study presents a methodology for simulating long-term differential track settlement in railway transition zones using time-domain analyses of vertical dynamic vehicle–track interaction. The approach accounts for the formation of voided sleepers, the redistribution of sleeper–ballast contact pressure across the transition zone, and the progressive evolution of vertical track irregularities. The computational framework integrates a two-dimensional (2D) vehicle model, a three-dimensional (3D) non-linear finite element (FE) model of the track superstructure, and a linear 3D FE model of the layered subgrade. Mitigation measures installed within the substructure to alleviate stiffness gradients between track forms are also represented. Rails and sleepers are modelled using Euler–Bernoulli beam finite elements. To enhance computational efficiency, the subgrade model is reduced via static condensation to form a reduced-order model (ROM). The methodology is applied to transition zones between ballasted track on embankment and slab track on a bridge or in a tunnel. In each iteration step, a short-term dynamic analysis determines the contact pressure distribution at each sleeper–ballast interface, providing the basis for calculating ballast settlement increments. Simultaneously, the depth-dependent deviatoric stress distribution is evaluated to estimate permanent deformation within the subgrade layers. Iterations proceed until the specified cumulative traffic load is reached, enabling prediction of long-term settlement development. The results show that differential settlement develops predominantly on the ballasted side of the transition, with a local maximum occurring between sleepers 5 and 10 from the transition. The precise location of the maximum depends on traffic direction. For traffic moving from softer ballasted track towards stiffer slab track, the maximum occurs at sleeper 8, whereas the reverse direction produces a maximum at sleeper 5, with slightly larger magnitude in the former case. Over time, settlement accumulation stabilises due to hardening behaviour in the settlement models. The proposed framework provides a robust basis for evaluating the long-term performance of railway transition zones and for assessing the effectiveness of substructure mitigation measures.