Dynamic vehicle–track interaction and differential settlement in a transition zone on railway ballast—An integrated 3D discrete–continuum model

A numerical methodology for simulating the mechanisms during the initial phase of differential settlement in a railway transition zone using an integrated discrete–continuum approach is presented. The methodology involves the coupling of the Discrete Element Method (DEM), the Finite Difference Method (FDM), and the Finite Element Method (FEM) to model the vertical dynamic interaction between vehicle and transition zone. Specifically, the extensive three-dimensional (3D) DEM model captures the discrete granular behaviour of the ballast and sub-ballast layers, while the continuum-based FDM model is employed to represent the rail structure and the subgrade layer. Based on a time-domain representation of vertical dynamic vehicle–track interaction, the nonlinear two-dimensional (2D) FEM model of the track, together with a multi-body system (MBS) model of the vehicle, is used to calculate the contact forces between wheels and rails. These forces are subsequently used as input to the DEM–FDM simulation for evaluating the non-uniform permanent displacements that will evolve within the granular layers. The support stiffness for each sleeper that is used as input in the FEM model is precomputed during the DEM–FDM coupling stage by applying a static load to each sleeper and calculating the resulting displacement. The developed methodology effectively simulates the progressive formation of voids beneath the sleepers, the redistribution of sleeper-ballast contact force between adjacent sleepers, and the evolving irregularity in vertical track alignment due to the accumulated traffic loading. The approach is demonstrated for a transition zone involving a stiffness gradient between a softer track on ballast and a stiffer track form, and accumulated settlements are calculated for a total of 500 axle passages. The proposed hybrid DEM–FDM–FEM framework provides critical insights into track degradation mechanisms, emphasising the importance of designing a gradual variation in track stiffness to mitigate dynamic loading leading to long-term differential track settlement, thereby reducing maintenance requirements in railway transition zones.