Application of three-dimensional shakedown solutions in railway structure under multiple Hertz loads

Abstract Shakedown analysis is a robust approach to solve the strength problem of a structure under cyclic or repeated loading, e.g. railway structure subjects to rolling and sliding loads. A three-dimensional analytical shakedown solution is developed in this paper for the strength analysis of cohesive-frictional materials under multiple Hertz loads, which is then applied into the shakedown analysis of railway structure. Different to the pavement, the railway structure subjects to multiple wheel loads acting on the surface of two parallel rails, and then transformed to the substructure, which leads to different elastic and residual stress fields. For the application of Melan's shakedown theorem, the residual stress field in railway strucuture is rationally simplified to find the most critical plane that influence the most on the shakedown limit. The elastic stresses is then analytically derived for a single layer half-space under multiple Hertz loads, and is obtained numerically by means of the finite element technique for the multi-layered railway structure. Finally, the shakedown limit can be obtained in a direct way. Parametric study indicates how wheel loads, material properties such as frictional coefficient, friction angle and Poisson's ratio, and multi-layers influence the shakedown limit and stability condition of railway structure. It is found that the shakedown load increases with the increase of cohesion ratio, ballast frictional angle, and thickness ratio. However, the increase of the shakedown load ceases once the failure location moves from ballast layers to sub-ballast layers. This indicates that there is an optimum combination of material properties and layer thickness which provides the maximum resistance to the train loads. The obtained results give a useful reference for the engineering design of the railway structure.