Shakedown limit analysis for heavy-haul railway tracks

The lower-bound shakedown theorem provides a useful framework for evaluating the long-term stability of structures subjected to cyclic loading by defining both the shakedown limit and critical depth. However, its application in freight railway engineering remains relatively limited. To overcome this gap, shakedown theory has been integrated into the design of heavy-haul railway trackbed systems, enabling assessment of substructure stability under repeated loading. The stress distributions along the longitudinal and transverse axes of the sub-ballast surface, induced by a four-axle loading pattern, were quantified and validated through Gaussian curve fitting. Additionally, a methodology based on the Mohr–Coulomb yield criterion was developed to estimate the shakedown limit of the subgrade, employing the corresponding shakedown axle load as the primary evaluation index. Parametric analyses examined the effects of three key design parameters: the internal friction angle of the sub-ballast, the elastic modulus of the engineered subgrade, and the thickness ratio between the sub-ballast and engineered subgrade. Findings consistently showed that increases in these parameters lead to higher shakedown axle loads. Among them, the internal friction angle of the sub-ballast has the most pronounced influence, whereas the thickness ratio plays a relatively minor role. For example, elevating the internal friction angle from 25° to 40° produces a significant 47.5% rise in the shakedown axle load, highlighting its pivotal contribution to enhancing the substructure's resilience against cyclic loads.