Closed-form solution for dynamic response of Pasternak visco-elastic foundation under moving loads: An analytical approach for transition zone

Transition zones in railways, marked by abrupt changes in foundation stiffness, lead to differential settlements and amplified dynamic loads, accelerating track degradation. This necessitates a method that accurately relates stiffness ratios while accounting for shear interaction within the soil. Traditional models like the Winkler foundation, which neglect soil interaction, often overestimate dynamic responses in such scenarios. This paper presents a closed-form solution for analyzing transition zones, incorporating shear interaction effects modeled through a Visco-elastic Pasternak foundation of abrupt stiffness. The study extends to examine the dynamic responses of a series of axle loads under varying stiffness ratios and speeds, comparing the behavior of Winkler and Pasternak foundations. Results indicate that the Winkler model overestimates the dynamic response by 24%–28% for displacement and 5%–10% for acceleration, approximately. Additionally, the inclusion of the shear layer in the Pasternak model leads to a higher predicted critical velocity compared to the Winkler model. These findings emphasize the substantial impact of foundation resistance as well as resistance due to shear strain on the dynamic performance of the transition zone. Hence, such models can replicate ground behavior more accurately than the Winkler model, offering improved accuracy over traditional Winkler-based solutions. By analyzing various values of the stiffness ratio $r$, designers and engineers can make informed decisions regarding the selection and placement of materials with tailored stiffness properties in transition zones. This strategic use of materials with varying stiffness enables the design of effective mitigation measures aimed at smoothing out the abrupt changes in structural rigidity.