Semi-analytical solution for calculating effective length of infinite axially loaded beam on two-parameter Pasternak foundation under a moving load

Using a finite beam with infinite boundary conditions to analogize the dynamic response of an infinite foundation beam under moving load is considered as a compromise approach in railway engineering applications. However, determining the optimal length of the finite beam to simulate the infinite beam has always been a pressing problem to be solved, especially for some complex infinite foundation beams. For this, this paper innovatively proposes a semi-analytical solution for calculating the effective length of infinite axially stressed two-parameter Pasternak foundation beams under moving loads. By assuming the displacement response of infinite foundation beams to be in exponential form, the formula for the effective length under different damping conditions is derived. The feasibility of the proposed theory is validated through the Fourier transform and residue theorem, and its accuracy is verified by comparing with existing methods. The parametric analysis shows that: (1) the derived closed-form solutions on different damping levels are highly accurate, and the proposed solution outperforms the residue theorem -based method as it eliminates the interference of imaginary numbers; (2) a supercritical-velocity moving load substantially extends the displacement - response transfer distance, greatly increasing the effective lengths of the front and back waves, while a subcritical - velocity load has a more modest effect; (3) foundation-related factors such as damping ratio, stiffness, and shear modulus, as well as axial loading, all have distinct impacts on the effective length. The study not only enriches the theoretical study of foundation beam effective lengths but also provides a more accurate and efficient calculation method, which has potential applications in high - speed railway engineering design and analysis.