Longitudinal Rail Load Distribution: An Analytical Solution
Authors: Marquis Brian , Liu Shushu , Stuart Cameron 
Published: August 2020

Abstract
Cut spikes have been used to hold rails directly to wood ties and thus provide lateral restraint to rails and maintain rail gage. For tightly curved and/or steeply graded wood tie tracks, particularly those undergoing heavy loads, elastic clips have been increasingly used to fasten rails securely to tie plates, which in turn are held down by cut spikes or lag screws to the crossties. However, several recent derailment accidents were attributed to the breakage of significant numbers of spikes used in association with the elastic fastening systems. A commonly observed failure mode is the development of fatigue cracking across spike surfaces located approximately 1.5 inches below the top surface of the tie. Previous Finite Element (FE) studies have shown that the presence of longitudinal force transmitted into the tie-spike interface is significant to accelerate wood damage and cause spike failure initiation as the strength of wood along the track longitudinal direction is weaker than the other directions. In the field, the magnitude of longitudinal load between a tie and fastener is very difficult to measure relying on the current sensor technology. A literature review conducted so far indicates that longitudinal-load-distribution related computational models are very limited. Therefore, the distribution mechanism of longitudinal load has not been well-established. To better understand how the longitudinal load is transferred from wheels to ties and fasteners, this paper developed an analytical model to investigate the longitudinal tie forces using the finite difference method.
In this paper an analytical model is firstly introduced to calculate the longitudinal track response under a single static wheel load. The solution calculates diverse sets of responses such as: (1) rail axial displacements; (2) rail axial forces; (3) force distribution to the tie; and (4) tie forces. The effect of multiple axle loads from a train is then computed using superposition theory. To validate the model, a field test was performed at the Horseshoe Curve in Altoona, Pennsylvania to investigate longitudinal rail forces and longitudinal displacements of rail and ties under train passages. The model predictions were compared and validated with the test results. This model applies to all train types on various track layouts. Key factors, e.g. train braking versus train traction, fastener stiffness, etc., that have a significant effect on longitudinal force distributions are presented in the paper. The model estimates the longitudinal forces during uniform train braking and traction events, and shows that locomotives in traction lead to more significant longitudinal forces distributed to ties. The results indicate the longitudinal force was distributed over a large span (or influence zone) of over one hundred ties whereas the vertical tie load was only distributed over three to five ties. Within the influence zone of the longitudinal load, the vertical tie load ranged dramatically from maximum compressive forces (negative forces) to uplifting tensile forces (positive forces), indicating load configuration was very tie-dependent under a train passage. Significant longitudinal forces could be existing on uplifted ties. The results also show that larger longitudinal stiffness of fastener led to higher maximum forces distributed to ties and smaller influence zones.