The discontinuity between rail ends at a joint creates dynamic wheel-rail forces (i.e. high impact forces and wheel unloading) that can result in a range of problems including wear, deterioration, and early failure of the track structure, its components, and passing equipment. The response and magnitude of the dynamic wheel-rail forces generated at joints depend upon the form of the discontinuity (e.g. battered rail ends, ramps, gaps, mismatches, etc.) and the support condition. Joints with battered rail ends, which result from degradation due to repeated impact loading, have been extensively analyzed using closed form expressions developed by Jenkins  to estimate P1 and P2 impact forces. While appropriate for analyzing joints with battered rail ends, P1 and P2 forces are not directly applicable to other forms of discontinuity at joints such as mismatches in which the rail ends are offset vertically when installed.
Under certain circumstances, railroads are introducing ramps (by grinding or welding) to reduce the mismatch discontinuity and produce a smoother transition in order to mitigate these dynamic wheel-rail forces. In this paper, analyses are conducted to estimate dynamic wheel-rail forces at joints having ramps and mismatches of various sizes using simplified models along with detailed NUCARS models for comparative purposes. The Federal Railroad Administration (FRA) Track Safety Standards (49 CFR Part213)  limit the maximum mismatch at joints by Track Class in order to minimize the impact forces which deteriorate the track structure, its components, and equipment, and may ultimately lead to derailment. Parametric studies are conducted to examine the effects of ramp length, direction of travel, mismatch height, and equipment speed (track class). Plots of primary shock-response-spectrum (maximum impact force on the ramp), residual shock-response-spectrum (maximum impact force after the ramp), and minimum wheel force (i.e. wheel unloading) are developed to provide guidelines on ramp length (H-rule) in order to control the maximum force by track class.