Equivalence and Comprehensive Guidance for Vertical Curves on Railroads

This paper aims to resolve confusion among track professionals regarding the shape of vertical curves on railroads, specifically whether they should be considered parabolic or circular. Through mathematical, geometric demonstrations, and CAD drawing the paper establishes the equivalence between parabolic and circular vertical curves on railroads. It derives a general formula for vertical curves that serves as the foundational source for existing formulas in reputable literature, such as the Transit Cooperative Research Program (TCRP) report and the American Railway Engineering and Maintenance-of-Way Association (AREMA) manual. The paper reviews these vertical curve formulas and recommends a maximum acceptable value for vertical acceleration. It also provides installation criterion of a vertical curve. The minimum radius required to ensure stability of a vertical curve under thermal load under unloaded condition is analyzed; this important safety aspect is ignored in literature. The paper then suggests the absolute minimum values for the radius of a vertical curve on both ballasted and direct fixation track. By addressing these aspects, the paper aims to offer a comprehensive guidance for professionals involved in designing and implementing vertical curves on railroads, ensuring safety and efficient operation of the rail network.