Simulation links hyperbolic highway curves to sharper acceleration, lower sag loads

A vehicle-dynamics simulation comparing two forms of highway vertical curve found that hyperbolic geometry produced substantially higher peak vertical acceleration than the conventional parabolic design, while lowering wheel loads and suspension movement in the sag-curve cases examined. Across the modeled crest and sag conditions, peak vertical-acceleration responses reached as high as 0.590 m/s² for hyperbolic curves, compared with approximately 0.030 m/s² for their parabolic counterparts.

The study compared crest and sag vertical curves designed according to stopping sight distance and passing sight distance criteria. Both curve formulations were evaluated under identical geometric boundary conditions, creating a common framework for comparing their simulated vehicle responses. Researchers used a CarSim-based multi-body vehicle model and assessed maximum vertical acceleration, dynamic wheel load, pitch angle and suspension deflection. Vertical acceleration proved to be the most sensitive of the four indicators, showing much larger differences between the hyperbolic and parabolic profiles than the other measures.

For the investigated sag-curve conditions, the hyperbolic profiles reduced peak wheel loads by approximately 20–30% relative to the parabolic alternative. They also reduced suspension deflections by approximately 15–20%. Those results represent a trade-off rather than a uniform improvement: the same hyperbolic configurations that produced lower wheel loads and less suspension movement were associated with substantially greater vertical acceleration. Pitch-angle responses differed only negligibly between the two formulations, indicating that the principal modeled contrasts concerned acceleration, wheel loading and suspension behavior rather than the vehicle’s pitch response.

The findings offer a simulation-based case for examining highway vertical-curve geometry through vehicle-dynamics measures alongside traditional geometric and sight-distance requirements. They do not establish that either formulation is safer or preferable in real-world highway design, because the reported results came from a computational vehicle model rather than field trials or an evaluation of road-user outcomes. The authors present the framework as a performance-based method for assessing alternative vertical-curve formulations and supporting future highway-design research that incorporates vehicle dynamics. Its scope remains limited to the modeled curve types, design criteria, vehicle behavior and response indicators, so the results do not by themselves determine which geometry should be used on a particular road.

Subject of Research: Vehicle-dynamics simulation of hyperbolic and parabolic highway vertical curves

Article Title: Performance-based evaluation of hyperbolic and parabolic vertical curves for highway geometric design

Article References: Shahvarani, M., Kordani, A. A., Shirini, B., & Moradi, F. (2026). Performance-based evaluation of hyperbolic and parabolic vertical curves for highway geometric design. Scientific Reports. https://doi.org/10.1038/s41598-026-75446-8

Image Credits: AI Generated

DOI: 10.1038/s41598-026-75446-8

Keywords: highway design, vertical curves, vehicle dynamics, CarSim, vertical acceleration, dynamic wheel load, suspension deflection, geometric design, hyperbolic curves, parabolic curves, Performance-based, evaluation