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

Vertical curves play a fundamental role in highway geometric design by influencing vehicle dynamics, ride comfort, and safety. Although parabolic curves are widely adopted in current practice, alternative curve formulations may exhibit different dynamic-performance characteristics. To the authors’ knowledge, limited research has examined the vehicle-dynamics implications of hyperbolic vertical curves using multi-body simulation under standard highway-design constraints. Therefore, this study compares the dynamic performance of hyperbolic and parabolic vertical curves using a CarSim-based vehicle model. The analysis covered crest and sag curves designed according to stopping sight distance (SSD) and passing sight distance (PSD) criteria under identical geometric boundary conditions. Vehicle responses were evaluated using four indicators: maximum vertical acceleration, dynamic wheel load, pitch angle, and suspension deflection. The results revealed substantial differences between the two curve formulations. Vertical acceleration was the most sensitive indicator, with hyperbolic curves producing peak responses of up to 0.590 m/s², compared with approximately 0.030 m/s² for the corresponding parabolic curves. Conversely, under sag-curve conditions, hyperbolic curves reduced peak wheel loads by approximately 20–30% and suspension deflections by approximately 15–20% relative to the parabolic alternative. Pitch-angle responses showed only negligible differences between the two formulations. The findings demonstrate a clear performance trade-off between the investigated curve geometries. While the parabolic profile consistently produced lower vertical acceleration, the hyperbolic profile reduced peak wheel load and suspension deflection under the investigated sag-curve conditions. These benefits were accompanied by substantially increased vertical acceleration. The proposed framework provides a performance-based approach for evaluating alternative vertical-curve formulations and supports the integration of vehicle-dynamics considerations into future highway geometric-design studies.