Development of transitioned vertical curve I Properties

Introducing spiral curves before and after horizontal circular curves has been widely accepted to enhance traffic safety, highway esthetics, sight distance, and driver comfort. Though, vertical curves are still designed as parabolic curves that are connected directly to the tangent (without transitions). In this paper, a cubic polynomial is used to develop a vertical transition curve before and after the parabolic vertical curve. The resulting curve, called transitioned vertical curve, consists of transition-parabolic-transition segments. Detailed mathematical formulation and derivation of the instantaneous elevation, grade, rate of curvature, and offset from the first tangent at any point are presented. The highest (or lowest) point on a transitioned crest (or sag) vertical curve, where the instantaneous grade equals 0, is determined as it is of particular importance in highway drainage design. The minimum length of a transition curve is derived based on the criterion of driver comfort. In addition, guidelines are provided to identify the conditions where the drainage of surface water on transitioned curves can be a concern. Finally, the layout of the transitioned vertical curve is described and illustrated using two numerical examples. The new transitioned vertical curve, which exhibits striking similarities to the spiraled horizontal curve, should enhance the design of highway vertical alignments.