Single Reverse and Unsymmetrical Vertical Curve for Highways Utilizing Quintic Polynomial Equation of Odd Powers

Abstract

The main goals of reasonable geometric design of unsymmetrical vertical highway curves are the fulfillment of the two main aspects: sufficient value of the sight distance and avoidance of the sudden change in vertical acceleration, i.e. rider's comfort. Existing asymmetrical vertical highway curves consist of either one or two or even three curves takes the form of parabolic or cubic equations. In the favor of maintaining more sufficient sight distance and curve smoothness, we introduce a new single reverse and unsymmetrical vertical highway curve employing a quintic polynomial equation of odd powers. Equation parameters were determined exploiting the given beginning and end grades, and elevations of the points of vertical curvature and vertical tangency. The comparative study presented showed increment ranges between the values of 6.1% to 20.8% of the sight distance. The proposed curve proves smoothness particularly at the beginning of the curve, i.e. improvement of the rider’s comfort along the range of length up to 200 m and greater than 650 m along the curve. Finally, the study demonstrated the suitability of using the curve for different values of beginning and end grades which is impossible to be connected using the other existing curves. Geometric properties and relationships of the curve are presented and justified numerically.