28.7: Vertical Curve: Problem Solving

Vertical curves provide the transition between two roadway grades, ensuring safety, comfort, and functionality. Calculating elevations at specific stations along the curve involves several systematic steps based on the curve's geometry and provided design parameters.

The vertical curve is defined by its length, grades, Point of Vertical Intersection (P.V.I.) location, and P.V.I. elevation. The stations of the Point of Vertical Curvature (P.V.C.), where the curve begins, and the Point of Vertical Tangency (P.V.T.), where the curve ends, are determined by subtracting and adding half the curve length to the P.V.I. station. The elevations at P.V.C. and P.V.T. are calculated using the grades and P.V.I. elevation.

The grade line elevation is computed for each station along the curve using the grade at the P.V.C. or P.V.T. and the horizontal distance to the station. This calculation provides the linear elevation profile before accounting for the curve's parabolic shape. The elevation at the midpoint of the long chord and the curve’s midpoint are determined to calculate the offset at the P.V.I. This offset, derived from the curve's parabolic geometry, represents the vertical distance between the grade line and the curve at the P.V.I.

Using the P.V.I. offset and the horizontal distance from the P.V.C. or P.V.T., tangent offsets are computed for each station. These offsets reflect the deviation from the grade line caused by the curvature of the roadway. The curve elevation at each station is then obtained by subtracting the tangent offset from the grade line elevation. The calculated elevations at specified stations form the vertical profile of the curve, which is critical for designing smooth transitions, ensuring proper drainage, and maintaining safety standards. These computations enable engineers to create roadways that balance functionality with driver comfort.