Vertical Curve: Problem Solving

A vertical curve is defined by four parameters: its length, the grades of the two connecting roadway sections, the Point of Vertical Intersection (P.V.I.) location, and the P.V.I. elevation. These parameters establish the curve's geometry and enable calculation of elevations at any station along the transition between the two grade lines.

The Point of Vertical Curvature (P.V.C.), where the curve begins, is found by subtracting half the curve length from the P.V.I. station. The Point of Vertical Tangency (P.V.T.), where the curve ends, is calculated by adding half the curve length to the P.V.I. station. These stations mark the boundaries of the vertical curve.

The grade line elevation represents the linear elevation profile before accounting for the curve's parabolic shape. It is calculated for each station using the grade at the P.V.C. or P.V.T. and the horizontal distance to the station. This provides the baseline from which tangent offsets are subtracted to obtain the actual curve elevation.

The offset at the P.V.I. is determined by calculating the midpoint elevation of the long chord and comparing it to the curve's midpoint elevation. This offset, derived from the curve's parabolic geometry, represents the vertical distance between the grade line and the actual curve at the P.V.I., which is essential for computing elevation of intermediate points on vertical curves.

Tangent offsets represent the deviation from the grade line caused by the roadway's curvature. They are computed for each station using the P.V.I. offset and the horizontal distance from the P.V.C. or P.V.T., scaled by the curve length. The curve elevation at each station is obtained by subtracting the tangent offset from the grade line elevation.

Calculating vertical curve elevations is critical for designing smooth transitions between grades, ensuring proper drainage, and maintaining safety standards. The computed elevations form the vertical profile of the curve, enabling engineers to create roadways that balance functionality with driver comfort and meet design specifications.

The process involves six steps: determine P.V.C. and P.V.T. stations, calculate elevations at these points, compute grade line elevations for each station, find the P.V.I. offset using midpoint elevations, calculate tangent offsets for each station, and subtract tangent offsets from grade line elevations to obtain final curve elevations at specified stations.