Vertical Curve Calculator

Engineering Smooth Transitions: The Vertical Curve Calculator

In civil engineering, particularly in road and railway design, a vertical curve is used to provide a smooth transition between two different gradients (slopes). Without these curves, abrupt grade changes would create an uncomfortable and unsafe experience for vehicles. The Vertical Curve Calculator is a specialized tool for engineers and designers to calculate the properties and elevations along these essential parabolic curves.

Types of Vertical Curves

• Crest Curve: A convex curve that transitions between an uphill grade and a downhill grade, or a less steep uphill grade. Think of the crest of a hill.
• Sag Curve: A concave curve that transitions between a downhill grade and an uphill grade. Think of the bottom of a valley.

This calculator helps determine key points on the curve, such as the highest or lowest point, and the elevation at any station along the curve.

How to Use the Vertical Curve Calculator

To define a vertical curve, you need to provide its fundamental geometric parameters.

1. Initial Grade (g1): The slope of the tangent roadway leading into the curve, expressed as a percentage (e.g., 2 for 2%).
2. Final Grade (g2): The slope of the tangent roadway leading out of the curve, also as a percentage (e.g., -3 for -3%).
3. Length of Curve (L): The horizontal distance from the beginning of the vertical curve (PVC) to the end of the vertical curve (PVT).
4. PVI Station and Elevation: The station (horizontal position) and elevation (vertical position) of the Point of Vertical Intersection (PVI), where the two grades would meet if not for the curve.

The calculator uses these inputs to compute the elevations at regular intervals (stations) along the curve and identifies the exact location and elevation of the highest or lowest point.

The Mathematics of a Parabolic Curve

Vertical curves are typically designed as parabolas because they provide a constant rate of change of grade, which ensures a smooth and comfortable ride.

The fundamental formula for the elevation at any point (x) along the curve is:

Y(x) = Y_PVC + g1*x + (A / 2L) * x²

Where:

• Y(x) is the elevation at distance 'x' from the start of the curve.
• Y_PVC is the elevation at the start of the curve.
• g1 is the initial grade.
• L is the total length of the curve.
• A is the algebraic difference in grades (g2 - g1).
• x is the horizontal distance from the start of the curve.

The calculator applies this formula to generate the curve profile.

Frequently Asked Questions (FAQ)