Table of Contents
What Is a Vertical Curve?
A vertical curve is a parabolic curve used in road and highway design to provide a smooth transition between two different grades (slopes). When a road changes from an uphill grade to a downhill grade (or vice versa), a vertical curve is required to ensure driver comfort, adequate sight distance, and proper drainage. Without vertical curves, the abrupt change in grade would create a dangerous and uncomfortable driving experience.
Vertical curves are defined by the Begin Vertical Curve (BVC) station and elevation, the End Vertical Curve (EVC) station and elevation, and the Point of Vertical Intersection (PVI) where the two tangent grades meet. The curve follows a parabolic equation that provides a constant rate of grade change throughout the curve length, which is essential for smooth vehicle dynamics.
Highway engineers use vertical curve calculations to determine the minimum curve length required for stopping sight distance, the elevation at any point along the curve, and the location of the highest or lowest point (critical for drainage design). These calculations are fundamental to the AASHTO road design standards used throughout the United States.
Vertical Curve Formulas
Where g1 and g2 are in percent (%), L and x are in feet, and elevations are in feet. The high/low point only exists within the curve when x is between 0 and L.
Curve Types
| Type | Condition | Description | Critical Feature |
|---|---|---|---|
| Crest Curve | g1 > g2 | Uphill transitioning to less uphill or downhill | Sight distance (over the hill) |
| Sag Curve | g1 < g2 | Downhill transitioning to less downhill or uphill | Headlight illumination at night |
Design Criteria
AASHTO provides minimum vertical curve lengths based on design speed and required stopping sight distance. The K-value (L/A) is used as a design control parameter, where L is the curve length and A is the algebraic difference of grades in percent.
| Design Speed (mph) | K (Crest) | K (Sag) | Min. Stopping Sight Distance (ft) |
|---|---|---|---|
| 30 | 19 | 37 | 200 |
| 40 | 44 | 64 | 305 |
| 50 | 84 | 96 | 425 |
| 60 | 151 | 136 | 570 |
| 70 | 247 | 181 | 730 |
Frequently Asked Questions
What is the K-value in vertical curve design?
The K-value is the ratio of curve length (L) to the algebraic difference of grades (A): K = L/A. It represents the horizontal distance in feet required for a 1% change in grade. Higher K-values produce flatter, longer curves. AASHTO specifies minimum K-values based on design speed to ensure adequate sight distance.
How do I find the high or low point of a vertical curve?
The high or low point occurs where the slope of the curve equals zero. The distance from the BVC to this point is x = -g1 x L / A. If this value falls between 0 and L, the turning point is on the curve. For crest curves, this is the highest point (critical for sight distance). For sag curves, this is the lowest point (critical for drainage).
Why are vertical curves parabolic?
Parabolic curves provide a constant rate of grade change, which means the vehicle experiences a uniform vertical acceleration as it traverses the curve. This produces the smoothest ride for drivers and passengers. A circular curve would produce an abrupt change in vertical acceleration at entry and exit, which is undesirable for comfort and vehicle dynamics.