Vertical Curve Calculator

Vertical curves in civil engineering are parabolic curves used to connect two different roadway grades (slopes) smoothly. The mathematical model relies on the initial gradient ($g_1$), the final gradient ($g_2$), the curve length ($L$), and the starting elevation at the BVC ($Y_{bvc}$).

The general equation for the elevation ($Y$) at any given horizontal distance ($x$) from the BVC is expressed as:
Y = Y_bvc + (g₁ × x / 100) + [x² × (g₂ - g₁)] / (200 × L)

Additionally, critical geometric points are determined using specific formulas: the elevation of the PVI is found halfway across the curve, while the EVC elevation accounts for the full length of the curve impact.

Consider a road design project with the following parameters: an initial upward gradient ($g_1$) of +2.0%, a final downward gradient ($g_2$) of -1.0%, a total curve length ($L$) of 200 meters, and a BVC starting elevation ($Y_{bvc}$) of 100.0 meters at distance 0.

First, let us find the elevation of the Point of Vertical Intersection (PVI):
Elevation_pvi = 100.0 + (2.0 × 200 / 200) = 102.0 meters

Next, let us calculate the elevation at a custom distance of 50 meters into the curve using our parabolic formula:
Y = 100.0 + (2.0 × 50 / 100) + [50² × (-1.0 - 2.0)] / (200 × 200)
Y = 100.0 + 1.0 + [2500 × (-3.0)] / 40000
Y = 101.0 - 0.1875 = 100.8125 meters

Always double-check your sign conventions for gradients: uphill slopes are positive (+) and downhill slopes are negative (-). Pay close attention to your units, ensuring that curve length and custom distances use the exact same unit of measurement (feet or meters) to prevent scaling errors.

Verify that your algebraic difference in grades aligns with local Department of Transportation (DOT) standards for sight distance and drainage before finalizing field staking data.