Vertical Curve Calculator

Vertical Curve Calculator finds the K-value, PVC, PVT and high or low point of a crest or sag curve. It also lists stakeout elevations and checks K against any AASHTO design speed.

Reading the Vertical Curve Calculator Results

The Vertical Curve Calculator needs six entries. They are the curve length, the entering grade, the exiting grade, the PVI station, the PVI elevation and the unit system. A grade that falls gets a minus sign. Stations are typed as plain distance, so 10+00 goes in as 1000.

The headline result is the K-value, the horizontal length the curve uses for each 1% of grade change. The cards then give the PVC and PVT, the high or low point, the elevation at any station you ask for, and a check of K against a design speed. A stakeout table lists every station along the curve.

With the default entries, a 400 ft curve joins a +2% grade to a -1.5% grade at PVI station 10+00, elevation 500 ft. The result is a crest curve with K at 114.29 ft per percent and a high point of 498.29 ft at station 10+28.57.

Vertical Curve Elevation Equation

The Vertical Curve Calculator treats a road vertical curve as a parabola with equal tangents. Its sharpness is described by K, the curve length divided by the algebraic difference between the two grades in percent.

$$K=\frac{L}{|g_1-g_2|}$$

The curve starts at the PVC, half the curve length before the PVI, and ends at the PVT, half the length after it. The elevation at any point comes from the PVC elevation, the entering grade and the distance x from the PVC.

$$y=y_{PVC}+\frac{g_1}{100}x+\frac{g_2-g_1}{200L}{x}^2$$

Grades are in percent and lengths are in feet or meters. The last term is the offset between the curve and the entering tangent. The high or low point is where the grade passes through zero, and its distance from the PVC has a simple form.

$$x_{turn}=\frac{g_{1}L}{g_1-g_2}$$

Vertical Curve Calculator With Steps

The default crest curve can be followed by hand in five steps. Each number below appears on a card or in the table of the Vertical Curve Calculator, so you can check your own work against it.

1. Grade difference A is 2 minus -1.5, or 3.5%. K is 400 ÷ 3.5, or 114.29 ft per percent.
2. The PVC is 200 ft before the PVI, at station 8+00. Its elevation is 500 minus 2% of 200 ft, or 496.00 ft.
3. The PVT is 200 ft after the PVI, at station 12+00. Its elevation is 500 minus 1.5% of 200 ft, or 497.00 ft.
4. The high point is 2 × 400 ÷ 3.5, or 228.57 ft past the PVC, at station 10+28.57. Its elevation is 496 + 4.571 - 2.286, or 498.29 ft.
5. At station 9+00, x is 100 ft. The elevation is 496 + 2.00 - 0.44, or 497.56 ft, and the grade there is +1.13%.

The 0.44 ft in the last step is the gap between the curve and the tangent at that station. At the PVI itself the gap is largest, A × L ÷ 800, which is 1.75 ft here. The curve passes 1.75 ft below the PVI elevation of 500 ft.

Sag Vertical Curve Example

When the exiting grade is higher than the entering grade, the Vertical Curve Calculator labels the result a sag curve and reports a low point. Enter -3% and +2% with the same 400 ft length and PVI. A is 5% and K is 80 ft per percent.

The PVC sits at 506.00 ft and the PVT at 504.00 ft. The low point is 240 ft past the PVC, at station 10+40, with an elevation of 502.40 ft. That station is where water collects, so it is where an inlet or a culvert belongs.

AASHTO K Values by Design Speed

The fourth card in the Vertical Curve Calculator compares K with the minimum that gives stopping sight distance at the design speed you pick. The values are the design controls in the AASHTO Green Book, as reproduced in the MassDOT design guide and the WSDOT Design Manual.

Crest values come from the driver's line of sight over the hill. Sag values come from how far headlights reach at night. With S as the stopping sight distance in feet, the two controls are these.

$$K_{crest}=\frac{S^2}{2158}$$

$$K_{sag}=\frac{S^2}{400+3.5S}$$

At 50 mph the stopping sight distance is 425 ft, which gives 83.7 and 95.7 before rounding to 84 and 96. The shortest curve is K times A. The default crest curve needs 84 × 3.5, or 294 ft, for 50 mph. At 400 ft its K of 114.29 also meets the 55 mph value of 114.

The sag example does not do as well. Its K of 80 meets 45 mph but falls short of the 96 needed for 50 mph, which would take a 480 ft curve. AASHTO also gives a minimum length of three times the design speed in mph, and the tool notes when a curve is shorter than that.

Drainage Near the High or Low Point

A long, flat curve can hold water. The TxDOT Roadway Design Manual gives the usual test. Drainage is rarely a problem if the grade reaches 0.30% about 50 ft from the level point, and that corresponds to a K of 167.

The grade changes by 1% over every K feet, so it stays under 0.30% for 0.3 × K on each side of the turning point. On the default curve that is 34 ft each side. Above K 167 the Vertical Curve Calculator shows a drainage notice, which matters most on curbed streets.

Stakeout Elevations and Station Format

Open Station Check and Design Speed in the Vertical Curve Calculator to set the stakeout interval. The table then lists the elevation and grade at every interval between the PVC and the PVT. It adds rows for the high or low point and for the station you asked about.

In US units a station is 100 ft, so 1,028.57 ft reads 10+28.57. In metric the format changes to kilometers, and 1,028.57 m reads 1+028.57. A station outside the curve is still answered. The card then gives the tangent elevation and the distance before the PVC or past the PVT.

Copy Results places the whole table on the clipboard as station, elevation and grade separated by commas. Paste it into Excel and split the text by comma. That turns the Vertical Curve Calculator output into an Excel sheet without typing the formulas.

If you only have two surveyed elevations for a tangent, the Elevation Grade Calculator gives the percent grade to enter. The Grade Slope Calculator converts a ratio or an angle to percent.

Vertical Curve FAQ

What if there is no high or low point on the curve?

That happens when both grades have the same sign. A +3% grade easing to +1% climbs the whole way, so the grade never reaches zero inside the curve. The Vertical Curve Calculator then reports which end is higher and where zero grade would fall.

For that example with a 400 ft curve, zero grade would fall 600 ft past the PVC, well beyond the PVT. The highest point of the curve is the PVT itself. Drainage is not a concern on a curve like this, because water keeps moving along the grade.

Does it work for unequal tangent curves?

No. The equations assume an equal-tangent parabola, with the PVI at the middle of the curve. An unequal tangent curve is two parabolas joined at a common point, each with its own length and rate of change.

You can still check each half as its own curve. Enter the length and the two grades of one half, with the grade at the joining point as the exiting grade. The K-value and elevations then apply to that half only.