8.3 Vertical Curves, Slopes, and Grades

Grades and Parabolic Vertical Curves

Slopes and grades are listed in the official FS Survey Computations and Computer Applications area, and they appear in both pure math and field-layout contexts. Grade is vertical change divided by horizontal distance. A 2.50 percent grade rises 2.50 ft per 100 ft of horizontal run, or 2.50 m per 100 m. A negative grade drops as stationing increases.

Vertical curves smooth the transition between two tangent grades. The common highway and site-design model is a parabola. For an equal-tangent vertical curve, the PVC is the beginning of curve, the PVI is the intersection of the tangent grades, and the PVT is the end of curve. The curve length L is measured horizontally along stationing. The algebraic grade difference A equals g2 - g1 when grades are expressed as decimals or percent consistently.

One common form for elevation on a parabolic vertical curve is E = E_PVC + g1 x + (A x squared)/(2L), when grades are decimals and x and L are in the same length units. If grades are in percent, divide percent grades by 100 before using the formula, or use a version of the formula designed for percent. Many wrong answers are off by a factor of 100.

Suppose PVC elevation is 420.00 ft, g1 is +2.00 percent, g2 is -1.00 percent, L is 600 ft, and x is 200 ft. Convert grades to decimals: g1 = 0.0200 and A = -0.0300. Elevation is 420.00 + 0.0200(200) + (-0.0300)(200 squared)/(2 x 600). That is 420.00 + 4.00 - 1.00 = 423.00 ft.

A crest curve changes from steeper uphill to flatter or downhill, so A is often negative. A sag curve changes from downhill to flatter or uphill, so A is often positive. The labels are helpful, but the formula signs still control the answer. If the point elevation on a crest curve keeps climbing too steeply after the curve midpoint, inspect the sign of A.

Slope questions may also use ratio notation. A 3H:1V fill slope has one vertical unit for three horizontal units. The percent grade is 1/3 x 100 = 33.33 percent if expressed as vertical over horizontal. Do not confuse a side slope ratio with a route centerline grade. Construction drawings may use both on the same sheet.

A vertical-curve calculation workflow is:

1. Convert stations to horizontal distances from PVC.
2. Convert grade percent to decimal if the formula requires it.
3. Compute A as outgoing grade minus incoming grade.
4. Apply the parabolic elevation formula.
5. Check whether the point lies between PVC and PVT.
6. Compare the result with tangent elevations for reasonableness.
7. Round only after the final elevation or grade answer.

The FS exam does not require inventing design criteria from memory when the problem supplies needed data. It tests whether you can use stationing, grades, and curve geometry cleanly. Keep the signs visible, and the arithmetic becomes manageable.