Vertical Curve Design: K-Values and High/Low Points

Vertical Curve Design: K-Values and High/Low Points

Vertical curves provide a smooth, constant-rate transition between two roadway grades meeting at a PVI (Point of Vertical Intersection). AASHTO specifies a simple parabolic curve for this transition because it produces a constant rate of grade change along its length, which simplifies both the mathematics and the resulting ride quality.

Crest vs. Sag Curves

A curve is a crest when the algebraic difference in grades (A = g₂ − g₁) is negative — the road humps upward then back down, as when transitioning from an uphill grade to a downhill grade. It is a sag when A is positive — the road dips before rising again. Crest curves are governed by stopping sight distance (a driver must see far enough over the crest); sag curves are governed by headlight sight distance at night and rider comfort during the vertical acceleration change.

The K-Value

K is defined as curve length (L) divided by the total grade change (A, in percent): K = L / A. It represents the curve length required per 1% of grade change, and AASHTO tabulates minimum K by design speed separately for crest and sag conditions. A higher K produces a longer, flatter, more forgiving curve; designers typically select L first based on the controlling K-value for the design speed, then solve for the remaining geometry.

Locating the High or Low Point

The point of maximum (crest) or minimum (sag) elevation along the curve rarely falls at the curve's midpoint unless the two grades are equal in magnitude — it shifts toward whichever tangent has the flatter grade. Locating it precisely matters for two practical reasons: on sag curves, it is the low point where storm drain inlets must be placed to prevent ponding; on crest curves, it is the critical station for stopping sight distance verification against roadside or median obstructions.