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.
Why do crest and sag curves use different minimum K-values at the same speed?
They're governed by entirely different criteria — crest curves by stopping sight distance over the hump, sag curves by headlight sight distance at night and rider comfort — so AASHTO tabulates separate K-value tables for each.
What if my high or low point calculation returns a value outside the curve's length?
That means the algebraic difference in grades doesn't produce an interior high or low point — it happens when one grade is zero or the math places the extremum beyond the PVC or PVT, in which case the curve's elevation is monotonic and the endpoint controls.
Is a longer curve always safer than the AASHTO minimum?
Generally yes for sight distance, but excessively long curves can create drainage problems on sag curves and can be more expensive in cut/fill — AASHTO minimums represent a balance, not just a floor to exceed as much as possible.
Does K-value directly give me the curve length I need?
Yes — once you select the controlling K-value for your design speed and curve type from the AASHTO table, curve length is simply L = K × A, where A is the absolute algebraic difference between the two grades in percent.
Why does a sag curve's low point matter for storm drain design specifically?
It's the single point along the curve where surface water collects and can't drain away by gravity in either direction — inlet structures are placed at or very near the computed low point to intercept ponding before it becomes a hazard.