8.3 Vertical Curves, Slopes, and Grades

Grades and the Equal-Tangent Parabola

Slopes and grades appear in the FS Survey Computations area in both pure-math and field-layout contexts. Grade (g) is rise over horizontal run, almost always written as a percent: a line rising 6.0 ft over 300 ft has g = (6.0/300)×100 = +2.0%. Uphill in the direction of stationing is positive; downhill is negative — sign control is the heart of vertical-curve problems.

A vertical curve smoothly joins an incoming grade g1 to an outgoing grade g2. The standard is a parabola, chosen because its rate of grade change is constant, giving a constant vertical acceleration. The curve runs from the PVC (point of vertical curvature) through the PVI (point of vertical intersection of the two tangent grades) to the PVT (point of vertical tangency). An equal-tangent curve has the PVI exactly midway in station between PVC and PVT, so the PVC is L/2 before the PVI and the PVT is L/2 after it (L = total curve length, measured horizontally).

A crest curve has the road grade decreasing (A negative); a sag curve has it increasing (A positive).

Elevation Along the Curve

The elevation of any point on an equal-tangent parabolic curve, measured by horizontal distance x in feet from the PVC, is:

y = elev_PVC + g1·x + (A / 200L)·x²

where g1 and A are in percent and L is in feet. The term g1·x is the elevation the back tangent would have at x; the (A/200L)x² term is the parabolic offset of the curve from that tangent. (The 200 arises from dividing the percent grades by 100 and the average-grade-change factor of 2.)

Worked example

A crest curve has g1 = +3.00%, g2 = −2.00%, L = 600 ft, with the PVC at station 30+00 and elevation 248.00 ft. Then A = g2 − g1 = −2.00 − 3.00 = −5.00%.

Elevation at station 33+00 (x = 300 ft from PVC):

• Tangent term: g1·x = (3.00/100)(300) = +9.00 ft
• Parabola term: (A/200L)x² = (−5.00 / (200·600))(300²) = (−5.00/120000)(90000) = −3.75 ft
• y = 248.00 + 9.00 − 3.75 = 253.25 ft

The rate of grade change is r = A/L = −5.00/600 = −0.00833% per ft, constant along the parabola — a defining property the exam may test directly.

High and Low Points and K Value

On a crest curve the high point (and on a sag curve the low point) is where the curve grade equals zero. Setting the derivative of the elevation equation to zero gives the turning-point station, measured from the PVC:

x_turn = −g1·L / A (equivalently x_turn = g1·K, with K = L/A)

For the example (g1 = +3.00, A = −5.00, L = 600): x_turn = −(3.00)(600)/(−5.00) = 360 ft from the PVC → station 33+60. The high-point elevation is found by substituting x = 360 ft into the elevation equation. Note the turning point exists on the curve only when g1 and g2 have opposite signs; if both grades are the same sign there is no interior high or low point.

The K value, K = L/A, is the horizontal length needed per 1% of grade change. A K of 100 means 100 ft of curve for each 1% of A. K directly links curve geometry to stopping sight distance in highway design: longer K means a flatter, safer crest or sag. Designers select a minimum K from AASHTO tables for the design speed, then set L = K·A.

Common traps

• Forgetting that A = g2 − g1 carries signs (a +3% to −2% transition gives A = −5%, not +1%).
• Using x in stations when the formula expects feet (or vice versa) — keep units consistent.
• Assuming the high point is at mid-curve; it only lands at L/2 when g1 = −g2.

PVI, Tangent-Offset, and Profile Checks

A second, equivalent way to think about the parabola is the tangent-offset view, which the FS handbook also presents. The vertical offset e from the tangent at the PVI (the midpoint of an equal-tangent curve) is the maximum offset, equal to:

e = A·L / 800 (with A in percent, L in feet)

and the offset at any distance x from the PVC scales as the square of the fractional distance: offset = e·(x/(L/2))² along the first half. For the earlier crest curve (A = −5.00%, L = 600 ft), e = (−5.00)(600)/800 = −3.75 ft — the curve lies 3.75 ft below the tangent intersection at mid-curve, matching the elevation computation.

Elevation of the PVI and PVT

The PVI elevation is found by projecting the back tangent: elev_PVI = elev_PVC + g1·(L/2). The PVT elevation is elev_PVI + g2·(L/2), or equivalently elev_PVC + g1·(L/2) + g2·(L/2). These give independent endpoints to check a full profile.

Why surveyors compute these

Vertical curve elevations feed construction stakeout of finished grade, drainage checks (a sag low point is where water collects, so its elevation sets inlet placement), and clearance verification on crests. A surveyor laying out a profile computes elevations at each 50-ft or 25-ft station, not just the endpoints, so fluency with y = elev_PVC + g1·x + (A/200L)x² across a whole station list is the practical skill the exam samples with a single representative point.