11.1 Vertical Curve Geometry and Grades

11.1 Vertical Curve Geometry and Grades

Quick Answer: An equal-tangent parabolic vertical curve joins incoming grade g1 to outgoing grade g2. A = g2 − g1. Curve length L is the horizontal length from PVC to PVT. From the PVC, offset y = (A/(2L)) x² and elev = elev_PVC + g1 x + (A/(2L)) x². On the worked crest (g1 = +2.0%, g2 = −1.5%, L = 400 ft, PVC 520.00 ft): PVI = 524.00 ft, PVT = 521.00 ft, midpoint offset 1.75 ft below the PVI.

This section is CES Domain III item F at the geometry layer: rate of grade and the parabola you will use for high/low points and intermediate elevations in §11.2. Domain III as a whole is the heaviest official weight (30%). The professional activity “determine line and grade from plans and profiles” is §11.3. Construction staking of the same curve is Chapter 15 — preview only here.

California notes may say BVC/EVC (begin/end of vertical curve) for PVC/PVT. Same three points: PVC (point of vertical curvature), PVI (point of vertical intersection of the two tangents), PVT (point of vertical tangency).

Equal-Tangent Parabola

On an equal-tangent curve the two tangent lengths are equal:

PVC to PVI = PVI to PVT = L/2 (horizontal).

The curve is a parabola in elevation versus station. That is the civil-engineering default on roadway, channel, and pipe profiles. CES items that give one length L and two grades, and that never say the tangent lengths differ, are equal-tangent.

The parabola is not a circular vertical curve. Do not bring Chapter 10’s radius, tangent T = R tan(Δ/2), or chord definitions into a profile. Horizontal-curve L is an arc; vertical-curve L is a horizontal length along the stationing.

Grades g1 and g2, and Algebraic Difference A

g1 is the incoming (back) tangent grade. g2 is the outgoing (forward) tangent grade. Both are slopes: rise over run.

Write them two ways and pick one system for the whole solution:

A = g2 − g1 is the algebraic difference in grade. Use the same unit system as the grades:

• Percent: A = −1.5 − (+2.0) = −3.5%
• Decimal: A = −0.015 − (+0.020) = −0.035 ft/ft

A is not |g1| + |g2| with the sign guessed afterward, and it is not g1 − g2. Subtract incoming from outgoing: g2 minus g1. A sign error here flips crest versus sag and sends every offset to the wrong side of the tangent.

A plus grade climbs as station increases. A minus grade falls. Rate of grade on a tangent is constant: Δelev = g × Δx with matching units (decimal g with x in feet, or percent g with x in stations: +2.0% × 2.00 stations = +4.00 ft).

Length L — Stations or Feet, Not Both at Once

L is the horizontal PVC-to-PVT length. 1 station = 100 ft.

The worked curve uses L = 400 ft = 4.00 stations. If PVC is at 20+00:

• PVI station = 20+00 + 2+00 = 22+00
• PVT station = 20+00 + 4+00 = 24+00

Stay consistent inside each formula. If x and L are in feet and A is decimal, y comes out in feet. If x and L are in stations and A is in percent, y = A x² / (2L) also comes out in feet. The exam trap is A in percent with x in feet stuffed into A/(2L) as if A were decimal — offsets 100 times too large.

Percent-and-feet form of the same offset (A in percent, x and L in feet):

y = (A% × x²) / (200 L)

Because A_decimal / (2L) = (A%/100) / (2L) = A% / (200L).

Offset and Elevation from the PVC

Measure x along the curve from the PVC (horizontal station difference). The offset from the incoming tangent is

y = (A / (2L)) x²

The elevation on the curve is the incoming-tangent elevation plus that offset:

elev = elev_PVC + g1 x + (A / (2L)) x²

At x = 0 (PVC), y = 0 and elev = elev_PVC. At x = L (PVT) the parabola meets the outgoing tangent.

On an equal-tangent curve the midpoint offset (x = L/2), which is also the offset from the PVI down (crest) or up (sag) to the curve, is

E = A L / 8

Same units warning as above. E is not the high or low point unless g1 = −g2 (§11.2).

The PVI elevation is not on the curve. It is the intersection of the two tangents:

elev_PVI = elev_PVC + g1 × (L/2)

elev_PVT = elev_PVI + g2 × (L/2)

Those two tangent equations are how you check the parabola at x = L.

Crest versus Sag

A crest that goes from upgrade to downgrade is the usual picture. A curve can still be a crest if both grades are positive but g2 is smaller than g1 (the climb flattens): A is still negative and the parabola still sits below the tangents. Sag is the opposite. Read A, not the cartoon of a hill.

Unequal-Tangent Curves (Know They Exist)

An unequal-tangent vertical curve has PVC-to-PVI ≠ PVI-to-PVT. The PVI is not at station PVC + L/2, and the single-parabola E = AL/8 does not apply at that PVI. CES numbers that give one L and two grades are equal-tangent unless the stem states unequal tangents or gives two different tangent lengths. If a stem does split the tangents, compute each tangent elevation with its own horizontal length; do not force L/2.

Worked Equal-Tangent Crest

Given: g1 = +2.0% = +0.020 ft/ft, g2 = −1.5% = −0.015 ft/ft, L = 400 ft, PVC elevation 520.00 ft. Hold PVC at station 20+00 for stationing.

A = −0.015 − 0.020 = −0.035 ft/ft (or −3.5%). A < 0 → crest, offsets below the tangents.

PVI at x = 200 ft, station 22+00:
elev_PVI = 520.00 + 0.020 × 200 = 520.00 + 4.00 = 524.00 ft

PVT at x = 400 ft, station 24+00:
elev_PVT = 524.00 + (−0.015) × 200 = 524.00 − 3.00 = 521.00 ft

Parabola check at x = L:
elev = 520.00 + 0.020 × 400 + (−0.035 / (2 × 400)) × 400²
= 520.00 + 8.00 + (−0.035 × 400) / 2
= 528.00 − 7.00 = 521.00 ft. Matches.

Midpoint offset at x = 200 ft:
E = A L / 8 = (−0.035)(400) / 8 = −14.00 / 8 = −1.75 ft
Percent-and-feet check: y = (−3.5)(200)² / (200 × 400) = −140,000 / 80,000 = −1.75 ft.

Elevation on the curve at the PVI station: 524.00 + (−1.75) = 522.25 ft. That 522.25 is the curve at 22+00, not the high point (the incoming 2% is steeper than the outgoing 1.5% fall, so the summit sits past the PVI — §11.2).

K = L/A as Optional Highway-Design Context

Highway designers often store K = L / |A| with L in stations and A in percent. For this curve L = 4.00 stations and |A| = 3.5, so K ≈ 1.14. In AASHTO and Caltrans Highway Design Manual practice, K is a convenience for picking L from a design speed and a sight-distance or headlight chart.

The 2022 CES test plan names high/low point, intermediate point, and rate of grade. It does not name HDM K values. If you brought a bound HDM in the Prometric box, a K chart is optional professional context — not a CES-required table to memorize. Compute L, A, offsets, and elevations from the parabola. Do not hunt a K table to answer a high-point item.

Exam Traps

1. A = g1 − g2 — reverses crest/sag and the sign of every y.
2. PVI elevation treated as a curve elevation — 524.00 is on the tangents; the curve at that station is 522.25.
3. Percent A in a decimal formula — A = −3.5 stuffed into A/(2L) with L in feet yields offsets 100 times too big.
4. Circular-curve formulas on a profile — no radius on this parabola.
5. Unequal-tangent instinct on an equal-L stem — PVI is at L/2 unless the problem gives two tangent lengths.