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Road Vertical Curve Calculator

Q: What standards are used for vertical curve design?

AASHTO "Green Book" (A Policy on Geometric Design of Highways and Streets) is the primary standard for the US. IRC:SP:23 covers Indian practice. Both use the parabolic curve model with K-value-based sight distance checks.

Calculate vertical curve geometry for highway design including crest and sag parabolic curves, PVC/PVT stations, high/low points, and sight distance checks per AASHTO 2018.

Highway & Transportation Highway engineers, transportation planners Commercial Intent: HIGH

Engineering Formulas

Vertical Curve Geometry

L = K × A
A = |G₂ – G₁|
y(x) = yPVC + G₁x + (A/2L)x²
PVC station = PVI – L/2
PVT station = PVI + L/2

A: Algebraic difference in grade (%)

K: Rate of curvature (m per % change)

L: Curve length (m)

G₁: Entering grade (%)

G₂: Exiting grade (%)

y(x): Elevation at distance x from PVC (m)

High / Low Point Location

xcrit = –G₁ × L / A
(from PVC, for crest = high, for sag = low)

x_crit: Distance to high/low point from PVC (m)

Sight Distance Checks

Crest: Kmin = SSD² / (200(√h₁+√h₂)²) = SSD² / 404
Sag: Kmin = SSD² / (200(0.6 + SSD tan(1°)))

SSD: Stopping sight distance (m)

h₁: Driver eye height = 1.07 m

h₂: Object height = 0.15 m

Worked Example

Crest Vertical Curve on Rural Highway

curveType: crestenteringGrade: 3exitingGrade: -2pviStation: 500pviElevation: 150designSpeed: 80curveLength: 260kValue: 0

Algebraic diff

A = |–2 – 3| = 5%

Curve length

L = 260 m (provided)

K-value

K = 260 / 5 = 52.0 m/%

PVC/PVT

PVC @ 370.000 m, el 146.100 m | PVT @ 630.000 m, el 147.400 m

High point

x_crit = –3 × 260/(–5) = 156 m → Sta 526.000 m, el 148.440 m

SSD check

SSD = 130 m → K_min = 130²/404 = 41.8 → K=52.0 PASS

Result: L=260 m, K=52.0, high point @ 526.000 m elev 148.440 m — SSD check PASS

Engineering Notes

Always check both enter and exit grades for drainage: minimum 0.3% grade recommended for paved ditches.

For sag curves under bridges, verify adequate headroom — structural depth may reduce clearance at low point.

Compound vertical curves (multiple parabolas) require careful speed and comfort analysis at the junction.

Consider truck braking performance on long downgrades (check runaway ramp spacing).

Vertical curves should be designed in conjunction with horizontal alignment for 3D consistency.

In rolling terrain, balance cut-and-fill by adjusting PVI elevation within reasonable limits.

Assumptions

• Parabolic curve shape (constant rate of change of grade)

• Driver eye height: 1.07 m

• Object height: 0.15 m (crest)

• Headlight height: 0.6 m (sag)

• Headlight beam divergence: 1° upward (sag)

Common Mistakes

✕ Using grade values as decimals instead of percent (3% as 0.03) in incorrect formula positions

✕ Forgetting that G₁ and G₂ are signed (+ for uphill, – for downhill)

✕ Placing PVI at the midpoint of the curve when it is actually at the grade intersection

✕ Checking K against crest tables when the curve is a sag, or vice versa

✕ Not verifying that the high/low point falls within the curve limits (0 to L)

✕ Confusing vertical curve K-value with horizontal curve degree of curvature

Frequently Asked Questions

What is a vertical curve?

A vertical curve is a parabolic curve used to transition between two roadway grades (slopes). Crest curves are convex (hilltop) and sag curves are concave (valley). The parabola provides constant rate of change of grade for driver comfort.

What is K-value in vertical curve design?

K is the rate of curvature, defined as L/A where L is curve length and A is the algebraic difference in grades. A higher K means a longer, flatter curve. Minimum K values are specified by AASHTO to ensure adequate sight distance.

How is stopping sight distance (SSD) checked on crest curves?

For crest curves, the driver must see a 0.15 m tall object over the crest from an eye height of 1.07 m. The minimum K = SSD² / 404 ensures the line of sight is above the curve. If K is insufficient, increase curve length.

How is sight distance checked on sag curves?

For sag curves at night, headlight beam (0.6 m above road with 1° upward angle) must illuminate the road ahead. Minimum K = SSD² / (200(0.6 + SSD tan(1°))). Bridge structures overhead must also be checked.

What is the minimum vertical curve length?

AASHTO recommends minimum curve length L_min = A × K_min, never less than the design speed in km/h (e.g., for 80 km/h, minimum L ≈ 80 m). Very short curves cause passenger discomfort and poor appearance.

What does the high/low point represent?

The high point (crest) is where drainage diverges and elevation is maximum. The low point (sag) is where drainage converges — critical for drainage design. Location is found at x = –G₁L/A from PVC.

How do I choose between crest and sag curves?

Use crest when the grade change goes from uphill to downhill (G₁ > G₂). Use sag when going from downhill to uphill (G₁ < G₂). The parabola opens downward for crest and upward for sag.

What standards are used for vertical curve design?

AASHTO "Green Book" (A Policy on Geometric Design of Highways and Streets) is the primary standard for the US. IRC:SP:23 covers Indian practice. Both use the parabolic curve model with K-value-based sight distance checks.