Learn the geometry of horizontal road curves — radius, tangent length, curve length, superelevation, and sight distance requirements from AASHTO standards — with step-by-step calculation examples.
Why Horizontal Curve Design Matters
When a road changes direction, a horizontal curve transitions the alignment to maintain vehicle stability and driver safety. A curve that is too sharp at highway speeds causes vehicles to slide outward (centrifugal effect), potentially leaving the roadway. AASHTO's "Policy on Geometric Design of Highways and Streets" (the Green Book) defines minimum curve radii, superelevation rates, and sight distance requirements to ensure safe operation at the design speed.
Simple Circular Curve Geometry
The most basic horizontal curve is a simple circular arc. Key geometric elements:
- Δ (Delta) — the intersection angle; the angle between the two tangents
- R — radius of the curve (feet or meters)
- T — tangent length: distance from the PI (Point of Intersection) to the PC (Point of Curvature) or PT (Point of Tangency)
- L — arc length: the length of the curve from PC to PT along the centerline
- M — middle ordinate: the distance from the midpoint of the chord to the midpoint of the arc
Key formulas:
- T = R × tan(Δ/2)
- L = π × R × Δ / 180 (when Δ is in degrees)
- Degree of Curve D = 5729.578 / R (US highway practice)
Minimum Radius and Design Speed
AASHTO sets minimum horizontal curve radii based on design speed and maximum superelevation rate (e_max). The formula is:
R_min = V² / (127 × (e_max + f_s)) (metric, where V is km/h)
or for US customary: R_min = V² / (15 × (e_max + f_s))
Where e_max is the maximum superelevation rate (typically 0.06–0.10 for highways) and f_s is the maximum side friction factor (varies by speed — higher at lower speeds). At 60 mph design speed with e_max = 0.06, minimum radius is approximately 1,000 feet.
Superelevation
Superelevation is the transverse banking of the roadway cross-section through a curve — the outside edge is raised relative to the inside edge, counteracting centrifugal force. On highway curves, superelevation is developed over a transition length (runoff) before the curve begins and removed after it ends. The maximum rate (e_max) varies by highway type:
- Rural highways: e_max = 0.08 or 0.10
- Urban streets with curbs: e_max = 0.04 or 0.06
- Very flat terrain, high-frequency curves: e_max = 0.06
Stopping Sight Distance on Curves
A driver must be able to see far enough ahead to stop before hitting an obstacle. On horizontal curves, sight distance may be restricted by objects on the inside of the curve (cut slopes, walls, guardrail, trees). AASHTO provides the formula for the middle ordinate (M) needed to provide adequate stopping sight distance (SSD):
M = R × (1 − cos(28.65 × SSD / R))
If the available clear zone inside the curve is less than M, objects must be removed or the design speed must be reduced. This calculation drives clearing limits in roadway design and is critical for safety during the design phase.