Simple circular curves: elements and setting out by Rankine's method

A simple circular curve joins two straights with an arc of constant radius R. The angle between the straights is the deflection angle Δ.

Elements of the curve

Tangent length: T = R tan Δ2

Curve length: L = π R Δ180°

Long chord: C = 2 R sin Δ2

External distance: E = R ( sec Δ2 − 1 )

Mid-ordinate: M = R ( 1 − cos Δ2 )

Chainages

• Chainage of the tangent point T₁ = chainage of the intersection point − T.
• Chainage of T₂ = chainage of T₁ + L (not + T).

Rankine's deflection angle method

Pegs are set at full chords (often 20 m or 30 m) from T₁, with a short first sub-chord to reach the first round chainage and a last sub-chord at the end. Each chord c has a tangential angle:

δ = 1718.9 cR minutes

The theodolite at T₁ is turned through the cumulative deflection angle for each peg, and the chord length is measured from the previous peg.

Worked example

R = 200 m, Δ = 40°, chainage of intersection point = 1250.00 m, 20 m pegs.

T = 200 × tan 20° = 72.79 m L = π × 200 × 40180 = 139.63 m

T₁ is at 1177.21 m and T₂ at 1316.84 m. The first sub-chord is 1180 − 1177.21 = 2.79 m.

δ₁ = 1718.9 × 2.79200 = 23.98′ δ for 20 m = 171.89′ = 2° 51′ 53″

Check: the final cumulative angle must equal Δ/2 = 20°.

Lay out curves with deflection tables in our road curves tool.