Rankine’s Method: The Definitive Guide to Setting Out Circular Curves

Overview

Horizontal circular curves are essential for connecting two straight sections of a road or railway 13. Rankine’s Method, also known as the tangential angle method, is the most popular field technique for establishing these curves using a theodolite and tape 14.

Why This Matters

Precise curve setting ensures that vehicles can navigate changes in direction safely at design speeds. Rankine’s method is versatile because it allows you to set out points on the curve even if the Intersection Point (IP) is inaccessible 14, 15.

Theory

Rankine’s method is based on the geometric principle that the angle between a tangent and a chord is equal to half the angle subtended by the chord at the center of the circle 16.

Mathematical Principles

To calculate the deflection angle ($\delta$) for a chord of length $c$ on a curve of radius $R$:

1. In Radians:$\delta \text{ rad} = \frac{\text{chord}}{2R}$

2. In Degrees (The Surveying Standard):$\delta^\circ = \frac{28.6479 \times \text{chord}}{R}$ 16

Field Workflow

Formula Breakdown

• Tangent Length: $T = R \tan(\Delta/2)$ 17
• Curve Length: $L = R\Delta \text{ (with } \Delta \text{ in radians)}$ 17
• Apex Distance: $IA = R(\sec \Delta/2 - 1)$ 18

Step-by-Step Example

Problem: Two straights meet at $\Delta = 30^\circ$. $R = 200 \text{ m}$. Calculate the deflection angle for a $20 \text{ m}$ standard chord 16.

1. Formula: $\delta^\circ = 28.6479 \times \frac{C}{R}$
2. Plug in values: $\delta^\circ = 28.6479 \times \frac{20}{200}$
3. Calculation: $\delta = 2.86479^\circ$
4. Convert to DMS: $2^\circ 51' 53''$ 19

Practical Tips

• Through Chainage: Always calculate your chords based on through chainage to ensure pegs land on round numbers (e.g., $2200, 2220, 2240$) 20.
• EDM Advantage: If using an EDM, you can set out the curve by measuring the total distance from $T_1$ to each peg ($2R \sin \delta_n$) rather than measuring chord-to-chord 21.

Common Mistakes

• Ignoring the Sub-chord: The first and last chords are rarely a full $20 \text{ m}$. Failing to calculate the specific deflection angle for these "sub-chords" will throw off the entire curve 19, 20.
• Cumulative Errors: In manual taping, errors in one peg affect all subsequent pegs. Always check into $T_2$ to verify the total deflection angle equals $\Delta/2$ 19.

FAQ

Conclusion

Rankine’s method is the "gold standard" for manual curve setting. Even in the age of GPS, understanding these geometric foundations is vital for verifying automated designs on-site.

References

Schofield, W. (2001). Engineering Surveying. 5th ed. Butterworth-Heinemann. 13-17, 19-21, 23-28.