If R is the radius of the circle and Δ is deflection angle, what is the formula for external distance (E) in circular curve for use in design and setting out ?
- AE = R (sec Δ/2 − 1)Correct
- BE = R (1 − sec Δ/2)
- CE = R (cos Δ/2 − 1)
- DE = R (1 − cos Δ/2)
Solution
Concept
External distance of a simple curve
Formula
E = R (sec Δ/2 − 1)
Step-by-step solution
1. In the triangle formed by the centre, the intersection point and the tangent point: OI = R sec(Δ/2). 2. External distance E = OI − R = R (sec Δ/2 − 1).
Explanation
The external distance is the distance from the intersection point to the midpoint of the curve.
Why the other options are wrong
R(1 − sec Δ/2) is negative; R(1 − cos Δ/2) is the versine (mid-ordinate) formula.
Shortcut
E = R(sec Δ/2 − 1); versine = R(1 − cos Δ/2).
Common mistake
Using the versine formula.
Correct answer: A) E = R (sec Δ/2 − 1)