Transition (clothoid) curve setting out: A 90 m transition is introduced between a straight and a circular curve of radius R = 500 m. What is the maximum deflection angle (from the tangent) used to locate the junction point (end of transition)?

Introduction / Context:
For a clothoid (spiral) transition, the tangent deflection used in field setting-out increases with the square of the distance along the transition. The final or maximum deflection at the junction with the circular arc is a standard quantity required to peg the end point precisely.

Given Data / Assumptions:

• Transition length L = 90 m.
• Circular-curve radius R = 500 m.
• Standard spiral relations apply (A^2 = R * L).

Concept / Approach:

For a clothoid transition laid from the tangent, the maximum deflection at the junction point (end of transition) from the original tangent is given by the well-known approximate field formula: θ_max (in radians) = L / (6R). This comes from integrating the linear curvature law along the length of the spiral and is sufficiently accurate for highway and railway work.

Step-by-Step Solution:

Verification / Alternative check:

Using A^2 = R L gives A = √(500 * 90) ≈ 212.13. The spiral angle at end = L^2 / (2 A^2) = 8100 / (2 * 45000) = 8100 / 90000 = 0.09 rad? Correction: with correct clothoid relations, θ_end = L / (2R) for total change in tangent direction, while the field deflection from tangent to chord for setting-out is θ_max = L / (6R). The smaller value (1°43′08″) corresponds to the setting-out deflection used at the junction, matching highway practice tables.

Why Other Options Are Wrong:

The other three angles differ by 10″ increments but do not match the computed 0.03 rad conversion. They reflect near values but are not exact for L = 90 m, R = 500 m.

Common Pitfalls:

Mixing up the total spiral angle (change in tangent direction) with the setting-out deflection to the junction; using degrees directly in the formula without converting minutes/seconds correctly.

Final Answer:

1° 43' 08''