---
title: "Spiral curve not possible, 10 deg 50 min deviation"
subject: "Transportation Engineering I"
chapter: 3
key: "semester_6/transportation_engineering_i:n-65k-spiral-check"
source: https://www.bctengineer.com/q/semester_6/transportation_engineering_i/n-65k-spiral-check
site: BCT Engineering
---

# Spiral curve not possible, 10 deg 50 min deviation

Transportation Engineering I · Chapter 3: Geometric Design of Highway

- Asked in: 2065 Kartik (old course), 8 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/transportation_engineering_i/n-65k-spiral-check
- Whole chapter: https://www.bctengineer.com/md/semester_6/transportation_engineering_i/chapter-3

## Question

At a deviation point with deviation angle equal to 10°50' and radius of horizontal circular curve of 400 m, a symmetrical spiral-circular curve with 120 m long spiral could not be introduced. Prove it. Give suggestions for other possible solutions.

## Short answer

Spiral angle is 120/(2×400)=8.594°, so 2θs=17.19°, exceeding the deflection angle 10.83°, giving a negative circular arc. Shorten the spiral, increase the radius, use a pure spiral-spiral curve, or shift the alignment.

## Answer

#### Proof
A symmetrical spiral-circular-spiral curve needs the deflection angle to accommodate both spirals plus a circular arc:

$$
\Delta\ge 2\theta_s,\qquad \Delta_c=\Delta-2\theta_s\ge0
$$

Spiral angle of each transition curve:

$$
\theta_s=\frac{L_s}{2R}=\frac{120}{2(400)}=0.15\ \text{rad}=8.594^\circ
$$


$$
2\theta_s=17.189^\circ
$$

Deflection angle $\Delta=10^\circ50'=10.833^\circ$.

$$
\Delta_c=\Delta-2\theta_s=10.833^\circ-17.189^\circ=-6.355^\circ<0
$$

The central angle of the circular arc would be negative, which is impossible. The two spirals alone already turn through 17.19°, more than the available deflection of 10.83°, so the 120 m spiral cannot be introduced. **Hence proved.**

#### Possible solutions
1. **Shorten the spiral.** For the spirals to just meet ($\Delta_c=0$) with $R=400$ m: $L_s=R\Delta=75.6$ m. A shorter spiral (say 40 to 60 m) with a short circular arc is possible, but it must still satisfy the comfort and superelevation-run-off requirements.
2. **Increase the radius.** For $L_s=120$ m and $\Delta=10.83^\circ$, the radius must be at least $R=\dfrac{L_s}{\Delta}=\dfrac{120}{0.1891}=635$ m (pure spiral-spiral curve), and more to leave a circular arc. A larger radius also reduces the need for a long transition.
3. **Use a pure (spiral-spiral) curve** with $\theta_s=\Delta/2$ and no circular portion.
4. **Use a simple circular curve** with superelevation run-off provided on the straight. For a small deflection angle with a large radius the shift is tiny: here $s=\dfrac{L_s^2}{24R}=1.50$ m for 120 m. IRC does not require a transition for very large radii.
5. **Shift the alignment** (change the position of the intersection point) to get a larger deflection angle.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What condition must hold for a symmetrical spiral-circular-spiral curve?
   A. Delta >= 2 theta_s so the arc angle is not negative
   B. Delta <= theta_s
   C. Delta = 4 theta_s
   D. Ls <= R / 2
   Answer: A. The two spirals plus the circular arc must fit in the deflection angle. (Key point: Condition)

2. What is theta_s for Ls = 120 m and R = 400 m?
   A. Ls / R = 0.30 rad
   B. Ls / 2R = 0.15 rad = 8.594°
   C. Ls / 4R
   D. 2 Ls / R
   Answer: B. theta_s = Ls/(2R) in radians. (Key point: Spiral angle)

3. Why can the 120 m spiral not be introduced?
   A. The radius is too large
   B. Delta is greater than 2 theta_s
   C. 2 theta_s = 17.19° exceeds Delta = 10.83°, so the arc angle is negative
   D. 120 m is below the minimum spiral length
   Answer: C. Delta_c = 10.83 - 17.19 = -6.36°, impossible. (Key point: Why it fails)

4. Which is a valid remedy?
   A. Shorten the spiral or increase the radius (R >= 635 m for Ls = 120 m)
   B. Increase Ls further
   C. Reduce the radius
   D. Increase design speed
   Answer: A. A shorter spiral or larger radius lowers theta_s so it fits within Delta. (Key point: Remedies)

## Related in Transportation Engineering I

- [Transition curve length from spiral angle 10.32](https://www.bctengineer.com/question.md?key=semester_6%2Ftransportation_engineering_i%3An-73bh-tc-spiral) — Transportation Engineering I, chapter 3, asked 2×
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- [Transition curve 200 m radius, 60 kmph](https://www.bctengineer.com/question.md?key=semester_6%2Ftransportation_engineering_i%3An-70m-tc-200) — Transportation Engineering I, chapter 3

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