---
title: "Minimum radius of valley curve"
subject: "Transportation Engineering I"
chapter: 3
key: "semester_6/transportation_engineering_i:valley-radius-derive"
source: https://www.bctengineer.com/q/semester_6/transportation_engineering_i/valley-radius-derive
site: BCT Engineering
---

# Minimum radius of valley curve

Transportation Engineering I · Chapter 3: Geometric Design of Highway

- Asked in: 2065 Kartik (old course), 10 marks; 2064 Shrawan (old course), 6 marks (asked 2 times)
- Video: [Valley curve in highway engineering | Lecture 15](https://www.youtube.com/watch?v=sC3UnMoi37c) — Self study goals
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/transportation_engineering_i/valley-radius-derive
- Whole chapter: https://www.bctengineer.com/md/semester_6/transportation_engineering_i/chapter-3

## Question

Derive an expression for the minimum permissible radius of valley curve from two considerations (comfort and night visibility).

## Short answer

From comfort criterion, minimum radius R = √(v³/(C N)) with allowable rate of change of centrifugal acceleration C = 0.6 m/s³. For night visibility with headlight height h and beam angle α, length L = N S²/[2(h + S tan α)] when L > S, and R = L/N; the larger radius governs.

## Answer

A valley (sag) curve is a vertical curve with the centre of curvature above the road. There is no obstruction to vision during the day, so the curve is designed from two criteria: **comfort** and **headlight sight distance at night.** The larger length (and radius) is adopted. The curve has length $L$ and algebraic difference of grades $N$ (as a ratio). For a simple parabola the radius of curvature is constant, $R \approx L/N$.

#### 1. Comfort criterion
A vehicle moving at speed $v$ (m/s) on the vertical curve feels a centripetal acceleration $v^2/R$, which should rise gradually from 0 at the start. The allowable rate of change of this acceleration is $C = 0.6$ m/s³.

For this, the valley curve is treated as two transition curves (each of length $L/2$ and deflection $N/2$), so the radius decreases from infinity at the tangent points to a minimum $R$ at the middle of the curve. For a spiral transition curve, the deflection angle is $\dfrac{L_s}{2R}$ and $L_sR = \dfrac{v^3}{C}$. Here, $L_s = L/2$:

$$
\frac{N}{2} = \frac{L/2}{2R}\ \Rightarrow\ R = \frac{L}{2N}
$$

$$
\frac{L}{2}\cdot\frac{L}{2N} = \frac{v^3}{C}
\ \Rightarrow\
L = 2\sqrt{\frac{N\,v^3}{C}}
$$

The minimum radius from comfort (at the middle of the curve):

$$
R_{min} = \frac{L}{2N} = \sqrt{\frac{v^3}{C\,N}}
$$

Here, $N = |g_1 - g_2|$ as a ratio and $v = V/3.6$.

#### 2. Night visibility (headlight) criterion
Let the headlight height above the road be $h$ and the upward beam angle $\alpha$ (usually $h = 0.75$ m, $\alpha = 1^\circ$). The driver must be able to see up to the stopping distance $S$. The curve is a parabola with $R = L/N$.

**Case (i): $L > S$.**
The beam meets the road at distance $S$ from the vehicle, where the curve is at height $y$ above the tangent:
$$
h + S\tan\alpha = \frac{N S^2}{2L}
\ \Rightarrow\
L = \frac{N S^2}{2\,(h + S\tan\alpha)}
$$
Radius: $R = \dfrac{L}{N} = \dfrac{S^2}{2(h + S\tan\alpha)}$.

**Case (ii): $L < S$.**
$$
L = 2S - \frac{2\,(h + S\tan\alpha)}{N},\qquad R = \frac{L}{N} = \frac{2S}{N} - \frac{2\,(h + S\tan\alpha)}{N^2}
$$

#### Design
Find $L$ from the comfort and headlight criteria and adopt the greater. Then check that the radius provided is not less than the minimum radius from each criterion.

## Check yourself

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

1. Why are comfort and night visibility the two criteria for a valley curve?
   A. Daytime vision is obstructed by the curve
   B. There is no obstruction to vision by day, so length is set by comfort and headlight sight distance
   C. Drainage is the only concern
   D. Summit curves use the same criteria
   Answer: B. A valley curve has no daytime obstruction; the greater of the comfort and headlight lengths is adopted. (Key point: Design criteria)

2. In the comfort criterion, the allowable rate of change of centripetal acceleration is taken as
   A. 0.06 m/s^3
   B. 6 m/s^3
   C. 0.3 m/s^3
   D. 0.6 m/s^3
   Answer: D. C = 0.6 m/s^3 is used. (Key point: Comfort rate)

3. The comfort length of the valley curve is
   A. L = 2 sqrt(N v^3 / C)
   B. L = N v^3 / C
   C. L = v^3 / (C N)
   D. L = 2 N v^2 / C
   Answer: A. Treating the curve as two spirals gives (L/2)(L/2N) = v^3/C, so L = 2 sqrt(N v^3/C). (Key point: Comfort length)

4. For L > S, the headlight condition is h + S tan(alpha) =
   A. N S / L
   B. N S^2 / (2L)
   C. N S^2 / L
   D. 2 N S / L
   Answer: B. The beam meets the road at S, where the curve is N S^2/(2L) above the tangent. (Key point: Headlight criterion)

5. How is the final length chosen?
   A. The shorter of the two
   B. The average of the two
   C. Always the comfort value
   D. The greater of the two
   Answer: D. The longer length satisfies both criteria. (Key point: Final design)

## Related in Transportation Engineering I

- [Vertical curve at station 6+525, night visibility](https://www.bctengineer.com/question.md?key=semester_6%2Ftransportation_engineering_i%3An-74bh-vc-6525) — Transportation Engineering I, chapter 3
- [Valley curve 1/35 and 1/45, 80 kmph](https://www.bctengineer.com/question.md?key=semester_6%2Ftransportation_engineering_i%3An-70bh-valley-1-35) — Transportation Engineering I, chapter 3
- [Valley curve design, 100 kmph, 1 in 40](https://www.bctengineer.com/question.md?key=semester_6%2Ftransportation_engineering_i%3An-valley-100-312) — Transportation Engineering I, chapter 3, asked 2×
- [Valley curve 2.5% and 3.5%, 80 kmph](https://www.bctengineer.com/question.md?key=semester_6%2Ftransportation_engineering_i%3An-63k-valley-80) — Transportation Engineering I, chapter 3
- [Vertical curve -3.5% and +4%, 250 m night](https://www.bctengineer.com/question.md?key=semester_6%2Ftransportation_engineering_i%3An-78b-vc-night) — Transportation Engineering I, chapter 3, asked 2×
- [Valley curve 1 in 30 and 1 in 25](https://www.bctengineer.com/question.md?key=semester_6%2Ftransportation_engineering_i%3An-67m-valley-1-30) — Transportation Engineering I, chapter 3

## Related in other subjects

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- [Night visibility, road marking, traffic islands](https://www.bctengineer.com/question.md?key=semester_7%2Ftransportation_engineering_ii%3Anight-visibility-road-marking-traffic-island) — Transportation Engineering II, chapter 1
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