---
title: "Valley curve 2.5% and 3.5%, 80 kmph"
subject: "Transportation Engineering I"
chapter: 3
key: "semester_6/transportation_engineering_i:n-63k-valley-80"
source: https://www.bctengineer.com/q/semester_6/transportation_engineering_i/n-63k-valley-80
site: BCT Engineering
---

# Valley curve 2.5% and 3.5%, 80 kmph

Transportation Engineering I · Chapter 3: Geometric Design of Highway

- Asked in: 2063 Kartik (old course), 12 marks
- Video: [Length of Valley Curve | Numerical | Transportation Engineering](https://www.youtube.com/watch?v=QFe8Fht6ZuM) — ONLINETYARI CLASSES
- 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-63k-valley-80
- Whole chapter: https://www.bctengineer.com/md/semester_6/transportation_engineering_i/chapter-3

## Question

Design the total length of valley curve at the junction of a descending gradient 2.5% and the ascending gradient of 3.5% if the design speed is 80 kmph so as to fulfill both comfort condition and head light sight distance for night driving. Locate the lowest point and determine its elevation if the elevation of beginning of the curve is 415.5 m. Assume other suitable data if necessary.

## Short answer

L = 165 m; lowest point 68.8 m from BVC, RL = 414.64 m.

## Answer

A valley (sag) curve has no sight restriction by day, so its length is governed by (1) the **comfort condition** (limiting the rate of change of centrifugal acceleration) and (2) **headlight sight distance at night**. The larger length is adopted (IRC SP:23 / NRS 2070).

#### Data and assumptions
$g_1=-2.5\%,\ g_2=+3.5\%$, $N=0.025+0.035=0.06$. $V=80$ km/h ($22.22$ m/s). Assumed: $t=2.5$ s, $f=0.35$, $C=0.6$ m/s³, headlight height $h_1=0.75$ m, beam angle $\alpha=1^\circ$.

#### Stopping sight distance

$$
SSD=0.278(80)(2.5)+\frac{80^2}{254(0.35)}=127.6\ \text{m}
$$

#### (1) Comfort condition

$$
L=2\sqrt{\frac{NV^3}{C}}=2\sqrt{\frac{0.06(22.22)^3}{0.6}}=66.3\ \text{m}
$$

#### (2) Headlight sight distance (assume $L>S$)

$$
L=\frac{NS^2}{2(h_1+S\tan\alpha)}=\frac{NS^2}{1.5+0.035S}=\frac{0.06(127.6)^2}{1.5+0.035(127.6)}=163.7\ \text{m}
$$

$L>S$ (164 m > 128 m), so the assumption is valid.

#### Design length
Headlight criterion governs: **L = 165 m**.

#### Lowest point
Distance from BVC where the grade is zero:

$$
x=\frac{g_1L}{N}=\frac{0.025(165)}{0.06}=68.75\ \text{m}
$$

Elevation, with RL of BVC = 415.50 m:

$$
y=RL_{BVC}-g_1x+\frac{Nx^2}{2L}=415.50-0.025(68.75)+\frac{0.06(68.75)^2}{2(165)}=414.641\ \text{m}
$$

Also, RL of EVC $=415.50-0.025(165)+\dfrac{0.06(165)}{2}=416.325$ m, and the apex offset $E=\dfrac{NL}{8}=1.238$ m.

**Answer:** L = 165 m; lowest point is 68.8 m from the beginning of the curve at RL = 414.64 m.

## Check yourself

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

1. Which two conditions fix valley curve length?
   A. Comfort and headlight sight distance at night
   B. Daytime SSD and OSD
   C. Drainage and comfort
   D. Overtaking and headlight
   Answer: A. A valley curve has no daytime sight restriction. (Key point: Governing conditions)

2. Which formula gives the comfort length?
   A. L = N V^2 / C
   B. L = 2 sqrt(N V^3 / C) = 66.3 m
   C. L = N S^2 / 4.4
   D. L = 2 N V / C
   Answer: B. Limits rate of change of centrifugal acceleration to C = 0.6 m/s^3. (Key point: Comfort length)

3. After L = 163.7 m from the headlight formula, why is the assumption valid?
   A. L < S
   B. L equals V
   C. L > S (164 > 128 m), as assumed
   D. L is below the comfort value
   Answer: C. The L > S formula only applies when L exceeds S. (Key point: Assumption check)

4. Where is the lowest point?
   A. x = g1 L / N = 68.75 m from BVC
   B. x = L/2 = 82.5 m
   C. x = g2 L / N = 96.25 m
   D. x = NL/8
   Answer: A. Grade is zero at x = g1 L / N, giving RL 414.64 m. (Key point: Lowest point)

## Related in Transportation Engineering I

- [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 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
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