---
title: "Vertical curve with horizontal curve, 45 degrees"
subject: "Transportation Engineering I"
chapter: 3
key: "semester_6/transportation_engineering_i:n-77c-vc-horizontal"
source: https://www.bctengineer.com/q/semester_6/transportation_engineering_i/n-77c-vc-horizontal
site: BCT Engineering
---

# Vertical curve with horizontal curve, 45 degrees

Transportation Engineering I · Chapter 3: Geometric Design of Highway

- Asked in: 2077 Chaitra, 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-77c-vc-horizontal
- Whole chapter: https://www.bctengineer.com/md/semester_6/transportation_engineering_i/chapter-3

## Question

A section of highway has vertical and horizontal curves with the same design speed. A horizontal curve on this highway with a ruling radius and a deviation angle of 45° is 180 m long. Design the length of a vertical curve on this highway connecting a +3% grade with -1.5% grade so as to fulfill the stopping sight distance criteria. Assume coefficient of longitudinal friction as 0.35.

## Short answer

Design speed is 80 km/h, SSD = 127.6 m. Length of summit vertical curve is170 m.

## Answer

#### Step 1: ruling radius and design speed
The horizontal curve has deflection $\Delta=45^\circ$ and length 180 m:

$$
R=\frac{L}{\Delta\ (\text{rad})}=\frac{180}{0.7854}=229.2\ \text{m}
$$

A ruling (minimum) radius uses $e_{max}=0.07$ and $f_{max}=0.15$ (IRC):

$$
V=\sqrt{127R(e+f)}=\sqrt{127(229.2)(0.22)}=80.0\ \text{km/h}
$$

So the design speed for both curves is **80 km/h**.

#### Step 2: stopping sight distance

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

#### Step 3: summit curve
$g_1=+3\%,\ g_2=-1.5\%$, so $N=0.03+0.015=0.045$ (summit). With eye height 1.2 m and object height 0.15 m (IRC/NRS), assuming $L>S$:

$$
L=\frac{NS^2}{2(\sqrt{H}+\sqrt{h})^2}=\frac{NS^2}{4.4}=\frac{0.045(127.6)^2}{4.4}=166.5\ \text{m}
$$

$L>S$, so the assumption holds.

**Answer:** adopt a summit vertical curve of length **L = 170 m** (design speed 80 km/h, SSD = 127.6 m).

## Check yourself

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

1. How is the radius of the horizontal curve obtained?
   A. R = L/Δ with Δ in radians = 180/0.7854 = 229.2 m
   B. R = L x Δ in degrees
   C. R = V^2/127f
   D. R = 180/45 = 4 m
   Answer: A. Arc length = R x Δ(rad), so R = 229.2 m. (Key point: Radius from deviation)

2. How is the design speed found from the ruling radius?
   A. V = sqrt(R g f) with f = 0.35 giving 60 km/h
   B. V = sqrt(127 R (e + f)) with e = 0.07, f = 0.15, giving 80 km/h
   C. V = 0.278 R t
   D. V = sqrt(225 R e) = 60 km/h
   Answer: B. Using maximum e and f for the ruling minimum radius gives V = 80 km/h for both curves. (Key point: Design speed)

3. What is the SSD at 80 km/h with f = 0.35?
   A. SSD = V^2/(254f) only = 72 m
   B. SSD = 2 x 0.278Vt = 111 m
   C. SSD = 0.278Vt + V^2/(254f) = 127.6 m
   D. SSD = 0.278Vt + V^2/(127f) = 183 m
   Answer: C. Lag 55.6 m plus braking 72.0 m = 127.6 m. (Key point: SSD)

4. Which formula gives the summit curve length for SSD with L>S?
   A. L = N S^2 / 9.6
   B. L = N S^2 / (2(h + S tanα))
   C. L = 2S - 4.4/N
   D. L = N S^2 / 4.4 for eye height 1.2 m and object height 0.15 m
   Answer: D. N = 0.045, so L = 0.045 x 127.6^2 /4.4 = 166.5 m; adopt 170 m. (Key point: Summit length)

## Related in Transportation Engineering I

- [Safe speed from stopping criterion on curve](https://www.bctengineer.com/question.md?key=semester_6%2Ftransportation_engineering_i%3An-78c-speed-stopping) — Transportation Engineering I, chapter 3
- [Vertical curve 35 m headlight and momentum grade](https://www.bctengineer.com/question.md?key=semester_6%2Ftransportation_engineering_i%3An-80c-vc-headlight-35m) — Transportation Engineering I, chapter 3
- [Vertical curve with 300 m grade sections](https://www.bctengineer.com/question.md?key=semester_6%2Ftransportation_engineering_i%3An-65k-vc-300) — 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
- [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
- [Design speed from setback on 650 m curve](https://www.bctengineer.com/question.md?key=semester_6%2Ftransportation_engineering_i%3An-76bh-setback-650) — Transportation Engineering I, chapter 3

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