---
title: "Setback distance expressions"
subject: "Transportation Engineering I"
chapter: 3
key: "semester_6/transportation_engineering_i:setback-derive"
source: https://www.bctengineer.com/q/semester_6/transportation_engineering_i/setback-derive
site: BCT Engineering
---

# Setback distance expressions

Transportation Engineering I · Chapter 3: Geometric Design of Highway

- Asked in: 2079 Jestha, 2+6 marks
- Video: [SET BACK DISTANCE - Highway Geometric Design - Transportation Engineering - GATE](https://www.youtube.com/watch?v=3M_1e89fnUU) — Ekeeda GATE & ESE
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/transportation_engineering_i/setback-derive
- Whole chapter: https://www.bctengineer.com/md/semester_6/transportation_engineering_i/chapter-3

## Question

Define set back distance with factors affecting set back distance. Derive expressions for the setback distance on a horizontal curve of single lane and multiple lane roads.

## Short answer

Setback distance is the clearance from the road centre line on a curve so sight distance S is unobstructed. For S<Lc, m = R - (R-d) cos(S/[2(R-d)]), with d=0 for single lane, d=(N-1)w/2 for multi-lane. For S>Lc, add (S-Lc)/2 sin(α/2). Obstructions inside m must be removed.

## Answer

**Setback distance** ($m$) is the clear distance, measured from the centre line of the road towards the inner side of a horizontal curve, within which obstructions (buildings, cut slopes, vegetation) must be removed so that the required sight distance is available around the curve.

#### Factors affecting setback
1. **Required sight distance** $S$ (stopping, intermediate or overtaking).
2. **Radius of the curve** $R$ and the length of the curve $L_c$.
3. **Width and number of lanes,** i.e. the position of the vehicle (line of sight) relative to the centre line.
4. Height of the line of sight and of the obstruction (for a cut slope).
5. Design speed (which fixes $S$).

#### Derivation
Let $R$ = radius of the road centre line, $d$ = distance from the road centre line to the centre of the inner lane (the vehicle path), $S$ = sight distance (measured along the lane centre), $\alpha$ = angle subtended at the centre of the curve by the sight distance along the lane arc, and $L_c$ = length of the curve.

```text
        chord = line of sight
   A ____________________ B
     \        |m         /
      \      -|-        /
       \  .   |   .    /   <- lane arc (radius R-d)
        '-.__ | __.-'
              O (centre, far below)
```

**Case 1: $S < L_c$ (sight distance within the curve)**
$$
\alpha = \frac{S}{R-d}\ \text{(radian)}
$$
The line of sight is a chord of the lane arc. Its distance from the centre $O$ is $(R-d)\cos\dfrac{\alpha}{2}$. The setback from the road centre line is

$$
m = R - (R-d)\cos\frac{\alpha}{2}
$$

**Case 2: $S > L_c$ (sight line extends on to the tangents)**
$$
\alpha = \frac{L_c}{R-d},\qquad
m = R - (R-d)\cos\frac{\alpha}{2} + \frac{S - L_c}{2}\,\sin\frac{\alpha}{2}
$$

#### Single-lane road
The vehicle runs on the centre line, so $d = 0$:
$$
m = R\left(1 - \cos\frac{\alpha}{2}\right),\quad \alpha = \frac{S}{R}
$$
(approximately $m \approx S^2/8R$ for $S<L_c$).

#### Multi-lane road
The sight line is along the centre of the inner lane, $d = \dfrac{(N-1)\,w}{2}$ for $N$ lanes of width $w$ each (for a two-lane road, $d = w/2$):
$$
m = R - (R-d)\cos\frac{\alpha}{2},\quad \alpha = \frac{S}{R-d}
$$
On wider multi-lane roads, the sight line of the innermost lane is the critical one.

## Check yourself

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

1. Setback distance is
   A. the minimum distance from the centre line of the inner side of a curve to the obstruction to be cleared so required sight distance is available
   B. the distance between the two tangent points
   C. the width of the shoulder
   D. the length of the transition curve
   Answer: A. It is the clearance from the road centre line to obstructions on the inside of the curve needed for sight distance. (Key point: Definition of setback)

2. For S < Lc on a curve, the setback is given by
   A. m = R - (R-d) sin(alpha/2)
   B. m = (R-d) - R cos(alpha/2)
   C. m = R - (R-d) cos(alpha/2), with alpha = S/(R-d)
   D. m = S^2/R
   Answer: C. The line of sight is a chord of the lane arc at distance (R-d)cos(alpha/2) from the centre, so m = R - (R-d)cos(alpha/2). (Key point: Case S < Lc)

3. When S > Lc, what extra term is added to the setback?
   A. (S - Lc)/2 x sin(alpha/2), for the straight parts of the sight line
   B. (S + Lc) x cos(alpha/2)
   C. Lc x tan(alpha)
   D. No extra term is added
   Answer: A. The sight line extends onto the tangents, adding ((S - Lc)/2) sin(alpha/2), with alpha = Lc/(R-d). (Key point: Case S > Lc)

4. How does d differ between a single-lane road and a multi-lane road?
   A. d = 0 for single lane; d = (N-1)w/2 for N lanes (sight line along the inner lane centre)
   B. d = w for both
   C. d = 0 for multi-lane and w/2 for single lane
   D. d is the radius of the curve
   Answer: A. A single-lane vehicle runs on the centre line (d = 0), while on a multi-lane road the sight line is along the inner lane centre. (Key point: Single versus multi-lane)

5. Which factor affects the setback distance?
   A. Colour of the pavement
   B. Required sight distance and radius of the curve
   C. Volume of traffic only
   D. Thickness of the sub-base
   Answer: B. Setback depends on sight distance S, radius R, curve length, number and width of lanes, and line-of-sight height. (Key point: Factors affecting setback)

## Related in Transportation Engineering I

- [Setback on 400 m curve, 1.95 m lane distance](https://www.bctengineer.com/question.md?key=semester_6%2Ftransportation_engineering_i%3An-68m-setback-1-95) — Transportation Engineering I, chapter 3, asked 2×
- [Setback on six lane highway, 450 m curve](https://www.bctengineer.com/question.md?key=semester_6%2Ftransportation_engineering_i%3An-73m-setback-six-lane) — Transportation Engineering I, chapter 3
- [Setback on four lane carriageway, 400 m radius](https://www.bctengineer.com/question.md?key=semester_6%2Ftransportation_engineering_i%3An-70bh-setback-four-lane) — Transportation Engineering I, chapter 3
- [Setback on six lane highway, 550 m radius](https://www.bctengineer.com/question.md?key=semester_6%2Ftransportation_engineering_i%3An-72a-setback-350) — Transportation Engineering I, chapter 3
- [Setback on six lane road, 500 m radius](https://www.bctengineer.com/question.md?key=semester_6%2Ftransportation_engineering_i%3An-65c-setback-500) — Transportation Engineering I, chapter 3
- [Factors affecting stopping sight distance](https://www.bctengineer.com/question.md?key=semester_6%2Ftransportation_engineering_i%3Assd-factors-only) — Transportation Engineering I, chapter 3, asked 3×

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