Setback distance expressions

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.

        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.