Hair pin bend elements

Hairpin bend

A hairpin bend is a sharp bend of about 180 degrees on a hill road that is used to gain height when the road has to turn back along a spur. It consists of a central circular curve (radius $R$, minimum about 14 m as per IRC:SP:48) with a transition curve of length $L_s$ at each end.

Symmetrical hairpin with transitions: elements

Let the two straight approaches meet at the apex (intersection point) $I$ with an interior angle $\alpha$. The deflection angle is

$$\Delta = 180^\circ - \alpha$$

For a hairpin $\Delta$ is large (about 120 to 180 degrees).

1. Angle turned through by each transition

On a spiral the curvature grows linearly with length $l$: $\dfrac{1}{r}=\dfrac{l}{RL_s}$. Since $\dfrac{d\phi}{dl}=\dfrac{1}{r}$, integrating gives $\phi=\dfrac{l^2}{2RL_s}$. At the end of the spiral ($l=L_s$), the spiral angle is

$$\theta_s = \frac{L_s}{2R} \ \text{(radian)}$$

2. Central circular arc angle

The two transitions together turn through $2\theta_s$, so the circular arc turns through the rest:

$$\Delta_c = \Delta - 2\theta_s$$

3. Shift of the circular curve

The spiral offset at its end is $y_s \approx \dfrac{L_s^2}{6R}$. The circle, produced back, is at $R(1-\cos\theta_s)\approx \dfrac{R\theta_s^2}{2}=\dfrac{L_s^2}{8R}$ from the tangent. The shift is the difference:

$$s = \frac{L_s^2}{6R}-\frac{L_s^2}{8R}=\frac{L_s^2}{24R}$$

4. Tangent length (from apex $I$ to the start of the transition, $T_s$)

The shifted circle (radius $R$, centre at $R+s$ from each straight) has a tangent length of $(R+s)\tan\frac{\Delta}{2}$ from $I$. Its point of tangency lies $x_s - R\sin\theta_s \approx L_s - \dfrac{L_s}{2}=\dfrac{L_s}{2}$ ahead of the start of the spiral, so

$$T_s = (R+s)\tan\frac{\Delta}{2} + \frac{L_s}{2}$$

5. Length of the circular arc and total curve length

$$L_c = R\,\Delta_c = R\,(\Delta - 2\theta_s)$$ $$L = L_c + 2L_s = R\Delta - L_s + 2L_s = R\Delta + L_s$$

(with $\Delta$ in radians).

6. External distance (apex distance)

$$E = (R+s)\sec\frac{\Delta}{2} - R$$

7. Chainages

$$\text{Start of curve} = \text{Ch}(I) - T_s,\qquad \text{End of curve} = \text{Start} + L$$

   straight          apex I
 -----------\       /------
             \  __/
   TS ~~~~~~~ SC  CS ~~~~ ST
              ( R )

Note: as $\Delta$ approaches 180°, $\tan\frac{\Delta}{2}$ and $\sec\frac{\Delta}{2}$ become very large, so the apex lies far away. For such bends the curve is set out from its centre or from the tangent points rather than from the apex.

Design checks (IRC:SP:48)

Superelevation up to 10% (1 in 10), developed over $L_s$; widening provided on the inner side; gradient on the curve not more than 1 in 40; minimum radius of the inner curve 14 m (IRC); and at least 60 m between consecutive hairpins.