Function of transition curve, deflection angle

Functions of a transition curve

A transition (easement) curve is introduced between a straight and a circular curve. Its functions are:

1. To change the radius gradually from infinity (straight) to $R$ (circular curve), so the centrifugal force builds up gradually.
2. To allow gradual introduction of super-elevation (and widening), so there is no sudden jerk.
3. To keep the rate of change of radial acceleration within a comfortable limit.
4. To improve the appearance and safety of the alignment, and to reduce the risk of overturning or side-slip.

Derivation of $\alpha = \dfrac{573\,l^2}{RL}$ minutes

A transition curve is designed so that the radius at any point is inversely proportional to the distance $l$ from the start (TS): $r \propto 1/l$. At the end of the transition ($l = L$) the radius is $R$, so

$$r\,l = R\,L \quad\Rightarrow\quad r = \frac{RL}{l}$$

Let $\phi$ be the angle between the tangent at TS and the tangent at a point P at distance $l$. For a small element $dl$, $d\phi = dl/r = l\,dl/(RL)$. Integrating from 0 to $l$:

$$\phi = \frac{l^2}{2RL}\ \text{radians}$$

Taking x along the initial tangent, $dy/dx = \tan\phi \approx \phi = x^2/(2RL)$ for small angles and $x \approx l$. Integrating,

$$y = \frac{x^3}{6RL}\approx \frac{l^3}{6RL}\qquad(\text{cubic parabola})$$

The deflection angle $\alpha$ of the chord from TS to P, measured from the initial tangent, satisfies

$$\tan\alpha \approx \alpha = \frac{y}{x} = \frac{l^2}{6RL}\ \text{radians} = \frac{\phi}{3}$$

Converting radians to minutes (1 rad = 3437.75′):

$$\alpha = \frac{3437.75}{6}\,\frac{l^2}{RL} = \frac{573\,l^2}{RL}\ \text{minutes}$$

At the end of the transition, $\alpha_s = 573L/R$ minutes $= \theta_s/3$ (one-third of the spiral angle).