Length of transition curve (criteria)

A transition curve is introduced between a straight and a circular curve so that the centrifugal acceleration and the superelevation are introduced gradually. Its length is found from the following criteria, and the largest length is adopted.

1. Rate of change of centrifugal acceleration

When a vehicle moves at speed $v$ on a circular curve of radius $R$, the centrifugal acceleration is $v^2/R$. On the transition curve it grows from 0 to $v^2/R$ in time $t = L_s/v$. If the allowable rate of change of acceleration is $C$ (m/s³):

$$C = \frac{v^2/R}{L_s/v} = \frac{v^3}{L_s R} \quad\Rightarrow\quad L_s = \frac{v^3}{C R}$$

With $v = V/3.6$ (V in km/h): $L_s = 0.0215\,\dfrac{V^3}{C R}$.

IRC: $C = \dfrac{80}{75+V}$, limited to $0.5 \le C \le 0.8$ m/s³.

2. Rate of introduction of superelevation

The pavement is raised from the normal camber to the full superelevation $e$ over the transition length at a rate of 1 in $N$. Let $E$ = total rise of the outer edge relative to the inner edge = $e\,(W + W_e)$ when the pavement is rotated about the inner edge. Then

$$L_s = N\,e\,(W+W_e)$$

(For rotation about the centre line, the rise of the outer edge is $\dfrac{(e+n)(W+W_e)}{2}$ and $L_s = N\dfrac{(e+n)(W+W_e)}{2}$, with $n$ = camber.)

Typical $N$ (IRC): 1 in 150 for plain and rolling terrain, 1 in 60 for mountainous and steep terrain (NRS 2070 uses similar values).

3. Empirical formula (IRC)

$$L_s = \frac{2.7\,V^2}{R}\ \text{(plain and rolling)},\qquad L_s = \frac{V^2}{R}\ \text{(mountainous and steep)}$$

Design procedure

1. Find the length by each of the three methods.
2. Adopt the largest value, rounded up to a multiple of 5 m.
3. Check the minimum (not less than about 30 m, or the distance travelled in 2-3 s) and make sure the length is not so large that no circular arc is left between the transitions (shift check).

Shape of transition

The most common type is the spiral (clothoid), in which the radius $r$ at a distance $l$ along the curve satisfies $r\,l = R\,L_s$ (the radius varies inversely with the length). It is set out by the cubic parabola $y = \dfrac{x^3}{6RL_s}$. The shift of the circular curve is $s = \dfrac{L_s^2}{24R}$.