Minimum radius of valley curve

A valley (sag) curve is a vertical curve with the centre of curvature above the road. There is no obstruction to vision during the day, so the curve is designed from two criteria: comfort and headlight sight distance at night. The larger length (and radius) is adopted. The curve has length $L$ and algebraic difference of grades $N$ (as a ratio). For a simple parabola the radius of curvature is constant, $R \approx L/N$.

1. Comfort criterion

A vehicle moving at speed $v$ (m/s) on the vertical curve feels a centripetal acceleration $v^2/R$, which should rise gradually from 0 at the start. The allowable rate of change of this acceleration is $C = 0.6$ m/s³.

For this, the valley curve is treated as two transition curves (each of length $L/2$ and deflection $N/2$), so the radius decreases from infinity at the tangent points to a minimum $R$ at the middle of the curve. For a spiral transition curve, the deflection angle is $\dfrac{L_s}{2R}$ and $L_sR = \dfrac{v^3}{C}$. Here, $L_s = L/2$:

$$\frac{N}{2} = \frac{L/2}{2R}\ \Rightarrow\ R = \frac{L}{2N}$$ $$\frac{L}{2}\cdot\frac{L}{2N} = \frac{v^3}{C} \ \Rightarrow\ L = 2\sqrt{\frac{N\,v^3}{C}}$$

The minimum radius from comfort (at the middle of the curve):

$$R_{min} = \frac{L}{2N} = \sqrt{\frac{v^3}{C\,N}}$$

Here, $N = |g_1 - g_2|$ as a ratio and $v = V/3.6$.

2. Night visibility (headlight) criterion

Let the headlight height above the road be $h$ and the upward beam angle $\alpha$ (usually $h = 0.75$ m, $\alpha = 1^\circ$). The driver must be able to see up to the stopping distance $S$. The curve is a parabola with $R = L/N$.

Case (i): $L > S$. The beam meets the road at distance $S$ from the vehicle, where the curve is at height $y$ above the tangent:

$$h + S\tan\alpha = \frac{N S^2}{2L} \ \Rightarrow\ L = \frac{N S^2}{2\,(h + S\tan\alpha)}$$

Radius: $R = \dfrac{L}{N} = \dfrac{S^2}{2(h + S\tan\alpha)}$.

Case (ii): $L < S$.

$$L = 2S - \frac{2\,(h + S\tan\alpha)}{N},\qquad R = \frac{L}{N} = \frac{2S}{N} - \frac{2\,(h + S\tan\alpha)}{N^2}$$

Design

Find $L$ from the comfort and headlight criteria and adopt the greater. Then check that the radius provided is not less than the minimum radius from each criterion.