Design controls for valley curves

A valley (sag) curve is a vertical curve with the centre of curvature above the road, formed where a downgrade meets an upgrade (or a lesser downgrade). A parabola is generally used.

Basic design controls

1. Comfort: the centrifugal acceleration due to the vertical curvature adds to the weight; its rate of change must be limited (0.6 m/s³) to avoid jerk.
2. Headlight sight distance at night: the beam must light up the road ahead for at least the stopping distance (in day time there is no obstruction to vision).
3. Overhead clearance under bridges and structures (flyovers) - minimum 5.0 m headroom (as per IRC) with the sight line below the structure.
4. Appearance and drainage: the curve should look smooth, and the lowest point needs special drainage (inlets and drains), the gradient near it should not be flat.
5. Safety and stopping distance: SSD must be available.

Length of valley curve

$N$ = algebraic difference of grades (ratio), $v$ = speed in m/s.

1. Comfort criterion ($C = 0.6$ m/s³):

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

2. Headlight sight distance criterion ($h = 0.75$ m, beam angle $\alpha = 1^\circ$):

• If $S < L$: $\ L = \dfrac{N S^2}{2\,(h + S\tan\alpha)}$
• If $S > L$: $\ L = 2S - \dfrac{2\,(h + S\tan\alpha)}{N}$

3. Overhead clearance criterion (under a structure with clearance $H_c$, driver's eye height $h_1$ and object height $h_2$), for $S<L$: $L = \dfrac{N S^2}{8\,\left(H_c - \tfrac{h_1+h_2}{2}\right)}$.

The longest of these values is adopted for design.