Valley curve 2.5% and 3.5%, 80 kmph

A valley (sag) curve has no sight restriction by day, so its length is governed by (1) the comfort condition (limiting the rate of change of centrifugal acceleration) and (2) headlight sight distance at night. The larger length is adopted (IRC SP:23 / NRS 2070).

Data and assumptions

$g_1=-2.5\%,\ g_2=+3.5\%$, $N=0.025+0.035=0.06$. $V=80$ km/h ($22.22$ m/s). Assumed: $t=2.5$ s, $f=0.35$, $C=0.6$ m/s³, headlight height $h_1=0.75$ m, beam angle $\alpha=1^\circ$.

Stopping sight distance

$$SSD=0.278(80)(2.5)+\frac{80^2}{254(0.35)}=127.6\ \text{m}$$

(1) Comfort condition

$$L=2\sqrt{\frac{NV^3}{C}}=2\sqrt{\frac{0.06(22.22)^3}{0.6}}=66.3\ \text{m}$$

(2) Headlight sight distance (assume $L>S$)

$$L=\frac{NS^2}{2(h_1+S\tan\alpha)}=\frac{NS^2}{1.5+0.035S}=\frac{0.06(127.6)^2}{1.5+0.035(127.6)}=163.7\ \text{m}$$

$L>S$ (164 m > 128 m), so the assumption is valid.

Design length

Headlight criterion governs: L = 165 m.

Lowest point

Distance from BVC where the grade is zero:

$$x=\frac{g_1L}{N}=\frac{0.025(165)}{0.06}=68.75\ \text{m}$$

Elevation, with RL of BVC = 415.50 m:

$$y=RL_{BVC}-g_1x+\frac{Nx^2}{2L}=415.50-0.025(68.75)+\frac{0.06(68.75)^2}{2(165)}=414.641\ \text{m}$$

Also, RL of EVC $=415.50-0.025(165)+\dfrac{0.06(165)}{2}=416.325$ m, and the apex offset $E=\dfrac{NL}{8}=1.238$ m.

Answer: L = 165 m; lowest point is 68.8 m from the beginning of the curve at RL = 414.64 m.