Vertical curve +3.5% and -0.5%, 1266 m at 780

An equal-tangent parabolic vertical curve is set out. Because the curve passes through the given level at the IP chainage, the vertical gap between the IP and the curve there is the mid-ordinate $e$ of the curve.

Step 1: Length of the curve

The curve is a summit curve, so the curve lies below the IP:

$$e = 1266 - 1267 = -1\ \text{m}\ \Rightarrow\ |e| = 1\ \text{m}$$

For an equal-tangent parabola, $e = \dfrac{L\,|g_2-g_1|}{800}$ (grades in %), hence

$$L = \frac{800\,e}{|g_2-g_1|} = \frac{800\times 1}{|-0.5-(3.5)|} = \frac{800}{4} = 200\ \text{m}$$

Step 2: BVC, EVC and their levels

The curve is a summit (crest) curve with equal tangent lengths $L/2 = 100$ m.

$$\begin{aligned} \text{Ch. of BVC} &= 780 - 100 = 680\ \text{m}\\ \text{Ch. of EVC} &= 780 + 100 = 880\ \text{m}\\ \text{RL of BVC} &= 1267 - \frac{(3.5)}{100}\times 100 = 1263.5\ \text{m}\\ \text{RL of EVC} &= 1267 + \frac{(-0.5)}{100}\times 100 = 1266.5\ \text{m} \end{aligned}$$

Step 3: RL of curve points

Using the tangent-offset form of the parabola ($x$ measured from BVC, grades in %):

$$y_x = \text{RL}_{BVC} + \frac{g_1 x}{100} + \frac{(g_2-g_1)x^2}{200L}$$

with $g_1 = 3.5\%$, $g_2 = -0.5\%$, $L = 200$ m, so the offset is $y = \frac{(-4)x^2}{200\times 200}$ and pegs are taken at multiples of 30 m.

Check: the last curve RL (1266.5 m) equals the RL of EVC (1266.5 m), and the curve at the IP chainage lies 1 m below the IP, at RL 1266 m.

Step 4: Highest point

At the highest point the grade is zero, so $x = \dfrac{-g_1 L}{g_2-g_1} = \dfrac{-3.5\times 200}{-4} = 175$ m from BVC.

• Chainage = 680 + 175 = 855 m
• RL = 1263.5 + (3.5/100)(175) + (-4)(175)²/(200 × 200) = 1266.562 m

Answer: $L = 200$ m, BVC at 680 m (RL 1263.5 m), EVC at 880 m (RL 1266.5 m); curve levels as tabulated.