Vertical curve -3.5% and +0.5%, IP 1300 m

In the question, the elevation and chainage of the given curve point appear to be interchanged (a chainage of 1300 m would be far away from the IP). The data are therefore taken as: IP at chainage 2600 m with RL 1300 m, and the curve passes through RL 1301 m at the IP chainage (2600 m).

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 sag curve, so the curve lies above the IP:

$$e = 1301 - 1300 = 1\ \text{m}\ \Rightarrow\ |e| = 1\ \text{m}$$

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 sag (valley) curve with equal tangent lengths $L/2 = 100$ m.

$$\begin{aligned} \text{Ch. of BVC} &= 2600 - 100 = 2500\ \text{m}\\ \text{Ch. of EVC} &= 2600 + 100 = 2700\ \text{m}\\ \text{RL of BVC} &= 1300 - \frac{(-3.5)}{100}\times 100 = 1303.5\ \text{m}\\ \text{RL of EVC} &= 1300 + \frac{(0.5)}{100}\times 100 = 1300.5\ \text{m} \end{aligned}$$

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

Check: the last curve RL (1300.5 m) equals the RL of EVC (1300.5 m), and at the IP chainage the curve lies 1 m above the IP, at RL 1301 m.

Step 4: Lowest point

At the lowest 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 = 2500 + 175 = 2675 m
• RL = 1303.5 + (-3.5/100)(175) + (4)(175)²/(200 × 200) = 1300.438 m

Answer: $L = 200$ m, BVC at 2500 m (RL 1303.5 m), EVC at 2700 m (RL 1300.5 m), lowest point at chainage 2675 m (RL 1300.438 m); curve levels as tabulated.