Short note: Setting out vertical curve

Vertical curves are provided at changes of grade on roads and railways to give a smooth transition, safe sight distance and comfort. A parabola is used: the rate of change of grade is constant.

Types

• Summit (crest) curve: the grade changes from rising to falling or less rising.
• Sag (valley) curve: the grade changes from falling to rising or less falling.

   Summit curve            Sag curve
      ___                 \       /
   __/   \__               \_   _/
                              ‾‾

Data required

• Grades $g_1$ and $g_2$ (in %), chainage and RL of the intersection point.
• Length $L$ of the curve, either given or found from the permissible rate of change of grade $r$: $L = \dfrac{|g_2-g_1|}{r}\times(\text{chord length})$, or from sight distance requirements.

Computations (equal tangent parabola)

1. Chainage of BVC = IP − $L/2$, of EVC = IP + $L/2$.
2. RL of BVC $= \text{RL}_{IP} - g_1L/200$; RL of EVC $= \text{RL}_{IP} + g_2L/200$.
3. RL of any point at distance $x$ from BVC:

$$y = \text{RL}_{BVC} + \frac{g_1x}{100} + \frac{(g_2-g_1)x^2}{200L}$$

4. Mid-ordinate (offset at the IP): $e = \dfrac{L\,|g_2-g_1|}{800}$. Level of mid-point of curve $= \text{RL}_{IP} \pm e$ (+ for sag, − for summit).
5. Turning point (highest or lowest): $x = \dfrac{g_1L}{g_1-g_2}$ from BVC.

Setting out in the field

1. Fix the IP, BVC and EVC by measuring $L/2$ along the two tangents.
2. Peg the curve at regular intervals (usually 20 or 30 m) from BVC.
3. Compute the offsets $y_x = (g_2-g_1)x^2/(200L)$ and the tangent levels, then the curve RL (tabulated).
4. Using a level and staff, set each peg so that its top is at the computed RL (or measure the offset from the tangent level).
5. Check that the RL at EVC agrees with the computed value.