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Vertical Curves
Grading
The Shape Of A Vertical
Curve
Properties Of A Parabola
Highest (Or Lowest Point)
On A Crest Curve
Lengths Of Vertical
Curves
Centrifugal Effects
Sight Distance
Setting Out Vertical
Curves
Example
Properties of a Parabola
The vertical through the intersection of the tangents, 0, is a diameter and bisects AB.
AO = BO, and MO = MN. M is the vertex of the parabola.
Offsets (y) from the tangent AO are proportional to the square of the distance (x) from A. These offsets should be at right angles to the tangent, but as flat gradients are usually involved it is sufficiently accurate to take them vertically.
Let the equation of the parabola be
| y | = Kx2 | ||
| then | dy | = 2Kx | (gradient) |
| dx | |||
| d2y | = 2K | (rate of change of gradient is constant, i.e. the parabola gives an even rate of gradient). | |
| dx2 | |||
For flat gradients it is accurate enough to treat the length along the tangent to be equal to the horizontal projection of the tangent. To find K, at x = L,
| y = BB' = | p | x | L | + | q | x | L |
| 100 | 2 | 100 | 2 | ||||
| = | p + q | L = KL2 | |||||
| 200 | |||||||
| K = | (p + q) | ||||||
| 200L | |||||||
| So | y = | (p + q) | x2 | (General form of parabola equation) |
| 200L | ||||
| At | x = | L | , y = MO | |
| 2 | ||||
| = | (p + q) | x | L2 |
| 200L | 4 | ||
| = | (p + q) | L | |
| 800 | |||
| also | NO = | BB' | = | (p + q) | L = 2MO | i.e. MO = MN = | (p + q) | L |
| 2 | 400 | 800 |