Highway Verical Curve alingment following a quadratic equation

[CivilG]

Could someone please tell me why highway vertical curve alignments follow a quadratic equation? (ax^2+bx+c=y). Is it because this curve is the most smoothest between to tangent lines? I build a lot of highways but am not sure why this is the common convention by my DOT and engineers.

Thanks,

 

[beej67]

I'm not a transpo guy, but I would presume it goes back to G forces. The effective gravity you feel is due to your acceleration in the vertical direction. Think about it in terms of rates of change.

Vertical velocity is rate of change (integral) of elevation. Vertical acceleration is rate of change (integral) of vertical velocity. If you want a constant G force, you want a constant vertical acceleration, which would mean a 1st order (linear) relationship with velocity, and a 2nd order (quadratic) relationship with elevation.

(did I do that right?)

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[TerryScan]

It is a parabolic curve.

 

[BUGGAR]

I was told by a very old man that it was to eliminate any changes to the vertical g forces on the cars. vertical g forces flexed the cars suspensions and early cars (Model T's, et al) had solid axles with steering drag links that were shorter than the axle control arms resulting in bump steer. I believe most early cars were designed to pull to the right on bump and bounce.

 

[LincolnPE]

(did I do that right?)

Acceleration is the first derivative of the velocity function.

 

[LincolnPE]

The second derivative of the parabolic position function gives 2a. Thus constant grade change per incremental distance.