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Vertical Alignment

Vertical Alignment - Geometric Design of Highways Vertical...

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Geometric Design of Highways Vertical Alignment: Crest vertical curves final tangent grade +G 2 -G 1 +G PVI A -G PVC PVT initial tangent L grade +G 1 -G 2 A = absolute value of the difference in grades (expressed in percent) L = curve length PVC = point of the vertical curve (initial point of curve) PVI = point of the vertical intersection PVT = point of vertical tangent (final point of curve) Sag vertical curves +G 2 -G 1 -G 1 +G 2 -G 2 +G 1
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Vertical Curve Fundamentals: The general form of the parabolic EQ, as applied to vertical curves is as follows: y = ax 2 + bx + c EQ. 1 y = roadway elevation at distance x (in stations or ft) from the beginning of the vertical curve (from PVC) c = elevation of the PVC (when x = 0 y = c at PVC) To determine a and b: Take the first derivative of y to get the slope of the curve EQ. 2 at PVC, x = 0, thefore G 1 = initial slope (ft/ft) EQ. 3 The second derivative of y gives the rate of change of slope a 2 dx y d 2 2 EQ. 4 However, the average rate of change of slope, by observation can be written as: +G 1 -G 2 L L G G dx y d 1 2 2 2 EQ. 5 Equating EQ. 4 & 5 a G G L 2 1 2 EQ. 6 Where, L = length of the curve (ft) c x G x L 2 G G y 1 2 1 2 EQ. 7 dy dx ax b 2 b dy dx G 1
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The low point (or high point) of the curve is where the first derivative of y = 0.
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