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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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View Full Document 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
The low point (or high point) of the curve is where the first derivative of y = 0.
K
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This note was uploaded on 02/09/2011 for the course ECON 2250 taught by Professor Schultz during the Spring '11 term at UNO.
 Spring '11
 schultz

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