Vertical Alignment
CE 5720
Spring 2011
Originally Created by Chris McCahill
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Components of The AlignmentHorizontal AlignmentVertical AlignmentCross-section
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Vertical Alignment & TopographyTexas DOT
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Today’s ClassMaximum/minimum grade
Properties of vertical curves (parabolic)
Technical design of vertical curves
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Crest CurveSag CurveG1G2G3Vertical Alignment
Tangents and CurvesLike the horizontal alignment, the vertical alignment is made up of tangent and curves
In this case the curve is a parabolic curve rather than a circular or spiral curve
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Maximum GradeHarlech, Gwynedd, UK (G = 34%)www.geograph.org.uk
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Maximum Gradewww.nebraskaweatherphotos.org
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Maximum GradeDee747 at picasaweb.google.com
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Maximum and Minimum GradeOne important design consideration is the determination of the maximum and minimum grade that can be allowed on the tangent section
The minimum grade used is typically 0.5%
The maximum grade is generally a function of the
Design Speed
Terrain (Level, Rolling, Mountainous)
On high speed facilities such as freeways the maximum grade is generally kept to 5% where the terrain allows (3% is desirable since anything larger starts to affect the operations of trucks)
At 30 mph design speed the acceptable maximum is in the range of 7 to 12 %
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Properties of Vertical CurvesBVCEVCLG2G1Change in grade: A = G2 - G1
where G is expressed as % (positive /, negative \)
For a crest curve, A is negative
For a sag curve, A is positiveL/2L/2PI
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Properties of Vertical CurvesBVCEVCLG2G1Rate of change of curvature: K = L / |A|
Which is a gentler curve - small K or large K?L/2L/2PI
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Properties of Vertical CurvesBVCEVCLG2G1L/2L/2Rate of change of grade: r = (g2 - g1) / L
where,
g is expressed as a ratio (positive /, negative \)
L is expressed in feet or meters
Note – K and r are both measuring the same characteristic of the curve
but in different waysPI
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Properties of Vertical CurvesBVCEVCPILG2G1Equation for determining the elevation at any point on the curve
y = y0 + g1x + 1/2 rx2
where,
y0 = elevation at the BVC
g = grade expressed as a ratio
x = horizontal distance from BVC
r = rate of change of grade expressed as ratioElevation = y
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Properties of Vertical Curves
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Properties of Vertical CurvesBVCEVCPIG2G1Example:
G1 = -1% G2 = +2%
Elevation of PI = 125.00 m
Station of EVC = 25+00
Station of PI = 24+00Length of curve?
L/2 = Sta. EVC – Sta. PI
L/2 = 2500 m - 2400 m = 100 m
L = 200 m
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Properties of Vertical CurvesBVCEVCPIG2G1Example:
G1 = -1% G2 = +2%
Elevation of PI = 125.00 m
Station of EVC = 25+00
Station of PI = 24+00r - value?
r = (g2 - g1)/L
r = (0.02 - [-0.01])/200 m
r = 0.00015 / meter
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Properties of Vertical CurvesBVCEVCPIG2G1Example:
G1 = -1% G2 = +2%
Elevation of PI = 125.00 m
Station of EVC = 25+00
Station of PI = 24+00Station of low point?
x = -(g1/r)
x = -([-0.01] / [0.00015/m])
x = 66.67 m
Station = [23+00] + 67.67 m
Station 23+67
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Properties of Vertical CurvesBVCEVCPIG2G1Example:
G1 = -1% G2 = +2%
Elevation of PI = 125.00 m
Station of EVC = 25+00
Station of PI = 24+00Elevation at low point?
y = y0 + g1x + 1/2 rx2
y0 = Elev. BVC
Elev. BVC = Elev. PI - g1L/2
Elev. BVC = 125 m - [-0.01][100 m]
Elev. BVC = 126 m
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Properties of Vertical CurvesBVCEVCPIG2G1Example:
G1 = -1% G2 = +2%
Elevation of PI = 125.00 m
Station of EVC = 25+00
Station of PI = 24+00Elevation at low point?
y = y0 + g1x + 1/2 rx2
y = 126 m + [-0.01][66.67 m] + 1/2 [0.00015/m][66.67 m]2
y = 125.67 m
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Properties of Vertical CurvesBVCEVCPIG2G1Example:
G1 = -1% G2 = +2%
Elevation of PI = 125.00 m
Station of EVC = 25+00
Station of PI = 24+00Elevation at station 23+50?
y = 126 m + [-0.01][50 m] + 1/2 [0.00015/m][50 m]2
y = 125.69 m
Elevation at station 24+50?
y = 126 m + [-0.01][150 m] + 1/2 [0.00015/m][150 m]2
y = 126.19 m
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Design of Vertical Curves
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Design of Vertical CurvesThe first step in the design is to determine the minimum length (or minimum K) for a given design speed.
Factors affecting the minimum length include
Sufficient sight distance
Driver comfort
Appearance
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Design of Vertical CurvesCrest Vertical Curve
If sight distance requirements are satisfied then safety, comfort, and appearance requirements are also satisfied.h1 = height of driver’s eyes, in fth2 = height of object, in ft
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Design of Vertical CurvesCrest Vertical CurveEquation relating sight distance to minimum lengthFrom AASHTO:
h1 ≈ 3.5 ft
h2 ≈ 0.5 ft (stopping sight distance)
h3 ≈ 4.25 ft (passing sight distance)