1. Dr. Wa'el M. Albawwab albawwab@gmail.com
ECGD4107
Pavement Engineering
Summer 2008
Sat. 15:30-18:30 PM
K004
Dr. Wa'el M. Albawwab

2. Dr. Wa'el M. Albawwab albawwab@gmail.com
Lecture 3
Vertical Alignments
Geometric Characteristics
Sight Distances: Crests & Sags
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3. Types of Vertical Alignment
Crest vertical curves
Sag vertical curves
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4. Dr. Wa'el M. Albawwab albawwab@gmail.com
Crest Vertical Curves
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5. Dr. Wa'el M. Albawwab albawwab@gmail.com
Crest Vertical Curves
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6. Dr. Wa'el M. Albawwab albawwab@gmail.com
Sag Vertical Curves
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7. Dr. Wa'el M. Albawwab albawwab@gmail.com
Sag Vertical Curves
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8. Elements of Vertical Curves
G1 = initial roadway grade in percent
G2 = final roadway grade in percent
A = Absolute value of difference in grades
(initial minus final, usually in percent)
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9. Elements of Vertical Curves
PVC = point of the vertical curve (the initial point of the curve)
PVI = point of vertical intersection (the intersection of initial and final grades)
PVT = point of vertical tangent, (the final point of the vertical curve)
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10. Elements of Vertical Curves
L = length of the curve measured in a constant elevation horizontal plane
Ascending = +ve grading
Descending = -ve grading
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11. Presentation of Vertical Curves
Parabolic mathematical expression
Provides a constant sloping rate
y = ax2 + bx + c
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12. Presentation of Vertical Curves
y = elevation at a corresponding distance x from the
beginning of the vertical curve (PVC)
x = distance from the beginning of the vertical curve
a & b = constant coefficients
c = elevation at the beginning of the vertical curve
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13. Calibration of Coefficients
The first derivative is the slope at location x
dy/dx = 2ax + b
At the PVC, x = 0 and slope = G1
b = G1
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14. Calibration of Coefficients
The second derivative is the rate of slope change
d2y/dx2 = 2a = (G2 – G1)/L
a = (G2 – G1)/2L
At the PVC, x = 0 and y = y0
c = y0
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15. Calibration of Coefficients
Then the parabolic vertical curve can be defined
in terms of G1, G2, L, and y0 as:
Dr. Wa'el M. Albawwab

16. Dr. Wa'el M. Albawwab albawwab@gmail.com
Significance
For algebraic grade differences of 2% and greater, and design speeds equal to or greater than 60 km/h, the minimum length of vertical curve in meters should not be less than twice the design speed.
For algebraic grade differences of less than 2%, or design speeds less than 60 km/h, the vertical curve length should be a minimum of 60 m.
Dr. Wa'el M. Albawwab

17. Dr. Wa'el M. Albawwab albawwab@gmail.com
Significance
Vertical curves are not required where the algebraic difference in grades is 0.5% or less.
Since flat vertical curves may develop poor drainage at the level section, adjusting the gutter grade or shortening the vertical curve may overcome any drainage problems.
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18. Dr. Wa'el M. Albawwab albawwab@gmail.com
Example1
A 600 ft long equal-tangent sag vertical curve has the PVC at station with an elevation of 1000 ft. The Initial grade is -3.5% and the final grade is +0.5%. Determine the stationing and elevation of PVI, PVT and the lowest point on this curve.
Dr. Wa'el M. Albawwab

19. Dr. Wa'el M. Albawwab albawwab@gmail.com
Example1 / Solution
Since the curve is equal tangent, the PVI will be 300 ft or 3 stations from the PVC, the PVT will be 600 ft or 6 stations from PVC, therefore, the stationing of PVI and PVT are: and respectively.
Elevation of PVI is:
(-3.5/100)(3)(100) = ft
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20. Dr. Wa'el M. Albawwab albawwab@gmail.com
Example1 / Continued
The elevation of the PVT is:
(0.5/100)(3)(100) = ft
Coefficients of the curve:
b = G1 = -3.5/100 =
a = (G2 – G1)/2L = [0.5-(-3.5)]/[(100)(2)(600)] = 1/30000
y0 = 1000 ft
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21. Dr. Wa'el M. Albawwab albawwab@gmail.com
Example1 / Continued
The function of the curve:
y = x2/30000 – x
The lowest point on the curve:
dy/dx = 0 at x = ft
The level of this point is:
y = ft
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22. Offsetting of Vertical Curves
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23. Offsetting of Vertical Curves
Offset (Y): the vertical distance from the initial tangent to the curve line
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24. Offsetting of Vertical Curves
Substituting for the middle offset at x = L/2
Substituting for the final offset at x = L
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25. Offsetting of Vertical Curves
K: the horizontal distance required to cause a 1% change in the slope of the vertical curve
K-value can be used to calculate the highest and lowest point location in crest and sag curves
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26. SSD for Vertical Curves
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27. SSD for Vertical Curves
s = SSD
H1 = height of driver’s eye above road surface
H2 = height of object above road surface
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28. SSD for Vertical Curves
From the property of parabola for an equal tangent curve,
for S < L:
For S > L:
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29. SSD for Vertical Curves
To determine the minimum length of curve required to provide adequate SSD (recommended by AASHTO’s Green Book)
Use: H1 = 3.5 ft and H2 = 2.0 ft
Use:
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30. SSD for Vertical Curves
For L > SSD
For L < SSD
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31. SSD for Vertical Curves
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32. SSD for Vertical Curves
The critical concern for sag vertical curve design is the actual length of roadway illuminated by the vehicle headlights during nighttime
In day light, the driver's sight distance on a sag vertical curve is unrestricted.
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33. SSD for Vertical Curves
L can be expressed by the parabola properties as:
For L > S
For L < S
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34. SSD for Vertical Curves
To determine the minimum length of curve required to provide adequate SSD (recommended by AASHTO’s Green Book)
Use H = 2.0 ft and  = 1.0 degrees
Use
Dr. Wa'el M. Albawwab

35. SSD for Vertical Curves
For L > SSD
For L < SSD
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36. SSD for Vertical Curves
For crest vertical curves:
For sag vertical curves:
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37. SSD for Vertical Curves
In practical calculations of Lm for both crest and sag vertical curves
The assumption that L > SSD is always the dominant decision made
When computing SSD, G is always ignored
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38. PSD for Vertical Curves
Passing Sight Distances on crest vertical curves:
For L > PSD:
For L< PSD:
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39. PSD for Vertical Curves
As was the case for stopping sight distance,
It is typically assumed L > PSD
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40. PSD for Vertical Curves
Underpass distance on sag vertical curves:
For L > S:
For L < S:
Hc: is the net clearance height of overpass structure above road surface
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41. Dr. Wa'el M. Albawwab albawwab@gmail.com
Due to next lecture:
Study this lecture (AASHTO 2001 – Ch. 3)
Review geometric alignments (Practical)
Dr. Wa'el M. Albawwab