1. Vertical Alignment CE 453 Lecture 20
Sources: A Policy on Geometric Design of Highways and Streets (The Green Book). Washington, DC. American Association of State Highway and Transportation Officials, th Edition, and FHWA’s Flexibility in Highway Design

2. Coordination of Vertical and Horizontal Alignment
Curvature and grade should be in proper balance
Avoid
Excessive curvature to achieve flat grades
Excessive grades to achieve flat curvature
Vertical curvature should be coordinated with horizontal
Sharp horizontal curvature should not be introduced at or near the top of a pronounced crest vertical curve
Drivers may not perceive change in horizontal alignment esp. at night
Image source:

3. Coordination of Vertical and Horizontal Alignment
Sharp horizontal curvature should not be introduced near bottom of steep grade near the low point of a pronounced sag vertical curve
Horizontal curves appear distorted
Vehicle speeds (esp. trucks) are highest at the bottom of a sag vertical curve
Can result in erratic motion

4. Coordination of Vertical and Horizontal Alignment
On two-lane roads when passing is allowed, need to consider provision of passing lanes
Difficult to accommodate with certain arrangements of horizontal and vertical curvature
need long tangent sections to assure sufficient passing sight distance

5. Coordination of Vertical and Horizontal Alignment
At intersections where sight distance needs to be accommodated, both horizontal and vertical curves should be as flat as practical
In residential areas, alignment should minimize nuisance to neighborhood
Depressed highways are less visible
Depressed highways produce less noise
Horizontal alignments can increase the buffer zone between roadway and cluster of homes

6. Coordination of Vertical and Horizontal Alignment
When possible alignment should enhance scenic views of the natural and manmade environment
Highway should lead into not away from outstanding views
Fall towards features of interest at low elevation
Rise towards features best seen from below or in silhouette against the sky

7. Coordination of Horizontal and Vertical Alignment
Coordination of horizontal and vertical alignment should begin with preliminary design
Easier to make adjustments at this stage
Designer should study long, continuous stretches of highway in both plan and profile and visualize the whole in three dimensions

8. Coordination of Horizontal and Vertical Alignment

9. Coordination of Horizontal and Vertical Alignment
Should be consistent with the topography
Preserve developed properties along the road
Incorporate community values
Follow natural contours of the land

10. Good Coordination of Horizontal and Vertical Alignment
Does not affect aesthetic, scenic, historic, and cultural resources along the way
Enhances attractive scenic views
Rivers
Rock formations
Parks
Historic sites
Outstanding buildings

16. Vertical Curves Connect roadway grades (tangents)
Grade (rise over run)
10% grade increases 10’ vertically for every 100’ horizontal

17. Vertical Curves Ascending grade:
Frequency of collisions increases significantly when vehicles traveling more than 10 mph below the average traffic speed are present in the traffic stream

18. Example
If a highway with traffic normally running at 65 mph has an inclined section with a 3% grade, what is the maximum length of grade that can be used before the speed of the larger vehicles is reduced to 55 mph?

19. Example
a 3% grade causes a reduction in speed of 10 mph after 1400 feet

20. Climbing lanes
When flatter grades cannot be accommodated, consider climbing lane when all 3 of the following criteria are met (AASHTO):
Upgrade traffic flow rate in excess of 200 vehicles per hour.
Upgrade truck flow rate in excess of 20 vehicles per hour.
One of the following conditions exists:
A 15 km/h or greater speed reduction is expected for a typical heavy truck.
Level-of-service E or F exists on the grade.
A reduction of two or more levels of service is experienced when moving from the approach segment to the grade.

21. Descending Grades
Problem is increased speeds and loss of control for heavy trucks
Runaway vehicle ramps are often designed and included at critical locations along the grade
Ramps placed before each turn that cannot be negotiated at runaway speeds
Ramps should also be placed along straight stretches of roadway, wherever unreasonable speeds might be obtained
Ramps located on the right side of the road when possible

22. Maximum Grades
Passenger vehicles can easily negotiate 4 to 5% grade without appreciable loss in speed
Upgrades: trucks average 7% decrease in speed
Downgrades: trucks average speed increase 5%

23. Vertical Curves Parabolic shape VPI, VPC, VPT, +/- grade, L
Types of crest and sag curves

24. Vertical Curves Crest – stopping, or passing sight distance controls
Sag – headlight/SSD distance, comfort, drainage and appearance control
Green Book vertical curves defined by K = L/A = length of vertical curve/difference in grades (in percent) = length to change one percent in grade

25. Vertical Curve Equations
Parabola
y = ax2 + bx + c
Where:
y = roadway elevation at distance x
x = distance from beginning of vertical curve
a = G2 – G1 L
b = G1
c = elevation of PVC

26. Vertical Curve AASHTO Controls (Crest)
Minimum length must provide stopping sight distance S
Two situations (both assume h1=3.5’ and h2=2.0’)
Source: Transportation Engineering On-line Lab Manual,

27. Assistant with Target Rod (2ft object height)
Observer with Sighting Rod (3.5 ft)

28. Vertical Curve AASHTO Controls (Crest)
Note: for passing sight distance, use 2800 instead of 2158

29. Example: Try SSD > L,
Design speed is 60 mph
G1 = 3% and G2 = -1%,
what is L?
(Assume grade = 0% for SSD)
SSD = 570feet ( see: Table 3.4 of text)
Lmin = 2 (570’) – 2158’ = ’
|(-1-3)|
S < L, so it doesn’t match condition

30. Example: Assume SSD < L,
Design speed is 60 mph
G1 = 3% and G2 = -1%,
what is L?
Assuming average grade = 0%
SSD = 570 feet - ( Table 3.4 of text)
Lmin = |(-3 - 1)| (570 ft)2 = 602 ft
2158
SSD < L, equation matches condition

31. Evaluation of example:
The AASHTO SSD distance equations provided the same design length from either equation in this special case. (600 compared to this is not typical)
Garber and Hoel recommend using the most critical grade of - 1% for SSD computation.
Resulting SSD would be: d = 573 ft
Resulting minimum curve: L = 608 ft
Difference between 602 and 608 is too small to worry about

32. Text example : g1 = + 3% g2 = -3% Design speed of 60 mph
If SSD = 570’ (AASHTO – no grade consideration)
Resulting minimum curve: L = 903 ft (S < L)
Consider grade per Garber and Hoel (p )
SSD, using - 3% grade, ’
Resulting minimum curve L = 994 ft

33. Assessment of grade adjustment
If sight distance is less than curve length, the driver will be on an upgrade a greater portion of the distance than on a down grade
(for eye ht = 3.5’ and object ht = 2.0 ft, 68% of the distance between eye and object will be on + grade.)
For crest vertical curve, selecting a curve length based on down grade SSD may produce an overly conservative design length.

34. AASHTO design tables
Vertical curve length can also be found in design tables
L = K *A
Where
K = length of curve per percent algebraic difference in intersecting grade
Charts from Green Book

35. From Green book

36. From Green book

37. Vertical Curve AASHTO Controls (Crest)
Since you do not at first know L, try one of these equations and compare to requirement, or use L = KA (see tables and graphs in Green Book for a given A and design speed)

38. Chart vs computed From chart V = 60 mph K = 151 ft / % change
For g1 = 3 g2 = - 1
A = |g2 – g1| = |-1 – 3| = 4
L = ( K * A) = 151 * 4 = 604

39. Sag Vertical Curves
Sight distance is governed by night- time conditions
Distance on curve illuminated by headlights need to be considered
Driver comfort
Drainage
General appearance

40. Vertical Curve AASHTO Controls (Sag)
Headlight Illumination sight distance
S < L: L = AS2
400 + (3.5 * S)
S > L: L = 2S – ( S) A

41. Vertical Curve AASHTO Controls (Sag)
For driver comfort use:
L > AV2
46.5
(limits g force to 1 fps/s)
To consider general appearance use:
L > 100 A

42. Sag Vertical Curve: Example
A sag vertical curve is to be designed to join a –3% to a +3% grade. Design speed is 40 mph. What is L?
Skipping steps: SSD = feet S > L
Determine whether S<L or S>L
L = 2( ft) – ( x ) = ft
[3 – (-3)]
< , so condition does not apply

43. Sag Vertical Curve: Example
A sag vertical curve is to be designed to join a –3% to a +3% grade. Design speed is 40 mph. What is L?
Skipping steps: SSD = feet
L = x (313.67)2 = ft x
< , so condition applies

44. Sag Vertical Curve: Example
A sag vertical curve is to be designed to join a –3% to a +3% grade. Design speed is 40 mph. What is L?
Skipping steps: SSD = feet
Testing for comfort:
L = AV2 = (6 x [40 mph]2) = feet
Testing for appearance:
L = 100A = (100 x 6) = 600 feet

45. Vertical Curve AASHTO Controls (Sag)
For curb drainage, want min. of 0.3 percent grade within 50’ of low point = need Kmax = 167 (US units)
For appearance on high-type roads, use min design speed of 50 mph (K = 100)
As in crest, use min L = 3V

46. Other important issues:
Use lighting if need to use shorter L than headlight requirements
Sight distance at under crossings

48. Example: A crest vertical curve joins a +3% and –4% grade
Example: A crest vertical curve joins a +3% and –4% grade. Design speed is 75 mph. Length = ft. Station at VPI is , elevation at VPI = 250 feet. Find elevations and station for VPC (BVC) and VPT (EVC).
L/2 = ft
Station at VPC = [ ] - [ ] =
Vertical Diff VPI to VPC: x (2184/2) = feet
ElevationVPC = 250 – = feet
Station at VPT = [ ] + [ ] =
Vertical Diff VPI to VPT = x (2184/2) = feet
Elevation VPT = 250 – = feet

49. Example: A crest vertical curve joins a +3% and –4% grade
Example: A crest vertical curve joins a +3% and –4% grade. Design speed is 75 mph. Length = ft. Station at VPI is , elevation at VPI = 250 feet. Station at VPC (BVC) is , Elevation at VPC: feet.
Calculate points along the vertical curve.
X = distance from VPC
Y = Ax2
200 L
Elevationtangent = elevation at VPC + distance x grade
Elevationcurve = Elevationtangent - Y

50. Example: A crest vertical curve joins a +3% and –4% grade
Example: A crest vertical curve joins a +3% and –4% grade. Design speed is 75 mph. Length = ft. Station at VPI is , elevation at VPI = 250 feet. Find elevation on the curve at a point 400 feet from VPC.
Y = A x 2 = - 7 x (400 ft)2 = feet
200L (2814)
Elevation at tangent = (400 x 0.03) =
Elevation on curve = – 2.56 feet = ’

51. Y
Calculating x from VPC, calculating tangent elevation along +3% tangent

53. Y
Calculating x from VPT, calculating tangent elevation along +4% tangent

54. Source: Iowa DOT Design Manual

55. Source: Iowa DOT Design Manual

56. Note: L is measured from here to here
Not here
Source: Iowa DOT Design Manual