1. VERTICAL ALIGNMENT
Spring 2015

2. Geometric Elements of Vertical Curves
Vertical Alignment
Geometric Elements of Vertical Curves
Passing Lanes
Vertical Grades
Sight Distance

3. Vertical Alignment
Highway engineers generally separate the characteristics of variations in typography according to the terrain:
Level terrain: highway sight distances, as governed by both horizontal and vertical restrictions, are usually long or can be made without construction difficulty.
Rolling terrain: natural slopes consistently rise above or fall below the road grade, and occasional steep slopes offer some restriction to normal alignment.
Mountainous terrain: longitudinal and transverse changes in the elevation of the ground are usually abrupt, and benching and side hill excavation are frequently needed.

4. Vehicle Operational Characteristics
Passenger cars: Grades as steep as 4% to 5% generally do not affect speed of most vehicles (may affect some compact/subcompact vehicles)
Trucks:
Effects on speed much more important
Maximum speed on upgrade is determined length and steepness of the grade, and the truck’s weight/power ratio (gross weight/engine power)

5. Speed distance Curves for Trucks
Crawl Speed

6. Operational Characteristics of Trucks
Travel time (and, therefore speed) of trucks on grades is directly related to the weight/power ratio
Trucks with same weight/power ratios have similar operating characteristics
Units are kg/kW (metric) or lb/hp (U.S.)
Trucks with a ratio above 200 lb/hp have acceptable operating characteristics from the standpoint of highway users
Improve performance: lower weight and/or increase power

7. Control Grades for Design
Maximum Grades:
5% is considered adequate for design speed of 70 mph
7% to 12% is considered acceptable for design speed of 30 mph; if more important highways 7% or 8% should be used as the max
Values should fall between these extremes for other design speeds
Can use 1% steeper if the upgrade length is below 500ft
Use maximum design grade very infrequently

8. Control Grades for Design
Minimum Grades:
The minimum grade is provided for drainage purposes
Typical 0.5% to 0.3% (for high-type pavement)
Particular attention should be given to the design of storm water inlets and their spacing

9. Climbing Lanes
Climbing lanes are increasingly used to decrease the amount of delay and improve safety (especially for 2-lane highways)
There are not designated as three-lane highways, but as a two-lane highways with an added lane
Climbing lanes are designed for each direction independently of each other
Where climbing lanes are provided, there has been a high degree of compliance by drivers

10. Climbing Lanes

11. Climbing Lanes Criteria for two-lane highways:
Upgrade traffic flow rate in excess of 200 vehicles per hour
Upgrade truck flow rate in excess of 20 vehicles per hour
A reduction in 10 mph or greater in operational speed of trucks
Level-of-service E or F (computed by HCM)
A reduction of two or more LOS between entrance and upgrade segments

12. Climbing Lanes Location of climbing lane
Depends on the where the truck will reduce its speed by 10 mph
Should extend the climbing lane beyond the crest vertical curve to allow trucks to accelerate to previous speed
Make sure the climbing lane is wide enough
Make use of signs “Slower Traffic Keep Right” or “Trucks Use Right Lane”
Optimum length: 0.5 to 2.0 miles
Taper length at the end: L=WS (W=width in ft, S=speed in mph)
Note: climbing on multilane highways usually not justified (usually based on capacity analysis)

13. Vertical Curves There are two types of vertical curves:
Crest curves
Sag curves
Design controlled by stopping sight distance
On occasion, decision sight distance may be needed
Vertical curves should result in a design that is safe, comfortable in operation, pleasing in appearance, and provide adequate drainage

14. Vertical Curves

15. Vertical Curves Some design issues:
The rate of change of grade (defined as the K- Value: K = L /|A| ) should be kept within tolerable limits
Appearance can be important: short vertical curves may give the appearance of sudden break in profile
For sag curves: should retain a grade no less than 0.30% within 50 ft from the level point (i.e., changes from negative to positive grades)
The use of asymmetrical vertical curves may be required (e.g., sight restriction in sag curve)

16. Vertical Curve A = G2 – G1 K = L / |A| Rate of change of grade Offset
y = 4E(x/L)2
Offset
External Distance
E = M = A L / 800
Vertical Point of Intersection
Ele. of P = [ele. Of VPC + (G1 / 100) x] + y
X = L|G1|/ (|G1 – G2|) (high or low point)
Vertical Point of Tangency
Vertical Point of Curvature

17. Vertical Curve
A 600-ft vertical curve connects a +4% grade to a -2% at station and elevation ft. Calculate the location and elevation of the PVC, the middle of the curve, the VPT, and the curve elevation at stations and

18. Vertical Curve Partial answer: A = -2 - (+4) = -6% K = 600/|-6| = 100
E = -6 x 600 / 800 = -4.5 ft
Elevation of curve in the middle = – 4.5 =
Using the curve elevation equation:
Ele P = [ele VPC + (G1/100) x] + y
ft = [ele VPC + 4/100 x 300] – 4.5
Ele VPC =
Station

19. Vertical Curve

20. Vertical Crest Curve

21. Vertical Crest Curve
Equation below is used for h1=3.5 ft and h2=2.0 ft

22. Vertical Crest Curve
For a design speed of 50 mph, determine the minimum length of a crest vertical curve with A=-4%. Assume h1 = 3.5 ft and h2 = 2.0 ft.

23. Vertical Crest Curve
For a design speed of 50 mph, determine the minimum length of a crest vertical curve with A=-4%. Assume h1 = 3.5 ft and h2 = 2.0 ft.
Stopping Sight Distance for 50 mph:
From Exhibit 3-1, SSD = 425 ft
S < L
L = 4 x 4252 / 2158 = 334 ft
S > L
L = 2 x 425 – 2158 / 4 = 310 ft
The length of curve is 310 ft since S > L

24. Vertical Crest Curve Design controls for crest vertical curves:
The minimum lengths of vertical curve for different values of A (G2 – G1) to provide the minimum stopping sight distance is provided in Exhibit 3-75
K = L /|A| (K= rate of vertical curvature, L=length of curve, A=difference in grade)
K is used to compare different curves; it covers all combinations of A and L for any one design speed
Minimum length:
Lmin = 3V (V = mph, L = ft)

25. Vertical Crest Curve

26. Vertical Crest Curve
Previous example:
310 ft

27. Vertical Crest Curve

28. Vertical Crest Curve Passing Sight Distance
Same principle as SSD, but with different values: h2 = 3.5 ft and S=PSD (Exhibit 3-7)
The lengths are 7 to 10 times higher

29. Vertical Sag Curve The lengths are established based on four criteria:
Headlight sight distance (h= 2 ft, 1o divergence angle)
Passenger comfort (max acc = 1 ft/s2)
Drainage control (minimum 0.3%)
General appearance (same as crest curve)

30. Vertical Sag Curve

31. Vertical Sag Curve

32. Vertical Sag Curve
Minimum Length for comfort:

33. Vertical Sag Curve
For a design speed of 60 mph, determine the length of the sag vertical curve with A = +10%

34. Vertical Sag Curve
For a design speed of 60 mph, determine the length of the sag vertical curve with A = +10%
Stopping Sight Distance: 570 ft
For S < L
L = 10 x 5702 / ( x 570) = 1,357 ft
Check for comfort:
L = 10 x 602 / 46.5 = 774 ft (L = 1,357 ft is higher)

35. Vertical Sag Curve

36. Vertical Sag Curve
1,357 ft

37. Vertical Sag Curve

38. Vertical Sag Curve
Since many sag vertical curves are used at grade separated locations (overpass, bridges), highway engineers need to determine if the under passing structure creates a visibility impediment
Can use a graphical method or the equations provided in the text

39. Vertical Sag Curve

40. Vertical Sag Curve
S > L:

41. Vertical Sag Curve
S < L:

42. General Controls
Smooth gradeline with gradual changes is the preferable design alignment
“Roller-coaster” type of profile should be avoided (old roads)
A “broken-back” gradeline should be avoided
Sag vertical curves should be avoided in cuts unless adequate drainage can be provided

43. General Controls Combinations of horizontal and vertical alignment:
Horizontal and vertical alignments should not be designed independently
Highway cost is an important issue; thus, you often need to design them together
Vertical curvature superimposed on horizontal curvature generally results in more pleasing facilities
Sharp horizontal curvature should not be introduced near the top of a pronounced crest vertical curve
See other design considerations in AASHTO Green Book