VERTICAL ALIGNMENT

1. VERTICAL ALIGNMENT Spring 2014

2. 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 Rate of change of grade K = L / |A| Offset y = 4E(x/L)2 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 25+60.55 and elevation 648.64 ft. Calculate the location and elevation of the PVC, the middle of the curve, the VPT, and the curve elevation at stations 24+00 and 27+ 00.

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 = 648.64 – 4.5 = 644.14 Using the curve elevation equation: Ele P = [ele VPC + (G1/100) x] + y 644.14 ft = [ele VPC + 4/100 x 300] – 4.5 Ele VPC = 636.64 Station 22+60.55

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 is310 ftsince 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 / (400 + 3.5 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