ECGD4107 Pavement Engineering Summer 2008 Sat. 15:30-18:30 PM K004

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

2. Lecture 3 • Vertical Alignments • Geometric Characteristics • Sight Distances: Crests & Sags Dr. Wa'el M. Albawwab albawwab@gmail.com

3. Types of Vertical Alignment • Crest vertical curves • Sag vertical curves Dr. Wa'el M. Albawwab albawwab@gmail.com

4. Crest Vertical Curves Dr. Wa'el M. Albawwab albawwab@gmail.com

5. Crest Vertical Curves Dr. Wa'el M. Albawwab albawwab@gmail.com

6. Sag Vertical Curves Dr. Wa'el M. Albawwab albawwab@gmail.com

7. Sag Vertical Curves Dr. Wa'el M. Albawwab albawwab@gmail.com

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) Dr. Wa'el M. Albawwab albawwab@gmail.com

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) Dr. Wa'el M. Albawwab albawwab@gmail.com

10. Elements of Vertical Curves • L = length of the curve measured in a constant elevation horizontal plane • Ascending = +ve grading • Descending = -ve grading Dr. Wa'el M. Albawwab albawwab@gmail.com

11. Presentation of Vertical Curves • Parabolic mathematical expression • Provides a constant sloping rate • y = ax2 + bx + c Dr. Wa'el M. Albawwab albawwab@gmail.com

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 Dr. Wa'el M. Albawwab albawwab@gmail.com

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 Dr. Wa'el M. Albawwab albawwab@gmail.com

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 Dr. Wa'el M. Albawwab albawwab@gmail.com

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 albawwab@gmail.com

16. 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 albawwab@gmail.com

17. 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. Dr. Wa'el M. Albawwab albawwab@gmail.com

18. Example1 A 600 ft long equal-tangent sag vertical curve has the PVC at station 170+00 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 albawwab@gmail.com

19. 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: 173 + 00 and 176 + 00 respectively. • Elevation of PVI is: 1000 + (-3.5/100)(3)(100) = 989.5 ft Dr. Wa'el M. Albawwab albawwab@gmail.com

20. Example1 / Continued • The elevation of the PVT is: • 989.5 + (0.5/100)(3)(100) = 991.0 ft • Coefficients of the curve: • b = G1 = -3.5/100 = -0.035 • a = (G2 – G1)/2L = [0.5-(-3.5)]/[(100)(2)(600)] = 1/30000 • y0 = 1000 ft Dr. Wa'el M. Albawwab albawwab@gmail.com

21. Example1 / Continued • The function of the curve: • y = x2/30000 – 0.03.5x + 1000 • The lowest point on the curve: • dy/dx = 0 at x = 525.0 ft • The level of this point is: • y = 990.8125 ft Dr. Wa'el M. Albawwab albawwab@gmail.com

22. Offsetting of Vertical Curves Dr. Wa'el M. Albawwab albawwab@gmail.com

23. Offsetting of Vertical Curves • Offset (Y): the vertical distance from the initial tangent to the curve line Dr. Wa'el M. Albawwab albawwab@gmail.com

24. Offsetting of Vertical Curves • Substituting for the middle offset at x = L/2 • Substituting for the final offset at x = L Dr. Wa'el M. Albawwab albawwab@gmail.com

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 Dr. Wa'el M. Albawwab albawwab@gmail.com

26. SSD for Vertical Curves Dr. Wa'el M. Albawwab albawwab@gmail.com

27. SSD for Vertical Curves • s = SSD • H1 = height of driver’s eye above road surface • H2 = height of object above road surface Dr. Wa'el M. Albawwab albawwab@gmail.com

28. SSD for Vertical Curves • From the property of parabola for an equal tangent curve, • for S < L: • For S > L: Dr. Wa'el M. Albawwab albawwab@gmail.com

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: Dr. Wa'el M. Albawwab albawwab@gmail.com

30. SSD for Vertical Curves • For L > SSD • For L < SSD Dr. Wa'el M. Albawwab albawwab@gmail.com

31. SSD for Vertical Curves Dr. Wa'el M. Albawwab albawwab@gmail.com

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. Dr. Wa'el M. Albawwab albawwab@gmail.com

33. SSD for Vertical Curves • L can be expressed by the parabola properties as: • For L > S • For L < S Dr. Wa'el M. Albawwab albawwab@gmail.com

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 albawwab@gmail.com

35. SSD for Vertical Curves • For L > SSD • For L < SSD Dr. Wa'el M. Albawwab albawwab@gmail.com

36. SSD for Vertical Curves • For crest vertical curves: • For sag vertical curves: Dr. Wa'el M. Albawwab albawwab@gmail.com

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 Dr. Wa'el M. Albawwab albawwab@gmail.com

38. PSD for Vertical Curves • Passing Sight Distances on crest vertical curves: • For L > PSD: • For L< PSD: Dr. Wa'el M. Albawwab albawwab@gmail.com

39. PSD for Vertical Curves • As was the case for stopping sight distance, • It is typically assumed L > PSD Dr. Wa'el M. Albawwab albawwab@gmail.com

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 Dr. Wa'el M. Albawwab albawwab@gmail.com

41. Due to next lecture: • Study this lecture (AASHTO 2001 – Ch. 3) • Review geometric alignments (Practical) Dr. Wa'el M. Albawwab albawwab@gmail.com