1. HORIZONTAL ALIGNMENT
Spring 2015

2. Horizontal Alignment Geometric Elements of Horizontal Curves
Transition or Spiral Curves
Superelevation Design
Sight Distance

3. Simple Curve Circular Curve Tangent PC PT Point of Tangency
Point of Curvature

4. Curve with Spiral Transition
Circular Curve
Spiral
Tangent CS SC ST TS
Curve to Spiral
Spiral to Curve
Spiral to Tangent
Tangent to Spiral

5. Design Elements of Horizontal Curves
Deflection Angle
Also known as Δ
Deflection Angle

6. Design Elements of Horizontal Curves
Larger D = smaller Radius

7. Design Elements of Horizontal Curves
E=External Distance
M=Length of Middle Ordinate

8. Design Elements of Horizontal Curves
LC=Length of Long Cord

9. Basic Formulas
Basic Formula that governs vehicle operation on a curve:
Where,
e = superelevation
f = side friction factor
V = vehicle speed (mph)
R = radius of curve (ft)

10. Basic Formulas Minimum radius: Where, e = superelevation
f = side friction factor
V = vehicle speed (mph)
R = radius of curve (ft)

11. Minimum Radius with Limiting Values of “e” and “f”

12. Superelevation Design
Desirable superelevation:
for R > Rmin
Where,
V= design speed in ft/s or m/s
g = gravity (9.81 m/s2 or 32.2 ft/s2)
R = radius in ft or m
Various methods are available for determining the desirable superelevation, but the equation above offers a simple way to do it. The other methods are presented in the next few overheads.

13. Methods for Estimating Desirable Superelevation
Superelevation and side friction are directly proportional to the inverse of the radius (straight relationship between 1/R=0 and 1/R =1/Rmin)
Method 2:
Side friction is such that a vehicle traveling at the design speed has all the acceleration sustained by side friction on curves up to those requiring fmax
Superelevation is introduced only after the maximum side friction is used

14. Method 3: Method 4: Method 5:
Superelevation is such that a vehicle traveling at the design speed has all the lateral acceleration sustained by superelevation on curves up to those required by emax
No side friction is provided on flat curves
May result in negative side friction
Method 4:
Same approach as Method 3, but use average running speed rather than design speed
Uses speeds lower than design speed
Eliminate problems with negative side friction
Method 5:
Superelevation and side friction are in a curvilinear relationship with the inverse of the radius of the curve, with values between those of methods 1 and 3
Represents a practical distribution for superelevation over the range of curvature
This is the method used for computing values shown in Exhibits 3-25 to 3-29

15. Five Methods fmax e = 0 emax f M2 M1 M5 M3 1/R M4 Side Friction Factor
Reciprocal of Radius
1/R M4

16. Design of Horizontal Alignment
Important considerations:
Governed by four factors:
Climate conditions
Terrain (flat, rolling, mountainous)
Type of area (rural vs urban)
Frequency of slow-moving vehicles
Design should be consistent with driver expectancy
Max 8% for snow/ice conditions
Max 12% low volume roads
Recurrent congestion: suggest lower than 6%

17. Method 1 Centerline

18. Method 2 Inside Edge

19. Method 3 Outside Edge

20. Method 4 Straight Cross Slope

21. Which Method?
In overall sense, the method of rotation about the centerline (Method 1) is usually the most adaptable
Method 2 is usually used when drainage is a critical component in the design
In the end, an infinite number of profile arrangements are possible; they depend on drainage, aesthetic, topography among others

22. Example where pivot points are important
Bad design
Pivot points
Good design
Median width
15 ft to 60 ft

23. Transition Design Control
The superelevation transition consists of two components:
The superelevation runoff: length needed to accomplish a change in outside-lane cross slope from zero (flat) to full superelevation
The tangent runout: The length needed to accomplish a change in outside-lane cross slope rate to zero (flat)

24. Transition Design Control
Tangent Runout

25. Transition Design Control
Superelevation Runoff

26. Transition Design Control

27. Transition Design Control

28. Minimum Length of Superelevation Runoff

29. Minimum Length of Superelevation Runoff
 = relative gradient in previous overhead

30. Minimum Length of Superelevation Runoff
Values for n1 and bw in equation

31. Minimum Length of Tangent Runout
See Exhibit 3-32 for values of Lt and Lr

32. Superelevation Runoff
Location: 1/3 on curve
Location: 2/3 on tangent

33. Superelevation Runoff

34. Transition Curves -Spirals
All motor vehicles follow a transition path as it enters or leaves a circular horizontal curve (adjust for increases in lateral acceleration)
Drivers can create their own path or highway engineers can use spiral transitional curves
The radius of a spiral varies from infinity at the tangent end to the radius of the circular curve at the end that adjoins the curve

35. Transition Curves -Spirals
Need to verify for maximum and minimum lengths

36. Transition Curves Superelevation runoff should be accomplished on the
entire length of the spiral curve transition
Equation for tangent runout when Spirals are used:

37. Sight distance on Horizontal Curve
The sight distance is measured from the centerline of the inside lane
Need to measure the middle-ordinate values (defined as M)
Values of M are given in Exhibit 3-53
Note: Now M is defined as HSO or Horizontal sightline offset.

39. Included for your benefit
Example Application
Included for your benefit

40. Selection of fdesign and edesign (Method 5)
fmax (for the design speed)
Side Friction Factor
e = 0
fdesign
emax (for the design speed)
Reciprocal of Radius
1/R

41. Selection of fdesign and edesign
Rf = V2/(gfmax)
Rmin = V2/[g(fmax + emax)]
fmax
Side Friction Factor
e = 0
fdesign
emax
Ro = V2/(gemax)
Reciprocal of Radius
1/R
R0: f = 0, e = emax

42. Selection of fdesign and edesign
fdesign = α(1/R)+β(1/R)2
fmax (for the design speed)
Side Friction Factor
α = fmaxRmin[1-{Rmin/(R0-Rmin)}]
e = 0
fdesign
β = fmaxRmin3/(R0-Rmin)
emax (for the design speed)
Reciprocal of Radius
1/R

43. Superelevation Design for High Speed Rural and Urban Highways

44. Example:
Design Speed: 100 km/h
fmax = 0.128
emax = 0.06
Question?
What should be the design friction factor and design superelevation for a curve with a radius of 600 m?

45. 1. Compute Rf, R0, and Rmin:
Rf = V2/(gfmax) = / (9.81 x 0.128) = 615 m
R0 = V2/(gemax) = / (9.81 x 0.06) = 1311 m
Rmin = V2/[g(fmax + emax)] = / [9.81( )]
Rmin = 418 m

46. Selection of fdesign and edesign (example)
fmax = 0.128
Side Friction Factor
e = 0
fdesign
emax = 0.06
1 / 1311
1 / 615
1 / 418
1/R

47. 2. Compute α and β:
α = x 418 x [1 – 418 / (1311 – 418) ] = m
β = x 4183 / (1311 – 418) = m2
3. Compute fdesign and edesign :
First, estimate the right-hand side of equation for designing superelevation
e + f = V2/(gR) = / (9.81 x 600) = 0.131
Then,
fdesign = / / 6002 = 0.076
edesign = – = (< emax = 0.06)

48. Selection of fdesign and edesign (example)
fmax = 0.128
Side Friction Factor
e = 0
fdesign
emax = 0.06
0.076
1 / 1311
1 / 615
1 / 418
1/R
1 / 600

49. Selection of fdesign and edesign (example)
R=600 ft