1. GEOMETRIC DESIGN: HORIZONTAL ALIGNMENT
CE331 Transportation Engineering

2. Objectives Describe components of horizontal alignment
Determine design parameters for circular curve
Understand the impact of superelevation and stopping sight distance on the design of horizontal curve

3. General Concepts Components Design values
Tangents
Curves (circular)
Design values
Function of design speed and superelevation
Stopping sight distance at any point
Length measured along centerline of the curve

4. Horizontal Curves Sta PC = Sta PI – T Sta PT = Sta PC +L Arc L =Δ
Cord C =
2R sin(Δ/2)
Tangent T =
R tan(Δ/2) T
M = R [1 – cos (Δ/2)] M
Δ/2 PT PC C R
Sta PC = Sta PI – T
Sta PT = Sta PC +L Δ

5. Example Given: R = 1800 ft, Δ = 30 o, Sta PI Find: L, Sta PC, Sta PT?
1257.3
Sta PC = Sta PI - T
Sta PT = Sta PC + L PC PT
L = π R Δ/180 = π(1800)(30)/180 = ft
T = R tan(Δ/2) = 1800 tan(30/2) = ft

6. Example
A highway has a design speed of 70mph and a superelevation rate of If fs = 0.15, What should be the radius of the curve?
R = V2/[15(fs+e)]
= 702/[15*( )] = 2042 (ft)

7. Example (cont’d)
If the curve is fitted through two tangents with central angle Δ = 25°, How long should the curve be?
L = πRΔ/180 = π*2042*25/180 = 891 (ft)

8. Sight Distance on Horizontal Curves
Clearance from roadside obstruction
M’ measured from the CL of the inside lane
M’ = R’ {1 – cos[90SSD/(πR’)]}
SSD C L M’ C L R R’

9.  Clearance from centerline of the road = 34.0 + 12/2 = 40 ft
Example
Min. distance to wall from centerline to ensure SSD?
Given: R=1000 ft; tr= 2 sec; V=60mph; a=11.2ft/sec2;
G=0%; lane width (W) = 12ft C L M’ 2
60 mph
11.2
0.00
SSD = 1.47Vtr + V2/[30(a/32.2+G)]
= (ft)
M’ = R’ {1 – cos[90SSD/(π R’)]} = 34.0 (ft) /2
 Clearance from centerline of the road = /2 = 40 ft

10. Example (cont’d) What if the clearance from the centerline is 35 ft?
Change V. C L M’
M’ = 35 – 12/2 = 29 ft
Find SSD associated with M’:
M’ = R’ {1 – cos[90SSD/(π R’)]}
 SSD = πR’ cos-1(1- M’/R’)/90 = (ft)
SSD = 1.47Vtr + V2/[30(a/32.2+G)]
V = 57.1 (mph)
Speed limit = 55 mph

11. Deflection Angles Used in laying out curves Incremental central angle
δi = 180 xi /(Rπ)
Subtended angles θi = δi/2=180 xi /(2Rπ)
Deflection angles
Cumulative of subtended angles
Chords for laying out curve
Ci = 2R sin(δi/2) = 2R sin(θi)

12. Example (1/4) R = 600 m Deflection angles, cords? 30o PC PT
Sta
30o
Sta
Sta PC PT

13. Laying Out a Curve (2/4) 30o PC 1+450.00 1+500.00 1+400.00 θ2 θ1 θ1 θ3 PC

14. Example (3/4) 1st Station 2nd Station 1400-1396.14 = 3.86 m
C1 = 2 (600)sin(0.18) = 3.86 m
2nd Station
= m
θ2 = 180 (50)/[2 (600) π] = 2.39o
C2 = 2 (600)sin(2.39) = m

15. Curve Layout Data (4/4) Station xi θi Deflect. Ci 1+396.14 1+400.00
3.86
0.18o
50.00
2.39o
2.57o
49.99
4.96o
7.35o
9.73o
12.12o
14.51o
10.30
0.49o
15.00o

16. Used to help surveyors stakeout curve
Arc -> Roadways
Chord -> Railroads

17. Superelevation Pavement cross slope to keep vehicles on road
e + fs = V2/(15R)
e = tan(α), superelevation rate
fs: coefficient of side friction
V: design speed, mph
R: radius of the curve, ft

18. Superelevation Issues
emax – Lower in Maine than in Florida – Why?
fs is a function of driver comfort and safety
ranges from 0.17 for 20mph to 0.08 at 80mph
Max superelevation sets Min radius
Practice: AASHTO Green Book tables

19. Superelevation Transition
From To
Tangent Runout 0% 2% 0% e%
Superelevation Runoff
Pavement rotation rate 1:200
L = 200 W e

20. Superelevation Transition0% PC e%
Example
W = 3.30 m
TR = 200(3.3)(0.02)=13.2 m
SR = 200(3.3)(0.08)=52.8 m
A-PC = SR(2/3) = 35.2 m A 2%

21. Where do we rotate the roadway?
Rotate pavement about the centerline
most common for undivided roadways
Others mainly for drainage or terrain
Rotate pavement about the inner edge
Rotate pavement about the outside edge
Rotate about the center of the median

22. Super-elevation Transition

23. Superelevation
Road Section View
Road Plan View CL 2% 2%

24. Superelevation
Road Section View
Road Plan View CL
1.5% 2%

25. Superelevation
Road Section View
Road Plan View CL 1% 2%

26. Superelevation
Road Section View
Road Plan View CL
0.5% 2%

27. Superelevation
Road Section View
Road Plan View CL
-0.0% 2%

28. Superelevation
Road Section View
Road Plan View CL
-0.5% 2%

29. Superelevation
Road Section View
Road Plan View CL
-1% 2%

30. Superelevation
Road Section View
Road Plan View CL
-1.5% 2% 2%

31. Superelevation
Road Section View
Road Plan View CL
-2% 2%

32. Superelevation
Road Section View
Road Plan View CL
-3% 3%

33. Superelevation
Road Section View
Road Plan View CL
-4% 4%

34. Superelevation
Road Section View
Road Plan View CL
-3% 3%

35. Superelevation
Road Section View
Road Plan View CL
-2% 2%

36. Superelevation
Road Section View
Road Plan View CL
-1.5% 2%

37. Superelevation
Road Section View
Road Plan View CL
-1% 2%

38. Superelevation
Road Section View
Road Plan View CL
-0.5% 2%

39. Superelevation
Road Section View
Road Plan View CL
-0.0% 2%

40. Superelevation
Road Section View
Road Plan View CL
0.5% 2%

41. Superelevation
Road Section View
Road Plan View CL 1% 2%

42. Superelevation
Road Section View
Road Plan View CL
1.5% 2%

43. Superelevation
Road Section View
Road Plan View CL 2% 2%

44. How do we transition into a super-elevated curve?
No Spiral
Spiral