This document provides an outline and content for a course on geometric design. It covers concepts of horizontal and vertical alignment, including fundamentals of stationing, vertical curves, horizontal curves, and superelevation. Examples are provided for calculating elements of vertical curves, horizontal curve properties, and minimum curve radii. Design controls are presented from AASHTO and WSDOT sources. Additional topics covered include cross sections, superelevation transition, stopping sight distance, and spiral curves, but these are noted as not being testable.

1. CEE
320
Spring
2008
Geometric Design
CEE 320
Anne Goodchild

2. CEE
320
Spring
2008
Outline
1. Concepts
2. Vertical Alignment
a. Fundamentals
b. Crest Vertical Curves
c. Sag Vertical Curves
d. Examples
3. Horizontal Alignment
a. Fundamentals
b. Superelevation
4. Other Non-Testable Stuff

3. CEE
320
Spring
2008
Concepts
• Alignment is a 3D problem broken
down into two 2D problems
– Horizontal Alignment (plan view)
– Vertical Alignment (profile view)
• Stationing
– Along horizontal alignment
– 12+00 = 1,200 ft.
Piilani Highway on Maui

4. CEE
320
Spring
2008
Stationing
Horizontal Alignment
Vertical Alignment

5. From Perteet Engineering

6. CEE
320
Spring
2008
Vertical Alignment

7. CEE
320
Spring
2008
Vertical Alignment
• Objective:
– Determine elevation to ensure
• Proper drainage
• Acceptable level of safety
• Primary challenge
– Transition between two grades
– Vertical curves
G1 G2
G1
G2
Crest Vertical Curve
Sag Vertical Curve

8. CEE
320
Spring
2008
Vertical Curve Fundamentals
• Parabolic function
– Constant rate of change of slope
– Implies equal curve tangents
• y is the roadway elevation x stations
(or feet) from the beginning of the curve
c
bx
ax
y 

 2

9. CEE
320
Spring
2008
Vertical Curve Fundamentals
G1
G2
PVI
PVT
PVC
L
L/2
δ
c
bx
ax
y 

 2
x
Choose Either:
• G1, G2 in decimal form, L in feet
• G1, G2 in percent, L in stations

10. CEE
320
Spring
2008
Relationships
Choose Either:
• G1, G2 in decimal form, L in feet
• G1, G2 in percent, L in stations
G1
G2
PVI
PVT
PVC
L
L/2
δ
x
1
and
0
:
PVC
At the G
b
dx
dY
x 


c
Y
x 
 and
0
:
PVC
At the
L
G
G
a
L
G
G
a
dx
Y
d
2
2
:
Anywhere 1
2
1
2
2
2







11. CEE
320
Spring
2008
Example
A 400 ft. equal tangent crest vertical curve has a PVC station of
100+00 at 59 ft. elevation. The initial grade is 2.0 percent and the final
grade is -4.5 percent. Determine the elevation and stationing of PVI,
PVT, and the high point of the curve.
PVI
PVT
PVC: STA 100+00
EL 59 ft.

12. PVI
PVT
PVC: STA 100+00
EL 59 ft.

13. CEE
320
Spring
2008
Other Properties
G1
G2
PVI
PVT
PVC
x
Ym
Yf
Y
2
200
x
L
A
Y 
800
AL
Ym 
200
AL
Yf 
2
1 G
G
A 

•G1, G2 in percent
•L in feet

14. CEE
320
Spring
2008
Other Properties
• K-Value (defines vertical curvature)
– The number of horizontal feet needed for a 1%
change in slope
A
L
K 
1
.
/ G
K
x
pt
low
high 


15. CEE
320
Spring
2008
Crest Vertical Curves
G1
G2
PVI
PVT
PVC
h2
h1
L
SSD
 
 2
2
1
2
2
2
100 h
h
SSD
A
L

  
 
A
h
h
SSD
L
2
2
1
200
2



For SSD < L For SSD > L
Line of Sight

16. CEE
320
Spring
2008
Crest Vertical Curves
• Assumptions for design
– h1 = driver’s eye height = 3.5 ft.
– h2 = tail light height = 2.0 ft.
• Simplified Equations
 
2158
2
SSD
A
L   
A
SSD
L
2158
2 

For SSD < L For SSD > L

17. CEE
320
Spring
2008
Crest Vertical Curves
• Assuming L > SSD…
2158
2
SSD
K 

18. CEE
320
Spring
2008
Design Controls for Crest Vertical Curves
from AASHTO’s A Policy on Geometric Design of Highways and Streets 2004

19. CEE
320
Spring
2008
Design Controls for Crest Vertical Curves
from
AASHTO’s
A
Policy
on
Geometric
Design
of
Highways
and
Streets
2004

20. CEE
320
Spring
2008
Sag Vertical Curves
G1
G2
PVI
PVT
PVC
h2=0
h1
L
Light Beam Distance (SSD)
 
 

tan
200 1
2
S
h
SSD
A
L

    
 
A
SSD
h
SSD
L

tan
200
2 1 


For SSD < L For SSD > L
headlight beam (diverging from LOS by β degrees)

21. CEE
320
Spring
2008
Sag Vertical Curves
• Assumptions for design
– h1 = headlight height = 2.0 ft.
– β = 1 degree
• Simplified Equations
 
 
SSD
SSD
A
L
5
.
3
400
2

    





 


A
SSD
SSD
L
5
.
3
400
2
For SSD < L For SSD > L

22. CEE
320
Spring
2008
Sag Vertical Curves
• Assuming L > SSD…
SSD
SSD
K
5
.
3
400
2



23. CEE
320
Spring
2008
Design Controls for Sag Vertical Curves
from AASHTO’s A Policy on Geometric Design of Highways and Streets 2004

24. CEE
320
Spring
2008
Design Controls for Sag Vertical Curves
from
AASHTO’s
A
Policy
on
Geometric
Design
of
Highways
and
Streets
2004

25. CEE
320
Spring
2008
Example 1
A car is traveling at 30 mph in the country at night on a wet road
through a 150 ft. long sag vertical curve. The entering grade is -2.4
percent and the exiting grade is 4.0 percent. A tree has fallen across
the road at approximately the PVT. Assuming the driver cannot see
the tree until it is lit by her headlights, is it reasonable to expect the
driver to be able to stop before hitting the tree?

26. CEE
320
Spring
2008
Example 2
Similar to Example 1 but for a crest curve.
A car is traveling at 30 mph in the country at night on a wet road
through a 150 ft. long crest vertical curve. The entering grade is 3.0
percent and the exiting grade is -3.4 percent. A tree has fallen across
the road at approximately the PVT. Is it reasonable to expect the driver
to be able to stop before hitting the tree?

27. CEE
320
Spring
2008
Example 3
A roadway is being designed using a 45 mph design speed. One
section of the roadway must go up and over a small hill with an
entering grade of 3.2 percent and an exiting grade of -2.0 percent.
How long must the vertical curve be?

28. CEE
320
Spring
2008
Horizontal
Alignment

29. CEE
320
Spring
2008
Horizontal Alignment
• Objective:
– Geometry of directional transition to ensure:
• Safety
• Comfort
• Primary challenge
– Transition between two directions
– Horizontal curves
• Fundamentals
– Circular curves
– Superelevation
Δ

30. CEE
320
Spring
2008
Horizontal Curve Fundamentals
R
T
PC PT
PI
M
E
R
Δ
Δ/2
Δ/2
Δ/2
R
R
D

 000
,
18
180
100








2
tan

 R
T
D
R
L




100
180

L

31. CEE
320
Spring
2008
Horizontal Curve Fundamentals










 1
2
cos
1
R
E





 


2
cos
1
R
M
R
T
PC PT
PI
M
E
R
Δ
Δ/2
Δ/2
Δ/2
L

32. CEE
320
Spring
2008
Example 4
A horizontal curve is designed with a 1500 ft. radius. The tangent
length is 400 ft. and the PT station is 20+00. What are the PI and PT
stations?

33. CEE
320
Spring
2008
Superelevation cp
f
p F
F
W 




 cos
sin
cos
sin
2
2
v
v
s
gR
WV
gR
WV
W
f
W 










α
Fc
W 1 ft
e
≈
Rv

34. CEE
320
Spring
2008
Superelevation



 cos
sin
cos
sin
2
2
v
v
s
gR
WV
gR
WV
W
f
W 










 

 tan
1
tan
2
s
v
s f
gR
V
f 


 
e
f
gR
V
f
e s
v
s 

 1
2
 
e
f
g
V
R
s
v


2

35. CEE
320
Spring
2008
Selection of e and fs
• Practical limits on superelevation (e)
– Climate
– Constructability
– Adjacent land use
• Side friction factor (fs) variations
– Vehicle speed
– Pavement texture
– Tire condition

36. CEE
320
Spring
2008
Side Friction Factor
from AASHTO’s A Policy on Geometric Design of Highways and Streets 2004

37. CEE
320
Spring
2008
Minimum Radius Tables

38. CEE
320
Spring
2008
WSDOT Design Side Friction Factors
from
the
2005
WSDOT
Design
Manual,
M
22-01
For Open Highways and Ramps

39. CEE
320
Spring
2008
WSDOT Design Side Friction Factors
from
the
2005
WSDOT
Design
Manual,
M
22-01
For Low-Speed Urban Managed Access Highways

40. CEE
320
Spring
2008
Design Superelevation Rates - AASHTO
from AASHTO’s A Policy on Geometric Design of Highways and Streets 2004

41. CEE
320
Spring
2008
Design Superelevation Rates - WSDOT
from the 2005 WSDOT Design Manual, M 22-01
emax = 8%

42. CEE
320
Spring
2008
Example 5
A section of SR 522 is being designed as a high-speed divided
highway. The design speed is 70 mph. Using WSDOT standards,
what is the minimum curve radius (as measured to the traveled vehicle
path) for safe vehicle operation?

43. CEE
320
Spring
2008
Stopping Sight Distance
Rv
Δs
Obstruction
Ms
 
v
s
R
SSD

180


D
R
SSD s
s
v




100
180

SSD
















v
v
s
R
SSD
R
M

90
cos
1













 
 
v
s
v
v
R
M
R
R
SSD 1
cos
90


44. CEE
320
Spring
2008
Supplemental Stuff
• Cross section
• Superelevation Transition
– Runoff
– Tangent runout
• Spiral curves
• Extra width for curves
FYI – NOT TESTABLE

45. CEE
320
Spring
2008
Cross Section
FYI – NOT TESTABLE

46. CEE
320
Spring
2008
Superelevation Transition
from the 2001 Caltrans Highway Design Manual
FYI – NOT TESTABLE

47. CEE
320
Spring
2008
Superelevation Transition
from AASHTO’s A Policy on Geometric Design of Highways and Streets 2001
FYI – NOT TESTABLE

48. CEE
320
Spring
2008
Superelevation Runoff/Runout
from
AASHTO’s
A
Policy
on
Geometric
Design
of
Highways
and
Streets
2004
FYI – NOT TESTABLE

49. CEE
320
Spring
2008
Superelevation Runoff - WSDOT
from the 2005 WSDOT Design Manual, M 22-01
FYI – NOT TESTABLE

50. CEE
320
Spring
2008
Spiral Curves
No Spiral
Spiral
from AASHTO’s A Policy on Geometric Design of Highways and Streets 2004
FYI – NOT TESTABLE

51. CEE
320
Spring
2008
No Spiral
FYI – NOT TESTABLE

52. CEE
320
Spring
2008
Spiral Curves
• WSDOT no longer uses spiral curves
• Involve complex geometry
• Require more surveying
• Are somewhat empirical
• If used, superelevation transition should
occur entirely within spiral
FYI – NOT TESTABLE

53. CEE
320
Spring
2008
Desirable Spiral Lengths
from AASHTO’s A Policy on Geometric Design of Highways and Streets 2004
FYI – NOT TESTABLE

54. CEE
320
Spring
2008
Operating vs. Design Speed
85th Percentile Speed
vs. Inferred Design Speed for
138 Rural Two-Lane Highway
Horizontal Curves
85th Percentile Speed
vs. Inferred Design Speed for
Rural Two-Lane Highway
Limited Sight Distance Crest
Vertical Curves
FYI – NOT TESTABLE

55. CEE
320
Spring
2008
Primary References
• Mannering, F.L.; Kilareski, W.P. and Washburn, S.S. (2005).
Principles of Highway Engineering and Traffic Analysis, Third
Edition. Chapter 3
• American Association of State Highway and Transportation
Officials (AASHTO). (2001). A Policy on Geometric Design of
Highways and Streets, Fourth Edition. Washington, D.C.