Transcript Vertical Alignment - Housing, Building & Planning

CEE 320
Winter 2006
Geometric Design
CEE 320
Steve Muench
Outline
1. Concepts
2. Vertical Alignment
a.
b.
c.
d.
Fundamentals
Crest Vertical Curves
Sag Vertical Curves
Examples
3. Horizontal Alignment
a. Fundamentals
b. Superelevation
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Winter 2006
4. Other Non-Testable Stuff
Concepts
• Alignment is a 3D problem broken
down into two 2D problems
– Horizontal Alignment (plan view)
– Vertical Alignment (profile view)
• Stationing
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– Along horizontal alignment
– 12+00 = 1,200 ft.
Piilani Highway on Maui
Stationing
Horizontal Alignment
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Vertical Alignment
From Perteet Engineering
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Vertical Alignment
Vertical Alignment
• Objective:
– Determine elevation to ensure
• Proper drainage
• Acceptable level of safety
• Primary challenge
– Transition between two grades
– Vertical curves
Sag Vertical Curve
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Winter 2006
G1
G2
Crest Vertical Curve
G1
G2
Vertical Curve Fundamentals
• Parabolic function
– Constant rate of change of slope
– Implies equal curve tangents
y  ax  bx  c
2
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• y is the roadway elevation x stations
(or feet) from the beginning of the curve
Vertical Curve Fundamentals
G1
PVC
PVI
δ
G2
PVT
L/2
L
x
y  ax  bx  c
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2
Choose Either:
• G1, G2 in decimal form, L in feet
• G1, G2 in percent, L in stations
Choose Either:
• G1, G2 in decimal form, L in feet
• G1, G2 in percent, L in stations
Relationships
At the PVC : x  0 and Y  c
dY
 b  G1
dx
At the PVC : x  0 and
d 2Y
G2  G1
G2  G1
Anywhere:
 2a 
a
2
dx
L
2L
G1
PVC
PVI
δ
G2
PVT
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L/2
L
x
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
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Winter 2006
PVC: STA 100+00
EL 59 ft.
PVI
PVT
PVC: STA 100+00
EL 59 ft.
•G1, G2 in percent
•L in feet
Other Properties
G1
x
PVT
PVC
Y
Ym
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Winter 2006
A  G1  G2
A
Y
x2
200 L
G2
PVI
AL
Ym 
800
Yf
AL
Yf 
200
Other Properties
• K-Value (defines vertical curvature)
– The number of horizontal feet needed for a 1%
change in slope
L
K
A
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Winter 2006
high/ low pt.  x  K G1
Crest Vertical Curves
SSD
PVI
Line of Sight
PVC
G1
PVT
G2
h2
h1
L
For SSD < L
ASSD
For SSD > L
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Winter 2006
2
L

100 2h1  2h2

2

200 h1  h2
L  2SSD 
A

2
Crest Vertical Curves
• Assumptions for design
– h1 = driver’s eye height = 3.5 ft.
– h2 = tail light height = 2.0 ft.
• Simplified Equations
For SSD < L
ASSD
L
2158
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2
For SSD > L
2158
L  2SSD  
A
Crest Vertical Curves
• Assuming L > SSD…
2
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SSD
K
2158
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Design Controls for Crest Vertical Curves
from AASHTO’s A Policy on Geometric Design of Highways and Streets 2001
from AASHTO’s A Policy on Geometric Design of Highways and Streets 2001
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Winter 2006
Design Controls for Crest Vertical Curves
Sag Vertical Curves
Light Beam Distance (SSD)
G1
headlight beam (diverging from LOS by β degrees)
PVT
PVC
h1
G2
PVI
h2=0
L
For SSD < L
ASSD
L
200h1  S tan  
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2
For SSD > L
200 h1  SSD  tan  
L  2SSD  
A
Sag Vertical Curves
• Assumptions for design
– h1 = headlight height = 2.0 ft.
– β = 1 degree
• Simplified Equations
For SSD < L
ASSD
L
400 3.5SSD
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2
For SSD > L
 400 3.5SSD 
L  2SSD  

A


Sag Vertical Curves
• Assuming L > SSD…
2
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Winter 2006
SSD
K
400 3.5SSD
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Design Controls for Sag Vertical Curves
from AASHTO’s A Policy on Geometric Design of Highways and Streets 2001
from AASHTO’s A Policy on Geometric Design of Highways and Streets 2001
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Design Controls for Sag Vertical Curves
Example 1
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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?
Example 2
Similar to Example 1 but for a crest curve.
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Winter 2006
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?
Example 3
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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?
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Horizontal
Alignment
Horizontal Alignment
• Objective:
– Geometry of directional transition to ensure:
• Safety
• Comfort
• Primary challenge
– Transition between two directions
– Horizontal curves
• Fundamentals
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– Circular curves
– Superelevation
Δ
Horizontal Curve Fundamentals
PI
Δ
T

T  R tan
2
E
M
PC
L
Δ/2
PT

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100 
L
R 
180
D
 180
100

  18,000

D

R
 R
R
R
Δ/2 Δ/2
Horizontal Curve Fundamentals
PI
Δ
T
E
M
PC
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Winter 2006
 1

E  R
 1
 cos  2 


M  R1  cos 
2

L
Δ/2
R
PT
R
Δ/2 Δ/2
Example 4
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Winter 2006
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?
Wp  Ff  Fcp
Rv
≈
Superelevation
Fc
e
W
1 ft
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α

 WV 2
WV 2
W sin   f s W cos 
sin   
cos
gRv

 gRv
Superelevation

 WV 2
WV 2
W sin   f s W cos 
sin   
cos
gRv

 gRv
V2
1  f s tan 
tan  f s 
gRv
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Winter 2006
V2
1  f s e
e  fs 
gRv
V2
Rv 
g  f s  e
Selection of e and fs
• Practical limits on superelevation (e)
– Climate
– Constructability
– Adjacent land use
• Side friction factor (fs) variations
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– Vehicle speed
– Pavement texture
– Tire condition
New Graph
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Winter 2006
Side Friction Factor
from AASHTO’s A Policy on Geometric Design of Highways and Streets 2004
New Table
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Minimum Radius Tables
New Table
WSDOT Design Side Friction Factors
from the 2005 WSDOT Design Manual, M 22-01
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For Open Highways and Ramps
New Graph
WSDOT Design Side Friction Factors
from the 2005 WSDOT Design Manual, M 22-01
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For Low-Speed Urban Managed Access Highways
New Graph
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Design Superelevation Rates - AASHTO
from AASHTO’s A Policy on Geometric Design of Highways and Streets 2004
New Graph
Design Superelevation Rates - WSDOT
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emax = 8%
from the 2005 WSDOT Design Manual, M 22-01
Example 5
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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?
Stopping Sight Distance

100  s
SSD 
Rv  s 
180
D
180SSD
s 
Rv

 90SSD 

M s  Rv 1  cos
 Rv 

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Winter 2006
Rv 
 Rv  M s 

SSD 
cos 
90 
 Rv 
1
SSD
Ms
Obstruction
Rv
Δs
FYI – NOT TESTABLE
Supplemental Stuff
• Cross section
• Superelevation Transition
– Runoff
– Tangent runout
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• Spiral curves
• Extra width for curves
FYI – NOT TESTABLE
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Cross Section
FYI – NOT TESTABLE
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Superelevation Transition
from the 2001 Caltrans Highway Design Manual
FYI – NOT TESTABLE
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Winter 2006
Superelevation Transition
from AASHTO’s A Policy on Geometric Design of Highways and Streets 2001
Superelevation Runoff/Runout
from AASHTO’s A Policy on Geometric Design of Highways and Streets 2001
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FYI – NOT TESTABLE
FYI – NOT TESTABLE
New Graph
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Superelevation Runoff - WSDOT
from the 2005 WSDOT Design Manual, M 22-01
FYI – NOT TESTABLE
Spiral Curves
No Spiral
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Spiral
from AASHTO’s A Policy on Geometric Design of Highways and Streets 2001
FYI – NOT TESTABLE
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Winter 2006
No Spiral
FYI – NOT TESTABLE
Spiral Curves
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•
•
•
•
•
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
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Winter 2006
Desirable Spiral Lengths
from AASHTO’s A Policy on Geometric Design of Highways and Streets 2001
FYI – NOT TESTABLE
Operating vs. Design Speed
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Winter 2006
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
Primary References
• Mannering, F.L.; Kilareski, W.P. and Washburn, S.S. (2005).
Principles of Highway Engineering and Traffic Analysis, Third
Edition. Chapter 3
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Winter 2006
• American Association of State Highway and Transportation
Officials (AASHTO). (2001). A Policy on Geometric Design of
Highways and Streets, Fourth Edition. Washington, D.C.