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CE
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Geometric Design
CEE 320Steve Muench
CE
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Outline
1. Concepts2. Vertical Alignment
a. Fundamentalsb. Crest Vertical Curvesc. Sag Vertical Curvesd. Examples
3. Horizontal Alignmenta. Fundamentalsb. Superelevation
4. Other Non-Testable Stuff
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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
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Stationing
Horizontal Alignment
Vertical Alignment
From Perteet Engineering
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Vertical Alignment
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Vertical Alignment
• Objective: – Determine elevation to ensure
• Proper drainage• Acceptable level of safety
• Primary challenge– Transition between two grades
– Vertical curves
G1 G2G1
G2
Crest Vertical Curve
Sag Vertical Curve
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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
cbxaxy ++= 2
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Vertical Curve Fundamentals
G1
G2
PVI
PVT
PVC
L
L/2
δ
cbxaxy ++= 2
x
Choose Either:• G1, G2 in decimal form, L in feet• G1, G2 in percent, L in stations
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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 Gbdx
dYx ===
cYx == and 0 :PVC At the
L
GGa
L
GGa
dx
Yd
22 :Anywhere 1212
2
2 −=⇒−==
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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.
G1=2.0%
G2= - 4.5%
PVI
PVT
PVC: STA 100+00EL 59 ft.
G1=2.0%
G2= -4.5%
PVI
PVT
PVC: STA 100+00EL 59 ft.
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Other Properties
G1
G2
PVI
PVTPVC
x
Ym
Yf
Y
2
200x
L
AY =
800
ALYm =
200
ALY f =
21 GGA −=
•G1, G2 in percent•L in feet
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Other Properties
• K-Value (defines vertical curvature)– The number of horizontal feet needed for a 1%
change in slope
A
LK =
1./ GKxptlowhigh =⇒
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Crest Vertical Curves
G1G2
PVI
PVTPVC
h2h1
L
SSD
( )( )221
2
22100 hh
SSDAL
+= ( ) ( )
A
hhSSDL
2
212002
+−=
For SSD < L For SSD > L
Line of Sight
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Crest Vertical Curves
• Assumptions for design– h1 = driver’s eye height = 3.5 ft.
– h2 = tail light height = 2.0 ft.
• Simplified Equations
( )2158
2SSDAL = ( )
ASSDL
21582 −=
For SSD < L For SSD > L
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Crest Vertical Curves
• Assuming L > SSD…
2158
2SSDK =
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Design Controls for Crest Vertical Curves
from AASHTO’s A Policy on Geometric Design of Highways and Streets 2001
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Design Controls for Crest Vertical Curves
fro
m A
AS
HT
O’s
A P
olic
y o
n G
eo
me
tric
De
sig
n o
f H
igh
wa
ys a
nd
Str
ee
ts 2
00
1
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Sag Vertical Curves
G1 G2
PVI
PVTPVC
h2=0h1
L
Light Beam Distance (SSD)
( )( )βtan200 1
2
Sh
SSDAL
+= ( ) ( )( )
A
SSDhSSDL
βtan2002 1 +−=
For SSD < L For SSD > L
headlight beam (diverging from LOS by β degrees)
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Sag Vertical Curves
• Assumptions for design– h1 = headlight height = 2.0 ft.
– β = 1 degree
• Simplified Equations
( )( )SSD
SSDAL
5.3400
2
+= ( ) ( )
+−=
A
SSDSSDL
5.34002
For SSD < L For SSD > L
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Sag Vertical Curves
• Assuming L > SSD…
SSD
SSDK
5.3400
2
+=
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Design Controls for Sag Vertical Curves
from AASHTO’s A Policy on Geometric Design of Highways and Streets 2001
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Design Controls for Sag Vertical Curves
fro
m A
AS
HT
O’s
A P
olic
y o
n G
eo
me
tric
De
sig
n o
f H
igh
wa
ys a
nd
Str
ee
ts 2
00
1
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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?
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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?
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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?
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Horizontal Alignment
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Horizontal Alignment
• Objective: – Geometry of directional transition to ensure:
• Safety• Comfort
• Primary challenge– Transition between two directions– Horizontal curves
• Fundamentals– Circular curves– Superelevation
Δ
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Horizontal Curve Fundamentals
R
T
PC PT
PI
M
E
R
Δ
Δ/2Δ/2
Δ/2
RRD
ππ 000,18
180100
=
=
2tan
∆= RT
DRL
∆=∆= 100
180
π
L
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Horizontal Curve Fundamentals
−
∆= 1
2cos
1RE
∆−=
2cos1RM
R
T
PC PT
PI
M
E
R
Δ
Δ/2Δ/2
Δ/2L
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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?
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Superelevation cpfp FFW =+
αααα cossincossin22
vvs gR
WV
gR
WVWfW =
++
α
α
Fcp
Fcn
Wp
Wn F f
F f
α
Fc
W 1 fte
≈Rv
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Superelevation
αααα cossincossin22
vvs gR
WV
gR
WVWfW =
++
( )αα tan1tan2
sv
s fgR
Vf −=+
( )efgR
Vfe s
vs −=+ 1
2
( )efg
VR
sv +
=2
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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
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Side Friction Factor
from AASHTO’s A Policy on Geometric Design of Highways and Streets 2004
New Graph
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Minimum Radius Tables
New Table
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WSDOT Design Side Friction Factors
from
the
20
05 W
SD
OT
Des
ign
Man
ual,
M 2
2-01
New Table
For Open Highways and Ramps
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WSDOT Design Side Friction Factors
from
the
20
05 W
SD
OT
Des
ign
Man
ual,
M 2
2-01
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
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Design Superelevation Rates - WSDOT
from the 2005 WSDOT Design Manual, M 22-01
emax = 8%
New Graph
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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?
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Stopping Sight Distance
Rv
Δs
Obstruction
Ms( )v
s R
SSD
π180=∆
DRSSD s
sv
∆=∆= 100
180
π SSD
−=
vvs R
SSDRM
π90
cos1
−= −
v
svv
R
MRRSSD 1cos
90
π
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Supplemental Stuff
• Cross section• Superelevation Transition
– Runoff
– Tangent runout
• 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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Superelevation Transition
from AASHTO’s A Policy on Geometric Design of Highways and Streets 2001
FYI – NOT TESTABLE
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Superelevation Runoff/Runout
from
AA
SH
TO
’s A
Pol
icy
on G
eom
etric
Des
ign
of
Hig
hway
s an
d S
tree
ts 2
001
FYI – NOT TESTABLE
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Superelevation Runoff - WSDOT
from the 2005 WSDOT Design Manual, M 22-01
FYI – NOT TESTABLENew Graph
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Spiral Curves
No Spiral
Spiral
from AASHTO’s A Policy on Geometric Design of Highways and Streets 2001
FYI – NOT TESTABLE
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No Spiral
FYI – NOT TESTABLE
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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
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Desirable Spiral Lengths
from AASHTO’s A Policy on Geometric Design of Highways and Streets 2001
FYI – NOT TESTABLE
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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
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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.