Post on 14-Dec-2015
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Geometric Design
CEE 320Anne Goodchild
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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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RelationshipsChoose 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 2004
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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
4
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Sag Vertical Curves
G1 G2
PVI
PVTPVC
h2=0h1
L
Light Beam Distance (SSD)
tan200 1
2
Sh
SSDAL
A
SSDhSSDL
tan2002 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 2004
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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
4
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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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Sag Vertical Curve
• Assume S<L, try both, but this is most often the case
• Equation specific to sag curve which accommodates headlight beam
• L and S in horizontal plane and comparable (150 and 146 ft)
• Required SSD = 196.53 ft assumes 0 grade
• Text problem versus design problem.
SSD
SSDAL
5.3400
2
rtV
Gga
g
VSSD 1
21
2
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Sag Vertical Curves
G1 G2
PVI
PVTPVC
h2=0h1
L
Light Beam Distance (S)
diverging from horizontal plane of vehicle by β degrees
Daytime sight distance unrestricted
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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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Crest Vertical Curve
• Assume S<L, try both, but this is most often the case
• Equation specific to crest curve which accommodates sight over hill
• L and S in horizontal plane and comparable (150 and 243 ft)
• Required SSD = 196.53 ft assumes 0 grade
• Text problem versus design problem.
2158
2SSDAL
A
SSDL2158
2
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Crest Vertical Curves
G1G2
PVI
PVTPVC
h2h1
L
SSD
Line of Sight
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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
000,18
180100
2tan
RT
DRL
100
180
L
D = degree of curvature (angle subtended by a 100’ arc)
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Horizontal Curve Fundamentals
1
2cos
1RE
2
cos1RM
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
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
cossincossin22
vvs gR
WV
gR
WVWfW
tan1tan2
sv
s fgR
Vf
efgR
Vfe s
vs 1
2
efg
VR
sv
2This is the minimum radius that provides for safe vehicle operation
Rv because it is to the vehicle’s path
e = number of vertical feet of rise per 100 ft of horizontal distance = 100tan
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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
Design values of fs are chosen somewhat below this maximum value so there is a margin of safety
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Minimum Radius Tables
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WSDOT Design Side Friction Factors
from
the
200
5 W
SD
OT
Des
ign
Man
ual,
M 2
2-01
For Open Highways and Ramps
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WSDOT Design Side Friction Factors
from
the
200
5 W
SD
OT
Des
ign
Man
ual,
M 2
2-01
For Low-Speed Urban Managed Access Highways
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Design Superelevation Rates - AASHTO
from AASHTO’s A Policy on Geometric Design of Highways and Streets 2004
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Design Superelevation Rates - WSDOT
from the 2005 WSDOT Design Manual, M 22-01
emax = 8%
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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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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?
For the minimum curve radius we want the maximum superelevation.WSDOT max e = 0.10
For 70 mph, WSDOT f = 0.10
efg
VR
sv
2
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Stopping Sight Distance
Rv
Δs
Obstruction
Ms
DRSSD s
sv
100
180
SSD (not L)•Looking around a curve•Measured along horizontal curve from the center of the traveled lane•Need to clear back to Ms (the middle of a line that has same arc length as SSD)
Assumes curve exceeds required SSD
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Stopping Sight Distance
Rv
Δs
Obstruction
Ms v
s R
SSD
180
SSD (not L)
vvs R
SSDRM
90
cos1
v
svv
R
MRRSSD 1cos
90
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Example 6
vvs R
SSDRM
90
cos1
A horizontal curve with a radius to the vehicle’s path of 2000 ft and a 60 mph design speed. Determine the distance that must be cleared from the inside edge of the inside lane to provide sufficient stopping sight distance.
rtV
Gga
g
VSSD 1
21
2
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Superelevation Transition
from the 2001 Caltrans Highway Design Manual
FYI – NOT TESTABLE
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Spiral Curves
No Spiral
Spiral
from AASHTO’s A Policy on Geometric Design of Highways and Streets 2004
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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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