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CEE 320 Spring 2008 Geometric Design CEE 320 Anne Goodchild.

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

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Geometric Design

CEE 320Anne Goodchild

Page 2: CEE 320 Spring 2008 Geometric Design CEE 320 Anne 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

Page 3: CEE 320 Spring 2008 Geometric Design CEE 320 Anne Goodchild.

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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

Page 5: CEE 320 Spring 2008 Geometric Design CEE 320 Anne Goodchild.

From Perteet Engineering

Page 6: CEE 320 Spring 2008 Geometric Design CEE 320 Anne Goodchild.

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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

Page 8: CEE 320 Spring 2008 Geometric Design CEE 320 Anne Goodchild.

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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

Page 10: CEE 320 Spring 2008 Geometric Design CEE 320 Anne Goodchild.

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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.

Page 12: CEE 320 Spring 2008 Geometric Design CEE 320 Anne Goodchild.

G1=2.0%

G2= -4.5%

PVI

PVT

PVC: STA 100+00EL 59 ft.

Page 13: CEE 320 Spring 2008 Geometric Design CEE 320 Anne Goodchild.

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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

Page 14: CEE 320 Spring 2008 Geometric Design CEE 320 Anne Goodchild.

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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

Page 15: CEE 320 Spring 2008 Geometric Design CEE 320 Anne Goodchild.

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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

Page 16: CEE 320 Spring 2008 Geometric Design CEE 320 Anne Goodchild.

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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

Page 17: CEE 320 Spring 2008 Geometric Design CEE 320 Anne Goodchild.

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Crest Vertical Curves

• Assuming L > SSD…

2158

2SSDK

Page 18: CEE 320 Spring 2008 Geometric Design CEE 320 Anne Goodchild.

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Design Controls for Crest Vertical Curves

from AASHTO’s A Policy on Geometric Design of Highways and Streets 2004

Page 19: CEE 320 Spring 2008 Geometric Design CEE 320 Anne Goodchild.

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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

Page 20: CEE 320 Spring 2008 Geometric Design CEE 320 Anne Goodchild.

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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)

Page 21: CEE 320 Spring 2008 Geometric Design CEE 320 Anne Goodchild.

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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

Page 22: CEE 320 Spring 2008 Geometric Design CEE 320 Anne Goodchild.

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Sag Vertical Curves

• Assuming L > SSD…

SSD

SSDK

5.3400

2

Page 23: CEE 320 Spring 2008 Geometric Design CEE 320 Anne Goodchild.

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Design Controls for Sag Vertical Curves

from AASHTO’s A Policy on Geometric Design of Highways and Streets 2004

Page 24: CEE 320 Spring 2008 Geometric Design CEE 320 Anne Goodchild.

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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

Page 25: CEE 320 Spring 2008 Geometric Design CEE 320 Anne Goodchild.

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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?

Page 26: CEE 320 Spring 2008 Geometric Design CEE 320 Anne Goodchild.

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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

Page 27: CEE 320 Spring 2008 Geometric Design CEE 320 Anne Goodchild.

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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

Page 28: CEE 320 Spring 2008 Geometric Design CEE 320 Anne Goodchild.

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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?

Page 29: CEE 320 Spring 2008 Geometric Design CEE 320 Anne Goodchild.

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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

Page 30: CEE 320 Spring 2008 Geometric Design CEE 320 Anne Goodchild.

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Crest Vertical Curves

G1G2

PVI

PVTPVC

h2h1

L

SSD

Line of Sight

Page 31: CEE 320 Spring 2008 Geometric Design CEE 320 Anne Goodchild.

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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?

Page 32: CEE 320 Spring 2008 Geometric Design CEE 320 Anne Goodchild.

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Horizontal Alignment

Page 33: CEE 320 Spring 2008 Geometric Design CEE 320 Anne Goodchild.

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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

Page 34: CEE 320 Spring 2008 Geometric Design CEE 320 Anne Goodchild.

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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)

Page 35: CEE 320 Spring 2008 Geometric Design CEE 320 Anne Goodchild.

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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

Page 36: CEE 320 Spring 2008 Geometric Design CEE 320 Anne Goodchild.

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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?

Page 37: CEE 320 Spring 2008 Geometric Design CEE 320 Anne Goodchild.

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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

Page 38: CEE 320 Spring 2008 Geometric Design CEE 320 Anne Goodchild.

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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

Page 39: CEE 320 Spring 2008 Geometric Design CEE 320 Anne Goodchild.

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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

Page 40: CEE 320 Spring 2008 Geometric Design CEE 320 Anne Goodchild.

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Minimum Radius Tables

Page 41: CEE 320 Spring 2008 Geometric Design CEE 320 Anne Goodchild.

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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

Page 42: CEE 320 Spring 2008 Geometric Design CEE 320 Anne Goodchild.

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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

Page 43: CEE 320 Spring 2008 Geometric Design CEE 320 Anne Goodchild.

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Design Superelevation Rates - AASHTO

from AASHTO’s A Policy on Geometric Design of Highways and Streets 2004

Page 44: CEE 320 Spring 2008 Geometric Design CEE 320 Anne Goodchild.

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Design Superelevation Rates - WSDOT

from the 2005 WSDOT Design Manual, M 22-01

emax = 8%

Page 45: CEE 320 Spring 2008 Geometric Design CEE 320 Anne Goodchild.

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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?

Page 46: CEE 320 Spring 2008 Geometric Design CEE 320 Anne Goodchild.

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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

Page 47: CEE 320 Spring 2008 Geometric Design CEE 320 Anne Goodchild.

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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

Page 48: CEE 320 Spring 2008 Geometric Design CEE 320 Anne Goodchild.

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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

Page 49: CEE 320 Spring 2008 Geometric Design CEE 320 Anne Goodchild.

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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

Page 50: CEE 320 Spring 2008 Geometric Design CEE 320 Anne Goodchild.

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Superelevation Transition

from the 2001 Caltrans Highway Design Manual

FYI – NOT TESTABLE

Page 51: CEE 320 Spring 2008 Geometric Design CEE 320 Anne Goodchild.

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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

Page 52: CEE 320 Spring 2008 Geometric Design CEE 320 Anne Goodchild.

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No Spiral

FYI – NOT TESTABLE

Page 53: CEE 320 Spring 2008 Geometric Design CEE 320 Anne Goodchild.

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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

Page 54: CEE 320 Spring 2008 Geometric Design CEE 320 Anne Goodchild.

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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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