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Geometric Design

CEE 320

Steve Muench

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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

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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 G2

G1 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.

PVI

PVT

PVC: STA 100+00

EL 59 ft.

PVI

PVT

PVC: STA 100+00

EL 59 ft.

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Other Properties

G1

G2

PVI

PVT PVC

x

Ym

Yf

Y

2

200x

L

AY =

800

ALYm=

200

ALYf=

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

G1 G2

PVI

PVT PVC

h2 h1

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

2SSDA

L = ( )A

SSDL2158

2 −=

For SSD < L For SSD > L

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Crest Vertical Curves

• Assuming L > SSD@

2158

2SSD

K =

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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 Policy on Geometric Design of Highways and Streets 2001

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Sag Vertical Curves

G1 G2

PVI

PVT PVC

h2=0 h1

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

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A Policy on Geometric Design of Highways and Streets 2001

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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

∆

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Horizontal Curve Fundamentals

R

T

PC PT

PI

M

E

R

∆

∆/2 ∆/2

∆/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

∆

∆/2 ∆/2

∆/2 L

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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

vv

sgR

WV

gR

WVWfW =

++

α

Fc

W 1 ft

e

≈

Rv

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Superelevation

αααα cossincossin22

vv

sgR

WV

gR

WVWfW =

++

( )αα tan1tan2

s

v

s fgR

Vf −=+

( )efgR

Vfe s

v

s −=+ 12

( )efg

VR

s

v +=

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

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WS

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T Design Manual, M

22

-01

New Table

For Open Highways and Ramps

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WSDOT Design Side Friction Factors

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WS

DO

T Design Manual, M

22

-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

∆s

Obstruction

Ms ( )v

sR

SSD

π180

=∆

DRSSD s

sv

∆=∆=

100

180

π SSD

−=

v

vsR

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

fro

m A

AS

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A Policy on Geometric Design of Highways and Streets 2001

FYI – NOT TESTABLE

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Superelevation Runoff - WSDOT

from the 2005 WSDOT Design Manual, M 22-01

FYI – NOT TESTABLE

New 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.