lec 2 _ vertical alignement

7/29/2019 Lec 2 _ Vertical Alignement

1/31

//

Vertical Alignment

The 1st Procedure in Alignment

was Stationing

Horizontal Alignment

Vertical Alignment


2/31

//

Highway Components

Highway plan and profile

PLAN

PROFILE

Definition:

It is the elevation or the profile of thecenter line of the road

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


3/31

//

1- Gradients

./

% = tan

G = +ve )( G = -ve )(

:

..

..

%-

%.


4/31

//

Maximum Grades for

Urban and Rural Highways

Maximum Grades for

Local Road and Street


5/31

//

2- Vertical Curves

Design Cri ter iaDesign Cri ter ia

1. Provision of minimum sight distance

2. Adequate drainage

3. Comfortable in operation

4. Pleasant appearance

The first criterion is only associated withcrest curves, whereas all four criteria are

associated with sag vertical curves.

Types of Vertical Curves

G1 G2

G1

G2

Crest Vertical Curve Sag Vertical Curve

G1

G2

G2

G1

G1

G2

G1 G2


6/31

//

Vertical Curve Fundamentals

Parabolic function

Constant rate of change of slope

Implies equal curve tangents

cbxaxy ++=2

Vertical Alignment - General

Parabolic shape as applied to vertical curves

y = ax2 + bx + c

Where:

y = roadway elevation at distance x

x = distance from beginnning of

vertical curve

a, b = coefficients that define shape

c = elevation of PVC


7/31

//

Vertical Curve Fundamentals

G1G2

PVI

PVT

PVC

L

L/2

cbxaxy ++= 2

x

Choose Either: G1, G2 in decimal form, L in feet

G1, G2 in percent, L in stations

Source: Iowa DOTDesign Manual

Note: L is measured from here to here

Not here


8/31

//

Crest Vertical Curves

G1 G2

PVI

PVTPVC

h2h1

L

SSD

( )

( )221

2

200 hh

SSDAL

+

= ( )( )

A

hhSSDL

2

21200

2+

=

For SSD < L For SSD > L

Line of Sight

Crest Vertical Curves

Assumptions for design

h1 = drivers eye height = 3.75 ft.

h2 = object height = 0.5 ft.

Simplified Equations

( )

1329

2

SSDAL = ( )

ASSDL

13292 =



9/31

//

Other Properties

K-Value (defines vertical curvature)

The number of horizontal feet needed for a 1%

change in slope

A

LK=

Design Controls for Crest Vertical Curves

from AASHTOsA Policy on Geometric Design of Highways and Streets 2001


10/31

//

Design Controls for Crest Vertical Curves

from

AASHTOsAPolicyonGeometricDesignofHighwaysandStreets2001

Sag Vertical Curves

G1 G2

PVI

PVTPVC

h2=0h1

L

Light Beam Distance (SSD)

( )

( )tan2001

2

Sh

SSDAL

+= ( ) ( )( )

A

SSDhSSDL

tan2002

1+

=


headlight beam (diverging from LOS by degrees)


11/31

//

Sag Vertical Curves

Assumptions for design

h1 = headlight height = 2.0 ft.

= 1 degree

Simplified Equations

( )( )SSD

SSDAL

5.3400

2

+

= ( )( )

+=

A

SSDSSDL

5.34002


Other Properties

G1

G2

PVI

PVTPVC

x

ym

yf

y

2

200x

L

Ay =

800

ALym =

200

ALyf =

21GGA =

G1, G2 in percent

L in feet


12/31

//

Design Controls for Sag Vertical Curves

from AASHTOsA Policy on Geometric Design of Highways and Streets 2001

Design Controls for Sag Vertical Curves

from

AASHTOsAPolicyonGeometricDesignofH

ighwaysandStreets2001


13/31

//

ExampleA 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.


14/31

//

Example 1A 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.

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 driverto be able to stop before hitting the tree?


15/31

//

Example 3A 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?

Source:A Policy on Geometric

Design of Highways and Streets

(The Green Book). Washington, DC.

American Association of State

Highway and Transportation

Officials,2001 4th Ed.P. 284

Coordination of Vertical and Horizontal

Alignment

Curvature and grade should be in properbalance

Avoid

Excessive curvature to achieve flatgrades

Excessive grades to achieve flat

curvature Vertical curvature should be coordinated

with horizontal

Sharp horizontal curvature should not beintroduced at or near the top of apronounced crest vertical curve

Drivers may not perceive change inhorizontal alignment esp. at night

Image source:

http://www.webs1.uidaho.edu/niatt_labmanual/Chapters/ge

ometricdesign/theoryandconcepts/DescendingGrades.htm
http://www.webs/http://www.webs/


16/31

//

Source:A Policy

on Geometric

Design ofHighways and

Streets (The

Green Book).

Washington, DC.

American

Association of

State Highway

and

Transportation

Officials,2001 4th

Ed.

Coordination of Vertical and Horizontal

Alignment

Sharp horizontal curvature should not be

introduced near bottom of steep grade

near the low point of a pronounced sag

vertical curve

Horizontal curves appear distorted

Vehicle speeds (esp. trucks) are highest at the

bottom of a sag vertical curveCan result in erratic motion

Coord inat ion of hor izontal and vert ical al ignm ent

shou ld begin wi th pre l iminary design

Easier to make adjus tments at this stage

Designer should stud y long, cont inuous stre tches of

Coordination of Horizontal and Vertical

Alignment

highw ay in both p lan and

prof i le and vis ual ize the

wh ole in three

dimensions


17/31

//


Alignment

Source: FHWA,

Chapter 5


Alignment

Source: FHWA,

Chapter 5

Should be consistent with the

topography

Preserve developed properties along

the road

Incorporate community values

Follow natural contours of the land


18/31

//

Good Coordination of Horizontal and

Vertical Alignment

Source: FHWA,

Chapter 5

Does not affectaesthetic, scenic,historic, and culturalresources along theway

Enhances attractivescenic viewsRivers

Rock formations

Parks

Historic sites

Outstanding buildings

Source:A Policy

on Geometric

Design of

Highways and

Streets (The

Green Book).

Washington, DC.

American

Association of

State Highway

and

Transportation

Officials,2001 4th

Ed.


19/31

//

Source:A Policy

on GeometricDesign of

Highways and

Streets (The

Green Book).

Washington, DC.

American

Association of

State Highway

and

Transportation

Officials,2001 4th

Ed.

Source:A Policy

on Geometric

Design of

Highways and

Streets (The

Green Book).

Washington, DC.

American

Association of

State Highway

and

Transportation

Officials,2001 4th

Ed.

There are 2 problems with this alignment.

What are they?


20/31

//

Source:A Policy

on GeometricDesign of

Highways and

Streets (The

Green Book).

Washington, DC.

American

Association of

State Highway

and

Transportation

Officials,2001 4th

Ed.

Source:A Policy

on Geometric

Design of

Highways and

Streets (The

Green Book).

Washington, DC.

American

Association of

State Highway

and

Transportation

Officials,2001 4th

Ed.

Maybe we want this if we are trying to slow people down???


21/31

//

Source:A Policy

on Geometric


Streets (The

Green Book).

Washington, DC.

American

Association of

State Highway

and

Transportation

Officials,2001 4th

Ed.

Source:A Policy

on Geometric

Design of

Highways and

Streets (The

Green Book).

Washington, DC.

American

Association of

State Highway

and

Transportation

Officials,2001 4th

Ed.


22/31

//

Source:A Policy

on Geometric


Streets (The

Green Book).

Washington, DC.

American

Association of

State Highway

and

Transportation

Officials,2001 4th

Ed.

Source:A Policy

on Geometric

Design of

Highways and

Streets (The

Green Book).

Washington, DC.

American

Association of

State Highway

and

Transportation

Officials,2001 4th

Ed.


23/31

//

Source:A Policy

on Geometric


Streets (The

Green Book).

Washington, DC.

American

Association of

State Highway

and

Transportation

Officials,2001 4th

Ed.

Source:A Policy

on Geometric

Design of

Highways and

Streets (The

Green Book).

Washington, DC.

American

Association of

State Highway

and

Transportation

Officials,2001 4th

Ed.

A

B


24/31

//

Source:A Policy

on Geometric

Design of

Highways and

Streets (TheGreen Book).

Washington, DC.

American

Association of

State Highway

and

Transportation

Officials,2001 4th

Ed.

Sag Vertical Curves

Sight distance is governed by nighttime

conditions

Distance of curve illuminated by headlights

need to be consideredDriver comfort

Drainage

General appearance


25/31

//

Vertical Curve AASHTO Controls (Sag)

Headlight Illumination sight distance with S < L

S < L

L = AS2

S > L

L = 2S (400 + 3.5S)A

Headlight Illumination sight distance with S > L

Source:A Policy on Geometric

Design of Highways and Streets

(The Green Book). Washington, DC.

American Association of State

Highway and Transportation

Officials,2001 4th Ed.

400 + (3.5 * S)

Sag Vertical Curve: Example

A sag vertical curve is to be designed to join a 3% to a

+3% grade. Design speed is 40 mph. What is L?

Skipping steps: SSD = 313.67 feet S > L

Determine whether SL

L = 2(313.67 ft) (400 + 2.5 x 313.67) = 377.70 ft

[3 (-3)]

313.67 < 377.70, so condition does not apply


26/31

//

Sag Vertical Curve: ExampleA sag vertical curve is to be designed to join a 3% to a


Skipping steps: SSD = 313.67 feet

L = 6 x (313.67 ft)2 = 394.12 ft

400 + 3.5 x 313.67 ft

313.67 < 394.12, so condition applies

Sag Vertical Curve: Example

A sag vertical curve is to be designed to join a 3% to a



Testing for comfort:

L = AV2 = (6 x [40 mph]2) = 206.5 feet

46.5 46.5


27/31

//

Sag Vertical Curve: ExampleA sag vertical curve is to be designed to join a 3% to a



Testing for appearance:

L = 100A = (100 x 6) = 600 feet

Example: A crest vertical curve joins a +3% and 4% grade.

Design speed is 75 mph. Length = 2184.0 ft. Station at PVI is

345+ 60.00, elevation at PVI = 250 feet. Find elevations and

station for PVC and PVT.

L/2 = 1092.0 ft

Station at PVC = [345 + 60.00] - [10 + 92.00] = 334 + 68.00

Distance to PVC: 0.03 x (2184/2) = 32.76 feet

ElevationPVC = 250 32.76 = 217.24 feet

Station at PVT = [345 + 60.00] + [10 + 92.00] = 357 + 52.00

Distance (vertical) to PVT = 0.04 x (2184/2) = 43.68 feet

Elevation PVT = 250 43.68 = 206.32 feet


28/31

//

Example: A crest vertical curve joins a +3% and

4% grade. Design speed is 75 mph. Length = 2184.0ft. Station at PVI is 345+ 60.00, elevation at PVI =

250 feet. Station at PVC is 334 + 68.00, Elevation at

PVC: 217.24 feet.

Calculate points along the vertical curve.

X = distance from PVC

y = Ax2

200 L

Elevationtangent = elevation at PVC + distance x grade

Elevationcurve = Elevationtangent - y

Example: A crest vertical curve joins a +3% and 4%

grade. Design speed is 75 mph. Length = 2184.0 ft.

Station at PVI is 345+ 60.00, elevation at PVI = 250 feet.

Find elevation on the curve at a point 400 feet from PVC.

Y = A x 2 = 6 x (400 ft)2 = 4.40 feet

200L 200 (2814)

Elevation at tangent = 217.24 + (400 x 0.03) = 229.24

Elevation on curve = 229.24 4.40 feet = 224.84


29/31

//

Tunnel floor elevation 100

ft

Bridge deck elevation 126.67 ft

Station 0+00

(PVC of Sag)

Station 12+00

(PVT of Crest)


30/31

//

Critical Lengths of Grade for Design

The term critical length of grade is used toindicate he maximum length of a designatedupgrade on which a loaded truck can operatewithout an unreasonable reduction in speed.

In the past, the general practice has been to usea reduction in truck speed of 25 km/h below theaverage running speed of all traffic to identify the

critical length of grade. It is recommended that a 15 km/h reduction be

used.

Critical Lengths of Grade for Design

The length of any given grade that will cause the speed of arepresentative truck (120 kg/kw entering the grade at 110 km/h to bereduced by various amounts below the average running speed.Exhibit 3-63

Where an upgrade is approached on a momentum grade, heavy

trucks often increase speed on order to make the climb. increase ofabout 10 km/h can be considered for moderate downgrades and aspeed increase of 15 km/h for steeper grades of moderate length orlonger.

The critical of grade in Exhibit 3-63 is derived as the length oftangent grade.

Some downgrades are long and steep enough that some heavyvehicles travel at crawl speeds to avoid loss of control on the grade.


31/31

//

SOME DECISIONS ARE EASY

lec 2 _ vertical alignement

Documents