3. Loads and Load Distribution

12007 PCA Bridge Professors' Seminar

LOADS AND LOAD DISTRIBUTION

Harry A. Cole, PhD, PE Department of Civil Engineering

Mississippi State University

A t 2 d 3 2007

August 2007

H.A.ColeMiss.State.Univ.

1

August 2 and 3, 2007

Material properties

Loads Available h

Structural "Analysis" / "Design" Overview

Load combinations

shapes

Philosophy/methodology LRFD, ASD, LFD, etc.

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Models

"Analysis" "Design"

OBJECTIVE: To assure the safe and economical

design of bridge structures

LOADS RESISTANCEThe effect of loads The structure's resistance

on the structure to those loads

In order for this to work both sides of the statement must

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In order for this to work, both sides of the statement must refer to the same condition. For any particular load effect, the resistance must be the resistance to that effect.

LIMIT STATES

A limit state is a condition beyond which a system (or a component

of a system) ceases to fulfill the function for which it was designed.

The system or component is loaded beyond its capability to resist.

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2Types and examples of common limit states

Type: Consequence: Example:

STRENGTH Collapse Exceed crushing strength of concrete Exceed breaking strength of PS strands Buckling of compression component Fatigue failure of component

SERVICE Unacceptablebehavior not involving

collapse

Excessive deflection at working loads Cracking of PS concrete beams

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Slip of steel bolted connectionsOther "improper behavior" Excessive foundation settlement

Squashing of bearing pads

AASHTO designation Limit state objective Loads

Li it i t i i d F ll l f i

AASHTO LRFD limit states most applicable to prestressed-girder/slab bridge girders

Service ILimit compressive stress in girder

and deck to maintain adequate

factor of safety against concrete

crushing

Full value of service

( unfactored ) dead and

live loads

Service IIILimit tensile stress in girder to

maintain factor of safety against

concrete tension cracking

Full service dead load, but

reduced service live load

Provide adequate resistance to Factored live and dead

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Strength Iq

girder "breaking" failure loads

FatigueLimit stresses caused by repetitive

vehicle live load

Loads produced by "fatigue

truck"

"Perhaps the most difficult part of any structural

Regardless of limit state:

design is determining the design loads ..... "

Anonymous

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LOADS AND LOAD COMBINATIONS

LOAD MODELS

Regardless of structure type, models are used to define design loads:

Buildings: ASCE 7 - Minimum Design Loads for Buildings and Other Structures ASCE 7 is a standard that is referencedOther Structures . ASCE 7 is a standard that is referenced in all major material performance specifications ( ACI, AISC, NDS, etc ) and building codes ( IBC 2006, etc. )

Examples:

Dead load - volume density

Live load - load per unit area; concentrated loads

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Forces of nature - replace wind effect, seismic effect, etc. with "equivalent" static loads

3Bridges: AASHTO LRFD Bridge Design Specifications. Containsboth load models and material performance criteria

LOAD MODELS ( Continued )

The following pages will look at AASHTO:

Load classifications

Models used to define loads

Load combinations

Application of load effects to components

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Application of load effects to components

AASHTO LRFD LOAD DEFINITIONS AND CLASSIFICATIONS

S3.3.2

Permanent loads

DD = downdrag DC = dead load of structural components and

non-structural attachmentsDW = dead load of wearing surfaces and utilitiesEL = accumulated locked-in force effects resulting

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from construction processEH = horizontal earth pressure loadEV = vertical pressure from dead load on earth fill

Transient loadsBR = vehicle breaking force CE = vehicular centrifugal forceCR = creepCT = vehicular collision forceCV = vessel collision forceEQ = earthquake forceFR = frictionIC = ice loadIM = impactLL = vehicular live loadLS = live load surchargePL = pedestrian live loadSE = settlementSH = shrinkage

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SH = shrinkageTG = thermal gradientTU = uniform temperatureWA = water load and stream pressureWL = wind on live loadWS = wind on structure

Loads normally used for designing prestressed girder / slab bridge superstructures:

Permanent loads ( "dead loads" ) :

DC loads - Components and attachments whose weights can becomputed with reasonable accuracycomputed with reasonable accuracy

Girder

Slab, haunch, stay-in-place forms

Diaphragm

Railings ( "parapet" ) / barriers

act on girder

acts on composite section

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DW loads - Components and attachments whose weights can notbe determined as accurately as DC loads

Future wearing surface ( FWS )

Utilities and other future loadsact on composite section

4slabSIP form haunch

Service I, III (elastic analyses)

Use girder section modulii to compute stresses in girder due

DC / DW dead loads

girder

diaphragm

compute stresses in girder dueto these loads, plus prestress

Strength I - DC loads

Use load factor = 1.25

FWS

railingService I, III (elastic analyses)

Use composite section modulii

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to compute stresses in girderand slab

Strength I - DW loads

Use load factor = 1.50

Transient loads - AASHTO S3.6.1.2.1

Loads normally used for designing prestressed girder / slab bridge superstructures ( continued ):

Include: Vehicular loads ( "live loads" )

Forces of nature

Extreme events ( catestrophic loads, such as truck-railing collisions )

The rest of this presentation will look at the AASHTO models used to

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The rest of this presentation will look at the AASHTO models used to

define and apply vehicular ("live") loads to girders and slabs, and

truck-rail collision extreme event loads.

"NOTIONAL" LOADS

AASHTO uses the concept of notional loads to define model live loads:

Notional loads are ficticious ("model") loads that have been created to produce the same load effects ( bending moment, shear ) as observed in real bridges caused by real traffic. g y

The AASHTO notional loads have been calibrated (optimized) based on strength. Thus, use of these loads in a girder Strength I analysis gives results that most closely match those that would produce strength failurein real bridge components using factored real traffic loads.

The notional loads also happen to give girder compressive stresses at service loads that reasonably match those produced by real traffic on real bridges. Therefore, the stresses produced by the notional loads are used i th S i I l ti l i

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in the Service I elastic analysis

But, the notional loads predict girder tensile stresses at service loads that are greater than those produced by real traffic on real bridges. Therefore, the stresses produced by the notional loads are adjusted ( multiplied by 0.80 ) for use in the Service III elastic analysis.

Design Truck Load ( S3.6.1.2.2)

32 kips

8 kips

32 kips

Notional vehicular loads ( S3.6.1.2.1 )

Design Tandem Load ( S3.6.1.2.3 )

14' to 30' 14'

4'

25 kips 25 kips

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Design Lane Load ( S3.6.1.2.4 )

0.64 kips / foot

5The Design Truck and Design Tandem loads are axle loads:

32 kips

32 kips

8 kitrailing axle

25 kips

25 ki

8 kips

travel direction14' - 30'

14'

6'

leading axleDesign Truck

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25 kips

4' 6'

Design Tandem

Why the variable spacing between center and trailing axles?

Simply-supported spans:

Strength and service limit states: 14' spacing ( loads closely grouped ) produces greatest design truck load moment, shear and deflection ( S3 6 1 2 2 )deflection ( S3.6.1.2.2 )

Fatigue limit state: 30' axle spacing ( S3 6 1 4 1 )

32 32 8 kips

14' 14'

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Fatigue limit state: 30 axle spacing ( S3.6.1.4.1 )

32 32 8 kips

30' 14'

Continuous spans: Example - truck load placement to cause maximum negative moment at center support in a two-span continuous bridge ( Service or Strength limit states )

50'32k 32k 8k 32k 32k 8k

14' 14' 14' 14'

32k 32k 8k50' 50'

long spans

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30' 14'short spans

Application of vehicular live loads: Service and Strength limit states ( S3.6.1.3 )

Design Truck plus Design Lane

OROR

Design Tandem plus Design Lane

Use whichever causes greater load effect

32 kip 32 kip 8 kip 25 kip 25 kip

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0.64 k/ft 0.64 k/ft

Note that the design lane load is not interrupted to "provide space" for the axle loads

6Application of vehicular live loads: Fatigue limit state; Impact

Fatigue (S3.6.1.4.1): Apply to Fatigue Truck only ( do not use lane load )

32 32 8 kips

30' 14'

Impact - "Dynamic impact allowance" (S3.6.2): Applies to truck / tandem loads only ( does not apply to lane load )

From AISC Table 3.6.2.1-1: IM

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All components except deck joints:

Service and Strength limit states: 33% Fatigue limit state: 15%

Deck joints (all limit states ): 75%

DESIGN LANE WIDTH

The Design Lane loads are applied over a10-foot lane width. The Design Truck load and the Design Tandem load occur anywhere within a 10-foot lane width:

32 kips32 kips32 kips

8 kips6'

10' lane

6'

10'

Design Truck Load or Design Tandem

( S3.6.1.3.1 )

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10' lane

0.64 k/ftDesign Lane Load

( S3.6.1.2.4 )

How are model vehicular live loads used to produce design live load shear and moment diagrams in a typical bridge girder ?

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USING MODEL VEHICULAR LIVE LOADS TO PRODUCE LIVE LOAD SHEAR AND MOMENT DIAGRAMS FOR INDIVIDUAL GIRDERS

A two-step process:

Step 1 - Use model vehicular live loads to draw moment and shear"diagrams" for imaginary 10'-wide bridge:

Moment diagram for Lane load: MLane

Shear envelope for Lane load: VLane

Moment/shear envelopes for Truck load: MTruck , VTruck( Similar for Dual Tandem load )

Apply impact factor (IM) to truck moment/shear, then combine:

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MLL = MLane + IM MTruck

VLL = VLane + IM VTruck

Step 2 - AASHTO Simplified Method: Use moment and shear distribution factors to obtain moment and shear "diagrams" for individual girders

7Step 1: Live load moment diagrams and envelopes

Simply-supported single-span bridge

32 32 8 kips

14' 14'

ctr. of brg.

14' 14'0.64 kips/ft

L ( ctr. of brg. )

Lane load moments - Computed at 1/10th points ( 1 i 11 ):

L

Lw 1Lw2

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= )xL(2Lw

)1i(

1011)1i(

20Lw

i = 1 i = 6 i = 11xi

i

Simply-supported single-span bridge ( continued )

Truck load moment envelope ( Similar for Dual Tandem load ):

Obtain moment envelope by computing the maximum moment at each 1/10th point caused by "marching" axles through that point ( use symmetry to obtain moments that would be found if truck were run across the bridge in the opposite direction )

32 32 8 kips32 32 8 kips

14' 14'

32 32 8 kips

14' 14'

32 32 8 kips

i

Compute: Mi1

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32 32 8 kips

14' 14'

32 32 8 kips

14' 14'

Compute: Mi2

Compute: Mi3

Use largest: Mi


32 32 8 kips

14' 14'

8 32 32 kips

14'14'

i = 6i = 1 i = 11

M1

M2M3

M4M5 M6 M7 M8

M9M10

M11

Left-to-right Right-to-left

Using left-to-right travel only: M2(env) = M10(env) = larger ( M2 , M10 ) , etc.

M M MTruck load moment envelope

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M1

M2

M3

M4

M5

M6

M7

M8

M9 M10

M11

The moment envelope "looks like" ( and is used like ) a moment diagram. The midspan moment M6is within 1% of the "absolute maximum moment" ; the longer the span, the smaller this difference.


32 32 8 kips

14' 14'0.64 kips/ft

LL

Lane load shear envelope - Computed at 1/10th points ( 1 i 6 ):

Vi =2

101i1

2Lw

i 1 i i = 6 i = 11

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i = 1 i 6 i = 11xi

2Lw

V )env(i 1.0 0.81 0.64 0.49 0.36 0.25

i = 1 i = 6

8Simply-supported single-span bridge ( continued )

Truck load shear envelope ( Similar for Dual Tandem load ):

Use shear influence lines to compute maximum shear at each 1/10th point

32 32 8 kips

32 32 8 kips

14' 14'V1

32 32 8 kips

14' 14'i = 1

i = 2

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32 32 8 kips

14' 14'V2


Truck load shear envelope ( Similar for Dual Tandem load )

32 32 8 kips

14' 14'

i = 1 i = 6 i = 11

V1 V2 V3 V4V5

Truck load shear envelope

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V6

Simply-supported single-span bridge: Numerical example

32 32 8 kips

14' 14'0.64 kips/ft

Vehicle load moments

140'

i = 1 i = 6 i = 11

564.51003.5 1317.1

1505.3 1568.0 sym.

Lane load moment diagram: MLane

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840.01478.4

1915.22172.8 2240.0 sym.

i = 1 i = 6 i = 11

Truck load moment envelope: MTruckSee next page for computations

( Similar for Dual Tandem load )

32 32 8 kips

14' 14'

Simply-supported single-span bridge: Numerical example ( continue )

Truck load moment envelope computations ( Similar for Dual Tandem load )

140'

i x Position 1 Position 2 Position 3 Maximum Envelopeft-k

1 0.00 M1 = 0.00 M2 = 0.0 M3 = 0.0 MM = 0.0 0.02 14.00 M1 = 100.80 M2 = 492.8 M3 = 840.0 MM = 840.0 840.03 28.00 M1 = 537.60 M2 = 1232.0 M3 = 1478.4 MM = 1478.4 1478.4

2 00 1 11 6 00 2 1 69 6 3 191 2 191 2 1915 2

Left-to-right travel

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4 42.00 M1 = 1176.00 M2 = 1769.6 M3 = 1915.2 MM = 1915.2 1915.25 56.00 M1 = 1612.80 M2 = 2105.6 M3 = 2150.4 MM = 2150.4 2172.86 70.00 M1 = 1848.00 M2 = 2240.0 M3 = 2184.0 MM = 2240.0 2240.07 84.00 M1 = 1881.60 M2 = 2172.8 M3 = 2016.0 MM = 2172.8 2172.88 98.00 M1 = 1713.60 M2 = 1904.0 M3 = 1646.4 MM = 1904.0 1915.29 112.00 M1 = 1344.00 M2 = 1433.6 M3 = 1075.2 MM = 1433.6 1478.410 126.00 M1 = 772.80 M2 = 761.6 M3 = 403.2 MM = 772.8 840.011 140.00 M1 = 0.00 M2 = 0.0 M3 = 0.0 MM = 0.0 0.0

9Simply-supported single-span bridge: Numerical example ( continued )

32 32 8 kips14' 14'

0.64 kips/ft

140'

MLL= live load moment

Similar for Dual Tandem + Lane loads

i = 1 i = 6 i = 11

564.51003.5 1317.1 1505.3 1568.0 sym.

1117.21966.3

2547.22889.8 2979.2 sym.

i = 1 i = 6 i = 11

MLane

IM MTruck = 1.33 MTruck

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i = 1 i = 6 i = 11

1681.72969.8

3864.34395.1

4547.2sym.

i = 1 i = 6 i = 11

MLL = MLane + 1.33 MTruck

Simply-supported single-span bridge: Numerical example ( continued )

32 32 8 kips

14' 14'0.64 kips/ft

Vehicle load shears:

140'

Lane load shear envelope: VLane

i = 1 i = 6 i = 11

44.8 36.3 26.7 21.9 16.1 mirror image sym.11.2

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Truck load shear envelope: VTruckSee next page for computations

( Similar for Dual Tandem load )

i = 1 i = 6 i = 11

67.2

mirror image sym.

60.0 52.8 45.6 38.4 31.2

Truck load shear envelope example ( Similar for Dual Tandem load )


32 32 8 kips

14' 14'

140'i = 1

i = 6

i = 11

V1 = 67.2 k

V2 = 60.0 k

V3 = 52.8 k

i 1

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V4 = 45.6 k

V5 = 38.4 k

V6 = 31.2 k


32 32 8 kips14' 14'

0.64 kips/ft

140'

VLL= live load shear

Similar for Dual Tandem + Lane loads

VLane

89.4

mirror image sym.

79.8 70.2 60.6 51.1 41.5IM VTruck = 1.33 VTruck

i = 1 i = 6 i = 11

44.8 36.3 26.7 21.9 16.1 mirror image sym.11.2

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i = 1 i = 6 i = 11

VLL = VLane + 1.33 VTruck

i = 1 i = 6 i = 11

134.2 116.196.9

mirror image sym.

82.5 67.252.7

10

Step 1: Live load moment diagrams and envelopes ( continued )

Multi-span indeterminate bridges

32 32 8 kips

14' 14'14' 14'0.64 kips/ft

L1 L2

Loads:

Truck load ( similar for tandem load ) - trucks on one or more spans

L l d l d ll l t d t

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Lane load - load on all or selected span segments

Computer analysis generally required:

Recommend QConBridge*, available at no cost from the Washington State DOT:

www.wsdot.wa.gov/eesc/bridge/software/index.cfm?fuseaction=download&software_id=48

Multi-span indeterminate bridges - overview ( continued )

For for negative moment between points of contraflexure caused by a uniform load on all spans, and reactions at interior piers only, use :

Application of Vehicular Live Loads:

Case 1 ( S3.6.1.3.1 )

90 percent of the effect of two design trucks spaced a minimum of 50.0 ft. between the lead axle of one truck and the rear axle of the other truck ( the distance between the 32-kip axles of each truck shall be taken as 14.0 ft.) ,

PLUS

90 percent of the effect of the design lane load

Case 2 ( not stated in S3.6.1.3.1 ) :

100 t f d i t k ( i b t 32 ki l )

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100 percent of one design truck ( vary spacing between 32-kip axles ),

PLUS

100 percent of the design lane load

For all other effects, use one truck per span plus lane load.

32 32 8 kips

14' 14'


Use the two-span continuous bridge shown below to illustrate these requirements:

14 140.64 kips/ft

L L

For the uniformly-distributed load on both spans:

Points of contraflexure

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0.25L 0.25L

Region 2

Region 1


Positive moment ( same for Regions 1 and 2 ):

Example: Mx+ ( 0 x L )

32 k 32 k 8 k 8 k 32 k 32 k

xL

Influence line for Mx

32 k 32 k 8 k

14' 14' 14' 14'

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Positive live load moment:

1.33 maximum moment produced by moving truck through IL-peak ( both directions )

PLUS

moment caused by uniform load over full span length L

11


Negative moment ( Region 1 ):

Example: Mx- ( 0 x 0.75L )

Influence line for M 32 k 32 k 8 k 8 k 32 k 32 k

xL

Influence line for Mx 32 k 32 k 8 k

14' 14' 14' 14'

L

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Negative live load moment:

1.33 maximum moment produced by moving truck ( both directions )

PLUS




Example: ML- ( 0.75L < x L )

Case 1

x = L

Influence line for Mx=L

32 k 32 k 8 k

14' 14'

L

32 k 32 k 8 k

14' 14'spacing 50'

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1.33 maximum moment produced by moving truck train ( vary spacing )

90% PLUS




Example: ML- ( 0.75L < x L )

32 k 32 k 8 k

14' - 30' 14'Case 2

x = L

Influence line for Mx=L

L


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1.33 maximum moment produced by truck ( vary axle spacing )

100% PLUS


Note: The only time that this case may control the negative support moment and/or the interior pier reaction is if the two spans L are very short.

Multi-span indeterminate bridges: Example

14' 14'

32 k 32 k 8 k

Positivemoment

MT1 = moment caused byone design truck

2-span continuous bridge span

14' 14'

0.64 k/foot

Vary spacingfrom 50' to140' in tensteps of 9'

32 k 32 k 8 k 32 k 32 k 8 k

EA B C D

Negative momentin BCD:

90% *( 1.33MT2

+ MLANE )

momentin ABCDE

andNegativemoment inAB and DE:

1.33MT1+ MLANE

MT2 = moment caused bytwo design trucks

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A B C D

0.25L 0.25L

140' 140'

QConBridge moment envelope on next slide

12

Multi-span indeterminate bridges: Example ( continued )

From QConBridge:

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Step 2 - Girder moments and shears

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The Design Truck ( or, alternately, the Design Tandem ) and the Design Lane loads are defined to act in a 10-ft-wide Design Lane. They do not account for:

Girder moments and shears

Where the design lane is placed within the roadway width of the bridge

Where the design lane is placed relative to the girders

The number of lanes that fit within the roadway width of the bridge

The probability that two or more adjacent lanes will be loaded simultaneously

The ability of the bridge deck to laterally distribute the load in one or more lane(s) to more than one girder

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or more lane(s) to more than one girder

Distributing lane loads to girders depends on several things:

Girder spacing

Gi d l t thGirders close together -

shorter direct load path to girders; stiffer slab more girders involved

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Girders far apart -

longer load path to girders; more flexible slab fewer girders involved

13

Load position relative to girders:

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Slab stiffness - ability to transfer loads to adjacent girders

Very stiff slab - load is distributed equally to girders

Very flexible slab - load is carried by only one girder

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Usual case- load is distributed between girders, but girders under load carry greatest share.

Girder flexural and torsional stiffness - functions of girder length, moment of inertia ( flexure ) and area (torsion ):

Long girders are more flexible than short girders, which tends to increase load distribution between girders

Girders with small moments of inertia deflect vertically more than girders with large moments of inertia, which tends to increase load distribution between girders

Girders with small areas twist more than girders with large areas, which tends to increase load distribution between girders

Example: Single load symetrically placed over interior girder

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deflect + twist

deflect + twist

deflect

Number of adjacent loaded lanes:

The AASHTO loading model assumes that there can be distribution of vehicles on a bridge at any given time ( a "vehicle" is represented by a combination of a truck or tandem load, plus a lane load ):

The Design Vehicle loads ( A ) are the nominal ( reference ) loads

There can be an occasional single vehicle load ( B ) greater than the Design Vehicle loads

Some vehicle loads ( C, D ) will be less than the Design Vehicle loads

A = Design Vehicle loadB

A AAA

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A AAC

DC

A

D DC

14

AA

D C

CC

C

DCD

A

A

A

Two adjacent loads @ A

D A

Single load B = 1.2A

AD A

C

C

D

Multiple presence factor for adjacent loaded lanes:

A D

DAC

DC A C

Four adjacent loads @ 0.65A

DA

C

C

C

D

D

CA

CD A C

C

Three adjacent loads @ 0.85A

DD

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CCD

C A CD A C

C

CC D

D

D C

C

DD

D

CC

B = 1.20 AA = Design Value ( reference )

Multiple presence factor for adjacent loaded lanes - AASHTO load model:

C = 0.85 A D = 0.65 A

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The AASHTO Specification ( S3.6.1.1.2 ) uses multiple presence factors to account for the probability that vehicles of these four load classes will occur in adjacent lanes.

Multiple presence factors :

Table 3 6 1 1 2-1 - Multiple Presence Factors m

Number ofloaded lanes

Multiple presencefactors, m

1 1.202 1.003 0.85> 3 0.65

The AASHTO load models assume that there is the same probability that there can be:

One vehicle that is 120% heavier than the Design Vehicle in one lane

Table 3.6.1.1.2 1 Multiple Presence Factors, m

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One vehicle that is 120% heavier than the Design Vehicle in one lane

Two Design Vehicles in two adjacent lanes

Three vehicles that are each 85% of the Design Vehicle load in three adjacent lanes

Four or more vehicles that are each 65% of the Design Vehicle load in adjacent lanes

For most prestressed girder/slab bridges, permits "distribution" of live load per lane moments and shear to girders through the use of distribution factors.

Girder Moments and Shears by the AASHTO "Simplified Method": Distribution Factors

14'14'

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56

15

L l d t di

1. Total live load moment for 10' lane ( from previous slides )

Lane load moment diagram

Truck load moment envelope ( Dual tandem similar )

Total live load moment for 10' lane

MLL = MLane + ( 1 + IM ) MTruck

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Shear similar

Live load moment for 10' lane:

MLL = MLane + ( 1 + IM ) MTruckML(int) = MLL DFM(int)ML(ext) = MLL DFM(ext)

2. Use distribution factors (DF) obtain live load moment or shear in individual girders

"Distribute"ML(int) = design live load

moment for interior girders

ML(ext) = design live load moment for exterior girder

DFM(int) , DFM(ext) = "distribution factors" for moment - AASHTO Tables 4.6.2.2.2b-1, 4.6.2.2.2d-1

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4.6.2.2.2d 1

( DF labeled "g" in Spec.)

Note: "Distribution" assigns a portion of the live load moment MLL to individual girders ( it does not dividethe live load moment between girders ).

The AASHTO Simplified Method ( distribution factors ) may be used when:

The deck width is constant

There are at least four girders

The girders are parallel and have approximately the same stiffnesses

Bridge curvature is limited ( see S4.6.1.2 )

The roadway part of the overhang , de 3.0 ft:

The bridge cross-section is one of those shown in Table 4.6.2.2.1-1

de

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Multiple presence factors used in Distribution Factor tables

The distribution factors include the following multiple presence factors :

Interior girders

m = 1.20m = 1.00

Exterior girders

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m = 1.20m = 1.00

16

Distribution factor tables applicable to prestressed-girder/slab bridges:

Table 4.6.2.2.2b-1 Moments in Interior Beams

Table 4.6.2.2.2d-1 Moments in Exterior Beams

Table 4.6.2.2.3a-1 Shear in Interior Beams

T bl 4 6 2 2 3b 1 Sh i E t i B

Notes on using Distribution Factors from these tables:

Service and Strength limit state analyses ( moment , shear ):

Compute Distribution Factor for both "One Design Lane Loaded" and "Two Design Lanes Loaded"

Use larger DF to compute girder moment or shear

Table 4.6.2.2.3b-1 Shear in Exterior Beams

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Fatigue limit state ( moment only ) - single truck, single lane, multiple presence factor = 1

Obtain lane live load moments for truck only, rear axle spacing = 30'

Compute Distribution Factor for "One Design Lane Loaded" only

Divide computed Distribution Factor by 1.20 to eliminate multiple presence factor

Example:

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The "One Design Lane Loaded" distribution factor includes the 1.20 multiple presence

factor shown earlier for a single loaded lane.

Multiple presence factor reference condition ( multiple presence factor = 1.0 ).

DFM(int) ( moment, interior girder, two lanes loaded ):

2ts

eg = yt + th + 2ts

yt

ts

th

c.g. (girder)

L = girder length, ft

DFM(int) = 0.075 + 1.0

3s

g2.06.0

tL0.12

KLS

5.9S

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63

S = center-center girder spacing, ft ts = slab thickness, in A = girder area , in2 I = girder moment of inertia, in2n = modular ratio ( girder E / slab E ) eg = distance between centers of gravity of girder and deck, in

Kg = n ( I + A eg2 )

Numerical example: Span: L = 140' Girder spacing: S = 8.0' Slab thickness: ts = 7.5" Haunch thickness: th = 1.5" Girder: Eg = 4,800 ksi Slab: Es = 4,000 ksiBT-72 girder: Ag = 767 in2 yt = 35.40 in. Ig = 545,850 in4 Interior girder ( two loaded lanes )

2ts

7.5"

1 5"

= 3.75"

eg = yt + th + 2ts

c.g. (girder)

35.4"

1.5"= 35.4" + 1.5" + 3.75" = 40.65"

n = 20.1ksi000,4ksi800,4

EE

s

g ==

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64

Kg = n ( I + A eg2 ) = 1.20 ( 545,850 in4 + 767 in2 ( 40.65 in )2 ) = 2,175,910 in4

DFM(int) = 0.075 + 1.0

3g

2.06.0

tL0.12

KLS

5.9S

= 0.075 +1.0

3

2.06.0

)5.7(1400.12910,175,2

1400.8

5.90.8

= 0.6443

17

DFM(int) ( moment , interior girder, one lane loaded ):

DFM(int) = 0.06 + 1.0

3g

3.04.0

tL0.12

KLS

14S

N i l l ti d ( i lid )Numerical example - continued ( see previous slide ):

n = 20.1ksi000,4ksi800,4

EE

s

g ==

Kg = n ( I + A eg2 ) = 1.20 ( 545,850 in4 + 767 in2 ( 40.65 in )2 ) = 2,175,910 in4

DFM(int) = 0.06 +1.0

3

3.04.0

)5.7(1400.12910,175,2

1400.8

140.8

= 0.4390

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65

For two loaded lanes ( previous slide ): DFM(int) = 0.6443 > 0.4390

Use

NOTE: Do not apply multi-presence factor = 1.20 ( it is included in the DF expression )

Numerical example - conclusion:

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66

DFM(ext) ( moment , exterior girder, one lane loaded ):

Lever rule:P

2P

2P

P

R

2P

2P

1.0'truck positioned at outside edge of lane

10' lane

6'3' 1'

S + de 7' :2P

S + de > 7' : P

Assumed hinge

R

S de

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67

R

S de

DFM(ext) =

+=

S21dS

PR e

R

S de

DFM(ext) =

+=

S4dS

PR e

Numerical example: Girder spacing: S = 8.0 ft de = 1' - 9" = 1.75 ft Exterior girder, one lane loaded

S + de = 8.0 ft + 1.75 ft = 9.75 ft > 7.0 ft

P1 0'

+ 4dSR e + 475.10.8 0 7188

R

8.0'1.75

1.0'

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68

DFM(ext) =

+=

S4dS

PR e

+=

0.8475.10.8 = 0.7188

18

DFM(ext) ( moment,exterior girder, two lanes loaded ):de

DFM(ext) =

+1.9

d77.0 e DFM(int)

1.75

DFM(int) = 0.6443

Numerical example ( continued from previous slides ):

DFM(ext) =

+1.9

75.177.0 0.6443 = 0.6200

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69

For one loaded lane ( previous slide ): DFM(int) = 0.7188 > 0.6200 Use

Live load moment for 10' lane:

MLL = MLane + ( 1 + IM ) MTruck = 4547.2 ft-k

Numerical example ( concluded ):

DFN(int) = 0.6443 ( two loaded lanes )DFN(ext) = 0.7188 ( two loaded lanes )

ML(int) = 0.6443 ( 4547.2 ft-k ) = 2930.0 ft-k

ML(ext) = 0.7188 ( 4547.2 ft-k ) = 3268.5 ft-k

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70

140 ft8.0 ft

Date post:	06-Jan-2016
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3. Loads and Load Distribution

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