3. loads and load distribution
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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
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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
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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
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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
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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
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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
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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
Simply-supported single-span bridge ( continued )
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.
Simply-supported single-span bridge ( continued )
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
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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
Simply-supported single-span bridge ( continued )
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
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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 )
Simply-supported single-span bridge: Numerical example ( continued )
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
Simply-supported single-span bridge: Numerical example ( continued )
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
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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'
Multi-span indeterminate bridges - overview ( continued )
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
Multi-span indeterminate bridges - overview ( continued )
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
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11
Multi-span indeterminate bridges - overview ( continued )
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
moment caused by uniform load over full span length L
Multi-span indeterminate bridges - overview ( continued )
Negative moment ( Region 2 ):
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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Negative live load moment:
1.33 maximum moment produced by moving truck train ( vary spacing )
90% PLUS
moment caused by uniform load over full span length L
Multi-span indeterminate bridges - overview ( continued )
Negative moment ( Region 2 ):
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
Negative live load moment:
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1.33 maximum moment produced by truck ( vary axle spacing )
100% PLUS
moment caused by uniform load over full span length L
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
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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
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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
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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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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
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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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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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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
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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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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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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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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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DFM(ext) =
+=
S4dS
PR e
+=
0.8475.10.8 = 0.7188
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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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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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140 ft8.0 ft
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