LOAD DISTRIBUTION FACTORS FOR CURVED CONCRETE SLAB-ON-STEEL I-GIRDER BRIDGES
BY
Mohammed A. Al-Hashimy, P. Eng. B.Sc., M.A.Sc., Baghdad U niversity, IRAQ
A Thesis Presented to Ryerson University
In partial fulfillment of the Requirement for the degree of
Master of Applied Science In the program of Civil Engineering
Toronto, Ontario, Canada, 2005PROPERTY OF
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■‘Load Distribution Factors for Curved Concrete Slab-on-Steel I-Girder Bridges’ By Mohammed A. Al-Hashimy, P. Eng., M.A.Sc.
Ryerson University - Civil Engineering Toronto, Ontario, Canada, 2005
ABSTRACT
The use of complex interchanges in modem highway urban systems have increased
recently in addition to the desire to conform to existing terrain; both have led to increase the
demand for horizontally curved bridges. One type of curved bridges consists o f composite
poncrete deck over steel I-girders which has been the preferred choice due to its simplicity in
fabrication, transportation and erection. Although horizontally curved steel bridges
constitute roughly one-third of all steel bridges being erected today, their structural behavior
still not well understood. Due to its geometry, simple presence o f curvature in curved
bridges produces non uniform torsion and consequently, lateral bending moment (warping
or bi-moment) in the girder flanges. The presence of the lateral bending moments would
significantly complicate the analysis and the design of the structure. Hence, a parametric
study is required to scmtinize a simplified method in designing horizontally curved steel I-
girder bridges.
A parametric study is conducted, using the finite-element analysis software
“SAP2000”, to examine the key parameters that may influence the load distribution on the
curved composite steel girders. Based on the data generated from the parametric study, sets
of empirical equations are developed for the moment and shear distribution factors for
straight and curved steel I-girder bridges when subjected to the Canadian Highway Bridge
Design Code (CHBDC) truck loading.
IV
ACKNOWLEDGEMENTS
The author wishes to express his deep appreciation to his advisor Dr. K. Sennah, for hi^
constant support and valuable supervision during the development of this research. Dr. Sennah
devoted his time and effort to make this study a success. His most helpful guidance is greatly
appreciated. Also, the author is very grateful to his father, mother, wife, son, and daughters for
their great support and encouragement during the course of this study.
The financial support from the Natural Scientific Research Council of Canada,
NSERC, as well as Ryerson University, is greatly appreciated.
TO MY FAMILY
VI
TABLE OF CONTENT
A B S T R A C T .........................;........................................................................................................................................................... IV
A C K N O W L E D G E M E N T S ........................................................................................................................... V
N O T A T IO N S ..................................................................................................................................................................................... X
L IST O F T A B L E S .................................................................................................................................................................... X V IB
L IST O F F IG U R S ........................................................................................................................................................................X IX
C H A PT E R 1.......................................................................................................................................................................................... 1
IN T R O D U C T IO N .............................................................................................................................................................................. 1
1.1 G e n e r a l ...................................................................................................................................................................................11.2 T h e Pr o b l e m ..........................................................................................................................................................................31.3 Ob je c t iv e s ............................................................................................................................................................................. 4
■ 1.4 Sc o p e ........................................................................................................................................................................................ 41.5 C o n ten ts a n d A rra n g em en t o f th is s t u d y ........................................................................................................5
C H A PT E R I I ....................................................................................................................................................................................... 6
LIT E R A T U R E R E V IE W ................................................................................................................................................................ 6
2.1 C o n c e p t o f L a t e r a l L o a d D i s t r ib u t io n F a c t o r ................................................................................................. 62.2 R ev iew o f P rev io u s Resea rch on Lo a d D is t r ib u t io n .....................................................................................11
2.2.7 -J Review o f Study on Distribution Factors fo r Straight Bridges................................................ 112.2.1.1 Lever Rule M ethod (Yao, 1990) ................................................................................................................. 122.2.1.2 Eccentric Compression M ethod (Yao, 1990)............................................................................................ 122.2.1.3 Hinged Joint Method (Yao, 1990)..............................................................................................................14
2.2.1.3.1 Hinged Joint Method for Slab Bridges........................................................................................................142.2.1.3.2 Hinged Joint Method for T-Shaped Girder Bridge...................................................................................16
2 .2 .1.4 Fixed Joint Girder method (Yao, 1990)...................................................................................................162.2.1.5 Orthotropic Plate Analogy (Guyon-Massonnet or G-M M ethod)....................................................162.2.1.6 AASHTO M ethods ...........................................................................................................................................18
2.2.1.6.1 AASHTO Standard Method 1996................................................................................................................ 192.2.1.6.2 AASHTO LRFD Method..............................................................................................................................20
2.2.1.7 Canadian Highway Bridge Design Code, 2000 (CH BD C)................................................................. . 212.2.1.8 Other Studies.................................................................................................................................................232.2.2 Distribution Factors fo r Curved Bridges ....................................................................................................232.2.2.1 Heins and Siminou 's Study...........................................................................................................................242.2.2.2 AASHTO M ethods .......................................................................................................................................... 26
2.2.2.2.1 AASHTO Guide Commentary Method....................................................................................................... 262.22.2.2 AASHTO Guide Method............................................................................................................................... 262.2.2.2.3 AASHTO with V-LoadModification Method............................................................................................ 27
2.2.2.3 Heins and Jin 's Method................................................................................................................................ 302.2.2.4 Yoo and Littrell ’s Study .................................................................................................................................302.2.2.5 Brockenbrough's S tudy .................................................................................................................................312.2.2.6 Schelling, Namini, Fu ’s S tudy .....................................................................................................................312.2.2.? Davidson, Keller, and Yoo’s study ............................................................................................................. 322.2.2.S Eissa, et. al. study .......................................................................... 332.2.2.9 Zhang’s study ............................................................................................ 332.2.2.10 W assefs study ..................................................................................................................................................35
2.3 Re v ie w o f L inear E la stic B eh a v io u r o f C u rv ed I- G ird er S y s t e m .............................................. 362.4 Rev ie w o f A na ly s is M eth o d s for C urv ed Sy s t e m ................................................................................ 40
2.4.1 Approximate methods.................................................................................................................... 412.4.1.1 The Plane-grid or Grillage method ...............................................................................................................41
V ll
2.4.1.2 The Space-frame method .................................................. 422.4.1.3 The V-Load method (Grubb 1984)..................... 422.4.2 Refined methods................................................................................................................................................ 432.4.2.1 The Finite-strip method ..................................................................................................................................442.4.2.2 The Finite-difference m ethod .......................................................................................................................442.4.2.3 Analytical solution to differential equations....................................................................................442.4.2.4 The Slope deflection method .........................................................................................................................452.4.2.5 The Finite-Element Method (FEM), By Logan (2002)..........................................................................45
2.4.2.5.1 Three-Dimensional Method.................................................................................................... 46
CHAPTER I I I ................................................................................................................................................................................48
FIN ITE-ELEM EN T A N A L Y SIS ............................................................................................................................................ 48
3.1 G en er a l .....................................................................................................................................................................483.2 F inite-Elem en t A ppr o a c h ................................................................................................................................. 503.3 SAP2000 C om puter Pr o g r a m .......................................................................................................................... 513.4 X H B D C Design Lo a d in g .................................................................................................................................... 523.5 T ruck Lo ading C a s e s .......................................................................................................................................... 533.6 C om posite B ridge C o n fig u r a t io n s .............................................................................................................. 553.7 R esearch A ssu m ptio n s .......................................................................................................................................573.8 F inite-Elem en t B ridges m o d e l in g ............................................................................................................... 58
3.8.1 Geometric Modeling ...............................................................................................................................................583.8.2 Aspect Ratio .....................................................................................................................................................593.8.3 Boundary conditions.............................................................................................................................................59
3.9 C alculation o f th e Shear D istribution F a c t o r s .................................................................................603.10 C alculation o f th e M om ent D istribution Fa c t o r s ............................................................................ 623.11 C alculation o f th e D eflection D istribution Fa c t o r s ......................................................................643.12 W a rp in g to -b e n d in g s t r e s s r a t i o ................................................................................................................ 66
CH A PTER IV ................................................................................................................................................................................68
RESULTS FROM THE PARAM ETRIC S T U D Y ..............................................................................................................68
4.1 G e n e r a l .....................................................................................................................................................................684.2 Shear D istribution in S im ply Su pported Co m posite Curved Brid g es ..........................................69
4.2.1 Effect o f Curvature....................................................................... 694.2.2 Effect o f Number o f Girders............................................................................................................................ 714.2.3 Effect o f Girders Spacing ................................................................................................................................. 724.2.4 Effect o f Span Length ........................................................................................................................................ 734.2.5 Effect o f Loading Conditions.......................................................................................................................... 734.2.6 Effect o f Number o f Lanes................................................................................................................................744.2.7 Effect o f Number o f Bracing Intervals..........................................................................................................754.2.8 Effect o f Load C ases ......................................................................................................................................... 75
4.3 Shear D istribution B etw een Bridge G ir d e r s .........................................................................................764.4 SDF E ffect with T ype o f the Su ppo r t ......................................................................................................... 774.5 Shear D istribution Equations Com parison betw een C H B D C .........................................................774.6 M om en t and Deflection D istributions in S imply S u ppo r ted .......................................................... 78Com po site C urved B rid g es .............................................................................................................................. 78
4.6.1 Effect o f Curvature.............................................................................................................................................784.6.2 Effect o f Number o f Girders............................................................................................................................ 794.6.3 Effect o f Girders Spacing .................................................................................................................................804.6.4 Effect o f Loading Conditions.................... 81
4.7 W arping Stress D istribution in S imply Supported C urved Brid g es ............................................814.8 Com parison B etw een CHBDC M o m ent D istribution E q u a t io n s ................................................... 834.9 D evelopm ent o f N ew Load D istribution F actor E qu a tio n s ........................................................... 84
4.9.1 Shear Distribution Factor Equations.......................................................................................................... 844.9.1.1 Shear Distribution Factors fo r Straight I-Girder Bridges........................................................................ 844.9.1.2 Shear Distribution Factors fo r Curved 1-Girder Bridges......................................................................... 85
Vlll
4.9.2 Moment Distribution Factor Equations............................................................................................. S54.9.2.1 Moment Distribution Factors for Straight I-Girder Bridges..............................................................854.9.3 Deflection Distribution Factor Equations.......................................................................................... 864.9.3.1 Deflection Distribution Factors for Straight I-Girder Bridges................................................ 86
CHAPTER V ...................................................................................................................................................................88
SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS............................................................................88
5.1 S u m m a r y ..................................................................................................................................................................... 885.2 C o n c l u s i o n s ..............................................................................................................................................................885.3 R eco m m en d a tio n s fo r F u tu r e R e se a r c h ....................................................................................................90
REFERENCES................................................................................................................................................................91
APPENDEX (A): SAP 2000 INPUT FILE FOR A STRAIGHT BRIDGE (SDF)...............................................214
APPENDEX (B): SAP 2000 INPUT FILE FOR A CURVED BRIDGE (SDF)..................................................219
APPENDEX (C): EXCEL DATA SHEET FOR SECTION AND GIRDER PROPERTIES............................. 227
IX
NOTATIONS
A Bridge width
B The clear spacing between girders
Be Effective concrete slab width
E Modulus of Elasticity
F Width dimension factor
Fv Shear distribution factor
It The moment of inertia of the composite girder
[K] The global stiffness matrix
L Centre line span of a simply supported bridge
M d l The mid-span moment for a straight simply supported girder due to a singlegirder dead load
Mt The mid-span moment for a straight simply supported girder due to a singleCHBDC truck loading
n Number of design lanes
N Number of girders
[P] Applied loads vector at the nodes
R Radius of curvature of the centre span of the curved bridge
Rl Multi-lane factor based on the number of the design lanes
R l' Multi-lane factor based on the number of the loaded lanes
S Girders spacing
[U] Displacement vector at the nodes
Wc Deck width
We Width of design lane
Yb The distance from the neutral axis to the bottom flange
( R r ) d l Right Support reaction due to Dead Load
( R l ) d l Left Support reaction due to Dead Load
( R r ) f l Right Support reaction due to Fully Loaded lanes
( R l ) f l Left Support reaction due to Fully Loaded lanes
( R r ) p l i Right Support reaction due to Partially Loaded lanes type 1
( R l ) p l i Left Support reaction due to Partially Loaded lanes type 1
( R r ) p l 2 Right Support reaction due to Partially Loaded lanes type 2
( R l ) p l 2 Left Support reaction due to Partially Loaded lanes type 2
(RR)pat Right Support reaction due to Fatigue Loading
(RÜFat Left Support reaction due to Fatigue Loading
(Rstraight)DL Maximum shear forces calculated for straight simply supported beam due toDead Load
(Rstraight)truck, Maximum shear forces calculated for straight simply supported beam due to truck loading
( R f e . ext)oL The greater reaction at the exterior girder supports found from the finite-element analysis due to dead load
(Rfe. ext)FL The greater reaction at the exterior girder supports found from the finite-element analysis due to Fully loaded lanes
( R f e . ext)pL The greater reaction at the exterior girder supports found from the finite-element analysis due to Partially loaded lanes
(RFE.ext)Fat The greater reaction at the exterior girder supports found from the finite-element analysis due to Fatigue loading
XI
(Rfe. mid)DL The greater reaction at the middle girder supports found from the finite- element analysis due to dead load
(Rfe. mid)FL The greater reaction at the middle girder supports found from the finite- element analysis due to Fully loaded lanes
( R f e . mid)pL The greater reaction at the middle girder supports found from the finite- element analysis due to Partially loaded lanes
(RpE.mid)Fat The greater reaction at the middle girder supports found from the finite-element analysis due to Fatigue loading
(Rfe. int)DL The greater reaction at the interior girder supports found from the finite-element analysis due to dead load
( R f e . int)FL The greater reaction at the interior girder supports found from the finite-element analysis due to Fully loaded lanes
( R f e . int)pL The greater reaction at the interior girder supports found from the finite-element analysis due to Partially loaded lanes
(RFE.int)Fat The greater reaction at the interior girder supports found from the finite-element analysis due to Fatigue loading
(SDF)DLext Shear distribution factor for the exterior girder due to Deal Load
(SDF)pL ext Shear distribution factor for the exterior girder due to Fully Loaded lanes
( S D F ) pl ext Shear distribution factor for the exterior girder due to Partially Loaded lanes
(SDF)pat ext Shear distribution factor for the exterior girder due to Fatigue Loading
( S D F ) d l mid Shear distribution factor for the middle girder due to Deal Load
(SDF)fl mid Shear distribution factor for the middle girder due to Fully Loaded lanes
( S D F ) p l mid Shear distribution factor for the middle girder due to Partially Loaded lanes
(SDF)pat mid Shear distribution factor for the middle girder due to Fatigue Loading
(SDF)dl int Shear distribution factor for the interior girder due to Deal Load
(SDF)PL int Shear distribution factor for the interior girder due to Fully Loaded lanes
( S D F ) pl int Shear distribution factor for the interior girder due to Partially Loaded lanes
(SDF)pat int Shear distribution factor for the interior girder due to Fatigue Loading
XU
(cJe i )d l Maximum flexural stresses in bottom flange fibers at point 1 of exteriorgirder, for the dead load case, obtained from finite-element analysis
(<7e3)dl Maximum flexural stresses in bottom flange fibers at point 3 of exteriorgirder, for the dead load case, obtained from finite-element analysis
(O e i ) f l Maximum flexural stresses in bottom flange fibers at point 1 of exteriorgirder, for the full lane loading case, obtained from finite-element analysis
(ctEi)pL Maximum flexural stresses in bottom flange fibers at point 1 of exteriorgirder, for the partial lane loading case, obtained from finite-element analysis
(cte3)fl Maximum flexural stresses in bottom flange fibers at point 3 of exteriorgirder, for the full lane loading case, obtained from finite-element analysis
(< E3)pL Maximum flexural stresses in bottom flange fibers at point 3 of exteriorgirder, for the partial lane loading case, obtained from finite-element analysis
(CTEi)Fat Maximum flexural stresses in bottom flange fibers at point 1 of exteriorgirder, for the fatigue case, obtained from finite-element analysis
(OE3)pat Maximum flexural stresses in bottom flange fibers at point 3 of exteriorgirder, for the fatigue case, obtained from finite-element analysis
(GM])oL Maximum flexural stresses in bottom flange fibers at point 1 of middlegirder, for the dead load case, obtained from finite-element analysis
(0'M3)dl Maximum flexural stresses in bottom flange fibers at point 3 of middlegirder, for the dead load case, obtained from finite-element analysis
(cTmOfl Maximum flexural stresses in bottom flange fibers at point 1 of middlegirder, for the full lane loading case, obtained from finite-element analysis
(o’m3) f l Maximum flexural stresses in bottom flange fibers at point 3 of middlegirder, for the full lane loading case, obtained from finite-element analysis
(t Mi)Fat Maximum flexural stresses in bottom flange fibers at point 1 of middlegirder, for the fatigue case, obtained from finite-element analysis
(CTM3)Fat Maximum flexural stresses in bottom flange fibers at point 3 of middlegirder, for the fatigue case, obtained from finite-element analysis
(OiOdl Maximum flexural stresses in bottom flange fibers at point 1 of interiorgirder, for the dead load case, obtained from finite-element analysis
X lll
(cTi3)dl Maximum flexural stresses in bottom flange fibers at point 3 o f interior girder, for the dead load case, obtained from finite-element analysis
(Gii)FL Maximum flexural stresses in bottom flange fibers at point 1 of interiorgirder, for the full lane loading case, obtained from finite-element analysis
(Gii)pL Maximum flexural stresses in bottom flange fibers at point 1 of interiorgirder, for the partial lane loading case, obtained from finite-element analysis
(Gi3)pL Maximum flexural stresses in bottom flange fibers at point 3 of interiorgirder, for the full lane loading case, obtained from finite-element analysis
(0'i3)pL Maximum flexural stresses in bottom flange fibers at point 3 of interiorgirder, for the partial lane loading case, obtained from finite-element analysis
(aii)Fat Maximum flexural stresses in bottom flange fibers at point 1 of interiorgirder, for the fatigue case, obtained from finite-element analysis
(ctD)Fat Maximum flexural stresses in bottom flange fibers at point 3 of interiorgirder, for the fatigue case, obtained from finite-element analysis
(a straight) DL Maximum flexural stresses in bottom flange fibers, for the straight simplysupported beam due to Deal Load
(o stndght) truck Maximum flexural stresses in bottom flange fibers, for the straight simply supported beam due to CHBDC truck loading
(o fe. ext)oL the bigger flexural stresses of points 1 and 3 of exterior girder due to dead load case
(o fe. ext)pL the b igger flexural stresses o f points 1 and 3 o f exterior girder due to Fullyloaded lanes case
( o f e . ext)pL the bigger flexural stresses of points 1 and 3 of exterior girder due to Partially loaded lanes case
(o fe. ext)pat the bigger flexural stresses of points 1 and 3 of exterior girder due to Fatigueloading case
(o f e . mid)DL the bigger flexural stresses of points 1 and 3 of middle girder due to dead load case
(o f e . mid)FL the bigger flexural stresses of points 1 and 3 of middle girder due to Fully loaded lanes case
XIV
(o FE. mid)pL the bigger flexural stresses of points 1 and 3 of middle girder due to Partiallyloaded lanes case
(o FE. mid)Fat the bigger flexural stresses of points 1 and 3 of middle girder due to Fatigueloading case
(o f e . int)DL the bigger flexural stresses of points 1 and 3 of interior girder due to deadload case
(o FE. int)pL the bigger flexural stresses of points 1 and 3 of interior girder due to Fullyloaded lanes case
(o f e . int)pL the bigger flexural stresses of points 1 and 3 of interior girder due to Partiallyloaded lanes case
(CT FE. int)Fat
( M D F ) d l ext
( M D F ) f L ext
( M D F ) p l ext
( M D F ) F a t .e x t
( M D F ) d l mid
( M D F ) f l mid
( M D F ) p l mid
( M D F ) p a t .m i d
( M D F ) d l int
( M D F ) f l int
( M D F ) p L i n t
( M D F ) F a t . i n t
(^simpIe)DL
the bigger flexural stresses of points 1 and 3 of interior girder due to Fatigue loading case
the moment distribution factor of exterior girder for dead load case
the moment distribution factor of exterior girder for full load case
the moment distribution factor of exterior girder for partial load case
the moment distribution factor of exterior girder for fatigue case
the moment distribution factor of middle girder for dead load case
the moment distribution factor of middle girder for full load case
the moment distribution factor of middle girder for Partial load case
the moment distribution factor of middle girder for fatigue case
the moment distribution factor of interior girder for dead load case
the moment distribution factor of interior girder for full load case
the moment distribution factor of interior girder for partial load case
the moment distribution factor of interior girder for fatigue case
mid-span deflection in bottom flange fibers, for a straight simply supported girder subject to dead load
XV
(A sim p ie) truck mid-span deflection in bottom flange fibers, for a straight simply supportedgirder subject to CHBDC truck loading
(Afe ext)oL mid-span deflection in bottom flange fibers at point 2 of exterior girder, forthe dead load case, obtained from finite-element analysis
(AFEext)pL mid-span deflection in bottom flange fibers at point 2 of exterior girder, forthe full lane loading case, obtained from finite-element analysis
(ApEext)pL mid-span deflection in bottom flange fibers at point 2 of exterior girder, forthe partial lane loading case, obtained from finite-element analysis
(A f e ext)pat mid-span deflection in bottom flange fibers at point 2 of exterior girder, forthe fatigue ease, obtained from finite-element analysis
(ApEmidjoL mid-span deflection in bottom flange fibers at point 2 of middle girder, forthe dead load case, obtained from finite-element analysis
(AFEmid)FL mid-span deflection in bottom flange fibers at point 2 of middle girder, forthe full lane loading case, obtained from finite-element analysis
(ApEmidjpL mid-span deflection in bottom flange fibers at point 2 of middle girder, forthe partial lane loading case, obtained from finite-element analysis
( A f e mid)Fat mid-span deflection in bottom flange fibers at point 2 of middle girder, forthe fatigue case, obtained from finite-element analysis
(A f e int)DL mid-span deflection in bottom flange fibers at point 2 of interior girder, forthe dead load case, obtained from finite-element analysis
(A f e int)FL mid-span deflection in bottom flange fibers at point 2 of interior girder, forthe full lane loading case, obtained from finite-element analysis
(AFEint)pL mid-span deflection in bottom flange fibers at point 2 of interior girder, forthe partial lane loading case, obtained from finite-element analysis
(A fe int)pat mid-span deflection in bottom flange fibers at point 2 of interior girder, forthe fatigue case, obtained from finite-element analysis
(DDF)DLext the deflection distribution factor of exterior girder for dead load case
(DDF)FLext the deflection distribution factor of exterior girder for full load case
(DDF)pLext the deflection distribution factor of exterior girder for partial load case
( D D F ) p a t .e x t the deflection distribution factor of exterior girder for fatigue case
XVI
( D D F ) d l mid the deflection distribution factor of middle girder for dead load case
(DDF)FLmid the deflection distribution factor of middle girder for full load case
(DDF)Fat.mid the deflection distribution factor of middle girder for fatigue case
( D D F ) o L int the deflection distribution factor of interior girder for dead load case
(DDF)FLint the deflection distribution factor of interior girder for full load case
( D D F ) p L i n t the deflection distribution factor of interior girder for partial load case
( D D F ) p a t . i n t the deflection distribution factor of interior girder for fatigue case
( W B R ) d l ext the warping bending stress ratio of exterior girder for dead load case
( W B R ) p L e x t the warping-to-bending stress ratio of exterior girder for full load case
( W B R ) p L e x t the warping-to-bending stress ratio of exterior girder for partial load case
( W B R ) p a t .e x t the warping-to-bending stress ratio of exterior girder for fatigue case
(WBR)DLmid the warping-to-bending stress ratio of middle girder for dead load case
( W B R ) p L m i d the warping-to-bending stress ratio of middle girder for full load case
( W B R ) p L m i d the warping-to-bending stress ratio of middle girder for Partial load case
(WBR)pat.mid the warping-to-bending stress ratio of middle girder for fatigue case
( W B R ) d l int the warping-to-bending stress ratio of interior girder for dead load case
(WBR)FLint the warping-to-bending stress ratio of interior girder for full load case
(WBR)pLint the warping-to-bending stress ratio of interior girder for partial load case
( W B R ) p a t . i n t the warping-to-bending stress ratio of interior girder for fatigue case
xvu
LIST OF TABLES
Table Page
Table 2 .1 Coefficient, C, for Various Multi-Girder Systems Assuming Equal Girder Spacing (Grubb, 1984)............................................................................................. 96
Table 3. 1 Bridge Configurations Considered in the Parametric Study.............................. 96
Table 3. 2 Number of Design Lanes........................................................................................97
Table 3. 3 Modification Factors for Multilane Loading....................................................... 97
Table 4. 1 Table 4.1 Effect of loading conditions, extreme cases......................................98
Table 4.2 Value of F for longitudinal shear for Straight Bridges (ULS and..SLS).........99
Table 4.3 Value of F for longitudinal shear for Straight Bridges (ELS)........................... 100
Table 4.4 Value of F for longitudinal shear for Curved Bridges (ULS and...SLS).........101
Table 4.5 Value of F for longitudinal shear for Curved Bridges (ELS)............................ 102
Table 4.6 Value of F for longitudinal moment for Straight Bridges (ULS and SLS) 103
Table 4.7 Value of F for longitudinal moment for Straight Bridges (ELS)...................... 104
Table 4.8 Value of F for Deflection for Curved Bridges (ULS and SLS).........................105
Table 4.9 Value of F for Deflection for Curved Bridges (ELS)..........................................106
xvm
LIST OF FIGURS
F ig u r e 1 .1 T y p ic a l I -G ir d e r B r id g e C r o s s -S e c t i o n ..................................................................................................107
F ig u r e 2 . 1 S in g le a n d M u lti-g ir d er S ystem under Co n cen tra ted L ive L o a d P .......................................108F ig u r e 2 .2 L a ter a l L o a d D istribu tio n o f T ruck A x le Lo a d ...............................................................................108F ig u r e 2 .3 G ird er D eflectio n w ith D iffer en t T ra n sv er se St if f n e s s .............................................................109F ig u r e 2 .4 F ree B o d y D iagram o f L ever R u le m e t h o d ...........................................................................................109F ig u r e 2 .5 Lo a d D istribu tio n un der E ccen tric L o a d ............................................................................................110F ig u r e 2 . 6 F r e e B o d y D ia g ra m o f a H in g e d s l a b B r id g e u n d e r C o n c e n t r a t e d L o a d ............................ 111F ig u r e 2 .7 F r e e B o d y D ia g ra m o f a H in g e d s l a b B r id g e u n d e r S in u s o id a l L o a d ................................... 111F ig u r e 2 . 8 F r e e B o d y D ia g ra m f o r H in g e d T -sh a p e d G i r d e r B r i d g e ..............................................................112F ig u re 2 .9 F ree B o d y D iagram o f F ixed Jo in t G irder B r id g e ............................................................................ 113F ig u r e 2 .1 0 R ea l Str u c tu r e an d O rth o tro pic P late A n a l o g y .........................................................................113F ig u r e 2 .1 1 V -L o a d o n G i r d e r ............................................................................................................................................ 114F ig u re 2 .1 2 E ffec t o f W arping M o m en t A pplied to I-G ir d e r ............................................................................. 114
F ig u r e 3 .1 C a n a d ia n H ig h w a y B r id g e D e s ig n C o d e s ’ T r u c k a n d L a n e l o a d in g s .....................................115F ig u r e 3 .2 M a x im u m S h e a r L o c a t io n s ........................................................................................................................... 116F ig u re 3 .3 M a xim um M om en t L o c a t io n s ...................................................................................................................... 117F ig u r e 3 .4 L iv e l o a d in g c a s e s fo r O n e -l a n e B r id g e ...................................................................................................... 118F ig u r e 3 .5 L iv e l o a d in g c a s e s fo r T w o -l a n e B r id g e ..................................................................................................... 119F ig u re 3 .6 L ive lo a d in g ca ses for T h r ee -lane B r id g e ...........................................................................................121F ig u re 3 .7 L ive lo a d in g ca ses for F o u r -lan e B r id g e ............................................................................................. 123F ig u re 3. 8 Cr o ss-Sectio n o f a C o m po site I-G irder B r id g e .................................................................................. 126F ig u r e 3 .9 P l a n o f t h e St e e l - G ir d e r A r r a n g e m e n t ............................................................................................. 127F ig u re 3 .1 0 F in ite - elem en t represen ta tio n o f brid g e cross - s e c t io n ........................................................128F ig u re 3 .11 V iew s o f SA P2000 F in ite-E lem en t M o d el - W ith o u t S lab pa n e l s ...........................................129FIGURE 3 .1 2 V iew s o f S A P 2 0 0 0 F in ite-E l e m e n t M o d e l - W ith Slab Pa n e l s ..................................................129F ig u re 3 .1 3 V iew s o f SA P2000 F in ite-E l e m e n t M o d el - Bou n d a ry Co n d it io n s ........................................ 130F ig u r e 3 .1 4 C r o s s -S e c t io n D im e n s io n s o f t h e s t e e l g ir d e r ................................................................................. 131F ig u r e 3 .1 5 N o r m a l S t r e s s D is t r ib u t io n in C u r v e d I -G ir d er F l a n g e s .......................................................... 131
F ig u r e 4. l E f f e c t o f C u r v a t u r e o n t h e S h e a r D is t r ib u t io n F a c t o r fo r t h e E x t e r io r G ir d e r d u e t o
D e a d L o a d ......................................................................................................................................................................... 132F ig u r e 4 .2 E f f e c t o f C u r v a t u r e o n t h e S h e a r D is t r ib u t io n F a c t o r fo r t h e E x t e r io r G ir d e r d u e t o
F u l l y L o a d e d L a n e s ...................................................................................................................................................... 132F ig u r e 4 .3 E f f e c t o f C u r v a t u r e o n t h e S h e a r D is t r ib u t io n F a c t o r fo r t h e E x t e r io r G ir d e r d u e t o
F a t ig u e L o a d in g ............................................................................................................................................................. 133F ig u r e 4 .4 E f f e c t o f C u r v a t u r e o n t h e S h e a r D is t r ib u t io n F a c t o r f o r t h e M id d l e G ir d e r d u e t o
D e a d L o a d ...........................................................................................................................................................................133
XIX
F igure 4 .5 E ffect o f C urva ture on th e Shear D istribution Factor for th e M iddle G irder due to
F ully Load ed L a n e s .................................................................................................................................................F ig u re 4 . 6 E f f e c t o f C u r v a t u r e o n t h e S h e a r D is t r ib u t io n F a c t o r f o r t h e M id d le G ir d e r d u e t o
Fatig ue Lo a d in g .......................................................................................................................................................... 134F igure 4 .7 E ffect o f C urvature on th e Shear D istribution Factor for th e Interior G irder du e to
D ead Lo a d ...................................................................................................................................................................... 135F ig u re 4 . 8 E f f e c t o f C u r v a t u r e o n t h e S h e a r D is t r ib u t io n F a c t o r f o r t h e I n t e r i o r G ir d e r d u e t o
F ully Loaded L a n e s .................................................................................................................................................. 135F igure 4 .9 E ffect o f C urvature on th e Shear D istribution Factor for th e Interior G irder d u e to
Fatigue L o a d in g .......................................................................................................................................................... 136F igure 4 .1 0 E ffect o f C urvature on th e Shear D istribution F actor for the Exterior G irder d u e
TO D ead Lo a d ................................................................................................................................................................ 136F igure 4 .1 1 E ffect of C urvature on the Shear D istribution Factor for th e Exterio r G irder due
TO F ully Loaded La n e s ............................................................................................................................................ 137F igure 4 .1 2 E ffect of C urvature on the Shear D istribution F actor for th e Exterio r G irder du e
TO Partially Loaded La n e s .................................................................................................................................... 137F igure 4 .1 3 E ffect of C urvature on the Shear D istribution Factor for the Exterior G irder d u e to
Fatigue Lo a d in g .......................................................................................................................................................... 138F igure 4 .1 4 E ffect of C urvature on the Shear D istribution F actor for th e Interior G irder due to
Dead Lo a d ...................................................................................................................................................................... 138F igure 4 .1 5 E ffect o f Curvature on the Shear D istribution F actor for th e Interior G irder due to
F ully Loaded L a n e s .................................................................................................................................................. 139F igure 4 .1 6 E ffect of C urvature on the Shear D istribution F actor for th e Interior G irder due to
Partially Load ed La n e s ..........................................................................................................................................139F igure 4 .1 7 Effect of N um ber o f G irders on the Shear D istribution Factor for the E xterior
G irder due to D ead Lo a d .........................................................................................................................................140F igure 4 .1 8 Effect o f N um ber o f G irders on the Shear D istribution Factor for the E xterior
G irder due to F ully Loa ded Lanes .....................................................................................................................140F igure 4 .1 9 E ffect of N um ber of G irders on the Shear D istribution Factor for the E xterior
G irder due to P artially Loaded La n e s ........................................................................................................... 141F igure 4 .2 0 Effect of N um ber of G irders on the Shear D istribution Factor for the E xterior
G irder due to Fatigue Lo a d in g ............................................................................................................................141F igure 4 .21 Effect of N um ber of G irders on the Shear D istribution Factor for the M iddle girder
DUE TO D ead Lo a d ........................................................................................................................................................142F igure 4 .2 2 Effect o f N umber o f G irders on the Shear D istribution Factor for the M iddle G irder
DUE TO Fully Loaded La n e s ....................................................................................................................................142F igure 4 .2 3 E ffect of N um ber of G irders on the Shear D istribution F actor for th e M iddle girder
DUE TO Fatigue Lo a d in g ............................................................................................................................................ 143F igure 4 .2 4 E ffect of N um ber of G irders on th e Shear D istribution F actor for the interior
G irder due to D ead Lo a d .........................................................................................................................................143F igure 4 .2 5 E ffect of N umber of G irders on the Shear D istribution F actor for the Interior
G irder due to F ully Loaded Lan es .....................................................................................................................144F igure 4 .2 6 E ffect of N um ber of G irders on the Shear D istribution Factor for the Interior
G irder due to Partially Loaded L a n e s ............................................................................................................144F igure 4 .2 7 E ffect of N um ber of G irders on the Shear D istribution Factor for the Interior
G irder due to F atigue Lo a d in g ............................................................................................................................145F igure 4 .2 8 E ffect of G irder Spacing on the Shear D istribution Factor for the E xterior G irder
DUE TO D ead Lo a d ........................................................................................................................................................ 145F igure 4 .2 9 Effect of G irder Spacing on the Shear D istribution Factor for the E xterior G irder
DUE TO Fully Loaded La n es .................................................................................................................................... 146F igure 4 .3 0 E ffect o f G irder Spacing on the Shear D istribution Factor for th e Exterior G irder
DUE TO Partially Loaded L a n e s ...........................................................................................................................146F igure 4 .31 E ffect of G irder Spacing on th e Shear D istribution Factor for the E xterior G irder
DUE TO Fatigue Lo a d in g ........................................................................................................................................... ..F igure 4 .3 2 E ffect o f G irder Spacing on the Shear D istribution Factor for th e M iddle G irder due
TO FULLY Loa ded L a n e s ........................................................................................................................................... ..
XX
F ig u r e 4 .3 3 E ffe c t o f G irder Spa cin g o n the Shear D istribution F a cto r for th e M iddle G irder du eTO F a tig u e L o a d in g ........................................................................................................................................................148
F ig u r e 4 .3 4 E ffec t o f G irder Spacing on the Shear D istribution F acto r for th e Interio r G irderDUE TO FULLY LOADED LANES.......................................................................................................................................148
F ig u r e 4 .3 5 E ffe c t o f G irder Spacing on the Shear D istribution F acto r for th e Interio r G irderDUE TO P a r tia lly L oa d ed L a n e s ............................................................................................................................. 149
F ig u r e 4 .3 6 E ffe c t o f Span Len g th o n th e Shear D istribution F actor for th e E xterio r G irder du eTO D e a d Lo a d ................................................................................................................................................................... 149
F ig u r e 4 .3 7 E ffec t o f Span L eng th on th e Shear D istribution Factor for th e E x terio r G irder dueTO F u lly Lo a d ed L a n e s ............................................................................................................................................... 150
F ig u r e 4. 38 E ffect o f Span L ength on th e Shear D istribution F actor for th e M iddle G irder d u e toF ully Lo a d ed L a n e s ..................................................................................................................................................... 150
F ig u r e 4 .3 9 E ffe c t o f Span Length o n th e Shear D istribution Factor for th e Interio r G irder dueTO D e a d L o a d ................................................................................................................................................................... 151
F ig u r e 4 .4 0 E ffe c t o f Span L ength on the Shear D istribution Factor for th e Interio r G irder du eTO F u lly Lo a d ed L a n e s ............................................................................................................................................... 151
F ig u r e 4 .4 1 E ffe c t o f Load ing Con d itio n s on the Shear D istribution F acto r for th e E xterio rG ird er o f t h e 10-m -span B r id g e s ............................................................................................................................152
F ig u r e 4 .4 2 E ffe c t o f Lo a d in g Con d itio n s on the Shear D istribution F actor for th e E xteriorG ird er o f t h e 15-m -span B r id g e s ............................................................................................................................152
F ig u r e 4 .4 3 E ffec t o f Lo a d in g Con d itio n s on th e Shear D istribution Factor for th e E xteriorG ird er o f t h e 25-m -span B r id g e s ............................................................................................................................153
F ig u r e 4 .4 4 E ffe c t o f Lo a d in g Con d itio n s on the Shear D istribution F actor for th e E xteriorG ird er o f t h e 35-m -span B r id g e s ............................................................................................................................153
F ig u r e 4 .4 5 E ffe c t o f Lo a d in g Co n d itions on th e Shear D istribution Factor for th e Interio r
G ird er o f th e 10-m -span B r id g e s ............................................................................................................................154F ig u re 4 .4 6 E ffe c t o f Lo a d in g Con d itio n s on the Shear D istribution F actor for th e Interio r
G irder o f th e 15-m -span B r id g e s ............................................................................................................................154F ig u r e 4 .4 7 E ffec t o f Lo a d in g Co n d itions on th e S hear D istribution F actor for th e Interio r
G ird er o f th e 25-m -span B r id g e s ............................................................................................................................155F ig u r e 4 .4 8 E ffec t o f L o ad in g C o n d itions on the Shear D istribution F actor for th e Interio r
G irder o f t h e 35-m -span B r id g e s ............................................................................................................................155F ig u r e 4 .4 9 E ffec t o f N u m ber o f L anes on th e Shear D istribution F actor for th e E xterio r G irder
DUE TO F u lly Loajded L a n e s .......................................................................................................................................156F ig u r e 4 .5 0 E ffe c t o f N u m ber of L anes on th e Shear D istribution Factor for th e E xterio r G irder
DUE TO P artia lly Loa d ed L a n e s ............................................................................................................................. 156F ig u r e 4 .5 1 E ffec t o f N u m ber o f L anes on the Shear d istr ib u tio n F actor for th e M idd le G irder
d u e to F ully L o a d ed L a n e s .......................................................................................................................................157F ig u r e 4 .5 2 E ffe c t o f N u m ber o f l a n e s on th e Shear d ist r ib u t io n F actor for th e M iddle G irder
DUE TO P a rtia lly L oa d ed La n e s ............................................................................................................................. 157F ig u r e 4 .5 3 E ffe c t o f N u m ber o f L anes on the Shear D istribution F actor for th e Interio r G irder
DUE to F ully L o a d ed L a n e s .......................................................................................................................................158F ig u re 4 .5 4 E ffe c t o f N u m ber o f Lanes on th e Shear D istribution F actor for th e Interio r G irder
d u e to Pa r tia lly L o a d ed La n e s ............................................................................................................................. 158F ig u re 4 .5 5 E ffe c t o f th e N um ber o f C ro ss-B racing Intervals on the Shear D istribution Factor
FOR THE E x terio r G ir d e r .............................................................................................................................................159F ig u re 4 .5 6 E ffe c t o f th e Loa d C ases N um ber on the S hear D istribution F actor for th e E xterior
G ird er o f a tw o -la n e B r id g e ................................................................................................................................... 159F ig u re 4 .5 7 E ffe c t o f th e L oa d C ases N um ber on th e S hear D istribution Facto r for t h e M iddle
G ird er o f a t w o -la n e B r id g e ................................................................................................................................... 160F ig u r e 4 .5 8 E ffe c t o f th e Loa d C ases N um ber on th e Shear D istribution F acto r for th e Interio r
G irder o f a t w o -lan e B r id g e ................................................................................................................................... 160F ig u re 4 .5 9 E ffe c t o f th e Lo a d C a ses N um ber o n th e Shear D istribution F acto r for th e E xterior
G irder o f a 3-lan e B r id g e .......................................................................................................................................... 161F ig u re 4 .6 0 e f f e c t o f th e Lo a d C ases N um ber o n th e Shear D istribution F a cto r for t h e M iddle
G irder o f a 3-lan e B r id g e .......................................................................................................................................... 161
XXI
F igure 4 .6 1 E ffect o f th e Load C ases N um ber on th e Shear D istribution F a ctor for th e Interio r
G irder o f a 3-lan e B r id g e ........................................................................................................................................^62F igure 4 .6 2 E ffect o f the Load Cases N um ber on th e Shear D istribution F actor for th e E xterior
GIRDER OF A 4-LANE BRIDGE........................................................................................................................................162F igure 4 .6 3 E ffect o f the Load C ases N um ber on th e Shear D istribution F actor for th e M iddle
G irder o f a 4 -lane B rid g e ........................................................................................................................................163F igure 4 .6 4 E ffect o f the Load Cases N um ber on th e Shear D istribution F actor for th e Interior
GIRDER OF A 4-LANE BRIDGE........................................................................................................................................163F ig u re 4 .6 5 E f f e c t o f G ir d e r L o c a t io n o n t h e S h e a r D is tr ib u t io n F a c t o r d u e t o D e a d L o a d in g . 164 F ig u re 4 . 6 6 E f f e c t o f G ir d e r L o c a t io n o n t h e S h e a r D is tr ib u t io n F a c t o r d u e t o F u l l y L o a d e d
La n e s ................................................................................................................................................................................. 164F ig u re 4 .6 7 E f f e c t o f G ir d e r L o c a t io n o n t h e S h e a r D is tr ib u t io n F a c t o r d u e t o D e a d L o a d in g . 165 F ig u re 4 . 6 8 E f f e c t o f G ir d e r L o c a t io n o n t h e S h e a r D is tr ib u t io n F a c t o r d u e t o F u l l y L o a d e d
La n e s ................................................................................................................................................................................. 165F igure 4 .6 9 E ffect of the T ype o f Support on the Shear D istribution Factor due to D ea d Loading
(H inge Support L ine) .................................................................................................................................................. 166F igure 4 .7 0 E ffect of the T ype of Support on the Shear D istribution Factor du e to F ully L oaded
Lanes (H inge S upport L ine) .................................................................................................................................... 166F igure 4 .7 1 E ffect of the T ype o f Support on the Shear D istribution F actor due to Dead Loading
(Roller Support L ine) ...............................................................................................................................................167F igure 4 .7 2 Effect of the T ype o f Support on the Shear D istribution F actor du e to F ully Loaded
Lanes (Roller Support L in e) ..................................................................................................................................167F igure 4 .7 3 Com parison betw een th e Shear D istribution Factors of the E xterior G irder du e to
Truck Loading as Specified in the CHBDC and from the C urrent St u d y .........................................168F igure 4 .7 4 C om parison betw een th e Shear D istribution Factors of the E xterior G irder du e to
F atigue Loading as Specified in th e CHBDC and from the Current St u d y ..................................... 168F igure 4 .7 5 Com parison betw een th e Shear D istribution Factors of the M iddle G irder d u e to
Truck Loading as Specified in the CHBDC and from the C urrent St u d y .........................................169F igure 4 .7 6 Com parison betw een th e Shear D istribution Factors of th e M iddle G irder d u e to
Fatigue due to T ruck L oading as Specified in the CHBDC and from the C urrent St u d y 169F igure 4 .7 7 Effect of C urvature on the M om ent D istribution Factor for th e E xterior G irder due
to D ead Lo a d .................................................................................................................................................................170F igure 4 .7 8 Effect of C urvature on the M om ent D istribution Factor for th e E xterior G irder due
to F ully Load ed La n e s .............................................................................................................................................170F igure 4 .7 9 Effect of C urvature on the M om ent D istribution Factor for the E xterior G irder due
to Partially Load ed La n e s .................................................................................................................................... 171F igure 4 .8 0 Effect of C urvature on the M om ent D istribution Factor for the E xterior G irder du e
TO Fatigue Lo a d in g .....................................................................................................................................................171F igure 4 .81 Effect of C urvature on the M om ent D istribution Factor for the M iddle G irder due
to D ead Lo a d ................................................................................................................................................................ 172F igure 4 .8 2 Effect of C urvature on the M om ent D istribution Factor for the M iddle G irder due
TO Fully Loa ded La n e s ............................................................................................................................................ 172F igure 4 .8 3 Effect o f C urvature on the M om ent D istribution Factor for th e M iddle G irder due
TO Fatigue Lo a d in g ................................................................................................................................................... 173F igure 4 .8 4 Effect o f C urvature on the M o m ent D istribution Factor for th e Interior G irder due
TO D ead L o a d ............................... 173F igure 4 .8 5 Effect of C urvature on the M om ent D istribution F actor for th e Interior G irder due
TO F ully Loa ded La n e s ............................................................................................................................................ 174F ig u re 4 . 8 6 E f f e c t o f C u r v a t u r e o n t h e M o m e n t D is tr ib u t io n F a c t o r f o r t h e I n t e r i o r G ir d e r d u e
to Partially Loaded L a n es .................................................................................................................................... 174F igure 4 .8 7 E ffect of C urvature on the M om ent D istribution Factor for th e Interior G irder due
TO F atigue Lo a d in g .....................................................................................................................................................175F ig u re 4 . 8 8 E f f e c t o f N u m b er o f G ird e r s o n t h e M o m e n t D is tr ib u t io n F a c t o r f o r t h e E x te r io r
G irder due to D ead L o a d .........................................................................................................................................175F igure 4 .8 9 E ffect of N um ber of G irders on the M om en t D istribution F actor for the E xterior
G irder due to Fully Loa ded L a n es .....................................................................................................................176
x x i i
F ig u r e 4 .9 0 E ffect o f N u m b er o f G irders o n th e M o m en t D istribution F acto r fo r t h e E xterio rG irder d u e to Pa r tia lly Lo a d ed L a n e s ............................................................................................................. 176
F igure 4 .9 1 E ffect o f N u m ber o f G irders o n th e M om en t D istribution F acto r for th e E xteriorG irder d u e TO F a tig u e Lo a d in g .............................................................................................................................. 177
F ig u r e 4 .9 2 E ffect o f N u m ber o f G irders on th e M om en t D istribution F actor for th e M iddleG ir d e r d u e t o D e a d L o a d ............................................................................................................................................177
F ig u re 4 .9 3 E ffect o f N u m ber o f G irders on th e M o m en t D istribution Factor for th e M iddleG irder d u e to F u lly Lo a d ed L a n e s ....................................................................................................................... 178
F ig u re 4 .9 4 E ffect o f N u m ber o f G irders on th e M o m en t D istribution Factor for th e M iddleG irder d u e to P a r tia lly L oad ed L a n e s ..............................................................................................................178
F ig u r e 4 .9 5 E ffect o f N u m ber o f G irders on th e M om en t D istribution F actor for th e M iddleG irder d u e to F a tig u e Lo a d in g .............................................................................................................................. 179
F ig u re 4 .9 6 E ffe c t o f N u m ber o f G irders o n th e M om en t D istribution F actor for th e Interio rG ird er d u e to D ea d L o a d ............................................................................................................................................179
F ig u re 4 .9 7 E ffect o f N u m ber o f G irders on the M o m en t D istribution Factor for th e InteriorG ir d e r d u e t o F u l l y L o a d e d L a n e s ....................................................................................................................... 180
F ig u r e 4 .9 8 E ffect o f N u m ber o f G irders on th e M o m en t D istribution Factor for th e Interior
G irder d u e to P artia lly L o a d ed La n e s ............................................................................................................. 180F ig u re 4 .9 9 E ffect o f N u m ber o f G irders on th e M o m en t D istribution Factor for th e Interior
G ird er d u e to Fa tig u e Lo a d in g .............................................................................................................................. 181F ig u r e 4 .1 0 0 E ffe c t o f G irder S pacing on th e M o m en t D istribution F actor for th e E xterio r
G irder du e to D ea d Lo a d ............................................................................................................................................181F ig u r e 4 .1 0 1 E ffec t o f G irder Spacing on the M om en t D istribution Factor for th e E xterio r
G irder d u e to F ully lo a d e d La n e s ....................................................................................................................... 182F ig u re 4 .1 0 2 E ffect o f G irder S pacing on the M om en t D istribution F actor for th e E xterio r
G ird er d u e to Pa r tia lly L oa d ed L a n e s ............................................................................................................. 182F ig u r e 4 .1 0 3 E ffec t o f G irder S pacing o n th e M o m en t D istribution F actor for th e E xterio r
G irder d u e to F a tig u e Lo a d in g .............................................................................................................................. 183F ig u re 4 .1 0 4 E ffe c t o f G irder S pacing on th e M o m en t D istribution Facto r for th e M iddle G irder
DUE TO Fu lly L oa d ed L a n e s .......................................................................................................................................183F ig u r e 4 .1 0 5 E ffe c t o f G irder Spacing on th e M o m en t D istribution Factor for th e M iddle G irder
DUE to F a tig u e L o a d in g ............................................................................................................................................... 184F ig u re 4 .1 0 6 E ffe c t o f G irder Spacing o n th e M o m en t D istribution Factor for th e Interio r G irder
DUE TO F u lly Loa d ed L a n e s .......................................................................................................................................184F ig u r e 4 .1 0 7 E ffect o f G irder Spacing o n th e M o m en t D istribution Facto r for th e Interio r G irder
d u e to F a tig u e L o a d in g ............................................................................................................................................... 185F ig u r e 4 .1 0 8 E ffe c t o f L o a d in g C on d itio n s on th e M o m en t D istribution F acto r for th e E xterior
G irder o f th e 10-m -span B r id g e s ............................................................................................................................ 185F ig u re 4 .1 0 9 E ffe c t o f Loa d in g C on d itio n s on th e M o m en t D istribution F acto r for th e M iddle
G ird er o f t h e 10-m -span B r id g e s ............................................................................................................................186F ig u r e 4 .1 1 0 E ffec t o f Loa d in g Con d itio n s on the M o m en t D istribution F actor for th e Interior
G ird er o f th e 10-m -span B r id g e s ............................................................................................................................186F ig u re 4. i l l E ffe c t o f C u rv a tu re on th e D eflection D istribution F actor for th e E xterio r G irder
d u e to D ea d Lo a d ........................................................................................................................................................... 187F ig u r e 4 .1 1 2 E ffec t o f C u rv a tu re on th e D eflection D istribution F actor for t h e E xterio r G irder
DUE TO F ully Lo a d ed L a n e s ....................................................................................................................................... 187F ig u re 4 .1 1 3 E ffe c t o f C u rv a tu re on th e D eflection D istribution F actor for th e E xterior G irder
d u e to Pa r tia lly L o a d ed L a n e s ............................................................................................................................. 188F ig u r e 4 .1 1 4 E ffect o f C u rv a tu re on th e D eflectio n D istribution F actor for t h e E xterio r G irder
DUE TO F a tig u e L o a d in g ............................................................................................................................................... 188F ig u re 4 .1 1 5 E ffec t o f C u r v a tu r e on th e D eflectio n D istribution F actor for th e M idd le G irder
DUE TO D ea d Lo a d ........................................................................................................................................................... • 89F ig u r e 4 .1 16Effect o f C u r v a tu r e on th e D eflection D istribution Facto r for th e M iddle G irder
DUE TO F ully Lo a d ed L a n e s .......................................................................................................................................189F ig u r e 4 .1 1 7 E ffect o f C u r v a tu r e on th e D eflectio n D istribu tio n F a ctor for t h e M id dle G irder
DUE TO Pa r tia lly L o a d ed L a n e s ............................................................................................................................. 190
X X lIl
F igure 4 .1 1 8 Effect o f C urvature on the D eflection D istribution Facto r for th e M iddle G irder
DUE to F atig ue L o a d in g .............................................................................................................................................190F igure 4 .1 1 9 Effect o f C urvature on the D eflection D istribution Factor for th e Interior G irder
DUE TO D ead Lo a d ........................................................................................................................................................ 191F igure 4 .1 2 0 E ffect o f C urvature on the d eflectio n D istribution Factor for the Interior G irder
DUE TO F ully Loa ded La n e s .................................................................................................................................... 191FIGURE 4 .121 E ffect OF C urvature on the D eflection D istribution Factor for the Interior G r d e r
DUE TO F atigue Lo a d in g .............................................................................................................................................192F igure 4 .1 2 2 E ffect o f N um ber o f G irders on the D eflection D istribution Factor for the Exterior
G irder due to D ead Lo a d ......................................................................................................................................... 192F igure 4 .1 2 3 E ffect o f N um ber o f G irders on the D eflection D istribution Factor for the Exterior
G irder due to F ully L oaded La n es ..................................................................................................................... 193F igure 4 .1 2 4 E ffect o f N um ber o f G irders on the Deflection D istribution Factor for the Exterior
G irder due to P artially L oaded L a n e s ........................................................................................................... 193F igure 4 .1 2 5 E ffect o f N umber o f G irders on the Deflection D istribution Factor for the E xterior
G irder due to F atigue Lo a d in g ............................................................................................................................194F igure 4 .1 2 6 E ffect of N um ber of G irders on the Deflection D istribution Factor for the M iddle
G irder due to D ead Lo a d .........................................................................................................................................194F igure 4 .1 2 7 E ffect o f N um ber of G irders on the Deflection D istribution Factor for the M iddle
G irder du e to F ully L oa ded L a n es .....................................................................................................................195F igure 4 .1 2 8 E ffect o f N um ber o f G irders on the Deflection D istribution F actor for the M iddle
G irder due to Partially L oaded La n e s ........................................................................................................... 195F igure 4 .1 2 9 E ffect o f N um ber o f G irders on the D eflection D istribution F actor for the M iddle
G irder du e to Fa tig ue Lo a d in g ............................................................................................................................196F igure 4 .1 3 0 E ffect o f N um ber o f G irders on the D eflection D istribution Factor for the Interior
G irder due to D ead Lo a d .........................................................................................................................................196F igure 4 .131 E ffect o f N um ber of G irders on the Deflection D istribution F actor for the Interior
G irder du e to Fully Loaded La n es .....................................................................................................................197F igure 4 .1 3 2 E ffect o f G irder Spacing on the D eflection D istribution Factor for the E xterior
G irder due to D ead Lo a d .........................................................................................................................................197F igure 4 .1 3 3 E ffect o f G irder S pacing on the D eflection D istribution Factor for the E xterior
G irder due to Fully Loaded La n es .....................................................................................................................198F igure 4 .1 3 4 E ffect o f G irder Spacing on the D eflection D istribution F actor for the E xterior
G irder due to Partially L oaded La n e s ........................................................................................................... 198F igure 4 .1 3 5 E ffect o f G irder Spacing on the D eflection D istribution F actor for th e E xterior
G irder due to Fatigue Lo a d in g ............................................................................................................................199F igure 4 .1 3 6 E ffect o f G irder Spacing on the D eflection D istribution F actor for th e M iddle
G irder due to D ead Lo a d .........................................................................................................................................199F igure 4 .1 3 7 E ffect o f G irder Spacing on th e D eflection D istribution F actor for the M iddle
G irder due to Fully Loaded La n es .................................................................................................................... 200F igure 4 .1 3 8 E ffect o f G irder Spacing on th e D eflection D istribution F actor for th e M iddle
G irder due to Fatigue Lo a d in g ........................................................................................................................... 200F igure 4 .1 3 9 E ffect o f G irder Spacing on th e D eflection D istribution F actor for th e Interior
G irder due to D ead Lo a d ........................................................................................................................................ 201F igure 4 .1 4 0 E ffect of G irder Spacing on the D eflection D istribution F actor for th e Interior
G irder due to F ully L oaded L a n es .................................................................................................................... 201F igure 4 .141 E ffect o f Loading Conditions on the D eflection D istribution Factor for the
Exterior G irder o f 10-m -span Br id g e s ............................................................................................................. 202F igure 4 .1 4 2 E ffect o f Loading Conditions on the D eflection D istribution Factor for the M iddle
G irder o f 10-m -span Br id g e s ................................................................................................................................. 202F igure 4 .1 4 3 E ffect o f Loading Conditions on the D eflection D istribution Factor for the
Interior G irder o f 10-m -span B r id g e s ...............................................................................................................203F igure 4 .1 4 4 E ffect o f C urvature on the W arping-to-B ending ratio for th e E xterior G irder d u e
TO D ead Lo a d ................................................................................................................................................................ 204F igure 4 .1 4 5 E ffect o f C urva ture on th e W arping-to -B enddmg ratio for th e E xterior G irder d u e
TO FULLY L oad ed L a n e s ............................................................................................................................................204
XXIV
F ig u r e 4 .1 4 6 E ffect o f C u r v a tu r e o n t h e W a rpin g -to -B end in g ratio for th e E x terio r G irder dueTO P a r tia lly Lo a d ed L a n e s ......................................................................................................................................205
F ig u r e 4 .1 4 7 E ffect o f C urv a tu re o n th e W a rpin g -to -B ending ratio for th e E xterio r G irder dueTO F a tig u e Lo a d in g ....................................................................................................................................................... 205
F ig u r e 4 .1 4 8 E ffec t o f C u r v a tu r e o n th e W a rpin g -to -B ending ratio for the M iddle G irder due toD ea d L o a d ........................................................................................................................................................................ 206
F ig u r e 4 .1 4 9 E ffect o f C urv a tu re on th e W a rpin g -to -B ending ratio for th e M iddle G irder du e to
F ully L oa d ed L a n e s .................................................................................................................................................... 206F ig u re 4 .1 5 0 E ffect o f C u r v a tu r e on th e W a rpin g -to -B ending ratio for th e M iddle G irder du e to
P a r tia lly Lo a d ed L a n e s ............................................................................................................................................ 207F ig u r e 4 .1 5 1 E ffect o f C u r v a tu r e on th e W a rpin g -to -B ending ratio for th e M iddle G irder d u e to
F a tig u e Lo a d in g ............................................................................................................................................................. 207F ig u r e 4 .1 5 2 E ffect o f C u r v a tu r e o n th e W a r pin g -to-B ending ratio for th e Interio r G irder d u e
TO D ea d L o a d ................................................................................................................................................................... 208F ig u r e 4 .1 5 3 E ffect o f C u r v a tu r e on th e W a rpin g -to -B ending ratio for th e Interio r G irder d u e
TO F u lly L oa d ed L a n e s ...............................................................................................................................................208FIGURE 4 .1 5 4 E ffect OF C u r v a tu r e ON th e W a r pin g -to -B ending ratio for th e Interior G irder d u e
TO P a r tia lly L oa d ed L a n e s ...................................................................................................................................... 209F igure 4 .1 5 5 E ffect o f C u rv a tu re on th e W a rpin g -to -B ending ratio for th e Interio r G irder due
TO F a tig u e L o a d in g ....................................................................................................................................................... 209F ig u r e 4 .1 5 6 C o m pa riso n betw een th e M o m en t D istribution F actors o f th e E xterior G irder due
TO T ru ck Lo a d in g as Specified in th e CHBDC and from the C u rrent St u d y ...................................210F igure 4 .1 5 7 Com pa riso n b etw een th e M o m en t D istribution Factors o f th e E xterio r G irder due
TO F a tig u e L o a d in g as Specified in th e CH BD C and from th e C u rrent St u d y ............................... 210F ig u re 4 .1 5 8 C om pa riso n betw een th e M om en t D istribution F actors o f th e M iddle G irder d u e to
T r u c k L o a d in g a s S p e c if ie d in t h e CH BD C a n d fr o m t h e C u r r e n t St u d y ............................................211F igure 4 .1 5 9 Com pa riso n betw een th e M o m en t D istribution Factors o f th e M iddle G irder due to
F a tig u e L o a d in g as Spec ified in th e CH BD C and from th e C u rrent St u d y ..................................... 211F ig u r e 4 .1 6 0 C o r r e l a t io n b e t w e e n M o m e n t a n d D e f l e c t io n D is t r ib u t io n F a c t o r s fo r t h e E x t e r io r
G irder o f th e Stu d ied brid ges due to T ruck Lo a d in g ................................................................................ 212F ig u r e 4 .1 6 1 Co r r ela tio n betw een M o m en t and D eflection D istribution Fa ctors for th e E xterior
G ird er o f th e Stu d ied brid g es d u e t o F a tig u e Lo a d in g .............................................................................212F ig u re 4 .1 6 2 C o r r ela tio n betw een M o m en t and D eflectio n D istribution F acto rs fo r th e M iddle
G ird er o f t h e Stu d ied brid g es d u e to T ruck Lo a d in g ................................................................................ 213F ig u r e 4 .1 6 3 C o r r ela tio n b etw een M o m en t a n d D eflectio n D istribution F a ctors for t h e M idd le
G ird er o f t h e Stu d ied brid g es d u e to F a tig u e Lo a d in g .............................................................................213
XXV
CHAPTER I
INTRODUCTION
1.1 General
The widely use of elevated highways to alleviate traffic congestions at existing at-grade
signalized junctions in urban areas became the common answer to ensure uninterrupted
smooth traffic flow in modern heavy traffic networks. One of the major components of the
elevated highways is the horizontally curved bridges, which in many cases are located in on-
and off-ramps with very tight radii of curvature and are characterized by complex verticalI ;■
and horizontal geometries. For this application, curved steel I-girder system is the preferred
{ choice for most designers for the following reasons:
1. Simplicity of fabrication and construction,
2. Less land is needed during erection,
3. Shallower sections can be designed,
4. Impose lighter weights on bridge foundation when compared to
precast/prestressed beams or segmental prestressed concrete box girder deck,
5. Excellent serviceability performance.
Generally, the horizontally curved girder bridges are more attractive to conform to existing
terrain. However, from the design point of view, the simple presence of curvature in curved
steel girders complicates, to a great extent, their behavior and design considerations over
those of straight girders.
There are two fabrication methods of the horizontally curved girders: (a) the flanges are cut
from plates to the stipulated curve (cut curve) and are then welded to the web plate, or (b)
fabrication of straight girder is carried out in the conventional manner and then the entire
girder is curved to the specific curve by applying heat in a controlled continuous or
intermittent manner to the edges of the flanges (heat curve). Figure 1.1 shows typical cross
section of a four-girder bridge. It consists of concrete deck slab supported over steel I-
girders. To enhance the bridge structural integrity during construction, cross-bracings as
well as top and bottom chords are used at equal inteiwals between bridge support lines.
In designing highway bridges, bridge codes require dead, live and other types of
loads to be considered. For live load design, the codes define a standard truck loading with
concentrated wheel loads. Both the longitudinal position and the transverse distribution of
the wheel loads are of great importance. In other words, the truck load must be positioned
longitudinally to produce the maximum positive and negative bending moments, shear and
deflection in the girders. North American Bridge codes define lateral-distribution factors for
different materials of construction that specify the fraction of each wheel load that must be
applied to each girder. For straight girder bridges, this allows each girder to be designed as
an isolated single girder. Thus, the distribution factor is of fundamental importance in bridge
design.
2
1.2 The Problem \ '
Although horizontally curved bridges constitute roughly one third of all steel
bridges, the structural behavior still not well understood (Zureick and Naqib 1999).
Curvature greatly complicates the behavior of curved steel used in bridges. It introduces out-
of-plane “bulging” displacement of the curved web resulting in an increase in stresses.
Based on literature review, the investigation of load distribution characteristics of such
bridges is needed. Currently, the Canadian Highway Bridge Design Code (CHBDC, 2000),
as well as the American bridge codes (AASHTO-LRFD, 2004; AASHTO, Guide, 2003;
AASHTO, 1996) specify load distribution factor equations for the design of straight
composite I-girder bridges and provide geometrically defined criterion when horizontally
curved bridges may be treated the same as straight bridges. In both cases, there is no
practical design method in the form of expressions for moment, shear and deflection
distribution factors for composite concrete-steel I-girder bridges with significant curved
alignment.
The CHBDC (2000) and the AASHTO Guide Specifications for Horizontally Curved
Bridges, (Guide, 2003) have recommended the use of grillage analogy, folded plate, finite-
strip, and finite-element methods for analysis of the curved steel bridges. Among the above
methods is the finite element method (FEM) which is a matter of concern in this study and is
frequently employed for accurate final analysis and results. Unfortunately, most engineers
are not familiar with the FEM and are reluctant to use this technique, especially in the
preliminary designs because of its time consuming in terms of modeling, results
interpretation, when the girders dimensions are not known. Consequently, ± e need of
simplified equations and methods are required to facilitate the design procedure. Hence, a
practical design method in the form of empirical expressions for moment, shear, and
deflection distribution factors need to be developed for curved composite steel I-girder
bridges to fill the gabs found in the bridge codes.
This study is an endeavor to achieve the abovementioned need. The results and the formulas
developed in this study will assist bridge engineers in predicting the live load distribution in
the horizontally curved composite steel I-girder bridges more reliably and economically.
1.3 Objectives
The objectives of this study are:
1. Identifying the key parameters that influence the lateral distribution of loads in
straight and curved composite concrete-steel I-girder bridges and calculating the
load distribution factors,
2. Providing accurate database that can be used for developing simplified design
method for curved composite concrete-steel I-girder bridges,
3. Developing simplified formulas for shear, moment, and deflection distribution
factors for straight and curved composite concrete-steel I-girder bridges.
1.4 Scope
The scope of this study includes the following;
1. A literature review of previous researches, text books, and codes of practice related
to the study,
2. Conducting a parametric-design-oriented study on the effect of key parameters on
load distribution among girders. The range of studied parameters include: The radius
of curvature, span length, number of girders, girders spacing, number of designs
lanes, cross-bracing intervals, and truck loading conditions, The parametric study
was performed using commercially available Finite Element Software “SAP2000”
on 240 curved and 80 straight composite I-girder bridge prototypes subjected to
CHBDC truck loading and bridges’ own weight,
3. Preparation of database that can be used to develop simplified design method,
4. Developing shear, moment, and deflection distribution factor formulas for curved I-
girder bridges.
1.5 Contents and Arrangement of this study
Following this introductory chapter, chapter II contains the literature review which is
thorough explanation of lateral load distribution factor concept and review of previous
works of distribution factors for straight and curved bridges. Chapter III describes the finite-
element method and “SAP2000” software used in the analysis, modeling, bridge
configurations, loading cases, and the methodological and theory-based approach for
predicting the lateral load distribution faetors pertained to the topic of this study. Chapter IV
presents the outcomes of the parametric study performed on the bridge prototypes, and
proposed empirical equations for load distribution factors. Lastly, Chapter V is the summary
and conclusions drawn from this study.
CHAPTER II
LITERATURE REVIEW
2.1 Concept of Lateral Load Distribution Factor
Dead loading and live truck loading are imposed on Bridges. The calculation of dead
load, in case of straight bridges, is simple. The distribution of deck slab, wearing surface,
and curbs or traffic barriers can be considered evenly among girders. Due to the fact that
curbs or traffic barriers are constructed after the concrete deck is cured, for better accuracy,
these dead loads can also be considered as the live load applied to the girders. However, this
approach is not valid in case of horizontally curved bridges due to the torsion effects
resulting from curvature. The curvature tends to increase the longitudinal moment in the
outside girder, decrease the longitudinal moment in the inside girder, and have an
intermediate effect on the remaining girders. Furthermore, curvature causes torsion and
consequently, lateral bending (warping) in the flanges of the girders that must be considered
(Brockenbrough, 1986).
In order to calculate the live load carried by each girder in case of a straight bridge,
lateral load distribution factor is a key element and important in analyzing existing bridges
and designing new ones. For simplicity, the concept of lateral load distribution factor can be
introduced herein by using a straight single girder and multi-girder bridge. Figure (2.1a)
shows the free body diagram of a straight single girder under live load P. Let q (x) be the
influence line of a certain section of the girder, then the internal force at this section can be
calculated as F = P x t] (x). It can be inferred from the above figure that this is a simple two-
dimensional problem since both the load and the girder deformation are in the (x-z) plane.
However, the mechanism is totally different for multi-girder straight bridge subjected to live
load P, as shown in Figure (2.1b). Lateral rigidity makes the live load P to distribute in the
lateral direction (y direction) as well as in the longitudinal direction (x direction). As a
result, the live load on the bridge is shared among the girders, and each girder is subjected to
different magnitude of the live load. In such case, the live load position and structural
deformation are three-dimensional (planes x-y-z) and consequently three-dimension theory
is needed to solve the internal forces of the structure. The common characteristic of three-
dimension theory is that the internal forces and deformation at any point of the structure can
be solved directly. Alternatively, the internal forces can be calculated using the influence
surface, in the same manner of using influence line to determine the internal forces in the
single girder. Let r](x,y) be the influence surface of a certain section of the structure under
live load, the response of the structure is then F=P x r\ (x,y). Since the live loads on the
bridge are multiple concentrated wheel loads, which can move both longitudinally and
transversely, the determination of maximum internal forces using influence surface is still
tedious and complicated. Accordingly, the influence surface method is not widely used in
practice.
To simplify the analysis, a frequently used method is to convert the complex three-
dimensional problem Figure (2.1b) into a simple two-dimensional problem Figure (2.1a).
The principle of this method is to break down and simplify the two-variable that influence
surface function t| (x,y) into the product of two single-variable functions, that is, r| (x,y) = r\
(x) X 'H (y)- Then the internal force at the section can be expressed as,
F = P x r i ( x , y ) = P x r i ( x ) X T i ( y ) ( 2 . 1 )
Where r\ (x) is the longitudinal influence line of that section for a single girder (Figure 2.1a),
T| (y) is the live load distributed to one certain girder when a unit load moves transversely
across the bridge, tj fyj is referred to as the transverse influence line for that girder and P x r|
(y) is the load distributed to that girder when live load P is at point a(x, y) (Figure 2.1b). As
a result, the internal forces at a certain section for a specific girder can be determined using
the longitudinal and transverse influence lines, which simplifies the three-dimensional
problem. In reality, actual truck loads are multiple wheel loads moving on the bridge. Figure
(2.2a) shows a multi-girder bridge subjected to truck loads. The rear, middle and front axle
loads of the truck are PI, P2 and P3, respectively. To determine the maximum response at
point (k) of girder No. 3, for example, the transverse influence line of girder No. 3, and the
worst loading position to determine the maximum magnitude of each axle load distributed to
girder No. 3 are first obtained. Secondly, the maximum response at section (k) of girder No.
3 using the longitudinal single girder influence line at section (k) is determined. Obviously,
if the positions of the truck wheels on the bridge are fixed, the load distributed value to
girder No. 3 is then fixed. In practice, the product of a factor (g) and the axle load expresses
this fixed value. Hence, the loads distributed to girder No. 3 of the rear, middle, and front
truck axle loads can be expressed as gP l, gP2, and gP3, respectively, as shown in Figure
(2.2b). The factor (g) is referred to'as lateral load distribution factor. It is important to know
that the maximum load distributed to one certain girder (here is girder No. 3) is a fraction of
each axle load (usually less than one).
It is noted that it is an approximate approach to convert the three-dimensional
problem into a two-dimensional problem, since the paths of the load being distributed to the
adjacent girders are eomplex. The concentrated load at one girder would no longer be
concentrated load at the same longitudinal position after being distributed to the adjacent
girders. However, (Yao, 1990) investigated theoretical and experimental research and
showed that the error was relatively small for lateral load distribution by lateral deformation
relationship. Moreover, the actual truck load on the bridge is not one single concentrated
load, but several wheel loads distributed at different longitudinal positions. Therefore, the
error would be even smaller for truck loading. Field testing has shown that distribution
factors to be conservative (Mounir, 1997).
The distribution factor (g) is different for each girder within the same bridge. It
varies with the variation of truck configuration, tmck longitudinal location on the bridge,
and the bridge lateral rigidity. The effect of truck longitudinal location is insignificant and
usually the distribution factor at girder maximum response location is used for design. The
bridge lateral rigidity is related to the relative stiffness of the girders and the deck. The load
distribution between girders is poor for transversely flexible bridges and is even for
transversely stiff bridges (Zhang, 2002). Figure (2.3) shows various deflections of the bridge
girders in a cross section when subjected to applied load P. In this Figure, E/j- is the bridge
transverse or lateral rigidity. Figure (2.3a) shows the deformation of the bridge structure
when the bridge transverse rigidity EIt is zero (very poor rigidity) and the middle girder is
subjected to load P . Since the load can only transfer to the middle girder, the distribution
factor {g) for the middle girder is one and for other girders is zero. But, when the bridge
transverse rigidity is infinity (full rigid section) and the same load P is applied at the middle
girder position, “every girder has the same deflection and the same magnitude of the load as
shown in figure (2.3c). As a result, the distribution factor (g) is identical for every girder is
equal to 1/A, where N is the numbers of girders. For a five- girder bridge shown in Figure
2.3c, the distribution factor {g) is 0.2. While for concrete, reinforced concrete, and steel
girder bridges, the transverse rigidity is between zero and infinity. When the middle girder is
subjected to load P, as shown in Figure (2.3b), the distribution factor for each girder is
between 1/A and one. The determination of the exact magnitude of the live load distributed
to each girder is a matter of interest and it is a key issue in bridge analysis and has been
studied by many researchers.
The analysis of structures under different types of loadings witnessed great
advancement due to the development of fast speed and high capacity computers, many
< computer programs were evolved and can be used to accurately analyze the structures under
dead and live loads and would promote more realistic calculation of the actual bridge
capacity. The availability of powerful personal computers and finite element analysis
software will assist bridge engineers in performing detailed and realistic modeling of the
deck geometry and boundary conditions and then in performing the analysis. However the
engineer needs to be familiar with the finite element method, mesh generation and its
density, accurate modeling of the geometry and support conditions, selection of proper
10
element types and their material properties, and loading positions that produce the maximum
response in the bridge. One way to estimate the maximum moment and shear response in
individual bridge girder would be to model the entire bridge in a three dimensional problem
which can be solved by using the finite element method (FEM) or other analytical methods
and determine all the internal forces and deflections in any individual member. The loading
in this case has to be varied, in both longitudinal and transverse positions, to find the worst
loading positions. For bridges with very complex configurations, this method might be the
only way to determine the accurate maximum moment and shear under live load for each
girder. However, for many types of bridges, this process could be very cumbersome and
unnecessary.
Distribution factors and empirical methods are still the major methods used in design
of modem bridges in North America. Empirical formulas were obtained by analyzing many
bridge systems and models. The development of the curved girder load distribution factors
procedure followed the same technique employed in determining the straight girder load
distribution equations. The difference is that to analyze a curved system is more complicated
than a straight one. For this reason, the following sections will review available literatures
for both straight bridges and curved bridges.
2.2 Review of Previous Research on Load Distribution
2.2.1 Review of Study on Distribution Factors for Straight Bridges
According to the level of bridge lateral rigidity, different methodologies are
implemented in practice, including lever mle, eccentric compression method, hinged joint
11
method, fixed joint method, orthotropic plate analogy, AASHTO Standard, AASHTO-
LRFD and CHBDC simplified method.
2.2.1.1 Lever Rule Method (Yao, 1990)
The lever rule is one of the most frequently used methods for calculation of
distribution factors. In this method the deck between the girders is assumed to acts as a
simply supported beam or cantilever beam, as shown in Figure (2,4). In this case, the load
on each girder shall be taken as the reaction of the wheel loads. Lever rule is very accurate
for two girder bridges. Lever rule can also be used for shear distribution near support, since
the load would pass to the pier or abutment mostly through the adjacent two girders. Lever
rule can also give very good results when the bridge transverse stiffness is relatively
flexible. However, the results usually would be slightly conservative for the interior girders,
and unconservative for the exterior girders.
2.2.1.2 Eccentric Compression Method (Yao, 1990)
This method can be applied to “narrow bridge” with adequate diaphragms along
bridge span. “Narrow Bridge” is defined as that the ratio of bridge width, B, to span length,
L, is less than 0.5, or for B/L is equal to 0.5 satisfying the ratios of bridge longitudinal
rigidity per unit length, Dy, to transverse rigidity per unit width, Dx, is greater than 0.48. The
deflection of a narrow bridge with adequate diaphragms under truck load is similar to that of
an eccentric compression member, as shown in Figure 2.5.
From the theory of mechanics, when girder k is subjected to load P, the load
distributed to girder i is:
12
Rj,= ( . . ! . . + - . j . : .— ) * p (2.2)n n
2 Saillii=l i=l
Where n is number of girders, /, is the moment of inertia of girder No. i; a,- and a* are the
distances from bridge centerline to girder No. i and k respectively. Therefore, the transverse
influence line can be obtained from Equation 2.2 when load P is equal to one. If all the
girders have the same cross section or the same moment of inertia, the control values of the
transverse influence line for girder No. 1 are simplified as (note that ai=a5):
+ - Ê l i . ) (2.3a)N n
I ai"/=/
ni5 = ( - L - _ ) (2.3b)N
'»iI a,"i=l
Where N is the number of girders. Once the two control values r\ 11 and t| 15 are determined,
the transverse influence line for girder No 1 is determined. The distribution factor can then
be obtained by arranging the trucks transversely on the bridge to get the worst situation.
Note that in the above procedure, the girder torque is ignored. When considering the girder
torque. Equations (2.3a) and (2.3b) become:
1 ai^n ii = ( l v " — ) (2.4a)
I aii=l
13
) (2.4b)
I ain
■
i=Iwhere
j6= ------- 1-----L < 1
1+ t / ' (2 .5)
12E l a h ,i=l
and G is modulus of elasticity in shear or modulus of rigidity; E is modulus of elasticity in
tension or compression; L is the bridge span length; 7, is the torsional inertia of girder No. i.
2.2.1.3 Hinged Joint Method (Yao, 1990)
2.2.1.3.1 Hinged Joint Method for Slab Bridges
This method can be utilized for slab bridges with pre-cast members connected by
tongue-and-groove joint. The deflection of a slab bridge under concentrated wheel load is
shown in Figure (2.6a). Figure (2.6b) illustrates the general internal forces occurred at the
tongue and groove joint, which are vertical shear g(x), transverse moment m(x), longitudinal
shear t(x), and normal force n(x). It was noted that the longitudinal shear t(x) and the normal
force n(x) are relatively small compared with vertical shear g(x) when the bridge is subjected
to truck load. Because of the joint is relatively short in configuration and very flexible in
resisting moment, the transverse moment m(x) as well as the longitudinal shear t(x) and
normal force n(x) can be neglected in analysis. Consequently, the joint can be simplified and
14
considered as a hinge; assuming only vertical shear force g(x) exists, as shown in Figure
(2.6c).
To convert the three-dimensional problem into a two-dimensional problem, the ratio
of the deflection, moment, shear, and applied load, in any two strips or girders must be equal
and equal to a constant C,
w I (x) M i (x) Qj (x) P i (x) = ---------- = ---------- = ----------- = C (2.6)W2 (x) M 2 (x) Q2 (x) P2 (x)
Sinusoidal load is assumed to meet this requirement and the sinusoidal load is in the form of
JVC
P(x) = Po sin ----- (2.7)
The free body diagram of a slab strip under sinusoidal load is shown in Figure (2.7).
The error of the sinusoidal load assumption is very small in view of the fact that along the
bridge span there will be many wheel loads. In order to obtain the distribution factor, the
girder transverse influence line must be obtained first. For a bridge with n strips, an
indeterminate problem of (n-1) order is to be solved to obtain the influence line. For
convenience, transverse influence line control values for bridges with 3 to 10 slab strips are
tabulated and the tables can be found in Bridge Engineering (Yao, 1990). After the
transverse influence line is obtained, trucks can then be arranged transversely across the
bridge to find the worst situation and the maximum distribution factors.
15
2.2.1.3.2 H ingedjoin t Method fo r T-Shaped Girder Bridge
The hinged joint method can also be used for small span concrete T-shaped girdeiL.
bridges without intermediate diaphragms. Figures (2.8a) and (2.8b) demonstrate the free
body diagrams of unit length section at bridge middle span of the hinged T-shaped girder
bridge under unit sinusoidal load. Unlike the case of slab bridges, the deflection of the T-
shaped girder flanges must be considered, as shown in Figures (2.8c) and (2.8d). When the
cantilever length is within 0.80 m and the span length is greater than 10 m, the tables for
calculating transverse influence line values for hinged slab bridges can also be used for
hinged girder bridges. For better accuracy, detailed calculation is required for bridges
beyond this range.
2.2.1.4 Fixed Joint Girder method (Yao, 1990)
In case when the lateral connection between girders is stiffer, the joint can be
considered as a fixed joint. In addition to shear force at the joint, moment must also be
considered, as shown in Figure (2.9). For n-girder Bridge, a 2(n-l) order of indeterminate
problem is to be solved to obtain the shear and moment at each joint. However, only
shearing force gi is considered for calculating distribution factor. Once g, is known, the same
procedure as in hinged joint method can be followed to obtain the transverse influence line
as well as the distribution factors.
2.2.1.5 Orthotropic Plate Analogy (Guyon-Massonnet or G-M Method)
In the case of concrete bridges with continuous slab, intermediate diaphragms and
when the bridge width to span length ratio B/L greater than 0.5, grillage system may be used
16
to simulate the bridge system. Alternatively, the bridge may be analogized to a rectangular
thin plate, which is called orthotropic plate analogy or known as Guyon-Massonnet (G-M)
method (Yao, 1990). Orthotropic plate is referred to when a plate possesses elastic
properties which are different in x and y directions. Figure (2.10a) shows the longitudinal
and transverse configuration of a bridge structure. In this case, the girder spacing is
considered as S, girder moment of inertia and torsional inertia are Ix and Itx, respectively,
diaphragm spacing is and diaphragm moment of inertia and torsional inertia are ly and Ijy
respectively. For very small values of S and Sc compared to the bridge width and span
length, and for fully composite action, we can distribute girder moment of inertia and
torsional inertia Ix and hx to the distance S and distribute diaphragm moment of inertia and
torsional inertia ly and hy to the distance Sc- Thus, the real grid system in Figure (2.10a) is
analogized to an imaginary plate as shown in Figure (2.10b). In Figure (2.10b), the thickness
in the x direction is shown in dashed line, which indicates that, the equivalent thickness in
the X and y directions are different for the analogized plate. The moment of inertia and
torsional inertia per unit width in the j: and y directions for the analogized plate are
considered as follows:
I t x l y I l y
Jx — > Jj x ~ j Jy ~ > Jj y — (2 .8 )
S S Sc Sc
For beam and slab concrete bridges and pre-stressed concrete bridges. Poisson’s ratio
V can be neglected for simplicity. In that case, the bridge can be analogized to an orthotropic
plate with rigidity per unit width EJx, Gx Jtx, Ex Jy. and Ox Jry The analogized orthotropic
(in configuration) plate differential equilibrium with Ex=Ey=E and V;c=Vy=v is:
17
3^W ' 9*w g VE J x + G(Jtx + Jry) + EJy--------- = p(x,y) (2,9)
dx^ dy^
Let Dx = EJx, Dy = EJy and H = G(Jtx + JTy)/ 2E, Equation 2.9 becomes:
4 4 49 W d W 9 W
D x + 2H ---------- +-Dy----------= p(x,y) (2.10)9 x ‘ d x ^ 9 y ^ 9 y ^
This is identical to the differential equation for orthotropic plate (in material elastic
properties). This means that analogized orthotropic (in configuration) plate can be solved the
same way as orthotropic (in material properties) plate, except that the stiffness constants
contained in the equations are different.
The internal forces can be obtained by solving the above equation for displacement w
under applied load. Directly solving the partial differential equation is difficult. For
convenience, solution charts had been developed by Guyon and Massonnet, which can be
found in Bridge Engineering (Yao, 1990) and can be used to easily obtain the transverse
influence line. Once the transverse influence line is obtained, the distribution factors can be
obtained by arranging the trucks transversely on the bridge.
2.2.1.6 AASHTO Methods
AASHTO introduced empirical methods which are more convenient to use as
Compared with the theoretical methods mentioned above. AASHTO defines the distribution
factor as the ratio o f the moment (or shear) obtained from the bridge system to the moment
18
(or shear) obtained from a single girder loaded by one truck wheel line {AASHTO Standard
1996) or the axle loads {AASHTO-LRFD 2004). It should be noted that AASHTO Standard
Specifications and AASHTO LRFD Specifications define the live load differently. The live
load in the Standard specifications consists of an HS 20 truck or a lane load. While, the live
load in the LRFD specifications consists of an HS 20 truck in conjunction with a lane load.
Since both trucks have a 1.8 m axle (gauge) width, it is assumed that the difference in the
live load configuration does not affect the lateral load distribution (Wassef, 2004).
2.2.1.6.1 AASH TO Standard Method 1996
AASHTO Standard specifications (1996) contain simple procedures used in the
analysis and design of highway bridges. AASHTO adopted the simplified formulas for
distribution factors based on the work done in the 1940s by Newmark (1948). AASHTO
typical procedure is used to calculate the maximum bending moment based on a single line
of wheel loads from the HS20 design truck or lane loading. This calculated bending moment
is then multiplied by the load distribution factor (S/5.5) or in the format of {S/D), where S is
the girder spacing in feet and D is a constant based on the bridge type to obtain the moment
in an individual girder. This method is applicable to straight and right (non-skewed) bridges
only. It was proved to be accurate when girder spacing was near 1.8m and span length was
about 18 m (Zokaie, 2000). For relatively medium or long bridges, these formulas would
lose accuracy.
19
2.2.1.6.2 AASHTO LRFD Method
The specifications outlined in Load and Resistance Factor Design, LRFD Design
specifications (AASHTO, 1994b) were adopted. This code introduced another load
distribution factors based on a comprehensive research project, National Cooperation
Highway Research Program (NCHRP) 12-26 which was entitled “Distribution of Live
Loads on Highway Bridges” and initiated in 1985, consequently the guide specification for
Distribution of Loads for Highway Bridges (AASHTO, 1994a) was found. This guide
recommends the use of simplified formulas, simplified computer analysis, and/or detailed
finite-element analysis (FEA) in calculating the actual distribution of loads in highway
bridges. It was noted that those new formulas were generally more complicated than those
recommended by the Standard Specifications for Highway Bridges (the above AASHTO
1996), but their use is associated with a greater degree of accuracy (Munir, 1997). For
example the lateral load distribution factor for bending moment in interior girders of
concrete slab on steel girder bridge superstructure is:
g = 0.15 + (S /3 f^ (S /L f^ (K g/12L t\f^ (2.11)
Where g = wheel load distribution factor; S = girder spacing (ft, 3.5 < S < 16); L = span
length of the beam (ft, 20 < L < 200); tg = concrete slab thickness (in., 4.5 < t < 12); Kg =
longitudinal stiffness parameter = n(l + .A e \); n = modular ratio between beam and deck
material; 1 = moment of inertia of beam (in." ); A = cross-sectional area of beam (sq in.) and
Cg = distance between the center of gravity of the basic beam and deck (in.).
20
AASHTO LRFD Specifications have become highly attractive for bridge engineers
because of its incentive permitting the better and more economical use of material. The
rationality of LRFD and its many advantages over the Allowable Stress Design method,
ASD, are indicative that the design philosophy will downgrade ASD to the background in
the next few years (Salmon and Johnson, 1996). The research results were first adopted by
AASHTO Standards in 1994 and were then officially adopted by AASHTO-LRFD in 1998.
More parameters, such as girder spacing, bridge length, slab thickness, girder longitudinal
stiffness, and skew effect are considered in the developed formulas which earned them
sound accuracy. The AASHTO-LRFD formulas were evaluated by Shahawy and Huang
(2001), their evaluation showed a good agreement with test results for bridges with two or
more loaded design lanes, provided that girder spacing and overhang deck did not exceed
2.4 m and 0.9 m, respectively. Outside of these ranges, the error could be as much as up to
30%. For one loaded design lane, the relative error was less than 10% for interior girders
and could be as high as 100% and as low as -30% for exterior girders. Shahawy and Huang
(2001) presented modification factors for the AASHTO LRFD formulas and the results of
the modified formulas showed good agreement with their test results.
2.2.1.7 Canadian Highway Bridge Design Code, 2000 (CHBDC)
The Canadian Highway Bridge Design Code, CAN/CSA-S6-00 is the 9'*’ edition of
the CSA Standard CAN/CSA-S6. It amalgamates and supersedes both the CAN/CSA-S6-88,
Design of Highway Bridges, and the OHBDC-91-01, Ontario Highway Bridge Design Code,
3" Edition. Earlier Editions of the Ontario Highway Bridge Design Code (OHBDC) were
21
published in 1983, and 1979 by the Ministry o f Transportation of Ontario. Earlier editions of
the CSA Standards were published in 1978, 1974, 1966, 1952, 1938, 1929, and 1922.
The above CHBDC used the limit state design philosophy and introduced simplified
methods for the analysis of different types of bridges after satisfying certain conditions. In
its simplified methods of analysis, CHBDC defines the lateral load distribution factors as
amplification factors that used to account for the transverse variation in maximum
longitudinal moments and shear intensities. The moment distribution factor is Fm and
defined as:
SN_\iCf
F „ , = — ---- - 3: ^ > 1.05 (2.12)F\ 1 +l 100.
And the shear distribution factor is Fy and defied as:
Fv = — (2.13)
Where S is center to center girders spacing in meters, N is number of girders, F is a width
dimension that characterize load distribution for a bridge, Cf is a correction factor, in %
obtained from tables in CHDBC, and p is;
_ —1:2 h u t < 1.0 (2.14)^ 0.6
Where We is the width of the design lane in meters.
2 2
2.2.1.8 Other Studies
Besides the AASHTO formulas, numerous papers have been published for load
distribution factors since 1950. Their findings are invaluable for further studies. Kostem and
DeCastro (1977) showed that the contribution of diaphragms to lateral load distribution was
marginal regardless of the loading pattern. Many graphical and simplified computer based
methods are also available for calculating wheel load distribution. Ontario Highway Bridge
Design Code (OMTC, 1983) used a one popular technique of design charts, However, Hayes
et al. (1986) developed a grillage analysis and simple computer program, SALOD
[developed at the University of Florida], to evaluate the lateral load distribution of simple-
span bridges in flexure, which have become more acceptable than monographs and design
charts. Bakht and Moses (1988) presented a procedure to calculate the constant D in the
AASHTO load distribution formula {S/D). Tarshini and Frederick (1992), using FEM, they
studied the effect of various parameters on wheel load distribution for I-girder highway
bridges and found that composite and non composite construction showed a negligible
effect; the effect of the most common types of channel diaphragm and cross bracing
between beams had negligible effect.
2.2.2 Distribution Factors for Curved Bridges
The first treatment of the analysis of curved beams was presented in 1843 by Barré
de Saint Venant as referred by Zureik (1998, 1999). McManus et al. (1969) presented the
first survey of the most published works related to horizontally curved bridges. His
bibliography list contained 202 references. After that his paper was discussed by Ketcheck
23
(1969); Pandit et al, (1970) who added additional references to the original list. Even though
thousands of articles on the subject have appeared in the literatures, however, serious studies
pertaining to the analysis and design of horizontally curved bridges begun only in 1969
when the Federal Highway Administration (FHWA) in the United States formed the
Consortium of University Research Teams (CURT). This team consisted of Carnegie
Mellon University, University of Pennsylvania, University of Rhode Island, and Syracuse
University, whose research efforts, along with those at University of Maryland, resulted in
the initial development of working Stress Design (WSD) or Allowable Stress Design (ASD)
criteria and tentative design specifications. The American Society of Civil Engineers
(ASCE) and the AASHTO Task Committee on flexural members (1977) compiled the
results of most of the research efforts prior to 1976 and presented a set of recommendations
pertaining to the design of curved I-girder bridges. The CURT research activity was
followed by the development of Load Factor Design (LFD) criteria (Stegmann and
Galambos 1976, Galambos 1978) adopted by AASHTO to go along with the ASD criteria.
These provisions appeared in the first Guide (1980) as well as the Guide (1993). It is
worthwhile to mention that the AASHTO guide specification for horizontally curved
highway bridges (1993) is primarily based upon research work conducted prior to 1978.
2.2.2.1 Heins and Siminou’s Study
In their study, Heins and Siminou (1970) presented and explained a simplified
method for evaluating the internal forces and deformations in radial curved girder system.
They introduced equations and factors that permit the determination of required cross-
sectional properties in a single, two, and three-span curved girder system, which are
24
necessary in utilizing various computer programs. A series of factors were developed by
comparing single straight; single curved; and curved system. They used AASHTO HS20-44
truck loading, and utilized two, three, and four trucks for four, six and eight girder system
respectively. The introduced factors were:
fscAmplification Factor, ^ 1 ,
Distribution Factor, K if s c
fcs ’Reduction Factor, K 3 = —-
f s c
Where f i e is the reaction on a single curved girder, fiss is the reaction on a single straight
girder, and fay is the reaction on a system of curved girders. The studies, which were
conducted, resulting in design equations, have the following limitations:
1. Girder spacing may be 2.1, 2.4, 2.7or 3 m.
2. Individual girder span lengths varied from 15 to 30m.
3. The girders of the system must have a constant curvature limited to radii of 30 to 180 m.
4. The number of girders in the system may be 4, 6 or 8.
5. Only two-and three-span continuous bridges were examined, with all interior end spans of
equal length.
Heins and Siminou’s concluded their study by introducing design charts for modification
factors of moment, shear, deflection, rotation, and warping torsion.
25
2.2.2.2 AASHTO Methods
2.2.2.2.1 ^^SH TO Guide Commentary Method
The Guide Commentary of the AASHTO Specifications 1993, (Guide, 1993), which
as mentioned earlier adopted the research results of Heins and Siminou (1970), gives the
distribution factors for bending moment as:
g =5.5
— L ( N + 3) — +0.7
4R(2.15)
where
S = girder spacing in ft. (7 < S < 12),
- RN = (R > 100 ft.),
100
L = span length in ft., and
R = radius of curvature in ft. (R > 100ft).
It is interesting to note that Equation 2.15 is analogous to AASHTO Standard
equation S/D. This equation is intended to present moment distribution of the exterior girder
and would be increasingly conservative for other girders across the bridge. Heins-Siminou’s
result is too conservative, mainly because of that the entire deck in the three-dimensional
model was not included. This method was omitted from the current version of the AASHTO
Guide (Guide, 2004)
2.2.2.2.2 AASHTO Guide Method
The effect of lateral bracing was considered in AASHTO Guide Specifications for
Horizontally Curved Bridges (1993). The AASHTO Guide equations take into account the
26
effect of lateral bracing, connecting the bottom flanges of the girders. The distribution
factors for both ASD and LRFD, in terms of the resulting maximum live load (normal stress
component + warping stress component) bottom flange stress in the girder, can be calculated
as:
For outside exterior girder:
3 .0 -0 .0 6 (L)S e b f—
ë ebf —
3/2
3 .0 -0 .0 6 (L)3/2
L
R
‘ l
R
+ 0.9 for all bays with bottom lateral bracing (2.16)
+ 0.95 or bottom lateral bracing in every other bay (2.17)
For inside exterior girder:
8 ibf — ê ebf
8 ib f — 8 ebf
'— " L "- 0.366 X + 0.944
R -—
L- 0.473 X R + 0.934
for all bays with bottom lateral bracing (2.18)
for bottom lateral bracing in every other bay (2.19)
All the dimensions in the equations (2.16) through (2.19) are in feet. Where, L is the outside
exterior girder span length, S is the girder spacing, and R is the radius of curvature of the
outside exterior girder. The maximum live load flange stress is obtained by multiplying the
distribution factor with the maximum stress based on grid analysis.
2.2.2.2.3 AASHTO with V-Load Modification Method
The history of the V-load method started in 1963 when it was presented by USX in
their report entitled USS Stmctural report, Analysis and Design of Horizontally Curved
Steel Bridge Girders. It was a simplified analysis of two-girder curved bridge. And it was
presented in 1965 as a chapter in U.S. Steel’s Highway Structures Design Handbook. The V-
27
Load method was a computer programmed about 1966 which was used for the analysis and
design of many curved bridges in the 1970 (Poellot, 1987). However, a limitation was that
the method and the program were applicable only to structures with radial support lines. In
the early o f 80s, the V-Load method continued to serve as a state of the art design tool when
it was applied to skewed bridges by USS Research. And it was first recognized by AASHTO
through some provisions included in ASSHTO Guide Specifications for Horizontally
Curved Bridges (1980). Figure (2.11), shows the interaction forces in two girder system
between girder and floor-beam consist of lateral reactive H forces along with vertical shears
V. If the cross fames are spaced at a distance d, then the radial flange reaction at girder (1) is
H i = M idi/hR i and at girder (2) is % = Midi/hRz. And with concentric girders and radial
cross frames dl/R l = d2/R2, hence the vertical shear force V between the cross frame and
the girder is obtained from the static’s:
V=(H1+H2)h/D = (Ml+M2)/(RD/d) = (Ml+M2)/K (2.20)
Where K=RD/d is a constant. Those shear forces then react on the girders resulting
in a set o f self-equilibrating girder shears. The net effect of the shears is to shift the total
load on the curved bridge toward the outside girder because they produced moments in the
girders outside of the bridge centerline which add to those that would exist in a straight
girder of the same developed length, and moments in the girders inside of the bridge
centerline which subtract from those that would exist in a straight girder of the same
developed length. These girder shears, which are applied as the external loads to the
equivalent straight structure to account for the curvature, are known as the V- loads.
28
Application of the external V-loads ensures that the internal forces in the straight structure
be nearly the same as those that exist in the curved structure under applied vertical loads. In
the V-Load analysis of a Multi-girder system, the bending moments caused by the applied
vertical loads at the cross bracings in each isolated developed straight girder are first
determined by applying the loads to straight girders. These vertical-bending moments will
hereafter be referred to as primary moments. The corresponding V-load moments, which are
caused by the V-load and are referred to as secondary moments, are then determined by
applying the V- loads in the proper directions to the straight girders at the cross bracings.
The final moments in the curved girder are then obtained by simply summing the respective
straight-girder primary and secondary moments. The V-load is calculated by (Grubb, 1984):
V = Z M p / ( C x K ) ( 2 . 2 1 )
Where
X Mp = summation of the primary moments in each girder at a particular cross bracing,
C =coefficient depending on the number of girder in the system (see Table 2.1),
K = (R X D )/Sd, R and Sd are for the outside girder
Where R = radius of curvature in feet,
Sd = diaphragm spacing, and
D = girder spacing.
The distribution factor can then be calculated as:
S primary + secondary momentgB = — X :----------------- (2.22)
5.5 primary moment
29
This method is referred to as the AASHTO with modified V-load Method. The results were
proved to agree with those from the FEM analysis (Brockenbrough, 1986) for exterior girder
and to be conservative for the interior girders.
2.2.2.3 Heins and Jin’s Method
The effect of cross bracing spacing on curved bridge distribution factors was studied
by Heins and Jin (1984). They investigated single-girder system and multi-girder system.
They developed a space model frame idealization and used grid analysis for obtaining their
results. They produced number of graphs and equations that correlate the effect of the cross
bracing and distribution factor, one major finding was as follows:
S5.5
0 .0 0 8 3 -hl.OS R
(0.4) (2.23)
Where g is the distribution factor. Sc is the cross bracing spacing in feet, S is girder spacing
in feet. The above distribution factor included the warping effect, and hence the equation
2.23 showed a good agreement with Brockenbrough’s, 1986 Study.
2 2.2.4 Y og and Littrell’s Study
Yoo and Littrell (1985) studied the response of a system of 5 girders horizontally
curved, connected by a slab and cross bracing under dead load and live load using three-
dimensional finite-element method (Sap6). There study considered the following different
value parameters: radius; length; and number of braced intervals. They developed empirical
30
design equations to predict the ratio of: (a) Maximum bending stress; (b) maximum warping
stress; and (c) maximum deck deflection. Their findings were (1) maximum bending stresses
and maximum deck deflections stabilized with minimal bracing; (2) warping stresses were
sensitive to the number of braced intervals. They also observed that partially loaded lanes; 2
trucks out of 3 trucks (deck width capacity) located near the outside edge of the bridge
produced higher stresses and deflection in the curved bridges due to the tilting of the bridge
deck created by the nonsymmetrical load distribution.
2.2.2.S Brockenbrough's Study
Brockenbrough (1986) utilized a 3-dimensinal model to study the effect of various
parameters on the load distribution for 4-girder curved bridge using FEM. His model was 2-
span continuous structure comprised of composite concrete deck with steel I-shape girder
and intermediate transverse cross frames between girders. His findings were as follows: (1)
the central angle per span, which includes the combined effect of curvature and span length
had larger effects, (2) the girder spacing, had larger effects identified; (3) variation in girder
stiffness and cross-frame spacing had relatively small effects on live-load distribution
factors. Brockenbrough also provided charts depicting the variation of the distribution
factors with the variation of these parameters.
2.2.2.6 Schelling, Namini, Fu’s Study
Schelling and Namini, (1989) studied the response of simple and continuous span
horizontally curved steel I-girder bridges with and without cross bracing, during
construction phase due to self own weight. They utilized 3-dimensional space frame model
31
and developed empirical equations of moment distribution factors for two, four, and six
girders bridges to determine the effect of the lateral bracing on curved bridges during
construction. Their analysis revealed that the results from the simple-span can be applied
conservatively to the continuous span bridges provided that the supports and radial and the
span length ratios do not differ greatly from unity. The empirical equation they developed
had a drawback because it can be used in conjunction with the results given by the 2-
dimentional grid analysis method. Another thing, is the range of the bridges spans
considered in their models were (36m to 90m) which are not practical for slab on steel I-
girder bridges.
2 .2 .2 .7 Davidson, Keller, and Y o g ’s study
Davidson et al. (1996) studied the effect of a number of parameters on the behavior
of the horizontally curved steel I-girder bridges connected by cross bracings. Their model
comprised of 3-girder Bridge and used shell elements to model the concrete deck; and
girders webs, whereas they used beam elements to model flanges, shear connectors and
cross frames. Finite-element, ABAQUS software was utilized for their investigation. The
above study resulted with the conclusions of that the (1) span length, (2) radius of curvature,
(3) flange width, and (4) cross frame spacing; have the greatest effect on the warping-to-
bending stress ratio. Based on this information, a regression analysis was performed to
predict the effect of these parameters on the warping-to-bending stress ratio. Consequently,
an equation was developed from this regression and proposed for the preliminary cross
frame spacing design:
32
Rbf “(---------- - ) (2.24)_ 2000L^ J
Where Smax is the maximum bracing spacing in meters, L is the span length in
meters, R is the radius of curvature in meters, and bf is the flange width in millimeters.
2.2.2.S Eissa, et. al. study
The shear distribution in curved steel I-girder bridges at construction phase (due to
self weight loading) was studied by Eissa et. al, (2000). They utilized finite-element
software SAP2000 to carry out bridge modeling. A three-dimensional, four-node shell
element was used to model the steel web, and two-node element was used to model the top
and bottom steel flanges. They studied the effect of number of parameters on the behavior of
shear distribution in curved steel I-girder bridge system, these parameters were; span-to
radius ratio (L/R), span length, number of girders, girder spacing and type of supports. For
3, 4, and 5 girders bridge models and for 15m, 25m, and 35m span lengths. Part of their
findings was proposing empirical expressions to determine the reaction forces at the
supports (hinges or rollers). Other parts of their findings were deduced that the degree of
curvature (L/R) ratio, number of girders, span length, and girder spacing were considered as
key parameters affecting the structural response of such bridges.
2.2 2.9 Zhang’s study
Zhang H., (2002) studied the load distribution factors for curved I-girder bridges
using finite-element method. A total of 111 bridge models with radius of curvature less than
33
450m were selected in his study, subjected to AASHTO truck loading. Mr. Zhang
considered the following range of parameters in his study:
> Radius of Curvature: 45 to 450 m;
> Girder spacing: 1.8 to 5.0 m;
> Span length: 15 to 70 m;
> Slab thickness: 170 to 300 mm;
> Longitudinal stiffness: 32122 to 72226 cm* ;
> Number of girders: 3 to 7;
> Cross frame spacing: 2 to 7 m;
He concluded that radius of curvature, girder spacing, and number of girders had
significant effect on the load distribution. Whereas, span length, slab thickness, and
longitudinal stiffness had slight effect. Moreover the effect of cross frame spacing and girder
torsional inertia could be neglected. He also concluded that the shear distribution factors have
similar trend as that of moment distribution factors. The difference is that the shear distribution
factor of the outside exterior girder and the inside exterior girder are close. Mr. Zhang
developed simplified equations for positive moment, negative moment, and shear distribution
factors for exterior and interior girders due to one-lane loading and multiple-lane loading. It
was found that the distribution factors of the outside exterior girder positive moment
obtained from AASHTO Guide Commentary method (1993) for multiple-lane loading were
less conservative compared with the results of FEM analysis. However, the results obtained
from AASHTO Guide Commentary were too conservative for other cases. AASHTO-LRFD
34
formulas for straight bridges led to either larger or smaller results when used for curved
bridges. The Heins and Jin’s formula was too conservative for all cases.
2.2.2.10 WassePs study
Wassef, (2004) conducted a theoretical study on 192 simply supported straight and
curved concrete slab-on-steel I-girder bridge prototypes to evaluate their structural response.
He examined the influence of several parameters on the moment, deflection, and warping
stress distribution in such bridges using commercially available finite-element computer
software SAP2000. In the study the bridge models were subjected to the Canadian Highway
Bridge Design Code (CHBDC) CL-625 truck and lane loading and dead loading. The study
considered the following parameters;
> Span length; 15, 25, and 35 m;
> Girder spacing; 2, 2.5, and 3 m
> Number of girders; 3, 4, 5, 6, & 7 for 2 m girder spacing; 3, 4, 5, & 6 for 2.5 m
girder spacing; 3, 4 & 5 for 3 m girder spacing;
> Span-to-radius ratio, L/R; 0.0, 0.1, 0.2, & 0.3 for span L=15 m; 0.0, 0.1, 0.3, & 0.5
for span L=25 m; 0.0,0.1,0.4, & 0.7 for span L=35 m
The study findings were; (1) curvature is the most critical parameter that influence the
design of curved bridges; (2) other parameters like span length, number of girders, and
girder spacing affect the values of the moment and deflection distribution factors in general;
(3) Full bridge live loading does not necessary produce the extreme design values of the
moment or deflection distribution factors; (4) warping-to-bending stress ratio values were
35
acceptable and within the limits, except for bridge with L/R ratio 0.7 and span length 35m;
(5) CHBDC moment and deflection distribution factors overestimate the structural response
of straight bridges.
2.3 Review of Linear Elastic Behaviour of Curved I- Girder System
The behavior of thin-walled members of open cross-section under flexure and torsion
has been established for a long time and has been reviewed in many books on elementary
mechanics. A recent comprehensive presentation of the basic theory of thin-walled beams,
including flexure, torsion, distortion, and stress distribution, can be found in “Analysis and
Design o f Curved Steel Bridges” (Nakai and Yoo, 1988). In curved bridge, the curvature
makes the cross bracings (or diaphragms) to resist torsional loads, which are of importance
for curved bridge stability. Correspondingly, cross bracings introduce restoring torques to
the girders and therefore cause non-uniform torsions in the girders. The torsions are resisted
partly by St.-Venant torsion and to some extent by warping torsion. The warping causes
lateral bending moment of the top and bottom flanges. The product of the lateral flange
moment and lever arm of the couple (less than girder depth) is often referred to as bimoment
(in the unit of force x length^). This bimoment causes twisting of the curved girders about
their longitudinal axes. For compression flange, the axial flange force tends to accentuate
curvature while the lateral flange bending moment tends to reduce it. However, the net
effect is always to increase curvature of the compression flange. For tension flange, the axial
force tends to reduce the curvature and the lateral flange bending moment tends to increase
it. The net effect can be either to increase or decrease the curvature of the tension flange,
depending on flange stress and stiffness.
- 36
Two approximate methods: AASHTO Guide (1993) and V-load method presented
below, can be used to estimate the flange lateral bending moment, M l a t
1) AASHTO Guide (1993) method
Mlat =Ms x DFb x DFbi x(0.35 L - 15) L
---------------------------- X —0.108L - 1.68 DR
D (2.25)
Where Ms is the equivalent straight girder moment due to truck load, which straight girder
will have a length equal to the arc length of the curved girder; DFb is the distribution factor
for bending moment; DFbi is the distribution factor for bimoment; D is the girder depth in
feet; R is the radius of curvature in feet; L is the span length in feet; and Z/ is the arm from
the centroid of girder top flange to the centroid of girder bottom flange in feet.
This equation should satisfy that the radius of curvature is greater than 100 feet.
2) V-Load Method
M l a t — Mv X
r2L' UN
10 DR(2.26)
Where Mv is the vertical moment of curved girder, and Lun is the unbraced length. The
exact solution of lateral flange moment is discussed in the following sections.
• 37
From the classic strength o f material theory, St.-Venant torque, Tp, is commonly
expressed in terms of the torsional rotation, 0 at any cross section as
d 0Tp = G J (2.27)
dx
Where J is the St. Venant torsion rigidity; G is the elastic modulus in shear; x is measured
along the member.
From warping theory, the warping torque, 7^ , can be expressed as;
=Vh (2.28)
Where V is the lateral shearing force in the flanges as shown in Figure 2.11; and h is the
distance from the top flange-shearing center to the bottom flange-shearing center. The
equation of equilibrium for torsion of a thin-walled member is then
d 0G J + Vh = T (2.29)
dx
Where T is the total torsion at the cross section.
From the elastic curve equation, lateral bending moment in the lateral direction of the upper
flange in Figure 2.12 is
— =- M (2.30)2 cb^
38
in which the X and Y axes are chosen with positive directions as shown in Figure 2.12; M is
the lateral bending moment in the flange at any section producing lateral bending in the
flange; E is the modulus of elasticity in tension or compression; and ly is the moment of
inertia of the entire cross section of the beam with respect to the axis of symmetry in the
web so that Vi ly closely approximate the value of the moment of inertia of a flange cross
section. In Figure 2.12, the deflection of the flange at section AB is
y = (2.31)
Differentiation of Equation 2.31 twice with respect to x gives
d^y h d 0
dx 2 dx(2.32)
Substituting this value of d^y/dx into Equation 2.30 gives
E lyhd^0 r— = - M (2.33)
4 dx^
Since dM / dx = V by differentiating both sides of Equation 2.33 with respect to x we obtain
Ely hd 0
Substituting the value of V in Equation 2.34 into Equation 2.29, which then becomes
39
d 0 Ely ( f 0(2 3 5 )
Let /w = Elyl^ /4, the warping torque can be written as
^ 0=- Ely, - - - - - (236)
dx
And Equation 2 3 5 can be rewritten as
d‘ 0 ê 0EI^ — — - GJ = ( (2.37)
dx dx
Where t is the distributed torque applied to the member; and EIw is warping rigidity.
Equation 2.37 along with two boundary conditions at each end can be used to describe the
behavior of a thin-walled member subject to torsion. The boundary conditions at each end
may be the rotation 0 and warping d0/dx.
2.4 Review of Analysis Methods for Curved System
In reality the exact solution of the above Equation 2 3 7 is difficult and complex.
Therefore for practical purposes, it is required to adopt other types of solutions which should
be easier and sufficiently accurate. The analysis methods found in the literature can be
classified into two major categories: approximate and refined methods as follows, Zureick
and Naqib, (1999):
40
2.4.1 Approximate methods '
From its name, approximate methods require a least modeling effort from the
designer, and therefore, are adequate for preliminary analysis and design purposes. The
following are some most frequently used in the analysis of curved bridges:
• The Plane-grid method
• The Space-frame method
• The V-load method
2.4.1.1 The Plane-grid or Grillage method
This method was first introduced in year 1965 by Lavelle and Boick and further
developed in 1971 by Lavelle et al. and in 1975 by Lavelle and Laska. This method model
the structure as an assemblage of two-dimensional grid members with one translational and
two rotational degrees of freedom. It is advisable to mention that this method does not
account for warping (Zureick and Naqib, 1999). The advantages of this method are: (1)
integration of stresses is not required, shear and moment values on girders are directly
obtained; (2) simple beam theory can be used to distribute the wheel load to adjacent nodes
when loads are applied between the nodal points; (3) it utilizes a less computer running time
with a plain grid idealization and only moderate effort is required for modeling. Whereas the
disadvantages of this method are: (1) the method is non rigorous and does not exactly
converge to the exact solution of the mathematical model; (2) Experience is required with
the grillage method (the mesh design and refinement can be artistic) in order to obtain good
solutions; (3) some discretion is required for assigning the cross section properties.
41
2.4.1.2 The Space-frame method
This method was first introduced in year 1973 by Brennan and Mandel for the
analysis of open and closed curved members. This method idealized the curved members as
three-dimensional straight members. In this method diaphragms and lateral bracing are
assumed as truss-like members that can carry only axial loads. The warping effect is not
usually included in this analysis (Zureick and Naqib, 1999).
2.4.1.3 The V-Load method (Grubb 1984)
Developed in the early of 1960’s, the V-Load method is a simplified
approximate analysis method for curved I-girder bridges. The United State Steel Corp had
initially extended the V-Load method to multi-girder bridges. The V-Load method evaluates
curved girders as a system of straight girders with additional loads externally applied (V-
Load) to account the effect of curvature and can be considered as a two-step process. First,
equivalent straight girders with span lengths equal to the arc lengths instead of the individual
curved girders are used so that the applied vertical loads are assumed to induce only
longitudinal girder stresses. Second, self-equilibrating external vertical shear forces (acting
on diaphragm location) are applied to the straight structure so that the resulting internal
forces are the same as those existing in the curved structure subjected to only vertical load
(refer to Figure 2.11). Thus, in the V-Load development, the curvature forces on the
equivalent straight structure are treated as externally applied load. These loads are dependent
on the radius of curvature, the bridge width, and diaphragm spacing (refer to Equation 2.24).
The V-load method was found suitable for approximate analysis of composite sections,
variable radius of curvature, and skewed supports. The effects of bracing in the plane of the
42
bottom flange are not considered. The dead load results obtained from the V-load methc
were proved to be very close to those obtained from the FEM analysis. For live load, th
lateral load distribution factor used in the V-load analysis has a significant influence on th
results. However; the V-load method has some drawbacks as follows: the V-Load method i
not valid when lateral bracing was present, furthermore, it is not accurate in predicting
diaphragm shear forces (McElwain and Laman, 2000); also it underestimate the innermos
girder stresses, does not consider bracing effect in the plane of the bottom flange, and its
reliability depends on the selection of the proper live-load distribution factors. Thus the V-
load method can only be recommended for preliminary analysis (Zureiek et. al. 1998).
2.4.2 Refined methods
The refined methods are more reliable, elaborate, computationally intensive, and
time eonsuming in terms of modeling. And hence these methods are used for final or
detailed analysis. Some major refined methods are stipulated as follows:
*t* The Finite-strip method
❖ The Finite-difference method
❖ Analytical solution to differential equations
♦Î* The Slope deflection method
❖ The Finite-element method
43
2.4.2.1 The Finite-strip method
The curved bridge is divided in this numerical method into many narrow strips in the
circumferential direction that are supported in their radial direction. The analysis of this
method considers bending, membrane action, warping, and distortional effect. This method
has been successfully utilized to analyze composite curved box and plate girders with
complete and incomplete interaction, using curved strip elements for the concrete slab and
steel girder and spring elements for shear connectors (Arizumi et al., 1982). This method
provides some simplicity (since only one single variable in the circumferential direction is
considered in the function, i.e. smaller unknown required) over the finite-element method,
however it does not offer the flexibility and the versatility of the latter method (Zureick and
Naqib, 1999).
2.4.2.2 The Finite-difference method
This method was utilized in the dynamic analysis of curved bridges with large
deflections and small rotations (Tene al et. 1975; Sheinman 1982). The basic of this method
is superimposing a grid on the structure and the governing differential equations are replaced
by algebraic difference equations that are solved for each grid point.
2.4.2.3 Analytical solution to differential equations
An analytical solution to the Governing Differential Equations (GDE) is obtained in
the method. The solution is usually a closed form or a convergent series solution, such as a
44
Fourier series. This method was used in studying curved bridge dynamic response (Culver
1967; Montaivao e Silva and Urgueira 1988).
2.4.2A The Slope deflection method
The partial differential equations are established in terms of slope-deflectior
equations, and the solution is assumed to be a Fourier series. The effects of curvature, non-
uniform torsion, and diaphragms are included the above analysis. The COBRA (Curved
Orthotropic Bridge Analysis) program (Bell and Heins 1969), developed by University ol
Maryland, is based on analytical techniques of the slop-deflection-Fourier series and it is
recommended by AASHTO Guide Specifications of 1993 to study composite and non
composite girder-slab action. This method was proved by experiment to be an accurate
analytical method of curved orthotropic deck bridge systems (Heins and Bell 1972).
2.4.2.S The Finite-Element Method (FEM), By Logan (2002)
This is the most famous and widely used method in many engineering applications.
The principal of this numerical method is discretizing the structure into small divisions, or
elements, where each element is defined by specific number of nodes (hence this process of
modeling a body by dividing it into an equivalent system of smaller bodies or units called
finite elements). The finite-element method is a numerical acceptable solution, it
formulation of the problem results in a system of simultaneous algebraic equations for
solution, rather than requiring analytical solutions (solutions of ordinary or differential
equations), which because of the complicated geometries, loadings, and material properties,
45
are not usually obtainable. The behavior of each element, and ultimately the structure, is
assumed to be a function of its nodal quantities (displacements and/or stresses), which
considered as the primary unknown of its nodal quantities. The modern development of the
finite-element method began by Hrennikoff in the 1941 and McHenry in 1943 using (one
dimensional) elements (bars and beams) in the field of structural engineering. In 1947 Levy
developed the flexibility or force method, and in 1953 he suggested that another method (the
stiffness or displacement method) could be a promising alternative for use in analyzing
statically redundant aircraft structures. However his equations were cumbersome to solve by
hand, and hence it only became popular after the advent of the high speed computers. Turner
et al. was the first who introduced the treatment of two-dimensional elements in 1956, they
derived stiffness matrices for truss elements, beam elements, and two-dimensional triangular
and rectangular elements in plane stress. The finite-element method extended to cover three-
dimensional problems only after the development of tetrahedral stiffness matrix which was
done by Martin in 1961.
2.4,2.5.1 Three-D im ensional M ethod
University o f Pennsylvania developed the STACRE (Shore and Wilson, 1973)
computer program which is characterized by a fully compatible three-dimensional flat plate
circular element. Many different elements and shape functions have been studied since then,
including using segmental and quadrilateral element for plate bending, annular conforming
and fully compatible four-node segment element for thin plates, horizontally curved three-
node isoparametric beam element, three-dimensional beam element with axial and
transverse displacements or arbitrary polynomial order, and so on. General finite element
46
packages, such as ABAQUS, ADINA, ALGOR, SAP, ANASYS and MSC/NASTRAN ar
also frequently used for curved bridges. The rapid advancement of computer technolog;
representing in producing high-speed and high capacity computer allows the three
dimensional modeling be possible. The bridge deck is usually modeled as shell element
including membrane and bending effects. Girder flanges are usually modeled as bean
elements to include axial and bending strains in two directions and torsional effects. Girdei
web can be modeled as shell element to account for the bending stiffness. Rigid beams are
usually used to connect the deck slab to girder flange and simulate the composite action with
slab. Cross bracings and wind bracings can be modeled as hinged bar element.
Three-dimensional plate/shell models can consider unusual geometry and complex
configuration and can get the most accurate results. The disadvantages are: (a) since most of
the programs do not allow loads to be placed at any point on the elements, equivalent nodal
loads must be calculated with care and the mesh must be fine enough to minimize errors that
may arise because of load approximations; (b) since the programs report stresses and strains
other than shear and moment values, calculation of shear and moment values from the
stresses must be carefully performed through integration over the beam section, and (c)
integration of stresses at node points is normally less accurate and may lead to inaccurate
results.
47
CHAPTER III
FINITE-ELEMENT ANALYSIS
3.1 General
The advancement of computers in terms of speed and storage capacity has led
the engineering researches to enter a new era. More extensive and approximate numerical
solutions to complicated engineering problems were initiated due to the wide use of the
finite element method. Finite element analysis has proven to give reliable results when
compared to experimental findings; this built up trust encouraged the designers and code
writers to allow the implementation of the finite element method in the analysis and design
o f different engineering structures. The Canadian Highway Bridge Design Code (CHBDC
2000), section 5.9, permits the use of six different refined methods of analysis for short and
medium span bridges. The finite element method is one of the methods recognized by
CHBDC. Of all the above-mentioned six permitted methods, the finite element method is
considered to be the most powerful, and versatile. Stipulated below the most important
advantages of the finite element method:
1- Solutions can be without the use of governing differential equations,
2- It permits the combination of various structural elements such as plates, beams, and
shells,
3- It is able to analyze structures having arbitrary geometries with any material
variations thereof,
4- It is possible to automate every step involved in the method.
48
As a result, the finite-element method is very suitable for the analysis of curve»
composite I-girder bridges. The recent development in finite-element methods has facilitate»
to model a bridge in a very realistic manner and to provide a full description of its structura
response within the elastic and post-plastic stages of loading.
In this chapter a brief description of finite-element approach will be reviewed as wel
as descriptions of modeling the different components of the composite I-girder bridges. The
models that intended to be analyzed by the finite-element method comprised of the
reinforced concrete deck, steel top flanges, steel webs, steel bottom flanges, and cross
bracing as described in subsequent sections in this chapter. The available commercial finite-
element program, SAP2000 (Wilson and Habibullah, 2002), was utilized through this study
to determine the structural response of the modeled bridge prototypes. A general description
of this software is presented later in this chapter. The procedure to perform an extensive
parametric study on selected straight and curved bridge prototypes, loading cases, and
different bridge configurations, to evaluate loads distribution characteristics is explained
also in this chapter.
As mentioned earlier, the presence of curvature induces non-uniform torsion in the
curved girders which as a result produces lateral bending moment (warping or bi-moment)
in the top and bottom flanges that must be considered. Hence, the design of such girders
becomes complicated. The methodology of how to obtain the warping-to-bending stress
ratio is presented at the end of this chapter.
49
3.2 Finite-Element Approach
The finite-clement method is a numerical method for solving problems of
engineering and mathematical physics. It is the best solution for problems involving
complicated geometries, loading, and material properties, when it is generally not possible to
obtain analytical mathematical solution. This numerical method of analysis which begins by
dividing a body into an equivalent system of smaller bodies or units (finite-elements)
interconnected at points (nodes) common to two or more elements and/or boundary lines
and/or surfaces is called discretization. Hence, instead of solving the problem for the entire
body in one operation, it facilitates the formation of equations for each finite-element and at
the end; it will combine them to obtain the solution of the whole body. For the purpose of
simplifying the formulation of the above elements equations, matrix methods are
implemented. Matrix methods are considered as an important tools used to structure the
program of the finite-element methods to facilitate their computation process in high-speed
computers.
In general there are two approaches associated with the finite-element; (1) force or
flexibility method, and (2) displacement or stiffness method. It has been shown that for
computational purposes, the latter method is more desirable because its formulation is
simpler for most structural analysis problems; moreover a vast majority of general-purpose
finite-element programs have incorporated the displacement formulation for solving
structure problems. The finite-element method uses different types of elements; (1) one
dimensional element or so called linear element; (2) two-dimensional element which can be
in the forms of plane element or triangular and quadrilateral shape elements; and (3) three-
dimensional solid shape elements.
50
Selecting the most appropriate element type should be to model the most closely to th(
actual physical behavior. An equation is then formulated combining all the elements to obtain :
solution for one whole body. Using a displacement formulation, the stiffness matrix of eact
element is derived and the global stiffness matrix of the entire structure can be formulated bj
the direct stiffness method. This global stiffness matrix, along with the given displacemeni
boundary conditions and applied loads is then solved, thus that the displacements and stresses
for the entire system are determined. The global stiffness matrix represents the nodal force-
displacement relationships and is expressed in a matrix equation form as follows:
[P] = [KJ[U] (3.1)
Where:
[P] = nodal load vector;
[K] = the global stiffness matrix;
[U] = the nodal displacement vector;
3.3 SAP2000 Computer Program
The available commercial software “SAP2000” is a structural analysis program that
employs the finite-element method in the analysis and designs of complicated structures.
This program has a range of capabilities depending on the version used. SAP2000 is also
capable of analyzing structures in static and/or dynamic modes. Its finite-element library
consists of six elements.
1- Two-dimensional PLANE element
2- Three-dimensional FRAME element.
3- Three-dimensional SHELL element
51
4- Two-dimensional SOLID element
5- Three-dimensional SOLID element
6- Three- dimensional NLLINK element
In addition, subsets of these elements with varying degrees o f freedom are available
in the form of truss, frame, membrane, beam, strain, gap, and hook elements.
3.4 CHBDC Design Loading
The design of Highways and Bridges in Canada has its own criteria in terms of the
critical live loads selected in the design. Two types of live loads were specified in the
Canadian Highway Bridge Design Code (CHBDC, 2000); namely: truck loading and lane
loading. Both above mentioned loads were investigated in this study. Figure 3.1 shows a
view the abovementioned CHBDC live truck and lane loads namely; CL-W truck loading
and the CL-W lane loading. The CL-W truck is an idealized five-axle truck, the number
”W ” indicates the gross load (625) of the CL-W truck in KN. Wheel and axle loads are
shown in terms of W, and are also shown specifically for CL-625 truck. Whereas the CL-W
lane loading consists of CL-W truck loading, with each axle load reduced to 80% of its
original value, and superimposed within a uniformly distributed load of 9 KN/m over 3.0 m
width. For the purpose of this study, the following different CHBDC truck loading
configurations were considered:
(1) For studying the shear distribution between the girders, four levels of loading
were employed in this study, namely: Level 1, Level 2, Level 3 and Level 4
truck. Figure 3.2 shows the configurations of each of these load levels.
52
(2) For studying the moment and deflection distributions between the girders
one level is employed in this study, namely: Level 1 as shown in Figure 3.3,
since bridges with 10 m spans were considered in this study as a continuation
of Wassef s research work on moment distribution. Level 2, Level 3, and
Level 4 shown in Figure 3.3 were studied by (Wassef, 2004).
Levels 1, 2, 3 and 4 refer to the different bridge span lengths, 10, 15, 25 and 35 m ,
respectively. In studying the shear, moment and deflection distributions, the loading on the
bridge prototypes was applied in such a way to produce maximum forces and longitudinal
stresses.
3.5 Truck Loading Cases
The selection between the two different CHBDC types of live loads (CL-625 truck
and CL-625 lane) depends on whichever gives the greatest design values. A sensitivity study
was carried out in this regard showed that the CL-625 truck loading is governing the
extreme design values for the single girder of 10, 15, 25, and 35 m span lengths.
Accordingly, the CL-625 lane loading was not utilized in this study. The CHBDC requires
considering three limit states in bridge designs; namely:
i. The Ultimate Limit State (ULS), that involve failure, including rupture, overturning,
sliding, and other instability,
ii. The Serviceability Limit State (SLS), at which the effect of vibration, permanent
deformation, and cracking on the usability or condition of the structure are
considered,
53
iii. The Fatigue Limit State (FLS), at which the effect of fatigue on the strength or
condition of the structure are considered.
Dead load and truck load cases were considered for each of the above three CHBDC
requirements. Different loading configurations were also considered in this study
represented by: one-lane, two-lane, three-lane and four-lane bridges. As a result, a total of
48 different load cases assembled in 4 groups were employed on each level of the above
mentioned design requirements. Figures 3.4, 3.5, 3.6 and 3.7 show schematic diagrams of
the loading cases considered in determining the structural response of the exterior girder,
middle girder, and interior girder. It is essential to mention that the exterior girder here refers
to the girder which is far away from the centre of curvature in the bridge, whereas the.
interior girder is the one which is closest to the centre of the curvature. The middle girder
considered herein is any girder between the exterior and interior girders. A brief description
might be required here to give an idea on how these load cases were assembled in the above
diagrams. Diagrams in Figure 3.5 are taken as an example, the first loading case. Load case
(1) in each figure is always for the dead load of the structure. For the exterior girder, two
truck loading cases were considered. Load case (2) included a single truck load located in
the outer lane far away from the centre of curvature, on which the outer wheel load was
located at 0.6 m from the barrier. Load case (3) included two trucks one in each lane with
the outer wheel load located at 0.6 m from the barrier for the first truck and at 0.6 m from
the outer edge of the inner lane for the second truck. Load case (4) was intended to provide
the maximum load on the middle girder, which is at or very close to the centroid of the
bridge cross-section. In this case one truck was considered in each lane and located as
shown in Figure 3.5. Load cases (5) and (6) were intended to provide the maximum load in
54
the interior girder. Whereas Load cases (7) and (8) were intended to provide the maximun
load in the girders for fatigue design. In this case, CHBDC specifies only one truck loading
located at the centre of the actual lane, the width of this actual lane is determined aftei
deducting the width of both side shoulders from the total bridge width and dividing the
outcome by the number of design lanes. In this study the shoulder width taken to be in the
range of 1.0 to 1.5 m.
3.6 Composite Bridge Configurations
320 simply supported straight and curved concrete slab-on-steel I-girder bridge
prototypes were considered for finite-element analysis in this parametric study. Several
major parameters were considered as follows:
❖ Span length (L): 10, 15, 25, and 35 m;
❖ Girder spacing (S): 2, 2.5, and 3 m
❖ Number of girders (N): 3, 4, 5, 6, & 7 for 2 m girder spacing; 3, 4, 5, & 6 for 2.5 m
girder spacing; 3 ,4 & 5 for 3 m girder spacing;
❖ Span-to-radius of curvature ratio (L/R): 0.0, 0.1, 0.2, & 0.3 for spans L=10 m and
L=15 m; 0.0, 0.1, 0.3, & 0.5 for span L=25 m; and 0.0, 0.1, 0.4, & 0.7 for span
L=35 m.
Based on CHBDC code which specifies number of design lanes as a basis for bridge
width (see Tables 3.1 & 3.2), some of the above diversity of parameters were determined.
Other bridge configurations are listed as below:
> The deck slab thickness of was taken as 225 mm, -
55
> The deck slab width (Wc) was taken equal to the total bridge width minus 1.0 m to
consider the parapet thickness of 0.5 m on each side of the bridge,
> The depth of the girder webs was taken (1/20) of the centre line span except for
spans length =10 m, the depth was taken 0.75 m,
> The girder web thickness was considered equal to 16 mm,
> The over-hanged slab length was considered equal to half the girder spacing,
> The bottom and top steel flanges width and thickness were maintained 300 mm, and
20 mm, respectively.
Table 3.1 summarizes the straight bridge configurations considered in this study. Figure
3.8 visualizes the details of the typical composite deck steel I-girder bridge cross-section
used in this study.
X-type cross-bracings with top and bottom chords were utilized in this study as shown in
Figure 3.8. These bracings were spaced at equal intervals between the support lines and
were made of single steel angles dimensioned (L150xL150x25mm) and of 0.0075 m cross
sectional area. The equal intervals spacing between these cross-bracings were based on
equation 2.24, which was developed by Davidson et al. (1996) to reduce and limit the
warping-to-bending stress ratio. Typical plan of straight and curved girders with the
distribution of the transverse bracings are shown in Figure 3.9.
56
3.7 Research Assumptions
This study was based on the following assumptions;
(1) The reinforced deck slab had complete composite action with the top steel flange of
the girders (fully restrained, 100% shear interaction),
(2) All the bridges were simply supported,
(3) All materials were elastic and homogenous,
(4) The effect of road superelevation, and curbs were ignored;
(5) Bridges had constant radii of curvature between support lines.
Regarding the first assumption, Wassef, (2004) performed a sensitivity study to verify the
full composite action between the concrete deck and top girder flange. He used 35-m span
bridge with 7 girders spaced at 2.0 m, the concrete deck connected with the steel top flange
by M22 studs. The bridge was first analyzed using the finite-element method for a case
representing the M22 studs of 0.5 m spacing as frame elements and for a case representing
the M22 studs as shell element with equivalent area. Wassef then verified the results
manually using flexural beam theory, he concluded that the shell element revealed better
agreement with the manual calculation than the frame element, hence the latter was ignored.
Other design values were taken as follows; the modulus of elasticity of concrete material
was taken 28 GPa with Poisson’s ratio of 0.20, while these design values for steel material
were taken; 200 GPa and 0.30, respectively.
57
3.8 Finite-Element Bridges modeling
3.8.1 Geometric Modeling
Finite element modeling is partly an art guided by visualizing physical interactions
taking place within the body (Logan, 2002). One can appear to acquire good modeling
techniques through experience and by working with experienced people. To analyze all the
above mentioned composite bridges and to determine their structural response, a three-
dimensional finite-element model was adopted. To facilitate the analysis, the structure was
divided into major components as follows: a concrete deck slab, top steel flange, steel web,
bottom steel flange, and cross-bracings. In modeling, choosing the proper type of element or
elements to match as closely as possible the physical behavior of the problem is sometimes
difficult task if not guided by an educated judgment. In general an element yield best results
if its shape is compact and regular. Hence, in this study, four-node shell elements with six
degree of freedom at each node were used to model the concrete deck slab, the top and
bottom girder flanges, and finally the girder web. Whereas frame elements, pinned at both
ends, were used to model the cross-bracings with the top and bottom chords. Based on
previous work on finite-element modeling, four vertical shell elements were used in each
web, and another four were used horizontally for the deck slabs between the webs, whereas
two shell elements were used for the overhanged deck slab, and for the upper and lower steel
flanges. Figure 3.10 shows a finite-element discretization of four-girder cross section.
Figures 3.11 and 3.12 visualized views from the SAP2000 finite-element models for 6-
girder curved bridge.
58
3.8.2 Aspect Ratio : C-’;
In finite-element modeling, the aspect ratio is defined as the ratio of the longesi
dimension to the shortest dimension of a quadrilateral element. In many cases, as the aspect
ratio increases, the inaccuracy of the solution increases (Logan, 2002). Logan presented a
graph showing that as the aspect ratio raises above 4, the percentage of error from the exact
solution increases greater than 15%. Consequently, in this study, 72 elements in the
longitudinal direction were considered for 15, 25, and 35-m bridge span lengths, whereas
only 36 elements were considered for the 10-m bridge span length to keep down the aspect
ratio in acceptable ranges.
3.8.3 Boundary conditions
Setting proper boundary conditions to suit the nature of the problem and type of
structure is sometimes a complicated mission. Experience and previous work are also called
herein to choose carefully proper boundary conditions. In modeling the bridge supports in
this study, the lower nodes of the web ends were restrained against translation in such way
to simulate temperature-free bridge superstructure. In Figure 3.13, the interior support at the
right end of the bridge was restrained against movements in all direction. The middle
supports and the exterior support at the same right end of the bridge were restrained against
the vertical movement and against the movement in y-direction (towards the bridge
longitudinal direction). On the other end of the bridge (left end), all the supports were
restrained only against vertical movement, except for the interior support which in addition
to the vertical restraining, it was restrained in x-direction (towards the bridge transverse
59
direction) and as shown in Figure 3.13. Appendices A and B demonstrate samples o f
SAP2000 input files used in this study.
3.9 Calculation of the Shear Distribution Factors
In determining the shear distribution factor (SDF) for curved girder, the maximum
shear forces, (Rstraight)tnick, and (Rstraight)oL, were calculated for straight simply supported
beam due to a single CHBDC truck loading and own dead load, respectively, as shown in
Appendix C. It is important to mention that the span length for the above straight simply
supported beam is taken equal to the centerline length of the curved bridges. And from the
finite-element modeling, the maximum shear forces for dead load, fully loaded lanes,
partially loaded lanes, and fatigue loading were determined. Consequently, the shear
distribution factors (SDF) were calculated as follows:
For Exterior girders:
( S D F ) d l ext ~ (RfE. ext)DL / (Rstraight)oL , (3*^)
( S D F ) p L ext — (R-FE. ext)PL X N / ((Rstraight)truck X II) ( 3 . 3 )
( S D F ) p L e x t — (RpE. ext)PL X N X Rp / ((Rstraight)truck X II X Rl) (3.4)
( S D F ) p a te x t = (RpE.ext)Fat X N / (Rstraight)truck ( 3 . 5 )
For Middle girders:
(SDF)dL mid = (RFE.mid)DL / (R straight)oL ( * )
(SDF)pl mid = (RFE.mid)pL X N / ((Rstraight)truck X II) (3.7)
60
( S D F ) p L m id — (R F E .m id )pL X N X K ] J / ((R straigh t)truck X H X R l ) ( 3 . 8 )
( S D F ) p a t mid = (R F E .m id)F at X N / (R straight)truck ( 3 - 9 )
For Interior girders:
(SD F )d l int = (R F E .in t)D L / (R stra ig h t)ü L (3.10)
(S D F )fl int — (R F E .in t)F L X N / ((R straigh t)truck X II) (3.11)
( S D F ) p L i n t = (R pE .in t)P L X N X Rl7 ((R straigh t)truck X n X R l ) (3.12)
( S D F ) p a t in t — (R F E .in t)pat X N / (R straight)truck (3.13)
Where (SD F)dl , (SD F)fl , (SD F)pl , and (SDF)pat are the shear distribution
factors for dead load, fully loaded lanes, partially loaded lanes, and fatigue loading
respectively. And the symbols ext, mid, and int. refer to the exterior, middle, and interior
girders respectively. (RpE.ext)DL , (RFE.ext)pL , (RpE.ext)pL , and (RpE.ext)Fat are the greatest
reaction at the exterior girder supports found from the finite-element analysis due to dead
load, fully loaded lanes, partially loaded lanes, and fatigue loading respectively. The same
can be said for (RpE.mid)DL . (RFE.mid)FL , (RFE.mid)pL . (RFE.mid)Pat , (RpE.inOoL ,
(RpEinOpL, (RpE.int)pL, and (RpE.inJpat , but for the middle and interior girders supports.
n : number of design lanes, as listed in Table 3.2,
Rl ; multi-lane factor based on the number of the design lanes; as shown in Table 3.3,
R l' : multi-lane factor based on the number of the loaded lanes; as shown in Table 3.3,
N : number of girders. . v
61
3.10 Calculation of the Moment Distribution Factors
In the same manner mentioned above, the moment distribution factors (MDF) for
curved girders were determined. The maximum flexural stresses (o straight) tmck, ( o straight) d u
were calculated for the straight simply supported beam due to CHBDC truck loading and
own dead load, respectively. It is important here to determine the effective concrete slab
width, Be, in order to calculate the moment of inertia of the idealized girder. CHBDC
specifies the following two equations for calculating Be:
Be
B 15B
Lfor — < 1 5
B(3.14)
Be
B= 1 ,
Lfor — >15
B(3.15)
Where B is the clear spacing between girders = (S - 2bf); bf is the steel flange width;
L is the girder span length; and S is the girder spacing (all the units are in meters). The
following flexural formula was used to calculate the flexural stress of the idealized girder:
(^straight) truck ” M 7 (Yb) / It (3.16)
where Mt = the mid-span moment for a straight simply supported girder due to a
single CHBDC truck loading.
yb = the distance from the neutral axis to the bottom flange.
It = the moment of inertia of the composite girder.
62
In likewise manner, the flexure stress of the idealized girder due to dead load was
calculated as in the following flexural stress equation;
(^s tra igh t) DL ~ ^ D L (Yb) / ( 3 -1 7 )
where Mdl = the mid-span moment for a straight simply supported girder due to a
single girder dead load.
Also the results of the above equations were tabulated in Appendix C and were
verified by SAP2000 program. And from the finite-element modeling, the maximum
longitudinal moment stresses along the bottom flange for dead load, fully loaded lanes,
partially loaded lanes, and fatigue loading were calculated. Consequently, the moment
distribution factors (MDF) were calculated as follows:
For Exterior girders:
( M D F ) d l ext = (Ct FE. ext)0L / ( o straight)oL (3.18)
( M D F ) F L e x t ~ FE. ext)FL X N / ( ( (7 straight)truck X I I ) (3.19)
( M D F ) p L e x t — (c t FE. ext)pL X N X R l^ / ( ( c straight)truck X Ü X R ]_ ) (3.20)
( M D F ) F a te x t — ( t^ FE. ext)Fat X N / ( o straight)truck (3.21)
For Middle girders:
( M D F ) d l fnjfj — (<J FE. mid)DL I ( t^ straight)DL (3.22)
(MDF)pLniid ~ (tJ FE. mid)FL X N / ((o straight)truck X II) (3.23)
( M D F ) p l mid = (G FE. tnid)pL X N X Rl7 ((o straight)truck XllXRJ (3.24)
(MDF)Fat mid ~ ( g FE. mid)Fat X N / (c t straight)truck (3.25)
63
F o r In te r io r g irders:
(MDF)dl int = (a FE. in t)o L / straight)DL (3.26)
(MDF)fl int = (CJ FE. int>FL X N / ((CJ straight)truck X n) (3.27)
(MDF)pl int = (cJ FE. int)pL X N X Rl7 ((o straight)truck X n X R J (3.28)
( M D F ) F a t in t — FE. int)pat X N / (q straight)tnick (3.29)
Where (MDF)dl > (MDF)fl , ( M D F ) p ^ , and ( M D F ) F a t are the moment
distribution factors for dead load, fully loaded lanes, partially loaded lanes, and fatigue
loading respectively. And the symbols ext, mid, and int. refer to the exterior, middle, and
interior girders respectively, (a f e . e x t)o L , ( o f e . e x t ) p L , ( o f e . ex i)pL , and ( c t f e . ex t)p a t are
the maximum longitudinal stresses which are the greater at bottom flange points 1 and 3, as
shown in Figure 3.14, found from the finite-element analysis for the exterior girder due to
dead load, fully loaded lanes, partially loaded lanes, and fatigue loading, respectively. In the
same criteria, (a f e . mid)DL , (o f e . mid)pL , (< f e . mid)pL , FE. m id )F a t, FE. int)oL , FE.
in t)FL , (o FE. in t)pL , and (o PE. int)Fat are the maximum stresses which are the greater of
points 1 and 3 but for the middle and interior girders under the same above types of loading.
While R l, Rl', N, and n are defined as before.
3.11 Calculation of the Deflection Distribution Factors
To determine the deflection distribution factor (DDF) for curved girder, the
mid-span deflection, (A stra ig h t)tru ck . (A stra igh t)D L , were calculated for a straight simply supported
64
girder subjected to CHBDC truck loading, and dead loads, respectively. Similar to the above
SDF and MDF cases, the span of the straight simply supported girder is taken as the curved
length of the bridge centre-line. The deflections values of the idealized girder due to truck
loading and dead load, were calculated using SAP2000 software, and then verified by
manual calculations. Results of these calculations are presented in Appendix C. And from
the finite-element modeling, the mid-span deflection values at the middle of the bottom
flange due to dead load, fully loaded lanes, partially loaded lanes, and fatigue loading were
determined. Consequently, the deflection distribution factors (DDF), were calculated from
the following relationships;-
For exterior girders:
(DDF)dl ext = ( A f e ex t)o L / (Astraight) DL (3.30)
(^^P)FLext ~ (ApE ext)pL X N / truck X II) (3.31)
(DDF)pl ext = (A p E ext)pL X N X R i / ((A gtraight) truck X II X R l) (3.32)
(D D F )p a t .e x t = (A p E ext)Fat X N /(Agtraight) truck (3.33)
For middle girders:
(DDF)dl mid ~ (A p E mid)DL/ (Astraight) DL (3.34)
(DDF)pl mid = (ApE mid)FL X N / ((Agtraight) truck X II) (3.35)
(DDF)pl mid = (ApE mid)pL X N X R l / ((A jtraight) truck X II X R l ) (3.36)
(D D F )p a t .m id = (A p E mid)pat X N /(A jtraight) truck ( 3 . 3 7 )
65
F o r in te r io r g irders:
( D D F ) d L int = ( A f e in t)o L / (A s tra ig h t) DL ( 3 . 3 8 )
( D D F ) f l int = ( A f e in t)p L X N / ( (A s tra ig h t) truck X II) ( 3 . 3 9 )
( D D F ) p l int = (A fe in t)p L X N X R \ / ( (A s tra ig h t) truck X R X R l ) ( 3 . 4 0 )
( D D F ) F a t - i n t = (A F E in t)p a t X N /(A s tra ig h t) truck ( 3 . 4 1 )
Where ( D D F ) d l , ( D D F ) f l , ( D D F ) p l , and ( D D F ) p a t are the deflection
distribution factors for dead load, fully loaded lanes, partially loaded lanes, and fatigue
loading respectively. And the symbols ext, mid, and int. refer to the exterior, middle, and
interior girders respectively. ( A f e ext)oL , ( A p e ext)pL , ( A f e ext)pL , and ( A f e ex t)p a t are the
deflections at point 2, refer to Figure 3.14, found from finite-element analysis for the
exterior girder due to dead load, fully loaded lanes, partially loaded lanes, and fatigue
loading respectively. In the same manner, ( A f e m id)D L , ( A f e m id)pL , ( A f e m id )pL , ( A f e
mid)pat , (ApE int)oL , (Afe int)pL , (ApE int)pL , and (ApE int)pat are the finite-element
deflections for the middle and interior girders under the same above types of loading. While
Rl, Rl', N, and n are defined as before.
3.12 Warping to-bending stress ratio
As previewed in chapter 2, curved bridges are subjected to lateral bending moments
acted at the top and bottom flanges due to curvature as shown in Figure 3.15. This lateral
bending moment also called “bi-moment” or “torsional warping moment which induces
warping of the girder cross-section. This bi-moment will increase the longitudinal flexural
6 6
stress in flanges, and hence is called “warping stress”. To study the effect of this warpin
stress and its changes with different bridges parameters and load cases, the ratio betwee
warping stress to the average bending stress in the bottom flange is examined in this stud]
with the following relationship:
WBR = <iw/CTb = (O i-a3)/(cf3 + cyi) (3.42)
where CTi and 0 3 : the corresponding mid-span stresses at points 1 and 3 shown in Figun
3.14; Gw is the warping stress; and Gy is the average bending stress in the bottom stee
flange.
67
CHAPTER IV
RESULTS FROM THE PARAMETRIC STUDY
4.1 General
As mentioned in the previous chapter the finite element “SAP2000” program was
chosen to conduct the current research. A parametric study on 256 simply supported straight
and curved concrete slab-on-steel I-girder bridge prototypes was conducted to investigate
the shear distribution factors. In addition to that; another parametric study conducted on 64
composite bridge prototypes to investigate the moment and deflection distribution factors as
a continuation of W assefs research work (2004). All the above-mentioned bridge
prototypes were analyzed to evaluate their structural response against a total of 48 different
combinations of load cases (dead loading as well as the Canadian Highway Bridge Design
truck loading, CHBDC truck). This chapter presents the results of the above parametric
studies in terms of ( 1 ) shear distribution among composite girders, (2 ) moment distribution
among composite girders, (3) deflection distribution, and (4) warping-to-bending stress
ratios in steel flanges. In this study the following major key parameters were considered:
(i). Span-to-radius of curvature ratio (L/R),
(ii). Number of girders (N),
(iii). Girder spacing (S),
(iv). Bridge span length (L),
(v). Number of cross-bracing intervals between support lines.
6 8
As mentioned in Chapter (I) and based on the results from this research, the objectives
this parametric study were to:
1- Scrutinize the influence of the major parameters on the structural response
composite I-girder bridges.
2- Generate a database for the maximum longitudinal bending stresses in bottc
flanges and shear forces at support locations.
3- Developing new simplified formulas for Shear and Moments Distribution Factors f
curved composite concrete steel I-girder bridges that help the design engineers and
code authors.
4.2 Shear Distribution in Simply Supported Composite Curved Bridge;
4.2.1 Effect of Curvature
Curvature is considered one of the major factors that influencing the curved bridge behavic
and controlling the stress distribution among their girders. The results of current parametri
study conformed this fact. Figures 4.1, 4.2 and 4.3 show the variation in the she£
distribution factors with increase in span-to-radius of curvature ratio (L/R), for the exteric
girder of one-lane bridge comprised of 3 girders spaced at 2 m under dead load, fully loade(
lanes with truck loading, and fatigue loading, respectively. It can be observed that as L/I
ratio increases, the shear distribution factor for the exterior girder increases, for example ii
Figure 4.2, as the L/R ratio increased from 0.1 to 0.4 for the 35 m bridge span length, the
69
shear distribution factor increased from 1.87 to 2.43. Also, the rate of increase of the shear
distribution factor is observed to generally increases with the increase in span length.
The above trend however, is considerably changed in the behavior of the shear distribution
factors o f the middle girder as shown in Figures 4.4, 4.5, and 4 . 6 due to dead load, fully
loaded lanes, and fatigue loading, respectively. Except for the dead load case, it can be
observed that the shear distribution factor slightly decreases as the L/R ratio increases for
bridge span lengths 10 and 15 m, whereas it follows the same pattern of the exterior girder
for the span lengths 25 and 35 m. On the other hand, an inverse relation between the shear
distribution factors and the curvature become more dominant in the behavior of the interior
girder (the closest to the central of curvature) as shown in Figures 4.7, 4.8 and 4.9 for dead
load, fully loaded lanes, and fatigue loading, respectively. This is due to the fact that the
torsion stresses induced from the curvature tend to increase longitudinal bending stresses in
the exterior girders and to reduce them in the interior girders. A two-lane bridge, with 4
girders spaced at 2.5 m was also investigated to broaden the findings with respect to the
effect o f curvature. The effects of the span-to-radius ratio (L/R) on the shear distribution
factor found to be similar to those of the above-mentioned one-lane, 3 girder system. This
can be observed in Figures 4.10, to 4.16, for different load cases of the exterior and interior
girders.
CHBDC considers curved bridge to be treated as a straight one if / (b.R) is not
greater than 1, where L is the curved span length, b is half of the bridge width, and R is the
radius of curvature. Considering the above CHBDC limitations in this study, the resulting
L/R ratio for the bridges considered in this study are:
70
(1) For 10-m bridge span; L/R ratio equal 0.3 and 0.7 for 6 -m and 14-m bridge wic
respectively;
(2) For 15-m bridge span; L/R ratio equal 0.2 and 0.47 for 6 -m and 14-m bridge wid
respectively;
(3) For 25-m bridge span; L/R ratio equal 0.12 and 0.28 for 6 -m and 14-m bridge wid
respectively;
(4) For 35-m bridge span; L/R ratio equal 0.09 and 0.2 for 6 -m and 14-m bridge widl
respectively;
When the above limiting values are applied to Figures 4.1 to 4.6 for the exterior and midd
girders of a one-lane bridge and Figures 4.10 to 4.13 for the exterior girders of a two-lai
bridge, it can be clearly seen that CHBDC significantly underestimates the shear distributic
factors for the exterior and middle girders of the curved system bridges.
4.2.2 Effect of Number of Girders
A bridge of 35-m span with 2.0 m girder spacing was selected to investigate the effect of th
number of girders on the shear distribution factors. Results on the shear distribution factor c
the exterior girder due to dead load revealed a reduction in the shear distribution factor witl
the increase in number of girders for span-to-radius of curvature ratio greater than 0 . 1, a
shown in Figure 4.17. However, different trend is noticed in Figures 4.18 and 4.19 for fulb
loaded lanes and partially loaded lanes, respectively. It can be observed that the shea
distribution factor decreases (from 2.43 to 1.60 as shown in Figure 4.18 for L/R=0.4) as
number of girders increase to 4, then the above relation is reversed (from 1.6 to 1.92) foi
number of girders more than 4. Figure 4.20 showed another trend in the relation between the
71
shear distribution factors and the number o f girders for fatigue loading case, the shear
distribution factors fluctuate with increase in number of girders. Figures 4.21, 4.22, and 4.23
represent the above-mentioned relationship for the middle girder under dead load, fully
loaded lanes, and fatigue loading respectively. While Figures 4.24, 4.25, 4.26, and 4.27
show the change of the shear distribution factor for the interior girder under dead load, fully
loaded lanes, partially loaded lanes and fatigue loading, respectively, with increase in
number of girders. In general, the shear distribution factors for the middle and the interior
girders increase with the increase in number of girders except for the middle girder dead
loading case where insignificant effect is witnessed.
4.2.3 Effect o f Girders Spacing
Figures 4.28, 4.29, 4.30, and 4.31 show the relationship between the shear distribution
factors and girder spacing for the exterior girder of two-lane, 25-m span bridge having 4
girders under dead load, fully loaded lanes, partially loaded lanes, and fatigue loading,
respectively. Generally, it can be noticed that the shear distribution factor increases as the
girder spacing increase ( from Figure 4.29, as the girder spacing increased from 2 to 3m, the
shear distribution factor increased by 23.5% for L/R = 0.3) except for the deal load case,
where the above relation is slightly reversed. Similar behavior is observed in case o f the
middle girder (Figures 4.32, and 4.33) for fully loaded lanes and fatigue loading,
respectively, and for the interior girder (Figures 4.34, and 4.35) for fully loaded lanes, and
partially loaded lanes, respectively.
72
4.2.4 Effect of Span Length /-K
Figures 4.36 and 4.37 show the effect of span length on the shear distribution factors for t
exterior girder of one-lane, 3-girder bridge due to dead load and fully loaded lan(
respectively. In case of dead load, insignificant effect of the span length on the she
distribution factors is observed except for the span-to-radius ratio (L/R) of 0.3, at tho
values the shear distribution factor increases as the span length increases. However, tl
shear distribution factor of the exterior girder under fully loaded lanes increases as the spt
length increases (from Figure 4.37, as the span length increased from 10 to 25 m, the she;
distribution factor increased from 1.73 to 2.10 for L/R = 0.3) . It is also observed that th
rate of increase in the shear distribution factor increases with increase in the L/R ratio. Th
effect of the span length on the shear distribution factor for the middle girder is observed t
be insignificant for spans greater than 25 m, as shown in Figure 4.38. However, the shea
distribution factor for the interior girder decreases with the increase of the span length ii
curved bridges as shown in Figures 4.39 and 4.40 for dead load and fully loaded lanes
respectively. Once again the rate of the slop decrease of the shear distribution factor line i:
increases as the L/R ratio increase.
4.2.5 Effect of Loading Conditions
When analyzing bridge prototypes to determine the shear distribution factors, many loading
cases for CHBDC truck loading were considered. These loading cases can be divided into
two main groups; namely: bridges with fully loaded lanes and bridges with partially loaded
lanes. To investigate the effect of these two loading cases on the shear distribution factors.
Figures 4.41, 4.42, 4.43, and 4.44 represent the relationship between the shear distribution
73
factors o f the exterior girder of 10, 15, 25, and 35-m span bridges, respectively, due to fully
loaded lanes and partially loaded lanes. The horizontal axis represents the shear distribution '
factor due to fully loaded lanes while the vertical axis represents the corresponding shear
distribution factor for the case of partially loaded lanes. These plotted relationships are for
all bridges regardless on number of girders, girders spacing, and number of lanes. It can be
observed that shear distribution factors obtained for partially loaded lanes are usually higher
than those for fully laded lanes, especially for shorter spans. Figures 4.45, 4.46, 4.47, and
4.48 show similar relationship but for the interior girder. Results revealed that in some
exterior and middle girders cases of 10-m span bridges with 4 design lanes; partially loaded
lanes with two trucks only produced higher shear distribution factors than the case of
partially loaded lanes with 3 trucks and fully loaded lanes as shown in Table 4.1.
4.2.6 Effect of Number of Lanes
Figures 4.49, and 4.50 illustrate the relationship between the shear distribution factors and
the number of lanes for the exterior girder of a bridge with 2 m girder spacing and 35-m
span length due to fully loaded lanes and partially loaded lanes respectively. It can be
noticed that as the number of lanes increases, the shear distribution factors decreases.
Example o f that can be observed in Figure 4.49, as the number of lanes increased from 1 to
3 lanes, the shear distribution factor decreased from 3.07 to 1.6 for L/R=0.7. Middle and
interior girders demonstrate different relationship between the shear distribution factors and
the number o f lanes for the same bridge configuration as shown in Figures 4.51 to 4,54.
74
4.2.7 Effect of Number of Bracing Intervals
It was mentioned in chapter (II) that the number of bracing intervals used in this study w
based on equation 2.24. To verify the reliability of this equation and to study the effect
number of bracing interval on the shear distribution factor, a sensitivity study was conduch
on a two-lane Bridge of 35-m span length, with 4 girders, and span-to-radius of curvatu
ratio (L/R) of 0.7, against various numbers of bracing intervals; (2, 3, 4, 6 , 8 , 9, and 12
Figure 4.55 shows the effect of the number of bracing intervals on the shear distributic
factor of the exterior girder when the bridge is subjected to dead load, fully loaded lane:
and partially loaded lanes. It can be observed that the shear distribution factor decreases wit
increase in number of cross-bracing intervals up to 4. No further reduction is observe
beyond 4 bracing intervals. This means that the outcome of equation 2.24 is too conservativ
in case of shear distribution, while it is required for limiting the warping-to-bending stresse
ratios.
4.2.8 Effect of Load Cases
As mentioned earlier in chapter III, this research investigated different patterns of load cases
shown in Figures 3.4 to 3.7 to produce the maximum shear distribution factors. Figures 4.56,
4.57, and 4.58 show the shear distribution factor obtained from each loading case for the
exterior, middle, and interior girders, respectively, of two-lanes, 25-m span, bridge with 4
girders spaced at 2 m. As expected, in the exterior girder case number 2 governs the shear
distribution factor value for L/R ratio of 0.1 and below, and case number 3 governs it for the
L/R ratio greater than 0.1, whereas cases number 4 and 5 govern the shear distribution factor
75
for the middle and interior girders, respectively. It’s important to mentions that in the
exterior girders (when L/R <= 0.1) and interior girders, the governing cases are those for the
partially loaded lanes (one truck loaded in a two-lane bridge). Figures 4.59, 4.60, and 4.61
show SDF obtained from each load case for 3-lane, 25-m span bridge, with 7 girders spaced
at 2 m for the exterior, middle, and interior girders, respectively. It can be observed that for
exterior and interior girders, loading cases 3 and 10, respectively, (two trucks loaded in 3-
lane bridge) provide the design values, whereas for middle girder, cases 3 and 5 (two trucks
loaded in 3-lane bridge) govern the design shear distribution factor. To show the diversity in
the outcome from loading cases, results from a 4-lane bridge with 35 m span are shown in
Figures 4.62, 4.63, and 4.64. Two cases (4 & 9), (4 & 9), and (10 & 15) show thp peak SDF
values for the exterior, middle, and interior girders, respectively. All these loading cases
represent partially loaded lanes of (3 trucks loaded in 4-lane bridge) except case 15 of 2
trucks loaded in 4-lane bridge.
4.3 Shear Distribution Between Bridge Girders
Figures 4.65, and 4.66 investigate the distribution of the shear between the girders o f the
composite bridges to understand how the loads are distributed between the girders with
increase in L/R ratios. A two-lane bridge of 35-m span with 5 girders spaced at 2.0 m, was
analyzed under dead loading and fully loaded lanes, respectively. It was observed that the
shear distribution factor increases as the span-to-radius ratio increases for the girders located
between bridge centre-line and the exterior girder. However it decreases as the span-to-
radius ratio increase for the rest of the girders between the inner (interior) girder and the
centre-line o f the bridge. This is attributed to the high torsional moments associated with
76
curvature. Similar trend was observed in case of a 4-lane bridge with girders, spaced at 2
m as shown in Figures 4.67 and 4.68.
4.4 SDF Effect with Type of the Support
A two-lane bridge of 35-m span and 5 girders spaced at 2 m was chosen to investigat
whether there is a change in shear distribution at the roller and hinge support line
mentioned in the finite-element modeling of support conditions in chapter HI. Figures 4.6!
and 4.70 show the values of shear distribution factor along the hinged support line unde
dead load and fully loaded lanes, respectively, while Figures 4.71 and 4.72 show simila
results along the roller support line under the same above load configuration. No significan
effect was witnessed; this means that calculation of shear forces or reactions along the rollei
or the hinge support lines does not effect the design value.
4.5 Shear Distribution Equations Comparison between CHBDC and those obtained from Current Finite Element Analysis
The Canadian Highway Bridge Design Code specifies equations for calculating the shear
distribution factors for straight slab-on-girder bridges, including reinforced concrete girders,
pre-stressed concrete girders and composite concrete-steel girders. CHBDC applied some
conditions to use these equations, one of them related to the type of girders’ lateral supports,
identified as diaphragms which are required to be provided at support lines and at equal
intervals between the supports to enhance girders stability. There are different types of
diaphragms that can be used to satisfy CHBDC requirements. As mentioned in chapter (III),
this study utilized, cross-bracing type diaphragm with top and bottom chords, which is
77
considered as lateral supports to the steel girders. This study conducted a comparison
between the shear distribution factors obtained from the empirical expressions in the
CHBDC for straight slab-on-girder bridges and the results acquired from the current study
using the finite-element analysis for the same types o f bridges. Figures 4.73 and 4.74 show
the correlation between the shear distribution factors of the exterior girder due to truck
loading and fatigue loading, respectively as obtained from CHBDC equations and from the
current study. In a similar manner, Figures 4.75 and 4.76 show similar correlation for the
middle girder under the same loading configurations. It can be concluded that the CHBDC
equations always overestimate the structural response except of very few cases for truck
loading as shown in Figure 4.75. This may be attributed to the presence of cross-bracings
between support lines that was ignored when developing CHBDC equations.
4.6 Moment and Deflection Distributions in Simply Supported Composite Curved Bridges
4.6.1 Effect of Curvature
Once again the results o f current parametric study revealed that the curvature of bridges is
considered as a major parameter which significantly effects the distribution of bending
moments among girders. Figures 4.77, 4.78, 4.79, and 4.80 present the moment distribution
factor for the exterior girder of a two-lane bridge with 7 girders equally spaced at 2 m, under
dead load, fully loaded lanes, partially loaded lanes, and fatigue loading, respectively. It can
be observed that the moment distribution factor increases as the span-to-radius of curvature
ratio (L/R) increases. Similar trend was observed in case of the middle girder, as shown in
Figures 4.81, 4.82, and 4.83 for dead load, fully loaded lanes, and fatigue loading.
78
respectively. Figures 4.84, 4.85, 4.86, and 4.87 show the effect of curvature ratio on tl
moment distribution factor for the interior girder due to dead load, fully loaded lane
partially loaded lanes, and fatigue loading, respectively. It can be observed for the case <
fully loaded lanes that the moment distribution factor follows the above-mentione
dominant behavior for the exterior girder, (i.e. increase in moment distribution factor as th
curvature ratio increase). However, in case of dead loading, the moment distribution factc
increases as the curvature ratio increases till reaching a certain value of L/R (0.2), beyon
which the moment distribution factor starts to decrease as shown in Figure 4.84. SimiL
behavior but with less effect is witnessed in case of fatigue load as shown in Figure 4.81
The case of partially loaded lanes shown in Figure 4.86, shows insignificant effect of L/I
ratio on the MDF. This behavior can be attributed to the fact that the increase in L/R rath
and the decrease in the span length of the interior girder. So, at low L/R values the increast
of curvature dominates the decrease in span length, but when the L/R increases the decreast
in span length become dominant. Going back to clause 4.2.1 to in the CHBDC for a curvec
bridge to be treated as a straight one, results in Figures 4.77 to 4.87 revealed that CHBDC is
significantly underestimate the moment distribution factors in curved bridges that meet the
requirement of L“ / (b.R) being less than 1. The effect of curvature ratio on deflection factors
was also studied parallel to the MDF study and similar trend was observed as shown in
Figures 4.111 to 4.121.
4.6.2 Effect of Number of Girders
Figures 4.88, 4.89, 4,90, and 4.91 show the effect of number of longitudinal girders on the
moment distribution factor for the exterior girder of a two-lane, 1 0 -m span bridge with 2 . 0 m
79
girder spacing under dead load, fully loaded lanes, partially loaded lanes, and fatigue
loading respectively. Although insignificant effect is observed between the number of
girders and moment distribution factor for the dead load case as shown in Figure 4.88,
significant increase in the moment distribution factor was found with increase in number of
girders for fully loaded and partially loaded lane cases as shown in Figures 4.89, and 4.90
respectively. Fatigue load case revealed another pattern as shown in Figure 4.91; following a
pattern other than a straight line. For the middle girder, the effect of number of girders on
the moment distribution factor revealed that the moment distribution factor increases as the
number of girder increases except for the dead load case, where insignificant behavior is
found as shown in Figures 4.92 to 4.95. Figures 4.96 to 4.99 show the moment distribution
factor of the interior girder under dead load, fully loaded lanes, partially loaded lanes, and
fatigue loading, respectively, with the increase in number of girders. A trend similar to that
in case o f the exterior girder is observed. The effect of number of girders on the deflection
distribution factor was also investigated in parallel to the study on moment distribution
factor. A similar trend was observed as shown in Figures 4.122 to 4.131.
4.6.3 Effect of Girders Spacing
Figures 4.100 to 4.103 for two-lane, 10-m span, bridge with 4 girders show the effect of the
girders spacing on moment distribution factor it can be observed that, the moment
distribution factor increases as the girder spacing increases except for the dead load case
when insignificant relationship was found. Similar trend is observed in case o f the middle
and interior girders as shown in Figures 4.104 to 4.107. Similar behavior was observed in
case of the deflection distribution factors as shown in Figures 4.132 to 4.140.
80
4.6.4 Effect of Loading Conditions
As examined in the case of the shear distribution factor study, it is important to examine tf
effect of number of loaded lanes on the moment distribution factor to establish the critic
cases that produce extreme values of moment distribution factors. Accordingly, two loadir
cases were considered; fully loaded lanes with truck loading and partially loaded lanes wil
truck loading. Figures 4.108, 4.109, and 4.110 show the relationship between result
obtained from the case of fully loaded lanes and the case that provides the maximui
moment distribution factor of all the partially loaded cases for the exterior, middle, an
interior girders, respectively. It is worthwhile to reiterate that these plotted values are for a
bridges of 1 0 -m spans regardless of number of lanes, or number of girders or girder
spacing. It can be infer from the above Figures that sometimes even though the partial!
loaded lanes are almost half of the live load of the fully loaded lanes, still they can provid
extreme design values. However, the results from the fully loaded lanes generally dominât
the design value when compared to the partially loading. Similar trend was observed for thi
deflection distribution factors as shown in Figures 4.141 to 4.143.
4,7 Warping Stress Distribution in Simply Supported Curved Bridges
To study the effect of the warping stress distribution in simply supported curved bridge, 10
m span bridge was selected. As mentioned earlier in chapter III, in order to investigate the
change in warping stresses with the changes in bridge geometry and loading conditions, the
ratio of the warping stress-to-the-average bending stress in the bottom flanges (WBR as
calculated in equation 3.42) was calculated in this study. The AASHTO Guide specification
81
for Horizontally Curved Bridges (Guide, 2003) acknowledged a limitation value for the
warping-to-bending stress ratio (WBR) of 0.5 to ensure the stability of the structure. In this
parametric study, the values for WBR were calculated for the exterior, middle, and interior
girders due to different load cases.
Figures 4.144, 4.145, 4.146, and 4.147 illustrate the WBR values for the exterior girder of
all the curved bridges of 1 0 -m span considered in this study in ascending order, due to dead
load, fully loaded lanes, partially loaded lanes, and fatigue loading, respectively. The
negative values observed in these figures refer to the orientation of the lateral stresses at the
bottom flange which is not affecting the absolute value of the warping-to-bending stress
ratio. It can be inferred from the above figures that the WBR increases with the increase of
the span-to-radius of curvature (L/R) ratio. Another important remark is that the extreme
values o f WBR shown in the above all figures did not exceed 0.2 which is considered
complying with the requirements of AASHTO Guide 2003. Figures 4.148, 4.149, 4.150, and
4.151 illustrate the warping-to-bending stress ratio for the middle girder of all the curved
bridges considered in this study in ascending order, due to dead load, fully loaded lanes,
partially loaded lanes, and fatigue loading respectively. Similar pattern of WBR to that of
the exterior girder was observed; however there was slight increase in the upper limit of the
WBR values for the middle girder as compared to those of the exterior girder. Nevertheless
in all cases the WBR values did not exceed 0.25 which is still below the AASHTO
requirements. The warping-to-bending stress ratio in the interior girder followed the same
trend to that o f the exterior and the middle girders. Figures 4.152, 4,153, 4.154, and 4.155
illustrate the warping-to-bending stress ratio for the interior girder of all the curved bridges
considered in this study in ascending order, due to dead load, fully loaded lanes, partially
8 2
loaded lanes, and fatigue loading, respectively. There was slight increase in the upper lim
of the WBR values in the interior girder as compared to those of the middle girder (0.4 <
shown in Figure 4.152), which still below the AASHTO limits.
4.8 Comparison Between CHBDC Moment Distribution Equations and those obtained from Current Finite Element Analysis
As mentioned earlier in this chapter, the Canadian Highway Bridge Design Code (CHBDC
specifies empirical equations for the moment distribution factors in case of straight bridges
including different types of bridges supported laterally by diaphragms at the support lines. Ii
order to study the correlation between the CHBDC empirical equations and the results fron
this study, the outcome from this research and the corresponding values obtained fron
CHBDC equations were drawn against each other in graphical format. The produced graph:
were for the exterior and middle girders due to truck loading and fatigue loading
respectively, as shown in Figures 4.156, 4.157, 4.158, and 4.159. It was observed that th(
CHBDC moment distribution equations always overestimate the structural response of the
studied bridges except for few cases for the exterior girders due to fatigue loading. In ordei
to investigate the correlation between the moment distribution factors and the deflection
distribution factors, the results from the finite-element analysis for moment distribution
factors and deflection distribution factors were plotted against each other in Figures 4.160,
4.161, 4.162, and 4.163 for the exterior girder under truck loading and fatigue loading and
for the middle girders under the same type of loading, respectively. For straight bridges, it
can be observed that the moment distribution factor correlates very well with the deflection
distribution factor, however in curved bridges the deflection distribution factors are always
83
bigger than the corresponding moment distribution factors. As a result, the deflection
distribution factors for curved bridges should not be taken as the moment distribution factors
and new deflection distribution factors should be developed for curved bridges in addition to
the moment distribution factors.
4.9 Development of New Load Distribution Factor Equations
4.9.1 Shear Distribution Factor Equations
Two types o f equations were developed from the results of this study; namely: (1)
shear distribution factors for straight composite concrete-steel I-girder bridges, and (2 ) shear
distribution factors for curved composite-steel I-girder bridges.
4.9.1.1 Shear Distribution Factors for Straight I-Girder Bridges
The general equation for the shear distribution factor for straight I-girder bridges
takes the following form:
Fv = S X N X F
F = (a + b x L " /
Where
Fv : is the shear distribution factor,
S : is the girder spacing in meters,
N : is the number of girders,
F : is a width dimension factor that characterizes load distribution for a bridge on which a,
b, c, and d are equation variables and shall be obtained from Tables 4,2, and 4.3.
L : is bridge span length in meters.
84
4.9.1.2 Shear Distribution Factors for Curved I-Girder Bridges "
The general equation for the shear distribution factor for curved I-girder bridges
takes the following form:
Fv = S X N X F
F = ( a - hbxL"/ x[ l +e (L/R) + g { L / R f ]
Where;
e, f g, and h are equation variables and shall be obtained from Tables 4.4, and 4.5,
L/R : is span-to-radius of curvature ratio.
4.9.2 Moment Distribution Factor Equations
Following the criteria in building the moment distribution factor equation as stated in
the CHBDC with the difference of developing new MDF values generated from this study.
4.9.2.1 Moment Distribution Factors for Straight I-Girder Bridges
The general equation of the moment distribution factor for straight I-girder bridges
takes the following form;
F.. ”
85
F = a + b / L
Where
Fm : is the moment distribution factor,
S ; is the girder spacing in meters,
N : is the number of girders,
F : is a width dimension factor that characterizes load distribution-for a bridge, a ,and b are
equation variables and shall be obtained from Tables 4.6, and 4.7.
JU = but < 1.0^ 0.6
We : is the width of a design lane in meters, calculated with CHBDC clause 3.8.2;
Cf ; is a correction factor, in %, obtained from Table A 5.7.1.2.1 (CHBDC);
4.9.3 Deflection Distribution Factor Equations
4.9.3.1 Deflection Distribution Factors for Straight I-Girder Bridges
The general equation of the moment distribution factor for straight I-girder bridges
takes the following form:
F . . "
V 100 ;
F = a + b / L
8 6
Where
Fa : is the Deflection distribution factor;
S : is the girder spacing in meters,
N : is the number of girders,
F : is a width dimension factor that characterizes load distribution for a bridge, a ,and b are
equation variables and shall be obtained from Tables 4.8, and 4.9.
We- 3 3= but < 1 .0^ 0.6
We : is the width of a design lane in meters, calculated with CHBDC clause 3.8.2;
C f ; is a correction factor, in %, obtained from Table A5.7.1.2.1 ( C H B D C ) ;
X 87
CHAPTER VSUMMARY, CONCLUSIONS, AND RECOMMENDATIONS
FOR FUTURE RESEARCH
5.1 Summary
This study investigated the effect of the following parameters on the lateral load
distribution factors for straight and horizontally curved composite concrete-steel I-girder
bridges using the finite-element analysis:
❖ Span length (L): 10, 15, 25, and 35 m;
❖ Girder spacing (S): 2, 2.5, and 3 m;
❖ Number of girders (N): 3, 4, 5, 6 , & 7 for 2 m girder spacing; 3, 4, 5, & 6 for 2.5 m
girder spacing; 3, 4 & 5 for 3 m girder spacing;
❖ Span-to-radius of curvature ratio, (L/R): 0.0, 0.1, 0.2, & 0.3 for spans L=10 m and
L=15 m; 0.0, 0.1, 0.3, & 0.5 for span L=25 m; and 0.0, 0.1, 0.4, & 0.7 for span
L=35 m;
❖ Loading conditions: fully loaded lanes and partially loaded lanes;
❖ Cross-bracing intervals, 2, 3, 4, 6, 8, 9, and 12;
❖ Number of lanes: 1, 2, 3, and 4
5.2 Conclusions
Based on the results from the parametric study, the following conclusions are drawn:
8 8
(1) Curvature is the most significant factor which is playing an important role
determining the lateral load distribution factors. The increase in the span-to-radius
curvature ratio (L/R) leads to significant increase in the shear distribution factors f
the exterior and middle girders and a decrease in the SDF for the interior girde
whereas the increase in the curvature ratio results in significant increase in tl
moment distribution factors and deflection distribution factors for all the girders. Tl
same conclusion can be drawn for the effect of curvature on warping-to-bendin
stress ratio.
(2) The number of girders and girders spacing are other important parameters that affei
the lateral load distribution factors. In general, the increase in the number of girder:
as well as in girders spacing results in an increase in the shear distribution facto:
moment distribution factor, and deflection distribution factor.
(3) In general, the span length slightly affects the shear distribution factors; howevei
span length shows significant effect when the L/R ratio exceeds 0.10.
(4) This study revealed that the partially loaded lane cases almost govern the extremi
values of the shear distribution factors. However, the moment distribution factor:
and deflection distribution factors showed rebellious results compared to those of th<
shear distribution factors, when the fully loaded lane cases govern the design.
(5) The number of cross-bracing intervals over 3, proved to have less significant effect
on the shear distribution factors.
(6 ) In general, this study revealed that CHBDC overestimates the structural response ol
the straight composite I-girder bridges.
89
(7) This study showed that CHBDC significantly underestimates the structural response
o f curved bridges by treating them as straight bridges when / (b.R) is not greater
than 1. This provision should be investigated.
(8 ) Sets o f empirical expressions for shear, moment and deflection distribution factors
were developed for both straight and curved bridges, yielding more economical
bridge construction.
5.3 Recommendations for Future Research
The author recommends future research in the following areas:
1- More experimental verification studies to gather field response data for on site
bridges having the same configuration and geometry of the bridges considered in this
study to evaluate the compatibility of the results.
2- The study of load distribution in continuous curved composite I-girder bridges.
3- Study of the effect of dynamic loads on curved composite I-girder bridges.
4- The study of the non-linear behavior on the structural response of curved composite
I-girder bridges and evaluates the critical stresses at failure.
5- Investigation on the safe limiting curvature ratio to treat a curved bridge as a straight
one.
6 - Investigate the effect of curvature on bracing axial forces.
90
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Engineering Mechanics Division”, ASCE: 108(4): 636-647, 1982.
Shore S., and Wilson J. L., “Users’ Manual for Static analysis of Curved Bridges (STACRB)”,
CURT Report No. T0173, Research Project HPR-2(111), Graduate Division of Civil &
Environmental Engineering, University of Pennsylvania, 1973.
Stegmann T.H., and Galambos T.V., “Load factor design criteria fo r curved steel girders o f
open section”. Research Report 23, Civil Engineering Department, Washington University,
St. Louis, 1976.
Tarshini K.M., and Frederick G.R., “Wheel load distribution in 1-girder highway bridges".
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94
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95
Table 2.1 Coefficient, C, for Various Multi-Girder Systems Assuming Equal Girder Spacing (Grubb, 1984)
Number of Girders
2 3 4 5 6 7 8 9 10
CoefficientC
1 1 10/9 5/4 7/5 14/9 12/7 15/8 165/81
Table 3.1 Bridge Configurations Considered in the Parametric StudyBridgeWidth
Deck Width Wc
Number of Girders
NumberSpacing
Number of Design Lanes
6 5 3 2 1-lane7.5 6.5 3 2.5 2-lane9 8 3 3 2-lane
8 7 4 2 2-lane10 9 4 2.5 2-lane12 11 4 3 2-lane & 3-lane
10 9 5 2 2-lane12.5 11.5 5 2.5 2-lane & 3-lane15 14 5 3 4-lane
12 11 6 2 2-lane & 3-lane15 14 6 2.5 4-lane
14 13 7 2 2-lane & 3-lane
96
Table 3.2 Number of Design Lanes
Wc n
6.0 m or less 1Over 6.0 m to 10.0 m incl.Over 10.0 m to 13.5 m incl. 2 or 3Over 13.5 m to 17.0 m incl. 4Over 17.0 m to 20.5 m incl. 5Over 20.5 m to 24.0 m incl. 6Over 24.0 m to 27.5 m incl. 7Over 27.5 m 8
Table 3.3 Modification Factors for Multilane Loading
Number of Loaded Design Lanes Modification Factor
1 1.00-2 0.903 0.804 0.705 0.606 or more 0.55
97
Table 4.1 Effect of loading conditions, extreme cases
Bridge Geometry Shear Distribution Factors
spanNo. of girders
girderspacing L/R
No.of
LanesExterior girder Middle girder
No. L N S n (LDF)FL.ext (LDF)pLl.exi (LDF)pL2exl (LDF)pLext (LDF)pLl.exl (LDF)pL2.ext1 10 5 3 0 4 1.005 1.285 1.324 1.302 1.625 1.7272 10 5 3 0.1 4 1.029 1.326 1.358 1.286 1.584 1.6413 10 5 3 0.2 4 1.049 1.356 1.377 1.275 1.562 1.5834 10 5 3 0.3 4 1.071 1.381 1.395 1.278 1.558 1.5835 10 6 2.5 0 4 0.891 1.176 1.216 1.448 1.869 1.8836 10 6 2.5 0.1 4 0.940 1.206 1.247 1.401 1.818 1.7947 10 6 2.5 0.2 4 0.974 1.255 1.287 1.372 1.783 1.7408 10 6 2.5 0.3 4 0.995 1.279 1.303 1.376 1.792 1.7469 15 5 3 0 4 1.081 1.274 1.272 1.229 1.360 1.31010 15 5 3 0.1 4 1.115 1.312 1.292 1.216 1.334 1.27011 15 5 3 0.2 4 1.134 1.327 1.292 1.205 1.314 1.24212 15 5 3 0.3 4 1.158 1.351 1.307 1.216 1.300 1.218
13 15 6 2.5 0 4 0.894 1.064 1.070 1.354 1.551 1.50814 15 6 2.5 0.1 4 0.931 1.107 1.097 1.343 1.534 1.47515 15 6 2.5 0.2 4 0.940 1.105 1.087 1.360 1.561 1.491
16 15 6 2.5 0.3 4 0.976 1.146 1.112 1.341 1.531 1.458
FL: Full load {4 trucks loaded in 4 lanes) PLl: Partial load 1 (3 trucks loaded in 4 lanes) PL2: Partial load 2 (2 trucks loaded in 4 lanes)
98
Table 4.2 Value of F for Longitudinal Shear for Straight Bridges (ULS and SLS)
Number of Lanes, n Exterior girder Eq. No.
1 (1.69 +4.74 4.1
■2 (3.0 + 0.0024 L 0.92) 4.2
3 (4.9 +2.4 L -0.6) 4.3
4 (5.7 - 0.54 L -3) 4.4
Number of Lanes, n Middle girder Eq. No.
1 (2.17 + 4.1 L' - ) 4.5
2 (2.38 + 1.21 L' ' ) 4.6
3 (2.545 + 1.99 4.7
4 (2.89 + 1.951 : ^ ) 4.8
99
Table 4.3 Value of F for Longitudinal Shear for Straight Bridges (FLS)
Number of Lanes, n Exterior girder Eq. No.
1 (1.935 + 5 4.9
2 (1.68 + 4.04 4.10
3 (1.918 + 0.62 L 4.11
4 (1.88 - 0.63 4.12
Number of Lanes, n Middle girder Eq. No.
1 (2.17 + 4.1 4.13
2 (1.8 + 3.4 4.14
3 (1.95 + 9 L"*) 4.15
4 (2.14 + 8.9 L'*) 4.16
100
Table 4.4 Value of F for Longitudinal Shear for Curved Bridges (ULS and SLS)
No. of Lanes,
nExterior girder Eq,
1 (1.69 + 4.74 [1+ 2.15 (L/R) ' -1.1 (L/R) 4.
2 (3.0 + 0.0024 L [1+ 4 (L/R) ' - 3.1 (L/R) 4.
3 (4.9 + 2.4 L -**) -1 2% [1 + 3 (l/R)®** - 2.4 (L/R) ®’®] 4.
4 (5.7 - 0.54 L - ) 4% [1 + 2.9 (L/R)®’’ - 2.51 (L/R) ®'®] 4.
No. of Lanes,
nMiddle girder Eq.
1 (2.17 + 4.1 L' ®) [1+ 1.3 (L/R) - - 0.178 (L/R) ®*' ] 4.:
2 (2.38 + 1.21 L'^ ) -1*2 X [1+ 1.2 (L/R) -2 _ 9 .1 9 (L/R) ®- i] 4.:
3 (2.545 + 1.99 L'"'®) '2 i*®x [1+ 1.6 (L/R) i - 0.16 (L/R) ®' ®] 4 . :
4 (2.89 + 1.95 L'®) -2x [1+ 0.67 (L/R)® ®® - 0.75 (L/R) ® i2] 4 , :
No. of Lanes,
nInterior girder Eq. ]
1 (0.255 + 14 ""x [1- 7.7 (L/Rf^ + 5.1 (L/R) 4.2
2 (-3.403 + 5.78 L “"“) [1- 3.9 (L/R)*'” + 3.79 (L/R) 4.2
3 (-2.82 + 6.33 L"“’) x [1- 4.05 (L/R)*“ + 3.71 (L/R) *”®] 4.24 (-2.33 + 6 .6 L*"") ‘-“ x [1- 3.62 (IVR)'"" + 3.62 (L/R) 4.2
101
Table 4.5 Value of F for Longitudinal Shear for Curved Bridges (FLS)
No. of Lanes,
nExterior girder Eq. No.
1 (1.935 + 5 " X [1+ 18.66 (L/R) * + 1.7 (L/R) 4.292 (1.68 + 4.04 L' ®) X [1- 0.449 (L/R) - - 10.16 (L/R) 4.30
3 (1.918 + 0.62 L'®-® ) -^*^x [1- 2.6 (L/R)'*' + (L/R) ' 4.31
4 (1.88 - 0.63 L'®'® ) - ®x [1- 0.95 (L/R) ' ® + 1.045 (L/R)* *' ] 4.32
No. of Lanes,
nMiddle girder Eq. No.
1 (2.17 + 4.1 L’ - ) -*73% [I. 0.13 (L/R)®- + 0.2 (L/R) ^ 4.33
2 (1.8 + 3.4 L’ - ) [1- 0.22 (L/R)®' + 400 (L/R) 4.34
3 (1.95 + 9 L'*) i*®x [1+ 0.99 (L/R)®®’ -1.06 (L/R) ®®'’] 4.35
4 (2.14 + 8.9 L' ) '*’ x [1- 0.2 (L/R)®'* + 0.02 (L/R) ®' ] 4.36
No. of Lanes,
nInterior girder Eq. No.
1 (0.33 + 13.95 L “ ) [1-188.6 (L/R)" ’’ + 223.65 (L/R) “ "l 4.37
2 (2.69 + 3.5 L"’") X [1- 8.73 (L/R)’ ""* + 7.38 (L/R) 4.38
3 (2.1 + 6.1 L"' ") X [1+ 3.49 (L/R)'-” - 3.6 (L/R) 4.39
4 (3.06 + 3.3 L'‘ ‘) X [1+ 3.45 (L/R)' " - 3.9 (L/R)’ ’'*] 4.40
102
Table 4.6 Value of F for Longitudinal Moment for Straight Bridges (ULS and SLS)
Number of Lanes, n Exterior girder Eq. No.
1 (3.89-2.38/L) 4.41
2 (7.4 - 6.4 / L) 4.42
3 (8.01 +10.4/L) 4.43
4 (10.2 +4.5/L) 4.44
Number of Lanes, n Middle girder Eq. No.
1 (6.06-6.83/L) 4.45
2 (8.7-32.5/L) 4.46
3 (11.95-21.1/L) 4.47
4 (15.0-19.5/L) 4.48
103
Table 4.7 Value of F for Longitudinal Moment for Straight Bridges (FLS)
Number of Lanes, n Exterior girder Eq. No.
1 (5.58 +3.46/L ) 4.49
2 (3.9 - 20.5 / L) 4.50
3 (2.8 - 2.7/L ) 4.51
4 (4.35 -1 9 /L ) 4.52
Number of Lanes, n Middle girder Eq. No.
1 (6.02-5.98/L ) 4.53
2 (9.0 - 43.21 / L) 4.54
3 (8.6-25.3/L ) 4.55
4 (16.3-95.8/L ) 4.56
104
Table 4.8 Value of F for Deflection for Straight Bridges (ULS and SLS)
Number of Lanes, n Exterior girder Eq. No.
1 (4.15-4.07/L) 4.57
. 2 (7.6 - 7.7/L) 4.58
3 (8.5 + 5.5/L) 4.59
4 (10.1-2.1/L) 4.60
Number of Lanes, n Middle girder Eq. No.
1 (6.05-5.94/L) 4.61
2 (8.8-28.1/L) 4.62
3 (12.2-21.5/L) 4.63
4 (15.25-23.2/L) 4.64
105
Table 4.9 Value of F for Deflection for Straight Bridges (FLS)
Number of Lanes, n Exterior girder Eq. No.
1 (5.78 +2.23/L) 4.65
2 (3.15 -12.5 / L) 4.66
3 (3.0 - 4.0/L ) 4.67
4 (4.0 - 22 / L) 4.68
Number of Lanes, n Middle girder Eq. No.
1 (6.05-5.94/L ) 4.69
2 (9.0-38.2/L ) 4.70
3 (10.6-60.0/L ) 4.71
4 (16.4-93.5/L) 4.72
106
bridge width
deck width
concrete deck
top-chord
top flange
web
bottom chordcross-bracing
Figure 1.1 Typical I-Girder Bridge Cross-Section
a) Single Girder b) M ulti-girder bridge
Figure 2 .1 Single and Multi-girder System under Concentrated Live Load P
Figure 2.2 Lateral Load Distribution of Truck Axle Load
108
L AT T T T I
__ — Original shape
^-D eform ed shape
a)Eir=0.0, g=t.O
J
L
I T Tb)0<Eff<x. 0.2<g<1.0
P
I JIT T T
c)Eir->«, g=0.2
Figure 2 .3 Girder Deflection with Different Transverse Stiffness
Figure 2.4 Free Body Diagram of Lever Rule method
109
a) Ps— !- ..............e --------------- ^
b) Ps
, 3 4 5
L----02 04 ----
u 1 - - Ub 1
P^M =P*e
I, . . U [S
c) P \
1
1
]
; i
J
W 1 w's
L à
r T T "
d)L
W I____ w_2_____ -
fT1f
R 2
I I j
e)
Figure 2 .5 Load Distribution under Eccentric Load
110
Figure 2 .6 Free Body Diagram of a Hinged slab Bridge under Concentrated Load
Figure 2.7 Free Body Diagram of a Hinged slab Bridge under Sinusoidal Load
111
o)
b)P=1
9' 92 qi
9' 92 93
9*
9+
p (x )* l sin itx/L
di/3f{x)=f sin w /L
- — d i — J
d) t
f
• 4 hi
-----V\\\
Figure 2. 8 Free Body Diagram for Hinged T-shaped Girder Bridge
1 1 2
o)P=1
1 2 i 4
' ? • ------- b-------' --------- b--------- -------- b--------- -P=1
b)r y " v “
%
II. II.. II.
c) =1
11
Figure 2 .9 Free Body Diagram of Fixed Joint Girder Bridge
h-S-M
a) Real Structure b) Analogized Equivalent Orthotropic Plate
Figure 2.10 Real Structure and Orthotropic Plate Analogy
113
^ gîrder 1 (outside) (in s id e)
o-
c r o s s - f r a m es h e a r
Hi Hz
Figure 2 .11 V-Load on Girder
Y
B V
\
Figure 2.12 Effect of Warping Moment Applied to I-Girder
114
15025
2 3 125 125 625 625
J Li4
175875
L*~3i6~i*2l' 6,6
•18J)0 m-
6.6
5 Axle number 150 Axle load CkN>
75 Wheel load (kN>
Gross load 623 kN
-B-B-
-B-B-Trgvel l.Jn
_ J,3m
tru ck width
(a> CHBDC tru ck loading
32 80 80
1 1112 96 Axle load CkN>
Uniform load 9 kN/n'/ —UrWfor /includlnQ the.DLA
I— 3.6 L •6.6 *
-18.00 n-
(b) CHBDC lane loading
Figure 3.1 Canadian Highway Bridge Design Codes’ Truck and Lane loadings
115
Level 1<L
125tiN I25&N I 175M
1QJQQlS '
Level 2150kN )75HN 12SkN 125M
0 .60 , 6.60
Level 3ISOkN 175kN l25kN 125kN 50kN
A0. 60, 6.60 6.60
Level 4
150kN l75kN l25kN 125kN
6.60i_ , 1.20
15.00
1.2Q[ M
25.00
SOkN
" A
1 J r 1 f 1f
n. . . 6 ..60 . . 6 .6 0 1.20A
35.00
Dimensions are in meters
Figure 3 .2 Maximum Shear Locations
116PROPpnr/OF
RVERS i LZAAR*
I
Level 1
Level 2\
Level 3
50kN
].20, 3.80,1.40
ISOkN125kN
A&60 6.60
Aa.90
15.00
50kN 125kN 125kN ISOkN175kN
2.20 160 6.605.50 4ao2100
Level 4 SOkN 125kN 125kN 175kN ISOkN
150 160
Dimensions are in meters
Figure 3.3 Maximum Moment Locations
117
Case (1): Dead load R
Case (2): Loading case for exterior girder
/
R
0 ,5 1 0 .6 . 1.8kl : f . . . ' ' ......... ...
Case (3): Loading case for middle girderR
.0 .9 ,0 .9 r\ ; r J
R
1 . 1.8 o ,s |\ -1
<0.5
Case (5): Loading case for fatigueh o u l d e r
R
0 9 ) 0 . 9 .
l o u l d e r
Dimensions are in meters
Figure 3. 4 Live Loading Cases for One-lane Bridge
118
Case (I): Dead loadK
Case (2); Exterior girder-Partia^^load R
0.5i
.61 1.8
Case (3): Exterior girder-FuII load ^ j
0.5 L ib. 61 1£ A
Case (4); Middle girder
eO.5
0.5 p.5
Case (5); Interior girder-Partial load R
d“1 ~ — — .
^ X — Zi ■
Case (6);Interior girder-FulI load n
D.6A 0.5
3.6 «0.5
Dimensions are in m eters
Figure 3 .5 Live Loading Cases for Two-lane Bridge
119
Case (7): Exterior girder-Fatique loading
sh o u ld e r
Rshoulder
1 rI ■ }
Case (8): Interior girder—Fatigue loading
sh o u ld e r
R
sh o u ld e r
Figure 3.5 Live Loading Cases for Two-lane Bridge - Continue
120
Case (1): Dead load
0.5
R
n
Case (2); Exterior girder-Partial ioadl pj
0.6
0.5
0.5 R I T
Case (3): Exterior girder-Partial load2
0.6 0.6
0.5
0.5 a 1
o-- X —
R
L Y X T s X Y t Y lCase (4): Exterior girder-Fuli load
0.5
0.5 a0.6 0.6-----
R
-7^
t Y X Y s X Y X Y l
0.5
Case (5): Middle girders0.6 0.6 h -----------7/ ^ I X
0.5 r r-rrr-,\p â r ] \ . l / | ~ T ï ï
t > < X 2 s X > < X Y l
A0.5
Case (6): Middle girders
0.5
&6 0.6
r j
Ri r r
a #l > s X > < X > < X Y l
Case (7): Middle girders
# —0.5
R-0Æ crs-
m f t
T F < X > < X Y t Y l
0.5
Dimensions are in m eters
Figure 3 .6 Live Loading Cases for Three-lane Bridge
121
Case (8): Middle girders | R
« ^ 1 — . r 04' I
0.5
Case (9): Interior girder-Partial Ioadl R
0.5f i -
%6
1.8
t > < X Y 5 X > < X Y < 2
0.5
Case (10): Interior girder-Partial lcad2 R<h
0.5fl'
0.6
l.t n o[_____ V t
Case (11): Interior girder-FuII load
0.6
0.5
R
1.8 W -
Case (12): Fatigue, Exterior girder
sh o u ld erR
sh o u ld ert— ---- 7^----- ----------- 7i"------------=------------ 7^-----------' -4
1.8/
!l > A X T s X > < X Y < 2
Case (13): Fatigue, middle girder ^sh o u ld er
f- - - - - - 7 - - - - - f-- - - 7/- - - - - - - T*'- - - - - '— 4L 1.8S ' . ' J
Case (14): Fatigue, interior girder R
sh ou ld
Rer sh o u ld er
1.8
Figure 3.6 Live Loading Cases for Three-lane Bridge ■ Continue
1 2 2
fîSÉ4iK-:ÿ;4;i.
Case (1): Dead load0.5
R 0.5
T i mCase (2): E x te rio r g ird e r -P a r t ia l load □
---------^ ---------- j---------- ----------0.5 0.6 0.5
t e . rCase (3): E x te rio r g ird e r -P a r ia l load pj
0.5 0.6 0.6 --------------- /' X/-1M 33-1 - / p n g
0.5
— — 'SkL'— ___
Case {+): E x te rio r g ird e r -P a r ia l load p
05 Oa OA ___ __ 0.5
JL
Case (5): E x te rio r g ird e r-F u li load
0-5 0.6 .,R
/r4 -M- 06 oTO" o sf j z q ^ y -ü ü l i^ p ü c i : 4 T c ^ n
T i m r n r n i m nCase (6): Middle g i rd e r -P a r ia l load
0.5 0.6 0.6 ^ ---------------------i------\\---- 0.5
Case (7); Middle g i rd e r -P a r ia l load 0.5 #------ 0.5
T F < T i 5 T _ t e I < : t > < 1D im ensions a re in m e te rs
Figure 3 .7 Live Loading Cases for Four-lane Bridge
123
Case (8); Middle girder-Paria! load ■0.5
-fr -A lA
Case (9): Middle girder-Parial load
i l l 1.8
0.5
[
-7^1.8
E H I B
R0.5
Case (10); Middle girder-Parial load p
JL
0.5
Z > < t T < X > jCase (11); Middle girder-Fuli load
: i > i TR
u.:
3— ^ I p
1.8 y i . 1.8 1.8 V./ 1.8 1J' % '' .. ,$
j.j
3C H X B l
Case (12): Middle girder-Fuli load
QJR
0.5
¥ m rZ > A X > < X H X > < X > < 1
Case (13): Middle girder-Fuli load ^
«■5------ ^ 0.6
3 1.8 .LA
0.5
1
Case (14); Interior girder-Parial load ^
0.5 -7#0.5
—2 : ^j_L8_-^.
T H I H E T H I H TDimensions are in meters
Figure 3.7 Live Loading Cases for Four-lane Bridge - Continue
124
Case (15): Interior girder-Parial load
0.5
Z > < X Y < t ? <Case (16): Interior girder-Parial load
0.6 0.5
m m
n r 0.51 :- - -^ - - - - - - 1- - - - - - - n — —
j1h 1 1.6 1.8'
Case (17): Interior girder-Full load p0:6' ■ 0.6 0.5
m
x m r < z > Y x m y < iCase (18): Fatigue, Exterior girder
shouldershoulder — ----- # ------ -----# ----------- # -----1.8
' '' 1—T—
z m r s T H X > < X Y < lCase (19): fatigue, Middle girder ^
shoulder-----7 ------1-----# -- -----7 ----------- 7 -----
1.8 7L ' ' /
shoulder
Case (20): Fatigue, Middle girder
shoulder shoulder
ICase (21): Fatigue, interior girder
shoulder shoulder-----7 ------ ----- 7 ----- -----7f -- ---7 --, 1.8
1\ '
X > < X > s X ? Y X > < X Y < XDimensions are in meters
Figure 3.7 Live Loading Cases for Four-lane Bridge - Continue
125
e x t e r n a l g i r d e r c r o s s - b r a c I n g ^ ' M r î I d d l e g i r d e r s in t e r n a l g i r d e r— s — L— S— L i — I — S—
Figure 3.8 Cross-Section of a Composite I-Girder Bridge
126
verticalbracing
/ su p p ortsupport \
/\\ /
a) I-Girder with Radial Cross-Bracings
G5
G4
03
G2
G1
-v ertica lbracing
1 7 - n 1
1 1
1 1
1 1
b) I-Girder with Transverse Cross-Bracings
Figures. 9 Plan of the Steel Girder Arrangement
127
S ie l l e l e m e n t s for d eck s lab
hell e le m e n t
Shell e le r r e n t s for f a n g e s
T r js s e l e m e n t s for bracings and top and b o t:o m ch ord s
Shell e l e m e n t s for web
Figure 3.10 Finite Element Representation of Bridge Cross section
128
Figure 3.11 View of a SAP2000 non-composite Finite-Element Model
Figure 3.12 View of a SAP2000 composite Finite-Element Model
129
SAP20008 Æ 5 /0 S 2 2 :4 5 :0 1
A
SAP2000 V7.40 - File:L25N4S20R5 - 3-D View - KN-m Units
Figure 3 .13 View of a SAP2000 Finite-Element Model with support Conditions
130
0.3m
-> ‘ - 16m m
oCNJ
O
point 1
Figure 3.14 Cross-Section Dimensions of the Steel Girder
a) Major Axis Bending Stress
b) Warping Stress Combined Bending and Warping Stress
Figure 3. IS Normal Stress Distribution in Curved I-Girder Flanges
131
3.5
3.0
0> 2.5
# LrnlOm — L-15m —A—L«35m —X—L=35m
2.0 •
1.0Exterior Girder, Dead Load, N=3, S=2rri, n=1
0.00 0.1 0.2 0.3 0.4 0.5 0.6 0.80.7
span-to-radius of Curvature ratio (UR)
Figure 4.1 Effect of Curvature on the Shear Distribution Factor for the Exterior Girder dueto Dead Load
3.5
3.0 -
u.W 2.5
• LbIOiti ■ -L«15m
—A—L“25m —X—L«35m1.5
Exterior Girder, Fully loaded lanes, N=3, S=2m, n=1
0.5
0.0 0.80.70.60.50.40.30.20.10span-to-radius of Curvature ratio (L/R)
Figure 4 .2 Effect of Curvature on the Shear Distribution Factor for the Exterior Girder dueto Fully Loaded Lanes
132
2.0
Exterior Girder, Fatigue Loading. N=3, S=2m, n=10.5
0.00.6 0.7 0.80 0.2 0.4 0.50.1 0.3
span-to-radius of Curvature ratio (L/R)
Figure 4.3 Effect of Curvature on the Shear Distribution Factor for the Exterior Girder dueto Fatigue Loading
U.otaios — L=10m
—■ —1=15m A L=25m
—X—L=35m•cI
Middle Girder, Dead Load, N=3, S=2m, n=10.2 -
0.00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
span-to-radius of Curvature ratio (L/R)
Figure 4.4 Effect of Curvature on the Shear Distribution Factor for the Middle Girder due toDead Load
133
0.6 -
<n 0.4 - Middle Girder, Fully loaded lanes, N=3, S=2m, n=1
0.2 ■
0.00 0.1 0.2 0.3 0.4 0.5 0.6 0.60.7
span-to-radius of Curvature ratio (L/R)
Figure 4 .5 Effect of Curvature on the Shear Distribution Factor for the Middle Girder due to FullyLoaded Lanes
1.6
.■S 0.8
0.6
m 0.4 Middle Girder, Fatigue Loading, N=3, S=2m, n=1
0.2
0.00.5 0.6 0.80.70,40.30,20.10
span-to-radius of Curvature ratio (L/R)
Figure 4 .6 Effect of Curvature on the Shear Distribution Factor for the Middle Girder due toFatigue Loading
134
1.5Interior Girder, Dead Load, N=3, S=2m, n=1
1.0
1 0.0
.6
•1.0 0.80.6 0.70.50.40.30.1 0.20span*to-radiu8 of Curvature ratio (L/R)
Figure 4.7 Effect of Curvature on the Shear Distribution Factor for the Interior Girder due tDead Load
2.0
IL.
n
£ 0.5
0.0
" a L=10m ■ L=l5m
—A—L=25m —X—Lss35m
Interior Girder, Fully loaded lanes, N=3, S=2m, n=1
0.1 0.2 0.3 0.4 0.5 0.6
span-to-radius of Curvature ratio (L/R)0.7 0.8
Figure 4.8 Effect of Curvature on the Shear Distribution Factor for the Interior Girder due toFully Loaded Lanes
135
1.5
interior Glider, Fatigue trading, N=3, S=2m, n=1
1.0u.
L»10m — L»15m - A L-aSm —X—L«35m
-0.5
- 1.00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
span-to-radius of Curvature ratio (L/R)
Figure 4.9 Effect of Curvature on the Shear Distribution Factor for the Interior Girder due toFatigue Loading
2.5
2.0 -
0.5 • Exterior girder, Dead Load N=4, S=2.5, n=2
0.00.7 o.g0.60.50.40.30.20 0.1
span-to-radius of Curvature ratio (L/R)
Figure 4.10 Effect of Curvature on the Shear Distribution Factor for the Exterior Girder dueto Dead Load
136
C 1.6
L=10
- # - L = 1 5 - A —L=25 -X -L = 3 5
TS
Exterior girder, Fully Loaded Lanes N=4, S=2.5, n=2
0.7 0.80.60.50.40.2 0.30 0.1
span-to-radius of Curvature ratio (L/R)
Figure 4.11 Effect of Curvature on the Shear Distribution Factor for the Exterior Girder dueto Fully Loaded Lanes
Exterior girder. Partially Loaded Lanes N=4, S=2.5, n-2
1.00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
span-to-radius of Curvature ratio (L/R)
Figure 4.12 Effect of Curvature on the Shear Distribution Factor for the Exterior Girder dueto Partially Loaded Lanes
.137
3.0
%C 2.0 -
1.5
1.0
Exterior girder. Fatigue Loading N=4, S=2.5, n=2
0.5
0.00.10 0.2 0.3 0.4 0.5 0.80.6 0.7
span-to-radius of Curvature ratio (L/R)
Figure 4.13 Effect of Curvature on the Shear Distribution Factor for the Exterior Girder dueto Fatigue Loading
1.6
1.4
0.4
0.2
0.0
- 0.2
Interior girder. Dead Load N=4, 5=2.5. n=2
0.1 0.2 0.3 0.4 0.5
span-to-radius of Curvature ratio (L/R)
0.6 0.7 0,8
Figure 4.14 Effect of Curvature on the Shear Distribution Factor for the Interior Girder dueto Dead Load
138
1.4
Interior girder, Fully Loaded Lanes N=4, S=2.5, n=2
g 0.8
0.6
3 0.4 '
02-
0.00.6 0.80.4 0.5 0.70.30.1 0.20
span-to-radius of Curvature ratio (L/R)
Figure 4.15 Effect of Curvature on the Shear Distribution Factor for the Interior Girder dto Fully Loaded Lanes
•a 0.8
0.6
S 0.4
Interior girder, Partially Loaded Lanes N=4, S=2.5, n=20.2 -
0.00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
span-to-radius of Curvature ratio (L/R)
Figure 4.16 Effect of Curvature on the Shear Distribution Factor for the Interior Girder dto Partially Loaded Lanes
139
3.5
Exterior Girder, Dead Load L=35m, S=2m
s
II
2.5
1•cCO
0.5
2 3 4 5 86 7
Number of Girders
Figure 4.17 Effect of Number of Girders on the Shear Distribution Factor for the ExteriorGirder due to Dead Load
3.5
Exterior Girder, Fully Loaded Lanes L=35m, S=2m
LLQ“ 2 5
£
5•I
I — FVL-O.O —m-R/L-0.1 -A -F V L -0 .4 —X— R /L ^.7
cn0.5
Number of Girders
Figure 4.18 Effect of Number of Girders on the Shear Distribution Factor for the ExteriorGirder due to Fully Loaded Lanes
140
3.6
Exterior Girder, Partially Loaded Lanes L=35m, S=2m
Q" 2.5
!
I■o
I R/L»0.0 -* -R /L = 0 .1 —&-R/L=0.4 —X— R/U0.7
0.5 •
8762 3 4 5
Number of Girders
Figure 4.19 Effect of Number of Girders on the Shear Distribution Factor for the ExteriorGirder due to Partially Loaded Lanes
2.5
IOI -♦ -R /L = 0 .0
— FVL=0.1 A R/L=0.4
—X— Fl/L=0.7
c
III(A
Exterior Girder, Fatigue Loading L=35m, S=2m
0.5-
2 3 4 5 6 7 8Number of Girders
Figure 4.20 Effect of Number of Girders on the Shear Distribution Factor for the ExteriorGirder due to Fatigue Loading
141
2.5
Middle Girder, Dead Load L=35m, S=2m
u.
o§u.g1€<0
’*5
I 0.5 •
2 3 4 5 6 87
Number of Girders
Figure 4.21 Effect of Number of Girders on the Shear Distribution Factor for the Middlegirder due to Dead Load
<n
S
Middle Girder, Fuliy Loaded Larws L=35m, S=2m
0,75
0.5
Number of Girders
Figure 4.22 Effect of Number of Girders on the Shear Distribution Factor for the MiddleGirder due to Fully Loaded Lanes
142
i 2.5-
i
I— R/L=0.0 H3—R/L=0.1
R/L=0.4 —X— FVL=0.7
II(0 Middle Girder, Fatigue Loading
L=35m, S=2m0.5 ■
872 64 53
Number of Girders
Figure 4.23 Effect of Number of Girders on the Shear Distribution Factor for the Middlegirder due to Fatigue Loading
0.4
Interior Girder, Dead Load, L=35m, S=2m•0.4
- 0.62 3 4 5 6 7 8
Number of Girders
Figure 4.24 Effect of Number of Girders on the Shear Distribution Factor for the interiorGirder due to Dead Load
143
1.75
1.5
i 1.25O
i2 'g 0.75
I0.5
♦ R/L"0.0-a-R/L-0.1
—X— R/L-0.7
0.25
Interior Girder, Fully Loaded Lanes L=35m, S=2m
-0.25
-0.52 3 4 5 6 87
Number of Girders
Figure 4.25 Effect of Number of Girders on the Shear Distribution Factor for the InteriorGirder due to Fully Loaded Lanes
2.5
2.25 -
€« 1.5
•2 1.25
I ’<5 0.75 £
—♦-R/L-0.0 —Q—R/L-0.1
—X— R/L-0.7
Interior Girder, Partially Loaded Lanes L=35m, S=2m
0.5 ■
0.25 ■
876542 3Number of Girders
Figure 4.26 Effect of Number of Girders on the Shear Distribution Factor for the InteriorGirder due to Partially Loaded Lanes
144
2JB
U.8 1.5
I 'O0.5
S
w•0.5
Interior Girder, Fatigue Loading L=35m, S=2m
Number of Girders
Figure 4.27 Effect of Number of Girders on the Shear Distribution Factor for the InteriorGirder due to Fatigue Loading
2.75
2.5
Q 2.25
Û: 1.75 — L/H=0.0 — L/R=0.1 —Ar-L/R=0.3 —X-L/R=0.5
£ 0.75
0.5Exterior girder, Dead Load L=25m, N=4,0.25
1.5 2 2.5 3 3.5Girders Spacing, (m)
Figure 4.28 Effect of Girder Spacing on the Shear Distribution Factor for the Exterior Girdeidue to Dead Load
145
1.75i€» 1.5
§— L/R-0.0 — L/R-0.1 -*-UR-0.3 —X— L/RaO.S
IJs 1.25
■I
IwExterior girder, Fully Loaded lanes L=25m, N=4,
0.751.5 2 2.5 3 3.5
Girders Spacing, (m)
Figure 4.29 Effect of Girder Spacing on the Shear Distribution Factor for the Exterior Girderdue to Fully Loaded Lanes
1.75 ■
11.
1.25
Exterior girder, Partially Loaded lanes L=25m, N=4,
0.753.532,521.5
Girders Spacing, (m)
Figure 4.30 Effect of Girder Spacing on the Shear Distribution Factor for the Exterior Girderdue to Partially Loaded Lanes
146
2,75
O 2.25
S» 1.75
CO 1.25
Exterior girder, Fatigue Loadeding L=25m, N=4,
0.753.532.521.5
Girders Spacing, (m)
Figure 4.31 Effect of Girder Spacing on the Shear Distribution Factor for the Exterior Girderdue to Fatigue Loading
1.75
S
!II
— L/R=0.0 — iyR=o.i -A -L/R=0.3 —X—L/R=0.51.25
CO
Middie girder, Fuiiy Loaded lanes L=25m, N=4,
0.751.5 2 2.5 3 3.5
Girders Spacing, (m)
Figure 4.32 Effect of Girder Spacing on the Shear Distribution Factor for the Middle Girderdue to Fully Loaded Lanes
147
2.5
2.25 ■
U.
u- 1.75 -
S 1.25
Middle girder, Fatigue Loadeding L=25m, N=4,
0.751.5 2 2.5 3 3.5
Girders Spacing, (m)
Figure 4.33 Effect of Girder Spacing on the Shear Distribution Factor for the Middle Girderdue to Fatigue Loading
1.75
1.5
O 1.25u.
.2 0.75
Inferior girder, Fully Loaded lanes L=25m, N=4,0.25
3.632.521.5Girders Spacing, (m)
Figure 4.34 Effect of Girder Spacing on the Shear Distribution Factor for the Interior Girderdue to fully Loaded Lanes
148
1.75
— L/R=0.0 -m-L/R=0.1 —A—UR=0.3 —X—L/R=0.5
£ 1.25
Interior girder, Partially Loaded lanes L=25m, N=4,
0.753.532.51.5 2
Girders Spacing, (m)
Figure 4.35 Effect of Girder Spacing on the Shear Distribution Factor for the Interior Girdeidue to Partially Loaded Lanes
Exterior girder. Dead Load, N=3, S=2, n=l
5 10 15 20 25 30 35 40
Span Length, (m)
Figure 4.36 Effect of Span Length on the Shear Distribution Factor for the Exterior Girderdue to Dead Load
149
2.5
- ♦ “-ITR-O — Lm-0,1 —X— lTR-0.3
Exterior girder, Fully Loaded Lanes, N=3, S=2, n=l
5 10 15 20 3025 35 40
Span Length, (m)
Figure 4.37 Effect of Span Length on the Shear Distribution Factor for the Exterior Girderdue to Fully Loaded Lanes
2
Middle girder, Fully Loaded Lanes, N=3, S=2, n=l
1.5
i
1
0.5
010 20 25
Span Length, (m)30 35 40
Figure 4.38 Effect of Span Length on the Shear Distribution Factor for the Middle Girder duetn Fully Loaded Lanes
150
Interior girder, Dead Load, N=3, S=2, n=l
II
0.5
I
— L/R=0 H i — L/R=0,I
—X— iyR=0.3
040353015 20 25105
Span Length, (m)
Figure 4.39 Effect of Span Length on the Shear Distribution Factor for the Interior Girderdue to Dead Load
— UR=0 -» -L /R = 0 .1 —X— UR=0.3
a 0.5-
Interior girder, Fully Loaded Lanes, N=3, S=2, n=l
5 10 15 20 25 30 35 40
Span Length, (m)
Figure 4.40 Effect of Span Length on the Shear Distribution Factor for the Interior Girderdue to Fully Loaded Lanes
; 151
2.0
Shear Distribution factors of Exterior girder for full and partial loads, 10m Bridges
1.6
■■■1.4
1.2
I 1.0
0.8
0.6
0.4
0.2
0.00,0 0.2 0.4 0.6 0.8 1.0 2.01.2 1.4 1.6 1.6
(SDF)FL
Figure 4. 41 Effect of Loading Conditions on the Shear Distribution Factor for the ExteriorGirder of the 10-m-span Bridges
2.5
Shear Distribution factors of Exterior girder for full and partial loads, 15m Bridges2.0
1.5
81.0
0.5
0.0 2.S2.01.51.00.50.0(SDF)n_
Figure 4.42 Effect of Loading Conditions on the Shear Distribution Factor for the ExteriorGirder of the 15-m-span Bridges
152
2 .5
S h ear Distribution factors of Exterior girder for full and partial loads, 25m Bridges2.0
■I
1 .5
i m1.0
0 .5
0.02.0 2 .51 .50.0 0 .5 1.0
(SDF)pl.
Figure 4.43 Effect of Loading Conditions on the Shear Distribution Factor for the ExteriorGirder of the 25-m-span Bridges
3.0
Shear Distribution factors of Exterior girder for full and partial loads, 35m Bridges
2.5
2.0
k 1.5
1,0
0.5
0.00.0 0.5 1.0 1.5 2.0 2.5 3.0
(SDF)fl
Figure 4.44 Effect of Loading Conditions on the Shear Distribution Factor for the ExteriorGirder of the 35-m-span Bridges
153
2.5
Shear Disiribution factors of Interior girder for full and partial loads, 10m Bridges
■■■2.0 -
I0.5
0.00.0 0.5 1.0 2.S1.5 2.0
(SDF)fl
Figure 4.45 Effect of Loading Conditions on the Shear Distribution Factor for the InteriorGirder of the 10-m-span Bridges
2.5
Shear Distribution factors of Interior girder for full and partial loads. 15m Bridges2.0 -
i m"
0.5 -
0.0 2.62.01.61.00.50,0(SDF)fl.
Figure 4.46 Effect of Loading Conditions on the Shear Distribution Factor for the InteriorGirder of the 15-m-span Bridges
154
2.5
Shear Distribution factors of Interior girder for full and partial loads, 25m Bridges2.0
i1.0
o.s
0.02.52.01,51.00.50.0
(SDF)pL.
Figure 4.47 Effect of Loading Conditions on the Shear Distribution Factor for the InteriorGirder of the 25-m-span Bridges
2.5
Shear Distribution factors of interior girder for full and partial loads, 35m Bridges
2.0
1.5
I1.0
M M0.5
0.0
•0.5 0.0 0.6 1.0 1.5 2.52.0(SDF)pl.
Figure 4.48 Effect of Loading Conditions on the Shear Distribution Factor for the InteriorGirder of the 35-m-span Bridges
155
$ ' ' L/R«0.0 — UR-0.& —A —L/R«0.4 —X—Lm«0.7
N 2.5
1.5
0.5 E x terio r g irder, fully loaded lanesSpan 35m, S=2m
0 2 43
Number of lanes
Figure 4.49 Effect of Number of Lanes on the Shear Distribution Factor for the ExteriorGirder due to Fully Loaded Lanes
S 2.5
E x terio r g irder, Partially loaded lanes Span 35m, S=2m0.5
Number of lanes
Figure 4.50 Effect of Number of Lanes on the Shear Distribution Factor for the ExteriorGirder due to Partially Loaded Lanes
156
2J
III£
IMiddle girder, fully loaded lanes Span 35m, S=2m
4320Number of lanes
Figure 4.51 Effect of Number of Lanes on the Shear Distribution Factor for the MiddleGirder due to Fully Loaded Lanes
2.5
,a
0.5 Middle girder. Partially loaded lanes Span 35m, S=2m
Number of lanes
Figure 4.52 Effect of Number of Lanes on the Shear Distribution Factor for the MiddleGirder due to Partially Loaded Lanes
157
2.5
UR->0.0 -•-L/R-0.1 — L/R=0.4 —X— UR-0.7&
g 1.5
! ■J% 0.5
12 0
Interior girder, fully loaded lanes Span 35m, S=2m-0.5 •
0 I 2 3 4
Number of lanes
Figure 4.53 Effect of Number of Lanes on the Shear Distribution Factor for the InteriorGirder due to Fully Loaded Lanes
2.5
0.5 - Interior girder. Partially loaded lanes Span 35m, S=2m
420Number of lanes
Figure 4.54 Effect of Number of Lanes on the Shear Distribution Factor for the InteriorGirder due to Partially Loaded Lanes
158
3.5
Span Length=35m, N=4 S=2m, L/R=0.7, n=2
3.0-
2.0
Due to Dead Load Due to Full Load Due to Partial Load
14121080 4 62Number of X-bracing Intervals
Figure 4.55 Effect of the Number of Cross-Bracing Intervals on the Shear Distribution Eachfor the Exterior Girder
III
.5
0.5
Exterior Girder, L=25m N=4, S=2, n=20
4 5
Load Case Number
Figure 4.56 Effect of the Load Cases Number on the Shear Distribution Factor for theExterior Girder of a two-lane Bridge
159
1.5
— L / R. 0
L /R -O .I
—àr—L /R -0 .3
—X — IJR -0 .5
Middle Girder, L=25m N=4, S=2, n-2
I 2 3 4 5 6 87
Load Case Number
Figure 4.57 Effect of the Load Cases Number on the Shear Distribution Factor for the MiddleGirder of a two-lane Bridge
1,5
•Ç 0.5
Interior Girder, L»25mN=4, S=2, n=2
-0.5 65432Load Case Num ber
Figure 4.58 Effect of the Load Cases Number on the Shear Distribution Factor for theInterior Girder of a two-lane Bridge
160
Exterior Girder, L=25m N=7, S=2, n=3
-0.510 II 12984 71 2 3 5 6
Load Case Number
Figure 4.59 Effect of the Load Cases Number on the Shear Distribution Factor for theExterior Girder of a 3-lane Bridge
2
1
II55 0.5 Middle Girder, L=25m
N=7, S=2, n=3
021 3 4 5 6 7 8 9 10 11 12
Load Case Number
Figure 4.60 Effect of the Load Cases Number on the Shear Distribution Factor for the MiddleGirder of a 3-lane Bridge
161
2
Interior Girder, L=25m N=7. S=2, n=31.5
I
0.5
0
-0.5
12 3 4I 5 6 7 8 9 1210 11
Load Case Num ber
Figure 4. 61 Effect of the Load Cases Number on the Shear Distribution Factor for theInterior Girder of a 3-lane Bridge
2
1.5
0.5
Exterior Girder, L=35m N=5, S=3, n=40
-0.51 2 3 4 5 6 7 8 9 10 I I 12 13 14 15 Ifi 17 18
Load Case Num ber
Figure 4.62 Effect of the Load Cases Number on the Shear Distribution Factor for theExterior girder of a 4-lane Bridge
162
Middle Girder, L=35m N=5, S=3, n=4
17 1815 1613 149 10 123 4 7 8 112 5 6Load Case Number
Figure 4.63 Effect of the Load Cases Number on the Shear Distribution Factor for the MiddlGirder of a 4-lane Bridge
Interior Girder, L=35m N=5, S=3, n=4
2 3 4 6 7 8 9 10 11 12 13 14 15 16 17 18
Load Case Number
Figure 4.64 Effect of the Load Cases Number on the Shear Distribution Factor for theInterior girder of a 4-lane Bridge
163
2J
Spanlength=35m, N=S S=2, ie=2. Dead load
Interior girder 1 Girder 2 Girder 3 —X—Girder 4 —X—Exterior Girder 5
0 0.1 0.1 0.5 0.7 0.1
span-to-radiiis of Curvature ratio (L/R)
Figure 4. 65 Effect of Girder Location on the Shear Distribution Factor due to Dead Loading
2.5
Span lenglh=35m, N=5 S=2, n=2. Fully loaded Lanes
Girder 3 —X—Girder 4 —X—Exterior girder 5Girder 2Interior girder
02span-to-radius of Curvature ratio (L/R)
Figure 4.66 Effect of Girder Location on the Shear Distribution Factor due to Fully Loaded Lanes
164
23
Span lenglh=35m, N=6
8=2.5, n=4, Dead load
Exterior G irder 6G irder 3 —X— G irder 4 —X— Girder 5Girder 2Interior girder 1
0.7 0.80.60.50.40.2 0.30.10span-to-radius of Curvature ratio (L/R)
Figure 4.67 Effect of Girder Location on the Shear Distribution Factor due to Dead Loadin;
Spanlength=35m, N=6
S=2.5. n=4. Fully loaded Lanes
- O — Interior girder G irder 2 Girder 3 —X—Girder 4 —X—Girder 5 Exterior G irder 6
0.10 0.2 0.3 0.4 0.5 0.7
span-to-radius of Curvature ratio (L/R)
Figure 4.68 Effect of Girder Location on the Shear Distribution Factor due to Fully Loade Lanes
165
Hinge Support, Span leng*h=35m, N=5
S=2, n=2, D ead loadCO
In te rio r g irder G ird er 2 G irder 3 —X— G irder 4 —X— E xterio r g irde r 5
0 0.1 03 0.4 0.7
span-to-radius of Curvature ratio (L/R)
Figure 4. 69 Effect of the Type of Support on the Shear Distribution Factor due to Dead Loading (Hinge Support Line)
Î.5
Hinge Support, Span length=35m, N=5 S=2, n=2, Fnily loaded Lanes
G irder 3 —X— G irder 4 —X— E xterior g irder 5G ird e r 2In te rio r ginder I
oj0,70.50 0.1span-to-radius of Curvature ratio (L/R)
Figure 4. 70 Effect of the Type of Support on the Shear Distribution Factor due to Fully Loaded Lanes (Hinge Support Line)
166
Roller Support, Span lepgth=35m, N=5 S=2. n=2. Dead load
e 0.5
■OSG irder 3 —X— G irder 4 —X— Exterior girder 5Interior girder I G irder 2
0.6 07
span-to-railius of Curvature ratio (L/R)
Figure 4.71 Effect of the Type of Support on the Shear Distribution Factor due to Dead Loading (Roller Support Line)
Roller Support, Span length=35m, N=5 S=2, n=2. Fully loaded Lanes
Interior girder 1 Girder 2 Girder 3 —X— Girder 4 —X— Exterior girder 5
0 0.1 0.3 0.5 0.7
span-to-radius of Curvature ratio (L/R)
Figure 4.72 Effect of the Type of Support on the Shear Distribution Factor due to Fully Loaded Lanes (Roller Support Line)
167
2.50
Exterior girder, fully loaded lanes2.00
0.50
0.000.00 0.50 1.00 1.50 2.00 2.50
(LDF) per CHBDC
Figure 4. 73 Comparison between the Shear Distribution Factors of the Exterior Girder due to Truck Loading as Specified in the CHBDC and from the Current Study
4.50
4.00
E xterio r g irder, fa tigue loading3.50
^ 3.00 Be]
t 2.50 a ,
g 2.00 Q
1.50
1.00
0.50
0.004.504.003503.002.502.001.501.000.500.00
(LDF) per CHBDC
Figure 4.74 Comparison between the Shear Distribution Factors of the Exterior Girder due to Fatigue Loading as Specified in the CHBDC and from the Current Study
168
2.50
Middle fdrder, fiiliy loaded lanes
2.00
g 1.50
Q.a 1.00
0.50
0.002.00 2.501.501.000.500.00
(LDF) per CHBDC
Figure 4.75 Comparison between the Shear Distribution Factors of the Middle Girder due tc Truck Loading as Specified in the CHBDC and from the Current Study
4.50
4.00 M iddle girder, fatigue loading
^ 3.00
% 2.50CL
gZM»^ 1.50
1.00
0.50
0.000.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.504.00
(LDF) per CHBDC
Figure 4.76 Comparison between the Shear Distribution Factors of the Middle Girder due ti Fatigue due to Truck Loading as Specified in the CHBDC and from the Current Study
169
2.5
2,0
IL
Q 1,0
Exterior girder. Dead Load, N»7, S=2m, fi=20.5
0.00 0.1 0.2 0.40.3
8pan-to-radiu8 of curvature ratio (L/R)
Figure 4.77 Effect of Curvature on the Moment Distribution Factor for the Exterior Girderdue to Dead Load
3.0
2.5
IΣI 2.0
I1i Exterior girder, Fully Loaded Lanes,
N=7, S=2m, n«2
1.00.40.30.2
8pan-to-radiU8 of curvature ratio (L/R)0.1
Figure 4. 78 Effect of Curvature on the Moment Distribution Factor for the Exterior Girderdue to Fully Loaded Lanes
170
3.0
2.6-
2.0
1O
£ Exterior girder, Partially Loaded Lanes, N=7, S=2m» n=2
1.00.40.30.2
span-to-radius of curvature ratio (L/R)0.10
Figure 4.79 Effect of Curvature on the Moment Distribution Factor for the Exterior Girderdue to Partially Loaded Lanes
3.5
3.0 •
2.5 •
2.0
Exterior girder. Fatigue Loading, N=7, S=2m, n=21.5
1.0
0.2span-to-radius of curvature ratio (L/R)
0.3 0.4
Figure 4.80 Effect of Curvature on the Moment Distribution Factor for the Exterior Girderdue to Fatigue Loading
171
2.5
2.0IIfi1.0
Mfddte ginder, Dead Load, N«7, S=2m, n=2
0.50 0.1 0.2 0.40.3
8pan-to-radiu8 of curvature ratio (UR)
Figure 4. 81 Effect of Curvature on the Moment Distribution Factor for the Middle Girderdue to Dead Load
2.5
Urn: 2.0
Middle girder, Fully Loaded Lanes, N=7, S=2m, n»2
1.00.40.30.20
span-to -rad ius of curvatu re ratio (UR)
Figure 4.82 Effect of Curvature on the Moment Distribution Factor for the Middle Girderdue to Fully Loaded Lanes
172
4.0
* 3.0
2.5
2.0Middle girder, Fatigue Loading, N” 7, S=2m, n=2
1.5
1.00.40.30.20.10
8pen>t0-radiu8 of curvature ratio (UR)
Figure 4.83 Effect of Curvature on the Moment Distribution Factor for the Middle Girderdue to Fatigue Loading
1.8
1.6
I 1.4
Li.
j 1.2
O1.0
Interior girder. Dead Load, N=7, S=2m, n=2
0,8
0.60.1 0.2
8pan-to-iadiu8 of curvature ratio (L/R)0.3 0.4
Figure 4.84 Effect of Curvature on the Moment Distribution Factor for the Interior Girderdue to Dead Load
173
2.1
2.0 •
i
i1 1.7
Interior girder. Fully Loaded Lanes, N=7,S=2m ,n=2
1.50.10 0.2 0.40.3
span-to-radius of curvature ratio (L/R)
Figure 4.85 Effect of Curvature on the Moment Distribution Factor for the Interior Girderdue to Fully Loaded Lanes
2.0 -
U. 1.8 •a
I
ae 1.4
1.2-
1.0 ■
Interior girder. Partially Loaded Lanes, N=7, Ss2m , n=2
0.1 0.2 0.3
span-UHradius of curvature ratio (L/R)0.4
Figure 4.86 Effect of Curvature on the Moment Distribution Factor for the Interior Girderdue to Partially Loaded Lanes
174
Z5
“ 20
Interior girder, Fatigue Loading, N=7, S=2m, n=2
1.00.2 0.30 0,1 0.4
span-t6-radius of curvature ratio (UR)
Figure 4.87 Effect of Curvature on the Moment Distribution Factor for the Interior Girderdue to Fatigue Loading
2.5
O 1
0.5
-*-UB=0-*-L/R=0.1-*-L /R =0.2-* -U R = 0 .3
Exterior girder, Dead Load, L=10m, S=2fri
5 6Number of Girders, (N)
Figure 4.88 Effect of Number of Girders on the Moment Distribution Factor for the ExterioiGirder due to Dead Load
175
2.5
u.
1.5
Exterior girder, Fully Loaded Lanes, L=10m. S«2m0.5
3 4 5 6 87Number of G irders, (N)
Figure 4.89 Effect of Number of Girders on the Moment Distribution Factor for the ExteriorGirder due to Fully Loaded Lanes
2.5
u_
Exterior girder, Partially L oaded L anes, l_=10m, S»2m
0.5 -
Number of G irders, (N)
Figure 4.90 Effect of Number of Girders on the Moment Distribution Factor for the ExteriorGirder due to Partially Loaded Lanes
176
3.5
5 2.5
1.5
Exterior girder. Fatigue Loading, L=10m, S=2m
0.5
3 4 S 6 7 8Number of Girders, (N)
Figure 4.91 Effect of Number of Girders on the Moment Distribution Factor for the ExterioiGirder due to Fatigue Loading
2.5
S
0.5- Middle girder, Dead Load, L=10m, S=2m
Number of Girders, (N)
Figure 4.92 Effect of Number of Girders on the Moment Distribution Factor for the MiddleGirder due to Dead Load
177
2.5
u.§
s
Iss
Middle gilder. Fuliy Loaded L anes, L=lOm, S s2 m
0.5 •
3 4 5 6 a7Number of Girders, (N)
Figure 4.93 Effect of Number of Girders on the Moment Distribution Factor for the MiddleGirder due to Fully Loaded Lanes
2.5
Li.
§
I8I1Q
IMiddle girder, Partially L oaded Lanes, L=lOm, S=2m0.5 ■
Number of Girders, (N)
Figure 4.94 Effect of Number of Girders on the Moment Distribution Factor for the MiddleGirder due to Partially Loaded Lanes
178
3.5-
II 2.6-
L/R=017R=0.1L/R=0.2L/R=0.3
1
Middle girder, Fatigue Loading. L=10m, S=2m
0.5-
876543Number of Girders, (N)
Figure 4.95 Effect of Number of Girders on the Moment Distribution Factor for the MiddleGirder due to Fatigue Loading
II!a
3
Interior girder, Dead Load, L=10m,S=2m
0.63 4 5 6 7 8
Number of Girders, (N)
Figure 4.96 Effect of Number of Girders on the Moment Distribution Factor for the InterimGirder due to Dead Load
179
2.5
u.
1.6
Interior girder, Fully Loaded Lanes, L=10m, S=2m
0.5 -
3 4 s 6 7 SNumber of Girders, (N)
Figure 4.97 Effect of Number of Girders on the Moment Distribution Factor for the InteriorGirder due to Fully Loaded Lanes
2.5
Ii£ UR-O
L/R-0.1UR-0.2L/R.0.3
I1I
S
Interior girder. Partially L oaded thanes, L=10m, S=2m0.5 ■
Number of G irders, (N)
Figure 4.98 Effect of Number of Girders on the Moment Distribution Factor for the InteriorGirder due to Partially Loaded Lanes
180
2.5
II
1.6Interior girder, Fatigue Loading, L=10m, S=2m
86 7543Number of Girders, (N)
Figure 4.99 Effect of Number of Girders on the Moment Distribution Factor for the InterioGirder due to Fatigue Loading
I
-X
-A
-e-U R =o-*-UR=0.1-* r-U R = 0 .2
Exterior girder. Dead Load, L=10, N=4, n=2
1.5 2.5
G irder Spacing, (m)
3.5
Figure 4.100 Effect of Girder Spacing on the Moment Distribution Factor for the ExterioGirder due to Dead Load
181
2.5
Cî
1E x te rio r g ird e r . F u lly L o a d e d Lane# , L&IO, Nss4, n= 2
0.51.5 2 2.5 3 53
Girder Spacing, (m)
Figure 4.101 Effect of Girder Spacing on the Moment Distribution Factor for the ExteriorGirder due to Fully loaded Lanes
2.5
b
II
E x te rio r g ird e r . P a rtia lly L o ad ed Lane#, L a lO , N » 4 , n = 2
0.53.S32.521.5
Girder Spacing, (m)
Figure 4.102 Effect of Girder Spacing on the Moment Distribution Factor for the ExteriorGirder due to Partially Loaded Lanes
182
3.5
I
IsExterior girder, Fatigue Loading, L=10, N=4, d=2
32 2.5 3.51.5
Girder Spacing» (m)
Figure 4.103 Effect of Girder Spacing on the Moment Distribution Factor for the ExteriorGirder due to Fatigue Loading
2.5
b
UR=0L/R=0.1L/R=0.2UR=0.3
.1
Middle girder, Fully Loaded Lanes, L=10, N=4, n=2
0.5
2.5
G irder Spacing, (m)3.5
Figure 4.104 Effect of Girder Spacing on the Moment Distribution Factor for the MiddleGirder due to Fully Loaded Lanes
183
2.5
I
IÈiII
M iddle girder. Fatigue Loading, LslO , N »4, n»2
0.5 •
1.5 2 2.5 3 3.5
Girder Spacing, (m)
Figure 4.105 Effect of Girder Spacing on the Moment Distribution Factor for the MiddleGirder due to Fatigue Loading
2.5
I
5
1Interior girder, Fully Loaded Lanea, LslO . Ns=4, n=2
0.53.92.5
Girder Spacing, (ro)
Figure 4.106 Effect of Girder Spacing on the Moment Distribution Factor for the InteriorGirder due to Fully Loaded Lanes
184
3
2.5
IgI
1.5
1
Interior girder, F atigue Loading, L=IO, N =4, n=2
0.5
02.5 3 3.51.5 2
Girder Spacing, (m)
Figure 4.107 Effect of Girder Spacing on the Moment Distribution Factor for the Interior Girdeidue to Fatigue Loading
3.0
Moment Distribution factors of Exterior girder for full and partial loads, 10m Bridges
2.5
2.0 ■ ■■
S§
0.5
0.00.0 0.5 1.0 1.5 2.0 2.5
(MDF)fl
Figure 4,108 Effect of Loading Conditions on the Moment Distribution Factor for theExterior Girder of the lO m-span Bridges
185
3.0
M om ent Distributkm factors of Middle g irder for full an d partial loads, 10m Bridges
2.5 -
2.0-
S■ ■
S"1.0
o.s
0.00.0 0.5 1.0 1.5 a.o2.5
(MDF)fl.
Figure 4.109 Effect of Loading Conditions on the Moment Distribution Factor for the MiddleGirder of the 10-m-span Bridges
2.5
M om ent Distribution factors of Interior g irder for full Eind partial loads, 10m Bridges2.0 ■■
éS
■■
0.5
0.0 2.01.51.00.50.0
Figure 4.110 Effect of Loading Conditions on the Moment Distribution Factor for the InteriorGirder of the 10-m-span Bridges
186
4.0
3.5
3.0
2.5iZ
2.0
1.5
Exterior girder, Dead Load,N=7, S=2m, n=2
1.0
0.5
0.00.40.30.2
span-to-fadius of curvature ratio (L/R)0.10
Figure 4. I ll Effect of Curvature on the Deflection Distribution Factor for the ExteriorGirder due to Dead Load
4.0
as
S 3.0
2.5
2.0
Exterior girder, Fuily Loaded Lanes, N=7, S=2m, n=2
1.5
1.00 0.1 0.2 0.3 0,
8pan-to-radiu8 of curvature ratio (UR)
Figure 4.112 Effect of Curvature on the Deflection Distribution Factor for the ExteriorGirder due to Fully Loaded Lanes
187
4.0
3.5
2.0
Exterior girder, Partially Loaded Lanes, N=7, Ss2m, n=2
1.00 0.1 0.2 0.40.3
spaiv to-rad ius of curvature ratio (UR)
Figure 4.113 Effect of Curvature on the Deflection Distribution Factor for the ExteriorGirder due to Partially Loaded Lanes
5.0
4.5 -
4.0 -
t> 3.5
Exterior girder, Fatigue Loading, Ns=7, S=2m, n=22.0
1.5
1.0 0.40.2sparv-toHradlua of curvature ratio (L/R)
0.1
Figure 4.114 Effect of Curvature on the Deflection Distribution Factor for the ExteriorGirder due to Fatigue Loading
188
3.0
Q 2.5
2.0
1.0 Middle girder, Dead Load, N=7, S=2m, n=2
0.5
0.00.40.30.20.10
spaivto-radius of curvature ratio (L/R)
Figure 4.115 Effect of Curvature on the Deflection Distribution Factor for the Middle Girdedue to Dead Load
3.0
2.0
1.5
Middle girder, Fully Loaded Lanes, N=7, S=2m, n=2
1.0
0.50 0.1 0.2 0.3 0.
8pan-to-radlu8 of curvature ratio (L/R)
Figure 4 .116Effect of Curvature on the Deflection Distribution Factor for the Middle Girdedue to Fully Loaded Lanes
189
3.0
2.5
f 2.0
1.5
Middle girder. Partially Loaded Lanes, N=7, S=2m, n*2
0.50 0.1 0.2 0.40.3
sparvto-radius of curvature ratio (L/R)
Figure 4.117 Effect of Curvature on the Deflection Distribution Factor for the Middle Girderdue to Partially Loaded Lanes
4.5
4.0
Q 3.5
3.0
M 2.5
5 2.0 Middle girder, Fatigue Loading, N=7, S=2m, n=2
1.00.40.30.2
apan-to-radius of curvature ratio (L/R)0.1
Figure 4.118 Effect of Curvature on the Deflection Distribution Factor for the Middle Girderdue to Fatigue Loading
190
2.0
g
81.0
Interior girder, Dead Load, N=7, S=2m, n=2
0.50.3 0.40.20.10
8pan-to-fadiu8 of curvature ratio (L/R)
Figure 4.119 Effect of Curvature on the Deflection Distribution Factor for the Interior Girderdue to Dead Load
2.0
8
b8
Interior girder, Fully Loaded Lanes, N=7, S=2m, n=2
1.00.2
span-toHvdius of curvature ratio (L/R)0.3 0.4
Figure 4.120 Effect of Curvature on the Deflection Distribution Factor for the Interior Girdeidue to Fully Loaded Lanes
191
2.5
IL
2.0
Interior girder, Fatigue Loading, Ns=7, S=2m, n=2
1.00 0.1 0.2 0.40.3
sp a tv to ra d iu s of curvature ratio (L/R)
Figure 4.121 Effect of Curvature on the Deflection Distribution Factor for the Interior Girderdue to Fatigue Loading
3.5
U. 2.5
Exterior girder, Dead Load,L=10m, S=2m0.5
Number of Girders, (N)
Figure 4.122 Effect of Number of Girders on the Deflection Distribution Factor for theExterior Girder due to Dead Load
192
4
3,5
I'iZ 2.5
§2
1.5O
1Exterior girder, Fully Loaded Lanes, L=10m, S=2m0.5
087653 4
Number of Girders, (N)
Figure 4.123 Effect of Number of Girders on the Deflection Distribution Factor for the Exterior Girder due to Fully Loaded Lanes
3.5
IL 2.5■1JR=0■L/R=C.1■L/R=0.2L/R=0.3
Exterior girder, Partially Loaded Lanes, L=10m, S=2m
0.5
Number of Girders, (N)
Figure 4.124 Effect of Number of Girders on the Deflection Distribution Factor for theExterior Girder due to Partially Loaded Lanes
193
4.5 ■
§
I§ UR-0
UR-0.1UR-0.2U R-0.3I
2.5 ■
s
I%a Exterior girder. Fatigue Loading,
L=10m, S=2m0-5
3 4 5 6 7 6Number of Girders, (N)
Figure 4.125 Effect of Number of Girders on the Deflection Distribution Factor for theExterior Girder due to Fatigue Loading
3.5
u.
u.UR-0UR-0.1UR-0.2UR-0.3
Middle girder, Dead Load, L=10m, S=2m
0.5 ■
87653 4
Number of Girders, (N)
Figure 4.126 Effect of Number of Girders on the Deflection Distribution Factor for theMiddle Girder due to Dead Load
194
3.5
i 1.5So
& Middle girder, Fully Loaded Lanes, L=10m, S=2m
0.5
7 863 4 5
Number of Girders, (N)
Figure 4.127 Effect of Number of Girders on the Deflection Distribution Factor for the Middle Girder due to Fully Loaded Lanes
U. 2.5
i 1-S
0.5Middle girder, Partially Loaded Lanes, L=1 Om, S=2tn
Number of Girders, (N)
Figure 4.128 Effect of Number of Girders on the Deflection Distribution Factor for theMiddle Girder due to Partially Loaded Lanes
195
4.5
U.3.5 ■
UR -0UR-0.1UR-0.2UR-0.3
2.5
1.5
Middle girder, Fatigue Loading, 1=1 Om, S=2m0.5 •
3 4 5 6 87
Number of Girders, (N)
Figure 4.129 Effect of Number of Girders on the Deflection Distribution Factor for theMiddle Girder due to Fatigue Loading
2
u.
1.5LL U R -0
UR-0.1UR-0.2U R-0.3
1
Interior girder. Dead Load, L=10m, S=2m
0.5
Number of Girders, (N)
Figure 4.130 Effect of Number of Girders on the Deflection Distribution Factor for theInterior Girder due to Dead Load
196
g
! L/R=0L/R=0.1L/R=0.2UR=0.3
1.2
|o.e I 0.61
0.4
Interior girder, Fully Loaded Lanes, L=10m, S=2m0.2
8763 4 5Number of Girders, (N)
Figure 4.131 Effect of Number of Girders on the Deflection Distribution Factor for the Interior Girder due to Fully Loaded Lanes
3.5
2.5L/R=0
UR=0.1
L/R=0.2
L/R=0.3
I
0.5 Exterior girder. Dead Load, L=10. N=4, n=2
1.5 2.5 3.5
Girder Spacing, (m)
Figure 4.132 Effect of Girder Spacing on the Deflection Distribution Factor for the ExteriorGirder due to Dead Load
197
17R-0L/R-O.IXyR-0.2IÆ«0.3
I
Exterior girder, Fully Loaded Lanes. L=10, N=4, n=2
0.5 41.5 2 2.5 3 3.5
Girder Spacing, (m)
Figure 4.133 Effect of Girder Spacing on the Deflection Distribution Factor for the ExteriorGirder due to Fully Loaded Lanes
IIeIrQ
1Exterior girder. Partially Loaded Lanes, L=10, N=4, n=2
0.5 3.J2.52Girder Spacing, (m)
Figure 4.134 Effect of Girder Spacing on the Deflection Distribution Factor for the ExteriorGirder due to Partially Loaded Lanes
198
c 3 J
Exterior girder, Fatigue Loading, L=10, N=4, n=2
3.52.5 31,5 2Girder Spacing, (m)
Figure 4.135 Effect of Girder Spacing on the Deflection Distribution Factor for the ExteriorGirder due to Fatigue Loading
I2.5
1.5
0.5
X-&-
-X
-4
—♦—UR=0 L/R=0.1
-* UR=0.2 - X - L /R = 0 .3
Middle girder, Dead Load L=10, N=4, n=2
1.5 2.5 3.5
Girder Spacing, (m)
Figure 4.136 Effect of Girder Spacing on the Deflection Distribution Factor for the MiddleGirder due to Dead Load
199
2.5
2I
o1.5
1
Middle girder. Fully Loaded Lanes, L=IO, N=4, n=2
0.51.5 2 2.5 3 3.5
G irder Spacing, (m)
Figure 4.137 Effect of Girder Spacing on the Deflection Distribution Factor for the MiddleGirder due to Fully Loaded Lanes
3.5
3
II 2-5 £
I 'g 1.5
I1 Middle girder. Fatigue Loading,
L=iO, N=4, n=2
0.53.332.521.5
G irder Spacing, (m)
Figure 4.138 Effect of Girder Spacing on the Deflection Distribution Factor for the MiddleGirder due to Fatigue Loading
2 0 0
u
I '— UR= 0
— UR=0. 1
—A — L/R=0.2
- H - U R = 0 .3'C
I I
.1
2 0.3
Interior girder. Dead Load, L=10,N=4, n=2
03.532.521.5
Girder Spacing, (m)
Figure 4.139 Effect of Girder Spacing on the Deflection Distribution Factor for the InteriorGirder due to Dead Load
I :
— L/R=0 —» - L /R = 0 .1
-T&—UR=0.2 —X — UR=0.3
I
Interior girder. Fully Loaded Lanes, L=10, N=4, n=2
2.5 3.5
Girder Spacing, (m)
Figure 4.140 Efliect of Girder Spacing on the Deflection Distribution Factor for the InteriorGirder due to Fully Loaded Lanes
201
4.0
3.5 ■ Deflection Distribution factors of Exterior girder for full and partial loads, 10m Bridges3.0 ■
2.5
U.QO2.0
f >1.5
0.5 -
0.00.0 0.5 1.0 1.5 2.0 4 .02.5 3.0 3.5
(DDF)fl.
Figure 4.141 Effect of Loading Conditions on the Deflection Distribution Factor for theExterior Girder of 10-m-span Bridges
3.5
Deflection Distribution factors of Middle girder for full and partial loads, 10m Bridges
3.0
2.5
s’ 2.0
1.5
1.0
0.5
0.03.53.02.52.01.00.0
Figure 4.142 Effect of Loading Conditions on the Deflection Distribution Factor for theMiddle Girder of 10-m-span Bridges
2 0 2
2.5
Deflection Distribution factors of Interior girder for full and partial loads, 10m Bridges
2.0
1.5
éQO■ mm
1.0
0.5
0.00.0 0.5 1.0 1.6 2.0 2.5
(DDF)fl.
Figure 4.143 Effect of Loading Conditions on the Deflection Distribution Factor for theInterior Girder of 10-m-span Bridges
203
0.20
TO
f■3 0.00
5 -0.10
-0.20
Exterior girder - Dead Load • RyL=0.1, L = 10m
■ R /L = 0.2 , L = 10m
A R / L ^ .3 , L = 10m
10
t t t 1
10m Span Bridge
Figure 4.144 Effect of Curvature on the Warping-to-Bending ratio for the Exterior Girderdue to Dead Load
0.20
Exterior girder - Fully Loaded Lanes• R /L = 0 .1 ,L = 1 0 m
■ R /L = 0.2 , L = 10m
A R /L = 0 .3 , L = 10m
0.10 ■
I•f"9 0.00
I12 14 16 i ;
■0 .10 • É É A
• • É A
-0.20 ■
10m span Bridge
Figure 4.145 Effect of Curvature on the Warping-to-Bending ratio for the Exterior Girderdue to Fully Loaded Lanes
204
020
0.10
I
î
0.00
-0.10
-0.20
[Exierior goder - Partially Loaded Lanes• R/L=0.1, L=10m ■ R/L=0.2,L=10m A R/L=0.3, L=10m
• • •
1 I ■
• •■ ■
a a
10m Span Bridge
Figure 4.146 Effect of Curvature on the Warping-to-Bending ratio for the Exterior Girderdue to Partially Loaded Lanes
0.20
0.10
I
{0.00
- 0 .10
-0.20
Exterior girder - Fatigue Loading• R/L=0.1, L=10m ■ RyL=0.2, L=10tn ▲ R7L=0.3, L=10m
10#
• #
e
É
y 16 H
10m Span Bridge
Figure 4.147 Effect of Curvature on the Warping-to-Bending ratio for the Exterior Girderdue to Fatigue Loading
205
0^0
0.10
II3 0.00
I-0.10
[Middle girder - Dead Load | #R/L=0.1.L=10m ■ R/LsO.2, LslOm a R/L=0,3, L=10m
10 12 14 15
I ■ ■i à É
A A A
10m Span Bridge
Figure 4.148 Effect of Curvature on the Warping-to-Bending ratio for the Middle Girder dueto Dead Load
0.20 ■
pg 0.10 ■
I* 0.00
g
®pÔ -0.10
Î ,> -0.20 H
-0 .30 ■
M iddle girder - Fully Loaded Lanes• R/L=0.1, L=10m ■ R/L=0.2, L=10m A R/L=0.3, L=10m
10 12 14 16
• • • • • * . . A A
■ AA
10m Span Bridge
Figure 4.149 Effect of Curvature on the Warping-to-Bending ratio for the Middle Girder dueto Fully Loaded Lanes
206
OlOi
g g 0.10
Middle grder - Partially Loaded Lanes• R/L=0.1,L=10m ■ R/L=0.2, L=10m AR/L=0.3,L=10m
Ic2 0.00'
f6 -0.10
f> 4).20
4 6 8 10 12 14
# • • • •
• •
## # • •
■ I 1 ■ ■ ■ ■ ■
I 1
-0.30 ^
10m Span Bridge
Figure 4.150 Effect of Curvature on the Warping-to-Bending ratio for the Middle Girder dueto Partially Loaded Lanes
0.20
0.10
I 0.00
Ii -0.10
^ -0.:,20
-0.30
Middle girder - Fatigue Loading• R/L=0.1,L=10m ■ R/L=0.2, L=10m A R/L=0.3, L=10m
10 12 14 15
• • * ■ ■ 1 1 1 1 I ■ I 1
■ AA
- 1 IA
1 I
10m Span Bridge
Figure 4.151 Effect of Curvature on the Warping-to-Bending ratio for the Middle Girder dueto Fatigue Loading
207
0.20
0.10
> 0.00
i-0.10
Isa -0.20
.1p. 4).30
-0.40
-0.50
(interior girder - Dead Load *R/L=0.1, L=IOm ■ R/L=0.2, L=IOm A R/L=0.3, L=IOm
10
A A
A A A AA A
10m Span Bridge
Figure 4.152 Effect of Curvature on the Warping-to-Bending ratio for the Interior Girder dueto Dead Load
0.20 ■
Interior girder - Fully Loaded Lanes0.10 ■
• R/L=0.1,L=I0m ■ R/L=0.2, L=10m AR/L=0.3, L=10m
> 0.00 -
i -0.10 •
1CQ -0.20 -
g . -0 .30 ■
I-0 .40 •
10 12 14 16•
• • • • • •
■
■ ■ ■ A
■ ■ ■
A AA
A A A
A A
-0.50
10m Span Bridge
Figure 4.153 Effect of Curvature on the Warping-to-Bending ratio for the Interior Girder dueto Fully Loaded Lanes
208
a2o
0.10
IK 0.00
Iè -o ">
I> 41.20
-0.30
Interior girder - Partially Loaded Lanes• R/L=0.1,L=10ra ■ R/L=0.2, L=10m AR/L=0.3, L=10m
A A
tf . • • fi •
_ ■ ■ ■ ! !■ A A AA
16
 !
A A
10m Span Bridge
Figure 4.154 Effect of Curvature on the Warping-to-Bending ratio for the Interior Girder dueto Partially Loaded Lanes
0.20
0.10
> 0.00
I6B -0.10I» -0.20
Sg . -0.30
-0,40
Interior girder - Fatigue Loading• R/L=0.1,L=10m ■ R/L=0.2, L=10m AR/L=0.3, L=10m
A A A A * A
• «
: : : : : :
• ft
: ;
lOm Span Bridge
Figure 4.155 Effect of Curvature on the Warping-to-Bending ratio for the Interior Girder dueto Fatigue Loading
209
2.50
Moment Distribution Factors of Exterior Girders For Live Load Case according to the Finite Element Analysis (FEA) and Canadian Code(CHBDC) For Straight Bridges
2.25 ■
2.00
1.75 ■
LL 1.50 -
1.25 -dQ 1.00E
0.75 -
0.50
0.25 -
0.000.00 0.25 0.50 0.75 1.00 2.251.25 1.50 2.501.75 2.00
(“ DF)a..xi - CHBDC
Figure 4.156 Comparison between the Moment Distribution Factors of the Exterior Girder due to Truck Loading as Specified in the CHBDC and from the Current Study
3.50
Moment Distribution Factors o f Exterior Girders For Fatigue Load Case according to the Finite Element Analysis (FEA) and Canadian Code(CHBDC) For Straight Bridges
3.25 -
3.00 ■
2.75 ■
2.50 -■ ■§ 2.25 -
2.00 ■
1 1.75 ■
1.50 ■
1.25 ■
1.00 -
0.75 -
0.50 -
0.25
0.003.503.263.002.50 2.752.252.001.751.50
(MDF)Frt..xi-CHBDC1.00 1.250.750.500.250.00
Figure 4.157 Comparison between the Moment Distribution Factors of the Exterior Girder due to Fatigue Loading as Specified in the CHBDC and from the Current Study
210
3.00
2.75Moment DistritMitioD FactofS o f Middle Girders For Live Load Case according to the Finite Element Analysis (FEA) and Canadian Code(CHBDC) For Straight Bridges
2.50
< 2.00
1.50
1.25Oz 1.00
0.75
0.50
0.25
0.002.25 2.50 2.75 3.002.001.751.501.250.75 1.000.500.00 0.25
Figure 4.158 Comparison between the Moment Distribution Factors of the Middle Girder due to Truck Loading as Specified in the CHBDC and from the Current Study
4.003.75
M om ent D istribu tion F ac to rs o f M iddle
G irders F o r F atigue L oad C ase acco rd ing to
the F in ite E lem ent A naly sis (F E A ) and C anad ian C ode(C H B D C ) F o r S traight B ridges
3.503.253.00
4j}l 2.50 ■ 2.25I 2.00
1.751.501.251.00
0.750.500.250.00
3.00 3.25 3.50 3.75 4.00
Figure 4.159 Comparison between the Moment Distribution Factors of the Middle Girder due to Fatigue Loading as Specified in the CHBDC and from the Current Study
211
4
Exterior girder,
Fully loaded lanesë3.5
CD3Oeeb
2.5
2
.2 1.5oaa 1 ♦ Straight bridge, L/R=0
□ Curved bridge, L/R>00.5
0.5 1.5 2 2.5 3 3.5 4
Moment Distribution Factor, MDF
Figure 4.160 Correlation between Moment and Deflection Distribution Factors for the Exterior Girder of the Studied bridges due to Truck Loading
Exterior girder,
Fatigue loading
□ □♦ Straight bridge, L/R=0
0 Curved bridge, L/R>0
5.5 654.543.532.521.50.5
Moment Distribution Factor, MDF
Figure 4.161 Correlation between Moment and Deflection Distribution Factors for theExterior Girder of the Studied bridges due to Fatigue Loading
212
Middle giider. Fully loaded lanes3.5
.2 1.5
♦ Straight bridge, L/R=0
0 Curved bridge, L/R>0
43.532.521.5
Moment Distribution Factor, MDF
Figure 4.162 Correlation between Moment and Deflection Distribution Factors for the Middle Girder of the Studied bridges due to Truck Loading
Middle girder, Fatigue loading
4.5
□□
«, 3.5-
•= 2.5 ■
♦ Straight bridge, UR=0
□ Curved bridge, L/R>0
0.5 1.5 2 2.5 3 3.5 4 4.5 5
Moment Distribution Factor, MDF
Figure 4.163 Correlation between Moment and Deflection Distribution Factors for the Middle Girder of the Studied bridges due to Fatigue Loading
213
APPENDEX (A): SAP 2000 input file for a straightbridge (SDF)
214
•.CASE L=10/STRAIGHT NB=3 NXBS=2 BS=2m # OF ELEMENTS=36 SYSTEMDOF=ALL LENGTH=M FORCE=KN
JOINTS1 X=0 Y=0 2=0.122537 X=0 Y=10 2=0.1225445 X=6 Y=0 2=0.1225481 X=6 Y=10 2=0.1225 /DeckLgen=l,445,37,37,l
2500 X=0. 85 Y=0 2=02536 X=0. 85 Y=10 2=04500 X=4. 85 Y=0 2=04536 X=4. 85 Y=10 2=0 ;U. FlangeLgen=2500 ,4500,1000,2536,1
2700 X=l. 15 Y=0 2=02736 X=l. 15 Y=10 2=04700 X=5. 15 Y=0 2=04736 X=5. 15 Y=10 2=0 ;U. FlangeLgen=2700 ,4700,1000,2736,1
3100 X=0. 85 Y=0 2=-0.753136 X=0. 85 Y=10 2=-0.755100 X=4.85 Y=0 2=-0.755136 X=4. 85 Y=10 2=-0.75 ;B. FlangeLgen=3100 ,5100,1000,3136,1
3300 X=l. 15 Y=0 2=-0,753336 X=l. 15 Y=10 2=-0.755300 X=5. 15 Y=D 2=-0.755336 X=5. 15 Y=10 2=-0.75 ;B. FlangeLgen=3300 ,5300,1000,3336,1
2600 X=1 Y=0 2=02636 X=1 Y=10 2=04600 X=5 Y=0 2=04636 X=5 Y=10 2=0 ;Top joint of WebsLgen=2600 ,4600,1000,2636,1
2800 X=1 Y=0 2=-0.18752836 X=1 Y=10 2=-0 .18754800 X=5 Y=0 2=-0.18754836 X=5 Y=10 2=-0.1875 /middle joint ofLgen=2800 ,4800,1000,2836,1
2900 X=1 Y=0 Z=-0.3752936 X=1 Y=10 Z=-0.3754900 X=5 Y=0 2=-0.3754936 X=5 Y=10 Z=-0.375 /middle joint ofLgen=2900 ,4900,1000,2936,13000 X=1 Y=0 2=-0.56253036 X=1 Y=10 Z=-0.56255000 X=5 Y=0 2=-0.56255036 X=5 Y=10 2=-0.5625 /lower joint of ’Lgen=3000 ,5000,1000,3036,13200 X=1 Y=0 Z=-0.75
215
3236 X=1 Y=10 Z=-0.755200 X=5 Y=0 Z=-0.755236 X=5 Y=10 Z=-0.75Lgen=32 0 0,52 0 0,1 0 0 0 ,323 6 , 1
PatternName=Default
;bottom joints of Webs
RESTRAINTS Add=4200,5200,1000 Add=4236,5236,1000 Add=3200 Add=3236
MaterialName=steel W=78.5
E=200000E3 U=0.3Name=concrete W=24
E=28000E3 U=0.2
Dof= Uy,Uz, Dof= Uz,Dof=Ux,Uy,Uz, Dof=Ux, Uz,
Shell Section Name=slab Name=web Name=flange Name=studs
Type=Shell Type=She11 Type=She11 Type=Shell
Mat=concreteMat=steelMat=steelMat=steel
T h = 0 . 2 2 5 T h = 1 . 6 E - 0 2 Th=2E-02 T h = 7 . 7 4 E - 0 4
SHELL Local=31 Pldir=0
1 J=l,2,38,39 Sec=sGen=l 36 1 397 36 Jinc=l 37
1009 J=2600,2601,2800,2801Gen=1009 1044 1 1045 J=2800,2801,2900,2901Gen=1045 1080 1 1081 36 Jinc
1117 8=3000,3001,3200,3201Gen=1117 1152 1
1153 8=3600,3601,3800,3801Gen=1153 1188 1
1189 8=3800,3801,3900,3901Gen=1189 1224 1 1225 36 8inc
1261 8=4000,4001,4200,4201Gen=1261 1296 1
1297 8=4600,4601,4800,4801Gen=1297 1332 1
1333 8=4800,4801,4900,4901Gen=1333 1368 1 1369 36 8inc
1405 8=50 0 0,5001,5200,52 01Gen=1405 1440 1
2017 8=2500,2501,2600,2601Gen=2017 2052 1 2053 36 8inc
2089 8=3100,3101,3200,3201Gen=2089 2124 1 2125 36 8inc
2161 8=3 500,3501,3600,3601Gen=2161 2196 1 2197 36 8inc
2233 8=4100,4101,4200,4201Gen=2233 2268 1 2269 36 8inc
lab ;DBCK
Sec=web ;WEB1
Sec=web ;WEB1 = 1 100Sec=v?eb ;WEB1
Sec=web ;WEB2
Sec=web ;WEB2 = 1 1 0 0Sec=web ;WEB2
Sec=web ;WEB3
Sec=web ;WEB3 = 1 10 0Sec=web ;WEB3
Sec=flange /UPPER PLANGEl = 1 100Sec=flange /LOWER PLANGEl
=1 10 0Sec=flange /UPPER PLANGE2
= 1 100Sec=flange /LOWER PLANGE2 =1 100
216
2305 J=4500,4501,4600,4601 Sec=flange /UPPER FLANGESGen=2305 2340 1 2341 36 Jinc=l 100
2377 J=5 1 0 0 ,5101,5200,5201 Sec=flange /LOWER FLANGESGen=2377 2412 1 2413 36 Jinc=l 100 3100 J=75,76,2600,2601 Sec=studs /studslGen=3100 3135 13200 J=223,224,3600,3601Gen=3200 3235 1 3300 J=371,372,4600,4601Gen=3300 3335 1
Frame Section Name=studs Name=dummy Name=bracing
Mat=steelMat=steelMat=steel
Sec=studs /studs2
Sec=studs /studsS
1=11922.9E-12 A=387E-6 AS=387E-6 1=11922.9E-12 A=387E-6 AS=387E-6 1=0 A=7500E-6
FRAMELocal=13 Pldir=+Z +Y ; SAP90 default1 J=2600,3600 Sec=bracing Irel=R3,R2Gen=l,9,4 Iinc=18 Jinc=182 J=2600,4200 Sec=bracing Irel=R3,R2Gen=2,10,4 Iinc=18 Jinc=183 J=3200,3600 Sec=bracing Irel=R3,R2Gen=3,11,4 Iinc=18 Jinc=18
4 J=3200,4200 Sec=bracing Irel=R3,R2Gen=4,12,4 Iinc=18 Jinc=18101 J=3600,4600 Sec=bracing Irel=R3,R2Gen=101,109,4 Iinc=18 Jinc=18
102 J=3600,5200 Sec=bracing Irel=R3,R2Gen=102,110,4 Iinc=18 Jinc=lB
103 J=4200,4600 Sec=bracing Irel=R3,R2Gen=103,111,4 Iinc=18 Jinc=18104 J=4200,5200 Sec=bracing Irel=R3,R2Gen=104,112,4 Iinc=18 Jinc=181300 J=75,76 Sec=dummy ; dummyGen=1300,1335,1 Iinc=l Jinc=l 1400 J=112,113 Sec=dummy ; dummyGen=1400,1435,1 Iinc=l Jinc=l
1500 J=149,150 Sec=dummy /dummyGen=1500,1535,1 Iinc=l Jinc=l 1600 J=186,187 Sec=dummy /dummyGen=1600,1635,1 Iinc=l Jinc=l
1700 J=223,224 Sec=dummy /dummyGen=1700,1735,1 Iinc=l Jinc=l 1800 J=260,261 Sec=dummy /dummyGen=1800,1835,1 Iinc=l Jinc=l
1900 J=297,298 Sec=dummy /dum m yGen=1900,1935,1 Iinc=l Jinc=l
2000 J=334,335 Sec=dummy /dummyGen=2000,2035,1 Iinc=l Jinc=l
2100 J=371,3 72 Sec=dummy /dummyGen=2100,2135,1 Iinc=l Jinc=l
valuesJral=R3,R2,R1 Nseg=4 /Xbracingl
Jrel=R3,R2,R1,/Xbracingl
Jrel=R3,R2,R1, / Xbracingl
Jrel=R3,R2,Rl,/Xbracingl
Jrel=R3 , R2 , R1, ,-Xbracing2
Jrel=R3,R2,Rl, ,-Xbracing2
Jrel=R3,R2,Rl,/Xbracing2
Jrel=R3,R2,Rl,/Xbracing2
LoadName=owType=Gravity Elem=Frame
.217
Add=* Uz=-1, Type=Gravity Elem=Shell Add=* Uz=-l
EXTERIOR GIRDER 1 lanes, Itruck
NAME=EXT1L1T ADD=1702 ADD=1706 ADD=1730
TYPE=CONCENTRATED CSYS=0 D=0.044 UZ=-125.0/2D=0.133 UZ=-125.0/2D=0.066 UZ=-175.0/2
RX=-125.0/2*-0.10 RX=-125.0/2*-0.10 RX=-175.0/2*-0.10
ADD=2102ADD=2106ADD=2130
D=0.044 D=G.133 D=0.066
UZ=-125.0/2 UZ=-125.0/2 UZ=-175 . 0/2
RX=-125.0/2*0.10 RX=-125.0/2*0.10 RX=-175.0/2*0.10
MIDDLE GIRDER lianas
NAME=MID+fatADD=1502ADD=1506ADD=1530
TyPE=CONCENTRATED CSYS=0D=0.044 D=0.133 D=0.066
UZ=-125.0/2UZ=-125.0/2UZ=-175.0/2
RX=-125.0/2*-0.10 RX=-125.0/2*-0.10 RX=-175.0/2*-0.10
ADD=1902ADD=1906ADD=1930
D=0.044 D=0.133 D=0.066
UZ=-125 . 0/2 UZ=-125.0/2 UZ=-175.0/2
RX=-125.0/2*0.10 RX=-125.0/2*0.10 RX=-175.0/2*0.10
INTERIOR GIRDER 1 lanes
NAME=INT1L1TADD=1302ADD=1306ADD=1330
TYPE=CONCENTRATED CSYS=0 D=0.044 UZ=-125.0/2D=0.133 UZ=-125.0/2D=0.066 UZ=-175.0/2
RX=-125.0/2*-0.10RX=-125.0/2*-0.10RX=-175.0/2*-0.10
ADD=1702ADD=1706ADD=1730
D=0.044 D=0.133 D=0 . 066
UZ=-125.0/2 UZ=-125.0/2 UZ=-175.0/2
RX=-125.0/2*0.10 RX=-125.0/2*0.10 RX=-175.0/2*0,10
OutputELEM=JOINT TYPE=DISP,REAC LOAD=* ELEM=SHELL TYPE=FORCE LOAD=*ELEM=SHELL TYPE=STRESS LOAD=* ELEM=FRAME TYPE=JOINTF LOAD=*
END
218
APPENDEX (B): SAP 2000 input file for a curvedbridge (SDF)
219
/CASE L=10 R= 100 L/R=0. 1 NB=4 NXBS=4 BS=2mSYSTEMDOF=ALL LENGTH=M FORCE=KNJOINTS9998 X=-50 Y=0 Z=-5 /Axis of Rotation9999 X=-50 Y=0 Z=5 /Axis of Rotation1 X=46 Y=0 Z=0.122 5 /DeckCgen=l,37,1 Da=0.159 Axvec=9998,9999,
38 X=46.5 Y=0 Z=0.1225 /DeckCgen==3 8 , 74,1 Da=0.159 Axvec=9998,9999,
75 X=47 Y=0 Z=0.1225 /DeckCgen=75,111,1 Da=0.159 Axvec=9998,9999,
112 X=47.5 Y=0 Z=0.1225 /DeckCgen=112,148, 1 Da=0.159 Axvec=9998,9999,
149 X=48 Y=0 Z=0.1225 /DeckCgen=
186Cgen=
223Cgen=
260Cgen=
297Cgen=
334Cgen=
371Cgen=
408Cgen=
445Cgen=
482Cgen=
519Cgen=
556Cgen=
593Cgen=
2500Cgen=
2700Cgen=
3100Cgen=
3300Cgen=
= 149,185,1 Da=0.159 Axvec = 9998, 9999,X=48.5 Y=0 Z=0.1225 ;Deck= 186,222,1 Da=0.159 Axvec = 9998,9999 ,X=49 Y=0 Z=0.1225 ;Deck=223,259,1 Da=0.159 Axvec = 9998,9999,X=49.5 Y=0 Z=0.1225 ;Deck=260,296,1 Da=0.159 Axvec=9998,9999,X=50 Y=0 Z=0.1225 ;Deck=297,333,1 Da=0.159 Axvec=9998,9999, X=50.5 Y=0 Z=0.1225 ;Deck=334,370,1 Da=0.159 Axvec = 9998,9999,X=51 Y=0 Z=0.1225 /Deck=371,407,1 Da=0.159 Axvec=9998,9999, X=51.5 Y=0 Z=0.1225 /Deck=408,444,1 Da=0.159 Axvec=9998,9999,X=52 Y=0 Z=0.1225 /Deck=445,481,1 Da=0.159 Axvec = 9998,9999,X=52.5 Y=0 Z=0.1225 /Deck=482,518,1 Da=0.159 Axvec=9998,9999,X=53 Y=0 Z=0.1225 /Deck519.555.1 Da=0.159 Axvec=9998,9999,X=53.5 Y=0 Z=0.1225 /Deck556.592.1 Da=0.159 Axvec=9998, 9999,X=54 Y=0 Z=0.1225 /Deck593.629.1 Da=0.159 Axvec=9998,9999,X=46.85 Y=0 Z=0 /U.FLANGl2500.2536.1 Da=0.159 Axvec=9998,9999,X=47.15 Y=0 Z=0 /U.FLANGl2700.2736.1 Da=0.159 Axvec=9998,9999 ,X=46.85 Y=0 Z=-0.75 /L.FLANGl3100.3136.1 Da=0.159 Axvec=9998, 9999,X=47.15 Y=0 Z=-0.75 /L.FLANGl3300.3336.1 Da=0.159 Axvec=9998,9999,
3500 X=48.85 Y=0 Z=0 /U.FLANG2Cgen=3500,3536,1 Da=0.159 Axvec=9998, 9999,
220
3700Cgens
4100Cgen=
4300Cgen=
4500Cgen=
4700Cgen=5100Cgen=
5300Cgen=5500Cgen=5700Cgen=6100Cgen=6300
X=49. 3700, X=48. 4100, X=49. '4300, X=50. 4500, X=51. '4700, X=50 . 5100, X=51. 5300, X=52 . 5500, X=53 . 5700, X=52 . 6100, X=53 .
15 Y=0 Z=0 ;U.PLANG23736.1 Da=0.159 Axvec=9998,9 9 9 9 ,8 5 y=0 Z = - 0 . 7 5 ;L.FLANG24136.1 Da=0.159 Axvec=9998,9999,1 5 y=0 Z = - 0 . 7 5 ;L.FLANG24336.1 Da=0.159 A3cvec=9998, 9999,85 Y=0 Z=0 ;U.FLANG34536.1 Da=0.159 Axvec=9998,9999,15 Y=0 Z=0 ;U.FLANG34736.1 Da=0.159 Axvec=9998,9999, 85 Y=0 Z=-0.75 ;L .FLANG35136.1 Da=0.159 Axvec=9998,9999,15 Y=0 Z=-0.75 ;L.FLANG35336.1 Da=0.159 Axvec=9998,9999,85 Y=0 Z=0 ;U.FLANG45536.1 Da=0.159 Axvec=9998,9999,15 Y=0 Z=0 ;U.FLANG45736.1 Da=0.159 Axvec=9998,9999,85 Y=0 Z=-0.75 ;L.FLANG46136.1 Da-0.159 Axvec=9998,9999,15 Y=0 Z--0.75 ;L.FLANG4
Cgen=6300,6336,1 Da 2600 X=47 Y=0Cgen=2600,2636,1 Da
3600 X=49 Y=0Cgen=3600,3636,1 Da
4600 X=51 Y=0Cgen=4600,4636,1 Da
5600 X=53 Y=0Cgen=5600,5636,1 Da
2800 X=47 Y=0Cgen=2800,2836,1 Da 3800 X=49 Y=0Cgen=3800,3836,1 Da
4800 X=51 Y=0Cgen=4800,4836,1 Da 5800 X=53 Y=0Cgen=5800,5836,1 Da
2900 X=47 Y=0Cgen=2900,2936,1 Da
3900 X=49 Y=0Cgen=3900,3936,1 Da
4900 X=51 Y=0Cgen=4900,4936,1 Da
5900 X=53 Y=0Cgen=5900,5936,1 Da
3000 X=47 Y=0Cgen=3000,3036,1 Da
4000 X=49 Y=0Cgen=4000,4036,1 Da
5000 X=51 Y=0Cgen=5000,5036 , 1 Da
6000 X=53 Y=0Cgen=6000,6036,1 Da
3200 X=47 Y=0Cgen=3200,3236,1 Da
0.159 Axvec=9998,9999,Z=0 ;WEB1
0.159 Axvec=9998,9999, Z=0 ;WEB2
0.159 Axvec=9998,9999, Z=0 ;WEB3
0.159 Axvec=9998,9999,Z=0 ;WEB4
0.159 Axvec=9998,9999,Z--0.1875 ;WEB1
0.159 Axvec=9998,9999, Z--0.1875 ;WEB2
0.159 Axvec=9998,9999, Z--0.1875 ;WEB3 159 Axvec=9998,9999,Z--0.1875 ;WEB4159 Axvec=9998,9999, Z--0.375 ;WEB1159 Axvec=9998,9999, Z--0.375 ;WEB2
0.159 Axvec=9998,9999, Z--0.375 ;WEB3
0.159 Axvec=9998,9999, Z--0.375 ;WEB4
0.159 Axvec=9998,9999, Z--0.5625 ;WEB1
0.159 Axvec=9998,9999, Z--0.5625 ;WEB2
0.159 Axvec=9998,9999, Z--0.5625 ;WEB3
0.159 Axvec=9998,9999, Z--0.5625 ;WEB4
0.159 Axvec=9998,9999, Z--0.75 ;WEB1
0.159 Axvec=9998,9999,
= 0
= 0
=0
221
4200 X=49 Y=0 Z=-0.75 ;WEB2Cgen=4200,4236 , 1 Da=0.1S9 Axvec=9998, 9 9 9 9 ,
5200 X=51 Y=0 Z=-0.75 ;WEB3Cgen=5200,5235 , 1 Da=0 .159 Axvec=9998,9 9 9 9 ,
6200 X=53 Y=0 Z=-0.75 ;WEB4Cgen=62 0 0 , 623 6 , 1 Da=0.159 A3cvec=9998, 9 9 9 9 ,
Pattern Name=Default
RESTRAINTS Add=4200,6200,1000 Add=4236,6236,1000 Add=3200 Add=3236
MaterialName=steel W=78.5
E=200000E3 U=0.3Name=concrete W=24
E=28000E3 U=0.2
Dof= Uy,Uz, Dof= Uz,Dof=Ux,Uy,Uz, Dof=Ux, Uz,
Shell Section Name=slab Name=web Name=flange Name=studs
Type=She11 Type=She11 Type=Shell Type=Shell
Mat=concreteMat=steelMat=steelMat=steel
Th=0.22STh=1.6E-02Th=2E-02Th=7.74E-04
SHELL Local=31 Pldir=0
1 J=l,2,38,39 Sec=sGen=l 36 1 541 36 Jinc=l 37
1009 J=2600,2601,2800,2 8 01Gen=1009 1044 1
1045 J=2800,2801,2900,2901Gen=1045 1080 1 1081 36 Jinc
1117 J=30 0 0,3 001,3200,32 01Gen=1117 1152 1
1153 J=3600,3601,3800,3801Gen=1153 1188 1
1189 J=380 0,3 8 01,3 900,3 901Gen=1189 1224 1 1225 36 Jinc
1261 J=4000,4001,4200,4201Gen=1261 1296 1
1297 J=4600,4601,4800,4801Gen=1297 1332 1
1333 J=4800,4801,4900,4901Gen=1333 1368 1 1369 36 Jinc
1405 J=5000,5001,5200,5201Gen=1405 1440 1
1441 J=5600,5601,5800, 5801Gen=1441 1476 1
1477 J=5800,5801,5900,5901Gen=1477 1512 1 1513 36 Jinc
1549 J=6000,6001,6200,6201Gen=1549 1584 1
2017 J=2500,2501,2600,2601
lab ;DECK
Sec=web ;WEB1
Sec=web ;WEBl = 1 100 Sec=web ;WEBl
Sec=web ;WEB2
Sec=web ;WEB2 = 1 100Sec=web ;WEB2
Sec=web ;WEB3
Sec=web ;WEB3 = 1 1 00 Sec=web ;WEB3
Sec=web ;WEB4
Sec=web ;WEB4 =1 100 Sec=web ;WEB4
Sec=flange ;UPPER PLANGEl
222
Gen=2017 2052 1 2053 36 Jinc=l 100 2089 J = 3 1 0 0 ,3101,3200,3201 Sec=flange /LOWER PLANGElGen=2089 2124 1 2125 36 Jinc=l 100
2161 J=3500,3501,3600,3601 Sec=flange /UPPER FLANGE2Gen=2161 2196 1 2197 36 Jinc=l 100 2233 J = 4 1 0 0 , 4101,4200,4201 Sec=flange /LOWER FLANGE2Gen=2233 2268 1 2269 36 Jinc=l 100
2305 J = 4 5 0 0 , 4501,4600,4601 Sec=flange /UPPER FLANGE3Gen=2305 2340 1 2341 36 Jinc=l 100
2377 J= 5 1 0 0 ,5 1 0 1 ,52 0 0,52 01 Sec=flange /LOWER FLANGE3Gen=2377 2412 1 2413 36 Jinc=l 100
2449 J=5 5 0 0 ,5501,56 0 0,5601 Sec=flange /UPPER FLANGE4Gen=2449 2484 1 2485 36 Jinc=l 100
2521 J=61 0 0 ,6101,62 00,6201 Sec=flange /LOWER FLANGE4Gen=2521 2556 1 2557 36 Jinc=l 100
3100 J=75,76,2600,2601 Sec=studs /studslGen=3100 3135 13200 J=223,224,3600,3601 Sec=studs /Studs2Gen=3200 3235 13300 J=371,372,4600,4601 Sec=studs /studs3Gen=3300 3335 13400 J=519,520,5600,5601 Sec=studs /studs4Gen=3400 3435 1
Frame Section Name=studs Name=dummy Narae=bracing
Mat=steel 1=11922.9E-12 A=387E-6 AS=387E-6Mat=steel 1=11922.9E-12 A=387E-6 AS=387E-6Mat=steel 1=0 A=7500E-6
FRAMELocal=13 Pldir=+Z +Y SAP90 default values
1 J=2600,3600 Sec=bracing Irel=R3,R2Gen=l,17,4 Iinc=9 Jinc=92 J=2600,4200 Sec=bracing Irel=R3,R2Gen=2,18,4 Iinc=9 Jinc=9
3 J=3200,3600 Sec=bracing Irel=R3,R2Gen=3,19,4 Iinc=9 Jinc=94 J=3200,4200 Sec=bracing Irel=R3,R2Gen=4,20,4 Iinc=9 Jinc=9101 J=3600,4600 Sec=bracing Irel=R3,R2Gen=101,117,4 Iinc=9 Jinc=9102 J=3600,5200 Sec=bracing Irel=R3,R2Gen=102,118,4 Iinc=9 Jinc=9
103 J=4200,4600 Sec=bracing Irel=R3,R2Gen=103,119,4 Iinc=9 Jinc=9104 J=4200,5200 Sec=bracing Irel=R3,R2Gen=104,120,4 Iinc=9 Jinc=9
201 J=4600,5600 Sec=bracing Irel=R3,R2Gen=201,217,4 Iinc=9 Jinc=9
202 J=4600,6200 Sec=bracing lrel=R3,R2
Gen=202,218,4 Iinc=9 Jinc=9203 J=5200,5600 Sec=bracing Irel=R3,R2Gen=203,219,4 Iinc=9 Jinc=9
204 J=5200,6200 Sec=bracing Irel=R3,R2
Jrel=R3,R2,Rl Nseg=4 /Xbracingl
Jrel=R3,R2,Rl,/Xbracingl
Jrel=R3,R2,Rl,/Xbracingl
Jrel=R3,R2,Rl,/Xbracingl
Jrel=R3,R2,Rl,/Xbracing2
Jrel=R3,R2,Rl,/Xbracing2
Jrel=R3 , R2 , Rl, ,-Xbracing2
Jrel=R3, R2, Rl, ,-Xbracing2
Jrel=R3,R2,Rl,/Xbracing3
Jrel=R3,R2,Rl,/XbracingS
Jrel=R3 , R2, Rl, ,-Xbracing3
Jrel=R3,R2,Rl,/XbracingS
223
Gen:1300Gen=
1400Gen=
1500Gen:
1600Gen:
1700Gen:1800Gen:
1900Gen=
2 0 0 0Gen=
2 1 0 0Gen=
2200Gen=
2300Gen=
2400Gen=
2500Gen=
:204,220,4 line=9 Jinc=9 J=7 5,76 Sec=dummy ; dummy
:1300,1335,1 Iinc=l Jinc=l J=112,113 Sec=dummy ;dummy
=1400,1435,1 Iinc=l Jinc=l J=149,150 Sec=dummy ; dummy
=1500,1535,1 Iinc=l Jinc=l J=186,187 Sec=dummy ; dummy
=1600,1635,1 Iinc=l Jinc=l J=223,224 Sec=dummy ; dummy
=1700,1735,1 Iinc=l Jinc=l J=260,261 Sec=dummy ;dummy
=1800,1835,1 Iinc=l Jinc=l J==:297, 298 Sec=dummy ; dummy
=1900,1935,1 Iinc=l Jinc=l J=3 34,3 3 5 Sec=dummy ; dummy
2000.2035.1 Iinc=l Jinc=l J=371,3 72 Sec=dummy ; dummy
2100.2135.1 Iinc=l Jinc=l J=408,409 Sec=dummy ; dummy
2200.2235.1 Iinc=l Jinc=l J=445,446 Sec=dummy ; dummy
2300.2335.1 Iinc=l Jinc=l J=482,483 Sec=dummy ; dummy
2400.2435.1 Iinc=l Jinc=l J=519,520 Sec=dummy ; dummy
2500.2535.1 Iinc=l Jinc=l
Load Name=ow Type=Gravity Elem=Frame Add=* Uz=-1,
Type=Gravity Elem=Shell Add=* Uz=-1,
EXTERIOR GIRDER 2 lanes, 2trucks
NAME=EXT2L2T ADD=1402 ADD=1406 ADD=143 0
TYPE=CONCENTRATED CSYS=0D=0.044 UZ=-125.0/2D=0.133 UZ=-125.0/2D=0.066 UZ=-175.0/2
RX=-125.0/2*-0.10 RX=-125.0/2*-0.10 RX=-175.0/2*-0.10
ADD=1802ADD=1806ADD=1830
D=0.044 D=0.133 D=0.066
UZ=-125.0/2 UZ=-125.0/2 UZ=-175.0/2
RX=-125.0/2*0.10 RX=-125.0/2*0,10 RX=-175.0/2*0.10
ADD=2102ADD=2106ADD=2130
D=0.044 D=0.133 D=0.066
UZ=-125.0/2 UZ=-125.0/2 UZ=-175.0/2
RX=-125.0/2*-0.10RX=-125.0/2*-0.10RX=-175.0/2*-0.10
ADD=2502 ADD=2506 ADD=253 0
D=0.044 D=0.133 D=0.066
UZ=-125.0/2 UZ=-125.0/2 UZ=-175.0/2
RX=-125.0/2*0.10 RX=-125.0/2*0.10 RX=-175.0/2*0.10
EXTERIOR GIRDER 2 lanes. Itruck
NAME=EXT2L1T TYPE=CONCENTRATED CSYS=0
224
ADD=2102 D=0.044 UZ=-125.0/2ADD=2106 D=0.133 UZ=-125.0/2ADD=2130 D=0.066 UZ=-175.0/2
ADD=2502 D=0.044 UZ=-125.0/2ADD=2506 D=0 .133 UZ=-125.0/2ADD=2530 D=0.066 UZ=-175.0/2
; MIDDLE GIRDER ; 2lanes
NAME=MID2L TYPE=CONCENTRATED CSYSADD=1402 D=0.044 UZ=-125.0/2ADD=1406 D=0.133 UZ=-125.0/2ADD=1430 D=0.0 66 UZ=-175.0/2
ADD=1802 D=0.044 UZ=-125.0/2ADD=1806 D=0 .133 UZ=-125.0/2ADD=1830 D=0 . 066 UZ=-175.0/2
ADD=2002 D=0.044 UZ=-125.0/2ADD=2006 D=0 .133 UZ=-125.0/2ADD=2030 D=0.0 66 UZ=-175.0/2
ADD=2402 D=0 . 044 UZ=-125.0/2ADD=2406 D=0 .133 UZ=-125.0/2ADD=2430 D=0.066 UZ=-175.0/2
; INTERIOR GIRDER ; 2 lanes, 2trucks
NAME=INT2L2T TYPE=CONCENTRATED CSYS^ADD=1302 D=0.044 UZ=-125.0/2ADD=1306 D=0.133 UZ=-125.0/2ADD=1330 D=0.066 UZ=-175.0/2
ADD=1702 D=0.044 UZ=-125.0/2ADD=1706 0=0.133 UZ=-125.0/2ADD=1730 D=0 .066 UZ=-175.0/2ADD=2002 D=0 .044 UZ=-125.0/2ADD=2006 0=0.133 UZ=-125.0/2ADD=2030 D=0.066 UZ=-175.0/2ADD=2402 D=0.044 UZ=-125.0/2ADD=2406 0=0.133 UZ=-125.0/2ADD=2430 D=0.066 UZ=-175.0/2
; INTERIOR GIRDER ; 2 lanes, Itruck
NAME=INT2L1T TYPE=CONCENTRATED CSY&ADD=1302 0=0.044 UZ=-125.0/2ADD=1306 0=0.133 UZ=-125.0/2ADD=1330 D=0.066 UZ=-175.0/2ADD=1702 0=0.044 UZ=-125.0/2ADD=1706 0=0.133 UZ=-125.0/2
225
RX=-125.0/2*-0.10RX=-125.0/2*-0.10RX=-175.0/2*-0.10
RX=-125.0/2*0.10 RX=-125.0/2*0.10 RX=-175.0/2*0.10
RX=-125.0/2*-0.10RX=-125.0/2*-0.10RX=-175.0/2*-0.10
RX=-125.0/2*0.10 RX=-125.0/2*0.10 RX=-175.0/2*0.10
RX=-125.0/2*-0.10RX=-125.0/2*-0.10RX=-175.0/2*-0.10
RX=-125.0/2*0.10 RX=-125.0/2*0.10 RX=-175.0/2*0.10
RX=-125.0/2*-0.10RX=-125.0/2*-0.10RX=-175.0/2*-0.10
RX=-125.0/2*0.10 RX=-125.0/2*0.10 RX=-175.0/2*0.10
RX=-125.0/2*-0.10RX=-125.0/2*-0.10RX=-175.0/2*-0.10
RX=-125.0/2*0.10 RX=-125.0/2*0.10 RX=-175.0/2*0.10
RX=-125.0/2*-0.10RX=-125.0/2*-0.10RX=-175.0/2*-0.10
RX=-125.0/2*0.10 RX=-125.0/2*0.10
ADD=1730 D=0.066 U Z = - 1 7 5 . 0 / 2 R X = - 1 7 5 . 0 / 2 * 0 . 1 0
FATIGUE EXTERIOR 2 lanes, Itruck
NAME=FTEXT2L ADD=2002 ADD=2006 ADD=2030
TYPE=CONCENTRATED CSYS=0 0=0.044 UZ=-125.0/20=0.133 UZ=-125.0/20=0.066 UZ=-175.0/2
RX=-125.0/2*-0.10RX=-125.0/2*-0.10RX=-175.0/2*-0.10
A00=2402 AOO=2406 AOO=243 0
0=0.044 0=0.133 0=0.066
UZ=-125.0/2 UZ=-125.0/2 UZ=-175.0/2
RX=-125.0/2*0.10 RX=-125.0/2*0.10 RX=-175.0/2*0.10
FATIGUE INTERIOR 2 lanes, Itruck
NAME=FTINT2L AOO=1402 AOD=1406 AOO=1430
AOO=1802AOD=1806AOD=1830
TYPE=CONCENTRATEO CSYS=0 0=0.044 UZ=-125.0/20=0.133 UZ=-125.0/20=0.066 UZ=-175.0/2
0=0.044 0=0.133 0= 0.066
UZ=-125.0/2 UZ=-125.0/2 UZ=-175.0/2
RX=-125.0/2*-0.10RX=-125.0/2*-0.10RX=-175.0/2*-0.10
RX=-125.0/2*0.10 RX=-125.0/2*0.10 RX=-175.0/2*0.10
OutputEIjEM=JOINT t y p e =d i s p ,r e a c l o a o =* ELEM=SHELL TYPE=FORCE L0AD=* ELEM=SHELL TYPE=STRESS LOAO=* ELEM=FRAME TYPE=JOINTF LOAO=*
ENO
226
APPENDEX (C): Excel data sheet for section andgirder properties
idge 35ms p a c i n g 2 m
The reactions of single girder for D.L,L.L
A(t)Xts n Area(t) Yi Yi Y*(t) Ys l(t) Is le w(kN/m3) Wi(kN/m)
2 0.225 7.14 0.06303 1.8825 0.11864 1.49712 0.885 0.00963 24 10.80.3 0.02 1 0.006 1.76 0.01056 1.49712 0.885 0.00041 0.00459 78.5 0.471
.016 1.73 1 0.02768 0.885 0.0245 1.49712 0.885 0.01727 0.0069 78.5 2.172880.3 0.02 1 0.006 0.01 0.00006 1.49712 0.885 0.01327 0.00459 78.5 0.471
0.10271 0.15376 1.49712 0.04059 0.01609 0.03691 13.9149
' spacing 2.5m
ts n Area(t) YiA(t)XYi Y*(t) Ys l{t) Is le w(kN/m3) Wi(kN/m)
2.5 0.225 7.14 0.07878 1.8825 0.14831 1.54838 0.885 0.00913 24 13.50.3 0.02 1 0.006 1.76 0.01056 1.54838 0.885 0.00027 0.00459 78.5 0.471
1.016 1.73 1 0.02768 0.885 0.0245 1.54838 0.885 0.01908 0.0069 78.5 2.172880.3 0.02 1 0.006 0.01 0.00006 1.54838 0.885 0.0142 0.00459 78.5 0.471
0.11846 0.18342 1.54838 0.04268 0.01609 0.03869 16.6149
spacing 3.0m
ts n Area(t) YiA(t)XYi Y*(t) Ys l(t) Is le w(kN/m3) Wi(kN/m)
3 0.225 7.14 0.09454 1.8825 0.17797 1.5876 0.885 0.00862 24 16.20.3 0.02 1 0.006 1.76 0.01056 1.5876 0.885 0.00018 0.00459 78.5 0.471
0.016 1.73 1 0.02768 0.885 0.0245 1.5876 0.885 0.02057 0.0069 78.5 2.172880.3 0.02 1 0.006 0.01 0.00006 1.5876 0.885 0.01493 0.00459 78.5 0.471
0.13422 0.21308 1.5876 0.0443 0.01609 0.04007 19.3149
span (Rr)dl (Rl)dl
244.6119 244.6119
span (Rr)c (Rl)dl
291.8619 291.8619
span (Rr)ol (Rl)dl
339.1120 339,1119
228
idge 25mspacing 2m
ts n Area(t) YiA(t)XYi Y*(t) Ys l(t) Is le w(kN/m3) Wi(kN/m)
2 0.225 7.14 0.06303 1.3825 0.08713 1.13245 0.635 0.00421 24 10.80.3 0.02 1 0.006 1.26 0.00756 1.13245 0.635 9.8E-05 0.00234 78.5 0.471
.016 1.23 1 0.01968 0.635 0.0125 1.13245 0.635 0.00735 0.00248 78.5 1.544880.3 0.02 1 0.006 0.01 0.00006 1.13245 0.635 0.00756 0.00234 78.5 0.471
0.09471 0.10725 1.13245 0.01922 0.00717 0.01741 13.2869
spacing 2.5m
ts n Area(t) YiA(t)XYi Y*(t) Ys l(t) Is le w(kN/m3) Wi(kN/m)
2.5 0.225 7.14 0.07878 1.3825 0.10892 1.16812 0.635 0.00395 24 13.50.3 0.02 1 0.006 1.26 0.00756 1.16812 0.635 5.1E-05 0.00234 78.5 0.471
016 1.23 1 0.01968 0.635 0.0125 1.16812 0.635 0.00807 0.00248 78.5 1.544880.3 0.02 1 0.006 0.01 0.00006 1.16812 0.635 0.00805 0.00234 78.5 0.471
span (Rr)dl (Rl)dl
166.873 166.873
span (Rr)dl (Rl)dl
0.11046
spacing 3.0m
0.12903 1.16812
A(t)X
0.02013 0.00717
ts n Area(t) Yi Yi Y*(t) Ys l(t) Is3 0.225 7.14 0.09454 1.3825 0.1307 1.19488 0.635 0.00373
0.3 0.02 1 0.006 1.26 0.00756 1.19488 0.635 2.6E-05 0.00234016 1.23 1 0.01968 0.635 0.0125 1.19488 0.635 0.00865 0.002480.3 0.02 1 0.006 0.01 0.00006 1.19488 0.635 0.00842 0.00234
0.12622 0.15082 1.19488 0.02083 0.00717
0.01818 15.9869 25
le w(kN/m3) Wi(kN/m) span24 16.2
78.5 0.47178.5 1.5448878.5 0.471
200.622803 200.62282
(Rr)d (Rl
0.01878 18.6869234.372809 234.37282
idge 15mspacing 2m
A(t)Xts n Area(t) Yi Yi Y*(t) Ys l(t) Is le w(kN/m3) Wi(kN/m)
2 0.225 7.14 0.06303 0.8825 0.05562 0.74663 0.385 0.00143 24 10.80.3 0.02 1 0.006 0.76 0.00456 0.74663 0.385 1.3E-06 0.00084 78.5 0.471
.016 0.73 1 0.01168 0.385 0.0045 0.74663 0.385 0.00205 0.00052 78.5 0.916880.3 0.02 1 0.006 0.01 0.00006 0.74663 0.385 0.00326 0.00084 78.5 0.471
0.08671 0.06474 0.74663 0.00673 0.00221 0.00605 12.6589
spacing 2.5m
ts n Area(t) YiA(t)XYi Y*(t) Ys i(t) Is le w(kN/m3) Wi(kN/m)
2.5 0.225 7.14 0.07878 0.8825 0.06952 0.76752 0.385 0.00137 24 13.50.3 0.02 1 0.006 0.76 0.00456 0.76752 0.385 5.4E-07 0.00084 78.5 0.471
1.016 0.73 1 0.01168 0.385 0.0045 0.76752 0.385 0.00223 0.00052 78.5 0.916880.3 0.02 1 0.006 0.01 0.00006 0.76752 0.385 0.00344 0.00084 78.5 0.471
0.10246 0.07864 0.76752 0.00705 0.00221 0.00632 15.3589
) spacing 3.0m
ts n Area(t) YiA(t)XYi Y*(t) Ys l(t) Is le w(kN/m3) Wi(kN/m)
3 0.225 7.14 0.09454 0.8825 0.08343 0.78285 0.385 0.00134 24 16.20.3 0.02 1 0.006 0.76 0.00456 0.78285 0.385 3.3E-06 0.00084 78.5 0.471
0.016 0.73 1 0.01168 0.385 0.0045 0.78285 0.385 0.00237 0.00052 78.5 0.916880.3 0.02 1 0.006 0.01 0.00006 0.78285 0.385 0.00358 0.00084 78.5 0.471
0.11822 0.09255 0.78285 0.00729 0.00221 0.00653 18.0589
span ( R r) iDL ( R l) dl
95.413667 95.413663
span (Rr)dl (Rl)dl
15.663678 115.66365
span (Rr)dl (Rl)dl
135.913683 135.91366
230
idge 10mspacing 2m
A(t)Xts n Area(t) Yi Yi Y*(t) Ys l(t) Is le w(kN/m3) Wi(kN/m)
2 0.225 7.14 0.06303 0.8825 0.05562 0.74663 0.385 0.00143 24 10.80.3 0.02 1 0.006 0.76 0.00456 0.74663 0.385 1.3E-06 0.00084 78.5 0.471
.016 0.73 1 0.01168 0.385 0.0045 0.74663 0.385 0.00205 0.00052 78.5 0.916880.3 0.02 1 0.006 0.01 0.00006 0.74663 0.385 0.00326 0.00084 78.5 0.471
0.08671 0.06474 0.74663 0.00673 0.00221 0.00605 12.6589
spacing 2.5m
ts n Area(t) YiA(t)XYi Y*(t) Ys 1(0 Is le w(kN/m3) Wi(kN/m)
2,5 0.225 7.14 0.07878 0.8825 0.06952 0.76752 0.385 0.00137 24 13.50.3 0.02 1 0.006 0.76 0.00456 0.76752 0.385 5.4E-07 0.00084 78.5 0.471
.016 0.73 1 0.01168 0.385 0.0045 0.76752 0.385 0.00223 0.00052 78.5 0.916880.3 0.02 1 0.006 0.01 0.00006 0.76752 0.385 0.00344 0.00084 78.5 0.471
0.10246 0.07864 0.76752 0.00705 0.00221 0.00632 15.3589
spacing 3.0m
ts n Area(t) YiA(t)XYi Y*(t) Ys 1(0 Is le w(kN/m3) Wi(kN/m)
3 0.225 7.14 0.09454 0.8825 0.08343 0.78285 0.385 0.00134 24 16.20.3 0.02 1 0.006 0.76 0.00456 0.78285 0.385 3.3E-06 0.00084 78.5 0.471
016 0.73 1 0.01168 0.385 0.0045 0.78285 0.385 0.00237 0.00052 78.5 0.916880.3 0.02 1 0.006 0.01 0.00006 0.78285 0.385 0.00358 0.00084 78.5 0.471
0.11822 0.09255 0.78285 0.00729 0.00221 0.00653 18.0589
span (Rr)dl (Rl)dl
63.6092 63.6092
span (Rr)dl (Rl)dl
span
77.1091 77.1091
(Rr)dl (Rl)dl
90.6091 90.6091
