load distribution for composite steel–concrete horizontally curved box girder bridge

8182019 Load Distribution for Composite SteelndashConcrete Horizontally Curved Box Girder Bridge

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[14] In their investigations the effect of different parameters such as

span length curvatureratiocross-framespacing and numberof loading

lanes on load distribution factors of horizontally curved bridges has

been studied It has been found that the curvature ratio and span length

are the most in1047298uential parameters that affected these distribution

factors

AASHTO [14] hasalso given loaddistribution factors forthese curved

bridge structures However these distribution factors are found to be

conservative according to some researchers [8915ndash

17] who veri1047297

edthis with physical tests analytical and numerical results Moreover

the loading con1047297guration (number of loads their values and place-

ments) recommended by any other codes such as Australian Bridge

Design Code AS 51002-2004 [18] is quite different from that of

AASHTO Thus the load distributionfactor accordingto AASHTOloading

con1047297guration cannot simply be used for designing a bridge in other

countries such as Australia Hence there is a need for determination of

load distribution factors for the Australian bridge loading (AS 51002-

2004)

In this study load distribution factors of horizontally curved steelndashconcrete composite box girder bridges are determined based on the

Australian bridge loading [18] First of all a detailed 1047297nite element

analysis is carried out on a large number of curved bridge decks All

the bridge components are modelled as shell elements except the top

1047298anges (narrow strips of steel plates used to connect the concrete slab

with the steel web plates) which is modelled as a beam element Initial-

ly the maximum spacing of the cross bracing is determined which will

not allow any local deformation and retain the maximum torsional

rigidity of the curved box girder deck slab systems The numerical

results obtained from the 1047297nite element analysis are validated with

the experimental results [4] to assess the performance of the 1047297nite

element model After getting the con1047297dence on the reliability of the

1047297nite element modelling technique an extensive parametric study is

performed on 180 different curved bridge models by varying different

parameters such as curvature ratio span length number of cells and

number of loading lanes using Australian bridge loading con1047297guration

From these analyses the maximum stresses or stress resultants at

the individual components or girders are obtained On the other hand

the closed form analytical solution for the in1047298uence line diagram of curved beams which has been developed by the authors [13] is used

to calculate the stress resultant acting over the entire bridge deck

section due to the same loading con1047297guration The stress resultants

obtained from the 1047297nite element analysis for the individual girders

and the total stress resultant obtained from the analytical solution are

used to calculate the load distributionfactors as theratio between them

Based on the huge amount of data generated from these analyses a

multivariate regression analysis is carried out to develop empirical

expressions to determine load distribution factors A computer code is

also developed in MATLAB [19] to implement the regression analysis

These expressions will be conveniently used by designers for the

preliminary design of individual bridge girders The same modelling

technique is also used to determine the load distribution factors of

straight steelndashconcrete composite box girder bridge decks Finally theload distribution mechanismof horizontally curved bridges is compared

with those obtained for straight box girder deck slabs Those bridge

decks are subjected to two different bridge design load con1047297gurations

such as AASHTO and AS 51002-2004 It is shown that there is a signi1047297-

cant difference between the obtained load distribution factors for

curved and straight bridge decks

2 Analytical solutions for horizontally curved beams

For the analysis of the statically indeterminate horizontally curved

beam the closed form analytical solution has recently been developed

by the authors [13] which has been used to generate the in1047298uence line

diagrams to predict the critical loading condition for the idealized

horizontally curved beam subjected to moving loads (P ) The detail of

the analytical model is not presented here as it has been reported in

[13] In this section only the 1047297nal equations which are used to calculate

the total shear force (V ) bending moment (M ) and torsion (T ) at any

critical section within a curved beam (Fig 1) are presented below

V frac14 V 1minusP θminusα h i eth1THORN

M frac14 V 1R sinθthorn T 1 sinθminusPR sin θminusα eth THORN θminusα h i eth2THORN

T frac14 T 1 cosθminusV 1R 1minus cosθeth THORN thorn P RminusR cos θminusα eth THORNfrac12 θminusα h i eth3THORN

where V 1 and V 2 arethe vertical support reactionsat A and B respective-

ly T 1 and T 2 arethe torsions at thesupports A and B respectively R isthe

radius of the curvature of the beam and the angle θ (0 le θ le ϕ) is

measured from support A (Fig 1) The moving load (P ) acting vertically

downward on thebeam islocated atθ=α InEqs 1ndash3 langθminusα rang equals to

one if θ ge α or langθminus α rang equals to zero if θ b α and the support reactions

(V 1 V 2 T 1 and T 2) are expressed as follows

V 1 frac14 P minusV 2 eth4THORN

V 2 frac14 T 2 sinϕthorn PR sinα

R sinϕ eth5THORN

T 1 frac14 V 2R 1minus cosϕeth THORN thorn T 2 cosϕminusPR 1minus cosα eth THORN eth6THORN

Fig 1 Simply supported horizontally curved beam

Fig 2 Variation of mid-span de1047298ection with respect to the element sizes for different

girders

20 SJ Fatemi et al Journal of Constructional Steel Research 116 (2016) 19ndash 28



T 2 frac14 minusΔT

f tt

eth7THORN

where

ΔT frac14

R2

GJ sinα cosα minus

cosϕ

sinϕ sin2

α thornα

sinα

sinϕminusα

thorn R2

GJ

sinϕ cosα minus sinα cosϕthorn ϕ sinα

sinϕminus sin ϕminusα eth THORN

thorn cosϕ

sinϕ sin2

α minus sinα cosα minusα sinα

sinϕ

eth8THORN

and

f tt frac14 Rϕ

GJ eth9THORN

It should be noted that the in1047298uence line for shear force has its

maximum value at the two ends (supports) of the beam while the

maximum value of the in1047298uence line for the bending moment will

occur at the middle of the beam [13]

3 Numerical modelling of curved bridge decks

The numerical simulation of the horizontally curved box girder

bridge systems is conducted by using a commercially available 1047297nite

element software (ABAQUS) [20] In the process of modeling of the

box girder bridges some assumptions are made to make the numerical

model a reasonable representation of the actual bridge for the present

purpose These assumptions are 1) the bridge materials are homoge-

neous and isotropic and behave elastically 2) no member will fail

due to local buckling 3) there is a complete composite action between

the concrete deck and the top 1047298ange of the steel girders by providing

adequate shear connection and 4) to simplify the analysis the contri-

bution of curbs and railings on the bridge deck are ignored

4 Convergence study

As a part of numerical analysis convergence studies are conducted

on a curved bridge model to 1047297nd a suitable 1047297nite element mesh size

and to identify an appropriate brace spacing to prevent any undesirable

premature failure due to local and global buckling Therefore it is

proposed to investigate the sensitivity of 1047297nite element models to the

mesh sizes and bracing con1047297gurations

Different element sizes ranging from 3000 mm down to 150 mm

(Fig 2) are tried for a 60 m span of composite curved bridge with a

curvature ratio of 12 and considering six lane traf 1047297c loading to observe

the effects of element size on the accuracy of the model and to 1047297nd a

convergence point It is very important to maintain the aspect ratio as

close to one as possible while changing the size of the elements to

achieve the most accurate results while performing the 1047297nite element

analysis

The results obtained from mesh size convergence study are present-

ed in Fig 2 and it is observed since the element size decreases from

3000 mm to 250 mm the corresponding de1047298ections converged when

the element with a size of 250 mm is used in the model There are no

further signi1047297cant changes in de1047298ections when the element size isreduced to a value below 250 mm In addition for the element size

smaller than 250 mm the numberof elements and nodes may be overly

large and reduce the ef 1047297ciency of the model For example for the

models with the element sizes of 500 mm 250 mm and 150 mm the

numbers of elements are 9891 55355 and 220071 respectively

On the other hand cross bracing systems in curved box girder

bridges stiffen the box girder internally to resist rotations of girders

about their longitudinal axes which prevent the structure from

Table 1

Analysis of mid-span de1047298ection for 60 m horizontally curved box girder bridge

Brace

spacing

Mid-span de1047298ection

(m) W1

(m)

Δ1

()

W2

(m)

Δ2

()

W3

(m)

Δ3

()

W4

(m)

Δ4

()

4 0209 ndash 0239 ndash 0270 ndash 0302 ndash

45 0209 000 0239 000 0270 000 0302 000

5 0210 048 0240 042 0271 037 0302 000

55 0211 047 0241 041 0271 000 0304 066

6 0212 047 0242 041 0273 073 0306 065

66 0214 093 0243 041 0274 036 0307 033

75 0215 047 0245 082 0276 072 0309 065

85 0218 138 0247 081 0278 072 0310 032

10 0233 644 0264 644 0295 576 0329 578

Table 2

Comparison of reaction distribution under concentrated load between experimental and

1047297nite element results

Models Location W1

(kN)

Location W2

(kN)

Location W3

(kN)

Location W4

(kN)

Experimental test minus611 635 1255 2220

Kennedys model minus738 650 1183 2264

Current model minus759 640 1188 2286

Fig 3 Box girder bridge model

21SJ Fatemi et al Journal of Constructional Steel Research 116 (2016) 19ndash 28



premature failure due to torsional buckling and distortional warping

and retain the maximum torsional rigidity of the structure The objec-

tiveof thebracing analysis is to identify an appropriate cross brace spac-

ing to estimate the bracing contribution to the reduction of the

de1047298ection while controlling torsional effects such as distortion and

warping of the cross sections To achieve this goal the design load con-

1047297guration of AS 51002-2004 [18] is used in a numerical investigation to

obtain the maximum cross brace spacing Afterwards the cross bracing

systems are located at different spacing on the horizontally curvedbridge model Table 1 shows the results of the brace spacing conver-

gence study which is conducted for themodel In this table the analysis

of mid-span de1047298ection for different girders such as W1 W2 W3 and W4

is done using different bracing spaces varying from 40 m to 10 m In

addition in the same table Δ represents the deviations of the mid-span

de1047298ection due to various brace spacing (eg Δ1 frac14 W 1eth5mTHORNminusW 1eth45mTHORNW 1eth5mTHORN

100)

According to Table 1 the obtained results from the brace spacing

convergence study indicated that there are signi1047297cant differences in

de1047298ection at the mid-span when the cross brace spacing is reduced

from 10 m to 5 m However for the brace spacing smaller than 5 m

the de1047298ection variations at mid-span are insigni1047297cant This means the

convergence starting point of brace spacing for the model occurred at

5 m This conforms to AASHTO guidelines [14] and theHanshin Express-

way Public Corporation of Japan [21] that prescribe that the maximumspacing of the intermediate diaphragms in these structural systems

should not exceed 75 m and 6 m respectively AASHTO guidelines

[14] specify the maximum spacing of the intermediate diaphragms

(LD) as

LD le L R

200Lminus7500

12

le 25 ft eth10THORN

where L and R denote the span length (in feet) and the radius of

curvature (in feet) respectively On the other hand the Hanshin

Expressway Public Corporation of Japan [21] speci1047297es the maximum

spacing of the intermediate diaphragms (LD) in horizontally curved

box girder bridges as

LD frac14 LDS k ϕ Leth THORN eth11THORN

in which

LDS frac146m014Lminus24eth THORNm

20m

L b 60meth THORN

60m le L le 160meth THORN

L N 160meth THORN

8lt eth12THORN

k ϕ Leth THORN frac1410

1minus

ffiffiffiffiϕ

p Lminus60eth THORN

100 ffiffiffi

2p

L b60meth THORN


8lt eth13THORN

where LDS k(ϕ L) and ϕ indicate the spacing of the intermediate

diaphragms in straight box girder bridges (in m) the reduction factor

and the central angle (in rad) respectively [22]

5 Validation of the 1047297nite element models

To ensure the accuracy of the 1047297nite element modelling methods a

curved bridge model is developed according to the published experi-

mental work [4] In the experimental study different horizontally

curved bridge prototypes by varying curvature ratio and loading

positions have been considered to obtain support reactions for each

case The curved bridge deck utilized herein for validation purposes is

a simply supported 26 m span of curved composite box girder bridge

including 1047297ve cross bracings The width of concrete deck is 1 m with

50 mm thickness where the span to radius of curvature ratio is 10

and has a central angle of 573deg

To model the bridge components such as concrete deck steel web

girders bottom 1047298ange and end diaphragms four node shell elements

type S4R with six degrees of freedom are used The end diaphragms

are located on each side of the bridge models between the web girders

Top 1047298ange and cross bracing systems are modelled as a two node beam

element type B31H (Fig 3) The modelling of the top 1047298ange as a beam

element makes the connections between the concrete deck slab to the

top 1047298ange much easier In addition tie constraints are de1047297ned to

connect the concrete deck slab to the top 1047298ange and top 1047298ange to the

box girders The element with a size of 250 mm is used in the model

as described in Section 4

Material properties of the steel and concrete such as the modules of

elasticity (200 GPa for steel and 27 GPa for concrete) and the Poisson

ratio (025 for steel and 02 for concrete) are de1047297ned [4] The support

conditions are considered as simply supported and to ensure full

interactions among the components of the bridge cross bracingsbox girders and diaphragms are merged together Finally reaction

distributions for this bridge model are measured at supports for all

four web girders (W1 W2 W3 and W4) due to four concentrated loads

(4 times 1675 kN) which are applied at mid-span of each bridge girder

(Fig 3) The results obtained from the numerical simulation are

compared with the experimental test data [4] and it is found that

there is a very good agreement between them (Table 2)

Table 3

Geometries of bridge models in parametric study

Bridge type Cross-section dimensions (mm)

Deck Top 1047298ange Cell depth Girder depth Top 1047298ange Web Bottom Deck Diaphragm

Loading span A C D F t1 t2 t3 t4 t5

2 lane L = 20 m 9300 300 800 1025 16 10 10 225 10

2 lane L = 40 m 9300 300 1600 1825 28 32 50 225 12

2 lane L = 60 m 9300 300 2400 2625 40 54 90 225 15

4 lane L = 20 m 16800 300 800 1025 16 10 10 225 10

4 lane L = 40 m 16800 300 1600 1825 28 32 50 225 12

4 lane L = 60 m 16800 300 2400 2625 40 54 90 225 15

6 lane L = 20 m 24300 300 800 1025 16 10 10 225 10

6 lane L = 40 m 24300 300 1600 1825 28 32 50 225 12

6 lane L = 60 m 24300 300 2400 2625 40 54 90 225 15




6 Parametric study using detailed 1047297nite element analysis

Havingbeen satis1047297ed thatthe1047297nite element modelsdescribed in the

preceding section yielded accurate results an extensive parametric

study is carried out to determine the stress resultants of the individual

components or girders of the horizontally curved bridge structures

subjected to Australian bridge load 180 different curved bridge models

aresimulated using the1047297nite element modelling techniqueas described

before by varying different parameters consisting of curvature ratio

(K = LR) span length (L) number of loading lanes (N L) and number

of cells (N C ) In the parametric study different span lengths (20 m

40 m and 60 m) with various curvatures (02 04 08 and 12) and the

number of cells ranging from two to four for a two-lane loading three

to 1047297ve for a four-lane loading and four to six for the six-lane loading

are considered in the modelling of these structural systems The width

of the bridge models also varied according to the number of loading

lanes these widths are 9300 mm 16800 mm and 24300 mm for two

four and six lanes respectively The detailed geometries of bridge

models used in the parametric study are presented in Table 3

61 Loading con 1047297 guration

To conduct the parametric study the highway bridge loading asprescribed by AS 51002-2004 [18] is applied on bridge models Accord-

ing to AS 51002-2004 S1600 (stationary loads) and M1600 (moving

loads) are two main systems of loading con1047297gurations to which the

bridge might be subjected Based on the in1047298uence line diagram [13] it

is found that the M1600 moving traf 1047297c load may cause the critical

effects on the curved bridge deck models To obtain the highest bending

moment of the models M1600 moving traf 1047297c load are applied at the

mid-span of the bridge models while to determine the highest shear

reactions M1600 moving traf 1047297c load are applied at the extreme end of

the bridge models The con1047297guration of M1600 moving load is shown

in Fig 4a which consists of 6 kNm uniformly distributed load over

a 32 m width and four wheel groups of six wheels which each wheel

counted as 60 kN of concentrated load Figs 4b 4c and 4d show

moving traf 1047297c loads for a 40 m span with four cells subjected to

two four and six lane loading respectively For example as can be

seen from Fig 4 for single lane loading the total uniformly distributed

load is 187 kNm2 and the total concentrated load is 1800 kN

The self-weight of the curved bridge decks is also included in the 1047297-

nite element models A uniformly distributed loadis applied to concrete

deck to account for self-weight of the deck slabby considering the value

of 24 kNm3 as density of concrete and also a uniformlydistributed load

is appliedto thesteel bottom1047298anges to account forself-weightof bridge

girders (including webs top and bottom 1047298anges) and cross bracing

systems by considering the value of 78 kNm3 as density of steel For

example for a 40 span a total of 63 kNm2 uniformly distributed load

as permanent (dead) load are applied to the bridge model to account

for the self-weight of this structure

Moreover as stated in the AS 51002-2004 [18] accompanying

lane factors of 1 08 and 04 are applied to the 1047297rst second and third(subsequent) loading lanes respectively In addition the magnitude of

13 is applied to eachloading laneas a dynamic load allowanceto de1047297ne

the interaction of moving vehicles and the bridge structures [18]

62 Load distribution factor

In the current researcha large amount of results from 1047297nite element

analysis of curved bridge decks are obtained and these results are

collated further to develop the distribution factors for both bending

moment and shear force of horizontally curved box girder bridges The

moment distribution factor is de1047297ned as

Dm frac14 M max

M eth14THORN

Fig 4 (a)M1600moving loads Curvedbox girder bridge model subjected to (b)two lane

loading (c) four lane loading and (d) six lane loading

Fig 5 Variation of momentdistributionfactor(Dm) withrespect tonumberof cellsfordif-

ferent numbers of lanes




where M max is a bending moment carried by each girder at mid-span

(which is determined by using the 1047297nite element analysis) and M is

total momentwhich is calculated from the analysis of simply supported

curved beam using analytical solutions [13] when subjected to M1600

moving traf 1047297c load [18] and dead load as a line load per meter long of

the bridge The shear distribution factor also is de1047297ned as

Ds frac14 V max

V eth15THORN

where V max is a reaction force under each girder at supports (which is

determined using the1047297nite element analysis) and V is the total reaction

force which is calculated from the analysis of simply supported curved

beam using analytical solutions [13] when subjected to M1600 moving

traf 1047297cload(V = V 1 obtained from Eq 4) and deadload asa lineload per

meter longof the bridge In thefollowingsections sampleresults of load

distribution factors are presented

621 Distribution factor for bending moment

The effects of number of loading lanes on moment distribution

factors of the horizontally curved bridges are investigated Based on

Fig 5 it is found out that as the number of lanes increases the value

of moment distribution factor also increases This 1047297gure shows the

change in moment distribution factors for outer girder of a 60 m spanof curved bridge with the curvature ratio of 08 when subjected to

different numbers of loading lanes The effect of the number of cells

on moment distribution factors is also studied According to Fig 5 as

the number of cells increases the moment distribution factor decreases

because more webs in the box girder can provide greater stiffness to

resist the torsion and distortion in curved bridge decks Moreover it

can be observed in Fig 6 that the intermediate girders carried the

great amount of moment compared to outer girders

Curvature ratio and span length also have a signi1047297cant impact on

moment distribution factors As can be seen from Fig 7 the moment

distribution factor decreases with an increase in the curvature ratios

Fig 8 shows that the longer spans have greater distribution factors

compared to the shorter spans For example a 60 m span has the

greatest moment distribution factor compared to that of 40 m and

20 m spans This trend can be explained by in longer span lengths

more loads are accommodate throughout the length

622 Distribution factor for shear forces

In the current study the investigation on shear distribution factors

revealed that the behaviour of intermediate girders are quite different

from outer girders due to position of moving loads as they are located

close to supports Accordingto Fig 9 shear distribution factors for inter-

mediategirdersof a 40m spanwiththecurvature ratio of02 are greater

than its outer girders Based on the same 1047297gure one can conclude that

as the number of cells increases the shear distribution factor decreases

because more loads are distributed among more girders which resulted

in decreasing the load distribution factors

The effects of curvature ratio and span length on shear distribution

factors of the outer left girders (Fig 10a) of horizontally curved bridge

decks are presented in Figs 10b and 11a respectively It is observed

that the shear distribution factor increases with an increase in the

curvature ratio as well as span lengths due to the presence of torsional

moments On the other hand Figs 10c and 11b present the shear

distribution factor trends for outer right girders due to curvature ratio

and span length respectively It is observed that as the curvature ratio

and the span length increase the shear distribution factor decreases It

is worth to mention that for the curvature ratio greater than 08 there

is an uplift to the right supportFig 12 shows the variation of shear distribution factors for different

numbers of lanes of a 40 m span with the curvature ratio of 06 Accord-

ing to this 1047297gure it is observed that for the same number of cells the

shear distribution factor for the outer girder decreases with an increase

in the number of lanes It is worth to note that although by increasing

the number of loading lanes the amount of load increases but the

number of cells is also increasing due to the bridge width Hence by

increasing the number of cells more loads are distributed among

more girders whichresulted in decreasingthe sheardistribution factors

7 Development of design guideline formulas

Based on the results obtained from the parametric study the design

guideline in the form of some expressions is developed using a

Fig 6 Variation of moment distribution factor (Dm) with respect to number of girders for different numbers of cells

Fig 7 Variation of momentdistribution factor(Dm) withrespect tonumberof cellsfordif-

ferent curvature ratios

Fig 8 Variationof momentdistributionfactor(Dm) with respectto numberof cellsfordif-

ferent span lengths




multivariate regression analysis The regression analysis is based on the

power functions which are implemented by a computer programme

code written in MATLAB [19] The parameters considered in the study

are curvature ratio (K frac14 LR) span length (L) number of lanes (N L) and

number of cells (N C ) for the determination of load distribution factor

expressions of horizontally curved bridges when subjected to the

Australian bridge load These empirical expressions are derived

separately for the inner outer and intermediate girders The obtained

results from numerical simulations are plotted against the results fromproposed expressions (Figs 13 and 14) which show a strong agreement

between them In addition it should be noted that to determine the

accuracy of the expressions correlation coef 1047297cient (R 2) is calculated

for each expression and all the corresponding R2 values are greater

than 092 which con1047297rm the reliability and accuracy of the expressions


preliminary design of individual bridge girders

71 Design expressions for moment distribution factor

Expressions for moment distribution factors of curved bridge decks

are generated for different girders such as outer right girders outer

left girders and intermediate girders The outer girders are de1047297ned as

two far end outside girders and the intermediate girders are consideredas all inner and central girders for the purpose of calculating the

moment distribution factors The moment distribution factor

Fig 9 Variation of shear distribution factor (Ds) for different numbers of cells

Fig 10 (a) Bridge model with different girders variation of shear distribution factor (Ds)

for different curvatures (b) outer left girder (c) outer right girder

Fig 11 Variation of shear distribution factor (Ds) for different span lengths (a) outer left

girder (b) outer right girder




expressions for the outer right girders (DMoR) and the outer left girders

(DMoL) are expressed respectively as follows

DMoR frac14 03N minus095c L023N 02

L K 0016N 2Lminus112 eth16THORN

DMoL frac14 007N minus108c L047N 042

L K 014N Lminus154 eth17THORN

The developed moment distribution factor expression for the

intermediate girders (DMi) which is obtained from regression analysis is

DMi frac14 016N minus106c L055N 013


72 Design expressions for shear distribution factor

Different formulas are deduced to account for shear distribution

factors of curved box girder bridges by considering the investigated

parameters Due to the variety of distribution factors 1047297ve different

empirical expressions are developed for various girders including

outer right girders outer left girders inner right girders inner left

girders and central girders It should be noted that the inner girders

arelocated between the outer andthe central girders Thetwo following

(shear distribution factor) expressions are derived for the outer left

girders (DSoL) and the outer right girders (DSoR) respectively

DSoL frac14 023N minus04c L036N minus063

L 1 thorn 254K minus174K 2 thorn 149K 3

eth19THORN

DSoR frac14 39N minus03c Lminus033N minus092

L 1minus322K thorn 389K 2

eth20THORN

The next two following expressions are deduced for determination

of shear distribution factors at the inner left girders (DSiL) and the

inner right girders (DSiR) respectively

DSiL frac14 038N minus052c L028N minus044


eth21THORN

DSiR frac14 0013N minus117c L013N 248


Thelast following formula is derived to determine shear distribution

factors at the central girders (DSc )

DSc frac14 06N minus11c L01N 036

L K 015minus0024N L eth23THORN

8 Assessment of moment distribution factors of horizontally curved

bridges due to loading recommended by AASHTO and AS 51002-

2004

A comparativeanalysis is conducted on load distributionmechanism

of horizontally curved steelndashconcrete composite box girder bridges

subjected to AASHTO [14] and AS 51002-2004 [18] bridge codes along

with similar bridgeparameters andproperties (such as material proper-

ties girder dimensions and span lengths) so that a general comparison

could be made This analysis revealed some similarities and distinctive

differences between the obtained load distribution factors according

to adopted bridge design loading con1047297gurations Both bridge design

codes provided various load factors outlining maximum and minimum

values used to produce the more critical combinations of bridge loading

which results in producing different distribution factors for both bend-

ing moment and shear force

This study is performed by considering different horizontally curved

box girder bridges subjected to AASHTO HS20-44 design loading [14]

The bridges are 40 m long with a curvature ratio of 04 including differ-

ent numbers of cells ranged between 2ndash4 cells The con1047297guration of HS20-44 consists of 934 kNm uniformly distributed load over a 3 m

width plus a single concentrated load which is taken as 80 kN for

moment distribution study [23] The moment distribution factors

obtained for these bridges for AASHTO loads are compared with the

moment distribution factors which are developed in the present study

based on the AS 51002-2004 (M1600 design loading) The obtained

results are presented in Fig 15 As can be seen from this 1047297gure the

moment distribution factors for AASHTO loads [14] are about 25 less

than that due to AS 51002-2004 [18] On the other hand the same

analysis techniques are used for these bridges when subjected to

AASHTO loading to calculate shear distribution factors for the purpose

of comparing the predicted distribution factors corresponding to AS

51002-2004 Fig 16 indicated that the shear distribution factors for

AASHTO loads [14] are 10 greater than that for AS 51002-2004 [18]Hence it can be concluded that due to signi1047297cant differences in the

value of load distribution factors of these structures when subjected to

various bridge design loading con1047297gurations it is not reasonable to

Fig 12 Variation of shear distribution factor (Ds) with respect to number of cells for

different numbers of lanes

Fig 13 Comparison between the obtained moment distribution factor from proposed

expressions and1047297

nite element analysis results

Fig 14 Comparison between the obtained shear distribution factor from proposed

expressions and 1047297nite element analysis results

Fig 15 Comparisonof momentdistribution factorbetween AASHTO andAS 51002-2004




use AASHTO load distribution factors for the design of curved bridges

subjected to Australian bridge design load

9 Comparison of load distribution factors of horizontally curved

bridges with straight bridges due to loading recommended by

AASHTO and AS 51002-2004

A comparative analysis is also performed on the load distribution

mechanism of horizontally curved and straight box girder deck slabs

according to AASHTO [14] and AS 51002-2004 [18] The same 1047297nite

element modelling techniques as described in the previous sections

are used to model straight box girder bridges The bridge models are

subjected to AS 51002-2004 loading The load distribution factors

obtained from this analysis for AS 51002-2004 loading with the

AASHTO load distribution factors which are obtained from literature

[523] (for both straight and curved box girder bridges) are presented

in Figs 17 and 18 These 1047297gures show that the load distribution factors

for horizontally curved box girder bridges are greater than straight

box girder bridges for both AASHTO [14] and AS 51002-2004 [18]

According to Fig 17 for example in the case of a 5 cell of box girder

bridge with a span of 60 m subjected to four-lanes of traf 1047297c loading of

AS 51002-2004 [18] the value of moment distribution factor is 02 at

the critical location whereas for a curved bridge with the same bridge

characteristic (N c = 5 N L = 4 and L = 60 m) as mentioned above

with curvature of 08 the corresponding value of moment distribution

factor is 03 In other words the maximum moment measured at the

mid-span of the curved bridge (17630 kNm) is much greater thanthat of the straight bridge (2130 kNm) This signi1047297cant difference

between the obtained moments at critical section of those bridges can

be explained by presence of torsional moment due to member

curvature at the curved bridges Hence designing of the curved bridge

structures based on the load distribution factors of straight bridges

may not be a safe and reliable design of these structures due to a signif-

icant difference between the obtained load distribution factors for

curved and straight bridge deck systems

10 Conclusion

In the current study the load distribution factors of horizontally

curved steelndashconcrete composite box girder bridges subjected to

Australian bridge load are determined from a large number of resultswhich are generated from detailed 1047297nite element analysis of these

structures for a large number of cases These large number of results

are obtained by varying different parameters such as curvature ratio

span length number of loading lanes and number of cells of these

structures The accuracy and reliability of the above mentioned 1047297nite

element models are con1047297rmed using experimental data available in

the literature The design guideline for load distribution factor is also

determined in the form of some expressions which can be used to

conveniently calculate bending moment and shear force of different

components of these structures in a designof 1047297ce Based on the observa-

tion in the current study the following conclusions can be drawn

bull The maximum spacing of the cross brace spacing of the horizontallycurved box girder bridges should not be more than 5 m to avoid any

undesirable premature failure due to local and global buckling of

these structures and to retain the maximum torsional rigidity of them

bull The curvature ratio and the number of cells are found to have signi1047297-

cant effects on load distribution factor for bending moment of

horizontally curved bridges The maximum value of the load distribu-

tion factor for bending moment (found in the outer girder) is always

reduced with the increase of the number of cells as well as the

curvature ratios

bull The values of load distribution factors for both bending moment and

shear force for AASHTO loads [14] of a curved bridge are greater

than those for Australian bridge load [18]

bull The maximum moment found at the mid-span of a curved bridge is

always found to be more than that of a straight bridge Thus it is notrecommended to treat a curved bridge as an equivalent straight

bridge in order to avoid unreliable design

References

[1] JFHajjarD Krzmarzick L PallareacutesMeasuredbehaviorof a curved compositeI-girderbridge J Constr Steel Res 66 (3) (2010) 351ndash368

[2] DG Linzell JF Shura Erection behavior and grillage model accuracy for a large ra-dius curved bridge J Constr Steel Res 66 (3) (2010) 342ndash350

[3] SS Dey AT Samuel Static analysis of orthotropic curved bridge decks ComputStruct 12 (2) (1980) 161ndash166

[4] K Sennah J Kennedy Shear distribution in simply-supported curved compositecellular bridges J Bridg Eng 3 (2) (1998) 47ndash55

[5] K Sennah J Kennedy Simply supported curved cellular bridges simpli1047297ed designmethod J Bridg Eng 4 (2) (1999) 85ndash94

[6] K Sennah J Kennedy S Nour Design for shear in curved composite multiple steelbox girder bridges J Bridg Eng 8 (3) (2003) 144ndash152

[7] G Issa-El-KhouryDG Linzell LF Geschwindner Computational studies of horizon-tally curved longitudinally stiffened plate girder webs in 1047298exure J ConstrSteel Res93 (2014) 97ndash106

[8] D DePolo D Linzell Evaluation of live-load lateral 1047298ange bending distribution for ahorizontally curved I-girder bridge J Bridg Eng 13 (5) (2008) 501ndash510

[9] P Barr et al Live-load analysis of a curved I-girder bridge J Bridg Eng 12 (4)(2007) 477ndash484

[10] W Kim J Laman D Linzell Live load radial moment distribution for horizontallycurved bridges J Bridg Eng 12 (6) (2007) 727ndash736

[11] Fiechtl Fenves Frank Approximate Analysis of Horizontally Curved Girder BridgesCenter for Transportation Research Bureau of Engineering Research The Universityof Texas at Austin 1987

[12] MA Grubb Horizontally curved I-girder bridge analysis V-load method TranspRes Rec 982 (1984) 26ndash35

[13] SJ Fatemi AH Sheikh MSM Ali Development and application of an analyticalmodel for horizontally curved bridge decks Adv Struct Eng 18 (1) (2015)

107ndash

118

Fig 16 Comparison of shear distribution factor between AASHTO and AS 51002-2004

Fig 17 Comparison between load distribution factor of straight and curved box girder

bridges according to AS 51002-2004


bridges according to AASHTO




[14] American Association of State Highway and Transportation Of 1047297cials (AASHTO) G-SfHCHB Washington DC 1993

[15] C Yoo et al Bending strength of a h orizontally curved composite I-girder bridge JBridg Eng 18 (5) (2013) 388ndash399

[16] D Huang Field test and rating of Arlington curved-steel box-girder bridge Jacksonville Florida Transp Res Rec 1892 (2004) 178ndash186

[17] D Huang Full-scale test and analysis of a curved steel-box girder bridge J BridgEng 13 (5) (2008) 492ndash500

[18] Standards Australia Bridge Design Part 2 Design Loads 2004 (AS 51002-2004 Syd-ney (Australia))

[19] MATLAB version 810 (R2013a) The MathWorks Inc 2013[20] ABAQUS Analysis Users Manual 612 2012[21] Hanshin Expressway Public Corporation GftDoHCGB Japan 1988[22] N-H Park Y-J Choi Y-J Kang Spacing of intermediate diaphragms in horizontally

curved steel box girder bridges Finite Elem Anal Des 41 (9ndash10) (2005) 925ndash943[23] K Sennah J Kennedy Load distribution factors for composite multicell box girder

bridges J Bridg Eng 4 (1) (1999) 71ndash78




[14] In their investigations the effect of different parameters such as

span length curvatureratiocross-framespacing and numberof loading

lanes on load distribution factors of horizontally curved bridges has

been studied It has been found that the curvature ratio and span length

are the most in1047298uential parameters that affected these distribution

factors

AASHTO [14] hasalso given loaddistribution factors forthese curved

bridge structures However these distribution factors are found to be

conservative according to some researchers [8915ndash

17] who veri1047297

edthis with physical tests analytical and numerical results Moreover

the loading con1047297guration (number of loads their values and place-

ments) recommended by any other codes such as Australian Bridge

Design Code AS 51002-2004 [18] is quite different from that of

AASHTO Thus the load distributionfactor accordingto AASHTOloading

con1047297guration cannot simply be used for designing a bridge in other

countries such as Australia Hence there is a need for determination of

load distribution factors for the Australian bridge loading (AS 51002-

2004)

In this study load distribution factors of horizontally curved steelndashconcrete composite box girder bridges are determined based on the

Australian bridge loading [18] First of all a detailed 1047297nite element

analysis is carried out on a large number of curved bridge decks All

the bridge components are modelled as shell elements except the top

1047298anges (narrow strips of steel plates used to connect the concrete slab

with the steel web plates) which is modelled as a beam element Initial-

ly the maximum spacing of the cross bracing is determined which will

not allow any local deformation and retain the maximum torsional

rigidity of the curved box girder deck slab systems The numerical

results obtained from the 1047297nite element analysis are validated with

the experimental results [4] to assess the performance of the 1047297nite

element model After getting the con1047297dence on the reliability of the

1047297nite element modelling technique an extensive parametric study is

performed on 180 different curved bridge models by varying different

parameters such as curvature ratio span length number of cells and

number of loading lanes using Australian bridge loading con1047297guration

From these analyses the maximum stresses or stress resultants at

the individual components or girders are obtained On the other hand

the closed form analytical solution for the in1047298uence line diagram of curved beams which has been developed by the authors [13] is used

to calculate the stress resultant acting over the entire bridge deck

section due to the same loading con1047297guration The stress resultants

obtained from the 1047297nite element analysis for the individual girders

and the total stress resultant obtained from the analytical solution are

used to calculate the load distributionfactors as theratio between them

Based on the huge amount of data generated from these analyses a

multivariate regression analysis is carried out to develop empirical

expressions to determine load distribution factors A computer code is

also developed in MATLAB [19] to implement the regression analysis


preliminary design of individual bridge girders The same modelling

technique is also used to determine the load distribution factors of

straight steelndashconcrete composite box girder bridge decks Finally theload distribution mechanismof horizontally curved bridges is compared

with those obtained for straight box girder deck slabs Those bridge

decks are subjected to two different bridge design load con1047297gurations

such as AASHTO and AS 51002-2004 It is shown that there is a signi1047297-

cant difference between the obtained load distribution factors for

curved and straight bridge decks

2 Analytical solutions for horizontally curved beams

For the analysis of the statically indeterminate horizontally curved

beam the closed form analytical solution has recently been developed

by the authors [13] which has been used to generate the in1047298uence line

diagrams to predict the critical loading condition for the idealized

horizontally curved beam subjected to moving loads (P ) The detail of

the analytical model is not presented here as it has been reported in

[13] In this section only the 1047297nal equations which are used to calculate

the total shear force (V ) bending moment (M ) and torsion (T ) at any

critical section within a curved beam (Fig 1) are presented below

V frac14 V 1minusP θminusα h i eth1THORN

M frac14 V 1R sinθthorn T 1 sinθminusPR sin θminusα eth THORN θminusα h i eth2THORN

T frac14 T 1 cosθminusV 1R 1minus cosθeth THORN thorn P RminusR cos θminusα eth THORNfrac12 θminusα h i eth3THORN

where V 1 and V 2 arethe vertical support reactionsat A and B respective-

ly T 1 and T 2 arethe torsions at thesupports A and B respectively R isthe

radius of the curvature of the beam and the angle θ (0 le θ le ϕ) is

measured from support A (Fig 1) The moving load (P ) acting vertically

downward on thebeam islocated atθ=α InEqs 1ndash3 langθminusα rang equals to

one if θ ge α or langθminus α rang equals to zero if θ b α and the support reactions

(V 1 V 2 T 1 and T 2) are expressed as follows

V 1 frac14 P minusV 2 eth4THORN

V 2 frac14 T 2 sinϕthorn PR sinα

R sinϕ eth5THORN

T 1 frac14 V 2R 1minus cosϕeth THORN thorn T 2 cosϕminusPR 1minus cosα eth THORN eth6THORN

Fig 1 Simply supported horizontally curved beam

Fig 2 Variation of mid-span de1047298ection with respect to the element sizes for different

girders




T 2 frac14 minusΔT

f tt

eth7THORN

where

ΔT frac14

R2


cosϕ

sinϕ sin2

α thornα

sinα

sinϕminusα

thorn R2

GJ



thorn cosϕ

sinϕ sin2


sinϕ

eth8THORN

and

f tt frac14 Rϕ

GJ eth9THORN

















4 Convergence study














analysis













Table 1


Brace

spacing


(m) W1

(m)

Δ1

()

W2

(m)

Δ2

()

W3

(m)

Δ3

()

W4

(m)

Δ4

()


45 0209 000 0239 000 0270 000 0302 000

5 0210 048 0240 042 0271 037 0302 000

55 0211 047 0241 041 0271 000 0304 066

6 0212 047 0242 041 0273 073 0306 065

66 0214 093 0243 041 0274 036 0307 033

75 0215 047 0245 082 0276 072 0309 065

85 0218 138 0247 081 0278 072 0310 032

10 0233 644 0264 644 0295 576 0329 578

Table 2



Models Location W1

(kN)

Location W2

(kN)

Location W3

(kN)

Location W4

(kN)






















100)











(LD) as

LD le L R

200Lminus7500

12

le 25 ft eth10THORN







in which


20m

L b 60meth THORN


L N 160meth THORN

8lt eth12THORN


1minus

ffiffiffiffiϕ

p Lminus60eth THORN

100 ffiffiffi

2p

L b60meth THORN


8lt eth13THORN





































Table 3





2 lane L = 20 m 9300 300 800 1025 16 10 10 225 10

2 lane L = 40 m 9300 300 1600 1825 28 32 50 225 12

2 lane L = 60 m 9300 300 2400 2625 40 54 90 225 15

4 lane L = 20 m 16800 300 800 1025 16 10 10 225 10

4 lane L = 40 m 16800 300 1600 1825 28 32 50 225 12

4 lane L = 60 m 16800 300 2400 2625 40 54 90 225 15

6 lane L = 20 m 24300 300 800 1025 16 10 10 225 10

6 lane L = 40 m 24300 300 1600 1825 28 32 50 225 12

6 lane L = 60 m 24300 300 2400 2625 40 54 90 225 15




























































Dm frac14 M max

M eth14THORN














Ds frac14 V max

V eth15THORN

































































ferent span lengths























































eth19THORN



eth20THORN






eth21THORN









2004











































































10 Conclusion






















curvature ratios







References














107ndash

118




















T 2 frac14 minusΔT

f tt

eth7THORN

where

ΔT frac14

R2


cosϕ

sinϕ sin2

α thornα

sinα

sinϕminusα

thorn R2

GJ



thorn cosϕ

sinϕ sin2


sinϕ

eth8THORN

and

f tt frac14 Rϕ

GJ eth9THORN

















4 Convergence study














analysis













Table 1


Brace

spacing


(m) W1

(m)

Δ1

()

W2

(m)

Δ2

()

W3

(m)

Δ3

()

W4

(m)

Δ4

()


45 0209 000 0239 000 0270 000 0302 000

5 0210 048 0240 042 0271 037 0302 000

55 0211 047 0241 041 0271 000 0304 066

6 0212 047 0242 041 0273 073 0306 065

66 0214 093 0243 041 0274 036 0307 033

75 0215 047 0245 082 0276 072 0309 065

85 0218 138 0247 081 0278 072 0310 032

10 0233 644 0264 644 0295 576 0329 578

Table 2



Models Location W1

(kN)

Location W2

(kN)

Location W3

(kN)

Location W4

(kN)






















100)











(LD) as

LD le L R

200Lminus7500

12

le 25 ft eth10THORN







in which


20m

L b 60meth THORN


L N 160meth THORN

8lt eth12THORN


1minus

ffiffiffiffiϕ

p Lminus60eth THORN

100 ffiffiffi

2p

L b60meth THORN


8lt eth13THORN





































Table 3





2 lane L = 20 m 9300 300 800 1025 16 10 10 225 10

2 lane L = 40 m 9300 300 1600 1825 28 32 50 225 12

2 lane L = 60 m 9300 300 2400 2625 40 54 90 225 15

4 lane L = 20 m 16800 300 800 1025 16 10 10 225 10

4 lane L = 40 m 16800 300 1600 1825 28 32 50 225 12

4 lane L = 60 m 16800 300 2400 2625 40 54 90 225 15

6 lane L = 20 m 24300 300 800 1025 16 10 10 225 10

6 lane L = 40 m 24300 300 1600 1825 28 32 50 225 12

6 lane L = 60 m 24300 300 2400 2625 40 54 90 225 15




























































Dm frac14 M max

M eth14THORN














Ds frac14 V max

V eth15THORN

































































ferent span lengths























































eth19THORN



eth20THORN






eth21THORN









2004











































































10 Conclusion






















curvature ratios







References














107ndash

118


































100)











(LD) as

LD le L R

200Lminus7500

12

le 25 ft eth10THORN







in which


20m

L b 60meth THORN


L N 160meth THORN

8lt eth12THORN


1minus

ffiffiffiffiϕ

p Lminus60eth THORN

100 ffiffiffi

2p

L b60meth THORN


8lt eth13THORN





































Table 3





2 lane L = 20 m 9300 300 800 1025 16 10 10 225 10

2 lane L = 40 m 9300 300 1600 1825 28 32 50 225 12

2 lane L = 60 m 9300 300 2400 2625 40 54 90 225 15

4 lane L = 20 m 16800 300 800 1025 16 10 10 225 10

4 lane L = 40 m 16800 300 1600 1825 28 32 50 225 12

4 lane L = 60 m 16800 300 2400 2625 40 54 90 225 15

6 lane L = 20 m 24300 300 800 1025 16 10 10 225 10

6 lane L = 40 m 24300 300 1600 1825 28 32 50 225 12

6 lane L = 60 m 24300 300 2400 2625 40 54 90 225 15




























































Dm frac14 M max

M eth14THORN














Ds frac14 V max

V eth15THORN

































































ferent span lengths























































eth19THORN



eth20THORN






eth21THORN









2004











































































10 Conclusion






















curvature ratios







References














107ndash

118












































































Dm frac14 M max

M eth14THORN














Ds frac14 V max

V eth15THORN

































































ferent span lengths























































eth19THORN



eth20THORN






eth21THORN









2004











































































10 Conclusion






















curvature ratios







References














107ndash

118


























Ds frac14 V max

V eth15THORN

































































ferent span lengths























































eth19THORN



eth20THORN






eth21THORN









2004











































































10 Conclusion






















curvature ratios







References














107ndash

118







































































eth19THORN



eth20THORN






eth21THORN









2004











































































10 Conclusion






















curvature ratios







References














107ndash

118




















































10 Conclusion






















curvature ratios







References














107ndash

118


















load distribution for composite steel–concrete horizontally curved box girder bridge

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