Research ArticleDistortional Buckling Analysis of Steel-Concrete CompositeGirders in Negative Moment Area

Zhou Wangbao12 Jiang Lizhong2 Kang Juntao1 and Bao Minxi3

1 School of Civil Engineering and Architecture Wuhan University of Technology Wuhan 430070 China2 School of Civil Engineering Central South University Changsha 410075 China3 School of Civil Engineering University of Birmingham Birmingham B15 2TT UK

Correspondence should be addressed to Kang Juntao jtkang163com

Received 4 July 2014 Revised 9 October 2014 Accepted 9 October 2014 Published 10 November 2014

Academic Editor Ting-Hua Yi

Copyright copy 2014 Zhou Wangbao et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Distortional buckling is one of the most important buckling modes of the steel-concrete composite girder under negative momentIn this study the equivalent lateral and torsional restraints of the bottom flange of a steel-concrete composite girder under negativemoments due to variable axial forces are thoroughly investigated The results show that there is a coupling effect between theapplied forces and the lateral and torsional restraint of the bottom flange Based on the calculation formula of lateral and torsionalrestraints the critical buckling stress of I-steel-concrete composite girders and steel-concrete composite box girders under variableaxial force is obtained The critical bending moment of the steel-concrete composite girders can be further calculated Comparedto the traditional calculation methods of elastic foundation beam the paper introduces an improved method which considerscoupling effect of the external loads and the foundation spring constraints of the bottom flange Fifteen examples of the steel-concrete composite girders in different conditions are calculated The calculation results show a good match between the handcalculation and the ANSYS finite element method which validated that the analytic calculation method proposed in this paper ispractical

1 Introduction

The steel-concrete composite girders are a type of importantlateral-load-carrying composite element A concrete flooror concrete deck and a steel girder are combined by shearconnections andhence the steel girder and concrete slab carryloads togetherThe existence of the concrete slab can improvethe entire and local stability The steel-concrete compositegirder has light self-weight strong lateral restraint good fireresistance and durability In terms of strength ductility andstability this type of component is of high compressive stressresistance benefitting from the concrete and excellent tensileresistance because of the steel Besides this steel-concretecomposite girder is an ecofriendly structure With effectivesteel recycling and high construction speed steel-concretecomposite girders have shown promising potential in thefuture construction market [1ndash4]

The negative bending moment area of the steel girderin a steel-concrete composite girder will be subjected to theconstraint caused by the concrete slab and hence experiencebuckling Chen and Jia [5] studied the ultimate resistance of acontinuous composite beam and the investigations indicatedthat the ultimate resistance was governed by either distor-tional lateral buckling or local buckling or an interactivemode of the two Svensson [6] improved themethod of elasticfoundation beamunder constant axial force whichwas basedon the assumption that the concrete slab was totally rigidThe method also introduced variable axial elastic foundationstruts so as to consider the bending gradient effect HoweverWilliams and Jemah [7] found that Svenssonrsquos method is notsafe enough and suggested increasing the involved area ofthe web Goltermann and Svensson [8] further developedWilliamsrsquo models by solving the eigenvalue of a four-step

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 635617 10 pageshttpdxdoiorg1011552014635617

2 Mathematical Problems in Engineering

differential equation to understand the buckling of steel-concrete composite girders in the negative moment areacaused by variable axial force In 1982 Swedish code forlight-gauge metal structures first simplified the issue [9] bydeeming the buckling analysis of steel-concrete compositegirder in negative moment area as a stability study of theelastic foundation beam under constant axial force that isthe method of elastic foundation beam under constant axialforce British Bridge Standard (BS5400) [10] also employsthis method to the design of continuous composite girdersBritish Steel Structure Institute [11] obtained a calculationformula of the critical stress 119872cr in the buckling analysis ofsteel-concrete composite girders in negative moment area byusing energy method Jiang et al [12] presented a stabilityanalysis calculation model of composite box beam consid-ering rotation of steel beam top flange and established thecritical bending moment calculation formula of distortionalbuckling by employing energy method Due to the limitedcomputation capacity at that time the articles reviewed abovedid not carry on a detailed analysis on the applicabilityof the elastic foundation beam method It requires furtherinvestigation especially on whether the variable axial strutis equivalent to the steel-concrete composite girder whenconsidering the real bending gradient Based on Svenssonrsquoselastic foundation strut model Ye et al [13 14] made animprovement on the lateral and torsional restraints of theelastic foundation strut by considering the involved part ofthe web and pointed out that the elastic foundation beammethod was more reasonable than the energy method Thebuckling analysis of a multispan steel-concrete compositegirder via a three-step simplification can be carried outHowever thismethod cannot be applied to composite girdersunder complex loads Zhou et al [15 16] undertook a researchon the equivalent lateral and torsional restraints of the bottomflanges in negative moment areas of I-steel-concrete girdersand steel-concrete composite box girders Correspondingcalculation formulaewere proposed and the results indicateda coupling relation between the external loads and thetorsional and lateral restraints of the bottom flange

In this paper the calculation formulae of the lateral andtorsional restraints under variable axial force are proposed byconsidering the coupling effect of restraint and external loadsThe critical buckling stress and critical bending moment ofthe steel-concrete composite girder are further developedFinally the precision analysis of the proposed formula isconductedwith an exampleThe calculationmethod providesa theoretical basis for further studying of the ultimate resis-tance of the steel-concrete composite girder under variableaxial force

2 Basic Assumptions

The cross-section dimensions of a steel-concrete compositegirder are shown in Figure 1 The distortional buckling modeof the steel girder in a composite girder is different fromthat of an unconstraint steel girder The top flange of thesteel girder in the composite girder is inserted into theconcrete slab which has greater lateral and torsional restraint

Mx0

z

l

xy

0

Mx1

bcbf

tw tf hchw

ycx

y

0

Figure 1 Cross-section dimensions of steel-concrete compositegirders and axes

stiffness Therefore both lateral deformation and torsionaldeformation of the steel girder are restrained by the concreteslab The lateral buckling of the composite girder happenswith the torsional bucking of the lateral distortion of thesteel web as shown in Figure 2 To simplify calculation thefollowing assumptions are made

(1) The lateral bending stiffness and torsional stiffness ofthe concrete slab are relatively greater The top flangeof the steel girder is restricted by the concrete slabso that the lateral distortion and torsional distortioncannot take place

(2) Tensile resistance of the concrete slab is ignored(3) Since no vertical deformation corresponding to flex-

ural buckling occurs when the distortional bucklinghappens [5ndash8 13ndash17] the vertical restraint stiffness ofthe bottom flange is deemed to be infinity that is119896119910 = infin

3 Restraining Stiffness Analysis of the Web ofSteel-Concrete Composite Girder

According to the above assumptions the compression stressat the edge of the bottom flange by considering the rein-forcement within the flanges of concrete slabs under negativebending moment is expressed as

1205901 (120585) =119872119909 (120585) 119910119888

119868 (1)

where 120585 = 119911119897 is a normalization parameter 0 le 120585 le 1 119897 is thelength of the composite girder119872119909(119911) is the negative bendingmoment acting on the composite girder and minus119910119888 is the centerposition of the equivalent cross-section in the vertical axisand can be expressed by (3)

The varying compression stress in order to take intoaccount the moment gradient is expressed as

1205901 (120585) = 1205900 (1198860 + 1198861120585 + 11988621205852) (2)

where 1205900 is the maximum compression stress of the bottomflange Here by definition positive 1205901 denotes compressionstress and coefficients 1198860 1198861 and 1198862 represent different loadconditions (1) 1198860 = 1 1198861 = 0 and 1198862 = 0 stand for the purebending moment (2) 1198860 = 0 1198861 = 1 and 1198862 = 0 representtriangle negative bending moment (3) 1198860 = 0 1198861 = 4 and1198862 = minus4 are uniform distributed loads

Mathematical Problems in Engineering 3

Original shapeShape after deformation

Figure 2 Distortional buckling of steel-concrete composite girdersunder negative moments

Consider

119910119888 =119860 119904119910119904 + 119860 119905ℎ119908 + 05119860119908ℎ119908

119860 119904 + 119860 119905 + 119860119908 + 119860119891

(3)

where 119860119891 is the area of the bottom flange 119860 119905 is the area ofthe top flange119860119908 is the area of the steel web119860 119904 is the area ofreinforcements within concrete slab and 119910119904 is the distance ofthe center position of the equivalent cross-section to the edgeof steel flange

31 The Torsional Restraint of the Steel Web The simplifiedmodel of the steel web is shown as in Figure 3 Two transverseedges are simply supported The junction of the web and topflange is fixed while the junction of web and bottom flange issimply supportedThe boundary condition of the buckling ofthe steel web is [15 16] given as follows

[119908]119911=0119897 = 0 [119908]119910=0minusℎ119908

= 0 [119908119910]119910=minusℎ119908

= 0

[minus119863119908 (119908119911119911 + 120583119908119910119910)]119911=0119897= 0

(4)

where 119863119908 = 1198641199053

11990812(1 minus 120583

2) 120583 is Poissonrsquos ratio of steel

119864 is the elasticity modulus of steel 119908(119910 119911) is the bucklingdeformation function of web 119905119908 is the thickness of the steelgirder web and ℎ119908 is the height of the steel girder web

Based on the boundary conditions the buckling deforma-tion function of the steel web is

119908 = [119910

ℎ119908

+ 2(119910

ℎ119908

)

2

+ (119910

ℎ119908

)

3

](

119899

sum

119894=1

119888119894 sin119894120587119911

119897) (5)

According to the principle of stationary potential energy[18ndash20] the buckling characteristic equation is given as fol-lows

(B0 + 1198961205931T minus 1205900N0)C = 0 (6)

Longitudinal edge of web

yc

1205901y

z

m(z)

tw

hw

1205901

yc

l

Figure 3 Rectangular plate under compression and moments

where1198610119894119894 = 119897119863119908((2ℎ3

119908)+(ℎ119908119894

41205732210)+(2119894

212057315ℎ119908)) 119879119894119894 =

1198972ℎ2

119908 1198610119894119895 = 119879119894119895 = 0 (119894 = 119895) 1198730119894119894 = (120587

2119892119894119897)((119905119908ℎ119908210) minus

(119905119908ℎ2

119908560119910119888)) 120573 = 120587

21198972 1198730119894119895 = (119905119908ℎ119908119887119894119895119897)((1105) minus (ℎ119908

280119910119888)) (119894 = 119895) 119887119894119895 = [(minus1)119894+119895

minus 1]1198861 + 2(minus1)119894+1198951198862(119894119895(119894

2+

1198952)(1198942minus 1198952)2

) 119892119894 = 1198942[1198860+(11988612)+1198862((13)+(12119894

21205872))] 1198961205931

is the lateral restraint stiffness of theweb by the bottomflangeand C = 1198881 1198882 119888119899

119879 is general coordinates and representsthe buckling distortion

According to the elastic plate theory the lateral distribu-tion force of the web is given as follows [21]

119891119909120593 = minus119863119908 [1205973119908

1205971199103+ (2 minus 120583)

1205973119908

1205971199112120597119910]

119891119909120593

10038161003816100381610038161003816119910=0= minus

6119863119908

ℎ3119908

(

119899

sum

119894=1

119888119894 sin119894120587119911

119897)

+ 119863119908120573(2 minus 120583)

ℎ119908

(

119899

sum

119894=1

1198881198941198942 sin 119894120587119911

119897)

(7)

32 The Lateral Restraint of the Steel Web The simplifiedmodel of the steel web is shown as in Figure 4 Two transverseedges are simply supported and the junction of the web andtop flange is fixed The junction of web and bottom flange isfree in the transverse direction The boundary condition ofthe buckling of the steel web is [15 16] given as follows

[119908]119911=0119897 = 0 [119908]119910=minusℎ119908

= 0 [119908119910]119910=0minusℎ119908

= 0

[minus119863119908 (119908119911119911 + 120583119908119910119910)]119911=0119897= 0

(8)

According to the boundary condition the buckling defor-mation function of the steel web can be expressed as

119908 = [1 minus 3(119910

ℎ119908

)

2

minus 2(119910

ℎ119908

)

3

](

119899

sum

119894=1

119889119894 sin119894120587119911

119897) (9)

According to the principle of stationary potential energy[18 20] the buckling characteristic equation is given asfollows

(H0 + 1198961199091R minus 1205900S0)D = 0 (10)

4 Mathematical Problems in Engineering

Longitudinal edge of web

yc

1205901y

z

f (z)

tw

hw

1205901

yc

l

Figure 4 Rectangular plate under compression and lateral stress

where 119877119894119895 = 1198670119894119895 = 0 (119894 = 119895) 119877119894119894 = 1198972 1198670119894119894 = 119897119863119908((6ℎ3

119908) +

(13ℎ11990870)12057321198944+ (65ℎ119908)120573119894

2) 1198780119894119895 = (119905119908ℎ119908119887119894119895119897)((1335) minus

(3ℎ11990835119910119888)) (119894 = 119895) 1198780119894119894 = (119905119908ℎ1199081205872119892119894119897)((1370) minus (3ℎ119908

70119910119888)) 1198961199091 is the lateral restraint stiffness of the web bythe bottom flange and D = 1198891 1198892 119889119899

119879 is generalcoordinates and represents the buckling distortion of thebottom flange

According to the elastic plate theory the lateral dis-tributed bending moment of the web is [21] given as follows

119891120593119909 = minus119863119908 (1205972119908

1205971199102+ 120583

1205972119908

1205971199112)

119891120593119909

10038161003816100381610038161003816119910=0= 119863119908

6

ℎ2119908

(

119899

sum

119894=1

119889119894 sin119894120587119911

119897)

+ 119863119908120583120573(

119899

sum

119894=1

1198891198941198942 sin 119894120587119911

119897)

(11)

33 Restraint Analysis of the Steel Web 1198961205931 and 1198961199091 can bedetermined by the following equations

10038161003816100381610038161003816B + 1198961205931T minus 1205900N

10038161003816100381610038161003816= 0

1003816100381610038161003816H + 1198961199091R minus 1205900S1003816100381610038161003816 = 0

(12)

It can be found from (12) that there is a couplingrelation between external loads and torsionallateral restraintstiffness It indicates that both the torsional and lateralrestraints of the bottom flange are not only determined bythe cross-section features of the composite girder but theyalso depended on the external loads Therefore it may notbe appropriate to take the restraint stiffness as a constantmaterial feature in the traditional elastic foundation beammethod

4 Buckling Analysis of I-Steel-ConcreteComposite Girders

According to the assumptions made upon the bucklingmodel of the I-steel-composite girder can be simplified asthe model depicted in Figure 5 The horizontal and torsionaldirections of the thin plate are restricted by springs while thevertical direction is rigidly restricted

kx

bf

kyk120593

tf

x

y

0

Figure 5 Simplified calculation model of steel-concrete compositegirders

In Figure 5 the thin plate is symmetric about both 119909-axisand 119910-axis The centroid of the plate is set to be the originpoint Assuming the horizontal lateral displacement of thebottom flange is 119906(119911) and the torsional angle is 120593(119911) theneutral equilibrium differential equation of an elastic thin-walled bar under variable axial force can be expressed as [17]

119864119868119910119906119868119881+ [119875 (119906

1015840+ 119910119886120593

1015840)]1015840

+ 119896119909 [119906 minus (119910119889 minus 119910119886) 120593]

minus 119891119909120593 = 0

119864119868119909V119868119881+ [119875 (V1015840 minus 119909119886120593

1015840)]1015840

+ 119896119910 [V + (119909119889 minus 119909119886) 120593] = 0

119864119868119908120593119868119881+ [(119875119903

2

0minus 119866119869) 120593

1015840]1015840

minus 119896119909 [119906 minus (119910119889 minus 119910119886) 120593]

times (119910119889 minus 119910119886) + 119891120593119909 minus 119909119886(119875V1015840)1015840

+ 119910119886(1198751199061015840)1015840

+ 119896119910 [V + (119909119889 minus 119909119886) 120593] (119909119889 minus 119909119886) + 119896120593120593 = 0

(13)

where 119868119910 = 1199051198911198873

11989112 119868119909 = 119887119891119905

3

11989112 119869 = 119887119891119905

3

1198913 11990320= 1199092

119886+

1199102

119886+ (119868119909 + 119868119910)119860 119904 and 119909119886 is center position of the curved

bottom flange in the horizontal axis here 119909119886 = 0 119910119886 is thecenter position of the curved bottom flange in the verticalaxis here 119910119886 = 0 119909119889 is the rotation axis of the bottom flangein the horizontal axis here 119909119889 = 0 119910119889 is the rotation axis ofthe bottom flange in the vertical axis here 119910119889 = 0 119868119908 is thesectorial inertia moment of bottom flange here 119868119908 = 0 119866is shear modulus of the steel 119875 is the pressure of the bottomflange119875 = 1198601198911205900(1198860+1198861120585+1198862120585

2) 119896120593 = minus1198961205931 and 119896119909 = minus1198961199091

Plugging119910119886 = 0 119910119889 = 0 119909119886 = 0 119909119889 = 0 V = 0 and 119868119908 =0 into (13) leads to

(1198751199032

0minus 119866119869) 120593

10158401015840+ 1199032

011987510158401205931015840+ 119896120593120593 + 119891120593119909 = 0

119864119868119910119906119868119881+ 11987510158401199061015840+ 11987511990610158401015840+ 119896119909119906 minus 119891119909120593 = 0

(14)

By combining (5) and (9) the torsional angle and lateraldisplacement of the bottomflange of the composite girder canbe obtained

120593 (119911) =

119899

sum

119894=1

119888119894

ℎ119908

sin 119894120587119911119897

119906 (119911) =

119899

sum

119894=1

119889119894 sin119894120587119911

119897

(15)

Mathematical Problems in Engineering 5

According to the Galerkin method [15 22] we have

(B1 + 119896120593T minus 1205900N1 Q

M H1 + 119896119909R minus 1205900S1)(

CD) = 0 (16)

where 1198611119894119895 = 119876119894119895 = 1198671119894119895 = 119872119894119895 = 0 (119894 = 119895) 1198611119894119894 =119866119869(119894212058722119897ℎ2

119908) 119872119894119894 = (3119863119908119897ℎ

3

119908)minus119863119908((2minus120583)119894

21198971205732ℎ119908) 119876119894119894 =

3(119863119908119897ℎ3

119908)+(119863119908120583119897120573119894

22ℎ119908) 1198731119894119894 = (119860119891119903

2

012058721198921198942119897ℎ

2

119908) 1198731119894119895 =

(1198601198911199032

0119887119894119895119897ℎ2

119908) (119894 = 119895) 1198671119894119894 = 119864119868119910120573

2(11989441198972) 1198781119894119894 = (119860119891120587

2119892119894

2119897) and 1198781119894119895 = (119860119891119887119894119895119897) (119894 = 119895)The combination of (6) (10) and (16) leads to

[(B QM H) minus 1205900 (

N 00 S)](

CD) = 0 (17)

where B = B0 + B1 N = N0 + N1 H = H0 + H1 and S =

S0 + S1The deformation vector C119879D119879119879 cannot be zero when

buckling happens Therefore the buckling of the compositegirder can be solved by the generalized eigenvalue of thecharacteristic matrix shown as follows

10038161003816100381610038161003816100381610038161003816

(B QM H) minus 1205900 (

N 00 S)

10038161003816100381610038161003816100381610038161003816

= 0 (18)

By solving (18) 2119899 generalized eigenvalue can beobtained 120590119905119894 (119894 = 1 2 2119899) let 120590cr = min120590119905119894 (119894 =

1 2 2119899) 120590cr is the critical buckling stress of the compos-ite girder The critical buckling moment of composite girdercan be calculated by the following equation

119872cr =120590cr (1198860 + 1198861120585 + 1198862120585

2) 119868

119910119888

(19)

5 Buckling Analysis of the Steel-ConcreteComposite Box Girder

The dimensions of the composite box girder are shownin Figure 6 According to the assumptions made abovethe buckling model of the composite box girder can besimplified as a thin-plate model that is restricted by springsin horizontal and torsional directions rigidly restricted invertical directionThe simplified model is shown in Figure 7

As the derivation in Section 3 the following can beobtained

(B0 + 1198961205931198971T minus 1205900N0)C119897 = 0 (20)

(B0 + 1198961205931199031T minus 1205900N0)C119903 = 0 (21)

(H0 + 1198961199091R minus 1205900S0)D = 0 (22)

Mx

x y

z

L

Mx hwyctw

tt hc

tfy

bc

x00

Figure 6 Cross-section dimensions of steel-concrete composite boxgirder

k120593l

kxl

ky = infin

y

bf

x

k120593r

kxr

0

ky = infin

Figure 7 Simplified calculation model of steel-concrete compositegirders

119891119909120593119897

10038161003816100381610038161003816119910=0= minus

6119863119908

ℎ3119908

(

119899

sum

119894=1

119888119897119894 sin119894120587119911

119871)

+ 119863119908120573(2 minus 120583)

ℎ119908

(

119899

sum

119894=1

1198881198971198941198942 sin 119894120587119911

119871)

(23)

119891119909120593119903

10038161003816100381610038161003816119910=0= minus

6119863119908

ℎ3119908

(

119899

sum

119894=1

119888119903119894 sin119894120587119911

119897)

+ 119863119908120573(2 minus 120583)

ℎ119908

(

119899

sum

119894=1

1198881199031198941198942 sin 119894120587119911

119897)

(24)

119891120593119909

10038161003816100381610038161003816119910=0=6119863119908

ℎ2119908

(

119899

sum

119894=1

119889119894 sin119894120587119911

119897)

+ 119863119908120583120573(

119899

sum

119894=1

1198891198941198942 sin 119894120587119911

119897)

(25)

where 1198961205931198971 is the torsional restraint stiffness of left web by

the bottom flange C119897 = 1198881198971 1198881198972 119888119897119899119879 is buckling general

coordinates of the left web 1198961205931199031 is torsional restraint stiffness

of the right web by the bottom flange C119903 = 1198881199031 1198881199032 119888119903119899119879

is buckling general coordinates of the right web 1198961199091 is lateralrestraint stiffness of the steel web by the bottom flange andD = 1198891 1198892 119889119899

119879 is buckling general coordinates of thebottom flange

As Figure 7 shows the lateral displacement bucklingfunction of the horizontal buckling of the bottom flange is119906(119911) the out-plane buckling deformation function of the

6 Mathematical Problems in Engineering

bottom flange is V(119909 119911) the left rotational angle is 120593119897(119911) andthe right rotational angle is 120593119903(119911) The boundary condition ofthe bottom flange is given as follows

V 1003816100381610038161003816119909=0 = 0 V11990910038161003816100381610038161003816119909=0

= 120593119897 (119911)

V100381610038161003816100381610038161003816119909=119887119891

= 0 V119909100381610038161003816100381610038161003816119909=119887119891

= 120593119903 (119911)

(26)

According to compatibility of deformation the displace-ment function of the bottom flange is

V =119887119891

2120587sin 120587119909

119887119891

(120593119897 minus 120593119903) +

119887119891

4120587sin 2120587119909

119887119891

(120593119897 + 120593119903) (27)

According to the principle of stationary potential energythe buckling characteristic equations are given as follows

119863119891(51205872120593119897

8119887119891

+31205872120593119903

8119887119891

minus

11988711989112059310158401015840

119897

2+

51198873

1198911205931015840101584010158401015840

119897

321205872minus

31198873

1198911205931015840101584010158401015840

119903

321205872)

+ (

51205901198601198911198872

1198911205931015840

119897

321205872)

1015840

minus (

31205901198601198911198872

1198911205931015840

119903

321205872)

1015840

+ 119896120593119897

120593119897 + 119891120593119909 = 0

2119896119909119906 + (1205901198601198911199061015840)1015840

+ 1198641198681199101199061015840101584010158401015840

minus 119891119909120593119897

minus 119891119909120593119903

= 0

119863119891(51205872120593119903

8119887119891

+31205872120593119897

8119887119891

minus

11988711989112059310158401015840

119903

2+

51198873

1198911205931015840101584010158401015840

119903

321205872minus

31198873

1198911205931015840101584010158401015840

119897

321205872)

+ (

51205901198601198911198872

1198911205931015840

119903

321205872)

1015840

minus (

31205901198601198911198872

1198911205931015840

119897

321205872)

1015840

+ 119896120593119903

120593119895 + 119891120593119909 = 0

(28)

where 119868119910 = 1199051198911198873

11989112 119905119891 is the thickness of bottom flange 119887119891

is the width of the bottom flange119863119891 = 1198641199053

11989112(1 minus 120583

2) 119896120593

119897

=

minus1198961205931198971 is torsional restraint stiffness of the bottom flange edge

by the left steel web 119896120593119903

= minus1198961205931199031 is torsional restraint stiffness

of the bottomflange edge by the right steel web and 119896119909 = minus1198961199091is lateral restraint stiffness of the bottom flange edge by thesteel web

According to the Galerkin method [15 16] we have

(

B1+ 119896120593119897

T minus 1205900N1 F + 06120590

0N1 Q

F + 061205900N1

B1+ 119896120593119903

T minus 1205900N1 Q

M M H1minus 1205900S1+ 2119896119909R)

times 120578 = 0

(29)

where 1198611119894119895 = 119865119894119895 = 119876119894119895 = 119872119894119895 = 1198671119894119895 = 0 (119894 = 119895) 1198611119894119894 =(5119863119891120587

211989716119887119891ℎ

2

119908) + (119863119891119887119891120573119897119894

24ℎ2

119908) + (5119863119891119887

3

11989112057321198971198944641205872ℎ2

119908)

119865119894119894 = (3119863119891120587211989716119887119891ℎ

2

119908) minus (3119863119891119887

3

11989112057321198971198944641205872ℎ2

119908) 119876119894119894 =

3(119863119908119897ℎ3

119908)+(1198971198631199081205831205732ℎ119908)119894

2 1198731119894119895 = (51198601198911198872

11989111988711989411989532120587

2ℎ2

119908119897) (119894 =

119895) 1198731119894119894 = (51198601198911198872

11989111989211989464ℎ

2

119908119897) 119872119894119894 = (3119863119908119897ℎ

3

119908) minus 119863119908((2 minus

120583)11989421198971205732ℎ119908) 1198671119894119894 = (119894

411986411986811991012057321198972) 1198781119894119894 = (119860119891120587

21198921198942119897) 1198781119894119895 =

(119860119891119887119894119895119897) (119894 = 119895) and 120578 = (C119879119897C119879119903D119879)119879

Table 1 Basic geometric size of I-steel composite girder cases

Examplenumber

Cross-sectionnumber

ℎ119908mm 119887

119891mm 119905

119891mm 119905

119908mm 119897mm

1 45002 1 600 200 12 12 72003 96004 30005 2 600 120 10 10 54006 84007 60008 3 719 268 25 16 102009 1440010 300011 4 300 100 3 3 540012 840013 420014 5 450 150 4 4 720015 10200

Since the constraint in the theoretical model is higherthan the real scenario the critical buckling stress is increasedTherefore the theoretical buckling deformation functions ofthe web and bottomflange cannot accurately describe the realbuckling deformation curves In order to eliminate errorsthe paper gives a reduction factor on the torsional restraintstiffness of the bottom flange and the reduction factor isfound to be 05 Combining (20) (21) (22) and (29) leadsto

[[[[

[

(

B F QF B QM M H

) minus 1205900(

N minus3N15

0minus3N15

N 00 0 S

)

]]]]

]

120578 = 0 (30)

where B = B02 + B1 N = N02 + N1 H = 2H0 +H1 and S = 2S0 + S1

The deformation vector 120578 cannot be zero when the buck-ling of the composite girder happensTherefore the bucklingof the composite girder can be solved by the generalizedeigenvalue of the characteristic matrix shown as follows

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(

B F QF B QM M H

)minus 1205900(

N minus06N1 0minus06N1 N 0

0 0 S)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

= 0 (31)

3119899 general eigenvalues can be obtained from (31) whichare 120590119905119894 (119894 = 1 2 3119899) let 120590cr = min120590119905119894 (119894 =

1 2 3119899) 120590cr is the critical buckling stress of the com-posite girderThe following equation can calculate the criticalbuckling moment

119872cr =120590cr (1198860 + 1198861120585 + 1198862120585

2) 119868

119910119888

(32)

Mathematical Problems in Engineering 7

Table 2 Basic geometric size of steel composite box girder cases

Example number ℎ119908mm 119887

119891mm 119887

119905mm 119905

119908mm 119905

119891mm 119905

119905mm 119897mm

1 40002 400 600 100 10 10 10 80003 120004 40005 400 500 100 10 10 10 80006 120007 40008 300 600 100 10 10 10 80009 1200010 400011 400 600 120 10 10 10 800012 1200013 400014 400 600 100 12 12 12 800015 12000

Table 3 Critical distortional buckling moment of I-steel compositegirder under negative uniform moment

Examplenumber

Distortional buckling critical moment119872crkNsdotmANSYS Williams Goltermann Svensson (19)

1 11291 10606 11936 15401 113832 12123 11480 12344 16647 120393 11497 10770 11971 15617 115504 39098 35402 40588 6196 404405 41019 37512 43686 6566 421616 40197 36603 46919 6406 414127 36727 31264 35211 38991 372008 38365 33904 40143 42281 387459 39051 32268 42454 40242 3932710 34155 33710 35235 489 3263211 35235 34884 36464 506 3358712 34729 34304 35951 497 3310613 94611 94325 101817 1368 8971314 107307 107811 102911 1563 1013115 91161 90153 93920 1307 86372

6 Analysis of Examples

Thegeometric dimensions of each example are listed inTables1 and 2 By means of the calculation method introduced inthis paper and the finite elementmethod the critical bucklinganalysis of the composite girder under uniform negativebending moment triangle bending moment and uniformloads can be carried out Svenssonrsquos method Williamsrsquomethod and Goltermannrsquos method are also employed in thecalculation of various I-steel-concrete composite girders soas to validate the calculation method proposed in this paperThe finite element analysis is conducted by using ANSYScommercial software Element SHELL43 is adopted to model

Table 4 Critical distortional buckling moment of I-steel compositegirder under negative triangular moment

Examplenumber

Distortional buckling critical moment119872crkNsdotmANSYS Williams Goltermann Svensson (19)

1 18686 18388 19714 26654 192522 17264 16805 17740 24367 173583 15909 15288 16213 22162 159744 6599 6327 7132 11072 703155 5866 5506 6056 9633 601566 5257 4846 5357 8481 538847 61252 55304 61444 68972 635348 55235 49455 54387 61674 560759 50604 44350 49232 55310 5119110 55183 59346 61236 860 5616711 51574 51935 53366 753 4946912 46366 46184 47412 670 4422413 131762 170748 181683 2476 1336114 129117 147906 150336 2144 1299815 127803 131099 134420 1901 12408

the steel girder The concrete slab of the composite girder isreplaced with constraints in the numerical simulation Themotions in 119909 and 119910 directions of the top flange edge arerestrained to represent the lateral and torsional restrictionscaused by the concrete slab in practice The results of eachsimplified calculation method are listed in Tables 3 4 5 and6 and the error analyses of each simplifiedmethod are shownin Figures 8 9 10 and 11

The following conclusions can be drawn based on theresults presented in Tables 3 to 6 and Figures 8 to 11

(1) Under uniform negative bending moment the crit-ical bending buckling moment in the same cross-section of the composite girder is rarely affected by

8 Mathematical Problems in Engineering

Table 5 Critical distortional buckling moment of I-steel compositegirder under uniformly distributed load

Examplenumber

Distortional buckling critical moment119872crkNsdotmANSYS Williams Goltermann Svensson (19)

3 15512 15394 16388 22321 161606 5272 4988 5508 8729 551109 50383 45093 50001 56234 5235111 497 52610 54675 76289 5031712 459 47925 47925 67500 4453313 1299 198896 204768 288401 1291014 1267 165281 165645 239639 1269515 1255 131099 135351 190107 12454

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1605060708091011121314151617

Example number

WilliamsGoltermann

SvenssonThis paper

Mcr

of an

alyt

ical

met

hods

Mcr

of A

NSY

S

Figure 8 Precision analysis of simplified methods under negativeuniform moment

the length The critical bending buckling moment isnot obviously changed with the increased length ofthe structural component

(2) Under triangle bending moment the critical bendingbuckling moment is greatly affected by the lengththat is the value decreased quickly when the lengthincreases

(3) Under uniform negative bending moment trianglebending moment and uniform loads the resultsyielded by the calculation method in this papermatch well the finite element analysis results Thediscrepancy is limited within 5 which validates theaccuracy and applicability of this method

(4) Traditional calculation methods such as Svens-sonrsquos method Williamsrsquo method and Goltermannrsquosmethod have considerable deviations from the finiteelement method

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Example number

WilliamsGoltermann

SvenssonThis paper

05060708091011121314151617

Mcr

of an

alyt

ical

met

hods

Mcr

of A

NSY

S

Figure 9 Precision analysis of simplified methods under negativetriangular moment

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Example number

05060708091011121314151617

Mcr

of an

alyt

ical

met

hods

Mcr

of A

NSY

S

WilliamsGoltermann

SvenssonThis paper

Figure 10 Precision analysis of simplifiedmethods under uniformlydistributed load

Therefore the traditional elastic foundation beammethod taking into account the moment gradient needs tobe improved It is also suggested that the constant lateral andtorsional restraints in the traditional methods may lead tothe relative deviations

7 Conclusions

In this paper the traditional elastic foundation beammethodsare improved by considering the coupling effect of the exter-nal loads and the foundation spring constraints Based on thisimprovement a simplified calculation method computing thecritical buckling loads of steel-concrete composite girders is

Mathematical Problems in Engineering 9

Table 6 Critical distortional buckling moment of composite box girder under negative moment

Examplenumber

Distortional buckling critical moment119872crkNsdotmUniform negative bending moment Triangle negative bending moment Uniform loadANSYS (32) ANSYS (32) ANSYS (32)

1 72605 72214 79236 81741 79282 786742 72714 72214 76467 78046 76122 755313 72797 72214 75492 76616 75146 742624 90593 90722 97811 10125 97547 974365 90824 90650 94867 97143 94390 939166 90957 90610 93864 95600 93434 927857 53751 53575 58654 60316 58284 580398 53852 53407 56622 57636 55978 556429 53917 53411 55906 56601 55267 5487510 74221 73610 81013 83322 80861 8019511 74333 73610 78178 79555 77685 7678812 74417 73610 77181 78097 76702 7569813 12536 12479 13676 14125 13659 1359514 12556 12479 13201 13486 13114 1301715 12607 12479 13035 13239 12947 12833

090

092

094

096

098

100

102

104

106

108

110

Negative uniform moment Negative triangular momentUniformly distributed load

Mcr

of th

is pa

perM

crof

AN

SYS

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Example number

Figure 11 Precision analysis of simplified methods

developed The method is compared with various traditionalmethods The following conclusions are obtained

(1) There is a linear coupling relation between bothtorsional and lateral restraints and vertical loads

(2) Under uniformnegative bendingmoment the criticalbending buckling moment in the same cross-sectionof the composite girder is rarely affected by thelength The critical bending buckling moment is notobviously changed with the increased length of thestructural component

(3) Under triangle bending moment the critical bendingbuckling is influenced to a great extent by the lengththat is the value decreased quickly when the lengthincreases

(4) Under uniform negative bending moment trianglebending moment and uniform loads the calculationmethod proposed in this paper matches well thefinite element calculation methodThe discrepancy islimited within 5 which validates the applicability ofthis method

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research described in this paper was financially sup-ported by National Natural Science Foundation of China(51408449 and 51378502) and the Fundamental ResearchFunds for the Central Universities of China (2014-IV-049)

References

[1] S Kim andU Lee ldquoEffects of delamination on guidedwaves in asymmetric laminated composite beamrdquoMathematical Problemsin Engineering vol 2014 Article ID 956043 12 pages 2014

[2] D Champenoy A Corfdir and P Corfdir ldquoCalculating thecritical buckling force in compressed bottom flanges of steel-concrete composite bridgesrdquo European Journal of Environmen-tal and Civil Engineering vol 18 no 3 pp 271ndash292 2014

[3] T-H Yi H-N Li and M Gu ldquoOptimal sensor placement forstructural health monitoring based on multiple optimization

10 Mathematical Problems in Engineering

strategiesrdquo The Structural Design of Tall and Special Buildingsvol 20 no 7 pp 881ndash900 2011

[4] T-H Yi H-N Li and X-D Zhang ldquoA modified monkeyalgorithm for optimal sensor placement in structural healthmonitoringrdquo Smart Materials and Structures vol 21 no 10Article ID 105033 2012

[5] S Chen and Y Jia ldquoNumerical investigation of inelasticbuckling of steel-concrete composite beams prestressed withexternal tendonsrdquoThin-Walled Structures vol 48 no 3 pp 233ndash242 2010

[6] S E Svensson ldquoLateral buckling of beams analysed as elasticallysupported columns subject to a varying axial forcerdquo Journal ofConstructional Steel Research vol 5 no 3 pp 179ndash193 1985

[7] F WWilliams and A K Jemah ldquoBuckling curves for elasticallysupported columns with varying axial force to predict lateralbuckling of beamsrdquo Journal of Constructional Steel Research vol7 no 2 pp 133ndash147 1987

[8] P Goltermann and S Svensson ldquoLateral distortional bucklingpredicting elastic critical stressrdquo Journal of Structural Engineer-ing vol 114 no 7 pp 1606ndash1625 1988

[9] Swedish Institute of Steel Construction Swedish Code for Light-GaugeMetal Structures Swedish Institute of Steel ConstructionStockholm Sweden 1982

[10] British Standards Institution Code of Practice for Design of SteelBridge BS5400 Part 3 BSI London UK 1982

[11] RM Lawson andW J RackhamDesign of haunched compositebeams in buildings [MS thesis] Steel Construction InstitutionAscot UK 1989

[12] L Jiang J Qi A Scanlon and L Sun ldquoDistortional andlocal buckling of steel-concrete composite box-beamrdquo Steel andComposite Structures vol 14 no 3 pp 243ndash265 2013

[13] J-H Ye and W Chen ldquoElastic restrained distortional buck-ling of steel-concrete composite beams based on elasticallysupported column methodrdquo International Journal of StructuralStability and Dynamics vol 13 no 1 Article ID 1350001 pp 1ndash29 2013

[14] W Chen and J Ye ldquoElastic lateral and restrained distortionalbuckling of doubly symmetric I-beamsrdquo International Journalof Structural Stability and Dynamics vol 10 no 5 pp 983ndash10162010

[15] W-B Zhou L-Z Jiang G-Q Shao and Z-W Yu ldquoElastic dis-tortional buckling analysis of steel-concrete composite beamsin negative moment regionrdquo Journal of Central South University(Science and Technology) vol 43 no 6 pp 2316ndash2323 2012

[16] W B Zhou L Z Jiang and ZW Yu ldquoThe distortional bucklingcalculation formula of the steel-concrete composite beams inthe negative moment regionrdquo Chinese Journal of ComputationalMechanics vol 29 no 3 pp 446ndash450 2012

[17] J Ye andW Chen ldquoElastic restrained distortional buckling of I-steel-concrete composite beamsrdquo Journal of Building Structuresvol 32 no 6 pp 82ndash91 2011

[18] W Zhou L Jiang Z Liu et al ldquoClosed-form solution for shearlag effects of steel-concrete composite box beams consideringshear deformation and sliprdquo Journal of Central South Universityvol 19 no 10 pp 2976ndash2982 2012

[19] W-B Zhou L-Z Jiang andZ-W Yu ldquoAnalysis of free vibrationcharacteristic of steel-concrete composite box-girder consider-ing shear lag and sliprdquo Journal of Central South University vol20 no 9 pp 2570ndash2577 2013

[20] W-B Zhou L-Z Jiang Z-J Liu et al ldquoClosed-form solutionto thin-walled box girders considering effects of shear deforma-tion and shear lagrdquo Journal of Central South University vol 19no 9 pp 2650ndash2655 2012

[21] A Teodor A Guran and G Ardbeshir Theory of Elasticity forScientists and Engineers Springer New York NY USA 2012

[22] C Bi and V Ginting ldquoTwo-grid discontinuous Galerkinmethod for quasi-linear elliptic problemsrdquo Journal of ScientificComputing vol 49 no 3 pp 311ndash331 2011

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

differential equation to understand the buckling of steel-concrete composite girders in the negative moment areacaused by variable axial force In 1982 Swedish code forlight-gauge metal structures first simplified the issue [9] bydeeming the buckling analysis of steel-concrete compositegirder in negative moment area as a stability study of theelastic foundation beam under constant axial force that isthe method of elastic foundation beam under constant axialforce British Bridge Standard (BS5400) [10] also employsthis method to the design of continuous composite girdersBritish Steel Structure Institute [11] obtained a calculationformula of the critical stress 119872cr in the buckling analysis ofsteel-concrete composite girders in negative moment area byusing energy method Jiang et al [12] presented a stabilityanalysis calculation model of composite box beam consid-ering rotation of steel beam top flange and established thecritical bending moment calculation formula of distortionalbuckling by employing energy method Due to the limitedcomputation capacity at that time the articles reviewed abovedid not carry on a detailed analysis on the applicabilityof the elastic foundation beam method It requires furtherinvestigation especially on whether the variable axial strutis equivalent to the steel-concrete composite girder whenconsidering the real bending gradient Based on Svenssonrsquoselastic foundation strut model Ye et al [13 14] made animprovement on the lateral and torsional restraints of theelastic foundation strut by considering the involved part ofthe web and pointed out that the elastic foundation beammethod was more reasonable than the energy method Thebuckling analysis of a multispan steel-concrete compositegirder via a three-step simplification can be carried outHowever thismethod cannot be applied to composite girdersunder complex loads Zhou et al [15 16] undertook a researchon the equivalent lateral and torsional restraints of the bottomflanges in negative moment areas of I-steel-concrete girdersand steel-concrete composite box girders Correspondingcalculation formulaewere proposed and the results indicateda coupling relation between the external loads and thetorsional and lateral restraints of the bottom flange

In this paper the calculation formulae of the lateral andtorsional restraints under variable axial force are proposed byconsidering the coupling effect of restraint and external loadsThe critical buckling stress and critical bending moment ofthe steel-concrete composite girder are further developedFinally the precision analysis of the proposed formula isconductedwith an exampleThe calculationmethod providesa theoretical basis for further studying of the ultimate resis-tance of the steel-concrete composite girder under variableaxial force

2 Basic Assumptions

The cross-section dimensions of a steel-concrete compositegirder are shown in Figure 1 The distortional buckling modeof the steel girder in a composite girder is different fromthat of an unconstraint steel girder The top flange of thesteel girder in the composite girder is inserted into theconcrete slab which has greater lateral and torsional restraint

Mx0

z

l

xy

0

Mx1

bcbf

tw tf hchw

ycx

y

0

Figure 1 Cross-section dimensions of steel-concrete compositegirders and axes

stiffness Therefore both lateral deformation and torsionaldeformation of the steel girder are restrained by the concreteslab The lateral buckling of the composite girder happenswith the torsional bucking of the lateral distortion of thesteel web as shown in Figure 2 To simplify calculation thefollowing assumptions are made

(1) The lateral bending stiffness and torsional stiffness ofthe concrete slab are relatively greater The top flangeof the steel girder is restricted by the concrete slabso that the lateral distortion and torsional distortioncannot take place

(2) Tensile resistance of the concrete slab is ignored(3) Since no vertical deformation corresponding to flex-

ural buckling occurs when the distortional bucklinghappens [5ndash8 13ndash17] the vertical restraint stiffness ofthe bottom flange is deemed to be infinity that is119896119910 = infin

3 Restraining Stiffness Analysis of the Web ofSteel-Concrete Composite Girder

According to the above assumptions the compression stressat the edge of the bottom flange by considering the rein-forcement within the flanges of concrete slabs under negativebending moment is expressed as

1205901 (120585) =119872119909 (120585) 119910119888

119868 (1)

where 120585 = 119911119897 is a normalization parameter 0 le 120585 le 1 119897 is thelength of the composite girder119872119909(119911) is the negative bendingmoment acting on the composite girder and minus119910119888 is the centerposition of the equivalent cross-section in the vertical axisand can be expressed by (3)

The varying compression stress in order to take intoaccount the moment gradient is expressed as

1205901 (120585) = 1205900 (1198860 + 1198861120585 + 11988621205852) (2)

where 1205900 is the maximum compression stress of the bottomflange Here by definition positive 1205901 denotes compressionstress and coefficients 1198860 1198861 and 1198862 represent different loadconditions (1) 1198860 = 1 1198861 = 0 and 1198862 = 0 stand for the purebending moment (2) 1198860 = 0 1198861 = 1 and 1198862 = 0 representtriangle negative bending moment (3) 1198860 = 0 1198861 = 4 and1198862 = minus4 are uniform distributed loads

Mathematical Problems in Engineering 3

Original shapeShape after deformation

Figure 2 Distortional buckling of steel-concrete composite girdersunder negative moments

Consider

119910119888 =119860 119904119910119904 + 119860 119905ℎ119908 + 05119860119908ℎ119908

119860 119904 + 119860 119905 + 119860119908 + 119860119891

(3)

where 119860119891 is the area of the bottom flange 119860 119905 is the area ofthe top flange119860119908 is the area of the steel web119860 119904 is the area ofreinforcements within concrete slab and 119910119904 is the distance ofthe center position of the equivalent cross-section to the edgeof steel flange

31 The Torsional Restraint of the Steel Web The simplifiedmodel of the steel web is shown as in Figure 3 Two transverseedges are simply supported The junction of the web and topflange is fixed while the junction of web and bottom flange issimply supportedThe boundary condition of the buckling ofthe steel web is [15 16] given as follows

[119908]119911=0119897 = 0 [119908]119910=0minusℎ119908

= 0 [119908119910]119910=minusℎ119908

= 0

[minus119863119908 (119908119911119911 + 120583119908119910119910)]119911=0119897= 0

(4)

where 119863119908 = 1198641199053

11990812(1 minus 120583

2) 120583 is Poissonrsquos ratio of steel

119864 is the elasticity modulus of steel 119908(119910 119911) is the bucklingdeformation function of web 119905119908 is the thickness of the steelgirder web and ℎ119908 is the height of the steel girder web

Based on the boundary conditions the buckling deforma-tion function of the steel web is

119908 = [119910

ℎ119908

+ 2(119910

ℎ119908

)

2

+ (119910

ℎ119908

)

3

](

119899

sum

119894=1

119888119894 sin119894120587119911

119897) (5)

According to the principle of stationary potential energy[18ndash20] the buckling characteristic equation is given as fol-lows

(B0 + 1198961205931T minus 1205900N0)C = 0 (6)

Longitudinal edge of web

yc

1205901y

z

m(z)

tw

hw

1205901

yc

l

Figure 3 Rectangular plate under compression and moments

where1198610119894119894 = 119897119863119908((2ℎ3

119908)+(ℎ119908119894

41205732210)+(2119894

212057315ℎ119908)) 119879119894119894 =

1198972ℎ2

119908 1198610119894119895 = 119879119894119895 = 0 (119894 = 119895) 1198730119894119894 = (120587

2119892119894119897)((119905119908ℎ119908210) minus

(119905119908ℎ2

119908560119910119888)) 120573 = 120587

21198972 1198730119894119895 = (119905119908ℎ119908119887119894119895119897)((1105) minus (ℎ119908

280119910119888)) (119894 = 119895) 119887119894119895 = [(minus1)119894+119895

minus 1]1198861 + 2(minus1)119894+1198951198862(119894119895(119894

2+

1198952)(1198942minus 1198952)2

) 119892119894 = 1198942[1198860+(11988612)+1198862((13)+(12119894

21205872))] 1198961205931

is the lateral restraint stiffness of theweb by the bottomflangeand C = 1198881 1198882 119888119899

119879 is general coordinates and representsthe buckling distortion

According to the elastic plate theory the lateral distribu-tion force of the web is given as follows [21]

119891119909120593 = minus119863119908 [1205973119908

1205971199103+ (2 minus 120583)

1205973119908

1205971199112120597119910]

119891119909120593

10038161003816100381610038161003816119910=0= minus

6119863119908

ℎ3119908

(

119899

sum

119894=1

119888119894 sin119894120587119911

119897)

+ 119863119908120573(2 minus 120583)

ℎ119908

(

119899

sum

119894=1

1198881198941198942 sin 119894120587119911

119897)

(7)

32 The Lateral Restraint of the Steel Web The simplifiedmodel of the steel web is shown as in Figure 4 Two transverseedges are simply supported and the junction of the web andtop flange is fixed The junction of web and bottom flange isfree in the transverse direction The boundary condition ofthe buckling of the steel web is [15 16] given as follows

[119908]119911=0119897 = 0 [119908]119910=minusℎ119908

= 0 [119908119910]119910=0minusℎ119908

= 0

[minus119863119908 (119908119911119911 + 120583119908119910119910)]119911=0119897= 0

(8)

According to the boundary condition the buckling defor-mation function of the steel web can be expressed as

119908 = [1 minus 3(119910

ℎ119908

)

2

minus 2(119910

ℎ119908

)

3

](

119899

sum

119894=1

119889119894 sin119894120587119911

119897) (9)

According to the principle of stationary potential energy[18 20] the buckling characteristic equation is given asfollows

(H0 + 1198961199091R minus 1205900S0)D = 0 (10)

4 Mathematical Problems in Engineering

Longitudinal edge of web

yc

1205901y

z

f (z)

tw

hw

1205901

yc

l

Figure 4 Rectangular plate under compression and lateral stress

where 119877119894119895 = 1198670119894119895 = 0 (119894 = 119895) 119877119894119894 = 1198972 1198670119894119894 = 119897119863119908((6ℎ3

119908) +

(13ℎ11990870)12057321198944+ (65ℎ119908)120573119894

2) 1198780119894119895 = (119905119908ℎ119908119887119894119895119897)((1335) minus

(3ℎ11990835119910119888)) (119894 = 119895) 1198780119894119894 = (119905119908ℎ1199081205872119892119894119897)((1370) minus (3ℎ119908

70119910119888)) 1198961199091 is the lateral restraint stiffness of the web bythe bottom flange and D = 1198891 1198892 119889119899

119879 is generalcoordinates and represents the buckling distortion of thebottom flange

According to the elastic plate theory the lateral dis-tributed bending moment of the web is [21] given as follows

119891120593119909 = minus119863119908 (1205972119908

1205971199102+ 120583

1205972119908

1205971199112)

119891120593119909

10038161003816100381610038161003816119910=0= 119863119908

6

ℎ2119908

(

119899

sum

119894=1

119889119894 sin119894120587119911

119897)

+ 119863119908120583120573(

119899

sum

119894=1

1198891198941198942 sin 119894120587119911

119897)

(11)

33 Restraint Analysis of the Steel Web 1198961205931 and 1198961199091 can bedetermined by the following equations

10038161003816100381610038161003816B + 1198961205931T minus 1205900N

10038161003816100381610038161003816= 0

1003816100381610038161003816H + 1198961199091R minus 1205900S1003816100381610038161003816 = 0

(12)

It can be found from (12) that there is a couplingrelation between external loads and torsionallateral restraintstiffness It indicates that both the torsional and lateralrestraints of the bottom flange are not only determined bythe cross-section features of the composite girder but theyalso depended on the external loads Therefore it may notbe appropriate to take the restraint stiffness as a constantmaterial feature in the traditional elastic foundation beammethod

4 Buckling Analysis of I-Steel-ConcreteComposite Girders

According to the assumptions made upon the bucklingmodel of the I-steel-composite girder can be simplified asthe model depicted in Figure 5 The horizontal and torsionaldirections of the thin plate are restricted by springs while thevertical direction is rigidly restricted

kx

bf

kyk120593

tf

x

y

0

Figure 5 Simplified calculation model of steel-concrete compositegirders

In Figure 5 the thin plate is symmetric about both 119909-axisand 119910-axis The centroid of the plate is set to be the originpoint Assuming the horizontal lateral displacement of thebottom flange is 119906(119911) and the torsional angle is 120593(119911) theneutral equilibrium differential equation of an elastic thin-walled bar under variable axial force can be expressed as [17]

119864119868119910119906119868119881+ [119875 (119906

1015840+ 119910119886120593

1015840)]1015840

+ 119896119909 [119906 minus (119910119889 minus 119910119886) 120593]

minus 119891119909120593 = 0

119864119868119909V119868119881+ [119875 (V1015840 minus 119909119886120593

1015840)]1015840

+ 119896119910 [V + (119909119889 minus 119909119886) 120593] = 0

119864119868119908120593119868119881+ [(119875119903

2

0minus 119866119869) 120593

1015840]1015840

minus 119896119909 [119906 minus (119910119889 minus 119910119886) 120593]

times (119910119889 minus 119910119886) + 119891120593119909 minus 119909119886(119875V1015840)1015840

+ 119910119886(1198751199061015840)1015840

+ 119896119910 [V + (119909119889 minus 119909119886) 120593] (119909119889 minus 119909119886) + 119896120593120593 = 0

(13)

where 119868119910 = 1199051198911198873

11989112 119868119909 = 119887119891119905

3

11989112 119869 = 119887119891119905

3

1198913 11990320= 1199092

119886+

1199102

119886+ (119868119909 + 119868119910)119860 119904 and 119909119886 is center position of the curved

bottom flange in the horizontal axis here 119909119886 = 0 119910119886 is thecenter position of the curved bottom flange in the verticalaxis here 119910119886 = 0 119909119889 is the rotation axis of the bottom flangein the horizontal axis here 119909119889 = 0 119910119889 is the rotation axis ofthe bottom flange in the vertical axis here 119910119889 = 0 119868119908 is thesectorial inertia moment of bottom flange here 119868119908 = 0 119866is shear modulus of the steel 119875 is the pressure of the bottomflange119875 = 1198601198911205900(1198860+1198861120585+1198862120585

2) 119896120593 = minus1198961205931 and 119896119909 = minus1198961199091

Plugging119910119886 = 0 119910119889 = 0 119909119886 = 0 119909119889 = 0 V = 0 and 119868119908 =0 into (13) leads to

(1198751199032

0minus 119866119869) 120593

10158401015840+ 1199032

011987510158401205931015840+ 119896120593120593 + 119891120593119909 = 0

119864119868119910119906119868119881+ 11987510158401199061015840+ 11987511990610158401015840+ 119896119909119906 minus 119891119909120593 = 0

(14)

By combining (5) and (9) the torsional angle and lateraldisplacement of the bottomflange of the composite girder canbe obtained

120593 (119911) =

119899

sum

119894=1

119888119894

ℎ119908

sin 119894120587119911119897

119906 (119911) =

119899

sum

119894=1

119889119894 sin119894120587119911

119897

(15)

Mathematical Problems in Engineering 5

According to the Galerkin method [15 22] we have

(B1 + 119896120593T minus 1205900N1 Q

M H1 + 119896119909R minus 1205900S1)(

CD) = 0 (16)

where 1198611119894119895 = 119876119894119895 = 1198671119894119895 = 119872119894119895 = 0 (119894 = 119895) 1198611119894119894 =119866119869(119894212058722119897ℎ2

119908) 119872119894119894 = (3119863119908119897ℎ

3

119908)minus119863119908((2minus120583)119894

21198971205732ℎ119908) 119876119894119894 =

3(119863119908119897ℎ3

119908)+(119863119908120583119897120573119894

22ℎ119908) 1198731119894119894 = (119860119891119903

2

012058721198921198942119897ℎ

2

119908) 1198731119894119895 =

(1198601198911199032

0119887119894119895119897ℎ2

119908) (119894 = 119895) 1198671119894119894 = 119864119868119910120573

2(11989441198972) 1198781119894119894 = (119860119891120587

2119892119894

2119897) and 1198781119894119895 = (119860119891119887119894119895119897) (119894 = 119895)The combination of (6) (10) and (16) leads to

[(B QM H) minus 1205900 (

N 00 S)](

CD) = 0 (17)

where B = B0 + B1 N = N0 + N1 H = H0 + H1 and S =

S0 + S1The deformation vector C119879D119879119879 cannot be zero when

buckling happens Therefore the buckling of the compositegirder can be solved by the generalized eigenvalue of thecharacteristic matrix shown as follows

10038161003816100381610038161003816100381610038161003816

(B QM H) minus 1205900 (

N 00 S)

10038161003816100381610038161003816100381610038161003816

= 0 (18)

By solving (18) 2119899 generalized eigenvalue can beobtained 120590119905119894 (119894 = 1 2 2119899) let 120590cr = min120590119905119894 (119894 =

1 2 2119899) 120590cr is the critical buckling stress of the compos-ite girder The critical buckling moment of composite girdercan be calculated by the following equation

119872cr =120590cr (1198860 + 1198861120585 + 1198862120585

2) 119868

119910119888

(19)

5 Buckling Analysis of the Steel-ConcreteComposite Box Girder

The dimensions of the composite box girder are shownin Figure 6 According to the assumptions made abovethe buckling model of the composite box girder can besimplified as a thin-plate model that is restricted by springsin horizontal and torsional directions rigidly restricted invertical directionThe simplified model is shown in Figure 7

As the derivation in Section 3 the following can beobtained

(B0 + 1198961205931198971T minus 1205900N0)C119897 = 0 (20)

(B0 + 1198961205931199031T minus 1205900N0)C119903 = 0 (21)

(H0 + 1198961199091R minus 1205900S0)D = 0 (22)

Mx

x y

z

L

Mx hwyctw

tt hc

tfy

bc

x00

Figure 6 Cross-section dimensions of steel-concrete composite boxgirder

k120593l

kxl

ky = infin

y

bf

x

k120593r

kxr

0

ky = infin

Figure 7 Simplified calculation model of steel-concrete compositegirders

119891119909120593119897

10038161003816100381610038161003816119910=0= minus

6119863119908

ℎ3119908

(

119899

sum

119894=1

119888119897119894 sin119894120587119911

119871)

+ 119863119908120573(2 minus 120583)

ℎ119908

(

119899

sum

119894=1

1198881198971198941198942 sin 119894120587119911

119871)

(23)

119891119909120593119903

10038161003816100381610038161003816119910=0= minus

6119863119908

ℎ3119908

(

119899

sum

119894=1

119888119903119894 sin119894120587119911

119897)

+ 119863119908120573(2 minus 120583)

ℎ119908

(

119899

sum

119894=1

1198881199031198941198942 sin 119894120587119911

119897)

(24)

119891120593119909

10038161003816100381610038161003816119910=0=6119863119908

ℎ2119908

(

119899

sum

119894=1

119889119894 sin119894120587119911

119897)

+ 119863119908120583120573(

119899

sum

119894=1

1198891198941198942 sin 119894120587119911

119897)

(25)

where 1198961205931198971 is the torsional restraint stiffness of left web by

the bottom flange C119897 = 1198881198971 1198881198972 119888119897119899119879 is buckling general

coordinates of the left web 1198961205931199031 is torsional restraint stiffness

of the right web by the bottom flange C119903 = 1198881199031 1198881199032 119888119903119899119879

is buckling general coordinates of the right web 1198961199091 is lateralrestraint stiffness of the steel web by the bottom flange andD = 1198891 1198892 119889119899

119879 is buckling general coordinates of thebottom flange

As Figure 7 shows the lateral displacement bucklingfunction of the horizontal buckling of the bottom flange is119906(119911) the out-plane buckling deformation function of the

6 Mathematical Problems in Engineering

bottom flange is V(119909 119911) the left rotational angle is 120593119897(119911) andthe right rotational angle is 120593119903(119911) The boundary condition ofthe bottom flange is given as follows

V 1003816100381610038161003816119909=0 = 0 V11990910038161003816100381610038161003816119909=0

= 120593119897 (119911)

V100381610038161003816100381610038161003816119909=119887119891

= 0 V119909100381610038161003816100381610038161003816119909=119887119891

= 120593119903 (119911)

(26)

According to compatibility of deformation the displace-ment function of the bottom flange is

V =119887119891

2120587sin 120587119909

119887119891

(120593119897 minus 120593119903) +

119887119891

4120587sin 2120587119909

119887119891

(120593119897 + 120593119903) (27)

According to the principle of stationary potential energythe buckling characteristic equations are given as follows

119863119891(51205872120593119897

8119887119891

+31205872120593119903

8119887119891

minus

11988711989112059310158401015840

119897

2+

51198873

1198911205931015840101584010158401015840

119897

321205872minus

31198873

1198911205931015840101584010158401015840

119903

321205872)

+ (

51205901198601198911198872

1198911205931015840

119897

321205872)

1015840

minus (

31205901198601198911198872

1198911205931015840

119903

321205872)

1015840

+ 119896120593119897

120593119897 + 119891120593119909 = 0

2119896119909119906 + (1205901198601198911199061015840)1015840

+ 1198641198681199101199061015840101584010158401015840

minus 119891119909120593119897

minus 119891119909120593119903

= 0

119863119891(51205872120593119903

8119887119891

+31205872120593119897

8119887119891

minus

11988711989112059310158401015840

119903

2+

51198873

1198911205931015840101584010158401015840

119903

321205872minus

31198873

1198911205931015840101584010158401015840

119897

321205872)

+ (

51205901198601198911198872

1198911205931015840

119903

321205872)

1015840

minus (

31205901198601198911198872

1198911205931015840

119897

321205872)

1015840

+ 119896120593119903

120593119895 + 119891120593119909 = 0

(28)

where 119868119910 = 1199051198911198873

11989112 119905119891 is the thickness of bottom flange 119887119891

is the width of the bottom flange119863119891 = 1198641199053

11989112(1 minus 120583

2) 119896120593

119897

=

minus1198961205931198971 is torsional restraint stiffness of the bottom flange edge

by the left steel web 119896120593119903

= minus1198961205931199031 is torsional restraint stiffness

of the bottomflange edge by the right steel web and 119896119909 = minus1198961199091is lateral restraint stiffness of the bottom flange edge by thesteel web

According to the Galerkin method [15 16] we have

(

B1+ 119896120593119897

T minus 1205900N1 F + 06120590

0N1 Q

F + 061205900N1

B1+ 119896120593119903

T minus 1205900N1 Q

M M H1minus 1205900S1+ 2119896119909R)

times 120578 = 0

(29)

where 1198611119894119895 = 119865119894119895 = 119876119894119895 = 119872119894119895 = 1198671119894119895 = 0 (119894 = 119895) 1198611119894119894 =(5119863119891120587

211989716119887119891ℎ

2

119908) + (119863119891119887119891120573119897119894

24ℎ2

119908) + (5119863119891119887

3

11989112057321198971198944641205872ℎ2

119908)

119865119894119894 = (3119863119891120587211989716119887119891ℎ

2

119908) minus (3119863119891119887

3

11989112057321198971198944641205872ℎ2

119908) 119876119894119894 =

3(119863119908119897ℎ3

119908)+(1198971198631199081205831205732ℎ119908)119894

2 1198731119894119895 = (51198601198911198872

11989111988711989411989532120587

2ℎ2

119908119897) (119894 =

119895) 1198731119894119894 = (51198601198911198872

11989111989211989464ℎ

2

119908119897) 119872119894119894 = (3119863119908119897ℎ

3

119908) minus 119863119908((2 minus

120583)11989421198971205732ℎ119908) 1198671119894119894 = (119894

411986411986811991012057321198972) 1198781119894119894 = (119860119891120587

21198921198942119897) 1198781119894119895 =

(119860119891119887119894119895119897) (119894 = 119895) and 120578 = (C119879119897C119879119903D119879)119879

Table 1 Basic geometric size of I-steel composite girder cases

Examplenumber

Cross-sectionnumber

ℎ119908mm 119887

119891mm 119905

119891mm 119905

119908mm 119897mm

1 45002 1 600 200 12 12 72003 96004 30005 2 600 120 10 10 54006 84007 60008 3 719 268 25 16 102009 1440010 300011 4 300 100 3 3 540012 840013 420014 5 450 150 4 4 720015 10200

Since the constraint in the theoretical model is higherthan the real scenario the critical buckling stress is increasedTherefore the theoretical buckling deformation functions ofthe web and bottomflange cannot accurately describe the realbuckling deformation curves In order to eliminate errorsthe paper gives a reduction factor on the torsional restraintstiffness of the bottom flange and the reduction factor isfound to be 05 Combining (20) (21) (22) and (29) leadsto

[[[[

[

(

B F QF B QM M H

) minus 1205900(

N minus3N15

0minus3N15

N 00 0 S

)

]]]]

]

120578 = 0 (30)

where B = B02 + B1 N = N02 + N1 H = 2H0 +H1 and S = 2S0 + S1

The deformation vector 120578 cannot be zero when the buck-ling of the composite girder happensTherefore the bucklingof the composite girder can be solved by the generalizedeigenvalue of the characteristic matrix shown as follows

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(

B F QF B QM M H

)minus 1205900(

N minus06N1 0minus06N1 N 0

0 0 S)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

= 0 (31)

3119899 general eigenvalues can be obtained from (31) whichare 120590119905119894 (119894 = 1 2 3119899) let 120590cr = min120590119905119894 (119894 =

1 2 3119899) 120590cr is the critical buckling stress of the com-posite girderThe following equation can calculate the criticalbuckling moment

119872cr =120590cr (1198860 + 1198861120585 + 1198862120585

2) 119868

119910119888

(32)

Mathematical Problems in Engineering 7

Table 2 Basic geometric size of steel composite box girder cases

Example number ℎ119908mm 119887

119891mm 119887

119905mm 119905

119908mm 119905

119891mm 119905

119905mm 119897mm

1 40002 400 600 100 10 10 10 80003 120004 40005 400 500 100 10 10 10 80006 120007 40008 300 600 100 10 10 10 80009 1200010 400011 400 600 120 10 10 10 800012 1200013 400014 400 600 100 12 12 12 800015 12000

Table 3 Critical distortional buckling moment of I-steel compositegirder under negative uniform moment

Examplenumber

Distortional buckling critical moment119872crkNsdotmANSYS Williams Goltermann Svensson (19)

1 11291 10606 11936 15401 113832 12123 11480 12344 16647 120393 11497 10770 11971 15617 115504 39098 35402 40588 6196 404405 41019 37512 43686 6566 421616 40197 36603 46919 6406 414127 36727 31264 35211 38991 372008 38365 33904 40143 42281 387459 39051 32268 42454 40242 3932710 34155 33710 35235 489 3263211 35235 34884 36464 506 3358712 34729 34304 35951 497 3310613 94611 94325 101817 1368 8971314 107307 107811 102911 1563 1013115 91161 90153 93920 1307 86372

6 Analysis of Examples

Thegeometric dimensions of each example are listed inTables1 and 2 By means of the calculation method introduced inthis paper and the finite elementmethod the critical bucklinganalysis of the composite girder under uniform negativebending moment triangle bending moment and uniformloads can be carried out Svenssonrsquos method Williamsrsquomethod and Goltermannrsquos method are also employed in thecalculation of various I-steel-concrete composite girders soas to validate the calculation method proposed in this paperThe finite element analysis is conducted by using ANSYScommercial software Element SHELL43 is adopted to model

Table 4 Critical distortional buckling moment of I-steel compositegirder under negative triangular moment

Examplenumber

Distortional buckling critical moment119872crkNsdotmANSYS Williams Goltermann Svensson (19)

1 18686 18388 19714 26654 192522 17264 16805 17740 24367 173583 15909 15288 16213 22162 159744 6599 6327 7132 11072 703155 5866 5506 6056 9633 601566 5257 4846 5357 8481 538847 61252 55304 61444 68972 635348 55235 49455 54387 61674 560759 50604 44350 49232 55310 5119110 55183 59346 61236 860 5616711 51574 51935 53366 753 4946912 46366 46184 47412 670 4422413 131762 170748 181683 2476 1336114 129117 147906 150336 2144 1299815 127803 131099 134420 1901 12408

the steel girder The concrete slab of the composite girder isreplaced with constraints in the numerical simulation Themotions in 119909 and 119910 directions of the top flange edge arerestrained to represent the lateral and torsional restrictionscaused by the concrete slab in practice The results of eachsimplified calculation method are listed in Tables 3 4 5 and6 and the error analyses of each simplifiedmethod are shownin Figures 8 9 10 and 11

The following conclusions can be drawn based on theresults presented in Tables 3 to 6 and Figures 8 to 11

(1) Under uniform negative bending moment the crit-ical bending buckling moment in the same cross-section of the composite girder is rarely affected by

8 Mathematical Problems in Engineering

Table 5 Critical distortional buckling moment of I-steel compositegirder under uniformly distributed load

Examplenumber

Distortional buckling critical moment119872crkNsdotmANSYS Williams Goltermann Svensson (19)

3 15512 15394 16388 22321 161606 5272 4988 5508 8729 551109 50383 45093 50001 56234 5235111 497 52610 54675 76289 5031712 459 47925 47925 67500 4453313 1299 198896 204768 288401 1291014 1267 165281 165645 239639 1269515 1255 131099 135351 190107 12454

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1605060708091011121314151617

Example number

WilliamsGoltermann

SvenssonThis paper

Mcr

of an

alyt

ical

met

hods

Mcr

of A

NSY

S

Figure 8 Precision analysis of simplified methods under negativeuniform moment

the length The critical bending buckling moment isnot obviously changed with the increased length ofthe structural component

(2) Under triangle bending moment the critical bendingbuckling moment is greatly affected by the lengththat is the value decreased quickly when the lengthincreases

(3) Under uniform negative bending moment trianglebending moment and uniform loads the resultsyielded by the calculation method in this papermatch well the finite element analysis results Thediscrepancy is limited within 5 which validates theaccuracy and applicability of this method

(4) Traditional calculation methods such as Svens-sonrsquos method Williamsrsquo method and Goltermannrsquosmethod have considerable deviations from the finiteelement method

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Example number

WilliamsGoltermann

SvenssonThis paper

05060708091011121314151617

Mcr

of an

alyt

ical

met

hods

Mcr

of A

NSY

S

Figure 9 Precision analysis of simplified methods under negativetriangular moment

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Example number

05060708091011121314151617

Mcr

of an

alyt

ical

met

hods

Mcr

of A

NSY

S

WilliamsGoltermann

SvenssonThis paper

Figure 10 Precision analysis of simplifiedmethods under uniformlydistributed load

Therefore the traditional elastic foundation beammethod taking into account the moment gradient needs tobe improved It is also suggested that the constant lateral andtorsional restraints in the traditional methods may lead tothe relative deviations

7 Conclusions

In this paper the traditional elastic foundation beammethodsare improved by considering the coupling effect of the exter-nal loads and the foundation spring constraints Based on thisimprovement a simplified calculation method computing thecritical buckling loads of steel-concrete composite girders is

Mathematical Problems in Engineering 9

Table 6 Critical distortional buckling moment of composite box girder under negative moment

Examplenumber

Distortional buckling critical moment119872crkNsdotmUniform negative bending moment Triangle negative bending moment Uniform loadANSYS (32) ANSYS (32) ANSYS (32)

1 72605 72214 79236 81741 79282 786742 72714 72214 76467 78046 76122 755313 72797 72214 75492 76616 75146 742624 90593 90722 97811 10125 97547 974365 90824 90650 94867 97143 94390 939166 90957 90610 93864 95600 93434 927857 53751 53575 58654 60316 58284 580398 53852 53407 56622 57636 55978 556429 53917 53411 55906 56601 55267 5487510 74221 73610 81013 83322 80861 8019511 74333 73610 78178 79555 77685 7678812 74417 73610 77181 78097 76702 7569813 12536 12479 13676 14125 13659 1359514 12556 12479 13201 13486 13114 1301715 12607 12479 13035 13239 12947 12833

090

092

094

096

098

100

102

104

106

108

110

Negative uniform moment Negative triangular momentUniformly distributed load

Mcr

of th

is pa

perM

crof

AN

SYS

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Example number

Figure 11 Precision analysis of simplified methods

developed The method is compared with various traditionalmethods The following conclusions are obtained

(1) There is a linear coupling relation between bothtorsional and lateral restraints and vertical loads

(2) Under uniformnegative bendingmoment the criticalbending buckling moment in the same cross-sectionof the composite girder is rarely affected by thelength The critical bending buckling moment is notobviously changed with the increased length of thestructural component

(3) Under triangle bending moment the critical bendingbuckling is influenced to a great extent by the lengththat is the value decreased quickly when the lengthincreases

(4) Under uniform negative bending moment trianglebending moment and uniform loads the calculationmethod proposed in this paper matches well thefinite element calculation methodThe discrepancy islimited within 5 which validates the applicability ofthis method

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research described in this paper was financially sup-ported by National Natural Science Foundation of China(51408449 and 51378502) and the Fundamental ResearchFunds for the Central Universities of China (2014-IV-049)

References

[1] S Kim andU Lee ldquoEffects of delamination on guidedwaves in asymmetric laminated composite beamrdquoMathematical Problemsin Engineering vol 2014 Article ID 956043 12 pages 2014

[2] D Champenoy A Corfdir and P Corfdir ldquoCalculating thecritical buckling force in compressed bottom flanges of steel-concrete composite bridgesrdquo European Journal of Environmen-tal and Civil Engineering vol 18 no 3 pp 271ndash292 2014

[3] T-H Yi H-N Li and M Gu ldquoOptimal sensor placement forstructural health monitoring based on multiple optimization

10 Mathematical Problems in Engineering

strategiesrdquo The Structural Design of Tall and Special Buildingsvol 20 no 7 pp 881ndash900 2011

[4] T-H Yi H-N Li and X-D Zhang ldquoA modified monkeyalgorithm for optimal sensor placement in structural healthmonitoringrdquo Smart Materials and Structures vol 21 no 10Article ID 105033 2012

[5] S Chen and Y Jia ldquoNumerical investigation of inelasticbuckling of steel-concrete composite beams prestressed withexternal tendonsrdquoThin-Walled Structures vol 48 no 3 pp 233ndash242 2010

[6] S E Svensson ldquoLateral buckling of beams analysed as elasticallysupported columns subject to a varying axial forcerdquo Journal ofConstructional Steel Research vol 5 no 3 pp 179ndash193 1985

[7] F WWilliams and A K Jemah ldquoBuckling curves for elasticallysupported columns with varying axial force to predict lateralbuckling of beamsrdquo Journal of Constructional Steel Research vol7 no 2 pp 133ndash147 1987

[8] P Goltermann and S Svensson ldquoLateral distortional bucklingpredicting elastic critical stressrdquo Journal of Structural Engineer-ing vol 114 no 7 pp 1606ndash1625 1988

[9] Swedish Institute of Steel Construction Swedish Code for Light-GaugeMetal Structures Swedish Institute of Steel ConstructionStockholm Sweden 1982

[10] British Standards Institution Code of Practice for Design of SteelBridge BS5400 Part 3 BSI London UK 1982

[11] RM Lawson andW J RackhamDesign of haunched compositebeams in buildings [MS thesis] Steel Construction InstitutionAscot UK 1989

[12] L Jiang J Qi A Scanlon and L Sun ldquoDistortional andlocal buckling of steel-concrete composite box-beamrdquo Steel andComposite Structures vol 14 no 3 pp 243ndash265 2013

[13] J-H Ye and W Chen ldquoElastic restrained distortional buck-ling of steel-concrete composite beams based on elasticallysupported column methodrdquo International Journal of StructuralStability and Dynamics vol 13 no 1 Article ID 1350001 pp 1ndash29 2013

[14] W Chen and J Ye ldquoElastic lateral and restrained distortionalbuckling of doubly symmetric I-beamsrdquo International Journalof Structural Stability and Dynamics vol 10 no 5 pp 983ndash10162010

[15] W-B Zhou L-Z Jiang G-Q Shao and Z-W Yu ldquoElastic dis-tortional buckling analysis of steel-concrete composite beamsin negative moment regionrdquo Journal of Central South University(Science and Technology) vol 43 no 6 pp 2316ndash2323 2012

[16] W B Zhou L Z Jiang and ZW Yu ldquoThe distortional bucklingcalculation formula of the steel-concrete composite beams inthe negative moment regionrdquo Chinese Journal of ComputationalMechanics vol 29 no 3 pp 446ndash450 2012

[17] J Ye andW Chen ldquoElastic restrained distortional buckling of I-steel-concrete composite beamsrdquo Journal of Building Structuresvol 32 no 6 pp 82ndash91 2011

[18] W Zhou L Jiang Z Liu et al ldquoClosed-form solution for shearlag effects of steel-concrete composite box beams consideringshear deformation and sliprdquo Journal of Central South Universityvol 19 no 10 pp 2976ndash2982 2012

[19] W-B Zhou L-Z Jiang andZ-W Yu ldquoAnalysis of free vibrationcharacteristic of steel-concrete composite box-girder consider-ing shear lag and sliprdquo Journal of Central South University vol20 no 9 pp 2570ndash2577 2013

[20] W-B Zhou L-Z Jiang Z-J Liu et al ldquoClosed-form solutionto thin-walled box girders considering effects of shear deforma-tion and shear lagrdquo Journal of Central South University vol 19no 9 pp 2650ndash2655 2012

[21] A Teodor A Guran and G Ardbeshir Theory of Elasticity forScientists and Engineers Springer New York NY USA 2012

[22] C Bi and V Ginting ldquoTwo-grid discontinuous Galerkinmethod for quasi-linear elliptic problemsrdquo Journal of ScientificComputing vol 49 no 3 pp 311ndash331 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

Original shapeShape after deformation

Figure 2 Distortional buckling of steel-concrete composite girdersunder negative moments

Consider

119910119888 =119860 119904119910119904 + 119860 119905ℎ119908 + 05119860119908ℎ119908

119860 119904 + 119860 119905 + 119860119908 + 119860119891

(3)

where 119860119891 is the area of the bottom flange 119860 119905 is the area ofthe top flange119860119908 is the area of the steel web119860 119904 is the area ofreinforcements within concrete slab and 119910119904 is the distance ofthe center position of the equivalent cross-section to the edgeof steel flange

31 The Torsional Restraint of the Steel Web The simplifiedmodel of the steel web is shown as in Figure 3 Two transverseedges are simply supported The junction of the web and topflange is fixed while the junction of web and bottom flange issimply supportedThe boundary condition of the buckling ofthe steel web is [15 16] given as follows

[119908]119911=0119897 = 0 [119908]119910=0minusℎ119908

= 0 [119908119910]119910=minusℎ119908

= 0

[minus119863119908 (119908119911119911 + 120583119908119910119910)]119911=0119897= 0

(4)

where 119863119908 = 1198641199053

11990812(1 minus 120583

2) 120583 is Poissonrsquos ratio of steel

119864 is the elasticity modulus of steel 119908(119910 119911) is the bucklingdeformation function of web 119905119908 is the thickness of the steelgirder web and ℎ119908 is the height of the steel girder web

Based on the boundary conditions the buckling deforma-tion function of the steel web is

119908 = [119910

ℎ119908

+ 2(119910

ℎ119908

)

2

+ (119910

ℎ119908

)

3

](

119899

sum

119894=1

119888119894 sin119894120587119911

119897) (5)

According to the principle of stationary potential energy[18ndash20] the buckling characteristic equation is given as fol-lows

(B0 + 1198961205931T minus 1205900N0)C = 0 (6)

Longitudinal edge of web

yc

1205901y

z

m(z)

tw

hw

1205901

yc

l

Figure 3 Rectangular plate under compression and moments

where1198610119894119894 = 119897119863119908((2ℎ3

119908)+(ℎ119908119894

41205732210)+(2119894

212057315ℎ119908)) 119879119894119894 =

1198972ℎ2

119908 1198610119894119895 = 119879119894119895 = 0 (119894 = 119895) 1198730119894119894 = (120587

2119892119894119897)((119905119908ℎ119908210) minus

(119905119908ℎ2

119908560119910119888)) 120573 = 120587

21198972 1198730119894119895 = (119905119908ℎ119908119887119894119895119897)((1105) minus (ℎ119908

280119910119888)) (119894 = 119895) 119887119894119895 = [(minus1)119894+119895

minus 1]1198861 + 2(minus1)119894+1198951198862(119894119895(119894

2+

1198952)(1198942minus 1198952)2

) 119892119894 = 1198942[1198860+(11988612)+1198862((13)+(12119894

21205872))] 1198961205931

is the lateral restraint stiffness of theweb by the bottomflangeand C = 1198881 1198882 119888119899

119879 is general coordinates and representsthe buckling distortion

According to the elastic plate theory the lateral distribu-tion force of the web is given as follows [21]

119891119909120593 = minus119863119908 [1205973119908

1205971199103+ (2 minus 120583)

1205973119908

1205971199112120597119910]

119891119909120593

10038161003816100381610038161003816119910=0= minus

6119863119908

ℎ3119908

(

119899

sum

119894=1

119888119894 sin119894120587119911

119897)

+ 119863119908120573(2 minus 120583)

ℎ119908

(

119899

sum

119894=1

1198881198941198942 sin 119894120587119911

119897)

(7)

32 The Lateral Restraint of the Steel Web The simplifiedmodel of the steel web is shown as in Figure 4 Two transverseedges are simply supported and the junction of the web andtop flange is fixed The junction of web and bottom flange isfree in the transverse direction The boundary condition ofthe buckling of the steel web is [15 16] given as follows

[119908]119911=0119897 = 0 [119908]119910=minusℎ119908

= 0 [119908119910]119910=0minusℎ119908

= 0

[minus119863119908 (119908119911119911 + 120583119908119910119910)]119911=0119897= 0

(8)

According to the boundary condition the buckling defor-mation function of the steel web can be expressed as

119908 = [1 minus 3(119910

ℎ119908

)

2

minus 2(119910

ℎ119908

)

3

](

119899

sum

119894=1

119889119894 sin119894120587119911

119897) (9)

According to the principle of stationary potential energy[18 20] the buckling characteristic equation is given asfollows

(H0 + 1198961199091R minus 1205900S0)D = 0 (10)

4 Mathematical Problems in Engineering

Longitudinal edge of web

yc

1205901y

z

f (z)

tw

hw

1205901

yc

l

Figure 4 Rectangular plate under compression and lateral stress

where 119877119894119895 = 1198670119894119895 = 0 (119894 = 119895) 119877119894119894 = 1198972 1198670119894119894 = 119897119863119908((6ℎ3

119908) +

(13ℎ11990870)12057321198944+ (65ℎ119908)120573119894

2) 1198780119894119895 = (119905119908ℎ119908119887119894119895119897)((1335) minus

(3ℎ11990835119910119888)) (119894 = 119895) 1198780119894119894 = (119905119908ℎ1199081205872119892119894119897)((1370) minus (3ℎ119908

70119910119888)) 1198961199091 is the lateral restraint stiffness of the web bythe bottom flange and D = 1198891 1198892 119889119899

119879 is generalcoordinates and represents the buckling distortion of thebottom flange

According to the elastic plate theory the lateral dis-tributed bending moment of the web is [21] given as follows

119891120593119909 = minus119863119908 (1205972119908

1205971199102+ 120583

1205972119908

1205971199112)

119891120593119909

10038161003816100381610038161003816119910=0= 119863119908

6

ℎ2119908

(

119899

sum

119894=1

119889119894 sin119894120587119911

119897)

+ 119863119908120583120573(

119899

sum

119894=1

1198891198941198942 sin 119894120587119911

119897)

(11)

33 Restraint Analysis of the Steel Web 1198961205931 and 1198961199091 can bedetermined by the following equations

10038161003816100381610038161003816B + 1198961205931T minus 1205900N

10038161003816100381610038161003816= 0

1003816100381610038161003816H + 1198961199091R minus 1205900S1003816100381610038161003816 = 0

(12)

It can be found from (12) that there is a couplingrelation between external loads and torsionallateral restraintstiffness It indicates that both the torsional and lateralrestraints of the bottom flange are not only determined bythe cross-section features of the composite girder but theyalso depended on the external loads Therefore it may notbe appropriate to take the restraint stiffness as a constantmaterial feature in the traditional elastic foundation beammethod

4 Buckling Analysis of I-Steel-ConcreteComposite Girders

According to the assumptions made upon the bucklingmodel of the I-steel-composite girder can be simplified asthe model depicted in Figure 5 The horizontal and torsionaldirections of the thin plate are restricted by springs while thevertical direction is rigidly restricted

kx

bf

kyk120593

tf

x

y

0

Figure 5 Simplified calculation model of steel-concrete compositegirders

In Figure 5 the thin plate is symmetric about both 119909-axisand 119910-axis The centroid of the plate is set to be the originpoint Assuming the horizontal lateral displacement of thebottom flange is 119906(119911) and the torsional angle is 120593(119911) theneutral equilibrium differential equation of an elastic thin-walled bar under variable axial force can be expressed as [17]

119864119868119910119906119868119881+ [119875 (119906

1015840+ 119910119886120593

1015840)]1015840

+ 119896119909 [119906 minus (119910119889 minus 119910119886) 120593]

minus 119891119909120593 = 0

119864119868119909V119868119881+ [119875 (V1015840 minus 119909119886120593

1015840)]1015840

+ 119896119910 [V + (119909119889 minus 119909119886) 120593] = 0

119864119868119908120593119868119881+ [(119875119903

2

0minus 119866119869) 120593

1015840]1015840

minus 119896119909 [119906 minus (119910119889 minus 119910119886) 120593]

times (119910119889 minus 119910119886) + 119891120593119909 minus 119909119886(119875V1015840)1015840

+ 119910119886(1198751199061015840)1015840

+ 119896119910 [V + (119909119889 minus 119909119886) 120593] (119909119889 minus 119909119886) + 119896120593120593 = 0

(13)

where 119868119910 = 1199051198911198873

11989112 119868119909 = 119887119891119905

3

11989112 119869 = 119887119891119905

3

1198913 11990320= 1199092

119886+

1199102

119886+ (119868119909 + 119868119910)119860 119904 and 119909119886 is center position of the curved

bottom flange in the horizontal axis here 119909119886 = 0 119910119886 is thecenter position of the curved bottom flange in the verticalaxis here 119910119886 = 0 119909119889 is the rotation axis of the bottom flangein the horizontal axis here 119909119889 = 0 119910119889 is the rotation axis ofthe bottom flange in the vertical axis here 119910119889 = 0 119868119908 is thesectorial inertia moment of bottom flange here 119868119908 = 0 119866is shear modulus of the steel 119875 is the pressure of the bottomflange119875 = 1198601198911205900(1198860+1198861120585+1198862120585

2) 119896120593 = minus1198961205931 and 119896119909 = minus1198961199091

Plugging119910119886 = 0 119910119889 = 0 119909119886 = 0 119909119889 = 0 V = 0 and 119868119908 =0 into (13) leads to

(1198751199032

0minus 119866119869) 120593

10158401015840+ 1199032

011987510158401205931015840+ 119896120593120593 + 119891120593119909 = 0

119864119868119910119906119868119881+ 11987510158401199061015840+ 11987511990610158401015840+ 119896119909119906 minus 119891119909120593 = 0

(14)

By combining (5) and (9) the torsional angle and lateraldisplacement of the bottomflange of the composite girder canbe obtained

120593 (119911) =

119899

sum

119894=1

119888119894

ℎ119908

sin 119894120587119911119897

119906 (119911) =

119899

sum

119894=1

119889119894 sin119894120587119911

119897

(15)

Mathematical Problems in Engineering 5

According to the Galerkin method [15 22] we have

(B1 + 119896120593T minus 1205900N1 Q

M H1 + 119896119909R minus 1205900S1)(

CD) = 0 (16)

where 1198611119894119895 = 119876119894119895 = 1198671119894119895 = 119872119894119895 = 0 (119894 = 119895) 1198611119894119894 =119866119869(119894212058722119897ℎ2

119908) 119872119894119894 = (3119863119908119897ℎ

3

119908)minus119863119908((2minus120583)119894

21198971205732ℎ119908) 119876119894119894 =

3(119863119908119897ℎ3

119908)+(119863119908120583119897120573119894

22ℎ119908) 1198731119894119894 = (119860119891119903

2

012058721198921198942119897ℎ

2

119908) 1198731119894119895 =

(1198601198911199032

0119887119894119895119897ℎ2

119908) (119894 = 119895) 1198671119894119894 = 119864119868119910120573

2(11989441198972) 1198781119894119894 = (119860119891120587

2119892119894

2119897) and 1198781119894119895 = (119860119891119887119894119895119897) (119894 = 119895)The combination of (6) (10) and (16) leads to

[(B QM H) minus 1205900 (

N 00 S)](

CD) = 0 (17)

where B = B0 + B1 N = N0 + N1 H = H0 + H1 and S =

S0 + S1The deformation vector C119879D119879119879 cannot be zero when

buckling happens Therefore the buckling of the compositegirder can be solved by the generalized eigenvalue of thecharacteristic matrix shown as follows

10038161003816100381610038161003816100381610038161003816

(B QM H) minus 1205900 (

N 00 S)

10038161003816100381610038161003816100381610038161003816

= 0 (18)

By solving (18) 2119899 generalized eigenvalue can beobtained 120590119905119894 (119894 = 1 2 2119899) let 120590cr = min120590119905119894 (119894 =

1 2 2119899) 120590cr is the critical buckling stress of the compos-ite girder The critical buckling moment of composite girdercan be calculated by the following equation

119872cr =120590cr (1198860 + 1198861120585 + 1198862120585

2) 119868

119910119888

(19)

5 Buckling Analysis of the Steel-ConcreteComposite Box Girder

The dimensions of the composite box girder are shownin Figure 6 According to the assumptions made abovethe buckling model of the composite box girder can besimplified as a thin-plate model that is restricted by springsin horizontal and torsional directions rigidly restricted invertical directionThe simplified model is shown in Figure 7

As the derivation in Section 3 the following can beobtained

(B0 + 1198961205931198971T minus 1205900N0)C119897 = 0 (20)

(B0 + 1198961205931199031T minus 1205900N0)C119903 = 0 (21)

(H0 + 1198961199091R minus 1205900S0)D = 0 (22)

Mx

x y

z

L

Mx hwyctw

tt hc

tfy

bc

x00

Figure 6 Cross-section dimensions of steel-concrete composite boxgirder

k120593l

kxl

ky = infin

y

bf

x

k120593r

kxr

0

ky = infin

Figure 7 Simplified calculation model of steel-concrete compositegirders

119891119909120593119897

10038161003816100381610038161003816119910=0= minus

6119863119908

ℎ3119908

(

119899

sum

119894=1

119888119897119894 sin119894120587119911

119871)

+ 119863119908120573(2 minus 120583)

ℎ119908

(

119899

sum

119894=1

1198881198971198941198942 sin 119894120587119911

119871)

(23)

119891119909120593119903

10038161003816100381610038161003816119910=0= minus

6119863119908

ℎ3119908

(

119899

sum

119894=1

119888119903119894 sin119894120587119911

119897)

+ 119863119908120573(2 minus 120583)

ℎ119908

(

119899

sum

119894=1

1198881199031198941198942 sin 119894120587119911

119897)

(24)

119891120593119909

10038161003816100381610038161003816119910=0=6119863119908

ℎ2119908

(

119899

sum

119894=1

119889119894 sin119894120587119911

119897)

+ 119863119908120583120573(

119899

sum

119894=1

1198891198941198942 sin 119894120587119911

119897)

(25)

where 1198961205931198971 is the torsional restraint stiffness of left web by

the bottom flange C119897 = 1198881198971 1198881198972 119888119897119899119879 is buckling general

coordinates of the left web 1198961205931199031 is torsional restraint stiffness

of the right web by the bottom flange C119903 = 1198881199031 1198881199032 119888119903119899119879

is buckling general coordinates of the right web 1198961199091 is lateralrestraint stiffness of the steel web by the bottom flange andD = 1198891 1198892 119889119899

119879 is buckling general coordinates of thebottom flange

As Figure 7 shows the lateral displacement bucklingfunction of the horizontal buckling of the bottom flange is119906(119911) the out-plane buckling deformation function of the

6 Mathematical Problems in Engineering

bottom flange is V(119909 119911) the left rotational angle is 120593119897(119911) andthe right rotational angle is 120593119903(119911) The boundary condition ofthe bottom flange is given as follows

V 1003816100381610038161003816119909=0 = 0 V11990910038161003816100381610038161003816119909=0

= 120593119897 (119911)

V100381610038161003816100381610038161003816119909=119887119891

= 0 V119909100381610038161003816100381610038161003816119909=119887119891

= 120593119903 (119911)

(26)

According to compatibility of deformation the displace-ment function of the bottom flange is

V =119887119891

2120587sin 120587119909

119887119891

(120593119897 minus 120593119903) +

119887119891

4120587sin 2120587119909

119887119891

(120593119897 + 120593119903) (27)

According to the principle of stationary potential energythe buckling characteristic equations are given as follows

119863119891(51205872120593119897

8119887119891

+31205872120593119903

8119887119891

minus

11988711989112059310158401015840

119897

2+

51198873

1198911205931015840101584010158401015840

119897

321205872minus

31198873

1198911205931015840101584010158401015840

119903

321205872)

+ (

51205901198601198911198872

1198911205931015840

119897

321205872)

1015840

minus (

31205901198601198911198872

1198911205931015840

119903

321205872)

1015840

+ 119896120593119897

120593119897 + 119891120593119909 = 0

2119896119909119906 + (1205901198601198911199061015840)1015840

+ 1198641198681199101199061015840101584010158401015840

minus 119891119909120593119897

minus 119891119909120593119903

= 0

119863119891(51205872120593119903

8119887119891

+31205872120593119897

8119887119891

minus

11988711989112059310158401015840

119903

2+

51198873

1198911205931015840101584010158401015840

119903

321205872minus

31198873

1198911205931015840101584010158401015840

119897

321205872)

+ (

51205901198601198911198872

1198911205931015840

119903

321205872)

1015840

minus (

31205901198601198911198872

1198911205931015840

119897

321205872)

1015840

+ 119896120593119903

120593119895 + 119891120593119909 = 0

(28)

where 119868119910 = 1199051198911198873

11989112 119905119891 is the thickness of bottom flange 119887119891

is the width of the bottom flange119863119891 = 1198641199053

11989112(1 minus 120583

2) 119896120593

119897

=

minus1198961205931198971 is torsional restraint stiffness of the bottom flange edge

by the left steel web 119896120593119903

= minus1198961205931199031 is torsional restraint stiffness

of the bottomflange edge by the right steel web and 119896119909 = minus1198961199091is lateral restraint stiffness of the bottom flange edge by thesteel web

According to the Galerkin method [15 16] we have

(

B1+ 119896120593119897

T minus 1205900N1 F + 06120590

0N1 Q

F + 061205900N1

B1+ 119896120593119903

T minus 1205900N1 Q

M M H1minus 1205900S1+ 2119896119909R)

times 120578 = 0

(29)

where 1198611119894119895 = 119865119894119895 = 119876119894119895 = 119872119894119895 = 1198671119894119895 = 0 (119894 = 119895) 1198611119894119894 =(5119863119891120587

211989716119887119891ℎ

2

119908) + (119863119891119887119891120573119897119894

24ℎ2

119908) + (5119863119891119887

3

11989112057321198971198944641205872ℎ2

119908)

119865119894119894 = (3119863119891120587211989716119887119891ℎ

2

119908) minus (3119863119891119887

3

11989112057321198971198944641205872ℎ2

119908) 119876119894119894 =

3(119863119908119897ℎ3

119908)+(1198971198631199081205831205732ℎ119908)119894

2 1198731119894119895 = (51198601198911198872

11989111988711989411989532120587

2ℎ2

119908119897) (119894 =

119895) 1198731119894119894 = (51198601198911198872

11989111989211989464ℎ

2

119908119897) 119872119894119894 = (3119863119908119897ℎ

3

119908) minus 119863119908((2 minus

120583)11989421198971205732ℎ119908) 1198671119894119894 = (119894

411986411986811991012057321198972) 1198781119894119894 = (119860119891120587

21198921198942119897) 1198781119894119895 =

(119860119891119887119894119895119897) (119894 = 119895) and 120578 = (C119879119897C119879119903D119879)119879

Table 1 Basic geometric size of I-steel composite girder cases

Examplenumber

Cross-sectionnumber

ℎ119908mm 119887

119891mm 119905

119891mm 119905

119908mm 119897mm

1 45002 1 600 200 12 12 72003 96004 30005 2 600 120 10 10 54006 84007 60008 3 719 268 25 16 102009 1440010 300011 4 300 100 3 3 540012 840013 420014 5 450 150 4 4 720015 10200

Since the constraint in the theoretical model is higherthan the real scenario the critical buckling stress is increasedTherefore the theoretical buckling deformation functions ofthe web and bottomflange cannot accurately describe the realbuckling deformation curves In order to eliminate errorsthe paper gives a reduction factor on the torsional restraintstiffness of the bottom flange and the reduction factor isfound to be 05 Combining (20) (21) (22) and (29) leadsto

[[[[

[

(

B F QF B QM M H

) minus 1205900(

N minus3N15

0minus3N15

N 00 0 S

)

]]]]

]

120578 = 0 (30)

where B = B02 + B1 N = N02 + N1 H = 2H0 +H1 and S = 2S0 + S1

The deformation vector 120578 cannot be zero when the buck-ling of the composite girder happensTherefore the bucklingof the composite girder can be solved by the generalizedeigenvalue of the characteristic matrix shown as follows

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(

B F QF B QM M H

)minus 1205900(

N minus06N1 0minus06N1 N 0

0 0 S)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

= 0 (31)

3119899 general eigenvalues can be obtained from (31) whichare 120590119905119894 (119894 = 1 2 3119899) let 120590cr = min120590119905119894 (119894 =

1 2 3119899) 120590cr is the critical buckling stress of the com-posite girderThe following equation can calculate the criticalbuckling moment

119872cr =120590cr (1198860 + 1198861120585 + 1198862120585

2) 119868

119910119888

(32)

Mathematical Problems in Engineering 7

Table 2 Basic geometric size of steel composite box girder cases

Example number ℎ119908mm 119887

119891mm 119887

119905mm 119905

119908mm 119905

119891mm 119905

119905mm 119897mm

1 40002 400 600 100 10 10 10 80003 120004 40005 400 500 100 10 10 10 80006 120007 40008 300 600 100 10 10 10 80009 1200010 400011 400 600 120 10 10 10 800012 1200013 400014 400 600 100 12 12 12 800015 12000

Table 3 Critical distortional buckling moment of I-steel compositegirder under negative uniform moment

Examplenumber

Distortional buckling critical moment119872crkNsdotmANSYS Williams Goltermann Svensson (19)

1 11291 10606 11936 15401 113832 12123 11480 12344 16647 120393 11497 10770 11971 15617 115504 39098 35402 40588 6196 404405 41019 37512 43686 6566 421616 40197 36603 46919 6406 414127 36727 31264 35211 38991 372008 38365 33904 40143 42281 387459 39051 32268 42454 40242 3932710 34155 33710 35235 489 3263211 35235 34884 36464 506 3358712 34729 34304 35951 497 3310613 94611 94325 101817 1368 8971314 107307 107811 102911 1563 1013115 91161 90153 93920 1307 86372

6 Analysis of Examples

Thegeometric dimensions of each example are listed inTables1 and 2 By means of the calculation method introduced inthis paper and the finite elementmethod the critical bucklinganalysis of the composite girder under uniform negativebending moment triangle bending moment and uniformloads can be carried out Svenssonrsquos method Williamsrsquomethod and Goltermannrsquos method are also employed in thecalculation of various I-steel-concrete composite girders soas to validate the calculation method proposed in this paperThe finite element analysis is conducted by using ANSYScommercial software Element SHELL43 is adopted to model

Table 4 Critical distortional buckling moment of I-steel compositegirder under negative triangular moment

Examplenumber

Distortional buckling critical moment119872crkNsdotmANSYS Williams Goltermann Svensson (19)

1 18686 18388 19714 26654 192522 17264 16805 17740 24367 173583 15909 15288 16213 22162 159744 6599 6327 7132 11072 703155 5866 5506 6056 9633 601566 5257 4846 5357 8481 538847 61252 55304 61444 68972 635348 55235 49455 54387 61674 560759 50604 44350 49232 55310 5119110 55183 59346 61236 860 5616711 51574 51935 53366 753 4946912 46366 46184 47412 670 4422413 131762 170748 181683 2476 1336114 129117 147906 150336 2144 1299815 127803 131099 134420 1901 12408

the steel girder The concrete slab of the composite girder isreplaced with constraints in the numerical simulation Themotions in 119909 and 119910 directions of the top flange edge arerestrained to represent the lateral and torsional restrictionscaused by the concrete slab in practice The results of eachsimplified calculation method are listed in Tables 3 4 5 and6 and the error analyses of each simplifiedmethod are shownin Figures 8 9 10 and 11

The following conclusions can be drawn based on theresults presented in Tables 3 to 6 and Figures 8 to 11

(1) Under uniform negative bending moment the crit-ical bending buckling moment in the same cross-section of the composite girder is rarely affected by

8 Mathematical Problems in Engineering

Table 5 Critical distortional buckling moment of I-steel compositegirder under uniformly distributed load

Examplenumber

Distortional buckling critical moment119872crkNsdotmANSYS Williams Goltermann Svensson (19)

3 15512 15394 16388 22321 161606 5272 4988 5508 8729 551109 50383 45093 50001 56234 5235111 497 52610 54675 76289 5031712 459 47925 47925 67500 4453313 1299 198896 204768 288401 1291014 1267 165281 165645 239639 1269515 1255 131099 135351 190107 12454

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1605060708091011121314151617

Example number

WilliamsGoltermann

SvenssonThis paper

Mcr

of an

alyt

ical

met

hods

Mcr

of A

NSY

S

Figure 8 Precision analysis of simplified methods under negativeuniform moment

the length The critical bending buckling moment isnot obviously changed with the increased length ofthe structural component

(2) Under triangle bending moment the critical bendingbuckling moment is greatly affected by the lengththat is the value decreased quickly when the lengthincreases

(3) Under uniform negative bending moment trianglebending moment and uniform loads the resultsyielded by the calculation method in this papermatch well the finite element analysis results Thediscrepancy is limited within 5 which validates theaccuracy and applicability of this method

(4) Traditional calculation methods such as Svens-sonrsquos method Williamsrsquo method and Goltermannrsquosmethod have considerable deviations from the finiteelement method

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Example number

WilliamsGoltermann

SvenssonThis paper

05060708091011121314151617

Mcr

of an

alyt

ical

met

hods

Mcr

of A

NSY

S

Figure 9 Precision analysis of simplified methods under negativetriangular moment

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Example number

05060708091011121314151617

Mcr

of an

alyt

ical

met

hods

Mcr

of A

NSY

S

WilliamsGoltermann

SvenssonThis paper

Figure 10 Precision analysis of simplifiedmethods under uniformlydistributed load

Therefore the traditional elastic foundation beammethod taking into account the moment gradient needs tobe improved It is also suggested that the constant lateral andtorsional restraints in the traditional methods may lead tothe relative deviations

7 Conclusions

In this paper the traditional elastic foundation beammethodsare improved by considering the coupling effect of the exter-nal loads and the foundation spring constraints Based on thisimprovement a simplified calculation method computing thecritical buckling loads of steel-concrete composite girders is

Mathematical Problems in Engineering 9

Table 6 Critical distortional buckling moment of composite box girder under negative moment

Examplenumber

Distortional buckling critical moment119872crkNsdotmUniform negative bending moment Triangle negative bending moment Uniform loadANSYS (32) ANSYS (32) ANSYS (32)

1 72605 72214 79236 81741 79282 786742 72714 72214 76467 78046 76122 755313 72797 72214 75492 76616 75146 742624 90593 90722 97811 10125 97547 974365 90824 90650 94867 97143 94390 939166 90957 90610 93864 95600 93434 927857 53751 53575 58654 60316 58284 580398 53852 53407 56622 57636 55978 556429 53917 53411 55906 56601 55267 5487510 74221 73610 81013 83322 80861 8019511 74333 73610 78178 79555 77685 7678812 74417 73610 77181 78097 76702 7569813 12536 12479 13676 14125 13659 1359514 12556 12479 13201 13486 13114 1301715 12607 12479 13035 13239 12947 12833

090

092

094

096

098

100

102

104

106

108

110

Negative uniform moment Negative triangular momentUniformly distributed load

Mcr

of th

is pa

perM

crof

AN

SYS

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Example number

Figure 11 Precision analysis of simplified methods

developed The method is compared with various traditionalmethods The following conclusions are obtained

(1) There is a linear coupling relation between bothtorsional and lateral restraints and vertical loads

(2) Under uniformnegative bendingmoment the criticalbending buckling moment in the same cross-sectionof the composite girder is rarely affected by thelength The critical bending buckling moment is notobviously changed with the increased length of thestructural component

(3) Under triangle bending moment the critical bendingbuckling is influenced to a great extent by the lengththat is the value decreased quickly when the lengthincreases

(4) Under uniform negative bending moment trianglebending moment and uniform loads the calculationmethod proposed in this paper matches well thefinite element calculation methodThe discrepancy islimited within 5 which validates the applicability ofthis method

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research described in this paper was financially sup-ported by National Natural Science Foundation of China(51408449 and 51378502) and the Fundamental ResearchFunds for the Central Universities of China (2014-IV-049)

References

[1] S Kim andU Lee ldquoEffects of delamination on guidedwaves in asymmetric laminated composite beamrdquoMathematical Problemsin Engineering vol 2014 Article ID 956043 12 pages 2014

[2] D Champenoy A Corfdir and P Corfdir ldquoCalculating thecritical buckling force in compressed bottom flanges of steel-concrete composite bridgesrdquo European Journal of Environmen-tal and Civil Engineering vol 18 no 3 pp 271ndash292 2014

[3] T-H Yi H-N Li and M Gu ldquoOptimal sensor placement forstructural health monitoring based on multiple optimization

10 Mathematical Problems in Engineering

strategiesrdquo The Structural Design of Tall and Special Buildingsvol 20 no 7 pp 881ndash900 2011

[4] T-H Yi H-N Li and X-D Zhang ldquoA modified monkeyalgorithm for optimal sensor placement in structural healthmonitoringrdquo Smart Materials and Structures vol 21 no 10Article ID 105033 2012

[5] S Chen and Y Jia ldquoNumerical investigation of inelasticbuckling of steel-concrete composite beams prestressed withexternal tendonsrdquoThin-Walled Structures vol 48 no 3 pp 233ndash242 2010

[6] S E Svensson ldquoLateral buckling of beams analysed as elasticallysupported columns subject to a varying axial forcerdquo Journal ofConstructional Steel Research vol 5 no 3 pp 179ndash193 1985

[7] F WWilliams and A K Jemah ldquoBuckling curves for elasticallysupported columns with varying axial force to predict lateralbuckling of beamsrdquo Journal of Constructional Steel Research vol7 no 2 pp 133ndash147 1987

[8] P Goltermann and S Svensson ldquoLateral distortional bucklingpredicting elastic critical stressrdquo Journal of Structural Engineer-ing vol 114 no 7 pp 1606ndash1625 1988

[9] Swedish Institute of Steel Construction Swedish Code for Light-GaugeMetal Structures Swedish Institute of Steel ConstructionStockholm Sweden 1982

[10] British Standards Institution Code of Practice for Design of SteelBridge BS5400 Part 3 BSI London UK 1982

[11] RM Lawson andW J RackhamDesign of haunched compositebeams in buildings [MS thesis] Steel Construction InstitutionAscot UK 1989

[12] L Jiang J Qi A Scanlon and L Sun ldquoDistortional andlocal buckling of steel-concrete composite box-beamrdquo Steel andComposite Structures vol 14 no 3 pp 243ndash265 2013

[13] J-H Ye and W Chen ldquoElastic restrained distortional buck-ling of steel-concrete composite beams based on elasticallysupported column methodrdquo International Journal of StructuralStability and Dynamics vol 13 no 1 Article ID 1350001 pp 1ndash29 2013

[14] W Chen and J Ye ldquoElastic lateral and restrained distortionalbuckling of doubly symmetric I-beamsrdquo International Journalof Structural Stability and Dynamics vol 10 no 5 pp 983ndash10162010

[15] W-B Zhou L-Z Jiang G-Q Shao and Z-W Yu ldquoElastic dis-tortional buckling analysis of steel-concrete composite beamsin negative moment regionrdquo Journal of Central South University(Science and Technology) vol 43 no 6 pp 2316ndash2323 2012

[16] W B Zhou L Z Jiang and ZW Yu ldquoThe distortional bucklingcalculation formula of the steel-concrete composite beams inthe negative moment regionrdquo Chinese Journal of ComputationalMechanics vol 29 no 3 pp 446ndash450 2012

[17] J Ye andW Chen ldquoElastic restrained distortional buckling of I-steel-concrete composite beamsrdquo Journal of Building Structuresvol 32 no 6 pp 82ndash91 2011

[18] W Zhou L Jiang Z Liu et al ldquoClosed-form solution for shearlag effects of steel-concrete composite box beams consideringshear deformation and sliprdquo Journal of Central South Universityvol 19 no 10 pp 2976ndash2982 2012

[19] W-B Zhou L-Z Jiang andZ-W Yu ldquoAnalysis of free vibrationcharacteristic of steel-concrete composite box-girder consider-ing shear lag and sliprdquo Journal of Central South University vol20 no 9 pp 2570ndash2577 2013

[20] W-B Zhou L-Z Jiang Z-J Liu et al ldquoClosed-form solutionto thin-walled box girders considering effects of shear deforma-tion and shear lagrdquo Journal of Central South University vol 19no 9 pp 2650ndash2655 2012

[21] A Teodor A Guran and G Ardbeshir Theory of Elasticity forScientists and Engineers Springer New York NY USA 2012

[22] C Bi and V Ginting ldquoTwo-grid discontinuous Galerkinmethod for quasi-linear elliptic problemsrdquo Journal of ScientificComputing vol 49 no 3 pp 311ndash331 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

Longitudinal edge of web

yc

1205901y

z

f (z)

tw

hw

1205901

yc

l

Figure 4 Rectangular plate under compression and lateral stress

where 119877119894119895 = 1198670119894119895 = 0 (119894 = 119895) 119877119894119894 = 1198972 1198670119894119894 = 119897119863119908((6ℎ3

119908) +

(13ℎ11990870)12057321198944+ (65ℎ119908)120573119894

2) 1198780119894119895 = (119905119908ℎ119908119887119894119895119897)((1335) minus

(3ℎ11990835119910119888)) (119894 = 119895) 1198780119894119894 = (119905119908ℎ1199081205872119892119894119897)((1370) minus (3ℎ119908

70119910119888)) 1198961199091 is the lateral restraint stiffness of the web bythe bottom flange and D = 1198891 1198892 119889119899

119879 is generalcoordinates and represents the buckling distortion of thebottom flange

According to the elastic plate theory the lateral dis-tributed bending moment of the web is [21] given as follows

119891120593119909 = minus119863119908 (1205972119908

1205971199102+ 120583

1205972119908

1205971199112)

119891120593119909

10038161003816100381610038161003816119910=0= 119863119908

6

ℎ2119908

(

119899

sum

119894=1

119889119894 sin119894120587119911

119897)

+ 119863119908120583120573(

119899

sum

119894=1

1198891198941198942 sin 119894120587119911

119897)

(11)

33 Restraint Analysis of the Steel Web 1198961205931 and 1198961199091 can bedetermined by the following equations

10038161003816100381610038161003816B + 1198961205931T minus 1205900N

10038161003816100381610038161003816= 0

1003816100381610038161003816H + 1198961199091R minus 1205900S1003816100381610038161003816 = 0

(12)

It can be found from (12) that there is a couplingrelation between external loads and torsionallateral restraintstiffness It indicates that both the torsional and lateralrestraints of the bottom flange are not only determined bythe cross-section features of the composite girder but theyalso depended on the external loads Therefore it may notbe appropriate to take the restraint stiffness as a constantmaterial feature in the traditional elastic foundation beammethod

4 Buckling Analysis of I-Steel-ConcreteComposite Girders

According to the assumptions made upon the bucklingmodel of the I-steel-composite girder can be simplified asthe model depicted in Figure 5 The horizontal and torsionaldirections of the thin plate are restricted by springs while thevertical direction is rigidly restricted

kx

bf

kyk120593

tf

x

y

0

Figure 5 Simplified calculation model of steel-concrete compositegirders

In Figure 5 the thin plate is symmetric about both 119909-axisand 119910-axis The centroid of the plate is set to be the originpoint Assuming the horizontal lateral displacement of thebottom flange is 119906(119911) and the torsional angle is 120593(119911) theneutral equilibrium differential equation of an elastic thin-walled bar under variable axial force can be expressed as [17]

119864119868119910119906119868119881+ [119875 (119906

1015840+ 119910119886120593

1015840)]1015840

+ 119896119909 [119906 minus (119910119889 minus 119910119886) 120593]

minus 119891119909120593 = 0

119864119868119909V119868119881+ [119875 (V1015840 minus 119909119886120593

1015840)]1015840

+ 119896119910 [V + (119909119889 minus 119909119886) 120593] = 0

119864119868119908120593119868119881+ [(119875119903

2

0minus 119866119869) 120593

1015840]1015840

minus 119896119909 [119906 minus (119910119889 minus 119910119886) 120593]

times (119910119889 minus 119910119886) + 119891120593119909 minus 119909119886(119875V1015840)1015840

+ 119910119886(1198751199061015840)1015840

+ 119896119910 [V + (119909119889 minus 119909119886) 120593] (119909119889 minus 119909119886) + 119896120593120593 = 0

(13)

where 119868119910 = 1199051198911198873

11989112 119868119909 = 119887119891119905

3

11989112 119869 = 119887119891119905

3

1198913 11990320= 1199092

119886+

1199102

119886+ (119868119909 + 119868119910)119860 119904 and 119909119886 is center position of the curved

bottom flange in the horizontal axis here 119909119886 = 0 119910119886 is thecenter position of the curved bottom flange in the verticalaxis here 119910119886 = 0 119909119889 is the rotation axis of the bottom flangein the horizontal axis here 119909119889 = 0 119910119889 is the rotation axis ofthe bottom flange in the vertical axis here 119910119889 = 0 119868119908 is thesectorial inertia moment of bottom flange here 119868119908 = 0 119866is shear modulus of the steel 119875 is the pressure of the bottomflange119875 = 1198601198911205900(1198860+1198861120585+1198862120585

2) 119896120593 = minus1198961205931 and 119896119909 = minus1198961199091

Plugging119910119886 = 0 119910119889 = 0 119909119886 = 0 119909119889 = 0 V = 0 and 119868119908 =0 into (13) leads to

(1198751199032

0minus 119866119869) 120593

10158401015840+ 1199032

011987510158401205931015840+ 119896120593120593 + 119891120593119909 = 0

119864119868119910119906119868119881+ 11987510158401199061015840+ 11987511990610158401015840+ 119896119909119906 minus 119891119909120593 = 0

(14)

By combining (5) and (9) the torsional angle and lateraldisplacement of the bottomflange of the composite girder canbe obtained

120593 (119911) =

119899

sum

119894=1

119888119894

ℎ119908

sin 119894120587119911119897

119906 (119911) =

119899

sum

119894=1

119889119894 sin119894120587119911

119897

(15)

Mathematical Problems in Engineering 5

According to the Galerkin method [15 22] we have

(B1 + 119896120593T minus 1205900N1 Q

M H1 + 119896119909R minus 1205900S1)(

CD) = 0 (16)

where 1198611119894119895 = 119876119894119895 = 1198671119894119895 = 119872119894119895 = 0 (119894 = 119895) 1198611119894119894 =119866119869(119894212058722119897ℎ2

119908) 119872119894119894 = (3119863119908119897ℎ

3

119908)minus119863119908((2minus120583)119894

21198971205732ℎ119908) 119876119894119894 =

3(119863119908119897ℎ3

119908)+(119863119908120583119897120573119894

22ℎ119908) 1198731119894119894 = (119860119891119903

2

012058721198921198942119897ℎ

2

119908) 1198731119894119895 =

(1198601198911199032

0119887119894119895119897ℎ2

119908) (119894 = 119895) 1198671119894119894 = 119864119868119910120573

2(11989441198972) 1198781119894119894 = (119860119891120587

2119892119894

2119897) and 1198781119894119895 = (119860119891119887119894119895119897) (119894 = 119895)The combination of (6) (10) and (16) leads to

[(B QM H) minus 1205900 (

N 00 S)](

CD) = 0 (17)

where B = B0 + B1 N = N0 + N1 H = H0 + H1 and S =

S0 + S1The deformation vector C119879D119879119879 cannot be zero when

buckling happens Therefore the buckling of the compositegirder can be solved by the generalized eigenvalue of thecharacteristic matrix shown as follows

10038161003816100381610038161003816100381610038161003816

(B QM H) minus 1205900 (

N 00 S)

10038161003816100381610038161003816100381610038161003816

= 0 (18)

By solving (18) 2119899 generalized eigenvalue can beobtained 120590119905119894 (119894 = 1 2 2119899) let 120590cr = min120590119905119894 (119894 =

1 2 2119899) 120590cr is the critical buckling stress of the compos-ite girder The critical buckling moment of composite girdercan be calculated by the following equation

119872cr =120590cr (1198860 + 1198861120585 + 1198862120585

2) 119868

119910119888

(19)

5 Buckling Analysis of the Steel-ConcreteComposite Box Girder

The dimensions of the composite box girder are shownin Figure 6 According to the assumptions made abovethe buckling model of the composite box girder can besimplified as a thin-plate model that is restricted by springsin horizontal and torsional directions rigidly restricted invertical directionThe simplified model is shown in Figure 7

As the derivation in Section 3 the following can beobtained

(B0 + 1198961205931198971T minus 1205900N0)C119897 = 0 (20)

(B0 + 1198961205931199031T minus 1205900N0)C119903 = 0 (21)

(H0 + 1198961199091R minus 1205900S0)D = 0 (22)

Mx

x y

z

L

Mx hwyctw

tt hc

tfy

bc

x00

Figure 6 Cross-section dimensions of steel-concrete composite boxgirder

k120593l

kxl

ky = infin

y

bf

x

k120593r

kxr

0

ky = infin

Figure 7 Simplified calculation model of steel-concrete compositegirders

119891119909120593119897

10038161003816100381610038161003816119910=0= minus

6119863119908

ℎ3119908

(

119899

sum

119894=1

119888119897119894 sin119894120587119911

119871)

+ 119863119908120573(2 minus 120583)

ℎ119908

(

119899

sum

119894=1

1198881198971198941198942 sin 119894120587119911

119871)

(23)

119891119909120593119903

10038161003816100381610038161003816119910=0= minus

6119863119908

ℎ3119908

(

119899

sum

119894=1

119888119903119894 sin119894120587119911

119897)

+ 119863119908120573(2 minus 120583)

ℎ119908

(

119899

sum

119894=1

1198881199031198941198942 sin 119894120587119911

119897)

(24)

119891120593119909

10038161003816100381610038161003816119910=0=6119863119908

ℎ2119908

(

119899

sum

119894=1

119889119894 sin119894120587119911

119897)

+ 119863119908120583120573(

119899

sum

119894=1

1198891198941198942 sin 119894120587119911

119897)

(25)

where 1198961205931198971 is the torsional restraint stiffness of left web by

the bottom flange C119897 = 1198881198971 1198881198972 119888119897119899119879 is buckling general

coordinates of the left web 1198961205931199031 is torsional restraint stiffness

of the right web by the bottom flange C119903 = 1198881199031 1198881199032 119888119903119899119879

is buckling general coordinates of the right web 1198961199091 is lateralrestraint stiffness of the steel web by the bottom flange andD = 1198891 1198892 119889119899

119879 is buckling general coordinates of thebottom flange

As Figure 7 shows the lateral displacement bucklingfunction of the horizontal buckling of the bottom flange is119906(119911) the out-plane buckling deformation function of the

6 Mathematical Problems in Engineering

bottom flange is V(119909 119911) the left rotational angle is 120593119897(119911) andthe right rotational angle is 120593119903(119911) The boundary condition ofthe bottom flange is given as follows

V 1003816100381610038161003816119909=0 = 0 V11990910038161003816100381610038161003816119909=0

= 120593119897 (119911)

V100381610038161003816100381610038161003816119909=119887119891

= 0 V119909100381610038161003816100381610038161003816119909=119887119891

= 120593119903 (119911)

(26)

According to compatibility of deformation the displace-ment function of the bottom flange is

V =119887119891

2120587sin 120587119909

119887119891

(120593119897 minus 120593119903) +

119887119891

4120587sin 2120587119909

119887119891

(120593119897 + 120593119903) (27)

According to the principle of stationary potential energythe buckling characteristic equations are given as follows

119863119891(51205872120593119897

8119887119891

+31205872120593119903

8119887119891

minus

11988711989112059310158401015840

119897

2+

51198873

1198911205931015840101584010158401015840

119897

321205872minus

31198873

1198911205931015840101584010158401015840

119903

321205872)

+ (

51205901198601198911198872

1198911205931015840

119897

321205872)

1015840

minus (

31205901198601198911198872

1198911205931015840

119903

321205872)

1015840

+ 119896120593119897

120593119897 + 119891120593119909 = 0

2119896119909119906 + (1205901198601198911199061015840)1015840

+ 1198641198681199101199061015840101584010158401015840

minus 119891119909120593119897

minus 119891119909120593119903

= 0

119863119891(51205872120593119903

8119887119891

+31205872120593119897

8119887119891

minus

11988711989112059310158401015840

119903

2+

51198873

1198911205931015840101584010158401015840

119903

321205872minus

31198873

1198911205931015840101584010158401015840

119897

321205872)

+ (

51205901198601198911198872

1198911205931015840

119903

321205872)

1015840

minus (

31205901198601198911198872

1198911205931015840

119897

321205872)

1015840

+ 119896120593119903

120593119895 + 119891120593119909 = 0

(28)

where 119868119910 = 1199051198911198873

11989112 119905119891 is the thickness of bottom flange 119887119891

is the width of the bottom flange119863119891 = 1198641199053

11989112(1 minus 120583

2) 119896120593

119897

=

minus1198961205931198971 is torsional restraint stiffness of the bottom flange edge

by the left steel web 119896120593119903

= minus1198961205931199031 is torsional restraint stiffness

of the bottomflange edge by the right steel web and 119896119909 = minus1198961199091is lateral restraint stiffness of the bottom flange edge by thesteel web

According to the Galerkin method [15 16] we have

(

B1+ 119896120593119897

T minus 1205900N1 F + 06120590

0N1 Q

F + 061205900N1

B1+ 119896120593119903

T minus 1205900N1 Q

M M H1minus 1205900S1+ 2119896119909R)

times 120578 = 0

(29)

where 1198611119894119895 = 119865119894119895 = 119876119894119895 = 119872119894119895 = 1198671119894119895 = 0 (119894 = 119895) 1198611119894119894 =(5119863119891120587

211989716119887119891ℎ

2

119908) + (119863119891119887119891120573119897119894

24ℎ2

119908) + (5119863119891119887

3

11989112057321198971198944641205872ℎ2

119908)

119865119894119894 = (3119863119891120587211989716119887119891ℎ

2

119908) minus (3119863119891119887

3

11989112057321198971198944641205872ℎ2

119908) 119876119894119894 =

3(119863119908119897ℎ3

119908)+(1198971198631199081205831205732ℎ119908)119894

2 1198731119894119895 = (51198601198911198872

11989111988711989411989532120587

2ℎ2

119908119897) (119894 =

119895) 1198731119894119894 = (51198601198911198872

11989111989211989464ℎ

2

119908119897) 119872119894119894 = (3119863119908119897ℎ

3

119908) minus 119863119908((2 minus

120583)11989421198971205732ℎ119908) 1198671119894119894 = (119894

411986411986811991012057321198972) 1198781119894119894 = (119860119891120587

21198921198942119897) 1198781119894119895 =

(119860119891119887119894119895119897) (119894 = 119895) and 120578 = (C119879119897C119879119903D119879)119879

Table 1 Basic geometric size of I-steel composite girder cases

Examplenumber

Cross-sectionnumber

ℎ119908mm 119887

119891mm 119905

119891mm 119905

119908mm 119897mm

1 45002 1 600 200 12 12 72003 96004 30005 2 600 120 10 10 54006 84007 60008 3 719 268 25 16 102009 1440010 300011 4 300 100 3 3 540012 840013 420014 5 450 150 4 4 720015 10200

Since the constraint in the theoretical model is higherthan the real scenario the critical buckling stress is increasedTherefore the theoretical buckling deformation functions ofthe web and bottomflange cannot accurately describe the realbuckling deformation curves In order to eliminate errorsthe paper gives a reduction factor on the torsional restraintstiffness of the bottom flange and the reduction factor isfound to be 05 Combining (20) (21) (22) and (29) leadsto

[[[[

[

(

B F QF B QM M H

) minus 1205900(

N minus3N15

0minus3N15

N 00 0 S

)

]]]]

]

120578 = 0 (30)

where B = B02 + B1 N = N02 + N1 H = 2H0 +H1 and S = 2S0 + S1

The deformation vector 120578 cannot be zero when the buck-ling of the composite girder happensTherefore the bucklingof the composite girder can be solved by the generalizedeigenvalue of the characteristic matrix shown as follows

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(

B F QF B QM M H

)minus 1205900(

N minus06N1 0minus06N1 N 0

0 0 S)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

= 0 (31)

3119899 general eigenvalues can be obtained from (31) whichare 120590119905119894 (119894 = 1 2 3119899) let 120590cr = min120590119905119894 (119894 =

1 2 3119899) 120590cr is the critical buckling stress of the com-posite girderThe following equation can calculate the criticalbuckling moment

119872cr =120590cr (1198860 + 1198861120585 + 1198862120585

2) 119868

119910119888

(32)

Mathematical Problems in Engineering 7

Table 2 Basic geometric size of steel composite box girder cases

Example number ℎ119908mm 119887

119891mm 119887

119905mm 119905

119908mm 119905

119891mm 119905

119905mm 119897mm

1 40002 400 600 100 10 10 10 80003 120004 40005 400 500 100 10 10 10 80006 120007 40008 300 600 100 10 10 10 80009 1200010 400011 400 600 120 10 10 10 800012 1200013 400014 400 600 100 12 12 12 800015 12000

Table 3 Critical distortional buckling moment of I-steel compositegirder under negative uniform moment

Examplenumber

Distortional buckling critical moment119872crkNsdotmANSYS Williams Goltermann Svensson (19)

1 11291 10606 11936 15401 113832 12123 11480 12344 16647 120393 11497 10770 11971 15617 115504 39098 35402 40588 6196 404405 41019 37512 43686 6566 421616 40197 36603 46919 6406 414127 36727 31264 35211 38991 372008 38365 33904 40143 42281 387459 39051 32268 42454 40242 3932710 34155 33710 35235 489 3263211 35235 34884 36464 506 3358712 34729 34304 35951 497 3310613 94611 94325 101817 1368 8971314 107307 107811 102911 1563 1013115 91161 90153 93920 1307 86372

6 Analysis of Examples

Thegeometric dimensions of each example are listed inTables1 and 2 By means of the calculation method introduced inthis paper and the finite elementmethod the critical bucklinganalysis of the composite girder under uniform negativebending moment triangle bending moment and uniformloads can be carried out Svenssonrsquos method Williamsrsquomethod and Goltermannrsquos method are also employed in thecalculation of various I-steel-concrete composite girders soas to validate the calculation method proposed in this paperThe finite element analysis is conducted by using ANSYScommercial software Element SHELL43 is adopted to model

Table 4 Critical distortional buckling moment of I-steel compositegirder under negative triangular moment

Examplenumber

Distortional buckling critical moment119872crkNsdotmANSYS Williams Goltermann Svensson (19)

1 18686 18388 19714 26654 192522 17264 16805 17740 24367 173583 15909 15288 16213 22162 159744 6599 6327 7132 11072 703155 5866 5506 6056 9633 601566 5257 4846 5357 8481 538847 61252 55304 61444 68972 635348 55235 49455 54387 61674 560759 50604 44350 49232 55310 5119110 55183 59346 61236 860 5616711 51574 51935 53366 753 4946912 46366 46184 47412 670 4422413 131762 170748 181683 2476 1336114 129117 147906 150336 2144 1299815 127803 131099 134420 1901 12408

the steel girder The concrete slab of the composite girder isreplaced with constraints in the numerical simulation Themotions in 119909 and 119910 directions of the top flange edge arerestrained to represent the lateral and torsional restrictionscaused by the concrete slab in practice The results of eachsimplified calculation method are listed in Tables 3 4 5 and6 and the error analyses of each simplifiedmethod are shownin Figures 8 9 10 and 11

The following conclusions can be drawn based on theresults presented in Tables 3 to 6 and Figures 8 to 11

(1) Under uniform negative bending moment the crit-ical bending buckling moment in the same cross-section of the composite girder is rarely affected by

8 Mathematical Problems in Engineering

Table 5 Critical distortional buckling moment of I-steel compositegirder under uniformly distributed load

Examplenumber

Distortional buckling critical moment119872crkNsdotmANSYS Williams Goltermann Svensson (19)

3 15512 15394 16388 22321 161606 5272 4988 5508 8729 551109 50383 45093 50001 56234 5235111 497 52610 54675 76289 5031712 459 47925 47925 67500 4453313 1299 198896 204768 288401 1291014 1267 165281 165645 239639 1269515 1255 131099 135351 190107 12454

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1605060708091011121314151617

Example number

WilliamsGoltermann

SvenssonThis paper

Mcr

of an

alyt

ical

met

hods

Mcr

of A

NSY

S

Figure 8 Precision analysis of simplified methods under negativeuniform moment

the length The critical bending buckling moment isnot obviously changed with the increased length ofthe structural component

(2) Under triangle bending moment the critical bendingbuckling moment is greatly affected by the lengththat is the value decreased quickly when the lengthincreases

(3) Under uniform negative bending moment trianglebending moment and uniform loads the resultsyielded by the calculation method in this papermatch well the finite element analysis results Thediscrepancy is limited within 5 which validates theaccuracy and applicability of this method

(4) Traditional calculation methods such as Svens-sonrsquos method Williamsrsquo method and Goltermannrsquosmethod have considerable deviations from the finiteelement method

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Example number

WilliamsGoltermann

SvenssonThis paper

05060708091011121314151617

Mcr

of an

alyt

ical

met

hods

Mcr

of A

NSY

S

Figure 9 Precision analysis of simplified methods under negativetriangular moment

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Example number

05060708091011121314151617

Mcr

of an

alyt

ical

met

hods

Mcr

of A

NSY

S

WilliamsGoltermann

SvenssonThis paper

Figure 10 Precision analysis of simplifiedmethods under uniformlydistributed load

Therefore the traditional elastic foundation beammethod taking into account the moment gradient needs tobe improved It is also suggested that the constant lateral andtorsional restraints in the traditional methods may lead tothe relative deviations

7 Conclusions

In this paper the traditional elastic foundation beammethodsare improved by considering the coupling effect of the exter-nal loads and the foundation spring constraints Based on thisimprovement a simplified calculation method computing thecritical buckling loads of steel-concrete composite girders is

Mathematical Problems in Engineering 9

Table 6 Critical distortional buckling moment of composite box girder under negative moment

Examplenumber

Distortional buckling critical moment119872crkNsdotmUniform negative bending moment Triangle negative bending moment Uniform loadANSYS (32) ANSYS (32) ANSYS (32)

1 72605 72214 79236 81741 79282 786742 72714 72214 76467 78046 76122 755313 72797 72214 75492 76616 75146 742624 90593 90722 97811 10125 97547 974365 90824 90650 94867 97143 94390 939166 90957 90610 93864 95600 93434 927857 53751 53575 58654 60316 58284 580398 53852 53407 56622 57636 55978 556429 53917 53411 55906 56601 55267 5487510 74221 73610 81013 83322 80861 8019511 74333 73610 78178 79555 77685 7678812 74417 73610 77181 78097 76702 7569813 12536 12479 13676 14125 13659 1359514 12556 12479 13201 13486 13114 1301715 12607 12479 13035 13239 12947 12833

090

092

094

096

098

100

102

104

106

108

110

Negative uniform moment Negative triangular momentUniformly distributed load

Mcr

of th

is pa

perM

crof

AN

SYS

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Example number

Figure 11 Precision analysis of simplified methods

developed The method is compared with various traditionalmethods The following conclusions are obtained

(1) There is a linear coupling relation between bothtorsional and lateral restraints and vertical loads

(2) Under uniformnegative bendingmoment the criticalbending buckling moment in the same cross-sectionof the composite girder is rarely affected by thelength The critical bending buckling moment is notobviously changed with the increased length of thestructural component

(3) Under triangle bending moment the critical bendingbuckling is influenced to a great extent by the lengththat is the value decreased quickly when the lengthincreases

(4) Under uniform negative bending moment trianglebending moment and uniform loads the calculationmethod proposed in this paper matches well thefinite element calculation methodThe discrepancy islimited within 5 which validates the applicability ofthis method

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research described in this paper was financially sup-ported by National Natural Science Foundation of China(51408449 and 51378502) and the Fundamental ResearchFunds for the Central Universities of China (2014-IV-049)

References

[1] S Kim andU Lee ldquoEffects of delamination on guidedwaves in asymmetric laminated composite beamrdquoMathematical Problemsin Engineering vol 2014 Article ID 956043 12 pages 2014

[2] D Champenoy A Corfdir and P Corfdir ldquoCalculating thecritical buckling force in compressed bottom flanges of steel-concrete composite bridgesrdquo European Journal of Environmen-tal and Civil Engineering vol 18 no 3 pp 271ndash292 2014

[3] T-H Yi H-N Li and M Gu ldquoOptimal sensor placement forstructural health monitoring based on multiple optimization

10 Mathematical Problems in Engineering

strategiesrdquo The Structural Design of Tall and Special Buildingsvol 20 no 7 pp 881ndash900 2011

[4] T-H Yi H-N Li and X-D Zhang ldquoA modified monkeyalgorithm for optimal sensor placement in structural healthmonitoringrdquo Smart Materials and Structures vol 21 no 10Article ID 105033 2012

[5] S Chen and Y Jia ldquoNumerical investigation of inelasticbuckling of steel-concrete composite beams prestressed withexternal tendonsrdquoThin-Walled Structures vol 48 no 3 pp 233ndash242 2010

[6] S E Svensson ldquoLateral buckling of beams analysed as elasticallysupported columns subject to a varying axial forcerdquo Journal ofConstructional Steel Research vol 5 no 3 pp 179ndash193 1985

[7] F WWilliams and A K Jemah ldquoBuckling curves for elasticallysupported columns with varying axial force to predict lateralbuckling of beamsrdquo Journal of Constructional Steel Research vol7 no 2 pp 133ndash147 1987

[8] P Goltermann and S Svensson ldquoLateral distortional bucklingpredicting elastic critical stressrdquo Journal of Structural Engineer-ing vol 114 no 7 pp 1606ndash1625 1988

[9] Swedish Institute of Steel Construction Swedish Code for Light-GaugeMetal Structures Swedish Institute of Steel ConstructionStockholm Sweden 1982

[10] British Standards Institution Code of Practice for Design of SteelBridge BS5400 Part 3 BSI London UK 1982

[11] RM Lawson andW J RackhamDesign of haunched compositebeams in buildings [MS thesis] Steel Construction InstitutionAscot UK 1989

[12] L Jiang J Qi A Scanlon and L Sun ldquoDistortional andlocal buckling of steel-concrete composite box-beamrdquo Steel andComposite Structures vol 14 no 3 pp 243ndash265 2013

[13] J-H Ye and W Chen ldquoElastic restrained distortional buck-ling of steel-concrete composite beams based on elasticallysupported column methodrdquo International Journal of StructuralStability and Dynamics vol 13 no 1 Article ID 1350001 pp 1ndash29 2013

[14] W Chen and J Ye ldquoElastic lateral and restrained distortionalbuckling of doubly symmetric I-beamsrdquo International Journalof Structural Stability and Dynamics vol 10 no 5 pp 983ndash10162010

[15] W-B Zhou L-Z Jiang G-Q Shao and Z-W Yu ldquoElastic dis-tortional buckling analysis of steel-concrete composite beamsin negative moment regionrdquo Journal of Central South University(Science and Technology) vol 43 no 6 pp 2316ndash2323 2012

[16] W B Zhou L Z Jiang and ZW Yu ldquoThe distortional bucklingcalculation formula of the steel-concrete composite beams inthe negative moment regionrdquo Chinese Journal of ComputationalMechanics vol 29 no 3 pp 446ndash450 2012

[17] J Ye andW Chen ldquoElastic restrained distortional buckling of I-steel-concrete composite beamsrdquo Journal of Building Structuresvol 32 no 6 pp 82ndash91 2011

[18] W Zhou L Jiang Z Liu et al ldquoClosed-form solution for shearlag effects of steel-concrete composite box beams consideringshear deformation and sliprdquo Journal of Central South Universityvol 19 no 10 pp 2976ndash2982 2012

[19] W-B Zhou L-Z Jiang andZ-W Yu ldquoAnalysis of free vibrationcharacteristic of steel-concrete composite box-girder consider-ing shear lag and sliprdquo Journal of Central South University vol20 no 9 pp 2570ndash2577 2013

[20] W-B Zhou L-Z Jiang Z-J Liu et al ldquoClosed-form solutionto thin-walled box girders considering effects of shear deforma-tion and shear lagrdquo Journal of Central South University vol 19no 9 pp 2650ndash2655 2012

[21] A Teodor A Guran and G Ardbeshir Theory of Elasticity forScientists and Engineers Springer New York NY USA 2012

[22] C Bi and V Ginting ldquoTwo-grid discontinuous Galerkinmethod for quasi-linear elliptic problemsrdquo Journal of ScientificComputing vol 49 no 3 pp 311ndash331 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

According to the Galerkin method [15 22] we have

(B1 + 119896120593T minus 1205900N1 Q

M H1 + 119896119909R minus 1205900S1)(

CD) = 0 (16)

where 1198611119894119895 = 119876119894119895 = 1198671119894119895 = 119872119894119895 = 0 (119894 = 119895) 1198611119894119894 =119866119869(119894212058722119897ℎ2

119908) 119872119894119894 = (3119863119908119897ℎ

3

119908)minus119863119908((2minus120583)119894

21198971205732ℎ119908) 119876119894119894 =

3(119863119908119897ℎ3

119908)+(119863119908120583119897120573119894

22ℎ119908) 1198731119894119894 = (119860119891119903

2

012058721198921198942119897ℎ

2

119908) 1198731119894119895 =

(1198601198911199032

0119887119894119895119897ℎ2

119908) (119894 = 119895) 1198671119894119894 = 119864119868119910120573

2(11989441198972) 1198781119894119894 = (119860119891120587

2119892119894

2119897) and 1198781119894119895 = (119860119891119887119894119895119897) (119894 = 119895)The combination of (6) (10) and (16) leads to

[(B QM H) minus 1205900 (

N 00 S)](

CD) = 0 (17)

where B = B0 + B1 N = N0 + N1 H = H0 + H1 and S =

S0 + S1The deformation vector C119879D119879119879 cannot be zero when

buckling happens Therefore the buckling of the compositegirder can be solved by the generalized eigenvalue of thecharacteristic matrix shown as follows

10038161003816100381610038161003816100381610038161003816

(B QM H) minus 1205900 (

N 00 S)

10038161003816100381610038161003816100381610038161003816

= 0 (18)

By solving (18) 2119899 generalized eigenvalue can beobtained 120590119905119894 (119894 = 1 2 2119899) let 120590cr = min120590119905119894 (119894 =

1 2 2119899) 120590cr is the critical buckling stress of the compos-ite girder The critical buckling moment of composite girdercan be calculated by the following equation

119872cr =120590cr (1198860 + 1198861120585 + 1198862120585

2) 119868

119910119888

(19)

5 Buckling Analysis of the Steel-ConcreteComposite Box Girder

The dimensions of the composite box girder are shownin Figure 6 According to the assumptions made abovethe buckling model of the composite box girder can besimplified as a thin-plate model that is restricted by springsin horizontal and torsional directions rigidly restricted invertical directionThe simplified model is shown in Figure 7

As the derivation in Section 3 the following can beobtained

(B0 + 1198961205931198971T minus 1205900N0)C119897 = 0 (20)

(B0 + 1198961205931199031T minus 1205900N0)C119903 = 0 (21)

(H0 + 1198961199091R minus 1205900S0)D = 0 (22)

Mx

x y

z

L

Mx hwyctw

tt hc

tfy

bc

x00

Figure 6 Cross-section dimensions of steel-concrete composite boxgirder

k120593l

kxl

ky = infin

y

bf

x

k120593r

kxr

0

ky = infin

Figure 7 Simplified calculation model of steel-concrete compositegirders

119891119909120593119897

10038161003816100381610038161003816119910=0= minus

6119863119908

ℎ3119908

(

119899

sum

119894=1

119888119897119894 sin119894120587119911

119871)

+ 119863119908120573(2 minus 120583)

ℎ119908

(

119899

sum

119894=1

1198881198971198941198942 sin 119894120587119911

119871)

(23)

119891119909120593119903

10038161003816100381610038161003816119910=0= minus

6119863119908

ℎ3119908

(

119899

sum

119894=1

119888119903119894 sin119894120587119911

119897)

+ 119863119908120573(2 minus 120583)

ℎ119908

(

119899

sum

119894=1

1198881199031198941198942 sin 119894120587119911

119897)

(24)

119891120593119909

10038161003816100381610038161003816119910=0=6119863119908

ℎ2119908

(

119899

sum

119894=1

119889119894 sin119894120587119911

119897)

+ 119863119908120583120573(

119899

sum

119894=1

1198891198941198942 sin 119894120587119911

119897)

(25)

where 1198961205931198971 is the torsional restraint stiffness of left web by

the bottom flange C119897 = 1198881198971 1198881198972 119888119897119899119879 is buckling general

coordinates of the left web 1198961205931199031 is torsional restraint stiffness

of the right web by the bottom flange C119903 = 1198881199031 1198881199032 119888119903119899119879

is buckling general coordinates of the right web 1198961199091 is lateralrestraint stiffness of the steel web by the bottom flange andD = 1198891 1198892 119889119899

119879 is buckling general coordinates of thebottom flange

As Figure 7 shows the lateral displacement bucklingfunction of the horizontal buckling of the bottom flange is119906(119911) the out-plane buckling deformation function of the

6 Mathematical Problems in Engineering

bottom flange is V(119909 119911) the left rotational angle is 120593119897(119911) andthe right rotational angle is 120593119903(119911) The boundary condition ofthe bottom flange is given as follows

V 1003816100381610038161003816119909=0 = 0 V11990910038161003816100381610038161003816119909=0

= 120593119897 (119911)

V100381610038161003816100381610038161003816119909=119887119891

= 0 V119909100381610038161003816100381610038161003816119909=119887119891

= 120593119903 (119911)

(26)

According to compatibility of deformation the displace-ment function of the bottom flange is

V =119887119891

2120587sin 120587119909

119887119891

(120593119897 minus 120593119903) +

119887119891

4120587sin 2120587119909

119887119891

(120593119897 + 120593119903) (27)

According to the principle of stationary potential energythe buckling characteristic equations are given as follows

119863119891(51205872120593119897

8119887119891

+31205872120593119903

8119887119891

minus

11988711989112059310158401015840

119897

2+

51198873

1198911205931015840101584010158401015840

119897

321205872minus

31198873

1198911205931015840101584010158401015840

119903

321205872)

+ (

51205901198601198911198872

1198911205931015840

119897

321205872)

1015840

minus (

31205901198601198911198872

1198911205931015840

119903

321205872)

1015840

+ 119896120593119897

120593119897 + 119891120593119909 = 0

2119896119909119906 + (1205901198601198911199061015840)1015840

+ 1198641198681199101199061015840101584010158401015840

minus 119891119909120593119897

minus 119891119909120593119903

= 0

119863119891(51205872120593119903

8119887119891

+31205872120593119897

8119887119891

minus

11988711989112059310158401015840

119903

2+

51198873

1198911205931015840101584010158401015840

119903

321205872minus

31198873

1198911205931015840101584010158401015840

119897

321205872)

+ (

51205901198601198911198872

1198911205931015840

119903

321205872)

1015840

minus (

31205901198601198911198872

1198911205931015840

119897

321205872)

1015840

+ 119896120593119903

120593119895 + 119891120593119909 = 0

(28)

where 119868119910 = 1199051198911198873

11989112 119905119891 is the thickness of bottom flange 119887119891

is the width of the bottom flange119863119891 = 1198641199053

11989112(1 minus 120583

2) 119896120593

119897

=

minus1198961205931198971 is torsional restraint stiffness of the bottom flange edge

by the left steel web 119896120593119903

= minus1198961205931199031 is torsional restraint stiffness

of the bottomflange edge by the right steel web and 119896119909 = minus1198961199091is lateral restraint stiffness of the bottom flange edge by thesteel web

According to the Galerkin method [15 16] we have

(

B1+ 119896120593119897

T minus 1205900N1 F + 06120590

0N1 Q

F + 061205900N1

B1+ 119896120593119903

T minus 1205900N1 Q

M M H1minus 1205900S1+ 2119896119909R)

times 120578 = 0

(29)

where 1198611119894119895 = 119865119894119895 = 119876119894119895 = 119872119894119895 = 1198671119894119895 = 0 (119894 = 119895) 1198611119894119894 =(5119863119891120587

211989716119887119891ℎ

2

119908) + (119863119891119887119891120573119897119894

24ℎ2

119908) + (5119863119891119887

3

11989112057321198971198944641205872ℎ2

119908)

119865119894119894 = (3119863119891120587211989716119887119891ℎ

2

119908) minus (3119863119891119887

3

11989112057321198971198944641205872ℎ2

119908) 119876119894119894 =

3(119863119908119897ℎ3

119908)+(1198971198631199081205831205732ℎ119908)119894

2 1198731119894119895 = (51198601198911198872

11989111988711989411989532120587

2ℎ2

119908119897) (119894 =

119895) 1198731119894119894 = (51198601198911198872

11989111989211989464ℎ

2

119908119897) 119872119894119894 = (3119863119908119897ℎ

3

119908) minus 119863119908((2 minus

120583)11989421198971205732ℎ119908) 1198671119894119894 = (119894

411986411986811991012057321198972) 1198781119894119894 = (119860119891120587

21198921198942119897) 1198781119894119895 =

(119860119891119887119894119895119897) (119894 = 119895) and 120578 = (C119879119897C119879119903D119879)119879

Table 1 Basic geometric size of I-steel composite girder cases

Examplenumber

Cross-sectionnumber

ℎ119908mm 119887

119891mm 119905

119891mm 119905

119908mm 119897mm

1 45002 1 600 200 12 12 72003 96004 30005 2 600 120 10 10 54006 84007 60008 3 719 268 25 16 102009 1440010 300011 4 300 100 3 3 540012 840013 420014 5 450 150 4 4 720015 10200

Since the constraint in the theoretical model is higherthan the real scenario the critical buckling stress is increasedTherefore the theoretical buckling deformation functions ofthe web and bottomflange cannot accurately describe the realbuckling deformation curves In order to eliminate errorsthe paper gives a reduction factor on the torsional restraintstiffness of the bottom flange and the reduction factor isfound to be 05 Combining (20) (21) (22) and (29) leadsto

[[[[

[

(

B F QF B QM M H

) minus 1205900(

N minus3N15

0minus3N15

N 00 0 S

)

]]]]

]

120578 = 0 (30)

where B = B02 + B1 N = N02 + N1 H = 2H0 +H1 and S = 2S0 + S1

The deformation vector 120578 cannot be zero when the buck-ling of the composite girder happensTherefore the bucklingof the composite girder can be solved by the generalizedeigenvalue of the characteristic matrix shown as follows

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(

B F QF B QM M H

)minus 1205900(

N minus06N1 0minus06N1 N 0

0 0 S)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

= 0 (31)

3119899 general eigenvalues can be obtained from (31) whichare 120590119905119894 (119894 = 1 2 3119899) let 120590cr = min120590119905119894 (119894 =

1 2 3119899) 120590cr is the critical buckling stress of the com-posite girderThe following equation can calculate the criticalbuckling moment

119872cr =120590cr (1198860 + 1198861120585 + 1198862120585

2) 119868

119910119888

(32)

Mathematical Problems in Engineering 7

Table 2 Basic geometric size of steel composite box girder cases

Example number ℎ119908mm 119887

119891mm 119887

119905mm 119905

119908mm 119905

119891mm 119905

119905mm 119897mm

1 40002 400 600 100 10 10 10 80003 120004 40005 400 500 100 10 10 10 80006 120007 40008 300 600 100 10 10 10 80009 1200010 400011 400 600 120 10 10 10 800012 1200013 400014 400 600 100 12 12 12 800015 12000

Table 3 Critical distortional buckling moment of I-steel compositegirder under negative uniform moment

Examplenumber

Distortional buckling critical moment119872crkNsdotmANSYS Williams Goltermann Svensson (19)

1 11291 10606 11936 15401 113832 12123 11480 12344 16647 120393 11497 10770 11971 15617 115504 39098 35402 40588 6196 404405 41019 37512 43686 6566 421616 40197 36603 46919 6406 414127 36727 31264 35211 38991 372008 38365 33904 40143 42281 387459 39051 32268 42454 40242 3932710 34155 33710 35235 489 3263211 35235 34884 36464 506 3358712 34729 34304 35951 497 3310613 94611 94325 101817 1368 8971314 107307 107811 102911 1563 1013115 91161 90153 93920 1307 86372

6 Analysis of Examples

Thegeometric dimensions of each example are listed inTables1 and 2 By means of the calculation method introduced inthis paper and the finite elementmethod the critical bucklinganalysis of the composite girder under uniform negativebending moment triangle bending moment and uniformloads can be carried out Svenssonrsquos method Williamsrsquomethod and Goltermannrsquos method are also employed in thecalculation of various I-steel-concrete composite girders soas to validate the calculation method proposed in this paperThe finite element analysis is conducted by using ANSYScommercial software Element SHELL43 is adopted to model

Table 4 Critical distortional buckling moment of I-steel compositegirder under negative triangular moment

Examplenumber

Distortional buckling critical moment119872crkNsdotmANSYS Williams Goltermann Svensson (19)

1 18686 18388 19714 26654 192522 17264 16805 17740 24367 173583 15909 15288 16213 22162 159744 6599 6327 7132 11072 703155 5866 5506 6056 9633 601566 5257 4846 5357 8481 538847 61252 55304 61444 68972 635348 55235 49455 54387 61674 560759 50604 44350 49232 55310 5119110 55183 59346 61236 860 5616711 51574 51935 53366 753 4946912 46366 46184 47412 670 4422413 131762 170748 181683 2476 1336114 129117 147906 150336 2144 1299815 127803 131099 134420 1901 12408

the steel girder The concrete slab of the composite girder isreplaced with constraints in the numerical simulation Themotions in 119909 and 119910 directions of the top flange edge arerestrained to represent the lateral and torsional restrictionscaused by the concrete slab in practice The results of eachsimplified calculation method are listed in Tables 3 4 5 and6 and the error analyses of each simplifiedmethod are shownin Figures 8 9 10 and 11

The following conclusions can be drawn based on theresults presented in Tables 3 to 6 and Figures 8 to 11

(1) Under uniform negative bending moment the crit-ical bending buckling moment in the same cross-section of the composite girder is rarely affected by

8 Mathematical Problems in Engineering

Table 5 Critical distortional buckling moment of I-steel compositegirder under uniformly distributed load

Examplenumber

Distortional buckling critical moment119872crkNsdotmANSYS Williams Goltermann Svensson (19)

3 15512 15394 16388 22321 161606 5272 4988 5508 8729 551109 50383 45093 50001 56234 5235111 497 52610 54675 76289 5031712 459 47925 47925 67500 4453313 1299 198896 204768 288401 1291014 1267 165281 165645 239639 1269515 1255 131099 135351 190107 12454

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1605060708091011121314151617

Example number

WilliamsGoltermann

SvenssonThis paper

Mcr

of an

alyt

ical

met

hods

Mcr

of A

NSY

S

Figure 8 Precision analysis of simplified methods under negativeuniform moment

the length The critical bending buckling moment isnot obviously changed with the increased length ofthe structural component

(2) Under triangle bending moment the critical bendingbuckling moment is greatly affected by the lengththat is the value decreased quickly when the lengthincreases

(3) Under uniform negative bending moment trianglebending moment and uniform loads the resultsyielded by the calculation method in this papermatch well the finite element analysis results Thediscrepancy is limited within 5 which validates theaccuracy and applicability of this method

(4) Traditional calculation methods such as Svens-sonrsquos method Williamsrsquo method and Goltermannrsquosmethod have considerable deviations from the finiteelement method

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Example number

WilliamsGoltermann

SvenssonThis paper

05060708091011121314151617

Mcr

of an

alyt

ical

met

hods

Mcr

of A

NSY

S

Figure 9 Precision analysis of simplified methods under negativetriangular moment

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Example number

05060708091011121314151617

Mcr

of an

alyt

ical

met

hods

Mcr

of A

NSY

S

WilliamsGoltermann

SvenssonThis paper

Figure 10 Precision analysis of simplifiedmethods under uniformlydistributed load

Therefore the traditional elastic foundation beammethod taking into account the moment gradient needs tobe improved It is also suggested that the constant lateral andtorsional restraints in the traditional methods may lead tothe relative deviations

7 Conclusions

In this paper the traditional elastic foundation beammethodsare improved by considering the coupling effect of the exter-nal loads and the foundation spring constraints Based on thisimprovement a simplified calculation method computing thecritical buckling loads of steel-concrete composite girders is

Mathematical Problems in Engineering 9

Table 6 Critical distortional buckling moment of composite box girder under negative moment

Examplenumber

Distortional buckling critical moment119872crkNsdotmUniform negative bending moment Triangle negative bending moment Uniform loadANSYS (32) ANSYS (32) ANSYS (32)

1 72605 72214 79236 81741 79282 786742 72714 72214 76467 78046 76122 755313 72797 72214 75492 76616 75146 742624 90593 90722 97811 10125 97547 974365 90824 90650 94867 97143 94390 939166 90957 90610 93864 95600 93434 927857 53751 53575 58654 60316 58284 580398 53852 53407 56622 57636 55978 556429 53917 53411 55906 56601 55267 5487510 74221 73610 81013 83322 80861 8019511 74333 73610 78178 79555 77685 7678812 74417 73610 77181 78097 76702 7569813 12536 12479 13676 14125 13659 1359514 12556 12479 13201 13486 13114 1301715 12607 12479 13035 13239 12947 12833

090

092

094

096

098

100

102

104

106

108

110

Negative uniform moment Negative triangular momentUniformly distributed load

Mcr

of th

is pa

perM

crof

AN

SYS

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Example number

Figure 11 Precision analysis of simplified methods

developed The method is compared with various traditionalmethods The following conclusions are obtained

(1) There is a linear coupling relation between bothtorsional and lateral restraints and vertical loads

(2) Under uniformnegative bendingmoment the criticalbending buckling moment in the same cross-sectionof the composite girder is rarely affected by thelength The critical bending buckling moment is notobviously changed with the increased length of thestructural component

(3) Under triangle bending moment the critical bendingbuckling is influenced to a great extent by the lengththat is the value decreased quickly when the lengthincreases

(4) Under uniform negative bending moment trianglebending moment and uniform loads the calculationmethod proposed in this paper matches well thefinite element calculation methodThe discrepancy islimited within 5 which validates the applicability ofthis method

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research described in this paper was financially sup-ported by National Natural Science Foundation of China(51408449 and 51378502) and the Fundamental ResearchFunds for the Central Universities of China (2014-IV-049)

References

[1] S Kim andU Lee ldquoEffects of delamination on guidedwaves in asymmetric laminated composite beamrdquoMathematical Problemsin Engineering vol 2014 Article ID 956043 12 pages 2014

[2] D Champenoy A Corfdir and P Corfdir ldquoCalculating thecritical buckling force in compressed bottom flanges of steel-concrete composite bridgesrdquo European Journal of Environmen-tal and Civil Engineering vol 18 no 3 pp 271ndash292 2014

[3] T-H Yi H-N Li and M Gu ldquoOptimal sensor placement forstructural health monitoring based on multiple optimization

10 Mathematical Problems in Engineering

strategiesrdquo The Structural Design of Tall and Special Buildingsvol 20 no 7 pp 881ndash900 2011

[4] T-H Yi H-N Li and X-D Zhang ldquoA modified monkeyalgorithm for optimal sensor placement in structural healthmonitoringrdquo Smart Materials and Structures vol 21 no 10Article ID 105033 2012

[5] S Chen and Y Jia ldquoNumerical investigation of inelasticbuckling of steel-concrete composite beams prestressed withexternal tendonsrdquoThin-Walled Structures vol 48 no 3 pp 233ndash242 2010

[6] S E Svensson ldquoLateral buckling of beams analysed as elasticallysupported columns subject to a varying axial forcerdquo Journal ofConstructional Steel Research vol 5 no 3 pp 179ndash193 1985

[7] F WWilliams and A K Jemah ldquoBuckling curves for elasticallysupported columns with varying axial force to predict lateralbuckling of beamsrdquo Journal of Constructional Steel Research vol7 no 2 pp 133ndash147 1987

[8] P Goltermann and S Svensson ldquoLateral distortional bucklingpredicting elastic critical stressrdquo Journal of Structural Engineer-ing vol 114 no 7 pp 1606ndash1625 1988

[9] Swedish Institute of Steel Construction Swedish Code for Light-GaugeMetal Structures Swedish Institute of Steel ConstructionStockholm Sweden 1982

[10] British Standards Institution Code of Practice for Design of SteelBridge BS5400 Part 3 BSI London UK 1982

[11] RM Lawson andW J RackhamDesign of haunched compositebeams in buildings [MS thesis] Steel Construction InstitutionAscot UK 1989

[12] L Jiang J Qi A Scanlon and L Sun ldquoDistortional andlocal buckling of steel-concrete composite box-beamrdquo Steel andComposite Structures vol 14 no 3 pp 243ndash265 2013

[13] J-H Ye and W Chen ldquoElastic restrained distortional buck-ling of steel-concrete composite beams based on elasticallysupported column methodrdquo International Journal of StructuralStability and Dynamics vol 13 no 1 Article ID 1350001 pp 1ndash29 2013

[14] W Chen and J Ye ldquoElastic lateral and restrained distortionalbuckling of doubly symmetric I-beamsrdquo International Journalof Structural Stability and Dynamics vol 10 no 5 pp 983ndash10162010

[15] W-B Zhou L-Z Jiang G-Q Shao and Z-W Yu ldquoElastic dis-tortional buckling analysis of steel-concrete composite beamsin negative moment regionrdquo Journal of Central South University(Science and Technology) vol 43 no 6 pp 2316ndash2323 2012

[16] W B Zhou L Z Jiang and ZW Yu ldquoThe distortional bucklingcalculation formula of the steel-concrete composite beams inthe negative moment regionrdquo Chinese Journal of ComputationalMechanics vol 29 no 3 pp 446ndash450 2012

[17] J Ye andW Chen ldquoElastic restrained distortional buckling of I-steel-concrete composite beamsrdquo Journal of Building Structuresvol 32 no 6 pp 82ndash91 2011

[18] W Zhou L Jiang Z Liu et al ldquoClosed-form solution for shearlag effects of steel-concrete composite box beams consideringshear deformation and sliprdquo Journal of Central South Universityvol 19 no 10 pp 2976ndash2982 2012

[19] W-B Zhou L-Z Jiang andZ-W Yu ldquoAnalysis of free vibrationcharacteristic of steel-concrete composite box-girder consider-ing shear lag and sliprdquo Journal of Central South University vol20 no 9 pp 2570ndash2577 2013

[20] W-B Zhou L-Z Jiang Z-J Liu et al ldquoClosed-form solutionto thin-walled box girders considering effects of shear deforma-tion and shear lagrdquo Journal of Central South University vol 19no 9 pp 2650ndash2655 2012

[21] A Teodor A Guran and G Ardbeshir Theory of Elasticity forScientists and Engineers Springer New York NY USA 2012

[22] C Bi and V Ginting ldquoTwo-grid discontinuous Galerkinmethod for quasi-linear elliptic problemsrdquo Journal of ScientificComputing vol 49 no 3 pp 311ndash331 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

bottom flange is V(119909 119911) the left rotational angle is 120593119897(119911) andthe right rotational angle is 120593119903(119911) The boundary condition ofthe bottom flange is given as follows

V 1003816100381610038161003816119909=0 = 0 V11990910038161003816100381610038161003816119909=0

= 120593119897 (119911)

V100381610038161003816100381610038161003816119909=119887119891

= 0 V119909100381610038161003816100381610038161003816119909=119887119891

= 120593119903 (119911)

(26)

According to compatibility of deformation the displace-ment function of the bottom flange is

V =119887119891

2120587sin 120587119909

119887119891

(120593119897 minus 120593119903) +

119887119891

4120587sin 2120587119909

119887119891

(120593119897 + 120593119903) (27)

According to the principle of stationary potential energythe buckling characteristic equations are given as follows

119863119891(51205872120593119897

8119887119891

+31205872120593119903

8119887119891

minus

11988711989112059310158401015840

119897

2+

51198873

1198911205931015840101584010158401015840

119897

321205872minus

31198873

1198911205931015840101584010158401015840

119903

321205872)

+ (

51205901198601198911198872

1198911205931015840

119897

321205872)

1015840

minus (

31205901198601198911198872

1198911205931015840

119903

321205872)

1015840

+ 119896120593119897

120593119897 + 119891120593119909 = 0

2119896119909119906 + (1205901198601198911199061015840)1015840

+ 1198641198681199101199061015840101584010158401015840

minus 119891119909120593119897

minus 119891119909120593119903

= 0

119863119891(51205872120593119903

8119887119891

+31205872120593119897

8119887119891

minus

11988711989112059310158401015840

119903

2+

51198873

1198911205931015840101584010158401015840

119903

321205872minus

31198873

1198911205931015840101584010158401015840

119897

321205872)

+ (

51205901198601198911198872

1198911205931015840

119903

321205872)

1015840

minus (

31205901198601198911198872

1198911205931015840

119897

321205872)

1015840

+ 119896120593119903

120593119895 + 119891120593119909 = 0

(28)

where 119868119910 = 1199051198911198873

11989112 119905119891 is the thickness of bottom flange 119887119891

is the width of the bottom flange119863119891 = 1198641199053

11989112(1 minus 120583

2) 119896120593

119897

=

minus1198961205931198971 is torsional restraint stiffness of the bottom flange edge

by the left steel web 119896120593119903

= minus1198961205931199031 is torsional restraint stiffness

of the bottomflange edge by the right steel web and 119896119909 = minus1198961199091is lateral restraint stiffness of the bottom flange edge by thesteel web

According to the Galerkin method [15 16] we have

(

B1+ 119896120593119897

T minus 1205900N1 F + 06120590

0N1 Q

F + 061205900N1

B1+ 119896120593119903

T minus 1205900N1 Q

M M H1minus 1205900S1+ 2119896119909R)

times 120578 = 0

(29)

where 1198611119894119895 = 119865119894119895 = 119876119894119895 = 119872119894119895 = 1198671119894119895 = 0 (119894 = 119895) 1198611119894119894 =(5119863119891120587

211989716119887119891ℎ

2

119908) + (119863119891119887119891120573119897119894

24ℎ2

119908) + (5119863119891119887

3

11989112057321198971198944641205872ℎ2

119908)

119865119894119894 = (3119863119891120587211989716119887119891ℎ

2

119908) minus (3119863119891119887

3

11989112057321198971198944641205872ℎ2

119908) 119876119894119894 =

3(119863119908119897ℎ3

119908)+(1198971198631199081205831205732ℎ119908)119894

2 1198731119894119895 = (51198601198911198872

11989111988711989411989532120587

2ℎ2

119908119897) (119894 =

119895) 1198731119894119894 = (51198601198911198872

11989111989211989464ℎ

2

119908119897) 119872119894119894 = (3119863119908119897ℎ

3

119908) minus 119863119908((2 minus

120583)11989421198971205732ℎ119908) 1198671119894119894 = (119894

411986411986811991012057321198972) 1198781119894119894 = (119860119891120587

21198921198942119897) 1198781119894119895 =

(119860119891119887119894119895119897) (119894 = 119895) and 120578 = (C119879119897C119879119903D119879)119879

Table 1 Basic geometric size of I-steel composite girder cases

Examplenumber

Cross-sectionnumber

ℎ119908mm 119887

119891mm 119905

119891mm 119905

119908mm 119897mm

1 45002 1 600 200 12 12 72003 96004 30005 2 600 120 10 10 54006 84007 60008 3 719 268 25 16 102009 1440010 300011 4 300 100 3 3 540012 840013 420014 5 450 150 4 4 720015 10200

Since the constraint in the theoretical model is higherthan the real scenario the critical buckling stress is increasedTherefore the theoretical buckling deformation functions ofthe web and bottomflange cannot accurately describe the realbuckling deformation curves In order to eliminate errorsthe paper gives a reduction factor on the torsional restraintstiffness of the bottom flange and the reduction factor isfound to be 05 Combining (20) (21) (22) and (29) leadsto

[[[[

[

(

B F QF B QM M H

) minus 1205900(

N minus3N15

0minus3N15

N 00 0 S

)

]]]]

]

120578 = 0 (30)

where B = B02 + B1 N = N02 + N1 H = 2H0 +H1 and S = 2S0 + S1

The deformation vector 120578 cannot be zero when the buck-ling of the composite girder happensTherefore the bucklingof the composite girder can be solved by the generalizedeigenvalue of the characteristic matrix shown as follows

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

(

B F QF B QM M H

)minus 1205900(

N minus06N1 0minus06N1 N 0

0 0 S)

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

= 0 (31)

3119899 general eigenvalues can be obtained from (31) whichare 120590119905119894 (119894 = 1 2 3119899) let 120590cr = min120590119905119894 (119894 =

1 2 3119899) 120590cr is the critical buckling stress of the com-posite girderThe following equation can calculate the criticalbuckling moment

119872cr =120590cr (1198860 + 1198861120585 + 1198862120585

2) 119868

119910119888

(32)

Mathematical Problems in Engineering 7

Table 2 Basic geometric size of steel composite box girder cases

Example number ℎ119908mm 119887

119891mm 119887

119905mm 119905

119908mm 119905

119891mm 119905

119905mm 119897mm

1 40002 400 600 100 10 10 10 80003 120004 40005 400 500 100 10 10 10 80006 120007 40008 300 600 100 10 10 10 80009 1200010 400011 400 600 120 10 10 10 800012 1200013 400014 400 600 100 12 12 12 800015 12000

Table 3 Critical distortional buckling moment of I-steel compositegirder under negative uniform moment

Examplenumber

Distortional buckling critical moment119872crkNsdotmANSYS Williams Goltermann Svensson (19)

1 11291 10606 11936 15401 113832 12123 11480 12344 16647 120393 11497 10770 11971 15617 115504 39098 35402 40588 6196 404405 41019 37512 43686 6566 421616 40197 36603 46919 6406 414127 36727 31264 35211 38991 372008 38365 33904 40143 42281 387459 39051 32268 42454 40242 3932710 34155 33710 35235 489 3263211 35235 34884 36464 506 3358712 34729 34304 35951 497 3310613 94611 94325 101817 1368 8971314 107307 107811 102911 1563 1013115 91161 90153 93920 1307 86372

6 Analysis of Examples

Thegeometric dimensions of each example are listed inTables1 and 2 By means of the calculation method introduced inthis paper and the finite elementmethod the critical bucklinganalysis of the composite girder under uniform negativebending moment triangle bending moment and uniformloads can be carried out Svenssonrsquos method Williamsrsquomethod and Goltermannrsquos method are also employed in thecalculation of various I-steel-concrete composite girders soas to validate the calculation method proposed in this paperThe finite element analysis is conducted by using ANSYScommercial software Element SHELL43 is adopted to model

Table 4 Critical distortional buckling moment of I-steel compositegirder under negative triangular moment

Examplenumber

Distortional buckling critical moment119872crkNsdotmANSYS Williams Goltermann Svensson (19)

1 18686 18388 19714 26654 192522 17264 16805 17740 24367 173583 15909 15288 16213 22162 159744 6599 6327 7132 11072 703155 5866 5506 6056 9633 601566 5257 4846 5357 8481 538847 61252 55304 61444 68972 635348 55235 49455 54387 61674 560759 50604 44350 49232 55310 5119110 55183 59346 61236 860 5616711 51574 51935 53366 753 4946912 46366 46184 47412 670 4422413 131762 170748 181683 2476 1336114 129117 147906 150336 2144 1299815 127803 131099 134420 1901 12408

the steel girder The concrete slab of the composite girder isreplaced with constraints in the numerical simulation Themotions in 119909 and 119910 directions of the top flange edge arerestrained to represent the lateral and torsional restrictionscaused by the concrete slab in practice The results of eachsimplified calculation method are listed in Tables 3 4 5 and6 and the error analyses of each simplifiedmethod are shownin Figures 8 9 10 and 11

The following conclusions can be drawn based on theresults presented in Tables 3 to 6 and Figures 8 to 11

(1) Under uniform negative bending moment the crit-ical bending buckling moment in the same cross-section of the composite girder is rarely affected by

8 Mathematical Problems in Engineering

Table 5 Critical distortional buckling moment of I-steel compositegirder under uniformly distributed load

Examplenumber

Distortional buckling critical moment119872crkNsdotmANSYS Williams Goltermann Svensson (19)

3 15512 15394 16388 22321 161606 5272 4988 5508 8729 551109 50383 45093 50001 56234 5235111 497 52610 54675 76289 5031712 459 47925 47925 67500 4453313 1299 198896 204768 288401 1291014 1267 165281 165645 239639 1269515 1255 131099 135351 190107 12454

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1605060708091011121314151617

Example number

WilliamsGoltermann

SvenssonThis paper

Mcr

of an

alyt

ical

met

hods

Mcr

of A

NSY

S

Figure 8 Precision analysis of simplified methods under negativeuniform moment

the length The critical bending buckling moment isnot obviously changed with the increased length ofthe structural component

(2) Under triangle bending moment the critical bendingbuckling moment is greatly affected by the lengththat is the value decreased quickly when the lengthincreases

(3) Under uniform negative bending moment trianglebending moment and uniform loads the resultsyielded by the calculation method in this papermatch well the finite element analysis results Thediscrepancy is limited within 5 which validates theaccuracy and applicability of this method

(4) Traditional calculation methods such as Svens-sonrsquos method Williamsrsquo method and Goltermannrsquosmethod have considerable deviations from the finiteelement method

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Example number

WilliamsGoltermann

SvenssonThis paper

05060708091011121314151617

Mcr

of an

alyt

ical

met

hods

Mcr

of A

NSY

S

Figure 9 Precision analysis of simplified methods under negativetriangular moment

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Example number

05060708091011121314151617

Mcr

of an

alyt

ical

met

hods

Mcr

of A

NSY

S

WilliamsGoltermann

SvenssonThis paper

Figure 10 Precision analysis of simplifiedmethods under uniformlydistributed load

Therefore the traditional elastic foundation beammethod taking into account the moment gradient needs tobe improved It is also suggested that the constant lateral andtorsional restraints in the traditional methods may lead tothe relative deviations

7 Conclusions

In this paper the traditional elastic foundation beammethodsare improved by considering the coupling effect of the exter-nal loads and the foundation spring constraints Based on thisimprovement a simplified calculation method computing thecritical buckling loads of steel-concrete composite girders is

Mathematical Problems in Engineering 9

Table 6 Critical distortional buckling moment of composite box girder under negative moment

Examplenumber

Distortional buckling critical moment119872crkNsdotmUniform negative bending moment Triangle negative bending moment Uniform loadANSYS (32) ANSYS (32) ANSYS (32)

1 72605 72214 79236 81741 79282 786742 72714 72214 76467 78046 76122 755313 72797 72214 75492 76616 75146 742624 90593 90722 97811 10125 97547 974365 90824 90650 94867 97143 94390 939166 90957 90610 93864 95600 93434 927857 53751 53575 58654 60316 58284 580398 53852 53407 56622 57636 55978 556429 53917 53411 55906 56601 55267 5487510 74221 73610 81013 83322 80861 8019511 74333 73610 78178 79555 77685 7678812 74417 73610 77181 78097 76702 7569813 12536 12479 13676 14125 13659 1359514 12556 12479 13201 13486 13114 1301715 12607 12479 13035 13239 12947 12833

090

092

094

096

098

100

102

104

106

108

110

Negative uniform moment Negative triangular momentUniformly distributed load

Mcr

of th

is pa

perM

crof

AN

SYS

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Example number

Figure 11 Precision analysis of simplified methods

developed The method is compared with various traditionalmethods The following conclusions are obtained

(1) There is a linear coupling relation between bothtorsional and lateral restraints and vertical loads

(2) Under uniformnegative bendingmoment the criticalbending buckling moment in the same cross-sectionof the composite girder is rarely affected by thelength The critical bending buckling moment is notobviously changed with the increased length of thestructural component

(3) Under triangle bending moment the critical bendingbuckling is influenced to a great extent by the lengththat is the value decreased quickly when the lengthincreases

(4) Under uniform negative bending moment trianglebending moment and uniform loads the calculationmethod proposed in this paper matches well thefinite element calculation methodThe discrepancy islimited within 5 which validates the applicability ofthis method

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research described in this paper was financially sup-ported by National Natural Science Foundation of China(51408449 and 51378502) and the Fundamental ResearchFunds for the Central Universities of China (2014-IV-049)

References

[1] S Kim andU Lee ldquoEffects of delamination on guidedwaves in asymmetric laminated composite beamrdquoMathematical Problemsin Engineering vol 2014 Article ID 956043 12 pages 2014

[2] D Champenoy A Corfdir and P Corfdir ldquoCalculating thecritical buckling force in compressed bottom flanges of steel-concrete composite bridgesrdquo European Journal of Environmen-tal and Civil Engineering vol 18 no 3 pp 271ndash292 2014

[3] T-H Yi H-N Li and M Gu ldquoOptimal sensor placement forstructural health monitoring based on multiple optimization

10 Mathematical Problems in Engineering

strategiesrdquo The Structural Design of Tall and Special Buildingsvol 20 no 7 pp 881ndash900 2011

[4] T-H Yi H-N Li and X-D Zhang ldquoA modified monkeyalgorithm for optimal sensor placement in structural healthmonitoringrdquo Smart Materials and Structures vol 21 no 10Article ID 105033 2012

[5] S Chen and Y Jia ldquoNumerical investigation of inelasticbuckling of steel-concrete composite beams prestressed withexternal tendonsrdquoThin-Walled Structures vol 48 no 3 pp 233ndash242 2010

[6] S E Svensson ldquoLateral buckling of beams analysed as elasticallysupported columns subject to a varying axial forcerdquo Journal ofConstructional Steel Research vol 5 no 3 pp 179ndash193 1985

[7] F WWilliams and A K Jemah ldquoBuckling curves for elasticallysupported columns with varying axial force to predict lateralbuckling of beamsrdquo Journal of Constructional Steel Research vol7 no 2 pp 133ndash147 1987

[8] P Goltermann and S Svensson ldquoLateral distortional bucklingpredicting elastic critical stressrdquo Journal of Structural Engineer-ing vol 114 no 7 pp 1606ndash1625 1988

[9] Swedish Institute of Steel Construction Swedish Code for Light-GaugeMetal Structures Swedish Institute of Steel ConstructionStockholm Sweden 1982

[10] British Standards Institution Code of Practice for Design of SteelBridge BS5400 Part 3 BSI London UK 1982

[11] RM Lawson andW J RackhamDesign of haunched compositebeams in buildings [MS thesis] Steel Construction InstitutionAscot UK 1989

[12] L Jiang J Qi A Scanlon and L Sun ldquoDistortional andlocal buckling of steel-concrete composite box-beamrdquo Steel andComposite Structures vol 14 no 3 pp 243ndash265 2013

[13] J-H Ye and W Chen ldquoElastic restrained distortional buck-ling of steel-concrete composite beams based on elasticallysupported column methodrdquo International Journal of StructuralStability and Dynamics vol 13 no 1 Article ID 1350001 pp 1ndash29 2013

[14] W Chen and J Ye ldquoElastic lateral and restrained distortionalbuckling of doubly symmetric I-beamsrdquo International Journalof Structural Stability and Dynamics vol 10 no 5 pp 983ndash10162010

[15] W-B Zhou L-Z Jiang G-Q Shao and Z-W Yu ldquoElastic dis-tortional buckling analysis of steel-concrete composite beamsin negative moment regionrdquo Journal of Central South University(Science and Technology) vol 43 no 6 pp 2316ndash2323 2012

[16] W B Zhou L Z Jiang and ZW Yu ldquoThe distortional bucklingcalculation formula of the steel-concrete composite beams inthe negative moment regionrdquo Chinese Journal of ComputationalMechanics vol 29 no 3 pp 446ndash450 2012

[17] J Ye andW Chen ldquoElastic restrained distortional buckling of I-steel-concrete composite beamsrdquo Journal of Building Structuresvol 32 no 6 pp 82ndash91 2011

[18] W Zhou L Jiang Z Liu et al ldquoClosed-form solution for shearlag effects of steel-concrete composite box beams consideringshear deformation and sliprdquo Journal of Central South Universityvol 19 no 10 pp 2976ndash2982 2012

[19] W-B Zhou L-Z Jiang andZ-W Yu ldquoAnalysis of free vibrationcharacteristic of steel-concrete composite box-girder consider-ing shear lag and sliprdquo Journal of Central South University vol20 no 9 pp 2570ndash2577 2013

[20] W-B Zhou L-Z Jiang Z-J Liu et al ldquoClosed-form solutionto thin-walled box girders considering effects of shear deforma-tion and shear lagrdquo Journal of Central South University vol 19no 9 pp 2650ndash2655 2012

[21] A Teodor A Guran and G Ardbeshir Theory of Elasticity forScientists and Engineers Springer New York NY USA 2012

[22] C Bi and V Ginting ldquoTwo-grid discontinuous Galerkinmethod for quasi-linear elliptic problemsrdquo Journal of ScientificComputing vol 49 no 3 pp 311ndash331 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

Table 2 Basic geometric size of steel composite box girder cases

Example number ℎ119908mm 119887

119891mm 119887

119905mm 119905

119908mm 119905

119891mm 119905

119905mm 119897mm

1 40002 400 600 100 10 10 10 80003 120004 40005 400 500 100 10 10 10 80006 120007 40008 300 600 100 10 10 10 80009 1200010 400011 400 600 120 10 10 10 800012 1200013 400014 400 600 100 12 12 12 800015 12000

Table 3 Critical distortional buckling moment of I-steel compositegirder under negative uniform moment

Examplenumber

Distortional buckling critical moment119872crkNsdotmANSYS Williams Goltermann Svensson (19)

1 11291 10606 11936 15401 113832 12123 11480 12344 16647 120393 11497 10770 11971 15617 115504 39098 35402 40588 6196 404405 41019 37512 43686 6566 421616 40197 36603 46919 6406 414127 36727 31264 35211 38991 372008 38365 33904 40143 42281 387459 39051 32268 42454 40242 3932710 34155 33710 35235 489 3263211 35235 34884 36464 506 3358712 34729 34304 35951 497 3310613 94611 94325 101817 1368 8971314 107307 107811 102911 1563 1013115 91161 90153 93920 1307 86372

6 Analysis of Examples

Thegeometric dimensions of each example are listed inTables1 and 2 By means of the calculation method introduced inthis paper and the finite elementmethod the critical bucklinganalysis of the composite girder under uniform negativebending moment triangle bending moment and uniformloads can be carried out Svenssonrsquos method Williamsrsquomethod and Goltermannrsquos method are also employed in thecalculation of various I-steel-concrete composite girders soas to validate the calculation method proposed in this paperThe finite element analysis is conducted by using ANSYScommercial software Element SHELL43 is adopted to model

Table 4 Critical distortional buckling moment of I-steel compositegirder under negative triangular moment

Examplenumber

Distortional buckling critical moment119872crkNsdotmANSYS Williams Goltermann Svensson (19)

1 18686 18388 19714 26654 192522 17264 16805 17740 24367 173583 15909 15288 16213 22162 159744 6599 6327 7132 11072 703155 5866 5506 6056 9633 601566 5257 4846 5357 8481 538847 61252 55304 61444 68972 635348 55235 49455 54387 61674 560759 50604 44350 49232 55310 5119110 55183 59346 61236 860 5616711 51574 51935 53366 753 4946912 46366 46184 47412 670 4422413 131762 170748 181683 2476 1336114 129117 147906 150336 2144 1299815 127803 131099 134420 1901 12408

the steel girder The concrete slab of the composite girder isreplaced with constraints in the numerical simulation Themotions in 119909 and 119910 directions of the top flange edge arerestrained to represent the lateral and torsional restrictionscaused by the concrete slab in practice The results of eachsimplified calculation method are listed in Tables 3 4 5 and6 and the error analyses of each simplifiedmethod are shownin Figures 8 9 10 and 11

The following conclusions can be drawn based on theresults presented in Tables 3 to 6 and Figures 8 to 11

(1) Under uniform negative bending moment the crit-ical bending buckling moment in the same cross-section of the composite girder is rarely affected by

8 Mathematical Problems in Engineering

Table 5 Critical distortional buckling moment of I-steel compositegirder under uniformly distributed load

Examplenumber

Distortional buckling critical moment119872crkNsdotmANSYS Williams Goltermann Svensson (19)

3 15512 15394 16388 22321 161606 5272 4988 5508 8729 551109 50383 45093 50001 56234 5235111 497 52610 54675 76289 5031712 459 47925 47925 67500 4453313 1299 198896 204768 288401 1291014 1267 165281 165645 239639 1269515 1255 131099 135351 190107 12454

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1605060708091011121314151617

Example number

WilliamsGoltermann

SvenssonThis paper

Mcr

of an

alyt

ical

met

hods

Mcr

of A

NSY

S

Figure 8 Precision analysis of simplified methods under negativeuniform moment

the length The critical bending buckling moment isnot obviously changed with the increased length ofthe structural component

(2) Under triangle bending moment the critical bendingbuckling moment is greatly affected by the lengththat is the value decreased quickly when the lengthincreases

(3) Under uniform negative bending moment trianglebending moment and uniform loads the resultsyielded by the calculation method in this papermatch well the finite element analysis results Thediscrepancy is limited within 5 which validates theaccuracy and applicability of this method

(4) Traditional calculation methods such as Svens-sonrsquos method Williamsrsquo method and Goltermannrsquosmethod have considerable deviations from the finiteelement method

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Example number

WilliamsGoltermann

SvenssonThis paper

05060708091011121314151617

Mcr

of an

alyt

ical

met

hods

Mcr

of A

NSY

S

Figure 9 Precision analysis of simplified methods under negativetriangular moment

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Example number

05060708091011121314151617

Mcr

of an

alyt

ical

met

hods

Mcr

of A

NSY

S

WilliamsGoltermann

SvenssonThis paper

Figure 10 Precision analysis of simplifiedmethods under uniformlydistributed load

Therefore the traditional elastic foundation beammethod taking into account the moment gradient needs tobe improved It is also suggested that the constant lateral andtorsional restraints in the traditional methods may lead tothe relative deviations

7 Conclusions

In this paper the traditional elastic foundation beammethodsare improved by considering the coupling effect of the exter-nal loads and the foundation spring constraints Based on thisimprovement a simplified calculation method computing thecritical buckling loads of steel-concrete composite girders is

Mathematical Problems in Engineering 9

Table 6 Critical distortional buckling moment of composite box girder under negative moment

Examplenumber

Distortional buckling critical moment119872crkNsdotmUniform negative bending moment Triangle negative bending moment Uniform loadANSYS (32) ANSYS (32) ANSYS (32)

1 72605 72214 79236 81741 79282 786742 72714 72214 76467 78046 76122 755313 72797 72214 75492 76616 75146 742624 90593 90722 97811 10125 97547 974365 90824 90650 94867 97143 94390 939166 90957 90610 93864 95600 93434 927857 53751 53575 58654 60316 58284 580398 53852 53407 56622 57636 55978 556429 53917 53411 55906 56601 55267 5487510 74221 73610 81013 83322 80861 8019511 74333 73610 78178 79555 77685 7678812 74417 73610 77181 78097 76702 7569813 12536 12479 13676 14125 13659 1359514 12556 12479 13201 13486 13114 1301715 12607 12479 13035 13239 12947 12833

090

092

094

096

098

100

102

104

106

108

110

Negative uniform moment Negative triangular momentUniformly distributed load

Mcr

of th

is pa

perM

crof

AN

SYS

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Example number

Figure 11 Precision analysis of simplified methods

developed The method is compared with various traditionalmethods The following conclusions are obtained

(1) There is a linear coupling relation between bothtorsional and lateral restraints and vertical loads

(2) Under uniformnegative bendingmoment the criticalbending buckling moment in the same cross-sectionof the composite girder is rarely affected by thelength The critical bending buckling moment is notobviously changed with the increased length of thestructural component

(3) Under triangle bending moment the critical bendingbuckling is influenced to a great extent by the lengththat is the value decreased quickly when the lengthincreases

(4) Under uniform negative bending moment trianglebending moment and uniform loads the calculationmethod proposed in this paper matches well thefinite element calculation methodThe discrepancy islimited within 5 which validates the applicability ofthis method

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research described in this paper was financially sup-ported by National Natural Science Foundation of China(51408449 and 51378502) and the Fundamental ResearchFunds for the Central Universities of China (2014-IV-049)

References

[1] S Kim andU Lee ldquoEffects of delamination on guidedwaves in asymmetric laminated composite beamrdquoMathematical Problemsin Engineering vol 2014 Article ID 956043 12 pages 2014

[2] D Champenoy A Corfdir and P Corfdir ldquoCalculating thecritical buckling force in compressed bottom flanges of steel-concrete composite bridgesrdquo European Journal of Environmen-tal and Civil Engineering vol 18 no 3 pp 271ndash292 2014

[3] T-H Yi H-N Li and M Gu ldquoOptimal sensor placement forstructural health monitoring based on multiple optimization

10 Mathematical Problems in Engineering

strategiesrdquo The Structural Design of Tall and Special Buildingsvol 20 no 7 pp 881ndash900 2011

[4] T-H Yi H-N Li and X-D Zhang ldquoA modified monkeyalgorithm for optimal sensor placement in structural healthmonitoringrdquo Smart Materials and Structures vol 21 no 10Article ID 105033 2012

[5] S Chen and Y Jia ldquoNumerical investigation of inelasticbuckling of steel-concrete composite beams prestressed withexternal tendonsrdquoThin-Walled Structures vol 48 no 3 pp 233ndash242 2010

[6] S E Svensson ldquoLateral buckling of beams analysed as elasticallysupported columns subject to a varying axial forcerdquo Journal ofConstructional Steel Research vol 5 no 3 pp 179ndash193 1985

[7] F WWilliams and A K Jemah ldquoBuckling curves for elasticallysupported columns with varying axial force to predict lateralbuckling of beamsrdquo Journal of Constructional Steel Research vol7 no 2 pp 133ndash147 1987

[8] P Goltermann and S Svensson ldquoLateral distortional bucklingpredicting elastic critical stressrdquo Journal of Structural Engineer-ing vol 114 no 7 pp 1606ndash1625 1988

[9] Swedish Institute of Steel Construction Swedish Code for Light-GaugeMetal Structures Swedish Institute of Steel ConstructionStockholm Sweden 1982

[10] British Standards Institution Code of Practice for Design of SteelBridge BS5400 Part 3 BSI London UK 1982

[11] RM Lawson andW J RackhamDesign of haunched compositebeams in buildings [MS thesis] Steel Construction InstitutionAscot UK 1989

[12] L Jiang J Qi A Scanlon and L Sun ldquoDistortional andlocal buckling of steel-concrete composite box-beamrdquo Steel andComposite Structures vol 14 no 3 pp 243ndash265 2013

[13] J-H Ye and W Chen ldquoElastic restrained distortional buck-ling of steel-concrete composite beams based on elasticallysupported column methodrdquo International Journal of StructuralStability and Dynamics vol 13 no 1 Article ID 1350001 pp 1ndash29 2013

[14] W Chen and J Ye ldquoElastic lateral and restrained distortionalbuckling of doubly symmetric I-beamsrdquo International Journalof Structural Stability and Dynamics vol 10 no 5 pp 983ndash10162010

[15] W-B Zhou L-Z Jiang G-Q Shao and Z-W Yu ldquoElastic dis-tortional buckling analysis of steel-concrete composite beamsin negative moment regionrdquo Journal of Central South University(Science and Technology) vol 43 no 6 pp 2316ndash2323 2012

[16] W B Zhou L Z Jiang and ZW Yu ldquoThe distortional bucklingcalculation formula of the steel-concrete composite beams inthe negative moment regionrdquo Chinese Journal of ComputationalMechanics vol 29 no 3 pp 446ndash450 2012

[17] J Ye andW Chen ldquoElastic restrained distortional buckling of I-steel-concrete composite beamsrdquo Journal of Building Structuresvol 32 no 6 pp 82ndash91 2011

[18] W Zhou L Jiang Z Liu et al ldquoClosed-form solution for shearlag effects of steel-concrete composite box beams consideringshear deformation and sliprdquo Journal of Central South Universityvol 19 no 10 pp 2976ndash2982 2012

[19] W-B Zhou L-Z Jiang andZ-W Yu ldquoAnalysis of free vibrationcharacteristic of steel-concrete composite box-girder consider-ing shear lag and sliprdquo Journal of Central South University vol20 no 9 pp 2570ndash2577 2013

[20] W-B Zhou L-Z Jiang Z-J Liu et al ldquoClosed-form solutionto thin-walled box girders considering effects of shear deforma-tion and shear lagrdquo Journal of Central South University vol 19no 9 pp 2650ndash2655 2012

[21] A Teodor A Guran and G Ardbeshir Theory of Elasticity forScientists and Engineers Springer New York NY USA 2012

[22] C Bi and V Ginting ldquoTwo-grid discontinuous Galerkinmethod for quasi-linear elliptic problemsrdquo Journal of ScientificComputing vol 49 no 3 pp 311ndash331 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

Table 5 Critical distortional buckling moment of I-steel compositegirder under uniformly distributed load

Examplenumber

Distortional buckling critical moment119872crkNsdotmANSYS Williams Goltermann Svensson (19)

3 15512 15394 16388 22321 161606 5272 4988 5508 8729 551109 50383 45093 50001 56234 5235111 497 52610 54675 76289 5031712 459 47925 47925 67500 4453313 1299 198896 204768 288401 1291014 1267 165281 165645 239639 1269515 1255 131099 135351 190107 12454

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1605060708091011121314151617

Example number

WilliamsGoltermann

SvenssonThis paper

Mcr

of an

alyt

ical

met

hods

Mcr

of A

NSY

S

Figure 8 Precision analysis of simplified methods under negativeuniform moment

the length The critical bending buckling moment isnot obviously changed with the increased length ofthe structural component

(2) Under triangle bending moment the critical bendingbuckling moment is greatly affected by the lengththat is the value decreased quickly when the lengthincreases

(3) Under uniform negative bending moment trianglebending moment and uniform loads the resultsyielded by the calculation method in this papermatch well the finite element analysis results Thediscrepancy is limited within 5 which validates theaccuracy and applicability of this method

(4) Traditional calculation methods such as Svens-sonrsquos method Williamsrsquo method and Goltermannrsquosmethod have considerable deviations from the finiteelement method

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Example number

WilliamsGoltermann

SvenssonThis paper

05060708091011121314151617

Mcr

of an

alyt

ical

met

hods

Mcr

of A

NSY

S

Figure 9 Precision analysis of simplified methods under negativetriangular moment

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Example number

05060708091011121314151617

Mcr

of an

alyt

ical

met

hods

Mcr

of A

NSY

S

WilliamsGoltermann

SvenssonThis paper

Figure 10 Precision analysis of simplifiedmethods under uniformlydistributed load

Therefore the traditional elastic foundation beammethod taking into account the moment gradient needs tobe improved It is also suggested that the constant lateral andtorsional restraints in the traditional methods may lead tothe relative deviations

7 Conclusions

In this paper the traditional elastic foundation beammethodsare improved by considering the coupling effect of the exter-nal loads and the foundation spring constraints Based on thisimprovement a simplified calculation method computing thecritical buckling loads of steel-concrete composite girders is

Mathematical Problems in Engineering 9

Table 6 Critical distortional buckling moment of composite box girder under negative moment

Examplenumber

Distortional buckling critical moment119872crkNsdotmUniform negative bending moment Triangle negative bending moment Uniform loadANSYS (32) ANSYS (32) ANSYS (32)

1 72605 72214 79236 81741 79282 786742 72714 72214 76467 78046 76122 755313 72797 72214 75492 76616 75146 742624 90593 90722 97811 10125 97547 974365 90824 90650 94867 97143 94390 939166 90957 90610 93864 95600 93434 927857 53751 53575 58654 60316 58284 580398 53852 53407 56622 57636 55978 556429 53917 53411 55906 56601 55267 5487510 74221 73610 81013 83322 80861 8019511 74333 73610 78178 79555 77685 7678812 74417 73610 77181 78097 76702 7569813 12536 12479 13676 14125 13659 1359514 12556 12479 13201 13486 13114 1301715 12607 12479 13035 13239 12947 12833

090

092

094

096

098

100

102

104

106

108

110

Negative uniform moment Negative triangular momentUniformly distributed load

Mcr

of th

is pa

perM

crof

AN

SYS

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Example number

Figure 11 Precision analysis of simplified methods

developed The method is compared with various traditionalmethods The following conclusions are obtained

(1) There is a linear coupling relation between bothtorsional and lateral restraints and vertical loads

(2) Under uniformnegative bendingmoment the criticalbending buckling moment in the same cross-sectionof the composite girder is rarely affected by thelength The critical bending buckling moment is notobviously changed with the increased length of thestructural component

(3) Under triangle bending moment the critical bendingbuckling is influenced to a great extent by the lengththat is the value decreased quickly when the lengthincreases

(4) Under uniform negative bending moment trianglebending moment and uniform loads the calculationmethod proposed in this paper matches well thefinite element calculation methodThe discrepancy islimited within 5 which validates the applicability ofthis method

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research described in this paper was financially sup-ported by National Natural Science Foundation of China(51408449 and 51378502) and the Fundamental ResearchFunds for the Central Universities of China (2014-IV-049)

References

[1] S Kim andU Lee ldquoEffects of delamination on guidedwaves in asymmetric laminated composite beamrdquoMathematical Problemsin Engineering vol 2014 Article ID 956043 12 pages 2014

[2] D Champenoy A Corfdir and P Corfdir ldquoCalculating thecritical buckling force in compressed bottom flanges of steel-concrete composite bridgesrdquo European Journal of Environmen-tal and Civil Engineering vol 18 no 3 pp 271ndash292 2014

[3] T-H Yi H-N Li and M Gu ldquoOptimal sensor placement forstructural health monitoring based on multiple optimization

10 Mathematical Problems in Engineering

strategiesrdquo The Structural Design of Tall and Special Buildingsvol 20 no 7 pp 881ndash900 2011

[4] T-H Yi H-N Li and X-D Zhang ldquoA modified monkeyalgorithm for optimal sensor placement in structural healthmonitoringrdquo Smart Materials and Structures vol 21 no 10Article ID 105033 2012

[5] S Chen and Y Jia ldquoNumerical investigation of inelasticbuckling of steel-concrete composite beams prestressed withexternal tendonsrdquoThin-Walled Structures vol 48 no 3 pp 233ndash242 2010

[6] S E Svensson ldquoLateral buckling of beams analysed as elasticallysupported columns subject to a varying axial forcerdquo Journal ofConstructional Steel Research vol 5 no 3 pp 179ndash193 1985

[7] F WWilliams and A K Jemah ldquoBuckling curves for elasticallysupported columns with varying axial force to predict lateralbuckling of beamsrdquo Journal of Constructional Steel Research vol7 no 2 pp 133ndash147 1987

[8] P Goltermann and S Svensson ldquoLateral distortional bucklingpredicting elastic critical stressrdquo Journal of Structural Engineer-ing vol 114 no 7 pp 1606ndash1625 1988

[9] Swedish Institute of Steel Construction Swedish Code for Light-GaugeMetal Structures Swedish Institute of Steel ConstructionStockholm Sweden 1982

[10] British Standards Institution Code of Practice for Design of SteelBridge BS5400 Part 3 BSI London UK 1982

[11] RM Lawson andW J RackhamDesign of haunched compositebeams in buildings [MS thesis] Steel Construction InstitutionAscot UK 1989

[12] L Jiang J Qi A Scanlon and L Sun ldquoDistortional andlocal buckling of steel-concrete composite box-beamrdquo Steel andComposite Structures vol 14 no 3 pp 243ndash265 2013

[13] J-H Ye and W Chen ldquoElastic restrained distortional buck-ling of steel-concrete composite beams based on elasticallysupported column methodrdquo International Journal of StructuralStability and Dynamics vol 13 no 1 Article ID 1350001 pp 1ndash29 2013

[14] W Chen and J Ye ldquoElastic lateral and restrained distortionalbuckling of doubly symmetric I-beamsrdquo International Journalof Structural Stability and Dynamics vol 10 no 5 pp 983ndash10162010

[15] W-B Zhou L-Z Jiang G-Q Shao and Z-W Yu ldquoElastic dis-tortional buckling analysis of steel-concrete composite beamsin negative moment regionrdquo Journal of Central South University(Science and Technology) vol 43 no 6 pp 2316ndash2323 2012

[16] W B Zhou L Z Jiang and ZW Yu ldquoThe distortional bucklingcalculation formula of the steel-concrete composite beams inthe negative moment regionrdquo Chinese Journal of ComputationalMechanics vol 29 no 3 pp 446ndash450 2012

[17] J Ye andW Chen ldquoElastic restrained distortional buckling of I-steel-concrete composite beamsrdquo Journal of Building Structuresvol 32 no 6 pp 82ndash91 2011

[18] W Zhou L Jiang Z Liu et al ldquoClosed-form solution for shearlag effects of steel-concrete composite box beams consideringshear deformation and sliprdquo Journal of Central South Universityvol 19 no 10 pp 2976ndash2982 2012

[19] W-B Zhou L-Z Jiang andZ-W Yu ldquoAnalysis of free vibrationcharacteristic of steel-concrete composite box-girder consider-ing shear lag and sliprdquo Journal of Central South University vol20 no 9 pp 2570ndash2577 2013

[20] W-B Zhou L-Z Jiang Z-J Liu et al ldquoClosed-form solutionto thin-walled box girders considering effects of shear deforma-tion and shear lagrdquo Journal of Central South University vol 19no 9 pp 2650ndash2655 2012

[21] A Teodor A Guran and G Ardbeshir Theory of Elasticity forScientists and Engineers Springer New York NY USA 2012

[22] C Bi and V Ginting ldquoTwo-grid discontinuous Galerkinmethod for quasi-linear elliptic problemsrdquo Journal of ScientificComputing vol 49 no 3 pp 311ndash331 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

Table 6 Critical distortional buckling moment of composite box girder under negative moment

Examplenumber

Distortional buckling critical moment119872crkNsdotmUniform negative bending moment Triangle negative bending moment Uniform loadANSYS (32) ANSYS (32) ANSYS (32)

1 72605 72214 79236 81741 79282 786742 72714 72214 76467 78046 76122 755313 72797 72214 75492 76616 75146 742624 90593 90722 97811 10125 97547 974365 90824 90650 94867 97143 94390 939166 90957 90610 93864 95600 93434 927857 53751 53575 58654 60316 58284 580398 53852 53407 56622 57636 55978 556429 53917 53411 55906 56601 55267 5487510 74221 73610 81013 83322 80861 8019511 74333 73610 78178 79555 77685 7678812 74417 73610 77181 78097 76702 7569813 12536 12479 13676 14125 13659 1359514 12556 12479 13201 13486 13114 1301715 12607 12479 13035 13239 12947 12833

090

092

094

096

098

100

102

104

106

108

110

Negative uniform moment Negative triangular momentUniformly distributed load

Mcr

of th

is pa

perM

crof

AN

SYS

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16Example number

Figure 11 Precision analysis of simplified methods

developed The method is compared with various traditionalmethods The following conclusions are obtained

(1) There is a linear coupling relation between bothtorsional and lateral restraints and vertical loads

(2) Under uniformnegative bendingmoment the criticalbending buckling moment in the same cross-sectionof the composite girder is rarely affected by thelength The critical bending buckling moment is notobviously changed with the increased length of thestructural component

(3) Under triangle bending moment the critical bendingbuckling is influenced to a great extent by the lengththat is the value decreased quickly when the lengthincreases

(4) Under uniform negative bending moment trianglebending moment and uniform loads the calculationmethod proposed in this paper matches well thefinite element calculation methodThe discrepancy islimited within 5 which validates the applicability ofthis method

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The research described in this paper was financially sup-ported by National Natural Science Foundation of China(51408449 and 51378502) and the Fundamental ResearchFunds for the Central Universities of China (2014-IV-049)

References

[1] S Kim andU Lee ldquoEffects of delamination on guidedwaves in asymmetric laminated composite beamrdquoMathematical Problemsin Engineering vol 2014 Article ID 956043 12 pages 2014

[2] D Champenoy A Corfdir and P Corfdir ldquoCalculating thecritical buckling force in compressed bottom flanges of steel-concrete composite bridgesrdquo European Journal of Environmen-tal and Civil Engineering vol 18 no 3 pp 271ndash292 2014

[3] T-H Yi H-N Li and M Gu ldquoOptimal sensor placement forstructural health monitoring based on multiple optimization

10 Mathematical Problems in Engineering

strategiesrdquo The Structural Design of Tall and Special Buildingsvol 20 no 7 pp 881ndash900 2011

[4] T-H Yi H-N Li and X-D Zhang ldquoA modified monkeyalgorithm for optimal sensor placement in structural healthmonitoringrdquo Smart Materials and Structures vol 21 no 10Article ID 105033 2012

[5] S Chen and Y Jia ldquoNumerical investigation of inelasticbuckling of steel-concrete composite beams prestressed withexternal tendonsrdquoThin-Walled Structures vol 48 no 3 pp 233ndash242 2010

[6] S E Svensson ldquoLateral buckling of beams analysed as elasticallysupported columns subject to a varying axial forcerdquo Journal ofConstructional Steel Research vol 5 no 3 pp 179ndash193 1985

[7] F WWilliams and A K Jemah ldquoBuckling curves for elasticallysupported columns with varying axial force to predict lateralbuckling of beamsrdquo Journal of Constructional Steel Research vol7 no 2 pp 133ndash147 1987

[8] P Goltermann and S Svensson ldquoLateral distortional bucklingpredicting elastic critical stressrdquo Journal of Structural Engineer-ing vol 114 no 7 pp 1606ndash1625 1988

[9] Swedish Institute of Steel Construction Swedish Code for Light-GaugeMetal Structures Swedish Institute of Steel ConstructionStockholm Sweden 1982

[10] British Standards Institution Code of Practice for Design of SteelBridge BS5400 Part 3 BSI London UK 1982

[11] RM Lawson andW J RackhamDesign of haunched compositebeams in buildings [MS thesis] Steel Construction InstitutionAscot UK 1989

[12] L Jiang J Qi A Scanlon and L Sun ldquoDistortional andlocal buckling of steel-concrete composite box-beamrdquo Steel andComposite Structures vol 14 no 3 pp 243ndash265 2013

[13] J-H Ye and W Chen ldquoElastic restrained distortional buck-ling of steel-concrete composite beams based on elasticallysupported column methodrdquo International Journal of StructuralStability and Dynamics vol 13 no 1 Article ID 1350001 pp 1ndash29 2013

[14] W Chen and J Ye ldquoElastic lateral and restrained distortionalbuckling of doubly symmetric I-beamsrdquo International Journalof Structural Stability and Dynamics vol 10 no 5 pp 983ndash10162010

[15] W-B Zhou L-Z Jiang G-Q Shao and Z-W Yu ldquoElastic dis-tortional buckling analysis of steel-concrete composite beamsin negative moment regionrdquo Journal of Central South University(Science and Technology) vol 43 no 6 pp 2316ndash2323 2012

[16] W B Zhou L Z Jiang and ZW Yu ldquoThe distortional bucklingcalculation formula of the steel-concrete composite beams inthe negative moment regionrdquo Chinese Journal of ComputationalMechanics vol 29 no 3 pp 446ndash450 2012

[17] J Ye andW Chen ldquoElastic restrained distortional buckling of I-steel-concrete composite beamsrdquo Journal of Building Structuresvol 32 no 6 pp 82ndash91 2011

[18] W Zhou L Jiang Z Liu et al ldquoClosed-form solution for shearlag effects of steel-concrete composite box beams consideringshear deformation and sliprdquo Journal of Central South Universityvol 19 no 10 pp 2976ndash2982 2012

[19] W-B Zhou L-Z Jiang andZ-W Yu ldquoAnalysis of free vibrationcharacteristic of steel-concrete composite box-girder consider-ing shear lag and sliprdquo Journal of Central South University vol20 no 9 pp 2570ndash2577 2013

[20] W-B Zhou L-Z Jiang Z-J Liu et al ldquoClosed-form solutionto thin-walled box girders considering effects of shear deforma-tion and shear lagrdquo Journal of Central South University vol 19no 9 pp 2650ndash2655 2012

[21] A Teodor A Guran and G Ardbeshir Theory of Elasticity forScientists and Engineers Springer New York NY USA 2012

[22] C Bi and V Ginting ldquoTwo-grid discontinuous Galerkinmethod for quasi-linear elliptic problemsrdquo Journal of ScientificComputing vol 49 no 3 pp 311ndash331 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

strategiesrdquo The Structural Design of Tall and Special Buildingsvol 20 no 7 pp 881ndash900 2011

[4] T-H Yi H-N Li and X-D Zhang ldquoA modified monkeyalgorithm for optimal sensor placement in structural healthmonitoringrdquo Smart Materials and Structures vol 21 no 10Article ID 105033 2012

[5] S Chen and Y Jia ldquoNumerical investigation of inelasticbuckling of steel-concrete composite beams prestressed withexternal tendonsrdquoThin-Walled Structures vol 48 no 3 pp 233ndash242 2010

[6] S E Svensson ldquoLateral buckling of beams analysed as elasticallysupported columns subject to a varying axial forcerdquo Journal ofConstructional Steel Research vol 5 no 3 pp 179ndash193 1985

[7] F WWilliams and A K Jemah ldquoBuckling curves for elasticallysupported columns with varying axial force to predict lateralbuckling of beamsrdquo Journal of Constructional Steel Research vol7 no 2 pp 133ndash147 1987

[8] P Goltermann and S Svensson ldquoLateral distortional bucklingpredicting elastic critical stressrdquo Journal of Structural Engineer-ing vol 114 no 7 pp 1606ndash1625 1988

[9] Swedish Institute of Steel Construction Swedish Code for Light-GaugeMetal Structures Swedish Institute of Steel ConstructionStockholm Sweden 1982

[10] British Standards Institution Code of Practice for Design of SteelBridge BS5400 Part 3 BSI London UK 1982

[11] RM Lawson andW J RackhamDesign of haunched compositebeams in buildings [MS thesis] Steel Construction InstitutionAscot UK 1989

[12] L Jiang J Qi A Scanlon and L Sun ldquoDistortional andlocal buckling of steel-concrete composite box-beamrdquo Steel andComposite Structures vol 14 no 3 pp 243ndash265 2013

[13] J-H Ye and W Chen ldquoElastic restrained distortional buck-ling of steel-concrete composite beams based on elasticallysupported column methodrdquo International Journal of StructuralStability and Dynamics vol 13 no 1 Article ID 1350001 pp 1ndash29 2013

[14] W Chen and J Ye ldquoElastic lateral and restrained distortionalbuckling of doubly symmetric I-beamsrdquo International Journalof Structural Stability and Dynamics vol 10 no 5 pp 983ndash10162010

[15] W-B Zhou L-Z Jiang G-Q Shao and Z-W Yu ldquoElastic dis-tortional buckling analysis of steel-concrete composite beamsin negative moment regionrdquo Journal of Central South University(Science and Technology) vol 43 no 6 pp 2316ndash2323 2012

[16] W B Zhou L Z Jiang and ZW Yu ldquoThe distortional bucklingcalculation formula of the steel-concrete composite beams inthe negative moment regionrdquo Chinese Journal of ComputationalMechanics vol 29 no 3 pp 446ndash450 2012

[17] J Ye andW Chen ldquoElastic restrained distortional buckling of I-steel-concrete composite beamsrdquo Journal of Building Structuresvol 32 no 6 pp 82ndash91 2011

[18] W Zhou L Jiang Z Liu et al ldquoClosed-form solution for shearlag effects of steel-concrete composite box beams consideringshear deformation and sliprdquo Journal of Central South Universityvol 19 no 10 pp 2976ndash2982 2012

[19] W-B Zhou L-Z Jiang andZ-W Yu ldquoAnalysis of free vibrationcharacteristic of steel-concrete composite box-girder consider-ing shear lag and sliprdquo Journal of Central South University vol20 no 9 pp 2570ndash2577 2013

[20] W-B Zhou L-Z Jiang Z-J Liu et al ldquoClosed-form solutionto thin-walled box girders considering effects of shear deforma-tion and shear lagrdquo Journal of Central South University vol 19no 9 pp 2650ndash2655 2012

[21] A Teodor A Guran and G Ardbeshir Theory of Elasticity forScientists and Engineers Springer New York NY USA 2012

[22] C Bi and V Ginting ldquoTwo-grid discontinuous Galerkinmethod for quasi-linear elliptic problemsrdquo Journal of ScientificComputing vol 49 no 3 pp 311ndash331 2011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of