Research Article Distortional Buckling Analysis of Steel ...Research Article Distortional Buckling Analysis of Steel-Concrete Composite Girders in Negative Moment Area ZhouWangbao,
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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
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
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
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
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
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
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]
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
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
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
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
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
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
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
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
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)
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)
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
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)
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
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
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
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
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
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
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
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]
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
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
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
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
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
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
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
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
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)
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)
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
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)
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
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
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
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
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]
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
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
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
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
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
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
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
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
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)
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)
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
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)
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
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]
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
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
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
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
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
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
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
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
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)
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)
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
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)
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
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
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
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
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
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
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
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
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)
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)
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
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)
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
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
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
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
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
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)
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)
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
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)
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
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)
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)
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
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)
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
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
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)
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
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
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