Research Article A Trigonometric Analytical …downloads.hindawi.com/journals/mpe/2016/2465025.pdfResearch Article A Trigonometric Analytical Solution of Simply Supported Horizontally

Research ArticleA Trigonometric Analytical Solution of SimplySupported Horizontally Curved Composite I-Beamconsidering Tangential Slips

Qin Xu-xi Liu Han-bing Wu Chun-li and Gu Zheng-wei

College of Traffic Jilin University Changchun 130025 China

Correspondence should be addressed to Wu Chun-li clwujlueducn

Received 27 March 2016 Revised 31 July 2016 Accepted 14 August 2016

Academic Editor Leonid Shaikhet

Copyright copy 2016 Qin Xu-xi et al This 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

This paper presents an analytical solution of the simply supported horizontally composite curved I-beam by trigonometric seriesconsidering the effect of partial interaction in the tangential direction Governing equations and boundary conditions are obtainedby using the Vlasov curved beam theory and the principle of minimum potential energyThe deflection functions and the Lagrangemultiplier functions are expressed as trigonometric series to satisfy the governing equations and the simply supported constraintsat both ends The numerical results of deflections and forces which are obtained by this method are compared with both FEMresults and experimental results and the inaccuracy between the analytical solutions in this paper and the FEM results is small andreasonable

1 Introduction

A composite beam can be defined as ldquopartially compositebeamrdquo when the number of shear connectors is less thanthe required number for fully composite design thereforethe interface shear force is limited by the strength of shearconnectors In contrast to fully composite beam slip betweenlayers can be significant and results in a decrease in theelastic stiffness of partially composite beam Compositebeam exhibiting partial shear interaction will stand largerdeflection than the beam exhibiting full shear interactionBy assuming complete shear interaction the calculationof deflection for partial shear interaction beam is maybeunderestimated Because serviceability issues often governthe structural design of composite section the accuratecalculation of deflection is critical The partial interaction isapplied not only in steel-concrete composite beam but alsoin other types of composite beams such as layered woodenbeams wood-concrete floor systems and other multilayeredlaminated composite structures [1 2]

Earlier studies on behavior of the partially compositebeam aremostly focused on the straight composite beamThefirst paper dealing with the analysis of composite beam with

partial interaction has been completed by Newmark et al[3] After that Goodman and Popkov [4 5] have conductedthe analytical and numerical research on the relative slipbetween layers and found that the relative slip between layershas a significant effect on the overall characteristics of thecomposite beam with the reduction of shear connectorsrsquostiffness Girhammar et al [6 7] have applied a partial shearinteraction theory for composite beam subjected to staticloads and dynamic loads Wang [8] has developed a methodto calculate deflection of partially composite beam basedon the stiffness of the shear connectors DallrsquoAsta [9] hasdeveloped a three-dimensional theory for composite beamwith partial shear interaction dealing with combination ofbending in the symmetry plane torsion and transversebending in the plane parallel to the shear connector interfaceNie and Cai [10] have studied the effects of shear slip onthe deflection of steel-concrete composite beam Ranzi andBradford [11] have presented analytical solutions for timedependent behavior of partially composite beam Liu etal [12] have found out the solution of shearing slip forsteel-concrete composite beam under the concentrated loadCampi and Monetto [13] have presented a new formulationto analyze two-layer linearly elastic Timoshenko beam with

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 2465025 12 pageshttpdxdoiorg10115520162465025

2 Mathematical Problems in Engineering

interlayer slip In the aspect of numerical simulations studiesdifferent kinds of numerical and finite element formulationsfor the analysis of composite beam with interlayer slip havebeen suggested [1 14ndash19]

Many significant researches have been accomplished inregard to the behavior of straight partially composite beamIn the same time various scholars have done researches onthe curved beam theory One of the earliest works dealingwith the stability behaviors of curved beam has been putforward by Vlasov [20] After that scholars covered differentextensions and enhancements to the Vlasov model [21ndash26] These researches are helpful to the development of thecomposite beam theory However very little literature hasfocused on the curved partially composite beam such ashorizontally composite curved steel I-beam bridge Theven-dran et al [27] and Shanmugam et al [28] have conductedexperiments on the steel-concrete composite curved beamto investigate the ultimate load behavior In their study thefinite element software ABAQUS was used to analyze thebehavior of test specimens Full composite action betweensteel beam and the concrete slab was assumed The resultsof deformations stress distributions and ultimate strengthsobtained by finite element analysis were found to be in goodagreement with the experimental results After that Topkayaet al [29] have conducted experimental and numericalstudies to establish the behavior of composite curved bridgeduring construction In their study two FEM models wereestablished by different kinds of software to predict thebehavior of the curved steel trapezoidal box-girder and theauthors have drawn a conclusion that the reasonable finiteelement model is able to accurately capture girder behaviorduring construction Giussani and Mola [30] have developedan analytical equation for elastic composite beam curved in-plane with the long term behavior But the partial interactionbetween the steel girder and concrete slab was not consideredin the study Erkmen and Bradford [31] have solved theequation of the composite curved beam considering thetwo-layer partial interaction by providing a highly efficient3D beam finite element The results demonstrate that thedeveloped formulation is accurate and effective in capturingthe behavior of composite beams curved in-plane

Even though someprevious researches of these evaluationmethodologies have been accomplished there is still lack ofthe fundamental understanding of the system-level behavioron the overall performance of composite curved beam withpartial shear interaction Besides although exact solutionscan be obtained by the 3D finite element model it is verycomplex and time consuming Therefore the FEM analysisprocedures are not suitable for the initial design This studyaims to provide an analytical theory of the horizontallycomposite curved beam considering the partial interactionin tangential direction The beam is assumed to be staticallydeterminatewith a constant radius of curvature along the lon-gitudinal axis Governing equations and boundary conditionsare obtained by using both the Vlasov curved beam theoryand the energy variation principleTheundetermined verticaldeflection torsional deflection and Lagrange multipliers willbe approximated by Fourier series to solve the governingequations of the partial interaction composite beam theory

q

y

x

120579

120593

z = R lowast 120593

m

Figure 1 Model of a horizontally composite curved I-beam

in the procedures The numerical results of deflections andforces obtained by using proposed theory are presentedand compared with FEM results and experimental resultsComparison results show that the calculation can be easilyand accurately handled which is a great advantage of thismethod

2 Basic Assumptions and Conditions

This study is on a horizontally composite curved I-beamwhere the model is shown as in Figure 1 and the noteworthyfeatures of this research are shown as follows

(1) The slab and I-girder are linear-elastic different mate-rials and the cross-sections made of both mate-rials are rigid in their plane The effects of sheardeformation warping deformation and distortiondeformation are neglected in this research Structuralanalysis of the beam is based on the Vlasov curvedbeam theory (for each part of the beam)

(2) The interlayer connectors between the slab and I-girder are flexible and they are continuous in tan-gential direction and rigid in radial direction Theload-slip behavior (per unit length) of connectors intangential direction is described in a linear-elasticrange with a constant slip modulus 119870[Nm2]

(3) The uplift between the slab and I-girder is neglectedThe radius of curvature is constant along the beam

The deflections in tangential direction vertical direc-tion and radial direction (119911-direction 119910-direction and 119909-direction) are indicated as 119906 119908 and V respectively TheI-girder has the same torsional deflection 120601 and verticaldeflection 119908 as the slab has In this paper subscript ldquo1rdquorefers to the cross-section of the slab and subscript ldquo2rdquo refersto the cross-section of I-girder such that the deflectionsof the slab and I-girder can be represented as (119908 120601 1199061 V1)

Mathematical Problems in Engineering 3

and (119908 120601 1199062 V2) respectively The geometric relationshipsbetween strains and deflections may be written as

120576119911119894 = 1199061015840119894 minus V119894119877 119896119910119894 = V10158401015840119894 + 1199061015840119894119877 119896119909119894 = 11990810158401015840 minus 120601119877119896119911119894 = 1206011015840 + 1199081015840119877 (119894 = 1 2)

(1)

where 120576119911119894 is the axial strain 119896119910119894 is the curvature in119910-direction119896119909119894 is the curvature in 119909-direction and 119896119911119894 is the curvature in119911-direction Where ()1015840 = 119889()119889119911 ()10158401015840 = 1198892()1198891199112The constitutive equations are as follows119873119894 = 119864119894119860 119894120576119911119894119872119910119894 = 119864119894119868119910119894119896119910119894119872119909119894 = 119864119894119868119909119894119896119909119894119879119894 = 119866119894119868119879119894119896119911119894 (119894 = 1 2)

(2)

where 119873119894 119872119910119894 119872119909119894 and 119879119894 are internal axial force in 119911-direction internal bending moment in 119910-direction inter-nal bending moment in 119909-direction and internal torsionmoment in 119911-direction respectively

The microunitrsquos force diagram of the horizontally com-posite curved I-beam is shown in Figure 2 which is under theexternal distribution force 1199020 and torsion moment 1198980 Thebalance equations of the slab are1198891198761199091119889119911 + 1198731119877 + 1199021199091 = 01198891198731119889119911 minus 1198761199091119877 + 1199021199111 = 01198891198721199101119889119911 + 1198761199091 = 01198891198761199101119889119911 minus 1199020 + 1199021199101 = 01198891198721199091119889119911 + 1198791119877 minus 1198761199101 + 1198981199091 = 01198891198791119889119911 minus 1198721199091119877 + 1198981199111 + 1198980 = 0

(3)

The balance equations of the I-girder are1198891198761199092119889119911 + 1198732119877 + 1199021199092 = 01198891198732119889119911 minus 1198761199092119877 + 1199021199112 = 01198891198721199102119889119911 + 1198761199092 = 01198891198761199102119889119911 + 1199021199102 = 01198891198721199092119889119911 + 1198792119877 minus 1198761199102 + 1198981199092 = 01198891198792119889119911 minus 1198721199092119877 + 1198981199112 = 0

(4)

where119898119909119894 is the distributed bendingmoment caused by shearforce 119902119911119894 119898119911119894 is distributed torque caused by shear force 119902119909119894119876119909119894 is the shear force in 119909-direction and119876119910119894 is the shear forcein 119910-direction For the shear forces 119902119911119894 119902119909119894 and 119902119910119894 there are1199021199111 + 1199021199112 = 01199021199091 + 1199021199092 = 01199021199101 + 1199021199102 = 0 (5)

By eliminating1198761199091 and1198761199092 from the first three equationsof (3) and (4) we can getminus (1198721199101 +1198721199102)10158401015840 + (1198731 + 1198732)119877 = 0

(1198731 + 1198732)1015840 + (1198721199101 +1198721199102)1015840119877 = 0 (6)

When there are no axial force and bending moment in 119910-direction of the I-beam caused by external forces we can get1198731 = minus11987321198721199101 = minus1198721199102 (7)

3 Equilibrium at the Interface

At the interface considering the deflections described inFigure 3 shear force 1199021199112 described in Figure 2 can be writtenas 1199021199112 = 119870Δ119906 = 119870(1199062 minus 1199061 + 1199081015840119887) = minus11987310158401 minus 11987210158401199101119877= 11987310158402 + 11987210158401199102119877 (8)

where 119887 = 1198871 + 1198872 and Δ119906 is the amount of the deflection atthe interface in the tangential direction So the equilibriumat the interface in the tangential direction can be written as119870(1199062 minus 1199061 + 1199081015840119887) + 11987310158401 + 11987210158401199101119877 = 0 (9)


q0

m0

Qy1

My1Mx1

Qx1T1

N1

Mx1 + dMx1

Qx1 + dQx1

My1 + dMy1

Qy1 + dQy1

T1 + dT1 N1 + dN1qy2

qx2qz2Qy2

My2

Mx2

Qx2T2

N2

Mx2 + dMx2

Qx2 + dQx2

My2 + dMy2

Qy2 + dQy2

T2 + dT2

N2 + dN2

Figure 2 A microunitrsquos force diagram of the horizontally composite curved I-beam

1

2

z

y

b

b1

b2w998400

w998400b1 w998400b2

Δu

u2 minus u1

Figure 3 Deflections of differential elements for the beam in 119910-119911plane

Considering deflection described in Figure 4 equilibriumat the interface in the radial direction can be written as

V2 minus V1 + 120601119887 = 0 (10)

In order to solve the problem under consideration itis convenient to rearrange (9) and (10) as detailed in whatfollows Firstly taking the derivative of (9) one time withrespect to 119911 component then eliminating V2 and 1199062 byinserting the derivative equation and (10) into the firstformula of (1) and at last using the first and second formulasof (2) one can obtain the following119891119911 = 11986411198601119870 (11990610158401 minus V1119877 )10158401015840 + 11986411198681199101119870119877 (V101584010158401 + 11990610158401119877 )10158401015840

minus 11987811986011986411198601 (11990610158401 minus V1119877 ) + 11990810158401015840119887 minus 120601119887119877 = 0 (11)

Analogously taking the derivative of differential equation(9) one time and taking the derivative of (10) two times with

cg 1

cg 2y

x

cg 0

b

b1

b2

120601b = 1 minus 2

Figure 4 Deflection of the composite curved I-beam in 119909-119910 plane

respect to 119911 component firstly and then eliminating V2 and 1199062by inserting these two derivative equations into the secondformulas of (1) using (2) one can obtain the following

119891119910 = 11986411198601119870119877 (11990610158401 minus V1119877 )10158401015840 + 119864111986811991011198701198772 (V101584010158401 + 11990610158401119877 )10158401015840minus 11987811991011986411198681199101 (V101584010158401 + 11990610158401119877 ) + 12060110158401015840119887 + 11990810158401015840119887119877 = 0 (12)

where 119878119860 = 111986421198602 + 111986411198601 and 119878119910 = 111986411198681199101 + 1119864211986811991024 Problem Formulation

The governing differential equations of the beam will bederived by the principle of minimum potential energy Thepotential energy of the beam takes the following formΠ = 119880119872119909 + 119880119879119911 + 119880119872119910 + 119880119886 + 119880119904 minus119882 (13)


119880119872119909 is the bending strain energy due to the internalbending moments1198721199091 and1198721199092

119880119872119909 = 12 int1198771205790 (11987211990911198961199091 +11987211990921198961199092) 119889119911= 12119864119868119909 int1198771205790 (11990810158401015840 minus 120601119877)2 119889119911 (14)

where 119864119868119909 = 11986411198681199091 + 11986421198681199092119880119879 is the torsional strain energy due to the internaltorsion moments 1198791 and 1198792

119880119879 = 12 int1198771205790 (11987911198961199111 + 11987921198961199112) 119889119911= 12119866119868119879int1198771205790 (1206011015840 + 1199081015840119877 )2 119889119911 (15)

where 119866119868119879 = 11986611198681198791 + 11986621198681198792119880119886 is the axial strain energy due to the internal axial forces1198731 and1198732119880119886 = 12 int1198771205790 (11987311205761 + 11987321205762) 119889119911

= 12 int1198771205790 119878119860 [11986411198601 (11990610158401 minus V1119877 )]2 119889119911 (16)


119880119872119910 = 12 int1198771205790 (11987211991011198961199101 +11987211991021198961199102) 119889119911= 12 int1198771205790 119878119910 [11986411198681199101 (V101584010158401 + 119906101584010158401119877 )]2 119889119911 (17)

119880119904 is the strain energy due to the connector deflections

119880119904 = 12 int1198771205790 1199021199111 sdot Δ119906 119889119911 = 12 int1198771205790 1119870 (11987310158401 + 11987210158401199101119877 )2 119889119911= 12 int1198771205790 1119870 [11986411198601 (11990610158401 minus V1119877 )1015840+ 11986411198681199101119877 (V101584010158401 + 11990610158401119877 )1015840]2 119889119911

(18)

119882 is the potential energy due to the external loading

119882 = int1198771205790

1198980120601119889119911 + int1198771205790

1199020119908119889119911 + (1198721199091199081015840)100381610038161003816100381610038161198771205790+ (119876119910119908)100381610038161003816100381610038161198771205790 + (119879119911120601)10038161003816100381610038161198771205790 (19)

where 119872119909 119876119910 and 119879119911 are the bending moment shearforce and torsion moment of the beam respectively Usingthe Lagrange multiplier method the principle of minimumpotential energy in augmented form now may be writtenas

minΠlowast = Π + int1198771205790

(120582119911 sdot 119891119911) 119889119911 + int1198771205790

(120582119910 sdot 119891119910) 119889119911 (20)

where the parameters 120582119911 and 120582119910 are the Lagrange multipliersand 119891119911 and 119891119910 are the equilibrium conditions at the interfacecorresponding to (11) and (12) respectively In (20) 120601 119908 V11199061 120582119911 and 120582119911 are all independent variables The variation ofΠlowast is120575Πlowast = int119877120579

0Γ1120575119908119889119911 + int119877120579

0Γ2120575120601 119889119911 + int119877120579

0Γ3120575120582119911119889119911

+ int1198771205790

Γ4120575120582119910119889119911 + int1198771205790

Γ5120575V1119889119911 + int1198771205790

Γ61205751199061119889119911+ 119867112057511990810038161003816100381610038161198771205790 + 11986721205751199081015840100381610038161003816100381610038161198771205790 + 119867312057512060110038161003816100381610038161198771205790 + 11986741205751206011015840100381610038161003816100381610038161198771205790+ 1198675120575119906110038161003816100381610038161198771205790 + 119867612057511990610158401100381610038161003816100381610038161198771205790 + 1198677120575V110038161003816100381610038161198771205790 + 1198678120575V10158401100381610038161003816100381610038161198771205790+ 1198679120575V101584010158401 100381610038161003816100381610038161198771205790+ 11986710120575 [[[[

11986411198681199101119877119870 (V101584010158401 + 11990610158401119877 )1015840 + 11986411198601119870 (11990610158401 minus V10158401119877 )1015840⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟Δ119906

]]]]1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816119877120579

0

(21)

where Γ119894 (119894 = 1sim6) and119867119894 (119894 = 1sim10) are calculated from thevariation ofΠlowast which are given in Appendix A Each term in(21) must be identically zero It makes Γ5 = 0 Γ6 = 01198675 = 01198676 = 01198677 = 01198678 = 0 and1198679 = 0 that is the underminedLagrange multipliers 120582119911 and 120582119910 are

120582119911 = 11986411198601 (11990610158401 minus V1119877 ) = 1198731120582119910 = 11986411198681199101 (V101584010158401 + 11990610158401119877 ) = 1198721199101 (22)

Equation (22) notes that the Lagrange multipliers and thedeflections (1199061 and V1) are related So there will be four ratherthan six independent variables in this problemWe can take120601119908 120582119911 and 120582119910 as independent variablesThe rest of governingequations are Γ1 = 0 Γ2 = 0 Γ3 = 0 and Γ4 = 0 which can berearranged and rendered in terms of matrix form as shownbelow


[[[[[[[[[[[[[

119864119868119909 11988941198891199114 minus 1198661198681198791198772 11988921198891199112 minus119864119868119909 + 119866119868119879119877 11988921198891199112 119887 11988921198891199112 119887119877 11988921198891199112minus119864119868119909 + 119866119868119879119877 11988921198891199112 minus119866119868119889 11988921198891199112 + 1198641198681199091198772 minus 119887119877 119887 11988921198891199112119887 11988921198891199112 minus 119887119877 1119870 11988921198891199112 minus 119878119860 1119877119870 11988921198891199112119887119877 11988921198891199112 119887 11988921198891199112 1119877119870 11988921198891199112 11198772119870 11988921198891199112 minus 119878119910

]]]]]]]]]]]]](119908120601120582119911120582119910)=(1199020119898000 ) (23)

We can get the pertaining boundary conditions[minus119864119868119909 (119908101584010158401015840 minus 1206011015840119877 ) minus 1198871205821015840119911 minus 1198871198771205821015840119910 + 119866119868119879(1206011015840119877 + 11990810158401198772)minus 119876119910] 1205751199081003816100381610038161003816100381610038161003816100381610038161198771205790 = 0[119864119868119909 (11990810158401015840 minus 120601119877) + 119887120582119911 + 119887119877120582119910 minus119872119909] 1205751199081015840100381610038161003816100381610038161003816100381610038161198771205790 = 0[119866119868119879(1206011015840 + 1199081015840119877 ) minus 1198871205821015840119910 minus 119879119911] 1205751206011003816100381610038161003816100381610038161003816100381610038161198771205790 = 0(119887120582119910) 1205751206011015840100381610038161003816100381610038161198771205790 = 0(120582119911 + 120582119910119877 )120575 (Δ119906)1003816100381610038161003816100381610038161003816100381610038161198771205790 = 0

(24)

It is evident that (23) is tenth-order ordinary differentialequationWehave ten boundary conditions for the beam thatis five boundary conditions for each end For the boundaryconditions (see (24)) to hold the following alternatives ateach end shown as in Table 1 are conducted

5 Analytical Solution by Trigonometric Series

In this section a closed form solution of the simply sup-ported horizontally composite curved I-beam is obtainedThe deflection functions and the Lagrange multiplier func-tions are expressed as undetermined coefficients and knowntrigonometric series to satisfy the governing equations andthe rigid torsion constraints at each end The deflection andthe Lagrangemultiplier are assumed to be the following form119908 = 119899sum

119894=1

119908119894 sin(119894120587119911119871 ) 120601 = 119899sum119894=1

120601119894 sin(119894120587119911119871 ) 120582119911 = 119899sum119894=1

120582119911119894 sin(119894120587119911119871 ) 120582119910 = 119899sum119894=1

120582119910119894 sin(119894120587119911119871 ) (25)

where119908119894 120601119894 120582119911119894 and 120582119910119894 are unknown Fourier coefficients tobe determined for each 119894 (119894 = 1 119899) and 119871 is span lengthThe applied distributed load 1199020 and1198980 are expanded in singletrigonometric series as

1199020 = 119899sum119894=1

119902119894 sin(119894120587119911119871 ) 1198980 = 119899sum

119894=1

119898119894 sin(119894120587119911119871 ) (26)

where 119902119894 and119898119894 are the Fourier coefficients Multiplying bothsides of (26) by each trigonometric series and then integratingthem over (0 119871) finally the trigonometric coefficients 119902119894 and119898119894 can be readily determined with the orthogonality prop-erties Substituting (25)-(26) into (23) and solving (23) theunknown parameters 119908119894 120601119894 120582119911119894 and 120582119910119894 can be determinedThe unknown Fourier coefficients are given as follows119908119894 = 119902119894 1205741199021 + 1205741199022120574 + 119898119894 1205741198981120574

120601119894 = 119902119894 1205741199023120574 + 119898119894 1205741198982 + 1205741198983120574 120582119911119894 = 119902119894 1205741199024120574 + 119898119894 1205741198984120574 120582119910119894 = 119902119894 1205741199025120574 + 119898119894 1205741198985120574

(27)

where 120574 120574119902119894 and 120574119898119894 (119894 = 1 5) are the relevant coefficientswhich are given in Appendix B

6 Numerical Example

A steel-concrete composite curved I-beam calculated byThevendran et al [27] is taken as a numerical example inorder to show the accuracy and reliability of the presenttrigonometric solution of the beam The beam is simplysupported at both ends as shown in Figure 1 And the beam issubjected to 119865 = 150KN vertical load at themidspan sectionThematerial properties and dimensions of the curved I-beam(SP4 beam) are shown in Table 2

The finite element software ANSYS is used to model thedescribed structure (Figure 5) In the FEM model both the


Table 1 Common support conditions

Type The boundary conditions

R

Free end

119876119910 = 0119872119909 = 0119879119911 = 0120582119911 = 0120582119910 = 0R

Fixed end

119908 = 0120601 = 01199081015840 = 01206011015840 = 0Δ119906 = 0R

Rigid torsion constraints end

119908 = 0120601 = 0119872119909 = 0120582119911 = 0120582119910 = 0R

Hinged end

119908 = 0119879119911 = 0119872119909 = 0120582119911 = 0120582119910 = 0Space

beam element Rigid link

Interface node and shear connector

Figure 5 Finite element model diagram of the composite curvedI-beam

concrete slab and the steel I-girder are modeled as spacebeam elements and the connection between the slab andI-girder is simulated by multiple-point constraints (MPC)which are modeled by two rigid links connected throughinterface nodes between the slab and I-girder The springelements are used in the tangential direction to allow for thepossibility of movement Coupling degrees of freedom 119908 areused to prevent the uplifting issue

The vertical deflection 119908 the tangential deflection 119906the radial deflection V and the torsional deflection 120601 arerestrained to satisfy the simply supported constraints atboth ends Although the BEAM4 element in ANSYS cannotconsider the warping effect the element meets the theory ofthis paper quite well So 120 BEAM4 elements are selected asthe spatial beam elements The element of MPC 184 with thenumber of 122 is modeled as the multiple-point constraintselement In addition the element of COMBIN 14 with thenumber of 61 is used when tangential slip is considered

For the full interaction case the slip parameters are takenas 119870 = 1010Nm2 Figure 6 shows the vertical deflection119908 based on the FEM the solution calculated by this paperand Thevendran et al experimental results respectively For

ExperimentFEMn10

n5

n1

10 20 30 40 50 600Distance along the tangential direction (dm)

minus0001000000010002000300040005000600070008000900100011

The v

ertic

al d

eflec

tion

(m)

Figure 6 Vertical deflection of the beam

FEMn10

n5

n1


minus0007

minus0006

minus0005

minus0004

minus0003

minus0002

minus0001

0000

0001Th

e tor

siona

l ang

le (r

ad)

Figure 7 Torsional angle of the beam

this paper solution the number of Fourier coefficients 119899 (in(25)) can be chosen as large as we can to achieve a requiredaccuracy As an illustration of convergence of the results weshow the results from this paper by taking the number 119899 as 15 and 10 terms respectively It can be seen in Figure 6 that thevertical deflections results solved from this paper and fromThevendran et al experiment are in reasonable agreement(the difference is within 12) and the solved results from thispaper and from FEM are in good agreement

Considering only vertical deflections are available fromthe experiments of Thevendran et al the other results ofthe solution calculated by this paper and by the FEM arecompared in Figures 7ndash12 The results are the torsional angle120601 the bending moment1198721199101 (1198721199101 = 120582119910) the axial force 1198731(1198731 = 120582119911) the bending moment119872119909 the shear force 119876119910 andthe torsion moment 119879119911 respectively


Table 2 Material properties and dimensions of the curved I-beam

Youngrsquos modulus(GPa)

Poissonrsquosratio

Density(kgm3)

Width(mm)

Thickness(mm)

Centralangle

Radius of curvature(m)

Steel girder 206 03 7850 Flange 332Web 172

Flange 13Web 8 120579 = 143∘ 24

Concrete slab 26 027 2400 1500 100

FEMn10

n5n1


00

02

04

06

08

10

The b

endi

ng m

omen

tMy1

(KN

m)

Figure 8 Bending moment1198721199101 of the beam

FEMn10

n5

n1


minus700

minus600

minus500

minus400

minus300

minus200

minus100

0

The a

xial

forc

eN1

(KN

)

Figure 9 Axial force1198731 of the beam

It can be seen from Figures 7ndash12 that the results solved bythis paper can be quickly convergent and one can get verygood results by taking more than 5 terms When the stiffnessof the shear connectors is changed the vertical deflectionand torsional angle at midspan section are shown in Figures13 and 14 respectively In Figure 13 it can be seen thatthe vertical deflections converge to the minimum when theslip parameter 119870 is greater than 1010Nm2 Moreover the

FEMn10

n5

n1


minus250

minus200

minus150

minus100

minus50

0

The b

endi

ng m

omen

tMx

(KN

m)


FEMn10

n5

n1


minus120minus100minus80minus60minus40minus20

020406080

100120

The s

hear

forc

eQy

(KN

)

Figure 11 Shear force 119876119910 of the beam

changes with slip parameter 119870 indicate the shear connecterrsquosstiffness in experimental model of Thevendran et al withthe range from 109Nm2 to 1010Nm2 As it can be seenfrom Figures 13 and 14 the performance of the curvedcomposite I-beam becomes significantly rigid when the shearconnecterrsquos stiffness is increased from 107 to 109Nm2 Thereduction of shear connectorsrsquo stiffness between layers has asignificant effect on the vertical deflection and torsional angleat midspan section



minus20

minus15

minus10

minus5

0

5

10

15

20

The m

omen

t of t

orqu

eTz

(KN

m)

FEMn10

n5

n1

Figure 12 Torsion moment 119879119911 of the beam

0 4 6 8 10 122The stiffness of the shear connecter log10

K

0008

0010

0012

0014

0016

0018

0020

0022

Mid

span

ver

tical

defl

ectio

n (m

)

FEMn10

n5

n1

Figure 13 Vertical deflection at midspan section when the stiffnessof the shear connecter is changed

The inaccuracy of the vertical deflection and torsionalangle between analytical solutions in this paper and the FEMresults in Figures 6ndash14 is small and reasonable Thus themodel created in this paper can be applied sufficiently forpractical purposes

7 Conclusions

In this paper an analytical solution has been developed andpresented for the simply supported horizontally compositecurved I-beam It is primarily used to solve a static problemof a two-layered composite curved beam with flexible shearconnection in the tangential direction The trigonomet-ric series are adopted in solution expression in terms ofspan coordinate Both governing equations and boundaryconditions are obtained by using the Vlasov curved beam

FEMn10

n5n1


K

minus00070

minus00069

minus00068

minus00067

minus00066

minus00065

minus00064

minus00063

Mid

span

tors

iona

l ang

le (r

ad)

Figure 14 Torsional angle atmidspanwhen the stiffness of the shearconnecter is changed

theory and the principle of minimum potential energy Theaccuracy of calculation results based on the theory presentedin this paper is verified by comparing with the FEM andthe Thevendran et al experimental results Therefore theanalytical expression and model established in this paperpossess certain actual engineering significance

Appendix

A Component of Γ119894 (119894 = 1sim6) and 119867119894 (119894 = 1sim10)ConsiderΓ1 = 1198641198681199091199081015840101584010158401015840 minus 119866119868119879119908101584010158401198772 minus 119864119868119909 + 119866119868119879119877 12060110158401015840 + 11988712058210158401015840119911

+ 11988712058210158401015840119910119877 minus 1199020Γ2 = minus119864119868119909 + 119866119868119879119877 11990810158401015840 minus 11986611986811988912060110158401015840 + 1198641198681199091198772 120601 minus 119887120582119911119877 + 11988712058210158401015840119910minus 1198980Γ3 = [11986411198601 (11990610158401 minus V1119877 )10158401015840 + 11986411198681199101119877 (V101584010158401 + 11990610158401119877 )10158401015840] 1119870minus 11987811986011986411198601 (11990610158401 minus V1119877 ) + 11990810158401015840119887 minus 120601119887119877 Γ4 = [11986411198601 (11990610158401 minus V1119877 )10158401015840 + 11986411198681199101119877 (V101584010158401 + 11990610158401119877 )10158401015840] 1119877119870

minus 11987811991011986411198681199101 (V101584010158401 + 11990610158401119877 ) + 12060110158401015840119887 + 11990810158401015840119887119877 Γ5 = 11986411198681199101119878119910 [11986411198681199101 (V101584010158401 + 11990610158401119877 ) minus 120582119910]10158401015840 minus 11986411198601119878119860


sdot 1119877 [11986411198601 (11990610158401 minus V1119877 ) minus 120582119911]+ 11986411198601119870119877 [11986411198601 (11990610158401 minus V1119877 ) + 11986411198681199101119877 (V101584010158401 + 11990610158401119877 )minus 120582119911 minus 120582119910119877 ]10158401015840 minus 11986411198681199101119870119877 [11986411198601 (11990610158401 minus V1119877 )+ 11986411198681199101119877 (V101584010158401 + 11990610158401119877 ) minus 120582119911 minus 120582119910119877 ]1015840101584010158401015840

Γ6 = 1119870 [11986411198601 (11990610158401 minus V1119877 ) + 11986411198681199101119877 (V101584010158401 + 11990610158401119877 ) minus 120582119911minus 120582119910119877 ]101584010158401015840 (11986411198601 + 119864111986811991011198772 )minus 11986411198601119878119860 [11986411198601 (11990610158401 minus V1119877 ) minus 120582119911]1015840minus 11987811991011986411198681199101119877 [11986411198681199101 (V101584010158401 + 11990610158401119877 ) minus 120582119910]1015840

1198671 = minus119864119868119909 (119908101584010158401015840 minus 1206011015840119877 ) minus 1198871205821015840119911 minus 1198871198771205821015840119910 + 119866119868119879(1206011015840119877+ 11990810158401198772) minus 1198761199101198672 = 119864119868119909 (11990810158401015840 minus 120601119877) + 119887120582119911 + 119887119877120582119910 minus1198721199091198673 = 119866119868119879(1206011015840 + 1199081015840119877 ) minus 1198871205821015840119910 minus 1198791199111198674 = 1198871205821199101198675 = 11986411198601119878119860 [11986411198601 (11990610158401 minus V1119877 ) minus 120582119911]

+ 11987811991011986411198681199101119877 [11986411198681199101 (V101584010158401 + 11990610158401119877 ) minus 120582119910]minus 1119870 [11986411198601 (11990610158401 minus V1119877 ) + 11986411198681199101119877 (V101584010158401 + 11990610158401119877 ) minus 120582119911

minus 120582119910119877 ]10158401015840 (11986411198601 + 119864111986811991011198772 ) 1198676 = 1119870 [11986411198601 (11990610158401 minus V1119877 ) + 11986411198681199101119877 (V101584010158401 + 11990610158401119877 )

minus 120582119911 minus 120582119910119877 ]1015840 (11986411198601 + 119864111986811991011198772 ) 1198677 = minus11986411198681199101119878119910 [11986411198681199101 (V101584010158401 + 11990610158401119877 ) minus 120582119910]1015840

minus 11986411198601119870119877 [11986411198601 (11990610158401 minus V1119877 )1015840 + 11986411198681199101119877 (V101584010158401 + 11990610158401119877 )1015840minus 1205821015840119911 minus 1205821015840119910119877 ] + 11986411198681199101119870119877 [11986411198601 (11990610158401 minus V1119877 )1015840+ 11986411198681199101119877 (V101584010158401 + 11990610158401119877 )1015840 minus 1205821015840119911 minus 1205821015840119910119877 ]10158401015840

1198678 = 11986411198681199101119878119910 [11986411198681199101 (V101584010158401 + 11990610158401119877 ) minus 120582119910]minus 11986411198681199101119870119877 [11986411198601 (11990610158401 minus V1119877 )1015840 + 11986411198681199101119877 (V101584010158401 + 11990610158401119877 )1015840minus 1205821015840119911 minus 1205821015840119910119877 ]1015840

1198679 = 11986411198681199101119870119877 [11986411198601 (11990610158401 minus V1119877 )1015840+ 11986411198681199101119877 (V101584010158401 + 11990610158401119877 )1015840 minus 1205821015840119911 minus 1205821015840119910119877 ]

11986710 = 120582119911 + 120582119910119877 (A1)

B Component of 120574 120574119902119894 and 120574119898119894 (119894 = 1sim5)Consider

120574 = (11987011987121198772 (119887211989421205872 + 1198712119864119868119879119878119910) (1198872 + 119864119868119909119878119860) + 11989421205872 (1198872119894212058721198772119864119868119909 + 1198712119864119868119879 (1198872 + 119864119868119909 (1198772119878119910 + 119878119860)))) 1205741199021 = 11987141198772 (1198872 (1198712 minus 119894212058721198772)2 + 1198712 (1198872119894211987012058721198774 + 119894212058721198772119864119868119879 + 1198712119864119868119909) 119878119860)(1198712 minus 119894212058721198772)2 1205741199022 = 11987161198774119878119910 (11988721198701198714 + 119894212058721198772119864119868119879 (11989421205872 + 1198701198712119878119860) + 119864119868119909 (119894211987121205872 + 1198701198714119878119860))(1198941198712120587 minus 119894312058731198772)2


1205741199023 = minus11987161198773 (11989421205872 (11988721198701198772 + 119864119868119879 + 119864119868119909) 119878119860 + 1198772119878119910 (11989421205872 (119864119868119879 + 119864119868119909) + 1198701198712 (1198872 + (119864119868119879 + 119864119868119909) 119878119860)))(1198712 minus 119894212058721198772)2 1205741199024 = 11988711987141198772 (11989421205872 (11988721198701198772 + 119864119868119909) + 119864119868119879 (11989421205872 + 11987011987121198772119878119910))1198712 minus 119894212058721198772 1205741199025 = 11988711987141198773 (11989421205872 (119864119868119879 + 119864119868119909) + 1198701198712 (1198872 + 119864119868119909119878119860))(minus1198712 + 119894212058721198772) 1205741198981 = minus11987161198773 (11989421205872 (11988721198701198772 + 119864119868119879 + 119864119868119909) 119878119860 + 1198772119878119910 (11989421205872 (119864119868119879 + 119864119868119909) + 1198701198712 (1198872 + (119864119868119879 + 119864119868119909) 119878119860)))(1198712 minus 119894212058721198772)2 1205741198982 = 1198942119871412058721198772 (1198712119864119868119879 + 1198772 (11988721198701198712 + 11989421205872119864119868119909)) 119878119860(1198712 minus 119894212058721198772)2 1205741198983 = 11987141198774119878119910 (119864119868119879 (119894211987121205872 + 1198701198714119878119860) + 119894212058721198772 (11988721198701198712 + 119864119868119909 (11989421205872 + 1198701198712119878119860)))(1198712 minus 119894212058721198772)2 1205741198984 = minus1198871198712119877 (119894212058721198772 (11988721198701198712 + 11989421205872119864119868119909) + 119864119868119879 (119894211987121205872 + 11987011987141198772119878119910))1198712 minus 119894212058721198772 1205741198985 = 1198871198942119871212058721198772 (1198712119864119868119879 + 119894212058721198772119864119868119909 + 11987011987121198772 (1198872 + 119864119868119909119878119860))1198712 minus 119894212058721198772

(B1)

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

The research described in this paper was financially sup-ported by the National Natural Science Foundation of China(51378236) and project supported by the transportationscience and technology plan of Jilin province (2012-4-1-4)

References

[1] A Chakrabarti A H Sheikh M Griffith and D J OehlersldquoAnalysis of composite beams with longitudinal and transversepartial interactions using higher order beam theoryrdquo Interna-tional Journal of Mechanical Sciences vol 59 no 1 pp 115ndash1252012

[2] E Onate A Eijo and S Oller ldquoSimple and accurate two-nodedbeam element for composite laminated beams using a refinedzigzag theoryrdquo Computer Methods in Applied Mechanics andEngineering vol 213-216 pp 362ndash382 2012

[3] N M Newmark C P Seiss and I M Veist ldquoTests and analysisof composite beams with incomplete interactionsrdquo Proceedingsof the Society for Experimental Stress Analysis vol 4 no 1 pp75ndash92 1951

[4] J R Goodman and E P Popkov ldquoLayered beam systems withinterlayer sliprdquo Journal of Structural Division vol 24 no 11 pp2535ndash2547 1968

[5] J R Goodman and E P Popov ldquoLayered wood systems withinter-layer sliprdquoWood Science vol 1 no 3 pp 148ndash158 1969

[6] U A Girhammar and V K A Gopu ldquoComposite beam-columns with interlayer slipmdashexact analysisrdquo Journal of Struc-tural Engineering vol 199 no 4 pp 1265ndash1282 1993

[7] U A Girhammar and D Pan ldquoDynamic analysis of compositememberswith interlayer sliprdquo International Journal of Solids andStructures vol 30 no 6 pp 797ndash823 1993

[8] Y CWang ldquoDeflection of steel-concrete composite beams withpartial shear interactionrdquo Journal of Structural Engineering vol124 no 10 pp 1159ndash1165 1998

[9] A DallrsquoAsta ldquoComposite beams with weak shear connectionrdquoInternational Journal of Solids and Structures vol 38 no 32-33pp 5605ndash5624 2001

[10] J Nie and C S Cai ldquoSteelndashconcrete composite beams consider-ing shear slip effectsrdquo Journal of Structural Engineering vol 129no 4 pp 495ndash506 2003

[11] G Ranzi and M A Bradford ldquoAnalytical solutions for thetime-dependent behaviour of composite beams with partialinteractionrdquo International Journal of Solids and Structures vol43 no 13 pp 3770ndash3793 2006

[12] H-B Liu W-H Liu and Y-L Zhang ldquoCalculation analysisof shearing slip for steel-concrete composite beam under con-centrated loadrdquo Applied Mathematics and Mechanics EnglishEdition vol 26 no 6 pp 735ndash740 2005

[13] F Campi and I Monetto ldquoAnalytical solutions of two-layerbeams with interlayer slip and bi-linear interface lawrdquo Interna-tional Journal of Solids and Structures vol 50 no 5 pp 687ndash6982013


[14] A Ayoub ldquoA two-field mixed variational principle for partiallyconnected composite beamsrdquo Finite Elements in Analysis andDesign vol 37 no 11 pp 929ndash959 2001

[15] G Ranzi M A Bradford and B Uy ldquoA direct stiffness analysisof a composite beam with partial interactionrdquo InternationalJournal for Numerical Methods in Engineering vol 61 no 5 pp657ndash672 2004

[16] B Cas M Saje and I Planinc ldquoNon-linear finite elementanalysis of composite planar frames with an interlayer sliprdquoComputers and Structures vol 82 no 23ndash26 pp 1901ndash19122004

[17] F Gara G Ranzi and G Leoni ldquoDisplacement-based for-mulations for composite beams with longitudinal slip andvertical upliftrdquo International Journal for Numerical Methods inEngineering vol 65 no 8 pp 1197ndash1220 2006

[18] WChung andED Sotelino ldquoThree-dimensional finite elementmodeling of composite girder bridgesrdquo Engineering Structuresvol 28 no 1 pp 63ndash71 2006

[19] Y Majdi C-T T Hsu and M Zarei ldquoFinite element analysisof new composite floors having cold-formed steel and concreteslabrdquo Engineering Structures vol 77 no 15 pp 65ndash83 2014

[20] V Z Vlasov Thin Walled Elastic Beams Israel Program forScientific Translation Jerusalem Israel 1961

[21] M-Y Kim S-B Kim and N-I Kim ldquoSpatial stability of sheardeformable curved beams with non-symmetric thin-walledsections I stability formulation and closed-form solutionsrdquoComputers and Structures vol 83 no 31-32 pp 2525ndash25412005

[22] M H Kim S B Kim and N I Kim ldquoSpatial stability of sheardeformable curved beams with non-symmetric thin walledsections II FE solutions and parametric studyrdquo Computers ampStructures vol 83 no 31-32 pp 2542ndash2558 2005

[23] A M Yu X G Yang and G H Nie ldquoGeneralized coordinatefor warping of naturally curved and twisted beams with generalcross-sectional shapesrdquo International Journal of Solids andStructures vol 43 no 10 pp 2853ndash2867 2006

[24] F N Gimena P Gonzaga and L Gimena ldquo3D-curved beamelement with varying cross-sectional area under generalizedloadsrdquo Engineering Structures vol 30 no 2 pp 404ndash411 2008

[25] F N Gimena P Gonzaga and L Gimena ldquoStiffness andtransfer matrices of a non-naturally curved 3D-beam elementrdquoEngineering Structures vol 30 no 6 pp 1770ndash1781 2008

[26] L Gimena F N Gimena and P Gonzaga ldquoStructural analysisof a curved beam element defined in global coordinatesrdquoEngineering Structures vol 30 no 11 pp 3355ndash3364 2008

[27] V Thevendran N E Shanmugam S Chen and J Y R LiewldquoExperimental study on steel-concrete composite beams curvedin planrdquo Engineering Structures vol 22 no 8 pp 877ndash889 2000

[28] N E Shanmugam M Mahendrakumar and V ThevendranldquoUltimate load behaviour of horizontally curved plate girdersrdquoJournal of Constructional Steel Research vol 59 no 4 pp 509ndash529 2003

[29] C Topkaya E B Williamson and K H Frank ldquoBehaviorof curved steel trapezoidal box-girders during constructionrdquoEngineering Structures vol 26 no 6 pp 721ndash733 2004

[30] F Giussani and F Mola ldquoService-stage analysis of curvedcomposite steel-concrete bridge beamsrdquo Journal of StructuralEngineering vol 132 no 12 pp 1928ndash1939 2006

[31] R E Erkmen andM A Bradford ldquoNonlinear elastic analysis ofcomposite beams curved in-planrdquo Engineering Structures vol31 no 7 pp 1613ndash1624 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


interlayer slip In the aspect of numerical simulations studiesdifferent kinds of numerical and finite element formulationsfor the analysis of composite beam with interlayer slip havebeen suggested [1 14ndash19]

Many significant researches have been accomplished inregard to the behavior of straight partially composite beamIn the same time various scholars have done researches onthe curved beam theory One of the earliest works dealingwith the stability behaviors of curved beam has been putforward by Vlasov [20] After that scholars covered differentextensions and enhancements to the Vlasov model [21ndash26] These researches are helpful to the development of thecomposite beam theory However very little literature hasfocused on the curved partially composite beam such ashorizontally composite curved steel I-beam bridge Theven-dran et al [27] and Shanmugam et al [28] have conductedexperiments on the steel-concrete composite curved beamto investigate the ultimate load behavior In their study thefinite element software ABAQUS was used to analyze thebehavior of test specimens Full composite action betweensteel beam and the concrete slab was assumed The resultsof deformations stress distributions and ultimate strengthsobtained by finite element analysis were found to be in goodagreement with the experimental results After that Topkayaet al [29] have conducted experimental and numericalstudies to establish the behavior of composite curved bridgeduring construction In their study two FEM models wereestablished by different kinds of software to predict thebehavior of the curved steel trapezoidal box-girder and theauthors have drawn a conclusion that the reasonable finiteelement model is able to accurately capture girder behaviorduring construction Giussani and Mola [30] have developedan analytical equation for elastic composite beam curved in-plane with the long term behavior But the partial interactionbetween the steel girder and concrete slab was not consideredin the study Erkmen and Bradford [31] have solved theequation of the composite curved beam considering thetwo-layer partial interaction by providing a highly efficient3D beam finite element The results demonstrate that thedeveloped formulation is accurate and effective in capturingthe behavior of composite beams curved in-plane

Even though someprevious researches of these evaluationmethodologies have been accomplished there is still lack ofthe fundamental understanding of the system-level behavioron the overall performance of composite curved beam withpartial shear interaction Besides although exact solutionscan be obtained by the 3D finite element model it is verycomplex and time consuming Therefore the FEM analysisprocedures are not suitable for the initial design This studyaims to provide an analytical theory of the horizontallycomposite curved beam considering the partial interactionin tangential direction The beam is assumed to be staticallydeterminatewith a constant radius of curvature along the lon-gitudinal axis Governing equations and boundary conditionsare obtained by using both the Vlasov curved beam theoryand the energy variation principleTheundetermined verticaldeflection torsional deflection and Lagrange multipliers willbe approximated by Fourier series to solve the governingequations of the partial interaction composite beam theory

q

y

x

120579

120593

z = R lowast 120593

m

Figure 1 Model of a horizontally composite curved I-beam

in the procedures The numerical results of deflections andforces obtained by using proposed theory are presentedand compared with FEM results and experimental resultsComparison results show that the calculation can be easilyand accurately handled which is a great advantage of thismethod

2 Basic Assumptions and Conditions

This study is on a horizontally composite curved I-beamwhere the model is shown as in Figure 1 and the noteworthyfeatures of this research are shown as follows

(1) The slab and I-girder are linear-elastic different mate-rials and the cross-sections made of both mate-rials are rigid in their plane The effects of sheardeformation warping deformation and distortiondeformation are neglected in this research Structuralanalysis of the beam is based on the Vlasov curvedbeam theory (for each part of the beam)

(2) The interlayer connectors between the slab and I-girder are flexible and they are continuous in tan-gential direction and rigid in radial direction Theload-slip behavior (per unit length) of connectors intangential direction is described in a linear-elasticrange with a constant slip modulus 119870[Nm2]

(3) The uplift between the slab and I-girder is neglectedThe radius of curvature is constant along the beam

The deflections in tangential direction vertical direc-tion and radial direction (119911-direction 119910-direction and 119909-direction) are indicated as 119906 119908 and V respectively TheI-girder has the same torsional deflection 120601 and verticaldeflection 119908 as the slab has In this paper subscript ldquo1rdquorefers to the cross-section of the slab and subscript ldquo2rdquo refersto the cross-section of I-girder such that the deflectionsof the slab and I-girder can be represented as (119908 120601 1199061 V1)



120576119911119894 = 1199061015840119894 minus V119894119877 119896119910119894 = V10158401015840119894 + 1199061015840119894119877 119896119909119894 = 11990810158401015840 minus 120601119877119896119911119894 = 1206011015840 + 1199081015840119877 (119894 = 1 2)

(1)


(2)



(3)


(4)



(1198731 + 1198732)1015840 + (1198721199101 +1198721199102)1015840119877 = 0 (6)






q0

m0

Qy1

My1Mx1

Qx1T1

N1

Mx1 + dMx1

Qx1 + dQx1

My1 + dMy1

Qy1 + dQy1


qx2qz2Qy2

My2

Mx2

Qx2T2

N2

Mx2 + dMx2

Qx2 + dQx2

My2 + dMy2

Qy2 + dQy2

T2 + dT2

N2 + dN2


1

2

z

y

b

b1

b2w998400

w998400b1 w998400b2

Δu

u2 minus u1



V2 minus V1 + 120601119887 = 0 (10)




cg 1

cg 2y

x

cg 0

b

b1

b2

120601b = 1 minus 2



119891119910 = 11986411198601119870119877 (11990610158401 minus V1119877 )10158401015840 + 119864111986811991011198701198772 (V101584010158401 + 11990610158401119877 )10158401015840minus 11987811991011986411198681199101 (V101584010158401 + 11990610158401119877 ) + 12060110158401015840119887 + 11990810158401015840119887119877 = 0 (12)





119880119872119909 = 12 int1198771205790 (11987211990911198961199091 +11987211990921198961199092) 119889119911= 12119864119868119909 int1198771205790 (11990810158401015840 minus 120601119877)2 119889119911 (14)


119880119879 = 12 int1198771205790 (11987911198961199111 + 11987921198961199112) 119889119911= 12119866119868119879int1198771205790 (1206011015840 + 1199081015840119877 )2 119889119911 (15)


= 12 int1198771205790 119878119860 [11986411198601 (11990610158401 minus V1119877 )]2 119889119911 (16)


119880119872119910 = 12 int1198771205790 (11987211991011198961199101 +11987211991021198961199102) 119889119911= 12 int1198771205790 119878119910 [11986411198681199101 (V101584010158401 + 119906101584010158401119877 )]2 119889119911 (17)



(18)


119882 = int1198771205790

1198980120601119889119911 + int1198771205790

1199020119908119889119911 + (1198721199091199081015840)100381610038161003816100381610038161198771205790+ (119876119910119908)100381610038161003816100381610038161198771205790 + (119879119911120601)10038161003816100381610038161198771205790 (19)



(120582119911 sdot 119891119911) 119889119911 + int1198771205790

(120582119910 sdot 119891119910) 119889119911 (20)


0Γ1120575119908119889119911 + int119877120579

0Γ2120575120601 119889119911 + int119877120579

0Γ3120575120582119911119889119911

+ int1198771205790

Γ4120575120582119910119889119911 + int1198771205790

Γ5120575V1119889119911 + int1198771205790

Γ61205751199061119889119911+ 119867112057511990810038161003816100381610038161198771205790 + 11986721205751199081015840100381610038161003816100381610038161198771205790 + 119867312057512060110038161003816100381610038161198771205790 + 11986741205751206011015840100381610038161003816100381610038161198771205790+ 1198675120575119906110038161003816100381610038161198771205790 + 119867612057511990610158401100381610038161003816100381610038161198771205790 + 1198677120575V110038161003816100381610038161198771205790 + 1198678120575V10158401100381610038161003816100381610038161198771205790+ 1198679120575V101584010158401 100381610038161003816100381610038161198771205790+ 11986710120575 [[[[

11986411198681199101119877119870 (V101584010158401 + 11990610158401119877 )1015840 + 11986411198601119870 (11990610158401 minus V10158401119877 )1015840⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟Δ119906

]]]]1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816119877120579

0

(21)


120582119911 = 11986411198601 (11990610158401 minus V1119877 ) = 1198731120582119910 = 11986411198681199101 (V101584010158401 + 11990610158401119877 ) = 1198721199101 (22)



[[[[[[[[[[[[[


]]]]]]]]]]]]](119908120601120582119911120582119910)=(1199020119898000 ) (23)


(24)




119894=1

119908119894 sin(119894120587119911119871 ) 120601 = 119899sum119894=1

120601119894 sin(119894120587119911119871 ) 120582119911 = 119899sum119894=1

120582119911119894 sin(119894120587119911119871 ) 120582119910 = 119899sum119894=1

120582119910119894 sin(119894120587119911119871 ) (25)


1199020 = 119899sum119894=1

119902119894 sin(119894120587119911119871 ) 1198980 = 119899sum

119894=1

119898119894 sin(119894120587119911119871 ) (26)


120601119894 = 119902119894 1205741199023120574 + 119898119894 1205741198982 + 1205741198983120574 120582119911119894 = 119902119894 1205741199024120574 + 119898119894 1205741198984120574 120582119910119894 = 119902119894 1205741199025120574 + 119898119894 1205741198985120574

(27)


6 Numerical Example






R

Free end

119876119910 = 0119872119909 = 0119879119911 = 0120582119911 = 0120582119910 = 0R

Fixed end

119908 = 0120601 = 01199081015840 = 01206011015840 = 0Δ119906 = 0R


119908 = 0120601 = 0119872119909 = 0120582119911 = 0120582119910 = 0R

Hinged end

119908 = 0119879119911 = 0119872119909 = 0120582119911 = 0120582119910 = 0Space







ExperimentFEMn10

n5

n1


minus0001000000010002000300040005000600070008000900100011

The v

ertic

al d

eflec

tion

(m)


FEMn10

n5

n1


minus0007

minus0006

minus0005

minus0004

minus0003

minus0002

minus0001

0000

0001Th

e tor

siona

l ang

le (r

ad)







Poissonrsquosratio

Density(kgm3)

Width(mm)

Thickness(mm)

Centralangle



Flange 13Web 8 120579 = 143∘ 24

Concrete slab 26 027 2400 1500 100

FEMn10

n5n1


00

02

04

06

08

10

The b

endi

ng m

omen

tMy1

(KN

m)


FEMn10

n5

n1


minus700

minus600

minus500

minus400

minus300

minus200

minus100

0

The a

xial

forc

eN1

(KN

)



FEMn10

n5

n1


minus250

minus200

minus150

minus100

minus50

0

The b

endi

ng m

omen

tMx

(KN

m)


FEMn10

n5

n1



020406080

100120

The s

hear

forc

eQy

(KN

)





minus20

minus15

minus10

minus5

0

5

10

15

20

The m

omen

t of t

orqu

eTz

(KN

m)

FEMn10

n5

n1



K

0008

0010

0012

0014

0016

0018

0020

0022

Mid

span

ver

tical

defl

ectio

n (m

)

FEMn10

n5

n1



7 Conclusions


FEMn10

n5n1


K

minus00070

minus00069

minus00068

minus00067

minus00066

minus00065

minus00064

minus00063

Mid

span

tors

iona

l ang

le (r

ad)



Appendix









minus 120582119910119877 ]10158401015840 (11986411198601 + 119864111986811991011198772 ) 1198676 = 1119870 [11986411198601 (11990610158401 minus V1119877 ) + 11986411198681199101119877 (V101584010158401 + 11990610158401119877 )





11986710 = 120582119911 + 120582119910119877 (A1)





(B1)

Competing Interests


Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120576119911119894 = 1199061015840119894 minus V119894119877 119896119910119894 = V10158401015840119894 + 1199061015840119894119877 119896119909119894 = 11990810158401015840 minus 120601119877119896119911119894 = 1206011015840 + 1199081015840119877 (119894 = 1 2)

(1)


(2)



(3)


(4)



(1198731 + 1198732)1015840 + (1198721199101 +1198721199102)1015840119877 = 0 (6)






q0

m0

Qy1

My1Mx1

Qx1T1

N1

Mx1 + dMx1

Qx1 + dQx1

My1 + dMy1

Qy1 + dQy1


qx2qz2Qy2

My2

Mx2

Qx2T2

N2

Mx2 + dMx2

Qx2 + dQx2

My2 + dMy2

Qy2 + dQy2

T2 + dT2

N2 + dN2


1

2

z

y

b

b1

b2w998400

w998400b1 w998400b2

Δu

u2 minus u1



V2 minus V1 + 120601119887 = 0 (10)




cg 1

cg 2y

x

cg 0

b

b1

b2

120601b = 1 minus 2



119891119910 = 11986411198601119870119877 (11990610158401 minus V1119877 )10158401015840 + 119864111986811991011198701198772 (V101584010158401 + 11990610158401119877 )10158401015840minus 11987811991011986411198681199101 (V101584010158401 + 11990610158401119877 ) + 12060110158401015840119887 + 11990810158401015840119887119877 = 0 (12)





119880119872119909 = 12 int1198771205790 (11987211990911198961199091 +11987211990921198961199092) 119889119911= 12119864119868119909 int1198771205790 (11990810158401015840 minus 120601119877)2 119889119911 (14)


119880119879 = 12 int1198771205790 (11987911198961199111 + 11987921198961199112) 119889119911= 12119866119868119879int1198771205790 (1206011015840 + 1199081015840119877 )2 119889119911 (15)


= 12 int1198771205790 119878119860 [11986411198601 (11990610158401 minus V1119877 )]2 119889119911 (16)


119880119872119910 = 12 int1198771205790 (11987211991011198961199101 +11987211991021198961199102) 119889119911= 12 int1198771205790 119878119910 [11986411198681199101 (V101584010158401 + 119906101584010158401119877 )]2 119889119911 (17)



(18)


119882 = int1198771205790

1198980120601119889119911 + int1198771205790

1199020119908119889119911 + (1198721199091199081015840)100381610038161003816100381610038161198771205790+ (119876119910119908)100381610038161003816100381610038161198771205790 + (119879119911120601)10038161003816100381610038161198771205790 (19)



(120582119911 sdot 119891119911) 119889119911 + int1198771205790

(120582119910 sdot 119891119910) 119889119911 (20)


0Γ1120575119908119889119911 + int119877120579

0Γ2120575120601 119889119911 + int119877120579

0Γ3120575120582119911119889119911

+ int1198771205790

Γ4120575120582119910119889119911 + int1198771205790

Γ5120575V1119889119911 + int1198771205790

Γ61205751199061119889119911+ 119867112057511990810038161003816100381610038161198771205790 + 11986721205751199081015840100381610038161003816100381610038161198771205790 + 119867312057512060110038161003816100381610038161198771205790 + 11986741205751206011015840100381610038161003816100381610038161198771205790+ 1198675120575119906110038161003816100381610038161198771205790 + 119867612057511990610158401100381610038161003816100381610038161198771205790 + 1198677120575V110038161003816100381610038161198771205790 + 1198678120575V10158401100381610038161003816100381610038161198771205790+ 1198679120575V101584010158401 100381610038161003816100381610038161198771205790+ 11986710120575 [[[[

11986411198681199101119877119870 (V101584010158401 + 11990610158401119877 )1015840 + 11986411198601119870 (11990610158401 minus V10158401119877 )1015840⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟Δ119906

]]]]1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816119877120579

0

(21)


120582119911 = 11986411198601 (11990610158401 minus V1119877 ) = 1198731120582119910 = 11986411198681199101 (V101584010158401 + 11990610158401119877 ) = 1198721199101 (22)



[[[[[[[[[[[[[


]]]]]]]]]]]]](119908120601120582119911120582119910)=(1199020119898000 ) (23)


(24)




119894=1

119908119894 sin(119894120587119911119871 ) 120601 = 119899sum119894=1

120601119894 sin(119894120587119911119871 ) 120582119911 = 119899sum119894=1

120582119911119894 sin(119894120587119911119871 ) 120582119910 = 119899sum119894=1

120582119910119894 sin(119894120587119911119871 ) (25)


1199020 = 119899sum119894=1

119902119894 sin(119894120587119911119871 ) 1198980 = 119899sum

119894=1

119898119894 sin(119894120587119911119871 ) (26)


120601119894 = 119902119894 1205741199023120574 + 119898119894 1205741198982 + 1205741198983120574 120582119911119894 = 119902119894 1205741199024120574 + 119898119894 1205741198984120574 120582119910119894 = 119902119894 1205741199025120574 + 119898119894 1205741198985120574

(27)


6 Numerical Example






R

Free end

119876119910 = 0119872119909 = 0119879119911 = 0120582119911 = 0120582119910 = 0R

Fixed end

119908 = 0120601 = 01199081015840 = 01206011015840 = 0Δ119906 = 0R


119908 = 0120601 = 0119872119909 = 0120582119911 = 0120582119910 = 0R

Hinged end

119908 = 0119879119911 = 0119872119909 = 0120582119911 = 0120582119910 = 0Space







ExperimentFEMn10

n5

n1


minus0001000000010002000300040005000600070008000900100011

The v

ertic

al d

eflec

tion

(m)


FEMn10

n5

n1


minus0007

minus0006

minus0005

minus0004

minus0003

minus0002

minus0001

0000

0001Th

e tor

siona

l ang

le (r

ad)







Poissonrsquosratio

Density(kgm3)

Width(mm)

Thickness(mm)

Centralangle



Flange 13Web 8 120579 = 143∘ 24

Concrete slab 26 027 2400 1500 100

FEMn10

n5n1


00

02

04

06

08

10

The b

endi

ng m

omen

tMy1

(KN

m)


FEMn10

n5

n1


minus700

minus600

minus500

minus400

minus300

minus200

minus100

0

The a

xial

forc

eN1

(KN

)



FEMn10

n5

n1


minus250

minus200

minus150

minus100

minus50

0

The b

endi

ng m

omen

tMx

(KN

m)


FEMn10

n5

n1



020406080

100120

The s

hear

forc

eQy

(KN

)





minus20

minus15

minus10

minus5

0

5

10

15

20

The m

omen

t of t

orqu

eTz

(KN

m)

FEMn10

n5

n1



K

0008

0010

0012

0014

0016

0018

0020

0022

Mid

span

ver

tical

defl

ectio

n (m

)

FEMn10

n5

n1



7 Conclusions


FEMn10

n5n1


K

minus00070

minus00069

minus00068

minus00067

minus00066

minus00065

minus00064

minus00063

Mid

span

tors

iona

l ang

le (r

ad)



Appendix









minus 120582119910119877 ]10158401015840 (11986411198601 + 119864111986811991011198772 ) 1198676 = 1119870 [11986411198601 (11990610158401 minus V1119877 ) + 11986411198681199101119877 (V101584010158401 + 11990610158401119877 )





11986710 = 120582119911 + 120582119910119877 (A1)





(B1)

Competing Interests


Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










q0

m0

Qy1

My1Mx1

Qx1T1

N1

Mx1 + dMx1

Qx1 + dQx1

My1 + dMy1

Qy1 + dQy1


qx2qz2Qy2

My2

Mx2

Qx2T2

N2

Mx2 + dMx2

Qx2 + dQx2

My2 + dMy2

Qy2 + dQy2

T2 + dT2

N2 + dN2


1

2

z

y

b

b1

b2w998400

w998400b1 w998400b2

Δu

u2 minus u1



V2 minus V1 + 120601119887 = 0 (10)




cg 1

cg 2y

x

cg 0

b

b1

b2

120601b = 1 minus 2



119891119910 = 11986411198601119870119877 (11990610158401 minus V1119877 )10158401015840 + 119864111986811991011198701198772 (V101584010158401 + 11990610158401119877 )10158401015840minus 11987811991011986411198681199101 (V101584010158401 + 11990610158401119877 ) + 12060110158401015840119887 + 11990810158401015840119887119877 = 0 (12)





119880119872119909 = 12 int1198771205790 (11987211990911198961199091 +11987211990921198961199092) 119889119911= 12119864119868119909 int1198771205790 (11990810158401015840 minus 120601119877)2 119889119911 (14)


119880119879 = 12 int1198771205790 (11987911198961199111 + 11987921198961199112) 119889119911= 12119866119868119879int1198771205790 (1206011015840 + 1199081015840119877 )2 119889119911 (15)


= 12 int1198771205790 119878119860 [11986411198601 (11990610158401 minus V1119877 )]2 119889119911 (16)


119880119872119910 = 12 int1198771205790 (11987211991011198961199101 +11987211991021198961199102) 119889119911= 12 int1198771205790 119878119910 [11986411198681199101 (V101584010158401 + 119906101584010158401119877 )]2 119889119911 (17)



(18)


119882 = int1198771205790

1198980120601119889119911 + int1198771205790

1199020119908119889119911 + (1198721199091199081015840)100381610038161003816100381610038161198771205790+ (119876119910119908)100381610038161003816100381610038161198771205790 + (119879119911120601)10038161003816100381610038161198771205790 (19)



(120582119911 sdot 119891119911) 119889119911 + int1198771205790

(120582119910 sdot 119891119910) 119889119911 (20)


0Γ1120575119908119889119911 + int119877120579

0Γ2120575120601 119889119911 + int119877120579

0Γ3120575120582119911119889119911

+ int1198771205790

Γ4120575120582119910119889119911 + int1198771205790

Γ5120575V1119889119911 + int1198771205790

Γ61205751199061119889119911+ 119867112057511990810038161003816100381610038161198771205790 + 11986721205751199081015840100381610038161003816100381610038161198771205790 + 119867312057512060110038161003816100381610038161198771205790 + 11986741205751206011015840100381610038161003816100381610038161198771205790+ 1198675120575119906110038161003816100381610038161198771205790 + 119867612057511990610158401100381610038161003816100381610038161198771205790 + 1198677120575V110038161003816100381610038161198771205790 + 1198678120575V10158401100381610038161003816100381610038161198771205790+ 1198679120575V101584010158401 100381610038161003816100381610038161198771205790+ 11986710120575 [[[[

11986411198681199101119877119870 (V101584010158401 + 11990610158401119877 )1015840 + 11986411198601119870 (11990610158401 minus V10158401119877 )1015840⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟Δ119906

]]]]1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816119877120579

0

(21)


120582119911 = 11986411198601 (11990610158401 minus V1119877 ) = 1198731120582119910 = 11986411198681199101 (V101584010158401 + 11990610158401119877 ) = 1198721199101 (22)



[[[[[[[[[[[[[


]]]]]]]]]]]]](119908120601120582119911120582119910)=(1199020119898000 ) (23)


(24)




119894=1

119908119894 sin(119894120587119911119871 ) 120601 = 119899sum119894=1

120601119894 sin(119894120587119911119871 ) 120582119911 = 119899sum119894=1

120582119911119894 sin(119894120587119911119871 ) 120582119910 = 119899sum119894=1

120582119910119894 sin(119894120587119911119871 ) (25)


1199020 = 119899sum119894=1

119902119894 sin(119894120587119911119871 ) 1198980 = 119899sum

119894=1

119898119894 sin(119894120587119911119871 ) (26)


120601119894 = 119902119894 1205741199023120574 + 119898119894 1205741198982 + 1205741198983120574 120582119911119894 = 119902119894 1205741199024120574 + 119898119894 1205741198984120574 120582119910119894 = 119902119894 1205741199025120574 + 119898119894 1205741198985120574

(27)


6 Numerical Example






R

Free end

119876119910 = 0119872119909 = 0119879119911 = 0120582119911 = 0120582119910 = 0R

Fixed end

119908 = 0120601 = 01199081015840 = 01206011015840 = 0Δ119906 = 0R


119908 = 0120601 = 0119872119909 = 0120582119911 = 0120582119910 = 0R

Hinged end

119908 = 0119879119911 = 0119872119909 = 0120582119911 = 0120582119910 = 0Space







ExperimentFEMn10

n5

n1


minus0001000000010002000300040005000600070008000900100011

The v

ertic

al d

eflec

tion

(m)


FEMn10

n5

n1


minus0007

minus0006

minus0005

minus0004

minus0003

minus0002

minus0001

0000

0001Th

e tor

siona

l ang

le (r

ad)







Poissonrsquosratio

Density(kgm3)

Width(mm)

Thickness(mm)

Centralangle



Flange 13Web 8 120579 = 143∘ 24

Concrete slab 26 027 2400 1500 100

FEMn10

n5n1


00

02

04

06

08

10

The b

endi

ng m

omen

tMy1

(KN

m)


FEMn10

n5

n1


minus700

minus600

minus500

minus400

minus300

minus200

minus100

0

The a

xial

forc

eN1

(KN

)



FEMn10

n5

n1


minus250

minus200

minus150

minus100

minus50

0

The b

endi

ng m

omen

tMx

(KN

m)


FEMn10

n5

n1



020406080

100120

The s

hear

forc

eQy

(KN

)





minus20

minus15

minus10

minus5

0

5

10

15

20

The m

omen

t of t

orqu

eTz

(KN

m)

FEMn10

n5

n1



K

0008

0010

0012

0014

0016

0018

0020

0022

Mid

span

ver

tical

defl

ectio

n (m

)

FEMn10

n5

n1



7 Conclusions


FEMn10

n5n1


K

minus00070

minus00069

minus00068

minus00067

minus00066

minus00065

minus00064

minus00063

Mid

span

tors

iona

l ang

le (r

ad)



Appendix









minus 120582119910119877 ]10158401015840 (11986411198601 + 119864111986811991011198772 ) 1198676 = 1119870 [11986411198601 (11990610158401 minus V1119877 ) + 11986411198681199101119877 (V101584010158401 + 11990610158401119877 )





11986710 = 120582119911 + 120582119910119877 (A1)





(B1)

Competing Interests


Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119880119872119909 = 12 int1198771205790 (11987211990911198961199091 +11987211990921198961199092) 119889119911= 12119864119868119909 int1198771205790 (11990810158401015840 minus 120601119877)2 119889119911 (14)


119880119879 = 12 int1198771205790 (11987911198961199111 + 11987921198961199112) 119889119911= 12119866119868119879int1198771205790 (1206011015840 + 1199081015840119877 )2 119889119911 (15)


= 12 int1198771205790 119878119860 [11986411198601 (11990610158401 minus V1119877 )]2 119889119911 (16)


119880119872119910 = 12 int1198771205790 (11987211991011198961199101 +11987211991021198961199102) 119889119911= 12 int1198771205790 119878119910 [11986411198681199101 (V101584010158401 + 119906101584010158401119877 )]2 119889119911 (17)



(18)


119882 = int1198771205790

1198980120601119889119911 + int1198771205790

1199020119908119889119911 + (1198721199091199081015840)100381610038161003816100381610038161198771205790+ (119876119910119908)100381610038161003816100381610038161198771205790 + (119879119911120601)10038161003816100381610038161198771205790 (19)



(120582119911 sdot 119891119911) 119889119911 + int1198771205790

(120582119910 sdot 119891119910) 119889119911 (20)


0Γ1120575119908119889119911 + int119877120579

0Γ2120575120601 119889119911 + int119877120579

0Γ3120575120582119911119889119911

+ int1198771205790

Γ4120575120582119910119889119911 + int1198771205790

Γ5120575V1119889119911 + int1198771205790

Γ61205751199061119889119911+ 119867112057511990810038161003816100381610038161198771205790 + 11986721205751199081015840100381610038161003816100381610038161198771205790 + 119867312057512060110038161003816100381610038161198771205790 + 11986741205751206011015840100381610038161003816100381610038161198771205790+ 1198675120575119906110038161003816100381610038161198771205790 + 119867612057511990610158401100381610038161003816100381610038161198771205790 + 1198677120575V110038161003816100381610038161198771205790 + 1198678120575V10158401100381610038161003816100381610038161198771205790+ 1198679120575V101584010158401 100381610038161003816100381610038161198771205790+ 11986710120575 [[[[

11986411198681199101119877119870 (V101584010158401 + 11990610158401119877 )1015840 + 11986411198601119870 (11990610158401 minus V10158401119877 )1015840⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟Δ119906

]]]]1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816119877120579

0

(21)


120582119911 = 11986411198601 (11990610158401 minus V1119877 ) = 1198731120582119910 = 11986411198681199101 (V101584010158401 + 11990610158401119877 ) = 1198721199101 (22)



[[[[[[[[[[[[[


]]]]]]]]]]]]](119908120601120582119911120582119910)=(1199020119898000 ) (23)


(24)




119894=1

119908119894 sin(119894120587119911119871 ) 120601 = 119899sum119894=1

120601119894 sin(119894120587119911119871 ) 120582119911 = 119899sum119894=1

120582119911119894 sin(119894120587119911119871 ) 120582119910 = 119899sum119894=1

120582119910119894 sin(119894120587119911119871 ) (25)


1199020 = 119899sum119894=1

119902119894 sin(119894120587119911119871 ) 1198980 = 119899sum

119894=1

119898119894 sin(119894120587119911119871 ) (26)


120601119894 = 119902119894 1205741199023120574 + 119898119894 1205741198982 + 1205741198983120574 120582119911119894 = 119902119894 1205741199024120574 + 119898119894 1205741198984120574 120582119910119894 = 119902119894 1205741199025120574 + 119898119894 1205741198985120574

(27)


6 Numerical Example






R

Free end

119876119910 = 0119872119909 = 0119879119911 = 0120582119911 = 0120582119910 = 0R

Fixed end

119908 = 0120601 = 01199081015840 = 01206011015840 = 0Δ119906 = 0R


119908 = 0120601 = 0119872119909 = 0120582119911 = 0120582119910 = 0R

Hinged end

119908 = 0119879119911 = 0119872119909 = 0120582119911 = 0120582119910 = 0Space







ExperimentFEMn10

n5

n1


minus0001000000010002000300040005000600070008000900100011

The v

ertic

al d

eflec

tion

(m)


FEMn10

n5

n1


minus0007

minus0006

minus0005

minus0004

minus0003

minus0002

minus0001

0000

0001Th

e tor

siona

l ang

le (r

ad)







Poissonrsquosratio

Density(kgm3)

Width(mm)

Thickness(mm)

Centralangle



Flange 13Web 8 120579 = 143∘ 24

Concrete slab 26 027 2400 1500 100

FEMn10

n5n1


00

02

04

06

08

10

The b

endi

ng m

omen

tMy1

(KN

m)


FEMn10

n5

n1


minus700

minus600

minus500

minus400

minus300

minus200

minus100

0

The a

xial

forc

eN1

(KN

)



FEMn10

n5

n1


minus250

minus200

minus150

minus100

minus50

0

The b

endi

ng m

omen

tMx

(KN

m)


FEMn10

n5

n1



020406080

100120

The s

hear

forc

eQy

(KN

)





minus20

minus15

minus10

minus5

0

5

10

15

20

The m

omen

t of t

orqu

eTz

(KN

m)

FEMn10

n5

n1



K

0008

0010

0012

0014

0016

0018

0020

0022

Mid

span

ver

tical

defl

ectio

n (m

)

FEMn10

n5

n1



7 Conclusions


FEMn10

n5n1


K

minus00070

minus00069

minus00068

minus00067

minus00066

minus00065

minus00064

minus00063

Mid

span

tors

iona

l ang

le (r

ad)



Appendix









minus 120582119910119877 ]10158401015840 (11986411198601 + 119864111986811991011198772 ) 1198676 = 1119870 [11986411198601 (11990610158401 minus V1119877 ) + 11986411198681199101119877 (V101584010158401 + 11990610158401119877 )





11986710 = 120582119911 + 120582119910119877 (A1)





(B1)

Competing Interests


Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










[[[[[[[[[[[[[


]]]]]]]]]]]]](119908120601120582119911120582119910)=(1199020119898000 ) (23)


(24)




119894=1

119908119894 sin(119894120587119911119871 ) 120601 = 119899sum119894=1

120601119894 sin(119894120587119911119871 ) 120582119911 = 119899sum119894=1

120582119911119894 sin(119894120587119911119871 ) 120582119910 = 119899sum119894=1

120582119910119894 sin(119894120587119911119871 ) (25)


1199020 = 119899sum119894=1

119902119894 sin(119894120587119911119871 ) 1198980 = 119899sum

119894=1

119898119894 sin(119894120587119911119871 ) (26)


120601119894 = 119902119894 1205741199023120574 + 119898119894 1205741198982 + 1205741198983120574 120582119911119894 = 119902119894 1205741199024120574 + 119898119894 1205741198984120574 120582119910119894 = 119902119894 1205741199025120574 + 119898119894 1205741198985120574

(27)


6 Numerical Example






R

Free end

119876119910 = 0119872119909 = 0119879119911 = 0120582119911 = 0120582119910 = 0R

Fixed end

119908 = 0120601 = 01199081015840 = 01206011015840 = 0Δ119906 = 0R


119908 = 0120601 = 0119872119909 = 0120582119911 = 0120582119910 = 0R

Hinged end

119908 = 0119879119911 = 0119872119909 = 0120582119911 = 0120582119910 = 0Space







ExperimentFEMn10

n5

n1


minus0001000000010002000300040005000600070008000900100011

The v

ertic

al d

eflec

tion

(m)


FEMn10

n5

n1


minus0007

minus0006

minus0005

minus0004

minus0003

minus0002

minus0001

0000

0001Th

e tor

siona

l ang

le (r

ad)







Poissonrsquosratio

Density(kgm3)

Width(mm)

Thickness(mm)

Centralangle



Flange 13Web 8 120579 = 143∘ 24

Concrete slab 26 027 2400 1500 100

FEMn10

n5n1


00

02

04

06

08

10

The b

endi

ng m

omen

tMy1

(KN

m)


FEMn10

n5

n1


minus700

minus600

minus500

minus400

minus300

minus200

minus100

0

The a

xial

forc

eN1

(KN

)



FEMn10

n5

n1


minus250

minus200

minus150

minus100

minus50

0

The b

endi

ng m

omen

tMx

(KN

m)


FEMn10

n5

n1



020406080

100120

The s

hear

forc

eQy

(KN

)





minus20

minus15

minus10

minus5

0

5

10

15

20

The m

omen

t of t

orqu

eTz

(KN

m)

FEMn10

n5

n1



K

0008

0010

0012

0014

0016

0018

0020

0022

Mid

span

ver

tical

defl

ectio

n (m

)

FEMn10

n5

n1



7 Conclusions


FEMn10

n5n1


K

minus00070

minus00069

minus00068

minus00067

minus00066

minus00065

minus00064

minus00063

Mid

span

tors

iona

l ang

le (r

ad)



Appendix









minus 120582119910119877 ]10158401015840 (11986411198601 + 119864111986811991011198772 ) 1198676 = 1119870 [11986411198601 (11990610158401 minus V1119877 ) + 11986411198681199101119877 (V101584010158401 + 11990610158401119877 )





11986710 = 120582119911 + 120582119910119877 (A1)





(B1)

Competing Interests


Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












R

Free end

119876119910 = 0119872119909 = 0119879119911 = 0120582119911 = 0120582119910 = 0R

Fixed end

119908 = 0120601 = 01199081015840 = 01206011015840 = 0Δ119906 = 0R


119908 = 0120601 = 0119872119909 = 0120582119911 = 0120582119910 = 0R

Hinged end

119908 = 0119879119911 = 0119872119909 = 0120582119911 = 0120582119910 = 0Space







ExperimentFEMn10

n5

n1


minus0001000000010002000300040005000600070008000900100011

The v

ertic

al d

eflec

tion

(m)


FEMn10

n5

n1


minus0007

minus0006

minus0005

minus0004

minus0003

minus0002

minus0001

0000

0001Th

e tor

siona

l ang

le (r

ad)







Poissonrsquosratio

Density(kgm3)

Width(mm)

Thickness(mm)

Centralangle



Flange 13Web 8 120579 = 143∘ 24

Concrete slab 26 027 2400 1500 100

FEMn10

n5n1


00

02

04

06

08

10

The b

endi

ng m

omen

tMy1

(KN

m)


FEMn10

n5

n1


minus700

minus600

minus500

minus400

minus300

minus200

minus100

0

The a

xial

forc

eN1

(KN

)



FEMn10

n5

n1


minus250

minus200

minus150

minus100

minus50

0

The b

endi

ng m

omen

tMx

(KN

m)


FEMn10

n5

n1



020406080

100120

The s

hear

forc

eQy

(KN

)





minus20

minus15

minus10

minus5

0

5

10

15

20

The m

omen

t of t

orqu

eTz

(KN

m)

FEMn10

n5

n1



K

0008

0010

0012

0014

0016

0018

0020

0022

Mid

span

ver

tical

defl

ectio

n (m

)

FEMn10

n5

n1



7 Conclusions


FEMn10

n5n1


K

minus00070

minus00069

minus00068

minus00067

minus00066

minus00065

minus00064

minus00063

Mid

span

tors

iona

l ang

le (r

ad)



Appendix









minus 120582119910119877 ]10158401015840 (11986411198601 + 119864111986811991011198772 ) 1198676 = 1119870 [11986411198601 (11990610158401 minus V1119877 ) + 11986411198681199101119877 (V101584010158401 + 11990610158401119877 )





11986710 = 120582119911 + 120582119910119877 (A1)





(B1)

Competing Interests


Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Poissonrsquosratio

Density(kgm3)

Width(mm)

Thickness(mm)

Centralangle



Flange 13Web 8 120579 = 143∘ 24

Concrete slab 26 027 2400 1500 100

FEMn10

n5n1


00

02

04

06

08

10

The b

endi

ng m

omen

tMy1

(KN

m)


FEMn10

n5

n1


minus700

minus600

minus500

minus400

minus300

minus200

minus100

0

The a

xial

forc

eN1

(KN

)



FEMn10

n5

n1


minus250

minus200

minus150

minus100

minus50

0

The b

endi

ng m

omen

tMx

(KN

m)


FEMn10

n5

n1



020406080

100120

The s

hear

forc

eQy

(KN

)





minus20

minus15

minus10

minus5

0

5

10

15

20

The m

omen

t of t

orqu

eTz

(KN

m)

FEMn10

n5

n1



K

0008

0010

0012

0014

0016

0018

0020

0022

Mid

span

ver

tical

defl

ectio

n (m

)

FEMn10

n5

n1



7 Conclusions


FEMn10

n5n1


K

minus00070

minus00069

minus00068

minus00067

minus00066

minus00065

minus00064

minus00063

Mid

span

tors

iona

l ang

le (r

ad)



Appendix









minus 120582119910119877 ]10158401015840 (11986411198601 + 119864111986811991011198772 ) 1198676 = 1119870 [11986411198601 (11990610158401 minus V1119877 ) + 11986411198681199101119877 (V101584010158401 + 11990610158401119877 )





11986710 = 120582119911 + 120582119910119877 (A1)





(B1)

Competing Interests


Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











minus20

minus15

minus10

minus5

0

5

10

15

20

The m

omen

t of t

orqu

eTz

(KN

m)

FEMn10

n5

n1



K

0008

0010

0012

0014

0016

0018

0020

0022

Mid

span

ver

tical

defl

ectio

n (m

)

FEMn10

n5

n1



7 Conclusions


FEMn10

n5n1


K

minus00070

minus00069

minus00068

minus00067

minus00066

minus00065

minus00064

minus00063

Mid

span

tors

iona

l ang

le (r

ad)



Appendix









minus 120582119910119877 ]10158401015840 (11986411198601 + 119864111986811991011198772 ) 1198676 = 1119870 [11986411198601 (11990610158401 minus V1119877 ) + 11986411198681199101119877 (V101584010158401 + 11990610158401119877 )





11986710 = 120582119911 + 120582119910119877 (A1)





(B1)

Competing Interests


Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














minus 120582119910119877 ]10158401015840 (11986411198601 + 119864111986811991011198772 ) 1198676 = 1119870 [11986411198601 (11990610158401 minus V1119877 ) + 11986411198681199101119877 (V101584010158401 + 11990610158401119877 )





11986710 = 120582119911 + 120582119910119877 (A1)





(B1)

Competing Interests


Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(B1)

Competing Interests


Acknowledgments


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article A Trigonometric Analytical …downloads.hindawi.com/journals/mpe/2016/2465025.pdfResearch Article A Trigonometric Analytical Solution of Simply Supported Horizontally