Analysis of Stress and Deflection about Steel-Concrete Composite Girders Considering Slippage and Shrink & Creep Under Bending

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1 Send Orders for Reprints to Te Open Civil Engineering Journal Open Access Analysis of Stress and Deflection about Steel-Concrete Composite Girders Considering Slippage and Srink Creep Under Bending Ceng Haigen Scool of Civil Engineering East Cina Jiaotong University Nancang JiangXi P.R. Cina Abstract: Steel-concrete composite beams are composed of concrete slabs and steel girders by sear connectors. Due to te limited rigidity of sear connector and te srink creep property of concrete relative slippage exists between te concrete slab and steel girder under bending and it is difficult to analyze te effect of tose factors by te ordinary beam teory te finite element metodfem) and so on. A differential equation of equilibrium is constituted corresponding to te compatibility of deformation and te equilibrium of forces of steel-concrete composite beams under particular assumed condition. Finite difference metod FDM) and variation principle are used to solve te differential equation. An example of steel-concrete composite T girder is given to analyze te effect of slippage and concrete srink creep on its stress and deflection. Te concrete slab stress increases wit increased rigidity in te sear connectors. Te stress of te steel girder and te deflection of te composite girder decrease wit increment in te rigidity of te sear connectors. Keywords: Differential equations relative slippage steel-concrete composite beams srink creep variation principle.. INTRODUCTION Composite beams are usually composed of concrete slab wic bears compressive stress and te steel beam wic bears te tensile stress. Tis combination takes full advantage of tensile capacity of steels and compression capacity of concretes respectively. For tis kind of beam tere is a larger difference between te stress calculated by te primary beam teory and te actual stress. Te discrepancy is observed not only in te distribution of stress along te transverse section but also in te relationsip between stress and time. In order to solve tis problem many scolars ave conducted some researces and put forward some feasible calculation teories and metods suc as energy-variational metod te finite plate strip metod te finite element metod and so on [-]. Some researcers ave performed corresponding model experimental researc [-7]. Kristek solved te problem by using te metod of series [8] independent of te condition of relative slip. However te fact is tat te steel beam and concrete plate of steel - concrete composite beam were combined togeter by sear connectors []. Because te stiffness of connecting bond is limited tere is a relative slip tendency between steel beam and concrete slab. Moreover because concrete as te property of srinkage and creep it is necessary to consider te influence of te property and relative slip of concrete. According to te principle of deformation compatibility and equilibrium conditions tis paper deduced te equilibrium differential equation of composite beam under te condition of considering te relative slippage and verified te reliability and te accuracy of te approximate metod by giving examples. Address correspondence to tis autor at te Scool of Civil Engineering East Cina Jiaotong University Nancang Jiangxi P.R. Cina; Tel: 8799; bridge7@.com. DIFFERENTIAL EQUATION OF THE COORDI- NATION OF BALANCE AND DEFORMATION.. Te Basic Assumptions Te steel concrete composite girders are sown in Fig. ). Several assumptions are as follows [9 ]: ) Tere is no vertical detacment between concrete slab and steel in te composite girder under bending. Tat is bot of tem ave te same vertical displacement. ) Te displacement does not take place at any cross section in x direction As sown in Fig. ). Namely lateral bending is not observed. ) Te maximum sear angle s difference function of concrete slab and flange plate of steel beam is f z t). Wile te distribution function of sear angle on transverse section is ψx). ) Te sear connectors are mainly distributed in te junction of flange plate and Web and te relative slip between te concrete slab and steel beam are mainly concentrated. Moreover slip will be linear wit respect to te sear force of sear connectors. ) Te normal strain εz and searing strain γxz of concrete slab are considered in te longitudinal direction z direction) ignoring te strain out of plane and transverse; [] Fig. ). In addition te normal strain εz of steel beam is considered in te longitudinal direction wile te oter strain is ignored. ) Sear force is linear wit respect to te strain on te steel beam. Te linear elastic constitutive relations for sear force and strain of concrete are satisfied. 87-9/ Bentam Open

2 7 Te Open Civil Engineering Journal Volume 9 Ceng Haigen Zk) Zk) H t tc O b t w b Gc Gs yc ys Xi) tc H O b t f t w b Gc b Gs yc ys Xi) t Yj) Yj) Fig. ). Cross section of typical composite girder. a) Composite box girder b) Composite T-girder.. Establising te Differential Equation and Solutions Wen te load is applied on te beam te deformation of cross section is sown in Fig. ). Te displacement component of concrete slab and steel beam on Z direction longitudinal direction) is wczt) and wszt) respectively. Te displacement components of bot of tem on y direction vertical direction) is vzt). Subsequently te displacement vector form of an arbitrary point on concrete slab and steel beam can be expressed as follows: x yzt) = v zt u c u s Steel beam x yzt) = v zt ys yc ) j + $ w c zt) y y c ) v" zt) + f zt ) j + $ zt) y y s ) v" zt) + f zt s vj )# x) )# x) Z i ) k ) k ) Fig. ). Deformation diagrams of composite girder Concrete slab. Were z means te distance in z direction from an arbitrary point to te origin of coordinates; y means te distance in x direction from an arbitrary point to te origin of coordinates; y c and y s respectively represent te distance from te part of concrete-steel beam s centroid axis to te reference axis. j and k are unit vectors in te reference coordinates y and z axisrespectively. According to te assumption of ) and ) te lateral distribution function ψx) of te maximum difference f z t) of te sear angle will satisfy te following conditions: ) At te midpoint and te side of flange plate te first derivative is zero wic ensures tat te sear stress is also zero; vj ) In te junction of flange slab and Web ψx)=. Tis condition simplifies te relative slip function between concrete slab and steel beam. According to te above conditions te relative slip function between concrete slab and steel beam can be obtained by formulas ) and ). zt) = zt) " w c zt) + v# q z zt) ) Were represents te distance between te concrete slab and steel beam. Centroid axis because of te relative slippage and te linear relationsip of interaction between contact surfaces based on constitutive relation of sear connectors can be obtained as follows: zt) = R s zt) = R " w c + v# ) ) Were R S is te stiffness of sear connectors kn/mm ). According to formulas ) and ) te strain and stress of an arbitrary point on concrete slab and steel beam respectively are: Concrete slab z x yzt) = w c " # y # y c ) v "" + f " $ xz x yzt) = f $ " ) ## ) z = E c " z = E c w c # $ E c y $ y c v + E c f # zx = G xz = Gf # Steel beam ) = " z x yz;t ) "" # y # y s z x yz;t) = E s " z = E s # $ E s ) + ) ) v + f " $ 7) ) ## y $ y s v + E s f # 8) Were E s is te elastic modulus of steel; E c is te elastic modulus of concrete; and G is te sear modulus of concrete. It was assumed tat te composite beam wasin a state of balance under te body load B z t) and te surface load s z t). Assuming tat te longitudinal displacement wcws) te vertical displacement v) and te maximum difference of sear and angle f z t) all of tem experienced a sligt

3 Analysis of Stress and Deflection about Steel-Concrete Composite Girders Te Open Civil Engineering Journal Volume 9 7 cange δvδwcδwsδf). According to te principle of virtual work virtual work done by te external force at constant distance equal to tat done by micro section stress resulted in deformation tat is: L $ [N c w " + N w" # M + M ) v"" + µ c s s c s c f " + c f + q z " w c + v #)]dz = p y v + p cz w c + p sz " m v # + b f )dz + L [T v + N c w c + N s " M v # + µ f ] L $ Were N c and N s represent te axial force wic is syntesized by te section stress of concrete slab and steel beam respectively; M c and Ms represent bending moment syntesized by te section stress of concrete slab and steel beamrespectively; µ c andηc represent double bending moment and double sear force respectively syntesized by te section stress of concrete slab; P y is te force resolved by resultant of forces in y direction; P CZ and P SZ represent te forces resolved by resultant of forces on concrete slab and steel beam in z direction respectively; M is te total bending moment caused by te force exerted on eac corresponding centroid axis and te force is resolved by te load on concrete slab and steel beam in z direction; b is te load resolved broadly in z direction under te condition of considering te sear lag effect of concrete slab; T is te resultant of forces syntesized by te surface load in y direction; N c and N s are te resultant forces syntesized by te surface load on steel beam and at te end of te section of concrete slab respectively resolved in z direction; M and µ are te bending moment and double bimoment respectively syntesized by load at te end of te section and te load is resolved by surface load in z direction. Eac resultant of forces is given as : N s M s N c M c Resultant of forces on section stress zt) = z "" da = E s # + E s S $ f # ; zt) = z y " y s )da = "E s I s v## $$ zt) = z da = "" E c w c # $ s ) + S f # zt) = z y " y c )da = "E c I c v# µ c zt $$ ) = z " da = E c [ w c # $ s ) + I " f # c zt ) = " xz $ # da = GI d$ f ) ] Resultant of forces on imposed load 9) p y p cz zt) = b y da + s y dl + b y da + s y dl "" " "" " zt) = b z da + s z dl "" " z c z c z s z t) = b y y ) da + s y y ) dl + b y y ) da + s y y ) m p sz Ac Ac As As zt) = b z da + s z dl b zt " z s dl ) = ## b z da + # s z dl Resultant of forces at te end section surface T = s y da + s y da N c = s z da N s = s z da M = "" s z y y c )da + "" s z y y s )da µ = "" s z da Were and I c are te area of concrete slab and te moment of inertia on te centroid axis of te cross section respectively; and I s are te area of steel beam and te moment of inertia on te centroid axis of te cross section respectively. S = "" da I = "" da Id = " da. Putting te above formulas into formula 9) by employing te teory of subsection integral te equilibrium differential equation and corresponding boundary conditions can be obtained as follows: ) + S $ f "" E c w c "" # s " E s "" S $ f "" + R s w c + v" E I + E I c c s s ) """" E c S $ w c "" " ## ) = p cz R w s s w c + v" ) = p sz ) = p y + m" v R s " w c " + v"" ) E c I $ "" # s f + E I c d$ +) Boundary conditions E c w c " # s ) + E c S $ E s + S $ f " N s ) w L = s ) f " N c ) w L = c " E c I c + E s I s v " M ) v L = ) " E c I c + E s I s v + R s " w c + v E c S $ w c " # s ) + E c I $ ) f = b ) " T + m f " µ ) f L = ) v L = " ) ) It is very difficult to obtain te analytical expression of formula ). Terefore numerical solution is usually used to solve tis problem. Among te numerical metods te difference metod is igly accurate. Tis paper will not introduce te difference metod. Readers can refer to te relevant content in te reference [].

4 7 Te Open Civil Engineering Journal Volume 9 Ceng Haigen. EXAMPLE As sown in Fig. ) eac end of te beam is consolidated using te corresponding model of te srinkage and creep of concrete in specification []. Assuming tat in te negative moment zone te beam as prestressed reinforcement terefore te tensile stress does not appear in te tension zone of concrete beam. Te strengt of concrete is f ck =MPa Poisson ratio isν=. te elastic modulus of steel is E=. MPa relative umidity in te environment is 7 and even load is.kn/m and te stiffness of sear connectors is.kn/mm and kn/mm. connectors is larger te magnitude of cange of deflection is larger. As sown in Fig. ) te effect of environmental umidity on te deflection of composite beam is not too obvious. If te stiffness of sear connector is larger te influence of environmental umidity on te deflection of composite beam will also be larger. 8 m) a) Beam deflection at Rs=.kN/mm Fig. ). A typical sample. a) Cross section b) Structural sketc As sown in Fig. ) it is te influence of sear lag effect tat causes deflection in te composite beam. If sear lag effect is considered te results of teoretical calculation will sow larger deflection. Wit te passage of time te deflection of composite beam is increased accordingly. Sear lag effect also as a certain influence on te deflection of composite beam. Moreover wit increase in te stiffness of sear connectors te degree of influence also gradually increases. Fig. ) and Fig. ) sow te influence of te stiffness of sear connector and relative environmental umidity on composite beam. Wit increase in te stiffness of sear connector te beam s deflection reduced accordingly. Wit te development of concrete srinkage and creep beam s deflection increased but te increase ad a relation wit te stiffness of te sear connectors. If te stiffness of sear 7 8 m) b) Beam deflection at Rs=kN/mm Fig. ). Effect on deflection about sear lag. mm) RH=7Rs=.kN/mm RH=Rs=.kN/mm RH=7Rs=kN/mm RH=Rs=kN/mm t=days 7 8 m) Fig. ). Effect on deflection about te rigidity of sear connectors.

5 Analysis of Stress and Deflection about Steel-Concrete Composite Girders Te Open Civil Engineering Journal Volume 9 7 As sown in Fig. 7) and Fig. 8) wit te manifestation of srinkage and creep of concrete te sectional internal force isredistributed and te stress of te concrete slab varies wit time. Wen load is applied on concrete slab initially te stress canges muc quickly. Tis can rapidly improve te srinkage and creep of concrete earlier wile after eigt monts te stress of te concrete slab canges very slowly. Because of te existence of sear connectors in composite beam it will ave an effect on te redistribution of sectional internal force after te beam applies load. Increased stiffness in te sear connector will benefit te distribution of cross-sectional stress. 7 8 Fig. ). Effect on deflection about RH of surrounding. MPa)... RH=7Rs=kN/mm. -. RH=7Rs=.kN/mm RH=7Rs=.kN/mm RH=7Rs=kN/mm Fig. 7). Stress course of concrete slab. MPa)... RH=Rs=kN/mm. -. Time RH=7Rs=kN/mm RH=Rs=kN/mm RH=7Rs=kN/mm Fig. 8). Te relationsip between concrete stress and RH of surrounding. m) CONCLUSION According to te above results of te sample te following conclusions are drawn:. If te lateral uneven distribution of sectional stress on composite beams is considered te value of deflection will be larger tan tatcalculated by ordinary beam teory and te deflection of composite beam will be increased accordingly over time. But wit increase in te stiffness of sear connectors its influence degree will gradually increase.. Te deflection of composite beam will decrease wit increase in te stiffness of sear connector and it will increase wit te manifestation of creep and srinkage in te concrete. But te increase corresponds to te stiffness of te sear connectors. If te stiffness of sear connectors is substantial te magnitude of te cange of deflection will be larger.. Wen te relative environmental umidity is ig te corresponding stress will cange slowly because of slow development of srinkage and creep of concrete. In summary wen te composite is bended te main influence factor about stress and deflection is te stiffness of te sear connectors. Te increase in te stiffness of sear connectors bond will ave advantages in decreasing te deflection of beam. Te cange in te umidity of te environment will ave little effect on te final value of stress and deflection. CONFLICT OF INTEREST Te autors confirm tat tis article content as no conflict of interest. ACKNOWLEDGEMENTS Te autors wis to acknowledge te support and motivation provided by te National Natural Science Foundation of Cina Grant No. 8). REFERENCES [] K.M. Senna X. Zang and J.B. Kennedy Impact factors for orizontally curved composite box girder bridges Journal of Bridge Engineering vol. 9 no. pp. - Jun.. [] J. Liu Z. Zang and W. Li Finite element anaiysis of concretesteel beam Journal of Sandong Jiaotong University vol. 7 no. pp. -9 Jan. 9. [] S. Zou A finite beam element considering sear lag effect in box girder Journal of Engineering Mecanics vol. pp. - Feb.. [] G. Sun R. Guan Y. Jiang C. Mu C. Huo and F. Xu Sunsineinduced temperature distribution on cross section of steel-concrete composite beams Engineering Mecanics vol. no. pp. -7 Nov.. [] Y. Jiang S. Xu and L. Huang Nonlinear FEM analysis of steel fiber reinforced concrete composite beam and te computer simulation of te entire failing process Journal of Sicuan University Engineering Sciences) vol. no. pp. 8-8 Jan.. [] Y. Yang and W. Xue Experinental study on te static beavior of reinforced concrete composite T-beams Cina Civil Engineering Journal vol. no. pp. -7 Marc. [7] T. Hu C. Huang X. Cen and Z. Liang Experinental study on te fatigue property of composite beams wit steel fiber reinforced

6 7 Te Open Civil Engineering Journal Volume 9 Ceng Haigen self-stressing concrete in te negative moment area Cina Civil Engineering Journal vol. no. pp. - Nov. 9. [8] V. Kristek ear lag analysis for composite box girder Journal of Construction Steel Researc vol. no. pp. -7 Jan. 99. [9] L. Dezi F. Gara G. Leoni and A.M. Tarantino Time-Dependent analysis of sear lag effect in composite beams Journal of Engineering Mecanics vol. 7 no. pp Jan.. [] Y. Long and S. Bao Structural Mecanics P.R.Cina Beijing 98 pp. -. [] L. Wang and J. Li Structural Analysis of te Finite-difference Metod P. R. Cina Beijing 98 pp. -8. [] A.-M. Tarantino and L. Dezi Creep effects in composite beams wit flexible sear connector Journal of Structural Engineering vol. 8 no. pp. -8 Jan. 99. Received: September 7 Revised: December 7 Accepted: December Ceng Haigen; Licensee Bentam Open. Tis is an open access article licensed under te terms of te Creative Commons Attribution Non-Commercial License ttp://creativecommons.org/licenses/ by-nc/./) wic permits unrestricted non-commercial use distribution and reproduction in any medium provided te work is properly cited.