Research Article Vibration Analysis of Steel-Concrete Composite Box Beams considering Shear Lag and Slip

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1 Mathematcal Problems n Engneerng Volume 215, Artcle ID 61757, 8 pages Research Artcle Vbraton Analyss of Steel-Concrete Composte Box Beams consderng Shear ag and Slp Zhou Wangbao, 1 Shu-jn, 1 Jang zhong, 2 and Qn Shqang 1 1 School of Cvl Engneerng and Archtecture, Wuhan Unversty of Technology, Wuhan 437, Chna 2 School of Cvl Engneerng, Central South Unversty, Changsha 4175, Chna Correspondence should be addressed to Zhou Wangbao; zhwb@whut.edu.cn Receved 21 January 215; Revsed 15 March 215; cepted 15 March 215 ademc Edtor: Govann Garcea Copyrght 215 Zhou Wangbao et al. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton cense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. In order to nvestgate dynamc characterstcs of steel-concrete composte box beams, a longtudnal warpng functon of beam secton consderng self-balancng of axal forces s establshed. On the bass of Hamlton prncple, governng dfferental equatons of vbraton and dsplacement boundary condtons are deduced by takng nto account coupled nfluencng of shear lag, nterface slp, and shear deformaton. The proposed method shows an mprovement over prevous calculatons. The central dfference method s appled to solve the dfferental equatons to obtan dynamc responses of composte beams subjected to arbtrarly dstrbuted loads. The results from the proposed method are found to be n good agreement wth those from ANSYS through numercal studes. Its valdty s thus verfed and meanngful conclusons for engneerng desgn can be drawn as follows. There are obvous shear lag effects n the top concrete slab and bottom plate of steel beams under dynamc exctaton. Ths shear lag ncreases wth the ncreasng degree of shear connectons. However, t has lttle mpact on the perod and deflecton ampltude of vbraton of composte box beams. The ampltude of deflecton and strans n concrete slab reduce as the degree of shear connectons ncreases. Nevertheless, the nfluence of shear connectons on the perod of vbraton s not dstnct. 1. Introducton The advantages of fully makng use of compressve strength of concrete and tensle strength of steel make composte box beams popular n the brdge engneerng. For composte box beams wth large flange wdth, however, there are shear lag effects n the top concrete slab and bottom plates of steel box beams due to the nonunform transverse dstrbuton of shear stress across secton. In addton, due to the fact that the shear connectors between the steel beam and concrete slab cannot be absolutely rgd, there exsts relatve slp between them even for fully shear connected beams. Therefore, the behavor of steel-concrete composte box beams suffers from coupled effects of both the shear lag and shear slp [1 5]. Gara [6, 7] presented a beam fnte element, n whch the warpng of the slab cross secton was consdered, for the long-term analyss of steel-concrete composte decks takng nto account the shear lag n the slab and the partal shear nteracton at the slab-beam nterface. By usng energy varaton method, Zhou et al. [5] derved the governng dfferental equatons and boundary condtons of the steel-concrete composte box beams by consderng the longtudnal warp caused by shear lag effects and slp between steel beams and concrete slabs. Morass and Dlena [8 1] nvestgated an experment on damage-nduced changes n modal parameters of steelconcrete composte beams subject to small vbratons, and the experments revealed that flexural frequences showed a rather hgh senstvty to damage and therefore can be consdered as a vald ndcator upon a dagnostc analyss. Therefore, t makes sense to study the vbraton characterstc of composte beam. Adam et al. [11] analyzed the dynamc flexural behavor of elastc two-layer beams wth nterlayer slp by assumng the Bernoull-Euler hypothess to hold for each layer separately and consderng a lnear consttutve equaton between the horzontal slp and the nterlayer shear force. Bscontn et al. [12] performed an expermental analytcal nvestgaton on the dynamc behavor of I-steelconcrete composte beams subject to small vbratons, and a

2 2 Mathematcal Problems n Engneerng one-dmensonal model of a composte beam was presented where the elements connectng the steel and renforced concrete slab were descrbed by means of a stran energy densty functon defned along the beam axs. Berczyńsk [13, 14]presentedasolutonoftheproblemoffreevbratonsofI-steelconcrete composte beams and found that the results obtaned on the bass of the Tmoshenko beam theory model acheved the hghest conformty wth the expermental results, both for hgher and for lower modes of flexural vbratons of the beam. Xu and Wu [15] nvestgated the statc, dynamc, and bucklng behavor of partal-nteracton T-composte members by takng nto account the nfluences of rotary nerta and shear deformatons and obtaned the analytcal expressons of the frequences of the smply supported composte beam. Shen et al. [16] studed the dynamc behavor of partalnteracton T-composte beams by usng state-space method, whch was properly establshed va selectng the approprate statevarables,andthecharacterstcequatonsoffrequency and the correspondng modal shapes of free vbraton under generalzed boundary condtons were then obtaned. Shen and Zhong [17] examned the deformaton of partally I-steelconcrete composte beams under dstrbuted loadng and free vbratons of partally I-steel-concrete composte beams under varous boundary condtons, where the weak-form quadrature element method was used. Consderable efforts have been put nto the nvestgaton of dynamc characterstcs of composte beams ncludng nterface slp effects [18 22]. Most of studes focus on the I- steel beam-concrete composte beams or T-type composte beams. The studes about the dynamc characterstcs of composte box beams are rare, especally the studes nvolvng boththenterfaceslpandshearlagnthedynamccharacterstcs. Based on a longtudnal warpng functon consderng self-balancng of axal forces and the Hamlton prncple, ths paper deduces the governng dfferental equatons of dynamc responses of composte box beams under arbtrarly dstrbuted loads. It takes nto account the nfluence of shear lag,nterfaceslp,andsheardeformaton.theequatonsare solved by central dfference methods. Numercal studes are carred out and a good agreement s acheved between results from the proposed method and fnte element method usng ANSYS. The nfluences of the shear lag effect and the degree of shear connectons on the dynamc responses of composte box beams are examned and meanngful conclusons for engneerng desgn are drawn. 2. Basc Assumptons Fgure 1 shows the secton sze and coordnate system of composte box beams. The parameters t 1, t 2, t 3, t 4,and t 4 are the thcknesses of the top concrete slab, cantlever plate, bottom plate, web, and top flange of steel box beam, respectvely; the parameters b 1, b 2, b 3, b 4,andb 4 are the wdths of half-top concrete slab, cantlever plate, half-bottom plate of steel beams, web of steel beams, and top flange of steel beams, respectvely; the parameters h c and h s are the dstances between the centrods of the concrete slab and steel beam to the slab-beam nterface, respectvely, and here b 2 b 1 b 1 b 2 t 4 t 4 t 3 t 1 b4 y, z, w b 3 b 3 Fgure 1: Schematc of secton of steel-concrete composte box beams. h=h c +h s.reasonableassumptonstosmplfytheanalyss model are made as follows. (1) cordng to the dsplacement compatblty, the longtudnal warpng functon of concrete top slab, cantlever plate, bottom flange, and web of steel beams s assumed as [5, 23, 24] g =ψ (y) U (x, t) = 1, 2, 3, 4, (1) ψ =α ( y2 b 2 1)+d =1,2,3,4. (2) Consderng self-balancng of axal forces produced by longtudnal dsplacement yelds [5, 23, 24] b 4 α 1 =1, α 2 = b2 2 b1 2, α 3 =z b b 2 3 (b 2 1 z t), α 4 =. Consderng self-balancng of axal forces produced by longtudnal warpng functon yelds [5, 23, 24] A ψda=. (4) Substtutng (2) nto (4) obtans the constant term of longtudnal warpng dsplacements as follows: d= 2(α 1A 1 /n + α 2 A 2 /n + α 3 A 3 ). (5) (3A ) For = 2,replaceywth y = b 1 +b 2 y. z t and z b are the z-coordnates of centrods of the concrete slab and bottom flange; z s the z-coordnate of the slab-beam nterface; d s the constant term of longtudnal warpng dsplacements; U(x, t) s the functon of the ampltude of warpng dsplacements; ψ s the warpng shape functon of the beam secton, as shown n Fgure 2; α s the self-balancng coeffcent of secton warpng; n=e s /E c,ande s and E c are the modulus of elastcty of steel and concrete, respectvely; A 1 =2b 1 t 1, A 2 =2b 2 t 2, A 3 =2b 3 t 3, A 4 =2(b 4 t 4 +b 4 t 4 ), A = (A 1 +A 2 ) n +A 3 +A 4. t 2 h c h s h (3) (6)

3 Mathematcal Problems n Engneerng 3 A 1 A 2 where ε ( = 1,2,3,4)are the longtudnal stran of the top concrete slab, cantlever plate, bottom flange, and web of steel beams, respectvely; γ ( = 1,2,3) are the shear stran of top concrete slab, cantlever plate, and bottom plate of steel beams, respectvely; γ xz s the shear stran of the web of steel beams; w(x, t) s the vertcal deflecton of composte box beams. A 3 Fgure 2: Schematc of warpng shape functon. (2) The longtudnal dsplacement of any pont n the transverse secton of composte box beams s assumed as the superposton of the longtudnal dsplacement based on the plan secton assumpton, the longtudnal warpng dsplacement due to the shear lag, and longtudnal dsplacement due to the relatve nterface slp. It can be expressed as [5, 23, 24] u =k c ξ (z z t ) θ+g = 1, 2, (7) u =k s ξ (z z s ) θ+g = 3, 4, (8) k c = A s A, k s = A c (na ), (9) ζ (x, t) = ξ + hθ, (1) where θ(x, t) s the rotaton of the beam secton, A w =2b 4 t 4, A s =A 3 +A 4 s the cross secton area of steel beams, A c = A 1 +A 2 s the cross secton area of concrete slabs, ξ(x, t) s the longtudnal dsplacement dfference between centrods of the concrete slab and steel beam, ζ(x, t) s slab-beam nterface slp, z s s the z-coordnate of the centrod of the steel beam, k s s the rato between steel beam s longtudnal dsplacement due to the relatve nterface slp and relatve nterface slp, and k c stheratobetweenconcreteslab s longtudnal dsplacement due to the relatve nterface slp and relatve nterface slp. (3) The vertcal compresson and transverse stran of concrete slabs and steel beams are gnored [5, 23]. 3. Vbraton Dfferental Equaton and Boundary Condtons 3.1. The Stran of the Cross Secton. The sectonal stran can be obtaned from the above longtudnal dsplacement of composte beam sectons as ξ ε x =k c x (z z t) θ x +ψ U = 1, 2, x ξ ε x =k s x (z z s) θ x +ψ U x γ xy = u y = ψ U = 1,2,3, y γ xz = w x θ, d = 3, 4, (11) 3.2. Total Potental Energy of the Composte Box Beam. The stran energy of composte box beams s defned as [5] V =.5 (E s ε A 2 +G s γ 2 )dadx +.5 k sl ζ 2 dx +.5G sa (w θ) 2. (12) Substtutng (1)-(11) nto (12) gves the stran energy of composte box beams as V =.5 [ Dξ 2 +FU 2 +2HU ξ +Iθ 2 +JU 2 [ 2SU θ +k sl ζ 2 + G sa (w θ) 2 ] dx, ] (13) where D=E c k 2 c A c +E s k 2 s A s, F=E c B cf +E s B sf, H=E c B ch + E s B sh, I=E c B c +E s B s, J=G c B cj +G s B sj, S=E c B cs +E s B ss, B cf = ψ 2 da, B sf = As ψ 2 da, B ch = k c ψda, B sh = As k s ψda, B c = (z z t ) 2 da, B s = As (z z s ) 2 da, B cj = ( ψ/ y) 2 da, B sj = A3 ( ψ/ y) 2 da, B cs = (z z t )ψ da, B ss = As (z z s )ψ da, and s the correcton coeffcent of shear deformaton. Consderng that the webs bear most of thevertcalshearforcensecton,herethevalueof s taken as A /(2b 4 t 4 ),and2b 4 t 4 s the secton area of webs; k sl s the slp stffness between the concrete slab and the steel beam; s the span of the composte box beam; G s s the shear modulus of the steel beam. The knetc energy of the composte box beam s [5] T T= 1 2 (ρu 2 +mw 2 )dadx. (14) A Substtutng of (7) and (8) yelds = 1 2 (mw 2 +D 1 ξ 2 +F 1 U 2 +2H 1 Uξ+I 1 θ 2 2S 1 Uθ) dx, (15) where D 1 =ρ c k 2 c A c+ρ s k 2 s A s, F 1 =ρ c B cf +ρ s B sf, H 1 =ρ c B ch + ρ s B sh, I 1 =ρ c B c +ρ s B s, S 1 =ρ c B cs +ρ s B ss, m=ρ c A c +ρ s A s, A=A s +A c,andρ c and ρ s are the densty of concrete slabs and steel beams, respectvely.

4 4 Mathematcal Problems n Engneerng The work done by the external loads can be expressed as W= q (x, t) wdx, (16) where q(x, t) s the dstrbuton functon of arbtrary load Vbraton Dfferental Equaton and Boundary Condtons. The governng equatons of vbraton of composte box beams and correspondng boundary condtons can be deduced basedonhamltonprncpleas FU +Hξ JU Sθ F 1 U H 1 ξ +S 1 θ =, (17) HU +Dξ k sl ζ H 1 U D 1 ξ=, (18) G s A (w θ ) mw+q(x, t) =, (19) G s A (w θ) +Iθ SU k l ζh + S 1 U I 1 θ=, (2) s (FU +Hξ Sθ )δu =, (21) (Dξ +HU )δξ =, (22) G s A (w θ) δw =, (23) (Iθ SU )δθ =. (24) Takng a smply supported beam as example, (21) (24) gve boundary condtons as U (, t) =ξ (, t) =θ (, t) =w(, t) =, U (, t) =ξ (, t) =θ (, t) =w(, t) = gven the ntal condtons as U (x, ) =U (x), ξ (x, ) =ξ (x), w (x, ) =w (x), θ (x, ) =θ (x), U (x, ) = U (x), ξ (x, ) = ξ (x), w (x, ) = w (x), θ (x, ) = θ (x). 4. Fnte Dfference Method of Vbraton Dfferental Equaton (25) (26) 4.1. Dfference Scheme. et soluton doman be σ={(x,t) x, t T}; T s the end tme of soluton; a rectangular mesh s made n the soluton area wth a tme step of τ and space step of,sothat where I = /, J = T/τ. x = (=,1,2,...,I), t j =jτ (j=,1,2,...,j), (27) et U j =U(x,t j ), ξ j =ξ(x,t j ), w j =w(x,t j ), θ j =θ(x,t j ). (28) The central dfference calculaton of governng dfferental equaton yelds F Uj +1 2Uj +U j 1 S θj +1 2θj +θ j 1 JU j ξ j+1 H 1 θ j+1 +S 1 H Uj +1 2Uj +U j 1 G s A G s A +H ξj +1 2ξj +ξ j 1 U j+1 F 1 2ξ j +ξ j 1 τ 2 2θ j +θ j 1 τ 2 = 2U j +U j 1 τ 2 (=1,...,I 1; j=1,2,...,j), +D ξj +1 2ξj +ξ j 1 k sl (ξ j +hθ j ) H U j+1 2U j +U j 1 1 τ 2 ξ j+1 D 1 ( wj +1 wj 1 2 2ξ j +ξ j 1 τ 2 = (=1,...,I 1; j=1,2,...,j), S Uj +1 2Uj +U j 1 θ j+1 I 1 θ j )+Iθj +1 2θj +θ j 1 U j+1 +S 1 2U j +U j 1 τ 2 2θ j +θ j 1 τ 2 k sl h(ξ j +hθ j )= ( wj +1 2wj +w j 1 +q j mwj+1 (=1,...,I 1; j=1,2,...,j), θj +1 θj 1 ) 2 2w j +w j 1 τ 2 = (=1,...,I 1; j=1,2,...,j). (29) (3) (31) (32)

5 Mathematcal Problems n Engneerng 5 The dfference calculaton of boundary condtons yelds U j I Uj I 1 ξ j I ξj I 1 θ j I θj I 1 = Uj 1 Uj = ξj 1 ξj = j =, 1,..., J, (33) = j=,1,...,j, (34) w j I =wj = j=,1,...,j, (35) = θj 1 θj = j =, 1,..., J. (36) The dfference calculaton of ntal condton yelds U =U, ξ =ξ, w =w, θ =θ, U 1 U τ ξ 1 ξ τ w 1 w τ θ 1 θ τ = U =, 1, 2,..., I, = ξ =, 1, 2,..., I, = w =, 1, 2,..., I, = θ =,1,2,...,I. (37) 4.2. Soluton Step. et U j ={U j,uj 1,...,Uj I }T, ξ j ={ξ j,ξj 1,...,ξ j I }T, w j ={w j,wj 1,...,wj I }T,andθ j ={θ j,θj 1,...,θj I }T. Gven U j 1, ξ j 1, w j 1, θ j 1, U j, ξ j, w j,andθ j,thesolvngof U j+1, ξ j+1, w j+1,andθ j+1 follows the steps below. (a) Calculate U, ξ, w, θ, U 1, ξ 1, w 1,andθ 1 from the ntal condtons gven n (37). (b) Gven U j 1, ξ j 1, θ j 1, U j, ξ j, w j,andθ j,calculate {U j+1 1,U j+1 2,...,U j+1 }, {ξj+1 I 1 1,ξ j+1 2,...,ξ j+1 },and I 1 {θ j+1 1,θ j+1 2,...,θ j+1 } from (29) (31). I 1 (c) Calculate {U j+1,u j+1 }, {ξ j+1 I,ξ j+1 }, and {θ j+1 I,θ j+1 } I from (33), (34), and(36), andthesolutonofu j+1, ξ j+1, and θ j+1 s obtaned n combnaton wth step (b). (d) Gven w j 1, w j,andθ j, determne {w j+1 1,w j+1 } from (32). w j+1 I 1 2,..., (e) Calculate {w j+1,w j+1 } from (35) and calculate w j+1 I combned wth step (d) Degeneraton of the Vbraton Dfferental Equaton. kewse, the governng equatons of vbraton of composte box beams and boundary condtons wthout shear lag effects can be deduced as Dξ k sl ζ D 1 ξ =, (38) G s A (w θ ) mw+q(x, t) =, (39) G s A (w θ) +Iθ k sl ζh I 1 θ =, (4) ξ δξ =, θ δθ =, (w θ)δw =. (41) For beams wth two ends smply supported, the boundary condtonscanbeexpressedfrom(41) as ξ (, t) =θ (, t) =w(, t) =, ξ (, t) =θ (, t) =w(, t) =. (42) The vbraton dfferental equaton of composte box beams, wthout consderng shear lag effects, s a degeneraton of the one consderng the shear lag effects, the soluton method of whch can be referred to n Sectons 4.1 and Analyss of Examples The valdty of the proposed method s verfed by comparng to numercal results from fnte element method. The comparsons are made on four smply supported composte box beams wth dfferent degree of shear connectons under suddenly mposed dstrbuted loads. The dynamc responses of beams wth and wthout shear lag effects are analyzed. The dstrbuted load s taken as q = 3 kn/m wthatme step of τ =.2 sandaspacestepof =.3 m. The mechancal and geometrcal parameters of composte box beams are taken as E s = MPa, E c = MPa, G s = MPa, μ s =.28, μ c =.18, ρ s = 79 kg m 3, ρ c = 24 kg m 3, b 1 =b 3 =2.5m, b 2 =2.m, b 4 = 3. m, b 5 =.2m, t 1 =t 2 =.3m, t 3 =t 5 =.6 m, t 4 =.9m, =3m, l =.3 m, n s =2,andf s = 3 MPa. The commercal fnte element software ANSYS s used n ths study. In the fnte element model, the concrete slab and steelbeamaremodeledbysoid65andshe43elements, respectvely. Shear connector s modeled by COMBIN14 elements, beng sprng elements [25]. The aspect rato of the mesh s kept close to one throughout, the mean mesh sze vares from.2 m to.3 m, and about 92 fnte elements n total are employed. The transverse dstrbuted loads are mposed on the top flanges. The translatonal x and y degrees of freedom of all nodes at two ends of beams are restraned.thetorsonatthebeamendssthusrestraned. The translatonal z DOF of the left end s restraned to meet the statc determned requrement and allow rotaton along the moment. The results are shown n Fgures 3 and 4. The k sl s calculated as [5, 23, 24] k sl = K 1 l. (43)

6 6 Mathematcal Problems n Engneerng Dsplacement at mdspan (mm) Dsplacement at mdspan (mm) Tme (s) Tme (s) (a) r =.25 (b) r= Dsplacement at mdspan (mm) Dsplacement at mdspan (mm) Tme (s) Tme (s) ANSYS Ths paper, λ=1 Ths paper, λ= ANSYS Ths paper, λ=1 Ths paper, λ= (c) r=1. (d) r=1.5 Fgure 3: Dsplacement tme hstory at mdspan of composte beams (λ =1represents ncluson of shear lag and λ=s for excluson of shear lag). The sprng parameter of the COMBIN14 elements can be calculated as [25] K 1 =.66n s V u. (44) Thedegreeofshearconnectonsscalculatedas[5] r= n sv u A s f s l, (45) where l s the spacng of shear connectors; r s the degree of shear connectons; f s s the tensle strength of steel beams; n s s the number of connectors across the beam secton; V u s the ultmate shear strength of a sngle shear connector. From Fgures 3 and 4, t can be seen that the results from the proposed method agree well wth those from ANSYS for dfferent degrees of shear connectons. Ths verfes the accuracyoftheproposedmethodandsomemeanngful conclusons for engneerng desgn can be drawn as follows. (1) For varous degrees of shear connectons, the vbraton perod of composte beams wth and wthout shear lag effects s almost the same whch ndcates that the shear lag and shear connecton degree have lttle mpact on the vbraton perod. (2) The larger the degree of shear connectons, the smaller the ampltude of deflecton. (3)Thenfluenceoftheshearlagontheampltudeof deflecton s not sgnfcant, ndcatng that t has lttle effect on the deflecton of beams. (4)Thelongtudnalstranntheconcreteslabandsteel beam bottom plate shows obvous shear lag effects. (5)Thelongtudnalstranntheconcreteslabreduces obvouslywthanncrementnthedegreeofshear

7 Mathematcal Problems n Engneerng Stran dstrbuton (1 6 ) Stran dstrbuton (1 6 ) ANSYS, r=1 Ths paper, λ=1, r=1 Ths paper, λ=, r=1 ANSYS, r =.5 Ths paper, λ=1, r = Y-coordnate (m) (a) Concrete plate Ths paper, λ=, r =.5 ANSYS, r =.25 Ths paper, λ=1, r =.25 Ths paper, λ=, r = ANSYS, r=1 Ths paper, λ=1, r=1 Ths paper, λ=, r=1 ANSYS, r =.5 Ths paper, λ=1, r = Y-coordnate (m) (b) Steel beam bottom plate Ths paper, λ=, r =.5 ANSYS, r =.25 Ths paper, λ=1, r =.25 Ths paper, λ=, r =.25 Fgure 4: Stran dstrbuton n the flanges of mdspan secton (λ =1represents ncluson of shear lag and λ=s for excluson of shear lag). connectons, whch ndcates that the shear connecton degree has a great effect on the longtudnal stran of concrete slab. Nevertheless, the longtudnal stran n the steel beam bottom plate changes lttle wth an ncrement n the degree of shear connecton. 6. Conclusons (1) Numercal analyses are carred out to verfy the accuracyoftheproposedtheory.thecomprehensve consderatons of shear lag, shear deformaton, and nterface slp yeld an accurate predcton of dynamc responses of composte box beams. (2) The degree of shear connectons has lttle mpact on the vbraton perod of composte beams but has sgnfcant mpact on the ampltude of deflecton of composte beams. Nevertheless, the shear lag effect has lmted contrbuton to the deflecton or the vbraton perod of composte beams. (3) The longtudnal stran n the concrete slabs and bottom plate of steel beams shows an obvous shear lag effect. (4) The longtudnal stran n the concrete slabs reduces greatly wth ncreasng shear connecton degree. Nevertheless, the longtudnal stran n the steel beam bottom plate changes lttle wth an ncrement n the degree of shear connecton. Conflct of Interests The authors declare that there s no conflct of nterests regardng the publcaton of ths paper. knowledgments The research descrbed n ths paper was fnancally supported by the Natonal Natural Scence Functon of Chna ( , ) and the Fundamental Research Funds for the Central Unverstes of Chna (214-IV-49). References [1] J. Ne and C. S. Ca, Steel-concrete composte beams consderng shear slp effects, Structural Engneerng, vol. 129, no. 4, pp , 23. [2] F.-F. Sun and O. S. Burs, Dsplacement-based and two-feld mxed varatonal formulatons for composte beams wth shear lag, Engneerng Mechancs, vol.131,no.2,pp , 25. [3] G. Ranz and A. Zona, A steel-concrete composte beam model wth partal nteracton ncludng the shear deformablty of the steel component, Engneerng Structures, vol. 29, no. 11, pp , 27. [4] Z. Wangbao, J. zhong, K. Juntao, and B. Mnx, Dstortonal bucklng analyss of steel-concrete composte grders n negatve moment area, Mathematcal Problems n Engneerng,vol.214, Artcle ID , 1 pages, 214. [5] W.-B. Zhou,.-Z. Jang, Z.-J. u, and X.-J. u, Closedform soluton for shear lag effects of steel-concrete composte box beams consderng shear deformaton and slp, Central South Unversty, vol. 19, no. 1, pp , 212. [6] F. Gara, G. Ranz, and G. eon, Smplfed method of analyss accountng for shear-lag effects n composte brdge decks, Constructonal Steel Research,vol.67,no.1,pp , 211. [7] F. Gara, G. eon, and. Dez, A beam fnte element ncludng shear lag effect for the tme-dependent analyss of steel-concrete

8 8 Mathematcal Problems n Engneerng composte decks, Engneerng Structures, vol. 31, no. 8, pp , 29. [8] A. Morass and. Rocchetto, A damage analyss of steelconcrete composte beams va dynamc methods: part I. Expermental results, Vbraton and Control, vol. 9, no. 5, pp , 23. [9] M. Dlena and A. Morass, A damage analyss of steel-concrete composte beams va dynamc methods: part II. Analytcal models and damage detecton, Vbraton and Control, vol. 9, no. 5, pp , 23. [1] M. Dlena and A. Morass, Vbratons of steel concrete composte beams wth partally degraded connecton and applcatons to damage detecton, Sound and Vbraton, vol. 32, no. 1-2, pp , 29. [11] C. Adam, R. Heuer, and A. Jeschko, Flexural vbratons of elastccompostebeamswthnterlayerslp, ta Mechanca, vol.125,no.1 4,pp.17 3,1997. [12] G. Bscontn, A. Morass, and P. Wendel, Vbratons of steelconcrete composte beams, Vbraton and Control, vol. 6, no. 5, pp , 2. [13] S. Berczyńsk and T. Wróblewsk, Vbraton of steel-concrete composte beams usng the Tmoshenko beam model, Journal of Vbraton and Control, vol. 11, no. 6, pp , 25. [14] S. Berczyńsk and T. Wróblewsk, Expermental verfcaton of natural vbraton models of steel-concrete composte beams, Vbraton and Control, vol. 16, no. 14, pp , 21. [15] R. Xu and Y. Wu, Statc, dynamc, and bucklng analyss of partal nteracton composte members usng Tmoshenko s beam theory, Internatonal Mechancal Scences,vol. 49, no. 1, pp , 27. [16] X. Shen, W. Chen, Y. Wu, and R. Xu, Dynamc analyss of partal-nteracton composte beams, Compostes Scence and Technology,vol.71,no.1,pp ,211. [17] Z. Shen and H. Zhong, Statc and vbratonal analyss of partally composte beams usng the weak-form quadrature element method, Mathematcal Problems n Engneerng, vol. 212,ArtcleID97423,23pages,212. [18] A. Chakrabart, A. H. Shekh, M. Grffth, and D. J. Oehlers, Dynamc response of composte beams wth partal shear nteracton usng a hgher-order beam theory, Structural Engneerng,vol.139,no.1,pp.47 56,213. [19] J. G. S. da Slva, S. A.. de Andrade, and E. D. C. opes, Parametrc modellng of the dynamc behavour of a steelconcrete composte floor, Engneerng Structures, vol. 75, pp , 214. [2] S. enc and J. Warmnsk, Free and forced nonlnear oscllatons of a two-layer composte beam wth nterface slp, Nonlnear Dynamcs,vol.7,no.3,pp ,212. [21] Q.-H. Nguyen, M. Hjaj, and P. e Grognec, Analytcal approach for free vbraton analyss of two-layer Tmoshenko beamswthnterlayerslp, Sound and Vbraton,vol. 331, no. 12, pp , 212. [22] W.-A.Wang,Q.,C.-H.Zhao,andW.-.Zhuang, Dynamc propertes of long-span steel-concrete composte brdges wth external tendons, Hghway and Transportaton Research and Development (Englsh Edton),vol.7,no.4,pp.3 38, 213. [23] W.-B. Zhou,.-Z. Jang, and Z.-W. Yu, Analyss of free vbraton characterstc of steel-concrete composte box-grder consderng shear lag and slp, Central South Unversty, vol. 2, no. 9, pp , 213. [24] W.-B. Zhou,.-Z. Jang, Z.-W. Yu, and Z. Huang, Free vbraton characterstcs of steel-concrete composte contnuous box grder consderng shear lag and slp, Chna Hghway and Transport,vol.26,no.5,pp.88 94,213. [25] J. Ne, J. Fan, and C. S. Ca, Stffness and deflecton of steelconcrete composte beams under negatve bendng, Structural Engneerng,vol.13,no.11,pp ,24.

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