Finite element model for nonlinear analysis of steel–concrete composite beams using Timoshenko's beam theory Dinh Huynh Thai2) Bui Duc Vinh1) and Le Van Phuoc Nhan1) 1) HCMUT, 268 Ly Thuong Kiet, Ho Chi Minh City, Viet Nam Hoang Vinh TRCC, 270A Tay Thanh, Ho Chi Minh City, Viet Nam 1) vinhbd@hcmut.edu.vn 2) ABSTRACT This paper presents an analytical model for steel-concrete composite beams with partial shear interaction and shear deformability of the two components The model is obtained by coupling the Timoshenko’s beam for the concrete slab and steel girder (T-T model) The nonlinear material of concrete slab, steel girder and shear connectors are taken into account The stiffness matrix of the composite element with 16 DOFs is derived by the displacement based finite element formulation The numerical solutions are verified on simply supported and continuous beams The analytical results show good agreement with experimental data, they are also compared with the difference models INTRODUCTION Steel - concrete composite beams (CB) have been widely used in the construction industry due to the advantages of combining the two materials Modeling and analysis of steel–concrete composite structures have been proposed in the literature (Spacone and El-Tawil 2004) Newmark et al (1951) analyzed CB with partial interaction The Newmark model couples two Euler–Bernoulli beams, i.e one for the reinforced concrete (RC) slab and one for the steel girder Since then, many researchers have been extended the Newmark’s model (Gattesco 1999, Dall’Asta and Zona 2002, Ranzi et al 2004) Recently, Ranzi and Zona (2007) introduced a beam model including the shear deformability of the steel component only This model was obtained by coupling an Euler–Bernoulli beam for the RC slab with a Timoshenko beam for the steel girder This parametric study was carried out using a locking-free finite element model under the assumption of linear elastic materials and considering the time-dependent behaviour of the concrete Schnabl et al (2007) presents an analytical solution and a FE formulation for CB with coupled Timoshenko beams for both components, the material models are limited on linear elastic behaviour The results showed that shear deformations are more important for high levels of shear connection degree, for short beams with small span-to-depth ratios, and for beams with high elastic and shear modules ratios In this work, w the prroposed moddel is formuulated by couupling Timoshenko beam ms for both the RC slab annd steel girdder, it is refe ferred as (T––T model) The T governinng equations of CB moodel with partiaal interactionn, based on kinematic assumptions a s substantiallly similar to o the analytiical solution was w reportedd by Schnaabl et al ((2007) Thee nonlinear behaviour of materiall is consideredd for all com mponents Numerical N soolutions are obtained byy displacemeent-based finnite element (F FE) Four nu umerical exaamples dealing with two simply suupported andd two two-sppan continuouss CB are preesented ANALY YTICAL MODEL M 2.1 Moddel assumptiions A typiccal steel–conncrete CB wiith prismaticc section is sh hown in Figg (Ranzi annd Zona 20007) An ortho normal n refereence system m {O; X, Y, Z Z} where i, j, j k are the uunit vectors of o axis X, Y,, Z The composite cross section s is foormed by thee concrete sllab, referredd to as Ac, and a by the stteel beam, refeerred to as As The com mposite actioon between the two com mponents is provided byy a continuouss deformablee shear conn nection at thhe interface between b the two layers, whose dom main consists off the points in the YZ plane with y = ysc and z ∈ [0, L] Thee main assum mptions for the T-T modell can be foun nd in work of o Schnabl ett al (2007) Fig.1 Typical compossite beam andd cross-sectiion a strain fieelds 2.2 Dispplacement and The dissplacement field f of a generic pointt P (x, y, z)) of the CB is defined by b vector d as shown in Eq E (1): ⎧dc ( y , z ) = v( z ) j + [ wc ( z ) + ( y − yc )ϕc ( z )]k ⎪ ∀( x, y ) ∈ Ac , z ∈ [0, L] ⎪ (1) d( y , z ) = ⎨ j d y z = v z + w z + − ϕ ( z )] k ( ) ( , ) ( ) [ ( ) y y s s s ⎪ s ⎪⎩ ∀( x, y ) ∈ As , z ∈ [0, L] where v( z ) representts the defleection of booth componeents; wc ( z ) and ws ( z ) are the axxial displacemeents of the reference r fibbers of the R RC slab and the steel girrder, locatedd at yc and ys , respectivelly; ϕc ( z ) and ϕs ( z ) arre the rotattions of the top and boottom layerss, respectiveely Translation ns and rotatiions are signned positive rrespectively y as in Fig (Ranzi and Zona 2007) The dissplacement field f can be grouped g in thhe vector: uT ( z ) = [ wc ( z ) ws ( z ) v( z ) ϕc ( z ) ϕ s ( z ) ] (2) The slip betweenn the two componentss, which reepresents thhe discontinnuity of axxial displacemeents at their interface, is given by veector s: s( z ) = s ( z )k = ds ( ysc , z ) − dc ( ysc , z ) = [w s ( z ) − w c ( z ) − h s ϕ s ( z ) − h c ϕc ( z )]k (3) where hc = ysc − yc annd hs = ys − ysc Fig Displacem ment field off the T–T com mposite beam m model Based on o the assum med displacem ment field, tthe non-zero componentss of the straiin field are: ⎧ε zzc ( y, z ) = w 'c + ( y − yc )ϕ 'c ⎪ ∀( x, y ) ∈ A , z ∈ [0, L] ∂d ⎪ c ε z ( y, z ) = k = ⎨ ε y z = w ( , ) ' ∂z s + ( y − ys )ϕ 's ⎪ zzs ⎪⎩ ∀( x, y ) ∈ As , z ∈ [0, L] γ yzzc = v '+ ϕc ⎧ ⎪∀( x, y ) ∈ A , z ∈ [0, L] ∂d ∂d ⎪ c γ yz ( y, z ) = j + k = ⎨ γ = v '+ ϕ s ∂z ∂y yzzs ⎪ ⎪⎩∀( x, y ) ∈ As , z ∈ [0, L] (4) (5) where ε zc , ε zs and γ yzc , γ yzs are the axial strains and the shear deformations of the two components, respectively The strain field can be presented in the vector: ε T ( z ) = ⎡⎣ε c ( z ) ε s ( z ) θ c ( z ) θ s ( z ) γ yzc ( z ) γ yzs ( z ) s ( z ) ⎤⎦ (6) (6) where ε c ( z ) = w 'c and ε s ( z ) = w 's are the axial strains at the levels of the reference fibres of the two components respectively, θ c ( z ) = ϕ 'c and θ s ( z ) = ϕ 's is the curvature of the RC slab and the steel girder The vector of strain functions can be obtained from the vector of displacement functions by means of the relation: (7) ε = Du where the matrix operator D is defined as: ⎡∂ ⎢0 ⎢ ⎢0 ⎢ D=⎢0 ⎢0 ⎢ ⎢0 ⎢ −1 ⎣ 0 ∂ 0 0 0 ∂ 0 ∂ ∂ − hc ⎤ ⎥⎥ ⎥ ⎥ ∂ ⎥ ⎥ ⎥ ⎥ − hs ⎥⎦ (8) being ∂ the derivative with respect to z 2.3 Balance conditions The principle of virtual work is utilized to obtain the weak form of the balance condition of the problem: ^ ∑ ∫∫ α L Aα = ∑∫ α σ zα ε zα dAdz + ∑ ∫ L L α ∫ ^ Aα b d dAdz + ∑ ∫ α ∂Aα ∫ ^ Aα ^ τ yzα γ yzα dAdz + ∫ g sc s dz L (9) ^ t d dsdz where b and t are the body and surface force respectively; ( α = c, s ) From Eq (9) in weak form, the stress resultant entities, which are duals of the kinematic entities derived from the assumed displacement field, can be identified and grouped in the vector r: rT = [ Nc in which Ns Mc M s Vc Vs g sc ] (10) Nα = ∫ σ zα dAα Aα M α = ∫ σ zα ( y − yα )dAα Aα (11) Vα = ∫ τ yzα dAα Aα Similarly, the external loads are written in the vector g: gT = ⎡⎣ g zc g zs gy mxs ⎤⎦ mxc (12) in which g zα = ∫ b.kdAα + ∫ ∂Aα Aα t.kds g y = ∑ ∫ b.jdAα + ∑ ∫ ∂Aα Aα t.jds mxα = ∫ b.k ( yα − y )dAα + ∫ ∂Aα Aα (13) t.k ( yα − y ) s Since Eq (9) can be rewritten in compact form as: ∫ L ^ ^ L r.D udz = ∫ g.H udz (14) with the matrix operator H defined as: ⎡1 ⎢0 ⎢ H = ⎢0 ⎢ ⎢0 ⎢⎣0 0 0 0 0 0⎤ ⎥⎥ 0⎥ ⎥ 0⎥ ⎥⎦ (15) MATERIAL MODELS 3.1 Concrete The stress-strain relationship suggested by the CEB-FIB Model Code (2010) is adopted in this paper for both compression and tension regions (Fig 3) The σ c − ε c relationship is approximated by the following functions: • For ε c < ε c ,lim : σc f cm ⎛ k η − η ⎞ = − ⎜⎜ ⎟⎟ ⎝ + ( k − ) η ⎠ (16) where: η = ε c / ε c1 and k = Eci / Ec1 • For σ ct ≤ 0.9 f ctm : σ ct = Eci ε ct • For 0.9 f ctm < σ ct ≤ f ctm : (17) ⎛ σ ct = f cm ⎜1 − 0.1 ⎝ ⎞ 0.00015 − ε ct ⎟ 00.00015 − 0.99 f ctm / Eci ⎠ (18) (b) (a)) Fig Sttress-strain diagram d for cconcrete: a) Compression, b) Tensio on (b) (a) m for steel, bb) Load-slip diagram forr stud shear connector c Fig a) Stress-strain diagram 3.2 Steeel In the study, s the steel is modelled as an elaastic-perfecttly plastic material m incorrporating strrain hardening Fig show ws the stress strain diagrram for steel in tension 3.3 Sheear connectors The connstitutive reelationship for f the stud shear conneector was prroposed by Ollgaard et al (1971), is given by: ( f scc = f max − e −β δ ) α with δ ≤ δ u (19) where fmaxx is the ultim mate strengthh of the studd shear connnector; and α , β are coefficients to be determinedd from test 4 FINITE E ELEMEN NT FORMU ULATION unctions to approximate a e displacemeent are chooose They arre must be the The poolynomial fu same ordeer in each displacement d t field, in faact that need d to avoid the occurreence of lockking problems: f (i) in axial strain (Eq 4), the first derivvative of thee axial dispplacement w and the first t rotationn ϕ must bbe polynom mials of the same orderr to avoid the derrivative of the ecccentricity isssue (Gupta and a Ma 19777, Erkmen annd Saleh 20112) (ii) in the shear deeformation (Eq ( 5), thee first derivaative of the ttransverse displacement d t v n ϕ must bee polynomialls of the sam me order in oorder to avoidd shear lockking andd the rotation (Yuunhua 1998,, Mukherjee and Prathapp 2001) (iii) in the interfacce slip (Eq 3), the axiaal displacem ments w and the rotatioon ϕ must be o in ordder to avoid slip and cuurvature lockking (Dall’A Asta pollynomials off the same order andd Zona 2004 4) 4.1 Thee displacemeent-based FE E The sim mplest elemeent (Fig 5a)) which can be derived for the T–T T model has 10 degrees offreedom (DOF) ( Thaat is the at least requuired DOFss for descrribing the problem p unnder considerattion Its shappe functions are linear fuunctions for the axial dissplacements,, deflection and a rotations of o componennts (Table 1) Fig.5 Fiinite elementts for the T––T CB model o the previo ous considerrations, this ssimple 10 DOF FE does not satisfy the t consistenncy Based on conditionss between th he different displacemennt fields couupled in thee problem The T use of this t element caan lead to poor p and unnsatisfactory results Th hus, the use of the 10D DOF T–T beeam element is discouragedd Table 1: Degrees D of shhape functionns for the prroposed T–T T finite elemeents 10D DOF 16D DOF wc ws v ϕc ϕs 2 2 The FE E fulfilling thhe consistenccy conditionns of the dispplacement fiield is the 166DOF depiccted in Fig 5b which enhannces the ordder of the approximated polynomialss to paraboliic functions for the axial displacement d ts, rotations and a to cubicc function forr transverse displacemennt (Table 1) 4.2 FE formulation The displacement of the FE with a polynomial approximation of the displacement field is written as: (20) u = Nd and relation of displacement and strain as in Eq (21): ε = DNd = Bd (21) r = Dε = DBd the Virtual Work Principle Eq (9) becomes: ∫ L ^ L ^ DBd.Bd dz = ∫ g H N d dz (22) Since, the following balance equation is obtained: K e d = fe (23) L where K e = ∫ BT DBdz is stiffness matrix; L and fe = ∫ ( H N)T gdz is the vector of the internal nodal forces The calculation of load vector, internal nodal forces vector and stiffness matrix is performed by means of numerical integration, using the trapezoidal rule through the thickness (the crosssection is subdivided into rectangular strips parallel to the x-axis) (Nguyen et al 2009) and by using the Gauss–Lobatto rule along the element length In computer code, five Gauss points are used in the 16DOF element The non-linear balance equation can be written in iterative form using the Newton–Raphson method NUMERICAL EXAMPLES The numerical solutions of the proposed model are compared against experimental data obtained by earlier experimental study In the fact that, a group of two CB which material limited in linear elastic range are investigated (Aribert et al 1983 and Ansourian 1981) Other group includes the simply-supported CB E1 tested by Chapman et al (1964) and the two spans CB CBI tested by Teraszkiewicz (1967) are considered for nonlinear analysis 5.1 Simply supported steel–concrete CB (Aribert et al 1983) The proposed 16DOF beam element model is used to predict the elastic deflection of the simply supported composite beam tested by Aribert et al.(1983) The geometric characteristics and the material properties of the beam are shown in Fig and Table As shown in the figure a steel plate 120 × mm is welded into bottom flange of the steel girder There are five rebars dia of 14mm, placing in at the mid-depth of the RC slab Fig.6 Geometriccal characterristics of CB (Aribert et al 1983) The beeam is moddeled using six elemennts in order to comparee the performance of the proposed model again nst the exissting EB-EB B 8DOF mo odel (Dall’A Asta and Zoona 2002) and a s fouur elements are used, annd two morre elements are experimenntal data Beetween the supports, placed at the t beam endds Tablee 2: Mechaniical characteeristics of CB B (Aribert ett al 1983) Param meter RC sllab Steell girder Distancee between thhe centroid of o layer and the layer interfaace Area hc = 50 mm hs = 187 mm m mm2 Ac = 82310 m As = 7220 mm2 Second moment of area a I c = 666.667 × 105 mm4 M Ec = 20000 MPa Pa Gc = 8333 MP I s = 1415 × 105 mm4 Es = 2000000 MPa Gs = 8000 00 MPa Elastic modulus m Shear modulus m Shear boond stiffnesss k sc = 4500 MPa s the looad–deflectiion curve unnder the poiint load It ccan be seen that numeriical Fig shows results of both modells are slighttly more flexxible than the t test dataa Howeverr, the proposed model is closer to thhe experimeental data thhan the EB EB 8DOF model, because the shhear on of the cross-section c n is taken innto accountt for each llayer Fig presents the deformatio comparisoon for the sliip distributio on along thee beam leng gth at load level l of 195 kN, the ressult shows both h models proovide almostt the identicaal slip distrib bution Fiig.7 Load–deflection currves Fig.8 Slip distribbution alongg the beam m deeflections obbtained withh the propossed model compared c w with Fig shows the mid-span those obtained with EB B-EB 8DOF F model for ddifferent spaan-to-depth rratios (L/H) and shear boond dicted by thee proposed model m is largger than the correspondding stiffness (kksc) The defflection pred one evaluaated accordiing to EB-E EB 8DOF model, m for an ny value of tthe ratio L/H H The curvves, related to cases of low wer shear stiiffness, monnotonically reeduce to thee case of abssent interacttion Pa) It can bee seen that partial p interaaction resultss in a reducttion (loose connnection withh ksc = MP of the effect of shear flexibility f off the connected memberss Fiig.9 Mid-spaan deflectionn versus the span-to-dept s th ratio 5.2 Twoo-span contiinuous compposite beam C CTB6 (Ansourian 1981) The prooposed modeel is now useed to simulatte a two-spaan continuouus steel–conccrete CB Beeam CTB6, whhich was a part of the experimenttal program carried outt by Ansourrian (1981),, is consideredd The geom metric definition of thee beam is illlustrated in Fig 10 The T RC slabb is longitudinnally reinforcced by rebarrs at the topp and bottom m with differrent reinforccement ratioo in the sagginng and hoggging region The distancces from the interface to t the bottoom and the top rebars are 25 mm and 75 mm, respectively T The shear bonnd stiffness is assumed of 10,000 M MPa (Nguyen et e al 2011) The materiaal parameterrs used in thee computer analysis are Es = 210 GPa; Gs = 80.76 GPa; Ec = 34 GPa; Gc = 14,167 GP Pa Fig 10 Geometrical G characteristtics of beam CTB6 (Anssourian 1981) Fig.11 Load versu us deflectionn curves Fig.12 Deflectionn curves alonng the beam Two annalyses havee been carried out using the proposeed model Thhe first one includes i an unu cracked an nalysis, in which w the cooncrete craccking in the slab is ignored The second analyysis comprises a ‘‘crackedd analysis’’, as suggested by Euro-ccode The concrete craacking is takken into accouunt by negleccting the ncrete contriibution along 15% of thhe span lengtth on each side s of the inteernal supporrt The mid-span deflecttions obtaineed by the prroposed model, using four fo elements for f the un-crracked analyysis and sixx elements for fo the crackked analysis,, are compaared against thhe experimenntal results in Fig 11 This figurre shows that the modeel predicts the deflection curve with the ‘‘crackeed analysis’’ rather well The resultss indicate thhat the concrrete cracking effects e must be b taken intoo account foor continuouss CB Thesee effects can be seen cleaarly in Fig 12 with deflection curves along a the beaam 5.3 Simp mply supporteed steel–conncrete CB E11 (Chapman et al 1964) The nonlinear anallysis of simpply supporteed beam E1 was carried out, based d on the tessted beam of Chapman C et al a (1964) Shear S connecctors are heaaded studs (112.7 mm diaameter) in paairs at 120 mm m pitch The geometric characteristi c cs, material properties and a constituttive coefficiient values of beam are shhown in Fig g 13, Tablee and Tab ble The beam b is mod deled using 22 elements: 20 elementss are used beetween the supports s and d two elemeents are placed at the beeam ends Fig.13 Geometrical characteristic c cs of beam E1 E The nu umerical sim mulation is compared thee performan nce of the prroposed model against the data of Gaattesco (1999 9) and experrimental dataa The load versus v mid-sspan deflectiion is plottedd in Fig 14 annd the valuess of slip at the t steel–conncrete interfface along thhe beam axiss are plottedd in Fig 15 foor various lo oading levelss The plots show the good g agreem ment between n the analytiical results andd the existinng data The small differrences in thee slip curvess are likely due d to the boond relationshiip Table 3: Geometrical G characteristiics of beam E1 Span length (mm) Concrete C slaab Steel beam Shear conneectors Longitudina L al reinforceme r nt Thickness (m T mm) W Width (mm) S Section A (mm2) Area K Kind of studs of studs D Distribution N Number of sttuds T (mm2) Top B Bottom (mm m2) 5490 152.4 1220 12” x 6”” x 44lb/ft BSB 8400 12.7 x 50 Uniform m in pairs 100 200 200 Tablee 4: Materiall properties aand constituttive coefficient values Material Concrete Comprressive stren ngth fc (MPa)) Tensilee strength fctt (MPa) Peak sttrain in com mpression ε c1 Peak sttrain in tensiion ε ct1 E1 32.7 3.07 0.0022 0.00015 CBI 466.7 3.89 0.0022 0.00015 Steel Yield stress s (MPa)) Ultimaate tensile stress (MPa) ( Strain– –harden strain ε sh Connectio on Elasticcity moduluss Es (MPa) Strain– –harden moduluus Esh (MPa) fmax (kN N) β (mm m-1) α Fig.14 Loaad versus miid-span defleection curvees Flang ge Web Reinfforcement Flang ge Web Reinfforcement Flang ge Web Reinfforcement 250 297 320 465 460 320 0.00267 0.00144 206000 3500 301 301 321 470 470 485 0.012 0.012 0.010 206000 2500 66 0.8 0.45 322.4 4.72 1.0 Fig.15 Slip distribuution along sppan at variouus loaad levels B CBI (Terasszkiewicz 19967) 5.4 Twoo-span contiinuous steel––concrete CB In orderr to verify thhe numericall model in thhe presence of o negative m moments, coontinuous beeam CBI, tested d experimen ntally by Terraszkiewicz (1967), werre simulated with the nuumerical moddel The geom metric characcteristics, maaterial propeerties and coonstitutive coefficient c v values of beeam CBI are sh hown in Fig 16, Table and Table 55 In these siimulations bbonding was not considered because th he experimenntal beams were w greasedd at the steel––concrete innterface to prrevent bondiing A total nuumber of 20 elements peer span weree used for beeam CBI Ass the beam was w symmetrric, only one half h of the beeam was mod deled Fig.16 Geeometrical chharacteristics of beam CBI metrical charracteristics of o continuouss beam Taable 5: Geom Span length (mm) Concrete C slaab Steel beam Shear conneectors Longitudina L al reinforceme r nt Thickness (m T mm) W Width (mm) S Section A (mm2) Area K Kind of studs P Pitch of studds (mm) N Number of sttuds H top (mm Hog m2) H bottom (mm2) Hog S top (mm Sag m2) S bottom ((mm2) Sag 3354 60 610 6” x 3” x 12lb/ft BSB 8400 9.5 x 50 146 96 445 - The com mparison beetween somee results of tthe simulatio on of beam C CBI and thee correspondding experimenntal results is i shown in Fig.17, Figg.18 and Figg.19 In com mpliance withh experimenntal results theese quantities were plotted at P = 122 kN,, which corrresponds too 81% of the experimenntal ultimate load at P = 150.5 kN Inn those figurre, the experrimental resuults of the right span are plotted p uponn the results of the left span to faccilitate compparison with h the numeriical results Thhe plots show w the good agreement a between the analytical a reesults and thee existing daata In Fig 17, it can be noted n that thee curve of thhe analyticaal results liess almost alw ways among the experimenntal results of the two hallves of the bbeam Fig.17 Deflected shappe at load levvel of 122 kN N Fig.18 Slip distribbution along span at loadd levell of 122 kN Fiig.19 Strain profile alongg the span inn the bottom flange at load level of 122 kN CONCLUSIONS A numeerical modell for the lineear analysis and nonlineear analysis of steel–conncrete CB with w partial sheear interactio on capable of o accountinng for the shhear deformaability of booth componeents has been presented p T proposed The d model is fformulated by b modelingg the RC slaab and the stteel girder by means m of thee Timoshenkko beam moodels The an nalytical forrmulation haas been derivved by means of o the princiiple of virtuaal work Thee numerical solution s has been obtained by meanss of the displaccement-based FE methodd The nuumerical-expperimental co omparisons validated thhe proposedd model reliability and the capacity too determine the behavio our of CB The T T-T mod del gives a bbetter agreem ment Based on these resuults, the effeects of shearr deformatioons need to be carefullyy evaluated for compossite 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