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Analysis of composite beams in the hogging moment regions using a mixed finite element formulation

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Analysis of composite beams in the hogging moment regions using a mixed finite element formulation
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   Journal of Constructional Steel Research 65 (2009) 737–748 Contents lists available at ScienceDirect  Journal of Constructional Steel Research  journal homepage: www.elsevier.com/locate/jcsr Analysis of composite beams in the hogging moment regions using a mixed finiteelement formulation Quang Huy Nguyen a,b , Mohammed Hjiaj a, ∗ , Brian Uy c , Samy Guezouli a a Department of Civil Engineering, INSA of Rennes, France b School of Civil, Mining and Environmental, University of Wollongong, Australia c School of Engineering, University of Western Sydney, Australia a r t i c l e i n f o  Article history: Received 19 October 2007Accepted 16 July 2008 Keywords: Composite beamsNegative bending momentTension stiffeningMixed F.E. formulationDiscrete shear connexion a b s t r a c t Cracking of the concrete slab in the hogging moment region decreases the global stiffness of compositesteel–concrete structures and also reduces the effect of continuity, thus making the structural behaviourhighly nonlinear even for low stress levels. In this paper, the behaviour of continuous composite beamswith discrete shear connection is investigated using a nonlinear mixed finite element model. The modelincludesappropriatenonlinearconstitutiverelationshipsfortheconcrete,thesteelandtensionstiffeningeffect.Furthermore,thediscretenatureoftheshearconnectionisembeddedinthemodelandthetensionstiffening effects are introduced in the analysis by using a concrete constitutive model proposed in theCEB-FIB Model Code 1990 which incorporates embedded steel. Special attention is paid to the hoggingmomentregions,wherecrackingoccurs.Comparisonsbetweenthenumericalanalysesandexperimentalresults in the current literature are undertaken to validate the accuracy of the model. Furthermore, aparametric study is carried out to study the influence of span length and degree of shear connection onthe strength and ductility of continuous composite beams. © 2008 Elsevier Ltd. All rights reserved. 1. Introduction For the last few decades, steel–concrete composite beams havebeen widely used in the construction industry (bridges and build-ings) because of the benefits of combining the two constructionmaterials. Reinforced concrete is inexpensive, massive and stiff,whilst steel is relatively strong, lightweight and easy to assem-ble. The best use is made of the two materials when concrete isused in the compressive zone where steel may experience buck-ling, whilst steel is used in the tensile zone where the concretewill crack. This is the case of simply supported steel-concretecomposite beam under positive bending. However in multistoreybuildingsandbridges,continuouscompositebeamsareoftenusedbecauseofthebenefitsatboththeultimateandserviceabilitylimitstates for long spans or heavily loaded members [1]. For these cases, there is a negative moment region, in which the concreteis cracked and the reinforcement carries the tensile forces, withthe steel component being subjected to a combination of negativebending and compression. Cracking of the slab decreases the stiff-ness of the structure, reduces the effects of continuity and makes ∗ Corresponding address: Department of Civil Engineering, INSA de Rennes,35043 CEDEX Rennes, France. Tel.: +33 2 23 23 87 11; fax: +33 2 23 23 84 48. E-mail address:  mohammed.hjiaj@insa-rennes.fr (M. Hjiaj). the structural behaviour highly nonlinear even at low stress levels(see references [2–4]). One of the main factors affecting the stiff- ness of cracked reinforced concrete slab is the bond that developsbetweenthereinforcementandtheconcrete.Itallowsthetransferof tensile stresses between the reinforcement and the uncrackedregions of concrete. This phenomenon is called tension stiffening.In flexure, the influence of tension stiffening is most important upto service loads and should be included in the deflection calcula-tions.A good deal of research has been devoted to develop modelsto analyse the behaviour of composite beams subjected tonegative bending. Lebet [5] developed a finite-difference modelto analyse composite beams under negative bending moment.In his model, the behaviour of steel and concrete is supposedto be linear but cracking is taken into account by assuminga different bending stiffness in cracked and uncracked regions.Manfredi et al. [6] presented a nonlinear analysis technique basedon the finite-difference method and on an array of generalisedmoment–curvaturerelationshipsrelatedtointerfaceslipandbondslip. Xu [7] and Gattesco [8] proposed a displacement-based finite elementmodelwithtwonodesandeightdegreesoffreedom.Theirnumerical procedure permits to consider nonlinear constitutivemodels for the steel, the concrete and the connexion. Recently Lohet al. [9] developed an iterative-based model using cross-sectional analysis technique. The model considered the concept of partial 0143-974X/$ – see front matter © 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.jcsr.2008.07.026  738  Q.H. Nguyen et al. / Journal of Constructional Steel Research 65 (2009) 737–748 Fig.1.  (a)Infinitesimalcompositebeamsegmentwithoutconnector;(b)Connectorelement. interaction allowing for the occurrence of slip at the slab-beaminterface.In this article, a finite element model is proposed to analysethe nonlinear flexural behaviour of a composite beam withdiscrete partial shear connection under negative bending. Thismodel is based on the two fields mixed force–displacementformulation [10] with nonlinear constitutive relationships for the components. The tension stiffening effect is taken into accountby using the relationship proposed in the CEB-FIB Model Code90 [11]. Local buckling is not considered. Comparisons between the numerical results and experimental results existing in theliterature are made in order to validate the accuracy of the model.Finally, a parametric study is carried out to consider the influenceof span length and degree of shear connection on the strength andductility of continuous composite beams. 2. Structural modelling  The main ingredients of the proposed formulation consistsof (a) mixed finite element formulation for the compositebeam; (b) nonlinear constitutive relationships for the componentmaterials; (c) model for steel embedded in concrete. 2.1. Basic assumptions The following assumptions are made: •  Preservationoftheplanecrosssectionforboththeslabandtheprofile. •  Steel beam cross-section is Class 1 or 2 according to the EC4[12]. •  Noupliftoccursbetweentheslabandtheprofile;thereforetwoparts of the composite section have the same rotation and thesame curvature. •  Slip can occur at the slab/profile interface. •  Theaxialstraindistributionoverthesectiondepthislinearwitha discontinuity at the slab/profile interface due to slip. •  Themembercross-sectionissubdividedintoconcreteandsteellayers (fibre beam element). •  When the reinforcement is in tension, all layers in the effectivearea  A c  , eff   arereplacedbyasinglelayerofsteelembeddedintheconcrete (see Fig. 7). 2.2. Field equations In this section, we recall the field equations for a compositebeam with discrete shear connection in a small displacementsetting.Allvariablessubscriptedwith c   belongtotheconcreteslabsection and those with  s  belong to the steel beam. Quantities withsubscript  sc   are associated with the shear connectors. 2.2.1. Equilibrium conditions Due to the discrete nature of the shear connexion, the internalforces (bending, normal force and shear force) distributions in theconcrete slab and in the steel profile are now discontinuous with jumps at each connector. To derive the equilibrium conditionsfor a composite beam with discrete shear connectors, we need toconsider first the equilibrium of an infinitesimal beam segmentwithout shear connector and the equilibrium at the cross-sectioncontainingashearconnector(seeFig.1).Thefirstsetofequilibriumequations, which apply between two consecutive connectors, isreadily obtained by expressing the equilibrium of a small elementof the composite beam, of length d  x , and subjected to internalforces(Fig.1(a)).Theequilibriumconditionsresultinthefollowingset of equations:d N  c  (  x ) d  x = d N  s (  x ) d  x =  0 (1) T  (  x )  = − d M  (  x ) d  x (2)d 2 M  (  x ) d  x 2  +  p 0  =  0 (3)where  T  (  x )  =  T  c  (  x )  +  T  s (  x )  and  M  (  x )  =  M  c  (  x )  +  M  s (  x ) .The equilibrium Eqs. (1)–(3) are rewritten in the followingcompact form ∂ D  +  P  =  0 (4)where D  =  N  c  (  x ) N  s (  x ) M  (  x )  is the internal stress resultant vector, P  =  00  p 0  the external loading vector and  ∂  a linear operator expressed as: ∂  =  dd  x 0 00 dd  x 00 0 d 2 d  x 2  .  (5)The above equations must be completed by equilibrium equationsat each cross-section containing a shear connector (Fig. 1(b)). Theresulting equation provides a relationship between the internalstress resultants and the shear force  R sc   (Fig. 1(b)): R sc   = − N  c   =  N  s  = − M  / H   (6)where  H   =  H  1  +  H  2 . It worth nothing that the internal stressresultants involved in Eq. (6) are acting at the cross-section level. 2.2.2. Compatibility conditions The curvature and the axial deformation at any section arerelated to the beam displacements through kinematic relations.Under small displacements and neglecting the relative transversedisplacement between the concrete slab and the steel beam, theserelationships are as follows (see Fig. 2): ε c  (  x )  = d u c  (  x ) d  x (7) ε s (  x )  = d u s (  x ) d  x (8) κ(  x )  = − d 2 v(  x ) d  x 2  (9) d sc  (  x )  =  u s (  x )  −  u c  (  x )  +  H  d v(  x ) d  x (10)where  u  is the longitudinal displacement,  v  the transversaldisplacement,  ε c   the strain at the concrete section centroid,  ε s  Q.H. Nguyen et al. / Journal of Constructional Steel Research 65 (2009) 737–748  739 Fig. 2.  Kinematic of composite beam. the strain at the steel section centroid,  κ  the curvature and  d sc  the relative slip between the concrete slab and the steel beam.The compatibility relationships (7)–(9) can also be rewritten in acompact form as ∂ ∗ d  −  e  =  0 (11)where  d  =  u c  (  x ) u s (  x )v(  x )  is the displacement vector;  e  =  ε c  (  x )ε s (  x )κ(  x )  is thesection deformation vector and the operator  ∂ ∗ is given by ∂ ∗ =  dd  x 0 00 dd  x 00 0  − d 2 d  x 2  .  (12) 2.3. Material models Thematerialbehaviourisdescribedusingexplicitrelationshipsbetween the total stress and the total strain with appropriateloading/unloading conditions. 2.3.1. Concrete The stress–strain relationship suggested by the CEB-FIB ModelCode 90 [11] is adopted in this paper for both compression andtension regions (Fig. 3). In compression regions, the stress–straincurve suggested by the CEB-FIB Model Code 90 [11] includes a monotonically increasing branch up to a peak value, followed bya descending part that gradually flattens to a constant value equalto zero. The initial portion of the ascending branch is linearlyelastic, but at about 30% of the ultimate strength, the presenceof microcracks leads to a nonlinear behaviour, with a reductionin tangent modulus. In the subsequent descending branch, theconcrete is severely damaged with prominent cracks.The  σ  c  – ε c   relationship is approximated by the followingfunctions: •  For  ε c   < ε c  , lim : σ  c   = −   ε c  1 ε c  , lim ξ   −  2   ε c  1 ε c  , lim  2   ε c  ε c  1  2 +   4 ε c  1 ε c  , lim −  ξ    ε c  ε c  1  − 1  f  cm Fig. 3.  Stress–strain diagram for concrete. •  For  ε c  , lim  ≤  ε c   ≤  0: σ  c   = −  1  +  E  ci E  c  1 −  2   ε c  ε c  1  − 1  E  ci E  c  1 ε c  ε c  1 −   ε c  ε c  1  2   f  cm  (13) •  For 0  < ε c   ≤  0 . 9  f  ctm / E  ci : σ  c   =  E  ci ε c  •  For 0 . 9  f  ctm / E  ci  < ε c   ≤  0 . 00015: σ  c   =  f  ctm  − 0 . 1  f  ctm 0 . 00015  −  0 . 9  f  ctm / E  ci ( 0 . 00015  −  ε c  ) •  For  ε c   >  0 . 00015: σ  c   =  0where ε c  1  = − 0 . 0022 ;  ξ   =  4  1  +  E  ci E  c  1 −  2   ε c  , lim ε c  1  − 2 ×  E  ci E  c  1 −  2  ε c  , lim ε c  1  2 +  2 ε c  , lim ε c  1 − E  ci E  c  1  . In the above relationships, the symbols have the followingmeaning:–  f  cm  is the mean compressive concrete strength;–  f  ctm  is the mean tension concrete strength;–  ε c  1  is the strain at the peak stress;–  ε c  , lim  is the strain at half the peak stress;–  E  ci  is the initial tangent modulus;–  E  c  1  is the secant modulus at the peak stress. 2.3.2. Steel In the present study, the steel is modelled as an elastic-perfectly plastic material incorporating strain hardening. Fig. 4shows the stress–strain diagram for steel in tension. Specifically,the relationship is linearly elastic up to yielding, perfectly plasticbetween the elastic limit and the commencement of strainhardening,linearhardeningoccursuptotheultimatetensilestressand the stress remains constant until the tensile failure strain isreached. 2.3.3. Steel embedded in concrete When uncracked concrete is in tension, the tensile force is dis-tributedbetweenthereinforcementandtheconcreteinproportionto their respective stiffness, and cracking occurs when the stressreaches a value corresponding to the tensile strength of the con-crete. In a cracked cross-section all tensile forces are balanced by  740  Q.H. Nguyen et al. / Journal of Constructional Steel Research 65 (2009) 737–748 Fig. 4.  Stress–strain diagram for steel. Fig. 5.  Stress–strain diagram for steel embedded in concrete in tension. the steel encased in the concrete only. However, between adja-centcracks,tensileforcesaretransmittedfromthesteeltothesur-roundingconcretebybondforces.Thecontributionoftheconcretemaybeconsideredtoincreasethestiffnessofthetensilereinforce-ment.Thiseffectiscalledtension-stiffening.Todescribethiseffect,anumberofmodelshavebeenproposed.Themajorityofthemod-els are based on the mean axial stress and the mean axial strain of the concrete member in the reinforced concrete, [13–16]. To take the tension stiffening effect into account, the stress-averagestrainrelationshipofsteelembeddedinconcreteproposedby the CEB-FIB model [11] is considered to describe the behaviour of the reinforced concrete members in tension. Fig. 5 shows thestress–strain diagram of steel embedded in concrete.According to the CEB-FIB Model Code 90 [11] the mean stress–strain relationship of embedded steel may be expressed as •  For  ε s , m  ≤  ε sr  1 : σ  s  =  1  + 1 αρ  E  s ε s , m •  For  ε sr  1  < ε s , m  ≤  ε srn : σ  s  =  σ  sr  1  + σ  srn  −  σ  sr  1 ε srn  −  ε sr  1  ε s , m  −  ε sr  1  •  For  ε srn  < ε s , m  ≤  ε sry : σ  s  =  σ  srn  +  f  sy  −  σ  srn ε sry  −  ε srn  ε s , m  −  ε srn   (14) •  For  ε sry  < ε s , m  ≤  ε sr  , sh : σ  s  =  f  sy Fig. 6.  Load–slip diagram for stud shear connector. •  For  ε sr  , sh  < ε s , m  ≤  ε sru : σ  s  =  f  sy  +  f  su  −  f  sy ε sru  −  ε sry  ε s , m  −  ε sry  •  For  ε s , m  > ε sru : σ  s  =  f  su where–  α  and  ρ  are the modular ratio and the geometric ratios of reinforcing steel, respectively;–  σ  sr  1  is the steel stress in the crack, when the first crack hasformed–  σ  srn  is the steel stress in the crack, when the last crack hasformed–  ε sr  1  and  ε sr  2  are the steel strains at the point of zero slip andat the crack when the cracking forces reach  f  tm –  ε srn  =  σ  srn / E  s  −  β t   (ε sr  2  −  ε sr  1 ) –  ε sry  =  ε sy  −  β t   (ε sr  2  −  ε sr  1 ) –  ε sr  , sh  =  ε sh  −  β t   (ε sr  2  −  ε sr  1 ) –  ε sru  =  ε sr  , sh  +  δ  1  −  σ  sr  1 /  f  sy   f  su  −  f  sy  / E  sh –  β t   =  0 . 4 for instantaneous loading, and  β t   =  0 . 25 for long-term and repeated loading–  δ  isacoefficienttotakeaccountthestressratio  f  su /  f  sy  andtheyield stress  f  sy  ( δ  =  0 . 8 was proposed in CEB-FIB Model Code90 [11]). 2.3.4. Stud shear connectors The constitutive relationship proposed by Ollgaard et al. [17] is considered for the stud shear connector. The analytical relation-ship between the shear force  R sc   and the slip  d sc   of a generic studis given by R sc   =  R max ( 1  −  e − β | d sc  | ) α (15)where  R max  is the ultimate strength of the stud shear connector; α  and  β  are coefficients to be determined from experimental re-sults [18]. By using the secant stiffness  k sc   (see Fig. 6) may berewritten as R sc   =  k sc  d sc  .  (16) 2.4. Cross sectional analysis The procedure followed to predict the flexural behaviourof steel–concrete composite beam sections is an incremental-iterative technique with secant stiffness formulation. For thepresent numerical procedure, the cross section is divided into afinite number of discrete layers (fiber model). The axial strain in  Q.H. Nguyen et al. / Journal of Constructional Steel Research 65 (2009) 737–748  741 Fig. 7.  Cross-sectional analysis by finite layer approach. any layer of the concrete slab section and the steel beam sectioncan be evaluated as ε cz   =  ε c   +  κ  z  c  ε sz   =  ε s  +  κ  z  s (17)To take into account the tension stiffening occurring when thereinforced layer is in tension, a single layer of embedded steel isconsidered in replacement of all layers in the effective concretearea  A c  , eff   (see Fig. 7).  A c  , eff   is the area of concrete surrounding thetensile reinforcement. According to the CEB-FIB Model Code [11],  A c  , eff   is calculated as  A c  , ef   =  2 . 5 ( c   +  φ/ 2 )  (18)where c   isthecoatinglengthoftheconcreteand φ  thediameterof the reinforcement.By using the secant stiffness of the constitutive relationships of materials for each layer, the axial resultant and bending resultantof the slab section can be expressed as N  c   =  ( EA ) c  ε c   +  ( EB ) c   κ  (19) M  c   =  ( EB ) c  ε c   +  ( EI  ) c   κ  (20)where ( EA ) c   = nlc  i = 1 E  ic   +  E  i + 1 c  2 b ic  t  ic   +  E  sr   A sr   (21) ( EB ) c   = nlc  i = 1  E  c  , i  (  z  i + 1  +  2  z  i )  +  E  c  , i + 1  ( 2  z  i + 1  +  z  i ) 6  b c  , i t  c  , i + E  sr   A sr   z  sr   (22) ( EI  ) c   = nlc  i = 1  E  c  , i ˜  z  i  +  E  c  , i + 1 ˜  z  i + 1 12  b c  , i t  c  , i  +  E  sr   A sr   z  2 sr   (23)with  ˜  z  i  =  z  2 i + 1  +  2  z  i  z  i + 1  +  3  z  2 i  and  ˜  z  i + 1  =  3  z  2 i + 1  +  2  z  i  z  i + 1  +  z  2 i  . E  c  , i  and  E  c  , i + 1  are the secant concrete modulus at the layer ends(trapezoidal integration scheme),  b c  , i  is the section width,  t  c  , i  thethickness of the concrete layer  i  and  nlc   is number of concretelayers in the slab section.Inthesamewaytheaxialresultantandbendingresultantofthesteel beam sections are given by N  s  =  ( EA ) s ε s  +  ( EB ) s  κ  (24) M  s  =  ( EB ) s ε s  +  ( EI  ) s  κ.  (25)Finally the force–deformation relationship of the cross section canbe written in matrix form as e (  x )  =  f  (  x ) D (  x )  (26)where  f  (  x )  =  ( EA ) c   0  ( EB ) c  0  ( EA ) s  ( EB ) s ( EB ) c   ( EB ) s  ( EI  ) c   +  ( EI  ) s  − 1 is the secant flexibilitymatrix of the cross section. 2.5. The mixed finite element formulation Several finite element formulations for composite beams withcontinuous shear interaction have been already proposed in theliterature (see for example [19–21]). However, little attention has been paid to composite beams with discrete shear connection asit will result in large number of degree of freedom. Nevertheless,discrete connection is more representative of the actual behaviourand provides accurate estimation on the load carried by the studsand the slip distribution along the beam. A displacement-basedformulation of composite beam with discrete shear connexionhas been proposed by Aribert et al. [22] and used to analysethe bending moment redistribution in continuous compositebeams [23]. In this paper, a new mixed finite element formulation [24] for composite steel–concrete beams with discrete shear connectionis used to investigate the nonlinear behaviour of continuouscomposite beams. A similar formulation has been proposed by[25] for composite beams with continuous shear connexion. The stiffness matrix is derived by combining the stiffness matrix of a single shear connector element and the stiffness matrix of acomposite beam element without connector.For the composite beam element without connector, thederivationfollowsthetwo-fieldmixedformulationwhichusestheintegral form of compatibility and equilibrium equations to derivethe matrix relation between the element generalized forces andthe corresponding displacements.In the two-field mixed formulation [10], both the displacement and the internal forces fields along the element are approximatedby independent shape functions. The displacement is assumed tobe continuous along the whole beam: d (  x )  =  a (  x ) q  (27)where a (  x )  is amatrix of 3 × n d  shape functionswith  n d  =  8 beingthe total number of displacement degrees of freedom and  q  thevector of element displacement (Fig. 8). Theinternalforcesfieldisassumedtobecontinuousalongeachbeam element (but not across adjacent elements): D (  x )  =  b (  x ) Q   +  D 0 (  x )  (28)where  b (  x )  is a matrix of 3  ×  n s  force interpolation functionswith  n s  =  4 being the total number of force degrees of freedom,  Q   the vector of element forces (Fig. 8) and  D 0 (  x )  =[  0 0 0 . 5  p 0  x ( L  −  x )  ] T isa internal forcevector accountingforthe effects of internal loading on the cross section forces.In the mixed formulation, the integral forms of compatibilityand equilibrium equations are enforced. These are then combinedto obtain the relationship between the element forces and thedisplacements.
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