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

Journal of Constructional Steel Research 65 (2009) 737–748

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Journal of Constructional Steel Research

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Analysis of composite beams in the hogging moment regions using a mixed finiteelement formulationQuang Huy Nguyen a,b, Mohammed Hjiaj a,∗, Brian Uy c, Samy Guezouli aa Department of Civil Engineering, INSA of Rennes, Franceb School of Civil, Mining and Environmental, University of Wollongong, Australiac 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 modelincludes appropriate nonlinear constitutive relationships for the concrete, the steel and tension stiffeningeffect. Furthermore, the discrete nature of the shear connection is embedded in themodel and the tensionstiffening 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 hoggingmoment regions, where cracking occurs. Comparisons between the numerical analyses and experimentalresults 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 multistoreybuildings and bridges, continuous composite beams are often usedbecause of the benefits at both the ultimate and serviceability limitstates for long spans or heavily loaded members [1]. For thesecases, 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:[email protected] (M. Hjiaj).

0143-974X/$ – see front matter© 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.jcsr.2008.07.026

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 developsbetween the reinforcement and the concrete. It allows the transferof 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 models

to 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–curvature relationships related to interface slip and bondslip. Xu [7] and Gattesco [8] proposed a displacement-based finiteelementmodelwith two nodes and eight degrees of freedom. 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-sectionalanalysis technique. The model considered the concept of partial

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738 Q.H. Nguyen et al. / Journal of Constructional Steel Research 65 (2009) 737–748

Fig. 1. (a) Infinitesimal composite beamsegmentwithout connector; (b) Connectorelement.

interaction allowing for the occurrence of slip at the slab-beaminterface.In this article, a finite element model is proposed to analyse

the 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 thecomponents. 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 betweenthe 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:

• Preservation of the plane cross section for both the slab and theprofile.• Steel beam cross-section is Class 1 or 2 according to the EC4[12].• No uplift occurs between the slab and the profile; therefore twoparts of the composite section have the same rotation and thesame curvature.• Slip can occur at the slab/profile interface.• The axial strain distribution over the section depth is linearwitha discontinuity at the slab/profile interface due to slip.• Themember cross-section is subdivided into concrete and steellayers (fibre beam element).• When the reinforcement is in tension, all layers in the effectivearea Ac,eff are replaced by a single layer of steel embedded in theconcrete (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. All variables subscripted with c belong to the concrete slabsection and those with s belong to the steel beam. Quantities withsubscript sc are associated with the shear connectors.

2.2.1. Equilibrium conditionsDue to the discrete nature of the shear connexion, the internal

forces (bending, normal force and shear force) distributions in theconcrete slab and in the steel profile are now discontinuous withjumps 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-sectioncontaining a shear connector (see Fig. 1). The first set of equilibriumequations, which apply between two consecutive connectors, isreadily obtained by expressing the equilibrium of a small elementof the composite beam, of length dx, and subjected to internalforces (Fig. 1(a)). The equilibrium conditions result in the followingset of equations:

dNc(x)dx

=dNs(x)dx= 0 (1)

T (x) = −dM(x)dx

(2)

d2M(x)dx2

+ p0 = 0 (3)

where T (x) = Tc(x)+ Ts(x) andM(x) = Mc(x)+Ms(x).The equilibrium Eqs. (1)–(3) are rewritten in the following

compact form

∂D+ P = 0 (4)

where D =[Nc (x)Ns(x)M(x)

]is the internal stress resultant vector, P =

[00p0

]the external loading vector and ∂ a linear operator expressed as:

∂ =

ddx

0 0

0ddx

0

0 0d2

dx2

. (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 Rsc (Fig. 1(b)):

Rsc = −Nc = Ns = −M/H (6)

where H = H1 + H2. It worth nothing that the internal stressresultants involved in Eq. (6) are acting at the cross-section level.

2.2.2. Compatibility conditionsThe curvature and the axial deformation at any section are

related 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):

εc(x) =duc(x)dx

(7)

εs(x) =dus(x)dx

(8)

κ(x) = −d2v(x)dx2

(9)

dsc(x) = us(x)− uc(x)+ Hdv(x)dx

(10)

where u is the longitudinal displacement, v the transversaldisplacement, εc the strain at the concrete section centroid, εs

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 dscthe 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 =[uc (x)us(x)v(x)

]is the displacement vector; e =

[εc (x)εs(x)κ(x)

]is the

section deformation vector and the operator ∂∗ is given by

∂∗ =

ddx

0 0

0ddx

0

0 0 −d2

dx2

. (12)

2.3. Material models

Thematerial behaviour is described using explicit relationshipsbetween the total stress and the total strain with appropriateloading/unloading conditions.

2.3.1. ConcreteThe stress–strain relationship suggested by the CEB-FIB Model

Code 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 amonotonically 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 σc–εc relationship is approximated by the following

functions:

• For εc < εc,lim:

σc = −

[(εc1

εc,limξ − 2

(εc1

εc,lim

)2)(εc

εc1

)2

+

(4εc1εc,lim

− ξ

)εc

εc1

]−1fcm

Fig. 3. Stress–strain diagram for concrete.

• For εc,lim ≤ εc ≤ 0:

σc = −

[1+

(EciEc1− 2

)εc

εc1

]−1 [ EciEc1

εc

εc1−

(εc

εc1

)2]fcm (13)

• For 0 < εc ≤ 0.9fctm/Eci:

σc = Eciεc• For 0.9fctm/Eci < εc ≤ 0.00015:

σc = fctm −0.1fctm

0.00015− 0.9fctm/Eci(0.00015− εc)

• For εc > 0.00015:

σc = 0

where

εc1 = −0.0022; ξ = 4[1+

(EciEc1− 2

)εc,lim

εc1

]−2×

[(EciEc1− 2

)(εc,lim

εc1

)2+ 2

εc,lim

εc1−EciEc1

].

In the above relationships, the symbols have the followingmeaning:– fcm is the mean compressive concrete strength;– fctm is the mean tension concrete strength;– εc1 is the strain at the peak stress;– εc,lim is the strain at half the peak stress;– Eci is the initial tangent modulus;– Ec1 is the secant modulus at the peak stress.

2.3.2. SteelIn 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, linear hardening occurs up to the ultimate tensile stressand the stress remains constant until the tensile failure strain isreached.

2.3.3. Steel embedded in concreteWhen uncracked concrete is in tension, the tensile force is dis-

tributed between the reinforcement and the concrete in proportionto 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


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-cent cracks, tensile forces are transmitted from the steel to the sur-rounding concrete by bond forces. The contribution of the concretemay be considered to increase the stiffness of the tensile reinforce-ment. This effect is called tension-stiffening. To describe this effect,a number of models have been proposed. Themajority of themod-els are based on the mean axial stress and the mean axial strain ofthe concrete member in the reinforced concrete, [13–16].To take the tension stiffening effect into account, the stress-

average strain relationship of steel embedded in concrete proposedby the CEB-FIB model [11] is considered to describe the behaviourof 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 εs,m ≤ εsr1:

σs =

(1+

1αρ

)Esεs,m

• For εsr1 < εs,m ≤ εsrn:

σs = σsr1 +σsrn − σsr1

εsrn − εsr1

(εs,m − εsr1

)• For εsrn < εs,m ≤ εsry:

σs = σsrn +fsy − σsrnεsry − εsrn

(εs,m − εsrn

)(14)

• For εsry < εs,m ≤ εsr,sh:

σs = fsy

Fig. 6. Load–slip diagram for stud shear connector.

• For εsr,sh < εs,m ≤ εsru:

σs = fsy +fsu − fsyεsru − εsry

(εs,m − εsry

)• For εs,m > εsru:

σs = fsu

where– α and ρ are the modular ratio and the geometric ratios ofreinforcing steel, respectively;

– σsr1 is the steel stress in the crack, when the first crack hasformed

– σsrn is the steel stress in the crack, when the last crack hasformed

– εsr1 and εsr2 are the steel strains at the point of zero slip andat the crack when the cracking forces reach ftm

– εsrn = σsrn/Es − βt (εsr2 − εsr1)– εsry = εsy − βt (εsr2 − εsr1)– εsr,sh = εsh − βt (εsr2 − εsr1)– εsru = εsr,sh + δ

(1− σsr1/fsy

) (fsu − fsy

)/Esh

– βt = 0.4 for instantaneous loading, and βt = 0.25 for long-term and repeated loading

– δ is a coefficient to take account the stress ratio fsu/fsy and theyield stress fsy (δ = 0.8 was proposed in CEB-FIB Model Code90 [11]).

2.3.4. Stud shear connectorsThe constitutive relationship proposed by Ollgaard et al. [17] is

considered for the stud shear connector. The analytical relation-ship between the shear force Rsc and the slip dsc of a generic studis given by

Rsc = Rmax(1− e−β|dsc |)α (15)

where Rmax 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 ksc (see Fig. 6) may berewritten as

Rsc = kscdsc . (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


Fig. 7. Cross-sectional analysis by finite layer approach.

any layer of the concrete slab section and the steel beam sectioncan be evaluated as

εcz = εc + κzcεsz = εs + κzs

(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 Ac,eff (see Fig. 7). Ac,eff is the area of concrete surrounding thetensile reinforcement. According to the CEB-FIB Model Code [11],Ac,eff is calculated as

Ac,ef = 2.5(c + φ/2) (18)where c is the coating length of the concrete and φ the diameter ofthe 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 asNc = (EA)cεc + (EB)c κ (19)

Mc = (EB)cεc + (EI)c κ (20)where

(EA)c =nlc∑i=1

E ic + Ei+1c

2bic t

ic + EsrAsr (21)

(EB)c =nlc∑i=1

(Ec,i (zi+1 + 2zi)+ Ec,i+1 (2zi+1 + zi)

6

)bc,itc,i

+ EsrAsrzsr (22)

(EI)c =nlc∑i=1

(Ec,iz̃i + Ec,i+1z̃i+1

12

)bc,itc,i + EsrAsrz2sr (23)

with z̃i = z2i+1 + 2zizi+1 + 3z2i and z̃i+1 = 3z

2i+1 + 2zizi+1 + z

2i .

Ec,i and Ec,i+1 are the secant concrete modulus at the layer ends(trapezoidal integration scheme), bc,i is the section width, tc,i thethickness of the concrete layer i and nlc is number of concretelayers in the slab section.In the sameway the axial resultant and bending resultant of the

steel beam sections are given byNs = (EA)sεs + (EB)s κ (24)

Ms = (EB)sεs + (EI)s κ. (25)Finally the force–deformation relationship of the cross section canbe written in matrix form ase(x) = f(x)D(x) (26)

where f(x) =[(EA)c 0 (EB)c0 (EA)s (EB)s

(EB)c (EB)s (EI)c + (EI)s

]−1is the secant flexibility

matrix 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 hasbeen 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. Thestiffness matrix is derived by combining the stiffness matrix ofa single shear connector element and the stiffness matrix of acomposite beam element without connector.For the composite beam element without connector, the

derivation follows the two-fieldmixed formulationwhich uses theintegral 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 a matrix of 3× nd shape functions with nd = 8 beingthe total number of displacement degrees of freedom and q thevector of element displacement (Fig. 8).The internal forces field is assumed to be continuous along each

beam element (but not across adjacent elements):

D(x) = b(x)Q+ D0(x) (28)

where b(x) is a matrix of 3 × ns force interpolation functionswith ns = 4 being the total number of force degrees offreedom, Q the vector of element forces (Fig. 8) and D0(x) =[ 0 0 0.5p0x(L− x) ]T is a internal force vector accounting forthe effects of internal loading on the cross section forces.In the mixed formulation, the integral forms of compatibility

and equilibrium equations are enforced. These are then combinedto obtain the relationship between the element forces and thedisplacements.


Fig. 8. (a) Beam element with rigid-body mode; (b) Beam element without rigid-body mode.

The weighted integral form of the compatibility equation (11)is∫LδDT(x)

(∂∗d(x)− e(x)

)dx = 0 (29)

where δD(x) are arbitrary (virtual) forces field fulfilling theequilibrium conditions and the integration extends over theelement length L. By substituting (26), (27) and into (29), oneobtains

δQT{(∫

LbT(x)B(x)dx

)q−

(∫LbT(x)f(x)b(x)dx

)Q

−

∫LbT(x)f(x)D0(x)dx

}= 0 (30)

where B(x) = ∂∗a(x) is the deformation interpolation matrix.From the arbitrariness of δQ, relation (32) reduces to the followingexpression

Gq− FQ = q0 (31)

where

G =∫LbT(x)B(x)dx (32)

F =∫LbT(x)f(x)b(x)dx (33)

q0 =∫LbT(x)f(x)D0(x)dx. (34)

A secant approximation of the flexibility matrix f(x) has been usedin Eq. (26). The weighted integral form of the equilibrium equation(4) is derived from the virtual displacement principle and takes theform∫LδdT(x)(∂D(x)+ P)dx = 0 (35)

where δd(x) are the displacement fields fulfilling the kinematicconditions. By integrating by parts (35) and substituting (27) andinto (35), one obtains

δqT{(∫

LBT(x)b(x)dx

)Q−

∫L

(aT(x)P− BT(x)D0(x)

)dx− Pe

}= 0 (36)

where Pe is the vector of applied nodal forces. Since Eq. (36) holdsfor arbitrary δq, it follows that

GTQ = Pe + Q0 (37)

where

Q0 =∫L

(aT(x)P− BT(x)D0(x)

)dx. (38)

The rearrangement and combination of Eqs. (31) and (37) resultsin[−F GGT 0

](Qq

)=

(q0

Q0 + Pe

). (39)

Eq. (39) represents inmatrix form the two fieldsmixed formulationfor the composite beam element without connector. If the firstequation in (39) is solved for Q and the result is substituted intothe second equation, the following expression is obtained:

Keq = P+Q̄0 (40)

where Ke = GTF−1G represents the secant stiffness matrix of thecomposite beam element without connectors and Q̄0 = Q0 +GTF−1q0 represents the nodal force vector produced by internalelement with transverse uniform loading p0.

2.6. Connector element

The slip dsc of the connector is defined as the relativedisplacement between the concrete slab and the steel beam at theinterface (see Eq. (11)). The connector element is a specific elementwith no length which allows a certain amount of sliding betweenthe concrete slab and the steel beam, according to its stiffness, butprevents any separation and rotation discontinuity.Substituting Eqs. (10) and (16) into Eq. (6) one obtains:

ksc(us − uc − Hθ) = −Nc = Ns = −M/H. (41)

Eq. (41) may be rewritten in matrix form as

Qsc = Kscqsc (42)

where Qsc =[NcNsM

]is the nodal force vector, qsc =

[ucusθ

]the nodal

displacement vector, and Ksc = ksc

[ 1 −1 H−1 1 −HH −H H2

]the secant

stiffness matrix of the connector element.The global stiffnessmatrix is obtained by assembling composite

beam elements matrices Ke and connector elements matricesKsc . To increase the accuracy of the results, it is possible touse several elements between two consecutive connectors. Asolution algorithm based on the secant stiffness method withappropriate loading/unloading conditions is used to solve theresulting nonlinear system of equations [24].

3. Comparison with experimental data

The predictions of the proposed model are compared againstexperimental data obtained by earlier experimental tests. Theaim of this model validation is to assess the capability ofthe proposed nonlinear technique to satisfactorily predict thestructural behaviour of composite beams with discrete partialshear connexion. Three composite beams are considered: thesimply-supported composite beam tested by Fabbrocino et al. [26]and the two span continuous beams tested by Ansourian [27] andTeraszkiewicz [28].


3.1. Fabbrocino’s beam Type C (2000)

The beam Type C was tested by Fabbrocino et al. [26] as a partof a series of tests aimed to investigate the influence of the ar-rangement and the mechanical properties of shear connectors onthe nonlinear behaviour of composite beams subjected to nega-tive bending moment. Fig. 9(a) shows the geometrical character-istics of the beam Type C. The beam is simply supported with aspan length of 3605 mm loaded by a single concentrated force atmidspan. Headed studs of 16mm diameter were uniformly spacedalong the beam. The steel section of beam Type C is HEB 180. Theslab is 800 mm wide and 120 mm thick, longitudinally reinforcedby 4 steel bars of 14mmdiameter at the centre. Thematerial prop-erties and the geometric characteristics for the beam Type C aregiven in Table 1.In terms of global parameters, the numerical–experimental

comparison for beam Type C is illustrated in Figs. 10 and 11, whichshow the load–deflection curve and load versus rotation curve atthe support. It should be noted that the numerical result obtainedby the proposed model using 72 elements without connectors and8 connectors are in very good agreement with the experimentaldata both in the elastic and post-elastic field.In terms of local parameters, the load versus slip at each

support curve is depicted in Fig. 12. The comparison confirms goodagreement between numerical and experimental results. Howeverthe model seems not to be able to describe the asymmetricphenomenon related to unexpected local failure mechanisms ofthe shear connection.

3.2. Ansourian’s beam CTB3 (1981)

The beam CTB3 was a part of the experimental program carriedby Ansourian [27] to investigate hogging hinges. The beam CTB3has two equal spans of 4500 mm and two concentrated loads areapplied at the centre of both spans. The steel section of beamTCB3 is IPBL 200. The slab is 1300 mm wide and 100 mm thick,longitudinally reinforced by steel at the top and the bottom.

Fig. 10. Numerical–experimental comparison of load–deflection curve for beamType C.

The distance from the interface to the bottom and the topreinforcement steel layers was 25 mm and 75 mm, respectively.Stud connectors with diameter of 19 mm were equally spaced intriples at 350mm along the beam except over the internal support(1050mmboth sides)where thepitch reduces to 300. Thedetails ofthe geometrical characteristics andmaterial properties of the beamare reported in Fig. 9(b) and Table 1.The numerical results of the proposed model using 108 beam

elements without connectors and 28 connector elements arecompared with Ansourian’s experimental results in Figs. 13–16.The model predicts quite well the load–deflection curve (seeFig. 13). The plots illustrating the load versus curvature at themidspan (sagging) and at 150 mm from the support (hogging)are shown in Figs. 14 and 15, respectively. It can be seen thatgood agreement between experimental and numerical results isachieved. In terms of localised behaviour, the load versus lower

Fig. 9. Geometrical characteristics of test beams.


Table 1Beam details for Type C, CTB3 and CBI

Beam identification Type C CTB3 CBI

Span length (mm) 3605Loading type Midspan point Midspan point Midspan pointConcreteThickness (mm) 120 100 60Width (mm) 800 1300 610Compressive strength fcm (N/mm2) 34 43 46.7Tensile strength fctm (N/mm2) 2.5 3.15 3.89SteelProfile section HEB 180 IPBI 200 6′′ × 3′′12 lb/ft BSBYield stress fsy (N/mm2)• Flange 375 220 292•Web 375 235 292• Reinforcement 540 430 310Ultimate tensile stress fsu (N/mm2)• Flange 475 390 470•Web 475 411 470• Reinforcement 635 533 485Elasticity modulus Es (N/mm2) 200000 200000 200000Hardening modulus Esh (N/mm2) 2666 5000 2500Rebar top area Asrt (mm2)• Sagging 616 360 445• Hogging 616 1230 445Rebar bottom area Asrb (mm2)• Sagging – 160 –• Hogging – 470 –ConnectorType of connector Stud φ16 mm Stud φ19 mm Stud φ9.5 mmNumber of stud 8 84 96Connector spacing @ (mm)• Sagging – 350 146• Hogging 315 300 146Shear strength (kN) 73.75 110 32.4Coefficient α 0.8 0.85 1Coefficient β (mm−1) 0.7 1.2 4.72

Fig. 11. Numerical–experimental comparison of load versus rotation at supportcurve for beam Type C.

flange strain at 150 mm from the support is plotted in Fig. 16. Itshould be noted that there is reasonable agreement between thenumerical and the experimental results.

3.3. Teraskiewicz’s beam CBI (1967)

The CBI beam tested by Teraszkiewicz [28] is a symmetric twospans composite beam with span length equal to 3304 mm. Thesteel section of beam CBI is 6′′ × 3′′ × 12 lb/ft BSB. The slab is610 mm wide and 60 mm thick, longitudinally reinforced by steelonly at the top. The beam was loaded by concentrated forces at

Fig. 12. Numerical–experimental comparison of load versus slip at support curvefor beam Type C.

the centre of both spans. The failure of the beam was attributedto connector failure in the sagging moment region. The estimatedultimate load is 151 kN. The details of geometrical characteristicsand material properties of the beam are reported in Fig. 9(c) andTable 1.To simulate the beam behaviour, 92 beam elements without

connectors and 47 connector elements were used. For a midspanload P equal to 122 kN (81% of the ultimate load), Figs. 17–19 show the numerical–experimental comparison of the beamdeflection, the slip distribution along the beam and the lowerflange strain distribution, respectively. It can be noted that


Fig. 13. Numerical–experimental comparison of load–deflection curve for beamCTB3.

Fig. 14. Numerical–experimental comparison of load versus midspan curvaturecurve for beam CTB3.

Fig. 15. Numerical–experimental comparison of load versus support curvaturecurve for beam CTB3.

although the experimental results were not perfectly symmetrical,the numerical curve is very close to the experimental results.

Fig. 16. Numerical-experimental comparison of load versus lower flange strain atsupport curve for Beam CTB3.

Fig. 17. Numerical–experimental comparison of the deflection distribution at P =122 kN for beam CBI.

Fig. 18. Numerical–experimental comparison of slip distribution at P = 122 kNfor beam CBI.

4. Parametric study

In this section, the proposed model which was successfullyvalidated above is used to conduct a parametric study to


Table 2Mechanical characteristics of materials

Concrete C30/35 according to CEB-FIB model [11]

Steel beam Es = 210 000 N/mm2; Esh = Es/33; fsy = 355 N/mm2; fsu = 510 N/mm; εsh = 1.69%; εsu = 5%Reinforcing steel Esr = 210 000 N/mm2; Esh = 0; fsry = 500 N/mm2; fsru = 500 N/mmεsru = 4%Stud shear connectors Rmax = 100 kN; α = 0.8; β = 0.7 mm−1; dsc,max = 7 mm

Fig. 19. Numerical–experimental comparison of strain distribution at the lowersteel flange for beam CBI (P = 122 kN).

Table 3Geometrical characteristics of 5 two-span continuous composite beam designed byEC4 [12]

Beam B1 B2 B3 B4 B5

Span length (mm) 6000 9000 12000 6000 6000Slab width (mm) 2000Slab thick (mm) 120Reinforcing 4Φ14 at the centre of slabSteel section HEB 260 HEB 400 HEB 800 HEB 260 HEB 260Degree connection 100% 100% 100% 70% 40%Number of connector 62 152 170 44 25

Class cross-section Under sagging bending: 1Under hogging bending: 1

investigate the effects of span length and degree of shearconnection on the global structural behaviour of symmetric twospan composite beams designed according to EC4 [12].In the present parametric study, the geometrical characteristics

of the slab were kept constant: 2000 mm wide and 120 mmthick. The slab is longitudinally reinforced by steel bars of 14 mmdiameter at the centre. The mechanical characteristics chosen forthis study are shown in Table 2. Five symmetrical two-span beams(Table 3) have been designed according to EC4 [12]. Each beam hasdifferent span length and different degree of shear connection.In fact, EC4 [12] suggests a minimal degree of shear connection

equal to 55% for the symmetrical beam of span length 10–15 m. Inthis study, we willingly decrease this to 40% to look profoundly atthe limitation of EC4 [12] as well as the effects of the variation ofdegree of shear connection on the deflection and the ultimate load.Table 4 shows the comparison between the ultimate load

predicted by the proposed model and the ultimate load calculatedusing analysis models given in EC4 [12] (uncracked analysis,cracked analysis and plastic analysis). It can be observed that theresults obtained by EC4 [12] are extremely conservative whencompared with the results obtained by the proposed model.

4.1. Effects of span length on the load–deflection response

The effect of different span length (6 m, 9 m and 12 m) onthe load–deflection and moment rotation response is shown in

Table 4Comparison of ultimate load between proposed model and EC4 [12]

Ultimate load (kN/m) calculated by EC4 Proposed modelUncrackedanalysis

Crackedanalysis

Plasticanalysis

Ultimate load(kN/m)

Type offracture

B1 165.4 183.6 187.8 204.0 RebarB2 348.4 351.5 370.6 420.0 RebarB3 379.3 373.3 447.7 516.0 ConnectorB4 – – 170.3 198.0 RebarB5 – – 152.8 192.0 Rebar

Fig. 20. Load-deflection response for different pan length of two-span compositebeam designed with EC4.

Fig. 21. Moment-rotation at 2 cm from support response for different span lengthof two-span composite beam designed to EC4.

Figs. 20 and 21, respectively. Because the beam is governed byserviceability, when the span length increases the steel beamsection should be larger, this is why, from Figs. 20 and 21, we canobserve that the increase in span length is found to increase theinitial stiffness and also increase the ultimate load.


Fig. 22. Load-end slip response for different span length of two-span compositebeams designed to EC4.

Fig. 23. Load–deflection response due to different degree of shear connection.

Fig. 22 provides the load–slip curves for the beams B1-3.The slip increases as the span length increases. This is tobe expected as slip is heavily dependent on span length. Itis comforting to know that all cases exceed the 4 mm sliprequirement of EC4 [12]. Hence if the ultimate load is to beconsidered at a slip level of 4mm, thenmost of beams need to havetheir load capacity reduced.

4.2. Effect of degree of shear connection

Three values of degree of shear connection corresponding tothe case of full shear connection (FSC 100%) and partial shearconnection (PSC70% and40%)were used to study their effect on theglobal response of symmetric two-span composite beams. Fig. 23demonstrates thatwhen the degree of shear connection is reduced,the ultimate load decreases (6% from FSC 100% to PSC 40%) and alsothe stiffness of the beam is lower but this brings about an increaseof ductility of the beam. This is also observed inmoment–curvatureat the mid-span curves shown in Fig. 24.Fig. 25 shows the load–slip relationship for different degrees

of shear connection. The slip increases as the degree of shearconnection decreases. Once again this is to be expected. Hence it isalso worthwhile noting that even for a level of shear connection of100%, one just exceeds the 4 mm EC4 [12] requirement. Reducingthe level of shear connection allows for amuch larger slip, but mayresult in a reduced capacity if one evaluates the flexural strengthat a slip level of 4 mm.

Fig. 24. Moment–curvature at the mid span response for different degree of shearconnection.

Fig. 25. Load-end slip response for different degree of shear connection.

5. Conclusions

A numerical model based on a nonlinear two-field mixed fi-nite element formulation for predicting the behaviour of contin-uous composite beams with discrete shear connectors has beenpresented. The model includes realistic nonlinear constitutive re-lationships for the concrete and the steel, taking into account thetension stiffening effect. The numerical–experimental comparisonshown validates the model reliability and the capacity to deter-mine the experimental behaviour of continuous composite beams.It is worth noting that the proposedmodel is able to perform a verygood prediction of the global behaviour of continuous compositesteel concrete beams subjected to monotonic loads.A parametric study has been undertaken and the effects of

span length and level of shear connection have been studied. Thestrength provision of EC4 [12] was compared with the proposedmodel and found that they were conservative. However, whenexamining the results of the parametric study one can see that theuse of the 4 mm slip requirement of EC4 [12] somewhat limitsour ability to take advantage of issues such as strain hardening.The 4 mm slip level is probably appropriate for a level of 100%shear connection; however perhaps a slip level of 6 mm is moreappropriate for cases when 40% shear connection is used. Furtherexperimental research needs to be conducted to confirm this.


References

[1] Oehlers D, Bradford MA. Composite steel and concrete structural members:Fundamental behaviour. Oxford: Pergamon; 1995. ed. Elsevier.

[2] Johnson RP, Allison RW. Shrinkage and tension stiffening in negative momentregions of composite beams. Structural Engineer 1981;59B:10–6.

[3] Cosenza E, Pecce M. Deflection and redistribution of moments due to thecracking in steel–concrete composite continuous beams. In: ICSAS’91. NewYork: Elsevier Science; 1991.

[4] Couchman G, Lebet JP. Design rules for continuous composite beams usingclass 1 and 2 steel sections: Applicability of EC 4. In: EPFL-ICOM Publ. 290.1993, Lausanne (Switzerland): EPFL-ICOM–Steel Structures.

[5] Lebet JP. Comportement des ponts mixtes acier-béton avec interactionpartielle de la connexion et fissuration du béton. Lausanne (Switzerland):Ecole Polytechnique Fédérale de Lausanne; 1987.

[6] Manfredi G, Fabbrocino G, Cosenza E. Modeling of steel–concrete compositebeams under negative bending. Journal of Engineering Mechanics 1999;125(6):654–62.

[7] Xu H. Modélisation numérique et étude du comportement de poutres mixtescontinues avec phénomène de semi-continuté et glissement. In: Génie civil.Rennes (France): Institut National des Sciences Appliquees des Rennes;1995.

[8] Gattesco N. Analytical modeling of nonlinear behavior of composite beamswith deformable connection. Journal of Constructional Steel Research 1999;52(2):195–218.

[9] Loh HY, Uy B, Bradford MA. The effects of partial shear connection in thehogging moment regions of composite beams Part II–Analytical study. Journalof Constructional Steel Research 2004;60(6):921–62.

[10] Zienkiewicz OC, Taylor RL. The finite element method. London: McGraw-Hill;1989.

[11] CEB-FIPmodel code 1990. Lausanne (Switzerland): Comité Euro-Internationaldu Béton; 1991.

[12] ENV 1994-1-1 Eurocode 4- Design of composite steel and concrete structures-Part 1-1: General rules and rules for buildings.

[13] ScanlonA,MurrayDW. Timedependent deflections of reinforced concrete slabdeflections. Journal of the Structural Division 1974;100(9):1911–24.

[14] Lin CS, Scordelis AC. Nonlinear analysis of RC shells of general form. Journal ofthe Structural Division 1975;101(3):523–38.

[15] Vebo A, Ghali A. Moment curvature relation of reinforced concrete slabs.Journal of the Structural Division 1977;103(3):515–31.

[16] Gilbert R, Warner R. Tension stiffening in reinforced concrete slabs. Journal ofthe Structural Division 1978;104(12):1885–900.

[17] Ollgaard JG, Slutter RG, Fisher JW. Shear strength of stud connectors inlightweight and normal weight concrete. AISC Engineering Journal 1971;8(2):55–64.

[18] Aribert JM, Labib AG. Modèle calcul élasto-plastique de poutres mixtes àconnexion partielle. Construction Metallique 1982;4:37–51.

[19] Salari MR, et al. Nonlinear analysis of composite beamswith deformable shearconnectors. Journal of Structural Engineering 1998;124(10):1148–58.

[20] Ayoub A. A two-field mixed variational principle for partially connectedcomposite beams. Finite Elements in Analysis and Design 2001;37(11):929–59.

[21] Dall’Asta A, Zona A. Non-linear analysis of composite beams by a displacementapproach. Computers & Structures 2002;80(27–30):2217–28.

[22] Aribert JM, Ragneau E, XuH. Développement d’un élément fini de poutremixteacier-béton intégrant les phénomènes de glissement et de semi-continuitéavec éventuellement voilement local. Revue Construction Métallique 1993;(2):31–49.

[23] Samy G, Jean-Marie A. Numerical investigation of moment redistributionin continuous beams of composite bridges. In: Roberto TL, Jorg L, editors.Composite construction in steel and concrete V. ASCE; 2006. p. 47–56.

[24] Nguyen QH. Modélisation numérique du comportement des poutres mixtesacier-béton. In: Génie civil. Rennes (France): Institut National des SciencesAppliquées des Rennes; 2008.

[25] Salari MR. Modelling of bond-slip in steel–concrete composite beams andreinforcing bars, Departement of Civil, Environmental and ArchitecturalEngineering, University of Colorado; 1999.

[26] Fabbrocino G,Manfredi G, Cosenza E. Analysis of continuous composite beamsincluding partial interaction and bond. Journal of Structural Engineering 2000;126(11):1288–94.

[27] Ansourian P. Experiments on continuous composite beams. Proceedings of theInstitution of Civil Engineers 1981;71:25–51.

[28] Teraszkiewicz J. Static and fatigue behavior of simply supported andcontinuous composite beams of steel and concrete. London: University ofLondon; 1967.

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

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