dimensionless formulation and comparative study of analytical models for composite beams in partial...

Journal of Constructional Steel Research 75 (2012) 21–31

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Journal of Constructional Steel Research

Dimensionless formulation and comparative study of analytical models for compositebeams in partial interaction

Enzo Martinelli a, Quang Huy Nguyen b, Mohammed Hjiaj b,⁎a Department of Civil Engineering, University of Salerno, Via Ponte don Melillo - 84084 Fisciano (SA), Italyb Structural Engineering Research Group/LGCGM, INSA de Rennes, 20 avenue des Buttes de Coësmes, CS 70839–35708 Rennes Cedex 7, France

⁎ Corresponding author. Tel.: +33 223238711; fax: +E-mail address: [email protected] (M

0143-974X/$ – see front matter © 2012 Elsevier Ltd. Aldoi:10.1016/j.jcsr.2012.02.016

a b s t r a c t
a r t i c l e i n f o
Article history:Received 30 September 2011Accepted 29 February 2012Available online 11 April 2012

Keywords:Composite beamsPartial interactionShear deformabilityModel classificationParametric studyDimensionless formulation

Steel–concrete composite beams are widely utilized as cost-effective structural solution in both buildings andbridges. Partial interaction through the possible occurrence of slips at the interface between the two con-nected members, strongly affects the behaviour of composite members and, therefore must be incorporatedin theoretical models dealing with composite members. In addition, shear deformability of the two connectedlayers cannot be ignored in stocky members. Several computational models simulating the behaviour of com-posite beams including partial interaction and shear deformability with various degree of sophistication arecurrently available in the scientific literature. The present paper focuses on the background and the mechan-ical assumptions adopted in these models as well as structural characteristics which actually govern theirpredictions. Based on the kinematic assumptions involved, a threefold classification is proposed. The paperfurther clarifies the hierarchy between the three groups of models. To do so, the governing equations foreach group of models are transformed into a proper dimensionless form by using mechanically sound dimen-sionless expressions of all functions of interest involved in the description of the mechanical response of thecomposite beam. A thorough parametrical study is presented which quantifies the influence of the identifieddimensionless parameters. Furthermore, the study clearly indicates possible threshold values beyond whichcertain effects become negligible.

© 2012 Elsevier Ltd. All rights reserved.

1. Introduction

Steel–concrete composite beams are widely used in both buildingsand bridges as a structurally-efficient and cost-effective solution [1].However, the analysis of steel–concrete composite beams is ratherchallenging from a computational point of view since the mechanicalbehaviour of the coupled system is complex and characterized by theoccurrence of slips at the interface. Such slip results in partial interac-tion even in cases of full shear connection, as defined by Eurocode 4[2] for design purposes at the Ultimate Limit States. Several theoreti-cal models, characterized by different levels of approximation, havebeen proposed to simulate the structural response of elastic compos-ite structures in partial interaction [3,4]. Such models have been alsothe basis for approximate solutions derived using either the finite dif-ference method [5,6] or the finite element method, the latter beingthe driving force behind various advanced formulations such as thedirect stiffness/flexibility method [7], force-based FE models [8]andmixed formulations [9,10,23]. These formulations have been used toinvestigate the time response as well as the inelastic response of com-posite beams with interlayer slips.

33 223238448.. Hjiaj).

l rights reserved.

Albeit the specific objectives of the various contributions (i.e., thelong-term behaviour of composite steel–concrete beams [11,12,22], theresponse under fire conditions [13], geometrically nonlinear analysis[24], analysis of frames [25] etc…) as well as the particular solutionmethods implemented therefore, commonassumptions of the above for-mulations can be easily recognized. Indeed, the transverse displacementof both members are generally considered equal (no uplift) all along thebeam. In fact, several studies have demonstrated that the overall struc-tural response is weakly influenced by considering possible relative dis-placements between the concrete slab and the steel beam [14]. Thereforeconsidering independent deflection for each layer unnecessarily compli-cates the model equations as we are dealing with highly nonlinear rela-tionswhich govern unilateral contact conditionswith orwithout friction.Consequently, a unique function describing the transverse displacementof both components is generally assumed for the sake of simplicity andwithout significant loss of accuracy [15,16,17]. Based on the kinematicalassumptionsunderlying thedifferent theoreticalmodels, the formulationsthat have been proposed for planar steel–concrete composite beams inpartial interaction can be basically classified in the following three groups:

1. Shear-rigid composite beam models (Group 1): these are modelswhere shear deformability is neglected for both connected layerseach of which is modeled according to the well-known BernoulliBeam Theory [3];

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Fig. 1. Free body diagram of an infinitesimal two-layer composite beam segment.

22 E. Martinelli et al. / Journal of Constructional Steel Research 75 (2012) 21–31

2. Constrained shear-flexible composite beam models (Group 2): theseare models that assume a single shear strain distribution for bothconnected layers and follow the well-known Timoshenko BeamTheory ( plane sections remain plane after the deformation) [18];

3. Shear-flexible composite beam models (Group 3): these are modelsthat introduce independent shear strains for each layer, both mod-eled according to the Timoshenko Theory [20].

For themodels belonging to the first group, the closed-form expres-sions of both the flexibility and the stiffness matrix [16] are readilyavailable. An “exact” stiffness matrix has been also derived for amodel belonging to the second group [19]. A general procedure for de-riving the stiffness matrix for a beam model belonging to the thirdgroup has been also recently reported [21].

The present paper intends to provide a comparative study to assessthe predictions of the existing models for steel–concrete compositebeams in partial interaction as classified above. Obviously models ofthe last group are themost sophisticated and so are expected to providethe most accurate solutions. However, this sophistication comes at aprice and it is not clear from the outset that it is needed in the firstplace. Hence, it is desirable to understand the limits and the accuracyof models belonging to Group 1 and Group 2 as this would give a guide-line to a proper selection of themodel under certain given circumstancesof loading, dimensions, material parameters etc. … The paper will fur-ther clarify the hierarchy between the aforementioned groups ofmodels. To do so, the governing equations for each group of modelsare transformed into a proper dimensionless formbyusingmechanicallysound dimensionless expressions of all functions of interest involved inthe description of the mechanical response of the composite beam. Aparametric study is carried out by considering a simply supportedbeam of span length L. The influence of key dimensionless parameterson the prediction of the overall structural behaviour is investigated.

The rest of the paper is organized as follows. Section 2 summarizesthe equilibrium and kinematic equations as well as the relevantstress–strain relationships. The main kinematic assumptions are for-mally introduced in Section 3 and the corresponding governing equa-tions are derived for each group of models. Key parameters whichdescribe the structural response of composite beams are identified. Aset of dimensionless parameters is defined in Section 4 and employedin Section 5 for transforming the governing equations in a more appro-priate dimensionless form. In Section 6, an extensive parametric studyis conducted based on the previously identified dimensionless parame-ters which actually control the predictions of the analytical models. Thekey outcomes of this study are then reported in Section 7. Based on theresults, the effect of shear deformability on steel–concrete compositebeams in partial interaction, which can be modeled either with a singleor two independent shear strains, is properly quantified. In addition,consistent threshold values of the mentioned non-dimensional param-eters are identified which result in practically negligible effects of theshear flexibility according to the two possible approaches correspond-ing to the models categorized in either Group 2 or 3.

2. Fundamental equations

The governing equations describing the geometrically linear be-haviour of an elastic shear-deformable steel–concrete compositebeam in partial interaction are briefly outlined in this section. All vari-ables subscripted with b belong to the concrete slab section and thosewith subscript a belong to the steel beam. Quantities with subscript scare associated with the shear connectors. The following assumptionsare commonly accepted in all models to be discussed in this paper:

∽ both connected members are made out of elastic, homogenousand isotropic materials;

∽ the transverse sections of both steel beamand concrete slabs remainplane after deformation, though relative slips can develop alongtheir interface;

∽ uplift is neglected so the transverse displacement of both the concreteslab and the steel profile are given by the same unique function w;

∽ discretely located shear connectors are regarded as continuous.

2.1. Equilibrium equations

A free body diagram of a differential element of a composite beamsubjected to a distributed transverse loading pz is shown in Fig. 1.

For the element to be at equilibrium, the following equations mustbe satisfied:

• For layer a:

∂xNa−Dsc ¼ 0 ð1Þ

∂xVa−Vsc ¼ 0 ð2Þ

∂xMa−Va þ haDsc ¼ 0 ð3Þ

• For layer b:

∂xNb þ Dsc ¼ 0 ð4Þ

∂xVb þ Vsc þ pz ¼ 0 ð5Þ

∂xMb−Vb þ hbDsc ¼ 0 ð6Þ

where

- ∂ xi •=di •/dxi;

– hi (i=a,b) is the distance between the centroid of the layer “i” andthe layers interface;

– Ni, Vi, Mi (i=a,b) are the axial forces, shear forces and bendingmoments in layer “i”;

- Dsc is the shear bond force per unit length;- Vsc represents the transverse component of the interface forcedistribution.

2.2. Kinematic relations

Kinematic equations relating the displacement components (ui,w,θi)with the corresponding strain components (εi,γi,κi) (Fig. 2) are derived,for both layers of the composite beam, based on the equal transverse dis-placement assumption and the Timoshenko beam theory.

For each layer, these equations read:

• For layer a:

εa ¼ ∂xua ð7Þ

γa ¼ ∂xwþ θa ð8Þ

Fig. 2. Kinematics of a shear-deformable two-layer beam with interlayer slip.

23E. Martinelli et al. / Journal of Constructional Steel Research 75 (2012) 21–31

κa ¼ ∂xθa ð9Þ

• For layer b:

εb ¼ ∂xub ð10Þ

γb ¼ ∂xwþ θb ð11Þ

κb ¼ ∂xθb ð12Þ

where

- εi and ui are the axial strain and the longitudinal displacement atthe centroid of layer “i”, respectively;

- γi is the shear strain of layer “i”;- w is the transverse displacement;- θi is the cross-section rotation of layer “i”;- κi curvature of layer “i”.

Basic geometric considerations provide the expression of theinterlayer slip in terms of the displacements of the layers:

dsc ¼ ua−ub−haθa−hbθb ð13Þ

2.3. Constitutive relationships

We adopt a linear stress–strain relationship at the material leveland deduce the following constitutive law for the cross-section ofeach layer:

• Layer a:

Na ¼ EaAaεa ð14Þ

Va ¼ kaGaAaγa ð15Þ

Ma ¼ EaIaκa ð16Þ

• Layer b:

Nb ¼ EbAbεb ð17Þ

Vb ¼ kbGbAbγb ð18Þ

Mb ¼ EbIbκb ð19Þ

where EiAi, kiGiAi, and EiIi denote the axial, shear and flexural stiffness ofeach component (i=a,b), respectively. ki is the shear stiffness factorthat depends on the cross-sectional shape. In what follows, the axialstiffness EiAi, the flexural stiffness EiIi and the shear stiffness kiGiAi will

be simply denoted as EAi, EIi and GAi (i=a,b), respectively. The aboverelations must be completed by the relationship between the shearbond force Dsc and the interlayer slip dsc. The assumption of linearand continuous shear connection can be expressed by the simplefollowing relationship between interface slips and shear flow:

Dsc ¼ ksc dsc ð20Þ

where ksc is the shear bond stiffness. Full composite action (infinite slipstiffness) and non-composite action (zero slip stiffness) represent upperand lower bounds for the partial composite action.

3. Model classification

Several models have been proposed in the past years to simulatekey features of the structural response of composite beams in partialinteraction. These models are based on the field equations presentedin the previous section. Nevertheless, depending on the way sheardeformability of the connected members is dealt with, these modelscould be gathered in three main groups. For the sake of simplicity,we shall address the case of composite beams in bending only (i.e.,Na=−Nb).

3.1. Group 1: shear-rigid composite beam models

This group of models completely ignores the shear deformabilityof both connected members and is devoted to slender compositebeam members. Accordingly, the two members behave according tothe Bernoulli's beam theory where the shear strain is equal to zero(γi=0) for both layers. Consequently, a direct relationship connectsboth cross-section rotations θi to the first derivative of the deflectionw. Since the latter is the same for both layers, the two members ex-hibit the same rotation θa=θb=θ and, based on Eqs. (8) and (9), aunique curvature κa=κb=κ. Thus, a second order differential equa-tion, relating the interface shear force to the total shear force, canbe derived [15]:

∂2xDsc−α2Dsc ¼ − kscdEIabs

V ð21Þ

where d=ha+hb is the distance between the centroids of the cross-sections of the slab and the profile, EIabs=EIa+EIb represents the bend-ing stiffness of the beamwith no shear interaction and V=Va+Vb is thetotal shear force. Moreover, α is a key mechanical parameter defined asfollows:

α2 ¼ kscEA�

EIfullEIabs

ð22Þ

where EIf ull ¼ EIabs þ EA�d2 is the bending stiffness of the cross-sectionof the composite beam as a whole and EA� ¼ EAa EAb

EAaþEAbis the relative axial

stiffness of the composite section. Eq. (21) can easily be solved in thecase of statically determinate structures where the expression of thetotal shear force V can be determined using static equilibrium equa-tions. In addition to the above differential equation, it is straightforwardto derive two other differential equations which relate the longitudinalshear force distribution by solving Eq. (21) to the transverse deflectionw:

∂2xw ¼ EA�d ∂xDsc−ksc Mksc EIfull

ð23Þ

and to the normal stress distribution at the interface Vsc:

Vsc

EI�¼ − pz

EIbþ ha

EIa–hbEIb

� �∂xDsc ð24Þ


whereM is the total bendingmoment distribution and EI� ¼ EIa EIbEIaþEIb

. Dueto length constraints, the complete derivation of these equations isomitted herein.

3.2. Group 2: constrained shear-flexible composite beam models

A second group of theoretical models can be defined by enhancingthe above group of models through the adoption of a shear-deform-able beam theory where a unique shear deformation is attributed toboth layers. It assumes that both connected members shear deformaccording to the Timoshenko's theory, but they have a unique shearstrain γa=γb=γ. Under the strictly mechanical standpoint, thismeans that the two connected members are subjected to a shear forceVi which is proportional to their shear stiffness. Moreover, in view ofthe kinematic Eqs. (8) and (9), the unique shear strain γ assumption re-sults in an identical cross-section rotation θi=θ=γ−∂xw for bothlayers since the deflection w is the same for both layers. Accordingly,Eq. (21), which related the shear flow Dsc to the total shear force V, ap-plies to this group of models as well. In view of Eqs. (15) and (18), theassumption of single shear strain γ provides a relationship between Vaand Vb which can be combined with Eqs. (2) and (6) to obtain the fol-lowing simple algebraic relationship for the normal interface stressesVsc:

Vsc

GAa¼ − pz

GAð25Þ

where GA=GAa+GAb is the shear stiffness of the section in full interac-tion. Finally, a relationship between the second derivative of the deflec-tion and other relevant mechanical variables can be derived (thealgebraic manipulations are omitted for conciseness):

∂2xw ¼ Vsc

GAaþ EA�d ∂xDsc−ksc M

ksc EIfullð26Þ

It is worth to point out that Eq. (26) is slightly different from Eq.(23) as a shear deformability related term is also involved in the for-mer one. Finally, rotations θ=θa=θb can be easily expressed in termsof the total shear force and the first derivative or the deflection func-tion as follows:

θ ¼ γ−∂xw ¼ VGA

−∂xw ð27Þ

3.3. Group 3: shear-flexible composite beam models

A third group of models can be defined by simply considering thethree general assumptions listed at the beginning of Section 2 andmodeling both connected members according to the Timoshenko'sbeam theory. In other words, this group of models accounts forshear deformability of the composite member by considering thateach layer shear-deforms independently. Since we have two shearstrain distributions γa and γb, it results in two different cross-section rotations. The analytical formulation of this model has beencompletely carried out in [21] and, the following governing equationshave been derived:

∂3xDsc− α2 þ ksc EI� ha

EIa− hb

EIb

� �2� �∂xDsc ¼

kschbEIb

pz−kschaEIa

− hbEIb

� �Vsc ð28Þ

haEIa

− hbEIb

� �∂xDsc ¼

pzEIb

þ Vsc

EI�−∂2xVsc

GA� ð29Þ

where GA� ¼ GAaGAb=ðGAa þ GAbÞ is the relative shear stiffness pa-rameter. The above equations can be combined in a single fifth

order differential equation in which the unknown function is Vsc.Once the expressions of interface stresses distributions are known,and in the case of statically determinate composite beams in purebending, the transverse displacement can be obtained by integratingthe following differential equation:

∂2xw ¼ Vsc

GAaþ EA�d ∂xDsc−ksc M

ksc EIfull− EIb þ EA�hbd

EIfull

Vsc

GA� þpzGAb

� �ð30Þ

Furthermore, subtracting Eq. (11) from Eq. (8) and replacing in theoutcome each shear strain γi with the corresponding shear force Vi(using Eqs. (15) and (18)) yields the following equation for the differencebetween the cross-section rotation of steel girder and concrete slab:

θa−θb ¼ Va

GAa− Vb

GAbð31Þ

The above equation can be further transformed by substituting theexpressions of Va and Vb which can be derived by solving equilibriumEqs. (3) and (6) for the variables Vi, respectively:

θa−θb ¼ haGAa

− hbGAb

� �Dsc þ

∂xMa

GAa–∂xMb

GAbð32Þ

Then, bending moments Ma and Mb can be expressed in terms ofthe curvatures κa and κb; since curvatures can be easily related tothe first derivative of rotations through kinematic Eqs. (9) and (12),the latter can be also expressed in terms of the shear strains and de-flections again through Eqs. (8) and (11). After a few tedious butstraightforward algebraic manipulations, the following relationshipwhich expresses the difference in cross-section rotations (the relativerotation of both members) in terms of the variables Vsc, Dsc andw (and their derivatives) can be easily obtained:

θa−θb ¼ ha

GAa− hb

GAb

� �Dsc þ

EIaGA2

a− EIb

GA2b

!∂xVsc−

EIaGAa

− EIbGAb

� �∂3xw ð33Þ

where the equilibrium relationships (2) and (5) have been also con-sidered in order to replace the derivatives of shear forces Va and Va

with those of Vsc. Consequently, the relative rotation of both memberscan be determined once the distributions Dsc, Vsc, and w have beenderived.

4. Definition of a relevant set of dimensionless quantities

The equations, given in Section 3 for the three group of models,highlights a clear hierarchy within these groups with increasing com-plexity as the kinematic assumptions become more general. Thiscomplexity can be “measured” in terms of the order of the differentialequations governing these three groups of models, the level of cou-pling of these differential equations in defining the key response pa-rameters and the mechanical parameters actually involved therein.For instance, the differential equation relating the second (or third)derivative of the deflection w to other mechanical variables becomesmore and more complex as one starts from Eq. (23) for Group 1models, to Eq. (26) for Group 2 and, finally, Eq. (30) for Group 3. Inparticular, a new term appears at the right hand of these equations,as one moves from a given group to the next one. A certain numberof geometrical and mechanical parameters are formally involved inEqs. (21)–(30) and, thus, the influence of those quantities cannot beevaluated by simply examining these equations. However, a clear un-derstanding of the role actually played by these parameters can beachieved by turning Eqs. (21)–(30) into a proper dimensionlessform by using mechanically sound dimensionless expressions of allfunctions of interest involved in the description of the mechanical re-sponse of the composite beam, (i.e., among the others, the interface


slip dsc, the shear flow Dsc,the normal interface stress distribution Vsc,the deflectionw). Thus, a dimensionless form of the same expressionsis derived with the aim of pointing out the key non-dimensional pa-rameters controlling the response of the structural model. First ofall, a normalized abscissa �x∈ 0;1½ � can is defined as follows:

x ¼ L �x ð34Þ

In principle, all other quantities can be defined as the product of afactor (which is aimed at reproducing the expected order of magni-tude of the quantity under consideration) by a dimensionless func-tion (which will always be denoted with the same symbol of thedimensional one with a superimposed bar). For instance, Eq. (34)has been actually introduced by following this principle. However, itis worth underlining that the transformation of the relevant functionsinvolved in Eqs. (21)–(30) into the corresponding dimensionlessones is not unique and should be carried out with a deep physical un-derstanding of the mechanical meaning of the various parameters in-volved. For instance, in this study bending moment M, shear force Vand transverse load pz are expressed by considering the following ex-pressions:

M ¼ EIfulld

�M ð35Þ

V ¼ EIfulld L

�V ð36Þ

pz ¼EIfulld L2

�pz ð37Þ

where �pz is just the above mentioned dimensionless transversal loaddistribution, whereas �M and �V are the dimensionless functions repre-senting bending moment and shear force distribution, respectively.They have been defined by considering that themaximumbendingmo-ment is in the order of magnitude of the quantity pzL

2, and the flexuralstrength is proportional to the ratio EIfull/d (see Eq. (35)). Following thesame approach, the interface slip can be turned into non-dimensionalform as follows:

dsc ¼EIfulld L2

d L3

EIabs�dsc ¼

EIfullEIabs

L �dsc ð38Þ

since the maximum interface slip in the case of absent interaction isproportional (according to both theories of Bernoulli and Newmark)to the ratio pzL

3/EIabs and the transverse load is proportional to the fac-tor already reported in Eq. (37). Moreover, again adopting a similar ap-proach (thoughomitting further comments for the sake of conciseness),the following transformations will be considered in the present paperfor the set of relevant functions involved in Eqs. (21)–(30):

Dsc ¼EA�

L�Dsc ð39Þ

Vsc ¼EIfullEIabs

EIbd L2

�V sc ð40Þ

w ¼ EIfullEIabs

L2

d�w ð41Þ

θi ¼EIfullEIabs

Ld

�θi ð42Þ

Moreover, the following relationship between the functions repre-senting the dimensionless shear flow and interface slip can be easilyrecognized by introducing Eqs. (38) and (39) in Eq. (20):

�Dsc ¼ αLð Þ2�dsc ð43Þ

Finally, the i-th derivative ∂ xiQ of the generic dimensional quantity

Q ¼ Qref�Q can be easily expressed in terms of its dimensionless form

∂i�x �Q as follows:

∂ixQ ¼ Qref

Li∂i�x �Q ð44Þ

For instance, the i-th derivative of the interface shear flow Dsc canbe expressed as follows in terms of the corresponding dimensionlessparameters:

∂ixDsc ¼EA�

Liþ1 ∂i�x�Dsc ð45Þ

5. Dimensionless form of the models governing equations

The main differential Eqs. (21)–(30) which represent the key fea-tures of the threemodel groups defined in Section 3 can be easily turnedinto a dimensionless form by just inserting into the governing Eqs.(21)–(30) the definitions (34)–(44) of the various functions involvedtherein. Thus, the three following subsections report the dimensionlessversions of Eqs. (21)–(30) and point out the key parameters which ac-tually govern the prediction for the different groups of models.

5.1. Shear-rigid composite beam model (Group 1)

The differential Eq. (21) which basically involves the interfaceshear flow Dsc (and its second derivative) and the shear force V canbe turned into a dimensionless form by replacing these mechanicalvariables with their dimensionless counterparts as defined in Eqs.(39) and (36), respectively, and applying the rule given in Eq. (44)for the non-dimensional derivative of Dsc. After some simple algebraictransformations, based on the following identity [15],

EIfull ¼ EIabs þ EA�d2 ð46Þ

the dimensionless form of Eq. (21) can be expressed as follows:

∂2�x �Dsc− αLð Þ2 �Dsc ¼ − αLð Þ2 �V ð47Þ

where αL is the first dimensionless parameter suggested by the non-dimensional transformation. It has been already considered in otherpapers (see [15,19]]) as interaction parameters ranging from zero(in the case of no shear interaction between the two connected mem-bers) to infinity (in the ideal case of full interaction). A similar proce-dure can be followed in order to transform Eq. (23) into an adequatedimensionless form by introducing the definition for the bending mo-mentM and the deflectionw given in Eqs. (35) and (41), respectively,and using the general rule (44) for calculating the derivatives of non-dimensional functions. The following dimensionless expression canbe obtained after some mathematical transformations:

∂2�x �w¼ − EIabsEIfull

�M þ 1−EIabsEIfull

!∂�x

�Dsc

αLð Þ2 ð48Þ

where a new non-dimensional parameter (namely, the absent-to-fullinteraction bending stiffness ratio EIabs/EIfull) emerges with the trans-formation of Eq. (23). It is not only a dimensionless quantity, but alsoa normalized one (namely, it ranges between zero and the unity) asthe bending stiffness EIabs of two unconnected members is either neg-ligible or significant with respect to the bending stiffness EIfull in fullinteraction. The key role of this parameter has been already pointedout in [15] and [16] for models which can be classified in Group 1according to the definition given in Section 3.1. Next, Eq. (24) isturned into its dimensionless form by introducing the definition of


pz, Dsc and Vsc; and given by Eqs. (37), (39) and (40), respectively; thefollowing expression can be found:

�V sc ¼ −pz þhad

had

EIaEIabs

� �−1−1

� �1− EIabs

EIfull

!∂�x

�Dsc ð49Þ

where twomore dimensionless parameters (namely, the ratios ha/d andEIa/EIabs), both ranging between zero and unity, emerge as relevantquantities controlling the interaction between the two connected mem-bers. Finally, dimensionless Eqs. (47)–(49) clearly indicates that theme-chanical behaviour of models belonging to Group 1 is basically governedby four non-dimensional parameters (i.e., αL, EIabs/EIfull, ha/d and EIa/EIabs).

5.2. Constrained shear-flexible composite beam model (Group 2)

As shown in Section 4, assuming the same shear strain distribution(and a unique deflection distribution) for both connected membersresults in a unique cross-section rotation and this basically leads tothe same differential Eq. (21) relating the interface shear flow Dsc

(or the slip distribution dsc) and the total shear force V. Thus, thesame dimensionless Eq. (47) can be considered for Group 2 modelsand the parameter αL still represents a relevant quantity for predict-ing interface shear flow and slips. The dimensionless expression ofthe normal interface interaction force distribution Vsc can be obtainedby transforming Eq. (25) according to the procedure explained above:

�V sc ¼ − EIabsEIa

GAa

GA�pz ð50Þ

where the steel-profile-to-composite-section shear stiffness ratioGAa/GA emerges as a relevant parameter for Group 2 models, alongwith the EIa/EIabs ratio already derived for Group 1 models. It isworth noting that assuming a finite shear flexibility for both membersand a kinematical constraint on their shear strains (enforced to beequal for both members) leads to a simple algebraic relationship forthe determination Vsc according to Eq. (25) and its non-dimensionalcounterpart through Eq. (50). On the contrary, a slightly more com-plicated dimensionless equation (with respect to the correspondingone derived for Group 1 models) is obtained by introducing the rele-vant definitions given in Section 4 within the differential Eq. (26)which after some simple mathematical simplifications can be pre-sented in the following dimensionless form:

∂2�x �w ¼ − EIabsEIfull

�M þ 1−EIabsEIfull

!∂�x �Dsc

αLð Þ2 −EIfullGA L2

EIabsEIfull

�pz ð51Þ

Again, the difference between the kinematic assumptions adoptedfor each group of models manifests itself in the previous equation. In-deed, comparing the previous equation to Eq. (48), clearly indicatesthe presence of an extra shear stiffness related term along with twomore dimensionless parameters. The bending-to-shear stiffness ratioEIfull/GA L2 has been already pointed out in [19] as a key parameter in-volved in the analytical closed-form expression of the “exact” stiffnessmatrix which has been derived for Group 2 models.

Finally, five parameters (i.e. αL, EIabs/EIfull, EIa/EIabs, GAa/GA and EIfull/GA L2) control themechanical response of composite beams that belongto Group 2. Thus, removing the constraint of zero shear strain results inintroducing one extra parameter needed for describing the mechanicalbehaviour of steel–concrete composite beams in partial interaction.

5.3. Shear-flexible composite beam model (Group 3)

The equations outlined in Section 3 for class 3 models can beturned to the corresponding dimensionless form by proceeding asexplained in Sections 1 and 2. First of all, the dimensionless form of

Eq. (28) can be expressed, after some mathematical manipulations,as follows:

∂3�x �Dsc−ξ1 ∂�x�Dsc ¼ ξ2 �pz−ξ3 �V sc ð52Þ

with

ξ1 ¼ αLð Þ2 1þ1− EIabs

EIfull

!had

− EIaEIabs

� �2

EIaEIabs

EIabsEIfull

1−had

� �266664

377775 ð53Þ

ξ2 ¼ αLð Þ21−ha

d

1− EIaEIabs

ð54Þ

ξ3 ¼ αLð Þ2had

− EIaEIabs

� �1−ha

d

� �EIabsEIfull

1− EIaEIabs

� � ð55Þ

Although the above equation is much more complicated that thecorresponding one (see Eq. (47) derived for Group 1 and 2 models), itcan be seen that Eq. (52) involves the same dimensionless parametersalready defined in Sections 1 and 2.Moreover, Eq. (52) reduces to an ex-pression which is similar to Eq. (47) when the two dimensionless pa-rameters ha/d and EIa/EIabs are equal. This condition has been alreadypointed out in [21], though in slightly different manner. However, Eq.(52) looks much more complicated from an analytical point of view asa result of the more general kinematic assumptions adopted in Group3 models compared to the more simpler ones considered for Group 1and 2. The same consequence can be observed by analyzing the dimen-sionless form of Eq. (29) which after several mathematical transforma-tions can be written as follows:

ξ4∂�x�Dsc ¼ ξ5 �pz þ �V sc

� �−ξ6∂

2�x�V sc ð56Þ

with

ξ4 ¼had

− EIaEIabs

EIaEIabs

1−had

� � ð57Þ

ξ5 ¼ 1

1− EIabsEIfull

!1−ha

d

� � ð58Þ

ξ6 ¼EIa

GAaL2 1− EIa

EIabs

� �

1−GAa

GA

� �1− EIabs

EIfull

! ð59Þ

Once more, even though the differential equation in dimension-less form appears to be rather complex, it can be seen that the samenon-dimensional parameters are involved in this differential equa-tion. Furthermore Eqs. (52) and (56) are coupled if ha/d and EIa/EIabsare not equal. In addition, Eq. (30) which relates the second derivativeof the deflection w to the other mechanical variables and parameters


can be turned into its dimensionless form and written, after severalmathematical manipulations, as follows:

∂2�x �w ¼ − EIabsEIfull

�M þ 1− EIabsEIfull

!∂�x

�Dsc

αLð Þ2 −ξ7 1−ξ8ð Þ�V sc þ ξ9 �pz ð60Þwith

ξ7 ¼ EIfullGAL2

EIaEIabs

EIabsEIfull

GAa

GA

� �−1ð61Þ

ξ8 ¼ EIabsEIfull

1− EIaEIabs

� �þ ha

d1− EIabs

EIfull

!" #1−GAa

GA

� �−1ð62Þ

ξ9 ¼ ξ7ξ8GAa

GAEIaEIabs

� �−1ð63Þ

Finally, the six non-dimensional parameters (i.e. αL, EIabs/EIfull, EIa/EIabs, ha/d, GAa/GA and EIfull/GA L2) control the equations derived forGroup 3 models defined in Section 3.

5.4. Comments about the dimensionless parameters involved in the threegroups of models

The dimensionless form of the governing equations from which aclosed-form solution can be derived for each group of models pointsout to a clear hierarchy between these models. Such a hierarchy ismade even clearer in Table 1 where the expressions of the dimension-less parameters for each group of models are reported. For instance, itcan be seen that the models in Group 3 are based on six dimension-less parameters, whereas those in Group 2 involve only five of theseparameters and four are sufficient for models of Group 1. Moreover,the order of the governing differential Eqs. (28)–(30) for Group 3models is higher than that of Eqs. (21)–(26) related to Group 1 and2 models which happen to be of the same order.

6. Parametric study

A comprehensive parametric study is presented in this sectionwith the aim of investigating the role and the influence of the key di-mensionless parameters as introduced in Section 5. This parametricstudy is carried out by considering a simply supported beam of spanlength L. The dimensionless form of the equations reported inSection 5 for the three groups are analytically solved taking into ac-count appropriate boundary conditions.

6.1. Boundary conditions for Group 1 and 2 models

The equations associated with Group 1 models can be analyticallysolved. The starting point is Eq. (47) which relates the interface shearflow Dsc to the total transverse shear force V. The boundary conditionNa=0 (at both x=0 and x=L) must then be formulated in terms ofDsc. To do so, dsc is eliminated from Eq. (13) using the interface constitu-tive law Eq. (20). The outcome is differentiated once which results in

1ksc

∂xDsc ¼Na

EA� −κ d ð64Þ

where the kinematic relationships (Eqs. (7) and (11)) and the generalizedconstitutive laws (Eqs. (14) and (17))havebeenused. Since the curvature

Table 1Relevant dimensionless parameters for each group of models.

Dimensionless parameter αL EIabs/EIfull ha/d EIa/EIabs GAa/GA EIfull/GAL2

Group 1 × × × × – –

Group 2 × × – × × ×Group 3 × × × × × ×

is equal to zero at both ends of the simply supported beam, the two fol-lowing boundary conditions can be derived in terms of the first derivativeof Dsc:

Na x¼0;L ¼1ksc

∂xDsc x¼0;L þ d κ �x¼0;L ¼ 0↦∂xDsc x¼0;L ¼ 0�� ð65Þ

which, in non-dimensional form, becomes

∂�x�Dsc �x¼0;1 ¼ 0�� ð66Þ

The above condition is employed to solve Eq. (47) in terms of �Dsc.Then Eq. (43) can be utilized to obtain the non-dimensional slip distribu-tion �dsc. Furthermore, the dimensionless equation for �w is easily solvedby introducing the expression for �Dsc in Eq. (48) and integrating theresulting second order differential equation with the usual boundaryconditions for simply supported beams:

�w �x¼0;1 ¼ 0�� ð67Þ

Finally, the normal interaction force �V sc can be easily obtained by in-troducing, in Eq. (49), thefirst derivative of the aboveobtained expressionfor �Dsc. Since Group 2models are governed by a similar differential equa-tion, a similar procedure can be carried out for a deriving a analytical ex-pressions of all mechanical variables.

6.2. Boundary conditions for Group 3 models

The analytical solution for Group 3 models is a more involving tasksince the model is, in general, governed by the two differential Eqs.(52) and (56). In the most general case, the two differential equationscan be combined in a single fourth-order differential equation whichrelates �V sc to the external loading. The expression of that fourth-orderdifferential equation is omitted herein for the sake of concisenesswhich can easily be derived by simply inserting Eq. (56) in Eq. (52).However, now four boundary conditions are needed for integratingsuch an equation. The first two of these conditions are obtained byinserting the condition (66) in Eq. (56):

ξ5 �pz þ �V sc �x¼0;1

�� −ξ6 ∂2�x �V sc �x¼0;1 ¼ 0

��ð68Þ

The remaining couple of boundary conditions can be derived bydifferentiating the equation obtained by subtracting Eq. (11) fromEq. (8) and making use of the constitutive relations (Eqs. (15) and(18)) to eliminate the shear strain variables:

∂xVa

GAa−∂xVb

GAb¼ κa−κb ð69Þ

Next, the equilibrium Eqs. (2) and (5) are employed to eliminatethe derivatives of the shear forces Vi and replace them with Vsc:

Vsc

GAa− Vsc

GAaþ pzGAb

¼ κa−κb ð70Þ

Since both curvatures vanish at the beam ends, the above condi-tion yields the following boundary conditions:

Vsc x¼0;L ¼ −GAa

GApz

�� ð71Þ

which can be easily turned into a non-dimensional form by employingEqs. (37) and (40). Likewise, the following expression can be derivedfor further two boundary conditions in terms of

�V sc �x¼0;1 ¼ −GAa

GAEIabsEIa

�pz

�� ð72Þ

0,950

0,975

1,000

1,025

1,050

0,000 0,002 0,004 0,006 0,008 0,010

Max

imum

slip

rat

io -

d sc,

i/dsc

,1

EIfull/GAL2

Group 2 models

Group 3 models

EIabs/EIfull=0.40ha/d=0.20EIa/EIabs=0.75GAa/GA=0.75

ααααL=1.0

Fig. 3. Maximum slip ratio versus EIfull/GA L2 for low interaction (EIa/EIabs=0.75).

0,000 0,002 0,004 0,006 0,008 0,010

EIfull/GAL2

1,025

1,050EIabs/EIfull=0.40ha/d=0.20EIa/EIabs=0.75GAa/GA=0.75

ααααL=10.0

0,950

0,975

1,000

Group 2 models

Group 3 models

Max

imum

slip

rat

io -

d sc,

i/dsc

,1

Fig. 4. Maximum slip ratio versus EIfull/GA L2 for high interaction (EIa/EIabs=0.75).


Thus, Eqs. (68) and (72) represent the two sets of boundary con-ditions to be considered for integrating Eq. (56) in terms of �V sc. It isworth mentioning that these conditions involve a number of the di-mensionless parameters suggested in Section 4. Once the solution interms of �V sc has been found, it can be inserted in Eq. (55) resultingin a third-order differential equation for �Dsc, which again can be inte-grated by considering the two conditions given in Eq. (66) and a fur-ther symmetry condition on both interface slips and shear flow:

�Dsc �x¼0:5 ¼ 0j ð73Þ

Finally, Eq. (55) can be easily integrated after inserting the alreadyderived expressions for �Dsc and �V sc, together with the boundary con-ditions given in Eq. (67). Thus, the integration procedure of the equa-tions describing the so-called Group 3 models is, in general, morecomplicated than the one required for models of Groups 1 and 2 asa result of the more general kinematic assumptions involved. Theclosed-form expressions have been obtained using the symbolic soft-ware Mathematica. However, in the particular case of EIa/EIabs=ha/d,the Eqs. (52) and (56) become uncoupled and therefore can be solvedvery much as in the case of Group 1 and Group 2 models.

6.3. Definition of the parametric field

Since the dimensionless form of the equations representing thethree groups of models for composite beams are now available, acomplete parametric study can be carried out to investigate the influ-ence of the six non-dimensional parameters (Table 1) on the predic-tion of the overall structural behaviour of composite beams in partialinteraction considering the aforementioned three groups of models.Because four out of the six parameters are normalized (i.e., EIabs/EIfull,EIa/EIabs, ha/d and GAa/GA) they will range between zero and unity(the extreme values 0 and 1 being actually excluded from the numer-ical analysis as they represent singular cases). Moreover, only twovalues, 1.0 and 10.0, will be considered for the shear interaction pa-rameter α L whose influence has been already investigated in severalpapers [15,16,19]. As a matter of fact, these two values represent thetwo cases of almost absent and rather strong shear interactions. Finally,the variation of the bending-to-shear stiffness ratio EIfull/GAL2 will be ex-amined by allowing this parameter to vary between 0.001 and 0.01,which should represent awide and realistic range of possiblemechanicalparameters.

7. Discussion of the main results of the parametric analysis

In this section, the key results of the parametric are presented anddiscussed in this section which is mainly devoted to point out the roleof the various non-dimensional parameters.

7.1. Predictions in terms of maximum interface slips

A first set of calculations has been performed in order to comparethe predictions of the three groups of models in terms of themaximuminterface slip dsc. In what follows, �dsc;i represents themaximum value ofthe dimensionless slip obtained for the i-th group ofmodels. According-ly, the ratio �dsc;i=

�dsc;1 provides a measure to relate the outcomes of allmodels to the prediction of model 1. Obviously the ratios �dsc;i=

�dsc;1

and dsc, i/dsc, 1 are equal. Fig. 3 shows the ratio dsc, i/dsc, 1 as a function ofthe bending-to-shear stiffness ratio EIfull/GAL2 for the following givenvalues of parameters:

- EIabs/EIfull=0.4,- EIa/EIabs=0.75,- ha/d=0.20,- GAa/GA=0.75,

and the case of low shear interaction characterized by αL=1. TheFig. 3 confirms that Group 1 and Group 2 models lead to exactly thesame predictions in terms of maximum interface slip (dsc, 2/dsc, 1 is al-ways equal to unity). However, this figure indicates a rather unex-pected result for the ratio dsc, 3/dsc, 1 being lower than unity for thecase of low shear interaction. It is related to the fact that a highershear deformability of the Group 3 models (due to the more generalkinematic assumptions) reduces the part of interface slips which aredirectly related to cross-section rotations (see Eq. (13)) and thereforeto the bending stiffness. Nevertheless, the value of the ratio dsc, 3/dsc, 1reported in Fig. 3 are only slightly lower than unity. This ratio be-comes significantly lower than unity in the case of higher shear inter-action (αL=10) as indicated in Fig. 4. This points out to the fact thatthe higher the shear interaction parameter is, the more important therole of shear flexibility becomes and so the difference between Group1 and Group 3 models widens. A similar couple of results are reportedin Figs. 5 and 6, whose results have been obtained with a value of EIa/EIabs equal to 0.25. Whereas no significant differences can be observedwhen comparing the results for the case of low shear interaction (seeFigs. 3 and 5), these differences become more pronounced in the caseof high shear interaction. Indeed, the values of the ratio dsc, 3/dsc, 1reported in Fig. 6 are closer to unity than the corresponding onesdepicted in Fig. 5. This confirms that the predictions in terms of inter-face slips based on Group 3 models become closer to both Group 1and 2 models as the difference between the dimensionless parame-ters EIa/EIabs and ha/d diminishes. In fact, it is possible to demonstratethat in these cases Eq. (28) reduces to the first derivative of Eq. (21)as the factor of Vsc vanishes in the former.

0,000 0,002 0,004 0,006 0,008 0,010

EIfull/GAL2

1,025

1,050

0,950

0,975

1,000

Max

imum

slip

rat

io -

d sc,

i/dsc

,1


ααααL=1.0

Group 2 models

Group 3 models

Fig. 5. Maximum slip ratio versus EIfull/GA L2 for low interaction (EIa/EIabs=0.25).

0,00 0,20 0,40 0,60 0,80 1,00

EIfull/GAL2

1,025

1,050

0,950

0,975

1,000

Max

imum

slip

rat

io -

d sc,

3/d s

c,1

1,010,0

EIabs/EIfull=0.40ha/d=0.20EIfull/GAL2=0.005GAa/GA=0.75

ααααL

Fig. 7. Maximum slip ratio versus EIa/EIabs (ha/d=0.2).

0,00 0,20 0,40 0,60 0,80 1,00

EIfull/GAL2

1,025

1,050

0,950

0,975

1,000M

axim

um s

lip r

atio

-d s

c,3/

d sc,

1


1,0

10,0

ααααL

Fig. 8. Maximum slip ratio versus EIa/EIabs (ha/d=0.4).


Finally, this general feature is clearly indicated in Fig. 7 which de-picts the ratio dsc, 3/dsc, 1 against the bending stiffness ratio EIa/EIabs fora given value of ha/d=0.20: the two curves which refer to the cases ofαL=1.0 and αL=10.0 intersect at EIa/EIabs=ha/d=0.20 with a valueof unity. This demonstrates once more the fact that the maximum in-terface slips obtained with Group 3 models coincides with the corre-sponding one obtained with Group 2 (and Group 1) models in thecase EIa/EIabs=ha/d. Fig. 8 confirms once more this property for ha/d=0.40.

7.2. Predictions in terms of maximum deflections

A similar comparison between the models predictions is now car-ried out in terms of maximum deflection wmax. To this end, the ratiowmax, i/wmax, 1 is considered to better understand the difference betweenthe prediction of the i-th Group ofmodels and the corresponding predic-tion of Group 1 models (namely, the Newmark's Theory). Fig. 9 reportsthe values of this ratio for both Group 2 and 3 models. It considers thesame parameters already adopted for deriving Fig. 3 which addressedthe issue of the maximum interface slips. Since it considers the case oflow shear interaction (αL=1.0) the maximum deflections predicted byboth Group 2 and 3models are rather close to one another and thereforethe values of wmax, i/wmax, 1(i=2,3) are not significantly higher than theunity. This mean that, in the case of low shear interaction, bending stiff-ness basically controls the structural behaviour and, thus, theNewmark'stheory leads to a rather accurate prediction of themaximum deflections.In fact, Fig. 10 shows a significant increase in both the predictions derivedby applying Group 2 and 3models and their distance with respect to the

0,000 0,002 0,004 0,006 0,008 0,010

EIfull/GAL2

1,025

1,050

0,950

0,975

1,000

Max

imum

slip

rat

io -

d sc,

i/dsc

,1


ααααL=10.0

Group 2 models

Group 3 models

Fig. 6. Maximum slip ratio versus EIfull/GA L2 for high interaction (EIa/EIabs=0.25).

corresponding values possibly deriving by the Newmark's Theory con-sidered as a reference. Thus, the shear flexibility plays a more significantrole in affecting deflection values of beams with high shear interaction.Since the importance of shear flexibility is more pronounced in thosecases, also the difference possibly stemming out between models (suchas those in Group 2 and 3) which assume different hypotheses aboutstrains developing in the two connected members. Another importanteffect emerges when comparing Figs. 9 and 11 which both refer to thecase of low shear interaction (αL=1.0). In particular, significantly higher

0,000 0,002 0,004 0,006 0,008 0,010

EIfull/GAL2

1,15

1,20

1,00

1,05

1,10

Mid

span

def

lect

ion

rati

o -

wm

ax,i/

wm

ax,1 Group 3 models

Group 2 models


ααααL=1.0

Fig. 9. Mid-span deflection versus EIfull/GA L2 for low interaction (EIa/EIabs=0.75).

0,000 0,002 0,004 0,006 0,008 0,010

EIfull/GAL2

1,15

1,20

1,00

1,05

1,10

Mid

span

def

lect

ion

rati

o -

wm

ax,i/

wm

ax,1 Group 3 models

Group 2 models


ααααL=10.0

Fig. 10. Mid-span deflection versus EIfull/GA L2 for high interaction (EIa/EIabs=0.75).

0,000 0,002 0,004 0,006 0,008 0,010

EIfull/GAL2

1,15

1,20

1,00

1,05

1,10

Mid

span

def

lect

ion

rati

o -

wm

ax,i/

wm

ax,1 Group 3 models

Group 2 models


ααααL=1.0

Fig. 11. Mid-span deflection versus EIfull/GA L2 for low interaction (EIa/EIabs=0.25).

-0,05

0,00

0,05

0,10

0,15

0,20

1,010,0


ααααL0,00 0,20 0,40 0,60 0,80 1,00

EIa/EIabs

Rel

ativ

e ro

tati

on r

atio

-(θ

a,3-

θ b,3

)/θ 2

Fig. 13. Relative rotation ratio versus bending stiffness ratio (ha/d=0.20).


ααααL

1,010,0

-0,05

0,00

0,05

0,10

0,15

0,20

0,00 0,20 0,40 0,60 0,80 1,00EIa/EIabsR

elat

ive

rota

tion

rat

io -

(θa,

3-θ b

,3)/

θ 2

Fig. 14. Relative rotation ratio versus bending stiffness ratio (ha/d=0.40).


values ofwmax, i/wmax, 1(i=2,3) can be observed in the latter as a result ofa much lower value of the ratio EIa/EIabs=0.25 which is also rather closeto the ha/d ratio, kept constant at 0.20. As shown in Figs. 7 and 8, the pre-diction in terms of maximum interface slips deriving by Group 3 modelstend to be close to the one deriving by applying Group 2 ones. Thismeans that, in the same cases, Group 3 models are much more flexiblethan Group 2 in terms of deflections, as they have the same contributionby the interface slip, but the former is less constrained in terms of shearstrains which can develop in both connected members. The same trendcan be, finally, observed by comparing Figs. 10 and 12 both referring to

0,000 0,002 0,004 0,006 0,008 0,010

EIfull/GAL2

1,15

1,20

1,00

1,05

1,10

Mid

span

def

lect

ion

rati

o -

wm

ax,i/

wm

ax,1 Group 3 models

Group 2 models


ααααL=10.0

Fig. 12. Mid-span deflection versus EIfull/GA L2 for high interaction (EIa/EIabs=0.25).

high shear interaction (αL=10.0). In this case, as a result of the highervalue ofαL, thewmax, i/wmax, 1 ratio reachesmuchhigher values, suggest-ing the general idea that the higher the two parameters EIfull/GAL2 andαL, themore influential the shear flexibility of the two connectedmem-bers. Finally, since Group 3 models consider two different rotations forthe connected members, whereas Group 2 restraint them to the samerotation, the ratio between the end rotations θa, 3−θb, 3 obtained bythe former and the unique one theta2 derived by the latter is a good pa-rameter for evaluating how far are from each other the predictionsobtained by the two Groups of models. Figs. 13 and 14 report this rela-tive rotation ratio against the value of the bending stiffness ratio for thesame sets of the other parameters already considered in Figs. 7 and 8,respectively. The show that the ratio ((θa, 3−θb, 3)/θ2 is higher for highinteraction levels (i.e.,αL=10.0) and lower for high values of the EIa/

EIabs/EIfull=0.20ha/d=0.20EIfull/GAL2=0.005EIa/EIabs=0.75

1,010,0

ααααL

-1,00

-0,05

0,00

0,50

1,00

Rel

ativ

e ro

tati

on r

atio

-(θ

a,3-

θ b,3

)/θ 2

0,00 0,20 0,40 0,60 0,80 1,00GAa/GA

Fig. 15. Relative rotation ratio versus shear stiffness ratio (ha/d=0.20).

-1,00

-0,05

0,00

0,50

1,00R

elat

ive

rota

tion

rat

io -

(θa,

3-θ b

,3)/

θ 2

0,00 0,20 0,40 0,60 0,80 1,00GAa/GA

1,010,0

EIabs/EIfull=0.40ha/d=0.40EIfull/GAL2=0.005EIa/EIabs=0.75

ααααL

Fig. 16. Relative rotation ratio versus shear stiffness ratio (ha/d=0.40).


EI ratios. The influence of the ratio ha/d is rather negligible. However,the relative rotation ratio ismuchmore influenced by the shear stiffnessratio GAa/GA. Actually, Figs. 15 and 16 clearly demonstrate that the dif-ference in terms of rotations (and, consequently, the difference be-tween simulations based on Group 2 and Group 3 model) tends tovanish for value of the shear stiffness ratio close to one half (basicallyin the case of similar shear stiffnesses for both connected members).On the contrary, the relative rotation ratio rises sharply as GAa/GA ap-proaches either zero or the unit and, in other words, the (θa, 3−θb, 3)/θ2ratio increases as the two connected members are characterized by sig-nificant differences in terms of shear stiffness.

8. Conclusions

In this paper, we summarized the kinematic assumptions generallyconsidered in formulating analyticalmodels for steel–concrete compositebeams in partial interaction with and without inclusion of the shear flex-ibility of the connected members. First of all, a general classification inthree groups of the various computational models currently available inthe literature has been proposed. This group classification is based onthe kinematical assumptions related to shear deformability. Group1 con-tains shear-rigidmodels, Group 2 encompassmodels with a unique sheardeformation while Group 3 deals with models allowing for independentshear deformation of each layer. Next the governing equations of modelgroups have been cast in dimensionless form suggesting structural pa-rameters which govern the behaviour. A variable number of dimension-less quantities (actually ranging between four and six) are needed fordescribing the structural response according to the three groups ofmodels. A comprehensive assessment of the roles played by these param-eters has been provided. This is based on a closed-form solution of thegoverning equations for each group considering a simply supported com-posite beam in partial interaction. The results of this parametric analysishave been finally reported in terms of both maximum interface slip anddeflections. General trends of the model predictions were identified foreach group.

Further investigations can be easily carried out by solving the di-mensionless equations for the values of the dimensionless parametersunder consideration in possible practical applications. Those solutions,

completely described in this paper, can drive the choice of themost con-venient model, i.e. the one leading to a reasonably good approximationof the structural response with the minimum computational effort(namely, by employing the simplest equations, or those involving theminimal set of relevant dimensionless parameters). Finally, it is worthnoting that the dimensionless equations derived herein can only be ap-plied in the linear-elastic range.

References

[1] Johnson RP. Composite structures of steel and concrete: beams, slabs, columns,and frames for buildings3rd Edition. Wiley-Blackwell; September 2004.

[2] EN 1994-1-1 Eurocode 4 - Design of composite steel and concrete structures - Part1–1: General rules and rules for buildings; 2004.

[3] Newmark MN, Siess CP, Viest IM. Tests and analysis of composite beams withincomplete interaction. Proc Soc Exp Stress Anal 1951;9(1):75–92.

[4] Adekola AO. Partial interaction between elastically connected elements of a compositebeam. Int J Solids Struct 1968;4:1125–35.

[5] Yam LCP, Chapman JC. The inelastic behaviour of simply supported compositebeams of steel and concrete. Proc Inst Civil Eng 1968;41(1):651–83.

[6] Aribert JM, Labib AG.Modèle de calcul élasto-plastiquede poutresmixtes à connectionpartielle. Construc Métall 1985;3:3–51 (in French).

[7] Dall'Asta A, Zona A. Non-linear analysis of composite beams by a displacementapproach. Comput Struct 2002;80:2217–28.

[8] Salari MR, Spacone E. Analysis of steel–concrete composite frames with bond-slip.J Struct Eng (ASCE) 2001;127(11):1243–50.

[9] Ayoub A, Filippou FC. Mixed formulation of nonlinear steel–concrete compositebeam element. J Struct Eng (ASCE) 2000;126(3):371–81.

[10] Dall'Asta A, Zona A. Three-field mixed formulation for the non-linear analysis of com-posite beams with deformable shear connection. Finite Elem Anal Des 2004;40(4):425–48.

[11] Dezi L, Tarantino AM. Creep in composite continuous beams. I: theoretical treatment.J Struct Eng (ASCE) 1993;119(7):2095–111.

[12] Amadio C, FragiacomoM. A finite element model for the study of creep and shrinkageeffects in composite beams with deformable shear connections. Costruzioni Metall1993;4:213–28.

[13] Ranzi G, Bradford MA. Analytical solutions for elevated temperature behaviour ofcomposite beams with partial interaction. J Struct Eng (ASCE) 2007;133(6):788–99.

[14] Abel-Aziz K, Aribert JM. Calcul des poutres mixtes jusqu'à l'état ultime avec un effetde soulèvement à l'interface acier-béton. Construct Métall 1985;4:3–36 (in French).

[15] Faella C, Martinelli E, Nigro E. Steel and concrete composite beams with flexible shearconnection: “exact” analytical expression of the stiffness matrix and applications.Comput Struct 2002;80(11):1001–9.

[16] Faella C,Martinelli E, Nigro E. Steel–concrete composite beams in partial interaction:closed-form “exact” expression of the stiffness matrix and the vector of equivalentnodal forces. Eng Struct 2010;32(9):2744–54.

[17] Ranzi G, Bradford MA, Uy B. A direct stiffness analysis of a composite beam withpartial interaction. Int J Numer Methods Eng 2004;61:657–72.

[18] Xu R, Wu Y. Static, dynamic, and buckling analysis of partial interaction compositemembers using Timoshenko's beam theory. Int J Mech Sci 2007;49:1139–55.

[19] Martinelli E, Faella C, Di Palma G. Shear-Flexible Steel–concrete composite beamsin partial interaction: closed form “Exact” expression of the stiffness matrix. J EngMech (ASCE) 2012;138(2):151–64.

[20] Schnabl S, Saje M, Turk G, Planinc I. Analytical solution of two-layer beam taking intoaccount interlayer slip and shear deformation. J Struct Eng (ASCE) 2007;133(6):886–94.

[21] NguyenQH,Martinelli E, HjiajM.Derivation of the exact stiffnessmatrix for a two-layerTimoshenko beam element with partial interaction. Eng Struct 2011;33(2):298–307.

[22] Nguyen QH, Hjiaj M, Uy B. Time-dependent analysis of composite beams withpartial interaction based on a time-discrete exact stiffness matrix. Eng Struct2010;32(9):2902–11.

[23] Nguyen QH, Hjiaj M, Uy B, Guezouli S. Analysis of composite beams in the hoggingmoment regions using a mixed finite element formulation. J Constr Steel Res2009;65(3):737–48.

[24] Battini JM, Nguyen QH, Hjiaj M. Non-linear finite element analysis of compositebeams with interlayer slips. Comput Struct 2009;87(13–14):904–12.

[25] Faella C, Martinelli E, Nigro E. Analysis of steel-concrete composite PR-frames inpartial shear interaction: A numerical model and some applications. Eng Struct2008;30(4):1178–86.

dimensionless formulation and comparative study of analytical models for composite beams in partial...

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