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

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 composite beams in partial interaction Enzo Martinelli a, Quang Huy Nguyen b, Mohammed Hjiaj b,⁎ a b

Department of Civil Engineering, University of Salerno, Via Ponte don Melillo - 84084 Fisciano (SA), Italy Structural Engineering Research Group/LGCGM, INSA de Rennes, 20 avenue des Buttes de Coësmes, CS 70839–35708 Rennes Cedex 7, France

a r t i c l e

i n f o

Article history: Received 30 September 2011 Accepted 29 February 2012 Available online 11 April 2012 Keywords: Composite beams Partial interaction Shear deformability Model classification Parametric study Dimensionless formulation

a b s t r a c t Steel–concrete composite beams are widely utilized as cost-effective structural solution in both buildings and bridges. Partial interaction through the possible occurrence of slips at the interface between the two connected members, strongly affects the behaviour of composite members and, therefore must be incorporated in theoretical models dealing with composite members. In addition, shear deformability of the two connected layers cannot be ignored in stocky members. Several computational models simulating the behaviour of composite beams including partial interaction and shear deformability with various degree of sophistication are currently available in the scientific literature. The present paper focuses on the background and the mechanical assumptions adopted in these models as well as structural characteristics which actually govern their predictions. Based on the kinematic assumptions involved, a threefold classification is proposed. The paper further clarifies the hierarchy between the three groups of models. To do so, the governing equations for each group of models are transformed into a proper dimensionless form by using mechanically sound dimensionless expressions of all functions of interest involved in the description of the mechanical response of the composite beam. A thorough parametrical study is presented which quantifies the influence of the identified dimensionless parameters. Furthermore, the study clearly indicates possible threshold values beyond which certain effects become negligible. © 2012 Elsevier Ltd. All rights reserved.

1. Introduction Steel–concrete composite beams are widely used in both buildings and bridges as a structurally-efficient and cost-effective solution [1]. However, the analysis of steel–concrete composite beams is rather challenging from a computational point of view since the mechanical behaviour of the coupled system is complex and characterized by the occurrence of slips at the interface. Such slip results in partial interaction even in cases of full shear connection, as defined by Eurocode 4 [2] for design purposes at the Ultimate Limit States. Several theoretical models, characterized by different levels of approximation, have been proposed to simulate the structural response of elastic composite structures in partial interaction [3,4]. Such models have been also the basis for approximate solutions derived using either the finite difference method [5,6] or the finite element method, the latter being the driving force behind various advanced formulations such as the direct stiffness/flexibility method [7], force-based FE models [8]and mixed formulations [9,10,23]. These formulations have been used to investigate the time response as well as the inelastic response of composite beams with interlayer slips.

⁎ Corresponding author. Tel.: + 33 223238711; fax: + 33 223238448. E-mail address: [email protected] (M. Hjiaj). 0143-974X/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.jcsr.2012.02.016

Albeit the specific objectives of the various contributions (i.e., the long-term behaviour of composite steel–concrete beams [11,12,22], the response under fire conditions [13], geometrically nonlinear analysis [24], analysis of frames [25] etc…) as well as the particular solution methods implemented therefore, common assumptions of the above formulations can be easily recognized. Indeed, the transverse displacement of both members are generally considered equal (no uplift) all along the beam. In fact, several studies have demonstrated that the overall structural response is weakly influenced by considering possible relative displacements between the concrete slab and the steel beam [14]. Therefore considering independent deflection for each layer unnecessarily complicates the model equations as we are dealing with highly nonlinear relations which govern unilateral contact conditions with or without friction. Consequently, a unique function describing the transverse displacement of both components is generally assumed for the sake of simplicity and without significant loss of accuracy [15,16,17]. Based on the kinematical assumptions underlying the different theoretical models, the formulations that have been proposed for planar steel–concrete composite beams in partial interaction can be basically classified in the following three groups: 1. Shear-rigid composite beam models (Group 1): these are models where shear deformability is neglected for both connected layers each of which is modeled according to the well-known Bernoulli Beam Theory [3];

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2. Constrained shear-flexible composite beam models (Group 2): these are models that assume a single shear strain distribution for both connected layers and follow the well-known Timoshenko Beam Theory ( plane sections remain plane after the deformation) [18]; 3. Shear-flexible composite beam models (Group 3): these are models that introduce independent shear strains for each layer, both modeled according to the Timoshenko Theory [20]. For the models belonging to the first group, the closed-form expressions of both the flexibility and the stiffness matrix [16] are readily available. An “exact” stiffness matrix has been also derived for a model belonging to the second group [19]. A general procedure for deriving the stiffness matrix for a beam model belonging to the third group has been also recently reported [21]. The present paper intends to provide a comparative study to assess the predictions of the existing models for steel–concrete composite beams in partial interaction as classified above. Obviously models of the last group are the most sophisticated and so are expected to provide the most accurate solutions. However, this sophistication comes at a price and it is not clear from the outset that it is needed in the first place. Hence, it is desirable to understand the limits and the accuracy of models belonging to Group 1 and Group 2 as this would give a guideline to a proper selection of the model under certain given circumstances of loading, dimensions, material parameters etc. … The paper will further clarify the hierarchy between the aforementioned groups of models. To do so, the governing equations for each group of models are transformed into a proper dimensionless form by using mechanically sound dimensionless expressions of all functions of interest involved in the description of the mechanical response of the composite beam. A parametric study is carried out by considering a simply supported beam of span length L. The influence of key dimensionless parameters on the prediction of the overall structural behaviour is investigated. The rest of the paper is organized as follows. Section 2 summarizes the equilibrium and kinematic equations as well as the relevant stress–strain relationships. The main kinematic assumptions are formally introduced in Section 3 and the corresponding governing equations are derived for each group of models. Key parameters which describe the structural response of composite beams are identified. A set of dimensionless parameters is defined in Section 4 and employed in Section 5 for transforming the governing equations in a more appropriate dimensionless form. In Section 6, an extensive parametric study is conducted based on the previously identified dimensionless parameters which actually control the predictions of the analytical models. The key outcomes of this study are then reported in Section 7. Based on the results, the effect of shear deformability on steel–concrete composite beams in partial interaction, which can be modeled either with a single or two independent shear strains, is properly quantified. In addition, consistent threshold values of the mentioned non-dimensional parameters are identified which result in practically negligible effects of the shear flexibility according to the two possible approaches corresponding to the models categorized in either Group 2 or 3.

∽ uplift is neglected so the transverse displacement of both the concrete slab 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 beam subjected to a distributed transverse loading pz is shown in Fig. 1. For the element to be at equilibrium, the following equations must be satisfied: • For layer a: ∂x Na −Dsc ¼ 0

ð1Þ

∂x V a −V sc ¼ 0

ð2Þ

∂x M a −V a þ ha Dsc ¼ 0

ð3Þ

• For layer b: ∂x Nb þ Dsc ¼ 0

ð4Þ

∂x V b þ V sc þ pz ¼ 0

ð5Þ

∂x M b −V b þ hb Dsc ¼ 0

ð6Þ

where - ∂ xi • = d i •/dx i; – hi (i = a, b) is the distance between the centroid of the layer “i” and the layers interface; – Ni, Vi, Mi (i = a, b) are the axial forces, shear forces and bending moments in layer “i”; - Dsc is the shear bond force per unit length; - Vsc represents the transverse component of the interface force distribution. 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 displacement assumption and the Timoshenko beam theory. For each layer, these equations read: • For layer a: ε a ¼ ∂x ua

ð7Þ

γ a ¼ ∂x w þ θa

ð8Þ

2. Fundamental equations The governing equations describing the geometrically linear behaviour of an elastic shear-deformable steel–concrete composite beam in partial interaction are briefly outlined in this section. All variables subscripted with b belong to the concrete slab section and those with subscript a belong to the steel beam. Quantities with subscript sc are associated with the shear connectors. The following assumptions are commonly accepted in all models to be discussed in this paper: ∽ both connected members are made out of elastic, homogenous and isotropic materials; ∽ the transverse sections of both steel beam and concrete slabs remain plane after deformation, though relative slips can develop along their interface;

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

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

23

be simply denoted as EAi, EIi and GAi (i = a, b), respectively. The above relations must be completed by the relationship between the shear bond force Dsc and the interlayer slip dsc. The assumption of linear and continuous shear connection can be expressed by the simple following relationship between interface slips and shear flow: Dsc ¼ ksc dsc

ð20Þ

where ksc is the shear bond stiffness. Full composite action (infinite slip stiffness) and non-composite action (zero slip stiffness) represent upper and lower bounds for the partial composite action. 3. Model classification Fig. 2. Kinematics of a shear-deformable two-layer beam with interlayer slip.

ε b ¼ ∂x ub

ð10Þ

Several models have been proposed in the past years to simulate key features of the structural response of composite beams in partial interaction. These models are based on the field equations presented in the previous section. Nevertheless, depending on the way shear deformability of the connected members is dealt with, these models could 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).

γ b ¼ ∂x w þ θb

ð11Þ

3.1. Group 1: shear-rigid composite beam models

κ b ¼ ∂x θb

ð12Þ

κ a ¼ ∂x θa

ð9Þ

• For layer b:

where - εi and ui are the axial strain and the longitudinal displacement at the 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 the interlayer slip in terms of the displacements of the layers:

This group of models completely ignores the shear deformability of both connected members and is devoted to slender composite beam members. Accordingly, the two members behave according to the Bernoulli's beam theory where the shear strain is equal to zero (γi = 0) for both layers. Consequently, a direct relationship connects both cross-section rotations θi to the first derivative of the deflection w. Since the latter is the same for both layers, the two members exhibit the same rotation θa = θb = θ and, based on Eqs. (8) and (9), a unique curvature κa = κb = κ. Thus, a second order differential equation, relating the interface shear force to the total shear force, can be derived [15]: 2

dsc ¼ ua −ub −ha θa −hb θb

ð13Þ

2.3. Constitutive relationships We adopt a linear stress–strain relationship at the material level and deduce the following constitutive law for the cross-section of each layer:

2

N a ¼ E a Aa ε a

ð14Þ

V a ¼ ka Ga Aa γ a

ð15Þ

M a ¼ Ea I a κ a

ð16Þ

• Layer b: N b ¼ E b Ab ε b

ð17Þ

V b ¼ kb Gb Ab γ b

ð18Þ

M b ¼ Eb I b κ b

ð19Þ

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

ksc d V EIabs

ð21Þ

where d = ha + hb is the distance between the centroids of the crosssections of the slab and the profile, EIabs = EIa + EIb represents the bending stiffness of the beam with no shear interaction and V = Va + Vb is the total shear force. Moreover, α is a key mechanical parameter defined as follows: α ¼

• Layer a:

2

∂x Dsc −α Dsc ¼ −

ksc EIfull EA EIabs

ð22Þ

where EIf ull ¼ EIabs þ EA d2 is the bending stiffness of the cross-section  EAa EAb of the composite beam as a whole and EA ¼ EA is the relative axial a þEAb stiffness of the composite section. Eq. (21) can easily be solved in the case of statically determinate structures where the expression of the total shear force V can be determined using static equilibrium equations. In addition to the above differential equation, it is straightforward to derive two other differential equations which relate the longitudinal shear force distribution by solving Eq. (21) to the transverse deflection w: 2

∂x w ¼



EA d ∂x Dsc −ksc M ksc EIfull

ð23Þ

and to the normal stress distribution at the interface Vsc:
 V sc pz ha hb þ – ∂ D  ¼− EI EIb EI a EIb x sc

ð24Þ

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EI b where M is the total bending moment distribution and EI ¼ EIEIaþEI . Due a b to length constraints, the complete derivation of these equations is omitted herein.

3.2. Group 2: constrained shear-flexible composite beam models A second group of theoretical models can be defined by enhancing the above group of models through the adoption of a shear-deformable beam theory where a unique shear deformation is attributed to both layers. It assumes that both connected members shear deform according to the Timoshenko's theory, but they have a unique shear strain γa = γb = γ. Under the strictly mechanical standpoint, this means that the two connected members are subjected to a shear force Vi which is proportional to their shear stiffness. Moreover, in view of the kinematic Eqs. (8) and (9), the unique shear strain γ assumption results in an identical cross-section rotation θi = θ = γ − ∂ xw for both layers 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, applies to this group of models as well. In view of Eqs. (15) and (18), the assumption of single shear strain γ provides a relationship between Va and Vb which can be combined with Eqs. (2) and (6) to obtain the following simple algebraic relationship for the normal interface stresses Vsc: V sc p ¼− z GAa GA

ð25Þ

where GA = GAa + GAb is the shear stiffness of the section in full interaction. Finally, a relationship between the second derivative of the deflection and other relevant mechanical variables can be derived (the algebraic manipulations are omitted for conciseness): 2

∂x w ¼

V sc EA d ∂x Dsc −ksc M þ ksc EIfull GAa

ð26Þ

V −∂x w GA

2

∂x w ¼


 V sc EA d ∂x Dsc −ksc M EIb þ EA hb d V sc pz þ − þ ksc EIfull EIfull GAa GA GAb

ð27Þ

ð30Þ

Furthermore, subtracting Eq. (11) from Eq. (8) and replacing in the outcome each shear strain γi with the corresponding shear force Vi (using Eqs. (15) and (18)) yields the following equation for the difference between the cross-section rotation of steel girder and concrete slab: θa −θb ¼

Va V − b GAa GAb

ð31Þ

The above equation can be further transformed by substituting the expressions of Va and Vb which can be derived by solving equilibrium Eqs. (3) and (6) for the variables Vi, respectively:
θa −θb ¼

ha h − b GAa GAb

 Dsc þ

∂x Ma ∂x M b – GAa GAb

ð32Þ

Then, bending moments Ma and Mb can be expressed in terms of the curvatures κa and κb; since curvatures can be easily related to the first derivative of rotations through kinematic Eqs. (9) and (12), the latter can be also expressed in terms of the shear strains and deflections again through Eqs. (8) and (11). After a few tedious but straightforward algebraic manipulations, the following relationship which expresses the difference in cross-section rotations (the relative rotation of both members) in terms of the variables Vsc, Dsc and w (and their derivatives) can be easily obtained:


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 former one. Finally, rotations θ = θa = θb can be easily expressed in terms of the total shear force and the first derivative or the deflection function as follows: θ ¼ γ−∂x w ¼

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 pure bending, the transverse displacement can be obtained by integrating the following differential equation:

θa −θb ¼

 ha h − b Dsc þ GAa GAb

!
 EI a EIb EIa EI 3 − − b ∂x w ð33Þ ∂x V sc − 2 2 GAa GAb GAa GAb

where the equilibrium relationships (2) and (5) have been also considered in order to replace the derivatives of shear forces Va and Va with those of Vsc. Consequently, the relative rotation of both members can be determined once the distributions Dsc, Vsc, and w have been derived. 4. Definition of a relevant set of dimensionless quantities

3.3. Group 3: shear-flexible composite beam models A third group of models can be defined by simply considering the three general assumptions listed at the beginning of Section 2 and modeling both connected members according to the Timoshenko's beam theory. In other words, this group of models accounts for shear deformability of the composite member by considering that each layer shear-deforms independently. Since we have two shear strain distributions γa and γb, it results in two different crosssection rotations. The analytical formulation of this model has been completely carried out in [21] and, the following governing equations have been derived: 
 
 h 2 k h ha h 3 2  ha ∂x Dsc ¼ sc b pz −ksc ∂x Dsc − α þ ksc EI − b − b V sc ð28Þ EI a EI b EI b EI a EI b




 2 ha h p V ∂ V − b ∂x Dsc ¼ z þ sc − x sc EIa EI b EIb EI GA

ð29Þ

where GA ¼ GAa GAb =ðGAa þ GAb Þ is the relative shear stiffness parameter. The above equations can be combined in a single fifth

The equations, given in Section 3 for the three group of models, highlights a clear hierarchy within these groups with increasing complexity as the kinematic assumptions become more general. This complexity can be “measured” in terms of the order of the differential equations governing these three groups of models, the level of coupling of these differential equations in defining the key response parameters 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 becomes more and more complex as one starts from Eq. (23) for Group 1 models, to Eq. (26) for Group 2 and, finally, Eq. (30) for Group 3. In particular, a new term appears at the right hand of these equations, as one moves from a given group to the next one. A certain number of geometrical and mechanical parameters are formally involved in Eqs. (21)–(30) and, thus, the influence of those quantities cannot be evaluated by simply examining these equations. However, a clear understanding of the role actually played by these parameters can be achieved by turning Eqs. (21)–(30) into a proper dimensionless form by using mechanically sound dimensionless expressions of all functions of interest involved in the description of the mechanical response of the composite beam, (i.e., among the others, the interface

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

slip dsc, the shear flow Dsc,the normal interface stress distribution Vsc, the deflection w). Thus, a dimensionless form of the same expressions is derived with the aim of pointing out the key non-dimensional parameters controlling the response of the structural model. First of all, a normalized abscissa x∈½0; 1 can is defined as follows: x ¼ L x

M¼

EIfull  M d

ð35Þ

V¼

EI full  V dL

ð36Þ

EIfull

pz ¼

d L2

p z

ð37Þ

where p z is just the above mentioned dimensionless transversal load  and V are the dimensionless functions repredistribution, whereas M senting bending moment and shear force distribution, respectively. They have been defined by considering that the maximum bending moment is in the order of magnitude of the quantity pzL2, and the flexural strength is proportional to the ratio EIfull/d (see Eq. (35)). Following the same approach, the interface slip can be turned into non-dimensional form as follows: dsc ¼

EIfull d L3  EIfull  d sc ¼ L d sc EIabs d L2 EIabs

ð38Þ

since the maximum interface slip in the case of absent interaction is proportional (according to both theories of Bernoulli and Newmark) to the ratio pzL3/EIabs and the transverse load is proportional to the factor already reported in Eq. (37). Moreover, again adopting a similar approach (though omitting further comments for the sake of conciseness), the following transformations will be considered in the present paper for the set of relevant functions involved in Eqs. (21)–(30): 

Dsc ¼

EA  D sc L

ð39Þ

V sc ¼

EIfull EIb  V EI abs d L2 sc

ð40Þ

2

w¼

Finally, the i-th derivative ∂ xiQ of the generic dimensional quantity  can be easily expressed in terms of its dimensionless form Q ¼ Q ref Q i  ∂x Q as follows: i

∂x Q ¼

Q ref Li

ð34Þ

In principle, all other quantities can be defined as the product of a factor (which is aimed at reproducing the expected order of magnitude of the quantity under consideration) by a dimensionless function (which will always be denoted with the same symbol of the dimensional one with a superimposed bar). For instance, Eq. (34) has been actually introduced by following this principle. However, it is worth underlining that the transformation of the relevant functions involved in Eqs. (21)–(30) into the corresponding dimensionless ones is not unique and should be carried out with a deep physical understanding of the mechanical meaning of the various parameters involved. For instance, in this study bending moment M, shear force V and transverse load pz are expressed by considering the following expressions:

EI full L  w EIabs d

EIfull L  θi ¼ θ EIabs d i

ð41Þ ð42Þ

Moreover, the following relationship between the functions representing the dimensionless shear flow and interface slip can be easily recognized by introducing Eqs. (38) and (39) in Eq. (20):  ¼ ðαLÞ2 d D sc sc

ð43Þ

25

i  ∂x Q

ð44Þ

For instance, the i-th derivative of the interface shear flow Dsc can be expressed as follows in terms of the corresponding dimensionless parameters: i

∂x Dsc ¼



EA i  ∂x D sc Liþ1

ð45Þ

5. Dimensionless form of the models governing equations The main differential Eqs. (21)–(30) which represent the key features of the three model groups defined in Section 3 can be easily turned into a dimensionless form by just inserting into the governing Eqs. (21)–(30) the definitions (34)–(44) of the various functions involved therein. Thus, the three following subsections report the dimensionless versions of Eqs. (21)–(30) and point out the key parameters which actually 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 interface shear flow Dsc (and its second derivative) and the shear force V can be turned into a dimensionless form by replacing these mechanical variables 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 algebraic transformations, based on the following identity [15],  2

EI full ¼ EIabs þ EA d

ð46Þ

the dimensionless form of Eq. (21) can be expressed as follows: 2 2 2 ∂x D sc −ðαLÞ D sc ¼ −ðαLÞ V

ð47Þ

where αL is the first dimensionless parameter suggested by the nondimensional transformation. It has been already considered in other papers (see [15,19]]) as interaction parameters ranging from zero (in the case of no shear interaction between the two connected members) to infinity (in the ideal case of full interaction). A similar procedure can be followed in order to transform Eq. (23) into an adequate dimensionless form by introducing the definition for the bending moment M and the deflection w given in Eqs. (35) and (41), respectively, and using the general rule (44) for calculating the derivatives of nondimensional functions. The following dimensionless expression can be obtained after some mathematical transformations: 2  ∂x w¼

EI  þ − abs M EI full

!  EIabs ∂x D sc 1− EI full ðαLÞ2

ð48Þ

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

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

pz, Dsc and Vsc; and given by Eqs. (37), (39) and (40), respectively; the following expression can be found: !  
 ha ha EIa −1 EI abs   −1 1− ∂ D V sc ¼ −pz þ d d EI abs EIfull x sc

ð49Þ

where two more dimensionless parameters (namely, the ratios ha/d and EIa/EIabs), both ranging between zero and unity, emerge as relevant quantities controlling the interaction between the two connected members. Finally, dimensionless Eqs. (47)–(49) clearly indicates that the mechanical behaviour of models belonging to Group 1 is basically governed by four non-dimensional parameters (i.e., αL, EIabs/EIfull, ha/d and EIa/ EIabs). 5.2. Constrained shear-flexible composite beam model (Group 2)

ð50Þ

2

!  EIfull EIabs EIabs  EI ∂ D M þ 1− abs x sc2 − p z EIfull EI full ðαLÞ GA L2 EIfull

ð51Þ

Again, the difference between the kinematic assumptions adopted for each group of models manifests itself in the previous equation. Indeed, comparing the previous equation to Eq. (48), clearly indicates the presence of an extra shear stiffness related term along with two more dimensionless parameters. The bending-to-shear stiffness ratio EIfull/GA L2 has been already pointed out in [19] as a key parameter involved in the analytical closed-form expression of the “exact” stiffness matrix which has been derived for Group 2 models. Finally, five parameters (i.e. αL, EIabs/EIfull, EIa/EIabs, GAa/GA and EIfull/ GA L 2) control the mechanical response of composite beams that belong to Group 2. Thus, removing the constraint of zero shear strain results in introducing one extra parameter needed for describing the mechanical behaviour 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 be turned to the corresponding dimensionless form by proceeding as explained in Sections 1 and 2. First of all, the dimensionless form of

ð52Þ

with 2 6 26 ξ1 ¼ ðαLÞ 6 61 þ 4

2

where the steel-profile-to-composite-section shear stiffness ratio GAa/GA emerges as a relevant parameter for Group 2 models, along with the EIa/EIabs ratio already derived for Group 1 models. It is worth noting that assuming a finite shear flexibility for both members and a kinematical constraint on their shear strains (enforced to be equal for both members) leads to a simple algebraic relationship for the determination Vsc according to Eq. (25) and its non-dimensional counterpart through Eq. (50). On the contrary, a slightly more complicated dimensionless equation (with respect to the corresponding one derived for Group 1 models) is obtained by introducing the relevant definitions given in Section 4 within the differential Eq. (26) which after some simple mathematical simplifications can be presented in the following dimensionless form:  ¼− ∂x w

3    ∂x D sc −ξ1 ∂x D sc ¼ ξ2 p z −ξ3 V sc

ξ2 ¼ ðαLÞ

As shown in Section 4, assuming the same shear strain distribution (and a unique deflection distribution) for both connected members results in a unique cross-section rotation and this basically leads to the same differential Eq. (21) relating the interface shear flow Dsc (or the slip distribution dsc) and the total shear force V. Thus, the same dimensionless Eq. (47) can be considered for Group 2 models and the parameter αL still represents a relevant quantity for predicting interface shear flow and slips. The dimensionless expression of the normal interface interaction force distribution Vsc can be obtained by transforming Eq. (25) according to the procedure explained above: EI GA V sc ¼ − abs a p z EIa GA

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

EI 1− abs EIfull EIa EIabs

!


ha EI − a d EIabs
 EI abs h 1− a EIfull d

ha d EIa 1− EIabs 1−

2 3 7 7 7 7 5

ð53Þ

ð54Þ



ha EI h − a 1− a d EIabs d 2
 ξ3 ¼ ðαLÞ EIabs EIa 1− EI full EIabs

ð55Þ

Although the above equation is much more complicated that the corresponding one (see Eq. (47) derived for Group 1 and 2 models), it can be seen that Eq. (52) involves the same dimensionless parameters already defined in Sections 1 and 2. Moreover, Eq. (52) reduces to an expression which is similar to Eq. (47) when the two dimensionless parameters ha/d and EIa/EIabs are equal. This condition has been already pointed out in [21], though in slightly different manner. However, Eq. (52) looks much more complicated from an analytical point of view as a result of the more general kinematic assumptions adopted in Group 3 models compared to the more simpler ones considered for Group 1 and 2. The same consequence can be observed by analyzing the dimensionless form of Eq. (29) which after several mathematical transformations can be written as follows:    ¼ ξ p þ V −ξ ∂2 V ξ4 ∂x D sc 5 z sc 6 x sc

ð56Þ

with ha EI − a d EI abs
 ξ4 ¼ EIa h 1− a EIabs d ξ5 ¼ 1−

EIabs EIfull

1 !
1−

ð57Þ

ha d




 EI a EI 1− a 2 EIabs GAa L ! ξ6 ¼
 GAa EI abs 1− 1− GA EIfull

ð58Þ

ð59Þ

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

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

can be turned into its dimensionless form and written, after several mathematical manipulations, as follows: !  EI 2  þ 1− EI abs ∂x D sc −ξ ð1−ξ ÞV þ ξ p  ¼ − abs M ∂x w ð60Þ 7 8 sc 9 z EI full EIfull ðαLÞ2 with
 EIfull EI a EI abs GAa −1 GAL2 EIabs EIfull GA " !#

 EIabs EIa ha EIabs GA −1 1− 1− þ ξ8 ¼ 1− a EI full EIabs d EI full GA ξ7 ¼

ξ9 ¼ ξ7 ξ8

ð61Þ

ð62Þ


 GAa EIa −1 GA EIabs

ð63Þ

Finally, the six non-dimensional parameters (i.e. αL, EIabs/EIfull, EIa/ EIabs, ha/d, GAa/GA and EIfull/GA L 2) control the equations derived for Group 3 models defined in Section 3. 5.4. Comments about the dimensionless parameters involved in the three groups of models The dimensionless form of the governing equations from which a closed-form solution can be derived for each group of models points out to a clear hierarchy between these models. Such a hierarchy is made even clearer in Table 1 where the expressions of the dimensionless parameters for each group of models are reported. For instance, it can be seen that the models in Group 3 are based on six dimensionless parameters, whereas those in Group 2 involve only five of these parameters and four are sufficient for models of Group 1. Moreover, the order of the governing differential Eqs. (28)–(30) for Group 3 models is higher than that of Eqs. (21)–(26) related to Group 1 and 2 models which happen to be of the same order. 6. Parametric study A comprehensive parametric study is presented in this section with the aim of investigating the role and the influence of the key dimensionless parameters as introduced in Section 5. This parametric study is carried out by considering a simply supported beam of span length L. The dimensionless form of the equations reported in Section 5 for the three groups are analytically solved taking into account appropriate boundary conditions. 6.1. Boundary conditions for Group 1 and 2 models The equations associated with Group 1 models can be analytically solved. The starting point is Eq. (47) which relates the interface shear flow Dsc to the total transverse shear force V. The boundary condition Na = 0 (at both x = 0 and x = L) must then be formulated in terms of Dsc. To do so, dsc is eliminated from Eq. (13) using the interface constitutive law Eq. (20). The outcome is differentiated once which results in 1 N ∂ D ¼ a −κ d ksc x sc EA

ð64Þ

where the kinematic relationships (Eqs. (7) and (11)) and the generalized constitutive laws (Eqs. (14) and (17)) have been used. Since the curvature Table 1 Relevant 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

× × ×

× × ×

× – ×

× × ×

– × ×

– × ×

27

is equal to zero at both ends of the simply supported beam, the two following boundary conditions can be derived in terms of the first derivative of Dsc:      1 Na x¼0;L ¼ ∂D  þ d κ x¼0;L ¼ 0↦∂x Dsc x¼0;L ¼ 0 ksc x sc x¼0;L

ð65Þ

which, in non-dimensional form, becomes    ∂x D sc x ¼0;1 ¼ 0

ð66Þ

 sc . The above condition is employed to solve Eq. (47) in terms of D Then Eq. (43) can be utilized to obtain the non-dimensional slip distribu is easily solved tion d sc . Furthermore, the dimensionless equation for w  sc in Eq. (48) and integrating the by introducing the expression for D resulting second order differential equation with the usual boundary conditions for simply supported beams:   x¼0;1 ¼ 0 w

ð67Þ

Finally, the normal interaction force V sc can be easily obtained by introducing, in Eq. (49), the first derivative of the above obtained expression  sc . Since Group 2 models are governed by a similar differential equafor D tion, a similar procedure can be carried out for a deriving a analytical expressions of all mechanical variables. 6.2. Boundary conditions for Group 3 models The analytical solution for Group 3 models is a more involving task since the model is, in general, governed by the two differential Eqs. (52) and (56). In the most general case, the two differential equations can be combined in a single fourth-order differential equation which relates V sc to the external loading. The expression of that fourth-order differential equation is omitted herein for the sake of conciseness which can easily be derived by simply inserting Eq. (56) in Eq. (52). However, now four boundary conditions are needed for integrating such an equation. The first two of these conditions are obtained by inserting the condition (66) in Eq. (56):

  2 ξ5 p z þ V sc x¼0;1 −ξ6 ∂x V sc x¼0;1 ¼ 0

ð68Þ

The remaining couple of boundary conditions can be derived by differentiating the equation obtained by subtracting Eq. (11) from Eq. (8) and making use of the constitutive relations (Eqs. (15) and (18)) to eliminate the shear strain variables: ∂x V a ∂x V b − ¼ κ a −κ b GAa GAb

ð69Þ

Next, the equilibrium Eqs. (2) and (5) are employed to eliminate the derivatives of the shear forces Vi and replace them with Vsc: V sc V p − sc þ z ¼ κ a −κ b GAa GAa GAb

ð70Þ

Since both curvatures vanish at the beam ends, the above condition yields the following boundary conditions:   GA V sc x¼0;L ¼ − a pz ð71Þ GA which can be easily turned into a non-dimensional form by employing Eqs. (37) and (40). Likewise, the following expression can be derived for further two boundary conditions in terms of   GA EI V sc x¼0;1 ¼ − a abs p z GA EI a

ð72Þ

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

 j D sc x ¼0:5 ¼ 0

ð73Þ

Finally, Eq. (55) can be easily integrated after inserting the already  sc and V sc , together with the boundary conderived expressions for D ditions given in Eq. (67). Thus, the integration procedure of the equations describing the so-called Group 3 models is, in general, more complicated than the one required for models of Groups 1 and 2 as a result of the more general kinematic assumptions involved. The closed-form expressions have been obtained using the symbolic software Mathematica. However, in the particular case of EIa/EIabs = ha/d, the Eqs. (52) and (56) become uncoupled and therefore can be solved very 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 the three groups of models for composite beams are now available, a complete parametric study can be carried out to investigate the influence of the six non-dimensional parameters (Table 1) on the prediction of the overall structural behaviour of composite beams in partial interaction 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 numerical analysis as they represent singular cases). Moreover, only two values, 1.0 and 10.0, will be considered for the shear interaction parameter α L whose influence has been already investigated in several papers [15,16,19]. As a matter of fact, these two values represent the two cases of almost absent and rather strong shear interactions. Finally, the variation of the bending-to-shear stiffness ratio EIfull/GAL2 will be examined by allowing this parameter to vary between 0.001 and 0.01, which should represent a wide and realistic range of possible mechanical parameters. 7. Discussion of the main results of the parametric analysis In this section, the key results of the parametric are presented and discussed in this section which is mainly devoted to point out the role of 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 compare the predictions of the three groups of models in terms of the maximum interface slip dsc. In what follows, d sc;i represents the maximum value of the dimensionless slip obtained for the i-th group of models. Accordingly, the ratio d sc;i =d sc;1 provides a measure to relate the outcomes of all models to the prediction of model 1. Obviously the ratios d sc;i =d sc;1 and dsc, i/dsc, 1 are equal. Fig. 3 shows the ratio dsc, i/dsc, 1 as a function of the bending-to-shear stiffness ratio EIfull/GAL2 for the following given values of parameters: - EIabs/EIfull = 0.4, - EIa/EIabs = 0.75, - ha/d = 0.20, - GAa/GA = 0.75,

1,050

Maximum slip ratio - dsc,i/dsc,1

Thus, Eqs. (68) and (72) represent the two sets of boundary conditions to be considered for integrating Eq. (56) in terms of V sc . It is worth mentioning that these conditions involve a number of the dimensionless parameters suggested in Section 4. Once the solution in terms of V sc has been found, it can be inserted in Eq. (55) resulting  sc , which again can be intein a third-order differential equation for D grated by considering the two conditions given in Eq. (66) and a further symmetry condition on both interface slips and shear flow:

1,025

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

αL=1.0

1,000

0,975 Group 2 models Group 3 models

0,950 0,000

0,002

0,004

0,006

0,008

0,010

EIfull/GAL2 Fig. 3. Maximum slip ratio versus EIfull/GA L2 for low interaction (EIa/EIabs = 0.75).

and the case of low shear interaction characterized by αL = 1. The Fig. 3 confirms that Group 1 and Group 2 models lead to exactly the same predictions in terms of maximum interface slip (dsc, 2/dsc, 1 is always equal to unity). However, this figure indicates a rather unexpected result for the ratio dsc, 3/dsc, 1 being lower than unity for the case of low shear interaction. It is related to the fact that a higher shear deformability of the Group 3 models (due to the more general kinematic assumptions) reduces the part of interface slips which are directly related to cross-section rotations (see Eq. (13)) and therefore to the bending stiffness. Nevertheless, the value of the ratio dsc, 3/dsc, 1 reported in Fig. 3 are only slightly lower than unity. This ratio becomes significantly lower than unity in the case of higher shear interaction (αL = 10) as indicated in Fig. 4. This points out to the fact that the higher the shear interaction parameter is, the more important the role of shear flexibility becomes and so the difference between Group 1 and Group 3 models widens. A similar couple of results are reported in 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 observed when comparing the results for the case of low shear interaction (see Figs. 3 and 5), these differences become more pronounced in the case of high shear interaction. Indeed, the values of the ratio dsc, 3/dsc, 1 reported in Fig. 6 are closer to unity than the corresponding ones depicted in Fig. 5. This confirms that the predictions in terms of interface slips based on Group 3 models become closer to both Group 1 and 2 models as the difference between the dimensionless parameters EIa/EIabs and ha/d diminishes. In fact, it is possible to demonstrate that in these cases Eq. (28) reduces to the first derivative of Eq. (21) as the factor of Vsc vanishes in the former. 1,050

Maximum slip ratio - dsc,i/dsc,1

28

1,025

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

αL=10.0

1,000

0,975 Group 2 models Group 3 models

0,950 0,000

0,002

0,004

0,006

0,008

0,010

EIfull/GAL2 Fig. 4. Maximum slip ratio versus EIfull/GA L2 for high interaction (EIa/EIabs = 0.75).

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

1,025

EIabs/EIfull=0.40 ha/d=0.20 EIa/EIabs=0.25 GAa/GA=0.75

1,050

αL=1.0

Maximum slip ratio - dsc,3/dsc,1

Maximum slip ratio - dsc,i/dsc,1

1,050

1,000

0,975 Group 2 models

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

1,025

1,000

0,975 αL 1,0

Group 3 models

0,950 0,000

0,002

0,004

0,006

0,008

0,950 0,00

0,010

10,0

0,20

A similar comparison between the models predictions is now carried out in terms of maximum deflection wmax. To this end, the ratio wmax, i/wmax, 1 is considered to better understand the difference between the prediction of the i-th Group of models and the corresponding prediction of Group 1 models (namely, the Newmark's Theory). Fig. 9 reports the values of this ratio for both Group 2 and 3 models. It considers the same parameters already adopted for deriving Fig. 3 which addressed the issue of the maximum interface slips. Since it considers the case of low shear interaction (αL =1.0) the maximum deflections predicted by both Group 2 and 3 models are rather close to one another and therefore the values of wmax, i/wmax, 1(i =2,3) are not significantly higher than the unity. This mean that, in the case of low shear interaction, bending stiffness basically controls the structural behaviour and, thus, the Newmark's theory leads to a rather accurate prediction of the maximum deflections. In fact, Fig. 10 shows a significant increase in both the predictions derived by applying Group 2 and 3 models and their distance with respect to the

Maximum slip ratio - dsc,i/dsc,1

EIabs/EIfull=0.40 ha/d=0.20 EIa/EIabs=0.25 GAa/GA=0.75

αL=10.0

1,000

0,975 Group 2 models Group 3 models

0,002

0,004

0,006

0,008

0,010

EIfull/GAL2 Fig. 6. Maximum slip ratio versus EIfull/GA L2 for high interaction (EIa/EIabs = 0.25).

Maximum slip ratio - dsc,3/dsc,1

7.2. Predictions in terms of maximum deflections

1,050

0,80

1,00

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

1,025

1,000

0,975

0,950 0,00

αL 1,0 10,0

0,20

0,40

0,60

0,80

1,00

EIfull/GAL2 Fig. 8. Maximum slip ratio versus EIa/EIabs (ha/d = 0.4).

corresponding values possibly deriving by the Newmark's Theory considered as a reference. Thus, the shear flexibility plays a more significant role in affecting deflection values of beams with high shear interaction. Since the importance of shear flexibility is more pronounced in those cases, also the difference possibly stemming out between models (such as those in Group 2 and 3) which assume different hypotheses about strains developing in the two connected members. Another important effect emerges when comparing Figs. 9 and 11 which both refer to the case of low shear interaction (αL = 1.0). In particular, significantly higher 1,20 Midspan deflection ratio - wmax,i/wmax,1

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

0,950 0,000

0,60

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

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

1,025

0,40

EIfull/GAL2

EIfull/GAL2

1,050

29

1,15

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

Group 3 models Group 2 models

1,10

1,05

1,00 0,000

αL=1.0

0,002

0,004

0,006

0,008

0,010

EIfull/GAL2 Fig. 9. Mid-span deflection versus EIfull/GA L2 for low interaction (EIa/EIabs = 0.75).

30

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

1,15

0,20

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

Group 2 models

1,10

1,05

1,00 0,000

0,002

0,004

0,006

0,008

Group 2 models

1,05

0,002

0,004

0,006

0,008

values of wmax, i/wmax, 1(i =2,3) can be observed in the latter as a result of a much lower value of the ratio EIa/EIabs =0.25 which is also rather close to the ha/d ratio, kept constant at 0.20. As shown in Figs. 7 and 8, the prediction in terms of maximum interface slips deriving by Group 3 models tend to be close to the one deriving by applying Group 2 ones. This means that, in the same cases, Group 3 models are much more flexible than Group 2 in terms of deflections, as they have the same contribution by the interface slip, but the former is less constrained in terms of shear strains which can develop in both connected members. The same trend can be, finally, observed by comparing Figs. 10 and 12 both referring to

Midspan deflection ratio - wmax,i/wmax,1

0,80

1,00

0,15

0,10

0,05

0,00 0,00

αL 1,0 10,0

0,20

0,40

0,60

0,80

EIa/EIabs 1,00

Group 3 models

1,05

αL=10.0

0,002

0,60

high shear interaction (αL =10.0). In this case, as a result of the higher value of αL, the wmax, i/wmax, 1 ratio reaches much higher values, suggesting the general idea that the higher the two parameters EIfull/GAL2 and αL, the more influential the shear flexibility of the two connected members. Finally, since Group 3 models consider two different rotations for the connected members, whereas Group 2 restraint them to the same rotation, the ratio between the end rotations θa, 3 − θb, 3 obtained by the former and the unique one theta2 derived by the latter is a good parameter for evaluating how far are from each other the predictions obtained by the two Groups of models. Figs. 13 and 14 report this relative rotation ratio against the value of the bending stiffness ratio for the same 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 high interaction levels (i.e.,αL = 10.0) and lower for high values of the EIa/

1,00

Group 2 models

1,10

1,00 0,000

0,40

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

0,010

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

EIabs/EIfull=0.40 ha/d=0.20 EIa/EIabs=0.25 GAa/GA=0.75

EIa/EIabs αL 0,20 1,0 10,0

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

-0,05

αL=1.0

EIfull/GAL2

1,15

0,00 0,00

0,20 Relative rotation ratio - (θa,3-θb,3)/θ2

Group 3 models

1,10

1,20

0,05

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

0,004

0,006

0,008

0,010

EIfull/GAL2 Fig. 12. Mid-span deflection versus EIfull/GA L2 for high interaction (EIa/EIabs = 0.25).

Relative rotation ratio - (θa,3-θb,3)/θ2

Midspan deflection ratio - wmax,i/wmax,1

EIabs/EIfull=0.40 ha/d=0.20 EIa/EIabs=0.25 GAa/GA=0.75

1,00 0,000

0,10

-0,05

0,010

2

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

1,15

0,15

αL=10.0

EIfull/GAL

1,20

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

Group 3 models

Relative rotation ratio - (θa,3-θb,3)/θ2

Midspan deflection ratio - wmax,i/wmax,1

1,20

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

0,50

0,00 0,00

0,20

0,40

0,80 GAa/GA 1,00

0,60

-0,05

-1,00

αL 1,0 10,0

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

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

Relative rotation ratio - (θa,3-θb,3)/θ2

1,00

completely described in this paper, can drive the choice of the most convenient model, i.e. the one leading to a reasonably good approximation of the structural response with the minimum computational effort (namely, by employing the simplest equations, or those involving the minimal set of relevant dimensionless parameters). Finally, it is worth noting that the dimensionless equations derived herein can only be applied in the linear-elastic range.

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

0,50

0,00 0,00

0,20

31

0,40

0,80 GAa/GA 1,00

0,60

References -0,05

αL 1,0 10,0

-1,00 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 is much more influenced by the shear stiffness ratio GAa/GA. Actually, Figs. 15 and 16 clearly demonstrate that the difference in terms of rotations (and, consequently, the difference between simulations based on Group 2 and Group 3 model) tends to vanish for value of the shear stiffness ratio close to one half (basically in the case of similar shear stiffnesses for both connected members). On the contrary, the relative rotation ratio rises sharply as GAa/GA approaches either zero or the unit and, in other words, the (θa, 3 − θb, 3)/θ2 ratio increases as the two connected members are characterized by significant differences in terms of shear stiffness. 8. Conclusions In this paper, we summarized the kinematic assumptions generally considered in formulating analytical models for steel–concrete composite beams in partial interaction with and without inclusion of the shear flexibility of the connected members. First of all, a general classification in three groups of the various computational models currently available in the literature has been proposed. This group classification is based on the kinematical assumptions related to shear deformability. Group1 contains shear-rigid models, Group 2 encompass models with a unique shear deformation while Group 3 deals with models allowing for independent shear deformation of each layer. Next the governing equations of model groups have been cast in dimensionless form suggesting structural parameters which govern the behaviour. A variable number of dimensionless quantities (actually ranging between four and six) are needed for describing the structural response according to the three groups of models. A comprehensive assessment of the roles played by these parameters has been provided. This is based on a closed-form solution of the governing equations for each group considering a simply supported composite beam in partial interaction. The results of this parametric analysis have been finally reported in terms of both maximum interface slip and deflections. General trends of the model predictions were identified for each group. Further investigations can be easily carried out by solving the dimensionless equations for the values of the dimensionless parameters under consideration in possible practical applications. Those solutions,

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