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Analysis of steel–concrete composite PR-frames in partial shear interaction: A numerical model and some applications
Engineering Structures 30 (2008) 1178–1186 www.elsevier.com/locate/engstruct
Analysis of steel–concrete composite PR-frames in partial shear interaction: A numerical model and some applications C. Faella a , E. Martinelli a,∗ , E. Nigro b a Department of Civil Engineering, University of Salerno, Via Ponte don Melillo – 84084 Fisciano (SA), Italy b Department of Structural Engineering, University of Naples “Federico II”, Via Claudio, 21 – 80125 Naples, Italy
Received 24 January 2007; received in revised form 1 June 2007; accepted 18 June 2007 Available online 21 August 2007
Abstract The combined effect of beam-to-column joint semi-rigidity and partial interaction in steel–concrete composite beams is considered in the present paper in order to quantify the relative importance of these two phenomena both affecting the global behavior of composite frames. A finite element model formulated by the authors in a previous paper is able explicitly to simulate partial interaction in composite beams and is extended herein with the aim also of analyzing framed structures, even those with semi-rigid beam-to-column joints otherwise called Partially-Restrained frames (PR-frames in the following). Final applications are devoted to point out the relative importance of partial interaction effect with respect to the well-recognized role played by joint semi-rigidity. Assessment of code provisions and alternative proposals are finally exposed for taking account of the effect of the former phenomenon on structural behavior. c 2007 Elsevier Ltd. All rights reserved.
Keywords: Composite frames; Partial interaction; Joint semi-rigidity; Finite element; Code provisions
1. Introduction Partial interaction between steel girder and concrete slab due to flexible shear connectors deeply affects the behavior of both steel–concrete composite beams and frames. The analytical solution of the bending problem of composite beams in partial interaction has been independently found by the three different researchers mentioned in [11]. Numerical methods have been proposed since the early ’70s for solving the equations of composite beams in partial interaction; the authors made in [7] a short review of the different contributions in this field, reporting proposals based on both finite difference and finite element methods. However, the same paper was basically focused on the formulation of a particular finite element which explicitly accounts for the effects of beam-to-slab relative displacements utilizing the “exact” solution of the well-known Newmark’s linear theory for composite beams in bending [13]. The same element has already been adopted for composite frame structures [8] and its quite straightforward evolution for ∗ Corresponding author. Tel.: +39 089 964098; fax: +39 089 964098.
E-mail address: [email protected] (E. Martinelli). c 2007 Elsevier Ltd. All rights reserved. 0141-0296/$ - see front matter doi:10.1016/j.engstruct.2007.06.006
mechanical and geometrical non-linearity has also been shown in [9]. Beam-to-column joint semi-rigidity plays a significant role in the global behavior of both bare steel and composite steel–concrete structures. Several models have been formulated and presented in the scientific literature for simulating joint behavior according to the well-known “Components Method”. A simplified formulation of this method has been adopted in [3,4] for bare steel joints whose main components are completely described in terms of stiffness, strength and ductility; the same formulation has also been generalized for composite joints in [1] and [5], introducing a further component for simulating concrete slab influence on the rotational behavior of composite joints. Hence, the moment–rotation relationship can be defined in terms of initial stiffness, maximum strength and rotation capacity according to the provisions of the mentioned codes of standard. In the present paper a numerical model is proposed with the aim of simulating the global behavior of composite structures, taking into account both beam-to-column joint semirigidity and partial interaction in composite beams. The explicit simulation of partial interaction represents the key feature of the
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model; indeed, composite frame analysis is usually carried out considering only the effect of joint semi-rigidity, but neglecting shear connection flexibility in composite beams [12]. A first parametric analysis accounting for partial shear interaction, beam-to-column joint semi-rigidity and concrete cracking, has been presented in [8] where each composite beam has been subdivided into three finite elements such as the one presented herein, with the aim of taking account of cracking in an RC slab that possibly arises throughout the hogging moment regions: the mechanical properties of the exterior regions have been determined by neglecting concrete contribution, as usual in the well known “cracked-analysis” procedure allowed by Eurocode 4 and conducted by means of the present finite element for simply reproducing also the partial interaction effect. The results of these analyses pointed out a remarkable increase in the global flexibility of composite frames when partial interaction of composite beams, beam-to-column semirigidity and concrete slab cracking are taken into account at the same time. The effect of beam-to-column joint semi-rigidity and partial shear interaction is even more deeply investigated in the present paper considering their combined influence on two important parameters that characterize the global structural behavior. Two such parameters are the fundamental elastic period T and the so-called stability coefficient γ whose influence has been already studied in [6] with reference to the case of bare steel structures. Both are related strictly to the transverse stiffness of framed structures and, consequently, are affected by partial interaction and joint semi-rigidity; a more restrictive lower bound for joint stiffness with respect to the one adopted for bare steel structure to be considered as Fully Restrained (FR) is also proposed with the aim of accounting implicitly for global flexibility increase due to partial interaction. Finally, simplified methods are available in the scientific literature [2] and code of standards [5] for considering shear connection flexibility as a further component (or a term affecting the traditional ones) of the beam-to-column joint. Therefore, an application is proposed specifically in the present paper to compare the results obtained by means of the proposed model (which accounts explicitly for interface slips) or by considering shear connection flexibility as a further component of beam-to-column joint according to EC4 provisions. 2. Key features of the present model The numerical model utilized in the next section has been already formulated by the authors in [8] and [9] and only its key characteristics will be outlined in the present paper for the sake of brevity. It is based on the so-called “exact” finite element, formulated in [7] according to the classical Newmark Theory of partial shear interaction for steel–concrete composite beams in bending. In particular, according to that approach the flexibility matrix Dr which relates the nodal forces Qr = (Mi , Fi , M j , F j )T and the corresponding nodal displacements δr = (φi , si , φ j , s j )T has been derived in closedform (Fig. 1). The following relationship, typical for the forcebased formulation of the finite element method, can be stated:
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Fig. 1. Nodal forces and displacements for simply supported composite beam element (formally force-based element).
δr = Dr Qr + δr,0
(1)
δr,0 being the vector of nodal displacements due to external actions. In particular, the flexibility matrix Dr involved in Eq. (1) has been obtained by solving the differential equation deriving by the Newmark model, as shown in [7] under suitable boundary conditions; the 4 × 4 matrix form is reported in Eq. (2) pointing out the mechanical meaning of the various terms which correspond to rotation ϕ or slip s, obtained in one of the two element nodes (either i or j) due to a unit moment M or interface force F applied on the node i or j: ϕi,Mi ϕi,Fi ϕi,M j ϕi,F j si,Mi si,Fi si,M j si,F j Dr 1 Dr 2 (2) Dr = = ϕ j,Mi ϕ j,Fi ϕ j,M j ϕ j,F j . Dr 3 Dr 4 s j,Mi s j,Fi s j,M j s j,F j In particular, Eq. (2) states that the k-th column of the flexibility matrix Dr collects the nodal displacement components (sorted as in vector δr ) due to a unit force value given to the k-th component of the nodal force vector Qr , while the other nodal forces are zero. Similarly, the end displacements (namely, rotations ϕ and slips s) collected in vector δr,0 directly depend upon the action distributed on the beam. For example, in [7] they have been expressed in terms of the uniformly distributed load q and shrinkage strain εsh affecting the concrete slab according to the following symbolic form: φoi,q φoi,sh soi,q soi,sh δ 0,r = δ 0,r,q + δ 0,r,sh = + . (3) φoj,q φoj,sh soj,q soj,sh Beam-to-column joint flexibility has to be considered for PR-frames and two springs per node can be introduced in order to model the elastic nodal restraints against nodal rotations and slips induced by joint semi-rigidity (Fig. 2). Consequently, the flexibility matrix Dr0 can be modified as follows: Dr0 = Dr + Dε
(4)
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Fig. 2. Nodal springs adopted for reproducing beam-to-column joint semirigidity.
where matrix Dε collects the flexibility parameters εφ,i , εs,i , εφ, j and εs, j as diagonal terms. The flexibility matrix Dr0 defined by Eq. (4) can also be utilized to obtain the “exact” finite element in the usual displacement formulation which involves the stiffness matrix K6 relating displacements δ 6 = (vi , φi , si , v j , φ j , s j )T and forces Q6 = (Ti , Mi , Fi , T j , M j , F j )T for an unrestrained composite beam (see Fig. 3). The mentioned stiffness matrix K6 and vector Q6 of nodal forces equivalent to the external loads can be obtained by the corresponding flexibility matrix Dr0 and the vector of nodal displacements δr,0 due to the external actions. Indeed, a reduced 4 × 4 stiffness matrix Kr can be obtained by inverting Eq. (1) (without distributed loads and, hence, δr,0 = 0) in which (4) the flexibility matrix Dr has been substituted by Dr0 with the aim of looking after the nodal semirigidity as well: δr,0 = 0 ⇒ Qr =
[Dr0 ]−1 δr
= Kr δr .
(5)
Moreover, the vector of equivalent nodal forces with reference to the four nodal degrees of freedom represented in Fig. 1 can be obtained by forcing nodal displacement δr to zero: δr = 0 ⇒ Qr,0 = −[Dr0 ]−1 δr,0 = −Kr δr,0 .
(6)
Consequently, a formally displacement-based relationship can be obtained for the simply supported beam by superposing the effect of nodal displacements (5) and external actions (6): Qr = Kr δr + Qr,0 .
(7)
The following relationship can be stated between the displacement vector δr , which deals with the simply-supported beam (Fig. 1), and the vector δ 6 defined for the unrestrained beam (Fig. 3a): δ r = P1 δ 6 , P1 being a transformation matrix defined as follows 1/L 1 0 −1/L 0 0 0 0 1 0 0 0 P1 = 1/L 0 0 −1/L 1 0 . 0 0 0 0 0 1
(8)
(9)
The corresponding vector of nodal forces Q6 can be easily obtained by imposing the equivalence of the external work expressed either in terms of nodal force Qr and displacement δr
(a) Element only suitable for beam analysis.
(b) Generalized composite element for frame analysis. Fig. 3. Nodal forces and displacements for unrestrained composite beam element (displacement-based element).
or in terms of the vector Q6 and δ 6 ; the following relationship holds between the force vectors as Eq. (8) applies between the displacement vectors: Q6 = PT1 Qr .
(10)
The stiffness matrix K6 which relates Q6 and δ 6 can be obtained as a function of Kr by introducing Eq. (8) in (7) and utilizing Eq. (10): K6 = PT1 Kr P1 .
(11)
Furthermore, the corresponding six-component vector of equivalent nodal forces Q6,0 can even be obtained through the same transformation matrix P1 . Addition of the terms which correspond to shear forces is also needed for the global equilibrium of the unrestrained beam in the transverse direction; in the case of uniformly distributed load q, the following expression can be assumed: T Q6,0 = PT1 Qr,0 − q L/2 0 0 q L/2 0 0 . (12) The composite beam element formulated above takes account of both partial interaction and beam-to-column joint semirigidity, but is not yet fit for composite frame analysis because it only deals with bending as usual and is sufficient for continuous beams. Since frame analysis is the aim of the present work, the finite element has to be provided with the missing degree of freedom related to the absolute axial displacements. Two alternative possibilities could be undertaken in doing that. On one hand two absolute displacements could be considered per each node with the aim of reproducing the values of axial displacements of girder and slab: in this case, an 8DOF finite element should be formulated and a joint
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at Eqs. (10) and (11):
element should be needed for connecting that 8-DOF composite beam element with the more traditional 6-DOF finite elements utilized for columns. On the other hand, only one effective axial displacement of the beam could be considered for each node by condensing interface slips within the element and leading once more to a 6DOF composite beam element with the absolute axial displacement u instead of the relative slip s at each node (Fig. 3b). This solution, which can even reproduce the behaviour of real composite frames where no continuity is usually guaranteed between slips of two beams connected to the same column, can be directly derived by the above model as presented in the following. Therefore, according to Eq. (4) nodal slips are considered as a sum of the end-slip of the beam (governed by the flexibility matrix Dr ) and a further contribution basically due to slab reinforcement deformations close to the joint and controlled by the terms εs,i and εs, j collected in Dε . Consequently, even if zero value is imposed on slips at the nodal level, beam–slab interface displacements could occur at the end of the composite beam depending on the flexibility terms εs,i and εs, j : they would vanish as such flexibility terms are placed to zero because no relative displacements can occur if springs become rigid bodies (Fig. 4). The transformation matrix P2 can be introduced for relating the displacement vector δ 6 (Fig. 3a) with the vector δ = (u i , vi , φi , u j , v j , φ j )T which collects the displacement components reported in Fig. 3b: (13)
with the following definition: P P2 = 2,b 0
0 P2,b
and
P2,b
(15)
K = PT2 K6 P2 Q0 = PT2 Q6,0 .
(16)
E Aa + E Ac . (18) L It is useful to precise that the symbol E Ac deals with the axial stiffness of the RC slab; consequently, it can be generally assumed according to the following relationship: [K]1,1 = [K]4,4 = −[K]4,1 = −[K]1,4 =
E Ac = E c Ac + E s As ,
1 0 0
(19)
where E c and Ac are Young’s modulus and section area of concrete slab, respectively, while E s and As refer to the corresponding properties of reinforcing bars. The concrete contribution at the second member of Eq. (19) can be neglected if the effect of concrete cracking is of concern. A similar definition can be assumed for the RC slab flexural stiffness E Ic which takes account of both concrete and rebar axial stiffness. The mechanical properties E Aa and E Ia of the steel girder can be obtained simply as a product of the Young’s modulus E a of structural steel and the transverse section Aa or the moment of inertia Ia . The above definitions easily apply within the framework of a linear elastic (either uncracked or cracked according to EC4 [5]) analysis. Nevertheless, a fully non-linear procedure based on the model formulated above has also been implemented by the authors [9], in which the stiffness parameters E Ac , E Aa , E Ic and E Ia can be determined depending on the actual stress state of the element taking account of various phenomena characterizing the behaviour of both concrete (cracking and possibly tension stiffening) and structural steel (plasticity, strain hardening and possibly local stability). Finally, the relationship between nodal forces Q = (Ni , Ti , Mi , Ni , T j , M j )T and displacements δ can be expressed according to the following usual relationship: Q = Kδ + Q0 .
0 = 0 0
(17)
Consequently, the stiffness matrix needs to be completed with the axial terms. Since no interface slips occur if axial forces are distributed between slab and beam on the basis of their axial stiffness, the terms related to axial stiffness can be approximately assumed as reported below:
Fig. 4. Resulting beam-to-column joint model.
δ 6 = P2 δ,
Q = PT2 Q6
(20)
0 1 . 0
(14)
However, axial displacement components u i and u j are not present in δ 6 , which has been introduced with exclusive reference to the bending problem of composite beams as approached in the Newmark model. Hence, no possible correspondence exists between u i and u j and any δ 6 component; moreover, the terms of the force vector Q, the stiffness matrix K and the equivalent nodal force vector Q0 related to the distributed loads can be defined through the transformation matrix P2 in the same way as follows to arrive
The proposed finite element inherits all the advantages derived from the “exact” solution of Newmark’s equation already utilized for the analysis of continuous beams, but can also be utilized for composite (plane) frame structures characterized by semi-rigid beam-to-column joints whose flexibility is considered by means of the terms εφ,i and εφ, j . 3. Applications Two applications will be discussed in this section for pointing out the combined role of partial shear interaction and beam-to-column joint semi-rigidity: the first one deals with the basic cell of a composite frame represented in Fig. 5a,
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k
is the shear connection stiffness. The following values will be assumed in the analyses:
– εϕ,i = εϕ, j = 1/K ϕ ; – εs,i = εs, j = 0, which result in restraining slips at the nodal level according to the models usually adopted for PRframes. 3.1. Application I: Partial interaction contribution to global flexibility
(a) Basic cell considered in Application I.
(b) Composite frame considered in Application II. Fig. 5. Structural schemes considered in the applications.
while the second one is focused on the global behavior of the frame represented in Fig. 5b. The following non-dimensional parameters are firstly introduced with reference to the symbols reported in Fig. 5a: ζ =
E Ib /L , E Icol / h
Kj =
Kϕ L L = . E Ib εϕ E I b
(21)
The former one takes into account the stiffness ratio between beam and column, while in the latter the joint rigidity K ϕ is related to the beam flexural stiffness E Ib /L: the resulting ratio K j is also utilized in EC3 and EC4 to classify joints in terms of stiffness which can range from pinned to rigid as K j varies between zero and infinity. These two parameters completely describe the behavior of bare steel semi-rigidity frames [6], at least within the elastic range. Partial interaction due to shear connection flexibility is considered as a further variable by means of the following non-dimensional parameter αL derived by the Newmark theory: s k E Ifull αL = · ·L (22) E A∗ E Iabs where: is the flexural stiffness of the beam with no interaction; it can be evaluated by summing the corresponding stiffnesses E Ia and E Ic of the steel profile and concrete slab section, respectively; E Ifull is the flexural stiffness of the beam section in full composite interaction (in the following the identity E Ib = E Ifull will be assumed); E A∗ = E Aa · E Ac /(E Aa + E Ac ) is a conventional axial stiffness (see [7]);
E Iabs
The behavior of a basic cell of the frame structure represented in Fig. 5a can be described by means of the nondimensional parameters defined by Eqs. (21) and (22); it can be easily modeled by considering four composite beam elements and just one beam-column element. The sway displacement δ due to a given value of the horizontal force F can be evaluated as well as the resulting lateral stiffness K l . For a given value of the ζ ratio, the stiffness K depends on joint semi-rigidity (related to K j ) and shear connection flexibility (controlled by αL): for this reason it is denoted by the symbol K K j ,αL . It is interesting to compare such a value with the corresponding one evaluated for rigid joints (Fully Restrained structure, namely K j → ∞) and complete interaction (αL → ∞): such a value is denoted herein by K F R,Full , with the subscripts meaning the same. The influence of the two mentioned parameters (K j and αL) on the lateral stiffness can also be observed in terms of period of vibration T ; since it is inversely proportional to the square root of the lateral stiffness K , the following relationship can be stated between T and K : s TK j ,αL K F R,Full = . (23) TF R,Full K K j ,αL A parametric analysis has been carried out considering a wide range of variation for the three relevant parameters αL, K j , ζ and even E Ifull /E Iabs ; the four diagrams of Fig. 6 represent the TK j ,αL /TF R,Full ratio depending on K j for different values of the beam-to-column stiffness ratio ζ and E Ifull /E Iabs = 2.0. The mentioned figure clearly shows that, as expected, the TK j ,αL /TF R,Full ratio is hugely affected by joint semi-rigidity, but even partial interaction controls the fundamental period of the structure in a way usually neglected by both code provisions and designers; the influence of both parameters increases with the beam-to-column relative stiffness ζ . Moreover, the limit value of TK j ,αL /TF R,Full for K j → ∞ depends on αL and is as great as the low interaction results (αL → 0) in composite beams. On the contrary, the curves represented in each diagram of Fig. 6 reduce to the following closed form solution proposed in [6] for bare steel structures as αL → ∞: s TK j ,Full K j · (1 + ζ ) + 6 = . (24) TF R,Full K j · (1 + ζ ) The reduction in lateral stiffness induced by partial interaction and beam-to-column semi-rigidity can also be regarded by
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Fig. 6. Influence of partial shear interaction and joint semi-rigidity on the period of vibration.
considering the so-called Stability Coefficient γ defined as follows: γ =
N , Kl · h
(25)
where N is the axial force, K l is the lateral stiffness and h is the inter-storey height. The Stability Coefficient measures how the structure is sensitive to P–∆ effects; the greater γ is (which is close to the inverse of the critical multiplier αcr ), the more sensitive to second order effects is the structure. Fig. 7, adopting for γ the same subscripts introduced for the period of vibration, shows the role played by K j and αL in the variation of such a parameter that is mainly related to K j , but is even affected by αL as partial interaction occurs. Clearly, Fig. 7 confirms the role of partial shear interaction which results in a further reduction of lateral stiffness of composite structures with respect to bare steel structures whose behavior is simply affected by beam-to-column joint semi-rigidity. Finally, Fig. 8 points out the effect of partial interaction as a further source of flexibility in addition to that induced by semi-rigid joints. It is worth noticing that sometimes partial interaction plays a role as important as that of joint semirigidity, even in cases of practical interest (αL = 3 ÷ 6). Eurocode 3 suggests for bare steel structures a lower threshold value of joint stiffness K j for semi-rigidity to be possibly neglected within analysis: EC3 states that K j has to be not smaller than 25 for looking after the given structure as a FR-frame. This lower bound cannot be directly extended to composite structures because partial interaction results
in a further increase of global flexibility of the structure. Consequently, as a matter of principle, a more restrictive limit K j,F R,lim has to be imposed on the non-dimensional joint stiffness parameter K j for looking after the partial interaction effect. The mentioned value K j,F R,lim can be determined on by imposing that the fundamental period TK j ,αL|(K j =25,αL→∞) evaluated for K j = 25 under the hypothesis of full interaction is equal to the value TK j ,αL|(K j =K j,F R,lim ) which can be obtained in partial interaction for a finite value of αL: TK j ,Full TK j ,αL = . (26) TF R,Full K j =25 TF R,Full K =K j,F R,lim j
Eq. (26) can be solved numerically on the basis of the results of the parametric analysis already represented in Fig. 6; solutions are graphically reported in Fig. 9 for the considered values of the relevant parameters describing partial interaction in composite beams. Analytical expression can be sought for K j,F R,lim by means of a multiple regression against the parameters E Ifull /E Iabs and αL in the following form: a(E Ifull /E Iabs ) K j,lim,F R = 25 · 1 + , (27) sinh[b(E Ifull /E Iabs ) · αL] which clearly reduces to 25 as full interaction (αL → ∞) arises. The functions a(E Ifull /E Iabs ) and b(E Ifull /E Iabs ) have been calibrated through a least-square minimization process against the numerical values and are reported below: E Ifull (28) a(E Ifull /E Iabs ) = 4.52 · −1 , E Iabs
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Fig. 7. Influence of partial shear interaction and joint semi-rigidity on the Stability Coefficient (log–log scale).
Fig. 8. Partial versus full shear interaction in Partially Restrained structures.
s b(E Ifull /E Iabs ) = 0.645 ·
3
E Iabs . E Ifull
(29)
Quite remarkable accuracy between the numerical results of Eq. (26), already reported in Fig. 9, and the closed form expression (27) of K j,F R,lim can be observed in Fig. 9. Finally, Fig. 10 shows a more direct way of determining K j,F R,lim depending upon E Ifull /E Iabs and αL according to Eq. (27) and the definitions of Eqs. (28) and (29); it is important to point out that Eq. (27) (and consequently the graph in Fig. 10) can be applied to determine the threshold K j,F R,lim whatever the value of the beam-to-column stiffness ratio ζ is. 3.2. Application II: Comparison with code provision
Fig. 9. Comparison between the numerical solution and the closed-form expression of K j,F R,lim .
The possibility of explicitly looking after the influence of partial interaction on the composite beam behavior is the characteristic feature of the present model for global
analysis of composite structures. Indeed, different proposals can be found in the scientific literature about the possibility of considering partial interaction by modifying the components
C. Faella et al. / Engineering Structures 30 (2008) 1178–1186
Fig. 10. K j,F R,lim depending upon the two parameters describing partial interaction in composite beams.
Fig. 11. Layout of the considered beam-to-column joint and corresponding flexible components.
defined for beam-to-column joints in composite structures. EC4 [5] basically introduces a further component related to the axial stiffness of reinforced concrete slab; indeed, a more refined provision is given in Annex A (with informative value) for evaluating the beam-to-column joint stiffness also accounting for the effect of the shear connection flexibility resulting in partial interaction. The method, derived by the usual Components Method adopted in bare steel structures, basically modifies the axial stiffness ks of the reinforcing bars embedded in the concrete slab which are regarded as a further joint flexibility component. In particular, a reduction factor kslip is considered for ks depending on the key characteristics of the shear connection (namely, the number Nl of shear connectors throughout a given distance l from the end of the beam, the secant stiffness ksc of the shear connectors, etc.) condensed in the parameter K sc whose complete expression is not reported herein for the sake of brevity and can be found in EC4 [5]: kslip =
1 1+
E s ks K sc
.
(30)
In the present application, the proposed method is utilized with reference to the framed structure represented in Fig. 5 in comparison with the simplified method proposed by EC4. The joint representation in Fig. 11, designed for full strength, is considered in the following linear elastic analysis in order to deal with the influence of partial shear connection as captured by the present model and the EC4 simplified method.
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Fig. 12. Top displacement δtop (with respect to the corresponding value δtop,F R,full determined for rigid joint and full interaction) versus shear connection degree N /N f .
A shear connection degree ranging from 0.1 to 1.0 is considered for the sake of completeness in the parametric analysis, even if a minimum 0.4 value is allowed by EC4. Moreover, the correspondence between the shear connection degree N /N f and the non-dimensional interaction parameter αL can be found once a particular shear connector P–s curve is adopted. In the following a φ19 Nelson Stud shear connector characterized by a P–s curve Type A according to the models proposed by [10] is considered. On the basis of these choices, αL parameter can be related to N /N f , assuming a 50% secant value for shear connector stiffness ksc . The linear elastic analyses, conducted for different values of the shear connection degree N /N f , are based upon the following three different structural models: – the first assumes full interaction in beams and no reduction in ks due to partial interaction according to the basic provision of EC4; – the second is derived according to the EC4-Annex A provisions briefly reported in Eq. (30) adopting full interaction in composite beams and decreasing values for joint stiffness K j as N /N f decreases; – the third is defined by means of the presented model, where K j stiffness is constant and shear connection stiffness k in Eq. (22) is as small as N /N f in order explicitly to look for partial interaction by means of the presented model for composite beams in PR-frames. The key results of the analyses have been represented within the plot of Fig. 12, which shows the top displacement δtop for the various considered shear connection degrees N /N f with respect to the reference value δtop,F R,full determined for rigid joints and full interaction. The proposed model provides δtop larger than those obtained by means of the analyses carried out according to the EC4 simplified method. Consequently, with reference to the beam-to-column joint considered for this example, EC4 provisions seem unsafe and could be enhanced considering partial interaction according to Annex A informative formulation. 4. Conclusions The combined effect of partial interaction and joint semirigidity in composite frames has been investigated by means of
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a finite element model developed for composite beams in partial interaction which has been formulated by the same authors in previous papers. The applications of such a model as presented herein dealt with measuring the increase in the global flexibility of framed structures due to the above-mentioned combined effect of partial interaction in composite beams and joint semi-rigidity. The parametric analysis pointed out that the partial interaction effect on the structural behavior of composite structures can be comparable even to that of joint semirigidity resulting in significant increases in the vibration period T (which basically controls seismic displacement demand) and the stability coefficient γ (related to P–∆ effect), which can achieve 15% even in cases of practical interest. A more restrictive proposal with respect to that provided by Eurocodes for bare-steel structures (or, indeed, for FRframes) has been calibrated for the lower bound value of joint stiffness possibly to assume rigid joints, even accounting for the effect of partial interaction in composite beams. Finally, a single application has been devoted to test the accuracy of the EC4 formula which accounts for partial interaction in composite beams possibly by relaxing the contribution of concrete slab rebars in affecting joint behavior. With reference to the above-mentioned case EC4 basic provision, which completely neglects partial interaction, underestimates the effects of shear connection flexibility; a possible enhancement could be obtained if the provision of Annex A (only informative in the current version) would become mandatory.
References [1] COST C1. Composite steel–concrete joints in braced frames for buildings. Brussels, Luxembourg: European Communities; 1997. [2] Crisinel M, Carretero A. Simple prediction method for moment–rotation properties of composite beam-to-column joints. In: Proceedings of an engineering conference about composite construction III. 1996. p. 823–35. [3] EUROCODE 3, 2001. Design of steel structures: General rules and rules for buildings, Part 1–1. prEN 1993-1-1: 2001, September 2001. [4] EUROCODE 3, 2002. Design of steel structures: Design of joint, Part 1–8. prEN 1993-1-1: 2002, February 2002. [5] EUROCODE 4, 2004. Design of composite steel and concrete structures: General rules and rules for buildings, Part 1–1. prEN, January 2004. [6] Faella C, Piluso V, Rizzano G. Structural steel semirigid connections. London: CRC Press; 1999. [7] Faella C, Martinelli E, Nigro E. Steel and concrete composite beams with flexible shear connection: “Exact” analytical expression of the stiffness matrix and applications. Computer & Structures 2002;80(11):1001–109. [8] Faella C, Martinelli E, Nigro E. Effects of partial shear interaction, cracking and joint flexibility on the behavior of composite frames. In: Proceedings of the 3rd EuroSteel conference, vol. 1. 2002. p. 381–90. [9] Faella C, Martinelli E, Nigro E. Non-linear-behavior of steel concrete composite frames with partial shear connection. In: Proceedings of the ASSCCA’03 conference, vol. 1. 2003. p. 555–61. [10] Johnson RP, Molenstra N. Partial shear connection in composite beams for buildings. Proceedings of the Institution of Civil Engineers, London, England 1991;Part. 2(91):679–704. [11] Leon RT, Viest IM. Theories of incomplete interaction in composite beams. In: Composite constructions in steel and concrete III: Proceedings of an engineering foundation conference. ASCE. 1996. p. 858–70. [12] Liew JYR, Looi KL, Uy B. Practical design guidelines for semicontinuous composite braced frames. Steel and Composite Structures 2001;1(2):213–30. [13] Newmark NM, Siess CP, Viest IM. Tests and analysis of composite beams with incomplete interaction. Proceedings, Society for Experimental Stress Analysis 1951;9:75–92.
Analysis of steel–concrete composite PR-frames in partial shear interaction: A numerical model and some applications C. Faella a , E. Martinelli a,∗ , E. Nigro b a Department of Civil Engineering, University of Salerno, Via Ponte don Melillo – 84084 Fisciano (SA), Italy b Department of Structural Engineering, University of Naples “Federico II”, Via Claudio, 21 – 80125 Naples, Italy
Received 24 January 2007; received in revised form 1 June 2007; accepted 18 June 2007 Available online 21 August 2007
Abstract The combined effect of beam-to-column joint semi-rigidity and partial interaction in steel–concrete composite beams is considered in the present paper in order to quantify the relative importance of these two phenomena both affecting the global behavior of composite frames. A finite element model formulated by the authors in a previous paper is able explicitly to simulate partial interaction in composite beams and is extended herein with the aim also of analyzing framed structures, even those with semi-rigid beam-to-column joints otherwise called Partially-Restrained frames (PR-frames in the following). Final applications are devoted to point out the relative importance of partial interaction effect with respect to the well-recognized role played by joint semi-rigidity. Assessment of code provisions and alternative proposals are finally exposed for taking account of the effect of the former phenomenon on structural behavior. c 2007 Elsevier Ltd. All rights reserved.
Keywords: Composite frames; Partial interaction; Joint semi-rigidity; Finite element; Code provisions
1. Introduction Partial interaction between steel girder and concrete slab due to flexible shear connectors deeply affects the behavior of both steel–concrete composite beams and frames. The analytical solution of the bending problem of composite beams in partial interaction has been independently found by the three different researchers mentioned in [11]. Numerical methods have been proposed since the early ’70s for solving the equations of composite beams in partial interaction; the authors made in [7] a short review of the different contributions in this field, reporting proposals based on both finite difference and finite element methods. However, the same paper was basically focused on the formulation of a particular finite element which explicitly accounts for the effects of beam-to-slab relative displacements utilizing the “exact” solution of the well-known Newmark’s linear theory for composite beams in bending [13]. The same element has already been adopted for composite frame structures [8] and its quite straightforward evolution for ∗ Corresponding author. Tel.: +39 089 964098; fax: +39 089 964098.
E-mail address: [email protected] (E. Martinelli). c 2007 Elsevier Ltd. All rights reserved. 0141-0296/$ - see front matter doi:10.1016/j.engstruct.2007.06.006
mechanical and geometrical non-linearity has also been shown in [9]. Beam-to-column joint semi-rigidity plays a significant role in the global behavior of both bare steel and composite steel–concrete structures. Several models have been formulated and presented in the scientific literature for simulating joint behavior according to the well-known “Components Method”. A simplified formulation of this method has been adopted in [3,4] for bare steel joints whose main components are completely described in terms of stiffness, strength and ductility; the same formulation has also been generalized for composite joints in [1] and [5], introducing a further component for simulating concrete slab influence on the rotational behavior of composite joints. Hence, the moment–rotation relationship can be defined in terms of initial stiffness, maximum strength and rotation capacity according to the provisions of the mentioned codes of standard. In the present paper a numerical model is proposed with the aim of simulating the global behavior of composite structures, taking into account both beam-to-column joint semirigidity and partial interaction in composite beams. The explicit simulation of partial interaction represents the key feature of the
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model; indeed, composite frame analysis is usually carried out considering only the effect of joint semi-rigidity, but neglecting shear connection flexibility in composite beams [12]. A first parametric analysis accounting for partial shear interaction, beam-to-column joint semi-rigidity and concrete cracking, has been presented in [8] where each composite beam has been subdivided into three finite elements such as the one presented herein, with the aim of taking account of cracking in an RC slab that possibly arises throughout the hogging moment regions: the mechanical properties of the exterior regions have been determined by neglecting concrete contribution, as usual in the well known “cracked-analysis” procedure allowed by Eurocode 4 and conducted by means of the present finite element for simply reproducing also the partial interaction effect. The results of these analyses pointed out a remarkable increase in the global flexibility of composite frames when partial interaction of composite beams, beam-to-column semirigidity and concrete slab cracking are taken into account at the same time. The effect of beam-to-column joint semi-rigidity and partial shear interaction is even more deeply investigated in the present paper considering their combined influence on two important parameters that characterize the global structural behavior. Two such parameters are the fundamental elastic period T and the so-called stability coefficient γ whose influence has been already studied in [6] with reference to the case of bare steel structures. Both are related strictly to the transverse stiffness of framed structures and, consequently, are affected by partial interaction and joint semi-rigidity; a more restrictive lower bound for joint stiffness with respect to the one adopted for bare steel structure to be considered as Fully Restrained (FR) is also proposed with the aim of accounting implicitly for global flexibility increase due to partial interaction. Finally, simplified methods are available in the scientific literature [2] and code of standards [5] for considering shear connection flexibility as a further component (or a term affecting the traditional ones) of the beam-to-column joint. Therefore, an application is proposed specifically in the present paper to compare the results obtained by means of the proposed model (which accounts explicitly for interface slips) or by considering shear connection flexibility as a further component of beam-to-column joint according to EC4 provisions. 2. Key features of the present model The numerical model utilized in the next section has been already formulated by the authors in [8] and [9] and only its key characteristics will be outlined in the present paper for the sake of brevity. It is based on the so-called “exact” finite element, formulated in [7] according to the classical Newmark Theory of partial shear interaction for steel–concrete composite beams in bending. In particular, according to that approach the flexibility matrix Dr which relates the nodal forces Qr = (Mi , Fi , M j , F j )T and the corresponding nodal displacements δr = (φi , si , φ j , s j )T has been derived in closedform (Fig. 1). The following relationship, typical for the forcebased formulation of the finite element method, can be stated:
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Fig. 1. Nodal forces and displacements for simply supported composite beam element (formally force-based element).
δr = Dr Qr + δr,0
(1)
δr,0 being the vector of nodal displacements due to external actions. In particular, the flexibility matrix Dr involved in Eq. (1) has been obtained by solving the differential equation deriving by the Newmark model, as shown in [7] under suitable boundary conditions; the 4 × 4 matrix form is reported in Eq. (2) pointing out the mechanical meaning of the various terms which correspond to rotation ϕ or slip s, obtained in one of the two element nodes (either i or j) due to a unit moment M or interface force F applied on the node i or j: ϕi,Mi ϕi,Fi ϕi,M j ϕi,F j si,Mi si,Fi si,M j si,F j Dr 1 Dr 2 (2) Dr = = ϕ j,Mi ϕ j,Fi ϕ j,M j ϕ j,F j . Dr 3 Dr 4 s j,Mi s j,Fi s j,M j s j,F j In particular, Eq. (2) states that the k-th column of the flexibility matrix Dr collects the nodal displacement components (sorted as in vector δr ) due to a unit force value given to the k-th component of the nodal force vector Qr , while the other nodal forces are zero. Similarly, the end displacements (namely, rotations ϕ and slips s) collected in vector δr,0 directly depend upon the action distributed on the beam. For example, in [7] they have been expressed in terms of the uniformly distributed load q and shrinkage strain εsh affecting the concrete slab according to the following symbolic form: φoi,q φoi,sh soi,q soi,sh δ 0,r = δ 0,r,q + δ 0,r,sh = + . (3) φoj,q φoj,sh soj,q soj,sh Beam-to-column joint flexibility has to be considered for PR-frames and two springs per node can be introduced in order to model the elastic nodal restraints against nodal rotations and slips induced by joint semi-rigidity (Fig. 2). Consequently, the flexibility matrix Dr0 can be modified as follows: Dr0 = Dr + Dε
(4)
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Fig. 2. Nodal springs adopted for reproducing beam-to-column joint semirigidity.
where matrix Dε collects the flexibility parameters εφ,i , εs,i , εφ, j and εs, j as diagonal terms. The flexibility matrix Dr0 defined by Eq. (4) can also be utilized to obtain the “exact” finite element in the usual displacement formulation which involves the stiffness matrix K6 relating displacements δ 6 = (vi , φi , si , v j , φ j , s j )T and forces Q6 = (Ti , Mi , Fi , T j , M j , F j )T for an unrestrained composite beam (see Fig. 3). The mentioned stiffness matrix K6 and vector Q6 of nodal forces equivalent to the external loads can be obtained by the corresponding flexibility matrix Dr0 and the vector of nodal displacements δr,0 due to the external actions. Indeed, a reduced 4 × 4 stiffness matrix Kr can be obtained by inverting Eq. (1) (without distributed loads and, hence, δr,0 = 0) in which (4) the flexibility matrix Dr has been substituted by Dr0 with the aim of looking after the nodal semirigidity as well: δr,0 = 0 ⇒ Qr =
[Dr0 ]−1 δr
= Kr δr .
(5)
Moreover, the vector of equivalent nodal forces with reference to the four nodal degrees of freedom represented in Fig. 1 can be obtained by forcing nodal displacement δr to zero: δr = 0 ⇒ Qr,0 = −[Dr0 ]−1 δr,0 = −Kr δr,0 .
(6)
Consequently, a formally displacement-based relationship can be obtained for the simply supported beam by superposing the effect of nodal displacements (5) and external actions (6): Qr = Kr δr + Qr,0 .
(7)
The following relationship can be stated between the displacement vector δr , which deals with the simply-supported beam (Fig. 1), and the vector δ 6 defined for the unrestrained beam (Fig. 3a): δ r = P1 δ 6 , P1 being a transformation matrix defined as follows 1/L 1 0 −1/L 0 0 0 0 1 0 0 0 P1 = 1/L 0 0 −1/L 1 0 . 0 0 0 0 0 1
(8)
(9)
The corresponding vector of nodal forces Q6 can be easily obtained by imposing the equivalence of the external work expressed either in terms of nodal force Qr and displacement δr
(a) Element only suitable for beam analysis.
(b) Generalized composite element for frame analysis. Fig. 3. Nodal forces and displacements for unrestrained composite beam element (displacement-based element).
or in terms of the vector Q6 and δ 6 ; the following relationship holds between the force vectors as Eq. (8) applies between the displacement vectors: Q6 = PT1 Qr .
(10)
The stiffness matrix K6 which relates Q6 and δ 6 can be obtained as a function of Kr by introducing Eq. (8) in (7) and utilizing Eq. (10): K6 = PT1 Kr P1 .
(11)
Furthermore, the corresponding six-component vector of equivalent nodal forces Q6,0 can even be obtained through the same transformation matrix P1 . Addition of the terms which correspond to shear forces is also needed for the global equilibrium of the unrestrained beam in the transverse direction; in the case of uniformly distributed load q, the following expression can be assumed: T Q6,0 = PT1 Qr,0 − q L/2 0 0 q L/2 0 0 . (12) The composite beam element formulated above takes account of both partial interaction and beam-to-column joint semirigidity, but is not yet fit for composite frame analysis because it only deals with bending as usual and is sufficient for continuous beams. Since frame analysis is the aim of the present work, the finite element has to be provided with the missing degree of freedom related to the absolute axial displacements. Two alternative possibilities could be undertaken in doing that. On one hand two absolute displacements could be considered per each node with the aim of reproducing the values of axial displacements of girder and slab: in this case, an 8DOF finite element should be formulated and a joint
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at Eqs. (10) and (11):
element should be needed for connecting that 8-DOF composite beam element with the more traditional 6-DOF finite elements utilized for columns. On the other hand, only one effective axial displacement of the beam could be considered for each node by condensing interface slips within the element and leading once more to a 6DOF composite beam element with the absolute axial displacement u instead of the relative slip s at each node (Fig. 3b). This solution, which can even reproduce the behaviour of real composite frames where no continuity is usually guaranteed between slips of two beams connected to the same column, can be directly derived by the above model as presented in the following. Therefore, according to Eq. (4) nodal slips are considered as a sum of the end-slip of the beam (governed by the flexibility matrix Dr ) and a further contribution basically due to slab reinforcement deformations close to the joint and controlled by the terms εs,i and εs, j collected in Dε . Consequently, even if zero value is imposed on slips at the nodal level, beam–slab interface displacements could occur at the end of the composite beam depending on the flexibility terms εs,i and εs, j : they would vanish as such flexibility terms are placed to zero because no relative displacements can occur if springs become rigid bodies (Fig. 4). The transformation matrix P2 can be introduced for relating the displacement vector δ 6 (Fig. 3a) with the vector δ = (u i , vi , φi , u j , v j , φ j )T which collects the displacement components reported in Fig. 3b: (13)
with the following definition: P P2 = 2,b 0
0 P2,b
and
P2,b
(15)
K = PT2 K6 P2 Q0 = PT2 Q6,0 .
(16)
E Aa + E Ac . (18) L It is useful to precise that the symbol E Ac deals with the axial stiffness of the RC slab; consequently, it can be generally assumed according to the following relationship: [K]1,1 = [K]4,4 = −[K]4,1 = −[K]1,4 =
E Ac = E c Ac + E s As ,
1 0 0
(19)
where E c and Ac are Young’s modulus and section area of concrete slab, respectively, while E s and As refer to the corresponding properties of reinforcing bars. The concrete contribution at the second member of Eq. (19) can be neglected if the effect of concrete cracking is of concern. A similar definition can be assumed for the RC slab flexural stiffness E Ic which takes account of both concrete and rebar axial stiffness. The mechanical properties E Aa and E Ia of the steel girder can be obtained simply as a product of the Young’s modulus E a of structural steel and the transverse section Aa or the moment of inertia Ia . The above definitions easily apply within the framework of a linear elastic (either uncracked or cracked according to EC4 [5]) analysis. Nevertheless, a fully non-linear procedure based on the model formulated above has also been implemented by the authors [9], in which the stiffness parameters E Ac , E Aa , E Ic and E Ia can be determined depending on the actual stress state of the element taking account of various phenomena characterizing the behaviour of both concrete (cracking and possibly tension stiffening) and structural steel (plasticity, strain hardening and possibly local stability). Finally, the relationship between nodal forces Q = (Ni , Ti , Mi , Ni , T j , M j )T and displacements δ can be expressed according to the following usual relationship: Q = Kδ + Q0 .
0 = 0 0
(17)
Consequently, the stiffness matrix needs to be completed with the axial terms. Since no interface slips occur if axial forces are distributed between slab and beam on the basis of their axial stiffness, the terms related to axial stiffness can be approximately assumed as reported below:
Fig. 4. Resulting beam-to-column joint model.
δ 6 = P2 δ,
Q = PT2 Q6
(20)
0 1 . 0
(14)
However, axial displacement components u i and u j are not present in δ 6 , which has been introduced with exclusive reference to the bending problem of composite beams as approached in the Newmark model. Hence, no possible correspondence exists between u i and u j and any δ 6 component; moreover, the terms of the force vector Q, the stiffness matrix K and the equivalent nodal force vector Q0 related to the distributed loads can be defined through the transformation matrix P2 in the same way as follows to arrive
The proposed finite element inherits all the advantages derived from the “exact” solution of Newmark’s equation already utilized for the analysis of continuous beams, but can also be utilized for composite (plane) frame structures characterized by semi-rigid beam-to-column joints whose flexibility is considered by means of the terms εφ,i and εφ, j . 3. Applications Two applications will be discussed in this section for pointing out the combined role of partial shear interaction and beam-to-column joint semi-rigidity: the first one deals with the basic cell of a composite frame represented in Fig. 5a,
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k
is the shear connection stiffness. The following values will be assumed in the analyses:
– εϕ,i = εϕ, j = 1/K ϕ ; – εs,i = εs, j = 0, which result in restraining slips at the nodal level according to the models usually adopted for PRframes. 3.1. Application I: Partial interaction contribution to global flexibility
(a) Basic cell considered in Application I.
(b) Composite frame considered in Application II. Fig. 5. Structural schemes considered in the applications.
while the second one is focused on the global behavior of the frame represented in Fig. 5b. The following non-dimensional parameters are firstly introduced with reference to the symbols reported in Fig. 5a: ζ =
E Ib /L , E Icol / h
Kj =
Kϕ L L = . E Ib εϕ E I b
(21)
The former one takes into account the stiffness ratio between beam and column, while in the latter the joint rigidity K ϕ is related to the beam flexural stiffness E Ib /L: the resulting ratio K j is also utilized in EC3 and EC4 to classify joints in terms of stiffness which can range from pinned to rigid as K j varies between zero and infinity. These two parameters completely describe the behavior of bare steel semi-rigidity frames [6], at least within the elastic range. Partial interaction due to shear connection flexibility is considered as a further variable by means of the following non-dimensional parameter αL derived by the Newmark theory: s k E Ifull αL = · ·L (22) E A∗ E Iabs where: is the flexural stiffness of the beam with no interaction; it can be evaluated by summing the corresponding stiffnesses E Ia and E Ic of the steel profile and concrete slab section, respectively; E Ifull is the flexural stiffness of the beam section in full composite interaction (in the following the identity E Ib = E Ifull will be assumed); E A∗ = E Aa · E Ac /(E Aa + E Ac ) is a conventional axial stiffness (see [7]);
E Iabs
The behavior of a basic cell of the frame structure represented in Fig. 5a can be described by means of the nondimensional parameters defined by Eqs. (21) and (22); it can be easily modeled by considering four composite beam elements and just one beam-column element. The sway displacement δ due to a given value of the horizontal force F can be evaluated as well as the resulting lateral stiffness K l . For a given value of the ζ ratio, the stiffness K depends on joint semi-rigidity (related to K j ) and shear connection flexibility (controlled by αL): for this reason it is denoted by the symbol K K j ,αL . It is interesting to compare such a value with the corresponding one evaluated for rigid joints (Fully Restrained structure, namely K j → ∞) and complete interaction (αL → ∞): such a value is denoted herein by K F R,Full , with the subscripts meaning the same. The influence of the two mentioned parameters (K j and αL) on the lateral stiffness can also be observed in terms of period of vibration T ; since it is inversely proportional to the square root of the lateral stiffness K , the following relationship can be stated between T and K : s TK j ,αL K F R,Full = . (23) TF R,Full K K j ,αL A parametric analysis has been carried out considering a wide range of variation for the three relevant parameters αL, K j , ζ and even E Ifull /E Iabs ; the four diagrams of Fig. 6 represent the TK j ,αL /TF R,Full ratio depending on K j for different values of the beam-to-column stiffness ratio ζ and E Ifull /E Iabs = 2.0. The mentioned figure clearly shows that, as expected, the TK j ,αL /TF R,Full ratio is hugely affected by joint semi-rigidity, but even partial interaction controls the fundamental period of the structure in a way usually neglected by both code provisions and designers; the influence of both parameters increases with the beam-to-column relative stiffness ζ . Moreover, the limit value of TK j ,αL /TF R,Full for K j → ∞ depends on αL and is as great as the low interaction results (αL → 0) in composite beams. On the contrary, the curves represented in each diagram of Fig. 6 reduce to the following closed form solution proposed in [6] for bare steel structures as αL → ∞: s TK j ,Full K j · (1 + ζ ) + 6 = . (24) TF R,Full K j · (1 + ζ ) The reduction in lateral stiffness induced by partial interaction and beam-to-column semi-rigidity can also be regarded by
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Fig. 6. Influence of partial shear interaction and joint semi-rigidity on the period of vibration.
considering the so-called Stability Coefficient γ defined as follows: γ =
N , Kl · h
(25)
where N is the axial force, K l is the lateral stiffness and h is the inter-storey height. The Stability Coefficient measures how the structure is sensitive to P–∆ effects; the greater γ is (which is close to the inverse of the critical multiplier αcr ), the more sensitive to second order effects is the structure. Fig. 7, adopting for γ the same subscripts introduced for the period of vibration, shows the role played by K j and αL in the variation of such a parameter that is mainly related to K j , but is even affected by αL as partial interaction occurs. Clearly, Fig. 7 confirms the role of partial shear interaction which results in a further reduction of lateral stiffness of composite structures with respect to bare steel structures whose behavior is simply affected by beam-to-column joint semi-rigidity. Finally, Fig. 8 points out the effect of partial interaction as a further source of flexibility in addition to that induced by semi-rigid joints. It is worth noticing that sometimes partial interaction plays a role as important as that of joint semirigidity, even in cases of practical interest (αL = 3 ÷ 6). Eurocode 3 suggests for bare steel structures a lower threshold value of joint stiffness K j for semi-rigidity to be possibly neglected within analysis: EC3 states that K j has to be not smaller than 25 for looking after the given structure as a FR-frame. This lower bound cannot be directly extended to composite structures because partial interaction results
in a further increase of global flexibility of the structure. Consequently, as a matter of principle, a more restrictive limit K j,F R,lim has to be imposed on the non-dimensional joint stiffness parameter K j for looking after the partial interaction effect. The mentioned value K j,F R,lim can be determined on by imposing that the fundamental period TK j ,αL|(K j =25,αL→∞) evaluated for K j = 25 under the hypothesis of full interaction is equal to the value TK j ,αL|(K j =K j,F R,lim ) which can be obtained in partial interaction for a finite value of αL: TK j ,Full TK j ,αL = . (26) TF R,Full K j =25 TF R,Full K =K j,F R,lim j
Eq. (26) can be solved numerically on the basis of the results of the parametric analysis already represented in Fig. 6; solutions are graphically reported in Fig. 9 for the considered values of the relevant parameters describing partial interaction in composite beams. Analytical expression can be sought for K j,F R,lim by means of a multiple regression against the parameters E Ifull /E Iabs and αL in the following form: a(E Ifull /E Iabs ) K j,lim,F R = 25 · 1 + , (27) sinh[b(E Ifull /E Iabs ) · αL] which clearly reduces to 25 as full interaction (αL → ∞) arises. The functions a(E Ifull /E Iabs ) and b(E Ifull /E Iabs ) have been calibrated through a least-square minimization process against the numerical values and are reported below: E Ifull (28) a(E Ifull /E Iabs ) = 4.52 · −1 , E Iabs
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Fig. 7. Influence of partial shear interaction and joint semi-rigidity on the Stability Coefficient (log–log scale).
Fig. 8. Partial versus full shear interaction in Partially Restrained structures.
s b(E Ifull /E Iabs ) = 0.645 ·
3
E Iabs . E Ifull
(29)
Quite remarkable accuracy between the numerical results of Eq. (26), already reported in Fig. 9, and the closed form expression (27) of K j,F R,lim can be observed in Fig. 9. Finally, Fig. 10 shows a more direct way of determining K j,F R,lim depending upon E Ifull /E Iabs and αL according to Eq. (27) and the definitions of Eqs. (28) and (29); it is important to point out that Eq. (27) (and consequently the graph in Fig. 10) can be applied to determine the threshold K j,F R,lim whatever the value of the beam-to-column stiffness ratio ζ is. 3.2. Application II: Comparison with code provision
Fig. 9. Comparison between the numerical solution and the closed-form expression of K j,F R,lim .
The possibility of explicitly looking after the influence of partial interaction on the composite beam behavior is the characteristic feature of the present model for global
analysis of composite structures. Indeed, different proposals can be found in the scientific literature about the possibility of considering partial interaction by modifying the components
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Fig. 10. K j,F R,lim depending upon the two parameters describing partial interaction in composite beams.
Fig. 11. Layout of the considered beam-to-column joint and corresponding flexible components.
defined for beam-to-column joints in composite structures. EC4 [5] basically introduces a further component related to the axial stiffness of reinforced concrete slab; indeed, a more refined provision is given in Annex A (with informative value) for evaluating the beam-to-column joint stiffness also accounting for the effect of the shear connection flexibility resulting in partial interaction. The method, derived by the usual Components Method adopted in bare steel structures, basically modifies the axial stiffness ks of the reinforcing bars embedded in the concrete slab which are regarded as a further joint flexibility component. In particular, a reduction factor kslip is considered for ks depending on the key characteristics of the shear connection (namely, the number Nl of shear connectors throughout a given distance l from the end of the beam, the secant stiffness ksc of the shear connectors, etc.) condensed in the parameter K sc whose complete expression is not reported herein for the sake of brevity and can be found in EC4 [5]: kslip =
1 1+
E s ks K sc
.
(30)
In the present application, the proposed method is utilized with reference to the framed structure represented in Fig. 5 in comparison with the simplified method proposed by EC4. The joint representation in Fig. 11, designed for full strength, is considered in the following linear elastic analysis in order to deal with the influence of partial shear connection as captured by the present model and the EC4 simplified method.
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Fig. 12. Top displacement δtop (with respect to the corresponding value δtop,F R,full determined for rigid joint and full interaction) versus shear connection degree N /N f .
A shear connection degree ranging from 0.1 to 1.0 is considered for the sake of completeness in the parametric analysis, even if a minimum 0.4 value is allowed by EC4. Moreover, the correspondence between the shear connection degree N /N f and the non-dimensional interaction parameter αL can be found once a particular shear connector P–s curve is adopted. In the following a φ19 Nelson Stud shear connector characterized by a P–s curve Type A according to the models proposed by [10] is considered. On the basis of these choices, αL parameter can be related to N /N f , assuming a 50% secant value for shear connector stiffness ksc . The linear elastic analyses, conducted for different values of the shear connection degree N /N f , are based upon the following three different structural models: – the first assumes full interaction in beams and no reduction in ks due to partial interaction according to the basic provision of EC4; – the second is derived according to the EC4-Annex A provisions briefly reported in Eq. (30) adopting full interaction in composite beams and decreasing values for joint stiffness K j as N /N f decreases; – the third is defined by means of the presented model, where K j stiffness is constant and shear connection stiffness k in Eq. (22) is as small as N /N f in order explicitly to look for partial interaction by means of the presented model for composite beams in PR-frames. The key results of the analyses have been represented within the plot of Fig. 12, which shows the top displacement δtop for the various considered shear connection degrees N /N f with respect to the reference value δtop,F R,full determined for rigid joints and full interaction. The proposed model provides δtop larger than those obtained by means of the analyses carried out according to the EC4 simplified method. Consequently, with reference to the beam-to-column joint considered for this example, EC4 provisions seem unsafe and could be enhanced considering partial interaction according to Annex A informative formulation. 4. Conclusions The combined effect of partial interaction and joint semirigidity in composite frames has been investigated by means of
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a finite element model developed for composite beams in partial interaction which has been formulated by the same authors in previous papers. The applications of such a model as presented herein dealt with measuring the increase in the global flexibility of framed structures due to the above-mentioned combined effect of partial interaction in composite beams and joint semi-rigidity. The parametric analysis pointed out that the partial interaction effect on the structural behavior of composite structures can be comparable even to that of joint semirigidity resulting in significant increases in the vibration period T (which basically controls seismic displacement demand) and the stability coefficient γ (related to P–∆ effect), which can achieve 15% even in cases of practical interest. A more restrictive proposal with respect to that provided by Eurocodes for bare-steel structures (or, indeed, for FRframes) has been calibrated for the lower bound value of joint stiffness possibly to assume rigid joints, even accounting for the effect of partial interaction in composite beams. Finally, a single application has been devoted to test the accuracy of the EC4 formula which accounts for partial interaction in composite beams possibly by relaxing the contribution of concrete slab rebars in affecting joint behavior. With reference to the above-mentioned case EC4 basic provision, which completely neglects partial interaction, underestimates the effects of shear connection flexibility; a possible enhancement could be obtained if the provision of Annex A (only informative in the current version) would become mandatory.
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