Improved FE model to simulate interfacial bond-slip behavior in composite beams under cyclic loadings

Computers and Structures 125 (2013) 164–176

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Improved FE model to simulate interfacial bond-slip behavior in composite beams under cyclic loadings Jin-Wook Hwang, Hyo-Gyoung Kwak ⇑ Department of Civil and Environmental Engineering, Korea Advanced Institute of Science and Technology, 335 Gwahangno, Yuseong-gu, Daejeon 305-701, Republic of Korea

a r t i c l e

i n f o

Article history: Received 8 October 2012 Accepted 20 April 2013 Available online 4 June 2013 Keywords: Partial interaction Steel–concrete composite beam Shear connector Beam element with bond-slip Concrete under cyclic loading

a b s t r a c t This paper describes an extension of a model for simulating the bond-slip at the interface in composite beams. The model extended in this paper is intended to simulate the non-linear cyclic behavior of composite beams on the basis of a beam element whose nodal degrees of freedom are lateral deflection and rotation, as in the original model. The hysteretic relations of concrete, steel, and shear connectors are used, and solution procedures including an iteration method to find a double neutral axis are introduced. Finally, correlation studies between numerical results and experimental studies have been conducted to validate the proposed model. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Composite beams, constructed by placing a concrete slab on a steel or concrete girder equipped with shear connectors, are widely used as structural members for bridges and high-rise building structures. Composite beams usually show partial composite action, which is induced from the bond-slip deformation along the interface between the slab and girder and represents an increase of lateral deflection and a decrease in the resisting capacity. Also, since the structural responses considering the slip effect are different from those obtained by ignoring the slip effect with the assumption that full connection is provided along the interface, the bond-slip effect should be accounted for in order to precisely evaluate the actual structural behavior of partially bonded composite beams. In addition to many experimental studies on the bond-slip behavior in composite beams [4,24], numerous analytical and numerical approaches have also been performed. Bradford and Gilbert conducted a section analysis on the basis of a static equilibrium condition [5], but additional time-consuming iterative processes are required to obtain the slip strain. Dezi et al. [9] and Jasim and Ali [17] also proposed closed form solutions. However, these solutions are limited in terms of direct implementation to FE analyses, which cover all the loading and boundary conditions, because the lateral deflection is basically evaluated by integrating the curvature under a single curvature assumption along the span and because closed form solutions are very dependent on the load⇑ Tel.: +82 42 350 3661; fax: +82 42 350 4546. E-mail addresses: [email protected], [email protected] (H.-G. Kwak). 0045-7949/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compstruc.2013.04.020

ing and boundary conditions. Many numerical analyses based on the double-node concept have also been performed [12,13,30– 32,38,42] but, by using double node, the slab and girder should be modeled as separate elements and connected by interfacial spring element. Modeling like this needs additional definition of nodes, elements and connectivity conditions and, thus, make mesh definition somewhat inconvenient. Furthermore, all the abovementioned analytical and numerical approaches are limited to only monotonic loading conditions. Since the slip behavior along the interface between the slab and girder becomes clearer with an increase of applied load, its effect on the structural response needs to be examined, even for the ultimate loading condition such as seismic loading or highly reversed cyclic loading accompanying severe inelastic deformation. Accordingly, the hysteretic behaviors for the composed materials of concrete and steel as well as the bond-slip phenomenon along the interface should be defined and considered in the numerical formulation of composite beams. Much research to date has concentrated on the behavior of shear connectors under cyclic loading conditions through experimental studies using push-out tests or numerical studies with commercial software [6,7,11,28,36]. The introduction of accurate load-slip relations considering the frictional slip phenomenon and the effect of stiffness degradation of shear connectors has followed [33,34]. Nevertheless, few studies have considered the bond-slip effect in the structural analysis of composite beams subjected to cyclic loading due to the complexity involved in numerical modeling of the structure. Kwak and Hwang proposed a slip model that can consider the partial bond-slip without adopting double-node concepts in composite beam structures subjected to arbitrary monotonic loadings

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[23]. The formulation in this model is based on a FE analysis using a conventional beam element defined by two nodal displacements and a usage of the proposed model makes modeling process simpler by defining only one beam element for expressing all the components of slab, girder and shear connector; however, the slip distribution in each element was evaluated on the basis of the governing equation constructed by assuming a linear load-slip relation. Upon consideration of the bond-slip effect in a beam element subjected to monotonic loadings, this paper introduces an extension and modification of the introduced model to cover arbitrary cyclic loading conditions. An incremental slip analysis, which considers the non-linearity of shear connectors and the pinching phenomenon that occurs at the frictional slip zone, is performed by adopting a non-linear cyclic load-slip relation. In addition to consideration of the cyclic behavior of shear connectors, hysteresis models for concrete and steel are implemented and a unique iteration method considering a double neutral axis in a discontinuous strain distribution across the section is proposed in this paper to determine the deformed configuration corresponding to the cyclic loading condition. The structural responses are analyzed by applying the proposed FE based slip model into composite structures under cyclic loadings, and the applicability of the model is verified by comparing the analysis results with experimental data.

conducting experiments for plain concrete subjected to cyclic loadings make it hard to develop a mathematical model of a cyclic stress–strain curve based on experimental results and, for this reason, related research has been limited [8,26]. The unloading–reloading behavior of cracked concrete is particularly complicated, and the cyclic behavior of concrete is generally defined by the unloading–reloading curves that connect several key points: the unloading point, the reloading point, the plastic strain, and the returning strain. Among material models representing these characteristics [10,20,27,41], a hysteresis material model in the compressive region proposed by Martínez-Rueda and Elnashai, where the variation of the plastic strain epl according to the unloading strain eun is elaborately considered, is applied in this paper. In the model, an unloading curve is a second degree parabola (see Fig. 1(a)) defined by the unloading point (eun, fun) and the plastic strain (epl, 0) determined by the focal point (ef, ff) in the high strain range, as delineated in Eq. (1). Note that equations determining focal points and plastic strain values in all strain ranges can be found in the original research paper [27]. A reloading branch represents a bi-linear curve where three points, the reloading point (ero, fro), degrading strength point (eun, fnew), and returning point (ere, fre), are connected (see Fig. 1(a)); detailed equations can be found elsewhere [27].

fc ¼ fun 2. Material models 2.1. Concrete Since concrete is used mostly under compression, the stressstrain relation in compression is of primary interest. Among the numerous mathematical models currently used in the analysis of RC structures, the monotonic envelope curve introduced by Kent and Park [18] and later extended by Scott et al. [35] is adopted in this paper because of its simplicity and computational efficiency. A detailed description of the monotonic stress-strain relation in compression can be found elsewhere [23]. On the other hand, concrete in the tensile region is assumed to be a linear elastic material, representing linear strain softening behavior beyond the tensile strength [22]. Furthermore, the unloading–reloading branches need to be defined in order to simulate the hysteretic behavior of concrete. Since a cyclic stress–strain curve describes the changing material properties of concrete under cyclic loadings, its exact definition is first required. However, unlike the envelope curves obtained from monotonic loading tests, the difficulties involved in

165



ec  epl eun  epl

2 ð1Þ

In the tensile region, a modification of the Giuffré–Menegotto–Pinto model, which was originally developed for reinforcing steel, is applied in this paper, as in other studies where numerical analyses under cyclic loading conditions were performed [1,3,10], because it represents the non-linear crack opening and closing in the tensile region appropriately. The model represents a smooth transition from one asymptote connecting the unloading point (eun, fun) and the origin to another one connecting the origin (0, 0) and focal point (ef, ff), assumed to be one-tenth of the compressive strength (see Fig. 1(b)). The detailed non-linear stress strain relationship with the definition of related constants can be found elsewhere [10]. 2.2. Steel The monotonic envelope of the stress-strain relation of steel is idealized with two straight lines on the basis of the linear elastic and linear strain-hardening behaviors of material with yield stress ry, and the shape of the curve is assumed to be identical in both the compression and tension regions. Upon defining the monotonic

Fig. 1. Hysteretic stress-strain relation of concrete.

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envelope curve, it is then necessary to implement the hysteretic curves. At load reversals, as shown in Fig. 2, the unloading stiffness is assumed to be the same as the initial stiffness. When the path continues unloading without any change in the direction of the load, the stress–strain curve exhibits the Bauschinger effect. This causes a non-linear stress–strain relation and a reduction in stiffness of the stress–strain curve before the stress reaches the yield stress in the reversed sign. Among a number of models developed to describe the cyclic stress–strain curve of steel, the most commonly used approach is the Giuffré–Menegotto–Pinto model, and it is also adopted in this paper, as in other studies [20,21]. The stress–strain relation can be expressed by

r ¼ be þ

ð1  bÞe 1

ð1 þ eR ÞR

;

ð2Þ

where e⁄ = (e  er)/(e0  er), r⁄ = (r  rr)/(r0  rr). Eq. (2) represents a curved transition from a straight line with a slope E0 to another asymptote with a slope E1, as represented by lines (a) and (b) in Fig. 2. The parameter b is the strain-hardening ratio between E0 and E1, e0 and r0 are the coordinates for the point at which the asymptote of the branch under consideration meets the other asymptote that starts from the strain reversal point with the initial stiffness, and er and rr are the strain and stress, respectively, at the point where the last strain reversal with stress of equal sign took place (see Fig. 2). e0, r0, er, and rr are updated at each strain reversal. R is a parameter that controls the shape of the transition curve and allows the representation of the Bauschinger effect. The expression for R is

R ¼ R0 

a1 n ; a2 þ n

ð3Þ

where R is a decreasing function of n, which is the unsigned normalization of the strain difference between the asymptote intersection point and the strain reversal point with the maximum or minimum strain by the yield strain, depending on whether the corresponding steel stress at reversal is positive or negative (Fig. 2). n is updated following a strain reversal, because the asymptote intersection point or the load reversal point with maximum or minimum strain may change. R0, a1, and a2 are experimentally determined parameters. In this paper, it is assumed that R0 = 20.0, a1 = 18.5, and a2 = 0.15, as used by Kwak et al. [20,21].

3. Load-slip relation Since composite beams are equipped with shear connectors between a concrete slab and girder to unify the behavior of the total structure, the flexural and slip behavior of these composite beams are greatly influenced by the shear connectors, which are characterized by their ductility and stiffness. Usually, the static behavior of the shear connectors, which govern the slip behavior at the interface, can be explained through the shear stiffness in the elastic region, the ultimate shear strength, and the corresponding ultimate slip. Many tests have been performed to investigate the mechanical behavior of shear connectors experimentally. Ciutina and Stratan performed a set of ten experimental tests on five different types of shear connectors subjected to cyclic and monotonic loading [6], Feldmann et al. investigated the fatigue behavior of shear studs by a cyclic displacement-controlled push-out test [11], Shariati et al. performed push-out tests for channel shear connectors under monotonic and fully reversed cyclic loading [36], and other researchers also investigated the cyclic behavior of shear studs [7,28]. As has been well established through numerous experimental studies, a typical example of the load-slip characteristic of a stud shear connector represents an almost linear variation up to half of the shear strength. Further increases in slip cause the stiffness to be reduced gradually until the shear strength is reached. On the other hand, constitutive models for shear connectors also have been proposed on the basis of experimental results. Shim et al. proposed tri-linear load-slip curves to define the shear stiffness of connectors in an elastic range [37], Topkaya et al. developed load-slip curves and strength expressions from 24 push-out tests [39], Pallarès and Hajjar proposed formulas for the limit states of the steel failure and concrete failure of headed studs [29], and Xue et al. proposed a new expression of the stud load-slip relationship with a calculation model of the bearing capacity of a stud [43]. Salari [33,34] proposed a bond constitutive law that effectively expresses the behavior of shear studs under cyclic shear, and in this paper, the load-slip relation proposed by Salari is used to represent the slip behavior. In this model, the monotonic envelope is divided into two parts of ascending and descending branches at the shear strength V1 in Fig. 3, and each part can be defined by Eqs. (4) and (5), respectively.

V ¼ V 1 a1

  a3  S S ; Exp a2 S1 S1

ð0 6 S 6 S1 Þ

ð4Þ

V

Cyclic Loading Path Monotonic Envelope (S2,V2)

V1 1

V2 = 0.95V1 V3 = 1.05Vfu

7 8

K1

15 6

Vfu

2

9

16

14 S1

5

13

Vf

10

3

4 12 K0 Fig. 2. Hysteretic stress-strain curve of steel.

11

Fig. 3. Cyclic load-slip relation of shear connectors.

(S3,V3)

S

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V ¼ V 1 b1 Exp b2



S 1 S1

b3 !

þ V fu ;

ðS1 < SÞ

ð5Þ

where Vfu is the ultimate frictional resistance, a1 = K0/K1, a2 = –Ln(1/ a1), a3 = 1/ a2, b1 ¼ 1  V fu , b2 ¼ ðLnðR2 ÞÞ=ðS2  1Þb3 , b3 ¼

    LnðR2 Þ Þ=ðLn SS2 1 Þ, ðLn LnðR 1 3Þ 3

Ri ¼ ðV i  V fu Þ=ð1  V fu Þ;

V fu ¼ V fu =V 1 ,

V i ¼ V i =V 1 , Si ¼ Si =S1 , and i = 2, 3. On the other hand, the behavior of shear connectors under cyclic loading is somewhat complicated. In the monotonic loading path 1 in Fig. 3, a shear connector is deformed in the same direction as the relative slab movement, and the deformation of the shear connector induces a gap in the concrete slab equivalent to the slip at the interface while the concrete at the opposite side of the gap is being compressed (see Fig. 4(a)). When the developed strain is larger than the strain value at the compressive strength, the crushing of concrete arises until the initiation of load reversing, which accompanies the unloading path with a very stiff slope of stiffness (see path 2 in Fig. 3). Generally, there is no remarkable change in this initial unloading stiffness value according to the unloading position on the monotonic envelope, and this has been confirmed in many studies [2,6,34,44]. In Fig. 4, the crushing of a concrete medium by the interface interaction degrades the stiffness, whose path moves back to the original unloading area by load reversal (see path 6 or 14 in Fig. 3). In most analytical studies, the initial unloading stiffness value is usually assumed to be constant regardless of the unloading position [2,34,44]. If the stiffness of the shear connector maintains a constant value without any decrease, the developed slip at loading path 2 will be small. However, when it enters the frictional slip zone, where the shear stiffness value is suddenly reduced, the gap begins to close and the slip change becomes very clear relative to the early stage of path 2. The gap almost closes at the frictional resistance Vf (see path 3 in Fig. 3), and similar mechanical phenomenon repeats in the opposite direction (see Fig. 4(b)). The tangential component of the load in the axial direction is mainly considered conventionally in this slip mechanism, and it is regarded as the load applied to a shear connector. In the cyclic load-slip relation, the unloading and reloading paths pass through the frictional resistance (0, Vf), and it is not greatly different from those defined for the cyclic bond stress-slip relation [19]. All the details for the hysteretic behavior and corresponding equations to define the cyclic load-slip relation can be found elsewhere [33,34], where a decrease of the envelope due to subsequent damage is not taken into account in this paper with an assumption that additional damage by only the repeated cycle at the same loading history is negligible. If shear connectors are assumed to be installed under the uniform spacing Ls and the load-slip curve of a shear connector is

linear, the slip S can be represented by Eq. (6), where q(x) is the shear force transmitted per unit length of the beam. This is known as the shear flow (q(x) = dF/dx), where F is the horizontal force at the interface between two materials in coordinate x and Ks is the stiffness of the load-slip relation.

S¼

VðxÞ qðxÞLS ¼ KS KS

ð6Þ

However, if shear connectors represent non-linear behavior, the stiffness Ks will not be constant but rather becomes variable, changing according to slip value S, and can be expressed by Eqs. (7) and (8), obtained by differentiating the load V in Eqs. (4) and (5) with respect to slip S. The total slip S can then be evaluated approxiP j mately by accumulating all the slip change (Sj) values S ¼ m j¼1 S , j V j ðxÞ qj ðxÞLS where S ¼ K S ¼ KðSj1 Þ and j and m mean the load step and the number of total load steps in the finite element analysis, respectively. KðSÞ ¼

dV dS

     a3   a3  V 1  S S  Exp a2   ; ¼ a1     1  a2 a3   S1 S1 S1 KðSÞ ¼

dV dS

(     b3 1   b3 ) V 1   S  S ¼ b1 b2 b3        1 ;  Exp b2    1 S1 S1 S1

Fig. 5 shows the strain and corresponding stress distribution across the concrete section of a composite beam with partial interaction. However, it is still assumed that there is no separation between the two elements, which means the curvatures of the two elements are identical. The axial forces F and moments M, induced in an arbitrary load step j, act through the centroid of the concrete slab at distance hs from the slab–girder interface, as shown in Fig. 5(c), and at the centroid of the girder at distance hg from the interface. The strain changes es,b at the bottom of the concrete slab and eg,t at the top of the girder, as shown in Fig. 5(b), can be given from the beam theory by

es;b ¼ 

Fs M s hs þ ; E s As Es I s

eg;t ¼

Fg M g hg  ; E g Ag Eg I g

ð9Þ

where As and Ag are the areas of the concrete slab and girder, respectively, while Is and Ig are the moments of inertia of the

Shear Connector

Concrete Slab

V V

V Girder Crushed Concrete Zone

(a) Loading path

ð8Þ

4.1. Construction of governing equation

Shear Crack

Slip

ðS1 < SÞ

4. Slip evaluation

(V: Shear Load applied to a connector)

V

ð0 6 S 6 S1 Þ ð7Þ

(b) Un-loading path

Fig. 4. Mechanical behavior of shear connectors under cyclic loadings.

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Reinforcement

M(x)

s,t

s

Ms Fs Fhorz.

hs Interface slip

s,b g,t

hg

V(x)

Ls

Mg Fg

Stirrup Reinforcement g,b

x (a) composite beam

b

(b) strains and curvature

(c) stresses and internal forces

Fig. 5. Slip behavior in a partially composite concrete beam.

cross-sectional area of each element with respect to their centroidal axes. Additionally, since there is no external longitudinal force being applied in conventional bending members, such as post-tensioning, horizontal force equilibrium requires that RF = 0, which implies that Fs = Fg = Fhorz. As mentioned above, when the composite section is subjected to the bending moment, the relative movement across the interface that is induced by the sliding action is referred to as the slip of S = ug – us, and the derivative of this relation with respect to the longitudinal distance x yields a slip strain change of eslip = dS/ dx = eg,t – es,b. Hence, from Eqs. (6) and (9), the following differential equation represented by the material properties and section dimensions can be obtained.





2 Ls d F horz: 1 1 M s hs M g hg  ¼ F þ þ horz: Es As Eg Ag K s dx2 Es I s Eg I g

ð10Þ

From the rotational equilibrium of the internal moment, the moment increment M(x) at the section being considered in Fig. 5(c) can be expressed by

MðxÞ ¼ Ms ðxÞ þ M g ðxÞ þ FðxÞhorz: ðhs þ hg Þ:

ð11Þ

Since the shear connection is required to prevent separation between the girder and slab, the curvature change j by the moment increment in the slab and girder are the same, as shown in Fig. 5(b), and thus

j¼

Ms Mg MðxÞ  FðxÞhorz: ðhs þ hg Þ ¼ ¼ : Es I s þ Eg I g Es I s Eg I g

ð12Þ

From Eqs. (11) and (12), the curvature j at a section located at distance x from the far end support can be expressed by j = {M(x) – F(x)horz.(hs + hg)}/REI, where REI = EsIs + EgIg. When applying this relationship, Eq. (10) yields the following ordinary linear differential equation for F(x)horz.: 2

d FðxÞhorz: 2

dx

where

1 EA

FðxÞ ¼ F h þ F p ¼ a coshðPxÞ þ b sinhðPxÞ þ

¼ Es1As þ Eg1Ag , EI ¼

P

EI þ EA ðhs þ hg Þ2 .

4.2. Numerical model for slip analysis

ð14Þ

ð15Þ

where D(x) (=dM(x)/dx) means the slope of the distribution of the moment increment and yields a constant value if the distribution of moment increment M(x) is assumed to be linear at each element, and a and b are constants to be determined by substituting the boundary conditions at the nodal points of each subdivided element. Furthermore, the horizontal force increment F(x) and the slip increment S(x) at the interface of the concrete slab and girder, given in Eqs. (14) and (15), can be represented in the ith element in the following matrix form:



F i ðxÞ



Si ðxÞ

¼

coshðPi xÞ

sinhðPi xÞ



T i sinhðPi xÞ T i coshðPi xÞ

ai



bi

þ

Qi P 2i

(

Mi ðxÞ Ls;i K s;i

Di

) ; ð16Þ

where Ti = (Ls,i/Ks,i)Pi and li = the element length. Since the horizontal force and slip increments must maintain continuity along the entire span, those values at the common node i of the (i  1)th and ith element must be the same; this means Fi = Fi1(li1/2) = Fi(li/2), Si = Si1(li1/2) = Si (li/2), where Fi and Si denote the horizontal force increment and corresponding slip increment at node i. Namely, superscript i is assigned to all the parameters related to node i. The substitution of the above compatibility conditions and subsequent rearrangement for the constant a and b, which will finally be determined, yields 1 1 1 Ai ¼ C1 2;i C1;i1 Ai1 þ C2;i ½Mi1  Mi  ¼ C2;i C1;i1 Ai1 þ C2;i DMi ;

ð17Þ

where (

ð13Þ

MðxÞ

    Ls dFðxÞ Ls Q aP sinhðPxÞ þ bP coshðPxÞ þ 2 DðxÞ ; ¼ dx Ks Ks P

SðxÞ ¼



Ks EI K ðhg þ hs Þ P FðxÞhorz: ¼  s  MðxÞ; Ls EA EI Ls ðEg Ig þ Es Is Þ

Q P2

Ai ¼

ai bi

) ;

( Ai1 ¼

8 9 i Q < M = Mi ¼ 2i Ls;i ; P i : Di ; K s;i

ai1 bi1

) ;

K s;i1

2 C1;i1

Mi1

8 9 i = Q i1 < M ¼ 2 ; L P i1 : s;i1 Di1 ;

6 ¼4

  3 sinh P i1 li1 2 7     5; li1 li1 T i1 cosh P i1 2 T i1 sinh P i1 2 cosh



P i1 li1 2

    31 sinh P i l2i cosh P i l2i 6 7 ¼4    5 T i cosh P i l2i T i sinh P i l2i    3 2 1 cosh P i l2i sinh P i l2i Ti 6 7 ¼4     5: li 1 sinh P i l2i cosh P i 2 Ti



2

For computational convenience, the differential equation in Eq. (13) can be rewritten in the form of F00 (x) – P2F(x) = – QM(x). The general solution of Eq. (13) representing F(x) is obtained by summing a particular solution to the associated homogeneous solution, and the corresponding distribution of slip increment S(x) is also obtained from the first derivation of the horizontal force increment. These steps lead to

C1 2;i

ð18Þ

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Table 1 Slip analysis formulations for ascending and descending branches. Ascending branch (KS > 0)

Descending branch (KS > 0)

Horizontal force and slip distributions FðxÞ ¼ F h þ F p ¼ a coshðPxÞ þ b sinhðPxÞ þ PQ2 MðxÞ

FðxÞ ¼ F h þ F p ¼ a cosðPxÞ þ b sinðPxÞ  PQ2 MðxÞ

Q P2

Ls Ks

SðxÞ ¼ KLss faP sinðPxÞ þ bP cosðPxÞ  PQ2 DðxÞg ( )

    M i ðxÞ ai F i ðxÞ sinðP i xÞ cosðP i xÞ  PQ2i Ls;i ¼ bi Si ðxÞ T i sinðP i xÞ T i cosðP i xÞ i K s;i Di

SðxÞ ¼ faP sinhðPxÞ þ bP coshðPxÞ þ DðxÞg ( )

    M i ðxÞ F i ðxÞ ai sinhðP i xÞ coshðP i xÞ þ QP2i Ls;i ¼ bi Si ðxÞ T i sinhðP i xÞ T i coshðP i xÞ i K s;i Di Applying compatibility conditions for continuity into Eq. (16)         ¼ F i  l2i ;Si ¼ Si1 li1 ¼ Si  l2i F i ¼ F i1 li1 2 2 1 1 1 Ai ¼ C1 2;i C1;i1 Ai1 þ C2;i ½Mi1  Mi  ¼ C2;i C1;i1 Ai1 þ C2;i DMi ;

1 1 1 Ai ¼ C1 2;i C1;i1 Ai1 þ C2;i ½Mi1  Mi  ¼ C2;i C1;i1 Ai1 þ C2;i DMi ;

where     ai ai1 Ai ¼ , Ai1 ¼ , bi bi1 ( ) ( ) i M Mi ; Mi ¼ PQ2i Ls;i ; Mi1 ¼ QP2i1 Ls;i1 i1 i K s;i1 Di1 K s;i Di   2 3 Þ cosh P i1 li1 sinhðP i1 li1  2   2  5; C1;i1 ¼ 4 T i1 cosh P i1 li1 T i1 sinh P i1 li1 2 2     31 2 2 li li cosh P coshðP i l2i Þ sinh P i i 4  2   2 5 ¼ 4   C1 2;i ¼ T i sinh P i l2i sinh P i l2i T i cosh P i 2li

  ai1 Ai1 ¼ , bi1 ( ) ( ) i M Mi Mi1 ¼  QP2i1 Ls;i1 ;Mi ¼  QP2i Ls;i ; i1 i K s;i1 Di1 K s;i Di   3 2 Þ sin P i1 li1 cosðP i1 li1  2   2  5; C1;i1 ¼ 4 li1 T T i1 sin P i1 li1 i1 cos P i1 2 2     31 2   2 cos P i l2i cos P i l2i sin P i l2i 1 4 5 4       ¼ C2;i ¼ T i sin P i l2i  sin P i l2i T i cos P i l2i

where 

Ai ¼

 3 sinh P i l2i  5 li 1 T i cosh P i 2 1 Ti

ai



bi

,

 3 sin P i l2i  5 li 1 T i cos P i 2 1 Ti

Applying boundary conditions into Eq. (16) Case I: Simply supported ( )     0 a1 0 Ls;1 Q 1 1 D1 ¼ C1 ¼ C1 A1 ¼ ¼ C1 2;1 2;1 2;1 f ðS1 Þ  D S b1 K s;1 P2 1 1 ( )     an ¼ C1 D ¼ C1 nþ1 0Ls;n Q 0 1 An ¼ n n 1;n 1;n  K s;n P2 Dn ¼ C1;n f ðSnþ1 Þ S bn n

Case I: Simply supported ( )     0 a1 0 Ls;1 Q 1 1 D1 ¼ C1 ¼ C1 ¼ C1 A1 ¼ 2;1 2;1 2;1 f ðS1 Þ þ D S b1 K s;1 P 2 1 1 ( )     an ¼ C1 D ¼ C1 nþ1 0Ls;n Q 0 1 An ¼ n n 1;n 1;n þ K s;n P2 Dn ¼ C1;n f ðSnþ1 Þ S bn n

Case II: Cantilever or propped cantilever 8 9 1 Q1 1   a1 ¼ C1 D ¼ C1 < F  P21 M = A1 ¼ 1 L 2;1 2;1 b1 :  S;1 Q21 D1 ; K S;1 P 1 ( )   0 a 1 1 n ¼ C1;n Dn ¼ C1;n Snþ1  LS;n Q n D An ¼ bn K S;n P 2n n

Case II: Cantilever or propped cantilever 8 9 1 Q1 1   a1 ¼ C1 D ¼ C1 < F þ P21 M = A1 ¼ 1 L 2;1 2;1 b1 : S;1 Q21 D1 ; K S;1 P 1 ( )   0 a 1 1 n An ¼ ¼ C1;n Dn ¼ C1;n Snþ1 þ LS;n Q n D bn K S;n P2n n

Case III: Two fixed ends 8 9   < F 1  Q21 M1 = a P1 1 1 1 A1 ¼ ¼ C2;1 D1 ¼ C2;1 b1 :  KLS;1 Q21 D1 ; S;1 P 1 8 9   < F nþ1  Q2n Mnþ1 = a Pn 1 1 n An ¼ ¼ C1;n Dn ¼ C1;n L bn :  KS;n Q2n Dn ;

Case III: Two fixed ends 8 9   < F 1 þ Q21 M 1 = a P1 1 1 1 A1 ¼ ¼ C2;1 D1 ¼ C2;1 b1 : KLS;1 Q21 D1 ; S;1 P 1 8 9   < F nþ1 þ Q2n M nþ1 = a Pn 1 1 n An ¼ ¼ C1;n Dn ¼ C1;n L S;n Q n bn : ; 2 Dn K

S;n

Pn

S;n

Since Eq. (17) refers to the vector of the solution coefficients of element i  1 and element i, the successive application of this equation from the first element 1 to the last element n produces a generalized transfer matrix relationship that expresses the relationship between the element 1 and element n as

" # ( " # ) n1 n n1 Y X Y 1 C An ¼ C1 Z A þ C Z D M 1;1 1 i j i ; 2;n 2;n i¼2

i¼2

ð19Þ

j¼i

Qn1 where Zi ¼ C1;i C1 2;i and i¼2 Zi ¼ Zn1  Zn2  Zn3 ; . . . ; Z2 . Two boundary conditions are required to solve Eq. (19). Since a multi-span continuous bridge usually has simply supported boundary conditions at both far end points, which implies that the horizontal force and moment value are zero, the following   boundary values can be introduced: F 1 ¼ F 1  l21 ¼ 0,     Mnþ1 ¼ Mn l2n ¼ 0, F nþ1 ¼ F n l2n ¼ 0, M 1 ¼ M 1  l21 ¼ 0,    S1 ¼ S1  l21 , Snþ1 ¼ Sn l2n . Substituting these boundary conditions into Eq. (16), the following relationships are obtained:

A1 ¼



a1 b1



( 1 ¼ C1 2;1 D1 ¼ C2;1

)

0 1

S 

Ls;1 Q 1 K s;1 P 2 1

D1

¼ C1 2;1



0 f ðS1 Þ

 ð20Þ

An ¼



an



bn

Pn

( 1 ¼ C1 1;n Dn ¼ C1;n

)

0 Snþ1 

Ls;n Q n K s;n P 2n

Dn

¼ C1 1;n



0 f ðSnþ1 Þ

 : ð21Þ

The substitution of Eqs. (20) and (21) into Eq. (19) produces the following system matrix equation in Eq. (22) related to the slip behavior at the interface of the slab and girder, and this equation can also be represented by the matrix form of Eq. (23).

Dn ¼ GD1 þ H 



0 f ðS

nþ1

Þ

ð22Þ

¼

G11 G21

G12 G22



0 1

f ðS Þ



 þ

H1 H2

 ;

ð23Þ

Q P Q where G ¼ ni¼1 Zi ¼ Zn  Zn1 . . . Z2  Z1 and H ¼ ni¼2 f½ nj¼i Zj DMi g . In the matrix formulation of Eq. (22), the two unknowns are the slip values at the two far ends (S1 and Sn+1), as all other values can be calculated from the coefficients of the governing equation and the nodal moment values. The formulations can also be applied in the case where the two far ends are not simply supported. Eqs. (14)–(19) are independent of the boundary conditions, and the changes are only in matrices D1 and Dn in Eqs. (20)–(23), because they include terms such as

J.-W. Hwang, H.-G. Kwak / Computers and Structures 125 (2013) 164–176

F1, Fn+1, M1, Mn+1, S1, and Sn+1 dependent on the boundary conditions. If the support is a fixed end, the horizontal force and moment at the boundary will take certain values; on the other hand, the slip is zero by restriction of the fixed end as in the following examples for a propped cantilever, where the left end is fixed and the right end is simply supported:

      l1 ln l1 ; F nþ1 ¼ F n ¼ 0; M 1 ¼ M 1  ; F1 ¼ F1  2 2 2       ln l1 ln ¼ 0; S1 ¼ S1  ¼ 0; Snþ1 ¼ Sn : M nþ1 ¼ M n 2 2 2

tion FðxÞ ¼ a cosðPxÞ þ b sinðPxÞ  PQ2 MðxÞ will then be obtained. All the related remaining solution procedures are the same as those for the loading path on the ascending branch. The related equations are summarized in Table 1 and this table is also describing the change in equations when Ks < 0.

5. Solution procedure

The unknown for the fixed end is not the nodal slip S1 but F1, unlike the simply supported end. The boundary conditions for the free end are the same as those for the simply supported end. Formulations for the other boundaries are summarized in Table 1. Above numerical formulations can be applied to any kinds of composite structural member with given boundary conditions for the member and, thus, the application of the model sufficiently can be extended to the structure like a multi-story building, which is composed of a lot of members, by applying solution procedures explained in Chapter 5. When the slip model is applied to the non-linear problem, where the stiffness of stud Ks is changed according to the slip value, all the procedures explained above can be repeated for every load step, and the total slip value should be computed by accumulating all the incremental values. If the path enters the descending branch in the accumulating process, where the stiffness Ks has a negative value, a slight modification for the solution procedure is required because P2 becomes a negative value and a hyperbolic function in Eqs. (14)–(16) includes an imaginary number P, which is meaningless. Thus, when the path is on the descending branch, P2 needs  to be replaced by P2 ¼  KLss EAEIP EI and the accompanying governing equation should be changed to F 00 ðxÞ þ P 2 FðxÞ ¼ QMðxÞ. The solu-

1st Iteration at 1st load increment (Step 1)

The FE analysis of composite beams is based on a beam element with four degrees of freedom, one lateral deflection, and one rotation at each node in an element, and all the constitutive equations are formulated upon the Timoshenko beam theory, which considers the shear deformation in its formulation. In this simple beam element, the separation between the slab and girder along the vertical direction is not considered, because the deformation by the vertical separation can be assumed to be negligibly small in a structural member subjected to dominant bending behavior. On the other hand, the longitudinal separation along the interface is reflected by the non-linear load-slip constitutive law. This law considers the ultimate frictional slip zone, where the resistance at the interface is mainly influenced by the friction between material surfaces. Details related to the derivation procedure of the governing equation in the finite element formulation can be found elsewhere [23] and the evaluation of deformation is based on the following basic assumptions: (1) the layered section method is accepted and each layer matches only one material; (2) plane sections remain plane to represent the linear strain distributions; and (3) all the stresses are assumed to act in a uni-axial direction. At the first iteration of each incremental load step, the section analysis is performed with the assumption of no additional bondslip between the slab and girder on the basis of the horizontal force determined at the previous loading step (see Fhorz. in Fig. 6). The non-linear iteration procedure for the bond-slip is

1st Iteration at arbitrary i th load increment (Step 1) Fs

Interface

Fhorz. Obtained at the last iteration of previous load increment

Criteria for perfect bond:

Single Neutral Axis

Double Neutral Axis

Fs + F g < Fg Axial Stress

Axial Strain

Axial Strain

Fs + Fhorz. < Fg - Fhorz. < Axial Stress

Steps (2) ~ (5) Steps (2) ~ (6)

Slip arises along the interface Slip analysis induces additional residuals.

Conversed status (by non-linear iteration with bond-slip) Fs (Compressive)

Fs

Fhorz.

Fhorz.

Double Neutral Axis

Fg Axial Strain

Axial Stress

(a) 1st load increment

Double Neutral Axis

Step (6)

Criteria for partial bond with slip: Fg (Tensile) Axial Strain

Fs + Fhorz. < Fg - Fhorz. < Axial Stress

(b) i th load increment

Fig. 6. Strain and stress distribution at a section according to iteration.

Increase loading step

170

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J.-W. Hwang, H.-G. Kwak / Computers and Structures 125 (2013) 164–176

then followed upon the evaluation of the internal forces corresponding to the external force increment, and the subsequent iterations are continued until reaching the converged values. The related sequential solution steps in a load step can be summarized as follows: (1) the strain and stress distributions across the section of each element are determined from the criteria (see Fig. 6) and element curvature; (2) coefficients in the governing equation of Eq. (13) (Pi, Qi and Mi(x)) are computed; (3) two nodal unknown values in Eq. (22) (among S1, Sn+1, F1, and Fn+1) at two far ends are computed; (4) solution coefficients ai and bi are computed for each element; (5) the horizontal force Fhorz. and slip S are computed; and (6) the convergence is checked and the non-linear iteration considering bond-slip is repeated, including steps (1)–(5) (see Fig. 6(b)). Especially after the first iteration at the first loading step, the continuous strain distribution is transformed into a discontinuous distribution through Eqs. (9) and (12), because bond-slip along the interface between the slab and girder has occurred (see Fig. 6(a)). Moreover, all the subsequent iterations up to the final iteration at the last loading step will be continued on the basis of the discontinuous strain distribution across a section determined from the element curvature and criterion with bond-slip (see Fig. 6). Details for the FE solution procedure are provided in Fig. 7.

6. Numerical application 6.1. Monotonic loading condition In addition to numerous composite beams tested under a monotonically increased loading condition in a previous study [23], a simply supported steel composite beam analyzed by Heidarpour and Bradford [15] was considered in this paper to verify the efficiency of the proposed bond-slip model. The variation of the structural behavior according to change in the stiffness of shear connector was studied in this example, because experimental studies of composite beams are usually conducted on the basis of a fixed value of shear stiffness and the variation of structural responses according to the change of the shear stiffness is difficult to assess by these conventional experimental studies. The considered beam has a span of L = 10 m and is composed of a concrete slab with a width and depth of 500 mm and 250 mm, respectively, and an I-shaped steel section with an overall depth of d = 612 mm, flange width of bf = 229 mm, flange thickness of tf = 19.6 mm, and web thickness of tw = 11.9 mm, as shown in Fig. 8. This beam is subjected to a uniformly distributed load of w = 20 kN/m. The elastic modulus of steel and initial elastic modulus of concrete are taken as Es = 200,000 MPa and Ec = 25,000 MPa, respectively, and the

i=1 to n

Start MATERIAL PROPERTIES

Increase or decrease nodal loading

- CONCRETE - STEEL - SHEAR CONNECTOR

The first iteration step ? GEOMETRY

- LOADING STEPS - MONOTONIC OR CYCLIC LOADINGS

STEP (4) Evaluate solution coefficient vector A1 (Eq. 20)

i=2 to n

Perform assembling

Evaluate solution coefficient vectors Ai (Eq. 17)

Perform Gauss elimination

Perform back-substitution process & obtain nodal displacement values STEP (1) Find neutral axis & compute strain, stress, moment and shear force

i=1 to n

STEP (5) Evaluate horizontal force Fi and slip Si (Eqs. 14 & 15)

Evaluate residual vector and convergence ratio No

STEP (2) i=1 to n

LOADING CONDITION

No

Formulate element stiffness matrix

Evaluate rigidity values EA* & EI* and equation coefficients Pi & Qi (Eq. 13)

Converge ? Yes

No

STEP (3) i=2 to n

- SIMPLY SUPPORTED END - FIXED END - FREE END

LOAD INCREMETN LOOP

BOUNDARY CONDITION

Evaluate matrix G, matrix H (Eq. 23) and unknown nodal values S1, Sn+1, F1 and Fn+1

Yes

ITERATION LOOP

- NODE - ELEMENT - SECTION DIMENSION

Evaluate matrices C1,i-1, C2,i (Eq. 18) and Zi (Eq. 19)

Evaluate nodal bending moment Mi

Fig. 7. Total FE analysis flow considering slip effect.

The last load step ? Yes End

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500mm

This Study (C=1) Heidapour et al. (C=1) This Study (C=10) Heidapour et al. (C=10)

14

Deflection (mm)

12

250mm

Shear Stud

Concrete Slab

10 8 6 4 2

11.9mm

612mm

0

Steel Girder

0

2

4 6 X Coordinate (m)

8

10

(a) Deflection

19.6mm 2.0

229mm

1.0

shear connectors are assumed to behave linearly, as was assumed by Heidarpour and Bradford, for comparison of the obtained results. Fig. 9 compares the obtained numerical results in this paper with the analytical results by Heidarpour and Bradford and shows good agreement between them. This figure shows that an increase of stiffness in shear connection induces less slip and lateral displacement. Here, the stiffness of a shear stud for unit length Ks = 6.6 N/mm2 for C = 1 and Ks = 660 N/mm2 for C = 10, according to Eq. (24), proposed by Girhammar and Pan [14]. In the equation to express the stiffness of the shear connection by a dimensionless constant, r is the summation of distances from the material interface to the center of gravity for the slab and girder, EI0 = E1I1 + E2I2, where E is the initial elastic modulus of the material, I is the moment inertia of each component for its own centroid, A is the section area, and the subscripts 1 and 2 are identifying numbers for the slab and girder, respectively.

ð24Þ

6.2. Cyclic loading condition To verify the applicability of the proposed numerical method under cyclic loading conditions, four steel composite beams were analyzed and their material properties are listed in Table 2. Moreover, in the case of no information for the definition of the cyclic slip behavior of studs such as the unloading branch, reloading branch and the frictional slip zone, appropriate values are evaluated and used on the basis of the configuration and layout of steel studs in the experiment. The first example is a simply supported steel composite beam SB1 tested by Humar [16]. This beam has a span length of 3.05 m and is subjected to a repeated concentrated load at mid-span. The geometry and cross-section of SB1 are shown in Fig. 10. The load-deflection curve for three cycles obtained in the test is used for correlation studies between experimental and numerical analyses. As shown in Fig. 11, the experimental results show a highly bended shape in both the unloading and reloading branches, because a pinching effect is induced from the frictional slip effect at the shear studs, where the slip effect becomes very evident. The numerical analysis considering the bond-slip effect through the proposed numerical model shows a clear pinching shape in the lower quadrants (see Fig. 11(a)), but the analysis results where

Slip (mm)

Fig. 8. Section dimensions of composite beam tested by Heidarpour et al.

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   1 1 r2 L C ¼ aL ¼ K þ þ E1 A1 E2 A2 EI0

This Study (C=1) Heidapour et al. (C=1) This Study (C=10) Heidapour et al. (C=10)

1.5

0.5

0.0 -0.5 -1.0 -1.5 -2.0

0

2

4 6 X Coordinate (m)

8

10

(b) Slip Fig. 9. Analysis results according to the change in the stud stiffness.

Table 2 Material properties of the steel composite beams SB1, CG3, SCB4, and CCB4. Shear stud Beams Concrete Steel strength strength Strength Stud Distance Girder Reinforcement Initial (MPa) (mm) (MPa) (MPa) stiffness (kN) (kN/mm) SB1 CG3 SCB4 CCB4

27.12 23.50 45.80 45.80

332.1 270.5 320.0 320.0

360.0 545.0 380.0 380.0

500.0 40.0 300.0 100.0

30.0 3.5 95.0 55.0

80.0 42.86 140.0 90.0/70.0/110.0

the bond-slip effect is not taken into account give an overestimated energy absorption capacity without representing any pinching behavior (see Fig. 11(b)). This indirectly illustrates that a conventional layered approach that is based on a perfect bond assumption along the interface has some difficulties in modeling composite beams affected by bond-slip. On the other hand, although the analysis results ignoring the bond-slip effect show overestimated ultimate resisting capacities relative to those obtained by considering the bond-slip effect in the upper quadrants in Fig. 11, no remarkable difference appears in the total structural behavior. The reason is that, in this example structure, the slip behavior induced from the friction slip along the interface between the slab and girder is more dominant when the structure is subjected to unloading paths rather than loading and reloading paths. The second example is a cantilevered steel composite beam CG3 [40] with a span length of 1143 mm (45 in) and a concrete slab with a width of b = 762 mm (30 in) and a height of h = 50.8 mm (2 in). The upper 25.4 mm part of the slab height is solid while

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P Concrete Slab Steel Girder

1524

1524 CL

1524

101.6 5/8"@7" c/c Concrete Slab

Shear Stud 6.35

304.8 9.525

W12 27 165

Fig. 10. Section dimensions and loading condition of the composite beam of SB1 (Unit: mm).

600

35 PARTIAL CONNECTION

PARTIAL CONNECTION

25

200

15

Load (kN)

Load (kN)

EXPERIMENT

EXPERIMENT

400

0 -200

5 -5

-400

Pinching behavior by the proposed slip model

-600 -30

-20

-10 0 Mid-span Deflection (mm)

10

(a) Considering bond-slip effect

-15 -25 -80

-60

-40

-20 0 20 Tip Deflection (mm)

40

60

80

Fig. 12. Load-deflection curve of the composite beam CG3.

600 FULL CONNECTION EXPERIMENT

Load (kN)

400 200 0 -200 -400 -600 -30

Overestimated structural behavior because of full connection

-20

-10 0 Mid-span Deflection (mm)

10

(b) Ignoring bond-slip effect Fig. 11. Load-deflection curve of the composite beam SB1.

the lower part is corrugated by a formed metal deck, and the cover depth of concrete above the top reinforcement in the slab is 9.5 mm. Meanwhile, the used steel girder is M6X4.4, which has an overall depth of 122.5 mm, flange thickness of 4.3 mm, web

thickness of 2.9 mm, and flange breadth of 46.7 mm. The local effect due to the metal deck is ignored in modeling the section, but the equivalent effective depth is considered to take into account the corrugated shape of the slab section along the span. When a concentrated cyclic loading is applied at the free end of the beam, the obtained hysteretic load-deflection curve can be found in Fig. 12. In the monotonic loading and the following first unloading branches, highly satisfactory agreement between the analysis and experiment is observed. The numerical results at the subsequent cyclic loadings accompanying an increase of structural deformation and bond-slip along the interface between the concrete slab and steel beam, however, represent a departure from the experimental data, and this digression is expected to intensify with an increase in structural deformation. Unlike the critical region located in the vicinity of the mid-span in a simply supported beam, the behavior of the critical region at the cantilevered end of relatively short-span beams may be greatly affected both by shear and also by the bond characteristics at the interface, similar to the anchorage details of reinforcements in RC beams. In particular, the slippage of a concrete slab at the cantilevered end accompanies the rotation of the beam at the fixed-end, and the yielding of a steel beam accompanies a sudden increase of bond-slip at the anchorage zone. This phenomenon can also be

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J.-W. Hwang, H.-G. Kwak / Computers and Structures 125 (2013) 164–176

P

P Concrete Slab Steel Girder

C L

C L

3900

3900 SCB4 CCB4 800

20

120 Shear Stud

200

11.4

16 16

7

Concrete Slab

I20a 100

350

600

250

450

150

300

Load (kN)

Load (kN)

Fig. 13. Section dimensions and loading condition of the composite beams, SCB4 and CCB4 (Unit: mm).

50 -50 -150

150 0 -150 -300

PARTIAL CONNECTION

-250 -350 -60

-45

-30

-15 0 15 30 Mid-span Deflection (mm)

45

60

EXPERIMENT PARTIAL CONNECTION

-450

EXPERIMENT

-600 -60

75

-45

(a) Considering bond-slip effect

-15 0 15 30 Mid-span Deflection (mm)

45

60

75

(a) Considering bond-slip effect

350

600

250

450 300

Load (kN)

150 Load (kN)

-30

50 -50 -150

150

0 -150 -300

FULL CONNECTION EXPERIMENT

-250 -350 -60

-45

-30

-15 0 15 30 Mid-span Deflection (mm)

45

60

EXPERIMENT FULL CONNECTION

-450 75

(b) Ignoring bond-slip effect

-600 -60

-45

-30

-15 0 15 30 Mid-span Deflection (mm)

45

60

75

(b) Ignoring bond-slip effect

Fig. 14. Load-deflection curve of the composite beam SCB4.

Fig. 15. Load-deflection curve of the composite beam CCB4.

found in the load-deflection relation of beam CG3 in Fig. 12, which shows a gradually increasing rotation in the load-deflection relation in proportional to an increase of the cyclic deformation. Since

the proposed numerical model in this paper has a limitation in considering this rigid body deformation due to the end rotation,

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0.02

-10 Slip S(x) (mm)

0.01

-20 S(x)

-30

0.00

-40 -50

-0.01 Fhorz.(x)

-0.02

0

0.5

1

1.5 2 2.5 X Coord. (m)

3

-60 3.5

Horz. force Fhorz.(x) (kN)

0

-70

(a) SCB4 0.04

25 S(x)

Slip S(x) (mm)

0.02

Fhorz.(x)

15

0.01

5

0.00

-5

-0.01

-0.02

-15

-0.03

-25

-0.04

0

1

2

3 4 5 X Coord. (m)

6

7

Horz. force Fhorz.(x) (kN)

35

0.03

-35

(b) CCB4 Fig. 16. Slip and horizontal force distributions of composite beams in a load step under monotonically increasing loads.

60 Proposed Model q (x) = V(x)·QT/IT

40

Shear Flow (N/mm)

the difference between the analysis and experiment will gradually increase in the case of cantilevered composite beams. The next examples are the composite beams of SCB4 and CCB4 tested by Lizhong et al. [25]. These two beams have the same crosssections, and the differences are the number of spans and stud distances. SCB4 is a simply supported one-span beam with uniform stud distance of 140 mm, but CCB4 is a two-span continuous beam, as shown in Fig. 13, and concentrated cyclic loads are applied at the middle of each span. Studs in CCB4 are equipped with the different distances for three areas: 90 mm from the end support to the position where a load is applied; 70 mm from the load-position to the position where the bending moment is zero; 110 mm from the zero-moment position to the interior support. Figs. 14 and 15 show the experimental and the analytical loaddeflection relations. The experimental load-deflection relations show pinched hysteretic loops, indicating the occurrence of bond-slip along the interface between the slab and girder. Notably, at the two-span continuous composite beam of CCB4, the slip effect appears to be more remarkable. The numerical results in Figs. 14 and 15(a), which are obtained by considering the bond-slip effect on the basis of the proposed numerical model, show excellent agreement with the experimental results through the entire response. If the bond-slip effect is not taken into account in the numerical analysis, however, there is a marked difference between the numerical results and experimental results, and the difference will become larger as the deformation increases. This tendency related with the influences of the bond-slip effect on the structural behavior is illustrated in Figs. 14 and 15(b). In the examples, the numerical results ignoring the bond-slip effect represent the overestimated energy absorption capacity as well as the ultimate resisting capacity, such as example SB1, and illustrate why the

20 0 -20 -40 -60

0

1

2

3 4 5 X Coord. (m)

6

7

Fig. 17. Horizontal shear flow distribution of CCB4.

interfacial bond-slip effect should be considered in the modeling of partially composite beams. Fig. 16 represents the slip and horizontal force distributions of SCB4 and CCB4 under the same monotonically increasing lateral load of upward 20 kN at each mid-span. For a simply supported beam SCB4, the element whose slip has an extremal value is positioned at the left or right end (see Fig. 16(a)). However, in the case of continuous beam CCB4, the slip values that developed at some points within the span are larger than those at the far end positions (see Fig. 16(b)). Although the densities of the shear connector differ for two composite beams, the existence of interior support also influences the horizontal force distributions. CCB4 has a maximum horizontal force value of 30 kN and three extremal values, even though the corresponding maximum value is about half of that developed at the one-span composite beam SCB4. These phenomena derive from the restraint for the lateral deflection at the interior support in a continuous beam. Fig. 17 shows that direct application of a linear elastic analysis for the horizontal shear flow q(x) = V(x)QT /IT assuming full connection along the interface to a partially composite beam may cause improper arrangement of the shear connectors, where the subscript T denotes the transformed section. A beam with partial shear connection or with flexural shear studs represents the four extremal values of horizontal shear flow at the far end supports and one point within each span, while a beam with full shear connections represents a constant distribution between the loading point and support. Since the partially composite beam gives very little horizontal shear flow at the interior support, the difference in horizontal shear force will be greatest at this position, and this will accompany an excessive arrangement of shear connectors at the region around the interior support in the case of partially composite beams. Accordingly, the slip behavior should be considered in order to reach a more reasonable shear design of partially composite beams.

7. Conclusion This paper introduces a numerical slip model that can simulate the bond-slip phenomenon at the interface between a concrete slab and girder in composite beams subjected to cyclic loadings. Instead of using a plane element with four nodes, which is usually considered to take into account the slip behavior through the double node concept, a beam element defined by two end nodes is used for the basis of deriving the governing equation for the bond-slip along the span. The introduced numerical slip model has been extended in this paper to cover the cyclic loading condition. In addition, solution procedures, from the determination of

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