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Modelling of continuous steel–concrete composite beams: computational aspects
Computers and Structures 80 (2002) 2241–2251 www.elsevier.com/locate/compstruc
Modelling of continuous steel–concrete composite beams: computational aspects G. Fabbrocino, G. Manfredi *, E. Cosenza Department of Structural Analysis and Design, University of Napoli Federico II, Via Claudio 21, 80125 Napoli, Italy Received 15 November 2000; accepted 4 July 2002
Abstract Steel–concrete composite members are an interesting option for structural designers, but the reliability of design procedures both in the case of gravity and seismic loads is in continuous development. The issue is very complex, since behaviour of continuous composite beams results from local phenomena of interaction such as partial shear connection and bond. Furthermore, composite beams in buildings generally are not characterised by a full continuity due to the beam to column connections; thus the analysis and the detailing of such parts have a key role in the development of suitable design procedures. In the present paper, some computational aspects related to the modelling of composite flexural members are discussed with reference to continuous and semi-continuous structural systems widely used in practice. 2002 Civil-Comp Ltd. and Elsevier Science Ltd. All rights reserved.
1. Introduction The research efforts in the last years have been devoted to analyse the mechanical behaviour of steel– concrete composite members subjected to gravity loads [1–7]. Thus codes concerning such structures, i.e. Eurocode 4 [8], provide rules only for vertical loading patterns. On the other hand, Eurocode 8 [9] deals with steel– concrete composite members, but the short part that is devoted to this specific type of structures is poor due to the lack of knowledge [10]. As a result, the extension of available code provisions for composite structures to seismic resistant structures is a very difficult task, since many aspects have to be analysed. In fact, a primary goal of such analyses is really a reliable assessment of the ductility of both composite
*
Corresponding author. Tel.: +39-81-768-3424/3488; fax: +39-81-768-3424/3491. E-mail address: [email protected] (G. Manfredi).
beams and columns, of the connection rotation capacity and also of the structural behaviour under cycling loading leading to strength softening [11]. Researches concerning each one of the above topics are complex, since composite behaviour results from many interaction phenomena that involve the steel component, the concrete component and the shear connection between the two parts [2,5,7]. As a consequence, the attention cannot be focussed on each component without taking into account the interaction with all the related aspects, as demonstrated by the effects of Northridge and Hyogoken-Nambu earthquakes on many composite floors-steel columns connections [12,13]. Furthermore, negative bending moments, that develop at the support regions of composite beams, modify the structural response of members due to tensile stresses acting on the slab and the consequent cracking of the concrete, so that steel reinforcement properties can influence the plastic rotation capacity of members [14]. In the present paper, some computational aspects related to the numerical simulation of composite beams are examined. In particular, the attention is focussed on
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the modelling of continuous beams and semi-continuous structural systems, that are meaningful of the composite mechanism governing the transfer of loads and the development of the rotation in the joint region of composite frames. Overall performances of composite beams with different end connections are compared pointing out some interesting remarks on the role of the slab reinforcement ductility and of deformation at the slab–profile interface due to shear connectors. 2. The continuity at intermediate supports The combination of the steel and the concrete leads to traditional composite flexural members that can sustain very well positive bending moment, so that the simply supported beams seem to be the more reliable structural pattern. However continuous composite beams are also frequently used, since a number of advantages can be obtained, i.e. reduction of the deflections and/or of the cross sections [15]. Furthermore in building systems, the fully continuity at internal support cannot be generally achieved (Fig. 1), since in framed structures the columns usually cross the joint regions, while the steel beams are not continuous and connected to the columns with different steelworks [2]. The main feature of the composite construction is the continuity of the slab in the nodal zone, so that the concrete slab can sustain loads in this area, resulting in a strongly different structural response of the joint if compared to bare steel connections. In fact, the continuous concrete slab leads to a composite action in the joint region that is the most important aspect in the behaviour and modelling of these structures [16]. In order to simplify the discussion, a very simple detailing of the joint, the so-called contact plate connection, is analysed and compared to the fully continuous composite beams from a computational point of view.
Fig. 1. The continuous beam at intermediate support.
Fig. 2. Semi-continuous beam in framed structures.
As shown in Fig. 2, the joint detail is characterised by the absence of any connection between the steel profile and the column. It is commonly used when braced frames are concerned, so that beams are designed basically to sustain gravity loads. It is easy to recognise that steel reinforcement of the slab has a relevant role in the development of the composite action and in the structural response of the connection.
3. The theoretical model of the composite cross section The present section summarises the main features of a theoretical model for composite flexural members that is able to take directly account of partial interaction and bond acting on continuous structural patterns, as demonstrated by the effective comparison between numerical and former experimental data [17–19]. The approach is based on a simplified kinematics of the cross section that enables a mono-dimensional approach. This assumption requires the reliable definition of the slab effective width depending on the type of loading, hogging or sagging, and on connection detailing at the beam end, but leads to a strong reduction of the computational effort. This circumstance has been checked both theoretically and experimentally pointing out that simple expression for effective width can be used if the non-linear response of materials is properly taken into account. The cross section of the beam is characterised by a slip at the rebar–concrete interface, s1 , and a slip at the slab–profile interface, s2 , as reported in Fig. 3. A linear pattern of strains is applied to each component of the cross section. The curvature and the rotation are the same for each component (concrete slab and steel profile), therefore the uplift between the slab and the profile is neglected. The slab is characterised by crack distance, dcrack , depending on the geometry of the slab and the me-
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Fig. 3. The model of the cross section under negative bending.
chanical properties and diameter of the reinforcing bars according to CEB provisions [20]. Furthermore, it is assumed that concrete between two subsequent cracks is able to bear tensile stresses; the related resultant tensile force on the concrete Tct is given by the following equation: Z rct ðy; zÞ dA ð1Þ Tct ¼ Aeff
where Aeff is the region of the slab where the distribution of strains is influenced by bond interaction. Since a constant level of strain is assumed on the effective area the relation (1) can be simplified as follows: Tct ¼ rct Aeff
ð2Þ
where rct is the tensile stress acting on the concrete; as a consequence the resultant of tensile stresses is applied at the centroid of the effective area of concrete. This static parameter influences directly the tensile strain levels in the steel bars and the tension stiffening effect and is equal to zero when cracked sections are concerned. Two slips allow taking directly account of local interaction due to bond and shear connection. They can be evaluated referring to the following kinematic equations: s1 ðxÞ ¼ wsc ðxÞ wct ðxÞ
ð3Þ
wup s ðxÞ
ð4Þ
s2 ðxÞ ¼
wlow c ðxÞ
where wsc ðxÞ is the displacement of the reinforcement; wct ðxÞ is the displacement of concrete in tension in the effective area Aeff ; wup s ðxÞ is the longitudinal displacement of the upper fibre of the steel profile; wlow c ðxÞ is the longitudinal displacement of the lower fibre of the concrete slab. The static parameter related to shear connection is the interaction force F, that can be expressed as follows: F ðxÞ ¼
n X Fj
ð5Þ
j¼1
where Fj is the force acting on the generic shear connector located before the reference cross section.
Equilibrium of the composite cross section is dependent upon the following three equations: • longitudinal equilibrium of the steel profile subjected to the interaction force due to shear connectors: Z Fs ¼ rs ðy; zÞ dA ¼ F ð6Þ As
• longitudinal equilibrium of the concrete slab subjected to the interaction force due to shear connectors: Z Fc ¼ Asc rsc þ Tct þ rcc ðy; zÞ dA ¼ F ð7Þ Acc
• rotational equilibrium of the composite section as a whole: M ¼ Ms þ Mc þ Fd
ð8Þ
where Z rs ðy ds Þ dA Ms ¼ As
Mc ¼
Z
rcc ðy dc Þ dA þ rsc Asc ðysc dc Þ
Acc
þ Tct ðyG;Aeff dc Þ are the components of the global bending moment due to the profile, Ms , and the concrete slab, Mc respectively. • dc , ds and ysc represent the distances between the centroids of the two parts of the section and of the reinforcing bars and the reference axis x; • rcc and rct are respectively the magnitude of compressive and tensile stresses of concrete; • As , Asc and Acc are respectively the area of the steel profile, of the rebars and of the concrete in compression. The above equations are not sufficient to define the distribution of strain along the y-axis, since bond and shear connection have to be directly introduced by
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means of additional kinematic equations that can be written as follows:
the positive bending regions and the extended model can be introduced in the negative bending zones, where the tension stiffening of the concrete component develops.
• compatibility equation for bond: 3.1. The moment–curvature relationship
ds1 ðxÞ ¼ esc ðxÞ ect ðxÞ dx
ð9Þ
• compatibility equation for shear connection: ds2 ðxÞ ¼ es ðxÞ ec ðxÞ þ vðxÞd dx
ð10Þ
• equation of equilibrium of the concrete in tension subjected to the bond stresses: dTct ðzÞ ¼ nsc PUsb ðzÞ dx
ð11Þ
The above equations need specific boundary conditions that lead to analyse at the same time the entire beam. Furthermore the constitutive relationships for phenomena of interaction have to be introduced [21,22]. The model discussed above represents an extension of the well known model by Newmark [23] and widely used [24–26]. NewmarkÕs kinematic model is reported in Fig. 4, where the stress and strain parameters are identified as already done in Fig. 3. It is easy to recognise that NewmarkÕs model is not dependent upon bond interaction, since concrete cannot bear tensile stresses according to traditional design approach of concrete structures. This circumstance leads to assume the tensile force Tct equal to zero as done in cracked sections. This remark is very useful since it allows to reduce the computational effort in the positive bending regions; in fact when positive bending moments act on the cross sections the interaction force F is basically associated to compressive stresses on the slab, so that the reinforcement is under compression and/or under moderate tension, thus the assumption of perfect bond between rebars and concrete can be satisfactory. In conclusion, the classical model for the composite section can be used to fit the behaviour of the beam in
The response of composite cross sections depends on two static parameters representing interactions due to bond and shear connection, as a consequence the traditional one to one function resulting from the BernoulliÕs assumptions cannot represent all the balanced solutions for composite sections. Conversely a generalised moment–curvature relationship can be defined and plotted using a family of curves depending on the interactions force F and the resultant of the tensile stresses on concrete, Tct . In the present section the main computational aspects related to the definition of the above relationship are discussed with reference to both hogging and sagging bending. A finite number of curves corresponding to given values of the tension force in concrete Tct and of the interaction level F , can be defined, since they can range between a minimum and a maximum value related only to the mechanical and geometrical properties of the cross section. A reliable representation of the generalised moment– curvature relationship can be performed using a reference system ðM; v; Tct Þ. From a geometrical point of view, the generalised moment–curvature relationship results in a 3D domain that can be drawn using the sections with plane surfaces corresponding to given values of the tensile forces Tct acting on the concrete and varying the value of the interaction force F between the concrete and the steel components. The numerical procedure to define the relationship is based on the strip method, as shown in Fig. 5. In fact, both the slab and the steel profile are divided into a finite number of strips that can be identified by the position of the centroid, yGi , evaluated respect to the upper fibre of the slab and the corresponding area Ai . It is also as-
Fig. 4. The model of the cross section under positive bending.
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• the distribution of the strains on the concrete slab that satisfies the longitudinal equilibrium of the concrete component with the given curvature vj must be evaluated. This step requires again the solution of a non-linear equation; to this end the strain of the lower fiber of the slab can be addressed as the control parameter. The longitudinal equilibrium of the slab can be written in the following form: slab
nstr X
j;l rj;l c;i ðec;low ÞAc;i þ
i¼1
nsc X
j;l p n rj;l sc;i ðec;low ÞAsc;i Tct F ¼ 0
i¼1
ð14Þ Fig. 5. Evaluation of the moment–curvature relationship.
sumed that the strip is affected by a constant level of strain and therefore of stress. In this way the equations of the longitudinal equilibrium of the slab, of the longitudinal equilibrium of the profile and the global rotational equilibrium of the cross section can be written as a sum of finite terms. The method is well known for compact sections, but can be modified for composite sections according to the following steps:
where rj;l c;i represents the stress acting on the generic strip of the slab, nslab str is the total number of the strips and nsc is the number of reinforcing bars. • the last step is the calculation of the global bending moment acting on the section and corresponding to the curvature vj . pro
M¼
nstr X
slab
rj;l s;i Ac;i yGi
i¼1
þ
nsc X
þ
nstr X
rj;l c;i Ac;i yGi
i¼1 p rj;l sc;i Asc;i yGsc þ Tct yGAeff
ð15Þ
i¼1
• a number of values of the resultant of the tensile stresses on the concrete Tctp is chosen and is sequentially imposed on the cross section; • a number of values of the interaction force F n (the apex identifies the generic moment–curvature curve) is chosen and the strain of the lower fibre of the steel profile, ejs;low , is given as a pivot parameter in compliance with the ultimate strain under compression and tension; • the distribution of the strain in the profile that satisfies the longitudinal equilibrium of the steel component is evaluated using as control parameter the strain of the upper fibre of the profile ej;k s;up (the apex j gives the generic increment of the pivot strain, the apex k gives the generic iteration of the process). This step requires an iterative process to solve the non-linear equation that can be written as follows: pro
nsrt X
j;k n rj;k s;i ðes;up ÞAi F ¼ 0
ð12Þ
i¼1
where Ai is the area of the single strip of the profile, pro rj;k s;i is the stress acting on the strip, nstr is the total number of strips of the profile. The iterative process gives the value of the strain of the pivot strain ejs;low as shown in Fig. 5 and the curvature of the cross section can be defined as follows: vj ¼
ejs;low ej;k s;up Hprofile
ð13Þ
where the lever arm of each term is evaluated respect to the upper fibre of the slab. Finally the flexural problem of the composite section is completely solved, since the distribution of strains are known and the derivative of the slips s1 and s2 can be evaluated as follows: dsj1 ¼ ejsc epct dx
ð16Þ
dsj2 ¼ ejc;low ejs;up dx
ð17Þ
The derivative of the function s2 can be easily defined using the roots of the non-linear equations of equilibrium (12) and (14), while the derivative of the slips s1 can be calculated using the strain of rebars and of the concrete in tension placed in the effective area. In fact: Tct epct ¼ eðrpct Þ ¼ e ð18Þ Aeff The above level of strain depends only on the value of the imposed force Tctp . Furthermore, the strain of the rebars are generally very high compared with the strain of the surrounding concrete in tension, thus it can be stated that the second term of Eq. (16) can be neglected. As a result the following relationship exists:
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dsj1 ffi ejsc dx
ð19Þ
The above calculations lead to the definition of an array containing the generalised moment–curvature relationship that can be stored and used to solve the problem of the composite beam [18]. The definition of the moment–curvature relationship of the cross section according to the classical NewmarkÕs approach can be easily carried out using the same above equations. In fact, the moment–curvature becomes a 2D domain, since Tct is equal to zero, so that all the related terms in the above equations disappear and the interaction force F is the key parameter [17].
iterative process is carried out. Rotations and displacements are then calculated by integration of the distribution of the curvature. The solution of the problem can be obtained using the shooting technique and the finite difference method. The specific boundary conditions concerning the slip at the slab–profile interface require the solution of the entire beam. In the following, the discussion is simplified referring to a simple structural system of the beam that is characterised by both geometrical and mechanical symmetry. The numerical procedure is based on the classical method of compatibility, so that the bending moment at the support is the main unknown of the problem and the beam is statically determined. The procedure is described in Fig. 6, and is based on the following steps:
4. The solution of the continuous composite beam The numerical procedure developed to solve continuous composite beams combines the main behavioural aspects of the different regions of the beam. In fact, the traditional NewmarkÕs model for positive bending is used when composite section is subjected to sagging bending moment and the extended one is introduced when cracked zones of the beam are concerned. The aim of the procedure is the definition of the actual moment–curvature relationship in each section of the beam among all the balanced solutions available in the generalised moment–curvature relationship, thus an
1. definition of discrete number of cracked sections along the beam depending on the properties of the slab and of reinforcement; 2. iterative process to ensure the compatibility at internal support. When a symmetrical structural scheme ðkÞ is considered the bending moment at support, mnþ1 , (n þ 1 is the number of integration nodes, k is the iteration number) can be assumed as pivot parameter ðkÞ and the corresponding rotation, unþ1 , as control parameter; 3. iterative process along the whole beam to ensure the compatibility at slab–profile interface for a given
Fig. 6. The numerical solution of the continuous beam.
G. Fabbrocino et al. / Computers and Structures 80 (2002) 2241–2251 ðkÞ
value of mnþ1 ; in this case the slip at the end of the beam, ðs2 ÞðjÞ 1 , governs the process and the check of compatibility is performed at the support, where the ðjÞ slip at the slab–profile interface, ðs2 Þnþ1 , must be zero due to symmetry; 4. when cracking occurs in the slab, according to a proper criterion, the last iterative process is oriented to define the distribution of the slip at the rebar–concrete interface. In this case the process must be performed for each part of the slab between two subsequent cracks (placed at the generic nodes icr , jcr ) until the control condition ðTct Þjcr ¼ 0 is satisfied. It is worth noting that the family of curves giving the moment–curvature relationship is calculated for each typical section of the beam, thus the number of the resolving equations to be used in the iterative procedure can be reduced. The remaining equations can be written in the following form: j
j Fiþ1 ¼ Fij þ F i
ðs1 Þjiþ1 ¼ ðs1 Þji þ
2247
transfer capacity; only the presence of a continuous concrete slab in the nodal region leads the joint to bear bending moments. This specific type of joint is generally subjected to negative bending since it is basically used in braced frames; in the following a symmetrical structural scheme is analysed in order to simplify the discussion. Therefore, the continuity of the slab allows to develop a tensile force in the reinforcement that is balanced horizontally by the compressive action applied at the lower centroid of the contact plate, as shown in Fig. 7a. The force T is applied at the centroid of the rebars, so that the geometry of the connection allows to define easily the lever arm d. It is worth noting that the strength of the nodal region is much lower than the strength of the composite cross section, since the whole contribution of the bending moment carried by the profile is equal to zero.
ð20Þ
j;h ðs2 Þj;h iþ1 ¼ ðs2 Þi þ
ds1 dx
j
ds2 dx
Dx
ð21Þ
i
j;h Dx
ð22Þ
i
j;h j;h ðTct Þj;h iþ1 ¼ ðTct Þi þ nsc pU Dx ðsb Þi
ð23Þ
i
where F is equal to zero if the shear connectors are not present in the considered section, while, if the connectors are present, it is given by the force acting on the shear connectors evaluated by the corresponding constitutive relationship. The procedure is quite complex, but an effective use of the moment–curvature relationships makes it reliable, as demonstrated by the validation against former experimental results [19].
5. The analysis of the semi-continuous beam 5.1. The modelling of the joint region The analysis of the semi-continuous composite beam requires the discussion of the mechanisms governing the strength and the deformation of the connection between the column and the profile. The considered joint is very simple; any steelwork is not needed, since the steel profile is only installed on a settlement welded to the column. In this way the connection at the steel profile level is able to bear only shear forces, but there is not moment
Fig. 7. The strength and the deformation of the contact plate beam–column connection.
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Even if the development of the strength of such a connection is very simple, the mechanism governing the joint rotation is characterised by complex phenomena. In fact, the relative rotation between the column and the beam, developed in the case of symmetrical structure, is related basically to: • the slip between the reinforcement and the concrete ssc;b at the end of the beam; • the slip between the reinforcement and the concrete ssc;j in the nodal zone; • the slip at the profile–concrete slab interface, sconn ; • the displacement of the web panel of the column subjected to the compressive action C, sweb . The calculation of the relative rotation resulting from the above sources of deformation can be performed assuming that the end section of the beam remains plane; furthermore the slab and the profile are characterised by the same rotation, in compliance with the assumptions made for the cross sections of the beam. In this way, referring to the symbols given in Fig. 7a, the relative rotation can be written as follows: ujoint ¼
ðsweb þ sconn þ ssc;j þ ssc;b Þ d
ð24Þ
The equation shows that the deformation of the nodal region is coupled with the response of the beam, since the slips at the slab–profile interface and at the rebar– concrete interface depends on phenomena of interaction developing along the beam. For what concerns the slip of the reinforcement placed in the joint region, it can be evaluated solving the problem of a concrete element subjected to a tensile force T, assuming a given distance between the cracked sections. It is easy to recognise that unbalanced bending moments lead to shear forces on the column in the joint region; in this case an additional source of deformation has to be introduced with a slight increase of the computational effort, but without relevant changes in the numerical approach. The solution of the problem is schematically reported in Fig. 8, where the part of the concrete slab placed in the nodal region is shown. The shooting technique can
Fig. 8. Solution of the slab under tension in the joint.
Fig. 9. Evaluation of the force–displacement relation of the web panel.
be used to determine the distribution of slips at the rebar–concrete interface complying with the boundary conditions. In this case only the bond related equations must be used to solve the problem. Therefore Eqs. (9) and (10) can be used; it is worth noting that Eq. (10) can be more effectively substituted by the following: drsc ðzÞ 4 ¼ sb ðzÞ dz U
ð25Þ
that represents the longitudinal equilibrium of the reinforcing bar subjected to the bond stress. The boundary conditions can be written in terms of stress of the rebar that can be easily evaluated according to the strength mechanism shown in Fig. 7a. On the other hand the relationship between the compressive force C and the related displacement sweb can be introduced using results of experimental tests [27] that take account of the influence of the axial force in the column, as shown in Fig. 9. 5.2. The procedure of solution The procedure of solution of the semi-continuous composite beams is based on the solution strategy described for continuous composite beams. In the following the beam is considered propped, so that the rotation required to the composite joint is maximised. On the analogy on the continuous beam, the procedure starts with the definition of a tentative value of the bending moment at support, as shown in Fig. 10. Due to the mechanism governing the resistance of the nodal region the bending moment leads to a known value of the interaction force at the support (Fig. 7a). Furthermore, the deformability of the nodal region modifies the compatibility conditions at the support, since the rotation in the last node of integration n þ 1 can be different from zero, due to a non-linear relationship between the bending moment and the relative rotation between the column and the beam exists. However, such a relationship is not known a priori since Eq. (24) contains some terms related to the problem of
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Fig. 10. The numerical solution of the semi-continuous beam.
the beam; in other words the problem is coupled, so that the rotation at support must be evaluated at each step. The iterative process is therefore scheduled as follows: • definition of a tentative value of the bending moment ðkÞ mnþ1 and of the related value of the interaction force ðkÞ Fnþ1 ; • iterative process to ensure the compatibility at the steel profile–concrete slab interface; the pivot parameter of this step of the procedure is the value of the profile–slab slip at the first node of integration, while the control parameter is the value of the interaction force at the last node of integration. In this way the process is oriented to solve the following non-linear equation: ðkÞ
Fnþ1 ððs2 Þj1 Þ ¼ Fnþ1
ð26Þ
• on the analogy with continuous beams, when cracking occurs in the slab, according to a proper criterion, the last iterative process is aimed to define the distribution of the slip at the rebar–concrete interface. In this case the process must be performed for each part of the slab between two subsequent cracks (located at the generic nodes icr , jcr ) and is oriented to solve the non-linear equation: Tctjcr ððs2 Þj1 ; ðs1 Þðj;hÞ icr Þ ¼ 0
remaining kinematic parameters given in Eq. (24) can be calculated according to the procedure discussed in Figs. 8 and 9. In addition, the knowledge of the distribution of the curvature along the beam allow to calculate the related ðkÞ ðkÞ rotation at the support unþ1 ðmnþ1 Þ. At this point the outer control condition: ðkÞ
ðkÞ
unþ1 ðmnþ1 Þ ¼ ujoint
ð28Þ
can be checked. Obviously the non-linear equations (26)–(28) have to be solved using proper tolerances. The procedure is very similar to the one developed for continuous beams, since the only different aspects are the boundary condition that controls the iterative procedure to solve the partial shear connection problem and the need to introduce a deformability at the end of the beam that is coupled with the behaviour of the beam as a whole. Eqs. (20)–(23) are the only used in the procedure based on the finite difference method, conversely the derivatives of the function s1 and s2 can be evaluated using the array of the generalised moment–curvature relationship. As a consequence, the solution of the flexural problem of the section has not to be solved at each node of integration and at each iteration.
ð27Þ 6. Conclusions
When the two above conditions are satisfied, the values of the profile–slab slip and the rebar–concrete slip at the end of the beam are known, so that the
An advanced modelling of steel–concrete composite beams requires the explicit introduction of local
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interaction phenomena i.e. shear connection force-slip relation and bond of reinforcement in hogging moment regions. To this end, a generalisation of the well known NewmarkÕs kinematic model of the composite cross section has been presented and the main computational problems related to local (cross section) and global (beam) non-linear analysis have been analysed. At local level, the attention has been focussed on the definition of a generalised moment–curvature relationship; at a global level the solution of simple but meaningful structural schemes has been analysed. The reliability and the effectiveness of the procedure for continuous beams, which allows to obtain both global parameters such as rotations and deflections as well as local quantities such as slips, curvature, interaction forces and rebar strains, can be extended also to semi-continuous beams. In this case, however, the key issue is the evaluation of a reliable modelling of the relationship between applied bending moment and relative rotation in the column–beam connection. The use of a generalised moment–curvature relationship is really useful and enables the numerical solution of continuous and semicontinuous beams according to the secant method. As a result, force and deformation pattern in composite elements can be identified and simplified models can be developed to enhance frame analyses under vertical and lateral loads.
References [1] Aribert JM, Xu H, Ragneau E. Theoretical investigation of moment redistribution in composite continuous beams of different classes. In: Engineering Foundation, Composite Construction in Steel and Concrete III. ASCE; 1996. [2] COST C1 semi-rigid behaviour of civil engineering structural connections composite steel–concrete joints in braced frames for buildings. European Communities, Brussels, 1997. [3] Nethercot DA. Semi-continuous building frames. Proceedings of the 1st International Conference on Steel and Concrete Composite Structures, Pusan, Korea, 2001. p. 13–22. [4] Kemp AR, Nethercot DA. Required and available rotations in continuous composite beams with semi/rigid connections. J Construct Steel Res 2001;57(4):375. [5] Bode H, Kronenberger HJ, Michaeli W. Composite joints–– further experimental results. Proceedings of the International Conference Composite Construction, Conventional and Innovative, IABSE Innsbruck, 1997. p. 433–8. [6] Kattner M, Crisinel. Finite element modelling of semi-rigid composite joints. Comput Struct 2001;78:341–53. [7] Kemp AR, Dekker NW. Available rotation capacity in steel and composite beams. Struct Eng 1991;69(5). [8] Commission of the European Communities. Eurocode 4, Common Unified Rules for Composite Steel and Concrete Structures, ENV 1994-1-1, 1992.
[9] Commission of the European Communities. Eurocode 8–– Design Provisions for earthquake resistance of structures, ENV 1998, 1994. [10] Plumier A. European research and code developments on seismic design of composite steel concrete structures. Proceedings of the 12th World Conference on Earthquake Engineering, WCEE, Paper 1147, Auckland, New Zealand, 2000. [11] Bursi OS, Caldara R. Composite substructures with partial shear connection: low cycle fatigue behaviour and analysis issues. Proceedings of the 12th World Conference on Earthquake Engineering, WCEE, Paper 0498, Auckland, New Zealand, 2000. [12] Matsuo A, Nakamura Y, Takamatsu T, Yoshimasa Matsui Y. Fracture and plastic deformation capacity of the welded joint of composite beam-to-steel column connections. Proceedings of the 12th World Conference on Earthquake Engineering, WCEE, Paper 0181, Auckland, New Zealand, 2000. [13] Yang IL, Tagawa Y. Effect of joint details for behaviour of moment resisting steel frames with composite girders-toSHS column. Proceedings of the 12th World Conference on Earthquake Engineering, WCEE, Paper 0181, Auckland, New Zealand, 2000. [14] Fabbrocino G, Manfredi G, Cosenza E. Ductility of composite beams under negative bending: an equivalence index for reinforcing steel classification. J Construct Steel Res 2001;57:185–202. [15] Cosenza E, Zandonini R. Composite construction––Handbook of structural engineering. In: Chen WF, editor Purdue University, CRC Press LLC; 1997. [16] Aribert JM, Dinga ON. Modelling and experimental investigation of bolted flush end-plate composite beamto-column joints. Proceedings of the UEF Conference Composite Construction IV, Banff, Alberta, Canada, 2000. [17] Fabbrocino G, Manfredi G, Cosenza E. Non-linear analysis of composite beams under positive bending. Comput Struct 1999;70:77–89. [18] Manfredi G, Fabbrocino G, Cosenza E. Modelling of composite beams under negative bending. ASCE J Eng Mech 1999;125(6):654–62. [19] Fabbrocino G, Manfredi G, Cosenza E. Analysis of continuous composite beams including partial interaction and bond. ASCE J Struct Eng 2000;126(11):1288– 94. [20] CEB, Bulletin dÕInformation no. 213/214, Model Code 90, 1993. [21] Ciampi V, Eligehausen R, Popov EP, Bertero VV. Analytical model for concrete anchorage of reinforcing bars under generalised excitations. UCB/EERC 82-23, University of California, Berkeley, 1982. [22] Ollgaard JG, Slutter RG, Fisher JW. Shear strength of stud connectors in lightweight and normal weight concrete. AISC Eng J 1971. [23] Newmark NM, Siess CP, Viest IM. Tests and analysis of composite beams with incomplete interaction. Proc Soc Exp Stress Anal 1951;(1):75–92. [24] Reza Salari M, Spacone E, Shing PB, Frangopol DM. Non-linear analysis of composite beams with deformable shear connectors. ASCE J Struct Eng 1998;(10):124.
G. Fabbrocino et al. / Computers and Structures 80 (2002) 2241–2251 lasto-Plastique [25] Aribert JM, Labib AG. Modele de CalculE de Poutres Mixtes a Connexion Partielle. Construct Metall 1982;(4):3–51. [26] Leon RT, Viest IM. Theories of incomplete interaction in composite beams. In: Proceedings of UEF Conference,
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Composite Construction in Steel and Concrete III. ASCE; 1996. p. 858–70. [27] Kuhlmann U. Influence of axial forces on the component web under compression. COST Document C1/WD2/01, Delft Meeting, 1999.
Modelling of continuous steel–concrete composite beams: computational aspects G. Fabbrocino, G. Manfredi *, E. Cosenza Department of Structural Analysis and Design, University of Napoli Federico II, Via Claudio 21, 80125 Napoli, Italy Received 15 November 2000; accepted 4 July 2002
Abstract Steel–concrete composite members are an interesting option for structural designers, but the reliability of design procedures both in the case of gravity and seismic loads is in continuous development. The issue is very complex, since behaviour of continuous composite beams results from local phenomena of interaction such as partial shear connection and bond. Furthermore, composite beams in buildings generally are not characterised by a full continuity due to the beam to column connections; thus the analysis and the detailing of such parts have a key role in the development of suitable design procedures. In the present paper, some computational aspects related to the modelling of composite flexural members are discussed with reference to continuous and semi-continuous structural systems widely used in practice. 2002 Civil-Comp Ltd. and Elsevier Science Ltd. All rights reserved.
1. Introduction The research efforts in the last years have been devoted to analyse the mechanical behaviour of steel– concrete composite members subjected to gravity loads [1–7]. Thus codes concerning such structures, i.e. Eurocode 4 [8], provide rules only for vertical loading patterns. On the other hand, Eurocode 8 [9] deals with steel– concrete composite members, but the short part that is devoted to this specific type of structures is poor due to the lack of knowledge [10]. As a result, the extension of available code provisions for composite structures to seismic resistant structures is a very difficult task, since many aspects have to be analysed. In fact, a primary goal of such analyses is really a reliable assessment of the ductility of both composite
*
Corresponding author. Tel.: +39-81-768-3424/3488; fax: +39-81-768-3424/3491. E-mail address: [email protected] (G. Manfredi).
beams and columns, of the connection rotation capacity and also of the structural behaviour under cycling loading leading to strength softening [11]. Researches concerning each one of the above topics are complex, since composite behaviour results from many interaction phenomena that involve the steel component, the concrete component and the shear connection between the two parts [2,5,7]. As a consequence, the attention cannot be focussed on each component without taking into account the interaction with all the related aspects, as demonstrated by the effects of Northridge and Hyogoken-Nambu earthquakes on many composite floors-steel columns connections [12,13]. Furthermore, negative bending moments, that develop at the support regions of composite beams, modify the structural response of members due to tensile stresses acting on the slab and the consequent cracking of the concrete, so that steel reinforcement properties can influence the plastic rotation capacity of members [14]. In the present paper, some computational aspects related to the numerical simulation of composite beams are examined. In particular, the attention is focussed on
0045-7949/02/$ - see front matter 2002 Civil-Comp Ltd. and Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 5 - 7 9 4 9 ( 0 2 ) 0 0 2 5 7 - 2
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the modelling of continuous beams and semi-continuous structural systems, that are meaningful of the composite mechanism governing the transfer of loads and the development of the rotation in the joint region of composite frames. Overall performances of composite beams with different end connections are compared pointing out some interesting remarks on the role of the slab reinforcement ductility and of deformation at the slab–profile interface due to shear connectors. 2. The continuity at intermediate supports The combination of the steel and the concrete leads to traditional composite flexural members that can sustain very well positive bending moment, so that the simply supported beams seem to be the more reliable structural pattern. However continuous composite beams are also frequently used, since a number of advantages can be obtained, i.e. reduction of the deflections and/or of the cross sections [15]. Furthermore in building systems, the fully continuity at internal support cannot be generally achieved (Fig. 1), since in framed structures the columns usually cross the joint regions, while the steel beams are not continuous and connected to the columns with different steelworks [2]. The main feature of the composite construction is the continuity of the slab in the nodal zone, so that the concrete slab can sustain loads in this area, resulting in a strongly different structural response of the joint if compared to bare steel connections. In fact, the continuous concrete slab leads to a composite action in the joint region that is the most important aspect in the behaviour and modelling of these structures [16]. In order to simplify the discussion, a very simple detailing of the joint, the so-called contact plate connection, is analysed and compared to the fully continuous composite beams from a computational point of view.
Fig. 1. The continuous beam at intermediate support.
Fig. 2. Semi-continuous beam in framed structures.
As shown in Fig. 2, the joint detail is characterised by the absence of any connection between the steel profile and the column. It is commonly used when braced frames are concerned, so that beams are designed basically to sustain gravity loads. It is easy to recognise that steel reinforcement of the slab has a relevant role in the development of the composite action and in the structural response of the connection.
3. The theoretical model of the composite cross section The present section summarises the main features of a theoretical model for composite flexural members that is able to take directly account of partial interaction and bond acting on continuous structural patterns, as demonstrated by the effective comparison between numerical and former experimental data [17–19]. The approach is based on a simplified kinematics of the cross section that enables a mono-dimensional approach. This assumption requires the reliable definition of the slab effective width depending on the type of loading, hogging or sagging, and on connection detailing at the beam end, but leads to a strong reduction of the computational effort. This circumstance has been checked both theoretically and experimentally pointing out that simple expression for effective width can be used if the non-linear response of materials is properly taken into account. The cross section of the beam is characterised by a slip at the rebar–concrete interface, s1 , and a slip at the slab–profile interface, s2 , as reported in Fig. 3. A linear pattern of strains is applied to each component of the cross section. The curvature and the rotation are the same for each component (concrete slab and steel profile), therefore the uplift between the slab and the profile is neglected. The slab is characterised by crack distance, dcrack , depending on the geometry of the slab and the me-
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Fig. 3. The model of the cross section under negative bending.
chanical properties and diameter of the reinforcing bars according to CEB provisions [20]. Furthermore, it is assumed that concrete between two subsequent cracks is able to bear tensile stresses; the related resultant tensile force on the concrete Tct is given by the following equation: Z rct ðy; zÞ dA ð1Þ Tct ¼ Aeff
where Aeff is the region of the slab where the distribution of strains is influenced by bond interaction. Since a constant level of strain is assumed on the effective area the relation (1) can be simplified as follows: Tct ¼ rct Aeff
ð2Þ
where rct is the tensile stress acting on the concrete; as a consequence the resultant of tensile stresses is applied at the centroid of the effective area of concrete. This static parameter influences directly the tensile strain levels in the steel bars and the tension stiffening effect and is equal to zero when cracked sections are concerned. Two slips allow taking directly account of local interaction due to bond and shear connection. They can be evaluated referring to the following kinematic equations: s1 ðxÞ ¼ wsc ðxÞ wct ðxÞ
ð3Þ
wup s ðxÞ
ð4Þ
s2 ðxÞ ¼
wlow c ðxÞ
where wsc ðxÞ is the displacement of the reinforcement; wct ðxÞ is the displacement of concrete in tension in the effective area Aeff ; wup s ðxÞ is the longitudinal displacement of the upper fibre of the steel profile; wlow c ðxÞ is the longitudinal displacement of the lower fibre of the concrete slab. The static parameter related to shear connection is the interaction force F, that can be expressed as follows: F ðxÞ ¼
n X Fj
ð5Þ
j¼1
where Fj is the force acting on the generic shear connector located before the reference cross section.
Equilibrium of the composite cross section is dependent upon the following three equations: • longitudinal equilibrium of the steel profile subjected to the interaction force due to shear connectors: Z Fs ¼ rs ðy; zÞ dA ¼ F ð6Þ As
• longitudinal equilibrium of the concrete slab subjected to the interaction force due to shear connectors: Z Fc ¼ Asc rsc þ Tct þ rcc ðy; zÞ dA ¼ F ð7Þ Acc
• rotational equilibrium of the composite section as a whole: M ¼ Ms þ Mc þ Fd
ð8Þ
where Z rs ðy ds Þ dA Ms ¼ As
Mc ¼
Z
rcc ðy dc Þ dA þ rsc Asc ðysc dc Þ
Acc
þ Tct ðyG;Aeff dc Þ are the components of the global bending moment due to the profile, Ms , and the concrete slab, Mc respectively. • dc , ds and ysc represent the distances between the centroids of the two parts of the section and of the reinforcing bars and the reference axis x; • rcc and rct are respectively the magnitude of compressive and tensile stresses of concrete; • As , Asc and Acc are respectively the area of the steel profile, of the rebars and of the concrete in compression. The above equations are not sufficient to define the distribution of strain along the y-axis, since bond and shear connection have to be directly introduced by
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means of additional kinematic equations that can be written as follows:
the positive bending regions and the extended model can be introduced in the negative bending zones, where the tension stiffening of the concrete component develops.
• compatibility equation for bond: 3.1. The moment–curvature relationship
ds1 ðxÞ ¼ esc ðxÞ ect ðxÞ dx
ð9Þ
• compatibility equation for shear connection: ds2 ðxÞ ¼ es ðxÞ ec ðxÞ þ vðxÞd dx
ð10Þ
• equation of equilibrium of the concrete in tension subjected to the bond stresses: dTct ðzÞ ¼ nsc PUsb ðzÞ dx
ð11Þ
The above equations need specific boundary conditions that lead to analyse at the same time the entire beam. Furthermore the constitutive relationships for phenomena of interaction have to be introduced [21,22]. The model discussed above represents an extension of the well known model by Newmark [23] and widely used [24–26]. NewmarkÕs kinematic model is reported in Fig. 4, where the stress and strain parameters are identified as already done in Fig. 3. It is easy to recognise that NewmarkÕs model is not dependent upon bond interaction, since concrete cannot bear tensile stresses according to traditional design approach of concrete structures. This circumstance leads to assume the tensile force Tct equal to zero as done in cracked sections. This remark is very useful since it allows to reduce the computational effort in the positive bending regions; in fact when positive bending moments act on the cross sections the interaction force F is basically associated to compressive stresses on the slab, so that the reinforcement is under compression and/or under moderate tension, thus the assumption of perfect bond between rebars and concrete can be satisfactory. In conclusion, the classical model for the composite section can be used to fit the behaviour of the beam in
The response of composite cross sections depends on two static parameters representing interactions due to bond and shear connection, as a consequence the traditional one to one function resulting from the BernoulliÕs assumptions cannot represent all the balanced solutions for composite sections. Conversely a generalised moment–curvature relationship can be defined and plotted using a family of curves depending on the interactions force F and the resultant of the tensile stresses on concrete, Tct . In the present section the main computational aspects related to the definition of the above relationship are discussed with reference to both hogging and sagging bending. A finite number of curves corresponding to given values of the tension force in concrete Tct and of the interaction level F , can be defined, since they can range between a minimum and a maximum value related only to the mechanical and geometrical properties of the cross section. A reliable representation of the generalised moment– curvature relationship can be performed using a reference system ðM; v; Tct Þ. From a geometrical point of view, the generalised moment–curvature relationship results in a 3D domain that can be drawn using the sections with plane surfaces corresponding to given values of the tensile forces Tct acting on the concrete and varying the value of the interaction force F between the concrete and the steel components. The numerical procedure to define the relationship is based on the strip method, as shown in Fig. 5. In fact, both the slab and the steel profile are divided into a finite number of strips that can be identified by the position of the centroid, yGi , evaluated respect to the upper fibre of the slab and the corresponding area Ai . It is also as-
Fig. 4. The model of the cross section under positive bending.
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• the distribution of the strains on the concrete slab that satisfies the longitudinal equilibrium of the concrete component with the given curvature vj must be evaluated. This step requires again the solution of a non-linear equation; to this end the strain of the lower fiber of the slab can be addressed as the control parameter. The longitudinal equilibrium of the slab can be written in the following form: slab
nstr X
j;l rj;l c;i ðec;low ÞAc;i þ
i¼1
nsc X
j;l p n rj;l sc;i ðec;low ÞAsc;i Tct F ¼ 0
i¼1
ð14Þ Fig. 5. Evaluation of the moment–curvature relationship.
sumed that the strip is affected by a constant level of strain and therefore of stress. In this way the equations of the longitudinal equilibrium of the slab, of the longitudinal equilibrium of the profile and the global rotational equilibrium of the cross section can be written as a sum of finite terms. The method is well known for compact sections, but can be modified for composite sections according to the following steps:
where rj;l c;i represents the stress acting on the generic strip of the slab, nslab str is the total number of the strips and nsc is the number of reinforcing bars. • the last step is the calculation of the global bending moment acting on the section and corresponding to the curvature vj . pro
M¼
nstr X
slab
rj;l s;i Ac;i yGi
i¼1
þ
nsc X
þ
nstr X
rj;l c;i Ac;i yGi
i¼1 p rj;l sc;i Asc;i yGsc þ Tct yGAeff
ð15Þ
i¼1
• a number of values of the resultant of the tensile stresses on the concrete Tctp is chosen and is sequentially imposed on the cross section; • a number of values of the interaction force F n (the apex identifies the generic moment–curvature curve) is chosen and the strain of the lower fibre of the steel profile, ejs;low , is given as a pivot parameter in compliance with the ultimate strain under compression and tension; • the distribution of the strain in the profile that satisfies the longitudinal equilibrium of the steel component is evaluated using as control parameter the strain of the upper fibre of the profile ej;k s;up (the apex j gives the generic increment of the pivot strain, the apex k gives the generic iteration of the process). This step requires an iterative process to solve the non-linear equation that can be written as follows: pro
nsrt X
j;k n rj;k s;i ðes;up ÞAi F ¼ 0
ð12Þ
i¼1
where Ai is the area of the single strip of the profile, pro rj;k s;i is the stress acting on the strip, nstr is the total number of strips of the profile. The iterative process gives the value of the strain of the pivot strain ejs;low as shown in Fig. 5 and the curvature of the cross section can be defined as follows: vj ¼
ejs;low ej;k s;up Hprofile
ð13Þ
where the lever arm of each term is evaluated respect to the upper fibre of the slab. Finally the flexural problem of the composite section is completely solved, since the distribution of strains are known and the derivative of the slips s1 and s2 can be evaluated as follows: dsj1 ¼ ejsc epct dx
ð16Þ
dsj2 ¼ ejc;low ejs;up dx
ð17Þ
The derivative of the function s2 can be easily defined using the roots of the non-linear equations of equilibrium (12) and (14), while the derivative of the slips s1 can be calculated using the strain of rebars and of the concrete in tension placed in the effective area. In fact: Tct epct ¼ eðrpct Þ ¼ e ð18Þ Aeff The above level of strain depends only on the value of the imposed force Tctp . Furthermore, the strain of the rebars are generally very high compared with the strain of the surrounding concrete in tension, thus it can be stated that the second term of Eq. (16) can be neglected. As a result the following relationship exists:
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dsj1 ffi ejsc dx
ð19Þ
The above calculations lead to the definition of an array containing the generalised moment–curvature relationship that can be stored and used to solve the problem of the composite beam [18]. The definition of the moment–curvature relationship of the cross section according to the classical NewmarkÕs approach can be easily carried out using the same above equations. In fact, the moment–curvature becomes a 2D domain, since Tct is equal to zero, so that all the related terms in the above equations disappear and the interaction force F is the key parameter [17].
iterative process is carried out. Rotations and displacements are then calculated by integration of the distribution of the curvature. The solution of the problem can be obtained using the shooting technique and the finite difference method. The specific boundary conditions concerning the slip at the slab–profile interface require the solution of the entire beam. In the following, the discussion is simplified referring to a simple structural system of the beam that is characterised by both geometrical and mechanical symmetry. The numerical procedure is based on the classical method of compatibility, so that the bending moment at the support is the main unknown of the problem and the beam is statically determined. The procedure is described in Fig. 6, and is based on the following steps:
4. The solution of the continuous composite beam The numerical procedure developed to solve continuous composite beams combines the main behavioural aspects of the different regions of the beam. In fact, the traditional NewmarkÕs model for positive bending is used when composite section is subjected to sagging bending moment and the extended one is introduced when cracked zones of the beam are concerned. The aim of the procedure is the definition of the actual moment–curvature relationship in each section of the beam among all the balanced solutions available in the generalised moment–curvature relationship, thus an
1. definition of discrete number of cracked sections along the beam depending on the properties of the slab and of reinforcement; 2. iterative process to ensure the compatibility at internal support. When a symmetrical structural scheme ðkÞ is considered the bending moment at support, mnþ1 , (n þ 1 is the number of integration nodes, k is the iteration number) can be assumed as pivot parameter ðkÞ and the corresponding rotation, unþ1 , as control parameter; 3. iterative process along the whole beam to ensure the compatibility at slab–profile interface for a given
Fig. 6. The numerical solution of the continuous beam.
G. Fabbrocino et al. / Computers and Structures 80 (2002) 2241–2251 ðkÞ
value of mnþ1 ; in this case the slip at the end of the beam, ðs2 ÞðjÞ 1 , governs the process and the check of compatibility is performed at the support, where the ðjÞ slip at the slab–profile interface, ðs2 Þnþ1 , must be zero due to symmetry; 4. when cracking occurs in the slab, according to a proper criterion, the last iterative process is oriented to define the distribution of the slip at the rebar–concrete interface. In this case the process must be performed for each part of the slab between two subsequent cracks (placed at the generic nodes icr , jcr ) until the control condition ðTct Þjcr ¼ 0 is satisfied. It is worth noting that the family of curves giving the moment–curvature relationship is calculated for each typical section of the beam, thus the number of the resolving equations to be used in the iterative procedure can be reduced. The remaining equations can be written in the following form: j
j Fiþ1 ¼ Fij þ F i
ðs1 Þjiþ1 ¼ ðs1 Þji þ
2247
transfer capacity; only the presence of a continuous concrete slab in the nodal region leads the joint to bear bending moments. This specific type of joint is generally subjected to negative bending since it is basically used in braced frames; in the following a symmetrical structural scheme is analysed in order to simplify the discussion. Therefore, the continuity of the slab allows to develop a tensile force in the reinforcement that is balanced horizontally by the compressive action applied at the lower centroid of the contact plate, as shown in Fig. 7a. The force T is applied at the centroid of the rebars, so that the geometry of the connection allows to define easily the lever arm d. It is worth noting that the strength of the nodal region is much lower than the strength of the composite cross section, since the whole contribution of the bending moment carried by the profile is equal to zero.
ð20Þ
j;h ðs2 Þj;h iþ1 ¼ ðs2 Þi þ
ds1 dx
j
ds2 dx
Dx
ð21Þ
i
j;h Dx
ð22Þ
i
j;h j;h ðTct Þj;h iþ1 ¼ ðTct Þi þ nsc pU Dx ðsb Þi
ð23Þ
i
where F is equal to zero if the shear connectors are not present in the considered section, while, if the connectors are present, it is given by the force acting on the shear connectors evaluated by the corresponding constitutive relationship. The procedure is quite complex, but an effective use of the moment–curvature relationships makes it reliable, as demonstrated by the validation against former experimental results [19].
5. The analysis of the semi-continuous beam 5.1. The modelling of the joint region The analysis of the semi-continuous composite beam requires the discussion of the mechanisms governing the strength and the deformation of the connection between the column and the profile. The considered joint is very simple; any steelwork is not needed, since the steel profile is only installed on a settlement welded to the column. In this way the connection at the steel profile level is able to bear only shear forces, but there is not moment
Fig. 7. The strength and the deformation of the contact plate beam–column connection.
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Even if the development of the strength of such a connection is very simple, the mechanism governing the joint rotation is characterised by complex phenomena. In fact, the relative rotation between the column and the beam, developed in the case of symmetrical structure, is related basically to: • the slip between the reinforcement and the concrete ssc;b at the end of the beam; • the slip between the reinforcement and the concrete ssc;j in the nodal zone; • the slip at the profile–concrete slab interface, sconn ; • the displacement of the web panel of the column subjected to the compressive action C, sweb . The calculation of the relative rotation resulting from the above sources of deformation can be performed assuming that the end section of the beam remains plane; furthermore the slab and the profile are characterised by the same rotation, in compliance with the assumptions made for the cross sections of the beam. In this way, referring to the symbols given in Fig. 7a, the relative rotation can be written as follows: ujoint ¼
ðsweb þ sconn þ ssc;j þ ssc;b Þ d
ð24Þ
The equation shows that the deformation of the nodal region is coupled with the response of the beam, since the slips at the slab–profile interface and at the rebar– concrete interface depends on phenomena of interaction developing along the beam. For what concerns the slip of the reinforcement placed in the joint region, it can be evaluated solving the problem of a concrete element subjected to a tensile force T, assuming a given distance between the cracked sections. It is easy to recognise that unbalanced bending moments lead to shear forces on the column in the joint region; in this case an additional source of deformation has to be introduced with a slight increase of the computational effort, but without relevant changes in the numerical approach. The solution of the problem is schematically reported in Fig. 8, where the part of the concrete slab placed in the nodal region is shown. The shooting technique can
Fig. 8. Solution of the slab under tension in the joint.
Fig. 9. Evaluation of the force–displacement relation of the web panel.
be used to determine the distribution of slips at the rebar–concrete interface complying with the boundary conditions. In this case only the bond related equations must be used to solve the problem. Therefore Eqs. (9) and (10) can be used; it is worth noting that Eq. (10) can be more effectively substituted by the following: drsc ðzÞ 4 ¼ sb ðzÞ dz U
ð25Þ
that represents the longitudinal equilibrium of the reinforcing bar subjected to the bond stress. The boundary conditions can be written in terms of stress of the rebar that can be easily evaluated according to the strength mechanism shown in Fig. 7a. On the other hand the relationship between the compressive force C and the related displacement sweb can be introduced using results of experimental tests [27] that take account of the influence of the axial force in the column, as shown in Fig. 9. 5.2. The procedure of solution The procedure of solution of the semi-continuous composite beams is based on the solution strategy described for continuous composite beams. In the following the beam is considered propped, so that the rotation required to the composite joint is maximised. On the analogy on the continuous beam, the procedure starts with the definition of a tentative value of the bending moment at support, as shown in Fig. 10. Due to the mechanism governing the resistance of the nodal region the bending moment leads to a known value of the interaction force at the support (Fig. 7a). Furthermore, the deformability of the nodal region modifies the compatibility conditions at the support, since the rotation in the last node of integration n þ 1 can be different from zero, due to a non-linear relationship between the bending moment and the relative rotation between the column and the beam exists. However, such a relationship is not known a priori since Eq. (24) contains some terms related to the problem of
G. Fabbrocino et al. / Computers and Structures 80 (2002) 2241–2251
2249
Fig. 10. The numerical solution of the semi-continuous beam.
the beam; in other words the problem is coupled, so that the rotation at support must be evaluated at each step. The iterative process is therefore scheduled as follows: • definition of a tentative value of the bending moment ðkÞ mnþ1 and of the related value of the interaction force ðkÞ Fnþ1 ; • iterative process to ensure the compatibility at the steel profile–concrete slab interface; the pivot parameter of this step of the procedure is the value of the profile–slab slip at the first node of integration, while the control parameter is the value of the interaction force at the last node of integration. In this way the process is oriented to solve the following non-linear equation: ðkÞ
Fnþ1 ððs2 Þj1 Þ ¼ Fnþ1
ð26Þ
• on the analogy with continuous beams, when cracking occurs in the slab, according to a proper criterion, the last iterative process is aimed to define the distribution of the slip at the rebar–concrete interface. In this case the process must be performed for each part of the slab between two subsequent cracks (located at the generic nodes icr , jcr ) and is oriented to solve the non-linear equation: Tctjcr ððs2 Þj1 ; ðs1 Þðj;hÞ icr Þ ¼ 0
remaining kinematic parameters given in Eq. (24) can be calculated according to the procedure discussed in Figs. 8 and 9. In addition, the knowledge of the distribution of the curvature along the beam allow to calculate the related ðkÞ ðkÞ rotation at the support unþ1 ðmnþ1 Þ. At this point the outer control condition: ðkÞ
ðkÞ
unþ1 ðmnþ1 Þ ¼ ujoint
ð28Þ
can be checked. Obviously the non-linear equations (26)–(28) have to be solved using proper tolerances. The procedure is very similar to the one developed for continuous beams, since the only different aspects are the boundary condition that controls the iterative procedure to solve the partial shear connection problem and the need to introduce a deformability at the end of the beam that is coupled with the behaviour of the beam as a whole. Eqs. (20)–(23) are the only used in the procedure based on the finite difference method, conversely the derivatives of the function s1 and s2 can be evaluated using the array of the generalised moment–curvature relationship. As a consequence, the solution of the flexural problem of the section has not to be solved at each node of integration and at each iteration.
ð27Þ 6. Conclusions
When the two above conditions are satisfied, the values of the profile–slab slip and the rebar–concrete slip at the end of the beam are known, so that the
An advanced modelling of steel–concrete composite beams requires the explicit introduction of local
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interaction phenomena i.e. shear connection force-slip relation and bond of reinforcement in hogging moment regions. To this end, a generalisation of the well known NewmarkÕs kinematic model of the composite cross section has been presented and the main computational problems related to local (cross section) and global (beam) non-linear analysis have been analysed. At local level, the attention has been focussed on the definition of a generalised moment–curvature relationship; at a global level the solution of simple but meaningful structural schemes has been analysed. The reliability and the effectiveness of the procedure for continuous beams, which allows to obtain both global parameters such as rotations and deflections as well as local quantities such as slips, curvature, interaction forces and rebar strains, can be extended also to semi-continuous beams. In this case, however, the key issue is the evaluation of a reliable modelling of the relationship between applied bending moment and relative rotation in the column–beam connection. The use of a generalised moment–curvature relationship is really useful and enables the numerical solution of continuous and semicontinuous beams according to the secant method. As a result, force and deformation pattern in composite elements can be identified and simplified models can be developed to enhance frame analyses under vertical and lateral loads.
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