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An Abridged Review of Instantaneous Analysis of Steel-Concrete Composite flexural members And a Novel Approach for Prediction of Deflection Subjected to Service Load
M. A. Modia, M. P. Ramnavasb, K. A. Patelc,*, and Sandeep Chaudharyd
*Corresponding Author
aPh.D. Research Scholar,
Department of Civil Engineering,
Sardar Vallabhbhai National Institute of Technology (SV-NIT)
Ichchhanath, Dumas road, Surat 395007, India.
E-mail: megh.modi20101995@gmail.com
cExecutive Director/Design,
Discipline of Civil Engineering,
“Metro Bhawan”,
East Highcourt Road (VIP Road)
In front of Dr. Babasaheb Ambedkar College
Near Dikshabhoomi, Nagpur 440010, India.
E-mail: ramnavas@mahametro.org
bAssistant Professor,
Department of Civil Engineering,
Sardar Vallabhbhai National Institute of Technology (SV-NIT)
Ichchhanath, Dumas road, Surat 395007, India.
E-mail: iitd.kashyap@gmail.com
Abstract
Steel-concrete composite structures found significance in the infrastructure sector due to the economy, ease, and speed of construction. Composite members are slender; however, they are susceptible to serviceability criteria like deflection and cracking. Therefore, it is required to review existing research works to bridge the gap between theoretical and practical assessment of serviceability criteria of steel-concrete composite flexural members. The present paper consists of an abridged literature review on serviceability criteria and technological aspects of composite flexural members at service load and ultimate load. Various methods are available in the existing literature for the prediction of deflection in composite flexural members. However, these methods are based on computing the effective moment of inertia of the span without considering cracked zones’ length. Therefore, in this paper, a novel approach is further proposed to derive a simplified equation for the prediction of deflection by idealising cracked and uncracked zones within the span. The derived equation takes into account the length of cracked zones in addition to the cracking and tension stiffening effects of the concrete slab. The results obtained from the equation are compared with the experimental and theoretical results available in the literature subjected to various loadings. The comparison shows a good agreement that is acceptable to civil engineering’s practical purposes.
Keywords: Composite members, Cracking, Deflection, Serviceability, Tension Stiffening,

1. Introduction
   Steel-concrete composite members are progressively being utilised in tall buildings and long-span bridges nowadays, owing to the economy, ease, and speed of construction. The steel-concrete composite beam/girder with profiled sheeting requires no false formwork and without profiled sheeting requires less false formwork, as shown in Fig. 1(a) and (b), respectively. Composite sections cause a reduction in-depth compared to conventional reinforced concrete sections because of the composite action of steel to resist tension and concrete to resist compression. Due to this, the number of floors could be increased for a given overall height of the building. In bridge superstructure, the composite sections have become prevalent due to the requirement of a large span-to-depth ratio. Composite members are slender; however, they are susceptible to serviceability criteria like deflection and cracking. Therefore, it is required to review existing research works to bridge the gap between theoretical and practical assessment of serviceability criteria of steel-concrete composite flexural members. The presented investigations would address the future scope of research in the domain of steel-concrete composite structures. The future scope includes a novel approach for the prediction of deflection in composite flexural members within the serviceability limit.

(a)

(b)
Fig.1 Steel-concrete Composite cross-section (a) with profiled steel sheeting deck and (b) without profiled steel sheeting deck
Steel-concrete composite flexural members are the combination of a steel beam/girder and reinforced concrete slab, as shown in Fig. 1. The monolithic action is achived by providing shear connectors between beam/girder and slab (Fig. 1), which leads to slippage resistance at their interface. Several researchers [Uy (1997)[1], Ramnavas et al.(2015)[2], Costa et al.(2021)[3]] have assumed that the stiffness of the connectors is infinite i.e. the connection at the interface is perfectly rigid in their studies. In fact, the slippage at the interface may usually occur, which significantly affects the serviceability aspects.
Various shear connectors (such as headed studs, channel, perfobond) are evolved and used. The headed studs are the most commonly used shear connectors (Fig. 2(a)). They resist horizontal shear as well as vertical uplift force subjected to external loads. After welding, the connectors provide adequate resistance to shear at the interface of steel beam/girder and concrete with the help of the head. Perfobond shear connectors consist of a connected steel plate with holes, as shown in Fig. 2(b). The concrete flows through these holes and acts as a dowel to provide shear resistance at the interface between steel joist and concrete. In channel shear connectors, the conventional channels are welded to the flange of the steel beam/girder, as shown in Fig. 2(c). The resistance provided by these shear connectors is comparatively more than that of headed stud connectors. This enables the replacement of many headed studs with a few channel connectors.
The above mechanical shear connectors requires welding of connector to the steel girder. Due to this, it becomes a tedious process to dismantle them from steel girder and reuse steel girder for further construction purpose. Nowadays, a lot of emphasis is laid on utilization of shear connectors which can be dismantled easily for reuse of steel girder. This has drawn the attention of many researchers in the area of demountable shear connectors. Demountable shear connectors are similar to headed stud shear connector with only difference being this type of shear connectors are connected to steel girder with bolted connection as shown in Fig. 2(d).
In addition to the above shear connectors (transfer systems), some other evolved shear connectors include reinforcement bars welded to steel section; providing indentations and corrugulations to steel deck; rectangular steel pipe; perforated with flange head; I-shaped, C-shaped, L- shaped, U-shaped, V-shaped, Hat-shaped (Oguejiofor and Hosain 1995; de Andrade et al. 2004; Ju et al. 2007; Vianna et al. 2009; Kim et al. 2011; Shariati et al. 2012; Mazoz et al. 2014; Su et al. 2014; Balasubramanian and Rajaram 2016; Zhang et al. 2017; Allahyari et al. 2018; Mansouri et al. 2019; Shariati and Mehrabi 2020; Zhu et al. 2021).

(a)

(b)

(c)

(d)
Fig.2 Types of Shear Connectors: (a) Headed Studs Connectors, (b) Perfobond shear connectors, (c) Channel shear connectors, and (d) Demountable connection
The above shear connectors caused high bearing stresses in surrounding concrete which led to concrete cracking. To overcome this phenomenan, an epoxy bonded connection is evolved which reduces damages to surrounding concrete and improves degree of interaction between steel and concrete sections. Epoxies are formed by polymerization of two components, the resin and the hardener. These components develop polymer chains at the interface and hence bonding occurs between two materials. (Kumar et al. 2017, 2018; Bhardwaj 2020; Bhardwaj et al. 2021, MMMM, one more paper)

2. Literature review
   Many research works have been carried out on behaviour of steel-concrete composite flexural members subjected to ultimate and service loads. Some key research areas of interest have been identified and presented under the following heads (Figure 3):
3. Degree of shear connection
4. Cracking of concrete and tension stiffening
5. Behaviour of composite flexural members/Structures
6. Provisions for cracking of concrete in Eurocode

2.1 Degree of shear connection
The steel-concrete girders consist of three parts: (1) Steel girder, (2) Concrete slab and (3) Shear connectors. The shear connector at the interface of concrete slab and steel girder plays a significant role in determining the degree of composite action between them. Design of shear connector is one of the most complex part in design of composite members. Partial interaction at the interface causes slippage and increase in deflection (Kalibhat et al, 2020)[4].
Headed studs are the most frequently used type of mechanical shear connectors. The codes used for design of shear stud connectors are BS 5950 (2000), AISC (2016) and Eurocode4 (2004). The shear capacity calculated by formulation given in Eurocode4 gave more accurate results than BS 5950 and AISC which overestimated the shear capacity of headed studs. The experimental push out test showed three modes of failure in composite members (1) Concrete cone failure in concrete surrounding the shear stud, (2) shear stud failure due to yielding and (3) Combination of concrete and shear stud failure. For concrete with lower compressive strength the failure is governed by concrete cone failure while for concrete with higher compressive strength the failure is governed by shear stud yielding (Lam and El-Lobody)[5].
Perfobond shear consist of a steel plate with holes through which reinforcement bars are passed(Su et al, 2016) [6]. This type of connectors have proved to be efficient in providing necessary shear resistance at the interface of steel and concrete in steel-concrete composite structure(Su et al, 2015[7], Su et al, 2012[8], Su et al, 2012[9]). Perfobond shear connector gives the advantage of ease in manufacturing and higher shear capacity in comparison to headed stud shear connectors(Su et al, 2014[10], Liu et al, 2003[11]). Zhang et al.(2017) examined perfobond shear connector groups and revealed that when the number of connectors in heavily loaded layers increases, the load carried by each connector in these layers decreases.
Channel shear connectors are gaining popularity for ease in construction as it utilizes conventional welding system in manufacturing. Channel shear connectors provide higher resistance compared to headed studs shear connector. Thus it can replace large number of shear connector with a single channel shear connector (Maleki and Bagheri, 2008[12]).Through FE modelling it was found that the main parameters affecting the shear resistance offered by channel shear connectors are concrete strength, web and flange thickness of the connector while changing the height of the connector had a negligible effect (Maleki and Bagheri, 2008[13]).
A novel hat shaped shear connector was proposed by Kim et al. (2011) that outperformed perfobond and standard headed stud shear connector in terms of shear resistance. The degree of shear interaction of the steel-concrete composite member consisting hat shaped shear connector is directly proportional to number of penetrating rebars through this shear connectors. Mazoz et al.(2013) examined the performance of I-shaped shear connectors by carrying out push out tests and obtained that I-shaped shear connectors remain ductile throughout the loading procedure. Su et al.(2014) introduced perforated shear connector with flange head (PSCFH) that showed higher shear capacity and improved ductility in steel-concrete composite members than perfobond shear connectors.
Balasubramaniam and Rajaram (2016) carried out push out tests on angle shear connectors with varying angle lengths from 50 to 90mm. The angle connectors with length 50 mm showed higher shear resistance than angle connectors with length 90mm. A novel numerical approach was developed by Allahyari et al. (2018) using artificial neural network (ANN) to estimate shear strength of perfobond rib shear connectors. Similarly, shear strength of V-shaped shear connector was computed by Mansouri et al. (2019) using artificial neural network (ANN) soft computing technique. Shariati et al. (2020) carried out an experimental investigation and found that C-shaped shear connector showed better shear resistance than L-shaped shear connector. Zhu et al. (2021) proposed a nonlinear finite element (FE) method considering nonlinear material properties and rectangular steel pipe shear connector-concrete interface.
Kozma et al. (2019) carried out a detailed discussion on failure pattern, types of damages and ability of demountable shear connectors for re-use. Suwaed and Karavasilis (2020) developed a demountable high friction based shear connectors which prevents slippage at the interface of steel and concrete (i.e. provide full degree of shear interaction) at service as well as ultimate load stage. Tzouka et al. (2021) developed a three-dimensional finite element model to study the behavior of innovative demountable shear connectors subjected to push-out tests.
A lot of research work is ongoing for utilization of epoxy based adhesives in order to enhance the load carrying capacity of the members by ensuring much lesser deflection. In the mechanical shear connectors there is high stress concentration in concrete surrounding the connectors. To overcome this limitation research has been done on adhesive based shear connection at the interface of steel and concrete.The instantaneous and long term bond strength of adhesive at the interface of steel and concrete depends on several parameters such as thickness of layer of adhesive, concrete placement duration, adhesive composition, temperature conditions and aspect ratio of bonded area (Kumar et al., 2017). These adhesive based connections have been found to be stiffer than shear stud connections and hence causes reduction in deflection (Kumar et al, 2018). Bhardwaj et al.(2021) exhibited a three-dimensional finite element model (FEM) to forecast the behaviour of bonded steel–concrete composite flexural components at ultimate load.
2.2 Cracking of Concrete and Tension Stiffening
In the case of concrete members, cracks occur, in the regions over which the tensile stress exceeds the tensile strength of the concrete. In reinforced concrete sections, owing to the presence of reinforcing bars and the distributed nature of cracking, the effective stiffness of the sections is higher than the stiffness of the cracked sections and this phenomenon is called tension stiffening (Massicotte et al, 1990)[14]. The tension stiffening effect may be significant in the service load performance of beams and slabs and the displacements may be overestimated if this effect is neglected(Ghali et al, 2002)[15].
Rots and Borst (1988)[16] have applied novel computational techniques and a continuum based elastic softening model to simulate the strain-softening behavior of concrete in a direct uniaxial (tension) test and observed non-symmetric deformations. Yankelevsky and Reinhardt (1989)[17] have conducted monotonic deformation controlled tensile tests and reported the behavior as linear upto 80% of its tensile strength and thereafter becoming nonlinear. After the peak stress, softening of concrete starts, signified by the negative slope of the descending branch of the stress strain curve. The softening happens due to the increase in microcracks. Most of the energy is absorbed in the descending zone and the stress drops to zero at a strain which is about 40 times that at the peak stress. Tension softening curves for normal strength concrete and high strength concrete were experimentally found by Chen and Su (2013)[18].
The tension stiffening decays as the load increases beyond the cracking load and proportionally is more significant for low reinforcement ratios than for higher ones (Khalfallah and Guerdouh 2014)[19]. Cracking and tension-stiffening parameters probably have the most significant effect on numerical results of concrete members subjected to short-term loading (Sokolov et al. 2010)[20]. Also, cracking in reinforced concrete is a major source of nonlinearity in the deformation analysis and hence it is important to model the cracking and the post cracking behavior adequately (Gribniak et al. 2012[21]; Sahamitmongkol and Kishi 2011[22]). Odrobinak et al. (2013)[23] has experimentally verified the steel concrete composite bridge with strain and deformation measurements and compared with the FEA. It is reported that concrete cracking and tension stiffening effects influence the stresses in the girders at intermediate supports. They also emphasized the need for nonlinear analysis for accounting for these effects.
Some researchers have taken into account the tension stiffening effect by using a complete stress strain curve for concrete in tension which includes the descending post peak branch (Rahman and Hinton 1986[24]; Carreira and Chu 1986[25]; Kaklauskas and Ghaboussi 2001[26]; Bischoff and Paixao 2004[27]; Torres et al. 2004[28]). A number of studies have been carried out to establish the stress strain curve of concrete in tension. An equation for stress-strain curve of concrete in tension has been proposed by Carreira and Chu (1986)[25] which includes the effect of cracking and slippage at cracks along the reinforcement. Kaklauskas and Ghaboussi (2001)[26] have proposed a method to determine the stress-strain curve for concrete in tension from the short-term flexural tests and have also identified a number of parameters which influence the stress strain curve of concrete in tension.
Neural network model has been developed for predicting the stress strain curve for concrete in tension using these parameters. Torres et al. (2004)[28] have proposed a methodology for obtaining the coefficients required in defining the stress-strain curve of concrete, in tension, taking into account the time-dependent behavior of reinforced and prestressed concrete members subjected to service load.
The tension stiffening effect has been taken into account in some other approaches by modifying the stress-strain curve of the reinforcement and thereby increasing the stiffness of the reinforcement or by assuming an area of concrete at the level of steel to be effective in providing the stiffening (Zou 2003). Some empirical approaches e.g. Branson (1977)[30], ACI 318-11 (ACI Committee 318 2011)[31], CEBFIP model code 1990 (CEB-FIP 1993), Eurocode 2 : Part 1-1 (2004)[32], Ghali et al. (2002)[15] have also been proposed to account for the tension stiffening effect. Branson (1977)[30] and ACI 318-11[31] suggest the use of effective moment of inertia for the cracked sections to incorporate the tension stiffening effect. The effective moment of inertia may be considered as a weighted average of gross moment of inertia of section neglecting the reinforcement and moment of inertia of transformed fully cracked section about its centroidal axis. Other empirical approaches i.e CEB-FIP Model Code 1990, Eurocode 2 : Part 1-1 (2004)[32] and Ghali et al. (2002)[15] are based on the use of mean values of stress, strain and displacements for the cracked sections. These mean values are obtained by interpolating between the values obtained by considering the section to be cracked and that obtained by considering the section to be uncracked. The value of interpolation coefficient (?) depends primarily on the ratio of the tensile stress in the extreme fiber of the section to the tensile strength of the concrete. The value of ? for no cracking and maximum cracking is equal to zero and one respectively.
Sokolov et al. (2010)[20] proposed a new tension stiffening law for short term deformational analysis of flexural reinforced concrete members. It was derived from experimental moment-curvature diagrams of RC beams. The method applied for deriving the relationship is based on Layer section model and combines direct and inverse techniques of analysis of RC members. A quantitative dependence has been established between the length of the descending branch and the reinforcement ratio. The length of the descending branch of the curves reflecting the tension stiffening effect was considerably more pronounced for beams with smaller reinforcement ratios.
The parametric study carried out by Khalfallah and Guerdouh (2014)[19] also confirms the increase of tension stiffening with decrease in reinforcement ratio and also reports that the tension stiffening contribution is more pronounced with light reinforcement with high quality of concrete. A discrete model for tension stiffening has been presented by Dai et al. (2012)[33]
Kang et al.(2017) studied the effect of reinforced engineered cementitious composites (ECC) on tension stiffening and calculated the minimum reinforcement ratio required to ensure the transfer of force in major cracked region. Sunayna and Barai (2018) deduced that utilization of recycled aggregates in combination with fly ash reduces tension stiffening effect and hence results in higher midspan deflection in flexural members. Zanuy and Ulzurrun (2019) proposed a tension chord model which represented tension stiffening characteristics at the interface of Ultra High Performance Fibre Reinforced Concrete and Reinforced Concrete. This model accurated predicted crack widths and curvature between cracks. Daud et al. (2021) carried out experimental and numerical investigation to study effect of utilizing steel fibres on tension stiffening characterisitcs of flexural members. Utilization of steel fibres caused a reduction of 3.5% in deflection of flexural members due to enhanced tension stiffening.
2.3 Behaviour of Composite flexural members/Structures
Extensive literature is available on the analysis of composite structures (composite beams and frames) up to ultimate load stage and at service load stage and is presented section wise.
2.3.1 Methods Applicable for Analysis upto Ultimate Load Stage
A general step by step model for the non-linear analysis of reinforced concrete, prestressed concrete and composite steel-concrete planar frame structures has been presented by Cruz et al. (1998)[38]. The model has been designed to simulate different kinds of structural changes expected for a structure. The nonlinear behavior of concrete has been incorporated by means of a damage model. The method requires discretisation of the structural members along the length.
A procedure for prediction of behavior of composite girder bridges has been presented by Kwak and Seo (2000)[39]. A nonlinear analysis algorithm has been presented which consists of four basic steps: (i) the formation of the current stiffness matrix, (ii) the solution of the equilibrium equations for the displacement increments, (iii) the state determination of all elements in the model, and (iv) the convergence check. A layered approach for the sections has been adopted and the beam is discretised, along the axis, in finite elements. Using the procedure, the effect of slab casting sequences and drying shrinkage of concrete slabs on the short-term and long-term behaviour of composite steel box girder bridges has been studied by Kwak et al. (2000)[40]. It has been concluded that the effect of slab casting sequence is negligible for both the short-term behaviour and long-term behavior of bridges.
A generalized procedure for nonlinear analysis of three dimensional reinforced, prestressed and composite frames has been presented by Marí (2000)[41]. The structural effects of the load and temperature histories, nonlinear behavior of material and relaxation of steel have been taken into account along with the nonlinear geometric effects. Possible changes on the structural geometry, boundary conditions and loading at any time have also been taken into account. The method uses three dimensional Hermitian thirteen degree of freedom finite elements.
Another model has been presented by Fragiacomo et al. (2004)[42]. The model employs finite element technique with every finite element having ten degrees of freedom. The concrete has been modeled through a constitutive law that takes into account cracking phenomenon. The concrete cracking and the tension stiffening effect have been modelled through a softening law for long term (time-dependent) analysis. For analyzing the non-linear problem, a new iterative procedure has been proposed which has been termed as modified secant stiffness method.
Park et al. (2013)[43] have recently presented a method of approximating the internal axial force by the Fourier series, for calculating the deflections and internal forces of various types of single span composite beams considering partial interactions. Gara et al. (2014)[44] have presented a novel higher-order beam model capable of capturing shearlag phenomenon and overall shear deformability of composite steel–concrete girders with partial shear interaction. An isogeometric approach based on non-uniform rational B-spline basis functions has been presented by Lezgy-Nazargah (2014)[45] for the analysis of composite steel–concrete beams. The theory satisfies all the kinematic and stress continuity conditions at the layer interfaces and considers effects of the transverse normal stress and transverse flexibility.
General purpose finite element software have been used for the analysis of composite beams (Baskar et al., 2002[46]; Baskar and Shanmugam 2003[47]; Barth and Wu 2006[48]; Zhao and Li 2008[49]; Liang et al. 2004[50], 2005[51]) and composite columns (Gupta et al. 2007[52]; Singh and Gupta 2013[53]; Gupta and Singh 2014[54]). Three-dimensional finite element models using ABAQUS has been used, by Baskar et al. (2002)[46] and Baskar and Shanmugam (2003)[47] to carry out the nonlinear analysis of steel-concrete composite plate girders under negative bending and shear loading, by Barth and Wu (2006)[48] to predict the ultimate load behavior of a four-span continuous composite steel bridge tested to failure, by Zhao and Li (2008)[49] for a new steel-concrete composite beam, by Liang et al. (2004[50], 2005[51]) to account for the geometric and material nonlinear behaviour of continuous composite beams and by Okasha et al. (2012)[55] for modelling a continuous five span bridge superstructure with four composite girders in structural reliability analysis.
The concrete is generally modelled in FEA by smeared crack model, cracking model for concrete or damaged plasticity model (ABAQUS 2011). In the case of smeared crack model, the concrete model is a smeared crack model in the sense that it does not track individual “macro” cracks. The presence of cracks enters into these calculations by the way in which the cracks affect the stress and material stiffness associated with the integration point. The model is intended as a model of concrete behaviour for relatively monotonic loadings under fairly low confining pressures.
The cracking model for concrete is most accurate in applications where the brittle behaviour dominates such that the assumption that the material is linear elastic in compression is adequate. The damaged plasticity model which is a continuum, plasticity-based, damage model for concrete, is designed for applications in which concrete is subjected to monotonic, cyclic, and/or dynamic loading under low confining pressures. It assumes that the main two failure mechanisms are tensile cracking and compressive crushing of the concrete material.
In the finite element suit ABAQUS (2011), failure ratios and tension stiffening options are used to define the smeared crack model (Barth and Wu 2006[48]; Baskar et al. 2002[46]; Baskar and Shanmugam 2003[47]; Sahamitmongkol and Kishi 2011[22]). The reinforcement in the concrete slab is modelled using the rebar option. The assumed linear tension stiffening model also accounts for the strain softening behaviour for the cracked concrete and local bond slip effects (Baskar et al. 2002[46]; Wahalathantri et al. 2011[56]). The predominant ratios are the ratio of ultimate biaxial compressive stress to ultimate uniaxial compressive stress and the ratio of uniaxial tensile stress to uniaxial compressive stress at failure (Zhao and Li 2008[49]). The value of tension stiffening defined as the total strain at which the tensile stress is zero, is usually taken as 10 times the strain at failure, for heavily reinforced concrete slabs and it has been found that this value is not adequate for concrete slabs in composite beams and a value of a total strain of 0.1 was used for reinforced concrete slabs in composite beams in some studies (Rex and Easterling 2000[57]; Liang et al. 2004[50], 2005[51]).
The geometrical modelling of steel beam can be done as wire element with the assigned cross sectional properties or as shell elements or as three dimensional elements. The slab is modelled as shell elements or as three dimensional elements. The connection between the steel beam and concrete slab can be with tie constraints or with springs (ABAQUS 2011).
Yang et al. (2017)[58] developed a higher order three-scale model for the analysis of composite structure having heterogeneous materials. Wang et al. (2018)[59] sformulated numerical method considering shrinkage stress and loading coupled with mechanical loading which can be applied in analysis of composite structures. Bilotta et al. (2019)[60] adopted a novel two level computational method that is composed of two levels: the frame level and the cross sectional level. The proposed procedure is suitable for analysing composite structures as it takes into consideration stress values obtained from the 3D analysis of element.
Uddin et al. (2020)[61] proposed the finite element model for steel concrete composite beams based on a higher order beam theory. This theory was developed by third order variation of longitudinal deformation with respect to beam depth.
Zhou et al. (2021)[62] prepared a nonlinear FEM model and validated it with experimental test results. It was concluded that the slab thickness had more impact on shear capacity of beam than reinforcement ratio and connection type. Hence, it is conservative to neglect the shear contribution of concrete slab for the non-compact sections that has generally application in bridge design.
Men et al. (2021)[63] performed a nonlinear finite element analysis for slender composite girders and validated with experimental results. It was observed that reinforcement ratio, concrete slab thickness and web thickness has significant effect on ultimate load of steel concrete composite girders.
In all these procedures, the division of the beam along the length and across the cross section is required to take into account the nonlinear behavior under ultimate load but this division leads to huge increase in the computational effort.
2.3.2 Methods Applicable for Analysis at Service Load Stage
A linear method of analysis for simply supported steel concrete composite beams has been presented by Bradford (1991)[64], Bradford and Gilbert (1992a)[65], Bradford (1997)[66]. Concrete is assumed to be uncracked as the concrete slab is mostly in compression in simply supported composite beams. Another simplified analytical model for simply supported composite beams has been presented by Amadio and Fragiacomo (1997)[67] using the closed form solutions. The analysis of simply supported composite beams is simpler than analysis of continuous composite beams since cracking at the supports is not involved.
A simplified procedure for two equal span continuous composite beams subjected to service load, taking into account cracking, has been proposed by Gilbert and Bradford (1995)[68]. It has been assumed that the steel component does not yield at the service load. The concrete has been assumed to be completely cracked once the stress in top fiber of a cross-section exceeds the tensile strength of concrete. The procedure may be considered as analytical in the absence of cracking. The transformed section approach has been used and the beam is taken as one element without subdivision along the length and across the cross-section. The tension stiffening has been neglected in the procedure. Slip at the interface of concrete slab and steel section has been neglected. It has been found from a previous study (Bradford and Gilbert 1992b)[69] and an experimental investigation (Bradford and Gilbert 1991)[70] that the effect of slip on the composite beams under sustained service loads is insignificant in comparison to the deformations and can be neglected provided the shear connectors are at sufficiently close spacing. This procedure (Gilbert and Bradford 1995)[68] has been further extended by Bradford et al. (2002)[71] making it applicable for two unequal span continuous composite beams subjected to different loads at the spans. The procedure though convenient for two-span beams, would tend to become tedious if extended to the composite beams having more than two spans.
Uy (1997)[1] presented the effect of creep and shrinkage on time dependent behavior of profiled composite slab using age-adjusted effective modulus and relaxation procedure in time domain. The comparison was drawn for the results of these procedure with BS 5950: Part 4 and concluded that there is need to consider the variation in stress and strain in bottom fibre of profiled steel sheeting and top fibre of concrete slab due to creep and shrinkage
Another method, a hybrid procedure (Chaudhary et al.2007a[72], 2007b[73]; Pendharkar 2007[74]), considers the effects of cracking and tension stiffening and is also computationally efficient; however, the formulations are cumbersome, and there is a scope for simplification in a manner similar to that reported in the case of RC beams (Patel et al. 2016)[75]. The procedure is not applicable for all types of loading, and also does not account for the settlement of supports in the case of continuous beams.
Wang et al. (2011)[76] proposed closed-form solutions for simply supported pre-stressed old new concrete composite beams and validated the expressions using test results reported in previous literature on steel-concrete composite beams.
Al-Deen et.al.(2011)[77] carried out an experimental study to evaluate the long term behavior of steel-concrete composite beams having partial shear connection at the interface of steel beam and concrete slab. They also modelled time-dependent behavior of beams using finite element method based on experiment results. From this research work, they concluded that there is need to consider the time dependent behavior of shear connector at the interface of steel joist and concrete in design for better accuracy.
Ramnavas et al. (2015)[2] in their research work proposed a cracked span length beam element. The beam element consists of three zones: one uncracked zone at the middle and two cracked zones at the ends. By utilization of this element, a hybrid analytical–numerical approach has been developed for composite bridges. This process considers tension stiffening effect due to presence of reinforcement in concrete slab. The process gives results in form of crack lengths and deflections along with redistributed moments. This process requires further modification for consideration of dependent effects like creep and shrinkage.
Ramnavas et al. (2017)[78] in their research work proposed a cracked span frame element. The derivation of analytical expressions have been done for stiffness and flexibility scoefficients, mid span deflections ,end displacements and load vector of the cracked span length frame element. The process uses an iterative method for generating the cracked region lengths and the distribution coefficients, and gives results in form of the redistributed moments and inelastic deflections. The process needs a portion of the computational effort that is needed for the numerical methods existing in literature and gives adequately correct results.
Lasheen et al. (2019)[79] in their research work proposed new equation for calculating effective slab width of steel-concrete composite structure at service load. The two parameters of steel section slenderness ratio and span to width ratio were taken into account for obtaining the equations.
Nie et al. (2019)[80] reviewed the utilization of Steel Plate Concrete Composite (SPCC) strengthen technique and its effect on service load requirements of structure. SPCC specimens has anti-permeability and anti-knocking property which considerably protects the specimen from corrosion and hence improve its durability to meet service load state requirements.
Costa et al. (2021)[3] analyzed the load-mid span deflection curves of steel concrete composite slabs as it is an essential relationship that is considered in serviceability limit checks. Three new equations for effective moment of inertia were developed using empirical approach to the data obtained from the experimental research work.
The above methods are simple, however do not consider all the aspects.
2.4 Provisions in Eurocode for Cracking of Concrete in Composite Beams
To account for the effects of cracking, Eurocode 4 : Part 1-1 (2004)[82] and Eurocode 4 : Part 2 (2005)[83] give guidelines for general cases and also for use in specific cases. In general cases, an ‘uncracked analysis’ with flexural stiffness of uncracked sections is to be performed, from which the stresses are calculated. In the regions in which the extreme fibre tensile stress in the concrete exceeds twice the tensile strength, the properties should be reduced to those of a cracked section, and a cracked analysis is to be performed. In specific cases of continuous beams with all of the ratios of adjacent spans (shorter / longer) not less than 0.6, the effect of cracking is accounted for by using the properties of cracked sections for 15% of the spans on either side of the internal supports and the properties of the uncracked sections elsewhere.
It can be seen that in the general cases in Eurocode 4 : Part 1-1 (2004)[82] and Eurocode 4 : Part 2 (2005)[83] above, the properties are to be reduced to those of the cracked section in regions in which the extreme fibre stress exceeds twice the tensile strength. In effect, this approach reduces the length of the cracked zones and compensates for the tension stiffening effect; however, it involves an approximation. Hence, the guidelines in Eurocode 4 : Part 1-1 (2004)[82] and Eurocode 4: Part 2 (2005)[83] in both general and specific cases, could lead to some error in the estimation of the cracked lengths and, hence, in the analysis, in some cases.

3. Methodology
   The methodology used in this research work is numerical integration. The deflection at mid-span dm can be expressed by following equation:

(eq.3.1)
3.1 Cross section properties
The cross section properties are derived with reference to work done by Costa et al.(2021)[3]
Fig3.1 Typical cross section of a steel deck composite slab
As shown in fig 3.1, the distance between neutral axis and most compressed is denoted ycf . CGT is the centroidal axis of the trspezoidal section and CGF is the centroidal axis of typical steel deck section.
The area of web’s trapezoidal section At will be given by following equation.

The distance from the centroidal axis of web’s trapezoidal section to the lower end of the typical cross-section is

The moment of inertia of trapezoidal section in relation to centroidal axis (CGT) will be given as:

The neutral axis of the uncracked section will be given as:

Where modular ratio ae is given as:

In which Ea is the Young’s Modulus of steel and Ecs is the secant modulus of elasticity of concrete.
The uncracked moment of inertia will be given as:

Where n is the number of typical cross section and I`sf is the moment of inertia of typical steel deck section.
The cracked moment of inertia will be calculated as follows:
For calculation of cracked moment of inertia ycf is converted to y2 which is the position of neutral axis of cracked section. y2 is calculated by following equation.

Where,

Thus, the cracked moment of inertia will be obtained as:

3.2 Derivation
Fig.3.2 Simply supported beam where K is Uncracked Length
In the cracked region, effective moment of inertia is to be considered which is given by following equation:
(Eq.3.2)
Where Ieff= Effective moment of inertia
Iun= Uncracked moment of inertia
Icr= Cracked moment of inertia
?,?= Interpolation coefficients of uncracked and cracked section respectively
Here, ? and ? is obtained from following equation:
? = 1-? = 1-(K ft/sun)2 (Eq.3.3)
where K is the coefficient representing influence of duration of application or repetition of
loading
ft is the tensile strength of concrete
sun is the stress at tensile face with considering cracked section
3.2.1 For one point symmetric loading

Fig.3.3 Simply Supported beam subjected to one point load
Here the loading is symmetric therefore eq.3.1 can be rewritten as:

On derivation the deflection equation for one point load was obtained as:

3.2.2 For two point symmetric loading

Fig.3.4 Simply Supported beam subjected to two point load
Here the loading is symmetric therefore eq.3.1 can be rewritten as:

On derivation the deflection equation for two point load and Ls=L/4 was obtained as:

On derivation the deflection equation for two point load and Ls=2.5L/8 was obtained as:

3.1.3 For uniformly distributed symmetric loading

Fig.3.5 Simply Supported beam subjected to uniformly distributed load
Here the loading is symmetric therefore eq.3.1 can be rewritten as:

On derivation the deflection equation for one point load was obtained as:

4. Validation Study
   Three validation studies have been carried out for above derived equations
   4.1 Case 1
   In this case deflection of one way concrete slab with steel decking subjected to two point loading is calculated using above derived equations and validated with experimental results obtained by Costa et.al. (2021).The properties of MF-50 steel deck slab are as follows:
   b2 = 175 mm (Longer width of trapezoidal section)
   bb = 130 mm (Shorter width of trapezoidal section)
   hF = 50 mm (Height of trapezoidal section)
   bn = 305 mm (Width of flange section)
   tc = 50 mm (Depth of flange section)
   dF =74 mm (Distance of centroidal axis of typical steel deck from top fibre)
   A`F,ef = 1452 mm2 (Effective area of typical steel deck cross section)
   tn = 1.25 mm (Thickness of typical steel deck section)
   fck = 30.39 MPa (Grade of concrete)
   Isf = 719469.33 mm4 (Moment of inertia typical Steel deck cross section)
   Ea = 201290 MPa (Youngs Modulus of elasticity of steel)
   Ecs = 27563 MPa (Secant modulus of elasticity of concrete)
   n = 3 (No. of typical steel deck cross sections)
   L= 1800 mm (span of slab)
   The comparison is shown by following graph:

4.2 Case 2
In this case deflection of one way concrete slab with steel decking subjected to uniformly distributed loading is calculated using above derived equations and validated with results obtained from numerical method developed by Gilbert R.I.(2013). The properties of the KF-70 slabs specimens are as follows:
L = 3100 mm
hT = 150 mm
AF,ef = 1320mm2 Ea = 212000 MPa Ecs = 30725MPa Iun = 278x106mm4 Icr = 102x106mm4 ? = 0.411 ? = 1-0.411 = 0.589 The comparison is shown in following table: Deflection from derived equation Deflection from method developed by R.I.Gilbert Difference(%) 1.89 1.85 2.16 4.3 Case 3 In this case deflection of one way concrete slab with steel decking subjected to combination of uniformly distributed loading and two point loading is calculated using above derived equations and validated with results obtained from numerical method developed by Gholamhoseini et.al.(2015). The properties of KF70 slab are as follows: L = 4800mm Ec = 30100 MPa AF,ef = 1467mm2
Iun = 413×106 mm4
Icr = 147×106 mm4
? = 0.5
? = 1-0.5 = 0.5
The comparison is shown in following table:
Deflection from derived equation Deflection from method developed by Gholamhoseini et.al. Difference(%)
11.04 11.15 0.99

5. Conclusions
   In this paper, a literature review and the serviceability criteria of deflection of steel-concrete composite members is studied. The equations of deflection are derived taking into consideration uncracked length and effective moment of inertia for cracked section is calculated considering tension stiffening effect. Three cases of validation are presented in this paper.
   In first case concrete slab with steel decking is subjected to increasing two point load. The graph plotted from the results of deflection using derived equation initially showed a maximum error of 44% with graph plotted from experimental results taken from literature of Costa et al. because of the proposed method used considered the moment of inertia to be constant i.e. Ieff while in real experimental case stiffness and hence moment of inertia varies with increment in load. But at serviceability limit of L/350 the results obtained from proposed equation on comparison with experimental results taken from literature of Costa et al. showed very good agreement. Then the crushing of concrete occurs which the derived equation does not take into account.
   In second case concrete slab with steel decking is subjected to uniformly distributed load of 7.68 kN/m including self-weight. The instantaneous deflection obtained using derived equation shows good agreement with deflection obtained by numerical method developed by Gilbert with a mere difference of 2.16%.
   In third case concrete slab with steel decking is subjected to uniformly distributed load of 3.8 kN/m including self-weight and two point load of 8kN each. The instantaneous deflection obtained using derived equation shows good agreement with deflection obtained by numerical method developed by Gholamhoseini et.al. with a mere difference of 0.99%.
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