Analytical model of composite floors with steel fibre reinforced concrete slab subjected to fire

AbstractUnder fire, membrane action plays an important role in the performance of slabs subjected to large deflections. In this paper, a new model is proposed based on a proper approximation of horizontal displacements for a simply supported composite slab. The novelty of the proposed approach consists in a special treatment of the system of shape functions for the “in-plane” displacements. Moreover, a load applied to the slab is divided into two components, so that one component is balanced by the membrane forces, while the second one is transmitted by the bending forces (including transfer of shear and moment). The deflection due to thermal elongations is replaced by the identical deflection caused by a fictitious load. Unknown parameters are calculated using the principle of virtual displacements. The effectiveness of the model is validated by the results obtained from experiments.


Introduction
It has long been recognised that the load applied to the slab is transmitted partly by the flexural rigidity and partly by the membrane action of the slab (Timoshenko, Woinowsky-Krieger 1959). In case of small displacements, design models do not allow for taking into account membrane forces (Timoshenko, Woinowsky-Krieger 1959). But in case of large deflections, which are acceptable in fire, membrane action can considerably enhance its load-carrying capacity (Bailey 2001). This behaviour is possible only in case of using the ductile material. It was proved that steel fibre reinforced concrete is sufficiently ductile for creating the membrane action (Fike, Kodur 2011;Meskenas et al. 2014;Marciukaitis et al. 2011). Partial differential equations describing the behaviour of the slab in the case of large deflections were introduced by Föppl and von Kármán as early as in the 1910s. Since the 1950s the interest to the membrane actions of slabs had increased. After Cardington experiments in the mid of the 1990s (Bailey et al. 1999(Bailey et al. , 2000, two different approaches have been developed to address the membrane action of slabs at elevated temperatures. They may be classified as: (i) analytical approaches, and (ii) FEM based approaches. There is a variety of both sim-plified and advanced analytical models developed since then (see, e.g. Bailey 2001;Usmani et al. 2001;Li et al. 2007;Wang 1997;Vassart, Zhao 2011). These models are proposed namely for reinforced concrete. A number of FEM models based on geometrically and physically nonlinear approach using commercial software packages like ABAQUS with FEAST subroutines  or ANSYS (Vassart, Zhao 2011;Kodur et al. 2013) have been introduced. Specialised software packages like Vulcan (Huang et al. 2003;Huang 2010), Safir (Franssen et al. 2002;Fike, Kodur 2011) or SlabFEM (Tesar 2008;Stadler, Mensinger 2014) have been developed to simulate composite slabs under elevated temperatures. When utilizing the FEM models, the slab is mostly modelled with layered shell elements. It is worth mentioning that just in Fike and Kodur (2011) the authors consider the use of steel fibre reinforced concrete to enhance the fire resistance of composite floor assemblies. The relevant thermal and mechanical properties of steel-fibre reinforced concrete for use in fire resistance calculations can be found in Lie and Kodur (1996) and Bednář et al. (2013).
Starting from the previous achievements in this field, the present paper proposes a new simplified model, which belongs to category (i). As an auxiliary tool for design-ing composite floors with steel fibre reinforced concrete slabs exposed to fire, the model should be simple and sufficiently quick in predicting the reasonable distribution of membrane forces and displacements during a fire. The proposed model covers both geometric and material non-linearity. It properly reflects the effect of thermal elongations governing the vertical displacements of the slab, namely at the stage of heating. In this stage, correct approximations of displacements and forces as well as the fulfilment of boundary conditions play a decisive role. Approximations of displacements in case of clamped edges were proposed by Timoshenko in Timoshenko and Woinowsky-Krieger (1959). A new model of the slab with simply supported edges is the subject of this paper. The effect of fire on a ceiling slab is twofold. It gives the rise to thermal elongations in the middle surface and causes a distinct reduction of the material stiffness.
The authors do not strive to predict the overall behaviour of the slab. Especially we are not aimed at the final stage preceding its collapse. Therefore, the model does not analyse in detail such phenomena as cracking in tensile zones, transient creep strains as well as development of plastic strains for concrete in compression, since their description at elevated temperatures might be dubious. Instead, the material properties are captured by the temperature-dependant overall characteristics, the stressstrain relationships, obtained from the experiments. The complexity of material modelling leaps to the eyes when comparing the results published in Lie and Kodur (1996) and Bednář et al. (2013). Recall that the resulting curves differ significantly.
The performance of the model is validated against a set of experiments carried out by the research team at the Czech Technical University in Prague. This paper is part of the complex research program including both the experiments on the response of composite floor slabs subjected to fire (Bednář et al. 2012(Bednář et al. , 2013 and numerical simulations.

Development of a new model
The model is based on the von Kármán theory of large deflections for plates and shallow shells. Displacements in the middle surface of the slab are crucial variables of the model. Unknown parameters are calculated using the principle of virtual displacements. A novelty value of the proposed model consists in a special selection of shape functions for the "in-plane" displacements u and v. They are classified into two sets. The first one serves to aptly cover the linear parts of expressions for membrane forces and the second set is used to satisfy the static boundary conditions (zero normal and shear forces prescribed on the boundary).

Basic ideas and assumptions of the model
A load p applied to the slab is divided into two yet unknown components p 1 and p 2 , so that p 1 is balanced by the membrane forces and p 2 is balanced by the bending forces (Timoshenko, Woinowsky-Krieger 1959). The load p 1 (membrane) generates a vertical displacement a 01 at the centre of the slab which is identical to the vertical displacement a 02 caused by the load p 2 (bending).
The deflection w 0 at the slab's centre brought about by thermal elongations is replaced by the identical deflection caused by a fictitious load p 0 , which is balanced by the membrane forces. Therefore, the large deflections of the slab affect only the membrane forces.
The underlying equation thus reads: where: p 1 is part of the load balanced by the membrane forces, p 2 is part of the load balanced by the bending forces, and p is the applied external load. Consequently, the second basic equation assumes this form: where: a 00 is the total vertical displacement at the centre of the slab, w 0 is the vertical displacement at the centre of the slab caused by a fictitious load p 0 (thermal effect), a 01 is the vertical displacement caused by p 1 (membrane), and a 02 is the vertical displacement caused by p 2 (bending). The slab in question is perceived to be simply supported ( Fig. 1). This assumption seems reasonable for common slabs subjected to a high temperature as the connections gradually lose their stiffness due to cracks arising in continuous slabs above the supports and the slab subsequently becomes discontinuous.
A common secant modulus of elasticity is adopted for the whole thickness of the slab (defined in Section 2, Fig. 3). Geometric characteristics of the structure were calculated by stratifying it into layers and reducing the width of layers with respect to the modulus of elasticity. This modulus is dependent on temperature. Iterations were used to obtain a proper value of the modulus for each temperature depending on the current mechanical strain attained in each layer.

Approximations of vertical displacements
The vertical displacements of the middle surface of the slab caused by the membrane forces (loads p 1 and p 0 ) are approximated as: where: L and B are the half of the length and the half of the width of the slab, respectively (see Fig. 1). The vertical displacements of the middle surface of the slab caused by bending forces (load p 2 ) are approximated as: The vertical displacements of the middle surface of the slab caused by fictitious membrane forces (fictitious load p 0 ) are approximated as: The vertical displacement at the centre of the slab w 0 , induced by thermal expansion, is evaluated by means of this simplified formula: where: ε θ is the overall thermal strain. The overall thermal strain, ε θ , of the homogenized cross-section is calculated as the weighted average satisfying the compatibility conditions along the thickness of the slab: where: ε θ,i is the thermal strain and E i is the modulus of elasticity of each layer, n -number of layers of stratified cross-section.

Approximations of horizontal displacements
In case of a simply supported slab the following boundary conditions for the membrane forces should be satisfied: The von Kármán strain-displacement equations written for the middle surface will be adopted as a stepping stone for the approximation of "in-plane" displacements u and v: The membrane forces are functions of strains: where: E is the modulus of elasticity, A x , A y and A xy are the cross-section areas of the slab per 1 m run at each direction (described in Section 2, Fig. 5) and ν is the Poisson ratio. Substituting Eqns (8)-(10) to Eqns (11)-(13) yields: The linear parts of Eqns (14) and (15)   Substituting Eqns (17), (18) and (3) to Eqns (14) and (15) and setting x = L for N x and y = B for N y gives: Solving differential Eqns (17) and (18) ( 3   ) ) ( ( . Since Eqns (21) and (22) 3  3   00  3  3   3  3  3 3  00  00  00  00  3  3 3 3 , where d 00 is the unknown parameter.
Recall that the nonlinear part of Eqn (16) satisfies both boundary conditions a priori. By differentiating Eqns (21) and (22) The compatibility condition requires that the shear strains calculated from Eqns (23) and (24) By solving them we eliminate parameters b 00 , b 01 , b 02 , b 03 .
Finally, inserting these parameters into Eqns (21) and (22) we arrive at final expressions for displacements u and v.

Determination of still unknown parameters
The remaining unknown parameters a 00 , c 00 , c 01 , c 02 , c 03 and d 00 are to be found using the principle of virtual displacements: The strain energy of membrane forces will be evaluated by: Analogically, the strain energy of bending forces will be calculated from: where I x , I y and I xy are the cross-section moments of inertia of the slab per 1 m run at each direction (described in Section 2, Fig. 5).
Considering just moderate strains due to bending and torsional effects, the bending moments in Eqn (28) have already been expressed by the following formulae: Equation (26) must be satisfied for any variation of each of the unknown parameters. In such a way we obtain the system of six Eqns (32) V Assuming that bending moments are not to be dependent on the initial deflection induced by temperature, allows us to determine parameter a 02 which is proportional to p 2 , see Eqns (38): To obtain the fictitious load p 0 as a function of w 0 , Eqns (32)-(37) can be used replacing w by w 1 and a 00 by w 0 .
The calculation proceeds in these steps: 1. Determine a 02 using Eqn (38). 2. Determine p 0 as a function of w 0 from the system of Eqns (32)-(37) replacing w by w 1 and a 00 by w 0 . 3. Determine c 00 , c 01 , c 02 , c 03 , d 00 as a function of a 00 by solving the system of Eqns (33) (39)

Validation of the model
The ribbed composite steel fibre reinforced concrete slab with steel beams displayed in Figure 2 was considered as an object of investigation. Experimental data of slab tests at elevated temperatures obtained at 21.10.2010 and 11.10.2011 in PAVUS laboratory in Veselí nad Lužnicí (Bednář et al. 2012(Bednář et al. , 2013 are used to validate the analytical model. The span and width of the structure is equal to 4.5 and 3.0 m, respectively. The slab was loaded by the load equal to 1.8 kN/m 2 . The distance of the steel beams IPE100 is equal to 1 m. The beams are connected to the slab using studs. The stress-strain curves of steel-fibre reinforced concrete used in the experiment at different temperatures are shown in Figure 3. Formulas for the stress-strain relationships of SFRC were obtained based on the experiments (Bednář et al. 2013) using the method presented in Gernay and Franssen (2011), by the first author as part of his PhD thesis and improved by the first author. The secant modulus of elasticity (E = tanα, see Fig. 3) dependent on temperature was used in computational simulations. Material characteristics of steel were applied according to EN 1993EN -1-2 (2005.
The nominal standard fire curve was used at the experiments. The experiment at 2011 was terminated before reaching the fire resistance after 126 min in order to follow the development of displacements in the course of cooling phase. A thermal analysis was carried out using the differential method of heat transfer (Incropera et al. 2006). The effective thickness of the slab for thermal analysis was taken as h 1 +h r /2 (EN 1994(EN -1-2 2005. Distribution of temperature through the thickness in time is shown in Figure 4. The temperature of steel beams and the remaining parts of the ribs was assumed to be equal to the temperature of the external layer of the calculated slab. The thermal characteristics of materials are used according to EN 1993-1-2 (2005) and EN 1992-1-2 (2004. Geometric characteristics of the structure were calculated by stratifying the structure into layers and reducing the width of layers according to the current modulus of elasticity (Fig. 5). Rigid connection is assumed between the beams and the slab.

Results and discussions
A stress distribution in the middle surface is shown in Figures 6 and 7. In the centre region of the slab tensile stresses are observed, while in the vicinity of edges compressive stresses appear. This phenomenon confirms the theory of tensile membrane at the centre of the slab and compressive ring near the boundaries. Shear stress distribution is shown in Figure 8.
The evolution of horizontal displacements u in the x-direction on one quarter of the slab is displayed in Figure 9. It is shown that the symmetry condition is satisfied. The utmost horizontal movements of individual points toward the centre of the slab appear at the corners of the slab. The evolution of membrane forces in the middle surface due to horizontal displacements u and v is shown in Figure 10. The evolution of the load taken by the membrane and by the bent slab, respectively, is shown in Figure 11. It is evident that at ambient temperature all the load is taken by the bending resistance of the slab, but increasing temperature causes the increase in deflections and after 60 minutes (in current experiment) the bending resistance of the slab becomes negligible.
The evolution of vertical displacements in time is shown in Figures 12 and 13. The thermal deflection represents the main component of total deflection.  Figures 14  and 15. Slight fluctuations of displacements at the initial phase of experiment can be ascribed to the fact that boundaries at the beginning of the experiment do not perform as simply supported, but provide a certain torsional rigidity, which, in the course of time, becomes negligible. This process is rather difficult to describe analytically. This phenomenon is typical for both analytical and FEMbased models; and at the initial stage of heating predominates over the remaining effects of material modelling.
The prediction of failure of structure could be performed by analysing the strains and defining limits. These results are under preparation and will be presented in a forthcoming journal paper.

Conclusions
The model with novel approximations of horizontal displacements, which satisfy the boundary conditions, allows predicting the overall performance of the slab. At the end of heating, the deflection caused by mechanical load exhibits not more than 4% from the total deflection. The model confirms the phenomena of appearance of a tension membrane at the centre of the slab and a compressive ring around the perimeter. The load transmitted by the bending resistance of the slab is less than 20% of the applied load after 30 min and is less than 12% of the applied load after 60 min according to the current experiment. The ratio of calculated to experimentally obtained deflections varies from 0.72 to 1.2 not considering the phase of the collapse of the structure.