Numerical analysis on axially-and-rotationally restrained cold-formed steel beams subjected to fire

Thin-Walled Structures 104 (2016) 1–16

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Numerical analysis on axially-and-rotationally restrained cold-formed steel beams subjected to fire Luís Laím, João Paulo C. Rodrigues n ISISE – Institute for Sustainability and Innovation in Structural Engineering, University of Coimbra, Portugal

art ic l e i nf o

a b s t r a c t

Article history: Received 26 October 2015 Received in revised form 25 February 2016 Accepted 4 March 2016

A numerical parametric investigation into the response of axially and rotationally restrained compound cold-formed steel beams in fire has been carried out. A suitable finite element model was validated against experimental fire tests. Some parameters that could have influence on the behaviour of these beams were evaluated, such as the section geometry, initial applied load, slenderness and influence of the axial and rotational restraints. The results showed that the critical temperature of axially restrained beams may drop significantly (reaching a 50% reduction) and, beyond a certain value of axial or rotational restraint it may be no longer possible to change its fire resistance. Furthermore, it was still concluded that the methods established in EN1993-1.2:2004 are not appropriate for the fire design of these beams (reaching relative differences in the order of 20% and some predictions can be unsafe). & 2016 Elsevier Ltd. All rights reserved.

Keywords: Cold-formed Steel Beams Fire Axial restraint Rotational restraint Finite element analyses Buckling

1. Introduction The structural performance of cold-formed steel (CFS) members under fire conditions has increasingly been studied over the last years. However, the great majority of them have been performed on single sections (with just one profile) [1–3] and assumed that the internal forces and moments at supports of the members remain unchanged throughout the fire exposure [4,5]. The more or less rigid connection between a member and the other elements of a structure restrains the thermal elongation of that member and also the rotation of its ends. This rotational restraint, as well as the axial restraint, has an elastic nature and might influence the fire behaviour of the members under fire conditions, and therefore their critical temperature and fire resistance. The imposed axial restraint can generate substantial unforeseen forces in the members during fire adding another hazard that may cause an unpredictable structural failure, in contrast to the imposed rotational restraint, which might avoid a sudden failure [6]. Note that this subject has only been studied on hotrolled steel members (class 1 or 2 cross-sections) [7–16]. Most of the conclusions from these works state that the fire resistance of identical compression members may be not significantly affected by the stiffness of the surrounding structure, due to the beneficial n

Corresponding author. E-mail address: [email protected] (J.P.C. Rodrigues).

http://dx.doi.org/10.1016/j.tws.2016.03.004 0263-8231/& 2016 Elsevier Ltd. All rights reserved.

effect of the rotational restraint at their ends [7–11]. On the other hand, local buckling near the beam ends might reduce the stiffness of the restrained beam greatly, whereas the restraint stiffness affected greatly the value of the axial force in the restrained beams [12]. In addition, if catenary action might be developed (i.e., the connections are able to sustain increased tensile forces), the beams will be able to survive to very high temperatures without collapse [13]. Rotational restraint will also improve the fire response of restrained beams by reducing the mid-span bending moment in the beam [14]. Still note that the higher the end axial restraint stiffness or the lower the end rotational restraint, the larger the beam's catenary force is [15,16]. Hence, if large deflection is not a design concern and beam design is based on catenary action, it is safe to assume complete end axial restraint stiffness and zero end rotational restraint. However, this may be very different on CFS members because they are usually classified as class 3 or 4 crosssections, according to EN1993-1.1:2004 [17], and have much lower rotational stiffness. Hence, their high susceptibility to buckling phenomena (including local, distortional, global buckling and their interactions) [18–20] may also cause an unpredictable structural failure, i.e., the degradation level of this type of beams may be already far too high even before the beam enters a catenary phase, especially for unbraced beams as it is the case study presented in this paper. As well as that connections in cold-formed steel structures are often only dimensioned not to fail in shear (i.e., the connections are not able to sustain increased tensile forces). Studies in this research field (CFS members in fire) are mostly

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Notation CFS CV FEA E L LL Mb, Rd Mcr MRd NA P P0 dL f0.2p f0.5 fp fu

cold-formed steel coefficient of variation finite element analysis elastic modulus of steel length of the member initial applied load level on the beams design value of the resistant buckling moment critical elastic moment for lateral-torsional buckling section moment capacity about the major axis axial restraining forces generated in the beam applied load on the beams initial applied load on a beam lateral displacement of beam at mid-span 0.2% yield strength of steel 0.5% yield strength of steel proportional stress limit of steel ultimate strength of steel

of a numerical nature. Some of them concluded that the increase of the magnitude of the local imperfections may lead to a relatively straightforward decrease of initial stiffness of the member and that, on the other hand, the magnitude of global imperfections may have more influence on the ultimate load of the member. For instance, Kaitila [21] observed that the compressive ultimate load of a C column may be reduced by 5.1% when the global imperfection magnitude increases from L/1000 to L/500 and may also be reduced by 3.9% when the local imperfection magnitude increases from h/200 to h/100 at 600 °C. Nevertheless, the failure by distortional buckling may still be further affected by the initial geometric imperfections. Ranawaka and Mahendran [22] noted that the maximum load capacity of a C column may be reduced by 20 and 30% when the distortional imperfection magnitude increases from zero to 2tn at 20 °C and 500 °C, respectively. In addition, it seems that the design method given in EN1993-1.2:2004 [23] is over-conservative for all the temperatures excepted for CFS beams with very high slenderness values [24]. With regard to the maximum temperature in CFS members with class 4 cross-sections, EN1993-1.2:2004 [23] has required a limit of 350 °C, which also seems to be overly conservative [25,26]. Therefore, there is a long trek to go in the clarification of this behaviour before developing simplified calculation methods that can be used in fire design. In what concerns to the residual stresses of CFS members, it is important to remember that coiling, uncoiling, cold bending to shape, and straightening of the formed member lead to a complicated set of initial stresses and strains in the cross-section [27,28]. It is also noticed that both residual stress and cold-work of forming effect (where the yield stress of the material in the corners is increased above the virgin yield stress) should not be modelled independent of one another because they are derived from the same process. However, this effect in the corners is generally small since the corners are usually just a small proportion of the overall cross-sectional area [22]. These residual stresses vary across the steel sheet thickness as membrane (constant) and bending (linear variation) components, in opposition to the hotrolled members. The membrane residual stresses are reasonably small and can be ignored. As well as that, Ranawaka and Mahendran [22] observed in their study that the influence of flexural residual stress in a press-braked C-section was negligible. This may occur because the members may buckle at a stress level lower than the yield point of steel and the residual stresses decrease

fy h kE,θ ka ka,b kr kr,b ky,θ tn

θB θcr θS λLT ̅

μ

s

yield strength of steel height of the cross-section reduction factor for the modulus of elasticity of steel at temperature θ axial restraining to the thermal elongation of the beam axial stiffness of the beam rotational stiffness of the beam supports rotational stiffness of the beam reduction factor for the yield strength of steel at temperature θ nominal thickness of the cross-section mean beam temperature critical temperature of the beam steel temperature non-dimensional slenderness for lateral-torsional buckling at ambient temperature mean value standard deviation

with increasing temperature. Hence, members with high values of steel yield strength and/or low values of the steel sheet thickness are little sensitive to residual stresses and, on the other hand, in those cases, the own residual stresses also have the tendency to be low [29]. This is why the residual stresses in CFS members are not often considered in the numerical simulations, especially at elevated temperatures [22]. The mesh density is another important parameter to take into account in numerical analysis. For example, an approximately 40 mm mesh size yields a 9% higher load capacity than an approximately 10 mm mesh size in case of the S4 element [30]. It is noticed that this element is doubly-curved, a four-node (4), quadrilateral and stress/displacement shell element. It is also essential to stress that, for distortional buckling, the lip itself of C sections undergoes bending and at least four linear elements or two quadratic elements are required to provide reasonable accuracy [31]. In order to acquire new knowledge to this scientific field, this paper intends to present a numerical parametric study on the structural behaviour of cold-formed galvanised steel beams under combined bending and fire conditions. Therefore, a suitable finite element model was first developed to compare with experimental results. Consequently, it is described in detail in this paper all parameters, considerations and assumptions took into account in a three-dimensional nonlinear finite element model to predict the behaviour of CFS beams in fire, such as, the beams previously tested in Laboratory by the authors [32]. After verifying the developed finite element model against the experimental results, it was performed a parametric study outside the bounds of the original experimental tests, concentrating particularly on variation in slenderness of the beams, level of initial applied load on the beams and stiffness of the surrounding structure. It is aimed to see how far the axial and rotational restraint at beam supports can affect the structural performance of the beams in case of fire. Finally, this paper is a continuation of previous works [20,32], which make it possible a better understanding of how the surrounding structure affects a CFS beam when is subjected to fire. Nevertheless, in the near future, these studies will be the basis of an analytical study for the development of simplified calculation methods for fire design of axially and rotationally CFS beams.

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2. Finite element model 2.1. Overview The finite element programme ABAQUS is a powerful computational tool for modelling structures with material and geometric nonlinear behaviour. Hence, ABAQUS version 6.10-1 (ABAQUS Analysis – User's Manual, 2010) [30] was used extensively by the authors to develop the finite element model, which aimed to simulate the structural behaviour of cold-formed steel beams under combined flexural loading and fire conditions and also to study the influence of the surrounding structure on the fire behaviour of these beams. In order to achieve these goals, the authors first performed 36 experimental tests on CFS beams (Fig. 1), composed of just one (C beam), two (lipped I and R beams) and four profiles (2R beam) with nominal thickness of 2.5 mm and connected by carbon steel self-drilling screws. Twelve of which were just simply supported beams, 12 others were the same beams but with restrained thermal elongation, and the others were beams with axial and rotational restraint. This experimental programme is also summarised in Table 1. Note that, the reference B_kaþkr-C_3 indicates the third test (3) of the C (C) beam (B) with axial (ka) and rotational (kr) restraint. The cross-sections of these beams were 250 mm tall and the beam span was 3.0 m. The beams were loaded at two points 1.0 m (one-third of the beam span) from the supports of the beam in such a way that between the two loading points the beam was under pure bending state (four-point bending test). This load was 50% of the design value of the load-bearing capacity of the beams at ambient temperature and calculated in accordance with the methods proposed in Eurocode 3 [17,33,34] and calculated for simply-supported boundary conditions, as it is common observed in design projects for this kind of structures. The specimens were then heated with a horizontal modular electric furnace and according to a fire curve as near as possible to the standard fire curve ISO 834 [35]. During the heating period, the load was kept constant until the specimen buckled. This furnace was 4500 mm  1000 mm  1000 mm in internal dimensions and capable to heat up to 1200 °C and to follow fire curves with different heating rates. These beams were supported by a roller and pinned support, which were made of refractory stainless steel, typically used for elevated temperature applications. Furthermore, the experimental system still comprised four restraining steel beams, two of them to simulate the axial restraint to the thermal elongation of the beam and the other two with the purpose of simulating the rotational stiffness of the beam supports. Hence,

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the axial restraining system was composed of two simply supported beams and the rotational restraining system of two cantilever beams, as it can be seen schematically in Fig. 2. Further detailed information or clarification regarding the experimental research can be easily found in the study of Laím et al. [32]. Next, it is then described in detail parameters, considerations and assumptions took into account in the developed three-dimensional nonlinear finite element model to predict the fire performance of such beams. 2.2. Finite element type All CFS beams were modelled by using shell elements (S4R) for the profiles and solid elements (C3D8R) for the screws in what concerns to the structural modelling, which can be seen in the literature as well [30]. It must be still pointed out that a 4-node linear heat transfer quadrilateral element (DC2D4) was used and a two-dimensional numerical model was developed to estimate the temperature distribution in the cross-sections of the beams. These 4-nodes elements with linear interpolation have only the temperature as a degree of freedom. As output the nodal values of the temperature were saved in order to be used as input data for the thermo-mechanical part of the problem in the parametric study. The S4R element [30] was chosen because it is one of the general-purposes from the ABAQUS programme library for elements of type shell, which also take transverse shear deformation into account as well as the thick shell elements (it uses a mixed finite element formulation) [37–41]. The S4R element is a fournode (4), quadrilateral and stress/displacement shell element (S) with reduced integration (R), a large-strain formulation, hourglass control and a first-order (linear) interpolation. The selected element type uses a reduced (lower-order) integration to form the element stiffness with only one integration location per element. Each node has three displacement and three rotation degrees of freedom. On the other hand, the C3D8R element [30] is defined as a three-dimensional (3D), continuum (C), hexahedral and an eight-node brick element with reduced integration (R), hourglass control and first-order (linear) interpolation. These finite elements have three degrees of freedom per node, referring to translations in the three directions X, Y and Z (global coordinates). 2.3. Finite element mesh The influence of the finite element size on the behaviour of CFS beams was first studied. It was found that good simulation results could be obtained by using finite element meshes of 5  5 mm,

Fig. 1. Scheme of the cross-sections of the tested beams.

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Table 1 Experimental test plan. Test reference

λ̅LT

MRd (kN m)

Mcr (kN m)

Mb, Rd (kN m)

ka,b (kN/mm)

kr,b (kN m/rad)

P0 (kN)

ka (kN/mm)

k r (kN m/rad)

B-C_i B-I_i B-R_i B-2R_i B_ka-C_i B_ka-I_i B_ka-R_i B_ka-2R_i B_ka þkr-C_i B_ka þkr-I_i B_ka þkr-R_i B_ka þkr-2R_i

1.61 1.22 0.81 0.58 1.61 1.22 0.81 0.58 1.61 1.22 0.81 0.58

15.87 32.15 28.93 57.89 15.87 32.15 28.93 57.89 15.87 32.15 28.93 57.89

6.09 21.71 43.63 173.10 6.09 21.71 43.63 173.10 6.09 21.71 43.63 173.10

3.93 11.88 16.52 41.96 3.93 11.88 16.52 41.96 3.93 11.88 16.52 41.96

61 124 120 241 61 124 120 241 61 124 120 241

1984 4018 3888 7814 1984 4018 3888 7814 1984 4018 3888 7814

3.93 11.88 16.52 41.96 3.93 11.88 16.52 41.96 3.93 11.88 16.52 41.96

0 0 0 0 15 15 15 15 15 15 15 15

0 0 0 0 0 0 0 0 150 150 150 150

i¼ 1, 2 or 3.

Fig. 2. General view of the experimental set-up for flexural tests of beams: 1 – specimen; 2 – roller support; 3 – pinned support; 4 and 5 – axial restraining beam; 6 and 7 – rotational restraining beam; 8, 9 and 10 – load cell; 11 – concrete slab.

10  10 mm or 20  20 mm, comparing with the experimental results [20]. To achieve an accurate and efficient finite element model with minimum computational time, finite element meshes of 10  10 mm for C, lipped I and R beams and of 15  15 mm for 2R beams were generated automatically by the ABAQUS programme [30] and used in all simulations. In relation to the screws, an approximately 2 mm mesh size was used. 2.4. Material modelling Material non-linearity in the specimens was modelled with von Mises criteria and isotropic hardening. The material behaviour provided by ABAQUS allows for a multi-linear stress–strain curve to be used. Stress–strain relationship of CFS profiles was described by a gradual yielding behaviour followed by a considerable period of strain hardening, whereas an elastic-perfectly plastic behaviour was assumed for the steel screws. Fig. 3 shows the stress–strain curve used in the finite element analysis (FEA) for the CFS profiles

based on tensile coupon test results and at the same time on other studies of literature [20,42]. So, it was tried to reproduce as faithful as possible the stress–strain relationship of the steel coupon specimens taken from the web of the tested beams in the longitudinal direction. A yield strength of 295 MPa, a tensile strength of 412 MPa and a modulus of elasticity of 208 GPa were obtained from those tests at ambient temperature (Table 2). The initial slope of the theoretical stress–strain curve was taken as the elastic modulus, E, of the material. The second, third and fourth slope (E1, E2 and E3) of the curve were defined by tangent modulus which were respectively 38%, 10% and 0.5% of the elastic modulus. Therefore, a gradual yielding behaviour was idealised by using a bilinear representation with tangent modulus E1 and E2 between 70% and 100% of the yield strength, fy, with an intermediate point at a stress of 0.875 fy [42]. All other components were modelled as elastic, i.e. the elastic modulus was equal to 210 GPa and the Poisson's ratio to 0.3 at ambient temperature. However, this last value was assumed to remain unchanged with increasing temperature [43]. The cold work of manufacturing process may cause a positive effect (i.e. the strength enhancement) and a negative effect (i.e. the reduction of the load-carrying capacity) on a cold-formed member, as a result of the combined effect of the residual stresses and equivalent plastic strains in the member. These (longitudinal and transverse) residual stresses caused by cold forming do not exist alone and are always accompanied by corresponding equivalent plastic strains which are responsible for the definition of the work hardened state. In addition, it is known that both transverse and longitudinal residual stress distributions are nonlinear through the thickness and depend on many parameters, such as, the type of steel (yield strength), the thickness of the steel sheet, the coil radius, among other ones [28], which make it difficult to predict these stresses properly. However, these stresses decrease with decreasing thickness of cross-section and with increasing temperature, as mentioned before. To sum up, residual stresses and cold-work of forming (where the apparent yield stress in the corners is increased) were ignored in these analyses, in other words, the mechanical properties of steel were assumed to be Table 2 Mechanical properties of structural steel S280GD at ambient temperature.

Fig. 3. Stress–strain relationship of the beam's and screw's steel.

Specimen

E (GPa)

fp (MPa)

f0.2p (MPa)

f0.5 (MPa)

fy (MPa)

fu (MPa)

A B C μ r CV (%)

202 213 209 208 5.5 2.66

187 199 189 192 6.7 3.47

284 295 294 291 6.1 2.09

290 297 297 295 4.3 1.47

290 298 297 295 4.4 1.49

411 414 411 412 1.8 0.44

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uniform across the cross-section, like other researchers in this field did [18,27,36,44]. The thermal properties of the CFS sections at elevated temperatures considered in the model (mass density, thermal conductivity and specific heat) were those given in EN1993-1-2:2004 [23], whereas the expansion was taken from Cheng and Young [45]. On the other hand, the reduction factors for the yield strength of steel at elevated temperatures were obtained from the annex E of EN1993-1-2:2004 [23], whereas the reduction factors for the modulus of elasticity were taken from Ranawaka [46], as it can be seen in Fig. 4. The reason for the choice of these parameters was due to the fact that these members are mostly class 4 crosssections and their manufacturing process is different from the hotrolled steel members and, as it can be seen further ahead on this paper, these parameters still allowed to well estimate the structural behaviour of these beams. Moreover, these reduction factors fit well with the experimental values obtained from tensile coupon tests at elevated temperatures previously carried out by the authors [47]. Finally, since the analysis of post-buckling may involve large inelastic strains, the nominal (engineering) static stress–strain curve was converted to a true stress and logarithmic plastic strain curve [4]. EN1993-1-2:2004 [23] steel properties are given as engineering stress–strain input, which should be converted to true stress and true (logarithmic) strain using Eqs. (1) and (2). Further information about the true stress and plastic true strain is available in ABAQUS User's Manual [30].

εtrue=ln (1+εeng )

(1)

σtrue = σeng (1+εeng )

(2)

2.5. Boundary, loading and contact conditions A three-dimensional numerical model was used to describe all buckling modes observed in the experimental tests, as stated above. The cross-sections of the different beams, the screws, the beam support system, the axial and rotational restraining system and the beam loading system were tried to reproduce with great accuracy in the numerical model. As it can be seen in Fig. 5a, the axis system of the model is such that Z axis lies in the longitudinal direction of the beam while X and Y axes lie in the major and minor axes of the beam's cross-section, respectively. Such as observed in the real test set-up, the beam supports and the loading

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were also applied on rigid plates attached to beams so as to distribute possible concentrated forces on them. Therefore, with regard to the loading on the beams, concentrated forces with the direction – Y were applied at the centre of those plates and each one was at a distance of 1 m from the nearest beam support, i.e., the forces were applied at one-third of the beam span (L/3). On the other hand, to simulate the pinned support all degrees of freedom of the nodes located on the bottom surface and at the middle of the respective rigid plate were constrained, whereas for the roller support only the translations in the directions X and Y were constrained. Finally, the translations in the direction X of all nodes located at each end of both supports were constrained in order to prevent their lateral deformation. All simulated beams were still modelled using the centre line dimensions, as it can be seen in Fig. 5b. Furthermore, two assumptions were introduced in these analyses for modelling the contact behaviour between the profiles and also between these ones and the screws. Thus, it was assumed a tangential friction coefficient of 0.2 for the contact behaviour in tangential direction and a hard contact (full transmission of compressive forces and no transmission of tensile forces) for the contact behaviour in normal direction between the profile surfaces. The surface-to-surface contact using the finitesliding tracking method [30] was used, because this one gives a good convergence rate and it is much less sensitive to the choice of master and slave surfaces. For surface-to-surface contact, contact conditions are enforced in an average sense, rather than at discrete points such as node-to-node discretization. Such averaging technique provides more accurate and smooth contact state transition [48]. As well as that the penalty method was defined as the contact property (both tangential and normal contact) between the steel profile surfaces. Finite sliding was also used in the contact tracking algorithm, which takes account for large relative movements between contact pairs compared to their element sizes and updates their contact tracking state for each contact iteration. So, it is well suited for models under fire conditions with large plastic deformations. A rough and hard contact between the profiles and the screws was also employed, which is similar to the methodology adopted in [36]. For the modelling of the axial and rotational restraining system, a linear spring model was used (Fig. 6). ABAQUS allows the user to define axial spring elements, connected to a node of the member and a support that have the appropriate stiffness coefficients. The springs were connected to the beams at the centre of the crosssection by means of the *Coupling Constraints option in ABAQUS [30]. The kinematic coupling constraint was employed in order to

Fig. 4. Reduction factors for yield strength (a) and modulus of elasticity (b) of steel at elevated temperatures.

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Fig. 5. Numerical model used in the three-dimensional finite element analysis: (a) perspective and (b) cross-sectional view.

transfer by convection in the cavity is neglected. As the beams are generally horizontal elements in the buildings and the cavities were very small, the last technique was adopted [53]. Therefore, a two-dimensional model was developed on the basis of the CFS cross-sections and the thermal resistances to heat conduction developed at steel–steel and steel–fresh air interfaces were respectively considered with thermal contact conductance coefficients equals to 200 and 10 W/(m2 K). Note that the emissivity coefficients, the heat transfer coefficients and the thermal contact conductance coefficients remained unchanged with increasing temperature. 2.6. Analysis method

Fig. 6. Axial and rotational restraining system introduced in the numerical model using linear axial springs.

constrain the motion of the end surfaces of the beams to the motion of a single point, in this case the centre point of the beam cross-section. Finally, the fire action was defined in ABAQUS programme by two types of surface, namely, “film condition” and “radiation to ambient”, corresponding respectively to heat transfer by convection and radiation. It is worth mentioning that before using the ISO 834 standard fire curve [35] the numerical heat transfer analyses were first performed with the furnace temperatures registered in the experimental tests in order to validate the thermal model. It is also important to stress that the thermal and structural models were calibrated separately, which means that the beam temperatures measured in the experimental tests were used to validate the three-dimensional structural model. Hence, the radiative heat flux was calculated using a steel emissivity value of 0.3 (due to the mirror surface of the zinc coating on the profiles [49]) and 0.7 for fire, and the Stefan-Boltzmann constant was 5.67  10  8 W/ m2 K4. On the other hand, convection was considered with heat transfer coefficient equal to 15 W/(m2 K) for the fire test curves [50] and 25 W/(m2 K) for the ISO 834 fire curve, as recommended by EN1991-1-2:2002 [51]. The heat transfer analysis of the builtup closed beams (R and 2R beams) has an additional challenge because of the voids between the profiles. This problem could be solved in a simplified way by three different technical approaches. One is to simply neglect the cavity, the other is to use the assumption of isothermal and iso-emissive cavity facets [52] and the last one is to consider the air as a solid material, i.e., the heat

Two types of analysis were employed by using the developed finite element model: elastic buckling and nonlinear static analyses. Elastic buckling analysis (also known as eigenvalue analysis) was performed to establish the buckling modes which were observed in the experimental tests [18,22,36], in contrast to the methodology adopted in practical applications, where usually only the lowest buckling mode predicted from the eigenvalue analysis is used. Then, the representative buckling modes were imported into the finite element model with the purpose of considering geometric imperfections in the load–displacement nonlinear analysis. After knowing the effects of the imperfections on structural response of this kind of beams and comparing with the experimental results, it was observed that a suitable maximum value for global imperfections was found to be approximately L/1000, for distortional imperfections t and for local imperfections h/200. Finally, the nonlinear structural analysis was undertaken with the purpose of modelling the performance of CFS beams under fire conditions until failure. The nonlinear geometric parameter (*NLGEOM¼ ON) was set to deal with the geometric nonlinear analysis, namely, with the large displacement analysis. Artificial damping was also applied in such a way that the viscous forces were sufficiently large to prevent instantaneous buckling or collapse, but small enough not to affect the behaviour significantly. This approach [36,40,54] was employed to ensure that the numerical model was able to go through all the phases of structural behaviour experienced in the experimental tests, achieving this way a better success in this research. Note that before the heating stage, a serviceability load was applied on the beams. This load level was a percentage (30%, 50% or 70%) of the design value of the load-bearing capacity of the beams at ambient temperature,

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calculated in accordance with the methods proposed in EN19931.1:2004 [17], EN1993-1.3:2004 [33] and EN1993-1.5:2006 [34] and calculated for simply-supported boundary conditions, as already stated. During the heating phase, the load was kept constant until the specimen buckled, where the beam deformation was too large, L2/(400h) (failure criteria in terms of deformation [35]), which corresponded approximately to the time when the beams no longer had any loadbearing capacity against the axial restraining forces (failure criteria in terms of strength), as it can be seen in [32]. Hence, the critical temperature of the beams was defined as the one which corresponded to that time. Note that the catenary action is beyond the scope of this study, because the degradation level of this type of beams is already far too high even before the beam enters a catenary phase [50]. In addition, the connections in cold-formed steel structures are often only dimensioned not to fail in shear.

3. Calibration of finite element model 3.1. Heat transfer analysis The purpose of this section was to check the suitability of the developed thermal model for this study and the reliability of the parameters adopted. As example, Figs. 7 and 8 shows respectively the comparison of the temperature-time curves obtained from the experimental tests and FEA for the C (Fig. 7a) and R (Fig. 8a) sections. Others presented similar results [53]. As the thermal action was assumed constant with height of the cross-section, only the readings of some thermocouples were compared with the numerical results. A symmetric temperature distribution in the beams was obtained. This can be clearly seen in Figs. 7b and 8b, where it is shown respectively the temperature distribution in the C and R sections at 9 min of test run. At the temperature measuring locations, the correlation between numerical and test results was quite good, the differences were around 5%. Therefore, the tools of ABAQUS programme can be used to simulate detailed temperature distribution in CFS members under high thermal actions, including in members with confined air in the interior of the built-up closed cross-sections.

Fig. 7. Comparison of the FEA and experimental temperature results in the C beam (a) and its temperature distribution at 9 min of test run (b).

Fig. 8. Comparison of the FEA and experimental temperature results in the R beam (a) and its temperature distribution at 9 min of test run (b).

3.2. Structural failure analysis Fig. 9 shows, for example, the comparison of the axial restraining force-temperature (NA  θB) curves of the axially and rotationally restrained beams obtained from the experimental tests and FEA used for the calibration of the model. Due to the effect of the thermal action, the axial force on the beams began to increase until it reached a maximum value. After this maximum it began to decrease as a result of the deterioration of the steel's mechanical properties with temperature (degradation of the beam strength), of the loss of beam stiffness and of the instability phenomenon. All curves from FEA (continuous red line) fit closely with the experimental curves, especially in what concerns to the critical temperature of the respective beams, as it can be seen in Table 3. The small detected difference at the peak axial load and in the post-peak range between the numerical and experimental results may be due to the existing friction between the (roller and pinned) beam supports and the respective beam profiles and also between the steel semi-spheres at both ends of the beams and the axial restraining beams despite the use of Teflon (PTFE) between the surfaces, which were not taken into account in the boundary conditions of the developed numerical model. So, this might be why there was a partially sudden decay in the numerical axial restraining forces after the peak force in contrast to the experimental axial forces. When the beams exhibited a sudden instability, this was not so relevant after all, as shown in Fig. 9d. In the end, the mean differences between experimental and numerical critical temperatures were less than 10% (Table 3). In the most cases, the predicted critical temperatures were on the safe side and not too conservative either. The highly complex postbuckling reserve of some beams and the change of the action line of the applied force during the tests may be also two of the reasons for the hard convergence of both type of results. Anyway, still concerning the maximum axial restraining force, the differences were almost none. Lastly, the general good agreement and accuracy between the experimental and numerical results ensures a strong validity of the developed finite element model and may also ensure reliable results obtained from parametric studies, especially the results needed for the development of simplified calculation methods for

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Fig. 9. Comparison of the FEA and experimental axial restraining forces in the axially and rotationally restrained C (a), lipped I (b), R (c) and 2R (d) beams.

fire design of restrained CFS beams. It is clear that predicting the structural behaviour of the CFS beams at ambient temperature was easier than under fire conditions [20], but it is also obvious that the number of variables which affect the structural response of the beams at high temperatures is much higher than at ambient temperature.

3.3. Failure mode analysis The FEA failure modes of the tested beams under fire conditions are also illustrated in Figs. 10–12, which can be contrasted to the experimental ones, in order to validate the performance of the developed finite element model in terms of deformations,

Table 3 Comparison of experimental test and finite element analysis results. Test reference

Test θcr (°C)

FEA θcr (°C)

Comparison (%)

Test reference

Test θcr (°C)

FEA θcr (°C)

Comparison (%)

B-C_1 B-C_2 B-C_3 μ B_ka-C_1 B_ka-C_2 B_ka-C_3 μ B_ka þkr-C_1 B_ka þkr-C_2 B_ka þkr-C_3 μ B-R_1 B-R_2 B-R_3 μ B_ka-R_1 B_ka-R_2 B_ka-R_3 μ B_ka þkr-R_1 B_ka þkr-R_2 B_ka þkr-R_3 μ

710 729 716 718 513 519 554 529 629 647 646 641 728 732 745 735 516 489 506 504 595 628 656 626

666

 6.2  8.6  7.0  7.3 1.7 0.5  5.9  1.4 4.3 1.3 1.5 2.4  10.5  11.0  12.4  11.3  15.4  10.7  13.8  13.4  1.6  6.7  10.7  6.5

B-I_1 B-I_2 B-I_3 μ B_ka-I_1 B_ka-I_2 B_ka-I_3 μ B_ka þ kr-I_1 B_ka þ kr-I_2 B_ka þ kr-I_3 μ B-2R_1 B-2R_2 B-2R_3 μ B_ka-2R_1 B_ka-2R_2 B_ka-2R_3 μ B_ka þ kr-2R_1 B_ka þ kr-2R_2 B_ka þ kr-2R_3 μ

693 709 672 691 544 544 – 544 567 567 – 567 715 732 744 731 696 676 620 664 675 677 659 670

647

 6.6  8.7  3.7  6.4  7.3  7.4 –  7.3  1.7  1.7 –  1.7  1.2  3.6  5.1  3.3  14.3  11.8  3.9  10.2 1.8 1.5 4.3 2.5

666 522

522 656

656 652

652 436

436 586

586

647 504

504 558

558 706

706 596

596 687

687

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Fig. 10. Experimental and numerical configuration of the deformed C (a), lipped I (b), R (c) and 2R (d) beams with no restraints after fire test.

especially the final configuration of the beams. It can be seen that most of them are identical to the experimental failure modes. Both kinds of figures show that lateral-torsional buckling was the main failure mode responsible for the collapse of the open sections (C and lipped I beams) and the distortional buckling for the collapse of the built-up closed sections (R and 2R beams). Anyway, all deformed configurations of the beams involved lateral-torsional, distortional and local buckling, as a result of their extensive lateral rotation. Nevertheless, some exceptions were observed regarding to the main failure modes responsible for the collapse of some beams, as

described above, involving the axially and rotationally open beams (Figs. 11 and 12), where shear buckling was the mode responsible for their collapse. Once again, it can be conclude that the developed finite element model predicted the behaviour of CFS beams with an acceptable precision. However, despite the minor influence, it should be still mentioned that the distortional buckling mode occurred on the compression flange near one loading point rather than in mid-length of the specimens (Figs. 11 and 12), the lower flange buckled inwards rather than outwards or the opposite (Fig. 11) and only one distortional buckling half-wave emerged on the U section rather than two or more (Fig. 10d).

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Fig. 13. Evolution of axial restraining forces in axially restrained C beams with low (a) and high (b) slenderness as a function of its mean temperature.

Fig. 11. Experimental (a) and numerical (b) configuration of the deformed C beam with axial and rotational restraint after fire test.

5 m) and, mostly, the axial restraint level to the thermal elongation of the beam (0, 0.45, 1, 7.5, 15, 30, 1 kN/mm) and the rotational restraint level at beam supports (0, 15, 150, 300, 1200 and 1 kN m/rad) were some of the parameters numerically evaluated in this study, covering hundreds of numerical simulations. Note that the sequence of the reference C-h-w-c-t-L corresponds respectively to the type of simulated beam (for this case, the C beam), its height (h), its width (w), the length of the edge stiffeners of the C profile (c), the thickness of the profiles (t) and the length of the beam (L), all in millimetres. It is also worth remembering again that a uniform temperature distribution along all beam length was intended for the parametric study, in contrast to the tested beams in the Laboratory, where only 2.5 m in length of the beam were heated [32]. This is one of the reasons why some results of the parametric study might be different from the experimental ones. 4.2. Effect of axial restraining forces

Fig. 12. Experimental (a) and numerical (b) configuration of the deformed lipped I beam with axial and rotational restraint after fire test.

4. Parametric study 4.1. General information An extensive parametric study using the developed finite element model was then carried out to investigate the fire response of such beams. The section geometry (C-, lipped I-, R- and 2Rshaped cross-sections), the level of initial applied load on the beams (30%, 50% and 70% of the design value of the buckling load of the beams at ambient temperature), the beam-span (2, 3, 4 and

The curves shown in Figs. 13 and 14 are typical of the axial restraining force and temperature curves for all the simulated CFS beams. As it can be seen, these curves had nearly the same trend but they may differ in the values and their rates depending on the studied parameters, especially on the slenderness, axial and rotational restraint. Each figure shows a set of curves corresponding to different values of axial restraint, ka (Fig. 13), and rotational restraint, kr (Fig. 14), ranging from the situation of a simply supported beam (ka¼ kr¼0) to the case of a fully fixed beam (ka¼ kr¼1), under always the same serviceability load (50% of the design value of the load-bearing capacity of the beams at ambient temperature). It is quite interesting to observe that the higher the axial restraint (ka) is, the higher the maximum axial restraining force becomes and the sooner it occurs, but the value of the critical temperature of the beam remains constant beyond a certain level of axial restraint. Moreover, as the slenderness of the beam increases, the reduction on the critical temperature decreases with increasing axial restraint. For example, when the span of the beam C-250-43-15-2.5 increases from 3 to 5 m, its critical temperature as a function of the axial restraint decreases respectively from 631

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Fig. 16. Lateral displacements of C beams at mid-span under different levels of rotational restraint.

slenderness of the beams did not introduce much changes in their critical temperature, in contrast to the changes in the axial restraining forces, especially the maximum values (Fig. 14). 4.3. Effect of axial restraint

Fig. 14. Evolution of axial restraining forces in axially and rotationally restrained C beams with low (a) and high (b) slenderness as a function of its mean temperature.

Fig. 15. Lateral displacements of C beams at mid-span under different levels of axial restraint.

to 454 °C (  177 °C) (Fig. 13a) and from 526 to 448 °C ( 78 °C) (Fig. 13b). This can be explained by the fact that the higher the slenderness of the beams is, the higher the lateral displacement of the beam is for the same beam temperature and axial restraint provided by the surrounding structure (Fig. 15). In addition, as the axial restraint imposed to beams increases, their lateral displacement rate increases for low values of temperature and decreases for high values of temperature. With regards to the influence of the rotational restraint imposed by the surrounding structure to the CFS beams, it is worth noting from Fig. 14 that the changes in the axial restraining forces and critical temperatures were only significant for very low levels of rotational restraint, kr. When this type of restraint grew from zero, the axial restraining forces generated in the beams increased as well as the corresponding critical temperatures. This is well understandable, because increasing the value of the rotational restraint makes the buckling length of the beams decreases. Consequently, the lateral displacements of the beams have the tendency to be relevant in their structural performance for higher levels of temperature (Fig. 16). However, for high values of kr these changes may be negligible (Fig. 14). It was clear that when the lateral displacement of the beam C-250-43-15-2.5 reached 180 mm, the failure of the beam was inevitable whatever the rotational restraint was (Fig. 16). Furthermore, it also seems that the

Identical graphs, as shown in Figs. 13 and 14, and same conclusions can be drawn for the other studied sections. From them it was possible to obtain their critical temperatures as a function of the corresponding relative slenderness, as shown in Figs. 17 and 18. Therefore, Fig. 17 presents, as example, the critical temperature of C (a), lipped I (b), R (c) and 2R (d) beams for the 50% load level and for different axial restraint levels to thermal elongation of the CFS beams, remaining the rotational restraint constant and equal to 0. Note that it was intended to study separately both parameters. Once again, it can be seen for all studied beams that as the slenderness of the beams increases the corresponding critical temperature decreases and it is clear that the critical temperature of the beams is affected by the axial restraint level, i.e. the critical temperature decreases with increasing axial restraint. It is worth pointing out that the effect of axial restraint depended strongly on the shape of the beam cross-section and that their structural response against the axial restraint to the thermal elongation as a function of the slenderness was not always linear. When the axial restraint level increased from 0 to infinite, the critical temperature of the C beam decreased by 38% (from 675 °C to 420 °C) for the 2000 mm span (non-dimensional slenderness equal to 1.11) and 15% (from 526 °C to 448 °C) for the 5000 mm span (non-dimensional slenderness equal to 2.47) (Fig. 17a). Regarding the lipped I, R and 2R beams, the critical temperatures decreased respectively by 52% (from 677 °C to 328 °C), 6% (from 648 °C to 608 °C) and 20% (from 724 °C to 319 °C) for the 2000 mm span and by 58% (from 357 °C to 151 °C), 5% (from 392 °C to 374 °C) and 43% (from 319 °C to 182 °C) for the 5000 mm span (Fig. 17b, c and d). So, the reduction on the critical temperatures with increasing slenderness caused by the axial restraint had a higher impact in the 2R beams, in contrast to the lipped I and R beams. In addition, the critical temperature of the beams dropped drastically even for low levels of axial restraint. For instance, beyond 7.5 kN/mm of axial restraint the critical temperature of the C beams was almost the same for any value of non-dimensional slenderness (especially between 1.11 and 2.47). 4.4. Effect of rotational restraint On the other hand, Fig. 18 shows, as example, the critical temperature of C (a), lipped I (b), R (c) and 2R (d) beams for the 50% load level and for different rotational restraint levels at their ends, remaining the axial restraint constant and equal to 15 kN/ mm. It is obvious that when there is rotational restraint in real situations it is quite probably that there is some axial restraint

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Fig. 17. Effect of axial restraint on the critical temperature of C (a), lipped I (b), R (c) and 2R (d) beams.

level as well. This figure depicts that as the rotational restraint imposed by the surrounding structure to the beams increases, their critical temperatures increases, as mentioned above. However, whereas for small values of the rotational restraint the critical temperature of the beams was highly affected by its slenderness, for large values of the rotational restraint the slenderness of the beams had almost a negligible effect on their critical temperature. This may result from the fact that the flexural buckling may become more relevant than the lateral torsional buckling for high rotational restraint levels and slenderness values. It is also worth remembering that none of this could have been possible without remaining constant the serviceability load on the beams with increasing rotational restraint. Here it can also be seen that the rotational restraint effect on the structural response of the beams depended on the shape of cross-section. For example, when the rotational restraint level increased from 0 to infinite, the critical temperature of the restrained C beam increased by 64% (from 445 °C to 729 °C) for the 2000 mm span (non-dimensional slenderness equal to 1.11) and 59% (from 450 °C to 715 °C) for the 5000 mm span (non-dimensional slenderness equal to 2.47) (Fig. 18a). Regarding the lipped I, R and 2R beams, the critical temperatures increased respectively by 45% (from 412 °C to 596 °C), 48% (from 470 °C to 696 °C) and 18% (from 648 °C to 765 °C) for the 2000 mm span and by 309% (from 156 °C to 637 °C), 162% (from 245 °C to 642 °C) and 134% (from 281 °C to 658 °C) for the 5000 mm span (Fig. 18b, c and d). So, the benefit of the rotational restraint to the critical temperature of the beams with increasing slenderness was higher in the open beams than in the closed beams, except for the C beams. This may be due to the fact that an axial restraint equal to 15 kN/mm or higher imposed to the simulated C beams has the same effect on their structural fire behaviour. This is why the full understanding of the behaviour of

CFS beams under fire conditions with restraint to thermal elongation should be acquired by means of two key parameters, such as the ratio between the axial restraint provided by the surrounding structure to the CFS beams and the axial stiffness of the CFS beams (ka/ka,b) and the ratio between the rotational restraint provided by the surrounding structure to the CFS beams and the rotational stiffness of the CFS beams (kr/kr,b), as it can be seen ahead in this paper (Fig. 20). 4.5. Effect of load level Fig. 19 provides a general idea of how the critical temperature of the studied beams is affected by the initial applied load level on the beams under different boundary conditions as a function of their non-dimensional slenderness. It is clear that the detrimental effect of the axial restraint to thermal elongation of the beams and the beneficial effect of the rotational restraint at their ends on the respective critical temperatures was higher for high initial load levels than for low levels. For instance, when the axial restraint level increased from 0 to 15 kN/mm, the critical temperature of the C-250-43-15-2.5-3000 beam (non-dimensional slenderness equal to 1.61) decreased by 17% (from 706 to 583 °C) for 30% of initial load level and 42% (from 538 to 309 °C) for 70% of initial load level (Fig. 19a). And when the rotational restraint level increased from 0 to 300 kN m/rad, the critical temperature of the same beam increased by 36% (from 583 to 791 °C) for 30% of initial load level and 119% (from 309 to 676 °C) for 70% of initial load level (Fig. 19b). However, it is important to emphasise that in general as the slenderness of these beams, their applied load level and restraint level increased, the influence on the critical temperature of the beams increased as well. For example, when the axial restraint level increased from 0 to 15 kN/mm and the load level was 30%,

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Fig. 18. Effect of rotational restraint on the critical temperature of C (a), lipped I (b), R (c) and 2R (d) beams.

the critical temperature of the R-250-43-15-2.5 beam decreased by 17% (from 711 °C to 587 °C) for the 2000 mm span (non-dimensional slenderness equal to 0.55) and 27% (from 630 °C to 458 °C) for the 5000 mm span (non-dimensional slenderness equal to 1.32). But when load level was 70%, the critical temperature decreased by 26% (from 585 °C to 430 °C) for the 2000 mm span and 38% (from 176 °C to 109 °C) for the 5000 mm span (Fig. 19c), hence higher than 17% and 27% respectively. An exception to this trend was for the axially restrained C beams.

critical temperature of the beams were much higher than in others with the same ka/ka,b ratio, but without rotational restraint at their ends (kr/kr,b ¼0). This suggests that the rotational restraint effect on the structural fire behaviour of beams is higher than the axial restraint effect. From the above mentioned in the paper, it is obvious that for each type of axially restrained beam there is a minimum ( θcr , min ) and a maximum ( θcr , max ) critical temperature, which depend both on the initial applied load level (LL) and on the non-dimensional slenderness of the beams ( λLT ̅ ). For rotationally restrained beams those parameters still depend on the axial re-

4.6. Effect of restraint ratio

straint to axial stiffness ratio ( k a ). One very important con-

k

a, b

Lastly, in order to clarify the behaviour of CFS beams with restrained to thermal elongation under uniform fire conditions, it is depicted in Fig. 20 the obtained critical temperatures of the different studied beams (C, lipped-I, R and 2R) with similar non-dimensional slenderness at ambient temperature for different values of ratios between the axial restraint imposed by the surrounding structure to the CFS beam and the axial stiffness of the beam (ka/ka, b) (Fig. 20a) and for different values of ratios between the rotational restraint imposed by the surrounding structure to the CFS beam and the rotational stiffness of the beam (kr/kr,b) (Fig. 20b). It is important to emphasise that the critical temperature of the beams was strongly affected by those ratios, but only for small values. It seems that the critical temperature of axially CFS beams remains almost unchanged for ratios between the axial restraint and the axial stiffness of the beam (ka/ka,b) higher than 0.40 (Fig. 20a) and that the critical temperature of axially and rotationally CFS beams remains almost unchanged for ratios between the rotational restraint and the rotational stiffness of the beam (kr/ kr,b) higher than 0.20 for a certain axial restraint level (Fig. 20b). As well as that when these ratios were equal (ka/ka,b ¼ kr/kr,b) the

sequence of this simple result is that the critical temperature method presented in EN1993-1.2:2004 [23] for hot rolled steel members should not be used for CFS beams since in this method at least the critical temperature almost does not change when the non-dimensional slenderness changes and that is far away from the truth as it can be observed, as example, in Fig. 21. Not to mention the limit temperature recommended by EN1993-1.2:2004 [23] for members with class 4 cross-sections. It is noticed that the simplified design methods presented in the EN1993-1.2:2004 [23] can be used for CFS members, according to its Annex E, but the area of the member cross-section must be replaced by the effective area and the section modulus by the effective section modulus, determined in accordance with EN1993-1.3:2004 [33] and EN1993-1.5:2006 [34], i.e. based on the material properties at 20 °C. From the authors' point of view, it is clear that the effective section modulus should be affected by the temperature of the steel member, i.e. the critical elastic local buckling stress of a compression plate decreases with increasing temperature as well as the critical elastic distortional buckling stress of an outstand compression element with an edge stiffener. Finally, the design

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Fig. 19. Effects of the initial applied load level on the critical temperature of the axially (a and c) and rotationally (b and d) restrained C (a and b) and R (c and d) beams.

Fig. 21. Comparison of FEA results with EN 1993-1-2:2004 predictions for simply supported C beams.

buckling curve established in EN1993-1.2:2004 [23] for hot rolled steel members might also not be the most appropriate for CFS members. 5. Conclusions Fig. 20. Impact of the non-dimensional axial (a) and rotational (b) restraint on the critical temperature of the studied beams.

This paper aimed to develop a three-dimensional nonlinear finite element model in order to simulate the response in flexion

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of elastically restrained single, open built-up and closed built-up CFS beams under uniform fire conditions, considering their interaction with the surrounding structure. The proposed model was first validated with respect to the experimental data from [32]. It has been shown that the beam strengths and failure modes predicted by finite element analysis are generally in good agreement with the experimental results. Therefore, the verified finite element model was used to perform extensive parametric studies to investigate especially the effect of axial and rotational restraint on the fire resistance of CFS beams, as well as the influence of the section geometry, initial applied load on the beam and slenderness on the critical temperature of the beams. The results of this study showed that in many cases, the benefit of the rotational restraint offered by the surrounding structure outweighed the detrimental effect of the axial restraint. Besides, the elastic restraint to the thermal elongation of the CFS beams and the elastic restraint to the rotation of the ends of the beams subjected to fire, imposed by the surrounding structure, affected deeply their critical temperature (reaching differences around 50%), but only when the ka/ka,b and kr /kr,b ratios approached to very small values (especially, for values lower than 0.15). Adding axial restraint to the thermal elongation of beams also had a great effect on the values of the generated restraint forces in the beams (reaching the design buckling resistance to compression for ka/ka,b ratios about 0.25), in contrast to the rotational restraint which had a relatively minor effect, but their failure temperatures were greatly increased under the same load (in some cases, doubling their critical temperature when the kr/kr,b ratio ranges from 0 to 0.10). However, it seems that for high values of ka/ ka,b and kr/kr,b (higher than 0.4), the surrounding structure to the CFS beams has a negligible influence on their fire resistance and the effect of the beam slenderness is lower. Results from this investigation also indicated that the initial load level applied on the beams was by far the parameter which most affected their structural fire behaviour, as expected. But, notice that the detrimental effect of the axial restraint provided by the surrounding structure to the CFS beam and the respective beneficial effect of the rotational restraint was significantly higher for high initial load levels than for low levels (in some cases, the reduction in the critical temperature doubled when the load level increased from 30% to 70%). Finally, at once, it can be still concluded that the methods established in EN1993-1.2:2004 [23] are not much appropriate for the fire design of CFS beams. Some predictions show relative differences in the order of 20% and others are unsafe. These simplified design guidelines did not even say anything concerning the fire design of flexural members under different types of restraining conditions (axial and/or rotational restraint). Hence, the developed finite element model should and will be used in the near future to further explore the flexural behaviour and predict the performance of a wide range of commercial CFS beams.

Acknowledgements The authors gratefully acknowledge to the Portuguese Foundation for Science and Technology – FCT (www.fct.mctes.pt) for its support under the framework of research Project PTDC/ECM/ 116859/2010 and for the Postdoctoral scholarship SFRH/BPD/ 94037/2013 given to the first author, as well as to the Human Potential Operational Programme (POPH), the European Social Fund and National Strategic Reference Framework (QREN). The authors would also like to especially thank the Portuguese coldformed steel profile maker PERFISA S.A. (www.perfisa.net) for their support.

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