NUMERICAL EVALUATION OF THE ROTATIONAL CAPACITY OF STEEL BEAMS AT ELEVATED TEMPERATURES

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1 8 th GRACM International Congress on Computational Mechanics Volos, 12 July 15 July 2015 NUMERICAL EVALUATION OF THE ROTATIONAL CAPACITY OF STEEL BEAMS AT ELEVATED TEMPERATURES Savvas Akritidis, Daphne Pantousa and Euripidis Mistakidis Laboratory of Structural Analysis and Design, Department of Civil Engineering University of Thessaly, Volos, GR-38334, Greece ; web page: Keywords: rotational capacity, elevated temperatures, steel beams Abstract. The global plastic analysis of steel structures requires that at the plastic hinge locations, the cross sections of the members should have rotational capacity greater than the required one. According to EN [1], this problem is handled through the classification of the cross sections. In the case of the fire design of steel structures, the classification of the cross sections is conducted in the same way as in room temperature, except that a reduction factor of 0.85 is applied to the coefficient ε. The main objective of this paper is the assessment of the rotational capacity of steel beams at elevated temperatures using numerical methods and the investigation of how accurate is this coefficient (0.85). Specifically, numerical models are developed using the non-linear finite element code MSC-Marc. Parametric analyses are conducted with respect to the amplitude of initial imperfections, the flange thickness and the length of the beam at ambient temperature and at elevated temperatures. 1 INTRODUCTION According to EN [2], in the case of the fire design of steel structures, the classification of the cross sections is conducted in the same way as in room temperature, except that a factor of 0.85 is used for the calculation of parameter i.e. it is: (1) where is the yield strength at 20⁰C. As it is stated in EN [2], the reduction factor considers influences due to the increased temperature. Moreover, this simplification implies that during fire-exposure the cross-sections are able to attain the plastic moment resistance and that at the plastic hinge locations the structural members have sufficient rotational capacity. This could be conservative or not [3] since the factor does not take into account several factors that affect the rotational capacity of steel members under fire conditions, as the lack of strain hardening in the stress-strain relationship after the temperature of 400⁰C, the effect of the initial imperfections etc. These parameters may lead to a premature occurrence of local or lateral torsional buckling in the plastic range, therefore limiting the available rotational capacity. The ductility of steel structural members in terms of the rotational capacity was the subject of numerous experimental, analytical and numerical studies. The scientific research is mainly conducted at ambient temperature. The research is focused on the geometric factors that affect the development of local buckling and lead to the deterioration of the ductility of the structural members and on the calculation of the required values for the rotational capacity. Unlike the extensive research activity related to the ductility of structural members at ambient conditions, the research at elevated temperatures is limited. A strain-based approach to local buckling of steel sections subjected to fire is proposed in [4]. This study states that the local buckling under fire conditions needs to be considered in the context of a wide range of cross-sectional slenderness than at ambient temperature. This can be attributed to the non-linear behaviour of steel at elevated temperatures and moreover to the fact that large strains are required for the achievement of the cross-sectional capacity. An extensive experimental program was conducted to investigate the rotational capacity of steel I-beams under fire conditions [5]. It is noted that the specific experimental program was the first one that was conducted worldwide for the investigation of the rotational capacity of steel beams at elevated temperatures. The main

2 objectives were to determine the effects of temperature on the rotational capacity and to identify the key parameters that affect it. This study highlights that the rotational capacity is reduced at elevated temperatures and that the primary difference of moment-rotation curve at elevated temperature compared to the corresponding one at ambient temperature is the non-linear pre-buckling stage, which takes place due to the non-linear stress-strain relationship of steel at elevated temperatures. In [6] numerical models were developed for the prediction of the rotational capacity at elevated temperatures corresponding to the beams tested in [5]. The aim of this paper is to examine the behavior of the steel I-beams at elevated temperature, and specifically the evolution of the rotational capacity with respect to the amplitude of the initial imperfections, the flange thickness and the length of the beam. Finally, the provisions of EN [2] about the cross-section classification are evaluated on the basis of the analysis results. 2 DEFINITION OF ROTATIONAL CAPACITY The definition of the rotational capacity that is used in this study is the ratio of the available inelastic rotation of the beam (which is the range of the rotation over which the 90% of the plastic moment resistance M of the cross section is retained) to the rotation that corresponds to the attainment of the 0.9M. An alternative definition for the inelastic rotation is used here. Figure 1a, illustrates the deformed shape of the beam during the loading. Configuration 1 corresponds to the first yielding of the cross-section (φ and φ are the related chord rotations at the support cross-section and at the theoretical point of the plastic hinge) while the second one is the deflected shape of the beam after yielding (φ and φ are the corresponding chord rotations). The available inelastic rotation φ is actually the pure rotation of the plastic hinge, defined as: φ Δφ Δφ (2) where Δφ φ φ and Δφ φ φ. This definition is based only on the chord rotations since the curvature of the beam remains the same after the formulation of the plastic hinge (configuration 1). Figure 1b, illustrates the Moment-Inelastic rotation curve that is used for the calculation of the rotational capacity, which is based on the inelastic rotation φ.. φ1y φ1 Plastic hinge A B Moment 0.9Μpl configuration 1 A φ2y B Δφy configuration 2 A φ2 B Δφ φ0.9μpl rotation (Δφpl) (a) (b) Figure 1: Deformation of the half-symmetric beam and the Moment Plastic rotation diagram. 3 DESCRIPTION OF THE PROBLEM Three different cross-sections are studied. The geometry and classes of the cross-sections according to EN [1] and EN [2] are given in Table 1. Three beams with different lengths are considered (6m, 7m and 9m respectively), for all the studied cross-sections. Initial imperfections are incorporated in the geometry of the steel beams in order to conduct parametric analyses with respect to the amplitude of the initial imperfections. In this study three different amplitudes of initial imperfections are used which are 0.5mm, 1mm, and 2mm. Table 1 indicates the classification of the cross sections at both ambient and elevated temperatures. It is noted that at elevated temperatures, the class changes for all the cross sections. The yield strength of steel is equal to 275Mpa in all cases. H B t f t w Classification Classification (mm) (mm) (mm) (mm) Ambient Temperature Elevated Temperature Section A Section B Section C Table 1. Geometry and classification according to EC3 for the considered beam cross-sections.

3 The rotational capacity of the beams is studied under static, monotonic loading at different temperature levels. The static problem that is modeled is a four point bending test. The problem is handled through geometric non-linear analysis taking into account the temperature dependent non-linear material properties. The numerical model that is developed considers the initial imperfections and parametric analyses are conducted taking into account different amplitudes of the initial geometrical imperfections. In order to simplify the already complex problem, the following assumptions are adopted for the assessment of the moment-rotation curves: The temperature is uniform and constant along the beam. No-thermal gradient is considered in the cross-section. The beam is free to expand longitudinally, which means that the study does not take into account the effect of thermal compressive forces. 4 NUMERICAL MODELLING The numerical models are developed using the non-linear finite element code MSC-Marc. The threedimensional numerical model utilizes four-node, thick-shell elements and takes into account the nonlinear elastic-plastic stress-strain relationship of steel at elevated temperatures. The yield stress, the proportionality limit and the elastic modulus are supposed to be temperature dependent according to EN [2]. The Von Mises yield criterion is used in the numerical analysis. Additionally, the analysis takes into account the geometric non-linearity. The numerical analysis that is conducted for the simulation of the four point bending has two different stages following the test procedure that is described in [6]. At the first stage the steel beam is heated until the desired temperature T is reached. It must be noticed that during the heating stage the temperature is supposed to be uniform along the member. At the second stage the temperature remains constant and the beam is submitted to loading at mid-span until failure (Figure 2). temperature displacement t 1 t 2 60 steps 200 steps Figure 2. Loading procedure pseudo-time There are many different ways to introduce initial geometric imperfections in structural members. A simple way in the context of finite element analysis is to extract the buckling eigenmodes and introduce them as imperfections with specific amplitude. More specifically, the normalized buckling mode is multiplied by a scale factor, leading to certain maximum amplitude and the resulting displacements are added to the initial coordinates of the structural member. The initial imperfections are generated from the first local buckling eigenmode for the upper flange of the beam. It is noted that the numerical analyses take place at both ambient and elevated temperatures ( ⁰ C). The structural boundary conditions at both supports are applied at the middle node of the web. The vertical y- displacement and the rotation about the longitudinal-x axis are restrained at both supports while the x- displacement is restrained only at the left support. The out-of-plane displacements are prevented at the location of the supports and at mid-span. Aiming to reduce the computational cost associated with the nonlinear three-dimensional modeling, only half of the total length of the beam is considered, using the appropriate symmetry boundary conditions. 5 RESULTS The moment plastic rotation diagrams for the 6m length beam for the three different cross sections are illustrated in Figures 3, 4, 5 and 6. More specifically, Figure 3 corresponds to the cross section with 15mm flange thickness, Figures 4, 5 to the cross section with 17mm flange thickness and Figure 6 to the cross section with 19mm flange thickness. The results of the analyses show that the incorporation of the initial imperfection of the beam geometry leads to the reduction of the ultimate load bearing capacity of the beam and this more

4 obvious at elevated temperatures. Additionally, it is interesting to notice the effect of the amplitude of the initial imperfections on the descending branch of the curve. The diagrams show that as the magnitude of the initial imperfection increases, the descending branch becomes steeper. This holds for all the imperfect models at both room conditions and at elevated temperatures. The perfect model simulates adequately the phenomenon, until the maximum moment is attained. The results concerning these analyses are not accurate concerning the unstable branch. In general, due to the local instabilities that arise during the physical phenomenon, it is more accurate to incorporate the initial imperfections into the beam geometry in order to obtain more realistic results using the finite element method. It can be observed that the gradient of the descending branch is milder, compared to the imperfect models. a) o C b) 200 o C c) 300 o C d) 400 o C e) 500 o C Figure 3. Moment Plastic rotation curves for span equal to 6m and flange thickness 15mm. a) o C b) 200 o C c) 300 o C Figure 4. Moment Plastic rotation curves for span equal to 6m and flange thickness 17mm ( o C)

5 d) 400 o C e) 500 o C f) 300 o C g) 700 o C Figure 5. Moment Plastic rotation curves for span equal to 6m and flange thickness 15mm ( o C) a) o C b) 200 o C c) 300 o C d) 400 o C e) 500 o C f) 600 o C g) 700 o C Figure 6. Moment Plastic rotation curves for span equal to 6m and flange thickness 19mm. The cross-section with the 15mm flange thickness, which is classified as Class 3 at elevated temperatures, loses its rotational capacity over 300 o C for amplitudes of initial imperfections of 1mm or more. The crosssection with the 17mm flange thickness is classified as Class 2 at elevated temperatures; therefore it should have

6 adequate rotational capacity. However, from the diagrams it is clear that the rotational capacity is rapidly decreased, especially for amplitude of initial imperfections greater than 1mm. The cross-section with the 19mm flange thickness is also classified as Class 2 at elevated temperatures. The behavior of this cross-section is similar to the one with the 17mm thick flange. Nevertheless, it maintains the rotational capacity for higher temperatures than the other cross-sections. Using the above diagrams for the 6m length beam and the corresponding ones for the 7m and the 8m length beams, the rotational capacity for each case is calculated. Figures 7, 8 and 9 illustrate the temperature evolution of the rotational capacity with respect both to the amplitude of the initial imperfections and to the flange thickness for beam lengths equal to 6, 7 and 9m respectively. It can be observed the rotational capacity is slightly increased at temperature equal to 500 o C. This is because of the stress-strain relationship of the steel at elevated temperatures. More specifically, at this temperature the yield stress starts dropping down and therefore the plastic moment resistance decreases. Moreover, it is clear that increasing the thickness of the flange, the rotational capacity increases as well. This behavior is expected, since the EN [2] provisions use the flange thickness in order to classify the cross-sections. Furthermore, increase of the amplitude of the initial imperfections force the rotational capacity to reduce. Finally, Figures 10 to 12 show the evolution of the rotational capacity with respect to the span of the beam for all the case studies that are considered. It is noted that the rotational capacity becomes zero for temperature higher than 300 o C, 600 o C and 700 o C for flange thickness equal to 15mm, 17mm and 19mm respectively. Therefore, the corresponding diagrams are not presented here. The results indicate that increasing the beam s length the rotational capacity decreases and this holds for all the cases that are considered in this study. As the length of the beam increases, the rotational capacity decreases and this is more obvious for higher amplitude of initial imperfections. As it is expected, the beam is more ductile as the thickness of the flange increases and as a result the beam can retain the rotational capacity for higher temperature levels. The above show that the classification of cross-sections according to EN is not accurate since the rotational capacity in all cases becomes zero for temperature higher than 600 o C. This indicates that, according to the findings of this study, the coefficient 0.85 that is used for the classification of the cross-sections at elevated temperatures should be reconsidered. In order to invigorate this remark, further research is required. Also, speculation arises for the effect of the span of the beam on the rotational capacity. Figure 7. Temperature evolution of rotational capacity for span equal to 6m

7 Figure 8. Temperature evolution of rotational capacity for span equal to 7m Figure 9. Temperature evolution of rotational capacity for span equal to 9m

8 Figure 10. Temperature evolution of rotational capacity with respect to the span of the beam (flange thickness equal to 15mm) Figure 11. Temperature evolution of rotational capacity with respect to the span of the beam (flange thickness equal to 17mm)

9 Figure 12. Temperature evolution of rotational capacity with respect to the span of the beam (flange thickness equal to 19mm) CONCLUSIONS This paper aims to develop numerical models for the determination of the rotational capacity of steal I-beams at elevated temperatures. The numerical models use four-node shell elements in order to take into account the effect of the geometric initial imperfections. The problem is handled through non-linear analysis taking into account both the geometric and the material non-linearity. Three different cross-sections are considered with respect to the thickness of the flange. The study reveals that the rotational capacity is decreased at elevated temperatures. The incorporation of the initial imperfections to the beam geometry leads to excessive decrease of the rotational capacity and this holds at both ambient and elevated temperatures. Furthermore, as the span of the beam increases, the beam loses the rotational capacity even for lower temperature levels. On the other hand the higher of the thickness of the flange of the beam (i.e. the more compact the beam), the values of the rotational capacity increase and the beam becomes more ductile at elevated temperatures. More research is required in order to get more safe and clear conclusions. The current study indicates that the coefficient 0.85 that is used for the classification of the cross-sections at elevated temperatures should be reconsidered. REFERENCES [1] European Committee for Standardization (CEN), (2005), EN , Eurocode 3: Design of steel structures Part 1-1. General rules and rules for buildings. [2] European Committee for Standardization (CEN), (2003), EN , Eurocode 3: Design of steel structures Part 1-2. General rules structural fire design. [3] Gioncu, V., Petcu, D. (2007), Available rotational capacity of wide flange beams and beam-columns, Part

10 2. Experimental and Numerical tests, Journal of Constructional Steel Research, Vol. 43, pp [4] Knobloch, M., Fontana, M. (2006), Strain-based approach to local buckling of steel sections subjected to fire, Journal of Constructional Steel Research, Vol. 62, pp [5] Dharma, R.B.,Tan, K. (2007), Rotational capacity of steel I-beams under fire conditions Part I: Experimental study, Engineering Structures, Vol. 29, pp [6] Dharma, R.B.,Tan, K. (2007), Rotational capacity of steel I-beams under fire conditions Part I: Numerical Simulation, Engineering Structures, Vol. 29, pp

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