# Parametric Study on the Buckling of Unbraced Steel Frames under Fire Situation
###### Abstract  
A parametric investigation of several unbraced steel frames with regular and irregular geometry subjected to elevated temperatures is carried out in this study to determine the most accurate procedure and buckling lengths to be considered during the structural design under a fire situation. In such conditions, the stiffness and strength of steel decrease considerably due to high temperatures, and uncertainty remains in the application of the fire design rules of Eurocode 3 Part 1-2 (EN 1993-1-2) for unbraced frames as no information is given regarding the treatment of the deformed geometry (the so-called second-order effects). More precisely, it is unclear in the norm whether the verification based on the buckling length concept could be used or if a second-order analysis to calculate the internal forces is sufficient to ensure the stability of the frame in case of fire. Based on the linear buckling analysis of the steel frames accounting for the temperature development during a fire, recommendations for the appropriate buckling lengths to be used are given. Finally, it is demonstrated that using the recommended buckling lengths together with the design rules of EN 1993-1-2 leads to results in favor of safety when compared to the results obtained with the finite element method. On the other hand, it is concluded that using second-order internal forces and the real length of the columns as the buckling length, as suggested for room temperature design, yielded results outside safety when compared with the finite element method, and this analysis and verification procedure should not be used for the case of fire. f 20  
In the case of fire, the reduction in strength and stiffness of steel presents an additional difficulty for each of the mentioned points, and different studies on the behavior of frames in the case of fire have been reported in the past [5; 6; 7; 8; 9; 10; 11]. Indeed, resulting from material degradation at elevated temperatures, the elements suffer large deformations, and there is also the effect of thermal expansion that increases the axial forces in beams and columns [8; 12; 13]. These effects would only be negligible if the structures were heated uniformly in all members, which is unrealistic [14]. Therefore, in the case of fire, the calculation of the internal forces must account with such effects through the concept of buckling lengths associated with the global instability mode of the frame [5]. However, following that this is a simplified approach to account for the second-order effects at elevated temperatures, it is necessary to check if the safety verification produces results in the favor of safety by comparison with more refined analysis, as in the present study. To calculate the buckling lengths in the case of fire, both EN 1993-1-2 [15] and ABNT NBR 14323 [16] standards recommend that normal temperature procedures are used. In addition, the EN 1993-1-2 defines that, for braced frames, the values to adopt are \(l_{fi}=l_{cr}=0.5l\) for columns belonging to intermediate stories and \(l_{fi}=l_{cr}=0.7l\) for columns on the upper stories. No guidance is provided, however, for columns belonging to unbraced frames. In this context, [PERSON] et al. [17] proposed equations for calculating buckling lengths of columns belonging to braced frames, while [PERSON] et al. [5] validated the recommended values suggested by EN1993-1-2 for the buckling lengths of columns in braced frames. In their work, it was recommended buckling lengths for columns of unbraced frames as well however, this study was limited to columns heated on four sides and beams on three sides, with generalized fire occurring in each story at a time, and only frames with regular geometry were considered. Although this work was later extended to natural fires [7], confirming the validity of the suggested recommendations, there is still the need to cover other fire design situations. A new method for evaluating the stability of unbraced steel frames under fire conditions was proposed in [9].