# Parametric Study on the Buckling of Unbraced Steel Frames under Fire Situation
## 2 Materials and Methods
### Methodology Used in the Design of the Structure  
Structural analysis of the frames to calculate the internal forces was carried out using the SAP 2000 [28] program. Second-order effects (P-\(\Delta\)) were included in the analysis and the  
Figure 2: Geometries, dimensions (in meters), and cross-sections of the studied frames.  
design, which was performed in the elastic range. In addition, global frame imperfections were modeled by the use of equivalent horizontal forces, as suggested in EN 1993-1-1 [2].  
The design of the beams and columns of the frames was performed in accordance with EN 1993-1-1. The columns are of type HEA and HEB and beams of type IPE. All the members were designed in S355 steel and were calculated with the buckling coefficient \(k=1.0\) in all members. It should be emphasized that the values of the actions and combinations considered are in accordance with the relevant Eurocodes [29; 30]. Figure 2 shows the final sections obtained.  
For the design in case of a fire, and because fire is classified as an accidental action, in the EN 1990 [29], accordingly the design effect of actions for the fire situation, \(E_{fi,d}\), must be obtained using the accidental load combination given by:  
\[\sum_{j\geq 1}G_{k,j}+(\psi_{1,1}or\psi_{2,1})\ Q_{k,1}+\sum_{i>1}\psi_{2,i}Q_{ k,j}+A_{d} \tag{1}\]  
Table 1 shows the load combinations used in this study at elevated temperatures, where \"Combination 1\" was used in the buckling analysis and \"Combinations 1 and 2\" were used in the calculation of the fire resistance, where, \(Q_{k}\) is the characteristic value of the variable action, \(G_{k}\) is the characteristic value of the permanent action, \(W_{k}\) is the characteristic value of the wind action, \(\psi_{1,1}\) and \(\psi_{2,1}\) are the combination factors affecting the leading action as defined in EN1991-1-2 and \(A_{d}\) represent the design value of the indirect thermal action due to the fire. In the load combinations of Table 1, the frequent value \(\psi_{1,1}Q_{k,1}\) of the leading variable action was used.  
Methodology Used to Determine the Critical Column of the Frame at Normal Temperature (20 \({}^{\circ}\)C) and in Fire  
At normal temperature, the critical column is the one that belongs to the critical story and has the highest ratio between axial compression force \(N_{Ed}\) and the Euler critical load \(N_{cr}\). Thus, the parameter \(\eta\) can be defined as:  
\[\eta=\frac{N_{Ed}}{N_{cr}} \tag{2}\]  
\[N_{cr}=\frac{\pi^{2}EI}{l^{2}} \tag{3}\]  
where \(l\) is the member length, \(E\) is the Young modulus at room temperature, and \(I\) is the section moment of inertia. Similarly, in the case of fire, the critical column corresponds to the one with the highest parameter \(\eta_{fi}\), where the subscript \"\(fi\)\" denotes the fire situation and can be calculated as:  
\[\eta_{fi}=\frac{N_{Ed,fi}}{N_{cr,fi}} \tag{4}\]  
\[N_{cr,fi}=\frac{\pi^{2}k_{E,\theta}EI}{l^{2}{}_{fi}} \tag{5}\]  
the \(k_{E,\theta}\) is the reduction factor for the Young modulus at elevated temperature according to EN1993-1-2.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Load Combination** & **Accidental Combination** \\ \hline Combination 1 & \(G_{k}+0.5Q_{k}\) \\ Combination 2 & \(G_{k}+0.2W_{k}+0.3Q_{k}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Load combinations.