# Parametric Study on the Buckling of Unbraced Steel Frames under Fire Situation
## 3 Results and Discussion
### Determination of the Buckling Coefficient
#### 3.1.2 Critical Column in Fire  
As defined previously in Section 2.2, the critical column in a fire situation corresponds to the column with the highest ratio between axial compression force (\(N_{Ed,fi}\)) and the Euler critical load as a function of temperature (\(N_{cr,fi}\)). For all the cases considered in this study, it is highlighted in Table 5 the critical columns in fire situations. Table 6 summarizes the findings.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Story**} & \multirow{2}{*}{**Column**} & \multirow{2}{*}{**Profile**} & **E** & **I\({}_{\mathbf{y}}\)** & **L** & **N\({}_{\mathbf{cr}}\)** & **N\({}_{\mathbf{Ed}}\)** & \(n\)** \\  & & & **[kN/mm\({}^{2}\)]** & **[mm\({}^{4}\)]** & **[mm]** & **[kN]** & **[kN]** & **[kN]** \\ \hline \multirow{4}{*}{0} & C1 & HE300A & 210 & 182,600,000 & 5000 & 15,138 & 518.4 & 3.42\% \\ \cline{2-8}  & C2 & HE300A & 210 & 182,600,000 & 5000 & 15,138 & 748.0 & 4.94\% \\ \cline{2-8}  & **C3** & **HE320A** & **210** & **229,300,000** & **5000** & **19,010** & **943.8** & **4.96\%** \\ \cline{2-8}  & C4 & HE300A & 210 & 182,600,000 & 5000 & 15,138 & 628.8 & 4.15\% \\ \hline \multirow{4}{*}{1} & C5 & HE200A & 210 & 36,920,000 & 3500 & 6247 & 409.6 & 6.56\% \\ \cline{2-8}  & **C6** & **HE200A** & **210** & **36,920,000** & **3500** & **6247** & **487.8** & **7.81\%** \\ \cline{2-8}  & C7 & HE220A & 210 & 54,100,000 & 3500 & 9153 & 637.9 & 6.97\% \\ \cline{2-8}  & C8 & HE200A & 210 & 36,920,000 & 3500 & 6247 & 383.2 & 6.13\% \\ \hline \multirow{4}{*}{2} & C11 & HE200A & 210 & 36,920,000 & 3500 & 6247 & 204.0 & 3.27\% \\ \cline{2-8}  & **C12** & **HE200A** & **210** & **36,920,000** & **3500** & **6247** & **242.6** & **3.88\%** \\ \cline{2-8}  & C13 & HE220A & 210 & 54,100,000 & 3500 & 9153 & 327.2 & 3.57\% \\ \cline{2-8}  & C14 & HE200A & 210 & 36,920,000 & 3500 & 6247 & 144.3 & 2.31\% \\ \hline \hline \end{tabular}
\end{table}
Table 4: Critical column of the unbraced frame P-3\(\times\)3 for normal temperature.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{2}{c}{**Critical Instability Mode**} \\ \hline
**Frame** & **Fixed** & **Pinned** \\ \hline P2-3\(\times\)3 & & & \\ \hline \hline \end{tabular}
\end{table}
Table 3: Critical mode of instability of frames with fixed and pinned support at time t = 0 min.  
During the fire, as the temperature increases, the Euler critical load decreases and, therefore, the \(\eta_{fi}\) parameter increases. Due to this variation of \(\eta_{fi}\), the critical column may change between normal temperature and fire. For example, for the frame P2-3\(\times\)3, the critical column at normal temperature is column C6 of story 1 and column C3 of story 0, respectively, for fixed and pinned supports, as shown in Table 3. However, for a generalized fire on story 2, after a certain temperature threshold, the critical column of the frame becomes column C12 of story 2, irrespectively of the supporting conditions, as shown in Table 5.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Frame**} & \multirow{2}{*}{**Generalized**} & \multicolumn{2}{c}{**Fixed**} & \multicolumn{2}{c}{**Pinned**} \\ \cline{3-6}  & & **Fire** & **Column** & **Profile** & **Column** & **Profile** \\ \hline \multirow{3}{*}{P2-3\(\times\)3} & 0 & C3 & HE320A & C3 & HE320A \\ \cline{2-6}  & 1 & C6 & HE200A & C6 & HE200A \\ \cline{1-1} \cline{2-6}  & 2 & C12 & HE200A & C12 & HE200A \\ \hline \hline \end{tabular}
\end{table}
Table 6: Critical Column of Frames in Fire.  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Generalized Fire**} & \multirow{2}{*}{**Cases**} & \multicolumn{2}{c}{**Critical Instability Mode**} \\ \cline{3-4}  & & **Fixed** & **Pinned** \\ \hline \multirow{3}{*}{**Story 2**} & \multirow{3}{*}{9 and 10} & C12 & C12 \\ \cline{3-4}  & & 1, 2, 5, 6, 11, and 12 \\ \cline{2-4}  & 3 and 7 & C6 \\ \cline{2-4}  & 9 and 10 \\ \hline \hline \end{tabular}
\end{table}
Table 5: _Cont._