# Parametric Study on the Buckling of Unbraced Steel Frames under Fire Situation
## 3 Results and Discussion
### Determination of the Buckling Coefficient
#### 3.1.4 Buckling Lengths  
The buckling length of a column depends on whether the instability of the frame is governed (or not) by the story for which the column belongs. In case of a fire, such a principle has a particular influence as the critical column may change from one story to another depending on the temperature evolution, as it is detailed in the next sections for the P2-3\(\times\)3 frame with fixed and pinned supports. Frame P2-3\(\times\)3 with Fixed Supports for All Fire Cases  
For the P2-3\(\times\)3 frame with fixed supports, the critical column at normal temperature (20 \({}^{\circ}\)C) is located in the story 1 (see Table 3). However, analyzing Figure 6, is observed that, when the temperature on story 0 reaches 485 \({}^{\circ}\)C, the instability of the frame becomes governed by a column in this story, making it the critical one. In conclusion, from 20 \({}^{\circ}\)C up to 485 \({}^{\circ}\)C, the critical column that governs the instability of the unbraced frame with fixed supports is in story 1. However, due to high temperatures in story 0 there is a loss of rigidity in that story. Consequently, when temperature reaches 485 \({}^{\circ}\)C, this story governs the overall stability of the frame. This makes the critical load parameter vary significantly only from that temperature onwards. The critical load parameter (\(\alpha_{cr,fi}\)) for this temperature is then used to determine values for Euler critical load \(N_{cr,fi}\) (Equation (6)), and the corresponding buckling lengths \(l_{fi}\) (Equation (7)) and buckling factor \(k_{fi}\) (Equation (8)) at elevated temperatures for the several fire cases. It is worth mentioning that the same phenomenon is observed for a generalized fire on story 2, i.e., when the temperature in this story reaches approximately 545 \({}^{\circ}\)C (see Figure 4e), it becomes the critical one. Furthermore, in Figure 6, it is shown that the buckling length at normal temperature (20 \({}^{\circ}\)C) is \(l_{fi}/l=1.21\) while for a generalized fire in the story 0 the value of the buckling length corresponding to a temperature of 485 \({}^{\circ}\)C is approximately \(l_{fi}/l=1.0\). When the fire is in story 1 or simultaneously in stories 0 and 1 and also simultaneously in stories 1 and 2, the critical column of the unbraced frame with fixed supports at 20 \({}^{\circ}\)C corresponds to the same as for the fire situation and thus governs the instability of the story both at 20 \({}^{\circ}\)C and at high temperatures. In these cases, it can be seen that the critical load parameter varies sharply since the beginning of the fire (Figure 4c,g,i) but the buckling lengths remain more or less constant during the fire, as shown in Figure 7. For both cases, whether the fire is developing on the critical story at normal temperature or not, it is concluded that it is conservative to use the buckling length determined at normal temperature when verifying the fire resistance of a column belonging to a steel frame as established by EN 1993-1-2 [15] and ABNT NBR 14323: 2013 [16]. Figure 5: Comparison between sections heated on 3 or 4 sides. Frame P2-3\(\times\)3 with Pinned Supports for All Fire Cases  
For the frame P2-3\(\times\)3 with pinned supports, the critical column at normal temperature (20 \({}^{\circ}\)C) is located in the story 0 (see Table 3). In this case, Figure 8 shows that, when the temperature on story 1 reaches 570 \({}^{\circ}\)C, the critical column of the frame now belongs to this story, and it governs the instability of the frame. In conclusion, from 20 \({}^{\circ}\)C up to 570 \({}^{\circ}\)C, the critical column that controls the instability of the pinned frame is in story 0 and due to high temperatures that develop in story 1, the critical column of this story gradually loses rigidity, and when the temperature reaches 570 \({}^{\circ}\)C in the story 1 it becomes the critical one. The same phenomenon is observed for story 2 when the temperature reaches approximately 670 \({}^{\circ}\)C (see Figure 4f).