# 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
As observed in Figure 8, the buckling length at normal temperature (20 \({}^{\circ}\)C) is \(l_{fi}/l=1.58\) and for a generalized fire in the story 1 the value of the buckling length corresponding to a temperature of 570 \({}^{\circ}\)C was approximately \(l_{fi}/l=1.0\). Thus, it is safe to use the buckling lengths calculated at normal temperature in the case of fire for frames with pinned supports. Such observation is also valid when the fire is in story 0 (Figure 4b) or simultaneously in stories-0 and 1 (Figure 4h), since the critical column of the pinned frame at 20 \({}^{\circ}\)C is the same as in fire. This fact is responsible for the instability of the story both at 20 \({}^{\circ}\)C and at high temperatures, with the buckling coefficient being calculated at normal temperature (20 \({}^{\circ}\)C), and buckling length remains more or less constant during the fire, as shown in Figure 9.  
Figure 6: Generalized fire in the story 0 of frame P2-3\(\times\)3 with fixed supports for a standard fire. Figure 7: Variation of the buckling factor with the temperature of the most critical column in frame P2-3\(\times\)3 with fixed supports for a standard fire. For cases 1-(CE,I,4S-B4L), 2-(CE,3S-CI,4S-B4S), 5-(CE,I,4S-B3S), 6-(CE,3S-CI,4S-B3S), 9-(CE,I,4S-Bcold), 10-(CE,3S-CI,4S-Bcold), 11-(CE,I,protection-B3S,15 min) and 12-(CE,I,3S, 15 min-Bprotection), the highest buckling length value was considered. Thus, for generalized fire in a story at a time or, simultaneously, in two adjacent stories in fire, the buckling coefficient of \(l_{fi}/l=1.0\) was considered for all columns. However, for columns belonging to story 0 of the frame with pinned supports, the coefficient to be used is \(l_{fi}/l=2.0\), (see Figures 10 and 11). Cases 3-(CE,cold-CI,4S-B4S) and 7-(CE,cold-CI,4S-B3S), where the external columns are at normal temperature (20 \({}^{\circ}\)C) and internal columns and beams are heated on 3 or 4 sides, presented lower buckling lengths. For these cases, a buckling factor of \(l_{fi}/l=0.5\) was considered for all columns except those belonging to story 0 of the frame with pinned supports, for which the value to be used is \(l_{fi}/l=1.0\). Finally, it was found that for cases 4 and 8, where the columns are at normal temperature (20 \({}^{\circ}\)C) and internal beams are heated, due to loss of stiffness provided by the heated beams, the buckling lengths were higher. One must pay attention to these cases, especially if the utilization factors of the columns are high; such an event may lead to the collapse of the columns even if they are protected and/or not heated. Figure 8: Generalized fire in story 1 of frame P2-3\(\times\)3 with pinned supports for a standard fire. Figure 10: Buckling lengths \(l_{fi}\) of columns in unbraced frames with pinned supports. Figure 9: Variation of the buckling factor with the temperature of the most critical column in frame P2-3\(\times\)3 with pinned supports for a standard fire. Buckling Lengths for All Different Frames for Fire Case-6  
In this section, all steel frames with fixed and pinned supports, as shown in Figure 2, were analyzed for the case of fire 6-(CE,3S-CI,4S-B3S), considering that the fire was located in a single story and also the situation where two stories are simultaneously exposed to fire. It was observed in Figures 12 and 13 that, for different geometries of steel frames (i.e., regular and irregular), the differences between the values of the buckling factors have not shown to be significant. We concluded that it is possible to propose approximate buckling lengths to verify the safety of regular and irregular unbraced steel frames fire was located in one story at a time or in two stories simultaneously.