# Parametric Study on the Buckling of Unbraced Steel Frames under Fire Situation
## 3 Results and Discussion
### Determination of the Buckling Coefficient
#### 3.1.3. Critical Load Parameter in Fire  
Figure 4 presents the evolution of the critical load parameter (\(\alpha_{cr,fi}\)) relative to temperature with different support conditions (columns with fixed or pinned supports at the base) for the various fire cases.  
As was the case in [PERSON] et al. (2018), during the fire, the \(\alpha_{cr,fi}\) decreases because of the stiffness reduction that occurs in the members with increasing temperature. In addition, the evolution of the critical load parameter as a function of the temperature depends on whether the most critical columns, i.e., those with the highest \(\eta\) at normal temperature, are affected by fire. In general, for a certain fire scenario, if the critical column in fire and normal temperature are the same, the \(\alpha_{cr,fi}\) reduces for lower temperature values, i.e., from the starting of the fire, as shown in Figure 3(b),c,g-i. Otherwise, the critical load parameter remains unchanged, as illustrated in Figure 3(a),d-f,j, until a certain temperature, and only after such temperature is reached does the story in fire begin to control the stability of the frame.  
Finally, it was found that the phenomena mentioned in the above paragraphs occur in a similar manner for fires isolated in one story or occurring, simultaneously, in two adjacent stories.  
Cases 1-(CE,1,4S-B4S), 2-(CE,3S-CI,4S-B4S), 5-(CE,1,4S-B3S), 6-(CE,3S-CI,4S-B3S), 9-(CE,1,4S-Bcold), 10-(CE,3S-CI,4S-Bcold), 11-(CE,1,protection-B3L,15 min) and 12-(CE,1,3L, 15 min-Bprotection) showed similar behavior, i.e., whether columns and beams are heated in three or four sides did not significantly influence the critical load parameter. This phenomenon is justified by the fact that there are no significant temperature differences between columns and beams heated on 3 (3S) or 4 (4S) sides (see Figure 5).  
It was observed that cases 3-(CE, cold-CI,4S-B4S) and 7-(CE,cold-CI,4S-B3S) showed better behavior, i.e., for the same temperature the critical load parameter of these two cases was higher than in the other mentioned in the previous paragraph, since the story in fire presents greater stiffness for these cases. Only the internal columns and beams were heated, while the external columns were at normal temperature. This phenomenon is more evident for cases where the heated story is not the critical story at normal temperature. Finally, it was also observed that frames with pinned supports are more sensitive to instability (see Figure 4).  
Meanwhile, cases 4-(CE,1,cold-B4S) and 8-(CE,1,cold-B3S) had better performance than all other cases, since only the beams are heated, and thus there was not much reduction of stiffness in the frames. For the cases where the frames have pinned supports and for which the beams in fire do not belong to the critical story at normal temperature, it was found that fire in this story had little influence (and in some cases no effect at all) on the instability of the frames (see Figure 3(d)-f).  
Figure 4: _Cont._Figure 4: Variation of the critical load parameter with changes in the temperature of unbraced frames. P2-3\(\times\)3.