# Fire Resistance of Ultra-High-Strength Steel Columns Using Different Heating Rates
## 2 Methodology and Validation
### Model Validation for CHS Columns under Fire
#### 2.3.3 Nonlinear Structural Analysis at Elevated Temperatures  
To validate the results in fire situations, where the temperature profile in the columns was provided through thermocouples attached to their surfaces, a nonlinear buckling analysis at elevated temperatures was conducted. The temperature profile from the thermocouples, which varied with time, was uniformly applied to the column, with the average temperature of the three central thermocouples (T2, T3, and T4), see Figure 2. A reference temperature of 20 \({}^{\circ}\)C was defined for zero thermal deformation. The force was applied with the desired load level, represented as a table in Ansys, as shown in Table 7, where \(t_{sim}\) corresponds to the simulation time. The Newton-Raphson method was employed with time increments. The criterion adopted was based on force and moment, with both having a tolerance of 0.001 and a minimum reference value of 1 N and 1 Nm, respectively. The validation of the thermo-mechanical model established using GMNIA models, using six three load levels and two different heating regimes. Each of these simulations was compared with its respective experimental and numerical results from [PERSON] et al. [11] and is presented in Figure 7. The rate of displacement (d\(C\)/d\(t\)) was also included as an addition criterion to find the fire resistance. \begin{table}
\begin{tabular}{c c} \hline \hline
**Time [s]** & **Force (N)** \\ \hline
0 & 0 \\
10 & Fz \\ \(t_{sim}\) & Fz \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Nonlinear analysis force at elevated temperatures. Figure 6: Validation of load-bearing capacity of CHS column at room temperature [11]. Figure 6(a) illustrates the test results with a heating rate of 5 \({}^{\circ}\)C/min, with a constant mechanical load corresponding to 25% of the load-bearing capacity. Figure 6(a) represents the displacement over time. The Ansys results for axial displacement (C) agree with the experimental tests up to the first 30 min, diverging slightly thereafter. The evolution of the displacement rate (dC/d\(t\)) is also represented. It is important to note that [PERSON] et al. [11] did not mention the displacement rate criterion. Figure 6(b) presents the results obtained with a heating rate of 20 \({}^{\circ}\)C/min using the same mechanical load (25%). Notably, the differences in axial displacement are much smaller in this case, indicating a higher level of agreement between the Ansys results and the experimental data. The accuracy of the approximation will be presented by the Root Mean Square Error (RMSE) and the relative error. In Figure 6(c), the results obtained with a heating rate of 5 \({}^{\circ}\)C/min using a mechanical load corresponding to 50% of the load-bearing capacity are represented. Once again, the  
Figure 6(a): Validation for CHS columns for high temperatures [11]. results with this heating rate exhibit good agreement during the initial 30 min, with a more noticeable difference observed in the later stages of the analysis. Figure 7d presents the results for a higher heating rate (20 \({}^{\circ}\)C/min), with the same load level of 50%. These results demonstrate better agreement when compared to those obtained with a smaller heating rate, showcasing a closer match between the Ansys simulations and the experimental results at higher heating rates. For the load level cases up to 50%, the displacement contraction reverts the direction with the effect of the thermal expansion but is able to revert the direction once again when the material starts to decrease the stiffness and strength with temperature. Figure 7e presents the results for a heating rate of 5 \({}^{\circ}\)C/min with a load level corresponding to 75%. In this case, the results coincide until the 40 min mark. However, ANSYS anticipates the column failure or collapse when compared to the experimental tests, suggesting some differences in the structural response under these conditions. Finally, Figure 7f presents the results for a load level of 75% and a heating rate of 20 \({}^{\circ}\)C/min.