# Fire Resistance of Ultra-High-Strength Steel Columns Using Different Heating Rates
## 2 Methodology and Validation
### Material Properties at Elevated Temperatures  
To describe the material's behaviour at high temperatures, stress-strain curves were generated using the Ramberg-Osgood model, as referenced in [11; 12]. This behaviour can be described by Equation (3) when \(\sigma\leq\sigma_{0,2,T}\) and by Equation (4) when \(\sigma_{0,2,T}<\sigma\leq\sigma_{u,T}\), where \(\varepsilon_{T}\) represents the strain at temperature \(T\), \(\sigma\) is the stress at temperature \(T\), \(E_{T}\) denotes the Young's modulus at that temperature \(T\), \(\sigma_{0,2,T}\) signifies the yield strength (0.2% proof stress), \(\varepsilon_{t0,2,T}\) corresponds to the strain for \(\sigma_{0,2,T}\), \(E_{0,2,T}\) is equivalent to the tangent modulus of \(\sigma_{0,2,T}\), \(n_{T}\) and \(n_{T}^{\prime}\) are the coefficients accounting for nonlinearity effects, and finally, \(\sigma_{u,T}\)  
Figure 3: Boundary conditions for: (**a**) CHS column; (**b**) Hybrid column. stands for the ultimate stress. For UHSS, \(\sigma_{u,T}\) represents the stress at 1.5% strain, while for SS, it represents the stress at 2% strain [17]. \[\varepsilon_{T}=\frac{\sigma}{E_{T}}+0.002\times(\frac{\sigma}{\sigma_{0,2,T}})^{ n_{T}} \tag{3}\]  
\[\varepsilon_{T}=\frac{\sigma-\sigma_{0,2,T}}{E_{0,2,T}}+\left(0.015-\varepsilon _{t0,2,T}-\frac{\sigma_{u,T}-\sigma_{0,2,T}}{E_{0,2,T}}\right)\times\left( \frac{\sigma-\sigma_{0,2,T}}{\sigma_{u,T}-\sigma_{0,2,T}}\right)^{n_{T}^{ \prime}}+\varepsilon_{t0,2,T} \tag{4}\]  
The properties of the UHSS tubes at ambient temperature are defined according to Table 2, while the properties of the SS plates are determined from Table 3. Using the reduction factors and nonlinear coefficients provided in [11] and [12], the stress-strain curves used for the constitutive model were generated for various temperatures, as depicted in Figure 4. Regarding the upper and bottom plates, only an elastic behaviour was considered, with an elastic modulus of 2000 GPa to simulate a rigid material. Another relevant mechanical property to consider is thermal expansion, which is defined by thermal strain. For UHSS tubes, the thermal strain was calculated as presented in [11] using Equation (5). Similarly, for SS plates, Equation (6) was retrieved from [12]. The UHSS material derives from a tempered martensitic microstructure, which is a microstructure responsible for being very strong but also brittle. In UHSS with martensitic microstructures, the strength depends on the concentration of carbon remaining in the solid solution (not presented in the cementite phase). From the analysis of phase diagrams of UHSS, it can be concluded that after exposure to temperatures below 600 \({}^{\circ}\)C, the UHSS will remain tempered martensite with ferrite, and the phases present in the microstructure of the material (ferrite, martensite, and cementite) do not change in the type of phases but changes only their sizes. During the heating process, at elevated temperatures, coincident with the 'tempering temperature of martensite', the tempering processes of this phase, with the subsequent carbon diffusion and removal from solid solution, will continue resulting in a significant strength reduction in the material. The heating effect on the material results in \begin{table}
\begin{tabular}{c c c} \hline \hline \(E_{T}\) **(GPa)** & \(\sigma_{0,2,T}\) **(MPa)** & \(\sigma_{u,T}\) **(MPa)** \\ \hline
198 & 1200 & 1375 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Mechanical properties room temperature of the UHSS tubes. Figure 4: Stress–strain graphs for: (**a**) UHSS; (**b**) SS. a different behaviour (reduction coefficients). The reduction in the yield strength on UHSS at elevated temperatures (heating for the first time) begins at a lower temperature level when compared to a preheated and cooled UHSS. At 600 \({}^{\circ}\)C, there could be a difference of 40% [6].