## 3 Results
### Numerical Model Assessment
#### 3.2.3 Validation of the Model  
Table 2 reports the final temperatures calculated by the FEM using the self-consistent equivalent coefficient H, calculated as reported in Equations (3)-(8). For comparison, the results of computation using the equivalent convection coefficients recommended by Eurocode 1 [24] equal to 9 W (m\({}^{2}\) K)\({}^{-1}\) are reported; these values largely overestimate the experimental results.  
As stated in the introduction, for a self-consistent methodology, a criterion for the validation must be applied. For this purpose, the condition reported in the following equation is therefore employed to numerically validate the results of the self-consistent model [27; 28; 29; 30].  
\[\left|\mathrm{E}\right|<\;\mathrm{U}, \tag{12}\]  
where E is the error between numerical (F\({}_{\mathrm{n}}\)) and experimental (F\({}_{\mathrm{e}}\)) measurements and U is the norm between experimental (e\({}_{\mathrm{e}}\)) and numerical error (e\({}_{\mathrm{n}}\)), as expressed in the following equations:  
Figure 5: Cell dimension vs. displacements calculated at point B. Comparison with the experimental displacements (RINA test report n. 2016 CS011021/14 2017).  
\[\mathrm{E}\ =\ \mathrm{F_{n}}-\mathrm{F_{e}}\, \tag{13}\]  
\[\mathrm{U}\ =\ \sqrt{\mathrm{e_{e}^{2}+e_{n}^{2}}}\, \tag{14}\]  
The numerical error can be calculated with the following equation:  
\[\mathrm{e_{n}}=\ 1.25\ |\mathrm{F_{0}}-\mathrm{F_{n}}|, \tag{15}\]  
where 1.25 is a security factor, suggested by [PERSON] [30], and \(\mathrm{F_{0}}\) is the ideal value calculated through the Richardson extrapolation [29] reported in the following equation:  
\[\mathrm{F_{0}}=\ \mathrm{F_{i}}+[(\mathrm{F_{i}}-\mathrm{F_{i+1}})/(\mathrm{r^{P} }-1)], \tag{16}\]  
For the thermal analysis, an experimental error of \(\pm\) 1.1 \({}^{\circ}\)C was considered, given by the accuracy of the thermocouples used. Figure 6 shows the temperature field on the fire-unexposed side of the door, whereas Table 2 compares the experimental temperatures and the numerical results. Equation (12) is satisfied in all of the test points, thus assessing the validity of the numerical method.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \multirow{2}{*}{**Point**} & **T Eurocode** & **T** & **T Measured \({}^{*}\)** & & \\  & **Model (\({}^{\circ}\)C)** & **Self-Consistent** & **(\({}^{\circ}\)C)** & **[E]** & **U** \\ \hline
1 & 206.0 & 100.0 & 99.0 & 1.0 & 1.4 \\
2 & 206.0 & 100.0 & 100.0 & 0.0 & 1.0 \\
5 & 204.0 & 99.0 & 99.0 & 0.0 & 1.0 \\
3 & 206.0 & 100.0 & 101.0 & 1.0 & 1.4 \\
4 & 206.0 & 100.0 & 101.0 & 1.0 & 1.4 \\ \hline \end{tabular}  
* Data reported in the RINA test report n. 2016 CS011021/14 2017. Ambient temperature 18 \({}^{\circ}\)C.  
\end{table}
Table 2: Comparison between modelled—using, respectively, Eurocode and self-consistent convection coefficient—and measured temperatures; error estimation and validation of the self-consistent model.  
Figure 6: Temperature distribution on the fire-unexposed side of the door at the final stage of the fire test (60 min); scale in (K).  
An experimental error of \(\pm\) 1.0 mm is reported for the structural analysis. Figure 7 shows the displacements field, whereas Table 3 compares the experimental and calculated displacements. The condition reported in Equation (12) is satisfied for all the measured displacements. The model is, therefore, validated according to this criterion.