## 2 Materials and Methods
### Numerical Modelling of Fire Resistance Test
#### 2.2.2 FEM Modelling  
Fire-test modelling is performed using MSC Nastran with Patran 2017 and it is developed in a two-step analysis: firstly, a thermal investigation is carried out in order to evaluate the temperature distribution, which is then used as an input for the subsequent structural analysis. The two analyses can be considered uncoupled as the thermal distortions do not significantly affect the temperature distribution during the test [2]. Both the analyses are performed as non-linear calculations, as the materials' properties depend on temperature. Figure 1: (**a**) Model geometry; (**b**) test measuring points: location of thermocouples (1, 2, 3, 4, 5) and displacements measurements point (A, B, C, D, I). The thermal analysis is performed by means of unsteady-state simulation which was shown to be necessary to obtain reliable fitting of the experimental results and to study the thermal evolutions (i.e., influence of the thermal bridge on fire resistance) during the fire-test [13,14]. Conversely, the structural modelling was performed using a steady-state simulation. Since the key constituent materials of the fire-door are not viscoelastic, the deformation history does not significantly influence the final deformations at the maximum temperature, which fully justifies the use of steady-state analysis in this case [10]. The model is validated on the basis of the RINA (Registro Navale Italiano--Italian Naval Register) certification [22] obtained from the FTP fire-test performed on the above quoted door. The geometry is discretized using solid elements for the insulating material, while shell elements are used for components that have thickness negligible compared to the other dimensions (e.g., door sheets, frame, bulkhead and reinforcements). Figure 2 reports an example of the door mesh. The model was bound by fixing rotations and translations at the edges of the bulkhead, since it was welded to the test machines. R-type elements were used to model both the lock and the hinges--in particular, rigid body elements (RBE2) that are multi-point constraints (MPCs) were adopted. These connectors are rigid elements that force the nodes to have the same displacements; they provide independent degrees of freedom (DOFs) at a grid point and dependent DOFs at one or more grid points. The thermal elongation of steel was adopted for the connectors. For the thermal analysis, radiation and convection coefficients were calculated for the fire-unexposed side of the door using the following equations [23]:  
\[\mathrm{h}_{\mathrm{r}}=\ \varepsilon\ \sigma\ \mathrm{(T_{2}^{2}+T_{0}^{2})(T_{2 }+T_{0})\ \ (W\ (m\ \mathrm{K})^{-1})} \tag{3}\]  
\[\mathrm{h}_{\mathrm{c}}=(\mathrm{Nu}\cdot\mathrm{k}_{0})/L\ \ (W\ (m\ \mathrm{K})^{-1}), \tag{4}\]  
where T\({}_{2}\) is the average temperature between T\({}_{0}\) and that calculated on the fire-unexposed side without considering radiation and convection, \(\varepsilon\) is the steel emissivity, \(\sigma\) is the Stefan-Boltzmann constant in (W m\({}^{-2}\) K\({}^{-4}\)), L is the height of the door in (m) (characteristic dimension) and k\({}_{0}\) is the thermal conductivity of air in (W (m K)\({}^{-1}\)). The Nusselt number \(\mathrm{Nu}\) is calculated for the vertical wall with turbulent and laminar flow [23] using the following equation:  
\[\mathrm{Nu}\ =\left\{0.825+\left[\left(0.387\ \mathrm{Ra}^{\frac{1}{5}} \right)/\left(1+\left(\frac{0.492}{\mathrm{Pr}}\right)^{\frac{9}{15}}\right)^{ \frac{8}{27}}\right]\right\}^{2} \tag{5}\]  
Figure 2: Particular of the door mesh. where Pr is the Prandtl number and Ra is the Rayleigh number reported in the following equations:  
\[\Pr\ =\ \
u/\alpha \tag{6}\]  
\[\mathrm{Ra}\ =\!\!\left[\beta\ (\mathrm{T}_{2}-\mathrm{T}_{0})\ g\,\mathrm{L}^{3} \right]/(\
u\ \alpha), \tag{7}\]  
with \(\
u\) as the kinematic viscosity of air in (m s\({}^{-1}\)), \(\alpha\) as the thermal diffusivity of air in (m s\({}^{-1}\)), \(\beta\) as the volumetric expansion coefficient of air in (K\({}^{-1}\)) and g as the gravity acceleration in (m s\({}^{-2}\)).