## 3 Thermo-Structural Numerical Model
### Nonlinear Structural FE Model  
The numerical structural solution requires non-linear material behavior, as the values of mechanical properties of concrete and steel depend on temperature. The results from the nonlinear heat transfer analysis consisted of the temperature-time curves for all the nodes within the three-dimensional model, which were subsequently applied as thermal loading to the mechanical model. Different mesh controls were used to ensure a regular mesh. After thermal analysis, thermal elements SHEL131 and SOLID90 were converted into structural elements. The steel docking was meshed with 8-node shell element (SHELL281) and a maximum size of 5 mm. This element is suitable when large non-linear deformations occur and temperature conditions are used. The concrete was meshed using a 20-node solid  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Temperature (\({}^{\circ}\)C)** & \(k_{CC}\) **(W/m\({}^{2}\) K)** \\ \hline
20 & 10 \\
604.5 & 10 \\
607 & 15 \\
673.7 & 15 \\
675.2 & 20 \\
827.1 & 20 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Relationship between ambient temperature and \(k_{CC}\) values.  
Figure 9: Thermal properties of LWC and NC: (**a**) thermal conductivity, and (**b**) specific heat.  
element (SOLID186). This element is used when plasticity, creep and large deformations and deflections occur. The contact between the steel decking and the concrete was defined as bonded [31]. The structural model had 90,509 nodes and 20,604 elements, with an average element quality of 0.9887 and an average orthogonal quality of 0.9928.  
The numerical model takes into account the plastic behavior of both materials, steel and concrete, using multilinear isotropic hardening laws. The design values for the steel decking were in accordance with experimental data previously reported [37] and followed the temperature dependence variation proposed by Eurocode 4. For all concretes, the stress-strain curves at ambient and elevated temperatures are based on Eurocode 4. For the case studied, the maximum experimental temperature reached was less than 400 \({}^{\circ}\)C.  
When composite slabs are mechanically loaded and simultaneously heated, the overall measured strain concrete is assumed as an additive combination of different components [38]. The stress-strain curves included in Eurocode 4 consider deformation as a combination of elastic deformation, plastic deformation, and creep deformation. However, there is an additional component to determine the total deformation to which steel decking and concrete are subjected. This component is deformation due to thermal expansion and has different values for steel and concrete. In recent decades, there have been deeper studies of the rheological behavior of both concrete and steel at high temperatures [39].  
The coefficient of thermal expansion of concrete varies with the type of cement and aggregates used, and the temperature reached. As the coefficients of thermal expansion of the various rocks that make up the aggregates and the cement paste are not the same, the variations in temperature cause differential thermal movements in the concrete mass, which can amplify its internal system of micro-fissures [40]. The deformation produced by temperature is shown in Equation (4).  
\[\varepsilon_{t}=\varepsilon_{te}+\varepsilon_{s}+\varepsilon_{mf} \tag{4}\]  
The \(\varepsilon_{t}\) can be simplified according to Equation (5), in which \(\varepsilon_{s}\) and \(\varepsilon_{mf}\) are defined in the \(\alpha\) coefficient [38]. This simplification was used in the coupled model developed in this study. Furthermore, as the maximum temperature measured within the concrete did not exceed 150 \({}^{\circ}\)C, the relationship between temperature and thermal expansion was defined as between 20 \({}^{\circ}\)C and 200 \({}^{\circ}\)C, as shown in Figure 10.  
\[\varepsilon_{t}=\alpha\times\Delta T \tag{5}\]  
Figure 10: Thermal expansion values of LWC and NC as a function of temperature.