# Geometrical Parametric Study on Steel Beams Exposed to Solar Radiation
## 4 Finite Element Modeling
### Finite Element Thermal Analysis  
COMSOL Multiphysics [45] was used in this research to carry out the three-dimensional finite element thermal analysis of the steel beams and modeling of all the boundary conditions, including solar radiation, convection, long-wave radiation, reflected radiation \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Sensor**} & \multirow{2}{*}{
\begin{tabular}{c} **x (mm) (from** \\ **Web Centerline**) \\ \end{tabular} } & **y (mm) (from** \\ **Bottom Surface**) & & **z (along the** \\ **Span)** & & **Location** \\ \hline TS & 0 & 500 & & Top surface \\ TF & 80 & 492 & & Top flange’s \\  & & 492 & & bottom surface \\ W1 & 4 & 470 & Mid-span & Web: 30 mm \\ W2 & 4 & 250 & & below TS \\ W3 & 4 & 30 & & Web: 30 mm \\ BF & 80 & 8 & & above BS \\ BS & 0 & 0 & & Bottom surface \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Sectional locations of virtual thermocouples. and convection. The volume of the identical geometry of the experimental steel beam was meshed using second order tetrahedral elements with a fine mesh that included 18,730 elements. To apply the radiation boundary conditions, including the built-in solar model, long wave radiation and surface-to-surface radiation, COMSOL automatically generates triangular surface elements. The triangular surface elements are automatically created by the COMSOL surface-to-surface radiation module and are used to apply the radiation boundary conditions including the built-in solar radiation model that account for solar radiation angles and quantities along the day hours, the surface-to-surface radiations and the reflected radiations. The number of generated triangular elements to cover the surfaces of the meshed steel beam was 12,700. As the analysis was conducted by coupling the heat transfer module with the mechanical module, the essential material parameters of both modules should be defined. In the current research, the typical material properties of constructional steel were adopted. The heat capacity, thermal conductivity and density were 475 J/kg-K, 44.5 W/m-K and 7850 kg/m\({}^{3}\), respectively, while the elastic modulus and coefficient of thermal expansion were 200 GPa and 12.3 \(\times\) 10\({}^{-6}\) 1/K, respectively. The input data (experimental records) of the environmental thermal loads were tabulated for each 30 min along the target day and the preceding two days then loaded into the model. On the other hand, the coordinates of the experimental field, the daily maximum solar radiation intensity and time were fed to the model as required solar model inputs. All runs were started 48 h before the target day to stabilize any possible effects due to the applied initial temperature. To further reduce the effect of the initial temperature, the analysis was started at midnight where the temperature of the beam is almost uniform. The analysis was conducted in two steps. In the first, the experimental beam was modeled with the exact geometry and orientation of the beam (Figure 5), where the measured solar radiation, air temperature and wind speed data of a selected sunny day from the test period, which was 14 June, were applied. The purpose of this analysis was to verify the conducted thermal analysis with the recorded surface temperatures. In the second step, the verified model was used to analyze two W12 and two W24 standard steel beams using the same conditions (boundary conditions, initial conditions, temperature and radiation loads, materials and so on). However, to evaluate the required geometrical parameters, only the geometry was different between the four beams as detailed in Table 2. Using the comparisons between the distributions and timely variations of temperatures and thermal stresses of the investigated beams, the effective geometrical parameters were investigated. These beams were selected because the beams of each pair have the same depth and different flange widths, which allows to study the shadowing effect on standard sections.