# Geometrical Parametric Study on Steel Beams Exposed to Solar Radiation
## 4 Finite Element Modeling
### Thermal Stresses  
The nonlinear temperature gradient distributions shown in Figure 13b, Figure 14b, Figure 15b result in the nonlinear stress distributions shown in Figure 16a-c. The uniform temperature change at the centroid of the beam leads to an axial deformation, which is translated into an axial force if restricted. On the other hand, the nonlinear temperature distribution along the depth of the section tends to deform the section nonlinearly, but it is restricted by the adjacent fibers that keep this deformation linear. The difference between the nonlinear free thermal strains, uniform thermal strains and restricted linear strains results in residual nonlinear distribution of what is called as self-equilibrating stresses. Figure 16 shows the distribution of self-equilibrating thermal stresses induced from the temperature gradients of the four investigated sections at sunrise, noon and sunset times. It is obvious that thermal stresses follow approximately similar trend of variation with depth as that of temperature gradients, but with reversed signs because the heated portions (hotter portions) try to elongate, therefore, compression thermal self-equilibrating stresses are induced to arrest this elongation. Oppositely, the tendency of shrinkage at colder portions is restricted by tensile equilibrating stresses. Figures 13b and 15b show that at sunrise and sunset, the webs are hotter than the top and bottom surfaces (negative gradient). Consequently, compression stresses arise in the web while the top and bottom surfaces are under tensile stresses as shown in Figure 16a,c. On the other hand, the opposite occurs for noon thermal stresses, where the hotter top and bottom surfaces exhibit compression stresses, while the maximum tensile stress occurs in the colder web as shown in Figure 16b. The distributions of stresses shown in Figure 16 confirm the discussed results and the conclusions obtained in the previous sections about the effect of geometrical aspects and shading from top flange. It is shown in Figure 16a that at sunrise, W12 sections are subjected to lower thermal stresses than W24 sections, and sections with larger surface area exhibited lower thermal stresses than those of smaller surface area. Thinner web also leads to faster heating and hence higher stresses at this time. The maximum thermal stress at sunrise in section W12 \(\times\) 50 equals 4.19 MPa, while that of W12 \(\times\) 58 equals 4.84 MPa. Similarly, the maximum sunrise thermal stresses of sections W24 \(\times\) 84 and W24 \(\times\) 104 are 4.82 and 5.26 MPa, respectively. The maximum stresses at sunrise are compression stresses that are located close to the mid-depth due to the negative temperature gradient at this time. The stress distribution at sunset (Figure 16c) has a similar variation trend to that at sunrise with maximum compression stresses in the web and smaller tensile stresses at top and bottom surfaces. The stresses are noticed to be influenced by the combined effect of the pre-heating of web through top flange during the hot hours before sunset period (as discussed in Figure 11) and the web thickness. Figure 16b shows the noon stress distribution which differs from those at sunrise and sunset, where it is the result of a positive temperature gradient. Thus, compression stresses are composed at the hotter top and bottom surfaces, while tensile stresses arise in the web. The effect of the flange width to thickness ratio is obvious in the figure, where sections with wider and thinner flanges exhibited larger stresses. It was discussed in Figure 9 that at noon hour, the geometrical ratio \(W_{\text{f}}/t_{\text{f}}\)\({}^{2}\) is the key parameter controlling the heating of the top surface. Hotter surfaces mean larger gradients with the colder webs resulting in higher self-equilibrating stresses.