# Geometrical Parametric Study on Steel Beams Exposed to Solar Radiation
## 4 Finite Element Modeling
### Vertical Temperature Distributions and Gradients  
The distributions of temperature along the depth of the four steel sections are presented at sunrise, noon and sunset times in Figure 13a, Figure 14a, Figure 15a, respectively, while their corresponding gradient distributions are shown in Figure 13b, Figure 14b, Figure 15b, respectively. The three figures show that the W12 sections are colder than W24 sections at sunrise and sunset hours, while hotter than W24 sections at noon hour. This is attributed to the thinner web and flanges of W12 sections compared to W24 sections, which facilitates their faster cooling and faster heating. Figure 12: Relationship between mid-depth temperature and \(W_{t}/H\) at afternoon hours. Figure 14: Vertical temperature distributions and temperature gradients at noon: (**a**) Temperature distributions; (**b**) Temperature gradients. Figure 13: Vertical temperature distributions and temperature gradients at sunrise: (**a**) Temperature distributions; (**b**) Temperature gradients. The effect of shadowing from the top flange is not clear at sunrise and sunset times due to the low inclination angle of sunrays, which reduces or eliminates any influence of overhanging flange portions on web radiation. The slight differences in temperatures and temperature gradients between the beams of each pair of sections shown Figure 13 are attributed to the effects of other geometrical aspects like the thickness of the web and to the already preserved temperatures along the night hours. On the other hand, the effect of shadowing from the flanges overhanging portions becomes more significant after 8:00 morning up to before midday and from approximately 14:00 to 17:00. In summer, the sun moves up in the sky within the early morning hours after sunrise with the angle between sunrays and the horizon increases as time passes until noon. With the increase of solar inclination angle, the amount of crossed radiation by the top flange increases leading to deeper shadow depth down along the web as discussed in Figure 1. At these hours, the top surface is being fully heated by solar radiation, while one side of the web is totally shaded and the other one is mostly shaded, which results in slow heating of the web. The different heating conditions between flanges and web lead to a higher temperature gradient, which increases with the accumulation of this process until midday. For beams with wider flanges compared to depth (flange width/section depth ratio), the effect of shadowing from the top flange becomes more pronounced as shown in Figure 2.  
Figure 14a shows that the temperature of W12 \(\times\) 58 is higher than that of W12 \(\times\) 50, which is attributed to the larger surface area of the former, where the surface area equals approximately 10,970 mm\({}^{2}\) for W12 \(\times\) 58 and 9420 mm\({}^{2}\) for W12 \(\times\) 50. Similarly, the temperature of W24 \(\times\) 104 is higher than that of W24 \(\times\) 84 for the same reason. The surface areas of W24 \(\times\) 84 and W24 \(\times\) 104 are approximately 15,940 and 19,740 mm\({}^{2}\), respectively. As all thermal loads are surface loads, then larger surface area leads to the absorbing of larger total amount of solar radiation and hence more heat. Figure 14b shows that section W12 \(\times\) 58 recorded higher temperature gradient than that of W12 \(\times\) 50, which can be attributed to the wider flange width and smaller thickness as discussed in Figure 9. The larger surface area leads to the increase of the whole body temperature, yet, the wider flange slowed down the heating of the web leading to higher difference with the top surface, which is recorded as larger temperature gradient. The flange widths of sections W12 \(\times\) 50 and W12 \(\times\) 58 are 205 and 254 mm, respectively. Since both sections have the same depth of 310 mm, then the flange width/section depth ratios of the two sections equal 0.66 and 0.82, respectively.