# Geometrical Parametric Study on Steel Beams Exposed to Solar Radiation

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

_{f}/H, W

_{f}/t

_{f}

^{2}and 2W

_{f}/Ht

_{f}were the most influential parameters on temperatures, temperature gradients and thermal stresses of steel beams subjected to solar radiation. The investigated section with the maximum W

_{f}/t

_{f}

^{2}value of 0.96 (W12 × 58) recorded the highest top-surface noon temperature, while section W24 × 84 with the lowest W

_{f}/t

_{f}

^{2}value of 0.60 exhibited the lowest temperature.

## 1. Introduction

## 2. Study Objective

## 3. The Experimental Steel Beam

^{2}with a maximum error of 5%. The three–cup NRG#40 anemometers from NRG system will be used to measure wind speed as shown in Figure 4a. This type of device recommended by Campbell Scientific. The sensor is able to read the wind speed ranging from 0 to 96 m/s (0 to 215 mph). The NRG#40 was the three conical-cups wind sensor used. It measures wind speeds up to 10 m/s with an estimated error of ±0.14 m/s. The three sensors are clearly visible next to the steel beam in Figure 4a.

## 4. Finite Element Modeling

#### 4.1. Heat Conduction and Thermal Boundary Conditions

_{p}) and the thermal conductivity (k) [16]:

_{c}) between the beam external surfaces and the moving air in the field environment is represented in Equation (2), where T

_{s}represents the temperature of the beam’s surface and T

_{a}represents the air temperature, while h is the convection coefficient, which is represented as a function of wind speed (v). It is important to address the used formula (Equation (3)) for the convection coefficient as there are several empirical formulas in the literature for I-beam sections. This formula was adopted from previous studies where it was used successfully to model the convection between air and beam surfaces:

#### 4.2. Finite Element Thermal Analysis

^{3}, respectively, while the elastic modulus and coefficient of thermal expansion were 200 GPa and 12.3 × 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.

#### 4.3. Verification with Experimental Records

## 5. Results and Discussion

#### 5.1. Temperature-Time Relations

_{f}) leads to higher accumulated absorbed energy. Thus, both flange width and flange thickness are effective. When the temperature variation behavior at the peak temperature hours (12:00 to 15:00) was compared with the ratio of flange width to flange thickness (W

_{f}/t

_{f}), no specific trend was clear, which is mainly due to giving equal weights for both flange width and flange thickness. The flange thickness should be given a higher influence weight as it is the main controller of temperature at this time of day. Therefore, the temperature variations were investigated with the ratio of flange width to the square of flange thickness (W

_{f}/t

_{f}

^{2}) as shown in Figure 9. It is obvious in the figure that the top surface temperature increases with the increase of the quantity W

_{f}/t

_{f}

^{2}. The section W12 × 58 with the highest W

_{f}/t

_{f}

^{2}value of 0.96 exhibited the highest top surface temperature at noon hours, while the section W24 × 84 with the lowest W

_{f}/t

_{f}

^{2}value of 0.60 showed the lowest temperatures at the same time. This means that the top surface temperature increases with the increase of top flange width and the decrease of its thickness but is more affected by flange thickness than flange width.

_{f}/t

_{f}× H/2 or more simply 2W

_{f}/Ht

_{f}, is a key geometrical parameter that shares the heating of this region at noon hour. Figure 11 shows that the temperature of the mid-depth section around noon hour increases with the increase of the quantity 2W

_{f}/Ht

_{f}. The increasing of flange width and decreasing of flange thickness leads to hooter flange, while shorter path (H/2) leads to faster heating of the mid-depth of the section via top surface. As shown in the figure, although W24 sections have wider flanges, their temperatures were the lowest at noon hours due to their significantly higher depth (H), while the compact depths of W12 sections make the path shorter to heat up their mid-depth.

_{f}/H). As shown in Figure 10, the slope of temperature variation after 15:00 is positive for the deep W24 sections, which means that the mid-depth is receiving direct solar radiation at this period. On contrary, the slope of the shallow W12 sections is negative at the same period reflecting a continuous cooling process, which means that the mid-depth is shaded at this time by the top flange. Figure 12 translate this behavior in terms of temperature variation with the geometrical quantity W

_{f}/H. the vertical axis of this figure represents the temperature difference between the subsequent hourly records from 15:00 to 18:00. It is shown in the figure that the slope of temperature variation is positive for the sections with lower shading effect (lower W

_{f}/H), which are the W24 sections, from 15:00 to 17:00 which reflects their heating process at this time. On the other hand, the slopes of the sections with higher shading effect (higher W

_{f}/H) is obviously negative, which refers to a cooling phase at this time due to the shading effect. The temperature variation slope is obvious to decrease with the increase of shading effect (W

_{f}/H) from section W24 × 84 with W

_{f}/H = 0.37 to section W12 × 58 with W

_{f}/H = 0.82.

#### 5.2. Vertical Temperature Distributions and Gradients

^{2}for W12 × 58 and 9420 mm

^{2}for W12 × 50. Similarly, the temperature of W24 × 104 is higher than that of W24 × 84 for the same reason. The surface areas of W24 × 84 and W24 × 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 × 58 recorded higher temperature gradient than that of W12 × 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 × 50 and W12 × 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. Thus, increasing this ratio for the same depth increases the vertical temperature gradients at noon hours, and thus increasing the daily maximum positive temperature gradient. This conclusion is confirmed by the results of the larger W24 sections. As it is clear in Figure 14b, the temperature gradient of W24 × 104 is higher than that of W24 × 84. The flange width/section depth ratios of the two sections are 0.53 and 0.37, respectively. It should be noticed that as discussed in the previous section, the noon temperature is affected by the shadowing effect during the preceding late morning hours (after 8:00) and the W

_{f}/t

_{f}

^{2}ratio during the noon hours.

_{f}/H ratio discussed in Figure 12, it is obvious in Figure 15b that W24 sections with lower W

_{f}/H ratio exhibited higher temperature gradients than W12 sections that are of higher W

_{f}/H ratio.

#### 5.3. Thermal Stresses

_{f}/t

_{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. Section W12 × 58 with the wider top flange (W

_{f}/t

_{f}= 15.6) exhibited a maximum stress of 10.72 MPa, which is higher that of W12 × 50 (9.39 MPa) that has W

_{f}/t

_{f}ratio of 12.62. Similarly, the W

_{f}/t

_{f}ratios of sections W24 × 104 and W24 × 84 are 17.07 and 11.71, respectively, and their maximum noon stresses are 11.16 and 9.99 MPa, respectively.

## 6. Conclusions

- (1)
- The smaller W12 sections are obviously colder than the bigger W24 sections at sunrise and sunset and hotter than W24 sections at noon hour. This is due to the thinner web and flanges of W12 sections compared to W24 sections, which allows for faster cooling and heating. The temperature gradient along the section depth is controlled by the time-dependent temperature variation, which is strongly affected by the geometrical properties of the steel sections.
- (2)
- The maximum temperature of the top surface at noon hour is mostly affected by the geometrical parameters of the top flange. The maximum temperature increases as the top flange width increases and its thickness decreases. However, this variation is more sensitive to flange thickness than flange width, where the maximum temperature of the top surface was found to be positively correlated to the ratio of flange width to the square of flange thickness (W
_{f}/t_{f}^{2}). Section W12 × 58 with the maximum W_{f}/t_{f}^{2}value among the four steel sections (0.96) recorded the highest top surface temperature at noon hour, while the temperature of W24 × 84, which has the lowest W_{f}/t_{f}^{2}value of 0.60, was the lowest. - (3)
- The temperature of the web’s mid-depth at noon hour is mainly due to the huge amount of solar radiation received by the top surface during this period. The influential geometrical parameters are the area and thickness of the heated part, which is the top flange, and the path to the target point, which is the mid-depth of the web. The temperature of the centroid of the web at noon hours increases with the increase of the quantity 2W
_{f}/Ht_{f}. The wider and thinner flange results in faster heating, while the shorter path (H/2) leads to faster heat conduction to the web’s centroid from the top surface - (4)
- The shading from the top flange on the web’s temperature becomes effective during afternoon hours. Wider flanges impose deeper shadows that reduce the temperature of the web, while deeper sections are less influenced by flange’s shading. The variation of temperature during afternoon hours affects that at sunset and is affected by the geometrical ratio of flange width to flange thickness (W
_{f}/H). - (5)
- The vertical thermal stress distributions are directly influenced by the nonlinear temperature gradients at the same times. In addition, their maximum values are also related to the geometrical parameters that affect the heat transfer and temperature variation during the heating and cooling phases. The thicknesses of web and flange and the ratio of flange width to flange thickness are the most influential parameters.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 4.**The experimental steel beam: (

**a**) Southern view and thermal loads sensors; (

**b**) Northern view and thermocouple locations.

**Figure 6.**Verification of predicted finite element temperatures with test records: (

**a**) TS and BS; (

**b**) W1 and BF.

**Figure 13.**Vertical temperature distributions and temperature gradients at sunrise: (

**a**) Temperature distributions; (

**b**) Temperature gradients.

**Figure 14.**Vertical temperature distributions and temperature gradients at noon: (

**a**) Temperature distributions; (

**b**) Temperature gradients.

**Figure 15.**Vertical temperature distributions and temperature gradients at sunset: (

**a**) Temperature distributions; (

**b**) Temperature gradients.

Sensor | x (mm) (from Web Centerline) | y (mm) (from Bottom Surface) | z (along the Span) | Location |
---|---|---|---|---|

TS | 0 | 500 | Mid-span | Top surface |

TF | 80 | 492 | Top flange’s bottom surface | |

W1 | 4 | 470 | Web: 30 mm below TS | |

W2 | 4 | 250 | Web: mid-height | |

W3 | 4 | 30 | Web: 30 mm above BS | |

BF | 80 | 8 | Bottom flange’s top surface | |

BS | 0 | 0 | Bottom surface |

Dimensions (mm) | W24 × 104 | W24 × 84 | W12 × 58 | W12 × 50 |
---|---|---|---|---|

Height (H) | 612.14 | 612.14 | 309.88 | 309.88 |

Web thickness (t_{w}) | 12.7 | 11.938 | 9.144 | 9.398 |

Flange width (W_{f}) | 325.12 | 229.108 | 254 | 205.232 |

Flange thickness (t_{f}) | 19.05 | 19.558 | 16.256 | 16.256 |

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**MDPI and ACS Style**

Abid, S.R.; Al-Gasham, T.S.; Xue, J.; Liu, Y.; Liu, J.; Briseghella, B.
Geometrical Parametric Study on Steel Beams Exposed to Solar Radiation. *Appl. Sci.* **2021**, *11*, 9198.
https://doi.org/10.3390/app11199198

**AMA Style**

Abid SR, Al-Gasham TS, Xue J, Liu Y, Liu J, Briseghella B.
Geometrical Parametric Study on Steel Beams Exposed to Solar Radiation. *Applied Sciences*. 2021; 11(19):9198.
https://doi.org/10.3390/app11199198

**Chicago/Turabian Style**

Abid, Sallal R., Thaar S. Al-Gasham, Junqing Xue, Yongjian Liu, Jiang Liu, and Bruno Briseghella.
2021. "Geometrical Parametric Study on Steel Beams Exposed to Solar Radiation" *Applied Sciences* 11, no. 19: 9198.
https://doi.org/10.3390/app11199198