# Solar Radiation Parameters for Assessing Temperature Distributions on Bridge Cross-Sections

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. State of Art of Estimation Models for Solar Radiation

#### 2.1. Estimation Models for Global Solar Radiation

#### 2.1.1. Estimation Models for Daily Global Solar Radiation

- Sunshine duration fraction models

_{C}= a + b(S/S

_{0})

_{C}is the clearness index, H is the daily global solar radiation (averaged over one month), H

_{C}is the average clear-day global solar radiation, S/S

_{0}is the sunshine duration fraction, S is the daily sunshine duration (averaged over one month), S

_{0}is the maximum daily sunshine duration (averaged over one month), and a and b are empirical coefficients.

_{0}) was proposed by Page to be used instead of H

_{C}, resulting in the Ångström–Page equation as follows [11]:

_{0}= a’ + b’(S/S

_{0})

- Non-sunshine duration models

- Artificial intelligence approach

#### 2.1.2. Estimation Model for Hourly Global Solar Radiation

_{T}= I/H = (π(cosω − cosω

_{s}))/(24(sinω

_{s}− ω

_{s}cosω

_{s}))

_{T}is the proportionality coefficient, I is the hourly global solar radiation, H is the daily global solar radiation, ω is the solar hour angle, and ω

_{s}is the sunset hour angle. However, some researchers have reported that a linear relationship only exists in a clear day [39]. The equation proposed by Liu and Jordan was modified by Collares-Pereira and Rabl as follows [40]:

_{T}= I/H = ((a + bcosω)(π(cosω − cosω

_{s})))/(24(sinω

_{s}− ω

_{s}cosω

_{s}))

_{s}− 60) and b = 0.6609 − 0.4767sin(ω

_{s}− 60). The validity of this equation was verified by measured data [41,42]. Although some modifications were proposed by other researchers [42,43,44], the Collares-Pereira and Rabl equation is the most frequently used model for estimating the hourly global solar radiation.

#### 2.2. Estimation Model for Beam and Diffuse Solar Radiation

_{b}and the transmission ratio of diffuse solar radiation τ

_{d}was proposed by Liu and Jordan in 1960 [38] under the assumption that the atmosphere is transparent.

_{d}= 0.2710 − 0.2939τ

_{b}

_{d}= I

_{d}/I

_{0}, τ

_{b}= I

_{b}/I

_{0}, I

_{d}is the hourly diffuse solar radiation, I

_{b}is the hourly beam solar radiation, and I

_{0}is the hourly extraterrestrial solar radiation on a horizontal surface.

_{b}= a

_{0}+ a

_{1}exp(−k/sinh)

_{0}, a

_{1}and k are empirical coefficients considering the altitude and meteorological conditions, and h is the solar altitude angle. Other meteorological factors, such as ambient temperature and relative humidity, as well as the artificial intelligence approach, were considered by some researchers to establish a numerical relationship for beam solar radiation and diffuse solar radiation [35,52,53]. However, a combination of the Liu and Jordan equation and Hottel equation has been used by researchers to estimate the hourly beam and diffuse solar radiation values.

## 3. Monitoring of Solar Radiation

^{2}) was the highest among all cities, because this city has the highest altitude (as shown in Figure 3a). However, the difference between maximum global solar radiation values for different cities was not large (about 100 W/m

^{2}). The maximum measured diffuse solar radiation and the calculated beam solar radiation were in Fu’an (about 250 W/m

^{2}) and Ninghua (about 900 W/m

^{2}), respectively, as illustrated in Figure 3b,c. The difference between maximum diffuse or beam solar radiation values for different cities was about 150 W/m

^{2}.

## 4. Estimation Model for Solar Radiation in Fujian

#### 4.1. Estimation Model for Daily Global Solar Radiation

_{0}= (2ω

_{s}/15) × (180/π)

_{s}is the sunset hour angle. The empirical coefficients a’ and b’ for different cities in Fujian can be calculated by using a linear regression analysis between the clearness index H/H

_{C}and the sunshine duration fraction S/S

_{0}.

#### 4.1.1. Influence of Time Scale on Empirical Coefficients

_{xy}) for the linear regression analysis using the monthly time scale is the largest among different time scales. Moreover, the root-mean-square error (RMSE) using the monthly time scale is the smallest. As a result, the monthly time scale was chosen to calculate the empirical coefficients in the Ångström–Page equation.

#### 4.1.2. Influence of Sunshine Duration on the Empirical Coefficients

#### 4.1.3. Verification of Ångström–Page Equation

#### 4.2. Estimation Model for Hourly Global Solar Radiation

#### 4.3. Estimation Model for Hourly Beam and Diffuse Solar Radiation

#### 4.3.1. Estimation Model for Hourly Beam Solar Radiation

#### 4.3.2. Estimation Model for Hourly Diffuse Solar Radiation

## 5. Solar Radiation in Fujian

^{2}) is the highest and the maximum global solar radiation for Anxi (about 970 W/m

^{2}) is the lowest among all cities, as shown in Figure 8a. The biggest difference among the global solar radiation maxima in different cities is about 240 W/m

^{2}. The maximum beam solar radiation for Zhouning (about 909 W/m

^{2}) is the highest, and the maximum beam solar radiation for Ningde (about 821 W/m

^{2}) is the lowest among all cities, as shown in Figure 8b. The biggest difference among the beam solar radiation maxima for different cities is about 88 W/m

^{2}. The maximum diffuse solar radiation for Xiapu (about 388 W/m

^{2}) is the highest and the maximum diffuse solar radiation for Dehua (about 102 W/m

^{2}) is the lowest among all cities, as shown in Figure 8c. The biggest difference among the diffuse solar radiation maxima for different cities is about 286 W/m

^{2}.

## 6. Influence of Solar Radiation on Temperature Distribution on Bridge Girder Cross-Sections

#### 6.1. Box Girder Bridge

#### 6.1.1. Finite Element Model

^{3}were used [60]. The finite element mesh size was set as 0.02 m. There were 21,000 and 19,337 nodes and elements, respectively, as illustrated in Figure 10. Each node had a single temperature degree of freedom.

#### 6.1.2. Influence of Solar Radiation on Temperature Distribution on Cross-Section of Box Girder

#### 6.2. Side-by-Side Box Girder Bridge

#### 6.2.1. Finite Element Model

^{3}was used. For the air (at 100 °C), a specific heat of 716.9 J/(kg °C), heat conductivity of 0.030 W/(m °C) and density of 0.946 kg/m

^{3}were used [61]. It can be found observed that the influence of temperature on the thermal parameters of the air is small. Therefore, the thermal parameters of the air (at 100 °C and air 0 °C) were input into the finite element models in summer and winter, respectively. The finite element mesh size was set as 0.02 m. There were 27,370 and 27,190 nodes and elements, respectively, as illustrated in Figure 16. Each node had a single temperature degree of freedom.

#### 6.2.2. Influence of Solar Radiation on Temperature Distribution on Cross-Section of Side-by-Side Box Girder

#### 6.3. T-Shaped Girder Bridge

#### 6.3.1. Finite Element Model

^{2}°C), which is larger than that in the box girders (3.5 W/(m

^{2}°C)) [62]. The finite element mesh size was set as 0.02 m. There were 16,165 and 15,149 nodes and elements, respectively, as illustrated in Figure 20. Each node had a single temperature degree of freedom.

#### 6.3.2. Influence of Solar Radiation on Temperature Distribution on Cross-Section of T-Shaped Girder

## 7. Discussion

- (1)
- The variation trends for all hourly solar radiation curves are similar, with the solar radiation appearing at about 6:00 a.m., reaching the maxima at about 12:00 p.m. and disappearing at about 6:00 p.m. The maximum measured global solar radiation for Ninghua (about 1100 W/m
^{2}) was the highest and the difference between maximum global solar radiation for different cities was about 100 W/m^{2}. The maximum measured diffuse solar radiation for Fu’an (about 250 W/m^{2}) and the calculated beam solar radiation for Ninghua (about 900 W/m^{2}) was the highest. The difference between the maximum diffuse or beam solar radiation for different cities were both about 150 W/m^{2}. - (2)
- The linear regression of Ångström–Page equation using the monthly time scale of data sample can predict the global solar radiation for different cities in Fujian. For the cities in Fujian that did not have actual data on sunshine duration, the empirical coefficients can be estimated using the available sunshine duration for the nearest city.
- (3)
- The Collares-Pereira and Rabl equation can estimate the hourly global solar radiation for different cities in Fujian based on the daily global solar radiation. The hourly beam solar radiation for different cities in Fujian can be predicted well using the Hottel equation. The hourly diffuse solar radiation for different cities in Fujian can be calculated by subtracting the hourly beam solar radiation from the corresponding hourly global solar radiation.
- (4)
- The maximum global solar radiation, beam solar radiation and diffuse solar radiation (for 21 June, the summer solstice) for 56 cities in Fujian were calculated. The maximum global solar radiation for Xiapu (about 1210 W/m
^{2}) is the highest and the maximum global solar radiation for Anxi (about 970 W/m^{2}) is the lowest. The biggest difference among the global solar radiation maxima in different cities is about 240 W/m^{2}. The maximum beam solar radiation for Zhouning (about 909 W/m^{2}) is the highest and the maximum beam solar radiation for Ningde (about 821 W/m^{2}) is the lowest. The biggest difference among the beam solar radiation maxima in different cities is about 88 W/m^{2}. The maximum diffuse solar radiation for Xiapu (about 388 W/m^{2}) is the highest and the maximum diffuse solar radiation for Dehua (about 102 W/m^{2}) is the lowest. The biggest difference among the diffuse solar radiation maxima in different cities is about 286 W/m^{2}. - (5)
- Comparisons of the measured and calculated temperature–time responses for the concrete box girder, side-by-side box girder and T-shaped girder with or without the consideration of solar radiation indicates that the influence of solar radiation should be considered in the analyses of the temperature distribution on the bridge girder cross-sections in summer. The accuracy of the solar radiation calculated using the estimation models can meet the engineering requirements. The highest calculated temperatures without the consideration of solar radiation were lower and had a time delay, especially for top flanges in the summertime. The vertical temperature variation when considering the effect of solar radiation was significantly larger than that without the consideration of solar radiation. The influence of solar radiation on the temperature distribution decreases as the distance from external surfaces. The influence of solar radiation on the temperature distribution of the bridge girder cross-sections is negligible in winter.
- (6)
- The solar radiation parameters for other cities and regions in China and elsewhere, as well as different bridge superstructure types, can be similarly developed based on more case studies to establish the relevant estimation models. This can serve as a prelude for future development of specifications related to temperature effects in bridge engineering.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 3.**Hourly solar radiation: (

**a**) measured global solar radiation; (

**b**) measured diffuse solar radiation; and (

**c**) calculated beam solar radiation.

**Figure 4.**Linear regression using different time scales: (

**a**) daily time scale; (

**b**) monthly time scale; and (

**c**) yearly time scale.

**Figure 5.**Comparison between measured and estimated hourly global solar radiation curves for eight cities in Fujian: (

**a**) Fu’an (10 August 2016); (

**b**) Jian’ou (7 August 2016); (

**c**) Fuzhou (2 August 2010); (

**d**) Ninghua (4 August 2016); (

**e**) Zhangping (27 July 2016); (

**f**) Anxi (2 August 2015); (

**g**) Hua’an (30 July 2016); and (

**h**) Zhangzhou (31 July 2014).

**Figure 6.**Comparison between measured and estimated hourly beam solar radiation curves for eight cities in Fujian: (

**a**) Fu’an (10 August 2016); (

**b**) Jian’ou (7 August 2016); (

**c**) Fuzhou (2 August 2010); (

**d**) Ninghua (4 August 2016); (

**e**) Zhangping (27 July 2016); (

**f**) Anxi (2 August 2015); (

**g**) Hua’an (30 July 2016); and (

**h**) Zhangzhou (31 July 2014).

**Figure 7.**Comparison between measured and estimated hourly diffuse solar radiation curves for eight cities in Fujian: (

**a**) Fu’an (10 August 2016); (

**b**) Jian’ou (7 August 2016); (

**c**) Fuzhou (2 August 2010); (

**d**) Ninghua (4 August 2016); (

**e**) Zhangping (27 July 2016); (

**f**) Anxi (2 August 2015); (

**g**) Hua’an (30 July 2016); and (

**h**) Zhangzhou (31 July 2014).

**Figure 8.**Maximum solar radiation for 56 cities in Fujian: (

**a**) maximum global solar radiation; (

**b**) maximum beam solar radiation; and (

**c**) maximum diffuse solar radiation.

**Figure 11.**Influence of solar radiation on temperature distribution on cross-section of box girder in summer (2 August 2010): (

**a**) east web (EW-1); (

**b**) east web (EW-2); (

**c**) west web (WW-1); (

**d**) west web (WW-2); (

**e**) bottom flange (B-1); and (

**f**) bottom flange (B-2).

**Figure 12.**Temperature contour plots of box girder at 15:00 on 2 August 2010: (

**a**) with solar radiation; and (

**b**) without solar radiation.

**Figure 13.**Influence of solar radiation on temperature distribution on cross-section of box girder in winter (16 December 2010): (

**a**) east web (EW-1); and (

**b**) bottom flange (B-1).

**Figure 14.**Temperature contour plots of box girder at 24:00 on 16 December 2010: (

**a**) with solar radiation; and (

**b**) without solar radiation.

**Figure 17.**Influence of solar radiation on temperature distribution on cross-section of side-by-side box girder in summer (31 July 2014): (

**a**) top flange (1-T); (

**b**) top flange (11-T); (

**c**) web (1-W); (

**d**) web (11-W); (

**e**) bottom flange (1-B); and (

**f**) bottom flange (11-B).

**Figure 18.**Temperature contour plots of side-by-side box girder at 16:00 on 31 July 2014: (

**a**) with solar radiation; and (

**b**) without solar radiation.

**Figure 21.**Influence of solar radiation on temperature distribution on cross-section of T-shaped girder in summer: (

**a**) top flange (T-1); (

**b**) web (1-W); (

**c**) web (2-W); and (

**d**) bottom flange (4-B).

**Figure 22.**Temperature contour plots of T-shaped girder at 15:00 on 23 August 2012: (

**a**) with solar radiation; and (

**b**) without solar radiation.

**Table 1.**Linear regression analysis of clearness index and sunshine duration fraction using different time scales.

Time Scale Types | Data Points | a’ | b’ | γ_{xy} | RMSE |
---|---|---|---|---|---|

Daily | 8030 | 0.213 | 0.505 | 0.86 | 0.10 |

Monthly | 264 | 0.196 | 0.553 | 0.92 | 0.03 |

Yearly | 22 | 0.242 | 0.422 | 0.79 | 0.01 |

**Table 2.**Linear regression analysis of clearness index and sunshine duration fraction using sunshine duration for four cities.

Solar Radiation Data Source | Sunshine Duration Data Source | a’ | b’ | γ_{xy} | RMSE |
---|---|---|---|---|---|

Hua’an | Nanping | 0.216 | 0.467 | 0.81 | 0.05 |

Hua’an | Fuzhou | 0.217 | 0.519 | 0.82 | 0.06 |

Hua’an | Yong’an | 0.212 | 0.519 | 0.85 | 0.06 |

Hua’an | Xiamen | 0.178 | 0.522 | 0.90 | 0.04 |

**Table 3.**Linear regression analysis of clearness index and sunshine duration fraction for eight cities.

City | Solar Radiation Data Source | Sunshine Duration Data Source | a’ | b’ | γ_{xy} | RMSE |
---|---|---|---|---|---|---|

Fu’an | Fu’an | Fuzhou | 0.211 | 0.546 | 0.84 | 0.05 |

Jian’ou | Jian’ou | Nanping | 0.180 | 0.537 | 0.84 | 0.05 |

Fuzhou | Fuzhou | Fuzhou | 0.196 | 0.553 | 0.92 | 0.03 |

Ninghua | Ninghua | Yong’an | 0.179 | 0.571 | 0.85 | 0.05 |

Zhangping | Zhangping | Yong’an | 0.213 | 0.520 | 0.84 | 0.05 |

Anxi | Anxi | Xiamen | 0.197 | 0.479 | 0.88 | 0.04 |

Hua’an | Hua’an | Xiamen | 0.178 | 0.522 | 0.90 | 0.04 |

Zhangzhou | Zhangzhou | Xiamen | 0.196 | 0.506 | 0.89 | 0.04 |

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Xue, J.; Lin, J.; Briseghella, B.; Tabatabai, H.; Chen, B.
Solar Radiation Parameters for Assessing Temperature Distributions on Bridge Cross-Sections. *Appl. Sci.* **2018**, *8*, 627.
https://doi.org/10.3390/app8040627

**AMA Style**

Xue J, Lin J, Briseghella B, Tabatabai H, Chen B.
Solar Radiation Parameters for Assessing Temperature Distributions on Bridge Cross-Sections. *Applied Sciences*. 2018; 8(4):627.
https://doi.org/10.3390/app8040627

**Chicago/Turabian Style**

Xue, Junqing, Jianhui Lin, Bruno Briseghella, Habib Tabatabai, and Baochun Chen.
2018. "Solar Radiation Parameters for Assessing Temperature Distributions on Bridge Cross-Sections" *Applied Sciences* 8, no. 4: 627.
https://doi.org/10.3390/app8040627