## 3 Theory of Solar Radiation Intensity and Sunshine Temperature Field  
### Solar Radiation Intensity Theory and Test Comparison  
#### 3.1.1 Ashrae Clear Sky Model  
The solar radiation intensity is a key factor in calculating the solar-induced temperature field of concrete bridges and is the direct cause of nonlinear temperature stress on structures. The ASHRAE clear sky model [33] is recommended for calculating solar radiation in civil engineering, noted for its simplicity, few parameters, and strong practicality. According to the ASHRAE clear sky model, the solar radiation absorbed by bridge structures includes direct radiation, scattered radiation, and reflected radiation.  
Direct radiation: Direct radiation is the solar radiation that reaches the ground directly, and its intensity is as follows:  
\[G_{D}=\frac{A}{\exp(B/\sin\beta)}C_{N}\cos\theta \tag{1}\]  
Figure 1: Field test of U-shaped girder bridge.  
Figure 2: Cross section of the U-shaped girder bridge (mm).  
where \(A\) is the solar radiation intensity when the atmospheric mass is 0, \(B\) is the atmospheric extinction coefficient, \(\beta\) is the solar elevation angle, \(C_{N}\) represents the atmospheric clarity, and \(\theta\) is the incidence angle of the sun on the structure surface.  
Scattered radiation: Scattered radiation is produced by the scattering of sunlight by molecules and suspended particles in the atmosphere. On the horizontal surface, the intensity of scattered radiation can be expressed as follows:  
\[G_{SH}=C\frac{A}{\exp(B/\sin\beta)}C_{N} \tag{2}\]  
where \(C\) is the ratio of scatter radiation to direct radiation on the horizontal surface.  
For the non-horizontal surface, the intensity of scattered radiation is as follows:  
\[G_{S}=C\frac{A}{\exp(B/\sin\beta)}C_{N}F_{ws} \tag{3}\]  
where \(F_{ws}\) is the angular coefficient between the structure surface and the ground.  
Reflected radiation: Assuming the surrounding environment is diffusely reflective, ground reflected radiation is the following:  
\[G_{R}=G_{HH}\rho_{g}F_{wg} \tag{4}\]  
where \(G_{HH}\) represents the radiation intensity reaching the ground, \(\rho_{g}\) is the reflectance of the surrounding environment, and \(F_{wg}\) is the angle factor for the ground.  
Total solar radiation: Total solar radiation is the sum of direct, scattered, and reflected radiation, and can be expressed as follows:  
\[G=G_{D}+G_{S}+G_{R} \tag{5}\]  
In practical applications, since the coefficients in the ASHRAE clear sky model are fitted based on actual solar radiation data from the United States, the calculation results cannot be applied directly to China. It is necessary to perform iterative regression on the model coefficients using meteorological data from specific areas to enhance the accuracy of the model. At the U-shaped girder bridge test site, a solar radiation monitoring system, specifically the TBS-YG5 automatic tracking solar radiation, was deployed to observe solar radiation intensity, as depicted in Figure 3. This system monitors and records three parameters continuously: direct radiation, scattered radiation, and the solar elevation angle.  
Figure 3: TBS-YG5 automatic solar radiation monitoring system.