## 3 Theory of Solar Radiation Intensity and Sunshine Temperature Field
### Sunshine Temperature Field Theory and Test Comparison  
#### 3.2.1 Finite Element Theory of Temperature Fields  
The theoretical analysis of the temperature field in concrete structures is complex, primarily involving the accurate depiction of the heat conduction process inside the medium. This process typically follows the Fourier law, which assumes that the material is homogeneous and isotropic in three-dimensional space, explaining the method of heat transfer within materials [35]. For U-shaped girder bridges, the differential equation of the temperature field based on Fourier Law can be expressed as follows:  
\[\upsilon\rho\frac{\partial T}{\partial t}=\frac{\partial}{\partial x}(\lambda \frac{\partial T}{\partial x})+\frac{\partial}{\partial y}(\lambda\frac{ \partial T}{\partial y})+\frac{\partial}{\partial z}(\lambda\frac{\partial T} {\partial z}) \tag{8}\]  
where \(T\) is the temperature of the U-shaped girder bridge, \(\lambda\) is the thermal conductivity of the medium, \(\upsilon\) is the specific heat of the medium, and \(\rho\) is the density of the medium.  
Assuming that both web sides and bottom surfaces of the U-shaped girder bridge exchange heat with the environment, the boundary conditions for the temperature field can be expressed as follows:  
\[-\lambda\frac{\partial T}{\partial n}=-q+\gamma(T-T_{f}) \tag{9}\]  
where \(q\) represents the rate of heat transfer, \(\gamma\) is the coefficient of thermal exchange, \(T_{f}\) is the specified ambient temperature, and \(\frac{\partial T}{\partial n}\) indicates the derivative of \(T\) in the direction of the unit normal \(n\) to the boundary.  
In numerical simulation, two-dimensional triangular thermal elements are often used to simulate the temperature field of bridges, as shown in Figure 6. The finite element equation for the temperature field of the U-shaped girder bridges is established by deriving the thermal equilibrium equation of the two-dimensional triangular thermal element [36].  
The node numbers for the triangular thermal element are set as \(I\), \(j\), and \(m\) in element \(e\), with each node having only one degree of freedom, which is the temperature \(a\). According to the consistency of the temperature differences at the nodes of element \(e\), the thermal equilibrium equation for element \(e\) can be expressed as follows:  
\[c^{e}\dot{a}^{e}+({k_{1}}^{e}+{k_{2}}^{e})a^{e}=p^{e} \tag{10}\]  
where \(c^{e}\) is the heat capacity matrix, \({k_{1}}^{e}\) is the element stiffness matrix, \({k_{2}}^{e}\) is the element boundary stiffness matrix, and \(p^{e}\) is the element load matrix.  
Figure 5: Comparison between actual and theoretical scattered radiation intensities: (**a**) 22 April; (**b**) 5 October.  
The total stiffness matrix is formed by aggregating the stiffness matrix of element e; that is, the total heat capacity matrix of the temperature field is \([c]=\sum c^{e}\), the total stiffness matrix of the temperature field is \([k]=[k_{1}]+[k_{2}]=\sum k_{1}{}^{e}+\sum k_{2}{}^{e}\) (the total stiffness matrix of the temperature field includes two parts: the element stiffness matrix and the element boundary stiffness matrix), and the load vector of the temperature field is \(p=\sum p^{e}\). The finite element equation for the transient temperature field of the U-shaped girder bridge is represented as follows [37]:  
\[[c]\dot{a}+[k]a=p \tag{11}\]  
where \([k]\) is the total stiffness matrix, \(a\) is the total temperature vector, and \(p\) is the total load vector.