## 4 Numerical Simulation of Temperature Field  
### Finite Element Model  
The temperature distribution of concrete box girders is commonly assumed to be uniform along the span direction due to natural environmental factors. When analyzing the temperature field using the general finite element program ABAQUS 6.14, it is presumed that the temperature remains constant across the longitudinal section. In other words, a two-dimensional heat conduction problem is established for conducting long-term numerical simulations on the test section. The differential equation for heat conduction can be expressed as follows:  
\[\frac{\partial T}{\partial t}=\frac{\lambda}{\rho c}\bigg{(}\frac{\partial^{2} T}{\partial x^{2}}+\frac{\partial^{2}T}{\partial y^{2}}\bigg{)} \tag{4}\]  
where \(T\) represents the variation in temperature with respect to the structural coordinates, \(\rho\) is the density of concrete, \(c\) is the specific heat capacity, \(\lambda\) is the thermal conductivity, \(t\) is the time, and \(x\), \(y\) are the structural coordinates. To obtain a finite solution for Equation (4), appropriate initial and boundary conditions are required. The initial condition refers to the temperature distribution of concrete at the beginning of analysis, which, in this study, was determined based on the daily average temperature at the location of the concrete box girder. On the other hand, boundary conditions represent external influences that affect heat conduction and temperature distribution within the box girder. Under solar radiation effects, three types of heat exchange between the bridge and the surrounding environment are considered: solar radiation heat flux, convective heat flux, and radiative heat flux. The thermal boundary conditions can be expressed as follows:  
\[\lambda\left.\frac{\partial T}{\partial n}\right|_{\Gamma}=q_{s}+q_{r}+q_{c} \tag{5}\]  
where \(q_{s}\), \(q_{r}\), and \(q_{c}\) represent the heat flux densities of solar radiation, radiative heat transfer, and convective heat transfer, respectively. The convective heat transfer between the box girder and the atmosphere is considered as the third boundary condition. The solar radiation heat transfer and long-wave radiative heat transfer between the structure itself and the surrounding environment are treated as the second boundary conditions. During the finite element analysis process, it is necessary to convert the mixed boundary conditions into the third boundary conditions to obtain comprehensive values for the heat transfer coefficient and atmospheric environment temperature. These values are then introduced into FEM calculations. Equation (5) is reformulated as follows:  
\[\lambda\left.\frac{\partial T}{\partial n}\right|_{\Gamma}=(h_{r}+h_{c})(T_{a} ^{*}-T),\ T_{a}^{*}=\frac{\alpha I_{\phi}}{h_{r}+h_{c}}+T_{a} \tag{6}\]  
where \(T_{a}^{*}\) represents the comprehensive atmospheric temperature, which is determined by the atmospheric temperature \(T_{a}\), the solar radiation intensity perpendicular to the irradiated surface \(I_{\phi}\), the radiative heat transfer coefficient \(h_{r}\), the convective heat transfer coefficient \(h_{c}\), and the solar radiation absorptivity \(\alpha\). Detailed calculations for radiation intensity and thermal boundary conditions can be found in reference [29]. For the finite element simulation of the box girder, a concrete box girder is modeled using four-node linear heat transfer quadrilateral elements (DC2D4). The mesh division of the model is illustrated in Figure 9. The thermal properties of the concrete in the segment model are listed in Table 3.