## 6 Establishment of Temperature Gradient Prediction Model and Test Comparison  
The effect of solar radiation on the temperature distribution of U-shaped girder bridges is important, influencing the mechanical properties and durability of the structure. Thus, it is essential for engineering design and maintenance to develop a prediction model that can accurately obtain the temperature gradients in U-shaped girder bridges. By analyzing the actual data in Figure 14, there is a difference in the variation rule of temperature at different parts of the U-shaped girder bridge. To perform rapid temperature gradient calculations, different fitting formulas were applied to each part of the U-shaped girder bridge to predict temperature gradients. For instance, the temperature distribution of the left web was fitted with an Allometricl function to the actual temperature distribution data, as depicted in Equation (13), with related fitting parameters detailed in Table 2. As seen in Figure 15a, the fitted curve closely matches the actual temperature distribution curve of the left web, and the correlation coefficient of the fitted curve is close to 1, with the standard error close to 0, indicating that the Allometricl fitted curve can effectively replace the actual curve for predicting the temperature gradient of the left web. In addition, the temperature distribution of the right web, the thickness direction of the bottom slab, and the lateral direction of the bottom slab can be fitted with the Boltzmann function, sine function, and the inverse transformation of the Gauss function, respectively, as shown in Equations (14)-(16), with these fitting parameters and fitting accuracy shown in Tables 3-6. Comparing the actual data to the fitting curves, these three functions all coincide well with the actual temperature distribution curves. However, there are differences in their fitting accuracy, with correlation coefficients of 0.95 for the right web, 0.99 for the thickness direction of the bottom slab, and 0.88 for the lateral direction of the bottom slab. This is due to the fact that the temperature distribution patterns along the right web and the thickness direction of the bottom slab are relatively simple, resulting in better fitting results. While the lateral temperature distribution of the bottom slab is irregular, making it harder to fit, the Gauss function still fits well in the areas with higher temperatures. Therefore, this study establishes temperature gradient prediction models based on different mathematical functions, which can provide effective computational tools for rapid temperature gradient prediction. \[y_{1}=a_{11}\times x_{1}^{b_{11}} \tag{13}\]  
\[y_{2}=a_{21}+\frac{a_{22}-a_{21}}{1+e^{(x_{2}-b_{21})/b_{22}}} \tag{14}\]  
\[y_{3}=a_{31}+a_{32}\times\sin\left[\frac{\pi\times(x_{3}-b_{31})}{b_{32}}\right] \tag{15}\]  
\[y_{4}=a_{41}\pm w\sqrt{-\frac{1}{2}\log\left[\frac{w(x-x_{0})}{a_{42}\sqrt{ \frac{2}{\pi}}}\right]} \tag{16}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & \(a_{11}\) & \(b_{11}\) \\ \hline Parameter value & \(4.32\times 10^{21}\) & \(-14.71\) \\ Standard error & \(1.50\) & \(1.02\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Allometricl function parameters. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & \(a_{21}\) & \(a_{22}\) & \(b_{21}\) & \(b_{22}\) \\ \hline Parameter value & \(0.04\) & \(1.72\) & \(31.23\) & \(0.68\) \\ Standard error & \(0.11\) & \(0.23\) & \(0.28\) & \(0.34\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Boltzmann function parameters. Figure 15: Comparison of temperature distribution curves: (**a**) left web; (**b**) right web; (**c**) thickness of the bottom slab; (**d**) lateral direction of the bottom slab.