## 7 Conclusions  
This study focused on a U-shaped girder bridge to analyze the effect of solar radiation on the distribution of its temperature field using the improved ASHRAE clear sky model and the finite element model. A temperature gradient prediction model for U-shaped girder bridges was developed through nonlinear fitting approaches. The simulation and fitting results of these models were compared with the actual temperature sensor data, and the following conclusions were drawn:  
1. An improved ASHRAE clear sky model is proposed to simulate the shadow areas of the structure under sunlight conditions, which lays the foundation for numerical simulations of the temperature field of U-shaped girder bridges.
2. A three-dimensional transient finite element model of the U-shaped girder bridge is established based on the heat exchange theory, which was used to perform a numerical simulation of the temperature field under solar radiation, comparing the results with actual data. The finite element model results match well with the actual data, verifying the accuracy of the model.
3. The finite element model of the U-shaped girder bridge was utilized to analyze the distribution and changing patterns of the temperature field under different solar radiation conditions. The longitudinal temperature change is minimal, while the transverse temperature distribution shows significant temperature gradient changes. Particularly, there are nonlinear changes in the temperature field along the height of the webs and lateral temperature distribution of the bottom slab, with the maximum temperature difference reaching 17 \({}^{\circ}\)C.
4. A practical calculation method for the temperature gradient is proposed, suitable for predicting the temperature gradient of a U-shaped girder bridge. The model not only has a good fit, but also displays a correlation coefficient with actual data greater than 88%, indicating high prediction accuracy.  
The study not only analyzed the characteristics of the temperature distribution of concrete U-shaped girder bridges under solar radiation but also provided an accurate temperature gradient prediction model. However, it is important to note that the applicability of this model is currently limited to the specific geometry of the U-shaped girder bridge. These results are crucial for understanding and predicting the temperature effects on concrete U-shaped girder bridges under solar radiation. Further studies are planned to validate and refine the prediction model proposed in this study under different regions and environmental conditions for a wide range of engineering applications.  
Conceptualization, Y.S. and J.Z.; Methodology, Y.S. and J.Z.; Software, J.L.; Validation, X.M. and J.L.; Investigation, J.Z.; Writing--original draft, Y.S. and J.Z.; Writing--review & editing, J.Z.; Supervision, J.Z.; Funding acquisition, Y.S. All authors have read and agreed to the published version of the manuscript.  
This research work was fully supported by the State Key Laboratory of Bridge Engineering Structural Dynamics and Key Laboratory of Bridge Earthquake Resistance Technology, Ministry of Communications, PRC (Project No.: 2022 NO.001).  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Function** & **Allometricl** & **Boltzmann** & **Sine** & **Gauss** \\ \hline \(R^{2}\) & 0.99 & 0.95 & 0.99 & 0.88 \\ Iterations & 68 & 6 & 60 & 9 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Fitting accuracy of different functions.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & \(\mathbf{a_{41}}\) & \(\mathbf{a_{42}}\) & \(\mathbf{w}\) & \(\mathbf{x_{0}}\) \\ \hline Parameter value & 2.25 & 53.16 & 3.81 & 22.69 \\ Standard error & 0.11 & 123.62 & 3.72 & 15.58 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Gauss function parameters.