## 1 Introduction  
Bridge structures generate significant temperature differences between the internal and external surfaces due to solar radiation, resulting in a nonlinear distribution of the internal temperature field [1, 2]. This nonlinear temperature distribution can induce thermal stresses in bridge structures, which sometimes exceed the stresses caused by loads. In extreme cases, excessive thermal stress can cause cracks in concrete bridges and even lead to structural failure [3]. U-shaped girder bridges are a novel form of urban rail transit structures. Compared to traditional box- and T-shaped girder bridges, U-shaped bridges are more sensitive to temperature changes under solar radiation [4, 5, 6]. Therefore, the study of temperature distribution and gradient prediction of U-shaped girder bridges under insolation radiation is of great significance to ensure the long-term safety and durability of their structures. In early studies on the temperature distribution in concrete structures, researchers typically used simplified calculation methods based on steady-state temperature fields or linear changes along the beam height [7; 8; 9; 10]. [PERSON] and [PERSON] [7; 8] introduced a formula for the maximum temperature difference within concrete beam sections based on the assumption of linear changes in beam height. As research continues, some researchers have realized that the assumption of a linear temperature change has certain limitations. [PERSON] [9] proposed a hyperbolic function to describe the one-dimensional nonlinear temperature change inside structures using measured temperature data. [PERSON] and [PERSON] [10] used an exponential function that varies along the thickness of the wall to simulate the nonlinear temperature changes within concrete bridge structures. While these simplified functions can reflect the nonlinear characteristics of temperature change to some extent, they often ignore factors such as solar intensity and environmental conditions, which can cause some errors in applications. As computer technology has advanced, numerical simulation methods have been increasingly applied to the study of temperature distributions in concrete structures [11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21]. [PERSON] et al. [12] used computational fluid dynamics (CFD) to simulate the temperature change of structural components under solar radiation, and conducted experimental validation and parameter analysis of the temperature field. [PERSON] [13] applied the finite difference method to solve the heat conduction differential equations and determined the temperature distribution in a concrete slab. [PERSON] and [PERSON] [14] utilized the stable implicit integration approach to determine the temperature distribution in bridge structures with asphalt paving layers. [PERSON] and [PERSON] [15] used the finite difference method to study the impact of cross-sectional geometry on temperature distribution and analyzed the temperature distribution in the corner areas of the cross section. [PERSON] [16] developed a finite element method for solving temperature fields in bridge structures and verified the accuracy of the method in the field. [PERSON] and [PERSON] [17] used the plane finite element method to analyze the temperature effects on bridge piers under solar radiation and proposed a temperature distribution model within concrete sections. [PERSON] and [PERSON] [18] studied the temperature differences in box-shaped girder bridges under various geographic latitudes and climate conditions, and determined the adverse meteorological conditions as boundary conditions for numerical simulation based on historical weather data from different geographical locations. The temperature distribution of the box girder was then simulated to obtain the temperature difference of the box-shaped girder bridges in different locations.