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]. Zuk and Erickson [
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. Churchward [
9] proposed a hyperbolic function to describe the one-dimensional nonlinear temperature change inside structures using measured temperature data. Priestley and Stephenson [
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]. Lin 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. Emerson [
13] applied the finite difference method to solve the heat conduction differential equations and determined the temperature distribution in a concrete slab. Hunt and Cooke [
14] utilized the stable implicit integration approach to determine the temperature distribution in bridge structures with asphalt paving layers. Mirabel and Agudo [
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. Berwnge [
16] developed a finite element method for solving temperature fields in bridge structures and verified the accuracy of the method in the field. Branco and Mendes [
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. Imbsen and Vandershaf [
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. Wang and An [
19] analyzed the characteristics of the temperature distribution in bridge structures and proposed a finite element mesh generation algorithm based on Delaunay triangulation for adaptive finite element analysis to calculate the temperature field of bridge structures. Liang et al. [
20] developed a finite element model for typical concrete casting blocks under solar radiation, analyzing the impact of solar radiation on the temperature field of low-heat concrete at the surface of casting blocks to monitor the temperature gradient on the concrete structure. Sun et al. [
21] measured the temperature of truss bridges and determined that solar radiation intensity was an important factor affecting the temperature difference distribution of truss bridges. In summary, the study of temperature distribution in concrete structures has progressed from simplified linear models to complex numerical simulations that take into account various factors. These studies not only improve the accuracy of the temperature field simulation in structures but also provide an important basis for engineering design and structural maintenance.
While there has been progress in studying the temperature fields of concrete structures, current research mainly focuses on the temperature distribution in commonly shaped bridge sections (like T-shaped, trough-shaped, I-shaped, and box-shaped, etc.). Hossain et al. [
22] obtained the temperature distribution of T-shaped girder bridges through field measurements and analyzed the impact of temperature gradients on bridges. That is, short-term temperature gradients will not lead to stresses that exceed the cracking limit of the beam cross section, but the long-term cumulative effect can surpass the tensile strength of concrete, leading to structural cracks. Lee et al. [
23] performed year-long temperature monitoring on a box-shaped girder bridge to study vertical and horizontal temperature gradients under varying environmental conditions and proposed a second-order curve model for temperature distribution. Xue et al. [
24] established a temperature simulation model for trough-shaped girder bridges by considering the atmospheric environment, bridge orientation, and bridge direction, and a temperature difference model for the box room and bottom slab of trough-shaped girder bridges was proposed by analyzing the variation pattern of the temperature field in trough-shaped girder bridges. Giussani [
25] studied the effects of seasonal temperature changes on the long-term performance of composite bridges, emphasizing the importance of accurately assessing temperature effects to prevent concrete cracking. Hagedorn et al. [
26] analyzed the temperature gradients of I-shaped girder bridges under solar radiation through field tests, identifying temperature fluctuations, low wind speeds, and lack of precipitation as key environmental factors affecting temperature gradients, and disclosed the phenomenon of nonlinear temperature gradients under uneven temperature conditions. Wei et al. [
27] studied the effects of temperature changes on the dynamic characteristics of T-shaped girder bridges and elucidated the mechanisms by which temperature affects the dynamic responses of structures. Ngo et al. [
28] monitored the temperature changes of three box-shaped girder bridges in different climatic regions and used data regression techniques to establish the prediction model for temperature distribution based on actual data. Yang [
29] studied the effects of temperature changes on the displacement of supports in long-span truss continuous bridges and demonstrated the nonlinear hysteresis effects between structural temperature and bridge support displacements through the analysis of field data. Although many studies have focused on the temperature distribution in commonly shaped bridge sections, research on the characteristics of temperature distribution in U-shaped girder bridges is relatively limited. Luo [
30] installed eight temperature sensors on a U-shaped girder bridge, located at the top surface of the flange, the middle of the web, and the bottom slab, and analyzed the data from these sensors to determine the temperature distribution of the U-shaped girder bridge. However, due to the limited number of sensors, these data could not capture the temperature distribution of any sections of the U-shaped girder bridge. Liu et al. [
31] established a finite element model of a U-shaped girder bridge considering factors such as climatic conditions, bridge location, structural dimensions, and thermal properties of materials. They selected the six days with the strongest radiation throughout the year for temperature field simulation to compare the temperature difference distribution of the bridge under different radiation intensity parameters. However, the model only simulates specific dates and fails to cover temperature changes throughout the year. Peng et al. [
32] established a refined finite element model of a 25-m U-shaped girder bridge and verified the accuracy of the model through field tests. Then, they analyzed the performance changes of the U-shaped girder bridge under different temperatures and loads, where the deflection caused by temperature changes could reach up to 2.91 mm. While the refinement of the finite element model improved the precision of the simulation, there are limitations in the robustness of the model due to the complexity of engineering applications. In summary, although existing studies have analyzed the temperature distribution in U-shaped girder bridges through field data and numerical simulations, there is still room for improvement in data comprehensiveness, model accuracy, and practical applicability. Therefore, it is urgent to study the temperature distribution of U-shaped girder bridges, and propose a simulation method to describe the temperature variation under solar radiation throughout the year. At the same time, developing a simple and effective prediction model of the temperature gradient is essential for ensuring the long-term stability and durability of the structure.
This study proposes a method for analyzing the temperature distribution of U-shaped girder bridges and develops a prediction model to estimate temperature gradients. An improved ASHRAE clear sky model is first used to simulate the shadowed areas of a U-shaped girder bridge under solar radiation. Then, a finite element model of the U-shaped girder bridge is established to simulate the temperature difference changes under different solar radiation effects. Finally, a prediction model of the temperature gradient is developed using nonlinear fitting approaches.
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
Table 3,
Table 4,
Table 5 and
Table 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.