The design of urban bridges requires not only the basic traffic function but also a certain degree of aesthetics in order to improve the appearance of the city. Incorporating elegant curve elements in design has become more prevalent due to the increasing demand for traffic flow and limited land availability. However, the curvature of curved bridges can cause a bending–torsion coupling effect due to the deviation of the center of gravity from the supporting line during service. Compared to straight bridges, curved counterparts exhibit complex stress characteristics and are more susceptible to damage, particularly under variable actions such as those induced by temperature [
1,
2,
3].
Temperature loads are considered one of the most critical variable load types. In particular, solar thermal actions generate a three-dimensional temperature field with uneven distribution within the structure, leading to temperature-induced secondary stresses [
4]. Scholars worldwide simplify the temperature field of the bridge into a two-dimensional distribution pattern along the longitudinal direction, comprising vertical and lateral temperature gradients. For small and medium-span bridges, there is little significant difference in lateral temperature distribution due to the relatively narrow width of the beam. Therefore, research on vertical temperature gradients is more widespread [
5]. Both domestic and international standards have specific regulations for concrete structure bridges regarding temperature effect, whereas there are distinct differences in distribution patterns for steel structure bridges, such as specifications in China lacking explicit definitions of the vertical temperature gradient distribution [
6]. Meanwhile, BS-5400 in the UK adopts a four-segment line to simulate this distribution [
7]. Extensive research has been carried out at home and abroad on the temperature gradient of steel box girders based on measured data. Zhu et al. [
8] and Teng et al. [
9] compared measured data from an actual project to multinational specifications and found that the four-segment nonlinear mode provided the best fit for the temperature gradient. Wang et al. [
10] used measured data from the steel box girder of a viaduct in Wuhan and discovered that the actual temperature distribution differed significantly from the specified linear distribution. The authors applied extreme value theory to derive the distribution pattern of an appropriate exponential function for the neighboring regions. Ding et al. [
11] utilized extreme value theory to analyze ten years of temperature monitoring data for the Runyang Bridge, and found that the estimated extreme values of vertical temperature differences were in excellent agreement with the UK code. Guo et al. [
12] analyzed in situ measurements of unpaved steel box girders and investigated the status of temperature gradient partitioning research in China [
13]. It is evident that the four-segment model in the UK BS-5400 code provides a superior description of the vertical nonlinear temperature distribution pattern in steel box girders compared to other code. The aforementioned research primarily concentrated on straight bridges, emphasizing the necessity for further in-depth research on the coupled stress characteristics of curved steel bridges under temperature effects.
In recent years, computer science has experienced booming development, leading to increased interest in deep learning among scholars worldwide. The backpropagation neural network (BPNN) is a fundamental algorithm of deep learning that has gained attention in various research fields due to its exceptional applicability, nonlinearity, and robustness. In civil engineering, neural networks are primarily used for predicting structural responses and identifying damages [
14]. Tian et al. [
15] investigated the effect of temperature on bridges by training a neural network with temperature data from a large-span concrete bridge. They compared the predictions of the neural network with finite element results and demonstrated its high accuracy simultaneously with a significant reduction in computational resource consumption. Ying et al. [
16] employed a BPNN to analyze and predict the temperature field distribution of a sea-crossing bridge, providing a reliable foundation for subsequent computations. Wang et al. [
17] performed a finite element analysis to simulate temperature gradients at various locations of concrete box girders. They formed a comprehensive dataset and established a neural network to predict the vertical and horizontal temperature gradients of the top and bottom plates of the box girder with R
2 exceeding 0.9. Recent research on temperature effects has primarily focused on predicting temperature fields using traditional BPNNs, while comparatively less attention has been paid to predicting structural responses. Furthermore, the training of the BPNNs heavily relies on the selected initial weight values. Inappropriate values can cause the network to become trapped in a local optimal solution and fail to converge. To address this issue, an optimization algorithm is often introduced to determine the initial weight values.
This paper introduces the concept of a “three-curved spatial bridge” composed of cross-section, plane, and profile curves based on an actual engineering case. The alignment variations influence the mechanical characteristics of the bridge in three directions. Therefore, finite element models with different scale elements are established for a detailed analysis. A comparative analysis is conducted on the temperature gradient effect of spatial three-curved steel box girder bridges using two typical temperature gradient models: multi-segment piecewise linear and exponential. Based on simulated results, a WOA algorithm-optimized BPNN serving as a surrogate model is trained to predict stresses under temperature gradients.