**1. Introduction**

Long-span bridge structures can be affected by various environmental factors during their long-term operations, among which the temperature effect is particularly significant [1–3]. In recent years, numerous studies on bridge temperature effects have been carried out [4–7], mostly considering uniform temperature fields. Some scholars [8,9] have studied the influence of boundary factors such as solar radiation on the bridge temperature field. Some scholars [10–13] studied the time-varying temperature distribution of bridges and established a temperature gradient distribution model. There have also been numerous scholars [14–20] who have studied the long-term measured cross-section temperatures of a large number of bridges with different cross-section forms, observed transverse temperature gradients and proposed two-dimensional temperature field calculation models.

**Citation:** Ma, W.; Wu, B.; Qin, D.; Zhao, B.; Yang, X. Statistical Analyses of the Non-Uniform Longitudinal Temperature Distribution in Steel Box Girder Bridge. *Buildings* **2023**, *13*, 1316. https://doi.org/10.3390/ buildings13051316

Academic Editor: Andrea Benedetti

Received: 12 April 2023 Revised: 8 May 2023 Accepted: 16 May 2023 Published: 18 May 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

<sup>1</sup> State Key Laboratory of Mountain Bridge and Tunnel Engineering, Chongqing Jiaotong University, Chongqing 400074, China; 611200080001@mails.cqjtu.edu.cn (W.M.); xianyi@yahoo.com (X.Y.)

It is worth noting that the above studies are based on the assumption that the temperature is uniformly distributed along the longitudinal span direction of the bridge. However, in actual engineering, the temperature field of large-scale structures, such as long-span bridges, is usually non-uniform. It has been shown that the stresses and deformations caused by non-uniform temperature distribution can be comparable to those caused by static and live loads, which may also cause cracks and excessive deformations [21–25].

Actually, the existing standards [26–30] have neglected the influence of the longitudinal non-uniform temperature distribution on bridges, assuming that the temperature field can be represented by one-point measuring data, i.e., a uniform distribution field. Although this assumption simplifies the analysis of bridges affected by temperature in the two-dimensional plane, it may overestimate the temperature effects at some spanwise locations and underestimate them at others. Moreover, the nonlinear effects caused by temperature effects are neglected by the uniform-distribution assumption [31,32]. Therefore, the temperature effects cannot be accurately evaluated when adopting the above assumption. Additionally, the inability to accurately assess the temperature effects of large-span bridges may bring uncertainty to the bridge during the design, construction and operation stages, thus affecting the structural safety of the bridge [33,34].

Therefore, the longitudinal temperature distribution is important for the accurate analysis of bridge temperature effects. Several studies have started to pay attention to this topic. Abid et al. [35] investigated the temperature distribution of bridge segmental models under the influence of solar radiation and air temperature variations through numerical simulations. Based on the health monitoring data of the Aizhai Bridge, Hu et al. [36] found that there were temperature differences at different measuring points along the longitudinal direction of the bridge span. Gu et al. [37] proposed a vertical and lateral temperature gradient model for different longitudinal positions along the bridge span. Liu et al. [38] studied the non-uniform longitudinal distribution temperature of concretefilled steel tubular bridges due to the change in component inclination. Although the above studies considered the variability of temperature distribution, it is limited to a single or several measuring points, and the conclusions may not be well applied to actual bridges. In this context, the longitudinal temperature distribution on the whole bridge and its statistical characteristics was insufficient. Moreover, the role of the non-uniform longitudinal distributions of temperature was not analyzed in depth.

Therefore, this work studies the longitudinal temperature distribution of steel box girders based on the field-measured temperature data. The main objective was to provide a more accurate description of the longitudinal temperature field of long-span bridges.

The paper is organized as follows. In Section 2, the bridge and its health monitoring system are introduced. In Section 3, the probability density characteristic analysis is carried out by selecting typical measuring points in the longitudinal direction of the bridge, and the probability density curves are fitted; next, the temperature of all measurement points is statistically analyzed, and longitudinal distribution curves are fitted; A correlation analysis was then performed, highlighting variability in the longitudinal direction. Finally, the frequency domain features were analyzed, and the space-time contour maps were given to study the mechanisms and processes of the non-uniform distribution of temperature.
