**3. Probability Density Characteristics of Temperature**

To display the annual variations in the girder surface temperature, the measuring point T4 at different longitudinal positions S2, S6 and S10 was chosen, for instance. Figure 5 shows the monitoring data from 1 September 2020 to 31 August 2021. It can be observed that the annual temperature changes all appeared to have obvious periodic patterns, and this could be approximately described by sinusoidal functions, with lower temperatures from December to February and higher temperatures from June to August. In addition to the low-frequency periods, the temperature data also contained high-frequency fluctuations. High-frequency fluctuations may be due to the effect of daily temperature changes. In addition, it was found that the overall temperature value of S6 and its variance amplitude were significantly larger than those of S2 and S10, and the difference was more obvious in summer. This was a preliminary indication of the non-uniformity of the temperature distribution along the longitudinal direction of the bridge, which is discussed later.

**Figure 5.** *Cont*.

**Figure 5.** Annual time-history of the girder surface temperature. (S-T: Sensor number T The section number S).

In previous studies, the Gaussian distribution model [44–46] has been commonly used for fitting the probability density function (PDF) of the temperature. However, it is obvious in Figure 6a that the PDF cannot be properly fitted by the Gaussian distribution model due to bimodal characteristics. By comparing the fitting results of several models, the weighted superposition of two normal distributions was selected for fitting the current data, with its mathematical expression as follows:

$$f(\mathbf{x}) = \gamma f\_1(\mathbf{x}) + (1 - \gamma) f\_2(\mathbf{x}) \tag{1}$$

$$f\_1(x) = \frac{1}{\sqrt{2\pi}\sigma\_1} e^{-\frac{(x-\mu\_1)^2}{2\sigma\_1^2}}\tag{2}$$

$$f\_2(\mathbf{x}) = \frac{1}{\sqrt{2\pi}\sigma\_2} \mathbf{e}^{-\frac{(\mathbf{x}-\mu\_2)^2}{2\sigma\_2^2}} \tag{3}$$

where *x* denotes the surface temperature of the steel box girder, *f*(*x*) denotes the probability density model of *x*, *fi*(*x*) (*i* = 1, 2) denotes the normal distribution function with the mean value *μ<sup>i</sup>* and the standard deviation *σ<sup>i</sup>* , *γ* is the weight of the two normal distribution functions with 0 ≤ *γ* ≤ 1.

**Figure 6.** Fitting of the PDFs of temperature at S2-T4 using different models: (**a**) Gaussian distribution model; (**b**) Superposition of two normal distribution models.

The goodness-of-fit for the different models is shown in Figure 6. The *R*<sup>2</sup> (coefficient of determination, COD) of the superposition of two normal distribution models was significantly higher than that of the Gaussian distribution model. Therefore, it was reasonable to choose the superposition of two normal distribution models to describe the probability distribution of temperature.

The probability density histograms of measuring points S2-T4, S6-T4 and S10-T4 and their fitted curves are shown in Figure 7. It can be seen that the fitted curves were in good agreement with the measured ones. Meanwhile, the fitting results all passed the K-S test with a significance level *α* = 0.05, indicating that the fitting curves could accurately describe the probability density characteristics of the surface temperature of the steel box girder. The results show that the probability density curves of each measuring point have bimodal characteristics. This was due to the bridge being located in the subtropical humid monsoon climate zone and its experience of a transition period between seasons. The temperatures at the two peaks represent the temperature during the spring-summer and summer-fall transitions and during the fall-winter and winter-spring transitions, respectively.

**Figure 7.** Probability density histograms and the fitting curves of S2-T4, S6-T4 and S10-T4.

To investigate the difference in temperature distribution along the longitudinal direction of the bridge, the probability densities of T4 and T5 (on the top and bottom plates, respectively) and at different longitudinal sections S2, S6 and S10 were fitted based on the abovementioned model. The estimated floating parameters are shown in Table 1, and the corresponding PDFs are given in Figure 8.


**Table 1.** Estimated fitting parameters for the probability density models of typical measuring points.

**Figure 8.** Fitted PDFs of the girder surface temperatures at different longitudinal sections. (**a**) Top plate measuring points; (**b**) Bottom plate measuring points.

From Table 1 and Figure 8, it can be seen that there was a significant difference between the PDFs of the mid-span and side-span for both the top and the bottom plates. Concerning the two peaks in the PDFs, the measuring point S6-T4 was mainly concentrated around 30 ◦C and 16 ◦C, while S10-T4 was mainly concentrated around 23 ◦C and 11 ◦C, indicating that there was a significant difference in the temperature distribution along the longitudinal direction of the bridge, and the temperature in the mid-span was significantly higher than that in the side span. Moreover, PDFs were not the same for the side-span measuring points, which were symmetrical along the bridge centerline, indicating a non-uniform longitudinal temperature distribution in the steel box girder. This phenomenon shows that the assumption of uniform temperature used in previous studies and specifications [14,20,30] is not reasonable.

#### **4. Statistical Analyses of Temperature Distribution along the Bridge**

## *4.1. Temperature Longitudinal Distributions*

To further analyze the longitudinal gradient of the bridge girder surface temperature, the annual temperature and annual temperature difference were statistically analyzed by selecting one year of the temperature time history data from 11 measurement points (S1–S11) in the longitudinal direction. The statistical analyses included the following four aspects: annual maximum temperature, annual minimum temperature, annual average temperature and annual maximum temperature difference for each measurement point. The annual maximum temperature difference was the difference between the annual maximum temperature and the annual minimum temperature.

By taking the top and bottom measuring points, including T4 and T5; for instance, the annual temperature and statistical analyses along the longitudinal direction of the main bridge are shown in Figure 9. An analysis of Figure 9 shows that the distribution of the temperature field along the longitudinal direction was non-uniform. The average temperature was approximately symmetrical about the mid-span. On the other hand, the annual average and maximum temperature curves showed the same patterns: the temperature was highest in the middle of the span and lower near the bridge tower while also rising again from there to the side span. In addition, the variation trends of the annual maximum temperature difference were similar to that of the annual maximum temperature. Although there was some variation in the minimum temperature between the different measurement points, this variation was not very significant compared to the maximum temperature. In addition, compared to the bottom plate, the temperature variation range from the top plate was larger. Measured data show that many cities in China have experienced extremely high temperatures over recent years. For example, the maximum temperature in Chongqing reached 43 ◦C in 2022, and the temperature of the steel box girder, in this case, may show a peak. We will continue to follow up on the study.

**Figure 9.** Distribution of annual temperature statistic values along the longitudinal direction. (**a**) Top plate measuring points T4; (**b**) Bottom plate measuring points T5.

To further investigate the longitudinal temperature distribution pattern, the annual measured data for all eight temperature measuring points (T1~T8) in each section were statistically analyzed, and the annual average temperature of each measuring point is shown in Figure 10. It can be seen that the surface temperature of the steel box girder had different distributions between the top and bottom plates, as well as the left and right sides of the mid-span. To clarify the distribution pattern, the polynomial fitting of the longitudinal temperature distribution curve was performed below.

**Figure 10.** Longitudinal distributions of annual average temperature.

First, the measured temperature of the top and bottom plates was, respectively, timeaveraged. Then, the average temperature values of different measuring points on the left and right sides of the mid-span (S6) were separately fitted using polynomial curves *y* = *Ax*<sup>2</sup> + *Bx* + *C*, where *y* denoted the temperature value, *x* denoted the Longitudinal distance from the mid-span, and the three parameters of the equation were A, B, and C. In order to conveniently compare the fitted formulas, the horizontal axis on the left side was set as *x*<sup>1</sup> and the horizontal axis on the right side was set as *x*2. The fitting results are shown in Figure 11. Meanwhile, the equations of the fitted curves are listed in Table 2. The temperature values at any longitudinal location on the steel box girder could be subsequently estimated from these fitted curve equations. By referring to the practice of the specifications and other studies [8,26,30,33] on the horizontal and vertical temperature gradient patterns, in our future research work, we should incorporate data from other similar bridges to obtain the longitudinal temperature distribution patterns of large-span suspension bridges.

**Table 2.** Longitudinal distribution of temperature fitting equation.


**Figure 11.** Fitting curves of the temperature's longitudinal distribution (**a**) Top plate; (**b**) Bottom plate.

From the fitted curves and the fitting parameters, it could be seen that: (1) For the top plate temperature, the two fitting curves along the left and right spans had significantly different parameters, with higher temperatures on the right side than on the left side, possibly due to variations in geographic position and material properties. (2) For the bottom plate temperature, the two fitting curves along the left and right spans had very similar parameters, indicating that the temperature of the bottom plate was longitudinally distributed and symmetric along the mid-span.
