*5.2. Contours of Daily Temperature*

To further analyze the generation mechanisms and processes of the longitudinal temperature gradient, taking into account the apparent daily periodicity of the temperature data, the data for one day were selected for analysis. The following is an analysis of the longitudinal temperature distribution pattern based on daily temperature curves. A total of 11 longitudinal sections were selected, and three measurement points on each of the top and bottom surfaces were taken for analysis. To better observe their regularity, the hourly temperature was averaged to obtain the time history of the measured daily temperature curves, as shown in Figure 14.

Figure 14 points out that, as summarized in the previous section, the temperature distribution along the longitudinal direction was always non-uniform. The temperature statistic values varied non-uniformly in spatial locations, and the temperature transients varied non-uniformly in space as well. Thus, the non-uniform longitudinal temperature gradient contained both spatial non-uniformity and temporal non-uniformity. Although previous studies have investigated the effect of temperature at different longitudinal locations on the temperature gradient of the cross-section through finite points, they have not investigated the mechanism of this non-uniform temperature field in depth.

**Figure 14.** Daily temperature-time histories for different measuring points at 11 sections.

Meanwhile, Sections S3, S4 and S8, S9 are, respectively, located on both sides of the east and west main towers. The amplitude of daily temperature variation in these cross-sections was smaller than that of other adjacent measurement points. This phenomenon is similar to the findings of Figures 10 and 12. Additionally, this was mainly due to the shading effect of the main tower, which reduced the temperature of this section. In addition, the temperature of the mid-span S6 section was the highest because it was directly affected by solar radiation. By comparing the temperature-time histories of the top and bottom plates between different sections, it is found that there was a difference in the longitudinal temperature distribution between the top and bottom surfaces.

If the surface temperature of any position on the box girder could be obtained, the accuracy of the temperature effect analysis could be improved. Therefore, Figure 15 shows the daily temperature contour maps of the top and bottom plate temperatures, respectively. The example time instant was chosen at 17:00 because the girder surface temperature at that time was the highest value in one day, and the longitudinal temperature distribution of the steel box girder had obvious non-uniformity.

It can be seen from Figure 15 that: (1) The longitudinal temperature contour on the top plate indicates that the high-temperature area slightly deviated to the right of the mid-span. This is because the bridge is east–west oriented, and the bridge deck on the right side of the mid-span was exposed to solar radiation for a longer time, so the temperature would be slightly higher than the bridge deck on the left side of the mid-span. (2) The longitudinal temperature distribution of the bottom plate was almost symmetrical about the mid-span because the bottom plate was not directly affected by solar radiation but was mainly affected by the environmental temperature. (3) The transverse temperature difference of the top plate was more obvious than that of the bottom plate because of solar radiation, and the temperature distribution had obvious three-dimensional spatial distribution characteristics.

After analyzing the temperature distribution characteristics on other days, we could also obtain a similar conclusion. The longitudinal distribution characteristics of daily and annual temperature were similar, while daily temperature also had a three-dimensional spatial distribution.

To summarize, the temperature of the long-span bridge structure was non-uniformly distributed not only in space but also in time. The non-uniformity of the longitudinal temperature distribution could be caused by the non-uniformity of the material and is also related to the environmental location where the measurement points are located. When performing temperature effect analysis, the temperature field of large-scale bridge structures is usually assumed to be a one-dimensional temperature field, which erases the longitudinal and transverse temperature gradients, resulting in large deviations between the calculated results and the real temperature field of the bridge. Further research into the effects of

these factors on the longitudinal temperature distributions could be useful in the design stage of long-span bridges to effectively reduce the risk caused by the temperature effects. Therefore, during the refinement of the temperature effect analysis for long-span bridges, the longitudinal distribution of temperature non-uniformity needed to be considered.

**Figure 15.** Contour maps of measured temperature at 17:00. (**a**) Top plate temperature contours; (**b**) Projection of (**a**); (**c**) Bottom plate temperature contours; (**d**) Projection of (**c**).

#### **6. Conclusions**

This study investigated the longitudinal distribution characteristics of the box girder surface temperature on a long-span suspension bridge. The probability density statistics, statistical values, cross-correlations coefficients, power spectrum densities, and time-space contours are detailed and analyzed. The main findings are as follows:

(1) The annual temperature's probability density curves at different longitudinal measuring points of the bridge all have bimodal characteristics, which could be fitted by the weighted sum of two normal distributions. This bimodal distribution was mainly caused by the transitions between different seasons.

(2) The statistical analyses show that the distribution of the girder surface temperature along the longitudinal direction was non-uniform. Moreover, the equations of the longitudinal distribution curves were obtained by polynomial fitting, and the distribution pattern was the highest in the mid-span, the lowest in the bridge tower, and increased along the side span. Therefore, the design phase should consider the non-uniform distribution of temperature.

(3) Using the correlation analysis between the temperature measured at the mid-span and other longitudinal sections, it was found that the correlation coefficient in the main span gradually decreased farther away from the mid-span section, with the lowest at the tower section. While in the side span, the correlation coefficient increased as the measurement points moved away from the tower. This revealed that the girder surface temperature was influenced by solar radiation and also the shedding effects of the tower.

(4) There were two prominent peaks in the frequency domain of the annual temperature, corresponding to the 24 h and 12 h time periods, respectively. This meant that the daily temperature could be considered a representative sample of the annual temperature.

(5) By comparing the daily time histories of temperature at different longitudinal sections, it was found that the temperature field was significantly three-dimensional and was not only non-uniformly distributed in space but also non-uniformly distributed in time.

(6) According to the time-space contour maps of temperature, we can gain insight into the underlying mechanisms of the generation of this non-uniformity distribution. Combined with the statistical analyses, the statistical values of the bridge temperature at any section could be obtained using the contour map.

In conclusion, the surface temperature of the steel box girder was non-uniformly distributed along the longitudinal direction of the bridge. Therefore, in analyzing the temperature effects, the traditional uniform-distribution assumption could lead to inestimable deviations from the real conditions. This work is helpful for a more accurate analysis of temperature effects on long-span bridges and can also provide a reference for the longitudinal distribution of temperature fields on other similar bridges.

**Author Contributions:** Conceptualization, W.M. and B.W.; Formal analysis, W.M. and B.W.; Methodology, W.M. and B.W.; Data curation, D.Q. and B.Z.; Validation, D.Q. and B.Z.; Investigation, D.Q. and X.Y.; Writing—original draft, W.M.; Writing—review and editing, B.W.; Project administration, B.W. and X.Y.; Funding acquisition, B.W. and X.Y. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by the National Natural Science Foundation of China (Grant No. 51978111), Chongqing Technology Innovation and Application Development Special Key Project (Grant No. CSTB 2022TIAD-KPX0205), Chongqing Transportation Science and Technology Project (Grant No. 2022-01), Natural Science Foundation of Chongqing, China (Grant No. cstc2021jcyjbshX0061), China Postdoctoral Science Foundation (Grant No. 2022MD713699), Special Funding of Chongqing Postdoctoral Research Project (Grant No. 2021XM1016) and Chongqing Zhongxian Science and Technology Plan Project (Grant No. zxkyxm202202) are greatly acknowledged.

**Data Availability Statement:** Data are available on request due to restrictions.

**Conflicts of Interest:** The authors declare no conflict of interest.
