*3.5. Deformation Data Analysis*

#### 3.5.1. Deformation Data Analysis since Closure of the Main Span

To verify the applicability of the method, the measured values, FEM calculated values and statistically calculated values (calculated by the MC-RSM method) of the vertical deformation data since closure of the main span were compared and analyzed. The measured deformation data was obtained from the health monitoring system (HMS) of Nanjimen Yangtze River Track Special Bridge. Using phase shift regression analysis, an empirical regression equation [28,29] was established to exclude the influence of temperature effects, and then compared with the calculation results. The results are shown in Figure 6.

As shown in Figure 6, the measured values of vertical deformation in the mid-span since closure of the main-span are relatively close to the FEM calculated values, with small fluctuations around the FEM calculated values. However, the engineering prediction values are low due to the fluctuation. At the same time, the statistically calculated values obtained by the MC-RSM method envelop the measured values. Although there is a gap between the statistically calculated and measured values, the statistically calculated values are more certain because they focus on the calculation and analysis of the sample data, and there is also some defect tolerance while maintaining accuracy and reliability. In addition, the statistically calculated values only consider some of the influencing factors and cannot include all the actual influencing factors, leading to partial deviation of the calculated results. If there were more factors studied, the statistically calculated results would be closer to the actual situation.

**Figure 6.** Comparison of vertical deformation at the mid-span of the main span.

3.5.2. Predictive Analysis of Deformation after Bridge Completion

When predicting deformation at the mid-span of the main span, the MC method was applied, selecting a sample number of 10,000. The shrinkage and creep of the Nanjimen Yangtze River Track Special Bridge were analyzed randomly based on the variability of the material parameters. The average value and standard deviation of vertical deformation at the mid-span of the main span were analyzed by selecting six time nodes from 1/2 year to 10 years after bridge completion. The results are shown in Table 5.


**Table 5.** Random analysis results for deformation at the mid-span of the main span.

Table 5 shows that the difference between the design expected values and the average values (obtained by sampling statistical analysis) is small, according to the parameters selected in this paper. It shows that the maximum vertical deformation at the mid-span of the main span can be well predicted. Furthermore, it can be seen that the standard value increases over time, indicating the discreteness of the deformation increment gradually increases.

The MC method obtained the probability distribution of vertical deformation at different time nodes, with the distribution results shown in Figure 7. The confidence levels

under five representative percentages were selected, which were 35%, 55%, 75%, and 95%, respectively.

**Figure 7.** Probability distribution of vertical deformation at different time nodes.

The vertical deformation predicted interval at each chosen time node for different random factors was plotted for these five different confidence levels. Figure 8 shows the results for the vertical deformation of different deformation intervals in 10 years.

Furthermore, the predicted values obtained by the FE calculation were compared with the confidence interval at the 95% confidence level. The predicted values for vertical deformation at the mid-span of the main span were obtained using MIDAS/Civil considering different time nodes after bridge completion. Under the same random parameters, the confidence intervals at the 95% confidence level were obtained by the MC method sampling, and the comparison results are shown in Figure 9.

Figure 9 illustrates that the predicted values obtained by the proposed model fall within the 95% confidence interval, demonstrating the high accuracy of the parameter distribution and analysis method. The 10-year deformation amplitude of the bridge calculated by the finite element software and the variation value of the lower limit of 95% confidence level are shown in Figure 10. It can be seen that the 10-year deformation interval calculated by the finite element software is (−86.5963, −21.3299) mm, and the variation value is 65.2664 mm. Using the uncertainty analysis method of MC-RSM, the variation interval of the lower limit of the 95% confidence level is (−133.64, −33.68) mm, and the variation amplitude of the lower limit of the 95% confidence level is 99.96 mm. The deformation difference between the two methods increased from 12.35 mm at half a year to 47.04 mm at 10 years. The deformation amplitude of the mid-span of the bridge at 10 years after completion (calculated by the FE method) and the lower limit of the 95% confidence level

(predicted by the MC-RSM method) and the deformation difference obtained by the two methods in 10 years are shown in Figure 10. The vertical deformation at the mid-span of the main span is relatively discrete, resulting in small calculation values by the FE method, which is due to the influence of parameter randomness. It can be concluded that environmental factors, material factors, and other factors should be considered comprehensively when analyzing the alignment of large-span track SCCB cable-stayed bridges.

**Figure 8.** Vertical deformation predicted interval at different confidence levels.

**Figure 10.** Comparison of 10-year deformation amplitude (unit: mm).

#### **4. Conclusions**

In this paper, an alignment prediction of a large-span track SCCB cable-stayed bridge was carried out by RSM considering the environmental factors, actual situation on site, applicable standards and specifications of the Nanjimen Yangtze River track Special Bridge. The deformation change of eight factors under the influence of randomness was discussed. The following main conclusions were obtained.


Due to the limited space, only the randomness of the eight influencing factors of the bridge was analyzed and studied in this paper, which resulted in the deviation between the predicted results and the actual results. To obtain more accurate results, further research on more influencing factors is required.

In addition, the time since the bridge completion is relatively short, so the deformation data for only one year after closure of the main span have been compared in this paper. As more extended deformation data becomes available, further verification is needed.

**Author Contributions:** Conceptualization, X.L. and X.C.; methodology, P.D.; software, P.D.; validation, X.L., X.C. and P.D.; formal analysis, X.L.; investigation, S.T. and H.L.; resources, P.D and S.T.; data curation, P.D. and S.T.; writing—original draft preparation, H.L.; writing—review and editing, X.L. and X.C.; visualization, H.L.; supervision, X.L.; project administration, X.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by the Open Fund of the Chongqing Key Laboratory of Energy Engineering Mechanics and Disaster Prevention and Mitigation (EEMDPM2021201) and the Chongqing Institute of Science and Technology Research Grant Program (CKRC2021074).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Apart from the original traffic data that were collected through WIM systems, all of the other data, models, and the code that was generated or used during the study are available from the corresponding author by request.

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