**5. Conclusions**

Structural reliability reflects the safety and stability of the entire practical structure subjected to permanent action and variable action [36], the calculation of which, through MCS, is restricted by the computational efficiency of the surrogate model. This work presents a framework for integrating the machine learning-based surrogate model into a Monte Carlo simulation to perform the reliability analysis with a satisfying accuracy and efficiency. An ML model is established and screened from four candidate ML models: as ANN, DT, RF, and XGBoost; the prediction performances of these are examined through three performance measures such as RMSE, MAE, and R2. Furthermore, the advantages of ML models are embodied by comparison with five empirical models. The final prediction model is used as the surrogate model of MCS, and an RC slab-column joint in an actual structure is introduced as the object of reliability analysis. The following conclusions can be drawn from this paper:

The punching shear resistance of RC slab-column joints is influenced mainly by seven influential factors: *s*, *A*, *d*, *f'c*, *fy*, *ρ*, and *λ* [38]. The capture of the mapping relationship between them can guarantee the construction of the ML model. With the help of the grid search method and 10-fold cross validation, four ML models with optimal hyperparameters are established. After comparison, XGBoost has the best prediction performance reflected in RMSE, MAE, and R2, and is selected as the final prediction model and used for reliability analysis.

To facilitate the understanding of the prediction process of ML, SHAP is utilized to quantify the contribution of input variables to punching shear resistance, and to visualize the prediction process. According to the importance sorting of input variables, *d* and *s* have the greatest and least impacts, respectively, on punching shear resistance. Furthermore, feature dependency plots display the specific impact of each input variable by marginalizing the impacts of other variables. The analysis of the influential factors provides not only the understanding of prediction process, but also the suitable optimization sorting in structural design.

The actual structure adopted for the case study is an RC slab-column shear wall office building. The punching shear resistance of 1,000,000 samples produced by random sampling is calculated through XGBoost. The reliability analysis of the interior joint selected from the prototype building is conducted through MCS, and the final reliability index *β* meets the requirement of the design provisions of GB 50068-2018 [55]. Moreover, the sensitivity analysis reveals the impact of the stochastic context and the values of stochastic variables on structural reliability. Based on these, the computational efficiency of the reliability analysis of the slab-column joints can be enhanced on the premise of high computational accuracy. In future reliability analysis, some advanced sampling methods, such as Latin hypercube sampling and importance sampling, can be used to reduce the number of simulations appropriately. Furthermore, a program with some input windows of influential factors can be designed as a practical tool for reliability analysis.

**Author Contributions:** Conceptualization, L.S. and S.L.; software, L.S. and Y.S.; validation, L.S., Y.S. and S.L.; formal analysis, L.S.; writing—original draft preparation, Y.S.; writing—review and editing, S.L.; visualization, Y.S.; supervision, L.S. and S.L.; project administration, S.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by Science Foundation of Zhejiang Province of China, grant number LY22E080016; National Science Foundation of China, grant number 51808499; Science Foundation of Zhejiang Sci-Tech University (ZSTU), grant number 19052460-Y; and the Education of Zhejiang Province, grant number 20050061-F.

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

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** This study is supported by the Engineering Research Centre of Precast Concrete of Zhejiang Province. The help of all members of the Engineering Research Centre is sincerely appreciated. We would also like to express our sincere appreciation to the anonymous referee for valuable suggestions and corrections.

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

#### **Appendix A**

To facilitate the acquirement of data, the entire database has been uploaded to GitHub: https://github.com/shenyx0126/Database-used-for-reliability-analysis.git (accessed on 3 October 2022).

#### **References**


**Lulu Shen 1, Bo Yang 1, Yingwu Yang 2,\*, Xuelin Yang 3, Wenwei Zhu <sup>3</sup> and Qingzhong Wang <sup>4</sup>**

<sup>1</sup> School of Civil Engineering and Architecture, Zhejiang Sci-Tech University, Hangzhou 310018, China


**Abstract:** Since monolithic movement is considered a promising technology to relocate historical buildings, corresponding real-time monitoring is of great interest due to the buildings' age and poor structural integrity. However, the related paperwork and practical applications are still limited. This paper describes a wireless sensor network (WSN)-based strategy as a non-invasive approach to monitor heritage curtilage during monolithic movement. The collected data show that the inclination of the curtilage is almost negligible. With the aid of finite element simulation, it was found that the crack displacement curves changed from −0.02 to 0.07 mm, which is affected by moving direction while the value is not enough to cause structural cracks. The deformation of the steel underpinning beam, which is used to reinforce masonry walls and wooden pillars, is obviously related to the stiffness in different directions. Additionally, the strain variations of the steel chassis, which bear the vertical loads from wooden pillars and masonry walls, are less than 0.04%. This indicates that they are kept within the elastic range during monolithic movement. This work has proved that the WSN-based approach has the potential to be applied as an effective route in real-time monitoring of the monolithic movement of an historic building.

**Keywords:** wireless sensor network; historic building; real-time monitoring
