**4. Reliability Analysis: RC Slab-Column Joint of an Office Building**

The prototype building used for reliability analysis is a 7-story, 5-span RC slab-column shear wall office building [53], and it was designed using GB 50010-2010 [11] and GB 50011-2010 [54]. The building itself contains 3 m storey height and is supported by a 7.5 m × 7.5 m column grid, and the interior joint shown in Figure 9 is selected as the study object. The selected joint consists of a slab with an effective depth of 209 mm and a square column with side length of 530 mm, which is subjected to the specified dead load of 7.0 kN/m2 and live load of 2.0 kN/m2. According to the requirement of GB 50068-2018 [55], the dead load and live load used for limit state design must be adjusted by multiplying the partial safety factors for the load, such as 1.3 and 1.5. Therefore, the limit state function Z of structure can be defined as:

$$Z = R - 1.3S\_G - 1.5S\_Q \tag{9}$$

where *R* is the punching shear resistance; *SG* is the dead load; *SQ* is the live load. Furthermore, the measured compressive strength of C50 concrete in the slab is 39.31 Mpa, and the measured yield strength of HRB400 reinforcement is 421 Mpa. The reinforcement ratio of the joint is 0.81%, and the main influential factors are listed in Table 4.

**Figure 9.** Prototype building [53]: (**a**) elevation; (**b**) plan.

**Table 4.** Main influential factors of the selected interior joint.


The statistic information and suitable probability density functions of the stochastic variables used for reliability analysis are listed in Table 5 [33,56], where COV is the coefficient of variance. According to the study conducted by Chojaczyk et al. [27], the COV of failure probability *Pf* calculated by MCS is accepted when its value is around 0.1; then the *Pf* around 10−<sup>4</sup> (the normal failure probability of an existing structure) can be calculated through the simulation based on *N* samples [57]:

$$\text{COV}\left(P\_f\right) = \frac{1}{P\_f} \sqrt{\frac{\left(1 - P\_f\right)P\_f}{N}}\tag{10}$$

where *N* signifying 1,000,000 can be determined according to the aforementioned conditions. Another method used in the study conducted by Hadianfard et al. [58] stipulates that the number of samples needed for MCS can be determined through:

$$N > \frac{-\ln(1-C)}{P\_f} \tag{11}$$

where *C* is the confidence level, with values of 0.95 in this paper. Equation (11) suggests that the number of samples should not be less than 30,000, so that the value range calculated by Equations (10) and (11) is determined between 30,000 and 1,000,000. In this range, the variation of COV of failure probability *Pf* within 10 simulations is shown in Figure 10. The COV of failure probability decreases with the increase of the sample size from 30,000 to 1,000,000, which means that the result of the reliability analysis increasingly stabilizes. Based on this knowledge, 1,000,000 samples are produced randomly and conduced for reliability analysis by XGBoost and MCS.

**Figure 10.** Effect of sample size on COV of failure probability.

**Table 5.** Stochastic variables used for Monte Carlo simulation.


#### *4.1. Results of Structural Reliability Analysis*

The efficient implementation of Monte Carlo simulation (MCS) is restricted by the sample size and the computational efficiency of the surrogate model [59], but this can be solved by XGBoost. The average computation time for 1,000,000 samples and the reliability analysis of the slab-column joint is 30 s. This is done by a laptop with four-core CPU and 8 GB memory, which demonstrates the efficiency of ML-MCS. Based on the regression prediction of punching shear resistance, the distribution and CDF of structural resistance are shown in Figure 11. The average and standard deviations of the distribution of structural resistance are 955.96 kN and 52.42 kN, respectively. MCS can estimate the failure probability of a structure effectively by calculating the probability of Z < 0 in Equation (9), and the related reliability analysis can also be realized. Table 6 displays the result of reliability analysis, where *Pf* is the failure probability of the structure; *β* is the reliability index; *α<sup>R</sup>* and *α<sup>S</sup>* are the sensitivity coefficients of resistance and load; *r*\* and *s*\* are the coordinates of the design point. The reliability index *β* indicates that the reliability and safety of the selected interior joint are good and meet the requirement of GB 50068-2018 [55].

**Table 6.** Results of reliability analysis.


**Figure 11.** Distribution of structural resistance.
