## 4 Practical Formula for Bearing Capacity and Reliability Analysis
### Reliability Analysis
#### 4.5.3 Statistical Parameters of Basic Random Variables  
The resistance of the function functions is influenced by basic random variables such as \(f_{c}\), \(D\), \(t\), and \(f_{y}\). This resistance typically follows a log-normal distribution.  
1. Statistical parameters of load  
For reliability analysis, two primary load combinations are taken into account: dead loads combined with office and residential live loads. Dead loads, which represent the structure's self-weight, are assumed to follow a normal distribution because of their constancy. Live loads, which are both permanent (e.g., building facilities) and temporary, are represented using an Extreme Value Type I distribution. The statistical characteristics for various combinations of loads are shown in Table 6.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Parameter** & **Dead Load (G)** & **Office Live Load (QO)** & **Residential Live Load (QR)** \\ \hline Mean Value (\(\mu\)) & 1.060 & 0.698 & 0.859 \\ Coefficient of Variation (\(\delta\)) & 0.070 & 0.288 & 0.233 \\ Distribution Type & Normal & Extreme Value Type I & Extreme Value Type I distribution \\ \hline \end{tabular}
\end{table}
Table 6: Statistical parameters of load combinations.  
Concrete Strength (\(f_{cu}\)): The reliability index (\(\beta\)) displays an incremental trend with the increase in \(f_{cu}\). This indicates a positive relationship, where higher-strength concrete enhances the structural reliability, with more pronounced improvements at increased steel yield strengths (Figure 14a).  
Figure 14: Influence of different parameters on the reliability index (\(\beta\)). (**a**) Concrete strength; (**b**) steel yield strength; (**c**) steel ratio; (**d**) load effect ratio; (**e**) load combination; (**f**) DS replacement ratio.  
Steel Yield Strength (\(f_{y}\)): Variations in \(f_{y}\) exhibited a relatively minor effect on the reliability index across the analyzed \(f_{cu}\). This suggests the dominant influence of \(f_{cu}\) over steel strength in the context of reliability within the examined parameters (Figure 14b).  
Steel Ratio (\(\rho\)): An inverse relationship is evident between the \(\rho\) and the \(\beta\). Higher \(\rho\) values lead to a reduction in \(\beta\), signaling a potential over-reliance on steel reinforcement that may not proportionately increase structural reliability, especially at lower concrete strengths (Figure 14c).  
Load Effect Ratio (\(\rho_{L}\)): The analysis indicates a negative correlation between the \(\rho_{L}\) and \(\beta\). An increase in \(\rho_{L}\) adversely affects the \(\beta\), underscoring the criticality of adept load management to maintain structural reliability above the target index (Figure 14d).  
Load Combination: Different load combinations affect the \(\beta\) differently. For instance, the combination QO+G results in a higher \(\beta\) compared to QR+G, suggesting that the latter may be more challenging for the structural system, potentially reducing the reliability index below the acceptable threshold (Figure 14d).  
DS Replacement Ratio (\(r\)): Figure 14f indicates a gradual increase in the \(\beta\) with an increase in the \(r\) for concrete strengths of 30 MPa and 50 MPa. The trend suggests that incorporating DS components positively influences structural reliability. However, the effect is less significant at higher concrete strengths, illustrating a complex relationship between DS replacements and material properties.  
In order to reach the desired reliability index of 3.2, our findings emphasize the need for a well-balanced design strategy that takes into account the interaction between material strengths, steel ratios, and loading circumstances. This research corroborates the reliability requirements specified in the Chinese standard GB50068-2018 [50] and underscores the need to consider all elements that contribute to guaranteeing sufficient safety margins against failure.