## 4 Practical Formula for Bearing Capacity and Reliability Analysis
### Comparison of Formulas  
In order to effectively demonstrate the advantages of the proposed equations and compare them with existing design code formulas, an assessment of their accuracy in predicting the ultimate load-bearing capacity of components was undertaken. Table 4 outlines the limitations and scope of application of these design codes. Meanwhile, Table 5 displays the formulas and expressions of the design codes, along with the \"Mean\" and \"CV\" columns, depicting the average ratio \(N_{\text{u,Exp}}\) to \(N_{\text{u,Eq}}\) of the observed ultimate load capacity value \(N_{\text{u,Exp}}\) to the computed values \(N_{\text{u,Eq}}\) based on the reference codes and formulas.  
Figure 12: Comparison of ultimate bearing capacity from experimental results and FE analysis using Equation (10). (**a**) Calculated values of Equation (10) versus experimental results; (**b**) Calculated values of Equation (10) versus FE results; (**c**) Calculated values of Equation (10) versus FE results (based on test data parameters); (**d**) Calculated values of Equation (10) versus FE results (based on FE parametric analysis).  
The comparison between the calculated results obtained using design code formulas \(N_{\text{u,Eq}}\) and the experimental results \(N_{\text{u,Exp}}\) is provided in Table 5, while the corresponding scatter plot is depicted in Figure 13. It is evident that the majority of design codes notably underestimate the ultimate load-carrying capacity of components, with the exception of the BS EN4 design code. Conversely, the proposed formula features a simpler structure and exhibits superior calculation accuracy. As a result, in comparison to the existing design codes, the proposed Equation (10) offers enhanced accuracy and a broader scope of applicability.  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Design Codes** & **ACI-318** & **AISC 360–16** & **EC 4** & **AIJ** \\ \hline \(f_{c}\),\(f_{cu}\) (MPa) & \(f_{c}\geq 17.2\) & \(21\leq f_{c}\leq 69\) & \(20\leq f_{c}\leq 50\) & \(f_{c}\leq 58.8\) \\ \(f_{y}\) (MPa) & - & \(\leq\)525 & 235–460 & \(235\leq f_{y}\leq 355\) \\ \(D/\text{t}\) & \(\leq\sqrt{8E_{s}/f_{y}}\) & \(\leq\frac{0.31E_{s}}{f_{y}}\) & \(\leq 90\frac{235}{f_{y}}\) & \(\leq 90\frac{2350}{F}\) \\ \hline \end{tabular}
\end{table}
Table 4: Limitations in the design code for CFST columns.  
Figure 13: _Cont._  
\begin{table}
\begin{tabular}{c c c c} \hline
**Ref.** & **Formula** & \multicolumn{3}{c}{\(N_{\text{u,Eq}}/N_{\text{u,Exp}}\)} \\ \cline{2-4}  & **Mean** & **CV** \\ \hline This paper & \(N_{u,r}=f_{c}A_{c}+k_{r}f_{y}A_{s},k_{r}=1.58\) & 0.972 & 0.054 \\ \hline ACI-318 [45] & \(N_{u,ACI}=A_{s}f_{y}+0.85f_{c}^{\prime}A_{c}\) & 1.262 & 0.055 \\ \hline AISC 360–16 [46] & \(N_{u,AISC}=A_{s}f_{y}+0.95f_{c}A_{c}\) & 1.180 & 0.058 \\ \hline \multirow{4}{*}{BS EN 1994 [47]} & \(N_{u,EC4}=\eta_{a}A_{s}f_{y}+A_{c}f_{c}^{\prime}(1+\eta_{c}\frac{t}{D}\frac{t}{ f_{c}})\) & \multirow{4}{*}{0.954} & \multirow{4}{*}{0.055} \\  & \(\eta_{a}=0.25(3+2\overline{\lambda})\leq 1.0\), & & \\ \cline{1-1}  & \(\eta_{c}=4.9-18.5\overline{\lambda}+17\overline{\lambda}^{2}\geq 1.0\); & & \\ \cline{1-1}  & \(\overline{\lambda}=\sqrt{\frac{N_{u,Eq}}{N_{\text{u,Exp}}}}\), \(N_{\text{pl,Rk}}=A_{s}f_{y}+A_{c}f_{c}^{\prime}\) & \multirow{4}{*}{0.955} \\ \cline{1-1}  & \(N_{cr}=\frac{\pi^{2}(\overline{\lambda}_{c},I_{c}+0.6E,I_{c})}{L^{2}}\) & & \\ \hline AIJ [48] & \(\text{N}_{\text{u,AIJ}}=\gamma_{c}f_{c}^{\prime}A_{c}+(1+\eta)f_{y}A_{s}\) & \multirow{4}{*}{1.135} & \multirow{4}{*}{0.054} \\ \cline{1-1}  & \(\eta=0.27,\gamma_{c}=0.85\) & & \\ \hline \end{tabular}
\end{table}
Table 5: Circular CFST stub column design code formulas.