## 4 Axial Compression Bearing Capacity
### Bearing Capacity Calculation Model
Based on assumption (2), the calculation formula for the bearing capacity of the unreinforced column and the filled concrete (\(N_{0}\)) was proposed as follows:  
\[N_{0}=0.9\varphi\Big{[}f_{co}A_{co}+f_{yo}^{\prime}A_{so}^{\prime}+\alpha(f_{ sc}A_{sc}+f_{y}^{\prime}A_{s}^{\prime})\Big{]} \tag{2}\]  
where \(\varphi\) is the stability coefficient of members, taking the value according to code [26]; \(f_{co}\) and \(f_{sc}\) are the strength of the column concrete and filled concrete, respectively; \(A_{co}\) and \(A_{sc}\) are the cross-sectional areas of the column and filled concrete, respectively;\(f_{yo}\)\({}^{\prime}\) and \(f_{y}\)\({}^{\prime}\) are the strength values of steel bars in the column and filled concrete, respectively; \(A_{yo}\)\({}^{\prime}\) and \(A_{y}\)\({}^{\prime}\) are the cross-sectional areas of steel bars in column and filled concrete, respectively; \(\alpha\) is the utilization coefficient of the filled concrete. According to [24] and [PERSON]'s study [30], the value of \(\alpha\) should be determined by the interface bonding between the column and filled concrete. Assuming that the influencing factors of the interface bonding are the roughness, concrete strength of the column, filled concrete strength, and the usage of an interface agent, the calculation formula of \(\alpha\) is as follows:  
\[\alpha=A(0.15+0.025\Delta)\ln f_{c}\zeta_{s} \tag{3}\]  
where \(A\) is the adjustment coefficient considering the preloading of concrete in the filled concrete, which is an undetermined constant; \(\Delta\) is the roughness of the interface when \(\Delta\geq 2.5\), taking \(\Delta=5\) mm;\(f_{sc}\) is the average strength of the column concrete and filled concrete; \(\zeta_{s}\) is the influence coefficient of the interfacial agent, which is an undetermined coefficient. Based on assumption (3), the calculation formula for the bearing capacity improved by the confinement (\(N_{0}\)) was proposed as follows:  
\[N_{s}=\beta\theta N_{0} \tag{4}\]  
\[\beta=B(\frac{f_{l}}{f_{co}})^{0.7} \tag{5}\]\[\theta=\frac{A_{gsc}f_{gt}+A_{gy}f_{gy}+A_{sc}f_{st}+A^{\prime}_{s}f^{\prime}_{y}}{ A_{co}f_{co}} \tag{6}\]  
where \(\theta\) is the confinement coefficient when \(\theta\geq 0.25\), taking \(\theta\) = 0.25; \(\beta\) is the adjustment coefficient related to the concrete strength of the SSAWC, the LCSS, and the column; \(B\) is an undetermined constant; \(f_{st},f_{gt}\), and \(f_{gy}\) are the tensile strengths of the SSAWC, the LCSS concrete, and steel bars reinforcement, respectively; \(A_{gc}\) and \(A_{gy}\) are the cross-sectional areas of the sleeve concrete and steel bars, respectively; the value of \(f_{l}\) takes the minimum of \(f_{sc}\) and \(f_{gc}\). Based on assumption (4), the calculation formula for the bearing capacity improvement caused by self-stress (\(N_{0}\)) was proposed as follows:  
\[N_{p}=\gamma N_{0} \tag{7}\]  
\[\gamma=C\eta^{3}+D\eta^{2}+E\eta \tag{8}\]  
where, \(\gamma\) is the improvement coefficient; \(\eta\) is the self-stress level, which is the ratio of the longitudinal self-stress value to the SSAWC strength; \(C\), \(D\), and \(E\) are coefficients of cubic equations to be solved. To sum up, the calculation formula of axial the compression bearing capacity of the components strengthened by the IPCSAM is as follows:  
\[N=0.9\varphi\Big{[}f_{co}A_{co}+f^{\prime}_{yo}A^{\prime}_{so}+\alpha(f_{sc} A_{sc}+f^{\prime}_{y}A^{\prime}_{s})\Big{]}(1+\beta\theta+\gamma) \tag{9}\]