## 4 Analysis of the Bearing Capacity and Crack Width of SCGCs
### Bearing Capacity
#### 4.2.2 Model Optimization
And \(OA\) corresponds to the parabolic portion of the theoretical stress block, while \(AB\) represents the rectangular portion of the theoretical stress block:  
\[OA=\frac{\varepsilon_{0}}{\varepsilon_{\text{cu}}}x=\frac{0.0024}{0.0028}x= \frac{6}{7}x \tag{9}\]  
\[AB=\left(1-\frac{\varepsilon_{0}}{\varepsilon_{\text{cu}}}\right)x=\frac{1}{7 }x \tag{10}\]  
The resultant compressive force \(D\) of the concrete can be determined from the area of the theoretical stress block:  
\[D=D_{1}+D_{2}=\frac{2}{3}f_{\text{c}}\frac{24}{28}xb+f_{\text{c}}\frac{4}{28} xb=0.714 xb \tag{11}\]  
Figure 27: Equivalent stress. (**a**) Equivalent compressive stress of concrete (**b**) Strain distribution of column section. According to the method of calculating the static moment of area, the distance from the resultant compressive force point of the concrete to the compression edge of the section can be determined:  
\[\frac{1}{2}\beta_{1}x=\frac{f_{\mathrm{c}}b\bigg{(}\frac{2}{3}.\frac{24}{28}x \bigg{)}\bigg{(}\frac{4}{28}x+\frac{3}{8}\frac{24}{28}x\bigg{)}+f_{\mathrm{c}}b \frac{4}{28}\bigg{(}\frac{1}{2}\cdot\frac{4}{28}x\bigg{)}}{0.714 xb}=0.38x \tag{12}\]  
The value of \(\beta_{1}\) is determined to be 0.76. The resultant compressive force \(D\) in the concrete, based on the area of the equivalent rectangular stress distribution, can be calculated as:  
\[D=\alpha_{1}f_{\mathrm{c}}\beta_{1}xb=0.76\alpha_{1}\varphi_{0}xb \tag{13}\]  
Equation (14) can be obtained from the equivalence principle. \[0.76\alpha_{1}f_{\mathrm{c}}xb=0.714f_{\mathrm{c}}xb \tag{14}\]  
Hence, the value of \(\alpha_{1}\) is determined to be 0.94. For subsequent calculations, \(\alpha_{1}\) is taken as 0.94, and \(\beta_{1}\) is taken as 0.76. On the other hand, as discussed in Section 3.3, only the SSR that is farther from the axial force reaches the yield point under the large eccentricity failure pattern of the SCGC. This differs from the assumption in the JGJ12-2006 [41] code, which states that both the tensile and compressive reinforcements yield. However, the mechanical behavior of the SSR is similar to that described in the JGJ12-2006 [41] code under the small eccentricity failure pattern of the SCGC. This paper derived Equations (15) and (16) based on the plane section assumption and force equilibrium to address this issue, in conjunction with Figure 27. The bearing capacity problem of large and small eccentric compression columns was resolved by solving these equations simultaneously with Equations (5) and (6). \[\frac{\varepsilon_{\mathrm{t}}}{\varepsilon_{\mathrm{cu}}}=\frac{h_{0}-x_{ \mathrm{c}}}{x_{\mathrm{c}}}\rightarrow\varepsilon_{\mathrm{t}}=\varepsilon_ {\mathrm{cu}}\bigg{(}\frac{h_{0}}{x_{\mathrm{c}}}-1\bigg{)} \tag{15}\]  
\[\frac{\varepsilon_{\mathrm{c}}}{\varepsilon_{\mathrm{y}}}=\frac{x_{\mathrm{c} }-a_{\mathrm{s}}^{\prime}}{h_{0}-x_{\mathrm{c}}}\rightarrow\varepsilon_{ \mathrm{c}}=\varepsilon_{\mathrm{y}}\bigg{(}\frac{x_{\mathrm{c}}-a_{\mathrm{s }}^{\prime}}{h_{0}-x_{\mathrm{c}}}\bigg{)} \tag{16}\]  
This paper utilized the JGJ12-2006 [41] code in conjunction with the modified Ramberg-Osgood model to calculate the bearing capacity (\(N_{\mathrm{u}}\)\({}^{\mathrm{S}}\)) of specimens SCGC1 to SCGC7, based on the aforementioned model revisions. The average ratio of \(N_{\mathrm{u}}\)\({}^{\mathrm{S}}\) to the measured bearing capacity \(N_{\mathrm{u}}\) was found to be 0.94, with a coefficient of variation of 0.034. The calculation results are shown in \[AB=\left(1-\frac{\varepsilon_{0}}{\varepsilon_{\text{cu}}}\right)x=\frac{1}{7 }x \tag{10}\]0 In summary, the revised bearing capacity calculation model, combined with the modified Ramberg-Osgood model, provides a reliable prediction of the bearing capacity of SCGC.