## 4 Analysis of the Bearing Capacity and Crack Width of SCGCs
### Bearing Capacity  
#### 4.2.1 Calculation and Analysis of \(N_{\rm u}\)  
The calculation of \(N_{\rm u}\) in this study is based on the following assumptions: (1) The SGCC satisfies the plane section assumption during loading (verified in Section 3.3); (2) The tensile strength of CAGC is not considered in the calculation; (3) The concrete stress in the compression zone of the SGCC section is calculated using an equivalent rectangular stress distribution; (4) With a slenderness ratio \(l/b=1000/250=4\), the test columns are considered short columns, and second-order effects are neglected in the calculation. This paper establishes formulas for different failure patterns and load-bearing capacity based on these assumptions. As shown in Figure 24, the failure patterns of the SGCC are classified into large eccentric compression and small eccentric compression. In these formulas, \(e_{\rm i}\), \(e_{\rm t}\), and \(e_{\rm c}\) represent the distances between the external load and the section center, the tensile SSR, and the compressive SSR, respectively; \(\sigma_{\rm s}\) and \(\sigma_{\rm s}\)\({}^{\prime}\) are the SSR stresses far from (\(A_{\rm s}\)) and near (\(A_{\rm s}\)\({}^{\prime}\)) the external load; \(\alpha_{1}\) is the rectangular stress coefficient; \(x_{\rm c}\), \(x_{\rm b}\), and \(x\) denote the actual height of the CAGC in compression, the limiting height of the compression zone, and the calculated height, where \(x=\beta_{1}x_{\rm c}\); \(\varepsilon_{\rm t}\) and \(\varepsilon_{\rm y}\) are the strains of the SSR far from the external load and its nominal yield strain, respectively; and \(\varepsilon_{\rm c}\) and \(\varepsilon_{\rm cu}\) are the strains of the SSR near the external load and at the section edge, respectively. Figure 23: Compression stress–strain curve of CAGC. Figure 24: Force diagram of column section. (**a**) Column section (**b**) Large eccentric compression mode (**c**) Small eccentric compression mode. When the external load acts on the SCGC with a small \(e_{0}\), the SCGC section will experience a small eccentric compression, as shown in Figure 24b. At this time, the concrete in the compression zone reaches its compressive strength, and the steel reinforcement on the side farther from the axial force has not reached its yield state, whether it is in compression or tension. Therefore, based on the plane section assumption and force equilibrium, the following formula can be derived:  
\[N_{\rm u}e_{\rm t}=\alpha_{1}f_{\rm c}bx(h_{0}-x/2)+f_{\rm y}^{\prime}A_{\rm s} ^{\prime}(h_{0}-a_{\rm s}^{\prime}) \tag{3}\]  
\[N_{\rm u}=\alpha_{1}f_{\rm c}bx+f_{\rm y}^{\prime}A_{\rm s}^{\prime}-\sigma_{ \rm s}A_{\rm s} \tag{4}\]  
where \(f_{\rm y}^{\prime}\) is the design value of the compressive strength of the SSR, in MPa. When \(e_{0}\) is large, the SCGC section will experience large eccentric compression, as shown in Figure 24b. The steel reinforcements that are farther from the axial force reach the yield state, and the concrete in the compression zone reaches its compressive strength. However, it is verified that the steel reinforcement that is closer to the axial force has not reached its yield state based on the strain measurements of the SSR. Therefore, based on the plane section assumption and force equilibrium, the following formula can be derived:  
\[N_{\rm u}e_{\rm t}=\alpha_{1}f_{\rm c}bx(h_{0}-x/2)+\sigma_{\rm s}^{\prime}A_ {\rm s}^{\prime}(h_{0}-a_{\rm s}^{\prime}) \tag{5}\]  
\[N_{\rm u}=\alpha_{1}f_{\rm c}bx+\sigma_{\rm s}^{\prime}A_{\rm s}^{\prime}-f_{ \rm y}A_{\rm s} \tag{6}\]  
where \(f_{\rm y}\) is the design value of the compressive strength of the SSR, in MPa. Determining the compression state directly influences the force calculations, as observed from the analysis. Additionally, the relative boundary compression zone height \(\xi_{\rm b}\) serves as a key criterion for distinguishing between large and small eccentric failures.