## 4 Analysis of the Bearing Capacity and Crack Width of SCGCs  
### Constitutive Relationship of CAGC  
The uniaxial constitutive model comprehensively reflects the mechanical behavior of concrete under uniaxial loading and plays a crucial role in the design and analysis of reinforced concrete structures [37]. When investigating the uniaxial constitutive relationship of CAGC, the compressive results of prismatic specimens from Section 2.1 were compared with studies by [PERSON] [38] and [PERSON] [39], as shown in Figure 21.  
Figure 21: The stress–strain curves of CAGC, CAC, OAC, and LAC. Refs. [36; 37].  
Figure 20: Concrete strain distribution diagram of midspan section.  
It can be observed that in the ascending segment, the curves for CAC, ordinary aggregate concrete (OAC), and lightweight aggregate concrete (LAC) are almost identical to the CAGC curve. However, in the descending segment, the CAGC curve is steeper than the OAC and LAC, but flatter than the CAC. Overall, the constitutive relationship curve of CAGC significantly differs from that of CAC, OAC, and LAC, indicating that their stress-strain relationships are not applicable to CAGC.  
Therefore, this paper proposes an optimized uniaxial constitutive model for CAC, based on the work of [PERSON] [38] and [PERSON] [39], as shown in Equation (1). A fitting analysis was conducted, and the results are presented in Figure 22.  
\[\frac{\sigma}{f_{\mathrm{c}}}=\left\{\begin{array}{ll}\left[a\left(\frac{ \varepsilon}{\varepsilon_{0}}\right)+\left(3-2a\right)\cdot\left(\frac{ \varepsilon}{\varepsilon_{0}}\right)^{2}+\left(a-2\right)\cdot\left(\frac{ \varepsilon}{\varepsilon_{0}}\right)^{3}\right]&0\leq\varepsilon<\varepsilon_ {0}\\ \varepsilon/\varepsilon_{0}&\varepsilon_{0}\leq\varepsilon\leq\varepsilon_{ \mathrm{cu}}\end{array}\right.\quad R^{2}=0.991 \tag{1}\]  
where \(\sigma\) is the stress of CAGC, in MPa; \(f_{\mathrm{c}}\) is the axial compressive strength of CAGC, with a measured value of \(f_{\mathrm{c}}\) = 29.0 MPa in this study; \(\varepsilon\) is the strain of CAGC; \(\varepsilon_{0}\) is the compressive strain of CAGC when the compressive stress reaches \(f_{\mathrm{c}}\), with \(\varepsilon_{0}\) = 0.0024; \(\varepsilon_{\mathrm{cu}}\) is the ultimate compressive strain of CAGC, taken as \(\varepsilon_{\mathrm{cu}}\) = 0.0028; and \(a\) and \(b\) are the control parameters for the ascending and descending segments of the curve, with values of \(a\) = 1.31 and \(b\) = 7.892, respectively.  
This constitutive model is considered for practical engineering applications, with attention to the safety reserve requirements of the project. Notably, the stress in the \(\varepsilon_{0}\) - \(\varepsilon_{\mathrm{cu}}\) range does not significantly decrease. This paper adopts a horizontal line for the descending segment and truncates it at \(\varepsilon_{\mathrm{cu}}\), referring to the provisions of GB50010-2010 [40] on post-peak concrete behavior, as shown in Figure 23. The resulting uniaxial constitutive relationship expression for CAGC is:  
\[\frac{\sigma}{f_{\mathrm{c}}}=\left\{\begin{array}{ll}\left[a\left(\frac{ \varepsilon}{\varepsilon_{0}}\right)+\left(3-2a\right)\cdot\left(\frac{ \varepsilon}{\varepsilon_{0}}\right)^{2}+\left(a-2\right)\cdot\left(\frac{ \varepsilon}{\varepsilon_{0}}\right)^{3}\right]&0\leq\varepsilon<\varepsilon_ {0}\\ f_{\mathrm{c}}&\varepsilon_{0}\leq\varepsilon\leq\varepsilon_{\mathrm{cu}} \end{array}\right. \tag{2}\]  
Figure 22: Normalized stress–strain curves.