## 4 Theoretical Calculation and Analysis
### Material Constitutive Model  
#### 4.1.1 Recycled Aggregate Concrete  
There are three kinds of constitutive model of concrete commonly used in ABAQUS: concrete diffusion crack model, concrete brittle cracking model and concrete damage plastic model [35]. The concrete damage plastic model considers the effects of damage has on the material and can adequately simulate the behavior and response of concrete under monotonic loading. Therefore, in this paper, the plastic damage model was used to simulate the concrete. The fracture properties will not be taken into account in the model. Figure 12: Effect of different parameters on the center deflection. (**a**) Effect of material types; (**b**) Effect of reinforcement ratio; (**c**) Effect of reinforcement form. 1. Tension and pressure injury  
The plastic damage model for concrete in ABAQUS assumes that the concrete material is mainly subjected to tensile cracking and crushing. The yield and failure were respectively determined by equivalent tensile plastic strain \(\widetilde{\varepsilon}_{t}^{pl}\) and equivalent compressive plastic strain \(\widetilde{\varepsilon}_{c}^{pl}\). The tensile damage curve of concrete transforms the cracking strain into plastic damage strain by the destroying stress and cracking strain, while the compression damage curve of concrete was calculated by further transforming inelastic strain into plastic strain. The two formulas are shown in Equations (1) and (2). \[\widetilde{\varepsilon}_{t}^{pl}=\widetilde{\varepsilon}_{t}^{ck}-\frac{d_{t}} {(1-d_{t})}\frac{\sigma_{t}}{E_{0}} \tag{1}\]  
\[\widetilde{\varepsilon}_{c}^{pl}=\widetilde{\varepsilon}_{c}^{in}-\frac{d_{c} }{(1-d_{c})}\frac{\sigma_{c}}{E_{0}} \tag{2}\]  
where \(\widetilde{\varepsilon}_{t}^{ck}\) is the cracking tensile strain, \(\widetilde{\varepsilon}_{t}^{ck}=\varepsilon_{t}-\varepsilon_{0t}^{el}\), \(\varepsilon_{0t}^{el}=\sigma_{t}/E_{0}\), among them \(\varepsilon_{0t}^{e1}\) is the elastic tensile strain of the non-destructive material, and \(\varepsilon_{t}\) is the total tensile strain. \(\widetilde{\varepsilon}_{c}^{in}\) is the inelastic compression strain, where \(\widetilde{\varepsilon}_{c}^{in}=\varepsilon_{c}-\varepsilon_{0c}^{el}\), \(\varepsilon_{0c}^{el}=\sigma_{c}/E_{0}\), among them \(\varepsilon_{0c}^{e1}\) is the elastic compression strain of the non-destructive material, \(\varepsilon_{c}\) is the total compression strain, \(d_{t}\) is the tension damage variable, and \(d_{c}\) is the compression damage variable. 2. Uniaxial compressive and tensile stress-strain curves  
In this paper, the compressive constitutive equation of RCA under uniaxial pressure proposed by [PERSON] [36] was used, as shown in Equation (3). Figure 13 shows the uniaxial compressive stress-strain curves of the recycled aggregate concrete. \[y= \begin{cases}\frac{ax+(3-2a)x^{2}+(a-2)x^{3},0\leq x<1\\ \frac{x}{b(x-1)^{2}+x}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \,x\geq 1 \end{cases}  \tag{3}\]  
where \(x=\varepsilon/\varepsilon_{0}\)\({}^{\prime}\), \(y=\sigma/\varepsilon_{c}\)\({}^{\prime}\); \(a\) is the coefficient at the ascending stage of the curve and \(a=2.2(0.748\ r^{2}-1.231\ r+0.957)\), taken as \(a=1.05\), among them \(r\) is the recycled coarse aggregate replacement ratio, where \(r=100\%\). \(b\) is the coefficient at the descending stage of the curve and \(b=0.8(7.6483\ r+1.142)\), taken as \(b=7.03\). The peak compressive strain of the recycled concrete prism is \(\varepsilon^{e}\)\({}_{0}\), \(1.2\) times the peak compressive strain of ordinary concrete, and \(f_{\rm c}^{r}\) is the compressive strength of the recycled concrete prism and taken as \(f_{\rm c}^{r}=0.67\,f_{\rm cu}\), where \(f_{\rm cu}\) is the measured value of the concrete cube compressive strength (Table 4). Figure 13: Compression stress-strain curve of recycled concrete. The model hypothesis of the tensile stress-strain principle is valid if the sample is without damage before reaching the peak stress. The descending curve can be calculated using Equation (4) [33].