# The Mechanical Properties of Reinforced Concrete Columns with Longitudinal Pre-Embedded Holes
## 2 Finite Element Model Building
### Material Constitutive Model  
#### 2.2.1 Concrete Constitutive  
This study employs the Concrete Damaged Plasticity (CDP) model in ABAQUS for numerical simulation analysis. The CDP model reduces the elastic stiffness matrix of concrete by introducing damage factors, representing a composite constitutive model that combines plasticity theory with linear damage mechanics. The CDP model primarily considers two failure modes of concrete: crushing failure under compression and cracking failure under tension. The constitutive relationship follows the stress-strain curves for tension and compression as outlined in Appendix C of the \"Code for Design of Concrete Structures\" (GB50010-2010) [25], as shown in Figure 2.  
The stress-strain curve for concrete under uniaxial compression is determined according to Equations (1) and (2):  
\[\sigma=(1-d_{c})E_{c}\varepsilon \tag{1}\]  
\[d_{c}=\left\{\begin{array}{ll}1-\frac{\rho_{t}n}{n-1+x^{n}}&x\leq 1\\ 1-\frac{\rho_{c}}{\alpha_{c}(x-1)^{2}+x}&x>1\end{array}\right. \tag{2}\]  
where \(x=\frac{\varepsilon}{\varepsilon_{c,r}}\); \(n=\frac{E_{c,r}}{E\varepsilon_{c,r}-f_{c,r}}\); \(\rho_{c}=\frac{f_{c,r}}{E\varepsilon_{c,r}}\).  
where \(f_{c,r}\)--the characteristic value of the uniaxial compressive strength of concrete;  
\(\varepsilon_{c,r}\)--the peak compressive strain of concrete corresponding to \(f_{c,r}\);  
\(\alpha_{c}\)--the parameter defining the descending branch of the uniaxial compressive stress-strain curve of concrete;  
\(d_{c}\)--the damage evolution parameter for concrete under uniaxial compression.  
The stress-strain curve for concrete under uniaxial tension is determined according to Equations (3) and (4):  
\[\sigma=(1-d_{t})E_{c}\varepsilon \tag{3}\]  
\[d_{t}=\left\{\begin{array}{ll}1-\rho_{t}\big{[}1.2-0.2x^{5}\big{]}&x\leq 1 \\ 1-\frac{\rho_{t}}{\alpha_{c}(x-1)^{1/2}+x}&x>1\end{array}\right. \tag{4}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Type** & **Diameter (mm)** & \begin{tabular}{c} **Yield Strength \(f_{y}\)** \\ **(MPa)** \\ \end{tabular} & \begin{tabular}{c} **Ultimate Strength \(f_{st}\)** \\ **(MPa)** \\ \end{tabular} & \begin{tabular}{c} **Elastic Modulus \(E_{s}\)** \\ **(MPa)** \\ \end{tabular} \\ \hline HRB400 & \begin{tabular}{c} 25 \\ 22 \\ 8 \\ 10 \\ \end{tabular} & \begin{tabular}{c} 400 \\ 300 \\ \end{tabular} & \
\begin{tabular}{c} 540 \\ 420 \\ \end{tabular} &
\begin{tabular}{c} 2.0 \(\times\) 10\({}^{5}\) \\ \(2.1\times\) 10\({}^{5}\) \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table}
Table 2: Mechanical properties of reinforcement.  
Figure 2: Stress–strain curve of concrete.  
where \(x=\frac{c}{\varepsilon_{t,r}}\); \(\rho_{t}=\frac{f_{t,r}}{E_{t,r}}\).  
where \(f_{t,r}\)--the characteristic value of the uniaxial tensile strength of concrete;  
\(\varepsilon_{t,r}\)--the peak tensile strain of concrete corresponding to \(f_{c,r}\);  
\(\alpha_{t}\)--the parameter defining the descending branch of the uniaxial tensile stress-strain curve of concrete;  
\(d_{t}\)--the damage evolution parameter for concrete under uniaxial tension.  
The remaining parameters of the CDP model are listed in the Table 3, where \(\varphi\) represents the dilation angle, \(e\) denotes the eccentricity of the plastic potential function, \(f_{\bar{\psi}0}/f_{\alpha}\) is the ratio of biaxial to uniaxial compressive strength, \(K\) is the ratio of the second stress invariant on the tensile meridian to that on the compressive meridian, and \(\mu\) signifies the viscosity parameter. These parameters are crucial for characterizing the plastic flow, failure surface, and damage evolution of concrete within the CDP model.