# Performance Analysis of an Improved Gravity Anchor Bold Expanded Foundation
## 3 Finite Element Model
### Concrete CDP Model  
The CDP model is a continuous, plastic-based damage model. It is often used for the damage analysis and crack distribution prediction of reinforced concrete structures. The model was proposed by [PERSON] et al. [42] and then improved by [PERSON] and [PERSON] [43]. The CDP model in ABAQUS can simulate the damage evolution of concrete conveniently and accurately. It is suitable for nonlinear compressive and tensile behavior. Therefore, we adopt this model for the foundation concrete in this study. It is assumed that the CDP model can determine the tensile and compressive behavior of concrete. Figure 7 shows the specific behavior of the model. The CDP model can be used to study the tensile cracking and \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Material** & \(\rho\) (kg/m\({}^{3}\)) & **E (MPa)** & \(\
u\) & \(\kappa\) & **Unit Type** & **Material Model** \\ \hline Concrete & 2450 & 32,500 & 0.2 & / & C3D8R & CDP \\ HSGL & 2450 & 38,000 & 0.2 & / & C3D8R & elasticity \\ Reinforcement & 7850 & 200,000 & 0.3 & / & T3D2 & elasticity \\ Anchor plate & 7850 & 206,000 & 0.3 & / & C3D8R & elasticity \\ Anchor bar & 7850 & 206,000 & 0.3 & \(1.2\times 10^{-5}\) & T3D2 & elasticity \\ Flange & 7850 & 206,000 & 0.3 & / & C3D8R & elasticity \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Material parameters. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Material** & \(\rho\) (kg/m\({}^{3}\)) & **E (MPa)** & \(\
u\) & **Angle of Dilation** & **Angle of Friction** & **Cohesive Yielding** & **Absolute Plastic** \\ \hline Soil & 1800 & 50 & 0.3 & 0.1 & 30 & 50 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters of the Mohr–Coulomb model in ABAQUS 2020.  
compression failure of wind turbine foundation concrete. Equations (2) through (10) give the stress--strain relationship of concrete under uniaxial tensile load and compressive load:  
\[\sigma_{\rm t}=(1-\rm d_{t})E_{\rm c}\epsilon \tag{2}\]  
\[\rm d_{t}=\left\{ \begin{array}{l}1-\rm\rho_{t}\big{[}1.2-0.2x^{5}\big{]},\;x \leq 1\\ 1-\frac{\rm\rho_{t}}{\alpha_{\rm t}(x-1)^{1-x}+x}\quad,x>1\end{array} \right. \tag{3}\]  
\[\rm\rho_{t}=\frac{f_{tr}}{E_{\rm c}\epsilon_{tr}} \tag{4}\]  
\[\rm x\,=\frac{\epsilon}{\epsilon_{tr}} \tag{5}\]  
\[\rm\sigma_{c}=(1-\rm d_{c})E_{c}\epsilon \tag{6}\]  
\[\rm d_{c}=\left\{ \begin{array}{l}1-\frac{\rm\rho_{c}n}{n-1+x^{n}},x\leq 1 \\ 1-\frac{\rm\rho_{c}}{\alpha_{\rm c}(x-1)^{2+x}},x>1\end{array} \right. \tag{7}\]  
\[\rm\rho_{c}=\frac{f_{c,r}}{E_{c}\epsilon_{cr}} \tag{8}\]  
\[\rm n\,=\frac{E_{c}\epsilon_{cr}}{E_{c}\epsilon_{cr}-f_{c,r}} \tag{9}\]  
\[\rm x\,=\frac{\epsilon}{\epsilon_{cr}} \tag{10}\]  
\(\rm\alpha_{t/c}\)--parameter of the descending section of concrete uniaxial tensile/compressive stress-strain curve, given in the Code for Design of Concrete Structures in the United States, China, and other countries [39];  
\(\rm f_{t/cr}\)--concrete uniaxial tensile/compressive strength;  
\(\rm\epsilon_{t/cr}\)--peak tensile/compressive strain of concrete corresponding to;  
\(\rm d_{t/c}\)--evolution parameter of concrete uniaxial tensile/compressive damage. According to the energy equivalence hypothesis [44], the stress-strain relationship of materials can be expressed in terms of Equations (11) and (12). By substituting Equation (11) into Equation (2), Equation (13) can be obtained. By substituting (12) into (6), Equation (14) can be obtained. Figure 8 shows the stress-strain relationship and damage-strain relationship of the C40 concrete under uniaxial compression and tension. The specific damage  
Figure 7: (**a**) Compressive behavior and (**b**) tensile behavior. evolution parameters \(\rm D_{t}\) and \(\rm D_{c}\) with the CDP model are input into ABAQUS for further calculation. The calculation is easier to converge when \(\rm D_{t}\) and \(\rm D_{c}\) are used [45]. Table 4 shows some of the properties defined in ABAQUS for establishing the CDP model.