## 4 FE Analysis and Result
### Material Model
#### 4.2.2 Concrete
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Steel Component** & **Density** & \begin{tabular}{c} **Young’s** \\ **Modulus,** \\ \(E_{s}\) **(GPa)** \\ \end{tabular} & \begin{tabular}{c} **Poisson’s** \\ **Ratio,** \(v\) \\ \end{tabular} & \begin{tabular}{c} **Yield** \\ **Stress,** \\ \(\sigma_{ys}\) **(MPa)** \\ \end{tabular} & \begin{tabular}{c} **Ultimate** \\ **Stress,** \\ \(\sigma_{us}\) **(MPa)** \\ \end{tabular} & \begin{tabular}{c} **Elastic** \\ **Strain,** \(\varepsilon_{us}\) \\ \end{tabular} & \begin{tabular}{c} **Plastic** \\ **Strain,** \(\varepsilon_{us}\) \\ \end{tabular} \\ \hline \begin{tabular}{c} Welded stud \\ (S1–S3) \\ Stud bolt \\ (S4) \\ \end{tabular} & \(7.8\times 10^{-6}\) & 205.51 & 0.3 & 248.81 & 429.59 & 0.0012 & 0.011 & 0.3966 \\ \begin{tabular}{c} Stud bolt \\ (S4) \\ \end{tabular} & \(7.8\times 10^{-6}\) & 210.00 & 0.3 & 330.00 & 540.00 & 0.0016 & 0.035 & 0.2500 \\ \
\begin{tabular}{c} Rebar \\ H beam \\ \end{tabular} & \(7.8\times 10^{-6}\) & 192.76 & 0.3 & 542.57 & 578.46 & 0.0024 & 0.011 & 0.1811 \\
\begin{tabular}{c} H beam \\ \end{tabular} & \(7.8\times 10^{-6}\) & 210.00 & 0.3 & 316.53 & 479.17 & 0.0015 & 0.011 & 0.3077 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Summary of steel component properties. Figure 14: Schematic stress–strain relationship for concrete material. concrete after cracking linearly decreased to 0. It is possible to obtain a numerical solution more easily by using the tension stiffening effect. [PERSON] and [PERSON] [4] used a strain of 0.005 (\(\beta=50\)) at a stress of 0 to avoid unstable behavior during analysis, and the same value was used in this study. To verify the push-out tests through FE analysis, we used the CDP model, which can be used in the ABAQUS material library. This material model is suitable for materials that have different tensile and compressive strengths. In this material model, the tension and compression of the yield part of the concrete stress-strain curve are handled separately. Table 5 shows each of the variable values in the CPD model that was used in this study. We utilized values that were well-suited to the overall push-out experiment results by performing additional variable analysis on the dilation angle and \(K\). In addition, Figure 15 depicts the damage model for concrete compression and tension in ABAQUS. The principle behind the damage parameters is described by the stress-strain diagram under non-elastic conditions, as shown in this figure. The compression failure conditions (\(\varepsilon_{c}^{in}\)) and amount of concrete strain during cracking (\(\varepsilon_{t}^{ck}\)) were calculated using Equations (4)-(7) based on Figure 15.  
\[\varepsilon_{c}^{in}=\varepsilon_{c}-\varepsilon_{c0}^{el}=\varepsilon_{c}- \frac{\sigma_{c}}{E_{c0}} \tag{4}\]  
\[\varepsilon_{c}^{el}=\varepsilon_{c}-\varepsilon_{c}^{pl}=\varepsilon_{c}-b_{ c}\varepsilon_{c}^{in} \tag{5}\]  
\[\varepsilon_{c}^{ck}=\varepsilon_{t}-\varepsilon_{t}^{ck}=\varepsilon_{t}- \frac{\sigma_{t}}{E_{c0}} \tag{6}\]  
\[\varepsilon_{t}^{el}=\varepsilon_{t}-\varepsilon_{t}^{pl}=\varepsilon_{t}-b_{ t}\varepsilon_{t}^{in} \tag{7}\]  
The concrete damage parameters are divided into compressive stress (\(d_{c}\)) and tensile stress (\(d_{l}\)), and they are each shown as functions of stress and plastic strain in Equations (8) and (9). Each damage parameter has a constant value that is \(d_{c}>0\) and \(d_{t}\leq 1.0\). Here, value 0 means that there is no concrete damage, and 1.0 means complete failure. Figure 15: Concrete damage parameters: (**a**) compression; (**b**) tension. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & **Dilation Angle** & **Eccentricity** & \(f_{b0}/f_{c0}\) & \(K\) & **Viscosity** \\ \hline Value & 36 & 0.1 & 1.16 & 0.59 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: ABAQUS/Explicit parameters in a CDP model.