## 4 Numerical Analysis  
In parallel to the experimental studies as illustrated above, finite element (FE) analyses have been carried out using ABAQUS software to support the experimental findings. FE model was developed to analyze the compressive behavior of confined and unconfined columns. FE model is shown in Figure 8. Solid element C3D8R is used for concrete and PVC pipe. The boundary conditions are shown in Figure 9. The load control (pressure load) simulates the compression load on columns. Contact interaction was defined between the inner surface of the PVC pipe and the external side surface of concrete columns, as shown in Figure 10. Hard contact is applied in the normal direction, and penalty friction is in the tangential direction with a friction coefficient of 0.3. The loading process of the sample was by applying a pressure load on the upper face of the sample. The pressure was applied by selecting smooth step amplitude in ABAQUS software (version 6.6). A concrete damage plasticity model is used for concrete. Equations (1)-(3) are used to develop the parameters of concrete damage plasticity, which are adopted from [34; 35]. The stress-strain curves for the tension and compression of concrete are shown in Figures 11 and 12. Compression and tension damage of concrete are shown in Figures 13 and 14.  
\[\varepsilon_{c}=(0.71\ f^{\prime}c+168)*10^{-5} \tag{1}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Sample’s Radius (mm)** & **Improvement Using Cc \%** & **Improvement Using Ccc** \\ \hline
110 & 8.3 & 14.9 \\
160 & 8.1 & 12.6 \\
220 & 6.0 & 12.2 \\
250 & 5.0 & 4.2 \\ \hline \hline \end{tabular}
\end{table}
Table 10: The improvement of compressive strength using PVC tubes at 28 days.  
Figure 7: Mode of failure (**a**) unconfined samples (**b**) fully confined samples (**c**) confined with cut ends.  
For \(0<\varepsilon<\varepsilon^{\prime}{}_{0}\),  
\[\frac{\sigma}{\sigma_{u}}=2\frac{\varepsilon}{\varepsilon^{\prime}{}_{0}}\left(1- \frac{\varepsilon}{2\varepsilon^{\prime}{}_{0}}\right) \tag{2}\]  
\[\frac{\sigma}{\sigma_{u}}=1-0.15\bigg{(}\frac{\varepsilon-\varepsilon^{\prime}{ }_{0}}{\varepsilon_{u}-\varepsilon^{\prime}{}_{0}}\bigg{)} \tag{3}\]  
Figure 8: Meshed model of the confined column with cut ends of PVC.  
Figure 9: Loading and boundary conditions.  
Figure 11: Stress-strain curve for compression stress in concrete.  
Figure 12: Stress-strain for tension stress in concrete.  
Figure 10: Interaction between PVC inner surface and side surface of the concrete.  
Figure 16: Relative error distribution for each diameter using the FE approach for all scenarios.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Sample’s Radius** & **Strength of Cu** & **Strength of Ce** & **Strength of Ccc** \\
**(mm)** & **(MPa)** & **(MPa)** & **(MPa)** \\ \hline
110 & 15.66 & 16.27 & 18.42 \\
160 & 16.77 & 17.61 & 18.58 \\
220 & 18.57 & 19.56 & 20.14 \\
250 & 20.33 &  
Figure 17: The failure mode of 110 mm samples with max principle stress. (**a**) before failure **(b**) after failure.  
Figure 18: The failure mode of 160 mm samples with max principle stress. (**a**) before failure (**b**) after failure.  
Figure 19: The failure mode of 220 mm samples with max principle stress. (**a**) before failure (**b**) after failure.