## 3 Results and Discussion
### Regression Analysis  
In this study, regression analyses were also employed to investigate the relationship between dependent and independent variables. A regression analysis based on numerical data was conducted for the safety factors of the autofrettaged cylinder without machining. The following equation defines the regression model. \[SF_{1}=0.4405+1.0040\;S_{y}-0.000192\;E+0.003416\;T \tag{19}\]  
A regression analysis based on numerical data was applied to the safety factors of the autofrettaged cylinder with only internal machining. The equation below defines the regression model. \[SF_{2}=0.43030+0.84555\;S_{y}-0.000073\;E+0.001499\;T \tag{20}\] \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Order**} & \multirow{2}{*}{\begin{tabular}{c} **Young’s** \\ **Modulus (GPa)** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Tangent** \\ **Modulus (GPa)** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Yield Strength** \\ **(GPa)** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Factor of Safety** \\ **(Numerical)** \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} **Factor of Safety** \\ **(Analytical)** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **Deviation** \\ **\%** \\ \end{tabular} } \\ \hline
8 & 200 & 10 & 1.3 & 1.64 & 1.58 & 3.36 \\ \hline
9 & 300 & 0 & 1.3 & 1.61 & 1.57 & 2.75 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Cont. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Order**} & \multirow{2}{*}{\begin{tabular}{c} **\(E\) (GPa)** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **\(T\)** \\ **(GPa)** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **\(S_{y}\)** \\ **(GPa)** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Without** \\ **Machine** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Internal** \\ **Machine** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **External** \\ **Machine** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Mechaning** \\ **Machine** \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} **Mechaning** \\ **\%** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **Mechaning** \\ **%** \\ \end{tabular} } \\ \hline
1 & 100 & 0 & 0.7 & 1.12 & 1.01 & 1.06 & \(-9.82\) & \(-5.36\) \\ \hline
2 & 200 & 20 & 0.7 & 1.17 & 1.04 & 1.11 & \(-11.11\) & \(-5.13\) \\ \hline
3 & 300 & 10 & 0.7 & 1.13 & 1.02 & 1.07 & \(-9.73\) & \(-5.31\) \\ \hline
4 & 100 & 10 & 1 & 1.46 & 1.29 & 1.39 & \(-11.64\) & \(-4.79\) \\ \hline
5 & 200 & 0 & 1 & 1.41 & 1.26 & 1.34 & \(-10.64\) & \(-4.96\) \\ \hline
6 & 300 & 20 & 1 & 1.45 & 1.28 & 1.38 & \(-11.72\) & \(-4.83\) \\ \hline
7 & 100 & 20 & 1.3 & 1.81 & 1.56 & 1.72 & \(-13.81\) & \(-4.97\) \\ \hline
8 & 200 & 10 & 1.3 & 1.72 & 1.52 & 1.64 & \(-11.63\) & \(-4.65\) \\ \hline
9 & 300 & 0 & 1.3 & 1.69 & 1.51 & 1.61 & \(-10.65\) & \(-4.73\) \\ \hline \hline \end{tabular}
\end{table} 
Table 12: The effect of internal and external machining processes on the safety factor. A regression analysis based on the numerical data was conducted for the safety factors of the autofrettaged cylinder with only external machining. The equation below defines the regression model. \[SF_{3}=0.4053+0.9607\;S_{y}-0.000161\;E+0.003426\;T \tag{21}\]  
Upon examining the equations derived from the regression analysis, it is observed that \(E\) is negatively signed in all equations. This indicates that \(E\) has a decreasing effect on the response. By evaluating the magnitudes of the coefficients preceding each parameter in the equations, we can determine the relative influence of each parameter on the response. Specifically, \(S_{y}\) has the most significant impact on the safety factor, followed by \(T\), and finally, \(E\). In practical terms, this means that increasing the yield strength of the material will result in the most substantial improvement in the cylinder's ability to withstand pressure, while changes in the tangent modulus and Young's modulus will have progressively lesser effects.