## 4 Results
The results presented in Table 2 are used to plot the variation in stress intensity factor and plastic limit pressure against crack length and pipe thickness ratio (\(a/t\)), with respect to the crack dimension ratio (\(c/a\)). These plots are illustrated in Figures 7 and 8. As expected, the observed plots demonstrate two trends: as the \(a/t\) ratio increases, the stress intensity factor increases, while the plastic limit pressure decreases. There is also a trend of an increased slope for the stress intensity factor with a higher \(c/a\) ratio, while for plastic limit pressure, the slope decreases with a higher \(c/a\) ratio. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(c/a\) & \(a\) (mm) & \(c\) (mm) & \(a/t\) & \(K_{\rm I}\) (MPa\(\sqrt{\rm mm}\)) & \(P_{\rm L}\) (MPa) & \(P_{\rm LS}\) (MPa) \\ \hline \multirow{4}{*}{0.5} & 0.84 & 0.42 & 0.15 & 49.811 & 17.7 & 19.62 \\  & 1.68 & 0.84 & 0.30 & 71.584 & 17.6 & 19.62 \\  & 2.52 & 1.26 & 0.45 & 89.724 & 17.35 & 19.60 \\  & 3.36 & 1.68 & 0.60 & 106.6 & 17.1 & 19.54 \\  & 4.2 & 2.1 & 0.75 & 123.044 & 16.9 & 19.36 \\ \hline \multirow{4}{*}{1} & 0.84 & 0.84 & 0.15 & 56.137 & 17.6 & 19.61 \\  & 1.68 & 1.68 & 0.30 & 82.088 & 17.5 & 19.60 \\  & 2.52 & 2.52 & 0.45 & 106.439 & 17.35 & 19.52 \\  & 3.36 & 3.36 & 0.60 & 131.995 & 17.15 & 19.29 \\  & 4.2 & 4.2 & 0.75 & 159.593 & 16.9 & 18.63 \\ \hline \multirow{4}{*}{1.5} & 0.84 & 1.26 & 0.15 & 61.258 & 17.75 & 19.62 \\  & 1.68 & 2.52 & 0.30 & 91.864 & 17.3 & 19.57 \\  & 2.52 & 3.78 & 0.45 & 120.197 & 17.15 & 19.39 \\  & 3.36 & 5.04 & 0.60 & 148.292 & 16.95 & 18.91 \\  & 4.2 & 6.3 & 0.75 & 183 & 16.55 & 17.59 \\ \hline \multirow{4}{*}{2} & 0.84 & 1.68 & 0.15 & 68.386 & 17.7 & 19.61 \\  & 1.68 & 3.36 & 0.30 & 103.613 & 17.6 & 19.53 \\  & 2.52 & 5.04 & 0.45 & 138.654 & 17.4 & 19.23 \\  & 3.36 & 6.72 & 0.60 & 175.518 & 17.2 & 18.42 \\  & 4.2 & 8.4 & 0.75 & 211. To obtain closed-form expressions from finite element analysis (FEA) results, a statistical symbolic regression analysis was performed by using TuringBot software v2.19 [47]. TuringBot is an advanced tool designed for symbolic regression. Unlike traditional linear regression, symbolic regression attempts to find the best-fitting equation by testing different mathematical expressions, which can be nonlinear or involve multiple interactions among the variables This makes it well suited for complex engineering applications where the relationships between variables, such as crack dimensions and stress responses, are not purely linear. Crack propagation and plastic limit pressure in materials often involve nonlinearity, especially in cases of complex geometries like semi-elliptical cracks. This approach enabled the generation of closed-form expressions for calculating the stress intensity factor \(K_{\text{I}}\) (MPa\(\sqrt{\text{mm}}\)) and plastic limit pressure \(P_{\text{L}}\) (MPa), where \(a\) and \(c\) are measured in millimeters (mm):  
\[K_{\text{I}}=(7.8071+1.49956\times a)\times(6.94365-(3.49515/(a\times(c+1.76901)) )+c) \tag{1}\]  
Figure 8: Plastic limit pressure plot regarding \(a/t\) and \(c/a\) ratio. Figure 7: Stress intensity factor plot regarding \(a/t\) and \(c/a\) ratio. \[P_{\rm L}=17.4989\,-\,(0.00745158\,\times\,(a\times a+(c-((-2.24832)/(-2.21352+(c/a)))) )\,\times\,c) \tag{2}\]  
The quality of the regression models was evaluated using two key parameters, the R-squared (\(R^{2}\)) and the root mean square (RMS) error. For stress intensity factor \(K_{\rm I}\) the \(R^{2}\) is 0.998518, showing that the regression model explains 99.85% of the variance in the FEA results. The RMS error for \(K_{\rm I}\) is 1.90855, indicating that there is an average deviation of the predicted values from the observed FEA results. For the limit pressure \(P_{\rm L}\), the regression model observed an R-squared value of 0.981582, meaning that the model explains 98.16% of the variance in the FEA results. The RMS error for \(P_{\rm L}\) is 0.1885232, showing a high level of accuracy in the predictions.