## 7 Conclusions  
As a result of this study, closed-form solutions for plastic limit pressure (\(P_{\text{L}}\)) and stress intensity factor (\(K_{\text{I}}\)) for elbows containing a part-through axial semi-elliptical crack were obtained. It can be concluded that as the \(c/a\) ratio increases, its impact on the plastic limit pressure and stress intensity factor becomes more significant. By analyzing the numerical results through a symbolic regression algorithm, closed-form solutions for the limit pressure and stress intensity factor were proposed (Equations (1) and (2)). These expressions enable fast calculation of stress intensity factor and limit pressure, which can be used for failure assessment procedures, eliminating the need for time-consuming finite element analysis (FEA) determination.  
The FAD diagram (Figure 11) was constructed by using SINTAP Standard option 1A, a material expected to display a yield plateau. The ligament yielding parameter, \(L_{\text{r}}\), representing the ratio of internal working pressure to plastic limit pressure held the value close to 0.2 across different crack dimensions. This is because the points in the safe domain of the FAD diagram have a similar limit pressure value, while the working pressure remains constant. Additionally, failure pressure was determined by identifying the points where the loading paths intersect the FAD function.  
Our approach goes beyond considering the elbow as an isolated case. Instead, it accounts for the interaction between elbows, pipe supports, and other pipeline components, ensuring a more realistic assessment of pipeline integrity. This comprehensive view provides greater safety margins and better reflects the complexities of real operating conditions.  
While the current FAD diagram was constructed using FEA results, the closed-form expressions developed here are expected to give comparable results, thus providing a reliable and efficient tool for assessing safe operational conditions. Although the regression models have shown a strong fit within the tested range, future work should explore the application of these methods to various pipeline configurations, including different dimensions, loading conditions, and materials. Further study could explore the models' behavior beyond the tested crack sizes and include parameters such as pipe thickness and radius, which may further increase their accuracy and applicability. It is also essential to validate these models against experimental data or other analytical solutions to ensure their robustness in different scenarios. The inclusion of other factors, such as temperature fluctuations, corrosion effects, and dynamic loading, can further improve the models' accuracy and extend their application to a wider range of realistic scenarios. Preparing experiments for pipe elbows with cracks is complex and poses significant challenges. Validating the FEA results through experiments on pipe elbows without cracks is a practical approach. Future work should focus on this direction, aiming to bridge the gap between theoretical predictions and practical validation. These findings are crucial for an accurate and efficient integrity assessment of the piping system, providing a reliable basis for safety evaluations and maintenance planning.  
Conceptualization, B.D. and P.K.; Investigation, B.D.; Methodology, P.K. and M.K.; Software, P.K.; Supervision, P.K. and M.K.; Validation, M.K.; Writing--original draft, B.D. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
Data Availability StatementData are available upon request to the corresponding author. The data are not publicly available due to privacy.  
The authors declare no conflicts of interest.