## 5 Conclusions  
1. Although in practice, bilinear behavior is mainly attributed to the interaction of two competitive failure modes, this paper focuses on the bilinear behavior of the S-N curve for the 43 CrMo4 steel material. The proposed methodology, however, can be applied to analyze any type of bilinear behavior.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & **Normal** & \(k_{a}\) & \(\eta\)**U** & \(\eta\)**L** & **CL** & **R(t)** \\ \cline{2-7}  & **Percentiles** & \(k_{a}\) & \(\eta\)**U** & \(\eta\)**L** & **CL** & **R(t)** \\ \hline
1\(\sigma\) & 0.6827 & 0.4752623 & 1480.6 & 1411.5 & 0.6742 & 0.9732 \\ Cpk & 0.9082 & 1.3297520 & 1545.6 & 1352.2 & 0.7480 & 0.9781 \\ Cp & 0.9525 & 1.6695926 & 1572.2 & 1329.3 & 0.7761 & 0.9798 \\
2\(\sigma\) & 0.9545 & 1.6901461 & 1573.8 & 1327.9 & 0.7777 & 0.9799 \\
3\(\sigma\) & 0.9973 & 2.7821505 & 1662.6 & 1257.0 & 0.8588 & 0.9846 \\ Test plan & 0.0880 & \(-1.3531742\) & 1547.4 & 1350.6 & 0.7500 & 0.9783 \\ \hline \hline \end{tabular}
\end{table}
Table 9: Normal and Weibull data with their corresponding \(R(t)\) indices.  
2. By applying the proposed methodology, the Weibull distribution modeling the linear behavior is identified and used to design the test plan to validate the required reliability index. For this analysis, the median stress is used but any value of stress can be used.
3. The application of the methodology allows us, for any desired percentile of the normal distribution, to determine the elements corresponding to the upper and lower limits of the Weibull scale parameter, their equivalent confidence level, and their reliability index.
4. The developed test plan highlights the importance of reliability and confidence level in determining the required sample size.
5. The proposed methodology enables the determination of the confidence level (\(CL\)) corresponding to any desired percentile, and vice versa.
6. It is very important to note that the normal and Weibull percentiles depend on the Weibull \(\beta\) value. Therefore, an accurate estimation of \(\beta\) is critical in the analysis. Therefore, once it is determined, the estimation of the predicted failure times will be efficient.
7. As part of future research, it is recommended to explore the sensitivity of the addressed Weibull family to more accurately determine the expected behavior of \(\beta\) and the variance of \(\eta\).  
Conceptualization, O.M.-Q., M.R.P.-M. and M.M.H.-R.; methodology, O.M.-Q., M.R.P.-M. and J.F.O.-Y.; formal analysis, O.M.-Q., M.R.P.-M. and J.F.O.-Y.; writing--original draft preparation, O.M.-Q. and M.R.P.-M.; writing--review and editing, O.M.-Q., M.R.P.-M. and M.M.H.-R.; supervision, M.R.P.-M.; funding acquisition, O.M.-Q. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Not applicable.  
The data presented in this study are available in [17].  
We appreciate the support of Conahcyt and the Autonomous University of Ciudad Juarez (UACJ) provided in this research.  
Author [PERSON] was employed by the company TED DE MEXICO SA DE CV. The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.