## 3 Steps of the Proposed Method  
The novelty of the proposed method consists of considering the probabilistic behavior in bilinear fatigue analysis and to generate its corresponding P-S-N field. The material used for the analysis is 42 CrMo4 steel. Since the material S-N curve represents median values, then here the analysis is on based on the median stress value of the collected data (see Table 1), and on the Weibull parameters in cycles that are equivalent to this median stress value, \(W(\beta=4.8031,\ \eta=1445.7\) cycles). Consequently, by considering \(\beta\) to be constant, the P-S-N percentiles were determined around \(\eta=1445.7\) cycles. However, because for each estimated P-S-N percentile, a unique eta (\(\eta_{i}\)) and \(R(t_{i})\) value exists, the confidence reliability percentile (CL) value that represents the difference between the \(\eta_{i}\) value and \(\eta_{i}=1445.7\) cycles was also determined. Additionally, the formulation to relate the \(CL\) value with the P-S-N percentile is also given. Therefore, a summary that shows the P-S-N percentile, the \(CL\) value, the \(R(t)\) index and the minimum and maximum \(\eta_{i}\) values for the one, two, and three sigma levels, as well as those corresponding to the capability \(Cp\) and ability \(Cpk\) indices is given. Step 0. Collect experimental fatigue data. Collected data must contain failure times and their corresponding stress value. Step 1. Separate data into two homogeneous sets \(s_{1}\) and \(s_{2}\). To ensure that the separation is accurate, you can perform the following steps to identify the knee-point, which will help you to confirm that the points in each set are correctly categorized. Step 1.1. Determine the optimal slopes associated with each set. To achieve this, the failure times must be arranged in ascending order based on the number of cycles to failure. Following this, the optimal slopes (\(A_{1}\) and \(A_{2}\)) and intersections (\(B_{1}\) and \(B_{2}\)) with the vertical axis for each data set are determined for each value using the least squares method. \[A_{1}=\frac{m\sum_{i=1}^{m}(\log(S_{i})\cdot\log(N_{i}))-\sum_{i=1}^{m}(\log( S_{i}))\cdot\sum_{i=1}^{m}(\log(N_{i}))}{m\sum_{i=1}^{m}\left(\log(N_{i})^{2} \right)-\left(\sum_{i=1}^{m}\log(N_{i})\right)^{2}}\]  
\[A_{2}=\frac{(n-m)\sum_{i=m+1}^{n}(\log(S_{i})\cdot\log(N_{i}))-\sum_{i=m+1}^{ n}(\log(S_{i}))\cdot\sum_{i=m+1}^{n}(\log(N_{i}))}{(n-m)\sum_{i=m+1}^{n} \left(\log(N_{i})^{2}\right)-\left(\sum_{i=m+1}^{n}\log(N_{i})\right)^{2}}\]  
\[B_{1}=\frac{\sum_{i=1}^{m}\log(S_{i})-A_{1}\sum_{i=1}^{m}\log(N_{i})}{m}\]  
\[B_{2}=\frac{\sum_{i=m+1}^{n}\log(S_{i})-A_{2}\sum_{i=m+1}^{n}\log(N_{i})}{n-m}\]  
where \(m\) is the total number of elements of \(s_{1}\) and \(n\) is the total number of elements performed. \begin{table}
\begin{tabular}{l c c c c c c c c c c c c c c c c} \hline \hline \multicolumn{1}{c}{Cycles} & \multicolumn{1}{c}{26} & 260 & 280 & 300 & 750 & 770 & 1050 & 1270 & 5000 & 7700 & 7500 & 17,230 & 18,400 & 27,400 & 81,400 & 80,000 & 152,000 & 156,000 & 290,2000 \\ \hline Status & 800.4 & 842.5 & 894.8 & 933.7 & 832.8 & 709.6 & 703.1 & 709.6 & 679.2 & 665.3 & 668.8 & 649.6 & 637.3 & 639.5 & 596.7 & 596.7 & 596.4 & 553.3 & 558.7 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Fatigue test of 42 CrMo4 steel. Step 1.2. Determine the knee-point. Using the values obtained in Step 1.1, identify the point where the slope changes by applying the equations:  
\[\log(N_{knee})=-\frac{B_{1}-B_{2}}{A_{1}-A_{2}}\]  
\[S_{knee}=A_{1}\cdot\log(N_{knee})+B_{1}\]  
where \(N_{knee}\) is in cycles and \(S_{knee}\) is in stress. The elements above the knee-point belong to \(s_{1}\), while those below are assigned to \(s_{2}\). Step 2. Determine the data set that provides the best fit between time and stress: For each set, determine the multiple regression coefficient \(R^{2}\), and select the data set with the highest \(R^{2}\) value as the basis for performing the P-S-N analysis. Step 3.