## 4 Conclusions  
1. Because the generated internal stresses are always related to the bending beam, and since the proposed method uses these stress values to determine the stress distribution, then the reliability of any structural element can always be determined by applying the proposed method. 2. Because the applied stresses are independent of the material strength, then the proposed method can be used as a complement to the classic design methodology of structural elements based on the reliability index to discriminate among the possible addressed alternatives. 3. Because the proposed method can be performed for any desired cycles, from the element selection to the reliability determination, then it can be used as a guideline in any structural analysis. Although here the analysis was performed at \(10^{6}\) cycles (or at the \(S_{e}\) value), it can be performed at any desired cycles. 4. Note that as in Table 5, when the \(S_{e}\) value is lower than the expected equivalent stress \(\sigma_{eq-i}\), then both the plastic and fracture of cracked mechanical components analysis should be performed. 5. Note because the less used modified factors the higher the \(S_{e}\) value, then for environments where fewer modifier factors are needed, the designed beam reliability index will be higher. 6. As the principal stress values on which the proposed method is based are the eigenvalues of a quadratic form represented by the stress matrix, the proposed method can be used to determine the reliability of an analyzed element in any field where the eigenvalues are known. As is the case for a principal components analysis and a quality analysis where they are performed using a second-order polynomial model. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Summary of W Beams Data** & **W Beam Selection** & **W Beam Selection** & **W Beam Selection** \\ \hline
**DATA** & **30X357** & **33X291** & **33X318** \\ \hline _Step 1_. Supported reaction \(\mathrm{Ay}\) (kN) & 451.89 & 443.22 & 446.77 \\ _Step 1_. Maximum bending moment (kN-m) & 2033.51 & 1994.50 & 2010.45 \\ _Step 2_. Max vertical displacement \(\delta_{max}\) (in) & 0.8192 & 0.8955 & 0.8030 \\ _Step 3_. Normal stress \(\sigma_{x}\) (MPa) & 91.17 & 105.74 & 95.91 \\ _Step 3_. Normal stress \(\sigma_{y}\) (MPa) & 64.84 & 67.47 & 68.79 \\ _Step 3_. Normal shear stress \(\tau_{xy}\) (MPa) & 1.98 & 1.94 & 1.95 \\ _Step 4_. Mean stress \(\sigma_{\mu}\) (MPa) & 78.00 & 86.61 & 82.35 \\ _Step 4_. Alternating stress \(\sigma_{a}\) (MPa) & 13.16 & 19.14 & 13.56 \\ _Step 4_. Principal stress \(\sigma_{1}\) (MPa) & 91.31 & 105.84 & 96.05 \\ _Step 4_. _Principal stress \(\sigma_{2}\) (MPa) & 64.69 & 67.37 & 68.65 \\ _Step 4_. Maximum shear stress \(\tau_{xy}\) (MPa) & 13.31 & 19.23 & 13.70 \\ _Step 5_. Time-dep modulus \(E_{i}\) (GPa) & 107.44 & 122.49 & 111.98 \\ _Step 5_. Time-dep strain \(\varepsilon_{i}\) at \(N_{f}=10^{6}\) & 0.000849 & 0.000864 & 0.0008577 \\ _Step 5_. Time-dep deflection \(\delta_{max}\) (m) & 0.04112 & 0.03955 & 0.03879 \\ _Step 5_. Defined time analysis (years) & 13.95 & 10.50 & 10.50 \\ _Step 6_. Fatigue strength limit \(S_{e}\) (MPa) & **117.30** & **119.89** & **118.74** \\ _Step 5_. _Equivalent stress_ (MPa) & **181.01** & **184.03** & **182.69** \\ _Step 10_. Stress-strength reliability & 0.15\% & 0.17\% & 0.16\% \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Selected structural elements based on the proposed stress-strength method. **Author Contributions:** Conceptualization, A.M., M.R.P.-M., S.T.d.l.C.-C., J.M.B.-C.; methodology, A.M., M.R.P.-M.; data analysis, A.M., S.T.d.l.C.-C.; writing-original draft preparation, A.M., M.R.P.-M.; writing-review and editing, A.M., M.R.P.-M., S.T.d.l.C.-C., J.M.B.-C.; supervision, M.R.P.-M.; funding acquisition, A.M., M.R.P.-M. All authors have read and agreed to the published version of the manuscript. **Funding:** This research received no external funding. **Institutional Review Board Statement:** Not applicable. **Informed Consent Statement:** Not applicable.