## 5 Fatigue Strength S-N Curve Application Case
### Stress-Strength Analysis for a Given Structural Component Based on the Probabilistic S-N curve  
For the stress-strength analysis application case, the selection's analysis of the W beam subjected to a uniform load is performed by using the available flexural strength verification method [20]. The selection case is supposed to be a left end fixed-right end free and continuously braced case. The variables for this analysis are the clear length \(L=9.00\) m, and the uniform load \(W=50.72\) KN m. After the selection process of the W beam, the selected structural component was the _W30X391_.  
Then, in order to perform the corresponding stress-strength analysis, from the static analysis and based on [PERSON] [23] methodology, the alternating stress was taken to be \(S_{a}=\sigma_{x}\), where \(\sigma_{x}\) represents the estimated normal stress, and the mean range stress value was taken to be \(S_{m}=\sigma_{\mu}\). Consequently, by considering the alternating stress \(S_{a}\) is the failure mode, and that it follows a normal distribution with arithmetic mean \(\mu_{s_{a}}=S_{a}=\sigma_{x}\) and standard deviation is given by:  
\[\sigma_{S_{a}}=0.10\ \mu_{s} \tag{18}\]  
The estimated maximum and minimum expected values are:  
\[M\text{aximum stress limit} =\ \mu_{s_{a}}+\left(\sigma_{s_{a}}\right) \tag{19}\] \[M\text{minimum stress limit} =\ \mu_{s_{a}}-\left(\sigma_{s_{a}}\right) \tag{20}\]  
Now, by using the above analysis based on the selected bending beam determination, the alternating stress values is \(\sigma_{x}=\mu_{s_{a}}=83.09\) MPa. Thus, the normal stress distribution used in the stress-strength analysis is \(N_{s}(83.09,\ 8.30)\). From Equations (19) and (20), the maximum and minimum alternating stresses are \(S_{a-max}=91.40\) MPa and \(S_{a-min}=74.78\) MPa. Hence, by using these maximum and minimum values in Equation (7), the Weibull scale parameter is \(\eta_{s}=82.68\) MPa. Similarly, by using \(\mu_{Y}=-0.545624\), and the maximum and minimum values in Equation (11), the Weibull shape parameter is \(\beta_{W}=11.14116988\). Therefore, the Weibull stress distribution used in the stress-strength analysis that represents the alternating stress behavior is \(W_{s}(11.14116988,\ 82.6829)\).  
From Section 5, the Weibull strength distribution is \(W_{S}(4.354995038,\ 170.27)\), which is found by using the reliability function for the Weibull/Weibull stress-strength analysis Pina-Monarrez [13] given by:  
\[R(t)=\frac{\eta_{S}^{\beta}}{\eta_{S}^{\beta}+\eta_{s}^{\beta}} \tag{21}\]  
where \(\eta_{S}\) and \(\beta\) are the Weibull strength parameters, and \(\eta_{s}\) is the Weibull stress eta parameter. Then, by considering the failure mode by fatigue, and by using \(\beta=4.354995038\), \(\eta_{S}=170.27\) MPa and \(\eta_{s}=82.6829\) MPa, the estimated reliability of the structural component is \(R(t)=95.87\%\).  
To summarize this application case, the selected structural component was selected by using the nominal flexural strength analysis. Then, the alternating stress was estimated, and the determination of the stress rank enabled us to determine the stress Weibull parameters, which where were used to perform the stress-strength analysis. Notice that, because the probabilistic analysis indicates that the structural component is subjected to the failure, then, in the next section, the failure theory analysis is given.