## 5 Fatigue Strength S-N Curve Application Case  
The objective of this section is performing the S-N curve for the A572 Gr. 50 steel and its corresponding probabilistic S-N curve analysis, based on the Weibull distribution. Therefore, the step by step numerical application is as follows. All strength data is given in _MPa_. _Step 1._ The selected material's features for A572 Gr. 50 [15; 17; 18] are \(S_{y}=344.73\), \(S^{\prime}_{f}=225.11\), \(S_{ut}=450.22\) and \(S_{e}=131.72\). _Step 2._ Based on the experimental data given by [PERSON] et al. (2011), the corresponding value of the strain hardening exponent parameter is \(n=0.1511\), which represents \(\epsilon_{ut}\) in the \(S_{ut}\) coordinate of the strain curve. _Step 3._ By using Equation (2), the corresponding true strain at the \(S_{ut}\) coordinate is \(\epsilon_{t}=0.1407\). _Step 4._ Based on the estimated true strain, the corresponding true stress \(\sigma_{t}=513.58\). After this true stress-true strain analysis, the numerical data for the standardized S-N curve is given below. _Step 5._ Based in the strength analysis given above, by using Equation (5), the fatigue slope value is \(b=-0.1115\). _Step 6._ Based on the determined true stress \(\sigma_{t}\) value of step 4, the corresponding proportion of the \(f\) value from Equation (6) is \(f=0.00337\). _Step 7._ Similarly, by using the determined true stress \(\sigma_{t}\) value in Equation (4), the corresponding strength for N = \(10^{3}\) and \(10^{6}\) are \(\left({S^{\prime}}_{f}\right)_{10^{3}}=220.10\) MPa and \(\left({S^{\prime}}_{f}\right)_{10^{6}}=S_{e}=131.72\) MPa. Here, by using the corresponding values of N = \(10^{3}\) and \(10^{6}\) the standardized S-N diagram is given in the Figure 2.  
Now, the probabilistic percentiles for the Weibull S-N diagram are given below. All strength data is given in _MPa_, and the log values are dimensional values based on the estimated _MPa_ data. _Step 8._ By using the estimated values in step 5 and 6, and based on the fact that the \(\left({S^{\prime}}_{f}\right)_{10^{3}}\) and \(\left({S^{\prime}}_{f}\right)_{10^{6}}\) values represent the \(S_{max}\) and \(S_{min}\) values respectively, by using Equation (7) the strength eta value is \(\eta_{S}=170.27\), and from Equation (8) the log strength mean value is \(\mu_{gS}=5.13\). _Step 9._ The desired reliability for the analysis is \(R(t)=95.35\%\). Thus, from Equation (9), \(n=21\). The generated \(Y\) elements, the corresponding mean and standard deviation are given in Table 1.  
Figure 2: S-N diagram for application case. _Step 11._ Based on the estimated \(\mu_{Y}\) and \(\sigma_{Y}\) values given in the Table 1, from Equation (11) and 12, the estimated Weibull parameters are \(\beta_{W}\) = 4.354995038 and \(\eta_{W}\) = 193.00 MPa, respectively. _Step 12._ Based on step 11 and by using Equation (14), the estimated log standard deviation value is \(\sigma_{g}\) = 0.269832. Based on the estimated Weibull \((\beta_{W},\eta_{W})\) parameters, the corresponding 95% and 5% percentiles are:  
_Step 13._ For this application case, the selected percentiles to the \((S^{\prime}{}_{f})_{10^{3}}\) values were the 95% and 5% percentiles.