## 3 Proposed Method
### Weibull Cycle Family Estimation and Its Random Behavior
#### 3.3.1 Estimation of the \(N\) Value that Correspond to the Equivalent Stress Value  
The Weibull analysis of the component is performed based on the dynamic loads. That is why finding the corresponding cycles to failure occurs as follows. Depending on the component's material, the yield stress \(S_{y}\) and ultimate stress \(S_{ut}\) values are obtained. Then the corresponding endurance limit \(S_{e}\) value is calculated as:  
\[S_{e}^{\prime}=0.5(S_{ut}) \tag{23}\]  
Then based on the correction factor values, the corrected endurance limit is determined as:  
\[S_{e}=K_{a}K_{b}K_{c}K_{d}K_{e}K_{f}S_{e}^{\prime} \tag{24}\]  
where, \(K_{a}\) = surface condition modification factor, \(K_{b}\) = size modification factor, \(K_{c}\) = load modification factor, \(K_{d}\) = temperature modification factor, \(K_{e}\) = reliability factor and \(K_{f}\) = miscellaneous-effects modification factor. After that, using Equations (25) and (26) and the values for principal stresses \(\sigma_{1}\) and \(\sigma_{2}\) from Section 2.1, the corresponding mid-range stress \(\sigma_{m}\) and the alternating stress \(\sigma_{a}\) values are set.  
\[\sigma_{m}=\ \frac{(\sigma_{1}+\sigma_{2})}{2} \tag{25}\]  
and,  
\[\sigma_{a}=\ \frac{(\sigma_{1}-\sigma_{2})}{2} \tag{26}\]  
Then, by using the \(S_{y}\), \(S_{e}\), \(\sigma_{a}\) and \(\sigma_{m}\) values in the used failure theory criterion, determine whether the designed element is safe or not. (Although as shown in Figure 1, there are several criteria, the one used here was the ASME Elliptic criterion). Thus, to determine the fatigue safety factor \(n_{f}\), in the case of the ASME elliptical theory criterion, the \(n_{f}\) function is:  
\[n_{f}=\sqrt{\frac{1}{\left(\frac{\sigma_{a}}{S_{e}}\right)^{2}+\left(\frac{ \sigma_{m}}{S_{y}}\right)^{2}}} \tag{27}\]If \(n_{f}\) is higher than one (\(n_{f}>1\)), the designed element is considered to have an infinite life (\(N=\infty\)). Otherwise, it is considered to have a finite life. However, because the aspect being worked with is the probabilistic behavior of the stress, what proceeds is to determine the corresponding reversed equivalent stress \(\sigma_{\alpha\beta}\) value [27]. This is obtained from the ASME elliptical failure theory criterion represented by:  
\[\left(\frac{\sigma_{\alpha}}{S_{e}}\right)^{2}+\left(\frac{\sigma_{m}}{S_{y}} \right)^{2}=1 \tag{28}\]  
The \(\sigma_{\alpha\beta}\) is placed instead of the \(S_{e}\) element into Equation (28); thus, its value is set as:  
\[\sigma_{\alpha\beta}=\frac{\sigma_{a}}{\sqrt{1-\left(\frac{\sigma_{a}}{S_{y}} \right)^{2}}} \tag{29}\]  
Since the calculation of the fatigue constant a and b parameters of Basquin's equation is required, such a calculation is undertaken by using the fatigue strength factor \(f\) that is taken from [6], the ultimate stress \(S_{ut}\), and the endurance limit \(S_{e}\) values.  
\[a=\ \frac{(fS_{ut})^{2}}{S_{e}} \tag{30}\]  
and,  
\[b=\ -\frac{1}{3}log\left(\frac{fS_{ut}}{S_{e}}\right) \tag{31}\]  
Then, by inserting the \(a\), \(b\) and \(\sigma_{\alpha\beta}\) estimated values into Equation (32), the expected average cycles to failure \(N\) value is obtained:  
\[N=\left(\frac{\sigma_{\alpha\beta}}{a}\right)^{\frac{1}{b}} \tag{32}\]  
From Equation (32), the \(N\) value that corresponds to the equivalent stress is set. That value will be used to determine the corresponding Weibull cycle parameters.