## 1 Introduction  
Fatigue is a random phenomenon [1; 2] that causes a mechanical component to fail [3] at a stress level lower than the material strength limit (\(S_{c}\)) [4]. Although fatigue is random and the mechanical component is subject to variable amplitude and cyclic load [\(\sigma_{1}\), \(\sigma_{2}\)], based on single equivalent stress \(\sigma_{\alpha\beta}\) value, its failure analysis can be performed by applying a failure theory such as the Goodman, Gerber, [PERSON], and ASME (American Society of Mechanical Engineers) elliptical theories [5] (pp. 185,186,188) [6] (pp. 313-316). Depending on the specific research, different statistical models based on (stress, strain, crack growth, etc.) have been proposed to determine the cycles to failure [7; 8; 9]. Therefore, because rather than using the random behavior the analysis uses only a stress value, its application does not enable either the reliability of the component or its expected cycles to failure (\(N_{i}\)) values to be found. However, because the expected \(N_{i}\) values can be determined from the related stress-cycles S-N curve of the used material [10], and since the ultimate strength (\(S_{ut}\)) and \(S_{e}\) values represent the strength of the material used, then based on the theory given in [11], and on the \(S_{ut}\) and \(S_{e}\) values, the Weibull shape \(\beta\) and scale \(\eta\) parameters can be determined. Therefore, the novelty of the proposed method lies in that it uses the Weibull parameters to determine both the component's reliability and the expected \(N_{i}\) values. Because the reliability of the component depends on both the applied stress and the inherent strength [12; 13] in the product to overcome it, then the proposed method to determine the random behavior of \(N\) can be based on the Weibull distribution used to model the stress, and on the Weibull distribution used to model the \(N_{i}\) values. Based on the theory given in [11], the Weibull stress \(\beta_{s}\) and \(\eta_{s}\) parameters are both fitted directly from the applied maximum and minimum principal stress \(\sigma_{1}\) and \(\sigma_{2}\) values, established here by static and finite element analyses (FEA). Thus, based on the fact that the reliability of the component is completely determined for either the applied stress or for the corresponding cycles to failure, the Weibull cycle to failure (\(\beta_{t}=\beta_{s}\), \(\eta_{t}\)) parameters are found based on both the addressed (\(\beta_{s}\), \(\eta_{s}\)) parameters and on the cycles to failure N value, where N is obtained from the S-N curve by inserting the \(\sigma_{\alpha}\) that corresponds to the applied failure theory in [PERSON]'s equation. This makes the proposed method highly efficient in determining the random behavior of N and the corresponding reliability indices. This efficiency is based on the fact that the Weibull stress and the Weibull cycle distributions both converge to the same reliability index [14]. To show numerically how the proposed method works, in the practical analysis the used principal stresses \(\sigma_{1}\) and \(\sigma_{2}\) values were determined by FEA and static analyses [15]. Because the proposed method depends only on the principal stress values, but not on the complex analysis used to determine them, then in the application a simple flat spring component made of steel AISI 4340 OQT 1300 subjected to tensile cycling stress was used. Since from the FEA simulation, the addressed principal stresses values are \(\sigma_{1}=491.75\) MPa and \(\sigma_{2}=184.8\) MPa, then by applying the proposed method, the corresponding Weibull stress parameters were (\(\beta_{s}=2.248519\), \(\eta_{s}=301.455469\) MPa).