Applying Goodman, Gerber, Soderberg and Elliptical failure theories does not make it possible to determine the span of failure times (cycles to failure-$N_i$) of a mechanical element, and so in this paper a fatigue-life/Weibull method to predict the span of the $N_i$ values is formulated. The input’s method are: (1) the equivalent stress (${\sigma }_{eq}$) value given by the used failure theory; (2) the expected $N_{eq}$ value determined by the Basquin equation; and (3) the Weibull shape β and scale η parameters that are fitted directly from the applied principal stress ${\sigma }_1$ and ${\sigma }_2$ values. The efficiency of the proposed method is based on the following facts: (1) the β and η parameters completely reproduce the applied ${\sigma }_1$ and ${\sigma }_2$ values. (2) The method allows us to determine the reliability index R(t), that corresponds to any applied ${\sigma }_{1i}$ value or observed $N_i$ value. (3) The method can be applied to any mechanical element’s analysis where the corresponding ${\sigma }_1$ and ${\sigma }_2$, ${\sigma }_{eq}$ and $N_{eq}$ values are known. In the performed application, the ${\sigma }_1$ and ${\sigma }_2$ values were determined by finite element analysis (FEA) and from the static stress analysis. Results of both approaches are compared. The steps to determine the expected $N_i$ values by using the Weibull distribution are given.

In this paper, a Weibull probabilistic methodology is proposed with an approach to model vibration fatigue damage accumulation using two parameters: Weibull distribution and a nonlinear fatigue damage accumulation model. The damage is cumulated based on the application of a vibration stress profile and is used to determine both the Weibull $\beta$ and $\eta$ parameters, and the corresponding component reliability R(t). The vibration fatigue damage is analyzed to accumulate the damage as a stress function for a fatigue life exponent derived with the assistance of the acceleration’s force response. The steps to determine the Weibull $\beta$ and $\eta$ parameters are estimated based only on the principal vibration stresses ${\sigma }_1$ and ${\sigma }_2$ that allow the reproduction of the vibration fatigue damage. The method’s efficiency is based on the probabilistic approach by using the vibration fatigue damage as the Y_i vector that covers the arithmetic mean as well as the $\beta$ parameter. Finally, the procedure proposed is applied in a practical case where a mechanical component is used as a support for telecommunication connections and is submitted to vibration stress. The results show that using the damage accumulated as the Y_i vector to estimate the parameters allows for the analysis of dynamic and individual applications.

The paper’s content allowed us to determine the fatigue life of a component that is being subjected to a random vibration environment. Its estimation is performed in the frequency domain with loading frequencies being closer to the system’s natural frequency. From loads’ amplitude and their interaction effect, we derive a nonlinear damage model to cumulate the generated fatigue damage. The exponent value of 0.4 from the Manson–Halford curve damage model was replaced by a vibration bending stress relation that considers the effect and interaction of loads. The analysis is performed from a progressive accelerated vibration spectrum to predict the fatigue life estimation. From this accelerated scenario, the accelerated coefficients and cumulated damage are both determined. The proposed nonlinear model is based on the following facts: (1) vibration and bending stress ${\sigma }_{vb}$ values are obtained from the response acceleration of power spectral density (PSD) applied and (2) the model can be applied to any mechanical component analysis where the corresponding acceleration responses $A_{res}$ and the dynamic load factor ${\sigma }_{dynamic}$ values are known. The steps to determine the expected fatigue damage accumulation D by using the curve damage are given.