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.

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.

In this paper a Weibull methodology to determine the probabilistic percentiles for the S-N curve of the A572 Gr. 50 steel is formulated. The given Weibull/S-N formulation is based on the true stress and true strain values, which are both determined from the stress-strain analysis. For the analysis, the Weibull β and η parameters are both determined directly from the maximum and minimum addressed stresses values. The S-N curve parameters are determined for 10³ and 10⁶ cycles. In the application, published experimental data for the CSA G40.21 Gr. 350W steel is used to derive the true stress and true strain parameters of the A572 Gr. 50 steel. Additionally, the application of the S-N curve, its probabilistic percentiles and the Weibull parameters that represent these percentiles are all determined step by step. Since the proposed method is flexible, then it can be applied to determine the probabilistic percentiles of any other material.

El problema en la determinación de los esfuerzos normales en una sección transversal utilizando como base la deflexión elástica, se basa en el hecho de que las metodologías existentes aún presentan carencias en su análisis. El artículo presenta un análisis de los esfuerzos normales desarrollados a partir de las cargas aplicadas sobre el elemento estructural y el desarrollo de un caso de aplicación. Asimismo, debido a que la deflexión de un elemento depende de las cargas aplicadas, entonces el análisis de los esfuerzos está basado en la deflexión elástica del componente estructural. Además, la selección del elemento estructural se basa en la normatividad de diseño de vigas para componentes estructurales. Por otro lado, el análisis del material para hacer un diseño de un componente estructural también se presenta en este artículo. Igualmente, el material presentará desgaste debido a las cargas aplicadas, entonces se realiza un análisis de fatiga basado en los esfuerzos normales.

Based on the principal stress values generated by the bending beam, the material’s strength required at ${10}^6$ cycles is determined depending on time. To determine the stress/strength reliability (R(t)), the stress distribution is determined directly from the range of the principal stresses values, and the strength distribution is determined based on the reduced tensile strength ($S{\prime }_e$) and fatigue strength ($S_e$) range. Therefore, based on the time-dependent stress and the material’s strength, a step-by-step method to determine the reliability R(t) of the structural element at ${10}^6$ cycles is provided. The R(t) index is used to select the best among the feasible beam alternatives of the static/elastic and plastic methodologies. The method’s efficiency is based on the time-dependent stress analysis performed by using the elastic modulus, and corresponding strain as time dependence variables. Because the time-dependent stress is related to the changes of the bending deflection through time, it is determined based on the addressed equivalent stress at ${10}^6$ cycles.