PSI - Issue 37

750 Sebastian Vetter et al. / Procedia Structural Integrity 37 (2022) 746–754 Sebastian Vetter / Structural Integrity Procedia 00 (2019) 000 – 000 5 The formulas for an analytical solution of this equation can be obtained from the literature (Liu, 2000) for the case of a planar stress state. The equivalent stress amplitude , according to the SIH given in equation (7) takes the mean stresses of the stress components into account using corresponding mean stress sensitivities for normal and shear stresses. If the mean stress sensitivity for a normal stress is known, the mean stress sensitivity for shear stress can be determined for ductile steels (Rennert et al., 2020). Also, the alternating fatigue limit ratio required for SIH can be determined according to / = 1/√3 for ductile steels. For the local strength verification, the strength must be compared with the determined stress. Therefore, the local hardness is included as a strength parameter based on the influence of the material condition. According to the literature (Garwood et al., 1951), the local hardness is transformed into the local fatigue limit using equation (8). ≅ 1.6 ± 0.1 ℎ 400 (8) The comparison of the local fatigue limit with the equivalent stress amplitude , according to equation (9) indicates whether a local failure occurs at the notch. , (9) 4. Probabilistic model Based on the influence parameters of the fatigue strength and the local strength approach, a probabilistic method for determining fatigue-strength scatter is explained using Fig. 2. This method is based on the principle of Monte Carlo simulations, in which many random experiments are conducted. First, the population of shafts for which the fatigue-strength scatter is to be estimated must be analyzed with regard to the influencing parameters. The parameters influencing the fatigue strengths of the shaft population are the external shape, the stress type, the surface condition and the material condition. The stress type is a parameter that does not influence scattering, and is consequently not considered stochastically. The scatter-influencing parameters can be described based on distributions and their location and scattering parameters. In the following, the influencing parameters are assumed to be normally distributed. In a first step, the scatters of the influencing parameters between individual shafts of the population are identified. For this purpose, distributions of the mean values of the individual shafts are described for all scattering input parameters required for the local strength approach. This means that distributions of the mean values for the geometry parameters, the residual stresses, the surface groove parameters and the local hardness are described. These distributions are estimated with the corresponding parameters of the mean of mean values ̅ ̅ and the standard deviation of the mean values ̅ . Based on these distributions, parameter values ̅ ℎ for the simulation of a single shaft are drawn at random. This single shaft can be simulated based on the randomly drawn values of the geometry parameters and the stress type using linear elastic FEA. The implemented finite-element model has a total of nodes in the failure-critical region of the notch surface, which lie on a curve in the longitudinal section of the shaft. For the calculation of the stress states of the nodes in the notch region, a nominal stress must be applied. Consequently, stress states due to external load are calculated for the nodes . In addition, each node can be assigned an area , which results from an arc length that can be assigned to each node and a circumferential length of the associated shaft cross-section. From the sum of the areas for all nodes, the shaft area ℎ that must be considered stochastically is calculated. In a parallel step, the distributions of the residual stress, the surface factor and the local hardness must be described within the shaft. For this purpose, the mean values ̅ ℎ of the residual stresses, surface groove parameters and local hardness values, which have already been randomly drawn for the single shaft, as well as the determined variation coefficients ℎ must be used. The variation coefficients of are assumed to be constant for all shafts. These parameters are assigned the areas , and , which represent the size of the corresponding measurement areas. In a further step, the considered shaft surface area ℎ is described stochastically. For this purpose, the distribution parameters of the scatter-influencing parameters of the shaft, their assigned measurement areas and the stress states obtained from the FEA as well as their assigned areas are transferred to a shaft random generator. These are stored in the vector , whereby the first element 1 contains the largest and the last element the smallest reference area. Each area fits several times into the next largest area −1 .