PSI - Issue 57

Lucas Carneiro Araujo et al. / Procedia Structural Integrity 57 (2024) 144–151 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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Variance Method (MVM) (Ferreira et al., 2022) in the history of the principal stresses, the amplitude of the principal stresses, , , can be computed.

, ,

Fig. 1. Path described by the principal stresses in combined out-of-phase loading.

The variance, a statistical measure, plays a crucial role in assessing the dispersion of events within a random process relative to their mean value. Several studies conducted by researchers have demonstrated that the fatigue damage is significantly influenced by the variance of the stress history (Ferreira et al., 2022). In this work the measure of the variance of the set of the two principal stresses, maximum and minimum, which will be called , is used to obtain the so-called amplitude of principal stresses, , , one of the determining values for the calculation of the fatigue parameter. In the following is shown the proposed methodology for calculating , , using the Maximum Variance Method. Considering the principal stresses histories (t), that can be decomposed in (t) and (t) that will be restricted to the reference plane (x,y) in plane stress state, we can define the covariance matrix expressed by Eq. 7. ∆ =[ , , ] = [ [ ( )] [ ( ) , ( )] [ ( ) , ( )] [ ( )] ] (7) In Eq. 7, [ ] represents the statistic calculation of the variance of the elements between the brackets and [ ] the calculation of the covariance of the elements between the brackets. The eigenvalues of ∆ define the maximum variance and can be obtained with Eq. 8, the eigenvectors define the orientation of the maximum variance in the analyzed plane. Finally, the equivalent value of the principal stresses amplitude can be obtained with Eq. 9. A detailed explanation of the MVM used can be found in the work of Ferreira (Ferreira et al., 2022). λ 1,2 = Δ11 + Δ22 2 ± √( Δ11 + Δ22 2 ) 2 + ( Δ12 ) 2 (8) , =√2(λ 1 +λ 2 ) (9)