PSI - Issue 75

Sgamma M. et al. / Procedia Structural Integrity 75 (2025) 709–718

713

Author name / Structural Integrity Procedia 00 (2025) 000–000

5

m γ 2 √ m γ 0 m γ 4

5( I − K 3 + K 2 R ) 4 K 1 ,

Q =

I =

,

and m γ k denotes the k -th spectral moment of the shear strain PSD:

m γ k =

∞

k df , k ∈ { 0 , 1 , . . . , 4 } .

G γ ( f ) f

(7)

0

(a)

(b)

Fig. 1: Observations on the joint distribution from multiaxial rainflow: The typical shape of the amplitude distribution of a Gaussian random process can be seen (a) and a normal distribution is observed towards the maximum stress values (b).

According to numerical observations performed via the multiaxial rainflow method, if the random load is narrow band and its components are fully correlated, each shear strain amplitude typically pairs with a single, simultaneous maximum normal stress. Conversely, broadband random loads exhibiting partial or no correlation produce a broader spread of possible maximum normal stress values for each strain amplitude, leading to a two-dimensional distribu tion of ( γ a ,σ n , max ) (see Figure 1). This distribution can be approximated by coupling the Dirlik-based PDF for strain amplitudes with a statistical model (e.g., a normal distribution) for the peak stresses:

p γσ ( γ a ,σ n , max ) = ϕ σ n , max ; ϕ µ ,ϕ σ p D ( γ a ; m γ ) ,

(8)

where ϕ ( σ n , max ; ϕ µ ,ϕ σ ) is the normal distribution with mean ϕ µ and standard deviation ϕ σ , and p D ( γ a ; m γ ) follows the Dirlik amplitude model. The mean and variance of σ n , max can change depending on factors such as the overall bandwidth and correlation level. For instance, an explicit representation might be: