PSI - Issue 38

424 Luca Vecchiato et al. / Procedia Structural Integrity 38 (2022) 418–427 L. Vecchiato et al./ Structural Integrity Procedia 00 (2021) 000–000 7 where is the Root Mean Square stress value of the random process. Accordingly, the number of exceeding cycles N of a stationary zero-mean narrow-band Gaussian random process can be obtained from the complementary cumulative distribution function, or exceedance, of the Rayleigh distribution:

2

1 2   −     a rms σ

N

σ

∞

∫

∫

1 = −

( ) p d

( ) p d e ξ ξ =

a

ξ ξ

=

(12)

N

0

σ

a

0

where N is the exceedance number of cycles, i.e. the number of cycles whose stress amplitude is higher than or equal to and N 0 is the total number of cycles, i.e. the length of the spectrum. Theoretically the Rayleigh distribution is defined for values ranging from 0 to infinity, but in practice peaks do not exceed a certain value of / called clipping ratio. Haibach et al. (Haibach et al. 1976) proposed to set the number of exceeding cycles N of the maximum stress level (clipping ratio) equal to one (in other words this means that the highest stress level is applied only once in the spectrum). Consequently, the clipping ratio becomes a function of the length of the spectrum according to Eq. (13). (13) By substituting the value from Eq. (13) into Eq. (12), the number of exceeding cycles N can be rewritten as a function of the relative stress amplitudes σ a / σ a,max or of the relative stress ranges ∆σ / ∆σ max as reported in Eq. (14) (Gaßner et al. 1964; Hanke 1970; Heuler et al. 2005): 2 ,max 1 2 0 1 a rms e N σ     −   = → ,max 0 2 ln N a rms σ = → ,max 0 2 ln a N = rms σ

2

   

   

σ  ∆ 

1

ln N

− 

 

0

σ

∆



N e =

max

(14)

In the present work, the length of the spectrum has been fixed to N 0 = 10

4 cycles (Fig. 4).

a)

b)

1.00

N

∆σ / ∆σ max

n

[-]

[cycles] [cycles]

0.75

1.000 0.864 0.727 0.591 0.455 0.318

5

5

72

77

∆σ / ∆σ max

0.50

569 2313 4416 2625

646 2959 7375 10000

Stepped Spectrum p = 0.25 p = 0 (Gaussian)

0.25

N 0

0.00

1E+00

1E+01

1E+02

1E+03

1E+04

Number of exceeding cycles, N

Fig. 4. Stress range spectrum: (a) comparison between a standard Gaussian spectrum (p = 0), a p-type spectrum with p = 0.25, and the applied stepped spectrum; (b) the applied relative nominal stress ranges ∆σ / ∆σ max with their corresponding number of cycles n and exceeding cycles N.

Then, once the exceedance cycle distribution is known, to obtain the p-type spectrum the amplitude of the stress ranges of the Gaussian spectrum (p = 0) has to be amplified by substituting p = 0.25 into the following expression (Gurney 2006):