PSI - Issue 24

Filippo Cianetti et al. / Procedia Structural Integrity 24 (2019) 526–540 Author name / Structural Integrity Procedia 00 (2019) 000–000

536

11

• the stress is Gaussian, so assume values that follow a normal distribution Braccesi et al. (2018a,b); Cianetti et al. (2018, 2017).

In strictly theoretical terms, the stationarity should also be mentioned, however this hypotheses is identically ver ified by the definition of the qualification test. The first two hypotheses were verified by calculating three statistical estimators: the mean, the skewness and the kurtosis, defined respectively by the relations (19, 20, 21), starting from the expected value Bendat and Piersol (2010); Dejong and Dejong (2001).

∞ i = 1

m X = E { X } =

x i p x ( x i )

(19)

3 }

E { ( X − m X )

(20)

γ =

σ 3

X

4 }

E { ( X − m X )

(21)

η =

σ 4

X

Fig. 12 (a) shows the results for di ff erent lengths of the time history generated as described in Par. 2.4. For con venience, the kurtosis coe ffi cient defined as η − 3 was plotted. It should be observed that as the length of the time history varies there is a substantial fluctuation of the indices around values very close to zero (3 for kurtosis) Braccesi et al. (2018a). This ensures the applicability of the method and the absence of errors due to neglecting any average components.

3.1. Frequency Domain: from Load Spectra to the Damage Accumulation

The estimation of the load spectrum poses two related questions: the evaluation of the multi-axiality of the stress state and and once the equivalent stress is obtained, the calculation of the cycles associated with each amplitude. The multi-axiality of the stresses was solved by tracing the steps described by Preumont, which extends the principle of Von Mises equivalent stress to the frequency domain, introducing the PSD of equivalent Von Mises stress Pitoiset and Preumont (2000); Preumont (2013). Let PSDs be the spectral power density of a biaxial random stress process (stress state on the component surface), then using: [ PSD eqv ] = trace [ Q PSD σ ] = i , j Q i , j PSD σ i , j (22) where 1 − 1 2 0 Q = − 1 2 1 0 0 0 3 a scalar that represents the state of stress in a random process is obtained. The number of cycles was calculated from the equivalent stress PSD, by applying the relationship due to Rice:

1 2 π

m 2 m 0

1 2

(23)

n +

0 =

where

m n =

∞

n d f

PSD eqv f

(24)

0

is defined as spectral moment of n-th order. Rice’s formula shows how the relationship between the spectral moments of the second and the first order is proportional to the number of zero crossing from positive values and therefore