Issue 78

F. Veschi et alii, Fracture and Structural Integrity, 78 (2026) 44-73; DOI: 10.3221/IGF-ESIS.78.04

 Spectral Parameters : Given the Gaussian nature of the input, it is assumed that the response is also Gaussian. Consequently, the zero up-crossing rate (Eqn. (4)) and the peak rate (Eqn. (5)) of the response are calculated. Additionally, the irregularity factor, as defined in Eqn. (7), is calculated.  Peak Distribution : Definition of the PDF of the response peaks. Although the distribution of rainflow counting cycles is essential for an accurate fatigue damage estimation, the response peaks PDF remains a fundamental tool for NB signals [27]. The response of an SDOF system to a Gaussian excitation is characterized by a Gaussian signal concentrated around the system's natural frequency. As the damping coefficient decreases, the response becomes increasingly NB. Under these conditions, the response   Z t can be considered a NB signal, allowing the peak distribution to be approximated by a Rayleigh distribution. Under such conditions, the FDS for a Gaussian random signal can be evaluated using the following expression:

m



T

m





  0

m p

m



1       2



D f

K

2

(12)

2

Z

C

where z  is the root-mean-square value of the response, T is the exposure time, and  is the gamma function. Evaluating the fatigue spectrum in the frequency domain provides the expected value of the damage, effectively eliminating the intrinsic statistical variability that arises from calculations based on individual time histories. This method ensures high computational efficiency while maintaining excellent agreement with time-domain results.

Figure 1: Overview of the FDS evaluation process in time domain.

The White Noise Approximation. A widely adopted simplification is the White Noise Approximation. By assuming that the input PSD,   ¨ ¨ xx G f , is locally constant around the natural frequency 0 f , the RMS stress response can be directly estimated as [19]:     ¨ ¨ 0 3 0 4 2 xx Z G f Q f    (13)

Substituting this into Eqn. (12) yields the analytical closed form for the FDS for the white noise case:

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