PSI - Issue 31

Damjan Čakmak et al. / Procedia Structural Integrity 31 (2021) 98– 104 Damjan Č akmak et al. / Structural Integrity Procedia 00 (2021) 000–000

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and f 2 = 10 Hz, are outlined with vertical dashed lines. Frequency width of left and right PSD column is Δ f = 1 Hz. Corresponding maximum stress PSDs, S PSD1 = 10 000 MPa 2 Hz -1 and S PSD2 = 2 500 MPa 2 Hz -1 , are outlined with horizontal dashed lines. This stress PSD can also be observed as a piecewise-constant conditional function. By viewing this PSD as a deterministic process, its time domain counterpart can be simply written as ( ) ( ) ( ) ( ) ( ) h,det PSD1 1 1 PSD2 2 2 2 cos 2 cos , S t S t S t Ω Ω Ω Ω = + (10) where total stress history S h signal is defined by adding/super-positioning two simple cosine functions with zero phase angle, and Ω = 2π f is circular excitation frequency. By adopting statistical approach and employing discrete inverse Fourier transform (DIFT) of S PSD,init ( f ), randomized stress time history S h,rand ( t ) is obtained, Park et al. (2014). This expression generally writes as max is the max number of discrete points used in simulation. For this case, the duration of stress process is set to t max = 2 048 s, and sample rate f sr = 1 024 Hz which gives over two million discrete points N max . It is shown later that with this setup, statistical relevance is readily achieved. This also results with small, i.e. fine frequency increment Δ Ω i and sufficiently fine time increment Δ t i . Small time increment is needed in order to sufficiently accurately capture peaks and valleys in time domain for precise RFC procedure. Relations (10,11) are simultaneously shown in Fig. 1b for small 2.5 second detail of stress history. It is interesting to note that Eqs. (10,11) are actually spectrally equivalent, i.e. they provide the same RMS value. However, Eq. (10) is not random/Gaussian, which excludes it from further RFC evaluation. Moreover, the validity of obtained random stress history S h,rand from Eq. (11) is cross-checked via Parseval’s theorem and random signal is transferred back to frequency domain via discrete Fourier transform (DFT). Stress PSD S PSD,rand ( f ) obtained from DFT of S h,rand ( t ) is shown in Fig. 1a with dashed line superimposed on original/initial PSD, i.e. S PSD,init . ( ) ( ) i ( ) max  h,rand S t PSD 1 2 cos t Ω Ω Ω γ Δ + , N i i i i i S = = (11) where γ i is i th uniformly random phase angle (between -2π and 2π) and N

Fig. 1. Bishop and Sherratt benchmark: (a) stress PSD S PSD ( f ); (b) stress history S h ( t ) detail

Furthermore, the generated random stress signal S h,rand from Eq. (11) is analyzed via RFC algorithm thoroughly described in ASTM E1049-85. Simplified RFC for repeating history is employed which ensures that there are no half cycles, i.e. residuals in the process. The parameters of the idealized Basquin’s power curve are adopted from Bishop and Sherratt (2000) where B ' rf = 10 15 MPa m and m = 4.2. Unified functions which compare results in time and frequency domain, in terms of only normalized ranges S rRMSn = S r /(2 S RMS ), are shown in Fig. 2. Ordinate axis in Fig. 2a shows