PSI - Issue 31

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

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

normalized probability density function of RFC cycles. It is scaled according to expression S rPDFn = S RMS p (PDF) in order to obtain dimensionless PDFs. It is observed that Dirlik’s PDF excellently describes RFC ranges in whole statistical RMS area. JB PDF also predicts time domain RFC cycles well. Moreover, TB PDF and Ding PDF, provide similar results. Match-up is rather poor in the area of 0 – ~1 RMS range. However, in the higher RMS ranges, i.e. “tail region” – prediction is well aligned with RFC. Since most damage occurs in the higher RMS ranges, this is a rather acceptable phenomenon. NB assumption from Eq. (3a), even with bandwidth γ -correction, expectedly over-predicts RFC ranges in higher RMS areas. By inspecting the most important results for this investigation, i.e. RL PDF and corresponding integral solutions from Eqs. (5) and (8,9) respectively, a few important points can be outlined. If γ -correction is not employed for RL distribution, PDF from Eq. (5) over-predicts both RFC and NB approximation from Eq. (3a). This is observed by poor matching of RFC cycles in both low and high RMS region. Furthermore, over-prediction of NB assumption implies non physics-based results since NB is upper bound of RFC damage, Tovo (2002). Hence, PA based RL PDF from Eq. (5) provides unfavorable results when compared to RFC without additional correction factor. However, by employing the γ -correction from Eq. (9), i.e. by multiplying the RL PDF with E [0] instead of E [P], drastic improvement is observed when compared to RFC PDF beyond 1 RMS range, i.e. probability of higher stress region. Moreover, ordinate axis in Fig. 2b denotes normalized cumulative damage as a function of RMS region. Damage is normalized according to expression D cn(…) = D (…) / D RFC , where total damage from RFC is adopted as referent, scaling value. Hence, normalized maximum RFC damage D nRFC always equals unity. This is emphasized by thin horizontal solid line in Fig. 2b which passes through one. Since discrete simulated data from RFC becomes almost horizontal at the far right, it might be concluded that simulation reached its saturation point and finally converged. NB cumulative integral from Eq. (4) shows this principle of cumulative damage D c . Upper bound of NB cumulative damage is obtained when S r → ∞, i.e. from Eq. (3b). This is emphasized by long-dashed horizontal line in Fig. 2b. Since D cnNB curve becomes almost horizontal at this point, one can conclude that predicted cumulative damage D c also converges. It can be seen that NB overestimates RFC damage by a factor of over two. Furthermore, RL without γ -correction overestimates RFC damage even more than NB which is non-physics based. This is shown with upper small-dashed horizontal curve. However, by applying γ -correction to RL PDF, it is noted that lower small-dashed horizontal curve coincides well with RFC. Same may be concluded for JB approximation which also agrees rather well with both RFC and RL. It is interesting to note that Dirlik, TB and Dg under-predict RFC damage for this particular case. However, all benchmarked spectral theories, besides NB and RL without γ -correction, predict RFC ranges rather well.

Fig. 2. Spectral fatigue comparison: (a) normalized stress range PDFs S rPDFn ( S r ); (b) normalized cumulative damage D cn ( S r )