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

in the time domain, Bouyssy et al. (1993). The most common and accurate approach for counting cycles of irregular ranges/amplitudes in the time domain is rainflow counting (RFC), Pastorčić et al. (2019). In that sense, the ongoing task is to establish an appropriate relationship between time and frequency domains, i.e. to propose a sufficiently accurate probability density function (PDF) to mimic RFC stress ranges, Bouyssy et al. (1993). One of the oldest spectral theories can be attributed to S.O. Rice (1944-45). It is based on peak approximation (PA), ASTM E1049-85 (2011), which may lead to an overly conservative fatigue life prediction when compared to RFC, Quigley et al. (2016). Moreover, it can predict even more conservative results compared to level crossing counting (LCC), ASTM E1049-85 (2011), which is in fact narrow-band (NB) assumption proposed by Miles (1954) and later Bendat (1964). Lalanne (2002,2014) is one of the main supporters of Rice PA in the context of statistical fatigue approach. He reasons that over a sufficiently long time period, PA tends to RFC ranges. The main drawback of utilizing Rice/Lalanne (RL) PDF is the need for numerical integration, or approximate expressions, since no true closed-form solution to Rice integral is currently known. Many researches proposed a variety of approximate expressions for RL integral, see Lü and Jiao (2000), but none of them is actually an analytical solution to the problem. This is addressed further on in the manuscript. Dirlik (1985) proposed a PDF which is considered as one of the most accurate empirical spectral fatigue approximations, Bishop and Sherratt (2000), Bouyssy et al. (1993), Quigley et al. (2016), even by today’s standard. It is based on fitting a large number of performed RFC Monte Carlo simulations, but lacks the true theoretical background. Almost twenty years later, Tovo (2002) and a few years later Benasciutti and Tovo (TB) (2005) proposed a true valid alternative to Dirlik PDF for RF ranges. They based it more firmly on theoretical assumptions and less on empirical corrections, via linearly combining NB and range counting (RC), ASTM E1049-85 (2011). TB solution may also be viewed as a sophisticated correction to NB approximation. JB Park et al. (2014) were highly influenced by Dirlik’s approach almost thirty years later and proposed an alternate version of Dirlik PDF by including higher order spectral moments. Moreover, Ding and Chen (2015) further built upon TB solution and derived an alternate PDF weight correction factor which is suitable for wind loading. The aim and novelty of this study is to provide an exact analytical closed-form solution for Rice/Lalanne PA. Furthermore, the accuracy of proposed expression and applicability to real-life engineering situations is demonstrated on Bishop and Sherratt spectral fatigue benchmark. Finally, it is reported that when compared to corresponding time domain RFC solution and concurrent Dirlik, TB, JB Park and Ding solutions; RL approximation may provide satisfyingly good match-up. 2. Vibration fatigue fundamentals Predicted damage intensity in the frequency domain, with respect to stress range S r , may be written as [ ] ( ) ( ) ( ) ( ) ( ) ( ) PDF r r r r r r PDF PDF PDF max rf 0 0 P d d , ' m m D E S p S S I S S p S S t B ∞ ∞ =  =   (1a,b) where D PDF is the damage predicted by quantity I (PDF) , i.e. integration of the PDF ranges p (PDF) (Sr), and t max is the duration of the random stress process. Moreover, B ' rf and m are the constants of Basquin’s high-cycle power law spectral fatigue curve and E [P] is designated as expected number of peaks per second. Important statistical properties of the stress process in the frequency domain write as

0  ∞

0  ∞

m m

m m

[ ] [ ] [ ] 0 0 P

[ ] P

( )

( ) S f f PSD d

d , m f S f f S = n

, m E

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,

E

=

=

=

=

2

4

PSD

RMS

0

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(2a-h)

0

2

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m

m

m

2 1 , γ

,

,

,

=

= =

= =

=

= −

n α

1 α δ

2 α γ

λ

1

2

n

m m

m m

m m

0 2

0 2

0 4

n