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

100

3

where m n is the n PSD , and S RMS is the root mean square of the stress process, i.e. standard deviation. Stress PSD( f ) is commonly the function of excitation frequency f . Furthermore, E [0] is defined as expected number of upward zeros per second. Finally, the most important statistical characteristics of the process: δ and γ (via using the notation of Bouyssy et al.), are special cases obtained from dimensionless spectral parameter α n for n = 1 and 2 respectively. Parameter γ is usually referred to as irregularity factor and λ is defined as spectral width parameter. They are related through Eq. (2h). After determining spectral properties via Eqs. (2), the task of predicting fatigue life comes down to integrating Eq. (1), i.e. obtaining quantity I (PDF) from approximate function p (PDF) . It is still a matter of debate which p (PDF) describes the RFC ranges most accurately, Bishop and Sherratt (2000), Bouyssy et al. (1993), Quigley et al. (2016). Expressions for Dirlik (1985) (Dr), Tovo and Benasciutti (2005) (TB), Park et al. (2014) (JB), and Ding and Chen (2015) (Dg) PDFs are omitted herein due to length, as they can be found in referent literature. NB and RL PDFs are introduced next. 2.1. Narrow-Band / Miles (1954) / Bendat (1964) PDF Even though original NB assumption was chronologically proposed after Rice PA, it is important to introduce it first due to simple and important concepts behind it. Rayleigh/NB PDF ( p NB ) and its corresponding integral solution according to Eq. (1b) write as where Γ(·) denotes the complete Euler Gamma function. It must be emphasized that Eq. (3) actually includes correction factor γ from Eq. (2g) introduced by Hancock and Gall, see Bouyssy et al. (1993). It is known that Eq. (3) represents an analytical solution when PSD is narrow, i.e. for γ → 1. For such conditions, NB/LCC described in the ASTM E1049-85 standard for counting cycles in fact coincides with RFC ranges, Tovo (2002), Benasciutti and Tovo (2005). In general case, i.e. broad-band (BB) stress process, NB always provides conservative results and represents an upper boundary for RFC predicted damage. By neglecting correction factor γ (i.e. setting it to unity), an overly conservative prediction expression is obtained which may lead to grossly erroneous fatigue life estimation. By setting the finite upper integration boundary in Eq. (3b) to arbitrary value of stress range S r , cumulative integral for NB assumption writes as th spectral moment of the stress power spectral density S ( ) ( ) ( ) ( ) 2 r 0 r 8 NB r p S NB r I S S p S S = r NB r r 0 0 0  e d 8 1 , Γ +     4 2 S m m m S m m m γ γ − ∞   =  = (3a,b)

S



  



  

m

m

S

(

)

 



r

m

( ) NB r I S S p S S = r NB r d m ( ) c

(4)

0 

8

1 Γ + − Γ +    

1 ,

.

m

=

γ

r

r

0

2

2 8

m

 



0

Expression (4) is used later on for an illustrative example comparison with an actual RFC.

2.2. Rice (1944)/Lalanne (2002) PDF RL PDF with respect to stress ranges S r writes as, Rice (1944-45), Lalanne (2002),

weighting coeff   Ra icient yleigh/NB

    

     

Gaussian/BB 

2

2



  

S

−

S

−

   

   

1

S

S

γ

γ

λ

r

r

(5)

( )

2

8

0 e , m  8

m

λ

e

1 erf

RL r p S

+ + 

=

r

r

0

2

2π

8 4 

m

m m

λ

 

0

0

0



where error function erf(·), with respect to upper dimensionless limit x , is the integral of the Gaussian distribution given by