PSI - Issue 3

F. Cianetti et al. / Procedia Structural Integrity 3 (2017) 176–190 Author name / Structural Integrity Procedia 00 (2017) 000–000

180

5

where Θ is the proportionality factor between displacement and stress. Under this hypothesis, the following relation can be used: ( ) ( ) 2 , , , , σ ξ ξ = i n z i n PSD f f Θ PSD f f (13) σ PSD ( i.e. to the elementary oscillator with natural frequency n f ) are known, the cumulable damage at each frequency can be computed as follow: Once the exposure time and the stress cycle probability density function (PDF) associated to the

m

∞

n

n

= ∫ K

( ) n

{ } m

(14)

D f

E

() f d

=

K

0

Once the () f is known by using Rayleigh (3) or Dirlik (4) formulation, it is possible to directly compute the fatigue damage spectrum by the previous equation. In the hypothesis of Rayleigh cycles distribution, the cumulated damage ( ) n D f (15) can be obtained by equation (8):

m

(

) 1  m

T f Θ

m



( ) n

( )

n

D f

PSD f

(15)

=

2  +   

z n

K

where T is the spectrum application duration (exposure). ( ) n D f represents the Fatigue Damage Spectrum (FDS). By the response spectrum of Miles [Miles (1954)] it is possible to obtain the following more common formulation for the cumulated damage:

m

( ) n

    



¨ Q PSD f

m

ξ

T f Θ

m

   

 

y

n

(16)

D f

( ) n

1

=

 + 

3

( 2 2

)

K

2

f

π

 

n

where ξ Q represents the dynamic amplification factor, a different representation of the percentage damping:

1 2 ξ ξ ξ = = k M Q

(17)

A qualitative evaluation of damage, but always related to its absolute evaluation, can be obtained. This is represented by the potential damage p D . The term potential states that, in the evaluation process, the evaluable signals (i.e. accelerations) are not always directly linkable to the stress state, obtaining an unreal value (potential) of the damage. This, under the hypothesis of proportionality between stress and those signals (displacement/acceleration) (12), can be used to compare different behaviors in terms of their damaging potential. Assuming an unitary value for the constant K of the S-N curve, for the Θ factor and for the exposure time T , it is possible to obtain (18).