PSI - Issue 4

Shun-Peng Zhu et al. / Procedia Structural Integrity 4 (2017) 3–10 S.P. Zhu et al. / Structural Integrity Procedia 00 (2017) 000 – 000

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the increasing loading cycles, while the variability of fatigue lives increases with decreasing stress levels [15, 21]. Thus, the variability of cumulative damage can be derived as a function of that in fatigue lives. Assuming that the variability in loading cycles is equal to zero at the initial stage, it increases to a certain value at fatigue life. Using the geometric reasoning method [15], the change rate of variability in loading cycles can be interpreted in Fig. 3.

PDF of fatigue life

D C

Damage, D

2 l

1 l

σ Nf

O

N f

Number of loading cycles, n

Fig. 3 Change rate of variability in loading cycles In Fig. 3, 1 l represents the mean trend line of cumulative damage and 2 l is the 1 -  curve of fatigue life distribution. Through converting the coordinate system in Fig. 3 into a double logarithmic coordinate system, the rate of change of standard deviation r  of loading cycles can be derived as



f N N

r



(11)



ln

f

The standard deviation of the given loading cycles n can be obtained as



  

   

f N N

n

(12)

ln

n 

 

ln

f

Meanwhile, the standard deviation of cumulative damage D is given by



  

   

f N N

a  

n

(13)

ln



D

ln

f

Note that Eq. (13) can be used to capture the variability of cumulative damage. Assuming that damage is accumulated stochastically and independently under each level stress loading, then the total variability of cumulative damage   D n under multi - level stress loading is derived as

2

   

   



   

   

j



N

a

n

ln





(14)

fi

D

i

i

N

ln

i

1



fi

where i represents the level of stress under multi - level stress loadings.

2.4 Fatigue reliability analysis

This section investigates the fatigue reliability assessment using the developed approach. Based on boundary conditions of fatigue, fatigue failure occurs when cumulative damage   D n reaches C D , where   1 C E D  . Assuming that the critical threshold damage has the same distribution as that of cumulative damage; and at the fatigue failure point, the variability of C D is equal to that of cumulative damage measure, i.e. 2 2 C D D    . Since the