PSI - Issue 39

Dmitry O. Reznikov / Procedia Structural Integrity 39 (2022) 236–246 Author name / Structural Integrity Procedia 00 (2019) 000–000

245

10

with the true fatigue crack growth curve ( ) tr a N ”. Once the posterior probabilities of various fatigue growth curves are estimated, means E'' ( x ) and standard deviations S'' ( x ) of the posterior probability distributions of the random model parameters can be evaluated:

Mn  

Mn 





2

,

,



( ) 0 { j P

0 { }

E

" a

a

a 

( ) 0 j

0 { }

0 { }

S" a

a

E" a

P





( ) j a a 

( ) j

}

{

}

a



tr

t

r

1

1

j

j





Mn 

Mn





2

( ) j j m m E P "    { 1 { }



( ) j m E" m P  { }

{ }

,

.

(17)

S" m





( ) j a a 

( ) j

{

}

}

a

a



t

r

t

r

1

j



Mn

Mn

( ) j       As the level of uncertainty in model parameters was reduced through accounting for the outcome of the in-service inspection, the posterior distributions of the random model parameters can now be assumed Gaussian:       0 0 0 ; a N E" a S a   ,       ; N E" m m S m   ,       ; N E" S      (18) Using the described above Matlab code, random numbers from the posterior probabilistic distributions (17) of the parameters a 0 , m , and ω were generated, and the procedure of multiple integration of the finite-difference equation (6) was performed. Then the posterior probability of failure can be estimated by equation (18) using Monte Carlo statistical simulation in a way that is quite similar to the estimation of prior probability of failure that was performed in the previous section according to equation (15): ( ) ( ) / F C n P N M N M    (19) Thus as soon as the outcome of the in-service inspection of the pipeline component become available in the form of measurement of the crack depth a meas after N i loading cycles, the posterior dependence of failure probability vs. number of loading cycles ( ) F P N  can obtained. If, for example, the measured crack depth a meas = 0.021 m is higher than the expected value of E' { a }=0.015m of the prior probabilistic distribution of the crack depth at N = N i number of cycles (Fig.6), the posterior curve ( ) F P N  shifts to the left from the position of the prior curve ( ) F P N  (Fig.7) . Due to the fact that the outcome of the inspection reduces the uncertainty in the crack size the posterior curve becomes steeper than the prior curve. ( ) j { } 1 { } t r a a j P E"        ,   ( ) j a a   2 ( ) j 1 { } { } { } t r j S" E" P 

Fig. 6. Bayesian updating using the outcome of in-service inspection.