PSI - Issue 39

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

244

9

For the numerical implementation of the described procedure, a file program was written in the Matlab environment, which allows forming samples of values of random parameters from the most frequently used probability distributions, including exponential, uniform, normal, logarithmically normal, Weibull, etc. Applying the Matlab algorithms for generating random numbers from the selected distributions of the parameters a 0 , m , and ω, the procedure of multiple solutions of the finite-difference equation (6) and estimation of failure probability by Monte Carlo method according to expression (15) is implemented.

a ) b ) Fig. 5. Failure probability vs. number of CA loading cycles. Overload applied after i OL =5,000 cycles and k OL =1.6

Fig. 5 presents the dependences of failure probability on the number of loading cycles in linear (a) and semi logarithmic (b) scale for the case of a plane-stress state (β = 2), when overload is applied after i OL =5000 cycles with the overload factor k OL =1.6. The dependence ( ) F P N  obtained according to equation (15) should be treated as the so called prior assessment of the component failure probability. This assessment is based on prior probabilistic distributions of random parameters (13) that were selected according to available general statistics on mechanical properties of structural materials and crack sizes that may be based on the literature review and on laboratory tests performed on compact specimens in loading conditions that differ substantially from the in-service conditions of the particular structural component. These probabilistic distributions can be updated through Bayesian inference as soon as data on crack size measurements from in-service inspections of the particular component become available. An original approach that combines Bayesian inference and random walk method for updating prior assessments of fatigue failure probability was developed by J.Karandikar and co-authors in (Karandikar et.al., 2012) and modified for the pipeline application in (Makhitov, Reznikov , 2021). This approach can be summarized as follows. The posterior probability of the event “Sampled j -th crack growth curve a ( j ) ( N ) coincides with the true crack growth curve a tr ( N ): { a ( j ) = a tr }” ( j = 1,2, ..., M n ) can be estimated using the Bayes rule by the following equation:

L

(16)





P

P



( ) j a a 

( ) j a a 

P

{

}

{

}

tr

tr

{ | } m a AI



| } 1 / P I 



( ) j

{

P

P

a

( ) N N a  ( )

M

n is the prior probability of the event “Sample curve ( ) ( ) j a N ( ) tr a N ” that is estimated according to equation (15) using the

where

tr

( ) j a a 

{

}

tr

coincides with the true fatigue crack growth curve

  2 a S

available prior information PI ;





2 ) /

} , PI

(

|

a a

{ L P a 



( ) j N a a N  ( ) ( )

exp

 is likelihood function, here is a normalizing constant;

  

meas

tr

meas

m a PI P

a maes is outcome of measurement of the crack depth;

{ | }





P

P

, )} is the posterior probability of the event “Sample curve ( ) j a coincides

( ) { ( ) ( ) | j N a N a PI a 

tr

m

eas

( ) { j a

}

a



tr