PSI - Issue 39

Dmitry O. Reznikov / Procedia Structural Integrity 39 (2022) 256–265 Reznikov D.O./ Structural Integrity Procedia 00 (2019) 000–000

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3. Assessment of the failure probability The probability of failure of the component under consideration is determined by the probability of fulfilling the failure condition (11), which can be estimated using the Monte Carlo statistical simulation method (Kroese, Taimre, Botev, 2013). The initial crack depth a 0 , parameters C and m of the Paris equation as well as the maximum value of the nominal hoop stress in the pipeline Σ max arising in a series of overloads are considered as probabilistic parameters in the failure condition (11). The choice of the distributions of the parameters a 0 , C , and m is determined by the amount of available information (non-destructive testing and experimental data). When the amount of available information is insignificant, the law of uniform distribution density U (α x ;β x ) can be used, where α x and β x are the left and right boundaries of the interval on which the distribution density function of x is grater then zero. If when solving the problem under consideration, there is a sufficiently large amount of information about the indicated quantities, then the normal distribution laws N ( E { x }; S ( x )), (where E { x } and S ( x ) are mean and standard deviation of the parameter x ) can be used. The estimation of the distribution parameters of the maximum hoop stresses Σ max is carried out as follows. According to the assumption made, the intensity of the maximum values of the internal pressure in the pipeline arising from overloads are independent random variables, equally distributed according to the normal law N ( E { p Σ }; S ( p Σ )), where E ( p Σ ) and S ( p Σ ) are mean and standard deviation of p Σ . Further, instead of p Σ , it is more convenient to use the associated maximum nominal hoop stress Σ max that occur in a series of N k overloads (Σ max =max{Σ 1 , Σ 1 , ..., Σ Nk }, where Σ i = P Σ i D /(2 δ ), i = 1 , 2, ..., N k ). Thus, the peak values of the nominal hoop stress arising from overloads are also random variables, equally distributed according to the normal law N ( E {Σ}; S {Σ}). The value of N k is assumed to be a deterministic and known number. Consider the value that is the maximum of the sequence of peak values of the nominal hoop stresses (Σ max =max{Σ 1 , Σ 1 ,…, Σ Nk }). This value is obviously also random, and its distribution function is determined by the expression:   max max 1 2 ( ) ( ) ( , ,..., ) ( ) k N Nk F x P x P x x x F x              , (12)

and the distribution density function can be written as:

( )

F x



1

N



.

  

 

k

(13)



( ) x

( ) m F x

( ) x

f

f



max

F

x







max

max



max

Since the original distribution of the quantity Σ obeys the normal law, according to the theorem of Fisher, Tippett, and Gnedenko, the distribution function of the quantity F Σmax ( x ) for sufficiently large N k ( N k → ∞) has an asymptotic form corresponding to an extreme distribution of the type I, at which the distribution function and probability density function have the form of a double exponential distribution (Ang, Tang,1984; Makhutov, Reznikov, 2014):







  ,

{ } / { } Nk x E S v

e  

    

 

( ) exp

F x

  



(14)

Nk



max





 

  

{ } / { } x E S v    

 

 

e  

(

)

x v 





,

( ) x

exp

f

e

Nk   



Nk

Nk

(15)

Nk

Nk

 



max

The governing parameters of the extreme value distribution are determined by the expressions:

  

   

ln ln ln 4 N 



,

(16)

{ } 2ln 

{ }

Nk v S

N

E  

  



2 2ln k

k

N

k