PSI - Issue 30

Dmitry O. Reznikov et al. / Procedia Structural Integrity 30 (2020) 128–135 Dmitry O. Reznikov/ Structural Integrity Procedia 00 (2020) 000–000

133

6

or in the short form:

( ) { } { } [ s ES

] [ L V V V     Comp Mat

] [ ] Str

(6)

where { L }={ P ( L 1 ) , P ( L 2 ) , …, P ( L n )} is the vector of loading regimes; { MD }={ P ( MD 1 ) ,P ( MD 2 ) , …, P ( MD m )} is the vector of material damages; { LD }={ P ( LD 1 ) ,P ( LD 2 ) ,P ( LD l )} is the vector of local damages of components; { ES ( s ) }={ P ( ES 0 ( s ) ) , P ( ES 1 ( s ) ) , …, P ( ES q ( s ) )} is the vector of end states of the system; [ V Mat ] is material vulnerability matrix: V Mat ( i,k ) =P ( MD i |L k ); [ V Comp ] is the matrix of component vulnerability: V Comp ( j,i ) =P ( LD j |MD i ); [ V Str ] is the structural vulnerability matrix: V Str ( q,j )= P ( ES q ( s ) | LD j ). The product of matrixes [ V Mat ], [ V Comp ] , and [ V Str ] is called the system vulnerability matrix [ V Sys ]: [V ] [ ] [ ] [ ] Sys Mat Comp Str V V V    . (7) It provides the relationship between the loading regime probabilities and probabilities of occurrence of various damaged end states of the system: ( ) { } { } [ ] s Sys ES L V   . (8) ( s ) ) of the CTS are determined one need to estimate direct consequences that correspond to each of the end states in terms of monetary values of loss U ( ES 0 ( s ) ) ,U ( ES 1 ( s ) ) , …, U ( ES q ( s ) ) that form the vector of direct losses { U dir } due to the system’s failure. Then the index of direct risks can be estimated using the matrix equation: As soon as the probabilities of various end states P ( ES 0 ( s ) ) , P ( ES 1 ( s ) ) , …, P ( ES q

( ) s

s

    

) LD U ES  

(

)

( P ES LD P ES  | ) (

    

( ) s

|

1 1 1 | ] P MD L P MD L P LD    [ | ] [ ( m

( P LD MD |

| MD )

)

(

)

    

    



1

q

0

1

l

1

1

1

0

   

(9)

        

R





P L P L

( ). . . ( )

. . .













  

dir

n

1

 



1    1 [ P MD L P MD L  | ] n [ | ] ( P LD MD P LD MD | ) ( | ) m n m m l

( ) s

( ) s

( ) s

( P ES LD P ES LD U ES  | ) ( | ) (

)

{ } L

l

l

q

q

0





{ } dir U

V

V

[

]

[

]

[ ] Str V

Mat

Comp

Or in the short form:

dir R  

.

(10)

{ } [ Str dir L V V V U    ] [ ] [ ] { } Mat Comp

Equations (6) or (7) provide the opportunity to estimate the mathematical mean of direct economic losses due to potential accidents at the CTS . One should bear in mind that expression (6) takes into account only direct losses due to the accident at the CTS and ignores indirect/secondary losses that can be inflicted upon the environment of the system (say losses to adjacent natural and/or manmade systems and infrastructures). In a more general formulation, each damaged end state of the system ES i ( s ) should be considered as a transient state or, in other words, as an initiating event IE i ( e ) that triggers a set of the secondary scenarios ( S 0 ( e ) , S 1 ( e ) ,…, S p ( e ) ) of disturbances in the environment of the system (Fig.3). Similar to what was done in the previous paragraph, one can write down a matrix equation that establishes the relationships between the probabilities of these interim states ( ES i ( s ) = IE i (e) ) and the probabilities of the end states of the scenarios of propagation of disturbances in the system environment ES 0 ( e ) , ES 1 ( e ) ,…, ES p ( e ) that were triggered by the system’s damage or failure:

( ) e P ES ES [ |

( ) e P ES ES [ |

   

    

]

]



p

0

0

0



  



( ) e

( ) e

( P ES

P ES

0 ( );. . .; ( ) P ES P ES  q

); ; ( 

)

. . .

 

p

0



( ) e P ES ES P ES ES   ( ) e 0 [ | ] [ | ] q p q

( ) { } s ES

( ) { } e ES

( ) e

V

[

]