Issue 57

A. Sadeghi et alii, Frattura ed Integrità Strutturale, 57 (2021) 138-159; DOI: 10.3221/IGF-ESIS.57.12

The reliability index ( )  is computed using the first - order approximation with regarding to Eqn. 3, where Φ -1 is the inverse cumulative distribution function of the standard normal distribution [8].

( )  − = − 1 f P

β

(3)

: i x i

In this regard, independent and identically distributed ( iid ) samples 

 =    1, , N

are generated according to PDF of

( ) x f and evaluating ( ) ( )  0 ( f g x x , MCS only provides an exact failure probability for any kind of structural systems [26]. Also, the partial derivative of the failure probability related to a statistical specification of the i-th random variable i.e.  i can be determined with Eqn. 4:

( )



 P ξ ξ f



( ) ( )   , x f

=

  

(4)

x dx

F

X

i

i

Then, Lebesgue dominated convergence theory and importance sampling ( IS ) are used and in the following, Eqn. 4 would be rewritten as Eqn. 5 [26, 28, 29]:

( )

( )  , x

( ) ( ) ( )     , , , X x f x f x



 P ξ ξ f





f

f

X

X

( ) x

( ) x

=

=

=





dx

dx

F

F



 i



i

i

X

( ) , x



   /

f

X

i

( ) x

( )  ,

=

=



(5)

f

x dx

F

X

( )  , x

f

X

( )  , x



log

f

X

( ) x

( )  ,

( ) ( .

)

=

=





f

x dx

[

X X

,

F

X

X F

i



 i

Therefore, the so - called score function presented as Eqn. 6:

( )  , x



log

f

X

( ) ,

=

i x ξ

(6)



 i

Also, it is recommended that researchers refer to the authentic papers [26, 29] for more information about MCS and score function. With regarding to high computational efforts of MCS , application of meta - model techniques in structural reliability problems has been developed and proposed [30, 31, 32]. Meta-models In recent years, meta - models are widely used among researchers over the world to reduce the computational costs of the reliability analyses by predicting the structural responses [33]. Meta - model techniques can provide an accurate representation of the PDF of the model response. Some meta - models, such as Kriging [34, 35], PRSM [36], and ANN [8] have been successfully applied for structural reliability problems that are briefly presented in the following sections. Kriging Kriging is a mathematical interpolation approach for constructing a surrogate model instead of the original prototype. Kriging surrogate model is widely applied in reliability evaluation and failure probability computation of problems. This method aims to minimize the error rate and fit the mean of the prediction errors to zero [37]. Also, considering support points and related response Y as ( ) ( ) =      1 , , T s Y G x G x , Kriging meta - model presents the definite response ( ) G x as a realization of a random function [36]. The function ( ) G x is presented as Eqn. 7:

( ) + T G x f x z x ( ) ( )  =

(7)

141