PSI - Issue 22

Abdelkader Guillal et al. / Procedia Structural Integrity 22 (2019) 201–210 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

204

4

( ) = ( + ( + ) ( 0 + ))

(7)

) 0.5

With = (1 + 1.61 2

(8) Where R o and R i are pipeline outer and internal radius, σ y is the yield strength referred as sigma in random variables. 2.3. Importance sampling for reliability based fatigue assessment The evaluation of failure probability in equation (3) involves multi-dimensional probability integration. For this purpose, Importance sampling ( IS ) method is applied. This method has the basic of crude Monte Carlo simulation method MCS Ben Seghier et al. (2018, el Amine et al. (2017). The domain of integration in equation (3) is changed to be in full space while using an indicator function I (LSF(x)). Equation (3) can be expressed as: = ∫ [ ( )] ( ) − + ∞ ∞ (9) Where: [ ( )] = { 1 ( ) ≤ 0 0 ( ) > 0 (10) From equation (10), the failure probability is equal to expected value of [ ( ) ] . P f is now easy to estimate, consider N realization of random variables X that was sampled according to the probability distribution of X Lee and Kim (2007). By conducting repeated deterministic evaluations of the performance function in equation (9), we can Define N f as the number of trials which are associated with negative values of the performance function. Then, the estimate P f by simulation becomes = (11) is the failure probability using Monte Carlo simulation a pproximates the failure probability when N tends to infinity. Monte Carlo simulation is simple and efficient method, however it requires a large number of simulation whitch is time consuming and expensive with implicit performance function problems. In order to avoid this problem and enhance the efficiency of MCS , several methods are proposed to overcome this problem such as Importance Simpling ( IS ). The basic of this method is to focus the simpling in a region of importance. The design point defined as the point with the shortest distance to the origin in the normal space and the highest probability density among all realizations in the failure region. First and second order reliability methods ( FORM and SORM ) are used to define this point. The region arround this point is the region of intersert. The simpling in this region is possible by changing the probability density function f(X) in equation (9) by an Importace Simpling probability density function f s (s) where: ∫ ( ) = 1 − + ∞ ∞ (12) Equation (11) can be writed as following: = 1 ∑ [ ( )] ( ) ( ) =1 (13)