PSI - Issue 44

Sourav Das et al. / Procedia Structural Integrity 44 (2023) 1680–1687 Das and Tesfamariam/ Structural Integrity Procedia 00 (2022) 000–000

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Similarly, ( , )  s t denotes the probability passing through a point s during unit time. The change of probability within a time interval [ t , t + Δ t ] can be written as (Li and Chen 2008): ( , ) ( , ) ( , ) ( ) ( )   ∂ ∆ = − ∆ + ∆ + ∆ = − ∆ + ∆   ∂   ∫    R L y K R L y y t P y t t y t t o t dy t o t y (4) Using the principle of probability conservation, it is written as: ( , ) ( , ) ∂ ∂ ∆ = ∆ = − ∂ ∂ ⇒  i Y J K p y t y t P P t y (5) where ( , )  y t is the flux of probability whose expression can be found in Li and Chen (2008). Finally, ( ) ( ) ( ) , , , , , 0 Θ Θ ∂ ∂ + = ∂ ∂  i i Y Y i p y t p y t Y t t y θ θ θ (6) Eq. 6 is also known as the generalized density evolution equation (GDEE). The joint PDF ( , , ) is estimated by solving Eq. 6. The PDF of Y i ( t ) can be obtained as ( , ) = ∫ ( , , ) . 2.2. Estimation of First-passage Reliability using PDEM The probability of failure for a first-passage problem can be expressed as: ( ) [ ] { } ( ) [ ] { } Thres Pr , 0, Pr , 0, = ∈Ω ∃ ∈ = > ∃ ∈ f F P Y t t T Y t Y t T (7) where Pr{ˑ} represents the probability operator, the failure domain is denoted by Ω F , and [0, T ] represents the time duration. In Eq. 7, Y Thres denotes the threshold value of Y ( t ). Also, the probability of failure, P f can be formulated using the principle of equivalent extreme-value event (Li et al. 2007), given as: { } ( ) EEV Thres EEV Thres Pr ∞ = ≥ = ∫ f Y Y P Y Y p y dy ; ( ) EEV [0, ] max ( ) ∈ = t T Y Y t (8) where EEV is the PDF of equivalent extreme-value (EEV) of Y ( t ) that is to be computed using the PDEM. To estimate it, a virtual stochastic process is formulated, which is expressed as: ( ) ( ) [ ] EEV sin ; 0,1 = ∈  Y τ ωτ τ (9) where ω = 5 π /2 (Li et al. 2007). Eq. 9 shows that when τ = 0, ( ) = 0 and when τ = 1, ( ) = EEV . Therefore, it is needed to compute the PDF of ( ) , i.e., ( , ) that is obtained by solving GDEE, as in Eq. 6: ( , ) ( , ) ( ) 0 ∂ ∂ + = ∂ ∂        p p τ τ τ τ (10) Once ( , ) is obtained, the PDF of Y EEV is calculated as: ( ) ( ) EEV 1 , = =   Y p y p τ τ (11) Eq. 10 is solved using finite difference method where representative points of random variables are generated using GF-discrepancy based method. Readers may refer Chen et al. (2016) for details of generation of sampling.