PSI - Issue 52

Mengke Zhuang et al. / Procedia Structural Integrity 52 (2024) 690–698 Author name / Structural Integrity Procedia 00 (2023) 000–000

696

7

Table 1. Details of the shell structure parameters and random variables used in the reliability analysis. Z i U i Parameter Description Distribution

Mean

COV

Z 1 Z 2 Z 3 Z 4 Z 5 Z 6 Z 7 Z 8 Z 9

U 1 U 2 U 3 U 4 U 5 U 6 U 7 U 8 U 9

W 2

Inner width Inner length Inner radius Thickness

Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal Lognormal

0 . 5 m

0 . 01 0 . 01 0 . 01 0 . 01 0 . 01

L 2 R 2

0 . 25 m 0 . 05 m 0 . 05 m 0 . 1 m −

h κ

1

Curvature

0 . 8 MNm − 1

N

Boundary traction Boundary moment Domain pressure Fracture toughness

0 . 1 0 . 1 0 . 1

M

0 . 01 MN

P

0 . 01 MNPa 29 MNPa

K IC

Deterministic Deterministic

U 10

a 0

Initial crack length at A

0 . 02 m

Z 10

the structure will have lower reliability. The e ff ective SIF was calculated vis the e ff ective energy release rate: K ef f = EG ef f G ef f = G 1 + α ( G 2 + G 3 + G 4 + G 5 ) α = | ∆ K Ib | | ∆ K Ib | + | ∆ K Im | and the corresponding derivatives can be evaluated as:

(16)

E 2 K ef f

K ef f , g =

G ef f , g

(17)

where the derivatives of the energy release rate components were obtained from: G 1 , m = 2 Eh 2 K 1 m K 1 m , m G 2 , m = 24 π Eh 4 K 1 b K 1 b , m G 3 , m = 72 π Eh 4 K 2 b K 2 b , m G 4 , m = 36(1 + ν ) π 5 Eh 2 K 3 b K 3 b , m G 5 , m = 2 Eh 2 K 2 m K 2 m , m

(18)

The results of the reliability index against the length of the crack are given in Fig.4. The reliability index decreases as the crack length extended which is within the expectation as the structure tends to be more unstable when a larger crack exists. The MPP search algorithm converged successfully in three iterations, and the computational time for evaluating one reliability index was 2475 . 2 s . As a validation, the results obtained from the IDM-FORM were com pared with the results obtained from 40,000 MCS. The computational time required for the MCS was 3 . 0788 × 10 7 s . The results are in good agreement with a maximum di ff erence of 2.99% found in a crack length of 0 . 0725 m . These outcomes highlight the reliability and accuracy of the proposed IDM-FORM approach, providing an e ffi cient alternative to MCS for evaluating reliability indices in shallow shell structures with cracks. In this research, we propose a novel methodology that employs the Dual Boundary Element Method (DBEM) to derive the sensitivity of the crack tip stress intensity factor in a shallow shell structure. The reliability analysis of a shallow shell structure with a crack initiated from the fuselage window corner was conducted. The fracture reliability of the structure was assessed, accounting for uncertainties in geometrical and loading parameters. By considering these uncertainties, the sensitivities of the crack tip stress intensity factors were derived. The reliability index was obtained by applying the Implicit Di ff erentiation Method (IDM)-based First-Order Reliability Method (FORM). The results were compared with those obtained from Monte Carlo Simulation (MCS). As expected, the reliability of the structure decreases as the crack propagates, indicating an increased probability of failure, aligning with our expecta tions. The maximum di ff erence between the results obtained from FORM and MCS was found to be 2.99% at a crack length of 0 . 0725 m, which demonstrates the reliability and accuracy of the proposed methodology. One of the significant advantages of using FORM is the substantial reduction in computational e ff ort. The compu tational time required to evaluate one reliability index using FORM is merely 2475 . 2 seconds, whereas performing 3.1. Conclusion