PSI - Issue 52

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

693

4

The sensitivity of the stress intensity factors with respect to some geometrical variable or curvature can be derived from the above equation. The derivatives of { K } can be written as: K 1 b , m = Eh 3 48 √ r π 2 ∆ ϕ 2 , m − r , m 2 r ∆ ϕ 2 K 2 b , m = Eh 3 48 √ r π 2 ∆ ϕ 1 , m − r , m 2 r ∆ ϕ 1 K 3 b , m = 5 Eh 24(1 + ν ) √ r π 2 ∆ w 3 , m − r , m 2 r ∆ w 3 K 1 m , m = Eh √ 2 π 8 √ r ∆ u 2 , m − r , m 2 r ∆ u 2 K 2 m , m = Eh √ 2 π 8 √ r ∆ u 1 , m − r , m 2 r ∆ u 1 (6) The sensitivities of the maximum stress intensity factors were considered, and the maximum SIF through the thickness of the shell is:

1 R 1 1 R 1 1 R 1 1 R 1 1 R 1 1 R 1

1 R 2 1 R 2 1 R 2 1 R 2 1 R 2 1 R 2

h 4 h 4 h 4 h 4 h 4 h 4

)] K max I

1 h K 1 m + 1 h K 2 m +

6 h 2 K 1 b 6 h 2 K 2 b

(

[1 +

=

+

)] K max II

(7)

[1 +

(

+

=

)] K max

3 2 h K 3 b

(

[1 +

+

III =

The derivatives of the maximum SIF can be found by:

1 h 1 h

6 h 2 6 h 2

)] K max

(

K 1 m , g +

K 1 b , g

[1 +

+

I , g =

)] K max

(8)

[1 +

K 2 b , g

(

K 2 m , g +

+

II , g =

3 2 h

)] K max

(

K 3 b , g

[1 +

+

III , g =

K 1 b ,ρ T κ + K 2 b ,ρ T κ +

K 1 b T κ,ρ K 2 b T κ,ρ

I ,ρ = II ,ρ = III ,ρ = I , h = II , h = III , h =

1 h 1 h

6 h 2 6 h 2

1 h 1 h

6 h 2 6 h 2

K max

K 1 m ,ρ +

K 1 m +

K max

K 2 m ,ρ +

K 2 m +

(9)

K 3 b ,ρ T κ +

K 3 b T κ,ρ

3 2 h

3 2 h

K max

K 1 b · T κ + K 2 b · T κ +

K 1 b · T κ, h K 2 b · T κ, h

1 h 1 h

6 h 2 6 h 2

1 h 2 1 h 2

12 h 3 12 h 3

1 h 1 h

6 h 2 6 h 2

K max

K 1 m , h +

K 1 m +

K 1 b , h − K 2 b , h −

K 1 m − K 2 m −

K max

K 2 m , h +

K 2 m +

(10)

K 3 b · T κ +

K 3 b · T κ, h

3 2 h

3 2 h 2

3 2 h

K max

K 3 b , h −

where terms () , g , () ,ρ and () , h represent the derivatives with respect to geometrical variables, curvatures and thickness respectively. Denote that T κ = [1 + h 4 ( 1 R 1 + 1 R 2 )] − 1 , the corresponding derivatives are T κ,ρ = − h 4 ( κ 11 + κ 22 ) − 2 h 4 κ 11 ,ρ + κ 22 ,ρ with respect to the curvature and T κ, h = ∂ T κ ∂ h = − h 4 ( κ 11 + κ 22 ) − 2 1 4 ( κ 11 + κ 22 ) with respect to the thickness. 2.3. First-order reliability analysis (FORM) The main purpose of the FORM is to evaluate the reliability index which can be thought of as a measurement of the ability that a structure remains its safety during the operation. The reliability index is usually associated with the probability of failure which quantifies the risk of structural failure. In the field of reliability, the region confined by the random variables Z can be separated into safe and failure regions