Issue 51
Y. Dubyk et alii, Frattura ed Integrità Strutturale, 51 (2020) 459-466; DOI: 10.3221/IGF-ESIS.51.34
2
3
3
1
1
0
2
3
2
s
2
2
s
2
3
1
1
[
]
0
3
3
(5)
L
MOD
2
2 s
2
2
s
3
3
3
2
1
13
1 2
3
2
2 s
2
s
2
2
s
2
2
2
2 N N N
N
0
x
x
2
2
x
s
s
2
2
2
3
1
2 N N N
[
]
0
3
(6)
L
INI
x
x
2
2
2 s
x
2
s
2
2
2
2 N N
2 N N N
N
x
x
x
2
2
x
s
x
s
The following notations are used:
2
h
Eh
D H
;
(7)
k
H
;
;
2 R R
2
2
12
1
R shell radius, h shell thickness, E Young’s modulus, Poisson ratio.
The complete explicit solution of Eq. (2) can be found only for simplest geometries, and loadings, for a single dent it can’t be obtained, thus a numerical procedure is developed below based on the accurate solution for the harmonic imperfection and Fourier series expansion. Harmonic imperfection A harmonic imperfection was considered as a base for further solution and the displacements representation can be found using: cos sin u mn u C n x R , sin cos v mn v C n x R , cos cos w mn w C n x R (8)
/ m R l , , n m wave number in circumferential and axial directions, l length of the shell, , , u v w mn mn mn C C C modes coefficients for corresponded directions. Substituting representations (8) in Eq. (2), we can get a simple algebraic set of equations:
2
2
2
2
k n kn n
2
N
n
1 2
u
v
w
2
2
2
2 C k n k kn
2 2
C
C
(9)
1
0
2
2
N N n
mn
mn
mn
H
2
2
x
2
n
2
H
2 2 2 3 2 k n
2
k
3
N
n
2 n
1 2
u
v
w C k mn
2
2 3 k
C
C
(10)
1
2 2
0
2 N N n
2
mn
mn
H
2
x
2
2
H
461
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