Issue 51

Y. Dubyk et alii, Frattura ed Integrità Strutturale, 51 (2020) 459-466; DOI: 10.3221/IGF-ESIS.51.34 461 2 3 3 2 3 2 2 3 2 2 3 3 3 2 3 2 2 2 1 1 0 2 2 1 1 [ ] 0 3 3 2 2 1 13 1 2 2 2 MOD s s s s s s s                                                                     L (5) 2 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 0 1 [ ] 0 2 3 2 2 2 x x INI x x x x x N N N N x s s N N N x s s N N N N N N s x x s                                                                                     L (6) The following notations are used: 2 2 2 2 ; ; ; 12 1 h Eh k H D H R R         (7) 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 2 2 2 2 2 2 2 1 1 2 2 2 0 2 2 2 2 u v w mn mn mn x k n kn n N n C C C k n k kn N N n H n H                                           (9)   2 2 2 2 2 2 2 2 2 3 3 2 1 1 3 2 2 0 2 2 2 2 u v w mn mn mn x k k n N n n C C C k k N N n H H                                           (10)