Issue 51

Y. Dubyk et alii, Frattura ed Integrità Strutturale, 51 (2020) 459-466; DOI: 10.3221/IGF-ESIS.51.34 462 2 2 2 2 2 2 2 2 2 4 2 2 4 2 2 2 2 3 2 2 2 2 2 2 1 2 u v mn mn x w mn xx N N n C k n k kn C k k H H N N n n Rl n R m C k k n kn kn k N N H m l                                                         (11) This set of equations can be easily solved using any mathematical software, like Maple or Mathematica, but its solution is too long to be given here. The solution of a harmonic imperfection is itself useful, as it can be used to approximate the long dent, but the solution for a single dent can be further obtained using Fourier series expansion. Solution for a single dent Dents can have different profiles, which depend on the mechanism of dent formation and working conditions. To work with a specific dent, the profile should be measured directly and afterwards approximated by a Fourier expansion but, to test the analytical procedure, the following smooth shape function was used to describe the dent profile: 2 2 0 0 1 1 ( , ) exp exp 2 2 x R x R x                                     (12) 0 x - length of the dent in the axial direction, 0  is the length in circumferential direction,  - dent depth. A very common way to deal with single imperfections is to use Fourier expansion. Thus, to find a solution for a dent, load coefficients (1) for every mode, using double integrals, have to be found: 2 2 2 , 0 2 0 0 0 0 1 1 1 1 1 exp exp 2 2 l m n x l x x p N dxd l x x x                                                           (13)   2 2 2 , 2 0 0 0 0 1 1 1 1 1 exp exp cos cos 2 2 l m n xx x l x x p N n x dxd l x x R x                                                                   (14)   2 2 2 , 2 0 0 0 0 1 1 1 1 1 exp exp cos cos 2 2 l m n l x p N n x dxd l x R                                                                        (15) These integrals can be computed numerically, so it's possible to deal with any dent form, thus, in contrast to other analytical solutions, the dent form influence can be further investigated. Also, it is very useful for analysing the dents found during the inspection. For a complete solution, internal forces and moments with a simple summing for each mode must be found:   2 cos cos u v w x mn mn mn n m H x N C C n C k n R R                (16)   2 1 cos cos u v w mn mn mn n m H x N C C n C kn k n R R               (17)   2 2 2 cos cos u v w x mn mn mn n m D x M C C n C n n R R                 (18)