Issue 51

Y. Dubyk et alii, Frattura ed Integrità Strutturale, 51 (2020) 459-466; DOI: 10.3221/IGF-ESIS.51.34 460 Equivalent loads method was proposed by [5], to study the stress behavior of thin–shell structures containing geometric imperfections. The stress behavior of a geometrically imperfect shell under load is equivalent to the sum of two stress fields [6]. The first stress field is produced by the original load acting on the geometrically perfect version of the shell. The second stress field results from applying to the perfect version of the shell a load pattern that results in perturbation stress behavior equivalent to that which would be induced by the geometric imperfection. It should be noted that the equivalent load does not produce the correct dent shape but instead produce a stress field that is similar to that which would have been produced by the dent. Equivalent load is given usually in the form [6]: 2 x xx x x p N N N           (1) , , x x N N N    membrane forces, that arise from the action of the initial load on the curved shell; , , x x       change curves in two main directions and the change curvature in twist. Eq. (1) is standard in the solution of non-ideal shell problems and is used by [7], who proposed to improve its accuracy by considering the additional terms for imperfections amplitudes larger than the shell thickness [8]. Modern regulatory documents [9] in the damage risk analysis consider only the permissible depth of such defects, neglecting other geometrical parameters and the load level. The main idea of a current study is to develop effective semi-analytical procedure for dent analysis based on equivalent loads method with second order terms to perform the convergency analysis and to investigate the dent shape effect and loading level on the stress concentrator factor. A NALYTICAL B ACKGROUND General equations he theory of the thin-shells is quite well-known and effective for pipelines application. General equation of cylindrical shell equilibrium under external forces:     [ ]  L u F (2)     T u v w  u is the displacement vector; , u v and w are the orthogonal components of displacements in the , x  and radial directions;   2 T x r R H p p p       F is the external loading vector; [ ] L is a matrix differential operator [10]: 1 [ ] [ ] [ ] [ ] D M MOD INI k H     L L L L (3) Differential matrix operator [ ] D M   L corresponds to Donnel-Mushtari shell theory, the simplest one; [ ] MOD  L modifying operator, for a different shell theory, further we will use Flugge shell theory; [ ] INI  L differential operator which implies the initial stress influence. These operators are written in explicit form below: 2 2 2 2 2 2 2 2 2 2 4 1 1 2 2 1 1 [ ] 2 2 1 D M s s s s s k s                                                                 L (4) T