Rationale of the use of the constructive anisotropy method in the calculation of shallow shells of double curvature, weakened holes

The paper presents the geometrically nonlinear mathematical model of deformation related to isotropic shallow shells with double curvature weakened by holes. The model is based on the hypotheses of Kirchhoff-Love theory of shells which is presented in the form of geometric and physical relationships and the functional of the total potential energy of deformation. Expressions for the forces and moments are also provided. We consider two methods of introducing holes: discrete and constructive anisotropy methods, which allow the most accurate "smudge" zero stiffness of holes through the shell field. Ritz method is used to minimize the functional; it reduces the problem to solving a system of nonlinear algebraic equations, which is solved by Newton's method. The algorithm is implemented in the medium 2015 Maple analytical calculations. The stability analysis of shallow shell structure with double curvature, made of steel, is performed when exposed to an external uniformly distributed lateral load using hinged-fixed method to place the shell contour. Calculations are carried out in for different numbers of holes, with a fixed ratio of the total area of holes to the shell area. Thus, on increasing the number of holes, their sizes decreased. Hole distribution along the shell is made in two different ways. The critical loads of buckling are given for all designs. We compare the values obtained with the discrete introduction of holes and those of the method of constructive anisotropy. For several variants of designs, the "load - deflection" graphs are shown. For one embodiment of the shell weakened by a large number of holes, we provide a deflection field before and after buckling; the intensity of the stress is also highlighted with colors. Based on the data, it is shown that the increase in the number of holes results in the lost of discreteness of their input; hence, it becomes possible to use a specially developed method of constructive anisotropy which justifies the use of this method in calculating the stability of shallow shells weakened by a large number of holes.