A method of solving problems of nonlinear deformation of anisotropic spherical shells with consideration of critical points and postcritical behavior is outlined. The method employs the method of incremental loading in which the load increment is specified with an unknown coefficient determined as an unknown function equivalent to the other ones. The algorithm is based on the numerical discrete-orthogonalization method, which allows analyzing the deformation path for a number of shells with different anisotropy parameters Introduction. Recent stability studies of anisotropic cylindrical shells [17, 18, 21, 22, 23, etc.] indicate that deviations of the mechanical properties of materials from orthotropy have a considerable effect on the critical loads. The causes of such deviations are of different origin. The factors reducing the order of symmetry in fibrous composites, which are laid layerwise with different fiber orientations to make shells, cannot all be eliminated. The main factor is that the layers are of finite length and limited in number. Even if a laminate is optimal [23], it is impossible to completely avoid the adverse effect of anisotropy on the deformation and stability of shells. Spherical shells, along with cylindrical ones, are of no less theoretical and practical interest. However, most publications on the stability of such structural elements [2-4, 8-11, 14-16, 19, 20] address only isotropic and orthotropic shells. Calculated results on anisotropic shells of revolution that are indicative of the necessity of taking into account the symmetry of composites in evaluating their load-bearing capacity are presented in [5,7].In what follows, we will solve a nonlinear boundary-value problem of axisymmetric deformation of shells of revolution made of layered materials with a single plane of elastic symmetry. The associated equations are linearized by the method of incremental loading modified so that regular and critical points are passed identically, without the need to change the leading parameter [19].1. Formulation of Boundary-Value Problem. If we are to use methods of analytical mechanics [12], then we need the expression of the potential energy of the shell-load system in terms of strains and the strain-displacement relationship to derive the governing system of differential equations. The energy function for thin anisotropic shells consists of the strain energy [1] and the potential of external loads A:
