Elastic-plastic analysis of shells by variational method on the basis of high-degree polynomials

The purpose of the research is to develop a variational method for calculation of three-dimensional structures based on approximating functions with finite carriers of an arbitrary degree of approximation. In the early papers of the authors, the method was presented in a linear formulation, and the possibility of calculating both three-dimensional compound structures and thin shells was shown. This paper proposes an algorithm for strength calculation of thick and thin shells with elastic-plastic deformations. The geometry of shells is described in a curvilinear orthogonal coordinate system, e.g., in cylindrical, spherical, or conical ones. The calculation method uses the basic equations of small elastic-plastic deformations for the curvilinear coordinate system. The calculation algorithm was based on a model of material with linear strengthening. To obtain a resolving system of nonlinear equations, the Lagrange variational principle is used. The problem is solved by means of iteration. The first iteration corresponds to a linear problem. At each iteration, after solving the system of equations, the intensities of deformations at each point of integration are calculated. These intensities of deformations are substituted into the matrices of elasticity at the following iterations. The process of iteration is characterized by recalculation of the elasticity matrix at each iteration in each integration point. The researche have shown a stable convergence of the process of iteration. A testing solution of elastic-plastic deformation problems of a thick pipe and a thin shell was carried out. The calculation results were in good agreement with the results obtained both by classical formulas for elastic plastic deformation and with the results of calculations in the Ansys Mechanical program.