Calculation of the spherical elements of non-uniform thickness for structures with holes based on the variational RVR-method

Abstract

This paper proposes a theoretically substantiated and universal new method to calculate the three-dimensional stressed-strained state of the statically loaded multi-link orthotropic shell of arbitrary thickness, made of heterogeneous material (a composite). The numerical-analytical RVR method used in this work is based on the Reissner principle, Vekua method, the R-function theory, as well as the algorithm of two-way assessment of the accuracy of approximate solutions to variational problems. In contrast to the classical principles by Lagrange and Castigliano, the application of the mixed variational Reissner principle yields an increase in the accuracy of solving boundary-value problems due to the independent variation of the displacement vector and the stress tensor. Vekua method makes it possible, as a result of expanding the desired functions into a Fourier series based on Legendre polynomials, to replace a solution to the three-dimensional problem with a regular sequence of solutions to the two-dimensional problems in the process of refining the models of shells. The R-function theory that takes into consideration, at the analytical level, the geometric information on boundary-value problems for multi-relationship regions is necessary to build the structures of solutions that accurately meet different boundary conditions. When studying spatial boundary-value problems, the constructed algorithm for a two-way integrated assessment of the accuracy of approximate solutions makes it possible to automate the search for such a number of approximations at which the process of solutions’ convergence becomes persistent. For an orthotropic spherical shell made from the material of non-uniform thickness and weakened by the pole holes, the RVR-method capabilities are shown on the numerical examples of solving the relevant boundary-value problems. The results of the reported research have been discussed, as well as the features typical of the new method, which could be effectively applied when designing responsible shell-type elements of structures in the different sectors of modern industry