Finite element calculation of an elliptical cylinder in a geometrically nonlinear formulation using the vector form of the interpolation procedure

Abstract

The article describes the developed algorithm for calculating the strength parameters of a thin shell in the form of an elliptical cylinder, taking into account shear deformations in a geometrically nonlinear setting. The numerical finite element method (FEM) was used as a research tool. An algorithm for the formation of the stiffness matrix and the column of nodal forces at the loading step was developed using two versions of the interpolation procedure. In the first version, the standard for FEM interpolation of individual components of the step vector of displacement and the components of the step vector of the angles of rotation of the normal was implemented through the nodal values of the corresponding components. In the second version, the developed vector form of the interpolation procedure was used, in which the interpolation expression was written directly for the step vector and the step vector of the normal rotation angles. As a result of the implementation of the vector form of the interpolation procedure, alternative standard interpolation expressions containing the parameters of the used curvilinear coordinate system were obtained. The element of discretization of the elliptical cylinder was a four-node fragment of the middle surface with nodal variable parameters in the form of components of the step vector of displacement, their first derivatives, as well as the component of the step vector of the angles of rotation of the normal. On the example of calculating an elliptical cylinder in a geometrically nonlinear formulation, loaded in the middle of the span with a concentrated force, a comparative analysis of two variants of the interpolation procedure was carried out. It was shown that when calculating elliptical cylinders in a curvilinear coordinate system in a geometrically nonlinear formulation, it is necessary to apply the developed vector form of the interpolation procedure. The use of the standard FEM form of interpolation of the unknown unknowns made it possible to obtain correct results only in the case of a circular cylinder.