Volumetric element with vector approximation of the desired values for nonlinear calculation of the shell of rotation

Abstract

The usage of traditional approximating functions directly to the desired displacement vector of the internal point of a finite element to determine it through nodal unknowns in the form of displacement vectors and their derivatives is described. To analyze the stress state of a geometrically non-linearly deformable shell of rotation at the loading step, the developed algorithm for forming the stiffness matrix of a hexagonal finite element with nodal values in the form of displacement increments and their derivatives was used. To obtain the desired approximating expressions, the traditional interpolation theory is used, which, when calculated in a curved coordinate system, is applied to the displacement vector of the internal point of a finite element for its approximation of class C(1) through nodal displacement vectors and their derivatives. For the coordinate transformation, expressions of the bases of nodal points are obtained in terms of the basis vectors of the inner point of the finite element. After the coordinate transformations, approximating expressions of class C(1) are found for the components of the displacement vector of the internal point of the finite element, leading in a curved coordinate system to implicitly account for the displacement of the finite element as a rigid whole. Using calculation examples, the results of the developed method of approximation of the required values of the FEM with significant displacements of the structure as an absolute solid are obtained.