DefinePK

Abstract

A finite elemental algorithm has been developed for calculating an arbitrarily loaded shell, the reference surface of which is determined by Cartesian coordinates , which are functions of curvilinear coordinates . For an arbitrary point of the shell located at a distance from the reference surface, the radius vector characterizing its position is determined by the sum of the radius vector of the corresponding point of the reference surface and the normal vector to the reference surface with the component . By differentiating the radius vector of an arbitrary shell point by curvilinear coordinates and by coordinate , three basis vectors of the considered arbitrary shell point are determined in the form of a function of the unit vectors of the Cartesian coordinate system. By differentiating the radius vector of an arbitrary shell point with respect to curvilinear coordinates and a normal coordinate , using the matrix relation between the Cartesian coordinate system orts and the basis vectors of the point under consideration, matrix expressions of the derivatives of the basis vectors of the considered point in the basis components of the same point are obtained. The specified load was determined in the components of the basis vectors of the point of application of the load. The displacement vector of an arbitrary point of the shell was represented by the components of its basis vectors. The derivatives of the displacement vector along the curvilinear coordinates and along the coordinate along the normal were determined taking into account the change in the basis vectors of the point under consideration. The components of the strain tensor at an arbitrary point of the shell were determined by the relations of continuum mechanics as the difference of the metric point tensors in the deformed and initial states. As volumetric finite elements, prisms with bases in the form of a triangle and a quadrangle with orientation of the bases in surfaces approximately parallel to the reference surface are taken. Nodal unknowns of volumetric finite elements accepted displacements and their first derivatives. The triangular prism was represented by three tetrahedra, the stiffness matrices of which were formed using Lagrange correction factors, which allow improved compatibility in the midpoints of the sides of the triangular bases of adjacent prismatic elements. To form the stiffness matrix of the tetrahedral element, we used the conditional Lagrange functional when adding terms with correction factors. On the example of calculating the strength of a fragment of a thin-walled tank for storing fuel and lubricants of the agro-industrial complex in the form of an elliptical cylinder, the efficiency of the developed finite element algorithm is shown.