The aim of the work is to perform a comparative analysis of the results of analyzing arbitrarily loaded shells of revolution using finite element method in various formulations, namely, in the formulation of the displacement method and in the mixed formulation. Methods. To obtain the stiffness matrix of a finite element a functional based on the equality of the actual work of external and internal forces was applied. To obtain the deformation matrix in the mixed formulation the functional obtained from the previous one by replacing the actual work of internal forces in it with the difference of the total and additional work was used. Results. In the formulation of the displacement method for an eight-node hexahedral solid finite element, displacements and their first derivatives are taken as the nodal unknowns. Approximation of the displacements of the inner point of the finite element was carried out through the nodal unknowns on the basis of the Hermite polynomials of the third degree. For a finite element in the mixed formulation, displacements and stresses were taken as nodal unknowns. Approximation of the target finite element values through their nodal values in the mixed formulation was carried out on the basis of trilinear functions. It is shown on a test example that a finite element in the mixed formulation improves the accuracy of the strength parameters of the shell of revolution stress-strain state.
The aim of the work - comparison of the results of determining the parameters of the stress-strain state of plane-loaded elastic bodies based on the finite element method in the formulation of the displacement method and in the mixed formulation. Methods. Algorithms of the finite element method in various formulations have been developed and applied. Results. In the Cartesian coordinate system, to determine the stress-strain state of an elastic body under plane loading, a finite element of a quadrangular shape is used in two formulations: in the formulation of the method of displacements with nodal unknowns in the form of displacements and their derivatives, and in a mixed formulation with nodal unknowns in the form of displacements and stresses. The approximation of displacements through the nodal unknowns when obtaining the stiffness matrix of the finite element was carried out using the form function, whose elements were adopted Hermite polynomials of the third degree. Upon receipt of the deformation matrix, the displacements and stresses of the internal points of the finite element were approximated through nodal unknowns using bilinear functions. The stiffness matrix of the quadrangular finite element in the formulation of the displacement method is obtained on the basis of a functional based on the difference between the actual workings of external and internal forces under loading of a solid. The matrix of deformation of the finite element was formed on the basis of a mixed functional obtained from the proposed functional by repla-cing the actual work of internal forces with the difference between the total and additional work of internal forces when loading the body. The calculation example shows a significant advantage of using a finite element in a mixed formulation.
The equations of the theory of the stress-strain state of an elastic body for practical engineering structures of the agro-industrial complex do not have analytical solutions. Therefore, the development of numerical methods for determining the strength parameters of agricultural facilities is an urgent task. Among the numerical methods of strength calculations, the finite element method is currently widely used. In a Cartesian coordinate system, a finite element of a hexahedral shape is used to determine the stress-strain state of an elastic body under bulk loading in two formulations: in the formulation of the method of displacements with nodal unknowns in the form of displacements and their derivatives, and in a mixed formulation with nodal unknowns in the form of displacements and stresses. Approximation of displacements through nodal unknowns in obtaining the finite element stiffness matrix was performed using a form function, the elements of which were Hermite polynomials of the third degree. When obtaining the deformation matrix, displacements and stresses of the internal points of the finite element were approximated in terms of nodal unknowns using bilinear functions. The calculation example shows a significant advantage of using a finite element in a mixed formulation.
Structures made of thin-walled shells are widely distributed elements in the environmental protection systems. In this connection, it is necessary to develop their effective and refined calculation. The point of the shell of the environmental protection structures is considered with a plane load in the initial position, deformed after j loading steps (displacement vector) and adjacent after (j + 1) loading step. The displacement increments and their derivatives are taken as nodal unknowns. The displacement vectors of the inner point of a finite element are represented in the initial basis, and their components are approximated through nodal unknowns using Hermite polynomials of the 3rd degree. To determine the deformed state of the shell, the algorithm of the method of discrete continuation with respect to the parameter in the vicinity of a singular point is used, in which the increment of loading at the step is the desired parameter. Based on the obtained dependencies, a step-by-step calculation procedure is organized, which allows you to get correct results if there is a special point.
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