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.
The article presents the data of the analysis of the methods of modeling the stress-strain state of hydraulic structures using computational systems. The quantitative evaluation was carried out using the RSCI database both in terms of the number of publications and their citation, qualitative analysis of abstract information and full-text publications. As a result, trends in the growth of interest of the scientific community on this subject, the most popular method of research, as the relevant conclusions. Directions of perspective researches in the field of the scientific developments connected with modeling of the stress-strain state of constructions of meliorative appointment are designated.
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