On the calculation of the rectangular finite element of the plate The article is devoted to the study of the thin plate bending by the finite element method. The application of the finite element method to solving the problem of the plate bending leads to the necessity of studying the rectangular finite element of the plate. All deformation and statics characteristics of the plate are functions of the displacement in the direction of the normal to the middle surface of the plate, which is determined by the deflection function. In the article, the formation of the plate deflection function in explicit form is carried out. The ways for finding the deflection function by division of the variables in the equilibrium equation of the plate, through an incomplete fourth-degree polynomial and in the form of Hermite polynomials are presented. The article is focused mainly on mechanics, engineers and scientific employees of technical specialties.

The article is devoted to the study of plate bending problems, which are of great applied importance and are found everywhere in various branches of science and technology. In this article the structure of the calculation methods is described, their main components are highlighted; the classical approach of calculating rectangular plates hinged supported on two parallel sides and with arbitrary boundary conditions on each of the other two sides is characterized. The mathematical apparatus of the method of trigonometric series is presented in the volume necessary for calculating the plates. Special cases of the calculation for the bending of a rectangular plate by the Levi method are given. This article is focused mainly on mechanics, physicists, engineers and technical specialists.

The structure of normal subsets of polyhedral cone The structure of normal convex subsets of polyhedral cone K in normalized space is in investigated. The normality of the subset Ω ⊂ K (in the sense of the cone K) is determined by the condition Ω − K ∩ K = Ω (a line over a set means taking a topological closure). The conical shell of finite number of rays mean the polyhedrons of the cone, which are extreme rays. The structure of normal sets were studied from the geometric point of view. It is shown that every normal subset Ω of a polyhedral cone can be divided into a sum of two subsets, one of which is a bounded normal subset (in the sense of some subcone in K) and the second-the subcone K contained in the set Ω (it is unbounded, if Ω is unbounded).

On the calculation of rectangular plates by the trigonometric series The article is devoted to the question of applying the method of single trigonometric series to solving the plate bending problems. In this article the structure of this method is described, its main components are highlighted, the classical approach of calculating rectangular plates hinged supported on two parallel sides and with arbitrary boundary conditions on each of the other two sides is characterized. The mathematical apparatus of the method of single trigonometric series is presented in the volume necessary for calculating the plates. The detailed example of calculating a rectangular plate by the stated method is given. The article is focused mainly on students and undergraduates engaged in research work in the field of mechanics and applied mathematics.