Features of derivation of formulas for calculation of nodal reactions and coefficients of matrix of rigidity of a finite element with averaged mechanical and geometrical parameters

Currently, the most widely used finite element method for the calculation of spatial structures, significant progress in the development of which is associated with the work of domestic and foreign scientists. In Ukrainian publications the problems of theoretical substantiation of the finite element method and its connection with other methods are considered, concrete types of finite elements and their application to various problems of mechanics of a continuous environment are studied. Much attention is paid to the choice of the appropriate shape of the finite element, the type and degree of approximating functions, as well as the development of methods for deriving stiffness matrices. The study of prismatic bodies with constants along one of the coordinates of mechanical and geometric parameters is most appropriate to carry out on the basis of the semi-analytical method of finite elements. Its essence is a combination of finite element sampling and decomposition of displacements in the characteristic direction by a system of trigonometric coordinate functions. The analysis of the literature shows that the issues related to the application of the semi-analytical finite element method to the calculation of thin-walled prismatic bodies in elastic-plastic, and massive even in elastic formulations, have not been properly reflected. In addition, there are no publications in this area devoted to the development of universal prismatic finite elements that allow you to explore massive, thin-walled and combined structures. The direction of this study is to create on the basis of the semi-analytical method of finite elements of an effective apparatus for numerical analysis of the stress-strain state of massive and thin-walled arbitrarily loaded properties of the material and solve a number of new practically important problems. Therefore, in this work, based on the moment diagram of finite elements, formulas for calculating nodal reactions and stiffness matrix coefficients of a finite element with averaged mechanical and geometric parameters for the study of massive, thin-walled and combined structures are derived.