EDGE STATE METHOD IN MECHANICS PROBLEMS CONCERNING ANISOTROPIC THIN PLATES

D A Ivanychev

doi:10.21822/2073-6185-2018-45-2-18-30

Abstract

ObjectivesThe aims of the study are to expand the edge state method for solving bending and torsion problems concerning anisotropic thin plates, develop the theory of construction of the bases of spaces of interior and edge states based on a general approximate solution to the plate bending problem, formulate the relationships determining the desired elastic state and implement the developed theory in solving specific problems.MethodsThe fulfilment of the tasks is assumed to be based on the edge state method. The state spaces comprising the methodological basis are formed according to the fundamental system of Weierstrass polynomials.ResultsAn isomorphism of interior and edge state spaces is demonstrated, allowing a correspondence between the elements of these spaces to be unambiguously established. The isomorphism of spaces allows the process of finding the internal state to be reduced to the study of the edge state isomorphic to it. The mechanical characteristics are represented in the form of a Fourier series. In the case of the first and second fundamental mechanics problems, the Fourier coefficients are represented by scalar products having an energy implication: in the space of edge states, this consists in the work of external forces; in the space of internal states, it is the internal energy of elastic deformation. In the case of mixed mechanical problems, the search for an elastic state in the terms of the edge state method is reduced to solving an infinite system of algebraic equations. ConclusionThe solution of the first-tested basic problem of bending with torsion for a rectangular fibreglass plate with corresponding conclusions is given, as well as the problems of torsion for a plate of nontrivial form. Commentaryis provided concerning the unreasonableness of the solution of the second fundamental problem, as well as the problem with mixed boundary conditions for a rectangular plate where twisting and bending forces are defined simultaneously on the one face, while the opposite face is squeezed. Both explicit and indirect signs of the convergence of the solution to the problem are presented along with a graphical visualisation of the results.

Highlights

Objectives The aims of the study are to expand the edge state method for solving bending and torsion problems concerning anisotropic thin plates, develop the theory of construction of the bases of spaces of interior and edge states based on a general approximate solution to the plate bending problem, formulate the relationships determining the desired elastic state and implement the developed theory in solving specific problems
The isomorphism of spaces allows the process of finding the internal state to be reduced to the study of the edge state isomorphic to it
In the case of the first and second fundamental mechanics problems, the Fourier coefficients are represented by scalar products having an energy implication: in the space of edge states, this consists in the work of external forces; in the space of internal states, it is the internal energy of elastic deformation

Summary

Introduction

In the case of the first and second fundamental mechanics problems, the Fourier coefficients are represented by scalar products having an energy implication: in the space of edge states, this consists in the work of external forces; in the space of internal states, it is the internal energy of elastic deformation. Для решения основных задач механики в теории изгиба анизотропных пластин предлагается метод граничных состояний (МГС). Метод граничных состояний [15] является энергетическим методом решения краевых задач уравнений математической физики. В пространстве граничных состояний Г скалярное произведение выражает работу внешних сил по поверхности тела D , например для 1-го и 2-го состояния:

Objectives

Results

Conclusion