Analytical study of the bending of isotropic plates, inhomogeneous in thickness

V P Plevako ,Viktor Kycenko,Ighor Lebedynecj,Iryna Pedorych,Володимир Олексійович Потапов

doi:10.15587/1729-4061.2016.75052

V P Plevako , Viktor Kycenko + Show 3 more

Open Access

https://doi.org/10.15587/1729-4061.2016.75052

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Abstract

A threedimensional problem of bending the plate, in which the parameters of elasticity of the material vary by thickness and are arbitrary integrable functions, was examined. And the plate itself is exposed to the action of mass forces while the action of surface loads is studied as a separate case. An analytical solution to the boundary problem by the operator methods was obtained in a case when the boundary conditions are satisfied exactly on the flat surfaces of the plates and on the lateral surface – in the Saint Venant approximation. It was theoretically proved that the exact, in the sense of Saint Venant, analytical solutions may be obtained if the plate is exposed to the action of the mass and surface forces, distributed on the plate and on its surface by the twodimensional polyharmonic law. In this case, the thinner the plate, the more exact the solution will be, since the corresponding solutions represent the series that contain a finite number of members. It was demonstrated that the obtained formulas for the calculation of bending the thin plates in the case of homogeneous material transfer to the classic formulas of the theory of bending thin plates. The obtained solutions allow using them as the approximate, “technical” theory for engineering calculations of the stressed and deformed state of nonhomogeneous plates.

Highlights

The heterogeneity is a characteristic feature of virtually all materials used in engineering and construction
If we find Ln, and Nm from the dependencies (15) and (16) and substitute them to the formulas (17), we will find functions Si, and Tj : ( ) ∫ ∫ ∫ S1
A three-dimensional problem of bending the plate, in which the parameters of elasticity of the material vary by thickness and are arbitrary integrable functions, was examined

Summary

Introduction

The heterogeneity is a characteristic feature of virtually all materials used in engineering and construction. It is due to a number of factors, which conditionally can be divided into three groups: 1. Action of environment (temperature fields, radioactive radiation, uneven humidity, etc.). 3. Project implementation plan (availability of the reinforcement, layers of other materials, etc.), which is an important source of reduction in weight, size and cost of the projected designs. The improvement of calculations of machine elements, constructions and structures is associated with taking into account the impact of heterogeneity of actual materials on the stressed and deformed states of elastic bodies. A relevant task is the search for the methods of determining the displacements and stresses in three-dimensional bodies of a relatively simple form – slabs and plates made of inhomogeneous materials

Analysis of scientific literature and the problem setting

The aim and the tasks of the research

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Conclusions