Stability of rectangular cantilever plates with high elasticity

Mikhail Sukhoterin,Tatiana Knysh,Sergey Baryshnikov,Elena Rasputina,V Breskich,S Uvarova

doi:10.1051/e3sconf/202124404004

Abstract

The problem of cantilever plate stability has been little studied due to the difficulty of solving the corresponding boundary problem. The known approximate solutions mainly concern only the first critical load. In this paper, stability of an elastic rectangular cantilever plate under the action of uniform pressure applied to its edge opposite to the clamped edge is investigated. Under such conditions, thin canopies of buildings made of new materials can be found at sharp gusts of wind in longitudinal direction. At present, cantilever nanoplates are widely used as key components of sensors to create nanoscale transistors where they are exposed to magnetic fields in the plate plane. The aim of the study is to obtain the critical force spectrum and corresponding forms of supercritical equilibrium. The deflection function is selected as a sum of two hyperbolic trigonometric series with adding special compensating summands to the main symmetric solution for the free terms of the decomposition of the functions in the Fourier series by cosines. The fulfillment of all conditions of the boundary problem leads to an infinite homogeneous system of linear algebraic equations with regard to unknown series coefficients. The task of the study is to create a numerical algorithm that allows finding eigenvalues of the resolving system with high accuracy. The search for critical loads (eigenvalues) giving a nontrivial solution of this system is carried out by brute force search of compressive load value in combination with the method of sequential approximations. For the plates with different side ratios, the spectrum of the first three critical loads is obtained, at which new forms of equilibrium emerge. An antisymmetric solution is obtained and studied. 3D images of the corresponding forms are presented.

Highlights

The classical problem of stability of a thin rectangular cantilever plate under the action of only compressive pressure applied to the free edge, parallel to the clamped edge, has no precise solution in a closed form
It is assumed that the plate material is perfectly elastic, so there is an infinite number of critical loads that change the form of equilibrium of the plate
In [11,12,13,14,15,16], for the solution of the stability problem of a rectangular plate clamped along the contour, Fourier series were used, which led to an infinite system of linear algebraic equations relative to the series coefficients

Summary

Introduction

The classical problem of stability of a thin rectangular cantilever plate under the action of only compressive pressure applied to the free edge, parallel to the clamped edge, has no precise solution in a closed form. This problem on eigenvalues is similar to the problem of determination of the spectrum of resonance frequencies and forms of oscillations of the cantilever plate [1, 2]. It is assumed that the plate material is perfectly elastic, so there is an infinite number of critical loads that change the form of equilibrium of the plate. In [11,12,13,14,15,16], for the solution of the stability problem of a rectangular plate clamped along the contour, Fourier series were used, which led to an infinite system of linear algebraic equations relative to the series coefficients

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