Buckling of an Elliptical Plate with Simple-Support Edge Conditions

Gerald Flanagan
Materials Sciences Corporation
Jan. 28, 2008

Introduction

The derivation of a buckling load estimate for an orthotropic plate with an elliptical boundary is presented. Thin-plate equations have been assumed. The solution uses an approximate displacement field in a Rayleigh-Ritz type approach. The assumed displacement field includes extra degrees of freedom that allow for buckling under inplane shear loads.

Basic Equations,Orthotropic

Plate Equations

The kinematic equations are

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where px and py are rotations about the midplane. The inplane strains are

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Transverse shear strains

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In order to compute the bending energy, extract the strains that are linear with respect to z

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The following vector is useful for computing the energy due to inplane loads interacting with the first-order finite deflection strains.

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bend is a bending stiffness matrix, and shear is the shear stiffness matrix.

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energy1 is the strain energy density

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energy2 is the energy related to the constant inplane loads Nx, Ny, and Nxy.

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Now introduce the assumed displacement field for w. c1-c3 are undetermined coefficients. The first set of terms assure that the w displacement is zero on the boundary of an ellipse. The terms containing c2 and c3 introduce non-symmetric terms that may be non-zero under a shear loading condition.

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At this point, a thin-plate theory is assumed. The mid-plane rotations are set equal to the derivatives of the w-displacement, as in a standard Kirkoff plate.

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Variable bounds of integration for an ellipse

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Integrate the strain energy density

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Integrate the work terms

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Solve for Coefficients

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Take derivatives wrt to each of the coefficients to minimize energy

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Find the coefficients with respect the coefficients c1-c3. This allows one to express the system of equations in matrix form.

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At this point, the most efficient numerical implimentation would probably be to use these matrices directly, and solve for the egenvalues. The minimum positive egenvalue would scale the applied loads to determine the critical buckling load vector. However, in Mathematica, the egenvalues can also be found in closed-form.

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The first two solutions (tmp[[1]] and tmp[[2]] involve the shear load Nxy. Examining only the shear buckling load from one of the eigenvalues.

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Plots of Displacement Fields

The basic bubble function (c2=c3=0)

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Now consider the case c1=0, c2=-c2

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Created by Wolfram Mathematica 6.0  (28 January 2008)