2. Benchmark solution#

2.1. Benchmark solution#

  • Critical loads

The analytical solution obtained with a thin plate theory in isotropic homogeneous linear elasticity [bib1] without taking into account the transverse shear energy determines the

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th critical load:

\({q}_{\mathit{cr}I}\mathrm{=}\frac{D{\pi }^{2}}{{L}^{2}}{(i+\frac{1}{i})}^{2}\)

with:

\(D=\frac{E{h}^{3}}{12(1-{\nu }^{2})}\): the flexural stiffness coefficient of the shell

\(h\): the thickness

\(L\): the length of the side of the square plate.

  • Membrane deformation

The analytical expression of membrane deformation along axis \(X\) is as follows:

\({e}_{\mathit{xx}}\mathrm{=}\frac{q\mathrm{\times }L}{(h\mathrm{\times }L\mathrm{\times }E)}\)

The reference result was calculated with \(q\mathrm{=}1.N\mathrm{/}\mathit{mm}\)

2.2. Benchmark results#

Some modes corresponding to the critical loads of the analytical solution are not symmetric and cannot be captured with the symmetry conditions for a quarter of a plate. The values of the critical loads obtained therefore correspond to the first 3 symmetric buckling modes:

  • Mode 1 of a quarter of the plate = Mode 1 of the entire plate

  • Mode 2 of a quarter of the plate = Mode 3 of the entire plate

  • Mode 3 of a quarter of the plate = Mode 5 of the entire plate

2.3. Uncertainty about the solution#

Exact solution for a plate theory without transverse shear.

2.4. Bibliographical references#

  1. J.G. EISLEY « Mechanics of Elastic Structures: Classical and Finite Element Methods ». Prentice Hall, Englewood Cliffs N.J. 07632 (19XX).

  2. « Stability of Square Plate Under Biaxial Loading ». The SAMCEF User’s Manuals V7.1. (1998).