Benchmark solution
=====================


Benchmark solution
---------------------


* Critical loads

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

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    :width: 9
    :height: 17


.. _RefImage_Object_3.svg:

th critical load:

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

with:

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

:math:`h`: the thickness

:math:`L`: the length of the side of the square plate.


* Membrane deformation

The analytical expression of membrane deformation along axis :math:`X` is as follows:

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

The reference result was calculated with :math:`q\mathrm{=}1.N\mathrm{/}\mathit{mm}`


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


Uncertainty about the solution
---------------------------

Exact solution for a plate theory without transverse shear.


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).