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


Calculation method used for the reference solution
--------------------------------------------------------

The analytical reference solution is based on the Love-Kirchhoff theory, commonly used for so-called "thin" plates [bia1].

Taking into account the problem and at any point on the plate, we have for the calculation of the arrow:

:math:`w\mathrm{=}\frac{{f}_{0}{a}^{4}}{4{\pi }^{4}D}\mathrm{sin}\pi \frac{x}{a}\mathrm{sin}\pi \frac{y}{a}`

with:

:math:`D\mathrm{=}\frac{E{h}^{3}}{12(1\mathrm{-}{\nu }^{2})}`, :math:`{f}_{0}\mathrm{=}1`, :math:`a\mathrm{=}1`, and :math:`\nu \mathrm{=}0.25`

For the calculation of moments, the theory leads to the following expressions:

:math:`\begin{array}{ccc}{M}_{\mathit{xx}}& \mathrm{=}& \alpha (1+\nu )\mathrm{sin}\pi \frac{x}{a}\mathrm{sin}\pi \frac{y}{a}\\ {M}_{\mathit{yy}}& \mathrm{=}& {M}_{\mathit{xx}}\\ {M}_{\mathit{xy}}& \mathrm{=}& \mathrm{-}\alpha (1\mathrm{-}\nu )\mathrm{cos}\pi \frac{x}{a}\mathrm{cos}\pi \frac{y}{a}\end{array}`

For the calculation of curvatures, the theory leads to the following expressions:

:math:`{\kappa }_{\mathit{xx}}\mathrm{=}\mathrm{-}\frac{{f}_{0}{a}^{2}}{4{\pi }^{2}D}\mathrm{sin}(\pi \frac{x}{a})\mathrm{sin}(\pi \frac{y}{a})`

:math:`{\kappa }_{\mathit{yy}}\mathrm{=}\mathrm{-}\frac{{f}_{0}{a}^{2}}{4{\pi }^{2}D}\mathrm{sin}(\pi \frac{x}{a})\mathrm{sin}(\pi \frac{y}{a})`

:math:`{\kappa }_{\mathit{xy}}\mathrm{=}\frac{{f}_{0}{a}^{2}}{4{\pi }^{2}D}\mathrm{cos}(\pi \frac{x}{a})\mathrm{cos}(\pi \frac{y}{a})`

with :math:`\alpha \mathrm{=}\frac{{f}_{0}{a}^{2}}{4{\pi }^{2}}`

For shear forces, we get:

:math:`\begin{array}{c}{T}_{x}\mathrm{=}\frac{{f}_{0}a}{2\pi }\mathrm{cos}\pi \frac{x}{a}\mathrm{sin}\pi \frac{y}{a}\\ {T}_{y}\mathrm{=}\frac{{f}_{0}a}{2\pi }\mathrm{sin}\pi \frac{x}{a}\mathrm{cos}\pi \frac{y}{a}\end{array}`

For a homogeneous plate, the plane stresses are given by:

:math:`(\begin{array}{c}{\sigma }_{\mathit{xx}}\\ {\sigma }_{\mathit{yy}}\\ {\sigma }_{\mathit{xy}}\end{array})\mathrm{=}z\mathrm{[}A\mathrm{]}(\begin{array}{c}{M}_{\mathit{xx}}\\ {M}_{\mathit{yy}}\\ {M}_{\mathit{xy}}\end{array})`

with :math:`\mathrm{[}A\mathrm{]}\mathrm{=}\frac{12}{{h}^{3}}\mathrm{[}I\mathrm{]}` and :math:`z` the position in the thickness of the plate

and the transverse shear stresses by:

:math:`(\begin{array}{c}{\sigma }_{x}\\ {\sigma }_{y}\end{array})\mathrm{=}\mathrm{[}{D}_{1}(z)\mathrm{]}(\begin{array}{c}{T}_{x}\\ {T}_{y}\end{array})`,

with :math:`\mathrm{[}{D}_{1}(z)\mathrm{]}\mathrm{=}\frac{6}{{h}^{3}}({(\frac{h}{2})}^{2}\mathrm{-}{z}^{2})`

For the flexural deformation energy, we obtain:

:math:`{E}_{\mathit{flexion}}\mathrm{=}\frac{1}{2}{\mathrm{\int }}_{S}\mathrm{[}({M}_{\mathit{xx}}\mathrm{.}{\kappa }_{\mathit{xx}}+{M}_{\mathit{yy}}\mathrm{.}{\kappa }_{\mathit{yy}}+{M}_{\mathit{xy}}\mathrm{.}{\kappa }_{\mathit{xy}})\mathrm{]}\mathit{dS}`


Benchmark results
----------------------

For each of the models, we calculate:


* in the center of the plate, the displacement,


* in the center of the plate and in the middle of the :math:`\mathit{AB}` side, the constraints :math:`{\sigma }_{\mathit{xx}}`, :math:`{\sigma }_{\mathit{yy}}`,

    .. image:: images/Object_32.svg
        :width: 25
        :height: 25


    .. _RefImage_Object_32.svg:

,

    .. image:: images/Object_33.svg
        :width: 25
        :height: 25


    .. _RefImage_Object_33.svg:

,

    .. image:: images/Object_34.svg
        :width: 25
        :height: 25


    .. _RefImage_Object_34.svg:

on the plans:


* lower, middle and upper part of the plate in the case of monolayer,


* lower, middle and upper of each slice in the case of multi-layer (5 layers),


* at the center, at the corners and in the middle of the sides :math:`\mathit{AB}` and :math:`\mathit{AD}`, the membrane forces

    .. image:: images/Object_35.svg
        :width: 25
        :height: 25


    .. _RefImage_Object_35.svg:

,

    .. image:: images/Object_36.svg
        :width: 25
        :height: 25


    .. _RefImage_Object_36.svg:

,

    .. image:: images/Object_37.svg
        :width: 25
        :height: 25


    .. _RefImage_Object_37.svg:

, the shear forces :math:`{T}_{x}`, :math:`{T}_{y}` and the moments :math:`{M}_{\mathit{xx}}`, :math:`{M}_{\mathit{yy}}`, and :math:`{M}_{\mathit{xy}}`,


* at the center (point O), at point A, the membrane deformations :math:`{e}_{\mathit{xx}}`, :math:`{e}_{\mathit{yy}}`, :math:`{e}_{\mathit{xy}}` and the curvatures :math:`{\kappa }_{\mathit{xx}}`, :math:`{\kappa }_{\mathit{yy}}`, :math:`{\kappa }_{\mathit{xy}}`,

For the V modeling, the deformation energy :math:`\mathit{TOTALE}`, :math:`\mathit{MEMBRANE}` and :math:`\mathit{FLEXION}` is calculated at the center (point O).

Expressing these quantities at points :math:`O,A,B,C,D` gives:


.. csv-table::

    "", ":math:`w` "," :math:`{M}_{\mathit{xx}}` "," :math:`{M}_{\mathit{yy}}` "," "," :math:`{M}_{\mathit{xy}}` "," "," :math:`{T}_{x}` "," :math:`{T}_{y}` "," :math:`{\kappa }_{\mathit{xx}}` "," :math:`{\kappa }_{\mathit{yy}}` "," :math:`{\kappa }_{\mathit{xy}}`"
    ":math:`O` "," :math:`\frac{3(1\mathrm{-}{\nu }^{2})}{{\pi }^{4}E{h}^{3}}` "," :math:`\alpha (1+\nu )` "," :math:`\alpha (1+\nu )` "," ", "0", "0", "0"," :math:`\frac{\mathrm{-}\alpha }{D}` "," :math:`\frac{\mathrm{-}\alpha }{D}` "," 0"
    ":math:`A` ", "-", "0", "0", "0"," 0", ":math:`\mathrm{-}\alpha (1\mathrm{-}\nu )` ", "0"," 0", "0"," :math:`\frac{\alpha }{D}`"
    ":math:`B` ", "-", "0", "0", "0"," 0", ":math:`\alpha (1\mathrm{-}\nu )` ", "0", "-", "-", "-"
    ":math:`\mathit{B1}` ", "-", "0", "0", "0", "0", "0"," :math:`1\mathrm{/}2\pi` ", "-", "-", "-"
    ":math:`C` ", "-", "0", "0", "0"," 0", ":math:`\mathrm{-}\alpha (1\mathrm{-}\nu )` ", "0", "-", "-", "-"
    ":math:`D` ", "-", "0", "0", "0"," 0", ":math:`\alpha (1\mathrm{-}\nu )` ", "0", "-", "-", "-"


Digital application:

:math:`\frac{3(1\mathrm{-}{\nu }^{2})}{{\pi }^{4}E{h}^{3}}\mathrm{=}1.154923`

:math:`\alpha (1+\nu )\mathrm{=}0.0316629`

:math:`\alpha (1\mathrm{-}\nu )\mathrm{=}0.0189972`

:math:`1\mathrm{/}2\pi \mathrm{=}0.159155`

The distribution of plane and shear stresses at points :math:`O` and :math:`\mathit{B1}` inside the plate is as follows:


.. csv-table::

    ":math:`O` "," :math:`{\sigma }_{\mathit{xx}}` "," :math:`{\sigma }_{\mathit{yy}}` "," "," :math:`{\sigma }_{\mathit{xy}}` "," :math:`{\sigma }_{\mathit{xz}}` "," :math:`{\sigma }_{\mathit{yz}}`"
    ":math:`h\mathrm{/}2` ", "18.9972", "18.9972", "18.9972", "0", "0"
    ":math:`\mathrm{3h}\mathrm{/}10` ", "11.3983", "11.3983", "11.3983", "0", "0"
    ":math:`h\mathrm{/}10` ", "3.7994", "3.7994", "0", "0", "0", "0"
    "0", "0", "0", "0", "0", "0"
    ":math:`\mathrm{-}h\mathrm{/}10` ", "—3.7994", "—3.7994", "0", "0", "0", "0"
    ":math:`\mathrm{-}\mathrm{3h}\mathrm{/}10` ", "—11.3983", "—11.3983", "0", "0", "0", "0"
    ":math:`\mathrm{-}h\mathrm{/}2` ", "—18.9972", "—18.9972", "0", "0", "0", "0"



.. image:: images/1000141E000069D50000615C2104F04402FACEA0.svg
    :width: 25
    :height: 25


.. _RefImage_1000141E000069D50000615C2104F04402FACEA0.svg:


.. csv-table::

    ":math:`\mathit{B1}` "," :math:`{\sigma }_{\mathit{xx}}` "," :math:`{\sigma }_{\mathit{yy}}` "," "," :math:`{\sigma }_{\mathit{xy}}` "," :math:`{\sigma }_{\mathit{xz}}` "," :math:`{\sigma }_{\mathit{yz}}`"
    ":math:`h\mathrm{/}2` ", "0", "0", "0", "0", "0"
    ":math:`\mathrm{3h}\mathrm{/}10` ", "0", "0", "0", "0", "1.5278"
    ":math:`h\mathrm{/}10` ", "0", "0", "0", "0", "2.3777"
    "0", "0", "0", "0", "0", "2.3873"
    ":math:`\mathrm{-}h\mathrm{/}10` ", "0", "0", "0", "0", "2.3777"
    ":math:`\mathrm{-}\mathrm{3h}\mathrm{/}10` ", "0", "0", "0", "0", "1.5278"
    ":math:`\mathrm{-}h\mathrm{/}2` ", "0", "0", "0", "0", "0"



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

Analytical solution.


Bibliographical references
---------------------------


1. BATOZ and DHATT. Modeling of structures by finite elements. Beams and Plates. Volume 2 HERMES, 1990.