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


Bibliographical reference
-------------------------

The reference solution comes from the results obtained in "Formulas for Stress and Strain" by ROARK'S (6th edition - McGraw-Hill International Editions), which can also be found in the fact sheet on the strength of materials by Jean Goulet and Jean-Pierre Boutin, page 192, combined with "Formulas for Stress, Strains and Structural Matrixes" by W. D. Pilkey, page 969.


Contactless reference analytical results
-----------------------------------------------

The results obtained were calculated on a plate in simple support subjected to vertical pressure (no contact).


* Calculation of the arrow at the center of the plate:

:math:`\mathrm{max}z=-\frac{\alpha p{l}^{4}}{E{e}^{3}}`


* Calculation of the stress at the center of the plate according to the width of the lower skin plate:


.. image:: images/Object_3.svg
    :width: 93
    :height: 43


.. _RefImage_Object_3.svg:

:math:`{\sigma }_{\mathrm{yy}}=+\frac{\beta p{l}^{2}}{{e}^{2}}`


* :math:`p` refers to the pressure applied to the plate,


* :math:`E` Young's modulus,


* :math:`L`, the length,


* :math:`l`, the width,


* :math:`e`, the thickness,


* :math:`\alpha ,\beta` being two coefficients obtained from the :math:`a/b` ratio.

:math:`\{\begin{array}{}\alpha =0.1110\\ \beta =0.6102\end{array}`

Either:

:math:`\{\begin{array}{}\mathrm{max}z=-0.69375\mathrm{cm}\\ {\sigma }_{\mathrm{yy}}=1.5255{10}^{10}N/{\mathrm{cm}}^{2}\end{array}`


Results obtained without contact with Code_Aster
----------------------------------------------


Modeling DKT
~~~~~~~~~~~~~~~~~

**Modeling:** **DKTpour test the contact between two plates.**

256 finite elements QUAD4 are arranged on the initial contact surface. The mesh has only one layer of elements in the thickness of the plate.


Characteristics of the mesh
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Number of knots: 289 knots

Number of meshes and type: 256 QUAD4


Aster reference values
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


.. csv-table::

    "**Identification**", "**Reference**", "**Aster** **** (DKT ****) **", "**Error**"
    ":math:`\mathrm{DZ}` in the center of the plate", "- 0.69375", "- 0.69138", "0.35%"
    ":math:`{\sigma }_{\mathrm{yy}}` in the center of the plate", "+ 1.5255 E+10", "+ 1.5298 E+10", "0.28%"



Modeling COQUE_3D
~~~~~~~~~~~~~~~~~~~~~~~

**Modeling:** **COQUE_3Dpour test the contact between two plates.**

256 finite elements QUAD9 are arranged on the initial contact surface. The mesh has only one layer of elements in the thickness of the plate.


Characteristics of the mesh
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Number of knots: 578 knots

Number of meshes and type: 256 QUAD9


Aster reference values
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


.. csv-table::

    "**Identification**", "**Reference**", "**Aster** **** (COQUE_3D) **", "**Error**"
    ":math:`\mathrm{DZ}` in the center of the plate", "- 0.69375", "- 0.65927", "4.97%"
    ":math:`{\sigma }_{\mathrm{yy}}` in the center of the plate", "+ 1.5255 E+10", "+ 1.41316 E+10", "7.36%"



Comments
------------

It can be seen that the results obtained without contact are very satisfactory for a modeling of the DKT type. The error obtained for a modeling of the COQUE_3D type can be explained by taking into account the effects of transverse shear which should not be negligible for this relatively thick plate since the :math:`l/e` ratio is equal to :math:`1/10`.