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


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

Displacement: analytical solution obtained by serial decomposition of the form:

:math:`w=\underset{i}{\Sigma }\underset{j}{\Sigma }{w}_{\mathrm{ij}}\mathrm{sin}(\frac{i\pi x}{a})\mathrm{sin}(\frac{j\pi y}{b})`

Constraints: numerical solution [:ref:`bib1 <bib1>`], [:ref:`bib2 <bib2>`]


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


.. image:: images/10001408000069D5000042AAA65542425923FD54.svg
    :width: 344
    :height: 214


.. _RefImage_10001408000069D5000042AAA65542425923FD54.svg:

The baseline results are as follows:


.. csv-table::

    ":math:`w(\mathrm{0,}\mathrm{0,}0)` "," :math:`0.01507m` ", "Move :math:`w` to the center of the plate (point :math:`A`),"
    ":math:`\mathrm{SIXX}(\mathrm{0,}\mathrm{0,}h/2)` "," :math:`2.4216{10}^{7}\mathrm{Pa}` ", "Stress :math:`{\sigma }_{\mathit{xx}}` on the upper skin of the upper skin from layer :math:`3` (:math:`z=h/2`) to the center of the plate (point A),"
    ":math:`\mathrm{SIYY}(\mathrm{0,}\mathrm{0,}h/6)` layer to :math:`90°` "," :math:`5.7810{10}^{6}\mathrm{Pa}` ", "Stress :math:`{\sigma }_{\mathit{yy}}` on the upper skin of the layer :math:`2` (:math:`z=h/6`) at the center of the plate (point :math:`A`),"
    ":math:`\mathrm{SIXY}(a/\mathrm{2,}a/\mathrm{2,}h/2)` "," :math:`1.2825{10}^{6}\mathrm{Pa}` ", "Stress :math:`{\sigma }_{\mathrm{xy}}` at point :math:`C` on the upper skin of the layer :math:`3`,"
    ":math:`\mathrm{SIXZ}(a/\mathrm{2,}\mathrm{0,}0)` "," :math:`2.3526{10}^{5}\mathrm{Pa}` ", "Stress :math:`{\sigma }_{\mathrm{xz}}` at point :math:`D` on the middle skin of the layer :math:`2` (:math:`z=0`),"
    ":math:`\mathrm{SIYZ}(\mathrm{0,}a/\mathrm{2,}0)` "," :math:`8.8950{10}^{4}\mathrm{Pa}` ", "Stress :math:`{\sigma }_{\mathrm{yz}}` at point :math:`B` on the middle skin of the layer :math:`2` (:math:`z=0`),"


Uncertainties about the solution
----------------------------


* The reference solution is given for a number of terms in the series equal to 25.


* The transverse shear correction factor used is 5/6.


* With significant slenderness (:math:`a/h=100`), the level of transverse shear is low and therefore difficult to obtain accurately. There is then uncertainty about the stress values :math:`{\sigma }_{\mathrm{ij}}` calculated during the validation of the :math:`\mathit{VPCS}` test, the differences obtained by the software on the shear components are of the order of :math:`\text{10\%}`.


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


1. VPCS: Software for calculating composite structures; Validation examples. Review of Composites and Advanced Materials, Volume 5 - special issue/ 1995. Hermes edition.


2. PUTCHA, N.S. and REDDY, J.N.: A mixed shear flexible finite element for the analysis of laminated plates, computer meth. in applied mech. Eng. 44 (1984).