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


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


Flexion problem
~~~~~~~~~~~~~~~~~~~~~

The reference solution to the flexure problem is the one given in sheet SDLS03 /89 of guide VPCS, which presents the calculation method as follows.

The formulation of M.V. BARTON for a rectangular plate, placed on its four sides, leads to:

:math:`{f}_{\mathit{ij}}=\frac{\pi }{2}\left[{\left[\frac{i}{a}\right]}^{2}+{\left[\frac{j}{b}\right]}^{2}\right]\sqrt{\frac{E{t}^{2}}{12\rho (1-{\nu }^{2})}}`

with:

:math:`i` = half-wavelength number according to :math:`y` (dimension :math:`a`),

:math:`j` = number of half-wave lengths according to :math:`x` (dimension :math:`b`).


Membrane problem
~~~~~~~~~~~~~~~~~~~~~~

For the search for the first vibration frequency, the problem treated with the membrane is equivalent to the following one-dimensional problem:


.. image:: images/100002A000001389000003D3590F44B9BA27389B.svg
    :width: 252
    :height: 49


.. _RefImage_100002A000001389000003D3590F44B9BA27389B.svg:

where:

:math:`k` is the stiffness of the springs,

:math:`m` is the mass of the plate.

So the frequency sought is: :math:`f=\frac{1}{2\pi }\sqrt{\frac{\mathrm{4k}}{m}}`


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

For the flexure problem, the first six vibration frequencies are calculated and for the membrane calculation, only the first frequency is calculated.


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

Since the solutions are analytical, there is no uncertainty.


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


1. M.V. BARTON "Vibrations of rectangular and skew cantilever plates" - Journal of Applied Mechanics, vol.18, p.129-134 (1951).