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


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

The reference formula is the one given in sheet SDLS02 /89 of the guide VPCS, which presents the calculation method as follows:

The formulation of M.V. BARTON, for a side plate a, led to:

:math:`{f}_{i}=\frac{1}{2\pi {a}^{2}}{\lambda }_{i}^{2}\sqrt{\frac{E{t}^{2}}{12\rho (1-{\nu }^{2})}}` :math:`i=\mathrm{1,2},\mathrm{...}`

where: :math:`{\lambda }_{i}^{2}=g(\alpha )`

with, for a Poisson's ratio :math:`\nu =0.3` and :math:`\alpha =30°`:


.. csv-table::

    "", ":math:`\alpha =30°`"
    ":math:`{\lambda }_{1}^{2}` ", "3.961"
    ":math:`{\lambda }_{2}^{2}` ", "10.19"



* M.V.Barton mentions the sensitivity of the result to the order of the mode and to angle :math:`\alpha`.


* This reference solution applies to thin plates such as: :math:`t/a<0.1`.


* The :math:`{\lambda }_{i}` coefficients were established with limited and insufficient development.


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

The first two eigenmodes given by:


* M.V.Barton's formula,


* the average of 5 software packages for calculating by the finite element method.


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

Semi-analytical solution :math:`\text{< 2\%}`.


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


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