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


Static calculation: embedded beam
----------------------------------

The theory of beams working under bending provides a reference value for maximum displacement. For a beam embedded at one end and free at the other under distributed loading, the maximum displacement at the free end, called an arrow, is given by:

:math:`f\mathrm{=}\frac{\mathrm{-}{\mathit{qL}}^{4}}{{\mathrm{8EI}}_{\mathit{eq}}}`


* :math:`q`: the force distributed in :math:`N/m`.


* :math:`L`: the length of the beam in :math:`m`.


* :math:`{\mathit{EI}}_{\mathit{eq}}`: in the mono-material case it is the product of the Young's modulus and the quadratic moment.

In the multi-material case, and with the configuration described above, :math:`{\mathit{EI}}_{\mathit{eq}}` is calculated as follows:

:math:`{\mathit{EI}}_{\mathit{eq}}\mathrm{=}{\mathrm{\int }}_{S}E(s){z(s)}^{2}\mathit{ds}`

In *Code_Aster* this calculation is approximated by a sum on the fibers:

:math:`{\mathit{EI}}_{\mathit{eq}}\mathrm{=}{\mathrm{\sum }}_{i\mathrm{=}1}^{{\mathit{nb}}_{\mathit{fibres}}}{E}_{i}{{z}_{i}}^{2}{A}_{i}`

where :math:`{z}_{i}` is the :math:`Z` coordinate of the center of the fiber :math:`i` and where :math:`{A}_{i}` is its surface.


Modal calculation: simple supported beam
-------------------------------------

For a beam with simple support, as defined in paragraph :ref:`1.3.1 <RefNumPara__4818_1648519106>`, the theory of beams provides reference values for the natural frequencies of the modal calculation.

The first natural frequency is: :math:`{f}_{1}\mathrm{=}\frac{\pi }{2{L}^{2}}\sqrt{\frac{{\mathit{EI}}_{\mathit{eq}}}{m}}`.

where :math:`L` is the length of the beam and :math:`m` is the linear mass of the beam.


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

None.