Benchmark solutions
======================


Calculation method used for reference solutions
----------------------------------------------------------

These are natural frequencies: solutions to the homogeneous problem without damping.

It is partially given in [:ref:`bib1 <bib1>`]. The resolution method, of the finite element type, concerns a POU_D_TG model. However, a series of results are provided in the event that the effects of warping twist are overlooked, bringing the modeling back to a POU_D_T.


.. csv-table::

    "Mode number", "1", "2", "2", "3", "4", "5"
    "Frequency (:math:`\mathrm{Hz}`)", "3,797", "7,788", "7,788", "11,74", "15,68", "19,62"


**Table 2.1-** A: Benchmark results according to [:ref:`bib1 <bib1>`]

Some confidence can be placed in these results published in a peer-reviewed journal. However, uncertainties exist if one wants to reproduce these calculations: the torsional constants :math:`{J}_{x}` and the shear constants :math:`{k}_{y}` are not provided in the article. They had to be recalculated based on the geometry of the section.


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

Natural frequencies of the girder without damping


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

Comparison between codes (CAILLOU [:ref:`bib2 <bib2>`] and ASTER), and analytical solution.


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


* WU J.S. & CHEN K.Z.: Dynamic Analysis of a channel beam due to a moving load. J. of Sound and Vibration, vol. 188, no. 3, pp 337-345, 1995.


* Code CAILLOU version 4 of October 30, 1996, IAT


* Report no. 2314/A from the Aerotechnical Institute "Proposal and implementation of new test cases missing the validation of beams ASTER"