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


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

The calculation of the critical discharge load is given in detail in [:ref:`bib1 <bib1>`].


.. csv-table::

    ":math:`{P}_{\mathrm{cr}}={\gamma }_{2}\frac{\sqrt{E{I}_{y}C}}{{L}^{2}}` ", "critical spill load"
    "with :math:`C=\mathrm{GJ}` ", "torsional stiffness"
    ":math:`J=((b-2{h}_{1}){h}_{2}^{3}+2a{h}_{1}^{3})` ", "torsional constant [:ref:`bib2 <bib2>`]"
    ":math:`{C}_{1}E{I}_{y}\frac{{b}^{2}}{2}` ", "warping stiffness corresponding to an I-beam"


**Digital application:**

:math:`C=8578.515{\mathrm{N.m}}^{2}`

:math:`\mathrm{C1}=5516.8{\mathrm{N.m}}^{4}`

:math:`\frac{{L}^{2}C}{{C}_{1}}=6.22`

The value of :math:`{\gamma }_{2}` depends on the :math:`\frac{{L}^{2}C}{{C}_{1}}` ratio. In our case :math:`{\gamma }_{2}` is equal to 8.617. This value is taken from an array given in [:ref:`bib1 <bib1>`]. Which gives us :math:`{P}_{\mathrm{cr}}=104797.82N`


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

Critical discharge load and associated mode.


.. image:: images/Object_13.svg
    :width: 336
    :height: 171


.. _RefImage_Object_13.svg:


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

Analytical solution


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


1. S.P. TIMOSHENKO, J.M. GERE: Elastic Stability Theory, Second Edition, DUNOD 1966.


2. S.P. TIMOSHENKO: Material Strength, Volume 2: DUNOD 1968.