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


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

The solution of the so-called "elastic" problem is presented in [:ref:`bib1 <bib1>`] by assuming the non-extension of the mean axis. The analytical solution is obtained by considering elliptical integrals.


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

The reference results selected for the checks are indicated in **bold** characters in the table below. Displacements are defined in the geometry definition coordinate system [:ref:`§1.1 <§1.1>`].


.. csv-table::

    ":math:`P\mathrm{/}{P}_{\mathit{cr}}` "," :math:`{u}_{B}\mathrm{/}L` "," :math:`{w}_{B}\mathrm{/}L` "," "," :math:`\theta °` "," "," :math:`{M}_{A}L\mathrm{/}\mathit{EI}` ", "", "Charge :math:`P`
    (:math:`N`)", ":math:`{u}_{B}`
    (:math:`m`)", ":math:`{w}_{B}`
    (:math:`m`)", ":math:`{M}_{A}`
    (:math:`\mathit{N.M}`)"
    "**1.015**", "**0.220**", "**0.030**", "**0.030**", "**0.56**", "", "**1141.07**", "**0.1100**", "**0.1100**", "**0.1100**", "**0.1100**", "**0.0150**", "**127.58**"
    "1.063", "0.422", "0.119", "0.119", "40°", "1.09", "", "1195.03", "0.2110", "0.0595", "0.0595", "248.32"
    "**1.152**", "**0.593**", "**0.593**", "**0.259**", "**1.67**", "**1295.09**", "**0.2965**", "**0.2965**", "**0.2965**", "**0.2965**", "**0.2965**", "**0.2965**", "**0.2965**", "**0.2965**", "**0.2965**", "**0.2965**", "**0.2965**", "**0.2965**",
    "1.293", "0.719", "0.440", "0.440", "80°", "80°", "2.28", "", "1453.60", "0.3595", "0.2200", "0.2200", "519.41"
    "**1.518**", "**0.792**", "**0.651**", "**0.651**", "**2.96**", "", "**1706.55**", "**0.3960**", "**0.3960**", "**0.3960**", "**0.3960**", "**0.3960**", "**0.3960**", "**0.3960**", "**0.3960**", "**0.3960**", "**0.3960**", "**0.3960**", "**0.3960**
    "1.884", "0.803", "0.877", "0.877", "120°", "120°", "3.73", "", "2118.01", "0.4015", "0.4385", "0.4385", "849.74"
    "**2.541**", "**0.750**", "**0.750**", "**1.107**", "**4.70**", "", "**2856.62**", "**0.3750**", "**0.3750**", "**0.3750**", "**0.3750**", "**0.3750**", "**0.3750**", "**0.3750**", "**0.3750**", "**0.3750**", "**0.3750**", "**0.3750**", "**0.3750**"
    "4.029", "0.625", "1.340", "1.340", "160°", "6.20", "", "4529.44", "0.3125", "0.6700", "0.6700", "1412.44"
    "**9,116**", "**0.421**", "**0.421**", "**1.577**", "**9.44**", "", "**10248.29**", "**0.2105**", "**0.2105**", "**0.2105**", "**0.2105**", "**0.2105**", "**0.2105**", "**0.2105**", "**0.2105**", "**0.2105**", "**0.2105**", "**0.2105**", "**0.2105**",



.. _1027492766:


.. image:: images/Object_6.png
    :width: 3.3374in
    :height: 2.9374in


.. _RefSchema_Object_6.png:


.. image:: images/Object_7.png
    :width: 3.2752in
    :height: 2.6689in


.. _RefSchema_Object_7.png:


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

Analytical solution


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


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


2. J.L. BATOZ: Large displacements and large rotations of thin elastic beams, Department of Mechanical Engineering University of Technology of Compiègne 1981.