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


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

Integration of the equilibrium equation in a deformed configuration by assuming no extension of the mean axis.


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

The balance is described by:

.. math::
 :label: eq-1

    {w} _ {x}\ text {'}\ text {'}\ text {'} (z) =\ frac {-M (z)} {{\ mathit {EI}}} _ {y}} =\ frac {- {F} _ {F} _ {F} _ {F} _ {F}} _ {F} _ {F} _ {F} _ {F} _ {F} _ {F} _ {F} _ {f} _ {f} _ {f} _ {f} _ {f} _ {f} _ {f} _ {f} _ {f} _ {f} _ {f} _ {f} _ {f} _ {f} _ {f} _ {f} _ {f} _ {f} _ {f} _ {f} _ {f} _ {{\ mathit {EI}} _ {y}}

The :math:`{w}_{x}(z)` move is therefore worth:

.. math::
 :label: eq-2

    {w} _ {x} (z) =\ frac {{F}} _ {f} _ {x}} {{F} _ {z}}\ left\ {-z+\ frac {1} {\ mathrm {\ alpha}}}\ left [\ mathrm {\ alpha}}}\ left [\ mathrm {\ alpha}}}\ left [\ mathrm {\ alpha}}}\ left [\ mathrm {\ alpha}}}\ left [\ mathrm {\ alpha}}}\ left [\ mathrm {\ alpha}}}\ left [\ mathrm {\ alpha}}}\ left [\ mathrm {\ alpha}}}\ left [\ mathrm {\ alpha}}}\ left [\ mathrm {\ alpha}}}\ left (1-\ mathrm {cos} (\ mathrm {\ alpha} z)\ right)\ right]\ right\ right\}

with :math:`\mathrm{\alpha }=\sqrt{\frac{{F}_{z}}{{\mathit{EI}}_{y}}}` and :math:`l\approx {l}_{0}`. This last lever arm approximation is acceptable for moderate :math:`{F}_{x}` efforts.


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

Analytical solution


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


1. F. VOLDOIRE, Y. BAMBERGER: Structural mechanics, 2008.