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


Calculation method
------------------


Static
~~~~~~~~

Travel in :math:`B`


* Single pull :math:`{u}_{x}\mathrm{=}\frac{{F}_{x}L}{ES}`


* Pure flex :math:`{u}_{z}\mathrm{=}\mathrm{-}\frac{{M}_{y}{L}^{2}}{2E{I}_{y}}` :math:`{\theta }_{y}\mathrm{=}\frac{{M}_{y}L}{E{I}_{y}}`


* Pure flex :math:`{u}_{y}\mathrm{=}\frac{{M}_{z}{L}^{2}}{2E{I}_{z}}` :math:`{\theta }_{z}\mathrm{=}\frac{{M}_{z}L}{E{I}_{z}}`

Maximum stress in :math:`A`


* Single pull :math:`{\sigma }_{x}\mathrm{=}\frac{{F}_{x}}{S}`


* Pure flex :math:`{\sigma }_{x}\mathrm{=}\mathrm{-}\frac{{M}_{y}}{\frac{2{I}_{y}}{h}}`


* Pure flex :math:`{\sigma }_{x}\mathrm{=}\mathrm{-}\frac{{M}_{z}}{\frac{{\mathrm{2I}}_{z}}{b}}`


Natural frequencies in bending
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Fashion 1: :math:`{f}_{1}\mathrm{=}\frac{3.516}{2{L}^{2}\pi }\sqrt{\frac{\mathit{EI}}{\rho S}}`

Mode 2: :math:`{f}_{2}\mathrm{=}\frac{22.0345}{2{L}^{2}\pi }\sqrt{\frac{\mathit{EI}}{\rho S}}`


Reference quantities and results
-----------------------------------


Static
~~~~~~~~


* Travel :math:`(m)`


.. csv-table::

    "Dot", ":math:`\mathit{DX}` "," :math:`\mathit{DY}` "," :math:`\mathit{DZ}`"
    ":math:`B` "," :math:`8.3333\mathrm{\times }{10}^{\mathrm{-}5}` "," :math:`1.6667\mathrm{\times }{10}^{\mathrm{-}4}` "," :math:`\mathrm{-}2.5\mathrm{\times }{10}^{\mathrm{-}4}`"



* Constraints :math:`(N\mathrm{/}{m}^{2})`


.. csv-table::

    "Point", ":math:`\mathit{SIXX}` "," :math:`\mathit{SIYY}` "," "," :math:`\mathit{SIZZ}` "," "," :math:`\mathit{SIXY}` "," :math:`\mathit{SIXZ}` "," :math:`\mathit{SIYZ}`"
    ":math:`\mathit{A1}(5.0\mathrm{,1}.5,\mathrm{-}1.0)` "," :math:`\mathrm{-}0.3333` "," :math:`0.` "," :math:`0.` "," "," :math:`0.` "," :math:`0.` "," :math:`0.`"
    ":math:`\mathit{A2}(5.0\mathrm{,1}.5\mathrm{,1}.0)` "," :math:`1.6667` "," :math:`0.` "," :math:`0.` "," "," :math:`0.` "," :math:`0.` "," :math:`0.`"



Natural frequencies in bending
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


.. csv-table::

    "Mode", "**Frequency** :math:`\mathit{Hz}`"
    ":math:`1` "," :math:`0.014449`"
    ":math:`2` "," :math:`0.090549`"



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

Analytical solution.