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


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


Distributed traction-compression loading
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

A straight beam of length :math:`L` working only under traction-compression is subjected to a loading distribution that is constant along :math:`x` but varies sinusoidally as a function of time. It is embedded at both ends.

:math:`\mathrm{\{}\begin{array}{c}\rho S\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{t}^{2}}\mathrm{-}\mathit{ES}\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^{2}}\mathrm{=}f(t)\\ u(0)\mathrm{=}\mathrm{0,}u(L)\mathrm{=}0\end{array}`

To solve, we apply the Fourier transform in time to the equation:

:math:`\frac{{\mathrm{\partial }}^{2}\stackrel{ˆ}{u}}{\mathrm{\partial }{x}^{2}}\mathrm{=}\mathrm{-}\frac{\rho }{E}4{\pi }^{2}{\omega }^{2}\stackrel{ˆ}{u}+\frac{1}{\mathit{ES}}\stackrel{ˆ}{f}(\omega )`


.. csv-table::

    ":math:`\stackrel{ˆ}{u}` ", ":", "Fourier transform of :math:`u`,"
    ":math:`\stackrel{ˆ}{f}` ", ":", "Fourier transform of :math:`f`."


So, for :math:`f(t)\mathrm{=}F\mathrm{cos}(2\pi {\omega }_{0}t)`, we have:


.. image:: images/Object_10.svg
    :width: 429
    :height: 145


.. _RefImage_Object_10.svg:

with: :math:`{a}^{2}\mathrm{=}\frac{E}{\rho }`.

Using the law of behavior gives us the tensile force compression:

:math:`N(x,t)=\frac{aF}{2\pi {\omega }_{0}}\left\{\left[1-\mathrm{cos}\left(\frac{2\pi {\omega }_{0}}{a}L\right)\right]\frac{\mathrm{cos}\left(\frac{2\pi {\omega }_{0}}{a}x\right)}{\mathrm{sin}\left(\frac{2\pi {\omega }_{0}}{a}L\right)}-\mathrm{sin}\left(\frac{2\pi {\omega }_{0}}{a}x\right)\right\}\mathrm{cos}\left(2\pi {\omega }_{0}t\right)`


Punctual loads
~~~~~~~~~~~~~~~~~~~~~~~

A console beam of length :math:`L` working only in tensile compression (or torsional) is subjected to a sinusoidal force in time, (or a moment) applied to its free end.


Traction
^^^^^^^^^


.. image:: images/Object_13.svg
    :width: 429
    :height: 145


.. _RefImage_Object_13.svg:

The resolution technique is equivalent to that in paragraph [:ref:`§2.1.1.1 <§2.1.1.1>`].

For :math:`f(t)\mathrm{=}F\mathrm{cos}(2\pi {\omega }_{0}t)`, we have:


.. image:: images/Object_15.svg
    :width: 429
    :height: 145


.. _RefImage_Object_15.svg:

and

.. image:: images/Object_16.svg
    :width: 429
    :height: 145


.. _RefImage_Object_16.svg:


Torsion
^^^^^^^^


.. image:: images/Object_17.svg
    :width: 429
    :height: 145


.. _RefImage_Object_17.svg:


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

Domestic efforts (:math:`N` and :math:`\mathrm{MT}`)


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

Analytical solution.


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


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