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


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

The reference solution comes from an analytical solution from [:ref:`Bib1 <Bib1>`]:

.. math::
 :label: eq-1

    {u} _ {x} (x, y) =\ frac {4 (1- {\nu} ^ {2})} {{\ mathrm {EL}}} ^ {2}}} ^ {2}}}\ mathrm {.} (\ mathrm {y.} ({x} ^ {2} -\ mathrm {2.Lx}) +\ frac {2+\nu (1-\nu)} {3}\ mathrm {.} y\ mathrm {.} y\ mathrm {.} (\ frac {{L} ^ {2}}} {4} - {y} ^ {2}))\ mathrm {.} {\ sigma} _ {d}

And next :math:`y`:

.. math::
 :label: eq-2

    {u} _ {y} (x, y) =\ frac {4 (1- {\nu} ^ {2})} {{\ mathrm {EL}} ^ {2}} ({\ mathrm {Lx}}} ^ {2}} ^ {2}} -\ frac {{x}} ^ {2})} {3} -\nu\ mathrm {.} (1-\nu}} {\ mathrm {.}} (1-\nu}} ^ {2}} (1-\nu)\ mathrm {.} (1-\nu)\ mathrm {.} (1-\nu}} m {.} {y} ^ {2}\ mathrm {.} (x-L) +\ frac {4+5\nu\ mathrm {.} (1-\nu)} {12}\ mathrm {.} {\ mathrm {xL}} ^ {2}))\ mathrm {.} {\ sigma} _ {d}


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

Applying, we find the following displacement :math:`x` at point :math:`C`: :math:`{u}_{x}(L,L/2)=-1.5\mathrm{mm}`

And by applying, we find the following displacement :math:`y` to point :math:`C`: :math:`{u}_{y}(L,L/2)=4.25\mathrm{mm}`


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

[:ref:`Bib1 <Bib1>`] Timoshenko & Woinowsky-Krieger, "Theory of Plates and Shells," McGrawhill, 1964.