Reference problem
=====================


Geometry
---------


.. image:: images/1000233A00002F0700001231480C68541A0B872D.svg
    :width: 606
    :height: 234


.. _RefImage_1000233A00002F0700001231480C68541A0B872D.svg:

Beam length: :math:`L=0.6m`

Consistent thickness: :math:`h=0.01m`

**Rectangular section:**


.. csv-table::

    "Initial right section:", "", "Section variation:"
    "width:", ":math:`{b}_{0}=0.03m` "," :math:`b={b}_{0}{e}^{-2\alpha x}` with :math:`\alpha =1.`"
    "do:", ":math:`{A}_{0}={3.10}^{-4}{m}^{2}` "," :math:`A={A}_{0}{e}^{-2\alpha x}`"
    "moment of inertia:", ":math:`{\mathrm{Iz}}_{0}=0.25{10}^{-8}{m}^{4}` "," :math:`\mathrm{Iz}={\mathrm{Iz}}_{0}{e}^{-2\alpha x}`"


**Point coordinates (** :math:`m` **) :**


.. csv-table::

    "", ":math:`A` "," :math:`B`"
    ":math:`x` ", "0. ", "0.6"
    ":math:`y` ", "0. ", "0."



Material properties
-----------------------

:math:`E={2.10}^{11}\mathrm{Pa}`

:math:`\nu =0.3`

:math:`\rho =7800.\mathrm{kg}/{m}^{3}`


Boundary conditions and loads
-------------------------------------

Points :math:`A` and :math:`B`: embedded, :math:`u=v=0.`, :math:`\theta =0`.