Introduction
====


Geometry
----

The study concerns a thick steel plate. Its geometry is simple. It is a simple flattened cube with the following dimensions [cf. :numref:`geometrie-vue-3D`]:

* length = :math:`\mathrm{2,0}` meters;

* width = :math:`\mathrm{2,0}` meters;

* thickness = :math:`\mathrm{0,003}` meters.

The plate is firmly embedded on one of its sides (ENCAS) and fixed on a vibrating table.

We want to calculate the displacement of a point located on the upper face in a corner of a plate, opposite to the embedment (point P).


.. figure:: images/geometrie_vue_3D.png
    :name: geometrie-vue-3D
    :width: 5.4917in
    :height: 2.672in


    Geometry (3D view)



.. figure:: images/geometrie_vue_dessus.png
    :width: 3.998in
    :height: 2.8354in
    :name: geometrie-vue-dessus

    Geometry (top view)


Materials
----

For linear analyses, steel is considered to be an isotropic, linear elastic material:


* Young's modulus: :math:`E\mathrm{=}200000.{10}^{+6}\mathit{Pa}`;


* Poisson's ratio: :math:`\nu \mathrm{=}\mathrm{0,3}`;


* density: :math:`\rho \mathrm{=}8000\mathit{kg}\mathrm{/}{m}^{3}`.

Post-elastic (plastic) behavior is idealized by an elastoplastic law with linear work hardening whose plastic modulus (the plastic slope) is 1\% of the elastic Young's modulus. The elastic stress threshold is 200 MPa.

Boundary conditions and loading
----

The plate is embedded on one of its sides.

A distracted (or somewhat facetious) engineer made a mistake in the vibrating table settings. Instead of the expected stress (a synthetic accelerogram), it sent a simple sinusoidal signal but of very high amplitude. The plate was not designed for such a shock and an attempt is being made to determine the movement that the plate may have undergone during this unfortunate test.

The input signal is a simple sine with frequency :math:`15\mathit{Hz}`.

The amplitude is :math:`\mathrm{30g}` horizontally and :math:`\mathrm{30g}` vertically downwards (:math:`g=10m/{s}^{2}`).

The duration of the signal is :math:`0.5s`.