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


Geometry
---------


.. image:: images/100002000000019C0000010CC4B9CE1295ACF8ED.png
    :width: 4.5071in
    :height: 3.4909in


.. _RefImage_100002000000019C0000010CC4B9CE1295ACF8ED.png:

Figure 1.1-1: Plate on spring supports subjected to imposed acceleration

The plate has a width of :math:`4m` (direction :math:`X`), a height of :math:`8m` (direction :math:`Y`), and a thickness of :math:`1m` in the :math:`Z` direction normal to its plane.

A double series of springs is assigned to the lower edge of the plate. A first series of 5 springs to represent the support of the plate and a second series of 5 springs interposed between the first and the lower edge of the plate to represent the contact springs by penalization.


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

For the material of the plate, we have:


.. csv-table::

    ":math:`E\mathrm{=}1.4\mathrm{\times }{10}^{8}\mathit{Pa}` "," :math:`\nu \mathrm{=}0.3` "," :math:`\rho =2.5\times {10}^{3}\mathrm{kg}/{m}^{3}`"


Proportional damping (RAYLEIGH):

:math:`C=\alpha K+\beta M` with :math:`\alpha =5.0\times {10}^{–3}s` and :math:`\beta =0.1{s}^{–1}`.

The first set of springs to represent the support of the plate has stiffness characteristics of :math:`1.0\times {10}^{8}\mathrm{N.m}` in the :math:`Y` horizontal direction. The other stiffness characteristics in the other directions as well as those of the penalization contact springs are worth :math:`1.0\times {10}^{15}\mathrm{N.m}`.


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

On the whole structure we impose :math:`\mathrm{DZ}=\mathrm{DRY}=0`.

The embedment condition is imposed at the bottom of the first series of springs in order to represent the support of the plate: :math:`\mathrm{DX}=\mathrm{DY}=\mathrm{DZ}=0`.

Gravity is applied in the vertical direction :math:`Y`.

An imposed single-press acceleration in the horizontal direction :math:`X` is also applied in the form of a Ricker signal with an amplitude :math:`0.23g` (cf.).


.. image:: images/100002000000033E0000027C4E52A814596E4452.png
    :width: 3.9764in
    :height: 2.9146in


.. _RefImage_100002000000033E0000027C4E52A814596E4452.png:

Figure 1.3-1: Imposed acceleration


Initial conditions
--------------------

The structure is initially at rest.