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


Geometry
---------

The system consists of a pole resting on the ground and subjected to seismic stress. It is modelled by a mass, its connection with the ground by a spring :math:`{k}_{0}` whose behavior relationship reflects a non-linearity of the effort-displacement type.


.. image:: images/100005CC0000082A000011E125935A3D1510106A.svg
    :width: 105
    :height: 230


.. _RefImage_100005CC0000082A000011E125935A3D1510106A.svg:

Characteristics of the pole:

length: :math:`L=\mathrm{2 }m`;

section: :math:`S=\mathrm{0,3}{m}^{2}`.


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

Pole mass: :math:`m=\mathrm{450 }\mathrm{kg}`.

Link spring stiffness: :math:`{k}_{\mathrm{0 }}={10}^{5}N/m`.


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

**Boundary conditions**

The only authorized movements are translations along the :math:`X` axis: :math:`\mathrm{dy}=\mathrm{dz}=0`.

The corrective force :math:`{F}_{c}` due to the nonlinearity of the ground is defined by the following relationship:

:math:`{F}_{c}(x)=\frac{f({x}_{\mathrm{seuil}})}{{x}_{\mathrm{seuil}}}-f(x)` with, if :math:`x>{x}_{\mathit{seuil}}`, :math:`f(x)={k}_{0}\left[1–\frac{\mid x\mid }{{x}_{0}}\right]x`

We take :math:`{x}_{\mathrm{seuil}}={10}^{-6}m`, :math:`{k}_{0}={10}^{5}N/m`, and :math:`{x}_{0}=\mathrm{0,1}m`.

Under the keyword RELA_EFFO_DEPL of the DYNA_VIBRA operator, we therefore impose the function: :math:`{F}_{c}(x)=\frac{{k}_{0}}{{x}_{0}}\mathit{x.}[\mid x\mid -{x}_{\mathit{seuil}}]` if :math:`∣x∣>{x}_{\mathit{seuil}}`

         :math:`{F}_{c}(x)=0` if :math:`∣x∣\le {x}_{\mathit{seuil}}`

**Loading**

The ground is subjected to an acceleration :math:`\gamma (t)` in the :math:`x` direction, constructed in such a way that the movement of the mass-spring system is sinusoidal :math:`x=\mathrm{a.sin}(\omega t)` with :math:`a=\mathrm{0,01}` and :math:`\omega =\pi /4`.


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

In the initial state, the system is released from its equilibrium position with a speed :math:`{v}_{\mathrm{0 }}`: to :math:`t=0`, :math:`\mathrm{dx}(0)=0`, :math:`{v}_{0}=\mathrm{dx}/\mathrm{dt}(0)=a\mathrm{.}\omega` *.*