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


Geometry
---------


.. image:: images/Object_1.svg
    :width: 627
    :height: 237


.. _RefImage_Object_1.svg:

:math:`L=\mathrm{50m}`; :math:`{R}_{f}=\mathrm{1m}`; :math:`{R}_{i}=\frac{5}{3}m`; :math:`{R}_{e}=\mathrm{3m}`; :math:`{k}_{1}={10}^{9}{\mathrm{N.m}}^{-1}`;; :math:`{k}_{2}=0.5{10}^{7}{\mathrm{N.m}}^{-1}`; :math:`{\rho }_{\mathrm{fluide}}=1000{\mathrm{kg.m}}^{-3}`; :math:`{\rho }_{s}=\mathrm{7800 }{\mathrm{kg.m}}^{-3}`; shell thickness: :math:`50\mathrm{cm}`.


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

**Fluid:** density :math:`{\rho }_{1}=1000{\mathrm{kg.m}}^{-3}`; :math:`{\rho }_{2}=750{\mathrm{kg.m}}^{-3}`.

**Structure:** :math:`{\rho }_{s}=7800\mathrm{kg}/{m}^{3}`; :math:`E=2.1{10}^{11}\mathrm{Pa}`;; :math:`\nu =0.3` (steel).


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

The outer cylinder is on the one hand connected to a fixed frame via the four springs of unitary stiffness :math:`{k}_{1}`, on the other hand connected to the middle cylinder by four springs of unitary stiffness :math:`{k}_{2}`. Both structures are rigid in this reference solution.