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

.. _Ref35354159:


Geometry and boundary conditions
-----------------------------------

The figure below represents the reference problem: a cylindrical cable immersed under the free surface in a column of water of height :math:`h=30m`. The ends of the beam are supported by two springs, which are themselves embedded at their ends.

Note :math:`{S}_{w}` the hydraulic section (seen by the fluid) transverse to the beam.


.. image:: images/100002010000041D0000037838E06D7DA834D231.png
    :width: 4.6783in
    :height: 3.9457in


.. _RefImage_100002010000041D0000037838E06D7DA834D231.png:


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

The material has an isotropic elastic behavior:

.. csv-table::

    "Young's module", ":math:`200\mathit{GPa}`"
    "Poisson's Ratio", ":math:`0.3`"
    "Cable density :math:`\mathrm{\rho }` "," :math:`7800\mathit{kg}/{m}^{3}`"
    "Water density :math:`{\mathrm{\rho }}_{w}` "," :math:`1000\mathit{kg}/{m}^{3}`"
    "Spring stiffness according to x, y and z", ":math:`50.0N/m`"
    "Drag coefficient :math:`{C}_{d}` "," :math:`1.0`"



.. _Ref35350656:


Wave kinematics
--------------------

We consider a regular monodirectional swell directed along the :math:`+x` axis, of order 1 (Airy swell), of height :math:`H`, of period :math:`T` and of wavelength :math:`\mathrm{\lambda }`, see diagram below.


.. image:: images/10000201000002F50000020D0162112424016B88.png
    :width: 3.4437in
    :height: 2.389in


.. _RefImage_10000201000002F50000020D0162112424016B88.png:

The free surface :math:`\mathrm{\eta }` as a function of space :math:`x` and time :math:`t` is expressed:

.. math::
 :label: eq-1

    \ mathrm {\ eta} (x, t) =\ frac {H} {2}\ mathrm {cos} (kx-\ mathrm {\ omega} t)

with :math:`k=\frac{2\mathrm{\pi }}{\mathrm{\lambda }}` the wave number and :math:`\mathrm{\omega }=\frac{2\mathrm{\pi }}{T}` the pulsation.

The speed of fluid :math:`{V}_{w}=(u,v,w)` is as follows:

.. math::
 :label: eq-2

    \ begin {array} {c} u (x, z, t) =0\\ v (x, z, t) =\ frac {H} {2}\ mathrm {\ omega}\ frac {\ mathrm {sinh} (k, z, t) =0\\ v (x, z, t) =\ frac {array} {c}}\ mathrm {sinh} (\ mathrm {.} x-\ mathrm {\ omega} t)\\ w (x, z, t) =\ frac {H} {2}\ mathrm {\ omega}\ frac {\ mathrm {sinh} (k)} (k (h+z))} {\ mathrm {sinh} (kh)}\ mathrm {sin} (k\ mathrm {sinh})} {\ mathrm {sinh} (kh)}\ mathrm {sin} (k\ mathrm {sinh}} (k\ mathrm {sinh})} {\ mathrm {sinh} (kh)}\ mathrm {sin} (k\ mathrm {sinh})} {\ mathrm {sinh} (kh)}} x-\ mathrm {\ omega} t)\ end {array}

The acceleration of the fluid is noted :math:`{a}_{w}=\frac{\partial {v}_{w}}{\partial t}`.


Loads
-----------

The cable is subject to drag forces induced by a field of regular waves of height :math:`H=3m` and period :math:`T=12s`.

Note that Archimedes' weight and thrust are not taken into account in this problem.

**Modeling A**

The cable is subject to Morison's efforts (drag term only):

.. math::
 :label: eq-3

    {F} _ {\ text {mor}} =\ frac {1} {2} {2} {2} {C} _ {d} {\ mathrm {\ rho}}} _ {w} _ {w} ({v} _ {w}} ^ {v} _ {w} ^ {v}} ^ {\ text {ortho}}}) ({v} _ {w}} ^ {v} _ {w} ^ {\ text {ortho}} - {v} _ {s} ^ {\ text {ortho}})

where :math:`{v}_{w}` and :math:`{v}_{s}` respectively designate the fluid speed and the solid speed (mechanical load input parameter FORCE_POUTRE). The symbol :math:`\text{ortho}` represents the projection in the direction normal to the neutral fiber of the beam [1] _
. It should be noted that these are linear forces that are applied in a distributed manner on the beam.

*Note: Morison's efforts depend on the position of the beam; they should be mentioned as follower loadings in the calculations.*


.. [1] This projection also requires the orientation of the beam to be passed as an input parameter of FORCE_POUTRE.