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


Geometry
---------


.. image:: images/100002010000019A000000EFE4218732F55AD697.png
    :width: 3.1492in
    :height: 1.8346in


.. _RefImage_100002010000019A000000EFE4218732F55AD697.png:

Fig. 1. Diagram of pipe geometry

Average radius: :math:`R=0.050m`

Pipe thickness: :math:`h=0.0025m`

Length: :math:`L=1m`

The vectors :math:`\vec{x}`, :math:`\vec{\mathrm{\theta }}`, and :math:`\vec{z}` represent the axial, circumferential, and radial directions of the pipe.

Displacements in the :math:`\vec{x}`, :math:`\vec{\mathrm{\theta }}`, and :math:`\vec{z}` directions are noted by :math:`u`, :math:`v`, and :math:`w`.


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

The properties of the material from which the pipe is made are:


.. csv-table::

    ":math:`E=2.{10}^{11}\mathrm{Pa}` ", "Young's module"
    ":math:`\mathrm{\mu }=0.3` ", "Poisson's ratio"
    ":math:`\mathrm{\rho }=7800\mathit{kg}/{m}^{3}` ", "Volume density"



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

     Eigenmodes are calculated under five different boundary conditions:


* FF: both ends of the pipe are free;


* SF: one end is free and the other is in simple support;


* SS: both ends are in simple support;


* CS: one end with simple support and the other embedded;


* CC: both ends are recessed.

     where F, S, and C represent *simply-supported*, *clamped*, and*free* in English.

     No load.


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

     Not applicable