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


Geometry
---------

The structure studied is similar to the curved zone of a steam generator tube. It is supposed to be embedded at its two ends :math:`A` and :math:`C`; the latter are supposed to correspond to the passage of the tube in the upper spacer plate of a steam generator. At point :math:`B` at the top of the hanger (also called apex), the structure is guided into a game support.


.. image:: images/10000F6A000069D500004F60C8098B1AA9E282FB.svg
    :width: 250
    :height: 153


.. _RefImage_10000F6A000069D500004F60C8098B1AA9E282FB.svg:

**Figure 1.1-a: Diagram of the curved zone of a GV tube,** **supposed to be embedded in the last spacer plate, guided into a game support**

The structure is similar to a beam with a hollow circular cross section with a total length of :math:`\mathrm{1,74329}m` and a straight :math:`\mathrm{250,43 }\mathrm{mm}` at each end of the tube.

Outer tube diameter: :math:`\mathrm{22,22 }\mathrm{mm}`

Inner tube diameter: :math:`\mathrm{19,68 }\mathrm{mm}`


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

The values of the characteristics of the various elements of the structure are as follows:

**Inconel 600 tube**


.. csv-table::

    ":math:`E={\mathrm{2,02 10}}^{11}N/{m}^{2}` "," :math:`\nu =\mathrm{0,3}` "," :math:`\rho =\mathrm{8330 }\mathrm{kg}/{m}^{3}`"



.. csv-table::

    "**Internal fluid:**", "the modelled internal fluid is water under pressure at high temperature; its density :math:`{\rho }_{i}` is assumed to vary linearly along the curved zone, between :math:`A` and :math:`C`, between and, from :math:`\mathrm{738,58 }\mathrm{kg}/{m}^{3}` to :math:`\mathrm{731,16 }\mathrm{kg}/{m}^{3}`."
    "**External fluid:**", "the tube is supposed to be immersed in a two-phase mixture over its entire length; the flow of the mixture is transverse to the hanger at all points. The equivalent density of the mixture is obtained using the formula:"


:math:`{\rho }_{e}\mathrm{=}\alpha {\rho }_{\mathit{gaz}}+(1\mathrm{-}\alpha ){\rho }_{\mathit{liquide}}`, where :math:`\alpha` refers to the volume of gas.

This density appears to be between :math:`84\mathrm{kg}/{m}^{3}` and :math:`150\mathrm{kg}/{m}^{3}`. An equivalent density is assigned to the dynamic system when calculating its modal fluid base at rest; this equivalent density includes the density of the internal fluid, that of the structure and that of the external fluid; the inertial effect of the latter is evaluated by means of an added mass coefficient.


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

The structure is embedded at points :math:`A` and :math:`C`. A game stand is positioned at point :math:`B` (:math:`\mathrm{jeu}=\mathrm{1,20 }\mathrm{mm}`). A distributed random load, transverse to the tube, is imposed on :math:`(A-C)`. This loading is defined, on the one hand using a speed profile along the excited zone, and on the other hand using a dimensionless excitation spectrum.


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

The tube is initially at rest, which results in zero displacements and velocities at the initial moment.