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

Geometry
---------


.. image:: images/Object_1.png
    :width: 5.5in
    :height: 3.85in


.. _RefSchema_Object_1.png:


.. csv-table::

    "Outside diameter", ":", ":math:`48.E-3m`"
    "Thickness", ":", ":math:`5.E-3m`"
    "Elbow radius", ":", ":math:`0.170m`"



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

.. csv-table::

    "Density", ":", ":math:`7960.{\mathrm{kg.m}}^{-3}`"
    "Young's module", ":", ":math:`1.9E+11{\mathrm{N.m}}^{-2}`"
    "Poisson's ratio", ":", ":math:`0.3`"
    "Mass concentrated at node 4", ":", ":math:`10.\mathrm{kg}`"



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

Boundary conditions
~~~~~~~~~~~~~~~~~~~~~~~~

At node 1: :math:`\mathrm{dx}=\mathrm{dy}=\mathrm{dz}=0` (ball joint)

At node 7: :math:`\mathrm{dx}=\mathrm{dy}=\mathrm{dz}=\mathrm{drx}=\mathrm{dry}=\mathrm{drz}=0` (embedding)

At node 2, support in direction :math:`z`

At node 3, support in direction :math:`y`

At node 5, support in direction :math:`y`

At node 6, support in the :math:`x` direction

The stiffness provided by each of the supports are:

:math:`{K}_{x}={K}_{y}={K}_{z}=80.E+3{\mathrm{N.m}}^{-1}` :math:`{K}_{\theta x}={K}_{\theta y}={K}_{\theta z}=J=1.2{\mathrm{N.m.deg}}^{-1}`


Loads
~~~~~~~~~~~

**Calculating static modes**

A first calculation makes it possible to validate the calculation of the static modes.

The entire model is subject to a uniform acceleration according to :math:`x` with a value of :math:`100\mathrm{\times }g` with :math:`g=9.81{\mathrm{m.s}}^{-2}`

**Spectral response**

The pipe line is subjected to an excitation following :math:`x`, defined by a response spectrum such as:

• its value in displacement for frequencies between 1 and 10. :math:`\mathrm{Hz}`, or :math:`d=48.E-\mathrm{3m}`

• its speed value for frequencies between 10. and 63. :math:`\mathrm{Hz}`, or :math:`v={\mathrm{3.m.s}}^{-1}`

• its acceleration value for frequencies between 63. and 1000. :math:`\mathrm{Hz}`, or :math:`\gamma =120\ast g`

Below is shown the acceleration spectrum, determined from the excitation for reduced damping :math:`\xi \mathrm{=}0`.


.. image:: images/100002000000027E000001C2B82D53D1ED537DBB.png
    :width: 3.6362in
    :height: 2.5646in


.. _RefImage_100002000000027E000001C2B82D53D1ED537DBB.png:

The characteristic values used are:

:math:`\gamma (1\mathrm{Hz})=\mathrm{1.92m}/{s}^{2}` :math:`\gamma (10\mathrm{Hz})=192m/{s}^{2}` :math:`\gamma (63\mathit{Hz})\mathrm{=}1000m\mathrm{/}{s}^{2}`

:math:`\gamma (1000\mathrm{Hz})=1000m/{s}^{2}`


Elbow flexibility
~~~~~~~~~~~~~~~~~~~~~~~~~

The flexibility coefficient, :math:`{C}_{\mathrm{flex}}`, of the elbows is given by the RCC -M regulation:

:math:`{C}_{\mathrm{flex}}=\frac{1.65}{h}` with :math:`h=\frac{\mathrm{ep}\ast {r}_{\mathrm{courb}}}{{r}_{m}^{2}}`

.. csv-table::

    ":math:`\mathrm{ep}` ", ":", "elbow thickness"
    ":math:`{r}_{\mathrm{courb}}` ", ":", "elbow radius of curvature"
    ":math:`{r}_{m}` ", ":", "mean elbow radius"


The stress intensification index, :math:`{I}_{\mathrm{sigm}}`, is given by:

:math:`{I}_{\mathrm{sigm}}=\frac{0.9}{{h}^{0.666}}`

According to regulation RCC -M, the flexibility coefficient and the stress intensification index are greater than or equal to one. This is not the case in this test case.