Reference problem
====

Geometry
----

The study concerns a pipe comprising two straight pipes and an elbow [:ref:`Figure 1.1-a <Figure 1.1-a>`].

The geometric data for the problem is as follows:


*   the length :math:`{L}_{G}` of the two straight pipes is :math:`3m`,


*   the radius :math:`\mathit{Rc}` of the elbow is :math:`0.6m`,


*   the angle :math:`\theta` of the elbow is :math:`90°`,


*   the :math:`\text{e}` thickness of the straight pipes and the elbow is :math:`0.02m`,


*   and the outside radius :math:`\text{Re}` of the straight pipes and the elbow is :math:`0.2m`.


.. image:: images/Object_1.svg
    :width: 522
    :height: 401


.. _RefImage_Object_1.svg:

Figure 1.1-a

**Note:**

*The geometry of the problem is symmetric with respect to the plane* :math:`(A,X,Y)` *.*


Material properties
----

For all models A, B, C and D:

Isotropic linear elastic material. Material properties are those of :math:`\mathrm{A42}` to :math:`20°C` steel:


*   Young's module :math:`E=204000.\times {10}^{+6}N/{m}^{2}`,


*   Poisson's ratio :math:`\mathrm{\nu }=0.3`.

For thermo-elastic calculation (modeling A):


*   the thermal expansion coefficient :math:`\mathrm{\alpha }=1.096\times {10}^{–5}/°C`,


*   heat conduction :math:`\mathrm{\lambda }=54.6W/m°C`,


*   volume heat :math:`\mathrm{\rho }\mathit{Cp}=3.71\times {10}^{6}J/{m}^{3}°C`,

For dynamic calculation (B, D models):

• density :math:`\rho =7800\mathrm{kg}/{m}^{3}`,

• the depreciation of clean modes will be taken to :math:`\text{5\%}` for modes.


Boundary conditions and loads
----

The boundary conditions for all models are as follows:

• embedding at the level of section :math:`A`,

• during static loads, embedding at the level of section :math:`A` and section :math:`B`.

As far as static calculations are concerned, the loads applied are of three types:

• internal pressure (on the inner side) :math:`P={15.10}^{+6}N/{m}^{2}` (shell or 3D modeling),

• thermo-mechanical loading with a temperature transient imposed on the inner face of the pipe (rise from :math:`20°C` to :math:`70°C` in :math:`\mathrm{10s}`) and a zero exchange condition on the outer face of the pipe (thermal insulation) (modeling A only).

As far as dynamic calculation is concerned, the load applied is a transitory force (in Newtons):

:math:`\mathit{FY}(t)=10000000.\mathrm{sin}(2\mathrm{\pi }\mathit{Freq}1\mathrm{.}t)` directed along the :math:`Y` axis and applied to section :math:`B` with :math:`\mathit{Freq}1=20\mathit{Hz}`.