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


Geometry
---------

Straight pipe of length :math:`80m`, modelled in shells between :math:`0` and :math:`\mathrm{40m}`, and in beams between :math:`40` and :math:`\mathrm{80m}`.

Outer radius: :math:`\mathrm{2m}`, thickness: :math:`\mathrm{0.1m}`.

The pipe axis is in plane :math:`\mathrm{Oxy}`, angled 30 degrees from :math:`\mathrm{Ox}`.


.. image:: images/Object_1.svg
    :width: 396
    :height: 180


.. _RefImage_Object_1.svg:


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

:math:`E={2.10}^{11}\mathrm{Pa}`

:math:`\nu =0.3`


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

"Beam-type" embedding in :math:`x=y=0`, made by a shell-beam connection between the :math:`\mathrm{C1}` edge of the shell and a point :math:`\mathrm{P1}` (located in :math:`O`). It is this point that is blocked.

4 unit load cases applied to point :math:`\mathit{P2}(80\mathrm{\times }\mathit{cos30},80\mathrm{\times }\mathit{sin30},0)`

Traction :math:`\mathrm{Fx1}=\mathrm{1N}` along the :math:`\mathrm{Ox1}` axis, i.e. :math:`\mathrm{Fx}=\mathrm{cos30.Fx}` and :math:`\mathrm{Fy}=\mathrm{sin30.Fx1}`

Torsional moment :math:`\mathrm{Mx1}=\mathrm{1Nm}` around :math:`\mathrm{Ox1}`, i.e. :math:`\mathrm{Mx}=\mathrm{cos30.Mx1}` and :math:`\mathrm{My}=\mathrm{sin30.Mx1}`

Bending moment :math:`\mathrm{My1}=\mathrm{1Nm}` around :math:`\mathrm{Oy1}` i.e. :math:`\mathrm{Mx}=-\mathrm{sin30.My1}` and :math:`\mathrm{My}=\mathrm{cos30.My1}`

Bending moment around :math:`\mathrm{Oz}`, which is :math:`\mathrm{Mz}=\mathrm{1Nm}`