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

Geometry
---------

A discrete element of zero size with 2 knots.

Local coordinate system = global coordinate system.

A K_ TR_D_L stiffness matrix assigned by default (associated with an DIS_TR_L element)

:math:`1.6N/\mathrm{mm}` in translation, :math:`1.9N/\mathrm{mm}` in rotation.

The stiffness characteristics in the local directions :math:`x` and rotation around :math:`y` are modified by a behavior relationship of type ASSE_CORN introduced by a characteristic material.


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

Linked to an incremental behavior ASSE_CORN including 2 mechanisms each requiring 5 characteristic parameters (see [:ref:`fig 1.2-a <fig 1.2-a>`] and [:ref:`fig 1.2-b <fig 1.2-b>`]):

:math:`\stackrel{ˉ}{{N}_{1}}=10050N`, :math:`\stackrel{ˉ}{{M}_{1}}=150000\mathrm{N.mm}`, :math:`\stackrel{ˉ}{{U}_{1}}=1\mathrm{mm}`, :math:`\stackrel{ˉ}{{\theta }_{1}}=6.7{10}^{-2}`, :math:`\stackrel{ˉ}{{C}_{1}}=0.95`

:math:`\stackrel{ˉ}{{N}_{2}}=50000N`, :math:`\stackrel{ˉ}{{M}_{2}}=750000\mathrm{N.mm}`, :math:`\stackrel{ˉ}{{U}_{2}}=10\mathrm{mm}`, :math:`\stackrel{ˉ}{{\theta }_{2}}=0.01`, :math:`\stackrel{ˉ}{{C}_{2}}=0.95`


.. image:: images/10000000000001AB0000015ECD5C4E0819456294.png
    :width: 2.9528in
    :height: 2.4098in


.. _RefImage_10000000000001AB0000015ECD5C4E0819456294.png:


.. image:: images/10000000000001AB00000162834B43707B1CED1D.png
    :width: 2.9528in
    :height: 2.4366in


.. _RefImage_10000000000001AB00000162834B43707B1CED1D.png:

**Figure** 1.2-a **: Assembly mechanisms under normal force and moment.**

:math:`\begin{array}{}R(p)=\sqrt{\mathrm{n²}+\mathrm{m²}}\\ \dot{p}\cdot (\begin{array}{}n\\ m\end{array})=R(p)\cdot (\begin{array}{}\dot{{U}_{r}}\\ \dot{{\theta }_{r}}\end{array})\end{array}`


+-----------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+
|                                                                                                                             |:math:`n=\mathrm{Nx}/\stackrel{ˉ}{N}`                                                                                                                                                    |
+ .. image:: images/1000000000000200000001B00B6DA04A010D3ADF.png                                                              +                                                                                                                                                                                         +
|    :width: 4.2575in                                                                                                         | :math:`m=\mathrm{My}/\stackrel{ˉ}{M}`                                                                                                                                                   |
+    :height: 3.5953in                                                                                                        +                                                                                                                                                                                         +
|                                                                                                                             | :math:`{U}_{r}=U/\stackrel{ˉ}{U}`                                                                                                                                                       |
+                                                                                                                             +                                                                                                                                                                                         +
|                                                                                                                             | :math:`\theta =\theta /\stackrel{ˉ}{\theta }`                                                                                                                                           |
+-----------------------------------------------------------------------------------------------------------------------------+-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------+


**Figure** 1.2-b **: Assembly behavior relationship**

with

:math:`\begin{array}{ccc}\dot{p}& =& \sqrt{\dot{{U}_{r}^{2}}+\dot{{\theta }_{r}^{2}}}\\ p& =& {R}^{-1}(p\text{'})=h(p\text{'})=\frac{1-c}{{c}^{2}}\mathrm{.}\frac{{p}^{2}\text{'}}{1-p\text{'}}\end{array}`


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

Embedding in one of the 2 knots.

Force imposed in the :math:`x` direction per unit of :math:`\mathrm{1 000 }N` and Moment imposed around the :math:`z` axis per unit of :math:`\mathrm{3 000 }N`. All on the second node, in load increments.


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

Zero displacements, efforts, and internal variables.