Benchmark solution
=====================

Modeling A
--------------

Benchmark results
~~~~~~~~~~~~~~~~~~~~~~~~

For the following displacement :math:`x`, it is given by the "normal" part of the discretes in parallel, which gives:

:math:`{F}_{x}={K}_{\mathit{el}}{u}_{x}+{K}_{n}({u}_{x}+{\mathit{DIST}}_{1})` if :math:`{u}_{x}<-{\mathit{DIST}}_{1}`,

:math:`{F}_{x}={K}_{\mathit{el}}{u}_{x}` otherwise.

We get:

.. csv-table::

    "**Time**", ":math:`{u}_{x}`"
    "0.5", "-0.5"
    "1.0", "-0.75"
    "2.0", "-0.75"


Table 2.1.1-a: Reference Solution

For the following displacement y, it is given by the "tangential" part of the discretes in parallel, which gives:

:math:`\begin{array}{c}{F}_{y}={k}_{t}\left({u}_{y}-{\delta }_{y}^{0}\right)+{k}_{\mathit{el}}{u}_{y}\\ \delta ={\delta }^{0}\end{array}` if :math:`∣{k}_{t}∣\left({u}_{y}-{\delta }^{0}\right)\le \mu ∣{F}_{x}∣`,

:math:`\begin{array}{c}{F}_{y}=\mu ∣{F}_{x}∣\mathit{sgn}\left({u}_{y}-{\delta }^{0}\right)+{k}_{\mathit{el}}{u}_{y}\\ \delta =u-\mathit{sgn}({u}_{y}-{\delta }^{0})\mu {K}_{n}({u}_{x}+{\mathit{DIST}}_{1})/{K}_{t}\end{array}` otherwise.

We noted the "sign" function :math:`\mathit{sgn}`.

We get:

.. csv-table::

    "**Time**", ":math:`{u}_{y}`"
    "1.05", "0.1333333"
    "1.50", "1.875"
    "1.55", "1.74166666"
    "2.00", "0.125"


Table 2.1.1-b: Reference Solution


Uncertainty about the solution
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

No uncertainty (analytical solution).


B & C models
-------------------

Benchmark results
~~~~~~~~~~~~~~~~~~~~~~~~

Charging paths 1 and 3
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

The formulas are identical to those of modeling A, with the characteristics of the discretes derived from the materials used for the B and C models.

For moments :math:`[\mathrm{1.5,}\mathrm{3.5,}4.0]` the normal effort in the discreet is

:math:`{N}_{1}=\mathit{RIGI}\underline{\phantom{\rule{2em}{0ex}}}\mathit{NOR}\phantom{\rule{2em}{0ex}}\mathrm{.}\phantom{\rule{2em}{0ex}}\mathit{Ux}=10000.0\phantom{\rule{2em}{0ex}}\mathrm{.}\phantom{\rule{2em}{0ex}}1.0`

For moments :math:`[\mathrm{9.0,}\mathrm{10.0,}11.0]` the normal effort in the discreet is

:math:`{N}_{2}=\mathit{RIGI}\underline{\phantom{\rule{2em}{0ex}}}\mathit{NOR}\phantom{\rule{2em}{0ex}}\mathrm{.}\phantom{\rule{2em}{0ex}}\mathit{Ux}=10000.0\phantom{\rule{2em}{0ex}}\mathrm{.}\phantom{\rule{2em}{0ex}}2.0`

This gives, with a coefficient of friction of :math:`0.3`

:math:`{\mathit{Seuil}}_{1}=\frac{0.3{N}_{1}}{\sqrt{2}},{\mathit{Seuil}}_{2}=\frac{0.3{N}_{2}}{\sqrt{2}}`


.. csv-table::

    "**INST**", "**N**", "**VY**", "**VZ**"
    "**1.50**", "-10000.0", "-Threshold 1", "-Threshold 1", "-Threshold 1"
    "**3.50**", "-10000.0", "Threshold 1", "Threshold 1"
    "**4.00**", "-10000.0", "Threshold 1", "Threshold 1"
    "**6.00**", "0.0", "0.0", "0.0"
    "**9.00**", "-20000.0", "0.0", "0.0"
    "**10.00**", "-20000.0", "-Threshold 2", "-Threshold 2"
    "**11.00**", "-20000.0", "-Threshold 2", "-Threshold 2"


Table 2.2.1.1-a: Reference solution, B and C models, paths 1 and 3


.. image:: images/1000020100000940000005DA07D6A23FBF2AB11C.png
    :width: 5.9047in
    :height: 3.2819in


.. _RefImage_1000020100000940000005DA07D6A23FBF2AB11C.png:

**Figure** 2.2.1.1-a **:**: **Discreet efforts as a function of time** . **


Charging path 2
^^^^^^^^^^^^^^^^^^^^^^^^^^

The forces in the tangential plane are verified for several angles:

:math:`[\mathrm{0,30,45,60,90,120,135,150,180,210,225,240,270,300,315,330,360}]`

The forces in the tangential plane are given by the following expressions:

:math:`{V}_{Y}=\text{RIGI\_NOR}\phantom{\rule{2em}{0ex}}.{D}_{x}\mathrm{\mu }\mathrm{sin}(\mathit{Angle})`

:math:`{V}_{z}=\text{RIGI\_NOR}\phantom{\rule{2em}{0ex}}.{D}_{x}\mathrm{\mu }\mathrm{cos}(\mathit{Angle})`


.. image:: images/1000020100000A54000005E8D53508B1176A96D8.png
    :width: 5.8701in
    :height: 3.2535in


.. _RefImage_1000020100000A54000005E8D53508B1176A96D8.png:

**Figure** 2.2.1.2-a **:**: **Discreet efforts as a function of time** . **


Uncertainty about solutions
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

No uncertainty (analytical solutions).