Modeling A
==============


Characteristics of modeling
-----------------------------------

We use a POU_D_EM model.

the beam is modelled with 1 finite element and the section is discretized with 8 fibers. The reference axis is deliberately chosen different from the barycenter, offset by h/2 downwards, with the keyword COOR_AXE_POUTRE from DEFI_GEOM_FIBRE.

The section is "cut" into 8 fibers (Figure 2). The coordinates of the fiber centers and their areas are given in table 1.


.. image:: images/1000020100000066000000987E9CF55EEACE648C.png
    :width: 1.4165in
    :height: 2.111in


.. _RefImage_1000020100000066000000987E9CF55EEACE648C.png:

Figure 2: Dividing the section into 8 fibers


.. csv-table::

    "Fibers", ":math:`{y}_{i}` "," :math:`{z}_{i}` "," "," :math:`{S}_{i}` ", "", "Fibers", ":math:`{y}_{i}` "," :math:`{z}_{i}` "," :math:`{S}_{i}`"
    "1", "0.1", "0.875", "0.75", "0.05", "", "5", "-0.1", "0.875", "0.05"
    "2", "0.1", "0.625", "0.05", "0.05", "", "6", "-0.1", "0.625", "0.05"
    "3", "0.1", "0.375", "0.05", "0.05", "", "7", "-0.1", "0.375", "0.05"
    "4", "0.1", "0.125", "0.05", "0.05", "", "8", "-0.1", "0.125", "0.05"


Table 1: Characteristics of fibers

Characteristics of the mesh
----------------------------

The mesh contains 1 elements of type SEG2.


Tested sizes and results
------------------------------

Numerical calculation of the quantities of the straight section gives:

:math:`S=0.4{m}^{2}` :math:`{A}_{G}=0.2{m}^{3}` :math:`{I}_{G}=0.13125{m}^{4}` and :math:`{I}_{{G}_{0}}=0.03125{m}^{4}`

The arrow at the loaded end of the beam is: :math:`f=0.00035555m`

At the level of embedding :math:`x=0` we have the following values:

:math:`M=-{10}^{-6}\mathit{Nm}` :math:`{\chi }_{s}=0.00106666{m}^{-1}` :math:`{ϵ}_{s}=-0.00053333{m}^{-1}`

At the level of the first Gauss point :math:`x=(1-\frac{1}{\sqrt{3}})/2=0.21132486540518503m` we have the following values: :math:`M=-788675.13\mathit{Nm}` :math:`{\chi }_{s}=0.00084125{m}^{-1}` :math:`{ϵ}_{s}=-0.0004206{m}^{-1}`

for fiber No. 1 (:math:`z=0.875m`): :math:`ϵ=0.00031547` and :math:`\sigma =9464101.6\mathit{Pa}` and for fiber No. 4 to :math:`z=0.125m` :math:`ϵ=-0.00031547` and :math:`\sigma =-9464101.6\mathit{Pa}`

Boom at the end of the beam (DEPL):

.. csv-table::

    "Point", "Component", "**Reference Value**", "**Tolerance**"
    "APPUI ", "DZ", ":math:`-3.5555555555555E-4` "," :math:`1.E-6`"


Generalized deformations during embedding (DEGE_ELNO):

.. csv-table::

    "Mesh", "Knot", "Component", "**Reference Value**", "**Tolerance**"
    "M1", "N1", "KY", ":math:`1.066666666666667E-3` "," :math:`1.E-6`"
    "M1", "N1"," EPXX "," :math:`-5.333333333333333E-3` "," :math:`1.E-6`"


Deformations and stresses in fibers (EPSI_ELGA and SIEF_ELGA):

.. csv-table::

    "Mesh", "Point", "Sub-Point", "Sub-Point", "Component", "**Reference Value**", "**Tolerance**"
    "M1", "1", "1"," EPXX "," "," :math:`3.15470053837926E-4` "," :math:`1.E-6`"
    "M1", "1", "1"," SIXX "," "," :math:`9.46410161513778E6` "," :math:`1.E-6`"
    "M1", "1", "4"," EPXX "," "," :math:`-3.15470053837926E-4` "," :math:`1.E-6`"
    "M1", "1", "4"," SIXX "," "," :math:`-9.46410161513778E6` "," :math:`1.E-6`"