B modeling (POU_D_EM)
=========================


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

Longitudinal mesh of the beam:

It is composed of 17 nodes and 16 pairs of POU_D_EM elements (16 elements for concrete and 16 for steel).

Beam cross section:

The concrete is modelled by a mesh (DEFI_GEOM_FIBRE/SECT) composed of :math:`2\times 20` quadrilaterals (40 fibers)


.. image:: images/100024E000003D2F00002B190F04B7B20931898D.svg
    :width: 182
    :height: 187


.. _RefImage_100024E000003D2F00002B190F04B7B20931898D.svg:

**Figure 3.1-a: Discretization of the section**

Steel is modelled by 4 point fibers (DEFI_GEOM_FIBRE/FIBRE)

Depreciation coefficients :math:`\alpha` and :math:`\beta` are calculated using the following formula

:math:`\left\{\begin{array}{c}\alpha \\ \beta \end{array}\right\}\mathrm{=}2\frac{{\omega }_{1}{\omega }_{2}}{{\omega }_{2}^{2}\mathrm{-}{\omega }_{1}^{2}}\left\{\begin{array}{cc}\frac{1}{{\omega }_{2}}& \frac{1}{{\omega }_{1}}\\ {\omega }_{2}& \mathrm{-}{\omega }_{1}\end{array}\right\}\left\{\begin{array}{c}{\xi }_{1}\\ {\xi }_{2}\end{array}\right\}`

where :math:`{\omega }_{1}` and :math:`{\omega }_{2}` are the first two natural pulsations :math:`(\omega =2\pi f)` and :math:`{\xi }_{1}` and :math:`{\xi }_{2}` are the desired damping on the first two modes.

With :math:`{f}_{1}=37.8\mathrm{Hz}` and :math:`{f}_{2}=149.2\mathrm{Hz}` (see paragraph [:ref:`§4 <§4>`]), for modal depreciations of :math:`\text{5\%}`, we find: :math:`\alpha =8.5{10}^{-5}` and :math:`\beta =18.985`.

For the calculation of the temporal response, the time step chosen is 1/100th of a second.

**3.2** **Tested quantities and results**

The reaction curves as a function of time and the arrow in the center as a function of time are shown in figures [:ref:`Figure 4-a <Figure 4-a>`] to [:ref:`Figure 4-d <Figure 4-d>`].


.. image:: images/100000000000025100000152F892C0EBCFC7E42C.png
    :width: 6.1756in
    :height: 3.5201in


.. _RefImage_100000000000025100000152F892C0EBCFC7E42C.png:

**Figure 4-a: Reaction to the first press as a function of time**


.. image:: images/100000000000025200000155982CAF89F16B7B05.png
    :width: 6.1866in
    :height: 3.5516in


.. _RefImage_100000000000025200000155982CAF89F16B7B05.png:

**Figure 4-b: Detail of the reaction between 2 and 3 seconds**


.. image:: images/100000000000024B0000017466CBF66FF4CD2263.png
    :width: 6.1138in
    :height: 3.8744in


.. _RefImage_100000000000024B0000017466CBF66FF4CD2263.png:

**Figure 4-c: Arrow in the center as a function of time**


.. image:: images/100000000000026200000179B9236EF183F6A763.png
    :width: 6.3535in
    :height: 3.9264in


.. _RefImage_100000000000026200000179B9236EF183F6A763.png:

**Figure 4-d: Detail of the arrow between 2.5 and 2.8 seconds**

Results tests (TEST_RESU) are carried out for the first three natural frequencies. The reaction on the first press is also tested and the arrow in the center is tested at the times :math:`\mathrm{1s}` (step 100) and :math:`\mathrm{2s}` (step 200), then for the first 2 extremes of the curves, at the times :math:`\mathrm{2,68}s` (step 268) and the arrow in the center are tested at the moments (step 268) and :math:`\mathrm{4,68}s` (step 468).


.. csv-table::

    "**Natural frequency**", "**ASTER Ref**", "**ASTER**", "**Relative error %**"
    "1", "37.80", "37.83", "0.07"
    "2", "149.20", "149.28", "0.05"
    "3", "200.30", "200.39", "0.04"



.. csv-table::

    "**REACTION**", "**ASTER Ref**", "**ASTER**", "****", "**E** **relative error %**"
    ":math:`\mathrm{1,00}s` ", "1,8878.104", "1,8479.104", "2,1"
    ":math:`\mathrm{2,00}s` ", "6,3393.104", "6,2184.104", "1,9"
    ":math:`\mathrm{2,68}s` ", "—2,3222.105", "—2,2443.105", "3,4"
    ":math:`\mathrm{4,68}s` ", "2,4692,105", "2,3979,105", "2,9"



.. csv-table::

    "**FLECHE**", "**ASTER Ref**", "**ASTER**", "**Relative error %**"
    "1.00 s", "—6,0694.10—4", "—5,9846.10—4", "1,4"
    "2.00 s", "—2,3507.10—3", "—2,3362.10—3", "0.6"
    "2.68 s", "8.5790.10—3", "8.3929.10—3", "2.2"
    "4.68 s", "—9,1084.10—3", "—8,9530.10—3", "1,7"