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

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 composed of :math:`2\mathrm{\times }20` quadrilaterals (40 fibers).

Steel is modelled by 4 point fibers.

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{\mathrm{-}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 \mathrm{=}2\pi f)` and :math:`{\xi }_{1}` and :math:`{\xi }_{2}` are the desired damping on the first two modes.

With :math:`{f}_{1}=39.9\mathrm{Hz}` and :math:`{f}_{2}=157.6\mathrm{Hz}` (calculated with *Code_Aster*), for modal depreciations of :math:`\text{5\%}`, we find: :math:`\alpha ={8.10}^{-5}` and :math:`\beta =20`.

The time step chosen is :math:`\mathrm{0,01}s`.

The law for the behavior of concrete is Mazars in version :math:`\mathrm{1D}`.


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

The support reaction curves as a function of time as well as the arrow in the center as a function of time are shown in figures to.


.. image:: images/10000200000003CA0000025C05B72637803C6408.png
    :width: 6.0634in
    :height: 3.7717in


.. _RefImage_10000200000003CA0000025C05B72637803C6408.png:

**Figure** 3.2-a: **Press reaction A as a function of time for the first three seconds.**


.. image:: images/10000200000003CA0000025C220D2F3718DDC6B7.png
    :width: 6.0634in
    :height: 3.7717in


.. _RefImage_10000200000003CA0000025C220D2F3718DDC6B7.png:

**Figure** 3.2-b: f **arrow in the center according to the time for the first three seconds.**


.. image:: images/10000200000003CA0000025CDD38CD0D6B398C1E.png
    :width: 6.0634in
    :height: 3.7717in


.. _RefImage_10000200000003CA0000025CDD38CD0D6B398C1E.png:

**Figure** 3.2-c: c **Reaction-arrow curve for the first three seconds.**

The tests are carried out for the reaction on the first press and the arrow in the center. These values are tested for a few moments in the first three seconds of the earthquake, i.e. at times :math:`\mathrm{1.76s}` (at the very beginning of the non-linear domain), then :math:`\mathrm{2.05s}`, :math:`\mathrm{2.68s}` and :math:`\mathrm{2.87s}` when the structure is already heavily damaged.


.. csv-table::

    "**Instant**", "**Size**", "**Location**", "**Reference Type**", "**Tolerance**"
    "1.76 s", "FORC_NODA: DY", "group:A", "NON_REGRESSION ", "1.0E-06", "1.0E-06"
    "1.76 s", "DEPL: DY", "Group:C", "NON_REGRESSION ", "1.0E-06", "1.0E-06"
    "2.05 s", "FORC_NODA: DY", "group:A", "NON_REGRESSION ", "1.0E-06", "1.0E-06"
    "2.05 s", "DEPL: DY", "Group:C", "NON_REGRESSION ", "1.0E-06"
    "2.68 s", "FORC_NODA: DY", "group:A", "NON_REGRESSION ", "1.0E-06", "1.0E-06"
    "2.68 s", "DEPL: DY", "group:C", "NON_REGRESSION ", "1.0E-06", "1.0E-06"
    "2.87 s", "FORC_NODA: DY", "group:A", "NON_REGRESSION ", "1.0E-06", "1.0E-06"
    "2.87 s", "DEPL: DY", "group:C", "NON_REGRESSION ", "1.0E-06", "1.0E-06"


The figures below show the evolution of the reaction on support A and of the arrow in the center, for 15 seconds.


.. image:: images/10000200000003CA0000025C924F67887268A3B6.png
    :width: 6.0634in
    :height: 3.7717in


.. _RefImage_10000200000003CA0000025C924F67887268A3B6.png:

**Figure** 3.2-d: **Press reaction in A as a function of time, for 15 seconds.**


.. image:: images/10000200000003CA0000025C4ED701C42BD135EB.png
    :width: 6.0634in
    :height: 3.7717in


.. _RefImage_10000200000003CA0000025C4ED701C42BD135EB.png:

**Figure** 3.2-e: f **arrow in the center as a function of time, for 15 seconds.**


.. image:: images/10000200000003CA0000025C772EC10119C21954.png
    :width: 6.0634in
    :height: 3.7717in


.. _RefImage_10000200000003CA0000025C772EC10119C21954.png:

**Figure** 3.2-f: c**reaction-arrow curve, for 15 seconds.**