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

This test case is inspired by the report cited in reference [:ref:`bib1 <bib1>`].


Geometry
---------

The building studied is composed of 3 floors and 9 columns embedded in the floors.


.. csv-table::

    "Section of columns :math:`A` to :math:`I` ", ":", ":math:`0.20m\times 0.40m`"
    "Area of columns :math:`A` to :math:`I` ", ", ":", ":math:`8.00{10}^{-2}{m}^{2}`"
    "Inertia of columns :math:`A` to :math:`I` ", ":", ":math:`\mathrm{Ix}=2.667{10}^{-4}{m}^{4}`"
    "(in the global coordinate system)", "", ":math:`\mathrm{Iy}=1.066{10}^{-3}{m}^{4}`"
    "", "", ":math:`J=7.45{10}^{-4}{m}^{4}`"
    "Reduced cross section coefficients", ":", ":math:`\mathrm{AY}=\mathrm{AZ}=1.2`"
    "Floor thickness", ":", ":math:`0.2m`"



.. image:: images/Object_1.svg
    :width: 302
    :height: 329


.. _RefImage_Object_1.svg:



.. image:: images/Object_2.svg
    :width: 302
    :height: 329


.. _RefImage_Object_2.svg:


.. image:: images/Object_3.svg
    :width: 302
    :height: 329


.. _RefImage_Object_3.svg:

Figure 1-a: Building diagram.


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

In order to obtain the center of mass :math:`\mathrm{Cm}`, eccentric with respect to the geometric center of :math:`\mathrm{0,3071}m`, a material of density :math:`{\rho }_{2}=\mathrm{1,848}{\rho }_{1}` is assigned to :math:`1/4` of the surface of each floor (:math:`\mathrm{PLAN21}`, :math:`\mathrm{PLAN22}` and :math:`\mathrm{PLAN23}`).

Columns and parts :math:`\mathrm{PLAN11}`, :math:`\mathrm{PLAN21}` and :math:`\mathrm{PLAN31}` of the floors:

+---------------+-------------------------------------------+----------------+--------------------------------+
|Young's module:|:math:`{E}_{1}=\mathrm{4,0}E+10\mathrm{Pa}`|Poisson's ratio:|:math:`{\nu }_{1}=\mathrm{0,15}`|
+---------------+-------------------------------------------+----------------+--------------------------------+
|Density:       |:math:`{\rho }_{1}=2500\mathrm{kg}/{m}^{3}`|                                                 |
+---------------+-------------------------------------------+----------------+--------------------------------+

Parts :math:`\mathrm{PLAN12}`, :math:`\mathrm{PLAN22}` and :math:`\mathrm{PLAN32}` of the floors:

+---------------+-------------------------------------------+----------------+--------------------------------+
|Young's module:|:math:`{E}_{2}=\mathrm{4,0}E+10\mathrm{Pa}`|Poisson's ratio:|:math:`{\nu }_{2}=\mathrm{0,15}`|
+---------------+-------------------------------------------+----------------+--------------------------------+
|Density:       |:math:`{\rho }_{2}=4620\mathrm{kg}/{m}^{3}`|                                                 |
+---------------+-------------------------------------------+----------------+--------------------------------+


.. image:: images/Object_4.svg
    :width: 302
    :height: 329


.. _RefImage_Object_4.svg:

Figure 1-b: Building diagram.


Boundary conditions and loading
------------------------------------

**Boundary condition**

The columns are embedded at the foundation level.

**Loading**

The earthquake is applied in the :math:`x` direction.

The response spectrum of oscillators in motion is obtained by superimposing four displacement spectra. Each of these displacement spectra corresponds to the response of an oscillator to a degree of freedom to the sinusoidal excitations defined in the table below:

:math:`\mathrm{SD}(f,\xi )=\underset{i=1}{\overset{4}{\Sigma }}\frac{{K}_{i}}{4{\pi }^{2}{f}^{2}\sqrt{{(1-\frac{{f}_{i}^{2}}{{f}^{2}})}^{2}+4{\xi }^{2}\frac{{f}_{i}^{2}}{{f}^{2}}}}`

In particular, the frequencies and damping selected are close to the first four modes of the structure.

.. csv-table::

    "", "**Frequency** :math:`{f}_{i}(\mathrm{Hz})` ", "**Amplitude** :math:`{K}_{i}(m)` ", "**Dampening** :math:`\xi`"
    "sine 1", "1.51", "0.15", "0.05"
    "sine 2", "2.05", "0.25", "0.05"
    "sine 3", "2.34", "0.25", "0.05"
    "sine 4", "4.86", "0.30", "0.05"


Table 1-a: Characteristics of the excitations used.

Neglected modes are represented by a pseudo-mode.