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


Geometry
---------

The chimney is a vertical beam of length :math:`10m`, embedded at its base and articulated at two altitude points :math:`4m` and :math:`8m`.


.. image:: images/Object_1.svg
    :width: 299
    :height: 302


.. _RefImage_Object_1.svg:

Beam cross section:

         Area: :math:`A\mathrm{=}3.4390{10}^{\mathrm{-}3}{m}^{2}`

         Moments of inertia: :math:`{I}_{y}\mathrm{=}\mathrm{1.3770.}{10}^{\mathrm{-}5}{m}^{4}`

                             :math:`{I}_{z}\mathrm{=}\mathrm{1.3770.}{10}^{\mathrm{-}5}{m}^{4}`

                             :math:`{J}_{x}\mathrm{=}2.7540{10}^{\mathrm{-}5}{m}^{4}`


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


.. csv-table::

    "Beam", "Young's module
    density
    Poisson's ratio", ":math:`E\mathrm{=}1.658{10}^{11}\mathit{Pa}`
    :math:`\rho \mathrm{=}1.3404106{10}^{4}\mathit{kg}\mathrm{/}{m}^{3}`
    :math:`\nu \mathrm{=}\mathrm{0,3}` ", ""



Boundary conditions and loads
-------------------------------------

Modeling A (3D)

Embedded point :math:`\mathit{N1}`: :math:`\mathit{DX}\mathrm{=}\mathit{DY}\mathrm{=}\mathit{DZ}\mathrm{=}\mathit{DRX}\mathrm{=}\mathit{DRY}\mathrm{=}\mathit{DRZ}\mathrm{=}0`

Points :math:`\mathit{N5}` and :math:`\mathit{N9}` attached: :math:`\mathit{DX}\mathrm{=}\mathit{DY}\mathrm{=}0`

Spectra of horizontal oscillators under acceleration applied to points :math:`\mathit{N1}`, :math:`\mathit{N5}` and :math:`\mathit{N9}` in the (:math:`x`) and (:math:`x` and :math:`y`) directions.

Modeling B (2D plane :math:`\mathit{XZ}`)

Problem plan :math:`\mathit{XZ}`: :math:`\mathit{DY}\mathrm{=}\mathit{DRX}\mathrm{=}\mathit{DRZ}\mathrm{=}0`

Embedded point :math:`\mathit{N1}`: :math:`\mathit{DX}\mathrm{=}\mathit{DZ}\mathrm{=}\mathit{DRY}\mathrm{=}0`

Points :math:`\mathit{N5}` and :math:`\mathit{N9}` attached: :math:`\mathit{DX}\mathrm{=}0`

Spectra of accelerating horizontal oscillators applied to points :math:`\mathit{N1}`, :math:`\mathit{N5}` and :math:`\mathit{N9}` in the (:math:`x`) direction.

Spectra of identical values for the 3 amortizations :math:`\text{0,5\%}`, :math:`\text{1\%}` and :math:`\text{1,5\%}`.


.. csv-table::

    "Frequency
    (:math:`\mathit{Hz}`)", "Pseudo-acceleration
    (:math:`{\mathit{m.s}}^{\mathrm{-}2}`) in :math:`x` ", "Pseudo-acceleration
    (:math:`{\mathit{m.s}}^{\mathrm{-}2}`) in :math:`y`"
    "1
    10
    30
    100
    10000", "1.962
    19.62
    19.62
    1.962
    1.962", "1.962
    19.62
    19.62
    1.962
    1.962"



.. image:: images/10000000000004B0000003859F35062EEE4E3B0F.png
    :width: 391
    :height: 294


.. _RefImage_10000000000004B0000003859F35062EEE4E3B0F.png:

For the calculation, a damping reduced by 3% is used, with an interpolation (LOG LOG) in frequency and (LIN LOG) in damping.

Multi-support case with different excitations:

point :math:`\mathit{N1}`: excitement :math:`1`

point :math:`\mathit{N5}`: excitement :math:`\mathrm{\times }1.5`

point :math:`\mathit{N9}`: excitement :math:`\mathrm{\times }2`


Initial conditions
--------------------

Not applicable for spectral analysis