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

Geometry
---------

Software MISS3D uses the frequency coupling method to take into account soil-structure interaction. This method, based on dynamic substructuring, consists in dividing the field of study into three sub-areas:


* the ground,


* the foundation,


* the building.

**The floor**

The ground corresponds to the 10-horizontal layer profile shown on [:ref:`Figure 1.1-a <Figure 1.1-a>`] below:


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++ .. image:: images/10000000000001FE0000013A7A6A3196310B57A9.jpg                                                             +
||    :width: 4.039in                                                                                                         |
++    :height: 3.0638in                                                                                                       +
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Figure 1.1-a: Configuration of buildings juxtaposed [:ref:`bib1 <bib1>`]


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    :height: 2.0882in


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Table 1.1-a: soil in buried foundation configuration [bia1]

**The foundation**

The surface foundation of the 2 buildings is represented on the [:ref:`Figure 1.1-b <Figure 1.1-b>`] below. Two surface models of the foundation complete two skew models of buildings ([:ref:`Figure 1.1-c <Figure 1.1-c>`] below). To the initial surface model of 128 plate elements representing the base of the double foundation, 320 very thin plate elements are added to represent the side walls of the :math:`5m` [:ref:`Figure 1.1-b <Figure 1.1-b>`] depression. The two configurations of the single-buried building type are juxtaposed, leaving a distance of :math:`60\mathit{cm}` between each floor.


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    :height: 1.0492in


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    :width: 3.5528in
    :height: 2.7311in


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**Figure 1.1-b: Surface mesh of the foundation**

**The building**

The building is modelled in 1D by a skewer model composed of 7 non-heavy beams of 5 types and 4 point masses as shown in [:ref:`Figure 1.1-c <Figure 1.1-c>`] below:


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    :height: 1.2752in


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    :height: 1.5146in


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**Figure 1.1-c: Modeling each of the buildings**

**Building characteristics:**

The characteristics of the beams and masses that were used to model each of the buildings are given in the tables below:


+----+-----------+-----------------------------+--------------------------------------------------+-------------------------+-------------------------+
|Mass|Altitude   |Mass                         |Mass inertia (:math:`{10}^{3}{\mathit{kg.m}}^{2}`)                                                    |
+----+-----------+-----------------------------+--------------------------------------------------+-------------------------+-------------------------+
|    |(:math:`m`)|(:math:`{10}^{3}\mathit{kg}`)|:math:`{J}_{\mathit{xx}}`                         |:math:`{J}_{\mathit{yy}}`|:math:`{J}_{\mathit{zz}}`|
+----+-----------+-----------------------------+--------------------------------------------------+-------------------------+-------------------------+
|M1  |10.38      |79.25                        |410.72                                            |482.34                   |893.06                   |
+----+-----------+-----------------------------+--------------------------------------------------+-------------------------+-------------------------+
|M2  |6.25       |104.09                       |574.75                                            |694.04                   |1268.79                  |
+----+-----------+-----------------------------+--------------------------------------------------+-------------------------+-------------------------+
|M3  |4.8        |156.71                       |1020.85                                           |1071.22                  |2092.07                  |
+----+-----------+-----------------------------+--------------------------------------------------+-------------------------+-------------------------+
|M4  |0.8        |316.97                       |1846.7                                            |1844.02                  |3690.72                  |
+----+-----------+-----------------------------+--------------------------------------------------+-------------------------+-------------------------+

**Table 1.1-b: characteristics of the weights of the skewer model provided by NUPEC** [:ref:`bib1 <bib1>`]


+----+----------------------+-----------------------------------+---------------+-----------------+---------------+----------------------------------+
|Beam|Area (:math:`{m}^{2}`)|Moment of inertia (:math:`{m}^{4}`)                |Shear coefficient                |Torsion constant (:math:`{m}^{4}`)|
+----+----------------------+-----------------------------------+---------------+-----------------+---------------+----------------------------------+
|    |:math:`A`             |:math:`{I}_{z}`                    |:math:`{I}_{y}`|:math:`{A}_{y}`  |:math:`{A}_{z}`|:math:`{J}_{x}`                   |
+----+----------------------+-----------------------------------+---------------+-----------------+---------------+----------------------------------+
|P1  |59.50                 |341.33                             |341.33         |1/0.93           |1/0.93         |682.70                            |
+----+----------------------+-----------------------------------+---------------+-----------------+---------------+----------------------------------+
|P2  |8.28                  |39.51                              |54.77          |2.94             |1.47           |94.30                             |
+----+----------------------+-----------------------------------+---------------+-----------------+---------------+----------------------------------+
|P3  |63.19                 |341.33                             |341.33         |1/0.99           |1/0.99         |682.70                            |
+----+----------------------+-----------------------------------+---------------+-----------------+---------------+----------------------------------+
|P4  |19.78                 |148.34                             |149.14         |2.13             |2.11           |297.50                            |
+----+----------------------+-----------------------------------+---------------+-----------------+---------------+----------------------------------+
|P5  |64.00                 |341.33                             |341.33         |1.00             |1.00           |682.70                            |
+----+----------------------+-----------------------------------+---------------+-----------------+---------------+----------------------------------+

Table 1.1-c: characteristics of the beams of the brochette model provided by NUPEC [:ref:`bib1 <bib1>`]

The geometry taken into account in *Code_Aster* is that of the structure of buildings as well as their foundation. Geometric and physical ground data are directly given to MISS3D.


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

**The floor**

The mechanical characteristics of the layers of the soil model that were used are those indicated in table 1.

**The foundation and the building**


.. csv-table::

    "E", ":math:`31000\mathit{MPa}`"
    "NUDE", "0.16"
    "RHO ", "0."
    "ALPHA ", "0"



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

Each of the 2 links between a 1D model and its foundation is achieved by a solid link condition between the foundation and the node common to the building model. This knot is blocked and a solid body movement is imposed on the floor.

We excite the top of the building model in the direction :math:`Y`, which is the direction of separation between the buildings, with a harmonic loading :math:`F\mathrm{=}{F}_{o}\mathrm{sin}\alpha t` whose force module :math:`{F}_{o}` is :math:`10\mathit{kN}` with a pulsation that varies from :math:`0` to :math:`20\mathit{Hz}` in steps of :math:`0.1\mathit{Hz}`.

For the buried configuration, the excitation is applied in the following way:


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    :height: 2.4098in


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Figure 1.3-a: harmonic excitation configuration in a buried foundation