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


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

DALLE

Y

A3

A4

A2

A1

LSYMY

LSYMX

LCONTX

LCONTY


.. image:: images/10000000000001E0000001E09B5895DB4FFF4EB0.png
    :width: 2.2083in
    :height: 2.2154in


.. _RefImage_10000000000001E0000001E09B5895DB4FFF4EB0.png:

Q4GG modeling (TRIA3)

- Boundary conditions:

. Side :math:`\mathrm{A2A4}`: :math:`\mathrm{DZ}=0`

- Symmetry conditions

. Side :math:`\mathrm{A1A2}`: :math:`\mathrm{DY}=\mathrm{DRX}=0`

. Side :math:`\mathrm{A1A3}`: :math:`\mathit{DX}\mathrm{=}\mathit{DRY}\mathrm{=}0`

X
The slab is symmetric with respect to planes :math:`(X\mathrm{=}0)` and :math:`(Y\mathrm{=}0)`, the calculations are carried out on a quarter of the slab.


Characteristics of the mesh
----------------------------

Number of knots: 169

Number of meshes and type: 288 TRIA3


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


.. csv-table::

    "**Identification**", "**Reference Type**", "**Reference**", "**Tolerance (%)**"
    ":math:`\mathit{DZ}(\mathit{A1})` ", "'ANALYTIQUE'", "6,926.10-5"," 5%"
    ":math:`\mathit{MXX}(\mathit{A1})` ", "'ANALYTIQUE'", "1550"," 8%"
    ":math:`\mathit{MYY}(\mathit{A1})` ", "'ANALYTIQUE'", "1550"," 8%"
    ":math:`\mathit{KXX}(\mathit{A1})` ", "'ANALYTIQUE'", "2,193.10-4"," 8%"
    ":math:`\mathit{KYY}(\mathit{A1})` ", "'ANALYTIQUE'", "2,193.10-4"," 8%"


The quantities are expressed in the coordinate system defined by the nautical angles :math:`\alpha \mathrm{=}33°` and :math:`\beta \mathrm{=}12°`


.. csv-table::

    "**Identification**", "**Reference Type**", "**Reference**", "**Tolerance**"
    ":math:`\mathrm{DZ}(\mathrm{A1})` ", "'ANALYTIQUE'", "6,926.10-5"," 5%"
    ":math:`\mathit{MXX}(\mathit{A1})` ", "'ANALYTIQUE'", "1550.0"," 8%"
    ":math:`\mathit{MYY}(\mathit{A1})` ", "'ANALYTIQUE'", "1550.0"," 8%"
    ":math:`\mathit{MXY}(\mathit{A1})` ", "'ANALYTIQUE'", "0. ", "2."
    ":math:`\mathit{KXX}(\mathit{A1})` ", "'ANALYTIQUE'", "2.193 10-4"," 8%"
    ":math:`\mathit{KYY}(\mathit{A1})` ", "'ANALYTIQUE'", "2.193 10-4"," 8%"
    ":math:`\mathit{KXY}(\mathit{A1})` ", "'ANALYTIQUE'", "0. ", "0.001"



+--------------------+---------------------+-----------------------+------------------+-------------+--------------+
|**Identification**                                                |**Reference type**|**Reference**|**Tolerance%**|
+--------------------+---------------------+-----------------------+------------------+-------------+--------------+
|:math:`\mathit{MXX}`|:math:`\mathit{M266}`|:math:`\mathit{Point}3`|'NON_REGRESSION'  |1445.794     |1.e-6         |
+--------------------+---------------------+-----------------------+------------------+-------------+--------------+
|:math:`\mathit{MYY}`|:math:`\mathit{M266}`|:math:`\mathit{Point}3`|'NON_REGRESSION'  |1447.847     |1.e-6         |
+--------------------+---------------------+-----------------------+------------------+-------------+--------------+
|:math:`\mathit{MXY}`|:math:`\mathit{M266}`|:math:`\mathit{Point}3`|'NON_REGRESSION'  |0.526        |1.e-6         |
+--------------------+---------------------+-----------------------+------------------+-------------+--------------+
|:math:`\mathit{KXX}`|:math:`\mathit{M266}`|:math:`\mathit{Point}3`|'NON_REGRESSION'  |2.14096 10-4 |1.e-6         |
+--------------------+---------------------+-----------------------+------------------+-------------+--------------+
|:math:`\mathit{KYY}`|:math:`\mathit{M266}`|:math:`\mathit{Point}3`|'NON_REGRESSION'  |2.14565 10-4 |1.e-6         |
+--------------------+---------------------+-----------------------+------------------+-------------+--------------+
|:math:`\mathit{KXY}`|:math:`\mathit{M266}`|:math:`\mathit{Point}3`|'NON_REGRESSION'  |1.2018 10-7  |1.e-6         |
+--------------------+---------------------+-----------------------+------------------+-------------+--------------+


notes
---------

The coefficients of the following elasticity matrices, used during the calculations, were calculated with :math:`{\nu }_{b}=\mathrm{0,22}`:


1. Membrane elasticity matrix: :math:`\left[\begin{array}{ccc}4832& \mathrm{990,4}& 0\\ \mathrm{990,4}& 4832& 0\\ 0& 0& 1756\end{array}\right]{10}^{6}\text{N/m}`


2. Flexural elasticity matrix: :math:`\left[\begin{array}{ccc}\mathrm{5,879}& \mathrm{1,188}& 0\\ \mathrm{1,188}& \mathrm{5,879}& 0\\ 0& 0& \mathrm{2,107}\end{array}\right]{10}^{6}\text{N/m}`

To be certain of staying within the elastic domain, the elastic limits, expressed in the orthotropy coordinate system, are set arbitrarily to a very high value:


1.
    * Elastic limits in positive flexure:

 Direction x: :math:`{1.10}^{10}\text{MNm/ml}`

 Direction y: :math:`{1.10}^{10}\text{MNm/ml}`


1.
    * Elastic limits in negative flexure:

 Direction x: :math:`\mathrm{-}{1.10}^{10}\text{MNm/ml}`

 Direction y: :math:`\mathrm{-}{1.10}^{10}\text{MNm/ml}`

As the structure remains in the elastic domain, the kinematic recall coefficient (Prager constant) can take any value.