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


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

Modeling A is*bidimensional* and*static nonlinear*. The calculation is carried out using *pure mechanics*, without hydromechanical coupling (equivalent to a perfectly drained soil).

We can first verify the coherence of the initial state (in particular the boundary conditions with the pre-consolidation state of the soil): mechanical balance must be established when only gravity acts, so the state of the system must not change.

The elastoplastic behavior law used is that of Cam-Clay, cf. [R7.01.14].

The vertical displacement is imposed on the GROUP_MA = 'APPUI' representing the interface between the foundation and the ground, and varies between :math:`0.` and :math:`–0.05m` in 20 time steps between :math:`t\mathrm{=}0.s` and :math:`t=1.0\phantom{\rule{2em}{0ex}}{10}^{+7}s`


.. image:: images/10000000000004FE000003AF3570FEDCCBE5B913.png
    :width: 5.1484in
    :height: 3.7201in


.. _RefImage_10000000000004FE000003AF3570FEDCCBE5B913.png:

**Figure 2**: mesh of the shooting foundation for modeling A.


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

The solutions are calculated at points :math:`O` and :math:`F` and compared to *FLAC* references. They are first given in terms of equivalent stress :math:`Q` as a function of the effective consolidation pressure :math:`P’`, and summarized in the following tables:

:math:`P\text{'}=\frac{1}{3}\mathrm{.}\mathit{trace}(\mathrm{\sigma }\text{'})`; :math:`Q=\sqrt{\frac{3}{2}\mathrm{s}\mathrm{:}\mathrm{s}}` (where :math:`\mathrm{s}=\mathrm{\sigma }\text{'}-P\text{'}\mathrm{.}\mathrm{Id}`)

At point :math:`O`, under the foundation in the center:

.. csv-table::

    ":math:`P’` [:math:`\mathit{Pa}`]", ":math:`Q` Code_Aster [:math:`\mathit{Pa}`]", ":math:`Q` FLAC [:math:`\mathit{Pa}`]", "relative error"
    "102000", "434", "450"," -0.035%"
    "110000", "19662", "20000"," -1.689%"
    "120000", "25837", "26060"," -0.855%"
    "130000", "29290", "29490"," -0.679%"
    "146000", "34006", "34040"," -0.100%"


At point :math:`F`:

.. csv-table::

    ":math:`P’` [:math:`\mathit{Pa}`]", ":math:`Q` Code_Aster [:math:`\mathit{Pa}`]", ":math:`Q` FLAC [:math:`\mathit{Pa}`]", "relative error"
    "101900", "227", "76"," +199%"
    "100000", "4945", "4950"," -0.110%"
    "98000", "9941", "10420"," -4.593%"
    "96000", "15283", "16830"," -9.190%"
    "94000", "20036", "21870"," -8.385%"


The resultant of the forces exerted on the foundation as a function of its sinking is then calculated. This is also compared to the solution given by *FLAC*:


.. csv-table::

    ":math:`\mathit{UY}` [:math:`m`]", "Code_Aster [:math:`N/m`]", "FLAC [:math:`N/m`]", "relative error"
    "-0.005", "-110105", "-108500"," +1.479%"
    "-0.02", "-129224", "-125800"," +2.722%"
    "-0.04", "-149470", "-144600"," +3.368%"
    "-0.06", "-167066", "-160900"," +3.832%"
    "-0.0875", "-188825", "-181100"," +4.266%"