Benchmark solution
=====================


Formal solution
-----------------

The aim is to define deformation :math:`\varepsilon` according to the main direction of a reinforcement sheet located in the plane :math:`({x}_{\mathrm{1 }};{y}_{1})`.

Taking into account the boundary conditions chosen, we can write:


.. image:: images/Object_1.svg
    :width: 178
    :height: 26


.. _RefImage_Object_1.svg:

with :math:`({u}_{\mathrm{x1}},{u}_{\mathrm{y1}})` the components of the displacement vector in the :math:`({x}_{\mathrm{1 }};{y}_{1})` plane and :math:`\theta` the angle between the main direction of the reinforcing sheet and :math:`{x}_{1}`.

To define the main direction of the water table, we use the nautical angles :math:`(\alpha ;\beta )` given by the keyword ANGL_REP. They define a vector :math:`v` whose projection :math:`{x}_{p}` on the tangential plane of the sheet fixes the main direction.


.. image:: images/Object_2.svg
    :width: 178
    :height: 26


.. _RefImage_Object_2.svg:

with :math:`(x,y,z)` the initial coordinate system. For our application, the displacement vector :math:`U` is written as:


.. image:: images/Object_3.svg
    :width: 178
    :height: 26


.. _RefImage_Object_3.svg:

For tablecloth :math:`\mathrm{GEOX}` (plan :math:`(y;z)`):


.. image:: images/Object_4.svg
    :width: 178
    :height: 26


.. _RefImage_Object_4.svg:

The main direction of the tablecloth is

.. image:: images/Object_5.svg
    :width: 178
    :height: 26


.. _RefImage_Object_5.svg:

(:math:`\theta =90°` *)*. The deformation is then written:


.. image:: images/Object_6.svg
    :width: 178
    :height: 26


.. _RefImage_Object_6.svg:

For tablecloth :math:`\mathrm{GEOY}` (plan :math:`(x;z)`):


.. image:: images/Object_7.svg
    :width: 178
    :height: 26


.. _RefImage_Object_7.svg:

The main direction of the tablecloth therefore makes an angle of :math:`40°` with the plane of the tablecloth. The deformation is then written:


.. image:: images/Object_8.svg
    :width: 178
    :height: 26


.. _RefImage_Object_8.svg:

For tablecloth :math:`\mathrm{GEOZ}` (plan :math:`(x;y)`):


.. image:: images/Object_9.svg
    :width: 178
    :height: 26


.. _RefImage_Object_9.svg:

The main direction of the tablecloth therefore makes an angle of :math:`15°` with the plane of the tablecloth. The deformation is then written:


.. image:: images/Object_10.svg
    :width: 178
    :height: 26


.. _RefImage_Object_10.svg:

These three values will be the analytical reference values for the validation of the calculations.