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


Calculation method
------------------

The densities of longitudinal steels are calculated according to the Capra and Maury method, for the directions :math:`\text{X}` and :math:`\text{Y}` of each element and according to the 2 faces, :math:`\text{I}` (Lower) and :math:`\text{S}` (Upper), defined by their position according to the elementary :math:`\text{Z}` normal. The transverse steel density is also calculated as described in the article entitled "Automatic calculation of the optimal reinforcement of reinforced concrete plates or shells" by Alain CAPRA and Jean-Francis MAURY.


Benchmark results
----------------------

The various results are published in the article by Capra and Maury in the form of graphs (for a quarter of a model only) are:


* the deformation of the tank,


* the iso-moments :math:`\text{Mxx}` and :math:`\text{Myy}`,


* reinforcement densities in the directions :math:`\text{X}` and :math:`\text{Y}` at the top and bottom.

Below is the excerpt from the article for the deformed and the moments :math:`\text{Mx}`, :math:`\text{My}`.


.. image:: images/10000000000003050000024FD8FBE987F962191D.png
    :width: 4.8307in
    :height: 3.6929in


.. _RefImage_10000000000003050000024FD8FBE987F962191D.png:


.. image:: images/10000000000003300000026F010E5D45883478DB.png
    :width: 5.0984in
    :height: 3.8937in


.. _RefImage_10000000000003300000026F010E5D45883478DB.png:


.. image:: images/100000000000031A00000258D593EFF1DB9EC2EA.png
    :width: 4.9602in
    :height: 3.752in


.. _RefImage_100000000000031A00000258D593EFF1DB9EC2EA.png:

Below is the excerpt from the article for reinforcement densities. For better readability, colors have been associated with the various reinforcement density values.


.. image:: images/1000000000000346000002664A9CB343054D4218.png
    :width: 5.2362in
    :height: 3.839in


.. _RefImage_1000000000000346000002664A9CB343054D4218.png:


.. image:: images/10000000000003300000026834DC93C18C0CDE03.png
    :width: 5.0984in
    :height: 3.8508in


.. _RefImage_10000000000003300000026834DC93C18C0CDE03.png:


.. image:: images/100000000000031F00000268EC12B01009F5563D.png
    :width: 4.9925in
    :height: 3.8508in


.. _RefImage_100000000000031F00000268EC12B01009F5563D.png:


.. image:: images/1000000000000329000002684BB849F21B2CA0F2.png
    :width: 5.0547in
    :height: 3.8508in


.. _RefImage_1000000000000329000002684BB849F21B2CA0F2.png:


Bibliographical references
---------------------------

     [:ref:`1 <1>`] Annals of the Technical Institute of Building and Public Works N° 367 December 1978 - Series: INFORMATIQUE APPLIQUEE - Article entitled "Automatic calculation of the optimal reinforcement of reinforced concrete plates or shells" by Alain CAPRA and Jean-Francis MAURY.