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


Calculation method used for the reference solution
--------------------------------------------------------

For this test, we will use two reference solutions, one experimental, resulting from the work of Schwaighofer and Microys [:ref:`bib2 <bib2>`], the other drawn from the work of Batoz in deep shell theory [:ref:`bib1 <bib1>`].


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

The baseline results are as follows:


.. csv-table::

    "**Short Cylinder** **(A and B)**", "Batoz [:ref:`bib1 <bib1>`]", "Experience [:ref:`bib2 <bib2>`]"
    "Move :math:`w` to point :math:`F` "," :math:`0.35{10}^{\mathrm{-}4}m` "," :math:`0.6{10}^{\mathrm{-}4}m`"
    "Move :math:`w` to point :math:`C` "," :math:`–0.710–3m` "," :math:`–0.6{10}^{\mathrm{-}3}m`"
    "Move :math:`w` to point :math:`D` "," :math:`0.25{10}^{\mathrm{-}4}m` "," :math:`0.1{10}^{\mathrm{-}3}m`"
    "Constraint :math:`{\sigma }_{\mathrm{xx}}` at point :math:`F` "," :math:`–0.35\mathit{MPa}` "," :math:`–0.325\mathit{MPa}`"
    "Constraint :math:`{\sigma }_{\mathrm{yy}}` at point :math:`F` "," :math:`0.50\mathit{MPa}` "," :math:`0.60\mathit{MPa}`"



.. csv-table::

    "**Long Cylinder** **(C and D)**", "Batoz [:ref:`bib1 <bib1>`]", "Experience [:ref:`bib2 <bib2>`]"
    "Move :math:`w` to point :math:`F` "," :math:`1.32{10}^{\mathrm{-}3}m` "," :math:`1.35{10}^{\mathrm{-}3}m`"
    "Move :math:`w` to point :math:`C` "," :math:`–2.45{10}^{\mathrm{-}3}m` "," :math:`–2.46{10}^{\mathrm{-}3}m`"
    "Move :math:`w` to point :math:`D` "," :math:`–0.35{10}^{\mathrm{-}3}m` "," :math:`–0.51{10}^{\mathrm{-}3}m`"
    "Constraint :math:`{\sigma }_{\mathrm{xx}}` at point :math:`F` "," :math:`1.68\mathit{MPa}` "," :math:`1.9\mathit{MPa}`"
    "Constraint :math:`{\sigma }_{\mathrm{yy}}` at point :math:`F` "," :math:`1.8\mathit{MPa}` "," :math:`1.55\mathit{MPa}`"



Uncertainties about the solution
----------------------------

Around 5% for the Batoz solution, probably much more — around 30% - for the experimental solution.


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


1. BATOZ J.L., DHATT G.: Modeling structures by finite elements, Vol. 3, Shells, HERMES.


2. SCHWAIGHOFER J., MICROYS H.F.: Orthotropic Cylindrical shells under line load, Journal of Applied Mechanics, June 1979, Vol. 46.


3. GEOFFROY P., Development and evaluation of a finite element for the static and dynamic nonlinear analysis of thin shells, PhD Thesis, Compiègne University of Technology, 27/04/83.