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


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

Numerical solution [:ref:`bib1 <bib1>`]: values of the control parameter (therefore of the force :math:`P`) as a function of time (therefore of the displacement :math:`\mathit{Uz}` of the point :math:`A`).


Reference quantities and results
-----------------------------------


.. code-block:: text

    Control coefficient (Multiplier coefficient of the force applied) as a function of the displacement :math:`\mathrm{Uz}` of the point :math:`A`.


.. code-block:: text

    Reference results obtained by modeling in 16x1x1 shell elements :math:`\mathrm{S4R}` from Abaqus.


.. csv-table::

    ":math:`P\mathrm{/}{P}_{\mathit{max}}` "," :math:`\text{-}{U}_{x}(m)` "," :math:`{U}_{z}(m)` "," :math:`P/{P}_{\mathrm{max}}` "," "," "," :math:`\text{-}{U}_{x}(m)` "," :math:`{U}_{z}(m)` "," :math:`P/{P}_{\mathrm{max}}` "," :math:`\text{-}{U}_{x}(m)` "," :math:`{U}_{z}(m)`"
    "0.05", "0.026", "0.663", "0.663", "0.4", "0.4", "1.184", "4.292", "0.75", "2.541", "6.031"
    "0.1", "0.103", "1.309", "1.309", "0.45", "0.45", "1.396", "4.631", "0.8", "2.705", "6.190"
    "0.15", "0.224", "1.922", "1.922", "0.5", "0.5", "1.604", "4.933", "0.85", "2.861", "2.861", "6.335"
    "0.2", "0.381", "2.493", "2.493", "0.493", "0.55", "1.807", "5.202", "0.9", "3.01", "3.01", "6.467"
    "0.25", "0.563", "3.015", "3.015", "0.6", "0.6", "2.002", "5.444", "0.95", "3.151", "3.151", "6.588"
    "0.3", "0.763", "3.488", "3.488", "0.65", "0.65", "2.190", "5.660", "1", "3.286", "6.698"
    "0.35", "0.971", "3.912", "3.912", "0.7", "2.370", "5.855", "", ""



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

Not applicable


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



.. _Ref208649747:

1. Sze K.Y, Liu X.H, and Lo S.H. Popular benchmark problems for geometric nonlinear analysis of shells. *Finite Elements in Analysis and Design,* Volume 40, Issue 11, Pages 1551-156, 2004.