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


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

     In both modes of stress, the membrane is in a state of uniform stress. It is therefore possible to calculate analytically the solution of this problem in both cases.


* **Traction solicitation**

In the case of a simple tensile load, it is simply shown that, in the global frame of reference:

:math:`\mathrm{\{}\begin{array}{c}{\varepsilon }_{\mathit{XX}}\mathrm{=}{F}_{X}\frac{{M}_{\mathit{LLLL}}}{({M}_{\mathit{TTTT}}{M}_{\mathit{LLLL}}\mathrm{-}{M}_{\mathit{LLTT}}^{2})}\\ {\varepsilon }_{\mathit{YY}}\mathrm{=}\mathrm{-}{F}_{X}\frac{{M}_{\mathit{LLTT}}}{({M}_{\mathit{TTTT}}{M}_{\mathit{LLLL}}\mathrm{-}{M}_{\mathit{LLTT}}^{2})}\\ {\varepsilon }_{\mathit{XY}}\mathrm{=}0\end{array}`


* **Shear load**

In the case of shear stress, it is shown that, in the global frame of reference:

:math:`\mathrm{\{}\begin{array}{c}{\varepsilon }_{\mathit{XX}}\mathrm{=}0\\ {\varepsilon }_{\mathit{YY}}\mathrm{=}0\\ {\varepsilon }_{\mathit{XY}}\mathrm{=}\frac{{\sigma }_{\mathit{XY}}}{{M}_{\mathit{LTLT}}}\end{array}`

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

Displacement, stress, and deformation are tested at summit :math:`\mathit{POINT}`. The quantities tested are summarized in the table below, for the two modes of stress.


.. csv-table::

    "**Size**", "**Component**", "**Simple Traction**", "**Shear**", "**Tolerance**"
    "Displacement", "DX", "3/8", "3/8", "1/2", "1.E-6"
    "", "DY", "-1/8", "1/2", "1.E-6"
    "Membrane deformations (local coordinate system)", "EXX "," ", "-1/8", "0", "1.E-6"
    "", "EYY ", "3/8", "0", "1.E-6"
    "", "EXY ", "0"," :math:`\sqrt{2}\mathrm{/}2` ", "1.E-6"
    "Membrane constraints (local coordinate system)", "NXX "," ", "0", "0", "1.E-6"
    "", "NYY ", "1", "0", "1.E-6"
    "", "NXY ", "0"," :math:`\sqrt{2}` ", "1.E-6"