Reference solution
=====================

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

Pure flexure - linear work hardening
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Analytical solution:


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:math:`{\varepsilon }_{\mathrm{xx}}=\varphi y` :math:`\varphi`: curvature

Calculation of the moment by:

:math:`M(u)={\int }_{s}{\sigma }_{\mathrm{xx}}(y)\mathrm{.}y\mathrm{ds}`

:math:`{\sigma }_{\mathrm{xx}}=E{\varepsilon }_{\mathrm{xx}}` for :math:`0\le ∣y∣\le u`

:math:`{\sigma }_{\mathrm{xx}}={\sigma }_{y}+H({\varepsilon }_{\mathrm{xx}}-\frac{{\sigma }_{y}}{E})`

For :math:`u<\mid y\mid \le v`

We get:

**for the rectangular section:**

:math:`\frac{M}{{M}_{e}}=(1-\frac{H}{E})(\frac{3}{2}-\frac{1}{2}{(\frac{{\varphi }_{e}}{\varphi })}^{2})+\frac{H\varphi }{E}/\varphi` with :math:`{\varphi }_{e}=\frac{{M}_{e}}{\mathrm{EI}}{M}_{e}=\frac{{I}_{z}\cdot {\sigma }^{y}}{v}`

for the circular section:

:math:`M(\mu )=\frac{{R}^{3}{\sigma }^{y}}{E}\left[\frac{\pi }{4}\frac{H}{\mu }+\frac{4}{3}(E-H){(1-{\mu }^{2})}^{3/2}+\frac{E-H}{2\mu }(\mathrm{Arc}\mathrm{sin}\mu -\mu (1-2{\mu }^{2})\sqrt{1-{\mu }^{2}})\right]`

with :math:`\mu =\frac{u}{R}=\frac{{\sigma }_{y}}{\mathrm{ER}\varphi }=\frac{{\varphi }_{e}}{\varphi }`

In discharge, after reaching the limit charge under charge, a limit charge with the opposite sign is obtained.

**for the tubular section:**

(Navier-Bernoulli beam hypothesis)

The limit load (:math:`H=0`) is equal to:

:math:`\frac{M}{{M}_{e}}=\frac{4}{\pi }`

The complete solution for a thin tube is [:ref:`bib1 <bib1>`]:

:math:`\frac{M(\mu )}{{M}_{e}}=\frac{\lambda }{\mu }+\frac{2(1-\lambda )}{\pi \mu }(\mathrm{arc}\mathrm{sin}\mu +\mu \sqrt{1-{\mu }^{2}})` with :math:`\lambda =\frac{{E}_{T}}{E}=\frac{H}{E+H}`


Traction - Linear work hardening
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Analytical solution: we immediately have :math:`N=S{\sigma }_{y}(1-\frac{H}{E})+\frac{\mathrm{HS}}{L}\mathrm{.}\mathrm{DX}`.


Pure traction - Elasticity
~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Analytical solution:

:math:`N=\frac{\mathrm{E.S.}\delta u}{L}` :math:`\sigma =\frac{\mathrm{E.}\delta u}{L}`, with :math:`\delta u=7.5E-03`


Pure Flexion - Elasticity
~~~~~~~~~~~~~~~~~~~~~~~~~~~

Analytical solution:

:math:`\mathrm{Mfz}=\frac{3.0\delta uEI}{{L}^{2}}` :math:`\sigma =\frac{{M}_{\mathrm{fz}}\mathrm{.}h}{I}`, with :math:`\delta u=7.5E-03`

with :math:`h=R` for the circular section, :math:`h=v` for the rectangular section.


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


1. J.H. LAU and T.T. LAU: Journal of Pressure Vessel Technology Vol. 106 p188-195 - May 1984.