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


Calculation method used for the reference solution
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Analytical solution, *isostatic structure*.

The elastic deflection, the elastic axial stresses and deformations, and the bending moment at any point on the :math:`x` axis are given by:


* Load 1: force :math:`{F}_{y}=–1`


.. csv-table::

    ":math:`{M}_{y}(x)={F}_{y}\mathrm{.}L(1-x/L)` ", "(= :math:`E\mathrm{.}{I}_{z}\mathrm{.}{u}_{y}\text{'}\text{'}(x)` in elasticity)"
    ":math:`{u}_{y}(x)={F}_{y}L\mathrm{.}{x}^{2}\mathrm{.}(3-x/L)/(6\mathrm{.}{\mathrm{E.I}}_{z})` ", "(in elasticity)"
    ":math:`{\varepsilon }_{\mathrm{xx}}(x,y)=–{F}_{y}\mathrm{.}L(1-x/L)\mathrm{.}y/({\mathrm{E.I}}_{z})` ", "(in elasticity)"
    ":math:`{\sigma }_{\mathrm{xx}}(x,y)=-{F}_{y}\mathrm{.}L(1-x/L)\mathrm{.}y/{I}_{z}` ", "(in elasticity)"



* Load 2: :math:`{C}_{z}=1` torque or :math:`{\mathrm{dr}}_{z}={C}_{\mathrm{z.L}}/({\mathrm{E.I}}_{z})` rotation


.. csv-table::

    ":math:`{M}_{y}(x)={C}_{z}` ", "(= :math:`E\mathrm{.}{I}_{z}\mathrm{.}{u}_{y}\text{'}\text{'}(x)` in elasticity)"
    ":math:`{u}_{y}(x)={C}_{\mathrm{z.}}{x}^{2}/(2\mathrm{.}E\mathrm{.}{I}_{z})` ", "(in elasticity)"
    ":math:`{\sigma }_{\mathrm{xx}}(x,y)=–{C}_{z}\mathrm{.}y/{I}_{z}` ", "(in elasticity)"
    ":math:`{\varepsilon }_{\mathrm{xx}}(x,y)=-{C}_{z}\mathrm{.}y/(E\mathrm{.}{I}_{z})` ", "(in elasticity)"


with:

:math:`L=\mathrm{L1}+\mathrm{L2}+\mathrm{L3}=30\mathrm{mm}`

:math:`{I}_{z}=a\mathrm{.}{h}^{3}/12=0.25{\mathrm{mm}}^{4}`

:math:`{\mathrm{dr}}_{z}=0.0006`


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

Deflections, axial stresses and deformations and bending moments at 4 points on the axis of the beam.


Uncertainty about the solution
---------------------------

Analytical solution.