Reference problem
=====================

Geometry
---------

Straight beam of length :math:`L`, direction :math:`x`. Dimensions are in meters, :math:`\text{[m]}`.


.. image:: images/100007FA000069D5000013D80209CAE30E6DE287.svg
    :width: 381
    :height: 71


.. _RefImage_100007FA000069D5000013D80209CAE30E6DE287.svg:

Three different types of cross sections are calculated simultaneously:


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+--------------------------------+-------------------------------------------------------------------------------------------------------------------+
|1 rectangular section           |                                                                                                                   |
+                                + .. image:: images/Shape1.gif                                                                                      +
|                                |                                                                                                                   |
+                                +                                                                                                                   +
|                                |                                                                                                                   |
+--------------------------------+-------------------------------------------------------------------------------------------------------------------+
|                                |                                                                                                                   |
+--------------------------------+-------------------------------------------------------------------------------------------------------------------+
|1 angle section with equal wings|                                                                                                                   |
+                                + .. image:: images/1000000000000128000000690B28797E35646899.png                                                    +
|                                |    :width: 296                                                                                                    |
+                                +    :height: 104                                                                                                   +
|                                |                                                                                                                   |
+                                +                                                                                                                   +
|                                |                                                                                                                   |
+--------------------------------+-------------------------------------------------------------------------------------------------------------------+
|                                |                                                                                                                   |
+--------------------------------+-------------------------------------------------------------------------------------------------------------------+
|1 circular section              |                                                                                                                   |
+                                + .. image:: images/Shape2.gif                                                                                      +
|                                |                                                                                                                   |
+                                +                                                                                                                   +
|                                |                                                                                                                   |
+--------------------------------+-------------------------------------------------------------------------------------------------------------------+


Material properties
-----------------------

Young's module: :math:`E\mathrm{=}2.{10}^{11}\mathit{Pa}`

Poisson's ratio: :math:`\nu \mathrm{=}0.3`


Boundary conditions and loads
-------------------------------------


Recessed case-free, unit loads at the end
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The length of the beam is :math:`L=2m`.

Embedding in :math:`O`.

Six unit loads in :math:`B`:

.. csv-table::

    ":math:`\mathrm{Fx}=1` "," :math:`\mathrm{Mx}=1`"
    ":math:`\mathrm{Fy}=1` "," :math:`\mathrm{My}=1`"
    ":math:`\mathrm{Fz}=1` "," :math:`\mathrm{Mz}=1`"


Combined flexure plus traction loading: :math:`\mathrm{Fx}=1`; :math:`\mathrm{My}=1`; :math:`\mathrm{Mz}=1`;

A load combined with shear forces plus twisting: :math:`\mathrm{Fy}=1` :math:`\mathrm{Fz}=1` :math:`\mathrm{Mx}=1`.

Simple support cases, unit loads at the end
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The length of the beam is :math:`L=6m`.

A linear distributed load: :math:`\mathit{Fy}=1000\mathrm{.}x` circular section, with radius :math:`0.1m` (F, G, H, I models) (with simple support in :math:`O` and :math:`B` in this case).


Notation of cross-sectional characteristics
--------------------------------------------------------

The geometric characteristics of the cross sections are noted:

.. csv-table::

    ":math:`A` ", "area of the section"
    ":math:`{I}_{y},{I}_{z}` ", "geometric moments of inertia with respect to the main axes of inertia of the section"
    ":math:`\mathrm{JX}` ", "torsional constant"
    ":math:`\mathrm{ay},\mathrm{az}` ", "shear coefficients in the :math:`\mathit{Gy}` and :math:`\mathit{Gz}` directions"
    ":math:`{A}_{y}^{\text{'}}=\frac{A}{\mathrm{ay}}` and :math:`{A}_{z}^{\text{'}}=\frac{A}{\mathrm{az}}` ", "reduced areas equivalent to the shear force"
    ":math:`{e}_{y},{e}_{z}` ", "eccentricity of the center of torsion"
    ":math:`\mathrm{JG}` ", "warping constant"