Introduction
============

A beam is a solid generated by a surface of area :math:`S` whose geometric center of inertia :math:`G` describes a curve :math:`C` called the mean fiber or neutral fiber. Area :math:`S` is the straight section (cross section) or profile, and it is assumed that while it is evolutionary, its evolutions (size, shape) are continuous and gradual when :math:`G` describes the mean line.

For the study of beams in general, the following hypotheses are made:

• the straight section of the beam is undeformable,

• the transverse displacement is uniform on the right section.

These hypotheses make it possible to express the displacements of any point in the section, as a function of the movements of the corresponding point located on the mean line, and as a function of an increase in displacement due to the rotation of the section around the transverse axes. The latter can be overlooked (POU_D_E) or be the subject of modeling (POU_D_T).

Discretization into "exact" beam elements is performed on a linear element with two nodes and six degrees of freedom per node. These degrees of freedom are the three translations :math:`u,v,w` and the three rotations :math:`{\theta }_{x},{\theta }_{y},{\theta }_{z}`.


.. image:: images/100000000000016C000000922BFCD7E8273910BB.png
    :width: 3.7819in
    :height: 1.5075in


.. _RefImage_100000000000016C000000922BFCD7E8273910BB.png:

**Figure** 2-a: **beam element and degrees of freedom.**

Since the deformations are local, a local base is constructed at each vertex of the mesh, depending on the element on which we are working. The continuity of the fields of travel is ensured by a change of base, bringing the data back into the global database.

In the case of straight beams, the mean line is traditionally placed on the axis :math:`x` of the local base, the transverse movements thus taking place in the plane :math:`(y,z)`.

Finally, when we order quantities related to the degrees of freedom of an element in an elementary vector or matrix (therefore of dimension :math:`12` or :math:`{12}^{2}`), we first order the variables for vertex 1 then those for vertex 2. For each node, we first store the quantities related to the three translations, then those related to the three rotations. For example, a displacement vector will be structured as follows:

.. math::

    (\ underbrace {{u} _ {1}, {1}, {v} _ {1}, {w} _ {1}, {\ theta} _ {1}}, {\ theta} _ {{1}}, {\ theta} _ {1}}}, {\ theta} _ {1}}}} _ {{x} _ {1}}}}, {\ theta} _ {1}}}, {\ theta} _ {1}}}, {\ theta} _ {1}}}, {\ theta} _ {1}}}, {\ theta} _ {1}}}, {\ theta} _ {vertex 1}}) (\ underbrace {{1}}) (\ underbrace {{1}}) (\ {u} _ {2}, {v} _ {2}, {w} _ {2}, {\ theta} _ {2}}, {\ theta} _ {y} _ {2}}, {{y} _ {2}}}, {\ theta}}}, {\ theta} _ {2}}}} _\ text {vertex 2}})




.. _RefNumPara__33352704: