.. _R3.08.08:

**r3.08.08** Multi-fiber beam element (right) POU_D_EM
=============================================================

**Summary**:

This document shows multi-fiber beam elements based on a solution to a beam problem in which each section of a beam is divided into several fibers. Each fiber then behaves like an Euler beam. Several materials can be assigned to a single finite element support (SEG2) which avoids having to duplicate the meshes (steel + concrete, for example).

The beams are straight (element POU_D_EM). The section can be of any shape, described by a "fiber mesh", see [:external:ref:`U4.26.01 <U4.26.01>`].

The hypotheses adopted are as follows:

• Euler hypothesis: transverse shear is neglected (this hypothesis is verified for strong movements),

• multifiber beam elements take into account the effects of thermal expansion, drying and hydration (terms of the second member) and in a simplified way twisting. The effort-normal flexure coupling is treated naturally, by integrating the behavioral models associated with each fiber group into the uniaxial responses section. An enrichment of the axial deformation, resolved by local condensation in the case of non-linear behaviors, allows good numerical results, regardless of the evolution in the section of the material center of gravity of the section.

**Table of Contents**

**Notes**

The correspondence between the notations in this document and those in the user documentation is given.

:math:`\mathit{DX},\mathit{DY},\mathit{DZ}` and :math:`\mathit{DRX},\mathit{DRY},\mathit{DRZ}` are in fact the names of the degrees of freedom associated with the components of displacement :math:`u,v,w,{\theta }_{x},{\theta }_{y},{\theta }_{z}`.


.. csv-table::

    ":math:`E` ", "Young's modulus", ":math:`E`"
    ":math:`\nu` ", "Poisson's ratio", ":math:`\mathit{NU}`"
    ":math:`G` ", "Coulomb's modulus = :math:`\frac{E}{2.(1+\nu )}` "," :math:`G`"
    ":math:`{I}_{y},{I}_{z}` ", "geometric moments of bending with respect to the axes :math:`y,z` "," :math:`\text{IY},\text{IZ}`"
    ":math:`{J}_{x}` ", "torsional constant", ":math:`\mathit{JX}`"
    ":math:`K` ", "stiffness matrix", ""
    ":math:`M` ", "mass matrix", ""
    ":math:`{M}_{x},{M}_{y},{M}_{z}` ", "moments around axes :math:`x,y,z` "," :math:`\mathit{MT},\mathit{MFY},\mathit{MFZ}`"
    ":math:`N` ", "normal effort at the section", ":math:`N`"
    ":math:`S` ", "section area", ":math:`A`"
    ":math:`u,v,w` ", "translations on the :math:`x,y,z` axes "," :math:`\mathit{DX},\mathit{DY},\mathit{DZ}`"
    ":math:`{V}_{y},{V}_{z}` ", "shear forces along the :math:`y,z` axes "," :math:`\mathit{VY},\mathit{VZ}`"
    ":math:`\rho` ", "density", ":math:`\rho`"
    ":math:`{\theta }_{x},{\theta }_{y},{\theta }_{z}` ", "rotations around axes :math:`x,y,z` "," :math:`\mathit{DRX},\mathit{DRY},\mathit{DRZ}`"
    ":math:`{q}_{x},{q}_{y},{q}_{z}` ", "external line forces", ""





.. toctree::
    :hidden:

    self

.. toctree::
    :maxdepth: 2
    :numbered:

    Introduction
    _l_ment_de_th_orie_des_poutres__rappels_
    Les__quations_du_mouvement_des_poutres
    _l_ment_de_poutre_droite_multifibre
    _l_ment_squelette_d_assemblage
    Cas_d_application
    Bibliographie