.. _R5.03.40:

**r5.03.40** Static and dynamic modeling of beams in large rotations
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**Summary**:

This note gives a mechanical formulation of beams in large displacements and rotations but with elastic behavior. The main difficulty in analyzing rotations lies in the fact that they are not switchable and do not constitute a vector space, but a manifold.

At any moment, the configuration of a straight section of a beam is defined by the displacement vector of its center of gravity and the rotation vector of the system of the main axes of inertia with respect to a reference position. As in classical beam theory, the internal forces are reduced to their resultant and their moment on the line of the centers of the sections. The associated deformations are defined.

The linearization of internal forces with respect to displacements leads to the usual stiffness matrix, which is symmetric, and, because of the large displacements and rotations, to the geometric rigidity matrix, which is arbitrary.

The linearization of inertial forces leads, for translational motion, to the usual mass matrix that is symmetric and, for rotational motion, to a much more complicated matrix with no symmetry at all.

The temporal integration scheme is that of Newmark.

This modeling was tested on five reference problems: three static problems and two dynamic problems.

This work was undertaken as part of the development of modeling tools for line and station components. The aim of modeling is the dynamic study of conductors equipped with **spacers** (for lines) or **descents on equipment** (for stations) and subjected to Laplace forces resulting from short circuit currents.




.. toctree::
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    self

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

    Notations
    Introduction
    Cin_matique_d_une_poutre_en_rotations_finies
    Vecteur_et_op_rateur_de_rotation
    Passage_des_axes_locaux_aux_axes_g_n_raux
    Efforts_int_rieurs__d_formations_et_loi_de_comportement
    Forces_d_inertie__l_mentaires
    _quation_du_mouvement_et_d_roulement_d_un_calcul
    Lin_arisation_des__quations_du_mouvement
    Mise_en__uvre_par__l_ments_finis
    Organisation_sch_matique_d_un_calcul
    Utilisation_par_le_Code_Aster
    Simulations_num_riques
    Bibliographie
    Description_des_versions_du_document