The purpose of this thesis is to develop corotational beam elements for the nonlinear dynamic analyse of flexible beam structures. Whereas corotational beam elements in statics are well documented, the derivation of a corotational dynamic formulation is still an issue. In the first journal paper, an efficient dynamic corotational beam formulation is proposed for 2D analysis. The idea is to adopt the same corotational kinematic description in static and dynamic parts. The main novelty is to use cubic interpolations to derive both inertia terms and internal terms in order to capture correctly all inertia effects. This new formulation is compared with two classic formulations using constant Timoshenko and constant lumped mass matrices. In the second journal paper, several choices of parametrization and several time stepping methods are compared. To do so, four dynamic formulations are investigated. The corotational method is used to develop expressions of the internal terms, while the dynamic terms are formulated into a total Lagrangian context. Theoretical derivations as well as practical implementations are given in detail. Their numerical accuracy and computational efficiency are then compared. Moreover, four predictors and various possibilities to simplify the tangent inertia matrix are tested. In the third journal paper, a new consistent beam formulation is developed for 3D analysis. The novelty of the formulation lies in the use of the corotational framework to derive not only the internal force vector and the tangent stiffness matrix but also the inertia force vector and the tangent dynamic matrix. Cubic interpolations are adopted to formulate both inertia and internal local terms. In the derivation of the dynamic terms, an approximation for the local rotations is introduced and a concise expression for the global inertia force vector is obtained. Four numerical examples are considered to assess the performance of the new formulation against two other ones based on linear interpolations. Finally, in the fourth journal paper, the previous 3D corotational beam element is extended for the nonlinear dynamics of structures with thin-walled cross-section by introducing the warping deformations and the eccentricity of the shear center. This leads to additional terms in the expressions of the inertia force vector and the tangent dynamic matrix. The element has seven degrees of freedom at each node and cubic shape functions are used to interpolate local transversal displacements and axial rotations. The performance of the formulation is assessed through five examples and comparisons with Abaqus 3D-solid analyses. / <p>QC 20131017</p>

L’objectif de cette thèse est de proposer des éléments finis poutres corotationnels 2D et 3D pour l’analyse du comportement dynamique non-linéaire des structures à barres. La contribution majeure de cette thèse est l’utilisation de fonctions d’interpolations cubiques à la fois pour la détermination de l’expression des efforts internes mais aussi celle des termes d’inertie. En négligeant le carré du déplacement transversal dans le repère local, une expression analytique des termes dynamiques en 2D est obtenue. Sur base d’une étude comparative approfondie sur la paramétrisation des rotations et les algorithmes d’intégration temporelle et d’une approximation des rotations locales, nous proposons deux éléments finis poutre 3D précis et robustes. Contrairement au premier, le second élément 3D prend en compte les déformations de gauchissement et l'excentricité du centre de cisaillement. Les diverses comparaisons réalisées démontrent la pertinence des formulations proposées. / The purpose of this thesis is to propose several corotational beam formulations for both 2D and 3D nonlinear dynamic analyse of flexible structures. The main novelty of these formulations is that the cubic interpolation functions are used to derive not only the internal force vector and the tangent stiffness matrix but also the inertial force vector and the dynamic matrix. By neglecting the quadratic terms of the local transversal displacements, closed-form expressions for the inertial terms are obtained for 2D problems. Based on an extensive comparative study of the parameterizations of the finite rotations and the time stepping method, and by adopting an approximation of the local rotations, two consistent and effective beam formulations for 3D dynamics are developed. In contrast with the first formulation, the second one takes into account the warping deformations and the shear center eccentricity. The accuracy of these formulations is demonstrated through several numerical examples.

The purpose of this thesis is to propose several corotational beam formulations for both 2D and 3D nonlinear dynamic analyse of flexible structures. The main novelty of these formulations is that the cubic interpolation functions are used to derive not only the internal force vector and the tangent stiffness matrix but also the inertial force vector and the dynamic matrix. By neglecting the quadratic terms of the local transversal displacements, closed-form expressions for the inertial terms are obtained for 2D problems. Based on an extensive comparative study of the parameterizations of the finite rotations and the time stepping method, and by adopting an approximation of the local rotations, two consistent and effective beam formulations for 3D dynamics are developed. In contrast with the first formulation, the second one takes into account the warping deformations and the shear center eccentricity. The accuracy of these formulations is demonstrated through several numerical examples.

Ce travail a pour cadre une collaboration entre le LMA et Thales Alenia Space. Nous nous intéressons au comportement des structures flexibles et plus particulièrement des mètres rubans qui ont la particularité de pouvoir, grâce à l’aplatissement de la section, s’enrouler ou développer des pliages localisés. Une première thèse a permis d’une part la mise au point d’un nouveau type de mètre ruban au déroulement maîtrisable thermiquement et d’autre part le développement d’un modèle plan de poutre à section flexible. Dans le travail de thèse présenté ici, nous proposons une version étendue de ce modèle adaptée à la simulation du comportement dynamique tridimensionnel des mètres rubans en grands déplacements et en grandes rotations. Ce modèle est dérivé de la théorie des coques et repose sur l’introduction d’hypothèses cinématiques et sthéniques adaptées. La déformation de la section est caractérisée par celle de sa ligne moyenne qui peut se déformer dans son plan par flexion et torsion mais non par extension, ainsi que hors de son plan par gauchissement de torsion. Les fortes variations de forme de la section dans son plan peuvent alors être décrites par une cinématique de type Elastica, tandis qu’une cinématique de type Vlassov est utilisée pour définir le gauchissement dans le repère local attaché à la section. Le modèle unidimensionnel est obtenu par intégration sur la section des expressions de la théorie des coques, une approche énergétique permet ensuite de formuler le problème associé qui est résolu grâce au logiciel de modélisation par éléments finis COMSOL. / This work was carried out within the framework of a collaboration between the LMA and Thales Alenia Space. We focus on the behaviour of flexible structures and more specifically of tape springs, whose particularity lies in their capacity to coil up or to form localized folds through the flattening of their cross-section. A first thesis led to the development of a new type of tape spring whose uncoiling is controlled thermically on one hand and of a planar rod model with a flexible thin-walled cross-section on the other hand. In this thesis, we offer an extended version of this model dedicated to the simulation of three-dimensional dynamic behavior of tape springs in large displacements and large rotations. This model is derived from shell theory and is based on the introduction of adapted kinematic and sthenic hypotheses. The deformation of the cross-section is characterized by that of its average line which can deform in its own plane by flexion and twisting but not by extension, as well as out of its plane through torsional warping. The large changes of the cross-section shape in its plane can then be described by an Elastica kinematics, whereas a Vlassov kinematics is used to define the warping in the local frame attached to the section. The unidimensionnal model is obtained by integration over the cross-section of the expressions of the shell theory, an energetic approach then allows to express the associated problem which is solved thanks to the finite element modeling software COMSOL.