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.

Este trabalho apresenta uma formulação tensorial genérica para parametrização das rotações do tipo vetorial destinada ao estudo de grandes rotações no espaço tridimensional. Esta formulação é compatível com as parametrizações de Euler e de Rodrigues. É dada ênfase aos que aqui se denominou parâmetros generalizados de Rodrigues, que fornecem expressões simples, computacionalmente mais eficientes que a parametrização clássica de Euler. A formulação apresentada é adequada para uso em métodos numéricos baseados nas projeções de Galerkin, como o método dos elementos finitos, podendo ser implementada com facilidade em programas já existentes de elementos finitos. Apresentam-se aqui expressões para o tensor das rotações e suas derivadas, bem como os tensores necessários à análise incremental. As formas fracas são construídas tanto com projeção ortogonal como não-ortogonal, correspondentes à aplicação do Teorema dos Trabalhos Virtuais e Teorema das Potências Virtuais, respectivamente. Os modelos propostos foram aplicados em um programa de elementos finitos utilizando formulações cinemáticas Lagrangiana total e Lagrangiana atualizada e foram resolvidos vários exemplos, dentre eles alguns clássicos da literatura, de forma a avaliar sua validade e aplicação. / This work presents a generic formulation of vector-type for the parameterization of large rotations in three-dimensional space. This formulation adapts to the Euler and the Rodrigues parameterization. Special distinction is made to the here named generalized Rodrigues parameters which result in very simple and computationally efficient expressions. The attained formulation is convenient for numerical procedures employing Galerkin projection like the finite element method and can be readily implemented in a FE code. The expressions of rotation tensor and its derivatives, which lead to a consistent linearization, are herein derived. The necessary tensor quantities employed in the derivation of the tangent bilinear weak form of incremental analysis are obtained too. The weak forms are constructed here with both orthogonal and non-orthogonal projections, corresponding to the application of the virtual work theorem or virtual power theorem respectively. The formulation is implemented within a finite element code in total Lagrangian and updated Lagrangian framework and assessment of the scheme is made by means of several numerical simulations.

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.

Este trabalho apresenta uma formulação tensorial genérica para parametrização das rotações do tipo vetorial destinada ao estudo de grandes rotações no espaço tridimensional. Esta formulação é compatível com as parametrizações de Euler e de Rodrigues. É dada ênfase aos que aqui se denominou parâmetros generalizados de Rodrigues, que fornecem expressões simples, computacionalmente mais eficientes que a parametrização clássica de Euler. A formulação apresentada é adequada para uso em métodos numéricos baseados nas projeções de Galerkin, como o método dos elementos finitos, podendo ser implementada com facilidade em programas já existentes de elementos finitos. Apresentam-se aqui expressões para o tensor das rotações e suas derivadas, bem como os tensores necessários à análise incremental. As formas fracas são construídas tanto com projeção ortogonal como não-ortogonal, correspondentes à aplicação do Teorema dos Trabalhos Virtuais e Teorema das Potências Virtuais, respectivamente. Os modelos propostos foram aplicados em um programa de elementos finitos utilizando formulações cinemáticas Lagrangiana total e Lagrangiana atualizada e foram resolvidos vários exemplos, dentre eles alguns clássicos da literatura, de forma a avaliar sua validade e aplicação. / This work presents a generic formulation of vector-type for the parameterization of large rotations in three-dimensional space. This formulation adapts to the Euler and the Rodrigues parameterization. Special distinction is made to the here named generalized Rodrigues parameters which result in very simple and computationally efficient expressions. The attained formulation is convenient for numerical procedures employing Galerkin projection like the finite element method and can be readily implemented in a FE code. The expressions of rotation tensor and its derivatives, which lead to a consistent linearization, are herein derived. The necessary tensor quantities employed in the derivation of the tangent bilinear weak form of incremental analysis are obtained too. The weak forms are constructed here with both orthogonal and non-orthogonal projections, corresponding to the application of the virtual work theorem or virtual power theorem respectively. The formulation is implemented within a finite element code in total Lagrangian and updated Lagrangian framework and assessment of the scheme is made by means of several numerical simulations.