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Large-amplitude vibration of thin plates and shells has been critical design issues for many engineering structures. The increasingly more stringent safety requirements and the discovery of new materials with amazingly superior properties have further focused the attention of research on this area. This thesis deals with the vibration problem of rectangular, circular and angle-ply composite plates. This vibration can be triggered by an initial vibration amplitude, or an initial velocity, or both. Four types of boundary conditions including simply supported and clamped combined with in-plane movable/immovable are considered. To solve the differential equation generated from the vibration problem, Lindstedt's perturbation technique and Runge-Kutta method are applied. In previous works, this problem was solved by Lindstedt's Perturbation Technique. This technique can lead to a quick approximate solution. Yet based on mathematical assumptions, the solution will no longer be accurate for large amplitude vibration, especially when a significant amount of imperfection is considered. Thus Runge-Kutta method is introduced to solve this problem numerically. The comparison between both methods has shown the validity of the Lindstedt's Perturbation Technique is generally within half plate thickness. For a structure with a sufficiently large geometric imperfection, the vibration can be represented as a well-known backbone curve transforming from soften-spring to harden-spring. By parameter variation, the effects of imperfection, damping ratio, boundary conditions, wave numbers, young's modulus and a dozen more related properties are studied. Other interesting research results such as the dynamic failure caused by out-of-bound vibration and the change of vibration mode due to damping are also revealed.

Theoretical formulations, analytical solutions, and finite element solutions for laminated composite plate and shell structures with smart material laminae are presented in the study. A unified third-order shear deformation theory is formulated and used to study vibration/deflection suppression characteristics of plate and shell structures. The von K??rm??n type geometric nonlinearity is included in the formulation. Third-order shear deformation theory based on Donnell and Sanders nonlinear shell theories is chosen for the shell formulation. The smart material used in this study to achieve damping of transverse deflection is the Terfenol-D magnetostrictive material. A negative velocity feedback control is used to control the structural system with the constant control gain. The Navier solutions of laminated composite plates and shells of rectangular planeform are obtained for the simply supported boundary conditions using the linear theories. Displacement finite element models that account for the geometric nonlinearity and dynamic response are developed. The conforming element which has eight degrees of freedom per node is used to develop the finite element model. Newmark's time integration scheme is used to reduce the ordinary differential equations in time to algebraic equations. Newton-Raphson iteration scheme is used to solve the resulting nonlinear finite element equations. A number of parametric studies are carried out to understand the damping characteristics of laminated composites with embedded smart material layers.

Theoretical formulations, analytical solutions, and finite element solutions for laminated composite plate and shell structures with smart material laminae are presented in the study. A unified third-order shear deformation theory is formulated and used to study vibration/deflection suppression characteristics of plate and shell structures. The von K??rm??n type geometric nonlinearity is included in the formulation. Third-order shear deformation theory based on Donnell and Sanders nonlinear shell theories is chosen for the shell formulation. The smart material used in this study to achieve damping of transverse deflection is the Terfenol-D magnetostrictive material. A negative velocity feedback control is used to control the structural system with the constant control gain. The Navier solutions of laminated composite plates and shells of rectangular planeform are obtained for the simply supported boundary conditions using the linear theories. Displacement finite element models that account for the geometric nonlinearity and dynamic response are developed. The conforming element which has eight degrees of freedom per node is used to develop the finite element model. Newmark's time integration scheme is used to reduce the ordinary differential equations in time to algebraic equations. Newton-Raphson iteration scheme is used to solve the resulting nonlinear finite element equations. A number of parametric studies are carried out to understand the damping characteristics of laminated composites with embedded smart material layers.

[pt] A formulação híbrida dos elementos finitos, proposta por Pian, com base no princípio variacional de Hellinger-Reissner, mostrou-se uma ótima alternativa para a construção de elementos finitos eficientes que atendessem a condições tanto de compatibilidade como de equilíbrio. O potencial de Hellinger-Reissner consiste na aproximação de dois campos: um campo tensões que satisfaz, a priori, as equações diferenciais homogêneas de equilíbrio do problema, e um campo de deslocamentos que atende a compatibilidade ao longo do contorno. O conjunto de funções não-singulares que satisfazem as equações governantes de um problema é conhecido como soluções fundamentais ou soluções de Trefftz, e é a base para a interpolação do campo de tensões no método híbrido de elementos finitos. O presente trabalho apresenta uma metodologia geral para a formulação de uma família de elementos finitos híbridos poligonais de membrana para problemas de elasticidade bidimensional, assim como elementos finitos híbridos simples e eficientes a para análise numérica de problemas de placa de Kirchhoff e Mindlin-Reissner. Algumas contribuições conceituais são introduzidas nas soluções fundamentais para a correta concepção dos elementos híbridos em problemas de placa espessa. O desempenho dos elementos é avaliado através de alguns exemplos numéricos, os quais os resultados são confrontados com os de outros elementos encontrados na literatura. / [en] The hybrid finite element formulation, proposed by Pian, on the basis of the Hellinger-Reissner variational principle, has proved to be a good alternative for the development of efficient finite elements that best attend compatibility and equilibrium conditions. The Hellinger-Reissner potential assumes two trial fields: a stress field that satisfies the equilibrium homogenous differential equation in the domain and a displacement field that attends the compatibility along the boundary. The set of nonsingular functions that satisfy the governing equations of the problem is known as Trefftz or fundamental solutions. This work presents a general methodology for the formulation of a family of polygonal hybrid elements for plane strain problems, as well as simple and efficient plate elements for the numerical evaluation of Kirchhoff and Mindlin-Reissner plate problems. Conceptual approaches are introduced for the correct use of fundamental solutions in the plate elements formulation. The performance of the proposed hybrid elements is assessed by means of several numerical examples from the literature.

Dans le cadre linéaire, les modèles de Mindlin-Reissner pour les plaques épaisses et de Naghdi pour les coques épaisses sont les plus utilisés. Il est connu que la discrétisation par éléments finis de ces modèles conduit à un phénomène de verrouillage numérique quand l’épaisseur tend vers zéro. Il s’agit du verrouillage en cisaillement dans le cas des plaques et du verrouillage en cisaillement et en membrane dans le cas des coques. Il existe quelques éléments finis qui permettent d’éviter ces difficultés ou du moins de les réduire. L’élément DKMQ pour les plaques et sa version DKMQ24 pour les coques, sont des éléments de bas ordre, basés sur une formulation mixte, qui ont été proposés il y a quelques années afin d’éviter ces phénomènes de verrouillage. Dans cette thèse, on s’est attaché à évaluer numériquement les performances de ces éléments. Outre les cas tests classiques, on s’est focalisé sur l’analyse de la condition inf-sup discrète pour l’élément DKMQ. Nous avons étudié également le test de la s-norme proposé par Bathe, pour l’élément DKMQ24. Enfin, nous avons effectué une analyse d’erreur a posteriori pour les éléments DKMQ et DKMQ24, en utilisant l’estimateur d’erreur Z2 (dû à Zienkiewicz et Zhu), associé aux techniques de recouvrement de la moyenne, de projection ou encore SPR. Les résultats obtenus ont permis de quantifier les performances de ces deux éléments finis pour les problèmes de verrouillage, et d’en dégager les limites. Deux applications importantes de ces éléments DKMQ et DKMQ24 ont été ensuite présentées, la première concerne la simulation des poutres à parois minces à section ouverte et la seconde le calcul des plaques composites. / In the linear case, the Mindlin-Reissner model for thick plates and the Naghdi model for thick shells are commonly used. The finite element discretization of these models leads to numerical locking phenomenon when the thickness approaches zero : shear locking for plates and both shear and membrane locking for shells. There are some finite elements that could reduce or even eliminate this phenomenon. DKMQ element for plates or DKMQ24 element for shells, are low-order elements, based on a mixed formulation, introduced a few years ago to prevent the numerical locking phenomenon. In this thesis, we concentrated on numerical evaluation of the performance of these elements. Besides the classical benchmark tests, we also focused on the analysis of discrete inf-sup condition for DKMQ element. We studied the s-norm test proposed by Bathe for DKMQ24 element. Finally, we performed a posteriori error estimation for DKMQ and DKMQ24 elements, using the error estimator Z2 (proposed by Zienkiewicz and Zhu), associated with the averaging, projection or SPR recovery methods. The results obtained have enabled us to quantify the performance of these two finite elements for locking problems, and to identify their limits. Two important applications of these elements DKMQ and DKMQ24 were then presented ; the first one concerns thin-walled beams with open cross-section and the second one composite plates.