Ce travail concerne le développement d’un outil numérique pour résoudre des problèmes de plaques et de coques mécaniques et piézo-électriques multicouches. En d’autres termes, nous développons des éléments finis basés sur le modèle de Reissner-Mindlin pour l’analyse des problèmes mécaniques et piézoélectriques de plaques et de coques multicouches. Cet outil doit être le moins coûteux en termes de degrés de liberté, simple à utiliser pour modéliser le problème, sans aucune pathologie numérique classique et satisfaisant :vitesse de convergence et efficacité.Dans le domaine des éléments finis, il existe une théorie appelée La « théorie des déformations du cisaillement du premier ordre »(FSDT), utilisée pour résoudre des problèmes purement mécanique prenant en compte l’effet des contraintes en cisaillement transverse. Dans cette théorie, cinq degrés de liberté (trois translations et deux rotations), sont à chercher.Prenant en compte l’hypothèse d’une approximation du déplacement au niveau de la surface moyenne, en utilisant la loi constitutive 2D.Dans ce travail, nous présentons l’extension piézoélectrique de cette théorie pour des éléments finis quadratiques à huit nœuds pour résoudre des problèmes de plaques et de coques multicouches. En effet, la nouveauté dans ce travail est celle de considérer une loi constitutive 3D afin de calculer non seulement le déplacement transversal au niveau du plan moyen de chaque couche mais aussi les déplacements transversaux en surface supérieure et en surface inférieure de chaque couche. Cette approximation de l’épaisseur fait intervenir deux nouveaux degrés de liberté, qui seront très importants dans l’étude des plaques et des coques épaisses et semi épaisses.Le déplacement mécanique est approximé en utilisant une approche Equivalent Single Layer(ESL) et le potentiel électrique est , quant à lui, approximé par une approche LayerWise (LW).Cette évolution est proposée afin d’acquérir un bon compromis entre le minimum de degrés de liberté et le maximum d’efficacité. D’une part, l’approximation par élément fini pour le potentiel électrique, respectant les coordonnées de l’épaisseur est représenté par une variation linéaire au niveau de chaque couche. D’autres part ce potentiel est constant par élément en chaque interface, ce qui réduit le nombre d’inconnus pour la recherche du potentiel électrique résultant. Plusieurs tests numériques sont présentés dans le but d’évaluer les éléments mécanique et piézoélectrique de plaque PQ8P7 et PQ8P7PZ et de coque CQ8P8 et CQ8P8PZ ainsi que leurs capacités de résoudre les problèmes physiques auxquels ils sont dédiés, en comparant nos résultats aux solutions de référence. / The aim of this work is to develop a computational tool for a multilayred piezolelectic plates and shells: a low cost tool, simple to use and very efficient for both convergence velocity and accuracy, without any classical numerical pathologies. In the field of finite elements, there is atheory, called "First-order-shear deformation theories (FSDT) used for the mechanical part in much reserches, taking in account the transverse shear stress effects. There are only five unknows generalized displacement, taking the hypothesis of the midsurface displacement approximation using a 2D constitutive law. In This work we present the piezo electric extension of the FSDT eight nodes plate/shell finite element using seven unknows genralized displacement. Two new mechanical unknowns are added to approximate the thickness in the top and the bottom of the /shell, considering a 3D constitutive law, wich is very interesting study especialy for the thick plate/shell modelisation. The mechanical displacement is approximated using the equivalent single layer approch (ESL) and the electric potential is approximated using the layerwise approch (LW).An evolution is proposed in order to access the best compromise between minimum number of degrees of freedom and maximum efficiency. On one hand, a finite element approximation for the electric potential with respect to the thickness coordinate are presented a linear variation in each layer. On the other hand, the in-plane variation is constant on the elementary domain at each interface layer. The use of a constant value reduces the number of unknown electric potentials. Furthermore, at the post processing level, the transverse shear stress are deduced using the equilibrium equations.Numerous tests are presented in order to evoluate that capability of these electric potential approximations to give accurate results with respect to piezoelasticity or finite element reference solutions.

Ingeniero Civil Mecánico / En la práctica de la ingeniería el método de elementos finitos o MEF se utiliza para realizar cálculos cuando resulta difícil o ineficiente obtener una solución analítica. En los últimos años se ha popularizado el uso de los métodos sin malla ya que estos entregan soluciones más exactas además de ser menos sensibles a las distorsiones de los elementos. Esto último se debe a que los métodos sin malla utilizan un vecindario de nodos para construir la aproximación de las variables, eliminando la necesidad de conexiones elementales (mallado) entre los nodos. La placa de Reissner-Mindlin (RM) se utiliza para el análisis de esfuerzos y deformaciones en placas gruesas. Este modelo supone que la deformación a lo largo del espesor varía en forma lineal además supone que la componente σzz del esfuerzo es nula.Sin embargo, la placa RM presenta el problema de bloqueo por corte, donde la rigidez de la placa aumenta drásticamente a medida que se disminuye el espesor. El objetivo de este trabajo es solucionar el bloqueo por corte mediante la proyección nodal, la cual ya ha sido utilizada para solucionar el bloqueo volumétrico en elasticidad incompresible (análogo al bloqueo por corte). Para este propósito se formula la proyección nodal para el problema de la placa RM. La proyección nodal se formula a partir de la formulación mixta eliminando los grados de libertad de corte S a partir de la discretización de los desplazamientos adyacentes disminuyendo el número de ecuaciones por resolver. Se implementa el esquema para la proyección nodal sin malla con funciones de base de máxima entropía utilizando el software Matlab. Se realizaron experimentos numéricos sobre problemas cuya solución analítica es conocida y se evaluó la convergencia y sensibilidad a distorsiones para la proyección nodal y métodos tradicionales. La proyección nodal converge óptimamente para la norma L2 del error y para la semi-norma H1 del error. La proyección nodal es menos sensible a distorsiones geométricas en la malla base que métodos MEF tradicionales. Se concluye que la proyección nodal soluciona el problema de bloqueo por corte utilizando métodos sin malla. Se consigue una mejor convergencia con respecto a los métodos tradicionales además de disminuir la sensibilidad a distorsiones geométricas.

Ce travail est consacré à l’étude d’estimateurs d'erreur a posteriori de type flux équilibrés et résiduels pour la résolution des équations de Reissner-Mindlin par la méthode des éléments finis. Le mémoire débute par l'introduction du problème aux limites et de son analyse de convergence a priori par la méthode des éléments finis. Nous construisons alors pour une discrétisation conforme un estimateur a posteriori de type flux équilibrés fiable, efficace et robuste en l'épaisseur de la plaque t. Nous obtenons finalement une constante multiplicative égale à 1 pour la fiabilité. Des tests numériques illustrent nos résultats pour différents maillages. Puis nous abordons le cas d’une discrétisation non-conforme, où nous proposons un estimateur a posteriori de type résiduel, utilisant une régularisation de la solution discrète. Des tests numériques illustrent également nos résultats. La suite du travail reprend la discrétisation conforme en construisant un estimateur a posteriori défini à partir de la résolution de problèmes localisés sur les patchs de la triangulation, menant à un choix plus consistant avec le problème aux limites. Le dernier chapitre est consacré à l'estimation a posteriori pour le problème aux valeurs propres de Reissner-Mindlin. L’estimateur obtenu est fiable et efficace pour la norme de l'erreur entre les vecteurs propres, permettant également de majorer l’erreur commise entre les valeurs propres. Des tests numériques illustrent nos résultats. / This work is devoted to the study of equilibrated fluxes and residual a posteriori error estimators for the finite element resolution of the Reissner-Mindlin system. This report begins by the introduction of the boundary value problem and of its a priori convergence analysis in the finite element method context. Then, an equilibrated fluxes a posteriori estimator is built for a conform discretization, which is proven to be reliable, efficient and robust on the plate thickness t. We finally obtain a multiplicative constant equal to 1 for the reliability. Numerical tests illustrate our results on different meshes. Then, we address the non-conforming discretization case, where a residual a posteriori estimator is proposed using a regularisation of the discrete solution. Numerical tests also illustrate our results. Next we come back to the conform discretization by building an a posteriori estimator defined from localised problems resolution on stars, leading to a consistent choice with the boundary value problem. The last chapter is devoted to an a posteriori estimation for the Reissner-Mindlin eigenvalues problem. The obtained estimator is reliable and efficient for the error norm between the eigenvectors, also allowing to evaluate the error between the eigenvalues. Numerical tests illustrate our results.

This thesis presents the first-known exact solutions for vibration of stepped circular Mindlin plates. The considered circular plate is of several step-wise variation in thickness in the radial direction. The Mindlin first order shear deformable plate theory is employed to derive the governing differential equations for the annular and circular segments. The exact solutions to these differential equations may be expressed in terms of the Bessel functions of the first and second kinds and the modified Bessel functions of the first and second kinds. The governing homogenous system of equations is assembled by implementing the essential and natural boundary conditions and the segment interface conditions. Vibration solutions are presented for circular Mindlin plates of different edge support conditions and various combinations of step-wise thickness variations. These exact vibration results may serve as important benchmark values for researchers to validate their numerical methods for such circular plate problems / Master of Engineering (Civil)

Submitted by ANA KARLA PEREIRA RODRIGUES (anakarla_@hotmail.com) on 2017-08-15T11:54:12Z No. of bitstreams: 1 arquivototal.pdf: 1761582 bytes, checksum: 7e797a75c54f45dbcc28cbeab246335d (MD5) / Made available in DSpace on 2017-08-15T11:54:12Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 1761582 bytes, checksum: 7e797a75c54f45dbcc28cbeab246335d (MD5) Previous issue date: 2016-12-15 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This thesis is concerned with the dynamics of Mindlin-Timoshenko system for beams and plates. We study issues relating to the asymptotic limit in relation to the parameters and decay rates. In the context of asymptotic limit, as the main result, we present a positive response to the conjecture made by Lagnese and Lions in 1988, where the Von-Kármán model is obtained as singular limit when k tends to infinity, the Mindlin-Timoshenko system. Introducing appropriate damping mechanisms (internal and boundary), we also show that the energy of solutions for the Mindlin-Timoshenko system has decay properties exponential and polynomial, with respect to the parameters. / Esta tese aborda a dinâmica do sistema de Mindlin-Timoshenko para vigas e placas. Estudamos questões relacionadas com o limite assintótico em relação aos parâmetros e as taxas de decaimento. No contexto do limite assintótico, como resultado principal, apresentamos uma resposta positiva à conjectura feita por Lagnese e Lions em 1988, onde o modelo de Von-Kármán é obtido como limite singular, quando k tende ao infinito, do sistema de Mindlin-Timoshenko. Introduzindo mecanismos de amortecimento apropriados (internos e de fronteira), também mostramos que, sob certas condições, a energia de solução do sistema de Mindlin-Timoshenko tem propriedades de decaimento exponencial e polinomial com relação aos parâmetros.

La présente thèse doctorale traite de la formulation et l'évaluation élément fini d'un model variationel de Mindlin discret et modifié pour l'analyse du comportement statique et dynamique, linéaire et non linéaire des plaques et coques composites. Incluant des termes additionnels de type zigzag, en vue d'améliorer la précision de contraintes, le modèle a été reformulé afin de prendre en considération la variation linéaire par couche du déplacement. En conséquence, deux éléments finis de plaques et coques à quatre nœuds, baptisés respectivement DMQP et DMQS (Discret Mindlin Quadrilateral Plates/ Shells), améliorés par un champ de rotations quadratique ont été développés et validés sur les codes de calcul REFLEX et ABAQUS. Dans une seconde version, les deux éléments avec un effet de zigzag ont été développés et validés à travers quelques tests statiques et dynamiques connus de la littérature. Les résultats montrent une indépendance vis-à-vis de la correction du cisaillement transverse et une précision des contraintes meilleure à celle obtenue par rapport le modèle initial (sans l'effet de zigzag).Les résultats satisfaisants de ce modèle constatés à travers les cas-tests linéaires de coque isotropes, nous ont motivés à étendre la présente approche aux applications non-linéaires géométriques. Un élément isoparamétrique courbe de coque a été développé à cet effet, avec l'hypothèse des petites déformations élastiques et grands déplacements et rotations modérées. Il est géométriquement simple et ne possède que quatre nœuds aux sommets et 6 ddl/nœud. Le calcul élémentaire de la matrice de rigidité tangente consiste à associer la partie linéaire du modèle de coque courbe (DMQS) avec celle non linéaire de l'élément standard Q4 de membrane. Une formulation lagrangienne actualisée à chaque itération (FLAI) a été utilisée avec la méthode de résolution de Newton-Raphson. Quelques tests standards non-linéaires des structures coques sont présentés, ils montrent un très bon comportement global et une convergence meilleure que celle d'éléments pareils. / This doctoral thesis deals with the finite element formulation and evaluation of a modified Mindlin's discrete variational model for static, dynamic, linear and non-linear composite plates and shells analysis. Including additional terms of zigzag type, in order to improve the accuracy of stress, the model has been reformulated to take into account the linear picewise of displacement variation. Consequently, two finite plate and shell elements with four nodes, called DMQP and DMQS (Discrete Quadrilateral Mindlin Plates and Shells respectively), enhanced by quadratic field rotations, have been developed and validated under REFLEX and ABAQUS codes.Both elements including the zigzag effect have been also developed in a second version, and validated through several static and dynamic test problems known from the literature, highlighting the independence towards the transverse shear correction and in particular the stress accuracy with respect to the initial model without the zigzag effect.The satisfactory results of this model found through cases of linear isotropic shell tests, motivated us to extend this approach to the non-linear geometric applications. An isoparametric curve element of shell has been developed for this purpose, where small elastic deformation assumptions of and large displacements and moderate rotations are adopted. It is geometrically simple and has only four nodes at corners and 6 DOF/node. The elementary calculation of the tangent stiffness matrix consists in combining the linear part of the curved shell element (DMQS) with that of the membrane Q4 non-linear part. An Updated Lagrangian Formulation at each Iteration (ULFI) is used with Newton-Raphson resolution Method. Some standard tests of nonlinear geometrical shell structures are presented; they show a very good convergence and global behavior better than such elements.

Este trabalho consiste em analisar os esforços em tabuleiros de pontes por meio de superfícies de influência. Para isto, o método dos elementos finitos (MEF) é utilizado e os resultados são comparados com os das tabelas de Rüsch. Os elementos finitos de barra, representando longarinas e transversinas, e placa, as lajes do tabuleiro, são implementados no código SIPlacas. Estes elementos finitos são formulados pelas teorias de viga Timoshenko e placa Reissner-Mindlin, respectivamente. Estes apresentam problema de travamento de força cortante (Shear Locking), que é contornado por duas propostas: o artifício matemático da integração reduzida e elementos finitos com campo assumido de deformação de força cortante (CADFC). Verifica-se que os elementos com aproximações quadráticas para os deslocamentos e com CADFC são os que melhor se adequam à proposta de análise da presente pesquisa. Tais elementos apresentam convergência de resultados considerando estruturas com baixa discretização. Os resultados analisados foram o deslocamento, momento fletor e força cortante. Posteriormente realiza-se um estudo de caso de uma ponte em viga. O tabuleiro da ponte é calculado utilizando-se as tabelas de Rüsch e o código SIPlacas. O cálculo dos esforços pelo SIPlacas é realizado de três maneiras. Na primeira consideram-se os painéis de lajes do tabuleiro isolados; na segunda o tabuleiro está sobre apoios não deslocáveis; e na terceira, o tabuleiro apresenta-se com vigas acopladas. Foi concluído que a terceira configuração, cuja representação melhor se aproxima da estrutura real de análise, apresentou os menores esforços internos. / This work aims at the analysis of bridge deck stresses through influence surfaces. The finite element method (FEM) is used and the results are compared with those of Rüsch\'s tables. The bar and plate finite elements represent stringers, cross beams and slabs bridge deck. These finite elements are implemented in the SIPlacas code and the theories of Timoshenko beam and Reissner-Mindlin plate are used to theirs formulation. The Shear Locking problem is solved by two proposals: reduced integration and definition of element with transversal shear strain assumed (TSSA). The elements with quadratic approximations for the displacements and TSSA are the best suited to the proposed analysis of this research. Such elements have convergence of results considering structures with low discretization. Displacement, bending moment and shear force were the results analyzed. Subsequently a case study on a beam bridge was carried out. The bridge deck is calculated using Rüsch\'s tables and SIPlacas code. The calculation of the internal forces by SIPlacas is performed in three ways. The first one considers the slabs isolated panels; the second, the slab deck is on a rigid support; and third, the slab deck is on deformable supports. It was concluded that the third configuration showed the lowest internal forces. This configuration is the optimum representation to the structure analysis.

Este trabalho consiste em analisar os esforços em tabuleiros de pontes por meio de superfícies de influência. Para isto, o método dos elementos finitos (MEF) é utilizado e os resultados são comparados com os das tabelas de Rüsch. Os elementos finitos de barra, representando longarinas e transversinas, e placa, as lajes do tabuleiro, são implementados no código SIPlacas. Estes elementos finitos são formulados pelas teorias de viga Timoshenko e placa Reissner-Mindlin, respectivamente. Estes apresentam problema de travamento de força cortante (Shear Locking), que é contornado por duas propostas: o artifício matemático da integração reduzida e elementos finitos com campo assumido de deformação de força cortante (CADFC). Verifica-se que os elementos com aproximações quadráticas para os deslocamentos e com CADFC são os que melhor se adequam à proposta de análise da presente pesquisa. Tais elementos apresentam convergência de resultados considerando estruturas com baixa discretização. Os resultados analisados foram o deslocamento, momento fletor e força cortante. Posteriormente realiza-se um estudo de caso de uma ponte em viga. O tabuleiro da ponte é calculado utilizando-se as tabelas de Rüsch e o código SIPlacas. O cálculo dos esforços pelo SIPlacas é realizado de três maneiras. Na primeira consideram-se os painéis de lajes do tabuleiro isolados; na segunda o tabuleiro está sobre apoios não deslocáveis; e na terceira, o tabuleiro apresenta-se com vigas acopladas. Foi concluído que a terceira configuração, cuja representação melhor se aproxima da estrutura real de análise, apresentou os menores esforços internos. / This work aims at the analysis of bridge deck stresses through influence surfaces. The finite element method (FEM) is used and the results are compared with those of Rüsch\'s tables. The bar and plate finite elements represent stringers, cross beams and slabs bridge deck. These finite elements are implemented in the SIPlacas code and the theories of Timoshenko beam and Reissner-Mindlin plate are used to theirs formulation. The Shear Locking problem is solved by two proposals: reduced integration and definition of element with transversal shear strain assumed (TSSA). The elements with quadratic approximations for the displacements and TSSA are the best suited to the proposed analysis of this research. Such elements have convergence of results considering structures with low discretization. Displacement, bending moment and shear force were the results analyzed. Subsequently a case study on a beam bridge was carried out. The bridge deck is calculated using Rüsch\'s tables and SIPlacas code. The calculation of the internal forces by SIPlacas is performed in three ways. The first one considers the slabs isolated panels; the second, the slab deck is on a rigid support; and third, the slab deck is on deformable supports. It was concluded that the third configuration showed the lowest internal forces. This configuration is the optimum representation to the structure analysis.

[pt] A historiografia da Arquitetura Moderna Brasileira tende a privilegiar a construção de uma coerência ficcional que determina grupos aos quais todos arquitetos atuantes naquele período são anexados. No entanto, uma análise mais detalhada traz à tona a existência de múltiplas formas de ação que não concorrem para tal unidade fictícia. Dentre eles estão os arquitetos que trabalham prioritariamente para o mercado da construção civil, que lhes impõe regras, meios, relações econômicas, que contaminam seu modo de operação projetual. Essa tese busca, através da análise de dois escritórios de arquitetura cariocas – MMMRoberto e Henrique Mindlin Arquitetos Associados - perceber de que maneiras as imposições do mercado alteram a obra de tais escritórios, fazendo-os adquirir características de concepção de projeto e de organização funcional que os torne distintos de outros escritórios atuantes no mesmo período. / [en] The historiography of Brazilian Modern Architecture tends to favor the construction of a fictional coherence; determining groups to which all architects that have worked in that period are attached to. However, a more detailed analysis brings to light the existence of multiple forms of action that do not compete for this fictitious unity. Among them, architects who work primarily for the construction market, which imposes rules, resources, economic relations, that contaminate their modus operandi. This thesis tries to determinate by the analyses of two modern architectural offices – MMMRoberto and Henrique Mindlin Arquitetos Associados - in which ways the market impositions do change the work of these architectural firms, leading them to acquire different design characteristics and organizational methods.

[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.