This thesis focuses on the development and convergence study of finite element methods for eigenvalue analysis of arbitrarily shaped domains with multi-material and material-void interfaces. Such configurations can be found in problems with evolving discontinuities and interfaces as in fluid-structure interaction or topology optimization problems. The differential equations considered in this thesis include the elliptic operator, Timoshenko beam and Mindlin plate. The compatibility conditions at the interface are weakly imposed using either Nitsche's method or Lagrange multipliers. The variational statements are derived for each case. The analysis results are benchmarked using Galerkin finite element discretization with bodyitted grids. Nitsche's method shows a direct dependence on a penalty term and for Lagrange multipliers method, additional degrees of freedom are added to the solution vector. The convergence rate of the discretized forms is computationally determined and is shown to be optimal for both Timoshenko beams and Mindlin plates.

A clássica teoria de Euler-Bernoulli para vibrações transversais de vigas elásticas é sabido não ser adequada para vibrações de altas freqüências, como é o caso de vibração de vigas curtas. Esta teoria assume que a deflexão deve-se somente ao momento fletor, uma vez que os efeitos da inércia de rotação e da força cortante são negligenciados. Lord Rayleigh complementou a teoria clássica demonstrando a contribuição da inércia de rotação e Timoshenko estendeu a teoria ao incluir os efeitos da força cortante. A equação resultante é conhecida como sendo a que caracteriza a chamada teoria de viga de Timoshenko. Usando-se a matriz de rigidez dinâmica, as freqüências naturais e a resposta dinâmica de estruturas de barras são determinadas e comparadas de acordo com resultados de quatro modelos de vibração. São estudados o problema de vibração flexional de vigas, pórticos e grelhas, bem como o problema de fundação elástica segundo o modelo de Winkler e também a versão mais avançada que é o modelo de Pasternak. / Classical Euler-Bernoulli theory for transverse vibrations of elastic beams is known to be inadequate to consider high frequency modes which occur for short beams, for example. This theory is derived under the assumption that the deflection is only due to bending. The effects of rotary inertia and shear deformation are ignored. Lord Rayleigh improved the classical theory by considering the effect of rotary inertia. Timoshenko extended the theory to include the effects of shear deformation. The resulting equation is known as Timoshenko beam theory. The natural frequencies and dynamic reponse of framed structures are determined by using the dynamic stiffness matrix and compered according to these theories. The flexional vibration problems of beams, plane frames and grids are analysed, as well problems of elastic foundation according the well known Winkler model and also the more general Pasternak model.

In this work, several beam finite element formulations are proposed for failure analysis of planar reinforced concrete beams and frames under monotonic static loading. The localized failure of material is modeled by the embedded strong discontinuity concept, which enhances standard interpolation of displacement (or rotation) with a discontinuous function, associated with an additional kinematic parameter representing jump in displacement (or rotation). The new parameters are local and are condensed on the element level. One stress resultant and two multi-layer beam finite elements are derived. The stress resultant Euler-Bernoulli beam element has embedded discontinuity in rotation. Bending response of the bulk of the element is described by elasto-plastic stress resultant material model. The cohesive relation between the moment and the rotational jump at the softening hinge is described by rigid-plastic model. Axial response is elastic. In the multi-layer beam finite elements, each layer is treated as a bar, made of either concrete or steel. Regular axial strain in a layer is computed according to Euler-Bernoulli or Timoshenko beam theory. Additional axial strain is produced by embedded discontinuity in axial displacement, introduced individually in each layer. Behavior of concrete bars is described by elastodamage model, while elasto-plasticity model is used for steel bars. The cohesive relation between the stress at the discontinuity and the axial displacement jump is described by rigid-damage softening model in concrete bars and by rigid-plastic softening model in steel bars. Shear response in the Timoshenko element is elastic. Finally, the multi-layer Timoshenko beam finite element is upgraded by including viscosity in the softening model. Computer code implementation is presented in detail for the derived elements. An operator split computational procedure is presented for each formulation. The expressions, required for the local computation of inelastic internal variables and for the global computation of the degrees of freedom, are provided. Performance of the derived elements is illustrated on a set of numerical examples, which show that the multi-layer Euler-Bernoulli beam finite element is not reliable, while the stress-resultant Euler-Bernoulli beam and the multi-layer Timoshenko beam finite elements deliver satisfying results.

Design denotes the transformation of an identified need to its physical embodiment in a traditionally iterative approach of trial and error. Conceptual design plays a prominent role but an almost infinite number of possible solutions at the outset of design necessitates fast evaluations. The traditional practice of empirical databases loses adequacy for novel concepts and an ever increasing system complexity and resource scarsity mandate new approaches to adequately capture system characteristics. Contemporary concerns in atmospheric science and homeland security created an operational need for unconventional configurations. Unmanned long endurance flight at high altitudes offers a unique showcase for the exploration of new design spaces and the incidental deficit of conceptual modeling and simulation capabilities. The present research effort evolves around the development of an efficient and accurate optimization algorithm for high aspect ratio wings subject to natural frequency constraints. Foundational corner stones are beam dimensional reduction and modal perturbation redesign. Local and global analyses inherent to the former suggest corresponding levels of local and global optimization. The present approach departs from this suggestion. It introduces local level surrogate models to capacitate a methodology that consists of multi level analyses feeding into a single level optimization. The innovative heart of the new algorithm originates in small perturbation theory. A sequence of small perturbation solutions allows the optimizer to make incremental movements within the design space. It enables a directed search that is free of costly gradients. System matrices are decomposed based on a Timoshenko stiffness effect separation. The formulation of respective linear changes falls back on surrogate models that approximate cross sectional properties. Corresponding functional responses are readily available. Their direct use by the small perturbation based optimizer ensures constitutive laws and eliminates a previously necessary optimization at the local level. The great economy of the developed algorithm makes it suitable for the conceptual phase of aircraft design.

A clássica teoria de Euler-Bernoulli para vibrações transversais de vigas elásticas é sabido não ser adequada para vibrações de altas freqüências, como é o caso de vibração de vigas curtas. Esta teoria assume que a deflexão deve-se somente ao momento fletor, uma vez que os efeitos da inércia de rotação e da força cortante são negligenciados. Lord Rayleigh complementou a teoria clássica demonstrando a contribuição da inércia de rotação e Timoshenko estendeu a teoria ao incluir os efeitos da força cortante. A equação resultante é conhecida como sendo a que caracteriza a chamada teoria de viga de Timoshenko. Usando-se a matriz de rigidez dinâmica, as freqüências naturais e a resposta dinâmica de estruturas de barras são determinadas e comparadas de acordo com resultados de quatro modelos de vibração. São estudados o problema de vibração flexional de vigas, pórticos e grelhas, bem como o problema de fundação elástica segundo o modelo de Winkler e também a versão mais avançada que é o modelo de Pasternak. / Classical Euler-Bernoulli theory for transverse vibrations of elastic beams is known to be inadequate to consider high frequency modes which occur for short beams, for example. This theory is derived under the assumption that the deflection is only due to bending. The effects of rotary inertia and shear deformation are ignored. Lord Rayleigh improved the classical theory by considering the effect of rotary inertia. Timoshenko extended the theory to include the effects of shear deformation. The resulting equation is known as Timoshenko beam theory. The natural frequencies and dynamic reponse of framed structures are determined by using the dynamic stiffness matrix and compered according to these theories. The flexional vibration problems of beams, plane frames and grids are analysed, as well problems of elastic foundation according the well known Winkler model and also the more general Pasternak model.

In wind turbines, flow pressure variations on the air-structure interface cause aerodynamic forces. Consequently the structure deforms and starts to move. The interaction between aerodynamic forces and structural deformations mainly concerns aeroelasticity. Since these two are coupled, they have to be considered simultaneously in cases which the deformations are not negligible in comparison to the other geometric dimensions. The purpose of this work is to improve the simulation model of a vertical axis wind turbine by modifying the structural model from undamped Euler-Bernoulli beam theory with lumped mass matrix to the more advanced Timoshenko beam theory with consistent mass matrix plus an additional damping term. The bending of the beam is then unified with longitudinal and torsional deformations based on a fixed shape cross-section assumption and the Saint-Venant torsion theory. The whole work has been carried out by implementing the finite element method using MATLAB code and implanting it in a previously developed package as a complement. Finally the results have been verified by qualitative comparisons with alternative simulations.

Rotating Euler-Bernoulli beams and non-homogeneous Timoshenko beams are widely used to model important engineering structures. Hence the vibration analyses of these beams are an important problem from a structural dynamics point of view. The governing differential equations of both these type of beams do not yield any simple closed form solutions, hence we look for the inverse problem approach in determining the beam property variations given certain solutions. Firstly, we look for a rotating beam, with pinned-free boundary conditions, whose eigenpair (frequency and mode-shape) is same as that of a uniform non-rotating beam for a particular mode. It is seen that for any given mode, there exists a flexural stiffness function (FSF) for which the ith mode eigenpair of a rotating beam with uniform mass distribution, is identical to that of a corresponding non-rotating beam with same length and mass distribution. Inserting these derived FSF's in a finite element code for a rotating pinned-free beam, the frequencies and mode shapes of a non-rotating pinned-free beam are obtained. For the first mode, a physically realistic equivalent rotating beam is possible, but for higher modes, the FSF has internal singularities. Strategies for addressing these singularities in the FSF for finite element analysis are provided. The proposed functions can be used as test functions for rotating beam codes and also for targeted destiffening of rotating beams. Secondly, we study the free vibration of rotating Euler-Bernoulli beams, under cantilever boundary condition. For certain polynomial variations of the mass per unit length and the flexural stiffness, there exists a fundamental closed form solution to the fourth order governing differential equation. It is found that there are an infinite number of rotating beams, with various mass per unit length variations and flexural stiffness distributions, which share the same fundamental frequency and mode shape. The derived flexural stiffness polynomial functions are used as test functions for rotating beam numerical codes. They are also used to design rotating cantilever beams which may be required to vibrate with a particular frequency. Thirdly, we study the free vibration of non-homogeneous Timoshenko beams, under fixed-fixed and fixed-hinged boundary conditions. For certain polynomial variations of the material mass density, elastic modulus and shear modulus, there exists a fundamental closed form solution to the coupled second order governing differential equations. It is found that there are an infinite number of non-homogeneous Timoshenko beams, with various material mass density, elastic modulus and shear modulus distributions, which share the same fundamental frequency and mode shape. They can be used to design non-homogeneous Timoshenko beams which may be required for certain engineering applications.

Les structures treillis constituées d’un nombre important de barres sont largement utilisées, notamment en génie civil. L’étude par éléments finis de telles structures se révèle très coûteuse dès que la maille répétitive du treillis est complexe. Il s’avère intéressant de réduire la taille du problème en définissant un milieu continu équivalent. L’objectif de la première partie de ce travail est de proposer, en se plaçant dans le cadre des méthodes d’homogénéisation des milieux périodiques, une poutre de Timoshenko équivalente à une structure périodique dont l’une des dimension est grande par rapport aux deux autres. Une des originalités réside dans l’étude de cellules de base non symétriques. Par ailleurs, on s’intéresse à la prise en compte de déformations libres (par exemple, d’origine thermique) apparaissant à l’échelle microscopique. La seconde partie est consacrée à l’étude de la structure axonémale du flagelle et des cils vibratiles. Il s’agit de proposer et valider un modèle pour cette structure biomécanique complexe et d’appliquer ensuite la méthode d’homogénéisation proposée / Lattice structures are widely used, especially in civil engineering. The finite element analysis of such structures might require a consequent amount of computational time when the periodical mesh of this lattice is complex. Defining an equivalent continuous medium in order to reduce the size of the problem appears to be interesting. The aim of the first part of this document is to apply a homogenization method in order to find a Timoshenko beam model macroscopically equivalent to a slender structure which is periodical in the longitudinal direction. One of the unusual aspects tackled reside in the study of structures with periodical cells having a longitudinal asymmetry. In addition, the case of periodical structures with free deformation (e.g. thermal dilatation) at microscopic scale is dealt. The second part is consecrated to the study of the axonemal structure of the biological cell flagella and Cilia. A shorten version of the axonemal structure is studied at first and homogenized afterward

Se presenta un modelo de elementos finitos que describe el comportamiento de vibración de libre de vigas compuestas laminadas. Se desarrolla el modelo utilizando el principio de Hamilton y la teoría de vigas Timoshenko que incluye deformaciones por corte. Se asume interpolaciones de alto orden para la aproximación de las variables fundamentales. Los laminados compuestos son ortotrópicos con fibras orientadas en diferentes direcciones. Se implementa un programa para materiales compuestos laminado en MATLAB. Se comparan resultados con otros obtenidos en la literatura para validar el modelo. Se realiza un estudio de convergencia y paramétrico con un mismo número de lámina y diferentes direcciones. Se verifica que la formulación que es bastante precisa con resultados satisfactorios en la investigación. / In this work, is presented a finite element model that describes the free vibration behavior of laminated composite beams. The model is developed by the Hamilton principle and the Timoshenko theory that includes shear deformations. Composite laminates are assumed to be orthotropic with fibers oriented in different directions, such as Angle Ply and Cross Ply cases. This investigation works out on a MAPLE program for laminated composites materials that will be completed all in MATLAB program. In order to validate the model, the results are compared with different literatures, also verify the formulation that is quite accurate and obtain quite satisfactory results in the investigation. High order interpolations are assumed to approximate fundamental variables. A convergence study and parametric study will be carried out with the same number of laminas in different directions. / Tesis

Piston rings are installed in the piston and cylinder wall, which does not have a perfect round shape due to machining tolerances or external loads e.g. head bolts tightening. If the ring cannot follow these deformations, a localized lack of contact will occur and consequently an increase in the engine blow-by and lubricant oil consumption. Current 2D computational methods can not implement such effects – more complex model is necessary. The presented master’s thesis is focused on the developement of a flexible 3D piston ring model able to capture local deformations. It is based on the Timoshenko beam theory in cooperation with MBS software Adams. Model is then compared with FEM using software ANSYS. The validated piston ring model is assembled into the piston/cylinder liner and very basic simulations are run. Finally, future improvements are suggested.