Beams are among the most commonly used structural members that are encountered in virtually all systems of structural design at various scales. Mathematical models used to determine the response of beams under external loads are deduced from the three-dimensional elasticity theory through a series of assumptions concerning the kinematics of deformation and constitutive behavior. The kinematic assumptions exploit the fact that such structures do not experience significant trans- verse normal and shear strains and stresses. For example, the solution of the three- dimensional elasticity problem associated with a straight beam is reformulated as a one-dimensional problem in terms of displacements whose form is presumed on the basis of an educated guess concerning the nature of the deformation. In many cases beam structures are subjected to compressive in-plane loads that may cause out-of-plane buckling of the beam. Typically, before buckling and during compression, the beam develops internal axial force that makes the beam stiffer. In the linear buckling analysis of beams, this internal force is not considered. As a result the buckling loads predicted by the linear analysis are not accurate. The present study is motivated by lack of suitable theory and analysis that considers the nonlinear effects on the buckling response of beams. This thesis contains three new developments: (1) the conventional beam theories are generalized by accounting for nonlinear terms arising from εzz and εxz that are of the same magnitude as the von K´arm´an nonlinear strains appearing in εxx. The equations of motion associated with the generalized Euler–Bernoulli and Timoshenko beam theories with the von K´arm´an type geometric nonlinear strains are derived using Hamilton’s principle. These equations form the basis of investigations to determine certain microstructural length scales on the bending, vibration and buckling response of beams used in micro- and nano-devices. (2) Analytical solutions of the conventional Timoshenko beam theory with the von K´arm´an nonlinearity are de- veloped for the case where the inplane inertia is negligible when compared to other terms in the equations of motion. Numerical results are presented to bring out the effect of transverse shear deformation on the buckling response. (3) The development of a nonlinear finite element model for post-buckling behavior of beams.

Made available in DSpace on 2015-05-08T14:59:59Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 2498464 bytes, checksum: f14d53abab590dc87f310472963c08ad (MD5) Previous issue date: 2014-09-29 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this work new solutions based on the direct Boundary Element Method (BEM) for static and dynamic stability beam problems are presented. Both Euler-Bernoulli and Timoshenko models are used to represent the beam responses. All discussions on mathematical steps to write down the BEM representation are presented. Alternative fundamental solutions for static and dynamic Euler-Bernoulli beam stability problems are proposed, resulting in the simpler forms than conventional fundamental solutions commonly used for the problems. In addition, the effects of Pasternak elastic foundations are incorporated into the expressions of proposed fundamental solutions. For the case of the Timoshenko static and dinamic stability, all the direct BEM representation (integral equations, fundamental solutions and algebraic equations) here proposed are inovative. Their fundamental solutions incorporate Pasternak foundation effects as well. A convenient strategy is also presented in order to deal with elastic end supports and discontinuities at beam domain such as abrupt change of cross section geometry (stepped beams), internetiated axial load, rigid or elastic supports at beam domain. Numerical examples incorporating various types of boundary conditions and domain discontinuities in order to validate the proposed BEM solution are presented. / Neste trabalho, novas soluções, baseadas no Método dos Elementos de Contorno (MEC) direto, são apresentadas para os problemas de estabilidade estática e dinâmica de vigas. Ambos modelos de Euler-Bernoulli e Timoshenko são usados para representar as respostas da viga. Todas as discussões sobre os passos matemáticos para escrever a representação do MEC são apresentadas. Soluções fundamentais alternativas são propostas para o problema da estabilidade estática e dinâmica de vigas de Euler-Bernoulli, resultando em formas mais simples que as comumente usadas para esses problemas. Além disso, os efeitos de fundações elásticas de Pasternak são incorporadas nas expressões das soluções fundamentais propostas. Para o caso da estabilidade estática e dinâmica de Timoshenko, toda a representação do MEC (equações integrais, soluções fundamentais e equações algébricas) aqui proposta é inovadora. Suas soluções fundamentais incorporam os efeitos da base elástica de Pasternak também. Uma estratégia conveniente é também apresentada para lidar com apoios elásticos no contorno e com discontinuidades no domínio tais como: mudança abrupta de geometria da seção transversal (viga escalonada), carga axial intermediária, apoios rígidos ou elásticos no domínio. Exemplos numéricos incorporando vários tipos de condições de contorno e discontinuidades no domínio são apresentadas para validar as soluções do MEC propostas.

[pt] Um procedimento simples, para a avaliação do aumento ou diminuição (sensibilidade) das cargas de flambagem estática ou dinâmica (flutter) de estruturas na presença de pequenas imperfeições geométricas, é apresentado. O modelo computacional considera estruturas planas elásticas modeladas por elementos de viga retos. Os sistemas de carregamento podem ser conservativos ou não- conservativos. A formulação matricial inclui as matrizes de rigidez elástica, de massa e geométrica usuais. O algoritmo implementado avalia a carga crítica clássica. Adicionalmente, as imperfeições são levadas em conta a partir de uma matriz de imperfeição, resultante da mudança na rigidez elástica decorrente de pequenos desvios na geometria (pequenas rotações nos elementos). São apresentados exemplos para ilustrar a técnica desenvolvida e investigar as cargas de flambagem de algumas estruturas básicas: pórtico de Roorda, coluna de Euler, coluna de Beck, arco circular e viga sobre base elástica (análogo à deformação axissimétrica de uma casca cilíndrica). Conclui-se que o uso da matriz de imperfeição proposta pode fornecer valiosa informação qualitativa sobre a sensibilidade da carga de flambagem em relação às imperfeições. / [en] A simple procedure for the evaluation of the increase or decrease (sensitivity) of static and dynamic (flutter) buckling loads of structures in the presence of small geometrical imperfections is presented. The computational model considers plane elastic structures modelled by straight beam elements. The loading systems cam be conservative or non-conservative. The matrix formulation includes the usual elastic stiffness, mass, and geometric matrices. The algorithm implemented herein evaluates the classical critical load. In addition, the imperfections are taken into account by an imperfection matrix, which results from the change in the elastic stiffness due to small deviations of geometry (small rotations of the elements). Examples are presented to illustrate the technique developed herein and to investigate the buckling loads of some basic structures: Roorda´s frame, Euler´s column, Beck´s column, circular arch, and beam on elastic foundation (analogous to axissimetric deformation of cylindrical shell). It is concluded that the use of the proposed imperfection matrix can provide valuable qualitative information on the sensitivity of the buckling load to imperfections. / [es] En este trabajo se presenta un procedimento simple para la evaluación del aumento o disminución (sensibilidad) de las cargas de flameo estática o dinámica (flutter) de extructuras en presencia de pequeñas imperfecciones geométricas. El modelo computacional considera extructuras planas elásticas modeladas por elementos de viga rectos. Los sistemasde carga pueden ser conservativos o no- conservativos. La formulación matricial incluye las matrices de rígidez elástica, de masa y geométrica usuales. El algoritmo implementado evalúa la carga crítica clásica. Adicionalmente, las imperfecciones se consideran a partir de una matriz de imperfección, resultante de la mudanza en la rígidez elástica consecuencia de pequeños desvios en la geometría (pequeñas rotaciones en los elementos). Se presentan ejemplos para ilustrar la técnica desarrollada e investigar las cargas de flameo de algumas extructuras básicas: pórtico de Roorda, columna de Euler, columna de Beck, arco circular y viga sobre base elástica (análogo a la deformación axisimétrica de una cáscara cilíndrica). Se concluye que el uso de la matriz de imperfección propuesta puede ofrecer una valiosa información cualitativa sobre la sensibilidad de la carga de flameo en relación a las imperfecciones.

We will create a class of generalized ellipses and explore their ability to define a distance on a space and generate continuous, periodic functions. Connections between these continuous, periodic functions and the generalizations of trigonometric functions known in the literature shall be established along with connections between these generalized ellipses and some spectrahedral projections onto the plane, more specifically the well-known multifocal ellipses. The superellipse, or Lam\'{e} curve, will be a special case of the generalized ellipse. Applications of these generalized ellipses shall be explored with regards to some one-dimensional systems of classical mechanics. We will adopt the Ramberg-Osgood relation for stress and strain ubiquitous in engineering mechanics and define a general internal bending moment for which this expression, and several others, are special cases. We will then apply this general bending moment to some one-dimensional Euler beam-columns along with the continuous, periodic functions we developed with regard to the generalized ellipse. This will allow us to construct new solutions for critical buckling loads of Euler columns and deflections of beam-columns under very general engineering material requirements without some of the usual assumptions associated with the Ramberg-Osgood relation.