In this study, Ritz method has been employed to analyze the following problems: free vibrations of plates with curvilinear stiffeners, the lowest 100 frequencies of thick isotropic plates, free vibrations of thick quadrilateral laminates and free vibrations and static deformations of rectangular laminates, and sandwich structures. Admissible functions in the Ritz method are chosen as a product of the classical Jacobi orthogonal polynomials and weight functions that exactly satisfy the prescribed essential boundary conditions while maintaining orthogonality of the admissible functions. For free vibrations of plates with curvilinear stiffeners, made possible by additive manufacturing, both plate and stiffeners are modeled using a first-order shear deformation theory. For the thick isotropic plates and laminates, a third-order shear and normal deformation theory is used. The accuracy and computational efficiency of formulations are shown through a range of numerical examples involving different boundary conditions and plate thicknesses. The above formulations assume the whole plate as an equivalent single layer. When the material properties of individual layers are close to each other or thickness of the plate is small compared to other dimensions, the equivalent single layer plate (ESL) theories provide accurate solutions for vibrations and static deformations of multilayered structures. If, however, sufficiently large differences in material properties of individual layers such as those in sandwich structure that consists of stiff outer face sheets (e.g., carbon fiber-reinforced epoxy composite) and soft core (e.g., foam) exist, multilayered structures may exhibit complex kinematic behaviors. Hence, in such case, C_z⁰ conditions, namely, piecewise continuity of displacements and the interlaminar continuity of transverse stresses must be taken into account. Here, Ritz formulations are extended for ESL and layerwise (LW) Nth-order shear and normal deformation theories to model sandwich structures with various face-to-core stiffness ratios. In the LW theory, the C⁰ continuity of displacements is satisfied. However, the continuity of transverse stresses is not satisfied in both ESL and LW theories leading to inaccurate transverse stresses. This shortcoming is remedied by using a one-step well-known stress recovery scheme (SRS). Furthermore, analytical solutions of three-dimensional linear elasticity theory for vibrations and static deformations of simply supported sandwich plates are developed and used to investigate the limitations and applicability of ESL and LW plate theories for various face-to-core stiffness ratios. In addition to natural frequency results obtained from ESL and LW theories, the solutions of the corresponding 3-dimensional linearly elastic problems obtained with the commercial finite element method (FEM) software, ABAQUS, are provided. It is found that LW and ESL (even though its higher-order) theories can produce accurate natural frequency results compared to FEM with a considerably lesser number of degrees of freedom. / Doctor of Philosophy / In everyday life, plate-like structures find applications such as boards displaying advertisements, signs on shops and panels on automobiles. These structures are typically nailed, welded, or glued to supports at one or more edges. When subjected to disturbances such as wind gusts, plate-like structures vibrate. The frequency (number of cycles per second) of a structure in the absence of an applied external load is called its natural frequency that depends upon plate's geometric dimensions, its material and how it is supported at the edges. If the frequency of an applied disturbance matches one of the natural frequencies of the plate, then it will vibrate violently. To avoid such situations in structural designs, it is important to know the natural frequencies of a plate under different support conditions. One would also expect the plate to be able to support the designed structural load without breaking; hence knowledge of plate's deformations and stresses developed in it is equally important. These require mathematical models that adequately characterize their static and dynamic behavior. Most mathematical models are based on plate theories. Although plates are three-dimensional (3D) objects, their thickness is small as compared to the in-plane dimensions. Thus, they are analyzed as 2D objects using assumptions on the displacement fields and using quantities averaged over the plate thickness. These provide many plate theories, each with its own computational efficiency and fidelity (the degree to which it reproduces behavior of the 3-D object). Hence, a plate theory can be developed to provide accurately a quantity of interest. Some issues are more challenging for low-fidelity plate theories than others. For example, the greater the plate thickness, the higher the fidelity of plate theories required for obtaining accurate natural frequencies and deformations. Another challenging issue arises when a sandwich structure consists of strong face-sheets (e.g., made of carbon fiber-reinforced epoxy composite) and a soft core (e.g., made of foam) embedded between them. Sandwich structures exhibit more complex behavior than monolithic plates. Thus, many widely used plate theories may not provide accurate results for them. Here, we have used different plate theories to solve problems including those for sandwich structures. The governing equations of the plate theories are solved numerically (i.e., they are approximately satisfied) using the Ritz method named after Walter Ritz and weighted Jacobi polynomials. It is shown that these provide accurate solutions and the corresponding numerical algorithms are computationally more economical than the commonly used finite element method. To evaluate the accuracy of a plate theory, we have analytically solved (i.e., the governing equations are satisfied at every point in the problem domain) equations of the 3D theory of linear elasticity. The results presented in this research should help structural designers.

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Employing the Finite Element Method (FEM), we numerically study three problems involving free and forced vibrations of linearly and nonlinearly elastic plates with a third order shear and normal deformable theory (TSNDT) and the three dimensional (3D) elasticity theory. We used the commercial software ABAQUS for analyzing 3D deformations, and an in-house developed and verified software for solving the plate theory equations. In the first problem, we consider trapezoidal load-time pulses with linearly increasing and affinely decreasing loads of total durations equal to integer multiples of the time period of the first bending mode of vibration of a plate. For arbitrary spatial distributions of loads applied to monolithic and laminated orthotropic plates, we show that plates' vibrations become miniscule after the load is removed. We call this phenomenon as vibration attenuation. It is independent of the dwell time during which the load is a constant. We hypothesize that plates exhibit this phenomenon because nearly all of plate's strain energy is due to deformations corresponding to the fundamental bending mode of vibration. Thus taking the 1st bending mode shape of the plate vibration as the basis function, we reduce the problem to that of solving a single second-order ordinary differential equation. We show that this reduced-order model gives excellent results for monolithic and composite plates subjected to different loads. Rectangular plates studied in the 2nd problem have points on either one or two normals to their midsurface constrained from translating in all three directions. We find that deformations corresponding to several modes of vibration are annulled in a region of the plate divided by a plane through the constraining points; this phenomenon is termed mode localization. New results include: (i) the localization of both in-plane and out-of-plane modes of vibration, (ii) increase in the mode localization intensity with an increase in the length/width ratio of a rectangular plate, (iii) change in the mode localization characteristics with the fiber orientation angle in unidirectional fiber- reinforced laminae, (iv) mode localization due to points on two normals constrained, and (iv) the exchange of energy during forced harmonic vibrations between two regions separated by the line of nearly stationary points that results in a beating-like phenomenon in a sub-region of the plate. This technique can help design a structure with vibrations limited to its small sub-region, and harvesting energy of vibrations of the sub-region. In the third problem, we study finite transient deformations of rectangular plates using the TSNDT. The mathematical model includes all geometric and material nonlinearities. We compare the results of linear and nonlinear TSNDT FEM with the corresponding 3D FEM results from ABAQUS and note that the TSNDT is capable of predicting reasonably accurate results of displacements and in-plane stresses. However, the errors in computing transverse stresses are larger and the use of a two point stress recovery scheme improves their accuracy. We delineate the effects of nonlinearities by comparing results from the linear and the nonlinear theories. We observe that the linear theory over-predicts the deformations of a plate as compared to those obtained with the inclusion of geometric and material nonlinearities. We hypothesize that this is an effect of stiffening of the material due to the nonlinearity, analogous to the strain hardening phenomenon in plasticity. Based on this observation, we propose that the consideration of nonlinearities is essential in modeling plates undergoing large deformations as linear model over-predicts the deformation resulting in conservative design criteria. We also notice that unlike linear elastic plate bending, the neutral surface of a nonlinearly elastic bending plate, defined as the plane unstretched after the deformation, does not coincide with the mid-surface of the plate. Due to this effect, use of nonlinear models may be of useful in design of sandwich structures where a soft core near the mid-surface will be subjected to large in-plane stresses. / Doctor of Philosophy / Plates and shells are defined as structures which have thickness much smaller as compared to their length and width. These structures are extensively used in many fields of engineering such as, designing ship hulls, airplane wings and fuselage, bodies of automobile, etc. Depending on the complexity of a plate/shell deformation problem, deriving analytical solutions is not always viable and one relies on computational methods to obtain numerical solutions of the problem. However, obtaining 3-dimensional (3D) numerical solutions of deforming plates/shells often require high computational effort. To avoid this, plate/shell theories are used for modeling these structures, which, based on certain assumptions, reduce the 3D problem into an equivalent 2-dimensional (2D) problem. However, quality of the solution obtained from such a theory depends on how suitable the assumptions are for the specific problem being studied. In this work, one such plate theory called as the Third Order Shear and Normal Deformable Theory (TSNDT) is used to model the mechanics of deforming rectangular plates under different boundary conditions (constraint conditions for the boundaries of the plate) and loading conditions (conditions of applied loads on the plate). We develop the TSNDT mathematical model of plate deformations and solve it using a computational technique called as the Finite Element Method (FEM) to analyze three different problems of mechanics of rectangular plates. These problems are briefly described below. vi In the first problem, we study vibrations of rectangular plates under time dependent (dynamic) loads. When a dynamic load acts on a plate, due to the effects of inertia, the plate continues to vibrate after the removal of the load. This is analogous to ringing of a bell long after the strike of the hammer on the bell. In this study we show that such vibrations of a rectangular plate can be varied by changing time dependencies of the applied load. We observe that under certain particular loading conditions, vibrations of the plate becomes miniscule after the load removal. We call this phenomenon as Vibration Attenuation and investigate this computationally in different problems of plate deformation using FEM solutions. In the second problem, we computationally investigate the effects of presence of internal fixed points (points within the volume of the plate restricted of motion) on the vibration characteristics of rectangular plate using TSNDT FEM solutions. We observe that when one or more points at locations inside a rectangular plate are fixed, vibration behavior of the plate significantly changes and the deformations are localized in certain regions of the plate. This phenomenon is called as Mode Localization. We study mode localization in rectangular plates under different boundary and loading conditions and analyze the effects of plate dimensions, locations of the internal fixed points and dynamic load characteristics on mode localization. In the third problem, we investigate the effects of introduction of nonlinearities into the TSNDT mathematical model of plate deformations. Simple models in mechanics consider materials to be linearly elastic, which means that the deformations of a body are proportional to the applied loads in a linear relation. However, most materials in nature undergoing large deformations (human tissues, rubbers, and polymers, for example) do not behave in this fashion and their deformation depends nonlinearly to applied loads. To investigate the effects of such nonlinearities, we study the behavior of nonlinearly elastic plates under different boundary and loading conditions and delineate the differences in the results of linearly elastic and nonlinearly elastic plates using the TSNDT FEM solutions. Findings of this study establishes that linear models overestimate the plate deformation under given boundary and loading conditions as compared to nonlinear models. This understanding may help in developing better design criteria for plates undergoing large deformations.

Em engenharia, a quantidade de problemas geométricos complexos que precisam ser resolvidos empregando teorias de placas ou cascas é notável. Esta é a razão por que há tantas teorias que buscam simplificar os problemas tridimensionais em outros menos custosos computacionalmente. Além disso, o aumento atual do uso de estruturas sanduíche requer que as formulações bidimensionais sejam mais precisas. Esta tese, num primeiro momento, compila a maioria das teorias de placa, comentando as principais diferenças, vantagens e desvantagens de cada uma. As formulações bidimensionais de placas laminadas são classificadas principalmente de acordo com o tratamento da coordenada na direção normal a superfície da mesma: Camada Única Equivalente (ESL), ESL refinada (teorias Zig-Zag) e Teorias Discretas ou de Camada (LW). Cada uma destas teorias é revista juntamente com as hipóteses de placas que são feitas para cada uma das camadas ou para o laminado como um todo. Para resolver tais problemas estruturais em engenharia, métodos numéricos são normalmente utilizados. Portanto, num segundo momento, alguns métodos de solução são citados e revisados, mas o foco é dado ao Método dos Elementos Finitos (MEF). A contribuição deste trabalho consiste na implementação de um novo método de solução de compósitos laminados e estruturas sanduíche com base em um sistema unificado de Formulação Generalizada (GUF) via MEF. Um elemento quadrilátero de 4 nós foi desenvolvido e avaliado com um código de Elementos Finitos desenvolvido pelo presente autor. Os requisitos para continuidade do tipo C-1 são respeitados para a variável de deflexão da placa. Esse método é nomeado de Formulação Generalizada do Caliri (CGF). Resultados para placas isotrópicas, placas de laminado compósito e estruturas sanduíche consideradas finas ou espessas são comparados com dados da literatura e soluções via Abaqus. Os resultados obtidos ao longo da espessura reforçam a necessidade de soluções de placa não-lineares para placas espessas (laminadas ou não). Mostrou-se que as soluções estáticas e dinâmicas empregando o método proposto fornecem resultados coerentes quando comparados com outros métodos de solução. Dentre os diversos estudos de caso investigados, verificou-se que é possível se obter resultados com alta concordância. Para uma estrutura sanduíche com núcleo macio, o resultado de deslocamento previsto para um carregamento estático chega a 99.8% de concordância e o resultado de uma análise modal da mesma estrutura mostra uma concordância de 99.5% com os resultados de um modelo feito com elementos 3D em um programa comercial de elementos finitos. / In engineering, the amount of complex geometrical problems, which need to be solved by using plates and shells theories, is remarkable. This is the reason why there are so many plate and shell theories which attempt to simplify three dimensional problems into ones with low computational cost. Additionally, the current increasing use of sandwich structures requires that the two dimensional formulations be accurate enough. First, this thesis compiles most of the plate theories from the literature and quotes the main differences, advantages and weaknesses of each one. The bi-dimensional laminated plate formulations are mainly classified according to the treatment of the variable in the normal direction of the plate surface: Equivalent Single Layer (ESL), Refined ESL (Zig-Zag theories) and Layer-Wise (LW) theories. Each one of these theories is reviewed along with the plate hypotheses which are made for each ply and/or laminate. To solve such complex structural engineering problems, numerical methods are normally used. Second, few solution methods are reviewed and quoted, but focus is given to the Finite Element Method (FEM). The contribution of this work is the implementation of a new solution method for laminated composites and sandwich structures based on a Generalized Unified Formulation (GUF) via FEM. A quadrilateral 4-node element was developed and evaluated using in-house Finite Element program. The C-1 continuity requirements is fulfilled for the transversal displacement field variable. This method is tagged as Caliri\'s Generalized Formulation (CGF). Results for isotropic plates, laminated composite plates and sandwich structures for thin and thick laminates are compared with literature data and solutions via Abaqus. The through-the-thickness profile results reinforce the need for non-linear plate (laminated or not) solutions. It was shown that the static and dynamic solutions employing the proposed solution method yield coherent results when compared with other solution methods. Among the different case studies investigated, it was verified that it is possible to obtain results with high agreement. For a soft-core sandwich structure, the displacement result for a static loading is reported as high as 99.8% and the result of a modal analysis of the same structure shows an accuracy of 99.5%, comparing to the results from a 3D finite element model built with a commercial software.

Em engenharia, a quantidade de problemas geométricos complexos que precisam ser resolvidos empregando teorias de placas ou cascas é notável. Esta é a razão por que há tantas teorias que buscam simplificar os problemas tridimensionais em outros menos custosos computacionalmente. Além disso, o aumento atual do uso de estruturas sanduíche requer que as formulações bidimensionais sejam mais precisas. Esta tese, num primeiro momento, compila a maioria das teorias de placa, comentando as principais diferenças, vantagens e desvantagens de cada uma. As formulações bidimensionais de placas laminadas são classificadas principalmente de acordo com o tratamento da coordenada na direção normal a superfície da mesma: Camada Única Equivalente (ESL), ESL refinada (teorias Zig-Zag) e Teorias Discretas ou de Camada (LW). Cada uma destas teorias é revista juntamente com as hipóteses de placas que são feitas para cada uma das camadas ou para o laminado como um todo. Para resolver tais problemas estruturais em engenharia, métodos numéricos são normalmente utilizados. Portanto, num segundo momento, alguns métodos de solução são citados e revisados, mas o foco é dado ao Método dos Elementos Finitos (MEF). A contribuição deste trabalho consiste na implementação de um novo método de solução de compósitos laminados e estruturas sanduíche com base em um sistema unificado de Formulação Generalizada (GUF) via MEF. Um elemento quadrilátero de 4 nós foi desenvolvido e avaliado com um código de Elementos Finitos desenvolvido pelo presente autor. Os requisitos para continuidade do tipo C-1 são respeitados para a variável de deflexão da placa. Esse método é nomeado de Formulação Generalizada do Caliri (CGF). Resultados para placas isotrópicas, placas de laminado compósito e estruturas sanduíche consideradas finas ou espessas são comparados com dados da literatura e soluções via Abaqus. Os resultados obtidos ao longo da espessura reforçam a necessidade de soluções de placa não-lineares para placas espessas (laminadas ou não). Mostrou-se que as soluções estáticas e dinâmicas empregando o método proposto fornecem resultados coerentes quando comparados com outros métodos de solução. Dentre os diversos estudos de caso investigados, verificou-se que é possível se obter resultados com alta concordância. Para uma estrutura sanduíche com núcleo macio, o resultado de deslocamento previsto para um carregamento estático chega a 99.8% de concordância e o resultado de uma análise modal da mesma estrutura mostra uma concordância de 99.5% com os resultados de um modelo feito com elementos 3D em um programa comercial de elementos finitos. / In engineering, the amount of complex geometrical problems, which need to be solved by using plates and shells theories, is remarkable. This is the reason why there are so many plate and shell theories which attempt to simplify three dimensional problems into ones with low computational cost. Additionally, the current increasing use of sandwich structures requires that the two dimensional formulations be accurate enough. First, this thesis compiles most of the plate theories from the literature and quotes the main differences, advantages and weaknesses of each one. The bi-dimensional laminated plate formulations are mainly classified according to the treatment of the variable in the normal direction of the plate surface: Equivalent Single Layer (ESL), Refined ESL (Zig-Zag theories) and Layer-Wise (LW) theories. Each one of these theories is reviewed along with the plate hypotheses which are made for each ply and/or laminate. To solve such complex structural engineering problems, numerical methods are normally used. Second, few solution methods are reviewed and quoted, but focus is given to the Finite Element Method (FEM). The contribution of this work is the implementation of a new solution method for laminated composites and sandwich structures based on a Generalized Unified Formulation (GUF) via FEM. A quadrilateral 4-node element was developed and evaluated using in-house Finite Element program. The C-1 continuity requirements is fulfilled for the transversal displacement field variable. This method is tagged as Caliri\'s Generalized Formulation (CGF). Results for isotropic plates, laminated composite plates and sandwich structures for thin and thick laminates are compared with literature data and solutions via Abaqus. The through-the-thickness profile results reinforce the need for non-linear plate (laminated or not) solutions. It was shown that the static and dynamic solutions employing the proposed solution method yield coherent results when compared with other solution methods. Among the different case studies investigated, it was verified that it is possible to obtain results with high agreement. For a soft-core sandwich structure, the displacement result for a static loading is reported as high as 99.8% and the result of a modal analysis of the same structure shows an accuracy of 99.5%, comparing to the results from a 3D finite element model built with a commercial software.