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.

We numerically analyze three mechanics problems described below. For each problem, the developed computational model is verified by comparing computed results for example problems with those available in the literature. Effective utilization of single wall carbon nanotubes (SWCNTs) and single layer graphene sheets (SLGSs) as reinforcements in nanocomposites requires their strong binding with the surrounding matrix. An effective technique to enhance this binding is to functionalize SWCNTs and SLGSs by covalent attachment of appropriate chemical groups. However, this damages their pristine structures that may degrade their mechanical properties. Here, we delineate using molecular mechanics simulations effects of covalent functionalization on elastic moduli of these nanomaterials. It is found that Young's modulus and the shear modulus of an SWCNT (SLGS), respectively, decrease by about 34% (73%) and 43% (42%) when 20% (10%) of carbon atoms are functionalized for each of the four functional groups of different polarities studied. A shell theory that gives results close to the solution of the corresponding 3-dimensional problem depends upon the shell geometry, applied loads, and initial and boundary conditions. Here, by using a third order shear and normal deformable theory and the finite element method (FEM), we delineate for a doubly curved shell deformed statically with general tractions and subjected to different boundary conditions effects of geometric parameters on in-plane and transverse stretching and bending deformations. These results should help designers decide when to consider effects of these deformation modes for doubly curved shells. Composite laminates are usually fabricated by curing resin pre-impregnated fiber layers in an autoclave under prescribed temperature and pressure cycles. A challenge is to reduce residual stresses developed during this process and simultaneously minimize the cure cycle time. Here, we use the FEM and a genetic algorithm to find the optimal cycle parameters. It is found that in comparison to the manufacturer's recommended cycle, for a laminate with the span/thickness of 12.5, one optimal cycle reduces residual stresses by 47% and the total cure time from 5 to 4 hours, and another reduces the total cure time to 2 hours and residual stresses by 8%. / Ph. D. / We analyze using computational techniques three mechanics problems described below. In the last three decades, two carbon nanomaterials (i.e., allotropes of carbon having length-scale of 10^-9 m), namely, single wall carbon nanotubes (SWCNTs) and single layer graphene sheets (SLGSs) have evolved as revolutionary materials with exceptional properties per unit weight that are superior to conventional engineering materials. A composite (i.e., a material made by combining two or more materials to attain desired properties which cannot be achieved by any of its constituents alone) made by using either of these carbon nanomaterials as reinforcements in a polymer could be a potential candidate for applications requiring high strength and light weight. However, the effective utilization of these composites for an application requires the strong binding between their constituents. An effective technique to enhance this binding is to modify the surface properties of SWCNTs and SLGSs by covalently bonding to them suitable chemical group that is usually called covalent functionalization. However, this damages their pristine structures that may degrade their mechanical properties. Here, it is found that the functionalization reduces elastic moduli of carbon nanomaterials, the reduction increases with an increase in the amount of functionalization and is essentially independent of the functionalizing chemical group. This study should help engineers interested in utilizing these materials to design novel nanocomposites. Composite laminates, made by stacking and binding together layers of fiber-reinforced composites, are widely used in aircraft, aerospace, marine, automobile, power generation, chemical and ballistic applications due to their high strength and stiffness per unit weight compared to that of conventional metallic materials. Shell theories are widely used to analyze deformations of composite laminates which reduces a 3-dimensional (3-D) problem to an equivalent 2-D problem by making certain assumptions related to the deformations of the laminate. This approach requires less computational effort to find a numerical solution (i.e., an approximate solution obtained using a computational technique) of the problem as compared to that needed for solving the full 3-D problem. However, the accuracy of the results predicted by a shell theory depends on the problem being studied, i.e., the shell geometry, applied loads, initial conditions (i.e., the motion of the laminate at the start of application of the load) and boundary conditions (i.e., constraints imposed on the deformations of the edges of the laminate). Here, we analyze effects of geometric parameters of the laminated shells on their deformations for different types of applied loads and various boundary conditions specified on the edges. The results should help designers find an optimal geometry of the composite laminates for a given mechanical application. Fiber-reinforced composite laminates are usually fabricated by curing (which involves heating and cooling in a prescribed manner under application of the pressure) resin preimpregnated fiber layers under prescribed temperature and pressure cycles. However, during this cure process the laminate deforms and the final product is not stress-free. Here, we find optimal parameters of the cure cycle that minimize stresses developed during the cure process as well as the time required to cure the laminate. It is found that for a laminate studied these optimal parameters reduce the stresses by 47% and the cure time from 5 to 4 hours in comparison to the standard cure cycle recommended by the laminate manufacturer. This study will provide manufacturing engineers with an approach to fabricate composite laminates of desired quality.