Plates (Engineering)

The response of fluid-loaded plates has been extensively studied in the past.
However, most of the work deals with either infinite plates or finite plates with particular
boundary conditions and the results are generally presented only in the limit of small
wavelengths compared with the dimensions of the plates. Furthermore, the problem of
coupled finite plates where both the acoustic interaction and structural interaction are
included in the solution has not been considered. In this dissertation the response of two
coupled finite plates set in two alternative configurations is considered. The plates are
simply supported on two edges, with arbitrary boundary conditions on the remaining two
edges. The solutions obtained for the response of the plates include both the structural
interaction at the common junction and the acoustic interaction due to the scattered
pressure from each of the two plates. The results are presented in terms of the vibrational
power flow into and out of each plate component. The solution is based on a formulation developed in the wavenumber domain
combined with the Mobility Power Flow method. Using this approach, different
substructural elements coupled under different boundary conditions to form a complex
global structure can be considered. The detailed spatial and temporal scales of the structure response are not lost when using this method.
In obtaining the solution for the scattering from the fluid-loaded plates, a modal
decomposition in the direction normal to the simply supported edge is used. A spatial
Fourier-transform decomposition is used in the other direction. Due to the finiteness of
the plate, eight unknowns parameters are obtained in the transformed result. The solution
for these eight unknown parameters is obtained from the boundary conditions and the
condition that the response must remain finite. Two analytical approaches are used to
solve the final plate integral equation. The first approach consists of an approximation
method which obtains a solution based on the solution of the corresponding infinite plate
problem. The second approach is a more accurate solution based on the Projection
Method for the solution of integral equations.
Both of the approaches used in the solution provide accurate predictions at high
frequencies. At low frequencies especially for low structural damping or for heavy fluid
loading, only the Projection Method gives reliable results. This is attributed to the fact
that at low frequencies, the influence of the edges of the plates on the scattering is
significant.
The overall results obtained from this analysis indicate that the fluid loading and
the plate characteristics have a significant influence on the acoustic scattering properties,
especially in the case of heavy fluid loading.
The application of the method to coupled fluid-loaded plates indicates that the junction
enhances the scattering properties. The acoustical interaction between the coupled plates
increases the contribution to scattering from subsonic wavenumber components. In the
absence of the interaction, only supersonic wavenumbers contribute to the scattering.
Inclusion of acousticlal interaction requires both supersonic and subsonic components.
The significance of the contribution from the subsonic wavenumber components is
dependent on the type of the fluid loading.

The effects of various nonuniform stress fields on the stress intensity factors for the semi-elliptic surface crack (three-dimensional problem) in a finite plate are determined using the weight function approach. The formulation satisfies the linear elastic fracture mechanics criteria and the principle of conservation of energy. Based on the knowledge of stress intensity solutions for the reference load/stress system, the expression for the crack opening displacement function for the surface crack is derived. Using the crack opening displacement function and the reference stress intensity factor, the three-dimensional weight functions and subsequently the stress intensity solutions for the surface crack subjected to nonuniform stress fields are derived. The formulation is then applied to determine the effects of linear, quadratic, cubic, and pure bending stress fields on the stress intensity factor for the surface crack in a finite plate. In the initial stage of the study a two-dimensional problem of an edge-crack emanating from the weld-toe in a T-joint is considered. The effect of parameters such as plate thickness, weld-toe radius, and weld-flank angle on the stress intensity factor for an edge-crack is studied. Finite element analyses of the welded T-joints are performed to study the effects of plate thickness, weld-toe radius and the weld-flank angle on the local stress distribution. The ratio of plate thickness to weld-toe radius ranging from 13.09 to 153.93, and the weld-flank angles of 30, 45, and 60 degrees are considered in the analyses. Based on the results from FEM analyses, a parametric equation for the local stress concentration factor and a polynomial expression for the local stress distribution across the plate thickness are derived using the method of least squares and the polynomial curve-fitting technique.

Natural frequencies of the double and triple-walled carbon nanotubes are determined exactly and approximately for both types. Approximate solutions are found by using Bubnov-Galerkin and Petrov-Galerkin methods. For the first time explicit expressions are obtained for the natural frequencies of double and triple-walled carbon nanotubes for different combinations of boundary conditions. Comparison of the results with recent studies shows that the above methods constitute quick and effective alternative techniques to exact solution for studying the vibration properties of carbon nanotubes. The natural frequencies of the clamped-clamped double-walled carbon nanotubes are obtained; exact solution is provided and compared with the solution reported in the literature. In contrast to earlier investigation, an analytical criterion is derived to establish the behavior of the roots of the characteristic equation. Approximate Bubnov-Galerkin solution is also obtained to compare natural frequencies at the lower end of the spectrum. Simplified version of the Bresse-Timoshenko theory that incorporates the shear deformation and the rotary inertia is proposed for free vibration study of double-walled carbon nanotubes. It is demonstrated that the suggested set yields extremely accurate results for the lower spectrum of double-walled carbon nanotube. The natural frequencies of double-walled carbon nanotubes based on simplified versions of Donnell shell theory are also obtained. The buckling behavior of the double-walled carbon nanotubes under various boundary conditions is studied. First, the case of the simply supported double-walled carbon nanotubes at both ends is considered which is amenable to exact solution.