Problems which require a study of the static and dynamic response of plates can be approached by first considering the plate to be a portion of an infinite plate, the prescribed boundary conditions being temporarily ignored. Once the plate's boundary has been defined in the infinite plate, a numerical solution is initiated by dividing this boundary into N segments of arbitrary length.
For the static case the desired loading can then be applied to the infinite plate, and its effect on the deflection and stresses at the midpoint of the N boundary segments computed. To satisfy the boundary conditions of elementary plate theory, a concentrated force and moment are applied at the midpoint of each boundary segment. The magnitudes of these N equivalent forces and moments are determined by specifying that their combined effects, together with the applied loading, satisfy the boundary conditions at the N boundary points. This yields a set of 2N simultaneous equations whose solution constitutes the solution to the problem.
A similar approach can be utilized for the vibrating plate. For the dynamic case the applied loading is assumed as zero, and a harmonically varying force and moment placed at the midpoint of each of the N boundary segments. The magnitudes of the N harmonically varying forces and moments are determined by specifying that their combined effects satisfy the boundary conditions at the N boundary points. This, coupled with the assumption of homogeneous boundary conditions, yields a set of 2N homogeneous equations. The frequency equation follows by setting the determinant of the coefficients equal to zero. The above approach to the solution of boundary value problems is formally known as the Reflection Method.
Application of the Reflection Method to the static plate was previously accomplished by placing the equivalent forces and moments in the infinite plate at a finite distance from the midpoint of each boundary segment. This finite distance was called a retracted distance, and the curve along which the equivalent forces and moments were applied, a retracted boundary. In this investigation, the magnitude of the retracted distance was found to influence the condition of the coefficient matrix, while the solution remained relatively independent.
The static response of plates by the Reflection Method as presented here applies the equivalent forces and moments directly to the boundary of the plate. This was found to impressively improve the condition of the coefficient matrix and reduce the number of significant figures necessary to obtain a numerical solution. With no increase in the number of boundary points, results were obtained comparable to those utilizing a retracted distance. The equations enabling the forces and moments to be applied directly to the boundary are developed and several examples presented.
Application of the Reflection Method to the problem of determining natural frequencies is first illustrated for beams and then extended to plates. In each case the necessary equations are developed and sample problems presented. / Ph. D.
| Identifer | oai:union.ndltd.org:VTETD/oai:vtechworks.lib.vt.edu:10919/104345 |
| Date | January 1965 |
| Creators | Ermold, Leonard Frederick |
| Contributors | Engineering Mechanics |
| Publisher | Virginia Polytechnic Institute |
| Source Sets | Virginia Tech Theses and Dissertation |
| Language | English |
| Detected Language | English |
| Type | Dissertation, Text |
| Format | 146 leaves, application/pdf, application/pdf |
| Rights | In Copyright, http://rightsstatements.org/vocab/InC/1.0/ |
| Relation | OCLC# 20297990 |
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