Date of Award

Spring 1968

Document Type

Thesis - Restricted

Degree Name

Master of Science (MS)

Department

Civil and Environmental Engineering

First Advisor

Nigro, Nicholas

Second Advisor

Richardson, B. L.

Third Advisor

Wackman, Peter

Abstract

Whenever a long rod or beam is dynamically loaded, the resulting disturbance travels from one end of the bar to the other in the form of an elastic wave. The problem discussed in this paper involves elastic wave propagation in an infinitely long bar having a rectangular cross section and composed of an isotropic, homogeneous material. The object of the investigation is to determine how the shape of the pulse traveling down the bar changes with time, i.e., to define the phenomenon known as dispersion. From a mathematical standpoint, this requires the solution to a three dimensional boundary value problem in elasticity. Originally, the object of the investigation was to formulate an exact solution to the wave propagation problem. The method of solution ultimately leads to six coupled transcendental boundary equations. The author could not find a way to uncouple these equations and, consequently, could not formulate the required system of equations which would have yielded the exact dispersion relationship. As a revised objective, it was decided to formulate an approximate solution utilizing as much of the original development as possible. In particular, the forms for the displacement functions developed in the original approach were used to approximately satisfy the boundary equations. The approximate solution developed in this paper has not been applied to general rectangular bars, but a form of the solution has been verified by an application to the flat plate. Many years after Columbus's famous voyage, an unimpressed acquaintance remarked to him; "Your deeds have been grossly overrated. People today discover more in one voyage than you have in your entire life." In reply, Columbus handed the man a boiled egg and asked him to balance it on end. After many attempts, the man gave up. Columbus then took the egg and plunked it to the table with enough force to crush the end of the egg so that it sat upright. The man protested. "That's easy! Anybody can do that." Columbus answered; "Certainly, once some one shows you how." The validity of this story is not of concern here, but there is an important analogy between it and this investigation. The results of this study are only a first step toward the solution of general rectangular bars.

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