Date of Award

Spring 1979

Document Type

Dissertation

Degree Name

Doctor of Engineering Science in Mechanical Engineering

Department

Mechanical Engineering

First Advisor

Arnold Allentuch

Second Advisor

Eugene B. Golub

Third Advisor

Murray Lieb

Fourth Advisor

Harry Herman

Fifth Advisor

Benedict C. Sun

Abstract

The problem of inplane and out of plane buckling of a thick circular ring subjected to a hydrostatic pressure is analyzed from first principals. The equations of equilibrium from nonlinear elasticity theory are used to describe two adjacent equilibrium positions, i.e., an initial and final state. The initial position problem is assumed to be governed by linear theory. The resulting incremental value equations obtained by subtracting the two equilibrium states, are also linear in nature. These problems are reduced to a one dimensional ring theory problem by the method of power series expansion of the displacements in the radial and axial directions and Fourier series expansions in the circumferential direction, and then by integrating through the thickness and depth of the ring. The coefficients of the terms in the power series displacement expansions are treated as unknown variables and the resulting eigenvalue problem is solved for the lowest root in each of the two perpendicular directions. The number of terms in the power series are reduced and a thin ring theory is defined, also in terms of unknown coefficients. These coefficients are determined and conpared with published equations for thin rings.

Several rings with rectangular cross sections were analyzed by these three methods, i.e., (a) thick ring theory with unknown coefficients, (b) thin ring theory with unknown coefficients, (c) thin ring theory with known coefficients. Both thin ring theories were found to agree with the thick ring theory to within ten percent for rings with diameter to thickness ratios of twenty or more. However, the thick ring theory does not become inaccurate until this ratio is less than five. In all of the analyzed cases of diameter to thickness ratios, the square cross sections had the largest critical buckling loads in both the inplane and out of plane directions. In each case the out of plane critical buckling load was the smaller. When the ratio of the radial thickness to the axial depth, called aspect ratio, was approximately equal to one half, the rings had equal critical buckling loads in both the inplane and out of plane directions. If the aspect ratio was greater than 0.5, the out of plane direction had a smaller critical load. If it was less than this value, the inplane direction would control.

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