Mechanical Engineering ETDs


Pen-Yuan Shih

Publication Date



The complexity inherent in the theory of elastic stability has resulted in the extensive use of approximate techniques for determining buckling loads. The Rayleigh quotient provides reasonable answers for a surprisingly broad category of problems. However, the quotient is an upper bound to the buckling load and for certain engineering structures, further it is difficult to estimate the error. A rigorous and conservative analysis requires that a lower bound be computed.

There are several techniques for finding lower bounds to eigenvalues but because of various complicating factors none is routinely used for elastic stability analyses. The purpose of this dissertation is to investigate the feasibility and applicability of an alternate theorem that was suggested by Isaacson and Keller for the standard matrix eigenvalue problem.

A useful extension of the theorem to the differential equations that describe the buckling phenomenon require the application of several formulations to simple problems. The formulation that is finally adopted is capable of handling all of the traditional boundary conditions with no modifications to the quadratic forms. The procedure involves the determination of the natural frequency as a function of a load parameter and requires the use of finite elements with high continuity properties for routine numerical calculations.

Results are given for several column and plate buckling problems to illustrate the procedure for representative combinations of boundary conditions, bending stiffness distributions and axial load distributions. Because of the high continuity requirements on the displacement field, plate problems utilizing the finite element approach are limited to two classical cases. vi

Degree Name

Mechanical Engineering

Level of Degree


Department Name

Mechanical Engineering

First Committee Member (Chair)

Howard D. Schreyer

Second Committee Member

Steven A. Pruess

Third Committee Member

Frederick Dsuin Ju

Fourth Committee Member

Youn-Chang Hsu

Fifth Committee Member

G. W. May

Document Type