The Elastic Instability of Thin Cantilever Struts on Elastic Supports with Axial and Transverse Loads at the Free End

Citation

Bell, Richard William (1958) The Elastic Instability of Thin Cantilever Struts on Elastic Supports with Axial and Transverse Loads at the Free End. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/5FEX-RT33. https://resolver.caltech.edu/CaltechETD:etd-10072004-093255

Abstract

An analysis is made of the elastic instability of thin, tapered cantilever struts subjected to a general concentrated load acting in the plane of the strut at its tip. The strut is supported at its root on a structure permitting elastic rotations of the root section in the buckled mode. The influence of the support on the minimum buckling load is one of the main points of interest. It is shown that the general linearized problem can be formulated in one second order differential equation with variable coefficients, and two associated boundary conditions. This homogeneous eigenvalue system constitutes a simplified statement of the problem which permits the easy extension of exact linear theory to a wide class of taper functions, including the effect of elastic supports. The solution emerges in terms of a generalized deflection parameter, rather than of either the torsional or the bending components of the coupled buckling mode, which are governed respectively by third and fourth order differential equations. Specific solutions are derived for some "natural" taper forms of the strut. The general solutions for the deflection mode are power series, which are rapidly convergent for certain limiting geometries. The problems of convergence of the series, some singular physical aspects associated with pointed tips, and the increasing numerical difficulty for large taper ratio are correlated with the behavior of the singular points of the equation. Numerical results showing the effects of the elastic supports on minimum buckling loads are presented for the uniform strut and for a simple case of the tapered strut. The series solutions for more general cases are given in a form which can be applied to digital computers.

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