Chapter 5: Flexure and Buckling

Overview

Galileo did not speculate on the shape that his cantilever might assume when loaded at the tip, although others were examining this question before the end of the seventeenth century. The problem may not have presented itself as important and indeed, from some viewpoints it is not but certainly the problem was difficult to solve with the mathematical tools available at the time. Some curves had been known since the time of the Greeks the conic sections, for example: that is, the ellipse, parabola and hyperbola but there was no general mathematics available to describe more complex shapes. The celebrated dispute as to whether Newton or Leibniz had invented the calculus came right at the end of the century; it was the lack of knowledge of this mathematics that prevented Hooke in 1675 from finding the shape of the hanging chain. Thus Pardies (for example) in 1673 had a clear understanding of the basic laws of statics the law of the lever, forces in windlasses, gear trains and so on but he had to assert, without mathematical proof, that the bent shape of the tip-loaded cantilever was a parabola. This seems to be the first discussion of the problem, and Pardies' assertion is wrong.

Newton's method of fluxions the calculus was well-developed by about 1670; with the method, he could discuss problems of motion the relation between velocity and distance, for example. The method had precedents; Cavalieri in 1635 gave mathematical results which were useful in mechanics. And Galileo's second new science of 1638 was precisely concerned...