The Dynamics of the Buckling of Elastic Columns

Abstract

Abstract The motion of a perfectly elastic initially slightly curved column is calculated when one of its ends is displaced axially at a constant speed in a perfectly rigid testing machine. A nondimensional quantity Ω is defined which depends only upon the slenderness ratio of the column and the ratio of the velocity of the propagation of sound in its material to the speed of the testing machine. Two elastic columns are dynamically similar if Ω and the initial deviations from straightness are the same for both of them. The dynamic transverse deflections of the column lag behind the static values at the outset and overshoot them later. At an advanced stage of the loading the dynamic deflections can be represented as oscillations superimposed upon the static deflections. The axial compressive force in the column increases proportionately to the displacement of the loading head of the testing machine in the first phase of the loading. When the deflections become large, the compressive force increases at a reduced rate, reaches a maximum, and finally begins to oscillate about the Euler load. The maximum force recorded on the testing machine can be a multiple of the Euler load when Ω and the initial deviations from straightness are small. At ordinary speeds and with the usual inaccuracies of routine testing the maximum load should differ little from the Euler load.