The Planar Buckling of Beams

Abstract

Armed with some understanding of the stability of discrete systems, we now move on to the stability of continuous systems. The equations that govern continuous systems are differential equations, and, hence, are considerably more complicated to solve than discrete systems. However, most of the issues of stability are the same. As mentioned previously, in order to investigate the stability of a system, we must work with the nonlinear equations that govern the behavior of that system. For mechanical systems, this nonlinearity can accrue from a variety of causes, as we discussed in Chapter 10, but we shall focus here on nonlinearity in the equilibrium and strain-displacement equations (and not constitutive nonlinearities). The description of a body in a deformed configuration requires that we work with nonlinear equations of the geometry of deformation and, thus, nonlinear equations of equilibrium. Without even considering the effects of nonlinear constitutive behavior, we are led to the interesting and important phenomenon of elastic buckling of structures, first discovered by the great mathematician Leonhard Euler centuries ago.