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  • Print publication year: 2009
  • Online publication date: June 2012

6 - Stability of the Equilibrium

Summary

An important property or quality of the equilibrium of a structure is the stability of the equilibrium; that is, its sensitivity to small disturbances. If, after the small disturbance has ended, the structure returns to its original position, then the equilibrium state is said to be stable; on the other hand, if the small disturbance causes an excessive response, then the equilibrium state is unstable. An important consideration is to where the unstable structure goes – this is called the postbuckling behavior. The postbuckling behavior is typically highly nonlinear, undergoing large displacements and sometimes incurring plasticity effects. Figure 6.1(a) shows an example of a collapsed frame.

In all stability analyses, there is an important parameter associated with the unfolding of the instability. For example, the axial compressive load in the buckling of a column or the velocity in an aeroelastic flutter problem. Imperfections of load or geometry also play a significant role in unfolding the instability. Identifying this parameter and observing its effect is one of the keys to understanding the stability of a system.

The explorations in this chapter consider the stability of both the static and dynamic equilibrium. The first exploration uses imperfections (of loading and geometry) to illustrate the concept of sensitivity to the unfolding parameter. The second exploration introduces eigenanalysis as a tool to determine the buckling loads and mode shapes of a perfect structure; Figure 6.1(b) shows the first three buckled mode shapes of a ring with uniform pressure around the circumference.