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General criteria for elastic stability


To establish general criteria for elastic stability and neutral equilibrium.


Structural design requires that the equilibrium configuration for the structure, under the prescribed loading, is determined and that this can be confirmed as stable. The analysis of stability problems is generally done using energy criteria. In this lecture, the Principle of Virtual Work and the Principle of Stationary Total Potential Energy are presented. The general energy criteria for elastic stability derived from these are established and the determination of critical loading corresponding to neutral equilibrium is explained. Only fully conservative systems are considered. The established criteria are illustrated by two basic examples of rod and spring systems.


The design of structures requires the determination of the internal equilibrium forces (moments, shears, etc.) in the structure, under given loads, and confirmation that the structure is stable under these conditions. It is of fundamental importance to be sure that a structure, slightly disturbed from an equilibrium position by forces, shocks, vibrations, imperfections, residual stresses, etc., will tend to return to it when the disturbance is removed. This required characteristic of elastic stability has become more and more critical with the increasing use of high strength steels resulting in lighter and more slender structures.

The theory of elastic stability (buckling) gives methods for determining the following:

  • the stability of an equilibrium configuration
  • the critical value of the loading under which the instability occurs

Most of these methods are derived from general energy criteria which come from energy principles of mechanics. Therefore, the purpose of this lecture is to briefly present the principles of mechanics required to understand the general criteria of elastic stability, thereby giving a better understanding of the methods used in buckling investigations.

The scope of this lecture is restricted to:

  • conservative loadings and adiabatic elastic systems (fully conservative systems)
  • systems whose configurations can be expressed as functions of a finite number of displacement parameters

It should be noted that only the static aspect of stability is considered.

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