Concepts of stable and unstable elastic equilibrium
To introduce the main concepts and definitions required for the understanding of stable and unstable elastic equilibrium in structures.
This lecture begins with a definition of the stable and unstable states of equilibrium for a mechanical system. The law of minimum potential energy and its relationship to the stability of a structure is introduced by means of non-mathematical considerations. The concepts of buckling by bifurcation, for perfect systems, and of buckling by divergence, for imperfect systems, are presented. The post-critical behaviour of a system and the erosion of the stability when coincidence of several stability modes occurs are also briefly discussed.
Stability theories are formulated in order to determine the conditions under which a structural system, which is in equilibrium, ceases to be stable. Instability is essentially a property of structures in their extremes of geometry; for example, long slender struts, thin flat plates, or thin cylindrical shells. Normally, one deals with systems having one variable parameter N, which usually represents the external load but which might also be the temperature (thermal buckling) or other phenomena. For each value of N, there exists only one unbuckled configuration.
In classical buckling problems, the system is stable if N is small enough and becomes unstable when N is large. The value of N for which the structural system ceases to be stable is called the critical value Ncr. More generally, the following should be determined:
- the equilibrium configurations of the structure under prescribed loadings
- which amongst these configurations are stable
- the critical value of the loadings and what behavioural consequences are implied at these load levels