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From:
http://www-mdp.eng.cam.ac.uk/web/library/enginfo/textbooks_dvd_only/DAN/buckling/intro/intro.html
The creator of the website cited above writes:
"When a structure ( subjected usually to compression ) undergoes visibly large displacements transverse to the load then it is said to buckle. Buckling may be demonstrated by pressing the opposite edges of a flat sheet of cardboard towards one another. For small loads the process is elastic since buckling displacements disappear when the load is removed.
"Local buckling of plates or shells is indicated by the growth of bulges, waves or ripples, and is commonly encountered in the component plates of thin structural members.
"Buckling proceeds in manner which may be either:
"stable -in which case displacements increase in a controlled fashion as loads are increased, ie. the structure's ability to sustain loads is maintained, or
"unstable -in which case deformations increase instantaneously, the load carrying capacity nose-dives, and the structure collapses catastrophically.
"Neutral equilibrium is also a theoretical possibility during buckling. This is characterised by deformation increase without change in load.
"Buckling and bending are similar in that they both involve bending moments. In bending these moments are substantially independent of the resulting deflections, whereas in buckling the moments and deflections are mutually inter-dependent - moments, deflections and stresses are not proportional to loads.
"If buckling deflections become too large then the structure fails. This is a geometric consideration, completely divorced from any material strength consideration. If a component or part thereof is prone to buckling then its design must satisfy both strength and buckling safety constraints.
"The slender elastic pin-ended column is the protoype for most buckling studies. It was examined first by Euler in the 18th century. The model assumes perfection - the column is perfectly straight prior to loading, and the load when applied is perfectly coaxial with the column.
"The behaviour of a buckling system is reflected in the shape of its load- displacement curve - referred to as the equilibrium path. The lateral or 'out-of-plane' displacement, δ, is preferred to the load displacement, q, in this context since it is more descriptive of buckling.
"Nothing is visible when the load on a perfect column first increases from zero - the column is stable, there is no buckling, and no out-of-plane displacement. The P-δ equilibrium path is thus characterized by a vertical segment - the primary path - which extends until the increasing load reaches the critical Euler load,
Pc = π^2 EImin/L^2, a constant characteristic of the column. (For a derivation of this equation, see Timoshenko & Gere, for example ).
"When the load reaches the Euler load, buckling suddenly takes place without any further load increase, and lateral deflections δ grow instanteously in either equally probable direction. After buckling therefore, the equilibrium path bifurcates into two symmetric secondary paths as illustrated. Clearly the critical Euler load limits the column's safe load capacity.
"Local buckling of an edge-supported thin plate does not necessarily lead to total collapse as in the case of columns, since plates can generally withstand loads greater than critical. However the P-q curve illustrates plates' greatly reduced stiffness after buckling. Therefore, plates cannot be used in the post-buckling region unless the behaviour in that region is known with confidence.
"It should be emphasised that the knee in the P-q curve is unrelated to any elastic-plastic yield transition; the systems being discussed are totally elastic. The knee is an effect of overall geometric rather than material instability."
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