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 - so 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 - that is why we now examine the subject of buckling.

Buckling has become more of a problem in recent years since the use of high strength material requires less material for load support - structures and components have become generally more slender and buckle- prone. This trend has continued throughout technological history, as is demonstrated by bridges in the following sequence :

The Pont du Gard in Provence was completed by the Romans in the first century AD as part of a 50km aqueduct to convey water from a spring at Uzès to the garrison town of Nemausus (Nimes). The bridge is constructed from limestone blocks fitted together without mortar and secured with iron clamps. The three tiered structure avoids the need for long compressive members. ( source Art images for College Teaching )
The Royal Border Bridge, Berwick upon Tweed, was built by Robert Stephenson whose father George built the Stockton and Darlington Railway (the first public railway) in 1825. Opened in 1850, the bridge continues today as an important link in the busy King's Cross (London) - Edinburgh line. The increased slenderness of the columns compared to the Pont du Gard reflect technological improvements over many centuries. ( source FreeFoto.com )
The Crymlyn Viaduct over the Ebbw Alley opened in 1857 as Welsh coal mining expanded. It was constructed of wrought and cast iron, and remained the highest railway viaduct in the UK until its closure in 1964 due to increased locomotive weights (1908 photo). The advance from masonary to the slender metal compressive members which make up each column requires substantial bracing to prevent buckling ( source John Croeso )
The Humber road bridge, opened in 1981, comprises a continuously welded closed box road deck suspended from catenary cables supported on reinforced concrete towers. Suspension bridges eliminate the need for struts other than the two towers, however avoiding buckles in other slender components becomes an issue ( source FreeFoto.com )
The dangers associated with over-slender build were tragically driven home by the collapse of the Tacoma Narrows road bridge over the Puget Sound in 1940. Although this failure was apparently due to wind- structure aerodynamic coupling rather than buckling as such, this film clip demonstrates graphically the ability of large structures to undergo significant elastic deflections. ( MoviePlayer or similar is required to view this .mov video ) ( source CamGuys.com )



Buckling of thin-walled structures
A thin-walled structure is made from a material whose thickness is much less than other structural dimensions. Into this category fall plate assemblies, common hot- and cold- formed structural sections, tubes and cylinders, and many bridge and aeroplane structures.
Cold- formed sections such as those illustrated are increasingly supplanting traditional hot- rolled I-beams and channels. They are particularly prone to buckling and in general must be designed against several different types of buckling.
It is not difficult to visualise what can happen if a beam is made from such a cold- rolled channel section. One flange is in substantial compression and may therefore buckle locally at a low stress ( ie. much less than yield ) thus reducing the load capacity of the beam as a whole. Buckling rather than strength considerations thus dictate the beam's performance.
Let us now look at typical examples of buckling.

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.