page_title: Buckling
section_title: Single-degree-of-freedom models
hierarchical_section_title: Buckling / Theory / Single-degree-of-freedom models
caption_attribution_description: English: By Khurram Wadee, created using tgif.
context_page_description: In engineering, buckling is the sudden change in shape of a structural component under load such as the bowing of a column under compression or the wrinkling of a plate under shear. If a structure is subjected to a gradually increasing load, when the load reaches a critical level, a member may suddenly change shape and the structure and component is said to have buckled.
  Buckling may occur even though the stresses that develop in the structure are well below those needed to cause failure in the material of which the structure is composed. Further loading may cause significant and somewhat unpredictable deformations, possibly leading to complete loss of the member's load-carrying capacity. However, if the deformations that occur after buckling do not cause the complete collapse of that member, the member will continue to support the load that caused it to buckle. If the buckled member is part of a larger assemblage of components such as a building, any load applied to the buckled part of the structure beyond that which caused the member to buckle will be redistributed within the structure.
context_section_description: Using the concept of total potential energy, , it is possible to identify four fundamental forms of buckling found in structural models with one degree of freedom.  We start by expressing
  where  is the strain energy stored in the structure,  is the applied conservative load and  is the distance moved by  in its direction.  Using the axioms of elastic instability theory, namely that equilibrium is any point where  is stationary with respect to the coordinate measuring the degree(s) of freedom and that these points are only stable if  is a local minimum and unstable if otherwise (e.g. maximum or a point of inflection).
  These four forms elastic buckling are the saddle-node bifurcation or limit point; the supercritical or stable-symmetric bifurcation; the subcritical or unstable-symmetric bifurcation; and the transcritical or asymmetric bifurcation.  All but the first of these examples is a form of pitchfork bifurcation.  Simple models for each of these types of buckling behaviour are shown in the figures below, along with the associated bifurcation diagrams.
caption: Link model with inclined spring
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