# Critical Buckling Compressive Loading of a Plate

## Description

In science, buckling is a mathematical instability that leads to a failure mode.

When a structure is subjected to compressive stress, buckling may occur. Buckling is characterized by a sudden sideways deflection of a structural member. This may occur even though the stresses that develop in the structure are well below those needed to cause failure of the material of which the structure is composed. As an applied load is increased on a member, such as a column, it will ultimately become large enough to cause the member to become unstable and it is said to have buckled. Further loading will cause significant and somewhat unpredictable deformations, possibly leading to complete loss of the member’s load-carrying capacity. 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.

In a mathematical sense, buckling is a bifurcation in the solution to the equations of static equilibrium. At a certain point, under an increasing load, any further load is able to be sustained in one of two states of equilibrium: a purely compressed state (with no lateral deviation) or a laterally-deformed state.

Buckling is a state which defines a point where an equilibrium configuration becomes unstable under a parametric change of load and can manifest itself in several different phenomena. All can be classified as forms of bifurcation.

There are four basic forms of bifurcation associated with loss of structural stability or buckling in the case of structures with a single degree of freedom. These comprise two types of pitchfork bifurcation, one saddle-node bifurcation (often referred to as a limit point) and one transcritical bifurcation. The pitchfork bifurcations are the most commonly studied forms and include the buckling of columns, sometimes known as Euler buckling; the buckling of plates, sometimes known as local buckling, which is well known to be relatively safe (both are supercritical phenomena) and the buckling of shells, which is well-known to be a highly dangerous (subcritical phenomenon). Using the concept of potential energy, equilibrium is defined as a stationary point with respect to the degree(s) of freedom of the structure. We can then determine whether the equilibrium is stable, as in the case where the stationary point is a local minimum; or unstable, as in the case where the stationary point is a maximum point of inflection or saddle point.

A plate is a 3-dimensional structure defined as having a width of comparable size to its length, with a thickness is very small in comparison to its other two dimensions. Similar to columns, thin plates experience out-of-plane buckling deformations when subjected to critical loads; however, contrasted to column buckling, plates under buckling loads can continue to carry loads, called local buckling. This phenomenon is incredibly useful in numerous systems, as it allows systems to be engineered to provide greater loading capacities.

Related formulas## Variables

N_{cr} | Critical buckling compressive load (J) |

k_{cr} | Buckling coefficient (dimensionless ) (dimensionless) |

π | pi |

E | Modulus of elasticity ( Young's modulus) (Pa) |

t | Thickness (m) |

ν | Poisson's ratio (dimensionless ) (dimensionless) |