# Linear Elastic Buckling Analysis

Linear Elastic Buckling Analysis of L bracket

Under certain loading, a structure continues to deform without an increase in the magnitude of loading. The structure has become unstable, it has buckled. Buckling is a failure mode characterized by the sudden loss of stability or stiffness and results are catastrophic. It is most commonly associated with compressive forces but it can also occur in elastic structures due to dead tensile loading.

The types of structures that can buckle are:

1) slender (Euler) columns under axial compressive load.

2) externally loaded thin walled pressure vessels

3) thin plates under edge pressure

4) deep, thin cantilevered beams under transverse loading on the top surface

For linear elastic buckling:

1) there is no yielding of the structure

2) direction of the applied force does not change

The equation used to model linear buckling is ([Ka] + λ [Kd]){U} = λ[Fa] where:

[Ka] is the system stiffness matrix

[Kd] is the differential stiffness matrix (This matrix is necessary to account for the change in      potential energy associated with rotation of continuum elements under load)

{U} is the displacement vector

λ is the eigenvalue

[Fa] is the applied load vector

It is worth noting, the differential stiffness matrix represents a linear approximation of softening (reducing) the stiffness matrix for a compressive axial load and stiffening (increasing) the stiffness matrix for a tensile load.

The output of interest from a linear buckling FEA analysis is the buckling load factor (value of λ for first buckling mode) and first buckling mode shape. The critical applied load which is the load the structure is predicted to buckle at is obtained by multiplying the actual applied load by the buckling load factor. The mode shape shows the instability corresponding to the buckling load factor.

It is very important to realize that the results of a linear buckling analysis is often non conservative. That is, it often over estimates the critical load. Buckling of the real structure may occur at a much lower value.