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Euler's Formula

Consider a long simply-supported column under an external axial load F, as shown in the figure to the left. The critical buckling load (elastic stability limit) is given by Euler's formula,

where E is the Young's modulus of the column material, I is the area moment of inertia of the cross-section, and L is the length of the column.

Note that the critical buckling load decreases with the square of the column length.

Extended Euler's Formula
In general, columns do not always terminate with simply-supported ends. Therefore, the formula for the critical buckling load must be generalized.

The generalized equation takes the form of Euler's formula,

where the effective length of the column Leff depends on the boundary conditions. Some common boundary conditions are shown in the schematics below: 

The following table lists the effective lengths for columns terminating with a variety of boundary condition combinations. Also listed is a mathematical representation of the buckled mode shape.

Boundary
Conditions
Theoretical Effective
Length

LeffT
Engineering Effective
Length

LeffE
Buckling Mode Shape
Free-Free L (1.2·L)
Hinged-Free L (1.2·L)
Hinged-Hinged
(Simply-Supported)
L L
Guided-Free 2·L (2.1·L)
Guided-Hinged 2·L 2·L
Guided-Guided L 1.2·L
Clamped-Free
(Cantilever)
2·L 2.1·L
Clamped-Hinged 0.7·L 0.8·L
Clamped-Guided L 1.2·L
Clamped-Clamped 0.5·L 0.65·L
 

In the table, L represents the actual length of the column. The effective length is often used in column design by design engineers.

Glossary