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.

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 L_eff 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.