Buckling: Critical Load

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 (1.2L)
Hinged-Free L (1.2L)
Hinged-Hinged
(Simply-Supported)
L L
Guided-Free 2L (2.1L)
Guided-Hinged 2L 2L
Guided-Guided L 1.2L
Clamped-Free
(Cantilever)
2L 2.1L
Clamped-Hinged 0.7L 0.8L
Clamped-Guided L 1.2L
Clamped-Clamped 0.5L 0.65L

 

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

 

 

Copyright efunda.com
 

mulberry outlet coach outlet burberry outlet coach factory outlet mulberry outlet coach outlet UGG Pas Cher cheap oakley sunglasses cheap nfl jerseys cheap oakleys wholesale nfl jerseys coach outlet canada black friday coach ugg boots on sale cheap uggs gucci outlet oakley outlet coach outlet coach outlet online