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Euler's Formula

Consider a long simply-supportedcolumn under an

external axial load F, as shown in the figure to the left. The

critical buckling load(elastic stability limit) is given byEuler's formula,

where Eis the Young's modulus of the column material, Iis

the area moment of inertia of the crosssection, and Lis

the length of the column.

!ote that the critical buc"ling load decreases with the

s#uare of the column length.

Extended Euler's Formula

$ds by %oogle

Custom Beam AttachmentsCustomi&able eam $ttachments for ie *uort $licationswww.iingtech.com

+n general, columns do not always terminate with simlysuorted ends.

Therefore, the formula for the critical buc"ling load must be generali&ed.

The generali&ed e#uation ta"es the form of Euler's formula,

where the effective length of the column Leffdeends on the boundary conditions.*ome common boundary conditions are shown in the schematics below

The following table lists the effective lengths for columns terminating with a varietyof boundary condition combinations. $lso listed is a mathematical reresentation of

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the buc"led mode shae.

Boundary

Conditions

Theoretical

Effective

engthLeff

T

Engineering

Effective

engthLeff

E

Buckling !ode "hape

-ree-ree L (1.2L)

inged-ree L (1.2L)

ingedinged

(*imly

*uorted)L L

%uided-ree 2L (2.1L)

%uidedinged 2L 2L

%uided%uided L 1.2L

Clamed-ree

(Cantilever) 2L 2.1L

Clamedinged 0.7L 0.8L

Clamed%uided L 1.2L

Clamed

Clamed 0.5L 0.65L

+n the table, Lreresents the actual length of the column. The effective length isoften used in column design by design engineers.