 Columns: Critical Load Formula Home Buckling of Columns Introduction Critical Load Elastic Buckling Inelastic Buckling Column Related Eccentric Loads Initially Curved Beam-Columns Calculators Critical Load Struc. Steel Columns Related Subjects Matl. Mechanics Beams Resources Bibliography  Login   Home Membership Magazines Forum Search Member Calculators  Materials  Design  Processes  Units  Formulas  Math 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.

 BoundaryConditions Theoretical EffectiveLengthLeffT Engineering EffectiveLengthLeffE 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.

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