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