Buckling: Critical Load  
Euler's Formula  
Consider a long simplysupported 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 crosssection, 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 simplysupported 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.
 
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
