Buckling of Long Slender Columns Under Compression 

(Euler 1707 - 1783)
This analysis involves defining the critical axial load.

If a small axial concentric compressive load is applied to a straight slender rod and the system is pulled sideways slightly and released it will spring back to the straight position. If a large axial compressive load is applied, any sideways movement of the system will result in the system collapsing.

The critical axial compressive load represents the boundary between these two conditions. The system is just held in the deflected position by the compressive load. This is the neutral equilibrium configuration.

Depending upon the value of n, different modal shapes are produced. In the case of a slowly applied load only the first modal shape is likely to be produced, however in the case of a high speed impact, higher modal shapes may be produced:

David J Grieve. Revised: 21st February 2010,. Original: 1st March 2004.