In the case of an ideal column under an axial load, the column remains straight until the critical load is reached. However, the load is not always applied at the centroid of the cross section, as is assumed in Euler buckling theory. This section analyzes a simply-supported column under an eccentric axial load.
Consider a column of length L subject to an axial force F. On one end of the column, the force F is applied a distance e from the central column axis, as shown in the schematic below.
Balance the moments on the free-body diagram on the right requires that,
The governing equation for the column's transverse displacement w can then be written as,
where M was eliminated using Euler-Bernoulli beam theory. The above equation contains a non-homogeneous term -Fe/EI and its general solution is,
where . The coefficients A and B depend on the boundary conditions. For a simply supported column the boundary conditions are,
The solution for the column's displacement is therefore,