Consider a curved column with an initial shape w0(x) that is under an axial load F,
Enforcing the moment equilibrium of the free-body yields the equation,
The governing equation for the transverse displacement w can then be expressed as,
The above equation contains a non-homogeneous term -Fw0/EI. To simplify the problem, we assume that the initial curve is a simple half sine wave with an amplitude a,
The general solution is found to be,
where is the critical load of a straight column.