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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.