Buckling: Initially Curved Columns 
Imperfections in Columns 
Consider a curved column with an initial shape w_{0}(x) that is under an axial load F,
Enforcing the moment equilibrium of the freebody yields the equation,
The governing equation for the transverse displacement w can then be expressed as,
The above equation contains a nonhomogeneous term Fw_{0}/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. 
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