Buckling of cylindrical shells with small curvature

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Abstract
We consider the bifurcation buckling of a rectangular plate with an imperfection of magnitude α \alpha under an applied lateral force of magnitude λ \lambda . The analysis allows the parameters ( λ , α ) \left ( {\lambda ,\alpha } \right ) to vary independently in a neighborhood of some ( λ 0 , 0 ) \left ( {{\lambda _0}, 0} \right ) , and describes all buckled states of small magnitude. If the plate is represented by the domain ( 0 , 2 ) × ( 0 , 1 ) \left ( {0, \sqrt 2 } \right ) \times \left ( {0, 1} \right ) in R 2 {R^2} , then the lateral force is applied to the edges x = 0 x = 0 , 2 \sqrt 2 , and the imperfection is a small vertical displacement of the form z = ( α / 2 ) y 2 ( σ x + τ ) z = \left ( {\alpha /2} \right ){y^2}\left ( {\sigma x + \tau } \right ) , where σ \sigma and τ \tau are fixed. Roughly, then, the plate has a small curvature in the y y -direction, of magnitude α ( σ x + τ ) \alpha \left ( {\sigma x + \tau } \right ) .

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