Buckling of cylindrical shells with small curvature

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