# 10.7: Effect of Initial Imperfection

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Plates may be geometrically imperfect due to the manufacturing process, welding distortion or mishandling during transportation. The shape of the imperfect plate can be measured as is defined by the function \(\bar{w}(x, y)\). In general the initial out-of-plane shape can be expanded in a Fourier series. The first fundamental mode grows more rapidly. Therefore it is sufficient to consider that imperfections are distributed in the first mode

\[\bar{w}(x, y) = \bar{w}_o \sin \frac{\pi x}{a} \sin \frac{\pi y}{a}\]

With the initial imperfection the definition of the curvatures and membrane strains must be modified

\[\kappa_{\alpha \beta} = −(w − \bar{w})_{,\alpha \beta} \]

\[\epsilon_{\alpha \beta} = \frac{1}{2} (u_{\alpha, \beta} + u_{\beta , \alpha}) + \frac{1}{2} w_{,\alpha} w_{\beta} − \frac{1}{2} \bar{w}_{,\alpha} \bar{w}_{,\beta} \]

which reduce to Equations (10.2.7) and (10.1.7), respectively, when \(\bar{w}(x, y) = 0\). The derivation presented in Section 10.5 is still valid and the expression for the total potential energy is the same, except all terms involving wo should now be replaced by \((w_o − \bar{w}_o)\). The structural imperfections are usually small and comparable to the thickness of the plate. A plot of the load-displacement curve for the geometrically perfect plate and the plate with two magnitudes of initial imperfections is shown in Figure (\(\PageIndex{1}\)). The load has been normalized with the critical buckling load and displacements by the critical buckling displacement.