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In the equations of the general theory of shells, the nonlinear terms accounting for membrane strains are accompanied by large coefficients (describing the extensional stiffness of the shell). Although formally infinitesimals of the second order, in the case of small but finite displacements these terms can be commensurate with, or even exceed in magnitude, the first-order terms. Disregarding such nonlinear terms can be debilitating to the linearized theory. For example, it will not reflect the fact that a plate or shell in bending prefers a deflection mode not accompanied by midsurface stretching (so-called isometric bending) and can switch to such a mode more or less abruptly, exhibiting the phenomenon of bending instability In this study, bending instability and vibration of plates and shallow shells are investigated by simple analytical and numerical means, having in mind certain types of layered, sandwich, and truss-like structures.