This paper deals with the Runge-Kutta numerical solution of the modified-Duffing ordinary differential equation with viscous damping. Accurate backbone curves for the finite-amplitude vibrations of geometrically imperfect rectangular plates and shallow spherical shells are presented. For a structure with a sufficiently large initial imperfection, the well-known soft-spring nature of the backbone curve is confirmed for small vibration amplitude. However, for large vibration amplitude, the backbone curves tend to exhibit the usual hard-spring behavior. The predominantly “inward” deflection response (as viewed from the center of curvature) of an imperfect system is found for undamped systems, but this is not necessarily true for a viscously damped structure. Both the initial-deflection and initial-velocity problems are examined.

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