Finite Element Multiple-Mode Approach to Nonlinear Free Vibrations of Shallow Shells

Two finite element (FE) modal formulations for large-amplitude free vibration of isotropic and arbitrary laminated composite shallow shells are presented. The system equation of motion is formulated first in the physical structural node degrees of freedom (DOF). Then the system is transformed into two distinctly different sets of general Duffing-type modal equations based on 1) modal amplitudes of coupled linear bending and in-plane modes, where in-plane inertia is included in the formulation, and 2) modal amplitudes of linear bending modes only, where the in-plane inertia is neglected. Multiple modes and the first-order transverse shear deformation are considered in the formulations. A shallow-shell finite element is developed as an extension from the triangular Mindlin (MIN3) plate element with the improved shear correction factor by Tessler. Time numerical integration is employed to determine the nonlinear periodic frequency characteristics. The inaccuracy in characterizing a shallow-shell response with coupled linear bending and in-plane modes is demonstrated and discussed by comparing with the FE solution in structural node DOF. Study cases include isotropic and composite panels of different shallow-shell geometries.