Dynamic Snap-Through and Snap-Buckling of Shear-Deformable Panels in a Random Environment

Abstract

Random vibrations are considered for the cases of shear-deformable plates with initial curvature and of elastically buckled plates. The nontrivial generalization of the flat plate vibrations is expressed by the fact that “small amplitude” vibrations exist about the curved equilibrium position together with the snap-through and snap-buckling type large amplitude vibrations about the flat position. The problem of snap-buckling in the large deflection random response, i.e., the loss of stability accompanied in simple and regular structures by the change from a symmetric to an antimetric configuration, is studied in detail. The geometrically nonlinear panel vibrations are treated by applying Berger’s approximation to the generalized von Karman-type plate equations considering hard hinged supports of the straight boundary segments of skew or even more generally shaped polygonal plates. Shear deformation is considered by means of Mindlin’s kinematic hypothesis. A distributed lateral force loading is applied, and additionally, the influence of thermal prestress is taken into account as well as a randomly fluctuating temperature. A multi-mode approach and the Galerkin procedure are applied to the boundary value problem. The result of the projection and of a transformation is a set of nonlinearly coupled ordinary differential equations (ODEs) driven effectively by random forces and with potential restoring forces. For reasons of convergence, a light viscous modal damping is added at this stage. By means of a nondimensional formulation and introducing the eigen-time of the basic mode of the associated linearized problem renders a unifying result with respect to the planform of the panel. With the simplifying approximation of the action of the random environment by effective forces modeled by uncorrelated, zero-mean wide-band noise processes, and by considering the set of modal equations to be finite, the Fokker-Planck-Kolmogorov (F.P.K.-) equation for the transition probability density of the generalized coordinates and velocities is derived. The probability of first dynamic snap-through is derived for a single mode approximation with the influence of higher modes taken into account. Using the two-mode expansion, the probability distribution of the asymmetric snap-buckling is also evaluated. All probabilistic results obtained for the complex panel structures to be considered hold independently (in the sense of similitude) of their special planform. They are cast in the form of graphs with a structural parameter varying in a wide range.