Nonlinear Dynamic Buckling and Stability of Autonomous Dissipative Discrete Structural Systems

Abstract

The Nonlinear dynamic buckling and stability of nonlinearly elastic autonomous discrete systems under conservative step loading (of infinite or finite duration), impact and impulsive loading is examined in detail. Discrete structural systems which under the same loading applied statically exhibit snap-through buckling are mainly considered. Emphasis is focused on the coupling effect of nonlinearities (geometric and/or material) and structural damping. A qualitative discussion of the dynamic buckling mechanism is comprehensively presented on the basis of energy and topological concepts in the light of recent progress of nonlinear dynamics and chaos. This leads to useful criteria for establishing exact, approximate and lower-upper bound dynamic buckling estimates without integrating the highly nonlinear equations of motion. It is shown that dynamic buckling can be defined as an escaped motion through a saddle of the (static) unstable postbuckling equilibrium path which leads either to an “unbounded” motion or to a point attractor associated with a remote stable equilibrium point. As byproducts of this analysis various phenomena such as restabilization (metastability), loading discontinuity, sensitivity to initial-conditions and to damping as well as chaoticlike Phenomena, are revealed. The theory is illustrated with analyses of single, two and three degrees of freedom models.