Imperfection behavior of elastic nonlinear systems illustrated by a three-degree-of-freedom model

Abstract

A simple three-degree-of-freedom spring model system is presented to study the instability behavior of perfect and imperfect systems. The model has all known types of critical points: the stationary point and symmetric and asymmetric bifurcation points. By changing three parameters of the model, the behavior of the system can be altered and investigated. Numerical results are presented to demonstrate instability behavior of perfect and imperfect elastic nonlinear systems. Major findings are 1) an equilibrium path may terminate at a point and jump to another point even when geometric change is continuous; 2) the optimum solutions for the mode of imperfection that gives the lowest carrying capacity may also terminate at a point and jump to another point even when norms of imperfections are changed continuously; 3) an iterative process most likely diverges when the aforementioned two conditions take place unless an initial guess solution closer to the jumped position is used; 4) even for the same mode of imperfection, the load carrying capacities are not necessarily the same when norms of initial imperfections approach zero, depending on the signs of the mode; and 5) when hidden critical points are present, the structure is extremely sensitive to imperfection.