Load carrying capacity of systems within a global safety perspective. Part I. Robustness of stable equilibria under imperfections

Abstract

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. 1 LOAD CARRYING CAPACITY OF SYSTEMS WITHIN Abstract:The problem of the practical stability of structures is addressed in a modern way by considering the effects of both static and dynamic perturbations. The major historical contributions, due to Euler, Koiter and Thompson, are reviewed and illustrated by an archetypal model permitting to highlight the main mechanical and dynamical points. It is found that a global approach is necessary for a reliable safety estimation, especially in the neighborhood of (static) critical loads. Considering that the admissible load threshold has to account for robustness to finite perturbations, the Koiter critical load must be lowered, obtaining the so called Thompson critical load. It is shown how these two thresholds share some properties (e.g. both depend in a sensitive way on imperfections, which must be known for practical calculations), while having a deep different meaning: the former is related to static imperfections, and requires only a local analysis, while the latter is related to dynamical imperfections, and requires a global analysis. It is shown that crit p Euler ≥ crit p Koiter ≥ crit p Thompson , i.e., that the advancement of knowledge leads to a lower estimation of the actual critical load.

The dynamical integrity concept for interpreting/ predicting experimental behaviour: from macro- to nano-mechanics

“…Clearly, by definition, the fractal parts are no longer present in the actual safe basin, which has a substantially compact shape. Note that the actual safe basin can be considered the equivalent domain, in the presence of dynamic excitations, to the safe basin also characterizing a static stability problem [28]. An example of nominal safe basin and of the associated actual one is reported in figure 1, which clearly shows the elimination of the fractal parts.…”

Jump and pull-in dynamics of an electrically actuated bistable MEMS device

“…They produce small, but finite perturbations, which may significantly affect and alter the system response. Taking them into [21,22], where all the basic aspects are illustrated and discussed, including the difference with respect to the classical stability concepts, the main current advances in analytical tools for quantifying the integrity of a system against disturbances, and the actual substantial developments in the direction of practical applications. Dynamical integrity predictions have been recently widely referred in the literature in micro and nanosystems, both for interpreting and predicting the experimental behavior [23,24], and for getting hints towards engineering design [25,26], and for controlling the global dynamics [27].…”

Load carrying capacity of systems within a global safety perspective. Part II. Attractor/basin integrity under dynamic excitations

“…In the companion paper [1] we put the main contributions to the determination of the practical load carrying capacity of structures into an historical perspective, with the aim of showing how it is reduced by the loss of robustness of stable equilibria. In summary, we have recalled the works of Euler [2], who discovered branching (pitchfork and transcritical) bifurcations in elastic systems (talking in modern language, of course); of Koiter [3], who discovered the structural instability of those bifurcations, i.e.…”

“…This has been done in Part I [1], where, after briefly dwelling on the Euler and Koiter theories, we focused on the reduction of load carrying capacity due to the loss of attractor robustness (Thompson approach). Note that, even if there is no external force, this analysis requires a truly dynamical approach, since it needs analyzing the effect of finite changes of initial conditions, or, equivalently, studying the evolution of safe basins for a growing governing parameters (e.g., the axial load of a beam).…”

“…1 of [1]), which is of course a crude approximation of real cases, but permits to highlight in a simple way the very basic underlying idea and the main mechanical issues. The simplicity of the model, and the absence of external excitation, have allowed us, for example, to address the matter without solving the equation of motion; not even the time history of the homoclinic orbit, which can be computed in principle (see Sect.…”