Finite element asymptotic analysis of slender elastic structures: A simple approach

Abstract

SUMMARYAn asymptotic method directly derived from Koiter's theory and suitable for the solution of elastic buckling problems and its natural adaptation to a numerical solution by means of a finite element technique are presented here. The order of the extrapolation of the equilibrium equations has been intentionally kept very low because attention has been entirely devoted to all those features (theoretical definitions, eigenproblem numerical techniques, suitable FEM implementation) which make such an approach competitive with respect to the classic step-by-step methods. For plane frames and 3D pin-jointed trusses, the performances of the algorithm (numerical accuracy and computational cost) are compared with those of Riks' arc-length method.

Nonlinear FEM analysis for beams and plate assemblages based on the implicit corotational method

“…Asymptotic analysis essentially corresponds to the implementation of Koiter's approach to nonlinear elastic stability [Koiter 1945;Budiansky 1974] into a FEM numerical context. While being less diffuse than path-following approaches within computational mechanics (maybe because of its high demands in terms of modeling accuracy) it has been described in detail in many papers (for example, [Casciaro et al 1992;Flores and Godoy 1992;Lanzo et al 1995;Pacoste and Eriksson 1995;Lanzo and Garcea 1996;Wu and Wang 1997;Poulsen and Damkilde 1998;Garcea et al 1999;Garcea 2001;Boutyour et al 2004;Casciaro 2005;Silvestre and Camoti 2005;Chen and Virgin 2006;Schafer and Graham-Brady 2006;Rahman and Jansen 2010] and references therein), so it only needs to be briefly summarized here.…”

A direct one-dimensional beam model for the flexural-torsional buckling of thin-walled beams

“…If we move in the frame of a mechanical theory of grade one (see, for example [Di Carlo 1996]), let us assume that the behavior of any substantial point on the (axis of the) beam is influenced by the substantial points contained in one of its neighborhoods. The interaction among different parts of the beam is then quantified for each test velocity field attainable by the beam, using a linear functional of the velocity fields and of their first derivatives with respect to x 1 .…”

“…This phenomenon was first investigated in [Wagner 1929] and [Kappus 1937], and since the publication of these pioneering works, many further studies have appeared on the subject, including [Vlasov 1961;Epstein 1979;Reissner 1983;Simo and Vu-Quoc 1991]. More recently, [Di Egidio et al 2003] investigated modelization aspects, and [Anderson and Trahair 1972;Casciaro et al 1991;Lanzo and Garcea 1996] presented a search of numerical results for standard elements. It is remarkable that most of the beam models presented in the literature are derived by projection of the results of three-dimensional continuum models on shell models (as in Vlasov 1961) or beam models (as in Simo and Vu-Quoc 1991).…”

“…This model describes the kinematics of the beam through the placement of the points of the beam axis, the rotation of the beam cross-sections (which are assumed to be plane in the reference configuration), and a coarse description of the warping of the beam cross-sections. Using a mechanical theory of grade one (see, for example Germain 1973aand 1973b, and Di Carlo 1996, the interaction of the beam with the surrounding environment is defined as a linear function of the velocities and of their first-order spatial approximations. In this way, the mechanical actions naturally appear as dual to the kinematic fields.…”

Direct calculation of critical points in parameter sensitive systems

“…Another technique with similar advantages and disadvantages, the Lyapunov-Schmidt-Koiter asymptotic approach, is based on regularizing the governing equations by a perturbation parameter [22,23,24]. More recently, researchers have introduced incremental-iterative numerical methods that directly calculate the buckling strength of parameterized (imperfect) structures [25,26].…”