Koiter's Analysis of Thin-Walled Structures by a Finite Element Approach

Abstract

SUMMARYThis paper refers to the analysis of the postbuckling behaviour of thin-walled structures by means of an asymptotic approach based on a finite element implementation of Koiter's non-linear theory of instability.The analysis has been accomplished by using the following assumptions: (i) the structure is described as an assemblage of flat slender rectangular panels; (ii) a non-linear Kirchhoff-type plate theory is used to model each panel; (iii) HC finite elements discretization is used; (iv) linear and quadratic extrapolations are assumed for the fundamental and the postbuckling paths, respectively; (v) multimodal buckling is considered; and (vi) imperfection sensitivity analysis is performed in both multimodal and monomodal form based on the steepest-descent path criterion.Several numerical results are presented and discussed. Comparisons with numerical solution obtained by standard incremental codes are given, which show the accuracy and reliability of the proposed approach.

The implicit corotational method and its use in the derivation of nonlinear structural models for beams and plates

“…The aim is to obtain objective models suitable for the nonlinear analysis of thin walled structures, such as those already analyzed in [Lanzo et al 1995;Lanzo and Garcea 1996;Garcea 2001].…”

“…However we must consider that, in some cases, a nonlinear solution can differ noticeably from the linear one, because of the activation of local buckling phenomena characterized by deformation patterns which play a negligible role in the linear solution and so are taken out in the linear modeling. Thin-walled beams or stiffened panels, for example, can be modeled as one-dimensional fibred continua or as equivalent orthotropic plates through homogenization criteria; this can be sufficient in linear analysis but is unable to deal with complex coupled buckling behaviors which can be better described by more detailed modeling like plates assemblage (for example, see [Lanzo and Garcea 1996;]). A certain care is then always needed in selecting the appropriate parent linear theory in order to obtain a reliable and accurate nonlinear modeling.…”

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.…”

“…Both use a mixed format and a separate interpolation of the displacement and stress fields and are free from interpolation nonlinear locking. The beam element represents a three-dimensional extension of the interpolation described in [Salerno and Lanzo 1997]; the plate element is based on the biquadratic spline interpolation already used in [Lanzo et al 1995;Lanzo and Garcea 1996;Garcea 2001] The availability of analytical results or known reference benchmarks and the possibility of comparing, in the case of thin-walled beams, the results obtained by two different models (such as a one-dimensional beam or an assemblage of two-dimensional plates) allow, in Section 6, a detailed discussion about the performance of the proposed elements and their effectiveness in test cases of technical interest. Final comments and remarks are made in the concluding section.…”

“…It provides the initial postbuckling behavior of the structure, including modal interactions and jumping-after-bifurcation phenomena. Moreover, once the preprocessor phase of the analysis has been performed (Steps (1)-(4)), the presence of small loading imperfections or geometrical defects can be taken into account in the postprocessing phase (Step (5)), by adding some additional imperfection terms in the expression of µ k [λ], with a negligible computational extra cost, so allowing an inexpensive imperfection sensitivity analysis (for example, see [Lanzo and Garcea 1996;]). From (6h) we can also extract information about the worst imperfection shapes [Salerno and Casciaro 1997;Salerno and Uva 2006] we can use to improve the imperfection sensitivity analysis or for driving more detailed investigations through specialized pathfollowing analysis (see [Casciaro 2005;Casciaro and Mancusi 2006] and references therein).…”