Perturbation Approach to Elastic Post-Buckling Analysis

Casciaro, Raffaele; Garcea, Giovanni; Attanasio, G.; Giordano, Francesco

doi:10.1016/s0045-7949(97)00112-0

Cited by 58 publications

(34 citation statements)

References 21 publications

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“…(ii) Perform the buckling search (6b); this corresponds to a nonlinear eigenvalue problem which can be, however, easily solved, as described in [Casciaro et al 1998], by an iterative scheme only requiring matrix K −1 0 be available in factorized form, as it already is from the previous step. (iii) Solve orthogonal equations (6f); they correspond to a linear system where the right-hand vectors p i j are obtained as a function of v i by an element-by-element assembling process similar to that used for obtaining s [u]; as K b ≈ K 0 within the orthogonal space ᐃ, its solution can be conveniently obtained, as described in [Casciaro et al 1998], through a modified Newton-like iteration scheme exploiting K 0 as an iteration matrix (see [Casciaro 2004] for further insights).…”

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“…(iii) Solve orthogonal equations (6f); they correspond to a linear system where the right-hand vectors p i j are obtained as a function of v i by an element-by-element assembling process similar to that used for obtaining s [u]; as K b ≈ K 0 within the orthogonal space ᐃ, its solution can be conveniently obtained, as described in [Casciaro et al 1998], through a modified Newton-like iteration scheme exploiting K 0 as an iteration matrix (see [Casciaro 2004] for further insights).…”

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

Garcea

Madeo

Casciaro

2012

J. Mech. Mater. Struct.

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In our previous paper the implicit corotational method (ICM) was presented as a general procedure for recovering objective nonlinear models fully reusing the information obtained by the corresponding linear theories.The present work deals with the implementation of the ICM as a numerical tool for the finite element analysis of nonlinear structures using either a path-following or an asymptotic approach. Different aspects of the FEM modeling are discussed in detail, including the numerical handling of finite rotations, interpolation strategies, and equation formats.Two mixed finite elements are presented, suitable for nonlinear analysis: a three-dimensional beam element, based on interpolation of both the kinematic and static fields, and a rotation-free thin-plate element, based on a biquadratic spline interpolation of the displacement and piece-wise constant interpolation of stress. Both are frame invariant and free from nonlinear locking.A numerical investigation has been performed, comparing beam and plate solutions in the case of thinwalled beams. The good agreement between the recovered results and the available theoretical solutions and/or numerical benchmarks clearly shows the correctness and robustness of the proposed approach as a general strategy for numerical implementations.

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confidence: 99%

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

Garcea

Madeo

Casciaro

2012

J. Mech. Mater. Struct.

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“…The details descriptions of this method can be found in the monographs: van der Heijden (2009), Thompson and Hunt (1973) or Kubiak (2013). Applicability of an asymptotic expansion for elastic buckling problems with mode interaction was discussed in many papers, for instance: Tvergaard (1973aTvergaard ( , 1973b, Koiter and Pignataro (1974), Byskov (1979Byskov ( , 1988, Sridharan (1983), Sridharan (1985a, 1985b), Pignataro and Luongo (1985, 1987a, 1987b, Casciaro et al (1998), Mollman (1989a, 1989b), Garcea et al (1999Garcea et al ( , 2009), Barbero et al (2014).…”

Section: Coupled Buckling Of Thin-walled Structuresmentioning

confidence: 99%

Coupled Static and Dynamic Buckling Modelling of Thin-Walled Structures in Elastic Range Review of Selected Problems

Kołakowski

Teter

2016

Acta Mechanica Et Automatica

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A review of papers that investigate the static and dynamic coupled buckling and post-buckling behaviour of thin-walled structures is carried out. The problem of static coupled buckling is sufficiently well-recognized. The analysis of dynamic interactive buckling is limited in practice to columns, single plates and shells. The applications of finite element method (FEM) or/and analytical-numerical method (ANM) to solve interaction buckling problems are on-going. In Poland, the team of scientists from the Department of Strength of Materials, Lodz University of Technology and co-workers developed the analytical-numerical method. This method allows to determine static buckling stresses, natural frequencies, coefficients of the equation describing the post-buckling equilibrium path and dynamic response of the plate structure subjected to compression load and/or bending moment. Using the dynamic buckling criteria, it is possible to determine the dynamic critical load. They presented a lot of interesting results for problems of the static and dynamic coupled buckling of thin-walled plate structures with complex shapes of cross-sections, including an interaction of component plates. The most important advantage of presented analytical-numerical method is that it enables to describe all buckling modes and the post-buckling behaviours of thin-walled columns made of different materials. Thin isotropic, orthotropic or laminate structures were considered.Key words: Interaction, Buckling, Thin-Walled Structures, FEM, Analytical-Numerical Method, Review COUPLED BUCKLING OF THIN-WALLED STRUCTURESThe theory of coupled or interactive buckling of thin walled structures has been already developed widely for over sixty years. Thin-walled structures, especially plates, columns and beams, have many different buckling modes that vary in quantitative and qualitative aspects. In these cases, nonlinear buckling theory should describe all buckling modes from global (i.e., flexural, flexural-torsional, lateral, distortional and their combinations) to local and the coupled buckling as well as the determination of their load carrying capacity taking into consideration the structure imperfection. Coupling between modes occur for columns of such length where two or more eigenvalues loads of a structure are nearly identical (Fig. 1). The local buckling takes place for the short columns. On the other hand, the long columns are subject to global buckling.The concept of coupled or interactive buckling involves the general asymptotic nonlinear theory of stability. Among all versions of the general nonlinear theory, the Koiter theory of conservative systems (Koiter, 1976; van der Heijden, 2009) is the most popular one, owing to its general character and development, even more so after Byskov and Hutchinson (1997) formulated it in a convenient way. The details descriptions of this method can be found in the monographs: van der Heijden (2009), Thompson and Hunt (1973) or Kubiak (2013). Applicability of an asymptotic expansion for elastic buck...

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“…As a result, the prebuckling, buckling and initial postbuckling behaviors are captured in just three solution steps compared to multiple iterations required in incremental algorithms. Koiter's approach was applied by Stein 33 for rectangular isotropic plates, Chandra & Raju 34 for symmetrically laminated orthotropic plates, and was extended in an optimization framework by Wu et al 35 Furthermore, Koiter's perturbation technique lends itself to robust implementation in numerical solution techniques such as the Finite Element Method (FEM) 36 and the Differential Quadrature Method (DQM). 37 Recently, Koiter's perturbation scheme was coupled with a Newton-Raphson iterative solver in order to combine the merits of both asymptotic and path-following techniques, thereby allowing efficient modeling of nonlinear prebuckling paths and limit-point buckling.…”

Section: Iib Koiter's Perturbation Approachmentioning

confidence: 99%

Mass Optimisation of Variable Angle Tow, Variable Thickness Panels with Static Failure and Buckling Constraints

Groh

2015

56th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference

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By taking advantage of curved fiber paths, Variable Angle Tow (VAT) laminates increase the design space for tailoring the structural behavior of thin-walled aerospace structures. In recent years, advancements in Automated Fiber Placement (AFP) and Continuous Tow Shearing (CTS) have facilitated the manufacture of these laminates. The CTS technique holds the advantage of reducing many of the manufacturing defects characteristic of the AFP process such as fiber wrinkling, tow gaps and tow overlaps, while also allowing for tighter steering radii. On the other hand, the CTS process features added complexity due to the coupling of fiber steering angle with tow thickness. In this study, a minimummass design of a typical aircraft wing panel under end-compression subject to pre-defined manufacturing, static failure and buckling load constraints is sought. The geometric effects of the asymmetric thickness distribution of the CTS panel on the critical buckling loads, postbuckling paths and static failure behavior are captured for the first time. A hybrid optimization scheme that couples a genetic algorithm with a pattern-search algorithm is used to define a VAT laminate that reduces the mass of both square and rectangular aircraft panels by 31% compared to a baseline straight fiber design. The optimization of the fiber paths is driven by two distinct requirements, namely local and global stiffness tailoring that influence the buckling performance and static strength, respectively. Finally, the initial postbuckling behavior of the optimized designs is investigated using Koiter's perturbation approach, which reveals that postbuckling stability should be considered when optimising VAT panels manufactured by the CTS technique.

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Perturbation Approach to Elastic Post-Buckling Analysis

Cited by 58 publications

References 21 publications

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

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

Coupled Static and Dynamic Buckling Modelling of Thin-Walled Structures in Elastic Range Review of Selected Problems

Mass Optimisation of Variable Angle Tow, Variable Thickness Panels with Static Failure and Buckling Constraints

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