GBT formulation to analyse the first-order and buckling behaviour of thin-walled members with arbitrary cross-sections

Gonçalves, Rodrigo; Dinis, Pedro Borges; Camotim, Dinar

doi:10.1016/j.tws.2008.09.007

Cited by 85 publications

(30 citation statements)

References 13 publications

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“…We also indicated the effect of different simplification choices and have shown that, by appropriate simplifying assumptions, we can recover existing nonlinear models (for example, the beam and plate models of [Simo 1986;Antman 1995]). The method could however easily be generalized to anisotropic materials and also applied in different contexts, such as, for instance, Vlasov thin-walled beam theory, generalized beam theories [Goncalves et al 2009], or laminated plate theory [Nayfeh and Pai 2004]. In all cases the advantage is that it provides fully objective nonlinear models by a black-box procedure which only needs the corresponding linear model to be already available.…”

Section: Resultsmentioning

confidence: 99%

“…The nonlinear ICM models obtained will obviously inherit all the approximations contained in these theories. More refined models can be obtained, by the same procedure, using more sophisticated theories, such as the thinwalled beam theory of Vlasov, the so-called generalized beam theories [Goncalves et al 2009], and the anisotropic theories for plates [Nayfeh and Pai 2004].…”

Section: Further Comments and Remarksmentioning

confidence: 99%

See 1 more Smart Citation

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

Garcea

Madeo

Casciaro

2012

J. Mech. Mater. Struct.

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What we call the implicit corotational method is proposed as a tool to obtain geometrically exact nonlinear models for structural elements, such as beams or shells, undergoing finite rotations and small strains, starting from the basic solutions for the three-dimensional Cauchy continuum used in the corresponding linear modelings.The idea is to use a local corotational description to decompose the deformation gradient in a stretch part followed by a finite rigid rotation. Referring to this corotational frame we can derive, from the linear stress tensor and the deformation gradient provided by linear elasticity, an accurate approximation for the nonlinear stress and strain tensors which implicitly assure frame invariance. The stress and strain fields recovered in this way as functions of generalized stress and strain resultants are then used within a mixed variational formulation allowing us to recover an objective nonlinear modeling directly suitable for FEM implementations through a black-box process which maintains the full details of the linear solutions, such as shear warping and other subtle effects.The method is applied to the construction of three-dimensional beam and plate nonlinear models starting from the Saint-Venant rod and Kirchhoff and Mindlin-Reissner plate linear theories, respectively.

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Section: Resultsmentioning

confidence: 99%

Section: Further Comments and Remarksmentioning

confidence: 99%

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

Garcea

Madeo

Casciaro

2012

J. Mech. Mater. Struct.

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“…It is worth mentioning that a more accurate recovery could be obtained by refined one-dimensional models which can be derived through the ICM procedure from more sophisticated linear beam theories, such as that of [Vlasov 1959] or even better the ones obtained by the so-called GBT approach (see [Goncalves et al 2009]) which allow a more realistic modeling of boundary conditions. An investigation of this would be of interest, but is outside the scope of the present paper.…”

Section: Numerical Validationmentioning

confidence: 99%

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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“…For the discretisation considered (involving 15 nodes), a total of = 3 × 15 = 45 deformation modes are obtained  the in-plane or out-of-plane configurations of the most relevant ones are depicted in Figure 3. They comprise: (i) the four classical rigid-body (or "global") modes (axial extension (1), major-and minor-axis bending (2-3) and torsion (4)), (ii) two distortional modes, associated with quasi-rigid body flange-lip motions (5-6), (iii) a sequence of local modes, involving transverse plate bending with increasing curvature (7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17), (iv) five global shear modes (18)(19)(20)(21)(22), consisting of the warping components of the Vlasov modes 2-6, (v) a set of local shear modes, (23)(24)(25)(26)(27)(28)(29)(30)(31), (vi) five global transverse extension modes (32)(33)(34)(35)(36) and (vii) the local transverse extension modes (37-45).…”

Section: Cross-section Analysismentioning

confidence: 99%

“…Recent progress concerning the cross-section analysis has made GBT applicable in the context of members exhibiting arbitrary flat-walled cross-sections [6][7] or circular/elliptical tubular cross-sections [8][9][10]. Concerning the member analysis, formulations/studies have been reported for various types of structural analysis, namely first-order [11,12], buckling [3,[13][14][15][16], vibration [17][18][19], post-buckling [20][21][22] and dynamic [23] analyses involving elastic members (mostly), frames and trusses. Recently, the second version of GBTUL [24], a GBT-based freeware code which performs linear buckling and vibration analyses of general thin-walled bars, has been released online [25].…”

Section: Introductionmentioning

confidence: 99%

GBT-Based Buckling Analysis Using the Exact Element Method

Bebiano

Eisenberger

Camotim

et al. 2017

Int. J. Str. Stab. Dyn.

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Generalised Beam Theory (GBT), intended to analyse the structural behaviour of prismatic thin-walled members and structural systems, expresses the member local and/or global deformed configuration as a combination of cross-section deformation modes multiplied by the corresponding longitudinal amplitude functions. The determination of the latter, usually the most computer-intensive step of the analysis, is almost always performed by means of GBT-based conventional 1D (beam) finite elements, using Hermite cubic polynomials as shape functions. This paper presents the formulation, implementation and application of a new GBT-based exact element, developed in the context of member (linear) buckling analyses. This exact element, originally proposed by , approximates the modal longitudinal amplitude functions by means of power series, whose coefficients are obtained by means of a recursive formulasince the higher-order coefficients tend to vanish, the method has the potential to become exact (up to computer precision). The buckling load parameters are the solutions of the (highly) non-linear characteristic equation associated with the buckling eigenvalue problem. A few numerical illustrative examples are presented, focusing mainly on the comparison between the combined accuracy and computational effort associated with the determination of buckling solutions with the exact and standard GBT-based (finite) elements. This comparison provides evidence that the exact element leads to equally accurate results with less degrees of freedom and, moreover, without the need to define a (longitudinal) mesh  the relative efficiency of the exact element is higher when the buckling modes exhibit larger half-wave numbers.

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GBT formulation to analyse the first-order and buckling behaviour of thin-walled members with arbitrary cross-sections

Cited by 85 publications

References 13 publications

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

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

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

GBT-Based Buckling Analysis Using the Exact Element Method

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