Nonlinear Generalized Beam Theory for open thin-walled members

2016 ◽  
Vol 22 (10) ◽  
pp. 1907-1921 ◽  
Author(s):  
Giuseppe Piccardo ◽  
Alberto Ferrarotti ◽  
Angelo Luongo
Keyword(s):  
Linear Theory ◽  
Cross Section ◽  
Beam Theory ◽  
Plate Elements ◽  
Nonlinear Galerkin Method ◽  
Out Of Plane ◽  
Thin Walled ◽  
Cross Section Profile ◽  
Nonlinear Elastic ◽  
Generalized Beam Theory

In the framework of the Generalized Beam Theory (GBT) a new cross-section analysis is proposed, specifically suited for nonlinear elastic thin-walled beams (TWB). The approach is developed according to the nonlinear Galerkin method (NGM), which calls for the evaluation of nonlinear (passive) trial functions, to be used in conjunction with linear (active) trial functions, in describing the displacement field. The set of (quadratic) trial functions is determined here by requiring that the classic Vlasov’s kinematic hypotheses of the linear theory (i.e. (a) transverse inextensibility and (b) unshearability) are satisfied also in the nonlinear sense. The linear field is described by the so-called conventional displacements, by neglecting non-conventional displacements, which violate Vlasov’s hypotheses. The nonlinear trial functions thus generated are innovative deformation fields, which describe extensional and shear displacements in a different way from that of the non-conventional fields of the linear theory. In particular, they consist of non-constant tangential and out-of-plane displacements of the cross-section profile, able to ensure inextensibility and unshearability of all the plate elements, by balancing the second-order strains induced by the conventional displacements. Since nonlinear trial functions do not increase the number of the unknowns, the GBT spirit, as a reduction method, is preserved. A very promising example is discussed, which shows that equilibrium paths can be determined by using few linear trial functions in conjunction with the corresponding nonlinear trial functions, supplying good results when compared with burdensome finite-element solutions.

scholarly journals Generalized Beam Theory for Thin-Walled Beams with Curvilinear Open Cross-Sections

2020 ◽  
Vol 10 (21) ◽  
pp. 7802
Author(s):  
Jarosław Latalski ◽  
Daniele Zulli
Keyword(s):  
Cross Section ◽  
Cross Sections ◽  
Beam Theory ◽  
Middle Surface ◽  
Fem Model ◽  
Out Of Plane ◽  
Thin Walled ◽  
Generalized Beam Theory ◽  
Load Conditions

The use of the Generalized Beam Theory (GBT) is extended to thin-walled beams with curvilinear cross-sections. After defining the kinematic features of the walls, where their curvature is consistently accounted for, the displacement of the points is assumed as linear combination of unknown amplitudes and pre-established trial functions. The latter, and specifically their in-plane components, are chosen as dynamic modes of a curved beam in the shape of the member cross-section. Moreover, the out-of-plane components come from the imposition of the Vlasov internal constraint of shear indeformable middle surface. For a case study of semi-annular cross-section, i.e., constant curvature, the modes are analytically evaluated and the procedure is implemented for two different load conditions. Outcomes are compared to those of a FEM model.

First and Second Order Elastoplastic Analyses of Thin-Walled Metal Members using Generalized Beam Theory

2018 ◽  
Author(s):  
Miguel Abambres
Keyword(s):  
Finite Element ◽  
Stainless Steel ◽  
Cross Section ◽  
Beam Theory ◽  
Second Order ◽  
Flow Rule ◽  
Non Linear ◽  
Thin Walled ◽  
Shell Finite Element ◽  
Generalized Beam Theory

Original Generalized Beam Theory (GBT) formulations for elastoplastic first and second order (postbuckling) analyses of thin-walled members are proposed, based on the J2 theory with associated flow rule, and valid for (i) arbitrary residual stress and geometric imperfection distributions, (ii) non-linear isotropic materials (e.g., carbon/stainless steel), and (iii) arbitrary deformation patterns (e.g., global, local, distortional, shear). The cross-section analysis is based on the formulation by Silva (2013), but adopts five types of nodal degrees of freedom (d.o.f.) – one of them (warping rotation) is an innovation of present work and allows the use of cubic polynomials (instead of linear functions) to approximate the warping profiles in each sub-plate. The formulations are validated by presenting various illustrative examples involving beams and columns characterized by several cross-section types (open, closed, (un) branched), materials (bi-linear or non-linear – e.g., stainless steel) and boundary conditions. The GBT results (equilibrium paths, stress/displacement distributions and collapse mechanisms) are validated by comparison with those obtained from shell finite element analyses. It is observed that the results are globally very similar with only 9% and 21% (1st and 2nd order) of the d.o.f. numbers required by the shell finite element models. Moreover, the GBT unique modal nature is highlighted by means of modal participation diagrams and amplitude functions, as well as analyses based on different deformation mode sets, providing an in-depth insight on the member behavioural mechanics in both elastic and inelastic regimes.

Global–Local-Distortional Vibration of Thin-Walled Rectangular Multi-Cell Beams

2015 ◽  
Vol 15 (08) ◽  
pp. 1540022 ◽  
Author(s):  
Rodrigo Gonçalves ◽  
Nuno Peres ◽  
Rui Bebiano ◽  
Dinar Camotim
Keyword(s):  
Cross Section ◽  
Plane Deformation ◽  
Beam Theory ◽  
Deformation Mode ◽  
Mode Shapes ◽  
Vibration Modes ◽  
Computationally Efficient ◽  
Out Of Plane ◽  
Thin Walled ◽  
Deformation Modes

This paper presents the results of an investigation concerning the free vibration behavior (undamped natural frequencies and vibration mode shapes) of thin-walled beams with rectangular multi-cell cross-section (assemblies of parallel rectangular cells in a single direction). Besides local (plate-type) and global (flexural, torsional and extensional) vibration modes, attention is paid to the relatively less-known distortional vibration modes, which involve cross-section out-of-plane (warping) and in-plane deformation, including displacements of the wall intersections. A computationally efficient semi-analytical Generalized Beam Theory (GBT) approach is employed to obtain insight into the mechanics of the problem. In particular, the intrinsic modal decomposition features of GBT — the fact that the beam is described using a hierarchical set of relevant cross-section deformation modes — are exploited to identify and categorize the most relevant vibration modes and deformation mode couplings.

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