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Influence of in-Plane Deformation in Higher Order Beam Theories

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Abstract

Comparing Euler-Bernoulli or Tismoshenko beam theory to higher order beam theories, an essential difference can be depicted: the additional degrees of freedom accounting for out-of plane (warping) and in-plane (distortional) phenomena leading to the appearance of respective higher order geometric constants. In this paper, after briefly overviewing literature of the major beam theories taking account warping and distortional deformation, the influence of distortion in the response of beams evaluated by higher order beam theories is examined via a numerical example of buckling drawn from the literature.
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... This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). section is also observed by Sapountzakis et al. [5,6], who consider mostly buckling analyses. ...
... After all, GBT is a two-step algorithm consisting of a cross-sectional analysis defining warping and distortional fields, followed by a member analysis, where those deformation fields are weighted along the beam's axis. According to Sapountzakis and Argyridi [5] most approaches in that field perform the cross-sectional analysis in two stages, first defining the warping modes and then the distortional modes found from Vlasov's zero shear stress conditions. Ranzi and Luongo [12] propose a reversed procedure starting with an eigenvalue decomposition of a planar discretized cross-section, where the eigenvectors define the distortional fields. ...
... In order to improve the accuracy of CBT in thin-walled box type cross-sections a Generalized Beam Theory (GBT) can be applied (see [1,5] for a comprehensive literature review and a discussion of main contributions in that field). As discussed in Section 2.1, GBT is a two-step algorithm consisting of a cross-sectional analysis defining warping and distortional fields, followed by a member analysis, where those deformation fields are weighted along the beam's axis. ...
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This paper proposes an efficient generalized beam theory (GBT) formulation, which accounts for cross-sectional deformations in slender prismatic structures. It was shown by the authors in a recent publication [1] that in-plane distortional deformations and accompanied out of plane warping deformations of the cross-section influence the accuracy of results in beam dynamics especially if thin-walled cross-sections are applied. The GBT formulation proposed in [1] overcomes the inaccuracies of classical beam mechanics, however, requires a two-dimensional plane discretization of the cross-section. The computational complexity can be reduced vastly, if thin-walled cross-sections can be discretized with one-dimensional elements. Consequently, this paper discusses an approach with a line mesh discretizing the cross-section and having six degrees of freedom at each node. The membrane part consists of massless micro-polar rotations (drilling rotations) and can be derived independently from the bending part, where a shear elastic formulation is selected.
... However, the authors showed recently that the accuracy of CBM deteriorates significantly especially if static or dynamic torsion analyses of shafts with thin-walled box-type cross-sections are performed (see Fig. 1 where torsional vibration is studied in a quite slender shaft, indicating that CBM does not even come close to continuum (or shell) results since in-plane distortions influence the mechanical properties significantly [1, 4] 1 ). These effects are also observed by Sapountzakis et al. [5,6] where also buckling analyses are considered. In order to circumvent the insufficient accuracy of CBM in thin-walled single or multi-cell box type cross-sections, however, maintaining a moderate number of global degrees of freedom, a Generalized Beam Theory (GBT) can be applied (see [1] and [5] for a comprehensive literature review and a discussion of main contributions in that field). ...
... These effects are also observed by Sapountzakis et al. [5,6] where also buckling analyses are considered. In order to circumvent the insufficient accuracy of CBM in thin-walled single or multi-cell box type cross-sections, however, maintaining a moderate number of global degrees of freedom, a Generalized Beam Theory (GBT) can be applied (see [1] and [5] for a comprehensive literature review and a discussion of main contributions in that field). After all, GBT is a two-step algorithm consisting of a cross-sectional analysis defining warping and distortional fields, followed by a member analysis, where those deformation fields are weighed axially along the beam. ...
... Deficiencies of CBM are typically reported in case of buckling problems of short members in connection with thin-walled cross-sections (see e.g. Sapountzakis and Argyridi, 2018, and the references therein). In order to motivate our following approach, we consider the torsional resonance curves of a prismatic shaft with a homogeneous rectangular box type cross-section (see Subsect. 11.4.2,Fig. ...
... The inclusion of in-plane distortions necessitates higher order beam theories or generalized beam theories (GBT)3. A comprehensively written paper discussing the state of the art in GBT is due to Sapountzakis and Argyridi (2018). There, it is pointed out that the majority of research focused on thin-walled profiles with pronounced distortion and warping. ...
Chapter
Mechanical properties of slender, prismatic structures are typically analyzed based on classical beam mechanics (Timoshenko’s shear force bending, Vlasov’s theory of warping torsion, …). There it is assumed that the cross-section remains rigid in its projection plane and in-plane distortional deformations of the cross-section are neglected. Such a model is predictive in case of static gradually distributed loading, and solid cross-sections, however, in case of thin-walled crosssections and dynamic loading severe deviations might occur. Therefore, a generalized beam theory is proposed, where warping fields and accompanied distortional fields of the cross-section are axially distributed each based on one generalized degree of freedom. The evaluation of pairs ofwarping and distortional fields in ascending order of importance is performed using a specific reference beam problem (RBP), where three-dimensional elasticity theory is applied in connection with semi-analytical finite elements (SAFE). Convergence of the resulting formulation is ensured by increasing the number of contributing pairs of warping and distortional fields. The resulting formulation yields significantly better results compared to classical beam mechanics especially in the dynamic regime.
... Functionally graded materials (FGM) can be characterized by the gradual variation of material properties in the thickness. A new type of composite materials is developed recently (Abdelbaki et al [1]; Arnab Choudhury et al [2]; Abdelbaki et al [3]; Ebrahimi and Barati [4]; Ebrahimi and Heidari [5]; Elmerabet et al [6]; Elmossouess et al [7]; Houari et al [8]; Karami et al [9]; Mahjoobi and Bidgoli [10]; Mohamed et al [11]; Mokhtar et al [12]; Mokhtar et al [13]; Sadoun et al [14]; Salari et al [15]; Shafiei and Setoodeh [16]; Shokravi [17]; Tlidji et al [18]; Tounsi et al [19]; Tu et al [20]; Bocko, J et al [21]; Jozef, B et al [22]; Stephan, K et al [23]; Murín, J et al [24]; Sapountzakis, E et al [25]). ...
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This paper shows an analysis of the free vibration of functionally graded simply supported nanoplate. The nonlocal four variables shear deformation plate theory is used to predict the free vibration frequencies of functionally graded nanoplate simply supported using non-local elasticity theory with the introduction of small-scale effects. The effect of the material properties, thickness-length ratio, aspect ratio, the exponent of the power law, the vibration mode is presented, the current solutions are compared to those obtained by other researchers. Equilibrium equations are obtained using the virtual displacements principle. P-FGM Power law is used to have a distribution of material properties that vary across the thickness. The results are in good agreement with those of the literature.
... On the other hand, EI Fatimi and Ghazouani [30] developed a higher order composite beam theory built on Saint-Venant solution and further analyzed built-in effects on the behavior of end-loaded cantilever beams with different cross-sections [31] . Sapountzakis and Argyridi [32] examined and verified the influence of distortion in buckling analysis of beams with higher order beam theories built on Saint-Venant solution. Paradisoet al. [33] proposed a 1D beam model from 3D Saint-Venant solution by enforcing the equivalence between 3D solid model and 1D beam model in terms of energy and displacements, and Kennedy and Martins [34] presented a homogenization-based theory for anisotropic beams, where the fundamental states are obtained from solutions to the Saint-Venant and the Almansi-Michell problems. ...
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