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- November 2018
- Strojnícky časopis - Journal of Mechanical Engineering 68(3):77-94

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- CC BY-NC-ND

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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. ...

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. ...

Conference Paper

Full-text available

- Jan 2021

... 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

- Jun 2020

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]). ...

Article

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- Dec 2019

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. ...

Article

- Feb 2022
- COMPUT STRUCT

In the analysis of thin-walled members, some types of boundary conditions met in practice cannot be correctly represented by conventional 1D models. Of course, both shell models or 1D generalised beam theory (GBT) can easily tackle this type of problems - at the cost, however, of the simple interpretation of results provided by conventional 1D models. This paper investigates the problem of combined bending and twisting of a doubly symmetric I-beam having only one flange supported, which precludes the application of the conventional Timoshenko's model, with its built-in assumption that the cross-sections do not distort in their own plane. A simple 1D model for this particular problem is developed, which accounts for the in-plane distortion of the web and for the relative rotation of the flanges. A mixed 3-field finite element was derived for the numerical implementation of this model and applied to three illustrative examples. Its principal merit is the ability to achieve high accuracy on very coarse meshes, particularly when it comes to the approximation of generalized stresses, which are of particular interest to designers.

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- Mar 2021

A simple and efficient method is proposed for the analysis of twist of rectangular box-girder bridges, which undergo distortion of the cross section. The model is developed in the framework of the Generalized Beam Theory and oriented towards semi-analytical solutions. Accordingly, only two modes are accounted for: (i) the torsional mode, in which the box-girder behaves as a Vlasov beam under nonuniform torsion, and, (ii) a distortional mode, in which the cross section behaves as a planar frame experiencing skew-symmetric displacements. By following a variational approach, two coupled, fourth-order differential equations in the modulating amplitudes are obtained. The order of magnitude of the different terms is analyzed, and further reduced models are proposed. A sample system, taken from the literature, is considered, for which generalized displacement and stress fields are evaluated. Both a Fourier solution for the coupled problem and a closed-form solution for the uncoupled problem are carried out, and the results are compared. Finally, the model is validated against finite element analyses.

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- Nov 2018

In this paper, results of numerical simulations and measurements are presented concerning the non-uniform torsion and bending of an angled members of hollow cross-section. In numerical simulation, our linear-elastic 3D Timoshenko warping beam finite element is used, which allows consideration of non-uniform torsion. The finite element is suitable for analysis of spatial structures consisting of beams with constant open and closed cross-sections. The effect of the secondary torsional moment and of the shear forces on the deformation is included in the local finite beam element stiffness matrix. The warping part of the first derivative of the twist angle due to bimoment is considered as an additional degree of freedom at the nodes of the finite elements. Standard beam, shell and solid finite elements are also used in the comparative stress and deformation simulations. Results of the numerical experiments are discussed, compared, and evaluated. Measurements are performed for confirmation of the calculated results.

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- Nov 2018

The buckling analysis of carbon nanotubes without and with hetero-junctions is described in this paper. The buckling behaviour was investigated by the finite element method and the carbon nanotubes were modelled as space frame structures. The results showed that the critical buckling force depends on the dimensions of carbon nanotubes. The critical buckling forces of hetero-junction carbon nanotubes are in range between critical buckling forces of carbon nanotubes of both used diameters with the same chiralities without hetero-junction.

Article

- Jan 2019
- THIN WALL STRUCT

In this paper, a higher order beam theory is developed for the analysis of beams of homogeneous cross-section, taking into account warping and distortional phenomena due to axial, shear, flexural and torsional behavior. The beam can be subjected to arbitrary axial, transverse and/or torsional concentrated or distributed load, while its edges are restrained by the most general linear boundary conditions. The analysis consists of two stages. In the first stage, where the Boundary Element Method is employed, a cross sectional analysis is performed based on the so-called sequential equilibrium scheme establishing the possible in-plane (distortion) and out-of-plane (warping) deformation patterns of the cross-section. In the second stage, where the Finite Element Method is employed, the extracted deformation patterns are included in the beam analysis multiplied by respective independent parameters expressing their contribution to the beam deformation. The four rigid body displacements of the cross-section together with the aforementioned independent parameters consist the degrees of freedom of the beam. The finite element equations are formulated with respect to the displacements and the independent warping and distortional parameters. Numerical examples of axially loaded beams are solved to emphasize the importance of axial mode. In addition, numerical examples of various loading combinations are presented to demonstrate the range of application of the proposed method.

Article

- Dec 2018
- ENG STRUCT

In this paper, a higher order beam theory is employed for linear local buckling analysis of beams of homogeneous cross-section, taking into account warping and distortional phenomena due to axial, shear, flexural, and torsional behavior. The beam is subjected to arbitrary concentrated or distributed loading, while its edges are restrained by the most general linear boundary conditions. The analysis consists of two stages. In the first stage, where the Boundary Element Method is employed, a cross-sectional analysis is performed based on the so-called sequential equilibrium scheme establishing the possible in-plane (distortion) and out-of-plane (warping) deformation patterns of the cross-section. In the second stage, where the Finite Element Method is employed, the extracted deformation patterns are included in the buckling analysis multiplied by respective independent parameters expressing their contribution to the beam deformation. The four rigid body displacements of the cross-section together with the aforementioned independent parameters constitute the degrees of freedom of the beam. The finite element equations are formulated with respect to the displacements and the independent warping and distortional parameters. The buckling load is calculated and is compared with beam and 3d solid finite elements analysis results in order to validate the method and demonstrate its efficiency and accuracy.

Article

- Jun 2016

The finite element method is employed for the flexural-torsional linear buckling analysis of beams of arbitrarily shaped composite cross-section taking into account generalized warping (shear lag effects due to both flexure and torsion). The contacting materials, that constitute the composite cross section, may include a finite number of holes. A compressive axial load is applied to the beam. The influence of nonuniform warping is considered by the usage of one independent warping parameter for each warping type, i.e. shear warping in each direction and primary as well as secondary torsional warping, multiplied by the respective warping function. The calculation of the four aforementioned warping functions is implemented by the solution of a corresponding boundary value problem (longitudinal local equilibrium equation). The resulting stress field is corrected through a shear stress correction. The equations are formulated with reference to the independent warping parameters additionally to the displacement and rotation components.

Article

- Oct 2017
- J ENG MECH-ASCE

This paper presents a general formulation for the distortional analysis of beams of arbitrary cross section under arbitrary external loading and general boundary conditions. The nonuniform distortional/warping distributions along the beam length are taken into account by employing independent parameters multiplying suitable deformation modes accounting for in-plane and out-of-plane cross-sectional deformation (distortional/warping functions). The paper proposes a novel procedure for cross-sectional analysis which results in the solution of separate boundary value problems for each resisting mechanism (flexure, torsion) on the cross-sectional domain instead of relying on eigenvalue analysis procedures encountered in the literature. These distortional and warping functions are computed employing a boundary element method (BEM) procedure. Subsequently, sixteen boundary value problems are formulated with respect to displacement and rotation components as well as to independent distortional/warping parameters along the beam length and solved using the analog equation method (AEM), a BEM-based technique. After the establishment of kinematical components, stress components on any arbitrary point of each cross section of the beam can be evaluated, yielding a prediction in good agreement with three-dimensional finite-element method (FEM) solutions, in contrast to conventional beam models.

Article

- Jun 1996

Based on the displacement variational principle a systematic semi-discrete method, called as spline finite member element method, is developed for buckling analysis of thin-walled compression members with arbitrary cross-sections in the present paper. The displacements at two ends of the member element are adopted as basic variables in the method. A transformed B3-spline function presented in this paper is used to simulate the warping displacements along the cross-section of the thin-walled member. The analysis takes into account the effect of shearing strains of the middle surface of walls on the buckling, which reflect the shear lag phenomenon. Compared with the results from classical theory and the “COSMOS/M” finite element analysis program, the numerical results proposed in this paper demonstrate the versatility, accuracy and efficiency of the proposed method. The fast convergency shown in numerical examples predicts the reliability of the results.

Article

- Oct 2016
- THIN WALL STRUCT

In this paper, an accurate and computationally efficient Generalised Beam Theory (GBT) finite element is proposed, which makes it possible to calculate buckling (bifurcation) loads of steel–concrete composite beams subjected to negative (hogging) bending. Two types of buckling modes are considered, namely (i) local (plate-like) buckling of the web, possibly involving a torsional rotation of the lower flange, and (ii) distortional buckling, combining a lateral displacement/rotation of the lower flange with cross-section transverse bending. The determination of the buckling loads is performed in two stages: (i) a geometrically linear pre-buckling analysis is first carried out, accounting for shear lag and concrete cracking effects, and (ii) an eigenvalue buckling analysis is subsequently performed, using the calculated pre-buckling stresses and allowing for cross-section in-plane and out-of-plane (warping) deformation. The intrinsic versatility of the GBT approach, allowing the incorporation of a relatively wide range of assumptions, is used to obtain a finite element with a reasonably small number of DOFs and, in particular, able to comply with the principles of the “inverted U-frame” model prescribed in Eurocode 4 [1]. Several numerical examples are presented, to illustrate the application of the proposed GBT-based finite element and provide clear evidence of its capabilities and potential.

Article

- Aug 2016
- COMPUT STRUCT

In this paper, we apply the asymptotic expansion method to the mechanical problem of beam equilibrium, aiming to derive a new beam model. The asymptotic procedure will lead to a series of mechanical problems at different order, solved successively. For each order, new transverse (in-plane) deformation and warping (out of plane) deformation modes are determined, in function of the applied loads and the limits conditions of the problem. The presented method can be seen as a more simple and efficient alternative to beam model reduction techniques such as POD or PGD methods. At the end of the asymptotic expansion procedure, an enriched kinematic describing the displacement of the beam is obtained, and will be used for the formulation of an exact beam element by solving analytically the arising new equilibrium equations. A surprising result of this work, is that even for concentrated forces (Dirac delta function), we obtain a very good representation of the beam’s deformation with only few additional degrees of freedom.