A non-linear model for the dynamics of open cross-section thin-walled beams—Part I: formulation (original) (raw)
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Journal of Sound and Vibration, 2015
An open-cross section thin-walled beam model, already developed by the authors, has been conveniently simplified while maintaining the capacity of accounting for the significant nonlinear warping effects. For a technical range of geometrical and mechanical characteristics of the beam, the response is characterized by the torsional curvature prevailing over the flexural ones. A Galerkin discretization is performed by using a suitable expansion of displacements based on shape functions. The attention is focused on the dynamic response of the beam to a harmonic force, applied at the free end of the cantilever beam. The excitation is directed along the symmetry axis of the beam section. The stability of the one-component oscillations has been investigated using the analytical model, showing the importance of the internal resonances due to the nonlinear warping coupling terms. Comparison with the results provided by a computational finite element model has been performed. The good agreement among the results of the analytical and the computational models confirms the effectiveness of the simplified model of a nonlinear open-cross section thin-walled beam and overall the important role of the warping and of the torsional elongation in the study of the one-component dynamic oscillations and their stability.
2019
This paper focuses on the dynamic response of thin-walled structural elements. A mixed three-dimensional (3D) beam formulation is adopted, that includes the effect of inertia forces under dynamic loading conditions and accounts for out-of-plane cross-section warping. This is introduced by adding a specific displacement field to those due to rigid body motions, and is interpolated in the element volume with the definition of specific shape functions. The element governing equations are derived by expressing the Lagrangian functional in terms of four independent fields, i.e. the material rigid displacements, the strains and stresses and the additional warping displacement field. Four Lagrange's equations of motion result, corresponding to the element compatibility condition enforced in weak form, the material constitutive law, and two sets of element equilibrium conditions associated to the rigid and warping displacements, respectively. The FE model has been implemented in a standard numerical code and used to investigate the effect of cross-section warping on the dynamic response of thin-walled structures. A T-shape beam is analyzed by performing modal decomposition and time-history analyses under linear elastic and nonlinear constitutive behavior. 3611 COMPDYN 2019 7 th ECCOMAS Thematic Conference on Computational Methods in Structural Dynamics and Earthquake Engineering M. Papadrakakis, M. Fragiadakis (eds.
International Journal of Solids and Structures, 2007
This two-part contribution presents a beam theory (BT) with a non-uniform warping (NUW) including the effects of torsion, and shear forces and valid for any homogeneous cross-section made of isotropic elastic material. In part I, the governing equations of the NUW-BT has been established and simplified-NUW-BT versions has been deduced, wherein the number of degrees of freedom is reduced. In this part II, these theories are used to analyze, for a representative set of cross-sections (CS) (solid-CS and thin-walled open/closed-CS, bi-symmetric or not), the elastic behavior of cantilever beams subjected to torsion or shear-bending. For bi-symmetrical-CS, torsion and shear-bending are analyzed separately: analytical and numerical results are given for the distributions along the beam axis of the cross-sectional displacements and stresses, for the NUW-BT and its simplified versions. Numerical results are also given for the three-dimensional stress distributions close to the embedded section: the stress predictions of the NUW-BT are compared to those obtained by threedimensional finite elements computations. It can be drawn from all these results indications that can help to decide when the simplified theories may be applied, and hence when the warping parameters may be reduced. As specified in NUW-BT, torsion and bending are coupled for non-symmetrical-CS, even if the bending moments refer to the centroid while the torsional moment refers to the shear center. To illustrate this coupling effect, the particular example of the channel-CS presented in Kim and Kim [Kim, N.-I., Kim, M.-Y., 2005. Exact dynamic/static stiffness matrices of non-symmetric thin-walled beams considering coupled shear deformation effects. Thin-Walled Structures 43, 701-734.] is analyzed and the results are compared.
OUT-OF-PLANE VIBRATIONS OF SHEAR DEFORMABLE CONTINUOUS HORIZONTALLY CURVED THIN-WALLED BEAMS
Journal of Sound and Vibration, 2000
This paper presents numerical study about the in#uence of the shear #exibility, due to either bending and warping, on the out-of-plane free vibration of continuous horizontally curved thin-walled beams with both open and closed sections. This study was made by means of a recently developed "nite element formulation for shear deformable curved thin-walled beams. The model is brie#y reviewed and is used to obtain natural frequencies of continuous I-beams and box beams. A parametric study was performed in order to elucidate the in#uence of shear deformability, on the dynamic behavior of continuous thin-walled curved beams, for di!erent slenderness ratios, cross-sectional characteristics and boundary conditions.
The Torsion Effects on the Non-Linear Behaviour of Thin-Walled Beams: A Finite Element Approach
Proceedings of the Eleventh International Conference on Computational Structures Technology, 2012
The authors have developed a beam finite element model for thin walled beams with arbitrary cross sections in the large torsion context . Circular functions of the torsion angle θ x (c=cosθ x -1 and s=sinθ x ) were included as variables. In this paper three other three-dimensional finite element beams are derived according to the three approximations of the circular functions c and s: cubic, quadratic and linear. A finite element approach of these approximations is carried out. Many comparison examples are considered. They concern non linear behaviour of beams under twist moment and post buckling behaviour of beams under axial loads or bending loads.
Non-uniform warping including the effects of torsion and shear forces. Part I: A general beam theory
International Journal of Solids and Structures, 2007
This two-part contribution presents a beam theory with a non-uniform warping including the effects of torsion and shear forces, and valid for any homogeneous cross-section made of isotropic elastic material. Part I is devoted to the theoretical developments and part II discusses analytical and numerical results obtained for torsion and shear-bending of cantilever beams made of different kinds of cross-section. The theory is based on a kinematics assuming that the cross-section maintains its shape and including three independent warping parameters associated to the three warping functions corresponding to torsion and shear forces. Starting from this displacement model and using the principle of virtual work, the corresponding beam theory is derived. For this theory, closed-form results are obtained for the cross-sectional constants and the three-dimensional expressions of the normal and shear stresses. Comparison with classical beam theories is carried out and additional effects due to the non-uniformity of the warping are highlighted. In particular, the contributions of primary and secondary internal forces and the effect of the non-symmetry of the cross-section on the structural behavior of the beam are specified. Simplified versions of this theory, wherein the number of degrees of freedom is reduced, are also presented. The analytical and numerical analyzes presented in part II give responses on the quality of this non-uniform beam theory and indicate also when its simplified versions could be applied.
Advances in Engineering Software, 2015
The authors have developed a beam finite element model in large torsion context for thin-walled beams with arbitrary cross sections [1]. In the model, the trigonometric functions of the twist angle h x (c = cos h x À 1 and s = sin h x ) were included as additional variables in the whole model without any assumption. In the present paper, three other 3D finite element beams are derived according to three approximations based on truncated Taylor expansions of the functions c and s (cubic, quadratic and linear). A finite element approach of these approximations is carried out. Finally, it is worth mentioning that the promising results obtained in [1,2] encourage the authors to extend the formulation of the model in order to include load eccentricity effects. Solution of the non-linear equations is made possible by Asymptotic Numerical Method (ANM) . This method is used as an alternative to the classical incremental iterative methods.
On free vibration analysis of thin-walled beams with nonsymmetrical open cross-sections
Computers & Structures, 2002
This work relates to the analysis of triply coupled vibrations of thin-walled beams having nonsymmetrical open cross-sections. The governing differential equations for coupled bending and torsional vibrations are derived and solved exactly. A recent study on the same subject is criticized and discussed in theoretical and numerical aspects.