A new finite element of spatial thin-walled beams (original) (raw)
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A beam finite element for non-linear analyses of thin-walled elements
Thin-Walled Structures, 2008
The aim of the present paper is to investigate a theoretical and numerical model which is able to study the behaviour of thin-walled beams with open cross section in presence of large torsion. The presented model takes into account for large torsion, linear and nonlinear warping currently named shortening effects, pre-buckling deformation and flexural-torsional coupling. In numerical analysis, a 3D beam with two nodes and seven degrees of freedom per node is adopted. The equilibrium equations and the material behaviour are derived in discrete form without assumption on torsion angle amplitude. Due to large torsion context, all the equilibrium equations are non-linear and highly coupled. The linear behaviour is made possible by disregarding non-linear terms. For non-linear behaviour and stability, the tangent stiffness matrix is carried out. Due to large torsion context, new matrices are present. The element is incorporated in a homemade finite element code. Newton-Raphson iterative methods are used with different control parameters. In order to prove the efficiency of the model many examples are presented in linear and non-linear behaviour with presence of bifurcations. r
A rational elasto-plastic spatially curved thin-walled beam element
International Journal for Numerical Methods in Engineering, 2007
Torsion is one of the primary actions in members curved in space, and so an accurate spatially curved-beam element needs to be able to predict the elasto-plastic torsional behaviour of such members correctly. However, there are two major difficulties in most existing finite thin-walled beam elements, such as in ABAQUS and ANSYS, which may lead to incorrect predictions of the elasto-plastic behaviour of members curved in space. Firstly, the integration sample point scheme cannot capture the shear strain and stress information resulting from uniform torsion. Secondly, the higher-order twists are ignored which leads to loss of the significant effects of Wagner moments on the large twist torsional behaviour. In addition, the initial geometric imperfections and residual stresses are significant for the elasto-plastic behaviour of members curved in space. Many existing finite thin-walled beam element models do not provide facilities to deal with initial geometric imperfections. Although ABAQUS and ANSYS have facilities for the input of residual stresses as initial stresses, they cannot describe the complicated distribution patterns of residual stresses in thin-walled members. Furthermore, external loads and elastic restraints may be applied remote from shear centres or centroids. The effects of the load (and restraint) positions are important, but are not considered in many beam elements. This paper presents an elasto-plastic spatially curved element with arbitrary thin-walled cross-sections that can correctly capture the uniform shear strain and stress information for integration, and includes initial geometric imperfections, residual stresses and the effects of the load and restraint positions. The element also includes elastic restraints and supports, which have to be modelled separately as spring elements in some other finite thin-walled beam elements. Comparisons with existing experimental and analytical results show that the elasto-plastic spatially curved-beam element is accurate and efficient. KEY WORDS: arbitrary thin-walled cross-section; curved-beam element; elastic restraints and supports; elasto-plastic; initial geometric imperfections; effects of load and restraint positions; residual stresses; Wagner effects 1. INTRODUCTION This paper develops a rational elasto-plastic spatially curved-beam element with arbitrary thinwalled cross-sections based on the elastic spatially curved-beam element derived by the authors [1]. The effects of initial geometric imperfections, residual stresses, elastic restraints and supports, and of load and restraint positions are incorporated in the element. Members curved in space are subjected to torsional, bending and compressive actions under loading [1, 2] . Hence, to predict the elasto-plastic behaviour of such members, a finite element (FE) needs to be able to deal with elasto-plastic torsional, bending and compressive responses accurately. However, most existing beam elements for three-dimensional analysis, such as the beam elements in space of the commonly used FE software packages ABAQUS [3] and ANSYS [4] cannot predict elasto-plastic torsional responses correctly, particularly when the twist rotations are large, because of two difficulties that have not been treated properly in the thin-walled beam elements in space in these FE packages .
Thin-walled Structures, 2021
This paper presents three beam Finite Element (FE) formulations developed for the analysis of thin-walled structures. These account for out-of-plane cross-section warping by removing the classical rigid body cross-section hypothesis and capture the interaction of axial/bending stress components with shear and torsion. The beam FE models rely on different kinematic assumptions to describe out-of-plane cross-section deformations. Indeed, warping displacement field is interpolated in the element volume according to different approaches, with increasing level of accuracy and detail. First two models adopt a coarse warping description, where warping displacement field is defined as the linear combination of assumed warping profiles and unknown kinematic parameters. In the first model, these are considered as equal to the generalized cross-section torsional curvature and shear strains and a classical displacement-based formulation is adopted to derive the element governing equations. In the second model, warping parameters are assumed as independent kinematic quantities and a mixed approach is considered to derive the FE formulation. Third model, also relying on a mixed formulation, independently interpolates warping introducing additional degrees of freedom on the cross-section plane, thus, resulting in a richer description of the outof-plane deformations. This latter is also adopted to propose a numerical procedure for the warping profile evaluation of thin-walled beams subjected to torsional and shear forces, for general cross-section geometry. The efficiency and accuracy of the proposed FE formulations are validated by simulating the response of thinwalled structures under torsion and coupled torsion/shear actions and the influence of the kinematic assumptions characterizing each formulation is discussed.
On the analysis of thin walled members in the framework of the Generalized Beam Theory
The analysis of thin walled members in the framework of the Generalized Beam Theory (GBT) is critically revised. Firstly, the classic form of the GBT is briefly explained along with its main advantages and problems. Subsequently, a new formulation that coherently accounts for shear deformation is presented along with a unique modal decomposition, the Cross Section analysis, that allows to recover classical shear deformable beam theories exactly. Furthermore, a stress recovery procedure for the finite element analysis of GBT beams is proposed as an improvement on the traditional elasto-kinematic approach. Performance is shown in numerical tests.
Effective Stiffness of Thin-Walled Beams with Local Imperfections
Materials
Thin-walled beams are increasingly used in light engineering structures. They are economical, easy to manufacture and to install, and their load capacity-to-weight ratio is very favorable. However, their walls are prone to local buckling, which leads to a reduction of compressive, as well as flexural and torsional, stiffness. Such imperfections can be included in such components in various ways, e.g., by reducing the cross-sectional area. This article presents a method based on the numerical homogenization of a thin-walled beam model that includes geometric imperfections. The homogenization procedure uses a numerical 3D model of a selected piece of a thin-walled beam section, the so-called representative volume element (RVE). Although the model is based on the finite element method (FEM), no formal analysis is performed. The FE model is only used to build the full stiffness matrix of the model with geometric imperfections. The stiffness matrix is then condensed to the outer nodes of...
AC0 finite element formulation for thin-walled beams
International Journal for Numerical Methods in Engineering, 1989
Timoshenko's and Vlasov's beam theories are combined to produce a Co finite element formulation for arbitrary cross section thin-walled beams. Section properties are generated using a curvilinear coordinate system to describe the cross section dimensions. The element includes both shear and warping deformations caused by the bending moments and the bimoment. A Gauss quadrature order is employed which exactly integrates the bending and warping stiffness matrices and provides a reduced integration order for the shear stiffness matrices. Numerical results are presented for a channel section cantilever beam. The influence of shear deformation is investigated and the calculated results are shown to be in excellent agreement with the classical solutions.
Research and Applications in Structural Engineering, Mechanics and Computation, 2013
A finite element formulation is developed for the coupled flexural-torsional analysis of thinwalled open members under general forces. Based on a generalized Timoshenko-Vlasov thin-walled beam theory, and the principle of minimum total potential energy, a two-node finite element is developed with four degrees of freedom per node. The element fully captures the effects of warping stiffness, shear deformation, and the torsional-flexural coupling and is based on shape functions which exactly satisfy the homogeneous form of the equilibrium conditions. The element is demonstrated to be free from shear locking and discretization errors occurring in conventional finite element solutions. The applicability of the finite beam element is illustrated through examples. The results based on the present formulation are compared against established analytical and finite element solutions and demonstrate the accuracy and efficiency of the solution.
Journal of Structural Engineering & Applied Mechanics, 2021
Starting with total potential energy variational principle, the governing equilibrium coupled equations for the torsional-warping static analysis of open thin-walled beams under various torsional and warping moments are derived. The formulation captures shear deformation effects due to warping. The exact closed form solutions for torsional rotation and warping deformation functions are then developed for the coupled system of two equations. The exact solutions are subsequently used to develop a family of shape functions which exactly satisfy the homogeneous form of the governing coupled equations. A super-convergent finite beam element is then formulated based on the exact shape functions. Key features of the beam element developed include its ability to (a) eliminate spatial discretization arising in commonly used finite elements, and (e) eliminate the need for time discretization. The results based on the present finite element solution are found to be in excellent agreement with ...
General Stiffness Matrix for a Thin-Walled, Open-Section Beam Structure
World Journal of Mechanics, 2021
This paper is to review the theory of thin-walled beam structures of the open cross-section. There is scant information on the performance of structures made from thin-walled beam elements, particularly those of open sections, where the behavior is considerably complicated by the coupling of tensile, bending and torsional loading modes. In the combined loading theory of thin-walled structures, it is useful to mention that for a thin-walled beam, the value of direct stress at a point on the cross-section depends on its position, the geometrical properties of the cross-section and the applied loading. This applies whether the thin-walled section is closed or open but this study will be directed primarily at the latter. Theoretical analyses of structures are fairly well established, considered in multi-various applications by many scientists. However, due to the present interest in lightweight structures, it is necessary to specify where the present theory lies. It does not, for example, deal with compression and the consequent failure modes under global and local buckling. Indeed, with the inclusion of strut buckling failure and any other unforeseen collapse modes, the need was perceived for further research into the subject. Presently, a survey of the published works has shown in the following: 1) The assumptions used in deriving the underlying theory of thin-walled beams are not clearly stated or easily understood; 2) The transformations of a load system from arbitrary axis to those at the relevant centre of rotation are incomplete. Thus, an incorrect stress distribution may result in; 3) Several methods are found in the recent literature for analyzing the behaviour of thin-walled open section beams under combined loading. These reveal the need appears for further study upon their torsion/flexural behaviour when referred to any arbitrary axis, a common case found in practice. This review covers the following areas: 1) Refinement to existing theory to clarify those observations made in 1-3 above; 2) Derivation of a general elastic stiffness matrix for combined loading; 3