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Higher Order Beam Theory in Linear Analysis of Beams - Axial Modes of Arbitrary Cross Sections
Abstract and Figures
Both Euler-Bernoulli and Timoshenko beam theories maintain the assumptions that neither out-of-plane (warping) nor in-plane (distortion) deformation contribute to beams response. To account for shear lag effects, the inclusion of non-uniform warping is necessary, relaxing the assumption of plane cross section. The shear flow associated with non-uniform warping leads also to in-plane deformation of the cross-section, relaxing the no-distortion assumption. For this purpose, the so-called higher order beam theories have been developed taking into account shear lag and distortional effects. In this paper, the higher order beam theory developed in a previous work of the authors is employed for linear analysis of beams of arbitrarily shaped, homogeneous cross-section, including warping and distortional phenomena due to axial, shear, flexural, and torsional behavior. The beam is subjected to general load and boundary conditions. The analysis consists of two stages. The first stage is a cross-sectional analysis, establishing the possible distortional and warping deformation patterns (axial, flexural and torsional modes by means of the sequential equilibrium scheme and the Boundary Element Method). The second stage is a longitudinal analysis where the four rigid body displacements along with the extracted deformation patterns multiplied by respective independent parameters expressing their contribution to the beam deformation are included in the beam analysis (Finite Element Method). Axial warping and distortional modes are examined. In addition, numerical examples with practical interest are presented in order to highlight the importance of axial modes in linear analysis of beams-cross-sections of class 4 (EN1993-1-1), i.e., vulnerable to axial load are analyzed.
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The computer program VABS (Variational Asymptotic Beam Section Analysis) uses the vari- ational asymptotic method to split a three-dimensional nonlinear elasticity problem into a two- dimensional linear cross-sectional analysis and a one-dimensional, nonlinear beam problem. This is accomplished by taking advantage of certain small parameters inherent to beam-like structures. VABS is able to calculate the one-dimensional cross-sectional stiffness constants, with transverse shear and Vlasov refinements, for initially twisted and curved beams with arbitrary geometry and material properties. Several validation cases are presented. First, an elliptic bar is modeled with transverse shear refinement using the variational asymptotic method, and the solution is shown to be identical to that obtained from the theory of elasticity. The shear center locations calculated by VABS for various cross sections agree well with those obtained from common engineering analyses. Comparisons with other composite beam theories prove that it is unnecessary to introduce ad hoc kinematic assumptions to build an accurate beam model. For numerical validation, values of the one-dimensional variables are extracted from an ABAQUS model and compared with results from a one-dimensional beam analysis using cross-sectional constants from VABS. Furthermore, point-wise three-dimensional stress and strain fields are recovered using VABS, and the correlation with the three-dimensional results from ABAQUS is excellent. Finally, classical theory is shown to be insuf- ficient for general-purpose beam modeling. Appropriate refined theories are recommended for some classes of problems.
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.
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.
In this paper, the influence of the variable axial force and of the Secondary Torsion-Moment Deformation-Effect (STMDE) on the deformations of beams due to torsional warping is investigated. The investigation is based on the second-order torsional warping theory of doubly symmetric beams with thin-walled open or closed cross-sections. The effect of the axial force on the torsional stiffness of thin-walled beams is considered according to the second-order torsional warping theory. The solutions of the underlying differential equations are used for setting up the relations, needed for application of the transfer matrix method. They are derived, considering both static and dynamic action. This enables stablishing the local element matrix of the twisted beam in the framework of the Finite Element Method (FEM). The numerical investigation comprises static and modal analyses of thin-walled beams with I cross-sections and rectangular hollow cross-sections. The results are compared with results obtained by the FEM, using solid and beam elements available in standard software.
For thin-walled sections, lateral and lateral-torsional buckling are often affected by distortion of the section and this can severely reduce the critical stress. A calculation method for the critical stress which takes account of distortional effects, based on Generalised Beam Theory (GBT), is presented in this paper. The limiting slenderness at which distortional effects begin to take effect is evaluated for channel- and hat-sections on the basis of parametric studies. Approximate formulae for the limiting slenderness are then given.
First-order generalized beam theory describes the behaviour of prismatic structures by ordinary uncoupled differential equations, using deformation functions for bending, torsion and distortion. In second-order theory, the differential equations are coupled by the effect of deviating forces. The basic equations for second-order generalized beam theory are outlined. Solutions for pin-ended supports are presented, demonstrating the coupling effect by modes and by loads. In the different ranges of length, the individual modes are sufficient approximations for the critical load. The application to a thin-walled bar with C-section under eccentric normal force demonstrates the quality of the single-mode compared to the exact solution.
Analysis of shear lag in box beams by the principle of minimum potential energy
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