Vlasov's Beam Theory Research Articles

Bimetallic Thin-Walled Box Beam Thermal Buckling Response.

A beam model for thermal buckling analysis of a bimetallic box beam is presented. The Euler-Bernoulli-Vlasov beam theory is employed considering large rotations but small strains. The nonlinear stability analysis is performed using an updated Lagrangian formulation. In order to account for the thermal effects of temperature-dependent (TD) and temperature-independent (TID) materials, a uniform temperature rise through beam wall thickness is considered. The numerical results for thin-walled box beams are presented to investigate the effects of different boundary conditions, beam lengths and material thickness ratios on the critical buckling temperature and post-buckling responses. The effectiveness and accuracy of the proposed model are verified by means of comparison with a shell model. It is revealed that all of the abovementioned effects are invaluable for buckling analysis of thin-walled beams under thermal load. Moreover, it is shown that the TD solutions give lower values than the TID one, emphasizing the importance of TD materials in beams.

Static analysis of thin-walled space frame structures with arbitrary closed cross-sections using transfer matrix method

This paper develops a new transfer matrix method for the static analysis of thin-walled space frame structures with arbitrary closed single-cell sections, that is, thin-walled beam transfer matrix method (TBTMM), which expands the advantages of the general transfer matrix method (TMM). In this method, the bimoment B and the rate of twist of the beam φ(x) are taken into account based on the Vlasov's beam theory and Kollbrunner-Hajdin assumption, and the bending and twisting can be calculated separately combined with the characteristic of Maillart's shear center. Using this method, the transfer matrix of single beam is obtained through Laplace transform and inverse Laplace transform at first, and then gets the coupling equations of the whole structure with respect to coordinate transformation matrices and joint coupling matrices. Some numerical examples are given and compared with the general TMM and the finite element method (B-FEM and S-FEM) results to validate the developed method.

Flexural-torsional vibration of a tapered C-section beam

Previous studies have shown that numerical models of tapered thin-walled C-section beams based on a stepped or piecewise prismatic beam approximation are inaccurate regardless of the number of elements assumed in the discretization. Andrade recently addressed this problem by extending Vlasov beam theory to a tapered geometry resulting in new terms that vanish for the uniform beam. (See One-Dimensional Models for the Spatial Behaviour of Tapered Thin-Walled Bars with Open Cross-Sections: Static, Dynamic and Buckling Analyses, PhD Thesis, University of Coimbra, Portugal, 2012, https://estudogeral.sib.uc.pt) In this paper, we model the coupled bending-twisting vibration of a cantilevered tapered thin-walled C-section using a Galerkin approximation of Andrade's beam equations resulting in an 8-degree-of-freedom beam element. Experimental natural frequencies and mode shapes for 3 prismatic and 2 tapered channel beams are compared to model predictions. In addition, comparisons are made to detailed shell finite element models and exact solutions for the uniform beams to confirm the validity of the approach. Comparisons to the incorrect stepped model are also presented.

Stability for Leipholz’s Type of Laminated Box Columns with Nonsymmetric Lay-Ups on Elastic Foundation

AbstractThe stability behavior of the Leipholz’s type of laminated box columns with nonsymmetric lay-ups resting on elastic foundation is investigated using the finite element method. Based on the kinematic assumptions consistent with the Vlasov beam theory, a formal engineering approach of the mechanics of the laminated box columns with symmetric and nonsymmetric lay-ups is presented. The extended Hamilton’s principle is employed to obtain the elastic stiffness and mass matrices, the Rayleigh damping and elastic foundation matrices, the geometric stiffness matrix due to distributed axial force, and the load correction stiffness matrix accounting for the uniformly distributed nonconservative forces. The evaluation procedures for the critical values of divergence and flutter loads with/without internal and external damping effects are briefly presented. Numerical examples are carried out to validate the present theory with respect to the previously published results. Especially, the influences of the fiber angle change and damping on the divergence and flutter loads of the laminated box columns are parametrically investigated.

Numerical analysis on springing and whipping using fully-coupled FSI models

This paper focuses on ship springing and whipping analysis using a three-dimensional (3-D) Rankine panel method combined with a beam-element-based 1-D structural model and a shell-element-based 3-D structural model. In addition, slamming loads are considered by 2-D generalized Wagner model (GWM). The beam model is a classical idealization of a ship structure, which is based on Timoshenko beam theory for bending and Vlasov beam theory for non-uniform torsion. The 3-D model consists of beam and shell elements, and its motions are approximated by a few of lower modes. Different coupling schemes are applied to couple the 3-D panel method and the different structure models. Whereas the 1-D beam model is coupled in a Cartesian coordinate system, the 3-D finite element (FE) model is coupled in a generalized coordinate system. The difference in coordinate systems leads to different numerical implementations of coupling. Agreement and discrepancy between the coupled models are discussed regarding results for a 60m barge, a 6500 TEU containership, and an experimental model of a virtual 10,000 TEU containership. The bulkheads in the barge and the 6500 TEU containership are properly considered in beam modeling according to relative stiffness between the bulkheads and hulls. In linear responses to waves, good agreement is obtained between all the models. However, differences between the models are found in nonlinear springing and whipping responses.

On the relationship of the shear deformable Generalized Beam Theory with classical and non-classical theories

The possibility to establish clear relationships between the results of the Generalized Beam Theory (GBT) and those of the classical beam theories is a crucial issue for a correct theoretical positioning of the GBT within the other existing beam theories as well as for the application of the GBT in the current engineering practice. With this in mind, the recovery of classical and non-classical beam theories within the framework of the GBT is presented in this paper. To this purpose, a new formulation of the GBT with shear deformation is conceived. Particularly, the formulation recently proposed by the authors is here modified by introducing new definitions of the kinematic parameters and of the generalized deformations, and extended to the dynamic case. Firstly, it is shown that a suitable choice of the flexural deformation modes allows recovering the Vlasov beam theory, both with and without shear deformation. Also, the analytical solution of the non-uniform torsion problem with shear deformation is given. Then, the recovery of the Capurso beam theory using the nonlinear warping deformation modes is illustrated.

Divergence and flutter behavior of Beck׳s type of laminated box beams

Abstract The divergence and flutter behavior of Beck׳s type of laminated box beams are investigated based on the finite element model. Mechanics of laminated box beam with non-symmetric lay-up is presented based on kinematic assumptions consistent with Vlasov beam theory. To obtain mass-, elastic stiffness-, geometric stiffness-, damping-, and load correction matrices, the extended Hamilton׳s principle is employed. Evaluation procedures for critical values of divergence and flutter with and without damping effects are briefly introduced. Through numerical examples, influences of fiber angle change, non-conservative parameter, external (viscous fluid) and internal (material) damping on divergence and flutter loads of laminated box beams subjected to a subtangential force are newly presented.

Automatic analysis of thin-walled laminated composite sections

In this paper a computer program is developed for the determination of geometrical and material properties of composite thin-walled beams with arbitrary open cross-section and any arbitrary laminate stacking sequence. Theory of thin-walled composite beams is based on assumptions consistent with the Vlasov's beam theory and classical lamination theory. The program is written in Fortran 77. Some numerical examples are given, with complete information about input and output.

A Rankine–Timonshenko–Vlasov beam theory for anisotropic beams via an asymptotic strain energy transformation

Abstract A refined shear deformable beam theory is developed to analyze and design anisotropic beams via the mixed variational theorem (MVT). The developed theory is referred to as a Rankine–Timoshenko–Vlasov (RTV) beam theory since it is able to account for the torsional warping restraint as well as the shear deformation. The MVT is employed to systematically blend the asymptotic stress field and the intuitive displacement field, so that one can derive a simple yet accurate beam model. The asymptotic stress field is adopted from the finite element-based asymptotic expansion method, whereas the displacement field is intuitively assumed to have the form of a traditional shear deformable and torsional warping restrained beam model. The two fields are synthesized by a strain energy transformation via the MVT, which yields the RTV beam model whose degrees of freedom are the same as the traditional refined beam theories. The resulting generalized beam stiffness matrix is different from those traditional theories qualitatively as well as quantitatively. This leads to the accurate prediction of torsional rigidity, bimoment stiffness, transverse shear stiffness, and composite coupling stiffness. The thin-walled composite beams with closed and open cross-sections are taken as illustrative examples to demonstrate the accuracy and validity of the developed RTV theory.

A distortional semi-discretized thin-walled beam element

Due to the increased consumption of thin-walled structural elements there has been increasing focus and need for more detailed calculations as well as development of new approaches. In this paper a thin-walled beam element including distortion of the cross section is formulated. The formulation is based on a generalized beam theory (GBT), in which the classic Vlasov beam theory for analysis of open and closed thin-walled cross sections is generalized by including distortional displacements. The beam element formulation utilizes a semi-discretization approach in which the cross section is discretized into wall elements and the analytical solutions of the related GBT beam equations are used as displacement functions in the axial direction. Thus the beam element contains the semi-analytical solutions. In three related papers the authors have recently presented the semi-discretization approach and the analytical solution of the beam equations of GBT. In this approach a full set of deformation modes corresponding to the homogeneous GBT equations are found. The deformation modes of which some are complex decouple the state space equations corresponding to the reduced order differential equations of GBT and allow the determination of the analytical solutions. Solutions of the non-homogeneous GBT differential equations and the distortional buckling equations have also been found and analyzed. Thus, these related papers are not dealing with an element but dealing with analytical solutions to the coupled differential equations.To handle arbitrary boundary conditions as well as the possibility of adding concentrated loads as nodal loads the formulation of a beam element is needed. This paper presents the formulation of such a generalized one-dimensional semi-discretized thin-walled beam element including distortional contributions. It should be noticed that we are only dealing with a basic generalized beam theory and not an extended finite element formulation of an approximate beam element, which allows the addition of special (transverse extension and shear lag) modes. Illustrative examples showing the validity and the accuracy of the developed distortional semi-discretized thin-walled beam element are given and it is shown how the novel approach provides accurate results making it a good alternative to the traditional and time consuming FE calculations.

Dynamic stability behavior of damped laminated beam subjected to uniformly distributed subtangential forces

Abstract The dynamic stability behavior of damped laminated beam with various boundary conditions subjected to the uniformly distributed subtangential forces is investigated using the finite element formulation. The formal engineering approach of the mechanics for the thin-walled laminated beam based on kinematic assumptions consistent with Vlasov beam theory is used. An extended Hamilton’s principle is employed to obtain the mass-, damping-, elastic stiffness-, geometric stiffness matrices, and the load correction stiffness matrix due to the subtangential forces, respectively. The method for the evaluation of critical values for divergence and flutter of the nonconservative systems is briefly introduced in case of considering and neglecting damping effects. Throughout numerical examples, the influence of various parameters on the dynamic stability behavior of the nonconservative laminated beam is newly investigated: (1) the variation of the divergence and flutter loads due to the nonconservativeness with respect to the fiber orientation, (2) the effect of boundary condition on the instability region of the divergence-flutter system, and (3) the influence of external and internal damping on the flutter load.

DYNAMIC STIFFNESS ANALYSIS FOR TORSIONAL VIBRATION OF CONTINUOUS BEAMS WITH THIN-WALLED CROSS-SECTION

An analytical method for determining natural frequencies and mode shapes of the torsional vibration of continuous beams with thin-walled cross-section is developed by using a general solution of the differential equation of motion based on Vlasov's beam theory. This method takes into account the effect of warping stiffness; it leads to an exact solution and is called the continuous mass method. Also, the approximate method based on the finite discrete element approach is presented. The mathematical relationship between the exact and the approximate methods is discussed, and the accuracy of the natural frequencies obtained by these analytical methods is investigated. Some typical continuous beams are analyzed to illustrate the applicability of the lumped, consistent, and continuous mass methods, and the computed results are given in tabular form.

A Vlasov beam element

A finite element for the analysis of thin-walled open section beam structures is presented. The element is based on Vlasov's beam theory. In linear analyses the element is very effective since it does not require the use of numerical integration and only one element can exactly model simple cases of nonuniform torsion.