Vlasov's Assumption Research Articles

Effect of a “Slight” Curvature of the Axis of a Thin-Walled Nonprismatic Beam on its Free Vibration

This paper presents an analysis of the effect of a “slight” curvature of the axis of a thin-walled beam on its eigenfrequencies and forms. By “slight” we understand such a beam curvature at which no theory of curved beams (arches, horizontally curved girders), but classical beam theory is applied. The subject of the analysis is the open nonprismatic thin-walled beams with any geometrical parameters, and prismatic beams. The thin-walled beam model used in the analysis was derived on the basis of the momentless theory of plates, using the Vlasov assumptions. The free vibrations of the beams were analysed. Calculations were performed for four groups of beams. Each of the groups comprised two beams with the same configuration of flanges and identical geometrical cross section characteristics, but differing in web geometry. One of the beams had a symmetrical web (a beam with a rectilinear axis in the web plane), whereas the web of the other beam was asymmetrical (a beam with its axis curved in the web plane). The algorithm based on Chebyshev polynomials was used to solve the variable coefficient equations describing the model’s vibration. According to this algorithm, solutions are sought in the form of Chebyshev series and the closed analytical recurrence formulas generated by the algorithm are used to calculate the coefficients of the equations. It has been shown that even a “slight” curvature of the beam’s axis has a significant effect on its eigenfrequencies and forms, as confirmed by a comparison of the results with the ones obtained using Finite Element Method (FEM).

Elastic buckling of stiffened panels in ships under bi-axial compression

ABSTRACTThis paper focuses on the elastic buckling strength of typical stiffened panels under bi-axial compression. Three levels of assumptions for stiffeners are investigated: (1) Vlasov assumption; (2) stiffeners’ web plate assumed deformable; (3) both stiffeners’ web and flange plates assumed deformable. The bi-axial buckling strengths for deck and bottom stiffened panels in typical merchant ships are obtained by minimum potential energy principle, and the results are compared with corresponding finite element analysis results. It shows that Vlasov assumption should be corrected, while the other two assumptions yield similar satisfactory results. Sensitivity analysis is made for scantlings of the stiffened panel. Above three assumptions all exhibit high efficiency.

Time domain springing analysis on a floating barge under oblique wave

This paper studies the springing response of a flexible floating barge under an oblique wave. A time domain Rankine panel method was used to represent fluid motion surrounding a flexible seagoing vessel, and a finite element method was used for structural response. For accurate prediction of the structural response under an oblique wave, special attention was given to the structural model, such as the effect of warping distortion and bending-torsion coupling. The Vlasov assumption was followed for a deformable beam element to take into account the effect of warping distortion so that the cross section of the beam deforms out of its original plane without changing its cross-section contour. The coupled equation for both the fluid and structural domain was solved by using the implicit iterative method. A fixed point iteration with a relaxation scheme was employed in this study with the aid of the Aitken δ2 process seeking acceleration of solution convergence. Accuracy of a developed computer program was verified through the comparison with experimental work done by Remy et al. (Experimental and numerical study of the wave response of a flexible barge, Hydroelasticity in Marine Technology, Wuxi, 2006) resulting in good correspondence between the two results.

Dynamic stiffness matrix of thin-walled composite I-beam with symmetric and arbitrary laminations

For the spatially coupled free vibration analysis of thin-walled composite I-beam with symmetric and arbitrary laminations, the exact dynamic stiffness matrix based on the solution of the simultaneous ordinary differential equations is presented. For this, a general theory for the vibration analysis of composite beam with arbitrary lamination including the restrained warping torsion is developed by introducing Vlasov's assumption. Next, the equations of motion and force–displacement relationships are derived from the energy principle and the first order of transformed simultaneous differential equations are constructed by using the displacement state vector consisting of 14 displacement parameters. Then explicit expressions for displacement parameters are derived and the exact dynamic stiffness matrix is determined using force–displacement relationships. In addition, the finite-element (FE) procedure based on Hermitian interpolation polynomials is developed. To verify the validity and the accuracy of this study, the numerical solutions are presented and compared with analytical solutions, the results from available references and the FE analysis using the thin-walled Hermitian beam elements. Particular emphasis is given in showing the phenomenon of vibrational mode change, the effects of increase of the modulus and the bending–twisting coupling stiffness for beams with various boundary conditions.

Numerical simulation of instability behaviour of thin-walled frames with flexible connections

A one-dimensional finite element formulation for numerical simulation of instability behaviour of thin-walled frames containing flexible connections is presented. Stiffness matrices of a conventional 14-degree of freedom beam element are derived by applying the linearized virtual work principle and Vlasov's assumption. The structural material is assumed to be homogeneous, isotropic and linear-elastic. Flexible connection behaviour and different warping deformation conditions are introduced into the numerical model by modifying stiffness matrices of a conventional beam element. For that purpose a special transformation matrix is derived. The effectiveness of the numerical algorithm discussed is validated through the test problem.

A Series Solution for the Free Vibrations of Thin-Walled Composite Beams

Based on series solutions, the free vibration analysis of thin-walled composite beam with channel sections is performed. For this, the dynamic stiffness matrix is presented using the homogeneous form of the simultaneous ordinary differential equations. The general theory for the vibration analysis of composite beam with arbitrary lamination including the restrained warping torsion is developed by introducing Vlasov's assumption. Next, the equations of motion and force-displacement relationships are derived from the energy principle and explicit expressions for displacement parameters are presented based on power series expansions of displacement components. Then the exact dynamic stiffness matrix is determined using force-displacement relationships. In addition, the finite element model based on Hermitian interpolation polynomial is developed. In order to verify the accuracy and validity of this study, numerical solutions are presented and compared with the analytical solutions from other researchers and the finite element solutions using the Hermitian beam elements. Particularly, the effects of modulus orthotropy and boundary conditions on the vibrational behavior of thin-walled composite beam with various fiber orientations are investigated.

Exact solutions for thin-walled open-section composite beams with arbitrary lamination subjected to torsional moment

The exact solutions for torsional analysis of thin-walled open-section composite beams with arbitrary lamination subjected to torsional moment are presented for the first time. For this, a general thin-walled composite beam theory with arbitrary lamination is developed by introducing Vlasov's assumption and the equilibrium equations and the force–deformation relations are derived from the energy principle. Applying the displacement state vector consisting of 14 displacement parameters and the nodal displacements at both ends of the beam, the displacement functions are derived exactly. Then, the exact stiffness matrix for torsional analysis is determined using the force–deformation relations. As a special case, the closed-form solutions for symmetrically laminated composite beams with various boundary conditions are derived. Finally, the finite element procedure based on Hermitian interpolation polynomial is developed. To demonstrate the validity and the accuracy of this study, the numerical solutions are presented and compared with the closed-form solutions and the finite element results using the Hermitian beam elements and ABAQUS's shell elements.

Exact dynamic stiffness matrix of non-symmetric thin-walled beams on elastic foundation using power series method

Exact dynamic element stiffness matrix for the flexural–torsional free vibration analysis of the shear deformable thin-walled beam with non-symmetric cross-section on two-types of elastic foundation is newly presented using power series method based on the technical computing program Mathematica. For this, the shear deformable beam on elastic foundation theory is developed by introducing Vlasov's assumption and applying Hellinger–Reissner principle. This beam includes the shear deformation effects due to the shear forces and the restrained warping torsion and due to the coupled effects between them, and rotary inertia effects and the flexural–torsional coupling effects due to the non-symmetric cross-sections. And then equations of motion and force–deformation relations are derived from the energy principle and explicit expressions for displacement parameters are derived based on power series expansions of displacement components and the exact dynamic element stiffness matrix is determined using force–deformation relationships. In order to verify the accuracy of this study, the numerical solutions are presented and compared with the analytical solutions and the finite element solutions using the isoparametric beam elements. Particularly the influences of the coupled shear deformation on the vibrational behavior of non-symmetric beam on elastic foundation are investigated.

Exact dynamic/static stiffness matrices of non-symmetric thin-walled beams considering coupled shear deformation effects

A general theory is proposed for the shear deformable thin-walled beam with non-symmetric open/closed cross-sections and its exact dynamic and static element stiffness matrices are evaluated. For this purpose, an improved shear deformable beam theory is developed by introducing Vlasov's assumption and applying Hellinger–Reissner principle. This includes the shear deformations due to the shear forces and the restrained warping torsion and due to the coupled effects between them, rotary inertia effects and the flexural–torsional coupling effects due to the non-symmetric cross-sections. Governing equations and force–deformation relations are derived from the energy principle and a system of linear eigenproblem with non-symmetric matrices is constructed based on 14 displacement parameters. And then explicit expressions for displacement parameters are derived and the exact dynamic and the static stiffness matrices are determined using force–deformation relationships. In order to verify the validity and the accuracy of this study, the numerical solutions are presented and compared with other numerical solutions available in the literature and results using the thin-walled beam element and the shell element. Particularly the influences of the coupled shear deformation on the vibrational and the elastic behavior of non-symmetric beams with various boundary conditions are investigated.

More general expression for the torsional warping of a thin-walled open-section beam

In this study, an analytical formulation for the torsional warping function of a thin-walled open-section beam is built based on the combination of the Vlasov assumption and the Kirchhoff assumption of plate/shell theory, essentially due to Goodier and Gjelsvik. Vlasov, Timoshenko and many authors (follow Vlasov) only consider the contour warping function as the real warping function. Goodier and Gjelsvik (follow Goodier) not only consider the contour warping but also the thickness warping. For some cross-sectional constants related to the warping function such as the torsional constant, the adoption of the warping function based on Vlasov's theory or Timoshenko's theory may cause them incorrectness. On the other hand, the torsional constant based on the Goodier's theory is consistent with the one based on the membrane analogy. Thus, Goodier's theory is a good approximation for the torsional warping of a thin-walled open-section beam. To the authors’ limited knowledge, the analytical expression for the complete torsional warping of a thin-walled open-section beam has not been found in the literature. In this study, a more general expression for the torsional warping of a thin-walled open-section beam is presented. The position formulas to determine the twist center are also given.

VIBRATION ATTENUATION VIA CONSTRAINED LAYER DAMPING AND DASHPOT ON A SPRING–MASS–PLATE SYSTEM

The vibrations and damping characteristics of an annular plate with constrained layer damping (CLD) treatment subject to a traveling spring–mass–damper (SMD) are investigated. The equations of the CLD-treated plate are first derived from the energy principle. These equations are simplified via the Donnell–Mushtari–Vlasov assumptions. The response equations are eventually uncoupled for each mode and are in terms of a single-degree-of-freedom (s.d.o.f.) linear oscillator with hysteretic damping. The receptance method follows to joint the plate and the SMD, and the resulting change of natural frequencies and damping ratios are investigated. Individual effects due to the inertia and the stiffness are illustrated as well. The results shows that the damping ratios resulted from the viscoelastic core are more significant than that from the viscous damper. In addition, there exists a best design on the thickness of the viscoelastic material core to have the maximum damping ratios. The results also show that the attachment of SMD bifurcated the plate's natural frequencies for every mode but n = 0. The bifurcation becomes more obvious with the rotational speed. These results provide useful information for vibration suppression in engineering design.

Vibration of a three-layered viscoelastic sandwich circular plate

The equations of motion for a three-layer sandwich circular plate with viscoelastic core are derived. The Donnell–Mushtari–Vlasov assumptions follow to simplify. For the case of iso-symmetric annular plate, five equations are further reduced to three equations. The assumed-mode method then yields the free and the forced solutions being characterized by six generalized coordinates for each circumferential wave number n. Due to the splitting of transversal and in-plane terms the study of vibration response can be reduced to the consideration of the SDOF linear oscillator with hysteretic damping. Numerical results first illustrated the frequency response function (FRF) for two different visco models and slight difference is observed. The effects of frequencies changes and damping due to viscoelastic material (VEM) core for several core-to-face thickness ratios are studied. The results, via the FRF's, show that a relatively thin visco (VEM) core can reduce vibration effectively, but as the core thickness exceeds it the damping gradually reduces. The embedded VEM core decreases the frequencies as expected. The damping effects upon individual modes are looked into as well. The results show that the n=0 mode is affected the most by VEM core.

Vibration attenuation of a cylindrical shell with constrained layer damping strips treatment

The damping effects of constrained layer damping treatment of strip type on a cylindrical shell are investigated. A thin shell theory in conjunction with the Donnell–Mushtari–Vlasov assumptions is employed in the derivation. With the assumed-mode method, discretized equations of motion in terms of shell’s transverse modal coordinates are derived. The frequency responses and damping effects under various compositions of viscoelastic core (VEM) thickness, constraining layer (CL) thickness, number of strips, and treatment length are discussed. Numerical examples show consistent results with our physical intuition. The results reveal that thicker CL provides better damping, but thicker VEM does not always guarantee better damping when CL exceeds a certain thickness. The treatment length shows significant damping effect at the beginning but it diminishes as the length exceeds forty percent of shell’s length.

Spatial Stability of Nonsymmetric Thin-Walled Curved Beams. I: Analytical Approach

An improved formulation for spatial stability of thin-walled curved beams with nonsymmetric cross sections is presented based on the displacement field considering both constant curvature effects and the second-order terms of finite-semitangential rotations. By introducing Vlasov's assumptions and invoking the inextensibility condition, the total potential energy is derived from the principle of linearized virtual work for a continuum. In this formulation, all displacement parameters and the warping function are defined at the centroid axis so that the coupled terms of bending and torsion are added to the elastic strain energy. Also, the potential energy due to initial stress resultants is consistently derived corresponding to the semitangential rotation and moment. Analytical solutions are newly derived for in-plane and lateral-torsional buckling of monosymmetric thin-walled curved beams subjected to pure bending or uniform compression with simply supported conditions. In a companion paper, finite-element procedures for spatial buckling analysis of thin-walled circular curved beams under arbitrary boundary conditions are developed by using thin-walled straight and curved beam elements with nonsymmetric sections. Numerical examples are presented to demonstrate the accuracy and the practical usefulness of the analytical and numerical solutions.

Spatial Stability of Nonsymmetric Thin-Walled Curved Beams. II: Numerical Approach

An improved formulation for spatial stability of thin-walled curved beams with nonsymmetric cross sections is presented based on the displacement field considering both constant curvature effects and the second-order terms of finite-semitangential rotations. By introducing Vlasov's assumptions and invoking the inextensibility condition, the total potential energy is derived from the principle of linearized virtual work for a continuum. In this formulation, all displacement parameters and the warping function are defined at the centroid axis so that the coupled terms of bending and torsion are added to the elastic strain energy. Also, the potential energy due to initial stress resultants is consistently derived corresponding to the semitangential rotation and moment. Analytical solutions are newly derived for in-plane and lateral-torsional buckling of monosymmetric thin-walled curved beams subjected to pure bending or uniform compression with simply supported conditions. In a companion paper, finite-element procedures for spatial buckling analysis of thin-walled circular curved beams under arbitrary boundary conditions are developed by using thin-walled straight and curved beam elements with nonsymmetric sections. Numerical examples are presented to demonstrate the accuracy and the practical usefulness of the analytical and numerical solutions.

Strength of compact steel beams with partial restraint

Compact steel beams with continuous and complete tension flange restraint are not affected by conventional lateral-torsional buckling, which is based on the Vlasov assumption of the cross-section of the beam remaining rigid when the beam buckles. Because of this, “conventional” theories for the strength of beams governed by overall buckling modes predict that the beam will reach its full plastic moment capacity. Beams with full restraint of the tension flange, however, buckle in a lateral-distortional mode, in which the web must distort in order for the compression flange to displace and twist during buckling. The buckling moment can therefore be less than that required to reach full plastification, significantly so in some cases. The paper addresses this issue by using a special-purpose inelastic finite element method of analysis to study the buckling of beams with continuous and complete tension flange restraint. For beams with low degrees of moment gradient, it is shown that the inelastic critical moment may be substantially less than the full plastic moment. Incorporating the results into the usual rational design methods allowed for in BS5950 and AS4100 is difficult, and a new design proposal that is generally conservative and based on AS4100 is suggested.

Vibrations of a cylindrical shell with partially constrained layer damping (CLD) treatment

This paper presents a mathematical model for a cylindrical shell with a partially constrained layer damping (CLD) treatment. A thin shell theory in conjunction with the Donnell–Mushtari–Vlasov assumptions is employed to yield the model. Employing the assumed-mode method, the discretized equations of motion in terms of shell’s transverse modal coordinates are derived. The effects of treatment length, of constraining layer (CL) thickness and stiffness, and of viscoelastic material core (VEM) thickness are then discussed. Numerical results show that thicker or stiffer CL warrants better damping. Thicker VEM does not always give better damping than thinner ones when CL exceeds a certain thickness.

SPATIAL STABILITY ANALYSIS OF THIN-WALLED SPACE FRAMES

A clearly consistent finite element formulation for spatial stability analysis of thin-walled space frames is presented by applying linearized virtual work principle and introducing Vlasov's assumption. The improved displacement field for unsymmetric thin-walled cross-sections is introduced based on inclusion of second-order terms of finite rotations, and the potential energy corresponding to the semitangential moments is consistently derived. In the present formulation, displacement parameters of axial and bending deformations are defined at the centroid axis and parameters of lateral and torsional deformations at the shear centre axis, and all bending-torsional coupling effects due to unsymmetric cross-sections are taken into account. For finite element analysis, cubic Hermitian polynomials for the flexural beam with four types of end conditions are utilized as shape functions of Hermitian space frame element. Also, load correction stiffness matrices for off-axis point loadings are derived based on the second-order rotation terms. Finite element solutions for the spatial buckling analysis of thin-walled space frames are compared with available solutions and other researcher's results.

Spatial stability and free vibration of shear flexible thin‐walled elastic beams. I: Analytical approach

AbstractAn improved formulation for spatial stability and free vibration analysis of thin‐walled elastic beams is presented by applying Hellinger–Reissner principle and introducing Vlasov's assumption. It includes shear deformation effects due to flexural shear and restrained warping stress, rotary inertia effects and bendirsg–torsional coupling effects due to unsymmetric cross sections. Closed‐form solutions for determining flexural–torsional buckling loads and natural frequencies of unsymmetric simply supported beam‐columns subjected to eccentric axial force are newiy derived and also, the tangent stiffness matrix and stability functions for symmetric thin‐walled beam elements subjected to axial force are presented. In a companion paper,26 these analytic solutions are compared with the finite element solutions according to the increase of shear deformation effects.

Dynamic response of elastic thin-walled structures influenced by coupling effects

The present paper describes the dynamic coupled vibrations of elastic thin-walled structures of arbitrary open cross-section influenced by various forms of coupling. The formulations employ Vlasov's assumptions and both warping and rotary inertia are included in the present developments. Because of the complex nature of such problems, the finite element method is employed in conjunction with an implicit self-starting unconditionally stable methodology for the dynamic computations. Three types of test examples dealing with triple, double, and no coupling are described to illustrate the effects of coupling and rotary inertia on the dynamic behavior of thin-walled structures.