Advances in nonlinear vibration analysis of structures. Part-I. Beams
01 Jun 2001-Sadhana-academy Proceedings in Engineering Sciences (Springer India)-Vol. 26, Iss: 3, pp 243-249
TL;DR: A review of work in each of these phases is very necessary in order to have a complete understanding of the process of evolution of nonlinear vibration formulations for beams in the literature can be seen to have gone through distinct phases as mentioned in this paper.
Abstract: The development of nonlinear vibration formulations for beams in the literature can be seen to have gone through distinct phases — earlier continuum solutions, development of appropriate forms, extra-variational simplifications, debate and discussions, variationally correct formulations and finally applications. A review of work in each of these phases is very necessary in order to have a complete understanding of the process of evolution of this field. This paper attempts to achieve precisely this objective.
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TL;DR: A critical review of literature on bending, buckling and free vibration analysis of shear deformable isotropic, laminated composite and sandwich beams based on equivalent single layer theories, layerwise theories, zig-zag theories and exact elasticity solution is presented in this paper.
327 citations
TL;DR: In this article, the free vibration of an orthotropic beam undergoing finite strain was studied and the effect of beam thickness and different boundary conditions were considered in finite strain formulation of the beam equations.
Abstract: In this paper, the free vibration of an orthotropic beam undergoing finite strain are studied. The second Piola-Kirchhoff stress tensor and Green-Lagrange strain tensor according to finite strain assumption were used to obtain Euler-Bernoulli beam governing equations. The Galerkin method and Generalized Differential Quadrature method were employed for solving the governing equations and boundary condition. The effect of beam thickness and different boundary conditions were considered in finite strain formulation of the beam equations. Natural frequencies of different composite materials are obtained and compared. The results revealed that by increasing the beams thickness, the difference between maximum vibration amplitude increased between von Karman and finite strain formulations. Also, in a beam with simply- simply supports, differences between linear and non linear mode shapes was remarkable.
34 citations
TL;DR: In this article, the authors highlight the necessity of formulating non-linear vibration problems in a variationally correct and consistent manner, and provide numerical computations using two different beam finite element models.
Abstract: When finite element formulations are used to study the non-linear vibration problems, some simplifications, which are not consistent with the governing variational principles, are commonly employed. Three such simplifications are critically reviewed here, through beam finite element models. The first one, ‘equivalent/ quasi-linearisation technique’ is shown to have a reduced non-linear stiffness. The second, where in ‘neglect of in plane displacements’ takes place, is seen to register an excessive non-linear stiffness. Thirdly, when both these simplifications are introduced together, they produce results closer to those of variationally correct ones,rather fortuitously. The objective of this paper is to highlight the necessity of formulating this class of problems in a variationally correct and consistent manner. Numerical computations are performed systematically, using two different beam finite element models for various commonly studied boundary conditions and suitable conclusions are drawn.
23 citations
TL;DR: In this article, a finite element model of large amplitude free vibrations of thin functionally graded beams with immovably supported ends is developed, and the material properties of these beams are investigated.
Abstract: A finite element model of large amplitude free vibrations of thin functionally graded beams with immovably supported ends is developed in this paper. The material properties of functionally graded ...
22 citations
Dissertation•
02 May 2007
TL;DR: Santillan et al. as mentioned in this paper presented an analysis of the ELASTICA with applications to VIBRATION ISOLATION and applied it to the problem of vibration separation.
Abstract: ANALYSIS OF THE ELASTICA WITH APPLICATIONS TO VIBRATION ISOLATION by Sophia Teresa Santillan Department of Mechanical Engineering and Materials Science Duke University
20 citations
Cites methods from "Advances in nonlinear vibration ana..."
...Marur outlined many of the analytical and numerical solution methods that have been used [44]....
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References
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TL;DR: In this paper, it was shown that the vibration of an extensible bar, carrying no transverse load and having the ends fixed at the supports, causes an axial tensile force with a period equal to the half-period of the vibration.
Abstract: It can be shown that the vibration of an extensible bar, carrying no transverse load and having the ends fixed at the supports, causes an axial tensile force with a period equal to the half-period of the vibration of the bar. This force modifies the process of the vibration to a nonlinear one and produces an increase of the frequency of vibration according to the increase of the amplitude.
547 citations
Abstract: A review of the recent developments in the analysis of laminated beams and plates with an emphasis on shear effects and buckling is presented. A discussion of various shear-deformation theories for plates and beams is given. The available theories are derived assuming a variation of either the in-plane displacement components or the stress components or both in the thickness coordinate. A review of the recently developed finite elements to analyze thin and thick laminated beams and plates is given next. These elements have been derived using the displacement methods, or the mixed methods or the hybrid methods. Recent studies on the buckling and postbuckling behavior of perfect and geometrically imperfect plates are described next. These behaviors have been studied using analytical, numerical, and experimental techniques. Finally, a review of the various studies on the delamination buckling and growth in beams and plates is given. Once again, the studies have been conducted using analytical, numerical, and experimental techniques. The energy release rates have been determined using closed-form solutions or using numerical differentiation. Mention also is made of studies on multiple delaminations and on dynamic response of composite laminates under impact loads.
490 citations
TL;DR: In this paper, the finite element equations for a variationally consistent higher-order beam theory are presented for the static and dynamic behavior of rectangular beams, which correctly accounts for the stress-free conditions on the upper and lower surfaces of the beam while retaining the parabolic shear strain distribution.
364 citations
"Advances in nonlinear vibration ana..." refers background or methods in this paper
...Perturbation solution with finite element Padovan 1980 Higher-order mixed finite element Reddy & Singh 1981 Higher-order C1 element Heyliger & Reddy 1988...
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...Heyliger & Reddy (1988) presented a higher order theory with C1 element formulation...
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TL;DR: A review of the recent developments in the analysis of laminated beams and plates with an emphasis on vibrations and wave propagations is presented in this paper, where a significant effort has been spent on developing appropriate continuum theories for modeling the composite materials.
Abstract: A summary of the recent developments in the analysis of laminated beams and plates with an emphasis on vibrations and wave propagations in presented. First, a review of the recent studies on the free-vibration analysis of symmetrically laminated plates is given. These studies have been conducted for various geometric shapes and edge conditions. Both analytical (closed-form, Galerkin, Rayleigh-Ritz) and numerical methods have been used. Because of the importance of unsymmetrically laminated structural components in many applications, a detailed review of the various developments in the analysis of unsymmetrical ly laminated beams and plates also is given. A survey of the nonlinear vibrations of the perfect and geometrically laminated plates is presented next. It is seen that due to the bending-stretching coupling, the nonlinear behavior of the unsymmetrically laminated perfect and imperfect plates, depending upon the boundary conditions, may be hardening or softening type. Similar behavior also is observed for imperfect isotropic and laminated plates. Lastly, the developments in studying the wave propagation in laminated materials are reviewed. It is seen that a significant effort has been spent on developing appropriate continuum theories for modeling the composite materials. Some recent studies on the linear and nonlinear transient response of laminated materials also are described.
288 citations
"Advances in nonlinear vibration ana..." refers background or methods in this paper
...Another survey of work on shear deformation theories, finite elements and buckling (Kapania & Raciti 1989a), along with those on free, forced, linear, nonlinear vibrations and wave propagation etc., with reference to laminated structures ( Kapania & Raciti 1989b ) was reported....
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...Structural vibrations Reddy 1979 Nonlinear analysis of beams Sathyamoorthy 1982a, 1982b Studies on laminated beams Kapania & Raciti 1989a, 1989b...
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TL;DR: In this article, the authors present expressions for incremental matrices that remain valid in the equilibrium equations and in the linear incremental equilibrium equations for truss elements, in-plane bending elements, membrane elements, and plate flexural elements.
Abstract: A common technique in geometrically nonlinear finite element analysis is to express the total potential in terms of Lagrangian displacement coordinates, differentiate the potential to obtain the equilibrium equations, and form the differentials of the equilibrium equations to obtain linear incremental equilibrium equations. The geometric nonlinearities in the strain-displacement equations give rise to incremental matrices in the preceding equations. The form of these matrices is not unique in the expression for the total potential. The paper presents expressions for incremental matrices that remain valid in the equilibrium equations and in the linear incremental equilibrium equations. The construction of such matrices is illustrated for truss elements, in-plane bending elements, membrane elements, and plate flexural elements. An examination of some of the recent literature indicates that some investigators have used inappropriate forms of these incremental matrices.
180 citations