Dynamic Analysis Of Plates Research Articles

A Novel Rigid Finite Element Formulation for Dynamic Analysis of Flexible Plates

This paper introduces a new rigid finite element method (RFEM) formulation for dynamic analysis of plates. In RFEM, a flexible body is divided into several rigid elements which are interconnected by spring and damping elements. Mainly, RFEM has been used to model systems with beam-like slender components, such as ropes and cables, and few RFEM formulations were developed to model plates and shells. In this study however, by means of the Timoshenko beam theory and cuboid rigid elements, a novel RFEM formulation is developed to model flexible flat plates with generic geometries of the surface, considering large deformation. For this purpose, an RFEM formulation is presented for straight beams with rectangular cross-section by means of the Timoshenko beam theory. Next, this formulation is expanded to model flexible rectangular plates with uniform thickness and then, it is elaborated on how to model non-rectangular plates with uniform thickness using only cuboid elements and consequently, the development of the proposed RFEM formulation is completed. After investigating various case studies, the competence of the proposed formulation is evaluated thoroughly. The results of RFEM came on a proper agreement with the FEM results in a way that the maximum difference between these results in dynamic analysis was about 3%.

Multicriteria optimization of variable thickness plates using adaptive weighted sum method

In this paper, a multicriteria design framework for variable thickness isotropic plates using the adaptive weighted sum method is developed. The design objectives are the minimization of weight and static displacement and the design variables are the elemental thicknesses of plates modelled using finite elements. Here, the multicriteria optimization framework is constructed by integrating the finite element method, analytical sensitivity technique along with optimization algorithms. The first-order shear deformation theory is used in the static and dynamic analyses of plates. Both single and multiobjective optimization studies are conducted to study the optimal thickness distributions of variable thickness plates under static and dynamic constraints. To study multicriteria optimization of plates, the weighted sum method is first applied which gives sparsely distributed Pareto optimal solutions. Then, the adaptive weighted sum method is employed where a coarser representation of Pareto optimal solutions is generated using the weighted sum method and less populated regions are identified for further refinement. The suboptimization problems are solved in these regions to determine a new set of Pareto optimal solutions. The Pareto optimal curves obtained using the adaptive weighted sum method are also compared with the conventional weighted sum method under different constraints. The effect of boundary conditions on the Pareto optimal solutions and thickness distributions of plates is also investigated.

Dynamic Analysis of Plates with Cut-Out Carrying Concentrated and Distributed Mass

An isoparametric plate bending element with nine nodes is used in this paper for dynamic analysis of isotropic cut-out plate having concentrated and uniformly distributed mass on the plate. The Mindlin’s first-order shear deformation theory (FSDT) is used in the present finite element formulation. Two proportionate mass lumping schemes are used. The effect of rotary inertia is included in one of the mass lumping schemes in the present element formulation. Dynamic analysis of rectangular isotropic plates with cut-out having different side ratio, thickness ratio and boundary condition is analysed using a finite element method. The present results are compared with the published results. Some new results on isotropic plates with cut-out having different side ratio, ratio of side-to-thickness of the plate, different position and size of cut-out in plates subjected to transversely concentrated and distributed mass are presented.

Dynamic analysis of plates with thick unconstrained layer damping

In this work the dynamic behaviour of free layer damping plates with thick unconstrained viscoelastic layer is analysed. With this aim, the Kirchhoff–Love thin plate formulation is adapted to take shear stiffness into account by using a frequency dependent equivalent flexural stiffness.To check the validity of the proposed formulation, it is implemented in a finite element model and compared to both the Oberst model and a reference 3D solid model in terms of eigenvalues and dynamic response across a wide range of boundary conditions – clamped in all edges, free, simply supported at edges, clamped in a single edge and simply supported at corners. In all the cases the material of the viscoelastic layer presents fractional damping, its modulus thus being complex and frequency dependent.The results show that the proposed model is in good agreement with the 3D model and that it provides better accuracy than the Oberst model, specially as the thickness of the damping layer, and consequently the effect of the shear, increases. Hence, the need of developing a 3D solid model can be avoided as well as the storage and computation time problems it arises, that are specially critical in industrial practical applications.

Numerical and Experimental Analysis of the Dynamic Behavior of Piezoelectric Stiffened Composite Plates Subjected to Airflow

In this paper, the authors present results on dynamic behavior analysis of the stiffened composite plate with piezoelectric patches under airflow by finite element method and experimental study. The first-order shear deformation plate theory and nine-noded isoparametric piezoelectric laminated plate finite element with five elastic degrees of freedom at each node and one electric degree of freedom per element per piezoelectric layer were used in the dynamic analysis of plates by finite element method. The modern equipment was used in the dynamic behaviors analysis of plates subjected to airflow load by experimental method. In this study, the results of the theoretical method have been compared with experimental studies.

Thermal effect on vibration and buckling analysis of thin isotropic/orthotropic rectangular plates with crack defects

The present research is concerned with the vibration analysis of thin isotropic and orthotropic rectangular plates with crack defects under thermal environmental conditions. In the literature, there are only few studies reported in this direction. Based on the classical plate theory, the governing equations of the isotropic and orthotropic cracked rectangular plates can be derived, in which a surface crack located at the plate center is formulated based on a line-spring model. Since the dynamic behavior of structural elements is significantly affected by thermal effects, a thermal buckling analysis for isotropic and orthotropic plates is also conducted. A uniform heating load on the cracked rectangular plates is considered and the critical buckling temperature of the plates with or without cracks is investigated. The discrete singular convolution (DSC) method is then applied to formulate the eigenvalue equations for the cracked rectangular plates under various thermal conditions. The DSC technique is an ingenious method in stability and dynamic analysis of plates, not only it is a flexible local method to handle complex geometries and boundary conditions, but also it performs as a global approach with a high degree of accuracy. To go beyond the limitation of the DSC method, the use of Taylor’s series expansion method is incorporated for the treatment of free boundary conditions. In addition, this is the first attempt to explore its application on the analysis of cracked rectangular plates under thermal effects. In this work, the vibration of isotropic and orthotropic cracked rectangular plates with various combinations of boundary conditions is studied. A special restrained manner of simply supported conditions that are permissible for in-plane movements is also analyzed. The obtained solutions herein are compared with the existing results to verify the accuracy and reliability. Besides, accurate first-known solutions are also presented.

Static and dynamic analysis of plates in cylindrical bending through displacement and mixed approaches

ABSTRACTThis work analyzes the influence of edge-boundary conditions on the static and dynamic behavior of isotropic and cross-ply plates in cylindrical bending conditions. The main concern of the analysis is addressed to dynamic problems (free vibrations) along with certain boundary conditions that seem to be mainly responsible for a low convergence of the eigenvalues to the exact values. The interest on the static behavior comes from the inherent sensitivity of related quantities, which can both be compared with previous studies and can better elucidate such a low convergence. The analysis is carried out within the frame of the three-dimensional theory by using different variational formulations based both on assumed displacements and/or stresses along with the attempt to unify the comparison of the different formulations. The numerical results/performance of such formulations are also compared to those achieved through alternative models.

Nonlinear Dynamic Analysis of Plates Stiffened by Parallel Beams with Deformable Connection

In this paper a general solution to the geometrically nonlinear dynamic analysis of plates stiffened by arbitrarily placed parallel beams of arbitrary doubly symmetric cross-section, subjected to dynamic loading, is presented. The plate-beam structure is assumed to undergo moderate large deflections and the nonlinear analysis is carried out by retaining nonlinear terms in the kinematical relations. According to the proposed model, the arbitrarily placed parallel stiffening beams are isolated from the plate by sections in the lower outer surface of the plate, making the hypothesis that the plate and the beams can slip in all directions of the connection without separation and taking into account the arising tractions in all directions at the fictitious interfaces. These tractions are integrated with respect to each half of the interface width resulting in two interface lines, along which the loading of the beams and the additional loading of the plate are defined. Six boundary value problems are formulated and solved using the analog equation method (AEM), a BEM-based method. Both free and forced transverse vibrations are considered and numerical examples with great practical interest are presented demonstrating the effectiveness, wherever possible, the accuracy, and the range of applications of the proposed method.

Nonlinear Transient Response of Laminated Composite Shells

This paper first compares the writers’ results of static and dynamic analyses of plates, cylindrical and spherical shells employing four-, eight-, and nine-noded elements with different integration rules with those of earlier investigators and including some of the recent composite theories. Thereafter, the nonlinear transient responses of laminated composite cylindrical and spherical shell panels with cutouts are investigated taking up additional examples that are yet to appear in the published literature. For these, the finite-element model is employed using eight-noded C0 continuity, an isoparametric quadrilateral element considering von Karman large deflection assumptions. In the time integration, the Newmark average acceleration method is used in conjunction with a modified Newton–Raphson iteration scheme. Important conclusions with respect to nonlinear transient responses are summarized for cylindrical and spherical shells with and without cutouts.

Vibration analysis of cracked plates using the extended finite element method

In the present paper, the extended finite element method (X-FEM) is adopted to analyze vibrations of cracked plates. Mindlin’s plate theory taking into account the effects of shear deformation and rotatory inertia is included in the development of the model. First, conventional FEM without any discontinuity is carried out, then the enrichment proposed by Moes et al. (Int J Numer Methods Eng 46, 131–150, 1999) of nodal elements containing cracks is added to the FEM formulation. Numerical implementation of enriched elements by discontinuous functions is performed, and thus dynamic equations (stiffness and mass matrices) are established. A FORTRAN computer code based on the X-FEM formulation is hence developed. Rectangular and square plates containing through-edge and central cracks with different boundary conditions are considered. The subspace iteration method is used to solve the eigenvalue problem. Natural frequencies as well as the corresponding eigenfunctions are consequently calculated as a function of the crack length. The obtained results show that the X-FEM is an efficient method in the dynamic analysis of plates containing discontinuities.

Exact convergence studies on static and dynamic analyses of plates by using the double U-transformation and the finite-element method

This paper presents exact static and dynamic analyses of simply supported rectangular plates. The analytical solutions for displacements, natural frequencies and dynamic responses are obtained by using the double U-transformation and the finite-element method. These explicit solutions then can easily be used to accurately study the convergence rates of the finite-element solutions. Two non-conforming plate elements are taken as examples to demonstrate the proposed procedure. The analytical finite-element solutions, the exact error expressions and convergence rates are derived. Moreover, the convergent speeds of the two elements are discussed.

Geometrically nonlinear analysis of laminate composite plates and shells using the eight-node hexahedral element with one-point integration

An eight-node hexahedral isoparametric element with one-point quadrature for the geometrically nonlinear static and dynamic analysis of plates and shells of laminate composite materials is formulated and implemented in this work. The element is free of volumetric and shear locking and spurious modes are not detected. Shear locking is avoided using a corotational system for strain and stress components, whereas volumetric locking is controlled using one-point quadrature for the dilatational (or spherical) part of the gradient matrix. The efficiency and potential of the three-dimensional element in the analysis of plates and shells of laminate composite materials undergoing large displacements and rotations are shown solving several numerical examples and comparing with results obtained by other authors using different plate and shells elements.

A semi-analytical solution for static and dynamic analysis of plates with piezoelectric patches

A modified mixed variational principle for piezoelectric materials is established and the state-vector equation of piezoelectric plates is deduced directly from the principle. Then the exact solution of the state-vector equation is simply given, and based on the semi-analytical solution of the state-vector equation, a realistic mathematical model is proposed for static analysis of a hybrid laminate and dynamic analysis of a clamped aluminum plate with piezoelectric patches. Both the plate and patches are considered as two three-dimensional piezoelectric bodies, but the same linear quadrilateral element is used to discretize the plate and patches. This method accounts for the compatibility of generalized displacements and generalized stresses on the interface between the plate and patches, and the transverse shear deformation and the rotary inertia of the plate and patches are also considered in the global algebraic equation system. Meanwhile, there is no restriction on the thickness of plate and patches. The model can be also modified to achieve a semi-analytical solution for the transient responses to dynamic loadings and the vibration control of laminated plate with piezoelectric patches or piezoelectric stiffeners.

Convergence studies on static and dynamic analyses of plates by using the U-transformation and the finite difference method

This paper presents the exact static and dynamic analyses of simply supported rectangular plates. The analytical solutions for displacements, buckling loads, natural frequencies and dynamic responses are obtained by using the double U-transformation method and the finite difference method. After an equivalent system with cyclic periodicity in two directions is established, the difference governing equation for such an equivalent system can be uncoupled by applying the double U-transformation. Then the exact analytical finite difference solutions, the exact error expressions and the exact convergence rates are derived. These results cannot be obtained if other methods are used instead.

Theories and Applications of Plate Analysis: Classical, Numerical and Engineering Methods

11R4. Theories and Applications of Plate Analysis: Classical, Numerical and Engineering Methods. - R Szilard. John Wiley and Sons, Inc, New York. 2004. 1056 pp. ISBN 0-471-42989-9. $225.00.Reviewed by AE Bogdanovich (R&D, 3TEX, Inc, 109 MacKenan Dr, Cary, NC 27511).This over 1000-page monumental volume can be, probably, best categorized as encyclopedia of plate theories, analysis methods, and applications. This is a completely reworked and greatly extended version of the same author’s 1974 book Theory and Analysis of Plates: Classical and Numerical Methods. As the book author states in the Preface, this work tries to satisfy the requirements to a textbook, reference book and manual for practicing engineers simultaneously. This objective is very well accomplished. Indeed, the four distinct parts of the book address (i) plate theories (static, dynamic, linear, nonlinear, elastic, inelastic), (ii) analytical solutions and numerical analysis methods, (iii) examples of specific problem solutions and numerical results, and (iv) practical design methods and engineering solution procedures. There are no other available sources treating such a scope in a single book. Besides, the scope covers thin, moderately thick and thick plates, as well as very thin membranes. The book is well illustrated by large number of high quality figures, contains hundreds of references and bibliography placed at the end of each section, provides problems for individual work at the end of each chapter, and has concise subject index. The book is based on the courses taught by the author and his research conducted at various universities in the US and Germany, as well as on his long-term experience as a practicing structural engineer. Accordingly, the book can be used as a text on plate theories, analysis methods, and their specific applications. At the same time, the volume can serve as a reference book for researchers and practicing engineers. Importantly, this volume is self-contained, i.e., no other sources are needed in most of the cases to fully understand the theories, analysis methods and specific problem solutions. In treating various plate theories and solutions of the pertinent plate problems, emphasis is placed on the fundamentals rather than on achieving exhaustive coverage of details. The majority of plate structures is analyzed by applying the linearly elastic approach. Consequently, the main part of the book presents various linearly elastic plate theories and suitable analytical and numerical solution techniques. Plates having isotropic, orthotropic mechanical properties and composed of layered materials are considered. Significant place is allocated to large-deflection plate theory, including pertinent solution techniques, and to nonlinear elasticity and plasticity of plate materials. However, the book only slightly touches the problems of stress/strain and failure analyses of plates. Numerical methods treated in this book include the finite difference, boundary element, gridwork, finite strip, and finite element methods. Importantly, the aforementioned numerical methods also applied to static, dynamic, and elastic stability problems. Practicing engineers, who must deal with “real-life” problems, need a much broader coverage of more complex and unique problems than one can usually find in textbooks and reference books on plate theories and analysis methods. The present book covers a large spectrum of plate problems and their solution procedures typical for mechanical and aerospace engineering, civil engineering and marine structures. In general, strong emphasis is placed on practical applications, as demonstrated by a large number of worked problems, many of them taken directly from the engineering practice. In addition to the main volume there is a plate analysis software provided on CD attached to the book. The computer programs enable for solving a broad spectrum of diverse plate analysis problems, by either plate formulas or numerical analysis tools. A collection of 170 most important plate formulas is given on the CD. The numerical analysis programs include the FORTRAN source codes, the executable forms of those computer codes for static and dynamic analyses of plates. The CD contains finite element code WinPlatePrimer, which not only solves static and dynamic plate problems, but also teaches it users how to write such programs by using readily available subroutines. With no hesitation this book is recommended to researchers and engineers whose work involves modeling, analysis and design of plates and plate-containing structures. The book is definitely a valuable text for graduate students studying plate theory, numerical analysis methods and their applications, as well as practical analysis and design of engineering structures. The book is highly recommended to be purchased by libraries and interested individuals.

Dynamic Analysis of Plates by Boundary Elements

A review of boundary element methods for the numerical treatment of free and forced vibrations of flexural plates is presented. The integral formulation and the corresponding numerical solution from the direct boundary element method viewpoint are described for elastic or inelastic flexural plates experiencing small deformations. When the material is elastic the formulation can be either in the frequency or the time domain in conjunction with the elastodynamic or the elastostatic fundamental solution of the corresponding flexural plate problem. When use is made of the elastodynamic fundamental solution, the discretization is essentially restricted to the perimeter of the plate, while an interior discretization in addition to the boundary one is needed when the elastostatic fundamental solution is employed in the formulation. However, the great simplicity of the elastostatic fundamental solution leads eventually to more efficient schemes. Besides, through dual reciprocity techniques one can again restrict the discretization to the plate perimeter. Free vibrations are solved by the determinant method when use is made of the elastodynamic fundamental solution, or by generalized eigenvalue analysis when use is made of the elastostatic fundamental solution. Forced vibrations are solved either in the frequency domain in conjunction with Fourier or Laplace transform or the time domain in conjunction with a step-by-step time integration. When the material is inelastic the problem is formulated incrementally in the time domain in conjunction with the elastostatic fundamental solution and the plate response is obtained through step-by-step time integration. Special formulations such as indirect, Green’s function, symmetric, dual and multiple reciprocity, or boundary collocation ones are also reviewed. Effects such as those of corners, viscoelasticity, anisotropy, inhomogeneity, in-plane forces, shear deformation and rotatory inertia, variable thickness, internal supports, elastic foundation and large defections are discussed as well. Representative numerical examples serve to illustrate boundary element methods and demonstrate their advantages over other numerical methods. This review article includes 150 references.

An analog equation solution to dynamic analysis of plates with variable thickness

The analog equation method (AEM) is applied to dynamic analysis of plates with variable thickness. Both free and forced vibrations are considered. The fourth order partial differential equation with variable coefficients governing the dynamic response of the plate is converted to a quasi-static plate bending problem under a fictitious time dependent load. The fictitious load is established using the AEM. Numerical examples are presented which illustrate the efficiency and accuracy of the method.

Static and dynamic analysis of plates using finite element and boundary element techniques — A bibliography (1992–1994)

This bibliography contains references to papers, conference proceedings and theses/dissertations dealing with finite element and boundary element methods applied in the analysis of plates that were published in 1992–1994.

A BEM approach to static and dynamic analysis of plates with internal supports

A boundary element approach is developed for the static and dynamic analysis of Kirchhoff's plates of arbitrary shape which, in addition to the boundary supports, are also supported inside the domain on isolated points (columns), lines (walls) or regions (patches). All kinds of boundary conditions are treated. The supports inside the domain of the plate may yield elastically. The method uses the Green's function for the static problem without the internal supports to establish an integral representation for the solution which involves the unknown internal reactions and inertia forces within the integrand of the domain integrals. The Green's function is established numerically using BEM. Subsequently, using an effective Gauss integration for the domain integrals and a BEM technique for line integrals a system of simultaneous, in general, nonlinear algebraic equations is obtained which is solved numerically. Several examples for both the static and dynamic problem are presented to illustrate the efficiency and the accuracy of the proposed method.

Finite panel for plate vibration

Finite panels are large finite elements which permit plate structures to be analyzed with minimum discretization. Data preparation and computer storage requirements are signicantly reduced with finite panels compared with conventional finite elements. A finite panel for the dynamic analysis of plates is presented, extending the existing application which treat the static case. Complete functions are used to represent displacements, with coefficients being chosen by using a modified variational principle. Panel matrices, which are presented in an explicit form, may be used to analyze structures by using standard finite element procedures. The panel has variable degrees of freedom, which enables convergence to be readily assessed. Four examples demonstrate the accuracy of the panel.