Analytical solution for free vibration analysis of composite plates with layer-wise displacement assumptions (original) (raw)
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Latin American Journal of Solids and Structures, 2017
In this paper, a new higher-order layerwise finite element model, developed earlier by the present authors for the static analysis of laminated composite and sandwich plates, is extended to study the free vibration behavior of multilayer sandwich plates. In the present layerwise model, a first-order displacement field is assumed for the face sheets, whereas a higher-order displacement field is assumed for the core. Thanks for enforcing the continuity of the interlaminar displacement, the number of variables is independent of the number of layers. In order to reduce the computation effort, a simply four-noded C 0 continuous isoparametric element is developed based on the proposed model. In order to study the free vibration, a consistent mass matrix is adopted in the present formulation. Several examples of laminated composite and sandwich plate with different material combinations, aspect ratios, boundary conditions, number of layers, geometry and ply orientations are considered for the analysis. The performance and reliability of the proposed formulation are demonstrated by comparing the author's results with those obtained using the three-dimensional elasticity theory, analytical solutions and other advanced finite element models. From the obtained results, it can be concluded that the proposed finite element model is simple and accurate in solving the free vibration problems of laminated composite and sandwich plates.
Journal of Sound and Vibration, 2015
The finite element vibration analysis of plates has become one of the classical problems over the past several decades. Different finite element plate models based on classical, standard and improved shear deformable plate theories, three-dimensional elasticity equations or their combinations have been developed. The ability and accuracy of each such model can be established by validating it against analytical models, if it is possible, or other numerical models. In this paper, a comparative study of different plate finite element models used for the free vibration analysis of homogeneous isotropic and anisotropic, composite laminated and sandwich thin and thick plates with different boundary conditions is presented. The aim of the study is to find out the weaknesses and strengths of each model used and to pick out their interchangeability for the finite element calculations. For comparisons, the plate models based on classical and first-order shear deformation theories within the framework of both single-layer and layer-wise concept and three-dimensional theory of elasticity are used. The models are created using the finite element package ABAQUS TM. Natural frequencies obtained by the authors are compared with results known in the literature from different analytical or approximate solutions and, then, the correlation between them is discussed in detail. At the end, conclusions are drawn concerning the utility of each model considered for vibration predictions of plates.
Free vibrations of laminated composite plates using layerwise displacement model
2012
In this paper Generalized Layer Wise Plate Theory of Reddy (GLPT) is used to formulate an isoparametric finite element model for free vibration of laminated composite plates. With the assumed displacement field, linear strain displacement relations and linear elastic orthotropic material properties for each lamina, virtual work statement is utilized in order to formulate isoparametric finite element model. The original MATLAB computer program is coded for finite element solution. Some new results using GLPT finite element model for soft core sandwich plate is presented, which may be used as the guideline for their optimal design in the laboratory
Equivalent Single Layer Models in Free Vibration Analysis of Laminated Multi-Layered Plates
International Journal of Structural Stability and Dynamics, 2020
The performance of selected equivalent single-layer (ESL) models is evaluated within several classical benchmark tests for small amplitude free vibration analysis of multi-layered plates. The authors elaborated their own Finite Element software based on the first-order shear deformation (FOSD) theory with some modifications incorporated including a correction of the transverse shear stiffness and an application of zigzag type functions. Seven different ESL models were considered in the study; beside the classical FOSD model, there were three FOSD models with various transverse shear corrections and three ESL models enhanced by the application of zigzag functions and based on Reissner’s Mixed Variational Theorem.
Composite Structures, 2015
A unique finite element model for static and free vibration analyses of thick and thin composite laminates is presented. The model is a combination of 3D mixed layerwise and equivalent single layer (ESL) theories. The ESL is employed in the global part of domain. On the other hand, a stack of 3D elements are used for estimation of the local parameters. The transverse inter laminar stresses are determined by using 18 noded 3D mixed layerwise element with 6 DOF per node in the local region. This mixed element incorporates the interlaminar stresses as the nodal DOF in addition to displacements for ensuring continuity of the transverse stresses in the thickness direction. Nine noded 2D elements with 12 DOF per node are used in the global domain. A transition has been developed for connection and compatibility of differently modelled sub-domains. Hamilton's variational principle has been used for the free vibration analysis. Present static and vibration analyses of laminates are in good agreement with the available elasticity and closed form solutions. The presented combined mesh modelling reduces number of elements to map the entire domain as compared to full 3D model. This results in substantial reduction of DOF and improves the computational efficiency.
Free vibration analysis of composite plates based on a variable separation method
Composite Structures, 2019
This work deals with the free vibration analysis of laminated composite plates through a variable separation approach. The displacement field is approximated as a sum of separated functions of the in-plane coordinates x y , and the transverse coordinate z. This choice yields to a non-linear problem that can be solved by an iterative process. That consists of solving a 2D and 1D eigenvalue problem successively. In the thickness direction, a fourth-order expansion in each layer is considered. For the in-plane description, classical Finite Element method is used. A wide range of numerical tests involving several representative laminated and sandwich plates is addressed to show the accuracy of the present LayerWise (LW) method. Different slenderness ratios and boundary conditions are also considered. By comparing with exact or 3D FEM solutions, it is shown that it can provide accurate results less costly than classical LW computations.
Computers & Structures, 2016
A method is presented to study the free vibrations of rectangular laminated composite plates with general layups and arbitrary boundary conditions. Based on the first-order shear deformation theory, the governing differential equations and boundary conditions are deduced via Hamilton's principle. Generalised displacements are expanded as series with Legendre polynomials as a base functions. A generalised eigenvalue problem is obtained by following a variational approach, where energy functional is extremised and boundary conditions are introduced by means of Lagrange multipliers. In order to overcome some difficulties in obtaining the natural frequencies and corresponding mode shapes, a new numerical strategy is proposed.
Composite Structures, 2001
Analytical formulations and solutions to the natural frequency analysis of simply supported composite and sandwich plates hitherto not reported in the literature based on a higher-order re®ned theory developed by the ®rst author and already reported in the literature are presented. The theoretical model presented herein incorporates laminate deformations which account for the eects of transverse shear deformation, transverse normal strain/stress and a nonlinear variation of in-plane displacements with respect to the thickness coordinate ± thus modelling the warping of transverse cross-sections more accurately and eliminating the need for shear correction coecients. In addition, few higher-order theories and the ®rst-order theory developed by other investigators and already available in the literature are also considered for the evaluation. The equations of motion are obtained using Hamilton's principle. Solutions are obtained in closed form using Navier's technique and by solving the eigenvalue equation. The comparison of the present results with the available elasticity solutions and the results computed independently using the ®rst-order and the other higher-order theories available in the literature shows that this re®ned theory predicts the fundamental and higher frequencies more accurately than all other theories considered in this paper. After establishing the accuracy of present results for composite plates, new results for sandwich laminates using all the theories considered in this paper are also presented which may serve as a benchmark for future investigations.
Journal of Sound and Vibration, 2001
Laminated composite plates are being increasingly used in the aeronautical and aerospace industry as well as in other "elds of modern technology. To use them e!eciently a good understanding of their structural and dynamical behaviour is needed. The Classicaļ aminate Plate ¹heory [1] which ignores the e!ect of transverse shear deformation becomes inadequate for the analysis of multilayer composites. The "rst order theories (FSDTs) based on Reissner [2] and Mindlin [3] assume linear in-plane stresses and displacements, respectively, through the laminate thickness. Since FSDTs account for layerwise constant states of transverse shear stress, shear correction coe$cients are needed to rectify the unrealistic variation of the shear strain/stress through the thickness and which ultimately de"ne the shear strain energy. In order to overcome the limitations of FSDTs, higher order shear deformation theories (HSDTs) that involve higher order terms in Taylor's expansions of the displacement in the thickness co-ordinate were developed. Hildebrand et al. were the "rst to introduce this approach to derive improved theories of plates and shells. Kant [5] was the "rst to derive the complete set of variationally consistent governing equations for the #exure of a symmetrically laminated composite plate incorporating both distortion of transverse normals and e!ects of transverse normal stress/strain by utilizing the complete three-dimensional generalized Hooke's law and presented results for isotropic plate only. Later Mallikarjuna [6], Mallikarjuna and Kant [7] and Kant and Mallikarjuna [8, presented a set of higher order re"ned theories and presented formulations and solutions for the free vibration analysis of general laminated composite and sandwich plate problems based on "nite element methods. In this investigation, analytical solutions for the free vibration analysis of laminated composite and sandwich plates based on two higher order re,ned theories already developed by the ,rst author for which analytical formulations and solutions were not reported earlier in the literature are presented. After establishing the accuracy of the present results with three-dimensional elasticity solutions for isotropic, orthotropic and composite plates, benchmark results and comparison of solutions using various theories are presented for multilayer sandwich plates.
The present article reviews the recent research done on the free vibration analysis of multilayered laminated composite and sandwich plates using various methods available for the analysis of plates. Displacement fields of various displacement based shear deformation theories have been presented and compared. Also, some numerical results related to fundamental flexural mode frequencies of laminated composite and sandwich plates are presented using a trigonometric shear and normal deformation theory. The theory involves six unknown variables and does not require problem dependent shear correction factor. Governing differential equations and associated boundary conditions of the theory are derived by employing the dynamic version of the principle of virtual work. Navier-type closed-form solutions are obtained for simply supported laminated composite and sandwich plates. The present results are compared with exact elasticity solution and other higher order shear deformation theories wherever applicable. This article cites 391 references.