Vibration and damping analysis of composite plates using finite elements with layerwise in-plane displacements

Finite element analysis is performed to study the effects of layerwise in-plane displacements on fundamental frequencies and specific damping capacity for composite laminated plates. The cross-ply and angle-ply composite laminated plates with simply-supported boundary conditions are considered. The strain energies of each stress component are computed to quantify the amount of the transverse shear deformations for thin and thick plates. The results show that the length-to-thickness ratios, cross-ply ratios, and fiber orientations have a great influence on the in-plane displacement responses. It is also shown that the layerwise in-plane displacements should be taken into account for the dynamic analysis of thick composite plate.     

Optimization of coupled thermal-structural problems of laminated plates with lamination parameters

The behaviour of a laminated plate with given boundary temperatures and displacement constraints is optimized and the optimization problem is expressed in terms of lamination parameters. Because the thermal conductivity and structural properties of a laminate depend on the lamination parameters of the laminate, the analysis of the plate consists of solving a coupled-field problem. The strain energy, or certain displacements of the laminated plate due to given boundary temperatures and displacement boundary conditions, is optimized with respect to in-plane lamination parameters, and also buckling of the plate is considered. The buckling factors for thermal loading are expressed as a function of four in-plane and four bending lamination parameters, and the smallest factor is maximized with respect to these parameters. In addition to these thermal problems, the natural frequencies of the laminated plate are studied. Since transverse shear deformations are taken into account,the natural frequencies can be expressed as functions of two in-plane and four bending lamination parameters, with respect to which the lowest natural frequency of the plate is maximized. The lay-up for the laminate, corresponding to four optimal in-plane or bending lamination parameters, consists of three layers at most and can be determined using explicit equations. Explicit equations are derived for creating a lay-up having optimal bending lamination parameters. Received May 12, 1999     

Adaptive analysis of plates and laminates using natural neighbor Galerkin meshless method

In this paper, natural neighbor Galerkin meshless method is employed for adaptive analysis of plates and laminates. The displacement field and strain field of plate are based on Reissner–Mindlin plate theory. The interpolation functions employed here were developed by Sibson and based on natural neighbor coordinates. An adaptive refinement strategy based on recovery energy norm which is in turn based on natural neighbors is employed for analysis of plates. The present adaptive procedure is applied to classical plate problems subjected to in-plane loads. In addition to that the laminated composite plates with cutouts subjected to transverse loads are investigated. Influence of the location of the cutout and the boundary conditions of the plate on the results have been studied. The results obtained with present adaptive analysis are accurate at lower computational effort when compare to that of no adaptivity. Further, the adaptive analysis provided accurate magnitude of maximum stresses and their locations in the laminate plates with and without cutout subjected to transverse loads. Additionally, failure prone areas in the geometry of the plates subjected to loads are revealed with the adaptive analysis.

     

Thermoelastic stress analysis of anisotropic composite sandwich plates by finite element method

In the present work a finite element approach is established for the analysis of sandwich plates with different anisotropic composite facings due to aerodynamic and thermal fields. The main contribution of this study is the consideration of the inherent coupling phenomenon of stretching and bending in such non-symmetric composite sandwich plates. Consequently, the stiffness matrix includes now sub-matrices which relate the “generalized” transverse thermal (or aerodynamic) loads to the in-plane displacements, and the in-plane forces to the “generalized” transverse displacements. These sub-matrices do not exist in symmetrically layered sandwich plates and their inclusion in the analysis is shown to be of primary importance. Three examples are included, indicating the sensibility of the stress and displacement fields to the class of heterogeneity and anisotropy of the considered sandwich plates.     

A refined analysis of laminated plates by finite element displacement methods—I. Fundamentals and static analysis

This and a companion paper (Computers and Structures 26, 915–923, 1987) present a local finite element model based on a refined approximate theory for thick anisotropic laminated plates. The three-dimensional problem is reduced to a two-dimensional case by assuming piecewise linear variation of the in-plane displacements u and ρ and a constant value of the lateral displacement w across the thickness. By using a substructuring technique the present model is demonstrated to be practical and economical. The static bending stresses, transverse shearing stresses and in-plane displacements are predicted in the present paper. The vibration and buckling analyses will be presented in the second paper. Comparison with both exact three-dimensional analysis and a high-order plate bending theory shows that this model provides results which are accurate and acceptable for all ranges of thickness and modular ratio.     

An equivalent layer sandwich panel model

An equivalent layer method for modeling sandwich panels with thin laminated composite facings and honeycomb cores is presented. It avoids the need for separate face sheet and core representations. The equivalent layer is given Mindlin plate behavior and it reproduces the face sheet midsurface strains and displacements, while matching the strain energy and work of external loads. The equivalent layer also matches the eight resultants of stress in the sandwich panel. This article starts with a derivation of the sandwich panel strain energy and the inherent bending, stretching, and transverse shearing stiffnesses. It is then shown that the available equivalent layer stiffness parameters can be used for exact matching if the sandwich has a special neutral surface. Therefore, the equivalent layer can be used for linear or geometrically nonlinear analyses for in-plane and out-of-plane loads. Examples are given illustrating two general types of sandwich panels for which exact equivalence is possible. Included are equivalent layer linear response calculations using finite element computer code ADINA.     

A simple displacement-based 3-node triangular element for linear and geometrically nonlinear analysis of laminated composite plates

A simple displacement-based 3-node, 18-degree-of-freedom flat triangular plate/shell element LDT18 is proposed in this paper for linear and geometrically nonlinear finite element analysis of thin and thick laminated composite plates. The presented element is based on the first-order shear deformation theory (FSDT), and the total Lagrangian approach is employed to formulate the element for geometrically nonlinear analysis. The deflection and rotation functions of the element boundary are obtained from the Timoshenko’s laminated composite beam functions, hence convergence to the thin plate solution can be achieved theoretically and shear-locking problem is avoided naturally. The plane displacement interpolation functions of the Airman’s triangular membrane element with drilling degrees of freedom are taken as the in-plane displacements of the element. Numerical examples demonstrate that the present element is accurate and efficient for linear and geometrically nonlinear analysis of thin to moderately thick laminated composite plates.     

Optimal design of sandwich and laminated composite plates based on the planning of experiments

Optimal design problems of sandwich plates with soft core and laminated composite face layers, and multilayered composite plates are investigated. The optimal design problems are solved by using the method of the planning of experiments. The optimization procedure is divided into the following stages: choice of control parameters and establishment of the domain of search, elaboration of plans of experiment for the chosen number of reference points, execution of the experiment, determination of simple mathematical models from the experimental data, design of the structure on the basis of the mathematical models discovered, and finally verification experiments at the point of the optimal solution. Vibration and damping analysis is performed by using a sandwich plate finite elements based on a broken line model. Damping properties of the core and face layers of the plate are taken into account in the optimal design. Modal loss factors are computed using the method of complex eigenvalues or the energy method. Frequencies and modal loss factors of the plate are constraints in the optimal design problem. There are also constraints on geometrical parameters and the bending stiffness of the plate. The mass of the plate is the objective function. Design parameters are the thickness of the plate layers. In the points of experiments computer simulation using FEM is carried out. Using this information, simple mathematical models for frequencies and modal loss factors for the plate are determined. These simple mathematical functions are used as constraints in the nonlinear programming problem, which is solved by random search and the penalty function method. Numerical examples of the optimal design of clamped sandwich and simply supported laminated composite plates are presented. A significant improvement of damping properties of a sandwich plate is observed in comparison with a simple plate of equal natural frequencies.     

Stress wave calculations in composite plates using the fast Fourier transform

The protection of composite turbine fan blades against impact forces has prompted the study of dynamic stresses in composites due to transient loads. The mathematical model treats the laminated plate as an equivalent anisotropic material. The use of Mindlin's approximate theory of crystal plates results is five two-dimensional stress waves. Three of the waves are flexural and two involve in-plane extenisonal strains. The initial value problem due to a transient distributed transverse force on the plate is solved using Laplace and Fourier transforms. A fast computer program for inverting the two-dimensional Fourier transform is used. Stress contours for various stresses and times after application of load are obtained for a graphite fiber-epoxy matrix composite plate. Results indicate that the points of maximum stress travel along the fiber directions.     

Optimal design of hybrid laminated composite plates with dynamic and static constraints

Optimization procedures are presented that consider the static and dynamic characteristic constraints for laminated composite plates and hybrid laminated composite plates subject to a concentrated load on the center of the plate. The design variables adopted are ply angle or ply thickness. Considered constraints are deflection, natural frequency and specific damping capacity. Using a recursive linear programming method, nonlinear optimization problems are solved, and by introducing the design scaling factor, the number of iterations is reduced significantly. Relating interactive optimization procedures with the finite element method analysis, various hybrid composite plates with arbitrary boundary conditions can be designed optimally. In the optimization procedure, verification of analysis and design of the laminated composite plates are compared with a previous paper. Various design results are presented on laminated composite plates and hybrid laminated composite plates.     

Transverse shear effect on vibration of laminated plate using higher-order plate element

An approach using a higher-order plate element to include the effect of transverse shear deformation on free vibration of laminated plate is presented. The total displacement of the element is expressed as the sum of the displacement due to bending and that due to shear deformation. The double-sized stiffness and mass matrices due to the separation of bending and shear displacements are then reduced to the size as if only the total deflection was considered. Numerical results for natural frequencies for a range of different isotropic and anisotropic plates with various thickness-to-length ratios are obtained and compared with solutions available in the literature. The effect of transverse shear deformation on natural frequencies of higher modes of laminated plates is also discussed.     

Shear flexible element based on coupled displacement field for large deflection analysis of laminated plates

A shear deformable theory accounting for the transverse-shear (in the sense of Reissner-Mindlin’s thick plate theory) and large deflections (in the sense of von Karman theory) is employed in the construction of variational statement. A four-node, lock-free, shear-flexible rectangular plate element based on the coupled displacement field is developed in this paper to carry out the large deflection analysis. The displacement field of the element is derived by making use of the linearized equations of static equilibrium. A bi-cubic polynomial distribution is assumed for the transverse displacement ‘w’. The field distribution for the in-plane displacements (u,v) and plate normal rotations (θ_x, θ_y) and twist (θ_xy) is derived using equilibrium of composite strips parallel to the plate edges. The displacement fields so derived are coupled through material couplings. The transverse shear strain fields of the proposed element do not contain inconsistent terms, so that the element predicts even shear-rigid bending accurately.The element is validated for a series of numerical problems and results for deflections and stresses are presented for rectangular composite plates with various boundary conditions, loading and lay-ups. The influence of the sign of the loading on the deflection of unsymmetrically laminated plates, in the large deflection regime is also investigated.     

A first-ply failure analysis of composite laminates

A finite-element computational procedure is developed for the first-ply failure analysis of laminated composite plates. The procedure is based on the first-order shear deformation theory and a tensor polynomial failure criterion that contains the maximum stress, maximum strain, the Hill, Tsai-Wu and Hoffman failure criteria as special cases. By specifying the desired criterion, a first-ply failure analysis of composite laminates subjected to in-plane and/or bending loads can be achieved. A number of problems are presented to evaluate these failure criteria when applied to laminates subjected to in-plane and or bending loads.     

Elastic-plastic dynamic analysis of anisotropic laminated plates

This paper presents the application of a refined finite element model to the elastic and elastic-plastic dynamic analysis of anisotropic laminated plates. Dynamic analysis is based on Newmark's algorithm used in conjunction with the Hughes and Liu predictor-corrector scheme resulting in an ‘effective static problem’ which is solved using a Newton-Raphson-type process. Flow theory is used in the inelastic range and the Huber-Mises yielding surface extended by Hill for anisotropic materials is adopted. Numerical results obtained for two categories of anisotropic structures, namely cross-ply laminated plates and angle-ply laminated plates, are presented and the effects of anisotropy and bending/ stretching coupling on the dynamic elastic and elastic-plastic responses are discussed. The effects of lamina stack sequences and lamina angle sequences on the dynamic responses are also considered.     

Nonlinear transient responses of initially stressed composite plates

The nonlinear transient response of initially stressed composite plates is investigated using the finite element method. A nine-node isoparametric quadrilateral element is developed to model laminated plates under initial deformation and initial stress according to the Mindlin plate theory and von Karman large deflection assumptions. In the time integration, the Newmark constant acceleration method in conjunction with an efficient and accurate iteration scheme is used. Numerical results for deflections and bending moments for isotropic and laminated plates are obtained.     

A new unconstrained third-order plate theory for Navier solutions of symmetrically laminated plates

Among the many higher order plate theories, the third-order shear deformation plate theory (TSDT) of Reddy is, perhaps, the most widely adopted model in the study of laminated plates. It, however, imposes a restriction that transverse shear stress vanishes on the top and bottom surfaces of the plate. Such requirement, although reasonable in many engineering applications, is not valid when the plate is subject to shear traction parallel to the surface. To account for such problems, the present plate model releases the constraints of vanishing transverse shear stress on the top and bottom plate surfaces. This unconstrained third-order shear deformation plate theory (UTSDT) is particularly useful for the study of a plate with contact friction or present in a flow field where the boundary layer shear stress is significant. The governing differential equations of UTSDT are of similar complexity as that of TSDT but it yields more accurate solutions. In addition, it is more flexible as it can be degenerated to the first-order shear deformation plate theory (FSDT) of Reissner and Mindlin if the higher-order rotation coefficients are neglected and a shear correction factor is considered, or to the TSDT if the relevant rotation coefficients are constrained. The present study further develops the unconstrained theory in composite laminates. Navier solutions for bending and stress analysis of multilayered and symmetrically laminated composite plates are presented. It is concluded that the present plate model provides more accurate solutions than that of TSDT, with similar level of analytical complexity, when compared with the 3D elasticity exact solutions.     

Finite element analysis of crack closure in plate bending

The effect of crack closure in plate bending is studied using the finite element method. Elastic plates containing through-wall-thickness stationary cracks under transverse pressure loading are considered with different plate thicknesses and boundary conditions, respectively. Crack closure on the compression side is modeled two different ways: line closure and surface closure models. A plate bending element degenerated from a three-dimensional solid element is used to model such crack closure. Effects of crack closure are compared using the line or surface closure model for different plate thicknesses and boundary conditions, respectively.     

Dynamics of a large scale rigid–flexible multibody system composed of composite laminated plates

A novel computational approach for the dynamic analysis of a large scale rigid–flexible multibody system composed of composite laminated plates is proposed. The rigid parts in the system are described through the Natural Coordinate Formulation (NCF) and the flexible bodies in the system are modeled via the finite elements of Absolute Nodal Coordinate Formulation (ANCF), which can lead to a constant mass matrix for the derived system equation of motion. For modeling composite laminated plates accurately, a new composite laminated plate element of ANCF is proposed and the corresponding efficient formulations for evaluating both the elastic force and its Jacobian of the element are derived from the first Piola–Kirchhoff stress tensor. To improve computational efficiency, the sparse matrix technology and graph theory are used to solve the huge set of linear algebraic equations in the process of integrating the equations of motion by using the generalized-a method, and an OpenMP based parallel scheme is also introduced. Finally, the effectiveness of the proposed approach is validated through two numerical examples. One is the static simulation of a single composite laminated plate under gravity and the other is the dynamic simulations of unfolding process of a satellite system with a pair of complicated antennas.     

Three-dimensional finite element analysis of thick laminated plates

A displacement-based, three-dimensional finite element scheme is proposed for analyzing thick laminated plates. In the present formulation, a thick laminated plate is treated as a three-dimensional inhomogeneous anisotropic elastic body. Particular attention is focused on the prediction of transverse shear stresses. The plane of a laminated plate is first discretized into conventional eight-node elements. Various through-thickness interpolation is then denned for different regions of the plate; layerwise local shape functions are used in the regions where transverse shear stresses are of interest, while an ad hoc global-local interpolation is used in the region where only the general deformation pattern is concerned. For satisfying the displacement compatibility between these two regions, a transition zone is introduced. The model incorporates the advantages of the layerwise plate theory and the single-layer plate theory. Details of formulation will be presented together with several numerical examples for demonstrating the proposed scheme.     

Fundamental frequencies of annular plates with internal cracks

The Rayleigh method with a simple sub-sectioning technique has been used to obtain fundamental frequencies of annular plates with internal concentric cracks. Single-term deflection functions which satisfy the respective boundary conditions for the bending of an annular plate are used as admissible functions. The predicted fundamental frequencies are in good agreement with the reported analytical results for annular plates without any internal cracks. Numerical results for fundamental frequencies are presented for annular plates with internal cracks simply supported and clamped on both the outside and inside edges of the plates.