Transverse vibration of symmetrically laminated rectangular composite plates

A general numerical method for estimation of the vibrational response of symmetrically laminated rectangular composite plates is presented. This is based on the Rayleigh-Ritz method using the admissible 2-D orthogonal polynomials to derive the governing eigenvalue equation. The natural frequencies and mode shapes for the laminated plates are obtained by solving this governing eigenvalue equation. Several test problems are solved to demonstrate the accuracy and flexibility of the proposed method. The present results, where possible, are verified with those available values from the literature. The effects of the material, number of layers and fibre orientation upon the frequencies and mode shapes are discussed. This study may provide valuable information for researchers and engineers in design applications.     

In-plane free vibrations of single-layer and symmetrically laminated rectangular composite plates

This work presents accurate upper-bound solutions for free in-plane vibrations of single-layer and symmetrically laminated rectangular composite plates with an arbitrary combination of clamped and free boundary conditions. In-plane natural frequencies and modes shapes are calculated by the Ritz method with a simple, stable and computationally efficient set of trigonometric functions. Reliability of the method is assessed by comparison with known accurate solutions for isotropic plates and specially orthotropic single laminates. The extensive results presented here for the first time can be considered benchmark data against which other methods may be compared and validated in the future. Influence of fiber orientation, stacking sequence, degree of orthotropy, aspect ratio and boundary conditions upon the in-plane vibration behavior are also discussed.     

Vibration of symmetrically laminated rectangular composite plates reinforced by intermediate stiffeners

An investigation of the free vibrations of symmetrically laminated rectangular plates with stiffeners based on the Rayleigh-Ritz method is presented. The assumed shape functions used in the formulation are the characteristic beam functions. It is assumed that the stiffeners and the plate are in contact at all times. The effects of the locations and orientations of the stiffeners and their relative bending and torsional stiffness to the plate on the natural frequencies and modes are examined. The effects of material properties and stacking sequence of the laminated plate are also studied.     

On the damping analysis of FRP laminated composite plates

This paper presents the damping analysis of fiber reinforced plastics laminated composite plates. For this purpose, the maximum strain and kinetic energies of a cross-ply laminated plate are evaluated analytically based on the three-dimensional theory of elasticity. The displacements of the simply supported rectangular plates are expanded into the polynomial forms with respect to a thickness coordinate, and then governing equations are formulated by using the Ritz's method. In the numerical calculations, natural frequencies and modal damping ratios are calculated for the plates with different stacking sequence and thickness ratios. The validity of the assumption of deformations and the applicability of the other plate theories (e.g. classical lamination theory (CLT), first-order shear deformation theory (FSDT) and higher-order shear deformation theory) to the laminated thick plates are discussed by comparing the numerical results obtained by the present method with the CLT and the FSDT solutions.     

Optimal design of symmetrically laminated plates for minimum deflection and weight

The minimum deflection and weight designs of laminated composite plates are given. The finite element method using Mindlin plate theory is used in conjunction with optimisation routines in order to obtain the optimal designs. Various boundary conditions are considered and results are given for varying aspect ratios and for different loading types. Comparative results are presented for minimum weight priority design as an alternative to minimum deflection/minimum weight priority design to investigate the effect of priority on the deflection and weight.     

On two mixed variational principles for thick laminated composite plates

The basic behavior of the shear deformation plate theories derived from the recently proposed Reissner's and Jing-Liao's functionals is studied in this paper. The difference between these two functionals is on the energy form of transverse normal stress. In Reissner's functional, the transverse normal stress has to be assumed as well as displacements and transverse shear. On the other hand, the independent variables in Jing-Liao's functional include only displacements and transverse shear. The comparison of these functionals is made through two examples, one with assumed transverse normal stress and the other without. It is found that for the cases studied in this paper Jing-Liao's functional provides a simpler, sometimes even better, alternative approach than Reissner's functional in formulating shear deformation theory of thick plates.     

An efficient meshfree method for vibration analysis of laminated composite plates

A detailed analysis of natural frequencies of laminated composite plates using the meshfree moving Kriging interpolation method is presented. The present formulation is based on the classical plate theory while the moving Kriging interpolation satisfying the delta property is employed to construct the shape functions. Since the advantage of the interpolation functions, the method is more convenient and no special techniques are needed in enforcing the essential boundary conditions. Numerical examples with different shapes of plates are presented and the achieved results are compared with reference solutions available in the literature. Several aspects of the model involving relevant parameters, fiber orientations, lay-up number, length-to-length, stiffness ratios, etc. affected on frequency are analyzed numerically in details. The convergence of the method on the natural frequency is also given. As a consequence, the applicability and the effectiveness of the present method for accurately computing natural frequencies of generally shaped laminates are demonstrated.     

Natural frequencies of symmetrically laminated rectangular plates with free edges

This paper is concerned with the estimation of natural frequencies of undamaged laminated rectangular plates having completely free edges. A possible application of the result is use as a reference in the determination of location and extent of defects in laminated plates from measurements of natural frequencies of damaged plates. The Rayleigh-Ritz energy approach is employed here to obtain the approximate natural frequencies of symmetrically laminated plates. The plates are considered to be thin and to undergo small deflections, satisfying Kirchhoff's hypothesis. A complete power series is used as admissible functions for in-plane and out-of-plane displacements. In the numerical examples the accuracy of the present results is compared with experimental and finite element analysis results. The effects of aspect ratio and fiber orientation on the natural frequencies of symmetrically laminated plates are documented. For each frequency, the associated mode shape is identified and tabulated in this paper.     

MITC finite elements for laminated composite plates

Within the framework of the first‐order shear deformation theory, 4‐ and 9‐node elements for the analysis of laminated composite plates are derived from the MITC family developed by Bathe and coworkers. To this end the bases of the MITC formulation are illustrated and suitably extended to incorporate the laminate theory. The proposed elements are locking‐free, they do not have zero‐energy modes and provide accurate in‐plane deformations. Two consecutive regularizations of the extensional and flexural strain fields and the correction of the resulting out‐of‐plane stress profiles necessary to enforce exact fulfillment of the boundary conditions are shown to yield very satisfactory results in terms of transverse and normal stresses. The features of the proposed elements are assessed through several numerical examples, either for regular and highly distorted meshes. Comparisons with analytical solutions are also shown. Copyright © 2001 John Wiley & Sons, Ltd.     

An analytical method for 3-D analysis of circular laminated composite plates

Summary In this paper, an analytical method is presented for 3-D (three-dimensional) researches of laminated composite plates. Using this method, the linear and nonlinear analytical 3-D solutions of high accuracy are obtained for circular laminated composite plates with rigidly clamped edges and under uniform transverse loading. Galerkin's method and the perturbation technique are used to obtained the solutions which satisfy the basic differential equations of linear and nonlinear 3-D problems. The geometric nonlinearity from a moderately large deflection is considered. Many numerical results of displacements and stresses are shown in figures. In the analytical method, there are not any restrictions regrading the stacking sequence of laminated composite plates, so the method may be used to analyze the 3-D problems of unsymmetrical laminated plates.     

用变分渐近法进行复合材料层合板仿真及三维场重构

为有效模拟和准确重构复合材料层合板三维应力/应变/变形场,基于变分渐近方法构建单斜对称的复合材料层合板渐近修正理论和重构关系。主要内容包括:基于旋转张量分解概念用一维广义应变和翘曲表示板的三维应变场,以考虑包括板翘曲变形在内的所有变形;基于变分渐近法将原三维问题分析严格拆分为非线性二维板分析(等效单层板模型)和沿法线方向的一维线性分析;通过层合板厚跨比和二维应变量阶数2个较小参数将应变能渐近修正到第二阶,并转换为Reissner形式以便于实际应用;利用生成的二维板变形和翘曲函数精确重构三维场。通过一具有20层复合层合板的柱形弯曲算例表明:基于该理论和重构过程开发的渐近变分程序VAPAS重构生成的三维应力场精确性较一阶剪切变形理论和古典层合理论更好,与三维有限元精确解相一致。     

Aerothermoelastic analysis of composite laminated plates

The flutter behavior of a thermally buckled composite laminated plate is investigated in the frequency and time domains using the finite element method. Von Karman large deformation assumptions and quasi-steady aerodynamic theory are employed for the analysis. The effects of temperature gradient, panel length-to-width ratio, fiber orientation, and stacking sequence on aerothermoelastic behavior of the plate are studied in detail. The flutter boundary, buckling boundary, time history response, and phase plane plots of cross-ply and angle-ply laminates are presented. The numerical results show that temperature gradient induces thermal moments and increases the overall stiffness of the plate, and thus may increase the flutter boundary significantly. When the buckle pattern of the plate changes, the eigenvalues of the natural modes are changed suddenly and the sequence of the natural modes may be altered. Therefore, the change in the buckle pattern postpones the coalescence and the flutter boundary may increase. The change in the coalescence pair may also postpone the coalescence and increase the flutter boundary.     

Thermal postbuckling analysis of laminated composite plates by the finite element method

The thermal postbuckling behavior of composite laminated plates subjected to a nonuniform temperature field is investigated by the finite element method. Based on the principle of minimum potential energy, the nonlinear stiffness matrix and geometry matrix are derived. The assumed displacement state over the middle surface of the plate element is expressed as a product of one-dimensional, first-order Hermitian polynomials. An iterative method is employed to determine the thermal postbuckling load. The results of the computations reveal that the thermal postbuckling behavior of composite laminated plates is influenced by lamination angle, plate aspect ratio, modulus ratio and the number of layers.     

Vibration analysis of laminated composite plates with damage using the perturbation method

The present work focuses on vibration characteristics of damaged laminated composite plates. Damage is considered as a local reduction of anisotropic plate stiffness, and three damage factors (representing the damage severity, damage anisotropy, and damage location/area, respectively) are defined to describe damage status in the laminated composite plates. The analytical solutions are obtained by the perturbation method. A numerical analysis is conducted on the vibration of damaged laminated composited plates, and the effect of damage factors on the vibration characteristics is discussed. Results indicate that three damage factors have different influences on the vibration characteristics. Also, the modal curvatures and strain energy show higher damage sensitivity than the natural frequencies and displacement mode shapes. The perturbation-based vibration analysis developed in this study can be used to effectively evaluate the effect of damage on the vibration behavior of anisotropic plates and potentially identify the damage in the laminated plates.     

Analysis of viscoelastic laminated composite plates

A micromechanics analysis is performed for the determination of the five independent elastic moduli of unidirectional fiber composites. By considering viscoelastic phases and by using the correspondence principle and the inversion of the Laplace transform, the five time-dependent functions which characterize the effective behavior of viscoelastic composites are established. The predicted time-dependent behavior is applied for the analysis of viscoelastic laminated plates. The resulting viscoelastic effects are shown, and comparison between the results obtained within the classical laminated plate theory and the first-order shear deformation theory is discussed.     

Theoretical modelling of laminated composite plates

Formulation of appropriate governing equations, simpler than the three-dimensional equations of elasticity yet capable of predicting, fairly accurately, all important response parameters such as stress and strain, is attempted in modelling a structural component. Several theoretical models are available in the literature for the analyses of plates. The emergence of fibre-reinforced plastics as an attractive form of structural construction, added a new complexity to the modelling considerations of laminates by requiring the estimation of the interlaminar stresses and strains. In this paper, modelling considerations of laminated composite plates are discussed. The classical laminated plate theory and higher-order shear deformation models are reviewed to bring out their interlaminar stress predictive capabilities, and some new modelling possibilities are indicated. This work has been supported by the Aeronautics Research and Development Board, Ministry of Defence, Government of India.     

Large deflections of laminated composite plates

A semi-energy approach is formulated for a stability study on laminated composite plates with simply supported edges subjected to uniaxial compression. The analytical approach incorporates the solution of von Karman's fundamental equation for large deflections of plates and consideration of the energy in the buckled-plate system. The effect of coupling responses on the initial buckling load and the immediate post-buckling region is discussed. The results for the initial buckling load are compared with results obtained by the well-known exact solution of Jones. The results for post-buckling behaviour are compared with the experimental data of Rahman.     

压电复合材料层合板的混沌动力学研究

压电复合材料具有独特的机电耦合性、比强度高、比刚度大和抗疲劳性能好等优点,越来越广泛地被应用于航空航天等工程领域中。根据Reddy的三阶剪切变形层合板理论,研究了受面内横向外激励力、面内纵向参数激励和面外参数激励以及压电参数激励联合作用下四边简支矩形复合材料层合板在1:2:4内共振情况下的六维平均方程。考虑平均方程存在一对双零特征值和两对纯虚特征值的情况,利用规范形理论进行化简,得到与原方程拓扑等价的最简规范形,然后在此基础上利用能量相位法从理论上分析得到系统会产生同宿的多脉冲跳跃现象;同时,基于平均方程,通过数值仿真,发现系统会产生混沌运动,具有跳跃现象。     

A semi-analytical finite element for laminated composite plates

This paper presents a semi-analytical finite element solution for laminated composite plates. The method is based on a mixed variational principle that involves both displacements and stresses. Finite element meshes are only used in the plane of plate, while the through thickness distributions of displacements and stresses are obtained using the method of state equations. Numerical results show that the rate of convergence of the new method is fast and the solutions can be very close to corresponding exact three-dimensional ones. The use of a recursive formulation of the state equations leads to an algebra equation system, from which solution are sought, whose dimension is independent of the numbers of layers of the plate considered.