Buckling of symmetric cross-ply square plates with various boundary conditions

The buckling analysis of cross-ply laminated square plates subjected to three types of in-plane forces and various edge boundary conditions is presented on the basis of a unified five-degree-of-freedom shear deformable plate theory. The employment of the appropriate “shear deformation shape functions” in the theory leads to certain shear deformable plate theories developed previously, also, fulfills the requirement of the continuity conditions among the layers. The governing equations of buckling behaviour of completely simply supported cross-ply laminated plates are solved analytically. For the plates with different combinations of free, clamped and simply supported boundary conditions at their edges, the Ritz method is applied by assuming the displacement components as the double series of simple algebraic polynomials. The numerical results obtained on the basis of various plate theories for uniaxial, biaxial compression and compression–tension types of loading and different length-to-thickness ratios are presented and compared with the ones available in the literature.     

Non-homogeneous response of cross-ply laminated elastic plates using a higher-order theory

A consistent formulation for the bending of cross-ply laminated composite plates that possess non-homogeneous elastic properties is presented. Based on a third-order shear deformation plate theory, the governing equations are obtained using the principle of virtual work. With the help of the small parameter method, a wide variety of results are presented for the symmetric and antisymmetric analysis of non-homogeneous rectangular laminated plates. The influence of non-homogeneity, lamination schemes, aspect ratio and material anisotropy on the deflections and stresses is investigated. The new results for non-homogeneous response of composite plates should serve as bench marks for future comparisons.     

Free vibration analysis of thick laminated plates using a mixed finite element

Abstract

A mixed finite element scheme based on assumed local high‐order displacements is proposed for the free vibration of thick laminated plates. The effects of transverse shear deformation, transverse normal stress and rotary inertia are considered in the formulation. Cross‐ply laminates with simple supports and angle‐ply laminates with clamped edges are presented as examples. The three dimensional elasticity solutions of cross‐ply laminates with simple supports are used to assess the accuracy of the present scheme. The effects of the span‐to‐thickness, aspect and material anisotropy ratio on the fundamental natural frequency are investigated. The present results are compared with the results in the published literature, and agree closely with the 3‐D elasticity solutions.     

Vibration and stability of cross-ply laminated composite plates according to a global higher-order plate theory

Natural frequencies and buckling stresses of cross-ply laminated composite plates are analyzed by taking into account the effects of shear deformation, thickness change and rotatory inertia. By using the method of power series expansion of displacement components, a set of fundamental dynamic equations of a two-dimensional higher-order theory for thick rectangular laminates subjected to in-plane stresses is derived through Hamilton's principle. Several sets of truncated approximate theories are applied to solve the eigenvalue problems of a simply supported thick laminated plate. In order to assure the accuracy of the present theory, convergence properties of the lowest natural frequency and buckling stress are examined in detail. Numerical results are compared with those of the published existing theories and FEM solutions. The modal displacement and stress distributions in the thickness direction are obtained and plotted in figures. It is noticed that the present global higher-order approximate theories can predict the natural frequencies, buckling stresses and stresses of thick multilayered composite laminates as accurately as three-dimensional solutions.     

Buckling response of laminated composite stiffened plates subjected to partial in-plane edge loading

This article presents the buckling analysis of laminated composite stiffened plates subjected to partial in-plane edge loading. The finite element method is used to carry out the analysis. The eight-noded isoparametric degenerated shell element with C⁰ continuity and first-order shear deformation and a compatible three-noded curved beam element are used to model the plate skin and the stiffeners, respectively. The eigen value analysis is carried out to track the buckling load. The convergence study is performed for some specific problems and the results are compared with the available results in the literature. It is observed that the convergence of results is very fast for this finite element model. Effect of different parameters like orientation of fibers, number of layers, and loading types are considered in the present investigation. It is also observed that all these parameters have significant effect on the buckling response of the composite stiffened plate.     

Finite strip buckling and free vibration analysis of stepped rectangular composite laminated plates

A general spline finite strip method is presented which allows the spline knots to be located arbitrarily along the plate strip and also facilitates the use of analytical integration in evaluating strip properties. The development takes place in the contexts of first‐order shear deformation plate theory and of classical plate theory, and encompasses composite laminated material. The prediction of natural frequencies and buckling stresses of stepped rectangular plates is considered using the new approach in which refinement of knot spacings is used local to a step change. The superstrip concept is used as part of an efficient solution procedure. A number of applications demonstrate the validity and practicability of the developed method. Copyright © John Wiley & Sons, Ltd.     

A higher-order polynomial shear deformation theory for geometrically nonlinear free vibration response of laminated composite plate

In this paper, a nonlinear analysis for large amplitude free vibration of laminated composite plates is developed using higher-order shear deformation theory. The effect of all higher-order terms arising from nonlinear strain-displacement relations are included in the formulation and present plate theory exhibits traction-free surface of the laminated plate in von-Karman sense. A finite element procedure considering a C° continuous isoparametric nine-node rectangular element is implemented for nonlinear model. The accuracy of the theory is validated with some available theory for different aspect ratio, modular ratio, number of layers, ply orientations, etc. through some numerical examples.     

Thermal buckling of cross-ply laminated composite and sandwich plates according to a global higher-order deformation theory

A two-dimensional global higher-order deformation theory is presented for thermal buckling of cross-ply laminated composite and sandwich plates. By using the method of power series expansion of continuous displacement components, a set of fundamental governing equations which can take into account the effects of both transverse shear and normal stresses is derived through the principle of virtual work. Several sets of truncated Mth-order approximate theories are applied to solve the eigenvalue problems of a simply supported multilayered plate. Modal transverse shear and normal stresses can be calculated by integrating the three-dimensional equations of equilibrium in the thickness direction, and satisfying the continuity conditions at the interface between layers and stress boundary conditions at the external surfaces. Numerical results are compared with those of the published three-dimensional layerwise theory in which both in-plane and normal displacements are assumed to be C⁰ continuous in the continuity conditions at the interface between layers. Effects of the difference of displacement continuity conditions between the three-dimensional layerwise theory and the global higher-order theory are clarified in thermal buckling problems of multilayered composite plates.     

Forced vibration and low-velocity impact of laminated composite plates

The layerwise theory of Reddy is used to study the low velocity impact response of laminated plates. Forced-vibration analysis is developed by the modal superposition technique. Six different models are introduced for representation of the impact pressure distribution. The first five models, in which the contact area is assumed to be known, result in a nonlinear integral equation similar to the one obtained by Timoshenko in 1913. The resulting nonlinear integral equation is discretised using a time-element scheme. Two different interpolation functions, namely: (i) Lagrangian and (ii) Hermite, are used to express the impact force. The Hermitian polynomial-based representation, obviously more sophisticated, is introduced to verify the Lagrangian-based representation. Due to its modular nature the present numerical technique is preferable to the existing numerical methods in the literature. The final loading model, in which the time dependence of the contact area is taken into account according to the Hertzian contact law, resulted in a relatively more complicated but more realistic, nonlinear integral equation. The analytical developments concerning this model are all new and are reported for the first time in this paper. Also a simple, but accurate, numerical technique is developed for solving our new nonlinear integral equation which results in the time-history of the impact force. Our numerical results are first tested with a series of existing example problems. Then a detailed study concerning all the response quantities, including the in-plane and interlaminar stresses, is carried out for symmetric and antisymmetric cross-ply laminates and important conclusions are reached concerning the usefulness and accuracy of the various plate theories. This paper, presented at a symposium to celebrate the Golden Jubilee of the Aerospace Department, Indian Institute of Science, Bangalore first appeared as a paper inComputational Mechanics (1994) 13: 360–379     

Buckling and free vibration finite strip analysis of composite plates with cutout based on two different modeling approaches

A Reddy type, third order shear deformation theory of plates is applied to the development of two versions of finite strip method (FSM), namely semi-analytical and spline methods, to predict the behavior of the moderately thick plates containing cutouts. The internal cutouts are modeled based on two different modeling approaches, and the effects of cutouts on the buckling critical stresses as well as natural frequencies are investigated.     

Buckling of fiber-reinforced viscoelastic composite plates using various plate theories

This paper studies the quasi-static stability analysis of fiber-reinforced viscoelastic composite plates subjected to in-plane edge load systems. The study is based on a unified shear-deformable plate theory. This theory enables the trial and testing of different through-thickness transverse shear-strain distributions and, among them, strain distributions that do not involve the undesirable implications of the transverse shear correction factors. Using the method of effective moduli solves the equations governing the stability of simply supported fiber-reinforced viscoelastic composite plates. The solution concerns the determination of the critical in-plane edge loads associated with the asymptotic instability of plates. In a study of this problem the general quasi-static stability solutions are compared with those based on the classical, first-order and sinusoidal transverse shear-deformation theories. Numerical applications using higher-order shear-deformation theory are presented and comparisons with the results of other theories are formulated.     

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.     

Free vibration and buckling analysis of composite laminated plates using layerwise models based on isogeometric approach and Carrera unified formulation

In this paper, layerwise (LW) theory has been utilized along with that of equivalent single layer (ESL) for free vibration and linearized buckling analysis of composite laminated plates. To this end, the isogeometric approach and Carrera unified formulation (CUF) have been combined. In particular, Taylor-like and Legendre-like polynomial expansions have been utilized in the framework of CUF to approximate the solution field in ESL and LW models, respectively. Indeed, CUF provides an ideal tool that facilitates the implementation of higher orders of the solution field expansion. As ESL model cannot inherently provide interlaminar continuity, they are not suitable for analyzing thick laminated plates. However, the LW model not only presents a three-dimensional (3D)-like accurate mathematical model using two-dimensionalplate theories but also considers interlaminar continuity requirements and obviates the need for the use of shear correction factor. In addition, the nonuniform rational B-spline basis functions have been employed to approximate the solution field, due to their interesting attributes in the analysis. These functions are able to describe the exact geometry of the structure and make it technically feasible to provide refinement process during analysis. The presented numerical results confirm the validity of the proposed methodology.     

A meshless local radial point collocation method for free vibration analysis of laminated composite plates

In this paper, a meshless local radial point collocation method based on multiquadric radial basis function is proposed to analyze the free vibration of laminated composite plates. This method approximates the governing equations based on first-order shear deformation theory using the nodes in the support domain of any data center. Natural frequencies of the laminated composite plates with various boundary conditions, side-to-thickness ratios, and material properties are computed by present method. The choice of shape parameter, effect of dimensionless sizes of the support domain on accuracy, convergence characteristics are studied by several numerical examples. The results are compared with available published results which demonstrate the accuracy and efficiency of present method.     

Buckling analysis of composite plates using moving least squares differential quadrature method

This work concerns with buckling and vibration analysis of composite plates based on a transverse shear theory. A numerical scheme is introduced to determine the angular frequencies and critical buckling loads of such plates. Moving least square differential quadrature method is employed to reduce the problem to that of eigen value problem. The accuracy and efficiency of the proposed scheme is examined with different computational characteristics, (radius of support domain, basis completeness order, and scaling factors). The obtained results agreed, at less execution time, with the previous ones. Further, a parametric study is introduced to investigate the influence of elastic and geometric characteristics, (Young's modulus gradation ratio, shear modulus gradation ratio, Poisson's ratio, loading parameter, and aspect ratio), of the composite on the values of critical buckling load, natural frequencies, and behavior of mode shape functions.     

Buckling analysis of trapezoidal composite sandwich plate subjected to in-plane compression

The results of the stability analysis of simply supported layered isosceles trapezoidal plate subjected to axial in-plane compression are presented. The layup configuration is confined to symmetric laminates. The solution has been obtained by means of the Galerkin orthogonalisation method combined with the proposed method of the coordinate system transformation. The results of the analytical solution are compared with the verifying FEM calculation. The computed results in the form of graphs of the buckling force as a function of material and geometrical parameters of the panel are included.     

对称层合折板结构自由振动分析的移动最小二乘无网格法

提出了一种求解对称层合折板结构自由振动问题的移动最小二乘无网格法。以作者提出的折板无网格模型为基础,将对称层合折板结构视为由不同平面上对称层合板组成的复合结构。先基于一阶剪切变形理论,由移动最小二乘近似推导出各对称层合板的刚度和质量矩阵,再利用板与板间的位移协调条件,将各板的刚度和质量矩阵叠加得到整个结构的刚度和质量矩阵,推导出描述层合折板结构自由振动行为的控制方程。文末算例表明由本文方法得到的解与采用壳单元得到的ANSYS有限元解吻合良好,证明了本文方法的准确性。     

Active aeroelastic flutter analysis and vibration control of supersonic composite laminated plate

The active aeroelastic flutter analysis and vibration control at the flutter bounds of the supersonic composite laminated plates with the piezoelectric patches are studied. The piezoelectric patches are bonded on the top and bottom surfaces of the composite laminated plate to act as the sensor and actuator so that the active aeroelastic flutter suppression and vibration control for the supersonic laminated plate can be conducted. The unsteady aerodynamic pressure in supersonic flow is computed by using the supersonic piston theory. Hamilton’s principle with the assumed mode method is used to develop the governing equation of the structural system. The controller is designed by the velocity feedback and proportional feedback control algorithm, and the active damping and stiffness are obtained. The solutions for the complex eigenvalue problem are obtained by using the generalized eigenvalue methodology. The natural frequencies and damping ratios are also gotten. The aeroelastic flutter bounds of the supersonic composite laminated plate are calculated to investigate the characteristics of the aeroelastic flutter. The impulse responses of the structural system are calculated by using the Houbolt numerical algorithm to study the active aeroelastic vibration control. The influences of ply angle of the laminated plate and the control method on the characteristic of flutter and active vibration control are analyzed. From the numerical results it is observed that the aeroelastic flutter characteristics of the supersonic composite laminated plate can be improved and that the aeroelastic vibration response amplitudes can be reduced, especially at the flutter points, by the proportional feedback or the velocity feedback control algorithm using the piezoelectric actuator/sensor pairs. The effectiveness of the flutter control by the two control algorithms is also compared. The results of this study are of great significance to the flutter analysis and aeroelastic design of the aircraft.     

On the vibration analysis of laminated composite parabolic arches with variable cross-section of various ply stacking sequences

Abstract

The main objective of this study is to perform the free and forced vibration analysis of transversely isotropic and laminated composite parabolic arches with a continuous cross-section variation. The anisotropy of the material of the arch, effects of the rotary inertia, and shear deformations are considered. An efficient unified numerical procedure of the Complementary Functions Method and Laplace transform is applied to solve the strong form of the differential equations that govern the dynamic response of the above structures. The validity and the accuracy of the presented scheme are tested by means of several comparisons with available literature and results of ANSYS. The presented approach has proven to be an accurate and stable numerical method. It is believed that derived results can be used as benchmark solutions for validation of related works in the future.     

Similitude invariants and scaling laws for buckling experiments on anti-symmetrically laminated plates subjected to biaxial loading

Establishing the similitude between the model and prototype rigorously is a necessary step in designing an experiment efficiently. So far, to the best of the authors’ knowledge, no one has ever derived the similitude invariant for anti-symmetric cross- and angle-ply laminated plates subjected to biaxial loading before. This research paper is the first to establish the similitude invariant of anti-symmetric cross- and angle-ply laminated plates by applying the similitude transformation to the governing differential equations of buckling directly. Then the scaling laws for buckling loads of laminated plates subjected to biaxial loads are derived. But in reality, either due to the complexity of the scaling laws or to economize on costly experiments, it may not be feasible to construct the model conforming to the scaling laws completely, therefore partial similitude is investigated theoretically and approximate scaling laws are recommended. The buckling loads of the prototype predicted from the scaling laws are then compared with the available theoretical values. The complete similitude cases show exact agreement between results predicted from the scaling laws and the available analytical solutions. For partial similitude cases, the models distorted in stacking sequences, number of plies, and material properties are studied and the approximate scaling laws which yield good agreement are recommended.