Thermomechanical buckling of laminated composite and sandwich plates using global–local higher order theory

In this paper, a global–local higher order theory has been used to study buckling response of the laminated composite and sandwich plates subjected to thermal/mechanical compressive loads. The present global–local theory satisfies the free surface conditions and the geometric and stress continuity conditions at interfaces, and the number of unknowns is independent of the layer numbers of the laminate. Based on this higher-order theory, a refined three-noded triangular element satisfying C¹ weak-continuity conditions has been also proposed. The present theory not only predicts accurately the buckling response of general laminated composite plates but also calculates the critical buckling loads of the soft-core sandwich plates. However, numerical results show that the global higher-order theories as well as first order theories encounter some difficulties and overestimate the critical buckling loads for the sandwich plates with a soft core.     

Hygrothermal effects on the postbuckling of shear deformable laminated plates

The influence of hygrothermal effects on the postbuckling of shear deformable laminated plates subjected to a uniaxial compression is investigated using a micro-to-macro-mechanical analytical model. The material properties of the composite are affected by the variation of temperature and moisture, and are based on a micro-mechanical model of a laminate. The governing equations of a laminated plate are based on Reddy's higher-order shear deformation plate theory that includes hygrothermal effects. The initial geometric imperfection of the plate is taken into account. Two cases of the in-plane boundary conditions are considered. A perturbation technique is employed to determine buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling behavior of perfect and imperfect, antisymmetric angle-ply and symmetric cross-ply laminated plates under different sets of environmental conditions. The influences played by temperature rise, the degree of moisture concentration, the character of in-plane boundary conditions, transverse shear deformation, plate aspect ratio, total number of plies, fiber orientation, fiber volume fraction and initial geometric imperfections are studied.     

Buckling studies for simply supported symmetrically laminated rectangular plates

Extensive and accurate numerical results are presented for the critical buckling loads of simply supported, rectangular, laminated composite plates subjected to five types of loading conditions: (1) uniaxial, (2) hydrostatic biaxial, (3) compression-tension biaxial, (4) positive shear and (5) negative shear. Considerably different results are found for the two types of shear loading for angle-ply composites. The Ritz method, along with displacements assumed in the form of a double sine series, is used to solve the problems. Convergence studies are presented to demonstrate the accuracy of the results. Contour plots of the buckled mode shapes are shown for some of the more interesting plate and loading configurations.     

Spline finite strip analysis of the buckling and vibration of composite prismatic plate structures

A spline finite strip capability is described for predicting the buckling stresses and natural frequencies of vibration of prismatic plate structures which may be of composite laminated construction with arbitrary lay-ups. The plate structures may have general boundary conditions. The capability embraces analyses based on the use of first-order shear deformation plate theory and of classical plate theory, and utilizes substructuring procedures which include the use of superstrips. The theoretical development is not detailed since the present paper reports a very direct extension of a theoretical study developed for the analysis of single plates in an earlier paper in this Journal. A considerable range of buckling and vibration applications is documented and comparison of spline finite strip numerical values of buckling stresses and frequencies is made with results generated using the semi-analytical finite strip method and, in some cases, the finite element method. Buckled and vibrational mode shapes are presented for some applications.     

Stability behaviour of arbitrarily laminated composite plates with free and elastically restrained unloaded edges

In this paper, the initial buckling loads and the corresponding buckling modes of symmetric rectangular laminated plates are investigated. The considered laminates are supposed to have a uniform thickness, are subjected to a linearly distributed inplane compressive normal load and are simply supported at the two loaded edges with one free unloaded plate edge and with one simply supported unloaded edge where elastic rotational restraints are considered. Unlike in many other investigations, the composite laminates presently under consideration may have arbitrary yet symmetric lamination schemes with bending–torsion coupling. The initial buckling loads of such plates are calculated using the RITZ-method for which some especially adjusted displacement shape functions are employed. Since a series expansion of the buckling shape is performed in the load direction only while in the perpendicular direction one single displacement function can be shown to be sufficient, the present approach is numerically very efficient when compared to approaches in which a series representation is chosen with respect to both inplane directions. Comparison with reference results and with finite element computations leads to an excellent agreement. Some new findings on the general stability behaviour of this class of laminated plates are presented as a closure.     

Generic nonlinear behavior of antisymmetric angle-ply laminated plates

A generic presentation of the large deflection, postbuckling and nonlinear flexural vibration behavior of antisymmetric angle-ply laminated plates is investigated. A transformation between on-axis and off-axis stiffnesses is presented such that all the stiffnesses, including bending-stretching coupling, in the nonlinear governing equations of angle-ply laminates can be expressed in terms of bounded global constants in orthotropic plates. The results presented are applicable to a large variety of composite materials and show that a higher generalized rigidity ratio will lead to smaller deflection of plates, higher buckling loads and lower postbuckling deflection, and higher relative large-amplitude natural frequencies. The information presented should be helpful to anyone seeking a better understanding of the correlation between composite material properties and nonlinear behavior of laminated plates.     

Thermal buckling behavior of laminated composite plates: a finite-element study

In this paper, the thermal buckling behavior of composite laminated plates under a uniform temperature distribution is studied. A finite element of four nodes and 32 degrees of freedom (DOF), previously developed for the bending and mechanical buckling of laminated composite plates, is extended to investigate the thermal buckling behavior of laminated composite plates. Based upon the classical plate theory, the present finite element is a combination of a linear isoparametric membrane element and a high precision rectangular Hermitian element. The numerical implementation of the present finite element allowed the comparison of the numerical obtained results with results obtained from the literature: 1) with element of the same order, 2) the first order shear deformation theory, 3) the high order shear deformation theory and 4) the three-dimensional solution. It was found that the obtained results were very close to the reference results and the proposed element offers a good convergence speed. Furthermore, a parametrical study was also conducted to investigate the effect of the anisotropy of composite materials on the critical buckling temperature of laminated plates. The study showed that: 1) the critical buckling temperature generally decreases with the increasing of the modulus ratio E _L/E _T and thermal expansion ratio α _T/α _L, and 2) the boundary conditions and the orientation angles significantly affect the critical buckling temperature of laminated plates.     

The buckling and post-buckling behaviour of composite plates under biaxial loading

The response to applied load of general composite plates exhibits a coupling between the bending and extensional modes of deformation which may be significant when shear or compressive loads are applied in-plane. The additional deformation modes may affect the nature of the buckling behaviour, reduce the buckling load or change the post-buckled stiffness. This paper considers the stiffness immediately after buckling of a rectangular panel of angle-ply type in which coupling effects occur and which undergoes bifurcational buckling when biaxial load is applied in directions parallel to the panel edges. Expressions are derived for the buckling loads and for the in-plane stiffness of the panel immediately after the instant of buckling, it being found that the coupling terms affect the stiffness at buckling mainly via the associated change in buckling mode shape.     

Analysis of sandwich plates with unbalanced cross-ply faces

The analysis of rectangular sandwich plates constructed of an orthotropic core and unbalanced cross-ply face plates is presented. A double Fourier series approach is used for simply supported sandwich plates under lateral loads. Results are compared to the corresponding sandwich plate results with orthotropic faces. The results indicate that the effect of bending-membrane coupling depends mainly upon the relative thicknesses of the core and faces; other factors include the shear stiffnesses of the core, the degree of anisotropy of the individual plies, the total number and lay-up of plies in the faces and the aspect ratio of the plate.     

Vibration and buckling of cross-ply laminated composite circular cylindrical shells according to a global higher-order theory

Natural frequencies and buckling stresses of cross-ply laminated composite circular cylindrical shells are analyzed by taking into account the effects of higher-order deformations such as transverse shear and normal deformations, 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 laminated composite circular cylindrical shells made of elastic and orthotropic materials is derived through Hamilton's principle. Several sets of truncated approximate higher-order theories are applied to solve the vibration and buckling problems of laminated composite circular cylindrical shells subjected to axial stresses. The total number of unknowns does not depend on the number of layers in any multilayered shells. In order to assure the accuracy of the present theory, convergence properties of the first natural frequency and corresponding buckling stress for the fundamental mode r=s=1 are examined in detail. The internal and external works are calculated and compared to prove the numerical accuracy of solutions. 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. It is noticed that the present global higher-order approximate theories can predict accurately the natural frequencies and buckling stresses of simply supported laminated composite circular cylindrical shells within small number of unknowns.     

Free vibration of a laminated composite shell of revolution: Effects of shear non-linearity

Effects of shear non-linearity on free vibration of a laminated composite shell of revolution are investigated using a semi-analytical method based on the Reissner–Mindlin shell theory. The coupling between symmetric and anti-symmetric vibration modes of the shell is considered in the shear deformable shell element employed in this study. The Hahn–Tsai non-linearly elastic shear stress–shear strain relation is adopted. Numerical examples are given for laminated composite circular cylindrical and conical shells with various boundary conditions. The numerical results indicate that shear non-linearity may reduce significantly the fundamental frequencies of cross-ply composite shells of revolution.     

Dynamic response with mass variation of laminated composite twisted plates

In this study, the dynamic response of free vibration of laminated composite twisted (skewed) plate with mass variation via cutout and additional mass is investigated. The mathematical model employs cubic variation of the thickness coordinates in the displacement field. The mathematical formulation also includes the effects of transverse shear and rotatory inertia. A realistic parabolic distribution of transverse shear strains through the plate thickness is assumed and the use of the shear correction factor is avoided. A C⁰ finite element formulation of the mathematical model is developed to analyze the dynamic behavior of laminated composite twisted plate with cutout and additional mass. Given that no results on the present problem of free vibration of laminated composite twisted plate with cutout and additional mass are available in literature, novel results for different twist angles of laminated composite plates along with mode shapes are presented by varying the geometry, boundary conditions, and ply orientations.     

Bending and buckling of thick symmetric rectangular laminates using the moving least-squares differential quadrature method

In this paper, the moving least-squares differential quadrature (MLSDQ) method is applied to the bending and buckling analyses of moderately thick symmetric laminates based on the first-order shear deformation theory (FSDT). The transverse deflection and two rotations of the laminate are independently assumed with the moving least-squares (MLS) approximation. The weighting coefficients used in the MLSDQ approximation are obtained through a fast computation of the MLS shape functions and their partial derivatives. Numerical examples are illustrated to study the accuracy, stability and convergence of the MLSDQ method. The typical displacements, stresses and critical buckling loads of various laminated plates are presented and compared with the analytical values. Effects of support size, order of the complete basis functions and node irregularity on the numerical accuracy are investigated.     

Analysis of rectangular laminated composite plates via FSDT meshless method

A meshless approach based on the reproducing kernel particle method is developed for the flexural, free vibration and buckling analysis of laminated composite plates. In this approach, the first-order shear deformation theory (FSDT) is employed and the displacement shape functions are constructed using the reproducing kernel approximation satisfying the consistency conditions. The essential boundary conditions are enforced by a singular kernel method. Numerical examples involving various boundary conditions are solved to demonstrate the validity of the proposed method. Comparison of results with the exact and other known solutions in the literature suggests that the meshless approach yields an effective solution method for laminated composite plates.     

Stability and free vibration analysis of thick piezoelectric composite plates using spline finite strip method

In the present study, a spline finite strip with higher-order shear deformation is formulated for stability and free vibration analysis of piezoelectric composite plates. At each knot, the electric potentials on the surfaces and middle plane of each piezoelectric layer are taken as nodal degrees of freedom. However, if a continuous electrode is installed on the surface of the layer, the electric potential on the electrode is changed to structural degree of freedom, so that the equipotential condition on the electrode is automatically satisfied. The analysis can be conducted based on Reddy's third-order shear deformation theory, Touratier's “Sine” model, Afaq's exponential model or Cho's higher-order zigzag laminate theory. Consequently, the shear correction coefficients are not required in the analysis, and an improved accuracy for thick plates over the first-order shear deformation theory is achieved at only little extra computational cost.The numerical results obtained based on different shear deformation theories are presented in comparison with the three-dimensional solutions. The effects of length-to-thickness ratio, fiber orientation, boundary conditions and electrical conditions on the natural frequency and critical buckling load of piezoelectric composite plates are investigated through numerical examples.     

Post-buckling and delamination propagation behavior of composite laminates with embedded delamination

Delamination occurred due to poor manufacturing process or in-service actions significantly affects the mechanical and failure behavior of laminated composite structures. In this study, the buckling and post-buckling delamination behavior of laminated composite with an embedded initial delamination under in-plane compression was studied experimentally and numerically. First, compression tests for laminated composite specimens with embeded initial delamination were performed and the buckling and delamination responses were obtained. Then the experimental test was numerically simulated using finite element methods with the progressive failure accounted for by using cohesive zone modeling. The load-displacement curve, strain behavior and delamination shapes of experimental specimens obtained from load cells, strain gages installed at different locations, and C scan images, respectively, were compared with the FEM results, and good agreements were attained. The effect of the buckling modes, laminate stacking sequence and shape of initial delamination on the buckling load and propagation behavior was studied by considering different ply stacking and shapes of initial delaminations. It was found that the buckling mode determined the growth direction of the delamination propagation, and the stacking sequence influenced the extent of the propagation area, while the orientation of the delamination affected the buckling loads.     

Buckling analysis of circular and annular composite plates reinforced with carbon nanotubes using FEM

The critical compressive load in the buckling of circular and annular composite plates reinforced with carbon nanotubes (CNTs) is calculated using finite element method. The developed model is based on the third-order shear deformation theory for moderately thick laminated plates. Effects of CNTs orientation angles and thickness-to-inner radius ratio on the buckling of composite plates are discussed. The results are compared with those obtained by analytical method based on classical plate theory. The finite element method shows lower values for critical buckling load because of the elimination of shear strain in the classical plate theory.     

Dynamic analysis of laminated composite plates with piezoelectric sensor/actuator patches using the FSDT mesh-free method

This paper investigates the active control of laminated composite plates with piezoelectric sensor/actuator patches using an efficient mesh-free method, i.e. the element-free Galerkin (EFG) method. The formulation of the problem is based on the first-order shear deformation plate theory (FSDT) and the principle of virtual displacements. A simple control algorithm coupling the direct and converse piezoelectric effect is used to control the dynamic response of the laminate plate with distributed sensor/actuator patches through a closed control loop. Several example problems are studied to show the influence of stacking sequence and position of sensor/actuator patches on the dynamic responses of the laminate plate. These simulations provide us with the best location of the sensor/actuator patches for active control of the laminate plate.     

Interlaminar stress analysis of general composite laminates

In this study, based on the reduced from of elasticity displacement field for a long laminate, an analytical method is established to exactly obtain the interlaminar stresses near the free edges of generally laminated composite plates under the extension and bending. The constant parameters, which describe the global deformation of a laminate, are properly computed by means of the improved first-order shear deformation theory. Reddy's layerwise theory is subsequently utilized for analytical and numerical examinations of the boundary layer stresses within arbitrary laminated composite plates. A variety of numerical results are obtained for the interlaminar normal and shear stresses along the interfaces and through the thickness of laminates near the free edges. Finally the effects of end conditions of laminates on the boundary-layer stress are examined.