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.     

A new hyperbolic shear deformation theory for buckling and vibration of functionally graded sandwich plate

A new hyperbolic shear deformation theory taking into account transverse shear deformation effects is presented for the buckling and free vibration analysis of thick functionally graded sandwich plates. Unlike any other theory, the theory presented gives rise to only four governing equations. Number of unknown functions involved is only four, as against five in case of simple shear deformation theories of Mindlin and Reissner (first shear deformation theory). The plate properties are assumed to be varied through the thickness following a simple power law distribution in terms of volume fraction of material constituents. The theory presented is variationally consistent, does not require shear correction factor, and gives rise to transverse shear stress variation such that the transverse shear stresses vary parabolically across the thickness satisfying shear stress free surface conditions. Equations of motion are derived from Hamilton's principle. The closed-form solutions of functionally graded sandwich plates are obtained using the Navier solution. The results obtained for plate with various thickness ratios using the theory are not only substantially more accurate than those obtained using the classical plate theory, but are almost comparable to those obtained using higher order theories with more number of unknown functions.     

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.     

Buckling of laminated sandwich plates subjected to partial edge compression

The buckling characteristics of sandwich plates having laminated stiff layers are studied for different types of partial edge loadings using a refined plate theory. With this plate theory, the through thickness variation of transverse shear stresses is represented by piecewise parabolic functions where the continuity of these stresses is satisfied at the layer interfaces by taking jumps in the transverse shear strains at the interfaces. The transverse shear stresses free condition at the plate top and bottom surfaces is also satisfied. It is quite interesting to note that this plate model having all these refined features requires unknown parameters only at the reference plane. To have a generality in the present analysis, finite element technique is adopted and it is carried out with newly developed triangular element, as existing finite elements cannot accommodate this plate model. So far, no solution exists in the literature for the problem of sandwich plate subjected to partial edge loading. The present analysis is first validated for the case of an isotropic plate subjected to partial edge compression and then it is extended to analyze sandwich plates. Few results are presented.     

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.     

Polyharmonic (thin-plate) splines in the analysis of composite plates

In the present study a layerwise theory for composite and sandwich laminated plates is discretized by polyharmonic (thin-plates) splines. A composite and a sandwich plate examples are presented and discussed. The combination of adequate shear deformation theory and thin-plate splines allows a very accurate prediction of displacements and stresses.     

Mixed-type finite element formulation of higher order shear deformation theory for the linear and nonlinear analyses of a laminated composite plate

For the linear and nonlinear analyses of a laminated composite plate structure, the mixed type finite element program is developed on the basis of higher order shear deformation theory of laminated plates. The accuracy of this program is checked by means of comparing with the existing results for laminated rectangular plates and is found to agree well with them. Deformations and interlaminar stresses of laminated plates are calculated according to the variation of layer numbers, fiber orientations, and plate thicknesses, so that the shear and nonlinear effects on their behaviors are studied. It is found that plate deformations are reduced by means of arranging the fiber direction into the angle-ply and increasing layer numbers.     

基于高阶改进方法的层合板固有频率分析

针对一般的计算层合板振动特性的方法过于繁复的问题,采用了一种高阶改进方法来计算复合材料层合板的固有频率。建立了高阶改进方法的层合板动能和势能表达式,利用哈密顿原理构建了层合板的自由振动方程并计算了层合板的固有频率。在此基础上,通过该方法计算了铺设角度、跨厚比和弹性模量比对层合板固有频率的影响。结果表明：该算法可以有效地计算出反对称铺设层合板和正交铺设层合板的固有频率,同时也可以计算任意形式铺设的层合板的固有频率。     

Postbuckling of shear deformable laminated plates under biaxial compression and lateral pressure and resting on elastic foundations

Postbuckling analysis is presented for a simply supported, shear deformable laminated plate subjected to biaxial compression combined with uniform lateral pressure and resting on an elastic foundation. The lateral pressure is first converted into an initial deflection and the initial geometrical imperfection of the plate is also taken into account. The formulations are based on the Reddy's higher-order shear deformation plate theory, and including the plate-foundation interaction. The analysis uses a perturbation technique to determine the buckling loads and the postbuckling equilibrium paths. Numerical examples are presented that relate to the performances of perfect and imperfect, antisymmetrically angle-ply and symmetrically cross-ply laminated plates under combined loading and resting on Pasternak-type or softening nonlinear elastic foundations from which results for Winkler elastic foundations are obtained as a limiting case. The effects played by foundation stiffness, transverse shear deformation, plate aspect ratio, total number of plies, fiber orientation, the biaxial load ratio and initial lateral pressure are studied.     

Deflections and free vibrations of laminated plates—Levy-type solutions

This study is concerned with the static deflections and natural frequencies of isotropic, orthotropic/laminated composite plates using a Levy-type solution. Mindlin plate theory is applied in conjunction with the state-space concept to find such solutions. A state-space formulation of such plates is composed of variables having physical meanings, such as moments, shear forces, displacements and rotations. The influences of aspect ratio, ratio, fiber orientation angle, laminate-layer arrangement and ratio of moduli have been investigated. Some numerical results from the present analyses are compared with published results and good agreement is found.     

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.     

A finite element formulation based on an enhanced first order shear deformation theory for composite and sandwich structures

A finite element formulation based on an enhanced first order shear deformation theory is developed to accurately and efficiently predict the behavior of laminated composite and sandwich structures. An enhanced first order shear deformation theory is systematically derived by minimizing the least-squared energy error between the first order shear deformable plate theory and a higher order shear deformable plate theory. In this way, the strain energy of a higher order theory is transformed to that of the Reissner-Mindlin plate theory. This minimization procedure yields a relationship between them that is also used to improve the accuracy of predicted stresses and displacements. The key feature of the proposed theory is in that it can be implemented to commercial FEM packages by simply changing the input, and the results obtained can be also enhanced by post-processing them via a differential quadrature method. Thus, a proposed finite element formulation can be widely used in various application problems. Through numerical examples, the accuracy and robustness of the present formulation are demonstrated.     

Nonlinear free and forced vibration of simply supported shear deformable laminated plates with piezoelectric actuators

This paper deals with the nonlinear vibration and dynamic response of simply supported shear deformable cross-ply laminated plates with piezoelectric actuators subjected to mechanical, electrical and thermal loads. The material properties are assumed to be independent of the temperature and electric field. Theoretical formulations are based on the higher order shear deformation plate theory and general von Kármán-type equation, which includes thermo-piezoelectric effects. Due to the bending and stretching coupling effects, a nonlinear static problem is first solved to determine the pre-vibration deformation caused by temperature field and control voltage. By adding an incremental dynamic state to the pre-vibration state, the equations of motion are solved by an improved perturbation technique to determine nonlinear frequencies and dynamic responses of hybrid laminated plates. The numerical illustrations concern nonlinear vibration characteristics of unsymmetric cross-ply laminated plates. The results presented show the effects of temperature rise, applied voltage and stacking sequence on the nonlinear vibration and dynamic response of the plates.     

Vibration and stability of angle-ply laminated composite plates subjected to in-plane stresses

Natural frequencies and buckling stresses of angle-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 fundamental natural frequency are examined in detail. Numerical results are compared with those of the published existing theories. The modal displacement and stress distributions in the thickness direction are obtained and plotted in figures. The present global higher-order approximate theories can predict the natural frequencies, buckling stresses and modal stresses of thick multilayered angle-ply composite laminates accurately within small number of unknowns which is not dependent on the number of layers.     

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.     

Free vibration of cantilevered symmetrically laminated thick trapezoidal plates

To account for the effect of transverse shear deformation, the p-Ritz method incorporating Reddy’s third-order shear deformation theory has been developed for the vibration analysis of cantilevered, thick, laminated, trapezoidal plates. In the p-Ritz method, a set of uniquely defined polynomial functions, consisting of the product of a two-dimensional function and a basic function, are used as the admissible trial displacement and rotation functions in the Ritz minimization procedure. The energy integral is formulated based on Reddy’s third-order shear deformation theory. From the p-Ritz method, the governing eigenvalue equation is derived which is used to compute the vibration frequency parameters and mode shapes of the laminated plate. Convergence and comparison studies have been presented to demonstrate and verify the accuracy of the results.     

On the thermal expansion effects in the transverse direction of laminated composite plates by means of a global-local higher-order model

In this paper, a new efficient global-local higher-order model is proposed for the thermoelastic analysis of laminated composite and sandwich plates. The proposed model takes into account explicitly the contribution of thermal expansion in the transverse displacement component. To satisfy the transverse displacement continuity along the thickness direction, the continuity condition of transverse displacement at interfaces, which is not satisfied in many other schemes, has been a priori enforced. This model fully satisfies the free surface conditions and the geometric and stress continuity conditions at interfaces. As the number of variables of the proposed model is independent of the number of layers of laminates, compared to the 1,2-3 theory proposed by Li and Liu (1997) [20], the present model offers some significant improvements, and is able to predict accurately thermoelastic response of laminated plates under uniform temperature without a corresponding increase in the number of unknowns. The governing equations of equilibrium are derived by means of the principle of virtual displacements involving the thermal strain field. Applying Navier's technique, analytical solutions in terms of a double trigonometric series for simply supported laminated plates are presented. Results of benchmark examples are compared with the three-dimensional thermoelastic solutions as well as other published works. Numerical results show that the proposed model is more rigorous and can better predict the thermoelastic response in comparison with the 1,2-3 theory and other two-dimensional models.     

General Model to Predict Power Flow Transmitted into Laminated Beam Bases in Flexible Isolation Systems

For estimating the vibration transmission accurately and performing vibration control efficiently in isolation systems, a novel general model is presented to predict the power flow transmitted into the complicate flexible bases of laminated beams. In the model, the laminated beam bases are simulated by the first-order shear deformation laminated plate theory, which is relatively simple and economic but accurate in predicting the vibration solutions of flexible isolation systems with laminated beam bases in comparison with classical laminated beam theories and higher order theories. On the basis of the presented model, substructure technique and variational principle are employed to obtain the governing equation of the isolation system and the power flow solution. Then, the vibration characteristics of the flexible isolation systems with laminated bases are investigated. Several numerical examples are given to show the validity and efficiency of the presented model. It is concluded that the presented model is the extension of the classical one and it can obtain more accurate power flow solutions.     

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.     

A generalized global-local high-order theory for bending and vibration analyses of sandwich plates subjected to thermo-mechanical loads

Although the global higher-order shear deformation theories may predict the gross responses of the sandwich plates sufficiently accurate, their results may show considerable errors in predicting the local effects. Layerwise and mixed layerwise theories are computationally expensive and generally, the interlaminar transverse stresses continuity conditions are not enforced in the former category of theories. Majority of the available zigzag and global-local theories suffer from the point that the transverse normal stress continuity that influences the transverse deformation significantly, especially in sandwich plates with soft-cores, is not satisfied at the layer interfaces.In the present paper, a generalized global-local theory that guarantees the continuity condition of all of the displacement and transverse stress components and considers the transverse flexibility under thermo-mechanical loads is introduced. One of the advantages of the present theory is that the number of unknown parameters is independent of the number of the layers. Furthermore, all stress components are considered in the formulations. Therefore, in contrast to the available works, the theory may be used for sandwich plates with stiff or soft cores. In contrast to the available global-local formulations, the present formulation is developed in a compact matrix form that makes it more desirable for computerized solutions. The present theory may be considered as a generalized layerwise theory with an optimized computational time. Compatible quadrilateral Hermitian elements are employed to further enhance the accuracy of the results. Validity, advantages, and efficiency of the present theory are investigated for different local and global behaviors of the layered composite and sandwich plates. Comparison of the present results with those of the three-dimensional theory of elasticity and the available plate theories confirms the efficiency and accuracy of the proposed theory. Results reveal that the global theories (e.g. the higher-order shear deformation theories) may encounter serious accuracy problems even in predicting the gross responses of the sandwich plates.