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 laminated composite plates using two variable refined plate theory

Free vibration of laminated composite plates using two variable refined plate theory is presented in this paper. The theory accounts for parabolic distribution of the transverse shear strains through the plate thickness, and satisfies the zero traction boundary conditions on the surfaces of the plate without using shear correction factors. Equations of motion are derived from the Hamilton's principle. The Navier technique is employed to obtain the closed-form solutions of antisymmetric cross-ply and angle-ply laminates. Numerical results obtained using present theory are compared with three-dimensional elasticity solutions and those computed using the first-order and the other higher-order theories. It can be concluded that the proposed theory is not only accurate but also efficient in predicting the natural frequencies of laminated composite plates.     

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.     

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.     

Free-edge stresses in general composite laminates

In the present study, by starting from the reduced form of elasticity displacement field for a long flat laminate, an analytical method is developed in order to accurately calculate the interlaminar stresses near the free edges of generally laminated composite plates under extension. The constant parameter appearing in the reduced displacement field, which describes the global rotational deformation of a laminate, is appropriately obtained by employing an improved first-order shear deformation theory. The accuracy and effectiveness of the proposed first-order theory are verified by means of comparison with the results of Reddy's layerwise theory as a three-dimensional benchmark. Reddy's layerwise theory is then utilized for analytical and numerical investigations of the boundary-layer stresses within arbitrarily laminated composite plates. Various numerical examples are presented for the interlaminar normal and shear stresses along the interfaces and through the thickness of laminates in the vicinity of the free edges. The effects of end conditions of laminates, fibers orientation angles as well as the stacking sequences of the layers within laminates, and geometric parameters on the boundary-layer stresses are presented and discussed.     

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.     

Effects of unsteady aerodynamic pressure load on the interlaminar stresses of laminated composite strips

The effects on the interlaminar stresses of an unsteady aerodynamic loading on a laminated composite strip are studied. A stress-function-based variational approach and the finite difference method are used to determine the interlaminar stresses in a multilayered strip subjected to arbitrary combinations of axial extension, bending and the unsteady aerodynamic loading of supersonic fluid flow over the upper surface of laminated composite strips. Symmetric four-layer, cross-ply and angle-ply laminates are considered in detail. Some numerical solutions by using a personal computer are obtained. The present results show that the unsteady aerodynamic loading has significant effects on the interlaminar stresses near the free-edge regions.     

Natural frequencies of shear deformable rhombic plates with clamped and simply supported edges

The natural vibrations of thick and thin rhombic plates with clamped and simply supported edges are analyzed, using assemblages of nine-node Lagrangian isoparametric quadrilateral C⁰ continuous finite elements based on a higher-order shear deformable thick plate theory. Here, additional nodal displacement degrees of freedom are derived by retaining higher-order powers of the thickness coordinate in the in-plane displacement fields, which in turn allows for the proper representation of the transverse shear strains of thick plates. Essential rotary inertia terms are derived and included in the present analysis. Nondimensional frequencies are calculated for thick and thin rhombic plates having various combinations of clamped and simply supported edge conditions, and skew angles. The efficacy of using higher-order shear deformable plate finite elements for predicting the in-plane vibration modes of rhombic plates is found to increase as the span-to-thickness ratio decreases and the skew angle increases. The present work shows that higher-order shear deformable finite elements essentially eliminate the transverse shear over-correction of thick rhombic plate frequencies that is produced when finite elements based on the widely used first-order Reissner-Mindlin plate theory are utilized.     

Modal characteristics of symmetrically laminated composite plates flexibly restrained at different locations

A method for determining modal characteristics (natural frequencies and mode shapes) of symmetrically laminated composite plates restrained by elastic supports at different locations in the interior and on the edges of the plates is presented. The classical lamination theory together with an appropriate set of characteristic functions are used in the Rayleigh-Ritz method to formulate the eigenvalue problem for determining the modal characteristics of the flexibly supported laminated composite plates. Sweep-sine vibration testing of several laminated composite plates flexibly restrained at different locations on the plates is performed to measure their natural frequencies. The close agreement between the experimental and theoretical natural frequencies of the plates has verified the accuracy of the proposed method. The effects of elastic restraint locations on the modal characteristics of flexibly supported laminated composite plates with different lamination arrangements and aspect ratios are studied using the present method. The usefulness of the results obtained for predicting sound radiation behavior of flexibly supported laminated composite plates is discussed.     

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.     

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.     

Modified stiffness formulation of unbalanced anisotropic sandwich plates

An approximate formulation is presented for the analysis of sandwich plates consisting of an orthotropic core and two unequal thickness anisotropic face plates. The method uncouples the membrane and bending actions, thereby significantly reducing the effort involved in accurately predicting displacements and stresses. Defining modified stiffnesses, the formulation is valid for relatively thick anisotropic laminated face plates and can be combined with a variety of analysis techniques. Using a series solution, results are generated herein for simply supported sandwich plates with unbalanced cross-ply and angle-ply faces.     

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.     

Nonlinear free vibration of laminated composite plates on elastic foundation with random system properties

The present investigation is concerned with free vibration analysis of laminated composite plates resting on elastic foundation undergoing large amplitude oscillation with random system properties. The lamina material properties and foundation stiffness parameters are modeled as basic random variables for accurate prediction of the system behavior. The basic formulation of the problem is based on higher-order shear displacement theory including rotatory inertia effects and von Karman-type nonlinear strain displacement relations. A C⁰ finite element is used for descretization of the laminate. A direct iterative method in conjunction with first-order Taylor series based perturbation technique procedure is developed to solve random nonlinear generalized eigenvalue problem. The developed probabilistic procedure is successfully used for the nonlinear free vibration problem with a reasonable accuracy. Typical numerical results (second-order statistics) are obtained for the composite plates resting on Winkler and Pasternak elastic foundations with different support conditions, side-to-thickness ratio, aspect ratio, oscillation amplitude ratio, stacking sequences and foundation parameters for symmetric and anti-symmetric cross-ply and angle-ply laminates. The results are validated with existing available results and independent Monte Carlo simulation.     

Modelling of cross-ply piezoelectric composite laminates in cylindrical bending with interfacial shear slip

An approximate theory for cross-ply piezoelectric composite laminates in cylindrical bending with interfacial shear slip is developed. This theory uses only 4 displacement and potential variables, the number of which is independent of the number of layers involved. The displacement and electric potential fields are depicted by the displacement and electric potential distribution functions through thickness, respectively. The two functions are formulated according to particular solutions to the three-dimensional elasticity equilibrium equations and electrostatics charge equation. In this shear slip modelling interfacial opening is neglected. The interfacial bonding conditions are characterised by a linear slip law and an electrically permeable assumption. A corresponding finite element is also developed to deal with piezoelectric laminates with local shear slip. The accuracy and effectiveness of the present theory are demonstrated in numerical examples.     

复合材料层合板阻尼特性有限元数值模拟

应用耗散能原理建立了复合材料层合板的阻尼预测分析模型,对复合材料层合板进行三维有限元模态分析,求出各个模态下的应力、应变分量。根据模态分析结果,从单向复合材料的阻尼性能参数出发,利用层合板应变能、耗散能和结构模态阻尼的关系求出各个模态对应的模态阻尼损耗因子。利用该方法,分别计算了单向层合板和对称层合板的结构模态阻尼损耗因子。数值计算结果与已有的理论分析和试验结果相比吻合较好,从而验证了该方法的合理性,该方法还可以比较三维应力分量对阻尼的贡献。     

Buckling of antisymmetric cross- and angle-ply laminated plates

A system of three well-known equations of equilibrium governing the buckling response of arbitrarily laminated composite plates is reduced to a single eighth order partial differential equation in terms of a displacement function. This equation is then solved in closed form to predict the buckling response of antisymmetric cross- and angle-ply plates for different boundary conditions. The effect of various plate parameters including the effect of coupling between inplane extension and out of plane bending upon the buckling response of composite plates is discussed. The results are presented in nondimensional graphical form.     

Stability of laminated composite plates subjected to various types of in-plane loadings

The aim of the present investigation is to examine the stability characteristics of laminated plates subjected to various types of in-plane loadings. Towards this, a rectangular four node finite element, having fourteen degrees of freedom per node, based on a simple higher-order shear deformation theory is developed. The theory employed herein involves four dependent variables. The element is found to be free of shear lock problems. A series of numerical problems are solved to study the effect of various parameters such as lay-up, side to thickness ratio, aspect ratio, type of loadings (uniaxial/biaxial/positive and negative shear/tension—compression/compression—tension) and boundary conditions. Some interesting observations regarding the considerable difference in the buckling resistance of angle-ply plates when subjected to positive and negative shear loading are made.     

Prestressed composite laminates featuring interlaminar imperfection

The third-order zigzag displacement model is improved to include inplane displacement jumps across each layer interface of composite laminates to enable interlaminar imperfection to be incorporated. The imperfection is characterized by a linear spring-layer model which includes perfect bonding as a special case. The principle of virtual work is used to derive a boundary value formulation for laminated composite plates initially in a prestress state. Bending, buckling and vibration problems are studied for the case of rectangular cross-ply laminated plates for illustrative purpose.     

Thermal stress analysis for laminated plates using actual temperature field

A global–local higher order model considering transverse normal deformation is proposed to predict the thermal response of laminated plates subjected to actual temperature field. The in-plane temperature field is expressed as a harmonic series expansion whereas the nonlinear temperature profile in the thickness direction is determined by solving the heat conduction equation. Validity of the proposed model is verified by comparing results with existing publications. Moreover, influence of the temperature fields along the thickness direction on the thermal behaviors of laminates is also investigated. Numerical results show that present model can reasonably predict thermal response of general laminated plates as well as laminated plates with arbitrary layouts under actual temperature loads. However, higher-order zigzag theory neglecting transverse normal deformation is inadequate for predicting thermal response of laminated plates subjected to actual temperature fields. A merit of the present model is that transverse shear stresses can be evaluated directly from the constitutive equations without stress smoothing.