Cross-ply Plates Research Articles

Vibration Analysis Of Laminated Plates Using A Refined Shear Deformation Theory

A previously published discrete-layer shear deformation theory is used to analyze free vibration of laminated plates. The theory includes the assumption that the transverse shear strains across any two layers are linearly dependent on each other. The theory has the same dependent variables as first order shear deformation theory, but the set of governing differential equations is of twelfth order. No shear correction factors are required. Free vibration of simply supported symmetric and antisymmetric cross-ply plates is calculated. The numerical results are in good agreement with those from three-dimensional elasticity theory.

IMPROVED HIGH-ORDER THERMOELASTIC RESPONSE THEORY FOR ANALYZING THERMAL BENDING OF LAMINATED COMPOSITE PLATES

A new high-order laminated plate thermoelastic responses theory, which is based on a new mixed variational principle proposed by Reissner (1984) and accounts for transverse shear and transverse normal strains, is developed to study the thermoelastic behavior of a laminated composite plate. To this end, a zig-zag-shaped Co function and Legendre polynomials are introduced into the approximate displacement distributions across the plate thickness. The thickness variation of temperature is considered to be linear. The theory is examined by applying it to the thermal bending problem of cross-ply rectangular plates. The numerical results of central deflection and thickness variation of in-plane and transverse thermal stresses are obtained for symmetric and antisymmetric cross-ply plates.

Transient Heat Conduction Analysis for Laminated Composite Materials.

Transient heat conduction analysis of hybrid laminated composite materials is very important because they are heated during the manufacturing process. Anisotropy of thermal conductivity must be considered for heat conduction analysis of such composite materials as FRP. Assuming that heat conducts uniformly in a normal direction in thin shells, shell structures can be modeled with membrane elements. This paper presents an FEM transient heat conduction analysis system for hybrid laminated shell structures. This program can treat the orthotropic materials whose principal direction is inclined to the coordinate axes of elements. The temperature distribution of cross section of an Aluminum-CFRP hybrid laminated plate is investigated. And the effect of fiber orientations for the heat conduction of CFRP cross-ply plates is discussed. Furthermore, the heat conduction of CFRP cylinders considering the heat transfer condition is analyzed.

Transverse deflection of cross-ply laminated plates under combined in-plane and lateral loads

The present work deals with the geometrically nonlinear behaviour of simply supported, square and rectangular, antisymmetrically laminated cross-ply plates. The formulation is based on the Mindlin plate theory. The finite element approach is used to solve nonlinear equations. Results are presented for plates subjected to combined uniaxial compression and lateral loads. The effect of the number of lay-ups and the aspect ratio on the load-deflection behaviour is studied and illustrated graphically.

Thermal post buckling behavior of rectangular antisymmetric cross-ply composite plates

The thermal post buckling behavior of simply-supported, antisymmetric cross-ply plates with immovable edges is investigated in this paper. For this purpose, a one term approximation for the inplane and transverse displacements is assumed and Rayleigh-Ritz method is used to obtain the equations of equilibrium. The question, whether the true thermal buckling (bifurcation buckling under thermal loads) exists for such plates is also addressed to. The conditions under which the thermal buckling in the classical sense occurs are derived. It is shown that nonlinear bending stiffness for such plates is direction dependent, because of an extra quadratic nonlinear term in addition to the usual cubic nonlinear term in the governing equilibrium equation. Results are presented for various plate configurations.

Large amplitude vibration of antisymmetric imperfect cross-ply laminated plates

In this paper the large amplitude vibration of an initially stressed, imperfect, cross-ply plate is studied. The non-linear equations of motion for an imperfect cross-ply plate in a general state of non-uniform initial stress are derived. The effects of both transverse shear deformation and rotatory inertia are included. A multi-mode solution is formulated for a simply supported imperfect cross-ply plate. The modal equations are obtained by performing the Galerkin procedure and then solved by the Runge-Kutta method. It is found that the existence of geometric imperfections may result in a drastic change on the large amplitude vibration behaviour. The effects of various parameters on the non-linear vibration frequencies are investigated.

Vibration of clamped moderately thick general cross-ply plates using a generalized Navier approach

A hitherto unavailable analytical solution to the free vibration problem of general cross-ply laminated rigidly clamped rectangular plates, incorporating first-order shear deformation, and rotatory and in-plane inertias into the formulation, is presented. A recently developed boundary continuous displacement-based generalized Navier solution technique is used to solve the five highly coupled linear second-order partial differential equations with constant coefficients, and the associated geometric boundary conditions. The assumed solution functions are in the form of double Fourier series, which satisfy the rigidly clamped boundary conditions a priori in a manner similar to the conventional Navier method. Convergence characteristics of the natural frequencies of both symmetric and antisymmetric cross-ply plates are numerically established. Other numerical results presented herein include (i) comparison with the corresponding available first-order shear deformation theory-based Galerkin and classical lamination theory-based boundary-discontinuous analytical solutions, and (ii) study of the effects of thickness and aspect ratio on the natural frequencies.

Bending analysis of laminated plates using a refined shear deformation theory

In this paper the refinement of a previously published discrete-layer shear deformation laminated plate theory by the assumption that the transverse shear strains across any two layers are linearly dependent on each other is briefly described. The theory contains the same dependent variables as first-order shear deformation theory, but the set of governing differential equations is of the twelfth order. No shear correction factors are required. Solutions for simply-supported symmetric and antisymmetric cross-ply plates are discussed. The numerical results are compared with three-dimensional elasticity solutions and first-order shear deformation theory solutions. The present theory gives a better estimate of the deflections and stresses even for fairly thick laminates. For laminates of the same thickness, it is shown that the solution accuracy improves as the number of layers increases.

Reinvestigation Of Non-linear Vibrations Of Simply Supported Rectangular Cross-ply Plates

Large amplitude free vibrations of simply supported rectangular cross-ply plates are examined here. The restoring force function in the equation of motion is found to be a cubic polynomial, giving a Duffing type equation, or a combination of quadratic and cubic terms. The exact solution of the equation of motion is presented in a form that can be evaluated numerically to any desired degree of accuracy. The results obtained for isotropic as well as six-layered antisymmtric cross-ply rectangular plates are found to be in good agreement with the existing results.

Buckling and post-buckling behavior of laminated plates using the generalized nonlinear formulation

In this paper the buckling and post-buckling analysis of symmetrically laminated cross-ply plates has been considered using the generalized formulation of which the von Karman type formulation is a special case. Unlike the von Karman type formulation, the governing equations of the present formulation are nonlinear in all the displacement parameters of the plate. This obviates the use of hitherto known solution techniques for nonlinear plate analysis. To overcome this difficulty the perturbation technique has been followed in this paper. Numerical results indicate that the two formulations differ by about 2.5% for thicker perfect plates and the difference ceases to exist as the imperfection value of the plate increases.

A Variational Approach to Three-Dimensional Elasticity Solutions of Laminated Composite Plates

The displacements in a laminated composite are represented as products of two sets of unknown functions, one of which is only a function of the thickness coordinate and the other is a function of the in-plane coordinates (i.e., separation of variables approach), and the minimization of the total potential energy is reduced to a sequence of iterative linear problems. Analytical solutions are developed for cross-ply and angle-ply laminated composite rectangular plates. The solution for simply-supported cross-ply plates under sinusoidal transverse load reduces to that of Pagano. Numerical results for stresses and displacements for antisymmetric angle-ply laminates are presented. The three-dimensional elasticity solutions developed are important because they can be used to study the behavior of composite laminates, in addition to serving as reference for approximate solutions by numerical methods and twodimensional theories.

Nonlinear Analysis of Thick Antisymmetric Cross-Ply Laminated Plates in Cylindrical Bending

The large deflection analysis (in the von Karman sense) is used to assess the effects of transverse shear deformation and bending-extension coupling on the behavior of antisymmetrically laminated, cross-ply plates under cylindrical bending. Solutions are presented for uniformly loaded simply supported and clamped plates.

Non-linear forced vibrations of antisymmetric rectangular cross-ply plates

A method of direct numerical integration of the frequency/time period expression is proposed to study the non-linear forced/free vibration behaviour of rectangular, antisymmetric cross-ply plates. The presence of bending-extension coupling in such plates results in a cubic non-linear term in addition to a fourth power term in the energy balance equation. The existence of this term does not allow two real and opposite roots at a maximum amplitude position unlike in isotropic, orthotropic, symmetric cross-ply, square antisymmetric cross-ply and symmetric and antisymmetric angle-ply plates, but such an energy balance equation definitely has two real roots, which are unequal in magnitude and different in sign. The extensively used perturbation method fails to predict the response of such plates, depending on the severity of the bending-extension coupling. The proposed direct integration method, even with single-term approximations for in-plane and transverse displacements, yields consistently accurate results. Response curves for several configurations of antisymmetric, simply supported rectangular cross-ply plates are presented.

Pulse propagation in a laminated composite plate and nondestructive evaluation

The surface response of a laminated composite plate with delamination cracks has been studied in this paper. The objective of this study is to analyze the surface response both in the time and frequency domains due to delamination cracks at different depths in a cross-ply composite plate. For reasons of simplicity, attention has been focused here on two-dimensional (plane-strain) motions. The source of excitation is taken to be a line vertical force acting perpendicular to the fibers in a lamina. Numerical results are presented, showing peak responses at certain resonance frequencies. A comparison of these results with those for a uniaxial graphite epoxy plate shows characteristic differences.

Influence of Lamination and Boundary Constraint on the Deformation of Moderately Thick Cross-Ply Rectangular Plates

Hitherto unavailable analytical solutions to boundary-value problems of moderately thick general cross-ply laminated rectangular plates, subjected to various boundary conditions, are presented. A recently developed double Fourier series-based method has been utilized to solve a system of five highly coupled linear second-order par tial differential equations (with constant coefficients), that emerge from the first-order shear deformation theory (FSDT) and the associated geometric and natural boundary con ditions. The convergence characteristics of the series solutions, especially their de pendence on lamination and boundary constraint, are numerically investigated in detail. Other numerical results presented here include (1) verification with the available analytical solutions based on the classical lamination theory (CLT) as well as FSDT, (2) investigation of the effect of length-to-thickness, length-to-width and modular ratios on the response of antisymmetric and symmetric cross-ply plates, with various boundary constraints and (3) spatial variation of displacements and moments.

Transient response of a laminated composite plate: Results from homogenization and discretization

Transient response of a multilayered laminated plate has been studied in this paper. The objective of this study is to analyze the effect of layering on the response of the laminated plate in both time and frequency domains to a line source on the surface of the plate. For simplicity of analysis, attention has been focused on the two-dimensional (plane strain) motion. It is shown that for a cross-ply plate when the number of plies is small the response of the plate is quite different than that of an equivalent homogeneous plate. For short pulses the response is fairly complicated due to the reflections from the interfaces between the plies. Dispersion of waves in the plate is also analyzed. It is found that the homogenized model predictions for the low-order modes agree with those of the layered model. However, they diverge when high-order modes are considered. This is consistent with the transient response comparisons

Elastic Wave Propagation in Laminated Composite Plates

A stiffness method and an analytical method have been used to study the dispersion characteristics of guided waves in laminated composite plates. Both cross-ply and angle-ply plates have been considered in the analysis. The objective of the study is to analyze the effect of fiber orientation, ply layout configuration, and number of layers on the dispersion characteristics. A Rayleigh-Ritz type of approximation of the through-thickness variation of the displacements that maintain continuity of displacements and tractions at the interfaces between the layers has been used in the stiffness method. The analytical method solves the exact dispersion relation of the laminated plate by using the Muller’s method with initial guesses obtained through the stiffness method. Both methods are applicable to plates with arbitrary number of layers having distinct mechanical properties. Numerical results presented show strong influence of anisotropy on the guided waves.

Random Response of Thick Laminated Rectangular Plates

The present paper deals with the random response analysis of clamped thick laminated rectangular plates subjected to white noise excitation. The refined plate theory postulated by Reddy has been used. The analysis has been done using an integral equation technique. The random response results obtained for istropic square plates based on different plate theories (viz.) classical theory, first order theory and refined theory, have been compared. A parametric study has been conducted for angle ply and cross ply plates by varying the lay up of layers and a/h ratios.

Large amplitude free vibration of simply supported antisymmetric cross-ply plates

The presence of bending-extension coupling in antisymmetric cross-ply plates results in the cubic nonlinear term in addition to the fourth-power term in the energy balance equation. A direct numerical integration method has been proposed to study the large amplitude free vibrations of such plates

Some observations on the large deflection bending of rectangular antisymmetric cross-ply plates

Large deflection bending analysis of antisymmetric rectangular cross-ply plates based on Von-Karman plate theory is investigated in this paper, with one-term approximation for the in-plane and transverse displacements, under sinusoidal loading. The presence of bending-stretching coupling in such plates results in an additional square nonlinear term which makes the solution (for example, displacements) load direction dependent, unlike isotropic, orthotropic, symmetric, square antisymmetric cross-ply and symmetric and antisymmetric angle-ply plates. The effect of this term on large deflection response, based on the Rayleigh-Ritz method is presented for various configurations and modular ratios of such plates. It is shown that the presence of this square term could alter the very nature of nonlinearity i.e. hardening or softening for certain values of the load parameter.