Antisymmetric Angle-ply Plates Research Articles

Exact solutions for bimorph cross-ply and antisymmetric angle-ply plate piezoelectric energy harvesters

This paper presents the exact solutions for the plate type piezoelectric bimorph energy harvesters composed of laminated cross-ply or angle-ply substrate plate. Both series and parallel connections of the piezoelectric layers of the bimorphs are considered for deriving the exact solutions. Since the exact solutions for the layers of the laminated substrate plate are available in the open literature exact solutions for the piezoelectric layers are only derived here for finding the exact solutions of the harvesters. The piezoelectric layers should be orthotropic or generally orthotropic for deriving the exact solutions. Three sets of exact solutions for the piezoelectric layers are required to be derived. The electric potential induced by the bimorph in which the piezoelectric layers are connected in series is significantly larger than that in case of the bimorph with piezoelectric layers connected in parallel. The effects of the variations of the off-axis angle in the generally orthotropic piezoelectric layer, the length over width ratio of the harvester and the thickness of the piezoelectric layers on the induced electric potential have been studied. The expressions for the exact solutions can be used in a straight forward manner for verifying the experimental and the numerical results.

Three-dimensional exact elasticity solutions for antisymmetric angle-ply laminated composite plates

This paper is concerned with the derivation of the benchmark three-dimensional exact solutions for the static analysis of rectangular antisymmetric angle ply plates. The expressions for the exact solutions of displacements and stresses can be used in a straight forward manner to find the numerical results for any fiber orientation angle. It has been explored that the expressions for the exact solutions of displacements and stresses in the square antisymmetric angle-ply plates with 45° fiber orientation angle cannot be used for other fiber orientation angles and vice versa. Comparison of the exact solutions with the analytical solutions employing the first order shear deformation theory (FSDT) reveals that the FSDT can be efficiently used for thin and two layered thick antisymmetric angle-ply plates.

Buckling load optimization of laminated plates resting on Pasternak foundation using TLBO

This paper deals with the maximization of the critical buckling load of simply supported antisymmetric angle-ply plates resting on Pasternak foundation subjected to compressive loads using teaching learning based optimization method (TLBO). The first order shear deformation theory is used to obtain governing equations of the laminated plate. In the present optimization problem, the objective function is to maximize the buckling load factor and the design variables are the fibre orientation angles in the layers. Computer programming is developed in the MATLAB environment to estimate optimum stacking sequences of laminated plates. A comparison also has been performed between the TLBO, genetic algorithm (GA) and differential evolution algorithm (DE). Some examples are solved to show the applicability and usefulness of the TLBO for maximizing the buckling load of the plate via finding optimum stacking sequences of the plate. Additionally, the influences of different number of layers, plate aspect ratios, foundation parameters and load ratios on the optimal solutions are investigated.

Vibration of antisymmetric angle-ply laminated plates under higher order shear theory

This paper deals with the analysis of vibration of antisymmetric angle-ply plates using spline method for higher order shear theory. Free vibration of laminated plates is addressed to show the capability of the present method in the vicinity of higher order shear deformation theory and simply supported edges of plates. The coupled differential equations are obtained in terms displacement and rotational functions. These displacement and rotational functions are approximated using cubic and quantic spline. A generalized eigenvalue problem is obtained and solved numerically for an eigenfrequency parameter and an associated eigenvector of spline coefficients. The antisymmetric angle-ply fiber orientation are taken as design variables. Numerical results enable us to examine the frequencies for various geometric and material parameters and accuracy and effectiveness of the proposed method is also verified by comparative study.

Free vibration of anti-symmetric angle-ply plates with variable thickness

Free vibration of anti-symmetric angle-ply plates with variable thickness is analysed using spline function approximation including the effect of shear deformation. The equations of motion for the plate are derived using the theory of Yang, Norris and Stavsky. Assuming the solution in a separable form, a system of coupled differential equations in displacement and rotational functions are obtained and these functions are approximated by Bickley-type splines of order three. A generalised eigenvalue problem is obtained and solved numerically for an eigenfrequency parameter and an associated eigenvector of spline coefficients. Two and four layered plates consisting of two different materials and plies comprising of same as well as different materials for two different boundary conditions are analysed. The effect of material properties, ply orientation, number of lay ups, aspect ratio and coefficients of thickness variations on the frequency parameter are presented. The accuracy of the result is ascertained by convergence and comparative study.

Dynamic Resonance of Curved Panels in Adverse Hygrothermal Environment

The present paper deals with the dynamic resonance of composite curved panels subjected to periodic dynamic loadings. The effects of various parameters of four-sided clamped composite curved panels at elevated temperatures and moisture concentrations on the principal instability regions are investigated by finite element method which is used to study the antisymmetric angle-ply square plates. The results show that instability of composite plates occurs for different parameters in adverse hygrothermal environment. The experimental and numerical investigation is also carried out for four-sided clamped boundary condition for vibration and buckling of curved panels in hygrothermal environment.

Active damping of geometrically nonlinear vibrations of laminated composite plates using vertically reinforced 1-3 piezoelectric composites

This paper addresses the analysis of active constrained layer damping (ACLD) of geometrically nonlinear transient vibrations of laminated composite plates using vertically reinforced 1-3 piezoelectric composite (PZC) as the material of the constraining layer of the ACLD treatment. The Von Karman type nonlinear strain-displacement relations and the first-order shear deformation theory (FSDT) are used for deriving the coupled electromechanical nonlinear finite element model. The Golla–Hughes–McTavish (GHM) method has been used to model the constrained viscoelastic layer of the ACLD treatment in the time domain. The numerical results indicate that the ACLD patches significantly improve the damping characteristics of the cross-ply and antisymmetric angle-ply plates for suppressing the geometrically nonlinear transient vibrations of the plates.

The Spline Finite Element Method for the Analysis of the Dynamic Response of Composite Material Plate

Based on Kirchhoff’s classical theory and adoptation the spline finite element method, three independent displacements are interpolated into the anti-symmetric angle-ply laminated plate by the spline base of the cubic spline B function, and the stiffness array and the quality array of the composite material laminated plate are derived. By using Rayleigh-Reeds method, the damping array is established; kinetic equation is derived from Lagrange equation; the response equation is expressed; the inherent frequency of the anti-symmetric angle-ply plate is found by calculation subject to the different proportions of the length to the width, the different number of the layers of the plate and different angles; and the dynamic response problems about the anti-symmetric angle-ply laminated plate are analyzed and solved. Dynamic response problems, such as the displacement, velocity and acceleration, are solved with spline collocation method, through choosing the sample strip functions as mode shape function for various boundary conditions in several numerical examples and the results are compared with Newmark’s method. And better result can be yielded. The example calculations prove the validity of the calculating ways introduced by this paper.

Active constrained layer damping of geometrically nonlinear transient vibrations of composite plates using piezoelectric fiber-reinforced composite

This paper addresses the analysis of active constrained layer damping (ACLD) of geometrically nonlinear transient vibrations of laminated thin composite plates using piezoelectric fiber-reinforced composite (PFRC) materials. The constraining layer of the ACLD treatment is considered to be made of the PFRC materials. The Golla–Hughes–McTavish (GHM) method has been used to model the constrained viscoelastic layer of the ACLD treatment in the time domain. A finite element model has been developed for the cross-ply and antisymmetric angle-ply plates integrated with the patches of ACLD treatment undergoing geometrically nonlinear vibrations. The Von Ka`rma`n-type nonlinear strain displacement relations and the first-order shear deformation theory (FSDT) are used for deriving this coupled electromechanical nonlinear finite element model. The numerical results indicate that the ACLD patches significantly improve the damping characteristics of the cross-ply and antisymmetric angle-ply plates for suppressing the geometrically nonlinear transient vibrations of the plates. Emphasis has also been placed on investigating the effect of variation of fiber orientation in the PFRC material on the control authority of the ACLD patches.

Semi-energy finite strip post-buckling analysis of laminated plates concerning the effects of mechanical coupling

In composite laminated plates the mechanical coupling is one of the significant properties which dominate the performance of a laminated plate structure in large out-of-plane deformations. Some effects of mechanical coupling on the post-buckling behavior of composite laminated plates have been studied in this paper by implementing a finite strip approach based on the concept of a rigorous post-buckling solution for composite plates and plate structures, namely the semi-energy approach, which has been originally developed by the authors of current paper. Moreover, to check the validity and soundness of presented semi-energy FSM the results obtained from developed approach are compared with those obtained from a fairly conventional and well exploited finite strip technique, namely the full-energy finite strip method as well as the obtained results from ANSYS (i.e. the general purpose finite element method software). In this paper, the post-buckling behavior of an anti-symmetric angle-ply square plate with the stacking sequence configuration as [±45]2 has been studied. Moreover, according to anti-symmetric nature of lamination in the mentioned plate there exists a mechanical coupling between in-plane membrane and out-of-plane twisting curvature; therefore, the results of this analysis have been referred as coupled solution in this paper. In order to evaluate the effects of mechanical coupling on the post-buckling performance of mentioned plates, the uncoupled solution has also been determined by precluding the coupling terms (i.e. B16, B26). The post-buckling load–end shortening and load-maximum out-of-plane deflection paths have been obtained for coupled as well as uncoupled solutions. The comparison between the obtained results by coupled and uncoupled solutions revealed markedly different behaviors due to presence of mechanical coupling in the anti-symmetric angle-ply laminates. The outcome of the paper can help structural designers have a better understanding of the effects of mechanical coupling on the post-buckling behavior of laminated composite plates at the design stage.

Higher Order Refined Computational Models for the Free Vibration Analysis of Antisymmetric Angle Ply Plates

Analytical formulations using two higher order refined displacement models have been developed and solutions presented for the first time to the natural frequency analysis of antisymmetric angle-ply composite and sandwich plates. These computational models already reported in the literature are based on Taylor's series expansion of the displacements in the thickness coordinate and consider the realistic parabolic distribution of transverse shear strains through the laminate thickness. One of them, with 12 degrees of freedom, considers the effects of both transverse shear and normal strain/stress while the other with 9 degrees of freedom includes only the effect of transverse shear deformation. In addition to above, a few higher order models and the first order model developed by other investigators and available in the literature are also considered for the evaluation. A simply supported plate is considered throughout as a test problem. The equations of motion are obtained using Hamilton's principle. Solutions are obtained in closed form using Navier's technique by solving the eigenvalue equation. Plates with varying slenderness ratios, number of layers, fiber orientations, degrees of anisotropy, edge ratios and thickness of core to thickness of face sheet ratios are considered for the analysis. Accuracy of the theoretical formulations and the solution method is first ascertained by comparing the results with those already available in the literature. After establishing the accuracy of the solutions, extensive numerical results are presented for the free vibration analysis of multilayer antisymmetric angle-ply composite and sandwich plates using all the models, which will serve as a benchmark for future investigations.

Analytical solutions using a higher order refined computational model with 12 degrees of freedom for the free vibration analysis of antisymmetric angle-ply plates

Abstract Analytical formulations and solutions to the natural frequency analysis of simply supported antisymmetric angle-ply composite and sandwich plates hitherto not reported in the literature based on a higher order refined computational model with 12 degrees of freedom already reported in the literature are presented. The theoretical model presented herein incorporates laminate deformations which account for the effects of transverse shear deformation, transverse normal strain/stress and a nonlinear variation of in-plane displacements with respect to the thickness coordinate thus modelling the warping of transverse cross sections more accurately and eliminating the need for shear correction coefficients. In addition, another higher order computational model with five degrees of freedom already available in the literature is also considered for comparison. The equations of motion are obtained using Hamilton’s principle. Solutions are obtained in closed-form using Navier’s technique by solving the eigenvalue equation. Plates with varying slenderness ratios, number of layers, degrees of anisotropy, edge ratios and thickness of core to thickness of face sheet ratios are considered for analysis. Numerical results with real properties using above two computational models are presented and compared for the free vibration analysis of multilayer antisymmetric angle-ply composite and sandwich plates, which will serve as a benchmark for future investigations.

An Elegant and Rigorous Analytical Solution for Anti-symmetric Angle-ply Rectangular Plates with any Combination of Simply Supported and Clamped Edges

A superposition approach based on exact untruncated infinite series solutions, developed recently for cross-ply plates, is extended herein to obtain a fast-convergent analytical solution for anti-symmetric angle-ply plates with any combination of S1, S3, C1, and C3 type edges under arbitrary loading. Numerical results are presented to highlight the differences between the various edge conditions. It is finally concluded that the results are significantly influenced by the in-plane conditions when the edges are simply supported, but not when they are clamped.

Higher order refined computational model with 12 degrees of freedom for the stress analysis of antisymmetric angle-ply plates – analytical solutions

Abstract Analytical formulations and solutions for the stress analysis of simply supported antisymmetric angle-ply composite and sandwich plates hitherto not reported in the literature based on a higher order refined computational model with twelve degrees of freedom already reported in the literature are presented. The theoretical model presented herein incorporates laminate deformations which account for the effects of transverse shear deformation, transverse normal strain/stress and a nonlinear variation of in-plane displacements with respect to the thickness coordinate thus modelling the warping of transverse cross sections more accurately and eliminating the need for shear correction coefficients. In addition, two higher order computational models, one with nine and the other with five degrees of freedom already available in the literature are also considered for comparison. The equations of equilibrium are obtained using Principle of Minimum Potential Energy (PMPE). Solutions are obtained in closed form using Navier’s technique by solving the boundary value problem. Accuracy of the theoretical formulations and the solution method is first ascertained by comparing the results with that already available in the literature. After establishing the accuracy of the solutions, numerical results with real properties using all the computational models are presented for the stress analysis of multilayer antisymmetric angle-ply composite and sandwich plates, which will serve as a benchmark for future investigations.

Elastoplastic Stress Analysis in a Woven Ni—Cr Reinforced Thermoplastic-matrix Composite Laminated Plates under Transverse Loading

In this study, an elastoplastic stress analysis is carried out on the symmetric and antisymmetric cross-ply and angle-ply thermoplastic-laminated plates for transverse loading. A low density polyethylene (LDPE) thermoplastic-matrix and woven Ni—Cr fibers are used to produce composite plates by using molds. A first-order shear deformation theory and a nine-node Lagrangian finite element are used. Mathematical formulation is given for the elastoplastic stress analysis of a laminated plate for small deformations. The yield points and stresses are obtained for symmetric and antisymmetric clamp supported laminated plates with a square hole. The residual stresses in the thermoplastic plates are given in the tables. The expansion of plastic zones is illustrated for 200, 350, and 500 loading steps. The yield points and expansion of plastic zones are compared for different plates. The mechanical properties of a layer are obtained experimentally. The intensity of the residual stress components is maximum near the square hole and clamped supports.

Free and Forced Vibration of Antisymmetric Angle-Ply Laminated Plate Strips in Cylindrical Bending

The dynamic behavior of antisymmetric angle-ply laminated plate strips in cylindrical bending is examined using the classical theory and the first-order shear deformation theory. Free vibration of the plate strip is explored, and accurate natural frequencies are determined for general boundary conditions. A generalized modal approach is presented to solve the dynamic response of antisymmetric angle-ply plate strips with arbitrary boundary conditions and for arbitrary loading. The exact analytical solutions obtained are illustrated numerically in a number of figures revealing the influences of the transverse shearing deformation, of the number of layers, of the ply angles, and of the character of boundary conditions in the fundamental frequencies and the dynamic response.

Stresses and Deflection under Transverse Loading of Antisymmetric Angle-Ply Clamped Plates Including the Effect of Shear Deformation

A mathematical formulation utilized the Ritz technique with Lagrange multipliers to predict the behavior of laminated composites under uniform transverse loading with particular focus on the Antisymmetric Angle Ply Clamped plates. The numerical results, with the effect of shear deformation theory, for both graphite/epoxy and boron/epoxy plates are presented. Cases of deflections and stresses of the laminate are analyzed for several geometric parameters of the plate. A brief examination to failure also studied using both Tsai Wu theory and the recently developed Yeh Stratton criterion.

Analytical Solutions for Bending of Clamped Layered Composite Plates Using a Higher-Order Theory

Analytical solutions are presented for the bending of clamped laminated plates using a higher-order shear deformation theory. Unlike the first-order shear deformation theory, the higher-order theory does not require a shear correction factor. The displacement field of the higher-order theory contains the same number of dependent variables as in the first-order shear deformation theory, and also accounts for the cubic variation of the in-plane displacements through the thickness. Closed-form solutions are not tractable for clamped composite plates due to the boundary conditions and anisotropy of the plate and hence approximate methods are required. In the present work, a method based on Lagrange multipliers is used to enforce the boundary conditions not satisfied by the assumed series. Results are presented for antisymmetric cross-ply and angle-ply plates to study the influence of anisotropy, aspect ratio, and loadings on the deflections and stresses of clamped plates.

Stability And Vibration Of Initially Stressed Thick Laminated Plates

Equations of motion for antisymmetric angle-ply plates in a general state of non-uniform initial stresses, where the effects of transverse shear and rotatory inertia are included, are derived by using Trefftz equations and the variational principle. These equations are then non-dimensionalized to a generic form by introducing global constants in orthotropic space and properly rearranging on-axis to off-axis stiffness relations for angle-ply laminates. Finally, the stability and vibration equations are solved for simply supported rectangular plates in a state of normal stress plus an edge twisting stress. The effects of twisting and the global material constants on the buckling and vibration of plates are studied.

Nonlinear vibrations of heated antisymmetric angle-ply laminated plates

Large amplitude vibrations, buckling and post-buckling analysis of heated angle-ply laminated plates have been considered using the parabolic shear deformation theory. Strains due to initial imperfections have also been retained using the von Karman type large deflection model. Numerical results are obtained by using the single mode approach to simply-supported plates, thus reducing five governing equations to a single nonlinear time differential equation involving quadratic and cubic nonlinearities. The effects of initial imperfections and the temperature loading on the response characteristics of the plate have been studied. Numerical results for isotropic and antisymmetric angle-ply plates have been presented and discussed.