A new 4-node quadrilateral FE model with variable electrical degrees of freedom for the analysis of piezoelectric laminated composite plates

A new 4-node quadrilateral finite element is developed for the analysis of laminated composite plates containing distributed piezoelectric layers (surface bonded or embedded). The mechanical part of the element formulation is based on the first-order shear deformation theory. The formulation is established by generalizing that of the high performance Mindlin plate element ARS-Q12, which was derived based on the DKQ element formulation and Timoshenko’s beam theory. The layerwise linear theory is applied to deal with electric potential. Therefore, the number of electrical DOF is a variable depending on the number of plate sub-layers. Thus, there is no need to make any special assumptions with regards to the through-thickness variation of the electric potential, which is the true situation. Furthermore, a new “partial hybrid”-enhanced procedure is presented to improve the stresses solutions, especially for the calculation of transverse shear stresses. The proposed element, denoted as CTMQE, is free of shear locking and it exhibits excellent capability in the analysis of thin to moderately thick piezoelectric laminated composite plates.     

A study of interlaminar stresses in variable stiffness plates

Variable stiffness composite panels have been in continuous development for the last two decades. Several studies have been carried out to evaluate their structural response under different hypotheses. It is known that correct prediction of the onset of delamination in multilayer composite laminates requires an accurate evaluation of interlaminar stresses. Although finite element codes provide results for interlaminar stresses, they are not continuous both across and along layer interfaces due to the use of C⁰ interpolated elements. In this work, a methodology for obtaining interlaminar stresses is extended and applied to the case of variable stiffness composite panels. Pagano’s three layer reference case is investigated for both constant stiffness and variable stiffness cases. A set of analyses are carried out to demonstrate the performance of the method and the variation in the stress response of the panel due to changes in composite layup. Results show that the distribution of shear stresses in the panel presents an significant variation depending on the stacking sequence. A debonding failure criteria is used to evaluate the performance of different variable stiffness configurations, obtaining improvements compared with constant stiffness panels.     

Vibration analysis of variable stiffness laminated composite sandwich plates

Abstract

This paper presents the free vibration analysis of a variable stiffness laminated composite sandwich plates. The fiber orientation angle of the face sheets (Skin) is assumed to vary linearly with the x-axis. The problem formulation is based on the higher-order shear deformation plate theory HDST C⁰ coupled with p-version of finite element method. The elements of the stiffness and mass matrices are calculated analytically. The sandwich plate is presented with a uniform mesh of four p-elements and the convergence properties are achieved by increasing the degree p of the hierarchical shape functions. A calculation program is developed to determine the fundamental frequencies for different physical and mechanical parameters such as plate thickness, core to face sheets thickness ratio, orientation angle of curvilinear fibers and boundary conditions. The results obtained show a good agreement with the available solutions in the literature. New comparison study of vibration response of laminated sandwich plate between the straight and curvilinear fibers is presented.     

Theoretical modelling of laminated composite plates

Formulation of appropriate governing equations, simpler than the three-dimensional equations of elasticity yet capable of predicting, fairly accurately, all important response parameters such as stress and strain, is attempted in modelling a structural component. Several theoretical models are available in the literature for the analyses of plates. The emergence of fibre-reinforced plastics as an attractive form of structural construction, added a new complexity to the modelling considerations of laminates by requiring the estimation of the interlaminar stresses and strains. In this paper, modelling considerations of laminated composite plates are discussed. The classical laminated plate theory and higher-order shear deformation models are reviewed to bring out their interlaminar stress predictive capabilities, and some new modelling possibilities are indicated. This work has been supported by the Aeronautics Research and Development Board, Ministry of Defence, Government of India.     

Optimal lay-up design of variable stiffness laminated composite plates by a layer-wise optimization technique

The optimal lay-up design for the maximum fundamental frequency of variable stiffness laminated composite plates is investigated using a layer-wise optimization technique. The design variables are two fibre orientation angles per ply. Thin plate theory is used in conjunction with a p-element to calculate the fundamental frequencies of symmetrically and antisymmetrically laminated composite plates. Comparisons with existing optimal solutions for constant stiffness symmetrically laminated composite plates show excellent agreement. It is observed that the maximum fundamental frequency can be increased considerably using variable stiffness design as compared to constant stiffness design. In addition, optimal lay-ups for the maximum fundamental frequency of variable stiffness symmetrically and antisymmetrically laminated composite plates with different aspect ratios and various combinations of free, simply supported and clamped edge conditions are presented. These should prove a useful benchmark for optimal lay-ups of variable stiffness laminated composite plates.     

Natural modes of vibration of variable stiffness composite laminates with curvilinear fibers

In this paper, natural frequencies and vibrational mode shapes of variable stiffness composite laminate (VSCL) plates with curvilinear fibers are studied. In each ply of this rectangular VSCL, the fiber-orientation angle changes linearly with respect to the horizontal coordinate. To define the modes of vibration of the laminates, a new p-version finite element, which follows third-order shear deformation theory (TSDT), is employed. The convergence properties of this new element are investigated. Taking manufacturing restrictions regarding the fiber curvatures into account, maps of natural frequencies as functions of tow-orientation angles are determined in demonstrative examples. It is verified that the use of curvilinear fibers instead of the traditional straight fibers introduces a greater degree of flexibility, which can be used to adjust frequencies and mode shapes.     

Analysis of isotropic and laminated plates by an affine space decomposition for asymmetric radial basis functions collocation

As a first endeavor, this paper applies an affine space decomposition, proposed by Ling and Hon, to the static analysis of laminated plates. The radial basis functions collocation method by Kansa is modified by this affine space decomposition. The present approach can be seen as an improvement of the original Kansa's method, producing better conditioned matrices and very stable solutions for the static analysis of laminated plates. A static analysis of isotropic and laminated plates is performed by considering a first-order shear deformation plate theory. The equilibrium equations and the boundary conditions are interpolated by collocation with radial basis functions.     

A two-step optimization scheme for maximum stiffness design of laminated plates based on lamination parameters

The purpose of this paper is to propose an effective solution scheme of simultaneous optimization design of layup configuration and fiber distribution for maximum stiffness design of laminated plates. Firstly, a numerical analysis of the lamination parameters feasible region for a laminated plate consisting of various given number of ply groups (each ply group may have different thickness and all the fibers in one ply group are orientated in an identical direction) is carried out, and it is found that the feasible region based on only a few ply groups is very close to the overall one determined by infinite plies. Therefore, it is suggested that the feasible region of lamination parameters of a laminated plate could be approximately determined by the layup configuration of least ply groups. Secondly, a two-step simultaneous optimization scheme of layup configuration and fiber distribution for maximum stiffness design of laminated plates is proposed. Accordingly, by using ply thickness, fiber orientation angle and fiber volume fraction in a laminated plate of least ply groups as design variables, the optimal lamination parameters for maximum stiffness is obtained. Then, taking the optimal lamination parameters as the design objective, a detailed layup design optimization is implemented by considering some limitations on manufacturing, such as preset ply thickness, and specific fiber orientation angle and a limited maximum number of consecutive plies in the same fiber orientation. Numerical examples are also presented to validate the proposed two-step optimization scheme.     

Geometrically nonlinear NURBS isogeometric finite element analysis of laminated composite plates

This research present the development of geometrically nonlinear NURBS isogeometric finite element analysis of laminated composite plates. First-order, shear-deformable laminate composite plate theory is utilized in deriving the governing equations using a variational formulation. Geometric nonlinearity is accounted for in Von-Karman sense. A family of NURBS elements are constructed from refinement processes and validated using various examples. k-refined NURBS elements are developed to study thin plates. Isotropic, orthotropic and laminated composite plates are studied for various boundary conditions, length to thickness ratios and ply-angles. Computed center deflection is found to be in an excellent agreement with the literature. For thin plate analysis, linear and k-refined quadratic NURBS element is found to remedy the shear locking problem. k-refined quadratic NURBS element provide stabilized response to distorted, coarse meshes without increasing the order of the polynomial, owing to the increased smoothness of solution space.     

Analysis of laminated composite beams and plates with piezoelectric patches using the element-free Galerkin method

 An efficient meshfree formulation based on the first-order shear deformation theory (FSDT) is presented for the static analysis of laminated composite beams and plates with integrated piezoelectric layers. This meshfree model is constructed based on the element-free Galerkin (EFG) method. The formulation is derived from the variational principle and the piezoelectric stiffness is taken into account in the model. In numerical test problems, bending control of piezoelectric bimorph beams was shown to have the efficiency and accuracy of the present EFG formulation for this class of problems. It is demonstrated that the different boundary conditions and applied actuate voltages affects the shape control of piezolaminated composite beams. The meshfree model is further extended to study the shape control of piezo-laminated composite plates. From the investigation, it is found that actuator patches bonded on high strain regions are significant in deflection control of laminated composite plates. Received: 23 October 2001 / Accepted: 29 July 2002     

Aerothermoelastic analysis of composite laminated plates

The flutter behavior of a thermally buckled composite laminated plate is investigated in the frequency and time domains using the finite element method. Von Karman large deformation assumptions and quasi-steady aerodynamic theory are employed for the analysis. The effects of temperature gradient, panel length-to-width ratio, fiber orientation, and stacking sequence on aerothermoelastic behavior of the plate are studied in detail. The flutter boundary, buckling boundary, time history response, and phase plane plots of cross-ply and angle-ply laminates are presented. The numerical results show that temperature gradient induces thermal moments and increases the overall stiffness of the plate, and thus may increase the flutter boundary significantly. When the buckle pattern of the plate changes, the eigenvalues of the natural modes are changed suddenly and the sequence of the natural modes may be altered. Therefore, the change in the buckle pattern postpones the coalescence and the flutter boundary may increase. The change in the coalescence pair may also postpone the coalescence and increase the flutter boundary.     

Multi-objective design optimization of variable stiffness composite cylinders

The bending-induced buckling improvement in a variable stiffness (VS) composite cylinder (made by fiber steering) is studied. For such a cylinder, the effect of the variation of the direction of the load on its buckling performance of the cylinder is also examined. Compromise programming, as a multi-objective optimization method, is used to design for buckling of the VS cylinder subjected to bending load in either of the two opposite directions. Different combinations of weight factors for the structural performance in the two opposite directions were also applied to obtain the Pareto frontier as the main decision making tool for the designers in a multi-objective design problem.     

Nonlinear vibrations of angle-ply laminated circular cylindrical shells: Skewed modes

Skewed modes in geometrically nonlinear forced vibrations of angle-ply laminated circular cylindrical shells are investigated in the present study by using the Amabili–Reddy higher-order shear deformation theory. An harmonic force excitation is applied in radial direction and simply supported boundary conditions are assumed. The equations of motion are obtained by using an energy approach based on Lagrange equations that retains dissipation. Numerical results are obtained by using the pseudo-arclength continuation method and bifurcation analysis.     

Thermal postbuckling analysis of laminated composite plates

The thermal postbuckling behavior of graphite/epoxy multi-layered rectangular plates of various boundary conditions is studied using the finite element method. Temperature dependent thermal and elastic properties of the material are used in the analysis. The nonlinear finite element equations are solved as a sequence of linear eigenvalue problems to trace the thermal postbuckling paths of 15-layered symmetric angle-ply plates. The presence of secondary instability with an unsymmetric deformation mode has been identified for symmetric laminates under uniform temperature rise. In the case of linearly varying temperature rise through the thickness of the plate, the nonlinear equilibrium equations are solved by the modified Newton–Raphson technique to get the temperature-displacement curves.     

Panel flutter analysis of general laminated composite plates

The problem of nonlinear aeroelasticity of a general laminated composite plate in supersonic air flow is examined. The classical plate theory along with the von-Karman nonlinear strains is used for structural modeling, and linear piston theory is used for aerodynamic modeling. The coupled partial differential equations of motion are derived by use of Hamilton’s principle and Galerkin’s method is used to reduce the governing equations to a system of nonlinear ordinary differential equations in time, which are then solved by a direct numerical integration method. Effects of in-plane force, static pressure differential, fiber orientation and aerodynamic damping on the nonlinear aeroelastic behavior of the plate are studied. Results show that the fiber orientation has significant effect on dynamic behavior of the plate and the asymmetric properties, changes the behavior of the limit cycle oscillation.     

A semi-analytical finite element for laminated composite plates

This paper presents a semi-analytical finite element solution for laminated composite plates. The method is based on a mixed variational principle that involves both displacements and stresses. Finite element meshes are only used in the plane of plate, while the through thickness distributions of displacements and stresses are obtained using the method of state equations. Numerical results show that the rate of convergence of the new method is fast and the solutions can be very close to corresponding exact three-dimensional ones. The use of a recursive formulation of the state equations leads to an algebra equation system, from which solution are sought, whose dimension is independent of the numbers of layers of the plate considered.     

Dynamic analysis of laminated composite plates subjected to a moving oscillator by FEM

This paper presents a finite element model based on the first order shear deformation theory to investigate the dynamic behavior of laminated composite plates traversed by a moving oscillator. The oscillator model is assumed to be consisting of two nodal masses that are connected by means of a spring-damper unit. The governing equations of motion of two sub-systems are separately integrated by applying the Newmark’s time integration procedure. Then, the obtained equations are coupled and the responses of system components are calculated in each time step. The accuracy of algorithm is verified by comparing the numerical results of static, free vibration and simplified moving force problems analysis with the available exact solutions and other numerical results in the literature. Also, the effects of mass ratio, damping ratio of system components, stiffness of suspension system, velocity and eccentricity of moving oscillator on dynamic responses is parametrically studied. This algorithm can be applied to various boundary conditions, lamination schemes and fiber angels.     

Analysis for buckling and vibrations of composite stiffened shells and plates

This paper deals with development of triangular finite element for buckling and vibration analysis of laminated composite stiffened shells. For the laminated shell, an equivalent layer shell theory is employed. The first-order shear deformation theory including extension of the normal line is used. In order to take into account a non-homogeneous distribution of the transverse shear stresses a correction of transverse shear stiffness is employed. Based on the equivalent layer theory with six degrees of freedom (three displacements and three rotations), a finite element that ensures C⁰ continuity of the displacement and rotation fields across inter-element boundaries has been developed. Numerical examples are presented to show the accuracy and convergence characteristics of the element. Results of vibration and buckling analysis of stiffened plates and shells are discussed.     

Free vibration of moving laminated composite plates

Two-dimensional axially moving materials have a wide range of industrial applications such as papers, plastics and composites in producing lines, power transmission and conveyor belts, etc. In many of these instances, the moving material is not isotropic, but is a single-layer orthotropic material or consists of several orthotropic layers.

In this article, free vibration of axially moving symmetrically laminated plates subjected to in-plan forces is studied by classical plate theory. This category includes symmetric cross-ply and angle-ply laminates and anisotropic plates. Firstly, an exact method is developed to analyze vibration of multi-span traveling cross-ply laminates, and then a semi-analytical finite strip method is extended for moving symmetric laminated plates in general, with arbitrary boundary conditions. By the finite strip method intermediate elastic or rigid supports can also be added to the model of the moving plate. The supports may be in the form of point, line or local distributed supports.     

Detection of stiffness reductions in laminated composite plates from their dynamic response using the microgenetic algorithm

In this study, we investigate a method to detect damage in a laminated composite structure by analyzing its dynamic response to impact loads. The combined finite element method (FEM) with seven degrees of freedom (DOF) and the advanced microgenetic algorithm described in this paper may allow us not only to detect the damaged elements but also to find their locations and the extent of damage. A high order shear deformation theory (HSDT) is used to predict the structural behavior and to detect damage of laminated composite plates. The effects of noise associated with the uncertainty of measurements due to the complex nature of composites are considered for [0/90]_s and [±45]_s layup sequences. The results indicate that the new method is computationally efficient in characterizing damage for complex structures such as laminated composites.