Material characterization of laminated composite materials using a three-point-bending technique

A three-point-bending technique is presented for identifying the elastic constants of laminated composite materials. In the proposed technique, three strains in the axial, lateral, and 45° directions on the bottom surface at the mid-span of a symmetric angle-ply beam subjected to three-point-bending testing are measured for elastic constants identification. The narrow beam theory together with the trial elastic constants is used to predict the theoretical strains of the beam. The theoretical and experimental strains of the beams are then used in a stochastic optimization method to identify the elastic constants of the beam. The accuracy and applications of the proposed technique are demonstrated by means of a number of examples on the elastic constants identification of graphite/epoxy (Gr/ep) or glass/epoxy (Gl/ep) laminated composite materials. The effects of specimen aspect ratio and thickness on the accuracy of the proposed method are investigated.     

Detection of stiffness degradation in laminated composite plates by filtered noisy impact testing

The purpose of this paper is to detect damage (stiffness degradation) of laminated composite plates from noisy impact response data. The combined finite element method (FEM) with five degrees of freedom (DOF) and the advanced noise filtering algorithm described in this paper may allow us not only to detect the deteriorated elements but also to find their locations and the extents. A first order shear deformation theory (FSDT) is used to predict the structural behavior and to detect damage of laminated composite plates. The filtering procedure is designed by means of a wavelet decomposition together with a selection of the measuring points, and the optimization criterion is constructed on an estimate of the probability of detection using genetic algorithms. All these techniques are applied for the first time to composites. The effects of filtered noise associated with the uncertainty of measurements due to the complex nature of composites are considered for different layup sequences, number of layers, and length–thickness ratios. Several numerical results show that the noise filtering system is computationally efficient in identifying stiffness degradation for complex structures such as laminated composites.     

Free vibration of laminated piezoelectric composite plates based on an accurate theory

An accurate theory for laminated piezoelectric composite plates in cylindrical bending is developed for free vibration analysis. The displacement and electric potential fields are depicted approximately by the accurate 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 the electrostatics charge equation. The complicated electromechanical coupling relations and the interfacial continuity conditions are enforced. Accordingly the two functions are coupled and make the displacement and potential fields coupled. The governing equations use only four displacement and potential variables, the number of which is independent of the number of layers involved. A corresponding finite element model is also developed. Natural frequencies of piezoelectric laminates subjected to different sets of boundary conditions are given and parameter studies are conducted in numerical examples. The high accuracy of this theory is demonstrated by comparing the present results with the existing exact three-dimensional solutions.     

Coupling effects in bending, buckling and free vibration of generally laminated composite beams

A closed form expression to determine the effective flexural modulus of a laminated composite beam is developed and presented in this contribution. This effective flexural modulus is applied to the bending, buckling and free vibration response of generally laminated composite beams with various boundary supports. The expression was developed using the combination of the Euler–Bernoulli beam and classical lamination theory. In addition the results of an extensive finite element analysis are used to validate the analytical model. The comparison of the analytical results, the finite element results and the experimental results showed good correlation. It is also observed that coupling response is an important variable that must be included in the computation of the effective flexural stiffness of generally laminated beam.     

Dynamic analysis of laminated composite plates traversed by a moving mass based on a first-order theory

The dynamic response of angle-ply laminated composite plates traversed by a moving mass or a moving force is investigated. For this purpose, a finite element method based on the first-order shear deformation theory is used. Stationary and adaptive mesh techniques have been applied as two different meshing schemes. The adaptive mesh strategy is then used to avoid off-nodal position of moving mass. In this manner, the finite element mesh is continuously adapted to follow and comply with the path of moving mass. A Newmark direct integration method is employed to solve the equations of motion. Parametric study is directed to find out how different parameters like mass of the moving object as well as the type of the angle-ply laminated composite plates affect the dynamic response. Numerical results show the significant effects of the stacking order on the dynamic responses of the composite structures under a moving mass. It is found that although [30/−60/−60/30] lamination shows the highest maximum vertical deflection but [−45/45/45/−45] lamination has the highest value of the dynamic amplification factor. The dynamic amplification factor for different stacking orders and mass velocities is less than 1.25.     

Stochastic progressive failure analysis of laminated composite plates using Puck's failure criteria

The present work deals with second-order statistics of the progressive failure response of laminated composite plates subjected to in-plane uniaxial and bi-axial loadings with random system properties. A stochastic finite element method based on higher-order shear deformation theory combined with first- and second-order perturbation technique is used for solution of random progressive failure equation. A Puck failure criterion is used for the evaluation of first ply and last-ply failure load. The results obtained using the present solution approach are validated with the results available in the literature and Monte Carlo simulation.     

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.     

The structural intensities of composite plates with a hole

The structural intensities in isotropic and orthotropic laminae and composite laminated plates with and without a hole of different rim conditions are studied using the finite element method. Patterns of superposition among structural stress fields, intensity vectors, mode shapes and streamlines of energy flow are obtained for the various cases. It is found that the hole completely changes the distributions of the stress and energy flow path in plates. Despite its nature of mode dependent the pattern of energy flow may be different from each other despite having similar structural mode shapes, moreover the position of maximum stress is not corresponding to the position of maximum structural intensity.     

Stochastic nonlinear failure analysis of laminated composite plates under compressive transverse loading

This paper present the second ordered statistics of first-ply failure response of laminated composite plate with random material properties under random loading. The basic formulation is based on higher order shear deformation plate theory (HSDT) with the geometrically nonlinearity in the von-Karman. The direct iterative based C⁰ nonlinear finite element method combined with mean centered first order perturbation technique developed by the authors are extended and successfully applied nonlinearity for failure problem with a reasonable accuracy to predict the second order statistics (standard deviation) of first-ply failure response using Tsai-Wu and Hoffman failure criterion with macroscopic analysis. Typical numerical results for various combinations of boundary conditions, plate thickness ratios, aspect ratios, laminates scheme and layers, elastic modulus ratios have been presented to illustrate the application of developed procedure. Some new results are presented and examined which clearly demonstrated the importance of the randomness in the system parameters in the failure response of the structures subjected to transverse loadings.     

Nonlinear static response of laminated composite plates by discrete singular convolution method

In this paper, large deflection analysis of laminated composite plates is analysed. Nonlinear governing equation for bending based on first-order shear deformation theory (FSDT) in the von Karman sense is presented. These equations have been solved by the method of discrete singular convolution (DSC). Regularized Shannon’s delta (RSD) kernel and Lagrange delta sequence (LDS) kernel are selected as singular convolution to illustrate the present algorithm. The effects of plate aspect ratio, fiber orientation, boundary conditions, thickness-to-side ratio, and applied load on the nonlinear static response of the laminated plate are investigated.     

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.     

Modeling of the dynamic delamination of L-shaped unidirectional laminated composites

One of the widely used geometrically complex parts in advanced commercial aircraft is the L-shaped composite. Due to the sharp curved geometry, interlaminar opening stresses are induced and delamination occurs under considerable mode-mixities in L-shaped beams. Dynamic phenomena during delamination initiation and propagation of L-shaped beams are investigated using dynamic (explicit) finite element analysis in conjunction with cohesive zone methods. The 2-D model consists of 24 plies of unidirectional CFRP laminate with an initial 1 mm crack at the center of the laminate at the bend. Loading is applied parallel to one of the arms quasi-statically. The loading type yields different traction fields and mode-mixities in the two sides of the crack in which delamination occurs under shear stress dominated loading on one crack tip and opening stress dominated loading on the other. The speed of the delamination under shear dominated loading at one side is 800 m/s and under normal stress dominated loading is 50 m/s. In addition radial compressive waves at the interface are observed. Finally, as the thickness is changed, a different failure mode is observed in which a secondary crack nucleates at the arm and propagates towards the center crack.     

Reliability predictions of laminated composite plates with random system parameters

Reliability predictions of laminated composite plates with random system parameters subjected to transverse loads are performed using different methods. System parameters such as material properties, layer thicknesses, and lamina strengths of a laminated composite plate are treated as base-line random variables and an appropriate failure criterion is used to construct the limit state equation of the plate in the reliability analysis. Based on the statistics of the base-line random variables obtained from experiments, different methods, namely, Monte Carlo method, β method, and first-order second moment method, are used to calculate the reliability of the laminated composite plates. In the first-order second moment method, the stochastic finite element method is used to derive for the statistics of the first-ply failure load of the laminated composite plates from those of the base-line random variables. The reliability of the laminated plate is then computed using the theoretically determined statistics together with an assumed probability distribution function of the first-ply failure load. The feasibility and accuracy of the different methods are studied by means of the experimental data of centrally loaded laminated composite plates with different lay-ups. The suitability of several commonly used failure criteria for reliability analysis of laminated composite plates is also investigated by means of several examples.     

A closed form solution for linear and nonlinear free vibrations of composite and fiber metal laminated rectangular plates

In this paper, the nonlinear equations of motion for laminated composite rectangular plates based on first order shear deformation theory, which include shear deformation and rotary inertia, have been derived. Then, through introducing a force function, these equations reduced to a set of coupled nonlinear partial differential equations and a compatibility equation. By using the Galerkin method, for the first time, a nonlinear ordinary differential equation is obtained, which includes nonlinear inertia and stiffness terms. By using the multiple time scales method, analytical relations for nonlinear frequency and transverse displacement have been obtained. Results are compared with the literature and good agreement is achieved for both linear and nonlinear frequencies. After proving the validity of our work for isotropic rectangular plates and laminated rectangular plates, linear and nonlinear free vibration of a Fiber Metal Laminate panel have been investigated. Also the effects of some system parameters on the nonlinear frequency have been investigated.     

Thermal buckling behavior of composite laminated plates

Thermal buckling behavior of composite laminated plates was studied by making the use of finite element method. The thermal buckling mode shapes of cross-ply and angle-ply laminates with various E₁/E₂ ratios, aspect ratios, fiber angle, stacking sequence and boundary condition were studied in detail. The results indicate that the high E₁/E₂ and α₂/α₁ ratios of AS4/3501-6 and T 300/5208 laminates produce higher bending rigidity along the fiber direction and higher in-plane compressive force in a direction perpendicular to the fiber direction. Therefore, the higher thermal buckling mode shapes are formed. The thermal buckling mode that composite laminated plate will buckle into is mainly dependent on the E₁/E₂ ratio, α₂/α₁ ratio, fiber orientation and aspect ratio of the plate.     

Finite element modeling of low-velocity impact on laminated composite plates and cylindrical shells

In this study, composite laminates and shell structures subjected to low-velocity impact are investigated by numerical analysis using ABAQUS finite element code. In order to model the impact phenomena by commercial finite element codes, various procedures are available. Accurate modeling requires the appropriate selection of element type, solution method, impactor modeling method, meshing pattern and contact modeling. In this investigation, by considering several case studies with various conditions, validity of the existed modeling processes is examined. In each case, by comparing the results of various methods with the related available experimental test results in existing literature, the best procedure is proposed which can serve as benchmark method in low-velocity impact modeling of composite structures for future investigations.     

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.     

In-plane free vibrations of single-layer and symmetrically laminated rectangular composite plates

This work presents accurate upper-bound solutions for free in-plane vibrations of single-layer and symmetrically laminated rectangular composite plates with an arbitrary combination of clamped and free boundary conditions. In-plane natural frequencies and modes shapes are calculated by the Ritz method with a simple, stable and computationally efficient set of trigonometric functions. Reliability of the method is assessed by comparison with known accurate solutions for isotropic plates and specially orthotropic single laminates. The extensive results presented here for the first time can be considered benchmark data against which other methods may be compared and validated in the future. Influence of fiber orientation, stacking sequence, degree of orthotropy, aspect ratio and boundary conditions upon the in-plane vibration behavior are also discussed.     

Boundary element formulation for the coupled stretching-bending analysis of thin laminated plates

The boundary integral equations for the coupled stretching-bending analysis of thin laminated plates involve an integral which will be singular when the field point approaches the source point. To avoid the singular problem occurring in the numerical programming, the boundary integral equations are modified in which the integrals of singular part are integrated analytically. The analytical solutions for the free term coefficients and singular integrals are obtained in explicit closed-form. By dividing the boundary into elements and using suitable interpolation polynomials for basic functions, the set of equations necessary for boundary element programming are written explicitly for regular nodes and corner nodes. The equations for the determination of displacements and stresses at internal points are also presented in this paper.     

A design of laminated composite plates using graded orthotropic fiber-reinforced composite plies

A new lamination scheme is proposed through the design of a graded orthotropic fiber-reinforced composite ply for achieving continuous variations of material properties along the thickness direction of laminated composite plates. First, a micro-structure of graded unidirectional fiber-reinforced composite ply is designed and its effective graded elastic properties are estimated using finite element procedure. Next, the new lamination scheme is demonstrated through the conversion of a conventional laminated composite plate (CLCP) into a conventional-graded laminated composite plate (CGLCP) utilizing presently designed graded orthotropic composite ply. The suitability of this conversion/proposed lamination scheme is substantiated through the bending analysis of both the plates (CLCP and CGLCP).