Dynamic investigation of laminated composite beams with shear and rotary inertia effect subjected to the moving oscillators using FEM

An algorithm based on the finite element method (FEM) has been developed to study the dynamic response of composite laminated beams subjected to the moving oscillator. The first order shear deformation theory (FSDT) is assumed for the beam model. The algorithm accounts for the complete dynamic interaction between the components of system. The proposed method can also be applied to the general moving mass and the simplified moving force problems. After deriving the governing equations of motion of beam and oscillator, the corresponding equations of motion are integrated by applying the Newmark’s time integration procedures to obtain the system responses in each time step. The numerical results of free vibration and moving force problems analysis of isotropic and composite laminated beams are presented and, whenever possible, compared to the available analytical solution and other numerical results in order to demonstrate the accuracy of the present method. In addition, parametric analysis is carried out over a wide range of velocities and mass, frequency and damping ratios of system components.     

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.     

Finite volume analysis of laminated composite plates

A numerical procedure for analysis of general laminated plates under transverse load is developed utilizing the Mindlin plate theory, the finite volume discretization, and a segregated solution algorithm. The force and moment balance equations with the laminate constitutive relations are written in the form of a generic transport equation. In order to obtain discrete counterparts of the governing equations, the plate is subdivided into N control volumes by a Cartesian numerical mesh. As a result, five sets of N linear equations with N unknowns are obtained and solved using the conjugate gradient method with preconditioning. For the method validation, a number of test cases are designed to cover thick and thin laminated plates with aspect ratio (width to thickness) from 4 to 100. Simply supported orthotropic, symmetric cross‐ply, and angle‐ply laminated plates under uniform and sinusoidal pressure loads are solved, and results are compared with available analytical solutions. The shear correction factor of 5/6 is utilized throughout the procedure, which is consistent with test cases used in the reviewed literature. Comparisons of the finite volume method results for maximum deflections at the center of the plate and the Navier solutions obtained for aspect ratios 10, 20, and 100 shows a very good agreement. Copyright © 2016 John Wiley & Sons, Ltd.     

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.     

Dynamic response of thick laminated annular sector plates subjected to moving load

The dynamic response of thick laminated annular sector plates with simply supported radial edges subjected to a radially distributed line load, which moves along the circumferential direction, is studied. A three-dimensional hybrid method composed of series solution, the layerwise theory and the differential quadrature method in conjunction with the finite difference method is employed. The fast rate of convergence and high accuracy of the method are demonstrated through different examples. Additionally, as a limit case, the out-of-plane dynamic responses of circular curved beams is obtained and compared with those of an unconstrained higher order shear deformation curved beam theory, which is formulated here. Then, the effects of different parameters such as the sector angle, thickness-to-outer radius ratio, ply lay out and the load velocity on the out-of-plane response of the symmetric and antisymmetric cross-ply laminated sector plates are investigated. The results can be used as benchmark solutions for future works.     

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 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.     

An enriched macro finite element for the static analysis of thick general quadrilateral laminated composite plates

This article presents the formulation of an enriched macro finite element based on the trigonometric shear deformation theory for the static analysis of symmetrically laminated composite plates. Shear correction factor is not required because this theory accounts for tangential stress-free boundary conditions on the plate boundary surfaces. The macro element is obtained using the principle of virtual work and Gram-Schmidt orthogonal polynomials as enrichment functions. The implementation of the obtained algorithm is simple and efficient, and allows studying general quadrilateral plates with a single macro element. Several examples are presented to show the capability and applicability of the developed formulation.     

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.     

Free vibration analysis of multilayered composite cylinder consisting fibers with variable volume fraction

In this paper, free vibration of a fiber reinforced composite cylinder in which volume fraction of its fibers vary longitudinally, is studied using a semi-analytical method. The distribution of volume fraction of fiber in base matrix is based on power law model. A micromechanical model is employed to represent its mechanical properties including elastic and physical properties of this composite cylinder. In addition, kinematically the first order shear deformation shell theory is employed for strain field. Then, weak form formulation and spatial approximations of variables are utilized to discretize the equations of motion. Different problems are solved in which primarily the validity of the results obtained for natural frequencies are evaluated by those similar results reported in the literature and with other commercial F.E. code for different boundary conditions. Furthermore, for different values of volume fractions and under various boundary conditions, computed natural frequencies of this composite cylinder with variable volume fraction of fiber are compared with the traditional one in which the volume fraction of fiber is constant throughout the structure. In spite the fact that average volume fraction of fiber and the layer fiber orientation are the same in both configurations, numerical results show that using variable volume fraction of fiber affects the shell natural frequencies and its mode shapes.     

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.     

Exact 3D stress analysis of laminated composite plates by sampling surfaces method

A paper focuses on the use of the efficient approach to exact 3D elasticity solutions of cross-ply and angle-ply laminated composite plates. This approach is based on the new method of sampling surfaces (SaS) developed recently by the authors. We introduce inside the nth layer I_n not equally spaced SaS parallel to the midsurface of the plate and choose displacements of these surfaces as fundamental plate unknowns. Such an idea permits the representation of the proposed higher order layer-wise plate theory in a very compact form. This fact gives in turn the opportunity to derive the exact 3D solutions of elasticity for thick and thin laminated composite plates with a prescribed accuracy by utilizing a sufficiently large number of SaS, which are located at interfaces and Chebyshev polynomial nodes.     

Dynamic stability of laminated FGM plates based on higher-order shear deformation theory

This paper conducts a dynamic stability analysis of symmetrically laminated FGM rectangular plates with general out-of-plane supporting conditions, subjected to a uniaxial periodic in-plane load and undergoing uniform temperature change. Theoretical formulations are based on Reddys third-order shear deformation plate theory, and account for the temperature dependence of material properties. A semi-analytical Galerkin-differential quadrature approach is employed to convert the governing equations into a linear system of Mathieu–Hill equations from which the boundary points on the unstable regions are determined by Bolotins method. Free vibration and bifurcation buckling are also discussed as subset problems. Numerical results are presented in both dimensionless tabular and graphical forms for laminated plates with FGM layers made of silicon nitride and stainless steel. The influences of various parameters such as material composition, layer thickness ratio, temperature change, static load level, boundary constraints on the dynamic stability, buckling and vibration frequencies are examined in detail through parametric studies.This work was fully supported by grants from the Australian Research Council (A00104534) and from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CityU 1024/01 E). The authors are grateful for this financial support.     

Bending analysis of thick laminated plates using extended Kantorovich method

This study is concerned with bending of moderately thick rectangular laminated plates with clamped edges. The governing equations, based on Reissner first-order shear deformation plate theory; in terms of deflection and rotations of the plate include a system of three second-order, partial differential equations (PDEs). Application of extended Kantorovich method (EKM) to the system of partial differential equations reduces the governing equations to a double set of three second-order ordinary differential equations in the variables x and y. These sets of equations were then solved in an iterative manner until convergence was achieved. Normally three to four iterations are enough to get the final results with desired accuracy. It is demonstrated that, unlike other weighted residual methods, in the extended Kantorovich method initial guesses to start iterations are arbitrary and not even necessary to satisfy the boundary conditions. Results of this study also reveal that the convergence of the EKM is rapid and the method is an efficient way to solve system of PDEs of the same type. To compare the results of this study, the problem was also analyzed using commercial finite element software, ANSYS. Results show reasonably good agreement with the finite element analysis.     

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.     

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.     

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.     

Buckling response of laminated composite stiffened plates subjected to partial in-plane edge loading

This article presents the buckling analysis of laminated composite stiffened plates subjected to partial in-plane edge loading. The finite element method is used to carry out the analysis. The eight-noded isoparametric degenerated shell element with C⁰ continuity and first-order shear deformation and a compatible three-noded curved beam element are used to model the plate skin and the stiffeners, respectively. The eigen value analysis is carried out to track the buckling load. The convergence study is performed for some specific problems and the results are compared with the available results in the literature. It is observed that the convergence of results is very fast for this finite element model. Effect of different parameters like orientation of fibers, number of layers, and loading types are considered in the present investigation. It is also observed that all these parameters have significant effect on the buckling response of the composite stiffened plate.     

Post-buckling analysis of composite plates containing embedded delaminations with arbitrary shape by using higher order shear deformation theory

The compressive post-buckling behavior of composite laminates containing embedded delamination with arbitrary shape is investigated analytically. For modeling the embedded delamination, the laminate is divided into three smaller regions. The higher order shear deformation theory is implemented and the formulation is based on the Rayleigh-Ritz approximation technique by the application of the simple/complete polynomial series for each region. The nonlinear equilibrium equations, which are achieved through the application of the principle of Minimum Potential Energy, are solved by employing the Newton-Raphson iterative procedure. Some interesting results are obtained and compared with those achieved by the finite element method of analysis using ANSYS commercial software. A good agreement is seen to exist between the results. This is while for a given level of accuracy in the results, ANSYS requires a markedly larger number of degrees of freedom compared to that needed by the developed method. Moreover, a considerable reduction in the load carrying capacity of laminate is noticed due to the presence of delamination.     

Static and free vibration analyses of carbon nanotube-reinforced composite plates using finite element method with first order shear deformation plate theory

This paper mainly presents bending and free vibration analyses of thin-to-moderately thick composite plates reinforced by single-walled carbon nanotubes using the finite element method based on the first order shear deformation plate theory. Four types of distributions of the uniaxially aligned reinforcement material are considered, that is, uniform and three kinds of functionally graded distributions of carbon nanotubes along the thickness direction of plates. The effective material properties of the nanocomposite plates are estimated according to the rule of mixture. Detailed parametric studies have been carried out to reveal the influences of the volume fractions of carbon nanotubes and the edge-to-thickness ratios on the bending responses, natural frequencies and mode shapes of the plates. In addition, the effects of different boundary conditions are also examined. Numerical examples are computed by an in-house finite element code and the results show good agreement with the solutions obtained by the FE commercial package ANSYS.