3D free vibration analysis of laminated cylindrical shell integrated piezoelectric layers using the differential quadrature method

This paper addresses the free vibration problem of multilayered shells with embedded piezoelectric layers. Based on the three-dimensional theory of elasticity, an approach combining the state space method and the differential quadrature method (DQM) is used. The shell has arbitrary end boundary conditions. For the simply supported boundary conditions closed-form solution is given by making the use of Fourier series expansion. Applying the differential quadrature method to the state space formulations along the axial direction, new state equations about state variables at discrete points are obtained for the other cases such as clamped or free end conditions. Natural frequencies of the hybrid laminated shell are presented by solving the eigenfrequency equation which can be obtained by using edges boundary condition in this state equation. Accuracy and convergence of the present approach is verified by comparing the natural frequencies with the results obtained in the literatures. Finally, the effect of edges conditions, mid-radius to thickness ratio, length to mid-radius ratio and the piezoelectric thickness on vibration behaviour of shell are investigated.     

Three-dimensional layerwise-finite element free vibration analysis of thick laminated annular plates on elastic foundation

Using a three-dimensional layerwise-finite element method, the free vibration of thick laminated circular and annular plates supported on the elastic foundation is studied. The Pasternak-type formulation is employed to model the interaction between the plate and the elastic foundation. The discretized governing equations are derived using the Hamilton’s principle in conjunction with the layerwise theory in the thickness direction, the finite element (FE) in the radial direction and trigonometric function in the circumferential direction, respectively. The fast rate of convergence of the method is demonstrated and to verify its accuracy, comparison studies with the available solutions in the literature are performed. The effects of the geometrical parameters, the material properties and the elastic foundation parameters on the natural frequency parameters of the laminated thick circular and annular plates subjected to various boundary conditions are presented.     

Buckling and free vibration analysis of thick rectangular plates resting on elastic foundation using mixed finite element and differential quadrature method

In this article, a combination of the finite element (FE) and differential quadrature (DQ) methods is used to solve the eigenvalue (buckling and free vibration) equations of rectangular thick plates resting on elastic foundations. The elastic foundation is described by the Pasternak (two-parameter) model. The three dimensional, linear and small strain theory of elasticity and energy principle are employed to derive the governing equations. The in-plane domain is discretized using two dimensional finite elements. The spatial derivatives of equations in the thickness direction are discretized in strong-form using DQM. Buckling and free vibration of rectangular thick plates of various thicknesses to width and aspect ratios with Pasternak elastic foundation are investigated using the proposed FE-DQ method. The results obtained by the mixed method have been verified by the few analytical solutions in the literature. It is concluded that the mixed FE-DQ method has good convergancy behavior; and acceptable accuracy can be obtained by the method with a reasonable degrees of freedom.     

Nonlinear transient analysis of moderately thick laminated composite sector plates

This study presents a simple formulation for the nonlinear dynamic analysis of shear-deformable laminated sector plates made up of cylindrically orthotropic layers. The non-axisymmetric formulation in cylindrical coordinates is discretized in space domain using two-dimensional Chebyshev polynomials. Houbolt time marching is used for temporal discretization. Quadratic extrapolation is used for linearization along with fixed-point iteration for obtaining the results. Several combinations of simply supported, clamped and free edge conditions are considered. Convergence study has been carried out and the results are compared with the results of square plates. Effects of boundary conditions, moduli ratio, lamination scheme, sector angle and annularity on the transient deflection response are plotted graphically. Transient responses are compared for step, saw-tooth and sinusoidal loadings.     

Accurate vibration analysis of skew plates by the new version of the differential quadrature method

An accurate free vibration analysis of skew plates is presented by using the new version of the differential quadrature method (DQM). Eight combinations of simply supported (S), clamped (C) and free (F) boundary conditions are considered. Detailed solution procedures are given and key points for success by using the DQM are emphasized. A way to simplifying the programming in using the DQM is proposed. Convergence study is made for the simply supported skew plate with a large skew angle. Good convergence of frequencies is observed. The DQ results agree very well with the existing first known accurate upper bound solutions, obtained by using Ritz method taking into considerations of the bending stress singularities occurred at corners having obtuse angles. Since slight discrepancy between the DQ data and the known accurate solutions is observed for plates with large skew angles, the DQ results are also compared with data obtained by using finite element method with very fine meshes to verify their accuracy.     

Unsteady seepage analysis using local radial basis function-based differential quadrature method

Numerical simulation of two-dimensional transient seepage is developed using radial basis function-based differential quadrature method (RBF-DQ). To the best of the authors’ knowledge, this is the first application of this method to seepage analysis. For the general case of irregular geometry and unstructured node distribution, the local form of RBF-DQ was used. The multiquadric type of radial basis functions was selected for the computations, and the results were compared with analytical, finite element method, and existing numerical solutions from the literature. Results of this study show that localized RBF-DQ can produce accurate results for the analysis of seepage. The method is meshfree and easy to program, but as with previous applications of RBFs, requires careful selection of suitable shape parameters. A practical method for estimating suitable shape parameters is discussed. For time integration, Crank–Nicolson, Galerkin and finite difference methods were applied, leading to stable results.     

Optimum design of laminated composite plates under dynamic excitation

An integrated model for optimum weight design of symmetrically laminated composite plates subjected to dynamic excitation is presented in this work. Optimum design procedure based on flexibility and strength criteria is presented. The objective is to determine the optimum thicknesses of the laminate layers and its optimum orientations without exhibiting any failure according to Tsai-Wu failure criterion. The finite element method, based on Mindlin plate theory, is used in conjunction with an optimization method in order to determine the optimum design. Newmark algorithm, as an implicit time integration scheme, is used to discretize the time domain and calculate the transient response of the laminated composite plate. Exterior penalty method is exploited for the constrained minimization procedure. Fletcher-Powell algorithm is used for the unconstrained minimization process. To verify the capability and efficiency of the proposed model, three examples are solved. The examples deal with flexibility and stress constraints for different boundary conditions under various dynamic excitations.     

Elastoplastic stress analysis of thick laminated metal-matrix composite plates by the finite-element method

The elastoplastic stress state of a laminated stainless-steel-fiber-reinforced aluminum-matrix plates, with or without a hole, subjected to a pressure on their top is examined by using the finite-element method. The analysis is carried out for three layouts: (0/90/0/90)_s, (45/-45/45/-45)_s, and (30/60/30/60)_s. The Newton-Raphson method is used to solve the nonlinear problem. The distributions of equivalent stresses and the plastic zones of the plates without a hole and with a hole of various diameters are determined. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 42, No. 4, pp. 531–544, July–August, 2006.     

A vibration analysis of composite laminated plates

A finite element formulation of the equations governing laminated anisotropic plates using Reddy's higher-order theory is presented. This simple higher-order shear deformable theory takes into account the parabolic distribution of the transverse shear deformation through the thickness of the plate and contains the same unknowns as in the first-order shear deformation theory. Finite element solutions are presented for rectangular plates of different layups, such as cross-ply, antisymmetric angle-ply, and sandwich plates with various material properties, boundaries, and plate aspect ratios. The numerical results are compared with the available closed-form results, the 3-D linear elasticity theory results, and the other available numerical results. A comparison is also made with test data from a laminated cantilever plate.     

A semi-analytic approach for the nonlinear dynamic response of circular plates

This paper presents a new semi-analytic perturbation differential quadrature method for geometrically nonlinear vibration analysis of circular plates. The nonlinear governing equations are converted into a linear differential equation system by using Linstedt–Poincaré perturbation method. The solutions of nonlinear dynamic response and the nonlinear free vibration are then sought through the use of differential quadrature approximation in space domain and analytical series expansion in time domain. The present method is validated against analytical results using elliptic function in several examples for both clamped and simply supported circular plates, showing that it has excellent accuracy and convergence. Compared with numerical methods involving iterative time integration, the present method does not suffer from error accumulation and is able to give very accurate results over a long time interval.     

A weak form quadrature element method for plane elasticity problems

A weak form quadrature element method is proposed and applied to analysis of plane elasticity problems. A variational formulation of plane elasticity problems is established and the differential quadrature analog of the derivatives in the functional is introduced. Several typical plane elasticity problems are studied to verify the proposed method. Results show that the method is highly efficient and promising. It is applied to the analysis of nearly incompressible materials and shown to be robust against volumetric locking. Similarities and dissimilarities, advantages and disadvantages as compared with other numerical methods, typically the p-version finite element method are discussed.     

Free vibration of centrifugally stiffened axially functionally graded tapered Timoshenko beams using differential transformation and quadrature methods

The free bending vibration of rotating axially functionally graded (FG) Timoshenko tapered beams (TTB) with different boundary conditions are studied using Differential Transformation method (DTM) and differential quadrature element method of lowest order (DQEL). These two methods are capable of modelling any beam whose cross sectional area, moment of inertia and material properties vary along the beam. In order to verify the competency of these two methods, natural frequencies are obtained for problems by considering the effect of material non-homogeneity, taper ratio, shear deformation parameter, rotating speed parameter, hub radius and tip mass. The results are tabulated and compared with the previous published results wherever available.     

Natural frequencies of a functionally graded cracked beam using the differential quadrature method

The present work is concerned with the free vibration analysis of an elastically supported cracked beam. The beam is made of a functionally graded material and rested on a Winkler–Pasternak foundation. The line spring model is employed to formulate the problem. The method of differential quadrature is applied to solve it. The obtained results agreed with the previous similar ones. Further, a parametric study is introduced to investigate the effects of the geometric and elastic characteristics of the problem on the values of natural frequencies and mode shape functions.     

Modeling of water surface profile in subterranean channel by differential quadrature method (DQM)

This study, investigates the hydraulic of flow in a subterranean channel headspring. The continuity and momentum equations of flow in porous media considering real conditions were used and the basic equation of flow in a subterranean channel was resulted. This equation is very similar to the spatially varied flow with increasing discharge. An equation, defining the hydraulic parameters of a subterranean channel section was adopted. Then differential quadrature method (DQM), was applied to the equation of flow in subterranean channel, consequently the water surface profile was resulted. To illustrate the rightness of model, the hydraulic parameters of flow in the Gavgard branch of the Joopar Goharriz Qanat were measured and the water surface profile was determined. This water surface profile was compared to the water surface profile computed by the model, which are in good agreement.     

Numerical analysis of free vibrations of laminated composite conical and cylindrical shells: Discrete singular convolution (DSC) approach

A numerical study on the free vibration analysis for laminated conical and cylindrical shell is presented. The analysis is carried out using Love's first approximation thin shell theory and solved using discrete singular convolution (DSC) method. Numerical results in free vibrations of laminated conical and cylindrical shells are presented graphically for different geometric and material parameters. Free vibrations of isotropic cylindrical shells and annular plates are treated as special cases. The effects of circumferential wave number, number of layers on frequencies characteristics are also discussed. The numerical results show that the present method is quite easy to implement, accurate and efficient for the problems considered.     

Numerical solution of two-dimensional reaction-diffusion Brusselator system

In this paper, polynomial based differential quadrature method (DQM) is applied for the numerical solution of a class of two-dimensional initial-boundary value problems governed by a non-linear system of partial differential equations. The system is known as the reaction-diffusion Brusselator system. The system arises in the modeling of certain chemical reaction-diffusion processes. In Brusselator system the reaction terms arise from the mathematical modeling of chemical systems such as in enzymatic reactions, and in plasma and laser physics in multiple coupling between modes. The numerical results reported for three specific problems. Convergence and stability of the method is also examined numerically.     

A random integral quadrature method for numerical analysis of the second kind of Volterra integral equations

In this paper, a novel meshless technique termed the random integral quadrature (RIQ) method is developed for the numerical solution of the second kind of the Volterra integral equations. The RIQ method is based on the generalized integral quadrature (GIQ) technique, and associated with the Kriging interpolation function, such that it is regarded as an extension of the GIQ technique. In the GIQ method, the regular computational domain is required, in which the field nodes are scattered along straight lines. In the RIQ method however, the field nodes can be distributed either uniformly or randomly. This is achieved by discretizing the governing integral equation with the GIQ method over a set of virtual nodes that lies along straight lines, and then interpolating the function values at the virtual nodes over all the field nodes which are scattered either randomly or uniformly. In such a way, the governing integral equation is converted approximately into a system of linear algebraic equations, which can be easily solved.     

Delamination growth of laminated circular plates under in-plane loads and movable boundary conditions

An investigation is made on interlaminar delamination growth of composite laminated circular plates under in-plane loads and movable delamination boundary conditions. A four-dissociated-region model is developed on the basis of von-Karman plate theory. The model is geometrically nonlinear and the laminated circular plate considered is subjected to axisymmetrical delamination. The effects of transverse shear deformation and contact effect of the delamination on the laminated plates are taking into account in the development of the governing equations of the laminated circular pates with random axisymmetrical delamination. The formulas for describing the total energy release rate and its individual mode components along the delamination front are also derived with considerations of Griffith criterion for fracture. Based on the model established, the delamination growth is numerically studied; and the influences of the parameters such as delamination radii and depths, together with material properties of the plates on the energy release rate are analyzed in detail.