Four-node Rectangular Element Research Articles

Random vibration analysis of FGM plates subjected to moving load using a refined stochastic finite element method

The article introduces a finite element procedure using the bilinear quadrilateral element or four-node rectangular element (namely Q4 element) based on a refined first-order shear deformation theory (r-FSDT) and Monte Carlo simulation (MCS), so-called refined stochastic finite element method to investigate the random vibration of functionally graded material (FGM) plates subjected to the moving load. The advantage of the proposed method is to use r-FSDT to improve the accuracy of classical FSDT, satisfy the stress-free condition at the plate boundaries, and combine with MCS to analyze the vibration of the FGM plate when the parameter inputs are random quantities following a normal distribution. The obtained results show that the distribution characteristics of the vibration response of the FGM plate depend on the standard deviation of the input parameters and the velocity of the moving load. Furthermore, the numerical results in this study are expected to contribute to improving the understanding of FGM plates subjected to moving loads with uncertain input parameters.

Nonlinear static analysis of functionally graded porous sandwich plates resting on Kerr foundation

The main purpose of this work is to explore the nonlinear static bending analysis of functionally graded porous sandwich plates with a ceramic core (FSCC) subjected to a transverse uniform load resting on Kerr foundation (KF) using the mixed interpolation of tensorial components for the four-node rectangular element (called MITC4). Porosity appearing in two skin layers is assumed according to the uneven porosity distribution with a porosity factor The advantage of this element is fast convergence, high accuracy and cancels the shear-locking phenomenon. The present study considers the geometrical nonlinearity resulting from mid-plane stretching based on the von Kármán geometrical nonlinearity. The first-order shear deformation theory (FSDT) is employed to approximate the displacement field of the plate due to its simplicity and efficiency. The current formulation’s authenticity is evaluated by comparing it with previously published works. For the first time, the nonlinear bending of FSCC subjected to a transverse uniform load resting on the Kerr foundation is investigated. Furthermore, the influence of input parameters on the nonlinear bending behavior of FSCC is fully provided.

Nonlinear static analysis of bi-directional functionally graded sandwich plates in thermal environments by a higher-order finite element model

This study addresses nonlinear static behaviours of bi-directional functionally graded (2D-FGM) sandwich plates lying on an elastic foundation in thermal environments for the first time. The plates comprise two 2D-FGM face sheets and a homogeneous core layer. The material properties of the face sheets change according to the power law in both the longitudinal and thickness directions. Furthermore, the material properties also possess temperature-dependent characteristics in the thermal environment. For the analysis purpose, a higher-order finite element model (FEM) based on Shi’s plate theory that takes into account the geometrically nonlinear von Kármán without regard to material nonlinearity is established. The minimum potential energy principle is used to establish the weak form of governing equations. The four-node rectangular element with the Lagrange and Hermitian interpolation functions is used to discretize the domain of the plate. The nonlinear equilibrium equations are solved by employing Newton–Raphson iterative procedure. The numerical examples comparing the obtained results with the published results in open literature have confirmed the accuracy and convergence of the proposed model. Finally, a few novel parametric studies are conducted to show the effects of the material properties, geometric parameters, temperature, elastic foundation stiffness, and boundary conditions on the nonlinear static behaviours of the 2D-FGM sandwich plates.

A novel trigonometric shear deformation theory for the buckling and free vibration analysis of functionally graded plates

In the current work, a four-node rectangular finite element with assumed natural shear strains, based on the exact neutral surface position and new higher-order shear deformation theory is employed to perform mechanical buckling and free vibration analysis of functionally graded plates. The present theory has only five unknowns and satisfies zero transverse shear stress at the top and bottom surfaces of the plate, therefore a shear correction factor is not necessary. The accuracy of the proposed formulation is evaluated by comparing the obtained results with those available in the literature.

Investigation of layered composite plates under acoustic emission using an appropriate finite element model

Structural damages generate acoustic emission (AE) in the media and cause extensional and flexural acoustic waves. Often in structures like plates, flexural mode is predominant. In this study, the flexural mode AE waveforms due to simulated damage are studied for multi-layered composite plates. A generalized refined 2D plate theory, which satisfies the transverse shear stress continuity at the layer interfaces, is proposed here for modelling the plates. This formulation is implemented through finite element method (FEM) where a four-node rectangular element, that satisfies C1 continuity, is used. Plates, having different thickness ratios, are studied through numerical examples using the model. Results are validated wherever applicable and some new results are obtained. The results indicate that the proposed model can simulate the flexural waveforms realistically for ‘very thin’ to ‘moderately thick’ plates. It is also found that the present model is as accurate as 3D FEM but it possesses much better computational efficiency.

Active vibration control of a piezoelectric functionally graded carbon nanotube-reinforced spherical shell panel

A finite-element model is presented based on the four-variable shear deformation refined theory for active vibration control of a functionally graded carbon nanotube-reinforced composite spherical panel with integrated piezoelectric layers, acting as an actuator and a sensor. The linear distribution of the electric potential across the thickness of the piezoelectric layer and different distribution types of carbon nanotubes through the thickness of the layers are considered. The weak form of the governing equation is derived using Hamilton's principle, and a four-node nonconforming rectangular element with eight mechanical and two electrical degrees of freedom per node is introduced for discretising the domain. A constant velocity feedback approach is utilised for the active control of the panel by closed-loop control with a piezoelectric sensor and actuator. The convergence and accuracy of the model are validated by comparing numerical results with data available in literature. Some new parametric studies are also discussed in detail.

Forced vibration of smart laminated viscoelastic plates by RPT finite element approach

In this study, a finite element formulation based on the four-variable refined plate theory (RPT) is presented for forced vibration analysis of laminated viscoelastic composite plates integrated with a piezoelectric layer. To the best of the authors’ knowledge, this is the first time that the proposed approach is extended for study of the dynamic behavior of the smart viscoelastic plate. The utilized RPT which works for both thick and thin plates predicts a parabolic variation for transverse shear stresses across the plate thickness. Considering a linear viscoelastic model for the substrate material, the relaxation module is predicted by the Prony series. Using Hamilton’s principle, the weak form equation is constructed and a four-node rectangular plate element is utilized for discretizing the domain. The Newmark scheme is employed for advancing the solution in time. A MATLAB code is developed based on the formulations and several benchmark problems are solved. Comparing the findings with existing results in previous studies confirms the accuracy and efficiency of the proposed method. The dynamic response of the smart viscoelastic plates under various electromechanical loads is investigated and the results show that the vibration can be passively controlled by adding and actuating the piezoelectric layer. The damping effects of viscoelastic parameters on the results are investigated, too.

Effects of the Radius of Curvature on Natural Frequency and Mode Shape

In this study, the effect of radius of curvature on the natural frequency of isotropic square thin plate is investigated. The models examined are obtained by simply changing the radius of curvature by keeping the length constant from the flat plate to the semicircle model. The free vibration analysis of the plates fixed on two straight edges is performed by the finite element method. As a finite element type, the four-node quadrilateral rectangular element type, which has a total of 24 generalized coordinates, is used. Out-of-plane theory and in-plane theory are used together to form the curve model. The accuracy and validity of the theory are controlled with the data obtained from the finite element program. The effect of the curvature radius on the first five natural frequencies and the mode shapes of these natural frequencies is given in tables and graphs.

A modified four-node rectangular element

The sensitized principle of virtual work is applied to modify the stiffness matrix of the ordinary four-node rectangular element by sensitizing terms. The sensitizing parameter values are determined by the single-element strain energy test. The reference solutions used are of bending mode types and their application removes the so-called parasitic shear behavior. A stiffness matrix of good quality is obtained corresponding exactly to an earlier formulation using incompatible modes.

Vibration analysis of a multifunctional hybrid composite honeycomb sandwich plate

In the present study, the free and forced vibration responses of the composite sandwich plate with carbon nanotube reinforced honeycomb as the core material and laminated composite plates as the top and bottom face sheets are investigated. The governing equations of motion of hybrid composite honeycomb sandwich plates are derived using higher order shear deformation theory and solved numerically using a four-noded rectangular finite element with nine degrees of freedom at each node. Further, various elastic properties of honeycomb core materials with and without reinforcement of carbon nanotube and face materials are evaluated experimentally using the alternative dynamic approach. The effectiveness of the finite element formulation is demonstrated by performing the results evaluated experimentally on a prototype composite sandwich plate with and without carbon nanotube reinforcement in core material. Various parametric studies are performed numerically to study the effects of carbon nanotube wt% in core material, core thickness, ply orientations, and various boundary conditions on the dynamic properties of composite honeycomb sandwich plate. Further, the transverse vibration responses of hybrid composite sandwich plates under harmonic force excitation are analyzed at various wt% of carbon nanotubes and the results are compared with those obtained without addition of carbon nanotubes to demonstrate the effectiveness of carbon nanotube reinforcement in enhancing the stiffness and damping characteristics of the structures. The study provides the guidelines for the designer on enhancing both the stiffness and damping properties of sandwich structures through carbon nanotube reinforcement in core materials.

Conforming shear-locking-free four-node rectangular finite element of moderately thick plate

AbstractAn outline of the modified Mindlin plate theory, which deals with bending deflection as a single variable, is presented. Shear deflection and cross-section rotation angles are functions of bending deflection. A new four-node rectangular finite element of moderately thick plate is formulated by utilizing the modified Mindlin theory. Shape functions of total (bending+shear) deflections are defined as a product of the Timshenko beam shape functions in the plate longitudinal and transversal direction. The bending and shear stiffness matrices, and translational and rotary mass matrices are specified. In this way conforming and shear-locking-free finite element is obtained. Numerical examples of plate vibration analysis, performed for various combinations of boundary conditions, show high level of accuracy and monotonic convergence of natural frequencies to analytical values. The new finite element is superior to some sophisticated finite elements incorporated in commercial software.

Finite Element Formulations for Buckling Analysis of Isotropic and Orthotropic Plates using Two-Variable Refined Plate Theory

In this paper, a finite element formulation based on two-variable refined plate theory is developed for buckling analysis of isotropic and orthotropic plates. The two-variable refined plate theory, which can be used for both thin and thick plates, predicts parabolic variation of transverse shear stresses across the plate thickness, satisfies the zero traction condition on the plate surfaces and does not need the shear correction factor. After constructing weak form equations using principle of minimum potential energy, a new four-node rectangular plate element with six degrees of freedom at each node is introduced for discretization of the domains. The uniaxial and biaxial buckling loads are obtained for simply supported isotropic plates. Also, the uniaxial buckling loads in two principal directions and the biaxial buckling loads are obtained for Levy-type orthotropic plates. Comparison of results with exact solutions and other common plate theories shows that besides the simplicity of presented finite element formulations, it presents accurate and efficient results. Also, the effects of orthotropy ratio, side-to-thickness ratio and types of boundary conditions on the buckling loads are studied.

Bending and free vibration analysis of functionally graded plates using new eight-unknown shear deformation theory by finite-element method

In this paper, a new eight-unknown shear deformation theory is developed for bending and free vibration analysis of functionally graded plates by finite-element method. The theory based on full 12-unknown higher order shear deformation theory simultaneously satisfies zeros transverse stresses at top and bottom surfaces of FG plates. A four-node rectangular element with 16 degrees of freedom per node is used. Poisson’s ratios, Young’s moduli, and material densities vary continuously in thickness direction according to the volume fraction of constituents which is modeled as power-law functions. Results are verified with available results in the literature. Parametric studies are performed for different power-law indices, side-to-thickness ratios.

Bending and free vibration analysis of functionally graded plates using new eight-unknown shear deformation theory by finite element method

In this paper, a new eight-unknown shear deformation theory is developed for bending and free vibration analysis of functionally graded plates by finite element method. The theory based on full twelve-unknown higher order shear deformation theory, simultaneously satisfy zeros transverse stresses at top and bottom surface of FG plates. A four-node rectangular element with sixteen degrees of freedom per node is used. Poisson’s ratios, Young’s moduli and material densities vary continuously in thickness direction according to the volume fraction of constituents which is modeled as power law functions. Results are verified with available results in the literature. Parametric studies are performed for different power law index, side-to-thickness ratios.

Application of the dual integrated force method to the analysis of the off-axis three-point flexure test of unidirectional composites

This paper shows the use of the dual integrated force method in the finite element method and its application to unidirectional composite laminates. This method was developed by SN Patnaik for isotropic materials, considering not only the equilibrium equations but also the compatibility conditions. The method is applied to obtain the stiffness matrix of a four-node rectangular plate element. Then, a three-point flexure test of a unidirectional off-axis composite has been analyzed. The results obtained with this method have been compared with those obtained from a previous analytic approach and with those obtained by commercial software of finite element method, which is based on the stiffness method.

Modal Analysis of a Plate Compliant in Transverse Shear

A finite-element solution of the problem on natural vibrations of a plate compliant in transverse shear is considered. A four-node rectangular finite element whose basic nodal kinematic parameters include the angles of transverse shear deformations is used. A comparative analysis of frequencies and modes of natural vibrations of composite and sandwich plates is performed for two variants of boundary conditions on their contour: the classical clamping and clamping with free transverse shear deformations.

Extended Penalty Coefficients For Elimination The Locking Effects In Moderately Thick Beam And Plate Finite Elements

AbstractThe present paper is dedicated to presentation and energy verification of the methods of stabilization the strain energy by penalty coefficients. Verification of the methods is based on the consistency and ellipticity conditions to be satisfied by the finite elements. Three methods of stabilization are discussed. The first does not satisfy the above requirements. The second is consistent but cannot eliminate parasitic energy terms. The third method, proposed by the author, is based on the decomposition of the element stiffness matrix. The method can help to eliminate locking of the finite elements. For two-noded beam element with linear shape functions and exact integration a stabilized free of locking (and elliptical) element is received (equivalent to reduced integration element). Two plate finite elements are analyzed: four-noded rectangular element and DSG triangle. A new method of stabilization with the use of four independent parameters is proposed. The finite elements with this kind of stabilization satisfy the consistency condition. In the rectangular element it was not possible to eliminate one parasitic term of energy which appears during the procedure. For DSG triangle all parasitic terms of energy are eliminated. The penalty coefficients depends on the geometry of the triangle.

Vibration and damping characteristics of sandwich plates with viscoelastic core

As an effective approach to suppress vibrations and noise, passive constrained layer damping (PCLD) treatments are widely used in engineering practice. However most of the studies concentrate on the one-dimensional beams with active/passive constrained damping layer, the analysis on the two-dimensional plates are relatively small. This research proposes an efficient sandwich modeling technique to deal with the vibration and damping characteristics of the PLCD plate structure. A type of three-layer four-node rectangular element with seven degrees of freedom on every node is used to simulate the PLCD plate structure. The displacement relationships of each layer are obtained based on the first-order shear deformation theory. The finite element equations of motions are derived by the Hamilton principle in variation form. The natural frequency and loss factor are discussed based on the eigenvalue problems. Numerical examples are provided to verify the accuracy and efficiency of the present finite element method. Finally, the influences of the layer thickness, the loss factors of the viscoelastic cores on the natural frequencies and loss factors of the PLCD plate structure are discussed as well.

A non-classical Mindlin plate finite element based on a modified couple stress theory

This paper presents a novel Mindlin plate element based on the framework of a modified couple stress theory for analyzing the static bending, free vibration and buckling behaviors of size-dependent Mindlin micro-plates. The element proposed is a four-node rectangular element which has 15-DOF (degrees of freedom) at C0 each node with considering both bending and stretching deformations, and 9-DOF with only considering bending deformation. Unlike the classical Mindlin plate element, this element satisfies the continuity and C1 weak continuity and contains a material length scale parameter. It can be directly used to predict the size effect on the Mindlin micro-plates. Finite element formulations are derived by applying the corresponding weak form equations. To illustrate the applicability and accuracy of the developed Mindlin plate element, the static bending, free vibration and buckling problems for rectangular Mindlin micro-plates with various boundary conditions are investigated. Convergence and comparison studies are carried out to examine the reliability of the numerical solutions. It is shown that the typical numerical results are in good agreement with those available results reported in previous literature. In addition, the numerical results illustrate that the size effect on Mindlin micro-plates can be effectively predicted by using the proposed plate element. These predicted trends agree with those observed in micro-torsion test of thin copper wires and micro-bending test of epoxy polymeric beams. Some results are believed to be the first known in the open literature and can be used as benchmark for further studies.

Stiffness matrix of the finite element of a plate compliant in transverse shear

A four-node rectangular finite element is elaborated for a plate calculated with account of transverse shear strains. On variational realization of the finite-element method, the stiffness matrix and the vector of equivalent nodal forces are obtained for the case of a surface load. The basic nodal kinematic parameters adopted also include the angles of transverse shear.