Plate Bending Problems Research Articles

Free vibration analysis of shear deformable rectangular plates with a line hinge

A discrete method is proposed for analysing the natural vibration problem of shear deformable rectangular plates with a line hinge. The fundamental differential equations and the solutions of these equations are derived for two parts of the plate, which are obtained by dividing the plate along the line hinge. By transforming these equations into integral equations, and using numerical integration and the continuous conditions along the line hinge, the solutions of the whole plate can be expressed by the unknown quantities on the boundary and the quantities of the rotation along the hinge. Green function which is the solution of deflection of the bending problem of plate is used to obtain the characteristic equation of the free vibration. The effects of the position of the line hinge, the aspect ratio, the thickness-to-length ratio and the boundary condition on the natural frequency parameters are considered. By comparing the numerical results obtained by the present method with those previously published, the efficiency and accuracy of the present method are investigated.

ＨＰＭによる薄板の離散化極限解析

In this paper, we have given the investigations of the thin plate bending problem by using hybrid-type penalty method (HPM). We apply Kirchhoff theory to the displacement field of the 3D case of HPM. For this purpose, we use quadratic form of the displacements which parameters are independently defined in each subdomain. We introduce penalty function that presents strong spring connecting each subdomain. We apply nonlinearity in penalty function such as spring system, which allow us to calculate hinge line. If hinge line makes mechanism then we can calculate limit load. We can calculate growing hinge line systematically using r-min method for the nonlinear analysis. In the first part of the paper, we have given brief formulation of proposed method, and in second part, we have given some numerical results to check accuracy of elastic solution and limit load.

Geometrically Nonlinear Analysis of Reissner-Mindlin Plate by Meshless Computation

In this paper, we perform a geo- metrically nonlinearanalysis of Reissner-Mindlin plate by using a meshless collocation method. The use of the smooth radial basis functions (RBFs) gives an advantage to evaluate higher order derivatives of the solution at no cost on extra-interpolation. Thecomputationalcost islow and requires neither the connectivity of mesh in the domain/boundary nor integrations of funda- mental/particular solutions. The coupled nonlin- ear terms in the equilibrium equations for both the plane stress and plate bending problems are treated as body forces. Two load increment schemes are developed to solve the nonlinear dif- ferential equations. Numerical verifications are given to demonstrate the efficiency and accuracy of the proposed method in comparing with exact solutionsand results from using thefinite element software (ABAQUS). Keyword: large deformation, Reissner-Mindlin plate theory, meshless collocation, radial basis functions.

An Adaptive Boundary Collocation Method for Plate Bending Problems

A boundary collocation method based on the least-square technique and a corresponding adaptive computation process have been developed for the plate bending problem. The trial functions are constructed using a series of the biharmonic polynomials, and the local error indicators are given by the residuals of the energy density on the boundary. In comparison with the conventional collocation methods, the solution accuracy in the present method can be improved in an economical and efficient way. In order to demonstrate the efficiency and advantages of the adaptive boundary collocation method proposed in this paper, two numerical examples are presented for circular plates subjected to uniform loads and restrained by mixed boundary conditions. The numerical results for the examples show good agreement with ones presented in the literature.

A relative quantity integral equation formulation for evaluation of boundary stress resultants in shear deformable plate bending problems

This paper presents new boundary element formulation for the shear deformable plate bending problems based on considering the relative displacement quantities. This can be done by subtracting the rigid body integral equations from the traditional displacement integral equations. The result is a new set of regularized integral equations, which could be used without any special integration considerations to compute the boundary generalized displacements. Using suitable constitutive relationships, new integral equations for computing boundary stress resultants are formulated. The behaviour of such equations are studied at smooth boundary points. Unlike the traditional formulation which contains kernels of O(1/ r 2), the newly derived equations are of O(1/ r), therefore they can be used to compute boundary values with the traditional quadratic continuous boundary elements. The main advantage of the present formulation is that it can be used to compute values anywhere within the plate domain or on the boundary, which gives the boundary element formulation a practical essence.

Stress intensity solutions for cracked plates by the dual boundary method

We describe the application of the dual boundary element method for the determination of stress intensity factors in plate bending problems. The loadings considered include internal pressure, and also combined bending and tension. Mixed mode stress intensity factors are evaluated by a crack surface displacement extrapolation technique and the J-integral technique. The boundary element results for the case studies considered in the paper have been compared with either analytical or finite element results and in all cases good agreement has been achieved.

Application of the quadrilateral area coordinate method: a new element for laminated composite plate bending problems

Recently, some new quadrilateral finite elements were successfully developed by the Quadrilateral Area Coordinate (QAC) method. Compared with those traditional models using isoparametric coordinates, these new models are less sensitive to mesh distortion. In this paper, a new displacement-based, 4-node 20-DOF (5-DOF per node) quadrilateral bending element based on the first-order shear deformation theory for analysis of arbitrary laminated composite plates is presented. Its bending part is based on the element AC-MQ4, a recent-developed high-performance Mindlin-Reissner plate element formulated by QAC method and the generalized conforming condition method; and its in-plane displacement fields are interpolated by bilinear shape functions in isoparametric coordinates. Furthermore, the hybrid post-processing procedure, which was firstly proposed by the authors, is employed again to improve the stress solutions, especially for the transverse shear stresses. The resulting element, denoted as AC-MQ4-LC, exhibits excellent performance in all linear static and dynamic numerical examples. It demonstrates again that the QAC method, the generalized conforming condition method, and the hybrid post-processing procedure are efficient tools for developing simple, effective and reliable finite element models.

Discrete singular convolution method for the analysis of Mindlin plates on elastic foundations

The method of discrete singular convolution (DSC) is used for the bending analysis of Mindlin plates on two-parameter elastic foundations for the first time. Two different realizations of singular kernels, such as the regularized Shannon's delta (RSD) kernel and Lagrange delta sequence (LDS) kernel, are selected as singular convolution to illustrate the present algorithm. The methodology and procedures are presented and bending problems of thick plates on elastic foundations are studied for different boundary conditions. The influence of foundation parameters and shear deformation on the stress resultants and deflections of the plate have been investigated. Numerical studies are performed and the DSC results are compared well with other analytical solutions and some numerical results.

Bordering method for solving systems of linear equations generated by the finite element method in the plate bending problem

A combined iteration algorithm based on the bordering and conjugate gradient methods is proposed to solve systems of linear equations generated by the finite element method in the plate bending problem. The numerical results for the analysis of the convergence rate of the iterative process are presented in the solution of model problems using a classical and modified algorithm of the method of conjugate gradients. The possibility of acceleration of the iterative algorithm is shown.

A new wavelet-based thin plate element using B-spline wavelet on the interval

By interacting and synchronizing wavelet theory in mathematics and variational principle in finite element method, a class of wavelet-based plate element is constructed. In the construction of wavelet-based plate element, the element displacement field represented by the coefficients of wavelet expansions in wavelet space is transformed into the physical degree of freedoms in finite element space via the corresponding two-dimensional C1 type transformation matrix. Then, based on the associated generalized function of potential energy of thin plate bending and vibration problems, the scaling functions of B-spline wavelet on the interval (BSWI) at different scale are employed directly to form the multi-scale finite element approximation basis so as to construct BSWI plate element via variational principle. BSWI plate element combines the accuracy of B-spline functions approximation and various wavelet-based elements for structural analysis. Some static and dynamic numerical examples are studied to demonstrate the performances of the present element.

Bilinear plate bending element for thin and moderately thick plates using Integrated Force Method

Using the Mindlin-Reissner plate theory, many quadrilateral plate bending elements have been developed so far to analyze thin and moderately thick plate problems via displacement based finite element method. Here new formulation has been made to analyze thin and moderately thick plate problems using force based finite element method called Integrated Force Method (IFM). The IFM is a novel matrix formulation developed in recent years for analyzing civil, mechanical and aerospace engineering structures. In this method all independent/internal forces are treated as unknown variables which are calculated by simultaneously imposing equations of equilibrium and compatibility conditions. In this paper the force based new bilinear quadrilateral plate bending element (MQP4) is proposed to analyze the thin and moderately thick plate bending problems using Integrated Force Method. The Mindlin-Reissner plate theory has been used in the formulation of this element which accounts the effect of shear deformation. Standard plate bending benchmark problems are analyzed using the proposed element MQP4 via Integrated Force Method to study its performance with respect to accuracy and convergence, and results are compared with those of displacement based 4-node quadrilateral plate bending finite elements available in the literature. The results are also compared with the exact solutions. The proposed element MQP4 is free from shear locking and works satisfactorily in both thin and moderately thick plate bending situations.

Bending behavior of FGM-coated and FGM-undercoated plates with two simply supported opposite edges and two free edges

Abstract This investigation analyzes laminated plates with two simply supported opposite edges and two free edges subjected to transverse loads. Two situations are considered. The first involves a two-layer plate in which an FGM layer is coated on a homogeneous substrate, simply called an FGM-coated plate; the other involves a three-layer plate in which an FGM is employed for the inter-medium layer and different homogeneous materials are in the top and bottom layers; this is called an FGM-undercoated plate. The Young’s modulus of FGM plates is assumed to vary in the thickness direction and the Poisson’s ratio to remain constant throughout the FGM plate. The material gradations in FGM portion of the laminate structures follow sigmoid functions. Even though the FGM-coated or undercoated plates are formed from composite materials, the bending and in-plane problems in these FGM plates with constant Poisson’s ratio and varying Young’s modulus in the thickness direction under transverse loads only are uncoupled. For the multi-layer structures of interest, the calculations of the quantities, A 11 , B 11 , C 11 , Q 11 , S 11 , defined herein become very complex. Therefore this study proposes an alternative approach for easily and efficiently calculating them. The results reveal that the analytical solution to the bending problem of FGM plates obtained using a Fourier series expanding is strongly consistent with the finite element calculation. The deflections, strains, and stresses of FGM-coated and undercoated plates are evaluated and under discussion. The differences between the flexibility behaviors of FGM-coated and undercoated plates are investigated.

THE LAYERED ELEMENT DKT_CST FOR ANALYSIS OF ANISOTROPIC PLATES

The new finite element of multilayered built up with an arbitrary series of layers plate for plate bending problem is formulated on the ground of widely used, effective finite element Discrete Kirchhof Theory (DKT). The material of each layer is supposed to be different and orthotropic. Triangular element has 6 d.o.f.’s at each of 3 nodal points: 3 displacements and 3 rotations about co‐ordinate axes. The element takes into account all the in‐plane/out‐of‐plane effects except for shear. The element could find an application in the slab bending problems or in the plate, where the shear influence could be neglected, bending problems. Numerical examples are presented. Present solutions are compared with available analytical and numerical solutions.

A BEM-based meshless method for plates on biparametric elastic foundation with internal supports

In this paper a BEM-based meshless method is developed for the analysis of plates on a biparametric elastic foundation which, in addition to the boundary supports, are also supported inside the domain on isolated points (a group of plies) and/or line supports (continuous plates). The presented method is achieved using the concept of the analog equation method (AEM) of Katsikadelis. According to this method the original governing differential equation is replaced by an equivalent problem for plates without internal supports not resting on an elastic foundation subjected to an “appropriate” fictitious load in addition to the transverse external loads under the same boundary conditions. The fictitious load is established using a technique based on BEM and approximated by radial basis functions series. The solution of the actual problem is obtained from the known integral representation of the solution for the classical plate bending problem, which is derived using the fundamental solution of the biharmonic equation. Thus, the kernels of the boundary integral equations are conveniently established and evaluated. The presented method has all the advantages of the pure BEM. To validate the effectiveness, accuracy as well as the applicability of the proposed method, numerical results of various example problems are presented.

An efficient wave based prediction technique for plate bending vibrations

The finite element method is a commonly used prediction method for dynamic simulations of plate bending problems. A major disadvantage of this method is its practical frequency limitation as its application involves prohibitively large computational efforts at higher frequencies. A newly developed wave based prediction technique pushes up the frequency limit through an enhanced computational efficiency. This paper discusses the principles of the wave based method for convex plate problems. The beneficial convergence rate of the wave based method as compared with the finite element method is verified for various validation examples, which include the most commonly encountered boundary conditions.

Local integral equation method for viscoelastic Reissner–Mindlin plates

A meshless local Petrov-Galerkin (MLPG) method is applied to solve static and dynamic bending problems of linear viscoelastic plates described by the Reissner–Mindlin theory. To this end, the correspondence principle is applied. A weak formulation for the set of governing equations in the Reissner–Mindlin theory with a unit test function is transformed into local integral equations on local subdomains in the mean surface of the plate. Nodal points are randomly spread on the mean surface of the plate and each node is surrounded by a circular subdomain to which local integral equations are applied. A meshless approximation based on the moving least-squares (MLS) method is employed in the numerical implementation.

A BEM-BASED MESHLESS METHOD FOR ELASTIC BUCKLING ANALYSIS OF PLATES

In this paper, a BEM-based meshless method is developed for buckling analysis of elastic plates with various boundary conditions that include elastic supports and restraints. The proposed method is based on the concept of the Analog Equation Method (AEM) of Katsikadelis. According to this method, the original eigenvalue problem for a governing differential equation of buckling is replaced by an equivalent plate bending problem subjected to an appropriate fictitious load under the same boundary conditions. The fictitious load is established using a technique based on BEM and approximated by using the radial basis functions. The eigenmodes of the actual problem are obtained from the known integral representation of the solution for the classical plate bending problem, which is derived using the fundamental solution of the biharmonic equation. Thus, the kernels of the boundary integral equations are conveniently established and evaluated. The method has all the advantages of the pure BEM. To validate its effectiveness, accuracy as well as applicability of the proposed method, numerical results of various problems are presented.

The flexibility of functionally graded material plates subjected to uniform loads

We analyze functionally graded material (FGM) plates with two opposite edges simply supported and the other two edges free subjected to a uniform load. Even though an FGM plate is a kind of composite material, if the Young’s modulus of the FGM plates varies along the thickness direction and the Poisson’s ratio is constant in the whole FGM plate, the bending and in-plane problems in FGM plates under transverse load only are uncoupled. Therefore, the analytical solution to the bending problem of FGM plates is obtained in this study by Fourier series expansions, which agrees very well with a finite element calculation. Results show that the maximum tensile stresses are located at the bottom of the FGM plates. However, the maximum compressive stresses move to the inside of the FGM plates. The coefficients A11, B11,C11 defined in this paper relate to the area and to the first and the second moments of the area under the E(z) curve from z = h/2 to z = h/2. The parameter Q11, representing the location of the centroid of the area under the E(z) curve, is related to the location of the neutral surfaces, and S11 represents the bending stiffness of the FGM plates.

A posteriori error estimates for the Morley plate bending element

A local a posteriori error indicator for the well known Morley element for the Kirchhoff plate bending problem is presented. The error indicator is proven to be both reliable and efficient. The technique applied is general and it is shown to have also other applications.

Superconvergence and postprocessing of MITC plate elements

In this paper, the MITC finite element methods for the Reissner–Mindlin plate bending problem are considered. A superconvergence result for the deflection is proved. Utilizing this property a local postprocessing is introduced and analyzed. The improved accuracy of the deflection is confirmed by numerical computations.