Plate Bending Problems Research Articles

On mixed finite element methods in plate bending analysis

Most of the existing convergence theory of mixed finite element methods for solving the plate bending problem converns the model case of a purely clamped or simply supported plate with sufficiently regular boundary. The extension of this analysis to more complicated situations encounters two major difficulties: first, the problem of verifying the stability of the schemes in the case of a partially free boundary and, second, the reduction of the solution's regularity in the presence of reentrant corners or changes in the type of the boundary conditions. In this paper these questions are studied for the approximation of the Kirchhoff plate model by one of the mixed finite element schemes due to L. R. Herrmann, the so-called “first Herrmann scheme”. It is shown that this method converges on any polygonal domain and for all usual boundary conditions. The proof is based on the fact that this particular mixed scheme is algebraically equivalent to a nonconforming displacement method.

Notch mechanics for plane and thin plate bending problems

This paper describes the summary of notch mechanics based on the linear elasticity. The V-shaped notch with sharp or round corner and symmetric to the bisector is the object of this study. The plane elastic problem and thin plate bending problem are considered for free and fixed boundary conditions. The following matters are investigated: the stress distributions near the sharp corner; the general expression of the stress concentration at the round corner; the relationship between the intensity of corner and the stress concentration; and the stress intensity factor of a crack initiating from the notch tip. The notch mechanics is connected with the crack mechanics by the expression of stress intensity factor.

引張り力を受ける２層斜交積層板のねじれ変形

It is well known that two-layered angle-ply laminates twist under tensile load. Twist angles of the laminates may be calculated by the utilization of so-called lamination theory. The lamination theory is based on Kirchhoff hypothesis for plate bending problems; therefore, restrictions on deformations are imposed. In the present paper twist phenomena of two-layered angleply laminates are analyzed by extending the theory of torsion of bars which is free from the restrictions. The results of numerical calculations show that the twist angles highly depend on the fiber directions and the direction of twist can be plus or minus for certain material properties.

Analysis of continuous plates by a combined boundary element-transfer matrix method

A structural analysis method based on a combined use of boundary element-transfer matrix method is proposed for continuous plate bending problems. A transfer matrix is evaluated from the system of equations based on the ordinary boundary element method. A technique for intermediate supports is proposed in order to apply this method to continuous plates. Some numerical examples of continuous plates subjected to uniform and concentrated loads are proposed and their results are compared with those obtained by other methods.

Geometrically nonlinear analysis of a Reissner type plate by the boundary element method

The boundary element method is used in the geometrically nonlinear analysis of laterally loaded isotropic plates taking into account the effects of transverse shear deformation. This paper presents the general equations for finite deformation of a Reissner type plate, and also gives an integral formulation of the Von Kármán type nonlinear governing equations which involve coupling between in-plane and out-of-plane deformation. The boundary and the domain of the plate are discretized to solve nonlinear plate bending problems. All unknown variables are at the boundary. An iterative procedure is applied to achieve linearization of the nonlinear equations. Some numerical results of the computation are compared with the analytical solutions and other numerical techniques, and good agreement is obtained.

Error estimates and adaptive refinement for plate bending problems

AbstractThe Zienkiewicz–Zhu error estimator is shown to be effective in problems of plate flexure. When used in conjunction with triangular elements and an adaptive mesh generator allowing a prescribed size of elements to be developed, very fast adaptive convergence for results of specified accuracy is achieved.

A new boundary integral equation‐finite‐element method for clamped orthotropic plate bending problems

AbstractIn this paper, a simplified boundary integral equation‐‐finite‐element method is proposed for the bending analysis of thin clamped orthotropic plates according to Kirchhoff's theory. The solution procedure is based on the direct boundary element method earlier developed for isotropic plate bending problems. The integral equations necessary for solving the orthotropic plate bending problem are derived after transformation of such a problem into an equivalent isotropic plate bending problem by introducing an unknown transverse loading distributed in the interior domain of the plate. A simple discretization scheme is described in this paper for the plate boundary and its interior domain. Some computational examples are presented in this paper, and the numerical results illustrate a satisfactory accuracy of the method.

An Optimal-Order Nonconforming Multigrid Method for the Biharmonic Equation

An optimal-order multigrid method for solving the biharmonic equation using Morley nonconforming finite elements is developed.

Bending, buckling and vibration of antisymmetrically laminated angle-ply rectangular simply supported plates

SummaryAffine transformation and similarity rules has been used to solve the classical bending, buckling, and vibration problems of orthotropic plates. The present work extends these concepts to antisymmetric angle-ply laminated plates by employing a simple transformation, utilising previous material constant definitions by Tsai and Brunelle, and a similarity parameter. With the help of certain defined characteristic and bounded quantities, comprehensive solutions are found to this problem; hence, the curves presented in the text are generic rather than specific. The results indicate that the frequency parameters and buckling coefficients increase with increasing generalised rigidity ratio D* and decrease with increasing generalised Poisson’s ratio ε, and the static deflection coefficients decrease with increasing D* and increase with increasing ε. The results of a reduced set of equations agree with those obtained by previous investigators.

On a mixed finite element method for clamped anisotropic plate bending problems

AbstractThe paper deals with a new mixed finite element method of solution of the bending problem of clamped anisotropic/orthotropic/isotropic plates with variable/constant thickness. This new mixed method gives simultaneous approximations to displacement u and bending and twisting moment tensor (ψij)1 ≤ i, j ≤ 2. Computer implementation procedures for this mixed method are given along with results of numerical experiments on a good number of interesting problems.

Bending of rectangular thin plates with free edges laid on tensionless Winkler foundation

In this paper, the bending problem of rectangular thin plates with free edges laid on tensionless Winkler foundation has been solved by employing Fourier series with supplementary terms. By assuming proper form of series for deflection, the basic differential equation with given boundary conditions can be transformed into a set of infinite algebraic equations. Because the boundary of contact region cannot be determined in advance, these equations are weak nonlinear ones. They can be solved by using iterative procedures.

Evaluation of a finite element for calculating binder wrap surfaces in sheet metal forming analysis

This paper presents a set of benchmark examples for evaluating finite elements to calculate the binder wrap surface of a draw die in sheet metal forming analysis and an extensive examination to evaluate the capability of a simple eight-node quadrilateral finite element in calculating binder wrap surfaces. The benchmarks consist of five cylindrical plate bending problems. They were selected for their strong geometric nonlinear deformations, typical of those of binder wraps, and by the existence of exact solutions. They are the best example problems available which resemble the binder wrap deformations and also have closed-form solutions. Numerical results obtained for the benchmarks from the quadrilateral element compare well with the exact solutions in all cases, thereby demonstrating the applicability of the element in binder wrap surface calculations.

Bending of Circular Plates Supported at Number of Points

There are many methods available in the literature to study the static behavior of plates with different boundary conditions. Except for a few simple cases, exact solutions for plate problems are rather difficult to obtain. In many cases, one may have to resort to various approximate and numerical methods. Each method has its own merits and demerits. In this note, a new method of solving plate bending problems known as charge simulation method is introduced. This method is somewhat similar to the boundary element method, but differs from it in that the source is placed outside the domain. The method is applied to a circular plate fixed at a number of points along its edge. Experiments have been carried out to check the results presented.

BENDING OF COMPOSITE LAMINATED PLATES WITH COMPLEX BOUNDARY CONDITIONS

In the theory of isotropic thin plates, the exact solutions of bending of thin plates with complex boundary conditions are obtained in Refs. [1] and [2] by applying the linear superposition principle and the idea of generalized simply-supported edge. This method has generality for solving bending problems of plates. The bending problems of thin plates with arbitrary boundary conditions, generally speaking, can be solved by using this method. In view of these advantages, we expand

A simple stress error estimator for Hybrid‐Trefftz p‐version elements

AbstractAn a posteriori error estimator is presented which allows a good pointwise evaluation of the error in predicted stresses and can easily be implemented in existing FE codes. Although this estimator has especially been developed for and tested on p‐version Hybrid‐Trefftz (HT) elements, it is anticipated that it can also be applied to conventional conforming p‐version elements. The practical efficiency of the estimator is illustrated through the solution of various plate bending problems by using the HT p‐version Kirchhoff plate elements.2

Large deflection analysis of bimodulus annular and circular plates using finite elements

In the present paper, the finite element method is used to solve the large-deflection bending problem of annular and circular bimodulus plates. The material model suggested by Jones is adopted to represent the bilinear stress-strain relations of bimodulus material. The resulting nonlinear equations are solved, using the Newton-Raphson iterative procedure. In the case of bimodulus composite structures, at any point, the position of neutral axis is a function of the state of stress at that point and is evaluated following iterative procedures. Numerical work has been done for two different types of boundary conditions, viz. clamped and simply supported cases. The results have been compared for particular cases and they are presented in the form of graphs and tables.

Stability of plates with step variation in thickness

The analytical strip method (ASM), which was recently developed for the bending problem of plates, is extended in this study to the buckling problem of rectangular plates. In this method, the plate is idealized as a system of plate strips and the system's behavior is derived by imposing the edge and continuity conditions on the closed-form solution of the individual plate strips. The advantage of this method lies in its versatility and its capability of dealing with plates which are partially compressed or have a step variation in their thickness. Results are presented for plates with various end conditions and for different step functions for the thickness.

Stress analysis of a crack at a juncture of a strip and a half-plane

An elastic plate with a crack at a juncture of a strip and a half-plane is analysed as a thin plate bending problem and as a plane elastic problem. Four and three loading conditions are considered for each problem. Stress distributions near the crack and the juncture are investigated. The stress intensity factors are obtained and compared with those of some similar shapes. The rational mapping function of a sum of fractional expressions and the complex variable method are used.

A finite difference scheme with arbitrary mesh systems for solving high-order partial differential equations

A finite difference technique for arbitrary irregular mesh systems is developed for numerical solutions of high-order boundary value problems. A successive differencing scheme is used to formulate the higher-order differencing equations. The details of the computational procedure for plate bending problems which are governed by a biharmonic equation are presented. In order to demonstrate the validity of the formulation procedure, numerical results obtained from the present technique were first compared with the analytical solutions for problems having regular and curved boundaries. For the regular boundary problem, the present results were also compared with that from the conventional finite difference method which is only suitable for treating regular meshes. Computations were then conducted to evaluate the accuracy of the solutions with different mesh arrangements. These studies have shown that, if well implemented, this method not only maintains the inherent simplicity of the finite difference method, but also puts it into a more programmable and flexible formulation so that the geometric generality, such as that in the finite element method, can be achieved.

On a highly accurate computational scheme for the approximate analysis of shear-flexible anisotropie plates

The paper deals with the analysis of shear-flexible anisotropic plates subjected to arbitrary boundary conditions. In order to solve the pertinent boundary value problem, a Ritz-type technique has been developed which, however, ist not based on the standard principle of minimum potential energy but a modification thereof. In this variational theorem, the boundary conditions are satisfied a posteriori as natural constraints, therefore relatively simple trial functions can be used to approximate the displacement variables in the interior of the plate. Extensive studies carried out on a series of relevant examples illustrate that the proposed method represents an inexpensive, reliable and highly efficient tool for solving a wide variety of plate bending problems.