Solution Of Plate Bending Problems Research Articles

A posteriori error majorants for numerical solutions of plate bending problems on a Winkler basis

A posteriori error majorants for numerical solutions of plate bending problems on a Winkler basis

Reduced-order strategy for meshless solution of plate bending problems with the generalized finite difference method

Abstract This paper presents some recent advances on the numerical solution of the classical Germain-Lagrange equation for plate bending of thin elastic plates. A meshless strategy using the Generalized Finite Difference Method (GFDM) is proposed upon substitution of the original fourth-order differential equation by a system composed of two second-order partial differential equations. Mixed boundary conditions, variable nodal density and curved contours are some of the explored aspects. Simulations using very dense clouds and parallel processing scheme for efficient neighbor selection are also presented. Numerical experiments are performed for arbitrary plates and compared with analytical and Finite Element Method solutions. Finally, an overview of the procedure is presented, including a discussion of some future development.

Analytical bending solutions of free orthotropic rectangular thin plates under arbitrary loading

This paper presents the analytical solutions for the bending of orthotropic rectangular thin plates by the double finite integral transform, which, as an effective tool in solving plate problems, should have received attention. As a representative and difficult problem in the theory of plates, free plates’ bending is successfully solved to demonstrate the accuracy of the method by comparing the present analytical solutions with those from the literature as well as those by the finite element method. With the proper integral transform kernels, the proposed solution procedure is applicable to the bending of orthotropic rectangular plates with all combinations of simply supported, clamped and free boundary conditions, which serves as an elegant approach to analytical solutions of plate bending problems.

Physical decomposition of thin plate bending problems and their solution by mesh-free methods

Abstract We develop and compare a number of mesh-free formulations for solution of plate bending problems using either the Moving Least Square-approximation or Point Interpolation Method-approximation. For solution of the original biharmonic problem, we develop only the local weak formulation to avoid the 4th order derivatives of deflections. The decomposition of the biharmonic operator leads to lower order derivatives of field variables in both the developed strong and local weak formulations which are applicable to arbitrary boundary value problems for thin plate bending. The modified differentiation technique is proposed, in order to increase the accuracy of higher order derivatives of field variables. The accuracy, convergence of accuracy and computational efficiency are studied in simple boundary value problems for circular plate. The discussed methods give reasonable numerical results when applied to decomposed problem, while the methods applied to original biharmonic problem either fail or give unreliable results.

Plate bending for frame analysis by coupling method of singular integral operators method with finite elements

The coupling of the Singular Integral Operators Method (SIOM) and the Finite Element Method (FEM) is proposed and investigated for the solution of plate bending problems. Such combination of the above two numerical methods gives more accurate results for the solution of plate bending problems in structural analysis. The plates are basic units of building frame structures, as well as to underground structures and thus the determination of bending moments around the plates is very important for the analysis of all kinds of structures.

A Posteriori Error Analysis of the Linked Interpolation Technique for Plate Bending Problems

We develop a posteriori error estimates for the so-called linked interpolation technique to approximate the solution of plate bending problems. We show that the proposed (residual-based) estimator is both reliable and efficient.

Explicit Expressions of the Fundamental Elasticity Matrices of Stroh-Like Formalism for Symmetric/Unsymmetric Laminates

AbstractBased upon our recent development of Stroh-like forma lism for symmetric/unsymmetric laminates, most of the relations for bending problems can be organized into the forms of Stroh formalism for two-dimensional problems. Through the use of Stroh-like formalism, the fundamental elasticity matrices Ni, S, H and L appear frequently in the real form solutions of plate bending problems. Therefore, the determination of these matrices becomes important in the analysis of plate bending problems. In this paper, by following the approach for two-dimensional problems, we obtain the explicit expressions of the fundamental elasticity matrices for symmetric and unsymmetric laminates, which are all expressed in terms of the extensional, bending and coupling stiffnesses of the composite laminates.

Interpolated boundary conditions in plate bending problems using C1-curved finite elements

The purpose of this paper is to construct high order C 1-curved finite elements which are useful for the approximate solutions of plate bending problems with clamp or simply support boundary conditions over nonpolygon domains. This kind of finite elements is defined over the exact domain of the problem, and the corresponding approximate solution is also obtained over the original domain. Moreover, the effect of the numerical quadrature schemes is studied.

Multiple reciprocity method in BEM formulations for solution of plate bending problems

Advanced multiple reciprocity boundary element method formulations are developed for solution of certain plate bending problems when either the fundamental solutions are not available or these are rather complex. The present formulation preserves a pure boundary character and the derived boundary integral equations are nonsingular. The fundamental solutions as well as all the integral kernels involved have been derived and are available.

Plate bending problems using the nonsingular boundary element formulation and C1-continuous elements

The numerical implementation of an advanced boundary element formulation for the solution of plate bending problems within the classical Kirchhoff theory is presented. A direct and nonsingular formulation is used which is applicable to planar plates of arbitrary shape with arbitrary boundary conditions and subjected to static or harmonically varying loads. Owing to the nonsingular formulation, the regular integration quadratures give sufficiently accurate results, but the use of C 1-continuous elements appears to be necessary. The Overhauser elements are employed on smooth parts of the boundary contour, while ‘corner’ elements are used to represent discontinuities (such as corners) in geometry. Several numerical solutions are presented and compared to existing exact solutions.

Alternative technique for the solution of plate bending problems using the boundary element method

The objective of this work consists of presenting a boundary element technique for analysis of bending plates where all algebraic representations of rotations for boundary nodes are avoided. The technique requires outside load points whose definition is extensively studied in this article.

A hierarchic family of conforming finite elements for the solution of plate bending problems

A family of finite elements of type p for plate bending problems is introduced. The elements are conforming, that is, the approximation spaces satisfy the C 1-continuity condition, and hierarchic, that is, when the degree of the polynomials is increased new shape functions are added to the previous ones. Several plate bending problems have been solved. The related numerical results are presented together with a description of the features of the finite elements.

On the implementation of an equilibrium finite element method for the numerical solution of plate bending problems

A finite element method of equilibrium type is used to solve plate bending problems. Continuity of displacemnts, bending moments and Kirchhoff shear forces at the interelement boundaries are required. Suitable choices for the approximation spaces allow the continuity requirements to be satisfied. The paper focuses on the implementation of the method. After recalling the main features of the equilibrium approximation we describe in details the numerical program. Finally the numerical results obtained solving a model problem are presented.

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 analysis of plates with variable thickness by boundary element-transfer matrix method

The combined boundary element-transfer matrix method is proposed for plate bending problems. In this method, a transfer matrix is obtained, from the sytem of equations derived by the procedure based on the boundary element method. This method permits the use of large number of elements, without getting involved with large matrices. A much smaller computer is therefore sufficient. Some numerical examples are presented to demonstrate the accuracy as well as the capability of the proposed method for solutions of plate bending problems.

A general boundary integral formulation for the numerical solution of plate bending problems

A direct approach is employed to obtain a general formulation of plate bending problems in terms of a pair of singular integral equations involving displacement, normal slope, bending moment and shear on the plate boundary. These equations are coupled with prescribed boundary conditions involving these same variables furnishing a convenient basis for numerical solution. A simple discretization scheme is described and two model problems treated to illustrate the nature and quality of some typical results.

Solution of plate bending problems in fracture mechanics using a specialized finite element technique

This paper deals with the development and application of a special crack-tip finite element to obtain the bending and shear intensity factors for thin elastic plates containing cracks. The bending and shear intensity factors are then used to compute the Strain Energy Density Factor and the direction of crack initiation. The solution procedure is illustrated through several numerical examples. The problem of an axial flow compressor blade containing a crack is solved using a combination of special crack tip plate bending and plane stress elements.

On the numerical solution of plate bending problems by hybrid methods

On the numerical solution of plate bending problems by hybrid methods

On the solution of plate bending problems for multiply connected regions

On the solution of plate bending problems for multiply connected regions

A comparison of approximate methods for the solution of plate bending problems.

It is well known that classical, exact methods of analytical solution cannot be applied to the plate bending problem in cases of irregular boundary shape and arbitrary transverse loading and constraint. The paper summarizes and compares results obtained from using nine approximate analytical methods: point matching, boundary point least squares, TrefftzMorley, interior collocation, interior least squares, subdomain, Galerkin, Ritz, and Kantorovich. For purposes of concrete comparison, each of the methods is applied to two problems for which exact, although intricate, solutions are known: 1) a uniformly loaded, simply supported elliptical plate; 2) a square plate having free edges, supported at four asymmetrically located interior points, and loaded by its own weight. Comparative results are presented. Each technique is rated good, fair, or poor according to 11 important technical criteria and the underlying rationale is explained.