The Galerkin element method applied to the vibration of rectangular damped sandwich plates

The Galerkin element method (GEM), which combines Galerkin orthogonal functions with the traditional finite element formulation, has previously been applied successfully to the vibration analysis of damped sandwich beams, and an improved iteration method was developed for its eigen solution. In the current paper, this promising method is extended to the vibration of damped sandwich plates. A quite different model is formulated which has both nodal coordinates and edge coordinates, while in the case of beams, there are only nodal coordinates. Displacement compatibility over the interfaces between the damping layer and the elastic layers is taken account of in order to ensure a conforming element and thereby guarantee good accuracy. The seed matrix method is proposed for simplifying the building of the element mass, stiffness and damping matrices. Numerical examples show that the application of the GEM to sandwich plate structures is computationally very efficient, while providing accurate estimates of natural frequencies and modal damping over a wide frequency range.     

Adaptive element-free Galerkin method applied to the limit analysis of plates

The implementation of an h-adaptive element-free Galerkin (EFG) method in the framework of limit analysis is described. The naturally conforming property of meshfree approximations (with no nodal connectivity required) facilitates the implementation of h-adaptivity. Nodes may be moved, discarded or introduced without the need for complex manipulation of the data structures involved. With the use of the Taylor expansion technique, the error in the computed displacement field and its derivatives can be estimated throughout the problem domain with high accuracy. A stabilized conforming nodal integration scheme is extended for use in error estimation and results in an efficient and truly meshfree adaptive method. To demonstrate its effectiveness the procedure is then applied to plates with various boundary conditions.     

Elastic-viscoplastic analysis of plates by the finite element method

In this paper the finite element technique is employed in the solution of elastic-viscoplastic plate bending problems. The method is applicable to both thick and thin plates and by attaining steady-state conditions the process offers an alternative method of solution for static elasto-plastic situations. A quadratic isoparametric element based on Mindlin plate theory is adopted and the Euler time-stepping scheme is employed in solution. Yield criteria are based on moment resultants and the Von Mises, Tresca and Johansen forms are included. Several numerical examples are presented and the results compared with those from other sources where available     

Nonlinear analysis of skew plates using the finite element method

In this paper finite element analysis of the large deflection behaviour of skew plates has been done. A high precision conforming triangular plate bending element has been used. The central deflection, bending and membrane stresses have been reported for simply supported and clamped rhombic plates. The variations of these quantities have been studied for different skew angles.     

Stability analysis for plates using the multivariable spline element method

The multivariable spline element method is used in this paper to solve the stability problems of plates and beams. The bicubic spline functions are employed to construct the bending moments, twisting moments and transverse displacements field. The spine eigenvalue equations with multiple variables are derived based on the Hellinger-Reissner mixed variational principle. Some numerical examples are given, the results are good agreement with other methods.     

An error analysis for the finite element method applied to convection diffusion problems

This paper analyzes the finite element method applied to a convection diffusion model problem. Linear elements are used for the trial space. The error is measured in a norm closely related to the L_p norm. When the test space is composed of linear elements with parabolic upwinding, the method is shown to be optimal when the input data is piecewise smooth—a condition which is usually observed in practice. Without these smoothness assumptions, the method is shown to be non-optimal, even if the class of test spaces is extended to include any elements which have a shape independent of the mesh size.     

Accuracy estimation in the finite element analysis of transverse bending of thin flat plates

As a basic study for the establishment of an accuracy estimation method in the finite element method, this paper deals with the problems of transverse bending of thin, flat plates. From the numerical experiments for uniform mesh division, the following relation was deduced, ε ∝ (h/a)^k, k 1, where ε is the error of the computed value by the finite element method relative to the exact solution and h/a is the dimensionless mesh size. Using this relation, an accuracy estimation method, which was based on the adaptive determination of local mesh sizes from two preceding analyses by uniform mesh division, was presented.

A computer program using this accuracy estimation method was developed and applied to 28 problems with various shapes and loading conditions. The usefulness of this accuracy estimation method was illustrated by these application results.     

Thermal stress analysis of skew plates by finite element method

Thermal stresses are induced in general due to nonuniform temperature distribution or due to the boundary restriction. Most of the work reported so far deals with either plates with edges clamped in plane of the plate or plates with stress free edges. While studying buckling or post-buckling problems, one should ideally analyse the plates with mixed in-plane boundary conditions. Hence, in the present analysis, thermal stress analysis of skew plates with mixed in-plane boundary conditions using finite element approach is attempted. In addition, the effect of in-plane boundary conditions on the thermal stresses is also discussed.     

Field/boundary element approach to the large deflection of thin flat plates

The problem of large deflections of thin flat plates is rederived here using a novel integral equation approach. These plate deformations are governed by the von Karman plate theory. The numerical solution that is implemented combines both boundary and interior elements in the discretization of the continuum. The formulation also illustrates the adaptability of the boundary element technique to nonlinear problems. Included in the examples here are static, dynamic and buckling applications.     

Postbuckling analysis of plates on an elastic foundation by the boundary element method

The postbuckling behavior of plates on an elastic foundation is studied by using the boundary element method (BEM). A new fundamental solution of lateral deflection is derived through the resolution theory of a differential operator, and a set of boundary element formulae in incremental form is presented. By using these formulae, the BEM solution procedure becomes relatively simple. The results of a number of numerical examples are compared with existing solutions and good agreement is observed. It shows that the proposed method is effective for solving the postbuckling problems of plates with arbitrary shape and various boundary conditions.     

Finite element analysis of stiffened plates

A finite element method is presented in which the constraint between stiffener and member is imposed by means of Lagrange multipliers. This is performed on the functional level, forming augmented variational principles. In order to simplify the initial development and implementation of the proposed method, two-dimensional stiffened beam finite elements are developed. Several such elements are formulated, each showing monotonic convergence in numerical tests. In the development of stiffened plate finite elements, the bending and membrane behaviors are treated seperately. For each, the stiffness matrix of a standard plate element is modified to account for an added beam element (representing the stiffener) and additional terms imposing the constraint between the two. The resulting stiffened plate element was implemented in the SAPIV finite element code. Exact solutions are not known for rib-reinforced plated structures, but results of numerical tests converge monotonically to a value in the vicinity of an approximate “smeared” series solution.     

A boundary integral method applied to plates of arbitrary plan form

An integral equation method for the solution of thin elastic plates of arbitrary plan form has been presented. The method involves embedding the real plate in a fictitious plate for which the Green's function is known. An unknown load vector is then introduced on the boundary of the real plate (line load and line normal moment). The deflection field due to both known transverse and unknown boundary loads can then be found everywhere by superposition. Satisfaction of the boundary conditions on the real plate results in a vector integral equation in the unknown boundary vector.In concept, any consistent set of boundary conditions will yield a solution. Practically, boundary conditions requiring higher derivatives of the deflection are both very cumbersome and yield singularities in the integral equations which cause numerical difficulties. For these reasons only clamped boundary conditions are treated numerically in the present paper.For interior bending moments and deflections (greater than distances of the order of one boundary subdivision from the boundary) the method is both highly accurate and inexpensive. Errors right on the boundary, e.g. the clamping moment in the clamped boundary condition case, can be appreciable, however. While this can be improved by a more sophisticated treatment of the unknown boundary vector in the numerical solution (increased expense) it is shown in the paper that a simple boundary extrapolation procedure gives excellent accuracy there.     

Bending analysis of rectangular moderately thick plates using spline finite element method

A bending analysis of rectangular, moderately thick plates with general boundary conditions is presented using the spline element method. The cubic B spline interpolate functions are used to construct the field function of generalized displacements w, φ_itxand φ_ity. The spline finite element equations are derived based on the potential energy principle. For simplicity, the boundary conditions, which consist of three local spline points, are amended to fit specified boundary conditions. The shear effect is considered in the formulations. A number of numerical examples are described for rectangular, moderately thick plates. Since the cubic B spline interpolate functions have sufficient continuity and are piecewise polynomial, so the present numerical solutions show not only that the method gives accurate results, but also that the unified solutions of thick and thin plates can be directly obtained; the trouble with the so-called shear locking phenomenon does not occur here.     

Thermoelastic stress analysis of anisotropic composite sandwich plates by finite element method

In the present work a finite element approach is established for the analysis of sandwich plates with different anisotropic composite facings due to aerodynamic and thermal fields. The main contribution of this study is the consideration of the inherent coupling phenomenon of stretching and bending in such non-symmetric composite sandwich plates. Consequently, the stiffness matrix includes now sub-matrices which relate the “generalized” transverse thermal (or aerodynamic) loads to the in-plane displacements, and the in-plane forces to the “generalized” transverse displacements. These sub-matrices do not exist in symmetrically layered sandwich plates and their inclusion in the analysis is shown to be of primary importance. Three examples are included, indicating the sensibility of the stress and displacement fields to the class of heterogeneity and anisotropy of the considered sandwich plates.     

Instability analysis of thin plates and arbitrary shells using a faceted shell element with loof nodes

The faceted representation is employed in the paper to derive a 24-dof triangular shell element for the instability analysis of shell structures. This element, without the deficiencies of displacement incompatibility, singularity with coplanar elements, inability to model intersections, and low-order membrane strain representation, which are normally associated with existing flat elements, has previously been found by the authors to perform well in linear static shell analyses. The total Lagrangian approach is used in the nonlinear formulation, and the results of the various numerical examples indicate that its performance is comparable to existing nonlinear shell elements. An extrapolation stiffness procedure, which will improve the convergence characteristics of the constant arc length solution algorithm used here, is also presented.     

Finite element buckling analysis of stiffened plates

An isoparametric stiffened plate bending element for the buckling analysis of stiffened plates has been presented. In the present approach, the stiffener can be positioned anywhere within the plate element and need not necessarily be placed on the nodal lines. The element, being isoparametric quadratic, can readily accommodate curved boundaries, laminated materials and transverse shear deformation. The formulation is applicable to thin as well as thick plates. The buckling loads for various rectangular and skew stiffened plates with varying skew angles and stiffness parameters have been indicated. The results show good agreement with those published.     

Boundary element method based on preliminary discretization

A new numerical method for solving wave diffraction problems is given. The method is based on the concept of boundary elements; i.e., the unknown values are the field values on the surface of the scatterer. An analog of a boundary element method rather than a numerical approximation of the initial (continuous) problem is constructed for an approximate statement of the problem on the discrete lattice. Although it reduces the accuracy of the method, it helps to simplify the implementation significantly since the Green functions of the problem are no longer singular. In order to ensure the solution to the diffraction problem is unique (i.e., to suppress fictitious resonances), a new method is constructed similarly to the CFIE approach developed for the classical boundary element method.     

基于边光滑有限元法的剪切变形板几何非线性分析

为改善在计算板的几何非线性问题时有限元法系统过硬的数值缺陷,提高计算精度,在考虑剪切变形的yon Karman假设下,基于全拉格朗日描述方法,将边光滑有限元法应用于板的几何非线性分析.计算公式基于1阶剪切变形理论,并采用离散剪切间隙有效地消除剪切自锁.在三角形单元的基础上进一步形成边界光滑域,在每个光滑域内对应变进行光...     

A numerical method for laterally loaded thin plates

A numerical method for dealing with laterally loaded thin plates is presented and compared to the similar but more basic numerical technique used in [1, 2]. This method considers the plate to be embedded in the infinite plane and uses point load sources external to the plate boundary to satisfy boundary conditions as in collocation. The exact solution for a constant lateral load acting over an arbitrary polygon in the infinite plane is first derived and then used to couple the effects of lateral loading to the collocation method described above. The results show that less than half the number of collocation points used in [1, 2] is needed for the same accuracy and that the computer execution time is reduced several hundred times. This method is not limited to particular plate shapes, boundary condition types or load distributions. Five examples are used to illustrate the method.     

Three-dimensional finite element analysis of thick laminated plates

A displacement-based, three-dimensional finite element scheme is proposed for analyzing thick laminated plates. In the present formulation, a thick laminated plate is treated as a three-dimensional inhomogeneous anisotropic elastic body. Particular attention is focused on the prediction of transverse shear stresses. The plane of a laminated plate is first discretized into conventional eight-node elements. Various through-thickness interpolation is then denned for different regions of the plate; layerwise local shape functions are used in the regions where transverse shear stresses are of interest, while an ad hoc global-local interpolation is used in the region where only the general deformation pattern is concerned. For satisfying the displacement compatibility between these two regions, a transition zone is introduced. The model incorporates the advantages of the layerwise plate theory and the single-layer plate theory. Details of formulation will be presented together with several numerical examples for demonstrating the proposed scheme.