A new algorithm for bending problems of continuous and inhomogeneous beam by the BEM

The conventional algorithms of BEM for the bending problems of continuous beam are inefficient and have several points to be improved upon. Although such defects may not become a serious problem as far as they are put to practical use for an individual calculation, once they are applied to a kind of optimal design with a certain optimization algorithm small disadvantages in the conventional algorithm are amplified greatly and the cost of design work becomes very high. It is because a great many repetitive calculations are required in such problems.From this point of view, we intend to develop a new algorithm. Main points of this study are: (1) to improve the composition of the simultaneous equations by introducing a new formulation process; (2) to establish a scheme without any variables at intermediate points; and (3) to establish a generalized solution scheme for an inhomogeneous beam. These new algorithms will greatly reduce the size of a matrix as well as the computing time and, therefore, will bring about high efficiency on the repetitive calculations. As a result, this will give a low cost for optimal design in daily work.     

A semianalytic solution for rectangular plate bending

By substituting the basic function satisfying boundary conditions along two opposite edges in one direction of the plate and then using suitable transformation, the plate bending equation is reduced to an ordinary differential equation. The resulting equation is solved by finite difference technique by using a small number of discrete variables. Examples have been presented for a variety of isotropic and orthotropic elastic rectangular plate of different boundary conditions under various loadings. Excellent accuracy has been obtained.     

A powerful finite element for plate bending

A powerful finite element formulation for plate bending has been developed using a modified version of the variational method of Trefftz. The notion of a boundary has been generalized to include the interelement boundary. All boundary conditions and the interelement continuity requirements (displacements, slopes, internal forces) have been obtained as natural conditions on the generalized boundary. Coordinate functions have been constructed to satisfy the nonhomogeneous Lagrange equation locally within the elements. Singularities due to isolated loads have been properly taken into account. For practical use a general quadrilateral element has been developed and its accuracy illustrated on several numerical examples. Work is in progress to extend the formulation to anisotropic and moderately thick plates and to vibration analysis.     

Mixed finite element models for plate bending analysis: A new element and its applications

With a few exceptions, finite element packages available in today's commercial software environment contain in their libraries displacement-type elements only. The present paper aims to demonstrate the feasibility that properly formulated mixed-type elements compete most favorably with displacement-type elements and should, therefore, be considered as potential candidates for inclusion in general purpose finite element packages. In doing so, the development of a new triangular doubly—curved mixed-hybrid shallow shell element and its extensive testing in carefully chosen example problems are reported on.     

A simple finite element model for elastic-plastic plate bending

A consistent finite element model for a plate is developed based on triangular elements and a piecewise-linear displacement field. The resulting generalized stresses are the average normal moments across the element interfaces. Equilibrium equations are derived for each node, and a simple constitutive equation is obtained for each generalized stress. Applications are made to some square plate problems.     

A hybrid displacement plate element for bending and stability analysis

A hybrid displacement plate element is derived from a modified energy functional based on a variational principle. The higher order curvature terms which generate high energy densities are filtered out by using independent interpolation of curvatures and moments. The inter-element compatibility requirements are relaxed by including element discontinuities in the variational formulation. The accuracy of the element is shown to be excellent in both plate bending and buckling analysis.     

板类结构动力检测与控制中的一种新方法

曲率模态在结构动力检测中具有对动力结构损伤部位非常敏感的特性，传统方法主要是运用中心差分法求解曲率模态，由于中心差分法的计算精度依赖于测点分布的紧密程度，这样就使动力检测结果具有很大的误差，本文利用函数的契贝雪夫多项式的展开式具有很高的逼近特性，提出了板类结构动力检测的曲率模态算法——契贝雪夫多项式算法，构造出了板类结构振型的契贝雪夫多项式函数，对该函数进行求二阶偏导得到x和y方向的曲率模态，进而求出结构损伤前后的曲率模态差，该方法可为结构损伤检测提供可靠的数据，从而达到良好的检测控制效果。     

A refined higher-order C° plate bending element

A general finite element formulation for plate bending problem based on a higher-order displacement model and a three-dimensional state of stress and strain is attempted. The theory incorporates linear and quadratic variations of transverse normal strain and transverse shearing strains and stresses respectively through the thickness of the plate. The 9-noded quadrilateral from the family of two dimensional C° continuous isoparametric elements is then introduced and its performance is evaluated for a wide range of plates under uniformly distributed load and with different support conditions and ranging from very thick to extremely thin situations. The effect of full, reduced and selective integration schemes on the final numerical result is examined. The behaviour of this element with the present formulation is seen to be excellent under all the three integration schemes.     

A simple and efficient shear-flexible plate bending element

A shear-flexible triangular element formulation, which utilizes an assumed quadratic displacement potential energy approach and is numerically integrated using Gauss quadrature, is presented. The Reissner/Mindlin hypothesis of constant cross-sectional warping is directly applied to the three-dimensional elasticity theory to obtain a moderately thick-plate theory or constant shear-angle theory (CST), wherein the middle surface is no longer considered to be the reference surface and the two rotations are replaced by the two in-plane displacements as nodal variables. The resulting finite-element possesses 18 degrees of freedom (DOF). Numerical results are obtained for two different numerical integration schemes and a wide range of meshes and span-to-thickness ratios. These, when compared with available exact, series or finite-element solutions, demonstrate accuracy and rapid convergence characteristics of the present element. This is especially true in the case of thin to very thin plates, when the present element, used in conjunction with the reduced integration scheme, outperforms its counterpart, based on discrete Kirchhoff constraint theory (DKT).     

An improved rectangular element for plate bending analysis

This paper presents a new, simple, rectangular finite element with twelve degrees of freedom for the bending analysis of thin plates. Three interpolation functions corresponding to the normal deflections and tangential slopes at the nodal points are written in parametric form. Convergence requirements are then used to find relationships among the parameters included in these functions. To identify the optimal values of the still undetermined parameters extensive comparisons are carried out using plate problems with different loading and boundary conditions. Certain values of the unknown parameters are found to produce displacement results with faster rate of convergence than those of other simple elements. When comparisons are based on a measure representing the actual computational effort rather than the mesh size the proposed element is found to excell higher-order elements as well. Stress results are also calculated for the proposed element and found to be fairly close to the exact values.     

Multivariable spline element analysis for plate bending problems

A multivariable spline element analysis for a plate bending problem is presented. The bicubic spline functions are used to construct the bending moments and transverse displacements field. The spline element equations with multiple variables are derived based on Hellinger-Reissner principle. Some numerical results are given and compared with other methods.     

A hybrid-stress quadratic serendipity displacement mindlin plate bending element

A hybrid-stress plate bending element is developed for thin and moderately thick plates based on a Serendipity-type quadratic displacement assumption. The through-thickness displacement behavior is consistent with Mindlin plate theory, and all components of stress are included. By proper choice of the transverse shear stress distributions, the element stiffness possesses correct rank, and results show no signs of locking in the thin plate limit; these and other considerations discussed herein suggest that the stress assumption used is optimum for the 8-node element. Results are compared with 8-node Serendipity and 9-node Lagrange assumed-displacement elements. The present element demonstrates high accuracy for all problems considered and has no apparent deficiencies.     

A discrete Kirchoff plate bending element with loof nodes

The majority of existing flat shell finite elements suffer from the deficiencies of displacement incompatibility, singularity when the elements are coplanar at a node, inability to model intersections and low-order membrane strain representation. In this paper, a plate bending element, labeled DKL (for Discrete Kirchoff element with Loof nodes), with the same nodal configuration as a triangular Semiloff plate element, but not formulated through the isoparametric concept is presented. This element when superposed with the linear strain triangle results in a faceted shell element free from the abovementioned deficiencies. Various numerical examples are tested using this plate element so as to demonstrate its reliability, accuracy and convergence characteristics.     

A review and catalogue of plate bending finite elements

A brief review of developments in the field of plate finite elements is presented. This review is followed by an extensive tabular listing of plate bending elements.     

A MLPG(LBIE) approach in combination with BEM

The local boundary integral equation (LBIE) approach is a promising meshless method, recently proposed as an effective alternative to the boundary element method (BEM), for solving non-homogeneous, anisotropic and non-linear problems. Since the LBIE method utilizes in its weak form fundamental solutions as test functions, it can be considered as one of the six meshless local Petrov–Galerkin (MLPG) methods proposed by Atluri and coworkers. This explains the use of the initials MLPG(LBIE) in the title of the present paper. This work addresses a coupling of a new MLPG(LBIE) method, recently proposed by the authors for elastodynamic problems, and the BEM. Because both methods conclude to a final system of linear equations expressed in terms of nodal displacement and tractions, their combination is accomplished directly with no further transformations as it happens in other MLPG/BEM formulations as well as in typical hybrid finite element method/BEM schemes. The coupling approach is demonstrated for static and frequency domain elastodynamic problems. Three representative examples are provided in order to illustrate the achieved accuracy of the proposed here MLPG(LBIE)/BE methodology.     

Mixed finite element models for plate bending analysis theory

The theoretical background of mixed finite element models, in general for nonlinear problems, is briefly reexamined. In the first part of the paper, several alternative “mixed” formulations for 3-D continua undergoing large elastic deformations under the action of time dependent external loading are outlined and are examined incisively. It is concluded that mixed finite element formulations, wherein the interpolants for the stress field satisfy only a part of the domain equilibrium equations, are not only consistent from a theoretical standpoint but are also preferable from an implementation point of view. In the second part of the paper, alternative variational bases for the development of thin-plate elements are presented and discussed in detail. In light of this discussion, it is concluded that the “bad press” generated in the past concerning the practical relevance of the so-called assumed stress hybrid finite element model is not justified. Moreover, the advantages of this type of elements as compared with the “assumed displacement” or alternative mixed elements are outlined.     

The “heterosis” finite element for plate bending

A new. high-accuracy, finite element for thick and thin plate bending is developed, based upon Mindlin plate theory. The element is a 9-node quadrilateral, which exhibits improved characteristics in comparison with the 8-node serendipity, or the 9-node Lagrange elements. In particular, the element stiffness possesses correct rank, and high accuracy is attainable for extremely thin plates. Due to the consistently good performance of the element, it is proposed as a candidate for inclusion in finite element programs available to the general user.     

Isoparametric finite difference energy method for plate bending problems

This paper incorporates the concept of isoparametry in finite difference energy method making it more powerful and versatile to tackle complex plate bending problems with curved boundaries. This approach overcomes the drawbacks of the finite difference energy method and its application to practical problems is now feasible. In order to estimate the accuracy and reliability of the present formulation several isotropic plates with a variety of planform are solved and the results are compared with the existing analytical and numerical solutions.