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.     

Semi-numerical analysis of rectangular plates in bending

Bending analysis of rectangular plates is proposed using a combination of basic functions and finite difference energy technique. The basic function satisfying the boundary conditions along the two opposite edges of the plate is substituted in the integral expression for the total potential energy of the plate thereby reducing a two dimensional functional into an unidirectional one. The discretized form of the total potential energy of the plate expressed as a functional of the displacement field is obtained by replacing the derivatives by the corresponding difference quotient. Using the principle of minimum potential energy a set of algebraic equations is obtained which is subsequently solved for unknown displacements. Examples have been presented for a variety of isotropic and orthotropic plates of rectangular plan-form with different edge conditions and loadings. Results have been compared with other numerical results and available analytical solutions.     

An automated,energy-based,finite-difference procedure for the elastic collapse of rectangular plates and panels

A FORTRAN IV, large capacity, computer program has been developed to determine collapse loads and bifurcation loads for linear and nonlinear prebuckling behavior for fiber-reinforced, laminated, rectangular plates and panels under general loading systems and boundary conditions. The program is based on the principle of total potential energy and uses finite-differences in the discretization process. Whole-station spacing has been used to calculate the strain energy associated with an area-element and an orthogonal finite-difference grid that provides for variable spacings in perpendicular directions is incorporated.Numerical results are presented that compare favorably with results obtained via the general computer program STAGS. Other numerical results are presented that illustrate the types of boundary conditions, applied loads, cut-outs and initial geometric imperfections that can be handled by the present program. A brief study of the effect of panel construction and initial geometric imperfections on the buckling behavior of fiber-reinforced panels is presented.     

Calculation of the natural frequencies and steady state response of thin plates in bending by an improved rectangular element

This paper presents an application of a new improved rectangular finite element to the problems of free and forced harmonic oscillations of thin plates in bending. The proposed element, called the parametric element, has been presented in a previous paper by the same authors and applied to the problem of static bending of plates. The shape functions corresponding to the various nodal movements are expressed in simple parametric forms which scan the space between the Adini-Clough-Melosh model and the Papenfuss model. The performance of the parametric element is found to be at its best in both the statical and dynamical applications when the parameters included in the shape functions assume certain values. Like what happened in the statical application, the optimal parametric element has shown remarkable superiority over other simple elements when used in the prediction of the natural frequencies and harmonic response of several plates having different boundary conditions.     

An efficient analysis for thin plates of general quadrilateral shape subject to bending stresses

Isotropic quadrilateral plates subject to bending are analysed by the Rayleigh-Ritz method. It is shown that, by defining the plate in a warped coordinate system, suitable global functions for the plate displacements which fit most boundary conditions of practical significance can be readily derived. The performance of the method is assessed by comparing the results with those obtained from different theoretical solutions and/or experiment. Good correlation is obtained between all sets of results, whilst those due to the proposed method indicate considerable advantages over the alternative solutions, both in terms of computational efficiency and the ease with which the requirements for an acceptably accurate solution can be pre-determined.     

Computation of elastic buckling loads of rectangular thin plates using the extended Kantorovich method

In this paper, the extended Kantorovich method proposed by Kerr is further extended to the eigenvalue problem of elastic stability of various rectangular thin plates. By taking advantage of the availability and reliability of state-of-the-art ordinary differential equation solvers, multi-term trial functions have been employed, which is a significant extension to Kerr’s single term approach. As a result, the accuracy is greatly improved and some special problems that a single-term trial function fails to solve are now accommodated. A large number of numerical experiments have been carried out and the computed buckling loads are either exact or more accurate than the best known results in the literature.     

Constraint variational schemes for analysis of rectangular plates on an elastic half space

The flexural interaction of a rectangular thin elastic plate resting in smooth contact with an isotropic homogeneous elastic half space is analysed by using constraint variational schemes. The deflected shape of the plate is represented by a double power series of spatial variables with a set of generalized coordinates. The contact stresses are expressed in terms of the generalized coordinates by discretizing the contact area into several rectangular regions and solving an appropriate flexibility equation based on generalized Boussinesq's solution. Using the representations adopted for displacement and contact stresses, a constraint energy functional is constructed to determine the generalized coordinates. The constraint term in the variational functional corresponds to plate edge boundary conditions and formulations corresponding to both Lagrange multiplier and penalty types are presented. It is noted that for the present class of problems, penalty type formulations are numerically efficient. The convergence and numerical stability of the solution scheme is confirmed. Selected numerical results are presented to illustrate the dependence of flexural response of plate on the governing parameters of the plate-half space system.     

Vibrations of thin rectangular plates with arbitrarily oriented stiffeners

Free vibration of plates with arbitrarily oriented stiffeners are studied using high precision plate bending and stiffener elements. Good convergence of frequency values for coarse mesh is demonstrated. Natural frequencies of square plates with various arrangement of stiffeners are determined for both simply supported and clamped boundary conditions.     

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.     

Mixed finite-difference scheme for analysis of simply supported thick plates

A mixed finite-difference scheme is presented for the stress and free vibration analysis of simply supported nonhomogeneous and layered orthotropic thick plates. The analytical formulation is based on the linear, three-dimensional theory of orthotropic elasticity and a Fourier approach is used to reduce the governing equations to six first-order ordinary differential equations in the thickness coordinate. The governing equations possess a symmetric coefficient matrix and are free of derivatives of the elastic characteristics of the plate. In the finite difference discretization two interlacing grids are used for the different fundamental unknowns in such a way as to reduce both the local discretization error and the bandwidth of the resulting finite-difference field equations. Numerical studies are presented for the effects of reducing the interior and boundary discretization errors and of mesh refinement on the accuracy and convergence of solutions. It is shown that the proposed scheme, in addition to a number of other advantages, leads to highly accurate results, even when a small number of finite difference intervals is used.     

作大范围运动FGM矩形薄板的动力学特性研究

对作大范围运动功能梯度材料矩形薄板的刚柔耦合动力学问题进行了研究.以连续介质力学为基础,在柔性薄板面内变形中考虑了传统建模方法忽略的二次耦合变形量,采用假设模态法对薄板变形位移进行离散,运用拉格朗日方程推导了大范围运动功能梯度材料板的刚柔耦合动力学方程.对旋转运动下取不同功能梯度指数的悬臂功能梯度板的动力学行为进行仿真,比较了本文建立的一次近似耦合模型和传统不计耦合变形项的零次近似模型.结果表明,随着转速的提高,传统零次模型发散,而本文模型收敛,能够较好地描述系统动力学行为.利用本文建立的一次近似模型,研究了功能梯度指数对转动功能梯度板变形的影响,研究表明,随着功能梯度指数的增大,板的横向变形也增大.通过求解旋转FGM板在恒定转速下的固有频率,进一步分析了功能梯度板材料组分变化对板振动特性的影响.     

矩形正交各向异性薄板弯曲受迫振动问题的分析解

通过恰当的辛内积定义,首先将矩形正交各向异性薄板弯曲受迫振动问题导入到辛对偶体系,并应用分离变量和辛本征展开的有效数学物理方法给出其受迫振动稳态解的一个解析求解方法.然后,具体讨论了对边简支和对边固支两种典型边界条件的正交各向异性薄板弯曲受迫振动问题的辛本征问题,并给出了对应的辛本征值超越方程和辛本征向量的解析表达式.最后,应用本文的方法分析求解了两个具体算例,并将受均布谐载作用的四边简支矩形板受迫振动稳态解的分析解与传统的Navier法进行了精度和收敛性的对比,结果表明本文的方法比传统的分析求解方法具有更好的精度和收敛性,尤其是对内力.     

An efficient algorithm for finite-difference modeling of mixed-boundary-value elastic problems

The paper describes an efficient numerical scheme for the solution of displacements and stresses in mixed-boundary-value elastic problems of solid mechanics. A new variable reduction scheme is used to develop the computational model. Solution of both two- and three-dimensional problems of linear elasticity is considered in the present paper. In the present approach, the problems are formulated in terms of a potential function, defined in terms of the displacement components. Compared to the conventional computational approaches, the present scheme is capable of providing numerical solution of higher accuracy with less computational effort. Application of the present finite-difference modeling scheme is demonstrated through the solutions of a number of practical stress problems of interest, and the results are compared with those obtained by the standard method of solution. The comparison of the results establishes the rationality as well as suitability of the present variable reduction scheme.     

四边简支矩形薄板的双Hopf分岔分析

基于奇异性理论,研究了主参数共振-1∶3内共振情形下参数激励与外激励联合作用下四边简支矩形薄板的双Hopf分岔问题.考虑弱阻尼和弱激励的情形,得到了四边简支矩形薄板的分岔方程,给出了四边简支矩形薄板在参数平面μ-σ1上的分岔图.对参数激励与外激励联合作用下四边简支矩形薄板的阻尼系数、外激励、参数激励以及调谐参数进行不同的取值,通过数值模拟得到了四边简支矩形薄板平衡解将发生Hopf分岔,并分岔出周期解,薄板系统的非线性振动形式为周期运动.当四边简支矩形薄板的参数满足给定条件时,我们得到薄板的1∶3共振双Hopf分岔.随后,四边简支矩形薄板将会呈现概周期振动.     

An efficient computer program for the large deflection analysis of rectangular orthotropic plates

A computer program is presented for the static analysis of rectangular specially orthotropic plates undergoing large deflections. In-plane loads and lateral pressure are both catered for, together with various combinations of boundary conditions. However, a restriction is that there must be symmetry about both plate axes. The plate can have an initial geometrical imperfection, which is represented by a double Fourier series with the same terms as that used for the lateral displacement. By specifying in the input data suitable values for the coefficients in the imperfection series, the effect of relatively complicated initial shapes can be examined. The program is called ELDAROP, a FORTRAN listing of which is given in Appendix A. Appendix B contains a listing of the associated subroutine STRRES, which calculates stress resultants. The analysis is based on equations derived in a recent paper (Comput. Struct. 26, 871–884, 1987), in which it was demonstrated that ELDAROP appears to be both accurate, and extremely rapid in its operation. Appendices C-E give examples of the simple arrangement of input data and output.     

An improved three-dimensional full-vectorial finite-difference imaginary-distance beam propagation method

1 Introduction Recently, the importance of the planar lightwave circuits or photonic integrated cir- cuits (PLCs/PICs) has been widely recognized because the monolithic integration of various kinds of photonic or optoelectronic devices can be achieved via…     

Vibration analysis of orthotropic rectangular plates using superelements

In recent years, development of the superelements has significantly influenced the analysis of structural dynamics. One of the common implementation of this element has been in the analysis of plates and shells. In this paper, application of superelements in the vibration analysis of orthotropic rectangular plates with different boundary conditions is considered, in which bending and in-plane displacements are also included. Results of this study indicate that there is a good agreement between single superelement and 625 elements using conventional methods     

Finite element bending analysis of nonuniform circular plates

The finite element method is applied to the small deflection bending analysis of nonuniform thin axisymmetric circular plates made of linear elastic material. Elements with annular and circular geometry with only 4 degrees of freedom are used in the analysis of both symetrically and nonsymmetrically loaded plates. Non-symmetric loads are expanded in Fourier series and elements restricted to deform with specified number of nodal diameters are used for each component of loading. The method is checked with several numerical examples. Although applicable to only axisymmetric plates, the method gives better results compared to other finite element methods besides offering savings in computer storage and time.