BOUNDARY INTEGRAL EQUATIONS FOR BENDING PROBLEM OF REISSNER'S PLATES ONTWO-PARAMETER FOUNDATION

I.IntroductionThickplatesonelastict'oundationarewidelyusedinengineering,suchasthebottomplatesofoffShorestructures,surfaceplatesonrunwayofairportsandfoundationsofhigh-risebuildingsandthelike.Itisextremelydifficulttoobtainanalyticalsolutiontarathickplatewithcomplicatedshapeorcomplicatedboundaryconditiononelasticfoundation.Inrecentyears,theboundaryelementmethod(BEM)hasbeensuccessl'ullyusedtoanalyzethebendingproblemofplatesoneverykindofelasticfoundation(Ref.[l,2.3]).Butthereareonlyfewreferences…     

A boundary integral equation formulation for the Reissner's plates resting on two-parameter foundation

In this paper a boundary integral equation formulation for the Reissner's plates resting on a two-parameter foundation is established. With the aid of the Hormander Operator method, the equations of the corresponding fundamental solutions are converted into a sixth order partial differential equation with a scale function as an unknown. In order to reduce the equation further, two auxiliary functions are introduced. They satisfy a second and a fourth order equation respectively. The expressions of the auxiliary functions can be derived easily. The fundamental solutions of the Reissnei's plates on the two-parameter foundation arc expressed by a linear combination of the auxiliary functions and their derivatives. The boundary integral equations are formulated by the use of the weighted residual procedure. The fundamental solutions obtained are taken as the kernel functions of the boundary integral equations. A few examples are studied. The numerical results show high accuacy and efficiency of the present formulation.This work was supported by the National Natural Science Foundation of China.     

The complex variable boundary element method for solving sandwich plate problems

The objective of this paper is to develop a new complex variable boundary element method for sandwich plates of Reissner's type and Hoff's type. The general solution of Helmhotz equation in complex field is given. Based on the Vekua's complex integral representation of the analytic function, the new boundary integral equations are formulated. The density function in the integral equation is determined directly by boundary element method. Some standard examples are presented, and the results of numerical solutions are accurate everywhere in the plate. The approach presented is only applicable for bounded simply connected regions. The project is supported by the National Science Foundation of China.     

Postbuckling behavior of rectangular moderately thick plates and sandwich plates

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Isotropicalized spline integral equation method for the analysis of anisotropic plates

In the view of Reissner’s and Kirchhoff’s theories,respectively,we formulate theisotropicalized governing equations for the anisotropic plates,and give the proof of theequivalence relation between these two plate-models for the simply-supported rectangularorthotropic plates.The well-known fundamental solutions of the isotropic plates are utlizedfor the spline integral equation analysis of anisotropic plates.Even with sparse meshes thesatisfactory results can be obtained.The analysis of plates on two-parameter elasticfoundation is so simple as the common case that only a few terms should be added to theformulas of fictitious loads.     

Boundary element analysis for bending elastic plates with three generalized displacements

In this paper, the two fundamental differential equations for bending elastic plates with three generalized displacements are transformed into a set of boundary integral equations by Green formula. Three kinds of boundary conditions on edges have been strictly derived. So this paper gives a satisfactory method of boundary element analysis for solving the problem of bending elastic plates.     

域内具有支承及集中质量的薄板自由振动的边界元分析

本文基于薄板小挠度弯曲问题的基本解,建立了任意边界条件、域内具有支承及附有征意个集中质量的薄板自山振动的边界积分方程,文中计算了若干算例,其精度是实际工程中所允许的。     

Boundary integral equations for the bending problem of plates on two-parameter foundation

By means of Fourier integral transformation of generalized fimction, the fimdamental solution for the bending problem of plales on two-parameter foundation is derived in this paper, and the fundamental solution is expanded into a uniformly convergent series.On the basis of the above work, two boundary integral eguations which are suitable to arhitrary shapes and arbitrary boundary conditions are established by means of the Rayleigh-Green identity. The content of the paper provides the powerful theories for the application of BEM in this problem.     

Boundary element method for orthotropic thick plates

The fundamental solutions of the orthotropic thick plates taking into account the transverse shear deformation are derived by means of Hörmander's operator method and a plane-wave decomposition of the Dirac δ-function in this papey. The boundary integral equations of the thick plates have been formulated which are adapted to arbitrary boundary conditions and plane forms. The numerical calculation of the fundamental solutions is discussed in detail. Some numerical examples are analyzed with BEM.     

不连续Reissner矩形板的自由振动

用分区加权残值法研究Ressner矩形板在几何形状,边界条件等有突变时的自由振动问题,将研究对象按照结构几何形状和边界条件的具体情况划分为若干子域,在每个子域内用不同的试函数代入该域的内的控制方程到内部残值,并代入板的边界条件和各子域的协调条件得到边界残值和连续性残值,然后用最小二乘法消除残值,得到特征方程,文中讨论了该方法的收敛性和计算精度,求解了开孔矩形板的固有频率,并与已有结果进行了比较,结果表明：？该方法收敛性好,精度较高,适用范围广。     

ANALYSIS OF SYMMETRIC LAMINATED RECTANGULAR PLATES IN PLANE STRESS

Symmetric laminated plates used usually are anisotropic plates. Based on the fundamental equation for anisotropic rectangular plates in plane stress problem, a general analytical solution is established accurately by method of stress function. Therefore the general formula of stress and displacement in plane is given. The integral constants in general formula can be determined by boundary conditions. This general solution is composed of solutions made by trigonometric function and hyperbolic function, which can satisfy the problem of arbitrary boundary conditions along four edges, and the algebraic polynomial solutions which can satisfy the problem of boundary conditions at four corners. Consequently this general solution can be used to solve the plane stress problem with arbitrary boundary conditions. For example, a symmetric laminated square plate acted with uniform normal load, tangential load and nonuniform normal load on four edges is calculated and analyzed.     

正交各向异性厚板的边界元解法

本文利用 Hormander 算子法和平面波分解法导出了计入剪切变形的正交各向异性厚板的基本解。建立了计入剪切变形的正交各向异性厚板的边界积分方程。文中详细地讨论了基本解的数值计算,并用边界元法分析了一些算例。     

Maximum frequency design of laminated plates with mixed boundary conditions

A layerwise optimization (LO) approach is extended to accommodate the finite element analysis for optimizing the free vibration behavior of laminated composite plates with discontinuities along the boundaries. The classical non-conforming finite element is modified to fit into the LO procedure and is used to calculate natural frequencies of symmetrically laminated plates. This combined LO-FEM approach is applied to laminated rectangular plates with combinations of free, simply supported and clamped edges along parts of the plate boundary. The fundamental frequency, an object function for the present study, is maximized by optimizing design variables that are a set of fiber orientation angles in the layers. For illustrative purpose nine examples of square and rectangular plates with various types of mixed boundary conditions are considered, and a comprehensive set of results are presented for the optimum fiber orientation angles and the maximum fundamental frequencies of the 8-layer and 24-layer plates.     

复变形式的各向异性板弯曲问题的基本解

提出了求解各向异性板弯曲问题基本解的新方法。得到的基本解简捷明了，相应的法向弯矩和相当剪力的表达式易求，故便于应用在一般边界条件的各向异性板弯曲问题的边界积分方程。     

THE APPLICATION OF WEINSTEIN-CHIEN’S METHOD——THE UPPER AND LOWER LIMITS OF FUNDAMENTAL FREQUENCY OF RECTANGULAR PLATES WITH EDGES ARE THE MIXTURE OF SIMPLY SUPPORTED POR-TIONS AND CLAMPED PORTIONS

In this paper, the method of relaxed boundary conditions is applied to rectangular plates with edges which are a sort of the mixture of simply supported portions and clamped portions, so that the lower limit of fundamental frequency of such plates is evaluated. A kind of polynomial satisfying the displacement boundary conditions is designed, os that it is enabled to evaluate the upper limit of fundamental frequency by Ritz’ method. The practical calculation examples solved by these methods have given satisfactory results. At the end of this paper, it is pointed out that the socalled exact solution of such plates usually evaluated by the force superposition method is essentially a kind of lower limit of solution, if the truncated error of series which occurs in actual calculation is considered.     

四边简支矩形中厚板的弯曲

本文采用Reissner中厚板理论求解了四边简支矩形中厚板的弯曲问题。文中首先对Reissner中厚板理论的控制方程进行了适当的变更,使之成为非耦联的二阶偏微分方程组,然后利用有限积分变换法求解所得新的控制方程,得到了四边简支矩形中厚板受均布载荷作用下的解析解。文中所述方法可用以求解具有其它边界条件和载荷的矩形中厚板的弯曲问题,同时还可移植应用于其它中厚板理论。     

集中载荷作用下不同边界条件的弯曲厚矩形板的受迫振动

运用边界积分法研究了四边简支、两对边固定另两对边简支、四边固定三种复杂边界条件下厚矩形板的受迫振动问题,求解过程清晰,从而给出了受迫振动控制方程和挠曲面方程。通过在Matlab平台上进行数值计算,得出了图表形式的计算结果,并与有限元模拟值进行对照。研究表明,边界积分法用于求解厚矩形板的受迫振动问题的准确性,本文推导的控制方程和挠曲面方程的正确性,进而对工程实际中的各种相关问题具有一定的现实意义,也为求解此类问题提供了一种新途径,可以直接运用到工程实际中。     

A series boundary integration method for the bending analysis of anisotropic plates

A series boundary integration method is given which exactly satisfies the fundamental equation of bending analysis of general anisotropic plates or laminated plates in Kirchhoffs sense. With a unified deflection series, the method may be applied to the plates having different planforms and support conditions. Several groups of representative examples are calculated. The examples include circular, square and triangular plates, and their boundaries include clamped edges, simply supported edges, free edges and free corner. Numerical results indicate rapid convergency for both deflection and stress resultants and demonstrate wide applicability of the method.This subject was supported by the Natural Science Foundation of Guangdong Province.     

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.     

Modified Kirchhoff's theory of plates including transverse shear deformations

A primary flexure problem defined by Kirchhoff theory of plates in bending is considered. Significance of auxiliary function introduced earlier in the in-plane displacements in resolving Poisson-Kirchhoff's boundary conditions paradox is reexamined with reference to reported sixth order shear deformation theories, in particular, Reissner's theory and Hencky's theory. Sixth order modified Kirchhoff's theory is extended here to include shear deformations in the analysis.