Reciprocal theorem method for the forced vibration of elastic thick rectangular plates

I.IntroductionTheproblelllsofthefol.cedvibrationofelasticthickrectallgularplatestobesolvedareofgreattheoreticalsignificanceandpractical\7altle.Asthevibratinggoverningequationofthethickplateismorecomplicatedthallthatofthinplate.itisdil'ficulttoso]\'einmathematics.Manyscholarshat,eresearchedintotheseproblemsalldadvancedvariousspecialapproximations.Suchassuperpositionmethod.initialfLinctiollnletllod.thecombinedseriesot'theeigenftlnctionsofdeepbeamabouttilecorrespondillgboulldarycollditions.eller…     

The bending of a thick rectangular plate with three clamped edges and one free edge

The exact solution of the bending of a thick rectangular plate with three clamped edges and one free edge under a uniform transverse load is obtained by means of the concept of generalized simply-supported boundary^[1] in Reissner’s theory of thick plates. The effect of the thickness h of a plate on the bending is studied and the applicable range of Kirchhoffs theory for bending of thin plates 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。     

角点悬空弯曲厚矩形板的边界积分法

在边界积分法中引用了拟基本系统矩形板,在该拟基本系统与实际系统之间应用功的互等定理,得到一挠曲面方程的积分表达式,只要对此表达式进行极简单的积分便可得到该挠曲面方程,这比直接求解Reissner挠度控制方程要简单,边界积分法的求解过程概念清晰,计算伊始便给出了挠曲面方程的总体表达式.以Reissner厚板理论为基础,应用边界积分法研究了角点悬空厚矩形板的弯曲问题,给出了在集中荷载作用下两邻边固定另两邻边自由且角点悬空弯曲厚矩形板的封闭解析解,并给出了相应的数据和图表以供工程上的应用和参考.     

Bending analysis of thin functionally graded plates using generalized differential quadrature method

In this paper, the differential quadrature (DQ) method is presented for easy and effective analysis of isotropic functionally graded (FG) and functionally graded coated (FGC) thin plates with constant Poisson’s ratio and varying Young’s modulus in the thickness direction. The bending of FG and FGC plates under transverse loading has been studied using the polynomial differential quadrature (PDQ) and the harmonic differential quadrature (HDQ) methods. A three-dimensional elasticity solution for a moderately thick FG plate with exponential Young’s modulus is used as the benchmark. Two examples, including a thin FG rectangular plate and a thin FGC rectangular plate with sigmoidal Young’s modulus, are investigated. The numerical results of PDQ and HDQ methods reveal good agreement with other solutions. Also, it is shown that the formulations for thin FG plates and homogeneous plates are similar, except that the plane strain components of the middle surface in FG plates are not zero.     

四边简支各向同性矩形厚板的插值矩阵法解

针对强厚度矩形板四边简支情况,论文根据状态变量法思想,基于三维弹性理论基本方程,以3个位移分量及3个应力分量按双三角级数展开,将三维弹性力学控制方程转化为常微分方程边值问题.尽管一些各向异性弹性矩形厚板早已由状态空间法获得分析解,可是各向同性厚板的分析解至今难以获得,因为状态空间解法中特征方程有重根问题而不易于收敛.论文提出采用插值矩阵法直接对常微分方程进行求解,获得各向同性矩形厚板在四边简支边界条件下三维理论的位移和应力解,并与有限元精细结果进行比较,证明了本文解的准确性.     

High-order shear theory for static analysis of functionally graded plates with porosities

The bending responses of porous functionally graded (FG) thick rectangular plates are investigated according to a high-order shear deformation theory. Both the effect of shear strain and normal deformation are included in the present theory and so it does not need any shear correction factor. The equilibrium equations according to the porous FG plates are derived. The solution to the problem is derived by using Navier's technique. Numerical results have been reported and compared with those available in the open literature for non-porous plates. The effects of the exponent graded and porosity factors are investigated.     

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.     

Winkler地基上四边自由正交各向异性矩形薄板弯曲问题的辛叠加解

研究Winkler地基上正交各向异性矩形薄板弯曲方程所对应的Hamilton正则方程, 计算出其对边滑支条件下相应Hamilton算子的本征值和本征函数系, 证明该本征函数系的辛正交性以及在Cauchy主值意义下的完备性, 进而给出对边滑支边界条件下Hamilton正则方程的通解, 之后利用辛叠加方法求出Winkler地基上四边自由正交各向异性矩形薄板弯曲问题的解析解. 最后通过两个具体算例验证了所得解析解的正确性.     

多层正交异性矩形薄板弯曲振动稳定解析解

构造了带有补充项的双重正弦傅里叶级数通解来求解各种边界条件的多层正交各向异性矩形薄板的弯曲、振动和稳定问题.将坐标轴取在中性面上,求出用挠度表示的应力表达式,然后由横截面上每单位宽度的应力合成板的内力;再将层合板的内力代入板的平衡方程中得到板的控制方程,将多层板的物理参数折算为等价的单层板物理参数;最后联立控制方程与边界条件,求得未知量的系数并代入本文的通解中.本文的通解不需要叠加即可求解各种边界条件的板的弯曲、振动和稳定问题;现有的对于单层板的研究都可以用本文的方法拓展到多层板领域;对于复杂边界条件的板,也可以使用该通解分析.     

Solution of bending of cantilever rectangular plates under uniform surface-load by the method of two-direction trigonometric series

The bending of a cantilever rectangular plate is a very complicated problem in thetheory of plates.For a long time,there have been only approximate solutions for thisproblem by energy methods and numerical methods.since 1979,Prof.F.V.Chang of Tsing Hua University obtained,by the method ofsuperposition,a series of analytic solutions for cantilever rectangular plates under uniformload and concentrated load.In this paper,the two-direction trigonometric series is used to obtain the solution forthe bending of cantilever rectangular plates under uniform load.The obtained results arecompared with the results by the method of superposition.The comparison shows that theresults of these two methods are in good agreement,hence they are mutually confirmed to becorrect.     

A general solution of rectangular thin plates in bending

In this paper a general solution of rectangular plates in bending is given. The integral constants are determined by means of boundary conditions. This method is simpler and easier than the method of superposition.The author is grateful to mr. Chen Guang-nan for his assistance in calculation.     

Nonlinear bending of simply supported symmetric laminated cross-ply rectangular plates

Based on the von Kármán-type theory of plates,nonlinear bending problems of simplysupported symmetric laminated cross-ply rectangular plates under the combined action ofpressure and inplane load are investigated in this paper.The solution which satisfies thegoverning equations and boundary conditions is obtained by using the double Fourier seriesmethod.     

混合边界各向异性矩形板的静力分析

采用一般解析解和配点法相结合的方法,求解混合边界各向异性矩形板的弯曲问题.先由弯曲挠度的微分方程求出各种类型的齐次解和特解,然后组成一般解析解,再将板的每个边等分 为很多微小的段,仅对每一微段的中点建立应满足的边界条件,由全部边界条件方程式即可求得全部积分常数.以每边一半边界为平夹、另一半边界为简支或自由的方板为例进行了计算,并与四边均为简支的方板进行了对比,表明理论简单,结果实用.     

On new symplectic approach for exact bending solutions of moderately thick rectangular plates with two opposite edges simply supported

The symplectic geometry method is introduced for exact bending solutions of moderately thick rectangular plates with two opposite edges simply supported. The basic equations for the plates are first transferred into Hamilton canonical equations. The whole state variables are then separated. Using the method of eigenfunction expansion in the symplectic geometry, typical examples for plates with selected boundary conditions are solved and exact bending solutions obtained. Since only the basic elasticity equations of the plates are used, this method eliminates the need to pre-determine the deformation function and is hence more reasonable than conventional methods. Numerical results were presented to demonstrate the validity and accuracy of this approach as compared to those reported in other literatures.     

中厚板屈曲的加权残值方法

用加权残值方法分析了矩形中厚板的屈曲问题。试函数和权函数依据短梁函数进行构造。结果良好。     

Bending of rectangular flat slabs supported by four columns

In this paper an analytical solution is proposed for the bending of uniformly loaded rectangular plates supported only by four intermediate columns, the edges and corners of which are all free. For several particular cases, the numerical results, which contain the column reaction and the values for the deflection and the bending moments at several points of the plate, are given.Calculations indicate that the method proposed in this paper is valid.     

CC series solution for bending of rectangular plates on elastic foundations

The analytical solution for the bending problem of the rectangular plates on an elastic foundation is investigated by using the Stockes' transformation of a double variables function. The numerical results for the rectangular plates with free edges on the elastic foundations under a concentrated force are given in the example. First Received Dec. 14 1993     

Theory and refined theory of elasticity for transversely isotropic plates and a new theory for tnick plates

A theory of elasticity for the bending of transversely isotropic plates has been developed from the basic equations of elasticity in terms of displacements for transversely isotropic bodies, which takes into account the loads distributed over the surfaces of the plates. Based on this theory, a refined theory of plates which can satisfy three boundary conditions along each edge of the plates and a new theory of thick plates are established. The solution of the refined theory for simply supported polygonal plates has been obtained; and its numerical result is very close to the exact solution of the three-dimensional theory of elasticity. A systematic comparison with the former theories of thick plates shows that the present theory of thick plates is closest to the result of the theory of elasticity.     

阶梯式变厚度两对边简支的矩形弹性薄板弯曲问题的解析解

讨论了阶梯式变厚度两对边为简支的矩形弹性薄极的弯曲问题，应用奇异函数及其傅立叶奇延拓展开获得了该问题的通解和一系列特解。文中应用所得公式对实例进行了分析和计算。