四边固支矩形弹性薄板的精确解析解

利用辛几何方法本文推导出了四边固支矩形弹性薄板弯曲问题的精确解析解.由于在求解过程中不需要事先人为的选取挠度函数,而是从弹性薄板的基本方程出发,首先将矩形薄板弯曲问题表示成Hamilton正则方程,然后利用分离变量和本征函数展开的方法求出可以完全满足四边固支边界条件的精确解析解.本文中所采用的方法突破了传统的半逆法的限制,使得问题的求解更加合理化.文中还给出了计算实例来证明推导结果的正确性.     

四边任意支承条件下弹性矩形薄板弯曲问题的解析解

利用辛几何法推导出了四边为任意支承条件下矩形薄板弯曲的解析解。在分析过程中首先把矩形薄板弯曲问题表示成Hamilton正则方程，然后利用辛几何方法对全状态相变量进行分离变量，求出其本征值后，再按本征函数展开的方法求出四边为任意支承条件下矩形薄板弯曲的解析解。由于在求解过程中并不需要人为的事先选取挠度函数，而是从弹性矩形薄板弯曲的基本方程出发，直接利用数学的方法求出问题的解析解，使得这类问题的求解更加理论化和合理化。文中的最后还给出了计算实例来验证本文方法的正确性。     

弹性矩形板问题的Hamilton正则方程

为了采用辛算法求出弹性矩形板问题的解析解，中直接从弹性矩形板的控制方程出发推导了弹性矩形板，其中包括弹性矩形薄板和厚板问题以及弹性地基上矩形薄板和厚板问题的Hamilton正则方程，为利用辛几何方法求出任意边界条件下这类问题的理论解奠定了基础．     

固支矩形Mindlin板弯曲问题的辛求解体系

本文利用二类变量广义变分原理推出了Mindlin板弯曲问题的Hamilton体系,利用辛几何方法对全状态向量进行分离变量,得到相应的横向本征问题,在求出其本征值后,按本征函数展开法导出了原问题的辛本征通解。给出了一个承受集中载荷的四边固支矩形薄板的算例,按本文求解体系得到的解与经典解吻合较好。本文直接从Mindlin板弯曲问题出发,在其Hamilton体系内使用辛几何方法给出了的一套新的求解体系,突破了传统解法的局限性,具有一般性及较高的理论推广价值。     

Maxwell模型薄板的自由振动

本文利用Ｍａｘｗｅｌ粘弹性模型建立了粘弹性薄板的振动微分方程，给出四边简支粘弹性矩形薄板的固有频率解析解．对粘弹性矩形薄板的振动特性进行了讨论     

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.     

四角点支承四边自由矩形薄板屈曲问题的新解析解

角点支承矩形薄板的屈曲问题是板壳力学的一类重要课题，控制方程和边界条件的复杂性导致寻求该类问题的解析解十分困难。虽然各类近似/数值方法可用于解决此类难题，但作为基准的精确解析解在公开文献中鲜有报道。本文基于近年来提出的辛叠加方法，解析求解了四角点支承四边自由矩形薄板的屈曲问题。首先将问题拆分为两个子问题，接着利用分离变量与辛本征展开推导出子问题的解析解，最后通过叠加获得原问题的解。由于求解过程从基本控制方程出发，逐步严格推导，无需假定解的形式，因此本文解法是一种理性的解析方法。数值算例给出了不同长宽比和不同面内载荷比情况下，四角点支承四边自由矩形薄板的屈曲载荷和典型屈曲模态，并经有限元方法验证，确认了解析解的正确性。     

三边简支一边自由混凝土矩形薄板的热弯曲

基于薄板的小挠度理论和叠加原理,考虑横向变温情况,将温度作用下的三边简支一边自由矩形薄板看作是面内温差作用下的四边简支矩形薄板和自由边上挠度作用下的三边简支一边自由矩形薄板的叠加,得到了温度作用下三边简支一边自由混凝土矩形薄板的挠度和弯矩解析解.首先通过在自由边界上试设具有待定参数的挠度函数,采用李维解法推导出三边简支一边自由矩形薄板在自由边界挠度作用下的挠度方程;其次利用横向变温作用下四边简支矩形薄板的求解得到待定参数;再采用叠加原理得出横向变温作用下三边简支一边自由矩形薄板的挠度和弯矩解析解;最后利用MATLAB编制程序得到了横向变温作用下三边简支一边自由矩形薄板的计算系数用表,为工程结构中三边简支一边自由混凝土矩形薄板在热环境下的设计计算提供了理论依据.     

弹性半空间地基与四边自由异性矩形板相互作用的研究

选用弹性半空间地基模型分析四边自由各向异性矩形地基板的弯曲和稳态振动解析解。将异性薄板控制微分方程与基于弹性半空间地基位移解建立的板与地基变形协调方程相结合,先按对称性分解,然后采用三角级数法得出了弹性半空间地基上四边自由各向异性矩形薄板的弯曲和稳态振动解析解,包括地基反力(幅值)、板的挠度(幅值)、板的内力(幅值)的解析表达式。克服了数值法的弊端,取消了对地基反力的假设,得到了板的内力(幅值)及地基反力(幅值)更切实际的分布规律。算例结果不但与文献结果吻合良好,而且表明对于异形板,对称载荷能引起反对称的内力和变形。该方法使得半空间地基上各向异性矩形薄板这一复杂的接触问题的求解统一化、简单化、规律化。     

弹性半空间地基与四边自由异性矩形板相互作用的研究

选用弹性半空间地基模型分析四边自由各向异性矩形地基板的弯曲和稳态振动解析解。将异性薄板控制微分方程与基于弹性半空间地基位移解建立的板与地基变形协调方程相结合,先按对称性分解,然后采用三角级数法得出了弹性半空间地基上四边自由各向异性矩形薄板的弯曲和稳态振动解析解,包括地基反力(幅值)、板的挠度(幅值)、板的内力(幅值)的解析表达式。克服了数值法的弊端,取消了对地基反力的假设,得到了板的内力(幅值)及地基反力(幅值)更切实际的分布规律。算例结果不但与文献结果吻合良好,而且表明对于异形板,对称载荷能引起反对称的内力和变形。该方法使得半空间地基上各向异性矩形薄板这一复杂的接触问题的求解统一化、简单化、规律化。     

弹性地基上矩形薄板问题的Hamilton正则方程及解析解

利用辛算法求出弹性地基上矩形薄板问题的解析解,将弹性地基视为双参数弹性地基,直接从弹性矩形薄板的控制方程推导出了问题的Hamilton正则方程,为求出任意边界条件下问题的理论解奠定了基础,并且通过算例验证了文中所采用方法的正确性．     

矩形悬臂薄板的解析解

首先把弹性薄板弯曲问题的控制方程表示成为Hamilton正则方程,然后利用辛几何方法对全状态相变量进行分离变量,求出其本征值后,再按本征函数展开的方法求出矩形悬臂薄板的解析解。由于在求解过程中不需要事先人为地选取挠度函数,而是从薄板弯曲的基本方程出发,直接利用数学的方法求出可以满足其边界条件的这类问题的解析解,使得问题的求解更加理论化和合理化。文中的最后还给出了计算实例来验证本文所采用的方法以及所推导出的公式的正确性。     

On new symplectic superposition method for exact bending solutions of rectangular cantilever thin plates

A novel superposition method based on the symplectic geometry approach is presented for exact bending analysis of rectangular cantilever thin plates. The basic equations for rectangular thin plate are first transferred into Hamilton canonical equations. By the symplectic geometry method, the analytic solutions to some problems for plates with slidingly supported edges are derived. Then the exact bending solutions of rectangular cantilever thin plates are obtained using the method of superposition. The symplectic superposition method developed in this paper is completely rational compared with the conventional analytical ones because the predetermination of deflection functions, which is indispensable in existing methods, is dispelled.     

On the symplectic superposition method for free vibration of rectangular thin plates with mixed boundary constraints on an edge

A novel symplectic superposition method has been proposed and developed for plate and shell problems in recent years. The method has yielded many new analytic solutions due to its rigorousness. In this study, the first endeavor is made to further developed the symplectic superposition method for the free vibration of rectangular thin plates with mixed boundary constraints on an edge. The Hamiltonian system-based governing equation is first introduced such that the mathematical techniques in the symplectic space are applied. The solution procedure incorporates separation of variables, symplectic eigen solution and superposition. The analytic solution of an original problem is finally obtained by a set of equations via the equivalence to the superposition of some elaborated subproblems. The natural frequency and mode shape results for representative plates with both clamped and simply supported boundary constraints imposed on the same edge are reported for benchmark use. The present method can be extended to more challenging problems that cannot be solved by conventional analytic methods.     

THEORETIC SOLUTION OF RECTANGULAR THIN PLATE ON FOUNDATION WITH FOUR EDGES FREE BY SYMPLECTIC GEOMETRY METHOD

The theoretic solution for rectangular thin plate on foundation with four edges free is derived by symplectic geometry method. In the analysis proceeding, the elastic foundation is presented by the Winkler model. Firstly, the basic equations for elastic thin plate are transferred into Hamilton canonical equations. The symplectic geometry method is used to separate the whole variables and eigenvalues are obtained simultaneously. Finally, according to the method of eigen function expansion, the explicit solution for rectangular thin plate on foundation with the boundary conditions of four edges frees are developed. Since the basic elasticity equations of thin plate are only used and it is not need to select the deformation function arbitrarily. Therefore, the solution is theoretical and reasonable. In order to show the correction of formulations derived, a numerical example is given to demonstrate the accuracy and convergence of the current solution.     

Symplectic system based analytical solution for bending of rectangular orthotropic plates on Winkler elastic foundation

This paper analyses the bending of rectangular orthotropic plates on a Winkler elastic foundation.Appropriate definition of symplectic inner product and symplectic space formed by generalized displacements establish dual variables and dual equations in the symplectic space.The operator matrix of the equation set is proven to be a Hamilton operator matrix.Separation of variables and eigenfunction expansion creates a basis for analyzing the bending of rectangular orthotropic plates on Winkler elastic foundation and obtaining solutions for plates having any boundary condition.There is discussion of symplectic eigenvalue problems of orthotropic plates under two typical boundary conditions,with opposite sides simply supported and opposite sides clamped.Transcendental equations of eigenvalues and symplectic eigenvectors in analytical form given.Analytical solutions using two examples are presented to show the use of the new methods described in this paper.To verify the accuracy and convergence,a fully simply supported plate that is fully and simply supported under uniformly distributed load is used to compare the classical Navier method,the Levy method and the new method.Results show that the new technique has good accuracy and better convergence speed than other methods,especially in relation to internal forces.A fully clamped rectangular plate on Winkler foundation is solved to validate application of the new methods,with solutions compared to those produced by the Galerkin method.     

New analytic solutions for static problems of rectangular thin plates point-supported at three corners

This paper deals with the bending of rectangular thin plates point-supported at three corners using an analytic symplectic superposition method. The problems are of fundamental importance in both civil and mechanical engineering, but there were no accurate analytic solutions reported in the literature. This is attributed to the difficulty in seeking the solutions that satisfy the governing fourth-order partial differential equation with the free boundary conditions at all the edges as well as the support conditions at the corners. In the following, the Hamiltonian system-based equation for plate bending is formulated, and two types of fundamental problems are analytically solved by the symplectic method. The analytic solutions of the plates point-supported at three corners are then obtained by superposition, where the constants are obtained by a set of linear equations. The solution procedure presented in this paper offers a rigorous way to yield analytic solutions of similar problems. Some numerical results, validated by the finite element method, are shown to provide useful benchmarks for comparison and validation of other solution methods.