Exact bending solutions of orthotropic rectangular cantilever thin plates subjected to arbitrary loads

Exact bending solutions of orthotropic rectangular cantilever thin plates subjected to arbitrary loads are derived by using a novel double finite integral transform method. Since only the basic elasticity equations for orthotropic thin plates are used, the method presented in this paper eliminates the need to predetermine the deformation function and is hence completely rational thus more accurate than conventional semi-inverse methods, which presents a breakthrough in solving plate bending problems as they have long been bottlenecks in the history of elasticity. Numerical results are presented to demonstrate the validity and accuracy of the approach as compared with those previously reported in the literature     

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.     

横观各向同性弹性半空间地基上正交异性矩形中厚板弯曲解析解

对横观各向同性体通解进行双重傅里叶变换,获得了直角坐标系下横观各向同性弹性半空间地基受任意竖向荷载作用下的位移积分变换解;在此基础上建立了板与地基的变形协调方程,并与三个广义位移变量描述的弹性地基上四边自由正交各向异性矩形中厚板的弯曲控制方程相结合,用三角级数法,得出横观各向同性弹性半空间地基上四边自由正交异性矩形中厚板受任意竖向荷载作用的弯曲解析解。相关算例分析表明,本文方法是有效的。     

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.     

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。     

Winkler地基上各向异性薄板弯曲的精确解-广义积分变换解

外界载荷作用下复合材料薄板的弯曲行为是工程重点关注的问题之一。针对各向同性和正交各向异性的薄板弯曲问题,研究人员已给出了经典数值解。由于计算的复杂性,针对各向异性薄板弯曲问题的解答较少。本文从薄板弯曲问题的控制方程出发,建立符合该问题的辅助特征方程,并确定相应的特征值和特征函数。利用广义积分变换的思想,建立了求解非正交铺层条件下各向异性薄板弯曲问题的数值算法,给出了各向异性薄板弯曲的精确解。与其他文献结果比较发现,该方法具有较好的收敛性和准确性。     

Boundary element analysis of anisotropic Kirchhoff plates

In this paper, the radial integration method is used to obtain a boundary element formulation without any domain integral for general anisotropic plate bending problems. Two integral equations are used and the unknown variables are assumed to be constant along each boundary element. The domain integral which arises from a transversely applied load is exactly transformed into a boundary integral by a radial integration technique. Uniformly and linearly distributed loads are considered. Several computational examples concerning orthotropic and general anisotropic plate bending problems are presented. The results show good agreement with analytical and finite element results available in the literature.     

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.     

矩形薄板弯曲问题的二维广义有限积分变换法

运用二维广义有限积分变换解法,本文推导出不同边界条件下矩形薄板弯曲问题的解析解．在推导过程中,选取满足边界条件的梁振型函数为广义积分变换的积分核,由此构造出广义有限积分变换对,通过对薄板弯曲问题的控制方程进行二维广义积分变换,可以将控制方程转换为易于求解的线性代数方程组．该方法无需预先选取位移函数,无需进行繁琐的叠加过程,求解过程思路清晰,说明该方法更加正确合理．最后通过计算实例对比,验证了该方法的合理性及所推导公式的正确性．     

Closed-form solutions for free vibration of rectangular FGM thin plates resting on elastic foundation

This article presents closed-form solutions for the frequency analysis of rectangular functionally graded material(FGM) thin plates subjected to initially in-plane loads and with an elastic foundation. Based on classical thin plate theory, the governing differential equations are derived using Hamilton's principle. A neutral surface is used to eliminate stretching–bending coupling in FGM plates on the basis of the assumption of constant Poisson's ratio. The resulting governing equation of FGM thin plates has the same form as homogeneous thin plates. The separation-ofvariables method is adopted to obtain solutions for the free vibration problems of rectangular FGM thin plates with separable boundary conditions, including, for example, clamped plates. The obtained normal modes and frequencies are in elegant closed forms, and present formulations and solutions are validated by comparing present results with those in the literature and finite element method results obtained by the authors. A parameter study reveals the effects of the power law index n and aspect ratio a/b on frequencies.     

层状横观各向同性地基上异性矩形薄板的弯曲解析解

选用更具广泛性的层状横观各向同性弹性地基模型,来分析四边自由各向异性矩形地基板的弯曲解析解。先基于直角坐标下横观各向同性体的静力胡海昌通解,借助双重傅里叶变换及矩阵传递法,获得层状横观各向同性地基的静力位移场和应力场;然后将异性薄板的弯曲控制方程,与基于层状横观各向同性弹性地基的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,再用三角级数法,得出层状横观各向同性弹性地基上四边自由各向异性矩形薄板的弯曲解析解,包括地基反力、板的挠度及板的内力的解析表达式。克服了数值法的弊端,取消了对地基反力的假设,且避免了矩阵指数函数的计算;同时考虑了地基的层状性及板和地基的各向异性,从而得到板的内力及地基反力更切实际的分布规律。算例结果与文献的有限元结果吻合良好,证明本文方法是切实可行的。     

Reciprocal theorem method for solving the problems of bending of thick rectangular plates

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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.     

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.     

Exact solutions for variable-thickness inhomogeneous elastic plates under various boundary conditions

In this paper, an exact solution to the governing equations of the bending of a variable-thickness inhomogeneous rectangular plate is presented. The procedure is applicable to variable-thickness inhomogeneous rectangular plates with two opposite edges simply supported. The remaining ones subjected to a combination of clamped, simply supported, and free boundary conditions and between these two edges the plate may have varying thickness. The procedure is valuable in view of the fact that tables of deflections and stresses cannot be presented for variable-thickness inhomogeneous orthotropic plates as for uniform-thickness homogeneous isotropic plates even for commonly encountered loads because the results depend on the inhomogeneity coefficient and the orthotropic material properties instead of a single flexural rigidity. Numerical results, useful for the validation or otherwise of approximate solutions, are tabulated. The influences of the degree of the inhomogeneity, aspect ratio, thickness parameter and degree of non-uniformity on the deflections and stresses are investigated. This paper is partially supported by the Deanship of Scientific Research at King AbdulAziz University (Grant no. 172/427).     

Free vibration analysis of a plate on foundation with completely free boundary by finite integral transform method

The theoretical solutions of eigenfrequencies and vibration modes of a rectangular thin plate on an elastic foundation with completely free boundary are derived by using a double finite cosine integral transform method. In the analysis procedure, the elastic foundation is regarded as a Winkler elastic foundation model. Because the basic dynamic elasticity equations of the thin plate on elastic foundation are only used, it is not needed to select the deformation function arbitrarily. Therefore, the solution developed in the present paper is more reasonable and more accurate. To prove the correctness of the solutions, numerical results obtained using the present solutions are compared with those in the literatures.     

On new symplectic elasticity approach for exact bending solutions of rectangular thin plates with two opposite sides simply supported

This paper presents a bridging research between a modeling methodology in quantum mechanics/relativity and elasticity. Using the symplectic method commonly applied in quantum mechanics and relativity, a new symplectic elasticity approach is developed for deriving exact analytical solutions to some basic problems in solid mechanics and elasticity which have long been bottlenecks in the history of elasticity. In specific, it is applied to bending of rectangular thin plates where exact solutions are hitherto unavailable. It employs the Hamiltonian principle with Legendre’s transformation. Analytical bending solutions could be obtained by eigenvalue analysis and expansion of eigenfunctions. Here, bending analysis requires the solving of an eigenvalue equation unlike in classical mechanics where eigenvalue analysis is only required in vibration and buckling problems. Furthermore, unlike the semi-inverse approaches in classical plate analysis employed by Timoshenko and others such as Navier’s solution, Levy’s solution, Rayleigh–Ritz method, etc. where a trial deflection function is pre-determined, this new symplectic plate analysis is completely rational without any guess functions and yet it renders exact solutions beyond the scope of applicability of the semi-inverse approaches. In short, the symplectic plate analysis developed in this paper presents a breakthrough in analytical mechanics in which an area previously unaccountable by Timoshenko’s plate theory and the likes has been trespassed. Here, examples for plates with selected boundary conditions are solved and the exact solutions discussed. Comparison with the classical solutions shows excellent agreement. As the derivation of this new approach is fundamental, further research can be conducted not only on other types of boundary conditions, but also for thick plates as well as vibration, buckling, wave propagation, etc.     

功能梯度矩形板的三维弹性分析

将功能梯度三维矩形板的位移变量按双三角级数展开，以弹性力学的平衡方程为基础．导出位移形式的平衡方程。引入状态空间方法，以三个位移分量及位移分量的一阶导数为状态变量，建立状态方程。考虑四边简支的边界条件，由状态方程得到了功能梯度三维矩形板的静力弯曲问题和自由振动问题的精确解。由给出的均匀矩形板自由振动问题的计算结果表明．与已有的理论解以及有限元方法的计算结果相吻合。假设功能梯度三维矩形板的材料常数沿板的厚度方向按照指数函数的规律变化．进一步给出了功能梯度三维矩形板的自由振动问题和静力弯曲问题的算例分析，并讨论了材料性质的梯度变化对板的动力响应和静力响应的影响。     

A refined finite element method for bending analysis of laminated plates integrated with piezoelectric fiber-reinforced composite actuators

This research presents a finite element formulation based on four-variable refined plate theory for bending analysis of cross-ply and angle-ply laminated composite plates integrated with a piezoelectric fiber-reinforced composite actuator under electromechanical loading. The four-variable refined plate theory is a simple and efficient higher-order shear deformation theory, which predicts parabolic variation of transverse shear stresses across the plate thickness and satisfies zero traction conditions on the plate free surfaces. The weak form of governing equations is derived using the principle of minimum potential energy, and a 4-node non-conforming rectangular plate element with 8 degrees of freedom per node is introduced for discretizing the domain. Several benchmark problems are solved by the developed MATLAB code and the obtained results are compared with those from exact and other numerical solutions, showing good agreement.     

Vibration and stability of hybrid plate based on elasticity theory

The governing equations of elasticity theory for natural vibration and buck- ling of anisotropic plate are derived from Hellinger-Reissner＇s variational principle with nonlinear strain-displacement relations. Simply supported rectangular hybrid plates are studied with a precise integration method. This method, in contrast to the traditional finite difference approximation, gives highly precise numerical results that approach the full computer precision. So the results for natural vibration and stability of hybrid plates presented in the paper can be riewed as approximate analytical solutions. Furthermore, several types of coupling effects such as coupling between bending and twisting, and coupling between extension and bending, when the layer stacking sequence is asymmetric, are considered by only one set of governing equations.