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.     

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.     

Three-dimensional vibration analysis of thick rectangular plates using Chebyshev polynomial and Ritz method

This paper describes a method for free vibration analysis of rectangular plates with any thicknesses, which range from thin, moderately thick to very thick plates. It utilises admissible functions comprising the Chebyshev polynomials multiplied by a boundary function. The analysis is based on a linear, small-strain, three-dimensional elasticity theory. The proposed technique yields very accurate natural frequencies and mode shapes of rectangular plates with arbitrary boundary conditions. A very simple and general programme has been compiled for the purpose. For a plate with geometric symmetry, the vibration modes can be classified into symmetric and antisymmetric ones in that direction. In such a case, the computational cost can be greatly reduced while maintaining the same level of accuracy. Convergence studies and comparison have been carried out taking square plates with four simply-supported edges as examples. It is shown that the present method enables rapid convergence, stable numerical operation and very high computational accuracy. Parametric investigations on the vibration behaviour of rectangular plates with four clamped edges have also been performed in detail, with respect to different thickness-side ratios, aspect ratios and Poisson’s ratios. These results may serve as benchmark solutions for validating approximate two-dimensional theories and new computational techniques in future.     

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.     

Thin plate spline radial basis functions for vibration analysis of clamped laminated composite plates

A meshless method based on thin plate spline radial basis functions and higher-order shear deformation theory are presented to analyze the free vibration of clamped laminated composite plates. The singularity of thin plate spline radial basis functions is eliminated by adding infinitesimal to the zero distance. Convergence characteristics of the present thin plate spline radial basis functions for the vibration analysis of the clamped laminated plates are investigated. The frequencies computed by the present method agree well with the available published results.     

COUPLED VIBRATORY CHARACTERISTICS OF A RECTANGULAR CONTAINER BOTTOM PLATE

The natural frequencies of an elastic thin plate placed into a rectangular hole and connected to the rigid bottom slab of a rectangular container filled with fluid having a free surface are studied. The fluid is assumed to be incompressible, inviscid and irrotational, and the effect of surface waves is neglected. An analytical-Ritz method is developed to study the vibratory characteristics of the plate in contact with the fluid. First of all, the exact expression of the motion of the fluid is obtained, in which the unknown coefficients are determined by using the method of separation of variables and the method of Fourier series expansion. Then, the Ritz approach is used to obtain the frequency equation of the system. The vibrating beam functions are adopted as the admissible functions for the wet-mode expansion of the plate, and the added virtual mass incremental (AVMI) matrices are obtained for plates with arbitrary boundary conditions. Finally, a convergence study is carried out and some numerical results are given. The accuracy of AVMI factor solutions is discussed by comparing with the more accurate analytical-Ritz solutions presented in this paper. Furthermore, It is seen that the present method is also suitable for the vibration analysis of rectangular plates in contact with infinite fluid by taking the finite, but larger size fluid domain as an approximation in the computation.     

Application of the reciprocal theorem for calculating the natural frequencies of rectangular elastic thin plates

This paper further extends the applications of the reciprocal theorem to calculating thenatural frequencies of rectangular elastic thin plates on the basis of [1]. Applying thepresented method. there is no need to solve governing differential equations. it is onlynecessary to solve a simple integral equation after using the reciprocal theorem between thebasic system and the actual system.Using the idea of the generalized edge simply supported and introducing the frequencymatrix. then all frequency equations of the rectangular plates with two opposite edgessimply supported and other two opposite edges variously suppored are obtained together.This is a simple convenient and general method for calculating the frequencyequations of the rectangular plates.     

Construction of rectangular displacement-based element for thick/thin plates

In this paper, a method of constructing displacement-based element for thick/thin plates is developed by using the technique of generalized compatibility, and a rectangular displacement based element with 12 degrees of freedom for thick/thin plates is presented. This method enjoys a good accuracy with simple formulation and is free of shear locking as the thickness of the plate approaches zero. The project supported by National Natural Science Foundation of China through Grant No. 59208075     

薄板小波有限元理论及其应用

利用样条小波尺度函数构造了常用的三角形和矩形薄板单元的位移函数,得到了利用小波函数表示的形函数。采用合理的局部坐标,对单元进行压缩,使单元在局部坐标区间上有其值,成功地推导出了分域的三角形和矩形薄板小波有限元列式。在此基础上,提出了弹性地基薄板的小波有限元求解方法。通过两个算例对薄板的挠度和弯矩进行了计算,数值结果表明,求解结果具有收敛快、精度高的特点。     

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     

功能梯度简支矩形板的非线性动力响应

研究了功能梯度简支矩形板在横向简谐激励作用下的非线性动力响应问题。采用幂律分布规律描述功能梯度材料的等效材料参数,基于Galerkin法建立了系统广义坐标的常微分控制方程。利用平均法得到了系统的幅频响应特性,分析了功能梯度矩形薄板的非线性主共振特性。数值算例验证了平均化方法的正确性,揭示了功能梯度平板主共振响应中的多值性和跳跃现象;同时分析发现初始条件会改变功能梯度平板主共振的响应幅值。最后讨论了功能梯度材料的梯度指数对系统幅值响应的影响。     

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.     

VIBRATIONS OF STEPPED RECTANGULAR THIN PLATES ON WINKLER’S FOUNDATION

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中面单向受拉（压）的阶梯式矩形薄板的振动

用奇异函数建立ｘ＝０与ｘ＝ａ两对边简支并受面内均布拉（压）力作用、加两边为任意支承、非单一材质的ｎ级阶梯式矩形薄板自由振动和强迫振动的微分方程并求得其通解,用Ｗ算子给出振型了函数的表达式及常见支承条件下板的方程。文中给出的固有频率表达式表明,面内均布拉（压）力对固有的数值有影响。此处导出的各种情况下的影响函数,对于求解相应民政部下的阶梯式矩形薄板的静力弯曲和稳定性问题,也是适用的。     

任意边界条件下双模量矩形薄板的弯曲

将双模量板等效为两个各向同性小矩形板组成的层合板,假定该层合板的中性面即为两个小矩形板的交界面。根据中性面上应力为零且薄板全厚度上应力的代数和为零,推导了双模量矩形薄板的中性面位置。本文采用严宗达提出的带补充项的双重正弦傅里叶级数通解,该通解可以适用于任意边界条件的矩形薄板且不需要叠加或者重新构造。联立边界条件和控制方程,求得通解中的待定系数并代入到通解中,即可得到任意边界条件下双模量矩形薄板的弯曲解析解。与有限元结果比较,本文结果符合工程精度要求。     

Energy theorems for solving equations of deflections

In this paper the unit-dummy-load method is generalized on the basis of Castigliano’s Theorem. On these grounds the general eguations of deflection surfaces of the structures, such as a kind of beams,plates and shells, are directly derived by the force method.We derived the eguations of the deflection surfaces of the rectangular thin plates and thick plates considering the effect of transverse shearing deformations with the inhomogeneous displacement boundary conditions. At the same time we give the equations of deflection axes of the corresponding straight beams.The applications of the reciprocal theorem are also generalized.Three simple calculated examples are given.     

中厚板的耦合多项式基径向点插值无网格法

采用Mindlin平板理论,通过最小位能原理建立了各向同性中厚板的伽辽金整体弱式方 程,形函数采用耦合多项式基的径向点插值法构造,可以直接施加本质边界条件. 算例表明, 用耦合多项式基的径向点插值无网格法分析中厚板问题,具有效率高、精度高和易于实现等 优点,可以避免薄板弯曲时的剪切自锁现象.     

VIBRATIONS OF ONE-WAY RECTANGULAR STEPPEDTHIN PLATES ON WINKLER'S FOUNDATION

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Vibrations of stepped one-way thin rectangular plates subjected to in-plane tensile/compressive force in y-direction on Winkler' s foundation

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基于物理信息的深度学习求解矩形薄板力学正反问题

建立基于物理信息的神经网络框架,利用深度学习求解矩形薄板力学正反问题.力学正问题为已知矩形薄板的基本参数、边界条件和受力情况,求薄板各点挠度;反问题为已知薄板部分点的挠度、基本参数和受力情况等,识别边界条件.基于物理信息的神经网络模型中,损失函数除基于数据驱动模型的挠度数据拟合部分以外,还引入薄板弯曲基本方程和应力应变...