Exact solutions for free flexural vibration of Lévy-type rectangular thick plates via third-order shear deformation plate theory

In this paper, exact closed-form solutions in explicit forms are presented for transverse vibration analysis of rectangular thick plates having two opposite edges hard simply supported (i.e., Lévy-type rectangular plates) based on the Reddy’s third-order shear deformation plate theory. Two other edges may be restrained by different combinations of free, soft simply supported, hard simply supported or clamped boundary conditions. Hamilton’s principle is used to derive the equations of motion and natural boundary conditions of the plate. Several comparison studies with analytical and numerical techniques reported in literature are carried out to demonstrate accuracy of the present new formulation. Comprehensive benchmark results for natural frequencies of rectangular plates with different combinations of boundary conditions are tabulated in dimensionless form for various values of aspect ratios and thickness to length ratios. A set of three-dimensional (3-D) vibration mode shapes along with their corresponding contour plots are plotted by using exact transverse displacements of Lévy-type rectangular Reddy plates. Due to the inherent features of the present exact closed-form solution, the present findings will be a useful benchmark for evaluating the accuracy of other analytical and numerical methods, which will be developed by researchers in the future.     

Accurate bending analysis of rectangular plates with two adjacent edges free and the others clamped or simply supported based on new symplectic approach

The symplectic geometry approach is introduced for accurate bending analysis of rectangular thin plates with two adjacent edges free and the others clamped or simply supported. The basic equations for rectangular plates are first transferred into Hamilton canonical equations. Using the symplectic approach, the analytic solution of rectangular thin plate with two adjacent edges simply supported and the others slidingly supported is derived. Accurate bending solutions of title problems are then obtained using the superposition method. The approach used in this paper eliminates the need to pre-determine the deformation function and is hence more reasonable than conventional methods. Numerical results are presented to demonstrate the validity and efficiency of the approach as compared with those reported in other literatures.     

Free vibration of functionally graded rectangular plates using first-order shear deformation plate theory

The main objective of this research work is to present analytical solutions for free vibration analysis of moderately thick rectangular plates, which are composed of functionally graded materials (FGMs) and supported by either Winkler or Pasternak elastic foundations. The proposed rectangular plates have two opposite edges simply-supported, while all possible combinations of free, simply-supported and clamped boundary conditions are applied to the other two edges. In order to capture fundamental frequencies of the functionally graded (FG) rectangular plates resting on elastic foundation, the analysis procedure is based on the first-order shear deformation plate theory (FSDT) to derive and solve exactly the equations of motion. The mechanical properties of the FG plates are assumed to vary continuously through the thickness of the plate and obey a power law distribution of the volume fraction of the constituents, whereas Poisson’s ratio is set to be constant. First, a new formula for the shear correction factors, used in the Mindlin plate theory, is obtained for FG plates. Then the excellent accuracy of the present analytical solutions is confirmed by making some comparisons of the results with those available in literature. The effect of foundation stiffness parameters on the free vibration of the FG plates, constrained by different combinations of classical boundary conditions, is also presented for various values of aspect ratios, gradient indices, and thickness to length ratios.     

Reissner厚板弹性弯曲的一般解析解

针对大型工程建设中的Reisner厚板弹性弯曲问题,本文采用复级数方法求解相应的常系数偏微分方程组的边值问题,并首次得到了任意边界条件下的一般解析解.该解形式简单,计算方便、可靠.以四边简支和三边固支一边自由两种支撑条件下厚板承受均布载荷为例进行了分析验算,与已有的计算结果相比,计算结果相当满意.同时本文还着重对解的收敛速度、正确性(合理性)及边界满足情况进行了考察.     

Natural frequencies of rectangular Mindlin plates coupled with stationary fluid

The present study is concerned with the free vibration analysis of a horizontal rectangular plate, either immersed in fluid or floating on its free surface. The governing equations for a moderately thick rectangular plate are analytically derived based on the Mindlin plate theory (MPT), whereas the velocity potential function and Bernoulli’s equation are employed to obtain the fluid pressure applied on the free surface of the plate. The simplifying hypothesis that the wet and dry mode shapes are the same, is not assumed in this paper. In this work, an exact-closed form characteristics equation is used for the plate subjected to a combination of six different boundary conditions. Two opposite sides are simply supported and any of the other two edges can be free, simply supported or clamped. To demonstrate the accuracy of the present analytical solution, a comparison is made with the published experimental and numerical results in the literature, showing an excellent agreement. Then, natural frequencies of the plate are presented in tabular and graphical forms for different fluid levels, fluid densities, aspect ratios, thickness to length ratios and boundary conditions. Finally, some 3-D mode shapes of the rectangular Mindlin plates in contact with fluid are illustrated.     

Modified mixed Ritz-DQ formulation for free vibration of thick rectangular and skew plates with general boundary conditions

Recently, the present authors proposed a simple mixed Ritz-differential quadrature (DQ) methodology for free and forced vibration, and buckling analysis of rectangular plates. In this technique, the Ritz method is first used to discretize the spatial partial derivatives with respect to a coordinate direction of the plate. The DQ method is then employed to analogize the resulting system of ordinary or partial differential equations. The mixed method was shown to work well for vibration and buckling problems of rectangular plates with simple boundary conditions. But, due to the use of conventional Ritz method in one coordinate direction of the plate, the geometric boundary conditions of the problem can only be satisfied in that direction. Therefore, the conventional mixed Ritz-DQ methodology may encounter difficulties when dealing with rectangular plates involving adjacent free edges and skew plates. To overcome this difficulty, this paper presents a modified mixed Ritz-DQ formulation in which all the natural boundary conditions are exactly implemented. The versatility, accuracy and efficiency of the proposed method for free vibration analysis of thick rectangular and skew plates are tested against other solution procedures. It is revealed that the proposed method can produce highly accurate solutions for the natural frequencies of thick rectangular plates involving adjacent free edges and skew plates using a small number Ritz terms and DQ sampling points.     

各向异性矩形板自由振动的一般解析解法

根据各向异性矩形薄板自由振动横向位移函数的微分方程建立了一般性的解析解．该一般解包括三角函数和双曲线函数组成的解,它能满足4个边为任意边界条件的问题．还有代数多项式和双正弦级数解,它能满足4个角的边界条件问题．因此,这一解析解可用于精确地求解具有任意边界条件的各向异性矩形卞的振动问题．解中的积分常数可由4边和4角的边界条件来确定．由此得出的齐次线性代数方程系数矩阵行列式等于零可以求得各阶固有频率及其振型,以四边平夹的对称角铺设复合材料迭层板为例进行了计算和讨论．     

Analytical solutions of refined plate theory for bending,buckling and vibration analyses of thick plates

Analytical solutions for bending, buckling, and vibration analyses of thick rectangular plates with various boundary conditions are presented using two variable refined plate theory. The theory accounts for parabolic variation of transverse shear stress through the thickness of the plate without using shear correction factor. In addition, it contains only two unknowns and has strong similarities with the classical plate theory in many aspects such as equations of motion, boundary conditions, and stress resultant expressions. Equations of motion are derived from Hamilton’s principle. Closed-form solutions of deflection, buckling load, and natural frequency are obtained for rectangular plates with two opposite edges simply supported and the other two edges having arbitrary boundary conditions. Comparison studies are presented to verify the validity of present solutions. It is found that the deflection, stress, buckling load, and natural frequency obtained by the present theory match well with those obtained by the first-order and third-order shear deformation theories.     

边缘有弹性点支矩形板的横向振动*

本文研究了两对边简支、另两对边任意支承的自由边有弹性点支的矩形板的横向振动问题,提供了一种求其固有频率和振型的高精度解法,自由边上弹性点支的个数及位置均可任意,本文用脉冲函数表示弹性点支的反力和力矩,利用Fourier级数将脉冲函数沿边缘展开,从而得到了满足全部边界条件的特征方程,可求得任意精度的任意阶固有频率及振型.     

Fundamental frequency analysis of microtubules under different boundary conditions using differential quadrature method

Microtubules (MTs) are a central part of the cytoskeleton in eukaryotic cells. The dynamic behaviors of MTs are of great interest in biomechanics. Many researchers have studied the vibration analysis of MTs by modeling them as an orthotropic cylindrical elastic shell and the exact solution to its displacements is investigated under simply supported boundary conditions. Other boundary conditions lead to some coupled equations, which there are no exact solution to them. Considering various boundary conditions requires implementing semi-analytic or numerical methods. In this study, the differential quadrature method (DQM) has been used to solve the nonlinear problem of seeking fundamental frequency. At first to verify the DQM results, this method has been applied to the equations of MTs under simply supported boundary condition. The coincidence of the exact solution results and the results of DQM shows the effectiveness and precision of this method. After verification, DQM has been used for the other boundary conditions. These boundary conditions are including clamped–clamped (CC), clamped–simply (CS), clamped–free (CF) and free–free (FF) constraints. Finally, the effect of edges boundary condition, radius of MTs and half wave numbers on the vibration behavior of MTs is considered.     

弹性厚矩形板受迫振动的功的互等定理法

本文将功的互等定理法（ＲＴＭ）推广应用于求解基于Ｒｅｉｓｓｎｅｒ理论的厚矩形板受迫振动问题·本文导出了厚矩形板动力基本解；给出了三边固定一边自由厚矩形板在均布简谐干挠力作用下稳态响应的精确解析解·这是计算厚矩形板受振动稳态响应的一个简便通用的方法·     

Hamiltonian system-based new analytic free vibration solutions of cylindrical shell panels

This paper deals with the classical challenging free vibration problems of non-Lévy-type cylindrical shell panels, i.e., those without two opposite edges simply supported, by a Hamiltonian system-based symplectic superposition method. The governing equations of a vibrating cylindrical panel are formulated within the Hamiltonian system framework such that the symplectic eigen problems are constructed, which yield analytic solutions of two types of fundamental problems. By the equivalence between the superposition of the fundamental problems and the original problem, new analytic frequency and mode shape solutions of the panels with four different combinations of boundary conditions are derived. Comprehensive benchmark results are tabulated and plotted, which are useful for validation of other numerical/approximate methods. The primary advantage of the developed approach that no pre-determination of solution forms is needed enables one to pursue more analytic solutions of intractable shell problems.     

弹性地基上自由边矩形厚板

弹性地基上自由边矩形厚板的分栀由于其难度较大,一直没有得到很好的解决.本文采用单三角级数和重三角级数相叠加的方法,求得该问题的精确解.文中所用方法简单明了.所得结果完全满足边界条件并与王克林等[2]的结果完全一致.     

Levy-type solution for buckling analysis of thick functionally graded rectangular plates based on the higher-order shear deformation plate theory

In this article, an analytical approach for buckling analysis of thick functionally graded rectangular plates is presented. The equilibrium and stability equations are derived according to the higher-order shear deformation plate theory. Introducing an analytical method, the coupled governing stability equations of functionally graded plate are converted into two uncoupled partial differential equations in terms of transverse displacement and a new function, called boundary layer function. Using Levy-type solution these equations are solved for the functionally graded rectangular plate with two opposite edges simply supported under different types of loading conditions. The excellent accuracy of the present analytical solution is confirmed by making some comparisons of the present results with those available in the literature. Furthermore, the effects of power of functionally graded material, plate thickness, aspect ratio, loading types and boundary conditions on the critical buckling load of the functionally graded rectangular plate are studied and discussed in details. The critical buckling loads of thick functionally graded rectangular plates with various boundary conditions are reported for the first time and can be used as benchmark.     

几种典型板考虑初始荷载效应的基频近似解

基于两组板考虑初始荷载效应的动力控制微分方程:一般形式的动力控制微分方程和极坐标形式的动力控制微分方程,运用Galerkin（伽辽金)法求解得到了简支矩形板、固支矩形板、简支等边三角形板、固支椭圆形板、简支圆形板和固支圆形板6种典型板考虑初始荷载效应的自由振动基频（第一阶频率）近似解．通过与相关文献提出的有限元法计算结果对比,验证了公式的正确性．基频近似解表达式简单明了,物理意义明确,清楚地说明了初始荷载及相关因素对板自由振动基频的影响,直观地说明了板的初始荷载效应这一概念．计算分析表明:初始荷载的存在增加了板的弯曲刚度,提高了板的自振频率．这种初始荷载效应对频率的影响主要受初始荷载大小、跨厚比及边界条件等因素的影响．在计算分析和设计中应考虑并重视这种初始荷载效应对板计算分析带来的影响.     

Levy-type solution for free vibration analysis of orthotropic plates based on two variable refined plate theory

Closed-form solutions for free vibration analysis of orthotropic plates are obtained in this paper based on two variable refined plate theory. The theory, which has strong similarity with classical plate theory in many aspects, accounts for a quadratic variation of the transverse shear strains across the thickness, and satisfies the zero traction boundary conditions on the top and bottom surfaces of the plate without using shear correction factors. Equations of motion are derived from the Hamilton’s principle. The closed-form solutions of rectangular plates with two opposite edges simply supported and the other two edges having arbitrary boundary conditions are obtained by applying the state space approach to the Levy-type solution. Comparison studies are performed to verify the validity of the present results. The effects of boundary condition, and variations of modulus ratio, aspect ratio, and thickness ratio on the natural frequency of orthotropic plates are investigated and discussed in detail.     

A two-variable first-order shear deformation theory considering in-plane rotation for bending,buckling and free vibration analyses of isotropic plates

This paper presents a two-variable first-order shear deformation theory considering in-plane rotation for bending, buckling and free vibration analyses of isotropic plates. In recent studies, a simple first-order shear deformation theory (S-FSDT) was developed and extended. It has only two variables by separating the deflection into bending and shear parts while the conventional first-order shear deformation theory (FSDT) has three variables. However, the S-FSDT provides incorrect predictions for the transverse shear forces on the insides and the twisting moments at the boundaries except simply supported plates since it does not consider in-plane rotation. The present theory also has two variables but considers in-plane rotation such that it is able to correctly predict the responses of plates with any boundary conditions. Analytical solutions are obtained for rectangular plates with two opposite edges that are simply supported, with the other edges having arbitrary boundary conditions. Numerical results of deflections, stress resultants, buckling loads and natural frequencies are presented with the FSDT, the S-FSDT and the present theory. Comparative studies demonstrate the effects of in-plane rotation and the accuracy of the present theory in predicting the bending, buckling and free vibration responses of isotropic plates.     

双参数地基上Reisner板弯曲问题的边界积分方程

本文应用广义函数的Ｆｏｕｒｉｅｒ积分变换，导出了双参数地基上Ｒｅｉｓｓｎｅｒ板弯曲问题的两个基本解·在此基础上，从虚功原理出发，依据胡海昌导出的Ｒｅｉｓｓｎｅｒ板弯曲理论，推导出适用于任意形状、任意荷载、任意边界条件情形的三个边界积分方程，为边界元法在这一问题中的应用提供了理论基础·文中给出了固支、简支、自由三类边界的算例，并与解析解比较，均得到满意的结果·     

Buckling mode localization in rib-stiffened plates with misplaced stiffeners – Kantorovich approach

Buckling mode localization in rib-stiffened plates with randomly misplaced stiffeners is studied in this paper. The method of Kantorovich on reducing a partial differential equation to a system of ordinary differential equations is employed to obtain the deflection surface of the rib-stiffened plates under axial compressive load. The edges of the plates normal to the stiffeners can be either simply supported or clamped. The solutions of the deflection surface are then expressed in the form of transfer matrices. The expressions of the solutions obtained for the case of one edge simply supported and one edge clamped and the case of two edges clamped are similar to those for the case of two edges simply supported. When the two edges are simply supported, the method of Kantorovich yields the exact results. Localization factors, which characterize the average exponential rates of growth or decay of amplitudes of deflection, are determined using the method of transfer matrix. The method of Kantorovich is a general approximate method, which is applicable for various support conditions.     

Free vibration of a functionally graded annular sector plate integrated with piezoelectric layers

Based on the first order shear deformation theory, free vibration behavior of functionally graded (FG) annular sector plates integrated with piezoelectric layers is investigated. The distribution of electric potential along the thickness direction of piezoelectric layers which is assumed to be a combination of linear and sinusoidal functions, satisfies both open and closed circuit electrical boundary conditions. Through a reformulation of governing equations and harmonic motion assumption, a novel decoupling method is suggested to transform the six second order coupled partial differential equations of motion into two eighth order and fourth order equations. A Fourier series method is then employed to present analytical solutions for free vibration of smart FG annular sector plates with simply supported radial edges and arbitrarily supported circular edges. The results, which can be used as a benchmark and suitable for design purposes, are verified with those reported in the literature. Finally, by presenting extensive ranges of frequencies, the effects of geometric parameters, power law index, FG and piezoelectric materials, electrical and mechanical boundary conditions as well as the piezoelectric layer thickness on vibration response of smart annular sector plates are discussed in detail.