Static analysis of rectangular thick plates resting on two-parameter elastic boundary strips

In this paper, the Generalized Differential Quadrature (GDQ) method is used to obtain bending solution of moderately thick rectangular plates. The plate is resting on two-parameter elastic (Pasternak) foundation or strips with a finite width. Various combinations of clamped, simply supported and free boundary conditions are considered. According to the first-order shear deformation theory, the governing equations of the problem consist of three second-order partial differential equations (PDEs) in terms of displacement and rotations of the plate. The governing equations and solution domain is discretized based on the GDQ method. It is demonstrated that the method converges rapidly while providing accurate results with relatively small number of grid points. Accuracy of the results is examined using available data in the literature for Pasternak foundation. Furthermore, due to lack of data for Pasternak strips, all predictions are verified by finite element analysis which can be used as benchmark in future studies.     

Vibration analysis of FG annular sector in moderately thick plates with two piezoelectric layers

Free vibration of functionally graded(FG) annular sector plates embedded with two piezoelectric layers is studied with a generalized differential quadrature(GDQ)method. Based on the first-order shear deformation(FSD) plate theory and Hamilton's principle with parameters satisfying Maxwell's electrostatics equation in the piezoelectric layers, governing equations of motion are developed. Both open and closed circuit(shortly connected) boundary conditions on the piezoelectric surfaces, which are respective conditions for sensors and actuators, are accounted for. It is observed that the open circuit condition gives higher natural frequencies than a shortly connected condition. For the simulation of the potential electric function in piezoelectric layers, a sinusoidal function in the transverse direction is considered. It is assumed that properties of the FG material(FGM) change continuously through the thickness according to a power distribution law.The fast rate convergence and accuracy of the GDQ method with a small number of grid points are demonstrated through some numerical examples. With various combinations of free, clamped, and simply supported boundary conditions, the effects of the thicknesses of piezoelectric layers and host plate, power law index of FGMs, and plate geometrical parameters(e.g., angle and radii of annular sector) on the in-plane and out-of-plane natural frequencies for different FG and piezoelectric materials are also studied. Results can be used to predict the behaviors of FG and piezoelectric materials in mechanical systems.     

Transient responses of magnetostrictive plates by using the GDQ method

We used the generalized differential quadrature (GDQ) method to compute the transient response of thermal stresses and center displacement in laminated magnetostrictive plates under thermal vibration. We obtained the GDQ solutions in a three-layer (0°^m/90°/0) and a 10-layer (0°^m/90°/0°/90°/0)_s laminated magnetostrictive plate with four simply supported edges. We presented the transient responses of thermal stress and center displacement with and without velocity feedback control, respectively. The advantage of the GDQ method used provide us with an efficient method to compute the results including shear deformation effect with a few grid points. These GDQ results had its potential that could be used and considered as basic data in the future magnetostrictive laminate studies.     

Free vibration analysis of moderately thick functionally graded plates on elastic foundation using the extended Kantorovich method

Free vibration analysis of moderately thick rectangular FG plates on elastic foundation with various combinations of simply supported and clamped boundary conditions are studied. Winkler model is considered to describe the reaction of elastic foundation on the plate. Governing equations of motion are obtained based on the Mindlin plate theory. A semi-analytical solution is presented for the governing equations using the extended Kantorovich method together with infinite power series solution. Results are compared and validated with available results in the literature. Effects of elastic foundation, boundary conditions, material, and geometrical parameters on natural frequencies of the FG plates are investigated.     

Non-linear analysis of functionally graded circular plates under asymmetric transverse loading

Based on the first-order shear deformation plate theory with von Karman non-linearity, the non-linear axisymmetric and asymmetric behavior of functionally graded circular plates under transverse mechanical loading are investigated. Introducing a stress function and a potential function, the governing equations are uncoupled to form equations describing the interior and edge-zone problems of FG plates. This uncoupling is then used to conveniently present an analytical solution for the non-linear asymmetric deformation of an FG circular plate. A perturbation technique, in conjunction with Fourier series method to model the problem asymmetries, is used to obtain the solution for various clamped and simply supported boundary conditions. The material properties are graded through the plate thickness according to a power-law distribution of the volume fraction of the constituents. The results are verified by comparison with the existing results in the literature. The effects of non-linearity, material properties, boundary conditions, and boundary-layer phenomena on various response quantities in a solid circular plate are studied and discussed. It is found that linear analysis is inadequate for analysis of simply supported FG plates which are immovable in radial direction even in the small deflection range. Furthermore, the responses of FG materials under a positive load and a negative load of identical magnitude are not the same. It is observed that the boundary-layer width is approximately equal to the plate thickness with the boundary-layer effect in clamped FG plates being stronger than that in simply supported plates.     

Free vibration of carbon nanotube reinforced composite plate on point Supports using Lagrangian multipliers

Chebyshev polynomial functions are used in the Lagrangian multipliers method to study the free vibration characteristics of rectangular moderately thick composite plates reinforced with carbon nanotubes (CNTs). Plate is resting on point supports. Distribution of CNTs across the plate thickness is considered to be either uniform or functionally graded. Properties of the plate are obtained using a refined rule of mixtures approach which includes the efficiency parameters to capture the size dependent characteristics of the composite plate. Using a Ritz solution method, an eigenvalue problem is established which results in natural frequencies and mode shapes of the plate. Based on the developed solution method, number and position of point supports are arbitrary and also various boundary conditions may be assumed for the four edges of the plate. After performing comparison studies for isotropic homogeneous plates on point supports, parametric studies are provided to explore the vibration characteristics of the carbon nanotube reinforced composite plates on point supports. It is shown that, frequencies of the plate increase as the volume fraction of CNTs increases.     

复合材料层合板自由振动分析的无网格自然邻接点Petrov-Galerkin法

基于一阶剪切变形理论,提出了复合材料层合板自由振动分析的无网格自然邻接点Petrov-Galerkin法。计算时在复合材料层合板中面上仅需要布置一系列的离散节点,并利用这些节点构建插值函数。在板中面上的局部多边形子域上,采用加权余量法建立复合材料层合板自由振动分析的离散化控制方程,并且这些子域可由Delaunay三角形方便创建。自然邻接点插值形函数具有Kronecker delta函数性质,因而无需经过特别处理就能准确地施加本质边界条件。对不同边界条件、不同跨厚比、不同材料参数和不同铺设角度的复合材料层合板,由本文提出的无网格自然邻接点Petrov-Galerkin法进行自由振动分析时均可得到满意的结果。数值算例结果表明,本文方法求解复合材料层合板的自由振动问题是行之有效的。     

Effect of boundary condition nonlinearities on free large-amplitude vibrations of rectangular plates

The effect of boundary condition nonlinearities on free nonlinear vibrations of thin rectangular plates is analyzed. The method for analysis of the plate vibrations with geometrical nonlinearity and the boundary condition nonlinearity is suggested. The nonlinear boundary conditions for membrane forces are transformed into linear ones using the in-plane stress function. Additional boundary conditions for the in-plane displacements vanishing on the clamped edge of the plate are imposed on the stress function. Simply supported and cantilever plates are analyzed. The backbone curves obtained by satisfying linear and nonlinear boundary conditions are compared. It is shown that the results of the calculations with nonlinear boundary conditions differ essentially from the data obtained without these boundary conditions.     

Bending of edge-bonded dissimilar rectangular plates

This study develops the extended Kantorovich method (EKM) to provide a closed form semi analytical solution for the bending analysis of two edge-bonded thin rectangular plates. The constituent plates could be different in thickness, length, material, loading conditions, and Winkler foundation’s stiffness. A combination of clamp, free, and simply supports are applied to the structure. The shared edge in the composite plate is assumed to be perfectly bonded. By applying the EKM together with the idea of weighted residual technique, two sets of ODEs are obtained. Bending is assumed to remain continuous on the bonded edge. The EKM procedure is modified by applying the coordinate of an arbitrary shared point in the boundary conditions for the shared edge, to relate the bending of the two plates. The ODEs are solved iteratively to obtain the deflection function in a fast convergence trend. Two examples of aluminium-steel plate and functionally graded material-steel plate are considered. The deflection results from the boundary modified EKM (BM-EKM) are in high agreement with the finite element solution results. The bending of stepped plates is a special case of the current study. The suggested BM-EKM strengthens the EKM’s ability for solving complex jointed/bonded structures in structural analyses.

     

Linear water wave propagation through multiple floating elastic plates of variable properties

We investigate the problem of linear water wave propagation under a set of elastic plates of variable properties. The problem is two-dimensional, but we allow the waves to be incident from an angle. Since the properties of the elastic plates can be set arbitrarily, the solution method can also be applied to model regions of open water as well as elastic plates. We assume that the boundary conditions at the plate edges are the free boundary conditions, although the method could be extended straightforwardly to cover other possible boundary conditions. The solution method is based on an eigenfunction expansion under each elastic plate and on matching these expansions at each plate boundary. We choose the number of matching conditions so that we have fewer equations than unknowns. The extra equations are found by applying the free-edge boundary conditions. We show that our results agree with previous work and that they satisfy the energy balance condition. We also compare our results with a series of experiments using floating elastic plates, which were performed in a two-dimensional wave tank.     

Maximum frequency design of laminated plates with mixed boundary conditions

A layerwise optimization (LO) approach is extended to accommodate the finite element analysis for optimizing the free vibration behavior of laminated composite plates with discontinuities along the boundaries. The classical non-conforming finite element is modified to fit into the LO procedure and is used to calculate natural frequencies of symmetrically laminated plates. This combined LO-FEM approach is applied to laminated rectangular plates with combinations of free, simply supported and clamped edges along parts of the plate boundary. The fundamental frequency, an object function for the present study, is maximized by optimizing design variables that are a set of fiber orientation angles in the layers. For illustrative purpose nine examples of square and rectangular plates with various types of mixed boundary conditions are considered, and a comprehensive set of results are presented for the optimum fiber orientation angles and the maximum fundamental frequencies of the 8-layer and 24-layer plates.     

Mechanical buckling and free vibration of thick functionally graded plates resting on elastic foundation using the higher order B-spline finite strip method

In this study, the mechanical buckling and free vibration of thick rectangular plates made of functionally graded materials (FGMs) resting on elastic foundation subjected to in-plane loading is considered. The third order shear deformation theory (TSDT) is employed to derive the governing equations. It is assumed that the material properties of FGM plates vary smoothly by distribution of power law across the plate thickness. The elastic foundation is modeled by the Winkler and two-parameter Pasternak type of elastic foundation. Based on the spline finite strip method, the fundamental equations for functionally graded plates are obtained by discretizing the plate into some finite strips. The results are achieved by the minimization of the total potential energy and solving the corresponding eigenvalue problem. The governing equations are solved for FGM plates buckling analysis and free vibration, separately. In addition, numerical results for FGM plates with different boundary conditions have been verified by comparing to the analytical solutions in the literature. Furthermore, the effects of different values of the foundation stiffness parameters on the response of the FGM plates are determined and discussed.     

Three-dimensional vibrations of annular thick plates with linearly varying thickness

The free vibration of annular thick plates with linearly varying thickness along the radial direction is studied, based on the linear, small strain, three-dimensional (3-D) elasticity theory. Various boundary conditions, symmetrically and asymmetrically linear variations of upper and lower surfaces are considered in the analysis. The well-known Ritz method is used to derive the eigen-value equation. The trigonometric functions in the circumferential direction, the Chebyshev polynomials in the thickness direction, and the Chebyshev polynomials multiplied by the boundary functions in the radial direction are chosen as the trial functions. The present analysis includes full vibration modes, e.g., flexural, thickness-shear, extensive, and torsional. The first eight frequency parameters accurate to at least four significant figures for five vibration categories are obtained. Comparisons of present results for plates having symmetrically linearly varying thickness are made with others based on 2-D classical thin plate theory, 2-D moderate thickness plate theory, and 3-D elasticity theory. The first 35 natural frequencies for plates with asymmetrically linearly varying thickness are compared to the finite element solutions; excellent agreement has been achieved. The asymmetry effect of upper and lower surface variations on the frequency parameters of annular plates is discussed in detail. The first four modes of axisymmetric vibration for completely free circular plates with symmetrically and asymmetrically linearly varying thickness are plotted. The present results for 3-D vibration of annular plates with linearly varying thickness can be taken as benchmark data for validating results from various plate theories and numerical methods.     

Near-Boundary Stress Wave Field in Anisotropic Plates under Impulsive Load

The stress wave field and the behavior of waves near the free boundary in an orthotropic plate and at the interface between two anisotropic media are studied. The results presented were obtained using the dynamic photoelastic method and optically sensitive fibrous models. Experimental data for impulsively loaded plates with various boundary conditions are analyzed     

A simple and accurate mixed FE-DQ formulation for free vibration of rectangular and skew Mindlin plates with general boundary conditions

A simple and accurate mixed finite element-differential quadrature formulation is proposed to study the free vibration of rectangular and skew Mindlin plates with general boundary conditions. In this technique, the original plate problem is reduced to two simple bar (or beam) problems. One bar problem is discretized by the finite element method (FEM) while the other by the differential quadrature method (DQM). The mixed method, in general, combines the geometry flexibility of the FEM and high accuracy and efficiency of the DQM and its implementation is more easier and simpler than the case where the FEM or DQM is fully applied to the problem. Moreover, the proposed formulation is free of the shear locking phenomenon that may be encountered in the conventional shear deformable finite elements. A simple scheme is also presented to exactly implement the mixed natural boundary conditions of the plate problem. The versatility, accuracy and efficiency of the proposed method for free vibration analysis of rectangular and skew Mindlin plates are tested against other solution procedures. It is revealed that the proposed method can produce highly accurate solutions for the natural frequencies of rectangular and skew Mindlin plates with general boundary conditions.     

带裂纹矩形板自由振动解析解

选取带有补充项的双重正弦傅里叶级数作为振型函数通解,来解析研究带裂纹矩形板的自由振动特性。先将带裂纹矩形板分割成若干小矩形板,利用各小矩形板的边界条件,并结合振型函数中待定常数的物理意义,简化得到各小矩形板的振型函数,再结合各板的控制方程、未使用的边界条件、公共边协调条件及本文提出公共自由角点的协调条件,建立求解频率的代数方程组,然后将其转化为广义特征值问题来求解带裂纹矩形板的无量纲频率;最后选取具体参数进行计算并与文献结果对比,吻合良好,证明了本文采用的研究方法以及所提出公共角点协调条件的正确性和合理性。由于该振型函数能满足矩形板的任意边界约束,且其中的待定常数具有明确的物理意义,所以可使矩形板问题的求解统一化、简单化和规律化。     

基于物理中面FGM中厚板自由振动的有限元分析

基于物理中面和一阶剪切变形板理论,研究了不同边界条件下功能梯度材料(FGM)中厚板的自由振动问题．假设功能梯度板的材料性质沿厚度方向按幂函数规律连续变化．根据哈密顿原理建立了FGM板有限元形式的自由振动方程,利用MATLAB软件编写程序进行了计算．通过数值算例,讨论了不同边界条件下FGM中厚板的无量纲频率随材料梯度指数和厚宽比的变化情况,并与经典板理论下的频率进行了比较．     

Flexural wave scattering in a quarter-infinite thin plate with circular scatterers

Scattering of flexural waves by circular scatterers in a quarter-infinite thin plate is formulated using the wave expansion method together with the method of images. The scattered waves are expressed as a summation series of wave functions and the unknown scattering coefficients are determined by enforcing boundary conditions at the scatterers. Both holes and rigid scatterers are studied. Simply-supported and roller-supported boundary conditions on the quarter-infinite thin plate are also considered. The analysis can be used to determine the stress concentration caused by circular scatterers in quarter-infinite thin plates.     

Vibration of continuous circular plates

The general development of the theory given here considers the material to be orthotropic and continuous over (n ? 1) elastic or rigid supports. The effect of rotatory inertia and in-plane loads are also included while formulating the equations of motion. Double and triple series solutions are given for orthotropic continuous plates. By matching the continuity conditions at the intermediate supports and satisfying the boundary conditions at the outer edge, the frequency determinant is obtained. For the purpose of numerical computations, an isotropic plate continuous over an intermediate-rigid or elastic-support and free and with no in-plane loads at the outer edge is considered. It is found that the influence of Poisson's ratio on the frequency parameter is significant only for the first symmetric or asymmetric modes. The rotatory inertia influences the frequency parameter when the radius to thickness ratio is less than 80, viz, when the plate is thick. Moreover, the elasticity of the support influences considerably the free vibration of plates.     

Exact characteristic equations for some of classical boundary conditions of vibrating moderately thick rectangular plates

The dimensionless equations of motion are derived based on the Mindlin plate theory to study the transverse vibration of thick rectangular plates without further usage of any approximate method. The exact closed form characteristic equations are given within the validity of the Mindlin plate theory for plates having two opposite sides simply supported. The six distinct cases considered involve all possible combinations of classical boundary conditions at the other two sides of rectangular plates. Accurate eigenfrequency parameters are presented for a wide range of aspect ratio η and thickness ratio δ for each case. The three dimensional deformed mode shapes together with their associated contour plots obtained from the exact closed form eigenfunctions are also presented. Finally, the effect of boundary conditions, aspect ratios and thickness ratios on the eigenfrequency parameters and vibratory behavior of each distinct cases are studied in detail. It is believed that in the present work, the exact closed form characteristic equations and their associated eigenfunctions, except for the plates with four edges simply supported, for the rest of considered six cases are obtained for the first time.