Vibration of circular Mindlin plates with concentric elastic ring supports

This paper studies the vibration behaviour of circular Mindlin plates with multiple concentric elastic ring supports. Utilizing the domain decomposition technique, a circular plate is divided into several annular segments and one core circular segment at the locations of the elastic ring supports. The governing differential equations and the solutions of these equations are presented for the annular and circular segments based on the Mindlin-plate theory. A homogenous equation system that governs the vibration of circular Mindlin plates with elastic ring supports is derived by imposing the essential and natural boundary and segment interface conditions. The first-known exact vibration frequencies for circular Mindlin plates with multiple concentric elastic ring supports are obtained and the modal shapes of displacement fields and stress resultants for several selected cases are presented. The influence of the elastic ring support stiffness, locations, plate boundary conditions and plate thickness ratios on the vibration behaviour of circular plates is discussed.     

Numerical differential quadrature method for Reissner/Mindlin plates on two-parameter foundations

Approximate solutions for the bending of moderately thick rectangular plates on two-parameter elastic foundations (Pasternak-type) as described by Mindlin's theory are presented. The plates are subjected to an arbitrary combination of clamped and simply-supported boundary conditions. An efficient computational technique, the differential quadrature (DQ) method, is employed to transform the governing differential equations and boundary conditions into a set of linear algebraic equations for approximate solutions. These resulting algebraic equations are solved numerically. In this study, the accuracy of the DQ method is established by direct comparison with results in the existing literature. The convergence properties of the method are illustrated for different combinations of boundary conditions. The deflections, moments and shear forces at selected locations are tabulated in detail for different elastic foundations. The efficiency and simplicity of the solution method are highlighted.     

DSC-element method for free vibration analysis of rectangular Mindlin plates

A novel DSC-element method is proposed to investigate the free vibration of moderately thick plates based on the well-known Mindlin first-order shear deformation plate theory. The development of the present approach not only employs the concept of finite element method, but also implements the discrete singular convolution (DSC) delta type wavelet kernel for the transverse vibration analysis. This numerical algorithm is allowed dividing the domain of Mindlin plates into a number of small discrete rectangular elements. As compared with the global numerical techniques i.e. the DSC-Ritz method, the flexibility is increased to treat complex boundary constraints. For validation, a series of numerical experiments for different meshes of Mindlin plates with assorted combinations of edge supports, plate thickness and aspect ratios is carried out. The established natural frequencies are directly compared and discussed with those reported by using the finite element and other numerical and analytical methods from the open literature.     

Analysis of annular Reissner/Mindlin plates using differential quadrature method

The axisymmetric flexure responses of moderately thick annular plates under static loading are investigated. The shear deformation is considered using the first-order Reissner/Mindlin plate theory and the solutions are obtained using the differential quadrature (DQ) method. In the solution process, the governing differential equations and boundary conditions for the problem are initially discretized by the DQ algorithm into a set of linear algebraic equations. The solutions of the problem are then determined by solving the set of algebraic equations. This study considers the plate subjected to various combinations of clamped, simply-supported, free and guided boundary conditions and different loading manners. The accuracy of the method is demonstrated through direct comparison of the present results with the corresponding exact solutions available in the literature.     

Pseudospectral analysis of buckling and free vibration of rectangular Mindlin plates

A study of buckling and free vibration of rectangular Mindlin plates is presented. The analysis is based on the pseudospectral method, which uses basis functions that satisfy the boundary conditions. The equations of motion are collocated to yield a set of algebraic equations that are solved for the critical buckling load and for the natural frequencies in the presence of the in-plane loads. Numerical examples of rectangular plates with SS-C-SS-C boundary conditions are provided for various aspect ratios and thickness ratios, which show good agreement with those of the classical plate theory when the thickness ratio is very small. This paper was recommended for publication in revised form by Associate Editor Eung-Soo Shin Jinhee Lee received B.S. and M.S. degrees from Seoul National University and KAIST in 1982 and 1984, respectively. He received his Ph.D. degree from the University of Michigan, Ann Arbor in 1992 and joined the Dept. of Mechanical and Design Engineering of Hongik University in Choongnam, Korea. His research interests include inverse problems, pseudospectral method, vibration and dynamic systems.     

Axisymmetric free vibration of thick annular plates

This paper presents a numerical analysis of the axisymmetric free vibration of moderately thick annular plates using the differential quadrature method (DQM). The plates are described by Mindlin’s first-order shear-deformation theory. The first five axisymmetric natural frequencies are presented for uniform annular plates, of various radii and thickness ratios, with nine possible combinations of free, clamped and simply supported boundary conditions at the inner and outer edges of the plates. The accuracy of the method is established by comparing the DQM results with some exact and finite element numerical solutions and, therefore, the present DQM results could serve as a benchmark for future reference. The convergence characteristics of the method for thick plate eigenvalue problems are investigated and the versatility and simplicity of the method is established.     

A novel approach for in-plane/out-of-plane frequency analysis of functionally graded circular/annular plates

An exact closed-form frequency equation is presented for free vibration analysis of circular and annular moderately thick FG plates based on the Mindlin's first-order shear deformation plate theory. The edges of plate may be restrained by different combinations of free, soft simply supported, hard simply supported or clamped boundary conditions. The material properties change continuously through the thickness of the plate, which can vary according to a power-law distribution of the volume fraction of the constituents, whereas Poisson's ratio is set to be constant. The equilibrium equations which govern the dynamic stability of plate and its natural boundary conditions are derived by the Hamilton's principle. Several comparison studies with analytical and numerical techniques reported in literature and the finite element analysis are carried out to establish the high accuracy and superiority of the presented method. Also, these comparisons prove the numerical accuracy of solutions to calculate the in-plane and out-of-plane modes. The influences of the material property, graded index, thickness to outer radius ratios and boundary conditions on the in-plane and out-of-plane frequency parameters are also studied for different functionally graded circular and annular plates.     

Forced axisymmetric response of linearly tapered circular plates

Forced axisymmetric response of a circular plate of linearly varying thickness, based on the classical theory, is analyzed by the eigen-function method. An exact solution for the free vibration mode shapes is obtained by the Frobenius method. Clamped and simply-supported plates subjected to symmetric uniformly distributed and concentrated impulsive ring and point loads are solved as example problems. Numerical results computed for transverse deflection and radial stress are plotted in the figures.     

Elastic small deflection analysis of annular sector Mindlin plates

An analytical method is developed for the bending response of annular sector Mindlin plates with two radial edges simply supported, and exact solutions are presented in the form of Levy-type series. Several different boundary conditions on the two circular edges are considered, viz. simply supported-simply supported, clamped-clamped and free-free. Numerical results for the case of uniform loading are presented to indicate the effect of shear deformation on the deflections and stress resultants at various points in the plate. Twisting stress couple and transverse shear stress resultant distributions along and near the edges of the plate are illustrated graphically, and the principal differences between the results predicted by Mindlin's plate theory and classical thin plate theory are discussed in detail. Results obtained with the present exact analysis may serve as references for approximate solutions and, especially, as a ‘shear locking’ test for thick plate finite element analysis.     

Deflection and stress-resultants of axisymmetric mindlin plates in terms of corresponding Kirchhoff solutions

The Kirchhoff plate theory, when used for the analysis of bending of plates that are relatively thick, underpredicts the deflections. This is because it does not account for the effect of transverse shear deformation which becomes significant in thick plates. A more refined plate theory proposed by Mindlin allows for this shear deformation effect by relaxing the condition that the normal to the plate midsurface must remain normal to the deformed midsurface. In this paper, new exact relationships are presented between the Kirchhoff and Mindlin solutions for deflection and stress-resultants for axisymmetric plates under general rotationally symmetric loading. These relationships enable engineers and designers to obtain readily the Mindlin solutions, of such loaded axisymmetric plates, from the abundantly available Kirchhoff solutions. Thus, the task of obtaining solutions from complicated shear deformable plate analysis using the Mindlin theory may be avoided.     

Three-dimensional free vibration of variable thickness thick circular and annular isotropic and functionally graded plates on Pasternak foundation

This paper describes a study of three-dimensional free vibration analysis of thick circular and annular isotropic and functionally graded (FG) plates with variable thickness along the radial direction, resting on Pasternak foundation. The formulation is based on the linear, small strain and exact elasticity theory. Plates with different boundary conditions are considered and the material properties of the FG plate are assumed to vary continuously through the thickness according to power law. The kinematic and the potential energy of the plate-foundation system are formulated and the polynomial-Ritz method is used to solve the eigenvalue problem. Convergence and comparison studies are done to demonstrate the correctness and accuracy of the present method. With respect to geometric parameters, elastic coefficients of foundation and different boundary conditions some new results are reported which may be used as benchmark solutions for future researches.     

用边界元法对受弯回转体进行优化设计

边界元法的主要优点是使问题降维。对于回转体，三维边界积分方程的维数通过坐标变换到极坐标系下还可以降维。这对工程计算费用的降低是大有用处的。本文给出了回转体受弯的边界元法弹性应力分析的格式和数值处理，对一个实际轴作了形状优化设计。     

Polynomial solutions for thin plates

A least-squares approach is used to obtain polynomial solutions for the deflected shape of thin plates in flexure. The method corresponds to the finite element method with an energy basis, except that a single “macro” element is used. Satisfactory results are obtained for both triangular and rectangular plates and the method is clearly widely applicable.     

Levy solutions for vibration of multi-span rectangular plates

This paper presents the Levy method to investigate the vibration behaviour of multi-span rectangular plates. The Levy method is applicable and analytical for rectangular plates with at least two parallel simply supported edges. The continuity at an interface between two spans is maintained by imposing both the essential and natural boundary conditions along the interface. The impact of the internal line supports on the vibration behaviour of the plates is investigated by varying both the number of internal lines and the line positions. Results for the vibration of two- and three-span rectangular plates are presented, in which the first-known exact solutions for plates involving free edges are included. The present results may serve as benchmark solutions for such plates.     

矩形薄板弯曲的严格简明解析解

解析解在理论上与数值计算上都有很高价值。根据历史已有的经典解的启发,对导出矩形薄板弯曲的严格简明解析解(无特殊函数与无穷级数)的方法,提出推导的新思路：在求导简明严格解析解时,应该改变已有办法,不是以外载荷的分布为给定参数,而是先考虑满足边界条件的薄板法向位移分布,再按基本方程求出外载荷与其余参数的应有分布。对于简支边界条件,为得出简明严格解析解,法向位移的解析函数在两个坐标上分别应该至少各有两个根,而且两个根值所在处同时也是函数的拐点。对此准则,以偶数多项式、概率函数与箕舌线函数作为法向位移函数为例,给出其应有的简明严格解析解。 同样,对于固定边界条件,类似的准则是：法向位移的解析函数在两个坐标上分别应该至少各有两个根,而且两个根值所在处同时也是函数极值所在。以奇次多项式与星型线函数为例,给出其法向位移函数和应有的简明严格解析解。上述思路与方法能再发展,例如用于不同或复合的边界条件中去。     

Differential cubature method for analysis of shear deformable rectangular plates on Pasternak foundations

In this paper, the differential cubature method (DCM) was applied to the bending analysis of shear deformable plates resting on Pasternak foundation. An attractive advantage of the DCM is that it can produce the acceptable accuracy of numerical results with very few grid points in the solution domain and therefore can be very useful for rapid evaluation in engineering design. The detailed procedures for discretizing the governing equations and boundary conditions of the title problems using the DCM are presented. Numerical solutions for rectangular thick plates on Pasternak foundation and subjected to different boundary conditions are obtained. The convergence studies are carried out to establish the minimal grid points needed for achieving accurate solutions. Next, the solutions for some selected cases are presented and verified by comparing them with the published values. It is observed that the DCM is able to furnish convergent solution with relatively fewer grid points than the more established differential quadrature method (DQM).     

Effect of transverse shear and rotatory inertia on the forced axisymmetric response of linearly tapered circular plates

The forced axisymmetric response of linearly tapered circular plates, based on the shear theory is analyzed by the eigenfunction method. Clamped and simply supported plates subjected to constant and half-sine pulse loads, uniformly distributed over a symmetric portion of the plate, are solved as example problems. Numerical results computed for transverse deflection and radial stress of the plate are compared with the corresponding results of classical theory. Results obtained for a plate of constant thickness, as a particular case, are compared with closed form solutions and a very good agreement is found.     

Dynamic response of a Mindlin elastic rectangular plate under a distributed moving mass

The elastodynamic response of a rectangular Mindlin plate subjected to a distributed moving mass is investigated. The set of governing characteristic partial differential equations that include the effects of shear deformation and rotary inertia is expressed in its dimensionless form. A finite difference algorithm is employed to transform the differential equations into a set of linear algebraic equations. Simply supported edge conditions were used as an illustrative example. The analysis is also valid for other edge conditions. It is found that the maximum shearing forces, bending and twisting moments occur almost the same time. Also, the values of the maximum deflections are higher for Mindlin plates than for non-Mindlin plates.     

Nonlinear bending of Reissner–Mindlin plates with free edges under transverse and in-plane loads and resting on elastic foundations

A nonlinear bending analysis is presented for a rectangular Reissner–Mindlin plate with free edges subjected to combined transverse partially distributed load and compressive edge loading and resting on a two-parameter (Pasternak-type) elastic foundation. The formulations are based on the Reissner–Mindlin plate theory considering the first-order shear deformation effect, and including the plate-foundation interaction. The analysis uses a mixed Galerkin-perturbation technique to determine the load–deflection curves and load–bending moment curves. Numerical examples are presented that relate to the performances of moderately thick rectangular plates with free edges subjected to combined loading and resting on Pasternak-type elastic foundations from which results for Winkler elastic foundations are obtained as a limiting case. The influence played by a number of effects, among them foundation stiffness, transverse shear deformation, loaded area, the plate aspect ratio and initial compressive load are studied. Typical results are presented in dimensionless graphical form.     

A new exact analytical approach for free vibration of Reissner-Mindlin functionally graded rectangular plates

An exact closed-form procedure is presented for free vibration analysis of moderately thick rectangular plates having two opposite edges simply supported (i.e. Lévy-type rectangular plates) based on the Reissner-Mindlin plate theory. The material properties change continuously through the thickness of the plate, which can vary according to a power law distribution of the volume fraction of the constituents. By introducing some new potential and auxiliary functions, the displacement fields are analytically obtained for this plate configuration. Several comparison studies with analytical and numerical techniques reported in literature are carried out to establish the high accuracy and reliability of the solutions. Comprehensive benchmark results for natural frequencies of the functionally graded (FG) rectangular plates with six different combinations of boundary conditions (i.e. SSSS-SSSC-SCSC-SCSF-SSSF-SFSF) are tabulated in dimensionless form for various values of aspect ratios, thickness to length ratios and the power law index. Due to the inherent features of the present exact closed-form solution, the present results will be a useful benchmark for evaluating the accuracy of other analytical and numerical methods, which will be developed by researchers in the future.