矩形Mindlin板振动分析的DSC方法

提出了一种离散奇异卷积（DSC：Discrete Singular Convolution）方法来对基于Mindlin剪切变形理论的矩形厚板进行自由振动分析。此方法采用了Gauss delta序列核作为基函数并结合pb-2Rayleigh-Ritz方法（pb-2指的是atwo-dimensional polynomial function（p-2）and a boundary function（b））的边界函数得到了一种新型的Ritz方法。数值结果表明此方法相当精确有效。     

DSC‐Ritz method for high‐mode frequency analysis of thick shallow shells

This paper addresses a challenging problem in computational mechanics—the analysis of thick shallow shells vibrating at high modes. Existing methods encounter significant difficulties for such a problem due to numerical instability. A new numerical approach, DSC‐Ritz method, is developed by taking the advantages of both the discrete singular convolution (DSC) wavelet kernels of the Dirichlet type and the Ritz method for the numerical solution of thick shells with all possible combinations of commonly occurred boundary conditions. As wavelets are localized in both frequency and co‐ordinate domains, they give rise to numerical schemes with optimal accurate, stability and flexibility. Numerical examples are considered for Mindlin plates and shells with various edge supports. Benchmark solutions are obtained and analyzed in detail. Experimental results validate the convergence, stability, accuracy and reliability of the proposed approach. In particular, with a reasonable number of grid points, the new DSC‐Ritz method is capable of producing highly accurate numerical results for high‐mode vibration frequencies, which are hitherto unavailable to engineers. Moreover, the capability of predicting high modes endows us the privilege to reveal a discrepancy between natural higher‐order vibration modes of a Mindlin plate and those calculated via an analytical relationship linking Kirchhoff and Mindlin plates. Copyright © 2004 John Wiley & Sons, Ltd.     

Vibration Analysis of Arbitrarily Shaped Sandwich Plates via Ritz Method

This article presents the Ritz method for the vibration analysis of sandwich plates having an orthotropic core and laminated facings. The planform of the plate may take on any arbitrary shape. On the basis of the Mindlin plate theory and the Ritz method, the governing eigenvalue equation for determining the natural frequencies was derived. The Ritz method was automated and made computationally effective for general-shaped plates with any boundary conditions by (1) adopting the product of polynomial functions and boundary equations that were raised to appropriate powers and (2) applying Green's theorem to transform the integration over the general-shaped domain into a closed line integration. The Ritz formulation and software were verified by the close agreement with vibration frequencies obtained by previous researchers for a wide range of subset plate problems involving isotropic, laminated, and sandwich plates of various shapes. Moreover, sample natural frequencies of sandwich plates with laminated facings are presented for some quadrilateral plate shapes. These frequencies should be useful as reference results to researchers who are developing new methods or software for vibration analysis of sandwich plates.     

Analysis of general quadrilateral orthotropic thick plates with arbitrary boundary conditions by the Rayleigh–Ritz method

A numerical method is developed for the analysis of general quadrilateral, moderately thick orthotropic plates having arbitrary boundary conditions. The procedure is based on the application of the Rayleigh–Ritz method in conjunction with the Reissner–Mindlin thick plate theory. A set of complete polynomials in terms of natural co‐ordinates comprising of boundary and domain terms, which satisfy the boundary conditions, is deployed as the basic functions to approximate the real displacement field. The generalized displacements and the eigenvalues are determined by imposing the principle of stationary potential energy. Although the procedure has the ability to solve problems involving thick quadrilateral plates, the numerical examples presented are mostly for skew plates. The results herein are compared with their counterparts determined by other investigators. Copyright © 2002 John Wiley & Sons, Ltd.     

A simple and accurate Ritz formulation for free vibration of thick rectangular and skew plates with general boundary conditions

The Ritz method is one of the most elegant and useful approximate methods for obtaining solutions for the natural frequencies and vibration modes of elastic plates. It is simple to use and also straightforward to implement. In conventional Ritz method, the geometric boundary conditions are only satisfied and hence the Ritz method is known as a method that can produce upper bound solution results for the natural frequencies of elastic plates. On the other hand, the accuracy of the Ritz method for the solution of differential equations with mixed natural boundary conditions at the boundary lines is not very satisfactory. To overcome this difficulty, this paper presents a simple and accurate Ritz formulation in which 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 to handle the mixed natural boundary conditions is simple to use and can produce highly accurate solutions for the natural frequencies of thick rectangular and skew plates involving free edges.     

Bending,Free Vibration and Buckling Analysis of Functionally Graded Plates via Wavelet Finite Element Method

Following previous work, a wavelet finite element method is developed for bending, free vibration and buckling analysis of functionally graded (FG) plates based on Mindlin plate theory. The functionally graded material (FGM) properties are assumed to vary smoothly and continuously throughout the thickness of plate according to power law distribution of volume fraction of constituents. This article adopts scaling functions of two-dimensional tensor product BSWI to form shape functions. Then two-dimensional FGM BSWI element is constructed based on Mindlin plate theory by means of two-dimensional tensor product BSWI. The proposed two-dimensional FGM BSWI element possesses the advantages of high convergence, high accuracy and reliability with fewer degrees of freedoms on account of the excellent approximation property of BSWI. Numerical examples concerning various length-to-thickness ratios, volume fraction indexes, aspect ratios and boundary conditions are carried out for bending, free vibration and buckling problems of FG plates. These comparison examples demonstrate the accuracy and reliability of the proposed WFEM method comparing with the exact and referential solutions available in literatures.     

Three‐dimensional vibration analysis of rectangular thick plates on Pasternak foundation

The free‐vibration characteristics of rectangular thick plates resting on elastic foundations have been studied, based on the three‐dimensional, linear and small strain elasticity theory. The foundation is described by the Pasternak (two‐parameter) model. The Ritz method is used to derive the eigenvalue equation of the rectangular plate by augmenting the strain energy of the plate with the potential energy of the elastic foundation. The Chebyshev polynomials multiplied by a boundary function are selected as the admissible functions of the displacement functions in each direction. The approach is suitable for rectangular plates with arbitrary boundary conditions. Convergence and comparison studies have been performed on square plates on elastic foundations with different boundary conditions. It is shown that the present method has a rapid convergent rate, stable numerical operation and very high accuracy. Parametric investigations on the dynamic behaviour of clamped square thick plates on elastic foundations have been carried out in detail, with respect to different thickness–span ratios and foundation parameters. Some results found for the first time have been given and some important conclusions have been drawn. Copyright © 2004 John Wiley & Sons, Ltd.     

Three-dimensional free vibration analysis of functionally graded annular plates using the Chebyshev–Ritz method

The aim of this paper is to investigate three-dimensional free vibration of functionally graded annular plates with different boundary conditions using the Chebyshev–Ritz method, in which a set of duplicate Chebyshev polynomial series multiplied by the boundary function satisfying the boundary conditions are chosen as the trial functions of the displacement components. Two kinds of variations of material properties in the thickness direction of the plates are considered. Convergence of the Chebyshev–Ritz method is checked. Numerical results are given and compared with the previously published solutions.     

Free vibration analysis of thin isotropic and anisotropic rectangular plates by the discrete singular convolution algorithm

This paper presents the free vibration analysis of thin isotropic and anisotropic rectangular plates with various boundary conditions by using the discrete singular convolution (DSC) algorithm. Based on Taylor's series expansion, a unique scheme is proposed to handle various boundary conditions, including the simply supported (S), clamped (C) and free (F) edge. To validate the proposed method, the non‐dimensional frequency parameters of isotropic, orthotropic and angle‐ply symmetric laminated rectangular plates with various combinations of boundary conditions are obtained by using the DSC algorithm and compared with the analytical and/or numerical solutions. Comparisons reveal that the proposed method can handle the zero bending moment and zero shear force conditions for the isotropic as well as anisotropic plates. The proposed method provides an alternative way for applying the simply supported boundary conditions in applications of the DSC algorithm to plate structures. This investigation extends the application range of the DSC algorithm to vibration analysis of anisotropic plates with various combinations of boundary conditions. Copyright © 2010 John Wiley & Sons, Ltd.     

A differential quadrature hierarchical finite element method and its applications to vibration and bending of Mindlin plates with curvilinear domains

A differential quadrature hierarchical finite element method (DQHFEM) is proposed by expressing the hierarchical finite element method matrices in similar form as in the differential quadrature finite element method and introducing interpolation basis on the boundary of hierarchical finite element method elements. The DQHFEM is similar as the fixed interface mode synthesis method but the DQHFEM does not need modal analysis. The DQHFEM with non‐uniform rational B‐splines elements were shown to accomplish similar destination as the isogeometric analysis. Three key points that determine the accuracy, efficiency and convergence of DQHFEM were addressed, namely, (1) the Gauss–Lobatto–Legendre points should be used as nodes, (2) the recursion formula should be used to compute high‐order orthogonal polynomials, and (3) the separation variable feature of the basis should be used to save computational cost. Numerical comparison and convergence studies of the DQHFEM were carried out by comparing the DQHFEM results for vibration and bending of Mindlin plates with available exact or highly accurate approximate results in literatures. The DQHFEM can present highly accurate results using only a few sampling points. Meanwhile, the order of the DQHFEM can be as high as needed for high‐frequency vibration analysis. Copyright © 2016 John Wiley & Sons, Ltd.     

Inelastic transient dynamic analysis of Reissner–Mindlin plates by the D/BEM

A direct domain/boundary element method (D/BEM) for dynamic analysis of elastoplastic Reissner–Mindlin plates in bending is developed. Thus, effects of shear deformation and rotatory inertia are included in the formulation. The method employs the elastostatic fundamental solution of the problem resulting in both boundary and domain integrals due to inertia and inelasticity. Thus, a boundary as well as a domain space discretization by means of quadratic boundary and interior elements is utilized. By using an explicit time‐integration scheme employed on the incremental form of the matrix equation of motion, the history of the plate dynamic response can be obtained. Numerical results for the forced vibration of elastoplastic Reissner–Mindlin plates with smooth boundaries subjected to impulsive loading are presented for illustrating the proposed method and demonstrating its merits. Copyright © 2000 John Wiley & Sons, Ltd.     

Buckling of triangular Mindlin plates under isotropic inplane compression

Summary This paper considers the elastic buckling of triangular Mindlin plates under isotropic inplane compression. The total potential energy functional for the considered buckling problem is derived and the newly developedpb-2 Rayleigh-Ritz method is applied for the analysis. The method features thepb-2 Ritz function defined by the product of a two-dimensional polynomial function (p-2) and a basic function (b) formed from taking the product of the boundary equations with each equation raised to appropriate powers. New buckling results for isosceles and right-angled triangular Mindlin plates with various apex angles, thickness to width ratios and different combinations of edge conditions are presented. Such tabulated results are not only valuable to designers but also important to researchers as reference values for validating their numerical techniques and software packages.     

Ritz analysis of vibrating rectangular and skew multilayered plates based on advanced variable-kinematic models

A variable-kinematic Ritz formulation based on two-dimensional higher-order layerwise and equivalent single-layer theories is described in this paper to accurately predict free vibration of thick and thin, rectangular and skew multilayered plates with clamped, free and simply-supported boundary conditions. The main result is the derivation at a layer level of so-called Ritz fundamental nuclei for the stiffness and mass matrices which are invariant with respect to both the assumed kinematic model and the type of Ritz functions. In this work, products of Chebyshev polynomials and boundary-compliant functions are chosen as admissible trial set. After studying the convergence of the method, its accuracy is evaluated, in terms of frequency parameters and through-the-thickness distribution of modal displacements, by comparison with some reference results available in the literature. Results for sandwich plates with soft core are given for the first time, which may serve as benchmark values for future research.     

点载静力梁函数在求解矩形薄板自振特性中的应用

本文选择一组点载静力梁函数作为基函数，应用李兹法近似求解矩形薄板的自振特性，计算了多种边界条件下各种长宽比的矩形薄板的自振频率，与其它方法相比，本文方法具有很好的精度和收敛性，且计算非常简单直接。     

Hybrid displacement function element method: a simple hybrid‐Trefftz stress element method for analysis of Mindlin–Reissner plate

In order to develop robust finite element models for analysis of thin and moderately thick plates, a simple hybrid displacement function element method is presented. First, the variational functional of complementary energy for Mindlin–Reissner plates is modified to be expressed by a displacement function F, which can be used to derive displacement components satisfying all governing equations. Second, the assumed element resultant force fields, which can satisfy all related governing equations, are derived from the fundamental analytical solutions of F. Third, the displacements and shear strains along each element boundary are determined by the locking‐free formulae based on the Timoshenko's beam theory. Finally, by applying the principle of minimum complementary energy, the element stiffness matrix related to the conventional nodal displacement DOFs is obtained. Because the trial functions of the domain stress approximations a priori satisfy governing equations, this method is consistent with the hybrid‐Trefftz stress element method. As an example, a 4‐node, 12‐DOF quadrilateral plate bending element, HDF‐P4‐11 β, is formulated. Numerical benchmark examples have proved that the new model possesses excellent precision. It is also a shape‐free element that performs very well even when a severely distorted mesh containing concave quadrilateral and degenerated triangular elements is employed. Copyright © 2014 John Wiley & Sons, Ltd.     

A unified formulation for free transverse vibration analysis of orthotropic plates of revolution with general boundary conditions

In this investigation, a general formulation for free transverse vibration analysis of orthotropic plates of revolution subjected to arbitrary boundary supports is presented by using the so-called spectro-geometric method (SGM) and the Rayleigh–Ritz technique. Under the current solution framework, the geometry of a structure can be accurately described in terms of mathematical or design parameters, rather than a computational grid or mesh. The unknown expansion coefficients are treated as the generalized coordinates and are determined using the Rayleigh–Ritz method. The accuracy and versatility of the current approach is fully demonstrated and verified through numerical examples involving plates with various shapes and boundary conditions.     

An enriched meshless method for non‐linear fracture mechanics

This paper presents an enriched meshless method for fracture analysis of cracks in homogeneous, isotropic, non‐linear‐elastic, two‐dimensional solids, subject to mode‐I loading conditions. The method involves an element‐free Galerkin formulation and two new enriched basis functions (Types I and II) to capture the Hutchinson–Rice–Rosengren singularity field in non‐linear fracture mechanics. The Type I enriched basis function can be viewed as a generalized enriched basis function, which degenerates to the linear‐elastic basis function when the material hardening exponent is unity. The Type II enriched basis function entails further improvements of the Type I basis function by adding trigonometric functions. Four numerical examples are presented to illustrate the proposed method. The boundary layer analysis indicates that the crack‐tip field predicted by using the proposed basis functions matches with the theoretical solution very well in the whole region considered, whether for the near‐tip asymptotic field or for the far‐tip elastic field. Numerical analyses of standard fracture specimens by the proposed meshless method also yield accurate estimates of the J‐integral for the applied load intensities and material properties considered. Also, the crack‐mouth opening displacement evaluated by the proposed meshless method is in good agreement with finite element results. Furthermore, the meshless results show excellent agreement with the experimental measurements, indicating that the new basis functions are also capable of capturing elastic–plastic deformations at a stress concentration effectively. Copyright © 2003 John Wiley & Sons, Ltd.     

Vibrations of skewed cantilevered triangular,trapezoidal and parallelogram Mindlin plates with considering corner stress singularities

Based on the Mindlin shear deformation plate theory, a method is presented for determining natural frequencies of skewed cantilevered triangular, trapezoidal and parallelogram plates using the Ritz method, considering the effects of stress singularities at the clamped re‐entrant corner. The admissible displacement functions include polynomials and corner functions. The admissible polynomials form a mathematically complete set and guarantee the solution convergent to the exact frequencies when sufficient terms are used. The corner functions properly account for the singularities of moments and shear forces at the re‐entrant corner and accelerate the convergence of the solution. Detailed convergence studies are carried out for plates of various shapes to elucidate the positive effects of corner functions on the accuracy of the solution. The results obtained herein are compared with those obtained by other investigators to demonstrate the validity and accuracy of the solution. Copyright © 2005 John Wiley & Sons, Ltd.     

T型耦合板结构振动特性研究

本文以T型耦合板为研究对象,在同时考虑面内振动和面外振动条件下采用改进傅立叶级数方法（Improved Fourier Series Method,IFSM）对其自由振动特性进行了计算分析。板结构的面内振动和面外振动位移函数表示为改进傅立叶级数形式,并引入正弦傅立叶级数以解决边界的不连续或跳跃现象。将位移函数的级数展开系数作为广义坐标,采用Rayleigh-Ritz方法对其进行求解。通过对不同边界条件及耦合连接情况下T型板自由振动特性进行计算,并将之与有限元法结果相比较,验证了本文方法的正确性和有效性,为耦合板结构的振动控制提供可靠的理论依据。