Three-dimensional static solutions of rectangular plates by variant differential quadrature method

An endeavor to exploit three-dimensional elasticity solutions for bending and buckling of rectangular plates via the differential quadrature (DQ) and harmonic differential quadrature (HDQ) methods is performed. Unlike other works, the priority of this paper is to examine the computational characteristics of the two methods; therefore, we focus our studies only on the simply supported and clamped rectangular plates. To start with, we first outline the basic equations and boundary conditions describing the bending and buckling of rectangular plates followed by normalizing and discretizing them according to the DQ and HDQ algorithms. The resulting algebraic equation systems are then solved to obtain the solutions. Based on these solutions, the computational characteristics of the DQ and HDQ methods are investigated in terms of their numerical performances. It is found that the DQ method displays obvious superior convergence characteristics over the HDQ method for the three-dimensional static analysis of rectangular plates.     

Accurate buckling loads of thin rectangular plates under parabolic edge compressions by the differential quadrature method

The buckling of thin rectangular plates with nonlinearly distributed loadings along two opposite plate edges is analyzed by using the differential quadrature (DQ) method. The problem is considerably more complicated since it requires that first the plane elasticity problem be solved to obtain the distribution of in-plane stresses, and then the buckling problem be solved. Thus, very few analytical solutions (the only one available in the literature is for rectangular plates with all edges simply supported) have been available in the literature thus far. Detailed formulations and solution procedures are given herein. Nine combinations of boundary conditions and various aspect ratios are considered. Comparisons are made with a few existing analytical and/or finite element data. It has been found that a fast convergent rate can be achieved by the DQ method with non-uniform grids and very accurate results are obtained for the first time. It has also been found that the DQ results, verified by the finite element method with NASTRAN, are not quite close to the newly reported analytical solution. A possible reason is given to explain the difference.     

Three-dimensional vibration, buckling and bending analyses of thick rectangular plates based on discrete singular convolution method

The main objective of this study is to give a numerical solution of three-dimensional analysis of thick rectangular plates. The analysis uses discrete singular convolution (DSC) method. Free vibration, bending and buckling of rectangular plates have been studied in this paper. Regularized Shannon's delta (RSD) kernel is selected as singular convolution to illustrate the present algorithm. In the proposed approach, the derivatives in both the governing equations and the boundary conditions are discretized by the method of DSC. The obtanied results are compared with those of other numerical methods. It is found that the convergence of the DSC approach is very good and the results agree well with those obtained by other researchers.     

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.     

In-plane vibration analyses of circular arches by the generalized differential quadrature rule

In-plane free vibrations of circular arches are investigated using the generalized differential quadrature rule (GDQR) proposed recently. The Kirchhoff assumptions for thin beams are considered, and the neutral axis is taken as inextensible. Several examples of arches with uniform, continuously varying, and stepped cross-sections are presented to illustrate the validity and accuracy of the GDQR. The necessary domain decomposition technique is used for some cases. The obtained frequencies are compared with those calculated from a number of other approaches from the Rayleigh–Ritz, Rayleigh–Schmidt and Galerkin methods to the finite element technique and the cell discretization method. The GDQR frequencies are always greater than those obtained from the cell discretization method that produces the lower bounds to the exact results, and are also in agreement with the upper bounds to the exact results.     

Frequency characteristics of a thin rotating cylindrical shell using the generalized differential quadrature method

In this paper, the generalized differential quadrature (GDQ) method is used for the first time to study the effects of boundary conditions on the frequency characteristics of a thin rotating cylindrical shell. The present analysis is based on Love-type shell theory and the governing equations of motion include the effects of initial hoop tension and the centrifugal and coriolis accelerations due to rotation. The displacement field is expressed as a product of unknown smooth continuous functions in the meridional direction and trigonometric functions along the circumferential direction so that the three-dimensional dynamic problem may be transformed mathematically into a one-dimensional problem. Based on this approach, the results are obtained for the effects of the boundary conditions on the frequency characteristics at different circumferential wave numbers and rotating speeds and various geometric properties; the effect of rotating speed on the relationship between frequency parameter and circumferential wave number is also discussed. To validate the accuracy and efficiency of the GDQ method, the results obtained are compared with those in the literature and very good agreement is achieved.     

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.     

Measurement of residual stress in thick section composite laminates using the deep-hole method

The deep-hole method is a method of measuring residual stress in large metallic components. In this paper, an extension to the deep-hole method is described to allow the residual stresses in thick section composite laminated plates to be evaluated. The method involves first drilling a small hole through the laminate perpendicular to the surface. The material around the hole is then machined away, resulting in a change in diameter of the hole due to the release of residual stress. This change in diameter is measured and used to calculate the residual stress. The calculation requires the evaluation of coefficients that depend on the properties of the composite. In this work, the finite element method is used to evaluate these coefficients. Using this method, the residual stresses in a 22 mm thick carbon/epoxy composite plate are measured and reported.     

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.     

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).     

The differential quadrature method for irregular domains and application to plate vibration

By its very basis, the differential quadrature method may be applied to domains having boundaries oriented along the coordinate axes. In this paper, it is shown that quadrature rules may also be formulated for irregular domains using the natural-to-Cartesian geometric mapping technique. The application of the technique is demonstrated through the vibration analysis of thin isotropic plates of general quadrilateral and sectorial planforms.     

硅微陀螺正交误差校正方案优化

本文意在寻求双质量硅微机械陀螺仪正交校正最优方案。首先介绍了带有正交校正和检测力反馈梳齿的双质量硅微机械陀螺结构,量化分析了正交误差对输出信号的影响并进行了仿真,结果显示解调相角变化为±2°,200(°)/s的正交误差等效输入角速率可引起15(°)/s的输出信号变化。然后,对目前3种比较主流的硅微机械陀螺仪正交校正方法(电荷注入法(CIM)、正交力校正法(QFCM)和正交耦合刚度校正法(QCSCM))进行了实验研究,从理论上证明了这3种方法的可行性。对未加入正交校正环节的陀螺进行了实验,结果显示其左、右质量块输出的正交误差信号峰峰值分别为150mV和300mV。针对两质量块正交误差不等的实际问题提出了质量块单独校正的方案。采用CIM、QFCM和QCSCM对校正前零偏及其稳定性分别为-4.589(°)/s和378(°)/h的陀螺进行了实验校正,结果显示3种方法均可有效消除检测通道中正交信号,3种方法的零偏及零偏稳定性结果分别为-8.361(°)/s和423(°)/h,2.419(°)/s和82(°)/h,1.751(°)/s和25(°)/h,证明了正交耦合刚度校正法为3种方法中的最优方案。     

Analysis of rectangular laminated composite plates via FSDT meshless method

A meshless approach based on the reproducing kernel particle method is developed for the flexural, free vibration and buckling analysis of laminated composite plates. In this approach, the first-order shear deformation theory (FSDT) is employed and the displacement shape functions are constructed using the reproducing kernel approximation satisfying the consistency conditions. The essential boundary conditions are enforced by a singular kernel method. Numerical examples involving various boundary conditions are solved to demonstrate the validity of the proposed method. Comparison of results with the exact and other known solutions in the literature suggests that the meshless approach yields an effective solution method for laminated composite plates.     

Free vibration analysis of arbitrary shaped thick plates by differential cubature method

In this paper, a new numerical solution technique, the differential cubature method, is applied to solve the free vibration problems of arbitrary shaped thick plates. The basic idea of the differential cubature method is to express a linear differential operation such as a continuous function or any order of partial derivative of a multivariable function, as a weighted linear sum of discrete function values chosen within the overall domain of a problem. By using the differential cubature procedure, the governing differential equations and boundary conditions are transformed into sets of linear homogeneous algebraic equations. This is an eigenvalue problem, of which the eigenvalues can be calculated numerically. The subspace iterative method is employed in search of the free vibration frequency parameters. Detailed formulations are presented, and the method is examined here for its suitability for solving the vibration problems of moderately thick plates governed by Mindlin shear deformation theory. The applicability, efficiency and simplicity of the method are demonstrated through solving some example plate vibration problems of different shapes. The numerical accuracy of the method is ascertained by comparing the vibration frequency solutions with those of existing literatures.     

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.     

Local adaptive differential quadrature for free vibration analysis of cylindrical shells with various boundary conditions

This paper presents the formulation and numerical analysis of circular cylindrical shells by the local adaptive differential quadrature method (LaDQM), which employs both localized interpolating basis functions and exterior grid points for boundary treatments. The governing equations of motion are formulated using the Goldenveizer–Novozhilov shell theory. Appropriate management of exterior grid points is presented to couple the discretized boundary conditions with the governing differential equations instead of using the interior points. The use of compactly supported interpolating basis functions leads to banded and well-conditioned matrices, and thus, enables large-scale computations. The treatment of boundary conditions with exterior grid points avoids spurious eigenvalues. Detailed formulations are presented for the treatment of various shell boundary conditions. Convergence and comparison studies against existing solutions in the literature are carried out to examine the efficiency and reliability of the present approach. It is found that accurate natural frequencies can be obtained by using a small number of grid points with exterior points to accommodate the boundary conditions.     

Exact solutions for vibration and buckling of an SS-C-SS-C rectangular plate loaded by linearly varying in-plane stresses

Exact solutions are presented for the free vibration and buckling of rectangular plates having two opposite edges (x=0 and a) simply supported and the other two (y=0 and b) clamped, with the simply supported edges subjected to a linearly varying normal stress σ_x=−N₀[1−α(y/b)]/h, where h is the plate thickness. By assuming the transverse displacement (w) to vary as sin(mπx/a), the governing partial differential equation of motion is reduced to an ordinary differential equation in y with variable coefficients, for which an exact solution is obtained as a power series (the method of Frobenius). Applying the clamped boundary conditions at y=0 and b yields the frequency determinant. Buckling loads arise as the frequencies approach zero. A careful study of the convergence of the power series is made. Buckling loads are determined for loading parameters α=0,0.5,1,1.5,2, for which α=2 is a pure in-plane bending moment. Comparisons are made with published buckling loads for α=0,1,2 obtained by the method of integration of the differential equation (α=0) or the method of energy (α=1,2). Novel results are presented for the free vibration frequencies of rectangular plates with aspect ratios a/b=0.5,1,2 subjected to three types of loadings (α=0,1,2), with load intensities N₀/N_cr=0,0.5,0.8,0.95,1, where N_cr is the critical buckling load of the plate. Contour plots of buckling and free vibration mode shapes are also shown.     

Effects of prebuckling deformations on the elastic flexural-torsional buckling of laterally fixed arches

The effect of the prebuckling in-plane deformations on the elastic flexural-torsional buckling of laterally fixed circular arches is studied in this paper. The finite strains and the energy equation for the flexural-torsional buckling of arches have been derived based on an accurate orthogonal rotation matrix. A closed form solution for the elastic flexural-torsional buckling resistance of laterally fixed arches in uniform bending, including the effects of the prebuckling deformations, is obtained. It is found that the notion that the prebuckling deformations increase the flexural-torsional buckling moment of an arch or of a beam is not necessarily correct for a laterally fixed arch or beam in uniform bending, in deference to a laterally pinned arch. When a laterally fixed arch is subjected to positive uniform bending, the effects of the prebuckling deformations decrease the buckling moment, and the reduction of the buckling moment increases with an increase of the included angle and of the out-of-plane slenderness ratio of the arch. When a laterally fixed arch is subjected to negative uniform bending, the effects of the prebuckling deformations decrease the absolute value of its buckling moment when the included angle is very small, but increase the absolute value of the buckling moment when the included angle exceeds a certain value. The increase in the absolute value of the buckling moment increases with an increase of the included angle and of the out-of-plane slenderness ratio of the arch. When the ratio of the out-of-plane to the in-plane second moments of area of the cross-section is not small, both the reduction of the buckling moment of a laterally fixed arch in positive uniform bending and the increase of the buckling moment of a laterally fixed arch in negative uniform bending, are substantial.     

Analytical studies for buckling and vibration of weld-bonded beam shells of rectangular cross-section

An approach to analytical solution is presented for vibration and buckling of thin-walled tubular beam shells typical of automotive structures, which are fabricated by joining sheet metal stampings along the two longitudinal edges with periodic spot welds, adhesive bonding, or combination of spot welds and bonding, known as weld bonding. Solutions are obtained for such beam shells of rectangular cross-section with two opposite ends simply supported. The beam shell is modeled as an assembly of the constituent walls and Levy-type formulation is used to obtain a series solution for the transverse displacement of each of the walls. The challenge of expressing the discrete point support conditions at the spot welds by a continuous function is addressed using the flexibility function approach used in literature. The flexibility function, used earlier to represent the flexibility distribution along weld-bonded edges of rectangular plates with periodic spot welds, is used here. The characteristic equations are obtained by satisfying the displacement, slope, shear, and moment equilibrium at the mating edges of the walls including the two weld-bonded edges and the compatibility conditions at the spot-weld locations. This approach to analytical solution, described here for thin-walled beam shells of rectangular cross-section, can be suitably adopted for more general cross-sections and joints along non-symmetric edges. A parametric study is undertaken to show the effect of aspect ratio of the beam shell, adhesive joint parameters, and the number of spot welds on the elastic buckling loads and the natural frequencies. Such parametric studies can be of use to designers in arriving at an optimal joint configuration of weld-bonded beam shells from buckling and vibration considerations.     

Generalized buckling analysis of laminated plates with random material properties using stochastic finite elements

A generalized layer-wise stochastic finite element formulation is developed for the buckling analysis of both homogeneous and laminated plates with random material properties. The pre-buckled stresses are considered in the derivation of geometric stiffness matrix and the effect of variation in these stresses on the mean and coefficient of variation of buckling strength is studied. The mean buckling strength of plates under uniform stress assumption exactly matches with those reported in the literature. However, it is shown that the actual mean buckling strength of plates can be significantly different based on the pre-buckled stress analysis which depends on boundary constraints, principal material directions, aspect and thickness ratios of plates. The statistics of buckling strength is determined using a Taylor series expansion based mean centered first order perturbation technique. The stochastic finite element solutions obtained using layer-wise plate theory is also validated with analytical solutions presented in this paper. Parametric studies are conducted for different aspect ratios, ply orientations and boundary conditions.