F.-L. Liu

Differential quadrature element method: a new approach for free vibration analysis of polar Mindlin plates having discontinuities

This paper presents the numerical development of the differential quadrature element method (DQEM) for free vibration analysis of the shear deformable plates in polar coordinates. This is an improvement on the global differential quadrature method (DQM). The formulations of the DQEM for polar plate vibration element are derived in detail. The convergence characteristics of the DQEM for solving the free vibration of polar plates are carefully examined. The reliability and flexibility of the DQEM are illustrated by solving several selected example polar plates having discontinuities which are not solvable directly by the DQM. The accuracy of the DQEM is evaluated and verified by comparing the present numerical results with the existing exact or approximate solutions, or the FEM solutions computed using the software package ANSYS (Version 5.3).

Free vibration analysis of Mindlin sector plates: Numerical solutions by differential quadrature method

Abstract In this paper, the free vibration analysis of moderately thick sector plates based on Mindlins first-order shear deformation theory is examined. The solutions to this problem are determined using the differential quadrature (DQ) method. For numerical evaluation, the DQ formulations for free vibration of the Mindlin sector plates in two-dimensional polar co-ordinate system are derived. To demonstrate the accuracy of the DQ method, convergence and comparison studies are carried out. In parametric study, the first eight frequency parameters are computed for sector plates with different boundary conditions, relative thickness ratios, and sector angles (30° ⩽ α ⩽ 360°). The effects of sector angle and relative thickness ratio on the frequency parameters are discussed.

Differential quadrature element method for static analysis of Reissner–Mindlin polar plates

Abstract In this paper, a new numerical technique, the differential quadrature element method (DQEM) , has been developed for static analysis of the two-dimensional polar Reissner–Mindlin plate in the polar coordinate system by integrating the domain decomposition method (DDM) with the differential quadrature method (DQM) . The detailed formulations for the sectorial DQEM plate bending element and the compatibility conditions between each element are presented. The convergence properties and the accuracy of the DQEM for bending of thick polar plates are investigated through a number of numerical computations. Consequently, the DQEM has been successfully applied to analyze several annular sector plates with discontinuous loading and boundary conditions and cutouts to illustrate the simplicity and flexibility of this method for solving Reissner–Mindlin plates in polar coordinate system which are not solvable directly using the differential quadrature method. The numerical results are verified by the existing exact solutions or the FEM solutions obtained using the software package ANSYS (Version 5.3) .

Static Analysis of Reissner-Mindlin Plates by Differential Quadrature Element Method

In this paper, a new numerical method, the differential quadrature element method has been developed for two-dimensional analysis of bending problems of Reissner-Mindlin plates, The basic idea of the differential quadrature element method is to divide the whole variable domain into several subdomains (elements) and to apply the differential quadrature method for each element. The detailed formulations for the differential quadrature element method and compatibility conditions between elements are presented. The convergent characteristics and accuracy of the differential quadrature element method are carefully investigated for the solution of the two-dimensional bending problems of Reissner-Mindlin plates. Finally, the differential quadrature element method is applied to analyze several bending problems of two-dimensional Reissner-Mindlin plates with different discontinuities including the discontinuous loading conditions, the mixed boundaries, and the plates with cutout. The accuracy and applicability of this method have been examined by comparing the differential quadrature element method solutions with the existing solutions obtained by other numerical methods and the finite element method solutions generated using ANSYS 5.3.

Vibration Analysis of Discontinuous Mindlin Plates by Differential Quadrature Element Method

A new numerical technique, the differential quadrature element method (DQEM), has been developed for solving the free vibration of the discontinuous Mindlin plate in this paper. By the DQEM, the complex plate domain is decomposed into small simple continuous subdomains (elements) and the differential quadrature method (DQM) is applied to each continuous subdomain to solve the problems. The detailed formulations for the DQEM and the connection conditions between each element are presented. Several numerical examples are analyzed to demonstrate the accuracy and applicability of this new method to the free vibration analysis of the Mindlin plate with various discontinuities which are not solvable directly using the differential quadrature method.

Differential cubature method for static solutions of arbitrarily shaped thick plates

This paper presents the first endeavor to exploit the differential cubature method as an accurate and efficient global technique for fundamental solutions of arbitrarily shaped thick plates. The method is examined here for its suitability for solving the boundary-value problem of thick plates governing by the first-order shear deformation theory. Using the method, the governing differential equations and boundary conditions are transformed into sets of linear algebraic equations. Boundary conditions are implemented through discrete grid points by constraining displacements, bending moments and rotations. Detailed discussion on the formulation and implementation of the method are presented. The applicability, efficiency and simplicity of the method are demonstrated through solving example plate problems of different shapes. The accuracy of the method is verified by direct comparison with the known values.

Rectangular thick plates on winkler foundation: differential quadrature element solution

Abstract This paper deals with the static analysis of homogenous isotropic rectangular plates on Winkler foundation on the basis of first-order shear deformation theory. An improved differential quadrature (DQ) method, called the differential quadrature element method (DQEM), has been developed for this analysis. The plates considered are subjected to a patch load or a concentrated line load, which are not solvable by the global DQ method. The convergence and comparison studies are carried out to establish the reliability of the DQEM results. Then the numerical results for different boundary conditions (i.e. SSSS, CCCC, S′S′S′S′ and SFSF) are presented showing the parametric effects of dimensions of loading area/line, relative thickness ratio and elastic foundation modulus on the deflection, bending and twisting moments, and shear forces at selected locations. Most of these data are new and due to the high accuracy of the DQ solution they can be useful for benchmarking future work.

Differential quadrature element method for buckling analysis of rectangular Mindlin plates having discontinuities

In this paper, a new solution approach, the differential quadrature element method is applied to the buckling analysis of discontinuous rectangular plates based on the Mindlin plate theory. The domain decomposition method is used to divide the solution domain into smaller elements according to the discontinuities contained in the plate. Then, differential quadrature procedures are applied to each element to formulate the discretized element governing equations. These discrete equations are then assembled into an overall equation system, using the compatibility conditions, and solved by a standard eigensolver. Detailed formulations for modeling of the plate and the compatibility conditions are derived. Convergence and comparison studies are carried out to examine the reliability and accuracy of the numerical solutions. Four rectangular Mindlin plates with different discontinuities (mixed boundary conditions and cracks) are analyzed to show the applicability and flexibility of the present methodology for solving a class of buckling problems. Due to the lack of published solutions for buckling of thick discontinuous plates and the high accuracy of the present approach, the solutions obtained may serve as benchmark values for further studies.

Static analysis of thick rectangular laminated plates : three-dimensional elasticity solutions via differential quadrature element method

This article presents the first endeavor to develop the differential quadrature element method for static solution of three-dimensional elasticity equations of thick rectangular laminated composite plates. The domain decomposition technique is employed to decompose a laminated plate into elements according to material layers. The differential quadrature (DQ) method is then applied to each element where the material properties are continuous to form the element weighting coefficient matrix and element force vector. The discretized element weighting coefficient matrices and element force vectors are assembled together to form the global weighting coefficient matrix and global force vector for the whole plate using connection conditions. The solution for the entire plate is obtained by solving the final algebraic equation system. Detailed formulations and numerical procedures are presented and the convergence characteristics of the method are investigated. The numerical results are then compared, where possible, with the analytical solutions to verify the present solutions. Consequently, some new numerical results are computed and analyzed using the present numerical method for laminated rectangular plates with different boundary conditions, which are not solvable directly by the global DQ method.