Xinwei Wang

Static and free vibrational analysis of rectangular plates by the differential quadrature element method

The differential quadrature (DQ) element method proposed by Wang and Gu in 1997 has been extended to analyse rectangular plate problems. The methodology is worked out in detail and some numerical examples are given.

Buckling analyses of anisotropic plates and isotropic skew plates by the new version differential quadrature method

A new version differential quadrature method (DQM) has been proposed to obtain buckling loads of thin anisotropic rectangular and isotropic skew plates. The essential difference from the old version DQM is the introduction of two degrees of freedom for boundary points and from the existing differential quadrature element method (DQEM) is the determination of the weighting coefficients. The methodology is worked out in detail and a variety of buckling problems shown slow convergence earlier by Rayleigh-Ritz method with beam functions, including isotropic skew plates with various skew angles and anisotropic rectangular plates with simply supported or clamped boundary conditions, are solved by the proposed DQM. Numerical results indicate that fast convergence is achieved and excellent results are obtained by the proposed DQM.

Differential quadrature for buckling analysis of laminated plates

Two new sets of grid points are proposed for applying differential quadrature (DQ) to the analysis of structural problems. The accuracy and convergence of differential quadrature for buckling analysis of laminated plates are discussed in this paper. A variety of buckling problems, including composite laminated plates with various boundary constrains under uniaxial, biaxial, and combined uniaxial and shear loadings, are solved by the DQ method with the proposed grid spacings. Numerical results indicate that excellent results are obtained by the DQ method on problems sensitive to grid spacings.

Differential Quadrature Method

A variety of different quadrature (DQ) formulations exist in literatures. This has often caused confusion for researchers and engineers and led to a difficulty in making a choice of a different quadrature method (DQM) for solving practical problems. This chapter presents the basic principle of the DQM and summarizes various DQ formulations, including the original DQM, the modified DQM, harmonic differential quadrature method (HDQM), local adaptive differential quadrature method (LaDQM), and the DQ-based time integration scheme. Existing explicit formulas to compute the weighting coefficients are given. Various grid distributions are summarized and their discrete error is briefly discussed. Some recommendations are made. Although Grid III is the most widely used grid spacing in literature, however, research shows that Grid V is the most reliable grid spacing and thus recommended, especially for dynamic analysis by using the DQM. The LaDQM is recommended if the number of grid points is large.

Accurate dynamic analysis of functionally graded beams under a moving point load

ABSTRACT Based on the physical neutral surface, an N-node novel weak form quadrature beam element is proposed and the explicit formulas for computing the stiffness and mass matrices are given. The proposed element is then used to analyze the dynamic behavior of the functionally graded material (FGM) beams under a moving point load. Both elasticity modulus and mass density vary exponentially across the thickness. Investigations show that the maximum dynamic magnification factors are independent of the power-law exponent k at a fixed nondimensional parameter α. This finding may be useful in design and engineering applications.

Differential Quadrature Element Method

To overcome the difficulty existing in the original DQM, such as the difficulties to apply the multiple boundary conditions and to deal with the load and geometric discontinuities, one type of the DQ-based element methods, the strong-form differential quadrature element method (DQEM), is proposed. The basic principle of the DQEM and two different formulations are presented. One formulation is based on the Hermite interpolation and the other is based on the Lagrange interpolation. For the formulation of a DQ rectangular plate element, the mixed Hermite interpolation with Lagrange interpolation can also be used. Assemblage procedures are given and several examples are worked out in detail for illustrations. Numerical results show that the DQEM can yield accurate results for beams and rectangular plates under discontinuous loads and beams with step changes in cross-sections. The DQEM can also be used for analysis of frame structures.

Quadrature Element Method

To overcome the difficulty existing in the DQM, such as the difficulties for applying the multiple boundary conditions and to dealing with the load and geometric discontinuities, another type of the DQ-based element methods, the weak-form differential quadrature element method (QEM), is presented. Different from the strong-form DQEM, the formulations of the QEM are the same as the high-order finite element method (FEM). However, differences from the FEM do exist and are illustrated. A simple way is proposed to formulate quadrature elements with nodes other than Gauss–Lobatto–Legendre (GLL) nodes. Thus, the explicit formulas to calculate the weighting coefficients can be directly used during formulations of the stiffness matrix. This makes the formulation and implementation of an N node quadrature element simpler and more flexible, since explicit expressions of the derivative of the shape functions are not needed. Various quadrature elements are presented.