Charles W. Bert

Differential Quadrature Method in Computational Mechanics: A Review

The differential quadrature method is a numerical solution technique for initial and/or boundary problems. It was developed by the late Richard Bellman and his associates in the early 70s and, since then, the technique has been successfully employed in a variety of problems in engineering and physical sciences. The method has been projected by its proponents as a potential alternative to the conventional numerical solution techniques such as the finite difference and finite element methods. This paper presents a state-of-the-art review of the differential quadrature method, which should be of general interest to the computational mechanics community.

Computational Models for Sandwich Panels and Shells

The focus of this review is on the hierarchy of computational models for sandwich plates and shells, predictor-corrector procedures, and the sensitivity of the sandwich response to variations in the different geometric and material parameters. The literature reviewed is devoted to the following application areas: heat transfer problems; thermal and mechanical stresses (including boundary layer and edge stresses); free vibrations and damping; transient dynamic response; bifurcation buckling, local buckling, face-sheet wrinkling and core crimping; large deflection and postbuckling problems; effects of discontinuities (eg, cutouts and stiffeners), and geometric changes (eg, tapered thickness); damage and failure of sandwich structures; experimental studies; optimization and design studies. Over 800 relevant references are cited in this review, and another 559 references are included in a supplemental bibliography for completeness. Extensive numerical results are presented for thermally stressed sandwich panels with composite face sheets showing the effects of variation in their geometric and material parameters on the accuracy of the free vibration response, and the sensitivity coefficients predicted by eight different modeling approaches (based on two-dimensional theories). The standard of comparison is taken to be the analytic three-dimensional thermoelasticity solutions. Some future directions for research on the modeling of sandwich plates and shells are outlined.

Two new approximate methods for analyzing free vibration of structural components

Two approximate methods, which have not previously been used for structural dynamics problems, are applied to the free vibration analysis of various structural components. The first method is a new version of the complementary energy method. It is shown to be considerably more accurate than the conventional Rayleigh and Rayleigh-Schmidt methods when applied to spatially one-dimensional free vibration problems: prismatic and tapered bars, prismatic beams, and axisymmetric motion of circular membranes. The second method is the differential quadrature method introduced by Bellman and his associates. It is applied successfully here to all of the problems mentioned plus square membranes and circular and square plates.

Effect of shear deformation on vibration of antisymmetric angle-ply laminated rectangular plates

Abstract The title problem is solved in closed form for the case of all edges simply supported. A displacement formulation of the heterogeneous shear-deformable plate theory originated by Yang, Norris and Stavsky is used. Material properties typical of a highly directional composite material (high-modulus graphite/epoxy) are used and numerical results are presented showing the parametric effects of aspect ratio, length/thickness ratio, number of layers, and lamination angle. The effects of deleting rotatory inertia and in-plane inertia, singly and in combination, were also investigated. The information presented should be useful to composite-structure designers, to researchers seeking to obtain better correlation between theory and experiment, and to numerical analysts in checking out finite-element programs.

Material damping: An introductory review of mathematic measures and experimental technique†

This paper begins with a discussion of various mathematical models which have been proposed to represent the damping behavior of solid materials, with emphasis on their usefulness in structural dynamic analyses. Next the various measures of damping which have been proposed for homogeneous materials are presented, along with derivations of interrelationships which exist among them. A wide variety of experimental techniques have been used to determine the damping characteristics; these are outlined in the next section. The paper concludes with a detailed discussion of excitation and data-reduction techniques for modal response of complicated structures. In an appendix (Appendix I) are presented two new extensions of the Kennedy-Pancu technique to structures made of materials represented by more realistic mathematical models than heretofore used.

Free vibrations of laminated rectangular plates analyzed by higher order individual-layer theory

Abstract A higher order plate theory is used in each individual layer to determine the natural frequencies and the relative stress and deflection distributions through the thickness of simply supported rectangular plates. This theory approximates the in-plane and normal displacements by third and second order functions of the thickness co-ordinate, respectively. The theory satisfies the displacement compatibility and stress equilibrium conditions along the interfaces between adjacent layers. Numerical results are calculated for the cases of isotropic and orthotropic homogeneous plates and for symmetric and non-symmetric cross-ply laminates. The present theory predicts more modes of free vibration than other approximate theories. Also, the relative stress and deflection distributions obtained by the present higher order theory are in very good agreement with results by three-dimensional elasticity theory.

DIFFERENTIAL QUADRATURE FOR STATIC AND FREE VIBRATION ANALYSES OF ANISOTROPIC PLATES

Abstract The differential quadrature method is used to analyse the deflection, buckling and free vibration behavior of anisotropic rectangular plates under various boundary conditions. The roots of Chebyshev polynomials are used to obtain grid-point locations and a new approach is used to apply boundary conditions. Results compare very well with existing numerical data, and less computational effort is required for the problems considered.

Static analysis of structures by the quadrature element method (QEM)

Abstract The differential quadrature method (DQM) is an alternative discrete approach to solving directly the governing equations of engineering and mathematical physics. Since the DQM does not require the derivation of the weak forms of the governing equations like the FEM, it can greatly reduce formulation efforts in higher order approximation. The DQM has been applied in the past to the analyses of various single structural components such as bars, beams, membranes and plates. All of the component analyses yielded good to excellent results. However, previous studies were limited to simple geometries and simple boundary conditions due to the limitation of using global basis functions. In the present study, a domain decomposition technique for the DQM is proposed to analyse truss and frame structures where the whole structural domain is represented by a collection of simple element subdomains connected together at specific nodal points. This method is named the quadrature element method (QEM).

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.

Simplified Analysis of Static Shear Factors for Beams of NonHomogeneous Cross Section

Using a simple mechanics-of-materials approach, a general expression is derived for the static correction factor for transverse shear in a beam having arbitrary nonhomogeneity in its cross section. The resulting expression is consistent with the variationally derived results of Reissner’s analysis t I ] when the latter are reduced from the two-dimensional (plate) case to the one-dimensional (beam) one. Also, when applied to an unsymmetric laminate considered by Whitney [2], the numerical result obtained is identical with his, even though the method of derivation and resulting mathematical form are entirely different.