Preface. Symbols. Chapter 1. Introduction: Basic Concepts and Terminology. Chapter 2. Vibration of Discrete Systems: Brief Review. Chapter 3. Derivation of Equations: Equilibrium Approach. Chapter 4. Derivation of Equations: Variation Approach. Chapter 5. Derivation of Equations: Integral Equation Approach. Chapter 6. Solution Procedure: Eigenvalue and Modal Analysis Approach. Chapter 7. Solution Procedure: Integral Transform Methods. Chapter 8. Transverse Vibration of Strings. Chapter 9. Longitudinal Vibration of Bars. Chapter 10. Torsional Vibration of Shafts. Chapter 11. Transverse Vibration of Beams. Chapter 12. Vibration of Circular Rings and Curved Beams. Chapter 13.Vibration of Membranes. Chapter 14. Transverse Vibration of Plates. Chapter 15. Vibration of Shells. Chapter 16. Elastic Wave Propagation. Chapter 17. Approximate Analytical Methods. A. Basic Equations of Elasticity. B. Laplace and Fourier Transforms. Index.

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Vibration of continuous systems

In the context of shape optimization, we seek minimizers of the sum of the elastic compliance and of the weight of a solid structure under specified loading. This problem is known not to be well-posed, and a relaxed formulation is introduced. Its effect is to allow for microperforated composites as admissible designs. In a two-dimensional setting the relaxed formulation was obtained in [6] with the help of the theory of homogenization and optimal bounds for composite materials. We generalize the result to the three dimensional case. Our contribution is twofold; first, we prove a relaxation theorem, valid in any dimensions; secondly, we introduce a new numerical algorithm for computing optimal designs, complemented with a penalization technique which permits to remove composite designs in the final shape. Since it places no assumption on the number of holes cut within the domain, it can be seen as a topology optimization algorithm. Numerical results are presented for various two and three dimensional problems.

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Shape optimization by the homogenization method

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.

Differential Quadrature Method in Computational Mechanics: A Review

Vibration of cracked structures: A state of the art review

This paper gives a historical review of the theories that have been developed for the analysis of multilayered structures. Attention has been restricted to the so-called Zig-Zag theories, which describe a piecewise continuous displacement field in the plate thickness direction and fulfill interlaminar continuity of transverse stresses at each layer interface. Basically, plate and shell geometries are addressed, even though beams are also considered in some cases. Models in which the number of displacement variables is kept independent of the number of constitutive layers are discussed to the greatest extent. Attention has been restricted to those plate and shell theories which are based on the so-called method of hypotheses or axiomatic approach in which assumptions are introduced for displacements and/or transverse stresses. Mostly, the work published in the English language is reviewed. However, an account of a few articles originally written in Russian is also given. The historical review conducted has led to the following main conclusions. 1! Lekhnitskii ~1935! was the first to propose a Zig-Zag theory, which was obtained by solving an elasticity problem involving a layered beam. 2! Two other different and independent Zig-Zag theories have been singled out. One was developed by Ambartsumian ~1958!, who extended the well-known Reissner-Mindlin theory to layered, anisotropic plates and shells; the other approach was introduced by Reissner ~1984!, who proposed a variational theorem that permits both displacements and transverse stress assumptions. 3 ! On the basis of historical considerations, which are detailed in the paper, it is proposed to refer to these three theories by using the following three names: Lekhnitskii Multilayered Theory, ~LMT!, Ambartsumian Multilayered Theory ~AMT!, and Reissner Multilayered Theory ~RMT!. As far as subsequent contributions to these three theories are concerned, it can be remarked that: 4! LMT although very promising, has almost been ignored in the open literature. 5! Dozens of papers have instead been presented which consist of direct applications or particular cases of the original AMT. The contents of the original works have very often been ignored, not recognized, or not mentioned in the large number of articles that were published in journals written in the English language. Such historical unfairness is detailed in Section 3.2. 6! RMT seems to be the most natural and powerful method to analyze multilayered structures. Compared to other theories, the RMT approach has allowed from the beginning development of models which retain the fundamental effect related to transverse normal stresses and strains. This review article cites 138 references. @DOI: 10.1115/1.1557614#

Historical review of Zig-Zag theories for multilayered plates 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.

Computational Models for Sandwich Panels and Shells

1.Introduction to Composite Materials 2. Macrochemical Behavior of a Lamina 3.Micromechanical Behavior of a Lamina 4.Macromechanical Behavior of a Laminate 5.Bending, Buckling, and Vibration of Laminated Plates 6.Other Analysis and Behavior Topics 7.Introduction to Design of Composite Structures Appendix A.Matrices and Tensors Appendix B.Maxima and Minima of Functions of a Single Variable Appendix C.Typical Stress-Strain Curves Appendix D.Governing Equations for Beam Equilibrium and Plate Equilibrium, Buckling, and Vibration Index

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Mechanics of composite materials

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.

Two new approximate methods for analyzing free vibration of structural components

The numerical technique of differential quadrature for the solution of linear and non-linear partial differential equations, first introduced by Bellman and his associates, is applied to the equations governing the deflection and buckling behaviour of one- and two-dimensional structural components. Separate transformations are used for higher-order derivatives, as suggested by Mingle, thus extending the method to treat fourth-order equations and to include multiple, boundary conditions in the respective co-ordinate directions. Results are obtained for various boundary and loading conditions and are compared with existing exact and numerical solutions by other methods. The application of differential quadrature to this class of problems is seen to lead to accurate results with relatively small computational effort.

Charles W. Bert

Papers

Differential Quadrature Method in Computational Mechanics: A Review

Computational Models for Sandwich Panels and Shells

Mechanics of composite materials

Two new approximate methods for analyzing free vibration of structural components

Application of differential quadrature to static analysis of structural components