Vibration and Nonlinear Dynamics of Plates and Shells - Applications of Flat Triangular Finite Elements E-Book
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- Herausgeber: Bentham Science Publishers
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
This e-book focuses on the vibrational and nonlinear aspects of plate and shell structure dynamics by applying the finite element model. Specifically, shell finite elements employed in the computational studies included in this book are the mixed formulation based lower order flat triangular shell finite elements. Topics in the book are covered over nine chapters, including the theoretical background for the vibration analysis of plates and shells, vibration analysis of plate structures, shells with single curvature, shells with double curvatures, and box structures (single-cell and double-cell) and the theoretical development for the nonlinear dynamic analysis of plate and shell structures. In addition to presenting the steps in the derivations of the consistent element stiffness and mass matrices, constitutive relations of elastic materials and elasto-plastic materials with isotropic strain hardening, yield criterion, return mapping, configuration and stress updating strategies, and numerical algorithms are presented and discussed. The book is a suitable reference for advanced undergraduates and post-graduate level engineering students, research engineers, and scientists working in the field of applied physics and engineering.
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Seitenzahl: 177
Veröffentlichungsjahr: 2014
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FOREWORD
Engineering Dynamics and Vibration are two foundation areas in many engineering fields. They are fundamental to the analysis and design of many dynamic engineering systems. The present volume, Vibration and Nonlinear Dynamics of Plates and Shells: Applications of Flat Triangular Finite Elements is a timely and unique addition to the literature in the two foundation areas in engineering. The authors, Professors Meilan Liu and C.W. Solomon To, have a combined experience of more than fifty years in the engineering dynamics and vibration analysis of plate and shell structures. The present volume has included the two foundation areas in a single book that applies the lower order flat triangular shell finite elements originated from their early research. Its main and important feature is the combination of vibration and nonlinear dynamics of plates and shells in a relatively comprehensive treatment employing the finite element method. Another feature of the present volume is the treatment of boxed or cell structures. It is believed that anyone working in the analysis and design of dynamic engineering systems will find it informative and an excellent reference.
Series Preface
The fields of engineering dynamics and vibration have advanced and expanded at an extremely impressive rate due, perhaps, to their high demand in applications in modern technologies. The main objectives of this series are three folds. The first objective is to be complimentary to existing books and handbooks in the fields of engineering dynamics and vibration. The second objective is to provide a common and single venue for the publication of books in both engineering dynamics and vibration fields. Books in the emerging area of engineering dynamics and vibration of nano-structural systems and devices are included. The third objective of the present series is to provide books suitable for use by advanced undergraduates and post-graduate level engineering students, research engineers, and applied scientists.
The series aims at keeping abreast of the modern developments and applications in the fields. Whenever new areas of development and application arise it is the intent of this series to invite leaders in the field to publish their work.
Volume Preface
The germ of this eBook was grown from the interests of the authors in engineering vibration and dynamics. The theoretical background and computational techniques adopted throughout this eBook were based on part of the doctoral degree thesis of the first author. While the fields of computational engineering dynamics and vibration are vast, and their applications have virtually no limits, the scope of the present eBook is confined to vibration and nonlinear dynamics of plates and shells. For computational studies, the versatile finite element method alone provides a multitude of impressive publications, such as the pioneered work, Finite Element Handbook published in 1987 by McGraw-Hill (Editor-in-Chief, H. Kardestunder and Project Editor, D. H. Norrie). Subsequently, there are various handbooks in finite element methods available in the literature. Thus, the subject matter and topics included in the present eBook are focused on the vibration and nonlinear dynamics aspects of plate and shell structures. While finite element analysis of plates and shells is generally regarded as a mature technology it seems that no single book that covers both vibration and nonlinear dynamics by applying the finite element method is currently available. Consequently, the present volume is a modest attempt to provide such a book, albeit a relatively limited one. The particular shell finite elements employed in the computational studies reported in this book are the mixed formulation based lower order flat triangular shell finite elements.
The present book has nine chapters. The brief introduction is included in Chapter 1. Chapter 2 is concerned with the theoretical background for the vibration analysis of plates and shells. In particular, the mixed formulation based three-node flat triangular shell elements are presented in this chapter. Vibration analysis of plate structures is considered in Chapter 3. In the latter the square, circular, and skew plates as well as membrane are treated. Vibration analysis of shells with single curvature is presented in Chapter 4 in which cylindrical panel with rectangular and trapezoidal projections, Scordelis-Lo roof, and cylindrical shell clamped at both ends with its effect of aspect ratio are included. Chapter 5 is concerned with the vibration analysis of shells of double curvatures. These structures include the spherical caps, spherical panel of square projection, hemispherical panel, and clamped hemispherical shell. Chapter 6 deals with the vibration analysis of box structures. Single-cell and double-cell box structures are studied.
Chapter 7 provides the theoretical development for the nonlinear dynamic analysis of plate and shell structures. The focus is on the mixed formulation based three-node flat triangular shell elements for nonlinear dynamic analysis. Aside from presenting the steps in the derivations of the consistent element stiffness and mass matrices, constitutive relations of elastic materials and elasto-plastic materials with isotropic strain hardening, yield criterion, return mapping, configuration and stress updating strategies, and numerical algorithms are presented and discussed. Nonlinear dynamics of flat-surface structures are treated in Chapter 8. The cantilevered, circular, and square plates under uniform pressures, rectangular plate subjected to a center load, and a cubic tube under internal and external pressures are considered in this chapter. Chapter 9 is concerned with the nonlinear dynamics of curved-surface structures. The cylindrical panel under a central point load and under a uniform pressure, hemispheres with and without a central hole and under alternating point loads, clamped and hinged spherical caps subjected to apex point loads and under pressures are examined in this chapter. Finally, it should be mentioned that no attempt has been made to include the important subject of chaotic dynamics of plate and shell structures applying the lower order flat triangular shell finite elements.
CONFLICT OF INTEREST
We, the authors, confirm that there is no conflict of interest in regard to contents of this book.
ACKNOWLEDGEMENT
There is none to declare.
Introduction
Abstract
This chapter consists of three sections. Objectives and scope of the book are given in Section 1.1. Section 1.2 outlines the organization. Notes on computer programming are included in the last section.
1.1.. OBJECTIVES AND SCOPE
There are numerous examples of engineering structures that are composed of shell segments, such as the body of an airplane, the hull of a ship or submarine, the roof of a domed structure, a pressure vessel, and so on. A shell structure, in general, refers to a body with one dimension much smaller than the other two. That is, its thickness is much smaller than the size of the curved mid-surface. The mid-surface can be of single curvature and double curvature. Examples of former are cylinders and cones. The latter may include spherical caps, for example. Special cases of shell structures include, plates whose mid-surfaces are flat, and beams whose length is much larger than the width and thickness.
The curvatures provide shells with significant advantages over plates and beams, making shell structures perhaps the most efficient light weight structures in terms of load-carrying capacity. However, the curvatures also pose challenges for the modeling and analysis of shell structures. As Ref. [1.1] pointed out, since the mid-1960s, “the published literature on modeling of plates and shells in the linear and non-linear regimes and their application to dynamic or vibration analysis of structures has grown extensively. There has been a tremendous interest on the part of researchers with sufficiently large amount of resources devoted to the subject, and there continues to be innovative activity in computational shell mechanics. In the last three decades, numerous theoretical models have been developed and applied to various practical circumstances. It may be fair to state that no single theory has proven to be general and comprehensive enough for the entire range of applications”.
In this book, the development of mixed formulation based, low-order, three-node flat triangular shell elements suitable for the linear and nonlinear analysis of thin to moderately thick shells is presented, together with their applications in the vibration characteristics and dynamic responses of complicated shell structures. It is the authors’ hope that this book, in a very small way, continues the “innovative activities in computational shell mechanics”.
Although as much details as needed regarding the development of the mixed formulation based three-node flat triangular shell elements are presented in two chapters, Chapters 2 and 7, this book is intended for those with some background in finite element analysis and numerical algorithms. Some familiarity with nonlinear mechanics is also assumed. As a result, the fundamental of finite element method is omitted. Readers may refer to [1.2, 1.3] for such topic.
1.2.. ORGANIZATION OF BOOK
The book is organized into 9 chapters. Chapters 2 to 6 are concerned with the linear version of the mixed formulation based three-node flat triangular shell elements and their application in investigating the vibration characteristics of linear elastic structures. Specifically,
Chapter 2 presents the mixed formulation based three-node flat triangular shell elements within the context of linear analysis. It also examines issues such as rigid body modes, patch tests and mesh topologies;Chapter 3 is concerned with the vibration analysis of plate structures;Vibration characteristics of shells of single curvature and double curvatures are included in Chapters 4 and 5, respectively; andChapter 6 demonstrates the application of the shell elements to single-cell and double-cell box structures.The remaining chapters, Chapters 7 to 9, deal with the general nonlinear dynamic analysis of shell structures by the mixed formulation based three-node flat triangular shell elements. The nonlinear formulation is given in Chapter 7, which is followed by Chapter 8 on the nonlinear dynamics of plate and box structures. Nonlinear dynamics of structures of single curvature and double curvatures are presented in Chapter 9. For the latter chapters, geometrical nonlinearity due to large deformation, material nonlinearity due to elastic-plastic material behaviour, and various loading situations including non-conservative pressure loads are investigated.
1.3.. NOTES ON COMPUTER PROGRAMMING
The linear and nonlinear mixed formulation based three-node flat triangular shell elements were initially programmed in the Fortran language and incorporated in NONSAP [1.4] which was modified and implemented on a SGI workstation for the computational results reported in the doctoral degree thesis of the first author [1.5]. The digital computer program has since been rewritten in the personal computer (PC) based MATLAB system [1.6]. All computations involved in this book are performed in the MATLAB environment. Plots such as mode shapes and dynamic responses are generated by appropriate MATLAB functions.
At the present time, the shell element formulations and associated functions, such as mesh generation, applying boundary conditions, direct time integration schemes, Newton-Raphson method and its variants, Riks-Wempner arc-length method, plotting of mode shapes and time histories, and so on, are combined into a package, written by the first author for academic and research purpose only.
REFERENCES
[1.1]Yang H.T., Saigal S., Masud A., Kapania R.K.. A survey of recent shell finite elements.Int. J. Numer. Methods Eng.200047101127[1.2]Cook R.B., Malkus D.S., Plesha M.E.. Concepts and Applications of Finite Element Analysis.New YorkJohn Wiley & Sons1989[1.3]Zienkiewicz O.C., Taylor R.L.. The Finite Element Method, I and II.4th edNew YorkMcGraw-Hill1991[1.4]Bathe K.J., Wilson E.L., Iding R.H.. NONSAP, a structural analysis program for static and dynamic response of nonlinear systems.Report No. SESM 74-3.Berkeley, CaliforniaStructural Engineering Laboratory, University of California1974[1.5]Liu M.L.. “Response statistics of shell structures with geometrical and material nonlinearities”, Ph.D. thesis, The University of Western Ontario, London, ON, 1993..[1.6]The MathWorks Inc., MATLAB The Language of Technical Computing, Massachusetts, U.S.A..Mixed Formulation Based Three-Node Flat Triangular Shell Elements for Vibration Analysis
Abstract
In order to investigate the vibration characteristics and dynamic responses of complicated shell structures with geometrical and material nonlinearities, it is essential to formulate shell finite elements that are easy to use, accurate, effective, and applicable to thin as well as moderately thick shells. This chapter presents the development of the mixed formulation or hybrid strain based three-node flat triangular shell elements, with a particular emphasis on the linear analysis of thin to moderately thick shells. Section 2.1 gives a brief introduction and an outline of the features of the shell elements. Section 2.2 deals with the derivation of consistent stiffness and mass matrices of a particular element. In Section 2.3, results and discussions pertaining to rigid-body modes, patch test, and mesh topology are presented. Concluding remarks are given in Section 2.4.
