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2011 | Buch

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The Finite Element Analysis of Shells - Fundamentals

verfasst von: Dominique Chapelle, Klaus-Jürgen Bathe

Verlag: Springer Berlin Heidelberg

Buchreihe : Computational Fluid and Solid Mechanics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book presents a modern continuum mechanics and mathematical framework to study shell physical behaviors, and to formulate and evaluate finite element procedures. With a view towards the synergy that results from physical and mathematical understanding, the book focuses on the fundamentals of shell theories, their mathematical bases and finite element discretizations. The complexity of the physical behaviors of shells is analysed, and the difficulties to obtain uniformly optimal finite element procedures are identified and studied. Some modern finite element methods are presented for linear and nonlinear analyses. In this Second Edition the authors give new developments in the field and - to make the book more complete - more explanations throughout the text, an enlarged section on general variational formulations and new sections on 3D-shell models, dynamic analyses, and triangular elements. The analysis of shells represents one of the most challenging fields in all of mechanics, and encompasses various fundamental and generally applicable components. Specifically, the material presented in this book regarding geometric descriptions, tensors and mixed variational formulations is fundamental and widely applicable also in other areas of mechanics.

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Inhaltsverzeichnis

Frontmatter

1. Introduction

Abstract

In this chapter, we briefly discuss shell structures – noting also that, actually, the analysis of shell structures gave the impetus for the development of finite element procedures – and we summarize the general approach of analysis of a shell problem. We then give the objectives of this book; namely, to present fundamentals regarding physical considerations, mathematical models and modern finite element procedures for the analysis of shells.

Dominique Chapelle, Klaus-Jürgen Bathe

2. Geometrical Preliminaries

Abstract

The description of the geometry is essential for the definition of a shell structure. Our objective in this chapter is to survey the main geometrical concepts, to introduce the related notation and to recall some essential results that will be needed in this book.

Dominique Chapelle, Klaus-Jürgen Bathe

3. Elements of Functional and Numerical Analysis

Abstract

A deeper understanding of finite element methods, and the development of improved finite element methods, can only be achieved with an appropriate mathematical and numerical assessment of the proposed techniques. The basis of such an assessment rests on identifying whether certain properties are satisfied by the finite element scheme and these properties depend on the framework within which the finite element method has been formulated.

In this chapter we first review fundamental concepts of functional analysis, and then present different basic frameworks of variational formulations and finite element discretizations that we will use in the later chapters for shell solutions. For completeness, we prove the stability and convergence properties of the abstract finite element discretizations for each of the frameworks of variational formulations considered. This chapter therefore provides the foundation used for the later assessment of the reliability and effectiveness of shell finite element schemes.

Dominique Chapelle, Klaus-Jürgen Bathe

4. Shell Mathematical Models

Abstract

In this chapter we describe and analyse the linear shell models that we consider in this book. We first describe the fundamental shell kinematics used. Then we discuss the “basic shell model” which is implicitly employed in general finite element solutions and from which other classical shell and plate models can be derived. We summarize the shell models that we call the “shear-membrane-bending model” and the “membrane-bending model”, and introduce the proper mathematical framework in which they define well-posed problems. As special cases of these shell models we obtain well-known plate models.

Dominique Chapelle, Klaus-Jürgen Bathe

5. Asymptotic Behaviors of Shell Models

Abstract

Implicit in the concept of a “shell” is the idea that the thickness is “small” compared to the other two dimensions. In practice, it is not unusual to deal with structures for which the thickness is smaller by several orders of magnitude, in which case the shell is said to be “thin” (consider, for example, the shell body of a motor car). Considering the role of the thickness parameter t in the shell models that we presented in the previous chapter (see for example Eqs. (4.36) and (4.51)), with different powers of t in the bilinear terms on the left-hand side, it is essential to determine how the mathematical properties and physical behaviors of the models are affected when this parameter becomes small.

Dominique Chapelle, Klaus-Jürgen Bathe

6. Displacement-Based Shell Finite Elements

Abstract

In this chapter, we describe and analyze the main strategies that have been proposed and used to formulate displacement-based finite element procedures for shells. By displacement-based we mean that the finite element solution is obtained by directly applying the variational principle in the finite element space which discretizes the space of admissible displacements for the structure. In particular, this implies that no “numerical trick” – such as reduced integration – is used in the formulation.

Dominique Chapelle, Klaus-Jürgen Bathe

7. Influence of the Thickness in the Finite Element Approximation

Abstract

The influence of the thickness in the finite element analysis of thin structures is a crucial issue, as it is deeply interrelated with the motivation of modeling a 3D continuum as a shell in engineering. Why, indeed, should we use shell models and finite elements – instead of 3D models – to analyze a given structure? The answer to this question seems obvious: firstly, the use of a shell model is to reduce the analysis cost, and secondly, the use of the shell model is to reduce the complexity of the analysis including the interpretation of the results for engineering design. Clearly, the motivation to use shell models rests upon the fact that shell mathematical models and finite elements incorporate kinematical assumptions pertaining to the displacement distribution across the thickness of the structure, see previous chapters.

Dominique Chapelle, Klaus-Jürgen Bathe

8. Towards the Formulation of Effective General Shell Elements

Abstract

In Chapter 7 we discussed the difficulties encountered in the formulation of reliable and effective shell elements. These difficulties are summarized in the synopsis of Figure 8.1 in correspondence with the various types of shell asymptotic behaviors that can be encountered, as addressed in Chapter 5. The objective of the present chapter is to propose some strategies to evaluate shell finite element discretizations in the search for improved schemes. With general analytical proofs not available for the convergence behavior, the numerical assessment is a key ingredient in these strategies. As an example we present the formulation of the MITC shell elements and demonstrate how the numerical assessment of these elements can be performed.

Dominique Chapelle, Klaus-Jürgen Bathe

9. On the Nonlinear Analysis of Shells

Abstract

The nonlinear analysis of shells is today clearly a very large field, in which much research and development has taken place, so that at present many nonlinear analyses can be performed with confidence in engineering practice, see for example (Bathe, 1999, 2001a; Ibrahimbegović & Krätzig, 2002).

Our objective in this chapter is to merely outline the process of nonlinear shell analysis, and to thus indicate that all the theory regarding the fundamentals of shell analysis presented in the previous chapters is directly applicable to the nonlinear analysis of shells as well.

Dominique Chapelle, Klaus-Jürgen Bathe

Backmatter

Titel: The Finite Element Analysis of Shells - Fundamentals
verfasst von: Dominique Chapelle
Klaus-Jürgen Bathe
Copyright-Jahr: 2011
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-642-16408-8
Print ISBN: 978-3-642-16407-1
DOI: https://doi.org/10.1007/978-3-642-16408-8

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