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2013 | Buch

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Solid Mechanics

A Variational Approach, Augmented Edition

verfasst von: Clive L. Dym, Irving H. Shames

Verlag: Springer New York

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

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Über dieses Buch

Solid Mechanics: A Variational Approach, Augmented Edition presents a lucid and thoroughly developed approach to solid mechanics for students engaged in the study of elastic structures not seen in other texts currently on the market. This work offers a clear and carefully prepared exposition of variational techniques as they are applied to solid mechanics. Unlike other books in this field, Dym and Shames treat all the necessary theory needed for the study of solid mechanics and include extensive applications. Of particular note is the variational approach used in developing consistent structural theories and in obtaining exact and approximate solutions for many problems. Based on both semester and year-long courses taught to undergraduate seniors and graduate students, this text is geared for programs in aeronautical, civil, and mechanical engineering, and in engineering science. The authors’ objective is two-fold: first, to introduce the student to the theory of structures (one- and two-dimensional) as developed from the three-dimensional theory of elasticity; and second, to introduce the student to the strength and utility of variational principles and methods, including briefly making the connection to finite element methods. A complete set of homework problems is included.

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Inhaltsverzeichnis

Frontmatter

1. Theory of Linear Elasticity

Abstract

In much of this text we shall be concerned with the study of elastic bodies. Accordingly, we shall now present a brief treatment of the theory of elasticity. In developing the theory we shall set forth many concepts that are needed for understanding the variational techniques soon to be presented.

Clive L. Dym, Irving H. Shames

2. Introduction to the Calculus of Variations

Abstract

In dealing with a function of a single variable, y = f (x), in the ordinary calculus, we often find it of use to determine the values of x for which the function y is a local maximum or a local minimum. By a local maximum at position x ₁, we mean that f at position x in the neighborhood of x ₁ is less than f (x ₁) (see Fig. 2.1). Similarly for a local minimum of f to exist at position x ₂ (see Fig. 2.1) we require that f (x) be larger than f (x ₂) for all values of x in the neighborhood of x ₂. The values of x in the neighborhood of x ₁ or x ₂ may be called the admissible values of x relative to which x ₁ or x ₂ is a maximum or minimum position.

Clive L. Dym, Irving H. Shames

3. Variational Principles of Elasticity

Abstract

This chapter is subdivided into four parts. In part A we shall set forth certain key principles which are related to or directly involve variational approaches. Specifically we set forth the principles of virtual work and complementary virtual work, and from these respectively derive the principles of total potential energy and total complementary energy. Reissner’s principle is also derived in this part of the chapter. Serving to illustrate certain of the aforementioned principles, we have employed a number of simple truss problems. These problems also serve the purpose of the beginnings of our work in this text in structural mechanics. In Part B of the chapter we concentrate on formulations which are derivable from the principles of part A and which are particularly useful for the study of structural mechanics. These are the well-known Castigliano theorems. We continue the discussion of trusses begun in Part A, thereby presenting in this chapter a reasonably complete discussion of simple trusses. In the following chapter we extend the structural considerations begun here to beams, frames, and rings.

Clive L. Dym, Irving H. Shames

4. Beams, Frames and Rings

Abstract

In the previous chapter, while developing as the primary effort certain variational principles of mechanics, we entered into a discussion of trusses in order both to illustrate certain aspects of the theory and to present a discussion of the most simple class of structures. We could take on this dual task at this stage because the stress and deformation of any one single member of a truss is a very simple affair. That is, the only stress on any section (away from the ends)

Clive L. Dym, Irving H. Shames

5. Torsion

Abstract

In this chapter we shall consider the Saint Venant theory of torsion for uniform prismatic elastic rods loaded by twisting couples at the ends of the rod (see Fig. 5.1). You will recall that for the special case of a circular cross section, it is assumed in strength of materials (and shown valid in the theory of elasticity) that cross sections of the rod merely rotate as rigid surfaces under the action of the twisting couples.

Clive L. Dym, Irving H. Shames

6. Classical Theory of Plates

Abstract

In Chap. 4 on Beams, Frames, and Rings we were concerned with structures having the distinguishing feature wherein one of the geometric dimensions dominated the configuration. This feature permitted us to make vast simplifications in that we could replace the three-dimensional body by a curve and thus sharply reduce the number of variables of the problem while still yielding important information with considerable accuracy.

Clive L. Dym, Irving H. Shames

7. Dynamics of Beams and Plates

Abstract

Up to this time we have considered only the case of structural bodies in static equilibrium. We now examine variational aspects of the dynamics of beams and plates. Our procedure will be first to present Hamilton’s principle since this principle underlies much of what we do in this chapter. In part A we shall center our attention on beams, first deriving the equations of motion from Hamilton’s principle, and then considering both exact and approximate solutions for free vibrations. For the latter calculations we shall feature the Rayleigh and Rayleigh–Ritz methods. The same pattern is then followed in Part B for the study of plates. In Part C of this chapter we examine the Rayleigh quotient, used in earlier parts of the chapter, in a more general manner and develop strong supporting arguments for some physically inspired assertions made earlier concerning the Rayleigh and the Rayleigh–Ritz methods. The very powerful maximum theorem and “mini-max” theorem from the calculus of variations are presented. Thus this section is a more theoretical exposition. It is supportive of earlier work as well as basic to new work on stability in Chap. 9.

Clive L. Dym, Irving H. Shames

8. Nonlinear Elasticity

Abstract

Up to this time we have restricted the discussion to so-called small deformation. This permitted the employment of the strain tensor of the form:

Clive L. Dym, Irving H. Shames

9. Elastic Stability

Abstract

This chapter will be devoted to considerations of certain of the foundations underlying the theory of elastic stability. We will examine the meaning of stability especially within the context of variational methods and discuss various methods of obtaining stability bounds for various problems. As a recurring example problem we shall make use of the simple Euler column problem to indicate a variety of approaches. Using variational techniques we shall set forth approximate techniques for solving stability problems involving columns and plates.

Clive L. Dym, Irving H. Shames

10. Finite Element Analysis: Preliminaries and Overview

Abstract

All of the work we have described so far has been about getting closed-form solutions, both exact and approximate, for a variety of problems in applied and structural mechanics. Much of that work is clearly “classical” as most of it was done long before the advent of computers. At the time the original edition of this book was being written, the late 1960s, engineers and mathematicians were beginning to develop discrete models of continuous structures that could be used to obtain algorithmic, computational solutions to such mechanics problems. These discretized approaches started with matrix methods for structures, which were extended into finite element analysis (FEA) of structures and of other continua. This discretization methodology now has a long, rich, and continuously growing body of literature that cannot be covered even remotely in a single book, never mind one or two chapters. Having said that, we will now provide an overview of how our general formulations can be cast in matrix representations. Then, we will describe how FEA emerges as a natural discretized extension, enabled by the advent of computers, of the classical variational approaches that have occupied us so far.

Clive L. Dym, Irving H. Shames

11. Finite Element Applications: Trusses and Beams

Abstract

We now turn to illustrating how we can implement the principles detailed in Chap. 10. In particular, and following to a large extent the historical development of these methods, we will now describe the implementation of matrix methods and FEA. Thus, we will first introduce finite elements (literally!) and matrix methods for trusses. We will use a matrix formulation of trusses to introduce the direct stiffness method as a way to solve truss problems, as well as to set the stage for formulating and solving structures problems generally. We then follow that with FEA of the bending of beams by expanding our notion of finite elements to describe, formulate, and solve beam problems. Our emphasis here will be on the hand calculations that, in practice, we would nowadays do with a finite element software package, because it is always important to keep in mind exactly what the computer is doing on our behalf: With this understanding, we are better able to both obtain valid and verifiable results and to interpret correctly whether we are actually solving the problem we set out to solve.

Clive L. Dym, Irving H. Shames

Backmatter

Titel: Solid Mechanics
verfasst von: Clive L. Dym
Irving H. Shames
Copyright-Jahr: 2013
Verlag: Springer New York
Electronic ISBN: 978-1-4614-6034-3
Print ISBN: 978-1-4614-6033-6
DOI: https://doi.org/10.1007/978-1-4614-6034-3

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