Statics of Deformable Solids
336
Statics of Deformable Solids
336eBook
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Overview
Divided into two parts, the book begins with an exposition of the fundamentals of solid mechanics and the principles of mechanics, statics, and simple statically indeterminate systems. The second half deals with strain and stress in three-dimensional solids, elementary elasticity, stress-strain relations for plastic solids, and energy principles in solid continuum. Each chapter concludes with a set of problems and a concise summary of important principles.
Product Details
| ISBN-13: | 9780486799407 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 09/20/2014 |
| Series: | Dover Books on Engineering |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 336 |
| File size: | 50 MB |
| Note: | This product may take a few minutes to download. |
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Read an Excerpt
Statics Of Deformable Solids
By Raymond L. Bisplinghoff, James W. Mar, Theodore H. H. Pian
Dover Publications, Inc.
Copyright © 1990 Raymond L. Bisplinghoff, James W. Mar, and Theodore H. H. KanAll rights reserved.
ISBN: 978-0-486-79940-7
CHAPTER 1
Foundation of Solid Mechanics
1.1 INTRODUCTION
A deformable body is distinguished from a rigid body by its susceptibility to changes in shape under the influence of forces. Strictly speaking, all of nature's bodies are deformable, but in problems of mechanics one may sometimes assume with sufficient precision that certain bodies retain their shape under load, that is, they are perfectly rigid. A deformable body may, however, be a solid or a fluid. The term "solid" is used to identify a solid deformable body in contrast to a fluid with the restriction that a solid possesses shear strength while a fluid does not. Our concern will be limited, furthermore, to the static behavior of solids, that is, to shape changes resulting from static loads only. Our interest in the behavior of solid bodies will extend to a full definition of the deformations, strains, and stresses throughout the volume and over the surface.
Why should the present-day engineer study solid mechanics? In the profession of engineering, for example in the field of aeronautics and astronautics, there is always an end product in mind—a supersonic transport, a launch vehicle, a spacecraft. All these systems, in order to fulfill their missions, must possess physical strength and rigidity. The provision of strength and rigidity can be accomplished relatively simply, but in the process the flight vehicle may become so heavy, so impractical, or so uneconomical that it will not be useful. For example, the structure of a supersonic transport airplane is subjected to widely fluctuating loads as well as temperature excursions of as much as 500° F during each flight. In addition, its users demand a lifetime of some 50,000 hours of flying. In the design of such an airplane, a structural weight of say 23 percent of the gross weight would ensure economic success, whereas a comparable figure only a few percent higher would result in economic failure. The difference between a spacecraft which achieves orbit and one which remains earthbound is sometimes measured in ounces. Thus the structural design of flight vehicles requires exotic materials, sophisticated methods of analysis, and ingenious structural design. Such demands for high structural efficiency are not peculiar to aeronautical and space systems. Because of the high temperatures usually required in energy conversion devices, solid mechanics plays a dominant role in their analysis and design. Higher structural efficiencies are increasingly demanded for civil engineering structures as well as for ground vehicles and marine vessels.
1.2 WAYS OF THINKING
A student of solid mechanics is at first confronted with a dichotomy in his way of thinking about the subject. There is first the physical way of thinking which relates directly to the real world in which we live and to the problems which we seek to solve. These problems are posed to us in a physical way: to predict the deformation of a bridge under load or to compute the stresses in a rocket booster case during launch. Then there is the mathematical way of thinking in which the theorems of mathematics are employed as tools for swift and accurate reasoning.
It is the purpose of a course in solid mechanics to interrelate in the student's mind these two ways of thinking, and to help him develop as second nature a facility for transition from the physical to the mathematical and back again. The student has already experienced this in its simplest manifestation during his study of geometry. Plane Euclidean geometry is a legitimate branch of mathematics regardless of whether there exist in the real world the physical entities of points, lines, and planes. The student, however, easily recognizes these as physical entities at the outset, and the axioms and theorems of geometry thereby provide him with tools for studying their behavior. Other branches of mathematics pose a more subtle transition between physical and mathematical ways of thinking, and a methodology is required to aid in our thinking.
1.3 METHODOLOGY IN SOLID MECHANICS
In general, five successive stages are involved in the study of solid mechanics and in the solution of problems.
(1) A physical system is presented as an object of analysis or design. This may range from a complete system such as a vehicle or a bridge to a single subsystem such as a beam, column, or fastener.
(2) An ideal model is conceived and sketched.
(3) Mathematical reasoning is applied to the ideal model which now becomes a mathematical model. The equations which result from this application are derived and solved.
(4) The mathematical results are interpreted physically in terms pertinent to the original object of the analysis or design.
(5) The results are compared with the results of physical observations. These observations may be made on a laboratory model or on the finished object.
Although it is the purpose of a large portion of the present book to discuss these five stages, certain remarks can be made about them at this point.
Stage 1 implies that man has a curiosity concerning the physical world about him and desires to infer information about it beyond that which he can immediately observe. This curiosity is generally motivated by a desire to construct an engineering system.
It is probably evident that the second stage, conception of the ideal model, is the key element in making the transition from physical to mathematical ways of thinking. Nature is, in general, so complex that it does not permit itself to be expressed exhaustively in a single model, equation, or thought. The scientist or engineer is invariably faced with a task of constructing simplified models of nature to represent levels of abstraction consistent with the results which are sought. If he seeks, for example, to describe the position of the atomic particles of a solid under load, the model he adopts must be vastly different from that required to describe the gross change of external shape. The most difficult task that confronts him is then one of introducing only those simplifying assumptions important to the physical process he is studying, and of neglecting all influences of lower order.
Applications of science at all levels involve conceptual models of nature which are used for mathematical or physical reasoning. The different divisions of basic science may be classified by their conceptual models. For example, to the student of solid mechanics a metal beam is a mass of homogeneous, isotropic, sometimes elastic and sometimes plastic material subjected to force and displacement boundary conditions; to the metallurgist, a collection of randomly oriented grains and crystals; to the chemist, a collection of atoms and molecules; and to the solid-state physicist, a swarm of nuclei and electrons. We can construct an approximate model of a portion of the physical world and trace the relations between its parts, but we cannot by such methods reveal the total intrinsic reality of nature. By applying the mathematical tools of reasoning to models, we can infer new information. If the model is consistent with experimental observation, it is regarded as a valid model. Pure scientists are engrossed in improving nature's models; engineers or applied scientists apply them to machines. In the student's professional life, his success in the use of mechanics will depend as much on his skill in the game of "mathematical make-believe" as on any other single factor. The art of stripping away physical complexities which are unimportant for the problem at hand, and of designing ideal models which yield to analysis is at the heart of successful engineering practice. The student will recognize that the development of these skills and viewpoints is, in fact, a requirement in all branches of the engineering sciences.
The third stage, application of mathematical reasoning, requires an application of physical laws to the ideal model. These laws, which in the present book rest upon the foundation of Newtonian mechanics, provide an experimentally verified basis for constructing mathematical equations to describe the behavior of the physical system. Mathematical reasoning is then brought into play by employing the axioms and theorems governing the mathematical equations which have been constructed.
The fourth stage, physical interpretation of mathematical results, presents no difficulty providing the physical laws introduced in the third stage are fully understood. Under these circumstances there should be a clear realization of the relationship between the parameters of the mathematical model and their counterparts in nature.
The fifth step, comparison with physical observations, is an important step in the process, although it will not be one of the objects of the present book to deal explicitly with the experimental sciences. The engineer or applied scientist must learn from experience. The tools he employs should be those which experience has shown will yield accurate, experimentally verified results. Mechanical, aeronautical, civil, and other engineers have learned from experience that certain ideal models provide accurate representations of physical systems with which they have traditionally dealt. The ideal models used to represent pulley systems, cantilever airplane wings, and bridge trusses are examples of accumulated experience based on experimental observation. The engineer working at the forefront of his technology may need to construct ideal models for systems which are entirely new and for which there is no accumulated experience. He may desire, in such cases, to construct experimental laboratory models or to make careful confirmatory measurements on the first version of the operational system in order to verify his design calculations. Thus he accumulates additional experience which gives him a unique advantage.
A simple example may serve to illustrate at this point the nature of the five stages. There are many structural problems associated with the design of an airplane. In principle, one ideal model or one gigantic mathematical equation covering the entire airplane (Fig. 1.1) is possible, but solving it would be impractical and even impossible. The engineer is forced to break the airplane down into its separate structural components and examine them individually. For instance, in the analysis and design of a wing structure to carry the flight loads, the five stages may be described in the following terms:
(1) The wing provides the lift necessary to sustain flight, and as such must be strong enough to resist the aerodynamic pressures created by the air flow over the wing and must be rigid enough to prevent excessive deflections and wing flutter.
(2) The wing is idealized as a one-dimensional structure rigidly attached to a wall. Such a structure, illustrated by Fig. 1.2, is called a cantilever beam. The bending-stiffness properties of the beam are condensed into a single parameter called the bending-stiffness factor, EI, and the airloads are condensed into a load per unit length, p(x).
(3) The equation which describes the bending behavior of the idealized model is
d2/dx2(EI[d2w/dx2) = p(x), (1.1)
where w (x) is the lateral deflection of the beam as a function of x. Since EI is also a function of x, the above equation is most easily integrated in numerical form.
(4) The lateral deflection of the wing is obtainable directly from w (x), and the strains and stresses may also be computed from the result by employing additional mathematical formulas.
(5) The structural designer often verifies his computations and obtains additional information from static tests in the laboratory of models or of a full-scale wing. The ultimate verification of the analysis and design, however, is the successful flight operation of the prototype airplane.
Other examples are abundant, and as the subject is gradually developed these will become evident. It will become increasingly clear to the student that mathematics plays an important role in the study of solid mechanics. Even though the problems are generally motivated by an engineering objective, and although the ultimate verification lies in the performance of a hardware object, a reasonably high proficiency in mathematics is required for participation in the challenging engineering projects which are at the forefront of new technologies. Of course, many of the fundamental principles of solid mechanics were laid down by men blessed with tremendous intuitive gifts and mental abilities. Fortunately, it is not necessary to possess these same attributes in order to understand the principles of solid mechanics and to apply them effectively to engineering problems.
CHAPTER 2Principles of Mechanics
2.1 INTRODUCTION
We shall find in subsequent developments of the present book that four fundamental laws provide a foundation for the subject of mechanics. These laws represent in concise form an acceptable mathematical model of a segment of nature. They are listed below for the sake of convenient reference, and they are discussed in detail in the following articles of the present chapter.
(a) Law of the Parallelogram of Forces (Section 2.3)
(b) Law of Transmissibility of Forces (Section 2.4)
(c) Law of Motion (Section 2.5)
(d) Law of Action and Reaction (Section 2.6)
These four laws provide an experimentally verified foundation for Newtonian mechanics upon which, by mathematical analysis, such vast subjects as particle, rigid-body, solid, and fluid mechanics may be erected. These laws can be fully understood only by applying them to diverse problems. It is quite important that the student adopt a philosophy of seeking an understanding of their full meaning by numerous applications to problems. Several examples are given in the present and later chapters to illustrate their application, and additional problems are listed at the ends of the chapters.
2.2 THE CONCEPT OF FORCE
All normal human beings are familiar in their day-to-day activities with the notion of a force. The opening and closing of doors, the lifting of weights, and the steering of an automobile are experiences which require one to exert force. It is evident from daily activities that the notion of a force involves both magnitude and direction; that is, a force is a vector quantity. Furthermore, a force is associated with a point of application. Forces are therefore represented in mechanics by vectors acting at specific points. In mechanics as in everyday experience, forces are usually produced by the action of one body on another. Since forces are vector quantities, they will be represented by boldface letters such as F, P, and Q. On the blackboard or in manuscript they are usually denoted by arrows over the letters, such as [??], [??] and [??].
The vector qualities of a force permit it to be represented in space by a directed line segment. In Fig. 2.1 a force F acting at a point p is schematically represented in three-dimensional space by the line segment pa. As explained in the Appendix, for the summation notation and tensor notation to be used in this text, the three rectangular Cartesian coordinates are represented by y1, y2, and y3, respectively.
The force F can be written as the vector sum of three independent components,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.2.1a)
where i1, i2, and i3 are the unit vectors associated with the rectangular Cartesian coordinates y1, y2, and y3. According to the summation notation (see Appendix, Section A.2) this is represented simply as
F = F mim (2.2.1b)
(Continues...)
Excerpted from Statics Of Deformable Solids by Raymond L. Bisplinghoff, James W. Mar, Theodore H. H. Pian. Copyright © 1990 Raymond L. Bisplinghoff, James W. Mar, and Theodore H. H. Kan. Excerpted by permission of Dover Publications, Inc..
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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Table of Contents
Well-written, thoughtfully prepared, and profusely illustrated, this text is the work of a trio of prominent experts. The treatment builds on the mechanics background obtained from an engineering curriculum's first course in physics, providing the foundations for a study of such advanced topics in solid mechanics as the theory of elasticity, structural analysis, plasticity, and shell theory.
Divided into two parts, the book begins with an exposition of the fundamentals of solid mechanics and the principles of mechanics, statics, and simple statically indeterminate systems. The second half deals with strain and stress in three-dimensional solids, elementary elasticity, stress-strain relations for plastic solids, and energy principles in solid continuum. Each chapter concludes with a set of problems and a concise summary of important principles.
Dover (2014) corrected republication of the edition published by Addison-Wesley Publishing Company, Inc., Reading, Massachusetts, 1965.
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