Elements of Elasticity PDF
Preview Elements of Elasticity
E L E M E N TS OF E L A S T I C I TY by D. S. DUG DALE, B.Sc, Ph.D. Mechanical Engineering Dept., University of Sheffield PERGAMON PRESS OXFORD . LONDON · EDINBURGH · NEW YORK TORONTO · SYDNEY · PARIS · BRAUNSCHWEIG Pergamon Press Ltd., Headington Hill Hall, Oxford 4 & 5 Fitzroy Square, London W.l Pergamon Press (Scotland) Ltd., 2 & 3 Teviot Place, Edinburgh 1 Pergamon Press Inc., 44-01 21st Street, Long Island City, New York 11101 Pergamon of Canada Ltd., 6 Adelaide Street East, Toronto, Ontario Pergamon Press (Aust.) Pty. Ltd., Rushcutters Bay, Sydney, New South Wales Pergamon Press S.A.R.L., 24 rue des icoles, Paris 5e Vieweg & Sohn GmbH, Burgplatz 1, Braunschweig Copyright © 1968 Pergamon Press Ltd. First edition 1968 Library of Congress Catalog Card No 67-28662 This book is sold subject to the condition that it shall not, by way of trade, be lent, resold, hired out, or otherwise disposed of without the publisher's consent, in any form of binding or cover other than that in which it is published. 08 103495 4 (Flexicover) 08 203495 8 (hardcover) AUTHOR'S PREFACE ENGINEERS have used an intuitive approach to stress analysis from early times. These methods have become known collec tively as Strength of Materials. Use was made of the principle that a structure is in equilibrium with its surroundings. Further, any part of a structure can be isolated and considered to be in equi librium with its adjoining parts. Hence an imaginary cut can be made through a load-bearing structure so that the force resisted at the site of this cut can be found. For example, a diametral cut may be made through a cylindrical boiler shell. Average stress is obtained by dividing this force by the cross-sectional area of the member. By assuming linear distributions of strain, this method can be extended to bending of beams and torsion of circular shafts. However, when the distribution of strain is not immediately obvi ous, a diiferent approach is needed. The theory of elasticity begins by examining the equilibrium of an element embedded within a solid body, without direct reference to the shape of the body or the load it sustains. It is then assumed that the material is elastic, with a fixed linear relationship between stress and strain. This restricts the possible ways in which stress may be distributed throughout the body. It remains to select the distribution that gives the required stresses, displacements and resultant forces at the boundaries of the body. It could be said that the theory of elasticity looks at load-bearing members from the inside rather than from the outside. Topics such as experimental stress analysis, materials testing techniques, criteria of failure, working stresses and design prac- χ AUTHOR'S PREFACE tice all come under the heading of strength of materials, but are outside the scope of the present subject. Theory of elasticity is often regarded as a special course for undergraduates in their final year. The reason for this may be that the subject, although not especially difficult, calls for a little mathematical perseverance. However, attempts are sometimes made to assimilate the subject into a general course on strength of materials. In post-graduate courses for engineers it should be possible to pursue some sec tions of the subject farther than is done here. The difficulties of the subject, which must be accentuated for those who approach it without attending a lecture course, are to some extent associated with gaining familiarity with the symbols used. A rather lavish use of subscript notations seems to be una voidable, but these are explained wherever they occur. In dealing with fundamental equations for stress and strain, the common symbols σ and e, each with a double subscript, are preferred. This emphasizes that shear stress is not a distinct kind of stress but merely a component of stress that can be changed at will by chang ing the reference axes. The symmetry of differential equations is often made more obvious by using this notation, which has the approval of many modern writers. When these considerations are a little more remote, a single subscript is used for direct stress in the interests of brevity. This book is not intended as a hand-book for the practical de signer. No formula is given unless it is derived from first princi ples. Elementary matters have been treated at length in the hope that the reader will go to specialized works for more advanced methods. For example, curvilinear coordinates have been examin ed in greater detail than would seem justified by the use made of them in subsequent chapters, as they are frequently encounter ed in present-day analytical work. In a book of this kind, it would not seem appropriate to give too many references to origi nal papers. While a few names and dates are given for their histo- AUTHOR'S PREFACE xi rical interest, references are limited to texts where alternative or more extended discussions may be found. The mathematics required is not of any very high standard, and an attempt has been made to maintain some degree of uniformity in this standard. The main requirement is to be able to differenti ate the usual functions with unfailing precision. The subject has some value as a general discipline, as the ideas and methods are easily carried over to allied subjects such as fluid dynamics. The text has been improved at a great many points as a result of being scrutinized by the Editor, Professor B. G. Neal, to whom I am sincerely grateful. However, the responsibihty for any remain ing obscurities must be mine alone. The book was written while I held a post at Shefiield University endowed by John Brown and Company Ltd. D.S.D. Sheffield January, 1967 CHAPTER 1 STRAIN 1.1. Coordinate systems Neither stress nor strain can be examined in a precise way without first defining a system of coordinates. Rectangular or Cartesian coordinates are suitable and sufficient for expressing the general physical relationships that hold at points in the inte rior of a solid body. Alternative coordinate systems may be pre ferable when one has to describe a boundary of some particular shape or to specify what happens at the boundary. Whatever the shape of the body may be, stresses and strains are known to be distributed within the interior of the body in accordance with certain differential equations derived from the physical properties of the material from which the body is made. The process of solution of a problem consists of selecting mathe matical expressions for the distribution of stress and strain that not only satisfy these differential equations, but also give the required values at the boundaries of the body. In the typical "boundary-value problem", stresses or displacements or both are specified over the surface of a body, or over part of its surface. When expressions for stress and displacement have been found, and it has been shown that these are satisfactory both on the boundaries and at interior points, the problem has been solved. In selecting a coordinate system for conveniently describing boundary conditions, it must be remembered that differential equations in rectangular coordinates must be re-cast into a new 2 ELEMENTS OF ELASTICITY form appropriate to the new coordinates. Useful coordinate sys tems are necessarily orthogonal, that is, one set of coordinate lines intersect^the other set at right angles at all points, but there is no further restriction on choice of coordinates. A more extended discussion of alternative coordinate systems, and the way in which differential equations are transformed, is deferred until stress and strain have been examined in terms of rectangular coordinates. Although stress and strain are necessarily three-dimensional quantities at points in the interior of a solid body, emphasis throughout this book is placed on two-dimensional states of elastic deformation. The resulting treatment is relatively simple, and yet permits the solution of many practical problems. Strain is consi dered before stress so that the signs of shear stresses can be settled by referring to the signs of shear strains rather than to some arbi trary convention. 1.2. Displacements When forces are applied to the edges of a flat sheet, the sheet may be stretched or otherwise distorted while still remaining flat. For the moment, the exact cause of the distortion is not being investigated. The problem in hand is to specify the distortion that has taken place. The result of applying loads is that all points on the sheet move in some way relative to fixed references axes χ and y. Some particular point Ρ as shown in Fig. 1.1 will move to a new position P'. It is assumed that displacements are small in rela tion to the overall dimensions of the body, so that the position of a given particle at any time is described to suflBcient accuracy by its coordinates measured before the distortion is imposed. More detailed restrictions on the size of displacements to ensure that strains remain infinitesimally small, will be mentioned later. Displacement at a given point on a sheet may be specified by giving the length m of the displacement vector together with its direction. Contours may be plotted on the sheet giving length m STRAIN and curves may be drawn in such a way that the tangent to the curve defines the direction of the vector. The example shown in Fig. 1.2 is for a sheet that is uniformly stretched in the j^-direction FIG. 1.1. Vector representing displacement at a point. FIG. 1.2. Displacements in a plane. Full lines are contours of con stant vector length m, and broken lines indicate direction of vector. and uniformly compressed to the same degree in the ^-direction. An alternative way of specifying the displacement vector at any point is to give its components measured parallel to some parti cular reference axes χ and These components are denoted u and V, though occasionally the sufllx notation and Uy is preferable. 4 ELEMENTS OF ELASTICITY Usually these components vary smoothly and continuously throughout the deformed body, but exceptions may occur. If the body initially has a rectangular shape and deforms in the central part only, while the upper and lower ends remain undeformed, as shown in Fig. 1.3a, the deformed part may be termed a kink band. It will be found later that this demands an impossible distribution of stress in an elastic body. If the upper part of the body slides over the lower part, as in Fig. 1.3b, the surface on which sliding occurs is called a dislocation. Horizontal displacements u change (a) (b) FIG. 1.3. (a) Kink band, (b) dislocation. discontinuously as this surface is crossed in a vertical direction, even though the two parts are imagined to be welded together after sliding has occurred. If it is found that vertical displacement υ changes discontinuously across a horizontal surface, it must be concluded that separation of the upper and lower parts has occur red, and that a gap or crack of finite width has developed. 1.3. Strain components Direct and shear strains are first considered separately, and expressed in terms of displacement components. From these ex pressions it will be seen that values of direct and shear strains depend entirely on the coordinates chosen, so that they cannot be regarded as existing independently from each other. STRAIN Direct strain This is defined as fractional increase in length. In the particular case of a long parallel-sided bar subjected to tensile loading, strains will be equal at all points along the bar, but in a more general kind of deformation, strain will vary from one point to another. Therefore it is necessary to consider a small element of length δχ initially at position A as shown in Fig. 1.4. After the -«—u 1 Β — -1 u+Su 1 A FIG. 1.4. Direct strain. body has been deformed, this element has moved to position B, The inner end of the element has moved a distance u while the outer end has moved a somewhat larger distance u+du. The final length of the element is therefore dx-\-du, so the fractional in crease in length is du/dx. An alternative definition of strain might be obtained by expressing the increase in length as a fraction of the final length, i.e. dul{dx+Su), However, attention is re stricted to strains that are very small in value in relation to unity. Hence the alternatively defined strains will differ from the ori ginal ones only by quantities of the second order of smallness, which may be .neglected, so that the distinction between these definitions disappears. Hence, the infinitesimal strain measured in the ^-direction at a particular point can be written e.. = ^. (1.1)
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