Theoryofelasticity-S.P.Timoshenko&J.N.Goodier.pdf

THEORY OF ELASTICITY

ENGINEERING SOCIETIES MONOGRAPHS

Bakhmeteff: Hydraulics of Open Channels
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Timoshenko and Goodier: Theory of Elasticity
Timoshenko and Woinowsky-Krieger: Theory of Plates and Shells

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A.I. Ch. E. Joseph F. Skelly Charles E. Reed

THEORY

OF

ELASTICITY

By S. TIMOSHENKO

And J. N. GOODIER Professors of Engineering Mechanics Stanford University

BIBLIOTECA CENTRALÃ UNIVERSITATEA "POLITEHNICA" TIMISOARA

THEORY OF ELASTICITY

Copyright, 1934, by the United Engineering Trustees, Inc. Copyright, 1951, by the McGraw-Hill Book Company, Inc. Printed in the United States of America. All rights reserved. This book, or parts thereof, may not be reproduced in any form without permission of the publishers.

XIII

64719

PREFACE TO THE SECOND EDITION

The many developments and clarifications in the theory of elasticity and its applications which have occurred since the first edition was written are reflected in numerous additions and emendations in the present edition. The arrangement of the book remains the same for the most part.

The treatments of the photoelastic method, two-dimensional problems in curvilinear coordinates, and thermal stress have been rewritten and enlarged into separate new chapters which present many methods and solutions not given in the former edition. An appendix on the method of finite differences and its applications, including the relaxation method, has been added. New articles and paragraphs incorporated in the other chapters deal with the theory of the strain gauge rosette, gravity stresses, Saint-Venant's principle, the components of rotation, the reciprocal theorem, general solutions, the approximate character of the plane stress solutions, center of twist and center of shear, torsional stress concentration at fillets, the approximate treatment of slender (e.g., solid airfoil) sections in torsion and bending, and the circular cylinder with a band of pressure.

Problems for the student have been added covering the text as far as the end of the chapter on torsion.

It is a pleasure to make grateful acknowledment of the many helpful suggestions which have been contributed by readers of the book.

PALO ALTO, CALIF. February, 1951

S. TIMOSHENKO J. N. GooDIER

THE MAPLE PRFSS COMPANY, YORK, PA.

V

PREFACE TO THE FIRST EDITION

During recent years the theory of elasticity has found considerable application in the solution of engineering problems. There are many cases in which the elementary methods of strength of materials are inadequate to furnish satisfactory information regarding stress distribution in engineering structures, and recourse must be made to the more powerful methods of the theory of elasticity. The elementary theory is insufficient to give information regarding local stresses near the loads and near the supports of beams. It fails also in the cases when the stress distribution in bodies, all the dimensions of which are of the same order, has to be investigated. The stresses in rollers and in balls of bearings can be found only by using the methods of the theory of elasticity. The elementary theory gives no means of investigating stresses in regions of sharp variation in cross section of beams or shafts. It is known that at reentrant corners a high stress concentration occurs and as a result of this cracks are likely to start at such corners, especially if the structure is submitted to a reversal of stresses. The majority of fractures of machine parts in service can be attributed to such cracks.

During recent years considerable progress has been made in solving such practically important problems. In cases where a rigorous solution cannot be readily obtained, approximate methods have been developed. In some cases solutions have been obtained by using experimental methods. As an example of this the photoelastic method of solving two-dimensional problems of elasticity may be mentioned. The photoelastic equipment may be found now at universities and also in many industrial research laboratories. The results of photoelastic experiments have proved especially useful in studying various cases of stress concentration at points of sharp variation of cross-sectional dimensions and at sharp fillets of reentrant corners. Without any doubt these results have considerably influenced the modern design of machine parts and helped in many cases to improve the construction by eliminating weak spots from which cracks may start.

Another example of the successful application of experiments in the solution of elasticity problems is the soap-film method for determining stresses in torsion and bending of prismatical bars. The difficult problems of the solution of partial differential equations with given boundary conditions are replaced in this case by measurements of slopes and deflections of a properly stretched and loaded soap film. The experiments show that in this way not only a visual picture of the stress distribution but also the necessary information regarding magnitude of stresses can be obtained with an accuracy sufficient for practical application. prata aα.

vii

Again, the electrical analogy which gives a means of investigating torsional stresses in shafts of variable diameter at the fillets and grooves is interesting. The analogy between the problem of bending of plates and the two-dimensional problem of elasticity has also been successfully applied in the solution of important engineering problems. ivo to engi-

In the preparation of this book the intention was to give to engineers, in a simple form, the necessary fundamental knowledge of the theory of elasticity. It was also intended to bring together solutions of special problems which may be of practical importance and to describe approximate and experimental methods of the solution of elasticity problems. eiasticity provim. -f -laatinit

Having in mind practical applications of the theory of elasticity, matters of more theoretical interest and those which have not at present any direct applications in engineering have been omitted in favor of the discussion of specific cases. Only by studying such cases with all the details and by comparing the results of exact investigations with the approximate solutions usually given in the elementary books on strength of materials can a designer acquire a thorough understanding of stress distribution in engineering structures, and learn to use, to his advantage, the more rigorous methods of stress analysis. use, to ms auvanvag, mothod

In the discussion of special problems in most cases the method of direct determination of stresses and the use of the compatibility equations in terms of stress components has been applied. This method is more familiar to engineers who are usually interested in the magnitude of stresses. By a suitable introduction of stress functions this method is also often simpler than that in which equations of equilibrium in terms of displacements are used. equrim -1-t:it nroblems

In many cases the energy method of solution of elasticity problems has been used. In this way the integration of differential equations is replaced by the investigation of minimum conditions of certain integrals. Using Ritz's method this problem of variational calculus is reduced to a simple problem of finding a minimum of a function. In this manner useful approximate solutions can be obtained in many practically important cases.

To simplify the presentation, the book begins with the discussion of two-dimensional problems and only later, when the reader has familiarized himself with the various methods used in the solution of problems of the theory of elasticity, are three-dimensional problems discussed. The portions of the book that, although of practical importance, are such that they can be omitted during the first reading are put in small type. The reader may return to the study of such problems after finishing with the most essential portions of the book.

The mathematical derivations are put in an elementary form and usually do not require more mathematical knowledge than is given in engineering schools. In the cases of more complicated problems all necessary explanations and intermediate calculations are given so that the reader can follow without difficulty through all the derivations. Only in a few cases are final results given without complete derivations. Then the necessary references to the papers in which the derivations can be found are always given.

In numerous footnotes references to papers and books on the theory of elasticity which may be of practical importance are given. These references may be of interest to engineers who wish to study some special problems in more detail. They give also a picture of the modern development of the theory of elasticity and may be of some use to graduate students who are planning to take their work in this field.

In the preparation of the book the contents of a previous book ("Theory of Elasticity," vol. I, St. Petersburg, Russia, 1914) on the same subject, which represented a course of lectures on the theory of elasticity given in several Russian engineering schools, were used to a large extent.

The author was assisted in his work by Dr. L. H. Donnell and Dr. J. N. Goodier, who read over the complete manuscript and to whom he is indebted for many corrections and suggestions. The author takes this opportunity to thank also Prof. G. H. MacCullough, Dr. E. E. Weibel, Prof. M. Sadowsky, and Mr. D. H. Young, who assisted in the final preparation of the book by reading some portions of the manuscript. He is indebted also to Mr. L. S. Veenstra for the preparation of drawings and to Mrs. E. D. Webster for the typing of the manuscript.

UNIVERSITY OF MICHIGAN December, 1933

S. TIMOSHENKO

viii

PREFACE TO THE FIRST EDITION

PREFACE TO THE FIRST EDITION

ix

CONTENTS

V
PREFACE TO THE SECOND EDITION.
vii
PREFACE TO THE FIRST EDITION.
. xvii
NOTATION.
CHAPTER 1. INTRODUCTION
1

1. Elasticity
   2
2. Stress. 3
3. Notation for Forces and Stresses 4. Components of Stress. 45
4. Components of Strain. 6
5. Hooke's Law. 10
   Problems
   CHAPTER 2. PLANE STRESS AND PLANE STRAIN
   11
6. Plane Stress 11
7. Plane Strain 13
8. Stress at a Point 17
9. Strain at a Point
   19
10. Measurement of Surface Strains
11. Construction of Mohr Strain Circle for Strain Rosette. 21
    21
12. Differential Equations of Equilibrium.
    22
13. Boundary Conditions
    23
14. Compatibility Equations.
    26
15. Stress Function. 27
    Problems
    CHAPTER 3. TWO-DIMENSIONAL PROBLEMS IN RECTANGULAR
    COORDINATES
    29
16. Solution by Polynomials. 33
17. Saint-Venant's Principle.
    34
18. Determination of Displacements
    35
19. Bending of a Cantilever Loaded at the End
    39
20. Bending of a Beam by Uniform Load .
    44
21. Other Čases of Continuously Loaded Beams
22. Solution of the Two-dimensional Problem in the Form of a Fourier
    46
    Series.
23. Other Applications of Fourier Series. Gravity Loading 53
    53
    Problems

xi

INSTITUTUL POLITEHNC TIMISOARA O IOTECA CENTRALA

CONTENTS

CHAPTER 4. TWO-DIMENSIONAL PROBLEMSIN POLAR COORDINATES
25. General Equations in Polar Coordinates. 55
26. Stress Distribution Symmetrical about an Axis. 58
27. Pure Bending of Curved Bars 61
28. Strain Components in Polar Coordinates. 65
29. Displacements for Symmetrical Stress Distributions. 66
30. Rotating Disks. 69
31. Bending of a Curved Bar by a Force at the End 73
32. The Effect of Circular Holes on Stress Distributions in Plates 78
33. Concentrated Force at a Point of a Straight Boundary 85
34. Any Vertical Loading of a Straight Boundary 91
35. Force Acting on the End of a Wedge 96
36. Concentrated Force Acting on a Beam. 99
37. Stresses in a Circular Disk. . 107
38. Force at a Point of an Infinite Plate. 112
39. General Solution of the Two-dimensional Problem in Polar Coordi
nates. 116
40. Applications of the General Solution in Polar Coordinates 121
41. A Wedge Loaded along the Faces. 123
Problems 125
CHAPTER 5. THE PHOTOELASTIC METHOD
42. Photoelastic Stress Measurement. 131
43. Circular Polariscope. 135
44. Examples of Photoelastic Stress Determination 138
45. Determination of the Principal Stresses 142
46. Three-dimensional Photoelasticity 143
CHAPTER 6. STRAIN ENERGY METHODS
47. Strain Energy 146
48. Principle of Virtual Work 151
49. Castigliano's Theorem. 162
50. Principle of Least Work. 166
51. Applications of the Principle of Least Work—Rectangular Plates. 167
52. Effective Width of Wide Beam Flanges 171
53. Shear Lag 177
Problems 177
CHAPTER 7. TWO-DIMENSIONAL PROBLEMS IN CURVILINEAR COOR-
DINATES
54. Functions of a Complex Variable. 179
55. Analytic Functions and Laplace's Equation 181
Problems 182
56. Stress Functions in Terms of Harmonic and Complex Functions .183
57. Displacement Corresponding to a Given Stress Function. 186
58. Stress and Displacement in Terms of Complex Potentials 187
59. Resultant of Stress on a Curve. Boundary Conditions 190
60. Curvilinear Coordinates. 192

CONTENTS

61. Stress Components in Curvilinear Coordinates 195
    Problems 197
62. Solutions in Elliptic Coordinates 197
63. Elliptic Hole in a Plate under Simple Tension 201
64. Hyperbolic Boundaries. Notches. 204
65. Bipolar Coordinates. . 206
66. Solutions in Bipolar Coordinates 208
    Other Curvilinear Coordinates 212
    CHAPTER 8. ANALYSIS OF STRESS AND STRAIN IN THREE DIMEN-
    SIONS
67. Specification of Stress at a Point 213
68. Principal Stresses. 214
69. Stress Ellipsoid and Stress-director Surface. 215
70. Determination of the Principal Stresses . 217
71. Determination of the Maximum Shearing Stress 218
72. Homogeneous Deformation. 219
73. Strain at a Point 221
74. Principal Axes of Strain. 224
75. Rotation. 225
    Problem. 227
    CHAPTER 9. GENERAL THEOREMS
76. Differential Equations of Equilibrium. 228
77. Conditions of Compatibility 229
78. Determination of Displacements 232
79. Equations of Equilibrium in Terms of Displacements 233
80. General Solution for the Displacements 235
81. The Principle of Superposition 235
82. Uniqueness of Solution 236
83. The Reciprocal Theorem. 239
84. Approximate Character of the Plane Stress Solutions 241
    Problems 244
    CHAPTER 10. ELEMENTARY PROBLEMS OF ELASTICITY IN THREE
    DIMENSIONS
85. Uniform Stress. 245
86. Stretching of a Prismatical Bar by Its Own Weight. 246
87. Twist of Circular Shafts of Constant Cross Section 249
88. Pure Bending of Prismatical Bars. 250
89. Pure Bending of Plates 255
    CHAPTER 11. TORSION OF PRISMATICAL BARS
90. Torsion of Prismatical Bars 258
91. Bars with Elliptical Cross Section. 263
92. Other Elementary Solutions 265
93. Membrane Analogy. 268
94. Torsion of a Bar of Narrow Rectangular Cross Section 272

CONTENTS

95. Torsion of Rectangular Bars. 275
96. Additional Results 278
97. Solution of Torsional Problems by Energy Method 280
98. Torsion of Rolled Profile Sections. 287
99. The Use of Soap Films in Solving Torsion Problems 289
100. Hydrodynamical Analogies. 292
101. Torsion of Hollow Shafts 294
102. Torsion of Thin Tubes 298
103. Torsion of a Bar in which One Cross Section Remains Plane 302
104. Torsion of Circular Shafts of Variable Diameter 304
     Problems 313
     HAPTER 12. BENDING OF PRISMATICAL BARS
105. Bending of a Cantilever. 316
106. Stress Function. 318
107. Circular Cross Section 319
108. Elliptic Cross Section 321
109. Rectangular Cross Section 323
110. Additional Results 329
111. Nonsymmetrical Cross Sections. 331
112. Shear Center. 333
113. The Solution of Bending Problems by the Soap-film Method. 336
114. Displacements 340
115. Further Investigations of Bending 341
     HAPTER 13. AXIALLY SYMMETRICAL STRESS DISTRIBUTION IN A
     SOLID OF REVOLUTION
116. General Equations 343
117. Solution by Polynomials. 347
118. Bending of a Circular Plate 349
119. The Rotating Disk as a Three-dimensional Problem. 352
120. Force at a Point of an Indefinitely Extended Solid 354
121. Spherical Container under Internal or External Uniform Pressure. 356
122. Local Stresses around a Spherical Cavity. 359
123. Force on Boundary of a Semi-infinite Body 362
124. Load Distributed over a Part of the Boundary of a Semi-infinite Solid 366
125. Pressure between Two Spherical Bodies in Contact 372
126. Pressure between Two Bodies in Contact. More General Case 377
127. Impact of Spheres 383
128. Symmetrical Deformation of a Circular Cylinder 384
129. The Circular Cylinder with a Band of Pressure. 388
130. Twist of a Circular Ring Sector. 391
131. Pure Bending of a Circular Ring Sector 395
     CHAPTER 14. THERMAL STRESS
132. The Simplest Cases of Thermal Stress Distribution . 398
133. Some Problems of Plane Thermal Stress. 404
134. The Thin Circular Disk: Temperature Symmetrical about Center 406
     CONTENTS xV
     408
135. The Long Circular Cylinder
     416
136. The Sphere 421
137. General Equations
     425
138. Initial Stresses 427
139. Two-dimensional Problems with Steady Heat Flow
     433
140. Solutions of the General Equations
     CHAPTER 15. THE PROPAGATION OF WAVES IN ELASTIC SOLID MEDIA
     438
141. 438
142. Longitudinal Waves in Prismatical Bars.
     444
143. Longitudinal Impact of Bars.
144. Waves of Dilatation and Waves of Distortion in Isotropic Elastic
     452
     Media. .
     454
145. Plane Waves.
146. Propagation of Waves over the Surface of an Elastic Solid Body . 456
     APPENDIX. THE APPLICATION OF FINITE DIFFERENCE EQUATIONS
     IN ELASTICITY
     461
147. Derivation of Finite Difference Equations
     465
148. Methods of Successive Approximation.
     468
149. Relaxation Method.
     473
150. Triangular and Hexagonal Nets.
     477
151. Block and Group Relaxation.
152. Torsion of Bars with Multiply-connected Cross Sections. 479
     480
153. Points Near the Boundary.
     483
154. Biharmonic Equation
155. Torsion of Circular Shafts of Variable Diameter 490
     495
     AuTHor INDEx.
     499
     SUBJECT INDEX.

xiv

NOTATION

x, y, z Rectangular coordinates.
r, θ Polar coordinates.
ξ, η Orthogonal curvilinear coordinates; sometimes rec
tangular coordinates.
R, ψ, θ Spherical coordinates.
N Outward normal to the surface of a body.
$\ell , m ,$ n Direction cosines of the outward normal.
A Cross-sectional area.
$I _ { x } , I _ { y }$ Moments of inertia of a cross section with respect
to x- and y-axes.
Ip Polar moment of inertia of a cross section.
g Gravitational acceleration.
ρ Density.
q Intensity of a continuously distributed load.
p Pressure.
X, Y, Z Components of a body force per unit volume.
X, Ý, Z Components of a distributed surface force per unit
area.
M Bending moment.
Mt Torque.
${ \pmb \sigma } _ { \pmb { x } } , { \pmb \sigma } _ { { \pmb u } } ,$ σz Normal components of stress parallel to $\pmb { x } \mathrm { - } ,$ y-, and
z-axes.
${ \pmb \sigma } _ { \pmb { \mathscr { B } } }$ Normal component of stress parallel to n.
$\sigma _ { \uparrow \mathrm { ~ , ~ } \sigma _ { \theta } }$ Radial and tangential normal stresses in polar
coordinates.
$\sigma _ { \xi } , \sigma _ { \eta }$ Normal stress components in curvilinear co
ordinates.
${ \sigma _ { r } , \sigma _ { \theta } , \sigma _ { z } }$ Normal stress components in cylindrical co
ordinates.
θ = σx + σy + σz = σr + σθ + σz
τ Shearing stress.
ν, xz, yz Shearing-stress components in rectangular co
ordinates.
Trθ Shearing stress in polar coordinates.
TEO Shearing stress in curvilinear coordinates.
Trθ, Tθz, Trz Shearing-stress components in cylindrical co
ordinates.
S Total stress on a plane.
u, v, w Components of displacements.
€ Unit elongation.
∈z, ∈y, ∈z Unit elongations in $x \cdot , y \cdot ,$ and z-directions.

$\epsilon _ { T , } \epsilon _ { \theta }$ Radial and tangential unit elongations in polar coordinates.

$e = \epsilon _ { x } + \epsilon _ { y } + \epsilon _ { z }$ Volume expansion.

γ Unit shear.

γxν, γzz, γyz Shearing-strain components in rectangular coordinates.

γre, γez, γrz Shearing-strain components in cylindrical coordinates.

E Modulus of elasticity in tension and compression.

G Modulus of elasticity in shear. Modulus of rigidity.

ν Poisson's ratio.

$\mu = G , \lambda = \frac { \nu E } { ( 1 + \nu ) ( 1 - 2 \nu ) }$ Lamé's constants.

φ 4 Stress function.

ψ(z), x(z) Complex potentials; functions of the complex variable $z = x + i y .$

IN The conjugate complex variable $x - i y .$

C Torsional rigidity.

θ Angle of twist per unit length.

${ \pmb { F } } = 2 G \pmb { \theta }$ Used in torsional problems.

V Strain energy.

V。 Strain energy per unit volume.

t Time.

T Certain interval of time. Temperature.

α Coefficient of thermal expansion.

CHAPTER 1

INTRODUCTION

1. Elasticity. All structural materials possess to a certain extent the property of elasticity, $i . e . ,$ if external forces, producing deformation of a structure, do not exceed a certain limit, the deformation disappears with the removal of the forces. Throughout this book it will be assumed that the bodies undergoing the action of external forces are perfectly elastic, $\pmb { i . e . , }$ that they resume their initial form completely after removal of forces.

The molecular structure of elastic bodies will not be considered here. It will be assumed that the matter of an elastic body is homogeneous and continuously distributed over its volume so that the smallest element cut from the body possesses the same specific physical properties as the body. To simplify the discussion it will also be assumed that the body is isotropic, $i . e . ,$ that the elastic properties are the same in all directions.

Structural materials usually do not satisfy the above assumptions. Such an important material as steel, for instance, when studied with a microscope, is seen to consist of crystals of various kinds and various orientations. The material is very far from being homogeneous; but experience shows that solutions of the theory of elasticity based on the assumptions of homogeneity and isotropy can be applied to steel structures with very great accuracy. The explanation of this is that the crystals are very small; usually there are millions of them in one cubic inch of steel. While the elastic properties of a single crystal may be very different in different directions, the crystals are ordinarily distributed at random and the elastic properties of larger pieces of metal represent averages of properties of the crystals. So long as the geometrical dimensions defining the form of a body are large in comparison with the dimensions of a single crystal the assumption of homogeneity can be used with great accuracy, and if the crystals are orientated at random the material can be treated as isotropic.

When, due to certain technological processes such as rolling, a certain orientation of the crystals in a metal prevails, the elastic properties of the metal become different in different directions and the condition of anistropy must be considered. We have such a condition, for instance, in the case of cold-rolled copper,

xviii

NOTATION

1