CONTENTS
Preface to the Second
Edition...... v
Preface to the First
Edition.......... vii
Notation..............
xvii
Chapter 1. INTRODUCTION
1. Elasticity... 1
2. Stress..............
2
3. Notation for Forces
and Stresses.......... 3
4. Components of
Stress.........4
5. Components of
Strain......... 5
6. Hooke's Law...... 6
Problems..... 10
Chapter 2. PLANE STRESS
AND PLANE STRAIN
7. Plane
Stress...................11
8. Plane Strain........
11
9. Stress at a Point....
13
10. Strain at a
Point...17
11. Measurement of
Surface Strains.... 19
12. Construction of
13. Differential
Equations of Equilibrium. 21
14. Boundary Conditions.
22
15. Compatibility
Equations. 23
16. Stress Function...
26 Problems.......27
Chapter 3.
TWO-DIMENSIONAL PEOBLEMS IN RECTANGULAR COORDINATES
17. Solution by
Polynomials.....29
18. Saint-Venant's
Principle... 33
19. Determination of
Displacements... 34
20. Bending of a
Cantilever Loaded at the End 35
21. Bending of a Beam by
Uniform Load... 39
22. Other Cases of
Continuously Loaded Beams. 44
23. Solution of the
Two-dimensional Problem in the Form of a Fourier Series................. 46
24. Other Applications
of Fourier Series. Gravity Loading 53 Problems.............53
Chapter 4.
TWO-DIMENSIONAL PROBLEMS IN 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 Coordinates.................. 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—Eectangular Plates.. 167
52. Effective Width of
Wide Beam Flanges....... 171
53. Shear Lag...... 177
Problems...... 177
Chapter 7.
TWO-DIMENSIONAL PROBLEMS IN CURVILINEAR COORDINATES
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
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 DIMENSIONS
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
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
Chapter 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
Chapter 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
135. The Long Circular Cylinder 408
136. The Sphere 416
137. General Equations 421
138. Initial Stresses 425
139. Two-dimensional Problems with Steady Heat Flow 427
140. Solutions of the General Equations 433
Chapter 15. THE PROPAGATION OF WAVES IN ELASTIC SOLID MEDIA
141............ 438
142. Longitudinal Waves in Prismatical Bars 438
143. Longitudinal Impact of Bars... 444
144. Waves of Dilatation and Waves of Distortion in Isotropic Elastic Media... 452
145. Plane Waves........... 454
146. Propagation of Waves over the Surface of an Elastic Solid Body 456
Appendix. THE APPLICATION OF FINITE DIFFERENCE EQUATIONS IN ELASTICITY
1. Derivation of Finite Difference Equations 461
2. Methods of Successive Approximation. 465
3. Relaxation Method....... 468
4. Triangular and Hexagonal Nets 473
5. Block and Group Relaxation. 477
6. Torsion of Bars with Multiply-connected Cross Sections 479
7. Points Near the Boundary. 480
8. Biharmonic Equation.... 483
9. Torsion of Circular Shafts of Variable Diameter 490
Author Index........... 495
(“THEORY OF ELASTICITY” by S. TIMOSHENKO and J. N. GOODIER 1951)