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### Strength of Materials Belyaev

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Strength of Materials

N.M. Belyaev

648 Pages

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Strength of Materials

Introduction:

The Science of Strength of Materials

In designing structures and machines, an engineer has to select the material and the cross-sectional area of each element of the structure or machine so that it enables the element to have strength to resist external forces transmitted to it by adjacent elements of the structure without failure of strength or distortion of shape, i. e. the element should function properly. Strength of materials provides the engineer with fundamentals for a proper solution of this problem.

Strength of materials deals with the behavior of various materials under the action of external forces and points out how to select the appropriate material and the cross-sectional area of each element of the structure so as to provide fully reliable functioning and the most economic design.

Sometimes, strength of materials has to deal with the problem in a modified form—to check the dimensions of a designed or existing structure.

The conditions for maximum economy in design and reliability of functioning are contradictory. The former demand minimum consumption of materials whereas the latter lead to increase in consumption. This contradiction forms the basis of the technique, which has facilitated the development of strength of materials.

Often the existing methods of checking the strength and the available materials are unable to meet the practical requirements for providing answers to new problems (for example, attaining high speeds in engineering in general and in aerostatics in particular, long-span structures', dynamic stability, etc.). This initiates a search for new materials and study of their properties, and inspires research for improving the existing methods of designing and devising the new ones. Strength of materials must keep pace with the general development of engineering and technology.

Sometimes, besides the chief requirements of maximum reliability and economy, an engineer has to ensure fulfillment of other conditions too, such as quick building (when restoring broken structures), minimum weight (in aircraft design), etc. These conditions influence the dimensions, the shape and the material of the various elements comprising the structure.

The emergence of strength of materials as a separate science dates back to 162® and is intimately connected with the works of Galileo Galilei, the great Italian scientist. Galileo was a professor of mathematics at Padua. Me lived in a period which saw the disintegration of the feudal system, the development of trade capital and international maritime transport,and the birth of mining and metallurgical industries.

The rapid economic developments of those times called for speedy solutions of new technological problems. Increase 'in international maritime trade perpetuated the need for bigger ships which in turn entailed changes in their design; at the same time it became necessary to reconstruct the existing and to build new internal waterways, including canals and sluices. These new technical problems could not be solved by simply copying the existing designs of ships; it became necessary to judge the strength of elements keeping in mind their size and the* forces acting upon them.

Galileo devoted a considerable part of his work to the study of the dependence between the dimensions of beams and bars and the loads they could withstand. He pointed out that the results of his experiments may prove very useful in building big ships, especially in strengthening the deck and covering because low weight is very important in structures of this type. Galileo’s works have been published in his book Discorsi e Dimoslrazioni Maiematiche . . . (“Dialogue on Two New Sciences . . . ”) (1638, Leiden, Holland).

Further development of strength of materials went on in step with the progress of mechanical and civil engineering, and materialized owing to the research work done by a large number of eminent scientists, mathematicians, physicists and engineers. Russian and Soviet scientists occupy an important place amongst them. Brief informative sketches about the role played by individual scientists in the development of some problems of strength of materials are given in corresponding chapters of the book.

TOC

PART 1. Introduction. Tension and Compression

Chapter 1. Introduction 17

1. The science of strength of materials 17

2. Classification of forces acting on elements of structures 18

3. Deformations and stresses 21

4. Scheme of a solution of this fundamental problem of strength of materials

5. Types of deformations 27

Chapter 2. Stress and Strain in Tension and Compression Within the Elastic Selection of Cross-sectional Area 27

6. Determining the stresses in planes perpendicular of the bar 27

7. Permissible stresses. Selecting the cross-sectional area

8. Deformations under tension and compression. Hooke's of the bar 27

9. Lateral deformation coefficient. Poisson's ratio

Chapter 3. Experimental Study of Tension and Compression in Various Materials and the Basis of Selecting the Permissible Stresses 40

10. Tension test diagram. Mechanical properties of materials 40

11. Stress-strain diagram 47

12. True stress-strain diagram 48

13. Stress-strain diagram for ductile and brittle materials 62

14. Rupture in compression of brittle and ductile materials. Compression test diagram 64

16. Comparative study of the mechanical properties of ductile and brittle materials 57

16. Considerations in selection of safely factors 59

17. Permissible stresses under tension and compression for various materials 64

PART 11. Complicated Cases of Tension and Compression

Chapter 4. Design of Statically Indeterminate Systems for Permissible Stresses 66

18. Statically indeterminate systems 66

19. The effect of manufacturing inaccuracies on the forces acting in the elements of statically indeterminate structures 73

20. Tension and compression in bars made of heterogeneous materials 77

21. Stresses due to temperature change 79

22. Simultaneous account for various factors 82

23. More complicated cases of statically indeterminate structures 85

Chapter 5. Account for Dead Weight In Tension and Compression. Design of Flexible Strings 86

24. Selecting the cross-sectional area with the account for the dead weight (in tension and compression) 86

25. Deformations due to dead weight 91

26. Flexible cables 92

Chapter 6. Compound Stressed State. Stress and Strain 99

27. Stresses along Inclined sections under axial tension or compression (uniaxial stress) 99

28. Concept of principal stresses. Types of stresses of materials 101

29. Examples of biaxial and triaxial stresses. Design of a cylindrical reservoir 103

30. Stresses In a biaxial stressed state 107

31. Graphic determination of stresses {Mohr's circle) 110

32. Determination of the, principal stresses with the help of the stress circle 114 !

33. Stresses in triaxial stressed state 117

34. Deformations In the compound stress 121

35. Potential energy of elastic deformation in compound stress 124

36. Pure shear. Stresses and strains. Hooke's law. Potential energy 127

Chapter 7. Strength of Materials In Compound Stress 132

37. Resistance to failure. Rupture and shear 132

38. Strength theories 136

39. Theories of brittle failure (theories of rupture) 138

40. Theories of tactile failure (theories of shear) 140

41. Reduced stresses according to different strength theories 147

42. Permissible stresses in pure shear 149

PART III. Shear and Torsion

Chapter 8. Practical Methods of Design on Shear 151

43. Design of riveted and bolted joints 151

44. Design of welded joints 158

Chapter 9. Torsion. Strength and Rigidity of Twisted Bars 164

45. Torque 164

46. Calculation of torques transmitted to the shaft 167

47. Determining stresses In a round shaft under torsion 168

48. Determination of polar moments of inertia and section moduli of a shaft section 174

49. Strength condition in torsion 176

50. Deformations in torsion. Rigidity condition 176

51. Stresses under torsion in a section inclined to the shaft axis 178

52. Potential energy of torsion 180

53. Stress and strain In close-coiled helical springs 181

54. Torsion in rods of non-circular section 187

PART IV. Bending. Strength of Beams

Chapter 10. Internal Forces In Bending. Shearing-force and Ben ding, moment Diagrams 198

55. Fundamental concepts of deformation In bending. Construction of beam supports 195

56. Nature of stresses in a beam. Bending moment and shearing force 200

57. Differential relation between the intensity of a continuous load, shearing force and bending moment 205

58. Plotting bending-moment and shearing-force diagrams 207

59. Plotting bending-moment and shearing-force diagrams for more complicated loads 214

60. The check of proper plotting of Q- and Ai-diagrams 221

61. Application of the principle of superposition of forces in plotting shearing-force and bending-moment diagrams 223

Chapter 11. Determination of Normal Stresses in Bending and Strength of Beams 225

62. Experimental investigation of the working of materials in pure bending 225

63. Determination of normal stresses In bending. Hooke's law and potential energy of bending 228

64. Application of the results derived above in checking the strength of beams 235

Chapter 12. Determination of Moments of Inertia of Plane Figures 239

65. Determination of moments of inertia and section moduli for simple sections 239

66. General method of calculating the moments of Inertia of complex sections 244

67. Relation between moments of inertia about two parallel axes one of which is the central axis 246

68. Relation between the moments of inertia under rotation of axes 247

69. Principal axes of inertia and principal moments of inertia 250

70. The maximum and minimum values of the central moments of inertia 254

71. Application of the formula for determining normal stresses to beams of rum-symmetrical sections 254

72. Radii of inertia. Concept of the momentum ellipse 256

73. Strength check, choice of section and determination of permissible load in bending 258

Chapter 13. Shearing and Principal Stresses In Beams 263

74. Shearing stresses in a beam of rectangular section 263

75. Shearing stresses in I-beams 270

76. Shearing stresses in beams of circular and ring sections 272

77. Strength check for principal stresses 275

78. Directions of the principal stresses 280

Chapter 14. Shear Center Composite Beams 283

79. Shearing stresses parallel to the neutral axis. Concept of shear center 283

80, Riveted and welded beams 289

PART V. Deformation of Beams due to Bending

Chapter 15. Analytical Method of Determining Deformations 292

81. Deflection and rotation hoi beam sections 292

82. Differential equation of the deflected axis 294

83, Integration of the differential equation of the deflected axis of a beam fixed at one end 296

84. Integrating the differential equation of the deflected axis of a simply supported beam 299

85. Method of equating the constants of integration of differential equations when the beam has a number of differently loaded portions 301

86. Method of Initial parameters for determining displacements in beams 304

87, Simply supported beam uasymmetrically loaded by a force 305

88, Integrating the differential equation for a hinged beam 307

89. Superposition of forces 310

90. Differential relations in bending 312

Chapter 16. Graph-analytic Method of Calculating Displacement in Bending 333

91. Graph-analytic method 313

92. Examples of determining deformations by the graph-analytic method 317

93. The graph-analytic method applied to curvilinear bending-moment diagrams 320

Chapter 17. Non-uniform Beams 324

94. Selecting the section in beams of uniform strength 324

95. Practical examples of beams of uniform strength 325

96. Displacements in non-uniform beams 326

PART VI. Potential Energy. Statically Indeterminate Beams

Chapter 18. Application of the Concept of Potential Energy in Determining Displacements 331

97. Statement of the problem 331

98. Potential energy In the simplest cases of loading 333

99. Potential energy or the case of several forces 334

100. Calculating bending energy using internal forces 330

101. Castigliano's theorem 337

102. Examples of application of Castigiiano's theorem 341

103. Method of introducing an external force 344

104. Theorem of reciprocity of works 346

105. The theorem of Maxwell and Mohr 347

106, Vereshchagin's method 349

107, Displacements In frames 351

108. Defection of beams due to shearing force 353

Chapter 19. Statically Indeterminate Beams 356

109. Fundamental concepts 356

110, Removing static indeterminacy via the differential equation of the deflected beam axis 357

111. Concepts of redundant unknown and base beam 359

112. Method of comparison of displacements 360

113. Application of the theorems of Castigliano and Mohr and Vereshcha¬ gin's method 362

114. solution of a simple statically Indeterminate frame 364

115. Analysis of continuous beams 366

116. The theorem of three moments 366

117. An example on application of the theorem of three moments 372

118. Continuous beams with cantilevers. Beams with rigidly fixed ends 375

PART VII. Resistance Under Compound Loading

Chapter 20. Asymmetrical Bending 378

119. Fundamental concepts 378

120. Unsymmetrlc bending. Determination of stresses 379

121. Determining displacements in unsymmetrlc bending 385

Chapter 21. Combined Bending and Tension or Compression 389

122. Deflection of a beam subjected to axial and lateral forces 389

123. Eccentric tension or compression 392

124. Core of section 396

Chapter 22. Combined bending and torsion 401

125. Determination of twisting and bending moments 401

126. Determination of stresses and strength check In combined bending and tension 404

Chapter 23. General Compound Loading 408

127. Stresses In a bar section subjected to general compound loading 408

128. Determination of normal stresses 410

129. Determination of shearing stresses 413

130. Determination of displacements 414

131. Design of a simple crank rod 417

Chapter 24. Curved Bars 423

132. General concepts 423

133. Determination of bending moments and normal and shearing forces 424

134. Determination of stresses due to normal and shearing forces 426

135. Determination of stresses due to bending moment 427

136. Computation of the radius of curvature of the neutral layer in a rectangular section 433

137. Determination of the radius of curvature of the neutral layer for circle and trapezoid 434

138. Determining the location of neutral layer from tables 436

139. Analysis of the formula for normal stresses In a curved bat 436

140. Additional remarks on the formula for normal stresses 439

141. An example on determining stresses in a curved bar 441

142. Determination of displacements in curved bars 442

143. Analysis of a circular ring 445 14 Contents

Chapter 25. Thick-walled and Thin-walled Vessels 446

144. Analysis of thick-walled cylinders 446

145. Stresses in thick spherical vessels 453

146. Analysis of thin-walled vessels 454

Chapter 26. Design for Permissible Loads. Design for Limiting States 467

147. Design for permissible loads. Application to statically determinate systems 457

146. Design or statically indeterminate systems under tension or compression by the method of permissible loads 458

149. Determination of limiting lifting capacity of a twisted rod 462 150. Selecting beam section for permissible loads 465

151. Design of statically indeterminate beams for permissible loads. The fundamentals. Analysis of a two-span beam 468

152. Analysis of a three-span beam 472

153. Fundamentals of design by the method of limiting states 474

PART VIII. Stability of Clements of Structures

Chapter 27. Stability of Ban Under Compression 477

154. Introduction. Fundamentals of stability of shape of compressed bars 477 4 155. Euler's formula for critical force 480

156. Effect of constraining the bar ends 484

157. Limits of applicability of Euler's formula. Plotting of the diagram of total critical stresses 488

158. 'Die stability check of compressed bars 494

159. Selection of the type of section and material 498

160. Practical importance of stability check 502 Chapter 28. More Complicated Questions of Stability In Elements of Structures 604

161. Stability of plane surface in bending of beams 504

162. Design of compressed-bent ban 512

163. Effect of eccentric compressive force and initial curvature of bar 517

PART IX. Dynamic Action of Forces Chapter 29. Effect of Forces of Inertia. Stresses due to Vibrations 521

164. Introduction 521 $ 165. Determining stresses in uniformly accelerated motion of bodies 523

166. Stresses in a rotating ring (flywheel rim) 524 1167. Stresses in connecting rods 525

168. Rotating disc of uniform thickness 529

169. Disc of uniform strength 533

170. Effect of resonance on the magnitude of stresses 535

17J. Determination of stresses in elements subjected to vibration 536

172. The effect of mass of the elastic system on vibrations 541 Chapter 30. Stresses Under Impact Loading 548

173. Fundamental concepts 548

174. General method of determining stresses under impact loading 549 Contents 15

175. Concrete cases of determining stresses and conducting strength checks under impact 554

178. Impact stresses in a non-uniform bat 559

177. Practical conclusions from the derived results 660

178. The effect of mass of the elastic system on impact 562

179. Impact testing for failure 565

180. Effect of various factors on the results of impact testing 568

Chapter 31. Strength Check of Materials Under Variable Loading 571

181. Basic ideas concerning the effect of variable stresses on the strength of materials 571

182. Cyclic stresses 573

183. Strength condition under variable stresses 575

184. Determination of endurance limit in a symmetrical cycle 576 1185. Endurance limit in an unsymmetrica) cycle 579

186. Local stresses 582

187. Effect of size of part and other factors on endurance limit 589

188. Practical examples of failure under variable loading. Causes of emergence and development of fatigue cracks 593 6 189. Selection of permissible stresses 597

190. Strength check under variable stresses and compound stressed state 600

191. Practical measures for preventing fatigue failure 602 Chapter 32. Fundamentals of Creep Analysis 605

192. Effect of high temperatures on mechanical properties of metals 605

193. Creep and after-effect 607

194. Creep and after-effect curves 609

195. Fundamentals of creep design 615

196. Examples on creep design 620