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Strength of Materials Belyaev

Structural Design and Analysis
Engineering Physics
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Strength of Materials
N.M. Belyaev
648 Pages

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Strength of Materials


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.


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