Strength of Materials

Citation preview

M IR P U B L IS H E R S M O S C O W

H. M SEJMEB COnPOTVIBJIEHME MATEPMAJIOB MSAAtUbCroO >HAyKA» MOCKBA

N. M. BELYAEV

Strength of Materials

translated from the Russian byN. R . Mehia

MIR PUBLISHERS MOSCOW

First published 1979 Revised from the 1976 Russian edition

Ha aHZAu&CKOM asuKe

© HaflareatcTBO cHaysa*, 1976 © English translation, Mir Publishers, 1979 Printed in the Union of Soviet Socialist Republics

Nikolai Mikhailovich Belyaev (1890—1944) Nikolai Mikhailovich Belyaev occupied a leading position among eminent Soviet scientists who worked on the technical application oT theory of elasticity and strength of materials and structures. After graduating from the St. Petersburg Institute of Railway Engi­ neering in 1916, Nikolai Mikhailovich Belyaev was invited to stay at the Strength of Materials Department, where he worked under S. P. Ti­ moshenko. Nikolai Mikhailovich Belyaev was associated with this institute (now the Leningrad Institute of Railway Engineering) throughout his life. At the institute he taught subjects like engineering structures, bridges, theoretical mechanics, strength of materials and theory of elasticity, and from 1924 to the end of his life was Head of the Strength of Materials Department. All his life Nikolai Mikhailovich Belyaev was a leading engineer and research worker. He was the first to formulate and solve the prob­ lem of stability of prismatic bars under variable axial loading—a problem interesting from the theoretical aspect and important from the point of view of applications. Simultaneously, Nikolai Mikhailo­ vich Belyaev worked on the problem of local stresses in bodies in conlact under compression. Here he considerably developed the works of Hertz. The work first published by Nikolai Mikhailovich Belyaev in 1924 has completely retained its value to this day. In the Soviet Union Belyaev was one of the first to undertake the study of the theory of plastic deformation, and he contributed a lot towards the development of this field. Nikolai Mikhailovich Belyaev spent the last years of his life in fruitful research on problems of creep and relaxation of metals under high temperatures. Nikolai Mikhailovich Belyaev was a rare talent who successfully combined theory with experimental research. In 1924 he took over as Head of the mechanical engineering laboratory of the Leningrad In­ stitute of Railway Engineering, and in the course of 16 years of admi­ nistration changed the laboratory into a leading scientific research centre. New technical specifications ensuring long and reliable performance of rails were compiled as a result of the research conducted at the laboratory under the guidance and with direct participation of Niko­ lai Mikhailovich Belyaev. These specifications with minor additions are in force to this day. Research done by Nikolai Mikhailovich Belyaev in the field of technology of concrete won wide acclaim all over the Soviet Union.

6

Nikolai Mikhailovich Belyaev

The pedagogical activity of Nikolai Mikhailovich Belyaev was not restricted to the Leningrad Institute of Railway Engineering. He worked at the Leningrad Technological Institute (1919-1926), Lening­ rad Institute of Civil Aviation (193M934), and from 1934 onwards was Head of the Strength of Materials Department at the Leningrad Polytechnical Institute—the biggest institute in the country. In 1939 Nikolai Mikhailovich Belyaev was elected Corresponding Member of the USSR Academy of Sciences, and from 1942 occupied the post of Deputy Director of the Institute of Mechanics of the Aca­ demy of Sciences of the USSR. His book Strength of Materials has won wide recognition in the

Preface to the Fifteenth Russian Edition

The new edition of Strength of Materials by N. M. Belyaev has been published after II years. In 33 years that lapsed between the publication by N. M. Belyaev of the first edition in 1932 and the last fourteenth edition in 1965 a total of 675 000 copies of the book were sold, testifying to its wide popularity. During this period the book was periodically enlarged and revised by N. M. Belyaev and, after his death in 1944, by a group of four of his co-workers. This group, which prepared from the filth to the fourteenth editions for publication, did not consider it proper to make substantia] changes in the original work of N. M. Belyaev. Additions were done at one time or another only when they became absolutely necessary due to changes in stan­ dards and technical specifications and in the light of recent research. In the present edition, prepared by the same group, a number of topics have been dropped either owing to their irrelevance to strength of materials or because they are rarely taught in the main course. The topics that have been dropped include Contact Stresses, Riveted Beams, Reinforced Concrete Beams, Approximate Methods for Calculating Deflection of Beams, Beams on Elastic Foundation, Design of Thinwailed Bars, all graphical methods, and a part of Complicated Prob­ lems of Stability Analysis, the other part of the last topic being presented in an abbreviated version. The reader may refer to the earlier editions of this book or special monographs in case information is required on these topics. Considering the availability of a large number of problem books (see, for instance, Problems on Strength of Materials edited by V. K. Kachurin) on the market, most of the examples have been dropped from the present edition. Only examples that are essential for the explana­ tion of theoretical part have been retained. For greater compactness the problem of design for safe loads has now been included in Chapter 26 For the first time the chapter inclu­ des the principles of design for limiting states, which though beyond the limits of the basic course of strength of materials are important enough to require an exposition of the basic concepts even at this stage of teaching. The problems of strength, which in the previous editions occupied two chapters, have been grouped into one. The part dealing with actual stresses has been transferred to Chapter 2, where it has been presented in a sufficiently detailed manner. The tables containing data on materials have been dropped from the appendices. A part of the data on materials has been transferred to

8

Preface to the Fifteenth Russian Edition

corresponding sections. The obsolete steel proliles grading has been replaced by new ones. As in the previous editions it was our endeavour to preserve Belyaev’s style and method of presentation of material. Therefore the author's text has in general been preserved. If Nikolai Mikhailovich Belyaev were alive today he would possibly write many things in a different way. However, since the book won wide popularity as written by N. M. Belyaev, we tried to preserve the original text as far as possible. The work involved In preparing the fifteenth edition for publica­ tion was distributed among the group as follows: Chapter 13, § 80 of Chapter 14, Chapters 15-19, 24-25—L. A. Belyavskii; Chapters 6, 8*12, 27-28-*-Ya. I. Kipnis; Chapters 1-5, 26 and appendices—N. Yu. Kushelev; Chapter 7, § 79 of Chapter 14, Chapters 20-23, 29-32— A- K. Sinitskii. A . /(. Sinitskii March 1976

Contents

Nikolai Mikhailovich Belyaev 5 Preface to the Fifteenth Russian Edition

7

PART 1. Introduction. Tension and Compression Chapter 1. Introduction § 1. 2.

!

3. 4. § 5.

17

The science of strength of materials 17 Classification of forces acting on elements of structures

18

Deformations and stresses 21 Scheme of a solution of the fundamental problem of strength of materials 23 Types of deformations 27

Chapter 2. Stress and Strain in Tension and Compression Within the Elastic Limit Selection of Cross-sectional Area 27 § 6. Determining the stresses in planes perpendicular io tile axis of the bar 27 §7. Permissible stresses. Selecting the cross-sectional area 30 1 8. Deformations under tension and compression. Hooke's law 32 §9 . Lateral deformation coefficient. Poisson's ratio 36 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 1 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. Compres­ sion test diagram 64 § 16. Comparative study of the mechanical properties of ductile and brittle materials 57 § 16. Considerations in selection of safety factoi 59 § 17. Permissible stresses under tension and compression for various materials 64

PART II. Complicated Cases of Tension and Compression Chapter 4. Design of Statically Indeterminate Systems for Permissible Stresses 66 $ 18. Statically indeterminate systems 66 1 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 mate­ rials 77 §21. Stresses due to temperature change 79 §22. Simultaneous account for various factors 82 §23. More complicated cases of statically indeterminate structures 85

Contents

10

Chapter 5. Account for Dead Weight In Tension and Compression* Design of Flexible Strings 88 § 24. §25. §26.

Selecting the cross-sectional area with the account for the dead weight (in tension and compression) 86 Deformations due to dead weight 81 Flexible cables 92

Chapter 6. Compound Stressed State. Stress and Strain 99 § 27. § 28. § 29. §30. §31. §32. 33. 34.

S

Stresses along Inclined sections under axial tension or compression (uniaxial stress) 99 Concept of principal stresses. Types of stresses of materials 101 Examples oil biaxial and triaxiaf stresses. Design of a cylindrical reservoir 103 Stresses In a biaxial stressed state 107 Graphic determination of stresses (Mohr’s circle) 110 Determination of the. principal stresses with the help of the stress circle 114 Stresses in triaxia! stressed state 117 Deformations In the compound stress 121

35. 36.

Potential energy of elastic deformation in compound stress 124 Pure shear. Stresses and strains. Hooke’s law. Potential energy 127 Chapter 7. Strength of Materials in Compound Stress 132 § 37. §38. §39. §40. §41. §42.

Resistance to failure. Rupture and shear 132 Strength theories 136 Theories of brittle failure (theories of rupture) 138 Theories of ductile failure (theories of shear) 140 Reduced stresses according to different strength theories Permissible stresses in pure shear 149

147

PART III. Shear and Torsion Chapter 8. Practical Methods of Design on Shear §43. §44.

151

Design of riveted and bolted joints Design of welded joints 158

151

Chapter 9. Torsion. Strength and Rigidity of Twisted Bars

164

45. 46.

Torque 164 Calculation of torques transmitted to the shaft

47. 48.

Determining stresses in a round shaft under torsion 168 Determination of polar moments of inertia and section moduli of a shaft section 174 Strength condition in torsion 176 Deformations in torsion. Rigidity condition 176 Stresses under torsion in a section inclined to the shaft axis 178 Potential energy of torsion 180 Stress and strain In dose-coiled helical springs 181 Torsion in rods of non-circular section 187

!

§49. §50. § 51. §52. §53. 1 54.

167

II

Contents

PART IV. Bending. Strength of Beams Chapter 10. Internal Forces In Diagrams 198 $ 55. §56. §57. §58. §59. 1 60. § 61.

Bending.

Shearing-force and

Fundamental concepts of deformation in bending. Construction of beam supports 195 Nature of stresses in a beam. Bending moment and shealine force 200 * Differential relation between the intensity of a continuous load, shearing force and bending moment 205 Plotting bending-moment and shearing-force diagrams 207 Plotting bending-moment and shearing-force diagrams for more complicated loads 214 The check of proper plotting of Qr and M-diagrams 221 Application of the principle of superposition of forces In plotting shearing-force and bending-moment diagrams 223

Chapter 1|. Determination of Normal Stresses in Bending Beams 225 §62. § 63. § 64.

Bendfng-momenl

and

Strength

of

Experimental investigation of the working of materials in pure bending 225 Determination of norma) stresses In bending. Hooke’s law and po­ tential energy of bending 228 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 oi the central moments of inertia 254 § 71. Application of the formula for determining normal stresses to beams of non-symmetricai sections 254 §72. Radii of inertia. Concept of the momenta! 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 1 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 the principal stresses 280 Chapter 14. Shear Centre. Composite Beams 283 § 79. Shearing stresses parallel to the neutral axis. Concept of shear centre 233 §80, Riveted and welded beams 289

Contents

12

PART V. Deformation of Beams due to Bending Chapter 15. Analytical Method of Determining Deformations 292 § 81. Deflection and rotation oi beam sections 292 $ 82. Differential equation of the deflected axis 294 § 83. Integration oi the differentia) 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 oi 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 unsymmettically loaded by a force 305 88. Integrating the differential equation for a hinged beam 307

!

89. 90.

Superposition of forces 310 Differential relations in bending 312

Chapter 16. Graph-analytic Method of Calculating Displacement in Bending 3)3 § 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 Displace­ ments 331 § 97. | 98. § 99. § 100. § 101. $ 102. $ 103. § 104. § 105. § 106. 1 107. 1 108.

Statement of the problem 331 Potential energy in the simplest cases of loading 333 Potential energy ior the case of several forces 334 Calculating bending energy using internal forces 336 Castigliano’s theorem 337 Examples of application of Castigiiano’s theorem 341 Method of introducing an external force 344 Theorem of reciprocity of works 346 The theorem of Maxwell and Mohr 347 Vereshchagin’s method 349 Displacements In frames 351 Deflection 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

Contents §111. 1 112. §113. § 114. § 1(5. 1 116. 1 117. §118.

13

Concepts of redundant unknown and base beam 359 Method of comparison of displacements 360 Application of the theorems of Castigliano and Mohr and Vereshcha­ gin's method 362 solution of a simple statically Indeterminate frame 364 Analysis of continuous beams 366 The theorem of three moments 366 An example on application erf the theorem of three moments 372 Continuous beams with cantilevers. Beams with rigidly fixed ends 375

PART VII. Resistance Under Compound Loading Chapter 20. Unsymmetrlc Bending 378 § 119. Fundamental concepts 378 £ 120. Unsymmetrlc bending.Determination ofstresses 379 § 121. Determining displacements inunsymmetrlcbending 365 Chapter 21. Combined Bending and Tension or Compression 389 § 122. Deflection of a beam subjected to axial and lateral forces 389 1 123. Eccentric tension or compression 392 § 124. Core of section 396 Chapter 22. Combined bending and torsion 401 § 125. Determination erf twisting and bending moments 401 § 126. Determination of stresses and strength check In combined bending and torsion 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

1

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 420 § 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 oi 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 1 142. 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 1 145. Stresses in thick spherical vessels 453 § 146. Analysis of thin-walled vessels 454 Chapter 26. Design for Permissible toads. Design for Limiting States 467 § 147. Design for permissible loads. Application to stattcally deter­ minate 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 Tor permissible loads 465 1 151. Design of statically indeterminate beams for permissible loads. The nindamentals. Analysis of a two-span beam 468 6 152. 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 ot Ban Under Compression 477 § 154. Introduction. Fundamentals of stability of shape of compressed bars 477 6 155. 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. 'The 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 b an 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. § 165. § 166. § 167. $ 168. 1 169. § 170. § 171. § 172.

Introduction 521 Determining stresses in uniformly accelerated motion of bodies 523 Stresses In a rotating ring (flywheel rim) 524 Stresses in connecting rods 525 Rotating disc of uniform thickness 529 Disc o f uniform strength 533 Effect of resonance on the magnitude of stresses 535 Determination of stresses in elements subjected to vibration 536 The effect of mass of the elastic system on vibrations 541

Chapter 30. Stresses Linder Impact Loading 548 § 173. Fundamental concepts 548 § 174. General method of determining stresses under impact loading

549

15

Contents § 175. $ 176. 177. 178.

1

179. 180.

Concrete cases of determining stresses and conducting strength checks under impact 5S4 Impact stresses in a non-uniform bat 559 Practical conclusions from tbe derived results 660 The effect of mass of the elastic system on impact 562 Impact testing for failure 565 Effect of various factors on the results of impact testing

568

Chapter 31. Strength Check of Materials Under Variable Loading 571 § 181. § 182. § 183. § 184. 1 185. 1 186. § 187. § 188. § 189. § 190. § 191.

Basic ideas concerning the effect of variable stresses on the -strength of materials 571 Cyclic stresses 573 Strength condition under variable stresses 575 Determination of endurance limit in a symmetrical cycle 576 Endurance limit in an unsymmetrlcsl cycle 579 Local stresses 682 Effect of size of part and other factors onendurance limit 589 Practical examples of failure undo: variableloading.Causes of emergence and development of fatigue cracks 593 Selection of permissible stresses 597 Strength check under variable stresses and compound stressed state 600 Practical measures for preventing fatigue failure 602

Chapter 32. Fundamentals of Creep Analysis 605 § 192. Effect of high temperatures on mechanical 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 Appendix 630 Name index 639 Subject index 641

properties oi

PART I Introduction. Tension and Compression

CHAPTER 1

Introduction § 1. 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 behaviour 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 prpyide 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 consump­ tion of materials whereas the latter lead to increase in consumption. This contradiction forms the basis of the technique, which has facili­ tated the development of strength of materials. Often the existing methods of checking the strength and the availab­ le materials are unable to meet the practical requirements for providing answers to new problems (for example, attaining high speeds in engi­ neering 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. 2-3310

16

introduction. Tension and Compression

[Part I

Sometimes, besides the chief requirements of maximum reliability and economy, an engineer has to ensure fulfilment of other conditions too, such as quick building (when restoring broken structures), mini­ mum weight (in aircraft design), etc. These conditions influence the dimensions, the shape and the material of the various elements compris­ ing 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 mathe­ matics 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 thebirth of mining and metallurgical indus­ tries. 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, in­ cluding 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 struc­ tures 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 scien­ tists, 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 develop­ ment of some problems of strength of materials are given in correspond­ ing chapters of the book. § 2. Classification of Forces Acting on Elements of Structures When in operation, the elements of structures and machines are subjected to external loads, which they transmit to one another. A dam bears its own weight and the pressure of water that it holds and trans­ mits these forces to the foundation. The steel trusses of bridges take

Ch. 11

Introduction,

19

the weight of the train through the wheels and rails and transmit it to the stone supports, and the latter, in turn, communicate this load to the foundation. The steam pressure in the cylinder of a steam engine is transmitted to a piston rod. The pulling force of the locomotive is transmitted to the train through a coupler which connects the tender with the wagons. Hence, the elements of structures are subject to either volume forces acting on each element of the structure (dead weight) or forces of interaction* between the element under consideration anti adjoining elements or between the element and the surrounding medi­ um (water, steam or air). In future, when we say that an external force is being applied to an element of the structure, this will imply the transmission of force of pressure (motion) to the element under consid­ eration from adjoining elements of the structure or the surrounding medium. The forces may be classified according to a number of criteria. We distinguish between the concentrated and distributed forces. A concentrated force is defined as the force of pressure transmitted to the element of structure through an area which is very small as compa­ red to the size of the element, for example, the pressure of the wheels of a moving train on the Tails. In practice the concentrated force is considered to be acting at a point owing fo the small area through which the pressure is transmitted. We must keep in mind that this is an approximation which has been introduced to simplify the calculations; actually, no pressure can be transmitted through a point. However, the error due to this approxi­ mation is so small that it may be generally ignored. A distributed force is defined as the force applied continually over a certain length or area of the structure. A layer of sand of uniform thick­ ness spread over the sidewalk of a bridge represents a force which is uniformly distributed over a certain area; if the thickness of the sand layer is not uniform we shall obtain a non-unifornily distributed load. The dead weight of a beam in the ceiling represents a load distributed over its length. The concentrated loads are measured in units of force (tons, kilog­ rams, newtons **); the loads distributed over an area are measured in terms of force per unit area