StrengthofMaterials

St r e n g t h of M a t e r ia l s

P a r t IElementary Theory and Problems

B y

S. TIMOSHENKOProfessor Emeritus of Engineering Mechanic•

Stanford University

THIRD EDITION

CBS

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First Indian Edition : 1986 R eprint: 2000, 2002

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In the preparation of the third edition of this book a con­siderable number of new problems were added, and answers to m any of the old problems inserted. The book was expanded by the addition of two new chapters; namely, Chapter V III which deals with bending of beams in a plane which is not a plane of sym m etry, and Chapter X II on the bending of curved bars. In Chapter V III the notion of shear center, which is of great practical importance in the case of thin walled struc­tures, is introduced. In Chapter X II is presented the m ate­rial on curved bars which previously appeared in the second volume of this book. T h a t m aterial has been entirely rewrit­ten and new m aterial added. I t is hoped with these m ajor changes, as well as the innumerable minor changes through­out the entire text, th a t the volume will be not only more complete, bu t also more satisfactory as a textbook in elemen­tary courses in strength of materials. The author wishes to thank Professor Jam es M. Gere of Stanford University, who assisted in revising the volume and in reading the proofs.

S. T im o s h e n k oSt a n f o r d U n iv e r s it y

a

In preparing the second edition of this volume, an effort has been made to adap t the book to the teaching requirem ents o f our engineering schools.

W ith this in view, a portion of the m aterial of a more advanced character which was contained in the previous edi­tion of this volume has been removed and will be included in the new edition of the second volume. At the same time, some portions of the book, which were only briefly discussed in the first edition, have been expanded with the intention of m aking the book easier to read for the beginner. For this reason, chapter I I , dealing with combined stresses, has been entirely rew ritten. Also, the portion of the book dealing with shearing force and bending m om ent diagram s has been ex­panded, and a considerable am ount of m aterial has been added to the discussion of deflection curves by the integration m ethod. A discussion of column theory and its application has been included in chapter V III , since this subject is usually required in undergraduate courses o f strength o f m aterials. Several additions have been m ade to chapter X dealing with the application of strain energy m ethods to the solution of statically indeterm ined problems. In various parts of the book there are m any new problems which m ay be useful for class and home work.

Several changes in the notations have been made to con­form to the requirem ents of American S tandard Symbols for M echanics of Solid Bodies recently adopted by The American Society of M echanical Engineers.

vI t is hoped th a t with the changes m ade the book will be found more satisfactory for teaching the undergraduate course of strength of m aterials and th a t it will furnish a better foundation for the study of the more advanced m aterial disfcussed in the second volume.

S. T im o s h e n k oP a lo A l t o , C a litoh w ia

June 13 , 1940

A t the present tim e, a decided change is tak ing place in the a ttitu d e of designers towards the application of analytical m ethods in the solution of engineering problems. Design is no longer based principally upon empirical formulas. The im­portance of analytical m ethods combined with laboratory experim ents in the solution o f technical problems is becoming generally accepted.

Types of m achines and structures are changing very rap­idly, especially in the new fields o f industry , and usually tim e does not perm it the accum ulation o f the necessary empirical da ta . T he size and cost of structures are constantly increas­ing, which consequently creates a severe dem and for greater reliability in structures. The economical factor in design under the present conditions of com petition is becoming of growing im portance. The construction m ust be sufficiently strong and reliable, and yet it m ust be designed with the greatest possible saving in m aterial. U nder such conditions, the problem of a designer becomes extrem ely difficult. Re­duction in weight involves an increase in working stresses, which can be safely allowed only on a basis of careful analysis of stress d istribution in the structu re and experim ental investi­gation o f the m echanical properties of the m aterials em­ployed.

I t is the aim of this book to present problems such th a t the s tu d e n t’s a tten tion will be focussed on the practical applica­tions of the subject. I f this is a tta ined , and results, in some measure, in increased correlation between the studies o f strength o f m aterials and engineering design, an im portan t forward step will have been m ade.

The book is divided into two volumes. T he first volume contains principally m aterial which is usually covered in required courses of strength of m aterials in our engineering

schools. The more advanced portions of the subject are of in terest chiefly to graduate students and research engineers, and are incorporated in the second volume of the book. This contains also the new developm ents of practical im portance in the field of s trength of m aterials.

In w riting the first volume of strength of m aterials, a tten ­tion was given to simplifying all derivations as much as possible so th a t a s tuden t with the usual preparation in m ath ­em atics will be able to read it w ithout difficulty. For example, in deriving the theory of the deflection curve, the area moment method was extensively used. In this m anner, a considerable simplification was made in deriving the deflections of beams for various loading and supporting conditions. In discussing statically indeterm inate system s, the method of superposition was applied, which proves very useful in treating such problems as continuous beams and frames. For explaining combined stresses and deriving principal stresses, use was made of the M ohrs circle, which represents a substan tia l simplification in the presentation of this portion of the theory.

Using these m ethods of sim plifying the presentation, the au thor was able to condense the m aterial and to discuss some problems of a more advanced character. For example, in discussing torsion, the tw ist of rectangular bars and of rolled sections, such as angles, channels, and I beams, is consider^ T he deform ation and stress in helical springs are discussed detail. In the theory of bending, the case of non-sym m etricai cross sections is discussed, the center of twist is defined and explained, and the effect of shearing force on the deflection of beams is considered. The general theory of the bending of beams, the m aterials of which do not follow H ooke’s law, is given and is applied in the bending of beams beyond the yielding point. The bending of reinforced concrete beams is given consideration. In discussing com binations of direct and bend­ing stress, the effect of deflections on the bending m om ent is considered, and the lim itation of the m ethod of superposition is explained. In treating combined bending and torsion, the cases of rectangular and elliptical cross sections are dis­

cussed, and applications in the design of crankshafts are given. Considerable space in the book is devoted to m ethods for solving elasticity problem s based on the consideration of the s tra in energy o f elastic bodies. These m ethods are ap­plied in discussing sta tically indeterm inate system s. The stresses produced by im pact are also discussed. All these problems of a m ore advanced character are prin ted in small type, and m ay be om itted during the first reading of the book.

T he book is illustrated w ith a num ber of problems to which solutions are presented. In m any cases, the problem s are chosen so as to widen the field covered by the tex t and to illustrate the application of the theory in the solution of design problems. I t is hoped th a t these problems will be of in terest for teaching purposes, and also useful for designers.

The au thor takes this opportun ity o f thanking his friends who have assisted him by suggestions, reading of nan u scrip t and proofs, particu larly Messrs. W. M . Coates and L. H . Donnell, teachers o f m athem atics and mechanics in the Engineering College o f the U niversity of M ichigan, and M r. F. L. E vere tt o f the D epartm ent of Engineering Research o f the U niversity o f M ichigan. H e is indebted also to M r. FJ C. W ilharm for the preparation of drawings, to M rs. E. D. W ebster for the typ ing of the m anuscript, and to the Van N ostrand Com pany for its care in the publication o f the book.

S. T im o s h e n k oA n n A r b o r , M i c h i g a n

M ay i, 1930

a ..................... Angle, coefficient of thermal expansion, numeri­cal coefficient

/3..................... Angle, numerical coefficienty ..................... Shearing strain, weight per unit volumeA.................... Unit volume expansion, distance5 ..................... Total elongation, total deflection, distancet ..................... Unit strain

ev, t z ........ Unit strains in x, y and z directionsd ..................... Angle, angle of twist per unit length of shaftn ..................... Poisson’s ratioa ..................... Unit normal stresscr1( cr2............. Principal stressesan ................... Unit normal stress on plane perpendicular to the

direction n<rx, (T„, <rz . . . . Unit normal stresses on planes perpendicular to

the x, y and z axes<7u.......... U ltim ate stress<jw .................. Working stress(7y.p................ Yield point stressr ..................... Unit shear stressr IV, tvz, tzx- ■ Unit shear stresses on planes perpendicular to

the x, y and z axes, and parallel to the y , z and x axes

TW................... Working stress in sheart y . p , ............... Yield point stress in shear<p..................... Angleoj..................... Angular velocityA .................... Cross sectional areaa, b, c, d ........ DistancesC .................... Torsional rigidity, constant of integrationD, d ............... DiametersE .................... M odulus of elasticity

G ..................... M odulus of elasticity in shearH ....................Horizontal force, horsepowerh ..................... H eight, thicknessI p ................... Polar moment of inertia of a plane area

. M oments of inertia of a plane area with respect to the y and z axes

I y, ..................P roduct of inertia of a plane area with respectto the y and z axes

K .................... Bulk modulus of elasticityk ..................... Spring constant, numerical factork y y k , ............. Radii of gyration of a plane area with respect to

the y and z axes/ ...................... Length, spanM ................... Bending momentM t ..................Torquen ..................... Factor of safety, revolutions per m inute, normal

to a plane P , Q ............... Concentrated forcesp . , ................. Pressure, steel ratio for reinforced concrete

beamsq ..................... Load per unit length, pressureR .................... Reaction, force, radiusr ........................Radius, radius of curvatureS ..................... Axial force in a bart ...................... Tem perature, thicknessU .................... Strain energyu ................. Deflection, distanceV .................... Volume, shearing forcev ...................... Velocity, deflection, distanceW ................... T otal load, weightw.................... W eight per unit length, strain energy per unit

volume» i ................... Strain energy per unit weightX , Y , Z ........ Axial forces in bars, unknown reactionsx ,y , z ............ Rectangular coordinatesZ .................... Section modulus

CONTENTSCHA PTER PAGE

I . T en sio n and C o m pression w it h in t h e E lastic L im it 11. E la s t i c i ty .................................................................................... 12. H ooke’s L a w .......................................................................... 23. T h e T ensile T e s t D i a g r a m .............................................. 64. W ork ing S t r e s s ...................................................................... 85. S tress an d S tra in P roduced in a B ar by I ts Ow n

W e ig h t .................................................................................... 146. S ta tica lly In d e te rm in a te P roblem s in T ension an d

C o m p r e s s io n ...................................................................... 207. A ssem bly an d T h erm a l S t r e s s e s ..................................... 268. E x tension o f a C ircu lar R i n g .......................................... 31

I I . A nalysis of Str ess and St r a i n .......................................... 379. S tress on Inclined P lanes for S im ple T ension an d

C o m p r e s s io n ...................................................................... 3710. M o h r’s C i r c l e ................................. ................................. 4011. T ension or C om pression in T w o P erpend icu lar D i­

rec tions ............................................................................... 4412. M o h r’s C ircle for C om bined S t r e s s e s ............................ 4613. P rincipal S t r e s s e s ................................................................. 4914. L a te ra l C o n t r a c t i o n ............................................................ 5315. S tra in in th e Case o f T ension or Com pression in T w o

P erpend icu lar D i r e c t io n s ..................................... 5416. P u re S hear. M odulus in S h e a r ..................................... 5717. W orking S tresses in S h e a r .............................................. 6218. T ension o r C om pression in T h ree P erpend icu lar D i­

rec tions ............................................................................... 65

I I I . B en d in g M om ent and Sh ea rin g F o r c e ............................ 7019. T y p es o f B e a m s ...................................................................... 7020. B ending M om en t an d S hearing F o r c e ....................... 7221. R e la tion betw een B ending M om en t a n d S hearing

F orce .................................................................................... 7622. B ending M om en t an d S hearing Force D iag ram s . . 78

IV . Str esses in L aterally L oaded Symmetrical B eams . 9223. P u re B e n d i n g .......................................................................... 9224. V arious S hapes o f Cross Sections o f B eam s . . . . 100

xi

xii C O N T E N T SCHAPTER

25. G eneral Case o f L atera lly L oaded S ym m etricalB e a m s ................................................................................... 105

26. Shearing Stresses in B e n d i n g ......................................... .....11327. D istrib u tio n o f S hearing S tresses in th e Case o f a

C ircu lar Cross S e c t i o n ................................................... .....12028. S hearing Stresses in I B e a m s ..........................................12229. P rincipal Stresses in B e n d i n g ..........................................12530. Stresses in B uilt-up B e a m s .............................................. 130

V. D eflec tio n of L aterally L oaded S ymmetricalB e a m s ........................................................................................ 137

31. D ifferential E q u a tio n o f the D eflection C urve . . . 13732. B ending o f a U niform ly L oaded B e a m ....................... .... 140

33. D eflection of a S im ply S upported Beam L oaded by aC o n cen tra ted L o a d ........................................................... 143

34. D eterm ination o f D eflections by the Use o f theB ending M om ent D iagram . A rea-M om entM ethod ................................................................................... 147

35. D eflection o f a C an tilever Beam by the A rea-M o­m en t M e t h o d ..........................................................................149

36. D eflection o f a S im ply S upported Beam by the A rea-M om en t M e t h o d ............................................................ 154

37. M eth o d o f S u p e r p o s i t io n ................................................... ....16238. T h e D eflection o f Beam s w ith O v e r h a n g s .................. 167

39. E ffect o f S hearing Force on the D eflection o f Beam s 170

V I. Statically I nd eterm in a te P roblems in B end ing . . 17640. R e d u n d an t C o n s tr a in ts ............................................................176

41. Beam B uilt In a t One E n d and S upported a t theO t h e r ................................................................................... ....179

42. Beam w ith B oth E nds B uilt I n ..................................... 18643. F r a m e s ........................................................................................ ....19044. Beam s on T hree S upports .............................................. ....198

45. C ontinuous B e a m s .................................................................202

V II. S ym metrical B eams of V ar ia b le C ross S ec tio n .B eams of T wo M a t e r ia l s ..........................................210

46. Beam s of V ariable Cross S e c t io n ..................................... 21047. S ym m etrical Beam s o f Tw o D ifferent M ateria ls . . 217

: 48. R einforced-C oncrete B e a m s 22149. Shearing Stresses in R einforced-C oncrete Beam s . . 225

V III . B ending of B eams in a P lane W hich Is N ot a P laneof Sy m m e t r y ..................................................................... 227

50. Pure Bending in a P lane W hich Is N o t a P lane ofS y m m e t r y .......................................................................... 227

C O N T E N T S xiiiCHAPTER PAGE

51. B ending o f Beam s H av in g T w o P lanes o f S ym m etry 23352. B ending of B eam s in a P rincipal P lane W hich Is

N o t a P lane o f S ym m etry ..........................................235

IX . C o m bined B end ing and A xial L oad . T heo ry ofC o l u m n s ............................ . ..............................................245

53. B ending A ccom panied by T ension or C om pression . 24554. E ccen tric L oading of a S hort S tru t ............................24955. T h e Core o f a S e c t i o n ........................................................254

56. E ccen tric C om pression o f a S lender S ym m etricalC olum n ...............................................................................258

57. C ritica l L o a d .......................................................................... 26358. C ritica l S tress; D esign o f C o lu m n s .................................268

59. D esign o f C olum ns on the Basis o f A ssum ed In ac ­curacies . . ■ . .................................................................274

60. E m pirical F orm ulas for C olum n D e s i g n ...................277

X . T orsion and C om bined B ending and T orsion . . . 28161. T orsion o f a C ircu lar S h a f t .............................................. 28162. T orsion o f a Hollow S h a f t .............................................. 287

63. S haft o f R ectangu lar Cross Section ............................ ....289 •64. H elical S pring , Close C o i le d .............................................. 290

65. C om bined Bending and T orsion of C ircular S hafts . 295

X I. Str a in E nergy and I mpact . . . ............................ 30166. E lastic S tra in E nergy in T e n s i o n .................................30167. T ension P roduced by I m p a c t ..........................................305

68. E lastic S tra in E nergy in S hear an d T orsion . . . . 31269. E lastic S tra in E nergy in B e n d i n g .................................316

70. D eflection P roduced by Im p ac t . . . . • . . . . . 32071. T h e G eneral Expression for S tra in E nergy . . . . 32572. T h e T heorem o f C a s tig lia n o .............................................. 32873. D eflection o f T ru s se s ..............................................................335

74. A pplication o f C astig liano T heorem in Solution ofS ta tica lly In d e te rm in a te P r o b le m s ............................340

75. T h e Reciprocal T h e o r e m ................................................... 35176. E xceptional C a s e s ................................................... .... . . 359

X II . C urved B ars ...............................................................................36277. P u re B ending o f C urved B a r s ..........................................362

78. B ending o f C urved B ars by Forces A cting in theP lane o f S ym m etry . . . . * ..................................... 366

79. P a r tic u la r Cases o f C urved B ars . * ............................36880. D eflection o f C urved B ars ...............................................378

81. A rch H inged a t the E n d s ................................................... ...394 .82. S tresses in a F l y w h e e l ..................................... ......................398

xiv CONTENTSCHAPTER

83. Deflection Curve for a Bar with a Circular CenterL i n e ..........................................................................

84. Bending of Curved T ubes...........................................85. Bending of a Curved Bar Out of Its Plane of Initial

C u r v a tu r e ...............................................................

A PPEND IX A

M oments of I n er tia of P lane A r e a s ............................I. The Moment of Inertia of a Plane Area with Respect to

an Axis in Its P l a n e ...............................................II. Polar Moment of Inertia of a Plane A rea ........................

III. Parallel-Axis T heorem .......................................................IV. Product of Inertia. Principal A x e s ...............................V. Change of Direction of Axes. Determination of the

Principal A xes..........................................................

A PPEN D IX E

T ables of Structural S h a p e s ..............................................

AUTHOR INDEX ................................................................................................................

PAGE

401405

410

417

417420422423

425

429

435

SUBJECT IN D EX 437

PA R T I

C H A PT E R I

TENSION AND COMPRESSION WITHIN THE ELASTIC LIMIT

1. Elasticity.—A m aterial body consists of small particles, or molecules, between which forces are acting. These molecu­lar forces resist the change in the shape of the body which external forces tend to produce. Under the action of external forces the particles of the body are displaced and the displace­m ents continue until equilibrium is established between the external and internal forces. The body is then in a state of strain. D uring deform ation the external forces acting upon the body do work, and this work is transform ed completely or partially into potential energy o f strain. A watch spring is an example of such an accumulation of potential energy in a strained body.I f the forces which produced the deform ation of the body are now gradually diminished, the body will return wholly or partly to its origi­nal shape and during this reversed deform ation the potential energy of strain which was accum ulated in the body m ay be recovered in the form of external work.

Consider, for instance, a prisma­tic bar loaded a t the end as shown in F ig ^ l .1 Under the action of this load a certain elongation of the bar will take place. The point of application of the load will then move in a downward direc­tion and positive work will be done by the load during this

1 I t is assumed tha t the load is acting along the axis of the bar, i.e., along the line passing through the centroids of the cross sections.

1

_Ldx

fa)t(t)

Fio. 1.

Strength Of Materials : ElementaryTheory And Problems

Publisher : CBS Publications ISBN : 9788123910307 Author : S Timoshenko

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