Advanced Mechanics of Materials 2

8/4/2019 Advanced Mechanics of Materials 2

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SIXTH EDITION

ADVANCED MECHANICSOF MATERIALS

ARTHUR P. BORESIProfessor EmeritusCivil and Architectural EngineeringThe University ofWyoming at Laramiean dProfessor EmeritusTheoretical and Applied MechanicsUniversity of Illinois at Urbana-Champaign

RICHARD J. SCHMIDTProfessorCivil and Architectural E ngineeringThe University ofWyoming at Laramie

JOHN WILEY & SONS , INC.


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CONTENTSCHAPTER 1 INTRODUCTION 11.1 Review of Elem entary Mech anics of Materials 1

1.1.1 Axially Loaded Mem bers 11.1.2 Torsionally Loaded Mem bers 31.1.3 Bending of Beam s 3

1.2 Me thods of Ana lysis 51.2.1 Method of Mecha nics of Materials 61.2.2 Method of Continuum Mecha nics and the

Theory of Elasticity 71.2.3 Deflections by Energ y Me thods 71.3 Stress-Strain Relations 81.3.1 Elastic and Inelastic Response of a Solid 81.3.2 Ma terial Prop erties 10

1.4 Failure and Lim its on Des ign 161.4.1 Mo des of Failure 19Problems 22References 24

CHAPTER 2 THEORIES OF STRESS AND STRAIN 252.1 Definition of Stress at a Poin t 252.2 Stress Nota tion 262.3 Symm etry of the Stress Array and Stress on an Arbitrarily

Oriented Plane 282.3.1 Symm etry of Stress Com ponents 282.3.2 Stresses Acting on Arbitrary Planes 292.3.3 Norma l Stress and Shear Stress on an Oblique

Plane 302.4 Transformation of Stress, Principal Stresses, and Other

Properties 31 ^2.4.1 Transformation of Stress 312.4.2 Principal Stresses 322.4.3 Principal Values and Directions 332.4.4 Octahedral Stress 362.4.5 Mean and Deviator Stresses 372.4.6 Plan e Stress 382.4.7 Mo hr's Circle in Two Dimen sions 402.4.8 Mohr's" Circles in Three Dimen sions 43

2.5 Differential Equations of Motion of a DeformableBody 502.5.1 Specialization of Equations 2.46 52

2.6 Deformation of a Deformable Body 54

2.7 Strain Theory, Transformation of Strain, and PrincipalStrains 552.7.1 Strain of a Line Elemen t 552.7.2 Final Direction of a Line Element 572.7.3 Rotation Between Two Line Elemen ts

(Definition of Shear Strain) 582.7.4 Principal Strains 60

2.8 Small-Displacem ent Theory 612.8.1 Strain Com patibility Relations 622.8.2 Strain-D isplacem ent Relations for Orthogonal

Curvilinear Coordinates 632.9 Strain Measurem ent and Strain Rosettes 70Problems 72References 78

CHAPTER 3 LINEAR STRESS-STRAIN-TEMPERATURERELATIONS 793.1 First Law of Therm odynam ics, Internal-Energy Density,

and Com plementary Internal-Energy Density 793.1.1 Elasticity and Internal-Energy Density 813.1.2 Elasticity and Com plementary Internal-Energy

Density 823.2 Hook e's Law: Anisotropic Elasticity 843.3 Hook e's Law: Isotropic Elasticity 85

3.3.1 Isotropic and Hom ogeneous Materials 853.3.2 Strain-Energy Density of Isotropic Elastic

Materials 853.4 Equations of Therm oelasticity for Isotropic

Materials 913.5 Hook e's Law : Orthotropic Materials 93

Problems 101References 103

CHAPTER 4 INELASTIC MATERIAL BEHAVIOR 1044.1 Limitations on the Use of Uniaxial Stress-Stra in

Data 1044.1.1 Rate of Loading 1054.1.2 Temperature Lower Than Room

Temperature 1054.1.3 Temperature Higher Than Room

Temperature 105

IX


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CONTENTS

4.1.4 Unloa ding and Load Reversal 1054.1.5 Multiaxia l States of Stress 106

4.2 Nonlinear Material Response 1074.2.1 Models of Uniaxial Stress-Strain Curves 108

4.3 Yield Criteria: Gen eral Con cepts 1134.3.1 Max imum Principa l Stress Criterion 1144.3.2 Maximu m Principal Strain Criterion 1164.3.3 Strain-Ene rgy Density Criterion 1164.4 Yielding of Du ctile Me tals 1174.4.1 Ma xim um Shear-Stress (Tresca) Criterion 1184.4.2 Distortional Energy Density (von Mises)Criterion 1204.4.3 Effect of Hyd rostatic Stress and theTT-Plane 122

4.5 Alternative Yield Criteria 1264.5.1 Moh r-Coulo mb Yield Criterion 1264.5.2 Drucke r-Prager Yield Criterion 1284.5.3 Hi ll's Criterion for Ortho tropic Ma terials 1284.6 Gene ral Yielding 1294.6.1 Elastic-Plastic Bending 1314.6.2 Fully Plastic Mom ent 1324.6.3 Shear Effect on Inelastic Bend ing 1344.6.4 Mod ulus of Ruptu re 1344.6.5 Com parison of Failure Criteria 1364.6.6 Interpretation of Failure Criteria for Gene ralYielding 137Problems 142References 146

CHAPTER 5 APPLICATIONS OF ENERGY METHODS 1475.1 Princip le of Stationary Poten tial Energy 1475.2 Cas tiglian o's Theorem on Deflections 1525.3 Castigliano's Theorem on Deflections for LinearLoad-Deflection Relations 155

5.3.1 Strain Energy U N for Axial Load ing 1565.3.2 Strain Energie s U M and U s for Bea ms 1585.3.3 Strain Energy U T for Torsion 1605.4 Deflections of Statically Determ inate Structures 1635.4.1 Curved Beam s Treated as Straight Beam s 1655.4.2 Dumm y Load Method and Dumm y Unit LoadMethod 170

5.5 Statically Indete rmin ate Structures 1775.5.1 Deflections of Statically Indete rmin ate

Structures 180Problems 187References 199CHAPTER 6 TORSION 2006.1 Torsion of a Prisma tic Bar of Circular Cross Section 2006.1.1 Design of Transm ission Shafts 2046.2 Saint-Vena nt's Sem iinverse Method 2096.2.1 Geom etry of Deform ation 209

6.2.2 Stresses at a Point and Equ ations ofEquilibrium 2106.2.3 Boun dary Cond itions 211

6.3 Linea r Elastic Solution 2136.3.1 Elliptical Cross Section 2146.3.2 Equilate ral Triangle Cross Section 2156.3.3 Other Cross Sections 216

6.4 The Prandtl Elastic-Mem brane (Soap-Film) Analogy 2166.4.1 Remark on Reentrant Corners 2196.5 Narrow Recta ngula r Cross Section 219

6.5.1 Cross Sections Made Up of Long NarrowRectangles 2216.6 Torsion of Rectangular Cross Section Mem bers 2226.7 Hollow Thin-Wall Torsion Mem bers and MultiplyConnected Cross Sections 228

6.7.1 Hollow Thin-Wall Torsion Mem ber HavingSeveral Com partments 2306.8 Thin-Wall Torsion Mem bers with RestrainedEnds 234

6.8.1 I-Section Torsion Mem ber Having One EndRestrain ed from Warping 2356.8.2 Various Loads and Suppo rts for Beam s inTorsion 239

6.9 Num erical Solution of the Torsion Problem 2396.10 Inelastic Torsion: Circular Cross Section s 243

6.10.1 Mo dulus of Rup ture in Torsion 2446.10.2 Elastic-Plastic and Fully PlasticTorsion 2446.10.3 Residu al Shear Stress 246

6.11 Fully Plastic Torsion: Gen eral Cross Sections 250Problems 254References 262

CHAPTER 7 BENDING OF STRAIGHT BEAMS 2637.1 Fundam entals of Beam Bending 263

7.1.1 Centroid al Coord inate Axes 2637.1.2 Shear Loading of a Beam and Shear CenterDefined 264

& 7.1.3 Symm etrical Bending 2657.1.4 Nonsymm etrical Bending 2687.1.5 Plane of Loads: Symmetrical andNonsymm etrical Loading 268

7.2 Bending Stresses in Beams Subjected to NonsymmetricalBending 2727.2.1 Equa tions of Equilibrium 2727.2.2 Geom etry of Deform ation 2737.2.3 Stress-Strain Relations 2737.2.4 Load -Stress Relation for Nonsym metricalBending 2737.2.5 Neutral Axis 2747.2.6 More Convenient Form for the Flexure Stress

azz 2757.3 Deflections of Straight Beam s Subjected toNonsymm etrical Bending 280


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CONTENTS X I

7.4 Effect of Inclined Loads 2847.5 Fully Plastic Load for Nonsym metrical Bending 285


CHAPTER 8 SHEAR CENTER FOR THIN-WALL BEAMCROSS SECTIONS 2958.1 Approximations for Shear in Thin-Wall Beam Cross

Sections 2958.2 Shear Flow in Thin-Wall Beam Cross Sections 2968.3 Shear Cente r for a Chan nel Section 2988.4 Shear Center of Com posite Beam s Formed from

Stringers and Thin Webs 3038.5 Shear Cente r of Box Beam s 306


CHAPTER 9 CURVED BEAMS 3199.1 Introductio n 3199.2 Circumferential Stresses in a Curved Beam 320

9.2.1 Location of Neutral Axis of Cross Section 3269.3 Radial Stresses in Curved Beam s 333

9.3.1 Curved Beam s Mad e from AnisotropicMaterials 334

9.4 Correction of Circumferential Stresses in Curved Beam sHaving I, T, or Sim ilar Cross Section s 3389.4.1 Bleich 's Correction Factors 340

9.5 Deflections of Curved Bea ms 3439.5.1 Cross Sections in the Form of an I, T, etc. 346

9.6 Statically Indeterminate Curved Beam s: Closed RingSubjected to a Concentrated Load 3489.7 Fully Plastic Loads for Curved Beam s 350

9.7.1 Fully Plastic Versus Maxim um Elastic Loads forCurved Beam s 351


CHAPTER 10 BEAMS ON ELASTIC FOUNDATIONS 3 5 7 "10.1 Gene ral Theory 35710.2 Infinite Beam Subjected to a Concentrated Load :Boundary Conditions 360

10.2.1 Me thod of Supe rposition 36310.2.2 Beam Supported on Equally Spaced DiscreteElastic Suppo rts 364

10.3 Infinite Beam Subjected to a Distributed LoadSegment 36910.3.1 Uniform ly Distribute d Load 36910.3.2 PL' 1 43112.3.3 The Imperfection Method 43212.3.4 The Energy Method 433


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X I I CONTENTS

12.4 Euler Buckling of Columns with Linearly Elastic EndConstraints 43612.5 Local Buck ling of Colum ns 44012.6 Inelastic Buckling of Colum ns 442

12.6.1 Inelastic Buc kling 44212.6.2 Two Formulas for Inelastic Buckling of anIdeal Column 44312.6.3 Tangent-Modulus Formula for an InelasticBuckling Load 44412.6.4 Direct Tangent-Modulus Method 446Problems 450References 455

CHAPTER 13 FLAT PLATES 45713.1 Introduc tion 45713.2 Stress Resu ltants in a Flat Plate 45813.3 Kinem atics: Strain-Displace men t Relations for

Plates 46113.3.1 Rota tion of a Plate Surface Elem ent 46413.4 Equilibrium Equations for Small-Di splacement Theoryof Flat Plates 466

13.5 Stress-Strain-T em perature Relations for IsotropicElastic Plates 46913.5.1 Stress Com ponents in Terms of Tractions

and Mom ents 47213.5.2 Pure Ben ding of Plates 472

13.6 Strain Energy of a Plate 47213.7 Bou ndary Con ditions for Plates 47313.8 Solution of Rectangular Plate Problems 476

13.8.1 Solution of V 2V 2w = for a RectangularPlate 477 D13.8.2 Westergaard Approx imate Solution forRectangular Plates: Uniform Load 479

13.8.3 Deflection of a Rec tangu lar Plate :Uniform ly Distributed Loa d 48213.9 Solution of Circu lar Plate Proble ms 486

13.9.1 Solution of V 2V 2w = for a CircularPlate 486 D13.9.2 Circular Plates with Simply SupportedEdges 48813.9.3 Circula r Plates with Fixed Edg es 48813.9.4 Circular Plate with a Circular Hole at the

Center 48913.9.5 Summ ary for Circular Plates with SimplySuppo rted Edg es 49013.9.6 Summ ary for Circular Plates with FixedEdges 49113.9.7 Sum mary for Stresses and Deflections inFlat Circular Plates with Central Holes 49213.9.8 Sum mary for Larg e Elastic Deflections ofCircular Plates: Clamped Edge andUniformly Distributed Load 492

13.9.9 Significant Stress When Edge s AreClamped 49513.9.10 Load on a Plate When Edges AreClamped 49613.9.11 Summ ary for Large Elastic Deflections ofCircular Plates: Simply Supported Edg e andUniform ly Distributed Load 49713.9.12 Rectangu lar or Other Shaped Plates withLarg e Deflections 498Problems 500References 501

CHAPTER 14 STRESS CONCENTRATIONS 50214.1 Nature of a Stress Concentration Problem and the Stress

Concentration Factor 50414.2 Stress Concentration Factors: Theory of Elasticity 507

14.2.1 Circula r Hole in an Infinite Plate Und erUnia xial Tension 507

14.2.2 Elliptic Hole in an Infinite Plate Stressed in aDirection Perpendicular to the Major Axis ofthe Ho le 508

14.2.3 Elliptica l Hole in an Infinite Plate Stressed inthe Direction Perpendicular to the Mino r Axisof the Hole 511

14.2.4 Crack in a Plate 51214.2.5 Ellipso idal Cavity 51214.2.6 Groove s and Hole s 513

14.3 Stress Concentration Factors: Combined Loads 51514.3.1 Infinite Plate with a Circu lar Hole 51514.3.2 Elliptic al Hole in an Infinite Plate Uniform ly

Stressed in Directions of Major an d MinorAxes of the Hole 51614.3.3 Pure Shear Parallel to Major and Minor Axes

of the Elliptica l Hole 51614.3.4 Elliptica l Hole in an Infinite Plate with

Different Loads in Two PerpendicularDirections 517

14.3.5 Stress Concentration at a Groove in a C ircular*" Shaft 520

14.4 Stress Concentration Factors: ExperimentalTechniques 52214.4.1 Photoelastic Method 52214.4.2 Strain-Gage Method 52414.4.3 Elastic Torsional Stress Concentration at a

Fillet in a Shaft 52514.4.4 Elastic Mem brane Method: Torsional Stress

Concentration 52514.4.5 Beam s with Rectangular Cross Sections 527

14.5 Effective Stress Con centra tion Factors 53014.5.1 Definition of Effective Stress Conc entration

Factor 53014.5.2 Static Load s 53214.5.3 Repe ated Loads 532


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CONTENTS XI I I

14.5.4 Resid ual Stresses 53414.5.5 r Very Abrupt Changes in Section: StressGradient 53414.5.6 Significance of Stress Grad ient 53514.5.7 Impact or Energy Loading 536

14.6 Effective Stress Conce ntration Factors: InelasticStrains 53614.6.1 Neu ber's Theorem 537Problems 539References 541

CHAPTER 15 FRACTURE MECHANICS 54315.1 Failure Criteria and Fracture 544

15.1.1 Brittle Fracture of Mem bers Free of Cracksand Flaws 545

15.1.2 Brittle Fracture of Cracked or FlawedMembers 54515.2 The Stationary Crack 551

15.2.1 Blunt Crack 55315.2.2 Sharp Crack 55415.3 Crack Propag ation and the Stress Intensity Factor 55515.3.1 Elastic Stress at the Tip of a Sharp

Crack 55515.3.2 Stress Intens ity Facto r: Definition andDerivation 55615.3.3 Derivation of Crack Extension Force G 55615.3.4 Critical Value of Crack Extension Force 558

15.4 Fractu re: Other Facto rs 56115.4.1 ElasticPlastic Frac ture Me chan ics 56215.4.2 Crack-Growth Analysis 56215.4.3 Load Spectra and Stress History 56215.4.4 Testing and Experimen tal DataInterpretation 563Problems 564References 565

CHAPTER 16 FATIGUE: PROGRESSIVE FRACTURE 56716.1 Fracture Resulting from Cyclic Loading 568 .

16.1.1 Stress Conc entrations 57316.2 Effective Stress Co ncen tration Facto'rs: Rep eated

Loads 57516.3 Effective Stress Concen tration Factors: Other

Influences 57516.3.1 Corrosion Fatigue 57516.3.2 Effect of Ran ge of Stress 57716.3.3 Metho ds of Redu cing Harmful Effects of

Stress Conc entrations 57716.4 Low Cycle Fatigue and the e-N Relation 580

16.4.1 Hysteresis Loop 580

16.4.2 Fatigue-Life Curve and the e-NRelation 581


CHAPTER 17 CONTACT STRESSES 58917.1 Introduction 58917.2 The Problem of Determ ining Contact Stresses 59017.3 Geom etry of the Contact Surface 591

17.3.1 Fundam ental Assump tions 59117.3.2 Co ntact Surface Shap e After Load ing 59217.3.3 Justification of Eq . 17.1 59217.3.4 Brief Discussion of the Solution 595

17.4 Notation and Meaning of Terms 59617.5 Expressions for Principal Stresses 59717.6 Method of Com puting Contact Stresses 598

17.6.1 Principal Stresses 59817.6.2 Max imum Shear Stress 59917.6.3 Max imum Octahedral Shear Stress 59917.6.4 Max imum Orthogo nal Shear Stress 59917.6.5 Curves for Com puting Stresses for Any Valueof B/A 605

17.7 Deflection of Bod ies in Point Con tact 60717.7.1 Significance of Stresses 61117.8 Stress for Two Bodies in Line Con tact: Loads Normal to

Contact Area 61117.8.1 Max imum Principal Stresses: k = 0 61317.8.2 Max imum Shear Stress: k = 0 61317.8.3 Max imum Octahedral Shear Stress:

k = 0 61317.9 Stresses for Two Bod ies in Line Contact: Loads Normaland Tangent to Contact Area 613

17.9.1 Roller on Plane 61417.9.2 Principal Stresses 61617.9.3 Max imum Shear Stress 61717.9.4 Max imum Octahedral Shear Stress 61717.9.5 Effect of Mag nitude of Friction

Coefficient 61817.9.6 Range of Shear Stress for One Load

Cycle 619Problems 622References 623

CHAPTER 18 CREEP: TIME-DEPENDENTDEFORMATION 62418.1 Definition of Creep and the Creep Curve 62418.2 The Tension Creep Test for Metals 62618.3 One-D imensional Creep Formulas for Metals Subjected

to Constant Stress and Elevated Temperature 626


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18.4 One-Dimensional Creep of Metals Subjected toVariable Stress and Temperature . 63118.4.1 Preliminary Concep ts 63118.4.2 , Similarity of Cree p Curv es 63318.4.3 Temperature Dependen cy 63518.4.4 Variable Stress and Tem perature 635

18.5 Creep Und er Multiaxial States of Stress 64018.5.1 General Discussion 64018.6 Flow Rule for Creep of Metals Subjected to Mu ltiaxialStates of Stress 64318.6.1 Steady-State Creep 64418.6.2 Non steady Creep 648

18.7 An App lication of Creep of Metals 64918.7.1 Summ ary 65018.8 Creep of Non metals 650

18.8.1 Asp halt 65018.8.2 Co ncrete 65118.8.3 Woo d 652References 654

APPENDIX A AVERAGE MECHANICAL PROPERTIES OFSELECTED M ATERIALS 657

APPENDIX B SECOND MOMENT (MOMENT OF INERTIAOF A PLANE AREA 660B.IB.2B.3

660oments of Inertia of a Plane AreaParallel Axis Theorem 661Transformation Equations for Moments and Products oInertia 664B.3.1 Principal Axes of Inertia 665Problems 666

APPENDIX C PROPERTIES OF STEEL CROSSSECTIONS 668

AUTHOR INDEX 673

SUBJECT INDEX 676

Advanced Mechanics of Materials 2

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