SEVENTHEDITIONMECHANICSOF MATERIALSJAMESM.GEREBARRYJ.GOODNO
CONVERSIONSBETWEENU.S.CUSTOMARYUNITSANDSIUNITSTimes conversionfactorU.S. Customary unitEquals SI unitAccuratePracticalAcceleration (linear)fi/s2m/s20.3048*0.305foot per second squaredmeter per second squaredin./s?2m/s20.0254*0.0254meter per second squaredinchpersecondsquaredAreaff2m20.09290304*squarefoot0.0929square meterin.2mm?645.16*645 square inchsquare millimeterDensity (mass)kg/m3slug/ft3515515.379slugper cubicfootkilogrampercubicmeterDensity (weight)Ib/ft3N/m3157157.087pound per cubic footnewton per cubic meterIb/in.3271271.447pound per cubic inchkilonewton per cubickN/m3meterEnergy: workJft-lb1.355821.36foot-poundjoule (N-m)Jin.-Ib0.1129850.113inch-poundjouleMJ3.6*3.6kilowatt-hourkWhmegajouleJ1055British thermal unitBtu1055.06jouleForce1bnewton (kg'm/s3)Npound4.448224.45k4.45kN4.44822kip (1000 pounds)kilonewtonForce per unit length1b/ft14.614.5939N/mpound per footnewtonpermeter175pound per inchIb/in.175.127N/mnewton permeterk/ftkip per foot14.593914.6kN/mkilonewton per meter175k/in.175.127kN/mkip per inchkilonewtonpermeterLengthft0.3048*0.305footmeterm25.4*inchin.25.4millimetermmmi1.609344*1.61milekilometerkmMass1b-s°/ft14.6kgslug14.5939kilogramMomentofaforce; torqueIb-ft1.355821.36Nmpound-footnewton meterIb-in.0.113 pound-inch0.112985N-mnewton meterk-ft1.36kip-foot1.35582kN-mkilonewton meter0.113kip-inchk-in.0.112985kN-mkilonewton meter
CONVERSIONS BETWEEN U.S. CUSTOMARY UNITS AND SI UNITS Times conversion factor U.S. Customary unit Accurate Practical Equals SI unit Acceleration (linear) foot per second squared ft/s2 0.3048* 0.305 meter per second squared m/s2 inch per second squared in./s2 0.0254* 0.0254 meter per second squared m/s2 Area square foot ft2 0.09290304* 0.0929 square meter m2 square inch in.2 645.16* 645 square millimeter mm2 Density (mass) slug per cubic foot slug/ft3 515.379 515 kilogram per cubic meter kg/m3 Density (weight) pound per cubic foot lb/ft3 157.087 157 newton per cubic meter N/m3 pound per cubic inch lb/in.3 271.447 271 kilonewton per cubic meter kN/m3 Energy; work foot-pound ft-lb 1.35582 1.36 joule (Nm) J inch-pound in.-lb 0.112985 0.113 joule J kilowatt-hour kWh 3.6* 3.6 megajoule MJ British thermal unit Btu 1055.06 1055 joule J Force pound lb 4.44822 4.45 newton (kgm/s2 ) N kip (1000 pounds) k 4.44822 4.45 kilonewton kN Force per unit length pound per foot lb/ft 14.5939 14.6 newton per meter N/m pound per inch lb/in. 175.127 175 newton per meter N/m kip per foot k/ft 14.5939 14.6 kilonewton per meter kN/m kip per inch k/in. 175.127 175 kilonewton per meter kN/m Length foot ft 0.3048* 0.305 meter m inch in. 25.4* 25.4 millimeter mm mile mi 1.609344* 1.61 kilometer km Mass slug lb-s2 /ft 14.5939 14.6 kilogram kg Moment of a force; torque pound-foot lb-ft 1.35582 1.36 newton meter N·m pound-inch lb-in. 0.112985 0.113 newton meter N·m kip-foot k-ft 1.35582 1.36 kilonewton meter kN·m kip-inch k-in. 0.112985 0.113 kilonewton meter kN·m
CONVERSIONSBETWEENU.S.CUSTOMARYUNITSANDSIUNITS(COntinued)TimesconversionfactorU.S. Customary unitEquals SI unitPracticalAccurateMoment of inertia (area)in.4416,231416,000inch tofourth powermillimeter to fourthmmpowerin.40.416231×10-60.416×10-6m4inch to fourth powermeter to fourth powerMoment of inertia (mass)slug-f?kg-m?1.355821.36slug foot squaredkilogram meter squaredPowerW1.36ft-1b/s1.35582foot-pound per secondwatt (J/s or N-m/s)w0.0226ft-Ib/min0.0225970wattfoot-pound per minuteWhp745.701746horsepower (550 ft-Ib/s)wattPressure; stresspascal (N/m3)psf47.9Pa47.8803pound per square footPapsi6894.766890pascalpound per square inchksf47.947.8803kilopascalkPakip per square footksi6.894766.89MPakip per square inchmegapascalSection modulusin.3mm16,40016,387.1inch tothird powermillimetertothird powerin. 316.3871 ×10~6m316.4 × 10~6inch to third powermeter to third powerVelocity (linear)ft/s0.3048*0.305m/sfoot per secondmeter per second0.0254*in./s0.0254m/sinch per secondmeter per second0.44704*mph0.447m/smile per hourmeter per second1.609344*1.61 km/h mile per hourmphkilometer per hourVolumem3ftcubicfoot0.02831680.0283cubic meterin.3m316.3871 × 10-616.4×10-6cubic inchcubic meterin.3cm316.4cubic inch16.3871cubic centimeter (cc)gallon (231 in.3)3.79Lgal.3.78541literm3gal.0.003785410.00379gallon (231 in.3)cubic meter*AnasteriskdenotesanexactconversionfactorNote:ToconvertfromSI unitstoUSCS units,dividebytheconversionfactorS(7(°F) - 32] = T(K) 273.15TemperatureConversionFormulasT(°C) =9 -2+21- 2T(K) = 9T(K)459.67T(C) +32 =T(°F) =55
Temperature Conversion Formulas T(°C) 5 9 [T(°F) 32] T(K) 273.15 T(K) 5 9 [T(°F) 32] 273.15 T(°C) 273.15 T(°F) 9 5 T(°C) 32 9 5 T(K) 459.67 CONVERSIONS BETWEEN U.S. CUSTOMARY UNITS AND SI UNITS (Continued) Times conversion factor U.S. Customary unit Accurate Practical Equals SI unit Moment of inertia (area) inch to fourth power in.4 416,231 416,000 millimeter to fourth power mm4 inch to fourth power in.4 0.416231 106 0.416 106 meter to fourth power m4 Moment of inertia (mass) slug foot squared slug-ft2 1.35582 1.36 kilogram meter squared kg·m2 Power foot-pound per second ft-lb/s 1.35582 1.36 watt (J/s or N·m/s) W foot-pound per minute ft-lb/min 0.0225970 0.0226 watt W horsepower (550 ft-lb/s) hp 745.701 746 watt W Pressure; stress pound per square foot psf 47.8803 47.9 pascal (N/m2 ) Pa pound per square inch psi 6894.76 6890 pascal Pa kip per square foot ksf 47.8803 47.9 kilopascal kPa kip per square inch ksi 6.89476 6.89 megapascal MPa Section modulus inch to third power in.3 16,387.1 16,400 millimeter to third power mm3 inch to third power in.3 16.3871 106 16.4 106 meter to third power m3 Velocity (linear) foot per second ft/s 0.3048* 0.305 meter per second m/s inch per second in./s 0.0254* 0.0254 meter per second m/s mile per hour mph 0.44704* 0.447 meter per second m/s mile per hour mph 1.609344* 1.61 kilometer per hour km/h Volume cubic foot ft3 0.0283168 0.0283 cubic meter m3 cubic inch in.3 16.3871 106 16.4 106 cubic meter m3 cubic inch in.3 16.3871 16.4 cubic centimeter (cc) cm3 gallon (231 in.3 ) gal. 3.78541 3.79 liter L gallon (231 in.3 ) gal. 0.00378541 0.00379 cubic meter m3 *An asterisk denotes an exact conversion factor Note: To convert from SI units to USCS units, divide by the conversion factor
MechanicsofMaterialsSEVENTH EDITIONJamesM.GereProfessorEmeritus,StanfordUniversityBarryJ.GoodnoGeorgia InstituteofTechnologyCENGAGELearningAustralia · Brazil - Japan · Korea · Mexico · Singapore + Spain · United Kingdom - United States
Mechanics of Materials SEVENTH EDITION James M. Gere Professor Emeritus, Stanford University Barry J. Goodno Georgia Institute of Technology Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States
1CENGAGELearning2009 Cengage LearningMechanics of Materials, Seventh EditionJames M. Gere and Barry J. GoodnoALL RIGHTS RESERVED. No part of this work covered by the copyright herein may bereproduced, transmitted, stored or used in any form or by any means graphic, electronic,Director, Global Engineering Program: Chris Carsonor mechanical,including but not limited to photocopying,recording,scanning, digitizing,Senior Developmental Editor: Hilda Gowanstaping, Web distribution, information networks, or information storage and retrievalPermissions: Nicola Winstanleysystems,except as permitted under Section107or108of the1976 United StatesCopyright Act, without the prior written permission of the publisher.Production Service: RPK Editorial Services, IncCopy Editor:RoseKernanFor product information and technology assistance, contact us atProofreader: Martha McMasterCengage Learning Customer & Sales Support,1-800-354-9706Indexer: Shelly Gerger-KnechtiFor permission to use material from this text or product,submit all requests online atCreative Director: Angela Cluercengage.com/permissionsFurther permissions questions can be emailed toText Designer: Peter Papayanakispermissionrequest@cengage.comCover Designer: Andrew AdamsCover Image: Guggenheim Bilbao Photo@graficart.net/Alamy,Library of Congress Control Number:2008923451Lee Chin Crystal Building Photo Courtesy of AlanISBN-13: 978-0-534-55397-5Marsh/First LightISBN-10: 0-534-55397-4Compositor: IntegraPrinter: Quebecor WorldCengage Learning1120Birchmount RoadTorontoONM1K5G4CanadaCengage Learning is a leading provider of customized learning solutions with officelocations around the globe, including Singapore, the United Kingdom, Australia, MexicoBrazil,and Japan.Locate your local office at: international.cengage.com/regionCengage Learning products are represented in Canada by Nelson Education Ltd.For your course and learning solutions, visit academic.cengage.comPurchase any of our products at your local college store or at our preferred online storewww.ichapters.comPrinted in the United States of America1234567111009.08
Mechanics of Materials, Seventh Edition James M. Gere and Barry J. Goodno Director, Global Engineering Program: Chris Carson Senior Developmental Editor: Hilda Gowans Permissions: Nicola Winstanley Production Service: RPK Editorial Services, Inc Copy Editor: Rose Kernan Proofreader: Martha McMaster Indexer: Shelly Gerger-Knechtl Creative Director: Angela Cluer Text Designer: Peter Papayanakis Cover Designer: Andrew Adams Cover Image: Guggenheim Bilbao Photo © graficart.net/Alamy; Lee Chin Crystal Building Photo Courtesy of Alan Marsh/First Light Compositor: Integra Printer: Quebecor World © 2009 Cengage Learning ALL RIGHTS RESERVED. No part of this work covered by the copyright herein may be reproduced, transmitted, stored or used in any form or by any means graphic, electronic, or mechanical, including but not limited to photocopying, recording, scanning, digitizing, taping, Web distribution, information networks, or information storage and retrieval systems, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without the prior written permission of the publisher. For product information and technology assistance, contact us at Cengage Learning Customer & Sales Support, 1-800-354-9706 For permission to use material from this text or product, submit all requests online at cengage.com/permissions Further permissions questions can be emailed to permissionrequest@cengage.com Library of Congress Control Number: 2008923451 ISBN-13: 978-0-534-55397-5 ISBN-10: 0-534-55397-4 Cengage Learning 1120 Birchmount Road Toronto ON M1K 5G4 Canada Cengage Learning is a leading provider of customized learning solutions with office locations around the globe, including Singapore, the United Kingdom, Australia, Mexico, Brazil, and Japan. Locate your local office at: international.cengage.com/region Cengage Learning products are represented in Canada by Nelson Education Ltd. For your course and learning solutions, visit academic.cengage.com Purchase any of our products at your local college store or at our preferred online store www.ichapters.com Printed in the United States of America 1 2 3 4 5 6 7 11 10 09 08
ContentsJamesMonroeGereixPhotoCreditsxPrefacexiSymbolsXVGreek Alphabetxvili1Tension,Compression,and Shear21.1IntroductiontoMechanicsof Materials51.2NormalStressandStrain71.3MechanicalPropertiesof Materials151.4Elasticity,Plasticity,andCreep241.527Linear Elasticity,Hooke'sLaw,and Poisson's Ratio1.6ShearStressandStrain32431.7AllowableStressesandAllowableLoads491.8DesignforAxial Loads andDirect ShearChapterSummary&Review55Problems572AxiallyLoadedMembers882.1Introduction912.2Changes in Lengths of Axially Loaded Members912.3Changes in Lengths Under NonuniformConditions1002.4StaticallyIndeterminateStructures1072.5116ThermalEffects,Misfits,andPrestrains2.6Stresseson InclinedSections1282.7StrainEnergy140*2.8ImpactLoading153*2.9Repeated Loading andFatigue162*2.10StressConcentrations164NonlinearBehavior170*2.11★ Specialized and/or advanced topics川
Contents James Monroe Gere ix Photo Credits x Preface xi Symbols xv Greek Alphabet xviii 1 Tension, Compression, and Shear 2 1.1 Introduction to Mechanics of Materials 5 1.2 Normal Stress and Strain 7 1.3 Mechanical Properties of Materials 15 1.4 Elasticity, Plasticity, and Creep 24 1.5 Linear Elasticity, Hooke’s Law, and Poisson’s Ratio 27 1.6 Shear Stress and Strain 32 1.7 Allowable Stresses and Allowable Loads 43 1.8 Design for Axial Loads and Direct Shear 49 Chapter Summary & Review 55 Problems 57 2 Axially Loaded Members 88 2.1 Introduction 91 2.2 Changes in Lengths of Axially Loaded Members 91 2.3 Changes in Lengths Under Nonuniform Conditions 100 2.4 Statically Indeterminate Structures 107 2.5 Thermal Effects, Misfits, and Prestrains 116 2.6 Stresses on Inclined Sections 128 2.7 Strain Energy 140 ★2.8 Impact Loading 153 ★2.9 Repeated Loading and Fatigue 162 ★2.10 Stress Concentrations 164 ★2.11 Nonlinear Behavior 170 iii ★ Specialized and/or advanced topics
ivCONTENTS*2.12ElastoplasticAnalysis175181ChapterSummary&Review182Problems3Torsion2203.1Introduction2223.2TorsionalDeformationsof a CircularBar2233.3CircularBars of LinearlyElastic Materials2263.4NonuniformTorsion2383.5Stresses and Strains in Pure Shear 2453.6252RelationshipBetweenModuliof ElasticityEandG3.7TransmissionofPowerbyCircularShafts2543.8StaticallyIndeterminateTorsionalMembers2593.9StrainEnergyinTorsionandPureShear2633.10Thin-WalledTubes270*3.11Stress Concentrations in Torsion279ChapterSummary&Review282Problems2834304Shear Forces and Bending Moments4.1Introduction3064.2TypesofBeams,Loads,andReactions3064.3ShearForces andBendingMoments3134.4RelationshipsBetweenLoads,ShearForces,andBendingMoments3204.5325Shear-Force and Bending-Moment DiagramsChapterSummary&Review337Problems3385Stressesin Beams (BasicTopics)3505.1Introduction3535.2353PureBending and NonuniformBending5.3Curvature of a Beam 3545.4LongitudinalStrainsinBeams3565.5361Normal Stresses inBeams (LinearlyElastic Materials)5.6Designof BeamsforBendingStresses3745.7NonprismaticBeams3835.8Shear Stresses inBeams of Rectangular Cross Section3875.9Shear Stresses inBeamsof CircularCross Section3975.10Shear Stresses in theWebs of Beams withFlanges400
★2.12 Elastoplastic Analysis 175 Chapter Summary & Review 181 Problems 182 3 Torsion 220 3.1 Introduction 222 3.2 Torsional Deformations of a Circular Bar 223 3.3 Circular Bars of Linearly Elastic Materials 226 3.4 Nonuniform Torsion 238 3.5 Stresses and Strains in Pure Shear 245 3.6 Relationship Between Moduli of Elasticity E and G 252 3.7 Transmission of Power by Circular Shafts 254 3.8 Statically Indeterminate Torsional Members 259 3.9 Strain Energy in Torsion and Pure Shear 263 3.10 Thin-Walled Tubes 270 ★3.11 Stress Concentrations in Torsion 279 Chapter Summary & Review 282 Problems 283 4 Shear Forces and Bending Moments 304 4.1 Introduction 306 4.2 Types of Beams, Loads, and Reactions 306 4.3 Shear Forces and Bending Moments 313 4.4 Relationships Between Loads, Shear Forces, and Bending Moments 320 4.5 Shear-Force and Bending-Moment Diagrams 325 Chapter Summary & Review 337 Problems 338 5 Stresses in Beams (Basic Topics) 350 5.1 Introduction 353 5.2 Pure Bending and Nonuniform Bending 353 5.3 Curvature of a Beam 354 5.4 Longitudinal Strains in Beams 356 5.5 Normal Stresses in Beams (Linearly Elastic Materials) 361 5.6 Design of Beams for Bending Stresses 374 5.7 Nonprismatic Beams 383 5.8 Shear Stresses in Beams of Rectangular Cross Section 387 5.9 Shear Stresses in Beams of Circular Cross Section 397 5.10 Shear Stresses in the Webs of Beams with Flanges 400 iv CONTENTS
VCONTENTS**5.11Built-UpBeamsandShearFlow408**5.12BeamswithAxialLoads412**5.13418Stress Concentrations inBending421ChapterSummary&ReviewProblems4246Stresses in Beams (Advanced Topics)4546.1Introduction4576.2457CompositeBeams6.3Transformed-Section Method14666.4DoublySymmetricBeamswithInclinedLoads4726.5Bendingof UnsymmetricBeamsS4796.6TheShear-CenterConcept4876.7Shear Stresses inBeams ofThin-WalledOpenCross Sections4896.8492Shear Stresses in Wide-FlangeBeams6.9496ShearCentersofThin-WalledOpenSections**6.10ElastoplasticBending504514ChapterSummary&ReviewProblems5167Analysis ofStressand Strain5367.1539Introduction7.2PlaneStress5407.3PrincipalStressesandMaximumShearStresses5487.4Mohr's CircleforPlaneStress5587.5575Hooke'sLawforPlaneStress7.6Triaxial Stress5807.7Plane Strain 584ChapterSummary&Review600602Problems8Applications of Plane Stress (Pressure Vessels,Beams,and Combined Loadings)6188.1621Introduction8.2Spherical PressureVessels6218.3Cylindrical Pressure VesselsS6278.4635MaximumStressesinBeams8.5Combined Loadings645661ChapterSummary&ReviewProblems 663** Advanced topics
★★5.11 Built-Up Beams and Shear Flow 408 ★★5.12 Beams with Axial Loads 412 ★★5.13 Stress Concentrations in Bending 418 Chapter Summary & Review 421 Problems 424 6 Stresses in Beams (Advanced Topics) 454 6.1 Introduction 457 6.2 Composite Beams 457 6.3 Transformed-Section Method 466 6.4 Doubly Symmetric Beams with Inclined Loads 472 6.5 Bending of Unsymmetric Beams 479 6.6 The Shear-Center Concept 487 6.7 Shear Stresses in Beams of Thin-Walled Open Cross Sections 489 6.8 Shear Stresses in Wide-Flange Beams 492 6.9 Shear Centers of Thin-Walled Open Sections 496 ★★6.10 Elastoplastic Bending 504 Chapter Summary & Review 514 Problems 516 7 Analysis of Stress and Strain 536 7.1 Introduction 539 7.2 Plane Stress 540 7.3 Principal Stresses and Maximum Shear Stresses 548 7.4 Mohr’s Circle for Plane Stress 558 7.5 Hooke’s Law for Plane Stress 575 7.6 Triaxial Stress 580 7.7 Plane Strain 584 Chapter Summary & Review 600 Problems 602 8 Applications of Plane Stress (Pressure Vessels, Beams, and Combined Loadings) 618 8.1 Introduction 621 8.2 Spherical Pressure Vessels 621 8.3 Cylindrical Pressure Vessels 627 8.4 Maximum Stresses in Beams 635 8.5 Combined Loadings 645 Chapter Summary & Review 661 Problems 663 CONTENTS v ★★ Advanced topics
viCONTENTS9Deflections of Beams6769.1Introduction6799.2DifferentialEquationsoftheDeflectionCurve6799.3Deflections byIntegrationoftheBending-MomentEquation6859.4DeflectionsbyIntegrationof theShear-Force696andLoadEquations9.5MethodofSuperposition7029.6Moment-Area Method17119.7NonprismaticBeams7209.8Strain Energy of Bending725**9.9Castigliano's Theorem731**9.10744Deflections Produced by Impact**9.11TemperatureEffects746749ChapterSummary&ReviewProblems75110Statically Indeterminate Beams77010.1773Introduction10.2773Types of Statically IndeterminateBeams10.3Analysis bythe Differential Equationsof theDeflection Curve77778410.4Methodof Superposition**10.5TemperatureEffects797801**10.6LongitudinalDisplacements attheEnds of a Beam805ChapterSummary&ReviewProblems80611Columns81681911.1Introduction11.2Bucklingand Stability81982311.3ColumnswithPinnedEnds11.4834ColumnswithOtherSupportConditions11.5s845ColumnswithEccentricAxialLoads11.6TheSecantFormulaforColumns85011.7Elasticand InelasticColumnBehavior856** Advanced topics
9 Deflections of Beams 676 9.1 Introduction 679 9.2 Differential Equations of the Deflection Curve 679 9.3 Deflections by Integration of the Bending-Moment Equation 685 9.4 Deflections by Integration of the Shear-Force and Load Equations 696 9.5 Method of Superposition 702 9.6 Moment-Area Method 711 9.7 Nonprismatic Beams 720 9.8 Strain Energy of Bending 725 ★★9.9 Castigliano’s Theorem 731 ★★9.10 Deflections Produced by Impact 744 ★★9.11 Temperature Effects 746 Chapter Summary & Review 749 Problems 751 10 Statically Indeterminate Beams 770 10.1 Introduction 773 10.2 Types of Statically Indeterminate Beams 773 10.3 Analysis by the Differential Equations of the Deflection Curve 777 10.4 Method of Superposition 784 ★★10.5 Temperature Effects 797 ★★10.6 Longitudinal Displacements at the Ends of a Beam 801 Chapter Summary & Review 805 Problems 806 11 Columns 816 11.1 Introduction 819 11.2 Buckling and Stability 819 11.3 Columns with Pinned Ends 823 11.4 Columns with Other Support Conditions 834 11.5 Columns with Eccentric Axial Loads 845 11.6 The Secant Formula for Columns 850 11.7 Elastic and Inelastic Column Behavior 856 vi CONTENTS ★★ Advanced topics
vilCONTENTS11.8InelasticBuckling85886311.9DesignFormulasforColumns882ChapterSummary&ReviewProblems88312 Review of Centroids and Moments of Inertia90012.1902Introduction12.2CentroidsofPlaneAreas90212.3CentroidsofCompositeAreas90512.4Momentsof Inertia of PlaneAreas90912.5912Parallel-AxisTheoremforMoments of Inertia12.6PolarMomentsof Inertia91612.7Productsof Inertia91812.8RotationofAxes92192312.9PrincipalAxes andPrincipal Momentsof InertiaProblems927References and Historical Notes935AppendixASystems of Units and ConversionFactors943A.1SystemsofUnits943A.2SIUnits944A.3U.S.Customary Units951A.4Temperature Units952A.5953Conversions Between UnitsAppendixBProblemSolving956B.1TypesofProblems956B.2957StepsinSolvingProblemsB.3958Dimensional HomogeneityB.4SignificantDigits959B.5Roundingof Numbers961AppendixCMathematical Formulas962AppendixDProperties of PlaneAreas966AppendixEProperties of Structural-Steel Shapes972
11.8 Inelastic Buckling 858 11.9 Design Formulas for Columns 863 Chapter Summary & Review 882 Problems 883 12 Review of Centroids and Moments of Inertia 900 12.1 Introduction 902 12.2 Centroids of Plane Areas 902 12.3 Centroids of Composite Areas 905 12.4 Moments of Inertia of Plane Areas 909 12.5 Parallel-Axis Theorem for Moments of Inertia 912 12.6 Polar Moments of Inertia 916 12.7 Products of Inertia 918 12.8 Rotation of Axes 921 12.9 Principal Axes and Principal Moments of Inertia 923 Problems 927 References and Historical Notes 935 Appendix A Systems of Units and Conversion Factors 943 A.1 Systems of Units 943 A.2 SI Units 944 A.3 U.S. Customary Units 951 A.4 Temperature Units 952 A.5 Conversions Between Units 953 Appendix B Problem Solving 956 B.1 Types of Problems 956 B.2 Steps in Solving Problems 957 B.3 Dimensional Homogeneity 958 B.4 Significant Digits 959 B.5 Rounding of Numbers 961 Appendix C Mathematical Formulas 962 Appendix D Properties of Plane Areas 966 Appendix E Properties of Structural-Steel Shapes 972 CONTENTS vii