MECHANICSOfMATERIALSSeventhEditionBeerJohnstonDeWolfMazurek
SeventhEditionMechanics of MaterialsFerdinandP.BeerLate of LehighUniversityE.Russell Johnston,Jr.Lateof Universityof ConnecticutJohnT.DeWolfUniversity of ConnecticutDavidF.MazurekUnited States CoastGuard AcademyMcGrawHillEducation
Seventh Edition Mechanics of Materials Ferdinand P. Beer Late of Lehigh University E. Russell Johnston, Jr. Late of University of Connecticut John T. DeWolf University of Connecticut David F. Mazurek United States Coast Guard Academy bee98233_FM_i-xvi_1.indd i 11/15/13 10:21 AM
McGrawEducationMECHANICSOFMATERIALS,SEVENTHEDITIONPublished by McGraw-Hill Education, 2 Penn Plaza, New York, NY 10121. Copyright 2015 byMcGraw-Hill Education.All rights reserved. Printed in the United States of America.Previous editions 2012, 2009, 2006, and 2002. No part of this publication may be reproduced or distributed in anyform or by any means, or stored in a database or retrieval system, without the prior written consentof McGraw-Hill Education, including, but not limited to, in any network or other electronic storage ortransmission, or broadcast for distance learningSome ancillaries, including electronic and print components, may not be available to customers outsidethe United States.This book is printed on acid-free paper1234567890QVR/QVR109876543210ISBN 978-0-07-339823-5MHID 0-07-339823-3Senior Vice President, Products &Markets:Kurt L. StrandVicePresident,GeneralManager:MartyLangeVice President, Content Production&Technology Services: Kimberly Meriuwether DavidEditorial Director:Thomas TimpGlobalBrandManager:RaghothamanSrinivasanBrand Manager: Bill StenquistMarketing Manager: Heather WagnerProduct Developer: Robin ReedDirector, Content Production: Terri SchieslContent Project Manager: Jolynn KilburgBuyer:NicholeBirkenholzMedia Project Manager: Sandra SchneePhoto Research:CarrieK.BurgerIn-HouseDesigner:MatthewBackhausCover Designer: Matt BackhausCover Image Credit:WalterBibikowCompositor:RPKEditorialServices,IncTypeface:9.5/12 Utopia StdPrinter:Quad/GraphicsAll credits appearing on page or at the end of the book are considered to be an extension of thecopyright page.The photo on the cover shows the steel sculpture "Venture" by Alex Liberman (1912-1999) in front ofthe Bank of America Building in Dallas, Texas. The building is supported by a combination of structuralsteel and reinforced concrete.Library of Congress Cataloging-in-PublicationData on FileThe Intermet addresses listed in the text were accurate at the time of publication. The inclusion of awebsite does not indicate an endorsement by the authors or McGraw-Hill Education, and McGraw-HillEducationdoesnotguaranteetheaccuracyoftheinformationpresentedatthesesites.www.mhhe.com
MECHANICS OF MATERIALS, SEVENTH EDITION Published by McGraw-Hill Education, 2 Penn Plaza, New York, NY 10121. Copyright © 2015 by McGraw-Hill Education. All rights reserved. Printed in the United States of America. Previous editions © 2012, 2009, 2006, and 2002. No part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without the prior written consent of McGraw-Hill Education, including, but not limited to, in any network or other electronic storage or transmission, or broadcast for distance learning. Some ancillaries, including electronic and print components, may not be available to customers outside the United States. This book is printed on acid-free paper. 1 2 3 4 5 6 7 8 9 0 QVR/QVR 1 0 9 8 7 6 5 4 3 2 1 0 ISBN 978-0-07-339823-5 MHID 0-07-339823-3 Senior Vice President, Products & Markets: Kurt L. Strand Vice President, General Manager: Marty Lange Vice President, Content Production & Technology Services: Kimberly Meriwether David Editorial Director: Thomas Timp Global Brand Manager: Raghothaman Srinivasan Brand Manager: Bill Stenquist Marketing Manager: Heather Wagner Product Developer: Robin Reed Director, Content Production: Terri Schiesl Content Project Manager: Jolynn Kilburg Buyer: Nichole Birkenholz Media Project Manager: Sandra Schnee Photo Research: Carrie K. Burger In-House Designer: Matthew Backhaus Cover Designer: Matt Backhaus Cover Image Credit: ©Walter Bibikow Compositor: RPK Editorial Services, Inc. Typeface: 9.5/12 Utopia Std Printer: Quad/Graphics All credits appearing on page or at the end of the book are considered to be an extension of the copyright page. The photo on the cover shows the steel sculpture “Venture” by Alex Liberman (1912-1999) in front of the Bank of America Building in Dallas, Texas. The building is supported by a combination of structural steel and reinforced concrete. Library of Congress Cataloging-in-Publication Data on File The Internet addresses listed in the text were accurate at the time of publication. The inclusion of a website does not indicate an endorsement by the authors or McGraw-Hill Education, and McGraw-Hill Education does not guarantee the accuracy of the information presented at these sites. www.mhhe.com bee98233_FM_i-xvi_1.indd ii 11/15/13 10:21 AM
About the AuthorsJohnTDeWolf, Professor of Civil Engineering attheUniversityof Connecticut, joined the Beer and Johnston team as an author on the secondedition of Mechanics of Materials. John holds a B.S. degree in civil engi-neering from the University of Hawaii and M.E. and Ph.D. degrees instructural engineering from Cornell University. He is a Fellow of the Amer-ican Society of Civil Engineers and a member of the Connecticut Academyof Scienceand Engineering.He isa registered Professional Engineer anda member of the Connecticut Board of Professional Engineers.He wasselected asaUniversityofConnecticutTeachingFellowin2006.Profes-sional interests include elastic stability, bridge monitoring, and structuralanalysis and design.DavidF.Mazurek,Professor of Civil Engineering at the United StatesCoast Guard Academy, joined the Beer and Johnston team as an authoron the fifth edition. David holds a B.S. degree in ocean engineering andan M.S. degree in civil engineering from the Florida Institute of Technol-ogy, and a Ph.D. degree in civil engineering from the University of Con-necticut.He is a registered Professional Engineer.He has served on theAmerican Railway Engineering & Maintenance of Way Association's Com-mittee 15-Steel Structures since 1991. He is a Fellow of the AmericanSociety of Civil Engineers, and was elected into the Connecticut Academyof Science and Engineering in 2013. Professional interests include bridgeengineering,structural forensics,andblast-resistant design.ii
iii About the Authors John T. DeWolf, Professor of Civil Engineering at the University of Connecticut, joined the Beer and Johnston team as an author on the second edition of Mechanics of Materials. John holds a B.S. degree in civil engineering from the University of Hawaii and M.E. and Ph.D. degrees in structural engineering from Cornell University. He is a Fellow of the American Society of Civil Engineers and a member of the Connecticut Academy of Science and Engineering. He is a registered Professional Engineer and a member of the Connecticut Board of Professional Engineers. He was selected as a University of Connecticut Teaching Fellow in 2006. Professional interests include elastic stability, bridge monitoring, and structural analysis and design. David F. Mazurek, Professor of Civil Engineering at the United States Coast Guard Academy, joined the Beer and Johnston team as an author on the fifth edition. David holds a B.S. degree in ocean engineering and an M.S. degree in civil engineering from the Florida Institute of Technology, and a Ph.D. degree in civil engineering from the University of Connecticut. He is a registered Professional Engineer. He has served on the American Railway Engineering & Maintenance of Way Association’s Committee 15—Steel Structures since 1991. He is a Fellow of the American Society of Civil Engineers, and was elected into the Connecticut Academy of Science and Engineering in 2013. Professional interests include bridge engineering, structural forensics, and blast-resistant design. bee98233_FM_i-xvi_1.indd iii 11/15/13 10:21 AM
ContentsPrefaceixGuided Tour3xiliListofSymbolsXVIntroductionConceptofStress31.1Reviewof TheMethodsofStatics41.2StressesintheMembersofaStructure:71.327StressonanObliquePlaneUnderAxial Loading1.4Stress Under General Loading Conditions; Componentsof Stress 281.531DesignConsiderationsReviewand Summary44Stress and Strain-AxialLoading552.1AnIntroductiontoStressandStrain572.2StaticallyIndeterminateProblems782.3Problems InvolvingTemperatureChanges822.4Poisson'sRatio942.5Multiaxial Loading:Generalized Hooke'sLaw95*2.6Dilatationand BulkModulus972.7Shearing Strain992.8DeformationsUnderAxial Loading-Relation Between E,v,andG102*2.9Stress-StrainRelationshipsForFiber-ReinforcedCompositeMaterials1042.10Stress and Strain Distribution Under Axial Loading:Saint-Venant's Principle1152.11Stress Concentrations 1172.12PlasticDeformations119*2.13Residual Stresses123133Review and Summary*Advanced or specialty topicsiv
iv Contents Preface ix Guided Tour xiii List of Symbols xv 1 Introduction—Concept of Stress 3 1.1 Review of The Methods of Statics 4 1.2 Stresses in the Members of a Structure 7 1.3 Stress on an Oblique Plane Under Axial Loading 27 1.4 Stress Under General Loading Conditions; Components of Stress 28 1.5 Design Considerations 31 Review and Summary 44 2 Stress and Strain—Axial Loading 55 2.1 An Introduction to Stress and Strain 57 2.2 Statically Indeterminate Problems 78 2.3 Problems Involving Temperature Changes 82 2.4 Poisson’s Ratio 94 2.5 Multiaxial Loading: Generalized Hooke’s Law 95 *2.6 Dilatation and Bulk Modulus 97 2.7 Shearing Strain 99 2.8 Deformations Under Axial Loading—Relation Between E, n, and G 102 *2.9 Stress-Strain Relationships For Fiber-Reinforced Composite Materials 104 2.10 Stress and Strain Distribution Under Axial Loading: SaintVenant’s Principle 115 2.11 Stress Concentrations 117 2.12 Plastic Deformations 119 *2.13 Residual Stresses 123 Review and Summary 133 *Advanced or specialty topics bee98233_FM_i-xvi_1.indd iv 11/15/13 10:21 AM
Contentsv147Torsion3.1CircularShaftsinTorsion1503.2AngleofTwistintheElasticRange1673.3StaticallyIndeterminate Shafts1703.4Designof Transmission Shafts1853.5StressConcentrations inCircularShafts187*3.6PlasticDeformations in CircularShafts195*3.7CircularShaftsMadeofanElastoplasticMaterial196*3.8Residual Stresses inCircularShafts199*3.9Torsion of Noncircular Members 209*3.10Thin-Walled Hollow Shafts211223Reviewand Summary237Pure Bending4.1Symmetric Members in Pure Bending2404.2StressesandDeformations intheElasticRange2444.3Deformations ina Transverse Cross Section 2484.4MembersMadeofCompositeMaterials2594.5Stress Concentrations263*4.6PlasticDeformations 2734.7Eccentric Axial Loading in a Plane of Symmetry2914.8Unsymmetric Bending Analysis3024.9307General CaseofEccentricAxial LoadingAnalysis*4.10CurvedMembers319334Review and SummaryAnalysisandDesignofBeams福345forBending5.1ShearandBending-MomentDiagrams3485.2Relationships Between Load, Shear,and Bending Moment3605.3DesignofPrismaticBeamsforBending371*5.4SingularityFunctions UsedtoDetermineShearandBendingMoment383*5.5NonprismaticBeams396Reviewand Summary407
v Contents 3 Torsion 147 3.1 Circular Shafts in Torsion 150 3.2 Angle of Twist in the Elastic Range 167 3.3 Statically Indeterminate Shafts 170 3.4 Design of Transmission Shafts 185 3.5 Stress Concentrations in Circular Shafts 187 *3.6 Plastic Deformations in Circular Shafts 195 *3.7 Circular Shafts Made of an Elastoplastic Material 196 *3.8 Residual Stresses in Circular Shafts 199 *3.9 Torsion of Noncircular Members 209 *3.10 Thin-Walled Hollow Shafts 211 Review and Summary 223 4 Pure Bending 237 4.1 Symmetric Members in Pure Bending 240 4.2 Stresses and Deformations in the Elastic Range 244 4.3 Deformations in a Transverse Cross Section 248 4.4 Members Made of Composite Materials 259 4.5 Stress Concentrations 263 *4.6 Plastic Deformations 273 4.7 Eccentric Axial Loading in a Plane of Symmetry 291 4.8 Unsymmetric Bending Analysis 302 4.9 General Case of Eccentric Axial Loading Analysis 307 *4.10 Curved Members 319 Review and Summary 334 5 Analysis and Design of Beams for Bending 345 5.1 Shear and Bending-Moment Diagrams 348 5.2 Relationships Between Load, Shear, and Bending Moment 360 5.3 Design of Prismatic Beams for Bending 371 *5.4 Singularity Functions Used to Determine Shear and Bending Moment 383 *5.5 Nonprismatic Beams 396 Review and Summary 407 bee98233_FM_i-xvi_1.indd v 11/15/13 10:21 AM
ContentsviShearing Stresses in Beams andThin-WalledMembers4176.1Horizontal Shearing Stress in Beams420*6.2Distribution of Stresses ina NarrowRectangular Beam4266.3Longitudinal Shearona BeamElementofArbitraryShape4376.4ShearingStresses inThin-WalledMembers439*6.5PlasticDeformations 441*6.6Unsymmetric Loading of Thin-Walled Members and ShearCenter454467Reviewand SummaryTransformationsofStressandStrain4777.1Transformationof Plane Stress4807.2Mohr's Circlefor Plane Stress4927.3General StateofStress5037.4Three-Dimensional Analysis of Stress504*7.5Theoriesof Failure5077.6Stresses in Thin-Walled Pressure Vessels520*7.7TransformationofPlaneStrain529*7.8534Three-Dimensional Analysis of Strain*7.9538MeasurementsofStrain;StrainRosetteReviewand Summary5468PrincipalStressesUnderaGiven557Loading8.1Principal Stresses ina Beam 5598.2DesignofTransmissionShafts5628.3Stresses UnderCombined Loads575ReviewandSummary591
vi Contents 6 Shearing Stresses in Beams and Thin-Walled Members 417 6.1 Horizontal Shearing Stress in Beams 420 *6.2 Distribution of Stresses in a Narrow Rectangular Beam 426 6.3 Longitudinal Shear on a Beam Element of Arbitrary Shape 437 6.4 Shearing Stresses in Thin-Walled Members 439 *6.5 Plastic Deformations 441 *6.6 Unsymmetric Loading of Thin-Walled Members and Shear Center 454 Review and Summary 467 7 Transformations of Stress and Strain 477 7.1 Transformation of Plane Stress 480 7.2 Mohr’s Circle for Plane Stress 492 7.3 General State of Stress 503 7.4 Three-Dimensional Analysis of Stress 504 *7.5 Theories of Failure 507 7.6 Stresses in Thin-Walled Pressure Vessels 520 *7.7 Transformation of Plane Strain 529 *7.8 Three-Dimensional Analysis of Strain 534 *7.9 Measurements of Strain; Strain Rosette 538 Review and Summary 546 8 Principal Stresses Under a Given Loading 557 8.1 Principal Stresses in a Beam 559 8.2 Design of Transmission Shafts 562 8.3 Stresses Under Combined Loads 575 Review and Summary 591 bee98233_FM_i-xvi_1.indd vi 11/15/13 10:21 AM
Contentsvii599DeflectionofBeams9.1602DeformationUnderTransverseLoading9.2StaticallyIndeterminateBeams611*9.3SingularityFunctions toDetermine Slopeand Deflection6239.4MethodofSuperposition635*9.5Moment-AreaTheorems649*9.6Moment-AreaTheoremsAppliedtoBeamswithUnsymmetricLoadings664679Review and Summary691Columns1010.1692StabilityofStructures*10.2Eccentric Loading and the Secant Formula70910.3Centric Load Design72210.4739EccentricLoadDesign750Review and Summary759Energy Methods11.1Strain Energy76011.2763Elastic Strain Energy11.3Strain Energyfora General Stateof Stress77011.4Impact Loads78411.5Single Loads788#11.6Multiple Loads802*11.7Castigliano's Theorem804*11.8Deflectionsby Castigliano'sTheorem806*11.9Statically Indeterminate Structures810823Reviewand Summary
vii Contents 9 Deflection of Beams 599 9.1 Deformation Under Transverse Loading 602 9.2 Statically Indeterminate Beams 611 *9.3 Singularity Functions to Determine Slope and Deflection 623 9.4 Method of Superposition 635 *9.5 Moment-Area Theorems 649 *9.6 Moment-Area Theorems Applied to Beams with Unsymmetric Loadings 664 Review and Summary 679 10 Columns 691 10.1 Stability of Structures 692 *10.2 Eccentric Loading and the Secant Formula 709 10.3 Centric Load Design 722 10.4 Eccentric Load Design 739 Review and Summary 750 11 Energy Methods 759 11.1 Strain Energy 760 11.2 Elastic Strain Energy 763 11.3 Strain Energy for a General State of Stress 770 11.4 Impact Loads 784 11.5 Single Loads 788 *11.6 Multiple Loads 802 *11.7 Castigliano’s Theorem 804 *11.8 Deflections by Castigliano’s Theorem 806 *11.9 Statically Indeterminate Structures 810 Review and Summary 823 bee98233_FM_i-xvi_1.indd vii 11/15/13 10:21 AM
ContentsviliAppendicesA1A5A2MomentsofAreasBTypical Properties of Selected Materials Used inEngineeringA13cProperties of Rolled-Steel ShapesA17DBeamDeflectionsand SlopesA29EFundamentals of Engineering ExaminationA30AnswerstoProblemsAN1Photo CreditsC1Index11
viii Contents Appendices A1 A Moments of Areas A2 B Typical Properties of Selected Materials Used in Engineering A13 C Properties of Rolled-Steel Shapes A17 D Beam Deflections and Slopes A29 E Fundamentals of Engineering Examination A30 Answers to Problems AN1 Photo Credits C1 Index I1 bee98233_FM_i-xvi_1.indd viii 11/15/13 10:21 AM
PrefaceObjectivesThe main objective of a basic mechanics course should be to develop in the engineering stu-dent the ability to analyze a given problem in a simple and logical manner and to apply to itssolution a few fundamental and well-understood principles.This text is designed for thefirstcourseinmechanicsofmaterials-orstrengthofmaterials-offeredtoengineeringstudentsinthe sophomore or junior year. The authors hope that it will help instructors achieve this goalin that particular course in the same way that their other texts may have helped them in staticsand dynamics.To assist in this goal, the seventh edition has undergone a complete edit ofthelanguagetomakethebook easierto read.GeneralApproachInthistextthestudyofthemechanicsof materials isbasedontheunderstanding ofafewbasicconcepts and on the use of simplified models.This approach makes it possible to develop allthe necessaryformulas in arational and logical manner, and to indicateclearly the conditionsunder which they can be safely applied to the analysis and design of actual engineering structures and machine components.Free-body Diagrams Are Used Extensively.Throughout the text free-body diagramsare used to determine external or internal forces.The use of"picture equations"will also helpthe students understand the superposition of loadings and the resulting stresses anddeformations.TheSMARTProblem-SolvingMethodology isEmployed.Newtothis editionofthetext, students are introduced to the SMART approach for solving engineering problems, whoseacronym reflects the solution steps of Strategy,Modeling,Analysis, and Reflect & Think.Thismethodology is used in all Sample Problems, and it is intended that students will apply thisapproachinthesolutionofall assignedproblems.Design ConceptsAreDiscussedThroughouttheTextWheneverAppropriate.Adis-cussion of the application of the factor of safety to design can befound in Chap.l, where theconcepts of bothallowablestress designand load and resistancefactor designarepresented.A Careful Balance Between Sl and U.S.Customary Units Is ConsistentlyMaintained.Becauseit isessential that studentsbeabletohandleeffectivelybothSI metricunitsand U.S.customaryunits,halftheconceptapplications,sampleproblems,andproblemstobeassigned have been stated in SI units and half in U.S.customary units.Sincea large numberof problems areavailable,instructors can assignproblemsusing each systemof units inwhateverproportiontheyfinddesirablefortheirclass.Optional Sections Offer Advanced or Specialty Topics.Topics such as residual stressestorsion of noncircular and thin-walled members, bending ofcurved beams, shearing stresses innon-symmetrical members,and failure criteriahave been included in optional sections foruse in courses ofvarying emphases.To preserve theintegrity of the subject, these topics arepresented in the proper sequence,wherever they logically belong.Thus, even when notix
ix Preface Objectives The main objective of a basic mechanics course should be to develop in the engineering student the ability to analyze a given problem in a simple and logical manner and to apply to its solution a few fundamental and well-understood principles. This text is designed for the first course in mechanics of materials—or strength of materials—offered to engineering students in the sophomore or junior year. The authors hope that it will help instructors achieve this goal in that particular course in the same way that their other texts may have helped them in statics and dynamics. To assist in this goal, the seventh edition has undergone a complete edit of the language to make the book easier to read. General Approach In this text the study of the mechanics of materials is based on the understanding of a few basic concepts and on the use of simplified models. This approach makes it possible to develop all the necessary formulas in a rational and logical manner, and to indicate clearly the conditions under which they can be safely applied to the analysis and design of actual engineering structures and machine components. Free-body Diagrams Are Used Extensively. Throughout the text free-body diagrams are used to determine external or internal forces. The use of “picture equations” will also help the students understand the superposition of loadings and the resulting stresses and deformations. The SMART Problem-Solving Methodology is Employed. New to this edition of the text, students are introduced to the SMART approach for solving engineering problems, whose acronym reflects the solution steps of Strategy, Modeling, Analysis, and Reflect & T hink. This methodology is used in all Sample Problems, and it is intended that students will apply this approach in the solution of all assigned problems. Design Concepts Are Discussed Throughout the Text Whenever Appropriate. A discussion of the application of the factor of safety to design can be found in Chap. 1, where the concepts of both allowable stress design and load and resistance factor design are presented. A Careful Balance Between SI and U.S. Customary Units Is Consistently Maintained. Because it is essential that students be able to handle effectively both SI metric units and U.S. customary units, half the concept applications, sample problems, and problems to be assigned have been stated in SI units and half in U.S. customary units. Since a large number of problems are available, instructors can assign problems using each system of units in whatever proportion they find desirable for their class. Optional Sections Offer Advanced or Specialty Topics. Topics such as residual stresses, torsion of noncircular and thin-walled members, bending of curved beams, shearing stresses in non-symmetrical members, and failure criteria have been included in optional sections for use in courses of varying emphases. To preserve the integrity of the subject, these topics are presented in the proper sequence, wherever they logically belong. Thus, even when not NEW bee98233_FM_i-xvi_1.indd ix 11/15/13 10:21 AM