Mechanics of material

1022 Pages·2020·16.438 MB·English

by  Bedford A., Liechti K.M

Anthony Bedford · Kenneth M. Liechti
Mechanics
of Materials
Second Edition
(cid:129)
Anthony Bedford Kenneth M. Liechti
Mechanics of Materials
Second Edition
AnthonyBedford KennethM.Liechti
UniversityofTexas UniversityofTexas
Austin,TX,USA Austin,TX,USA
ISBN978-3-030-22081-5 ISBN978-3-030-22082-2 (eBook)
https://doi.org/10.1007/978-3-030-22082-2
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Preface
Mechanics of materials is the study and the analysis of the internal forces within
materialsandthedeformationsthatresultfromthoseforces.Thisbookappearsina
timeoftransitionforeducationinthemechanicsofmaterials.Thetraditionalcourse
in “strength of materials” that long formed an important part of the engineering
curriculumhadasoneofitsprimarygoals,acquaintingthestudentswiththedetails
ofmanyanalyticalandempiricalsolutionsthatcouldbeappliedtostructuraldesign.
This reliance on a catalog of results has lessened as the finite element method has
becomecommonly available for stressanalysis. Another important development is
that current research in mechanics of materials is beginning to bring to reality the
dream of the merger of continuum solid mechanics and material science into a
unified field. For these reasons, the emphasis in the first course in mechanics of
materials is becoming oriented more toward helping students understand the theo-
retical foundations, especially the concepts of stress and strain, the stress-strain
relationsincludingthemeaningofisotropy,andthecriteriaforfailureandfracture.
In Chap. 1, we provide an extensive review of statics, with problems, that the
instructormaychoosetocoverorsimplyhavestudentsread.Inreviewingdistributed
loads,welaythegroundworkforourdefinitionsinChap.2ofthenormalandshear
stresses. Chapter 2 also introduces the longitudinal and shear strains in terms of
changesininfinitesimalmaterialelements.Chapters3and4coverbarssubjectedto
axialandtorsionalloadsandintroducethedefinitionsoftheelasticandshearmoduli.
InChaps.5and6,wediscusstheinternalforcesandmomentsandthestatesofstress
inbeams.WiththeexamplesinChaps.2,3,4,5,and6asmotivation,inChaps.7
and 8, we discuss the general states of stress and strain and their transformations.
Chapters 9 and 10 cover deformations of beams and the buckling of columns.
Energy methods are introduced in Chap. 11, and in Chap. 12, we discuss failure
criteriaforgeneralstatesofstressandintroducemodernfracturemechanics,which
v
vi Preface
we believe should become an integral part of the first course in mechanics of
materials.MostofthetopicsinChaps.1,2,3,4,5,6,7,8,9,and10and selected
topicsfromChapters11to12canbecoveredinatypicalone-semestercourse.
The first course in mechanics of materials prepares students for subsequent
courses in structural analysis, structural dynamics, and advanced mechanics of
deformablemedia.Incomparisonwith thecoursesinstatics anddynamics, itisan
interesting challenge for instructors and students. Engineering students begin the
study of statics and dynamics having had some prior experience with the basic
conceptsinhighschoolandcollegephysicscourses.Incontrast,thecoreconceptsin
mechanics of materials are new to most students. It is therefore essential that a
textbook introduce and explain these concepts with great care and reinforce them
withmanyexamples.Thishasbeenourobjectiveinwritingthisbook.Ourapproach
istopresentthematerial aswe dointheclassroom, emphasizingunderstandingof
the basic principles of mechanics of materials and demonstrating them with exam-
plesdrawnfromcontemporaryengineeringapplicationsanddesign.
Features
(cid:129) Examples That Teach We continue reinforcing the problem-solving skills
studentshavelearnedintheirintroductorymechanicscourses.Separatestrategy
sectionsthatprecedemostexamplesandselectedproblemsteachstudentshowto
approachandsolveproblemsinengineering:Whatprinciplesapply?Whatmust
be determined? And in what order? Many examples conclude with discussion
sections that discuss ways of checking and interpreting answers, point out
interestingfeaturesofthesolution,orsuggestalternativemethodsofsolution.
(cid:129) Free-Body Diagrams Correct and consistent use of free-body diagrams is the
mostessentialskillthatstudentsofmechanicsmustacquire.Wereviewthesteps
involved in drawing free-body diagrams in Chap. 1 and emphasize their use
throughout the book. Many of our figures are designed to teach how free-body
diagramsarechosenanddrawn.
A B C
F
B
P F P
AB BC
(cid:129) Design The Accreditation Board for Engineering and Technology (ABET) and
manypracticingengineersstronglyencouragetheintroductionofdesignthrough-
out the engineering curriculum. By expressing many of our examples and
Preface vii
problems in terms of design, we demonstrate the use of mechanics of materials
withinthelargercontextofengineeringpractice.Chapters3,4,6,and7contain
sectionsexplicitlyaddressingthedesignofparticularstructuralelements,andin
Chap.12,wediscussfailureandfracturecriteriausedinstructuraldesign.
Austin,TX,USA AnthonyBedford
KennethM.Liechti
Contents
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 EngineeringandMechanicsofMaterials. . . . . . . . . . . . . . . . . . . 1
1.2 Units. . . .. . . .. . . . .. . . .. . . . .. . . .. . . .. . . . .. . . .. . . . .. 2
1.2.1 SIUnits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2.2 USCustomaryUnits. . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 ReviewofStatics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3.1 Free-BodyDiagrams. . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3.2 Equilibrium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3.3 Structures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.4 Centroids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.3.5 DistributedForces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
ReviewProblems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2 MeasuresofStressandStrain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.1 Stresses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.1.1 Traction,NormalStress,andShearStress. . . . . . . . . . . . 38
2.1.2 AverageStresses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.1.3 ComponentsofStress. . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.2 Strains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
2.2.1 NormalStrain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
2.2.2 ShearStrain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
2.2.3 ComponentsofStrainandtheStress-StrainRelations. . . 78
ChapterSummary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
ReviewProblems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3 AxiallyLoadedBars.. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. . . . .. 99
3.1 StressesinPrismaticBars. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.1.1 TheStateofStress. . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
3.1.2 StressesonObliquePlanes. . . . . . . . . . . . . . . . . . . . . . 107
3.2 StrainsandtheElasticConstants. . . . . . . . . . . . . . . . . . . . . . . . . 124
3.2.1 AxialandLateralStrains. . . . . . . . . . . . . . . . . . . . . . . . 124
ix
x Contents
3.2.2 MaterialBehavior. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
3.2.3 Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
3.3 StaticallyIndeterminateProblems. . . . . . . . . . . . . . . . . . . . . . . . 145
3.4 NonprismaticBarsandDistributedLoads. . . . . . . . . . . . . . . . . . 164
3.4.1 BarswithGraduallyVaryingCrossSections. . . . . . . . . . 164
3.4.2 DistributedAxialLoads. . . . . .. . . . . .. . . . . .. . . . . . . 167
3.5 ThermalStrains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
3.6 DesignIssues. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
3.6.1 AllowableStress. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
3.6.2 OtherDesignConsiderations. . . . . . . . . . . . . . . . . . . . . 195
ChapterSummary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
ReviewProblems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
4 Torsion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
4.1 PureShearStress. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
4.1.1 ComponentsofStressandStrain. . . . . . . . . . . . . . . . . . 211
4.1.2 StressesonObliquePlanes. . . . . . . . . . . . . . . . . . . . . . 213
4.2 PrismaticCircularBars. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
4.2.1 StressesandStrains. . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
4.2.2 PolarMomentofInertia. . . . . . . . . . . . . . . . . . . . . . . . 223
4.2.3 PositiveDirectionsoftheTorqueandAngleofTwist. . . 225
4.3 StaticallyIndeterminateProblems. . . . . . . . . . . . . . . . . . . . . . . . 238
4.4 NonprismaticBarsandDistributedLoads. . . . . . . . . . . . . . . . . . 245
4.4.1 BarswithGraduallyVaryingCrossSections. . . . . . . . . . 245
4.4.2 DistributedTorsionalLoads. . . . . . . . . . . . . . . . . . . . . . 248
4.5 Elastic-PerfectlyPlasticCircularBars. . . . . . . . . . . . . . . . . . . . . 258
4.6 Thin-WalledTubes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
4.6.1 Stress. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
4.6.2 AngleofTwist. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
4.7 DesignIssues. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
4.7.1 CrossSections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
4.7.2 AllowableStress. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
ChapterSummary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
ReviewProblems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
5 InternalForcesandMomentsinBeams. . . . . . . . . . . . . . . . . . . . . . 301
5.1 AxialForce,ShearForce,andBendingMoment. . . . . . . . . . . . . 301
5.2 ShearForceandBendingMomentDiagrams. . . . . . . . . . . . . . . . 311
5.3 EquationsRelatingDistributedLoad,ShearForce,
andBendingMoment.. . . . .. . . . .. . . . . .. . . . .. . . . .. . . . .. 322
5.3.1 DerivationoftheEquations. . . . . . . . . . . . . . . . . . . . . . 322
5.3.2 ConstructionoftheShearForceDiagram. . . . . . . . . . . . 327
5.3.3 ConstructionoftheBendingMomentDiagram. . . . . . . . 330
Contents xi
5.4 SingularityFunctions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
5.4.1 DeltaFunction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
5.4.2 Dipole. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344
5.4.3 MacaulayFunctions. . . . . . . . . . . . . . . . . . . . . . . . . . . 347
ChapterSummary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
ReviewProblems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
6 StressesinBeams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
6.1 DistributionoftheNormalStress. . . . . . . . . . . . . . . . . . . . . . . . 361
6.1.1 GeometryofDeformation. . . . . . . . . . . . . . . . . . . . . . . 361
6.1.2 RelationBetweenNormalStressandBending
Moment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367
6.1.3 BeamsSubjectedtoArbitraryLoads. .. . . .. . . . .. . . .. 369
6.2 DesignIssues. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383
6.2.1 FactorofSafetyandSectionModulus. . . . . . . . . . . . . . 383
6.2.2 StandardizedCrossSections. . . . . . . . . . . . . . . . . . . . . 387
6.3 CompositeBeams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395
6.3.1 DistributionoftheNormalStress. . . . . . . . . . . . . . . . . . 396
6.3.2 TransformedAreaMethod. . . . . . . . . . . . . . . . . . . . . . . 401
6.4 Elastic-PerfectlyPlasticBeams. . . . . . . . . . . . . . . . . . . . . . . . . . 410
6.5 AsymmetricCrossSections. . . . . . . . . . . . . . . . . . . . . . . . . . . . 424
6.5.1 MomentExertedAboutaPrincipalAxis. . . . . . . . . . . . . 424
6.5.2 MomentExertedAboutanArbitraryAxis. . . . . . . . . . . 428
6.6 ShearFormula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436
6.6.1 RectangularCrossSection. . . . . . . . . . . . . . . . . . . . . . . 442
6.6.2 ShearStressonanObliqueElement. . . . . . . . . . . . . . . . 443
6.7 Built-UpBeams. . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . 456
6.8 Thin-WalledCrossSections. . . . . . . . . . . . . . . . . . . . . . . . . . . . 467
6.8.1 DistributionoftheShearStress. . . . . . . . . . . . . . . . . . . 467
6.8.2 TheShearCenter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474
ChapterSummary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 490
ReviewProblems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498
7 StatesofStress. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505
7.1 ComponentsofStress. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505
7.2 TransformationsofPlaneStress. . . . . . . . . . . . . . . . . . . . . . . . . 509
7.2.1 CoordinateTransformations. . . . . . . . . . . . . . . . . . . . . . 510
7.2.2 MaximumandMinimumStresses. . . . . . . . . . . . . . . . . 517
7.3 Mohr’sCircleforPlaneStress. . . . . . . . . . . . . . . . . . . . . . . . . . 538
7.3.1 ConstructingtheCircle. . . . . . . . . . . . . . . . . . . . . . . . . 539
7.3.2 WhyMohr’sCircleWorks. . . . . . . . . . . . . . . . . . . . . . 540
7.3.3 DeterminingMaximumandMinimumStresses. . . . . . . . 542
7.4 MaximumandMinimumStressesinThreeDimensions. . . . . . . . 557
7.4.1 GeneralStateofStress. . . . . . . . . . . . . . . . . . . . . . . . . 557
7.4.2 TriaxialStress. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 560
xii Contents
7.5 Application:BarsSubjectedtoCombinedLoads. . . . . . . . . . . . . 566
7.5.1 TheFundamentalLoads. . . . . . . . . . . . . . . . . . . . . . . . 566
7.5.2 CombinedLoads. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 569
7.6 Application:PressureVessels. . . . . . . . . . . . . . . . . . . . . . . . . . . 579
7.6.1 SphericalVessels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579
7.6.2 CylindricalVessels. . . . . . . . . . . . . . . . . . . . . . . . . . . . 582
7.6.3 DesignIssues. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 586
7.7 TetrahedronArgument. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 590
7.7.1 DeterminingtheTraction. . . . . . . . . . . . . . . . . . . . . . . . 590
7.7.2 DeterminingtheNormalandShearStresses. . . . . . . . . . 593
ChapterSummary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602
ReviewProblems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607
8 StatesofStrainandStress-StrainRelations. . . . . . . . . . . . . . . . . . . 611
8.1 ComponentsofStrain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611
8.2 TransformationsofPlaneStrain. . . . . . . . . . . . . . . . . . . . . . . . . 614
8.2.1 StrainGaugeRosette. . . . . . . . . . . . . . . . . . . . . . . . . . . 618
8.2.2 MaximumandMinimumStrains. . . . . . . . . . . . . . . . . . 621
8.3 Mohr’sCircleforPlaneStrain. . . . . . . . . . . . . . . . . . . . . . . . . . 639
8.3.1 ConstructingtheCircle. . . . . . . . . . . . . . . . . . . . . . . . . 639
8.3.2 DeterminingMaximumandMinimumStrains. . . . . . . . . 639
8.4 Stress-StrainRelations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646
8.4.1 LinearElasticMaterials. . . . . . . . . . . . . . . . . . . . . . . . . 647
8.4.2 IsotropicMaterials. . . . . . . . . . . . . . . . . . . . . . . . . . . . 648
ChapterSummary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664
ReviewProblems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669
9 DeflectionsofBeams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 671
9.1 TheSecond-OrderEquation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 671
9.1.1 DifferentialEquation. . . . . . . . . . . . . . . . . . . . . . . . . . . 671
9.1.2 BoundaryConditions. . . . . . . . . . . . . . . . . . . . . . . . . . 674
9.2 StaticallyIndeterminateBeams. . . . . . . . . . . . . . . . . . . . . . . . . . 688
9.3 SingularityFunctions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697
9.4 Moment-AreaMethod. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706
9.4.1 FirstTheorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706
9.4.2 SecondTheorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707
9.5 Superposition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715
ChapterSummary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724
ReviewProblems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 726
10 BucklingofColumns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 729
10.1 EulerBucklingLoad. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 729
10.2 OtherEndConditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745
10.2.1 AnalysisoftheDeflection. . . . . . . . . . . . . . . . . . . . . 745
10.2.2 EffectiveLength. . . . . . . . . . . . . . . . . . . . . . . . . . . . 757