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this watermark.( Reaffirmed 1995 )HANDBOOK FOR STRUCTURALENGINEERS
No.6 Please purchase PDF Split-Merge on www.verypdf.com to remove
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Intentionally Left BlankPlease purchase PDF Split-Merge on
www.verypdf.com to remove this watermark.SP: 6(6)-1972 HANDBOOK FOR
STRUCTURALENGINEERS 6.APPLICATIONOFPLASTICTHEORYIN
DESIGNOFSTEELSTRUCTURES BUREAUOFINDIANSTANDARDS
MANAKBHAVAN,9BAHADURSHAHZAFARMARG NEWDELHI110002 PriceRs.275.00
October1973 Please purchase PDF Split-Merge on www.verypdf.com to
remove this watermark.BUREAUOFINDIANSTANDARDS Edition1-1972
SirthReprint DECEMBER 1998 UDC624.014.2:624.04 SP:6(6)-1972
QCopyrtght1973 BUREAUOFINDIANSTANDARDS
ThispublicationisprotectedundertheIndianCopyrightAct(XIV
of1957)andreproduction inwholeorinpart byauymeansexcept
withwrittenpermissionofthepublishershallbedeemedtobean
infringementofcopyrightunderthesaidAct.
PrintedinIndiabyPrintsgraph,NewDelhiand
PublishedbyBureauofIndianStandards,NewDelhi110002 Please purchase
PDF Split-Merge on www.verypdf.com to remove this
watermark.CONTENTS PACE FOREWORD. . .. . .. .. . .. . . SYMBOLS. .
.. . .. . .. . .s.. SECTI ONAI NTRODUCTI ON 1.SCOPE...............
2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.
GENERAL. . .. . .. . .. . .. . . STRUCTURALSTRENGTH. . .. . .. . ..
. . MECHANICALPROPERTIESOF STEEL. . .. . .. . . MAXIMUM STRENGTHOF
SOME ELEMENTS. . .. . . HISTORICALDEVELOPMENT. . .. . .. . . SECTI
ON6J USTI FI CATI ONFORPLASTI CDESI GN WHYPLASTIC DESIGN. . .. . ..
. . INADEQUACY OF STRESS ASTHE DESIGN CRITERION
EXPERIMENTALVERIFICATION. . .. . . THECASE FOR PLASTICDESIGN. . ..
. . SECTI ONCFLEXUREOFBEAMS ASSUMPTIONSANDCONDITIONS. . .. . .
BENDING OF RECTANGULARBEAM. . .. . . BENDING OF WIDEFLANGE BEAM. .
.. . . PLASTIC HINGE. . .. . .. . . REDISTRIBUTIONOF MOMENT. . .. .
. SECTI ONDPLASTI CANALYSI S FUNDAMENTALPRINCIPLES. . .. . .
STATICALMETHOD OF ANALYSIS. . .. . . MECHANISMMETHOD OF ANALYSIS. .
.. . . FURTHER CONSIDERATIONS. . .. . . 5 . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . 7 II 15 15 16 19
21 23 25 26 32 33 35 35 40 43 45 49 53 56 66 Please purchase PDF
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watermark.8F:6(6)-1972 20. n. 22. P. 24. 25. 26. 27. 2%. 29. 30.
SECTI ONEAPPLI CATI ONTODESI GN GENERAL............
PRFUYINARYDESIGN. . .. . .. . . G.WB~AL DESIGNPROCEDURE. . .. . .
SECORDARYDESIGNCONSIDERATIONS . . . SECTl ONFDESI GNEXAMPLES
INTRODUCTION. . .. . .. . .. . . DESIGNEXAMPLESON CONTINUOUSBEAMS.
. . DESIGNEXAMPLESON IKDUSTRIALBUILDINGFRAMES DESIGNEXAMPLEON
MULTI-STOREYSTRUCTURES SECTI ONGSI MPLI FI EDPROCEDURES
INTRODUCTION. . .. . .. . .. . . CONTINUOUSBEAMS. . .. . .. . .
SINGLE-SPANFRAMES(SINGLESTOREY). . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 31. MULTI-SPANFRAMES. . .. . .. . .. .
. APPENDIXASELIXTEDREFERENCES. . .. . .. . . APPENDIXBSPACINGOF
LATERALBRACING. . .. . . APPENDIXCCHARTSAND FORMULASFOR
SIMPLEBEAMS. . . APPENDIXDCOMPOSITIONOFSTRUCTURALENGINEERINGSEG
TIONALCOMMITTEE,SMBDC7. . .. . .. . .
.APPENDIXEINDIANSTANDARDSRELATING TOSTRUCTURAL 77 77 78 81 125 125
136 178 187 187 188 191 195 198 203 216 ENGINEERING. . .. . .. . ..
. . 218 6 Please purchase PDF Split-Merge on www.verypdf.com to
remove this watermark.SP:6(6)-1972 FOREWORD
ThisHandbook,whichhasbeenprocessedbytheStructural
Eng!neeringSectionalCommittee,SMBDC7,thecompositionofwhich
isg;veninAppendixD,hadbeenapprovedforpublicationbythe
StrutturalandMetalsDivisionCouncilandtheCivilEngin&ring
DivisionCouncilofISI.
Steel,whichisaveryimportantbasicrawmaterialsforindus-
trialization,hadbeenr&ivingattentionfromthePlanningCommis-
sionevenfromtheveryearlystagesofthecountrysFirstFiveYear
Planperiod.ThePlanningCommissionnotonlyenvisagedanincrease
inproductioncapacityinthecountry,butalsoconsideredthequestion
ofevengreaterimportance,namely,takingofurgentmeasuresfor
theconservationofavailableresources.Itsexpertcommitteescame
totheconclusionthatagoodproportionofthesteelconsumedbythe
structuralsteelindustryinIndiacouldbesavedifhigherefficiency
procedureswereadoptedintheproductionanduseofsteel.The
PlanningCommission,therefore,recommendedtotheGovernmentof
IndiathattheIndianStandardsInstitutionshouldtakeupaSteel
EconomyProjectandprepareaseriesofIndianStandardspecifications
andcodesofpracticeinthefieldofsteelproductionandutilization.
Overfifteen.yearsofcontinuousstudyinTndiaandabroad,and
thedeliberationsatnumeroussittingsofcommittees.panelsandstudy
groupsresultedintheformulationofanumberofIndianStandards
inthefieldofsteelproduction,designanduse,alistofwhichisgiven
inAppendixE.
ThisHandbookwhichrelatestotheapplicationofplastictheory
indesignofsteelstructuresisintendedtopresenttheimportantprin-
ciplesandassumptionsinvolvedintheplasticmethodofstructural
analysis,andtoprovideillustrativeexamplesfortheguidanceofthe
designerintheanalysisofpracticaldesignproblems.
Thesubjectisintroducedbyconsideringthevariouslimitsof
usefulnessofasteelstructure,thelimitsthatarefunction(inpart)
ofthemechanicalpropertiesofsteel.Knowledgeoftheseproperties
isusedinSectionAtoshowhowthemaximumstrengthofsomesimple
structuresmaybecomputed. Thehistoricaldevelopmentofthe
plastictheoryofstructuresisalsodealtwithinbrief.
SectionBanswersthequestion Whyplasticdesign.Itisshown
thatstressisaninadequatedesigncriterionforalargenumberof 7 Please
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practicalengineeringstructures.Theexperimentalverificatiorroftha
plastictheory(whichbasesthedesignofstructuresonthemaximum
strength)hasalsobeenindicated.Thebasictheoreticalworkisdeait
withinSectionsCandD.Theconceptsofplasticbendingandredls-
tributionofmomentsaredescribedandthemethodsofanalysishas
beenindicated.SectionEcontainsgeneralcommentsondesign
procedures.Althoughthissectioncoversafewexamplesrelatingto
multistoreyframes,itisproposedtodealwiththesubjectindetailin
asupplementinduecourse.Thelimitations,modificationsanddesign
detailshavebeendescribedundertheheading SecondaryDesign
Consideration.Properattentionshouldbegiventotheeffectofshear
force,axialforce,localandlateralbuckling,etc.Further,thebeams,
columnsandconnectionsshouldbedesignedtomeettherequirements
ofplastichingeformation.
Thesectionondesignexamplestreatsanumberofbuildingframes
ofdifferentprofiles.Thesecondarydesignconsiderationsarechecked
throughout.Section7describessimplifiedproceduresofsolvingdesign
problemswiththeuseofformulas,chartsandgraphs.
InAppendixAisgivenalistofselectedreferencesforfurther
detailedinformationonplastictheoryofstructures.
Whatwillplasticdesignmean?Tothe sidewalksuperintendent,
itwillmeannothing.Thestructurewilllookjustthesameasacon-
ventionallydesignedstructure.Totheengineer,itwillmeanamore
rapidmethodofanalysis.Totheowner,itwillmeaneconomy,because
plasticdesignrequireslesssteelthanconventionaldesign.Forthe
buildingauthority,itwouldmeanmoreefficientoperationsbecause
designsmaybecheckedfaster.Tosteelindustry,itwouldmeanmore
efficientuseofitsproducts.Finally,toanation,itwillmeanbetter
useofhernaturalresources.
ThisHandbookisbasedonandrequiresreferencetothefollowing
publicationsissuedbyISI: IS: IS: IS: IS: IS:
226-1969Specificationforstructuralsteel(standardquality)
(fourt/zrevision)
800-1962Codeofpracticeforuseofstructuralsteelingeneral
buildingconstruction(revised)
875-1964Codeofpracticeforstructuralsafetyofbuildings:
Loadingstandards(revised)
2062-1969Specificationforstructuralsteel(fusionwelding
quality)(@Strevision)
4000-1967Codeofpracticeforassemblyofstructuraljoints
usinghightensilefrictiongripfasteners 8 Please purchase PDF
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watermark.SP:6(6)-1972
Inthepreparationofthishandbook,thetechnicalcommitteehas
derivedvahlableassistancefromDrLynnS.Bee&e,Professorof
StructuralEngineering,LehighUniversity,Bethlehem,USA.DrBcedle
preparedthepreliminarydraftofthishandbook.Thisassistance
wasmadeavailabletoIS1throughMessrsRamseyer&Miller,Inc,Iron,
andSteelIndustryConsultants,NewYork,bytheTechnicalCo-operation
MissiontoIndiaoftheGovernmentofIndiaundertheirTechnical
AssistanceProgrammc.
Nohandbookofthiskindmaybemadecompleteforalltimesto
comeattheveryfirstattempt.Asdesignersandcnginecrsbeginto USC
it,theywillbeabletosuggestmodificationsandadditionsfor
improvingitsutility.Theyarerequestedtosendsuchvaluable
suggestionstoISIwhichwillbereceivedwithappreciationand gratitude. 9
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watermark.As in the Original Standard, this Page is Intentionally
Left BlankPlease purchase PDF Split-Merge on www.verypdf.com to
remove this watermark.SP : 6(6) -1972 SYMBOLS AE AP= A,;= Aw= b = 5
ii= E= Symbolsusedinthishandbookshallhavethemeaningassigned
tothemasindicatedbelow: Areaofcross-section
ArcaofbothflangesofWFshape Areaofsplit-tee Areaofwebbetweenflanges
Flangewidth Distancefromneutralaxistotheextremefibre Depthofsection
Youngsmodulusofelasticity Strain-hardeningmodulus=2 rt
Tangentmodulus Eccentricity Loadfactorofsafety Es,= E,= s .;=f =f
=G = Gt= H = HB= I Iz I *= J = K= k = KL= L = L,= iu=
Shapefactor=M=s M,
Fixityfactorfor:seinevaluatingandrestraintcoefficient
Modulusofelasticityinshear
Modulusofelasticityinshearatonsetofstrain-hardening
Hingerotationrequiredataplastichinge
Portionofhingerotationthatoccursincritical(buckling)seg. mentofbeam
Momentofinertia Momentofinertiaofelasticpartofcross-section
Momentofinertiaofplasticpartofcross-section
Numberofremainingredundanciesiuastructurethatis
redundantatultimateload Eulerlengthfactor
Distancefromflangefacetoendoff2lct Effective(pinend)lengthofcolumn
Spanlength;actualcolumnlength Criticallengthforlateralbuckling
Moment 11 Please purchase PDF Split-Merge on www.verypdf.com to
remove this watermark.SP:6(6)-1972 m = Mb= MO= Mp= Mp= Mpc= Mps=
M,= M,= i&c= M,= N = = ;= PC,= P= P,= P,= Pt= P"= P,= P,= R = =
;= s,= T = tf= ts= t.,= v = vc=
Numberofplastichingesdevelopedinastructurethatis
redundantatultimateload Momentatthehaunchpoint
Endmoment;ausefulmaximummoment;hingemoment Plasticmoment
Plasticmomentcapacityofabeamsection
Plastichingemomentmodifiedtoincludetheeffectofaxial compression
Plastichingemomentmodifiedtoincludeeffectofshearforce
Maximummomentofasimply-supportedbeam
Momentatwhichyieldpointisreachedinflcxure
Momentatwhichinitialouterfibreyieldoccurswhenaxial thrustispresent
Momentattheworkingload Numberofpossibleplastichinges
Numberofpossibleindependentmechanisms Concentratedload
Usefulcolumnload.Aloadusedasthemaximumcolumn load Eulerbucklingload
Reducedmodulusload Stabilizingload Tangentmodulusload
Theoreticalultimateload Workingload
Axialloadcorrespondingtoyieldstresslevel;P=Aa, Rotationcapacity
Radiusofgyration Sectionmodulus,I /C
Sectionmodulusofelasticpartofcross-section Force Flangethickness
Stiffnerthickness Webt hickncss Shearforce
Shearcarryingcapacityofasection _. 2c,
v,w=Displacementsinx,y,andzdirections W =TotaldistributedJoad 12
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watermark.SF:6(6)-1972 W EXT= Externalworkduetovirtualdisplacement
WI NT= Internalworkduetovirtualdisplacement W =
Distributedloadperunitoflength Wd = Thicknessofthewetdoublers WU
=Totaluniformitydistributedload X =Numberofredundancies x
=Longitudinalcoordinate
x=Distancetopositionofplastichingeundrrdistributedload Y
=Transversecoordinate Y
=Distancefromneutralaxistocentroidofhalf-area 2 MtJ
=Plasticmodulus=- CY 2, = Plasticmodulusofelasticportion 2,
=Plasticmodulusofplasticportion 2=Lateralco-ordinate
AL=Equivalentlengthofconnection 6=DC flection d=Strain Gt
=Strainatstrain-hardening CY =
Straincorrespondingtofirstattainmentofyieldstresslevel
8=Measuredanglechange;rotation.Rotation p =Poissonsratio
=Radiusofcurvature d=Nomialstress QY = Loweryieldpoiat 9
=Proportionlimit 0, =Residualstress a,rt
=Ultimatetensilestrengthofmaterial QWY = Upperyieldpoint 0,
=Workingstress fJ Y I Yieldstresslevel =Shearstress
=Rotationperunitlength,oraverageunitrotation;curvature Y =
Curvaturecorrespondingtofirstyieldinflexure Please purchase PDF
Split-Merge on www.verypdf.com to remove this watermark.As in the
Original Standard, this Page is Intentionally Left BlankPlease
purchase PDF Split-Merge on www.verypdf.com to remove this
watermark.SECTI ONA I NTRODUCTI ON 1.SCOPE
1.1Itisthepurposeofthishandbooktopresentthefundamental
conceptsinvolvedinplasticdesignandtojustifyitsapplicationto
structuralsteelframes.Themethodsofplasticanalysiswillbedes-
cribedtogetherwiththedesignproceduresthathavesofarbeen
developed.Secondarydesignconsiderationsarealsoincluded.
1.2Specificapplicationmaybemadetostaticallyloadedframesof
structuralsteeltocontinuousbeams,tosingle-storeyedindustrialframes
andtosuchotherstructureswhoseconditionofloadingandgeometry
areconsistentwiththeassumptionsinvolvedinthetheory.Numerous
applicationswillundoubtedlybemadetoothertypesofstructures
suchasringsandarches,butforthetimebeingthescopeofapplication
islimitedtotheindicatedstructuraltypes. 2. GENERAL
2.1Steelpossessesductility,auniquepropertythatnootherstructural
materialexhibitsiuquitethesameway.Throughductilitystructural
steelisabletoabsorblargedeformationsbeyondtheelasticlimitwith-
outthedangeroffracture.
2.2Althoughthereareafewinstanceswhereconscioususehasbeen
madeofthisproperty,byandLarge theengineerhasnotbeenableto
fullyexploitthisfeatureofductilityinstructuralsteeLAsaresultof
theselimitationsitturnsoutthatconsiderablesacrificeofeconomyis
involvedintheso-calledconventionaldesignprocedures. 2.3
Engineershaveknownofthisductilityforyears,andsincethe
19206havebeenattemptingtoseeifsomeconscioususecouldbe
madeofthispropertyindesign.Plasticdesignistherealizationofthat
goal.Thisgoalhasbeenachievedbecausetwoimportantconditions
havebeensatisfied.First,thetheoryconcerningtheplasticbehaviour
ofcontinuoussteelframeshasbeensystematizedandreducedtosimple
designprocedures.Secondly,everyconceivablefactorthatmighttend
toLimit theload-carryingcapacitytosomethinglessthanthatpredicted 15
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bythesimpleplastictheoryhasbeeninvestigatedandruleshavebeen
formulatedtosafeguardagainstsuchfactors. 3.STRUCTURALSTRENGTH
3.1Thedesignofanyengineeringstructure,beitabridgeorbuilding,
issatisfactoryifitispossibletobuiltitwiththeneededeconomyand
ifthroughoutitsusefullifeitcarriesitsintendedloadsandotherwise
performsitsintendedfunction.Asalreadymentioned,intheprocess
ofselectingsuitablemembersforsuchastructure,itisnecessaryto
makeageneralanalysisofstructuralstrengthandsecondlytoexamine
certaindetailstoassurethatlocalfailuredoesnotoccur.
3.2Theabilitytocarrytheloadmaybetermedstructuralstrength.
Broadlyspeaking,thestructuralstrengthordesignloadofasteelframe
maybedeterminedorcontrolledbyanumberoffactors,factorsthat
havebeencalledlimitsofstructuralusefulness.Theseare:first
attainmentofyieldpointstress(conventionaldesign),brittlefracture,
fatigue,instability,deflections,andfinallytheattainmentofmaximum
plasticstrength. 3.3Strictlyspeaking,adesignba~sed
onanyoneoftheabove-mentioned sixfactorscouldberoferredtoasa
limitdesign,althoughtheterm
usuallyhasbeenappliedtothedeterminationofultimateloadaslimited
bybucklingormaximumstrengthl*.Plasticdesignasanaspectof
limitdesignandasappliedtocontinuousbeamsandframesembraces,
then,thelastofthelimits_theattainmentofmaximumplasticstrength.
3.4Thus,plasticdesignisfirstadesignonthebasisofthemaximum
loadthestructurewillcarryasdeterminedfromananalysisofstrength
intheplasticrange(thatisiaplasticanalysis).Secondlyitconsistsof
aconsiderationbyrulesorformulasofcertainfactorsthatmightother-
wisetendtopreventthestructurefromattainingthecomputedmaxi-
mumload.Someofthesefactorsmaybepresentinconventional
(elastic)design.Othersareassociatedonlywiththeplasticbehaviour
ofthestructure.Buttheuniquefeatureofplasticdesignisthatthe ultimaL
load ratherthantheyieldstressisregardedasthede&u Criterion.
3.5Itha;longbeenknownthatwhenevermembersarerigidlycon-
nected,thestructurehasamuchgreaterload-carryingcapacitythan
indicatedbytheelasticstressconcept.Continuousor rigidframes
areabletocarryincreasedloadsabovefirstyieldbecausestructural
steolhasthecapacitytoyieldinaductilemannerwithnolossinstrength;
*Thisnumberreferstotheserialnumberoftheselectedreferencesgivenin
AppendixA. 16 t Please purchase PDF Split-Merge on www.verypdf.com
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indeed,withfrequentincreaseinresistance.Althoughthephenomenon
willbedescribedincompletedetaillater,ingeneraltermswhathappens
isthis: Asloadisappliedtothestructure,thecross-sectionwiththe
greatestbendingmomentwilleventuallyreachtheyieldmoment.
Elsewherethestructureiselasticandthepeakmomentvalues
arelessthanyield.As!oadisadded,azoneofyieldingdevelops
atthefirstcriticalsection;butduetotheductilityofsteel,the
momentatthatsectionremainsaboutconstant.Thestructure,
therefore,callsuponitsless-heavilystressedportionstocarrythe
increaseinload.Eventually,zmesofyieldingareformed
atothersectionsuntilthemomentcapacityhasbeenexhausted
atallnecessarycriticalsections.Afterreachingthemaximumload
value,thestructurewouldsimplydeformatconstantload.
3.6Attheoutsetitisessentialtomakeacleardistinctionbetween
elasticdesignandplasticdesign.Incmventionalelasticdesignprac-
tice,amemberisselectedsuchthatthemaximumallowablebending
stressisequalto1 650kgf/cm2attheworkingload.Asshownin
Fig.1suchabeamhasareserveofstrengthof1.65iftheyieldpoint stressis2
400kgf/cm*.Duetotheductilityofsteelthereisan OESIBNBASIS
CONVENTIONALDESIGN 0 PLASTICDESIGN FIG.1PLASTIC DESIGNCOMPARED
WITHELASTIC DESIGN I_ ,\y .: .L 17 Please purchase PDF Split-Merge
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additionalreservewhichamountsto12to14percentforawideflange
shape.Thusthetotalinherentoverloadfactorofsafetyisequalto
1~65x1~12=1~85asanaveragevalue.
3.7Inplasticdesign,ontheotherhand,thedesigncommenceswith
thelcltimate load.(Aswillbeevidentlater,itismucheasiertoanalyze
anindeterminatestructureforitsultimateloadthantocomputethe
yieldload.)Thustheworkingload,P,,ismultipliedbythesameload
factor(1.85)andamemberisselectedthatwillreachthisfactored load.
3.8Theloadvdeflectioncurvefortherestrainedbeamisshownin
Fig.1.Ithasthesameultimateloadastheconventionaldesignofthe
simplebeamandthememberiselasticatworkingload.Theimportant
thingtonoteisthatthefactorofsafetyisthesameintheplasticdesign
oftheindeterminatestructureasitisintheconventionaldesignofthe
simplebeam.
3.9Whilethereareotherfeatureshere,theimportantpointtogetin
mindatthisstageisthatinconventionalproceduresonecomputesthe
maximummomentundertheworkingloadandselectsamembersuch
thatthemaximumstressisnotgreaterthan1650kgf/cm2onthe
otherhandinplasticdesignonemultiplestheworkingloadby F =
1.85andselectsamemberwhichwilljustsupporttheultimate load.
3.10Terminology-Plasticdesignnaturallyinvolvestheuseofsome
newterms.Actuallythesearefewinnumber,butforconvenienceare
listedbelow : LimitDesign-Adesignbasedonanychosenlimitofstructural
usefulness. PlasticDesign -Adesignbasedupontheultimateload-carrying
capacity(maximumstrength)ofthestructure.Theterm plastic
isderivedfromthefactthattheultimateloadiscomputedfrom
aknowledgeofthestrengthofsteelintheplasticrange.
UltimateLoad(P,,)orMaximumStrength -Thehighestloada
structurewillcarry.(Itisnottobeconfusedwiththetermas
appliedtotheultimateloadcarriedbyanordinarytensiletest
specimens.)InthedesignP,isdeterminedbymultiplyingthe
expectedworkingload(PW) bytheloadfactor(seebelow).
Plastijicath-Thedevelopmentoffullplasticyieldofthecross- section.
PlasticMoment(M&-Maximummomentofresistanceofafully-
yieldedcross-section. 18 Please purchase PDF Split-Merge on
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Plasticlodzclus(Z)-Combinedstaticmomentsabouttheneutral
axisofthecross-sectionalareasaboveandbelowtheneutralaxis.
PlasticHiqeeAyieldedsectionofabeamwhichactsasifit
werehinged,exceptwithaconstantrestrainingmoment.
ShapeFactor(fl-Theratioofthemaximumresistingmoment
ofacross-section(M,)totheyieldmoment(M,). Mechanism -A
hingesystem,asystemofmembersthatcan movewithoutanincreaseinload.
RedistributiorcofMoment 4Aprocesswhichresultsinthesuc-
cessiveformationofplastichingesuntiltheultimateloadis
reached.Byit,theless-highlystressedportionsofastructure
alsomayreachthe(M&-value.
LoadFactor(F)-Asafetyfactor.Thetermisselectedtoempha-
sizethedependenceuponload-carryingcapacity.Itisthe
numberbywhich*theworkingloadismultipliedtoobtainP,.
4.MECHANICALPROPERTIESOFSTEEL
4.1Anoutstandingpropertyofsteel,which(asalreadymentioned)
setsitapartfromotherstructuralmaterials,istheamazingductility
whichitpossesses.ThisischaracterizedbyFig.2whichshowsin
somewhatidealizedformthestress-strainpropertiesofsteelintheinitial
portionofthecurve.InFig.3areshownpartialtensilestress-strain
curvesforanumberofdifferentsteels.Notethatwhentheelastic
limitisreached,elongationsfrom8to15timestheelasticlimittake
placewithoutanydecreaseinload.Afterwardssomeincreaseinstrength
isexhibitedasthematerialstrainhardens.
4.2Althoughth;firstapplicationofplasticdesignistostructures
fabricatedofstructuralgradesteel,itisnotlessapplicabletosteelsof
STRAI N- STRESS-STRAINCURVE OF S~-42STEEL IDEALIZED 19 Please
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watermark.SP:6(6)-1972 LASTICRANGE--W&RAINHARDENINO 0 If0 $0
STR4IN 3dx10-2 C FIG.3STRESS-STRAINCURVEOF ST&~
ANDST-%STRUCTURALSTEELS higher strength aslong as they possess
thenecessary ductility. Figure 3 attests
totheabilityofawiderangeofsteelstodeformplastically
withcharacteristics similar tosteelconformingtoIS:226-1969*.
4.3Itis important tobear inmind thatthestrains shown inFig.3 are
reallyverysmall.AsshowninFig.4,forordinarystructural steel, final
failure byrupture occurs onlyafter aspecimen has stretched some
IMALIZEDCURVE .Od! o- 11 l b50 1525 PERCENTAGEELONGATION____c
FIG.4COMPLETESTRESS-STRAINCURVEOF STRUCTURALSTEEL -.__-
*Structuralsteel(standardquality)(fourthrcuision). 20 Please
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watermark.SP:6(6)-1972 15
to25timesthemaximumstrainthatisencounteredinplasticdesign.
Eveninplasticdesign,atultimateloadthecriticalstrainswillnothave
exceededpercentageelongationofabout1.5.Thus,theuseofultimate
loadasthedesigncriterionstillleavesavailableamajorportionofthe
reserveductilityofsteelwhichmaybeusedasanaddedmarginof
safety.Thismaximumstrainof1.5percentisastrainatultimate load>n
thestructurenotatworkingload.Inmostcasesunderworking
loadthestrainswillstillbebelowtheelasticlimit.
5.MAXIMUMSTRENGTHOFSOMEELEMENTS
5.1Onthebasisoftheductilityofsteel(characterizedbyFig.2)itis
nowpossibletocalculatequicklythemaximumcarryingcapacityof
certainelementarystructures.
Asafirstexampletakeatensionmembersuchasaneyebar(Fig.. 5).
ThestressisQ =P/A.
Thelo&dvdeflectionrelationshipwillbe\elasticuntiltheyield
pointisreached.AsshowninFig.5deflectionattheelasticlimitis
givenby6,=P,L/AE. t T- i A i 1 I L I I I UNRESTRICTED PLASTICFLOW
STRESS: 6p =- A DEFLECTION: $1PUL 6,.EL:7:AE P,,s$A FIG.
5MAXIMUMSTRENGTHOF AN EYEBAR (DETERMINATESTRUCTURE) 21, Please
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Sincethestressdistributionisuniformacrossthesection,unrest-
rictedplasticflowwillsetinwhentheloadreachedthevaluegivenby P,= ayA
Thisis,therefore,theultimateload.Itisthemaximumloadthe
structurewillcarrywithouttheonsetofunrestrictedplasticflow.
Asasecondexampleconsiderthethree-barstructureshownin
Fig.6.Itisnotpossibletoconsiderthestateofstressbystaticsalone
andthusitisindeterminate.Considertheelasticstate.Fromthe
equilibriumconditionthereisobtained: 2Tl+TI =P. . .. . .. . ..
..(l) whereT1istheforceinbars1and3andT,theforceinthebar2.
PARTIALLVPl .ASTl t PLASTIC EWL~IUY:21,+12-P VLLOLIMIT:pr212*2byh L
0 OEFLECTION- FIG. 6PLASTICAND ELASTICANALYSIS OF AN
INDETERMINATESYSTEM
Thenextconditiontoconsideriscontinuity.Forarigidcrossbar,
thetotaldisplacementofBar1 willbeequaltothatofBar2. Therefore:
l-14T&a AE-AE.**.**** . . . (2) T,= 2(asL,= 2L.J. . .. . ..
..(3) WiththisrelationshipbetweenT,andT,obtainedbythecontinuity
condition,usingEq(1)itisfoundthat: T&.. . .. . .. . .. . ..
..(4) 22 Please purchase PDF Split-Merge on www.verypdf.com to
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Theloadatwhichthestructurewilltirstyieldmaythenbedetermined
bysubstitutinginEq(4)themaximumloadwhichT,canreach, namely,oYA .
Thus, P,,=2T,=2a,A. . .. . .. ..(5)
Thedisplacementattheyieldloadwouldbedeterminedfrom: 8=q2&
Yor;22E ....... ..(6)
Now,whenthestructureispartiallyplasticitdeformsasifitwerea
two-barstructureexceptthataconstantforceequaltoCT,,Aissuppliedby
Bar2(thememberisintheplasticrange).Thissituationcontinues
untiltheloadreachestheyieldvalueinthetwoouterbars.Notice
howeasilyitispossibletocomputetheultimateload: P,=3ayA. . .. . ..
..(7) Thebasicreasonfort,hissimplicityisthatthecontinuitycondition
neednotbeconsideredwhentheultimateloadintheplasticrangeis
beingcomputed.
Theload-deflectionrelationshipforthestructureshowninFig.6
isindicatedatthebottom.Notuntiltheloadreachesthatvalue
computedbyaplasticanalysis(Eq7)didthedeflectionscommenceto
increaserapidly.Thedeflectionwhentheultimateloadisfirstreached
canbecomputedfrom: I.. . .. ..(8)
Thethreeessentialfeaturesofthissimpleplasticanalysisareasfollows:
a)Eachportionofthestructure(eachbar)reachedaplasticyield condition,
b)Theequilibriumconditionwassatisfiedatultimateload,and
c)Therewasunrestrictedplasticflowattheultimateload.
Thesesamefeaturesareallthatarerequiredtocompletetheplastie
analysisofanindeterminatebeamorframe,andinfact,thissimple
exampleillustratesalloftheessentialfeaturesofaplasticanalysis.
6.HISTORICALDEVELOPMENT
6.1Theconceptofdesignbasedonultimateloadasthecriterionis
morethan40yearsold!Theapplicationofplasticanalysistostructural
designappearstohavebeeninitiatedbyDrGaborKazinszy,aHunga-
rian,whopublishedresultsofhisTests2ofClampedGirdersasearlyas 23
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1914.Healsosuggestedanalyticalproceduressimilartothose
nowcurrent,anddesignsofapartment-typebuildingswereactually
carriedout.
6.2InhisStrengthofMaterialsa,Timoshenkoreferstoearlysuggestions
toutilizeultimateloadcapacityintheplasticrangeandstatesasfollows:
Suchaprocedureappearslogicalinthecaseofsteelstructures
submittedtotheactionofstationaryloads,sinceinsuchcasesa
failureowingtothefatigueofmetalisexcludedandonlyfailure
duetotheyieldingofmetalshastobeconsidered.
EarlytestsinGermanyweremadebyMaier-Leibnitzwhoshowedthat
theultimatecapacitywasnotaffectedbysettlementofsupportsof
continuousbeams.Insodoinghecorroboratedtheprocedurespre-
viouslydevelopedbyothersforthecalculationofmaximumloadcapa-
city.TheeffortsofVandenBroeklinUSAandJ.F.Baker6310 and
hisassociatesinGreatBritaintoutilizeactuallytheplasticreserve
strengthasadesigncriterionarewellknown.Progressinthetheory
ofplasticstructuralanalysis(particularlythatatBrownUniversity)
hasbeensummarizedbySymondsandNeal.
6.3FormorethantenyearstheAmericanInstituteofSteelConstnm-
tion,theWeldingResearchCouncil,theNavyDepartmentandthe
AmericanIronandSteelInstitutehavesponsoredstudiesatLehigh
University6**se.Thesestudieshavefeaturednotonlytheverification
ofthismethodofanalysisthroughappropriatetestsonlargestructures,
buthavegivenparticularattentiontotheconditionsthatshouldbe
mettosatisfyimportantsecondarydesignrequirements.
6.4Plasticdesignhasnowcomeofage.Itisalreadyapartofthe
BritishStandardspecificationsandnumerousstructuresbothinEurope
andNorthAmericahavebeenconstructedtodesignsbaseduponthe
plasticmethod.IS:800-1962*permitstheuseofPlasticTheoryin
thedesignofsteelstructures(see13.5.1ofIS:800-1962*).
*Codeofpracticeforuseofstructuralsteelingeneralbuildingconstruction
:>T.$ - (rcuised). ** 24 Please purchase PDF Split-Merge on
www.verypdf.com to remove this watermark.SECTI ONB J USTI FI CATI
ONFORPLASTI CDESI GN 7.WHYPLASTICDESIGN 7.1
Whatisthejustificationforplasticdesign?Onecouldreversethe
questionbyasking, whyuseelasticdesign?Ifthestructurewill suppor t
theloadandotherwisemeetitsintendedfunction,arethe
magnitudesofthestressesreallyimportant?
7.2Itistruethatinsimplestructurestheconceptofthehypothetical
yieldpointasalimitofusefulnessisrational.Thisisbecausethe
ultimateloadcapacityofasimplebeamisbut10to15percentgreater
thanthehypotheticalyieldpoint,anddeflectionsstartincreasingverv
rapidlyatsuchaload.Whileitwouldseemlogicaltoextendelastic
stressanalysistoindeterminatestructures,suchprocedureshavetended
tooveremphasizetheimportanceofstressratherthanstrengthasa
guideinengineeringdesignandhaveintroducedacomplexitythatnow
seemsunnecessaryforalargenumberofstructures.
7.3Actuallytheideaofdesignonthebasisofultimateloadratherthan
allowablestressisareturntotherealisticpointofviewthathadtobe
adoptedbyourforefathersinaverycrudewaybecausetheydidnot
possessknowledgeofmathematicsandstaticsthatwouldallowthem
tocomputestresses.
7.4Theintroductionofwelding,ofcourse,hasbeenaveryrealstimulus
tostudiesoftheultimatestrengthofframes.Byweldingitispossible
toachievecompletecontinuityatjointsandconnections-andtodo
iteconomically.Thefullstrengthofonemembermaythusbetrans-
mittedtoanother.
7.5Ithasoftenbeendemonstratedthatelasticstressanalysiscannot
predicttherealstress-distributioninabuildingframewithanything
likethedegreeofaccuracythatisassumedinthedesign.Thework
doneinEnglandbyProf.Bakerandhisassociatesasaforerunnerto
theirultimatestrengthstudiesclearlyindicatedthis. 7.6Examplesof
imperfectionsthatcausesevereirregularityin
measuredstressesare:differencesinbeam-columnconnectionfit-upand
flexibility,spreadingofsupports,sinkingofsupports,residualstresses,
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this watermark.SP:6(6)-1972
flexibilityassumedwhereactuallythereisrigidity(andvice-versa),
andpointsofstressconcentration.Suchfactors,however,usuallydo
notinfluencethemaximumplasticstrength.
7.7Assumingthatstressisnotthemostrationaldesigncriteria,inorder
tojustifyourfurtherconsiderationofmaximumstrengthasthedesign
criteriatheremustbeotheradvantages.Therearetwosuchadvantages:
economyandsimplicity.
7.8Sincethereisconsiderablereserveofstrengthbeyondtheelastic
limitandsincethecorrespondingultimateloadmaybecomputedquite
accurately,thenstructuralmembersofsmallersizewilladequately
supporttheworkingloadswhendesignisbasedonmaximumstrength.
Numerousdemonstrationsofthiswillbemadelaterinthisbandbook.
7.9Thesecondfeaturewas simplicity.Ananalysisbasedupon
ultimateloadpossessesaninherentsimplicitvbecausetheelasticcon-
ditionofcontinuityneednolongerbeconsidered.Thiswasevident
fromaconsiderationofthethree-bartrussinSectionA(Fig.6)andthe
examplesofSectionDwilldemonstratethisfurther.Alsothe imper-
fectionsmentionedaboveusuallymaybedisregarded.
7.10Asalreadymentionedtheconceptismorerational.Byplastic
analysistheengineercandeterminewithanaccuracythatfarexceeds
hispresentlyavailabletechniquestherealmaxirnumstrengthofa
structure.Therebythefactorofsafetyhasmorerealmeaningthanat
present.ItisnotunusualforthefactorofsafetytovaryfromI.65up
to3ormoreforstructuresdesignedaccordingtoconventionalelastic
methods. 7.11Thustheapplicationofplasticanalysisshouldbeconsidered
seriouslybecauseitprovidesaless-expensivestructure,itisasimilar
designofficetechnique,anditconstitutesarationaldesignbasis.Fur-
ther,theseconceptsareverifiedbytestsand(asweshallnowsee)they
havebeenusedconsciouslyorunconsciouslyinconventionaldesign
practice. 8.INA.DEQUACYOFSTRESSASTHEDESIGNCRITERION 8.1
Thequestionimmediatelyarises,willitnotbepossiblesimplyto
changetheallowablestressandretainthepresentstressconcept?While
theoreticallypossible,thepracticalansweris no.Itwouldmeana
differentworkingstressforeverytypeofstructureandwouldvaryfor
differentloadingconditions.
8.2Toagreaterextentthanwemayrealize,themaxnnumstrengthof
astructurehasalwaysbeenthedominantdesigncriteria.Whenthe 26 Please
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watermark.SP:6(6)-1972 permissibleworkingstressof1
400k&mahasledtodesignsthatwere
consistentlytooconservative,thenthatstresshasbeenchanged.Thus
thebenefitsofplasticityhavebeenusedconsciouslyorunconsciously
indesign.Itisalsoevidenttomostengineersthatpresentdesignpro-
cedurescompletelydisregardlocalover-stressingatpointsofstress-
concentrationlikeboltholes,notches,etc.Longexperiencewith
similarstructuressodesignedshowsthatthisisasafeprocedure.Thus,
thestressesthatarecalculatedfordesignpurposesarenottrue
maximumstressesatall,theysimplyprovideanindexforstructura design.
8.3Anumberofexampleswillnowbegiveninwhichtheductilityof
steelhasbeencountedupon(knowinglyornot)inelasticdesign.It
shouldbeborneinmindthatplasticanalysishasnotgenerallybeen
usedasabasisfordeterminingtheseparticulardesignrulesandasa
resulttheso-calledelasticstressformulashavebeendevisedinarather
haphazardfashion.Arationalbasisforthedesignofacomplete
steelframe(aswellasitsdetails)canonlybeattainedwhenthe
maximumstrengthintheplasticrangeisadopted.asthedesign criterion.
8.4Suchexamplesarethefollowingandarelistedintwocategories:
(a)factorsthatareneglected.becauseofthecompensatingeffectof
ductility;and(b)instancesinwhichtheworkingstresseshavebeen
revisedbecausethe normalvaluewastooconservative.Several
examplesofeacharegiven: a)F&tomthatareneglected: 1) 2) 3) 4) 5)
6) 7) 8) 9) Residualstresses(inthecaseofflexureduetocoolingafter
rolling); Residualstressesresultingfromthecamberingofbeams;
Erectionstresses; Foundationsettlements;
Over-stressatpointsofstress-concentration(holes,etc);
Bendingstressesinanglesconnectedintensionbyonelegonly;
Over-stressatpointsofbearing:
Non-uniformstress-distributioninsplices,leadingtodesign
ofconnectionsontbeassumptionofauniformdistribution
ofstressesamongtherivets,bolts,orwelds;
Differenceinstress-distributionarisingfromthe cantilever
ascomparedwiththe portalmethodofwindstress analysis; 27 Please
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watermark.SP: 6( 6) - 1972 10)IS:800-
1962specifiesthefollowingvaluesforbendingstresses: 1
650kgf/cmforrolledsections, 1 575kgf/cm*forplategirders,and 1
890kgf/cm2inflatbases.
b)Revisionsinworkingstressduetoreserveplasticstrength: 1) 2) 3) 4)
5) Bendingstressof2109kgf/cm2(or30ksi)inroundpins(in
AISCspecification); Bearingstressof2
812kgf/cms(or40ksi)onpinsindouble shear; Bendingstressof1
687kgf/cm2(or24ksi)inframedstruc- turesatpointsofinteriorsupports;
Bendingstressof1 65?kgf/cm2and.1 575.kgf/cm2forrol!ed
sae$onsandplategirdersrespectively(mIS:800-1962), Bendingstressof1
890kgf/cmsinslabbases(iuIS:800-1962*).
ConsiderItem(a)(1)forexample.Allrolledmemberscontain
residualstressesthatareformedduetocoolingafterrollingordueto
cold-straightening.Atypicalwideflangeshapewithatypicalresidual
stresspatternshowninFig.7.Whenloadcarryingbendingstresses
areapplied,theresultingstrainsareadditivetotheresidualstrains
alreadypresent.Asaresult,the finalstresscouldeasilyinvolve
yieldingatworkingload. IntheexampleofFig.7,suchyieldinghas
RESIDUALSTRESSAPPLIEDSTRESSFINALSTRESS (a)(b)(cl FIG.7RESIDUAL
STRESSESIN *Code ofpractice foruse ofstructural (mviscd). 28 (d)(0)
AROLLED BEAMSECTION steel ingeneral buildingconstruction Please
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occurredbothatthecompressionflangetipsandatthecentreofthe
tensionflange.Thus,itisseenthatcoolingresidualstresses(whose
influenceisneglectedandyetwhicharepresentinallrolledbeams)
causeyieldingintheflangetipsevenbelowtheworkingload.
8.5Structuralmembersexperienceyieldwhilebeingstraightenedin
themill,fabricatedinashoporforcedintopositionduringerection.
Actually,itisduringthesethreeoperationsthatductilityofsteel
beyondtheyieldpointiscalledupontothegreatestdegree.Having
permittedsuchyieldinginthemill,shopandfield,thereisnovalid
basistoprohibititthereafter,providedsuchyieldhasnoadverseeffect
uponthestructure.Asanillustrationofitem(a)(3)inthelistin8.4,
Fig.8showshowerectionforceswillintroducebendingmomentintoa
st.ructurepriortotheapplicationofexternalload(see firstlineforP =0)
.Althoughtheyield-pointloadisreducedasaresultofthese erection
moments(inthesecondlineofthefigure,theyield-pointloadhasbeen
reachedforcase2).themis#oefectwhateveronthemaximumstrength.
Thereasonforthisisthatredistributionofmomentfollowedtheonset
ofyieldingatthecorners(case2)untiltheplasticmomentwasreached
atthebeamcentre;therefore,theultimateloadmomentdiagramsfor
cases1and2areidentical.
8.6Consider.next.thedesienofarivetedorboltedjoint[Item(a)(8)
in8.41.Thecommonass&ptionismadethateachfastenercarriesthe p-0
P=Py(21 P=P FIG.8DEMONSTRATIONTHAT ERECTION STRESSESDO NOT
INFLUENCE ULTIMATE LOAD 29 Please purchase PDF Split-Merge on
www.verypdf.com to remove this watermark.SP:6(6)-1972
sameshearforce.Thisistrueonlyinthecaseoftwofasteners.When
moreareadded(Fig.9),thenaslongasthejointremainselastic,the
outerfastnersshouldcarrythegreaterportionoftheload.Forexample,
withfourrivets,ifeachrivettransmittedthesameload,then,between
rivetsCandDoneplatewouldcarryperhapsthreetimestheforcein
theother.Therefore,itwouldstretchthreetimesasmuchandwould
necessarilyforcetheouterrivetDtocarrymoreload.Theactual
forceswouldlooksomethingliketheseshownundertheheading Elastic.
Whateventuallyhappensisthattheouterrivetsyield,redistributing
forcestotheinnerrivetsuntilallforcesareaboutequal.Therefore,
thebasisfordesignofarivetjointisreallyitsultimateloadandnotthe
attainmentoffirstyield. 8.7A
revisedworkingstressexample[seeItem(b)(1)in8.41isshown
inFig.10andisconcernedwiththedesignofaroundpin.Inasimple
beamwithwideflanges,whenthemaximumstressduetobendingreaches
theyieldpoint,mostoftheusablestrengthhasbeenexhausted.How-
ever,forsomecross-sectionalshapes,muchadditionalloadmaybe
carriedwithoutexcessivedeflections.Therelationbetweenbending
momentandcurvatureforwideflangeandroundbeamsisshowninFig.10.
Theuppercurveisforthepin,thelowerforatypicalwideflangebeam,the
non-dimensionalplotbeingsuchthatthetwocurvescoincideinthe
elasticrange.Themaximumbendingstrengthofthewideflangebeam
isl-14MY,whereasthatofthepinisI.70My.Thepermissibledesign
stresses(forsteelwithyieldstress36ksi)accordingtospecifications
oftheAmericanInstituteofSteelConstructionare1550kgf/cmz(or
22ksi)forthewideflangebeamand2 320 kgf/cm2(or30ksi)forthe roundpin.
A BcD FIG.9REDISTRIBUTIONOF SHEARINTHE FASTENERSOF ALAP JOINT 30
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watermark.SP:6(6)-1972 FIG.10MAXIMUM STRENGTH OF OF AWIDE 5 A- *Y
lo AROUNDPINCOMPAREDWITH THAT FLANGE BEAM
Expressingthesestressesasratiosofyieldpoints+-:
Wideflange:0=F5g=0.61 QY Pin: =* 2320 -=20=0.91 =Y
Forasimplv-supportedbeamthestresses,moments,andloadallbear
alinearrelationshiptooneanotherintheelasticrangeandthus: PcM
p,=6=My Therefore,themomentatallowableworkingstress(M,)inthewide
flangebeamisO-61 My;forthepin,ontheotherhand,M,=0.91My.
Whatisthetrueloadfactorofsafetyforeachcase1 M WideRange:F=p~=M*=
?_!%=1.87 0 0.61 M, Pin: ~&_2!%=1.87 0.91My
Theexactagreementbetweenthetruefktorsofsafetywithrespectto
ultimateloadinthetwocases,whilesomewhatofacoincidence,isindi-
cativeoftheinfluenceoflongyearsofexperienceonthepartofengineers 31
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watermark.SP: 6(6) -1972
whichhasresultedindifferentpermissibleworkingstressesforvarious
conditionsresultinginpractice.Probablynosuchanalysisasthe
foregoinginfluencedthechoiceofdifferentunitstressesthatgiveidentical
factorsofsafetywithvarioussections;nevertheless,thechoiceofsuch
stressesisfullyjustifiedonthisbasis.Whenyearsofexperienceand
commonsensehaveledtocertainempiricalpracticesthesepracticesare
usuallyjustifiedonascientificbasis.
8.8Permittinga20percentincreaseintheallowableworkingstressat
pointsofinteriorsupportincontinuousbeamsrepresentsanothercase
inwhichbothexperienceada plasticanalysisjustifyarevisionin
workingstresses. 9. EXPERIMENTALVERIFICATION
9.1Inthepreviousclausessomeoftheimportantconceptsoftheplastic
theoryaredescribed.Howwelldoesstructuralbehaviourbearoutthe
theory?Dostructuresreailycontaintheductilityassumed?Ifwe testa
fullsizestructurewithrolledmemberswillitactuallycarrythe
loadpredictedbyplasticanalysis1
9.2Theimportantassumptionsmadewithregardtotheplasticbeha-
viourofstructuresarerecapitulatedinFig.11).InLecture4of Ref12(see
AppendixA),theexperimentalconfirmationoftheseassump-
tionsisgiven,demonstratingtheductilityofsteel,thedevelopmentof
plastichingesinbeamsand%oiinections,andredistributionofmoment.
Inthelastanalysis,themostimportantverificationofplastictheoryis
thatgivenbytheresultsoffull-scaletestsandsomeofthesewillnow
bepresented.
9.3TypicalstructuresweretestedbothinUSAandothercountries.
Thestructurecarriedthepredictedultimateload,theload-deflection
curvebeingshowninFig.12.
9.4Furthertestsconductedonframesfabricatedfromrolledsections
haveshownthattheactualstrengthofeventheweakeststructurewas
within5percentofitspredictedultimateloadanagreementmuchbetter
thanobtainedattheso-called elasticlimit~J6*20*a0@*u. Intestson
beamswiththreesupports,applyingtheverticalload,thecentralsupport
wasraiseduntiltheyieldpointwas:firstreached,withtheresultthat
applicationofthefirstincrementofexternalloadcausedthestructure
toyield.Inspiteofthis,thecomputedultimateloadwasattained.
Inthetestsconductedonpinnedandfixedbasisandwithflat,saw
toothant1gabledroofs,theultimateloadcomputedbytheplastic
theorywasreachedandinnumerouscasesitexceededa14s*44. 32 Please
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watermark.SP:6(6)-1972 FlASTCMOMENT-9 PlASTCHINGE -GMOMENTCAtXtTY=
UTIMATEWADHING&--i&ANISM FIG.11ASSUMPTIONSMADE INREGARD TO
PLASTIC BEHAVIOUROFSTRUCTURES 10.THECASEFORPLASTICDESIGN
10.1Assummarizedintheprecedingparagraphstheresultsoftestshave
verifiedthetheoryofplasticanalysis.Istheengineernowjustifiedin
givingfurtherattentiontothemethodofplasticanalysis,instudying
it,andinapplyingittotheappropriatedesignproblems7Theanswer isyes.
Thecaseforplasticdesignisillustratedbythefollowingobserva- tions :
4 Thereserveinstrengthaboveconventionalworkingloadsis
considerableinindeterminatesteelstructures.Indeed,insome
instancesasmuchload-carryingcapacityisdisregardedasis
usedinconventionaldesign. .. 33 k.. Please purchase PDF Split-Merge
on www.verypdf.com to remove this watermark.SP:6(6)-1972 016
0255075100125150175200225: DEFLECTION INcm-
FIG.12LOAD-DEFLECTIONCURVE OFA TESTFRAME b) d 4
Useofultimateloadasthedesigncriterionprovidesatleast
thesamemarginofsafetyasispresentlyaffordedintheelastic
designofsimplebeams(Fig.1).
Atworkingloadthestructureisstillintheso-calledelasticrange (Fig.1).
Inmostcases,astructuredesignedbytheplasticmethodwill
deflectnomoreatworkingloadthanwillasimply-supported
beamdesignedbyconventionalmethodstosupportthesame load(Fig.1). 4
Plasticdesigngivespromiseofeconomyintheuseofsteel,of
savingsinthedesignofficebyvirtueofitssimplicity,andof
buildingframesmorelogicallydesignedforgreaterover-all strength.
Itisimportanttobearinmindthatdependencemaybeplaced
uponthemaximumplasticstrengthonlywhenproperattentionisgiven to
details.Thesearethesecondarydesignconsiderationsmentioned
earlierandtreatedinSectionE. 34 Please purchase PDF Split-Merge on
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watermark.11.ASSUMPTIONSANDCONDITIONS SECTI ONC FLEXUREOFBEAMS
11.1Certainofthefundamentalconceptsofplasticanalysiswereprs
sentedinSectionA(see3and4).Theexamplesthere,however,were
limitedtocasesofsimpletensionandcompression.Thenextobjective
istodeterminehowabeamdeformsbeyondtheelasticlimitunderthe
actionofbendingmoments,thatis,todeterminethemoment-curvature
(M-9)relationship.
Theassumptionsandconditionsusedinthefollowingdevelopmentare:
a)strainsareproportionaltothedistancefromtheneutralaxis
(planesectionsunderbendingremainplaneafterdeformation).
b)thestress-strain relationshipis idealizedtoconsistof
twostraightlines: 4 4 u=E( O
