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TECHNICAL NOTE 3782
HANDBOOKOFSTRUCTURALSTABILITY
PART II - BUCKLINGOF COMPOSITEELEMENTS
ByHerbertBecker
NewYorkUniversity
vWashingtonJuly1957
II1
1,
.
. . . . . ____ . .
TECHLIBRARYKAFB,NM
IllllllllllllunlnNATIONALADVISORYCOMWITEEFORAERONAUTICS OUbLbqA—
TECHNICALNOTE3782
HMJDBOOKOFSTRUCTURALSTABILITY
PARTII - BWKH3VGOFCOMPOS~EHMINB3
ByHerbertBecker
SWMARY
Thelocalbucklingofstiffenersectionsandthebucklingofplateswitheturdystiffenersexereviewed,andtheresultsaresummarizedinchartsandtables.Numericalvaluesofbuckl~ coefficientsarepresentedforlongitudinallycompressedstiffenersectionsofvariousshapes,forStiffmeastM?fenedprimarilydiscussed
platesloadedinlongitudinalcompressionandinshear,andforcylhdersloadedintorsion.Althoughthedatapresentedconsistofelastic-bucklingcoefficients,theeffectsofplastici@arefora fewspecialcases.
RWRODUCTION
ThebucklingbehaviorofsimpleplateelementsisdescribedinpartsI andIIIofthis“HandbookofStructuralStability”(refs.1 and2). Structuralcomponentsoftenconsistoftwoormoresimpleplateelementssoarrangedthatthebucklingstressofeachisincreasedasaresultofthesupportprovidedbycontiguousneighbors.Suchcompositeelementsaretermedstiffenersbecausetheyarefrequentlyusedtostiffena plateinordertoincreasethebuc~ingstress.A compactstiffeneriSdescribedas “sturdy”whenitisnotsubjecttolocalbuckltngandthere-foreonlytheaxial,bending,andtorsionalrigiditiesofthestiffenerinfluencethebehavioroftheplate-stiffenercomb@ationundera specifiedloading.Thedatapresentedinthisreportonthebucklingofstiffenedplatespertaintosturdystiffeners.
Thereportbeginswitha discussionofcalculationoflocalbucklingstressofstiffeningelements.Stiffenerstructuralshapesincommonuse,suchasZ-,channel,andhatsections,havebeenanalyzedforbucKMngandchartsarepresentedtofacilitatebuckling-stresscomputations.Forsectionswhichbuckleelastically,failuremayoccuratloadsconsiderablyinexcessofbuckling.Failure,orcrippling,ofstiffeningelementsistreatedinreference3.
Whentheproportionsofa stiffeneraresuchthatitissturdywithrespecttotheplatewhichitissttifening,itactsessentiallyasan
.
.
. — —.— —.. .- —-- —- --
.
2 NACATN3782
elasticrestrainttotheplate.Itmayassistintheresistancetoload,asdothespanwisestiffenersina wingcover,oritmaybehaveprimarilyasa support,suchasa transverserib.“lheithercase,itisnecessarytoconsideronlyitsaxial,bending,androtationalspringpropertiesincalculatingthebucklingstressofa“stiffenedplate.Bucklingofthecompositewillthenoccureitherlocallyintheplateorgenerally,involvingboththeplatesndstiffener.Theinformationonbucklingofstiffenedplatesappearsinthesectionentitled“BucklingofStiffenedPlatesUnderImgitudinalCompression”foruniaxialloadandinthesectionentitled“BucklingofStiffenedPlatesUnderShearIoad”forshearload.
ThebucklingofstiffenedcurvedplatesinvolvesthecomplicationofplatecurvatureinadditiontoaXLthepsmmetersaffectingbucklingofstiffenedflatplates.ThebucklingofUnsttifenedcurvedplateshasbeendescribedh reference2,inwhichitwasshownthattheoryisingoodagreementwithtestdataforshearloadingandinpooragreementwithdataforaxialcompressionloading.Forcertainproportions,thecurvedplatesapproachcylinderbehavior,whichpermitsevaluationoftheunstiffened-plateresultsinthelimitingcase.Analysesofstiffenedcurvedplatesarereportedinthesectionentitled“BucklingofStiffenedPlatesUnderLongitudinalCompression”foraxialloadsandinthesectionentitled“BucMingofStiffenedPlatesHer ShearLoad”forshearloads.Znaddition,thelimitingcaseoftorsionalbucklingofa stiffenedcylinderisdescribedinthelattersection.Theresultsofthetheoryandtestdataarecomparedwiththeinformationonstiffenedcurvedplatesundershear.
Thebucklingstressofa stiffenerora stiffenedplatemaybefoundfromthegeneralrelationship
(1)
in Whichb pertainstoa general.dimension.Itmaybethewidthofaflangeonanangle,thedepthofthewebona channel,orthewidthofoneofthesidesofa rectsmgulsrtube.Thebucuingcoefficientkbisthecoefficienttobeusedtogetherwiththisdtiensionb equation(1).
TIM?parametersuponwhichkb dependsare a/b or A/b oftheplate,theamountofelasticrotationalrestratitalongtheunloadededgese,theratiooftheareaofthestiffenertothatoftheplateA/bt,theratioofthebendingrigidityofthestiffenertothatoftheplateEI/bD,andthecurvaturepsxameterforcurvedplates~. The
.
— . .
3NACAm 3782
figuresdiscussedtheseparameters.
in thefollowingsectionsshowkb asa functionof
Theeffectsofmaterialpropertiesonthebucklingofsimpleelementswerecoveredh ref=ence1,inwhichstress-straincurves,Poisson’sratio,andcladdingandplasticity-reductionfactorsarepresentedanddiscussed.Plasticity-reductionfactorsforcurvedplatesandshellsaredescribedinreference2. Forconvenience,a summaryofpertinentinfor-mationappearsinthe“ApplicationSection”andtables1 to3.
Thissurveywasconducteds%NewYorkWversityunderthesponsor-shipandwiththefinancialassistanceoftheNationalAdvisoryCommitteefor-Aeronautics.
A
a
b
D
d
E,Es,~
G
h
SYMlms
areaofst~enercrosssection,
.
,
Sqim
len~ ofunloadededgeonlongitudinallycompressedplatesands~le elementsorlongersideofplatesloadedinshear,in.
lengthofloadededgeonlongitudinallycompressedplatesandsimpleelementsorin.
flexuralrigidityofin-lb
shortersideofplatesloadedinshea”r,l
.
plateperinchofwidth,E+2(. - ,3],
widthofbulbofbulbflange
Young’smodulus,secantmodulus,andtangentmodulus,respectively,psi
shearmodlihls,pSi
widthofrectsmgubrtubestiffener(seefig.5(c))
%ymbolsa and b pertaintodimensionsbetweenstiffenersonstiffenedplatesortodistancesbetweenparalleledgesohunstiffenedplates.Thus,bucklingstressisfoundfora singleelementofa stiffaedplateandnotb termsofovemlldimensionsoftheplate,exceptforcurvedstiffenedplatesundershear.Inthislattercaseitismorecon-venienttoutilizea and b asoverallplatedimensionsforeaseofcomparisonwithcylinderdata.
.. ... . .. . ... . . _ --— — .. ——. —-—— .- . .. .
4 NACATN3782
bentlngmomentofinertiaofstiffenercrosssection,in.4
I4
J torsionalmomentofinertiaofstiffenercrosssection,in.
k buclsUngcoefficient
kb generalbucklingcoefficientofstiffenerpertainingtobucklingstressofelementofwidthb
%J% bucklhgcoefficientsforcompressionandshear,respectively
length ofcylinder,in.
momentappliedtoedgesofrotationallyrestrainedelement,in-lb
numberoflongitudinalstiffenersonplateoftotalwidthnb~ornumberofcircumferentialrtigsoncylinderoflengthL
numberoftransversebucklesinlongitudinallystiffenedplatelongitudinallycompressed .
radiusofcurvatureofcurvedplate,in.
correctionforpresenceofstiffenerononesideofplate
platestiffness(seefig.2 forclifferenttypes)
thickness,in.
platecurvatureparameter,(b21+ - ‘e2)1’2
cylindercurvatureparameter,
factorh r dependentupon
(L’ld (1 - ,e2)l/2
n and q (seefig.12)
distanceofstiffenercentroidfrommidsurfaceofplate,in.
ratioofrigidityofelasticrestrainttorotationalrigidityofplate;alsostrain
plasticity-reductionfactor
rotationofedgeofsimpleelement,radians
wavelengthofbuc=einshnpleelementorplate~ in.
.— . __— ——
NACATN3782 5
Ve Poisson’sratioinelasticrange
‘cr generalbucklingstress;also,bucklingstressofcompressedelement,psi
‘cr bucklingstressofelementloadedinshear,psi
Subscripts:
cr buckling
e effective
f flange
L lip
T topwebofhat-sectionstiffmers
v web
IOCAL13UCKLINGOFSKOWEMIMGELEMENTS
BehaviorofStiffeners
Whena plateunderlongitudinalloadissupportedbya stiffenerinthedtiectionoftheload,thestiffenerparticipatestiresist~thisload.Asa result,thepossiblebucklingmodesofthiscompositearelocalinstabilityoftheplatealone,localbucklingofoneormoresimpleelementsofthestiffener,generalWtabili& oftheplateinvolvingcolumnactionofthestiffener,orsomecombinationofthesemodes.The_ses perta~to stfifen~pl.atesa~lyto stm stiffenersonly,and,consequently,thesecondmodeisprecludedbytheanalysisinthosecasesdescribedh thelastsectionsofthepresentpaper.Howev=,inordertoinsurethesturdinessofthestiffener,itisfirstnecessarytodetermineitslocalbucklingstress.Thisisthesubjecttobedis-cussedinthepresentsection,whichpresentsthebackgroundforanalysis-oflocalbucklinginstiffenersandincludeschartsforrapidcalculationofbuckl~ stressforseveralcomonshapes.
Thelocalbucklhgstressofa stiffeneristhessmeasthatofitsweakestelement.Consequently,eachsimpleelementmustbeanalyzedforbucklingunder101@tUdb31 load. Oftentheweakestelementisreadilyevidentbyinspection.Theanalysisoftheelementinvolvesdetemdningthenatureofthesupportsandrotationalrestraintsalongtheedgesand
. .— ——.—._ -— . -—-.. —. ———__—— ——. ____ ._ _ _ ___ ._
6 NACATN3782
thencomputingthebucKlingstressoftheelementconsideredasa simpleplateunderlongitudinalloadwiththeappropriateboundaryconditions.
Iugeneral,however,thereismutualrestraintata longitudinaljointamongallthemembersmeetingalongsucha line.Ifthisrestraintcouldbeconverteddirectlyintoa valueofrotationalrestrainte forthesimpleelementbeinganalyzed,thenthebuckling-coefficientchartsofreferences1 and2 couldbeusedtofindthebucliMngstressofthesimpleelement,and,[email protected],the”bucldingstressofthestiffener.Becauseoftherotationalinteractionamongthesimpleelementsateachjotitline,whicharisesfromthepreservationofthecorneranglesbetweenelementpairs (el= t12= . . . = en),howev=,therestraintimposedbyeachupontheotherscannotbefoundimmediately.Itisnec-essarytoanalyzetheproblemasoneinthedistributionofmomentamongthemembersofa staticallyindeterminateSyst=. Whenthishasbeend~e, e canbefoundand Ucr canbecalculated.
librthemostpart,thestiffnessofoneelementh itsownplaneissufficienttoimpartsupporttoitsadjacentelementsperpendiculartotheirplanes,althoughthecorneranglesmqyclifferfromX“. Mostsimpleelementsofa stiffenerbehaveinthismanner.Ups andbulbsmaybetooweaktoprovidecompletetransversesupporttoanelement(invariablyaflange).Theyactascolumnsthattendtoresistelasticallythetrans-versedeflectionsoftheotherwisefreeedgesoffI.angesand,consequently,.cannotbeticludedintheusualmethodsofanalystsoftheinteractionbucklingproblem.However,a flangewitha liporbulbalongitsfreeedgemsytieanalyzedasa stiffenedplatetodeterminetherigidityofthiscomposite,whichthencanbeusedh theinteractionanalysis.
CalculationofBucklingStress
!l!hebucklingstressofeachsimpleelementofa stiffenermeybefoundfromegyation(1).Chertsof ~ forseveralstiffenersectionsticommonusearepresentedh thisreportendarediscussedbelowhthesectionon“NumericalValuesofBucklingStress.” ThegeneralmethodsofconstructingthesechartsandforftiMngthebucklingstressofa newstfffenersectiontivolvea successiveapproximationproceduresuchasthemoment-distributionmethodofLundquist,Stowell,andSchuette(ref.4)orthestep-by-stepprocedureofI@oK1.,Fisher,andHetierl(ref.5).
Thebasisforthemcnnent-distributionmethodisthejoint-stiffnesscriterion,whichrequiresthatatbuctiingthesumofthestiffnessesofthesimpleelementsmeetingata jotntlinemustbe zero.Thisispre-dictedupona distributionofstiffnessesemongthejointmemberssuchthatallhavethesamelongitudinalwavelength.Thevanishingofthejointstiffnessatbucklingfolhwsfromthefact-thatstiffnessiseqpal
NACAm 3782 7
,,
to M/e.Ehce e isthesameforallsimpleelementsatthejointline,stiffhessisproportionaltothemomentcarriedbyeachelement.However,sincethesemomentsmustvanishatbucld.ingforsmalldeflec-tionsoftheelements,thejoint-stiffnesscriterionfollows.
Themoment-distributionanalysisissimplified~ theuseofchsz%sofelementstiffnessandcarry-overfactorprepazedbyKrollfordifferenttypesofboundaryconditionsalongtheunloadededges(ref.6). Thesearedescribedinthefollowi.ngsectionoqnumericalvaluesofbucklingstress.
lhessence,thestep-~-stepprocedureforcalculatingthebuclClingstressofa simpleelementinvolvesthearbitraryselectionofa bucklingstresstogetherwithseveralarbitraryvaluesofbucklewavela@h. llbreachofthesevalues,Ucr iscalculatedfromequation(1)untilitsminimumvalueisfound.Ifthisisdifferentfromtheinitiallyassumedbucklingstress,theprocessisrepeateduntiltheassumedandcalculatedvaluesa~ee. Thisisthebucklingstressofthecompositeelement.
Chartsof ~ (X/b,~) areused(aspresentedinref.1)togetherwiththerigiditytablesof~ol.1(ref.6).
Thebucklingstressofa flangewitha liporbulbwasinvestigatedbyHuandMcCulloch(ref.7),Gbodmanand130yd(ref.8),andGoodman(ref.9)whoconsidereda largerangeoflip,bulb,andflsngepropor-tions. Gerardsimplifiedtheanalysisbyselectingthegeometriesusuallyencounteredindesignanddefinedtherangeofsectionproportionsinwhichtheelementundergoesthetransitionfroma flangetoa webastherigidityoftheedgestiffenerincreases(ref.10).
RoyandSchuettehavedemonstratedexperimentallythatthelocalbucklingstressofthesectionisunaffectedalthoughthesinglebetweenadjacentelementsisassmallas30°oraslargeas120°(ref.11).TheprincipaleffectofchangingthecorneranglefrcmW“ istodecrasethesectionmomentofinertia,tiichdiminishesitscolumnstrength.
NumericalValuesofBucklingStress
Thebucklingstressofa stiffenerisdeterminedusingthebreak-downschemeoffigure1. Eqpation(1)isutilizedtocomputethenun@r-icalvalueofthisstressfortheweakestelementafterthebuckUngcoefficienthasbeenfoundaccordingtoa methodsuchasthatdescribedintheprecedingsection.TheclifferentstiffnessesevaluatedbyRYollintabularform(ref.6)aredepictedinfigure2.
Theeffectsof-lipsorbulbsareobtainablefromfigures3(a)and3(b)whichpresentthechartsdevelopedbyGerard(ref.10).Thebuckling
—.. — .. ——------ —-— --- —..—.—. —— -——— ——. .——— —- —-—----
8 N/MATN3782
stratiisshownasa functionoftheflangeb/t andtheedge-stiffenerproportions,whichpermitdeterndnationoftherigidityofsucha com-positeforuseintheindetermhacyanalysis.Inthismanner,thesechartsserveasanadjuncttoKroll’stables.
Bucklingcoefficientsarepresentedforccmnnonstiffenershapessuchasshowninfigure4,inwhichthedimensionsofwebsandflangesareshownforbothformedandextrudedshapes.Thebuckliug-coefficientchartsforchannel,Z-,W, andrectanguhr-tubestiffenersappearinfigure5. TheyweretakenfromthereportofI&oll,Fisher,andHeimerl(ref.5). Thedashedlinesonthesechartsdeftiethesectionpropor-tionsatwhichbothwebandfkngebucklesimultaneously.Dataforhat-sectionstiffenersappearinfigure6. Thecurves,adaptedfromthoseofVanDerMaas(ref.12),covera rangeofflsnge”sizesforclifferentwidthsofcenterandlateralwebsofthehatsection.ItshouldbenotedthathatandlippedZ-andchaunelsectionsarestructurallyequivalent.
EffectsofPlasticity
Theinelastic-bucklingstressofa stiffenermaybecomputedbyamethodsuchasthemoment-distributionprocedureofLundqyist,Stowell,andSchuetteforelastic-buclWngproblems(ref.4). ThiswasdonebyStowellandPride(ref.u), whoobtainedgoalagreementwithexperi-mentaldata(fig.7),forH-sectionstiffeners.Theplasticity-reductionfactorforeachsimpleelementofthesectionwasemployedincomputingthebucklingstressforuseinthemoment-distributionprocedure,inwhichthejoint-stiffnesscriterioncontrolsthetheoreticalbucklingstressofthesection.
Itshouldbenotedthatthetestdataatthelargerstrainslieabout5 percentbelowthestress-straincurve,whilethetheorybandis3 percentbelowatthemost.Thisanalysisse- totidicatethattheuseoftheplasticity-reductionfactorfora clampedflangewouldbeconservative.Useofthesecantmodulusfora simplysupportedflangewouldbeslightlyoptimistic.
BWKLCNGOFS~ l?IATllSUNDERLONGITUDINALCOMPRESSION
GeneralBackground
Asdiscussedintheprecedingsection,thegeneralcaseofbucklingofstiffenedpanelsinvolveslocaltestabilityofthestiffenersaswellasthespringpropertiesincompression,bending,sndtorsion.h thissectionthespecialcaseofsturdystiffenersisdiscussed,anda briefdescriptionoftheinfluenceoftorsionalrigidityofthestiffeneris
●
-——..
Y NACATN3782
included.Thisisofsignificancesincethechartspertaintostiffenerswithno
9
thedesigndatapresentedintorsionalrigidi~.
A descriptionofthebucklingbehaviorofa supportedandrestrainedrectangularplatemaybefoundinreference1 forflatplatesandinref-erence2 forcurvedplates.Thestiffeningelements,@ose localbuclClingbehaviorwasdepictedinthepreced~sectionofthisr@port,providethesesupportsandrestraintstotheplatesatintermediatepositionsintheplatespans.Theeffectivenessofthesesupportsdependsupontheaxial,bending,andtorsionalrigiditiesoftheplateandstiffeners.Representativearrangementsofplate-stiffenercombhationsareshowntifigure8.
EehaviorofStiffenedPlatesHer IOngitudhalCompression
Thetwotestabilitymodestobeconsideredinthissectionarelocalbucklingoftheplatebetweenstiffenersandgeneralinstabi~tyofthecompositeelement.timostcasesthetorsionalrigidi~ofthestiff=erisassumedtobenegligiblysmall.,thusexcludingrotationalrestraintoftheplatealongthestiffenerline.
Thebehaviorofa platebucklingunderlongitudinalloadandsupportedbydeflectionalandrotationalspringsisshownschematicallyinfigure9.Waveformsforthethreelimitingcasesofperfectlyflexibleandperfectlyrigidspringsareshown.b general,thewaveformsforfinitespringrigiditiesdonotchangeshapesignificantly,althoughtheamplitudesofthewavesmayvary.Whenthestiffenerrigidityissufficienttoenforcea node,theplatewillreceivenoadditionalf1exuralsupportfromthestiffener.Thisdescriptionparallelsthatforcolumnswhichwaspresented.byBudiansky,Seide,andWeinberger(ref.14).
Thebucklingusuallyexpressedtheplatebetweenofthepsxameters
CalculationofBucklingStress
stressofa stiffenedplateunderlongitudinalloadisintheformofequation(1)whereb isthewidthofstiffeners.Thebubklingcoefficient~ isa functionofthecompositeelement:
kC= kC(a/b,A/W,EI/bD, Zb) (2)
Eoththeenergy-integralapproachandthedtiferential-eqpationmethodofsolutionhavebeenusedtosolvetheproblemofbuclClingofa stiffenedplate.Theessentialsofboththeseprocedureshavebeendescribedinreference1.
-.. - ----- .-—.—____ .. . ____ . -_ . -.—-— — —..—- .-
10
NumericalValuesofBucklingStress
NACATN3782
Thenumericalvaluesofbucklingstressforstiffenedflatplatesandcurvedplatesunderlongitudinalcompressionareasfollows:
Stiffenedflatplates.-SeideandSteincalculatedthebucklingcoef-ficientsforlongitudinallyloadedsimplysupportedflatplateswithone,two,three,andaninfinitenmberoflongitudinalsttifeners(ref.15).Theresultsappearh figure10inwhich~ isshownasa functionofa/b fora rangeofvaluesof EI/bD.A summaryofcoefficientsforinfinitelylongplatesispresentedinfigure11forconvenienceindeter-miningbuc~ingstressesforlongstiffenedplates.
ThecalculationsofSeideandSteinwerebasedupontheassumptionthatthestiffener-sectioncentroidwaslocatedatthemidsurfaeeoftheplate.Thisisnotusuallythecaseinactualpractice,inwhichthesttifmeriscommonlylocatedononesideoftheplate.Thisproblemwasinvestigatedingeneralterms~ ~wallaandNovak(ref.16).Seidealsoevaluatedthiseffect(ref.17)andevolveda correctionforthechartsoffigure10applicabletoplateswitione,two,orinfinitelyq stiffeners
@J-/~)e. ~~ IA~2Ir = (EI/bD) 1 + (Zm#’bq
(3)
fromwhichtheeffectivebendingrigidi~ratio(EI/bD)emaybeobtained.Thefunction&q = f(X/b,n,q) tifigure12,inwhichA/b (Mb = a/qb,whereq = 1,2,and3) mustmatchthevalueusedtoenterfigure10. Atrial-and-errora~roachmightberequiredsince(EI/bD)e~ occurata differentvalueof q infigure10thandoesEI/bDatthe a/b origi-nallyusedtogetherwithn (n= 1,2,and ~)toenterthesecharts.Whentherearethreestiffenersontheplate,itisnecessarytosatisfyanequation,otherthanequation(3),appearinginSeide’sreport.
Budians@ andSeideinvestigatedlongi~ compressivebucklingoftransverse~stiffenedsimqilysupportedplates(ref.18). me datawhichper&into a/b= 0.20,0.35,and0.50appearinfigure13. Thecurvescovera rangeofstiffenertorsionalrigidity,incontrastwiththecurvesforaxiallystiffenedplatesforwhichGJ= O inallcases.
Theprecedingdatapertaintogeneralinstabilityofstiffenedplates.GallaherandBoughan(ref.19)andBoughanandI?aab(ref.20)determinedlocalbucklingcoefficientsforidealizedweb-,Z-,andT-stfffenedplates.Thest~ener-webcompositeswereidealizedasshowninfigure14,h whichthebucklingcoefficientsszepresentedasfunctionsoftheproportionsof -theccunposite.
—. .-. — .- —- .--. —
NACAm 3782 11. .
Stiffenedcurvedplates.-Theinformationforstiffenedcurvedplatesrelatestothatobtainedfromsectionsofcircularcyltiers.%tdorfandSchildcroutinvestigatedthecompressivebucklingofa simplysupportedcurvedplatewitha centralcticumferentialstiffenerhaxdngnotorsionalrigidityoraxialstiffness(ref.21). b additiontodeterminingthethe-oreticalbucklingstress,whichwasdoneusingltieartheory,thepercent-ageincreaseinbuclil.ingstressoverthetheoreticalvaluewasobtainedandisshowninfigure15. Becauseofthelowqerimentalvaluesofbucklingstresscomparedwiththeresultsofthelineartheory,I!atdorfandSchildcroutrecommendeda@@ng thetheoreticalpercentageincreasetotheqerimentalbucklingstress,valuesofwhichmaybefoundinref-erence2. Themaximumpossibleincreasefora curvedplatewitha givenvalueof a/b and Zb isshowninfigure15(a),whileanincreaselessthanthemaximumisobtainablefromfigure15(b).Furthermore,thevalueof EI/bDrequiredtocausea bucklenodeatthestiffenerlineisobtain-ablefromthesefigures~ cross-plotting.
Notethatnogainisindicatelwhen a/b>0.7 orwhen ~ isgreaterthanthevaluesshowninthetablebelow.
9a/b 0.600 0.500 0.417 0.333 0.250 0.167
% 28.o 14.0 7.8 4.1 1.9 0.74
!Ihechartsoffigure15weredesignedtopermitanestimateoftheincreaseinbucklingstresstobeexpectedinanaxiallycompressedcurvedplatewhenthecentralcircumferentialstiffenerhaslessbendhgrigiditythanthatreq~ed toenforcea nodealongthestiffenerline.Whenthestiffenerhasthisminimumrigidity,thelengthoftheoriginalplatemaybeconsideredtobehalved,andthedatainreference2 shouldbeusedtoobtainthebucklingstress.
Thisapproachalsoappliestoplateswithaxialstiffeners,whichwereanalyzedbySchildcroutandStein(ref.22). Thecurvesforthistypeofpanelappearinfigure16,inwhich~ appearsasa functionof EI/bDfora rangeofvaluesof a/b and ~. Tnordertoaccountforthedisparityoftestdatawiththeoryforcurvedplates,SchildcroutandSteinrecommendthefollowingprocedure:
(1)Determinetheclifferencebetweenthebucklingstressofthestiffenedpanel(fig.16)andthatoftheunstiffened.panel(ref.2).
(2)Tothisdifference,sddthelargerofthetwofollowingstresses:
(a)Thebucklingstressoftheunstiffenedpanel
(b)Thebucklingstressofthecorrespondingflatplate
. . . . . . . ..— —.. - . ..— ----- — — -—- .—-c -.. — —-— . . -. -.
12
Whenthecurvedplatewidthexceedsthelength,usecylinders.Usethecurved-platebucklingdataonlyexceedsthewidth.
EffectsofPlasticity
NACATN3782
thecurvesforwhenthelength
Plastici@-reductionfactorsforstiffenedpanelsdependuponthefactorspertinenttoeachelementofthecomposite.Forexmqle,thefactorforsupportedplateswouldbe~ected toapplytotheplateele-mentsbetweenstiffeners,whereassturdystiffenersbehavingascolumnsshouldfollowthetsngentmodulus.Iftheseconditionsholdinthecom-posite,theelastic-qepsrameterEI/bDkuld becomelZ#/qbDintheinelasticrange.
Gallaherandl?Qu@ancomparedtestdataonZ-stiffenedpanelssubjecttolocalbucklingwithbucklingstressescomputedusingthesecantmodulusastheplastici~-reductionfactorandobtainedtheagreementshowniufigure17 (ref.19). Someofthedatapertaintoplateswithsturdystiffeners.However,a largeportionappliestocompositesinwhichthestiffenersbuckledlocally.
EffectofTorsionalRigidityofStiffen=”
Thebuckl~-coefficientchartsdiscussedintheprecedingparagraphswerepreparedforstiffenerswithnotorsionalrigidi~.Actuallyallstiffenershavesometorsionalrigidity,andclosedstiffeners,ofwhichthehatsectionistypical,mayfuuctionasfullyrigidstiffenersintorsionforscmeapplicatims.Inreference1 a chartbasedupontestdatawaspresenteddepictingtheeffectonbucklingstressasthetor-sionalrigiditychangesrelativetotherigidityoftheplate‘beingstiffenea. Thishasbeenreproducedhereb figure18,inwhichmaybeseenthecomparativeeffectsofstiffenerswithlargeandsmalltorsionalrigidi~.
Thegaininplatebucklingstressrealizablewithstiffenersoffinitetorsionalrigiditydependsupontherotationalrestrainte providedbythestiffener.~is isrelatedtotheclifferenceinbucklingstressbetweenstiffmerandplate,wherethestiffenerisnowconsideredtobea simpleelementofspecifiedelasticproperties,inordertosatisfythejotit-stiffnesscriterioniUEcuss&i=BucuingofstiffeningElements.“ Thus,
thesectionentitled“Loc&l.
)‘Crplate (4)
-———. —-——. .
d
NACATIV3782 u
.
Thisissubstantiatedbyfigure18,whichshowslittlegainoversimplesupportwhentheplaterigidityishigh(lowvaluesof b/t).
BUCKLINGOFSTIFFENEDPIATESmm smmLOAD
BehaviorofStiffenedPlatesUnderShear
Whentransversestiffenersareattachedtoa plateloadedinshear,.theymayberigidenoughtoenforcenodesattheattachmentlinesortheymaybesoweakastoexertvirtuallynoinfluenceontheplatebucklepattern.TheextremecaseofweakstiffenerswasexaminedbySchmieden(ref.23),Seydel(ref.24),andWang(ref.25)whilerigidstiffenerswereexsminedbyThoshenko(ref.26).
.Theintermediaterigidi@rangewasanalyzedbyCrateandIawho
demonstratedthemmnerinwhichtheshearbucklingstressofaninfin-itelylongflatplateisincreasedlongitudinallyasstiffenerrigidi~risesuntilitissufficienttoenforcenodesalongtheattachmentlines(ref.27). Duringthisprocessthebucklepatternofthepl-.techangesfromthewaveformforanunstiffenedplateofinfinitelengthandofwidth(n+ l)b tothatofa platewidthb. TestdataobtainedbyCrateandLofollowthetheoreticaltrendof ~ asa functionof E1/bD.Thescatterislargewithmostofthedatalyingbetweenthecurvesforsimplesupportandclampededges,asshowninfigure19(a).
SteinandFYalichanalyzedbucklingoflongflatplateswithtrans-versestiffenerssubjectedtoshearload.(ref..28). Thebehaviorisanalogoustothatofa longitudinallycompressedplatewithtransversestiffen=s. tifigure19(b)testdataareshowntoagreewiththethqqofSteinandl?ralich.
SteinandJaegeranalyzedbucklingofa curvedplatewitha centralstiffenerplacedeitheraxiallyorcirctierentially(ref.29). Althoughthegeneralbehaviorpatterncorrespondstothatforflatplates,theadditionalfactorofcurvaturemodifiesthebucklepattezm,whichtendstowardthatof a cylinderinwidecurvedplates.
A stiffenedcylinderrepresentsa limitingcaseofstiffenedcurvedplates.Mostoftheliteraturepertainingtothiscasecovers‘testresults,themajorportionOfwhichappliestowedcstiffeners’thatbucklelocallyortostiffenersrigidenoughtomforcenodesandtherebycausethecylindertobehaveasa groupofplates.Thesecasesareaiscusseain reference2,whichdealsspecificallywiththisproblem.
Stein,Sanders,andCrateinvestigatedthebucklj.ngofcylindersloadedintorsionandstiffenedbyringswithfiniterigidi@(ref.30).
.— . . . ——— .. . . . . . .—.— — .—— — .-— — - -- ——— --- —-— .- --
—.
14
A largerangeofvaluesof-e ofvaluesof EI/bD.theresultsshowninfigureoptimistic.
%The20,
NACATN3782
wascoveredfora correspondinglargetheorywascompsxedwithtestdatawithin whichthe
CalculationofBucklhg
Thebuckl@ stressisexpressedinthe
&eoryisseentobesliglrbly “
Stress
formofequation(1)inwhichthebucklingcoefficientks isa functionofgeometryandloading.As -h thecaseoflongitudinalload,thebasicparametersforflatplatesare a/b,A/bt,and EI/bD,whileZb isanadditimalparameterforCUPRd pkteS d ZL 13~lieStO CyliIlderS. h the theoreticalinvesti-gationsthestiffenerswereassumedtopossessnotorsionalrigidity,andthecentroidstiereassumedtolieh themidsurfaceoftheplate.
NmericalValuesofBucklingStress
Thenumericalvaluesofbucklingstressforstfifenedflatandcurvedplatesandcylindersintorsionundershearloadssxeasfollows:
Stiffenedflatplates.- ThetheoryofCrateandb forlongflatplatesloadedh shearandstiffenedlongitudinally(ref.27)ispresentedinfigure19(a),inwhichks isplottedasa functionof EI/bDforbothclampedandsimplysupportedplateswithcmeortwostiffeners.TheresultsofStetiandIYalich(ref.28)fortransverselystiffenedflatplatesappearh figure21. Fromthislatterfigureitmaybeseenthattheminimumvalueof EI/bDremind toaforcea nodeatthestiffener-attachmentlineincreasesrapi~ @th a/b.Approxtitevaluesareinthetablebelow.
shown
Illa/b. . . . . . . . . . . . . . . . ...1 2 5MinimumvalueofEI/bDfornode . . . . . . . . . . . . . ..lOl~ 700
tionbya
Stiffenedcurvedplates.- Theresultsofthetheoreticalinvestiga-ofStetiandYaegeroncurvedplatesloadedinshearandsupportedcentralstiffener(ref.29)appearinthechartsoffigure22. EOth
axialsmdcircumferentialstiffenersareconsideredtogetherwithwideplatesandlongplates.me resultsareplottedintheformof ks asa functionof EI/bD,inwhichb istheshortside.Thispermitscom-parisonwiththecurvesforunstiffenedplatespresentedinreference2.0
-——. — — .. .— — —
.
NACATN3782
Thecurvesareplottedforseveralvaluesof Zb .- a/band2 (where,forthiscase,a and b aretheoverallplateSions). Inaddition,thelimitingcurvesofinfinitea/b for
15
= 1,1.5,aimlen-long
platesandthecyl~er curvefor-wideplatesareinclud-ed.Thela~termaybecheckedagainstthecurvesforcylinderstobediscussedinthefollowingparagmphandpresentedinfigure23.
Stiffenedcylindersintorsion.-Stiffenedcylindersh torsionrepresenta limitingcase.ofstiffenedwideplatesinshear.Stein,Sanders,andCratecalculatedthebucklingstressasa functionofthecylinderandstiffenerparameters(r&.30).l?hecurvesappearinfig-ure23,inwhichks isshownasa fuuctionof EI/bDfora largerangeofvaluesof ~ andforone,two,three,andfourhxtemmxliaterings.Thecurvespertainingtoonerhg msybeseentoagreewiththe“cylindercurvesoffigure22(d)forwideplateswitha centralcticumferentialstiffener,
Theseresultswereobtainedforstiffenerswithnotorsionalrigidity,withthesectioncentroidinthemidsurfaceofthecylinder,Q1.
EffectsofPlasticity
Theplasticity-reductionfactorsforstiffenedplatesundershearmaybefoundinreferences1 and2 forflatandcurvedplateswithspecificboundaryconditions. Thisinformationshouldapplytoplateswithstiff-enersrigidenoughtoenforcenodesalongtheirattachmentlines.Nodataexist,however,forplasticity-reduction-factorsforplatesstiffenedbyelem&tsofrigidi~lessthanthatrequiredfora
been
AEPIJCATIONSEZ!KIDN
Theuseofbucklingchsrtsforstiffenersanddescribedintheprecedingsections.Inthis
theresultsaresmmarizedforrapidreferenceand
Table1 containsdataforstiffeningelements
node.
stiffenedplateshasapplicationsectionthetablesareexplained.
loadedincompression.Homation onstiffenedplatesloadedinlongitudinalcompressionappearsintable2,anddatarelatingtostiffenedplatesloadedinsheararefouudh table3. Inallcasesthebucklingstresscanbefouudfromequation(1):
. .. . .... .. . . -.— .—— —— —- .— -. -.——--————.——— - - —
16
Plasticity-reduction
.
NACATN3782
factorsappearinthetableswheretheyareappli-cable.Forfurtherinformationonplasticity-reductionfactorsandcladdingreductionfactorsalso,seereferences1 and2. l?brinformationonfailureofstiffenersseerefer=ce3.
I?&ring-stiffenedcylindersintorsion, .
acr=&~~
For
forfor
thiscase,~ dependsupon ~ tisteadof~,
Itshouldbenotedthata and b aretheoverallplatedimensionsstiffenedplatesundershear.Thispermitscomparisonwiththedataring-stiffenedcylindersintorsion.
ResearchDivision,CollegeofEngineer-,NewYorkUniversi@,
NewYork,N.Y.,April15,1955.
—----- .-—.
Y mm m 3782 17
1.Gerard,George,andBecker,Hqrbert:HandbookOf StructuralStability.PartI - BucklingofFlatPlates.NACA~ 3781,1957.
2.Gerard,George,andBecker,Herbert:HandbookofStructuralStability.~ III- BucklingofCurvedPlatesandShells.NACATM3783,195?.
3. Gerard,George:Hbook ofstructuralStability.Partn -l?ailweofPlatesandCompositeElements.NACATN3784,1937.
4.Lundquist,EugeneE.,Stowell,ElbridgeZ.,andSchuette,EvanH.:PrinciplesofMomentDistributionAppliedtoStabilityofStructuresComposedofBarsw plates.NACAWl?L-326,1943.(Former~NACA~ 3K06.)
5.Kkoll,W.D.,Fisher,GordonP.,&d Hetierl,GeorgeJ.: ChartsforCalculationoftheCriticalStressforIota.1hstabil.i@ofColmnswithI-,Z-,Channel,andRectangular-TubeSection.NACAWRL-429,1943.(FormerlyNACAARR3KD4.)
6. fiO~, W.D.: TablesofStiffnessaadCarry-OverFactorforFlatRectangularPlatesWnderCompression.NACAWRL-3g8,1943.(FormerlyNACAARR3H27.)
7. Hu,PaiC.,andMcCulloch,JamesC.: TheLocalBucklingStrengthofLippedZ-ColumnsWith9nallLipWidth.NACATN1335,15#+7.
8. Goodman,Stanley,andR@, Evelyn:InstabilityofOutstandingFlangesSimplySupportedatOne~ge andReinforcedbyBulbsatOtherEdge.NACATN1433,1947.
9. Goodman,Stanley: ElasticBucklingofOutstandingFlangesClampedatOneEdgeandReinforcedbyBulbsatOtherFdge.NACATN1985,1949.
10.Gerard,George: TorsionalInstabilityofHingedFlangesStiffenedbyLi.pSand Bulbs.NACATN3757,136.
11.Roy,J.Albert,andSchuette,Eva H.: TheEffectofAngleoflkndBetweenPlateElementsontheIocalInstaBili~ofFormedZ-Sections.NACAWRL-268,1944.(FormerlyNACARBL4126.)
1.2.vanDerMEU3S,ChristianJ.: ChartsfortheCalculationsoftheCriticalCompressiveStressforLocalIhstabili@ofColumnsWithHatSections.Jour.Aero.Sci.,vol.21,no.6,June1954,pp.399-403.
13.Stowell.,E1.bridgeZ.,and~ide,RichardA.: PlasticBucklingofExtrudedCompositeSectionsinCompression.NACATNlg71.,1949.
. -. .- -.-—- _______... —.—. — - ..— _ —~---- — -.— -.—— ._ ___ .. _ _
.
18
14.BuaisJls@,BucklingSprings.
NACATN3782
Bernard,Seide,Paul,andWeinberger,RobertA.: Theofa ColumnonEquallySpacedReflectionalandRotationalNACATN1519,lW.
15.Seide,Paul,andStein,Manuel:CompressiveBucklingofSimplySupportedPlateswithlongitudinalStiffeners.NACATN1825,1%9.
16.Chwalla,E., andNovak,A.: TheTheoryofOne-SidedWebStiffeners.R.T.P.TranslationNo.2501,I&itishMinistryofAircraftProduction.(FYomBautechnic- SuQP1.dermbEUl, vol. 10,no.10,MSy7, 1557,pp. 73-76.)
17.Seide,Paul:TheEffectofIOngitudinalStiffenersLocatedonOneSideofa PlateontheCompressiveBucklingStressofthePlate-StiffenerCombination.NACATN2873,1953.
18.hdiEUISky,Bermurd,and Seide,Paul:CompressiveBuckl@ ofSimplySupportedPlatesWithTransverseStiffeners.NACATN1557,1948.
19.Gallaher,GeorgeL.,andI@u@an,RollsB.: A MethodofCalculating “theCompressiveStrengthofZ-stiffenedPanelsThatDevelopIocalInstability.NACATN1482,1947.
20.Boughan,RollsB.,andBaab,GeorgeW.: ChartsforCalculaticmoftheCriticalCompressiveStressforIocalInstabilityofIdealizedWeb-andT-StiffenedPsnels.NACAWRL-204,1944.(FmmrlyNACAACRL4H29.)
21.Batdorf,S.B.,andSchildcrout,Murry:CriticalAxial-CompressiveStressofa CurvedRectangularF%nelWitha CentralChordwiseStiffener.NACATN1661,1948.
22.Schildcrout,Murry,and%ein,Manuel:CriticalAxial-CompressiveStressofa CurvedRectangularPanelWitha CentralLongitudinalstiffener. NACATN1879,1949.
23.Schmieden,c.: TheBucklingofStiffenedPlatesinShear.Transla-tion No; 31,U.S.lhsper~entalModelBasin,June1936.
24.Seydel,Edgar:WMnklingofReinforcedPlatesStresses.NACATM602,1931.
WkshingtxmNavyYard,
SubjettedtoShear
25● Wang,TsunKuei:BucklhgofTransverseStiffenedPlatesUnderShear.Jour.Appl.Mech.,vol.14,no.4,Dec.1947,p.A-269-A-274.
26.Timoshenlm,S.: TheoryofElasticStability.Firsted.,McGraw-HillEookCo.,MC., 1936.
-—. — -— .. . .
NACATN3782 19
27.Crate,Harold,andIo,Hsu:EffectoflongitudinalStiffenersontheBucklingLoadofIongFlatPlatesUnderShear.NACATN1589,19$8.
28.Stein,Manuel,andFYalich,RobertW.: CriticalShesrStressofInfinitelyLong,ShplySupportedPlateWithTransverseStiffeners.NACATN1851,1949.
29.Stein,Manuel,andYaeger,RectangularPanelWitha
30.Stein,Manuel,Sanders,J.StressofRing-Stiffened
DavidJ.: CriticalShearStressofa CurvedCentralSW?fener.MACATN1972,1949.
well,Jr.,andCrate,Harold:CriticalCylindersinTorsion.NACARep.989,1950.
----- ____ ________ .__, - ...— —.. —-- -- --—
20 NACATN3782
TABLE 1
STIFFENERELEMENTSIN COMPRESSION
[See fig. I for breakdownof typicalsectionsintotheir componentelements]
Fig. Section Buckling Plasticity-reducfioncoefficient factor
5(a)
12c ‘w ‘E;:;:J~]
t-i
bwj(b) kw Nonereported
j(c)
‘ D
bh kh Nonereported
6
JY
btk+ Nonereported
— ..- -— .. . —.—
NACATN3782
TABLE 2
STIFFENEDPLATESUNDERLONGITUDINALCOMPRESSION
[Seefig.8 for sketchesof plate-stiffener arrangements; GJ=O]
Pfg. Section - Plasticity-mductIonfactor(a)
Endview
10 x A A vw w A
n=l,2,3,a)
Endview WhenMfenersenforcenodes,
4(a) ~
InfinitelywideEndview ~=(i)(a
4(b) = ~+,[++’Infinitelywide
4(c)Endview
T T T T4(d) Infinitelywide
Endview Side Wew
15 - :=: .
Whenstiffenersenforcenodes,Endview
usedata in ref. 2
16 4
an, numberof stiffenerson plate; ●, sturdy~ffen%
~, transversesupportwithno restraintof lateralmovement.
. .__._- —- ..— .—
—. -- —-—————-———— --—— -.. ...— .-— -.-. .—..-.
NACATN3782
TABLE 3
STIFFENED PLATES UNDER SHEAR
1See fig.8 for sketches of plate-stiffener arrangements. GJ=O 1
Fig. Section(a) Plastlclty7eductIonfactor
Endview
9(a)
n=l,2, coLomplatesonly
Endview Sideview
21 :; iv ~ v1A 6~
LowplatesonlyEndview Whenstiffenersenforcenales,
:2(0)❑nd 4 7 = (%/E) (l~e2)/(i - V2~
2(b)
Endview Sideview:2(C)and42(d)=
. uw A
m
-
23
n=l,2,3,4SimplySUpprtedends
%, numberof stiffenersan plate; ●, sturdystiffener;~, transversesupportwithnorestraintof lateralmovement.
—-.—. ..
23
Figurel.- BreakdownofsngleandZ-stiffenersintoelements.as,simplysupported.
componentplate
——. .. .——.—— .- —. . —.—. .—— —- - -—- -—.
24 NACATN3782
s
s’
s“
s’”
‘c+.
M
M
.
S’vM M
JW3- 20- Rotationalstiffnessesofflatplateswithdifferentboundaryconditions. Momentivarysinusoidallyalongplatelangth.
.
.—. . ——._
Y NACATN3782
./c T4@ff
\s,
\
.
\
. /m
(a)Lipflanges.
Figure3.- Buckl@ strati.
%@2 t 2.Ecr=12(l - Ve)()z=’
ofhhgedflanges.Lfi = 3.5;
Ve=0.3 (dataofref.10).
-.—.—.—..... . _ __ _ .._ ___ __ _ —. —-. — ———— . .—
26
./0
“ a
Ce
Do/
.000<
NACATN3782
~
.
\\ Y\. . \ 1 I
~\\ “uNs77wEMmFLANGE
“/0 /00
(b)Bulb”flanges.
=e 3*- Concluded.
——
NACATN3782 27
rb
L‘1--Jb
r.+-m.-
(a)
r&f“
bw
LJbf.
Extrudedsections.
Figure 4.- Typical
(b)Formedsections.
formedandexlamdedstiffeners.
. . .. . .. . . . ____________ ___ —-—- —. ——- .—— —:. —— --
.
28
71 I 1 (
kw
WEB BUCKLESFIRST
y
//
.4 fZAME BUCKLESFIRST—
5
4
\\\\ \ \ \ \ \ \ \ I
\ .P’f[vFUNGE, bf
2 ~t
“ WEB,bw9
E-s‘&l-l
I 20’-O)IIIIIJ I
.2 4 .6 .8 ● 8!0 12
(a.)Channel-andZ-sectionstiffeners.acr=I$#E t#
12(1-Vez)r
lllgure5.-Bucklingcoefficientsfcmstiffeners(dataofref.5).
.
,
-—-—-—— —— —-———-—— —-. — —. . .— -.. .
NACATN3782
7
6
5
4
kw3
2
/
●
7
FLAh@&bf
o 1111111111o .2 # .8 10 /2
;
&w
(b)H-sectionstiffeners.Ucr=~2E %2 “-~e2 ~.12(1 )
Figure 5.- Continued.
0
. ..— ...- ___ ___ . ——. -.—— _ __ . ..—
NACATN3782
kh
l-+--i I I I
. (c)Rectangular-tube-section
Figure
stiffeners.am =*($
5.- Concltid.
0
-- —
.
1
i
III
6
5
4
k, 3
2
1
0
Figure 6.- Eucklingstcessfor hat-sectionstiffeners.t = %=%=%;~21J t2
% = —. (E&a of ref. 12.)U!(1 - Veq bq?
u41%
I
1
32 NACATN3782
85
75
7G
C*,
ksi65
6G
55
STRESS- STRAIN
o
.005 .0/0 .0/5
bucklingofofthecmyandtestdqtaforinelasticFigure 7.- COmpsrison
H-sectionstiffaers.ecr= %+%2 Q= 1.();+$=0.5~12(1-ve2)b#;~
0.8. (I%taofref.13.).
.
—- —. .—.. . . . .
5Y NACATN3782 33
LON61WWN44LSTIFFENER
TRANSVERSEST/F~AfEk?
AXIAL— —.
CA?CUMFEW6ALSTIFFENER STIF=NER
Figure8.- IIIYPiCalarrangementsofplatestiffenergitudinalcompression.
combinationsunderion-
._ —..-. .——. -.— ——-— ...— — — - - -.
34 NACATN3782
EDGE STIFFENER
El= o
EDGE
GPO
I
.
me 90- BucKlingbehaviorofaxiallycompressedflatplatesuppofiedbydeflectianalandrotation~springs.
___ ..
5
4
3
k.
2
/
\
‘2
-0 / 2 3 4 56 7 8
o“b
(a)One stiffener.A/bt = O.
Figure10.-Ccmqressive-bucklingcoefficientsfar shply sqpportedflet
plateswith lcmgitudinelEtiffenem● ... .*Y. (Data
.of ref. 15. )
kc
4 ‘%
3
\’+<:;—---J”\_ 40
2
/
/ - ~
~/b=a/b+ A/b=oA2b +A/b=o/3bFLllllllld [ I
‘o / 2 3 4 ~ 6 7 8
db
(b) One stiffener.A/bt= ().2.
m lo. - Continued.
,
.
5
4
3
kc
2
/
L,,l,l,ll.
I I,
I I I00 / 2 3 4 5 6 7 8
o/b
(c)onestiffener.A/bt=I
Figure10.- Chtin’ued.
0.4.
I
I
5
“4
3
kc
2
/
\
IIlllllil Axb=a/b-#-- b+=wa’0
.’
1“
0 / 2 3 4 5 6 7 8
w0)
(24)!3
M 6tiffener6.A/m =
F’Uwe 10.-Continued.
o.
, ,
.
5
4
3
k=2
/
o
—.
ti
s
T)’ A
\ 20—\G ‘/0
I \l hl~
1
I \ I \ I -I J
,’2”\
3 4 5 6 7 80 / 2
(e)0/’
Two stiffeners.A/bt=
Figure10.- Ccmtimed.
5
4
3
k=
2
/
o(
-T
\
2
11111111
/ 2 3 4 5 6 7 4
(7A5
(f)Twu Stiffeners.A/bt= 0.4.
Figure10.-Continued.
w-1aln)
,
, ,* J 5’
I,.I{i
!
5
4
3
kc
2
/
0(
“\i
/ 2
k)
3 4 5 6 7 8
oh
Threestiffeners.A/bt= O.
Figure10.- Continued.
I
1kc
5
4
3
2
/
q 11111111/ z 3 4 5
u/b
(h)Threestiffeners.A/bt.0.2.
-10”- -tfi~d”
6 7 8
4=Iv
>
II
,
5
4
3
4
2
/
o
El
o / 2 3 4 5 6 7 8.
Cvf!b
(i)ThreeStiffe?mra.A/bt.0.4.
F@u.’e lfl. - Contintid.. .
5
4
3
4
2
I
/
o0
Il\ I I I
/ 2 3 4 5 6 7 8
U/b ‘Mb
(J) Infinite nunberof stiffeners.
Fil!3m=10. - Continti.
A/bt = O.
. ,
* , ,
I
5
4
3
&
2
/
o0
\.1 \ 75
111,,11I ~/ 2 3
(k)Ihflnite
4
OA ‘ Mb
5 6
numberof stiffeners.A/bt = 0.2.
Figure10.- continued.
7
w4Is
g
I
II
5
4
3
2
I _
o / 2 3 4 5 6 7 8
O!!b ‘M
(2)IMtnite numberof stiffeners.A/bt -0.4.
Figme 10.- Concluded.
. ,
5
4
3
k
2
/
o./ / /0= /oa
.g
FigureIL- C!cmpresslve-buckJ.lngcoefficientsfor infinitelylongshlp~ S~ flat plateswith lcmgltudiw1 stM&ners. .—
k#E .&~o. (Date d ref. 15. )‘m=K!(1 - ve2) b
a34
4
3
/
z—w
2
c
UA!FAUZEDSTURDYS77ffEWRS
+
\
/“-
n=2,
n=l, q=f I
n=2, q4 rfpw
Ilgure12. - l/~q aa a functionof
J/b
stiffenedplate proportions(m/bD)epattern.r=—=(EI/bD)
1+* A ‘s”=’”crosssection. (Dataof ref. 17.j
stiffener
.
VI-a0)N
‘Y
.
.
NACA
4
m
75
50
e
.Figure13.-
ported
#
I I I [, ---u 100 Zm
(a) a/b= 0.20.
Longitudinal-compressive-b~coefficientsforsimply
(Dataofref.
withtransverse
18.)
&x2~2stiffeners.T= =
(J21 )- ~e2b2”
49
sup-
. . . ..— ------ —.—- -.—. ..— — ._. — -—. —— —.—— _—
50 NACATN3782
5C
-4
30
~
2C
/c
a
. .—
La w tE]n
(b) a/b= 0.35.
Figure13.- Conttiued.
.
—. —.
2(
k
/6
5
6
51
(c) a/b=O.50.
Figure13.- Concluded.
.— —— .-..-— .—. . .-—
52
?
6
5
4
k,
3
2
/
NACATN3782
IJlllllllll1 I I I00 2 4 .6 .8 D
bp/b*
(a)Websttifeners.0.5< ~/ts<2.0 (Dataofref.20.)
Figure14. - Compressive-local-buckMngcoefficientsforinfinitely
idealizedstiffenedfkt p~teS. am =
*:7”
.
.
wide
. .—— -. .— ..- .—. -- .-. . .-— —
.
. 53
“
.
..
.
7
6
5
4
k=
3
2
/
ou z .4 s .8 10 M
bw
~
Z-sectionstiffeners.%/% =0.50snd0.79.(Dataofref.19.)
Figure14.- Continued.
.. . ...— ..__ _ ___ — ——. ——.. —— —— ———.—.. .
7
6
5
4
k=
3
2
/
00
(c)
2 .6 .8 LO
b~~
Stiffalers. i#ts= 0.63 enaLO. (l)ataofref.19.)
Figme14.- Continued.
.
.
.
NACATN3782 55
.
.
.
7
6
5
4
k.
3
2
/
o0
d
BUCKUN6OFSKIN‘EWRAWEOBYSTIFFENER
8UCKLIN6WWIFFEWWRWRAINEOeY WW
bf bfb~
~
4
m’ TTr.2
.
.6 .8
(d)T-sectionstiffeners.bf %?t_& = 1.0;~> 10;~> 0.25.
s
(Dataofref.20.)
Figure14.- Continued.
L2
.-—. .—._ ----- . ——. . —— —.. . . . .—— —. .- ———.—— —.—- -. . - ---
56 NACATN3782
7
6
5
4
k,
3
2
I
o
m mm 1.6
,t,I I
o .2 4 .6 .8 40 E
b/b “Ws
(e)T-sectionstiffeners.tw/tf= 0=7;bf/tf> 10;%/bs>0.25.
(Dataofref.20.)
Figure14.- Concluded.
1
II
i
II
.
,
0 ./ .2 .3 .4
(~ti~)(~b)”(a)Maximumincrease. (b)IiuxeaseXorgivenstiffbms.
Figure15.- Increasein compressive-buckldngstressfor siqplysupportedcurvedplateswith acentercircumferentialstiffener. (Dataof ref. 21.)
III
58 NACATN3782
k.
6 I5 A/bf=O “
.
3
oo~4 /2 /6
6I
5 A/bt-+.4 _
4 —
~= 3 -
2
/
00 4 8 /2 A?
A/bt=C12 _
. .
048 12 16
t
A/W =0.6
{
048 /2 B
~gure 16.- Compressive-bucklingcoefficientsforsimplysupported,curved
.
.
plateswithcenterlongitudinal stiffener. am =*Y” -
(Data of ref.22.)
NACATN
.
. .
3782
oo~24 3?
6 ●
5 A/bt=O.4
.+
3
2
‘O 8 /6 24 32
IA/bt=O.2
\
08
I 1. IA/bf =0.6
\
f
<08 tf524 32
(b) a/b= 2.
Figure16.- Continued.
59
. . -. .——. -.--—- — —-—... —.— ——— ...— —. — .—— . ..— —
60 NACATN3782
6
5
4
kc 3
2
/
0
k.
2
I
I i I I) /6 32 48 64
DxzIl”
6‘ I I5 ‘ A/bt=O.4
4 1
3’
324w&?
A/bt=0.6
\
zg250
/
O /6 324864
.
(c) a/b= 3.
~gure 16.- continued.
.
.
61
6
5
4
kc 3
2
/
Ot
N.I
Zpfjo
f aio_///!//////zw/z/
=b- /ww///wzu’
I I I I I I~163?486480$W[
6 I5
\4
3
2 A
/
‘O16324864&U96
>1
/6 32 48 64 80 96
I I_ #Mbt=Q6
O /6 324864 8096
Elm(d) a/b= 4.
Figuqe16.- Concluded.
.. —.-— — .—.— —.. .— .—-— — .—. — ..-— .. . _. _.
62
50
40
30G5
ksi
20
/0
q
MAXIMUM– STRESS- STRAIN
CURVE.
.
STRESS-STRAIN-
.002 .004 .006 .Ot
.
.
.
?
Figure17.- Comparisonoftestdatawithsecaut-modulustheoryforinelastic-locsl-buckMngstressofZ-stiffenedflatplatesuuder .compressiveload.(Dataofref.19.)
.
I 8
7
6
5
4
3
I I I
TORS1O)WL Y UWG,EDGES CLA@’A?5D– RIG/D S7iFi5ENEf?
—
LONG EWES SYMRYw’PmrED&-4.oo)
—
I-o 50
ItLgure18.-
/50
Effdctof torsionalrigidityof stiffenerficiemk fm flatplateB. (Dataof ref.
A?50
on bua1.)
coef-300
m
m
40
If*30
m
/0
o
—m 7MWY I vo nmj -------- / /
n ~.p ———
n am
❑
o
m, O/ws77FFEm -
0
CLAMR5Z Em NY SU+0R7ED EDsm —
(a) C1.aqedand shply
EI/bD
Slqpcrtedplates;longitudinal stiffeners.
m 19.- Cmqparimn of theoryaudbucklingcoefficientsof longflatplates with stiffeners.
. test“datafor shear-
. , ,
/0
A“
5
0
0 on‘1.4
u
.2.4
.4.8
A
THEORY TEST DATA0/. = L4, 2.4,4.8
00A
E//bD(b)Wmplysupportedplates;trsnmersestiffeners.
----- ... . . . . ..— —— —— -.— ----——- -. ————— __— —_. — ..-
———— —- .— - 7q.~=o ___ --T———~— —=s
●4* - –––– ~= “-–f
137 N~OFUAWS
o Ch!w#n SEcmON M WOE
u Sro?lw● me ON ONEWQkl Mm-
. .0 .1 I 10 10= 10” 10’
E.I_/iiD
FigureZO.- Comparisonof theoryend testdatafm dmply sqpportedcircuhr cyMnd.ersstHfened by ringsand loadsdin torsion.
NACATN3782 67
.
m
u
—‘ S/m r bYJPmR48 KAn%s
4 G!ED+-00 4“ 8‘e :1
16 so m
E1/bD
(a) a/b= 1.
#
6 /k, 4
.m
J_z b #-
–-r
00 40“ “ ,11
80 L?o#oma
E1/bD
(b) a/b= 2.
E7/bD.,
(c) a/b=
Shear-buckkncoefficients.
59
forlongshplysupportedflat
@dH3 withtiEUM3VETSeStiffeI,IerS.Tm= lsf#r2E&2)()I-2(1- Ve2b “
(wta of ref. 28.)
.
.. - .—.—— .—— ——— —. -. — .. ..-—
— —.
68 NACA?CN3782
●
0w .-1 I II I I 11111111 I 1111111I I IlllllJ
/ - /0./ /d /03
El.-m .‘.+.%.>>, -. “:.. .,-*,-
, -.-
(a)Centersxial stiff~j axial lengthgreaky~hsncircbnferentialwidth. -j:..-
.,,
Figure22.- Shear-buddingcoefficientsfor”si&# &zppdr$edcurvedplateswithC(?31ti Stifftie)?● (DataofZ&b 29,.) .;
$
.
.
-.. .. --—— —---- .—— —-. —____ ____ ._ ._.
NACATN3782 69
tts
[ ‘‘ -7~b “z~
200 / WooCYUWER
/00
50 /z M 100
20 CYLL4D
/0I
30
! – “ -
/ 3020 L5
CYLJ!E10
30
! : “- “ ‘ -
I /020 48106
CYLINDER
0 J / .d /03
(b)Centeraxialstiffener;circumferentialwidthgreaterthau”sdsllength.
. . —.-. . . ..— .—— —. . . . . . — -———.—. — - —. ---
70 NACATN5782
200
/m60
3020
/0
k,
40
R ‘ = ‘
Ir 30
20 2 M
/0 m
30
i: “ “ “ 44
/ /020 45/0 2
6 m
“1:zz‘II:30 I
1 J20
10 ‘5 t--al6
———m
4 { /7-
UJ\
b
w——./0 .f / /0 fop /03
(c)Centercircumferential.sti&ner; axial lengthgreaterthancXrcum-ferentialwidth.
.
.
—.
Figure22.- Conttiued.
NACATN3782 n.
.
I‘‘ ‘Q300 b z~200 / .~
CYLINDERpoo
10060 f(~~
CYLINDER 10040
20
/0
(d)Centercircumferentialstiffener;ci&muferentialwidthgreaterthan-al length.
Figure22.- Concluded.
. . ... . -- - .——.--—. -- — ----- .. ——- ..—-
I
.I t #v /v- {U” I
EV20
Figure 23. - BuckMng coefficientsfor t3implysupportedcticulercylinders
stiffenedby r*6 and loadedin torsion. Ta =
&-#r”
(Ik&a of ref. 30.)
-1N
,
