NATIONALADVISORYCOMMITTEEFORAERONAUTICS

. I

TECHNICAL NOTE

No. 1728

SHEARLAGINAXIALLY LOADEDPANELS

ByPaulKuhnandJamesP. Peterson

LangleyAeronauticalLaboratoryLangleyField,Va.

WashingtonOctober1948

1

t

I

I

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https://ntrs.nasa.gov/search.jsp?R=19930082421 2018-05-24T08:54:40+00:00Z

TECHLIBWY KAFB.NM

NATIONALADVTSORYCOMMNYEE

TEC!HNIhUNOTEITO. 1728

LAGm AXIAIUmADED PANEW

By PaulKuhnandJamesP.Peterson

Themethodofcalculattigshear-lageffectsinaxiallyloadedpanelsbymeansofthepreviouslydevelopedconceptofthe“substitutesingle-strlngerpanel”is simplifiedby en empiricalexpressionforthewidthofthesubstitutepanelWch eliminatestheneedforsuccessiveapprox5ma–tions. Forsirqlety_pesof sin@~tringerpanels,a theorynotdependentontheassumptionofTnfin;tetransversestiffnessis developedthatcanbeusedto estimatetheeffectoftransversestiffnessonthestressesinpracticalpanels.Strainmeasurementsonfivepanelsindicatethatthetheoryshould%e adeqzatefordesignpurposesandthattheeffectoftransversestiffnessmaybe appreciable.

Theprollaof introductigconcentrateedforcesat oneendofalongitudinallystiffenedpanelisa ,$undamntalme intheshe~lagtheoryandhasbeentreatedby a numiberofauthors.Thesolutionsobtainedby standardmethotiofanalysisarequitecunibersomeevenfor~snelsofconstentcrosssecticm,andmostofthemarenotapplicativetothepracticalcaseofpemelbwitharbitrarilyvariablecrosssection.Moreover,almostallthesesolutionsarebasedontheassumptionofinfinitetrsmsversestiffluessofthepenel;thisassumptionleadstotheresultthatthemaximumshearstressis infinitewhenthereerenodiscretestringersattaohedtothesheet,a resultwhichis somch inerrorastohe uselesstothestressanalyst.Fora finitenumiberofstringers,theerrorbecomesfinitebutisstillappreciableh theusualrangeofstringernunibers.

Tnen efforttoprovidea practicelmethodofshear-laganalysis,anapprorlmate“substitutesfngk-stringermethcd”waspresentedinreference1. AlthoughthismethodisalsobasedontheassumptionofWinite transversestiffness,itdoesnotgiveinfiniteshearstressesasthemathematicallymorerigorousmethodsdo;infact,theagreementbetweenthistheoryandesrlytestswasfoundtobe fairlygood(refer-ence2). Furtherstudyoftheproblaindicated,however,thatsomeinvestigationofthe~luence offinitetransversestiffnesswasdesirable.Theresultsofthisinvestigationarepresentedinthispaper.

-— .— ...-. .=-—. — ___ ___ __ ..— — -——-. .. —— .~— —. —.—. –—.,,

2 NACATNNo● 1728

Reference1 describesa successiv=pproximationmethd forlocating *thesubstitutesinglestrtnger.~ viewoftheapprodmatenatureofthemethod,however,successiveapproximationsappeartobe anumarrantedcomplicatim.A “one-step”mdhod isthereforedevelopedto locatethesubstitutestr~r. Althou@thissubjectistheoreticallynotdirectlyrelatedtothatoff~te transversestiffluess,it.wasfoundnecessarytoinvestigatethetwosubjectssimultaneously,becausethOformulasdevelopedrestpartlyonan ampiricalbasis,andh teststhetwoproblemscemnotbe separatedentirely.

Thereferencematerial,particularlythatofa theoreticalnature,isscatteredamcmga numberofpapers.lh orderto elintnatethenecessitythatthereaderreferto allthesepapers,thepresentinvestigatimincorporatesa generaldiscussionoftheapproximatemethodandoftherelevantfeaturesoftherigorous~thOas.

SYMBOIS

A exea,scpareinches .

E Young!smodulus,poundspersquareinch

G shearmodulus,poundspersqzareinch

J#.g

1? efiemd loadonhalfpanel,pounds

II hdf+dth ofsingl~tri.ngerpenel,inches

b-! transversedistancefromcentroidofflangeto commoncentroidof stringersh half-panel

-b~ transversedistancefrcmflange

fblJ

= b~

to substitutesinglestringer

,~

n tier of str@ers inhalfpanel

t sheetthiclmess,inches(withoutsubscriptdenotesshearcsrryingsheet)

x distencefrcmtipofpanel,tichea .

—...__ —--— -–--‘.

IW4ATIv

a

T

NO.1728

directstress,poundspersquareinch

sheerstress,poundsyerqzsre inch

3

Subscripts:

F flemge

L longitudinalor stringer

R chordwiserib

s denotes

T total

o denotes

c denotes

substitutepanel

stationattip (x= O)

centroidof stringermaterial

MEcHoDsAlmlmAImEs

Theoryof‘singl*inger panel ofinfinitetransversesti&?ness.—Thesingl~trhgerpenelasvisualizedh thes@ilifiedshear-lagtheoryconsistsoftwoflangesF (fig.l(a)),a str@er L, a connectingsheetcapableofdevelopingonlyshearstiesses,anda systemoftransverseribs. Theribsareassumedtobe infinitelycloselyspaced;iftheyarealsoassumedtohaveinfiniteaxialstiffness,theydonotenterintothetheoryexplicitly@ willthereforenotbe showninthefigures.Tbrough–outthisTaper,symmtryaboutthelongitudhalaxisisassumedto e~stsothat@e analysiscembe confinedtothehalf~el.

Fora penelofconstmtcrosssectim,theeqyationsofequilibriumoftheelements(fig.l(b))meld therelatims

+F d~ =Tt&=ALduL (1)

H thetransversestiffnessisinfinite,theincrementalshearstresscausedby thedifferencebetweenq -

&=-. q~ %’- CFL)ax (2)

Differentiationofexpression(2)and.for dq and d% fromequation(1)

substituticmintoit of.thevaluesgivesthedifferential.equation

d2 T~–J@7=o (3)

———— —--. — .—. — —... .- —.. — ———._-—

4

where

ITMliTNNo.1728

●

Forthepresentpurpose,panels. Thesolutionof

attenti~maybe confinedto infinitelylongequation(3) isthen

(4)

(5)

A simplesolutionmayalsobe obtajnedfora panelinwhichtheflangeistaperedsoastomatitaticmstantflangestr&3s(figo2(a)). The -equilibriumequationsarethen(fig.2(b))

~d.AF=fi~=ALdcrL (6)

Relaticm(2)stillappliesandthedifferenti~equationforthiscasetakesagaintheformof equation(3) with %2

where

Thesolutiauis

%#L +xr= —etA~

o

where AFO denotesthecross-sectionalareaof

Thecross+ectionalarea AF necessaryto

m-bstitutedfor K’j

t

(7)

(8)

theflangeatthetip.

maintainq constant,obtainedby substitutingequation(8) @to eqpation(6) andintegrating,is

H AFO< +, a constsntvalueof q cannotbe obtahed.

(9)

RiRoroustheoryofthemmltistr@qerpanelof=inite transversestiffness.— Foran infigurel(a)buthavingseveral

idealizedpanelshilsrto that&ownstringers,relatimscorrespcmiLlngto

I

,

.

equations(1)

—

NACA93?No.1728 5

end(2)canlewTittenforeachbay. Forafigure3 (twostringersinthehalfqmnel),simultaneousdifferentialeq.zaticns

&T2

— - ‘2%!32+ ‘1%2&#

panelsuchasthatshownintheresultisa setoftwo

(lo)

wherethecoefficientsK areshilartiformtothecoefficientKgiveninequation(4)exceptthattheyinvolvethetidthoftheindividualsheetbayandtheareasoftheadjacentstringers.writtenintheform

Thesolutionmaybe

‘lx +C2e*‘1 = Cle 1‘lx + C4e+S‘2 = C3e

I

Theconstentscm be determinedby standerdmethodstithoutdifficulty,buttherathercumbersomeformulasarenotof sufficientinteredtobegivenherein.An equivalentsolutionmaybe foundh reference3 hslightly differentforrti(thedifferentialequationsarewrittenforthestringerforcesWeal. oftheshearstmesses).

Forpanelstithmorestr~ers (say3 to10 intheW-panel),thestendardmethodsfordeterminingtheconstantsbecomeverycumbersme,smdInatiematicalrefinamentsaredesirable.A largeamountofworkonthissubjecthasbeendone,chieflyb England.Reference4 3-srepresentativeoftheresultsobtainedandwasusedasbasisforthecomparativeCalculationtobe shownsubsequentlyherein.Thestressesareobtainedby summinga nuniberoftermsofsn infiniteseriesafterthecoefficientsfortheseserieshavebeenobtafnedby solvingatranscendentalequationforeachcoefficient;thecaqutaticmsarequitele@@, p.rticulerlyforpotntsneartheti~ofthepenelwheretheconvergenceisslow.

Whenthenumberof strhgersbecomesverylarge,themostconvenientmethodofapproachisto assumethatthestr~rs arespreadoutinto

. ..- -. —— —.-— .. . . ..-. —...- —----- . . ...— —. . .. ——-. .._ —.- .—

J!rAcA‘pvNo● 1728

a “stringersheet”ofuniformthickness;thesetof simultaneousordinary ,differentialequationsisthenreplacedby a partialclifferentialequation.Thisproblemwassolvedby tivestigatorsin severalcountriesduringthewaryears,tithresultswhichareeitherstrictlyequivalentorelsetier onlyinminordetails.Thesolution@ven.inreference5 isusedinthepresentpaperbecausethereferenceisreadilyavailableandcontainsmorenumericallyccqutedcasesthantheothers.Thestringe~sheetsolutionmaybeusedas anapproximationforpanelswitha finitenumberof stringers.Thestressina givenstr~er istakenas equalto thestressinthecorrespondhgfiber(or“elmentalstringer”)ofthestringersheet;thesheerstresstna sheetbaybetweentwostringersmaybe similarlytakenastheshearstressinthestringersheetalonga liuecorrespu tothemiddleofthesheetbay. ~ regimswheretheshearstresschangesrapidlyinthechordwisedirection,somewhatbetterresuitsareobtatiedby titegrat3ngthesheerstressinthestringersheetbetweentwolinescorrespmd3ngtothestringersboundingthesheetbayinquestion.Graphsad formulasbasedcmthesemethodsaregiveninreference6, andcomparisonstie by Britishinvestigatorsshowthatthereisfairlycloseagreementwiththeresultsobtainedbysolvingsetsof shmil.taneousclifferentialequationslikeeqzatione(10)whenthenumberofstr@ers inthehaJ.f_l isaslowasfive.

Veryfewattemptshavebeenmadeto extendanyofthesemathematicalmethodstopenelswithmlable crosssection,andthecomputationallabortivolvedistoolargeto considerthemaspracticalmethodsforgeneraluse.

Thesubstitutesingle-stringer methodofanalyzingm.iltistriqqer=“– D practice,theflangesofmltistringerpanelsarestronglytaperedinordertoreducetheweight.Becausethemorerigorousmethodsofanalyzingmultistringerpanelsdiscussedintheprecedingsectioncannotdealwithpanelsofarbitrarilyvariablesectionwithoutefcessivelabor,ifatall,a simplifiedmethodwasdevelopedandpresentedtnreference1. !l$ebasicideatithismethodisthatthedesignerneednotknowall.thedetailsofthestressdistribtiioninthepanel.Heneedstoknowprimarilytwoitems:themaximumshearstress,becauseitdeteminesthesheetthiclmessrequired,andtheshearflowalongtheflenge,becauseitdeteminestherivetdesign;inaddition,hemustbeableto computetheflangestressinorderto insurethattheflangeisnottaperedtoorapidly.‘15isinformationcanbe obtainedwitha fairdegreeofaccuracyby analyzinga shplified“substitutepanel”thatisidenticalwiththeactualpanelexceptthatallthestringerscontainedinthehalf+dth arecombtiedtitoa s-e stringer.Thechordwiselocationofthissubstitutestr@er hadtobe establishedby theoreticaloreqerhentaldata.

Theproceduregiveninreference1 wasasfollows.Infirstapproxi-mation,thesubstitutestringerislocatedatthecamuoncentroidofthestrin&s WWchstringerpanel”ofthestringer

it replaces.- 5e analysis ofthe“substitutesingle-givesa firstappro-tion forthechordwiseaverage ●

stressesatallstationsalongthespan.Thechordtise

...— .—-— —.--— —--— ——. _———

WA TNNO. 1728 7

distributionofthesestressesisthencaputedbyuseofemassumeddmplelawofdistrihqtion.Thechordwiselocationofthecentroidofthestringerforcesisnextcalculated&d.usedas secondapprox.lmationforthelocationofthesubstitutestringer,sadtheprocessisrepeated,ifnecessary,untilthechangesbecomenegligible.

Testresultsshowedreasauableagreementwiththosecalculatedbytheforegoingprdcedure;however,becauseoftheaypro-te natureofthemethod.,theuseof successiveappro-tions appearsmmewhatunjustified.A procedurew3JJ.thereforebe developedlaterinthispalerforestablishingthelocationofthesubstitutestr~er directly.

.Thesubstitutesingl-ringerpanelwitharbitraryvariationof

crosssectionalmg thespancenbe malyzedbymeansoftherecurrenceformla giveninreference7. Oneitemshouldbenotedthatisnotcoveredh thisreference.Theelementarysolutionisdef@edasthatgivingthenormelstresses

Now,if AF or AL (orboth)varyalongthespan,the“elementaryflangeforce”(theflangeforcegivenby theelementarytheory)

PAFl?F=—

%

endthe(total) “elementarystr@gerforce”

wiU.alsovaryalongthespan.For*atic equilibrium,thisvariationcallsfor“elementarysheerflows”

(12)

TheseelementaryshearflowsmustYe addedtothosearisingfrcuntheX-forcesoftheshemlag enalysismadeaccordingtoreference7.

Whenthearea AT (or AL) variesalongthespanly steps: .formula”(U) wouldgivean infiniteshearflowthatacts,however,onlyoveran infinitesimallysmalld.istancf3alongthespan;theelementaryshearforceisther8foremathematicallyhieteminate.Physical

—-. —_—— .—-..—. ———— ——— —.. .———

8 ,

considerationoftheproblemtionshouldbereplacedby a

suggeststhat theste~urve ofaxeamria- ,ccmtinuouscurveforthepurposeofevaluattng

f ormla (12). H-the stepsareclosetogether,or mail-a faircurvemaybe clxawntorepresentthe“effective”variationofsxea. E thestepssrenotclosetogethersndarelarge the“effective”qurvetill

~undoubtedlynotbe fair(thoughcontinuousbutthereisneithertheorynorexperimentalevidence@vailableatpresentto serveasa guideinestimatingthiscurve.Itwillbe advisable,therefore,to avoidthisuncertaintyby avoidinga largestepclosetothetipofthepanel,wherethesheaxstressisa msx3mum.Theelementaryshearflowinapanelwithconstant+tressflangeanda ratio~ /AL equal.tounitycmstitutes25percent-ofthetotalshearflow~t%e problemofestimating

()AZtheeffectivevalueof & ~ atthetipistherefcmeof someimportance.

.

Themaximumshearstressgobablys3.waysoccursh ti-esheetbayadjacenttotheflange;consequently,fordesignpurposes,thereisnoapparentneedforfindingthechordwisedistributionoftheshearstresses.Themaximumstringerstressmaybe eithertheuniformstressexistingata largedistancefrcmthetipofthepanelora localyeak .closetothetipinthefirststringer.Ifthis10CS2peakshouldbeofdesigninterest,thestringerstressesinthetipregioncan%eestimatedby theprocedureforchordwisedistributimgiveninreference1.

Theoryofsinglqringerpanelwithfinitetransversestiffness.-Thesubstituteshgl~inger methodofanalyzingmultistringerpanels,baseda thetheoryofthes-~ringer pnel ofinfinitetransversestiffness,hasbeenappliedquitesuccessfullyto a numberoftestpanels(references1, 2,andotherdata).Thisfactsuggetiedthatthesubstitutesingl~inger methodmightalsobeusedto developanapproximatetheoryforpanelstithfimite ~erse stiffness.

Thepanelisagainvisualizedas infigurel(a). TheaxialStiffneaqoftheribsisnowassumedtobe finite:Becausetheribsme assumedtobe infinitelycloselyspaced,theymaybe consideredasformingaribsheet;thethiclmess~ ofthissheetdefinestheextensicmalstiffnessoftheribs. (Theribsheethas,ofcourse,zerolongitudinalandsheerstiffness).At thetip,a specialribofcrossaectionalarea AR isassumedto efist(fig.k(a),wheretheribsareshownafinitedistanceapartforpracticalreasons).

A ribawayfrm thetiyisloadedby thedifferenceintheshearflowsto eithersideofit (fig.4(c));thesedifferencesaremnd.1andpracticallyvanishat somedistancefromthetip. Thetiprib however,isloadedby thew shearflowexistingatthetip(fig.k(b~)Wis,therefore,relativelyheavilystrained.Theeffectofftaitetremversestiffnessmayconsequentlybe expectedtobe chieflya tipeffect,enda theorydevelopedforlongpanelsshouldbe adequateformostpracticalneeds.

—. ——. .-—. — ---—— --—,, - .. -,.

NAC.ATN No. 1728 I 9

Theshear

wherethepart

stressT maybe consideredasmadeup oftwoprts

T=T?+T*? (13)

Tt is duetothelongitudinalstrains(flangeandstringerstrains)andthepart T rT is duetothet(ribstrains).

ransvers;strainsTheequilibriumeqpaticms(1)writtenforthepel

wifhWhite transversestiffnessremainunchanged,butequaticm(2)mustbe changedtored

Asmentionedbetweentheshear

before,anyelementalribisloadedby thedifferenceflowsto eithersideof it (fi~.4(c)).SWe the

shearfluwsareconstantbetweentheflange& ~e ~r&erj theribstressincreaseslinearlyfromzeroattheflangeto a maxbmmatthestringer.Forconvenience,let q designatetheaver~estressinarib;theribstressatthestringeristhen 2%. Theeqtiibriumequationfora ribthenyieldstheexpression

Thetotal.extensionofa ribistherefore

(15)

(16)

Thederivativeofthisextensiondefinesa shearstrainalongtheflange(fig.4(d))

.

Theshearstraindecreaseslinesrlyalongtheribto zeroatthestringer.Thetheoryofthesingl~ringerpanelusedherein,however,requirestheassumptionthatthesheerstressieconstsntalongtherib;theaveragevalue(1/2) oftheshearstrainisthereforeusedto calculatethepartoftheshearstresscausedby trensversestrainsas

.

—.—__._. .—_____ . . .__________ _ .— - - ... ——. —.,

10

Differentiatingtwice,andletting

G%q=a

d2Ttt 4d-r—.a -

yields

dx=- ~4

Differentiationofexpression(14)gives

NACATNNO● 1728

(18)

(19)

whichcanle ~omned tiththeaidofinto

-)duLax

theequilibriumeqpation~(1)

(20)dz~:—= K2T&

.where K hasthesamemean3ngas givenbeforetnforrmla(k). Hequatiq (19)and(20)areaddedandthedefiningexpression(13) isintroduced,a slightreacmmgemntoftermsgivesthedifferentialequatian

Thisequationreducesto equation(3) forstiffnessifitismultipliedthroughbyindefinitely.

= o

the psnel withaand~is

(21)

infinitetransversethenincreased

Thesolutionoftheclifferentialequatimfortheinfinitelylongpanelis

‘lx +C2e+-r= Cle

wheretheconstantsK aredeftiedby

K~2= &(l + /=)

(22)

(23)

.

.

.

—.. —— -- -.—. . —.,, ,,,, .’.- ..-

, NACATNNO.1728 IL

BecauseK’a isoftenwhentheslideruleisapproximateon \

The constants Cl

%’=*(d==)small,thecomputationofused;thedifficulty

““K4+W)

.(’4)

% W givetroublemay be ~voidedby usingthe

(’5)

and C2 sre determinedfromtheboundaryConditions. Oneconditionis,at x =

%P=—AF

y=o

Theotherconditionisthatthestrainthestrainintheadjacentedgeoftheadjacentelementslrib. Thestrainingivenby theexpression

,

0,

inthetipribmustbe equaltoribsheet,orthestraininthetheti~rib (fig.k(b)) is

CrD 1Obt.% =$=-G

andthestrainintheadjacentelementalribisobtainedbymodifyingexpression(16)as

bt()g

%=-- ~R~O

Withtheseboundaryconditions,andwiththeauxiliaryparameters

P=-EbAF

~ ?R+%$R‘$+~~

1

(26)

,,- .. . .. -

1.2 tiCATNMO.1728

theconstsatsarefoundto be

.

●

c2=–

Foren infinitelystdff

c1=–w~

(27)

(28) -

tiprib (*- CO),theexpressims@~fY to

%y==; (29)

‘ K@ - M&a

cl = –7C2

Whenthereisno tiprib (~ = 0),

(30)

(3)

71=

c1. .-C2

andconsequentlyTo = OSas itmustbebecauseno shearstresscanexist along a free edge.

Ihspecticmofthederivatiashowsthattheformulasareapplicableto a panelwitha constant+tressfbnge if ~ istierstotitobe AFOintheexpressionfor p end K isreplacedby Kt. IftheQheetcarriesdiscretetrens-versestiffenersofexea ~ d pitch d, methicknessoftheribsheet

t+

liesbetweenthelimits

Atr. Atr~>’~z~ (32)

Theupperlimitapplieswhenthesheetisnotbuclled,thelowerlmtwhenthesheetisfullybuckled.Fora ‘bucldedsheet,thevalueof Gmustalsobereduced.Becausetheeffectoffinitetremsversestiffness

.

.’

.,-——. — — —- . — ———

13NACATNNO. ;728

is localizednearthetip,transversestiffenersshould~robab~bedisregardedunlesstheirpitch d islessthen l~l. -

Iocationof substitutestriqq3r establishedby camperisontith“rigorous”methods.–Thelocationofthesubstitutesinglestringermay%e definedby theeqyession

where f isa factorlessthanunity”.hfactorona theoreticalbasis,comparativenumlerofnmltistringerpemelsasfol.lows.

\>>)

anattqmptto establishthiscalcul.atimsweremadeforaWee typesofpemelswere

selected,eachty-pehavtngtwo,six,orinfinitely&y str-&gershtheW+rtdth. Foreacht~e, ‘panelsoftwoproportionswereselected,onepanelinwhichtheflangearea AF wasa fractionofthetotalstringerarea AL andonepsneiinwhich AF wasa multipleof %“Theratios~/AL chosenwerenotthessmeforalltyyesofpanelslecauseavailableresultswereusedwheneverpossible.Foreachofthesixpenelsthu8selected,theshearstressalongtheedgewascomputed,by a “rigorous”method(infinitetransversestiffnessbeingassumed)endagainby thesubstitutesingl-ringermethodforthreeassumedvaluesofthefactorf. Thesheerstresswaschosenas.a basisofc-perisoninpreferenceto theflangestressbecauseit isa moresensitivecriterion.(Theflsmgestressislmownfromelementarystaticsatbothendsofthepanel;consequently,no theorycenerrverymuchontheflsngestress.)

Forthetwo-stringerpenel,equations(10)weresetup endsolved;forthesix=stringeryanel,themethodofreference4 wasused,and,forthestrhgersheet,themethodofreference5.

Theresultsareshowninfigure5. Forthetwo-stringerpenelwitha smallflsnge(fig.s(a)),f = 0.7 givesa veryclosea-pproxima–tion(withina fractionofa percent);withthelsrgeflenge(fig.5(b)),theerrorisabout4 percent,thesubstitutestigl~ringermethodgivingtheh@her shearstress.Forthesix-strtigerpsaelwithasmallflange(fig.5(c)),f = 0.5 givesthebestapproximaticm,andtheinspectionofthecurvesindicatesthattheagreementcouldbe brprovedbyuseofa smallervalue~of f. Forthesix-strtigerpaneltithalargeflemge(fig.5(d)), f = O.5 givesthebestapproximationforthemsxinmmshearstress,althou@notthebestoneforthestressat somedistanceawayfromthetip.

. Forthestringersheet,therigoroustheorygivesan infhitesheerstressatthetip. By thesubstitutestigl~ringer.method,thisvaluecannotbe obtainedifa reasonableapproxhationtotherigorousshearstressesatfinitedistancesawayfromthetipisalsotobe obtatied.Thesingle+tr~ermethodinwhicha finitevalueofthefactorf is

— .-—. .—— —. ——. .— ___ .

14

usediscertain

, NACATNNO.1728

capableanlyofapproximatingtherigorousshearstressesoveraregim awayfromthetip(figs.~(e)andS(f))andyieldsthen

finitevaluesofmaximumshearstress.Iftheinfinitesheerstressoftherigoroustheoryweretobe obtainedhythesingle+tringertheory,itwouldbenecessarytomakethefactorf eqyalto zero.

Ttieresultsmaybe.mmmaxizedasfollows:b orderto achievethebestpossibleagreementbetweenthemmdmumsh”earstresscalculatedbythesubstitutesinglsstringertheoryandthatcalculatedby therigoroustheoriesbasedontheassumptimof infinitetransversestiffness,thefactor f shouldbetakenasabout0.7fortwo-str5ngerpanelsandshouldbe progressivelydecreasedto zeroasthenmiberof stringersgoesto infinity.

Theprecedingcomparisonsareessentiallyofacademicratherthanpractical.interest.Actualpanelshaveonlyfin.itetransversestiffness,andthefactor f wouldthereforebe deteminedbestby comparisonswithrigoroustheoriesbasedon‘theassumptio?loffinitetransversestiffhasswhichwouldelimhatethedifficultyofdealingwiththetofiniteshearstressesencounteredintheli.mithgcaseoftnfiuitelymanystringers.Unfortunately,theonly~heorlesavailable(references8and9) requirelaboriouscalculatias,andexperhentslcheckswouldstillbe desirablebecausesimplifyingassumptionsaremadeevenhthesetheories.Forthesereasons,furtherworkonthetheoreticaldetemdnationofthefactorf wasabsadonedinfavorofa directempiricaldetermination.

Ehpiricd location of substitutestr@ eraandverificationoftheoryforfinitetransversestiffness.-Fortheempirical.deterdnationofthefactorf, shearstrainmeasurementsal.ongsi.detheflangesofthreepenilsof&m3tentsectionandtwopanelsofwriallesectionwereused. Theconstant-sectionpanels are shown in figure 6. PanelAhadbeentestedpreviously(reference2). PanelB waslxzilttothesamenominaldimensionsaspanelA, exceptthattheheavytipribwasreplacedby a verylightrib. Therigoroustheorybas+iontheaszmp-tionofhfinitetransversestiffnessindicatesthatthenuniberofstringersinthesepanelsissufficienttobe consideredas “large,”h thesensethatthestressdistributiondoesnotdifferappreciablyfromthat3na panelwithan inftaitenuniberof str~ers,themainclifferencebeingthefinitevalueofthepeakshearstress.However,panelC wasbuiltinorderto obtaina directcheckforthislimitindcasa.TheshearstressesinthesheetweremeasuredwithTuckermsnopticalstratigagesplacedas closetotheflangesasthegagelengthof2 incheswouldpermit.

Teatswerealsoavailableontwopanelswithtaperedflangesanda smallnuniberof stringers(fig.7). These~els differedmainlyinthatpanelD hadflangesmachinedfromonepiece,whi~e-theflangesofpanelE werebuiltup.

.— —T. . ..—. .— .—— ..= . ..—..,

NACATNNO.1728

Preklminmycalculationsforthemadeasfol.lows.Threevaluesofthe

constant-sectiontestpanelswerefactorf (thesamethreethat

wereusedforthecmperismsinfigure5) werechosen.Fortheresult*substitutesingl~inger panels,thesheerstresseswerecalculatedontheassumptionofWhite aswellasfinitetran&ersestiffnesswiththetheorydevelopedherein.

Preliminarycalculationsforthetapered-flangepnels D andE were., slightlymoreinvolved.Thefirststepwasthecalculationofshearstresses

basedontheassumptionofinfinitetransversestiffnesslymeansoftherecurrenceformulaendexpression(12).A “referencepanel”wasthenintroducedthatwasdmilertotheactualoneexceptthat,startingjustleyondthetip,taperwasincorporatedintotheflangein sucha mannerasto giveconstantflangestress.Forthisreferencepanel,shearstressesT1werecalculatedontheassumptionofinfinitetransversestif$nessandstressesT2 ontheassumptionoffhite stiffness.Theratio72/Tlwasthenusedto correctthesheers&essescalculatedtithefirststep.Thismethodwasjustifiedby thefaotsthattheflangeshadroughlyconstantstress* thatthecorrecticmfactorsdidnotditfergreatlyfromunity.

&pection offigures8(a),8(b), and8(c)showsthatemn thou~thefactorf isvariedoverquitea tidermge (frma0.5to 0.9),thecurvescontractintoa rathernarrowbandat somedistanceframtheti~;theyfenoutonlyinthetip r@on. !Chechoice of the factor must“thereforebebasedchieflyoncomparisonsbetwemexperhentelendcalculatedstressesh thetipregims,a procedurewhichisalsodesirablebecausethelargeststressesexistinthetiyregion.Someconsideratimshouldbe given,ofcourse,tothestrem3esintheremainderofthepanels.

Thepreliminarycompsrismsshowedthata factorf = 0.7 gavefairresultsforallfive eaels,althoughthreedifferentstringer

Tnumberswererepresentedn = 7 forpanelsA and B,n=~forpanelC,n = 3 forpanelsD and E). Ontheotherhand,theCOIU-prisonswithrigoroustheoriesshowninfigure5 indicatedthatthefactorshouldticreasewithdecreasingstringernumber,andfora (half)panelwitha singlestringer,thefactorshouldlogicallybe equaltounity,becausethesubstitutepanelshouldbe identicalwiththeactualoneh thislimitingcase. (The“actual”panelrefenedto is,ofcourse,an idealizedoneinwhichthesheetcaviesonlyshear.) Closercomparisonsbetweenthecurvesfor f = 0.7 emitheeqerimentalresultstidicatedthattheagreementcouldbe improvedsomewhatbymakingfvariableinagreamentwiththeseconsiderations.Thetestdataareinadequateto establiShf asa functionof n witha highdegreeofaccuracy,particularlywhen n isverysmall (n. 3 or 2). Fortunately,thecalculatimsindicatethattheresultsarenotsensitiveto changesin f,andpmelswithveryfewstringersareoflittlepracticalinterest.As a tentativesolutim,theexpression

f= 0.65+ ~ (34)

.-. —.— --- . .. . . .— ——.._ ._ _ — ..- .— __ __— _

16 NACA

waschosenafterconsiderationwasgivento suchdifferencesbetweenthetestresults-andthepreliminarycurvesbasedon

TNNo.1728

ae existedf = 0.70

{Somejudgmentshculdbeusedwhentheratioof str~r areatono@–stress~earingsheetareaisverymch lessthaninthetestpanels.Ifa verysmallsimingerwereattached-atthecenterlineofpanel C,thestressobvious~wouldchangeverylittle,and n shouldbe takenasinf~ty ratherthanunityh expression(34).)

Figures8 and$ showthatthesolid–linecurvescalculatedwithexpression(34)agreequitewellnotonlytiththemeasuredpeakstresses,butingeneralalsowiththemeamredstressesalongtieentirelengbhofthecurves.ThehighestmeasuredstressinpanelA (secondtestpointfromtip) isabout4 to 6 ~ercenthigherthanthecalculatedstress,butcomparisonwiththefirstpointindicatestheprobabilityofa localirre@lmityora testerror.of particularinterestisthecloseagree-mentbetweenmeapuredandcalculatedstressesinpanelB withtheverylighttiprib. Thedifferencebetwe=thecurvescalculatedforthispenelontheassumptionofeitherinfiniteorfinitetransversestiffnessisverymarkedandindicatesthata shear-..theorysatisfactoryovertheentirerangeofdesignproportionscannotbe obtainedifthetransversestiffnessisassumedtohe infinite.l%nelB hasa lightertipribthanislikelytobe encounteredinpractice;however,evenonpanelsA andC,whichhavetipribsconsiderablyheavierthanlfielytobe foundinpractice,we effectoffinitetraneversestiffnessonthepeakshearstressisappreciable(oftheorderof20percent).

,

,

,

Figure8(d) showstheflangestressesinpanelC. Thereisasurprisinglylargevariationof stressoverthewidthoftheflange,whichisonly1 inchwide;thev-ariatiadisappearsat a distancefrcmthetipequaltoabout6 timestheflangewidth.

OnpanelsD endE, thecalculatedeffectoffinitetransversestiffnessonthepeakshearstresseswasfairlyamedl(figs.g(a)andg(b)),andthecalculatedsimessesexceedthemeasuredstressesnearestthe-1 tipsby 4 perc~ta 10percent,respectively.Thediscrepanciescanprobablybe attributedlargelyto a simplifyingassumptionimpliedinthetheory.Thetran6verseribshavea finitebendingstiffnesswithintheplaneofthepanel;theyarethereforecapableoftransferringsomeloadfromtheflengetothestringers,andtheyrestrainthesheardeformationatthecornersofthepanels.Thisribeffectisneglectedby thepresenttheory;itwasmoreimportantinpanels,D andE theaintheotherpanelsbecausetheribswerestifferbyvirtueof -er .length,greatersection,orboth.

h thecalculatimsshownforpanelsD and.E,thetransverseribs .

(otherthanthetiprib)weredisregarded.Calculationswerealsomadeontheassumptionthatthematerialintheseribswasuniformlydistributedspanwiseto equaldistancesoneithersidefromtheactuallocationof >

eachrib,withtheresultthatthevalueof ~ (thicknessof “rib-sheet”)wasgreatlyincreased.ontheotherhand,thevalueof AR was,

I

- ——. .— ..— — -— ——. - –———. .

NACATNNo.3.728= , 17

.

decreasedlecausea partofthematerialintheactualtipribwasassumedtobe spreadouttoformtheribsheetintheoutbosrdhalfofthetiphay. (Thisprocedureap~earstobe themostlogicaloneandwasalsosu~sted h reference9). Theincreasein ~ andthedecreasein AR counteractedeachother,andthestresses calculatedinthismamnerwerepracticallyidenticalwiththoseshowninfigures9(a)~a g(b).Ik general,suchcloseagreementbetweenthetwomethods-ofcalculationcannotbe expected.Becausethefirsttransverseribliesata statimwherethesheartransferislargelycmpletedthefirst

)methodofcalculation(theribsleingentirelydisregsmiedisprobably 0moreappro~iate.IYthepitchoftheribswere,say,5 inchesorlessratherthem21 inches,thesecondmethodwouldseemmoreappropriate.

Figures9(c)and9(d)showtheflangestressesinpmels D andE.Themeasuredstressesshownarethoseonthetopsurfaces;the“featheredge”ofeachstrapcarriesmly a lowstressbecausethefirstrivetisnotstiffenoughtotransmitthefullloadtothestrap.Jmvestiga.tionofthis“she~lageffect”withinthepackon otherpanelshasshownthattheaveragestressinthepackagreeswellwithf+hecalculatedstres~;thedeficiencyof stressintheoutermoststrapsiscompensatedby anexcessintheinnermoststrapswhichhasbeenfoundtobe ashighas30percentinpacksofsmewhatsimilarproportims.

CONCLUSIONS

Themethodofcslculathgshear-lageffectsinaxiallyloadedpanelsbymeansofthepreviouslydevelopedconceptofthesubstitutesingl-stringerpanelisimprovedintworespects:

(a)Thewidthofthesubstitutepaneliscalculatedby anempiricalformulawhichelhdnatesthesuccessiveapproxh@ionprocedureusedpreviously.

(b)A methodfortakingtitoisintroduced.

Testresultson”threepanelsnumberof stringersagreedwithin

accountfinitetransversestiffness

ofconstmntsectionhav5nga “large”4 percentwiththecalculations.On

twopanelswithtaperedflangeshavingmil.y3 stringersinthehalf-panel,thecalculatedpeaksheerelressedexceededthemeasuredvaluesby4 percentand10percent,respectively.

LangleyAeronauticalLaboratoryNationalAdvisoryComitteeforAeronautics

LangleyField,Va.,August16,I-948

. . .. . —-. .-. ..— .— —— —— —. ---— —-—-—— _ —————-——-——

18

R.EFERmcEs

NACATNNO.1728 .

.

“1.Kuhn,Paul,andChiarito,PatrickT.: ShearLaginBoxBeams– MethodsofAnaljsisandIMper-ntallimeetigations.NACARep.No.739,’1942.

2.Duberg,JohnE.: CcqarisonofanApprodmateandanllmctMethodofShesr-IagAnalysis.NACAARRNO.4u8, 1944.

03. Duncsn,W. J. (withAppendixbyH. L. Cox):DiffusionofLoadinCertainSheeW%ringerConibinations.R.& M.No.1825,BritishA.R.C.,1938. ‘

4. Goodey,W. J.: StressDiffusiauProblems.AircraftE@@eering,vol.XKUI, no.213,Nov.1946,pp.385-389.

5. Hildebrand,FremcisB.: ThelkactSolutionofShear-LagProblemsinFlatPanelsandBoxBeamaAssumedRigidintheTknsverseDirection.NACA~NO, 894,1943.

6. Anon.: Stressed Skin Structm%s. Roy.Aero.Sot.DataSheets02.05, .1942.

7.Kuhn,Psul:A RecurrenceFormulaforShear+agProblems.NACATN .No.739,1939.

8. Elmer, H.,androller,H.: fierdenI&aftverlaufinl~s–undquerversteiftenScheihen.Iuftfahrtforshun.gjBd-15,Ifg.10/lJ-,Oct.10,1938,pp.527%42.

9. K&Uer,H.:SpsnnungsverlaufundmittragendeBreitebeiLasteinleitungh orthotropeBlechscheihen.Bericht119derLilienthal–%se~sc~t f%rLuftfahrtforschung,1939,PP.61--8I.

.

>

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P P

R

+b

TFL F

(a)

Flgwe L-’SWgle-stringer

Figure 3.- Two-strhger POWI.

panel.

(0)

+b+

‘=&G=(d .- (b)

Figure 2,- Panel with constant- stress flong-a.

figure 4.- Panel wllh finite imnsveree. stlffrwss.

G

.

20

\

NACATNNO.1728

“Rigorou<solution—–— SuMfufe- single-stririjer solution, ~ = 0.5~— - — Substitute-single-stringersu!uticm,bs = .7bc—-— EWbstitufe:shje - stringer solutirn, ~ = .9”bc-

01 Io .2 4 .6 .8 10

fLx

(o)

___~-=_=_=

2-

01 I05m152D 2.5311

@

(c)

//

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. :>..-h=+

t

.5 /L=g

0: .2 .4 .6 .8 U3/lx

(b)

2.0

1.5

LO

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px

(0

o

(f)

Fqw 5.-Compor-kcmsbetween solutions obtained by r.kjwausmethods (M assumptionofinfinitetransversestiffness) and substitute-single-stringermethod.

.

—. ..— —-, . . . .

. .

24 NIX+

P

Panel A

P.

A

Steel

Sectb AA

Panel B

&cmeach side of sheet

L 2A lxtx& SteeI

I

Panel C

F@re6--Test ponds of c~stant. section, Material is 24 S-T aluminum alloy unless ofherwk noted,

— _. .-—_.-. -—..—-- —.- ——-—

NACATNNo.172822

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Panel D

Figure7- Test panels

Panel E

with tapered flanges. Material is 24S-T aluminumalloy

unless otherwisenoted.)

—— ..-. ,-—— --. .—.--— - ---——.— _—-_— _ .—— ..— —-.. –-.

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i

,

—.. —

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“—

05 ~“J5 20 25 wWtmca frwll lip, h.

(a) am stresm m pmel A. P. 1.95ki~.

2 -3*.

1.

‘O 5 10 15 20 25 30C+stm fmn lip, h

(c) Shm stresses In prowl C. P- 2.0 kip

tR

Fin”&

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ha5bc

As mted

AS mtedQ9bc

OLstmx frrxn IQ, h

Io~O 5 10 15 20 25 :

Bstome fmll lb, Irl

[a Fb~ stresw In ponel C, P12.0 klps.

Figm 8: Experiment md calculated strews in test panels of ccmstont section,

)

)

.1

“.\

!

o Ex~”mental stress––– Stress cdcubted m assurrqticm of hflnlte trartsvefse stiffness— Stress cdudated m assurr@n of ftite tmnswrsa stiffness

10\

36 “

‘“ 4 .

2 .“0o

(a) Shear stresses h FKWl D. P E 7.66 I@

flaq3, top &7* view. I

I/

-0 KI 20 30 40 50 60Cis&G8 from tip, in.

(cl Flawe stresses In PIYWI D. P= 7.66 kips

[2

10 .

m6 Qx

‘-- 4 -

‘2 - n

o0 (o ~o ~ ~o ~o 6:Ck?arc8frmn lip, in,

(Q Shear stres% In pnei E. P * 7.5 kips-

10I A00

?K$y:0’

0102030405060Olstarm M tip, h

(d) Flaw stres-%s in panel E. P= 7.5 lips.

figure 9.- Experhntol and ckulated stressm in test pansk with tapered flonges.

I

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