~..,-. W. AD?-lSORYC(NIXiH TiL2 FORAEROPNJrIICS TECHNICAL NOTE 3781 =~- I - BUCKIJNG OF FLAT PLATES. q -e -~d andherbertbecker Xew YarkUniversity

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1 --e,----- :, -h , - - -, --- L --i: ~,- W AD?-lSORYC(NXiH TiL2 FORAEROPNJrCS TECHNCAL NOTE 3781 OAR TECHNCAL LBMY AFL * =~- - BUCKJNG OF FLAT PLATES q -e -~d andherbertbecker Xew YarkUniversity?* & s Wmngtoil my 1957, i- - i - -1

2 TECHLBRARY KAFB,NM llllllllll]llll[lkll~ll! E OLLL+ S!!mlw- -a 1,~ 1N3%C)XJCTOH * S!mKKJJ 13ASZ2PRJC?X L = GeneralRemarks * : * Dit<=z~alEquation s i *_~lntegrak dilutions *!3 -? - 10 ~coxmmcxs *2 lkthe=timl~sis -12 Jhticlastic e?lzz-ture l~ SZEESS-STRA12T ~Glii311 YELDRBHON s 5hree-Parameter kcription of Stress-Strain Cum=5=5 K inelasticlkdti * 17 me3asticpoi=$=*sratio 17 ~fx!ry-rxmnz%afxxms * Mastic-Ew&Lqj-Stress Equation - 19 J3mrparisoa of55koriesand&pertiintal&ta 2Q Assumptions of ZEelastic-BuckMng Theories - 20 nelastic-bu_!x%eo5es 23 FactorsUsedim C!mqwtations 24 CcLnstmction of 3hndizensionlQ Euckling-s 25 ~m-gremsctjee EwmRs - 33esicPrinciples -- lkrivationof W Stress-Strain Curve -= #2cmpzmL5un ofmeqrysmd Eqerhent kivations Of ~fied c~g Reduction ~CtC2ZZZ * a e - a- 28!sse1 Ong~y supporte~ pletesincompr~~~ - 26 Csse2 Pkteaums 2g Case3 supporte~ platesinshear - 2g ~ QFmAT E=2!2K?=ULAR -Pup UN!ERccMmEssm= X&ms 31 3fstoricalEse~ 31 ~cal Values=f C=pressive-13uckMng Coeffic =- =ur Ehtes ,

3 , 1 r- %16;-, n -: Sup~rtedPlate,E~cesElasiicQ>-?==T== -==== &=illst Rotation =- 32 PlatesWithUnequalFklgeRoLLLZZ-L ~ Supported klangeswithelastic:~ - ====%===-=int 33 Effectof ateralrestminton ~~? --~ % l?-:: EUCKLWJOF FiiTRECTAXXJLAR PJ== ===== ==== WADS = - - ~~ Historical hckground = 35 Symmetric andantisyrmetric M&-= %5 Eumerical Valuesof Shear-~cUq 2== --=: = - - Effectof PlateLengthon BuckltLK,i&kfEZL*2s?- 36 KXKGNGOFFATFWTANGULMPLATESEZZ=T LSXJ=XG 3JMDS Historical l?ackgroun d ~----% Numsrical Vah~esof Bending-Ewxz%l$-~ _Z~ ;&s*- e---* - 36 HJCHJN3OF Phil RECTMGULARPrA!i55Q3KK==--* ~--?mms ~ne~~~c~o~a --37 BiaxialCo-=pression t--3t Shearan&HoxmalS+xess % BendingandNormalStress Bending,and ShearStress =---$x Ber~ing,Shear,andTransverse -*** ---- kl Longittiinal Benin, Longitud+= S -==-, and Transverse Coxzpression 42 Combined~elasticStresses & EETK?l? OF PRESSUREON KHKDG = E%===== = -T-T= &3?kmgeofPublished Results ~~ Longitudiral%f Compressed aw ~!= ~Plates Lmgitudini CoqressedLcmz~ +iz%~ * SPECALC AS:3 * -, * * Useof Elastic-Eucklin6-%ress ~ %*=-Z- - & AxiallyCoqressedPlateWithW= ~~ and!mclmess # AxiallyCompressed P:lte Withm-i=+- 2 =3 Cond=nt l%ickne= o- Par&Llelogmm Panels4LnCmp~ :; ParaUelogram Platesm =-~ Triangular Plates -- a == APPEWDXA - APPlzT4ToNsEtmmN m--- = ntr&uction= - j ~ PhysicalProperties ofmaterial= -- Compressive Eucklinc < ----~~ -,

4 , - --:- m PRteE FlsrAes PlateColuzms 9xwrB~ckling _* %in~ Buckling C!cmbir~& Lading RL3RJCH TEtiAEs* , * = - -- FKXZELZS s --, -, -, -*- ** - & t -, - -

5 , ;,,-, 1- :, - : :,,,, :c-:r,-, T~A:~[!i ~~~ ~{ ~~ ~ fij~ HN21PC)OK OF osnjczjll SYYLKCLi Y PART1 - PLJCKLRG OFFATPL4TSS ~ GeorgzGerarda&EerLertBecker SUMMARY A Thevariousfactorsgoverninz bucklirgof flatplatesarecritically retie-~ed andtheresultsaresumarizedina coqrehensive series of charts acd+&blesnumerical valuesarepresented forbuc kling coefficients of flatplateswithvariousboundaryconditions andappliedloadingsthe effectsof plasticity areinco~ratedinnondimensional bucklingcharts utilizingthethree-parazeter description of stress-strain curves NTRODUCTON His kmdbook of StmcturalStability presentsa rathercoqrekensiveretiewandco~ilation of theories&ailexperi=iental datarelstingt-o thebucklingaridfailureof plateelementsencountered in theairf~ ~ =et theanticipateii ceedsof thosewhowouldusethisrewe~:aadcorripilation,it appearedbesttq adopta handbookstyleof presentation Thersterialisnotintendedas a often on ther~themticaldevelowentof dif~erent typesof relatedproblenz Neitheris it intendedto>orpetetiththef~ar aircraft-c&pany structures mcualswhichgenerdjypresentdesigninfo?nr~tion, empirical data,andmetkaisof extending resultsbeyondthescopeof theoriginal report Thishandbookattezptsto coverthegenerally neglected areabetween thetext-k andthestxwctures ranualnoattemptistie topresentan exhaustive coverageof mathematical techniques whichareof greatimportancen thesolutionof bucklingprablers This=terialhaBbeenwell presentedin severalexcellent booksandpaperswhichareincludedin the reference listthesubjectof colunnsis comprehensively treatedin severalbooksand,therefore, theinclusion of suchmaterialin this reviewdidnotappearto bewarranted Thispresentation primrilyconstitutes a criticalretiewof developwsntsconcerning bucklingandfailureof,plat ele=ntssincethe early1940 sthisdatehnebeenselectedsincethelastcomprehensive reviewof thisnature(ref1) appearedat tlwttime $ r _

6 ntheraintextof thi~report,thevariousfactorsappearingin thegeneral buckling-stress equation ucr(cm7==) = qij ha ( ~e2))b ( (1; arecritically examinedfromthes~dpointof theirtheoretical developmentanqtheagreement of-theory withtestda a n tbesectionentitled EasicPrinciples a briefreviewof the Lasicxmthematical principles involvedin solutionof bucklingproblem la giventheprbaryobjecti--e inpresenting thismaterial to acquahtthereaderwiththeapproximate methodsusedinordertobe ableta indicatetheaccuracyof tkeresultscfparticular solutions discussedinsubsequent sections Tnthesectionentitled%uudary Conditions theini lu=ce of the geometric &nm&ry conditio~s upontkebucklingstresstsdis~sed et somelengtht isindicated thattheuseofa freeunloadededgeina plateinvolvespoisson s ratiointhecompressive b??ckling coefficient Assan example,thebucklingcoefi icients forplatecolu!ms,flanges,and simplysupported platesaredetermice5 fromtheoryto dezmnstrate the effectafvariousboundaryconditions uponthebehatiorof suchelements Also,theth&e-parar&ermethodofrmthe-~tically describing stressstrainrelations ispresented inan introductory mannerinthesection entitled Stress-Strain Relations intheyieldregion Useof this methd affordsa considerable simplification inthepresentation of resultsof inelsticbuck ingtheories Theeffectsof exceeding theproportional limitofa rmterialare incorporated ina plasticity-reduction factorq Becauseof thenriouatheoriesthathavekeenrecently advancedtogetherwiththefact thatno onepublication hasreviewed theconflicting assumptions of *!

7 ,,,-:p - f: J k; Theeffectof claddin&u~n thelickling stressof flatplaheshas keentreate~hy an extension of inelastic-buckling theoryn tl:esectionentitled Cladding Reduction Factors a simplified treatmnt of bucklingof cladplatesispresented iriwhichvaluesforthecladding Correction factor~ arederived Thebaclc~~?dfordeterringtheelastic-kuckllng coefficientk haskeenwelldocumentedṫkerefore, thelastsectionsercconcerfied withthebucklingcoefficients fora largenumberof casesthepresentationconsi6ts, fortte?m6tpart,ofa straightforward cataloging of resultsintheformof buckling-coefficient charts Theappendixhasbeenorganized fcrunimpeded useinanalysisand designandforthisreasonro references appearinthisportionof the reportthereferences are examinedindetailin thepertinent part of themaintexttheliterature is reviewedanddiscussedbothas toconteutandapplication to theparticular problemexperimental evidence spresented whereittend6to Substantiate onethecryamng several whichmayhavebeenadvancedon a particular phaseof thebucklingproblem;plasticity-reduction factorsareperhapsthemastconspicuous e~mpleof thisthus,therecozzzendation fora particular theoqyis gene- supported byexperimental data The=& textalso containsomenewmterialdeveloped duringthe courseof thiscompilation Althoughsuchmaterialis important tothe unification of priorresults,ithasnotbeenconsidered of sufficient consequence tomeritseparatepublication Therefore, wkensuchmaterialdoesappearinthishandbookitisin a detailedform Thissurveywasconducted underthe sponsorship andwiththefinancialassistance of theeational Adviso~Committee foraeronautics Ar a b -S areaof ribcros sectioa,sq in longdimension ofplate,usuallyunloadededgefnuniaxial compression, in shortdimension ofplate,usuallyloadededgeinuniaxial compression, in # - ()

8 - cl c4 b ]! D D1 E platecross-cectioa rigidity,et3/12(1- V2),lb-h plasticplatecross-section rigidity, E6t39, / lb-in Young cdulus, psi secentmdulus, O/c tangentmdulus, du/de secantandtangentrzodulus forcladplates,respectively ratioof totalclaading thickness to totalplatethickness shearnmdulus cmuient ofinertia 3 = (%/E)(1- Vez)(1- Vq i i ;* e } d K k L M N n P P mdifiedbucklingcoefficient, - V2) bucklingcoefficie~t lengthofplate,in bendingmomentappliedinplaneofplate,in-lb axialload,lb/in numberoflongitudinal halfwavesinbuckledplate;also, shapeparameter forstress-strain curve norma loadapplieinplane of plate,atb,lb nornulpressure, psi J -,-T-,! i-,

9 , -; i;g-- veo+\)~ ~ shesrlouding,lb/in 6= E2+ ve(flb/a)2 R t strezs ratio thickness of plate,w u= - ks ( %+ - + ks )/( %+ - ) w w X,y,z yl+3pf a potentialenergy,in-lb displacement nomal toplaneofplate,in coordinates edgeangle,deg;also,12m/(pb+ 6M) & = fi(b/a)l/2 * $ ratioof cla~dingyieldstressto core stress, cl/acore$ also,loadingratioforplatewithvaryingaxialbad, Maxhmlload/14imim71 lead Y v shearstrain normalstrain;also,rati of rotational rigidity of plate e~gesti?fener toro~=tional rigidityofplate plasticity-reduction factor claddingreduction factor total-reduction factor,qfi bucklehalfwavelength, in inelastic Poisson s l%tioj V = Vp- ~p-j e)(%/e) for orthotropic solids -,- - -

10 ,,, ~o7jjo85 stressat secantmdulus, op=d 085E, respectively, psi T ~ Subscripts : A,B av b c c1 cr e P pl r B Sm X,y stressintensity, ~x2+ ~y2- ~ 1/2 axay+ 3T, ( ), psi shearstress>psi angleofdiagonalsupportaplatewidth, radiansor deg valuesat stationa andstationb; seefig30 average bending compression claddingproportional limit criticalorbuckling elastic plastic proportional limit intraverseribof compressed plate shea r shearon infinitely longplate directions of loading + loadings producing tension loadingsproducing compression

11 -- - ~u:,:if:c,::s: c F ch!!pd free Ss s iqiy suppurtei (hinged) nsketchesacccnyanying figures,supportedgeswithelusticrota- tiofial restraint areshwn shadedunshadedloadededgesaresimply sup~ortedunshadedunloadededgesarefree BASCPRNCPLES GeneralRenarks Thetheoretical bucklingstressofa flatstructural ele~~ntisthe stressatwhichan exchanged stablequilibrium configurations occurs tet-~enthestraightandtheslightlybentformtmarkstheregionin whichcontinued application of loadresultsinaccelerated growthof deflections perpendicular to theplaneofthe~:atetsimportance lies in thefactthatkucklinginitiates thephysi~-~l processes whichleadto eventualfailureof theplate 74< * Themathe=tical solutionofparticular kucklingproblemsreqtires thatequilibrium andboundaryconditions be satisfiedthiscanbe accozplishe~ by integration of theequilibrium partialdifferential equationof theflatplateor by useof mathematical n&thodswhichmaynot completely satis&theboundaryor equilibrium conditions Theformer solutions areexactwhereasthemethodsbasedgenerally on ecer~ integralsareapproximate althoughusuallyveryaccuratetheneedfor approximate rithodsarisesfromthefactthatexactsolutions canbe foundforonlyalimitednumberof buckling problemsof practical importance n thissection,a briefoutlineof themethodsof analysisof buckling problemsispresentedforextensive discussions of thevariousmethodsofanalysisandtheirapplication to a widevarietyof problems,reference to thelooksof Timshenko,Sokolrdkoff, anddleich (refs2tok) issuggested t / Equilibrium Differential Equation Thegenemlformofthedifferential equationdescribing theslightly bentequilibrium conflgumtion ofan initially flatplatewasderivedby StQvellin thefollowing form(ref5]: ,--

12 ,,, (2) inwhichtheconstants aredefinedas: cl= 1- (3/4)(a+=,) ~ - (%/ s] * C2 = 3u~Tai ( /2)~-(%, s] - (w ] s C=j=l- (3/4) py - )k (3) C4= 3ay7ai ( )p( a Thesedefinitions oftheconstmtsare&sed on theassumption thatno elasticunloading occursduringthebuckling process Furthermore, a valueof Misson sratioeqwl to 11 2wasassuredforboththeelastic andinelastic rangee n the elasticrange,%~% = 1,and,therefore, fora32loadings C1C3=C5= and C2 = C4 = O,andequation(2)reducesto the familiarequilibriaequationfortheelasticase: t t = &=&+2&+3 ax4 (4) - _, ~ ~

13 d 8 krdingrigidityof D = l?l3/9, whereasthe elasticvalueis D = Et3/12(1- Ve2) Thesolutionof individual buckllngprohle=scanbe mostreadily handledby selection ofappropriate solutions of equation(2),insertion of properboundaryconditions, andminimization to obtainthebuckling stressh thisconnection, supported pletecolumns,compressed flanges,andplatesareconsidered in some detailin thesectionentitled Boundary Conditions to illustrate the differences inb~cklingbehatiorof tkse structural elezients : Knergyntegrals Sinceexactsolutions to equations(2)and(4)canbefoundfor onlya llmitednumberof bucklidproblemsof practical importance approximate solutions genemllyutilizing ener~ integrals havefound tideapplication Thepotential energy of theplateanditsloadingsystemis representedby thedifference of twointegralsṭhefirstintegr~of equation(5)represents theincreasein strainenergyduetabendingand tuistingoftheplatedurir~thebucklingprocess,whereasthesecond integralrepresents energyassociated with rrembranestresses resulting fmm lateraldeflectionịf thepkte edgesarefixedduringbuckling, thelatterermesents themembranener~ f theedgesexperience a relati~shif~,thesecondintegralrep~esents thewo=koftheexternal loadingsystem t Thegeneralenergyintegralforpkteswithsimplysupportedges wasderivedby Stowell(ref5) fortheinelastic case: 1 - J + (5), ~ ~ #--

14 , restraints of magnitudec along t!o edgesof th!plate,tlien tt~estrnin enerm inthesercstrnintm isaddedtoequation(5) Thesetermshave theform oj \QY/y=yo where y stheedgecoordinate Fortheelasticase,equation(5)c-k simplified to dxdy- (6) Soluttons h principle, of au thedeflection functions satisfying thegeometricboundaryconditions of theproblem,thepotential ener~ AM will be zeroforthatfunctionwhichalsosatisfies theequilibrium differentialequationthisfunctionwouldbe an exactsolutionof theproblem Sinceexactsolutions cank-efoundinonlya lfmitednumberof cases, theener~integrals areof greatusefulness infindingapproximate solutionswhichsatisfythegeometric bcmndsryconditions exactlyandthe differential equationapproximately Thus,oftheseveral~ctions satisfying thegeometric boundarycotiitions butnotnecessarily thedifferential equation, thefunctionforwhichtheener~integralsa minixmnnconstitutes tkebeetapproximate solutionof thedifferential equation Probablythebestlmmunenergymethodfordetermining thebuckling stressof thinplatesis therayleigh-ritz procedureṭhemethodconsistsof thefo~ouingsteps: (1]Thedeflection surfaceofthebuckledplateisexpressed in expendedfom as thesumof an infinitesetof functions havingundeterminedcoefficients ngeneral,eachtermoftheexpmsionmst satisfy thegeometrical boundaryconditions of theproblem,

15 z ~, i f 2 4 (3)Thisminimizing procedure leadstoa set of lineur horzogeneous equations in theundetcri~c2 coefficients Thes equations hrvenollof theircoefficient vanishes vanishingsolutions onlyif thedeteralr%nt Thevanishing of thisstability determintint providestheequationthat rnybe solvedfortheb!!ckling stress Whentkesetof fuctionsuse~is a completesetcapablsof representingthedeflection, slope,aridcurvature of anypossibleplatedeformation,thesolutionobtainedis,inprinciple, exactsince,however, theexactstability de termlnant isusurllyinfinite, a finitedeterminant yieldingapproximate resultsisusedinstead Thebucklingstressesobtainedby theapproximate methodarealways higherthantkeexactsolutionalthoughtheynaybeveryaccuratethis is 8 resultof thefactthatthedeflection function approximates the truebuckleshapeandtherefore the~tentialeneraresulting fromuse of theapprox~tingfunctionisgreaterthanzerofthedeflection fumctianisthetrueone,thenanexactsolutionto thedifferential equationis obtained f a deflection functionischosenwhichsatisfies thegeometrical boundaryconditions approxtitely, ispossibletoobtaintuckling stresseswhichapproachtheexactsolution fromthelowersidethis canbeaccomplished bya revisionof therayleigh-ritz procedure known as thelagrangian multiplier method Theagmngianmultiplier methodfollowsthegeneralprocedureoutlinedfortherayleigh-ritz methodwithbutonesignificant cluzngethe restriction instep(1)thatthekaundaryconditions be satisfied by everytermof theexpmsionisdiscafied andisreplacedby thecondition, thattheexpansion as a wholesatisfies theboundsrjconditionsṭhis conditionismathematically satisfied in step(2),duringtheminimization process,by theuseof agrangian titipllers Theadvantage of theagrangian multiplier methodliesin thefact that,withtherejection of thenecessity of thefulfillment of boundnv conditions termby term,thechoiceof an expansion ismuchlessrestricted- Forexample,in theclamped-plate compression problem,a simplefourier expansionnayke usedinsteadof thecomplicated functions usuallyassured in therayleigh-ritz analysesof thisproblemftihermore,theorthogonalityproperties of thesimplefourierexpansion leadtoenergyexpressionsof a simplicity that1s instnmental inpermitting accurate computations c - ~ -r-~ ----

16 ,,,,, ThLslWi!Oi&tditSfi~~liCZLiO:l iospectticpro*l~vg isu,:: rit;wl by hdimlsky WA HU (Wr 6) l%cyhivetreatei~kckgranl~imul tl - plierr=tlxxl ina namer inwhichit ispossibletoobtuinupproxz&le solutions foriothupperandloxerbourdsas dewrmin:~nts of lli~!ler orderarz usedto o!:ain betterapproxlmitioas, taththeup>erandl~wcr boundsapproachthetme bucklingstrcscthu6,theagrangian mltipliermethodraybeusedto ohtsdn resultstithir anydesiredegreeof accuracy nadditiontotheaboveprocedures whichsrebasedonenergyirtcgrals,othermethodsof obtaining approximate solutions of bucklingproblemshavebeenuse~whichinvolvetheequilibrium differential equation ~CtiOnS which&&:lsfythegeorr~trical tiundaryconditions exactlyare usedto sat~sfyt~egoverning differential equationapproxkately by processes thatleadto integration of thesefucctionsgalerkln s =thod, finite-difference e~uations, relaxation techniques, anditeration aresome of thenumerical nethodsthatcanbeused KUNDARYCONDTONS Thenatureof thebuckleratternina relate dependsnotonlyupon thetypeof appliedloadingbutalsouponthe=%er in whichtheeds=s aresupportedthisis illustrated infigure1 inwhichthes&me-al compressive loadingisseen to generatethreetypesof bucklepatterns ona longrectangular platewithdifferent geo~ trical boucdaryconditionsthesinglewaveisrepresentative of coluznbehatior, thetwisted waveis representative of flangebehatior, andtkemultiple-buckle pattern 58representative of platetehatior N indicatethem=nne=in whichtinegeometric boundarycondltio-= mathe-tically influence thebuc kling behavioran~alsotodamnstrate thesolutionof theequilibrium differential equation(eq(4))forsome particular cases,theplate showninfigure1 cueanalyzed!o-f conditions vhichc~cterize simplysqrporte~ vi~ecolumns,~ges, andplatesareconsidered - d Wthemtical&ialysis Theequilibrium differential equationforelasticbucklingof a unlaxially compressed platecanbecbtainedfromequation(4)in the form (7)

17 -,,,, J;! ; T!,isES5UT*2,3 &ld~h~ refore w= ( c1 uhere (9) (lo) (m Thecoefficients cl to C4 areto be detenzined by thegeometrical bmndaryconditions alongthe &nloadei edgesof theplate Fortiewidecolumn,tkeunloadededgeslocatedat y = tb/2 are free,andconsequently theedgeemqent6andre-iuced she?~smust be zero Therefore, * i A ~ + 2(1 -Ve)zz- *O ax~ytib/2 1 (1 ) Fortheflange,theunloadededgeat y = O isassumedto he simply supported andthatat y b is free:

18 5=&2 - ve(xb/a)2 A:-,, ), ;:;,: (W)y=() =o (=+veg),=o,b=o &+2~- e)-]y=b=0 Theplateisassumedtobe simplysuppmtedalongtheunloadeded~es locatedai y = ~b/2: (w)y~b/2 =o () +,,22 a % * & ~2 ~*~/2 = 0 ncorporation of theseboundaryconditions intothesolution given by equation(8)leadsto thefollowing implicit expressions for kc Forthecolumn, d for the flange, #p sinh&cosp - ~%coshtisln~= O (16) 1 andfortheplate [itanh{~/2) +j3 tan(5/2~-1=o (17) where #- 1,- - _ _

19 !, ~lebucklingcoefficient fora sixplysupported flangewasderived (ref8) intheform kc = (6/Lr2) (1 e)i[fib/a)2/6]} { - (18) kc= ve+l34(a/fib)2+ 010(xb/A)2 (19) Forthe s*w supported plate kc= [m+(b/ajj2 (a Anticlastic-titure AS my be seenfromthesolutions inthepreceding section, the,,u:kling coefficient for ~d 1s independent ofrissontsratio,whilethecoefficients thesimplysupported platedependsupononly for J/A de columnandflmgearefunctions ofboth Ve and b/l ThiS ~5tufition isnotlimitedtotbecaseof 6fmplesupportalonebutper- ~tl,vtoa~ degreeof rotational restraint alongthew-a ed=s of ~ ~lfitec Theinfluence of v= upon ~ istraceable to thereduced- ~~lemr te- atthefreeedgesofflangesandcolumns Boundary conditil)rl~ suchas simplesupportdo notmposetherequirement of zero ~ettuc~she= ~ong theunloadededges, whicheliminates the Ve influ-,ntofrom therelationship for kc me valueof thecompressive bucklingcoefficient foranelement ~mtdinin6a free unloadededgedependsuponthedegreeofanticlastlc ~~lrv/ltul~ developedḟora verynarrowelementsuchas a be&m,complete,---

20 - enbiclastlc curutureoccurs anithelending rigiditiy iu slcplye?01 a relatively vitiebtrip,t?ma:iticlaztic CU-JatJreis suppressed sot?t~t thecrosssectiofi reraizsrelntlve>y flatexceptfora highlylaculize; Curlingat thefreeeigesvkre the~tressdistribution rermrapges itself to satisfythegeometrical bow-simycomiibions Therestraint of anticksticcurntureres-dtsinm increaseinbendingstiffmss Fora verywideele~ent,tkek::<irg stiff~ess approachese/(1- v2);this limitingcondition is~r-~:: as cylindrical kending Platecolumnsandflangesmy oft-en he relatively narrow,inwhich casethebendingstiffness lieslctweenthelimiting valuesdiscbssed Thiseffectcanteacccunted forby useoffigure2 STRESS-STRAN R!21ATONS NYELDRGON Three-tineterDescription ofstress-stmin Curves Stress-strain cumesareof f undanental iqcmtanceinthecosrputa- inelastic bucklingstressesṫhenumberofdesirechartsrewired tionof forthemny materials a~-ilable andthevariousallowabie stresses~or thesemterialsat nomalandelevate~temperatures canbetrewmdous;ly reducedbyuseof a nondimensional mathematical description of stressstrainrelations RambergandOsgod(ref9)haveproposeda three-pszaneter representation of stress-strain relations intheyieldregionwhichfrsfound wideapplication TYeir e~uationspecifies thestress-strain curveky theuseof threepar==ters:themdulueof elasticitye, thesecant fieldstressco-7 corres~nding to theintersection of thestressstralncurveanda secantof 07E,andtheshapeparametern which describes thecwxa:uzze of thekneeof thestress-strain curvethe ebapeparaceter isa function of Jo7~d ~085Jthelatterstress corresponding toa secantof055eas shownin figure3(a)theshape ~ter n ispresentti infigure3(b)as a functionoftheratio / - m Thethree-paraceter methodis base~on theexperimen~l observation thatformny mterialsa simple~wer lawdescribes therelationbetween theplasticandelasticomponents of straineyuseof this fact,tke following nondimensional equationcanbederived: # Ee= =07 () n +2A e 7U07 (a)

21 i!, nelastic Moduli ThequalitiesM -/ l?~yan: u / C(-J,7 tire nond~rwmio::al andccmsc~uefilly tk nondimensional stress-struin curveshorninfigureh canleplottei Therefore, thestress-strain curwxof ranyffi%erials my be foundwith theaidof figy?eh providinge, n, ahd =07 areknownforthespecificmterials Forinelastic-tickling problems, tilemodulusratiosee/e,et/e, and Et/EsappearTheseratioscanbe co~ ~tedinnondimensional form byuseof equation(21)sincees= u/c,itfollowsdirectlyfromequation(21)that ~~ = 1+ (22) Since~ = du)de,differentiation ofequation(21)leadstothe expression * E~ = (23) Fronequations (22)and(23) it Et/Es= followsthat (%%)/(%) 1+ (3m\u/%7)n-l ~+ (3/7) n(u/uo~)n-1 (24) Thesequantities areusedinsubsequent seetions concem,ed with inelastic buckling nelastic Poisson s Ratio Poissen s ratioforengineering mterialsusuallyhasa valuein theelasticregionof between1/4and1/3and,ontheassumption ofa plastically irlcompressible i$otropic solid,assuresa valueof 1/2in theplasticregionthetransition fromtheelasticto theplasticvalue isnxxtpronounced intheyieldregionof thestress-strain curvesince

22 is of sorei:uport:mce in irelastic-iucklit:~ prollc%s, GerardandWildhorn, w;::k ot~ers, havestudiejthisproblemon severalalu~dtuum alloysandhuveshornthatpoisson c ratioisseriously affectedby misotmpyof thematerial(ref10) ForrmterklsUMC canke considered tote orthotropic (eg,hatingthesmc properties alongthey-?u_d z-~e6iflotied810ngtkle x-is) the rehtion describes thetransition ntheyieldregion: v = P - e) (&j) - * ti thisrelation, P is thefullyplasticvalueofpoisson s ratio For isotropic rmterialsvp= 1/2, ukreas fororthotropic materialsvp is generally different froma valueof 1/2 tisevidentfrom thebucklingstressexpression thattworaterials whichdifferonlyintheirvaluesofpoisson s ratioshouldhavedifferent bucklingstressesas a rule,however,thevalueof v= istirtually constantfora material whoseproperties m3ychange as a resultofheat treatment, detailsof composition, orarcount of cold-work TheUSUEL1mge of ye for EOst technically inportant structural mterialsis #sween025 and035~lere are exceptions, howeverone of themostextrer~terialssberyllium, forwhichudy,shav,8nd Wulgerreportatiue of 002(refn)!, - - * ntheinelastic range,presmblybecauseof anisotropy, numerical valuesof v havebeenfoundwhichme considerably inexcessof the theoretical upperlimitof 05,whichisderivedon tke assumption of incompressibility of- isotropic Gerardand Wildhornobtainedvaluesof v as mge as070forseveralhigh-strength aluminum alloys (ref10),vhile~n andrussellreporteda value of 077forcanzercially puretitaniumsheetand062forfs-lh=gnesium alloy(ref12) Stang,Greenspan, andnewcanalsoobtaineddataat variancewiththetheoretical valueof 05forplasticstrains(ref13) Thesethreereports cover a large varietyofalloys,deformedbyvarious total strainsinbothbarandsheetstock,andshouldbe consulted for morecompletedata 0- ~- - _ , ~ ~

23 ,, PLASTCTY-RZHK TCW FACTURS nelastic-buckling-st -essequatson Theelastic buckltngstressofa flatrectangular platecanbe expressed in theform Cre= ~2E ~ 2 )() 12(1 2 b - Ve (26) Whenthebucklingstressexceedstheproportional limitof theplate material, thetermsinequation(26)whichareinfluenced are ~, E, and v Thebucklingcoefficient k dependsuponthetypeofloading, thebucklewavelengthas affectedby thegeometrical features of boundaryconditions andaspectratio,thestresslevel,andpoisson s ratio in thecaseofplateswithfreeedgestheelasticmuus E is altered by thereduction intendingstiffness associated withinelastic behatior Poisson s ratioin theyieldregionexhibitsa gradualtransition from theelasticvalueve to8 valueof 1/2for8 plastically incompressible isotropic material Forsimplicity of calculation alleffects of exceeding theproportiona limit-are &nera12y incorporated a singlecoefficient referred to as theplasticity-reduction factorq Bydefinition t ~ = aq~cre (2 7),Substituting equation(27)intoequation(26), 0- (28) Since q = 1 nthe elasticrange,equation(28)isperfectly general anditisnotnecesssxy to distinguish betweenelasticandplastic buckling!hevaluesof ve arealwaystheelasticvalues sincethecoefficient ~ containsallchangesin thosetermsresulting frominelastic behatior / _ - -

24 3J * L,, \ f, ij:, ; : -, -,? - s Ratherprecisexperimental dataexistforplasticbucklingof colurms,sinplysu~ortedflangesandplatesundercompressive loads, andelastically sup~rted platesundershearloadsforpractical alundnum-a~oy columsundercompression, itisa well-known facttk=t theexperimental failingstressis closelyapproximated by theeuler formulawiththetangentnmdulusubstituted fortheelasticmodulus n figure5, testdataforbucklingof slmply supported flanges undercompression areshownincomparison withthetheoretical values as derivedby Stowell(ref14)according themethodofgerard (ref15) Excellent agreent is obtained Thetheoretical andexperimental determinations of thevnluesof q appropriate tov trious typesof loadingsandtiund[lry conditioner lqvc reaulte~i3extensive literatu~eṫheassumptions underlying the v%riau$ theoriesdifferwithrespectoplaatlclty laws,btress-strain relaticws, andbuckling mdelsused nordert~avoidpossibleconfusionindiscussingthevarioustheories, itappearsdesirable toresor to tke expedient of comparing theorieswithtestdatafirst n figure6, test dataofprideandheimerl(ref16]andpeters (ref17)forplasticbucklingof sinplysupported platesundercovressionareshownin co,xparison withthetheoriesof Bijleard(ref18), HandelmandPrager(ref19),Hyushin(ref20),andStowell(ref5), andthe~thodof Gerard(ref15) Pooragreement isobtainedbetweea thetestdataandtheflowtheoryof Handehn andprager,whereasrelatively goodagree~entis obtainedforthedeformation theoriesof tke othqrswithstowell s theoryinbestagreer~ct nfigure7, testdataforplasticticklingof elastically supported platesundersheerareshownin comparison withthetheoriesof Bi~laard (ref18),gerard(ref21),andstowell(ref5) tcanbcobs_ed thatthemethodofgerard,whichis basedon themaximum-shear plasticity lawto transform axialstress-strain curveintia shearstress-strain curve,is in goodagreement withtestdataon aludnum alloys On the-basis of theagreercnt withtestdata,thevaluesof q recommended forusewithequation(28)appearintheappendixȧlso, nondimensional bucklingchartsderivedthroughtheuseof thesereduc- ~tionfactorsappearinfi~es 8, 9,and10foraxiallycompressed flangesandplates and for shear-loaded plates Assumptions of nelastic-buc?xng Theories Thestateof knowledge up to 1936concerning inelastic bucklingof platesandshellshasbeens~rfzed by Tiimmkenko (ref2) The=:Q

25 , i ei?or<s rc?crtc l tl:cv-ei~ Jereca[~c~~ccd vlthatl~iii~)ts t-o modifylk!variousl-erldiu~-rrfirest tcrl:s of td::~ e~:l~litriu~, dlfrcrentltil e~u~tione by t!c! useof suitak,lc plasticity coefficients determiried frome~erirental date on columns Ȧlt?ogh suchsenier~frical effortsrfitwitha reasonet~le degreeof success,thetkoreticaldeterntinatlon ofplasticity-re~uction factorsforflatplateshasbeenechlevedwithinrecentyearsas the resultof thedevelopzxmt of a Satisfactory inelestic-kuck~ing theory Becausesuchdeveloprxmts arerecentandbecauqethevarioustheoriesave notbeen,as yet,adequately treatedintextbooks,thefollowing dis- CUS6LOnconcerning theassu~ptions andreslilts of theveiriou6 theoriesis presented in souedetail Mathematical theoriesof plasticity arephenomsnological innature sincesuchtheoriesgenerally proceedfromtheexperirr~ntally determined stress-strain relations forsimpleuniaxialoadingsịntheelastic range,stressw strainarelinearlyrelatedby theelasticmdulus At strainsbeyondtheproportional lfm~t,a finitestress-strain relationcanbeusedin thefora or an incremental relationcanbeused h eitherelationthesec~tmoduluses or thetangentiulus % varieswithstressandappliesas longas theloadingcontinues to increaseunloading usually occursalongan elasticwe parallelto ttieinitiale-tic portionof thestress-strain curve nthebucklingprocess,forexample,thestresstateisc&siderablynrmecomplexthansimpleuniaxialoadingtherefore, formulation of suitablestress-strain lawsforthree-dimensional stresstatesbeyond theprofortiorsl limitformsoneof thebasicassumptions C* thevarious phsticitytheoriesease3on generalizations of equation(3) which involvefiniterelations, deformation t~s of stress-strain lawshave beenad-ted Similargeneralizations of equation(30)involving incrementalrelations arereferredtoas flow-type theoriesḥ boththeories, unloading occurselastically Theuseofthevariousplasticity theoriesisgreatlyfacilitated by theintroduction of rotationa~y invariant fimctions todefinethe three-dimensional stressandstrainstates;suchfunctions aretermed stressandstrainntensities Theassumption thatthestressintensity isa uniquelydefined,single-valued functionof thestrainintensity - - -

26 rzterialwtentheslressintensity incre%ses (loadin~) uri is elasticwtenitdecre-fies (unloading) is a secondof thefunda%entsl hypotkesefi ofplasticity theory Thedefinitions of,thestresswdstrainintensities tkeoretlcaliv canlechosenfrona f~all%y of rotationally invariant fi,anctiors Two suchfunctions referre~toas diemqximlk-shear lawend-octahedral-skar awhavebeenfoundtobe of considerable usefuheseforcorrelalin3 stress dataonductilezterialsthus,trothof theselawshave teen assumedtoapplyin variousolutions forinelastic buckling norderto obtainsolutions tovariousplasticity problerrs, adiiitionalassurzptions are~-nerally includethe axesof stressandstraincoincideandthe assunrption ofpkstictsotropyẇtherrmre,thevariation Poisson s ratiofromtheelasticvaluetothevalueof 05fora plastically incompressible, isotropic solidisnmstpronounced in theyieldregionsome solutions accountforthe instantaneous valueof Poisson s ratiowhereas othersassumea valueof05forboththeehsticardplasticregion The latterissumptioa servesto sirplifytheanalysisconsiderably Corrections fortheuseof thefullyplastic value of Poisson s ratio cangenerauybe incorporated thefinalresults From thestandpoint of classical stability theory,tttebucklingload is theloadatvhichan stablequilihriurr configurations occursbetweenthestraightformandthebentform Sticetheloadrezains conskntduringthisexcwnge, a strainreversalmustoccuron theco~vex sideand,therefore, tkebucklingnndelleadingtothereduced-mdulus conseptforcolumnsiscorrectheoretically Alltheforegoing assumptions formthebasisforsolutionof plasticityproblemsingeneti Forthespecificprobleaof inelastic buckling, t isnecessary tamakeanadditional assumption concerning thestressdistribu~idn attheinstantofbuckling of somesort,and,therefore, axialloadingandbendingproceed simultaneously n thisczse,thebentformis theonlystableconfi~uration Ṣincein thepresenceof relatively largeaxialcompressive stressesthetendingstresses aresmll,no strainreversaloccursand theincremental bendingstressesintheinelastic rangearegivenby equation(30) Sincefailingloadsobtained from testson ahuninum-alloy columns arecloselyapproxinuited bytheeulerbucklingequationwiththetangent modulusubstituted fortheelasticmodulus,certainof theinelastictucklingtheoriesassumetheno-strain-reversal, ~gent-nmdulus, modelas thebasicbuckling process andthenproceedto solutions byuse of equilibrium equations basedon classical stability concepts ,- ~ ,

27 ::;-;, x ;f;: 1 -, nelusiicikcklirrg i-~eori~s Different invectlgntcrs hnveusedvariousonesof t},ose assuwlions di~cussed above,n ordertoindicatethemajorassuimpkions underlyirl~ eachof tl:etheories, ELsunraryis present~a in table1 Historically, EiJlaardappezrstohavekeenthefirsttoarrive at 6Qtisfactor~ theoretim~solutions forinelastic-buck~ing theories (ref 18) Hisworkis thenestcomprehensive of allthoseconsidered inthatheconsiders kth incre~ntal anddeforrztion theoriesandconcludesthatthedeformation typeis correctsinceit leadsto lowerinelasticbucklingloadbth&nareobtainedfromincrerzntal theorieseis workwasfirstpublishe~ in1937!hispnperanclaterpublications includesolutions inelastic-buckling problemhovever,thisworkappearstohaveremainedunknoumtonestof thelater investigators lyushinbrieflyreferredto Bijkard sworkandthenproceeded to derivethebasicdifferential equationforinelastic bucklingof flat platesaccording tothestrain-reversal rdel (ref20) Thederivation of thisequationisratherelegantandwasusedby Stowell,who,however, usedtheno-strain-reversal mdel (ref5) Thedifferential equation obtainedby BiJlaardreducesto thatderivedby Stawellby setting v 1/2 intheformerhandelrxm andfrager,duringthistime,obtained solutions to severalfnelastic-buckling groblemsby useof incremental theory(ref19) Testdata,suchas shownin figure6, indicatethat theresultsof incremental theories, regardless of thebucklingmodel, aredefinitely unconsemtive,whereasdefomtion-typetheoriesarein relatively goodagreerent Alltheforegoing theoriesweretzscdon tteuseof theoctahedralshearlaw Eovever, testdataon theinelastic bucklingof aluminum-alloy platesin shearindicated thattheresultsof theabovetheorieswere unconservative Gerardusedthemaximum-shear lawin placeof the octahedral-shear lawto transform axialstress-strain curvesto shear stressandfoundgoodagreezent withthealuuinum-alloy-plate shearbucklingdata(ref21) Tc sumarize,then,theassuruptions whichleadtothebestagreement betweentheoryandtestdataon inelastic bucklingof aluminum-alloy flat platesundercompression andshearloadingsincludedeformation-type stre~s-strain laws,stressandstrainintensities definedby theoctahedralshearlaw,andtheno-strain-reversal modelof inelastic bucklingȧlthos$ theremaybetheoretical objectior-is todeformation theoriesas a classand theuseofa no-strain-reversal modelinconjunction withclassical stibilityconcepts, testdatado suggestheuseof resultsobtainedfroma theorybasedon theseassurrgtions inengineering applications Thechoice,oflawsto transform axials ress-strain datato shearstress-strain data! - - -,

28 *> -,} -, -:-,), :,;i,:l de~endsupontj:ede~reeof currc::,:l G:lObt::irtczl b:: dc%!rl eachor th,jjs L - lawswlihpolyaxial test,dntufor iriivldwd :LLtF:rlfLs ~ FactorsUsedinCo:qm&iticms As alreadyinilicated, thelnelastlc-kuckl~ng stressu~ybc coxputedbyuseof plasticity-reductfon fectorsappropriate theboundaryandloadin~conditionsṫhefaclar3incoqymatealleffectsof exceeding the proportion92 lizxit upon k,,e,a~d v Forconveaienee inpreparing desi~ chartsforinelastic buckding, thecriticalelastic straincanbe used: Frcmequations(28)end(31) k# tz Ecr = () 5 + 1#) (31) Ucr* V%r (32) - Therecommended valuesof q sregivenintable2 Forcowressive loads,thevaluesof q derivedby Stowellforinfinitely longplates exceptinthecaseofplatecolumns(seerefs5 and22)havebeencorrectedtoaccountforthetistant~leous valueof Poisson s ratioaccording to a methodsuggested by Stowe12 andpride(ref23) Thusj,=,8(+) k-v2) (33) where qs is theoriginalvaluegivenby Sti#ellEquation(33)is the formof theplasticity-reduction factorstkatappearsintable2 sndhas beenuse~to constmc thenondimensional bucklingchartsof figures8, 9, andlo Forlongsimplysupported platesundercoxzbined axialco~ression andbendingbijlaardfoundtheoretically, bya finite-difference approach (ref24),that - -

29 ,,,7,, *- :, -& ~? t}eplasticity-reduction factorfcraxialcompression Equation(Zl)) r~lucesto tms valueforaxialloudalofie, sincea = O forthiscase Forpurebendingu = 2 andequation(34)is equalto theplasticityreiiuction factorfora hingedfbnge Todetermine theinstanta~eous value ofpoisson s ratio,equation(25) canbeused Forthenond~nencional bucklingchartsthetheoretical fully plasticvalue of 05wasas3um4forPoissonts ratio,as wasassvcedby Stowellinhisdeterminations oftheplasticity-reduction factorsstowell andpridereported on computations tie usingequation(34)insteadof v = 05 andshowedthattherewaslittledifference betweenthetwocurves forflangesandsimplysupported pliates (ref23) Bijlaardtookexception to thisreport(ref25);however,thedifferences were slight,as wns pointedoutby StowellandRride,anditcanbe assuredforpractical purposesthattheplasticity-reduction factorshownintheappend~ix aresatisfactory forgeneral desi~ andanalysis &mstruction Of Nondimensional BUC~lg(%Sl_tS Thenondimensional buckling-stress chartsof figures8, 9,and10 wereconstructed fromthebasicnond~sionalstress-strain curvesof figureh andtheplasticity-reduction factorshownintheappendix, incorporating themethodof criticalstrainsas depictedthroughequations(31)and(32)sincethere islittledifference amongthenwzericalvaluesof thebucklingstressesthatwcxildbeobtainedforthe plasticity-reduction factorsapplicable toa longckuxpedflangeandto a hug platewithanyamuntof edgerotational restraint, tkesecases thereduction theaverageof thethreefactors CADDM3RRXCTONFACTORS Eaiic%inciples Thepresenceof claddingon thefacesofplatesmy haveanappreciableeffecton thebucklingstressincethecladdingraterial, which usually has lower mechanicalstrengththantheplatecore, is locatedat theextremefibersof thephte cros section(fig11)wherethebending strainsduringbucklingattaintheirhighestvalues Buchertdetermined buckling-stress-reduction factorsforcladplates whichincludeplasticity effectsaswellas reduction dueto cladding (ref26) However,it ispossibletodetezmine a reduction factorfor

30 -,,, L L-il ~ ;, claddind<a;: : -!Jlt lxiy& uciltli~ tli~-i L:(tl,einclzstic t)uc}:lic~ stress t~yiel~a i innl bucklifi~ strc:sforl1eclmipl;ite tl~:~t a~reesquite close~withtketestdata Thecladdir~g reductio~l i uckrsmaytlhc: kc _ usedwiththeexistin& inelastic-bucklix~ curvesof fig~es8, ~,ata13 Theformofbucklingeqmtionco~mdy useifor detercinin~ t!:c bucklingstressof a bureflatplatewithanytypeof loadingandlm~n~~ - arysuppcrtsisgivenas equation(28)forclaiplatesthisexprec~ian isusedtofindc nominalbucklingstress,wliere thethic!vess is tt~t of thetotl-l plateandtheraterialproperties arethoseof thecore Theactual bucklingstressof a thenmy be foundby applying a simplentczerical multiplierfi to thisstressthisrultlplier, termedthecladdingreduction factorbecauseitreducestheratioo: the nominalcorestressto thebuc kling stressof tkecladplate,is a fxnctionof therelativecoreandcladdingstresslevelsandtherespective mduli of thecoreandcladdingmaterialstkeclad-plate bucklingstress canbe foundfrom 1 acr = <ucr (35) E tinenominalbucklingstressexceedstkeproportional limitof thecore =terial,thenthenominalbucklingstressforthecladplate my be foundby usingtheappropriate valueof q, theplasticityreduction factorof thecoremterialvaluesof q maybe obtained fromtheclad-plate stress-strain curveshowninfigure12,thederivationofwhichisdiscussed belo-w, t shouldbe notedthattheplasticity-reduction factordepenti uponthestresslevelandconsequently requiresan estimteof thefinal bucklingstressof theplatebefor equation(28)canbeusedto fi~~ ffcrthecladdingreduction factorhasbeenfourdtobe of sucha -ture, however,thatlittl errnrisinmlvedinfirst~indingthenominalbucklingstressandthenmultiplying itdirectlyby thecladdingreductiaa fact-or to findtheactualbucklingstressof tlcecladplatetheproduct qfiis ~T$whichwasdetermined by Buckert Table3 containsalistingof the=riouscladdingreduction factorsdeteminedinsubsequent portionsof thissectionnthetable, allplatesarelor&andsti,ply supportedṛallcasesforwhichtke cladding proportional-limit stressucl exceedsthenoninalbuckling stressucr thecladdingreduction factorisequaltounitytheymntity p %sdefinedas P = aclci ~ cr,ad f istherati of thetotal claddingthickness to theclad-plate totalthi-kness

31 - :, :1 7 DcrfvaLion Or CoreStress-ZkrEln Curve Tkecorestress-strein curve My be derivedfroma skress-str8fn curvefortheentirecladp~%teus showninfi=ve12 Usingthenotitionof fl~e 11,inuhtcha sectiofi ofa cladplateisshown,thetotal axialloadactingon thesectionisdetermirtible ikon (36) Dividingthisexpression by tucoreyields where $ = cllacore, 5/ocore =1 -f+pf (37)!Ums,thecorestress-strain currecanbeconstructed by plotting tf&corestressdeterrzined fromequation(37)at eachvalueof strain forwhichthecorresponding clad-plate stresswasfound(seefig12) ~einitial slopeof thecorecurve,whichisthesarzeas theinitial slopeoftheclad-plate curve,istheelasticmdulustobe usedinthe noriln=1-buckling-stress equationsincethebucklingstress refersto thecoreruiterial, Ucorewasreplacedby itscounterpart Ucr inthe succeeding derivations ~ical valuesof f foralcladplateappearin tablek forseveralaluninumalloysduchertshoweda valueof acl= 10,~ psifor ll(x)-kl4alloy (ref26) However,thecladdingstresstillvarywith thecladding material, ofwhichdifferent typesareusedon different alloycores Comparison of Thetotal-reduction f8ctor,defined8s theproductoftheplasticityandcladding-reduction fnctors,hasbeenplot-ted in fi,-re 13asa functiffnof stressforboththetestdataandthetlfeory ir,thecaseof axially compressed plateswo mterialsarerepresented, eachwitha different percentage of claddingthlckness purthe~ore, sheet) isa sirrply supported platewhereasthesecond(~zk-~ sheet)isa long colum Plasticity-reduction factorsforthesetwocaseswere obtained frontable2 tis instructive tonoticetheclosecorrelation forthe colunncase,forwhichthetangentmdulusistheapplicable plasticityreduction mdulus Thisfo~owstheprediction of thesimplified theo~, ,,

32 ,/,,,, l \ :: 4 Derivations ofsimplified CladdingReduction Factors Buchertderivedexpressions forthetotal-reduction factorforfist simplysupported rectangular platesubjected to severaltypesof lceiings n thefollowing sections arepreriented derivations of simplified claiding reduction factorsthatyieldbuc kling stres6es atallstresslevelsz~ rely by mltlplyingthengzinalstress(elastic or inelastic) bythecladding reduction factorat tkt stress Ṭhisisdofieby separating thecladiing effectfromtketotal-reduction factorbyusingtherelationship fi= ~/~ Case1 LongSs!lySuppotiedplatesin coqression- Buchertderived theevressiori for ~ at a==> upl (ofthecore)(refx): (*) where ~ = (3f%/Es)[1A) + (3/4)(%/%] Fora bareplatef = O and ~= q,ufiich give (39) (cftable2) Then 3 ) 1+ (,&,E=jl 5= 1 l+3r +{[l+(mjes] 1+11A)+ (3/4) (%/%]1 2 p,,,+(,,4)(2,,++w]~f (Q) i -,- --- =

33 <= (l+3pf) +*p+39f)(4+313fp2} (41) 2(1 + 3f) { vhich JllSybe written -(8[+i~-[$vfm + 3,f)l} 2) f itis ass-d that 9W/(1+ 3Pf)<<4, thefouowingsiaplexpresforthecladdingreduction factor: sionisobtained * (42) (c)forlarge stresses,p-o mdtheretore 1?= (43) l+sf - (( Equations (42) ad (43) m=r infi~ 13 ~nthefo~of qt= Tfi> wherethey,maybe seentoagreecloselywiththetotal-reduction factor andthetestdata,- Case2 Platecolumns- Thederivations of forshortandlong platecolumnsfollowthefomnusedincase1 forthesupported plate uithoutanysimpli~ingassumptions Theresultsareshownin table3 W COlumncurveisplottedinfi~e 13 int~~efo~ qt= vi>whereit is seentoagreecloselywiththedataandwithf!uchert s theory Case~ Langsimplysupported platesinshearḇuchert(re&26) showsthat ~ forshearona longsimplysupported plateis - _

34 -,, wherethenodal-li[,e slopeof theshearbucklesisobtaine~ from equation The&nLmm-energy state occurs foruncladelssticplateswhen a-l ~, andthereislittleyeasontoexpecta significantly different valueforcladplatesconsequently, thisvalueof a isassuredin thefollowing development: M C5T EsE wpa+p3t/%)+*ij 4(l*3f) (45), (a)when acr< ~c13 pfi=lq=l - (b)theplasticity-reduct~on factorfor a=r>dcl isderivable fromthetotal-reductian factorin theform (E8,8E)f,= +(~ps) +-+]

35 - -,,, : where Y 1+ 3pf - := 1:3f 4+(qEJ+3/~-(%iq 1 (47) The expression in bracesdeviatesabout2 percentfromunityfor f 010 andfor $ >02,u~ch willbein theneighborhood of the proportional limitfortypicalstructuraluminumalloysconsequently, ittillnotintroduce an appreciable errorto consideritequalto unity, inwhichcaseequation(42)forthecompressed simplysupported plate F=ldstrue (c) Forlargestresses,$-+0,and therefore {= ~ l+3f KEKLNGOF FATLECTMGUAR PLATESUNDERCOMPRESSVE LOADS nthe precedingsectionsthemathe=ticalendphysicalbackground fortheflat-plate bucklingprobleahasbeenpresentedtwasshown thatbasicequation(1)canbe usedforthesolutionof bucklingproblems pertaining to flatrectan~ platesundervarioustypesof loadingsin theelasticandnelzstic rangesby suitablechoiceof reductionfactors andbucklingcoefficients Considerations thatinfluence thedeterminationof k havebeenanalyzedin thesectionsentitled EasicPrinciples and Boundary Conditions! SM claddingreduction factorswerediscussed in thesections Plasticity-Reduction Fac~ors and Cladding Reduction Factors h thissection,andinthoseto follow,t%e bucklingcoefficientk wi~ be discussed anditsnumerical valuesfor variousloadingandboundaryconditions ti12be presented Historical Background _ inves-kigated thebucklingofa simp~ys-~pported flatrec-tangularplateundertxialloadingintheelasticrangeusingtheenerw method(ref27) He obtainedtheexplicitformfor ~ forthistype of l~dingandsupport: kc = ~a/nb)+ (nb/a]2 (48)

36 ,,,,,,, : j: : : - - Tiw,sw2s tr~tcdt]wnerou~ niiitima,-nses ot h) l(t i 1,;; :;1 t?,u,~{uy cortd~tio15 Ut~liZf:lg bo~h bk ei2-~ up~rc:tct: ar,d Llk4:solul-iott 01 t~e differential equntion(ref2) liiilconstructed Q churtof kc covisy:ng thecompleterznge ofpossi~;?: til&x:~ ;~ conditiol]s k~il+i: simplyc j~port~~d, clampedor freecir;c: on one stie, awl simply Guppor!& or ck,peded~~son theother,withthelomhi C:WS eitherclumpedor simplysupported(ref28) Lundquist and%owe~ presented tk efirstunifiedtreatrent of tkc compressive-buckling problemintheiranalyses, by boththedifferentialequationandenergyrethods,of thecasesof supported platesandflanges withsimplysupported lomltiedgesandwithvaryingdegreesof elastic rotational restraint alongtkesupprkd unloadeii edges(refs8 and29) SteinandLibove,inconsidering co~binedlongitudimlandtransverse axialloads,coveredtheeffectsof c~~ing alongtheunloadededgesot rectangular plates(ref~) Numerical Valuesof Compressive-Buckling Coefficients * forpktes Figure14 isa sumarychartdepictingthevariation of ~ asa ~CtiOn Of a/b forvsriouslimithgconditions Of edgesupportand rotational restraint on a rectangular fht platet isapparenthat forvaluesof a/b greaterthanfcmrtheeffectof rotational restraint alongtheloadededgesbeccmesnegligible andthatthecl~ed plate wouldbuckleat tirha12y tinesamecompressive bad as a platewith siqply supported loadededges + Supported Plate,~~ws,, Elastically Restrained Against~otation Thebehaviorof compressed plateswithvariouamuntsof elastic rotational restraint alongtheunloadededgescanbe understood by exminingtherelationbetweenbucklingcoefficient andbucklewave lengthforplatesupported alongbothuriloaied edgesthecurves appearin figure15forrotational restraint fromfullclamping(e= =) tohingedsuppo1 ts (~= O) From thisfigure,which1s takenfromthe reportby Lundqulst andstove-l (ref29),it ispossibleto seether~ner inwhich~hebucklewavelengthdecreases as rotational restraint increases, andthevalueof A/b for a minimum value of ~ canbe seento increasefrom2/3fore-lampedgesto 100forhingededges Thelowerportionsof thesecurvesandtheportionsto theleftof tke mininum~ lineformthefirstarmsof thecurves of ~ as a function