The Ultimate Compressive Strength of Thin Sheet Metal Panels Citation Sechler, Ernest Edwin (1934) The Ultimate Compressive Strength of Thin Sheet Metal Panels. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/HX3P-3Y69. https://resolver.caltech.edu/CaltechETD:etd-05212003-155534 Abstract No abstract. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: compressive strength ; thin sheet metal panels ; Aeronautics Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Aeronautics Thesis Availability: Public (worldwide access) Research Advisor(s): von Kármán, Theodore Group: GALCIT Thesis Committee: Unknown, Unknown Defense Date: 1 January 1934 Record Number: CaltechETD:etd-05212003-155534 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-05212003-155534 DOI: 10.7907/HX3P-3Y69 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 1908 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 22 May 2003 Last Modified: 03 Oct 2019 23:12 Thesis Files Preview PDF - Final Version See Usage Policy. 6MB Repository Staff Only: item control page

The Ultimate Compressive Strength of' Thin Sheet Metal Panels Thesis by E.. E. Sech1er In parti&l Fulfillment of the Requirements tor the Degree of Doctor of Philosophy, CalifQ?'nia Institute of Technology~ Pasadena, California 193' on :!?:art III ... ..-Flat Sheet .Panels Beyond t.'he 8ta.bilit.y -:Wirst Approximate Theory fl' Par~ IV--•E.x.perimer1tal lt.pperatus ar:d Eesul·~s of First r;x- Pa:rt VIJ: ...... Application to C~.rrved; Sheet a VJ on References Appendix e.~ of repor~ o~ ot thin ehe~t metal panels a the gei1eral formulas for the equations hs:ve been obtained by tb,eoreti total load UDsti:tfened 11 simply supported panel can be g.i~ven by P= c JEc:rt,_ and ~ha't c~nstant. the b7 C ill C a f {A ) ~ ) ~ .{: (II t ) fr ~) E • the Young's medulus of of t • = o:r R llf rs41 us o'£ b of ted load& lous P~'"'"'''-:!·-~ tt bo tb round to :failure ot (2) The author Wishee to his appreeiation of the assistance !'urnlshed by members of the st.aff' of ·t.ne Guggenheim Gre:dnate School of' Aeronautries at t.ne Calif'ornia of Technology and also of' by the In parti eul ar he c. :B., llil1ika.n, and Donnell for their helpt'ul suggeetirinui and criticiem&., Also itr. Raymond 0£ the Douglas Aircraft co .. " who aid.ea by suggestions makir1g the work more a.ppli ... cable to deaign~ G:rati tude is also Institute or Technology, and in particuJ,a:r to Professor Newell of that institutioD for giving access to their ex~ perimental and Qa•i.~ .. Of the stu4enta aiding in :mention due to ][essrs. Hutc1'J.ne 11 and Childers~ for their ., (3) weri:: ~ partieUlar The subject. of t.he stability thin pla:tes :fi rat became of' praeti onl im.por tmu;,e in the :fi el de o:f :naval architecture civil engineering. In tecture,. the designer was interested in side plates of a ship archi- failure of they were to In this and shearing forces in the par ta ceuJe, "the problem watt l.ae:o eompli.catet! of the by the additi co:! o:f t-he water pressure acting normal to the plane of the plat~.. l engin.., eer f"aeed a simi lsr pr()>blem when d~si gn:».ng l~~:rg.e plate girders in which other dimensions o.:t t.he in compari e:on to thickness of were very la.rge m;:lter1 bea."Ils were .subjected to loads whi el1 sion" and shear loads which and it 1111"1!.~.s the deiii tha. t buckl i of ¢ ~ used.. These ten.si on,. compres..,. in the pl.ane o:f the web" pl'obl em ·to t::to eonstruct his beam did In both e>:f the above lure timS eo:nsi. clered to have occurred when the plate suffered any appreciable def'ormati 011 in a direettcn perpeudieulaI: to i.ts plane; since any loading above this 'lllcri ti cal l.oatP' cauaed a, rapid. increase in the ae:for.wati 011 in a direct.i o:n parallel to the plane of the sheet., This large defol"lnation, if accumulated ewer a large structure such as a ship or bridge t would cause total displacements which eollld net be tolerated ir1 the desing of such a structure., Whe!l :flat and curved sheet; vms first used i'.'.l aircraft construction. the tendency \"las to design the sheets on the same basis as tbat used. previously.e (4) i~e.,, on the basie of the stability limit et:r the critical. buckliug load., 'fhi s 'buckl.ing load, however wae ''ITer,y low :fo:,r t.h.e ·thin ma·t;.- erials that were used in aircraft construetiou and this cri ti load induced stresses in eh were al :far belaw ·the stresses that would cause tion in the sheet,. It was also noticed that if thin sheet metal p&p.els were lQaded above the stabili they continued to carry a certain a."llount of increased losd m thout suffering any perBuan~nt deformation., Consequently there arose the second problem~ that ot: daetermiuing the ma:ximwn permi as:i bl e load be:f ore the sheet suff'ered a perxna:uent de:fcrmationt or e limit of the mater- bef Q"re the stresses reached the is.l at any point. struetur·es,. such conditions as i.ndi permissible., above are quite The normal flying or l.a:m:'ii.:ug loads ot1 an air plane are tisua.lly quite eompar time.. ch ~Y loads be applied under extreme conditious 0 ods o:f Consequently it may be possible to design t.he strEtssed sheet. of such a structure so that i t is in the unbuckled st~te for !lo-rm.al lead conditions and Cfin eerry considerable ls.trger lo~ds :in the buckle-de eondi ticmsi;; In other s:t.~.te wne:;1 sub,j~'eted ,, the be based en t.he ultimate lead to extreme load should car1~yif1g caJpac:1ty of the sheet rat.her tha..?J upon the load at \'iit'..ich. the $heet re.lches the sta.bi li ty limit .. The det.ermi.uat.ion e:f the of flat sheet members .has e)~af!.1'i c l.i ty limit cally aud checked experimentally by a nUlllber of author$"" (5) The first theoretical. was by :1) and :bis method. or ot.he:r methods giving other~). r~Eu~r1 ts, are one m.ethod is given i:n II oi' this paper." e !he treatment ~f r,Jta,bilit;y limit is quit.e renent~ beyond lt been o1rliained, ex"" periJ&entalJ.y a nurrtber at times but was first .eon&idex"ld seriously in a publication· of Newell~s (Re:fe:rence 6) and i:u IL.A • C.,A,, Report llo. 356 (Re:f'ereuce 7) on l'irOrk done at ga.v e the experimental data tempt no These the hope that these results would be a guide ~o on m~.de by von aud it is this pc.nsion >dlieh f orma the b~eic s and pnrt of this 001 "1frl!iS its .ez;... p~.pe:r.., A semi- the:ore ti cal edge ·compression it! the: :rJ.ane of ·the sheet, aud this theory has been checked by a large nt!!nber of GXp~riir.ents,, Tr.ii s has t.hen beG"n mth and. witbeut stiff en.ers*' the personally by tb e o:f it and part of it uas performed at c .. I,. T ~ by gradtu~,te stude11ts in Aeronautics.., 1.£"ention of authors ru~d publictd.ione wtll be made as their work is discussed.:1 (6) II. The Stability Limit atabi li ty l:bd. 'the method is quite general , on othe1· and to x s stress o- dis=- Y • ot' y b., ii:· See Fig$ l,.. All edges of to simply supported"' • if the is the general dirreren•~lal equation (l) mld, since er\\ -= 7""°"J:. o t 1.E n_(,~"L) cr1 -: :. -er J'l-r (2) 6 tJ. i! =-er .J'1'\. which gives the differential ecruation for this particular (3) the ca.se of simply suppor·ted edges" be given by an equation of the !orm.: '" \ can where integers,. are and tlons es the ;l:.-:O EqtW.tiO!l {4} ~ , a.. f- J kl,.::;;: 0 'Jl=-01 c._ put ti the values into equ:ation (3) we get, after dividing through by the on t'E f C':)\ In"(~~:)~ i"Z.(l-;...").) ~ wllich ean be solved for l. 0-=:. +'2.. ll.b-,.;-•·-'·) (:rr-rJ- er( ¥)~=-o er , yi (5) ding lJ?.~ .)l. {.!r'( )~ l ~)"t.11.. b J (6} Vi G.... { '1) (8) .A plot of' K agah1st the b/a ratio is given in Fig., 2~ Thi a family of curves indi that the critical. the sheet has a length lo\ buckli to give r1cHial liwe& this i a the eune of -,, a.iUl1 . . ns t .. bl t:.ne ., ~ is for b/a. that1 l.$10 it suffi eiently accu:::rat e o ......~--~--1~~..,.._~_..~~..1,,,..~.....J..~~..i......~-'- to oonsi.der thgt o I Thie method very a~.tisfacto:ry for plates Qf the th:tckn;Jse used in naval and. civil engineering as it. was poseil•l.e to otit&.in stresses reasc;mahle close to the ela.etic liri:ii t of t-be material t':ith steel- !n thick could be used up to 30~ w.i.de and of any length thout having the buekling stress (Q\ tl1e elaatie limit (30,000 lbs stress falls of£ proportional ·to the of the thickness 'E"f:S appli of design of aircraft 120 materi to a a compressive get the buckli :ug stresei "Lrp for any b/a ratio equal to, or greater thau l~O,. This ob"" vi ously means that :m.:unerou.;; stif:fene:rs must be used hJ order to break up the sheet into a large nU!nber of very narrow panels., This leads to an difficult construe- ti on whi cb would not be used merely :rrom an eeonom:l c: staua- If a panel of the same material h1 made 4* wide, the buckling stress falls off to ch is too low for ecorrnmi cm this pauel is increased after buckl average stress nf t.1vc1"" 7" can {10) rea.ched this last streee is still far belc;;: the cla.sti,.c limit o'f the material,, the increase is· impert.ant enough ·to be giv£n serious cousiderati.011 for thit1 sheet :met.el pDnels... Part III will consider thi .s increase in allor,;:;;b.Le stres$ ~boV~! the &t.;;.bili ty li:a1i t iii detail"* {ll.) Sheet III,,. Stability pression (Re:ferenoes 6 and etriking obtained gave buckli;ug nearly independent of the width of the :::=s: ./~. . l it- '../ . I ((l) load ~ra-s carrie~ by that part was the the the buckled state" (C) o~ In other the ila:r to the dth (12) design 8} I w tb are of the :first of by VOil assumes th~t in th:J,s regi.on the plate can be she~t it ene -tl.E ( '1..l l?Ll..) and eqi11a1~1 on (9) (9) or er::: 1tl.E+71... \1.. ( '?~.<l,,."11..) l~ (3) J..r To find This gives~ --'..1 4w"L- JP- =o J! ~ l.w :from wb.i 1T l. E 't-a. ~i1t.·c.a. l : :solving tbi s !'or (ll) \1..(l-j4-\)W11.. effective {12) and the total load c-1:1rried by the sheet; 1 'PToT-= 2.wto"-::: ~n.~:_.,.> JEcr -t1.. (14\ ftq'.:ia,l to (l.3) def om- - "--.f JE 'j a-~ - r _j,,.l.... ~I!. ( ) with t.he- f'om:"tl1 side :free (Fi.g" 4c)~ mis an equation ried out by Doiinell (Re:ferem:ie S) and total 1oad of the same as (15) 0 Don• nell's equation is "D \ , (16) lr?J\~':it :ro11owea. since they the as a function of the ela.st-ic propert1es of the material" the sqttare of the of the lo&d was i.nde:penrlent o:f the Width of the sidering the value .of r for normal {metal) to he equal to o.3$1 equations (15) and {16) give us tp-a"->t = 'LG I i Ecr t "t. 1 1> '-.:;.~. .::. respectively~ r.:::"".:" \.1. Ii v E<r'1 t l... and one would expect to find that the exper .... !mental values 0 r:1 Sr. I R.o~ Molil i.,; ,fO(J mental values of Cr~ ealcula:ted f'r@m foutHi in Ref'erenoe 1 are shown plotted $eeE. that nearly elcl (a~%) of the Gf do lie within the range 1, and it iB val u.es !tn.,' cat.et! by order to more au exper-imental program ?nu; started of iree:hnol ogy in the Glagges.meim was felt faults California lnst1 tute Labo:rat<;,ry 4 l t c these a~ mt1ch e.a po~s;t bl., e in the tei::;tf ng !1.,.T*T. (l?) IV"' nperimental Appa.ra.tus of First Experimental Tests,, The testing machine is G. F1G. in Figs. B and 9, and Fig., 10 snows .~ photograpl'l of the ma.chine in use,. me upper head of the t.e.st:ing machine cons.ists of a 1 ten inch I-heam fastened to two. upright ten i:uch channels 11 ami the lower~ moving head ot" the machine is a simiJ,ar ten ineh I-beam pivoted, on ball bearings. with a which might a.rm o:f eeven feet., introduced from and lower beams were usetl in ]'.1g,. 10 order to eldn1.nate any deflection o:f the testing heads under (lS) load distribution over the edges o:f the specimen.,, The ho:rizont.al motio11 of the lower beamf' be:tug approximately 0.-01 inch for every inch of vertical motion, eo11ld be neglectedg foe the vertical motions obtained in the experiments wer~ in general less than a o:r an inch .. load was appl;t ed by meaJl~ o:r a seven and one• half' t.on hydrauJ.ic Jack a.ml tbe valu.e~ gJ: the J.gutl.:s were by meaus of and a low- range pressure gauge" Fig., ll"' The lowrange gauge read :\'.'rom :0-2000 pounds aud the gauge were per- Fig. ll iod1cally cal.1brat.e<1 against. a standard 'type t.es;t:i:ng m.achJne., fbe surf'aees of the I-beams next to the specimens were faced th a strip of heavy cold-rolled steel having a 1200 V groove 0 :t'\lmli:ng lengthwise the center of the beam<!" rbese twtl V-grooves could be ac• curately al iSJ1ed ll' one above the other., by means of ad_Justment$ on the testing machine~ 12 The sides of the spec.imens were al so supported i:n V•grooves, in thfs 'case 30°11' which could be adjusted for any flQ'\ wit\th of' sheet d~aired and Vihi ch could that they were so of A photograph o;f the edge s11pports first "sed 1 s 3.hown in manner I! lower. edges acting i:n the plane of the 'first series which were all 12fll long and which sheet, al:ominum, and brass. O~Ol2 to 0"067 inchtHl., th a (Part lll) Since the Ka.rma:in ii1dicated that the total load material~ a wide range or to check the equation!!' even though som,e of the materials st.:ructura.1 materials.., It; wa.s also expected large and 1~hat of' the 1n order ~o establish :for very o this limits of the material was ordered crated and shipped flat When purchased and was ueed as reeei ved t :l:"ej ecting however• any specimens whiqh showed :f ects.,, ci1nature or The ends were sheared iH.J.uare and the specimens were made loaded edges of -cinular as there was a te1uieucy for !lat. edges to ride on o:ue si.de of the upper and iower on tl1e cattsing it to fa:Ll prematurely" A number -Of scattered ene inch wide specirnens were taken eloee to the l.arger compressio11 (20) specimens and inese were used as 'Eor the determination of Tb e el asti. o p:roperti es were deter- ing mea.StU"ed with balanced HU:ggen.,,. berge);" tensiomete:rs.. See Fig,, 13., The compression tests were run ·on each specimen and the value of the ultimate load iu compression was tabulated,, From th1sac using equatio11 (15) .. the value ef Cf could be determined§ since E aud ~ were given by the Fig,. 13 stre.ss-strain curve of be :ror the detern1inat.ion of the of was this time it ~J:l definition of considered :suitable to check first set of e.xperimentt?.J. tests,. It that the a further analysis in o:rder to determi1ne whether or not this d.iffereuce was due to experimental inaccuracies or wlJ.et.ber C4\' was some ;J. function of' t.he experimental variables"' Consequently a closer investigation of the probable stress distribution was earri.ed c21> out and s second approximation to the prob£d::ile ultimate load carri {22) Second Approximate Theory The first assumption made in determining the ultimate load that could be earried by a thit1 sheet pm1el in compression was that the center of the sheet carried ~10 load t.he edge regions supported the by the sheet.. load However 1t an investigati.~n of' the action o! a sheet during .loading shows that this ass:umpt.ior1 needs to be mod.if'ied in order to.more closely approach the actual load At first, the sheet remains in its original pla.ne form ~s the load is applied. At a eertain load., the. critical buckling load .. which ea11 be determined by the equations rived in Part I!9' the sh-eet buckles in a which is is continued" the e.Qges remain restrained and eontin:ue to being in the buckled state,,. c~:n carry little or no increase in load. however they still carry the stress corresponding to the critical buckling .load on the panel., The second approx- imation to the ultimate compressive load is Ga.sea on t.b.e above assumption 8 i.,e. that the center p.ortion of the sheet carries a stress e:orrespot:uiing to the critical buckling stress o:r the given panel and that the edges of the sheet carry a stress o- distributed over the effective width Consider a tz:11in sheet panel of length I., width b, and thickness t. made o:f material with a Young's modulus E, e>.: .. ~., ..,., BQnpol' ~Pi v ;;:- f,~Q on a.ll i'ow~ and subjected to ~ eompressiou load on the .upper aud l.Oll'<·er edges.- Assume a load distribution aa in Fig., (23) 14~ in -wnich t.he ons over et:f'eetive on each side uniformly distributed stress er a11d the center portion of the sheet~ w 1- i ~ - I I w of width (b • 2w), afll"ries the critical buckling st.ri$as the sheet, b Ob • Ua1ng Ka,rman assun1ptionv equation (12} as (12} and the buckliug formulair equr:rtion (?) gives ll-i.. 0: ..... k - - - b IZ(I -.,µ.'!!.) E+t... b "- (7) (l, 7) (l2a) (7a) depending upi:m the L/b ratio given ends is given by: Tbe total l aud the total load carried by the center f!ac·t1on 1s : , 1'~(b-lw)tcrb= 1 ~l.:aEt}(\- ~ \~~) (.19) by the By adding the two. ve get the total load combinatio11 as:: '°l°'Tn -- P-t 'P~ ~ecr t\ ~;r [1..,. "Z.ot ~ VU [±..,. ~Et.?. l b oC 0- b'l.-_j 2 (20} Letting l ii - Ir t {21) ~~ b (20a) (22) tnis b:Y sueh a val·tte "O ( 20a) we _ r ~" - ~ +l. :z.ifr. ~ .\!\ _ ~t ",,_] -= r JEO:" t'l. "£11", ,. . ~ r. + l« ") al.?. .fl~ ._,.. ~ (20b) in which vl.i"t.h the value o.so and theu leaves 6f as a f.unotiou of the L/b o is greater u:ui ty and that. it is conservative to U:f.le this value for K f'o:r al.l values of L/n,,. Taking theu" eh 0 give.s a •slue o:f« a 3.,29, we set for cf~ cf = [1.£11 -t 3."r i- "· G'°Aj hol,d up to c2•> when the two e1feetive widtha would Join, or when (b - 2w) s 0.., From this point on, the failure would be purely a yield point fe,ilure and the total load carried wottl.d be equal, to 'f'::. b t<r ::: CJ.' -V E<T f''" (21}) (25) place,. of and~ we putting it into the ~c:rm f'or C.f ''-- k: olt. 1r1. fE t. - K Ti~ ~-=- 3,, 1.). {~ b cl :for the region i11 whieh :K :r: 4.,.0 an<:1 OC. = 3"'29., (27) This gives a straight line as shoml in Curve III in Fig,. 15. '?he clii"'- after buckling takes place. In a recent paper by Cox of England (Refere:uce 9) {26} e, ~lightly d.if:f erent method 7nii user.'l tc calculate the allow= able compressive strength. of' :flat sh~etll' using very nearly the sa!l'ie .iui tial assumption as used in this seeend a,:pproxi1mat1 on and a formula watil obt.a.ined which CQUld be ptit i11to the f'orm where and where l is defined as in the above derivation,; curve is show--.a as Curve IV in Fig_. J.5.,, This Also, for :referenceii the two values of Cr :foUlid from the first approximate theoxy Fig,., 16 shows the exper:bneatal value-a ot' Cf' :frmri F't.G. lb Table I plotted agaiust A 4 Th1 s r.lot ehOl'!S- that there is a defini~e rela.tionahip between Cr antt A even though the ch i& .taken a~ the :mean curve through the plotted poiut.sll' does not agree with any of the theoretical {27) Tnis first set of truat could be au.;pported by a flat panel could be given by the equation and. S'eeond p tha-t the e~f:fieiant Cf wa.e a function or th(~ parameter for a wide range o:f materials 11rlth different elastic proper• tigated. the affect of' the length of the specimen on the deter.minati o?l of the exact shape of' the C:r vs .. l made on flat sh~et panels., curve was Wlith the fact established that the value or Cf eould be plotted as a. :funct,ion of l > a, much larger series of panels were designed in order to better determine this d~penda:nce. In order to obtain a number of points over the whole range of' possible values of 1 a 11.:u:aries of specimens were choseu 0 am!t Oji\ O., 064" az:ui 0"' OS21t l 7ST Dural th wi :from l that the value of Cf w~..s more doubtful £or am.all values of A the ccncu:!ntre"tion o:f e~perimeutal points v-ras made greater as A decreased. This lead t.o speeimens ranging from 0,,8f' For the dete:rmi.:tia ti on or the dependence of Cf on the length of' the panel, seve:u differeut lengths o:r sjflecimens {!>A.\ to tt in This gave from 3 to 21 inches those used in the first group being of thix1 to leaves of these blocks was negli.gible compared to the The experimental resulta of these tests are given i:n Table Il plotted in. enn be seen from .l.O that there ii;, no c:r L/b ratios C'f ·a& L/b decreases as show by the dotted curves~ This is as would he expected as tl'le stress carried by the center :r·egions of the panel i:nereases as the L/b ratio decreaaes from 1.0.., lfowever !t for design purposes'!' it is :felt that the iu;ie of the soliG. curve in Pig,. 18 :for all values or J,.,""b will give good design values for the :!lat eheet eoeffi.cient~ In fable II the valtie of on the was tak.en as the,t point curve of the material ded to a 0,,.2% el onga ti on o:f the eorre5pc:n., Thi$ was donejc :first. to agree with the d:e~ini ti on of the yield point given in the because design :requirementst and second use of this value seemingly gave less &eat ter to the experimental point$. fhis scatter V;ms particularly noticeable in materials with a .stress-strain curve !!hieh departed slowly :from a straight linef failure seemingly bei:ng retarded until a st:rl!s.s had been reaehed which was considerably beyond the proportional limit of the material,. euasion of the sfil.a,pa of the next section. A s- of the equations fer flat sheet~ equat.ions {l6} (21)., This 0 L-~~...;._~~-l..~~~'--~~-'--~~--L.:-~~~~~~'.'""-~~-:-~~~~~~~0 .1 .z .3 ·"' .s It .e. ., .a .9 1.0 't6 was particularly :fortunate since the whole theoretical discussi on has been based on ~he assumpt.i ou that the small f!,e ... fleetio:n theories hold beyond the stabili.ty limit.II' This assum.ptio11 has made in the derivation of all of the curves i:u Fig. 15,. Obviously, after the sheet has gone i:nto flections in w~ich the stresseB introduced. eonai d.ered as due to direct load111 as tl:i.e stresses This: treat ... but as yet is not in a form t;hat F1'1. 17 would make it applicable to the {Sl l ca-n:fi:rmation o:f' J;n investigation o:r the exr-erimeutsl curve crf F,ig,,. lS it is the theor~ticel curves o.:f 16 :for nearly all valu~s of l . the loading over the ef'feetive width as cannot be trn:i ?orm the stress mu.st be a ma~x.i.mlillm ·~he in Fi would tion... The position of' the experimental curve shows this t.o be correct exeept ?ery small Tal ue.e of ) . . :r·he increase in the value o:f Cf at sm&J..J. val~ ues oi· "lralues of l that either t must be huUcate If b the value of the stress carried by the center.portion of the sheet proportionally,,, there is a tendency to increase for decreasing values of L/b is shown by the dotted curves th.at !'or de-flign purposee it is· felt that the use of t"'le solid curve"' neglecting the of L/b j,s advisable"" There is another e:r:reet, bowever,f which WE!S first natieed when testing de specimens of' thin brass., Ot"ling easily noticed and the bnelcli.ng action was_ visible thrtru&hout the loading_.. e~ 'fak:h1g,. fo.r a sheet 12" long" 14" .wid$ ~ and 0,,016* thick!>· corresponding to a value of ). =- 0,,025l> the ·buckling f'iret started at a very 1-ow load and was iu the !orm of one halt-wave~ as would be expected from the buckling {32) equation~ conti.nued tL'tltil a load. erqual to appl"Olt= Thi!:! of the ~ailure i:mately the edges t,he sheet bu.ekled in o·istt and then of five tb.e center 01.' number of small ~7a:Ves ( finally a Fig., was shows clea.rly the mult:lple wave fg_rm whieh thin.a wi.de sheets go int.o,.. :oot.!eed in ·a, ple wave This cf the Fig<!' eat! also Scpecimens shOtnl in 19~ The :e.bove action was observed :f o.r all. of the panels tested l?.nd the e:ffeet seemed to decrease 8.S the sh:eet~1 becanle narrowerq This wattl.d seem ~o 11.Hiieate toot the wider sheets the distribution o.f et.:ress on the lottdec! edges could not be accurately eXJ:iressed ev·t;m by ·the iaeeoru'1 met,hod waves probably gives a sti, f:feulng ef':feet which is shown by i:u.,. crea~e 111 the value e:f A a.s the V<::-~lue of }( decreases.,. more caref'ul investigation ot the actual wave torn .of the sheet . under load is being plam1ed in order to se-e i. f it will not give a better ide~ as to the aetual stress distribution acro;;;;s the loaded edges~ Fig.. 19 shows a number of photograp11.s of test panels taken after :failnre-t i'""n which the type of' wave ean clearly seen., Al so$' more rigid edge aupporta for the edge$ of the sheet C.?..:ll be seen ill these picturesi:- (33) i ! _I r ' l 't [- ,·r r <I), r,J1:1 -""___ _! (33a) on to VlI., Pa:nels method of Part V"' wil~l ntiw be expanded to cover a simply R \ T .load in t.he plane a- -0f the sheet and parallel to the s the cylin- dri l. l tnodu1 us of the and the yield l>oi:ut to be <r~ ... supported~ Since the Si•"ies of the sheet are simply they will undergo a stress condition sim:i.l.ar to that found for the at. sheet,. at the very edge be i~elll' at :failure the stres:ia a; al~.di if.hi"' atreco wil.l. decrease as the center of the sheet approached~ However~ for eimpl.if'ica.tiou, it will be asa~ed that the region very close to the edge will act as a flat sneet a:ud will strpport a stress <.r over an effective ~rom dth the e-quation for flat sheet1 (34) (29) (30} ,. Appendix B. total load carried by the tuo aide regions wil.1 then be L+ Je:.(J"' t ~ "P-==- zw+cr-= (31) specimens could be cl.osely approxin1at.ed by the r;:-- - UC. - A,,!'>. 1'- u•;JJ.._ .::t R {32 } an<l a recent paper by Redsl:uaw i:u J:ias theoretically that (Reference ll) 5a.in€ critical buckling as :for a complete eireru.lar cylinder., value oi the cie.nt in eqtw.tion (32) also agrees with the ax:i:.erimental d.ata gi:ven in an N.. A,,,C"'A .. Report by Lundquis·t (Ref~renee 12) .. 'llllill U'1t Ci;.I'ry ·t.he onal loe..d the aides whicn will enabl~ the side regions to tiomu stress stbove t.he critical stress V&;lue for the center regiottS On the basis o.f the al1ove ~iscu.esion,. it iii as- 9 that the st:rez.s distribution over v simply suppor·~ed curved sheet panel is as f'ollo1fie: a)OVer e region o! aeting a. dth ¢Cw~ en each siclel> there is a- ., b}OVer the cent!'al region of" i1r:i.dth (b - h) t,here j,s by Fig<:r 20., load carried by the side regions is given by e01t:ta1;1ou (Sl} the J.oad cal"ried by the Pc-= {b-lw}cr-"'t: bo-c.+- 2.wcrc.t {33) ( ) which!' if put into the :rorm used in fla.t sheet caleu:lati ons • becomes (35) where (36) which can he put irito the form where 1= ~+ <5 I? uo the equ.i:;ttion eau be ll\!Titteu aa a t·u:uetio:i:i of A a:ne as given below (36} 1 alone, e-_ c: - +-~- $ -b Jia.. =AV\ E + er -R = -b - R. ·a- 6 (38) \ (39} o.f' C plotted V) is given in Figia plate~ "\ t £.. ~ O,. a1ui C • Cf:'" which gives ths tI·l.lee.x:-µ~~r:i1J;riii,:111;:~J.. val.ue for the :flat plate; since iu equation (39) and Fig.. B-2 Cf i.s taken from the value o:f Cf as a :n.metion ef' te:rrnined e~perimentally., Thua 11 in Fi A as de.,,. :B-2 , the curve f 01' 1 = tJ (!!orresponds to sl"leet.s wi t11 in:fini t.e radius er flat sheet specimens. curves for the value of C are so limited by their intersection with the o.urve giving the yield point aud point& to the right. of thio curve ind.ioate that th'9 stress in the center of the sheet ail- by (32) is higher than the yield point stress cf t.he material aud t:.t.w.t the pa.nel would f:.dl as a yield point failµre at a value and Gale ou the streugth of' cu:L"V&d series of sheet panels in {37) C :from the experimental value of the total load by the equation ( ~ 'Pfitf (40) VEa-., t'l. 1n liihi ch 1 t has been assumed that E • 107 a.nd o-~ • 36, 000 lbs./sq.in11 The resul·ta1 of this calculation are give:u in Table III., ~rom o~ 'fable JII it. c~n be seen tha.t the above method calculation gives a very good check on the values of the curved sheet constant as determii1ed by experiment"' and could: be used ttireetly .for the calculation of the maximum compresei.on load that could be resisted by a curved sheet panel withou.t sti:f:feners.. IIowever 11 a cl.oser inspection of' the e.xperimental a.nd theoretical values seems t() indicate th.at there is a consistent varia.t.ion of the value of C with the length of the speci.me11, the shorter speeime11s g1vit1g higher, and longer specimens giving lower!!' va1ues than are obtained by equation (39)~ Taking the average ratio between the theoretical value of C and that obtained by experiment for each length, we obtain For L = 6" C(exp) • lit.l C(t.heory) C(exp) • i.o C(theory) C(exp) ~ 099 C(theory} There are two possible explru1ations o:f this var- iation th length .. the first being that the stress<»c. is a 'function of the length of the specimen .. &i,nce the last aeries ot: tests 011 flat sheet pro-ved t.hat there was 110 varia~iou of importance irt Cf with variations in length. This source of error., howeve:rt does not seem importaut as the value of .::Tc. has been checked for a large range of leugths of specimens a.nd there has never been :uoti ced any consistent variation"' The second possibil.i ty ot: error lies in the method of edge support used for the paneis4 The ai del:l of the aheet. were supported in V-grooves with in1 angle of 45° which is too large. These grooves initially hold the sheet in a straight line, but if any buckling takes place in the center of the sheete the edges are free to move out of the grooves~ This motion will allow the edge of the sheet to deflect perpem:Ucular 'to the plane et' tb.e plate aml to a.et as an Euler co1mn with a restricted range of' buckling,. Thi;;; would teud to decre.as.e the load carried by the edge region and this bu.ck.ling load wou1d very likely be a function of the length of the coltmu:i. panels,, An investigation of the sti:f:fened sheet ct.1sctu:H:u;~d lat.er~ seems to 1ud1 cate that t.he &econd source of error is the most probableo On the ba&is of the above discussion, i:u1d on the results of check with stiffei1ed. curved _sheet panelsie it is felt. that the value of C obtained from equation (39) or F1g. B•2 can sa~ely be used ~or design purposes. Tl.II~ Sheet Panels th Stiffeners.. Sheet pane.ls are :never useii in the construction of an airplane simply supported as those o:f the experime11tal. tests we have just been discussing, but always have stif- :fener shapes forming the bocu11da.ries of the panel. psrt.1 cularly interested tllen in a.re the load that can be carried by a thin sheet panel to which are attached ~tif'feners rtmni11g in t.he direction of t.lle applied eompressio:r1 for this determination a method suggested by Lut.u1qu1st (Ke:rere.uee 14) .ts it:~ed¢r The e!'recti ve widtll 0£ to act with the stiffener section as tbe sheet is ll fail u:nder ,a, certain eh will a ftmcti on of the type ot sti;ffener~ the e:ffecti·ve width o:f the sbeet. the eoltUn.- For the determination of the effective widt.h11. tha equation for f'lat sheets is used w~ C+ If T 2. ye;- Cf i.s a function of ).. where l~ ~t: It can readily be seen that. the determination of the true ations. fhis calculation is earried out in detail iu Appen ... dix A and consists in starting with some arbitrary value of cr;t- e.g. the value of. the :failing stress of the stiffener with nQ sheet attached, calculatiu.g the effective si d:th IAA\ combined sheet and etif:fene;r section, and trom. this obtaini:ng a. second value of erst and oSt .. This procedure 1a .carried out un1~il a.re ecmpa:tible, which usually invol-res not more than three approximations., Then, the total. load carried by a sti£fened .sheet panel (fl.at) is g:iVEm by the sum of the carried by these sheet•pl ua"!'stifi"ener eolt.mltlt1:L. This gives (41) where .a1 1a the area o"f the st:iffetier, ~ ii' -the area of the sheet attached to the stiffener'" and u;.7 is the final sheet panels In th s.tif'feners, the pan.els carry;, i:n a.ddi tion. tc the above load carried by pai-t of the sheet lltlxt to the · ati f':feners, a, load in the center of the panel given by the equation ~ I:. ~- ( b-2. W) t ( 42) load that can be carried by a stif:fened curved sheet. is given by the equation ~cT~ Ltr;t( ~,-t ll~) -t I_(b-2.w)t (43} where the first term indicates ·the contribution o.f the edges of the sheet with the attached sti ff.e:ner sections and the seeoud tei'm indicates the co:ut.ribution of the ce11ter portion of the sheet, or the part due to the curvature of the panel .. The calcula,tit>u of' the predicted load ou the basis of equations (41) and {43) bas been made f.or a number of sheet and stiffener combinations tested at results are given in 'fables IV s..ud Vo (41) I .. 11\, and the ~able IV gives the ca,lculatione for flat panels and Table V gives the ealenlated loads for curved panels. In Table V i~ will noticed two values are given for the predicted l.oad,. .cussed detail These are di&• it is Cfn1y Appendix A t10< mention l1ere that the speoimene giving these valties lie outside of the U'5ual range 0£ monocoque deeign, inas- much as they have a very small radit.is ~f curvature coupled with a. large sheet thickness. Di.sregardiug these values,. the remainder of the tests show good agreement between the experimen~al· values of the load and. the load: predicted by t.he met.llo4 Just ou~l-lned., In ~he case of stiffened pe.?1els z there arises anot(.her variable which as yet has not been discussed. This is tbe ty:pe of attachment between sheet and stiffener and the apa.oing of the attachment points. On the ba,sis of some recentf< Q11published work at C"l .. T~ on the .relative stren~rth of stif:fener and sheet a:ttacllment m&thotis (Reference lo) it has been :found that spot-welding, Jtf well done. gives the highest failure loads, rivets a.re from 5-16% lower and bolt• will give failure loads which are consid.erably below· the loads ee.rried. by the riveted specimen.Se Also_, as would be expected" the failure loads drop off as the spacing of the attaehment. points is increased., lt will bo no'tice-d in 'r~bl.e V that. there do~s not seem. to be any consistent. variation in the load carried by the panels for d.i:ff ere11t lengths of specimens, other than that expected :from a shortening of the column.. ln other words, the discrepaucies between the predie:ted aud actual loads are not a function of the length of t-he specimen., ( 4-2} the fact that values for taken throughout., owing to la.ck cf' more complete d~ta by the The a..2swnptio11 coarrect to :t • 107 lbs~/sq~in. i~ probably however 4 the a:1un11mption ol: G'"'': 36 ,000 lbs./ sq .. 1%1. is much more doubtful, since this value varies tlomparatively wide limits for dii'ferent sheets o! the same shee~s of different thickuees.ese It ie expected tha~ the experimental scatter could be appreciably decreased if' the aett~al values of' E and ~ were to be ueed for each separate panel tested. The values assumed,. however., are good. aiea11 values" anfl tbe use o:t t.11.e true va1uee for eaQ"h panel woulu appreciably alter the average correlation betll:een probably and theory,. Another possible souriC:.\2 or eca.tter for the stiffened panels lies the teating method. used. were made up. the e11ds were maehi:ned parallel and were then tested h1 a parallel bead testing as !lat ended columns. This tenes to give a "fiery doubtful value of the e:u.d fixity a.s the e:f:t'ective erui fixity will have a tendency \d th di:f:fere:r..t thicknesses of sheet~ the accuracy of the machining operation"' the type o:f J?or the above reasons,. i .. e. that the actual pbysiea.l prope:eties of the materia.ls were not availableir and "that tbe eud :fixity of the panel3 was aomerr.hat dollbt!ul :r it appears to the author that the obtained scatter between experimental and theoretical va.lues is not excessive, and that the method used to calculate the predicted load on stirrened panels is applicable to prsctical design., Conel usi en: it thought tb.at the treatm:e:nt as a special problem in ~all defleetions and tl1e pbysi eal a,11alysis of' a large fHlr!lber cf experim.ental tests equations the are to haa: l rp -: C JE ci t "'· c: = f-().,'1)-= 4f~ t \ Jr ':a) for C as a fuuetion cf A and 1\ whi are appli.eable to necessary ·~o correspond to go to a more elaborate m1alysis o:r the problem!' using the de:fl ons. The applieati on of the above analyai~ to flat and of an effective Y1idth of sheet acting with the stiffener sectio-n is jueti- fied by the agreement between ex:perimenttl tests a.nd cal.cu- there should be e011sidered the type a.nd spacing of the attachment Although the testis on curved sheet have all bee:n on " it is felt that the method ttsed will apply equally well to other materials., Sl.lpportee sides the above the has been 'l'he strength of the to given by equatious very sati s:if'ae:torily for any material and the allowable stress in the center section is taken from tests on complete cylinders which have covered quite a large range of materials. iippendix A gives in detail the method of ea.lculatittg Iii, stiffetH~d panel. ~.nd. Appendix :B contains the curvea o:f c and Cf as wel.l as several curves which simplify the calculations o'f: values needed in using this methoelo {46) J..,.....,G.,.HeBryaa=•Prec.,.. London Math. Soc~ vol<!> * 1891 : Part II'4' of El as ti city .. 6.w Age-·Nov.. and .Dec. l.Sl30 a 7....L-11Sehuman and G~Baclt••W'I' s.,...,Th., von Karman, Report No. Sechler 11. L"H,,Dounell ... -Applied Mech .• Trms.,. of t.he A,.S.,Jlit,.E.........ru:ne 1932. 1554., Report No .. l.3·~J' .s.)fe11iell "'Gele••"•"A Report on Aircrai"t nte::i..~ial& l4·•E,,::Ei..,.Lundqu:ist--]t,.J\.,e.. A.. fech't' note no.,. 455.., 15--C"'B":Erutehi.n;s--Tbe Relative Stre:ngth of Stiffener aud Sheet attachme11t Methods°' C.. I .. T ~ l934a. Appendix A The method or ealculgting the ultimate load tha:t the failing stress of the combined eol'Umnt- which is a :function ef the radius of gyratienir t.he leugthll and the end ealeulations. a eul"'Ve showing In ~rder to f aeili th c v a.r i gyration on of the a very th wi Of ca1culation . . I ,/i\ arrived at an e-quation of Since the feil.ing stress of a column is diree~l.y prop~rtional to the squar~ uf' the radius of gy.;ration,, the equa:tiou is w.ritten in the :ferro 1-a- Lr-r·( tYJ ·t ( t-u· where f - 1tr- ( ) ~.,- radius oi' gyration of th1e stiffeuer alone~ radius oi' g:-1r a ti o.n of the stif.fener plus the at·tached sheet., Ao- e ~rea of stifiener alone s • distance from the center of the Bheet to tb.e neutral axis of the &ti£fener~ t • t:r1i ckness of the sheet ~ • total width of sheet acting with the stiffener {48) iuto curve f"or.m ru1d is given in .Fig"" .B-3. I:u the above equ&J.ti<m J. is taken as the total w ,----w u u "I"i w ,..... ; Li b Fu~. A· I wiClth sf' sheet scti:P_g with the sti:ff'ener (See Fig .. A-1) .. In the case of a contiuuoue structure t.be value of .R woUld obviously t)e twiitce the e::fTecrtt1ve wid:tl'l.p or ending ~. stiffer1e:r section then 1-::. w+d Far the sheilt (J'l(erhang beyond the stiffener aitachment. point, the sheet is considered bo be acting as a panel aimply supported on three sides with the :fourth side free. The buckling stress of such a panel is given by atl equation ch ean be put into a form giviug the effective width of Whenever of aa found above is less t:1w1 the actual overhang is grea.ter than the overha:ng value., it is obvious that the actual measured distance is used~ Example l.--Cousider :first the :flat sheet p~t-iel Ii' No o ll 1 'l'a1ile IV. 1.1he known value:;:; are -1::-= ~01 ~'I Ao::: .oSt.'3 .., .•-... • ,o -:::: .1«000" 0 ~ :::') = . &S"lb-+ 1:/L"' •4t,S.f b-=S·"'U"' equati~n1s or the curves bx Appendix B ~ S/~ 0 : f.o;;..\ t/Ao= ,·!>1lS'1 A= .DC't.«i Cf Cf-= &.rz. . 0-1) . Repeat ,..{) = ,(..,.3.6 cr/o-o = • 'i ~ ' , • (j;"..1 . • (Al(l$t'1, II'\ = and the total area under thi$ stress is end stiffeuere is T"nen the load carried by Pe = . ,311 i l'l.~oo -::= :i.o<oo"'* dth of stif.feners ia acting the center The- saaond approximatic:n gives ~'=.01'li C.t-== M~ 2.. w =,J(,C/" )l ::. .Tl~" O'bo-=' 41.l .·.Cf'J.c::::•<j>'lt: Zit11..001tt-ta•1 The total area thi"s ~t1~ese the load carried by the center stiffener is 'PIC--:::: • 010.'5 :!( i.'l...fl..O Q -= f..firoel ~ rhis gives a total load ed as For a stiffened curved ie the smae except that there must is subjecteti to the ? (!. 9'\le stre.s~ ~ '!' added the load ce.rri ed This lo£:~d is (b-"lw)Tuc lj" ::: which is carried in addition to the loe.d st-iffener plus the attached, sheet., c~.:::rried by the This j_g the method that has been followed in calc:nuatiug the values gf Table V.. In Table V there will be noticed a number of specimens for which two ealeulated values l1ave 1:;eeu shovm., The :first value listed is that obtained by the method ou:t- liue4 above i in 'ti"hieh the stiffener~ take a ~tresecr.... si:ul )ii and ce11ter of' the sheet t,akea a stress CJstz.. respectively~ cr- .. Consider!! :for example, t.he l2ff long specimen with three stiffeners" made of 0"'052° sheet. e:ud ben~ to a radius (51) 'le. = 31200 lbs;!)' /sq,,, Calculated in the ordi:uary manner" the predicted which iliil Since, in the above case, the failing stress of all of the stt:ffene:rs is far beloc; the all.owable compressive stress of the curved sl'l~:Hst" the stiffeuers very likely buckle at their failing stress and precipitate :failure of sheet bet.iveen them .• lone; bf!'f'ore the maximum allowable stress. sh.~et curved has reached its For this reason, it is suggested that !o:r conservative deeign for this case~ the whole panel be assumed to fail as soon as stiff'euer wi value ofc.iSt reaches its failing stress 11 a.ud. that the sheet between st,iffe:uers ~an undergo no higher stresses than this value of C\+(""'"-<!U:)., If this method is usedt!c the:n in the above exwnple·t1 and the total load carried by the i e give11 by which is 18% below the val:ue obtained. by test., It is obvious from an inspect:iJJn of the results in Table V that the higher orit.i sheet tend to retard failure Tsilues of the the st1 it seems the a.mount of this. effect is largely ferable to ignore it altogether!' aud design as indicated. above to the rigbt of the limiting linew these points can never be reached~ we are a.ssurning that ( 52) TABLE. t-\o "'T"C-fa. l '1 fiY 1 'DuRAI.. 2. .3 "1 S" i::J l .., ,.. 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" :u 21 2JH:i3 2.2.0 10 Ul .11.'t, .s:s-o ,CHI b Mo 1110 '"!·00 20.0 A1.vM1M.uM ll. .oS'IO .o4''f't osoo l.Stl 2,;;.)l.ii 00 2"11 Z..00 l"fJ:!.l!IO 11.00 ~00 12¥"' i.S80 "· " I. 7'10 1.SS<.J . iat.a 38a5- J,o"'i "f 11;,s-0 3.01 515 01 10 ZSQ a.~sa z.a,qoa 2.aio l.t\1..0 Z.82.0 ~ 1 000 . oiled 290 lS1 ooo <!$ 1000 'f•'i·""' • ..I 1.SS"'i •?So '~ " I· 3"4.fD ,Z.il.d It, ... 1.~s:z -~"'-.o \,3bO hi~ • :>S•J'? .s101S 1.00.0 , &.S3'a l3 l h. ,01S~ 2fo ViS ""'1.6 V' 1 ,01~7 2.3o :Z.62.0 i.r,oao ti ooo ,O.SlB 2,AfZ.0 'l,.JJ.78 351 '100 lt~~I ~.o"l l·218 :klla 3$, o;oo as:.io .os Z!J Zif 3o 31 3Z 3.3 3A/ 35 D1.1R~ ... I 'Ir. "" I 2. .. . I Y~ ... 1'/-c 6 .If SRA SS . . 3 Ct 2,120 S10 g,"fl ... ,, ... . .:;.o"I 11.' 1.111 ,.41,70 3i'~ ~00 ~.0"4 l( /#" 1.o)q .•n10 (.. ,.;, ~ .'i1f1 .!008 \,318 !. Ol'iO ..., ..., 1.011 'C0 1 "f00 1, 1s:o f·~ol1 "-1.01"100 (., .04't10 3,830 1,03'1 <10,•-100 "·"41 3,lEtO ... "' ~ 2,Mo liA 1.. oo A , .,. 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