,:1 N65- 21

was theoreticalonly, and no actual computationswere carriedout.Thefinal step was made in 1963 by Huaug (reference7), who obtainednum ericalresultsfor a deformationprocessstartingwith a central dimple,followed by a circleof circumferentialdimplesoccurringat a pressureroughly80% of the value given by Zoellyfor a completesphere.These results were confirmedby Parmerter(reference6) a year later, in 1964.The existingexperimentalbucklingdata beforethe initiationof thisstudy, as depictedin Figurei.I shows a lack of consistencybetween various investigatorswhich complicatesany attemptedcorrelationwiththeoreticalpredictions.The reason for this large scatteringof testdata ms be attributedto the significantinfluencingfactorsof imperfections,boundaryconditions,residualand Imperfectionsof geometryconsistof deviationsof the shellmldsurfacefrom that of a perfectsphere and variationinthickness.Of these two, the most serious is midsurfacelesphericalof dimpleeeedeviationsof thesize or nswhich lack rotationalsymmetrycause edgedisturbanceswhich propagatedeeplyinto the shell interiorbeforedampingout.Regionsof dimplesize are stressedtovaluesconsiderablyabove the average pR/2t membranestressesassumedin theory and precipitateprematurefailure.Residualstressescan effect stabilityin two ways.First,by causinga releasein residualstrain energyoccasionedbythe bucklingchange of shape, and secondlyby causing overstressedregionsto becomeprematurelyplastic.Prebuckledbendingradii of curvatureeffectsinfluencethe local geometricof the shell in a similarfashion to thatof local flat spots with an accompanyingdrop in the criticalpressure.The ma nltudeof prebuckledbendingis greaterforvery thin shells(generallythose with large tratios).This aspect is incorporatedin the large deformationtheory,but not in the earlier"classical"theoryof Zoelly.As an exampleof the effectsof boundary(reference26) fabricateddome specimensconditions,Litle, at MIT,with hat-likebrims which werec dto the testingfixture.Since the materialwas highly elastic,the domes could be retestedwith the brim removed,and the edge cementedinto a ring.The second set of test pressureswere 100% higherthan thefirst.Any effortsto correlatesuch tests with imperfectiontheorywould be misleading.It could be equallymisleading,when residual

1CLASlCAL,tLEGENDc: L(R) 22E BELLINFANTE 12.3 (REF 11)O BELLINFANTE 38 40' (REF 9) HOMEWOOD 12 32' (REF 9)[3 HOMEWOOD (9 25 10' KLOPPEL 22033' (REF 12)Z KLOPPEL( 53 x KAPLAN( 7 - 12 (REF 10)z GOERNER8 27 45' (REF 13)RtXc .)zXX i, XXxo XX P ,,JXXOx0l"x0.2O" (9O13Xx1:3ZzzzOZzzzz!z7,X2ZlXZxX Z zXT,,13XZ0A dk2O4O601002003005007001,800R/tFIGURE 1.1 BUCKLINGOF MONOCOQUESPHERICALCAPS- EXPERIMENTALDATAN '2,0003,000

stresses are high, to compare domes made by essentially different processes, since the magnitude and distribution of residual stresses depend upon the fabrication technique.RESULTSOF OPTIMIZATION ANALYSISThe shape is considered to be optimized for the least weight dome configuration to support a given external pressure.Since the dome divides two common tandem tanks, no cylindrical material is considered inthe weight comparison.The shapes to be investigated will consist ofconstant thickness ellipsoids, spherical caps, torispheres and zerohoop-stress domes.The buckling criteria will2be taken as a modifiedform of the Zoelly equation, i.e., p 2CE (t) where C is an experimentally determined coefficient, and R is the maximum radius of curvature of the dome according to the theory of local stability developedby Mushtari and Gellmov (reference 8).The optimization analysis is effected by computing a weight index obtained by factoring out the dimensions, the buckling coefficient, andthe dome density from the dome weight vs. the a/b (base radius toheight) ratio of the dome for the shapes of interest.The results ofthis weight indexing are shown in Figure 1.2. From the figure, theminimum weight is obtaineS for spherical cap with an a/b 3,half-openlng angle @ 60 .at aAlthough it may appear that a dome shape which frames into a cylinderat an angle will require a large ring to accomodate the hoop thrust fromthe dome at the juncture with the cylinder, a more careful analysismatching the radial displacements of ring and dome shows that a Considerable portion of the hoop load is ta/ en by the dome itself.For thisreason, a small ring is adequate.See Chapter II, Figure 2.4.PHASE II - TESTS TO CONFIRM SHAPE OPTIMIZATIONMATEEIAL PROI:'EI 'Y TESTSTo establish basic material properties of the polyvinyl chloride plasticmaterial used in making experimental models, the following tests weremade at the start of the program:(a)Elastic Modulus.Standard tensile specimens were fabricatedfrom sheet material supplied by the manufacturer to obtainvalues in both directions of the sheet, from different areasof the sheet, and different thickness of sheets.The resultsgive an average Young's Modulus of 465,000 psi with a maximumdeviation from the mean of 3.5%- Figures 5.1 and 5.2 inChapter V show the variation in the modulus and a typicalstress-strain curve from the experimental results.

2.2W wp a 31.8!ELLIPSOIDAL-ZERO-HOOP--' ,r,x 1.4o/ ,,-"/1.0 , .---.6, ,u ,,,,. n GURE 1.2 WEIGHT INDEX OF MONOCOQUEDOMESUNDERUNIFORM EXTERNAL PRESSURE

(b)Poisson'sratio(c)Ratio.to beCreep.Constantat thverylittleloadsweremaintainedlevels belowcreep.Poisson'sscatter.the ONThe plasticsheets were pressureformed into a metal mold at 240 F,cooled,and finish machinedin the mold.They were then cementedintoa heavy plastlcring.The physicalstructureof the plasticis analogous to that of a sponge saturatedwith water and frozen.Heatingmeltsthe water so that the sponge may be formed.The water is then frozenand the resultingnew shape is virtuallyfree of residualstresses.This materialand fabricationtechniqueminimizedthe major factorscontributingto the test scatterand the reductionof the bucklingcoefficient.TESTDESCRIPTIONA wooden block rests betweenthe dome's inner surface and the base ofthe test fixture with a gap of approximately1/8" separatingthem.Thepurposeof the block isto preventcompletecollapseof the specimenso that it may be used for fUrthertesting.A vacuumpump evacuatesthe air beneaththe dome thus subjectsthe dome to an equivalentexternaldifferentialpressure.The plungersof six transformerdisplacementtransducersare alignednormal to the surfacealong a dome meridiantomeasurenormaldisplacement.The output of six pressuregages are recorded with each correspondingtransducerand plottedautomaticallyon anX-Y recorderto give pressurevs. displacementfor each of the six positions.At the instantof buckling,as recordedon the graphs,the sixpressurereadingsare averagedto obtain the s tested2.00, 3.33 and 4.78withRESULTShad base radil/helghtratios of 1.00,a constantbase aiameterof 16 inches.The bucklingcoefficientswere C 0.48; 0.50; 0.50; and 0.50respectively.The bucklingpatternson the models are shownin e that C is independentof the R/tangle of the dome in this range.Thesecorrelationwith the shallowshell,of Huang (reference7) extrapolatedin Figureand of thetests also1.4 anddemon-half-openingshow excellentclampededge bucklingtheoryto includedeep domes.It isthe authorsopinionthat the large numbe of dimplesappearingonthe deep domes are the reason for extrapolatedcorrelationsincea shallowshell slice from a deep dome demonstratesthe typicaldeformationpattern.

(b)Ellipsoida! shapes tested had base radii/height ratios of 2.00and 3.33- The buckling coefficients, based upon the maximumradius of curvature theory of Mushtari-Galimov (reference 8),were C 0.54 and 0.Mg. Onthe first dome, a single dimpleappeared at the apex and was followed by a ring of sausageshaped dimples surrounding it.Onthe seconddome, two adjacent,size, atcirculardimplesarreared,point equalof tangencythe apex.(Figure1.S) .with their common(c)A single torispherewith a sphericalradius of 11.50 in. anda knuckleradius of 1.71 in. was tested and gave a bucklingcoefficientC 0.BB, based upon the sphericalradius.Thisspecimen,with a reduced bucklingcoefficient,had a singlecircleof dimplesgirdlingthe Junctionbetweenthe sphericalcap and the toroldalbase, in the vicinitywhere discontinuitystresseswere high (Figure 1.3).It appearsthat the discontinuitygeometryhas to be includedin bucklingcorrelationof such domes, and that, in general,they will be less efficientthan the sphericalor ellipsoidalshapes based upon the shapeoptimizationanaysisshown in Figure 1.2.The zero-hoopstress dome was not testedin this programbecause of its apparentinefficiencybased upon the analyticalinvestigation(Figure 1.2) and the need to reduce the scopeof experimentationin this program.The resultsof these testsconfirm the optimnmshape analysis,showingthe lightestweightmonocoquedome shape to be the sphericalcap with an a/b 3.The reducedbucklingcoefficientobtainedfor the torlspherewillshift the curve shown in Figure 1.2 relativelyhigher,and makeit less competitivethan anticipated.The resultsof the fourmonocoquesphericaldome tests plottedon the theoreticalbuckling curve of Huang (Figure 1.4), togetherwith the experimentalresultsof Parmerter(reference6) shows the scatterof theParmertertests comparedwith the consistencyof the presentresults.Parmerter'scopper specimenshad some residualstressesand surface roughnesswhich probablyaccountsfor the scatter.Table I shows the summaryof the experimentalresultsfor themonocoquedomes tested in this program.Consideringthe mass of conflictingdata hithertoexistingrelating to monocoquedome buckling,it may be said that a remarkablecorrelationof test and theory has finally been achievedforsphericaldomes under MIZATIONHistorySince the bucklingphenomenaismechanisminvolvesa transferofto the bendingcondition,it wasmaterialto increasethe bendingconsideredas a bifurcationof equilibriumstrain energy from the membraneconditionearly appreciatedthat a redistributionofrigidityof the shell with no increasein

weight should increase the buckling pressure.The first analysis to include his effect was the stiffened cylinderdissertation of D. D. Dschouin 1935 (Reference 14). Subsequentanalysis and testing was confined to shapesof single curvature(cylinders and cones). The major reason for this emphasiswas the useof such shapes in aircraft construction.With the coming of age of the space industry, attention has nowbeenfocused upon shapes of double curvature forming end closures of largepressure vessels. Until very recently, the only attack on this problemwas experimental. In Germany,Ebner; Kloppel and Jungbluth; and Kloppeland Roos, (References 16, 12, 15 respectively) tested models stiffenedby meridional and meridional-circumferential ribs.Semi-empiricalanalysis was developed for flat meridian-stiffened spherical domesbytreating them as arches subjected to triangular loading. Stiffenedmodels of the circumferential, meridian, and waffle-type were alsotested by Krenzke at David-Taylor Model Basin. (Reference 17)StiffeninTheoryIt has been the custom to stiffensphericaldomes by placingribs inthe meridianand/or circumferentialdirectionin order to achieveanimprovementin the structuralweight efficiencycomparedto monocoquedomes.These stiffeningconfigurationssuffer from the defect thatall directionson a sphere are principaldirectionsand no orientationof the patterncan be assigned.Aware of these possibledrawbacks,semi-empiricalanalysesare developedin this study for meridianandcircumferentialstiffening,and appropriateoptimizationproceduresare applied.ChapterIII containsthe detailsof the analyticalinvestigation.Since a sphericalcap, which is the minimumweightshape for monocoqueconstruction,has homogeneous,isotropic,geometrical properties,the major analysiswas directedtowardsobtaininga stiffeningconceptthat is homogeneousand isotropicover the shellmid-surface.A geodesicstiffeningconfiguration,with equilateraltriangulargrids, meets this criteriaof homogeneityand asotropyifthe grid spacingis close and the elasticpropertiesare independentof the grid orientation.The increasein efficiencyinherentin thisgeodesicconceptis supportedby the experimentaltests conductedinthis study.A briefoutlineof the more importantaspects of the geodesicstiffeninganalysesis discussedin the proceedingparagraphs.detailedanalysisis presentedin bilityPressure- In terms ofthe generalinstabilitypressureis expressedt 2non-dlmensionalas

Lwhere1/2Y --[B (l )2 (1 a)(l 2)]bdthIn termssphericalof theshelldt'Zoellyequationfor the bucklingof a monocoque*2p c E ( --)wheret t,y IPanel Instability- It is assumedthat there is no couplingbetweenpanel and generalinstabilityand that the panels may beconservativelyapproximatedas flat with hingededges.The panelinstabilitypressureis then:tPl Cl (db) 2d 2Cl )[( 3 i]Rib Crippling- Rib cripplingassumes no coupling with eitherpanel or generalinstability,ignores rib curvatureand assumestheribs to be hinge connectedto both the panels and to the rib intersections.The pressurefor rib crippling,on this basis is:2tt)

OptimumDesi jn- If Po, PI' and P2 are continuous,strictlyincreasingfunctionsof the distributionsof materialto general,paneland rib cripplingmodes of failure,it can be shown that the leastweight solutionfor a given pressureoccurs whenPcr Po Pl P2Equatingthesethe solution:values,and2(h Pcr-E' ('"successively)2eliminating(i ) c32[-3* /9,'( , )unknowns,yieldsc32 7j g(a) ]2where2ClCC3and c is a gridC2sizeCl I e'ChparametergivenC2bytherelationh 2 RtForthe panelcto beapproximateditis necessarythat. h.OBy assumingvaluesfor c andthen given by the relationsI0-as a plate,a, p/Emay6 -3 JR (h ' ) ( )Y[3o:(1 6)2 (i a)(lbedetermined ot 62)] 1/2and6, V are

The ratioof stiffened weight/monocoqueweight, is:{qi t.3c,if the same value of the generalinstabilitycoefficient,Co 2C isassumedfor both monocoqueand geodesicstiffenedccastructlons.Byholdingp/E constantand varyingc in the previousequations,aminimumweightconstructionmay be found.Evaluationof StiffenedResults- The assessmentof thequalityof a monocoquedome is made by observinghow closely thegeneralinstabilitycoefficientC comes to the upper limit value.The larger C is, the lighter weightis the dome for a given pressure.In stiffeneddomes,one may compute twoC i from the equivalentweight thicknessness of the dome.coefficientsor effectiveof meritbucklingandthick--2p 2 E*p( )C*-- 2E,t* 2( --)where { t(l ) is the smearedout thickness,which may be computedfrom the above formulaor from the actual weigheddome, and t* t Vis the effectivethickness.ApparentlyC * should have the same upperlimit value as the monocoqueC found from the previouslydeterminedexperimentalThatresults.isC* C 0.80 0.50forFor equalgivingweightof monocoque andstiffeneddomes,I -3one hast {2 E(")2,p22 CEC(t)11

#2-S SPHERICAL DOMEC.50:i :ii:: ; :,.:.:.:.::BUCKLINGPRESS. 7.72 PSI#3-S SPHERICAL DOME#4-S SPHERICAL DOMEC .50C .50(a/b 3.33)(a/b 4.78)BUCKLINGPRESS.5.30 PSiBUCKLING PRESS. 3.87 PSIFIGURE 1.3 MONOCOQUEDOMECONFIGURATIONS

(I,NOO)SNOIIV IIIDI-INOC)3 0(] 3n )OOONO I 'I 3 1N91-1ISd 9L'E "SS:JHd9NI'IHON8ISd LE"[ "SS3 ld 9NI'IHON8ISd ;' "SS:IUd 9NI'INON8(00" q/e)( 'E q/e)(00" q/e)E ' O6tK 33WOO 7V31 J3HclSI J01 l"L#.::::;i:!:i ::%: ,?!": - .,::::: ::'::i: ':: :::::;,t.'i" ": WO(37V311d177: ::::::: :. 'f::::::::::::::::::::::::::::::":i?: i::iii::iii ii;i;iiiii::iii::ii?: i::iii:.iii.! s ! !i i::iiiiiilsiii:::: iii;i[ii::i::ii;ii}::i::i::;::i::i::;gi ::: :i:i:iIPg" 3::lWO(]7V311d177:l3" #

CONFIGURATIONa 8.0 IN. 4811.14.04092457.72.5010.20avgSPHERICAL- #1-S1.08.0SPHERICAL - #242.010.0SPHERICAL - #3-S3.3314.5.04902965.30.5010.70SPHERICAL- #444.7820.0.05803453.87.5012.60ELLIPSOIDAL .4910.8Z.011.5.04012873.76.3310.8J,ELLIPSOIDAL - #6-E--.a'/MAX[TORISPHERICAL - #7-T1.711TABLE 1.1SUMMARYOF EXPERIMENTAL RESULTSFOR MONOCOQUEDOMES UNDER EXTERNAL PRESSURE

PCLASS 1.21E ( t)2REXPERIMENTALPARAMETERPRESENTSTUDYx 1.2].0xx 0.8Ox PCRxPCLASS 0,6xx,HUANG.x0.4,0.2048121620242 1,,k:[12(I- .)]2832FIGURE 1:4 BUCKLING OF MONOCOQUE SPHERICAL CAPSCOMPARISON OF TESTS WITH HUANG'STHEORY364044

:iiiiiiiii!g !,i:,GEODESICSTIFFENING(a/b-

The geodesic rib-stiffened dc_e is the most favorable reinforcement arrangement for spherical domes subjected to external pressure. The geodesic rib-stiffened dome is approximately 30 - 40% heavier than an optimum honeycomb sandwich dome for an external loading condition. However, Other loading conditions should be investigated before a