1、1w.”.+=-= =*-”-=+“3;:-”G*. ._TECHNICALNOTESNATIONALADVISORYCOMMITTEEIOR AERCNAUT!ICS.-.N(2.847. _SQUARE PLATEWITH CLAi4PEDEDGES UNDER NORMALPRESSUREPRf3DUCIHGLARGEDEFLECTIONS .-By SamuelLevy .-NaiicnalBureau of Standards. DOCUMENT doaa.mentcontainsclaeaifiedinformationafkctingthe hkmnatDefenwof tbe
2、United StateswiCOMMTYE13;FQR AERONA-UTICS - .-. . -. .TECHNICAL.NOTE.,NO847 .: .- .=-.-4 “. “. -,-.-.,. . .-_ .=.- ,_+UARE FLATii WITHC”tiMED.Eii% Ul.PEESSURZI”- “ “:”- ;.L._._,. . .PRODUCINGR.E. ,.,. .,. By Sariue:.,DEFLEGTZONS : - . m.-., .z- -.- .-,.-m=.+.+Levy . - - -:;*. -.:-r,.“ :“- . .l.-. :-
3、“.SUMMARY “. :- . ,.- -.=- . . -f. . . ., .-” :-.- LA theor;plate with clampededgeswas givenby Henclcyin reference2 and an approximatesolutionfor largedeflectionswa.epresentedby Way in reference3. In a previous.pape5 “ “ -“-=(reference1) there iB presenteda solutionof the funds -mentalVon,K_-+sionan
4、d lateralloading. .-.=- In the presentpaper.atheor.et.ica”lan,a-ly”sisi“s-iv”e”“-”-for the stressesand deflectione”of a squarelateundernormalpressureyroducinglarge,de,flectio,na,. The edge . -, -supportsare assumed“t-oclamp the”l”a,t.erigidlyagainstrotationsand -. . . . . . . . . -.Provided by IHSNo
5、t for ResaleNo reproduction or networking permitted without license from IHS-,-,-2 IJACA!J?ecknicalNote No. 847simplysupportedrectangularplate. The resultsfc?rs-malldeflectionsobtainedby the analysisagree exactlywiththose o,fHenckyand for largedeflectionsdifferby Ies-sthan 5 percentfrom theapproxima
6、tesolutionnf Way.The workwas carried-on with thefinancialassistanceof the NationalAdvisoryCommitteefnr Aeronautics. Ac-knowledgmentis made to theBureau ofAeronautics,NavYDepartment,for its cooperationin a programof test-sofrectangularplatesunder normalpressurethatfurnishedthe backgroundfor the prepa
7、rationof this paper. Theauthor is gratefulfor theasistanceof membersof theEngineeringMechanicsSectionof the NationalBureau ofStandards,particularlythatGreenman.FUNDAM13NT!.LLof Dr. W. Rambergand Mr. S.EQUATIONSSymbolsConsideran initiallyflat squareplate of unif%rmthicknese(fig.1), and leta lengthof
8、sidesh thicknessP normalpressure, assumed”unifcrmw normaldisplacementof points of middlesurfaceh.a71.,E YoungrsmodulusL PoissonlsratioD Eh: flexuralrigidityof plat”e”12(1+2)X*Y coordinateaxes,lyingalong edges of platewiththeir originat one,cornermx,my edge,beti$”iqgmomentsPer“unitlengthabout x andY
9、r“esect-ively ,.“axes, .-,.a normalstress “T shearingstess.+Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TechnicalNote No. 847 3tensilestra-in, “ unit el”o”rigationshearngstra,in .-.T- . extreme-fiberstressesin dir”ecti”onsofaxes median-fiber
10、stressesin dir-e,ctionsof axesextremefiberbendingstresse,sin directionsofaxesdeflectioncoefficientsstressfunction . .stresscoefficientsaveragemedian-fi%erstressesin s$ andydirections,respectivelyauxiliarypressurereplacingedge momentsunif2rmn?rmalpressure P. expressedas a Fourierseries= Pa(x,y)+ P(x,
11、Y)coefficientinXourier seriesfor pressure,-PC(X,Y) .momentarm of auxiliarypressuredistribution,PJX,Y)momentcoefficientsExpressionsfor Stressesa“ndStrainsThe generalequationsfor stressesand strainsaredevelopedby imoshenkoin reference4 (ch. IX) and arealso given in reference1. The Stressesat the middl
12、esurfaceof the relateare related to the stressfunctionF by: ,.a2Fcrl.-Y ?)Xa I“.(1).Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-4 NACATechnicalMote N;-847the extreme-fiberbendingstressesin the plateare re-latediiothe deflectionsbyEh(-?JW)J32w .G;
13、=_2(1-W2) ax2 2;Eh(a2w a2wall= _ +lJ)/“(2)Y2(1-I.L2)bya “. x2 .Eh a2w()T1f=- Xy (l+M) axay(3)and the extreme-fiberbendingstressesat the edges of theplateare relatedto the bendingmomentsper unit lengthby:“l6my =(x=O,x=a)6myII0Y =w ha1 -“6mxQ11x =lfha .:.(y=O,y=a)6m=all= _.Y .ha J!l?hestrainsat the mi
14、ddlesurfaceof theplate are:.(4)Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TechnicalNobeIMb.847 58RelationsbetweenEdge Mdmentsand LateralPressureThe requirededge moments my=willbe replacedby an auxiliarypressuredistrib%;on” a(xSY)near theedg
15、es of the plateas shown in figure2. If thispressuredistributionis expressed%y a Fourierseries (reference5, p.295)and the value . 4TTmxpa-(x,y)= .: i Brlys sln (5) “-r=T,3,5. a S=j3#5”.a2 “a,.,.Express;m= and bj a Fourierseries,Kher8 ksYand kr are coefficientsto bedeterrninedand wherefor.,a squarepla
16、te ks = kr when s =r,4a2 ? “- rlrxx = PIJ k= sin n l?=1,3,5. a“co7.“4a2 ) -Y=- ks sinS=1,3,5.;.1- -(6) .Insertingequation (6) in equation(5)giv6s .-,=,4.,2Pa(x,Y)=;).p y (rks+skr)sinsin (7)r d=173,5.S=1,3,5.The uniformnormalpressure .p may also he ex-pressedby a Fourierseries (reference5, p. 295) as
17、,m co,4,2 “() . sin= s7Typb(x,y)( .P “(l-r, sin (8) -r rs a a=1,3,5 8=1,3.,Sc-c,The additionof the,tiniformnormalpr”essurep (xry)?and the auxiliarypressurereplacingthe edge mcmen s.Pa(x,Y) is obtainedby adding equatinns(7)an(16)Provided by IHSNot for ResaleNo reproduction or networking permitted wit
18、hout license from IHS-,-,-8 NACfLTechnicalNote.No.847The family of equationsrelatingthe pressurecoeffi-cients Pr,s and the deflectioncoefficients Wm,n arealso givenby the generalsolution(reference1). For thespecialcase a = b(squareplatej,presentedin thispaper, the.first22 termsin each of thss.eequ y
19、= 0, y=a wmn sin (19)m= sX,3,6,. an=l,3,5.,.Equations(18)or (19)are equivalentto the familyof equationsO=w1,1+3W1,3+5W1,5+7W1,7+. Cl=w,+3w,.J+5W3,5+7W3,7+. O=w5,1+3W5,3+5W5*5+7W7+.s J. ,. . a71 a71 a15(20)Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,
20、-,-NAc!ATechnical”NeteNo647 9.The deflectioncoefficients m.n mtistnow %e de-.-terminedfrom table 1 by”solvingeach equationfor thelinearterm in terms of the quhic termsand the.pressurecoefficients Pr,s- The deflectioncoefficients m.nthus obtainedare now substitutedintoequations(20);and, for the press
21、urecoefficientspr.sare substitutedtheirvaluesas given%y equation (lb)-The resultingequationsare,(1=2.835+7.66k1+0.324k3+0.0800k5+0.03Q3k7+0.0145k9+EKl)o=0.0523+0.324kl+l.713k3+0.140sk5+Oo0675kT.60k9+.K3Io=0.00680+().08()()k1+0.1405k3+o.96k5+.0690k7+o.0433k9+.+Kq aly)o=0.001767+o.03031cl+0.0675k3+0.0
22、690k5+.660k7+0.0402k9+.+K7O:000648+0.0145kl+0.0360k3+0.0433k5+0.0402k70.f3t15k9+.+K9J. . a71 a15 a15 a15 a15where K1.K are functios”of the pressure p and of the cubes of the deflection”-functiofim,n” The first 22terms in the equationsfor the first five coefficients Krare given in table2. As an examp
23、leof the use of table2,.SOLUTIONOK EQUATIOFSValuesof DeflectionCoefficients m,n andEdge MomentCoefficients kr.The methodof obtainingthe requiredvalues of the de-flectioncoefficients m,n and the edge momentcoeffi-wl-,-”-cients kr con ists of assumingvaluesfor -2 -h and thenpa W1,3 3t3solvingfor , kl,
24、 k3, k5, byXh4 h hProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-10 NACA TechnicalNot=-No. 847successiveapproximationfrom the simultaneousequationsin ta%le 1 and equations(10)and (21). These.calcula-W1l.tionshavebeen madefor-iOvalues of -W1,3 a) W1,
25、3 ,=.6”28i/h 3”2Qth ) (28a) .:the use of only the,first three terms in the first equation.l!7,3 W3,3of table 1, excluding cubicterms involving 9 ,-W1,5 h h, etc.,as factors, gives the equationh _ -Pi; la4 Wl,l wl,l 3- -+ “0.370 ()i- 0.490 - “(t28b)n4Eh4 h “h,. , ., .,and the use of”only the first tw
26、o t“ermsin the firstequation of table.1, excludingall cubic terms, givestheequation , , . pl,1a4 “-wl, l:0. = “-1- o.370-. 4Eh4. =,.,-(28C) .-.- -.,Itis evidentfrom table 6 thatthe”convergenceisrapid for thecenterdeflection. The”convergencei.ssome-what slower”in the case of the b“e.nd.ing stressat t
27、he rnidpoint of tiheedge. It is estirua”tedfram table 6 that thepossibleerror in table5 is less than 2 percent.,=.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-14 NACA TechnicalNo”teiJo.”-847COMPARISONWITH THE RESULTS OBTAINEDBY PIZ3VIOUSAUTHORSThe
28、 ClampedRectangularPlatewith SmsllDeflections!l!he.earliestwork on the problemof the clampedrec-tangularplateknown.to the author is that in 1902byKoialovich(reference7). Koialovichsolvedthepreblemby using trigonometricseries. In 1913Hencky (reference2), usinga methodwhich he creditstoM. Levy, madeat
29、horoughanalystsof the momentsand deflectionsforplateswith smalldeflections. In 1914Boobnov (seep.222 of reference4) extendedthescopeof oia.lovichlsearlierwork.Since that time”additionalwork on theproblemextendingthe analysistc differenttypesof load-ing and a widerrange of plate sizeshas been donebyN
30、adai (reference8), !Timoshenko(references4 and 9),I#ojtaszak(reference10),Evans,(reference11),Young(reference12)-;.andPickett reference13). The resultsof theseauthorsfor the squareplatewith clampededgesagree Czoselywith Henckysresultspresentedin reference 2. The presentpaper gives,for small deflecti
31、ons,a value of the centerdeflectionof 0.001263pa4/11ascomparedwith Henckytsvalueof 0.001265pa4/D;and.avalue of the bending“momentperpendiculartO the egeatmidpoint.of0.0512 a2p csGmparodwith-HenckyIsvalue of0.0513sap. .,The ClampedRectangularPlatewith LargeDeflectionsThe onlypreviousanalysisof square
32、plateswithclampededgesunder normalpressureproducinglargede-flectionsthat is known to the author iS the analysisby Way (reference3) in which theRitz energymethod“isused with polynomialssatisfyingtheboundaryconditions,andcontd,ining11 undeterminedconstant Althoughhiscalculationswere made for a Poisson
33、fsratio of 0.3, itappearsfrom Waysanalysisof circularplates (ref-er-ence 15) thatsmall changesin Poissonrsratio do “notappreciablyalter the solution. In”figures6 and 7 arecomparedtheresultsobtainedby Wtiyin reference.3 wit-hw = 03 and the resultsof the presentpaperwith p =0.316. The agreement“isexce
34、llent(within5percent)forboth thetotal stressat the middleof the sideand thecenterdeflection. Theagreementbetweenthe membranestressesis not so good. In no case,however,do themembranestressesdifferby more than4 percentof thetotal stress.Provided by IHSNot for ResaleNo reproduction or networking permit
35、ted without license from IHS-,-,-NACA TechnicalNote No. 847 15The InfinitePlate Stripand the CircularPlate. ,The values of the centerdeflectionand of the ex-tremefiberstressesat the centerof the sidesfor asquareclampedplatewith Ia,rgedeflectionare comparedin figures8 and 9 with thosefor a clampedcir
36、cularplate (reference14)lofdiametera and thosefor aclampedlongrectangularplate (references4 and 15) ofwidth a. As wouldbe expected,thesquare plate is morerigid than the long rectangular-plateand more flexiblethan the circularplate.NUMERICALE2CAMPLES 13mgle.1Calculatethe centerdeflectionand the maxim
37、ufiextrenefiberstressfor a 10by 10 by 0.05inchaluminumalloy plate (E= 107 lb/in2,jL= 0.316) with clampededges, subjectedto a normalpressureof 2 poundspersquare inch.The pressureratio is:pa4 2 x 104=Eh4 107 x (0.05)4,“= 320From figure3, the correspondingdeflectionratio iscenter.= 1.72hso that the cen
38、terdeflectionisw = 1.72X 0.05 = 0.0860inchcenterFrcm figure5, the maximumextremefiberstressratioforthe edge at its midpointisoaz = 65.0Eh2so that the maximumextreme-fiberstress isProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-16 NACATechnicalNote No
39、.107” X. (0.05)2a= 65.0 - = 16,300 pounds102 “Zxample2.-.per squareinchCalculatet-hepressurethatwill producea maximumextreme-fiberstressof 30,000pbundsper squareinch ina 15- by 15-by O.IO-inchaluminum-alloyplatewithclampededges. .-The maximumextreme-fiberstressratio is,Ua a 30000 x 152”.-= 67*5 = 10
40、7 x (0.10)2Eh2From figure5, the correspondingpres$ureratio ispa4 = 339Eh4so that the normalpressureis339 x 107 x (0.10)4P= = 6.70poundsper squareinch“154NationalBureau of Standards,Washington,D. C.,May 24, 1941.-. -. .1.,Provided by IHSNot for ResaleNo reproduction or networking permitted without li
41、cense from IHS-,-,-NACA TechnicalNote No. 847 17 .REFERENCES. . .1. Levy, S.: BendingofRe,ctngular.Plateswith LargeDeflections. !I?.N. No. 846,NACA, 1942.Hencky.,H.”hber.denSpannungszus.tandi-n:”echteckigen,-.-2. “.-ebenenPlatten. DoctorIS.diss.,Darmstadt, R.Oldenbourg(Munich),1913., .-3. Way, Stewa
42、rt: UniformlyLoaded,Clamped,RectangularPlateswith LargeDeflection. Proc,.,Fifth Int.Cong.Appl. Mech. (Cambridge,Mass., 1938). JohnWileypt. 4, of Ecyk.er Math.Wiss., 1910,art. 27, pp. 31185.?. Koialovich,Boris Mikhailovich: Ob odnumuravnenti. chastnfimirproizvodniimi chetvertagoporiadkaIloctorialdiss
43、.,Sankt.,(Petersburg),1902. . 8. Nddai,A.: ElastischeFlatten. JuliusSpringer (Berlin),1925,pp. 18084.9.Timoshenko,StephenP.: Bending of RectangularPlateswith ClampedEdges. Proc.,FifthInt.Cong,Appl.14ech.(Cambridge,Mass.,1938). John Wiley9lg4:o245:03M00402.0Canterdeflectioncentorho.237,471.695.9121.1
44、211.3231.5211.7141,902StressatmidpointLfedge-hzlna2o“5.3611. OJj16.9723,$30,63g.247.056.366,2o.12.471.06l.n2.924.235;7i37609.64o5.4su.5218.0325.3233.542.452.?363.975.8Stressatcenterofplate ,ind%mao2.54.66.7a.g9.911.112.913./315.1w direct(b) y=o, y=a.Y9cdistributionforapplyingtheedges(a)x = 0,Provide
45、d by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-. *A , dvtba” midpoint Of (06s9) .9, cd mipointOffD6SD)ic,m“+Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-*. . .tS, Uumped .squme /de nxwf piper)&iksons rut
46、io, 0.316.-iC. C/amDed clcubrbi? &eference .3)-PoAsonG rafio,”0.3.- /00 “./Pressure rofio, pa+/EhqFigure 9.- Variation of meximomextreme fiber stress at W&with pressure for square plate, oiraulu plate,aod long reotanguhr plate.a.28.R, Clmped ttnvg recfongulopl (ref/.Poi&s ru+io, 0.316.+o Cokxhted poinb1 1 1 1 1 I 1 1 I Io /00 200 300400Pressure ratio, pa+JEh4rwla 0.- Variation of deflection at oenter with prmmurefor quare pltbte, circtimr p18*e, and lemgreotangulu plate.IProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-
