NASA NACA-TN-847-1942 Square plate with clamped edges under normal pressure producing large deflections《夹边在正常压力下产生大挠度的方形板》.pdf
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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,
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