REG NACA-ARR-3K13-1943 Critical stress for an infinitely long flat plate with elastically restrained edges under combined shear and direct stress.pdf

1、- . -5 .+3 1176013455424 : .-.(%“ “. .-.NATIONAL ADVISORY COMMITTEERX AERONAUTICSORIGINALLYISSUEDNovember 1943 asAdvance Restricted Report 3X13CRITICAL STRESS FOR AN INFINITELYK4?GWITH ELASTICALLYRESTRAINED tilUlfXSFLAT PLATEUNDERCOMBINED SHEAR AND DIRECT SIIU2SSBy Elbridge Z. Stowell and Edward B.

2、SchwartzLangley Memorial AercmauticalLaboratoqLangley Field, Va., , !, ;,. ,WASHINGTONNACA WARTIME REPORTS are reprints of papers originaLly issued to provide rapid distribution ofadvance research results to au authorized group requiring them for the war effort. They were pre-vio-iisly held under a

3、security status but are now unclassified. Some of these reports were not tech-nically edited. All have been reproduced without change in order to expedite general distribution.L- 340E _Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-1 .WATIOl!?ALADVI

4、SORY 00M41TTEE FOR AltEONAUTICSJLDVAMCEElW5RICTE9RXIPORT.- , .,. . . . . . . . CRITIOAL STRESS FOR AN Il?FIl?ITILY LONG FLAT PLAiEg WITH ELASTICALLY. .RESTIE EDGIUS UNDERhCO14BINED SiEAR AHD DIRECT STRESSBy Elbridge Z. Stbwell and Edward B, SchwartzSU3HM.RY A simple Interaction curve “is.preented fo

5、r evaluat-ing the ooniiitiong of combined shear and direct etreseunder whioh an infinitely long, flat plate with equalelaetio reetrainte againnt rotation along the edges willbeaome unstable. The theoretical work that led to theinteraoion curve is presented In the form of appendixesINTRODUCTIOIV . .

6、In the deeigri of streOse(l.-ekin structure, considera-tion must eoeti;ee be given to.the oritioal etreeee fora sheet under a com%lnetion of shear and direct stress,The upper surface of a wing in fllght. for example, maYhe eu3Jected to oonl)ined akar anface may be taken ?.n the form -TrxiAw= Ye (A4)

7、where Y IS a notion of y oply and h .is the halfwave lengtli ofthe buokl”s I“nthe x-dlreotiong Substitution of expression (A4) into the differentialequation (A3) glvee as the ec-uation which determines YI - . - . . .- Provided by IHS Not for ResaleNo reproduction or networking permitted without lice

8、nse from IHS-,-,-Is -l6A solution of equation (A5) Is1+ “Y.gwhen m is a root of the characterimtlo equation.O (A6)Except for the substitution ofr+) “a (%J”C or (%sequation (A6) ie identical with eauatlon (A6) of referenoe2, in whioh equation (Al) of thle appendix wae eolyed withk. = Oo With this cha

9、nge, all the results oktained in .that appendix are applicable here. lhe etabtllty orlte-rlon for combined compression and shear is therefore for-mally the came a for ehear alone ae given by equation(Al?) of reference 2. This oriterion isa(y”-:Xoah2aC“s2- Os4)-a(!a-aa)-(a+aaY-(“a-a+Ylinh 2ain 2( a +

10、 ma+ p )a+C a 4Y ooeh 2m sin 2( )+ p 4ya - “ - pa sinh 2a cos 2P - 4a.Py sin 4Y1=O (A7)where the relation between kg and m, P, and Y Ie.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-.7the same as in referenoe 2, bquation (A23) “ . -. .but In the pr

11、esent report#a= J +.ii)a+h”+(%)aa+%(s)”0(A8)(A9)The reiatraint coefficient is a measure of the relativereslstanae to rotation of the restraining element at theedge of the plate; it Is discussed more fully in refer-enoe 29The prooedure for evaluating ks, after values ofk. and an expression for E have

12、 been chosen, is asfollows: A value of 11/7i Is asaumetl; a series of valuesof Y are taken until” one ie found which, together withthe oorrespondlng values of u and P as oomputed fromequations (A9), satisfies equation (A7); ks is then oom-puted from equation (A8). Another value of b/A Is as-sumed, a

13、 new set of values of Y, a, and B are foundthat sati8fy the stability oriterion, and a new value ofkg Is oomputed. The entire prooess is repeated until byplotting k against b/h the minimum value of ka isfound. In the oasei where e is a funotion of b/A, emust be re-evaluated each time a different val

14、ue Of b/hIe aamumed. T!hie minimum value of ks and the ohosenvalue of ko, when inserted In equatiens (42), give acritical aomblnation of shear and dlreot stress.,- .Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-8SOLUTION BY EMERGY METHOD FOR 00HSTA

15、19T tIn appendix B of referenoe 2, an energy solution waegiven for the etabillty of a plate under pure shear byuse of oblique ooordinatee. This solution may he extendedto cover the ease of combined ehear and” compression load-ing hy simply adding to the work done by the shear forcean additional term

16、 expreeelng the work done by the compres-sive force,llgure 3 shows the ooord!.pate system and the platedimensions. The procedure is to evaluate the terms in theequationT= + T* = VI + v= (El)whereTo work dons by the compressive force per half wave lengthTa work done by the shear force per half wae le

17、ngthVI strain energy In the plate per half wave lengthPa btrain energy per half wave length in the elasticrestraining members assumed to be present along theedges of the infinitely long plateThe deflection surface is taken to be the same as inreference 2, equation (S2); that 1s,(B2)where in th”e re=

18、train.t coefficient defined as in ref-ersnce 2, The values of T, Vz, and Pa may he takendirectly from eouations (B3), (B4), and (35) of reference2:Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-,9- 9 -VIn Woa4b1:00%”cm(%”+ :- 3)” (: - 3)+s+ 2(I + 2

19、Sins )(% - $)”+ that is,(98)(B9)If the shear stress T 5s considered a 8 given aonetant,and the restraint coeff#.clent 1s independent of b/h,the values of b/A ad resulting from the operationsindicated by eauatlons (738)and (39) are:(;)=K Coa (B1O).-.Provided by IHSNot for ResaleNo reproduction or net

20、working permitted without license from IHS-,-,-.11(Bll)Inasmuah as the value of b/A given by equation (B1O)and the value of given “by equation (Bll) are the val-uee which make the stress ax a m!nimum, it is neeessaryto eubetitute them in eacuaton (37) to obtain the orltbalCompressive stress. When th

21、is substitution is made andthe. result sxpreeeed In nondimensional form, it 1s foundthat . .In order to identify the denominator In equation(B12), It is neceesary to refer to the energy solutionsfor pure oompresion and for pure shear. .-lFromequation(B14) of reference 3,it is found thatoa!2+ c1kpc =

22、 +C=oba.Kwhere %0 is the value of k. for pure compression; or,c beingassumed constant and . minimized with respeotto b/A, :. .” - 2+Ca 1kpcm = (B13jl?rom equation (B6) in referenoe 2, it 1s found that. .I.-)aa=-J- A) c1 Cosq)+ + Ca(l + 2 ein%)sin 2$ IoosaO()b.ak: xL 1Provided by IHSNot for ResaleNo

23、reproduction or networking permitted without license from IHS-,-,-. . . . . . . -.- 12.where s Is the valae of kE for pure shear. If thevalue of c IS assumed cotistant and IS mlnimlsedwith respeot to b/A and , , beaomeeorSubstitution of relations (B13) and (B14) In eauation(B12) gives the combined l

24、oading formulaorwhere k.TwaD/bat sk. k a()+ =1kpc peand kg arerespectively.written for ax(315)andwaD/batEquation (15) may also be writ-ten in the formR. + Rea = 1 (see equation (1)whereRc = the test should reveal; however, whether” the formula iseuffioiently accurate for engineering oomputationeo. M

25、quation (Cl) was tetatedfor the follcwing vlues ofc: =oSElo6000Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-14 .The first three values of c are independerit of thewve length, and are therefore representative of the val-u9e of e upon whioh equation

26、 (Cl) ie based. The finalvaiue of c varies with the wave length in a manner typ-ioal of a sturdy stiffener (reference 4), and this varia-tion i.atherefore representative of a practioal oaee.For eaoh value of c a value of kc was oh”hsen,and the aeeooi.ated value of” kg required to cause the plate to

27、buokle wae oomputed aa deoribed in appendix A*our oases were computed with tension on the plate (k.negative). The values of kpc and se were then readfrom their reepeotive charts in references 2 and 3,.Forthe oaee of oonstant c, ths values of kpc and kp0were read at the minimums of their appropriate

28、c-ourve9.(The values of b/A at which the plate buckles under purecompression, pure shear, and under the oombi.ned loadingare, of course, all different. ) For” the oaee c+fvariablec; It is neoessary to shift from one c-ourve to another,In conformity with the assumed relationship between cand A, until

29、 nlninum values of wc and 5 s are found.The results of the numerical tet of the interactionformula are shown in table 1 and in figure 1. The finaloolumn of table 1 indlcatee the value of the left-hand mem-bers of equation (Cl). Yor the oaees in which E IS in-dependent of A the difference In this Pal

30、ue from unityis only a fraction of a percent, except when Ec has alarge negative value, in which case the differenob Is 2 to5 peroent; the values In the last column of table I will bemore in error for these values of R. because their oom- .putatlon Involves subtracting quantities of the same orderof

31、 ragnitude. l?or the case in which E varies with Athe dlfferenoe from unity is 1 or 2 peroent. The validityof the interaatlon formula is therefore established for 811engineering purposes.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-. . /- .,-. ,5r

32、.- . . . . . . .16g 1. Yimoshenko, S.: Theory of Elastic Stability, McGraw-Hill Book Co., Inc. , 1936.A2, Stowell , Elbrldge Z. : Critloal Shear Stress of anInfinitely Long lat Plate with Equal ElaetloRestraints against Rotation along the ParallelEdgee, lTACA ARR Ho. 3K12, 1943.3. Lundqulst, Eugene

33、Il.,.an”dStowell, I!llbrldge”i%.:Critioal Compreeslve S*res fir Ylat RectangularPlates Supported along All Edgee and ElasticallyRestrained aglnst Rotation along the UnloadedEdges. Rep. No. 733, NACA, 1943,4. Lundqulst, Eugene 3., and Stowell, Flbridge Z.:Restraint Proylded a Plat Rectangular plate b

34、Y a “Sturdy Stiffener along an Edge of the Plate. Rep.No, 735, RACA, 1942.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-. . - .-.16TABLE ITEST OBACCURACY Or IETERACTIOll FORMULAvalues oomputed by the method of appendix A. .k. Rc + Ra%)eE=o. .1.0000

35、 “ o o 1.0000a71 7500 2.682 .5027 1.00274.00 .5250 3.690 6.335 .6917 1.0034.2500 4.623 .8665 1.00080 6.335 1.0000 1.0000-5.0000 la.997 2.4362 .94684.003,002.101.000-20.00e = 101.0000.9993.9999.99661.0000.9949.98305.6056,002.eo1s30o-1.30-6.001.0000.9921.4996.23190-.2315-1.070602.Z905.1156.357?.2708.0

36、5210.4180.3274.7073.87441.00001.10761.43305.605 7.2706.986.403.491.800-18.001.0000.7736.6000.25790-2.578804.2726.3477.7378.?8013:9530.4757.70688.98 .86161.00001.88791.0000.9999/99961.0002:.0000.9854.6.98- 1 I=+($6.1195.603.001.0001s0000.a998.4903.163400 0 l.oqoo2.325 .3363 1.o1195.018 6.883 .7290 1.

37、05176.32tl .9194 1.ooe76.893 1.0000 1.00006.119Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Etav.10?-c-4Egjo a714 a718 “ 1.2I I 1o :A.lo: +:pk( ,()-bj b. IY, 1.-.-L _Pl-.!- - 1.8.4(a) Shear andcompreOOih .0(b) Shecoordinate systemused inappendixA.

38、rig. 2ttProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-. .Iirlm-. - -.” -. .,-t. .-, .I+ I 1.J btb- -. -41 2 2.FieJJre3.- Oblique coordinate eystan used in appendix B.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-Illlllllllfltiflnllllllllllllllll:3 1176013455424 ,.,:,:o!,. ,. .,-1. .Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-