1、!-) ,.,4 .4 . . .-.“n ,J”IIl;jlilll Illillllll3 1176 00139 1466_ _ . - . .3 $ ATIONAL ADVISORY COMMITTEE FOR AERONAUTICS-./ WAITIRII1 IUUBOIU!ORIGINALLYISSUEDOctober1943 asAdvanceRestrictedReport3J02STRESSES AROUND RECTANGULAR CUT-OUTS IN SKIN-STRINGERPANEISUNDERAXIALLOADS-11By PaulKuhn,JohnE.Duberg
2、,andSimonH.DiskinLangleyMemorialAeronauticalLaboratoryLangleyField,Va.NACA“W-ASHINGTONNACA WARTLME REPORTS are reprints of papers originally issued to provide rapid distribution ofadvance research results to an authorized group requiring them for the war effort. They were pre-viously held under a se
3、curity 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- 368,-,. . . -. , . .,. . .Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,
4、-NATIONAL ADVISORY COMMITTEE“I “-4iIVANCEflW?dTtiCTSDFOR AERONAUTICS.REPORT .STRESSES ARC)UNDRECTANQUMR CIh!-OUTSIN SKIN-STRINGER“PANELS UNIMR AXIAL LOADS = 11 “ Paul Kuhn, John E. Dtiberg,and Simon H. IMskin. WJMMARY “Cut-outs In wings or fuselages produce stress con-centrations that present a seri
5、ous problem to the stressanalyst. As a partial solution of the general problem,this paper presents formulas for calculating the stressdistribution arod rectangular cut-outs in axially loadedpane1s. The formulas are derived by means of the substitute-strlner method oishear-lag analysis.In a previous
6、paper published under the same title asthe present one, the analysis had been based on a substitutestructure containing only twc stringers. The presentsolution is basad on a substitute structure containing threestringers and is more complete as well as more accurate thanthe previous one. It was foun
7、d that the results could beused to improve the accuracy ofthe previous solutionwithout appreciably reducing the speed of calculation.Details are given of the three-stringerof the modified two-stringer solution.solution as well asI, mmnmmm-m . . . . m, - . - .- . .Provided by IHSNot for ResaleNo repr
8、oduction or networking permitted without license from IHS-,-,-In order to check the theory against experimentalresults, stringer stresses and shear stresses were meas-ured around a systematic series of cut-outs. In addi-tion, the stringer stressesmeasured in the previous in-vestigation were reanalyz
9、ed by the new formulas. “Thethree-stringermethod was found to give very good accuracyin predicting the stringer stresses. The shear stressescannot be predicted with a comparable degree of accuracy;the discrepancies are believed to be aused by loc . . .effeot.edby.qomb$n a ner of str:ngers into a s a
10、qd onqequently ts struoture neither hms the. .effect of length of cut-out nm gtyes a eoltitlonfor themtmb.zn strl.ngerstresses,These maxlmqm stresses must“.be pbtained by separate assupions. In addltlona there1.is no obvious relation between the phear stresses t.nthe” actualatructureaqd the shear at
11、remes In the #ubotiZtuWtwo-stringer scture 4s used , .fermee 2-,.In order to obtain a.rnQ?esat.sfqctorybasis of ana . . .than that of rmenoQ 2, fo.laa were developed for IR. .Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-, -I4 “skin-stringer struct
12、ure containing three stringers. At thesame the, a new experimental lnvesttgatlon was made c“on-slsting of strain surveys around a systematic series of “cut-outs. “ Stringer strains as well as shear strains inthe sheet were measured In these tests, whereas onlystringer strains had been measured In mo
13、st of the tests ofreference 2. A study of the three-stringermethod and ofthe new experimental results Indicated that the accuracy ofthe two-stringermethod could be improved by introducingsome modifications which have no appreciable effect on therapldlty of the calculations.The main body of the prese
14、nt paper describes the ap-plication to a panel with a cut-out of a stiplified three-strhger method of analysis as well as a modified two-stringer method, Comparisons are then shown betweencalculated and experimental results of the new tests andof the tests of reference 2. Appendixes A and B givemath
15、ematical details of the exact and of the simplifiedthree-stringermethods, respectively. Appendix C givesa numerical example solved by all methods.THEOFU3TICALANALYSIS OF CUT-UUTS IH AXIALI/YLOADED SKIN-STRINGER PM?!ZSGeneral Principles and AssumptionsThe general procedure of analysis is stillar to t
16、heprocedure developed for structures without cut-outs.-Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-. .- .(reference1).The actual sheet-stringerstructure Is -w. .-replaced by an idealtzed structure in wch” the sheetoarries only shear. “Theability
17、of the sheet to-carrynomal stresses is taken Into account”by adding a suitableeffective area of sheet to the cross-sectionalarea of eachstringer. The idealized structure I.sthen simplifiedbyconiblnhg groups of strlers into sine stringers,whichare termed tsubstltutestringersll;this substitution isana
18、logous to the use of “phantommemberstin truss analysls.The substitute stringers are assumed to be connected by asheet having the same properties as the actual shest. Thestresses In the substitute sheet-stringerstructure arecalculatedby formulas obtained by solving the differentialequations governing
19、the stresses In thethe stresses In thethe problem (Seeappendix A.) Finally,actual structure are calculated fromsubstitute structure.It will be assumed that the panel is symmetrical aboutboth axes; the analysts can then be confined to one quadrant,It is furthermore assumed that the cross-sectionalare
20、as ofthe stringers and of the sheet do not vary spanwise, thatthe panel is very long, and that the stringer stresses areuniform at large spanwise dtstances from the cut-out.- -. -.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-#.- .i6Al .A2AsribConv
21、entions “ :effective cress-seotlonalarea of all continuousstringers, exclusive of main stringer borderl .cut-out, quare inches (fig. 9) .“effective cross-sectionalarea of main 69ntinuoilsstringer bordering cut-out, sqaare inches (fig. 2)efiective cpos8-sectional area of all discontinuousstringers sq
22、mre fnohes (fig. 2) . .cross-sectionalarea of rib at edge of c-it-out,square inches (fi. 2) “ K12 + lb2 +.2RB=iA one substitute strhger replaces all the remlrdngcontinuous stringers, and another substitute strler re-places all the discontinuous stringers. The thre-stringersubstitute structure obtain
23、ed by this procedure is shown Infigure 2, which summarizes graphically the”alient featuresof the method. The figure shows the actual structure, thesubstitute tructure, and the distribution of the stressesin the actual structure.The maximnn stringer stress as well as the maximumshear stress occurs at
24、 the rlb station. The formulas givenhereinafter for the stresses at the rib station and in thenet section in other words, the lengtheffect can be related more directly to the length-widthratio L/a of the net section than to the proportions ofthe cut-out itself.The running shear in the continuous pan
25、el at the ribstation isProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,- . 12(3)The running shear in the discontinuouspanel at themribstation is. . K4( )12 mT2Rt2 = - FA2 l+RCO+ . .-(4)xin which the factor D mayThe stresses 02R andU2 and 72, respectiv
26、ely,the panel. The stress ocenter Mne of the cut-out.be obtained from figure 5.72R are the maximum values ofand are the maximum stresses inreaches its maximum at theThe stress ? reaches its. .maxtium in the gross section at the station whereal = 02 # CJo.Stresses in the net section of the substitute
27、 structure.-“Theformulas for the stresses in the not section are ob-tainedtancein thefrom the exact solution (appendixA). At a dis-x from the center line of the cut-out, the stressescontinuous stringers are( .)RCOA2 cosh Klx%=6 1“- A1 cosh KIL( )cosh KIX02=F 1 + Rco cosh KILlength of the cut-out - o
28、r, more precisely, theProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-13length of the net section - Increases, the magnitude of the. parameter KIL Increases and the stresses ul and the total force fn the remaining continuous stringers ofthe actual st
29、ructure is equal to the force In stringer 1 ofthe substitute structure, and the total force $n the cutstringers of the actual structure 1s equal to the force Instringer 3.In the shear-lag theory for beam without cut-outs(reference1), the force acti on a substitute stringeris distributed over the cor
30、respondingactual stringers onProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,- . -. 15 -the assumption that the chordwlse distribution follows a-. F, .-,., .-.-hyperbollc cosine law. Inspection of the test data forpanels with cut-outs indicated that e
31、lther this nor anyma other simple assmptlon fitted the data on the average asx- well as the assumption of unlfozm distribution. It is .therefore recommended, for the presents that the stressesIn all continuous stringers other than the main striner beassumed to equal U1 and that the stresses In all c
32、utstringers be assumed to equal tJa71 (See fig. 2,) Thevalidity of these assumptions will be discussed in con-nection with the study of the experimental data.Again, by the principles of the substitute structure,the shear stresses T in the substitute structure equalthe shear stresses tn the first con
33、tinuous sheet paneladjacent to the rin stringer. In order to be consistentwith the assunptlon that the chordwise distribution of thestringer stresses is uniform, the chordwise distributionof the shear stresses should be assumed to taper llnearlyfrom 71 to zero at tpe edge of the panel (fig. 2).Simil
34、arly, the chordwise distribution of the shearstresses in the cut sheet panels should be assumed tovary linearly frma T2 adjacent to the main stringer tozero at the center line of the panel. Inspection of thetest data Indicated that this assumption does not hold very- - . .Provided by IHSNot for Resa
35、leNo reproduction or networking permitted without license from IHS-,-,-*16well in thecrepancy isimmediate vicfni.tyof the cut-outs. The dis-of some practtcal importance because the maxi-mum stress In the rib depends on the chordwise distributionof the shear stress at the rib. By plottlng experimenta
36、lvalues, it was found that the law of chordwise distribution “of the shear stress T2 at the rib station could be approxi-mated quite well by a cubio parabola. The effect of thislocal variation may be assumed-to end at a spanwise distancefrom the rib equal to one-fourth the full width of the cut-outa
37、71 A straight line 1S sufficiently accurate to”repre-sent the spanwise varlatlon wlthln thfs distance (fig. 2).If the stress T2 is distributed according to cubiclaw, the stress In the rib caused by the shear In the sheet1sT2Rt2b 1 - Wlrib = 4 and all the uncutstringers,“inoludlng.the -in one, are al
38、so replacedby a .single stringer, Contrary to thg usual shear-lagmethod,however, the stringer substituted for the continuousstringers Is located not at the centpold of these stringersbut along the edge of the cut-out. The substitute structureis used to establish the shear-lag parameter K, whichdeter
39、mines the maximum shear stress, the spanwlse rate ofdecay of the shear stress, and the spanwise rate of changeof stringer stress. The maximum stringer stress must beobtained by an Independent assumption,because a singlestringer that Is substituted for all continuous stringersobviously cannot give an
40、y indication of the chordwisedistribution ofare obtained bystresses In thestress in these strlnEers. No solutionsthe two-stringermethod for the shearcontinuous panels, either in the netsection .orIn the gross section.Stresses In the substitute structure.-Throughout thelength of the net section, the
41、stress in the main stringerIsa2R = rl + 2R(C - 1)3 (11)where C is the stress-concentrationfactor derived in.- - -Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-18reference 2. Values of C may be obtained from figure 7,whloh Is reproduced from referen
42、ce 2 for convenience. Itmay be remarked here that reference 2 placed no explicitrestrtctlon on the use of the factor C; whereas the use informula (11) of the correction factor 2R, which varies from2 for short cut-outs to.1 for 10n.g cut-outs, implies trot.the factor C bysection Is long.In the grossi
43、tselt should be used only when the netsection, the stress In the main stringerdecreases withto the formulaTheThe03byincreasing distance from the rib according( ) -ILKU2 =cfo + 02 - 00 eR (12)stress in the discontinuous subst!.tutestringer is03 = O.(l - e-=) (13)stress cl may be obtained by formula (
44、7) when 02 andare known.The running shear in the discontinuous panel is givenT2t2 = - ooA3Ke-Kx (14)Stresses in the actual structure.-The stresses in theactual structure are obtained from the stresses in the sub-stitute structure under the same assumptions as in thethree-stringermethod.Provided by I
45、HSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-,.,19 “EXPERIMENTAL.WRII?ICATI(M!J(YFFORMULAS ANDW9kiPARXSbit03?BtiHS”: -” “Teat SpalWIs and TeaIIProcedure .zAAA In order to obtain experimental verification for.theComxmlas developed, a Mrgs akin-stringerpanel wad
46、 builtand tested. The p#nfiure 10(b) showsfor reference purposes the cross section of the panel testedpreviously (reference2). Straifiswere measured byTuckerman strain ages with a“gage length of 2 inches.The gages were used In pairs on both sides of the test panel.Stnains werequadrants.each plottedm
47、easured at correspondingpoints in all faurThe final figures are drawn as for one quadrant;point representip,therefore, the average offour statj.osor eight gages. .The load was applied $n three equal Mmements. Ifthe straight hrough the three points on the load-stressProvided by IHSNot for ResaleNo re
48、production or networking permitted without license from IHS-,-,- - - - . . -.20plot did not pass through the origin, the line was shiftedto pass throu however, if the necessary shiftwas more than 0.2 klp per square inch, a new set of read- -ings was t*en. :An effective value of Youngs modulus of X!.lb x IO? .ipsper.square Inch was derived by measuring the strains.1n”allstringers
