1、. . -. -. -. .,?. “+!mB-NATIONALADVISORY COMMITTEEFOR AERONAUTICSWliurm!m lumolrORIGINALLYISSUEDJanuary1945-AdvanoeRestrictedReportL5A05JUTI!MFIRICALFORMULAFORTHE CR13!ICALSHEARSTRESSOF CUKVEDSKEEI?SByLangleyPaulKuhn andL. ROSSMemorial Aeronauticaley Field,Va.LevinLabcmatoryNACAiti A C IByLANGLIY ME
2、IWOW AEROMULA.BOMTORYWASHINGTON buwleyField, v% .NACA WARTIME REPORTS are reprints of papersoriginally issued to provide rapid distribution ofadvance research resulk to an authorized group requiring them for the war effort. They were pre-viously held under a security status but are now unclassified.
3、 Some of these reports were not tech-nically edited. All have been reproduced without change in order to expedite general distribution.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA ARR NO. L5A05NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS-. . .
4、 , ., . . .“VANCE lWi9TilIC”AN EMPIRICAL FORMULA FOR THEMPORTCRITICAL SREARSTRESS OF CURVED .SHEETSBy Paul Kuhn and L. Ross LevlnSUMMARYTests were made to detemine the critical shearstress ofcurved sheets. The empirical formula derivedfrom these tests is applicable to panels with a ratioof radius to
5、 thickness of”300 or greater, a centralangle of 1 radian or less, and a ratio of arc lengthto axial length not greater than 1. m some panelswith faulty workmanship the critical shear stresseswere found to be much lower than predicted by theformula. The critical shear stress decreased withrepeated lo
6、ading, but no general lawsdete?mlnlng the amount of decrease.INTRODUCTIONwere found forA knowledge of the buckling stress of curvedsheet under shear Is of considerable importance inaircraft structural design. For complete circularcylinders, the problem has been attacked theoreticallyand experimental
7、ly by a number of authors. For a panelthat constitutes only a part of the circumference, thepubllshed theory appears to be limited to papers byLeggett (reference 1) and by ICromm(reference 2), whichgive approximate solutions for a panel very long In theaxial direction. Previous to the publication of
8、 refer-ences 1 and 2, Wagner had proposed a formula (refer-enoe 3) In which the buckling stress appears as the sumof a term expressing the effect of curvature and aterm expressing the buckllng strength of a flat late.tThis formula was modified slightly In reference by. . . ,. _Provided by IHSNot for
9、 ResaleNo reproduction or networking permitted without license from IHS-,-,-adding a term correcting the flat-plate term for finiteaspeot ratio. An analyais of miscellaneous publishedand unpublished test data to detez%nine the coefficientfor the curvature term was also given In reference 4.The test
10、data showed a large amount of scatter forreasons that could not be determined from the publishedevidence. The present paper gives the results of asystematic series of tests undertaken to obtain amore reliable formula than heretofore available.abEKl%RPcrtTCrcrlcmSYMBOLSlength of psnel in axial direct
11、ion, incheslength of panel in circumferential direction,inchesYoungls modulus of elasticity, psiceffiiet of c-mvature term in proposed formulafcr c:,lttcalshear stresscoefflcie=t of flat-plate term in proposedformlzlafor crltiicalshear stressradius of curvature of plate, inchescritical buckling load
12、, poundsthiclmess of date, inchcritical shear stress, psicritical shear stresscritical shear stressTEST SPECIMHSforforANDfirst loading, psinth loading, psiAPPARATUSThe test panels were made of 2-T aluminm alloy.Two identical panels formed opposite sides of a torsionbox (fig. 1). Pure shear was produ
13、ced in the psnels bysubjecting the box to torsion In the setup shown inm 1-1 -mm I m Imlm l-l -1 m II II mProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA A.thltnetne.of supportwas the center llne of the box,It Is very dlfflcult to realize in pra
14、otice astiple edge support or a clamped edge support. Onlythe edge conditionsnormally existing In actual structureswith stiffeners riveted to the edges of the sheet, andnot the theoretical edge conditions, were reproduced inthe tests. The longitudinal stiffeners were steelangles riveted to the outsi
15、de ofthe sheet a shortdistance from the edges of the box (fig. 1). Thetransverse stlffenes were the flanged edges of thebulkheads, The test section proper of the panel laybetween the longitudinal steel angles and bulkheads Band E. The panel ends between bulkheads A and B, orbetween E and F,served as
16、 cushion bays to smooth outirregularities of stress distribution caused by thenearness of the loaded end bulkheads. In a similarmanner, the strips of sheet lylng between the steelangles snd the adjacent edges of the box helped toisolate the test section from possible disturbingeffects of the edges.T
17、he thiclm-esses,radii of curvature, and aspectratios a/b of the curved test panels are given intable 1. addition, flat panels of O.0.O-inchthickness and aspect ratios of 1 and were built.Aspect ratios of 1 (square panels) wera obtained byriveting the panels to each bulkhead; aspect ratiosof3 were ob
18、tained by ri”vetingthe panels only tobulkheads A, B, E, and F. The panels with an aspectratio of 3 were actually resthg on the intermediatebulkheads, but thase buleads were bslieved to exertonly a negligible influence on the buckling stress.The curvature of each panel was.checked by meansof a dial g
19、age indicating to 0.0001 inch the rise betweentwo points b inches apart. A straightedge was used tocheck for sagging between bulkheads, and a carefulvisual check was made for surface irregularities suchas dimples around rimts or flat spots near the longi-tudinal stiffeners. These cheeks indicated “t
20、hat. .m.r .Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-1 lm -1- . I4bNACA ARR No. L5A05some panels had very serious Imperfections; Thesepanels with faulty workmanship were tested, but theresults were not considered In establishing the formulafor
21、orltioal shear stress. In order to ensure the samecurvature at all points, the panels had to be preformedaccurately before they were riveted to the side wallsOf the box.Each accepted test box comprised either twoidentical test panels with an aspect ratio of 3, orsix identical panels with an aspect r
22、atio of 1.Tuckerman strain gages of 2-inch gage length were placedIn the centers of all panels of eaoh box at right anglesto the expected direction of the buckle. The box wasthen loaded in small Increments to a load somewhatbeyond that necessary to produce buckling of the sheet.The strains read were
23、 plotted against load and thepoint at which the strain-load plot departed from theInitial straight line was taken as indicating thebuckling load. The torque corresponding to the bucklingloads was then used to compute the critical shear stressfor the sheet. TWO typical plots for this method ofdetemln
24、ing the buckling load are shown in figure 3.On one panel with the lowest radius-thtckness ratiotested (specimen 12-1-kO, table 1), buckllng occurredwith a snap-diaphragm act!on; the stress at which thisaction occurrd was taken to be the buckling stress,The longitudinal angles remained straight after
25、 bucklingoccurred and were therefore assumed to be adequate asfar as buckling resistance was concermed. order to obtain some information on the effectof repeated loading, a number of boxes were loaded 50 to60 ttmes. Buckllng stresses were determined on thefirst, second, and third loadins and thereaf
26、ter atIntervals of 10 loadings.RESULTS AND DISCUSSIONDerivation of formula for critical shear stress.-The fa for critical shear stress proposed byWagner as modified in reference 4.isProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-1:- NACA-”ARRNO. L5A
27、05 57or = Kl+ K2E(;)2 + O* (1),.,s -.,.,. ,.,.,.-,: . ,.,.-.- .The first term in this equatton expresses the effect ofcurvature; the second term expresses the bucklingstrength of a flat plate. lhoretical solutions for .the lat plate have been obtained for plates of variousaspect rations(references 5
28、 and 6). The second term ofequation (1) reresents an attempt to combine therasults of all these theoretical calculations into asingle simple expression. IfPoissons ratio is takenas 0.316 for aluminum alloy, the theoretical value ofthe constant K2 is 4.89 for simply supported edgesand 8.20 for clampe
29、d edges.The critical shear stresses for the test panelshaving zero curvature and aspsct ratios of 1 and 3corresponded to values of Q of 6.79 and 5.96,respectively, which are averages of all the individualpanels in each test box. These values are reasonablyconsistent with each other and fall about ha
30、lfwaybetween the theoretical values or simply supported andfor clamped edges. These results appear plauslble forriveted edges and are in line with the well-establishedfact that the experimental buckllng stress of flatplates under shear is in good agreement with the theoryif the tests are carefully m
31、ade. The results of theflat-plate tests may, therefore, be considered asjustifying the strain-gage method of determining thebuckling stress as well as establishing thecient K2 for the riveted-edge condition.The test results for the c“urvedplatesevaluated with the aid of formula (l). Forthe experimen
32、tal values of the coefficientfrom the flat plates ofaspect ratios 1 andcoeffi-wereSimplicity,Q obtained3wereaveraged, although the individual values might havebeen used with a negligible change in the final formula.With the average coefficient IQ = 6.38, the flat-plate term of equation (1) was calcu
33、lated for eachspecimen and subtracted from the experimental criticalstress Tcr to obtain the curvature term in equation (l). .The values of K1 calculated from the curvature termsare plotted in figyre 4.against the radius-thicknessratio. The points are for those specimens that wereconsidered to be of
34、 good workmanship. In spite of thisfactandtifhct that each point represents either an average ofProvided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-6 NACA ARR 0. L5A05six panels when the aspect ratio Is 1 or an average oftwo panels when the aspect ratio
35、Is 3, the pointsscatter considerably. Within the accuracy defined bythg width ofthe scatter band, however,Rthe coeffi-cient K1 1s independent of R/t for 600;for ; 3C0, the buckling stress of a curved-sheet panel that la bounded by riveted-on stiffenersmay therefore be expressed byFor ; 600, equation
36、 (3) reduces toTcr = 0.115E: + 6S(:)2F + “w (4)to a degree of accuracy appreciably better thanthat ofthe test results. Because these formulas areempirical, they should be anplied only tb panels havingdimensions falling within ths test range, that is, topanels havin an arc length no greater than the
37、axiallength cnd a central angle less than 1 radian.Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-MACA A2R No. L5A057DiscussIon offormula.- A comparison of thecalculated oritioal stresses based on formula (3)“-”wth”thd”amfitige”expe”rimentalstresses
38、 (table 1)shows that the error ranges from about -9 percent toabout 15 percent. .An idea of the scatter amongIdentical panels may be,obtained from figure 5, whichshows the average coefficient for eaoh test box aswell as the maximum and minimum values obtained forany one individual panel. TMs scatter
39、 is caused partlyby uncontrollable differences in the panels, partlyby the uncertainty in the determination of the bucklingstresses.The effect of poor workmanship on the panels isshown graphically by figure 6, which is identical withfigure L except that the test points for specimenswith faulty workm
40、anship have been added. For twospecimens having radius-thickness ratios larger than1000, the effect of faulty workmanship was sufficientnearly to eliminate the strengthening effect of curvature.Other panels with radius-thickness ratios larger andsmaller than 1000 showed that the buckling stress aspr
41、edicted by formula (3) may be materially decreased byfaulty workmanship.A comparison of the experimental results withKromms formula of reference 2(5)is shown In figure 7. This formula 1s applicable tobinfinitely long Panels for which -Ft F .4.3. It isobvious from the figure that EYomms formula Is ve
42、ryconservative even for the panels with an aspect ratioof 3, which may be considered as panels of Infinitelength.No comarison was made with Leggett?s fomula(reference 1) because the proportions of the testpanels of the present Investigation were outside therange for which results are given In Yefere
43、noe 1:Reference 7 gives test results obtained on aseries of complete cylinders subdivided into panels1 -Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-1 1 11 mm- . mmmmm-. m mm mm m mm , . J991-. .-m. -m,8 MACA ARR Nb. Z5A05by rings and longitudinal
44、 stiffeners. The bucklingstresses were determined from torque-twist plots andfrom observations of sudden changes In load while thecylinders were being twisted. Table 2 shows that theobserved buckling stresses taken from reference 7 exceedthose medioted by formula (3) by amounts varyingfrom 8 to 54 p
45、ercent, w!th an average difference of31 percent.Reference 8 describes tests or curved-web beams.mth9 oourse of these tests, the critical shear stressesof indivl.dualpanels were detetined by sual observa-tion. Comparison of the strasses given in table 3shows that the experimental stresses exceed thos
46、epredicted by formula (3) by amounbs ranging from 5 to79 percent, with an average difference of 37 percent.The exnerlmental stresses given in table 3 are nottaken directly from reference 8 but are the averagesfor th”pansls adjacent to the neutral axis of eachbeam; the other panels were excluded from
47、 the averagebecause their crltlcal stresses were changed by thepresence of tension or compression stresses.The.methods of determining critical stresses usedIn references 7 and 8 are probably less sen- .sltive than “themethods of the present Inmstlgation.This differee my be responsible for the fact t
48、hatthe experimental bucklhg stresses of references 7.and 8 average higher than those ofthe presentinvestigation.Effect of reneated loading.- The effect of repeatedloading on the buckling strasses is shown by tha curvasof figura 8. The first faw loadings decreased the“buckling-strms appreciably; additional loadingsgenerally caused a small but continued decrease,although some curves appear to level off. No permanent.set-was.noticed visually except in one panel havingR-= 300,t “but presumably yleldlng had taken Dlace in“localized regions in the other panel
